EXTENSION OF THE NUCLEAR MASS SURFACE FOR NEUTRON-RICH ISOTOPES
OF ARGON THROUGH IRON
By
Zachary Paul Meisel

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Physics - Doctor of Philosophy
2015

ABSTRACT
EXTENSION OF THE NUCLEAR MASS SURFACE FOR NEUTRON-RICH
ISOTOPES OF ARGON THROUGH IRON
By
Zachary Paul Meisel
Nuclear mass measurement has maintained an important position in the field of nuclear
physics for a little over a century. Nuclear masses provide key evidence of the structural
transformation of nuclei away from the valley of β-stability and are essential input for many
simulations of extreme astrophysical environments. However, obtaining these masses is often
a challenging endeavor due to the low production cross sections and short half-lives of the exotic nuclei which are of particular interest. To this end, the time-of-flight mass measurement
technique has been developed to obtain the masses of several nuclei at once to precisions of
1 part in 105 with virtually no half-life limitation.
This dissertation contains a description of the experiment, analysis, and results of the
second implementation of the time-of-flight nuclear mass measurement technique at the
National Superconducting Cyclotron Laboratory. 18 masses were obtained for neutron-rich
isotopes of argon through iron, where the masses of 48 Ar, 49 Ar, 56 Sc, 57 Sc, 64 Cr, 67 Mn,
and 69 Fe were measured for the first time. These newly obtained masses were applied to
outstanding problems in nuclear structure and nuclear astrophysics, resulting in significant
scientific advances.
The measurement results for 48 Ar and 49 Ar, which were found to have atomic mass
excesses of −22.28(31) MeV and −17.8(1.1) MeV, respectively, provide strong evidence for
the closed shell nature of neutron number N = 28 in argon. It follows that argon is therefore
the lowest even-Z element exhibiting the N = 28 closed shell. The masses of 64 Cr, 67 Mn, and

69 Fe,

which were found to have atomic mass excesses of −33.48(44) MeV, −34.09(62) MeV,

and −39.35(60) MeV, respectively, show signs of nuclear deformation occurring around the
N = 40 subshell. In addition, we find 64 Cr is substantially less bound than predicted by
global mass models that are commonly used in nuclear astrophysics simulations, resulting
in a significant reduction in the predicted strength and depth of electron capture heating
in the accreted neutron star crust due to the rather abundant A = 64 mass-chain. The
reported value for the atomic mass excess of 56 Sc, −24.85(59)+0
−54 MeV, which contains an
asymmetric systematic uncertainty due to potential isomeric contamination, results in a
smaller than expected odd-even mass staggering in the A = 56 mass chain. Depending on
the choice of theoretical models for electron capture transition strengths and energies, this
could lead to strong Urca cooling in accreted neutron star crusts, due to the large amount
of A = 56 material predicted to be present on the surface of accreted neutron stars.

To Jayda.

iv

ACKNOWLEDGMENTS

An undertaking such as the one presented in this dissertation provides regular opportunities
to become indebted to family, friends, and colleagues, and this project was no different.
Firstly, I would like to thank my advisor, Hendrik Schatz, who provided me with a
challenging but enriching dissertation project, as well as many opportunities to participate
in other interesting research activities. His insight and enthusiasm created an intellectually
stimulating work environment and, through regular discourse, he has taught me to learn how
to ask the right questions and to understand when I truly understand something. In addition, I owe thanks to the other members of my guidance committee for their various support
throughout the years. Fernando Montes has always provided a friendly and willing ear to discuss whatever scientific topic I may come bother him with. . . generally unannounced. Wolfi
Mittig kindly brought his encyclopedic knowledge and jolly disposition to the regularly held
data analysis meetings. Ed Brown frequently provided patient and thorough explanations of
complex topics. Chih-Wei Lai provided excellent mentorship early on in my graduate career
and also gave me an unexpected and appreciated level of freedom to perform my teaching
duties for his lab sections.
Throughout my stay at the NSCL I have had the good fortune of working with several
excellent colleagues, many of whom I count among my friends. The laboratory staff here
are second to none, though I would especially like to thank John Yurkon, Daniel Bazin, and
Andy Thulin for their willingness to impart some of their vast knowledge on a young graduate
student who they could have just as well avoided. The graduate students have proven to
be capable colleagues and generally fun people. I will always have fond memories of my
interactions both at and (especially) outside of work with Anthony Schneider, Steve Quinn,
v

Titus Morris, Jack Winkelbauer, and Adam Fritsch. The past and present graduate students
in the Schatz research group, in particular Karl Smith, Wei Jia Ong, Justin Browne, Alfredo
Estrad´e, Giuseppe Lorusso, and Ana Becerril, were a pleasure to work with and learn from.
I have learned a great deal from the many post-docs I have had the chance to work with,
including Tony Ahn, Antonios Kontos, Christoph Langer, Marcelo del Santo, and especially
Sebastian George, who was an endless source of both ideas and encouragement along the road
to completing my thesis project. I also appreciate the insightful input from Richard Cyburt,
who proved to be an ideal mentor during my early days as an undergraduate researcher.
I owe a great deal of thanks to my family for their support and entertainment value.
My father, ‘Paulski’, and my brother, Collin, made sure to keep my debating skills sharp,
particularly with regards to topics each of us knows little to nothing about. My sister,
Johnna, made sure I always had a vacation spot a short drive away in scenic northern Ohio
and kindly gave me many long-overdue haircuts. My mom, ‘the Beckmeister’, has always
been an endless source of encouragement, letting me know that I can do it, whatever ‘it’ is,
. . . even when all the evidence points to the contrary. I appreciate the loyal companionship
and excellent listening skills of Larry Fischer, Reggie Espinoza, and Sofi Bjorenthun, who
have been like family to me, though we are not related by blood (or species). Finally, I owe the
greatest thanks to my wife, Jayda, who has been a great friend and companion throughout
our time together. Her support, including (but certainly not limited to) reminders of what’s
important in life and willingness to listen to me ramble on about whatever may be on my
mind, mean a great deal to me.

vi

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

x

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

Chapter 1 Introduction . . . . . . . . . . . . .
1.1 Nuclear structure . . . . . . . . . . . . . .
1.2 Nuclear astrophysics . . . . . . . . . . . .
1.3 Overview of nuclear mass measurement . .
1.4 TOF mass measurement . . . . . . . . . .
1.4.1 TOF mass measurement principle .
1.4.2 Previous TOF mass measurements

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1
2
4
6
15
15
16

Chapter 2 Experimental setup . . . . . . . . . . . .
2.1 Overview of NSCL TOF mass measurement setup
2.2 Production and transport of nuclei . . . . . . . .
2.2.1 Fragment production . . . . . . . . . . . .
2.2.2 Fragment separation . . . . . . . . . . . .
2.2.3 Fragment transport . . . . . . . . . . . . .
2.3 Timing measurement . . . . . . . . . . . . . . . .
2.3.1 Method overview . . . . . . . . . . . . . .
2.3.2 Timing detector resolution . . . . . . . . .
2.3.3 Timing measurement efficiency . . . . . .
2.4 Magnetic rigidity measurement . . . . . . . . . .
2.4.1 Method overview . . . . . . . . . . . . . .
2.4.2 Electron production . . . . . . . . . . . . .
2.4.3 Electron transport . . . . . . . . . . . . .
2.4.4 MCP position determination . . . . . . . .
2.4.5 Position measurement efficiency . . . . . .
2.5 S800 focal plane . . . . . . . . . . . . . . . . . . .
2.5.1 Beam tracking . . . . . . . . . . . . . . . .
2.5.2 Isotope identification . . . . . . . . . . . .
2.5.3 Isomer detection . . . . . . . . . . . . . .

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20
20
23
23
24
25
26
27
30
32
36
36
41
43
45
47
50
50
53
54

Chapter 3 Data analysis . . . . . . .
3.1 Timing . . . . . . . . . . . . . . .
3.1.1 Linear time calibration . .
3.1.2 Nonlinear time calibration

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56
57
58
60

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vii

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64
68
69
72
74
77
78
80
80
82
82
85
91
97
97
102
107
110
116
120
126
126
130
135
145
157

Chapter 4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1 Comparison to global mass models . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Model descriptions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1.1 Liquid drop model . . . . . . . . . . . . . . . . . . . . . . .
4.1.1.2 FRDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1.3 WS3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1.4 Duflo-Zuker . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1.5 HFB-21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1.6 Density functionals . . . . . . . . . . . . . . . . . . . . . . .
4.1.2 Mass comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Nuclear structure applications . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Termination of the N = 28 closed shell . . . . . . . . . . . . . . . . .
4.2.2 Structure in the Cr isotopic chain approaching N = 40 . . . . . . . .
4.3 Applications to the accreted neutron star crust . . . . . . . . . . . . . . . . .
4.3.1 Quiescent cooling neutron stars and nuclear heating and cooling . . .
4.3.2 Heating and cooling in the neutron star crust associated with electron
captures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

165
168
169
169
172
174
176
178
180
180
194
194
200
204
204

3.2

3.3

3.4

3.5

3.6
3.7
3.8
3.9

3.1.3 Cuts on timing spectra . . . . . . . . . .
Event TOF construction . . . . . . . . . . . . .
3.2.1 Main event-TOF component . . . . . . .
3.2.2 Scintillator position correction . . . . . .
3.2.3 Event-TOF offset . . . . . . . . . . . . .
3.2.4 Selection of the best event-TOF . . . . .
Energy loss . . . . . . . . . . . . . . . . . . . .
3.3.1 Ionization chamber ∆E . . . . . . . . .
3.3.2 S800 fast-timing scintillator ∆E . . . . .
Particle identification . . . . . . . . . . . . . . .
3.4.1 PID verification . . . . . . . . . . . . . .
3.4.2 PID gates . . . . . . . . . . . . . . . . .
3.4.3 Charge state contamination . . . . . . .
Magnetic rigidity determination . . . . . . . . .
3.5.1 Position determination principle . . . . .
3.5.2 Position resolution . . . . . . . . . . . .
3.5.3 Identification of scattering on collimator
Local rigidity correction . . . . . . . . . . . . .
Global rigidity correction . . . . . . . . . . . . .
Final TOF determination . . . . . . . . . . . .
Mass fit . . . . . . . . . . . . . . . . . . . . . .
3.9.1 Reference nucleus selection . . . . . . . .
3.9.2 Mass fit procedure . . . . . . . . . . . .
3.9.3 Mass fit function selection . . . . . . . .
3.9.4 Mass fit uncertainty . . . . . . . . . . .
3.9.5 The mass fit as a diagnostic tool . . . . .

viii

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221

Chapter 5 Future prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix A Magnet-yoke assembly . . . . . . . . . . . . . . .
Appendix B Local TOF–YMCP corrections . . . . . . . . . . .
Appendix C Global TOF–YMCP corrections . . . . . . . . . .
Appendix D Potential TOF mass measurement improvements

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244
245
252
254
256

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

ix

LIST OF TABLES

Table 1.1

Table 3.1

Quantities of interest which describe main characteristics of the TOFI,
SPEG, and NSCL TOF mass measurement setups. . . . . . . . . . .

19

Expected time-of-flight for nuclei calculated from Equation 3.9 assuming Lpath = 60.57 m and Bρ = 3.95024 Tm. . . . . . . . . . . .

76

Table 3.2

Final mass-fit parameters for Equation 3.17 with reference masses
removed from the fit to test fit robustness. . . . . . . . . . . . . . . 145

Table 3.3

Mass-fit parameters for the set of ‘best’ fits identified in the mass fit
(See Section 3.9.3.) as identified by their highest-order in z = Z − Z ,
τ = TOF − TOF , or a combination of the two. The corresponding
fit functions are written-out in the captions of Figures 3.56, 3.57, 3.58,
3.59, and 3.60. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

Table 4.1

Atomic mass excesses (in keV) obtained in this experiment compared
to the adopted experimentally-based value in the 2012 Atomic Mass
Evaluation [1] and predictions from global mass models (See Section 4.1.1.) FRDM 1995 (FRDM) [2], HFB-21 [3], 10-parameter
Duflo-Zuker (DZ) [4], and WS3 [5]. The asymmetric uncertainty
included for the 56 Sc mass excess is an additional systematic uncertainty from potential isomeric contamination (See Section 4.3.1.).
Nuclei whose masses have not been previously measured are marked
with †. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

x

LIST OF FIGURES

Figure 1.1

Figure 1.2

Figure 1.3

Figure 1.4

Figure 1.5

Energy required to remove two neutrons from a nucleus as calculated
by mass differences S2n (See Equation 4.3.) for neutron-rich isotopes
of neon through titanium. Clear patterns emerge, particularly the
rapid drop-offs after N = 20 for phosphorous P through titanium Ti
and N = 28 for isotopes of potassium K through titanium. (From [6].)

3

Demonstration of the 68 Se–69 Br mass difference Sp impact on a calculated X-ray burst light curve. The black lines demonstrate the
uncertainty in the X-ray burst light curve for a 69 Br mass uncertainty of ±300 keV, where it is possible that significant two-proton
capture occurs on 68 Se to make 70 Kr. An uncertainty reduction to
±42 keV makes it clear that proton emission of 69 Br primarily occurs
rather than proton capture after the 68 Se(p, γ) reaction occurs in this
calculation of the rp-process reaction network. (Adapted from [7].)

6

Comparison of typical observation times for a single event and ultimately achievable precision for the main nuclear mass measurement
techniques currently employed for radioactive nuclei. Typical values are shown for Penning traps (PT) [8], storage rings employing
Schottky mass spectrometry (SR-Sch) [9], storage rings employing
isochronous mass spectrometry (SR-IMS) [10], multi-reflection timeof-flight devices (MR-TOF) [11], and direct time-of-flight measurements (TOF) [6]. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

The trend in exoticity ǫ, which quantifies the relative accessibility of a
nucleus for mass measurement, as a function of the ratio of nucleons
A to protons Z for neutron-rich isotopes of neon, titanium, iron,
and tin. When experimental values [12] are unavailable, theoretical
β-decay half-lives and the neutron drip-line used in [13] were used.
Filled shapes indicate nuclei with experimentally known masses [1].
(From [6].) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

Exoticity and measurement precision for Penning trap and TOF mass
measurements existing to date. Penning trap data are from the
ISOLTRAP [14], JYFLTRAP [15], SHIPTRAP [16], LEBIT [17],
TITAN [18], and CPT [19] setups, while TOF data are from the
TOFI [20], SPEG [21], and NSCL [22] setups (See Section 1.4.2.).
(From [6].) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

xi

Figure 1.6

Figure 1.7

Figure 1.8

Figure 2.1

Exoticity of nuclei present in the rp-process (a), r-process (b), and
neutron star crust (c) reaction networks, using the networks from [23],
[24], and [25], respectively. Nuclei with known mass as of [1] are represented by the red-filled regions, while nuclei with no experimentally
determined mass are represented by the cross-hatched regions. . . .

14

Schematics of the TOFI, SPEG, and NSCL TOF mass measurement
setups. Letters indicate the following components: C collimator, D
dipole magnet, ∆E energy loss detector, EF electrostatic filter, H hodoscope, MF magnetostatic filter, P position detector, PB primary
beam, Q quadrupole magnet, QT quadrupole magnet triplet, S sextupole magnet, ST stopper, T timing detector, TG production target,
TKE total energy detector, and W wedge. (From [6].) . . . . . . .

18

Nuclei measured by TOF mass measurement setups at TOFI, SPEG,
and NSCL (as of 2013) as indicated by neutron number N and proton
number Z. For reference, the neutron and proton closed shells at 8,
20, 28, and 50 are indicated. Note that several nuclei have been
measured by more than one facility. (From [6].) . . . . . . . . . . .

19

Schematic of the time-of-flight mass measurement set-up at the National Superconducting Cyclotron Laboratory (NSCL). The coupled
K500 and K1200 cyclotrons [26] provided a 140 MeV/u 82 Se32+ beam
which was alternately impinged on beryllium targets of 517 mg/cm2
and 658 mg/cm2 thickness. The secondary beam fragments were
transported through the A1900 fragment separator [27] where a 7.2 mg/cm2
Kapton wedge degrader removed the high flux of low-Z nuclei that
would otherwise have complicated fragment identification. The timeof-flight TOF was measured by fast-timing scintillators (See Section 2.3.) at the A1900 and S800 focal planes and a relative measurement of magnetic rigidity was performed at the base of the S800
spectrograph (See Section 2.4.). Beam fragments were identified using
the TOF and the energy-loss information from the ionization chamber located in the S800 focal plane [28] (See Section 2.5.2). Note
that the A1900 timing signal was the TOF ‘stop’ signal and the S800
timing signal was the TOF ‘start’ in the experiment data acquisition
(See Section 2.3.1.). (From [29].) . . . . . . . . . . . . . . . . . . . 21

xii

Figure 2.2

Diagram of electronics scheme used for the time-of-flight mass measurement (NSCL experiment E09039). Timing and charge signals
were obtained from photomultiplier tubes (PMTs) located in the
A1900 and S800 focal planes as well as from the microchannel plate
detector (MCP). The four charge signals were obtained from the MCP
corners were amplified by two different gains. Detection rates on all
detectors were monitored by scalers. An S800 timing signal was generally used as the event trigger, though the MCP was used as the
event trigger for the mask-run made to assess the beam positionmeasurement resolution. (Contact the author for an electronic form
of this figure.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

Figure 2.3

Particle identification (See Section 2.5.2.) plot of nuclei produced in
the time-of-flight (TOF) mass-measurement experiment, where the
color indicates production intensity (counts per 100 picoseconds×10 ionizationchamber-adc-units) and TOF was not rigidity-corrected (See Sections 3.6 and 3.7.). Nuclei located to the right of the red-line had no
known experimental mass prior to the mass-measurement reported
here; 50 Ca, 54 Ca, 65 Fe, and 69 Fe are labeled for reference. . . . . . . 26

Figure 2.4

Photograph of a single BC-418 ultra-fast timing scintillator from
Saint-Gobain Crystals [30] coupled to two R4998 head-on photomultiplier tubes (PMTs) via small plastic light-guides and optical grease.
An aluminum frame helped hold the PMTs to the scintillator and allowed for a collimator to be mounted on the upstream side to protect
the PMTs from stray beam. (Photo Credit: Milan Matoˇs and Alfredo
Estrad´e [31].) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

Simplified picture of four measured beam events of the same isotope
which would yield different times-of-flight TOFs without the scintillator position correction. Here four events (a, b, c, and d) of the same
isotope follow different paths from the scintillator in the A1900 to
the scintillator in the S800, where the flight path has been simplified
to a straight line. Assigning a direct flight time TOFdirect =50 ns
and noting that the light-travel time across the 1.5 cm scintilla1.5 = 79 ps, where n = 1.58 is the refractive index
tor is dt = c/n
of BC-418 [30] and c is the speed of light in vacuum, the directdown TOF TS3D−XFD for the four events would be TD,a = 50.000 ns,
TD,b = 49.921 ns, TD,c = 50.079 ns, and TD,d = 50.000 ns without
a scintillator position correction. Applying the correction described
in Section 3.2.2, +0.5 × (TXFU−XFD − TS3U−S3D ), corrects all four
flight times to 50.000 ns. . . . . . . . . . . . . . . . . . . . . . . . .

30

Figure 2.5

xiii

Figure 2.6

Figure 2.7

Figure 2.8

Figure 2.9

Cartoon of the dispersive plane position measurement system. Secondary electrons generated by the ion beam passing through a gold
foil were guided to a MCP by an electric and magnetic field to measure the beam position at the foil. (From [29]). . . . . . . . . . . . .

37

Photograph of the magnetic rigidity measurement setup. Two magnets were mounted on an steel yoke plane-parallel to an 8×10 cm
microchannel plate detector (MCP). A gold foil was mounted planeparallel to the MCP on a driver-arm so that it could be inserted from
below. A foil and hole-mask were also mounted on the driver arm
below the first foil, where hole-mask upstream of the foil in order to
only allow ions to generate electrons at certain positions for a MCP
position calibration . A collimator was located upstream to protect
the MCP during beam-alignment. A hole-mask with a distinctive
hole-pattern and a phosphorescent coating was mounted on a driver
arm further upstream so that it could be inserted for beam-alignment.
A camera was located within vacuum chamber further upstream and
off of the beam-axis to provide an image of the hole-mask during
beam-alignment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

Measured (solid circles) magnetic field as a function of distance from
the left magnet in the rigidity measurement set-up, using a magnetsurface separation of 17.5 cm, compared to the calculated field value
(See Equation 2.4.) for that magnet separation (solid red curve).
The dashed blue curve indicates the calculated magnetic field value
for the magnet separation used in the mass measurement experiment.
Vertical dashed lines indicate the position of various components used
in the position (rigidity) measurement with respect to the magnets.

39

Cartoon of the MCP position determination method. An electron
freed from the gold foil by a passing beam particle strikes the wall of
a microchannel (µchannel in the diagram), creating secondary electrons. Secondary electrons are accelerated by a voltage difference
(∆V) applied across the plate (MCP) and strike the microchannel
wall, creating more secondary electrons [32]. The avalanche of electrons ultimately reaches the resistive backplane, where charges are
free to drift to the four corners of the backplane. The position at
which charge is initially injected onto the backplane, which corresponds to the position of the microchannel and thus the position of
the beam on the gold foil, is determined by comparing the relative
amount of charge collected on each of the four backplane corners.
Note: A single plate with straight channels is pictured here for simplicity. This experiment employed two MCPs with angled channels
stacked in a way known as the chevron configuration [33]. . . . . .

49

xiv

Figure 2.10

Figure 2.11

Figure 2.12

Figure 3.1

Schematic of the detectors within the focal plane (a) of the S800
spectrograph (b) (Adapted from [28] and [34].). Following the rigidity measurement at the base of the S800 (See Section 2.4.), the beam
was steered ∼150◦ into the S800 focal plane. Cathode readout drift
counters (CRDC) were used for beam tracking (See Section 2.5.1.), a
fast-timing scintillator was used for the TOF-start (See Section 2.3.),
an ionization chamber was used to measure energy loss (See Section 2.5.2.), and an aluminum plate (‘ion catcher’) was used in conjunction with a hodoscope to detect isomers of isotopes with atomic
mass A 40 (See Section 2.5.3.). . . . . . . . . . . . . . . . . . . .

51

Photograph of detectors within the S800 spectrograph focal plane
(center) with insets of the ion catcher and hodoscope (left) and fasttiming scintillator mounted to two photomultiplier tubes (See Section 2.3.) downstream of the first cathode readout drift counter
(CRDC) (right). Note that the ion catcher and fast-timing scintillator were not installed in the focal plane at the time the center image
was taken. (Center photo credit: Shumpei Noji.) . . . . . . . . . . .

52

Demonstration of the energy-loss ∆E versus time-of-flight TOF particle identification method (‘PID’). Since ∆E ∝ A2 and T OF ∝ A/Z,
where A is the atomic mass number and Z is the atomic number, an
idealized ∆E vs. TOF PID is represented by plotting A2 versus A/Z
for several combinations of A and Z. It is apparent that each A,Z
combination occupies a unique location in the PID matrix. (For the
PID from this experiment, see Figure 2.3.) . . . . . . . . . . . . . .

55

Raw time spectra accumulated during the time-of-flight mass measurement experiment (gray-filled histograms) and linear time-calibration
spectra (red-filled histograms) for the corresponding TAC–ADC time
signal. The timing signals (a) through (i) are MCP–Clk, XFD–Clk,
S3U–Clk, S3D–Clk, S3U–S3D, XFU–XFD, S3U–XFU, S3D–XFD,
and XFU–Clk, respectively. The black-dashed vertical lines in (c)
indicate the peaks used for time calibration. Each spectrum is a histogram of counts per ADC channel (See Section 2.3 for information
on data acquisition.). Note that the full-scale range for (a) through
(f) and for (i) is ≈ 50 ns while the full-scale range for (g) and (h) is
≈ 100 ns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

xv

Figure 3.2

Figure 3.3

Figure 3.4

Figure 3.5

Spectrum of counts per channel for the S3D–Clk ADC for the full
set of experimental data. Red-dashed horizontal lines indicate the
number of mean counts per channel and the expected statistical deviation from the mean. The semi-periodic deviations from the mean
are evidence of nonlinearities in the time-to-channel relationship for
this TAC–ADC combination. . . . . . . . . . . . . . . . . . . . . .

60

Residual from the linear time-calibration for the linear (cyan circles)
and nonlinear (black crosses) time calibrations for the XFD–Clk (a),
S3U–Clk (b), S3D–Clk (c), and XFU–Clk (d) timing signals. It is
apparent that significant nonlinearities existed in the time-to-channel
response of the four TAC–ADC combinations, particularly in the case
of the XFU–Clk timing signal. Since the XFU–Clk spectrum was
discarded (See Figure 3.5.), this was not an issue. . . . . . . . . . .

62

Spectra used to identify a timing jitter for the Clock Down TOF
(See Equation 3.5) using the nonlinear time calibration. The left
panel shows the nonlinear time-calibrated Clock Down TOF vs the
Direct Down TOF and it is apparent that jitters on the order of
several nanoseconds are present. The right panel shows the linear
time-calibrated Clock Down TOF vs the Direct Down TOF and no
jitters are present, implying that the jitter in the left panel is due to
the nonlinear time-calibrated Clock Down TOF. . . . . . . . . . . .

63

Spectra used to identify a ∼1.25 ns timing jitter in the XFU–Clk
timing signal. The left panel contains the spectrum created by taking
the difference between the XFU–Clk and XFD–Clk timing signals for
events of 45 Ar, which in principle should contain only one peak with
a width of ≈180 ps. The left-panel inset shows the existence of a
∼1.25 ns jitter in one of these two timing signals. The right panel
and its inset demonstrate the lack of any jitter in the spectrum created
from the XFD–Clk and S3D–Clk timing signals, where the existence
of two peaks in the main figure in the right panel is due to the fact
that two different integer number of clock pulses could have elapsed
during the 45 Ar flight-time. In order to demonstrate the lack of a
jitter in the right panel, the globally determined rigidity correction
(See Section 3.7.) had to be applied to the spectrum to remove the
∼2 ns spread caused by the rigidity-spread of 45 Ar events. It is
apparent that the signal which contained the jitter must have been
the XFU–Clk timing signal. Note that the grass-like background are
random coincidences, which were much more prevalent in the A1900
signals due to the higher rate at that point in the beam line. . . . .

65

xvi

Figure 3.6

Figure 3.7

Figure 3.8

Figure 3.9

Figure 3.10

Spectrum used to identify the several nanosecond timing jitter in the
XFU–XFD timing signal. The inset highlights the unphysical timedifference recorded for ≈8% of non-background events, where lighttravel time in the plastic scintillator limited the possible time-window
for valid events. Note that the double-hump feature in the main
peak was not problematic, since it was counteracted by the opposite
double-hump feature present in the S3U–S3D spectrum; i.e. when the
XFU–XFD and S3U–S3D times were employed to make a scintillator
position correction to TOF (See Section 3.2.), their combined timeshift produced a single narrow-peaked distribution free of any doublehump-like features. . . . . . . . . . . . . . . . . . . . . . . . . . . .

67

Demonstration of the principle of clock pulse correction for ‘Clock’
times. Since the clock pulses arrived at random times with respect
to the scintillator signals, it is clear that different integer numbers of
clock periods could have elapsed during an ion’s time-of-flight. In this
example, either 12 or 13 clock pulses could have elapsed during an
ion’s 500 ns flight time. As such, two clock times would be possible
for the events of such an ion and a correction would be required to
obtain a single TOF for events of that ion. . . . . . . . . . . . . . .

70

Spectra employed for ‘Clock Down’ time clock pulse correction. The
time difference between a direct TOF, here ‘Direct Up’, and a clock
time, here ‘Clock Down’ (panel e), results in multiple peaks (panels
a–c) spaced by the clock period τ =40 ns. Narrow gates around the
peaks were used to remove background and to determine the clock
pulse correction that was to be added to a given event. Panel d
demonstrates the fact that events of an ion with a single direct TOF
could result in multiple clock TOFs. The black histograms in panels
a–c and e are gated on events of 45 Ar, while the red histograms are
for all events. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

Schematic illustrating the need for a scintillator position correction
to the main TOF component. The desired time difference (the true
TOF) is the difference in ion-impact times for the A1900 and S800
scintillators. However, for a Down TOF, the main component recored
for TOF would be the difference in signal generation times at the
A1900 and S800 scintillators. To obtain the true TOF from the
recorded TOF, a correction of +0.5 × (TXFU−XFD − TS3U−S3D ) had
to be applied to the main recorded TOF. . . . . . . . . . . . . . . .

73

Rigidity corrected ‘Direct Up’ TOF distribution for 45 Ar events before (red-filled histogram) and after (black unfilled histogram) application of the scintillator position correction to TOF given in Equation 3.8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75

xvii

Figure 3.11

Figure 3.12

Figure 3.13

Figure 3.14

Figure 3.15

Figure 3.16

Spectra used to identify a timing jitter for the Direct Down TOF
(See Equation 3.2). The left panel shows the Direct Up TOF vs the
linear time-calibrated Clock Down TOF, where it is apparent that no
jitters were present. The right panel shows the Direct Up TOF vs
the Direct Down TOF, where jitters are present, implying that the
jitter was due to the Direct Down TOF. . . . . . . . . . . . . . . .

78

Energy loss in individual ionization chamber segments versus the sum
of energy loss in the full ionization chamber for a subset of the full
data-set. Events in the region where surplus events were present in a
single ionization chamber energy loss segment(∆E1 segment < 150 or
> 3100) were excluded from the analysis. . . . . . . . . . . . . . . .

79

Energy loss in the ionization chamber and in the S800 fast-timing
scintillator for a subset of the full data-set. A subset of 45 Ar events
are shown for comparison. Events outside of the region bounded by
the two black lines were excluded. . . . . . . . . . . . . . . . . . . .

81

Particle identification matrix produced by LISE++ simulations using
the time-of-flight mass measurement experimental set-up, where the
results for the thick and thin targets (See Section 2.2.1.) have been
combined. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

A demonstration of one method of identifying nuclei within the energy
loss time-of-flight, ∆E–TOF particle identification matrix. Nuclei at
certain A/Z, such as 2.5 (solid black line) and 2.6 (dashed black
line), are expected to be vertically aligned as was demonstrated in
Figure 2.12, so an educated guess can be made as to which vertically
aligned nuclei correspond to which A/Z. Since ∆E ∝ A2 , with a
small Z and velocity dependence, it is expected that lines of constant
Z lay along a diagonal shallow slope (diagonal red lines). An educated
guess can be made as to which sloping line is which Z and then it
can be checked if the resulting identification matrix is consistent; e.g.
75
30
35
here 50
20 Ca and 30 Zn lay along A/Z = 2.5 (along with 12 Mg, 14 Si,
40 S, 45 Ar, 55 Ti, 60 Cr, 65 Fe, and 70 Ni) while 52 Ca and 78 Zn lay along
16 18
22
24
26
28
20
30
65 Mn). . . . . . . . . . . . . . . . .
A/Z = 2.6 (along with 39
P
and
15
25

84

Comparison of the three PID gates looked into for the mass measurement data analysis pipeline. A ‘generous’ hand-gate (red lines),
‘conservative’ hand-gate (black lines) which closely traced the main
outline of PID blobs, and automated gate (brown lines) drawn on a
PID transformed into Z and A/Z were each employed and the results
were compared. The inset shows the gates in more detail for isotopes
of argon through vanadium. . . . . . . . . . . . . . . . . . . . . . .

86

xviii

Figure 3.17

Figure 3.18

Figure 3.19

Figure 3.20

Impact of the choices of three gates on the rigidity-corrected TOF
distribution of 45 Ar (The gates are shown in Figure 3.16.). The ‘generous’ hand-cut (red thick-lined histogram), ‘conservative’ hand-cut
(blue medium-lined histogram), and automated cut (thin black-lined
histogram) had minor deviations in the total number of counts and
had no measurable difference in their mean or standard deviation.
The left panel highlights the slight increase in total counts the generous cut provided, while the log-scale on the right panel highlights
the small high-TOF tail which was removed by the conservative cut.

87

√
Demonstration of the proportionality between Z and ∆E/TOF
(right panel) and between A/Z and TOF (left panel). ∆E and TOF
were the mean values obtained from the TUpDir vs ionization chamber
∆E for the nuclei corresponding to each Z and A. . . . . . . . . .

89

Particle identification matrix where ∆E and TOF have been employed to obtain Z and A/Z. A projection onto the Z-dimension
yielded a one-dimensional histogram which could be gated on as a
Z-cut (See Figure 3.20.). . . . . . . . . . . . . . . . . . . . . . . . .

89

Projection onto the Z-dimension of the PID that was transformed
into Z vs A/Z coordinates (See Figure 3.19.). Gaussian fits to each
peak are delineated by color. The thick-dashed and thin-solid vertical lines of the same color indicate the ±2.5σ and ±3σ locations for
the Gaussian fit, respectively. The symbols of the same color for a
given peak indicates the locations at which the ratio of the number of
counts in that peak to the number of counts in the neighboring peak
(as determined by the Gaussian fits) is 10,000:1 (triangle), 1,000:1
(square), and 100:1 (circle) to give a feel for potential contamination
from neighboring-Z nuclei (Note that the majority of this contamination is rid of by the cuts on non-dispersive microchannel plate position
[See Section 3.5.3.] and the local rigidity correction procedure [See
Section 3.6.].). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

90

xix

Figure 3.21

Figure 3.22

Figure 3.23

Subsection of the non-rigidity corrected particle identification matrix
(with a cut on the non-dispersive position of the microchannel plate
detector described in Section 3.5.3) showing isotopes of iron and manganese. The black outline traces the majority of events attributed to
fully ionized 66 Fe, the dotted outline is the same shape shifted in
TOF to mimic the effect of changing to a 25+ charge state in the
gold foil at the S800 target position (Note that these events should
not have been within the acceptance of the spectrometer, since a large
change in p/q would be suffered due to the charge-change.), and the
dashed line mimics the same thing for electron-pickup prior to or at
the A1900 fast-timing scintillator. It is apparent that there was not
a significant amount of 25+ charge state production for 66 Fe. . . .

95

The left panel shows a subsection of the PID, where example PID
gates have been drawn around 69 Co (black-dotted line), 72 Ni (reddashed line), and 75 Cu (solid-blue line). The right panel shows the
same PID subsection, where events identified as 69 Co (black dots),
72 Ni (red dots), and 75 Cu (blue dots) have been corrected for their
magnetic rigidity, Bρ. It is apparent that the overlap between nuclei
in the non-Bρ corrected PID (gray ovals) leads to improperly Bρcorrected events in the Bρ-corrected PID, which may be interpreted
as charge states when they are in fact not. The misidentified events
wind up being located near events of the nucleus to which they properly belong because the TOF–Bρ relationship varies smoothly with
Z and A/Z (See Section 3.7); however, the correction for neighboring
nuclei is different enough that the improperly corrected events are not
quite in-line with the properly identified events. These misidentified
events are not problematic, since they are removed from the TOF
distribution, as described in Section 3.6. . . . . . . . . . . . . . . .

96

The upper four panels compare the high-gain and low-gain corner
signals for each event for the upper left (UL), upper right (UR),
lower right (LR), and lower left (LL) corners of the MCP, where the
pedestals have been subtracted. It is clear that the high-gain signal
is saturated for all but the weakest of low-gain signals. The lower
four panels show the low-gain and high-gain corner signals recorded
for each event on the four MCP corners, where the high-gain signal
(thin black-lined histogram) has been corrected to overlay the lowgain signal (thick red-dashed histogram). Figure 4.20 of [35] shows
similar MCP corner signal gain-matching results for that work. . .

99

xx

Figure 3.24

Panel (a) shows the mask with a distinctive hole pattern (5 mm holespacing) which was placed in between the incoming ion and gold foil
in order to only allow electrons to be created from certain locations
for calibration runs. Panel (b) shows the image created on the MCP
by electrons generated from a 232 Th α-source. Panels (c) and (d)
show the image created by the electrons generated by the 82 Se primary beam, where the beam was tuned to four separate positions to
achieve the mask-coverage shown (The mask position is indicated in
Figure 2.7.), where the low-gain corner signals were used for panel
(c) and the combined high-low gain signals were used for panel (d).
Since only the relative position was relevant, the effort was not made
to achieve the exact 5 mm hole-spacing of the mask in the MCP
image. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

Figure 3.25

Projection onto the Y (dispersive) dimension for MCP-positions obtained from the α-source hole-mask measurement (Figure 3.24b) for
the column of holes with < X >∼ 0 mm (left panel) and onto the
X (non-dispersive) dimension for the row of holes with < Y >∼
−15 mm (right panel). The peak resolution and statistics were affected by the size of the corresponding hole (seen in Figure 3.24b).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

Figure 3.26

Projection onto the Y (dispersive) dimension for MCP-positions obtained from the 82 Se primary beam hole-mask measurement (Figure 3.24c) for the column of holes with < X >∼ −8 mm (left panel)
and onto the X (non-dispersive) dimension for the row of holes with
< Y >∼ 0 mm (right panel). The peak resolution and statistics were
affected by the size of the corresponding hole (seen in Figure 3.24c).

103

Figure 3.27

Demonstration of the correlation between high energy-loss (∆E) PID
events and the microchannel plate (MCP) non-dispersive position.
The left panel shows a subset of the PID containing isotopes of calcium, scandium, and titanium, where ‘main’ events are within the
purple box and ‘top-hat’ events are within the red-dashed box. The
right panel shows the location of the ‘main’ (purple dots) and ‘tophat’ (red dots) events on the MCP, where it is clear that the relatively
high ∆E events corresponded to larger non-dispersive positions. . . 108

Figure 3.28

Side (a) and upstream (b) views of the collimator intended to protect
the microchannel plate detector (MCP) during beam tuning. We
determined that beam fragments likely scattered on the right-wall
(looking from the upstream direction) of the collimator opening (See
Figure 3.27.). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

xxi

Figure 3.29

Particle identification matrix before (red dots) and after (black dots)
the cut on the microchannel plate detector non-dispersive position,
X < −11 mm. The reduction in relatively high-∆E events for individual nuclei is apparent, though it is less clear for the high-statistics
events. The relationship between the high-∆E events and X is shown
in more detail for isotopes of calcium, scandium, and titanium in Figure 3.27. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

Figure 3.30

The five rows show the successive steps taken in the local rigidity
correction procedure to remove contamination from neighboring nuclei in the PID and determine the slope of the TOF vs YMCP of a
single nucleus, here 68 Fe. The upper left panel shows a histogram of
TOF vs YMCP for events identified as 68 Fe, where the black points
are the resultant graph obtained by applying ROOT’s TProfile class
to the histogram. The black line is a linear fit to the graph. The
upper middle panel shows the resultant rigidity-corrected TOF vs
YMCP histogram after removing the linear trend found in the upper
left panel, pivoting about YMCP =0. The upper right panel shows
the projections onto the TOF-dimension of the uncorrected (red histogram) TOF vs YMCP relationship, which was not easily visible on
the same scale for 68 Fe, and rigidity corrected (black histogram) TOF
vs YMCP relationship, where the blue line is a Gaussian fit to the
rigidity corrected histogram. The second row contains the same information as the first, but after applying a cut to only include data
within ±4σ of the mean of the rigidity corrected TOF distribution
determined in the first row. The following rows contain the same
information after applying cuts to only include data within 3, 2.5,
and 2σ, respectively, of the mean rigidity corrected TOF, where the
mean and σ were determined by the Gaussian fit in the previous row.
Appendix B contains similar plots for all isotopes of elements with
11 ≤ Z ≤ 29 observed in this experiment. . . . . . . . . . . . . . . 113

Figure 3.31

Residual between the TOF vs YMCP slope determined ‘locally’ (i.e.
by-isotope) and the ‘global’ value determined from the fit with Equation 3.15 to all local slopes with 18 ≤ Z ≤ 26 and A/Z > 2.44. It
is apparent that the majority of isotopes were fit within 1% (Note
that the average slope was ≈ 0.40 ps/mm, as seen in Figure 3.32.)
and that there was a significant statistically-distributed systematic
scatter for each element as a function of A/Z. . . . . . . . . . . . . 117

xxii

Figure 3.32

TOF vs YMCP slope as a function of mass number A for observed isotopes of argon, potassium, calcium, scandium, titanium, vanadium,
chromium, manganese, and iron (18 ≤ Z ≤ 26), respectively, as determined by ‘local’ by-nucleus fits (data points) (See Section 3.6.)
and fits to the locally-determined slopes that employed the ±2σ cutoff (See Figure 3.30.), where the black data points were included in
the fit and the blue points were not. It is apparent that in general
the locally determined slopes for the 3σ (circles), 2.5σ (squares), and
2σ (triangles) cut-offs were in agreement. The by-element fit along
a single isotopic chain as a cubic function of A is shown by the red
lines, where the upper and lower lines indicate the extremes obtained
for upper and lower limits of the fit-parameters, and the orange band
indicates the ±1σ confidence interval. The black line shows the trend
of the rigidity-correction slope along an isotopic chain as determined
by the global fit (See Equation 3.15.) to all locally-determined slopes
of nuclei with A/Z > 2.44 and 18 ≤ Z ≤ 26. . . . . . . . . . . . . . 119

Figure 3.33

Examples of global rigidity-corrected (See Section 3.7.) TOF distributions for some m
q (TOF)-calibration nuclides (unfilled histograms) and
nuclides whose mass was evaluated (filled histograms). The Gaussian
fits which are shown demonstrate the Z-dependent skewness present
in the TOF distributions. See Appendix C for final TOF distributions
of all nuclides involved in the mass-fit and mass evaluation. . . . . 121

Figure 3.34

‘Down Clock’ TOF mean values (See Section 3.2.) obtained with different rigidity corrections shown as a difference to the values obtained
with the global rigidity correction as functions of the atomic mass
number to nuclear charge ratio A/Z. Shown are TOFs obtained via
the local (black circles), by-element (cyan squares), and global (red
lines) rigidity corrections (See Sections 3.6 and 3.7.), using the ±3σ
range for observed nuclei with 18 ≤ Z ≤ 26 and A/Z > 2.44. The
average deviation between global and local TOFs and between global
and by-element TOFs was 2.3 ps and 0.3 ps, respectively. . . . . . . 122

xxiii

Figure 3.35

Difference in TOF between the global rigidity-corrected ‘Clock Down’
and ‘Direct Up’ TOFs, with their uncertainties added in quadrature, after correcting for the trivial offset between the two (See Section 3.2.), using the ±3σ range for observed nuclei with 18 ≤ Z ≤ 26
and A/Z > 2.44. The left, middle, and right panels show the TOF
difference as a function of the atomic mass number to nuclear charge
ratio A/Z, nuclear charge Z, and nuclear mass A, respectively. Note
that the TOF differences do not correspond to final mass differences,
since each mass-TOF surface was fit separately, ultimately yielding
similar mass results (See Section 3.9.5.), albeit with different uncertainties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

Figure 3.36

Number of observed events (indicated by the color) for observed nuclei
with 18 ≤ Z ≤ 26 and A/Z > 2.44, within the ±4σ TOF distribution
range that was used for the global rigidity correction (See Section 3.7.)
of the Down Clock time (See Section 3.2.). Note that only nuclides
with ≈ 500 events or more are plotted. . . . . . . . . . . . . . . . . 124

Figure 3.37

Mean TOF uncertainty in picoseconds (indicated by the color) of
observed nuclei with 18 ≤ Z ≤ 26 and A/Z > 2.44, using the global
rigidity correction (See Section 3.7.) for the Down Clock time (See
Section 3.2.) with the ±4σ TOF distribution range. . . . . . . . . . 125

Figure 3.38

Standard deviation of final TOF distributions in picoseconds (indicated by the color) of observed nuclei with 18 ≤ Z ≤ 26 and
A/Z > 2.44, using the global rigidity correction (See Section 3.7.) for
the Down Clock time (See Section 3.2.) with the ±4σ TOF distribution range. The average standard deviation of the TOF distributions
was 80.8 ps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

Figure 3.39

Map of nuclei observed in the TOF mass measurement analysis (with
sufficient statistics to obtain a TOF value) in terms of atomic mass
number to nuclear charge ratio A/Z and nuclear charge Z. Solid
black circles indicate reference nuclei, open blue circles indicate nuclei with masses known in the literature, but not to sufficient precision
to qualify as reference nuclei, and red stars indicate nuclei with unknown mass as of the completion of the data analysis. (Compare to
Figure 4.34 of [31].) . . . . . . . . . . . . . . . . . . . . . . . . . . 129

Figure 3.40

Mass–TOF surface of reference nuclei where the linear dependence of
mass over charge m/q on TOF has been removed. Solid white points
mark the nuclear charge Z and TOF of reference nuclei while the
color of the surface at that location indicates the linear fit residual in
MeV. (Note that the flat region occurs outside of the region bounded
by data points as a feature of ROOT’s ‘SURF’ drawing option.) . . . 131
xxiv

Figure 3.41

Residual of the fit to reference nuclei using the function m
q (τ ) =
a0 + a1 ∗ τ , where τ = TOF − TOF and z = Z − Z , with TOF
being the time-of-flight, Z being the nuclear charge, and the averages
of these being taken over the set of reference nuclei. Reference nuclei
are identified by element by their symbol and the text label indicates
the reference nucleus mass number A. Thick colored error bars show
the statistical uncertainties. Thin black error bars show the sum in
quadrature of the statistical uncertainty and the systematic uncertainty (9 keV/q) applied in the case of the best-fit (See Figure 3.47),
which was used for the mass evaluation. . . . . . . . . . . . . . . . 136

Figure 3.42

Same as Figure 3.41 but for the fit function m
q (τ ) = a0 + a1 ∗ τ + a2 ∗ z. 136

Figure 3.43

Same as Figure 3.41 but for the fit function m
q (τ ) = a0 + a1 ∗ τ +
a2 ∗ z + a3 ∗ τ 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

Figure 3.44

Same as Figure 3.41 but for the fit function m
q (τ ) = a0 + a1 ∗ τ +
2
2
a2 ∗ z + a3 ∗ τ + a4 ∗ z . . . . . . . . . . . . . . . . . . . . . . . . 137

Figure 3.45

Same as Figure 3.41 but for the fit function m
q (τ ) = a0 + a1 ∗ τ + a2 ∗
2
2
z + a3 ∗ τ + a4 ∗ z + a5 ∗ z ∗ τ . Note that this fit was one of the fits
used to evaluate the extrapolation ‘function-choice uncertainty’ (See
Section 3.9.4.). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

Figure 3.46

Same as Figure 3.41 but for the fit function m
q (τ ) = a0 + a1 ∗ τ +
a2 ∗ z + a3 ∗ τ 2 + a4 ∗ z 2 + a5 ∗ z ∗ τ + a6 ∗ z 3 . Note that this fit
was one of the fits used to evaluate the extrapolation ‘function-choice
uncertainty’ (See Section 3.9.4.). . . . . . . . . . . . . . . . . . . . 137

Figure 3.47

Same as Figure 3.41 but for the fit function m
q (τ ) = a0 + a1 ∗ τ +
a2 ∗ z + a3 ∗ τ 2 + a4 ∗ z 2 + a5 ∗ z ∗ τ + a6 ∗ z 4 . Note that this was the
fit function ultimately used for the mass evaluation. . . . . . . . . . 138

Figure 3.48

Same as Figure 3.41 but for the fit function m
q (τ ) = a0 + a1 ∗ τ +
2
2
4
a2 ∗ z + a3 ∗ τ + a4 ∗ z + a5 ∗ z ∗ τ + a6 ∗ z + a7 ∗ τ 3 . . . . . . . 138

Figure 3.49

Same as Figure 3.41 but for the fit function m
q (τ ) = a0 + a1 ∗ τ + a2 ∗
z + a3 ∗ τ 2 + a4 ∗ z 2 + a5 ∗ z ∗ τ + a6 ∗ z 4 + a7 ∗ τ 4 . Note that this fit
was one of the fits used to evaluate the extrapolation ‘function-choice
uncertainty’ (See Section 3.9.4.). . . . . . . . . . . . . . . . . . . . 138

xxv

Figure 3.50

Same as Figure 3.41 but for the fit function m
q (τ ) = a0 + a1 ∗ τ + a2 ∗
z + a3 ∗ τ 2 + a4 ∗ z 2 + a5 ∗ z ∗ τ + a6 ∗ z 4 + a7 ∗ z ∗ τ 2 . Note that this fit
was one of the fits used to evaluate the extrapolation ‘function-choice
uncertainty’ (See Section 3.9.4.). . . . . . . . . . . . . . . . . . . . 139

Figure 3.51

Residuals of the fit (with Equation 3.17) to the time-of-flight of calibration nuclei (See Section 3.9.1.) as a function of the mass number
to nuclear charge ratio A/Z. Isotopes are labeled with their mass
number and symbols indicate the elements (solid circle for argon,
solid square for potassium, solid triangle for calcium, open circle for
manganese, and open square for iron). Calibration masses were fit to
within 9 keV/q without any systematic trends. The gray band shows
the average systematic mass uncertainty included for reference nuclei
as described in Section 3.9.2.(From [36].) . . . . . . . . . . . . . . . 143

Figure 3.52

Residuals of the fit in keV (with Equation 3.17) to the time-of-flight
of calibration nuclei (See Section 3.9.1.) as a function of the mass
number to nuclear charge ratio A/Z and nuclear charge Z. Note
that Z is shifted by 1/2 so that the colored box corresponds to the
leftward-Z value. The text labels and color indicate mass-fit residual
in keV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

Figure 3.53

Panel (a) shows the residuals of the fit (with Equation 3.17) to the
time-of-flight of calibration nuclei (See Section 3.9.1.) as a function
of the mass number to nuclear charge ratio A/Z. Isotopes are labeled
with their mass number and symbols indicate the elements (solid
circle for argon, solid square for potassium, solid triangle for calcium,
open circle for manganese, and open square for iron). Panel (b) shows
the fit residuals obtained using the same fit function as panel (a), but
removing 47 Ar, 54 Ca, 51 K, 66 Mn, and 66 Fe (marked by yellow arrows)
from the set of reference nuclei in order to assess the robustness of
the mass-fit extrapolation in the TOF-dimension, since they were the
most exotic reference nucleus in their isotopic chain. Panel (c) shows
the fit residuals obtained using the same fit as panel (a), but removing
49−51 Ca (marked by yellow arrows) from the set of reference nuclei
to assess robustness of the mass-fit interpolation in the Z-dimension.
Fit parameters for the three fits are compared in Table 3.2. . . . . 146

xxvi

Figure 3.54

Mass distributions resulting from the Monte Carlo TOF-perturbation
procedure for isotopes of argon through vanadium (18 ≤ Z ≤ 23)
which were observed in the TOF mass measurement (See Figure 2.3.)
and which were not reference nuclei for the mass fit (See Section 3.9.1.).
The ‘RMS’ value was the standard deviation of the mass distribution
in keV and was chosen as the ‘Monte Carlo uncertainty’ for the evaluated masses. The vertical red line indicates the evaluated mass from
the mass fit function (Equation 3.17.) without perturbations to the
reference nuclei TOFs. . . . . . . . . . . . . . . . . . . . . . . . . . 149

Figure 3.55

Same as Figure 3.54 for isotopes of chromium through iron (24 ≤
Z ≤ 26). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

Figure 3.56

Mass–TOF surface, where the linear dependence of mass over charge
m/q on TOF has been removed, for reference nuclei (black-outlined
white-filled circles connected by the white planes) and the fit (color
indicates keV/q) to the reference nuclei with the mass-fit function,
m (τ ) = a + a ∗ τ + a ∗ z + a ∗ τ 2 + a ∗ z 2 + a ∗ z ∗ τ + a ∗ z 4 . Note
0
1
2
3
4
5
6
q
that the averages over Z and TOF, used to construct z = Z − Z and
τ = TOF − TOF , were taken over the set of reference nuclei (See
Section 3.9.1.). The red-outlined white-filled circles show the location
of nuclei whose mass were evaluated with the final mass-fit function,
Equation 3.17 (in this case, the same function as the fit shown), in
m/q–TOF–Z space. Note that all points lay on the function surface,
though the plotting program does not extend the graphing of the
surface far enough in some cases. . . . . . . . . . . . . . . . . . . . 152

Figure 3.57

Same as Figure 3.56 for the mass-fit function, m
q (τ ) = a0 + a1 ∗ τ +
a2 ∗ z + a3 ∗ τ 2 + a4 ∗ z 2 + a5 ∗ z ∗ τ + a6 ∗ z 3 . . . . . . . . . . . . 153

Figure 3.58

Same as Figure 3.56 for the mass-fit function, m
q (τ ) = a0 + a1 ∗ τ +
2
2
a2 ∗ z + a3 ∗ τ + a4 ∗ z + a5 ∗ z ∗ τ . . . . . . . . . . . . . . . . . . 154

Figure 3.59

Same as Figure 3.56 for the mass-fit function, m
q (τ ) = a0 + a1 ∗ τ +
2
2
4
a2 ∗ z + a3 ∗ τ + a4 ∗ z + a5 ∗ z ∗ τ + a6 ∗ z + a7 ∗ τ 4 . . . . . . . 155

Figure 3.60

Same as Figure 3.56 for the mass-fit function, m
q (τ ) = a0 + a1 ∗ τ +
a2 ∗ z + a3 ∗ τ 2 + a4 ∗ z 2 + a5 ∗ z ∗ τ + a6 ∗ z 4 + a7 ∗ z ∗ τ 2 . . . . . 156

xxvii

Figure 3.61

Statistical uncertainty in keV for nuclei whose mass was evaluated in
the time-of-flight mass measurement. Colored boxes indicate nuclei
whose mass was evaluated, with the color reflecting the uncertainty in
keV, boxes with red circles indicate reference nuclei used as calibrants
for the mass–TOF relationship (See Section 3.9.1.), boxes with ×’s
indicate the most exotic isotope for that element with a known mass
prior to this experiment, and the black boxes indicate stable nuclei.

158

Figure 3.62

Same as Figure 3.61, but with the color indicating the systematic
uncertainty of evaluated nuclei in keV. . . . . . . . . . . . . . . . . 158

Figure 3.63

Same as Figure 3.61, but with the color indicating the Monte Carlo
uncertainty of evaluated nuclei in keV. . . . . . . . . . . . . . . . . 159

Figure 3.64

Same as Figure 3.61, but with the color indicating the function-choice
uncertainty of evaluated nuclei in keV. . . . . . . . . . . . . . . . . 159

Figure 3.65

Same as Figure 3.61, but with the color indicating the total uncertainty of evaluated nuclei in keV, where the total is the sum in quadrature of the statistical, systematic, Monte Carlo, and function-choice
uncertainties. Note that 56 Sc has an additional systematic uncertainty due to the presence of a β-decaying isomer (See Section 4.3.1.)
which is not included in this figure. . . . . . . . . . . . . . . . . . . 160

Figure 3.66

Residual of the fit to reference nuclei (See Section 3.9.1.) using the
2
2
4
function m
q (τ ) = a0 +a1 ∗τ +a2 ∗z +a3 ∗τ +a4 ∗z +a5 ∗z ∗τ +a6 ∗z ,
where τ = TOF − TOF and z = Z − Z , with TOF being the timeof-flight, Z being the nuclear charge, and the averages of these being
taken over the set of reference nuclei, where the TOF was rigidity corrected by-element (See Section 3.7.). Reference nuclei are identified
by element by their symbol and the text label indicates the reference
nucleus mass number A. Thick colored error bars show the statistical
uncertainties. Thin black error bars show the sum in quadrature of
the statistical uncertainty and the systematic uncertainty (9 keV/q)
applied in the case of the best-fit (See Figure 3.47), which was used
for the mass evaluation. . . . . . . . . . . . . . . . . . . . . . . . . 163

Figure 3.67

Same as Figure 3.66 but using the by-isotope (local) rigidity correction to TOF (See Section 3.6.). Note that 54 Ca is not present in
the set of reference nuclei for this case since it did not have sufficient statistics (> 5, 000 counts) to have a locally-determined rigidity
correction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

xxviii

Figure 3.68

Same as Figure 3.66 but using the global rigidity correction(See Section 3.7.) for the ‘Direct Up’ TOF (See Section 3.2.). . . . . . . . . 164

Figure 3.69

Residual of the fit to reference nuclei (stars), where a reference nucleus
was defined as one with a literature mass with an uncertainty less
than 400 keV, using the function m
q (τ ) = a0 + a1 ∗ τ + a2 ∗ z +
2
2
a3 ∗ τ + a4 ∗ z + a5 ∗ z ∗ τ + a6 ∗ z 4 , where τ = TOF − TOF
and z = Z − Z , with TOF being the time-of-flight, Z being the
nuclear charge, and the averages of these being taken over the set of
reference nuclei, where the TOF was rigidity corrected globally (See
Section 3.7.). Reference nuclei are identified by element by their color,
where isotopes of the same element are connected by lines for clarity,
and the text label indicates the reference nucleus mass number A.
The uncertainty shown is the statistical plus the mass-perturbation
distribution from the Monte Carlo procedure. . . . . . . . . . . . . 164

Figure 4.1

Rigidity-corrected time-of-flight (TOF) distributions for reference num
clei (unfilled histograms) used to calibrate the rest
q (T OF ) relationship to obtain new masses from the TOFs of 48,49 Ar (green-filled
histograms), 52−57 Sc (red-filled histograms), 59−64 Cr (blue-filled histograms), 67 Mn (orange-filled histogram), and 67−69 Fe (gray-filled
histograms). Note that the TOF spectra for argon, scandium, and
chromium isotopes with respect to reference TOFs are highlighted
separately in Figures 4.20, 4.33, and 4.23, respectively. . . . . . . . 167

Figure 4.2

The Segr´e chart with the absolute value of the mass excess deviation in keV (indicated by color) between the 2012 Atomic Mass
Evaluation [1] and the liquid drop model as represented by the semiempirical mass formula given in [37]. The black boxes indicate the
stable nuclei for orientation. The root mean square RMS deviation
is indicated by the text box. . . . . . . . . . . . . . . . . . . . . . . 171

Figure 4.3

The Segr´e chart with the absolute value of the mass excess deviation
in keV (indicated by color) between the 2012 Atomic Mass Evaluation [1] and the 1995 Finite Range Droplet Model (FRDM) [2, 38].
The black boxes indicate the stable nuclei for orientation. The root
mean square RMS deviation is indicated by the text box. . . . . . . 173

Figure 4.4

The Segr´e chart with the absolute value of the mass excess deviation
in keV (indicated by color) between the 2012 Atomic Mass Evaluation [1] and the Weizs¨acker-Skyrme WS3 mass model [5, 39]. The
black boxes indicate the stable nuclei for orientation. The root mean
square RMS deviation is indicated by the text box. . . . . . . . . . 175

xxix

Figure 4.5

The Segr´e chart with the absolute value of the mass excess deviation
in keV (indicated by color) between the 2012 Atomic Mass Evaluation [1] and the 10-parameter Duflo-Zuker mass formula [4, 40]. The
black boxes indicate the stable nuclei for orientation. The root mean
square RMS deviation is indicated by the text box. . . . . . . . . . 177

Figure 4.6

The Segr´e chart with the absolute value of the mass excess deviation
in keV (indicated by color) between the 2012 Atomic Mass Evaluation [1] and the Hartree-Fock-Bogolyubov HFB-21 mass model [3, 41].
The black boxes indicate the stable nuclei for orientation. The root
mean square RMS deviation is indicated by the text box. . . . . . . 179

Figure 4.7

Difference between mass excesses obtained in this experiment and
global mass models FRDM (1995), HFB-21, WS3, and Duflo-Zuker
(10-parameter), as well as experimental values listed in the 2012
Atomic Mass Evaluation as a function of mass over charge A/Z. . . 183

Figure 4.8

Same as Figure 4.7, but as a function of atomic number Z. Note that
multiple nuclei are represented for each Z, so that experimental 1σ
uncertainties are overlapping. . . . . . . . . . . . . . . . . . . . . . 183

Figure 4.9

Same as Figure 4.7, but as a function of nuclear mass number A. Note
that multiple nuclei are represented for A = 67, so that experimental
1σ uncertainties are overlapping. . . . . . . . . . . . . . . . . . . . 183

Figure 4.10

Distributions of mass excess deviations, weighted by our one standard
deviation measurement uncertainty and binned in 32 σ-wide bins, from
masses obtained in this experiment for global mass models FRDM
(1995), HFB-21, WS3, and Duflo-Zuker (10-parameter), as well as
experimental values listed in the 2012 Atomic Mass Evaluation. . . 184

Figure 4.11

Mass excess deviation (indicated by color) from the 2012 Atomic Mass
Evaluation vs Z and N , weighted by our experimental uncertainty.
The same information projected into a one-dimensional histogram
presented in Figure 4.10. . . . . . . . . . . . . . . . . . . . . . . . . 185

Figure 4.12

Mass excess deviation (indicated by color) from the 1995 FRDM
global mass model vs Z and N , weighted by our experimental uncertainty. The same information projected into a one-dimensional
histogram presented in Figure 4.10. . . . . . . . . . . . . . . . . . . 185

Figure 4.13

Mass excess deviation (indicated by color) from the HFB-21 global
mass model vs Z and N , weighted by our experimental uncertainty.
The same information projected into a one-dimensional histogram
presented in Figure 4.10. . . . . . . . . . . . . . . . . . . . . . . . . 185
xxx

Figure 4.14

Mass excess deviation (indicated by color) from the WS3 global mass
model vs Z and N , weighted by our experimental uncertainty. The
same information projected into a one-dimensional histogram presented in Figure 4.10. . . . . . . . . . . . . . . . . . . . . . . . . . 186

Figure 4.15

Mass excess deviation (indicated by color) from the 10-parameter
Duflo-Zuker mass formula vs Z and N , weighted by our experimental
uncertainty. The same information projected into a one-dimensional
histogram presented in Figure 4.10. . . . . . . . . . . . . . . . . . . 186

Figure 4.16

Two-neutron separation energy S2n along isotopic chains of argon,
scandium, chromium, manganese, and iron for this experiment, the
2012 Atomic Mass Evaluation, and global mass models. The gray
lines and points indicate S2n for Z ± 1. . . . . . . . . . . . . . . . . 190

Figure 4.17

Empirical shell gap ∆n [42] along isotopic chains of argon, scandium,
chromium, manganese, and iron for this experiment, the 2012 Atomic
Mass Evaluation, and global mass models. . . . . . . . . . . . . . . 191

Figure 4.18

One-neutron separation energy Sn along isotopic chains of argon,
scandium, chromium, manganese, and iron for this experiment, the
2012 Atomic Mass Evaluation, and global mass models. The upper
and lower trends are for even and odd N isotopes, respectively. . . 192

Figure 4.19

Empirical pairing gap PGn [43] along isotopic chains of argon, scandium, chromium, manganese, and iron for this experiment, the 2012
Atomic Mass Evaluation, and global mass models. The gray points
indicate the pairing gap trend for one Z lower, for context. . . . . . 193

Figure 4.20

Rigidity-corrected time-of-flight distributions for reference nuclei (unm
filled histograms) used to calibrate the rest
q (T OF ) relationship to
48
49
obtain masses from TOFs of Ar and Ar (red-filled histograms).
(From [36].) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

xxxi

Figure 4.21

Dn [44] as a function of neutron number N near N = 28 for sulfur
(dot-dash line), argon (solid line), calcium (dotted line), and titanium
(dashed line). The previously known [1] argon trend (solid line, open
circles) is shown along with results from this experiment (solid line,
solid circles). E(2+
1 ) energies for argon isotopes [45–47] are shown for
comparison (crosses). The peak at N = 28 followed by a reduction
in Dn for N > 28 as compared to N < 28 indicates the presence of a
closed shell. From shell-model calculations we conclude the transition
from Dn ≈ 3 MeV for N < 28 to Dn ≈ 1.5 MeV for N > 28
corresponds to the transition from filling the f7/2 orbital to filling
the p3/2 orbital.(From [36].) . . . . . . . . . . . . . . . . . . . . . . 198

Figure 4.22

The Dn [44] trend near N = 28 for argon from currently known
masses [1] (open circles) and the masses presented in this dissertation
(solid circles) is shown along with shell-model calculations employing the SPDF-MU Hamiltonian [48] (solid squares) and the SDPF-U
Hamiltonian [49] (open squares). E(2+
1 ) energies [45–47] are shown
for comparison (crosses). (From [36].) . . . . . . . . . . . . . . . . 199

Figure 4.23

Rigidity-corrected time-of-flight distributions for reference nuclei (unm
filled histograms) used to calibrate the rest
q (T OF ) relationship to
obtain masses from TOFs of 59−64 Cr (red-filled histograms). . . . . 202

Figure 4.24

Trend along the chromium isotopic chain for S2n (upper panel) for
this experiment (red-filled circles) and the 2012 Atomic Mass Evaluation [1] (black-open circles), E(2+
1 ) [45, 50, 51] (middle panel), and
B(E2) [52, 53] (lower panel). Note that the full mass uncertainties
were used to calculate S2n , resulting in similar error bars for each
point since each chromium mass had the same systematic uncertainty
and similar interpolation uncertainties. . . . . . . . . . . . . . . . . 203

Figure 4.25

Schematic of an accreted neutron star cross section. The Urca cooling
(See Section 4.3.2.) layer separates heat generated deep in the crust
by nuclear processes from the shallow layer where carbon is thought
to be ignited for superbursts. See the text for a brief description of
the labeled layers. . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

xxxii

Figure 4.26

The upper panel shows the effective temperature of a neutron star as
a function of time, from the time an accretion outburst ends until the
neutron star crust has returned to thermal equilibrium with the core.
The central panel shows thermal profiles at various time snapshots
throughout the duration of an accretion outburst, where the time
associated with a given contour increases in the vertical direction,
since the crust is being heated out of equilibrium. The discontinuity
at low density for the contour just prior to accretion turning on is
due to the boundary condition set for the atmosphere temperature
during accretion. The small kink at ≈ 1010 g/cm3 is due to electron
capture heating and the large bump at ∼ 1013 g/cm3 is due to deep
crustal heating. The lower panel shows thermal profiles at various
time snapshots after an accretion outburst has ended, where the time
associated with a given contour decreases in the vertical direction,
since the crust is cooling back into equilibrium with the core. The
symbols on the thermal profiles in the central and lower panels indicate the thermal profile snapshot that belongs to a given point in
time along the cooling curve shown in the upper panel. . . . . . . . 212

Figure 4.27

Same as Figure 4.26, where an impurity parameter Qimp. = 30 was
used instead of Qimp. = 5. It is apparent that the increase in the
impurity parameter drastically slows heat diffusion. . . . . . . . . . 213

Figure 4.28

Same as Figure 4.26, where an Urca cooling layer [54] with a luminosity of 1037 erg/s (at 0.5 GK) has been included at a density of
2.94×1010 g/cm3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

Figure 4.29

Effective temperature as a function of time of the neutron star XTE
J1701-462 as observed [55, 56] after accretion turned off (at day 0)
as compared to dStar [57] simulations. Note that the observational
points labeled ‘?’ and ‘(?)’ have been called into question due to suspected brief accretion outbursts [58]. The baseline simulation (solid
black line) mimicked the parameters employed by [58], the shortdashed black line simulation added a 1036 erg/s (for 0.5 GK) Urca
cooling layer to the baseline simulation, the long-dashed black line
simulation added a 1037 erg/s (for 0.5 GK) Urca cooling layer to the
baseline simulation, the black dot-dash line changed the impurity parameter from the baseline simulation to 30, and the dotted red line
changed the impurity parameter to 30 and included a 3 × 1036 erg/s
(for 0.5 GK) Urca cooling layer. More details on input parameters
are provided in the text. . . . . . . . . . . . . . . . . . . . . . . . . 216

xxxiii

Figure 4.30

Effective temperature as a function of time of the neutron star EXO
0748-676 as observed [59] after accretion turned off (at day 0) as
compared to dStar [57] simulations. The only parameters varied
between the simulations shown were the core temperater Tcore , the
crust impurity parameter Qimp. , and the Urca cooling strength, with
the exception of the simulation corresponding to the thick green line
whose shallow crustal heating was reduced to 0.15 MeV/u (from the
default value of 0.3 MeV/u). Note that the blue diagonally-hashed
band is the inferred core temperature from observations of EXO0748676 after a long quiescent phase where it is thought the crust reached
thermal equilibrium with the core [59]. More details on input parameters are provided in the text. . . . . . . . . . . . . . . . . . . . . . 219

Figure 4.31

Schematic demonstrating the origin of electron capture heating for
odd-A (left panel) and even-A isobaric chains, where the vertical direction indicates the energy of the system, which rises with depth due
to the increasing electron chemical potential µe . Each panel considers
the system energy at the depth where µe = QEC (Z, A) = ME(Z, A)−
ME(Z − 1, A). In the odd-A case, QEC (Z, A) < QEC (Z − 1, A), so
once the initial electron capture happens (1), nothing further happens
until the nucleus sinks to the depth where µe = QEC (Z − 1, A). At
this point, it is possible that electron capture to the ground state is
strongly hindered, as it could have been for the first electron capture
in the sequence (for instance, due to a large difference in spin between
the initial and final state), so electron capture may not ensue until it
is energetically possible to electron capture into an excited state of
the nucleus with Z − 2 protons (2). Electron capture into the excited
state (3) will be followed by de-excitation (4) which deposits heat into
the crust. For the even-A case, heating can be more significant due
to the odd-even mass stagger that results from the pairing force (the
5th term of Equation 4.1). In this case, after electron capture onto
an even-even nucleus (1), it is immediately energetically favorable to
electron capture into the even-even nucleus with Z −2 protons, which
results in heat deposition into the outer crust. Often QEC (Z, A) is
greater than QEC (Z −1, A) by several MeV, so electron capture into a
high-lying excited state is possible (2). The subsequent de-excitation
(3) enhances the energy deposition into the neutron star crust, often
by several MeV [60]. . . . . . . . . . . . . . . . . . . . . . . . . . . 223

xxxiv

Figure 4.32

Integrated heat release (in units of MeV per accreted nucleon u) from
electron capture for an A = 64 mass element as it is buried to deeper
depths of the neutron star crust by accretion. Note that the time
required for a surface fluid element to reach these depths depends on
˙ 26, 400 g/cms2
the accretion rate; for our chosen accretion rate of M=
(1/3 the Eddington accretion rate), the time to sink from the surface
to the depths shown is on the order of centuries. The calculations corresponding to the black and red lines employed the 1995 FRDM [2]
and HFB-21 [3] global mass models for nuclei with unknown masses,
where the 2012 Atomic Mass Evaluation [1] was used otherwise. Calculations indicated by solid lines included the mass of 64 Cr presented
in this dissertation. The first two heating events are labeled by their
corresponding electron capture sequence. . . . . . . . . . . . . . . . 226

Figure 4.33

Rigidity-corrected time-of-flight distributions for reference nuclei (unm
filled histograms) used to calibrate the rest
q (T OF ) relationship to
52−56
obtain masses from TOFs of
Sc (red-filled histograms). . . . . 229

Figure 4.34

∆QEC (Z, A) for odd-odd A = 56 nuclei using ME(56 Sc) from this
experiment and ME(56 Ca) from FRDM’95 or HFB-21 (black stars),
compared to global mass models [2, 3] and mass-differences predicted
from recent energy density functional calculations [61, 62] (open shapes).
A low ∆QEC is essential for the existence of an Urca cooling pair in
an even-A mass chain. . . . . . . . . . . . . . . . . . . . . . . . . . 231

xxxv

Figure 4.35

(a) Net energy released from (positive values) or deposited into (negative values) the neutron star crust as a function of depth for a fluid
element made of A = 56 material as it sinks into the neutron star
crust. Lines from the energy curves to the reaction sequence indicate
which heating or cooling event corresponds to which nuclear reaction
when employing the HFB-21 [3] mass model for the 56 Sc and 56 Ca
masses. Differences between the amount of heating or cooling and the
depth at which it occurs are due to the different value of ME(56 Sc)
used in the calculation. ME(56 Sc) found in this experiment results
in Urca cooling for the A = 56 mass-chain, as given by the HFB-21
mass model but not the FRDM’95 [2] mass model. Note that the
56 V and 56 Ti masses used are from [1]. (b) Integrated energy per
accreted nucleon released from (negative values, blue-shading) or deposited into (positive values, red-shading) the neutron star crust for
an A = 56 fluid element as a function of depth. It is apparent that
cooling from one pair of nuclei overwhelms heating from electron captures. (c) Schematic of an accreted neutron star cross section. The
Urca cooling layer separates heat generated deep in the crust by nuclear processes from the shallow layer where carbon is thought to be
ignited for superbursts. . . . . . . . . . . . . . . . . . . . . . . . . . 232

Figure 4.36

Energy level diagram for the A = 56 mass-chain at a depth where
EF ≈ |QEC (56 Ti)|. The large low-lying Gamow-Teller (GT) transition strength for 56 Sc allows 56 Ti(e− , νe )56 Sc to occur for this condition. Whether or not electron capture EC onto 56 Sc directly follows
depends on the choice of mass model. EC on 56 Sc into the lowest energy level Exs in 56 Ca that can be entered via a GT transition occurs
at EF ≈ |QEC (56 Ti)| for the FRDM mass model [2], but not for the
HFB-21 mass model [3]. In the latter case, Urca cooling occurs for
the 56 Ti–56 Sc pair. . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

xxxvi

Figure 4.37

Schematic demonstrating the origin of Urca neutrino cooling in evenA isobaric chains, where the vertical direction indicates the energy
of the system, which rises with depth due to the increasing electron
chemical potential µe . The figure considers the system energy at the
depth where µe = QEC (Z, A) = ME(Z, A) − ME(Z − 1, A), where Z
and A are even. In order to maintain an Urca cycle, electron capture
from the Z to Z −1 isobar must be relatively strong while the electron
capture from the Z − 1 isobar to the Z − 2 isobar must be hindered.
Since even-even nuclei have a ground state spin-parity J π = 0+ , this
condition can only be achieved if the first allowed Gamow-Teller (GT)
transition into the Z −1 isobar is at a relatively low-lying excited state
energy, while the first allowed Gamow-Teller transition into the Z − 2
isobar is at a relatively high-lying excited state energy such that it
cannot be immediately captured into following the transition from
(Z, A) → (Z − 1, A) [54]. If the (Z − 1, A) excited state with spinπ is significantly thermally populated, nuclei will be drained
parity Jlow
from the Urca cycle by transition from this state to the (Z − 2, A) 0+
ground state if there is a strong GT transition. . . . . . . . . . . . 236

Figure 4.38

The relative probability of β − -decay occurring from the ground-state
of an odd-odd nucleus in an Urca pair (blue-dotted line) multiplied
by the relative thermal population of the ground state of that same
nucleus (black-dashed line) is coined here as the ‘Even-A Urca window’ (thick red-solid line) for a temperature of 0.5 GK. Also shown is
the ‘Urca window’ for a temperature of 0.1 GK (thin red-solid line).
The functional forms which were used are explained in the text. . . 238

Figure 5.1

Predicted FRIB production rate [63], where color indicates particles
per second, with respect to nuclei with known masses in the 2012
Atomic Mass Evaluation [1] and reaction network paths for the astrophysical rp-process [64], astrophysical r-process [24], and neutron
star crust processes [60]. For orientation, stable nuclei are shown as
black squares and the two-neutron drip-line calculated from FRDM
is indicated by the rightmost black-line contour. For reference, the
masses measured for the first time presented in this dissertation are
indicated by black stars. Nuclei produced at a rate of > 10−3 pps
would have sufficient statistics for a TOF mas measurement determination, while nuclei produced at a rate of > 1 pps could serve as
calibration nuclei, so long as they had a well-known mass (for instance, from Penning trap measurements). . . . . . . . . . . . . . . 243

Figure A.1

Cartoon of the frame which was clamped around the magnets used
in the rigidity measurement set-up in order to facilitate placement of
the magnets on the steel yoke. . . . . . . . . . . . . . . . . . . . . . 245
xxxvii

Figure A.2

Cartoon of the magnet being lowered onto the yoke piece that served
as the magnet base plate. Miscellaneous parts were used as spacers to
prevent the magnet from rapidly attaching to the base plate. Guide
rods were used to assist in placing the magnet in the proper location.
Bolts located in the frame were used to slowly lower the magnet onto
the base plate in a well-controlled manner. . . . . . . . . . . . . . . 247

Figure A.3

Cartoon of the magnet-yoke assembly process. The two magnets
mounted on base plates were connected to the jaws of a vice via
aluminum plates. The vice was slowly closed to align the connecting
plate with the base plate attached to the fixed vice jaw (“1” in the
figure). Finally the non-magnetic platform was lowered onto the base
plates and the aluminum feet, which were fastened to the platform,
were fastened to the magnet base-plates (“2” in the figure). . . . . . 249

xxxviii

Chapter 1
Introduction
This dissertation discusses the results of experiment e09039 performed at the National Superconducting Cyclotron Laboratory, a time-of-flight mass measurement that resulted in an
extension of the known nuclear mass surface for neutron-rich isotopes of argon through iron.
This chapter will serve to briefly motivate nuclear mass measurement and justify the use
of the time-of-flight technique. The second and third chapters will describe the methods
employed in the measurement and the data analysis, respectively. It was anticipated that
these chapters will be of the greatest potential use for future practitioners, and so an attempt
was made to provide a thorough explanation that could be understood by an introductory
graduate student. The fourth chapter contains the mass measurement results, including comparisons to theoretical predictions, along with detailed discussions of the implications that
the newly measured masses have for nuclear structure and nuclear astrophysics. This and
the fifth chapter, which provides a brief outlook on the future of time-of-flight mass measurements, were written to be interesting to specialists interested in the scientific results.
Nonetheless these two chapters contain sufficient introductory material so that non-experts
should be able to take away the salient details. Finally, the appendices contain more details
which are likely to only be of interest to future practitioners of the time-of-flight technique,
including a brief discussion of lessons learned and suggestions to take into consideration for
future measurements that are similar to the one presented in this dissertation.

1

1.1

Nuclear structure

In the early days of nuclear physics research, it was recognized that nuclear masses provided
important insight into the fundamental structure of nuclei. The father of nuclear mass spectrometry, Francis Aston 1 , remarked in 1927 [65],
“The accurate determination of these divergences [from the whole-number rule] is of fundamental importance since it is one of the few avenues by which the problem of the structure
of nuclei of atoms can be approached . . . ”.
Though the whole-number rule 2 today is a footnote in history, the role of nuclear masses as
an avenue to understand the problem of nuclear structure is as prominent as ever [66]. The
basic reason behind the fundamental connection between nuclear masses and nuclear structure is that the nuclear forces which bind nucleons together and reduce their summed mass
also describe the way in which constituent nucleons arrange themselves within a nucleus.
Shortly after the discovery of closed shells in nuclei [67], the mother of nuclear structure,
Maria Goeppert-Mayer, pointed out that nuclear masses, or more specifically their binding
energies, could be used to identify the presence of these closed shells [68]. A striking example
of this is provided by the so-called two-neutron separation energy, which is simply the mass
difference between two isotopes of an element which differ in constituency by two neutrons
(See Section 4.1.2 for a more in-depth explanation.), shown in Figure 1.1 for isotopes of neon
through titanium. The clear patterns which emerge in the two-neutron separation energy
can ultimately be traced back to the interplay between the forces at work within nuclei.
1 As

described in Section 1.3, though J.J. Thompson was the first to perform mass spectrometry, Aston was the first to systematically apply the technique towards the study of
nuclear masses.
2 The ‘whole number rule’ states that nuclear masses are integer multiples of the mass
mass of hydrogen.

2

Perhaps more intriguing than the presence of these trends in nuclear mass differences is
their disappearance for exotic nuclei. One prominent example of nuclear mass differences
indicating a shift in nuclear structure is the discovery of the ‘island of inversion’ at neutron
number N = 20, which was identified by an anomalous trend in the two-neutron separation
energy for the sodium isotopes [69] as compared to elements with higher atomic number
(which is apparent in Figure 1.1).

30

S2n [MeV]

25
20
15
Ar
Cl
S

10
Na

5
0

Ne

15

20

Al
Mg

25

Si

P

30

Ca

Ti

K
Sc

35

N
Figure 1.1: Energy required to remove two neutrons from a nucleus as calculated by mass
differences S2n (See Equation 4.3.) for neutron-rich isotopes of neon through titanium. Clear
patterns emerge, particularly the rapid drop-offs after N = 20 for phosphorous P through
titanium Ti and N = 28 for isotopes of potassium K through titanium. (From [6].)

The nuclear mass measurements presented in this dissertation were utilized to explore the

3

evolution of nuclear structure for neutron-rich isotopes of argon through iron, as discussed
in detail in the fourth chapter of this dissertation. In particular, Section 4.2 highlights the
advances which were made regarding the N = 28 shell closure and the island of inversion
near N = 40.

1.2

Nuclear astrophysics

One could argue that the birth of nuclear astrophysics can be traced back to the consideration
of nuclear masses. Informed by early experiments of Francis Aston, Sir Arthur Eddington
noted that the energy corresponding to the difference in mass between a helium nucleus
and four protons sufficed to provide the energy reserve required to sustain heating in stars
over their surmised lifetime [70]. The deep relation between stars and nuclei, in particular
the masses of nuclei, continues to present day, where precise nuclear mass data is essential
to accurately calculate the nucleosynthesis and nuclear energy release occurring in extreme
astrophysical environments, such as in supernovae and on neutron stars [71].
To date, nuclear masses have played an important role in understanding the microphysics
of the majority of the astrophysical sites of interest to the nuclear astrophysics community.
Examples can readily be found where the study of nuclear masses directly advances the
understanding of rapid neutron-capture (r-)process nucleosynthesis [72–74], nucleosynthesis
in neutrino-driven winds following core collapse supernovae [75], the energy generation and
abundance yield from the rapid proton-capture (rp-)process that takes place on the surface of
accreted neutron stars [23, 76, 77] , and the energy generation and general structure of both
accreted [78] and non-accreted [79] neutron star crusts, to name just a few recent examples.
The sensitive dependence of predictions produced by simulations of many astrophysical sce-

4

narios on nuclear masses in part derives from the fact that mass differences, namely Q-values,
determine the temperatures or densities at which nuclear reactions can ensue. Additionally,
the balance between forward and reverse rates in nuclear reaction networks depends exponentially on nuclear mass differences [71]. Figure 1.2 demonstrates the impact of varying the
mass-difference between the rp-process waiting-point nucleus 68 Se and its proton-capture
daughter 69 Br on the predicted light curve for type I X-ray bursts (See Section 4.3 for an
overview of the relevant astrophysics.) [7]. In this example the required uncertainty in the
nuclear masses involved is on the order of kT , where k is Boltzmann’s constant and T is the
environment temperature, as is the case for most reactions in the r- and rp-processes, where
reaction rates depend exponentially on the Q-value [71]. For the case of electron captures
in the neutron star crust, nuclear energy generation and the depth at which it happens are
of primary interest. Since mass differences in this case determine the depth at which energy
can be released and the maximum possible heat release (See Section 4.3.2.), masses must be
known with a precision of an order of magnitude or so less than the typical nuclear mass
difference. As such, uncertainties on the order of several hundred keV or better need to be
achieved.
As with the prior section, a more detailed discussion of the aforementioned topics is left
for the Results chapter (Chapter 4). There it is discussed how the nuclear masses resulting
from the measurement described in this dissertation impact results of model calculations of
electron capture reactions occurring within the accreted neutron star crust. Section 4.3.2
in particular demonstrates the implications of the mass measurement results reported in
this dissertation for the nuclear heating and cooling processes that take place in the outer
neutron star crust.

5

Sp= -150 keV
Sp= -450 keV

70Kr

Sp= -750 keV
Sp= -641 ± 42 keV

69Br

68Se

Figure 1.2: Demonstration of the 68 Se–69 Br mass difference Sp impact on a calculated X-ray
burst light curve. The black lines demonstrate the uncertainty in the X-ray burst light curve
for a 69 Br mass uncertainty of ±300 keV, where it is possible that significant two-proton
capture occurs on 68 Se to make 70 Kr. An uncertainty reduction to ±42 keV makes it clear
that proton emission of 69 Br primarily occurs rather than proton capture after the 68 Se(p, γ)
reaction occurs in this calculation of the rp-process reaction network. (Adapted from [7].)

1.3

Overview of nuclear mass measurement

The first indication that quantum systems were special with regard to mass came with
Einstein’s postulate of mass-energy equivalence [80], now codified by the well known relation
E = mc2 . This fundamental relation describes the fluidity with which mass can be converted
into energy, and thus it is so that the sum of the masses of nucleons that make up a nucleus
do not necessarily equal the mass of the nucleus itself, as first demonstrated by Aston [65].
This phenomenon is known as nuclear binding and the energy that is liberated due to the
reduced mass of a nucleus with respect to the sum of its nucleons is known as the ‘binding
energy’. Though early attempts to predict nuclear masses via the so-called ‘liquid drop’
6

model performed remarkably well [81], to date no mass model exists with the power to
accurately predict nuclear masses within sufficient precision and in many cases predictions
from various models are widely discrepant [82, 83].3 Thus experimental mass measurements
are crucial to provide data for applications that require accurate nuclear masses.
Experimental mass spectrometry began just over 100 years ago when J.J. Thomson first
observed ‘positive rays’ [84], which we now know to be positively charged ions [85]. In
the past century numerous methods have been employed to obtain nuclear masses [42, 86–
88] 4 . Currently, the leading mass measurement techniques which are recognized to give
accurate and precise masses of radioactive nuclei employ storage rings [9, 89, 90], Penning
traps [8, 91], multi-reflection time-of-flight [11, 92], and direct time-of-flight [6]. Each method
has its strengths and weaknesses, so the aforementioned techniques can be thought of as
complementary ways to approach the mapping of the nuclear mass surface.
Penning trap measurements obtain nuclear masses by measuring the resonant frequency of
the nucleus of interest orbiting within a few cubic-centimeter volume, confined by a strong
magnetic field and hyperbolic electrodes [8]. The resonant frequency is converted into a
nuclear mass for the nucleus of interest by comparison to the resonant frequency of an atom
or atomic cluster of known mass which is nearby in magnitude to the expected mass of the
nucleus of interest [93]. Penning trap mass spectrometry requires that nuclei of interest be
inserted into the trap at low velocity. This means that nuclei must either be produced with
a low initial velocity, as with the isotope separation online (ISOL) method, or be slowed
down from their initially high velocity and guided into the trap [8]. Regardless of the exact
3A

few leading global mass models are described and compared to experimental results
in Section 4.1.
4 Note that many of these techniques actually directly obtain ion masses and then the
nuclear mass is derived by correcting for electron masses and binding energies. However, for
simplicity we will refer to the nuclear mass as what is being measured.
7

technique used, Penning trap mass measurements tend to take on the order of hundreds of
milliseconds per measurement, where 8.8 ms is the shortest half-life measured thus far [94],
and, provided on the order of a hundred events survive the measurement before undergoing
radioactive decay, a typical precision of δm/m ∼ 10−7 is commonly achieved for radioactive
nuclei [8].
Storage rings are used to measure nuclear masses either via the isochronous mass spectrometry (IMS) method or the Schottky mass spectrometry (SMS) method. In both methods
several exotic nuclei are injected into a ring in which their orbit is confined via magnetic
fields. Nuclei with well known masses are injected simultaneously and are used for calibration [9]. IMS employs isochronous ion optics (See Section 2.2.3.), where the flight time per
turn is measured with fast-timing detectors and directly relates to the mass. This technique
takes on the order of milliseconds for a single event measurement and a typical precision of
the order δm/m ∼ 10−6 is commonly achieved [10] (Though, shorter times can be achieved at
the cost of precision [9].). SMS is used to obtain nuclear masses by making non-interfering
revolution-frequency measurements via capacitive pickups. In order to obtain a resolved
frequency measurement, SMS requires that the ion beam be electron cooled to reduce its
momentum spread. This process takes on the order of seconds, but is capable of yielding
typical precisions of δm/m ∼ 10−7 [9]. Both IMS and SMS require on the order of one hundred events per nucleus to produce a precise mass measurement, provided several reference
nuclei are also observed with higher statistics.
Multi-reflection time-of-flight (MR-TOF) mass measurements employ a method similar
to IMS, where the flight time of an ion is measured over a long flight path, however with
MR-TOF the path length consists of several turns within an electrostatic trap [92]. Ions are
injected into the trap and confined by electrostatic mirrors, typically for tens of milliseconds,
8

until they are finally allowed to exit and terminate their flight path on a microchannel plate
detector [95]. Nuclei with experimentally known masses [1] are used to calibrate the relationship between mass and time-of-flight. Roughly a thousand events are needed for nuclei
that are to be used for calibration purposes and of the order of a hundred events are needed
for a mass measurement [96]. The MR-TOF technique is relatively new, as it is typically
employed as a beam-purification mechanism for Penning trap mass measurements [11, 97].
A direct time-of-flight (TOF) mass measurement, the focus of this dissertation, employs
a method which is similar to IMS, where the flight time of an ion between two fast-timing
detectors is converted into a nuclear mass by comparison to the flight times of nuclei with
known masses. In contradistinction to IMS, the flight path for direct TOF measurements is
not a closed loop and thus only a single flight time is determined for a single beam particle.
This technique is very rapid, taking on the order of microseconds, but is limited to a typical
measurement precision of δm/m ∼ 10−5 [6]. Similar to storage ring mass measurements,
direct TOF requires hundreds of events for a single nucleus to yield a precise mass with
the caveat that several nuclei with well known masses must also be observed with higher
statistics (See Section 3.9.).
The choice of a mass spectrometry technique is determined by the precision requirement
to answer the scientific question at hand and by the applicability to the nucleus of interest
given its half-life, production mechanism, and associated beam intensity. To answer the
question of applicability, one must consider the limit in precision a given technique has due
to systematic considerations as well as the limit in precision due to statistical considerations,
i.e. the number of measurement-events available for the nucleus of interest and its calibrants.
While the systematic limit to precision is inherent to a technique, ample statistics are determined by the efficiency with which nuclei and calibrants are produced and transported
9

TOF
SR-IMS
MR-TOF

PT

SR-Sch

Figure 1.3: Comparison of typical observation times for a single event and ultimately achievable precision for the main nuclear mass measurement techniques currently employed for
radioactive nuclei. Typical values are shown for Penning traps (PT) [8], storage rings employing Schottky mass spectrometry (SR-Sch) [9], storage rings employing isochronous mass
spectrometry (SR-IMS) [10], multi-reflection time-of-flight devices (MR-TOF) [11], and direct time-of-flight measurements (TOF) [6].

to the measurement device. Transport efficiency is also impacted by the number of nuclei
which survive radioactive decay during transport. Though a direct TOF mass measurement
is clearly lacking in achievable precision (typically δm/m ∼ 10−5 ) with respect to other common techniques, it is a more efficient method as it allows for the simultaneous measurement
of hundreds of nuclei and accepts continuous beam with high transmission efficiency. As is
apparent in Figure 1.3, the swift measurement time renders decay losses to be negligible. Due
to the reliance of direct TOF mass spectrometry on reference nuclei with well-known masses
(See Section 3.9.), it must be noted that this technique is primarily useful for extending the

10

known nuclear mass surface beyond the limit achieved by other means.

6
4
2

ε

0
-2
Ne

-4

Ti

-6

Fe
Sn

-8
2.5

3

3.5

4

A/Z
Figure 1.4: The trend in exoticity ǫ, which quantifies the relative accessibility of a nucleus
for mass measurement, as a function of the ratio of nucleons A to protons Z for neutron-rich
isotopes of neon, titanium, iron, and tin. When experimental values [12] are unavailable,
theoretical β-decay half-lives and the neutron drip-line used in [13] were used. Filled shapes
indicate nuclei with experimentally known masses [1]. (From [6].)

In order to quantify the experimental reach of direct TOF mass measurement, hereinafter
referred to as TOF mass measurement, with respect to other techniques, we can assign to
each nucleus a quantity known as exoticity ǫ [6]5 . Exoticity accounts for the difficulty in
production of a nucleus, which is related to its isotopic distance from stability dNstab and
the drip-line dNdrip , and the time available to measure a nucleus, which is related to the
β-decay half-life Tβ :
ǫ = log10

dNstab
,
Tβ (dNdrip + 1)

5 An

(1.1)

alternative metric, but similar in spirit, the so-called “relative isobaric distance from
stablity”, is employed by [42].

11

where, when experimental data are not available, we adopt β-decay half-lives (in seconds)
and a drip-line defined in [13]. To give a feel for how ǫ evolves in the nuclear landscape, the
trends in ǫ for neutron-rich isotopes of neon, titanium, iron, and tin are shown in Figure 1.4.
It is apparent that nuclei with ǫ ≈ 1 are at the current frontier of mass spectrometry and
nuclei with ǫ ≈ 5 are on the neutron drip-line (as defined by [13]). Since Tβ ∼constant
for very exotic nuclei, typically around tens of milliseconds [12], ǫ is primarily dominated
by neutron-richness with respect to stability6 . The comparison of ǫ with the measurement
precision for Penning trap and TOF mass measurements in Figure 1.5 makes it clear that
though Penning trap measurements are orders of magnitude more precise, the TOF method
reaches the more exotic isotopes. Figure 1.6 shows the exoticity of nuclei participating in the
astrophysical rp-process, r-process, and neutron star crust nuclear reaction networks, where
a distinction is made between nuclei whose mass is either known or not known experimentally
as of [1]. From this figure it is evident that, in order to experimentally determine the masses
of nuclei participating in extreme astrophysical processes, mass measurement techniques that
probe very exotic nuclei such as the TOF mass measurement approach are required.

6 This

feature is well suited to describing the experimental reach of TOF mass measurement because such measurements are typically not limited by half-lives, but rather are
limited by the production of nuclei of interest and reference nuclei, which rapidly decreases
for increasing distance from stability [6].
12

Mass Uncertainty [keV]

103
102
10
1

TOF-Bρ
Penning Trap

10-1
-10

-5

0

5

10

ε
Figure 1.5: Exoticity and measurement precision for Penning trap and TOF mass measurements existing to date. Penning trap data are from the ISOLTRAP [14], JYFLTRAP [15],
SHIPTRAP [16], LEBIT [17], TITAN [18], and CPT [19] setups, while TOF data are from
the TOFI [20], SPEG [21], and NSCL [22] setups (See Section 1.4.2.). (From [6].)

13

30

45
40

30

rp, unknown mass

r, unknown mass

rp, known mass

r, known mass

NS, unknown mass
NS, known mass

25

25

35

(a)
25
20
15

20

Nuclei per 0.2ε

Nuclei per 0.2ε

Nuclei per 0.2ε

(c)

(b)
20

30

15

15

10

10

5

5

10
5
0
-1 -0.5

0

0.5

1

ε

1.5

2

2.5

0
-1 -0.5 0

0.5

1 1.5 2 2.5 3

ε

0
-10 -8

-6

-4

-2

0

2

4

ε

Figure 1.6: Exoticity of nuclei present in the rp-process (a), r-process (b), and neutron star
crust (c) reaction networks, using the networks from [23], [24], and [25], respectively. Nuclei
with known mass as of [1] are represented by the red-filled regions, while nuclei with no
experimentally determined mass are represented by the cross-hatched regions.

14

1.4
1.4.1

TOF mass measurement
TOF mass measurement principle

The relationship between time-of-flight TOF and nuclear rest mass mrest arises from the
equation of motion for a charged massive particle through a magnetic system. The two
counteracting forces that act upon the nucleus as it traverses a system with fixed magnetic
rigidity Bρ are the Lorentz force FL and the centripetal force Fc . Thus we obtain the
following relationship:

Fc = FL
γ(v)mrest v 2
= qvB
ρ
1 q(Bρ)
→ mrest =
v γ(v)
TOF q(Bρ)
,
mrest =
Lpath
Lpath
γ TOF

(1.2)

where the Lorentz factor γ is a function of velocity v, which is in turn the ratio of flight-path
length Lpath to flight time TOF. It follows that, in principle, the simultaneous measurement
of an ion’s TOF, charge q, and Bρ through a system of known Lpath yields mrest .
However, in practice neither Lpath nor the ion optical dispersion used to determine Bρ
(See Section 2.4.) are known with sufficient precision. Furthermore, only a measurement of
Bρ relative to the central ion optical axis is performed. Instead, the

mrest
q

(TOF) relationship

is determined empirically by measuring the TOF of calibration or reference nuclei [6]. The
reference nuclei must have well-determined masses, typically from previous Penning trap
mass measurements, and no isomeric states [98] that are so long-lived that they would

15

survive the flight-path (See Section 3.9.). Additionally, to reduce systematic errors, the
reference nuclei must have a similar charge to mass ratio m/q and nuclear charge Z and be
measured in the same experiment as the nuclei of interest. The dissimilarity in m/q and
Z between reference nuclei and nuclei of interest gives rise to systematic uncertainties that
must be accounted for, as discussed in Section 3.9.4.
Equation 1.2 provides an estimate for the precision with which TOF must be measured
in order to achieve a given mass uncertainty, independent of concerns regarding systematic
effects: δm/m ∝ δTOF/TOF→ δTOF∝ δm/m×TOF. For the typical values encountered
in the experiment described in this dissertation, TOF≈500 ns and m ≈50 GeV, to achieve
an uncertainty (not including systematic effects) of 100 keV for a given nucleus, its TOF
must be determined with a precision of δTOF≈1 picosecond. It is apparent that to achieve
a mass measurement uncertainty of a few hundred keV, effects must be controlled for that
systematically impact the TOFs of nuclei on the order of a picosecond. This fact provides a
substantial challenge for experiment design and data analysis (See Chapter 3.).

1.4.2

Previous TOF mass measurements

To date, three facilities have employed the time-of-flight TOF technique to obtain nuclear
masses [6]: the Time-of-Flight Isochronous Spectrometer (TOFI) at Los Alamos National
Laboratory (LANL) [20], the Spectrom`etre `a Perte d’Energie du Ganil (SPEG) at the Grand
Acc´el´erateur National d’Ions Lourds (GANIL) [21], and the TOF setup at the National Superconducting Cyclotron Laboratory (NSCL) which employs the S800 Spectrograph [22].
TOFI, which was in operation from 1987–1998 [99, 100], employed isochronous ion optics
(See Section 2.2.3.), obtaining the mass over charge m/q of ions from a precise TOF measurement. In addition to a precise TOF measurement, the mass measurements at GANIL
16

(which took place between 1986–2012 [101, 102]) and NSCL (which have taken place from
2006–present [29, 31, 36, 78]) included a precise measurement of beam particle’s magnetic
rigidity in order to cope with the large momentum spread of the nuclear beams produced
there. Together, these three facilities have provided over 280 nuclear masses [6].
TOFI measured the TOF of nuclei produced via spallation over a ∼14 m flight path for
typical flight times of ∼600 ns. Timing measurements were performed with microchannel
plate detectors and energy loss measurements for particle identification (See Section 2.5.2.)
were performed with either a solid-state detector telescope, a gas ionization detector, or a
Bragg spectrometer. SPEG measured the TOF of nuclei produced via projectile fragmentation over a ∼115 m flight path for typical flight times of ∼1 µs. Timing measurements
were performed with the accelerator RF or an avalanche counter and a plastic scintillator
and energy loss measurements for particle identification were performed with an ionization
chamber and plastic scintillator. NSCL measured the TOF of nuclei produced via projectile
fragmentation over a ∼60 m flight path for typical flight times of ∼500 ns. As described in
Chapter 2, timing measurements were performed with plastic scintillators and energy loss
measurements were performed with an ionization chamber. For comparison, a schematic of
each of the three measurement setups is given in Figure 1.7 and some of the main characteristics are given in Table 1.1. The nuclei whose mass was measured, with mass uncertainties
ranging from ∼100–1500 keV, by each of the three TOF setups (excluding the NSCL measurement presented in this dissertation) are shown in Figure 1.8.
The following chapters are devoted to the most recent TOF mass measurement at the
NSCL. A detailed description is given of the experimental technique in Chapter 2, the analysis
in Chapter 3, and the results in Chapter 4.

17

TOFI

SPEG

NSCL
Figure 1.7: Schematics of the TOFI, SPEG, and NSCL TOF mass measurement setups.
Letters indicate the following components: C collimator, D dipole magnet, ∆E energy loss
detector, EF electrostatic filter, H hodoscope, MF magnetostatic filter, P position detector,
PB primary beam, Q quadrupole magnet, QT quadrupole magnet triplet, S sextupole magnet, ST stopper, T timing detector, TG production target, TKE total energy detector, and
W wedge. (From [6].)

18

Facility

Solid Angle [msr]

δp/p [%]

Bρmax [T.m]

Lpath [m]

m/δmachieved

TOFI

2.5

4

0.9

14

2600

SPEG

4.9

6

1.44

116

4000

NSCL

20

1

4

60

5500

Table 1.1: Quantities of interest which describe main characteristics of the TOFI, SPEG,
and NSCL TOF mass measurement setups.

Figure 1.8: Nuclei measured by TOF mass measurement setups at TOFI, SPEG, and NSCL
(as of 2013) as indicated by neutron number N and proton number Z. For reference, the
neutron and proton closed shells at 8, 20, 28, and 50 are indicated. Note that several nuclei
have been measured by more than one facility. (From [6].)

19

Chapter 2
Experimental setup
The masses of several neutron-rich nuclei were determined for the first time via the time-offlight mass measurement technique. The contents of this chapter describe the experimental
technique and the following chapter discusses the data analysis procedure used to obtain
the masses. The experiment, NSCL experiment E09039, took place at the National Superconducting Cyclotron Laboratory (NSCL) 11–20 November 2011 and the data analysis was
concluded Fall of 2014.

2.1

Overview of NSCL TOF mass measurement setup

The time-of-flight mass measurement set-up at the NSCL [22, 31] consisted of measuring
the flight time between the A1900 and S800 focal planes (See Section 2.3.), the relative
magnetic rigidity at the S800 target position (See Section 2.4.), and energy loss and tracking
information in the S800 focal plane (See Section 2.5.) for ∼200 nuclides simultaneously over
the course of ∼100 hours [29]. The nuclei of interest, neutron-rich isotopes of argon through
iron, were produced by fragmenting a 82 Se beam alternately on one of two beryllium targets,
where a thin target generally produced nuclei with experimentally known masses and a thick
target generally produced the more neutron-rich nuclei of interest (See Section 2.2.1.). A
wedge degrader was employed in the intermediate image of the A1900 fragment separator
(See Section 2.2.2.) to remove low atomic number nuclei that were not of interest for this
20

experiment to limit the overall beam rate (See Sections 2.2.3 and 2.5.1.). A schematic of the
set-up is shown in Figure 2.1 and the scheme for the data-acquisition electronics is shown in
Figure 2.2. The following sections in this chapter describe components of the aforementioned
setup in more detail.

Figure 2.1: Schematic of the time-of-flight mass measurement set-up at the National Superconducting Cyclotron Laboratory (NSCL). The coupled K500 and K1200 cyclotrons [26]
provided a 140 MeV/u 82 Se32+ beam which was alternately impinged on beryllium targets
of 517 mg/cm2 and 658 mg/cm2 thickness. The secondary beam fragments were transported
through the A1900 fragment separator [27] where a 7.2 mg/cm2 Kapton wedge degrader removed the high flux of low-Z nuclei that would otherwise have complicated fragment identification. The time-of-flight TOF was measured by fast-timing scintillators (See Section 2.3.)
at the A1900 and S800 focal planes and a relative measurement of magnetic rigidity was
performed at the base of the S800 spectrograph (See Section 2.4.). Beam fragments were
identified using the TOF and the energy-loss information from the ionization chamber located in the S800 focal plane [28] (See Section 2.5.2). Note that the A1900 timing signal was
the TOF ‘stop’ signal and the S800 timing signal was the TOF ‘start’ in the experiment data
acquisition (See Section 2.3.1.). (From [29].)

21

Figure 2.2: Diagram of electronics scheme used for the time-of-flight mass measurement
(NSCL experiment E09039). Timing and charge signals were obtained from photomultiplier
tubes (PMTs) located in the A1900 and S800 focal planes as well as from the microchannel
plate detector (MCP). The four charge signals were obtained from the MCP corners were
amplified by two different gains. Detection rates on all detectors were monitored by scalers.
An S800 timing signal was generally used as the event trigger, though the MCP was used as
the event trigger for the mask-run made to assess the beam position-measurement resolution.
(Contact the author for an electronic form of this figure.)

22

2.2

Production and transport of nuclei

Nuclei were produced via in-flight projectile fragmentation [103]. In this process a stable
atom is impinged upon a stable target at high velocity, the reaction products, ‘fragments’,
of interest are collected and separated-out from all fragments by a beam analyzer, and the
fragments of interest are sent on to an experimental end-station. For the time-of-flight mass
measurement reported here, a 82 Se beam was accelerated by the National Superconducting
Cyclotron Laboratory (NSCL) coupled K500 and K1200 cyclotrons [26] to 140 MeV/u and
impinged upon a beryllium target. Fragments (See Figure 2.3.) were analyzed by the A1900
fragment separator [27] and transported to the focal plane of the S800 spectrograph [34], as
seen in Figure 2.1 1 .

2.2.1
82 Se

Fragment production

was chosen as the primary beam from the list of those available at the NSCL [104] since

it gave the highest yield for the neutron-rich isotopes of argon to iron that were the focus
of the time-of-flight mass measurement [105]. The electron cyclotron resonance (ECR) ion
source delivered 82 Se [106] in a 13+ charge-state to the K500 cyclotron, which accelerated
82 Se13+

to ∼12 MeV/u. The selenium beam was injected into the K1200 cyclotron, stripped

to a 32+ charge state by a carbon foil [107], and accelerated to ∼140 MeV/u. Together the
coupled cyclotrons [26] provided a 82 Se32+ primary beam on the production target with an
intensity of ∼30 pnA (particle nanoamps).
1 The

numbers following devices labeled K, A, and S in the above description denote the
maximum kinetic energy for a proton KEp,max as determined by the device’s maximum

(qp Bρmax )2
p2
=
,
magnetic rigidity Bρmax using the non-relativistic formula KEp,max = 2m
2mp
p
where mp and qp are the proton’s mass and charge, respectively, and p2 is it’s momentum
squared.

23

The primary beam was impinged upon a beryllium target to produce a cocktail beam of
various isotopic species, ‘fragments’, via a mechanism in which it is thought a primary beam
nucleus has some nucleons scraped-off to form a compound nucleus which then evaporatesoff several more nucleons during deexcitation 2 . Beryllium was used as the target material
because it has a relatively large fragment production cross section per energy lost through
atomic interactions, due to a small electrons/atom ratio, and a high melting point, which
means it can withstand the large energy density deposited by the primary beam [103, 113].
Two beryllium targets were alternately employed: a ‘thin’ target used to produce less-neutron
rich nuclei used as calibrants in the mass measurement (See Section 3.9.) and a ‘thick’ target
used to produce the more neutron-rich nuclei of interest. The thicker target on average
produced more exotic nuclei since a beam particle typically experienced more beam-target
interactions than in the thin target. The thin target was nominally 517 mg/cm2 , though
it was measured to be 515.813 mg/cm2 , and sustained beam-on-target for ≈11 hours. The
thick target was nominally 658 mg/cm2 (NSCL target ‘658a’), though it was measured to
be 675.057 mg/cm2 , and sustained beam-on-target for ≈91 hours. The combined yield of
both targets is shown in Figure 2.3. No appreciable target degradation was observed over
the duration of the experiment.

2.2.2

Fragment separation

Fragments, which exited the target with a typical energy of ∼ 90 − 100 MeV/u, were forward focussed and were collected in the A1900 fragment separator [27] with >90% collection
efficiency [103]. Fragment separation was achieved via the Bρ-∆E-Bρ method [114–116],
2 This

production mechanism is typically described via the abrasion-ablation model [108–
111], though it should be noted that recent studies show this model may be deficient in
describing the production of some neutron-rich nuclei via projectile fragmentation [112].
24

where the rigidity Bρ settings of the first and second halves of the A1900 were used for
selection of momentum over charge Bρ = p/q (slits were set to ±0.5% momentum acceptance) and a 7.2 mg/cm2 Kapton (C22 H10 N2 O4 ) wedge degrader [115, 116] was placed at
the intermediate image (A1900 ‘I2’ position [117]) to discriminate ions by energy loss ∆E.
The wedge degrader was chosen to be as thin as possible to maximize transmission of the
fragments of interest [117] and minimize Z-dependent TOF-perturbations, but thick enough
to limit the beam rate at the first S800 focal plane cathode readout drift counter (CRDC)
(See Section 2.5.1.) by removing low-Z fragments with a similar mass-to-charge m/q ratio
to the fragments of interest [103, 115].

2.2.3

Fragment transport

Achromatic ion optics were chosen to provide a focused beam on the timing scintillators at
the A1900 and S800 focal planes [22, 31, 103, 115, 118, ch.9,sec.2.3]. The total flight-path
for ions between the scintillators was ≈60.57 m. The ion optics of the S800 spectrograph
analysis line were set to ‘dispersion matching mode’, providing a momentum dispersion
just before the entrance of the S800 (the ‘target position’) of δp/p ≈ 1%/11 cm [34]. The
active height of the MCP detector located at this position was ≈10 cm (See Section 2.4.5.),
limiting the momentum acceptance to δp/p ≈ ±0.45%. Magnetic elements in the A1900,
transport line, and S800 were tuned to optimize transmission of 63 Cr and 65 Mn, which were
ultimately transmitted with ∼50% and ∼65% efficiency, respectively. The full set of nuclei
delivered over the course of the mass measurement is pictured in Figure 2.3. Isochronous
optics, which are set such that an ion’s time-of-flight TOF is independent of it’s pathlength (i.e. also it’s momentum) [118, 119, ch.9,sec.3.1-3.2], were also previously explored
however they were found to yield worse TOF resolution than the achromatic setting by
25

∼ ×2 [120]. The dispersion produced by the achromatic ion optics at the S800 target
position was ∼0.04 ns/mm (See Section 3.6.), which corresponded to ∼ 120 ns/T.m.

Figure 2.3: Particle identification (See Section 2.5.2.) plot of nuclei produced in the time-offlight (TOF) mass-measurement experiment, where the color indicates production intensity
(counts per 100 picoseconds×10 ionization-chamber-adc-units) and TOF was not rigiditycorrected (See Sections 3.6 and 3.7.). Nuclei located to the right of the red-line had no
known experimental mass prior to the mass-measurement reported here; 50 Ca, 54 Ca, 65 Fe,
and 69 Fe are labeled for reference.

2.3

Timing measurement

The dispersion matched ion optics mode [31, 121] was chosen to provide achromatic transport
(See Section 2.2.3.) of ions from the A1900 [27] focal plane to the S800 [34] focal plane (See
Figure 2.1.), where the foci at the timing detectors, due to the achromatic nature of the

26

ion optics [103], had a sub-centimeter spread 3 in the plane perpendicular to the beam line.
Flight times TOF were obtained for ∼200 nuclei, ranging from atomic number 14 ≤ Z ≤ 30
and atomic mass to atomic number ratio 2.35

A/Z

2.72. A typical TOF was ∼500 ns

and the total span of TOFs of observed nuclei was ∼ 50 ns (See Figure 2.3.).

2.3.1

Method overview

The method developed by [22, 31] was used to measure the time-of-flight TOF for nuclei
in the mass measurement experiment. Two 1 cm-tall×1.5 cm-wide×0.25 mm-thick BC-418
plastic ultra-fast timing scintillators from Saint-Gobain Crystals [30] were each coupled to
two Hamamatsu [122] R4998 1 in-diameter head-on photomultiplier tubes (PMTs) [123, ch.9]
(housed inside of their modified [31] H6533 assembly), which were biased to ∼ −2 kV, via
small plastic light guides and optical grease, as pictured in Figure 2.4. The signal from each
PMT was split, sending one signal through a Tennelec TC455 Quad CFD (constant fraction
discriminator) followed by a Phillips 710 leading edge discriminator (LED) to be used for
timing and sending the other signal directly to a Phillips 7166 QDC (charge-to-digital converter) to assess scintillator light-output for position and atomic number Z information. To
maintain the quality of signals generated by the PMTs in the A1900 as they were transported
to the data acquisition electronics on the top floor of the S800 vault, PMT signals were sent
through RG-8/U type coaxial cables from Belden (model 7810A) which provided a delay of
∼3.84 ns/m [124]. These particular coaxial cables, which were insulated and jacketed with
high-density polyethylene and shielded with a bonded aluminum foil that was surrounded
by tinned copper [124], were chosen as they were found to maintain the quality of timing
3 The

1σ spread in the dispersive position on the first CRDC in the S800 focal plane (See
Section 2.5.1.) was ≈0.7 mm.
27

signals with minimal signal attenuation [22, 31]. The length of the delay cables (∼520 ns)
was such that the timing signals from the A1900 PMTs reached the timing electronics after
the signals from the S800 PMTs so that the S800 signals could be chosen to provide the
‘start’ for an event and the A1900 signals could be chosen to provide the ‘stop’ for an event.
This inversion was necessary because beam transmission from the A1900 to the S800 was not
100% (See Section 2.2.3.), and this scheme prevented the electronics from having a ‘start’
that did not have a corresponding ‘stop’.

Figure 2.4: Photograph of a single BC-418 ultra-fast timing scintillator from Saint-Gobain
Crystals [30] coupled to two R4998 head-on photomultiplier tubes (PMTs) via small plastic
light-guides and optical grease. An aluminum frame helped hold the PMTs to the scintillator
and allowed for a collimator to be mounted on the upstream side to protect the PMTs from
stray beam. (Photo Credit: Milan Matoˇs and Alfredo Estrad´e [31].)

Various combinations were made of the timing signals to create several time differences
to be used in the data analysis. A single time-difference was recorded by using one timing
signal as the ‘start’ and another as the ‘stop’ in an Ortec 566 TAC (time-to-amplitude
converter) and sending the TAC output to an Ortec AD413A Quad ADC (analog-to-digital
converter). Direct time differences between timing detectors and timing differences with
respect to a stable clock were measured. Referring to the two PMT signals accompanying
a single scintillator as “up” (U) and “down” (D), the S800 PMTs with the prefix “S3”,
the A1900 PMTs with the prefix “XF”, and the clock (Ortec 462 Time Calibrator) used to
28

provide random ‘stop’ signals as “Clk”, the following time differences were recorded: S3U–
S3D, XFU–XFD, S3U–XFU, S3D–XFD, MCP–Clk (discarded due to an electronics glitch),
XFU–Clk, XFD–Clk, S3U–Clk, and S3D–Clk, where the first signal in the pair is the ‘start’
and the second is the ‘stop’. A veto was put in place to reject any TAC ‘stop’ signal from
the clock that arrived within 10 ns of the TAC ‘start’ in order to avoid a previously found
glitch in the electronics for start-stop time differences <10 ns [31].
The difference between two time-differences made with a clock-induced ‘stop’ signal directly yields the true time-difference with respect to the ‘start’ signals, offset by an integer
multiple of the clock period (40 ns). These so-called clock times are for each nuclide distributed randomly across the full-range of the ADCs in order to cancel-out any systematic
effects from nonlinearity in the ADC channel-to-time mapping. The direct TOF measurements were used to correct for the fact that a different number of clock pulses could have
occurred for multiple events of a single isotope’s transit from the A1900 to the S800. For
example, a TOF using the “down” PMT signals for the TAC ‘start’ signals and clock pulses
for the TAC ‘stop’ signals would be constructed as:

TD,Clk = TS3D−Clk − TXFD−Clk + ND τ,

(2.1)

where TS3D−Clk is the S3D–Clk time span, TXFD−Clk is the XFD–Clk time span, ND is the
number of clock pulses that occurred during the event’s TOF, and τ is the period of the
clock (for this experiment, 40 ns). The procedure for performing the clock-pulse correction,
as well as a description of other corrections, to generate the actual TOF for a given event
is described in Section 3.2. Time-differences were recorded between PMTs of the same
scintillator (e.g. S3U–S3D and XFU–XFD) to allow for a correction of the measured TOF

29

for different light transit times from different scintillator ion-impact positions (See Figure 2.5
and Section 3.2.). Details on the cuts and corrections made to timing signals and an estimate
for their dead-times are given in Section 3.1, and details of event-TOF construction are given
in Section 3.2.

Figure 2.5: Simplified picture of four measured beam events of the same isotope which would
yield different times-of-flight TOFs without the scintillator position correction. Here four
events (a, b, c, and d) of the same isotope follow different paths from the scintillator in
the A1900 to the scintillator in the S800, where the flight path has been simplified to a
straight line. Assigning a direct flight time TOFdirect =50 ns and noting that the light-travel
1.5 = 79 ps, where n = 1.58 is the refractive
time across the 1.5 cm scintillator is dt = c/n
index of BC-418 [30] and c is the speed of light in vacuum, the direct-down TOF TS3D−XFD
for the four events would be TD,a = 50.000 ns, TD,b = 49.921 ns, TD,c = 50.079 ns, and
TD,d = 50.000 ns without a scintillator position correction. Applying the correction described
in Section 3.2.2, +0.5 × (TXFU−XFD − TS3U−S3D ), corrects all four flight times to 50.000 ns.

2.3.2

Timing detector resolution

The choice of timing detectors, namely BC-418 scintillators coupled to Hamamatsu R4998
photomultiplier tubes (PMTs) in a Hamamatsu H6533 assembly, was made due to their
fast response time and superb timing resolution. The coupling [123, ch.8,sec.III] of the
scintillator, which nominally had a rise time of 0.5 ns [30], and PMT, which nominally had
a rise time of 0.7 ns and electron-transit time spread of 0.16 ns [122], were found to have a
combined rise time of 1.5 ns for a 140 MeV/u 136 Xe beam [22, 31]. The signal rise time was
30

expected to depend on the beam species, since scintillator response depended on the energy
and atomic number Z of the detected particle [125, 126], so timing signals were processed
through a constant fraction discriminator (CFD) to remove this effect. After transmission of
the discriminator output signal from the detector location to the central electronics setup,
a leading-edge-discriminator was used to restore fast-rising signal for input to the time-toamplitude converter (TAC).
The bias was optimized for each PMT to give the minimum time resolution at the lowest
current possible. A larger PMT bias decreased the relative spread in electron-transit time
within the PMT [123, ch.9,sec.IV B]. In addition, increasing the PMT bias increased the
timing-signal amplitude while keeping the rise time near-constant and thus created a steeper
signal. The steeper signal was less sensitive to electronic noise/jitter and thus produced
a higher-fidelity timing signal. (Factors which affected the signal/noise due to scintillator photon-count are discussed in detail in Section 2.3.3.). However, the voltage was not
increased beyond this optimum value in order to keep the operating current as low as possible to increase the lifetime of the PMT. The optimum voltages for the four PMTs were:
VS3U = −1900 V, VS3D = −2000 V, VXFU = −1800 V, and VXFD = −2000 V. After
the voltage adjustments, the CFD settings had to be altered to remove the timing-signal
dependence on the ion-impact position. This was achieved by changing the CFD-settings
(zero-crossing and threshold) until a linear relationship was obtained between the chargecollected from one PMT (via the QDC) and the time-difference between the two PMTs for
a given scintillator, as detailed in [31, sec.2.21]. Timing measurements were also found to be
sensitive to temperature changes, likely due to thermal-response of materials in the PMTs,
TACs, and ADCs [31, sec.3.3.2]. Thermocouples monitored vault temperatures and temperatures on the ADCs and reported them to the data acquisition event-by-event via the
31

Experimental Physics and Industrial Control System (EPICS). Ultimately we found that the
random variations in temperature only slightly broadened the TOF-resolution. To avoid the
introduction of systematic errors, a temperature correction to TOF was not applied.
The time-resolution of the timing detectors was found to be σtiming−detectors ≈ 30 ps [22,
31]. Note that the TOF resolution, ∼ 81ps (See Section 3.8.), was poorer due to the spread
introduced from the beam line ion-optics and from the finite position resolution of the microchannel plate detector (MCP) (See Sections 2.4.4 and 3.5.2.) used for the ion-rigidity
correction.

2.3.3

Timing measurement efficiency

The efficiency for detecting a timing-signal was primarily influenced by the scintillator lightoutput due to ion impacts, the light-collection efficiency of the scintillator and light guide
combination, and matching of the wavelength emission range of the scintillator with the
acceptance range of the photomultiplier tube (PMT). Since the scintillator efficiency is sensitively related to the timing signal strength, it is directly related to the timing resolution
(See Section 2.3.2.).
Scintillator light-output is primarily dependent on energy deposited within the scintillator by ionizing radiation [123, p.229]. Na¨ıvely this would lead one to choose a thick
scintillator for the timing measurement, however the scintillator thickness had to be kept
at a minimum to minimize energy loss, which introduces Z-dependent perturbations to
the ions’ TOF. Employing the ATIMA energy loss model in LISE++ [127], we found the
average energy loss within the fast-timing scintillator of ions produced in this experiment
( Z ≈ 23) was ≈ 0.5 MeV/µm, and thus a total energy loss of ∼125 MeV/event. The
photon yield of BC-418 is 67% that of anthracene [30] (C14 H10 ), meaning BC-418 yields
32

∼11,000 photons/MeV (See Tables 8.1 and 8.3 of [123].). However, this value is for energy
deposited by electrons; ions require more energy deposition to yield the equivalent number
of photons, where the typical reduction factor for plastic scintillators is ∼ ×0.1 [123, 128,
p.229], resulting in ∼140,000 photons/event. Long-term exposure to air and typical ambient
light (∼10’s of hours) can degrade scintillator performance due to coloration and breakdown of the scintillator polymer, reducing fluorescence by a factor of ∼ ×0.1; though this
is dependent on the scintillator material, exposure time, and fluorescence wavelength [129].
Since the scintillators used in this experiment were subjected to long-term oxygen and lightexposure (during testing), their light-output was likely reduced to ∼14,000 photons/event
prior to beam exposure. Previous studies [130–132] have demonstrated the rapid drop-off in
scintillator light-output that occurs after prolonged exposure to radiation. Tests of plastic
scintillators similar to BC-418 found scintillator light-output experienced no reduction after exposure to 50 Gray, ∼15%-reduction after 600 Gray, and nearly 100%-reduction after
14,000 Gray, where γ-rays were the ionizing radiation [132]. Separate tests of plastic scintillators similar to BC-418, where the radiation source was heavy ions, found a 50%-reduction
after exposure to 25,000 Gray [130, 131]. Our average beam-rate on the A1900 scintillator
during the experiment was ∼2×104 particles/second. Using estimates of scintillator degradation from [132] and [130, 131], the scintillator light-output would be reduced by 50% after
∼150 hours and ∼1,000 hours, respectively. Since the total beam-on-scintillator time was
<150 hours (See Section 2.2.), we estimate the scintillator light-output was at no time less
than 50% of the aforementioned estimated value of 14,000 photons/event; i.e. a conservative estimate for photons produced in the A1900 scintillator for a given beam-event would
be ∼7,000 photons/event. Since the S800 scintillator used for timing experienced a much
lower beam-rate, it was likely not subject to beam degradation and can be conservatively
33

estimated to have produced ∼14,000 photons/event.
Further considerations must be made as to the number of produced photons which would
ultimately generate a signal in the PMTs the scintillators were coupled to. Primary issues
which prevent photons from producing signals in the PMT are light escaping the scintillator surface, light attenuation within the scintillator, poor light transmission across the
scintillator-PMT coupling surface, matching between wavelength ranges for scintillator emission and PMT acceptance, and quantum efficiency of the PMT photocathode. Since scintillation light is emitted in all directions, some fraction of the light will escape from the
scintillator surface, since only light striking the scintillator surface at an angle greater than
n

the critical angle θc will be totally internally reflected, where θc = sin−1 n1 . In this case
0
the two indices of refraction were n0 = nscintillator = 1.58 [30] and n1 = nvacuum ≡ 1, so
θc = 39.3◦ . Defining the ratio of the two indices of refraction as 1/n ≡ n1 /n0 = 0.63, the
percentage emitted from each of the six sides of all light produced is

1
1
F1−side (n) = Ftotal−emitted (n) = {1 − [(n2 − 1)1/2 /n]} ≈ 11%,
6
6

(2.2)

where incredibly the fraction is independent of the shape of the scintillator body [133]4 ,
neglecting effects of light attenuation which in this case were minimal [30, 123, p.258]. Coating the scintillator surface with a diffuse reflector such as titanium oxide [30], magnesium
oxide, or aluminum oxide powder could have increased the light collection by reflecting light
that would otherwise have escaped [123, 133, p.259], however the improved light-collection
efficiency would have to be weighed against the increased variation in photon travel distance
caused by the varying number of reflections experienced by each photon. By coupling the
4 Note

that [133] accidentally drop a division sign moving from Equations 3 to 4 and that
Equation 2.2 is only valid for n > 1.5, as is evidenced by [133] Figure 2.
34

scintillator to the PMT with a light-guide of roughly the same index of refraction (∼1.5) and
a larger area than the scintillator edge (∼ 1.5) via an optical coupling grease with a lower
index of refraction (∼ 1.45), transmission of light to the PMT was expected to be near 100%.
Since the wavelength range for light emitted from BC-418, 355 < λ < 500 nm, [30] overlapped with the acceptance range for the R4998 PMT [122], no light was expected to be lost
due to lack of PMT sensitivity. The quantum efficiency for the R4998 PMT in the wavelength
range 355 < λ < 500 nm ranged between 10-20%, with a quantum efficiency of ∼20% for the
scintillator’s most probable wavelength of emission, ∼390 nm. Note that radiation damage
to the scintillator material was not expected to alter the emission wavelength spectrum from
the BC-418 scintillator [132]. Given the above considerations, it was expected that of the
photons produced by beam particles within the scintillator, 2 × 0.11 × 0.2 × 100 ≈ 4% of
light would create a timing signal in the PMT. Given a PMT gain of 5.7 × 106 [122], it was
anticipated that all detected photons would produce a measurable timing signal.
Taking into account the above considerations, each of the ∼14,000 photons/event in the
S800 fast-timing scintillator and ∼7,000 photons/event in the A1900 fast-timing scintillator
had a detection probability of ∼4%, meaning that the probability for obtaining a good time
signal from a given event was 100%. (Note that the probability for obtaining a good TOF
was only ∼50% due to imperfect beam transmission (See Section 2.2.3.).). The measured
timing signal efficiency for the A1900 and S800 scintillators was > 99.9% throughout the
mass measurement.

35

2.4

Magnetic rigidity measurement

The dispersion matched ion optics mode [31, 121] was chosen to provide achromatic transport
of ions from the A1900 [27] focal plane to the S800 [34] focal plane (See Figure 2.1.). The
dispersion matched mode provided a focused ion beam at the A1900 and S800 focal planes
with a beam position independent of momentum and a dispersive focus at the S800 target
position, referred to as the dispersive plane, where the vertical beam position corresponded to
an ion’s magnetic rigidity [22, 31]. The rigidity acceptance was limited to δBρ/Bρ ∼ ±0.5%
by the size of our position measurement detector due to the dispersion of ∼ 11cm/% [22].
For the ion optics used in our experiment, this dispersion resulted in spread in time-of-flight
of ∼40 ps/mm.

2.4.1

Method overview

The method developed by [134] was chosen to measure beam position at the dispersive plane
because it involved placing a small amount of material in the path of the beam, ultimately
minimizing the position measurement’s influence on the ions’ times-of-flight. As shown in
Figures 2.6 and 2.7, this method consisted of sending the ion beam through a foil and guiding
the secondary electrons generated in this process to the surface of an 8 cm-wide×10 cm-tall
microchannel plate detector (MCP). The foil was a 70 µg/cm2 polypropylene (C3 H6 ) film
sputtered with 1500 ˚
A of gold biased to -1 kV by a Canberra Model 3002D high voltage
power supply. The MCP [123, p.286] consisted of two lead glass plates from Quantar [135]
(model 3398A), oriented in the chevron configuration to avoid feedback from heavy ions
and to prevent electrons from passing directly through channels [33], biased to a 2.35 kV
voltage. Rectangular (2×4×4.74 in3 ) NdFeB 35 permanent magnets from Magnet Sales and

36

Figure 2.6: Cartoon of the dispersive plane position measurement system. Secondary electrons generated by the ion beam passing through a gold foil were guided to a MCP by an
electric and magnetic field to measure the beam position at the foil. (From [29]).

Manufacturing [136], which had a residual flux density of 1.23 T, were placed 4 cm behind the
MCP face and 5 cm behind the foil with their poles aligned and planes parallel to the MCP
and foil to create a region of near homogeneous magnetic field between the foil and MCP (See
Figure 2.8.). The magnets were kept at a constant spacing by a steel yoke, which also helped
to create a homogeneous magnetic field. The parallel electric and magnetic fields guided the
electrons in a helical trajectory with a small radius towards the MCP surface so that there
was an accurate mapping between the beam-foil interaction position and recorded MCP
position. Upon reaching the MCP, incoming electrons from the foil created an avalanche
of electrons, due to amplification by the MCP, which reached the MCP back-plane [33].
37

Figure 2.7: Photograph of the magnetic rigidity measurement setup. Two magnets were
mounted on an steel yoke plane-parallel to an 8×10 cm microchannel plate detector (MCP).
A gold foil was mounted plane-parallel to the MCP on a driver-arm so that it could be
inserted from below. A foil and hole-mask were also mounted on the driver arm below
the first foil, where hole-mask upstream of the foil in order to only allow ions to generate
electrons at certain positions for a MCP position calibration . A collimator was located
upstream to protect the MCP during beam-alignment. A hole-mask with a distinctive holepattern and a phosphorescent coating was mounted on a driver arm further upstream so
that it could be inserted for beam-alignment. A camera was located within vacuum chamber
further upstream and off of the beam-axis to provide an image of the hole-mask during
beam-alignment.

The back-plane consisted of a resistive layer in which electrons drifted to its four corners,
where the amount of charge collected in each corner was used to determine an MCP-impact
position [35, 137] and therefore a foil-impact position for the beam. Additionally a timing
signal was extracted from the MCP back-plane, however it was not ultimately used due
to defects in the electronics modules used for MCP timing during the beam-time, and so
it will not be discussed further. To assist in beam-tuning, the gold foil was mounted in a
ladder-configuration on a driver arm, where a mask with a known hole-pattern was located

38

on another ladder location, and another mask with a phosphorescent coating was located
on a driver arm upstream of the gold foil (which was observed by a camera located within
the vacuum chamber), and a collimator was placed upstream of the MCP to protect it from
stray beam (See Figure 2.7.).

Figure 2.8: Measured (solid circles) magnetic field as a function of distance from the left
magnet in the rigidity measurement set-up, using a magnet-surface separation of 17.5 cm,
compared to the calculated field value (See Equation 2.4.) for that magnet separation (solid
red curve). The dashed blue curve indicates the calculated magnetic field value for the
magnet separation used in the mass measurement experiment. Vertical dashed lines indicate
the position of various components used in the position (rigidity) measurement with respect
to the magnets.

One concern which should be raised about this method of position detection is that
the magnetic field from the permanent magnets may have deflected the beam fragments
and therefore impacted their time-of-flight. However, the angular deflection occurred at a
dispersive focus ion the ion optics. Since acrhomatic ion optics were used, the TOF of ions
after the dispersive focus was independent of the ion angle at this location. In any case, the
angular deflection would be rather small and it would depend smoothly on the charge-to-mass
39

ratio of ions. For a relatively small impulse, tan(θ) ≈ θ ≈ δp/p =

qvB(l/v) sin(α)
mv

=

qBl sin(α)
,
mv

where l is the distance over which an ion, moving at velocity v with charge q and mass m,
is subject to the magnetic field strength B from the MCP magnet-yoke which is oriented at
angle α with respect to the beam trajectory. As an example, the fragment 45 Ar, q = Ze ≈
2.9×10−18 C and m = Au ≈ 7.5×10−26 kg (where e is the electron charge and u is the atomic
mass unit), which had a velocity of ≈ 0.4c (from the beam energy stated in Section 2.2.2)
and traversed the ∼ 10 cm-wide ≈ 0.22 T magnetic field region (which was oriented ∼ 70◦
with respect to the beam), would have experienced an angular deflection of ∼ 0.7 mrad. To
roughly assess the impact on TOF, we can employ the ion optical transfer matrix calculated
for the previous NSCL mass measurement [31, Their Equation 3.1.], which stated that an
angular deflection at the A1900 focus in the non-dispersive direction was related to an ion
flight-path length change as −5 × 10−4 cm/mrad. If the corresponding relationship between
the non-dispersive deflection at the dispersive focus at the entrance of the S800 and the path
length is of the same order, then the expected change to the overall flight path would be
∼ 10−6 m, corresponding to a relative TOF change (since δl/l ∝ δTOF/TOF) of 10−7 . Note
that not only was the alteration to TOF over one order of magnitude less than systematic
TOF effects that were controlled for, the kick had a smooth behavior with respect to m/q 5
and therefore any impact would have likely been captured in the TOF-dependent terms of
the final mass-fit function (See Equation 3.17.).
5 Note

that the full set of nuclei used in the final mass analysis (reference nuclei and the
most exotic evaluated nuclei) had a relatively narrow range in A/Z: 2.44 ≤ A/Z ≤ 2.72
(See Section 3.9.), further mitigating the issue of the m/q-dependent kick from the MCP
magnet-yoke set-up.

40

2.4.2

Electron production

The number of electrons produced from the foil by passing ions is influenced by a number
of factors, such as beam-particle nuclear charge, beam-particle charge state, beam fluence,
beam energy, foil thickness, and foil-material nuclear charge [138–142]. The basic process of
fast-ion induced electron emission from a foil is characterized by two steps: energy lost by
the ion in the foil ionizes foil-atoms and some of the electrons freed in the ionization process
are able to escape the foil [123, 143, p. 41]. Thus, to first order, the electron yield is given
by the energy required to free an electron from the foil and the energy loss per unit length
in the foil.
At the high ion-energies employed in this experiment, beam nuclei are fully stripped
(Discussed in more detail in Section 3.4.3.) and ion-energy loss in the foil is well described
by the Bethe formula [123, p. 31]:

−

4πe4 zb2
dE
Nt Zt ln
=
dx
me vb2

2me vb2
It

v2
− ln 1 − 2b
c

v2
− 2b ,
c

(2.3)

where e is the electron charge, me is the electron rest mass, c is the speed of light, zb is
the beam particle charge (which is the beam nuclear charge Zb here, since the ions are fully
stripped), vb is the beam particle velocity, Nt is the number density of atoms in the foil,
Zt is the atomic number of atoms in the foil, and It is the average ionization potential of
the foil. It is apparent from this relation that the electron yield from a foil should scale
roughly as Zt2 , so a foil material with a large atomic number is desirable to produce enough
electrons to induce a strong signal in the MCP. However, a large Zt would also cause beam
ions to have a large energy loss, and thus a large perturbation to their time-of-flight TOF.
This perturbation was minimized by using as thin of a foil as could be readily manufactured
41

on-site. Additionally, the foil material chosen needed to be electrically conductive so that
it could be biased to generate an accelerating potential toward the MCP for the emitted
electrons. The aforementioned considerations lead to the choice of a 1500 ˚
A gold layer
(≈30µg/cm2 ) deposited on a 70 µg/cm2 polypropylene film as the foil.
Due to the electric field generated by space-charge separation created in the wake of
a fast-ion, the foil electron yield is reduced. Rather than scaling as zb2 , the electron yield
from gold scales as zb1.71 [138]. Since our beam species of interest had a charge range
zb = Zb = 14 − 30 and roughly the same ion velocity, the electron yields produced for
each species were within a factor of ∼4. The relatively low beam fluence used in this
experiment, ≈

5000 ions/sec
80 cm2

∼ 100 ions/cm2 , meant that the electron yield was not reduced

due to modification of the foil properties by beam fluence [142]. Previous studies showed
that fully-stripped nickel ions impinged at 74 MeV/u on a 30 µg/cm2 carbon foil would
produce ∼20 electrons/ion emitted at a backward angle (Note that the electrons emitted at
a forward angle will not be directed toward the MCP by the electric field in the setup, but will
rather be directed back into the foil). Using the scaling for beam energy (From Section 2.2:
∼100 MeV/u) and foil charge (79), and noting that the first ionization potentials for carbon
and gold are within 20% [144], ions studied in this experiment were expected to yield ∼50200 electrons/ion emitted toward the MCP. For comparison, a ∼5.4 MeV α-particle, as is
typically emitted from 228 Th, would yield ≈4 electrons [140].
As discussed in Section 2.4.3, electrons produced by the beam could potentially have had
cyclotron radii up to re ≈ 1.7 mm. Given that >50 electrons/ion were expected for a typical
beam particle, we expected several electrons with relatively large cyclotron radii, and thus
a relatively poor position resolution. For comparison, a ∼5.4 MeV α-particle emitted from
a 228 Th source would have typical electron total kinetic energies Te,t ≈0.1 keV [141] and
42

thus potentially cyclotron radii up to re ≈ 0.17 mm. It is apparent that tests of the position
measurement set-up with an α-source were expected to yield a better position resolution
than that obtained for fast-ions in the mass measurement beam-time, in agreement with the
findings of [145]. Na¨ıvely, this might lead one to conclude that a lower beam-energy should
have been used in order to lower the average electron kinetic energy and thus improve the
position resolution for the rigidity measurement. However, a lower beam energy would have
lead to the presence of multiple charge states for beam species (See Section 3.4.3.), adding a
significant complication to the data analysis. [146] investigated the dependence of electron
kinetic energy on the foil material, employing carbon and aluminized-mylar for 5.8 MeV
α-particles and 30 MeV oxygen-16 nuclei, and found no detectable difference.

2.4.3

Electron transport

The magnetic field and voltages were chosen to optimize transport of the electrons from
the foil to the MCP surface. Na¨ıvely, increasing the foil voltage would continually improve
electron transport, since the acceleration induced perpendicular to the foil would overcome
any non-perpendicular velocity components. However, due to the relatively broad electron
kinetic energy distribution, the improvement in position mapping ceased at a relatively
modest MCP-to-foil potential difference [146], in this case -1 kV. The magnetic field provided
the radial confinement for an electron’s helical trajectory, where the radius of the helix was
given by the electron’s relativistic cyclotron orbit, re = pe c/eBc [147], where pe is the
electron’s momentum, c is the speed of light, e is the electron charge, and B is the magnetic
field in which the electron orbits. Since the total electron energy Ee,total =

p2e c2 + m2e c4 =

Te,t + me c2 , where me is the electron mass and Te,t is the electron’s kinetic energy transverse
to the magnetic field, re =

2 +2T me c2
Te,t
e,t
eBc

. The transverse electron kinetic energy depends
43

not only on the angle at which the electron leaves the foil, but also on the ion charge and
ion kinetic energy which causes the electron emission from the foil [134, 141]. For the beam
energy and beam charge typical for this experiment, ≈100 MeV/u and Zbeam ≈ 23, we can
estimate from a similar measurement [141] that the average total kinetic energy for a given
electron was Te,total ≈ 10 keV. Note that the electron kinetic energy depends roughly linearly
on the ion energy expressed in energy per nucleon [141]. Using an electron kinetic energy of
10 keV in the direction transverse to the magnetic field, an electron mass of 511 keV, and
a magnetic field strength of 0.2 T, the resultant radius for an electron’s helical trajectory is
≈1.7 mm. This represents a worst-case for our beam-foil interaction-position measurement
accuracy, since in general several electrons will contribute to the position measurement (See
Section 2.4.2.) and most will likely not have the full initial kinetic energy in the component
transverse to the magnet and foil planes. We note our treatment is an oversimplification
since the dynamics of the helical trajectory gradually reduces the transverse component
of the electron’s velocity as it travels from the foil [134]. Additionally, the magnetic field
between the foil and MCP is not constant, as is shown in Figure 2.8, and instead is well
approximated by the formula for identical rectangular magnets facing each other in attracting
positions inside a ferromagnetic yoke from the Magnet Sales [136] catalog equation 8:

B(x) =

Br
AC
√
(tan−1
2
π
2x 4x + A2 + C 2
−

Br
tan−1
π
2(2L + x)

+

Br
tan−1
π
2(R − x)

−

Br
tan−1
π
2(2L + (R − x))

AC
4(2L + x)2 + A2 + C 2
AC
4(R − x)2 + A2 + C 2

44

AC
4(2L + (R − x))2 + A2 + C 2

(2.4)

where Br is the residual flux density (here 1.23 T), A is the magnet width (here 10.16 cm),
C is the magnet length (here 12.07 cm), L is the magnet thickness (here 5.08 cm), x is the
distance from the first magnet in the pair, and R is the distance between the two magnet
surfaces (14.9 cm for the experiment; 17.5 cm for the data points shown in Figure 2.8). It
is apparent that in our simple model, the electron cyclotron radius depends weakly on the
electron kinetic energy (which anyhow depends weakly on the beam energy and beam species
for the range of beam particles studied in this experiment [141]) , since the numerator in
the above relation for re is dominated by the electron mass. Thus, the primary influence
on re comes from the magnetic field strength. In principle stronger magnets would improve
position resolution 6 , but obvious limitations are cost, safety of assembly of the magnetic
yoke system (See Appendix A.), and undesirable effects on nearby electronics.

2.4.4

MCP position determination

As described in Section 2.4.3, electrons were guided directly from the beam-interaction point
on the foil to the surface of the MCP and then amplified as depicted in Figure 2.9, where
the voltage applied to the MCP was 2.35 kV (The unique secondary electron production
mechanism of MCPs, the secondary electron trajectories, and the principle by which the
continuous channels act as a fixed number of dynodes is described in detail by [32]). Charges
then freely drifted in the resistive backplane, until finally reaching one of the four backplane
corners. A large number of charges was produced in the backplane for a single incident
electron and so signals were registered at all four corners of the backplane for most events.
Corner signals were amplified by “NSCL Quad Fast Amps” ( manufactured by the Electronics
6 In

Section 3.5.2 it is shown that a factor of two increase in field strength would have
likely resulted in a factor of two improvement in position resolution, leaving the intrinsic
resolution of the MCP as the limiting factor.
45

Shop of the National Superconducting Cyclotron Laboratory) and sent to a CAEN V792
charge-to-digital converter (QDC). To ensure weak corner signals would be detected and
strong corner signals wouldn’t saturate the QDC, each corner signal was sent through both
a low-gain and a high-gain amplification. The low-gain amplification consisted of two stages
of NSCL Fast Amps and the high-gain amplification consisted of three stages, where a single
stage offered a gain of ∼ ×7. An alternative signal processing scheme, which sent corner
signals through Tennelec TC174 pre-amplifiers to Ortec AD143A analog-to-digital converters
(ADCs) was attempted, however it was found to produce position measurements of inferior
quality, so it was not used.
Runs triggered on background were used to obtain QDC pedestals to be subtracted from
the corner signals and runs using a pulser signal sent through an attenuator with various
levels of attenuation were used for gain-matching. Attempts were made to isolate the MCP
from electronic noise, primarily by adding additional shielding (aluminum foil) to cables.
This was necessary because the rather large MCP-face (∼ 8 × 10 cm2 ) acted as a capacitive
pick-up for nearby electronic noise. After pedestal subtraction and gain matching, the four
corner signals, upper left (UL), upper right (UR), lower left (LL), and lower right (LR), were
used to determine horizontal (X) and vertical (Y) positions via the following algorithms:

UR + LR − UL − LL
UL + UR + LL + LR
UL + UR − LL − LR
Y =
.
UL + UR + LL + LR

X=

(2.5)
(2.6)

Further considerations, such as pedestal determination, gain matching, and whether the
high or low-gain was chosen for a corner for a given event, are described in Section 3.5.
Aside from physical considerations discussed in Section 2.4.3, MCP position resolution was
46

primarily determined by data analysis and so the full discussion is left for Section 3.5

2.4.5

Position measurement efficiency

The efficiency of the position measurement depended on the electron production from the
gold foil, the electron-detection efficiency of the microchannel plate detector (MCP), and the
response time of the MCP and associated electronics.
The intrinsic detection efficiency of an MCP for an incident electron is difficult to quantify because previous studies have found it to depend on both electron energy and angle of
incidence [33, 148–150]. Perhaps unsurprisingly, there is a large range of measured MCP
detection efficiencies for incident electrons at a given energy. Typical values for the detection efficiency of an electron are ∼100% at 0.1 keV, ∼60-100% at 1 keV, and ∼25-80% at
10 keV [148, 149], however it has been shown at 1 keV that the angle of incidence of the electron on the microchannel plate can vary the relative efficiency from 1-100%, with the average
reduction in efficiency from non-optimum incidence angles being a factor of ∼0.5 [150]. For
our typical electron energy of 10 keV (See Section 2.4.3.), an expected intrinsic efficiency for
detection of a single electron is then ∼25%. Since we expected a beam-ion passing through
the gold foil to emit >50 electrons (See Section 2.4.2.), the efficiency of detecting an electron
on the MCP surface for a position measurement was expected to be 100%, independent of
signal processing considerations. This is in agreement with observations made by [134].
In principle detector dead-time could also be expected to lower intrinsic detection efficiency. However, the quoted maximum rate per microchannel from Quantar was ∼50 counts
per microchannel per second [135]. Since the microchannel plate consisted of ∼ 106 microchannels, each with ∼25 µm diameter, spread over an active area of 74 mm×92 mm, a
single channel would only become saturated by space-charge and cause detector dead-time
47

if the incident electrons came at a high-rate or in a small area. Since typical beam rates at
the MCP were ∼200 particles/second and on the order of ∼100 electrons were produced per
particle, yielding ∼20,000 electrons/second roughly spread over and area π(re2 ) ∼ 3 mm2 ,
we expected no dead-time due to physical properties of the MCP.
The position-detection efficiency of the MCP was measured during the mass-measurement
experiment by taking the ratio of events with good rigidity (MCP position) and time-of-flight
(TOF) measurements to events with good TOF measurements, noting that the expected
TOF-measurement efficiency was ≈100% (See Section 2.3.3.). We found an average MCP
efficiency of ≈96%. A possible explanation from the reduction from 100% is the condition
applied to detected MCP events that they have a ‘good’ position, i.e. signals from all four
corners of the resistive readout MCP-backplane that are above the threshold and below
saturation (where an algorithm is used to decide to use the high-gain or low-gain corner
signal, as described in Section 3.5). It is possible that events registered near the corners
of the MCP may have had corner signals which were below threshold or above saturation.
Alternatively, a coincident detection of a background photon could have driven a corner
signal over saturation.

48

Figure 2.9: Cartoon of the MCP position determination method. An electron freed from
the gold foil by a passing beam particle strikes the wall of a microchannel (µchannel in the
diagram), creating secondary electrons. Secondary electrons are accelerated by a voltage
difference (∆V) applied across the plate (MCP) and strike the microchannel wall, creating
more secondary electrons [32]. The avalanche of electrons ultimately reaches the resistive
backplane, where charges are free to drift to the four corners of the backplane. The position at
which charge is initially injected onto the backplane, which corresponds to the position of the
microchannel and thus the position of the beam on the gold foil, is determined by comparing
the relative amount of charge collected on each of the four backplane corners. Note: A single
plate with straight channels is pictured here for simplicity. This experiment employed two
MCPs with angled channels stacked in a way known as the chevron configuration [33].

49

2.5

S800 focal plane

The S800 spectrograph focal plane [28, 34] served as the end-point of the beam line in
the time-of-flight mass measurement experiment and contained several detectors which were
used to measure timing, energy loss, angular trajectory, and isomeric content of the beam
fragments, as pictured in Figures 2.10 and 2.11. In order of upstream to downstream, the
S800 focal plane contained a cathode readout drift counter (CRDC) (See Section 2.5.1.),
a fast-timing scintillator coupled to two photomultiplier tubes (See Section 2.3.), a second
CRDC, an ionization chamber with sixteen segments (See Section 2.5.2.), an aluminum plate
(‘ion catcher’), and an array of cesium-iodide scintillators coupled to photomultiplier tubes
(‘hodoscope’) (See Section 2.5.3.).

2.5.1

Beam tracking

The trajectories of ions within the S800 focal plane, which were correlated with their momentum and path length, were obtained by position measurements at each cathode readout
drift counter (CRDC) [34]. The CRDCs [151], which were spaced 1 m apart and capable
of 0.4 mm position resolution, had an active volume of 30 cm×59 cm×1.5 cm filled with
a mixture of 80% tetrafluoromethane (CF4 ) and 20% isobutane (C4 H10 ) (by volume) and
horizontal (non-dispersive direction) parallel anodes with 2.54 mm pitch [28]. The vertical
(dispersive) position measurement was determined by the induced charge distribution on
CRDC anodes and the horizontal (non-dispersive) position measurement was given by the
drift-time of secondary electrons, created in an ionization avalanche near the anode wire,
to the cathode. The reliance on electron drift time for the horizontal position measurement meant that this position measurement fluctuated with the gas pressure within the

50

Hodoscope
Ion Catcher
Ionization Chamber
CRDC 2
TOF Start
CRDC 1

Focal Plane

Beam

(a)

(b)

Figure 2.10: Schematic of the detectors within the focal plane (a) of the S800 spectrograph
(b) (Adapted from [28] and [34].). Following the rigidity measurement at the base of the
S800 (See Section 2.4.), the beam was steered ∼150◦ into the S800 focal plane. Cathode
readout drift counters (CRDC) were used for beam tracking (See Section 2.5.1.), a fasttiming scintillator was used for the TOF-start (See Section 2.3.), an ionization chamber was
used to measure energy loss (See Section 2.5.2.), and an aluminum plate (‘ion catcher’) was
used in conjunction with a hodoscope to detect isomers of isotopes with atomic mass A 40
(See Section 2.5.3.).

CRDC [123, ch.6,sec.IC]. Since CRDC fill-gas pressures were not incorporated into the data
readout, position drifts in the CRDCs would have to be determined by gating on nuclei
produced with high statistics and monitoring their position drift. This was ultimately not
done because it was found [31, ch.4,sec.3.2] that the S800 focal plane angle corrections to
TOF were redundant with respect to the S800 fast-timing scintillator position correction
(See Sections 2.3.1 and 3.2.).
The CRDCs limited the overall event rate that could be sustained in the S800 focal
plane. Due to the focused beam at the first CRDC, it was determined that rates in excess
of ∼500 particles/second would produce high enough space charge density to damage the
51

Hodoscope

Ionization
Chamber
Hodoscope

CRDC2

Ion
Catcher

CRDC1

Figure 2.11: Photograph of detectors within the S800 spectrograph focal plane (center)
with insets of the ion catcher and hodoscope (left) and fast-timing scintillator mounted to
two photomultiplier tubes (See Section 2.3.) downstream of the first cathode readout drift
counter (CRDC) (right). Note that the ion catcher and fast-timing scintillator were not
installed in the focal plane at the time the center image was taken. (Center photo credit:
Shumpei Noji.)

CRDC7 . Na¨ıvely one may conclude that had the CRDCs not been used, the experiment
could have been run at higher rates, i.e. the wedge placed in the A1900, which stopped lowZ particles from being transmitted but also limited the overall rate of nuclei of interest (See
Section 2.2.2.), could have been removed. However, higher rates would have caused event
pile-up within the ionization chamber and possible degradation of the fast-timing scintillator
(See Section 2.3.3.) located in the A1900 focal plane.
7 High

space-charge density can cause impurities to build-up on CRDC anode wires and
can also lead to excessive currents within the readout electronics.

52

2.5.2

Isotope identification

Particle identification was achieved using the time-of-flight energy-loss technique [152–154,
sec.3], where the time-of-flight was provided by fast timing scintillators (See Section 2.3.) and
energy-loss was provided by an ionization chamber [123, ch.5]. The ionization chamber had
an active area of 30 cm×59 cm with sixteen independent 2.54 cm-thick segments filled with
P10 gas (90% argon and 10% methane (CH4 ) by volume), allowing for redundant energy-loss
measurements with an achievable energy resolution of ∆E/E∼10−4 [28].
Upon examination of the Bethe formula (See Equation 2.3.) it is apparent that energy
loss ∆E of a fully stripped nucleus with atomic charge Z and velocity v is proportional to the
2

square of the charge divided by the velocity ∆E ∝ Z2 . Since velocity is equal to momentum
v
p over mass m 8 and momentum over charge q = Z is equal to magnetic rigidity Bρ, it
2
q 2 (Bρ)2
Z 2 (Bρ)2
follows that v 2 = p 2 =
=
. Substituting this relation into the relation for
m
m2
m2

energy loss, and taking into account that Bρ ∼constant due to the limited acceptance of the
beam line (See Section 2.4.), it is apparent that

∆E ∝

Z 2 m2
∼ m2 .
Z 2 (Bρ)2

(2.7)

This near proportionality between energy loss and nuclear mass-squared is readily apparent
in the non-rigidity corrected particle identification plot shown in Figure 2.3.
As described in Section 1.4.1, absent of relativistic corrections, a fully-stripped nucleus
with atomic charge Z and nuclear mass m in a magnetic system has an equation of motion
Bρ
described by m
Z = v , where velocity is simply the path length L divided by the time-of-flight
8 Relativistic

corrections are ignored here, but would be on the order of ∼16% since
v ≈ 0.4c. The impact of the relativistic correction on the result shown would be mitigated
by the small velocity spread of ions of interest.
53

TOF along that flight path. It follows that

T OF ∝

m
,
Z

(2.8)

as is apparent in Figure 2.3.
Since Z is an integer value and m (in GeV) is nearly equal to the atomic mass number
A produces a
A, which is an integer, the two-dimensional plot of ∆E ∝ A2 versus T OF ∝ Z

matrix where each point is a nucleus with a unique A,Z combination. This is demonstrated
for a hypothetical idealized case in Figure 2.12 and employed for this experiment in Figure 2.3. The uniqueness of the PID matrix allowed nuclei to be readily identified, where
the identification was confirmed by comparison to LISE++ [127] simulations and detection of
known isomers (See Section 2.5.3.).

2.5.3

Isomer detection

A system was developed to measure the decay from isomeric states of nuclei whose halflives were an order of magnitude less than the average time of ∼2 ms that elapsed between
focal plane event rates, but at least as long as half of the typical flight time of ∼500 ns.
At the most downstream location of the S800 focal plane, an ∼8 cm× ∼12 cm× ∼1 cm
aluminum plate intended to stop ions with atomic mass A

40 (as determined by SRIM [155]

calculations) referred to as the ‘ion catcher’ was located ∼3 cm in front of the hodoscope,
as seen in the left panel of Figure 2.11. The hodoscope is a sodium-doped cesium-iodide,
CsI(Na), 8-crystal×4-crystal array coupled to photomultiplier tubes, where each crystal is
7.6 cm×7.6 cm and 5.1 cm thick [156, 157]. The hodoscope was expected to have a resolution
of ∼ 7% for 662 keV γ-rays [156] and a relatively high efficiency due to its nearly 2π geometric
54

5000

4000

A2

3000
2000

1000

1

1.5

2

2.5

3

A/Z
Figure 2.12: Demonstration of the energy-loss ∆E versus time-of-flight TOF particle identification method (‘PID’). Since ∆E ∝ A2 and T OF ∝ A/Z, where A is the atomic mass
number and Z is the atomic number, an idealized ∆E vs. TOF PID is represented by plotting
A2 versus A/Z for several combinations of A and Z. It is apparent that each A,Z combination occupies a unique location in the PID matrix. (For the PID from this experiment, see
Figure 2.3.)

coverage with respect to the ion catcher and the relatively large γ-ray absorption coefficient
of CsI(Na) scintillators [123, ch.8,sec.IIB2]. The well known microsecond isomers of 65 Fe
and 67 Fe [98] were used confirm the assignment of nuclei in the PID (See Figure 2.3.).

55

Chapter 3
Data analysis
The data was collected in the manner described in Chapter 2, placed into so-called ‘.evt’ files,
and processed on-line during the experiment via SpecTcl [158]. Following the experiment,
the data were converted 1 to ROOT [160] ‘trees’ so that the majority of the data analysis was
carried out within the ROOT 2 framework.
The following sections describe the analysis procedure used to obtain the masses presented in Chapter 4. Development of the steps within the analysis was done in an iterative
process, by necessity. Since, as stated in Section 1.4, no effect could be left uncontrolled
for that impacted the time-of-flight of nuclei systematically by a picosecond or more, a full
analysis pipeline had to be put in place before details of each analysis step could be fully
investigated. This pipeline, whose initial form was heavily influenced by [31], consisted of
the full set of analysis procedures, from processing raw data to producing masses and uncertainties. Briefly stated, these steps were: creating timing signals, constructing by-event
time-of flight TOF, processing energy loss information, particle identification, checking for
charge-state contamination, determining the by-event magnetic rigidity, performing a rigidity (momentum) correction to the event-TOF, obtaining a global relationship for the rigidity
1 The

evt-to-ROOT code was created by Kathrin Wimmer [159] and modified by the author
and Karl Smith to accommodate the microchannel plate detector and fast-timing scintillator
electronics.
2 Version 5.34.09 was ultimately used, though tests provided the same results with version
5.26.00, as tested on the NSCL fishtank and seaside clusters, as well as the Max-PlanckInstitut f¨
ur Kernphysik cluster.
56

correction, determining a final TOF for each nucleus, performing a fit to obtain the massTOF relationship, and the propagation of uncertainty. The following sections describe in
detail the pipeline that was employed and its methods, as well as alternative methods that
were investigated.

3.1

Timing

As discussed in Section 2.3, timing signals were collected from two fast-timing scintillators,
one at the A1900 extended focal plane and the other at the S800 spectrograph focal plane,
which were each coupled to two photomultiplier tubes PMTs. The various signals are referred
to by their point of origin. For example, the signal from the PMT located on the ‘up’ side of
the scintillator in the A1900 extended focal plane is ‘XFU’, while the signal from the ‘down’side PMT of the S800 focal plane scintillator (located in the NSCL S3 vault) is ‘S3D’. The
signal from the clock, which came at regular 40 ns intervals, is referred to as ‘Clk’ and the
timing signal from the microchannel plate detector, which was not used due to an electronics
glitch, is referred to as ‘MCP’.
The timing signals were employed to create various time differences via a time-to-amplitude
converter TAC, which are referred to by (start signal)–(stop signal). The full list of timedifferences that were recorded in TACs is: MCP–Clk3 , XFD–Clk, S3U–Clk, S3D–Clk, S3U–
S3D, XFU–XFD, S3U–XFU, S3D–XFD, and XFU–Clk. The raw spectra for these time
differences are shown in Figure 3.1. The purpose of the S3U–S3D and XFU–XFD times was
to give a direct measurement of the difference in light-arrival time for the two PMTs attached
to a given scintillator, which could be used to ascertain the beam-particle impact location.
3 This

time-difference was discarded due to an electronics glitch that caused large jitters
in the recorded time.
57

The purpose of the S3U–XFU and S3D–XFD times was to give a direct measurement of the
time-of-flight for beam particles, which was essential in constructing the TOF for each event
(See Section 3.2.). The remaining timing signals, which relied on a clock for the stop-signal,
could be used to construct scintillator-position times or direct flight-times that would not be
systematically impacted by nonlinearities in the time-to-channel relationships for the TAC–
ADC combinations. The time-to-channel calibration was performed in two different ways
for TAC–ADC timing signals.

3.1.1

Linear time calibration

A linear time-calibration was obtained by recording data for known stop–start time-spans in
10 ns intervals (red-filled histograms in Figure 3.1) from an Ortec 462 Time Calibrator. The
time-to-channel calibration parameters determined from the linear time-calibration were,
in picoseconds per channel, MCP–Clk: 6.232, XFD–Clk: 6.309, S3U–Clk: 6.294, S3D–
Clk: 6.308, S3U–S3D: 6.238, XFU–XFD: 6.257, S3U–XFU: 12.586, S3D–XFD: 12.487, and
XFU–Clk: 6.304. Since the timing signals were recorded in units of channels, they were
discrete by nature. This discreteness favored certain times and thus could have introduced
artificial features into the timing signal spectra once converted into units of seconds. To
avoid this issue, for each event a uniform distribution whose width was equal to the ADC
bin-width in units of time was randomly sampled and the resulting value was added to the
time determined from the linear calibration. For example, given a time-to-channel slope s
(picoseconds per channel), a timing signal registered in ADC channel c would be converted
to a time t = [s × c] + [U nif orm{0, 1} × s] (picoseconds).

58

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

Figure 3.1: Raw time spectra accumulated during the time-of-flight mass measurement experiment (gray-filled histograms) and linear time-calibration spectra (red-filled histograms) for
the corresponding TAC–ADC time signal. The timing signals (a) through (i) are MCP–Clk,
XFD–Clk, S3U–Clk, S3D–Clk, S3U–S3D, XFU–XFD, S3U–XFU, S3D–XFD, and XFU–
Clk, respectively. The black-dashed vertical lines in (c) indicate the peaks used for time
calibration. Each spectrum is a histogram of counts per ADC channel (See Section 2.3 for
information on data acquisition.). Note that the full-scale range for (a) through (f) and for
(i) is ≈ 50 ns while the full-scale range for (g) and (h) is ≈ 100 ns.
59

Counts/Channel

7400
7200
7000
6800
6600
6400
6200
1000

2000

3000

4000

5000

6000

7000

8000

ADC Channel
Figure 3.2: Spectrum of counts per channel for the S3D–Clk ADC for the full set of experimental data. Red-dashed horizontal lines indicate the number of mean counts per channel
and the expected statistical deviation from the mean. The semi-periodic deviations from the
mean are evidence of nonlinearities in the time-to-channel relationship for this TAC–ADC
combination.

3.1.2

Nonlinear time calibration

A second technique was used in addition to the first technique for the ‘ –Clk’ times to obtain
a time calibration that took into account nonlinearities in the time-to-channel relationship.
Since these timing signals had a stop signal (from the clock) that was random with respect
to the start signal (from a PMT) their spectra should have been uniformly filled over the
full range of possible times. Aside from expected statistical scatter, any deviations from
uniformity indicated a nonlinearity in the time-to-channel relationship for that particular
TAC–ADC combination at the channel at which the deviation existed. Figure 3.2 shows,
for the full set of data, the counts per channel for the S3D–Clk ADC, where it is clear
that deviations from uniformity existed that were beyond the statistically expected scatter
and were possibly systematic in nature. For this type of spectrum, the time at a particular
channel T (c) was equal to the sum of time-widths of the bins up to and including that channel
60

c. The time width of a bin ∆t(c) was equal to the ratio of counts in that channel n(c) divided
by the total number of counts in the histogram N for the full time-span multiplied by the
full time span tspan . The full time span for randomly-filled spectra was equal to the clock
period, τClk = 40 ns±2 ps, since the ‘stop’ signal from the clock could have come any time
after the ‘start’ from the PMT signal. Therefore, the time-difference inferred for channel c
in a particular TAC–ADC combination was,
c

c

∆t(c) =

T (c) =
0

0

n(c)
tspan =
N

c
0

n(c)
τ .
N Clk

(3.1)

As with the case of the linear calibration, some small amount of time, randomly selected
from a uniform distribution bounded by zero picoseconds and the bin time-width, was added
to the event time in order to avoid favoring the exact times that corresponded to the mean
value of each channel.
The comparison between the channel-time derived from the linear time-calibration and
the nonlinear time-calibration is shown in Figure 3.3. It is apparent that the deviations from
linearity were significant, and thus the nonlinear time-calibration was favored over the linear
time calibration for these timing signals. For timing signals other than the ‘ –Clk’ times,
the linear time calibration was the only option and thus the nonlinearities were unavoidable
for these timing signals. Though, as shown in Figure 3.4, an issue was identified with the
nonlinear time-calibrated Down Clock time and thus the linear time-calibration was chosen
for this TOF. In principle higher-order calibrations could have been performed from the
time-calibrator spectra, however it was determined that at least a quartic fit would have
been required to adequately capture the nonlinearities and this would have resulted in a fit
with zero degrees of freedom.

61

(a)

(b)

(c)

(d)

Nonlinear cal.
Linear cal.

Nonlinear cal.

Nonlinear cal.

Nonlinear cal.

Linear cal.

Linear cal.

Linear cal.

Figure 3.3: Residual from the linear time-calibration for the linear (cyan circles) and nonlinear (black crosses) time calibrations for the XFD–Clk (a), S3U–Clk (b), S3D–Clk (c),
and XFU–Clk (d) timing signals. It is apparent that significant nonlinearities existed in the
time-to-channel response of the four TAC–ADC combinations, particularly in the case of the
XFU–Clk timing signal. Since the XFU–Clk spectrum was discarded (See Figure 3.5.), this
was not an issue.

62

3

550

1200
1000

500

800
600

450

400
200

400

×10
1800

550

Clock Down TOF, Lin. Tcal [ns]

Clock Down TOF, Nonlin. Tcal [ns]

3

×10
1400

1600
1400
1200

500

1000
800
600
400
200

450

0

400

400

450

450

0

Direct Down TOF [ns]

Direct Down TOF [ns]

Figure 3.4: Spectra used to identify a timing jitter for the Clock Down TOF (See Equation
3.5) using the nonlinear time calibration. The left panel shows the nonlinear time-calibrated
Clock Down TOF vs the Direct Down TOF and it is apparent that jitters on the order of
several nanoseconds are present. The right panel shows the linear time-calibrated Clock
Down TOF vs the Direct Down TOF and no jitters are present, implying that the jitter in
the left panel is due to the nonlinear time-calibrated Clock Down TOF.

63

3.1.3

Cuts on timing spectra

Crosschecks were performed using physical constraints to assess the fidelity of the timing
signals. The light-travel time across the scintillators was ∼79 ps (See Figure 2.5.), therefore
all events recorded for two PMTs of a given scintillator should have had an absolute timedifference between the two PMTs that was of this order. For example, a valid event striking
the S800 scintillator should have generated S3U–Clk and S3D–Clk time signals and the
quantity {TS3U−Clk − TS3D−Clk } should have been within a time range with a width of
∼ ±79 ps, taking into account different cable delay times. Any events which were outside
of this range must have suffered from a jitter in one of the two timing signals. Similarly,
when gated upon a single isotope in the particle identification plot (See Figure 2.3 and
Section 2.5.2.), the difference between clock times (e.g. {TS3U−Clk −TXFU−Clk }) should have
been within the reasonable time-spread expected for a nucleus’s time-of-flight, δTOF∼100 ps
after rigidity correction [31, Appendix A2]. Note that the method of gating on a single isotope
obviously required first using a timing signal or signals to generate a particle identification
plot so that a gate on an isotope could be applied; but such was the iterative nature of the
time-of-flight mass measurement data analysis. Additionally, it must noted that several clock
periods can have elapsed between registering the timing signals from the S800 scintillator
and the A1900 scintillator (See Section 3.2.). Therefore, there were several time-difference
ranges with widths of ∼100 ps in which good events were expected to lay in a spectrum.
By employing the physical constraints imposed on different combinations of the ‘ –Clk’
time differences, defects for single timing signals could be identified. Figure 3.5 shows how
the anomalous ∼1.25 ns jitter was identified in the XFU–Clk timing signal. The existence
of a jitter in the spectrum produced from the XFU–Clk and XFD–Clk time difference but

64

Figure 3.5: Spectra used to identify a ∼1.25 ns timing jitter in the XFU–Clk timing signal.
The left panel contains the spectrum created by taking the difference between the XFU–Clk
and XFD–Clk timing signals for events of 45 Ar, which in principle should contain only one
peak with a width of ≈180 ps. The left-panel inset shows the existence of a ∼1.25 ns jitter
in one of these two timing signals. The right panel and its inset demonstrate the lack of
any jitter in the spectrum created from the XFD–Clk and S3D–Clk timing signals, where
the existence of two peaks in the main figure in the right panel is due to the fact that two
different integer number of clock pulses could have elapsed during the 45 Ar flight-time. In
order to demonstrate the lack of a jitter in the right panel, the globally determined rigidity
correction (See Section 3.7.) had to be applied to the spectrum to remove the ∼2 ns spread
caused by the rigidity-spread of 45 Ar events. It is apparent that the signal which contained
the jitter must have been the XFU–Clk timing signal. Note that the grass-like background
are random coincidences, which were much more prevalent in the A1900 signals due to the
higher rate at that point in the beam line.

65

lack of jitter in the spectrum produced for the XFD–Clk and S3D–Clk time difference, when
gated on 45 Ar and rigidity corrected, demonstrated that the XFU–Clk had a timing jitter4 .
As such, the XFU–Clk timing signal was not used in the mass measurement data analysis.
The background from random coincidences is discussed here as an aside. The grass-like
background in the left and right panels of Figure 3.5 arose from random coincidences and also
had to be omitted from the data analysis. Since the rate at the A1900 was much higher than
the rate at the S800 due to imperfect beam transmission (See Section 2.2.), there were more
random coincidences for these signals. The random coincidences are thought to have been
caused by the dead-time (signal processing time) of the TACs. If a ‘ –Clk’ TAC processed
an event (i.e. just received a ‘stop’ signal) less than 1 µs (the reset cycle for the Ortec
566 TAC) before the arrival of a new ‘start’, then it misses that ‘start’ signal even though
the particle for that event would likely be correctly registered by other detectors. The next
event arriving within the data acquisition gate would provide the ‘start’ for that particular
‘ –Clk’ TAC while the other timing signal TACs would likely be busy. This could happen
even for two signals coming from the same scintillator due to the different delay-times each
signal was subject to. Using a rough estimate of a ∼50 kHZ rate at the A1900 scintillator
and a full time-of-flight of ∼500 ns, random coincidences would be expected for an A1900
TAC with a frequency of 100×(5×10−7 ns)/(2×10−5 µs)=2.5%. By taking the ratio of total
counts to counts within the peaks of the left panel of Figure 3.5, it is apparent that the
total dead-time for the XFU–Clk and XFD–Clk timing signals combined was ≈4.7%, well
in-line with expectations (since the 2.5% estimate was for a single TAC signal). A similar
background was observed in the analysis presented in [31] (e.g. Figure 3.6 of that work.).
4 Note

that this demonstration required the global rigidity correction (See Section 3.7.),
as explained in Figure 3.5, which demonstrates the iterative nature of the time-of-flight data
analysis procedure and the need for a full data analysis pipeline.
66

Figure 3.6: Spectrum used to identify the several nanosecond timing jitter in the XFU–
XFD timing signal. The inset highlights the unphysical time-difference recorded for ≈8% of
non-background events, where light-travel time in the plastic scintillator limited the possible
time-window for valid events. Note that the double-hump feature in the main peak was
not problematic, since it was counteracted by the opposite double-hump feature present in
the S3U–S3D spectrum; i.e. when the XFU–XFD and S3U–S3D times were employed to
make a scintillator position correction to TOF (See Section 3.2.), their combined time-shift
produced a single narrow-peaked distribution free of any double-hump-like features.

67

An additional defect in a timing signal was found by employing the physical constraint
on the light-travel time in the fast-timing scintillators. As stated previously, the timespread between two PMT timing signals for a single scintillator was restricted to a narrow
time-window for valid events. Examination of the direct time differences for the S800 and
A1900 scintillators, TS3U−S3D and TXFU−XFD , revealed unphysical values for the XFU–
XFD timing signal. It is clear in Figure 3.6 that roughly 8% of events (that did not belong
to the background created by random coincidences) displayed a scintillator-position time
that was well outside the physical bound imposed by the time-difference that the lighttravel time within the scintillator would have produced. The double-hump feature present
in the main XFU–XFD peak, which contained events that were not cut-out from the data
analysis, was not problematic since an equal and opposite feature was present for events
recorded in the S3U–S3D spectrum5 . Thus, when the two times were combined to make a
scintillator-position correction to the event TOF (See Section 3.2.), the resultant correction
was a single-peaked distribution.

3.2

Event TOF construction

Several methods were available to construct the time-of-flight TOF for each recorded event.
This redundancy was by design so as to provide the ability to cross-check TOF results, as
well as to provide multiple TOFs to choose from, with the reasoning being that one TOF
would contain the least systematic issues6 . Aside from TOFs which were discarded due to the
involvement of problematic timing signals, the primary method of evaluating a given eventTOF was by using it throughout the analysis pipeline to produce mass-results. The quality
5 This
6 The

was presumed to be an ion optical effect.
success of this method as employed in [31] motivated its use.
68

of the mass-fit was used to choose which of the available TOFs was best. The components of
the TOF measured for a single event, the main TOF component, the scintillator correction,
and the TOF offset, are described in the following subsections.

3.2.1

Main event-TOF component

The main component of an event’s TOF was the full time-of-flight of a nucleus, absent any
other corrections. The times to choose from were the direct times, i.e. the S3D–XFD and
S3U–XFU timing signals or some combination of these two, and the clock times, which
required taking the difference between two or more clock timing signals, e.g. {TS3D−Clk −
TXFD−Clk }, and entailed a correction for the number of clock pulses that elapsed during an
event’s TOF. The main TOF components that were constructed are,

Direct Down ≡ TDownDir = TS3D−XFD

(3.2)

Direct Up ≡ TUpDir = TS3U−XFU

(3.3)

Direct Combo ≡ TCombDir = 0.5 × (TUpDir + TDownDir )

(3.4)

Clock Down ≡ TDownClk = TS3D−Clk − TXFD−Clk + Nd τ

(3.5)

Clock Up ≡ TUpClk = TS3U−Clk − TXFU−Clk + Nu τ

(3.6)

Clock Combo ≡ TCombClk = 0.5 × (TUpClk + TDownClk ),

(3.7)

where Nd,u are the number of clock pulses that elapsed during the TOF time-span created
by Down,Up clock times and τ is the period of the clock, 40 ns.
The ‘Direct’ TOFs were not favored because they were subject to substantial nonlinearities in the time-to-channel conversion process, as discussed in Section 3.1, where the

69

Clock alignment 1: 12 pulses
Clock alignment 2: 13 pulses
Particle hits
A1900 Scintillator

Particle hits S800
Scintillator

0

500
40 ns

Event Time [ns]

Figure 3.7: Demonstration of the principle of clock pulse correction for ‘Clock’ times. Since
the clock pulses arrived at random times with respect to the scintillator signals, it is clear
that different integer numbers of clock periods could have elapsed during an ion’s time-offlight. In this example, either 12 or 13 clock pulses could have elapsed during an ion’s 500 ns
flight time. As such, two clock times would be possible for the events of such an ion and a
correction would be required to obtain a single TOF for events of that ion.

‘Clock’ TOFs uniformly sampled the range of ADC channels, thereby removing the systematic impact of time-to-channel nonlinearities. This deficiency of Direct TOFs was ultimately
confirmed by employing the Direct Up TOF in the data analysis and comparing the resultant
mass-fit residuals (See Section 3.9.5.) to that obtained for the ultimately preferred TOF,
‘Clock Down’ (See Section 3.2.4.). However, as will be shown, the Direct TOFs were essential
in the process of determining Nd and Nu .
The concept that corrections Nd and Nu , which corrected for the number of clock pulses
that elapsed during an event’s TOF, were required to construct ‘Clock’ times is illustrated
in Figure 3.7. The method to determine Nd and Nu relied on taking the difference in

70

(a)

(b)

(c)

(b)

(d)

(a)

(c)

(e)

(c)
(b)
(a)

Figure 3.8: Spectra employed for ‘Clock Down’ time clock pulse correction. The time difference between a direct TOF, here ‘Direct Up’, and a clock time, here ‘Clock Down’ (panel
e), results in multiple peaks (panels a–c) spaced by the clock period τ =40 ns. Narrow gates
around the peaks were used to remove background and to determine the clock pulse correction that was to be added to a given event. Panel d demonstrates the fact that events of an
ion with a single direct TOF could result in multiple clock TOFs. The black histograms in
panels a–c and e are gated on events of 45 Ar, while the red histograms are for all events.

71

time between a Direct time and the Clock time for which N was required. As is clear
from Figures 2.3 and 3.8d, the majority of valid events recorded fell within an ∼85 ns-wide
time window. As such, when comparing directly measured TOFs to clock TOFs, which were
stopped at random intervals by pulses from the 40 ns-period clock, three different possibilities
existed as to the number of clock pulses that could have elapsed during an event’s TOF.
Though, as is apparent from Figure 3.8d, a single ion generally had only two clock times
available. The clock pulse correction was applied by gating on the peaks in the ‘Clock’ time
minus ‘Direct’ time spectrum, e.g. Figure 3.8e for the Clock Down time, and applying a shift
to the event Clock time corresponding to the peak it populated in the difference spectrum,
e.g. +40 ns for events in the peak in Figure 3.8a, +0 ns for events in the peak in Figure 3.8b,
and -40 ns for events in the peak in Figure 3.8c7 . The shifts were chosen such that the main
peak was unaltered to minimize the overall correction applied to the data. This clock-pulse
correction method had the added benefit of removing the majority of random coincidences,
though gates were made to err on the side of inclusion of good events (i.e. accepting the
inclusion of some background) in order to not impose an artificial cut-off on TOF.

3.2.2

Scintillator position correction

The main component of a single event TOF required a small correction to account for the
systematic TOF shift associated with an ion’s scintillator impact-position (An example is
given in Figure 2.5.). As illustrated for the case of a Down time in Figure 3.9, the lighttravel time in the A1900 and S800 scintillators modified the main TOF component that was
7 [31]

employed an algorithm instead (which is not cited in the thesis but is in the analysis
pipeline code used for that study), however this was found to be problematic as it occasionally
applied the wrong correction to ions whose flight time was within a few picoseconds of the
40 ns clock pulse period τ .

72

XFU
tspan

S3U

thitA
δtgapA
tAd

XFD

thitS
δtgapS

tSd

S3D

t# = true time
= initial light from nucleus
= light travelling down
scintillator
= nucleus’s flight path
= scintillator + 2 pmts

Figure 3.9: Schematic illustrating the need for a scintillator position correction to the main
TOF component. The desired time difference (the true TOF) is the difference in ion-impact
times for the A1900 and S800 scintillators. However, for a Down TOF, the main component recored for TOF would be the difference in signal generation times at the A1900
and S800 scintillators. To obtain the true TOF from the recorded TOF, a correction of
+0.5 × (TXFU−XFD − TS3U−S3D ) had to be applied to the main recorded TOF.
recorded and therefore this modification needed to be undone.
In the framework of this schematic, the time we wanted to record was the difference in
ion-impact times for the A1900 and S800 scintillators, TOFtrue = thit,A − thit,S . However,
the time we recorded as the main TOF component, for the case of a Down TOF, was the time
difference in signal-generation times for the A1900 and S800 down PMTs, TOFrecorded =
tXFD − tS3D . From Figure 3.9, it is evident that the A1900 PMT signal generation time
was equal to the A1900 scintillator ion-impact time plus the light-travel time across the
A1900 scintillator, tXFD = thit,A + δtgap,A . Considering the fact that the difference between
the signal arrival time at the Up and Down A1900 PMTs was equal to the distance in
time of the ion-impact from the center of the A1900 scintillator, it is clear that for the
maximum time difference between the Up and Down timing signals δtgap,A was equal to the
full light travel time across the A1900 scintillator tspan and for the minimum time difference
between the Up and Down timing signals δtgap,A was (1/2)tspan . Therefore, δtgap,A =
73

(1/2)(tspan +(tXFD −tXFU )), where one of the measured timing signals appears, TXFU−XFD =
−(tXFD − tXFU )8 . With the aforementioned relationship for tXFD , the ion-impact time at
the A1900 scintillator thit,A = tXFD −(1/2)(tXFD −tXFU )−(1/2)tspan , which can be rewritten
as thit,A ≡ tXFD + (1/2)TXFU−XFD − (1/2)tspan . The same line of reasoning provides the
ion-impact time at the S800 scintillator, thit,S ≡ tS3D + (1/2)TS3U−S3D − (1/2)tspan . Finally,
from the original relationship for the desired Down time-of-flight, TOFtrue = TS3D−XFD +
1/2(TXFU−XFD −TS3U−S3D ),

where the correction to TOF from the scintillator position was,

δTScint.Corr. = 1/2(TXFU−XFD − TS3U−S3D ).

(3.8)

Application of the scintillator impact-position correction to TOF generally altered the TOF
distributions’ means and widths on the order of tens of picoseconds, as seen for example in
Figure 3.10. This correction removed the undetectable systematic bias due to the dependence
of the main TOF component on the scintillator ion-impact position.

3.2.3

Event-TOF offset

The main component of the event-TOF was constructed with timing signals that were affected by the relative cable delay each timing signal experienced. This means that the main
event-TOF component times weren’t equal to the actual flight times experienced by beam
nuclei. Thus, the choice was made to add a constant offset to all event-TOFs so that they
would have the proper physical magnitude. This was effectively a cosmetic choice, but one
that was performed nonetheless, so it will be described here.
From Equation 1.2 it is clear that
8 Of

mrest
q

=

Bρ
Lpath TOFtrue

1−

L2
path
2,
TOF2
true c

course either ‘ -Clk’ or direct timing signals could be used here.
74

where mrest is

1000

45

Ar

800
600
400
200

433.5

434

434.5

TOF [ns]
Figure 3.10: Rigidity corrected ‘Direct Up’ TOF distribution for 45 Ar events before (redfilled histogram) and after (black unfilled histogram) application of the scintillator position
correction to TOF given in Equation 3.8.

the ion rest mass, q is the ion charge, Lpath is the ion flight-path length, Bρ is the magnetic
rigidity of the ion optics, and c is the speed of light. It follows that the value of TOFtrue
expected for a nucleus of a given mass and charge along a flight path with a given magnetic
rigidity is

mrest
Bρ
=
T OFtrue
q
Lpath
→

mrest
Bρ
=
q
Lpath
→

TOF2true

TOFtrue =

=

1−

L2path
TOF2true c2

TOF2true −
L2path
c2

L2path
c2

+
+

L2path
c2

m2rest L2path
q2 (Bρ)2
m2rest L2path
q2 (Bρ)2

.

(3.9)

Using Lpath =60.57 m, given in LISE++ for the distance between the A1900 extended focal

75

Isotope

Z

A

Expected TOF [ns]

45 Ar

18

45

446

49 Ar

18

49

477

50 Ca

20

50

445

54 Ca

20

54

474

57 V

23

57

442

61 V

23

61

467

64 Fe

26

64

440

69 Fe

26

69

468

Table 3.1: Expected time-of-flight for nuclei calculated from Equation 3.9 assuming Lpath =
60.57 m and Bρ = 3.95024 Tm.

plane and S800 focal plane, and Bρ=3.95024 Tm 9 , from the NSCL Barney Readout [162] for
kg
, 1 u=1.66053892×10−27 kg,
S800 dipole 1 (‘I265DS’), along with the conversion factors 1 T=1 Cs

and |qe | = 1.602176565 × 10−19 C, the expected flight times given in Table 3.1 are generated.
As expected, these TOFs agree with calculations performed in LISE++. Having obtained the
true time of flight TOFtrue , the constant offset is given by toffset =TOFtrue -TOFrecorded . For
the final data analysis the offset which was chosen was that for the ‘Clock Combo’ time
(Equation 3.7), toffset =484 ns. Though this was not the time ultimately used in the mass
fit (See Section 3.9.), this is not a major issue since, as was mentioned previously, the offset
was for purely cosmetic purposes10 .
9 Note

that studies of the rigidity calibration of the NSCL A1900 fragment separator
magnets indicate Bρ was accurate to ±0.5% [161]. Therefore this is the expected uncertainty
of Bρ.
10 A constant offset in no way affects the fits performed involving TOF later in the analysis
pipeline.

76

3.2.4

Selection of the best event-TOF

The event-TOFs to choose from for the analysis were the six TOFs given by Equations
3.2–3.7 with the scintillator position correction (See Equation 3.8.) applied. However, four
of these six TOFs were disqualified on the basis of issues with their timing signals. The
‘Clock Up’ time (Equation 3.6), and therefore the ‘Clock Combo’ time (Equation 3.7), were
discarded due to the timing jitter found in the XFU–Clk timing signal (See Figure 3.5.). A
jitter was also identified in the ‘Direct Down’ TOF (Equation 3.2), as shown in Figure 3.11,
which therefore ruled out this and the ‘Direct Combo’ TOF (Equation 3.4). Therefore the
only remaining TOFs without obvious problems were the ‘Direct Up’ TOF (Equation 3.3)
and the ‘Clock Down’ TOF (Equation 3.5). Due to the issue identified with the nonlinear
time-calibration for the ‘Clock Down’ TOF (See Figure 3.4.), the linear time calibration was
chosen for this TOF. Each of these event-TOFs were employed throughout the analysis and
ultimately the ‘Clock Down’ TOF was chosen over the ‘Direct Up’ TOF as superior based
on the smaller relative systematic scatter present in the final mass fit (See Section 3.9.5).
Note that the masses obtained by using these two event-TOFs in the analysis pipeline agreed
within their uncertainties.
Thus the ‘best’ event-TOF, which was used throughout the remainder of the analysis,
was

TOFevent = toffset − TDownClk + TScint.Corr

(3.10)

= TXFD−Clk − TS3D−Clk + Nd τ + 1/2(TXFU−XFD − TS3U−S3D ) + toffset ,
where toffset = 484 ns, τ =40 ns, Nd was determined as described in Section 3.2.1, and all
timing signals employed the linear time-calibration described in Section 3.1.1. Note that the
77

480

460
440
420

Direct Up TOF [ns]

Direct Up TOF [ns]

480

460
440
420
400

400
440 460 480 500 520

400

Clock Down TOF, Lin. Tcal [ns]

420

440

460

Direct Down TOF [ns]

Figure 3.11: Spectra used to identify a timing jitter for the Direct Down TOF (See Equation
3.2). The left panel shows the Direct Up TOF vs the linear time-calibrated Clock Down
TOF, where it is apparent that no jitters were present. The right panel shows the Direct Up
TOF vs the Direct Down TOF, where jitters are present, implying that the jitter was due
to the Direct Down TOF.

event-TOF was not the same as the TOF used to ultimately obtain the mass of the nucleus
from the mass fit (See Section 3.9.). Those TOFs are described in Section 3.8.

3.3

Energy loss

Energy loss measurements, necessary for particle identification (See Sections 2.5.2 and 3.4.),
were provided by the 16-segments of the S800 focal plane ionization chamber. A redundant
measurement was also provided by the S800 fast-timing scintillator used for the TOF ‘start’,
which was useful for eliminating events that experienced light-particle pile-up in a portion
of the ionization chamber.

78

Figure 3.12: Energy loss in individual ionization chamber segments versus the sum of energy
loss in the full ionization chamber for a subset of the full data-set. Events in the region where
surplus events were present in a single ionization chamber energy loss segment(∆E1 segment
< 150 or > 3100) were excluded from the analysis.

79

3.3.1

Ionization chamber ∆E

The energy loss reading in the 16 segments of the ionization chamber were generally expected
to be in agreement for a single event and therefore, prior to gain-matching, a plot of energy
loss in two segments should produce a straight line for all valid events. Though each segment
obviously reduced the total energy of the ion, the expected energy loss for ions, using beam
energies calculated from the expected TOF (See Equation 3.9.) and specifications of the
ionization chamber (See Section 2.5.2.) in the Bethe formula (See Equation 2.3.), was ≈2–
4 MeV over the whole length of the ionization chamber for isotopes of argon through iron.
Since total ion energies were ≈100 MeV/u (See Section 2.2.2.), the energy loss per segment
amounted to a reduction in energy of ∆E/E ∼ 10−3 , and was thus negligible for these
purposes. As such, the only deviations expected from agreement between energy loss in each
segment would have been due to undesired effects such as an electronics glitch or a nuclear
reaction occurring within an ionization chamber segment. Figure 3.12 shows the comparison
between the energy loss reading for single ionization chamber segments with respect to the
total energy loss within the whole ionization chamber for a subset of the full data-set 11 .
Events with anomalous (too low or too high) energy loss readings for a single ionization
chamber segment were excluded from the analysis.

3.3.2

S800 fast-timing scintillator ∆E

Additionally, agreement between ion energy loss obtained from the ionization chamber and
from the fast-timing scintillator in the S800 focal plane was enforced. A relative energy loss
measurement was extracted from the charge-to-digital converter (QDC) readings from the
11 ∼ 1/45

of the full data-set for ease of plotting, since large files are time-consuming to
process. The cuts discussed were determined using the full data-set.
80

Scintillator ∆ E [arb. units]

2000

1500

1000

500

0

0

10000

20000

30000

40000

50000

Ionization Chamber ∆ E [arb. units]
Figure 3.13: Energy loss in the ionization chamber and in the S800 fast-timing scintillator
for a subset of the full data-set. A subset of 45 Ar events are shown for comparison. Events
outside of the region bounded by the two black lines were excluded.

81

two photomultiplier tubes PMTs attached to the S800 fast-timing scintillator. The amplitude of the signal recorded in the QDC was directly proportional to the amount of light that
reached the PMT. This amount of light N was attenuated with respect to the number of photons generated at the beam-interaction point NPMT = Norig. ×e−λ∆xi , where λ is the scintillator attenuation coefficient and ∆xi is the distance light traveled from the beam-interaction
point to the PMT. However, this effect could be cancelled out by combining the QDC read2 × e−λ(∆x1 +∆x2 )) = N 2 ×Constant,
ings from both PMTs: NPMT,1 × NPMT,2 = Norig
orig

because ∆x1 +∆x2 is just the length of the scintillator. Therefore, the energy loss in the
S800 fast-timing scintillator was given by ∆E ∝ Nphotons,orig. ∝

NPMT,1 × NPMT,2 . The

energy loss from the scintillator and from the ionization chamber are compared in Figure 3.13,
along with the agreement-cut that was applied.

3.4

Particle identification

Nuclei were identified via the method described in Section 2.5.2, where a particle identification (PID) matrix was formed by plotting the energy loss (∆E) and time-of-flight (TOF)
of each event in a two-dimensional histogram (See Figure 2.3 or 3.15.). A gate was made
for each nucleus identified, where care had to be taken to not systematically alter the TOF
distributions of the nuclei.

3.4.1

PID verification

To assign nuclides in the particle identification matrix, several methods are available. One
approach was to compare the observed PID matrix with the one calculated in LISE++ (See
Figure 3.14.). A second method employed was to gate on a nucleus and check its observed

82

Energy Loss [arb. units]

Time-of-flight [ns]
Figure 3.14: Particle identification matrix produced by LISE++ simulations using the timeof-flight mass measurement experimental set-up, where the results for the thick and thin
targets (See Section 2.2.1.) have been combined.

83

Energy Loss [arb. units]

ܼ ൌ ͵Ͳ

‫ܣ‬Ȁܼ ൌ ʹǤͷ

‫ܣ‬Ȁܼ ൌ ʹǤ͸

ܼ ൌ ʹͲ

Figure 3.15: A demonstration of one method of identifying nuclei within the energy loss
time-of-flight, ∆E–TOF particle identification matrix. Nuclei at certain A/Z, such as 2.5
(solid black line) and 2.6 (dashed black line), are expected to be vertically aligned as was
demonstrated in Figure 2.12, so an educated guess can be made as to which vertically aligned
nuclei correspond to which A/Z. Since ∆E ∝ A2 , with a small Z and velocity dependence,
it is expected that lines of constant Z lay along a diagonal shallow slope (diagonal red lines).
An educated guess can be made as to which sloping line is which Z and then it can be
75
checked if the resulting identification matrix is consistent; e.g. here 50
20 Ca and 30 Zn lay along
35
40 45
55
60
65
70
52
A/Z = 2.5 (along with 30
12 Mg, 14 Si, 16 S, 18 Ar, 22 Ti, 24 Cr, 26 Fe, and 28 Ni) while 20 Ca and
78 Zn lay along A/Z = 2.6 (along with 39 P and 65 Mn).
30
15
25

84

isomeric decays against isomeric states of known nuclei. However, few known isomers were
expected in our PID (See Section 2.5.3) so it is not clear a full matrix could have been
constructed. Additionally, this method would have left open the (very remote) possibility of
misidentifying a nucleus due to a coincidental near-equality between the energies of known
and unknown isomeric states. A more reliable method however, was a guess-and-check
technique demonstrated in Figure 3.15 that takes advantage of the uniqueness of the PID
matrix (See Section 2.5.2.) and the large number of isotopes observed in this experiment.
In a ∆E–TOF particle identification matrix, nuclei at certain A/Z, such as 2.5 and 2.6, are
expected to be vertically aligned as was demonstrated in Figure 2.12, so an initial guess can
be made as to which vertically aligned nuclei corresponded to which A/Z. Since ∆E ∝ A2 ,
with a small Z and velocity dependence, it was expected that lines of constant Z lay along a
shallow diagonal slope. A guess could also be made as to which diagonal line was which Z.
Cross checking the two guesses with each other, and checking the consistency of all other Z
75
and A/Z chains leads then to a unique identification. For example, here 50
20 Ca and 30 Zn lay
78
along A/Z = 2.5 while 52
20 Ca and 30 Zn lay along A/Z = 2.6. Once one nuclide is identified,

the remaining nuclei could then be identified by ‘walking’ in the Z and A dimensions in the
PID matrix. Detection of known isomeric decays provides an additional cross check to verify
the final particle identification scheme. In this case a few counts of the 0.43 µs γ-decay from
the 364 keV state of 65 Fe sufficed.

3.4.2

PID gates

Having identified nuclides in the PID matrix, the task remained to create a software gates
that identified which events corresponded to which nuclei. To ensure no systematic bias
was introduced to the TOF distributions, three PID gates (compared in Figure 3.16) were
85

58V
56Ti

55Ti
53Sc

51Ca

50Ca
48K

45Ar

Figure 3.16: Comparison of the three PID gates looked into for the mass measurement data
analysis pipeline. A ‘generous’ hand-gate (red lines), ‘conservative’ hand-gate (black lines)
which closely traced the main outline of PID blobs, and automated gate (brown lines) drawn
on a PID transformed into Z and A/Z were each employed and the results were compared.
The inset shows the gates in more detail for isotopes of argon through vanadium.

86

Figure 3.17: Impact of the choices of three gates on the rigidity-corrected TOF distribution
of 45 Ar (The gates are shown in Figure 3.16.). The ‘generous’ hand-cut (red thick-lined
histogram), ‘conservative’ hand-cut (blue medium-lined histogram), and automated cut (thin
black-lined histogram) had minor deviations in the total number of counts and had no
measurable difference in their mean or standard deviation. The left panel highlights the
slight increase in total counts the generous cut provided, while the log-scale on the right
panel highlights the small high-TOF tail which was removed by the conservative cut.

explored: a ‘generous’ hand-drawn gate, a more restrictive ‘conservative’ hand-drawn gate,
and, to potentially reduce systematic errors, an automated gate drawn on a PID transformed
into Z vs A/Z space. The ‘conservative’ hand-cut was drawn to tightly conform to the
contour made by the main-body of a PID-blob formed by a single nucleus. The ‘generous’
hand-cut was drawn to include all counts which appeared to belong to a nucleus in the PID.
Practically speaking relatively few extra events were included by the ‘generous’ hand-cut
when compared to the ‘conservative’ hand-cut, and as a result no measurable change was
detected in the mean or standard deviation of TOF distributions. The automated gate also
resulted in the same 12 mean and standard deviation for TOF distributions, as seen for 45 Ar
in Figure 3.17, however it was chosen over the two other methods because it had a more
rigorous definition.
12 Note

that ‘same’ here means identical down to the 0.1 ps level, as determined by calculations of the mean. This level of precision obviously cannot be seen in the TOF spectra.

87

Creation of the automated PID gates required a transformation of the PID from ∆E–
TOF space to Z–A/Z space. The physical relationship which guided this conversion are the
proportionalities between ∆E and A2 and between TOF and A/Z described in Section 2.5.2.
√

∆E (See Figure 3.18.), and thus ∆E and
From these relationships, it is clear that Z ∝ T OF

TOF could be used to produce a Z–A/Z PID matrix. While the proportionality between
A/Z and TOF was found to hold true, it was determined that a higher order correction was
√
required for the relationship between Z and ∆E/TOF. The corrections ultimately used to
obtain a Z vs A/Z PID for the ‘Direct Up’ TOF 13 were determined empirically to be

A/Z = (TOF − 75.8)/143.1

(3.11)

√
Z = Zmain − Zcorr = ( ∆E/TOF − 0.007)/0.0159 − Zcorr ,

(3.12)

2
Zcorr = (1.73 − 0.0064 ∗ Zmain + 0.0053 ∗ Zmain
)
2
+(−4.1 × 10−3 + 1. × 10−4 ∗ Zmain − 1.2 × 10−5 ∗ Zmain
) ∗ TOF,

where TOF is expressed in nanoseconds and ∆E is expressed in channels. The resultant
PID is shown in Figure 3.19.
A projection was made onto the Z-dimension of the Z–A/Z PID so that a gate could be
made in the Z-dimension. The upper and lower limits for the Z gates were the ±3σ obtained
from a Gaussian fit 14 to each Z-projection peak, which are shown in Figure 3.20. This gate
was then transformed back into a gate in ∆E–TOF space via Equations 3.11 and 3.12.
The upper and lower limits in TOF for the automated gate were given by the upper and
lower TOF limits for the ‘generous’ hand-cut. The resultant automated gates are shown in
13 A

linear relationship existed between the ‘Direct Up’ and ‘Clock Down’ TOFs, so the
same sets of PID gates could be used after applying a linear transformation.
14 Information on ROOT’s fitting technique is provided in 3.9.2.
88

0.5

∆ E/TOF [arb. units]

480

TOF [ns]

460
440
420

0.4

0.3

0.2

400
2.2

2.3

2.4

2.5

2.6

2.7

2.8

10

15

A/Z

20

25

30

Z

√
Figure 3.18: Demonstration of the proportionality between Z and ∆E/TOF (right panel)
and between A/Z and TOF (left panel). ∆E and TOF were the mean values obtained from
the TUpDir vs ionization chamber ∆E for the nuclei corresponding to each Z and A.

30

104

Z

25

3

10

20

102

15
10

10
2.2

2.3

2.4

2.5
A/Z

2.6

2.7

2.8

1

Figure 3.19: Particle identification matrix where ∆E and TOF have been employed to obtain
Z and A/Z. A projection onto the Z-dimension yielded a one-dimensional histogram which
could be gated on as a Z-cut (See Figure 3.20.).

89

25000
20000
15000
10000
5000
0

10

15

20

25

30

Z
Figure 3.20: Projection onto the Z-dimension of the PID that was transformed into Z vs A/Z
coordinates (See Figure 3.19.). Gaussian fits to each peak are delineated by color. The thickdashed and thin-solid vertical lines of the same color indicate the ±2.5σ and ±3σ locations
for the Gaussian fit, respectively. The symbols of the same color for a given peak indicates
the locations at which the ratio of the number of counts in that peak to the number of counts
in the neighboring peak (as determined by the Gaussian fits) is 10,000:1 (triangle), 1,000:1
(square), and 100:1 (circle) to give a feel for potential contamination from neighboring-Z
nuclei (Note that the majority of this contamination is rid of by the cuts on non-dispersive
microchannel plate position [See Section 3.5.3.] and the local rigidity correction procedure
[See Section 3.6.].).

Figure 3.16. As was mentioned previously, the automated PID gates were chosen for the
remainder of the analysis since they had a rigorous definition, however the results were not
systematically impacted by this choice 15 (See Figure 3.17.).
Examination of the gates shown in Figures 3.16 and 3.20 may lead one to believe that
significant contamination existed within each PID gate which was supposed to contain only
events from a single nucleus. However, as will be shown in Section 3.6, additional rejection
was provided by the expected relationship between magnetic rigidity Bρ and TOF for each
nucleus. Contamination from the relatively high ∆E events from a lower-Z nucleus was pri15 I.e.

the mean and standard deviation of each TOF distribution was not systematically
altered on the 0.1 ps-level.

90

marily removed by applying a cut on the non-dispersive microchannel plate position XMCP ,
as described in Section 3.5.3.
After this constraint, the only worrisome contamination which could enter a PID gate
would be from a different nucleus that had nearly the same central value for its TOF distribution, e.g. the set of nuclei with A/Z=2.5 (See Figure 3.15.). The impact of contaminants with the same A/Z on the analysis depends on the difference in rest mass over
charge mrest /q. For the example of contamination in the 50 Ca PID gate from 45 Ar, after converting from atomic mass excess to nuclear mass (See Equation 3.16.), one obtains
M 45
50
∆M = Q × M
Q ( Ca) − Q ( Ar)

− 41878270.1
= 20 × 46524902.3
20
18

keV = 6.5 MeV. There-

fore to keep the systematic shift induced by contamination below 100 keV, the ratio of con5
N
taminant (45 Ar) counts to good (50 Ca) counts must be kept below Ncontam < 1×10 6eV =
6.5×10 eV
good

1/65. . . a not so rigid constraint. Since a ±3σ cut was applied in the Z-projection histogram and, as seen in Figure 3.20, this limit nearly corresponded to the bounds given
by Ncounts,neighbor /Ncounts,peak < 100, contamination for nuclei with the same A/Z was
determined to be below the acceptable limit 16 . The potential existence of charge state contamination is addressed in the following subsection, since these could have also systematically
shifted TOF distributions.

3.4.3

Charge state contamination

As was described in the previous subsection, contamination in the PID was a major concern
as it could have dramatically altered the results for the TOF distributions of nuclei. One form
of contamination would be nuclides that were not fully ionized upon entering the fast-timing
16 Note

that TOF gates drastically reduced the overlap observed between Z-peaks in Fig-

ure 3.20.

91

scintillator at the A1900 which maintained their charge state throughout the full flight path.
Each time an ion passes through material, it has the chance to exchange electrons with the
material and alter its charge state, where a distribution of final charges is possible depending
on the beam energy, ion nuclear charge, target nuclear charge, and target thickness [163].
Only events which maintained the same charge state from the A1900 to S800 were among
the set of possible contaminants, since a charge change after the A1900 fast timing scintillator
would have placed an ion outside of the S800 rigidity acceptance. Since an ion’s magnetic
rigidity is given by Bρ = p/q, the change in rigidity due to picking-up an electron on a fully
ionized nucleus would be δBρ
Bρ =

(p/Z)−(p/(Z−1))
p/Z

=

(1/Z)−(1/(Z−1))
1/Z

1 . Therefore,
= Z−1

for the highest Z isotopes measured in this experiment, Z = 30, a rigidity change of ≈ 3.5%
would be experienced due to a single electron pick-up. Since this is the smallest possible
rigidity change of all measured ions expected due to electron pick-up in the MCP foil and
the acceptance of the S800 spectrograph is 5% (i.e. ±2.5%) [34], no ions changing charge
state at this position would make it to the S800 focal plane 17 .
For an example of possible contamination, if 66 Fe were not fully ionized for all events, such
that there were some 66 Fe25+ events in the PID, then some contaminants may be present
for 66 Mn25+ since ∆E ∝ A2 and A/Q ∝TOF, as demonstrated in Section 2.5.2. The mass
66 25+ ) − M (66 Mn25+ )
difference for a misidentified event would be ∆M = Q × M
Q ( Fe
Q

= 25 × 61415271.3
− 61429098.0
25
25

keV =14 MeV. To keep the systematic shift caused by

contamination below 100 keV, the ratio of the number of 66 Fe25+ events which could
have been present in the 66 Mn gate with respect to 66 Mn25+ was N66 Fe25+ /N66 Mn25+ =
17 Fully-stripped

isotopes of elements with Z ≥ 39 experiencing electron pick-up within
the MCP foil would fall within the S800 spectrograph acceptance. However, the resulting
rigidity change would cause the ion to miss the fast-timing scintillator at the S800 focal plane
focus.

92

1×105 eV
1.4×107 eV

= 1/1400.

As discussed in [163], to first order the beam energy required to produce a given charge
state after passing through material is given by the Bohr criterion. This criterion states that
in order to remove a given electron from an atom, the projectile must pass through material
at a velocity vprojectile that is comparable to the classical velocity for that electron
Zαc
vprojectile = vclassical =
,
n

(3.13)

where Z is the number of protons in the projectile, c is the speed of light, n is the principal
quantum number of the electron to be removed, and α ≈ 1/137 is the fine structure constant.
The concept behind the criterion is that a perturbed electron in the orbital given by n
wouldn’t be able to catch up to the nucleus as it continues relatively unperturbed through
a medium. Thus, to first order, removal of the most bound electron from 66 Fe (n = 1; a
‘K-shell’ electron) would require a projectile velocity of ∼ 0.2c, which for 66 Fe is a kinetic
energy of ∼ 20 MeV/u. Note that this is a much lower energy than the beam-fragment
energies for this experiment 18 .
A more careful semi-empirical/semi-analytic calculation, such as that presented in Appendix A5 of [115], could be used to estimate charge-state production. However, as detailed by [164], the production of charge states is sufficiently complicated that bulk calculations of charge-state production are best performed by empirically calibrated models.
We explored the possibility of using the CHARGE and GLOBAL [164] models as implemented in
18 LISE++

calculations predicted a fragment energy of ∼90 MeV/u for iron isotopes. For
this beam energy the Bohr criterion relation, vprojectile > vclassical is fulfilled up to 137 Ba.
Results of calculations for the beam energy required to provide fully-stripped fragments up
to a given Z for aluminum and niobium primary beams are shown as an example in Figure 8
of [9].
93

LISE++ [165] to estimate expected charge-state distributions for nuclei observed in this experiment to confirm our simple calculation 19 . For these charge-changing events the predicted
transmission was roughly 10% of the transmission for non-charge-changing ions. However,
these charge-state production models are known to be inaccurate for beam energies below 100 MeV/u [164], especially for thin materials such as the wedge-shaped degrader (See
Section 2.2.2.), A1900 fast-timing scintillator (See Section 2.3.2.), and gold foil (See Section 2.4.2.) used in this experiment [166].
To be sure no charge states were present in significant quantities, the particle identification matrix was investigated for features in between the main ∆E–TOF distributions
for nuclei with high Z, since, as is apparent from the Bohr criterion (Equation 3.13), these
would be the most susceptible to having charge states that were not fully ionized. Figure 3.21
shows a subsection of the PID, designating the locations 66 Fe25+ ions would be expected
to be located if the +1e charge state was created from electron pickup at any point up to
the A1900 fast-timing scintillator or from the gold foil at the S800 target position 20 . Since
66 Fe26+

had ≈ 205, 500 counts, it follows from the lack of signal above background in the

∆E space directly above 64 Mn25+ that charge state production in the gold foil for iron isotopes was below the 0.0005% level. Similarly, considering ≈ 14, 300 66 Fe26+ events had an
energy loss greater than any 66 Mn25+ events (∆E

35, 200 channels) and there is a lack

of signal above background in the ∆E space directly above 66 Mn25+ , it follows that charge
state production from any point upstream of the A1900 fast-timing scintillator was below
the 0.007% level. Considering the previously demonstrated contamination required to affect
19 LISE++

only predicted charge-state changes to occur within the gold foil located opposite the MCP at the level of roughly 1/100 events for iron, 1/1000 for vanadium, and
1/10000 for calcium.
20 As previously explained, events undergoing a charge change at the foil location would
have been outside of the S800 spectrograph acceptance.
94

66Fe26+
66Mn25+

64Mn25+

Figure 3.21: Subsection of the non-rigidity corrected particle identification matrix (with a cut
on the non-dispersive position of the microchannel plate detector described in Section 3.5.3)
showing isotopes of iron and manganese. The black outline traces the majority of events
attributed to fully ionized 66 Fe, the dotted outline is the same shape shifted in TOF to
mimic the effect of changing to a 25+ charge state in the gold foil at the S800 target position
(Note that these events should not have been within the acceptance of the spectrometer,
since a large change in p/q would be suffered due to the charge-change.), and the dashed line
mimics the same thing for electron-pickup prior to or at the A1900 fast-timing scintillator.
It is apparent that there was not a significant amount of 25+ charge state production for
66 Fe.

95

75Cu

75Cu
72Ni

72Ni
69Co

69Co

Figure 3.22: The left panel shows a subsection of the PID, where example PID gates have
been drawn around 69 Co (black-dotted line), 72 Ni (red-dashed line), and 75 Cu (solid-blue
line). The right panel shows the same PID subsection, where events identified as 69 Co (black
dots), 72 Ni (red dots), and 75 Cu (blue dots) have been corrected for their magnetic rigidity,
Bρ. It is apparent that the overlap between nuclei in the non-Bρ corrected PID (gray ovals)
leads to improperly Bρ-corrected events in the Bρ-corrected PID, which may be interpreted
as charge states when they are in fact not. The misidentified events wind up being located
near events of the nucleus to which they properly belong because the TOF–Bρ relationship
varies smoothly with Z and A/Z (See Section 3.7); however, the correction for neighboring
nuclei is different enough that the improperly corrected events are not quite in-line with the
properly identified events. These misidentified events are not problematic, since they are
removed from the TOF distribution, as described in Section 3.6.

the mass of a given nucleus was one contaminant for every few thousand ‘good’ events, we
can safely rule out the possibility of charge state contaminants in the PID.
It should be noted that the analysis of the previous NSCL TOF mass measurement [31]
found what they interpreted as evidence of charge states (See their Section 4.1.). However,
given the analysis presented in this subsection, it seems unlikely such charge states were
actually present. The signature they interpreted as charge states could have resulted from
applying the rigidity correction for a nucleus with a given Z, A/Z to events from the neighboring nuclei with Z ± 1, above or below in terms of ∆E, that had a similar A/Z and leaked
into that PID gate, as demonstrated in Figure 3.22. This issue of misidentification would be
particularly prevalent for regions of the non-rigidity corrected PID that are crowded, such as

96

the high-Z regions. The misidentified events are not problematic in the TOF determination,
since they are shifted in TOF very far from the main TOF distribution, and are thus easily
removed (See Section 3.6.).

3.5

Magnetic rigidity determination

The magnetic rigidity, Bρ, measurement, required to correct the time-of-flight, TOF, for
momentum dependence, was performed by detecting the dispersive position of the ions at
the S800 target position via the technique discussed in Section 2.4. Ions passed through a
gold foil to produce electrons, which were then guided via electric and magnetic fields to a
resistive readout microchannel plate detector, MCP. The electrons were amplified and the
total charge was collected on a resistive backplane and allowed to drift freely to the four
corners of the MCP. This charge information was then used to infer an ion-impact position
on the gold foil, and therefore a relative Bρ.

3.5.1

Position determination principle

The basic concept behind position detection on the MCP was that more charge would be
collected on the MCP back-plane corners closest to the point at which the electron impacted
the MCP face [35, 137, 145]. For example, if the upper half collected more charge than
the lower half and the right side collected more charge than the left side, then the electron
likely impacted the MCP in its upper-right quadrant. Therefore one uses the simple relations
stated in Equations 2.5 and 2.6, which calculate the position from using the four MCP corner
signals UR, UL, LR, and LL. For these relations to hold true, the corner signals as recorded
in the charge-to-digital converter (QDC) had to be pedestal-subtracted and gain-matched

97

so that the same signal strength corresponded to the same amount of charge.
The pedestal to be subtracted from a corner signal was determined in dedicated runs (i.e.
runs which were not used to gather TOF data) by triggering the data acquisition system when
no valid signal was provided to the MCP and recording the highest QDC channel for which
a value was registered. The pedestals subtracted from the 4096 channel QDC spectra were,
130, 158, 183, and 163 for the high-gain UL, UR, LR, and LL signals, respectively, while the
pedestals for the corresponding low-gain signals were 185, 166, 112, and 143, respectively. In
order to be able to mix high-gain and low-gain corner signals to obtain an MCP position, a
linear calibration was obtained to convert high-gain signals to match-up with low-gain signals
(as was done in [35], shown in their Figure 4.20), so that the two would agree in the signal
strength region where both gave valid readings (above the pedestal and below saturation).
Such a conversion looked like: ULhigh,match low = offset + slope × ULhigh . The offset and
slope obtained for each corner signal were: 3.861 and 0.132 for UL, -5.633 and 0.141 for UR,
0.991 and 0.119 for LR, and 0.312 and 0.123 for LL, using the pedestal-subtracted corner
signals. These values were obtained from linear fits 21 to the portions of the graphs in the
upper half of Figure 3.23 where the high-gain signal was not saturated. The lower half of
Figure 3.23 shows the agreement between low-gain and high-gain corner signals obtained
after applying the linear conversion to the high-gain signals. For the positions that mixed
high-gain and low-gain signals, the converted high-gain signals (e.g. ULhigh,match low ) were
chosen over the low-gain signals for corners when the low-gain corner signals were below the
following thresholds: 100, 500, 150, and 100, for UL, UR, LR, and LL, respectively.
The position calibration was carried out by placing a mask with a known hole-pattern 22
21 Information

on ROOT’s fitting technique is provided in 3.9.2.

22 Holes were spaced in the horizontal and vertical dimensions by 5 mm.

The large, medium,
and small holes had diameters of 2.00 mm, 1.52 mm, and 0.79 mm, respectively.
98

4000

4000

4000

4000

102

2000
1

1000

10

3000

2000
1

1000

3000

1

2000

1000

10-1

LL, high-gain [chan.]

10

LR, high-gain [chan.]

3000

UR, high-gain [chan.]

UL, high-gain [chan.]

10

3000
10

2000

1

1000

10-1

10-1

1000 2000 3000 4000

1000 2000 3000 4000

UL, low-gain [chan.]

1000 2000 3000 4000

UR, low-gain [chan.]

1000 2000 3000 4000

LR, low-gain [chan.]

LL, low-gain [chan.]

102
2

10

102

102

10
10
10

10

1

1

1
1
1000

2000

3000

UL [chan.]

4000

1000

2000

3000

4000

1000

UR [chan.]

2000

3000

LR [chan.]

4000

1000

2000

3000

4000

LL [chan.]

Figure 3.23: The upper four panels compare the high-gain and low-gain corner signals for
each event for the upper left (UL), upper right (UR), lower right (LR), and lower left (LL)
corners of the MCP, where the pedestals have been subtracted. It is clear that the high-gain
signal is saturated for all but the weakest of low-gain signals. The lower four panels show
the low-gain and high-gain corner signals recorded for each event on the four MCP corners,
where the high-gain signal (thin black-lined histogram) has been corrected to overlay the
low-gain signal (thick red-dashed histogram). Figure 4.20 of [35] shows similar MCP corner
signal gain-matching results for that work.

99

(a)

(b)

(c)

(d)

Figure 3.24: Panel (a) shows the mask with a distinctive hole pattern (5 mm hole-spacing)
which was placed in between the incoming ion and gold foil in order to only allow electrons
to be created from certain locations for calibration runs. Panel (b) shows the image created
on the MCP by electrons generated from a 232 Th α-source. Panels (c) and (d) show the
image created by the electrons generated by the 82 Se primary beam, where the beam was
tuned to four separate positions to achieve the mask-coverage shown (The mask position is
indicated in Figure 2.7.), where the low-gain corner signals were used for panel (c) and the
combined high-low gain signals were used for panel (d). Since only the relative position was
relevant, the effort was not made to achieve the exact 5 mm hole-spacing of the mask in the
MCP image.
100

upstream of the MCP detector foil [145]. The mask, shown in the upper left panel of
Figure 3.24, was used off-line with a collimated 228 Th α-source at a distance of roughly
20 cm from the foil to perform a simple test of the foil-impact to MCP-position mapping
(upper-right panel of Figure 3.24.), where the mask was located directly in front of the foil on
the source side 23 . During the experiment, the 82 Se primary beam was used to produce the
MCP position maps shown in the lower-left and lower-right panels of Figure 3.24, where the
lower-left panel used only the low-gain MCP corner signals and the lower-right panel used
the combined high and low gain corner signals. To obtain the primary beam mask images,
the primary beam was defocused at the foil position (but still focused at the S800 fast-timing
scintillator) and centered on four separate locations (in x,y-space) sequentially in order to
provide adequate coverage over the MCP-mask, where the positions were chosen based on
the observed MCP impact-positions during the TOF measurement events. Additionally, the
fast-timing scintillator in the A1900 was retracted in order to avoid damage due to the high
rate of the primary beam. A beam attenuator was used to ensure the CRDC rate did not
exceed the maximum allowable rate. The trigger employed for the primary beam mask run
was a valid signal for the S800 ‘Up’ fast-timing scintillator. Ultimately no improvement was
seen with the inclusion of high-gain signals and thus, since the high-to-low gain conversion
process provided more opportunities for mistaken position readings (e.g. from a faulty corner
signal), only the low-gain corner signals were used in the final analysis for the MCP position
determination.
Based on the known hole-positions (5 mm spacing) from the mask, the MCP positions
obtained using the low-gain corner signals in Equations 3.14 and 3.15 were converted to
23 Collimation

ensured that α’s did not directly impact the MCP surface and the distance
of 20 cm was chosen so that the source would not alter the magnetic field between the foil
and MCP or block traveling electrons.
101

physical horizontal and vertical positions by the following scaling factors (obtained by comparing the average hole-spacing to the known hole-spacing): Xactual = 37 ∗ Xlow−gain mm
and Y actual = 47 ∗ Ylow−gain mm 24 . A non-linear mapping between the recorded and the
actual X and Y positions was not performed, as a significant non-linearity was not observed.
Due to the acceptance of the beam line, the only region of the MCP which detected electrons
due to ion-impacts on the gold foil corresponded directly to the hole-mask region imaged in
Panels (c) and (d) of Figure 3.24.

3.5.2

Position resolution

The MCP-position resolution (and therefore rigidity resolution) was determined by assessing
peak widths for single columns and rows of of the hole-mask image (See Figure 3.24.).
The position measurements obtained for single rows and columns of holes are shown for
the α-source measurement in Figure 3.25 and for the measurement with the 82 Se primary
beam in Figure 3.26. As was discussed in Section 2.4.2, the achieved resolution for the
α-source measurement was much better than the achieved resolution for the primary-beam
measurement 25 . The assignment of a position resolution directly from the width of a holemask peak was complicated by the convolution of the peak-resolution with the size of the
corresponding hole in the mask.
MCP position resolutions for the α-source and 82 Se primary beam measurements were
determined by fitting 26 single-peaks obtained from one hole in the mask and removing
24 Note

that here X is the non-dispersive (horizontal) position and Y is the dispersive
(vertical) position at the S800 target position, in contradiction to the usual terminology of
X as the dispersive axis.
25 Recall that the lower kinetic energy of electrons generated by α-particles passing through
the foil led to a cyclotron radius that was 10× smaller than electrons generated by primary
beam particles passing through the gold foil (See Section 2.4.3.).
26 Information on ROOT’s fitting technique is provided in 3.9.2.
102

Figure 3.25: Projection onto the Y (dispersive) dimension for MCP-positions obtained from
the α-source hole-mask measurement (Figure 3.24b) for the column of holes with < X >∼
0 mm (left panel) and onto the X (non-dispersive) dimension for the row of holes with
< Y >∼ −15 mm (right panel). The peak resolution and statistics were affected by the size
of the corresponding hole (seen in Figure 3.24b).

Figure 3.26: Projection onto the Y (dispersive) dimension for MCP-positions obtained from
the 82 Se primary beam hole-mask measurement (Figure 3.24c) for the column of holes with
< X >∼ −8 mm (left panel) and onto the X (non-dispersive) dimension for the row of holes
with < Y >∼ 0 mm (right panel). The peak resolution and statistics were affected by the
size of the corresponding hole (seen in Figure 3.24c).

103

the contribution to the width perceived to be from the hole-width. The (arguably crude)
relationship for position resolution used was

σMCP =

′2
2 ,
σpeak
− σhole

(3.14)

σ

peak
′
′
),
where σpeak
is the peak width corrected for image magnification (σpeak
= magnification
diameter . The concept was that the hole contributed a Gaussian profile
and σhole = hole2∗2.35

to the peak with a standard deviation that was related to the full-width at half maximum by the usual relation (FWHM=2.35σ), where FWHM was chosen to be the hole
radius. The magnification was given by the average deviation from 5 mm of the holespacing in that region. For example, the magnification used for a peak i, as determined
by the relative spacing from the neighboring peaks i − 1 and i + 1, in the Y -projection
was, magnification = 1/2

|<Y>i −<Y>i−1 |
5

+

|<Y>i −<Y>i+1 |
5

, using mean peak positions

in dimensions of millimeters.
Choosing the < Y >∼ −1.6 mm peak obtained from the α-source measurement in the
left-panel of Figure 3.25, σpeak =0.70 mm, σhole =0.32 mm, and magnification=1.01, yielding σMCP (Y ) = 0.6 mm. Choosing instead the < Y >∼ −21.2 mm peak from the same
spectrum, σpeak =0.41 mm, σhole =0.17 mm, and magnification=0.91, yielding σMCP (Y ) =
0.4 mm. The same procedure for the < X >∼ 10.2 mm peak obtained from the α-source
measurement in the right-panel of Figure 3.25 resulted in σpeak =0.53 mm, σhole =0.43 mm,
and magnification=0.83, yielding σMCP (X) = 0.5 mm. For the < X >∼ −7.4 mm peak
from the same spectrum, σpeak =0.36 mm, σhole =0.17 mm, and magnification=0.92, yielding σMCP (X) = 0.4 mm. Therefore, from this rather rough method of position resolution
assessment, we determined the position resolution of the MCP for the α-source test was

104

σMCP (Y ) ≈ 0.5 mm, σMCP (X) ≈ 0.4 mm. Since the cyclotron radius for electrons produced
by the α-source was ∼0.17 mm (See Section 2.4.2.), we concluded that for the α-source
measurement the resolution of the MCP was the primary contribution to the position uncertainty. This is consistent with the comparison to the approximate spatial resolution quoted
by the manufacturer for this series of MCP (Model 3398A): full width at half maximum
FWHM ≈ 0.79 mm [135], i.e. σMCP ≈ 0.34 mm.
Choosing the < Y >∼ 21.8 mm peak obtained from the 82 Se primary beam measurement
in the left-panel of Figure 3.26, σpeak =1.11 mm, σhole =0.17 mm, and magnification=0.80,
yielding σMCP (Y ) =1.4 mm. Choosing the < Y >∼ −1.1 mm peak from the same spectrum,
σpeak =1.02 mm, σhole =0.17 mm, and magnification=0.94 , yielding σMCP (Y ) =1.1 mm.
The same procedure for the < X >∼ −13.1 mm peak obtained from the 82 Se primary beam
measurement in the right-panel of Figure 3.26 resulted in σpeak =0.83 mm, σhole =0.17 mm,
and magnification=0.83, yielding σMCP (X) = 1.0 mm. Choosing the < X >∼ −17.1 mm
peak from the same spectrum, σpeak =0.88 mm, σhole =0.17 mm, and magnification=0.82,
yielding σMCP (Y ) =1.1 mm. Therefore, from this rough method, we determined the position
resolution of the MCP for the 82 Se primary beam measurement was σMCP (Y ) ≈ 1.3 mm,
σMCP (X) ≈ 1.1 mm. Since the cyclotron radius for electrons produced by the 82 Se was
∼1.7 mm (See Section 2.4.3.) and since the α-source measurement was able to achieve a
×2 better resolution, we concluded that for the time-of-flight mass measurement the large
spread in electron trajectories was the primary contribution to the position uncertainty.
Therefore potential room for improvement in a future experiment would be a reduction of
the electron trajectory spread to the order of the position resolution achieved for the α-source
by choosing permanent magnets twice as strong as those used for the current experiment
(See Section 2.4.3.).
105

The ≈ 0.5 mm position resolution for the α-source test and ≈1 mm position resolution for
the 82 Se primary beam was compared to previous studies [31, 35, 137, 145]. [137] achieved
position resolutions of σMCP (X) = 0.35 mm and 27 σMCP (Y ) = 0.7 mm (See their Figure 5.4.) for a similar α-source test for a circular MCP with an active radius rMCP = 2 cm2
and a magnetic field strength of ∼0.05 T. While this field is much smaller than the ∼0.25 T
field used in this experiment (See Figure 2.8.), the ∼ 20% smaller active area (recall the
experiment presented here had an 8 × 10 cm2 MCP) likely helped compensate by incurring
less noise from capacitive pickup. For nearly the same conditions 28 , [35, 145] found an
average position resolution of ∼0.6 mm (See Figure 4.22 of [35].) and [31] found an average
position resolution of ∼ 0.4 mm. When using a primary beam instead, [35, 145] found a
position resolution of ∼ 1.1 mm using electrons produced from a gold foil by 72 MeV/u
70 Se.

As such we concluded our MCP position resolution compared favorably with previous

results 29 .
We used the relationship between time-of-flight (TOF) and the MCP dispersive, Y , position (See Section 3.6.) to estimate the overall contribution of the MCP position resolution
to the final TOF resolution. Since δT OF/δY ≈ 0.04 ns/mm (See Figure 3.32.), the MCP
position resolution contributed ∼ 40 ps to the overall TOF resolution. This roughly agrees
with the expected overall contribution, considering that the intrinsic TOF resolution of the
timing detectors was found to be ≈ 30 ps [22, 31] and the total TOF resolution was found
to be ≈81 ps (See Section 3.8.) 30 .
27 The

worsened resolution in the Y -dimension was a feature of the foil-MCP geometry.
MCP-magnet set-ups employed by [137], [31], [35], and [145] were all provided by
NSCL’s HiRA group.
29 In particular, considering the fact that the manufacturer expected a factor of two better
resolution for the smaller MCP (Model 3394A) [135].
30 Therefore an increase in field strength of the permanent magnets (used in the MCPyoke setup) by a factor of two could have improved the TOF resolution, and therefore the
28 The

106

3.5.3

Identification of scattering on collimator

Though the dispersive (Y ) position of the microchannel plate detector (MCP) was of interest
for the correction of the time-of-flight (TOF) for ions’ magnetic rigidity (See Section 3.6.),
the non-dispersive position wound up being critical for the identification of background
from collimator scattering. Revisiting the ∆E–TOF particle identification (PID) plot (See
Figures 2.3 and 3.27.), it is apparent that individual nuclei in the non-rigidity corrected PID
had high-∆E tails extending from the main group of events. This feature, shown in the
left-panel of Figure 3.27, was referred to as the ‘top-hat’ for convenience. Since this feature
was present even after enforcing agreement between the ionization chamber and S800 fasttiming scintillator energy loss (See Section 3.3.2.), it could not be attributed to an issue with
the energy loss detection such as event pile-up. After an extensive search, a correlation was
found between the non-dispersive position on the microchannel plate detector X and the
relatively high-energy loss ∆E events in the PID, as seen in Figure 3.27.
The reason for the correlation between high-∆E and X was attributed to scattering
of the beam fragments on the collimator that was upstream of the MCP-foil set-up (See
Figure 2.7.), which was intended to protect the MCP during beam tuning. From the photographs shown in Figure 3.28, one can see how some fragment events could have scattered
on the collimator opening. The scattered events (apparently from the right wall opening,
in terms of the upstream side, from Figure 3.27) would have lost energy in the scattering event and would have therefore deposited more energy in the ionization chamber (and
S800 fast-timing scintillator), as is apparent from the Bethe formula (See Equation 2.3.). In
principle this scattering was supposed to be avoided during beam tuning, where a camera
was located inside the S800 target position vacuum chamber. However, no phosphorescent
contribution to the final mass uncertainty from statistics (See Section 3.9.4.), by ∼25%.
107

Figure 3.27: Demonstration of the correlation between high energy-loss (∆E) PID events
and the microchannel plate (MCP) non-dispersive position. The left panel shows a subset
of the PID containing isotopes of calcium, scandium, and titanium, where ‘main’ events are
within the purple box and ‘top-hat’ events are within the red-dashed box. The right panel
shows the location of the ‘main’ (purple dots) and ‘top-hat’ (red dots) events on the MCP,
where it is clear that the relatively high ∆E events corresponded to larger non-dispersive
positions.

108

(a)

(b)
collimator

collimator

BEAM

Figure 3.28: Side (a) and upstream (b) views of the collimator intended to protect the microchannel plate detector (MCP) during beam tuning. We determined that beam fragments
likely scattered on the right-wall (looking from the upstream direction) of the collimator
opening (See Figure 3.27.).

109

Figure 3.29: Particle identification matrix before (red dots) and after (black dots) the cut
on the microchannel plate detector non-dispersive position, X < −11 mm. The reduction
in relatively high-∆E events for individual nuclei is apparent, though it is less clear for the
high-statistics events. The relationship between the high-∆E events and X is shown in more
detail for isotopes of calcium, scandium, and titanium in Figure 3.27.

material was located on the collimator, so scattering could have gone unnoticed, and the
camera overheated prematurely so the beam position wasn’t as fine-tuned as was originally
desired. Investigation of the collimator scattering effect for a large range of nuclei lead to the
determination that a cut was required on the MCP non-dispersive position to only accept
events in the analysis with XMCP < −11 mm. The PID before and after the cut in the
non-dispersive direction is shown in Figure 3.29.

3.6

Local rigidity correction

The relationship between the time-of-flight TOF and magnetic rigidity Bρ, which needs to
be determined to momentum-correct TOF, was first 31 determined individually for each
31 As

the following section will explain, these local rigidity corrections were used to fit for
a global function of rigidity correction as a function of nuclear mass A and charge Z.
110

nucleus. Since Bρ was proportional to the MCP dispersive position Y (See Section 2.4.),
this amounted to determining and correcting for the TOF-YMCP relationship. The details
of this procedure are contained in this section.
Estimates for the TOF–YMCP slope were obtained with LISE++ simulations, however the
simulated TOF values were too coarse to obtain realistic results. Though the TOF precision
limit in the LISE++ simulations was 10 ps, simulations were found to have a large variability
in predictions. For instance, different Monte Carlo runs with the same number of events
would often yield mean TOF values that were discrepant by over 100 ps 32 . Therefore, the
rigidity corrections could not be performed on the basis of simulations.
An empirical approach was used to determine the relationship between time-of-flight TOF
and the dispersive position on the microchannel plate detector YMCP (which was a proxy for
magnetic rigidity Bρ and therefore momentum, since Bρ = p/q). This procedure consisted
of linearly fitting 33 the TOF–YMCP trend nuclide-by-nuclide to determine the correction
required to remove the momentum-dependence from TOF. As will be shown, an iterative
approach was applied to remove contamination from neighboring nuclides in the PID which
was present within gates for each nuclide, and which can be separated once a correction is
applied.
The procedure for determining the slope of the TOF–YMCP relationship consisted of fitting the relationship between TOF and YMCP for all events identified as belonging to a given
nuclide according its PID gate (See Section 3.4.), projecting the Bρ-corrected relationship
onto the TOF dimension, fitting the projected TOF distribution with a normal distribution
32 Additionally,

the more realistic Monte Carlo calculations took exceedingly long for the
large set of nuclei observed, so that obtaining sufficient statistics to obtain LISE++ estimates
for an ion’s TOF and rigidity correction proved time-consuming.
33 Information on ROOT’s fitting technique is provided in 3.9.2.

111

to get the mean and standard deviation σ, making a cut in TOF to only include events within
N σ of the mean rigidity-corrected TOF, and repeating the procedure for the data after the
N σ cut was applied, for N =4, 3, 2.5, and 2. Note that the ‘pivot point’ of the rigidity correction was about YMCP = 0, though we found the mass-fit results to be insensitive to this
choice. This procedure is shown graphically for the example of 68 Fe in Figure 3.30 (Similar
figures for all isotopes of sodium through copper, 11 ≤ Z ≤ 29, which were observed are contained in Appendix B.). The iterative procedure was required due to the contamination from
neighboring nuclei, which is present in the upper left panel of Figure 3.30. The origin of the
contamination is explained by Figure 3.22. Briefly summarized, relatively low-TOF events of
higher-Z nuclei and relatively high-TOF events of lower-Z nuclei could leak into the PID cut
for a nucleus with a given Z, A, but the contaminants are separated in the TOF–YMCP plane
since relatively high-TOF events had relatively low-Bρ (high-YMCP ) and relatively low-TOF
events had relatively high-Bρ (low-YMCP ). The slope generally converged after the first
decontamination step was taken (i.e. the 4σ cut). However, the tighter cut was necessary
for isotopes along A/Z = 2.5, since for this case nuclei with Zcontaminant = Zgate ± 1 would
have a very similar TOF–YMCP distribution as compared to the nucleus of interest.
As a technical point, the TOF vs YMCP data was fit by filling a two-dimensional histogram
for each event assigned to a given nucleus, transforming that histogram into a graph via
ROOT’s ‘TProfile’ class [167], and fitting the resultant graph 34 . TProfile generates a
graph by assigning a data point to each histogram bin along the horizontal axis, where the
vertical position of the data point is the geometric mean of the vertical axis histogram bins
for that horizontal axis bin and the error bar is given by the standard error of the mean
(i.e. the standard deviation divided by the number of events) for that same set of bins. The
34 Information

on ROOT’s fitting technique is provided in 3.9.2.
112

496
-40

502

-20

0

20

YMCP[mm]

40

±3σ cut

499
496
-40

502

-20

0

YMCP[mm]

20

40

±2.5σ cut

499
496
-40

502

-20

0

YMCP[mm]

20

40

±2σ cut

499
496
-40

-20

0

YMCP[mm]

20

40

-40

-20

0

20

YMCP[mm]

40

498.6

Slope: 0.038±0.000 ps
-40

-20

0

20

YMCP[mm]

40

498.6

Slope: 0.038±0.000 ps
-40

-20

0

YMCP[mm]

20

40

498.6

Slope: 0.038±0.000 ps
-40

-20

0

20

YMCP[mm]

40

498.6

Slope: 0.038±0.000 ps
-40

-20

0

498.7

498.9

2000
1000
0

498.4

498.8

499.2

2000
1000
0

498.4

498.7

499.0

20

YMCP[mm]

2000
1000
0

498.4

498.7

TOFBρ-corrected [ns]

498.8

498.4

498.5

TOFBρ-corrected [ns]

498.9

498.3

0

TOFBρ-corrected [ns]

498.9

498.3

1000

TOFBρ-corrected [ns]

499.0

498.2

Counts per 6ps

498.4

Counts per 6ps

±4σ cut

499

Slope: 0.038±0.000 ps

Counts per 6ps

40

498.6

2000

Counts per 6ps

20

TOFBρ-corrected [ns]

0

YMCP[mm]

498.8

40

Counts per 6ps

502

-20

TOFBρ-corrected [ns]

-40

TOFBρ-corrected [ns]

494

TOFBρ-corrected [ns]

no cut

500

TOFBρ-corrected [ns]

TOFdown-clock[ns]
TOFdown-clock[ns]
TOFdown-clock[ns]
TOFdown-clock[ns]
TOFdown-clock[ns]

506

499.0

2000

1000
0

498.5

498.7

498.9

TOFBρ-corrected [ns]

Figure 3.30: The five rows show the successive steps taken in the local rigidity correction
procedure to remove contamination from neighboring nuclei in the PID and determine the
slope of the TOF vs YMCP of a single nucleus, here 68 Fe. The upper left panel shows
a histogram of TOF vs YMCP for events identified as 68 Fe, where the black points are
the resultant graph obtained by applying ROOT’s TProfile class to the histogram. The
black line is a linear fit to the graph. The upper middle panel shows the resultant rigiditycorrected TOF vs YMCP histogram after removing the linear trend found in the upper left
panel, pivoting about YMCP =0. The upper right panel shows the projections onto the TOFdimension of the uncorrected (red histogram) TOF vs YMCP relationship, which was not
easily visible on the same scale for 68 Fe, and rigidity corrected (black histogram) TOF vs
YMCP relationship, where the blue line is a Gaussian fit to the rigidity corrected histogram.
The second row contains the same information as the first, but after applying a cut to only
include data within ±4σ of the mean of the rigidity corrected TOF distribution determined
in the first row. The following rows contain the same information after applying cuts to only
include data within 3, 2.5, and 2σ, respectively, of the mean rigidity corrected TOF, where the
mean and σ were determined by the Gaussian fit in the previous row. Appendix B contains
similar plots for all isotopes of elements with 11 ≤ Z ≤ 29 observed in this experiment.
113

‘Approximate’ option [167] was used to assign as an uncertainty to horizontal bins which
contained only one data point the average uncertainty of all other horizontal bins with more
than one count. This was inconsequential here, since very few cases had a single count per
horizontal bin, however it is an important practical point for someone performing a similar
analysis since omission of ‘Approximate’ will result in that bin being completely discarded.
Instead of generating a graph by applying TProfile to a two-dimensional histogram, a
graph could have been directly created using TOF and YMCP information from each event.
However, directly creating a graph made it hard to visualize how the TOF–YMCP space
was populated and it was found that the results obtained were the same even when only
100 events were present within the two-dimensional histogram. A test was performed where
events were randomly generated along a linear function within a Gaussian perturbation from
linearity. The generated data was fit as a graph and also fit as a TProfile applied to a twodimensional histogram of the data. It was found that both methods agreed even for 100
counts. An interesting result from this simulation was that ROOT underestimated the fit
uncertainty for fewer than ∼5,000 counts. As will be discussed in Section 3.7, nuclei with so
few events were not factored into the final rigidity-correction determination.
The linear relation between TOF and YMCP is apparent in the left panel of Figure 3.30,
though a higher-order dependence is apparent in the middle column of the same figure, which
contains a two-dimensional histogram of the rigidity-corrected TOF vs YMCP relationship for
a single nucleus. Uncorrected higher orders are of concern as they may worsen the resolution
of TOF, and therefore increase the statistical uncertainty of the final masses, and they may
cause TOF distributions to deviate from a Gaussian shape. The latter concern is potentially
more problematic since different isotopes populate different sections of the TOF-Bρ space
and therefore are affected differently by nonlinearities. Detector or electronics induced non114

linearities in the microchannel plate position measurement were ruled-out since nonlinear
behavior of the significance required was not observed in the MCP position determination
(See Section 3.5.). The likely origin of non-linearities are therefore ion optics. Simulations
using 5th -order ion-optics from COSY [168] as implemented in LISE++ demonstrated that a
qualitatively similar feature was induced in the TOF–YMCP relationship from inclusion of
material in the path of the beam (i.e. the wedge degrader, fast-timing scintillators, and gold
foil at the MCP location), however the feature was much less pronounced in the simulations
than in the data.
Fits were performed to the TOF–YMCP trends to assess the presence of quadratic and
cubic terms and in all cases the quadratic and cubic fit-parameters were found to have
large uncertainties and values consistent with zero. Given that statistics were not sufficient
to determine higher order deviations, and because of the potential to introduce additional
systematic errors with weakly constrained analysis parameters, it was decided not to correct
for the higher order dependencies. The final rigidity corrected TOF distributions are very
nearly Gaussian (See Figure 3.30 and Appendix B.) justifying this choice 35 . Isotopes of
elements with Z < 18 were found to span a small range on the lower portion of the MCP,
resulting in a large uncertainty in the locally-determined TOF vs YMCP slope. Ultimately
this resulted in these nuclei being discarded from the analysis (See Section 3.7.).
The slopes of the TOF–YMCP trends for isotopes of argon through iron ranged from
∼ 35.0 − 45.0 ps/mm (See Figure 3.32.) Note that, as will be shown in the following
section, the slope did not simply increase linearly in magnitude as a function of A/Z (See
Figure 3.32.).
35 Additionally,

a mass-fit was performed where the TOFs were corrected to third order in
YMCP and an increase in the statistically-distributed systematic uncertainty was observed
(See Section 3.9.2.).
115

3.7

Global rigidity correction

In order to reduce systematic biases that affect the rigidity correction in a pathological way
and to ensure smooth behavior of the rigidity correction, the locally determined dependencies of TOF on YMCP were fit to determine a global rigidity correction for TOF. The global
correction had the added benefit that it could be employed to low-statistics nuclides, whose
local rigidity correction has a large uncertainty in the TOF vs YMCP slope. This was particularly important since the lower-statistics nuclides are the most exotic, and therefore the
most interesting ones. Motivated by the success of the approach employed in the prior NSCL
time-of-flight mass measurement data analysis [22, 31, 78], the locally determined slopes of
the TOF–YMCP trends (See Section 3.6.) were fit 36 with a polynomial with nuclear mass
A and charge Z as independent variables.
The optimum global fit was found by trying numerous functions that contained various
combinations of polynomials in A, Z, and A/Z, where up to fourth-order in each variable
was explored. Additionally, multiple sets of Z-ranges were explored for the global slope fit,
e.g. omitting low-Z and/or high-Z nuclei. The procedure used to determine the local TOF
vs YMCP slope was found to be relatively unreliable for isotopes of elements with Z < 17
since their distributions in TOF–YMCP space were concentrated at the lower-limit of the
MCP (See Appendix B.). Including these nuclides made a ‘good’ global-slope fit (lacking
in systematic trends in fit residuals) unattainable. Nuclei with Z > 26 were excluded from
the analysis for the same reason. Finally, nuclei with A/Z < 2.44 were excluded from the
global slope determination because their inclusion required a more than 3rd -order A/Z term
in the correction (and still the fits were very poor), for which there was no evidence in mid36 Information

on ROOT’s fitting technique is provided in 3.9.2.

116

Local-Global TOF vs. YMCP slope [ns/mm]

×10-3
0.4

Z=18

0.3

Z=19

0.2

Z=20
Z=21

0.1

Z=22

0

Z=23
-0.1

Z=24

-0.2

Z=25

-0.3

Z=26

-0.4
2.45

2.5

2.55

2.6
2.65
A/Z

2.7

2.75

2.8

Figure 3.31: Residual between the TOF vs YMCP slope determined ‘locally’ (i.e. by-isotope)
and the ‘global’ value determined from the fit with Equation 3.15 to all local slopes with
18 ≤ Z ≤ 26 and A/Z > 2.44. It is apparent that the majority of isotopes were fit within
1% (Note that the average slope was ≈ 0.40 ps/mm, as seen in Figure 3.32.) and that there
was a significant statistically-distributed systematic scatter for each element as a function
of A/Z.

to-high A/Z (∼ 2.5 − 2.7) nuclei. Since the mid-A/Z nuclei (e.g. ∼ 2.5) had the highest
statistics and the high-A/Z (e.g. ∼ 2.7) nuclei were the object of the mass measurement, it
was determined that exclusion of the A/Z < 2.44 nuclei provides more reliable neutron-rich
masses.
The optimum global rigidity-correction function had the same terms, albeit different

117

coefficients, as in [31] (See their Equation 4.6.) for the fit to linear local rigidity corrections 37 ,

A
dTOF
= a0 + a1 + a2
dYMCP global
Z

A 2
+ a3 Z + a4 Z 2 + a5 A,
Z

(3.15)

where a0 = 0.288±0.003, a1 = −0.130±0.002, a2 = 0.0202±0.0005, a3 = −0.00911±0.00005,
a4 = 0.00189 ± 0.00001, a5 = 10.0 ± 1.5. As compared to the values obtained by [31] (See
their Table 4.1.), the relative magnitude of parameter a5 , which captured the A-dependency,
determined here was three orders of magnitude larger. It is suspected that this is related
to the extra energy loss of ions (as compared to the measurement described in [31]) in the
A1900 wedge degrader, since ∆E ∝ A2 .
The residual of the global fit to locally determined rigidity correction slopes is shown
in Figure 3.31. Though all slopes were fit to within

1% and no systematic trends were

observed as a function of A/Z or Z, a significant statistically-distributed scatter was present.
A direct comparison between the local rigidity-correction slopes and the global fit function
is given, along with a comparison to fits of the local correction along an isotopic chain,
for isotopes of argon through iron in Figure 3.32. It is apparent that the by-element and
global rigidity-correction slope fits were generally in agreement, and each generally fit the
locally-determined slopes well. All three rigidity-corrections were employed through the
remainder of the analysis pipeline to decide which correction was best and it was found
that the global rigidity correction minimized the final systematic uncertainty in the mass-fit
(See Section 3.9.5.). Unless noted otherwise, the following data presented used the global
rigidity-corrected TOFs.
37 Note

that [31] ultimately chose a quadratic local rigidity correction, since this produced
a slightly smaller scatter in their mass-fit residual (See their Section 4.2.1.). As discussed in
Section 3.6, no evidence was found in this analysis for quadratic behavior of TOF vs YMCP .
118

0.04

46
A

48

50

52

44

46

48 50
A

52

54

0.04

0.045
Slope [ns/mm]

0.045

48

50

52 54
A

56

58

0.035

0.04

3 σ Local Slope, Fit
3 σ Local Slope, NOT Fit
2.5σ Local Slope, Fit
2.5σ Local Slope, NOT Fit
2 σ Local Slope, Fit
2 σ Local Slope, NOT Fit
2 σ Fit by Element w/Error
68.2% conf. interval
Global-Fit

56

0.04

3 σ Local Slope, Fit
3 σ Local Slope, NOT Fit
2.5σ Local Slope, Fit
2.5σ Local Slope, NOT Fit
2 σ Local Slope, Fit
2 σ Local Slope, NOT Fit
2 σ Fit by Element w/Error
68.2% conf. interval
Global-Fit

54 56 58 60 62 64 66
A

0.045

3 σ Local Slope, Fit
3 σ Local Slope, NOT Fit
2.5σ Local Slope, Fit
2.5σ Local Slope, NOT Fit
2 σ Local Slope, Fit
2 σ Local Slope, NOT Fit
2 σ Fit by Element w/Error
68.2% conf. interval
Global-Fit

0.04

0.035

0.035
56 58 60 62 64 66 68
A

54

Z:26

0.045

Slope [ns/mm]

0.04

50 52
A

0.035

Z:25
3 σ Local Slope, Fit
3 σ Local Slope, NOT Fit
2.5σ Local Slope, Fit
2.5σ Local Slope, NOT Fit
2 σ Local Slope, Fit
2 σ Local Slope, NOT Fit
2 σ Fit by Element w/Error
68.2% conf. interval
Global-Fit

0.045

50 52 54 56 58 60 62
A

60

Z:24
0.045

0.04

3 σ Local Slope, Fit
3 σ Local Slope, NOT Fit
2.5σ Local Slope, Fit
2.5σ Local Slope, NOT Fit
2 σ Local Slope, Fit
2 σ Local Slope, NOT Fit
2 σ Fit by Element w/Error
68.2% conf. interval
Global-Fit

0.035

0.035

48

Z:23

Z:22
3 σ Local Slope, Fit
3 σ Local Slope, NOT Fit
2.5σ Local Slope, Fit
2.5σ Local Slope, NOT Fit
2 σ Local Slope, Fit
2 σ Local Slope, NOT Fit
2 σ Fit by Element w/Error
68.2% conf. interval
Global-Fit

46

Slope [ns/mm]

44

Slope [ns/mm]

42

0.04

0.035

0.035

Z:21
Slope [ns/mm]

0.045

3 σ Local Slope, Fit
3 σ Local Slope, NOT Fit
2.5σ Local Slope, Fit
2.5σ Local Slope, NOT Fit
2 σ Local Slope, Fit
2 σ Local Slope, NOT Fit
2 σ Fit by Element w/Error
68.2% conf. interval
Global-Fit

0.045
Slope [ns/mm]

0.04

0.035
40

3 σ Local Slope, Fit
3 σ Local Slope, NOT Fit
2.5σ Local Slope, Fit
2.5σ Local Slope, NOT Fit
2 σ Local Slope, Fit
2 σ Local Slope, NOT Fit
2 σ Fit by Element w/Error
68.2% conf. interval
Global-Fit

0.05
Slope [ns/mm]

Slope [ns/mm]

3 σ Local Slope, Fit
3 σ Local Slope, NOT Fit
2.5σ Local Slope, Fit
2.5σ Local Slope, NOT Fit
2 σ Local Slope, Fit
2 σ Local Slope, NOT Fit
2 σ Fit by Element w/Error
68.2% conf. interval
Global-Fit

0.045

Slope [ns/mm]

Z:20

Z:19

Z:18

58 60 62 64 66 68 70
A

60 62 64 66 68 70 72 74
A

Figure 3.32: TOF vs YMCP slope as a function of mass number A for observed isotopes
of argon, potassium, calcium, scandium, titanium, vanadium, chromium, manganese, and
iron (18 ≤ Z ≤ 26), respectively, as determined by ‘local’ by-nucleus fits (data points) (See
Section 3.6.) and fits to the locally-determined slopes that employed the ±2σ cut-off (See
Figure 3.30.), where the black data points were included in the fit and the blue points were
not. It is apparent that in general the locally determined slopes for the 3σ (circles), 2.5σ
(squares), and 2σ (triangles) cut-offs were in agreement. The by-element fit along a single
isotopic chain as a cubic function of A is shown by the red lines, where the upper and lower
lines indicate the extremes obtained for upper and lower limits of the fit-parameters, and
the orange band indicates the ±1σ confidence interval. The black line shows the trend of
the rigidity-correction slope along an isotopic chain as determined by the global fit (See
Equation 3.15.) to all locally-determined slopes of nuclei with A/Z > 2.44 and 18 ≤ Z ≤ 26.

119

3.8

Final TOF determination

The final analysis step before fitting the mass–TOF surface to obtain a calibrated relationship
(See Section 3.9.) was to determine the final TOFs and their uncertainties of observed nuclei.
This consisted of identifying all events belonging to given nucleus, correcting TOF for its
magnetic rigidity dependence 38 , and determining the mean of the final TOF distribution
and its uncertainty.
The means of rigidity-corrected TOF distributions and their uncertainties were obtained
via a Gaussian fit 39 . The Gaussian was chosen since this provided a good fit to the data,
as can be seen in Appendix C, however, as seen in the same appendix, a slight Z-dependent
skewness was present in observed TOF distributions. This is also apparent for the example
TOF spectra shown in Figure 3.33. The Gaussian fit to obtain the mean was performed
with limiting ranges of ±4σ, ±3σ, and ±2σ of the mean TOF, where the mean and σ were
determined for each case in an iterative procedure, i.e. setting σ to determine the mean,
then determining σ, and repeating until convergence. The mean TOFs of the ±4σ and ±3σ
cases were found to agree within less than 1 ps, while the mean TOF from the ±2σ case was
found to deviate from the other two by several picoseconds without any clear trend in A, Z,
or A/Z. It was determined that the ±2σ case cut too deeply into the TOF distributions and
therefore it was discarded. Ultimately the ±4σ case was chosen for the mass-fit, though it
did not give significantly different results than the ±3σ case.
The final TOF values obtained for observed nuclei with 18 ≤ Z ≤ 26 and A/Z > 2.44
using the local, by-element, and global rigidity-corrections (See Section 3.7.) are compared
38 The

pivot-point for the rigidity correction was YMCP =0 mm, however other positions
were explored and were, as expected, found to only result in a global systematic shift of all
TOFs. Since all TOFs were shifted equally, there was no impact to the final mass fit.
39 Information on ROOT’s fitting technique is provided in 3.9.2.
120

Figure 3.33: Examples of global rigidity-corrected (See Section 3.7.) TOF distributions for
some m
q (TOF)-calibration nuclides (unfilled histograms) and nuclides whose mass was evaluated (filled histograms). The Gaussian fits which are shown demonstrate the Z-dependent
skewness present in the TOF distributions. See Appendix C for final TOF distributions of
all nuclides involved in the mass-fit and mass evaluation.

121

60
40
20
0
-20
-40
2.5

2.6

A/Z

2.7

80
60
40
20
0
-20
-40
18

20

22

Z

24

26

TOFDown Clock - TOFUp Direct [ps]

80

TOFDown Clock - TOFUp Direct [ps]

TOFDown Clock - TOFUp Direct [ps]

Figure 3.34: ‘Down Clock’ TOF mean values (See Section 3.2.) obtained with different
rigidity corrections shown as a difference to the values obtained with the global rigidity
correction as functions of the atomic mass number to nuclear charge ratio A/Z. Shown are
TOFs obtained via the local (black circles), by-element (cyan squares), and global (red lines)
rigidity corrections (See Sections 3.6 and 3.7.), using the ±3σ range for observed nuclei with
18 ≤ Z ≤ 26 and A/Z > 2.44. The average deviation between global and local TOFs and
between global and by-element TOFs was 2.3 ps and 0.3 ps, respectively.

80
60
40
20
0
-20
-40
45 50 55 60 65 70

A

Figure 3.35: Difference in TOF between the global rigidity-corrected ‘Clock Down’ and
‘Direct Up’ TOFs, with their uncertainties added in quadrature, after correcting for the
trivial offset between the two (See Section 3.2.), using the ±3σ range for observed nuclei with
18 ≤ Z ≤ 26 and A/Z > 2.44. The left, middle, and right panels show the TOF difference
as a function of the atomic mass number to nuclear charge ratio A/Z, nuclear charge Z, and
nuclear mass A, respectively. Note that the TOF differences do not correspond to final mass
differences, since each mass-TOF surface was fit separately, ultimately yielding similar mass
results (See Section 3.9.5.), albeit with different uncertainties.

122

in Figure 3.34. It is apparent that the TOF values did not systematically differ. Each TOF
set was ultimately employed in the mass fit and it was found that, while the resultant mass
values agreed, the global rigidity-corrected TOF resulted in a significantly smaller scatter
in the fit residuals (See Section 3.9.5.). A smaller scatter in the mass-fit residuals was also
achieved for the global rigidity corrected ‘Clock Down’ TOF as compared to the ‘Direct
Up’ TOF (See Section 3.2.), which are compared in Figure 3.35. Note that even though
a systematic deviation is apparent in these TOFs, this did not correspond to a systematic
deviation in the final mass results, since the difference in the TOF surface (i.e. with respect
to A/Z, Z, and A) was generally captured by the mass-fit (See Section 3.9.). Recall that the
large scatter of the ‘Direct Up’ time with respect to the ‘Clock Down’ time was expected,
since it was expected that the former was subject to timing nonlinearities that the latter
mitigated (See Section 3.2.).
The final statistical uncertainties for the mean TOFs were generally below 1 picosecond,
as seen in Figure 3.37, and the average resolution of TOF distributions for observed nuclei
was 81 picoseconds, as seen in Figure 3.38. The statistics for calibration and evaluated
nuclides are shown in Figure 3.36, where the minimum number of counts required to obtain
a reasonable statistical TOF uncertainty was ≈ 500. The TOF resolution roughly agreed
with the sum of expected contributions, where a ∼30 ps resolution had been previously
attributed to the timing detection set-up [22, 31] and a ∼40 ps resolution was found to
be due to the finite position resolution of the MCP used for the rigidity correction (See
Section 3.5.2.). The remaining contribution to the TOF resolution likely can be attributed
to the deviation from linearity in the TOF–YMCP relationship (See Section 3.6.). The overall
TOF resolution achieved was comparable to that obtained from the previous NSCL time-offlight mass measurement data analysis (See Figure 4.31 of [31].).
123

26
25
105

Z

24
23
22
21
20
19
18

104

103
26

28

30

32

34

36

38

40

42

44

N
Figure 3.36: Number of observed events (indicated by the color) for observed nuclei with
18 ≤ Z ≤ 26 and A/Z > 2.44, within the ±4σ TOF distribution range that was used for
the global rigidity correction (See Section 3.7.) of the Down Clock time (See Section 3.2.).
Note that only nuclides with ≈ 500 events or more are plotted.

124

Z

26
25
24
23
22
21
20
19
18

1

10-1
26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44

N
Figure 3.37: Mean TOF uncertainty in picoseconds (indicated by the color) of observed nuclei
with 18 ≤ Z ≤ 26 and A/Z > 2.44, using the global rigidity correction (See Section 3.7.)
for the Down Clock time (See Section 3.2.) with the ±4σ TOF distribution range.

Z

26
25

100
95

24
23
22

90
85
80

21
20
19
18

75
70
26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44

65

N
Figure 3.38: Standard deviation of final TOF distributions in picoseconds (indicated by the
color) of observed nuclei with 18 ≤ Z ≤ 26 and A/Z > 2.44, using the global rigidity
correction (See Section 3.7.) for the Down Clock time (See Section 3.2.) with the ±4σ TOF
distribution range. The average standard deviation of the TOF distributions was 80.8 ps.

125

3.9

Mass fit

The mass fit was the final step of the data analysis pipeline where, as described in Section 1.4.1, nuclides with well known masses and well determined times of flight were used to
obtain a calibrated relationship between time of flight and mass. This consisted of choosing
a set of reference nuclides, finding the optimum fit function to capture the mass–TOF relationship, assessing the mass-fit uncertainty, and, in keeping with the iterative nature of TOF
mass measurement data analysis (See the introduction to this chapter.), using the mass-fit
results as a diagnostic tool to choose between otherwise equally favored options that were
available earlier in the data analysis pipeline.

3.9.1

Reference nucleus selection

Reference nuclides needed to have a well-determined mass and a lack of long-lived isomeric
states (which could have been confused for the ground state). Defining ‘well-determined’
for a mass amounted to picking a cut-off in mass uncertainty above which a nucleus would
not qualify based on the experimentally-based atomic mass excess uncertainties listed in the
2012 Atomic Mass Evaluation [1]. In addition the more recent measurements of 53,54 Ca [96]
were also included. Defining ‘long-lived’ for an isomeric state amounted to picking an upperlimit on the allowable half-life, which necessarily had to be much shorter than the overall
flight-time, where information on isomeric states was obtained from the National Nuclear
Data Center [12]. We chose 100 ns for this upper limit. Practicality imposed the constraint
that the reference mass criterion could not be so rigid as to not allow for adequate mapping
of the mass-TOF phase-space 40 .
40 A compromise between this constraint and the ultimately chosen reference mass criterion

resulted in the omission of titanium and vanadium nuclei from the set of results to be
126

The masses 41 of the reference nuclei within the final mass-fit range, 18 ≤ Z ≤ 26
and A/Z > 2.44, are from mass measurements performed with the ISOLTRAP Penning
trap [43, 169], the LEBIT Penning trap [170, 171], the TITAN Penning trap [172, 173], the
ISOLDE multi-reflection time-of-flight spectrometer [96], the Experimental Storage Ring
(ESR) [174, 175], TOF analysis with the SPEG spectrometer [176], TOF analysis with the
TOFI spectrometer [100, 177, 178], and the previous TOF analysis at the NSCL with the
S800 spectrograph [78] (See Section 1.3 for a brief overview of the different measurement
techniques.). Potential reference masses resulting from the TOFI mass measurements were
excluded because of systematic issues identified by several follow-up studies. More recent
Penning trap mass measurements found a systematic deviation from the TOFI masses for
72−74 Cu

[179] and for neutron-rich isotopes of cobalt and iron [171], and systematic devi-

ations were also observed between TOFI and the ESR for neutron-rich isotopes of calcium
through nickel [180, footnote on their pg. 58]. Furthermore, as repeated in [42], [181] demonstrated that TOFI produced inconsistent mass results for the same nuclei when employing
different spectrometer settings. Presumably these revelations and others have resulted in an
arbitrary increase of the TOFI mass uncertainties by a factor of ×1.5 as listed by the 2012
Atomic Mass Evaluation [1] and outright omission of TOFI masses from the 2012 Atomic
Mass Compilation [182]. Potential reference masses resulting from ESR mass measurements
were excluded because they were not published in refereed journals [174, 175]. Finally, potential reference masses resulting from the previous NSCL TOF mass measurement analysis [78]
were excluded to avoid amplification of any unidentified systematic biases in the NSCL TOF
published since the distance in Z from reference nuclei was too great (See Section 3.9.4.).
41 Note that for all nuclei but 53 Ca and 54 Ca the experimentally-based evaluated atomic
mass excesses from the 2012 Atomic Mass Evaluation [1] were chosen, where [96] was used
for 53,54 Ca because publication of these values followed the release of the evaluation.

127

set-up that would likely occur via this sort of bootstrapping.
The resulting set of reference nuclides had literature mass uncertainties of δMlit. ≤
50 keV 42 . The inclusion of additional nuclides whose Atomic Mass Evaluation mass excesses were primarily derived from storage ring and TOF mass measurements was explored
by relaxing the reference mass criterion to δMlit. ≤ 400 keV 43 . The resultant scatter in the
mass-fit was substantially increased (See Section 3.9.5.).
As in the previous NSCL TOF mass measurement analysis [31], nuclei with isomeric
states that had longer than 100 ns half-lives and an excitation energy larger than 200 keV
were excluded as reference nuclei, where isomeric state half-lives were obtained from [12].
Given the constraints above, the 20 remaining nuclei available as reference masses were
44−47 Ar, 47−51 K, 49−54 Ca, 63,65,66 Mn,

and 64,66 Fe. The location of these reference nuclei

with respect to the other nuclei observed with 18 ≤ Z ≤ 26 and A/Z > 2.44 in terms of
A/Z and Z is shown in Figure 3.39.
The published atomic mass excesses of the reference nuclei had to be converted into nuclear masses. For each nucleus with atomic mass number A and atomic number Z, this process consisted of converting atomic mass excess MEatom. (Z, A) to atomic mass Matom. (Z, A)
and then converting the atomic mass to nuclear mass Mnucl. (Z, A) by subtracting the mass
of the electrons me and adding back the energy given by their binding BEe (Z). This resulted
42 In

practice this meant reference nuclides had masses which were primarily derived from
Penning trap or multi-reflection time-of-flight measurements. Therefore nuclides whose mass
excesses, as listed in the 2012 Atomic Mass Evaluation, were primarily derived from TOF
or storage ring measurements were omitted as calibration nuclides.
43 This less restrictive reference mass uncertainty cutoff was chosen in the previous NSCL
TOF mass measurement analysis [31], however this was motivated by the scarcity of observed
nuclei with higher-precision masses.

128

27
26
25
24

Z

23
22
21
20
19
18
17
2.4

2.45

2.5

2.55

2.6

2.65

2.7

2.75

A/Z
Figure 3.39: Map of nuclei observed in the TOF mass measurement analysis (with sufficient
statistics to obtain a TOF value) in terms of atomic mass number to nuclear charge ratio
A/Z and nuclear charge Z. Solid black circles indicate reference nuclei, open blue circles
indicate nuclei with masses known in the literature, but not to sufficient precision to qualify
as reference nuclei, and red stars indicate nuclei with unknown mass as of the completion of
the data analysis. (Compare to Figure 4.34 of [31].)

129

in the following conversion,

Mnucl. (Z, A) = Matom. (Z, A) − Z ∗ me + BEe (Z),

(3.16)

where me = 510.99 keV and Matom. = MEatom. + A ∗ u, with u = 931494.01 keV being
the atomic mass unit. The total electron binding energy of each element was obtained by
summing over the individual binding energies of each electron. The individual electron binding energies were taken from Table II of [144], which provides ‘binding energies (ionization
energies) of electrons in free atoms in the ground state’ to a ∼2 eV precision 44 . The electron
binding energy correction was between ≈ 9 − 21 keV for 18 ≤ Z ≤ 26.
The resultant mass–TOF surface to fit to obtain the calibrated m
q (TOF) relationship
(See Section 1.4.) is shown in Figure 3.40, where the linear dependence of TOF on mass
over charge m/q has been removed to better highlight the shape of the mass–TOF surface.

3.9.2

Mass fit procedure

The mass fit to obtain the calibrated relationship between mass and TOF was carried out
over the mass–TOF surface created by the reference nuclei (See Figure 3.40.) after making
two minor modifications to the data points. During the fitting routine both statistical and
systematic uncertainties were included.
First a relativistic correction was made to the TOF of nuclei to take into account the impact of time-dilation 45 . Referring to the measured TOF as TOFmeas. , the TOF to include in
44 [183]

was useful for converting between the older notation used in [144] to the more popular spectrscopic notation, i.e ‘K’=1s1/2, ‘L1’=2s1/2, ‘L2’=2p1/2, ‘L3’=2p3/2, ‘M 1’=3s1/2,
‘M 2’=3p1/2, ‘M 3’=3p3/2, ‘M 4’=3d3/2, ‘M 5’=3d5/2, ‘N 1’=4s1/2, and ‘N 2’=4p1/2 .
45 Equivalently the relativistic correction could have been applied to the reference masses,
but then this would have required undoing the correction for masses obtained by evaluating
130

Figure 3.40: Mass–TOF surface of reference nuclei where the linear dependence of mass over
charge m/q on TOF has been removed. Solid white points mark the nuclear charge Z and
TOF of reference nuclei while the color of the surface at that location indicates the linear
fit residual in MeV. (Note that the flat region occurs outside of the region bounded by data
points as a feature of ROOT’s ‘SURF’ drawing option.)

131

the mass-fit was TOFfit =TOFmeas. ×

1 − L2path/ TOF2meas. c2

, where Lpath =60.57 m

is the flight-path length and c = 299792458 m/s is the speed of light [147]. At first it may be
worrisome that the correction relies on knowing the absolute path-length of the ion flightpath, however we found that the relativistic correction had no significant impact on the
mass-fit. The reason for the lack of sensitivity to the relativistic correction is that it is
absorbed well in the mass fit function. Nonetheless, the correction was included.
The second modification to the data was to fit an effective nuclear charge, z ≡ Z − Z ,
and effective time-of-flight, τ ≡ TOF − TOF , where Z and TOF were the average
nuclear charge Z and time-of-flight TOF of the set of reference nuclei. This modification was
performed to reduce the multicollinearity of fit-parameters [22]. This amounted to reducing
the changes in fit-parameters caused by minor shifts in data-point positions, thereby avoiding
erratic behavior, which was particularly important for the mass-fit extrapolation uncertainty
determined via a Monte Carlo technique (See Section 3.9.4.).
The two-dimensional graph of mass over charge m/q, effective time-of-flight τ , and effective nuclear charge z, where τ and z were the independent variables, was fit via ROOT’s
standard ‘Fit’ command [184] which utilizes the MINUIT function minimization package via
ROOT’s TMinuit class [185]. The ‘M’ option [184] was employed for the mass-fit, activating
the ‘IMPROVE’ command of TMinuit, which searches for other minima after a minimum
is found in an attempt to find the global minimum as opposed to a local minimum [185].
The initial uncertainty ascribed to the data points was the literature mass uncertainty added
in quadrature to the statistical uncertainty, where the latter used standard propagation of
uncertainty to translate uncertainty in TOF into uncertainty in mass over charge 46 . This stathe final mass-fit function with measured TOFs of non-reference nuclei.
46 Anyhow, even if the choice was made to leave the TOF uncertainty in the TOFdimension, ROOT converts uncertainties in the independent variable dimensions of data points
132

m (TOF)
∂
tistical uncertainty depended on the fit function itself, δMstat. = (δTOF)× ∂TOF
q

where m
q (TOF) is the mass–TOF calibration function and δTOF is the one standard deviation uncertainty of the mean TOF for a nucleus (data point). Therefore the final statistical
uncertainty assigned to each data point was determined in an iterative procedure where the
data was fit to obtain a mass–TOF calibration function, statistical uncertainties were calculated for each of the data-points (corresponding to reference nuclei), and the process was
repeated until convergence.
Upon completion of the mass-fit, including literature and statistical uncertainties, the
reduced χ2 of the fit was typically much larger than one. This indicated that the uncertainty
of the reference nuclide data-points was underestimated and that some additional heretofore
unaccounted for uncertainty was present. The approach outlined in [22] was followed, where
the missing uncertainty was treated as a statistically-distributed systematic uncertainty, i.e.
one that accounted for a uniform scatter in the mass-fit residual as a function of m/q 47 . Such
an effect could have been created by many uncontrolled factors in the measurement, such as
time-dependent magnetic fields of the dipole magnets along the beam line, time-dependent
variations in the response of the timing electronics due to variations in temperature, or
unidentified biases present in the data analysis pipeline. To include this additional systematic
uncertainty, the uncertainty of reference nuclide data-points was increased uniformly, i.e.
each data point had the same systematic uncertainty δMsyst. (in keV/q), until χ2red. =
1. The mass-fit was then repeated and the statistical uncertainty was recalculated to be
consistent with the current parameters of the fit function 48 . This process was then repeated
into an uncertainty in the dependent variable dimension [184].
47 Analyses of storage ring mass measurements employing isochronous mass spectrometry
have employed a similar technique to assess their systematic uncertainty (See Equation 11
of [186].).
48 The results of the mass-fit with and without inclusion of the systematic uncertainty
133

iteratively until it converged. The fit-function resulting from this procedure was the mass–
TOF calibration function which was used to obtain masses for nuclides with unknown masses
whose TOF was measured (See the following chapter for final mass values.). This function
is presented in Section 3.9.3.
It should be noted that legitimate grievances with this approach to assessing the systematic uncertainty of a fit are raised by [187]. Firstly, they rightly allege that this procedure
erases any measure of goodness-of-fit that χ2red. is intended to provide; however we based
our goodness-of-fit on the value of χ2red. prior to inclusion of the systematic uncertainty.
Secondly, they maintain that for this procedure to be valid the systematic uncertainty must
be statistically distributed (i.e. independent of the value of the dependent variable), the
model must be linear in all fit parameters, and that the model must be ‘correct’ (i.e. must
accurately describe the data). Regarding the first concern, it will be shown in Section 3.9.3
that the additional scatter in the mass-fit residual (that was unaccounted for by statistical
and literature uncertainties) appeared to be independent of m/q, so this condition was met.
Regarding the second concern, Section 3.9.3 will demonstrate that the only fit functions under consideration were polynomials of the independent variables, and therefore were linear
in all fit parameters. The third concern, which demanded that the model be ‘correct’ was
arguably not satisfied since no formal proof was made of the analytic form of the mass–TOF
relationship. However, we addressed this by adding a so-called ‘function-choice uncertainty’
to the final mass uncertainties which accounted for our ignorance of which of the set of
best or close-to-best fits was the ‘correct’ function (See Section 3.9.4.). For what it’s worth,
Numerical Recipes endorses the procedure of normalizing χ2red. = 1 in order to obtain some
were compared to make sure it did not drastically effect the fit results. The evaluated
mases resulting from each fit agreed within the final one standard deviation uncertainty, i.e.
including the extrapolation uncertainties (See Section 3.9.4.).
134

measure of systematic uncertainty when its origins are unknown (See Section 15.1 of [188].).
The additional concern of errors in the interpolation and extrapolation from the mass-fit
function that was done to evaluate masses from TOFs is addressed in Section 3.9.4.

3.9.3

Mass fit function selection

Since the relationship between mass and TOF at the precision level required to make a meaningful mass measurement was a priori unknown, the approach of [31] was followed, where
several fit functions were tried, each of which was a combination of polynomials in TOF,
nuclear charge Z, and/or a combination of these variables. The goal of this approach is to
find the minimum number of terms that reproduce the calibration mass surface without any
systematic trends in the residuals. This ensures maximum robustness against interpolation
and small-distance extrapolation. The complex nature of the mass–TOF surface (See Figure 3.40.) clearly necessitated higher orders in both TOF and Z; a step-by-step procedure
was taken to justify the inclusion of each term in the fit-function. As might be expected,
some degeneracy existed as to the benefit of including certain terms in the fit-function. This
set of ‘best’ fits was used to inform the uncertainty of masses evaluated from the mass-fit
function present from extrapolation-from and interpolation-between the mass-fit calibration
points (See Section 3.9.4.).
The approach to finding the ‘best’ mass-fit function was to start simple and to increase
complexity in a way that was motivated by the simpler fit’s fit-residuals. Since to first order
the mass over charge only depended on TOF (See Equation 1.2.), the initial fit function
49
chosen was m
q (τ ) = a0 + a1 ∗ τ , where ai indicates a fit-parameter and τ = TOF − TOF

(where the average over TOF was taken over the set of reference nuclei). As seen in Fig49 See

Section 3.9.2 for an explanation of using variables τ and z instead of TOF and Z.
135

10000

44
45

Mfit-Mliterature [keV]

5000

47

49

0

48

49

50

-5000

47

46

52
65

51
63 66

64

Z=18
Z=19
Z=20

-10000
-15000

2.45

2.5

50
6653

2.55

A/Z

2.6

51

54

Z=25
Z=26
2.65

2.7

Figure 3.41: Residual of the fit to reference nuclei using the function m
q (τ ) = a0 +a1 ∗τ , where
τ = TOF − TOF and z = Z − Z , with TOF being the time-of-flight, Z being the nuclear
charge, and the averages of these being taken over the set of reference nuclei. Reference
nuclei are identified by element by their symbol and the text label indicates the reference
nucleus mass number A. Thick colored error bars show the statistical uncertainties. Thin
black error bars show the sum in quadrature of the statistical uncertainty and the systematic
uncertainty (9 keV/q) applied in the case of the best-fit (See Figure 3.47), which was used
for the mass evaluation.

Mfit-Mliterature [keV]

4000

66

44
45

2000
0

47
4964

63 48
50

-2000
-4000

Z=18
Z=19
Z=20

-6000
-8000
2.45

6547

46

66

51
50

49

54

53

52

51

Z=25
Z=26

2.5

2.55

A/Z

2.6

2.65

2.7

Figure 3.42: Same as Figure 3.41 but for the fit function m
q (τ ) = a0 + a1 ∗ τ + a2 ∗ z.
66

Mfit-Mliterature [keV]

4000 44

45

2000

-2000

66

47

0

4964

-4000
-6000
2.45

6547

46

50

52

51

Z=18
Z=19
Z=20

Z=25
Z=26

2.5

2.55

A/Z

51

50

49

48
63

2.6

53

2.65

54

2.7

2
Figure 3.43: Same as Figure 3.41 but for the fit function m
q (τ ) = a0 + a1 ∗ τ + a2 ∗ z + a3 ∗ τ .

136

66

Mfit-Mliterature [keV]

4000

65

2000 44
0

47

49

45
48
50 63

46
51
66

49 5247 50 53

-2000

64

-4000
2.45

2.5

2.55

A/Z

Z=18
Z=19
Z=20

Z=25
Z=26

2.6

2.65

51 54

2.7

Figure 3.44: Same as Figure 3.41 but for the fit function m
q (τ ) = a0 + a1 ∗ τ + a2 ∗ z + a3 ∗
τ 2 + a4 ∗ z 2 .
1500

63

Mfit-Mliterature [keV]

1000
500
0

49
44 64

47

50
45

48

49

46
51

47
65
52

66
50
53

51

-500

Z=18
Z=19
Z=20

-1000

66

-1500
-2000

2.45

2.5

2.55

A/Z

2.6

54

Z=25
Z=26
2.65

2.7

Figure 3.45: Same as Figure 3.41 but for the fit function m
q (τ ) = a0 + a1 ∗ τ + a2 ∗ z +
a3 ∗ τ 2 + a4 ∗ z 2 + a5 ∗ z ∗ τ . Note that this fit was one of the fits used to evaluate the
extrapolation ‘function-choice uncertainty’ (See Section 3.9.4.).

Mfit-Mliterature [keV]

1000
500

4964 47
0 44
-500

Z=18
Z=19
Z=20

-1000
-1500

49
50 48
51
63
46
45
66

2.45

2.5

66
47 50
65
53
52

Z=25
Z=26
2.55

A/Z

51

54

2.6

2.65

2.7

Figure 3.46: Same as Figure 3.41 but for the fit function m
q (τ ) = a0 + a1 ∗ τ + a2 ∗ z + a3 ∗
2
2
3
τ + a4 ∗ z + a5 ∗ z ∗ τ + a6 ∗ z . Note that this fit was one of the fits used to evaluate the
extrapolation ‘function-choice uncertainty’ (See Section 3.9.4.).

137

Mfit-Mliterature [keV]

1000
500

4964
0 44
47
-500

Z=18
Z=19
Z=20

-1000
-1500

50
51
48 66 46
45 63

2.45

2.5

47
66
50 53
65
52

49

51

Z=25
Z=26
2.55

54

2.6

A/Z

2.65

2.7

Figure 3.47: Same as Figure 3.41 but for the fit function m
q (τ ) = a0 + a1 ∗ τ + a2 ∗ z + a3 ∗
τ 2 + a4 ∗ z 2 + a5 ∗ z ∗ τ + a6 ∗ z 4 . Note that this was the fit function ultimately used for the
mass evaluation.

Mfit-Mliterature [keV]

1000
500

4964
0 44
47

47

49
52

65

66
50 53

51
54

-500

Z=18
Z=19
Z=20

-1000
-1500

50
46
48 6651
45 63

2.45

2.5

Z=25
Z=26
2.55

A/Z

2.6

2.65

2.7

Figure 3.48: Same as Figure 3.41 but for the fit function m
q (τ ) = a0 + a1 ∗ τ + a2 ∗ z + a3 ∗
2
2
4
3
τ + a4 ∗ z + a5 ∗ z ∗ τ + a6 ∗ z + a7 ∗ τ .

Mfit-Mliterature [keV]

1000
500
0

4964
44
47

49

47
66
65
50
53
52

51
54

-500

Z=18
Z=19
Z=20

-1000
-1500

51
50
48 66 46
45 63

2.45

2.5

Z=25
Z=26
2.55

A/Z

2.6

2.65

2.7

Figure 3.49: Same as Figure 3.41 but for the fit function m
q (τ ) = a0 + a1 ∗ τ + a2 ∗ z + a3 ∗
2
2
4
4
τ + a4 ∗ z + a5 ∗ z ∗ τ + a6 ∗ z + a7 ∗ τ . Note that this fit was one of the fits used to
evaluate the extrapolation ‘function-choice uncertainty’ (See Section 3.9.4.).

138

Mfit-Mliterature [keV]

1000
500

49
44
64
0
47
-500

Z=18
Z=19
Z=20

-1000
-1500

50
46
63 6651
45 48

2.45

2.5

49

47
65
52

506653

Z=25
Z=26
2.55

A/Z

51

54

2.6

2.65

2.7

Figure 3.50: Same as Figure 3.41 but for the fit function m
q (τ ) = a0 + a1 ∗ τ + a2 ∗ z + a3 ∗
2
2
4
2
τ + a4 ∗ z + a5 ∗ z ∗ τ + a6 ∗ z + a7 ∗ z ∗ τ . Note that this fit was one of the fits used to
evaluate the extrapolation ‘function-choice uncertainty’ (See Section 3.9.4.).

ure 3.41, this simple function reproduces the masses of reference nuclei within 10 MeV, i.e. a
−4
precision of δm
m ∼ 10 . However, it is apparent that a strong Z-dependence is present in the

fit residuals. Therefore, the fit function was augmented to become m
q (τ ) = a0 +a1 ∗τ +a2 ∗z,
where z = Z − Z (where the average over Z was taken over the set of reference nuclei). The resultant factor of ∼ ×2 reduction in the mass-fit residual scatter is apparent
in Figure 3.42, however a trend is still present with a quadratic A/Z-dependence. Since
A/Z ∝ m/q ∝ TOF, a second order term in TOF was added to the function, resulting
2
in the form m
q (τ ) = a0 + a1 ∗ τ + a2 ∗ z + a3 ∗ τ . The improvement to the fit residu-

als, though less dramatic than the improvement from the addition of the previous term,
is seen in Figure 3.43. To remove the remaining Z-dependence, the mass-fit function was
further augmented to include a second order term in Z, expanding the function to the form
m
q

(τ ) = a0 + a1 ∗ τ + a2 ∗ z + a3 ∗ τ 2 + a4 ∗ z 2 . The dramatic improvement resulting from the

addition of the higher-order Z term is shown in Figure 3.44, but also present in this figure is
the Z-dependent trend in the fit-residual as a function of A/Z. Therefore another term was
2
2
added to result in the function m
q (τ ) = a0 + a1 ∗ τ + a2 ∗ z + a3 ∗ τ + a4 ∗ z + a5 ∗ z ∗ τ ,

139

ultimately achieving the drastic improvement to the mass-fit residual seen in Figure 3.45 50 .
Though improvements were realized beyond this functional form, it was decided that the
quality of the fit merited this functional form’s inclusion into the set of ‘best’-fits which were
used to evaluate the mass evaluation extrapolation uncertainty present due to the ambiguity
in the optimum function choice (See Section 3.9.4.).
Since the mass-fit residual for the aforementioned 6-parameter form lacked major systematic trends, beyond this point the addition of more parameters was explored with the
condition that a marked improvement in the overall mass-fit residual scatter had to be
realized without the addition of any systematic biases (beyond the normally-distributed
systematic scatter extant in the fit residuals). The metric used for the quality of the
fit was the systematic uncertainty δMsyst. which had to be added to all data points to
achieve χ2red. = 1 (See Section 3.9.2.). For the 6-parameter function shown in Figure 3.45,
δMsyst. = 22.7 keV/q. To remove the potential Z-dependent scatter remaining in the 6parameter function fit residual, the next highest order in Z was added to the fit, yielding
2
2
3
the function m
q (τ ) = a0 + a1 ∗ τ + a2 ∗ z + a3 ∗ τ + a4 ∗ z + a5 ∗ z ∗ τ + a6 ∗ z , which

resulted in a systematic uncertainty δMsyst. = 11.2 keV/q, as demonstrated graphically in
Figure 3.46. Given the demonstrated improvement, this functional form was included in
the set of ‘best’ fits used to determine the extrapolation uncertainty due to function choice
(See Section 3.9.4.). Since the behavior of odd-powered polynomials is very different from
that of even powers, a Z 4 term was explored in lieu of the 3rd -order Z-term, resulting in
50 It

should be noted that since TOF ∝ A/Z, z ∗ τ ∝ A. Recalling that ∆E ∝ A2 (See
Section 2.5.2.), it seems likely that this term accounts for the alteration of the ions’ flight
path due to energy loss at some point. Considering that this term was not necessary in the
prior NSCL TOF mass measurement analysis (See Equation 4.22 of [31].), and that set-up
lacked a wedge degrader in the A1900, it seems likely that the need for the z ∗ τ term was
caused by the use of the A1900 wedge degrader.

140

2
2
4
the function m
q (τ ) = a0 + a1 ∗ τ + a2 ∗ z + a3 ∗ τ + a4 ∗ z + a5 ∗ z ∗ τ + a6 ∗ z . In this

case the scatter of the mass-fit, shown in Figure 3.47, was reduced to a minimum for the set
of 7-parameter fit functions, with a systematic uncertainty δMsyst. = 9.0 keV/q, ultimately
making this the final mass-fit function.
The addition of an 8th term to the mass-fit function was also explored, keeping the prior
condition that the addition of an extra parameter must yield a substantial improvement
to the fit. Motivated by the concern that additional trends may be present in the A/Z
dimension, the additional terms focused on addressing any remaining TOF-dependence. In
the same vein as the investigation of higher than 2nd order Z-dependence, both 3rd order
and 4th order TOF terms were explored, as shown in Figures 3.48 and 3.49, respectively.
The fit with the τ 3 -term actually yielded a worse fit, where δMsyst. = 10.0 keV/q, which was
unexpected since it would seem more parameters generally should improve a fit or leave the
goodness of fit the same. At present it is not clear why the fitting algorithm does not find the
true minimum, which in this case would be a coefficient of zero for the τ 3 term. The fit with
the τ 4 -term produced a marginal improvement, resulting in δMsyst. = 8.5 keV/q, and, since
this higher-order TOF-dependence couldn’t be ruled out, this functional form was included
in the set of ‘best’ fit functions used to produce the function-choice extrapolation uncertainty
(See Section 3.9.4.). Finally, Z-dependence of the 2nd -order TOF term was also explored by
2
2
4
2
trying the fit function m
q (τ ) = a0 +a1 ∗τ +a2 ∗z+a3 ∗τ +a4 ∗z +a5 ∗z∗τ +a6 ∗z +a7 ∗z∗τ .

The improvement to the fit, evident from the systematic uncertainty δMsyst. = 8.2 keV/q,
was not deemed substantial enough to merit the inclusion of this 8th term in the final fit
function, but this function was still included in the set of ‘best’ fit functions used to evaluate
the extrapolation uncertainty from the ambiguity as to which function was truly the best.

141

In the spirit of Occam’s razor 51 , the mass-fit function used to obtain the central value
of the masses in the final mass evaluation was the function

m
(τ ) = a0 + a1 ∗ τ + a2 ∗ z + a3 ∗ τ 2 + a4 ∗ z 2 + a5 ∗ z ∗ τ + a6 ∗ z 4 ,
q

(3.17)

which had the mass-fit residual shown in Figure 3.47 and, in a slightly more appealing form,
Figure 3.51. The same fit-residual in two dimensions is shown in Figure 3.52. In principle
this function provides a more reliable extrapolation than the higher-order functions which
were explored.
The robustness of the results obtained with the final mass-fit function (Equation 3.17)
was investigated by removing reference nuclides and repeating the mass fit. The metrics
used for robustness were whether or not the masses of the removed reference nuclides were
reproduced and whether the new mass results were significantly changed. The robustness
of the mass-fit extrapolation in the TOF-dimension was tested by removing the most exotic
reference nuclide from each isotopic chain (47 Ar, 51 K, 54 Ca, 66 Mn, and 66 Fe) and repeating
the mass-fit (See Figure 3.39 to see where these nuclides are located in A/Z–Z-space with
respect to the nuclides for which we obtain new masses.). The mass-fit residuals for this case
are shown in Figure 3.53b, where it is apparent that the changes to the mass-fit residual were
minimal. To assess the robustness of the mass-fit interpolation in the Z-dimension, reference
nuclei from the calcium isotopic chain (49−54 Ca) were removed from the set of reference
nuclei and the mass-fit was repeated (See Figure 3.39 to see where these nuclides are located
in A/Z–Z-space with respect to nuclides for which we obtain new masses.). The mass-fit
51 Note

that the method used to select the final mass-fit function, that of choosing the
minimal set of parameters to yield a good fit and exploring fits with one extra level of
complexity, is akin to the more formal process of evaluating the ‘evidence ratio’ or ‘Bayes
factor’ [189].
142

Mfit-Mliterature [keV]

1000
500

49 64
0

44 47

50

-500

Z=18

-1000

Z=19

-1500

Z=20
2.45

63

45

48 51
66

49
46

65 47
52

66
53
50

Z=25

51

54

Z=26

2.5

2.55

A/Z

2.6

2.65

2.7

Figure 3.51: Residuals of the fit (with Equation 3.17) to the time-of-flight of calibration
nuclei (See Section 3.9.1.) as a function of the mass number to nuclear charge ratio A/Z.
Isotopes are labeled with their mass number and symbols indicate the elements (solid circle
for argon, solid square for potassium, solid triangle for calcium, open circle for manganese,
and open square for iron). Calibration masses were fit to within 9 keV/q without any
systematic trends. The gray band shows the average systematic mass uncertainty included
for reference nuclei as described in Section 3.9.2.(From [36].)

143

Figure 3.52: Residuals of the fit in keV (with Equation 3.17) to the time-of-flight of calibration nuclei (See Section 3.9.1.) as a function of the mass number to nuclear charge ratio
A/Z and nuclear charge Z. Note that Z is shifted by 1/2 so that the colored box corresponds
to the leftward-Z value. The text labels and color indicate mass-fit residual in keV.

144

Parameter

Full set of Refs.

Most Exotic Refs. Removed

Ca-Chain Refs. Removed

a0

238378±5

2360298±4

237793±12

a1

5966.9±0.2

5970.9±0.2

5967±0.3

a2

83±2

101±2

76±2

a3

-0.44±0.02

-0.46±0.02

-0.46±0.02

a4

-25±1

-23±1

-27±2

a5

-1.6±0.1

-1.6±0.1

-1.6±0.1

a6

0.37±0.05

0.27±0.05

0.48±0.07

Table 3.2: Final mass-fit parameters for Equation 3.17 with reference masses removed from
the fit to test fit robustness.

residuals for this case are shown in Figure 3.53c, where it is apparent that the changes to
the mass-fit residual were minimal, except that the evaluated masses for the calcium nuclei
themselves were somewhat poorly reproduced. This result motivated the limitation of the
mass-fit interpolation to nuclides with Zref. ± 1 (i.e. masses were evaluated for scandium and
chromium, but not titanium and vanadium). Fit parameters resulting from the robustness
tests with the final mass fit function, Equation 3.17, are compared in Table 3.2, where it
is seen that the majority of variation was in the Z-dependent parameters. This variation
provided further motivation to limit the mass-fit interpolation to Zref. ±1. These tests were
repeated for each of the ‘best’ fit functions which were used to assess the function-choice
extrapolation uncertainty (See Section 3.9.4.) and the qualitative conclusions were the same.

3.9.4

Mass fit uncertainty

The philosophy behind the uncertainty assignment was to be conservative and to thoroughly
investigate the ways in which systematic bias could have affected the results 52 . As was
52 To

quote an expert on TOF mass measurement, “There are no wrong answers, just
wrong error bars.”.

145

(a)

(b)

(c)

Figure 3.53: Panel (a) shows the residuals of the fit (with Equation 3.17) to the time-of-flight
of calibration nuclei (See Section 3.9.1.) as a function of the mass number to nuclear charge
ratio A/Z. Isotopes are labeled with their mass number and symbols indicate the elements
(solid circle for argon, solid square for potassium, solid triangle for calcium, open circle for
manganese, and open square for iron). Panel (b) shows the fit residuals obtained using the
same fit function as panel (a), but removing 47 Ar, 54 Ca, 51 K, 66 Mn, and 66 Fe (marked by
yellow arrows) from the set of reference nuclei in order to assess the robustness of the mass-fit
extrapolation in the TOF-dimension, since they were the most exotic reference nucleus in
their isotopic chain. Panel (c) shows the fit residuals obtained using the same fit as panel
(a), but removing 49−51 Ca (marked by yellow arrows) from the set of reference nuclei to
assess robustness of the mass-fit interpolation in the Z-dimension. Fit parameters for the
three fits are compared in Table 3.2.
146

mentioned in Section 3.9.2, the statistical uncertainty was taken into account via standard
error analysis. An additional uncertainty was added to each mass result to take into account
the additional statistical scatter found in the fit residuals which was not accounted for by
the original statistical error bars. The remaining uncertainty to be accounted for is the
uncertainty of the mass-fit function, which is determined empirically from a statistically
limited data set. The primary contributors were the uncertainties of the fit coefficients
that were a result of the uncertainties in the calibration mass values and TOFs (referred to
as Monte Carlo uncertainty, motivated by the way it was calculated), and the uncertainty
from the choice of the general form of the fit function (referred to as the function choice
uncertainty).
The measured TOF of a reference nucleus has a statistical uncertainty. Therefore, a
calculation was performed to determine how this uncertainty propagates through the fit
function and the final results. The procedure consisted of randomly perturbing the mass of
each reference nucleus within an amount that was consistent with its systematic uncertainty
(See Section 3.9.2.), performing the mass fit, placing the fit result for each evaluated mass in
a histogram, and repeating 10,000 times. The standard deviations of the resultant evaluatedmass distributions were referred to as the Monte Carlo uncertainty.
In order to perturb the TOF values of reference nuclei within an amount commensurate
with their full statistical scatter (which was mainly due to the systematic mass uncertainty),
the m
q (τ ) relationship had to be inverted so that the mass uncertainty could be translated
into a TOF uncertainty. This was accomplished by grouping the terms of the mass-fit
function by their respective powers in TOF and employing the quadratic, cubic, or quartic
equation (depending on the highest order of TOF that was included in the mass-fit) to
obtain the solution to the equation τ

mperturbed
q

147

. While the quadratic equation is likely

familiar to most readers, the cubic and quartic equations are somewhat more complicated 53
and the reader is directed to [190] and [191] for their solutions 54 . Note that, to perform
the translation from mass to TOF, the roots of the aforementioned functions were solved
with the the perturbed mass over charge of the nucleus added to the constant offset term
(since we were finding the root at

mperturbed
q

and not at 0). The root which provided the

physical answer (of the two, three, or four from the quadratic, cubic, or quartic equations,
respectively) was found by evaluating the function for TOF using the true mass, obtaining
TOF, and evaluating the original mass-fit function with that TOF to obtain the mass. If
the input of that process matched the output then the proper root was selected.
Thus, for each reference nucleus for each Monte Carlo event, the known reference mass
value would be randomly changed by an amount selected from a Gaussian distribution,
with the systematic uncertainty as the standard deviation, and this mass value would be
translated into the corresponding shift in TOF. A graph would then be generated with all
of the reference nuclei’s known masses preserved, but with their TOFs shifted from their
measured values by their perturbations. The mass fit was then performed. Repeating this
10,000 times for the final mass-fit function (Equation 3.17.) yielded the distributions shown
in Figures 3.54 and 3.55. Note that the ‘RMS’ indicated in these plots is the standard
deviation of the mass distribution in keV and was dubbed the ‘Monte Carlo uncertainty’.
In addition a ‘function-choice uncertainty’ was included in the uncertainty of the extrapolated masses, which took into account the evaluated mass results for each of the ‘best’-fits
identified in Section 3.9.3. The approach which was taken was to compare the evaluated
53 I.e.

they are difficult to program/debug.
that a general analytic solution for the roots of a fifth-order polynomial does not
exist, i.e. their is no ‘quintic equation’, so in this case a numerical solution would have to
be devised.
54 Note

148

200

250
200
150
100

6

700

400
300
200

Mass [keV]

Mean
RMS

×10

5.306e+07
2.083e+02

400
300
200
0

Z=22 A=54
400

dN/dMass

600

53.066

Mean
RMS

0

100

50.244

50.246

5.399e+07
1.154e+02

53.05

53.051

×10

5.304e+07
2.266e+02

200

53.985

0
51.179

100
50

6

53.039

Mass [keV]

53.04

×10

×10

5.398e+07
1.859e+02

51.191

Mean
RMS

5.118e+07
1.470e+02

51.18

Mass [keV]

Mean
RMS

51.181

500

Mass [keV]

300
200
6

0

52.128

Mass [keV]

Mean
RMS

52.13

×10

5.211e+07
1.309e+02

5.492e+07
1.612e+02

54.921

54.922

Mass [keV]

Mean
RMS

400
300
200
100

×10

6

0

52.113

Z=22 A=60
300

200

Z=23 A=59

×10

5.213e+07
1.237e+02

400

600

300

0
54.92

6

53.976

×10

500

Z=22 A=56

52.114

Mean
RMS

5.586e+07
2.668e+02

250
200
150
100
6

0

55.854

55.856

×10

Mass [keV]

5.491e+07
1.573e+02

Z=23 A=60
600
500

400
300
200

Mean
RMS

5.585e+07
1.364e+02

400
300
200
100

6

0

×10

Mass [keV]

50

×10
54.923

100
53.975

Mean
RMS

48.388

100
6

×10
51.192

400

Mass [keV]

Mean
RMS

48.387

600

6

dN/dMass

200
150

400
350
300
250
200
150
100
50
0

53.986

200

Z=21 A=56

6

6

300

0

100

Z=23 A=58

250

5.119e+07
1.017e+02

200

500

300

0

dN/dMass

Mean
RMS

Mean
RMS

300

Z=22 A=59

400

400

500

Mass [keV]

Mean
RMS

500

Mass [keV]

400

×10

500

Mass [keV]

dN/dMass

500

100
6

0
53.038

Z=22 A=55

600

200

350
300

5.024e+07
1.862e+02

4.839e+07
1.203e+02

6

×10

100

700

300

Mass [keV]

48.408

6

Z=22 A=58

400

Z=23 A=57

Mean
RMS

48.406

Mass [keV]

200

×10

5.305e+07
1.177e+02

800
700
600
500
400
300
200
100
0

Mass [keV]

6

500

0

50.257

×10

300

dN/dMass

700

6

50.256

48.404

Z=21 A=55

600
500

Mass [keV]

Z=22 A=57

5.026e+07
1.078e+02

Mean
RMS

100
6

0

100
53.064

100

dN/dMass

400
350
300
250
200
150
100
50
0

49.321

dN/dMass

dN/dMass

Z=21 A=57

49.3205

Mean
RMS

100

6

49.32

150

×10

dN/dMass

100
0

45.618

dN/dMass

500

600

200

6

45.616

Z=21 A=54

600

700

dN/dMass

4.932e+07
1.103e+02

Z=21 A=52

250

Mass [keV]

dN/dMass

dN/dMass

700

0

Mass [keV]

Mean
RMS

Z=21 A=53

×10

4.841e+07
2.378e+02

50

50
44.681

300

300

100
44.68

350

dN/dMass

300

Mean
RMS

Z=19 A=52

dN/dMass

400

4.562e+07
2.241e+02

dN/dMass

350

500

0

Mean
RMS

Z=18 A=49

dN/dMass

4.468e+07
1.311e+02

dN/dMass

dN/dMass

600

dN/dMass

Mean
RMS

Z=18 A=48

54.909

54.91

Mass [keV]

×10
54.911

6

0
55.845

55.846

×10

Mass [keV]

Figure 3.54: Mass distributions resulting from the Monte Carlo TOF-perturbation procedure
for isotopes of argon through vanadium (18 ≤ Z ≤ 23) which were observed in the TOF
mass measurement (See Figure 2.3.) and which were not reference nuclei for the mass fit
(See Section 3.9.1.). The ‘RMS’ value was the standard deviation of the mass distribution in
keV and was chosen as the ‘Monte Carlo uncertainty’ for the evaluated masses. The vertical
red line indicates the evaluated mass from the mass fit function (Equation 3.17.) without
perturbations to the reference nuclei TOFs.

149

300

5.583e+07
2.521e+02

dN/dMass

dN/dMass

250
200
150
100
50
6

0

×10
55.832

55.83

Mass [keV]

Mean
RMS

Z=24 A=64
400

Mean
RMS

Z=24 A=61

5.957e+07
1.910e+02

400
350
300
250
200
150
100
50
0

5.677e+07
2.059e+02

Z=24
500A=62

Mean
RMS

5.770e+07
1.710e+02

500

300
200
100

Mass [keV]

Mean
RMS

Z=24 A=65
300

0
57.699

Z=24 A=66

6.051e+07
2.837e+02

200

57.7

Mass [keV]

Mean
RMS

×10
57.701

6.144e+07
4.233e+02

100

150
100
50

300

Mean
RMS

Z=25 A=64

5.956e+07
1.451e+02

0

60.506

Z=25 A=67
300

400

dN/dMass

dN/dMass

500

×10

300
200
100

×10
60.51

60.508

Mass [keV]

Mean
RMS

6.236e+07
2.575e+02

150
100

59.562

Mean
RMS

Z=26 A=65

0

62.364

62.366

6.048e+07
1.653e+02

Z=26 A=67
350
300

400
300
200
100

Mean
RMS

6.235e+07
2.536e+02

150
100

dN/dMass

150

Mean
RMS

6.515e+07
5.128e+02

140

50
6

0
65.152

0
62.351

Z=26 A=71

100

65.154

65.157

Mass [keV]

×10
65.16

Mean
RMS

6.328e+07
3.245e+02

62.352

62.353

Mass [keV]

Mean
RMS

6.609e+07
6.410e+02

100

100
80
60
40

×10

160
140
120
100
80
60
40
20
0
64.23

Mean
RMS

6.423e+07
5.403e+02

6

64.233

64.237

Mass [keV]

Mean
RMS

×10
64.24

6.422e+07
4.092e+02

150
100

×10

63.286

Mass [keV]

Mean
RMS

6

0

64.22

64.222

Mass [keV]

×10
64.224

6.700e+07
7.944e+02

80
60
40
20

20
0

63.284

Z=26
120 A=72

120

57.692

Mass [keV]

50
6

0

100

200

50
×10
62.354

150

Z=26 A=69

100

6

×10

dN/dMass

Z=26 A=70

60.485

Mass [keV]

×10
63.304

150

dN/dMass

60.484

Mass [keV]

63.302

200

50
6

60.483

63.3

250

200

200

Z=25 A=69

6

0
63.297

Z=26 A=68

250

250

0
57.69

50

Mass [keV]

dN/dMass

dN/dMass

500

6.330e+07
3.817e+02

6

dN/dMass

59.561

Mean
RMS

100

×10

5.769e+07
2.528e+02

6

×10
61.445

150

50
×10

Mass [keV]

0

Mass [keV]

200

200

6

0

61.44

Z=25 A=68

250

×10

50

0

dN/dMass

Mass [keV]

59.572

6

dN/dMass

59.57

100

58.637

Mean
RMS

350

150

6

dN/dMass

0

58.636

Mass [keV]

50

6

200

0

dN/dMass

dN/dMass

dN/dMass

dN/dMass

200

200

300

Z=25 A=62

250
300

400

6

6

×10
56.768

5.864e+07
1.570e+02

100

6

56.766

Mean
RMS

Z=24 A=63

400

dN/dMass

350

dN/dMass

Mean
RMS

Z=24 A=60

6

66.09

Mass [keV]

×10
66.095

6

0

67

67.005

Mass [keV]

67.01

×10

Figure 3.55: Same as Figure 3.54 for isotopes of chromium through iron (24 ≤ Z ≤ 26).

150

Parameter

z4

z3

z2

τ4

z ∗ τ2

a0

238378±5

2383359±6

238370±12

2383382±5

2383379±5

a1

5966.9±0.2

5966.9±0.3

5966.5±0.5

5966.9±0.2

5966.9±0.2

a2

83±2

65±5

91±4

84±2

85±2

a3

-0.44±0.02

-0.44±0.02

-0.41±0.04

-0.53±0.06

-0.46±0.02

a4

-25±1

-21±1

-18±1

-25±1

-25±1

a5

-1.6±0.1

-1.6±0.1

-2.1±0.2

-1.6±0.09

-1.7±0.1

a6

0.37±0.05

2.1±0.3

–

0.36±0.04

0.36±0.04

a7

–

–

–

0.0003±0.0002

-0.012±0.007

Table 3.3: Mass-fit parameters for the set of ‘best’ fits identified in the mass fit (See
Section 3.9.3.) as identified by their highest-order in z = Z − Z , τ = TOF − TOF , or a
combination of the two. The corresponding fit functions are written-out in the captions of
Figures 3.56, 3.57, 3.58, 3.59, and 3.60.

mass result for each nucleus obtained from each of the ‘best’ fits and to take the difference
between the two most extreme predictions (i.e. highest and lowest masses) of the set as
the ‘function-choice uncertainty’. Figures 3.56, 3.57, 3.58, 3.59, and 3.60 show the behavior
of the best-fit functions (where the linear m/q dependence on TOF has been removed for
clarity) with respect to the reference nuclei that were fit (black points) and the evaluated
nuclei, where the evaluated residual shown is always that for the final mass-fit function,
Equation 3.17. The large deviation seen in these functions for large distances from the reference nuclei, particularly at high-Z, ultimately provide the basis for limiting extrapolation
from calibration masses, even for nuclides with reasonable statistical uncertainties. Table 3.3
shows the fit parameters resulting from mass-fits with each of the best-fit functions.
The results for the Monte Carlo and function-choice uncertainties, as well as the robustness test (See Section 3.9.3.), were used to determine at which point the mass-fit extrapolation and interpolation should be cut-off; i.e. how far in TOF and Z from the reference nuclei
were the results deemed trustworthy and worth reporting. Given the deviation observed in

151

800

800
600

600

400
200

400

0

-200
-4

200
-2
0

Z-

0

<Z

2

>

4
6
8 -30

-20

-10

0

TOF-<TOF>

10

20

30

[ns]

Figure 3.56: Mass–TOF surface, where the linear dependence of mass over charge m/q on
TOF has been removed, for reference nuclei (black-outlined white-filled circles connected by
the white planes) and the fit (color indicates keV/q) to the reference nuclei with the mass-fit
2
2
4
function, m
q (τ ) = a0 + a1 ∗ τ + a2 ∗ z + a3 ∗ τ + a4 ∗ z + a5 ∗ z ∗ τ + a6 ∗ z . Note that
the averages over Z and TOF, used to construct z = Z − Z and τ = TOF − TOF ,
were taken over the set of reference nuclei (See Section 3.9.1.). The red-outlined white-filled
circles show the location of nuclei whose mass were evaluated with the final mass-fit function,
Equation 3.17 (in this case, the same function as the fit shown), in m/q–TOF–Z space. Note
that all points lay on the function surface, though the plotting program does not extend the
graphing of the surface far enough in some cases.

152

/q vs TOF linear-fit residual [keV/q]

1000

M

Mnucl./q vs TOF linear-fit residual [keV/q]

1000

1200

500

500
0

0

-500

-1000
-500
-4
-2
0

Z-

-1000

2

<Z

>

4
6
8 -30

-20

-10

0

10

20

30

[ns]
TOF-<TOF>

Figure 3.57: Same as Figure 3.56 for the mass-fit function, m
q (τ ) = a0 + a1 ∗ τ + a2 ∗ z +
a3 ∗ τ 2 + a4 ∗ z 2 + a5 ∗ z ∗ τ + a6 ∗ z 3 .

153

/q vs TOF linear-fit residual [keV/q]

1000

M

Mnucl./q vs TOF linear-fit residual [keV/q]

1000

1500

/q vs TOF linear-fit residual [keV/q]

-200

M

Mnucl./q vs TOF linear-fit residual [keV/q]

0

1000

1000

800

800
600

600

400
200

400

0

-200

200

-4
-2
0

Z-

<Z

2

>

4
6
8 -30

-20

-10

0

10

[ns]
TOF-<TOF>

20

30

Figure 3.58: Same as Figure 3.56 for the mass-fit function, m
q (τ ) = a0 + a1 ∗ τ + a2 ∗ z +
a3 ∗ τ 2 + a4 ∗ z 2 + a5 ∗ z ∗ τ .

154

1000
800
600
400

-4

200

-2

Z-

0
2

<Z

>

0
4
6
8 -30

-20

-10

0

10

[ns]
TOF-<TOF>

20

30
-200

Figure 3.59: Same as Figure 3.56 for the mass-fit function, m
q (τ ) = a0 + a1 ∗ τ + a2 ∗ z +
a3 ∗ τ 2 + a4 ∗ z 2 + a5 ∗ z ∗ τ + a6 ∗ z 4 + a7 ∗ τ 4 .

155

/q vs TOF linear-fit residual [keV/q]

1200

M

Mnucl./q vs TOF linear-fit residual [keV/q]

1400

1600
1400
1200
1000
800
600
400
200
0
-200

400

500

200
0

0

-500

-200

-1000

-400

-1500
-4

-600
-800

-2

-1000

0

Z-

<Z

2

>

4
6
8 -30

-20

-10

0

10

20

30

-1200

[ns]
TOF-<TOF>

Figure 3.60: Same as Figure 3.56 for the mass-fit function, m
q (τ ) = a0 + a1 ∗ τ + a2 ∗ z +
a3 ∗ τ 2 + a4 ∗ z 2 + a5 ∗ z ∗ τ + a6 ∗ z 4 + a7 ∗ z ∗ τ 2 .

156

/q vs TOF linear-fit residual [keV/q]

600

1000

M

Mnucl./q vs TOF linear-fit residual [keV/q]

800

the results for calcium isotopes when they were removed as reference masses, the choice was
made to only report results for nuclei which were within one atomic number of a reference
nucleus. This eliminated titanium and vanadium isotopes from the results. These nuclei
were particularly sensitive to the highest-order Z dependence used in the mass-fit, which
provided further justification for not reporting mass results for these nuclei. Given the deviation of the results at high TOF due to different function choices, the set of reported masses
was limited to nuclei whose TOF was no greater than 10% of the full reference TOF spread
(≈ 36 ns; see the Results chapter) beyond the highest-TOF reference nucleus (≈ 511 ns; see
the Results chapter) and no more than three nuclei (in terms of the nuclear chart) away from
a reference nucleus. The uncertainty components for evaluated nuclei, whose masses will be
reported in the following chapter are shown in Figures 3.61, 3.62, 3.63, and 3.64, while their
sum in quadrature, which is the final total uncertainty, is given in Figure 3.65.

3.9.5

The mass fit as a diagnostic tool

Given the sensitivity of the mass results to nearly imperceptible differences in TOFs resulting from choosing different but seemingly equivalent cuts and corrections in the analysis
pipeline 55 , the mass-fit residuals themselves had to be used to distinguish between many
available options in the analysis pipeline. To compare the impact of different choices in
the analysis pipeline, the mass-fit residuals will be shown using the final mass-fit function
(Equation 3.17) to highlight the different mass-surfaces that certain analysis choices resulted
in. While the best fit function for the final mass-fit wasn’t necessarily the best fit function
for the mass surface resulting from a different analysis pipeline choice, using the same fit
55 Recall

that a 1 ps shift in TOF corresponded to a ∼100 keV shift in the mass (See
Section 1.4.1.).

157

Z

Mass Excess Statistical Uncertainty [keV]
26

450

25

400

24

350

23

300

22

250
200

21

Reference Nuclei

150

20
Stable Nuclei

100

19
Literature Limit

50

18
26

28

30

32

34

36

38

40

42

0

N
Figure 3.61: Statistical uncertainty in keV for nuclei whose mass was evaluated in the timeof-flight mass measurement. Colored boxes indicate nuclei whose mass was evaluated, with
the color reflecting the uncertainty in keV, boxes with red circles indicate reference nuclei
used as calibrants for the mass–TOF relationship (See Section 3.9.1.), boxes with ×’s indicate
the most exotic isotope for that element with a known mass prior to this experiment, and
the black boxes indicate stable nuclei.
Mass Excess Systematic Uncertainty [keV]
26

220

25

200
180

Z

24

160

23

140

22

120
100

21

Reference Nuclei

20

80
60

Stable Nuclei

40

19
Literature Limit

18
26

28

30

32

34

36

38

40

42

20
0

N
Figure 3.62: Same as Figure 3.61, but with the color indicating the systematic uncertainty
of evaluated nuclei in keV.

158

Mass Excess Monte Carlo Uncertainty [keV]
400

Z

26
25

350

24

300

23

250

22

200

21

Reference Nuclei

150

Stable Nuclei

100

Literature Limit

50

20
19
18
26

28

30

32

34

36

38

40

42

0

N
Figure 3.63: Same as Figure 3.61, but with the color indicating the Monte Carlo uncertainty
of evaluated nuclei in keV.
Mass Excess Function Uncertainty [keV]
1200

26
25

1000

24
800

Z

23
22

600

21

Reference Nuclei

400

20
Stable Nuclei

19

200
Literature Limit

18
26

28

30

32

34

36

38

40

42

0

N
Figure 3.64: Same as Figure 3.61, but with the color indicating the function-choice uncertainty of evaluated nuclei in keV.

159

Mass Excess Total Uncertainty [keV]
26

1200

25
1000

24

Z

23

800

22

600

21

Reference Nuclei

400

20
Stable Nuclei

19

200
Literature Limit

18
26

28

30

32

34

36

38

40

42

0

N
Figure 3.65: Same as Figure 3.61, but with the color indicating the total uncertainty of
evaluated nuclei in keV, where the total is the sum in quadrature of the statistical, systematic,
Monte Carlo, and function-choice uncertainties. Note that 56 Sc has an additional systematic
uncertainty due to the presence of a β-decaying isomer (See Section 4.3.1.) which is not
included in this figure.

160

function allows the mass–TOF surfaces to be compared directly. It should be noted that
significantly better fit residuals for these other analysis pipeline choices were not obtained in
the exploration of their ‘best’ fit function when using 7-parameters or less for the fit. In each
case the resultant mass-fit residual should be compared to the best-fit, shown in Figure 3.47.
Distinguishing between which of the TOF rigidity corrections was best, given the choices
of by-isotope, by-element, or global (See Sections 3.6 and 3.7), was not possible at a prior
location in the analysis pipeline. As such, the minimum achievable scatter in the mass-fit was
used as a metric to distinguish between the three options. The mass-fit residuals using the
by-element rigidity correction are shown in Figure 3.66, where the systematic uncertainty
added to compensate for the scatter of reference nuclei was 24.9 keV/q. The by-isotope
(local) rigidity correction fit residuals are shown in Figure 3.67, where the added systematic
uncertainty was 38.6 keV/q. Inspection of each of these mass-fit residuals shows that the
mass–TOF surface was more rapidly changing with respect to the global rigidity corrected
mass–TOF surface, as evidenced by the rapid change in TOF-dependence of the fit residual
with changing Z. As such each of these resultant mass–TOF surfaces were more difficult to
fit than the global rigidity corrected TOF mass–TOF surface, and thus the global rigidity
correction was ultimately chosen. It should be noted that the mass results obtained with
the final best-fits for each of the rigidity corrections agreed within uncertainties, though, as
noted, those uncertainties were rather large.
Finding the best of the two promising event-TOFs identified in Section 3.2 was only
achievable by inspection of the mass-fit residual. Figure 3.68 shows the mass-fit residual
when using the global rigidity corrected ‘Direct Up’ TOF (See Section 3.2.), where the
systematic uncertainty added was 18.6 keV/q. It is apparent that there was a larger overall
scatter in the fit-residuals which was a result of a more erratic mass–TOF surface (likely due
161

to non-linearities in the TAC–ADC signal processing, discussed in Section 3.1), and thus a
good mass-fit was difficult to obtain. The mass results from the mass-fit to the ‘Direct Up’
TOF agreed with the mass results from the ‘Clock Down’ TOF within uncertainties, though
the uncertainty from the former was rather large.
As mentioned in Section 3.9.1, the criterion for inclusion of a nucleus as a reference mass
included the condition that its literature mass had to have an uncertainty less than 100 keV.
The mass-fit for the case when the condition was relaxed to δMliterature < 400 keV is shown
in Figure 3.69. Here a systematic offset is apparent for the elements with atomic number
21 ≤ Z ≤ 24, whose reference masses were derived from prior storage ring and time-offlight mass measurements, which ultimately required an inclusion of 14.3 keV/q systematic
uncertainty to account for the scatter in the residuals. Given the less reliable nature of the
extra reference nuclei included in this fit, they were excluded from the final mass-fit.

162

Mfit-Mliterature [keV]

1000

49
46

500

64

0
-500

47

44

2.45

49

48
6651

2.5

65

66
53

54

50

52

63

Z=18
Z=19
Z=20

-1000
-1500

45
50

47

51

Z=25
Z=26
2.55

A/Z

2.6

2.65

2.7

Figure 3.66: Residual of the fit to reference nuclei (See Section 3.9.1.) using the function
m (τ ) = a + a ∗τ + a ∗z +a ∗τ 2 + a ∗z 2 + a ∗z ∗τ + a ∗z 4 , where τ = TOF− TOF and
0
1
2
3
4
5
6
q
z = Z − Z , with TOF being the time-of-flight, Z being the nuclear charge, and the averages
of these being taken over the set of reference nuclei, where the TOF was rigidity corrected
by-element (See Section 3.7.). Reference nuclei are identified by element by their symbol
and the text label indicates the reference nucleus mass number A. Thick colored error bars
show the statistical uncertainties. Thin black error bars show the sum in quadrature of the
statistical uncertainty and the systematic uncertainty (9 keV/q) applied in the case of the
best-fit (See Figure 3.47), which was used for the mass evaluation.
2000

51

Mfit-Mliterature [keV]

1500
1000

49

45

500

50

0

64 47

-500

-1000

44

-1500
-2000

2.45

Z=18
Z=19
Z=20
2.5

486651
46
63

6547
49

Z=25
Z=26
2.55

A/Z

52

66
50
53

2.6

2.65

2.7

Figure 3.67: Same as Figure 3.66 but using the by-isotope (local) rigidity correction to
TOF (See Section 3.6.). Note that 54 Ca is not present in the set of reference nuclei for this
case since it did not have sufficient statistics (> 5, 000 counts) to have a locally-determined
rigidity correction.

163

Mfit-Mliterature [keV]

1000
500 44
0

49

-500

2.45

65

51

47

50
45

Z=18
Z=19
Z=20

-1000
-1500

64

2.5

46

48
63

53
52
47 50
49
66

66

54

Z=25
Z=26
2.55

A/Z

2.6

51

2.65

2.7

Mfit-Mliterature [keV]

Figure 3.68: Same as Figure 3.66 but using the global rigidity correction(See Section 3.7.)
for the ‘Direct Up’ TOF (See Section 3.2.).

A/Z

Figure 3.69: Residual of the fit to reference nuclei (stars), where a reference nucleus was
defined as one with a literature mass with an uncertainty less than 400 keV, using the
2
2
4
function m
q (τ ) = a0 + a1 ∗ τ + a2 ∗ z + a3 ∗ τ + a4 ∗ z + a5 ∗ z ∗ τ + a6 ∗ z , where
τ = TOF − TOF and z = Z − Z , with TOF being the time-of-flight, Z being the nuclear
charge, and the averages of these being taken over the set of reference nuclei, where the TOF
was rigidity corrected globally (See Section 3.7.). Reference nuclei are identified by element
by their color, where isotopes of the same element are connected by lines for clarity, and
the text label indicates the reference nucleus mass number A. The uncertainty shown is the
statistical plus the mass-perturbation distribution from the Monte Carlo procedure.

164

Chapter 4
Results
The atomic mass excesses obtained from the time-of-flight mass measurement described in
this dissertation are presented in Table 4.1. 18 masses were obtained, where the masses of 7
nuclei, 48,49 Ar 1 , 56,57 Sc, 64 Cr, 67 Mn, and 69 Fe, were measured for the first time. The timeof-flight of these nuclei with respect to reference nuclei is shown in Figure 4.1. The following
sections will discuss the comparison to results obtained from global mass model calculations
(also presented in Table 4.1), as well as discuss the advances made by applying these masses
to outstanding problems in nuclear structure and nuclear astrophysics. In principle TOF
data exists to obtain new masses for 54−60 Ti, 57−63 V, 68 Mn, and 70,71 Fe, however these
masses are not reported due to lack of reliable calibration masses. Note that at present
no masses exist in the literature for 58−60 Ti, 62,63 V, 68 Mn, and 70,71 Fe. As discussed in
Chapter 5, masses may be obtained for these nuclides in a future re-evaluation of the data
once more reference masses are available.

1 Published

in [36].
165

Isotope

This experiment

AME2012

FRDM

HFB-21

DZ

WS3

48 Ar†

-22 280(310)

–

-21 240

-21 900

-21 889 -22 044

49 Ar†

-17 800(1100)

–

-14 880

-16 110

-15 648

-16 406

52 Sc

-40 300(520)

-40 167(142)

-39 360

-40 110

-39 241

-40 400

53 Sc

-38 170(570)

-38 107(270)

-36 840

-38 480

-37 148

-38 861

54 Sc

-33 750(630)

-33 599(363)

-32 030

-33 980

-32 623

-34 139

55 Sc

-30 520(580)

-29 977(463)

-29 170

-31 320

-29 597

-30 758

56 Sc†

–

-23 840

-25 230

-24 425

-25 146

57 Sc†

-24 850(590)+0
−540
-21 000(1300)

–

-20 440

-22 550

-20 627

-21 115

59 Cr

-48 540(440)

-47 891(244)

-48 680

-49 160

-48 013

-48 451

60 Cr

-47 440(460)

-46 504(213)

-47 910

-48 200

-46 732

-46 779

61 Cr

-43 080(510)

-42 455(129)

-42 700

-43 710

-42 534

-42 461

62 Cr

-40 890(490)

-40 895(148)

-41 180

-41 960

-40 630

-40 445

63 Cr

-35 940(430)

-35 722(459)

-36 030

-37 290

-35 962

-35 773

64 Cr†

-33 480(440)

–

-34 950

-34 730

-33 545 -33 347

67 Mn†

-34 090(620)

–

-34 480

-34 960

-33 141 -33 294

67 Fe

-45 190(430)

-46 069(218)

-46 530

-46 940

-45 991

-45 577

68 Fe

-43 620(430)

-43 825(365)

-45 360

-45 170

-43 853

-43 665

69 Fe†

-39 350(600)

–

-40 230

-40 390

-39 156 -39 380

Table 4.1: Atomic mass excesses (in keV) obtained in this experiment compared to the
adopted experimentally-based value in the 2012 Atomic Mass Evaluation [1] and predictions
from global mass models (See Section 4.1.1.) FRDM 1995 (FRDM) [2], HFB-21 [3], 10parameter Duflo-Zuker (DZ) [4], and WS3 [5]. The asymmetric uncertainty included for the
56 Sc mass excess is an additional systematic uncertainty from potential isomeric contamination (See Section 4.3.1.). Nuclei whose masses have not been previously measured are
marked with †.

166

105
4

Counts/10ps

10

103
102

44

Ar

47

K

45

Ar 48
53 K
Sc
50
61
Ca

Ar 62Cr 68
Fe
49
65
Cr
K Mn 63
Sc
51 54
69
Cr
66
60 63Mn66Fe Ca 67Sc 47 55
Fe
Mn
Cr
Fe 52CaAr Sc50 64Cr
K
59
48 67Mn
Cr
56Ar
64
Sc
Fe
51
53
K
Ca
57

49

Ca 52

46

54

Sc
49
Ca Ar

10
1

475

480

485

490

495

500

505

510

515

TOF[ns]
Figure 4.1: Rigidity-corrected time-of-flight (TOF) distributions for reference nuclei (unfilled
m
histograms) used to calibrate the rest
q (T OF ) relationship to obtain new masses from the
48,49
TOFs of
Ar (green-filled histograms), 52−57 Sc (red-filled histograms), 59−64 Cr (bluefilled histograms), 67 Mn (orange-filled histogram), and 67−69 Fe (gray-filled histograms).
Note that the TOF spectra for argon, scandium, and chromium isotopes with respect to
reference TOFs are highlighted separately in Figures 4.20, 4.33, and 4.23, respectively.

167

4.1

Comparison to global mass models

As described in Sections 1.1 and 1.2, nuclear masses play an important role in the fields of
nuclear structure and nuclear astrophysics. Research at the forefront of each area depends
on accurate masses of nuclei which are very far from stability [6, 66, 71]. Given the myriad
of challenges posed by measuring the masses of exotic nuclei (See Section 1.3.), calculations
often must rely on global mass models to fill the void that exists beyond the known mass
landscape 2 . Though, as will be shown, global mass models have yet to reach a predictive
power better than a precision of ∼ 10−5 , it is interesting to note that no reason exists a
priori that future models couldn’t achieve precisions of

10−6 [193]. This section contains

a detailed comparison of the mass measurement results presented in this dissertation to
frequently used global mass models, where the comparison is preceded by a brief description
of the models (See Table 4.1 for a direct comparison.). The models are presented in a rough
order of more macroscopic in form to more microscopic in form, though it should be noted
that the distinction isn’t always clear 3 .
2 While

the Atomic Mass Evaluation (2012 being the most recent version [1]) is the nearly
universally accepted source for experimentally measured masses, its releases are few and
infrequent. A compilation of recently measured (but unevaluated) masses can be found
here: [192].
3 For example, the Duflo-Zuker mass model is described as microscopic in the original
manuscript [4], macroscopic-microscopic in a later paper in which Andr´es Zuker is a coauthor [194], and neither microscopic nor macroscopic but ‘a functional of orbital properties’
in a later paper by Zuker [195] (available at [40]).

168

4.1.1

Model descriptions

4.1.1.1

Liquid drop model

The 4 semi-empirical mass formula (SEMF), originally proposed by Carl Friedrich von
Weisz¨acker in 1935 [81], provided the first global model of nuclear masses. The form of
the SEMF was motivated by considering the nucleus as a drop of water, i.e. a nearly
constant-density body consisting of microscopic particles that interact over a short range
and cohesively bond into a nearly spherical shape, and thus it is also often referred to as the
liquid drop model. The conceptual basis for the drop-like part of the formula is that there
should be some bulk amount of binding resulting from the cohesive attraction of nucleons
(the ‘volume term’), a penalty in binding for the surface area since these nucleons have a
neighbor-less side where they do not benefit from cohesive binding (the ‘surface term’), and
a penalty in binding from Coulomb repulsion of protons within the nucleus (the ‘Coulomb
term’). Nuclear physics considerations bring about additional corrections, where an ‘asymmetry term’ must account for the tendency of nuclei to prefer similar numbers of protons and
neutrons N = Z (ultimately due to the fact the proton-neutron strong interaction is generally stronger than the proton-proton interaction or neutron-neutron interaction, as well as a
contribution from the Pauli principle) and a ‘pairing term’ which accounts for the tendency
of nucleons to anti-align spins to become a spin-0 pair 5 in order that they can occupy the
same spatial state and thereby be bound more tightly.
The spherical nuclear shape assumed in the liquid drop assumption implies that the
volume of the nucleus scales with the number of nucleons as V ∝ A, therefore the radius of
the nucleus scales as A1/3 and the surface area scales as A2/3 . The combination of these
4 This
5 Said

discussion is primarily based on [37] and [196].
in a more fancy way, this is ‘nucleonic superfluidity’ [197].
169

considerations (along with comparison to measured masses) results in the following formula 6
for the nuclear binding energy BE(Z, A),

BE(Z, A) = avol. A − asurf. A2/3 − aCoul.

Z(Z − 1)
A1/3

− aasym.

(Z − A/2)2
+ i ∗ apar. A−1/2 , (4.1)
A

where i = +1 for even-Z and even-N (=A − Z), i = −1 for odd-Z and odd-N , and i = 0
otherwise. Typical parameters are avol. = 15.56, asurf. = 17.23, aCoul. = 0.697, aasym. =
93.14, and apar. = 12 , in units of MeV/c2 [37]. The nuclear binding energy can be converted
to atomic mass excess ME(Z, A) via the relations

BE(Z, A) ≡ Z(mp + me ) + N mn − M (Z, N )
ME(Z, A) ≡ M (Z, N ) − (Z + N )mu ,

(4.2)

where M (Z, N ) is the mass of a nucleus with Z protons and N neutrons, mp = 938.27200 MeV/c2
is the proton mass, me = 0.51100 MeV/c2 is the electron mass, mn = 939.56533 MeV/c2
is the neutron mass, and mu = 931.494013 MeV/c2 is the atomic mass unit. Using these
parameters one obtains a root mean square (RMS) deviation from the 2012 Atomic Mass
Evaluation of 7.7 MeV (See Figure 4.2.), which is rather remarkable for a 5-parameter fit
considering the fact that nuclear masses are on the order of [A]*GeV/c2 . Furthermore, it
should be noted that the best available global mass models (See the following descriptions.)
at the present only outperform this model from 1935 by a factor of ∼10!

6 Note

that several variations exist of the SEMF. The form from [37] is shown here.
170

|MEame2012 - MEliq.drop| [keV]
100

RMS: 7702keV

104

Z

80

103

60
102

40
10

20
20

40

60

80

N

100

120

140

160

1

Figure 4.2: The Segr´e chart with the absolute value of the mass excess deviation in keV
(indicated by color) between the 2012 Atomic Mass Evaluation [1] and the liquid drop model
as represented by the semi-empirical mass formula given in [37]. The black boxes indicate
the stable nuclei for orientation. The root mean square RMS deviation is indicated by the
text box.

171

4.1.1.2

FRDM

The 7 Finite Range Droplet Model (FRDM) built on the success of the liquid drop model
for the macroscopic description of nuclear binding energy with both macroscopic and microscopic refinements. The macroscopic refinements primarily consist of the inclusion of nuclear
compressibility (hence, ‘drop’→‘droplet’), for instance taking into account the reduction in
Coulomb repulsion and increase in surface energy from shape deformation, and modifying the
surface energy by substituting an implicit zero-range nucleon-nucleon force with an explicit
finite-range force (diffuseness). The microscopic refinements include corrections to account
for the closed shells, nuclear pairing (to account for the odd-even mass stagger), and the
Wigner effect (which is the observed enhancement in binding of nuclei with N = Z beyond
what is expected from the asymmetry term). The closed shell correction is performed via
the method of Strutinsky [199], where, roughly speaking, single particle energies are calculated with the shell model, these single particle energies are summed to get a bulk binding
energy, and the difference between the bulk binding energy for that particular nucleus and
other nuclei with similar numbers of nucleons is taken as the shell correction. The FRDM
model published in 1995 [2] has an RMS deviation from the 2012 Atomic Mass Evaluation of
654 keV (See Figure 4.3.). Note that an update exists for 2012 [200] 8 ,but it is not compared
here to other models since it is not yet publicly available. The 1995 FRDM mass table is
presently available here: [38].

7 This
8 They

discussion is primarily based on [42] and [198].
cite an RMS deviation from the 2003 AME of 570 keV.
172

|MEame2012 - MEfrdm95| [keV]
100

RMS: 654keV

103

Z

80
102

60
40

10

20
20

40

60

80

N

100

120

140

160

1

Figure 4.3: The Segr´e chart with the absolute value of the mass excess deviation in keV
(indicated by color) between the 2012 Atomic Mass Evaluation [1] and the 1995 Finite Range
Droplet Model (FRDM) [2, 38]. The black boxes indicate the stable nuclei for orientation.
The root mean square RMS deviation is indicated by the text box.

173

4.1.1.3

WS3

The 9 Weizs¨acker-Skyrme (WS) formula mass model WS3 [5] is very similar to the FRDM
mass model in concept, but differs in terms of details of the execution. The proponents
state that it has 13 independent parameters to FRDM’s 31 [5]. However, what one counts
as a parameter is somewhat difficult to define due to the complicated nature of the mass
models, particularly the very detailed microscopic corrections. The WS3 model has yet to
see widespread use in the literature, likely because of its recency, though it has an impressive
RMS deviation from 2012 Atomic Mass Evaluation of 378 keV (See Figure 4.4.). The WS3
mass table is presently available here: [39].

9 This

discussion is primarily based on [201].
174

|MEame2012 - MEws3| [keV]
100

RMS: 378keV

103

Z

80
102

60
40

10

20
20

40

60

80

N

100

120

140

160

1

Figure 4.4: The Segr´e chart with the absolute value of the mass excess deviation in keV
(indicated by color) between the 2012 Atomic Mass Evaluation [1] and the Weizs¨ackerSkyrme WS3 mass model [5, 39]. The black boxes indicate the stable nuclei for orientation.
The root mean square RMS deviation is indicated by the text box.

175

4.1.1.4

Duflo-Zuker

The 10 Duflo-Zuker mass model is comprised of a phenomenologically-motivated formula
that consists of either 10, 28, or 31 parameters. Here the focus is on the 10-parameter form,
as this is what is compared to the data presented in this dissertation. The 10-parameter
formula contains six parameters which capture the macroscopic liquid-drop aspects of the
nucleus, three parameters which microscopically account for shell corrections, and one parameter which accounts for nuclear deformation for nuclides with Z > 50 11 . The parameters
themselves are determined by a fit to known masses. The 10-parameter form, which was
based on the known masses as of 1995, achieves an RMS deviation from the 2012 Atomic
Mass Evaluation of 639 keV 12 (See Figure 4.5.). The mass-table for the 28 parameter form
and for the 10-parameter form (used for comparison to data), as well as the Fortran code
used to generate the 10-parameter table, can currently be found here: [40].

10 This

discussion is primarily based on [42] and [194].
The relatively simple form, as compared to other global mass models, lends itself to
rapid computation, i.e. the mass table is calculated with 150 lines of Fortran code [40].
12 The RMS for isotopes with Z ≥ 8 and A ≥ 16, to compare to FRDM fairly, since that
table only begins with 16 O, is 595 keV.
11

176

|MEame2012 - MEdz10| [keV]
100

RMS: 639keV
103

Z

80
60

102

40
10

20
20

40

60

80

N

100

120

140

160

1

Figure 4.5: The Segr´e chart with the absolute value of the mass excess deviation in keV
(indicated by color) between the 2012 Atomic Mass Evaluation [1] and the 10-parameter
Duflo-Zuker mass formula [4, 40]. The black boxes indicate the stable nuclei for orientation.
The root mean square RMS deviation is indicated by the text box.

177

4.1.1.5

HFB-21

The 13 Hartree-Fock-Bogolyubov (HFB)-21 mass model obtains nuclear masses by solving the many-body Schr¨odinger equation for nucleons interacting within a mean-field that
is generated by the nucleons themselves. The nuclear many-body wave function is represented as a Slater determinant of single-particle wave functions Φ = det{φi (xi )}. The
non-relativistic Schr¨odinger equation is solved for each of the single-particle wave functions
φi (xi ), Hφi = εi φi . Since the potential energy portion of the Hamiltonian is a mean-field
potential, it depends on the single-particle wave functions, and thus the Schr¨odinger equation for each of the single-particle wave functions must be solved in an iterative fashion until
convergence is realized. The ground-state energy of the nucleus is obtained by calculating the
expectation value of the final Hamiltonian for the final wave function E = Φ|H|Φ . Since
the single-particle wave functions are non-interacting within this framework, the nucleonnucleon forces included in the Hamiltonian must be modified from the forms obtained by
comparison to nucleon-nucleon scattering data to parameterized effective forces, such as the
Skyrme force (zero-range) or the Gogny force (finite-range).
To now this description applies to the Hartree-Fock (HF) method. This method, though
it reproduces the broad features of the nuclear mass surface, misses some of the empirically
observed features, such as the odd-even mass staggering and the Wigner effect (whereby N=Z
nuclei are more bound by ∼2 MeV than expected from macroscopic approaches), which are
a product of correlations between nucleons. The Hartree-Fock-Bogolyubov (HFB) method
is one avenue for dealing with these features within the HF framework by including pairing
correlations into the Slater determinant trial function. The parameters used to account for
the pairing correlations depend on the single-particle wave functions themselves. Ultimately
13 This

discussion is primarily based on [42] and [202].
178

|MEame2012 - MEhfb21| [keV]
100

RMS: 572keV

103

80

Z

102

60
40
10

20
20

40

60

80

N

100

120

140

160

1

Figure 4.6: The Segr´e chart with the absolute value of the mass excess deviation in keV
(indicated by color) between the 2012 Atomic Mass Evaluation [1] and the Hartree-FockBogolyubov HFB-21 mass model [3, 41]. The black boxes indicate the stable nuclei for
orientation. The root mean square RMS deviation is indicated by the text box.

the parameters are determined by fitting to observed masses. The current state-of-the-art
HFB mass model is HFB-21, which has a 572 keV RMS 14 deviation from the 2012 AME
mass excesses [3]. Details on the HFB-21 parameters and the resultant mass table itself can
currently be found here: [41].
14 It

is the same for nuclei with Z ≥ 8, in order to fairly compare with the 1995 FRDM
table, which begins at 16 O.

179

4.1.1.6

Density functionals

As 15 with HFB, density functional theory (DFT) also aims to solve for the mass of the
many-body system, but recasts the many-body problem into an effectively one-body problem. Rather than dealing with a many-body wave function, the fundamental quantity is the
energy density, which is a function of space and time. The functional which describes the
energy density is employed in HFB calculations in the place of the Slater determinant of
single-particle wave functions. Obtaining the optimized nuclear energy density functional,
and solving the resultant HFB calculations, are outstanding problems whose solution is the
primary goal of the “Universal Nuclear Energy Density Functional” (UNEDF) collaboration [204] 16 . The UNEDF0 mass model has been quoted to have an RMS deviation from
known nuclear masses (as of 2010) of even-even nuclei only of 1.45 MeV [61]. Mass tables
from state-of-the-art DFT calculations are presently available here: [205].

4.1.2

Mass comparison

Here we graphically compare the masses obtained in this experiment to the masses predicted
by the FRDM, HFB-21, WS3, and Duflo-Zuker mass models, where physical parameters derived from these masses are also compared (See Table 4.1 for a direct numerical comparison.).
Note that the uncertainties shown are only experimental. Though a prescription exists for
calculating the theoretical uncertainty for global mass model predictions [2], in practice this
uncertainty does not substantially deviate (i.e.

15 keV) from the root mean square devia-

tion from known masses. The magnitude of the observed deviations can be put into context
15 This

discussion is primarily based on [203].
recent work [197] made an extensive comparison between mass predictions obtained
by various DFT calculations in the context of the limits of the nuclear landscape.
16 A

180

via comparison to the deviations observed from all known masses listed in the 2012 Atomic
Mass Evaluation (See previous subsection.). Using the root mean square deviations between
global mass models and known masses as a benchmark, it is apparent that deviations on the
order of several hundreds of keV are to be expected, while deviations greater than an MeV
occur for nuclear masses that are particularly poorly calculated by global mass models.
The deviations of the masses measured in this experiment from measured values listed in
the 2012 Atomic Mass Evaluation and from the global mass models FRDM(1995), HFB-21,
WS3, and 10-parameter Duflo-Zuker, are shown as a function of mass over charge A/Z,
nuclear charge Z, and nuclear mass A in Figures 4.7, 4.8, and 4.9, respectively. The same
mass deviations are shown projected onto the mass-deviation axis, weighted by the one
standard deviation uncertainty of our measured value in Figure 4.10. The same weighted
mass deviations are shown with respect to reference nuclei versus atomic number Z and
neutron number N for the 2012 Atomic Mass Evaluation and the global mass models FRDM,
HFB-21, WS3, and 10-parameter Duflo-Zuker in Figures 4.11, Figures 4.12, Figures 4.13,
Figures 4.14, and Figures 4.15, respectively.
From these comparisons it is apparent that the masses obtained in the experiment described in this dissertation generally agree with the present literature values, with the exception of the chromium isotopes, for which we obtain more-bound values with the deviation
becoming smaller for increasing A/Z (The implications for this deviation are discussed in
Section 4.2.2.). The deviation from the 1995 FRDM global mass model has a marked Adependence, with our values being more-bound at low-A and less-bound at high-A. The
HFB-21 global mass model seems to produce systematically over-bound values compared to
our results, with the exception of the scandium isotopic chain, which is reproduced relatively
well (The implications for this are discussed in Section 4.3.2.). The WS3 global mass model
181

does a remarkably good job of reproducing the masses measured in this experiment, where
nearly all values agree within one standard deviation. The 10-parameter Duflo-Zuker mass
formula does well for the higher-Z measured masses, but performs rather poorly at low-Z.
The physical implications for these mass deviations are perhaps better showcased by comparing mass differences 17 . Arguably two of the most useful mass differences used to examine
the nuclear mass surface of neutron rich nuclides are the one and two neutron separation
energies and the quantities derived from these. The one and two neutron separation energies
are the energetic cost of removing one or two neutrons from a nucleus to form a less neutronrich isotope of a given element. When the energetic cost becomes <0 MeV for an isotope
of a given element, that isotope is unbound with respect to (one or two) neutron emission
and therefore does not exist for any length of time. The location where neutron separation
energies transition from positive to negative values is known as the ‘neutron drip-line’ 18 .
The two-neutron separation energy S2n (Z, A) of a nucleus with Z protons and A nucleons
is obtained by taking the difference in masses between isotopes of an element with A and
A − 2 nucleons after correcting for the binding of the two extra neutrons,
S2n (Z, A) = 2 ∗ MEneutron + ME(Z, A − 2) − ME(Z, A),

(4.3)

where ME is the atomic mass excess. Considered along an isotopic chain, S2n is relatively
high for a neutron-magic/closed-shell nucleus and drops to a relatively low value when two
more neutrons are added. Thus, S2n provides a good model-independent signature of a
closed neutron shell and it has the advantage that it is not subject to effects from odd-even
17 Note

that the most interesting physics conclusions obtained from mass differences that
employed the experimental results presented in this dissertation are showcased in Sections 4.2
and 4.3.
18 The drip-line as predicted by several mass models is discussed in [197].
182

MEmodel - MEthis expt. [keV]

Our 1σ Uncertainty
AME 2012
FRDM '95
HFB-21
WS3
DZ 10param.

3000
2000
1000
0
-1000
-2000
2.45

2.5

2.55

2.6

2.65

2.7

A/Z

MEmodel - MEthis expt. [keV]

Figure 4.7: Difference between mass excesses obtained in this experiment and global mass
models FRDM (1995), HFB-21, WS3, and Duflo-Zuker (10-parameter), as well as experimental values listed in the 2012 Atomic Mass Evaluation as a function of mass over charge
A/Z.
Our 1σ Uncertainty
AME 2012
FRDM '95
HFB-21
WS3
DZ 10param.

3000
2000
1000
0
-1000
-2000
17

18

19

20

21

22

23

24

25

26

27

Z

MEmodel - MEthis expt. [keV]

Figure 4.8: Same as Figure 4.7, but as a function of atomic number Z. Note that multiple
nuclei are represented for each Z, so that experimental 1σ uncertainties are overlapping.
Our 1σ Uncertainty
AME 2012
FRDM '95
HFB-21
WS3
DZ 10param.

3000
2000
1000
0
-1000
-2000
50

55

60

65

70

A

Figure 4.9: Same as Figure 4.7, but as a function of nuclear mass number A. Note that
multiple nuclei are represented for A = 67, so that experimental 1σ uncertainties are overlapping.
183

10
9
8

dN/dσ

7

AME 2012
FRDM '95
HFB-21
WS3
DZ 10param.

6
5
4
3
2
1
0
-4

-3

-2

(ME

model

-1

0

1

- MEthis expt.)/dME

2

this expt.

3

4

[Nσ]

Figure 4.10: Distributions of mass excess deviations, weighted by our one standard deviation
measurement uncertainty and binned in 23 σ-wide bins, from masses obtained in this experiment for global mass models FRDM (1995), HFB-21, WS3, and Duflo-Zuker (10-parameter),
as well as experimental values listed in the 2012 Atomic Mass Evaluation.

mass staggering. However, S2n can sometimes be difficult to interpret as a signature for
a closed shell since it depends on assessing the change of the S2n slope-trend relative to
the local pattern (The parameter Dn has been recently introduced [44] to overcome this
shortcoming; it is employed to discuss the N = 28 shell closure in Section 4.2.1.).
The S2n trends for isotopic chains observed in this mass measurement are compared
for our values, the 2012 Atomic Mass Evaluation, and global mass models (1995 FRDM,
HFB-21, WS3, and 10-parameter Duflo-Zuker) in Figure 4.16. It is apparent that the new
argon masses (48,49 Ar) from this experiment provide evidence for the N = 28 shell closure
in argon (This topic is explored further in Section 4.2.1.). In principle the new scandium
masses (56,57 Sc) provide information on the possible presence of the N = 34 subshell closure,
which is known to be present for calcium [206] and not titanium [207]. Evidence for this
subshell closure was found to be inconclusive for scandium [208]. However, our mass excess
184

Figure 4.11: Mass excess deviation (indicated by color) from the 2012 Atomic Mass Evaluation vs Z and N , weighted by our experimental uncertainty. The same information projected
into a one-dimensional histogram presented in Figure 4.10.

Figure 4.12: Mass excess deviation (indicated by color) from the 1995 FRDM global mass
model vs Z and N , weighted by our experimental uncertainty. The same information projected into a one-dimensional histogram presented in Figure 4.10.

Figure 4.13: Mass excess deviation (indicated by color) from the HFB-21 global mass model
vs Z and N , weighted by our experimental uncertainty. The same information projected
into a one-dimensional histogram presented in Figure 4.10.

185

Figure 4.14: Mass excess deviation (indicated by color) from the WS3 global mass model vs
Z and N , weighted by our experimental uncertainty. The same information projected into
a one-dimensional histogram presented in Figure 4.10.

Figure 4.15: Mass excess deviation (indicated by color) from the 10-parameter Duflo-Zuker
mass formula vs Z and N , weighted by our experimental uncertainty. The same information
projected into a one-dimensional histogram presented in Figure 4.10.

186

uncertainties are too large at present to provide closure on this question 19 . There is an
apparent discrepancy between the S2n trend for the chromium isotopes resulting from our
mass values (59−64 Cr) and those from the 2012 Atomic Mass Evaluation, which is discussed
further in Section 4.2.2. The new manganese mass from this experiment (47 Mn) provides
a hint of evidence for the N = 40 subshell closure in that isotopic chain, in contradiction
to prior conclusions [43], however the rather large present uncertainty makes our result
somewhat inconclusive 20 . The iron masses presented in this dissertation (67−69 Fe) suggest
interesting behavior following N = 40 for the iron isotopic chain (which, as seen in the figure,
potentially mirrors the behavior of cobalt) that may be a sign of deformation in this region,
but our present uncertainties do not allow such firm conclusions 21 .
The trend in S2n along an isotonic chain can be used to quantify the size of the shell gap
associated with that neutron number via the empirical shell gap ∆n [42],

∆n (Z, A) = S2n (Z, A) − S2n (Z, A + 2)
= −2 ∗ ME(Z, A) + ME(Z, A + 2) + ME(Z, A − 2),

(4.4)

where Equation 4.3 was used to go from the first line of the equation to the second. While
the trend in ∆n provides information about the disappearance or appearance of a magic
neutron number, it is unfortunately convoluted with the change in binding due to closed
proton shells. Nonetheless, the new masses for 48,49 Ar provide a data point for ∆n along
19 As

discussed in the final chapter of this dissertation, the addition of reference nuclei
from future mass measurements may reduce this uncertainty to a level for which the N = 34
subshell closure could be confirmed or denied.
20 As with the N = 34 subshell closure for the scandium isotopic chain, our data may yet
provide closure on this question once more reference nuclei become available.
21 Again, the addition of reference nuclei to the mass fit may provide the necessary reduction in uncertainty.
187

N = 28 which falls nicely along the already present trend, as seen in Figure 4.17. It is
interesting to note that for all isotonic chains shown the ∆n trend from the 10-parameter
Duflo-Zuker mass formula is largely featureless, while the trends for the 1995 FRDM mass
model and HFB-21 mass model seem to behave rather erratically. In contrast, the WS3 mass
model demonstrates a smooth behavior which generally represents the data well.
In analogy to the two-neutron separation energy, the one-neutron separation energy
Sn (Z, A) is obtained by inspecting the difference in binding between isotopes of an element
with A and A − 1 nucleons after correcting for the binding of the extra neutron,
Sn (Z, A) = MEneutron + ME(Z, A − 1) − ME(Z, A).

(4.5)

As with S2n , Sn is higher for closed neutron-shell isotopes of an element, but in contrast
to S2n it is also sensitive to the odd-even mass staggering present due to pairing (i.e. the
5th term in Equation 4.1). Since odd-N nuclei have an unpaired neutron, the Sn trend for
odd-N isotopes of an element lays below the corresponding trend for the even-N isotopes, as
seen in Figure 4.18. The gap between the odd-N and even-N Sn trends provides and avenue
to quantify the strength nuclear binding due to pairing 22 . One method of quantifying this
pairing force is the empirical pairing gap [43] PGn ,
(−1)(N +1)
[Sn (Z, A + 1) − 2 ∗ Sn (Z, A) + Sn (Z, A − 1)]
(4.6)
4
(−1)(N +1)
[3 ∗ ME(Z, A) − 3 ∗ ME(Z, A − 1) + ME(Z, A − 2) − ME(Z, A + 1)] ,
=
4

PGn (Z, A) =

22 In addition,

the change of Sn along an isotopic chain (via Dn [44]) can be used to identify
the presence of magic neutron numbers and even indicate the transition between filling one
neutron shell to another (See Section 4.2.1.).

188

where N = A − Z is the number of neutrons and Equation 4.5 was used to go from the
first line to the second. While this quantity’s power to directly probe the pairing interaction
within nuclei is intriguing, the requisite involvement of several mass excesses means that
relatively high precisions are required for all or most-all of the involved nuclei to reach any
strong conclusions. It can be seen in Figure 4.19 that our masses provide evidence for a
rather low pairing gap for 48 Ar, but the large uncertainty makes this conclusion somewhat
weak. Similarly, the reduction in the pairing gap after N = 40 for manganese is interesting
but inconclusive. Perhaps even more intriguing is the opposite conclusion for the pairing
gap trend following N = 40 for the 2012 Atomic Mass Evaluation as opposed to the values
obtained in this experiment. Various global mass models exhibit rather different pairing gap
behavior (See Figure 4.19.).

189

16
14
12

15

16
14
12

10

10

FRDM 1995
HFB-21
WS3
Duflo-Zuker
AME 2012
This expt.
AME 2012 for Z ± 1

Z=24

S2n [MeV]

S2n [MeV]

20

FRDM 1995
HFB-21
WS3
Duflo-Zuker
AME 2012
This expt.
AME 2012 for Z ± 1

Z=21

S2n [MeV]

FRDM 1995
HFB-21
WS3
Duflo-Zuker
AME 2012
This expt.
AME 2012 for Z ± 1

Z=18
25

10

8
8
6

5
20

22

24

26

28

30

32

28

34

16

S2n [MeV]

14

32

34

FRDM 1995
HFB-21
WS3
Duflo-Zuker
AME 2012
This expt.
AME 2012 for Z ± 1

36

38

12

30

32

34

36

38

40

42

44

N
FRDM 1995
HFB-21
WS3
Duflo-Zuker
AME 2012
This expt.
AME 2012 for Z ± 1

Z=26
18
16

S2n [MeV]

Z=25

30

N

N

14
12

10

10

8

8
32 34 36 38 40 42 44 46 48

N

32 34 36 38 40 42 44 46 48

N

Figure 4.16: Two-neutron separation energy S2n along isotopic chains of argon, scandium,
chromium, manganese, and iron for this experiment, the 2012 Atomic Mass Evaluation, and
global mass models. The gray lines and points indicate S2n for Z ± 1.

190

2
1
0

FRDM 1995
HFB-21
WS3 (2012)
Duflo-Zuker (1996)
AME 2012
Our results (w/AME 2012)

3
2
1
0

FRDM 1995
HFB-21
WS3 (2012)
Duflo-Zuker (1996)
AME 2012
Our results (w/AME 2012)

N=30

N=31

N=32

N=33
Shell Gap, ∆ n [MeV]

3

4

Shell Gap, ∆ n [MeV]

4

5

Shell Gap, ∆ n [MeV]

3.5
3

3

2.5

2
1
0
-1

2
1.5

FRDM 1995
HFB-21
WS3 (2012)
Duflo-Zuker (1996)
AME 2012
Our results (w/AME 2012)

1
0.5
0

-0.5

FRDM 1995
HFB-21
WS3 (2012)
Duflo-Zuker (1996)
AME 2012
Our results (w/AME 2012)

3
2
1
0

FRDM 1995
HFB-21
WS3 (2012)
Duflo-Zuker (1996)
AME 2012
Our results (w/AME 2012)

3
2
1
0
-1

FRDM 1995
HFB-21
WS3 (2012)
Duflo-Zuker (1996)
AME 2012
Our results (w/AME 2012)

-1
16 18 20 22 24 26 28

16 18 20 22 24 26 28

Z

Z

Z

Z

Z

Z

1
0.5
0
-0.5
-1
-1.5

FRDM 1995
HFB-21
WS3 (2012)
Duflo-Zuker (1996)
AME 2012
Our results (w/AME 2012)

N=36

2
1
0
-1

-2

FRDM 1995
HFB-21
WS3 (2012)
Duflo-Zuker (1996)
AME 2012
Our results (w/AME 2012)

N=37

3
2

1
0
-1

-2
-3

FRDM 1995
HFB-21
WS3 (2012)
Duflo-Zuker (1996)
AME 2012
Our results (w/AME 2012)

Shell Gap, ∆ n [MeV]

N=35

2.5
2
1.5

Shell Gap, ∆ n [MeV]

N=34
3

3
2
1
0
-1

-2

FRDM 1995
HFB-21
WS3 (2012)
Duflo-Zuker (1996)
AME 2012
Our results (w/AME 2012)

-4
16 18 20 22 24 26 28

-3
16 18 20 22 24 26 28

Z

Z

Z

Z

N=41

2
1
0
-1

-2
-3

FRDM 1995
HFB-21
WS3 (2012)
Duflo-Zuker (1996)
AME 2012
Our results (w/AME 2012)

N=42

2
1
0
-1

-2
-3

FRDM 1995
HFB-21
WS3 (2012)
Duflo-Zuker (1996)
AME 2012
Our results (w/AME 2012)

2
1
0
-1

-2
-3

N=39

3
2

1
0
-1

-2

FRDM 1995
HFB-21
WS3 (2012)
Duflo-Zuker (1996)
AME 2012
Our results (w/AME 2012)

2
1
0
-1

-2
-3

FRDM 1995
HFB-21
WS3 (2012)
Duflo-Zuker (1996)
AME 2012
Our results (w/AME 2012)

16 18 20 22 24 26 28

16 18 20 22 24 26 28

Z

Z

N=43

FRDM 1995
HFB-21
WS3 (2012)
Duflo-Zuker (1996)
AME 2012
Our results (w/AME 2012)

Shell Gap, ∆ n [MeV]

N=40
3

Shell Gap, ∆ n [MeV]

-3
16 18 20 22 24 26 28

Shell Gap, ∆ n [MeV]

-2
16 18 20 22 24 26 28

N=38

Shell Gap, ∆ n [MeV]

16 18 20 22 24 26 28

Shell Gap, ∆ n [MeV]

16 18 20 22 24 26 28

Shell Gap, ∆ n [MeV]

16 18 20 22 24 26 28

Shell Gap, ∆ n [MeV]

-1
16 18 20 22 24 26 28

Shell Gap, ∆ n [MeV]

Shell Gap, ∆ n [MeV]

5

Shell Gap, ∆ n [MeV]

N=29

6

Shell Gap, ∆ n [MeV]

N=28

2
1
0
-1

-2
-3
-4

FRDM 1995
HFB-21
WS3 (2012)
Duflo-Zuker (1996)
AME 2012
Our results (w/AME 2012)

16 18 20 22 24 26 28

16 18 20 22 24 26 28

16 18 20 22 24 26 28

16 18 20 22 24 26 28

Z

Z

Z

Z

Figure 4.17: Empirical shell gap ∆n [42] along isotopic chains of argon, scandium, chromium,
manganese, and iron for this experiment, the 2012 Atomic Mass Evaluation, and global mass
models.

191

16

Sn [MeV]

14
12
10
8

FRDM 1995, Even N
HFB-21, Even N
WS3 (2012), Even N
Duflo-Zuker (1996), Even N
AME 2012, Even N
Our results (w/AME 2012), Even N
FRDM 1995, Odd N
HFB-21, Odd N
WS3 (2012), Odd N
Duflo-Zuker (1996), Odd N
AME 2012, Odd N
Our results (w/AME 2012), Odd N

Z=21
18
16
14
12
10
8

16
14
12
10
8

6

6

4

4

4

2

2

20

22

24

26

28

30

32

2
22 24 26 28 30 32 34 36 38

N

12
10
8

30

35

40

45

N

FRDM 1995, Even N
HFB-21, Even N
WS3 (2012), Even N
Duflo-Zuker (1996), Even N
AME 2012, Even N
Our results (w/AME 2012), Even N
FRDM 1995, Odd N
HFB-21, Odd N
WS3 (2012), Odd N
Duflo-Zuker (1996), Odd N
AME 2012, Odd N
Our results (w/AME 2012), Odd N

Z=26
16
14

Sn [MeV]

16
14

25

N

FRDM 1995, Even N
HFB-21, Even N
WS3 (2012), Even N
Duflo-Zuker (1996), Even N
AME 2012, Even N
Our results (w/AME 2012), Even N
FRDM 1995, Odd N
HFB-21, Odd N
WS3 (2012), Odd N
Duflo-Zuker (1996), Odd N
AME 2012, Odd N
Our results (w/AME 2012), Odd N

Z=25

Sn [MeV]

18

6

18

FRDM 1995, Even N
HFB-21, Even N
WS3 (2012), Even N
Duflo-Zuker (1996), Even N
AME 2012, Even N
Our results (w/AME 2012), Even N
FRDM 1995, Odd N
HFB-21, Odd N
WS3 (2012), Odd N
Duflo-Zuker (1996), Odd N
AME 2012, Odd N
Our results (w/AME 2012), Odd N

Z=24

Sn [MeV]

18

Sn [MeV]

FRDM 1995, Even N
HFB-21, Even N
WS3 (2012), Even N
Duflo-Zuker (1996), Even N
AME 2012, Even N
Our results (w/AME 2012), Even N
FRDM 1995, Odd N
HFB-21, Odd N
WS3 (2012), Odd N
Duflo-Zuker (1996), Odd N
AME 2012, Odd N
Our results (w/AME 2012), Odd N

Z=18

12
10
8
6

6

4

4

2
25

30

35

N

40

45

25

30

35

40

45

50

N

Figure 4.18: One-neutron separation energy Sn along isotopic chains of argon, scandium,
chromium, manganese, and iron for this experiment, the 2012 Atomic Mass Evaluation,
and global mass models. The upper and lower trends are for even and odd N isotopes,
respectively.

192

4
3.5
3
2.5
2
1.5
1

3.5
3
2.5
2
1.5
1
0.5

0.5
18

20

22 24

26

28 30

32

0

34

3
2.5
2
1.5
1
0.5
0

2
1.5
1

25

30

35

40

45

N

FRDM 1995
HFB-21
WS3 (2012)
Duflo-Zuker (1996)
AME 2012
Our results (w/AME 2012)
AME 2012 for Z-1

Z=26
4
Pairing Gap [MeV]

Pairing Gap [MeV]

3.5

3
2.5

0

20 22 24 26 28 30 32 34 36 38

N

FRDM 1995
HFB-21
WS3 (2012)
Duflo-Zuker (1996)
AME 2012
Our results (w/AME 2012)
AME 2012 for Z-1

3.5

0.5

N

Z=25
4

FRDM 1995
HFB-21
WS3 (2012)
Duflo-Zuker (1996)
AME 2012
Our results (w/AME 2012)
AME 2012 for Z-1

Z=24
4
Pairing Gap [MeV]

Pairing Gap [MeV]

4.5

FRDM 1995
HFB-21
WS3 (2012)
Duflo-Zuker (1996)
AME 2012
Our results (w/AME 2012)
AME 2012 for Z-1

Z=21
4
Pairing Gap [MeV]

FRDM 1995
HFB-21
WS3 (2012)
Duflo-Zuker (1996)
AME 2012
Our results (w/AME 2012)
AME 2012 for Z-1

Z=18

3.5
3
2.5
2
1.5
1
0.5

25

30

35

N

40

45

0

25

30

35

40

45

50

N

Figure 4.19: Empirical pairing gap PGn [43] along isotopic chains of argon, scandium,
chromium, manganese, and iron for this experiment, the 2012 Atomic Mass Evaluation,
and global mass models. The gray points indicate the pairing gap trend for one Z lower, for
context.

193

4.2

Nuclear structure applications

While Section 4.1.2 provided a broad comparison of the nuclear structure features probed
by the masses presented in this dissertation, the following subsections will delve into two
specific cases. Namely, the progress made in understanding the N = 28 shell closure and the
exploration of nuclear structure features, including the apparent onset of the N = 40 island
of inversion, in the chromium isotopes will be discussed. Note that the former topic resulted
in the publication [36] 23 .

4.2.1

Termination of the N = 28 closed shell

Note that this subsection (4.2.1) is a modified excerpt from the publication [36]. The author
of this dissertation was the primary author of that work, however coauthors 24 contributed
to its writing.
As discussed in Section 1.1, the “magic” numbers of protons and neutrons, which enhance
nuclear binding for isotopes near the valley of β-stability, can evolve for more neutron-rich
or neutron-deficient nuclei [66, 209, 210]. The neutron magic number N = 28 has been
the subject of extensive recent experimental and theoretical investigations [48, 49, 211–213].
Since neutron-rich N = 28 nuclei are within experimental reach and are computationally
tractable for shell-model calculations, they are ideal candidates for illuminating the funda23

As such, the discussion of the corresponding Section closely follows the discussion of
the published manuscript.
24 The full list of coauthors of [36] is:
Zach Meisel, Sebastian George,
Sunghoon (Tony) Ahn, Justin Browne, Daniel Bazin, B. Alex Brown, J. Fiore Carpino,
Hank Chung, Richard H. Cyburt, Alfredo Estrad´e, Mike Famiano, Alexandra Gade,
Christoph Langer, Milan Matoˇs, Wolfi Mittig, Fernando Montes, Dave J. Morrissey,
Jorge Pereira, Hendrik Schatz, Jule Schatz, Mike Scott, Dan Shapira, Karl Smith,
Jeremy Stevens, Wanpeng Tan, Oleg Tarasov, Sarah Towers, Kathrin Wimmer,
Jack R. Winkelbauer, John Yurkon, and Remco G.T. Zegers. See [36] for their affiliations at the time of that work.
194

mental forces at work in exotic nuclei. It is known that the N = 28 shell gap, which stabilizes
42
doubly magic 48
20 Ca28 , is absent in the Z = 14 and Z = 16 isotopic chains at 14 Si28 [214–217]
40
and 44
16 S28 [218–222]. Experimental information on the structure of 12 Mg28 suggests it has a

prolate deformed ground state [223], which would be consistent with the absence of a neutron
shell gap.
The existence of the N = 28 shell gap for argon is a matter of some controversy. Several
previous experimental studies have assessed the shell structure of neutron-rich argon [46,
47, 224–232]. Investigation of the energy of the lowest excited states of 45
18 Ar27 via β-decay
spectroscopy of 45
17 Cl28 suggested a weakened, but still present, N = 28 shell closure for
argon [226]. The first 2+ state energies E(2+
1 ) along the argon isotopic chain [45–47] and
information on neutron single-particle structure from transfer [228, 229] and knockout [227]
reactions are consistent with the presence of an N = 28 shell gap in 46
18 Ar28 . Though,
disagreement exists as to the inferred nuclear structure from measurements of the 46
18 Ar28
+
quadrupole excitation strength, B(E2, 0+
1 → 21 ), written as B(E2) hereafter for brevity.

Three projectile Coulomb excitation measurements, two at intermediate energies [224, 225]
and one at Coulomb-barrier beam energy [232], deduce a low B(E2), corresponding to a
reduced quadrupole collectivity. In this case quadrupole collectivity reflects a propensity for
neutrons to be excited across the N = 28 shell gap, and thus a low B(E2) may be expected
for a semi-magic nucleus. State-of-the-art shell-model calculations that properly account for
the breakdown of the N = 28 magic number in silicon and sulfur isotopes predict a markedly
46
higher B(E2) for 46 Ar [231]. A low-statistics lifetime measurement of the 2+
1 state of Ar

deduced a high B(E2) value in agreement with theory [230], but at odds with the three
consistent, independent Coulomb excitation measurements [224, 225, 232].
However, B(E2) measurements are not necessarily unambiguous probes of neutron shell
195

structure, since they are sensitive to proton degrees of freedom and proton-neutron interactions. In contrast, mass measurements, and the neutron separation energies derived from
them, directly probe the neutron shell gap in a model-independent way. The masses of 48 Ar
and 49 Ar reported in this dissertation, which are the first experimentally determined masses
for these nuclei 25 , provide robust evidence for the persistence of the N = 28 shell gap for
argon. The times-of-flight of the newly measured argon isotopes with respect to reference
nuclides are shown in Figure 4.20.
The atomic mass excesses obtained for 48 Ar and 49 Ar, which are reported in Table 4.1,
were used to calculate binding energy differences along the argon isotopic chain in order to
examine the N = 28 shell closure. Given the drawbacks of the quantity which is usually
used to asses neutron shell closures (See Section 4.1.2), namely the two-neutron separation
energy S2n (See Equation 4.3.), instead a recently introduced quantity Dn [44] was used.

Dn (Z, A) = (−1)N +1 [Sn (Z, A + 1) − Sn (Z, A)],

(4.7)

where Sn is defined in Equation 4.5, provides a more readily recognizable signature of a
shell closure. In a given mass region, Dn indicates the number of orbital angular momentum
projection “m” states that participate in pairing for a given nucleus. A peak in Dn at a
certain neutron number along with a change in the Dn level before and after that neutron
number indicates a shell gap [44]. The change in the Dn level is a crucial element since it
indicates a transition from filling one “m” state to filling another.
25 The

review of shell structure in the N = 28 region [211] lists the mass of 48
18 Ar30 as
measured, though the link they cite pointing to an AME version from 2011 is no longer in
operation. The more recent 2012 Atomic Mass Evaluation [1] lists the mass of 48
18 Ar30 as
unmeasured and we are unable to find experimental data for this mass after a detailed search
of the literature.

196

105

Counts/10ps

4

10

44

Ar

47

K

45
50

Ca

103

48

K

46

Ar 65
49
K Mn
Ca
66
63
66
47
Mn Fe
Ar Mn
52
Ca 50K

Ca

49

64

Ar

51

48

Ar

Fe

53

Ca

2

10

51

K
54

49

Ca Ar

10
1

480

490

500

510

TOF[ns]
Figure 4.20: Rigidity-corrected time-of-flight distributions for reference nuclei (unfilled hism
48
tograms) used to calibrate the rest
q (T OF ) relationship to obtain masses from TOFs of Ar
and 49 Ar (red-filled histograms). (From [36].)

197

6
Ti, AME 2012
Ca, AME 2012
Ar, AME 2012
S, AME 2012
E2+ Ar, NNDC
Ar, This expt.

Dn [MeV]

5
4
3
2
1
0

24

26

28

30

32

34

N
Figure 4.21: Dn [44] as a function of neutron number N near N = 28 for sulfur (dot-dash
line), argon (solid line), calcium (dotted line), and titanium (dashed line). The previously
known [1] argon trend (solid line, open circles) is shown along with results from this experiment (solid line, solid circles). E(2+
1 ) energies for argon isotopes [45–47] are shown for
comparison (crosses). The peak at N = 28 followed by a reduction in Dn for N > 28 as
compared to N < 28 indicates the presence of a closed shell. From shell-model calculations
we conclude the transition from Dn ≈ 3 MeV for N < 28 to Dn ≈ 1.5 MeV for N > 28 corresponds to the transition from filling the f7/2 orbital to filling the p3/2 orbital.(From [36].)

198

6

SDPF-MU
SDPF-U
AME 2012
This expt.
E2+, NNDC

Dn [MeV]

5
4
3
2
1
0

24

26

N

28

30

Figure 4.22: The Dn [44] trend near N = 28 for argon from currently known masses [1]
(open circles) and the masses presented in this dissertation (solid circles) is shown along with
shell-model calculations employing the SPDF-MU Hamiltonian [48] (solid squares) and the
SDPF-U Hamiltonian [49] (open squares). E(2+
1 ) energies [45–47] are shown for comparison
(crosses). (From [36].)

The Dn values for argon isotopes from this dissertation show a clear signature for an
N = 28 shell closure (Figure 4.21). With the new mass excesses for 48,49 Ar, it is apparent
that neutron-rich argon displays the same systematics in Dn as calcium and titanium, which
are known to exhibit an N = 28 shell gap [211]. As seen in Figure 4.21, sulfur does not peak
at N = 28 [219], which is consistent with prior conclusions that sulfur does not exhibit the
N = 28 closed shell [218]. Based on our experimental data we can therefore conclude that
argon is the lowest even-Z element with a closed neutron shell for N = 28. Furthermore,
the value of Dn at the N = 28 peak in Figure 4.21 indicates that the value of the neutron
shell gap for 46 Ar is ≈ 4.5 MeV. The halving of the value of Dn for N > 28 as compared
to N < 28 is a signature of the transition between filling the f7/2 orbital, where m = 8, to
filling the p3/2 orbital, where m = 4.
This conclusion is supported by the favorable comparison to shell model calculations,
199

shown in Figure 4.22. Here we compare the experimental Dn trend to the Dn trends for local
mass predictions obtained from shell-model calculations using the SDPF-U [49] and SDPFMU [48] interactions. In both cases there is excellent agreement between experiment and
theory. This indicates current shell-model calculations adequately describe the interaction
between core and valence neutrons around N = 28 for argon.
Therefore our new masses for 48 Ar and 49 Ar unequivocally show that the N = 28 closed
shell is present for argon, which makes argon the lowest even-Z element that exhibits an
N = 28 shell gap. Based on this result we can conclude that the problems of shell model
calculations in describing electromagnetic observables in argon isotopes near N = 28 are not
related to the neutron shell gap, but instead point to issues with the interaction of valence
neutrons and core protons 26 .

4.2.2

Structure in the Cr isotopic chain approaching N = 40

As discussed in Section 4.2.1 in the context of the argon isotopes, the first 2+ state energies
E(2+
1 ) and the quadrupole excitation strengths B(E2) in even-even nuclei, and the masses
along an isotopic chain provide complementary information about the forces at work within
a nucleus. While masses provide model independent information regarding neutron magicity,
E(2+
1 ) and B(E2) convolute this information with effects such as collective excitations of
nucleons within a nucleus. Nonetheless, the general trends of these two observables are
expected to roughly mirror each other, particularly near neutron closed shells. This is
because a neutron-magic nucleus is relatively stable with respect to its isotopic neighbors (by
definition) and it should therefore have a weaker excitation from the ground state (B(E2))
26 To

bring the electromagnetic observables in agreement with the values predicted by shell
model calculations, the shell-model calculations discussed here would need to use weaker
proton-neutron interaction strengths in their Hamiltonians [233].
200

and cost more energy, relatively speaking, to excite the magic nucleus from it’s ground state
(E(2+
1 )) or to remove neutrons (S2n ). This mirrored behavior is apparent in the case of
argon (See Figure 4.21), where the relationship is strong due to the presence of neutron
magic number 28.
New masses for the neutron-rich chromium isotopic chain were obtained in this experiment (See Figure 4.23.), which can be used to obtain the trend in two-neutron separation
27
energy S2n (See Equation 4.3.). The trends for S2n , E(2+
1 ) [45, 50, 51], and B(E2) [52, 53]

for the chromium isotopes are compared in Figure 4.24. It is apparent that the S2n trend
is different for the experimentally-based masses obtained from the 2012 Atomic Mass Evaluation (AME) [1] and this experiment, particularly at N = 36 and N = 38 (though they
agree within 2σ uncertainties). The discrepancy in the S2n trends primarily stems from
the ∼750 keV, ∼1 MeV, and ∼500 keV differences between our reported masses and the
experimentally-based AME values for 59 Cr, 60 Cr, and 61 Cr, respectively. In particular, the
difference between our 60 Cr mass excess and the adopted AME value causes the S2n trend to
pivot about N = 37. As seen in Table I of [98], the AME values for these nuclei are primarily
based on three separate measurements from the TOFI facility [100, 177, 178], between which
there is a ∼500 keV discrepancy for reported masses of 59,60 Cr and a ∼1.5 MeV discrepancy
for the reported 61 Cr masses 28 .
A rigorous comparison to shell model calculations has yet to be performed to ascertain
which of the two S2n trends is consistent with expectations and to determine what the
27 To

convert from B(2+ → 0+) in Weisskopf units from [52] to B(0+ → 2+) to be
(2∗2+1)
compatible with [53], the values from [52] were multiplied by (2∗0+1) (0.0594)A4/3 , where A
is the nuclear mass number [196].
28 Internal discrepancies between TOFI measurements as well as discrepancies between
high-precision follow-up studies to TOFI mass measurements are discussed in more detail in
Section 3.9.1
201

105
4

Counts/10ps

10

44

Ar

47

45

Ar

K

48

49

Ca

Ca
60

10

46

Cr
51

Ar

62
49

Ca
66
Mn Fe

Cr
65

K Mn

66

Mn
63
47
Ar Cr
52
50
Ca
K 64

63

Cr

3

K
61

50

59

Cr

Cr

64

Fe
53

Ca

2

10

51

K
54

Ca

10
1

475

480

485

490

495

500

505

510

515

TOF[ns]
Figure 4.23: Rigidity-corrected time-of-flight distributions for reference nuclei (unfilled hism
tograms) used to calibrate the rest
q (T OF ) relationship to obtain masses from TOFs of
59−64 Cr (red-filled histograms).

implications of the new masses are for the structural evolution of neutron-rich chromium
isotopes. However, a cursory examination of the S2n trend obtained from this experiment
seems to suggest a subshell closure for N = 36. Additionally, the leveling-off of S2n at N = 40
is reminiscent of the signature of the N = 20 ‘island of inversion’ provided by the sodium
S2n trend [69] 29 . The likely explanation for this enhancement in binding near N = 40
for chromium is a transition from filling the f p-shell to filling the g9/2 orbital [233]. The
astrophysical implications for the newly obtained 64 Cr mass are described in Section 4.3.2.

29 Recent

evidence for the N = 40 island of inversion has been provided by other mass
measurements [43].
202

AME-2012
This expt.

S2n [MeV]

14
12
10
8
32

34

36

38

40

N

800

1

E(2+) [keV]

1000

600
400
32

34

36

38

40

32

34

36

38

40

N

B(E2) [e2fm4]

1800
1600
1400
1200
1000
800
600

N
Figure 4.24: Trend along the chromium isotopic chain for S2n (upper panel) for this experiment (red-filled circles) and the 2012 Atomic Mass Evaluation [1] (black-open circles),
E(2+
1 ) [45, 50, 51] (middle panel), and B(E2) [52, 53] (lower panel). Note that the full
mass uncertainties were used to calculate S2n , resulting in similar error bars for each point
since each chromium mass had the same systematic uncertainty and similar interpolation
uncertainties.

203

4.3

Applications to the accreted neutron star crust

As discussed in Section 1.2, nuclear masses provide critical input to calculations of nucleosynthesis and nuclear energy generation in extreme astrophysical environments. The masses
of neutron-rich nuclei measured in the experiment presented in this dissertation can be used
to improve calculations of processes occurring within accreted neutron star crusts. As an
aside, it is interesting to note that the majority these very neutron-rich low-Z nuclei are too
neutron-rich to participate in the astrophysical rapid neutron-capture (r-)process, since nucleosynthesis at early times in these environments proceeds near N = Z [234, 235]. The first
of the following subsections will discuss nuclear processes occurring in the accreted neutron
star crust with respect to the impact on astronomical observables. The second subsection
will focus on the application of our mass measurements to calculations of electron captures
in the neutron star crust, focusing on the implications our 64 Cr mass has for crustal heating
and the potential impact our 56 Sc mass may have on the newly discovered Urca neutrino
cooling layer in the outer neutron star crust.

4.3.1

Quiescent cooling neutron stars and nuclear heating and
cooling

Shortly after the discovery of X-ray bursting systems [236–238], it was proposed that they
were associated with neutron stars accreting from a binary companion [239]. While some of
the bursts from the initially observed X-ray bursting system could be explained by variable
accretion rates, the ‘type-II’ bursts with durations of seconds and recurrence times of tens of
seconds, most of the bursts, the ‘type-I’ X-ray bursts which had durations of tens of seconds
and recurrence times of several hours to days, were attributed to thermonuclear runaway of
204

material accreted onto the neutron star surface [240–242]. The ultimately accepted conclusion is that type-I X-ray bursts are powered by explosive nuclear burning which is triggered
by the triple-α reaction (whereby three helium nuclei fuse to become a carbon nucleus via its
Hoyle state), quickly followed by breakout from the CNO-cycle (likely via the 15 O(α, γ) reaction), and afterward proceeding through the rapid proton capture (rp-)process [243–247].
Aside from type-I X-ray bursts, accreted neutron stars exhibit a host of other observable
phenomena including X-ray superbursts [248–252], formerly accreting (cooling) quiescent
neutron stars (a.k.a. quasipersistent transients or quiescent transients) [55, 58, 59, 253–257],
quasiperiodic oscillations following X-ray bursts and magnetar giant flares [258–262], and
possibly gravity (g)-mode oscillations [263, 264]. These phenomena can be used to probe the
physical properties of neutron stars, for instance providing data which can be juxtaposed to
models of the neutron matter equation of state [54, 58, 257, 265–268].
The very neutron-rich nuclei whose mass measurement has been described earlier in this
dissertation naturally occur in the outer crust 30 of accreted neutron stars. As ashes of the
rp-process accumulate on the neutron star surface, previously deposited material is buried,
sinks into the crust, and is transformed by electron captures which are enabled due to the
rising electron Fermi energy with depth [60, 78, 269–272]. As the ashes sink further they
ultimately become so neutron-rich that neutrons can ‘drip’ out of nuclei. It is around this
point that neutron capture and neutron emission, along with pycnonuclear (density-driven)
fusion, further transform the crust composition [272–274]. The result of these processes 31
is a stratified compositional structure that is accompanied by various amounts of nuclear
30 Defined

in the following paragraphs.
that, even for a high accretion rate (i.e. around the Eddington limit), a matter
element takes on the order of centuries to sink to the outer crust and millennia to reach the
inner crust.
31 Note

205

energy deposition (heating) or release (cooling) from the neutron star crust [54, 60], each
of which, so long as accretion continues, drastically alter the thermal profile of the outer
layers of the neutron star from the isothermal state expected for an old isolated neutron
star [58, 257, 272].
The location of these heating and cooling processes (whose origins are discussed later on)
are indicated in Figure 4.25, where [275] and [25] were used to identify the radial span of the
various labeled zones. In this figure the ‘atmosphere’ refers to the gas-like (i.e. no long-range
correlations of matter) outer layer, which has densities up to ∼ 106 g/cm3 , where the rpprocess occurs. The ‘ocean’ is the subsequent liquid-like (i.e. has short-range correlations of
matter) layer, which has densities up to ∼ 109 g/cm3 , where carbon ignition for superbursts
and electron captures occur. The ‘outer crust’ follows, which has nuclei arranged in a latticelike structure with densities up to ∼ 1011 g/cm3 , where electron capture heating occurs
along with Urca neutrino cooling and deep crustal heating from pycnonuclear fusion of light
elements that survive thermonuclear burning near the surface. The ‘inner crust’ begins at
the point at which neutron-drip is reached and pycnonuclear fusion of heavier elements setsin, ultimately allowing for the existence of the exotic so called ‘pasta phases’ and reaching
densities up to the nuclear saturation density ∼ 1014 g/cm3 (i.e. 0.16 nucleons/fm3 ). The
depths beyond the nuclear saturation density are referred to as the core and are posited
to consist of mixtures of nucleons, electrons, and muons, and maybe even hyperons, pions,
and/or kaons.
The compositional and thermal structures of accreted neutron star crusts directly impact
the light curve observed by transiently accreting neutron stars which have gone into quiescence [257]. Therefore, computational models of these environments can be used to map the
phase space of unknowns involving the neutron star structure [58]. As will be shown, the role
206

ocean
(~100m)

atmosphere (~10m)

ν

ν

ν

ν

ν

ν

ν

ν

ν

outer crust
(~0.5km)

inner crust
(~0.5km)

core
(~9km)

ν

ν

ν

Carbon ignition layer
Urca cooling layer
Deep crustal heating

Figure 4.25: Schematic of an accreted neutron star cross section. The Urca cooling (See
Section 4.3.2.) layer separates heat generated deep in the crust by nuclear processes from
the shallow layer where carbon is thought to be ignited for superbursts. See the text for a
brief description of the labeled layers.

207

of nuclear physics in these models is to remove free parameters (or reduce the freedom with
which the parameters can be tuned) by providing the nuclear data required to accurately
model the nuclear reactions occurring within the neutron star crust.
Software such as the open source program dStar [57] can be used to model quiescent
cooling neutron stars (with run times on the order of seconds! ) and determine which set of
inputs, e.g. composition, heating strength/depth, and cooling strength/depth, are compatible with observations. In a nutshell, dStar models the neutron star crust as a slab of material
divided into discrete zones 32 with pressure and density as a function of depth defined by the
Tolman-Oppenheimer-Volkoff equation supplemented by various equations of state depending on the density. The crust is brought out of thermal equilibrium by accretion-induced
heating for some time and then thermally relaxes after accretion turns off. The heat sources
primarily come from the gravitational energy release of accreted material, nuclear reactions
such as electron capture and pycnonuclear fusion of material buried by accretion, and an
unknown shallow heat source which is required to match the initially observed temperatures
after accretion turns off [257]. Since both electron capture and pycnonuclear fusion only
occur for material which has been forced into a new layer, they are only ‘on’ during accretion phases. The newly discovered crustal cooling source resulting from Urca cycling [54],
whose nuclear physics origin is described more in Section 4.3.2, continually operates. The
crust primarily cools via heat diffusion, whose timescale is governed by the specific heat and
thermal conductivity, which themselves depend on the crust composition [58]. The outer
crust typically cools from a heated state back to thermal equilibrium after hundreds of days,
while the inner crust takes on the order of thousands of days to reach thermal equilibrium
with the core. Note that the neutron star core is highly conductive and cools over a very
32 The

default zone-size is log(P erg/cm3 ]) = 0.05 where P is the pressure.
208

long time scale, so it’s temperature can be considered fixed during the thermal relaxation of
the crust.
dStar itself has a host of free parameters whose values are generally only loosely bounded
by physical considerations for any given system. General system information can be specified such as the core mass and radius (left fixed at 1.6 M⊙ and 11 km for the simulations
shown hereafter), core temperature, accretion rate, length of the accretion outburst, atmosphere temperature during accretion 33 , pressure at the top and bottom of the crust, and
the pressure above which there is a ‘light element atmosphere’ [276]. Heat sources and
sinks can be specified by the pressure bounds for an outer heat source (meant to approximate electron-capture heating [60]), the pressure bounds for an inner heat source (meant to
approximate deep crustal heating caused by a complex relation between electron capture,
neutron emission, and pycnonuclear fusion [272]), and the pressure at which the recently
postulated [54] shallow Urca neutrino cooling layer exists, as well as by the total amount
of heat deposited/released by these processes. At present, compositional information is
only changed by modifying the so called impurity parameter Qimp. , which is effectively the
weighted variance of the nuclear charge of nuclei making up the crust.

2
Qimp. = n−1
ion Σi ni (Zi − Z ) ,

(4.8)

where ni is the number density of nuclei with atomic number Zi , Z is the average atomic
number of all species at the location where Qimp. is defined, and nion is the total ion number
density. The impurity parameter has a significant impact on the crust conductivity [58].
However, the utility exists within dStar to specify the nuclear abundances as a function of
33 This,

in effect, accounts for the mysterious shallow heat source.

209

depth in order to consistently calculate the impurity parameter. In principle, this abundance
information could also be used to provide consistent depths at which the heating and cooling
processes take place, since they originate from nuclear reactions. Note that it is at this
interface that nuclear data, especially nuclear masses, are able to contribute to reducing the
parameter space (For an example, see the discussion in Section 4.3.2 on the impact of nuclear
masses on Urca cooling.).
∞ (gravitationally red-shifted,
The primary outputs of dStar are effective temperature Teff.

as seen by a distant observer) as a function of time after accretion shuts off and the crust
thermal profile as a function of depth (normally specified by pressure or density) at time-steps
during and after the accretion outburst. Figure 4.26 shows the cooling curve and thermal
profiles during and after an accretion outburst for an example case, where symbols are used
to indicate which thermal profiles correspond to which moments on the cooling curve. The
central panel shows the neutron star crust being heated out of thermal equilibrium with
the core by accretion and the electron capture heating and deep crustal heating which are
turned on by accretion, since accretion buries matter deeper into the star, allowing nuclear
reactions to proceed for the buried nuclei. The lower panel shows the neutron star crust
returning to thermal equilibrium with the core (‘relaxing’). Since the outer layers are less
dense, they conduct heat faster and therefore cool quicker, which means the cooling curve
allows us to effectively see deeper into the neutron star with increasing time. The final
shape of the cooling curve is a result of the effectiveness of the heat transport along with the
location and strength of heating and cooling sources. Figure 4.27 demonstrates the effect of
reducing the neutron star crust conductivity by choosing an impurity parameter Qimp. = 30
instead of 5, as was chosen in the previous case. For this case the crust has not yet reached
thermal equilibrium with the core after 10,000 days of cooling and the knee in the cooling
210

curve around 1000 days is more pronounced, since the temperature gradients are in general
steeper (due to less efficient heat transport) just prior to accretion shutoff. Figure 4.28
demonstrates the affect of introducing a strong Urca cooling layer [54], 1037 ergs/sec at
0.5 GK, at a relatively shallow depth (a density of 2.94×1010 g/cm3 ) within the neutron star
crust. It is apparent that the Urca layer reduces heat transport from the surface to the inner
layers, ultimately increasing the temperature gradient before the Urca layer and decreasing
the temperature gradient after the Urca layer, with the net effect of more pronounced kneelike feature in the cooling curve at ∼ 100 days. Note the interesting feature (which is perhaps
a simulation artifact) that the shallow region of the crust actually becomes cooler than the
core at late times.

211

∞
kTeff.
[eV]

180
160
140
120
100
1

10

Time [days]

104

0 days
1 days
2 days
4 days
8 days
16 days
30 days
58 days
111 days
216 days
318 days
584 days

0.5

Temp. [GK]

103

102

0.4
0.3
0.2
0.1
1010

1011

1012 -3

Density [g cm ]

0 days
1 days
3 days
9 days
24 days
53 days
112 days
224 days
423 days
777 days
1518 days
2472 days
3246 days
4441 days
6382 days
10000 days

0.5

Temp. [GK]

1013

0.4
0.3
0.2
0.1
1010

1011

1012

-3

Density [g cm ]

1013

Figure 4.26: The upper panel shows the effective temperature of a neutron star as a function
of time, from the time an accretion outburst ends until the neutron star crust has returned
to thermal equilibrium with the core. The central panel shows thermal profiles at various
time snapshots throughout the duration of an accretion outburst, where the time associated
with a given contour increases in the vertical direction, since the crust is being heated out of
equilibrium. The discontinuity at low density for the contour just prior to accretion turning
on is due to the boundary condition set for the atmosphere temperature during accretion.
The small kink at ≈ 1010 g/cm3 is due to electron capture heating and the large bump at
∼ 1013 g/cm3 is due to deep crustal heating. The lower panel shows thermal profiles at
various time snapshots after an accretion outburst has ended, where the time associated
with a given contour decreases in the vertical direction, since the crust is cooling back into
equilibrium with the core. The symbols on the thermal profiles in the central and lower
panels indicate the thermal profile snapshot that belongs to a given point in time along the
cooling curve shown in the upper panel.
212

∞
kTeff.
[eV]

180
160
140
120
100
1

10

102

Time [days]

Temp. [GK]

0.5

104

0 days
1 days
3 days
6 days
12 days
23 days
45 days
88 days
174 days
347 days
584 days

0.4
0.3
0.2
0.1

1010

1012 -3 1013

1011

Density [g cm ]

0.5

Temp. [GK]

103

0 days
1 days
3 days
10 days
27 days
61 days
134 days
275 days
545 days
1028 days
1875 days
3750 days
7165 days
10000 days

0.4
0.3
0.2
0.1

1010

1012 -3 1013

1011

Density [g cm ]

Figure 4.27: Same as Figure 4.26, where an impurity parameter Qimp. = 30 was used instead
of Qimp. = 5. It is apparent that the increase in the impurity parameter drastically slows
heat diffusion.

213

∞
kTeff.
[eV]

180
160
140
120
100
1

10

102

Time [days]

104

0 days
1 days
2 days
4 days
8 days
16 days
30 days
58 days
111 days
213 days
308 days
579 days
584 days

0.5

Temp. [GK]

103

0.4
0.3
0.2
0.1
1010

1011

1012

-3

1013

Density [g cm ]
0 days
1 days
3 days
9 days
22 days
47 days
101 days
192 days
350 days
639 days
1226 days
2010 days
2666 days
3734 days
5409 days
8563 days
10000 days

Temp. [GK]

0.5
0.4
0.3
0.2
0.1
1010

1011

1012 -3

Density [g cm ]

1013

Figure 4.28: Same as Figure 4.26, where an Urca cooling layer [54] with a luminosity of
1037 erg/s (at 0.5 GK) has been included at a density of 2.94×1010 g/cm3 .

214

dStar calculations of the surface effective temperature can then be compared to observational data of quiescent transients. The input parameters have been varied so that the
calculation results approximate observational data (See Figure 4.29.). The goal was to get
a feel for the parameter space that plausibly reproduces quiescent transient light curves 34
in order to indirectly motivate the need for nuclear data to reduce this parameter space
by, for instance, using nuclear masses to consistently calculate the crust impurity profile
and the location and strength of crustal heating and cooling processes. The sources XTE
J1701-462 [55, 56] and EXO 0748-676 [59] are used as example cases 35 .
Figure 4.29 compares several cooling curves produced with dStar to observational data of
XTE J1701-462. The baseline simulation (solid black line) attempts to mimic the parameter
set employed for model A of [58] 36 . The parameters for the neutron star core were: 11 km
radius, 1.6 M⊙ 37 mass, 0.1 GK temperature. The accretion rate was 1018 g/s for an outburst
of 584 days (to match observations), where the atmospheric temperature during accretion
was fixed to 0.5 GK. The crust extended from pressures P of log(P [erg/cm3 ]) = 27 − 32.5
with an impurity parameter Qimp. = 5 (See Equation 4.8.), with 0.3 MeV per accreted
nucleon of heating (mimicking electron capture heating) extended over the pressure range
log(P [erg/cm3 ]) = 26.86 − 30.13 and 1.5 MeV per accreted nucleon of heating (mimicking
deep crustal heating) extended over the pressure range log(P [erg/cm3 ]) = 30.42−31.20. The
‘light element atmosphere’ [276] began at a column depth of 1010 g cm−2 (which corresponded
to a density of ∼ 1024 g/cm3 ). The neutron superfluid pairing gap (important for determin34 A

forthcoming publication by A.T. Deibel will systematically explore this parameter
space in a more quantitative manner. Detailed comparisons to observational data are also
available in [277], [58], and [257].
35 There are currently six known quasipersistent sources which have been monitored for
∼1000 days or more after accretion shut-off. Each are compared in Figure 5 of [278].
36 This parameter set was provided by E.F. Brown.
37 M = one solar mass≈ 2 × 1033 g.
⊙
215

240
mimic D.P.& S.R. PRL 2013
36
include 10 erg/sec Urca
37
include 10 erg/sec Urca
change to Q =30
imp.
change to Q =30, include 3×1036erg/sec Urca
imp.
XTEJ1701, Observation, J.F. ApJ 2011

220

∞

Teff. [eV]

200
180
160

?

140

(?)

120
100
80
1

10

102

103

Time since accretion outburst [days]

104

Figure 4.29: Effective temperature as a function of time of the neutron star XTE J1701-462 as
observed [55, 56] after accretion turned off (at day 0) as compared to dStar [57] simulations.
Note that the observational points labeled ‘?’ and ‘(?)’ have been called into question
due to suspected brief accretion outbursts [58]. The baseline simulation (solid black line)
mimicked the parameters employed by [58], the short-dashed black line simulation added a
1036 erg/s (for 0.5 GK) Urca cooling layer to the baseline simulation, the long-dashed black
line simulation added a 1037 erg/s (for 0.5 GK) Urca cooling layer to the baseline simulation,
the black dot-dash line changed the impurity parameter from the baseline simulation to 30,
and the dotted red line changed the impurity parameter to 30 and included a 3 × 1036 erg/s
(for 0.5 GK) Urca cooling layer. More details on input parameters are provided in the text.

216

ing the conductivity of the deep inner crust) was the model given by [279]. For simulations
which include an Urca cooling layer, this was placed at a pressure of log(P [erg/cm3 ]) = 28.5
using cooling strengths given by Table 1 of [54] as a benchmark. Table 1 of [54] lists neutrino luminosities for several Urca pairs for a temperature of 0.5 GK, with cooling strengths
ranging between (0.46 − 24) × 1036 ergs/s for the Urca pairs identified in that study. Figure 3 of [54] predicts neutrino luminosities of ∼ 1035 − 3 × 1036 ergs/s for Urca cooling from
electron captures occurring on typical accreted neutron star ashes.
In Figure 4.29 it is apparent that the baseline simulation reproduces the data relatively
well, though it possibly undershoots the last observational point. It is seen that adding Urca
cooling with a strength of 1036 erg/s (for 0.5 GK) has very little impact on the resultant
light curve 38 . For demonstrative purposes, the Urca cooling was turned up to 1037 erg/s
(for 0.5 GK), where a clear impact on the light curve was shown, though it clearly disagrees
with the observational data (given the initial set of assumed parameters). For the case
where Urca cooling was omitted but the impurity parameter was changed to 30 and the core
temperature was raised to 0.13 GK, there is significant extension of the cooling curve, which
seems to overshoot the data somewhat. Visually, the best reproduction of the data is given
by the red-dotted case which employs Qimp. = 30, Tcore = 0.13 GK, and 3 × 1036 erg/s (for
0.5 GK) Urca cooling. The qualitative conclusion to be drawn here is that a large impurity
parameter only plausibly reproduces the XTE J1701-462 data if the extended light curve is
compensated by a reasonably strong Urca cooling layer. Additional data points extending up
to a few thousand days should be sufficient to distinguish whether this case or the baseline
best reproduces that observational data.
38 Reducing shallow heating by half produced nearly identical results,

from the plot for clarity.

217

so this case is omitted

As an aside, it should be noted that the XTE J1701-462 data shown was obtained from
observations starting in the middle of 2007 and extending through late 2010 and it appears [280] no accretion outburst has happened since. Therefore, it is possible in principle
to extend the quiescent cooling data out to ∼ 2600 days if a follow-up observation were
made in the near future (so long as accretion doesn’t turn back on in this time). This would
be sufficient to distinguish between the high and low impurity cases. In addition, prior to
accretion turn-off, three type-I X-ray bursts were observed for XTE J1701-462, two of which
were photospheric radius expansion bursts [281] 39 , where the burst rise-time was used to
infer that the bursts were powered by helium burning. In principle, one could attempt to
reproduce these X-ray burst light curves with simulations (as done with GS 1826-24 by [266])
in order to get an estimate for the surface ash composition 40 . The ash composition could
then be evolved with a neutron star crust nuclear reaction network (e.g. as employed in [60]
and [54]) to obtain the crust composition profile along with the location and strength of
heating and cooling sources. Though tedious, this exercise would reduce the phase space
available to reproduce the quiescent cooling curve for XTE J1701-462.
Figure 4.30 compares several cooling curves produced with dStar to observational data of
EXO 0748-676. Aside from the core temperature, atmosphere temperature during accretion,
impurity parameter, and Urca cooling strength, the parameters chosen were the same as
for the XTE J1701-462 simulations, with the exception of the simulation represented by the
thick-green line for which shallow crustal heating was reduced by 1/2 from the default value 41 .
39 Recall

that these bursts can be used to obtain rough neutron star mass and radius
information [244].
40 Note that the crust is composed of matter which was accreted over the past hundreds of
years, so the ashes from the bursts observed prior to accretion shutoff would not have been
buried to the outer crust depth yet.
41 Note that little physical motivation existed for choosing to use most of the same parameters as for XTE J1701-462. For instance, the accretion outburst on EXO 0748-676 was
218

Tcore=0.12GK, Qimp.=5, No Urca
36
Tcore=0.12GK, Qimp.=5 ,10 erg/sec Urca
Tcore=0.12GK, Q =5 , 1037erg/sec Urca
imp.
Tcore=0.10GK, Qimp.
=40, No Urca
Tcore=0.10GK, Q =40, 5×1036erg/sec Urca
Tcore=0.12GK, Qimp.
=5, No Urca, Halve Shallow Heating
imp.
EXO0748, Observation, N.D. ApJ 2014
EXO0748, 1980 Einstein data

180

∞
Teff.
[eV]

160
140
120
100
80
1

10

102

103

104

Time since accretion outburst [days]

Figure 4.30: Effective temperature as a function of time of the neutron star EXO 0748-676
as observed [59] after accretion turned off (at day 0) as compared to dStar [57] simulations.
The only parameters varied between the simulations shown were the core temperater Tcore ,
the crust impurity parameter Qimp. , and the Urca cooling strength, with the exception of the
simulation corresponding to the thick green line whose shallow crustal heating was reduced
to 0.15 MeV/u (from the default value of 0.3 MeV/u). Note that the blue diagonallyhashed band is the inferred core temperature from observations of EXO0748-676 after a long
quiescent phase where it is thought the crust reached thermal equilibrium with the core [59].
More details on input parameters are provided in the text.

219

It is apparent that the parameter variations have a much smaller impact on the overall cooling
curve, which is due to the smaller initial temperature gradient between the atmosphere and
the core. As before, the addition of Urca cooling appears to fit the data well, so long as
the impurity parameter is increased to higher values. For a fixed impurity parameter, the
results for including strong Urca cooling and reducing shallow heating have a similar impact
on the cooling curve. In this case it is unclear if observations at later times would allow the
different cases to be distinguished, since the variation in the predictions is roughly within
observational uncertainties. Note that X-ray bursts for EXO 0768-676 were observed just
prior to its quiescent phase [282], so in principle the same exercise described previously for
XTE J1701-426 (reproducing burst observations with simulations and combining this with
crust composition evolution simulations to obtain Qimp. , heating, and cooling versus depth)
could be performed. However, as was noted previously, the crust is composed of matter
which was accreted over the past hundreds of years, so the ashes from the bursts observed
prior to accretion shutoff would not have been buried to the outer crust depth yet.
These two cases have served to illustrate the presently existing degeneracy between sets of
suitable crustal cooling simulation parameters. Rather than remove the degeneracy, nuclear
data could serve to limit the available parameter space so that more information could
be gleaned from observations. It has been demonstrated that nuclear masses are crucial to
determine the location and strength of crustal heating and cooling [54, 60, 78]. The following
section will discuss the role nuclear masses play as input to simulations which evolve the
crustal composition of accreted neutron star ashes as they are buried by accretion.
observed to last for 24 years, as opposed to 584 days.

220

4.3.2

Heating and cooling in the neutron star crust associated
with electron captures

Note that this subsection (4.3.2) is a modified excerpt from the upcoming publications [283,
284]. The author of this dissertation is the primary author of those works, however coauthors 42 contributed to the writing of [283] and will contribute to the writing of [284].
As briefly mentioned in the previous section, the ashes from thermonuclear burning occurring on the surface of accreted neutron stars are buried as further matter is accreted
and forced into lower depths of the neutron star. As the depth increases, the density increases, therefore the electron number density increases, and therefore the electron Fermi
energy EFermi rises 43 . Eventually the electron chemical potential µe = EFermi reaches
42 The

full list of coauthors for [283] and [284] is: Zach Meisel, Sebastian George,
Sunghoon (Tony) Ahn, Justin Browne, Daniel Bazin, B. Alex Brown, J. Fiore Carpino,
Hank Chung, Richard H. Cyburt, Alfredo Estrad´e, Mike Famiano, Alexandra Gade,
Christoph Langer, Milan Matoˇs, Wolfi Mittig, Fernando Montes, Dave J. Morrissey,
Jorge Pereira, Hendrik Schatz, Jule Schatz, Mike Scott, Dan Shapira, Karl Smith,
Jeremy Stevens, Wanpeng Tan, Oleg Tarasov, Sarah Towers, Kathrin Wimmer,
Jack R. Winkelbauer, John Yurkon, and Remco G.T. Zegers. See [36] for their affiliations
at the time of these works.
43 This is roughly demonstrated for a degenerate electron gas by considering the electron as
a particle in a box. The energy of a relativistic electron (which we can assume is relativistic
due to its small mass) is E = pc. From the uncertainty principle, we know ∆x∆p ≥ /2,
where ∆p is the characteristic momentum spread (uncertainty) and ∆x is the characteristic
position spread (uncertainty). By considering the momentum spread to be on the order of
the momentum, p ∝ ∆p, p ∼ /∆x, the electron energy is given by E ∼ c/∆x. Since
the environment under consideration is electron degenerate, the electrons will fill orbitals
corresponding to quantum numbers l, where the most energetic electrons (the ones at the
Fermi energy EF ) will occupy the outermost orbital whose corresponding quantum number
is lF . The total number of electrons N is given by the sum of the total number of electrons
3 , where the factor of 2 accounts
which can occupy each of the orbitals, N = 2 ∗ 1/8 ∗ 4π/3 ∗ lF
for the fact that electrons with opposite spin can occupy the same location, the factor
of 1/8 accounts for the fact that the quantum number corresponding to the orbital radius
lF is represented by orbitals in the x, y, and z dimensions whose quantum numbers can
only be positive, and 4π/3 is the typical prefactor for the volume of a sphere. Therefore
the most energetic electrons have quantum number lF = (3N/π)1/3 . A rough estimate for
the uncertainty of the electron position is given by the radius of the space in which the
221

the point where it is equal to the electron capture Q-value for a nucleus at a given depth,
µe ≈ QEC (Z, A) = ME(Z, A) − ME(Z − 1, A), where ME(Z, A) is the mass excess of a nucleus with Z protons and A − Z neutrons. At this point it is energetically favorable for an
electron to be forced into the nucleus, combining with a proton to become a neutron and a
neutrino, transmuting that nucleus to the more neutron-rich isobar with Z − 1 protons and
A nucleons. [60] show that this process often results in heat deposition within the neutron
star crust.
A rough schematic demonstrating the origin of electron capture heating is shown in
Figure 4.31. In odd-A mass chains, electron captures occur one at a time at the point
where µe ≈ QEC (Z, A). In some cases selection rules will hinder an electron capture from
ground-state to ground-state, since large changes in spin from initial to final states are not
favorable, so the chemical potential may need to rise to an energy larger than the electron
capture Q-value. In these cases electron capture into an excited state occurs and the energy
released in de-excitation is deposited into the crust. Due to the large number of low-lying
states for odd-A nuclei, the heating from these electron capture chains would in general
be modest. The even-A mass chains, however, can deposit significantly more energy into
the neutron star crust since electron capture into an excited state is often energetically
possible. The binding energy penalty for not having and bonus for having paired nucleons
(represented by the 5th term of Equation 4.1) causes the so-called odd-even mass stagger,
electrons could occupy a divided by the minimum number of orbitals which the electrons
fill lF , ∆x ∼ a/lF . Employing this relation for ∆x in our prior relation for EF results in
EF ∼ ( ∗ c ∗ lF )/a ∼ ( cN 1/3 )/a. Since the volume of the space in which the electrons are
located scales as the radius cubed, a ∝ V 1/3 , EF ∼ cN 1/3/V 1/3 . Recognizing the number
1/3

density of electrons ne = N/V , we finally arrive at the conclusion EF ∼ ( c)ne . Noting
that the chemical potential is equal to the Fermi energy µe = EF , our relation between the
Fermi energy and the electron number density is consistent with the more carefully calculated
result presented in Equation 2.4 of [269].
222

(a) A odd

(b) A even
3
Exs+Egs

2

Energy

μe

μe

4

Egs(Z-2,A)

Energy

EC

1
μe

EC
Egs(Z-1,A)

Egs(Z,A)

EC

Exs+Egs

3

1
μe

2

Egs(Z-2,A)

EC
Egs(Z-1,A)

Egs(Zeven,A)

Figure 4.31: Schematic demonstrating the origin of electron capture heating for odd-A (left
panel) and even-A isobaric chains, where the vertical direction indicates the energy of the
system, which rises with depth due to the increasing electron chemical potential µe . Each
panel considers the system energy at the depth where µe = QEC (Z, A) = ME(Z, A) −
ME(Z − 1, A). In the odd-A case, QEC (Z, A) < QEC (Z − 1, A), so once the initial electron
capture happens (1), nothing further happens until the nucleus sinks to the depth where
µe = QEC (Z − 1, A). At this point, it is possible that electron capture to the ground state
is strongly hindered, as it could have been for the first electron capture in the sequence (for
instance, due to a large difference in spin between the initial and final state), so electron
capture may not ensue until it is energetically possible to electron capture into an excited
state of the nucleus with Z −2 protons (2). Electron capture into the excited state (3) will be
followed by de-excitation (4) which deposits heat into the crust. For the even-A case, heating
can be more significant due to the odd-even mass stagger that results from the pairing force
(the 5th term of Equation 4.1). In this case, after electron capture onto an even-even nucleus
(1), it is immediately energetically favorable to electron capture into the even-even nucleus
with Z − 2 protons, which results in heat deposition into the outer crust. Often QEC (Z, A)
is greater than QEC (Z − 1, A) by several MeV, so electron capture into a high-lying excited
state is possible (2). The subsequent de-excitation (3) enhances the energy deposition into
the neutron star crust, often by several MeV [60].

223

whereby binding energy differences do not monotonically increase from the stable isobar by
ever-larger amounts but increase in alternating large and small steps. This stagger results in
the case QEC (Zeven , Aeven ) > QEC (Zodd , Aeven ), so electron captures proceed in a two-step
process. Ground-state capture of electrons with Ee < EFermi deposits EFermi − Ee worth of
heat into the crust. The second electron capture in the pair can often proceed through an
excited state, which enhances the amount of energy deposited into the crust via de-excitation
of the electron capture daughter. Nuclear reaction network calculations of rp-process ashes
sinking into the neutron star crust demonstrate that even-A mass chains in general produce
more heat in the neutron star crust than odd-A mass chains [60]. The largest amount of heat
is released in regions of the nuclear chart near magic neutron and proton numbers, where
shell effects can enhance the relative binding energy differences.
From Figure 4.31 it is clear that nuclear masses play a fundamental role in determining
at which depth (µe ) electron captures ensue and the quantity of heat released 44 . To this end
the previous NSCL time-of-flight mass measurement determined the mass of 66 Mn for the
first time and found an unexpectedly small QEC for 66 Fe. This resulted in a shallower depth
for this heating source than was previously assumed; a significant fact considering that a
considerable amount of A = 66 material is produced from X-ray bursts, X-ray superbursts,
and stable burning during accretion [64, 243, 285, 286]. Motivated by this study, we explore
the impact of the mass measurements presented in this dissertation on electron capture
heating in the accreted neutron star crust.
ME(64 Cr) and electron capture heating
Note that this subsection is a modified excerpt from the upcoming publication [284].
44 Note

that the energies and spin-parity of excited states are also critical inputs, since
these determine the rates of energetically allowed electron capture transitions.
224

The author of this dissertation is the primary author of that work, however coauthors 45 will
contribute to its writing.
The newly obtained mass of 64 Cr presented in this dissertation is of special interest with
regards to electron capture heating in the outer neutron star crust. Given the waiting-point
nature of 64 Ge in the rp-process 46 , it is expected that a large amount of A = 64 material is
present on the surface of accreted neutron stars [64, 243, 285, 286]. Additionally, we note that
the 64 Cr mass excess obtained from our measurement differs from the values given by the
HFB-21 and1 1995 FRDM global mass models by ∼ 3σ (See Table 4.1.). Therefore neutron
star crust nuclear reaction network calculations (employed in [54, 60, 78]) were performed to
see the resultant change in crustal heating in the A = 64 mass chain, where the results are
shown in Figure 4.32. Here it is seen that the reduction in binding of 64 Cr 47 as compared
to global mass models reduces the shallow heating due to the 64 Fe →64 Mn →64 Cr and
64 Cr

→64 V →64 Ti electron capture sequences, while also shifting the latter electron capture

sequence to significantly shallower depths. Note that the mass of 64 V must be determined
experimentally to finally determine the depth of the 64 Cr →64 V transition.

45 The

full list of coauthors for [284] is:
Zach Meisel, Sebastian George,
Sunghoon (Tony) Ahn, Justin Browne, Daniel Bazin, B. Alex Brown, J. Fiore Carpino,
Hank Chung, Richard H. Cyburt, Alfredo Estrad´e, Mike Famiano, Alexandra Gade,
Christoph Langer, Milan Matoˇs, Wolfi Mittig, Fernando Montes, Dave J. Morrissey,
Jorge Pereira, Hendrik Schatz, Jule Schatz, Mike Scott, Dan Shapira, Karl Smith,
Jeremy Stevens, Wanpeng Tan, Oleg Tarasov, Sarah Towers, Kathrin Wimmer,
Jack R. Winkelbauer, John Yurkon, and Remco G.T. Zegers. See [36] for their affiliations
at the time of this work.
46 [287] claim their mass measurement of 65 As establishes that 64 Ge is not an rp-process
waiting-point, but X-ray burst calculations demonstrate that their measurement precision
does not merit this claim [288].
47 This is likely due to the models taking the intrusion of the g
9/2 orbital into account
differently, as discussed in Section 4.2.2.
225

Integrated Heating [MeV/u]

0.25
FRDM'95+Lit.
FRDM'95+Lit+This expt.
HFB-21+Lit.
HFB-21+Lit+This expt.

0.2
64

0.15

Fe→64Mn→64Cr
64

0.1
100

150

Cr→64V→64Ti
200

Depth from NS surface [m]
Figure 4.32: Integrated heat release (in units of MeV per accreted nucleon u) from electron
capture for an A = 64 mass element as it is buried to deeper depths of the neutron star crust
by accretion. Note that the time required for a surface fluid element to reach these depths
˙ 26, 400 g/cms2 (1/3 the
depends on the accretion rate; for our chosen accretion rate of M=
Eddington accretion rate), the time to sink from the surface to the depths shown is on the
order of centuries. The calculations corresponding to the black and red lines employed the
1995 FRDM [2] and HFB-21 [3] global mass models for nuclei with unknown masses, where
the 2012 Atomic Mass Evaluation [1] was used otherwise. Calculations indicated by solid
lines included the mass of 64 Cr presented in this dissertation. The first two heating events
are labeled by their corresponding electron capture sequence.

226

ME(56 Sc) and Urca cooling
Note that this subsection is a modified excerpt from the upcoming publication [283].
The author of this dissertation is the primary author of that work, however coauthors 48
contributed it its writing.
It is important to note that to this point we have ignored the possibility of β − -decay
occurring after electron capture, due to the electron degenerate environment. Recently it was
shown that the finite temperature of the neutron star crust opens up considerable phase space
for β-decay to occur, leading to the presence of so called Urca cycles at the compositional
boundaries where electron captures occur [54]. This is the dominant cooling mechanism in
the outer crust when the temperature exceeds 108 K 49 .
At finite temperature, the compositional transition between one nucleus to its nextmost neutron-rich isobar takes place over a thin-shell of finite thickness (∼ 1 m). Within
this shell, both electron-capture parent and daughter nuclei are present, and both electron
capture and its inverse, β − -decay, can occur (Demonstrated pictorially in Figure 4.37.).
Alternating electron captures and β − -decay between these nuclei can then lead to a rapid
neutrino emission known as Urca cooling [290]. Whether such an Urca process is efficient
at compositional boundaries depends strongly on nuclear masses and electron capture/β − decay transition energies and strengths [54, 60, 78].
48 The

full list of coauthors for [283] is:
Zach Meisel, Sebastian George,
Sunghoon (Tony) Ahn, Justin Browne, Daniel Bazin, B. Alex Brown, J. Fiore Carpino,
Hank Chung, Richard H. Cyburt, Alfredo Estrad´e, Mike Famiano, Alexandra Gade,
Christoph Langer, Milan Matoˇs, Wolfi Mittig, Fernando Montes, Dave J. Morrissey,
Jorge Pereira, Hendrik Schatz, Jule Schatz, Mike Scott, Dan Shapira, Karl Smith,
Jeremy Stevens, Wanpeng Tan, Oleg Tarasov, Sarah Towers, Kathrin Wimmer,
Jack R. Winkelbauer, John Yurkon, and Remco G.T. Zegers. See [36] for their affiliations
at the time of this work.
49 Note that this process is highly temperature dependent, where the cooling luminosity
Lν,Urca ∝ T 5 [54, 289].
227

The most important Urca shell cooling identified in [54] is associated with the transition
from 56 Ti to 56 Sc and is predicted to occur at a depth of 5.6×1010 g/cm3 (≈ 160 m from the
surface). This transition is of particular importance because it is the only one associated with
cooling in the A = 56 mass chain, and is one of the few cases where Urca shell cooling was
predicted to occur for an even mass number A. Even-A nuclei, and in particular A = 56 nuclei
tend to be produced in large quantities by X-ray bursts and superbursts, and are therefore
predicted to be present in the outer crust in large quantities [243, 285, 291]. However, the
existence of the 56 Ti–56 Sc Urca pair (and therefore A = 56 Urca cooling) depends strongly
on the unknown masses of 56 Sc and 56 Ca. In [54], cooling only occurs when employing the
HFB-21 [3] global mass model and not when employing the 1995 FRDM [2] global mass
model.
While the HFB-21 and FRDM predictions for the atomic mass excess of 56 Ca ME(56 Ca)
agree within ≈ 500 keV, there is a nearly 2 MeV discrepancy for the case of ME(56 Sc) [2, 3].
When using the FRDM mass for 56 Sc, the odd-even staggering of electron capture Q-values,
∆QEC (56 Sc) = QEC (56 Ti) − QEC (56 Sc), is very strong. This leads to a low threshold for
electron capture on 56 Sc, which results in a fast electron capture that removes nuclei from the
56 Ti–56 Sc

Urca cycle. The HFB-21 mass model predicts a much weaker odd-even staggering

of QEC , resulting in a higher threshold for electron capture on 56 Sc that suppresses this
reaction, thereby confining material to the 56 Ti–56 Sc Urca cycle. The mass determined for
56 Sc

in the experiment presented in this dissertation was used to compute ∆QEC (56 Sc) to

resolve the question of whether or not 56 Ti–56 Sc could be an Urca pair in the accreted
neutron star crust.
As was described previously in this dissertation, the mass of 56 Sc was determined experimentally for the first time by a relative measure of time-of-flight (See Figure 4.33.). Aside
228

Counts/10ps

105

44

Ar

47

45

65
K Ar 48K 46
Mn 47Ar
53
Ar49
52 50Ca
Sc
4 49Ca
Sc
51 54 K
50
10
K
63 66 Ca Sc 55
Mn Fe
52
Sc
Ca
3

10

64

66

Mn

56

Fe

53

Sc

Ca

2

10

51

K
54

Ca

10
1

480

490

500

510

TOF[ns]
Figure 4.33: Rigidity-corrected time-of-flight distributions for reference nuclei (unfilled hism
tograms) used to calibrate the rest
q (T OF ) relationship to obtain masses from TOFs of
52−56 Sc (red-filled histograms).

229

from the standard set of mass uncertainties that were assigned to the measured masses (See
Section 3.9.4.), an additional uncertainty for ME(56 Sc) originates from the presence of an
experimentally known β-decaying isomeric state [207, 208] that may be populated in the
fragmentation reaction producing 56 Sc. 56 Sc has a β-decaying low-spin (1+ ) state and a βdecaying high-spin (5+ or 6+ ) state, but it is not known which is the ground state and which
is the isomeric state. Shell-model calculations predict an excitation energy of the isomer of
540 keV. The resolution of the 56 Sc TOF peak is 100 ps, corresponding to a mass resolution
of 10 MeV, and can therefore not be used to constrain the relative population of the ground
and isomeric states. Thus, the atomic mass excess obtained in this work represents a leastbound limit for the 56 Sc ground state and we add an asymmetric uncertainty of +0
−540 keV to
our result to account for the unknown population ratio. The resulting atomic mass excess of
56 Sc

determined in this work is −24.85(59)(+0
−54 ) MeV. As seen in Table 4.1, the atomic mass

excess of 56 Sc presented here is consistent with the prediction from the HFB-21 [3] global
mass model, but is more bound than the prediction from the FRDM [2] global mass model.
With our result for the 56 Sc mass, QEC (56 Ti) =ME(56 Ti)−ME(56 Sc)= −14.4(+1.3
−0.7 ) MeV
is now determined exclusively from experimental data. For QEC (56 Sc) = ME(56 Sc) −
ME(56 Ca) we combine our new 56 Sc mass with the 56 Ca mass predicted by either the FRDM
+0.6 ) MeV and −12.3(+0.6 ) MeV,
or the HFB-21 mass models and find similar values of −12.0(−1.1
−1.1

respectively (uncertainty is experimental only). For the two choices of 56 Ca mass, this re+0.7 ) MeV and
sults in a QEC staggering of ∆QEC (56 Sc) = QEC (56 Sc) − QEC (56 Ti)=2.3(−1.3

2.1(+0.7
−1.3 ) MeV, respectively. Fig. 4.34 shows evolution of ∆QEC in the A = 56 mass chain
for odd-Z nuclei as a function of Z, where we have included both of the aforementioned
∆QEC (56 Sc) in an attempt to capture the contribution of the theoretical mass uncertainty
of 56 Ca. Clearly our data neither support the strong decrease towards the neutron drip
230

∆QEC [MeV]

8
6

FRDM'95

UNEDF0

HFB-21
AME'12

UNEDF1
56
This expt. with ME( Ca)

20

22

Global

4
2
0

19

21

Z

23

24

25

Figure 4.34: ∆QEC (Z, A) for odd-odd A = 56 nuclei using ME(56 Sc) from this experiment
and ME(56 Ca) from FRDM’95 or HFB-21 (black stars), compared to global mass models [2,
3] and mass-differences predicted from recent energy density functional calculations [61, 62]
(open shapes). A low ∆QEC is essential for the existence of an Urca cooling pair in an
even-A mass chain.
line (lower Z) predicted by HFB-21, nor the strong increase predicted by FRDM. Rather,
our data confirm the predictions of recent energy density functional based binding energy
calculations [61, 62] of a fairly constant ∆QEC along A = 56.
The absolute value of ∆QEC (56 Sc) obtained here agrees with the HFB-21 value used
in [54] that led to strong Urca cooling for A = 56 and excludes the FRDM value that
led to a suppression of Urca cooling. This can be understood by examining the energy
thresholds for electron captures, shown graphically in Fig. 4.36. The 56 Ti(e− ,νe )56 Sc reaction is predicted by the QRPA model used in [54] to proceed through a 10 keV 3+
excited state (consistent in energy with the ground state) of 56 Sc 50 . Our experimental
50 It

should be noted that this 3+ state is not presently seen in experimental data, but
231

ocean
(~100m)

atmosphere (~10m)

ν

ν

ν

ν

ν

ν

ν

ν

ν ν

outer crust
(~0.5km)

inner crust
(~0.5km)

core
(~9km)

(c)

ν

ν

ν

Carbon ignition layer
Urca cooling layer
Deep crustal heating

Figure 4.35: (a) Net energy released from (positive values) or deposited into (negative values)
the neutron star crust as a function of depth for a fluid element made of A = 56 material as
it sinks into the neutron star crust. Lines from the energy curves to the reaction sequence indicate which heating or cooling event corresponds to which nuclear reaction when employing
the HFB-21 [3] mass model for the 56 Sc and 56 Ca masses. Differences between the amount of
heating or cooling and the depth at which it occurs are due to the different value of ME(56 Sc)
used in the calculation. ME(56 Sc) found in this experiment results in Urca cooling for the
A = 56 mass-chain, as given by the HFB-21 mass model but not the FRDM’95 [2] mass
model. Note that the 56 V and 56 Ti masses used are from [1]. (b) Integrated energy per
accreted nucleon released from (negative values, blue-shading) or deposited into (positive
values, red-shading) the neutron star crust for an A = 56 fluid element as a function of
depth. It is apparent that cooling from one pair of nuclei overwhelms heating from electron
captures. (c) Schematic of an accreted neutron star cross section. The Urca cooling layer
separates heat generated deep in the crust by nuclear processes from the shallow layer where
carbon is thought to be ignited for superbursts.

232

System Energy [MeV]

EC

-10
-20

EC
ME, AME 2012
ME, HFB-21
ME, FRDM'95
QRPA Exs(1st allowed G.T.)
56
QEC( Ti→56Sc), FRDM'95
56
QEC( Ti→56Sc), HFB-21

-30
-40
56

Ti

56

Sc

56

Ca

Figure 4.36: Energy level diagram for the A = 56 mass-chain at a depth where EF ≈
|QEC (56 Ti)|. The large low-lying Gamow-Teller (GT) transition strength for 56 Sc allows
56 Ti(e− , ν )56 Sc to occur for this condition. Whether or not electron capture EC onto 56 Sc
e
directly follows depends on the choice of mass model. EC on 56 Sc into the lowest energy
level Exs in 56 Ca that can be entered via a GT transition occurs at EF ≈ |QEC (56 Ti)| for
the FRDM mass model [2], but not for the HFB-21 mass model [3]. In the latter case, Urca
cooling occurs for the 56 Ti–56 Sc pair.
+1.3 ) MeV indicates therefore, that the transition occurs at a depth where
QEC (56 Ti)=−14.4(−0.7
+0.7 ) MeV. This is a shallower depth with respect to the calculation
EF is slightly below 14.4(−1.3

that employs the FRDM mass model mass model for ME(56 Sc). What happens next with
the resulting 56 Sc nucleus depends on the threshold for the 56 Sc(e− ,νe )56 Ca reaction. Our
+0.6 ) MeV or −12.3(+0.6 ) MeV indicate that the transition to the ground
QEC (56 Sc)= −12.0(−1.1
−1.1
+0.7 ) MeV, with
state of 56 Ca would be energetically possible for electrons with EF ≡ 14.4(−1.3
+0.7
an excess energy (with respect to the 56 Ca ground state) of 2.3(+0.7
−1.3 ) MeV or 2.1(−1.3 ) MeV

the only 56 Sc level scheme information that is available is from the isomeric γ-decay of its
775 keV excited state, where all spin-parity assignments were based on the assumption that
the ground state was J π = 1+ .
233

that corresponds to ∆QEC . However, the transition is predicted to go to an excited state
in 56 Ca at 3.4 MeV [292], which is not energetically possible. As a consequence, the only
destruction mechanism for 56 Sc at EF ≈ 14.4 MeV is β − decay and a strong 56 Ti–56 Sc Urca
cycle ensues at this depth, 6.0×1010 g/cm3 . The resulting strength of the cooling or heating as a function of depth in the neutron star crust model of [54] when employing different
masses for 56 Sc is shown in Fig. 4.35. For context, Fig. 4.35 shows the location of the Urca
cooling layer with respect to the proposed locations of the carbon ignition layer that powers
superbursts [249, 268] and the general region of deep crustal heating [272, 277].
We note that shell-model Gamow-Teller strength distributions calculated with the GXPF1A
effective interaction predict a strong transition from the 56 Sc ground state to the 56 Ca ground
state. However, this conclusion assumes a spin-parity J π =1+ ground state for 56 Sc while,
as noted previously, experiment has yet to determine whether the low-spin or high-spin
long-lived state in 56 Sc is the ground state [207, 208].
Therefore, the mass measurement of 56 Sc presented in this dissertation removes the mass
model ambiguity in the conclusions of [54] concerning the possible existence of a 56 Ti–56 Sc
Urca neutrino cooling layer in hot accreted neutron star crusts. With our new 56 Sc mass,
strong Urca neutrino cooling occurs at the 56 Ti–56 Sc boundary within the model framework
of [54], regardless of which mass model is used for the remaining unknown masses. A strong
56 Ti–56 Sc Urca neutrino cooling layer makes shallow neutrino cooling in the crust of accreting

neutron stars a strong and robust effect due to the copious production of A = 56 material
in superbursts and other thermonuclear burning processes at the neutron star surface. In
addition, with the new experimental mass of 56 Sc, the depth of the 56 Ti electron capture
transition, and the associated location of a 56 Ti–56 Sc Urca neutrino cooling layer, can now be
determined on the basis of experimental data. The remaining nuclear uncertainties related
234

to a 56 Ti–56 Sc Urca cooling layer in accreting neutron stars are the 56 Ca mass excess, the
excitation energy of the final state populated in the 56 Sc(e− , νe )56 Ca reaction, and, most
importantly, the spin-parity of the 56 Sc ground state.
However, an important consideration regarding even-A Urca pairs is the possible thermal
population of excited states, which is not taken into account in the model of [54]. While
the finite temperatures open-up phase space for β-decay, as quantified by the Saha equation
they should also lead to the thermal population of excited states. These excited states could
have a much stronger electron capture transition strength, thereby rapidly draining nuclei
from the Urca cycle. In fact, as shown in Figure 4.37, it is virtually a requirement that there
be a low-lying excited state in the electron-capture daughter of the Urca pair that has a
significant electron capture strength (Since the electron capture transition in the figure from
π is strong, it is presumed that the transition from J π → 0+ will be strong as
0+ → Jlow
low

well.). Thus, whether or not Urca neutrino cooling is present for an even-A pair not only
depends on ∆QEC as discussed previously, but also on the competition between Urca cycling
and draining of nuclei from the cycle via electron capture on a thermally populated excited
state in the odd-odd nucleus.
The criterion for a strong even-A Urca pair then has an inherent contradiction. A low
excited-state energy (relative to kT , where T is the environment temperature and k is Boltzmann’s constant), in the odd-odd nucleus of the Urca pair is preferred to open-up sufficient
phase space to increase the rate of β − -decay. However, a high excited-state energy of the
low-J π state (through which electron capture into the odd-odd nucleus proceeds) is preferred
so that this state will not be significantly thermally populated so that it can undergo electron
capture and drain the Urca cycle. Here a rough attempt is made to find the possible excited
state energy window in which these two criteria could be balanced, while neglecting time-scale
235

Energy

μe

μe

EC

EC

Exs + Egs

π

J
π
J
Egs(Z-1,A)

EGT
xs + Egs

π

(±0,1+J )
high

+
0
Egs(Z-2,A)

low

high

β-

0+

Egs(Zeven,Aeven)

Figure 4.37: Schematic demonstrating the origin of Urca neutrino cooling in even-A isobaric
chains, where the vertical direction indicates the energy of the system, which rises with depth
due to the increasing electron chemical potential µe . The figure considers the system energy
at the depth where µe = QEC (Z, A) = ME(Z, A) − ME(Z − 1, A), where Z and A are even.
In order to maintain an Urca cycle, electron capture from the Z to Z − 1 isobar must be
relatively strong while the electron capture from the Z − 1 isobar to the Z − 2 isobar must be
hindered. Since even-even nuclei have a ground state spin-parity J π = 0+ , this condition can
only be achieved if the first allowed Gamow-Teller (GT) transition into the Z − 1 isobar is
at a relatively low-lying excited state energy, while the first allowed Gamow-Teller transition
into the Z − 2 isobar is at a relatively high-lying excited state energy such that it cannot
be immediately captured into following the transition from (Z, A) → (Z − 1, A) [54]. If the
π is significantly thermally populated, nuclei will
(Z − 1, A) excited state with spin-parity Jlow
be drained from the Urca cycle by transition from this state to the (Z − 2, A) 0+ ground
state if there is a strong GT transition.

236

considerations (discussed later on). Figure 4.38 shows what is coined here as the ‘Even-A
Urca window’, which is just the product of the relative probability of the (Z −1, A) → (Z, A)
β − -decay and the relative thermal population of the ground state of (Z − 1, A) as a function of the excited state energy through which the (Z, A) → (Z − 1, A) electron capture
proceeds, where Z and A are even. The relative probability of β − -decay used was the dimensionless portion of the rate for β − -decay in an electron-degenerate environment given by
Equation 14 of [289] 51 . The relative population of the ground state is obtained by transforming the Saha equation (which ultimately results in the Fermi distribution). It is apparent
that their product peaks at higher excited state energies for higher temperatures. Note that
this simple model does not take into account the time-scales involved in the Urca cycling
and the electron capture which drains the Urca cycle. Since the finite temperature continually replenishes the thermal population of the excited state, Urca cycling which is much
slower than the electron capture onto the thermally populated excited state would be rather
ineffective. In fact, for the case of 56 Ti–56 Sc, where QEC (56 Ti) = 14.4 MeV at 0.5 GK, it
appears the actual window found in a full reaction network peaks around Exs ≈ 400 keV,
with a substantial reduction in the overall cooling [293].
As the prior discussion has demonstrated, nuclear masses play a prominent role in the
determination of the heating and cooling associated with electron captures in the outer
neutron star crust. However, as was also demonstrated, precise knowledge of the spin-parity
and energy of excited states of nuclei involved in the electron capture process is required
to accurately calculate the strength and location of heating and cooling processes within
the neutron star crust. Though theoretical models exist to obtain information on electron
|E −Q

(Z,A)|

EC
e F
), where EF is the electron Fermi energy, me is
∝ exp( m
kT
QEC (Z,A)
the electron mass, k is Boltzmann’s constant, and T is the temperature.
51 P (β − )

Z−1,A

237

1
∝ G.S. population
-

Relative Probability

∝ β -decay prob.

0.8

'Urca Window', T=0.5GK
'Urca Window', T=0.1GK

0.6
0.4
0.2

14.4

14.6

14.8

15

15.2

15.4

QEC(Zeven,Aeven)+Ex.s. [MeV]
Figure 4.38: The relative probability of β − -decay occurring from the ground-state of an oddodd nucleus in an Urca pair (blue-dotted line) multiplied by the relative thermal population
of the ground state of that same nucleus (black-dashed line) is coined here as the ‘Even-A
Urca window’ (thick red-solid line) for a temperature of 0.5 GK. Also shown is the ‘Urca
window’ for a temperature of 0.1 GK (thin red-solid line). The functional forms which were
used are explained in the text.

238

capture thresholds and transition strengths, they are often found wanting when confronted
with data [294]. Nonetheless, the local calculations provided by the shell model (particularly
calculations using the GXFP1A interaction) tend to perform relatively well, and a large-scale
set of calculations for the nuclei involved electron captures in the neutron star crust would
provide valuable information for reaction network calculations. Since the maximum amount
of heating/cooling possible for any given electron capture transition is determined by mass
differences, in principle this information could be used to narrow the set of nuclei for which
high-quality excited state information is required.

239

Chapter 5
Future prospects
The results presented in this dissertation are proof of the potential time-of-flight TOF mass
measurement has as a tool for operating at the frontiers of nuclear physics. TOF mass
measurements are a vital tool for extending the limits of the known nuclear mass surface,
providing over 280 nuclear masses to date (See Section 1.4.2.). By operating in a complementary fashion with Penning trap mass spectrometry, this powerful technique is poised
contribute to the solution of unsolved problems in both nuclear structure and nuclear astrophysics. As current generation facilities push to more exotic nuclei with Penning trap
measurements, the TOF technique can be used to push a few nucleons further by leveraging
the Penning trap results as calibration masses. In fact, previously collected TOF data can
be re-analyzed to reduce extrapolation uncertainties and provide masses for nuclei whose
masses were previously too poorly constrained to be worth reporting. When next generation
facilities come online, TOF mass measurement will be one of the essential methods employed
to explore the limits of the nuclear landscape.
When provided with additional calibration nuclei due to future Penning trap mass measurement results, the data set described in this dissertation can be re-analyzed to improve
the precision of nuclei whose mass was reported and obtain masses of nuclei whose current uncertainty was too large to be worth reporting. In fact, as discussed in the previous

240

chapter 1 , an improvement in precision for our 57 Sc mass could settle the question as to
whether the N = 34 subshell terminates at this isotopic chain or in calcium. A reduction
in the uncertainty for the 67 Mn mass reported here would allow firmer conclusions to be
drawn regarding the strength of the N = 40 subshell closure for manganese. Improvements
in the precision of the reported masses of iron isotopes, and the ability to report masses
of the even more neutron-rich species observed, could allow quantitative conclusions to be
drawn about the apparent signature of nuclear deformation observed in the present data set.
Furthermore, the availability of calibrants in the Z = 21 − 24 isotopic chains would allow
useful information to be extracted for the observed titanium and vanadium isotopes. For
instance, the mass surface near 62 Ti could be investigated directly, where a previous study
of production cross section systematics inferred an enhancement in binding [295]. Each of
these exciting prospects may be realized in the near future, given the active state of the
Penning trap community.
The advent of the Facility for Rare Isotope Beams (FRIB) promises to provide a wealth
of previously unobserved nuclei. As with current radioactive beam facilities, TOF mass measurement can be employed to provide masses for some of the most scarcely produced nuclear
species in order to help answer outstanding questions in nuclear physics. Figure 5.1 shows
the predicted FRIB production rate 2 for nuclei beyond the currently known mass landscape,
along with proposed astrophysical reaction paths. So long as charge state contamination can
be overcome for higher-Z nuclei 3 and more radiation-resistant detectors can be developed
1 See

the figures in Section 4.1.2 which demonstrate the points made in this paragraph.
that the production rate prediction shown employed the KTUY mass model [296],
the EPAX 2.15 fragmentation cross-section parameterization [297], the LISE++3EER model
for production cross sections from the projectile fission of 238 U, and LISE++v9.2.68 for
beam transmission efficiency [127].
3 A currently approved experiment at the NSCL contains a proposed method for ridding
of unwanted charge states whose main concepts are briefly discussed in the final appendix
2 Note

241

to cope with high beam intensities, significant gains are expected in astrophysics, regarding
nuclei involved in neutron star crust processes and the astrophysical r-process, as well as
nuclear structure, particularly regarding the evolution of the N = 82 and N = 126 shell closures. However, a necessary condition is that higher precision mass measurement techniques
must also make gains towards more exotic nuclei 4 , as their measurements provide critical
calibration points. Together these complementary methods are likely to expand the known
mass surface several more nucleons beyond the current limit for each element, potentially
doubling the size of the set of known masses. Such a large expansion will undoubtedly be
a significant aid to the development and improvement of theoretical mass models, which, as
can be seen in Figure 5.1, will still be relied upon to bridge the gap between known masses
and the neutron drip-line.

of this dissertation.
4 The primary concern for the Penning trap method is its long measurement time. The
present length of a single ion measurement, ∼ 100 ms, means that only roughly one of every
thousand nuclei with ∼10 ms half-lives make it to the Penning trap. The present record
for the shortest lived nuclide measured by Penning trap mass measurement has an 8.8 ms
half-life [94].
242

100
80

Z

60
40

New, This Expt.
Known in AME 2012
r-process path
rp-process path
NS crust reactions
FRDM 2n-drip-line

20
0

0

20

40

60

80

100

120

140

109
108
107
106
105
104
103
102
10
1
10-1
10-2
10-3

N
Figure 5.1: Predicted FRIB production rate [63], where color indicates particles per second,
with respect to nuclei with known masses in the 2012 Atomic Mass Evaluation [1] and
reaction network paths for the astrophysical rp-process [64], astrophysical r-process [24],
and neutron star crust processes [60]. For orientation, stable nuclei are shown as black
squares and the two-neutron drip-line calculated from FRDM is indicated by the rightmost
black-line contour. For reference, the masses measured for the first time presented in this
dissertation are indicated by black stars. Nuclei produced at a rate of > 10−3 pps would
have sufficient statistics for a TOF mas measurement determination, while nuclei produced
at a rate of > 1 pps could serve as calibration nuclei, so long as they had a well-known mass
(for instance, from Penning trap measurements).

243

APPENDICES

244

Appendix A
Magnet-yoke assembly
The following is the procedure that was used to mount the permanent magnets, used to
guide electrons from the foil to the MCP (See Section 2.4.), to the steel yoke that holds the
magnets in position (See Figure 2.7.):
1. Mounting the frame around the magnet:
• A two-piece aluminum frame was made to act as a holder for the magnet. The
frame had three threaded holes for non-magnetic ‘lowering bolts’ and two smooth
holes for non-magnetic ‘guide rods’, as seen in Figure A.1.

Figure A.1: Cartoon of the frame which was clamped around the magnets used in the rigidity
measurement set-up in order to facilitate placement of the magnets on the steel yoke.

245

• The following steps were taken for both magnets.
(a) The two pieces of the frame were placed on either side of the magnet, centered
longitudinally.
(b) The two pieces of the frame were bolted together to tightly clamp the magnet
between the two frame pieces.
(c) ‘Lowering bolts’ were inserted fully into lowering holes with bolt-heads on the
side of the magnet which was opposite that of the side that would ultimately be
placed flush to the yoke. Care had to be taken to make sure that the two
magnets were oriented correctly with respect to each other so that they
produced a nearly uniform magnetic field in the electron-drift region.
• A possible improvement would have been to add a guide-rod hole to the frame
piece which lacked one. This likely would have helped in the frame lowering
procedure since three points make a plane.
• The lowering bolts initially had feet on the side opposite to the bolt-head, however
these were removed since they added the complication that the lower-bolt could
turn within the foot, ultimately working its way out of the foot.
2. Mount magnet with frame to base:
• The magnet-yoke primarily consisted of three iron plates: a cross-piece which was
used to connect the two other plates and two plates with shallow indents that
had roughly the area of the magnet and were used as the magnets’ base plates,
as seen in Figure 2.7.

246

• The challenge was to gently lower the magnet onto the base plate. As seen in
Figure A.2, spacers were employed to assist in safely lowering the magnet to the
base plate.

Figure A.2: Cartoon of the magnet being lowered onto the yoke piece that served as the
magnet base plate. Miscellaneous parts were used as spacers to prevent the magnet from
rapidly attaching to the base plate. Guide rods were used to assist in placing the magnet in
the proper location. Bolts located in the frame were used to slowly lower the magnet onto
the base plate in a well-controlled manner.

• The following steps were taken for both magnets.
(a) Guide rods were threaded into the base plate.
(b) Spacers were placed on the base plate at the location where the magnet would
ultimately be placed.
(c) The frame was lowered so that the guide rods went through the guide rod holes in
the frame and the magnet was rested on the spacers. Note: There were several
spacers so the lowering could take place incrementally, with a small gap

247

between the magnet’s starting location and it’s closest possible distance
to the base-plate. As seen in Figure A.2 there were also spacers employed below
the lowering bolts. This was because we did not have bolts that were long enough
to touch the base plate while the magnetic attraction to the base plate was weak.
(d) With the magnet at rest on the spacers at a sufficient distance from the base
plate, the lowering bolts were lowered so that their feet rested on the base plate.
(e) The top spacer was then removed and the lowering bolts were turned (so that they
started to be removed from the frame) until the magnet was nearly touching the
next spacer. The bolts were loosened in an order that avoided tilting the frame,
so that it did not bind on the guide holes. This procedure was repeated until all
spacers were removed.
(f) Finally the magnet was lowered onto the base plate by loosening the lowering
bolts all the way. Note that final alignment had to take place just prior
to lowering the magnet fully onto the base plate.
• Note of caution: The magnets are fragile! (e.g. We could not hammer them
into place for final adjustments [without breaking a piece off].)
3. Connecting the base plates by connecting plate:
• The full magnet-yoke consisted of the two magnets mounted to their base plates,
an iron connecting plate to which the base plates were mounted, two aluminum
L-plates which acted as the feet for the yoke, and a non-magnetic platform which
was the base of the entire yoke.
• The yoke assembly employed a vice, where the usual jaws were replaced by alu248

minum plates, to slowly bring together the magnets on their base plates so they
could be connected by an iron connecting plate an non-magnetic platform, as seen
in Figure A.3.

Figure A.3: Cartoon of the magnet-yoke assembly process. The two magnets mounted on
base plates were connected to the jaws of a vice via aluminum plates. The vice was slowly
closed to align the connecting plate with the base plate attached to the fixed vice jaw (“1”
in the figure). Finally the non-magnetic platform was lowered onto the base plates and the
aluminum feet, which were fastened to the platform, were fastened to the magnet base-plates
(“2” in the figure).

(a) The flat aluminum plate was attached to the vice’s movable jaw.
(b) The aluminum ‘L-plate’ was mounted to one of the magnet’s base plate and the
L-plate was fastened to the outside of the vice’s fixed jaw, where a wooden spacer
separated the magnet from the vice body, as seen in Figure A.3.
249

(c) With the vice fully opened and wood spacers in place, the connecting plate was
mounted to the flat aluminum plate on the vice’s movable jaw. Note: The
second base plate and magnet were not mounted to the connecting
plate at this time.
(d) The connecting plate was slowly put into place below the first magnet base plate
by closing the vice jaws. Once in position, the connecting plate was fastened to
the first base plate.
(e) The L-plate was removed from the first magnet-base plate and the connecting
plate, now connected to the first magnet-base plate, was lifted away from the vice
set-up.
(f) The second magnet base plate was mounted to the L-plate, where a wood spacer
separated the magnet from the vice body.
(g) The connecting plate, attached to the first magnet-base plate, was fastened to the
aluminum flat plate on the movable vice jaw, with the vice fully opened and with
wood spacers in place.
(h) The vice jaw was slowly closed to move the connecting plate in place underneath
the second magnet-base plate (“1” in Figure A.3).
(i) The connecting plate was fastened to the second magnet-base plate.
(j) The platform, with aluminum feet attached, was lowered into place and the feet
were fastened to the magnet-base plates to complete the yoke assembly. The
completed yoke is pictured in Figure 2.7.
(k) Finally, the magnetic field in between the magnets was checked to assure the
magnets were positioned in an aligned configuration, to assure a nearly uniform
250

field in the region between the two magnets.

251

Appendix B
Local TOF–YMCP corrections
Note: These figures are available online (https: // people. nscl. msu. edu/ ~ meisel/
files/ LocalLinSlopeCorrAll. pdf ) (or by request) separately since there are so many.
Each figure takes one pdf page, so inclusion of all nuclei makes for a ∼200 page appendix
section! However, I would like them to be available since they may be of use for a future
similar data analysis.
The figures show the step-by-step process followed for the local rigidity correction for
each isotope of the elements 11 ≤ Z ≤ 29 observed in this experiment sorted alphabetically
by symbol: aluminum (Al), argon (Ar), calcium (Ca), chlorine (Cl), cobalt (Co), chromium
(Cr), copper (Cu), iron (Fe), potassium (K), magnesium (Mg), manganese (Mn), sodium
(Na), nickel (Ni), phosphorous (P), sulfur (S), scandium (Sc), silicon (Si), titanium (Ti),
and vanadium (V). These figures correspond to Figure 3.30 within Section 3.6, where the
sub-plots of the figure and the analysis procedure which generates these sub-plots is described
in more detail.
The five rows show the successive steps taken in the local rigidity correction procedure
to remove contamination from neighboring nuclei in the PID and determine the slope of
the TOF vs YMCP of a single nucleus, here 68 Fe. The upper left panel shows a histogram
of TOF vs YMCP for events identified as 68 Fe, where the black points are the resultant
graph obtained by applying ROOT’s TProfile class to the histogram. The black line is a

252

linear fit to the graph. The upper middle panel shows the resultant rigidity-corrected TOF
vs YMCP histogram after removing the linear trend found in the upper left panel, pivoting
about YMCP =0. The upper-right panel shows the projections onto the TOF-dimension of
the uncorrected (red histogram) TOF vs YMCP relationship, which was not easily visible on
the same scale for 68 Fe, and rigidity corrected (black histogram) TOF vs YMCP relationship,
where the blue line is a Gaussian fit to the rigidity-corrected histogram. The second row
contains the same information as the first, but after applying a cut to only include data
within ±4σ of the mean of the rigidity corrected TOF distribution determined in the first
row. The following rows contain the same information after applying cuts to only include
data within 3, 2.5, and 2σ, respectively, of the mean rigidity-corrected TOF, where the mean
and σ were determined by the Gaussian fit in the previous row.
To save space, these figures are now located here: https://people.nscl.msu.edu/
~meisel/files/LocalLinSlopeCorrAll.pdf, or they can be obtained by request from Zach
Meisel.

253

Appendix C
Global TOF–YMCP corrections
Note: These figures are available online (https: // people. nscl. msu. edu/ ~ meisel/
files/ FinalTOFprojectionByElementRigidityCorrection. pdf and https: // people.
nscl. msu. edu/ ~ meisel/ files/ FinalTOFprojectionGlobalRigidityCorrection. pdf
) (or by request) separately since there are so many. Each figure takes one pdf page, and
there are two figures per nucleus, so inclusion of all nuclei makes for a ∼200 page appendix
section! However, I would like them to be available since they may be of use for a future
similar data analysis.
These figures contain the final TOF distribution of nuclei with 18 ≤ Z ≤ 26 alphabetically
by element symbol (argon (Ar), calcium (Ca), chromium (Cr), iron (Fe), potassium (K), manganese (Mn), nickel (Ni), scandium (Sc), titanium (Ti), and vanadium (V).) after the globally
determined magnetic rigidity correction (https://people.nscl.msu.edu/~meisel/files/
FinalTOFprojectionGlobalRigidityCorrection.pdf) and after the magnetic rigidity correction determined by fitting to local rigidity correction slopes along an isotopic chain. The
left panel shows the mean and standard deviation of the TOF distribution and their uncertainties using a 4, 3, and 2 σ range from the mean TOF, the central panel shows the TOF vs
YMCP distribution before (black histogram) and after (colored histogram) the rigidity correction, and the right panel shows the rigidity corrected TOF distribution and the Gaussian
fit using the 2σ range. See Section 3.7 for more details.

254

To save space, these figures are now located here: https://people.nscl.msu.edu/
~meisel/files/FinalTOFprojectionGlobalRigidityCorrection.pdf and here https://
people.nscl.msu.edu/~meisel/files/FinalTOFprojectionGlobalRigidityCorrection.
pdf , or they can be obtained by request from Zach Meisel.

255

Appendix D
Potential TOF mass measurement improvements
Though the experiment described in this dissertation was planned with great care, the gift of
hindsight makes it possible to suggest alternative choices which could have been made for the
experimental set-up (described in Chapter 2) that could have possibly yielded improvements.
The emphasis here is on “possibly”, as most of the following statements, though they are
given justifications, have not been verified by an experimental test and are more speculation
than fact. As such, each of the suggestions should be given careful consideration before
they are implemented in a future time-of-flight mass measurement to be sure they do in fact
have the potential to improve the measurement results. The improvements generally fall
into the categories of improving the timing measurement technique, improving the rigidity
measurement technique, and employing an alternative solution to cope with high beam rates
as opposed to the inclusion of the A1900 wedge degrader.
The timing measurement as it was performed is described in Section 2.3. It was noted in
that section that one concern for the timing measurement was the degradation of the BC-418
scintillators due to prolonged beam bombardment. This concern could be addressed in future
measurements by either taking precautions to begin the measurement with pristine BC-418
scintillators, performing a detailed study of the degradation of BC-418 scintillators under
beam bombardment, or replacing the scintillator-photomultiplier tube timing detection set-

256

up with a more radiation-damage resistant one. The first solution would entail obtaining new
scintillators from the manufacturer [30], in order to ensure maximum light output and thereby
reduce the contribution of signal noise to the amplified timing signal, and then storing the
scintillators in an oxygen-free (e.g. in dry-nitrogen or under vacuum) dark environment in
order to prevent fluorescence degradation, which would reduce light output and increase the
relative contribution of signal noise to the amplified timing signal. The second solution would
consist of sacrificing several new scintillators in a careful study of light-output degradation of
BC-418 under bombardment of high-energy heavy-ions 5 , in order to determine the maximum
sustainable beam rate so that maximum statistics could be achieved. The third solution
would be to consider employing thin diamond detectors as timing detectors instead, which
generally have competitive timing resolution and superior radiation hardness. However, a
single-crystal detector would have to be used since the timing detector must be as uniform
in thickness as possible, so as to not introduce systematic effects in the time-of-flight due
to variable energy loss, and it would have to be large enough in area to accommodate the
beam-spot. Such diamond detectors are potentially an expensive proposition, assuming they
are available at all.
The magnetic rigidity measurement as it was performed is described in Section 2.4. Perhaps the most certain improvement to the rigidity measurement would be a factor of two
increase in the field strength of the yoke-magnets which, as described in Section 3.5.2, would
have improved the microchannel plate position resolution by ∼ ×2 and thereby improved
the ultimate timing resolution by ∼ 25% 6 and therefore reduced the Monte Carlo mass
5 Section

2.3.3 discusses similar studies, however none were performed at the high-energy,
relatively high-Z conditions experienced in the described time-of-flight mass measurement.
6 This calculation uses a nominal time-spread due to position-spread of 36 ps, obtained
from the time-of-flight magnetic-rigidity slope (See Section 3.6.), and a total final time-spread
of 80 ps (See Section 3.8.).
257

uncertainty (See Section 3.9.4.) as well as the statistical TOF uncertainty. Another improvement would be the inclusion of diagnostic features of the rigidity measurement set-up
that would help identify the issue of beam scattering on the collimator that went undetected
in this experiment, as described in Section 3.5.3, and ultimately cost valuable statistics. This
issue could be mitigated by painting the collimator with a fluorescent coating, such that it
would shine during beam-tuning should the beam steer astray and begin scattering on the
collimator slit inner-wall. Alternatively, an active collimator, such as a thick scintillator
with a metal backing on the downstream side, could be employed so that scattering on the
collimator is easily identified. Finally, a systematic study could be performed to find the
minimum foil thickness that would still conduct electricity, so as to hold a negative bias and
repel electrons which are freed by passing beam particles, but would induce the smallest
achievable perturbation to the ion time-of-flight via energy loss.
The final and likely most substantial improvement would be to eliminate the wedge
degrader from the A1900, which was used to remove low-Z contaminants, as described in
Section 2.2.2. The low-Z contaminant removal was necessary in order to avoid damaging
the cathode readout drift counters and, as a secondary consideration, avoid signal pile-up
in the ionization chamber that would have caused confusion in the particle identification
(See Section 2.5.). However, as was mentioned in Sections 3.7 and 3.9.3, inclusion of the
wedge degrader complicated the relationships between time-of-flight, magnetic rigidity, and
nuclear mass. Rather than employing a wedge degrader, a beam attenuator could have
been used, though this would have reduced the rate for all beam species and not just the
rate of the undesired low-Z nuclei. However, perhaps the aformentioned improvement in
the statistical uncertainty expected from an increased yoke-magnet field would compensate

258

for the loss in statistics 7 . Alternatively, the reason for the extant rate limitation could
be removed; i.e. the gas-detectors in the S800 focal plane could be removed or be left
unused (without fill-gas or bias). This is not unrealistic, given that the cathode readout
drift counters ultimately went unused in the data analysis 8 , and alternative detectors could
be employed for the energy loss that was provided by the ionization chamber. For instance,
a silicon detector could sustain a somewhat higher rate and avoid pile-up (though these
detectors are not without their rate limitations). The silicon detector would have the added
advantage of providing charge state information, if used within a stack of silicon detectors,
via ∆E–E charge state discrimination [298] 9 . Though, an additional complication would be
identifying nuclei that underwent a charge change betwen the fast-timing scintillators since,
as described in Section 3.4.3, isotopes of elements with Z ≥ 39 could change charge at the
foil and fall within the acceptance of the S800 spectrograph (though it is unlikely they would
be focussed on the proper location, i.e. the fast-timing scintillator, at the S800 focal plane
focus). These events could possibly be identified by observing the change in the relative
fraction of an ion’s total TOF that the MCP to S800 scintillator time-span contributed.
Charge state information at the MCP location could possibly be obtained via a more recent
method which was developed to identify beam charge states. For this recent technique the
ion charge is gleaned from the number of secondary electrons ejected from a foil through
which the ion passed [299].
To reiterate, each of the above suggestions have not been vetted in detail to assure that
7 Given

the fact that the statistical uncertainty made the smallest contribution to the
overall mass uncertainty in all cases (See Section 3.9.4.), sacrificing statistics to mitigate a
source of systematic uncertainty would likely yield a net improvement
8 The small beam-spot on the drift chambers’ front faces removed the possibility of performing a position calibration via use of a hole mask with a known hole-pattern.
9 Use of this technique is planned for and described in detail in the accepted NSCL experiment proposal e12022 (M. Famiano).
259

they would provide an overall improvement to the time-of-flight mass measurement set-up.
However, it is the author’s belief that each of the ideas presented in this appendix are worth
considering prior to performing the next time-of-flight mass measurement at the National
Superconducting Cyclotron Laboratory.

260

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