SPECTROSCOPY OF NEUTRON-UNBOUND FLUORINE
by
Gregory Arthur Christian

A DISSERTATION
Submitted
to Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
PHYSICS
2011

ABSTRACT
SPECTROSCOPY OF NEUTRON-UNBOUND FLUORINE
by
Gregory Arthur Christian
Neutron-unbound states in 27 F and 28 F have been measured using the technique of invariant mass
spectroscopy, with the unbound states populated from nucleon knock out reactions, 29 Ne(9 Be, X ).
Neutrons resulting from the decay of unbound states were detected in the Modular Neutron Array
(MoNA), which recorded their positions and times of flight. Residual charged fragments were
deflected by the dipole Sweeper magnet and passed through a series of charged particle detectors
that provided position, energy loss, and total kinetic energy measurements. The charged particle
measurements were sufficient for isotope separation and identification as well as reconstruction
of momentum vectors at the target. In addition to the neutron and charged particle detectors, a
CsI(Na) array (CAESAR) surrounded the target, allowing a unique determination of the decay
path of the unbound states.
In 27 F, a resonance was observed to decay to the ground state of 26 F with 380 ± 60 keV relative
energy, corresponding to an excited level in 27 F at 2500 ± 220 keV. The 28 F relative energy
spectrum indicates the presence of multiple, unresolved resonances; however, it was possible to
determine the location of the ground state resonance as 210+50 keV above the ground state of 27 F.
−60

This translates to a 28 F binding energy of 186.47 ± 0.20 MeV.

Comparison of the 28 F binding energy to USDA/USDB shell model predictions provides insight into the role of intruder configurations in the ground state structure of 28 F and the low-Z limit
of the “island of inversion” around N = 20. The USDA/USDB calculations are in good agreement
with the present measurement, in sharp contrast to other neutron rich, N = 19 nuclei (29 Ne, 30 Na,
and 31 Mg). This proves that configurations lying outside of the sd model space are not necessary
to obtain a good description of the ground state binding energy of 28 F and suggests that 28 F does
not exhibit inverted single particle structure in its ground state.

ACKNOWLEDGEMENTS

As with any project, a large number of people have made contributions to the work presented in
this dissertation, and each one of them deserves recognition and thanks. First, I thank my guidance
committee members—Artemis Spyrou, Michael Thoennessen, Alex Brown, Carl Schmidt, Carlo
Piermarocchi, and Hendrik Schatz—for their time and effort in reviewing my work. In particular,
Michael and Artemis have both served as top-notch academic advisors at various points in my time
as a graduate student, giving useful insight, advice, assistance, and motivation.
A number of people made direct contributions to the experiment and its analysis, including the
entire MoNA Collaboration, the NSCL Gamma Group, and NSCL operations and design staff. I
will single a few of them out, and I hope I do not miss anyone. Nathan Frank wrote the original
experiment proposal and was a close collaborator during the preparation, running, and ongoing
analysis. The MoNA graduate students—Michelle Mosby, Shea Mosby, Jenna Smith, Jesse Snyder, and Michael Strongman—were all expert shift-takers during the run. In particular, Jesse took
every night shift, which I think we all appreciated. Thomas Baumann and Paul DeYoung designed
the electronics logic for the timestamp setup, without which the experiment would not have run.
Alexandra Gade, Geoff Grinyer, and Dirk Weisshaar provided much assistance in the assembly
and running of CAESAR, and Alissa Wersal, NSCL REU student, performed much of the hard
labor necessary for CAESAR calibration and analysis. Daniel Bazin kept the Sweeper magnet and
its associated detectors tuned and running throughout the experiment, and Craig Snow and Renan
Fontus put a lot of work into designing and assembling a magnetic shield for CAESAR. On the
theory side, Angelo Signoracci provided the matrix elements and binding energy corrections for
the IOI shell model calculations.
For each person who made direct contributions to this work, there are many more who contributed indirectly. First among these are my parents, Paul and Jane, who have always provided
support and encouragement. All of my extended family and friends at various stages in life have

iii

contributed in some way, but they are too numerous to list individually. I should also thank my
teachers at all levels of education, in particular the late Owen Pool, my first physics teacher, and
John Wood, who introduced me to nuclear physics. I am also grateful to everyone in the NSCL and
the MSU Department of Physics and Astronomy for creating an environment that is, by and large,
a pleasure to work in as a graduate student. There are plenty of horror stories out there regarding
graduate student life, but I have found little of this to be true in my time at MSU and the NSCL.
Along these lines, I am very grateful that I was able to take a two year break from my graduate
studies and be allowed to return and pick up right where I left off, with no negative impact to my
progress.
Finally, I must acknowledge the funding agency which made this work possible, the National
Science Foundation, under grants PHY-05-55488, PHY-05-55439, PHY-06-51627, and PHY-0606007.

iv

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
CHAPTER 1

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

CHAPTER 2 MOTIVATION AND THEORY .
2.1 Evolution of Nuclear Shell Structure . . .
2.2 Theoretical Explanation . . . . . . . . . .
2.3 Correlation Energy . . . . . . . . . . . .
2.4 The Neutron Dripline . . . . . . . . . . .
2.5 Previous Experiments . . . . . . . . . . .

1

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. 4
. 4
. 9
. 11
. 13
. 15

CHAPTER 3 EXPERIMENTAL TECHNIQUE
3.1 Invariant Mass Spectroscopy . . . . . . . .
3.2 Beam Production . . . . . . . . . . . . . .
3.3 Experimental Setup . . . . . . . . . . . . .
3.3.1 Beam Detectors . . . . . . . . . . .
3.3.2 Sweeper . . . . . . . . . . . . . . .
3.3.3 MoNA . . . . . . . . . . . . . . . .
3.3.4 CAESAR . . . . . . . . . . . . . .
3.3.5 Electronics and Data Acquisition . .

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22
22
24
26
29
31
33
34
36

CHAPTER 4 DATA ANALYSIS . . . . . . .
4.1 Calibration and Corrections . . . . . . .
4.1.1 Sweeper . . . . . . . . . . . . .
4.1.1.1 Timing Detectors . . .
4.1.1.2 CRDCs . . . . . . . .
4.1.1.3 Ion Chamber . . . . .
4.1.1.4 Scintillator Energies .
4.1.2 MoNA . . . . . . . . . . . . . .
4.1.2.1 Time Calibrations . .
4.1.2.2 Position Calibrations .
4.1.2.3 Energy Calibrations .
4.1.3 CAESAR . . . . . . . . . . . .
4.2 Event Selection . . . . . . . . . . . . .
4.2.1 Beam Identification . . . . . . .
4.2.2 CRDC Quality Gates . . . . . .
4.2.3 Charged Fragment Identification
4.2.3.1 Element Selection . .
4.2.3.2 Isotope Selection . . .

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39
39
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41
51
52
57
57
63
65
65
67
68
69
70
70
71

v

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4.3

4.4

4.5

4.2.4 MoNA Cuts . . . . . . . . . . . . .
4.2.5 CAESAR Cuts . . . . . . . . . . .
Physics Analysis . . . . . . . . . . . . . .
4.3.1 Inverse Tracking . . . . . . . . . .
4.3.1.1 Mapping Cuts . . . . . .
4.3.2 CAESAR . . . . . . . . . . . . . .
Consistency Checks . . . . . . . . . . . . .
4.4.1 23 O Decay Energy . . . . . . . . .
4.4.2 Singles Gamma-Ray Measurements
Modeling and Simulation . . . . . . . . . .
4.5.1 Resonant Decay Modeling . . . . .
4.5.2 Non-Resonant Decay Modeling . .
4.5.3 Monte Carlo Simulation . . . . . .
4.5.3.1 Verification . . . . . . . .
4.5.4 Maximum Likelihood Fitting . . . .

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101

CHAPTER 5 RESULTS . . . . .
5.1 Resolution and Acceptance
5.2 27 F Decay Energy . . . . .
5.3 28 F Decay Energy . . . . .
5.4 Cross Sections . . . . . . .

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106
106
106
114
120

CHAPTER 6 DISCUSSION .
6.1 Shell Model Calculations
6.2 27 F Excited State . . . .
6.3 28 F Binding Energy . . .

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123
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126
128

CHAPTER 7

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SUMMARY AND CONCLUSIONS . . . . . . . . . . . . . . . . . . . . 136

APPENDIX A CROSS SECTION CALCULATIONS . . . . . . . . . . . . . . . . . . 140
A.1 27 F Excited State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
A.2 28 F Ground State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

vi

LIST OF TABLES

2.1 Bound state gamma transitions observed in [45]. . . . . . . . . . . . . . . . . . . . . . 18
3.1 List of charged particle detectors and their names. Detectors are listed in order from
furthest upstream to furthest downstream. . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2 Numeric designations for the PMTs in the thin and thick scintillators. . . . . . . . . . 33
3.3 Logic signals sent between the Sweeper and MoNA subsystems and the Level 3 XLM.
A valid time signal is one which surpasses the CFD threshold. . . . . . . . . . . . . . 36
4.1 List of the bad pads for each CRDC detector. Pads are labeled sequentially, starting
with zero. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.2 Slope values for each pad on the ion chamber. Pad zero was malfunctioning and is
excluded from the analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.3 Slope values for thin and thick scintillator energy signals. . . . . . . . . . . . . . . . . 53
4.4 List of gamma sources used for CAESAR calibration. Energies are taken from Refs.
[68–72]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.6 Final correction factors used for isotope separation. The numbers in the right column
are multiplied by the parameter indicated in the left and summed; this sum is then
added to ToFTarget→Thin to construct the final corrected time of flight. . . . . . . . . . 77
4.7 Incoming beam parameters. Each is modeled with a Gaussian, with the widths and
centroids listed in the table. Additionally, x and θx are given a linear correlation of
0.0741 mrad/mm, and the beam energy is clipped at 1870 MeV. . . . . . . . . . . . . . 100
A.1 Average values used in calculating the cross section to 27 F∗ . . . . . . . . . . . . . . . 140
A.2 Run-by-run values used in calculating the cross section to 27 F∗ . . . . . . . . . . . . . 140
A.3 Average values used in calculating the cross section to 28 F. . . . . . . . . . . . . . . . 144
A.4 Run-by-run values used in calculating the cross section to 28 F. . . . . . . . . . . . . . 144

vii

LIST OF FIGURES

1.1 Level spacings in the nuclear shell model (up to number 50), from a harmonic oscillator
potential that includes a spin-orbit term [1]. . . . . . . . . . . . . . . . . . . . . . . .

2

1.2 The nuclear chart up to Z = 12. The white circle indicates the fluorine isotopes under
investigation in the present work. For interpretation of the references to color in this
and all other figures, the reader is referred to the electronic version of this dissertation.

3

2.1 Difference between measured and calculated sodium (Z = 11) masses, adapted from
Ref. [8]. A high value of Mcalc − Mexp indicates stronger binding than predicted by
theory, as is the case for 31,32Na. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

2.2 Two neutron separation energies for isotopes of neon (Z = 10) through calcium (Z =
20). The most neutron rich isotopes of Mg, Na and Ne do not demonstrate a dramatic
drop in separation energy at N = 20, indicating a quenching of the shell gap. The top
panel is a plot of the difference in two neutron separation energy between N = 21 and
N = 20 as a function of proton number. S2n values are calculated from Ref. [12]. . . .

6

2.3 2+ first excited state energies for even-even N = 20 isotones, 10 ≤ Z ≤ 20. The excited
state energy drops suddenly to below 1 MeV for Z ≤ 12, indicating a quenching of the
N = 20 shell gap. The 2+ energy for neon is taken from Refs. [14, 15]; all others are
taken from the appropriate Nuclear Data Sheets [16–19]. . . . . . . . . . . . . . . . .

7

2.4 N = 20 shell gap for even-Z elements, Z ≤ 8 ≤ 20, calculated using the SDPF-M
interaction. The gap size is large (∼ 6 MeV) for calcium (Z = 20) and remains fairly
constant from Z = 18 to Z = 14. Below that, the gap size begins to diminish rapidly,
reaching a value of ∼ 2 MeV for oxygen (Z = 8). Adapted from Ref. [21]. . . . . . .
2.5 Left panel: Feynman diagram of the tensor force, resulting from one-pion exchange
between a proton and a neutron [27]. Right panel: Diagram of the collision of a spin′
′
flip nucleon pair { j− , j+ } (left) and a non spin-flip pair { j+ , j+ } (right). In the
spin-flip case, the wave function of relative motion is aligned parallel to the collision
direction, resulting in an attractive interaction, while in the non spin-flip case the wave
function of relative motion is aligned perpendicular to the collision direction, resulting
in a repulsive interaction [27]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

viii

8

9

2.6 Schematic illustrating the role of the tensor force in driving changes in nuclear shell
structure. Thick arrows represent a strong interaction and thin arrows a weak one. In
the case of stable nuclei near N = 20 (left panel), there is a strong tensor force attraction
between 0d3/2 neutrons and 0d5/2 protons, as well as a strong repulsion between 0 f7/2
neutrons and 0d5/2 protons. These interactions lower the 0ν d3/2 and raise the ν 0 f7/2 ,
resulting in a large gap at N = 20. In contrast, neutron rich nuclei near N = 20 (right
panel) have a deficiency in 0d5/2 protons, weakening the attraction to 0ν d3/2 and the
repulsion to ν 0 f7/2 . This causes the 0ν d3/2 to lie close to the 0ν f7/2 , reducing the
gap at N = 20 and creating a large gap at N = 16 [27]. . . . . . . . . . . . . . . . . . 10
2.7 Sources of correlation energy. Thick grey lines represent a strong correlation; thin
grey lines a weak one. The top panels [(a) and (b)] demonstrate the case of nuclei
with closed neutron shells, while the bottom panels [(c) and (d)] show nuclei without a
closed shell. In both cases, the intruder configurations [(b) and (d)] produce a stronger
correlation energy, with the greatest energy gain coming from the configuration in
(b) [30]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.8 Left panel: Two neutron separation energy as a function of proton number, for N = 20
isotones, 9 ≤ Z ≤ 14. The dashed line is a shell model calculation truncated to 0p-0h,
while the solid line is the same calculation without truncations. The triangle markers
are experimental data, with the square marker being a more recent datum for 30 Ne
[33]. The cross at 29 F is the result of a 2p-2h calculation which incorrectly predicts
an unbound 29 F. Right panel: Occupation probabilities of 0p-0h (dotted line), 2p-2h
(solid line), and 4p-4h (dashed line) configurations. Figure adapted from Ref. [34]. . . 12
2.9 Chart of stable and bound neutron rich nuclei up to Z = 12. Note the abrupt shift in the
neutron dripline between oxygen (Z = 8) and fluorine (Z = 9) . . . . . . . . . . . . . . 14
2.10 Particle identification plot from the reaction of 40 Ar at 94.1 AMeV on a tantalum
target. Eight events of 31 F were observed in the experiment, while no events were
observed for 26 O or 28 O. Figure adapted from Ref. [37]. . . . . . . . . . . . . . . . . 16
2.11 Gamma decay spectra of bound excited states in 27,26,25F, along with sd shell model
predictions on the right of the figure. For 27 F, the 1/2+ excited state prediction from
a calculation done in the full sd p f model space is also included in red. Adapted from
Ref. [45]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.12 Comparison of experimental and theoretical levels for 25 F from Ref. [46]. The superscripts next to the experimental level energies denote references numbers within [46],
and the measurement of [46] is labeled “this work” in the figure. . . . . . . . . . . . . 18
2.13 30 Ne gamma de-excitation spectrum from Ref. [14]. Experimental level assignments
and a variety of theoretical predictions are also included. . . . . . . . . . . . . . . . . 20
3.1 Decay of an unbound state via neutron emission. . . . . . . . . . . . . . . . . . . . . . 23

ix

3.2 Illustration of the two possible decay processes of an unbound state which lies higher
in energy than a bound excited state of the daughter. The state can either decay through
direct neutron emission to the ground state of the daughter (grey arrow) or by neutron
emission to an excited state in the daughter and subsequent gamma emission (black
arrows). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.3 Beam production. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.4 Experimental setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.5 Dramatized view of the operation of a CRDC. Charged fragments interact in the gas,
releasing electrons. The electrons are subjected to a drift voltage in the non-dispersive
direction and are collected on the anode wire, in turn causing an induced charge to
form on a series of aluminum pads. Position in the dispersive direction is determined
by the charge distribution on the pads, while position in the non-dispersive direction is
determined from the drift time of the electrons. . . . . . . . . . . . . . . . . . . . . . 30
3.6 Location of MoNA detectors. Each wall is sixteen bars tall in the vertical (y) direction,
and the center of each wall in the vertical direction is equal to the beam height. The
black lines in the figure indicate the central position in z of the corresponding MoNA
bar. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.7 Arrangement of CAESAR crystals. The left panel shows a cross-sectional view perpendicular to the beam axis of an outer and an inner ring. The right panel shows a
cross sectional view parallel to the beam axis of the nine rings used in the experiment.
Figure taken from Ref. [53]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.8 Diagram of the timestamp trigger logic. . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.1 Example calibrated timing spectra before (left panel) and after (right panel) jitter subtraction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.2 Pedestals for each of the CRDC detectors. Color represents the number of counts per
bin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.3 Example histogram of the signal on a single pad in a CRDC pedestal run. The blue
curve is the result of an unweighted Gaussian fit and is used to calculate the pedestal
value. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.4 Example of the plots used in the gain matching procedure. The left panel is a two
dimensional histogram of charge on the pad versus ∆, while the right panel shows the
Gaussian centroids of the y axis projection of each x axis bin in the plot on the left.
The curve in the right panel is the result of an unweighted fit to the data points, with
the fit function a Gaussian centered at zero. . . . . . . . . . . . . . . . . . . . . . . . 43

x

4.5 Difference in signal shape between a bad pad (red histogram) and a normal one (blue
histogram). The bad pad in this figure is pad 24 of CRDC2, while the normal pad is
number 60 in CRDC2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.6 Charge distributions for an event near the middle of CRDC2 (left panel) and one near
the edge (right panel). The blue curve is a Gaussian fit to the data points, and the blue
vertical line is the centroid of that fit. The red vertical line is the centroid of a gravity
fit to the points. The Gaussian and gravity centroids are nearly the same in the case
of events near the middle, but on the edge the gravity fit is skewed towards lower pad
number. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.7 Gaussian versus gravity centroids for CRDC2. The two fits disagree near the edge of
the detector, with the gravity fit skewed lower. . . . . . . . . . . . . . . . . . . . . . . 46
4.8 Example of calibrated position in both planes for CRDC1, with the tungsten mask in
place. The blue open circles denote the position of the mask holes, and the blue vertical
lines denote the position of slits cut into the mask. . . . . . . . . . . . . . . . . . . . . 47
4.9 Drift of CRDC calibration parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.10 Gaussian centroids in pad space for TCRDC2. . . . . . . . . . . . . . . . . . . . . . . 49
4.11 Plot of TCRDC2 x position versus TCRDC1 x position. The plot is used to determine
a correlation of x2 = 1.162 · x1, shown by the black line in the figure. The position at
TCRDC2 determined by this correlation is what is used in the final analysis. . . . . . . 50
4.12 Upper left: Ion chamber ∆E signal versus CRDC2 x position; the black curve is a 3rd
order polynomial fit used to correct for the dependence of ∆E on x. Upper right: Result
of the x−correction: the dependence of ∆E on x is removed. Lower Left: x−corrected
∆E versus CRDC2 y position; the black curve is a linear fit used to correct for the
dependence of ∆Excorr on y. Lower right: Final position corrected ion chamber ∆E
versus CRDC2 y position. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.13 Position correction of the thin and thick scintillator energy signals. Top left: thin
scintillator ∆E vs. x position, with a third order polynomial fit. Bottom left: thin scintillator ∆E (corrected for x dependence) vs. y position, with a second order polynomial
fit. Top right: thick scintillator Etotal vs. x position, with a third order polynomial fit.
Bottom right: thick scintillator Etotal (corrected for x dependence) vs. y position, with
a first order polynomial fit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.14 Results of the position correction of the thin and thick scintillator energy signals. The
panels correspond to those of Fig. 4.13, displaying the final position corrected energy
signals. As seen in the upper-left panel, the correction for thin ∆E could be improved
by the use of a higher order polynomial; however this signal is not used in any of the
final analysis cuts, so the correction is adequate as is. . . . . . . . . . . . . . . . . . . 55

xi

4.15 Thin (top panel) and thick (bottom panel) scintillator energies for 29 Ne unreacted beam
in production runs. Gaps along the x axis correspond to non-production runs taken at
various intervals during the experiment. The drift seen in the figure is corrected by
setting Gaussian centroid of each run’s 29 Ne energy signal to be constant throughout
the experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.16 Example of the two types of muon tracks used in calculating independent time offsets
for MoNA. The left panel illustrates a nearly vertical track, used to determine the time
offsets within a single wall. The right panel shows an example of a diagonal track used
to calculate offsets between walls. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.17 Time of flight to the center of the front wall of MoNA. The events in the figure are
collected during production runs. The prompt γ peak used to set the global timing
offset is clearly identifiable and separated from prompt neutrons. The plot includes a
number of cuts, which are listed in the main text. . . . . . . . . . . . . . . . . . . . . 59
4.18 Time of flight to the front face of MoNA versus deposited energy, for prompt gammas.
The plot includes cuts similar to those used in generating Fig. 4.17. The dependence
of time of flight on deposited energy, as indicated in the figure, demonstrates the presence of walk for low signal size. The function in the figure is partially used for walk
corrections, as explained in the text. . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.19 Inset: mean time vs. absolute value of x position for cosmic-ray data collected in
standalone mode, where the trigger is the first PMT to fire. The time axis on the inset
is determined by the mean travel time of light from the interaction point to the PMTs.
If this parameter is corrected for the x position, using the line drawn on the inset, then
a theoretically constant ToF is obtained. Plotting this constant ToF versus deposited
energy reveals walk, as shown in the main panel of the figure. The function drawn in
the main panel is used for walk correction of production data at high deposited charge,
as explained in the text. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.20 Walk correction functions. The blue curve is from a fit of time of flight vs. deposited
energy, as shown in Fig. 4.18, while the red curve is from a fit of position-corrected
time vs. deposited energy for cosmic ray data, shown in Fig. 4.19. The black curve
is the final walk correction function, using the γ peak correction function (blue curve)
below the crossing point (q = 1.8 MeVee) and the cosmic correction function (red
curve) above. Note that the y axis in this figure represents the actual correction applied
to the time of flight, which is the reason for the offset difference between the curves in
this figure and those of Figs. 4.18–4.19. . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.21 Illustration of MoNA x position measurement: the time it takes scintillation light to
travel to each PMT is directly related to the distance from the PMT. By taking the time
difference between the signals on the left and right PMTs, the x position in MoNA can
be calculated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

xii

4.22 Example time difference spectrum for a single MoNA bar in a cosmic ray run. The
1
3 · max crossing points, indicated by the red vertical lines in the figure, are defined to
be the edges of the bar in time space; these points are used to determine the slope and
offset of Eq. (4.23). The asymmetry in the distribution is the result of the right side of
MoNA being closer to the vault wall, thus receiving a larger flux of room γ -rays. . . . 63
4.23 Example raw QDC spectrum for a single MoNA PMT. The pedestal and muon peak
are indicated in the figure, along with the peak associated with room γ −rays interacting in MoNA. After adjusting voltages to place the muon bump of every PMT close
to channel 800, a linear calibration is applied to move the pedestal to zero and the
Gaussian centroid of the muon bump to 20.5 MeVee. . . . . . . . . . . . . . . . . . . 64
4.24 Example source calibrations for a single CAESAR crystal (J5). The solid black line
is a linear fit to the data points (R2 = 0.999933). All other crystals are similarly well
described by a linear fit, so in the final analysis the calibration is done using the two
88 Y gamma lines at 898 keV and 1836 keV. . . . . . . . . . . . . . . . . . . . . . . . 66
4.25 Beam components, including 29 Ne gate. . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.26 Flight time from the K1200 cyclotron (measured by the cyclotron RF) to the A1900
scintillator versus flight time from the A1900 scintillator to the target scintillator. The
double peaking in ToFRF→A1900 is due to wraparound of the RF. By selecting only
events which are linearly correlated in these two parameters, the contribution of wedge
fragments to the beam is reduced. These selections are indicated by the black contours
in the figure, with the final cut being an OR of the two gates. . . . . . . . . . . . . . . 68
4.27 CRDC quality gates. Each plot is a histogram of the sigma value of a Gaussian fit to
the charge distribution on the pads versus the sum of the charge collected on the pads.
The black contours indicate quality gates made on these parameters. . . . . . . . . . . 69
4.28 Energy loss in the ion chamber versus time of flight through the Sweeper. Each band
in the figure is a different isotope, with the most intense band composed primarily of
Z = 10 unreacted beam. The fluorine (Z = 9) events of interest are circled and labeled
in the figure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.29 Energy loss in the ion chamber versus total kinetic energy measured in the thick scintillator. As in Fig. 4.28, the bands in the figure are composed of different elements,
and the most intense band is Z = 10. fluorine (Z = 9) events are circled and labeled. . . 73
4.30 Left panel: focal plane dispersive angle vs. time of flight for magnesium isotopes in
the S800, taken from Ref. [73]. In this case, isotopes are clearly separated just by
considering these two parameters. Right panel: focal plane dispersive angle vs. time
of flight for fluorine isotopes in the present experiment. The plot shows no hint of
isotope separation, as three dimensional correlations between angle, position and time
of flight need to be considered in order to distinguish isotopes. . . . . . . . . . . . . . 74

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4.31 Three dimensional plot of time of flight through the Sweeper vs. focal plane angle
vs. focal plane position (color is also representative of time of flight). The figure is
composed of Z = 9 events coming from the 32 Mg contaminant beam, and each band
in the figure is composed of a different isotope. . . . . . . . . . . . . . . . . . . . . . 75
4.32 Profile of the three dimensional scatter-plot in Fig. 4.31. The solid black curve is a fit
to lines of iso-ToF; this fit is used to construct a reduced parameter describing angle
and position simultaneously. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.33 Histogram of the emittance parameter, constructed from the fit function in Fig. 4.32,
vs. time of flight. The bands in the plot correspond to different isotopes of fluorine.
A corrected time of flight parameter can be constructed by projecting onto the axis
perpendicular to the line drawn on the figure. . . . . . . . . . . . . . . . . . . . . . . 77
4.34 Main panel: Corrected time of flight for fluorine isotopes resulting from reactions on
the 29 Ne beam. The isotopes if interest, 26,27F, are labeled. The black curve in the
figure is a fit to the data points with the sum of five Gaussians of equal width. Based on
this fit, the cross-contamination between 26 F and 27 F is approximately 4%. The inset
is a scatter-plot of total energy measured in the thick scintillator vs. corrected time
of flight. These are the parameters on which 26,27F isotopes are selected in the final
analysis, and the cuts for each isotope are drawn in the figure. . . . . . . . . . . . . . 78
4.35 Neutron time of flight to MoNA for all reactions products produced from the 29 Ne
beam. The first time-sorted hit coming to the right of the vertical line at 40 ns is the
one used in the analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.36 Doppler corrected energy vs. time of flight for gammas recorded in CAESAR. To
reduce the contribution of background, only those events falling between the solid
black curves are analyzed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.37 Comparison of forward tracked and inverse tracked parameters for unreacted 29 Ne
beam. The upper right panel is a comparison of kinetic energies, with the x axis being
energy calculated from ToFA1900→Target and the y axis energy calculated from inverse
tracking in the sweeper, c.f. Eq. (4.37). The lower left and lower right panels show a
similar comparison for θx and θy , respectively. In these plots, the x axis is calculated
from TCRDC measurements and forward tracking through the quadrupole triplet. . . . 84
4.38 Left panel: Focal plane dispersive angle vs. position for unreacted beam particles
swept across the focal plane. These events display positive correlation between angle
and position, and they define the region of the emittance for which the magnetic field
maps of the Sweeper are valid. Right panel: Focal plane dispersive angle vs. position
for reaction products produced from the 32 Mg beam. A significant portion of these
reaction products fall in the region of positive position and negative angle, due to
taking a non-standard track through the Sweeper. The rectangular contour drawn on
the plot is defined by the “sweep band” of the left panel, and only events falling within
this region are used in the final analysis. . . . . . . . . . . . . . . . . . . . . . . . . . 86
xiv

4.39 Inset: focal plane angle vs. position for unreacted beam particles swept across the
focal plane. Unlike the left panel of Fig. 4.38, here the A1900 optics were tuned to
give a beam that is highly dispersed in angle. causing the emittance region of negative
angle and positive position to be probed. The main panel is a plot of incoming beam
(fp)
angle for all events (unfilled histogram) and events with +x(fp) and −θx (orange
(fp)
filled histogram). The plot reveals that the +x(fp) , −θx events have a large positive
angle as they enter the Sweeper. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.40 Left panel: fragment kinetic energy, calculated from the partial inverse map, vs. time of
flight through the Sweeper. The events in the figure are 26 F + n coincidences produced
from the 29 Ne beam. Events with extreme values of Efrag also fall outside of the
expected region of inverse correlation between Efrag and ToF. Right panel: Efrag vs.
CRDC1 y position. This plot reveals an unexpected correlation between Efrag and
CRDC1 y, with the extreme Efrag events also having a large absolute value of CRDC1
y. This is likely due to limitations of the Sweeper field map, so the events with | y |>
20 mm are excluded from the analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.41 Decay energy for 23 O∗ → 22 O + n events produced from the 32 Mg beam. The spectrum displays a narrow resonance at low decay energy, consistent with previous measurements that place the transition at 45 keV. The inset is a relative velocity vn − v f
histogram for the same events. The narrow peak around vrel = 0 is also consistent with
the 45 keV decay. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.42 Left panel: Doppler corrected gamma energies from inelastic excitation of 32 Mg. The
blue vertical line indicates the evaluated peak location of 885 keV [16]. Right panel:
Doppler corrected gamma energies for 31 Na, produced from 1p knockout on 32 Mg.
The most recent published measurement of 376 (4) keV [84] is indicated by the blue
vertical line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.43 Schematic of the process by which non-resonant background is observed in coincidence with 26 F. First a highly excited state in 28 F is populated from 29 Ne. This state
then decays to a high excited state in 27 F, evaporating a neutron (thick arrow). The
excited state in 27 F then decays to the ground state of 26 F by emitting a high-energy
neutron (thin arrow), which is not likely to be observed. The evaporated neutron (thick
arrow) has fairly low decay energy and is observed in coincidence with 26 F, giving
rise to the background. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.44 Comparison of simulation to data, with unreacted beam sent into the center of the focal
plane. The black squares are data points and the solid blue lines are simulation results.
It should be noted that the dispersive angle in the focal plane, shown in the upper-right
panel, does not match unless the beam energy is clipped at 1870 MeV. . . . . . . . . . 101

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4.45 Comparison of simulation to data for for 26 F reaction products in the focal plane. In
each panel, the parameter being compared is indicated by the x axis label. In the
lower-right panel, δ E refers the the deviation from the central energy of the Sweeper
magnet. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.46 Same as Fig. 4.45 but for 27 F in the focal plane. . . . . . . . . . . . . . . . . . . . . . 103
5.1 Simulated acceptance curve for 26 F + n coincidences. The inset is a histogram of
measured neutron angles for fragment-neutron coincidences produced from the 32 Mg
beam, demonstrating the limited neutron acceptance that is a result of neutrons being
shadowed by the beam pipe and the vacuum chamber aperture. . . . . . . . . . . . . . 107
5.2 Demonstration of the experimental resolution as a function of decay energy for 26 F + n
coincidences. The bottom panel shows a variety of simulated decay energy curves; in
each, the un-resolved decay energy is a delta function, with energies of 0.1 MeV (red),
0.2 MeV (blue), 0.4 MeV (green), 0.8 MeV (orange), and 1.5 MeV (navy). The top
panel is a plot of the Gaussian σ of simulated decay energy curves as a function of
input energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.3 Decay energy histogram for 26 F + n coincidences. The filled squares are the data
points, and the curves display the best fit simulation results. The red curve is a simulated 380 keV Breit-Wigner resonance, while the filled grey curve is a simulated
non-resonant Maxwellian distribution with Θ = 1.48 MeV. The black curve is the sum
of the two contributions (resonant:total = 0.33). The inset is a plot of the negative log
of the profile likelihood versus the central resonance energy of the fit. Each point in
the plot has been minimized with respect to all other free parameters. The minimum
of − ln (L) vs. E0 occurs at 380 keV, and the 1σ (68.3% confidence level) and 2σ
(95.5% confidence level) limits are indicated on the plot. . . . . . . . . . . . . . . . . 109
5.4 Histogram of the Doppler corrected energy of γ -rays recorded in coincidence with 26 F
and a neutron. Only two CAESAR counts were recorded, giving strong indication that
the present decays are populating the ground state of 26 F. . . . . . . . . . . . . . . . . 111
5.5 Comparison between simulation and data for intermediate parameters (opening angle,
relative velocity, neutron ToF, and fragment kinetic energy) used in calculating the
decay energy of 26 F + n coincidences. The red curves are the result of a 380 keV
resonant simulation, and the filled grey curves are the result of the simulation of a nonresonant Maxwellian distribution (Θ = 1.48 MeV). Black curves are the sum of the
two contributions (resonant:total = 0.33). . . . . . . . . . . . . . . . . . . . . . . . . 112

xvi

5.6 Summary of experimentally known levels in 26,27F. The presently measured decay of
a resonant excited state in 27 F to the ground state of 26 F is indicated by the arrow.
The grey boxes represent experimental uncertainties in the absolute placement of level
energies relative to the 27 F ground state. The dashed lines surrounding the present
measurement correspond to the ±60 keV uncertainty on the 380 keV decay energy, and
the grey error box includes both this 60 keV uncertainty and the 210 keV uncertainty of
the 26,27 F mass measurements. The bound excited states measured in [45] are placed
assuming that all transitions feed the ground state as the authors of [45] do not state
conclusively whether their observed γ -rays come in parallel or in cascade. . . . . . . . 113
5.7 Decay energy spectrum of 28 F (filled squares with error bars), along with the best
single resonance fit (red curve). The inset is a plot of the negative log-likelihood vs.
central resonance energy, demonstrating a minimum at 590 keV. . . . . . . . . . . . . 115
5.8 Decay energy spectrum for 28 F, with the best two resonance fit results superimposed.
The filled squares with error bars are the data, and the red and blue curves are the
(ex)
(gs)
lower resonance (E0 = 210 keV) and upper resonance (E0 = 770 keV) fits, respectively. The black curve is the superposition of the two individual resonances. The
(gs)
inset shows a plot of the negative log-likelihood vs. E0 , demonstrating a minimum
+50,+90

at 210 keV and 1, 2σ confidence regions of −60,−110 keV. Each point on the inset like(ex)

(ex)

lihood plot has been minimized with respect to the nuisance parameters E0 , Γ0 ,
and the relative contribution of the two resonances. . . . . . . . . . . . . . . . . . . . 116
5.9 Comparison of simulation and data for intermediate parameters used in constructing
the decay energy of 28 F. The parameters being compared are noted as the x axis labels
on the individual panels. The filled circles with error bars are the data, and the red,
blue and black curves represent the 210 keV resonance simulation, 770 keV resonance
simulation and their sum, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5.10 Experimental level scheme of 28 F and 27 F. The black lines represent the nominal placement of states relative to the ground state of 27 F, and the grey boxes represent 1σ
errors on the measurements. As described in the text, the placement of the 560 keV
(770 keV decay energy) excited state in 28 F is extremely uncertain, and it is only included in the figure for consistency. . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.1 Summary of experimental and theoretical (USDA, USDB, IOI) excited levels in 27 F.
The grey boxes surrounding the experimental values represent the 1σ uncertainties
on the measurements. Experimental values are taken from Refs. [97] (ground state
J π ), [45] (bound excited states), and the present work. Both possible placements of
the 504 keV transition observed in Ref. [45] are included, with the cascade placement
shown as the dashed line at 1282 keV. . . . . . . . . . . . . . . . . . . . . . . . . . . 127

xvii

6.2 Excitation energies of theoretical p shell proton hole states in even-N fluorine isotopes.
Adapted from Ref. [104], with the 25,27F 1/2− candidate states of Ref. [45] added,
assuming parallel gamma emission. . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.3 Difference between experimental [12,96] and theoretical (USDB) binding energies for
sd shell nuclei, with positive values meaning experiment is more bound than theory.
Adapted from Ref. [99] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6.4 Average ℏω excitations in the ground states of nuclei with 8 ≤ Z ≤ 17 and 15 ≤ N ≤ 24,
as calculated in the IOI interaction and truncated sd p f model space. Taken from Ref.
[100]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.5 Theoretical predictions for the ground state binding energies of N = 19 isotones from
oxygen (Z = 8) through Chlorine (Z = 17). Calculations are shown for three interactions: IOI (black inverted triangles), USDA (red squares), and USDB (blue circles). To
put the binding energies on roughly the same scale, the calculation results have been
shifted by subtracting 16 · Z from their original values. . . . . . . . . . . . . . . . . . . 132
6.6 Binding energy difference between experiment (Refs. [12, 96] and the present work
for Z = 9) and theory (IOI, USDA, USDB) for N = 19 isotones with 9 ≤ Z ≤ 17. As
in Fig. 6.3, positive values indicate experiment being more bound than theory. Error
bars are from experiment only, and the shaded grey region represents the 370 keV
RMS deviation of the IOI interaction, while the horizontal red and blue lines denote
the respective 170 keV and 130 keV RMS deviations of USDA and USDB. . . . . . . 133
6.7 Experimental (Refs. [12, 96] and the present work for A = 28) and theoretical (IOI,
USDA, USDB) binding energy predictions for fluorine isotopes, 24 ≤ A ≤ 31. The
USDA and USDB calculations end at A = 29 (N = 20), as their corresponding model
spaces cannot accommodate more than 20 neutrons. The top panel is a plot of binding
energy differences (experiment minus theory) for nuclei whose mass has been measured. Error bars in the top panel are from experiment only, and the shaded grey
region and red/blue horizontal lines are the same as in Fig. 6.6. . . . . . . . . . . . . . 135

xviii

Chapter 1
INTRODUCTION

The fundamental question of nuclear structure is a simple one: what happens when protons and
neutrons are combined to form a compound system? Despite its seemingly simple nature, the
atomic nucleus is a varied and complex object. Construction of a complete and accurate model of
all nuclei would require a detailed understanding of the strong nuclear force as well as the ability
to exactly solve an n−body quantum mechanical problem. At present, neither of these is tractable,
but it is nevertheless possible to construct models which reproduce a variety of experimental observations relevant to nuclear structure. One of the most successful models used to describe the
structure of light to medium mass nuclei is the “shell model,” which treats nuclei as collections of
independent nucleons (protons and neutrons) moving in a mean field.
Beginning in 1933, W. Elsasser noted that certain “magic” numbers of protons and neutrons
result in enhanced nuclear stability [2]. This led to the development of the shell model, in which
nuclei are modeled as systems of non-interacting nucleons sitting in a potential well. Initially, this
idea was disregarded as it was not believed that strong inter-nucleon fores could average to form
such a well. Moreover, little experimental data was available to support such a hypothesis [3].
Over a decade later, M. Goeppert-Mayer revisited the nuclear shell model with the benefit of a
large number of experimental data, noting the particular stability of nuclei at proton or neutron
numbers 8, 20, 50, 82 and 126 [4]. However, using simple potential wells to construct the mean
field, it was only possible to reproduce gaps at numbers 8 and 20, so the independent particle model
of the nucleus continued to be disregarded [3].
The major breakthrough for the shell model came in 1949, when Mayer and the group of
Haxel, Suess, and Jensen independently demonstrated that large shell gaps at 8, 20, 28, 50, 82 and
126 are reproduced theoretically in a mean field model by adding a spin-orbit term to a harmonic
oscillator potential [5, 6]. The energy spacings resulting from such a potential are illustrated in

1

4ℏω

0g

1p
0f

(10) [50]
(2) [40]
(6) [38]
(4) [32]

50

0 f7/2

3ℏω

0g9/2
1p1/2
0 f5/2
1p3/2

(8) [28]

28

0d3/2
1s1/2
0d5/2

(4) [20]
(2) [16]
(6) [14]

20

2ℏω

1s
0d

1ℏω

0p

0p1/2
0p3/2

(2) [8]
(4) [6]

8

0ℏω

0s

0s1/2

(2) [2]

2

Figure 1.1: Level spacings in the nuclear shell model (up to number 50), from a harmonic
oscillator potential that includes a spin-orbit term [1].
Fig. 1.11 . Later refinements were made to the shell model, including the use of a more realistic
Woods-Saxon potential [7], but for stable nuclei the basic picture is similar to that of Fig. 1.1.
Early studies involving nuclei away from stability indicated a breakdown in the large shell gap
at N = 20 [8–11]. The loss of magicity at N = 20 and at other magic numbers has since been
demonstrated in a wide variety of experiments, and a great amount of experimental and theoretical
effort has been put forth to understand the evolution of nuclear shell structure when going from
stable nuclei to those with large neutron to proton ratio. In the present work, these efforts are
continued through the study of neutron-unbound states in 27 F and 28 F, circled on a nuclear chart in
Fig. 1.2. These nuclei lie close to the traditional magic number N = 20 and are some of the most
neutron rich N ∼ 20 systems presently available for experimental study.
1 Throughout this document, the shell labeling scheme of Fig. 1.1, beginning with n = 0, is used. Other

sources may use a scheme that begins with n = 1, but the distinction is typically clear from context.

2

Proton Number

19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 –Mg

12

18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 –Na
16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 –Ne

10

14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 –F
12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 –O

8

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 –N
8

9 10 11 12 13 14 15 16 17 18 19 20 21 22 –C

6

7

8

9 10 11 12 13 14 15 16 17 18 19 –B

5
4

6
5

7
6

8
7

9 10 11 12 13 14 15 16 –Be
8 9 10 11 12 13 –Li

3

4

5

6

7

2

3

4

5

6 –H

6
4
2
1

8

p–rich (unbound)
p–rich (bound)
stable

9 10 –He

n–rich (bound)
n–unbound

1 –n

0
0

5

10

15

20

25

30
Neutron Number

Figure 1.2: The nuclear chart up to Z = 12. The white circle indicates the fluorine isotopes under investigation in the present work. For
interpretation of the references to color in this and all other figures, the reader is referred to the electronic version of this dissertation.

3

Chapter 2
MOTIVATION AND THEORY

2.1 Evolution of Nuclear Shell Structure
Since the earliest studies indicating a change in shell structure for neutron rich nuclei [8–11], a
wide variety of data have been collected to support this hypothesis. The first indications came in
Ref. [8], in which mass measurements of sodium (Z = 11) isotopes, ranging from 26 Na to 31 Na,
were reported. The sodiums were produced in reactions of a 24 GeV proton beam on a thick
Uranium target, with the reaction products separated and their masses measured in a single-stage
magnetic spectrometer at CERN [13]. The authors found that the heavier isotopes 31,32Na were
significantly more bound than predicted by theoretical calculations using a closed N = 20 shell.
This deviation is shown in Fig. 2.1, adapted from the original article. The masses of these isotopes
could only be reproduced theoretically if neutrons were allowed to be promoted from the 0d3/2
to the 0 f7/2 level, a configuration which was not in line with previous assumptions of a closed
N = 20 shell.
Extending the mass measurements of [8] to include other isotopes in the N = 20 region, one
can see clear evidence for quenching of the N = 20 shell gap by plotting two-neutron separation
energies as a function of mass number, as is done for isotopes of neon (Z = 10) through calcium
(Z = 20) in Fig. 2.2. A significant decrease in two neutron separation energy is expected to follow
a large shell gap since nuclei below the gap will be much more tightly bound than those above it.
As shown in the figure, a large decrease in separation energy is present for stable and nearly stable
isotopes (aluminum through calcium, 13 ≤ Z ≤ 20). This decrease becomes diminished for the
neutron rich isotopes of magnesium, sodium and neon (Z = 12, 11, 10) .
Further confirmation of N = 20 shell gap quenching comes from consideration of the 2+ first
excited state energies of even-even nuclei in the N ∼ 20 region. For nuclei with a closed N = 20
neutron shell, reaching the 2+ configuration requires the promotion of neutrons across the large

4

Mcalc − Mexp (MeV)

MASSES OF SODIUM ISOTOPES
5
4
3
2
1
0
-1
-2
-3
-4
-5

GK II
SM

GK I
MS

19

21

23

25

27

29

31

33

35
A

Figure 2.1: Difference between measured and calculated sodium (Z = 11) masses, adapted from
Ref. [8]. A high value of Mcalc − Mexp indicates stronger binding than predicted by theory, as is
the case for 31,32Na.
shell gap. This requirement leads to a high-lying 2+ first excited state, on the order of 2 MeV
or more. On the other hand, if the gap is quenched, the next available level is lower, resulting
in a decreased 2+ excitation energy. Fig. 2.3 shows the 2+ first excited state energies of eveneven N = 20 isotones from neon (Z = 10) through calcium (Z = 20) . As expected, the 2+ energy
is large for the nuclei with Z ≥ 14, but it drops dramatically to less than 1 MeV for neon and
magnesium, indicating a quenching of the N = 20 gap.
In addition to masses and excited state energies, a number of other data have been collected
to indicate the onset of deformation for neutron rich N ∼ 20 nuclei. Deformation in nuclei is the
result of collective modes, and its presence indicates the lack of a strong shell closure. Some of
the experimental signatures of deformation include B (E2) values for the transition from the 2+
first excited state to the 0+ ground state (for even-even nuclei); charge radii; and electromagnetic
moments. A detailed overview of previous measurements of these observables is not relevant in
the present work; however, a concise summary can be found in Ref. [20].
Prompted by the experimental discoveries outlined above, theoretical models were adjusted to

5

∆S2n (MeV)

5
4
3
2
1
0

S2n (MeV)

20

19

18

17

16

15

14

13

12

11 10
Proton Number

17

18

19

20

21

22

23

24 25 26
Neutron Number

40
35
30
25
20
15
10
5

Ca
K
Ar
Cl
S
P
Si
Al
Mg
Na
Ne

0
-5
16

Figure 2.2: Two neutron separation energies for isotopes of neon (Z = 10) through calcium (Z =
20). The most neutron rich isotopes of Mg, Na and Ne do not demonstrate a dramatic drop in
separation energy at N = 20, indicating a quenching of the shell gap. The top panel is a plot of the
difference in two neutron separation energy between N = 21 and N = 20 as a function of proton
number. S2n values are calculated from Ref. [12].

6

E ∗ (2+ ) (MeV)

N = 20 Isotones
4

Si

3

Ca

S

2

Ar

1
Ne

Mg

0
10

12

16

14

18

20
Proton Number

Figure 2.3: 2+ first excited state energies for even-even N = 20 isotones, 10 ≤ Z ≤ 20. The
excited state energy drops suddenly to below 1 MeV for Z ≤ 12, indicating a quenching of the
N = 20 shell gap. The 2+ energy for neon is taken from Refs. [14, 15]; all others are taken from
the appropriate Nuclear Data Sheets [16–19].

better reproduce the anomalies observed for neutron rich nuclei around N = 20. One important
adjustment is the extension of the shell model space to include neutron p f shell components,
although in the interest of conserving computing power the models were—and often still are—
truncated to include only a subset of p f shell. Further truncations include the allowance of only
certain modes of excitation—for example, allowing a maximum of only two or four neutrons to be
promoted into p f levels.1
Another major adjustment made to shell models was the development of effective interactions
which more accurately reproduce single particle energies for cross-shell nuclei. A widely used
interaction is the SDPF-M interaction [22], which consists of three parts. The first part, for the
1 Such excitations also leave neutron holes in the sd shell and are often referred to as “multi-particle,

multi-hole” (or np-nh) excitations.

7

Gap size (MeV)

N = 20 Shell Gap

7
Conventional (40 Ca)
6
5
SDPF-M Interaction
4
3
2
1
0
8

10

12

14

16

18
20
Proton number

Figure 2.4: N = 20 shell gap for even-Z elements, Z ≤ 8 ≤ 20, calculated using the SDPF-M
interaction. The gap size is large (∼ 6 MeV) for calcium (Z = 20) and remains fairly constant
from Z = 18 to Z = 14. Below that, the gap size begins to diminish rapidly, reaching a value of
∼ 2 MeV for oxygen (Z = 8). Adapted from Ref. [21].
sd shell, is basically the Universal SD (USD) interaction [23], which consists of two-body matrix elements (TBME) that have been fit to reproduce experimental data on stable sd shell nuclei.
The second component, relevant to the p f shell, is the Kuo-Brown (KB) interaction [24]. The
KB interaction is obtained from the renormalized G-matrix, which is based on realistic nucleonnucleon scattering. The final part, pertaining to the sd-p f cross-shell region, was originally developed in Ref. [25]. It begins with Millener-Kurath (MK) interaction [26] and adjusts the TBME
0 f7/20d3/2 | V | 0 f7/2 0d3/2 JT , J = 2-5 and T = 0-1. These TBME have all been scaled by a
factor A−0.3, similar to what is done for the USD interaction. As demonstrated in Fig. 2.4, the
SDPF-M interaction reproduces the quenching of the N = 20 shell gap for neutron rich nuclei.
The figure is a plot of the N = 20 gap size for even-Z elements from oxygen (Z = 8) to calcium

8

attractive

repulsive

p
n

π
n

j−

p

spin

′
j+

j+

′
j+

wave function
of relative motion

Figure 2.5: Left panel: Feynman diagram of the tensor force, resulting from one-pion exchange
between a proton and a neutron [27]. Right panel: Diagram of the collision of a spin-flip nucleon
′
′
pair { j− , j+ } (left) and a non spin-flip pair { j+ , j+ } (right). In the spin-flip case, the wave
function of relative motion is aligned parallel to the collision direction, resulting in an attractive
interaction, while in the non spin-flip case the wave function of relative motion is aligned
perpendicular to the collision direction, resulting in a repulsive interaction [27].

(Z = 20) . The predicted gap size is large for calcium, around 6 MeV, and stays fairly constant for
nuclei with Z = 18, 16, 14. Below Z = 14, the gap size steadily decreases, reaching a value of
around 2 MeV for oxygen (Z = 8).

2.2 Theoretical Explanation
Changes in nuclear shell structure are driven by the tensor component of the effective nucleonnucleon interaction [27–29]. This force is the result of one-pion exchange between nucleons, as
illustrated in the Feynman diagram in the left panel of Fig. 2.5. Between protons and neutrons,
the tensor force can be either attractive or repulsive, depending on the orbital and total angular
momenta of the particles. If the orbital angular momenta of the proton and neutron are denoted
′
by ℓ and ℓ′ , respectively, the total angular momenta will be j± = ℓ ± 1/2 and j± = ℓ′ ± 1/2. In

′
the case of “spin-flip” partners { j± , j∓ }, the tensor force is attractive. This is illustrated for

′
nucleons with j− and j+ in the right panel of Fig. 2.5. The two colliding nucleons have a high

relative momentum, causing the spatial wave function of relative motion to be narrowly distributed

9

Stable

Neutron
Rich

′
j+

ν 0 f7/2

′
j+
′
j−

N = 20
′
j−

π 0d5/2

ν 0 f7/2
ν 0d3/2
N = 16

ν 0d3/2
ν 1s1/2

ν 1s1/2
π 0d5/2

j+

j+

Figure 2.6: Schematic illustrating the role of the tensor force in driving changes in nuclear shell
structure. Thick arrows represent a strong interaction and thin arrows a weak one. In the case of
stable nuclei near N = 20 (left panel), there is a strong tensor force attraction between 0d3/2
neutrons and 0d5/2 protons, as well as a strong repulsion between 0 f7/2 neutrons and 0d5/2
protons. These interactions lower the 0ν d3/2 and raise the ν 0 f7/2 , resulting in a large gap at
N = 20. In contrast, neutron rich nuclei near N = 20 (right panel) have a deficiency in 0d5/2
protons, weakening the attraction to 0ν d3/2 and the repulsion to ν 0 f7/2 . This causes the 0ν d3/2
to lie close to the 0ν f7/2 , reducing the gap at N = 20 and creating a large gap at N = 16 [27].

in the direction of the collision. This results in an attractive tensor force, analogous to the case of a
′
′
deuteron. In the opposite case of { j± , j± }, illustrated for { j+ , j+ } in the right panel of Fig. 2.5,

the wave function of relative motion is stretched perpendicular to the direction of the collision,
resulting in repulsion. Thus the tensor force is attractive for proton-neutron pairs with { j± , j∓ }
and repulsive for pairs with { j± , j± }.
The role of the tensor force in driving changes in nuclear shell structure is illustrated in Fig. 2.6.
In stable nuclei near N = 20, the proton 0d5/2 level ( j+) is full or nearly filled. The result is a
′
strong tensor force attraction to 0d3/2 neutrons ( j− ), as well as a strong tensor force repulsion to

′
0 f7/2 neutrons ( j+ ). These interactions serve to raise the ν 0 f7/2 and lower the ν 0d3/2, producing

a large shell gap at N = 20. In contrast, neutron rich nuclei near N = 20 are deficient in 0d5/2 protons. This weakens both the π 0d5/2-ν 0d3/2 attraction and the π 0d5/2-ν 0 f7/2 repulsion, resulting
in a raising of the ν 0d3/2 and a lowering of the ν 0 f7/2 relative to stable nuclei. In this configuration, ν 0d3/2 and ν 0 f7/2 are close in energy, quenching the large gap at N = 20. Furthermore, the

ν 0d3/2 is raised in energy relative to the ν 1s1/2 , forming a large gap at N = 16.

10

Normal

Intruder

(a)

p f shell

(b)
sd shell

semi
magic

(c)

sd shell

(d)

p f shell

sd shell

open
shell

π

ν

sd shell

π

ν

Figure 2.7: Sources of correlation energy. Thick grey lines represent a strong correlation; thin
grey lines a weak one. The top panels [(a) and (b)] demonstrate the case of nuclei with closed
neutron shells, while the bottom panels [(c) and (d)] show nuclei without a closed shell. In both
cases, the intruder configurations [(b) and (d)] produce a stronger correlation energy, with the
greatest energy gain coming from the configuration in (b) [30].

2.3 Correlation Energy
Reduction of the shell gap at N = 20 greatly enhances the contribution of neutron multi-particle,
multi-hole (np-nh) excitations across the 0d3/2-0 f7/2 gap. Such arrangements are often referred
to as “intruder configurations,” the idea being that the p f shell component is intruding on the more
conventional sd shell arrangement. As pointed out in 1987 by Poves and Retamosa [31], intruder
configurations can result in enhanced binding due to correlation interactions between nucleons.
Correlation interactions include the proton-neutron quadrupole interaction and pairing between
like nucleons [32], with the binding gain of proton-neutron interactions being much greater than
that of n-n or p-p pairing [30]. If the binding energy to be gained from correlation interactions is

11

full
calculation

10

0p–0h

5

Probability (%)

S2n (MeV)

15

100
80

0p–0h

60

2p–2h

40

4p–4h

20

0
8

9

10

11

0

12 13 14 15
Proton Number

8

9

10

11

12 13 14 15
Proton Number

Figure 2.8: Left panel: Two neutron separation energy as a function of proton number, for N = 20
isotones, 9 ≤ Z ≤ 14. The dashed line is a shell model calculation truncated to 0p-0h, while the
solid line is the same calculation without truncations. The triangle markers are experimental data,
with the square marker being a more recent datum for 30 Ne [33]. The cross at 29 F is the result of
a 2p-2h calculation which incorrectly predicts an unbound 29 F. Right panel: Occupation
probabilities of 0p-0h (dotted line), 2p-2h (solid line), and 4p-4h (dashed line) configurations.
Figure adapted from Ref. [34].

similar to that which is lost by exciting the nucleons into the p f shell, intruder configurations will
play a significant role in the low-lying structure of the nucleus in question.
Fig. 2.7, adapted from Ref. [30], schematically illustrates the role of correlations in nuclei with
normal [panels (a) and (c)] and intruder [panels (b) and (d)] configurations. In the case of a normal configuration in a semi-magic (closed N = 20 neutron shell, open proton shell) nucleus, the
energy to be gained from correlations is small, limited to pairing interactions between protons.
In contrast, for intruder configurations in semi-magic nuclei a large amount of energy is gained
from correlations between protons and p f shell neutrons as well as protons and sd shell neutron
holes. Similarly, there is an energy gain due to correlations when considering normal versus intruder configurations in open shell (no N = 20 magic closure) nuclei; however, the energy gain
is smaller as the normal configuration already receives a significant amount of correlation energy
from interactions between protons and sd shell neutron holes.

12

2.4 The Neutron Dripline
Perhaps one of the simplest and yet most intriguing problems of nuclear structure is that of existence: to determine which nuclear systems are bound, energetically able to exist as compound
objects, and which ones are not. For neutron rich nuclei, this problems manifests itself as understanding the location of the neutron dripline, the line beyond which it is not possible to add
more neutrons and still maintain a bound system. One of the most striking features of the neutron
dripline is its abrupt shift towards neutron rich nuclei when transitioning from oxygen to fluorine.
As shown in Fig. 2.9, adding a single proton to oxygen allows for the binding of at least six more
neutrons (although 33 F is likely unbound, this has not yet been experimentally verified [35]).
The reason for this abrupt shift in the fluorine dripline was explored theoretically by Utsuno,
Otsuka, Mizusaki, and Honma [34] with large-scale shell model calculations done using the Monte
Carlo Shell Model (MCSM) [36]. The authors argue that quenching of the N = 20 shell gap
allows for significant mixing between normal (0p-0h), intruder (2p-2h), and higher intruder (4p4h) configurations, with the intruder configurations serving to increase the binding energy of the
heaviest fluorines. In their calculations, 29 F is only predicted to be bound when 4p-4h and higher
excitations are included, as shown in the left panel of Fig. 2.8. The calculations further reveal
that the contribution of 4p-4h excitations to the ground state of 29 F is nearly 30% (right panel of
Fig. 2.8). However, even the inclusion of high intruder configurations is not enough to bind 31 F.
Regarding this point, the authors note that it is possible to bind 31 F by lowering the energy of the
neutron 0p3/2 by 350 keV. They argue that a lowered ν 0p3/2 could result from a neutron halo or
halo-like structure.

13

Proton Number

24 25 26 27 28 29 30 31 32 33 34 35 36 37 38

12

23 24 25 26 27 28 29 30 31 32 33 34 35
20 21 22 23 24 25 26 27 28 29 30 31 32

10

19 20 21 22 23 24 25 26 27

29

40 –Mg

37 –Na

34 –Ne

31 –F

16 17 18 19 20 21 22 23 24 –O

8

14 15 16 17 18 19 20 21 22 23 –N
6

22 –C

12 13 14 15 16 17 18 19 20
10 11 12 13 14 15

4
6
3

2
1

4

2

7

17

19 –B

9 10 11 12
14 –Be
8 9
11 –Li
8 –He

6

3 –H

stable
n–rich

1 –n

0
0

5

10

15

20

25

30
Neutron Number

Figure 2.9: Chart of stable and bound neutron rich nuclei up to Z = 12. Note the abrupt shift in the neutron dripline between oxygen
(Z = 8) and fluorine (Z = 9) .

14

2.5 Previous Experiments
A simple yet important test of theoretical predictions is that of determining which nuclei are bound
and which nuclei are not. As outlined in the previous section, theoretical reproduction of the
fluorine neutron dripline is a difficult and enlightening problem—one which cannot be fully solved
without ad hoc additions to the calculations. Experimentally, the neutron dripline around fluorine
and oxygen has been explored a number of times [37–43], and it was determined in Ref. [37] that
the fluorine dripline extends at least to 31 F. This experiment, performed at RIKEN, impinged a
94.1 AMeV beam of 40 Ar on a 690 mg/cm2 tantalum target, separating reaction products in the
RIPS fragment separator [44]. As shown in Fig. 2.10, eight counts of 31 F were observed in the
experiment, confirming its bound nature. No events were observed for oxygens with A ≥ 26, and
the authors argue that 26,27,28O are unbound based on their non-observation and interpolation of
observed yields for other isotopes.
Another useful test of theory is the determination of excited state energies. By tuning shell
model calculations to reproduce observed energies, one can better understand the makeup of the
nuclei in question. In 2004, Elekes et al. [45] measured bound excited states in 25,26,27F using the
reactions 1 H(27 F, 25,26,27Fγ ) and measuring de-excitation gamma-rays in an array of 146 NaI(Tl)
detectors surrounding the target. The observed (confidence level ≥ 2σ ) gamma transitions are
shown in Fig. 2.11 along with sd shell model predictions; they are also summarized in Table 2.1.
The authors note that the 727 keV transition in 25 F and the 681 keV transition in 26 F are reasonably
well reproduced by sd shell model predictions which place the respective excited levels at 911 keV
(J π = 1/2+ ) and 681 keV (J π = 2+ ). However, the sd calculations fail to reproduce the energies
of the higher excited states in 25,26F. Furthermore, they fail to reproduce the energies of either of
the observed transitions in 27 F, regardless of whether the transitions are placed in parallel or in
cascade. In all cases, the observations are significantly lower in energy than the sd shell model
predictions.
The authors of [45] continue by presenting the results of a 27 F shell model calculation in the
full sd p f model space. This calculation predicts the first excited state to be a 1/2+ at 1.1 MeV.
15

Z

12
11

N = 2Z + 2
32 Ne

N = 2Z + 4

10
29 F

31 F

9
28 O

8
25 N

23 N

7
20 C

22 C

6
19 B

17 B

5
3.1

3.2

3.3

3.4

3.5

3.6

3.7

3.8 3.9
A/Z

Figure 2.10: Particle identification plot from the reaction of 40 Ar at 94.1 AMeV on a tantalum
target. Eight events of 31 F were observed in the experiment, while no events were observed for
26 O or 28 O. Figure adapted from Ref. [37].
Although still ∼ 300 keV higher than experiment, the lowering of the 1/2+ energy to better agree
with experiment indicates that the inclusion of cross-shell excitations is necessary for a complete
description of neutron rich fluorines. The authors also speculate that the 504 keV transition observed in 27 F is from the decay of 1/2− intruder state arising from simultaneous and correlated
proton-neutron excitations across the Z = 8 and N = 20 shell closures. They do not present any
calculations to support this assertion.
Recently, a measurement of unbound excited states in 25,26F was performed using the MoNASweeper setup of the present work [46]. Unbound excited states in 25,26F were populated using
proton knockout (25 F) and charge exchange (26 F) reactions on a 26 Ne beam. A resonance in 25 F

16

25

1 H(27 F, 27 Fγ )

504 keV

20
counts / 48 keV

(a)

777 keV

15

1/2+ 1997
10
1/2+ 1100 [sdpf ]
5
5/2+
0
18

1 H(27 F, 26 Fγ )

0

(b)

468 keV
counts / 48 keV

15
665 keV

12
9
6

2+
4+
1+

3
0

1 H(27 F, 25 Fγ )

681
353
0

(c)

30
counts / 48 keV

727 keV
3/2+ 3073
20
1753 keV
1/2+

911

5/2+

10

0

0
0

500 1000 1500 2000 2500
Eγ (keV)

Figure 2.11: Gamma decay spectra of bound excited states in 27,26,25F, along with sd shell model
predictions on the right of the figure. For 27 F, the 1/2+ excited state prediction from a calculation
done in the full sd p f model space is also included in red. Adapted from Ref. [45].

17

Table 2.1: Bound state gamma transitions observed in [45].
Nucleus Peak Position (keV)
25 F
26 F

1/21/2, 3/2+
5/2+
9/2+
3/2+

468(17)
665(12)

27 F

5/2+

727(22)
1753(53)

504(15)
777(19)

4800 keV
This work (28 keV)

4296 keV

Sn = 4226 keV30
3700 keV28

3753 keV

3300 keV28

3230 keV
3074 keV

1753 keV15
1/2+

727 keV15

911 keV

5/2+ (g.s.)
Theory

Experiment

Figure 2.12: Comparison of experimental and theoretical levels for 25 F from Ref. [46]. The
superscripts next to the experimental level energies denote references numbers within [46], and
the measurement of [46] is labeled “this work” in the figure.

18

was observed with 28 (4) keV decay energy, corresponding to an excited level at 4254 (112) keV.
The authors also claim to observe a resonance in 26 F with decay energy 206+36 keV; such a reso−35
nance would correspond to an excited level in 26 F at 1007 (119) keV. The 25 F result is compared
to shell model calculations that allow for 2p-2h neutron excitations across the sd-p f shell gap, as
well as proton excitations between the p and sd shell. The calculations predict a 1/2− excited
state at 4296 keV in 25 F. If this is indeed the resonant state observed in the experiment, then the
calculations would be in excellent agreement. Based on calculated spectroscopic overlap with the
ground state of 26 Ne, the authors argue that their observed resonance is likely the 1/2− . If they are
correct, it would indicate that a shell model including neutron 2p-2h and proton p-sd excitations is
sufficient to describe the level structure of 25 F, at least up to and around the one-neutron separation
energy. No comparison to theory is made for the 26 F resonance as experimental resolution and
background contributions make the interpretation of this decay spectrum extremely difficult.
The role of p f shell configurations in the binding of 29,31F can also be investigated by measuring properties of neighboring elements in the same mass region as the heavy fluorines. Along
these lines, measurements of two-proton knockout cross sections to excited states in 30 Ne have
recently been performed with the goal of understanding the contribution of intruder configurations
to the structure of 30 Ne and the implications for the binding of 29,31F [14]. In this work, a beam
of 32 Mg was impinged on a beryllium target, with reaction products separated and identified in
the S800 spectrometer [47] and gammas collected in an array of 32 segmented germanium detectors. Two transitions were observed in coincidence with 30 Ne: a strong one at 792(4) keV and a
weaker one at 1443(11) keV, as shown in Fig. 2.13. The 792 keV transition was also measured in
Ref. [15] with far lower statistics. The authors of [14] place their observed transitions in cascade,
assigning the lower transition to the 2+ first excited state in 30 Ne and the higher transition to the
4+ second excited state at 2235 keV. The authors then compare their measured excited state energies to three separate shell model calculations: one in the sd shell only; another including 2p-2h
cross-shell excitations; and the final one, done in the MCSM, including mixed 0p-0h, 2p-2h, and
4p-4h configurations. As with 27 F, the sd shell calculation greatly overpredicts the first excited

19

40
2596 2+

4+ 2500 4+
792

Counts / 10 keV

30

(4+) 2235

0+ 2710
4+ 2520

957

2+ 1000

2558
2+ 1800

1443
2+

(2+) 792
792
0+

0+

0+

Experiment

20

(a) USD

(b) 2p2h

0+
(c) SDPF-M

10
1443

0
500

1000

1500

2000

2500

3000

3500

Energy (keV)
Figure 2.13: 30 Ne gamma de-excitation spectrum from Ref. [14]. Experimental level assignments
and a variety of theoretical predictions are also included.

state energy, placing it ∼ 1 MeV too high at 1800 keV. The 2p-2h and MCSM calculations are
closer to experiment, lowering the energy to 957 keV and 1000 keV, respectively.
Following the discussion of excited state energies, the authors of [14] discuss their results for
two-proton knockout cross sections from 32 Mg to the 0+ , 2+ , and 4+ states in 30 Ne. In all cases,
their results are smaller than the 2p-2h and MCSM predictions by a factor of four or greater (the
sd shell model predicts even higher cross sections). The authors argue that the mismatch between
theory and experiment is the result of 30 Ne having a large 4p-4h component (∼ 50% in the ground
state), resulting in a smaller than predicted overlap with 32 Mg, whose ground state was measured to
have 2p-2h components at a level of 90% or greater [48]. This mismatch is not accounted for in the
shell models, resulting in the overprediction of 2p knockout cross sections. The authors speculate

20

that extending the sd p f shell model space to include the 0 f5/2 and 1p1/2 levels—as opposed
including only the lower lying 0 f7/2 and 1p3/2 as presently done—could predict a greater 4p-4h
composition of 30 Ne (and also of 29,31 F). They also explore the possibility that dripline effects
such as weak binding can increase the 4p-4h occupation of 30 Ne. Weak binding allows the wave
function for a low-ℓ orbital to significantly extend beyond the average nuclear radius. This causes
the level energy to be less sensitive to changes in radius (mass number), altering the density of
states near the Fermi surface and causing multi-particle, multi-hole configurations to be favored.
Shell models are not able to account for these changes due to their use of harmonic oscillator wave
functions. The authors simulate these effects by performing an ad hoc lowering of the 0 f7/2 and
1p3/2 energies by 800 keV and making shell model calculations. The calculations predict the 30 Ne
ground state to be 49% 4p-4h, in agreement with the authors’ observation. The 4p-4h components
of the ground states of 29 F and 31 F are predicted to be 51% and 36%, respectively. This lowering
of the 0 f7/2 and 1p3/2 energies is similar to the 350 keV lowering of the 1p3/2 in Ref. [34], which
allows for a prediction of bound 31 F.
The present work deals with neutron decay spectroscopy performed on unbound states in 27 F
and 28 F. In particular, measurements of the ground state binding energy of 28 F and the excitation
energy of a neutron-unbound excited state in 27 F are presented and interpreted via comparison
with shell model predictions. These measurements provide experimental data in a region where
very little is known, and they will help to improve upon the current understanding of the evolution
of nuclear structure in neutron rich nuclei.

21

Chapter 3
EXPERIMENTAL TECHNIQUE

3.1 Invariant Mass Spectroscopy
The neutron-unbound states under investigation in this work decay primarily through neutron emission, a process which happens on a very short timescale (∼ 10−21 s). The decay products are the
emitted neutron, n, and the residual charged fragment, f , as illustrated in Fig. 3.1. The decay
energy of the de-excitation is calculated using the technique of invariant mass spectroscopy. This
technique is derived from the conservation of relativistic four-momentum,
P = (E, p) ,

(3.1)

where P is the relativistic four-momentum of the particle in question; E is its total energy; and
p is its Euclidean momentum vector.1 Conservation of P can be expressed as

Pi = P f + Pn ,

(3.2)

where the subscripts i, f , and n refer to the initial nucleus, residual nucleus and emitted neutron,
respectively.
Squaring both sides of Eq. (3.2) yields
P2 = P f + Pn
i

2

≡ W 2,

(3.3)

where W is defined to be the invariant mass of the system, a constant. In Euclidian terms, this is
expressed as
W2 =

E f + En

2

− p f + pn 2

= m2 + m2 + 2 E f En − p f · pn .
n
f
1 Here we use natural units, c ≡ 1.

22

(3.4)
(3.5)

pn(lab)

(cm)
pn

θ

p f(lab)

p f(cm)

Lab Frame

Center of Mass Frame

Figure 3.1: Decay of an unbound state via neutron emission.

Taking the square root of Eq. (3.5) and expanding the momentum dot product gives an expression for W :

W=

m2 + m2 + 2 E f En − p f pn cos θ ,
n
f

(3.6)

where θ is the opening angle between the fragment and the neutron in the lab frame. Finally,
subtracting the masses of the decay products results in an expression for the decay energy:

Edecay =

m2 + m2 + 2 E f En − p f pn cos θ − m f − mn .
n
f

(3.7)

In order to make a measurement of the decay energy, as expressed in Eq. (3.7), it is necessary
to measure the energy and angle of each decay product as it leaves the reaction target. The way in
which these quantities are calculated will be explained in Section 4.3.
One limitation of invariant mass spectroscopy is that it is only able to measure the difference
in energy between the initial unbound state and the state to which it decays. In some cases, this

23

S1n

n
n

γ
A Z
NX

A−1 Z
N−1 X

Figure 3.2: Illustration of the two possible decay processes of an unbound state which lies higher
in energy than a bound excited state of the daughter. The state can either decay through direct
neutron emission to the ground state of the daughter (grey arrow) or by neutron emission to an
excited state in the daughter and subsequent gamma emission (black arrows).

causes an ambiguity in the absolute placement of the level in question. In particular, if the daughter
nucleus has bound excited states, the unbound state has the option to neutron decay by one of two
processes, as illustrated in Fig. 3.2. The first is direct neutron emission to the ground state of the
daughter, and the second is a two step process: the unbound state first neutron decays to a bound
excited state of the daughter which then de-excites by gamma emission. It is possible to distinguish
between the two processes experimentally by measuring de-excitation γ −rays in coincidence with
the neutron and the charged fragment, and this is the approach taken in the present work.

3.2 Beam Production
The reaction used to populate unbound states in 28 F is single proton knockout on a 29 Ne beam,
while excited states in 27 F are likely populated in a two step process, starting with proton knockout
to a highly excited state in 28 F. This highly excited state then decays into the unbound state of
27 F

for which the breakup is observed. It is also plausible that unbound excited states in 27 F are

populated by direct 1p-1n knockout. The desired beam nucleus, 29 Ne, is β −unstable (t1/2 ≃ 15 ms
24

e
29 N

(62

V/
Me

u)

K500 Cyclotron

pl e
dC
yc
l ot

ron

s

Final Focus

MeV/u)

Co
u

48 Ca (12

9
A1

00

m
ag
Fr

en

t or
ara
ep
tS

Aluminum Wedge
Production Target
(1316 mg/cm2 Be)
48 Ca

(140 MeV/u)

K1200 Cyclotron

Figure 3.3: Beam production.

[49]), so it cannot be accelerated directly; instead the method of fast fragmentation [50] is used for
beam production.
A diagram of the beam production mechanism is shown in Fig. 3.3. A beam of stable 48 Ca is
first accelerated to an energy of 140 MeV/u in the NSCL coupled K500 and K1200 cyclotrons [51],
exiting the K1200 fully ionized. It is then impinged on a beryllium target with a thickness of
1316 mg/cm2 . The beam undergoes fragmentation and other reactions in the target, producing a
wide variety of nuclear species. These reaction products pass through the A1900 fragment separator [52] which selects 29 Ne fragments based on magnetic rigidity, Bρ = p/q. The selection is

25

accomplished by tuning the four dipole magnets of the A1900 to the expected rigidities of the 29 Ne
reaction products, with the final dipole being at 3.469 Tm. An achromatic aluminum wedge is also
included after the second dipole to selectively disperse reaction products and improve separation.
This wedge also has the undesired effect of creating tertiary reaction products which can have
the same magnetic rigidity as the desired 29 Ne. Finally, slits are located at the intermediate focal
plane, allowing the user to tune the total momentum acceptance of the device. Since the expected
production rate of 29 Ne fragments is low, the slits are opened to a full momentum acceptance of
3.93%. The final energy of the 29 Ne beam delivered to the experimental vault is 61.9 ± 4.7 MeV/u.
Since the A1900 is only able to make selections based on Bρ , it is not possible to obtain a pure
beam of 29 Ne. Isotopes of other elements will end up with roughly the same rigidity after passing
through the device, and, as mentioned before, tertiary reaction products created in the wedge can
also compose a portion of the beam. The A/Z of these contaminants is different from that of the
beam, causing their velocity to also differ. As a result, the contaminants can be separated in off-line
analysis as explained in Section 4.2.1.

3.3 Experimental Setup
A diagram of the experimental setup is shown in Fig. 3.4. After exiting the A1900, the beam
is passed through a pair of plastic timing scintillators which provide a measurement of its time
of flight. It also passes through a pair of position sensitive Cathode Readout Drift Chambers
(CRDCs), which allow a calculation of its incoming position and angle. After the second CRDC,
the beam is focused by a quadrupole triplet magnet onto the reaction target, which is made of 9 Be
and is 288 mg/cm2 in thickness.
After undergoing reactions in the Be target, there are up to three types of particles which
need to be measured: charged fragments, neutrons and gammas. Gammas are recorded in the
CAEsium-iodide ARray (CAESAR) [53], an array of 182 CsI(Na) crystals that surrounds the target
and measures the gamma energies. Neutrons continue to travel at close to beam velocity and
are recorded in the Modular Neutron Array (MoNA) [54, 55] which is located at zero degrees
26

behind the target and measures the neutrons’ positions and times of flight. Charged fragments
also continue to travel at close to beam velocity but are deflected away from zero degrees by the
Sweeper magnet [56], a dipole. After deflection, the charged particles are passed through a series of
detectors whose measurements make it possible to identify their nuclear species and to reconstruct
their four-momenta at the target. Further details regarding the operation of these experimental
systems are presented in the following subsections.

27

Beam tracking
detectors

CsI(Na) Array
Quadrupole triplet (CAESAR) Sweeper magnet

Ionization
chamber

Reaction Target
(288 mg/cm2 Be)
Timing scintillators

Charged fragment
tracking detectors

x

y

z

DAQ

Modular Neutron
Array (MoNA)

Timestamp

“Thin” (5 mm) scintillator (t, ∆E)
“Thick” (150 mm) scintillator (E)
x
DAQ

y
z

Figure 3.4: Experimental setup.

28

3.3.1 Beam Detectors
As mentioned, the time of flight of the beam is measured in a pair of plastic scintillators. The
first is located at the focal plane of the A1900, and the second is located 44.3 cm upstream of
the reaction target, resulting in a total flight path of 10.44 m. When a charged particle passes
through one of the plastic scintillators, it creates electron-hole pairs which recombine, emitting
photons. These photons are collected in a photo-multiplier tube (PMT) that is optically coupled to
the plastic. The PMT serves to amplify the luminous signal and convert it to an electrical signal that
can be recorded. The detection process happens on a fast timescale, allowing a time measurement
with good resolution (< 1 ns). The scintillator located at the A1900 focal plane is 1008 µ m thick,
while the one close to the reaction target is 254 µ m. Each scintillator is made of Bicron BC-404
material (H10 C9 ) [57] and is coupled to a single PMT. In addition to the timing scintillators, a time
measurement is also taken from the cyclotron radio-frequency (RF) signal.
The emittance of the incoming beam particles is measured with a pair of position sensitive
CRDCs. A dramatization of the operation of a CRDC is shown in Fig. 3.5. Each detector is filled
with a gas mixture of 80% CF4 and 10% iso-butane at a pressure of 50 Torr. When charged particles
pass through the gas, they ionize some of its molecules, releasing electrons. These electrons are
subjected to a −250 V drift voltage, causing them to travel in the non-dispersive2 direction towards
an anode wire which collects the charge. The anode wire is held at a potential of +1100 V. Located
near the anode wire are 64 aluminum pads with a 2.54 mm pitch, and the charge collected on the
anode wire in turn induces a charge on these pads. Additionally, a Frisch Grid is used to remove
any dependence of the pulse amplitude on where the charged fragment hits the gas.
The position of the charged fragment in the dispersive direction is determined from the distribution of the charge on the aluminum pads. The charge collected on each pad is plotted as a
function of pad number, and the centroid is determined by fitting with a Gaussian. This centroid
can be converted from pad space to position space by a linear transformation, using the pad pitch
2 Through this document, the dispersive plane will be referred to as x and the non-dispersive plane as y.

z refers to the beam axis.

29

-V
Charged Fragment Track

-E

Gas Ionization
e− Drift

Frisch Grid

Anode Wire

Cathode Pads

Figure 3.5: Dramatized view of the operation of a CRDC. Charged fragments interact in the gas,
releasing electrons. The electrons are subjected to a drift voltage in the non-dispersive direction
and are collected on the anode wire, in turn causing an induced charge to form on a series of
aluminum pads. Position in the dispersive direction is determined by the charge distribution on
the pads, while position in the non-dispersive direction is determined from the drift time of the
electrons.

30

of 2.54 mm as the slope. The offset is determined from data taken when a tungsten mask, with
holes drilled at known locations, shadows the detector. In the non-dispersive direction, the fragment position is determined from the time it takes the electrons to drift from the interaction point to
the anode wire. The drift time is converted to absolute position using data taken with the tungsten
mask in place.
Located after the beam tracking CRDCs is a quadrupole triplet magnet which serves to focus
the beam particles onto the target. The field of this magnet is mapped, allowing the position and
angle on the reaction target to be calculated from the position measurements of the beam tracking
CRDCs. The outer quadrupole magnets have an effective length of 22.8 cm, while the inner quad
has an effective length of 41.6 cm. Each quad is separated by a free drift of 22.8 cm. The optics of
the triplet are tuned to produce a beam-spot which is narrow in angle and wide in position. Such a
parallel beam is desired, as it improves the transmission of reaction products through the Sweeper.

3.3.2 Sweeper
The Sweeper magnet is a dipole with a bending angle of 43° and a radius of 1 meter. It has a large
vertical gap of 14 cm to allow neutrons to pass on to MoNA uninhibited. Its maximum rigidity is
4 Tm, and for the present experiment it was set to a rigidity of 3.3065 Tm. This setting was chosen
because it optimizes the transmission of 27 F reaction products.
Following the Sweeper is a pair of position sensitive CRDC detectors. The operation of these
CRDCs is identical to that of the beam tracking CRDCs, as explained in Section 3.3.1. However,
the specifications of the detectors are different. Each detector measures 30 × 30 cm2 and has 128
pads (2.54 mm pitch) in the dispersive direction. The gas pressure for each detector is 50 torr,
while the anode and drift voltages are set at +950 V and −800 V, respectively.
Following the downstream CRDC is an ionization chamber. It serves to measure energy loss,
which is useful for element identification as explained in Section 4.2.3. Similar to the CRDCs, the
ionization chamber is gas-filled, here the gas being a mixture of 90% argon and 10% methane at
300 torr. Charged particles traveling through the gas release electrons which travel via a −800 V
31

Table 3.1: List of charged particle detectors and their names. Detectors are listed in order from
furthest upstream to furthest downstream.
Detector Name

Detector Type

RF
A1900 Scintillator
TCRDC1
TCRDC2
Target Scintillator
CRDC1
CRDC2
Ion Chamber
Thin Scintillator
Thick Scintillator

Time Measurement
Timing Scintillator
Cathode Readout Drift Chamber
Cathode Readout Drift Chamber
Timing Scintillator
Cathode Readout Drift Chamber
Cathode Readout Drift Chamber
Ionization Chamber
Timing/Energy Loss Scintillator
Total Kinetic Energy Scintillator

drift voltage to sixteen charge collection pads held at +1100 V. The total amount of charge collected on the pads is summed to give a measurement of energy deposited in the gas.
Downstream of the ionization chamber are two plastic (BC-404) scintillators. These detectors
operate in the same way as the timing scintillators described in Section 3.3.1. Due to their larger
area of 40 × 40 cm2 , the scintillators are coupled to four PMTs, with each PMT being located near
a corner of the detector. The upstream scintillator is 0.5 cm in thickness, and its primary purpose
is to measure the time of flight of the charged fragments. Additionally, the charge deposited in the
detector can be used as an energy loss measurement, complimenting that of the ionization chamber.
The downstream scintillator is 15 cm thick and stops the beam. Thus the charge collected in the
detector gives an indication of the total kinetic energy of the fragment.
For ease of reference, each charged particle detector in the setup is given a unique name. These
are listed in Table 3.1. Furthermore, each of the four PMTs in the thick and thin scintillators
is given a numeric designation, outlined in Table 3.2. The point at which reaction products are
narrowest in dispersive position is referred to as the “focal plane” or “focus,” despite not being a
true focus in the ion-optical sense. Similarly, the chamber housing all of the detectors which are
downstream of the Sweeper is called the “focal plane box.”

32

Table 3.2: Numeric designations for the PMTs in the thin and thick scintillators.
Number
0
1
2
3

PMT Location
Upper-Left
Lower-Left
Upper-Right
Lower-Right

C D

A B

Target

E F

GHI

Beam
//

MoNA
658 cm

93 cm

93 cm

71 cm

x
0
y

1m

z

Figure 3.6: Location of MoNA detectors. Each wall is sixteen bars tall in the vertical (y)
direction, and the center of each wall in the vertical direction is equal to the beam height. The
black lines in the figure indicate the central position in z of the corresponding MoNA bar.

3.3.3 MoNA
MoNA is an array of 144 organic plastic scintillator bars, each bar being made of BC-408 material,
which has an H:C ratio of 1.104 [57]. The bars measure 200×10×10 cm3 and are coupled through
light guides to a PMT on either end. Due to the modular design of the detector, it can be arranged
in a number of different configurations. In the present experiment, the array was arranged in nine
walls, each 16 detectors high. Each wall is labeled with a letter from A–I, with wall A being
the most upstream. Within a wall, the bars are labeled numerically from 0—15, with bar 0 being
closest to the floor. The walls are arranged into four groups: the first three are two walls deep,

33

while the final one is three walls deep. Within a group, the detectors are placed flush against one
another. The spacing between groups and overall distance from the reaction target are indicated in
Fig. 3.6.
Since neutrons lack electrical charge, they cannot directly excite atoms in the MoNA bars to
release scintillation light. Instead the neutrons interact via the strong interaction with protons in
the hydrocarbon. When a neutron strikes a proton directly, it knocks the proton out of the lattice.
This recoil proton can then generate scintillation light in a process similar to the one described in
Section 3.3.1. This light is then collected in the PMTs and converted into electrical signals. The
anode signal is sent to a constant fraction discriminator (CFD), where the exact time of the pulse
is determined from the signal shape. The output of the CFD is fed into a time to digital converter
(TDC), which, along with a common stop signal from the target scintillator, measures the time of
flight of the neutron. The dynode signal is fed into a charge to digital converter (QDC), which
measures the signal size, giving an indication of the amount of energy deposited in the plastic. The
light deposition signal is calibrated into units of MeV-electron equivalent (MeVee), as explained in
Section 4.1.2.3.

3.3.4 CAESAR
CAESAR is an array of 192 CsI(Na) crystals, situated in ten rings which surround the target. The
rings are labeled A–J, with A being the most upstream. An outline of the detector arrangement
is shown in Fig. 3.7. The crystals in the four outer rings have dimensions of 3 × 3 × 3 in3 , while
those in the inner rings have dimensions of 2×2×4 in3 . The outermost rings each hold ten crystals,
and their neighbors each hold fourteen. The remaining inner rings contain 24 crystals each. The
crystals are encased in an aluminum housing 1 mm in thickness to shield them from water, and the
1.5 mm gap between the crystal and the aluminum is filled with reflective material. Each crystal is
coupled to a PMT for collection of scintillation light. As PMTs rely on the flow of free electrons to
operate, their operation is highly sensitive to the presence of magnetic fields. The Sweeper magnet
produces large fringe fields, on the order of hundreds of Gauss, so a steel magnetic shield was

34

Beam

Figure 3.7: Arrangement of CAESAR crystals. The left panel shows a cross-sectional view
perpendicular to the beam axis of an outer and an inner ring. The right panel shows a cross
sectional view parallel to the beam axis of the nine rings used in the experiment. Figure taken
from Ref. [53].

constructed and placed between the Sweeper and CAESAR.
Due to space limitations, only nine of the ten rings were used in the present experiment; the
ring selected for omission was the one most upstream of the target. This ring was chosen because
the Lorentz boost causes γ −ray emission to be forward focused. In its full configuration, the
efficiency of CAESAR for γ −rays with v/c = 0.3 is approximately 30%, and its in-beam resolution
is approximately 10% [53].
As CsI(Na) is an inorganic material, the process for producing scintillation light differs from
that of organic plastic scintillators discussed previously. When a γ −ray enters the crystal, it can
excite an electron from the valence band into a higher energy band where it is able to drift through
the material. When the drifting electron encounters an impurity in the crystal (here a sodium atom),
it ionizes it. The hole created in this ionized atom can then be filled by another electron, releasing a
photon [58]. CsI(Na) is well suited for gamma detection, as it has good stopping power for γ −rays,
as well as good intrinsic energy resolution.

35

Table 3.3: Logic signals sent between the Sweeper and MoNA subsystems and the Level 3 XLM.
A valid time signal is one which surpasses the CFD threshold.
Logic Signal

Description

Sweeper Trigger
MoNA Trigger
MoNA Valid
System Trigger
Busy

Valid time signal in the thin scintillator upper-left PMT.
Valid time signal in any MoNA PMT.
Valid time signal in at least two PMTs on the same bar.
Coincidence condition satisfied in Level 3.
The system in question is working to process event data.

3.3.5 Electronics and Data Acquisition
Data from MoNA and from Sweeper/CAESAR were recorded on separate data acquisition (DAQ)
machines. Each event was tagged with a unique 64-bit number, and data from the two systems
were merged off-line before being analyzed. This is referred to as running in “timestamp mode,”
although the event tags are not timestamps in the strict sense as the clock generating them does
not run continuously. The experiment was set up to require coincidences between the Sweeper and
MoNA, and CAESAR was essentially a passive add on to the Sweeper system, not factoring into
the trigger logic.
The trigger logic was handled by programmable Xilinx Logic Modules (XLMs), grouped into
three levels depending on their function. “Level 1” and “Level 2” deal with the determination of
whether or not an event in MoNA is valid, with a valid event defined to be one for which—at a
minimum—each PMT on a single bar produces a valid time signal in its respective CFD. “Level 3”
deals with the trigger conditions, involving both the Sweeper and MoNA, necessary for an event
to be deemed a coincidence. Level 3 also contains a clock which runs whenever it is not busy
processing an event; the signal from this clock serves as the unique event tag. In this section,
the trigger logic specific to the timestamp setup is detailed as this mode of operation is unique
to the present experiment. The logic and electronics of the individual subsystems are identical to
previous experiments, and their details can be found in Refs. [53, 59–61].
Fig. 3.8 outlines the coincidence trigger logic. The two subsystems, Sweeper and MoNA, run
independently but only record events when they are told to do so by the Level 3 XLM. A number

36

of signals are sent back and forth between Level 3 and the subsystems, and a description of each
signal is given in Table 3.3. When an event triggers the Sweeper subsystem, it sends a trigger
signal and a busy signal to Level 3. Receiving the busy signal from the Sweeper causes Level 3 to
also go busy, stopping the clock. In parallel, if an event triggers MoNA, it sends a trigger and busy
signal to Level 3 as well as a “valid” signal if at least two PMTs in the same bar have fired.
Upon receiving a trigger signal from the Sweeper, Level 3 opens a coincidence gate of 35 ns,
waiting for a valid signal from MoNA. If it receives one, it sends a “system trigger” signal to each
subsystem, telling them to go ahead and process the event. It also sends a clock signal to scaler
modules that are part of each subsystem’s data acquisition. Once the subsystems have finished
processing, their busy signals to Level 3 cease, readying the system for the next event.
If Level 3 fails to receive a valid signal from MoNA before the coincidence gate closes, it will
never send a system trigger signal to the Sweeper. Failing to receive the system trigger from Level
3, the Sweeper fast clears itself and ceases its busy signal to Level 3.
In the case of an event in MoNA that is not coincident with one in the Sweeper, MoNA will
send, at a minimum, trigger and busy signals to Level 3. This causes Level 3 to go busy and reject
signals from the Sweeper. Without a signal from the Sweeper, the coincidence gate is never opened
and thus Level 3 cannot produce a system trigger. As it will not receive a system trigger signal
from Level 3, MoNA fast clears itself and stops sending a busy signal to Level 3.

37

Scaler
Clock

Sweeper Busy
Level 3

Sweeper
Sweeper Trigger

System Trigger

DAQ

MoNA

Level 1

Level 2
Scaler

Figure 3.8: Diagram of the timestamp trigger logic.

38

MoNA Valid

Merging Program

MoNA Busy

MoNA Trigger

DAQ

Chapter 4
DATA ANALYSIS

4.1 Calibration and Corrections

4.1.1 Sweeper
4.1.1.1 Timing Detectors
Each of the timing scintillators in the Sweeper setup records the time of the interaction as a channel
number in its corresponding TDC. The channel number is then multiplied by a slope of +0.1 ns/ch
to convert into physical units (nanoseconds). This slope is taken simply from the full range of
each TDC divided by the total number of channels. The time measurement also includes a 20 ns
jitter, introduced by the use of Field-Programmable Gate Array (FPGA) delays. To remove the
jitter, the nanosecond time value of the upper-left PMT in the thin scintillator is subtracted from
the nanosecond time value of each individual PMT. For example, the calibrated time signal of the
target scintillator is
(raw)
(raw)
(cal)
ttarget = ttarget · 0.1 − tthin_lu · 0.1 .

(4.1)

It should be noted that the signals are only subtracted after applying the 0.1 ns/ch slope, to
account for any situation where the slope of the thin left-up TDC might be different from that
of other TDCs in the system. Fig. 4.1 shows a sample timing spectrum before and after jitter
subtraction.
After jitter subtraction, each time signal is given an offset to place it at the correct point in
absolute time, with t = 0 defined to be the time at which the beam passes through the target. The
offsets are determined by considering a run in which the target is removed and the Sweeper is tuned
to match the rigidity of the incoming beam. In this case, the velocity of beam particles is known
and can be used to calculate the appropriate time offsets. Once the fully calibrated (including
offsets) time signal has been determined, time of flight between various detectors is calculated
39

Counts / 0.3 ns

Counts / 0.3 ns

1000
800
600
400
200
0
30

35

40

45

50

55 60
Time (ns)

×103
12
10
8
6
4
2
0
30

35

40 45 50 55 60
Jitter Subtracted Time (ns)

Figure 4.1: Example calibrated timing spectra before (left panel) and after (right panel) jitter
subtraction.
by subtracting the calibrated time signal of the upstream detector from that of the downstream
detector. In the case of the thin scintillator, the average signal of all four PMTs is used for time of
flight calculations. For example, the time of flight from the target scintillator to the thin scintillator
is calculated as
(0)

(1)

(2)

(3)

t +t +t +t
ToFtarget→thin = thin thin thin thin − ttarget,
4

(4.2)

where t refers to a calibrated time value, and the numeric superscripts refer to the corresponding
PMT in the thin scintillator, as introduced in Table 3.2.
At two points during the experiment, the voltages on the thin and thick scintillators tripped due
to fluctuations in the vacuum level of the Sweeper focal plane box. In order to protect the PMTs
from sparking, their high voltage controllers were equipped with a safeguard that caused them to
turn off if the vacuum level became too low. Although the PMT voltages were returned to their
previous settings after the trips, the changes caused a noticeable shift in the time measurement of
the thin scintillator PMTs. To account for this, the offset values of the thin scintillator PMTs were
modified after each trip. The offset values were changed such that the central time of unreacted
beam particles remains constant throughout the experiment.

40

Raw Charge (channel)

1500
1000
500
0
0

20

40

60

80 100 120
Pad Number

2000

Raw Charge (channel)

Raw Charge (channel)
Raw Charge (channel)

2000

1500
1000
500
0

0

10

20

30

40 50 60
Pad Number

2000
1500
1000
500
0
0

20

40

60

80 100 120
Pad Number

10

20

30

40 50 60
Pad Number

2000
1500
1000
500
0

0

Figure 4.2: Pedestals for each of the CRDC detectors. Color represents the number of counts per
bin.
4.1.1.2 CRDCs
As explained in Section 3.3.1, CRDC position in the x direction is calculated from the distribution
of charge on the pads. Before doing this, the raw charge values are pedestal suppressed and gain
matched. Pedestal suppression is done using data taken while no signal was present in the CRDCs.
The signals on each CRDC pad are summarized in Fig. 4.2, and an example histogram of the signal
on a single pad is shown in Fig. 4.3. To determine pedestal values, a histogram of the signal on
each pad is fit with a Gaussian. Since the histogram shape is skewed slightly, each bin is given
equal weight in the fit1 , which puts the Gaussian centroid at a value close to the true centroid of the
histogram. This centroid is then divided by the number of samples recorded during the runs (eight
in each case) to determine the pedestal offset for the pad in question. In the analysis, the pedestal
1 Throughout the document, such a fit is referred to as “unweighted.”

41

1000

800

600

400

200

0

750

800

850

900

950

1000

1050

Raw Charge (channel)
Figure 4.3: Example histogram of the signal on a single pad in a CRDC pedestal run. The blue
curve is the result of an unweighted Gaussian fit and is used to calculate the pedestal value.

offset is subtracted from each sample before proceeding with further calculations.
After pedestal suppression, the CRDCs are gain matched to account for differences in charge
collection or amplification between pads. Gain matching is done by considering a run where unreacted beam is swept across the focal plane2 . This illuminates every pad in the active area of all
four CRDCs and ensures that the signal size on all pads should be the same. Instead of applying
a simple linear slope, a more active technique is used for gain matching since the necessary amplification factor varies with absolute signal size. The procedure used to gain match the CRDC
detectors is as follows:
1. For each pad, make a two dimensional plot of charge on the pad versus ∆, where ∆ is the
distance of the pad in question from the pad registering maximum charge for that event. Such
2 Hence referred to as a “sweep run.”

42

×103

400

5
4

300

3

200

2

4

-10

-5

0

5 10
∆ (pad number)

×103

3
2

100

1
0
-15

Charge Centroid

Charge (arb. units)

6

1

0

0

-15 -10 -5

0

5 10 15
∆ (pad number)

Figure 4.4: Example of the plots used in the gain matching procedure. The left panel is a two
dimensional histogram of charge on the pad versus ∆, while the right panel shows the Gaussian
centroids of the y axis projection of each x axis bin in the plot on the left. The curve in the right
panel is the result of an unweighted fit to the data points, with the fit function a Gaussian centered
at zero.
a plot is shown in the left panel of Fig. 4.4.
2. For each bin along the x axis, fit the y axis projection with a Gaussian. Similar to the
pedestals, the fit should be unweighted to account for skew.
3. Plot each of the centroids from Step 2 versus ∆. Fit (unweighted) this plot with a Gaussian
centered at zero. An example is shown in the right panel of Fig. 4.4.
4. Choose a reference pad near the center of the detector and match the signal sizes of all other
pads to it by applying the following transformation:
q′j = q j

fref (∆)
,
f j (∆)

(4.3)

where q′ and q denote the gain matched and non-gain matched signals, respectively; j denotes the pad being gain matched; ref denotes the reference pad; and f is the Gaussian
function determined from the fits in Step 3.
The procedure outlined above is applied to each pad in the active area of all four CRDC detectors. For the focal plane CRDCs, the reference pad is chosen to be pad 60, while for the beam
tracking CRDCs, the reference pad is number 50.
43

Normalized Counts

×10−3
20

Normal Pad
Bad Pad

18
16
14
12
10
8
6
4
2
0
0

500

1000 1500 2000 2500 3000 3500 4000 4500 5000
Charge (arb. units)

Figure 4.5: Difference in signal shape between a bad pad (red histogram) and a normal one (blue
histogram). The bad pad in this figure is pad 24 of CRDC2, while the normal pad is number 60 in
CRDC2.
Certain pads in each of the CRDCs display pathological features and need to be rejected in the
analysis. Most often these pads are overly sensitive to electronic noise, causing them to display a
signal which does not reflect the real amount of charge deposited on the pad. To determine which
pads are bad, the charge signal of each pad in a sweep run is examined. Those pads displaying
unusual charge distributions are labeled as bad pads and ignored in further analysis. This procedure
is subjective, but as demonstrated in Fig. 4.5, the difference in signal shape between a bad pad and
a normal one is fairly obvious. Table 4.1 lists the bad pads for each detector.
Once the pads in each CRDC have been pedestal suppressed and gain matched and all bad pads
have been determined, it is possible to use the charge distribution on the pads to calculate the x
position at which the charged particle hit the detector. The method for calculating the interaction
position in pad space is to fit a plot of charge versus pad number with a Gaussian. The Gaussian

44

Table 4.1: List of the bad pads for each CRDC detector. Pads are labeled sequentially, starting
with zero.
Detector

Bad Pads

CRDC1
CRDC2
TCRDC1
TCRDC2

66
24, 89, 105, 126
30, 31, 58, 59, 63
14, 21, 62, 63

centroid is then taken as the pad space interaction position. The fitting is done event-by-event using
the GNU Scientific Library (GSL) [62] implementation of the Levenberg–Marquardt minimization
algorithm [63, 64]. Starting values for the minimization are taken from the results of a center of
gravity fit, and each data point in the fit is weighted equally. It should be noted that for events
in the center of a detector, the difference in centroid values between a Gaussian fit and a center
of gravity fit is negligible. However, for events near the edge where the full charge versus pad
distribution is clipped, the Gaussian fit does a significantly better job of finding the true centroid.
This effect is demonstrated in Fig. 4.6, which shows example charge versus pad distributions in
the center and near the edge of CRDC2, as well as in Fig. 4.7, which is a scatter-plot of gravity
centroids versus Gaussian centroids. The two values agree well until the edge of the detector is
approached, at which point they begin to diverge.
To convert the position in pad space to one in real space, a simple linear transformation is used.
The same is true for conversion of drift time to position in the y direction. In the x direction, the
slope is taken simply from the pad pitch: ±2.54 mm/pad, with the sign of the slope depending
on the orientation of the detector. To determine the x offset and the y slope and offset, a tungsten
mask with holes drilled at known locations inserted into the beam line, shadowing the detector. A
sample masked position distribution for CRDC1 is shown in Fig. 4.8. By determining the centroid
of the holes in time versus pad space and comparing with their known locations, the correct linear
factors can be determined. During the experiment, the mask drive for CRDC2 was malfunctioning,
causing the mask to only be inserted partially. As the drive operates in the y direction, the x offset
value was not affected by this malfunction. Likewise, the y slope could still be determined using

45

3
2.5
2
1.5
1
0.5
0

Charge (arb. units)

Charge (arb. units)

×103

58

60

62

64 66
Pad Number

3.5
3
2.5
2
1.5
1
0.5
0

×103

118

120 122 124 126 128
Pad Number

Gravity Centroid (pad number)

Figure 4.6: Charge distributions for an event near the middle of CRDC2 (left panel) and one near
the edge (right panel). The blue curve is a Gaussian fit to the data points, and the blue vertical line
is the centroid of that fit. The red vertical line is the centroid of a gravity fit to the points. The
Gaussian and gravity centroids are nearly the same in the case of events near the middle, but on
the edge the gravity fit is skewed towards lower pad number.

120
125
100

120
115

80

110
110 114 118 122 126
60
40
20
0
0

20

40

60

80
100
120
Gaussian Centroid (pad number)

Figure 4.7: Gaussian versus gravity centroids for CRDC2. The two fits disagree near the edge of
the detector, with the gravity fit skewed lower.

46

y (mm)

100
80
60
40
20
0
-20
-40
-60
-80

-100
-150

-100

-50

0

50

100

150
x (mm)

Figure 4.8: Example of calibrated position in both planes for CRDC1, with the tungsten mask in
place. The blue open circles denote the position of the mask holes, and the blue vertical lines
denote the position of slits cut into the mask.

the mask, as the spacing between holes remains identical. However, since the absolute position of
the mask holes in the y direction was not known, the mask could not be used to determine the y
offset. Instead, a beam of 25 Ne was sent down the focal plane, centered, with the vertical position
of the beam defined to be y ≡ 0. The y offset for CRDC2 is then set from the location of the beam
centroid.
Due to the possibility of detector drift, mask runs were taken approximately once per day
during the experiment. The changes in the calibration parameters are shown in Fig. 4.9. As any
drifts are fairly small, the calibration parameters are simply updated after each mask run to reflect
their new values.
A further issue related to CRDCs is the poor performance of TCRDC2. As shown in Fig. 4.10,
the x position tends to cluster around certain pads, creating the “spike” features seen in the plot.

47

x-offset (mm)

-179.2
-179.4
-179.6
-179.8

CRDC1
CRDC2
TCRDC1
TCRDC2

187.5
187
186.5
186
-115
-115.5
-116
115.18
115.16
115.14
115.12
115.1

-0.175

126
124
122
120

-0.18

-0.168
-0.17
-0.172
-0.174

y-offset (mm)

y-slope (mm/channel)

-0.185

-0.062
-0.064
-0.066

-0.065
-0.066
-0.067
-0.068
20:02
22:28
22-Jan 2010 24-Jan

128
126
124
28.5
28
27.5

21.5
21
20.5
00:55
27-Jan

03:21
29-Jan

20:02
22:28
22-Jan 2010 24-Jan

Date and Time (EST)
Figure 4.9: Drift of CRDC calibration parameters.

48

00:55
27-Jan

03:21
29-Jan

Counts / 0.1 pad

×103
3
2.5
2
1.5
1
0.5
0

35

40

45

50
55
60
Gaussian Fit Centroid (pad)

Figure 4.10: Gaussian centroids in pad space for TCRDC2.

This is the result of inhomogeneities in the drift field that cause electrons to be preferentially
attracted to specific points on the anode wire. The spikes are not physical and are a result of
detector malfunction; hence the x position measurement of TCRDC2 is not used event-by-event
in the analysis. Instead, a plot of TCRDC2 x position versus TCRDC1 x position, as shown in
Fig. 4.11, is used to determine a linear correlation between the two parameters,
x2 = 1.162 · x1.

(4.4)

Such a correlation is expected based on the optics of the A1900. In the final analysis, the x
position of beam particles at TCRDC2 is calculated simply from this linear function. Due to the
small angular spread of the incoming beam, the influence of using this technique on the overall
resolution is fairly minor.

49

TCRDC2 x (mm)

103

50
x2 = 1.162 · x1

40
30
20

102

10
0
-10
10

-20
-30
-40
-50
-50

-40

-30

-20

-10

0

10

20

30
40
50
TCRDC1 x (mm)

1

Figure 4.11: Plot of TCRDC2 x position versus TCRDC1 x position. The plot is used to
determine a correlation of x2 = 1.162 · x1, shown by the black line in the figure. The position at
TCRDC2 determined by this correlation is what is used in the final analysis.

Table 4.2: Slope values for each pad on the ion chamber. Pad zero was malfunctioning and is
excluded from the analysis.
Pad

Slope

0
1
2
3
4
5
6
7

n/a
1.218959
0.941313
0.864580
0.925724
0.725158
0.971480
0.852719

8
9
10
11
12
13
14
15

50

1.034799
0.979036
0.809853
0.817919
0.853119
0.928206
0.850262
0.836822

4.1.1.3 Ion Chamber
Similar to the CRDCs, the pads on the ion chamber must be gain matched to account for differences
in signal collection and amplification. The gain matching is done by multiplying each pad’s signal
by a slope factor, with the factors determined from a run where a beam of 25 Ne is centered in the
focal plane. Since each pad should measure the same amount of energy loss, the slopes are set
such that the signals match. Slope factors for each pad are listed in Table 4.2. The most upstream
pad (pad zero) was malfunctioning during the experiment, so it is excluded from the analysis.
After gain matching, the signals from the fifteen working pads are averaged to form an energy loss
parameter.
As shown in Fig. 4.12, there is a dependence of the average ion chamber signal on both x and y
position in CRDC2. The figure is generated from a sweep run with unreacted beam, so the energy
loss should be uniform. To correct for this dependence, the plot of ∆E versus x is first fit3 with a
second order polynomial; the result of the fit is
f (x) = 399.762 − 0.0583744 · x − 0.0008205 · x2,

(4.5)

and the ∆E signal is corrected as follows:
∆Excorr = 399.762 ·

∆E
.
f (x)

(4.6)

From here, ∆Excorr is plotted against CRDC2 y position and fit with a first order polynomial:

f (y) = 399.551 − 0.305093 · y.

(4.7)

The same method is used to correct for the y−dependence:
∆Exycorr = 399.551 ·

∆Excorr
.
f (y)

(4.8)

3 The procedure used for fitting the two dimensional histogram is as follows: 1) Fit (unweighted) the

y axis projection of each x axis bin with a Gaussian. 2) Plot the Gaussian centroids versus the x axis bin
centers and fit this plot with the desired function. Unless otherwise noted, this is always the method used to
fit two dimensional histograms.

51

400
350

∆Excorr (arb. units)

300
-200

-100

500
450
400
350
300
-100

-50

450
400
350
300
250
200
150
100
50
0
0
50
100
CRDC2 y (mm)

∆Excorr (arb. units)

450

180
160
140
120
100
80
60
40
20
0
0
100
200
CRDC2 x (mm)

500
450
400
350
300
-200

∆Exycorr (arb. units)

∆E (arb. units)

500

-100

180
160
140
120
100
80
60
40
20
0
0
100
200
CRDC2 x (mm)

500

500

450

400
300

400

200

350
300
-100

100
-50

0

0
50
100
CRDC2 y (mm)

Figure 4.12: Upper left: Ion chamber ∆E signal versus CRDC2 x position; the black curve is a
3rd order polynomial fit used to correct for the dependence of ∆E on x. Upper right: Result of the
x−correction: the dependence of ∆E on x is removed. Lower Left: x−corrected ∆E versus
CRDC2 y position; the black curve is a linear fit used to correct for the dependence of ∆Excorr on
y. Lower right: Final position corrected ion chamber ∆E versus CRDC2 y position.

4.1.1.4 Scintillator Energies
The calibration procedures for the thin scintillator ∆E and thick scintillator E signals are identical.
First each of the four PMTs is gain matched using data from a sweep run, with the requirement
that | xscint |< 10 mm and | yscint |< 10 mm, where xscint and yscint are the vertical and horizontal
positions on the scintillator, calculated using ray tracing and the CRDC position measurements.
It is necessary to use only events that hit near the center of the scintillator as this ensures that
light attenuation is equal for all PMTs. Because the detector trip mentioned in Section 4.1.1.1
significantly impacts the energy signal of the scintillators, two sets of gain factors were determined:
one for before the trip and one for after, with the gain factors set such that signal size is equal

52

Table 4.3: Slope values for thin and thick scintillator energy signals.
PMT

Slope Pre-Trip Slope Post-Trip

Thin 0
Thin 1
Thin 2
Thin 3

1.136
1.158
1.360
1.073

2.167
2.174
2.437
2.169

Thick 0
Thick 1
Thick 2
Thick 3

2.860
1.438
2.483
2.155

4.771
4.888
3.272
3.174

throughout the experiment. Table 4.3 list the gain factors for each scintillator PMT.
After gain matching, a total energy signal for the detector is calculated as follows:
e + e2
,
etop = 0
2

(4.9)

e + e3
,
ebottom = 1
2

(4.10)

and

etotal =

e2 + e2
top
bottom
2

.

(4.11)

As the ∆E and Etotal signals are used only to determine relative differences between the various
reaction products present in the focal plane, an absolute energy calibration is not needed. Hence
the scintillator energy signals are left in arbitrary units.
The scintillator energy measurements, as calculated from Eqs. 4.9–4.11, have a dependence
on the position at which the particle hits the scintillator. The reason is light attenuation: particles
striking near an edge of the detector will produce a stronger signal in PMTs near that edge than
in those on the opposite side, as the scintillation light becomes diminished when it traverses the
plastic. This effect is corrected for empirically in the same way as the ion chamber, described in
Section 4.1.1.3. The correction functions for the thin scintillator are:
(thin)

fcorr (x) = 710.805 + 0.295609 · x − 6.28603 × 10−3 · x2 − 2.22623 × 10−5 · x3
53

(4.12)

120
100

1.5

80
1

60

∆Excorr (arb. units)

2

-100

1.5
1

0.5
0

-40

-20

200
1.5

0
20
40
Thin y position (mm)

160
120

1

0
0
100
200
Thin x position (mm)

×103

×103

0.5

20

0
-200

2

80

40

0.5

E (arb. units)

×103

40

0
-200

140
120
100
80
60
40
20
0

Excorr (arb. units)

∆E (arb. units)

2

2

-100

0
0
100
200
Thick x position (mm)

×103

300
250

1.5
1

0.5
0

-40

-20
0
20
40
Thick y position (mm)

200
150
100
50
0

Figure 4.13: Position correction of the thin and thick scintillator energy signals. Top left: thin
scintillator ∆E vs. x position, with a third order polynomial fit. Bottom left: thin scintillator ∆E
(corrected for x dependence) vs. y position, with a second order polynomial fit. Top right: thick
scintillator Etotal vs. x position, with a third order polynomial fit. Bottom right: thick scintillator
Etotal (corrected for x dependence) vs. y position, with a first order polynomial fit.

and

(thin)

fcorr (y) = 1001 + 1.362 · y + 0.01158 · y2,

(4.13)

while the correction functions for the thick scintillator are:
(thick)

fcorr

(x) = 1346.74 + 1346.74 · x − 1.61185 × 10−3 · x2 + 4.09761 × 10−6 · x3

(4.14)

and

(thick)

fcorr

(y) = 999 + 1.166 · y.
54

(4.15)

100
1.5

60
40

0.5

∆Ecorr (arb. units)

2

-100

×103

1
0.5
0

-40

-20

0
20
40
Thin y position (mm)

250
200

1

150
100
50

0
-200

0
0
100
200
Thin x position (mm)

1.5

×103

0.5

20

0
-200

2

1.5

80

1

Ecorr (arb. units)

×103

140
120
100
80
60
40
20
0

Ecorr (arb. units)

∆Ecorr (arb. units)

2

2

×103

-100

0
0
100
200
Thick x position (mm)

1.5
1

0.5
0

-40

-20
0
20
40
Thick y position (mm)

350
300
250
200
150
100
50
0

Figure 4.14: Results of the position correction of the thin and thick scintillator energy signals.
The panels correspond to those of Fig. 4.13, displaying the final position corrected energy signals.
As seen in the upper-left panel, the correction for thin ∆E could be improved by the use of a
higher order polynomial; however this signal is not used in any of the final analysis cuts, so the
correction is adequate as is.

The results of the position correction are shown in Fig. 4.14. It should be noted that the x
position correction to thin ∆E signal produces a sharp kink at large positive x position. The reason
is that a third order polynomial cannot describe the shape of ∆E versus x across the entire face of
the scintillator. This can be seen in the upper-left panel of Fig. 4.13 where the fit curve fails to
reproduce the histogram shape at large positive x. The correction could be improved by using a
higher order polynomial; however, since the thin ∆E signal is only used for intermediate analysis
checks and not in any of the final cuts or calculations, the third order correction presented here is
sufficient.
As seen in Fig. 4.15, both the thin and thick scintillator energy signals drift throughout the

55

100

150

Thin ∆E (arb. units)

Thick Etotal (arb. units)

1400
1200
1000
800
600
400
200
0

1400
1200
1000
800
600
400
200
0

200

250

300
350
Run Number

250

300
350
Run Number

Voltage Trip

100

150

200

Figure 4.15: Thin (top panel) and thick (bottom panel) scintillator energies for 29 Ne unreacted
beam in production runs. Gaps along the x axis correspond to non-production runs taken at
various intervals during the experiment. The drift seen in the figure is corrected by setting
Gaussian centroid of each run’s 29 Ne energy signal to be constant throughout the experiment.
experiment. The events in the figure are from production runs, gated on 29 Ne unreacted beam
particles that make it into the focal plane (gaps along the x axis correspond to calibration and other
non-production runs). The ∆E and Etotal measurements should be constant, so the observed drift is
corrected run-by-run. The ∆E and Etotal measurements, gated on 29 Ne, are fit (unweighted) with a
Gaussian, and the corrected energy is calculated such that the centroids are constant:
E
Eruncorr = 0 ,
Ec

(4.16)

where E0 is an arbitrarily chosen value (1200 for thin ∆E and 500 for thick Etotal), and Ec is
the centroid of the Gaussian fit. Each run is approximately one hour in length, and the drift within
a run is negligible.

56

y (cm)

y (cm)

100
50

100
50

0

0

-50

-50

-100
-100

0

-100
600

100
x (cm)

700

800

900

1000
z (cm)

Figure 4.16: Example of the two types of muon tracks used in calculating independent time
offsets for MoNA. The left panel illustrates a nearly vertical track, used to determine the time
offsets within a single wall. The right panel shows an example of a diagonal track used to
calculate offsets between walls.
4.1.2 MoNA
4.1.2.1 Time Calibrations
MoNA TDCs signals are calibrated with a linear slope and offset, and from there, the signals on
each end of a bar are averaged to give a time of flight measurement. The slope of each TDC is
determined using a time calibrator which sends a signal to the modules at a regular frequency of
40 ns−1 . For each TDC, the slope is determined as such:

m=

40 ns
,
∆ch

(4.17)

where ∆ch is the average spacing between pulses in channel number. Slopes measured in a
previous MoNA-Sweeper experiment were used, as the MoNA TDCs remained identical between
the two runs.
Timing offsets are divided into two parts: a global offset for the entire array and individual
offsets of each TDC relative to the others. The individual time offsets are determined using muons
produced by the interaction of cosmic rays with the Earth’s upper atmosphere. The muons travel
to earth and pass through the detector, often interacting multiple times. They travel at a known

57

velocity close to the speed of light, 29.8 cm/ns, so by selecting for specific tracks through the
detector, the time offset of each bar relative to the others can be determined. To calculate time
offsets within a given wall, muon tracks which are nearly vertical are selected, as illustrated in the
left panel of Fig. 4.16. The expected travel time between two bars in the same wall is then:
t = n·

10.27 cm
,
29.8 cm/ns

(4.18)

where n is the number of bars between the two, and 10.27 cm is the nominal vertical distance between the center of two bars. Appropriate time offsets are determined by comparison of
measured times with the expected time of Eq. (4.18). Time offsets between walls are determined
similarly, except using diagonal tracks, as illustrated in the right panel of Fig. 4.16. All offsets are
set relative to bar A8, which is at beam height and in the front wall of MoNA. Offsets within the
front wall are calculated directly relative to bar A8, using vertical tracks. From here, offsets are
propagated from wall to wall using diagonal tracks.
The global offset is calculated from the time of flight of γ −rays made at the target during
production runs. The flight time of gammas to the front and center of MoNA is

ToFγ = xa8 /c

(4.19)

where xa8 = 658 cm is the distance to the center of the front wall of MoNA (the center of bar
A8), and c is the speed of light. Fig. 4.17 is a histogram of the time of flight to the center of the
front wall of MoNA, with the prompt γ peak clearly identified and separated from prompt neutrons.
To increase statistics, the entire front wall of MoNA is used for determining the global offset, with
the time of flight scaled to account for each bar’s distance from the target. Furthermore, when
setting the timing offset a number of cuts are used to enhance the presence of target gammas: hit
multiplicity must be equal to one; charge deposited must be less than 6 MeVee; and the absolute
value of the interaction position in MoNA must be less than 30 cm in both the x and y planes.
Despite the use of constant fraction discriminators, there is a walk present in the MoNA timing
at low signal size. This is demonstrated in Fig. 4.18, which shows a plot of time of flight for prompt

58

Counts / ns

180
60
160
Prompt neutrons

40

140
120

20

Prompt γ −rays

0
15

100

20

25

30

80
60
40
20
0
-50

0

50

100

150
200
Time of Flight (ns)

Figure 4.17: Time of flight to the center of the front wall of MoNA. The events in the figure are
collected during production runs. The prompt γ peak used to set the global timing offset is clearly
identifiable and separated from prompt neutrons. The plot includes a number of cuts, which are
listed in the main text.
gammas versus energy deposited in MoNA. As seen in the figure, there is a clear dependence of
time of flight on deposited energy. The function indicated in the figure,
f (q) = 23.5531 − 2.56625e−0.62272/q,

(4.20)

is a fit to the histogram, with the functional form taken from Ref. [65].
In order to appropriately use the prompt γ −ray measurements to set the global MoNA timing
offset, a correction must be made to account for the the CFD walk. Additionally, the walk correction must be included in the final neutron analysis in order to have an accurate measurement
of neutron energy. An attempt was made to do the correction using Eq. (4.20), but the range of
deposited energies probed by the γ peak is too small for this to be sufficient. Higher deposited
energies can be reached, however, by considering runs in which MoNA records cosmic rays in
59

ToF (ns)

30

5
4.5

28

4
26

3.5

24

3
23.5531 − 2.56625e

22

−0.622715
x

2.5
2

20

1.5

18

1
0.5

16
0

1

2

3

4

5

6

7
8
9
10
Deposited Energy (MeVee)

0

Figure 4.18: Time of flight to the front face of MoNA versus deposited energy, for prompt
gammas. The plot includes cuts similar to those used in generating Fig. 4.17. The dependence of
time of flight on deposited energy, as indicated in the figure, demonstrates the presence of walk
for low signal size. The function in the figure is partially used for walk corrections, as explained
in the text.
standalone mode, with the trigger being the first PMT in the array to fire. When run in this mode,
a plot of the average time signal for a given bar versus the absolute value of x position, as shown
in the inset of Fig. 4.19, represents the mean time it takes light to travel from the interaction point
to each PMT. By applying a linear correction to the average time, one obtains a time measurement which should be constant. This time is plotted against deposited energy in the main panel of
Fig. 4.19, and this histogram can then be fit and used to correct the walk. The fit function used is
this case is:
f (q) = 229.4 +

2.861
,
q

(4.21)

with the functional form taken from Ref. [66]. As seen in Fig. 4.19, this function blows up
at low deposited charge, making it a poor choice for walk correction in that region. Instead, a
60

Average Time (ns)

Position Corrected Average Time (ns)

×103

255
250
245
240

250
245
240
235
230
225
220

235

×103
1
0.8
0.6
0.4
0.2
0
0 20 40 60 80 100120
| x − position | (cm)
229.4 + 2.861
x

230

3
2.5
2
1.5
1
0.5

225

0
0

5

10

15

20

25

30

35
40
45
50
Deposited Energy (MeVee)

Figure 4.19: Inset: mean time vs. absolute value of x position for cosmic-ray data collected in
standalone mode, where the trigger is the first PMT to fire. The time axis on the inset is
determined by the mean travel time of light from the interaction point to the PMTs. If this
parameter is corrected for the x position, using the line drawn on the inset, then a theoretically
constant ToF is obtained. Plotting this constant ToF versus deposited energy reveals walk, as
shown in the main panel of the figure. The function drawn in the main panel is used for walk
correction of production data at high deposited charge, as explained in the text.

piecewise function is used to do the walk correction, with Eq. (4.20) used when q ≤ 1.8 MeVee
and Eq. (4.21) used otherwise. The transition point of 1.8 MeVee is taken from the crossing point
of the two functions, as shown in Fig. 4.20. The final form of the walk corrected time of flight is
then
tcorr = t −



 −2.56625e−0.62272/q + 1.6531

 2.861


q − 1.761

61

: q < 1.8
(4.22)
: q ≥ 1.8.

Walk Correction (ns)

10
Gamma peak correction function
Cosmic ray correction function
Final piecewise correction function

8
6
4
2
0
-2
-4
0

1

2

3

4

5

6

7

8

9

10

Charge Deposited (MeVee)
Figure 4.20: Walk correction functions. The blue curve is from a fit of time of flight vs. deposited
energy, as shown in Fig. 4.18, while the red curve is from a fit of position-corrected time vs.
deposited energy for cosmic ray data, shown in Fig. 4.19. The black curve is the final walk
correction function, using the γ peak correction function (blue curve) below the crossing point
(q = 1.8 MeVee) and the cosmic correction function (red curve) above. Note that the y axis in this
figure represents the actual correction applied to the time of flight, which is the reason for the
offset difference between the curves in this figure and those of Figs. 4.18–4.19.

tright

-150

-100

-50

tleft

0

50

100
150
x (cm)

Figure 4.21: Illustration of MoNA x position measurement: the time it takes scintillation light to
travel to each PMT is directly related to the distance from the PMT. By taking the time difference
between the signals on the left and right PMTs, the x position in MoNA can be calculated.

62

Counts / ns

14

×103

12
10
8
6
4
2
0
-30

-20

-10

0

10

20
30
tleft − tright (ns)

Figure 4.22: Example time difference spectrum for a single MoNA bar in a cosmic ray run. The
1 · max crossing points, indicated by the red vertical lines in the figure, are defined to be the edges
3
of the bar in time space; these points are used to determine the slope and offset of Eq. (4.23). The
asymmetry in the distribution is the result of the right side of MoNA being closer to the vault
wall, thus receiving a larger flux of room γ -rays.

4.1.2.2 Position Calibrations
In order to accurately calculate neutron energy, as well as the angle of neutrons as they leave the
target, the interaction position of a neutron within MoNA needs to be known. In the y and z planes,
calculation of the interaction position is straightforward: it is simply taken to be at the center of the
bar in which the neutron interacts. In the x direction, the interaction position is determined from
the time difference between the left and right PMT signals. The time difference is directly related
to the x interaction point since it is the result of scintillation light traveling a larger distance to
one PMT versus the other, illustrated in Fig. 4.21. The time difference is related to the interaction
position via a linear calibration:

63

Counts / channel

106
Pedestal

105
104
103

γ peak

102
Muon peak
101
100
0

500

1000

1500

2000

2500

3000
3500
4000
QDC Signal (channel)

Figure 4.23: Example raw QDC spectrum for a single MoNA PMT. The pedestal and muon peak
are indicated in the figure, along with the peak associated with room γ −rays interacting in
MoNA. After adjusting voltages to place the muon bump of every PMT close to channel 800, a
linear calibration is applied to move the pedestal to zero and the Gaussian centroid of the muon
bump to 20.5 MeVee.

x = m · tleft − tright + b.

(4.23)

The slope m and offset b are determined from cosmic ray data: muons interacting near the edge
of a bar are used to find its ends in time space, and these two points are sufficient to determine the
linear factors. The edge of a bar in time space is defined to be at 1/3 of the maximum height of a
histogram of time difference measurements, based on GEANT3 [67] simulations. Fig. 4.22 shows
an example time difference spectrum, with the edges of the bar indicated.

64

Table 4.4: List of gamma sources used for CAESAR calibration. Energies are taken from
Refs. [68–72].
Source

Gamma Energies (keV)

133 Ba

356
662
1275
898, 1836
517, 846, 1771, 2034, 2598, 3272

137 Cs
22 Na
88 Y
56 Co

4.1.2.3 Energy Calibrations
To account for differences in light collection and amplification, the PMTs in MoNA must be gain
matched. This is done using cosmic ray muons, which deposit an average energy of 20.5 MeVee
into a MoNA bar. The gain matching is done in two steps. The first step is to adjust the bias
voltages on the the PMTs until their signals are approximately equal. Fig. 4.23 shows an example
histogram of the raw QDC signal for a single PMT; the bump at around channel 800 is from muons
interacting in MoNA. For each PMT, this bump is fit with a Gaussian, and the voltage on the PMT
is adjusted to move the peak value close to 800. This procedure is iterated until the muon peak
centroid of every PMT is within a few (∼ 5 or less) channels of 800.
After adjusting the voltages to approximately line up the muon peaks, a software correction is
applied to exactly match the peak locations. This is done in an automated routine which finds the
location of the pedestal indicated in Fig. 4.23, as well as the Gaussian centroid of the muon bump.
A linear slope and offset are then applied to place the pedestal at zero and the centroid of the muon
bump at 20.5 MeVee.

4.1.3 CAESAR
The CAESAR crystals are calibrated using a variety of standard γ −ray check sources, listed in
Table 4.4. Prior to beginning the experiment, the gain on each CAESAR photo-tube was adjusted
to roughly align the peaks from the 88 Y source. Post experiment, a series of runs were taken
with each source at the target location. For each of the gamma transitions listed in Table 4.4, the

65

Channel Number

3500
3000
2500

– 133Ba
– 137 Cs
– 22 Na
– 88 Y
– 56 Co

2000
1500
1000
500
200

400

600

800

1000

1200

1400
Eγ (keV)

Figure 4.24: Example source calibrations for a single CAESAR crystal (J5). The solid black line
is a linear fit to the data points (R2 = 0.999933). All other crystals are similarly well described by
a linear fit, so in the final analysis the calibration is done using the two 88 Y gamma lines at
898 keV and 1836 keV.
peak values in channel number were determined and plotted as a function of the known transition
energy. This was done separately for each crystal in the array. As demonstrated in Fig. 4.24, the
response is well described by a linear fit.
Since the response of CAESAR photo-tubes is influenced by the Sweeper’s fringe fields, the
calibration needs to be updated any time the Sweeper’s current setting is changed, as hysteresis
effects could potentially change the fringe field, in turn altering the calibration. During the experiment, approximately ten minutes of data were collected from the 88 Y source any time the Sweeper
current was changed. Since the response function of CAESAR crystals is linear, the two data
points from the 88 Y source Eγ = 898 keV and Eγ = 1836 keV are used to set the calibration:
E = m · ch + b,
66

(4.24)

Scintillator ∆E (arb. units)

2000
32 Mg

103

1500
29 Ne

102

Wedge Fragments
1000

10

500

0
80

85

90

95

100

105
110
Beam ToF (ns)

1

Figure 4.25: Beam components, including 29 Ne gate.

where ch is channel number, and m and b are determined from a linear fit of channel number
versus energy for the two 88 Y transitions. The fit parameters are re-calculated after every 88 Y
source run, with the updated parameters applied to the next block of production data.

4.2 Event Selection
In the course of the experiment, far more events are recorded than those of interest. This section4
details the cuts used to select the events which are the result of the decay processes of interest:
28 F → 27 F + n

and 27 F∗ → 26 F + n.

67

ToFRF→A1900 (arb. units)

103

0

-20
102
-40

10

-60

-80
1
98

100

102

104

106

108
ToFA1900→Target (ns)

Figure 4.26: Flight time from the K1200 cyclotron (measured by the cyclotron RF) to the A1900
scintillator versus flight time from the A1900 scintillator to the target scintillator. The double
peaking in ToFRF→A1900 is due to wraparound of the RF. By selecting only events which are
linearly correlated in these two parameters, the contribution of wedge fragments to the beam is
reduced. These selections are indicated by the black contours in the figure, with the final cut
being an OR of the two gates.

4.2.1 Beam Identification
As mentioned in Section 3.2, the incoming beam is made up of a number of different nuclear
species. The 29 Ne beam particles are selected from measurements of energy loss in the target
scintillator and time of flight from the A1900 scintillator to the target scintillator. Fig. 4.25 shows
a histogram of these two parameters, with the 29 Ne events circled. The overall contribution of
29 Ne

to the beam is approximately one percent, with the remainder primarily composed of 32 Mg,

as well as a variety isotopes created by reactions in the aluminum wedge.
4 Cuts pertaining to inverse reconstruction of tracks through the Sweeper magnet will be presented in

Section 4.3.1.1, after that subject has been properly introduced.

68

CRDC2

5
103

4
3

102

2
10
1
0

0

1
10000 20000 30000 40000
Total Charge (arb. units)

Gaussian σ (pad number)

Gaussian σ (pad number)

CRDC1
5

103

4
102

3
2

10
1
0

0

1
10000 20000 30000 40000
Total Charge (arb. units)

Figure 4.27: CRDC quality gates. Each plot is a histogram of the sigma value of a Gaussian fit to
the charge distribution on the pads versus the sum of the charge collected on the pads. The black
contours indicate quality gates made on these parameters.

In addition to the ∆ETarget versus ToFA1900→Target cut, a cut on time of flight from the K1200
cyclotron to the A1900 scintillator versus time of flight from the A1900 scintillator to the target
scintillator is used to improve the incoming beam selection. Here the time at the K1200 cyclotron
is measured from the cyclotron RF signal. This cut has the effect of removing wedge fragments
from the beam, as particles produced from reactions in the aluminum wedge will have a more
dramatic change in velocity than those which pass through the wedge unreacted. Thus by selecting
only events which are linearly correlated in ToFRF→A1900 versus ToFA1900→Target, the presence
of wedge fragments is reduced. This cut is indicated in Fig. 4.26.

4.2.2 CRDC Quality Gates
Often a CRDC detector will record an event for which the x position measurement is not reliable.
The contribution of such events can be reduced by applying quality gates to each of the focal
plane CRDCs. Application of CRDC quality gates will also remove any events that do not pass
through the active area of both detectors. Events that have an unreliable x position measurement
usually result from a pathological charge distribution on the CRDC pads. They can be identified

69

event-by-event by considering a plot of the σ value of the Gaussian fitting procedure explained in
Section 4.1.1.2 versus the sum of the charge on all CRDC pads; the events for which the x position
measurement is not reliable will be anomalous in such a plot. The plots of σ versus total charge
are shown for CRDC1 and CRDC2 in Fig. 4.27, with the quality gates indicated in the figure.

4.2.3 Charged Fragment Identification
A major requirement in selecting the events of interest is charged particle separation and identification. This is done in two steps: element selection and isotope selection.

4.2.3.1 Element Selection
Element separation is achieved by measurement of the fragment’s energy loss in the ion chamber,
as well as its velocity. Energy loss in the ion chamber gas is given by the Bethe-Bloch formula [58]:
−

dE
Z z2
2me γ 2 v2Wmax
2
= 2π Na re me c2 ρ
ln
dx
A β2
I2

− 2β 2 ,

with
2
2π Na re me c2 = 0.1535 MeVcm2 /g

re :

classical electron radius = 2.187 × 10−13 cm

me :

electron mass

Na :

Avagadro’s number = 6.022 × 1023 mol−1

I:

mean excitation potential

Z:

atomic number of absorbing material

A:

atomic weight of absorbing material

ρ:

density of absorbing material

z:

charge of incident particle in units of e

β:

v/c of the incident particle

γ:

1/

1−β2

Wmax : maximum energy transfer in a single collision.
70

(4.25)

From Eq. (4.25), it is clear that energy loss is related to the charge number of the incident
particle, as well as its velocity:
z2
∆E ∝ 2 · f (β ) .
β

(4.26)

Thus elements can be separated by plotting energy loss in the ion chamber versus a parameter
which is indicative of fragment velocity. There are two velocity indicator parameters available in
the experiment: time of flight through the Sweeper ToFTarget→Thin and total kinetic energy measured in the thick scintillator. As demonstrated in Figs. 4.28 and 4.29, plotting ion chamber energy
loss versus either of these parameters reveals well-separated bands, with each band corresponding
to a different element. In both plots, the most intense element is unreacted 29 Ne beam (Z = 10) .
Hence the band directly below this one is composed of the fluorine (Z = 9) events of interest. In
the final analysis, the element cuts indicated in Figs. 4.28 and 4.29 are both used.

4.2.3.2 Isotope Selection
The magnetic rigidity of a charged particle is equal to its momentum:charge ratio,5
Bρ =

p mv
=
.
q
q

(4.27)

If isotopes of the same element (constant q) and equivalent Bρ are sent through a dipole magnet,
then their mass number, A, can be related to their time of flight, t, as follows:

t =A

Bρ q
m

(4.28)

Lmu
,
Bρ q

v=

(4.29)

5 Ignoring relativity.

71

Ion Chamber ∆E (arb. units)

600

103
Z = 11

500

Z = 10

400

102
Z=9

300
Z=8
200

10

Z=7

100
0
46

48

50

52

54

56

58

60

1

ToFTarget→Thin (ns)
Figure 4.28: Energy loss in the ion chamber versus time of flight through the Sweeper. Each band
in the figure is a different isotope, with the most intense band composed primarily of Z = 10
unreacted beam. The fluorine (Z = 9) events of interest are circled and labeled in the figure.

where mu is the average mass of a nucleon in the nucleus and L is the track length. Thus in the
case of constant Bρ (constant momentum), isotopes of the same element can be mass-separated
simply by considering their time of flight.
In practice, charged particles produced in nuclear reactions have a large spread in momentum.
If the magnetic elements used for separation accept a reasonably large range of momenta, then the
assumption of Eq. (4.29) that Bρ is constant is no longer valid. Moreover, differing momenta result
in variable L, as the track of a charged particle passing through a dipole depends on its rigidity.
However, the Bρ and L values of the charged particles are reflected in their emittance (dispersive
position and angle) as they exit the device. Magnetic spectrometers, such as the NSCL’s S800 [47],
can be tuned such that the fragments exiting the device are highly focused in position. In this case,
it is possible to see isotopic separation simply by plotting angle at the focal plane versus time of

72

Ion Chamber ∆E (arb. units)

600

103

500
Z = 11
400

102

Z = 10
300

Z=9
Z=8

200

10

Z=7
100
0
0

200

400

600

800

1000

1

Thick Scintillator Etot (arb. units)
Figure 4.29: Energy loss in the ion chamber versus total kinetic energy measured in the thick
scintillator. As in Fig. 4.28, the bands in the figure are composed of different elements, and the
most intense band is Z = 10. fluorine (Z = 9) events are circled and labeled.

flight. The left panel of Fig. 4.30 is an example of the use of this technique in the S800. The events
in the figure are magnesium (Z = 12) , and each band is composed of a different isotope [73].
In the case of the Sweeper magnet, the technique outlined above is not sufficient to separate
isotopes. This is demonstrated in the right panel of Fig. 4.30, in which a plot of focal plane angle
versus time of flight shows no hint of separation. The main reason is that the Sweeper lacks
focusing elements; hence angle and position at the focal plane are correlated to a large degree.
Furthermore, the magnetic field of the Sweeper is highly nonuniform, due to its large vertical gap
of 14 cm. This leads to a significant degree of nonlinearity in the emittance. In the case of the
Sweeper, the full correlation between angle, position, and time of flight needs to be considered for
isotope separation to be visible. This is demonstrated in Fig. 4.31, which is a three dimensional
plot of time of flight versus angle and position, from the present experiment. The events in the plot

73

50
0
-50

Sweeper Magnet
Focal Plane θx (mrad)

Focal Plane θx (mrad)

S800 Spectrometer
100
50
0
-50

-100
40

200 250 300 350 400 450 500
Time of Flight (channel)

45

55
50
60
Time of Flight (ns)

45
40
35
30
25
20
15
10
5
0

Figure 4.30: Left panel: focal plane dispersive angle vs. time of flight for magnesium isotopes in
the S800, taken from Ref. [73]. In this case, isotopes are clearly separated just by considering
these two parameters. Right panel: focal plane dispersive angle vs. time of flight for fluorine
isotopes in the present experiment. The plot shows no hint of isotope separation, as three
dimensional correlations between angle, position and time of flight need to be considered in order
to distinguish isotopes.
are fluorine isotopes produced from reactions on the 32 Mg beam, and the bands which can be seen
in the figure are each composed of a different isotope.
From the picture of Fig. 4.31, a systematic method has been developed to correct the time of
flight for angle and position at the focal plane, resulting in a parameter which can be used for
isotope selection. The corrections are first determined for fluorine elements from the 32 Mg beam,
to take advantage of higher statistics. The same corrections can then be used for the isotopes of
interest: fluorines produced from the 29 Ne beam.
The first step in correcting the time of flight is to construct a single parameter which describes
the dispersive-plane emittance, both angle and position. To do this, the three dimensional plot in
Fig. 4.31 is profiled in the following way:6
1. Slice the x and θx axes into a square grid.
2. For the events in each each slice, make a one dimensional projection of the ToF axis.
6 In practice, this is simply done using the ÌÀ¿

ÈÖÓ 
Ø¿ ÈÖÓ Ð method of the ROOT [74, 75] data

analysis package.

74

ToFTarget→Thin (ns)

32 Mg →

Fluorine

58
56
54
52
50
48
46
Fo 40
cal
Pla 0 -4
ne
0
θx
(m -80
rad
)

0 -20
-80 -60 -4

0

60
20 40
(mm)
Focal Plane x

Figure 4.31: Three dimensional plot of time of flight through the Sweeper vs. focal plane angle
vs. focal plane position (color is also representative of time of flight). The figure is composed of
Z = 9 events coming from the 32 Mg contaminant beam, and each band in the figure is composed
of a different isotope.

3. Find the gravity centroid of the projection of (2).
4. Plot the centroid from (3) versus the central x and θx positions of the slice.
The result is shown in Fig. 4.32, with the color axis representing the gravity centroids of the
ToF projections. Breaks in color indicate lines of iso-ToF. From here, one determines a function
which describes the location of the iso-ToF lines throughout the figure:
f (x) = 0.010397 · x2 + 0.84215 · x + c,

(4.30)

where c is a constant offset. This function, for a given c, is drawn as the solid black curve
in Fig. 4.32. The effect of varying c is to move the curve vertically along the θx axis, and an
appropriate function will fall on the iso-ToF lines independent of c.
75

Focal Plane θx (mrad)

80
60

Fluorine
58

Fit along lines of iso-ToF:
f (x) = 0.010391 · x2 + 0.84215 · x + c

56
54

40
20

52

0
-20

50

-40

Central ToFTarget→Thin (ns)

100

32 Mg →

48

-60
46

-80
-100
-100

44
-80

-60

-40

-20

0

20

40

60

80

100

Focal Plane x (mm)
Figure 4.32: Profile of the three dimensional scatter-plot in Fig. 4.31. The solid black curve is a fit
to lines of iso-ToF; this fit is used to construct a reduced parameter describing angle and position
simultaneously.

From Eq. (4.30), a parameter describing both angle and position is constructed:

e (x, θx ) = θx − 0.010397 · x2 + 0.84215 · x .

(4.31)

As shown in Fig. 4.33, a plot of the appropriately constructed e (x, θx ) versus time of flight
demonstrates bands, with each band corresponding to a different isotope.
From the picture of Fig. 4.33, it is possible to improve isotope separation significantly. The
first step is to create a corrected time of flight parameter by determining the nominal slope, m, of
the bands, represented by the black line in the figure. The time of flight is corrected by simply
projecting onto the axis perpendicular to this line:

tcorr = t + m−1 · e (x, θx ) .
76

(4.32)

e (x, θx ) = θx − f (x) (arb. units)

32 Mg →

Fluorine

50

25

0

20

-50

15

-100

10

-150

5

-200

0
42

44

46

48

50

52

54

56

58

ToFTarget→Thin (ns)
Figure 4.33: Histogram of the emittance parameter, constructed from the fit function in Fig. 4.32,
vs. time of flight. The bands in the plot correspond to different isotopes of fluorine. A corrected
time of flight parameter can be constructed by projecting onto the axis perpendicular to the line
drawn on the figure.

Table 4.6: Final correction factors used for isotope separation. The numbers in the right column
are multiplied by the parameter indicated in the left and summed; this sum is then added to
ToFTarget→Thin to construct the final corrected time of flight.
Parameter

Correction Factor

x
x2
x3
θx
2
θx
3
θx
xθx
x2 θx
2
xθx

−5.0595 × 10−2
−8.97 × 10−4
−3.0 × 10−6
8.0 × 10−2
−1.0 × 10−5
2.0 × 10−6
−1.5 × 10−4
−2.0 × 10−6
−6.0 × 10−6

77

2
x2 θx
y2
θy
ytrgt.
Ethick
∆Ei.c.
xtcrdc1
ToFbeam

1.4 × 10−7
1.0 × 10−3
−3.0 × 10−3
4.0 × 10−3
1.3 × 10−3
4.0 × 10−3
1.7 × 10−2
1.0 × 10−1
—

Fluorine
800

120

600

100

400

80

200

60

40 42 44 46 48 50 52
Corrected ToF (arb. units)

Ethick (arb. units)

Counts / Bin

29 Ne →

26 F

27 F

0

40
20
0
40

42

44

46

50
52
48
Corrected ToF (arb. units)

Figure 4.34: Main panel: Corrected time of flight for fluorine isotopes resulting from reactions on
the 29 Ne beam. The isotopes if interest, 26,27F, are labeled. The black curve in the figure is a fit to
the data points with the sum of five Gaussians of equal width. Based on this fit, the
cross-contamination between 26 F and 27 F is approximately 4%. The inset is a scatter-plot of total
energy measured in the thick scintillator vs. corrected time of flight. These are the parameters on
which 26,27 F isotopes are selected in the final analysis, and the cuts for each isotope are drawn in
the figure.

The separation is then improved by iteratively plotting corrected time of flight versus angle or
position and removing the correlations in a manner similar to that of Eq. (4.32). This is done up
n
to fourth order in x and θx , as well as for cross terms (xn · θx ) . Additionally, correlations between

corrected time of flight and any other parameter available in the experiment are searched for and,
if present, removed. Table 4.6 lists all of the factors used in constructing the final corrected time of
flight. It should be noted that the corrections for non-dispersive angle θy and dispersive position
at TCRDC1 (xtcrdc1 ) significantly improve the quality of the separation.
The main panel of Fig. 4.34 shows the corrected time of flight, using the factors listed in
Table 4.6, for fluorine isotopes produced from 29 Ne. As can be seen in the figure, the fluorine
78

Counts / 5 ns

300
250
200
150
100
50
0
-100

-50

0

50

100

150

200

250

300

ToFMoNA (ns)
Figure 4.35: Neutron time of flight to MoNA for all reactions products produced from the 29 Ne
beam. The first time-sorted hit coming to the right of the vertical line at 40 ns is the one used in
the analysis.
isotopes are well separated, with a cross contamination between 26 F and 27 F of approximately
4%. The cross-contamination is determined by fitting the histogram with the sum of five equalwidth Gaussians, shown as the solid black curve in the figure, and calculating the overlap between
the gate for a given isotope and the Gaussian fit function of its neighbor(s). Fluorine isotopes
are identified simply by noting that 27 F is the heaviest fluorine species that can be produced from
29 Ne.

In the final analysis, a two dimensional cut on corrected time of flight and total energy from

the thick scintillator is used for isotope selection. A scatter-plot of these two parameters is shown
in the inset of Fig. 4.34, along with the gates used in the final analysis.

79

4.2.4 MoNA Cuts
To avoid biasing the neutron energy measurement, cuts on MoNA parameters are avoided. However, since MoNA often records multiple hits for a given event, it is necessary to decide which hit
to use in the final analysis. Hits in MoNA can arise from a number of sources in addition to prompt
neutrons: prompt γ −rays; random background (muons and gammas); and multiple detection of
the same event due to scattering of neutrons within the array. Thus a scheme is devised to ensure
that the analysis is being performed on the hit which is most likely to be the result of a prompt
neutron interacting in MoNA for the first time: hits are time-sorted, and the first hit with time of
flight greater than 40 ns is selected. The reason for requiring that the hit come at ToF > 40 ns is
that it is not possible for neutrons produced in the target to make it to MoNA any earlier than this.
This can be seen in Fig. 4.35, in which the prompt neutron peak begins abruptly at around 50 ns.
To avoid cutting any early neutron events, the opening of the neutron window is conservatively
placed at 40 ns.

4.2.5 CAESAR Cuts
Fig. 4.36 demonstrates a cut used to reduce the presence of background events in CAESAR. Events
which are correlated with the γ decay of a beam nucleus will come at a specific time in CAESAR.
Thus by eliminating events which fall outside of a certain time window, the signal to noise ratio
is improved. Since CAESAR uses leading edge discriminators for its time measurements, there is
significant walk in the time signal. However, by plotting Doppler corrected7 gamma energy versus
time of flight, as in Fig. 4.36, a two dimensional gate can be drawn to select only beam-correlated
events. This gate is outlined by the solid black curves drawn in the figure.
7 The Doppler correction procedure will be presented in Section 4.3.2

80

Doppler Corrected Energy (MeV)

4

12

3.5

10

3
8

2.5
2

6

1.5

4

1
2

0.5
0

0
0

200

400

600

800

1000

1200

1400

1600

1800

Time of Flight (arb. units)
Figure 4.36: Doppler corrected energy vs. time of flight for gammas recorded in CAESAR. To
reduce the contribution of background, only those events falling between the solid black curves
are analyzed.

4.3 Physics Analysis
The purpose of the present experiment is to measure the decay energy of neutron-unbound states
using the invariant mass equation, Eq. (3.7):
Edecay =

m2 + m2 + 2 E f En − p f pn cos θ − m f − mn .
n
f

This requires measurement of the kinetic energies and angles at the target of both the neutron,
n, and the fragment, f , involved in the breakup of the unbound state. In the case of the neutron,
calculation of these quantities is straightforward. The angle is taken from simple ray-tracing between the target location and the interaction point in MoNA, while the kinetic energy is calculated
from the time of flight and total distance traveled, using relativistic kinematics:

81

x2 + y2 + z2
t

vn =

(4.33)

vn 2
c

(4.34)

γn = 1/ 1 −

E n = γn mn .

(4.35)

In the case of the fragment, calculation of kinetic energy and target angle is more involved. It
requires reconstruction of tracks through the Sweeper, as described below.

4.3.1 Inverse Tracking
From knowledge of the Sweeper’s magnetic field and ion-optical quantities of a charged particle
at the reaction target, it is possible to calculate the ion-optical quantities of the particle as it exits
the magnet:













x(crdc1)















y(crdc1)  = M 






(crdc1)


θy


L
(crdc1)
θx

x(trgt)








y(trgt)  ,


(trgt) 

θy

(trgt)
E
(trgt)
θx

(4.36)

where L is the track length of the fragment through the Sweeper and M is a third-order transformation matrix calculated from magnetic field measurements. The transformation matrix is produced using the ion-optical code COSY INFINITY [76]. COSY takes as input the magnetic field
of the Sweeper in the central plane—the plane where the only existing vertical components of the
field are those perpendicular to the horizontal plane. To measure the central plane field, seven
Hall probes were mounted vertically, evenly spaced, on a movable cart and stepped through the
magnet. The field in the central plane was constructed from interpolation of the seven Hall probe
measurements. More details about the mapping procedure can be found in Ref. [60].
82

(trgt)

The quantities in Eq. (4.36) which need to be known for invariant mass spectroscopy are θx
(trgt)

θy

,

, and E (trgt) . As each of these quantities is influenced by the nuclear reaction taking place in

the target, none is measured directly. The parameters which are measured, however, are x(crdc1),
(crdc1) (crdc1) (crdc1) (trgt)
θx
,y
, θy
,x
, and y(trgt) . Thus it is desirable to come up with a transformation

which takes as input some combination of the known quantities
(crdc1) (crdc1)
(crdc1) (trgt) (trgt)
,y
, θy
,x
,y
x(crdc1), θx

and gives as output the desired quantities
(trgt)

θx

(trgt)

, θy

, E (trgt) .

Such a transformation cannot be calculated from direct inversion of M , as the track length L
is not known a priori. The approach taken by COSY to calculate an inverse matrix is to assume
that x(trgt) = 0. This allows elimination of the row concerning x(trgt) and the column concerning L,
leading to a form of M that is invertible. Such an approach is valid when the beam is narrowly
focused in x, which is not the case in the present experiment: the beam spot size is on the order of
2 cm.
To construct an appropriate inverse transformation matrix, a procedure has been developed to
perform a partial inversion of M [77]. The partially inverted matrix, Mpi, takes as input the positions and angles at CRDC1 (behind the Sweeper), as well as the x position on the target (measured
(trgt)

from the tracking CRDCs8 ). Its output includes all of the desired quantities: θx

(trgt)

, θy

, and

E (trgt) :
8 It should be noted that a transformation similar to Eq. (4.36) is used to calculate x(trgt) , as the tracking

CRDCs are located upstream of the quadrupole triplet. In this case, the E and L terms are ignored, as the
dependence of tracks through the triplet on beam energy is negligible.

83

E (itrack) (MeV/u)

75
70
65
60
55

-20
-40
-40

-20

0

20
(ftrack)
θx

40
(mrad)

55

60

65
70
75
(ToF) (MeV/u)
E

20

30
25
20

0

(mrad)
(itrack)

20

18
16
14
12
10
8
6
4
2
0

θy

40

0

θx

(itrack)

(mrad)

50
50

7
6
5
4
3
2
1
0

15

-20

10
5
0

40

-40
-40

-20

0

20
(ftrack)
θy

40
(mrad)

Figure 4.37: Comparison of forward tracked and inverse tracked parameters for unreacted 29 Ne
beam. The upper right panel is a comparison of kinetic energies, with the x axis being energy
calculated from ToFA1900→Target and the y axis energy calculated from inverse tracking in the
sweeper, c.f. Eq. (4.37). The lower left and lower right panels show a similar comparison for θx
and θy , respectively. In these plots, the x axis is calculated from TCRDC measurements and
forward tracking through the quadrupole triplet.














(trgt)
θx










y(trgt) 




(trgt)  = M 
θy
pi 







L


(trgt)
E

x(crdc1)








y(crdc1)  .


(crdc1) 

θy

(trgt)
x
(crdc1)
θx

(4.37)

The procedure for calculating the partial inverse transformation matrix Mpi is to perform a series of matrix operations which exchange a coordinate on the right hand side of Eq. (4.36) with one
on the left. This is done until the form of Eq. (4.37) is reached. The coordinates to be exchanged

84

must be entangled to a large degree, i.e. they must share a large first order matrix element in M .
The procedure for coordinate-swapping is detailed in Ref. [77].
To check that the inverse mapping procedure gives the correct results, a comparison is made between forward-tracked target parameters and their counterparts calculated with Eq. (4.37), for data
taken with the reaction target removed. The comparison was done for a variety of magnet settings,
and good agreement was found except when the beam is on the extreme edge of the Sweeper’s
acceptance. An example comparison, when the beam is near the center of the acceptance, is shown
in Fig. 4.37. The reason for disagreement between forward and inverse tracking when the beam
is near the edge of the Sweeper’s acceptance is that the magnetic field of the Sweeper is poorly
understood in this region. However, during production runs the Sweeper is tuned such that the
reaction products of interest lie near the center of the acceptance, in a region where there is good
agreement between forward and inverse tracking.

4.3.1.1 Mapping Cuts
As mentioned in Section 4.2, some cuts have to be made to ensure that the inverse tracked parameters of Eq. (4.37) are being calculated correctly. The first involves the dispersive plane emittance.
As shown in the left panel of Fig. 4.38, when unreacted beam particles are swept across the focal
plane, they maintain a positive correlation between angle and position. This is not always the case
for reaction products, which are far more dispersed in angle. The right panel of Fig. 4.38 reveals
that a significant number of reaction products exit the Sweeper with negative angle and positive
position. Such an emittance is the result of the fragments entering the Sweeper with large positive
angle, as demonstrated in Fig. 4.39. These large-angle reaction products are off the standard acceptances of the Sweeper, and they are only observed because they take a non-standard path through
the magnet. Such paths are not well described by the magnetic field maps of the Sweeper. As such,
the target parameters of reaction products falling in this region cannot be faithfully reconstructed.
Hence in the final analysis only those fragments which fall within the positively correlated region
of θx versus x, defined by the sweep run, are included. The cut is indicated by the rectangular

85

100

Focal Plane θx (mrad)

Focal Plane θx (mrad)

100
400
50
300
0
200
-50

-100
-100

100
-50

0
0
50
100
Focal Plane x (mm)

400
50
300
0
200
-50

-100
-100

100
-50

0
0
50
100
Focal Plane x (mm)

Figure 4.38: Left panel: Focal plane dispersive angle vs. position for unreacted beam particles
swept across the focal plane. These events display positive correlation between angle and
position, and they define the region of the emittance for which the magnetic field maps of the
Sweeper are valid. Right panel: Focal plane dispersive angle vs. position for reaction products
produced from the 32 Mg beam. A significant portion of these reaction products fall in the region
of positive position and negative angle, due to taking a non-standard track through the Sweeper.
The rectangular contour drawn on the plot is defined by the “sweep band” of the left panel, and
only events falling within this region are used in the final analysis.

contour in the right panel of Fig. 4.38.
The second mapping cut involves position in the non-dispersive plane. As revealed by the
left panel of Fig. 4.40, a number of events are reconstructed with a kinetic energy that deviates
significantly from the mean. Furthermore, these events do not possess the expected inverse correlation between Efrag and ToFTarget→Thin , indicating that the accuracy of the inverse reconstruction
may be questionable. Plotting Efrag versus y position on CRDC1, as shown in the right panel
of Fig. 4.40, reveals that the events with extreme Efrag values also hit CRDC1 far from the center. Additionally, the two parameters demonstrate a parabola-like correlation, enhanced at large y
position, which is not expected. Most likely, the correlations seen in the figure are not real and
are instead a result of deficiencies in the Sweeper field map. COSY only considers field values
in the central plane when constructing maps, so it is plausible that the full correlations between
energy and non-dispersive parameters are not reproduced. There seems to be no way to improve
the situation, at least not while using COSY’s central plane method of field map construction. As

86

100

3.5

50
3

0
-50

2.5
2

F.P. θx (mrad)

Counts / 2 mrad

×103

-100
-100 -50 0 50 100
Focal Plane x (mm)

1.5
1
0.5
0
-50

-40

-30

-20

-10

0

10

20

30
40
50
Target θx (mrad)

Figure 4.39: Inset: focal plane angle vs. position for unreacted beam particles swept across the
focal plane. Unlike the left panel of Fig. 4.38, here the A1900 optics were tuned to give a beam
that is highly dispersed in angle. causing the emittance region of negative angle and positive
position to be probed. The main panel is a plot of incoming beam angle for all events (unfilled
(fp)
histogram) and events with +x(fp) and −θx (orange filled histogram). The plot reveals that the
(fp)
+x(fp) , −θx events have a large positive angle as they enter the Sweeper.
the total number of events impacted is fairly small, those events with absolute value of CRDC1 y
greater than 20 mm are excluded from the analysis. The range of ±20 mm is chosen because it
corresponds to the approximate points where Efrag begins to correlate significantly with CRDC1
y, as seen in the right panel of Fig. 4.40.

4.3.2 CAESAR
Gamma energies are taken simply from the energy deposited in a crystal, using the calibration
procedure of Section 4.1.3. Often a single gamma will interact in multiple crystals due to Compton
scattering [78], depositing only a portion of its energy in each. In this case, the gamma energies

87

Efrag (MeV/u)

Efrag (MeV/u)

70
65
60
55

70
65
60
55

50

50

45

45

40
48

50

40
-60 -40 -20

52
54
56
58
ToFTarget→Thin (ns)

0

20 40 60
CRDC1 y (mm)

Figure 4.40: Left panel: fragment kinetic energy, calculated from the partial inverse map, vs. time
of flight through the Sweeper. The events in the figure are 26 F + n coincidences produced from
the 29 Ne beam. Events with extreme values of Efrag also fall outside of the expected region of
inverse correlation between Efrag and ToF. Right panel: Efrag vs. CRDC1 y position. This plot
reveals an unexpected correlation between Efrag and CRDC1 y, with the extreme Efrag events also
having a large absolute value of CRDC1 y. This is likely due to limitations of the Sweeper field
map, so the events with | y |> 20 mm are excluded from the analysis.
of up to three crystals are summed to calculate the total energy deposition for the event. The
summing procedure is only performed when the multiple hits are in neighboring crystals, as hits in
non-neighboring detectors are likely to be the result of random coincidences, not multi-scattering.
Gammas resulting from the de-excitation of a beam nucleus are emitted from a source moving at roughly one third the speed of light. As such, their energy measured in the lab frame is
significantly Doppler shifted [79]. Thus the gamma energies recorded in CAESAR are Doppler
corrected:
Edop = Elab

1 − β cos θ
1−β2

,

(4.38)

where Edop and Elab are the Doppler corrected and lab frame energies, respectively; β is the
relativistic beta-factor, vbeam /c; and θ is the angle between the point where the γ −ray is emitted
and the point where it is detected. To calculate θ , the detection point is assumed to be the center
of the crystal in which the γ −ray interacts. In the case of multi-scattering, the center of the first
interaction crystal is used. The emission point is assumed to be the center of the reaction target,

88

and the z position of the target is 7 cm upstream of the center of the array. The z position is verified
through comparison with known transitions, as explained in Section 4.4.2.

4.4 Consistency Checks
The present experiment is complicated, both in terms of the physical setup and data analysis.
Comparison of present results with those previously published can help to ensure that the present
measurement and analysis is performed accurately. In particular, comparison of present results
with previous ones is performed for two cases: decay energy reconstruction of 23 O∗ → 22 O + n
and gamma transitions in 32 Mg and 31 Na.

4.4.1 23 O Decay Energy
The unbound first excited state in 23 O has been measured to have a decay energy of 45 keV [80–
83], feeding the ground state of 22 O. In addition to being confirmed multiple times, the transition
from 23 O∗ → 22 O + n is narrow, making it a good candidate for a consistency check. The same
transition is observed in the present experiment, with relatively high statistics, from fragmentation
reactions on the 32 Mg beam. These events are analyzed in the same way as the data of interest,
with the results presented in Fig. 4.41. The results are consistent with the previous measurements,
lending credibility to the analysis.

4.4.2 Singles Gamma-Ray Measurements
To test the Sweeper-MoNA-CAESAR setup, data were taken using a MoNA singles trigger, where
a hit in MoNA is not necessary for the event to be recorded. This greatly enhances the collection of
de-excitation γ −rays since a triple coincidence event is no longer required. To reduce experimental
dead time, a tungsten beam blocker was inserted in front of CRDC1 to reject the majority of
unreacted beam particles.

89

→ 22 O + n

100
Counts / 0.2 cm/ns

Counts / 50 keV

23 O∗

80

60

40

140
120
100
80
60
40
20
0
-5 -4 -3 -2 -1 0 1 2 3 4 5
vn − v f (cm/ns)

20

0
0

0.5

1

1.5

2

2.5

3

Decay Energy (MeV)
Figure 4.41: Decay energy for 23 O∗ → 22 O + n events produced from the 32 Mg beam. The
spectrum displays a narrow resonance at low decay energy, consistent with previous
measurements that place the transition at 45 keV. The inset is a relative velocity vn − v f
histogram for the same events. The narrow peak around vrel = 0 is also consistent with the
45 keV decay.

In the singles runs, two previously measured gamma transitions were collected with good statistics. The first is the result of inelastic excitation of the 32 Mg beam, populating the first 2+ excited
state at 885 keV [16]. As shown in the left panel of Fig. 4.42, this transition is prominent on top of
random background. This well known transition was used to verify the z position of the reaction
target, as misplacement of the target in the Doppler correction algorithm will cause the peak to
shift and broaden. The peak is narrowest and located at 885 keV when the target location is set at
7 cm upstream of the center of CAESAR.
The other transition prominently observed in the singles data is in 31 Na. An excited level
at around 370 keV has been observed in three previous measurements [84–86], with the most
recent placing the transition at 376 (4) keV [84]. The same transition is observed in the present

90

31 Na + γ

Counts / 20 keV

Counts / 30 keV

32 Mg + γ

350
300
250
200
150
100
50
0

100
80
60
40
20

0

0.5

1

0
0

1.5
2
2.5
3
Gamma Energy (MeV)

0.5

1

1.5
2
2.5
3
Gamma Energy (MeV)

Figure 4.42: Left panel: Doppler corrected gamma energies from inelastic excitation of 32 Mg.
The blue vertical line indicates the evaluated peak location of 885 keV [16]. Right panel: Doppler
corrected gamma energies for 31 Na, produced from 1p knockout on 32 Mg. The most recent
published measurement of 376 (4) keV [84] is indicated by the blue vertical line.

experiment, as shown in the right panel of Fig. 4.42.

4.5 Modeling and Simulation

4.5.1 Resonant Decay Modeling
The breakup of an unbound resonant state is a two body process involving a neutron and the
residual nucleus. As such, it can be described as a neutron scattering off a nucleus, the neutron
impinging with variable energy and angle. The cross section as a function of energy, σ (E) , of
such a scattering process is well described by R-matrix theory [87], with the cross section for
resonances given by a Breit-Wigner distribution [88]. The particular form of the Breit-Wigner
used in this work has an energy-dependent width:

σ (E; E0 , Γ0 , ℓ) = A

Γℓ (E; Γ0 )
[E0 + ∆ℓ (E; Γ0 ) − E]2 + 1 [Γℓ (E; Γ0 )]2
4

,

(4.39)

where A is an amplitude; E0 is the central resonance energy; Γ0 parameterizes the resonance
width; ℓ is the orbital angular momentum of the resonance; and Γℓ and ∆ℓ are functions to be

91

explained. E0 , Γ0 and ℓ are parameters to be determined from the data (or from theoretical considerations). As mentioned, Eq. (4.39) is derived from R-matrix theory. A summary of the derivation
is given here, with the full details available in Ref. [87].
The radial Schrödinger equation for a neutron scattering off a nucleus is:
ℓ (ℓ − 1) 2M
d
−
− 2 (V − E) uℓ (r) = 0.
dr2
r2
ℏ

(4.40)

R-matrix theory is developed from the solution of Eq. (4.40) at the minimum approach distance
before the nuclear interaction becomes important:
1/3

a = r 0 An

1/3

+Af

,

(4.41)

where r0 parameterizes the nuclear radius (here we use 1.4 fm); and An and A f are the mass
number of the neutron and fragment, respectively. Since the nuclear force is effectively absent and
the neutron is unaffected by the Coulomb interaction, the potential term, V, in Eq. (4.40) is zero.
The solution is then a superposition of incoming and outgoing waves:
u(in) = (Gℓ − iFℓ )
ℓ

u(out) = (Gℓ + iFℓ ) ,
ℓ

(4.42)

where Fℓ and Gℓ are related to J−type Bessel functions:
Fℓ = (πρ /2)1/2 Jℓ+1/2 (ρ )
Gℓ = (−1)ℓ (πρ /2)1/2 J−(ℓ+1/2) (ρ ) .

(4.43)

√
In Eq. (4.43), ρ = a 2ME/ℏ, with M the reduced mass of the neutron-fragment system; E the
relative energy; and a the boundary distance of Eq. (4.41).
The R-matrix relates the incoming wave function, u(in) , to its derivative at the boundary. For a
ℓ
single resonance, it is given by
R=

2
γ0
ℏ2 |uℓ (a)|2
=
,
2Ma E0 − E
E0 − E

(4.44)

where E0 is the resonance energy, and γ0 is a reduced width representing the wave function at
the boundary a :
92

γ0 = √

ℏ
uℓ (a) .
2Ma

(4.45)

An outgoing collision matrix, Uℓ , is related to the R-matrix and the logarithmic derivative of
the external wave function, Lℓ (and its complex conjugate, L∗ ), by a phase factor:
ℓ
u(in) 1 − L∗ R
ℓ
ℓ = e2iδℓ ,
Uℓ = (out)
1 − Lℓ R
u

(4.46)

ℓ

with the logarithmic derivative given by
Lℓ =

ρ u′(out)
ℓ
u(out) r=a
ℓ

= Sℓ + iPℓ .

(4.47)

In Eq. (4.47), Sℓ and Pℓ are called the shift and penetrability functions, respectively, and are
related to the Fℓ and Gℓ of Eq. (4.43) via
S = ρ Fℓ Fℓ′ + Gℓ G′ / Fℓ2 + G2
ℓ
ℓ
P = ρ/

Fℓ2 + G2
ℓ

r=a

r=a

(4.48)

− φℓ ,

(4.49)

.

The phase shift, δℓ , is given by:

δℓ (E) = tan−1

1 Γ (E)
2 ℓ

E0 + ∆ℓ (E) − E

where φℓ is the hard sphere scatter phase shift. Γℓ and ∆ℓ are the functions presented in
Eq. (4.39) and are given by
2
Γℓ (E) = 2Pℓ (E) γ0
2
∆ℓ (E) = − [Sℓ (E) − Sℓ (E0 )] γ0 .

(4.50)

The outgoing collision matrix can then be expressed as:
1/2

Uℓ =

iΓℓ

(E)

i
E0 + ∆ℓ (E) − E − 2 Γℓ (E)

.

(4.51)

This is related to the scattering cross section by

σℓ =

π
σ (θ ) dx = 2 ∑ (2ℓ + 1) |1 −Uℓ|2 .
k ℓ
93

(4.52)

Combining Eq. (4.51) and Eq. (4.52) results in an expression for the cross section, up to a
normalization constant:

σ =A

Γℓ (E)
[E0 + ∆ℓ (E) − E]2 + 1 [Γℓ (E)]2
4

,

(4.53)

which is the same as Eq. (4.39).
Finally, it should be noted that the reduced width γ0 is related to the width parameter Γ0 of
Eq. (4.39) by
2
Γ0 = 2γ0 Pℓ (E0 ) .

(4.54)

4.5.2 Non-Resonant Decay Modeling
In addition to resonances, there may be non-resonant contributions to the data. The non-resonant
contribution comes from the decay of highly excited states in 28 F, which lie in a region where the
level density is large. These states de-excite by neutron emission, and in some cases the emitted
neutron is observed in coincidence with the final fragment.
In the case of 28 F → 27 F + n, the non-resonant contribution is expected to be negligible. The
reason is that states in 28 F are populated by direct proton knockout from 29 Ne. As such, only
neutrons which result from the decay of a state in 28 F to a bound state in 27 F are present in the
data. A non-resonant contribution could arise from the decay of continuum states in 28 F directly
to bound 27 F, but the probability of observing such decays is extremely low. The reason is that the
decay energy of such a transition would be large (on the order of 5 MeV or greater). As will be
explained in Section 4.5.3, the probability to observe a transition with such large decay energy is
low, due to geometric acceptances.
It is also plausible that a background contribution to 28 F → 27 F + n could arise from neutrons
that are removed from the beryllium target nuclei in the knockout reaction. However, observation
of such a neutron requires that it exit the target with close to beam velocity and with a transverse
momentum direction close to zero degrees. This requires that the proton knocked out of the beam
transfer nearly all of its momentum to a single neutron in beryllium in a head-on collision. Such a

94

29 Ne

26 F
28 F
27 F

Figure 4.43: Schematic of the process by which non-resonant background is observed in
coincidence with 26 F. First a highly excited state in 28 F is populated from 29 Ne. This state then
decays to a high excited state in 27 F, evaporating a neutron (thick arrow). The excited state in 27 F
then decays to the ground state of 26 F by emitting a high-energy neutron (thin arrow), which is
not likely to be observed. The evaporated neutron (thick arrow) has fairly low decay energy and is
observed in coincidence with 26 F, giving rise to the background.

process is expected to have a small enough cross section that the contribution of target neutrons to
the 28 F decay spectrum can be neglected in the present work.
In the case of 27 F∗ → 26 F + n, however, a non-resonant contribution is expected. The process
through which this background arises is illustrated in Fig. 4.43. A high excited state in 28 F is first
populated from 29 Ne. This state then decays to another highly excited state in 27 F, evaporating
a neutron. From here, the excited state in 27 F can decay to the ground state in 26 F by neutron
emission, with the 26 F observed in the Sweeper. The neutron from the final step is not likely to be
observed as the decay energy going from a high excited state in 27 F to the ground state of 26 F is
large; however, the evaporated neutron from the decay of highly excited 28 F to highly excited 27 F
will have relatively low decay energy, allowing it to be observed in coincidence with the 26 F.
The decay energy of non-resonant evaporated neutrons can be modeled as a Maxwellian distribution [89, 90]. The arguments for using such a model, as presented in Ref. [89], are summarized

95

here.
The kinetic energy of the evaporated neutron, relative to the beam, is given by:9

εn = E ∗ − E ∗′ ,
f
f

(4.55)

where E ∗ is the excitation energy of the original nucleus (in our case 28 F), and E ∗′ is the
f
f

excitation energy of the daughter (27 F). For ℓ = 0, the probability of emitting a neutron with εn

increases as εn becomes larger, due to better penetration of the angular momentum barrier. The
distribution of the number of neutrons having energy between ε and ε + dε is given by

∑

Gn (ε ) dε =

GC (β ) ,

(4.56)

ε <εβ <ε +dε

with GC (β ) the probability of decay through a specific channel β :
GC (β ) =

2
kβ σC (β )
2
∑ kα σC (α )

.

(4.57)

α

In Eq. (4.57), the σC are the cross sections for decay through a specific channel, and k = λ −1
is a wave number. The sum in the denominator is over all of the possible decay channels, α .
To determine the relative intensity distribution of emitted neutron energies, the denominator of
Eq. (4.57) is ignored:
In (ε ) dε = cεσC (ε ) w f ε f − ε dε .

(4.58)

Here the wave number k2 has been replaced by the factor ε . The function w f ε f − ε is related
to the level density w (E) , and its logarithm,

S (E) = ln [w (E)],

(4.59)

is analogous to the entropy of a thermodynamic system. Expanding S around the maximum
decay energy, ε f , gives
S εf −ε = S εf −ε
9 Neglecting the small recoil energy of the fragment.

96

dS
+ ....
dE E=ε
f

(4.60)

Using Eq. (4.60) to approximate w f ε f − ε and absorbing the factor coming from S ε f into
the constant, one obtains:
In (ε ) = cεσC (ε ) exp −

ε
Θ εf

dε .

(4.61)

The function σC (ε ) varies extremely slowly and can be absorbed into the leading constant, c.
The function Θ is given by
dS
1
=
,
Θ dE

(4.62)

the expression for the temperature of a thermodynamic system. Thus Θ can be interpreted as
the temperature of a thermal neutron source whose relative distribution is dominated by e−ε /Θ .
The functional form used to model background events differs slightly from that of Eq. (4.61):
f (ε ; Θ) = A ε /Θ3 e−ε /Θ ,

(4.63)

with the temperature Θ to be determined from the data. However, the important features,
namely the dominant e−ε /Θ term, are the same. Minor differences in shape as a result of the
leading terms become irrelevant when the effects of experimental resolution are taken into account.

4.5.3 Monte Carlo Simulation
If the decay energy distribution of an unbound state is given by the function f (E; µ ) , where µ
are parameters to be determined from the data (E0 , Γ0 , and ℓ), the observed distribution will be
smeared by experimental resolution and acceptance:
˜
F E; µ =

˜
R E, E f (E; µ ) dE.

(4.64)

˜
˜
Here R E, E is the smearing function, and F E; µ is what is observed by the experiment.
Thus to compare a model to data, the influence of resolution and acceptance must be taken into
account.
In the present work, the integral on the right hand side of Eq. (4.64) is not known analytically;
hence the influence of resolution and acceptance is introduced using Monte Carlo simulation. First
97

the breakup of an unbound state is modeled using some function f (E; µ ) . The neutron and fragment are then propagated through their respective experimental systems, and the measurements
made in various particle detectors are given appropriate resolutions, as described below. Any event
for which a particle falls outside of geometric acceptance limits is tagged appropriately. The result
is a simulated data set which is then analyzed in the exact same way as the real experimental data,
ignoring events which fail acceptance cuts. The decay energy of the simulated and analyzed data
˜
set represents F E; µ in Eq. (4.64) and is compared directly to experiment.
The simulation begins with the 29 Ne beam impinging on TCRDC1. The properties of the
incoming beam (emittance and energy) are free parameters, and they are constrained from comparison with data in which unreacted beam is sent down the focal plane (c.f. Section 4.5.3.1). The
x position on TCRDC1 is recorded with a Gaussian spread, σ = 1.3 mm, to simulate experimental
resolution. The σ value is taken from mask runs, comparing the width of recorded events to the
actual size of the mask holes.
From TCRDC1, the beam is forward tracked to the reaction target. The proton knockout reaction is treated in the Goldhaber model [91] with the inclusion of a friction term [92] to degrade the
beam energy by 0.6%. Inclusion of the friction term is necessary for simulated fragment energies
to match those of the data. Following the knockout reaction, the breakup of the unbound state is
modeled using one of the functions presented in Sections 4.5.1 and 4.5.2.
After the breakup reaction, the neutron and fragment are treated separately. The neutron is
propagated to the front face of MoNA where its measurement is simulated. Time of flight resolution is introduced as a Gaussian spread with σ = 0.3 ns. Position in the y direction is discretized in
the same way as the real measurement: the position is set to be at the center of the bar in which the
neutron interacts. In the z direction, only the front wall of MoNA is included in the simulation. The
z position discretization is introduced by giving the z position in the simulation a uniform 10 cm
spread and analyzing as if the neutron hit at the center of the bar. The x position resolution is given
by the convolution of two Laplacians:

98

p1 ·

e−|x/σ2 |
e−|x/σ1 |
+ (1 − p1 ) ·
,
2σ1
2σ2

(4.65)

with σ1 = 16.2 cm, σ2 = 2.33 cm, and p1 = 53.4%. Timing and x position resolutions are
based on GEANT3 simulations and shadow bar measurements [59].
In addition to resolutions, acceptance cuts concerning the neutron are imposed. There are two
points in the beam line where neutron events are lost. The first is the point where the beam pipe
intersects the Sweeper magnet gap, 1.06 m downstream of the reaction target. Neutrons passing
this point with a radius greater than the internal radius of the pipe, 7.3 cm, are flagged as failing
the acceptance cut. The second cut is only in the y direction, coming at the point where neutrons
exit the Sweeper vacuum chamber, 1.133 m downstream of the target. Here any neutrons with
y position greater than ±7.3 cm, the gap size, are flagged as being outside the acceptances. The
active area of MoNA is included as an acceptance cut in the simulation, but it has no effect as
MoNA is entirely shadowed by the beam pipe and vacuum chamber aperture.
Following the breakup, the charged fragment is propagated to CRDC1 using a COSY forward
transformation matrix as in Eq. (4.36). From here, the fragment is ray-traced to CRDC2 and the
thin scintillator; at each location an acceptance cut is imposed, requiring that the events fall within
the active region of the detector (±150 mm for CRDC2 and ±200 mm for the thin scintillator).
The positions and angles recorded at CRDC1 are given a Gaussian spread to simulate experimental
resolution, with σposition = 1.3 mm and σangle = 0.8 mrad in both planes. As with TCRDC1,
position and angle σ values are taken from CRDC mask runs.
As the x position on the target is used in calculation of decay energy, c.f. Eq. (4.37), an
appropriately resolved x(target) must also be simulated. This is accomplished by taking the resolved
(tcrdc1)
x position measurement on TCRDC1 and determining a nominal angle, ϑx
, using the method

of Section 4.1.1.2. The resolved x position at the target is then calculated by propagating the
(tcrdc1)
through the triplet using a COSY transformation matrix.
resolved x(tcrdc1) and ϑx

99

Table 4.7: Incoming beam parameters. Each is modeled with a Gaussian, with the widths and
centroids listed in the table. Additionally, x and θx are given a linear correlation of
0.0741 mrad/mm, and the beam energy is clipped at 1870 MeV.
Parameter

Centroid

Width

x
θx
y
θy
E

0.18 mm
0 mrad
0 mm
0 mrad
1800 MeV

11 mm
4 mrad
9 mm
1.1 mrad
50 MeV

4.5.3.1 Verification
As mentioned above, free parameters in the simulation are constrained by comparison with data.
The free parameters related to the incoming beam are its emittance in both planes and its kinetic
energy. Each of these is modeled as a Gaussian distribution, with the free parameters listed in
Table 4.7. The incoming position and angle in the dispersive plane are also given a linear correlation, with a slope of 0.0741 mrad/mm. In the case of incoming energy, the Gaussian distribution is
clipped on the high side, with a maximum of 1870 MeV.
The non-dispersive angle and position, as well as the dispersive position, are constrained by
TCRDC measurements. Because of the TCRDC2 malfunction (explained in Section 4.1.1.2), the
incoming dispersive angle cannot be constrained by the TCRDCs. Instead it is set from comparison
with data where unreacted beam is centered in the focal plane. The incoming beam energy is
also constrained in the same way. Furthermore, the beam energy was cross-checked against the
ion-optics program LISE++ [93] and found to be in reasonable agreement. To verify that the
incoming beam parameters are correct, simulated CRDC1 parameters—in the case of unreacted
beam centered in the focal plane—are compared to data. These comparisons are shown in Fig. 4.44
and indicate reasonable agreement.
In addition to unreacted beam data, intermediate parameters in the simulation are also verified
for the reactions of interest (26,27 F in the focal plane). This provides a check of the knockout
reaction model used in the simulation and probes a wider region of phase space. In addition to
parameters at CRDC1, target parameters, calculated using the inverse mapping procedure of Sec100

Counts / 2 mrad

Counts / 2 mm

200
150
100
50
-50

0

100

0
-100

50
100
CRDC1 x (mm)
Counts / mrad

Counts / 2 mm

150

50

0
-100

150
100
50
0

200

-50

0
50
100
CRDC1 θx (mrad)

-10

0
10
20
CRDC1 θy (mm)

150
100
50

-40

-20

0

0
-20

20
40
CRDC1 y (mm)

Figure 4.44: Comparison of simulation to data, with unreacted beam sent into the center of the
focal plane. The black squares are data points and the solid blue lines are simulation results. It
should be noted that the dispersive angle in the focal plane, shown in the upper-right panel, does
not match unless the beam energy is clipped at 1870 MeV.

tion 4.3.1, are compared. As shown in Figs. 4.45 and 4.46, the agreement between simulation and
data is reasonable. The deviations in non-dispersive angle and position are due to the deficiencies
of the field map explained in Section 4.3.1.1.

4.5.4 Maximum Likelihood Fitting
Simulation results are compared to data using an unbinned maximum likelihood technique. Given
˜
a resolved decay energy model F E; µ , the likelihood is constructed by taking the product of F
evaluated at each data point:

∏ F (εi; µ ) ,
i

101

(4.66)

Counts / 5 mrad

Counts / 8 mm

80
60
40
20
-50

0

100
50

Counts / 5 mm

40
20
-50

0

-40

-20
0
20
40
Inverse Tracked θx (mrad)

-40

-20
0
20
40
Inverse Tracked θy (mrad)

-40

-20
0
20
40
Inverse Tracked y (mm)

40
20

40
30
20
10
0

50
100
CRDC1 y (mm)

100

100
50
0

20

0

0
50
100
CRDC1 θx (mrad)

60

0
-100
Counts / 6 mrad

-50

Counts / 3%

Counts / 8 mm

0
-100

40

0

50
100
CRDC1 x (mm)
Counts / 5 mrad

Counts / 8 mrad

0
-100

60

-50

50
0

0
50
CRDC1 θy (mrad)

-20

0
20
Inverse Tracked δ E (%)

Figure 4.45: Comparison of simulation to data for for 26 F reaction products in the focal plane. In
each panel, the parameter being compared is indicated by the x axis label. In the lower-right
panel, δ E refers the the deviation from the central energy of the Sweeper magnet.

102

20
10

30
20
10
-50

20
10
0
-100

-50

0

40
20
0

10
-40

-20
0
20
40
Inverse Tracked θx (mrad)

-40

-20
0
20
40
Inverse Tracked θy (mrad)

-40

-20
0
20
40
Inverse Tracked y (mm)

30
20
10

30
20
10
0

50
100
CRDC1 y (mm)

60

20

0

0
50
100
CRDC1 θx (mrad)

30

30

0

50
100
CRDC1 x (mm)
Counts / 5 mrad

0

40

0
-100
Counts / 10 mm

-50

Counts / 5 mm

Counts / 10 mrad

0
-100

Counts / 6 mrad

Counts / 5 mrad

30

Counts / 3%

Counts / 10 mm

40

30
20
10

-50

0

0
50
CRDC1 θy (mrad)

-20

0
20
Inverse Tracked δ E (%)

Figure 4.46: Same as Fig. 4.45 but for 27 F in the focal plane.

103

where each εi is the decay energy of a recorded event in the experimental data, and the product
is over all recorded events. This is translated into a log-likelihood:
ln (L) = ∑ ln (F (εi ; µ )) .

(4.67)

i

The log-likelihood is then used as a test statistic [94]: the best estimate of the parameters µ is
at the point where ln (L) is maximized, and nσ confidence intervals are determined by finding the
region of µ where
ln (L)max − ln (L) ≤

n2
.
2

(4.68)

˜
In general, Eq. (4.67) requires a continuous form of F E; µ , which is not the case when
˜
F E; µ is obtained from Monte Carlo simulation. An appropriate log-likelihood can be constructed, however, by following the prescription of Ref. [95]. The procedure is outlined as follows:
1. Create a simulated data set, for a given set of parameters µ , and save the generated decay
energy values, E j (from the unsmeared f (E; µ )), along with the resolution-smeared ones,
˜
E j.
2. Form a small volume Vi centered around each experimental data point εi .
˜
3. For each εi , calculate the total number of events whose smeared decay energy, E j , falls
within the volume Vi .
4. Divide the sum in (3) by the size of Vi and, for normalization, the total number of generated
events.
5. Take the natural log of the quotient in (4) and sum over all of the experimental data points

εi .
In addition, the sums in Steps 3 and 4 can be over weighted events, with the weights being a
function of the generated decay energies. By appropriately weighting the points, one can calculate
the likelihood using a different decay model, f E; µ ′ , without having to generate a new set of

104

simulated data. To do this, the appropriate weights are
f E, µ ′
W E j, µ , µ′ =
.
f (E, µ )

(4.69)

Care must be taken to ensure that the weight values are not too large, as statistical errors are
determined by the sum of the square of the weights. In particular, generating functions, f (E, µ ) ,
which are too narrow are not a good choice for re-weighting.
The procedure for calculating the log-likelihood from Monte Carlo data can be summarized in
a single equation [95]:


ln (L) = ∑ ln 
i

∑E ∈V W E j , µ , µ ′
˜
j i
Vi ∑ j W E j , µ , µ ′



.

(4.70)

The log-likelihood calculation procedure outlined above can introduce systematic error from
two sources: nonlinearity in the smeared function, F E; µ ′ , over the volume size, Vi , and limited
Monte Carlo statistics within each Vi . In the present work, the volume size was chosen to be
0.05 MeV. It was demonstrated that this choice of Vi introduces negligible systematic error by
performing a likelihood fit to a known (simulated) data set and comparing the fit results to the actual
parameter values. To avoid systematic errors of the second type, the total number of generated
events is set to be large: approximately three million. This was shown to be sufficient by repeating
the fitting procedure using a variety of generator seeds, with the results nearly identical.

105

Chapter 5
RESULTS

5.1 Resolution and Acceptance
Before presenting the decay energy curves for 27 F∗ → 26 F + n and 28 F → 27 F + n, a brief discussion of the resolution and acceptance functions of the experimental setup is warranted. Fig. 5.1
is a plot of the simulated acceptance curve for 26 F + n coincidences as a function of decay energy
(the same plot for 27 F + n coincidences is virtually identical). As seen in the figure, the acceptance begins to drop off rather quickly after ∼ 400 keV, which is primarily a result of neutrons
being shadowed by the beam pipe and Sweeper vacuum chamber box. This shadowing is clearly
observed in a plot of the neutron θx versus θy , shown in the inset of the figure.
As with the acceptance, the resolution of the experimental setup also varies significantly as a
function of decay energy. This response function cannot be described analytically (hence the need
to perform Monte Carlo simulations), but a general idea of its shape can be obtained by plotting
the Gaussian σ of simulated decay energy curves as a function of input decay energy. Such a
plot for 26 F + n coincidences is shown in the top panel of Fig. 5.2, with the input for each point
being a delta function at the indicated relative energy. To give an idea of the actual shape of
the resolution function, the main panel of Fig. 5.2 shows the simulated decay energy curves for
a variety (0.1 MeV, 0.2 MeV, 0.4 MeV, 0.8 MeV, and 1.5 MeV) of delta function inputs. As
with the acceptance curves, the resolution for 27 F + n coincidences is virtually identical to that of
26 F + n.

5.2 27 F Decay Energy
The measured decay energy curve for 26 F + n coincidences, resulting from the decay of unbound
excited states in 27 F, is presented in Fig. 5.3. Comparison with the acceptance curve of Fig. 5.1

106

θy (degrees)

Acceptance (%)

80
70
60

10
5
0

50

-5

40

-10
-10

-5

0

30

5
10
θx (degrees)

20
10
0
0

0.5

1

1.5

2

2.5

3

Decay Energy (MeV)
Figure 5.1: Simulated acceptance curve for 26 F + n coincidences. The inset is a histogram of
measured neutron angles for fragment-neutron coincidences produced from the 32 Mg beam,
demonstrating the limited neutron acceptance that is a result of neutrons being shadowed by the
beam pipe and the vacuum chamber aperture.

reveals a clear enhancement around 400 keV in decay energy. As discussed in Section 4.5, an
appropriate model for the decay is that of a Breit-Wigner resonance on top of a non-resonant
Maxwellian background. The data are fit by constructing an unbinned log-likelihood test statistic,
c.f. Section 4.5.4. The parameters in the fit are the central energy of the Breit-Wigner resonance,
E0 ; the width of the resonance, Γ0 ; the orbital angular momentum, ℓ; the temperature of the nonresonant Maxwellian distribution, Θ; and the relative contribution of the two lineshapes. The
width of the measured curve is dominated by experimental resolution and acceptance cuts, and the
parameters pertaining to the non-resonant background are not of interest in the present experiment.
Hence the only parameter of interest is the central resonance energy, E0 , and all others are treated
as nuisance parameters using the profile likelihood method [94]. In this method, − ln (L) is plotted

107

σ (MeV)

0.6
0.4

Normalized Counts

0.2
0
E = 0.1 MeV

0.15
E = 0.2 MeV
0.1
E = 0.4 MeV
E = 0.8 MeV

0.05

E = 1.5 MeV
0
0

0.5

1

1.5

2

2.5
3
Decay Energy (MeV)

Figure 5.2: Demonstration of the experimental resolution as a function of decay energy for
26 F + n coincidences. The bottom panel shows a variety of simulated decay energy curves; in
each, the un-resolved decay energy is a delta function, with energies of 0.1 MeV (red), 0.2 MeV
(blue), 0.4 MeV (green), 0.8 MeV (orange), and 1.5 MeV (navy). The top panel is a plot of the
Gaussian σ of simulated decay energy curves as a function of input energy.

108

→ 26 F + n
− ln (L)

Counts / bin (normalized)

27 F∗

380 keV resonance
Non–resonant
Sum

2σ
1σ

104

0.4

103
102

0.3

101
0.2 0.3 0.4 0.5 0.6
E0 (MeV)

0.2

0.1

0
0

0.5

1

1.5

2

2.5

3

Decay Energy (MeV)
Figure 5.3: Decay energy histogram for 26 F + n coincidences. The filled squares are the data
points, and the curves display the best fit simulation results. The red curve is a simulated 380 keV
Breit-Wigner resonance, while the filled grey curve is a simulated non-resonant Maxwellian
distribution with Θ = 1.48 MeV. The black curve is the sum of the two contributions
(resonant:total = 0.33). The inset is a plot of the negative log of the profile likelihood versus the
central resonance energy of the fit. Each point in the plot has been minimized with respect to all
other free parameters. The minimum of − ln (L) vs. E0 occurs at 380 keV, and the 1σ (68.3%
confidence level) and 2σ (95.5% confidence level) limits are indicated on the plot.

109

as a function of the parameter of interest, with each point on the plot minimized with respect to
the nuisance parameters. The orbital angular momentum of the resonance is fixed at ℓ = 2 from
theoretical expectations, assuming the emission of a 0d3/2 neutron. Although this assumption may
be invalid if the state in question has significant p- f shell components, use of the other plausible ℓ
values (ℓ = 1 or ℓ = 3) to model the decay does not significantly alter the best fit E0 value.
The inset of Fig. 5.3 displays a plot of the negative log-likelihood of the data versus E0 , with
each point on the curve minimized with respect to Γ0 , Θ, and the relative contributions of the
resonant and non-resonant models. This constitutes the profile likelihood as mentioned in the
previous paragraph. The minimum is located at E0 = 380 keV, and the 1σ (68.3% confidence
level) and 2σ (95.5% confidence level) intervals1 are ±60 keV and +130 keV, respectively. At
−120
E0 = 380 keV, the nuisance parameters are minimized to Γ0 = 10 keV; Θ = 1.48 MeV; and a
resonant:total ratio of 33%. The best fit results are plotted as the curves in the main panel of
the figure, with the red curve representing the 380 keV resonance; the filled grey curve the nonresonant Maxwellian background; and the black curve their sum.
As mentioned in Section 3.1, the decay energy measurement provides no information to distinguish whether the decays are feeding the ground state or excited states of the daughter nucleus,
26 F.

To make this determination, one must look at γ -rays recorded in coincidence with 26 F and

a neutron. A histogram of the Doppler corrected energy of γ -rays recorded in coincidence with
26 F + n

is shown in Fig. 5.4. Only two triple coincidence events were recorded, which strongly

indicates that the presently observed decays feed the ground state of 26 F. In the extreme opposite
case of a decay which had a 100% branching to a bound excited state in 26 F, around 50 counts
would be expected in CAESAR assuming 30% efficiency for gamma detection. It is plausible
that a small branching fraction to bound excited states in 26 F could produce a gamma spectrum
that is similar to the one presently observed; however, the presence of multiple decay branches
would either manifest itself as multiple resonances in Fig. 5.3 or occur at such a low rate that the
1 As mentioned in Section 4.5.4, the nσ confidence interval is the region in which ln (L) − ln (L)
min ≤

n2 /2.

110

Counts / 10 keV

27 F∗

→ 26 F + n + γ

1

0.5

0
0

1

2

3
Eγ (MeV)

Figure 5.4: Histogram of the Doppler corrected energy of γ -rays recorded in coincidence with 26 F
and a neutron. Only two CAESAR counts were recorded, giving strong indication that the present
decays are populating the ground state of 26 F.

contribution of the excited state branch is negligible.
To check that the fit to the data presented in Fig. 5.3 is accurate, it is instructive to compare
simulation and data for intermediate parameters used in calculating the decay energy. Fig. 5.5
shows four such comparisons. The parameters being compared in the figure are the opening angle
between the neutron and the fragment; the relative velocity (neutron minus fragment) between
the two particles; the neutron time of flight to the front layer of MoNA; and the fragment kinetic
energy calculated from the inverse mapping procedure of Section 4.3.1. As with Fig. 5.3, red
curves represent the 380 keV resonant simulation, filled grey curves the non-resonant background,
and black curves their sum. As can be seen in the figure, there is reasonable agreement between
simulation and data for each of the four parameters.

111

→ 26 F + n
Counts / 0.5 cm/ns

Counts / 0.5°

27 F∗

30
20
10

0

2

30
20
10
0

40
30
20
10
0

4
6
8
10
Opening Angle (degrees)
Counts / MeV/u

Counts / 2 ns

0

380 keV resonance
Non–resonant
Sum

-4

-2

0

2
4
vn − v f (cm/ns)

40
30
20
10

40

0

60
80
100
ToF to Front of MoNA (ns)

50

60
70
Fragment KE (MeV/u)

Figure 5.5: Comparison between simulation and data for intermediate parameters (opening angle,
relative velocity, neutron ToF, and fragment kinetic energy) used in calculating the decay energy
of 26 F + n coincidences. The red curves are the result of a 380 keV resonant simulation, and the
filled grey curves are the result of the simulation of a non-resonant Maxwellian distribution
(Θ = 1.48 MeV). Black curves are the sum of the two contributions (resonant:total = 0.33).

112

E ∗ (MeV)

3

2

(2+ ) [45]
(?) [45]

eV
380 k

(1+ ) [96, 97]

1

(1/2+ ) [45]
(1/2− ) [45]

0

(5/2+ ) [97]
26F + n

27 F

Figure 5.6: Summary of experimentally known levels in 26,27F. The presently measured decay of
a resonant excited state in 27 F to the ground state of 26 F is indicated by the arrow. The grey boxes
represent experimental uncertainties in the absolute placement of level energies relative to the 27 F
ground state. The dashed lines surrounding the present measurement correspond to the ±60 keV
uncertainty on the 380 keV decay energy, and the grey error box includes both this 60 keV
uncertainty and the 210 keV uncertainty of the 26,27 F mass measurements. The bound excited
states measured in [45] are placed assuming that all transitions feed the ground state as the authors
of [45] do not state conclusively whether their observed γ -rays come in parallel or in cascade.
In order to place the present measurement into a level scheme of 27 F, the central resonance
energy of 380 keV must be added to the one neutron separation threshold of 27 F. The one neutron
separation threshold, Sn , of a nucleus with mass number A is the difference in mass2 between the
nucleus and the constituents of its neutron breakup (a lone neutron and the neighboring isotope
with mass number A − 1):
Sn = (mA−1 + mn ) − mA .

(5.1)

Experimental masses are often published in terms of the atomic mass excess, ∆, given by
∆A = mA − Au,
2 Atomic mass, including electrons.

113

(5.2)

where u is the atomic mass unit expressed in the same dimensions as mA . Thus it is convenient
to express Sn in terms of the ∆ values of the A and A − 1 isotopes; combining Eqs. 5.1 and 5.2
results in this expression:

Sn = [(A − 1) u + ∆A−1 + mn ] − (Au + ∆A ) .

(5.3)



=  − u + ∆A−1 + mn −  − ∆A
Au
Au

(5.4)

= −∆A + ∆A−1 − u + mn .

(5.5)

The most recent mass measurements of 26,27F were performed using the time of flight technique at GANIL and are reported in Ref. [96]. These measurements place the atomic mass excesses
of 27 F and 26 F at ∆27 = 24630 ± 190 keV and ∆26 = 18680 ± 80 keV, giving a 27 F one neutron
separation energy of Sn = 2120 ± 210 keV. Combining this Sn measurement with the present measurement of a 380 ± 60 keV resonant decay from 27 F∗ to the ground state of 26 F corresponds to
the measurement of a previously unobserved excited state in 27 F at 2500 ± 220 keV. It should be
noted that despite the relatively large uncertainty of the present measurement, the total uncertainty
on the absolute level placement is dominated by that of the 27 F one-neutron separation energy.
Fig. 5.6 shows the placement of this newly observed excited level in a level scheme summarizing
experimentally known levels in 27 F and 26 F.

5.3 28 F Decay Energy
The decay energy spectrum of 28 F is shown in Figs. 5.7 and 5.8. The simulated acceptance curve
is identical to that of Fig. 5.1, indicating a resonance in the data with a maximum around 500 keV.
As outlined in Section 4.5.2, no background contribution is expected in the decay energy spectrum
of 28 F; hence an attempt is made to fit the spectrum with a single ℓ = 2 Breit-Wigner resonance. As
with 27 F decays, the fit is done by minimizing the unbinned negative log-likelihood. The parameter
of interest is the central energy of the resonance, E0 , and the only nuisance parameter, Γ0 , is treated

114

590 keV resonance

0.4

− ln (L)

Counts / bin (normalized)

28 F → 27 F + n

2σ
1σ

73

72
0.3
71
0.5

0.6

0.2

0.7
E0 (MeV)

0.1

0
0

0.5

1

1.5

2

2.5

3

Decay Energy (MeV)
Figure 5.7: Decay energy spectrum of 28 F (filled squares with error bars), along with the best
single resonance fit (red curve). The inset is a plot of the negative log-likelihood vs. central
resonance energy, demonstrating a minimum at 590 keV.

with the profile likelihood method mentioned in Section 5.2. Additionally, Γ0 is restricted to values
≤ 1 MeV as the simulated lineshape is virtually identical for Γ0 ≥ 1 MeV.
The result of the single resonance fit is shown in Fig. 5.7. The best fit parameters are E0 =
590 keV and Γ0 = 1 MeV. As seen in the figure, the quality of the single resonance fit is rather
poor: the fit fails to reproduce the broad shape of the spectrum despite a width value that is over
an order of magnitude greater than the single-particle prediction of ∼ 60 keV.3 This suggests that
multiple resonances are present in the data and that they should be fit with a superposition of
independent resonances.
3 The single particle prediction for the width is obtained from the solution of the problem of a 0d
5/2

neutron moving in a Woods-Saxon potential, with the potential well depth adjusted until a resonance is
found at 590 keV.

115

210 keV resonance
770 keV resonance
Sum
61

− ln (L)

Counts / bin (normalized)

28 F → 27 F + n

2σ
1σ

60

0.3

59
58
57

0.2

0.1

0.2

0.3
E0 (MeV)

0.1

0
0

0.5

1

1.5

2

2.5

3

Decay Energy (MeV)
Figure 5.8: Decay energy spectrum for 28 F, with the best two resonance fit results superimposed.
The filled squares with error bars are the data, and the red and blue curves are the lower resonance
(ex)
(gs)
(E0 = 210 keV) and upper resonance (E0 = 770 keV) fits, respectively. The black curve is
the superposition of the two individual resonances. The inset shows a plot of the negative
(gs)
log-likelihood vs. E0 , demonstrating a minimum at 210 keV and 1, 2σ confidence regions of
+50,+90
−60,−110

keV. Each point on the inset likelihood plot has been minimized with respect to the
(ex)

nuisance parameters E0

(ex)

, Γ0

, and the relative contribution of the two resonances.

116

Because of coarse resolution and low statistics, it is not possible to precisely resolve the contribution of multiple individual resonances, and the main interest is in determining the location
of the ground state. To avoid a solution in which the ground state resonance is made artificially
(gs)

wide, the width of the lower resonance is fixed at Γ0

= 10 keV, which is approximately equal
(gs)

to the single-particle prediction for resonances in the expected region of E0
(ex)

upper resonance, Γ0

. The width of the

is allowed to take any value ≤ 1 MeV. The parameter of interest is the
(gs)

central energy of the lower resonance, E0
(ex)

likelihood method—are E0

(ex)

, Γ0

, and the nuisance parameters—treated with the profile

, and the relative contribution of the two resonances. Both

resonances in the fit have a fixed orbital angular momentum value of ℓ = 2.
The result of the two resonance fit is shown in Fig. 5.8. The best fit is found with a lower
(gs)

resonance at E0

= 210 keV and 1σ and 2σ confidence regions of +50 keV and +90 keV,
−110
−60
(ex)

respectively. The central energy and width of the upper resonance are E0

(ex)

= 770 keV and Γ0

=

1 MeV, and the ratio of the ground state resonance to the total area is 24.2%. As seen in the
figure, the two resonance model gives a reasonable agreement with the data, providing a much
better fit than the one resonance hypothesis. The likelihood ratio of the one and two resonance
hypotheses, D = −2 ln (L1 /L2 ) , is equal to 26.7. The two resonance hypothesis has two additional
free parameters relative to the one resonance one, so probability distribution of D should be χ 2
2
with two degrees of freedom [94]. Comparison of D with χndf=2 critical values indicates that the

one resonance hypothesis can be rejected at a confidence level of greater than 4σ .
(ex)

The large Γ0

of the upper resonance in the fit suggests that more than two resonances might

be present in the data, and this interpretation is certainly plausible based on the theoretical level
density of 28 F below 1 MeV. However, with the present resolution and statistics it is not possible to
accurately distinguish the contribution of multiple excited state resonances to the measured decay
spectrum. Attempts to fit the data with three independent resonances indicate that the location of
the ground state resonance is insensitive to the number of resonances used in the fit: there is no
way to match the data points below ∼ 300 keV unless the lowest resonance in the sum is placed
around 200 keV.

117

210 keV resonance
770 keV resonance
Sum

15

Counts / 0.5 cm/ns

Counts / 0.5°

28 F → 27 F + n

10
5

0

2

15
10
5
0

10

0
-4

4
6
8
10
Opening Angle (degrees)
Counts / MeV/u

Counts / 2 ns

0

20

-2

0

2
4
vn − v f (cm/ns)

20
15
10
5

40

0

60
80
100
ToF to Front of MoNA (ns)

50

60
70
Fragment KE (MeV/u)

Figure 5.9: Comparison of simulation and data for intermediate parameters used in constructing
the decay energy of 28 F. The parameters being compared are noted as the x axis labels on the
individual panels. The filled circles with error bars are the data, and the red, blue and black curves
represent the 210 keV resonance simulation, 770 keV resonance simulation and their sum,
respectively.

118

E ∗ (MeV)

3
[present]
2

1

0

(1/2+) [45]
(1/2−) [45]

770

(5/2+ ) [97]

keV

560 keV

210 keV

27F + n

28 F

Figure 5.10: Experimental level scheme of 28 F and 27 F. The black lines represent the nominal
placement of states relative to the ground state of 27 F, and the grey boxes represent 1σ errors on
the measurements. As described in the text, the placement of the 560 keV (770 keV decay energy)
excited state in 28 F is extremely uncertain, and it is only included in the figure for consistency.
As with 27 F decays, it is instructive to compare simulation and data for intermediate parameters
that contribute to the calculation of decay energy. This comparison is shown in Fig. 5.9 for decays
from 28 F. The parameters compared are opening angle between the neutron and the fragment;
relative velocity (neutron minus fragment); time of flight to the front and center of MoNA; and
the reconstructed fragment energy. As seen in the figure, reasonable agreement is achieved for all
parameters when comparing the data to the two resonance fit.
No gamma-rays were recorded in CAESAR in coincidence with 28 F + n, indicating that the
presently observed decays are feeding the ground state of 27 F. This places the ground state binding
energy of 28 F at Erel = 210+50 keV above the ground state of 27 F, as shown in the level scheme
−60

of Fig. 5.10. The level scheme also displays an excited state in 28 F at 560 keV, corresponding
to the upper 770 keV resonance of the fit; however, it should be emphasized that the placement
of this excited level in 28 F is extremely uncertain. As mentioned previously, it is possible to fit

119

the 28 F decay spectrum with more than two resonances, and although the location of the ground
state resonance is robust, the placement of excited levels will vary as additional resonances are
introduced. Hence the only reliable measurement for 28 F is that of its ground state energy relative
to 27 F.
Using the present measurement and that of the 27 F mass, the absolute binding energy of 28 F is
calculated as
B28 = B27 + Erel ,

(5.6)

with the binding energy of 27 F calculated from its atomic mass measurement via:
B27 = Zm p + Nmn − (∆27 − Zme ) − Au,

(5.7)

where Z, N, and A are the proton, neutron and mass numbers of 27 F; m p , mn and me are the
proton, neutron, and electron masses; and, as in Eq. (5.2), u is the atomic mass unit expressed in
the same dimensions as the various masses. Using the measurement of ∆27 = 24630 ± 190 keV
from Ref. [96], Eq. (5.7) evaluates to B27 = 186.26 ± 0.19 MeV. The binding energy of 28 F is then
calculated from Eq. (5.6) as B28 = 186.47 ± 0.20 MeV.
5.4 Cross Sections
In addition to measuring decay energies, it is possible to calculate cross sections for the population
of observed states in the reaction 9 Be(29 Ne, X). The cross section, in millibarns, is given by

σ=

nr
× 1027 ,
nt nb

(5.8)

where nr , nt , and nb are the numbers of reactions, target nuclei, and incoming beam particles,
respectively. The number of target nuclei is calculated from the target thickness, δ , and the atomic
mass of natural beryllium, MBe :
nt = NA δ /MBe ,

(5.9)

where NA is Avagodro’s number, 6.02 × 1023 mol−1 . Plugging δ = 0.288 g/cm2 and MBe =
9.012 g · mol−1 into Eq. (5.9), the number of target nuclei evaluates to nt = 1.92 × 1022. The number of incoming beam particles was recorded by a scaler module connected to the target scintillator
120

CFD, and to obtain the number of incoming 29 Ne beam particles, the total number of scaler counts
is divided by the live time of the experimental system and multiplied by the ratio of 29 Ne to total beam particles. The 29 Ne:total ratio is calculated by comparing the number of events falling
within the 29 Ne time of flight gate (c.f. Section 4.2.1) to the total number of events recorded in
the beam ToF spectrum; this ratio is calculated separately for each production run taken during the
experiment.
The number of reactions is given by
nr = f nc /ε ,

(5.10)

where nc is the number of recorded neutron-fragment coincidences; f is the fraction of the total
fit area taken up by the decay channel in question; and ε is the total efficiency of the experimental
system. The efficiency can be divided up into its constituent parts:

ε = εgeomεm ε¯s .

(5.11)

In the above equation, εgeom refers to the geometric efficiency of the Sweeper-MoNA system
(at the appropriate decay energy) and is obtained from Monte Carlo simulations. εm is the detection efficiency of MoNA and is calculated from GEANT4 simulations done with the appropriate
¯
incoming neutron energy. εs is the intrinsic efficiency of the charged particle detectors. To obtain

ε¯s , a run-by-run intrinsic efficiency is first calculated by comparing the number of fluorine elements
that register signals in all charged particle detectors to the total number that trigger the experimental system. This calculation excludes the ion chamber efficiency since the ion chamber is needed
to make a fluorine gate. To account for this, the efficiency of the ion chamber for Z ∼ 9 particles
is fist calculated from a target-out run, and this number is multiplied by the run-by-run fluorine
efficiency. Drift in the ion chamber efficiency between runs, as well as the difference in efficiency
between Z = 10 (unreacted beam) and Z = 9 is negligible, so the procedure described above is
sufficient to obtain a run-by-run charged particle detection efficiency for Z = 9 particles. From
here, ε¯s is calculated by taking a weighted average of the run-by-run charged particle efficiency

121

values:

ε¯s = ∑ wi εi ,

(5.12)

i

where the weights wi are equal to the number of incoming 29 Ne beam particles in run i divided
by the total number of incoming beam particles, and εi is the Z = 9 efficiency for run i.
Using the calculation procedure described above, the cross section for the population of the
ground state of 28 F is calculated to be 0.23 ± 0.06 mb, and the cross section for the population of
the 2500 keV excited in 27 F is calculated as 1.54 ± 0.68 mb. A detailed summary of the numbers
which go into the cross section calculations can be found in Appendix A. It should be noted that
there is a significant systematic error on the 28 F ground state cross section measurement resulting
from uncertainty in the number of excited state resonances to include in the fit to data. The total
number of resonances significantly alters the relative contribution of the ground state resonance
( f in Eq. (5.10)) which in turn alters the cross section value. Quantifying this systematic error in
terms of 1σ confidence intervals is not practical, but to give a sense of its magnitude, the cross
section would be calculated as 0.40 ± 0.18 mb ( f = 42.9 ± 18.3%) in the case of three resonances.

122

Chapter 6
DISCUSSION

res

6.1 Shell Model Calculations
In this chapter, the present experimental results are compared to theoretical predictions in order to
place them into a broader physical context. The theoretical predictions are made with large scale
shell model calculations, and in this section a brief description of these calculation methods is
presented.
As mentioned in Chapter 1, the nuclear shell model is built upon the idea of treating nuclei as a
collection of independent particles subjected to a mean field. The most realistic form for the mean
field is the Woods-Saxon potential:
V (r) = −

V0
,
(r−R)/a + 1
e

(6.1)

where V0 is the potential well depth and a represents the surface thickness of the nucleus. R
is the nuclear radius, which typically scales as R = r0 A1/3 , with the parameter r0 in the range of
1.2 to 1.4 fm. In addition to the potential well, a strong spin-orbit term is necessary for an accurate
description of nuclear properties:
1
Vso (r) = − V ′ (r) ℓ · s.
r

(6.2)

This spin-orbit term gives rise to the splitting of ℓ orbitals displayed in Fig. 1.1, with the
j = ℓ − 1/2 coupling raised in energy relative to the j = ℓ + 1/2 coupling.
The simple nuclear potential outlined above, consisting of a mean field and spin orbit term,
is only adequate for the description of very basic nuclear properties such as the reproduction of
large energy gaps at the “magic numbers” 2, 8, 20, 28, 50, 82, and 126. For a more complete

123

description of nuclei, the nuclear Hamiltonian, H = T + V, must also, at a minimum, take twobody interactions into account. These are typically introduced by splitting the Hamiltonian into
two parts:
H = H0 +Vres ,

(6.3)

where H0 = T +V0 includes the kinetic energy T and central potential V0 and Vres is the residual
potential given by
A

A

Vres = ∑ Vi j − ∑ Vc (ri ) .
i, j

(6.4)

i

In Eq. (6.4), Vi j represent the two-body interactions between nucleons and Vc represents the
central potential of Eqs. (6.1) and (6.2).
In principle, one can solve the Schrödinger equation using the potentials described above, with
Eq. (6.4) applied to all nucleons in the nucleus. In practice, however, this requires large amounts
of computation time for all but the lightest nuclei and is not practical for typical calculations.1
Instead, the nucleus is divided into an inert core and an active valence space for which the twobody contributions are calculated. Within this model space, the total wavefunction, |ψk , for a
state with quantum numbers k = {n, ℓ, j} is given by a sum of basis states:
|ψk = ∑ akα |ψα ,

(6.5)

α

where the sum is over the orbitals α in the active valence space. The basis states, |ψα ,
are typically defined to have either a definite magnetic quantum number M (referred to as the
m-scheme) or a definite total spin quantum number J (the j-scheme).
From the potential of Eqs. (6.3) and (6.4) and the wave function of Eq. (6.5), the time independent Schrödinger equation,
H |ψk = Ek |ψk ,

(6.6)

is expressed as
1 It should be noted that there is presently a great deal of effort being put into the “no core” shell model

[98] and other ab initio methods that attempt to describe nuclei by starting from first principles and treating
all nucleons on an equal footing. However, these calculations are currently only tractable up to A ∼ 20.

124

(H0 +Vres ) ∑ akα |ψα = Ek ∑ akα |ψα .
α

(6.7)

α

Taking the inner product of both sides of Eq. (6.7) with ψk | = ∑β akβ ψβ | yields

∑ ak α ak β

α ,β

ψβ | H |ψα = Ek

∑ akα akβ δαβ .

(6.8)

α ,β

The solution of Eq. (6.8) is then equivalent to solving the eigenvalue problem

H11 − Ek
H21
.
.
.
Hn1

···

H1n

H22 − Ek · · ·
..
.
···

H2n
.
.
.

H12

,

(6.9)

· · · Hnn − Ek

Hn2

which determines the energy eigenvalues, Ek , of the system as well as the matrix elements,
Hβ α =
=

ψβ | H |ψα

(6.10)

ψβ | H0 |ψα + ψβ |Vres |ψα .

(6.11)

The right hand side of Eq. (6.11) is divided into two parts: the single particle energies
(0)

(0)

ψβ | H0 |ψα = Eαβ δαβ ≡ Eα

(6.12)

and the two-body matrix elements (TBME)

ψβ |Vres |ψα ≡ ψβ |V12 |ψα .

(6.13)

From here, one can solve for the akα coefficients in Eq. (6.7), thus determining the nuclear
wavefunctions and allowing calculation of additional spectroscopic properties of the various states
as well as their overlap, ψi |ψ f .
In Eq. (6.13), the V12 represent the effective two-body interactions. In principle these can
be derived from the bare nucleon-nucleon interaction, but in practice they are often determined
by fitting to experimental data. The fitting procedure begins by choosing a core and valence
125

space for which the interaction is intended and then by collecting a wide variety of reliable experimental information (typically ground and low-lying excited state energies) on nuclei in the
intended region. The TBME are then adjusted until the deviation between experimental data
and theoretical predictions is at a minimum, with an approximate theoretical error determined
from the Root Mean Square (RMS) deviation between the two. In the present work, shell model
calculations are performed using three interactions: USDA and USDB [99], which operate in
the sd valence space (π 0d5/2π 1s1/2π 0d3/2ν 0d5/2 ν 1s1/2ν 0d3/2 ), and the recently developed
IOI interaction [100], which operates in a truncated sd-p f valence space (π 0d5/2π 1s1/2π 0d3/2

ν 1s1/2 ν 0d3/2ν 0 f7/2 ν 1p3/2 ν 1p1/2 ). The calculations are performed using the j-scheme code
NuShellX@MSU [101, 102].

6.2 27 F Excited State
Fig. 6.1 shows the experimental and theoretical (USDA, USDB, IOI) level schemes of 27 F, up
to 3.5 MeV. As seen in the figure, all three models predict a fairly high level density in the
region of the presently observed resonance at 2500 ± 220 keV. The observation of only a single
resonance could indicate selectivity of the reaction mechanism used to populate 27 F∗ , but as the
precise reaction mechanism is not known, any attempt to argue for the assignment to a specific
state based on reaction cross sections would be highly speculative. Moreover, the ground state
structure of the 29 Ne beam is somewhat uncertain. Its spin and parity have not been measured
experimentally, and there is also discrepancy in shell model predictions. MCSM calculations using
the SDPF-M interaction predict the 29 Ne ground state to be 3/2+ with a 100% 2p-2h configuration
[103], while calculations using the IOI interaction and model space predict a 3/2− ground state
with a configuration that is primarily 1p-1h. The lack of certainty about the 29 Ne ground state
structure further distorts any clarification that might be gained from consideration of population
cross sections.
Further distorting the interpretation of the present observation is the fact that shell model calculations fail to reproduce the bound state measurements of Ref. [45]. Of the calculations presented,
126

E ∗ (MeV)

3.5
5/2+
9/2+
7/2+
3/2+

3
2.5
2

9/2+
9/2−
5/2−
1/2−
11/2−
5/2+
7/2+
3/2+
7/2−
1/2+
3/2−
5/2−
1/2−
1/2+
5/2+

3/2+
9/2+
7/2+
1/2+

1.5

1/2−

1
0.5

1/2+
1/2−

0

5/2+

1/2+

Exp.

5/2+
USDA

5/2+
USDB

IOI

Figure 6.1: Summary of experimental and theoretical (USDA, USDB, IOI) excited levels in 27 F.
The grey boxes surrounding the experimental values represent the 1σ uncertainties on the
measurements. Experimental values are taken from Refs. [97] (ground state J π ), [45] (bound
excited states), and the present work. Both possible placements of the 504 keV transition observed
in Ref. [45] are included, with the cascade placement shown as the dashed line at 1282 keV.
all overpredict the 1/2+ first excited state energy by ∼ 700 keV or more, and the IOI calculation overpredicts the 1/2− excited state energy by either ∼ 1300 keV or ∼ 500 keV depending
on whether the experimental observation is placed in parallel or cascade with the 1/2+ . Additionally, the MCSM/SDPF-M calculations presented in [45] overpredict the 1/2+ energy by around
320 keV.
As mentioned in Ref. [45], proton excitations from the p shell to the sd shell likely play a role
in the excited state structure of 27 F. This has been explored theoretically in Ref. [104], which
deals with excited states in even-N fluorine isotopes that involve p to sd shell proton excitations
(or “proton hole” states). The predicted energies of bandhead proton hole states are summarized in
Fig. 6.2, taken from the reference. The proton holes are expected to couple to three types of neutron
configurations: pure sd (resulting in a 1/2− bandhead state), 1p-1h (3/2+ bandhead state), and

127

14

ground state

excitation energy [MeV]

12
1/2−
2

10

3/2+
2

8

3/2+
2

6

1/2−
1

4
2
0

1/2−
2

1/2−
1
1/2+
1

1/2−
1
1/2−
1
1/2−
1

19 F

proton hole state
without neutron in pf–shell
proton hole state
with one neutron in pf–shell
proton hole state
with two neutron in pf–shell
experiment (1/2− only)
3/2+
2
1/2−
2
1/2−
1

1/2−
2
3/2+
2
1/2−
1

1/2−
3

3/2+
2
1/2−
1

5/2+
1

5/2+
1

5/2+
1

5/2+
1

5/2+
1

21 F

23F

25 F

27 F

29F

Figure 6.2: Excitation energies of theoretical p shell proton hole states in even-N fluorine
isotopes. Adapted from Ref. [104], with the 25,27 F 1/2− candidate states of Ref. [45] added,
assuming parallel gamma emission.
2p-2h (1/2− bandhead). In the neutron rich fluorine isotopes, the proton hole states coupling to
1p-1h and 2p-2h neutron configurations are predicted to be relatively low in excitation energy as a
result of the diminished N = 20 shell gap, and it is interesting to note that the predicted energies of
the 1p-1h and 2p-2h states in 27 F are both close to the present observation. Regarding p-sd proton
excitations, it should also be noted that a new PSDPF interaction has recently been developed [105]
to treat configurations in the full p-sd-p f model space, and it would be interesting to compare the
predictions of this interaction to the present measurement.

6.3 28 F Binding Energy
As outlined in Chapter 2, it has been well established, both experimentally and theoretically, that
there is a region of nuclei in the vicinity of Z = 11 and N = 20 whose low-lying structures have

128

significant p f shell “intruder” components, in the form of ℏω excitations of neutrons from the sd
shell to the p f shell2 . This region is often referred to as the “island of inversion,” in reference to
the inverted filling of single particle levels (p f before sd). The region of inversion was originally
thought to encompass only those nuclei directly surrounding 32 Na, but in more recent years its
boundaries have become an open question. Approaching stability, where nuclei become more
experimentally accessible, it is generally agreed that the intruder components fade away for N
18 and Z

13. What is less certain is the importance of intruder configurations as one moves

away from stability and towards more neutron rich nuclei, either by adding neutrons or removing
protons. It was originally proposed that intruder configurations are significant only for N ≤ 22
[106], but recent experiments have contradicted this claim, collecting evidence that significant
intruder components are present in the ground states of nuclei with higher neutron number. For
example, the 9 Be(38 Si,36 Mg)X reaction has been used to establish that there is a significant 2ℏω
contribution to the ground state of 36 Mg, which has 24 neutrons [107].
Going away from stability in the other direction, i.e. removing protons, the contribution of
intruder configurations to the ground state structure of nuclei has been largely unexplored, limited to theoretical calculations such as those of Ref. [34], which were introduced in Section 2.4.
Experimentally, the fluorine isotopic chain is the only area in which the intruder structure of
Z < 10, N ≥ 19 nuclei can be examined in any detail. All lighter elements are unbound past
N = 16, making experimental investigation of isotopes with N ≥ 19 extremely difficult since all
of these nuclei will decay by the emission of three or more neutrons. The present measurement of
the ground state binding energy of 28 F is the first experimental investigation into the structure of
fluorine nuclei with N ≥ 19, and in this section its relevance to the intruder structure of 28 F will be
discussed.
A simple way to check for p f shell intruder components in the ground state of a nucleus is
2 Here a nℏω excitation means that n neutrons have been promoted from the sd shell to the p f shell,

in contrast to the “normal” filling of orbitals seen near stability. In general, however, it means that a harmonic oscillator gap has been crossed n times: for example 2ℏω could mean that either two nucleons were
promoted across a single oscillator gap or that a single nucleon was promoted across two oscillator gaps.

129

22
2.0
1.5
1.0
0.5
0
-0.5
-1.0
-1.5

20

Proton Number

18
16
14
12
10
8

28 F

6

6

8

10

12 14 16 18
Neutron Number

20

22

Figure 6.3: Difference between experimental [12, 96] and theoretical (USDB) binding energies
for sd shell nuclei, with positive values meaning experiment is more bound than theory. Adapted
from Ref. [99]

to compare its measured binding energy to the predictions of a shell model which includes only
sd orbitals in the active valence space. A significant deviation between experiment and theory
would likely indicate that configurations which extend outside of the sd shell are important to the
description of the nucleus in question. Such a comparison is shown in Fig. 6.3, which is a plot of
BEexp − BEth for sd shell nuclei, adapted from Ref. [99]. The theoretical calculations are done in
the sd valence space using the USDB interaction (the authors of [99] claim similar results using the
USDA interaction). The region circled on the plot is composed of traditional island of inversion
nuclei, and it is clear that the theoretical predictions deviate significantly from experiment, with
experiment more bound than theory by up to 2 MeV. The authors of [99] speculate that 28,29 F lie
outside of the island of inversion, but at the time of publication the only available experimental
binding energies for these nuclei were mass extrapolations, making this claim extremely tenuous.
Another way to examine p f shell intruder components theoretically is to use a shell model that

130

Proton Number

Legend
17
16
15
14
13
12
11
10
9
8

Stable
2.0
1.75
1.5
1.25
1.0
0.75
0.5
0.25
0.0

28 F

15 16 17 18 19 20 21 22 23 24
Neutron Number
Figure 6.4: Average ℏω excitations in the ground states of nuclei with 8 ≤ Z ≤ 17 and
15 ≤ N ≤ 24, as calculated in the IOI interaction and truncated sd p f model space. Taken from
Ref. [100].

includes the sd and p f shells in its active valence space to calculate the occupation probabilities of
p f shell orbitals. Such calculations were recently done for a wide variety of nuclei centered around
N = 20 [100]. These calculations utilize the IOI interaction and an active valence space composed
of the 1s1/2 0d3/20 f7/2 1p3/21p1/2 neutron orbitals. Fig. 6.4 is a plot of the average ℏω excitations
in the ground states of nuclei with 8 ≤ Z ≤ 17 and 15 ≤ N ≤ 24. Nuclei in the traditional island
of inversion region are clearly reproduced as having large ℏω contributions to their ground state
structure, as are nuclei with Z ≤ 12, N ≥ 22. This calculation also predicts all fluorine isotopes
with N ≥ 19 to have an average ℏω excitation of one or greater, placing them within the island of
inversion.
The theoretical analyses presented above give differing suggestions as to whether or not 28 F
is an island of inversion nucleus. To distinguish between the two assertions, it is instructive to
compare the predicted ground state binding energies given by the two models. Such a comparison
is shown in Fig. 6.5, which is a plot of the predicted binding energies of N = 19 isotones in the

131

N = 19
BE − 16 · Z (MeV)

IOI
USDA
USDB

50

45

40

35
8

9

10

11

12

13

14

15

16

17
Z

Figure 6.5: Theoretical predictions for the ground state binding energies of N = 19 isotones from
oxygen (Z = 8) through Chlorine (Z = 17). Calculations are shown for three interactions: IOI
(black inverted triangles), USDA (red squares), and USDB (blue circles). To put the binding
energies on roughly the same scale, the calculation results have been shifted by subtracting 16 · Z
from their original values.

IOI, USDA, and USDB shell models (to put the binding energies on roughly the same scale, the
calculation results have been shifted by subtracting 16 · Z from their original values). For heavier
nuclei, which the IOI calculation predicts to have minimal ℏω components, the binding energy
predictions are generally in good agreement. As expected, the calculations diverge significantly
for Z = 10, 11, 12, with the IOI calculations indicating significantly greater binding. For Z = 8
and Z = 9, the IOI calculations continue to predict greater binding than UDSA/USDB, albeit to a
lesser extent than for Z = 10, 11, 12.
To examine the binding energy systematics of N = 19 isotones further, theoretical predictions
must be compared to experiment. Fig. 6.6 displays this comparison, plotting the difference in bind-

132

BEexp − BEth (MeV)

N = 19

IOI
USDA
USDB

1.5
1

0.5

0
-0.5

-1
9

10

11

12

13

14

15

16

17
Z

Figure 6.6: Binding energy difference between experiment (Refs. [12, 96] and the present work
for Z = 9) and theory (IOI, USDA, USDB) for N = 19 isotones with 9 ≤ Z ≤ 17. As in Fig. 6.3,
positive values indicate experiment being more bound than theory. Error bars are from
experiment only, and the shaded grey region represents the 370 keV RMS deviation of the IOI
interaction, while the horizontal red and blue lines denote the respective 170 keV and 130 keV
RMS deviations of USDA and USDB.
ing energy between experiment and theory for N = 19 isotones from Z = 9 to Z = 17. The experimental value for 28 F is from the present work, and all others are from Refs. [12,96]3 . As in Fig. 6.5,
the theoretical calculations are done using the IOI, USDA, and USDB interactions and their associated model spaces. In this plot the experimental binding energies of isotones with Z = 10, 11, 12
clearly deviate from the USDA/USDB predictions. These nuclei are reasonably well reproduced
by the IOI calculations, with the differences between experiment and theory falling within the
3 Unless otherwise noted, the citation [12,96] means the following: if the mass in question was measured

in Ref. [96], the measurement of [96] is used. Otherwise the mass is taken from the 2003 Atomic Mass
Evaluation of Ref. [12].

133

370 keV RMS deviation of the IOI interaction. For 28 F, the difference between experiment and
USDA/USDB theory drops dramatically, with the USDA calculation giving exact agreement within
experimental error bars. Taken alone, this feature strongly suggests that p f shell components are
not necessary for a complete description of the 28 F ground state. However, this conclusion must
be weighed against the IOI calculation results which predict an average ℏω excitation of 1.22 in
the 28 F ground state. The difference in binding energy between theory (IOI) and experiment is less
than the theory’s 370 keV RMS deviation; hence the IOI calculation’s assertion of a 28 F ground
with significant (≥ 1ℏω ) p f shell components cannot be ruled out conclusively. Despite this, the
consistency of the present measurement with the USDA/USDB calculations demonstrates that p f
shell components are not necessary for reproduction of the presently observable properties of the
28 F

ground state. This is in sharp contrast to the N = 19 isotones whose inversion structure is

better established—29Ne, 30 Na, and 31 Mg: the ground state binding energies (and in the case of
31 Mg, spin-parity [108]) of these nuclei

cannot be reproduced in an sd-only calculation.

In addition to the N = 19 isotones, it is also instructive to examine the binding energy systematics of the fluorine isotopic chain. Fig. 6.7 plots the experimental (when available) and theoretical
(IOI, USDA, USDB) binding energies for fluorine isotopes from A = 24 (N = 15) to A = 31
(N = 22), with the top panel showing the difference between experiment and theory for the various models. The lightest two nuclei, 24,25F, are well-reproduced in the USDA/USDB shell models
but significantly under-bound by the IOI calculation, likely due to the IOI calculation’s exclusion
of the ν 0d5/2 orbital from its active valence space [100]. The calculations are in fairly good agreement with data and with each other for 26,27F, but as discussed previously they begin to diverge
at 28 F, with the IOI model predicting greater binding, and the level of divergence continues to
increase for 29 F. There is little to compare for the heaviest fluorine nuclei, but the only available
experimental information—namely the (non)existence of (30)31 F—is reproduced by the IOI shell
model within its 370 keV RMS deviation.

134

Binding Energy (MeV)

[exp] - [th]

Z=9
1
0
-1
188
186

Exp.
IOI
USDA
USDB

184
182
180
178
24

25

26

27

28

29

30

31
A

Figure 6.7: Experimental (Refs. [12, 96] and the present work for A = 28) and theoretical (IOI,
USDA, USDB) binding energy predictions for fluorine isotopes, 24 ≤ A ≤ 31. The USDA and
USDB calculations end at A = 29 (N = 20), as their corresponding model spaces cannot
accommodate more than 20 neutrons. The top panel is a plot of binding energy differences
(experiment minus theory) for nuclei whose mass has been measured. Error bars in the top panel
are from experiment only, and the shaded grey region and red/blue horizontal lines are the same
as in Fig. 6.6.

135

Chapter 7
SUMMARY AND CONCLUSIONS

In this work, the binding energy of 28 F and the excitation energy of a neutron-unbound state in 27 F
have been measured using the technique of invariant mass spectroscopy. The neutron-unbound
states were populated by reactions of a 62 MeV/u 29 Ne beam impinging on a 288 mg/cm2 beryllium target, with the radioactive 29 Ne beam produced at the NSCL Coupled Cyclotron facility at
Michigan State University. Neutrons resulting from the decay of the unbound states were detected
in a large area plastic scintillator array (MoNA), and the residual charged fragments were deflected
by a dipole magnet and analyzed in a variety of charged particle detectors. Gammas produced in
the reaction were also recorded in a CsI(Na) array surrounding the target, allowing for a unique
determination of the decay path of the neutron-unbound states. The observed resonances were fit
using a Breit-Wigner lineshape with energy dependent width, and the smearing of experimental
resolution and acceptance was accounted for in a Monte Carlo simulation of the experiment.
In 27 F, a resonant state was observed with 380 ± 60 keV decay energy, feeding the ground state
of 26 F. In addition to this resonant state, a non-resonant background was observed in the data; this
background is expected to arise from the neutron evaporation of high-lying states in 28 F, and it
is well described by a Maxwellian distribution of thermalized beam velocity neutrons. Combined
with the one neutron separation energy of 27 F, measured in Ref. [96] to be 2120 ± 210 keV, the
presently observed resonance corresponds to an excited level in 27 F at 2500 ± 220 keV.
In 28 F, the ground state was observed to decay to the ground state of 27 F with a relative energy
of 210+50 keV, corresponding to a 28 F binding energy of 186.47 ± 0.20 MeV. In addition to
−60
the ground state, the inclusion of at least one additional higher-lying resonance was necessary to
describe the data, with the location of the ground state peak insensitive to the number of additional
resonances included in the fit.
To interpret the present measurements, they are compared to shell model predictions. For 27 F,

136

shell model calculations in both the sd and truncated sd-p f model spaces predict a fairly high level
density in the neighborhood of 2500 keV in excitation energy, making it impossible to assign the
present observation to a specific state based on energies alone. Assignments based on calculated
cross sections are also not possible as the specific mechanism by which the observed excited state
was populated is not certain. Excited states with p shell proton hole structure are also predicted
in this energy region, making them plausible candidates as well. Either an improvement in the
predictive power of theoretical calculations or the collection of additional experimental data will
be needed before it is possible to assign the presently observed 2500 keV state in 27 F to a specific
level.
The measurement of the 28 F ground state energy is particularly interesting when interpreted
in terms of ground state mass systematics and the boundaries of the island of inversion. Until
now, there was no experimental data available to indicate whether the island of inversion extends
to lower element number than its traditional boundary of Z = 10. The present measurement of
the 28 F binding energy lies between the predictions of the IOI (truncated sd-p f model space) and
USDA/USDB (sd model space) shell models and is consistent with both calculations within their
RMS values. The merging of the USDA/USDB and IOI calculations—in contrast to 29 Ne, 30 Na,
and 31 Mg—suggests the presence of a “southern shore” of the island of inversion at 28 F, and this
is confirmed by the binding energy measurement of the present work.
The present results also suggest a number of additional studies which could be useful in clarifying the structure of fluorine nuclei in the vicinity of N = 20. Perhaps the most pertinent for
determining the bottom edge of the island of inversion is a mass measurement of 29 F. As shown
in Fig. 6.7, the deviation between USDA/USDB and IOI calculations for the ground state binding
of 29 Ne is large, so an experimental test of these predictions would provide significant insight into
the situation. Furthermore, a measurement of the first excited state energy in 29 F would provide
much needed information about the size of the N = 20 shell gap for fluorine isotopes. It is unlikely that 29 F has bound excited states (though a measurement would also be necessary to confirm
this assumption); hence measuring its first excited state would require neutron decay spectroscopy

137

involving two neutron emission.
There are also a few possibilities to expand upon the present measurement of unbound states
in 27 F and 28 F. Since the presently observed resonances have been shown to decay to the ground
state of their respective daughter nuclei, one possibility would be to repeat the present measurement without the CAESAR array. The main advantage would be an increase in statistics: removing
CAESAR would allow the reaction target to be moved forward, significantly decreasing the neutron
shadowing demonstrated in Fig. 5.1 and increasing the detection efficiency for high decay energies.
Additionally, inclusion of the recently commissioned Large multi-Institutional Scintillator Array
(LISA) [109] in the setup would allow for neutron detection at larger angles, further increasing
efficiency at high decay energy. Removal of CAESAR would also allow the Sweeper magnet to
be run at higher rigidity as the Sweeper’s field was limited in the present experiment to allow the
CAESAR PMTs to operate properly. In turn, this allows the incoming beam energy to be optimized
for maximum 29 Ne production, increasing overall statistics. Finally, it may be possible to populate unbound states in either 27 F or 28 F by other reaction mechanisms, such as 27 F(d, p)28F(∗),
27 F(p, p′ )27 F∗ ,

2p knockout, 1n knockout to 28 F, or fragmentation of an intense secondary beam.

Depending on specific cross sections and beam rates, each of these reaction mechanisms has the
potential to increase overall population rates for the states of interest. Additionally, using (d, p),
(p, p′ ), or 2p knockout to populate unbound states could increase the selectivity of the reaction,
making it possible to assign observed resonances to specific levels in 27 F or 28 F.

138

APPENDIX

139

Appendix A
CROSS SECTION CALCULATIONS

A.1 27 F Excited State
Table A.1: Average values used in calculating the cross section to 27 F∗ .
Number of Recorded Reactions:
Ion Chamber Efficiency:
Number of Target Nuclei:
Geometric Acceptence:
MoNA Efficiency:
Fit Fraction:
Average Sweeper Efficiency:
Actual Number of Reactions:
Cross Section (mb):

157 ± 12.5
98.93%
1.92 × 1022
11.66 ± 0.77%
77%
32.93 ± 14.00%
58.32 ± 0.37%
987.37 ± 432.10
1.54 ± 0.68

Table A.2: Run-by-run values used in calculating the cross section to 27 F∗ .
Run
1089
1090
1091
1093
1094
1095
1096
1097
1105
1106
1108
1109
1110
1111
1113
1114
1115

Target Scint.
Scaler

Live
Time

Sweeper
Efficiency

29 Ne

29 Ne

Counted

Actual

3001020 ± 1732
4040444 ± 2010
189262 ± 435
25275536 ± 5027
24541327 ± 4953
22791135 ± 4774
21890982 ± 4678
20407576 ± 4517
19950460 ± 4466
12628284 ± 3553
17691674 ± 4206
17867137 ± 4226
17716754 ± 4209
16059746 ± 4007
4750703 ± 2179
25168982 ± 5016
25160236 ± 5015

91.7%
91.6%
96.9%
91.1%
91.6%
92.0%
91.9%
92.4%
92.7%
93.8%
93.2%
93.8%
93.7%
94.3%
95.0%
90.7%
90.9%

39.9 ± 9.9%
44.3 ± 8.1%
49.5 ± 60.6%
41.7 ± 3.3%
46.9 ± 3.6%
51.1 ± 4.2%
47.7 ± 3.9%
48.4 ± 3.9%
54.8 ± 4.4%
56.0 ± 5.4%
57.3 ± 4.9%
55.2 ± 4.5%
62.7 ± 5.0%
60.1 ± 4.8%
59.1 ± 8.3%
52.0 ± 3.9%
60.4 ± 4.3%

200 ± 14.1
271 ± 16.5
6 ± 2.4
1575 ± 39.7
1582 ± 39.8
1392 ± 37.3
1304 ± 36.1
1297 ± 36.0
1225 ± 35.0
834 ± 28.9
1407 ± 37.5
1369 ± 37.0
1286 ± 35.9
1260 ± 35.5
354 ± 18.8
1836 ± 42.8
1689 ± 41.1

31317.9 ± 2225.2
42209.7 ± 2576.4
1975.8 ± 810.7
242605.4 ± 6139.9
244373.5 ± 6172.1
212834.0 ± 5729.2
200507.3 ± 5576.0
195929.6 ± 5464.6
184233.9 ± 5286.5
124168.9 ± 4319.5
199581.8 ± 5348.9
193730.4 ± 5262.7
186514.1 ± 5226.8
180875.0 ± 5122.8
49556.5 ± 2647.0
277433.6 ± 6507.3
257524.3 ± 6295.5

140

Table A.2 (cont’d)
1116 15548067 ± 3943
1117 24027531 ± 4901
1118 21661193 ± 4654
1119 17135263 ± 4139
1120 17113386 ± 4136
1121 14095999 ± 3754
1122 17641234 ± 4200
1123 19223693 ± 4384
1124 22330606 ± 4725
1125 18688958 ± 4323
1126 15837226 ± 3979
1127 19985301 ± 4470
1128 18085021 ± 4252
1129 15951827 ± 3993
1130 2492166 ± 1578
1140 16757226 ± 4093
1141 13926658 ± 3731
1143 21910209 ± 4680
1144 23719230 ± 4870
1145 21989507 ± 4689
1146 20372770 ± 4513
1147 21596655 ± 4647
1148 15650741 ± 3956
1149 5868992 ± 2422
1150 24269714 ± 4926
1151 19513034 ± 4417
1152 22430111 ± 4736
1153 19863414 ± 4456
1154 19033532 ± 4362
1155 20794627 ± 4560
1156 22451595 ± 4738
1157 20948687 ± 4576
1158 21333904 ± 4618
1159 20446442 ± 4521
1160 11803820 ± 3435
1161 20873358 ± 4568
1162 16189705 ± 4023
1163 20564885 ± 4534
1164 7878480 ± 2806
1178 19672213 ± 4435
1179 17834673 ± 4223
1180 19197414 ± 4381
1181 19140691 ± 4375
1182 20354920 ± 4511

91.6%
91.4%
92.8%
94.1%
93.7%
93.9%
92.9%
92.9%
91.8%
92.9%
92.3%
93.0%
93.2%
94.3%
95.0%
87.6%
88.3%
84.2%
84.3%
85.1%
85.9%
85.5%
83.4%
84.2%
83.8%
85.0%
84.6%
85.2%
86.2%
85.1%
84.3%
84.8%
84.5%
85.7%
87.3%
85.9%
87.1%
85.5%
85.9%
92.2%
92.6%
92.6%
92.4%
92.1%

54.0 ± 5.1%
59.3 ± 4.2%
58.7 ± 4.3%
54.5 ± 4.6%
55.7 ± 4.8%
54.0 ± 5.0%
54.8 ± 4.7%
55.8 ± 4.6%
59.1 ± 4.1%
54.8 ± 4.3%
50.4 ± 4.5%
51.9 ± 4.1%
55.1 ± 4.4%
57.9 ± 4.7%
38.6 ± 9.5%
53.1 ± 4.6%
56.2 ± 5.2%
54.5 ± 4.4%
54.8 ± 4.1%
51.5 ± 4.0%
54.7 ± 4.4%
55.8 ± 4.3%
53.1 ± 5.0%
57.8 ± 8.8%
60.5 ± 4.3%
57.1 ± 4.7%
60.0 ± 4.4%
63.7 ± 5.0%
60.9 ± 4.8%
63.5 ± 5.0%
63.1 ± 4.7%
57.7 ± 4.4%
64.5 ± 4.8%
66.1 ± 5.3%
56.8 ± 5.7%
63.7 ± 4.9%
63.3 ± 5.2%
60.1 ± 4.6%
60.8 ± 7.0%
57.5 ± 4.4%
59.0 ± 4.6%
64.8 ± 4.8%
57.5 ± 4.1%
60.9 ± 4.4%
141

926 ± 30.4
1575 ± 39.7
1479 ± 38.5
1214 ± 34.8
1182 ± 34.4
1013 ± 31.8
1292 ± 35.9
1366 ± 37.0
1522 ± 39.0
1417 ± 37.6
1176 ± 34.3
1416 ± 37.6
1420 ± 37.7
1273 ± 35.7
171 ± 13.1
1205 ± 34.7
1040 ± 32.2
1485 ± 38.5
1533 ± 39.2
1445 ± 38.0
1400 ± 37.4
1526 ± 39.1
1091 ± 33.0
440 ± 21.0
1775 ± 42.1
1552 ± 39.4
1694 ± 41.2
1524 ± 39.0
1431 ± 37.8
1500 ± 38.7
1586 ± 39.8
1477 ± 38.4
1563 ± 39.5
1444 ± 38.0
835 ± 28.9
1535 ± 39.2
1213 ± 34.8
1497 ± 38.7
602 ± 24.5
1733 ± 41.6
1556 ± 39.4
1807 ± 42.5
1818 ± 42.6
1923 ± 43.9

145404.8 ± 4798.9
237691.9 ± 6016.5
216643.1 ± 5659.6
173626.0 ± 5007.1
171097.1 ± 5000.0
144142.2 ± 4550.7
187542.4 ± 5243.5
195480.7 ± 5314.2
226377.7 ± 5829.8
203755.2 ± 5440.4
171328.9 ± 5021.1
204460.7 ± 5459.5
197721.8 ± 5273.9
172301.6 ± 4853.9
24404.5 ± 1875.0
193735.1 ± 5609.4
161555.7 ± 5035.4
237811.5 ± 6199.5
266120.7 ± 6829.1
245730.8 ± 6495.2
235622.1 ± 6328.7
261159.3 ± 6720.1
192036.5 ± 5843.8
77004.9 ± 3691.4
302168.8 ± 7209.8
254968.9 ± 6508.1
283908.0 ± 6935.1
253086.4 ± 6518.3
235644.5 ± 6262.6
250472.7 ± 6500.5
269563.8 ± 6803.2
251494.4 ± 6577.4
266312.9 ± 6771.8
242834.8 ± 6423.0
135507.9 ± 4713.0
254828.4 ± 6538.4
194190.1 ± 5604.9
246009.6 ± 6391.0
97552.5 ± 3997.2
214828.0 ± 5186.6
193200.4 ± 4922.5
216209.7 ± 5112.9
217485.0 ± 5127.7
232113.8 ± 5321.1

Table A.2 (cont’d)
1183 22614066 ± 4755
1184 20477828 ± 4525
1185 20578329 ± 4536
1186 19386935 ± 4403
1187 19010744 ± 4360
1188 15234289 ± 3903
1189 2945693 ± 1716
1190 20949406 ± 4577
1191 25329980 ± 5032
1192 25537387 ± 5053
1193 18188870 ± 4264
1200 23858454 ± 4884
1201 21881779 ± 4677
1202 21276721 ± 4612
1203 20828044 ± 4563
1204 20387850 ± 4515
1205 19828325 ± 4452
1206 21125139 ± 4596
1207 22280195 ± 4720
1208 26351043 ± 5133
1209 23918837 ± 4890
1210 6372790 ± 2524
1211 17853055 ± 4225
1212 20277572 ± 4503
1213 21140403 ± 4597
1214 22781767 ± 4773
1215 20588488 ± 4537
1216 20042146 ± 4476
1220 22596837 ± 4753
1221 16165191 ± 4020
1222 19070486 ± 4366
1223 19325690 ± 4396
1225 20909552 ± 4572
1233 18187898 ± 4264
1234 21761008 ± 4664
1235 23146474 ± 4811
1236 13516906 ± 3676
1237 21490463 ± 4635
1238 19105536 ± 4370
1239 23556654 ± 4853
1240 22511285 ± 4744
1241 21297564 ± 4614
1242 19179082 ± 4379
1243 24268564 ± 4926

90.9%
91.3%
91.5%
92.0%
92.6%
93.7%
94.5%
89.9%
89.7%
89.8%
90.1%
90.7%
90.7%
91.1%
91.6%
91.9%
91.9%
91.8%
91.9%
90.5%
90.7%
91.1%
90.4%
91.4%
91.5%
91.3%
91.8%
92.8%
90.5%
92.9%
92.3%
91.4%
91.3%
92.9%
91.8%
90.8%
90.8%
91.2%
91.1%
90.3%
90.7%
91.5%
92.2%
90.9%

59.1 ± 4.2%
62.5 ± 4.6%
55.0 ± 4.1%
55.5 ± 4.3%
58.4 ± 4.4%
62.4 ± 4.8%
68.1 ± 11.1%
57.5 ± 4.2%
57.2 ± 3.9%
56.0 ± 3.8%
55.0 ± 4.5%
59.8 ± 4.3%
61.1 ± 4.3%
59.6 ± 4.3%
60.6 ± 4.2%
60.6 ± 4.3%
62.6 ± 4.7%
59.4 ± 4.4%
62.7 ± 4.5%
57.2 ± 3.7%
58.0 ± 4.1%
61.7 ± 8.5%
57.3 ± 4.8%
60.0 ± 4.6%
57.3 ± 4.3%
58.6 ± 4.2%
62.0 ± 4.3%
61.0 ± 4.6%
58.8 ± 4.2%
59.8 ± 4.8%
60.5 ± 4.5%
60.1 ± 4.3%
63.0 ± 4.5%
61.6 ± 4.7%
61.8 ± 4.4%
58.6 ± 4.1%
61.1 ± 5.7%
54.9 ± 4.0%
57.3 ± 4.4%
56.6 ± 4.0%
59.4 ± 4.1%
59.0 ± 4.3%
57.6 ± 4.4%
57.7 ± 3.9%
142

2018 ± 44.9
1824 ± 42.7
1772 ± 42.1
1608 ± 40.1
1569 ± 39.6
1247 ± 35.3
232 ± 15.2
1657 ± 40.7
2029 ± 45.0
2016 ± 44.9
1505 ± 38.8
2088 ± 45.7
1998 ± 44.7
1833 ± 42.8
1636 ± 40.4
1655 ± 40.7
1665 ± 40.8
1762 ± 42.0
1768 ± 42.0
2000 ± 44.7
1797 ± 42.4
507 ± 22.5
1387 ± 37.2
1531 ± 39.1
1597 ± 40.0
1882 ± 43.4
1730 ± 41.6
1556 ± 39.4
2415 ± 49.1
1761 ± 42.0
2207 ± 47.0
1848 ± 43.0
1502 ± 38.8
1706 ± 41.3
1961 ± 44.3
1957 ± 44.2
1157 ± 34.0
1844 ± 42.9
1652 ± 40.6
1891 ± 43.5
1859 ± 43.1
1756 ± 41.9
1559 ± 39.5
1862 ± 43.2

253296.9 ± 5667.5
228056.2 ± 5367.2
222769.6 ± 5318.4
199332.9 ± 4994.6
194731.2 ± 4939.6
152597.1 ± 4341.7
29491.2 ± 1945.4
221344.2 ± 5463.6
269480.7 ± 6011.3
265923.2 ± 5950.4
196188.1 ± 5081.8
269477.9 ± 5927.8
258741.0 ± 5819.7
235270.5 ± 5523.1
213453.6 ± 5302.2
213027.4 ± 5261.7
212022.4 ± 5221.8
224502.0 ± 5374.6
226144.2 ± 5403.5
265713.6 ± 5968.8
239095.2 ± 5665.9
67020.7 ± 2990.8
186370.3 ± 5028.0
206020.4 ± 5289.9
208358.9 ± 5237.5
243934.9 ± 5650.6
220717.1 ± 5332.8
196403.2 ± 5001.8
289448.9 ± 5924.3
198652.3 ± 4761.0
249994.3 ± 5353.8
220047.3 ± 5145.6
261405.1 ± 6783.6
199869.8 ± 4863.9
239931.8 ± 5445.7
249185.6 ± 5660.5
149798.1 ± 4426.2
231587.1 ± 5419.7
207406.0 ± 5128.3
244735.4 ± 5654.5
238712.5 ± 5563.3
222533.5 ± 5336.0
194173.4 ± 4940.9
244984.7 ± 5703.6

Table A.2 (cont’d)
1244 24539032 ± 4953
1245 20324635 ± 4508
1246 19032078 ± 4362
1247 8933724 ± 2988
1248 19970394 ± 4468
1249 19232613 ± 4385
1250 18142414 ± 4259
1251 10099094 ± 3177
1258 28323971 ± 5322
1259 28388250 ± 5328
1260 22416359 ± 4734
1261 8040483 ± 2835
1262 24326748 ± 4932
1263 19628230 ± 4430
1264 22249854 ± 4716
1265 18994505 ± 4358
1266 21764049 ± 4665
1267 22692282 ± 4763
1268 19894163 ± 4460
1269 17044353 ± 4128
1270 1332497 ± 1154
1271 19664923 ± 4434
1272 19760737 ± 4445
1273 10796136 ± 3285
1301 22137675 ± 4705
1302 20878220 ± 4569
1303 24360464 ± 4935
1304 24321781 ± 4931
1305 7182742 ± 2680
1306 26656131 ± 5162
1307 25245755 ± 5024
1308 24413198 ± 4940
1309 16203154 ± 4025
1310 28670090 ± 5354
1311 22870406 ± 4782
1322 19534132 ± 4419
1323 20906242 ± 4572
1324 19428946 ± 4407
1325 28672792 ± 5354
1326 25794283 ± 5078
1327 26064005 ± 5105
1328 23805045 ± 4879
1329 11224701 ± 3350
1331 23538523 ± 4851

90.5%
92.0%
92.4%
94.0%
92.1%
92.4%
92.7%
93.4%
90.4%
90.2%
90.6%
90.7%
91.5%
92.0%
92.4%
92.9%
91.7%
91.7%
92.7%
93.0%
92.7%
91.8%
92.4%
93.5%
91.1%
92.0%
91.9%
91.1%
91.7%
92.2%
91.8%
91.9%
92.5%
91.8%
91.8%
92.0%
91.5%
92.0%
89.6%
90.0%
90.5%
90.6%
90.6%
92.2%

55.1 ± 3.8%
59.8 ± 4.4%
58.0 ± 4.3%
55.9 ± 5.9%
56.7 ± 4.2%
53.5 ± 4.1%
57.2 ± 4.4%
54.2 ± 5.6%
58.9 ± 3.8%
57.9 ± 3.7%
59.3 ± 4.2%
57.9 ± 6.9%
58.2 ± 4.0%
59.5 ± 4.4%
62.6 ± 4.3%
55.8 ± 4.2%
57.7 ± 4.4%
58.2 ± 4.0%
56.5 ± 4.3%
58.7 ± 4.5%
55.4 ± 18.5%
57.5 ± 4.3%
60.6 ± 4.5%
65.7 ± 6.4%
58.2 ± 4.3%
59.5 ± 4.3%
61.4 ± 4.1%
60.7 ± 4.1%
67.5 ± 7.7%
60.6 ± 3.8%
58.5 ± 3.9%
56.0 ± 3.8%
60.1 ± 5.2%
62.5 ± 3.9%
60.5 ± 4.2%
61.8 ± 4.3%
61.7 ± 4.2%
63.0 ± 4.3%
60.2 ± 3.5%
57.8 ± 3.8%
58.7 ± 3.9%
59.9 ± 4.1%
59.4 ± 5.8%
61.0 ± 4.1%
143

1906 ± 43.7
1580 ± 39.7
1525 ± 39.1
719 ± 26.8
1515 ± 38.9
1480 ± 38.5
1400 ± 37.4
717 ± 26.8
2346 ± 48.4
2250 ± 47.4
1851 ± 43.0
650 ± 25.5
1968 ± 44.4
1694 ± 41.2
1893 ± 43.5
1655 ± 40.7
1919 ± 43.8
1882 ± 43.4
1624 ± 40.3
1368 ± 37.0
102 ± 10.1
1622 ± 40.3
1607 ± 40.1
924 ± 30.4
1651 ± 40.6
1639 ± 40.5
1997 ± 44.7
1759 ± 41.9
568 ± 23.8
1993 ± 44.6
1974 ± 44.4
1870 ± 43.2
6217 ± 78.8
2021 ± 45.0
1570 ± 39.6
1651 ± 40.6
1828 ± 42.8
1619 ± 40.2
2362 ± 48.6
2150 ± 46.4
2111 ± 45.9
1914 ± 43.7
869 ± 29.5
1846 ± 43.0

251699.7 ± 5792.2
203183.3 ± 5135.3
193178.2 ± 4970.1
88158.2 ± 3303.1
195149.8 ± 5036.4
187825.3 ± 4904.5
176389.5 ± 4735.6
89891.9 ± 3371.1
299106.6 ± 6205.0
288505.4 ± 6110.3
236018.4 ± 5512.2
82188.5 ± 3238.7
255008.2 ± 5776.1
209904.9 ± 5125.2
229400.7 ± 5297.8
199158.9 ± 4919.5
237714.7 ± 5453.8
236056.2 ± 5467.5
200924.2 ± 5009.3
167024.0 ± 4536.5
13346.9 ± 1327.7
203356.0 ± 5073.4
198574.9 ± 4976.7
111120.8 ± 3673.3
213825.3 ± 5285.7
208679.3 ± 5178.4
255136.1 ± 5736.9
231993.1 ± 5555.7
75912.5 ± 3200.7
255867.0 ± 5756.9
253918.1 ± 5741.6
240004.4 ± 5575.3
795445.3 ± 10316.8
275219.7 ± 6149.2
211037.6 ± 5348.8
205993.0 ± 5094.4
230675.9 ± 5422.7
202705.8 ± 5062.1
312490.6 ± 6461.4
289287.6 ± 6270.6
279468.3 ± 6112.3
253670.2 ± 5826.4
116825.2 ± 3981.8
237044.3 ± 5542.9

Table A.2 (cont’d)
1332 25622670 ± 5061
1333 25451103 ± 5044
1334 22669137 ± 4761
1335 24981509 ± 4998
1336 23463093 ± 4843
1337 22452005 ± 4738
1338 22038360 ± 4694
1339 21424663 ± 4628
1340 18608744 ± 4313

91.5%
91.3%
91.4%
91.5%
92.0%
91.7%
91.9%
92.1%
92.3%

62.0 ± 3.7%
58.5 ± 3.9%
55.4 ± 3.8%
58.9 ± 3.8%
58.7 ± 4.0%
58.6 ± 4.2%
57.0 ± 4.3%
58.0 ± 4.4%
59.9 ± 4.4%

1979 ± 44.5
2022 ± 45.0
1809 ± 42.5
1910 ± 43.7
1827 ± 42.7
1836 ± 42.8
1816 ± 42.6
1717 ± 41.4
1501 ± 38.7

256977.1 ± 5803.3
259075.6 ± 5788.5
231646.2 ± 5471.9
245197.8 ± 5635.8
228916.5 ± 5379.8
231477.4 ± 5427.9
228725.5 ± 5393.1
214529.1 ± 5201.3
187062.9 ± 4850.9

A.2 28 F Ground State
Table A.3: Average values used in calculating the cross section to 28 F.
Number of Recorded Reactions:
Ion Chamber Efficiency:
Number of Target Nuclei:
Geometric Acceptence:
MoNA Efficiency:
Fit Fraction:
Average Sweeper Efficiency:
Actual Number of Reactions:
Cross Section (mb):

90 ± 9.5
98.93%
1.92 × 1022
33.39 ± 2.41%
77%
24.17 ± 6.10%
58.32 ± 0.37%
145.08 ± 41.05
0.23 ± 0.06

Table A.4: Run-by-run values used in calculating the cross section to 28 F.
Run
1089
1090
1091
1093
1094
1095
1096
1097
1105
1106
1108

Target Scint.
Scaler

Live
Time

Sweeper
Efficiency

29 Ne

29 Ne

Counted

Actual

3001020 ± 1732
4040444 ± 2010
189262 ± 435
25275536 ± 5027
24541327 ± 4953
22791135 ± 4774
21890982 ± 4678
20407576 ± 4517
19950460 ± 4466
12628284 ± 3553
17691674 ± 4206

91.7%
91.6%
96.9%
91.1%
91.6%
92.0%
91.9%
92.4%
92.7%
93.8%
93.2%

39.9 ± 9.9%
44.3 ± 8.1%
49.5 ± 60.6%
41.7 ± 3.3%
46.9 ± 3.6%
51.1 ± 4.2%
47.7 ± 3.9%
48.4 ± 3.9%
54.8 ± 4.4%
56.0 ± 5.4%
57.3 ± 4.9%

200 ± 14.1
271 ± 16.5
6 ± 2.4
1575 ± 39.7
1582 ± 39.8
1392 ± 37.3
1304 ± 36.1
1297 ± 36.0
1225 ± 35.0
834 ± 28.9
1407 ± 37.5

31317.9 ± 2225.2
42209.7 ± 2576.4
1975.8 ± 810.7
242605.4 ± 6139.9
244373.5 ± 6172.1
212834.0 ± 5729.2
200507.3 ± 5576.0
195929.6 ± 5464.6
184233.9 ± 5286.5
124168.9 ± 4319.5
199581.8 ± 5348.9

144

Table A.4 (cont’d)
1109 17867137 ± 4226
1110 17716754 ± 4209
1111 16059746 ± 4007
1113 4750703 ± 2179
1114 25168982 ± 5016
1115 25160236 ± 5015
1116 15548067 ± 3943
1117 24027531 ± 4901
1118 21661193 ± 4654
1119 17135263 ± 4139
1120 17113386 ± 4136
1121 14095999 ± 3754
1122 17641234 ± 4200
1123 19223693 ± 4384
1124 22330606 ± 4725
1125 18688958 ± 4323
1126 15837226 ± 3979
1127 19985301 ± 4470
1128 18085021 ± 4252
1129 15951827 ± 3993
1130 2492166 ± 1578
1140 16757226 ± 4093
1141 13926658 ± 3731
1143 21910209 ± 4680
1144 23719230 ± 4870
1145 21989507 ± 4689
1146 20372770 ± 4513
1147 21596655 ± 4647
1148 15650741 ± 3956
1149 5868992 ± 2422
1150 24269714 ± 4926
1151 19513034 ± 4417
1152 22430111 ± 4736
1153 19863414 ± 4456
1154 19033532 ± 4362
1155 20794627 ± 4560
1156 22451595 ± 4738
1157 20948687 ± 4576
1158 21333904 ± 4618
1159 20446442 ± 4521
1160 11803820 ± 3435
1161 20873358 ± 4568
1162 16189705 ± 4023
1163 20564885 ± 4534

93.8%
93.7%
94.3%
95.0%
90.7%
90.9%
91.6%
91.4%
92.8%
94.1%
93.7%
93.9%
92.9%
92.9%
91.8%
92.9%
92.3%
93.0%
93.2%
94.3%
95.0%
87.6%
88.3%
84.2%
84.3%
85.1%
85.9%
85.5%
83.4%
84.2%
83.8%
85.0%
84.6%
85.2%
86.2%
85.1%
84.3%
84.8%
84.5%
85.7%
87.3%
85.9%
87.1%
85.5%

55.2 ± 4.5%
62.7 ± 5.0%
60.1 ± 4.8%
59.1 ± 8.3%
52.0 ± 3.9%
60.4 ± 4.3%
54.0 ± 5.1%
59.3 ± 4.2%
58.7 ± 4.3%
54.5 ± 4.6%
55.7 ± 4.8%
54.0 ± 5.0%
54.8 ± 4.7%
55.8 ± 4.6%
59.1 ± 4.1%
54.8 ± 4.3%
50.4 ± 4.5%
51.9 ± 4.1%
55.1 ± 4.4%
57.9 ± 4.7%
38.6 ± 9.5%
53.1 ± 4.6%
56.2 ± 5.2%
54.5 ± 4.4%
54.8 ± 4.1%
51.5 ± 4.0%
54.7 ± 4.4%
55.8 ± 4.3%
53.1 ± 5.0%
57.8 ± 8.8%
60.5 ± 4.3%
57.1 ± 4.7%
60.0 ± 4.4%
63.7 ± 5.0%
60.9 ± 4.8%
63.5 ± 5.0%
63.1 ± 4.7%
57.7 ± 4.4%
64.5 ± 4.8%
66.1 ± 5.3%
56.8 ± 5.7%
63.7 ± 4.9%
63.3 ± 5.2%
60.1 ± 4.6%
145

1369 ± 37.0
1286 ± 35.9
1260 ± 35.5
354 ± 18.8
1836 ± 42.8
1689 ± 41.1
926 ± 30.4
1575 ± 39.7
1479 ± 38.5
1214 ± 34.8
1182 ± 34.4
1013 ± 31.8
1292 ± 35.9
1366 ± 37.0
1522 ± 39.0
1417 ± 37.6
1176 ± 34.3
1416 ± 37.6
1420 ± 37.7
1273 ± 35.7
171 ± 13.1
1205 ± 34.7
1040 ± 32.2
1485 ± 38.5
1533 ± 39.2
1445 ± 38.0
1400 ± 37.4
1526 ± 39.1
1091 ± 33.0
440 ± 21.0
1775 ± 42.1
1552 ± 39.4
1694 ± 41.2
1524 ± 39.0
1431 ± 37.8
1500 ± 38.7
1586 ± 39.8
1477 ± 38.4
1563 ± 39.5
1444 ± 38.0
835 ± 28.9
1535 ± 39.2
1213 ± 34.8
1497 ± 38.7

193730.4 ± 5262.7
186514.1 ± 5226.8
180875.0 ± 5122.8
49556.5 ± 2647.0
277433.6 ± 6507.3
257524.3 ± 6295.5
145404.8 ± 4798.9
237691.9 ± 6016.5
216643.1 ± 5659.6
173626.0 ± 5007.1
171097.1 ± 5000.0
144142.2 ± 4550.7
187542.4 ± 5243.5
195480.7 ± 5314.2
226377.7 ± 5829.8
203755.2 ± 5440.4
171328.9 ± 5021.1
204460.7 ± 5459.5
197721.8 ± 5273.9
172301.6 ± 4853.9
24404.5 ± 1875.0
193735.1 ± 5609.4
161555.7 ± 5035.4
237811.5 ± 6199.5
266120.7 ± 6829.1
245730.8 ± 6495.2
235622.1 ± 6328.7
261159.3 ± 6720.1
192036.5 ± 5843.8
77004.9 ± 3691.4
302168.8 ± 7209.8
254968.9 ± 6508.1
283908.0 ± 6935.1
253086.4 ± 6518.3
235644.5 ± 6262.6
250472.7 ± 6500.5
269563.8 ± 6803.2
251494.4 ± 6577.4
266312.9 ± 6771.8
242834.8 ± 6423.0
135507.9 ± 4713.0
254828.4 ± 6538.4
194190.1 ± 5604.9
246009.6 ± 6391.0

Table A.4 (cont’d)
1164 7878480 ± 2806
1178 19672213 ± 4435
1179 17834673 ± 4223
1180 19197414 ± 4381
1181 19140691 ± 4375
1182 20354920 ± 4511
1183 22614066 ± 4755
1184 20477828 ± 4525
1185 20578329 ± 4536
1186 19386935 ± 4403
1187 19010744 ± 4360
1188 15234289 ± 3903
1189 2945693 ± 1716
1190 20949406 ± 4577
1191 25329980 ± 5032
1192 25537387 ± 5053
1193 18188870 ± 4264
1200 23858454 ± 4884
1201 21881779 ± 4677
1202 21276721 ± 4612
1203 20828044 ± 4563
1204 20387850 ± 4515
1205 19828325 ± 4452
1206 21125139 ± 4596
1207 22280195 ± 4720
1208 26351043 ± 5133
1209 23918837 ± 4890
1210 6372790 ± 2524
1211 17853055 ± 4225
1212 20277572 ± 4503
1213 21140403 ± 4597
1214 22781767 ± 4773
1215 20588488 ± 4537
1216 20042146 ± 4476
1220 22596837 ± 4753
1221 16165191 ± 4020
1222 19070486 ± 4366
1223 19325690 ± 4396
1225 20909552 ± 4572
1233 18187898 ± 4264
1234 21761008 ± 4664
1235 23146474 ± 4811
1236 13516906 ± 3676
1237 21490463 ± 4635

85.9%
92.2%
92.6%
92.6%
92.4%
92.1%
90.9%
91.3%
91.5%
92.0%
92.6%
93.7%
94.5%
89.9%
89.7%
89.8%
90.1%
90.7%
90.7%
91.1%
91.6%
91.9%
91.9%
91.8%
91.9%
90.5%
90.7%
91.1%
90.4%
91.4%
91.5%
91.3%
91.8%
92.8%
90.5%
92.9%
92.3%
91.4%
91.3%
92.9%
91.8%
90.8%
90.8%
91.2%

60.8 ± 7.0%
57.5 ± 4.4%
59.0 ± 4.6%
64.8 ± 4.8%
57.5 ± 4.1%
60.9 ± 4.4%
59.1 ± 4.2%
62.5 ± 4.6%
55.0 ± 4.1%
55.5 ± 4.3%
58.4 ± 4.4%
62.4 ± 4.8%
68.1 ± 11.1%
57.5 ± 4.2%
57.2 ± 3.9%
56.0 ± 3.8%
55.0 ± 4.5%
59.8 ± 4.3%
61.1 ± 4.3%
59.6 ± 4.3%
60.6 ± 4.2%
60.6 ± 4.3%
62.6 ± 4.7%
59.4 ± 4.4%
62.7 ± 4.5%
57.2 ± 3.7%
58.0 ± 4.1%
61.7 ± 8.5%
57.3 ± 4.8%
60.0 ± 4.6%
57.3 ± 4.3%
58.6 ± 4.2%
62.0 ± 4.3%
61.0 ± 4.6%
58.8 ± 4.2%
59.8 ± 4.8%
60.5 ± 4.5%
60.1 ± 4.3%
63.0 ± 4.5%
61.6 ± 4.7%
61.8 ± 4.4%
58.6 ± 4.1%
61.1 ± 5.7%
54.9 ± 4.0%
146

602 ± 24.5
1733 ± 41.6
1556 ± 39.4
1807 ± 42.5
1818 ± 42.6
1923 ± 43.9
2018 ± 44.9
1824 ± 42.7
1772 ± 42.1
1608 ± 40.1
1569 ± 39.6
1247 ± 35.3
232 ± 15.2
1657 ± 40.7
2029 ± 45.0
2016 ± 44.9
1505 ± 38.8
2088 ± 45.7
1998 ± 44.7
1833 ± 42.8
1636 ± 40.4
1655 ± 40.7
1665 ± 40.8
1762 ± 42.0
1768 ± 42.0
2000 ± 44.7
1797 ± 42.4
507 ± 22.5
1387 ± 37.2
1531 ± 39.1
1597 ± 40.0
1882 ± 43.4
1730 ± 41.6
1556 ± 39.4
2415 ± 49.1
1761 ± 42.0
2207 ± 47.0
1848 ± 43.0
1502 ± 38.8
1706 ± 41.3
1961 ± 44.3
1957 ± 44.2
1157 ± 34.0
1844 ± 42.9

97552.5 ± 3997.2
214828.0 ± 5186.6
193200.4 ± 4922.5
216209.7 ± 5112.9
217485.0 ± 5127.7
232113.8 ± 5321.1
253296.9 ± 5667.5
228056.2 ± 5367.2
222769.6 ± 5318.4
199332.9 ± 4994.6
194731.2 ± 4939.6
152597.1 ± 4341.7
29491.2 ± 1945.4
221344.2 ± 5463.6
269480.7 ± 6011.3
265923.2 ± 5950.4
196188.1 ± 5081.8
269477.9 ± 5927.8
258741.0 ± 5819.7
235270.5 ± 5523.1
213453.6 ± 5302.2
213027.4 ± 5261.7
212022.4 ± 5221.8
224502.0 ± 5374.6
226144.2 ± 5403.5
265713.6 ± 5968.8
239095.2 ± 5665.9
67020.7 ± 2990.8
186370.3 ± 5028.0
206020.4 ± 5289.9
208358.9 ± 5237.5
243934.9 ± 5650.6
220717.1 ± 5332.8
196403.2 ± 5001.8
289448.9 ± 5924.3
198652.3 ± 4761.0
249994.3 ± 5353.8
220047.3 ± 5145.6
261405.1 ± 6783.6
199869.8 ± 4863.9
239931.8 ± 5445.7
249185.6 ± 5660.5
149798.1 ± 4426.2
231587.1 ± 5419.7

Table A.4 (cont’d)
1238 19105536 ± 4370
1239 23556654 ± 4853
1240 22511285 ± 4744
1241 21297564 ± 4614
1242 19179082 ± 4379
1243 24268564 ± 4926
1244 24539032 ± 4953
1245 20324635 ± 4508
1246 19032078 ± 4362
1247 8933724 ± 2988
1248 19970394 ± 4468
1249 19232613 ± 4385
1250 18142414 ± 4259
1251 10099094 ± 3177
1258 28323971 ± 5322
1259 28388250 ± 5328
1260 22416359 ± 4734
1261 8040483 ± 2835
1262 24326748 ± 4932
1263 19628230 ± 4430
1264 22249854 ± 4716
1265 18994505 ± 4358
1266 21764049 ± 4665
1267 22692282 ± 4763
1268 19894163 ± 4460
1269 17044353 ± 4128
1270 1332497 ± 1154
1271 19664923 ± 4434
1272 19760737 ± 4445
1273 10796136 ± 3285
1301 22137675 ± 4705
1302 20878220 ± 4569
1303 24360464 ± 4935
1304 24321781 ± 4931
1305 7182742 ± 2680
1306 26656131 ± 5162
1307 25245755 ± 5024
1308 24413198 ± 4940
1309 16203154 ± 4025
1310 28670090 ± 5354
1311 22870406 ± 4782
1322 19534132 ± 4419
1323 20906242 ± 4572
1324 19428946 ± 4407

91.1%
90.3%
90.7%
91.5%
92.2%
90.9%
90.5%
92.0%
92.4%
94.0%
92.1%
92.4%
92.7%
93.4%
90.4%
90.2%
90.6%
90.7%
91.5%
92.0%
92.4%
92.9%
91.7%
91.7%
92.7%
93.0%
92.7%
91.8%
92.4%
93.5%
91.1%
92.0%
91.9%
91.1%
91.7%
92.2%
91.8%
91.9%
92.5%
91.8%
91.8%
92.0%
91.5%
92.0%

57.3 ± 4.4%
56.6 ± 4.0%
59.4 ± 4.1%
59.0 ± 4.3%
57.6 ± 4.4%
57.7 ± 3.9%
55.1 ± 3.8%
59.8 ± 4.4%
58.0 ± 4.3%
55.9 ± 5.9%
56.7 ± 4.2%
53.5 ± 4.1%
57.2 ± 4.4%
54.2 ± 5.6%
58.9 ± 3.8%
57.9 ± 3.7%
59.3 ± 4.2%
57.9 ± 6.9%
58.2 ± 4.0%
59.5 ± 4.4%
62.6 ± 4.3%
55.8 ± 4.2%
57.7 ± 4.4%
58.2 ± 4.0%
56.5 ± 4.3%
58.7 ± 4.5%
55.4 ± 18.5%
57.5 ± 4.3%
60.6 ± 4.5%
65.7 ± 6.4%
58.2 ± 4.3%
59.5 ± 4.3%
61.4 ± 4.1%
60.7 ± 4.1%
67.5 ± 7.7%
60.6 ± 3.8%
58.5 ± 3.9%
56.0 ± 3.8%
60.1 ± 5.2%
62.5 ± 3.9%
60.5 ± 4.2%
61.8 ± 4.3%
61.7 ± 4.2%
63.0 ± 4.3%
147

1652 ± 40.6
1891 ± 43.5
1859 ± 43.1
1756 ± 41.9
1559 ± 39.5
1862 ± 43.2
1906 ± 43.7
1580 ± 39.7
1525 ± 39.1
719 ± 26.8
1515 ± 38.9
1480 ± 38.5
1400 ± 37.4
717 ± 26.8
2346 ± 48.4
2250 ± 47.4
1851 ± 43.0
650 ± 25.5
1968 ± 44.4
1694 ± 41.2
1893 ± 43.5
1655 ± 40.7
1919 ± 43.8
1882 ± 43.4
1624 ± 40.3
1368 ± 37.0
102 ± 10.1
1622 ± 40.3
1607 ± 40.1
924 ± 30.4
1651 ± 40.6
1639 ± 40.5
1997 ± 44.7
1759 ± 41.9
568 ± 23.8
1993 ± 44.6
1974 ± 44.4
1870 ± 43.2
6217 ± 78.8
2021 ± 45.0
1570 ± 39.6
1651 ± 40.6
1828 ± 42.8
1619 ± 40.2

207406.0 ± 5128.3
244735.4 ± 5654.5
238712.5 ± 5563.3
222533.5 ± 5336.0
194173.4 ± 4940.9
244984.7 ± 5703.6
251699.7 ± 5792.2
203183.3 ± 5135.3
193178.2 ± 4970.1
88158.2 ± 3303.1
195149.8 ± 5036.4
187825.3 ± 4904.5
176389.5 ± 4735.6
89891.9 ± 3371.1
299106.6 ± 6205.0
288505.4 ± 6110.3
236018.4 ± 5512.2
82188.5 ± 3238.7
255008.2 ± 5776.1
209904.9 ± 5125.2
229400.7 ± 5297.8
199158.9 ± 4919.5
237714.7 ± 5453.8
236056.2 ± 5467.5
200924.2 ± 5009.3
167024.0 ± 4536.5
13346.9 ± 1327.7
203356.0 ± 5073.4
198574.9 ± 4976.7
111120.8 ± 3673.3
213825.3 ± 5285.7
208679.3 ± 5178.4
255136.1 ± 5736.9
231993.1 ± 5555.7
75912.5 ± 3200.7
255867.0 ± 5756.9
253918.1 ± 5741.6
240004.4 ± 5575.3
795445.3 ± 10316.8
275219.7 ± 6149.2
211037.6 ± 5348.8
205993.0 ± 5094.4
230675.9 ± 5422.7
202705.8 ± 5062.1

Table A.4 (cont’d)
1325 28672792 ± 5354
1326 25794283 ± 5078
1327 26064005 ± 5105
1328 23805045 ± 4879
1329 11224701 ± 3350
1331 23538523 ± 4851
1332 25622670 ± 5061
1333 25451103 ± 5044
1334 22669137 ± 4761
1335 24981509 ± 4998
1336 23463093 ± 4843
1337 22452005 ± 4738
1338 22038360 ± 4694
1339 21424663 ± 4628
1340 18608744 ± 4313

89.6%
90.0%
90.5%
90.6%
90.6%
92.2%
91.5%
91.3%
91.4%
91.5%
92.0%
91.7%
91.9%
92.1%
92.3%

60.2 ± 3.5%
57.8 ± 3.8%
58.7 ± 3.9%
59.9 ± 4.1%
59.4 ± 5.8%
61.0 ± 4.1%
62.0 ± 3.7%
58.5 ± 3.9%
55.4 ± 3.8%
58.9 ± 3.8%
58.7 ± 4.0%
58.6 ± 4.2%
57.0 ± 4.3%
58.0 ± 4.4%
59.9 ± 4.4%

148

2362 ± 48.6
2150 ± 46.4
2111 ± 45.9
1914 ± 43.7
869 ± 29.5
1846 ± 43.0
1979 ± 44.5
2022 ± 45.0
1809 ± 42.5
1910 ± 43.7
1827 ± 42.7
1836 ± 42.8
1816 ± 42.6
1717 ± 41.4
1501 ± 38.7

312490.6 ± 6461.4
289287.6 ± 6270.6
279468.3 ± 6112.3
253670.2 ± 5826.4
116825.2 ± 3981.8
237044.3 ± 5542.9
256977.1 ± 5803.3
259075.6 ± 5788.5
231646.2 ± 5471.9
245197.8 ± 5635.8
228916.5 ± 5379.8
231477.4 ± 5427.9
228725.5 ± 5393.1
214529.1 ± 5201.3
187062.9 ± 4850.9

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