UNBOUND STATES IN THE LIGHTEST ISLAND OF INVERSION: NEUTRON
DECAY MEASUREMENTS OF 11 LI, 10 LI, AND 12 BE
By
Jenna Kathleen Smith

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Physics - Doctor of Philosophy
2014

ABSTRACT
UNBOUND STATES IN THE LIGHTEST ISLAND OF INVERSION:
NEUTRON DECAY MEASUREMENTS OF 11 LI, 10 LI, AND 12 BE
By
Jenna Kathleen Smith
The light nuclei in the N ∼ 8 region provide easily accessible systems for studying the
structure of exotic nuclei. The nuclide 11 Li is a two-neutron halo system, a Borromean
nucleus, and located in the N ∼ 8 island of inversion. The next lightest lithium isotope, 10 Li
is neutron unbound, located on the western edge of the island of inversion, and important
for the understanding of the 11 Li system. 12 Be is the heaviest N = 8 isotone in the island of
inversion. Neutron unbound states from all three systems have been measured and analyzed
in this work.
The experiment was conducted at the National Superconducting Cyclotron Laboratory.
Beams of exotic boron isotopes (12 B and 13 B) were provided by the Coupled Cyclotron
Facility and delivered to the MoNA-LISA-Sweeper setup, each at 71 MeV/u. The neutronunbound isotopes were created in a secondary beryllium target where they immediately
decayed into a charged daughter fragment and one or two neutrons. The charged daughter fragment was measured by a suite of charged particle detectors and the neutrons were
detected by MoNA-LISA.
Measurement of the 11 Li unbound system showed a state at 1210±80 keV excitation
energy with a width greater than 1200 keV, consistent with previous measurements. Correlations between the three decay products were examined with use of Jacobi spectra and Dalitz
plots, which revealed a correlation between the two neutrons stronger than that expected
from the decay of two neutrons uncorrelated in the nucleus. This is the first correlation

measurement attributable to a specific unbound state in 11 Li. Unbound states in 10 Li were
populated with a two-proton knockout from a 12 B beam, a reaction expected to populate
primarily positive-parity states. The decay energy spectra of 10 Li was fit with three peaks at
110±40 keV, 500±100 keV, and 1100±100 keV. The lowest two of these peaks are expected
to be a 1+ /2+ doublet. Calculation of the 0p1/2 single particle energy in the 10 Li system
and comparison with Faddeev calculations suggests that both states of the 2− /1− doublet
are below 50 keV, consistent with previous measurements of one

= 0 state at 50 keV or

below. In the 12 Be system, a measured decay energy of 1243±20 keV corresponds to an
excitation energy between 4400 and 4800 keV, depending on the combination of final states
populated in 11 Be. A lack of decays to 10 Be placed an upper limit of 5% on the branching
ratio to the ground state of 10 Be via 2n emission. This low branching ratio, as well as spin
and parity selectivity from the 1p-knockout production mechanism, suggests that the spin
and parity of this state is 1− or 2− .

In honor of all those who blazed the trail and made my path possible

iv

ACKNOWLEDGMENTS

Nuclear physics is a team sport and this thesis was no exception. There are scores of
people whose collaboration, experience, and friendship have influenced my time in graduate
school. My advisor Michael Thoennessen provided guidance and expertise during my five
years at Michigan State University and was quite adept at dispensing both words of encouragement and skepticism. The other members of my committee, Filomena Nunes, Artemis
Spyrou, Stuart Tessmer, and Kirsten Tollefson I thank for their time, attention, and interesting discussion during our meetings, especially Artemis who served as a secondary advisor
and mentor during my time here.
I have enjoyed working with the MoNA Collaboration from day one, both the local group
and the larger collaboration. Thomas Baumann is an extraordinary device physicist without
whom I’m not sure MoNA-LISA would have worked. These five years were made easier and
more enjoyable thanks to the support, collaboration, and comraderie of the other graduate
students and post-docs in our group: Michael Strongman, Greg Christian, Shea Mosby,
Michelle Mosby, Jesse Snyder, Zach Kohley, Michael Jones, and Krystin Stiefel. Thanks
are due to the external collaborators: for building both MoNA and LISA as well as serving
as a sounding board and mentors for the past five years. Without all your experience and
contributions, my experiment would not have been as successful as it was. I am especially
grateful for the opportunity to work closely with Nathan Frank on the development of the
CsI(Na) array. Paul DeYoung and Braden Marks made specific contributions to this analysis
by correcting the walk in the target scintillator timing and cleaning up the beryllium isotope
identification. Though not a member of the MoNA Collaboration, Alexander Volya’s work
was the basis for the two-neutron decay lineshapes used in this work and I’m also grateful
v

for the clarifying emails he exchanged with me.
The experiment that this work is based on began with a pure and intense secondary
beam that was produced by the Coupled Cyclotron Facility and the A1900 Fragment Separator. The high-quality beam they provided gave us a stable running status for which I am
eternally grateful. Without Daniel Bazin and Scott Williams and their assistance with the
charged-particle detectors, we would only have been able to make a half measurement.
Beyond this specific work, the lab community and graduate student community have
made the past five years truly enjoyable. To those graduate students that I have been honored
to call friends, thanks for the venting sessions, the laughs, the thoughtful suggestions, the
chocolate, and the Happy Hours. Sharing life’s highs with you has made them more enjoyable
and sharing the lows made them more bearable. I had fantastic office-mates, who served as
sounding boards and helped me deal with stress (or celebrate) via Nerf Gun fights. Thanks
for listening to my musings and encouraging me to be a better physicist. I’m thankful to
the NSCL Cycling and DALMAC crews, with whom I shared many hours in the saddle and
possibly just as many rehydrating and refueling. I wouldn’t be such an avid cyclist without
your company. The Grand Canonical Ensemble welcomed me to the group during my first
weeks in East Lansing and served as a needed outlet for my more dramatic tendencies over
the next few years. I’m honored they allowed me to direct them for the last two years.
Of course, one does not simply arrive at graduate school without help. I am indebted
to the many teachers I have had over the years who have nurtured my curiosity in scientific
observation, from worms in elementary school to magnetorheological fluid in college. Thanks
in particular to Mark Blachly for showing me that a PhD in physics was a possibility,
challenging me every single day in class, and making me earn every single Twix bar.
I would not be here without the full-throated encouragement and never-failing confidence
vi

supplied by my family, both immediate and extended. You facilitated and encouraged my
interest in science both before and after school. Thanks for teaching me long division over
breakfast, nagging me to work on my science projects, and always pushing me to follow my
passions, even when they were somewhat unconventional.
Finally, I thank Ragnar for being my rock. You have been a vocal cheerleader, a shoulder
to cry on, a neutral sounding board, and my most difficult jury. I am a better person and
better physicist because of you. I look forward to our next adventure.

vii

TABLE OF CONTENTS

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

xii

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiii
Chapter 1 Introduction . . . . . . . . . . . .
1.1 Chart of nuclides . . . . . . . . . . . . .
1.2 Shell model . . . . . . . . . . . . . . . .
1.3 Islands of inversion . . . . . . . . . . . .
1.4 Halo nuclei . . . . . . . . . . . . . . . .
1.5 Three-body correlations and decays . . .
1.6 Previous experiments . . . . . . . . . . .
1.6.1 Lithium isotopes: A = 11, 10, 9 .
1.6.2 Beryllium isotopes: A = 12, 11, 10

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1
1
3
5
8
11
14
14
22

Chapter 2 Theoretical background . . . . .
2.1 Single neutron decay . . . . . . . . . . .
2.1.1 R-matrix derivation . . . . . . . .
2.1.2 Saturation of resonance lineshape
2.2 Two neutron decay . . . . . . . . . . . .
2.2.1 Phase space decay . . . . . . . .
2.2.2 Sequential decay . . . . . . . . .
2.2.3 Dineutron decay . . . . . . . . .

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25
25
26
31
32
33
34
37

Chapter 3 Experimental techniques . . . .
3.1 Beam production . . . . . . . . . . . . .
3.2 Beam detector and target . . . . . . . .
3.3 Sweeper magnet . . . . . . . . . . . . . .
3.4 Charged particle detectors . . . . . . . .
3.4.1 Blocker . . . . . . . . . . . . . .
3.4.2 Cathode Readout Drift Counters
3.4.3 Ion chamber . . . . . . . . . . . .
3.4.4 Timing scintillator . . . . . . . .
3.4.5 CsI(Na) array . . . . . . . . . . .
3.5 MoNA-LISA . . . . . . . . . . . . . . . .
3.6 Electronics and Data Acquisition . . . .
3.7 Invariant mass spectroscopy . . . . . . .
3.8 Particle correlation techniques . . . . . .
3.8.1 Jacobi coordinates . . . . . . . .
3.8.2 Dalitz plots . . . . . . . . . . . .

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40
40
42
42
43
44
44
46
47
48
50
51
54
56
56
60

viii

Chapter 4 Data Analysis . . . . . . . . . . . . . . . . .
4.1 Calibrations and corrections . . . . . . . . . . . . .
4.1.1 Charged particle calibrations and corrections
4.1.1.1 CRDCs . . . . . . . . . . . . . . .
4.1.1.2 Ion chamber . . . . . . . . . . . . .
4.1.1.3 Thin scintillator . . . . . . . . . .
4.1.1.4 Other timing scintillators . . . . .
4.1.1.5 CsI(Na) array . . . . . . . . . . . .
4.1.2 MoNA-LISA calibrations and corrections . .
4.1.2.1 Charge calibration . . . . . . . . .
4.1.2.2 Position calibration . . . . . . . . .
4.1.2.3 Time calibrations . . . . . . . . . .
4.2 Event selection . . . . . . . . . . . . . . . . . . . .
4.2.1 Beam identification . . . . . . . . . . . . . .
4.2.2 Event quality gates . . . . . . . . . . . . . .
4.2.3 Isotope identification . . . . . . . . . . . . .
4.2.4 Neutron interaction selection . . . . . . . . .
4.3 Inverse tracking . . . . . . . . . . . . . . . . . . . .
4.4 Modeling and simulation . . . . . . . . . . . . . . .
4.4.1 Incoming beam parameters . . . . . . . . . .
4.4.2 Beam energy . . . . . . . . . . . . . . . . .
4.4.3 Reaction parameters . . . . . . . . . . . . .
4.4.4 Other parameters . . . . . . . . . . . . . . .
4.4.5 Additional cuts . . . . . . . . . . . . . . . .
4.4.6 Decay models . . . . . . . . . . . . . . . . .
4.4.6.1 Single neutron decay . . . . . . . .
4.4.6.2 Two neutron decay . . . . . . . . .
4.5 Fitting and error analysis . . . . . . . . . . . . . .

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63
63
63
63
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73
80
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89
90
91
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98
107
110
111
115
118
121
121
122
123
124
125

Chapter 5 Results and Discussion
5.1 11 Li . . . . . . . . . . . . . . .
5.2 10 Li . . . . . . . . . . . . . . .
5.3 12 Be . . . . . . . . . . . . . . .

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128
128
147
156

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Chapter 6 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . 166
REFERENCES

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

ix

LIST OF TABLES

Table 1.1

Table 1.2

Table 4.1

Previous measurements of unbound states in 11 Li below 5 MeV. The
energy shown here is the excitation energy. When papers reported
a decay energy, the two-neutron separation energy S2n =369.15 keV
[51] was added to produce the excitation energy. . . . . . . . . . . .

15

Previous measurements of low-lying unbound states in 10 Li under 2
MeV. The energy shown here is the decay energy. . . . . . . . . . .

20

CRDC1 mask calibration slopes and offsets from all CRDC1 mask
runs. Because there were so few holes in the y-direction (typically
only 3), errors were unreasonably small. Slopes and offsets have been
truncated at the same significance as their CRDC2 counterparts (Table 4.2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

Table 4.2

CRDC2 mask calibration slopes and offsets from all CRDC2 mask runs. 69

Table 4.3

Ion chamber position and time correction coefficient values. . . . . .

71

Table 4.4

Thin scintillator locations and gain-matching slopes and offsets. . . .

75

Table 4.5

Thin scintillator position and time correction coefficient values. . . .

76

Table 4.6

Calibration parameters for the timing signals from the A1900 and
target scintillators. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

80

Final time offsets for each table of MoNA-LISA, before and after the
electronics change. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

Table 4.7

Table 4.8

List of correction factors used to produce the lithium isotope separation parameter. The numbers in the right column were multiplied by
the parameter on the left, summed, and added to the time-of-flight
from the A1900 to the thin to produce a corrected time-of-flight parameter for the lithium isotopes from the 13 B beam. Both position
parameters, x and y, were calculated 12.59 cm downstream of CRDC1. 97

x

Table 4.9

List of correction factors used to produce the beryllium isotope separation parameter. The numbers in the right column were multiplied
by the parameter on the left, summed, and added to the time-of-flight
from the A1900 to the thin to produce a corrected time-of-flight parameter for the lithium isotopes from the 13 B beam. Both position
parameters, x and y, were calculated 12.59 cm downstream of CRDC1. 97

Table 4.10

List of simulation parameters used to recreate the incoming 13 B beam
profile. Part 1 recreated the horizontal peak around 30 mm in the
y-position of CRDC2, Part 2 recreated the square background, and
Part 3 recreated the slanted peak around -20 mm in the y-position. . 113

Table 4.11

List of simulation parameters used to recreate the incoming 12 B beam
profile. Part 1 recreated the horizontal peak around 30 mm in the
y-position of CRDC2, Part 2 recreated the square background, and
Part 3 recreated the slanted peak around -20 mm in the y-position. . 117

Table 4.12

Widths of the parallel and perpendicular kicks used to reproduce the
kicks of the knockout reaction. . . . . . . . . . . . . . . . . . . . . . 119

Table 5.1

Energies and widths for best fits of each two-neutron decay model.
Error bars are only statistical. . . . . . . . . . . . . . . . . . . . . . 137

Table 5.2

Energies and widths used for the best dineutron fit shown in Figures
5.9 and 5.10. Decay through a ∼200 keV = 1 state, shown in Figure
5.5, was minimized at zero contribution to the total spectrum. . . . 138

Table 5.3

Sources of systematic error in the 11 Li “dineutron” decay. Sources of
error in the data are in the top section and sources of error in the simulation are in the second section. Here, MoNA-LISA is abbreviated
with ML. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

Table 5.4

Best fit parameters as used in Figure 5.18. Quantities with error bars
were measured. Other parameters, such as relative contribution and
the width, are reported without error bars, simply for reference. . . 150

Table 5.5

Sources of systematic error in the 10 Li decay. Sources of error in the
data are in the top section. The sources of error from the simulation
tabulated in Table 5.3 (ML position resolution, ML time resolution,
and ML PMT threshold) were all dominated by the error from the
fitting. Here, MoNA-LISA is abbreviated as ML. . . . . . . . . . . . 152

xi

Table 5.6

Sources of systematic error in the 12 Be decay. Sources of error in the
data are in the top section and sources of error in the simulation are
in the second section. . . . . . . . . . . . . . . . . . . . . . . . . . . 158

Table 5.7

Parameters for potential decays from positive and negative parity
states in 12 Be, using Sn =3169±16 keV [117]. . . . . . . . . . . . . . 159

xii

LIST OF FIGURES

Figure 1.1

Chart of the nuclides, showing the principal decay mode of the nuclide. Data from [1]. . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

Wood-Saxon orbitals (left) and Wood-Saxon plus spin-orbit orbitals.
Numbers indicate the number of particles required to fill the levels
up to those large shell gaps, commonly called “magic numbers”. . .

4

Difference between 0p1/2 and 1s1/2 neutron single-particle energies
for the N = 7 isotones. Data from Ref. [6] and references therein. . .

6

Chart of the nuclides and the five predicted islands of inversion (red
circles) [7]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

Figure 1.5

Drawings of two halo nuclei: 11 Be (left) and 11 Li (right). . . . . . .

8

Figure 1.6

Deduced neutron (blue), proton (red), and matter (black) radii of
lithium and beryllium isotopes. Data from [19]. . . . . . . . . . . . .

9

Chart of the nuclides, showing the known neutron and proton halo
nuclei and the suspected neutron halo nuclei as reported by Tanihata,
Savajols, and Kanungo [19]. Red squares indicate confirmed proton
halo nuclei, dark blue squares indicate confirmed neutron halo nuclei,
and light blue squares indicate potential neutron halo nuclei. Black
squares indicate stable nuclei and the gray squares indicate bound
nuclei. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

Figure 1.8

Level diagram for two-neutron decay labeling Ek1 and ΓI . . . . . . .

12

Figure 1.9

Level diagram of states in 11,10,9 Li, relative to the ground state of
11 Li. Neutron separation energies are from Refs. [51–56]. Excited
state energies and parities for 10,11 Li are general summaries of measurements presented in Tables 1.2 and 1.1. Ground state spin of 11 Li
from Ref. [57]. Excited state spins in 10 Li are expected spins and parities based on 1s1/2 and 0p3/2 neutrons coupling with a 0p3/2 proton.
Energy, spin, and parity of first excited state in 9 Li from Refs. [58–60] 14

Figure 1.10

Previous measurements of 11 Li with excitation energies less than 3.5
MeV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 1.2

Figure 1.3

Figure 1.4

Figure 1.7

xiii

16

Previous measurements of unbound, positive-parity states in 10 Li.
Measurements reported without error have arrows for error bars.
Two different measurements in Chernyshev’s 2013 paper produced
two different measurements of 0.7±0.05 and 0.78±0.15 MeV, here
the weighted average is used [72]. a Both Gornov’s 1998 paper [97]
and Santi’s 2003 paper [101] found two possible fits for the data.
Details are in Table 1.2 and footnotes within. . . . . . . . . . . . . .

21

Level diagram of states in 12,11,10 Be, relative to the ground state of
12 Be. Neutron separation energies are from [117, 118]. Bound and
resonant state energies are from [107, 108, 111, 113–116]. . . . . . . .

23

Energy-dependent Breit-Wigner lineshapes with varying Γ0 . For all
lineshapes, = 1 and E0 = 1.0 MeV. All lineshapes have been normalized to the same peak height. . . . . . . . . . . . . . . . . . . . .

32

Figure 2.2

A diagram of the sequential decay described. . . . . . . . . . . . . .

34

Figure 3.1

Schematic of the Coupled Cyclotron Facility and A1900 Fragment
Separator with the production setting for this experiment. . . . . . .

41

Figure 3.2

Schematic of the experimental area. . . . . . . . . . . . . . . . . . .

43

Figure 3.3

Schematic of CRDC operation modified from [139] where the z-direction
has been expanded for detail. The field shaping wires mentioned in
the text are not included for visibility and simplicity. . . . . . . . . . 45

Figure 3.4

Schematic of ion chamber. . . . . . . . . . . . . . . . . . . . . . . .

46

Figure 3.5

A schematic of the thin scintillator, as seen from upstream of the
detector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48

Figure 3.6

A mechanical design drawing of the CsI(Na) array. . . . . . . . . . .

49

Figure 3.7

A schematic of a MoNA LISA bar. . . . . . . . . . . . . . . . . . . .

50

Figure 3.8

Schematic of the MoNA-LISA-Sweeper electronics set-up. . . . . . .

52

Figure 3.9

Three possible coordinate definitions for a three-body system. . . . .

56

Figure 3.10

T and Y systems and associated Jacobi coordinates for fragment+n+
n system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

Figure 1.11

Figure 1.12

Figure 2.1

xiv

Figure 3.11

Figure 3.12

Figure 4.1

Figure 4.2

Figure 4.3

Figure 4.4

Figure 4.5

Figure 4.6

Example Jacobi plots for a phase space decay (black), a dineutron
decay (red), and a sequential decay (blue). The left column shows
Jacobi coordinates from the T-system and the right column shows
Jacobi coordinates from the Y-system. The energy ratios are shown
on the top row and the angular distributions are shown in the bottom
row. All distributions are without any acceptances or resolutions
applied. In the cos θ-T system plot, the dineutron and phase space
lines overlap. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

Example Dalitz plots for a phase space decay (upper left), a dineutron
decay (upper right), and two sequential decays (bottom row). The
sequential decay in the bottom left panel shows a decay through an
intermediate state approximately halfway between the initial and final
states, while the sequential decay shown in the bottom right has a
more asymmetric energy distribution. All distributions are without
any acceptances or resolutions applied. . . . . . . . . . . . . . . . .

61

Examples of total charge spectra for two pads in CRDC2. Pad 64
(left) is an example of a good total charge spectrum and pad 24
(right) is an example of a bad total charge spectrum. . . . . . . . .

64

Example charge distribution on CRDC pads as a function of time.
A full charge integration measurement would integrate the shaded
area on the left. The sampling method adds the values of a discrete
number of points, as shown on the right. . . . . . . . . . . . . . . .

65

CRDC y-position versus number of samples collected for each pad
(left: CRDC1, right: CRDC2). Both plots are shown with the zaxis on a log scale. Each CRDC collects fewer samples at certain
regularly spaced intervals along the y-direction, visible in these plots
as enhancements at lower numbers of samples and vacancies at the
spectra for four samples. . . . . . . . . . . . . . . . . . . . . . . . .

65

An example of a charge distribution fit for CRDC2. The centroid of
the Gaussian fit, shown here by the vertical line, would be used as
the position of the particle in this detector. . . . . . . . . . . . . . .

67

Left: mechanical design drawing of the CRDC mask. Right: Example
of CRDC mask data from run 157. . . . . . . . . . . . . . . . . . . .

68

A schematic of the CRDC masks and their relation to the lab coordinate frame, the beam, and the pads in each detector. . . . . . . .

68

xv

Figure 4.7

An example of fitting used in gain-matching the ion chamber. Left:
points show raw charge signal from pad 4, thick solid line is Gaussian
fit to data, and thin vertical line indicates centroid of “ideal” pad 8.
Right: points show gain-matched charge signal from pad 4 and thin
vertical line indicates centroid of “ideal” pad 8. . . . . . . . . . . . .

71

Figure 4.8

Position dependence of the ion chamber. The left panel shows the
summed signal from the ion chamber as a function of CRDC2 x-position
when using a cocktail beam. The right panel shows the positioncorrected summed signal from the ion chamber. . . . . . . . . . . . . 72

Figure 4.9

Time dependence of the summed ion chamber signal. The top panel
shows the summed, position-corrected ion chamber signal as a function of run number. The red line is the fitted function 4.3 using
parameters from Table 4.3. The bottom panel shows the summed,
position-corrected, and time-corrected ion chamber signal as a function of run number. . . . . . . . . . . . . . . . . . . . . . . . . . . .

74

Position dependence of the charge signal from the thin scintillator.
The left panel shows the total signal from the thin scintillator as
a function of CRDC2 x-position when using a cocktail beam. The
right panel shows the position-corrected summed signal from the thin
scintillator. The correction was chosen so that the isotopes of interest
(lithium and beryllium) had no position dependence. The remaining
position dependence was primarily for the heavier isotopes, nitrogen
and carbon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77

Time dependence of the total thin scintillator charge signal. The top
panel shows the summed, position-corrected thin scintillator charge
signal as a function of run number. The red line is the fitted function
4.3 using parameters from Table 4.5. The bottom panel shows the
summed, position-corrected, and time-corrected charge signal as a
function of run number. . . . . . . . . . . . . . . . . . . . . . . . . .

78

Timing signal from the target scintillator before (left) and after (right)
the walk correction. . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

Target scintillator timing spectra from run 41 (black), taken while
the target exhibited double peaking, and from run 90 (blue), taken
after the double peaking was fixed. . . . . . . . . . . . . . . . . . . .

82

Figure 4.10

Figure 4.11

Figure 4.12

Figure 4.13

xvi

Figure 4.14

An example of MoNA-LISA’s charge calibration procedure. The raw
charge from each QDC channel is plotted (left). The cosmic peak
around channel 1200 is fit with a red Gaussian line. The inset shows
channels 500 to 2000 with a linear scale on the y-axis. Those values
are used to construct a linear fit whose slope and offset are applied
to produce a calibrated charge signal in MeVee (right). . . . . . . .

84

An example of the raw time spectrum during MoNA-LISA’s TDC calibration procedure. A pulser pulses the system every 40 ns, creating
a spectrum with peaks at intervals that correspond to 40 ns. . . . .

85

Figure 4.16

TDC slope values for MoNA and LISA. . . . . . . . . . . . . . . . .

86

Figure 4.17

X-position calibration example. The final x-position spectrum (right)
for one bar was calculated based on the time difference between the
two PMTs (left). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87

Schematics showing the differences used to calibrate the time measurement in MoNA-LISA. The view on the left is from the front of
the array and the schematic on the right shows the side view of the
array, looking head-on at the PMTs. . . . . . . . . . . . . . . . . . .

88

Calibrated timing spectra for each table. Impinging the beam on
a thick target produced many γ-rays and neutrons. Vertical lines
indicating the expected arrival of γ-rays and neutrons at the front of
that table. The left panel shows the time-of-flight to the MoNA table,
the center panel shows the time-of-flight to the first LISA table, and
the right panel shows the time-of flight to the second LISA table. . .

89

Beam component identification using energy loss through the ion
chamber (y-axis), and time-of-flight from the A1900 scintillator to
the target scintillator (x-axis). . . . . . . . . . . . . . . . . . . . . .

90

Time difference between the A1900 and thin scintillators, showing
“echos” from events in which the A1900 scintillator recorded the time
of a different particle than the thin scintillator. The vertical lines
indicate the limits of the gate used to eliminate these events. . . . .

92

Event quality gates applied to the CRDC1 parameters TAC (left)
and pad sum (right). Red vertical lines with arrows indicate the
requirements for an event to pass through the gate. . . . . . . . . .

92

Event quality gates applied to the CRDC2 parameters TAC (left),
pad sum (center), and χ2 (right). Thick red vertical lines indicate
the requirements for an event to pass through the gate. . . . . . . .

93

Figure 4.15

Figure 4.18

Figure 4.19

Figure 4.20

Figure 4.21

Figure 4.22

Figure 4.23

xvii

Figure 4.24

Element identification. . . . . . . . . . . . . . . . . . . . . . . . . .

94

Figure 4.25

Three-dimensional plot showing lithium isotope identification by plotting x-position, x-angle, and time-of-flight. . . . . . . . . . . . . . .

95

Lithium (top) and beryllium (bottom) particle identification from the
13 B beam and in coincidence with an interaction in MoNA-LISA. . .

96

Lithium (top) and beryllium (bottom) particle identification from the
12 B beam and in coincidence with an interaction in MoNA-LISA. . .

98

Figure 4.26

Figure 4.27

Figure 4.28

Top: Raw neutron time-of-flight spectrum. Bottom: Charge deposited vs. time-of-flight for the first interaction in MoNA-LISA. . . 100

Figure 4.29

A breakdown of the time-of-flight spectrum of Figure 4.28 by sections
(tables) of the detector. . . . . . . . . . . . . . . . . . . . . . . . . . 101

Figure 4.30

Top: Time-of-flight breakdown of events with two (and only two)
interactions in the section of LISA at 0 degrees. The blue line shows
the first interaction spectrum, while the green line shows the spectrum
of the second interaction. The black line is the sum of those two
components. Bottom: Comparison of the time-of-flight spectrum for
events with one interaction (red line) and the summed spectrum from
above (black line). The red line has been normalized to the flat
background before the gamma peak of the black line. . . . . . . . . 102

Figure 4.31

The γ-ray peak in MoNA-LISA during the 13 B production runs. . . 103

Figure 4.32

The arrival time of the large γ-ray peak, as observed by MoNA (left),
the LISA table at 0 degrees (center), and the LISA table at 30 degrees
(right). Vertical lines indicate expected arrival time of γ-rays from
the target area. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

Figure 4.33

Difference of time differences between two of the three detectors. The
top panel shows the comparison between LISA at 0◦ and LISA at 30◦ ,
and the bottom panel shows the comparison between MoNA and
LISA at 30◦ . Black lines indicate the possible locations of a point
gamma source to produce the measured time difference between the
two detectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

Figure 4.34

The two zero-lines from Figure 4.33 overlaid on the vault layout.
The intersection of these lines indicates the likely location of a point
gamma source. Within the error of the calculation, and given the
layout of the experimental vault, a gamma source was identified inside
the Sweeper box. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
xviii

Figure 4.35

Final time-of-flight spectrum for MoNA-LISA. This spectrum requires that each interaction not only have a valid charge but also
a valid time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

Figure 4.36

A flow chart of the simulation process and the various data that are
used as input. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

Figure 4.37

Incoming beam distributions in CRDC1 (top row) and CRDC2 (bottom row). The left column shows the two-dimensional positional profile, the middle column shows the x-projection, and the right column
shows the y-projection. . . . . . . . . . . . . . . . . . . . . . . . . . 112

Figure 4.38

Simulated (blue) and data (black) position and angular spectra for
the incoming 13 B beam profile. Top row: CRDC1 x-position, CRDC2
x-position, and x-angle. Bottom row: CRDC1 y-position, CRDC2
y-position, y-angle. . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

Figure 4.39

CRDC2 position spectra from data (left) and simulation (right) for
the incoming 13 B beam. . . . . . . . . . . . . . . . . . . . . . . . . . 115

Figure 4.40

Simulated (blue) and data (black) position and angular spectra for
the incoming 12 B beam profile. Top row: CRDC1 x-position, CRDC2
x-position, and x-angle. Bottom row: CRDC1 y-position, CRDC2
y-position, y-angle. . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

Figure 4.41

CRDC2 position spectra from data (left) and simulation (right) for
the incoming 12 B beam. . . . . . . . . . . . . . . . . . . . . . . . . . 117

Figure 4.42

Simulated (blue) and data (black) position and angular spectra for
the 9 Li fragments from the 13 B beam. Top row: CRDC1 x-position,
CRDC2 x-position, and x-angle. Bottom row: CRDC1 y-position,
CRDC2 y-position, y-angle. See Section 4.4.5 for information about
the cut on CRDC2 y-position. . . . . . . . . . . . . . . . . . . . . . 119

Figure 4.43

Total momentum of the 9 Li fragments in the data (black line) and
simulation (blue line). . . . . . . . . . . . . . . . . . . . . . . . . . . 120

Figure 4.44

CRDC1 x-angle versus CRDC1 x-position for a beam sweep run with
no target (left) and for production (right). The red polygon indicates
the “sweep” gate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

Figure 4.45

Full x-position spectrum for 11 Be, data (black line) and simulation
(blue line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

xix

Figure 5.1

Decay energy for the 9 Li+n+n system. No causality cuts are applied. 129

Figure 5.2

Number of real two neutron events (green, top line), and fake two
neutron events (red, bottom line) that pass the causality cut as a
function of the magnitude of the distance cut, based on a simulation
of a phase space decay from 11 Li. . . . . . . . . . . . . . . . . . . . 130

Figure 5.3

Decay energy for the 9 Li+n+n system. Causality cuts are applied. . 130

Figure 5.4

Jacobi plots for the 9 Li+n+n system. Causality cuts are applied, as
well as a requirement that the total three-body decay energy be less
than 1.6 MeV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

Figure 5.5

Possible decay paths from an ≈1 MeV state in 11 Li populated using
a 2p knockout from the 13 B beam. The red, blue, green, and magenta decay paths go through intermediate states in 10 Li, while the
orange and cyan decay paths explicitly bypass those states in favor
of correlated (dineutron) or uncorrelated (phase space) two-neutron
emission. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

Figure 5.6

Possible decay paths from states in 10 Li populated using a 2p1n
knockout from the 13 B beam. The gray path indicates population
of and decay from a much higher-lying state. . . . . . . . . . . . . . 134

Figure 5.7

Decay energy plots used when fitting simulation to data. The upper
left panel shows the two-body decay energy in coincidence with 9 Li
and one neutron. The upper right panel shows the same two-body
decay energy spectrum, but this time requiring only one interaction in
MoNA-LISA. The lower left panel shows the three-body decay energy
spectrum, as shown in Figure 5.1. The lower right panel shows the
same three-body decay energy spectrum but with causality cuts, as
shown in Figure 5.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

Figure 5.8

χ2 minimization plots for the 11 Li energy (left) and width (right), for
many different decay paths. . . . . . . . . . . . . . . . . . . . . . . . 136

Figure 5.9

The best fits to the decay energy spectra from Figure 5.7 with all four
decay models. The upper left panel shows the two-body decay energy
in coincidence with 9 Li and one neutron. The upper right panel shows
the same two-body decay energy spectrum, but this time requiring
only one interaction in MoNA-LISA. The lower left panel shows the
three-body decay energy spectrum. The lower right panel shows the
same three-body decay energy spectrum but with causality cuts. . . 137

xx

Figure 5.10

The Jacobi coordinate spectra (as shown in Figure 5.4) compared to
the same spectra from the best fits of the four decay models. . . . . 139

Figure 5.11

The best fit to the decay energy spectra of Figure 5.7 with the dineutron model. Different colored curves show the different components
as labeled in Figures 5.5 and 5.6. The black line shows the summed
total. The upper left panel shows the two-body decay energy in coincidence with 9 Li and one neutron. The upper right panel shows
the same two-body decay energy spectrum, but this time requiring
only one interaction in MoNA-LISA. The lower left panel shows the
three-body decay energy spectrum. The lower right panel shows the
same three-body decay energy spectrum but with causality cuts. . . 140

Figure 5.12

Previous measurements of 11 Li with excitation energies less than 3.5
MeV. The result from this work is shown by the box. . . . . . . . . 141

Figure 5.13

The Jacobi coordinate spectra (as shown in Figure 5.4) compared to
the same spectra from the best fit from the dineutron model. Different
colored curves show the different components as labeled in Figures
5.5 and 5.6. The black line shows the summed total. . . . . . . . . . 143

Figure 5.14

Dalitz plots for 11 Li. The top set is the data and the bottom set is
the best fit simulation using the dineutron model. The data only uses
events with a three-body decay energy less than 1.6 MeV that also
passed the causality cuts. . . . . . . . . . . . . . . . . . . . . . . . . 145

Figure 5.15

Two different dineutron models in comparison to the experimental
data. The solid line simulation used a scattering length of -18.7 fm
while the dashed line simulation used a scattering length of -100 fm,
implying stronger correlation. . . . . . . . . . . . . . . . . . . . . . . 146

Figure 5.16

Decay energy in coincidence with 9 Li and one neutron, populated by
two-proton knockout from a 12 B beam. . . . . . . . . . . . . . . . . 147

Figure 5.17

A fit of a single = 1 decay energy lineshape to the data from Figure
5.16 (top) and the residual (bottom). . . . . . . . . . . . . . . . . . 148

Figure 5.18

A fit of three = 1 decay energy lineshapes to the data from Figure
5.16. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

Figure 5.19

χ2 plots that show the minimization of the central energies of the
blue, green, and magenta lineshapes in Figure 5.18. Dashed lines
indicate statistical error and solid lines indicate the total error margin.150

xxi

Figure 5.20

Previous measurements of unbound, = 1 states in 10 Li (same as
Figure 1.11) with the limits of the new measurements overlaid in
blue, green, and magenta lines. Measurements reported without error
have arrows for error bars. Two different measurements in Chernyshev’s 2013 paper produced two different measurements of 0.7±0.05
and 0.78±0.15 MeV, here the weighted average is used [72]. a Both
Gornov’s 1998 paper [97] and Santi’s 2003 paper [101] found two possible fits for the data. Details are in Table 1.2 and footnotes within.

151

Figure 5.21

Calculations of 0p1/2 (red) and 1s1/2 single-particle levels of 10 Li,
11 Be, 12 B, and 13 C. Energies of single-particle levels and relevent
energy eigenstates are given relative to the calculated 0p1/2 single
particle energy level. Dashed line shows the extrapolation of the
linear fit to the 1s1/2 single-particle levels. 9 He levels are from Ref.
[155], 10 Li levels are from this work, 11 Be and 12 B levels are from
Ref. [156], and 13 C levels are from Ref. [157]. . . . . . . . . . . . . . 153

Figure 5.22

Calculations by Thompson and Zhukov show the relationship between
the 11 Li binding energy and the 10 Li 0p1/2 single-particle energy as
a function of the 1s1/2 scattering length [80]. The line at -0.369 MeV
shows the limits imposed by the measurements of Smith et al. [51]
and this thesis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

Figure 5.23

Decay energy of the 11 Be+n system after population of 12 Be via a
two-proton knockout from a 13 B beam. . . . . . . . . . . . . . . . . 157

Figure 5.24

Different simulated lineshapes fitted or compared to the 11 Be+n decay energy lineshape shown in Figure 5.23. Left panel: data is fit
with a single = 1 resonant lineshape and a background. Right
panel: data is compared to two lineshapes assuming total decay from
a previously measured state at 4580 keV in 12 Be to the excited 1/2−
state in 11 Be (left lineshape) and to the 1/2+ ground state (right
lineshape). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

Figure 5.25

χ2 plots for the energy (left) and width (right) of the single = 1
decay fit to the data in the upper left panel of Figure 5.24. Dashed
vertical lines show the statistical 1σ limits and the solid vertical lines
indicate the additional systematic error. . . . . . . . . . . . . . . . . 158

Figure 5.26

Fits of the 12 Be decay energy spectrum with an = 0 (left) or = 2
(right) lineshape. The fits are shown on the top row and the bin-bybin contributions to the χ2 value are shown in the bottom row. . . . 160

xxii

Figure 5.27

Various decay energy spectra in coincidence with 10 Be fragments.
Upper left panel: two-body decay energy. Upper right panel: twobody decay energy spectrum, requiring only one interaction per event
in MoNA-LISA. Lower left panel: three-body decay energy spectrum.
Lower right panel: three-body decay energy spectrum with causality
cuts applied. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

Figure 5.28

The data from Figure 5.27, fitted with simulated spectra. The green,
red, and purple lineshape show single-neutron decays from 11 Be as
used by Peters et al. [115]. The blue line shows a two-neutron decay.
The energies and widths of each decay were fixed, leaving only the
relative amplitudes to vary in the fitting. All spectra except the lower
right were included in the fitting. . . . . . . . . . . . . . . . . . . . . 163

Figure 5.29

Level diagram of measured states in 12 Be from Figure 1.12 compared
to a continuum shell model calculation of states in 12 Be [132]. The
state presented here is represented by the grey box. The width of the
box represents the error of the energy measurements. . . . . . . . . 165

xxiii

Chapter 1
Introduction

1.1

Chart of nuclides

The atomic nucleus is a many-body system of protons and neutrons. A nuclide is a species
of atomic nucleus defined by the number of protons (Z) and neutrons (N ) it has. Nuclear
scientists use a chart of nuclides to organize information (Figure 1.1). Like the periodic table,
this chart organizes the systems under study and presents an easy way to view patterns.
The chart of nuclides is organized by neutron number (along the x-axis) and proton
number (along the y-axis). Straight down a diagonal of the chart, denoted in Figure 1.1 by
the black squares, is the valley of stability. These are the common, stable nuclei, many of
which are naturally occuring. Nuclides on either side of the valley of stability tend to decay
toward that valley. The way a nuclide decays is called its decay mode and the time constant
that defines how long it will live is called its half-life. Figure 1.1 shows the most common
mode of decay for each nuclide from its ground state. On the proton-rich side of the valley
of stability, most nuclides decay by emitting a positron and turning a proton into a neutron
(β + decay, shown by the pink squares). On the neutron-rich side, the primary mode of decay
is β − decay (blue squares): turning a neutron into a proton by emitting an electron.
Far out from the valley of stability are the proton and neutron driplines. These are the
limits beyond which the nuclides are unbound with respect to neutron or proton emission and
decay by emission of a proton (purple squares in Figure 1.1) or a neutron (orange squares).
1

120

Proton number

100
80
Stable
Beta minus
Beta plus
Alpha
Neutron
Proton
Fission
Unknown

60
40
20
0

0

20

40

60

80
100
Neutron number

120

140

160

180

Figure 1.1: Chart of the nuclides, showing the principal decay mode of the nuclide. Data
from [1].

2

These limits are called driplines because beyond these lines, the protons and neutrons just
“drip” off the nucleus. Beyond the neutron dripline, the nuclides have a half-life on the
order of 10−21 s. Two other primary modes of decay employed by nuclei are alpha decay
(the emission of a 4 He nucleus) and fission (the splitting of the nucleus into two or more
fragments).

1.2

Shell model

The nuclear landscape is not a smooth, continuously changing landscape with the valley
of stability as the lowest point. Already visible in Figure 1.1 are signs of more intricate
structure: nuclides with even Z or even N seem to be more stable than their odd Z or N
neighbors and there are some “ridges” where the nuclei are more stable than their neighbors
(Z = 50 for example). In 1933, Elsasser noted that nuclides with particular numbers of
protons or neutrons appeared to be more stable [2]. Fifteen years later, Maria Goeppert
Mayer summarized the increased experimental evidence for the stability of certain nuclei
with neutron numbers of 50, 82, or 126 and proton numbers of 20, 50, or 82 [3]. One year
later, Mayer and the collaboration of Haxel, Jensen, and Suess independently published shell
models with strong spin-orbit splitting that resulted in increased stability via large shell gaps
at these numbers [4, 5].
The nuclear shell model shares some properties with the electronic shell model. Both
begin by placing particles in a single-particle potential. This potential has eigenstates which
can be characterized by their quantum numbers (n and ) and energies. A spin-orbit term
is then added to the original potential. This spin-orbit term lifts the degeneracies of some
eigenstates and defines single-particle orbitals, which are identified by their total angular

3

2s

1d 3/2
2s 1/2

1d

0g
7/2
1d 5/2

0g

50
0g

40

9/2

1p
1p
28

0f

1/2
0f 5/2
1p
3/2
0f 7/2

20

20
0d 3/2
1s 1/2

1s
0d

0d 5/2
8

8

0p
0p

0p
2

1/2
3/2

2

0s

0s 1/2
WS

WS+SO

Figure 1.2: Wood-Saxon orbitals (left) and Wood-Saxon plus spin-orbit orbitals. Numbers
indicate the number of particles required to fill the levels up to those large shell gaps,
commonly called “magic numbers”.

4

momentum (j) value. These eigenstates are filled consecutively by energy, beginning with
the lowest energy, and following the Pauli principle: no identical particles may be placed in
eigenstates with the same quantum numbers. Each model has gaps of varying size between
the orbitals. Gaps that are large correspond to “magic numbers”. Systems filled to these
numbers are observed to be more stable than others (i.e. noble gases, “magic” nuclei).
There are some differences between electronic and nuclear shell models. In the nuclear
shell model, the potential felt by each particle is generated by the other particles considered
in the system (other nucleons), rather than by a central body (for electrons, this would be
the atomic nucleus). The length scales are different, 10−15 m for the nuclear shell model
and 10−12 m for the atomic shell model. The spin-orbit term, while a small correction to
energy levels in the electronic shell model, plays a larger role in the nuclear shell model and
is observed to have a different sign. Also, while there is a clear physical reason for including
the spin-orbit term in the electronic shell model, the physical reason for including the nuclear
spin-orbit term is less clear.
Two common bases for the initial potential are the harmonic oscillator and a Wood-Saxon
potential. The harmonic oscillator is chosen because its basis states are simple and analytic,
while the shape of the Wood-Saxon potential is closer to the empirically observed shape of
the nuclear densities. The low-lying Wood-Saxon and Wood-Saxon plus spin-orbit orbitals
are shown in Figure 1.2, along with the “magic numbers”.

1.3

Islands of inversion

As the exploration of the nuclear chart has expanded beyond the stable isotopes, some of the
large shell gaps observed in the stable nuclei have disappeared and other gaps have appeared.

5

E − Ep1/2 (MeV)

6
5

N =7

4

Ep1/2

3

Es1/2

2
1
0

-1
2

3

4

5

6

7

Proton number
Figure 1.3: Difference between 0p1/2 and 1s1/2 neutron single-particle energies for the N = 7
isotones. Data from Ref. [6] and references therein.
The energies associated with each single-particle orbital are not universal. Nuclides that have
either protons or neutrons filled up to a large shell gap display increased stability because
excitations of protons or neutrons to the next orbital takes a significant amount of energy. As
single-particle orbitals on either side of a large gap approach each other, those excitations take
less energy and the “magic number” associated with that shell gaps becomes less “magic”. In
some cases, known as “islands of inversion”, states from either side of magic numbers cross
each other. One example of this is shown in Figure 1.3 that shows the energy difference
between the 0p1/2 and 1s1/2 neutron single-particle energies for the N = 7 isotones. As
protons are removed from 14 N (Z = 7), the difference between those levels decreases until
at 11 Be (Z = 4), when the levels invert.
These “islands of inversion” are located throughout the chart of the nuclides. Figure 1.4
shows the islands identified by B.A. Brown in 2010 [7] around the major shell closures of
N = 8 and 20, as well as the sub-shell closures of N = 14, 20 and 40. There is experimental

6

35

Proton number

30
25
20
15
10
Stable
Known

5
0

0

10

20
30
Neutron number

40

50

Figure 1.4: Chart of the nuclides and the five predicted islands of inversion (red circles) [7].
evidence for shell breaking in all of these islands centered around 12 Be, 20 C, 32 Mg, 42 Si, and
64 Cr

[8–14].

In the island of inversion centered around 12 Be, used as an example previously, the 1s1/2
level from the sd shell (the group of 0d3/2 , 1s1/2 , and 0d5/2 states shown in Figure 1.2)
drops down into the p shell (comprised of the 0p3/2 and 0p1/2 levels), breaking the N = 8
shell closure. The breaking of the N = 8 closure was first predicted by Talmi and Unna in
1960 [15] based on indications of a parity inversion in β-decay measurements by Wilkinson
and Alburger [16]. The first experimental confirmation of this inversion was a measurement
of the ground state parity of 11 Be performed by Alburger et al. four years later [8]. In
normal shell model ordering, the ground state of 11 Be should be 1/2− , with a 0s21/2 0p23/2
structure in the protons and a 0s21/2 0p43/2 0p11/2 structure in the neutrons. Alburger et al.,
however, found that the parity of the 11 Be ground state was positive. The inversion is also
evident in the ground state structure of 11 Li and 12 Be in which the two valence neutrons are
shared between the 0p21/2 and 1s21/2 configurations [17, 18].
7

Figure 1.5: Drawings of two halo nuclei: 11 Be (left) and 11 Li (right).

1.4

Halo nuclei

Far from stability and near the driplines, the binding energy of the last nucleon or two
becomes increasingly small. As the binding of that last nucleon decreases, the nucleon can
exist at greater distances from the core. It can exist at such a distance that instead of
modeling a nucleus with mass A as an A-body system, it can be considered a two-body
system, with an A − 1 core and one additional nucleon. These systems are commonly called
one neutron (or proton) halo nuclei. There are also two- and four- nucleon halo systems.
Figure 1.5 shows the known and suspected halo nuclei.
Halo nuclei are commonly associated with increased neutron or proton radii. Plots showing the neutron, proton, and matter radii of lithium and beryllium isotopes are presented in
Figure 1.6. The halo nuclei 11 Be, 14 Be and 11 Li are clearly indicated by the sudden increase
in neutron and matter radii but (relatively) constant proton radii. Experimentally, halo nuclei are identified by measurements of large reaction cross-section, small neutron (or proton)
separation energies, narrow momentum distributions, and a soft E1 mode of excitation [19].
Figure 1.7 shows the locations of the identified and potential halo nuclei in the chart of the
nuclides.
The first identified halo system was 11 Li [20,21]. Tanihata et al. calculated the radii of a
8

4

4
Li isotopes

3

2.5

2

Be isotopes

3.5
Radius (fm)

Radius (fm)

3.5

Rm
Rp
Rn

3

2.5

6

7

8
9
10
Mass number

2

11

7

8

9 10 11 12 13 14
Mass number

Figure 1.6: Deduced neutron (blue), proton (red), and matter (black) radii of lithium and
beryllium isotopes. Data from [19].
chain of lithium isotopes by measuring the interaction cross-sections and noticed a significant
increase in radius at A = 11 [20]. Kobayashi et al. later confirmed the halo identification
by measuring the transverse momentum distribution of a 9 Li fragment after a 2n removal
from 11 Li [22]. The measured distribution was fit by a combination of two Gaussians: one
Gaussian whose width was similar to the transverse momentum distributions measured from
bound nuclei, and another narrower one which indicated “the weak binding of the two outer
neutrons in the 11 Li system” [22]. The first measurement of an electromagnetic dissociation
(EMD) enhancement, predicted by Hansen and Jonson [23] was reported by Kobayashi et
al. [24]. When comparing interaction cross-sections on targets of differing Z, they wrote that
“a large enhancement due to the EMD process was found for high-Z targets” [24].
All presently known two-neutron halo nuclei are also identified as Borromean systems
[25, 26]. A Borromean system is a three-body system in which each two-body subsystem is
unbound. For a two-neutron halo nucleus of mass A, those systems include the dineutron

9

17 Ne 18 Ne 19 Ne 20 Ne 21 Ne 22 Ne 23 Ne 24 Ne 25 Ne 26 Ne 27 Ne

10

17 F 18 F 19 F 20 F 21 F 22 F 23 F 24 F 25 F 26 F

Proton number

9
8

13 O 14 O 15 O 16 O 17 O 18 O 19 O 20 O 21 O 22 O 23 O 24 O

7

12 N 13 N 14 N 15 N 16 N 17 N 18 N 19 N 20 N 21 N 22 N 23 N

6

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

5

8B

10 B 11 B 12 B 13 B 14 B 15 B

17 B

4

7 Be

9 Be 10 Be 11 Be 12 Be

Neutron halo

Stable

3

6 Li 7 Li 8 Li 9 Li

Proton halo

Bound

14 Be

11 Li

22 C

19 B

Potential neutron halo
2

4 He

6 He

8 He

2

4

6

8
10
Neutron number

12

14

16

Figure 1.7: Chart of the nuclides, showing the known neutron and proton halo nuclei and
the suspected neutron halo nuclei as reported by Tanihata, Savajols, and Kanungo [19]. Red
squares indicate confirmed proton halo nuclei, dark blue squares indicate confirmed neutron
halo nuclei, and light blue squares indicate potential neutron halo nuclei. Black squares
indicate stable nuclei and the gray squares indicate bound nuclei.

10

and the A − 1 system. Due to pairing effects (in which a pair of nucleons is more bound
than a single nucleon), all the identified two-neutron halo nuclei have an even number of
neutrons.

1.5

Three-body correlations and decays

The properties of a three-body system, such as a Borromean nucleus, depend on the correlations between the three bodies. These correlations can be extracted using a variety of
methods: radii measurements, correlation measurements based on intensity interferometry,
Coulomb excitation, and reaction measurements [19]. Radii measurements, only able to
be performed on bound nuclei, use charge and matter radii measurements to calculate the
average distance between the two neutrons as well as their distance from the core. Correlation measurements look at the intensity correlation of two observed particles after they
have been emitted from a bound state. The correlation function, which is a function of
the difference of momenta between two neutrons, is fit and from that fit, a source size is
extracted [27]. Coulomb excitation is used to excite a collective soft dipole mode in halo
nuclei [23]. These measurements excite a bound nucleus and measure the strength of the
E1 transition. From the integral of that strength, one can extract the distance between the
core and the center-of-mass of the two neutrons in the ground state. Reaction measurements
measure the kinematic properties of the fragments after break-up or decay and compare these
to model predictions. These measured values include perpendicular and parallel momentum
distributions, energy ratios, and angles between fragments. The work in this dissertation
uses reaction measurements to investigate the correlations from the decay of excited states
of 11 Li.

11

Ek1

AZ

ΓI

A−1 Z

A−2 Z

Figure 1.8: Level diagram for two-neutron decay labeling Ek1 and ΓI .
The kinematics of a two-body decay, such as the emission of a single neutron or proton,
are very clear: in order to conserve momentum, the two particles are emitted back-to-back
in the center-of-mass frame. The inclusion of a third particle makes this more complex.
Conserving momentum in a three-body decay means all three particles must be emitted in
the same plane. Within that plane, however, there are opportunities for various correlations
between the three particles. In addition, there is the additional factor of timing: could this
three-body decay be two sequential two-body decays?
Much more work has been done with two proton decays than with two neutron decays.
Goldansky was the first to predict two-proton radioactivity [28]. Since this paper, the definitions have gotten a bit more nuanced. Goldansky defined a “true 2p” decay as one where
one-proton decay is energetically forbidden and all products are emitted simultaneously [28].
The first discovery of two-proton radioactivity like this was in 45 Fe [29,30]. When there is an
unbound intermediate state, the decay can be either sequential or democratic. A sequential
decay is defined as one decay to an intermediate state followed by a second decay to a bound
state. The decay of 9 B to p + 2α is a good example, although in this case, one of the two
sequential steps is an alpha decay, not a proton decay [31]. In a democratic decay, there
are no strong kinematic correlations between particles [32]. The decay of the ground state
of 6 Be provides an example of this type of decay [31, 32]. The distinction between sequen12

tial and democratic is not cleanly defined, but Bochkarev suggested a ratio to quantify the
difference [32]:
Sequential:

ΓI
<< 1
Ek1

Democratic:

ΓI
>1
Ek1

where ΓI is the width of the potential intermediate state and Ek1 is the energy released
when decaying to that intermediate state, shown in Figure 1.8. When the decay is through
a narrow intermediate state, this ratio is much less than one and the decay is sequential.
When the ratio is greater than or equal to one, the decay is democratic.
Though these definitions were mostly developed when discussing two-proton decays, they
apply to two-neutron decays as well. For two-proton decays, the interaction between the
two protons as they leave the nucleus (called a final state interaction) is repulsive, due
to the Coulomb force. Without the Coulomb force, the potential between two neutrons
is repulsive at very short ranges and attractive at longer distances, with the change-over
distance being around 1 fm [33]. This attractive force can correlate the kinematics of the
neutrons during the decay even if they were not correlated while in the nucleus [34]. The
decay of strongly-correlated neutrons is sometimes called a dineutron model.
Experimentally, identification of these decay types is performed by measuring different
energy and kinetic distributions. Dalitz plots were introduced in the 1950s [35, 36], have
continued to be used in high-energy physics, and exist in a slightly modified form in nuclear
physics [37, 38]. Jacobi plots of radii, energies, and kinematic variables have been used in
understanding correlations in three-body decays, both proton and neutron [26, 31, 39–49].
When identifying a sequential decay of an unbound state in 24 O to 22 O, Hoffman et al. took
a slightly different approach and plotted two different two-body decay energies against each
13

1 2

Energy MeV

3

2

1

2
1
2

0

3 2

3 2
11

10

Li

Li n

9

Li 2n

Figure 1.9: Level diagram of states in 11,10,9 Li, relative to the ground state of 11 Li. Neutron
separation energies are from Refs. [51–56]. Excited state energies and parities for 10,11 Li
are general summaries of measurements presented in Tables 1.2 and 1.1. Ground state spin
of 11 Li from Ref. [57]. Excited state spins in 10 Li are expected spins and parities based on
1s1/2 and 0p3/2 neutrons coupling with a 0p3/2 proton. Energy, spin, and parity of first
excited state in 9 Li from Refs. [58–60]
other [50].

1.6
1.6.1

Previous experiments
Lithium isotopes: A = 11, 10, 9

The first halo nucleus, 11 Li, has been the subject of many experiments and many more
papers. The review of previous work presented here will focus on measurements of the energies, spin, and parities of low-lying unbound states as well as measurements or calculations
regarding correlations between the neutrons and the decay mode. In addition, some details
of previous measurements and calculations of 10,9 Li will be discussed.
After the first detection of 11 Li [20,21], Arnold et al. measured the 3/2− spin and parity of
the ground state [57], attributed to a single, unpaired proton in the 0p3/2 orbital. Kobayashi
14

Table 1.1: Previous measurements of unbound states in 11 Li below 5 MeV. The energy
shown here is the excitation energy. When papers reported a decay energy, the two-neutron
separation energy S2n =369.15 keV [51] was added to produce the excitation energy.
Reaction
Energy (MeV)
11 Li(p,p)11 Li
1.25 ± 0.15
11 Li(p,p)11 Li
3.0 ± 0.2
11 Li(p,p)11 Li
4.9 ± 0.25
11 Li(p,p)11 Li
1.3 ± 0.1
11 Li(Pb,X)9 Li+2n
1.1
11 Li(Pb,X)9 Li+2n
1.2 ± 0.1
11 Li(Pb,X)9 Li+2n
2.4 ± 0.2
11 Li(C,X)9 Li+2n
1.24 ± 0.05
11 Li(C,X)9 Li+2n
2.45 ± 0.27
14 C(π − ,pd)11 Li
1.02 ± 0.07
14 C(π − ,pd)11 Li
2.07 ± 0.12
14 C(π − ,pd)11 Li
3.63 ± 0.13
14 C(π − ,pd)11 Li
0.92 ± 0.15
14 C(π − ,pd)11 Li
2.29 ± 0.25
14 C(π − ,pd)11 Li
3.9 ± 0.25
11 B(π − ,π + )11 Li
1.2 ± 0.1
10 Be,14 C(14 C,X)11 Li
2.47 ± 0.10
10 Be,14 C(14 C,X)11 Li
4.85 ± 0.3

Width (MeV)

0.75 ± 0.6
0.8
0.7 ± 0.2
2.1 ± 0.6
0.26 ± 0.24
2.91 ± 0.72

≈ 0.3
≈ 0.7
< 0.2
0.3 ± 0.9
1.2 ± 0.2
< 0.1

15

Ref.
Korsheninnikov96
Korsheninnikov96
Korsheninnikov96
Korsheninnikov97
Sackett93 [63]
Zinser97 [64]
Zinser97 [64]
Simon07 [55]
Simon07 [55]
Gornov98 [65]
Gornov98 [65]
Gornov98 [65]
Gurov10 [66]
Gurov10 [66]
Gurov10 [66]
Kobayashi93 [67]
Bohlen95 [68]
Bohlen95 [68]

[61]
[61]
[61]
[62]

Kobayashi (1993) [67]
Sackett (1993) [63]
Bohlen (1995) [68]
Korsheninnikov (1996) [61]
Korsheninnikov (1997) [62]
Zinser (1997) [64]
Gornov (1998) [65]
Simon (2007) [55]
Gurov (2010) [66]
0

1

2

3
4
Excitation energy (MeV)

5

6

Figure 1.10: Previous measurements of 11 Li with excitation energies less than 3.5 MeV.
et al. then measured the parallel momentum distribution of the daughter 9 Li fragment
after impinging a 11 Li beam on carbon and lead targets [22]. Expanding the targets used
to include beryllium, aluminum, and copper allowed the same group to measure an EMD
enhancement, cementing its halo status [24].
Most of the experimental work of 11 Li has focused on the properties of the bound ground
state. The ground state wave-function is known to consist of an admixture of 0p21/2 and
1s21/2 configurations for the valence neutrons (see, for example, Refs. [69–71]). This reflects
the level inversion that breaks the N = 8 magic number, which was also seen in 11,12 Be.
With a two-neutron separation energy of 369.15±0.65 keV [51], all the excited states of
11 Li

are unbound. A compilation of previous measurements of the energies and widths of

unbound states, organized by reaction mechanism, are presented in Table 1.1. A plurality of
the measurements have used a 11 Li radioactive beam on stationary targets [55,61–64] and two
16

experiments have utilized the pions from the Los Alamos Meson Physics Facility [65, 66, 72].
One measurement used a pion double charge exchange reaction [67,73] and another used both
the 10 Be(14 C,13 N)11 Li and 14 C(14 C,17 F)11 Li reactions [68]. Besides a state seen at roughly
1.1 MeV in every experiment except [68], there is little consistency. Even the possible values
for that 1.1 MeV state range from 0.92 to 1.3 keV. Most of the measured widths have large
errors bars, if the authors attempted to place any at all. A compilation of the previously
measured energies is shown in Figure 1.10.
Beyond the spectroscopy of 11 Li, considerable interest has been shown in understanding
the correlation between the two valence neutrons. Experimentally, consensus on the correlation between the two neutrons is difficult to find. Refs. [19] and [27] do agree on the average
distance between the two-neutrons in the ground state. Kobayashi et al.’s 1992 pion charge
exchange reaction showed “hints of a sequential decay process”, but had no firm conclusions
regarding the correlation [67, 73]. An experiment at Michigan State University compared
data from a Coulomb excitation to models using a phase space decay as well as a dineutron
decay and found that the data was fit with a three-body phase space decay and not with
the dineutron model [63, 74]. Shimoura et al. performed the same experiment and found
that they could fit their data with a “direct Coulomb breakup mechanism with an extended
‘dineutron’ model” [75]. Ieki et al. fired back with the first publication of a spectrum of the
cosine of the angle between the two neutrons and claimed it was fit with a phase-space decay.
Zinser et al. also used a Coulomb excitation reaction and could not fit their neutron-neutron
kinetic energy spectrum with a very simple dineutron model [64]. Gornov et al. found that
they required break-up to 9 Li and a dineutron to fit the continuum portion of their missing
mass spectrum but found it difficult to pin down a stronger conclusion because the “reaction mechanism operative in [their] pion absorption measurements is not known” [65]. This
17

assessment was completely based on decay energy spectra, with no correlation spectra presented. Nakamura et al. were able to fit their B(E1) distribution with a three-body model
that included two-neutron correlations in the initial and final states [76], which implied a
two-neutron correlation in the ground state of 11 Li. Finally, in 2007, Simon et al. found
evidence for sequential decay through 10 Li by constructing “mixed-event” spectra and seeing
an enhancement after their Coulomb excitation [55]. It is important to note that the correlation results from Refs. [55, 63, 64, 74, 76, 77] all used Coulomb excitation of a 11 Li. Since
the E1 operator only acts on the center-of-mass coordinate of the dineutron, all their correlation measurements investigated the ground-state correlations. So far, no detailed Jacobi
analysis as a function of decay energy (like those performed for 6 Be in Ref. [47] or for 14 Be
in Ref. [78]) has been performed for 11 Li.
Most of the theoretical work regarding 11 Li has been devoted to understanding the properties of the ground state and describing both 10 Li and the 11 Li ground state in a consistent
manner. There have been three main strategies for calculating these properties: large basis shell model calculations [23, 79], three-body calculations in hyper-spherical coordinates
usually achieved by solving the Faddeev equations [26, 80–84], and by using a two-particle
Green’s function [85–89]. Most calculations predict a dineutron correlation in the ground
state of 11 Li. Reproduction of the 0p1/2 and 1s1/2 percentages in the ground state has
proven sensitive to coupling to the continuum and the presence of a low-lying
10 Li

= 0 state in

[80], but not to core excitation [83].

There have been few predictions of the unbound level structure of 11 Li. One 3/2− excited
state was predicted by Aoyama et al. [90]. Garrido and collaborators predicted 13 unbound
states below 2 MeV [82]. Predictions of the correlations of neutrons beyond the bound ground
state have also been rare. In 1992, Esbensen and Bertsch predicted that the neutrons released
18

from the E1 excitation of the soft dipole mode would be emitted with an anti-correlation
in the opposite directions, as did Hagino in 2009, using the same method [86, 89]. The
data have not shown this anti-correlation though. Sackett and Ieki argued there was no
correlation while the data from Shimoura showed a correlation between the two neutrons
emitted in the decay [63, 75, 77]. As one of the two sub-systems in the Borromean nucleus
11 Li,

an understanding of 10 Li is essential for three-body calculations. The locations of

states in 10 Li are used to create the n+9 Li potential used in those calculations [18, 87, 91].
= 0 state in 10 Li significantly increased the

As an example, the inclusion of a low-lying

percentage of 2s21/2 configuration in the ground state of 11 Li in a three-body calculation by
Esbensen, Bertsch, and Hencken [87].
The 10 Li nucleus was confirmed as unbound by Poskanzer et al. in 1966 [104]. The
first measurement of the mass was performed almost a decade later by Wilcox et al. who
measured the neutron separation energy at −0.81±0.25 MeV [92]. Fifteen years later, Amelin
et al. used the 11 B(π − ,p)10 Li reaction to measure the energy of 10 Li and measured a lower
separation energy of -0.15±0.15 MeV. This figure has since been revised to approximately
-50 keV based on many measurements of a lower-lying

= 0 virtual state [52–56]. A

compilation of previous measurements of low-lying states in 10 Li is given in Table 1.2. Many
of the measured energies disagree with one another and the spins have only been tentatively
assigned by one experiment where they were based on expected doublets from coupling a
1s1/2 or 0p1/2 neutron to the unpaired 0p3/2 proton [99]. A review of the 10 Li picture
generates the following two conclusions: the ground state decays by the emission of an = 0
neutron and a state around 580 keV decay energy decays by the emission of an = 1 neutron.
This puts the ground state as a negative parity state and at least one excited positive parity
state around 580 keV decay energy.
19

Table 1.2: Previous measurements of low-lying unbound states in 10 Li under 2 MeV. The
energy shown here is the decay energy.
Reaction
Energy (MeV or fm)
9 Be(9 Be, 8 B)10 Li
0.8 ± 0.25
11 B(π − ,p)10 Li
0.15 ± 0.15
11 B(7 Li,8 B)10 Li
≥ 0.1
11 B(7 Li,8 B)10 Li
0.538 ± 0.062
11 Li(C,X)10 Li
< −20 fm
11 Li((Pb,C),X)10 Li
0.21 ± 0.05
11 Li((Pb,C),X)10 Li
0.62 ± 0.1
11 Li(p,pn)10 Li
0.42
11 Li(p,n)11 Be*
−16+4
−7 fm
11 B(π − ,p)10 Li
0.48 ± 0.1a
11 B(π − ,p)10 Li
0.1 ± 0.1
11 B(π − ,p)10 Li
0.7 ± 0.2
14 C(π − , dd)10 Li
0.4 ± 0.07
9 Be(9 Be, 8 B)10 Li
0.5 ± 0.06
18 O(9 Be,X)10 Li
< −20 fm
10 Be(12 C,12 N)10 Li
0.24 ± 0.04
9 Be(13 C,12 N)10 Li
0.53 ± 0.06
9,10 Be(13,12 C,12 N)10 Li
1.4 ± 0.08
11,12 Be(9 Be,X)10 Li
< −20 fm
9 Li(d,p)10 Li
0.35 ± 0.11b
9 Li(d,p)10 Li
0.77 ± 0.24
9 Li(d,p)10 Li
< 0.2
11 Li(C,X)10 Li
0.68 ± 0.1
11 Li(C,X)10 Li
< −40 fm
9 Li(d,p)10 Li
0.38
9 Li(d,p)10 Li
−13 to −24 fm
11 Li(C,X)10 Li
−30+12
−31 fm
11 Li(C,X)10 Li
0.51 ± 0.044
11 Li(C,X)10 Li
1.486 ± 0.088
11 Li(p,pn)10 Li
0.566 ± 0.014
11 Li(p,pn)10 Li
−22.4 ± 4.8 fm
14 C(π − ,(dd,pt))10 Li
0.7 ± 0.05
12 C(π − ,pt)10 Li
0.7 ± 0.05
12 C(π − ,dd)10 Li
0.78 ± 0.15

Width (MeV)
Ref.
1.2 ± 0.3
Wilcox75 [92]
< 0.4
0 Amelin90 [93]
< 0.23
0,1 Young94 [94]
0.358 ± 0.023 1 Young94 [94]
0 Zinser95 [95]
+0.1
0.12−0.05
0 Zinser97 [64]
0.6 ± 0.1
1 Zinser97 [64]
1 Kobayashi97 [96]
0 Shimoura98 [52]
0.5 ± 0.1
0 Gornov98 [97]
0.4 ± 0.1
0 Gornov98 [97]
0.1 ± 0.1
1 Gornov98 [97]
0.3 ± 0.07
Gornov98 [97]
0.4 ± 0.06
1 Caggiano99 [98]
0 Thoennessen99 [53]
0.1 ± 0.07
1 Bohlen99 [99]
0.35 ± 0.08
1 Bohlen99 [99]
0.2 ± 0.07
0,2 Bohlen99 [99]
0 Chen01 [100]
1 Santi03 [101]
Santi03 [101]
Santi03 [101]
0.87 ± 0.15
1 Simon04 [102]
0 Simon04 [102]
0.2
1 Jeppesen06 [54]
0 Jeppesen06 [54]
0 Simon07 [55]
0.54 ± 0.16
1 Simon07 [55]
< 2.2
Simon07 [55]
0.548 ± 0.03
1 Aksyutina08 [56]
0 Aksyutina08 [56]
< 0.2
Gurov11 [103]
< 0.2
Chernyshev13 [72]
0.5
Chernyshev13 [72]

a A single resonance at 0.48±0.1 MeV and two resonances at 0.1±0.1 and 0.7±0.2 MeV were both used
to fit the data.
b A single resonance at 0.35±0.11 MeV and a pair of resonances at 0.77±0.24 and < 0.2 MeV fit the data
equally well.

20

Wilcox (1975) [92]
Young (1994) [94]
Kobayashi (1997) [96]
Zinser (1997) [64]
Gornov (1998)a [97]
Bohlen (1999) [99]
Caggiano (1999) [98]
Santi (2003)a [101]
Simon (2004) [102]
Jeppesen (2006) [54]
Simon (2007) [55]
Aksyutina (2008) [56]
Gurov (2011) [103]
Chernyshev (2013) [72]
0

0.5

1
1.5
Decay energy (MeV)

2

2.5

Figure 1.11: Previous measurements of unbound, positive-parity states in 10 Li. Measurements reported without error have arrows for error bars. Two different measurements in
Chernyshev’s 2013 paper produced two different measurements of 0.7±0.05 and 0.78±0.15
MeV, here the weighted average is used [72].
a Both Gornov’s 1998 paper [97] and Santi’s 2003 paper [101] found two possible fits for the
data. Details are in Table 1.2 and footnotes within.

21

Some measurements have indicated another state between the negative parity ground
state and the state around 580 keV [19, 64, 99, 101], but such a state hasn’t been measured
since 2003. Figure 1.11 shows previous measurements of low-lying positive parity states.
Zinser et al. asserted that this low-lying state was an

= 0 decay based on the trend of the

first four data points in the decay energy spectrum but then used an = 1 lineshape to fit the
data, remarking “We are aware that this may not be the ideal parametrization of a virtual
s state ... but have done so to be consistent with other papers on the subject” [64]. Other
measurements in this energy region may not have seen this state because of poor statistics,
overwhelming background, or significant population of the nearby (and broad) ground state.
If there were a positive parity state around 200 keV, it and the 580 keV state would
likely correspond to the 1+ , 2+ doublet that arises from coupling a p3/2 proton with a p1/2
neutron. Only one experiment has assigned spins and parities to these states [99]. These
assignments were based on expected spins and parities due to coupling and the selectivity
of the reaction mechanisms used and assigned the states at 240 and 530 keV with spins and
parities of 1+ and 2+ , respectively. Although this experiment was not expected to populate
the wide ground state that could overshadow the low-lying positive parity state, the statistics
were quite low and had significant background.
The ground state of 9 Li has been identified as 3/2− [59] while the first excited state, at
2.69 MeV has a spin and parity of 1/2− [58–60].

1.6.2

Beryllium isotopes: A = 12, 11, 10

The isotope 12 Be is located in the same island of inversion as 11 Li [10, 17, 105–110]. The
ground state is expected to have a significant contribution from two neutrons in the 1s1/2
orbital. A low-lying 2+ state and a low-lying isomeric 0+ state confirm a nearby 1s1/2 orbital
22

3 2

6
Energy MeV

2

3 2 ,5 2

7

3 2
5 2

5
3

4
3
2

0
1 2
1 2
1
0
2

11

Be n

10

Be 2n

1
0

0
12

Be

Figure 1.12: Level diagram of states in 12,11,10 Be, relative to the ground state of 12 Be.
Neutron separation energies are from [117, 118]. Bound and resonant state energies are
from [107, 108, 111, 113–116].
and high spectroscopic overlap with the ground state of 11 Be confirms a significant contribution from a 1s21/2 configuration [10, 108, 111]. In addition to the ground state structure,
studies have been performed on the three bound states: 2+ , 0+ , and 1− [107, 108, 111]. A
0− state is expected to complete the 0− , 1− doublet from a valence neutron configuration
of 0p11/2 1s11/2 , but a bound 0− has not yet been found [112]. Very little is known about the
higher-lying states, only two of which have been observed [113–116]. Both of those observed
states have been above the one and two-neutron separation energies. A level diagram of all
the measured 12 Be states and their relation to states in 11 Be and 10 Be is shown in Figure
1.12.
The lower of the two unbound states was first measured at 4559±25 keV with a (t, p)
reaction [119]. More than 15 years later, Fortune et al. repeated the reaction and reported a resonance energy and width of 4580±5 keV and 107±17 keV, respectively [114].
Since 1994, the state has been identified in four different reactions: 10 Be(14 N,12 N)12 Be,
9 Be(12 C,9 C)12 Be, 14 C(12 C,14 O)12 Be,

and 11 Be(d, p)12 Be [120].
23

Although the state has been identified in multiple reactions, the assigned spin and parity
have changed since the first measurement and have stirred recent debate. At the time of
the first measurement, Fortune et al. assigned this state a spin and parity of 2+ based
on a comparison of angular distributions to theoretical Distorted Wave Born Approximation
(DWBA) calculations [114]. More than fifteen years later, however, Fortune and Sherr revised
that assignment to 3− based on a private communication from J.D. Millener, who suggested
that the population of the state was too strong to be a second 2+ state [121]. After that
re-assignment, Garrido et al. performed three-body calculations which suggested that the
spin and parity were 0+ , though both 1− and 3− assignments also could fit [122,123]. Those
three-body calculations were an extension of previous calculations that had been performed
for the bound structure of 12 Be [84].
Since all these unbound states are unbound to both one and two neutron decay, some
understanding of 11 Be and 10 Be is needed. As mentioned previously, 11 Be is also located
in the island of inversion. Its ground state is 1/2+ [8, 124] with a dominant configuration
of 1s1/2 for the valence neutron [125, 126]. The only bound excited state, at 320 keV, is
the 1/2− state [124], when the valence neutron is primarily in the 0p1/2 state. There have
also been measurements of several unbound states [115, 116], shown in Figure 1.12. The
ground state of 10 Be is 0+ , while the first 2+ state is located at an excitation energy of 3.37
MeV [115].

24

Chapter 2
Theoretical background
The decay of an unbound resonance is defined by a central energy and a width. Those can
be measured directly from the data, but it can be difficult if not impossible to de-convolute
the effects of resolutions and geometric acceptances to arrive at a spectrum that could be
compared to other data sets. One solution to this is to perform a simulated experiment.
This allows for the fitting of the energy and width through manipulation of an input decay
energy distribution while properly accounting for experimental resolutions and acceptances.
This chapter describes the theoretical background behind the lineshapes used to describe
the single neutron and two neutron decays measured in this experiment.

2.1

Single neutron decay

Resonant decays, like those from neutron-unbound nuclei, can be described by a Breit-Wigner
lineshape. Though the justification for the use of this lineshape takes many forms [127–
130], the single neutron decay derivation here follows the R-matrix derivation presented in
Thompson and Nunes [131]. A full derivation is beyond the scope of this document, but an
outline of the derivation will be presented here. The reader is directed to Thompson and
Nunes [131] as well as Lane and Thomas [129] for more detail.

25

2.1.1

R-matrix derivation

R-matrix theory is commonly used to describe nuclear scattering events. In this case, the one
or two-proton knockout is assumed to be a direct reaction, populating the unbound state,
which promptly decays. This is mapped onto R-matrix theory as inelastic scattering from
one entrance channel to a different exit channel. Near the end, a simplification will be made
to factor out dependence on the entrance channel.
R-matrix theory divides the calculation into two sections corresponding to physical regions: the interior and the exterior region. In the interior region, the particles interact with
each other via the strong force and in the exterior region, they do not. The line of demarcation is the “channel radius” a, chosen at a distance where this can be assumed to be true. In
the interior, the wavefunctions are constructed from the nuclear Hamiltonian. The R-matrix
basis functions are constructed to have a fixed logarithmic derivative, β, at the radius a.
The wavefunctions resulting from the diagonalization of the Hamiltonian in the R-matrix
basis are matched at the radius a with the scattering wavefunctions in the exterior region.
Here, R-matrix phenomenology is used to derive a resonance lineshape that will be fit to the
data, rather than to make a prediction based on an assumed Hamiltonian. The lineshape
will have pole energies Ep and reduced width amplitudes γpα as its fitted parameters.
The elements of the R matrix at a scattering energy E for the uncoupled case are defined
as:
P

Rα α (E) =
p=1

γpα γpα
Ep − E

(2.1)

where α indicates the entrance channel, α is the exit channel, P is the number of poles, and

26

the subscript p specifies a specific pole. The scattering S matrix is defined as:

S=

t1/2 H− − aRt1/2 (H− − βH− )

(2.2)

t1/2 H+ − aRt1/2 (H+ − βH+ )

Here, the matrix t has diagonal elements tα = 2 /2µα where µα is the reduced mass of the
+/−

channel α. H− and H+ are also diagonal matrices with elements Hα

= Gα ± iFα where

Gα and Fα are the irregular and regular solutions to the Coulomb wave equation and related
to J-type Bessel functions. These functions are

and E dependent, but the notation for

those dependencies is included in the channel subscript α for simplicity. Equation 2.2 can
be rewritten as
S = (t1/2 H+ )

1 − aR(H− /H− − β)
.
1 − aR(H+ /H+ − β)

(2.3)

Now define a diagonal ‘logarithmic’ matrix, L, in terms of H+ as well as in terms of a shift
matrix, δ, and a penetrability matrix, P:

1
L ≡ H+ /H+ − β = (δ + iP − aβ)
a

The diagonal shift and penetrability matrices have elements

δα = (F˙ α Fα + G˙ α Gα )Pα

Pα =

kα a
+ G2α

Fα2

where the dot indicates a derivative with respect ρ = kα r instead of r. Though the logarithmic matrix is defined in terms of H+ , its complex conjugate can be expressed in terms

27

of H− :
L* = H− /H− − β.
Using these definitions, the elements of the S matrix can be rewritten as

S = Ω(tH− H+ )−1/2

1 − aRL*
(tH− H+ )1/2 Ω
1 − aRL

where the matrix Ω has diagonal elements Ωα = eiφα and φα are hard-sphere phase shifts.
The product of diagonal matrices tH− H+ can be rewritten in terms of the channel velocities
v, with elements vα = kα /µα :

(tH− H+ )α =

vα a
2Pα

The cross section for scattering from channel αi to channel α is proportional not to the
˜ which is constructed with a similarity transformation
matrix S, but the symmetric matrix S,
˜ ≡ v1/2 Sv−1/2 , and simplifies to
S

˜ = Ω[1 + 2iP1/2 (1 − aRL)−1 RP1/2 ]Ω.
S

This general equation simplifies with one assumption: a single pole (energy level, designated
by E0 ) and only two channels. Two channels must be considered because the desired description is one where the entrance and exit channels have the mass partitioned differently.
This means that the final cross-section calculation will be performed with S˜12 = S˜21 :

√
1/2 √
1/2
i 2γ1 P1
2γ2 P2
 eiφ2
S˜12 = eiφ1 
(E0 − E)(1 − aR11 L1 − aR22 L2 )


28

(2.4)

The following definitions for formal width Γα , δα0 , energy shift ∆α , total energy shift ∆T ,
and total formal width ΓT can be substituted in Equation 2.4:

Γα = 2γα2 Pα

δα0 = δα − aβ
∆α = −γα2 δα0
∆T = ∆1 + ∆2 = −γ12 δ10 − γ22 δ20
ΓT = Γ1 + Γ2 = 2γ12 P1 + 2γ22 P2 .
The value of −aβ can be set at any constant value and in this work it has been set to
−aβ = δα (E0 ) as suggested by Lane and Thomas [129]. Once those definitions have been
included, the expression for S˜12 is much simpler and already looks like a Breit-Wigner:

S˜12 = eiφ1 

1/2 1/2
iΓ1 Γ2

(E0 − E + ∆T ) + iΓT /2


 eiφ2 .

(2.5)

All that remains is to substitute Equation 2.5 into the following equation for the cross-section
to go from entrance channel αi to exit channel α:
π
2Jt + 1
σααi (E) = 2
|S˜12 |2
(2I
+
1)(2I
+
1)
ki
pi
ti
Γαi Γα
π
2Jt + 1
= 2
ki (2Ipi + 1)(2Iti + 1) (E − E0 + ∆T )2 + Γ2T /4

(2.6)
(2.7)

In this equation, ki is the wavenumber of the entrance channel, Jt is the total spin of the
populated state, and Ipi and Iti are the spins of the projectile- and target-like fragments in
29

the entrance channel.
The process of interest here is the decay of an unbound state. The population mechanism
is unimportant. Equation 2.7 can be rearranged to clearly display a factorization of these
two components, the population and the subsequent decay:

σααi (E) =

(2Jt + 1)Γαi
π
ki2 (2Ipi + 1)(2Iti + 1)

Γα
(E0 − E + ∆T )2 + Γ2T /4

When all that is desired is a lineshape of the E dependence, the first grouping related to
the population can be treated as a constant. In addition, during the decay process, the
probability for decay through the entrance channel is small. As a consequence, Γα >> Γαi
and ΓT ≈ Γα and ∆T ≈ ∆α . The lineshape for the decay of a neutron-unbound state then
is:
σ(E; E0 , Γ0 ) = A

Γ (E; E0 , Γ0 )
[E0 − E + ∆ (E; E0 , Γ0 )]2 + 14 [Γ (E; E0 , Γ0 )]2

(2.8)

where the channel subscripts have been dropped and the dependences on the angular momentum of the neutron, the decay energy, the central resonance energy, and the energy-dependent
width at E0 (Γ0 ) have been explicitly stated. Γ0 , the width of the decay at E0 is used as a
substitute parameter for the reduced partial width γ 2 :

Γ0 = 2γ 2 P (E0 )

which means that Γα (or Γ in the case of Equation 2.8) is now:

Γ (E; E0 , Γ0 ) = 2P (E)

30

Γ0
2P (E0 )

.

(2.9)

2.1.2

Saturation of resonance lineshape

The energy-dependent Breit-Wigner has as its input parameters a central energy, a width
at that central energy, and an angular momentum index. One peculiar feature of just using
the lineshape and ignoring the magnitude of the prefactor A from Equation 2.8 is that as
the width Γ0 increases, the lineshape becomes saturated. This can be seen by explicitly
expanding the dependence of ∆ on Γ0 :

∆ (E; E0 , Γ0 ) = −[δ (E) − δ (E0 )]

Γ0
2P (E0 )

and referring to a similar expansion in Equation 2.9. Substituting these into Equation 2.8
produces:

σ(E; E0 , Γ0 , ) =

A
P (E)/P (E0 )
Γ0 [ E0 −E − 1 (δ (E) − δ (E ))]2 + 1 [P (E)/P (E )]2
0
0
4
Γ0
2P

where Γ0 is present only twice. Once in the denominator of the prefactor, where it has a
solely multiplicative impact, and once when dividing the quantity (E0 −E). When the width
increases, Γ0 >> E0 −E and that term becomes small. Far from the peak, the δ (E)−δ (E0 )
term is large and so the Γ0 /(E0 − E) term is negligible.
When that term can be neglected, the only effect of increasing the width of the BreitWigner distribution is to decrease the overall magnitude of the calculation. When the lineshape itself changes very little, then the sensitivity to the width Γ0 is lost. An example
of this saturation can be seen in Figure 2.1. Especially once experimental resolutions and
acceptances are applied, the difference between the Γ0 = 10.0 and Γ0 = 100.0 distributions,
for example, is small.

31

Γ0 = 0.05 MeV
Γ0 = 0.50 MeV
Γ0 = 1.00 MeV
Γ0 = 10.00 MeV
Γ0 = 100.00 MeV

0

0.5

1

1.5

2

2.5

3

Decay energy, two-body (MeV)
Figure 2.1: Energy-dependent Breit-Wigner lineshapes with varying Γ0 . For all lineshapes,
= 1 and E0 = 1.0 MeV. All lineshapes have been normalized to the same peak height.

2.2

Two neutron decay

A single-neutron, two-body decay is entirely determined by the masses of the two particles
and the decay energy of the system. The addition of a third-particle in the decay process
increases the number of degrees of freedom as well as the complexity and options of the
simulations. The three-body decays in this work were modeled by three different types of a
decays: a phase space decay, a sequential decay, and a dineutron decay. Each uses a different
method for constraining these extra degrees of freedom. Details of the implementation of
the models in the simulations can be found in Section 4.4.6.2.

32

2.2.1

Phase space decay

Just as an isotropic distribution uniformly fills the spherical surface elements, in a phase space
decay of one particle into three daughter particles, the kinematics of the decay uniformly fill
the three-body phase space. The three-body phase space integral, R3 is defined as:
3

R3 =

δ 4 (P

0

−

3

δ(Pi2 − m2i )d4 Pi

Pj )
j=1

i=1

where P0 is the four-momentum of the initial state, Pi are the four-momenta of the three
particles, and mi are the rest masses of the three particles. This description includes kinematic factors such as the conservation of energy and momentum with the first delta function.
The products of the second delta function and the infintesimals also incorporate the density
of states:

δ(Pi2 − Mi2 )d4 Pi = δ(Ei2 − (p2i + m2i ))dEi p2i dpi d(cos θi )dφi
=

p2i
dp d(cos θi )dφi
2Ei i

where pi is the magnitude of the three-momentum, Ei is the energy, and θi and φi are
the angular coordinates of the daughter particles. When the momentum of a particle is
large, there are more kinematic options for its arrangement, as reflected in the factor of
p2i /(2Ei ). Energies and angles of the three decay products are constrained by distributions
that uniformly fill this three-body phase space.

33

E1 , Γ1
E2 , Γ2
E
1
2
AZ

A−1 Z

E3 = 0
A−2 Z

Figure 2.2: A diagram of the sequential decay described.

2.2.2

Sequential decay

The sequential decay model used in this document is based on previously published work by
A. Volya [132,133]. The goal of the continuum shell model is to naturally integrate both the
continuum and shell model descriptions of the nucleus, essential for accurately describing
neutron decays which connect the two spaces. Details beyond this outline are available in
Refs. [132, 133].
A diagram of a sequential decay is shown in Figure 2.2. An unbound initial state in a
nucleus of mass A has a distribution defined by a central energy E1 and a width, Γ1 . One
neutron, with kinetic energy 1 is emitted in the decay to the intermediate state, defined by
another central energy and width pair, E2 and Γ2 . A second neutron with kinetic energy 2 is
then emitted in a decay to a bound state in the A−2 nucleus. Here, the continuum shell model
is used to calculate a distribution for the relative energy of the two neutrons Er = 1 − 2 as a
function of total decay energy E (note that E is the continuous variable indicating the total
decay energy, as opposed to E1 , which is the central value of the initial state distribution).
This explicitly comes down to calculating dσ(E)/dEr . The energy-dependent Breit-Wigner

34

approximation for the cross-section due to a single resonance is

dσ
1
dΓ(E)
∝
2
2
dEr
(E − E1 ) + ΓT (E)/4 dEr
and the total width due to a sequential decay is

ΓT (E) =

dEr

dΓ(E)
.
dEr

(2.10)

The Fermi Golden Rule gives the partial decay width distribution for the sequential
process as:
dΓ(E)
= 2πδ(E − 1 − 2 )|AT ( 1 , 2 )|2
d 1d 2

(2.11)

where AT is the total amplitude for the decay. Assuming a spin anti-symmetric pair of
neutrons, the spatial part of AT should be symmetric. Define that total amplitude as
1
AT ( 1 , 2 ) = √
2

A1 ( 1 )A2 ( 2 )
A1 ( 2 )A2 ( 1 )
+
i
i
2 − (E2 − 2 Γ2 ( 2 ))
1 − (E2 − 2 Γ2 ( 1 ))

(2.12)

where A1 and A2 are the single-particle decay amplitudes for the two decays. This assumes
that the two neutrons are in a spin-antisymmetric singlet state with a symmetric position
component. Specifically, the implementation included here assumes that the two neutrons
came from the same orbital and are coupled to a J π of 0+ .
With a substitution of E = 1 + 2 and E2 = E + S, where S is the difference between
the energy of the intermediate state and the initial energy, Equation 2.12 can be rewritten

35

as
1 A1 ( 1 )A2 ( 2 )[S + 2 − 2i Γ2 ( 1 )] + A1 ( 2 )A2 ( 1 )[S + 1 − 2i Γ2 ( 2 )]
AT ( 1 , 2 ) = √
(2.13)
2
[S + 1 − 2i Γ2 ( 2 )][S + 2 − 2i Γ2 ( 1 )]
At this point, it is useful to know the energy dependence of A1 and A2 . The energy
dependence can be understood in the following way. Γi is defined as 2π times the square of
the single-particle amplitude and it can also be defined as an energy-dependent single-particle
decay width, γ ( ), multiplied by a spectroscopic factor, Si . Put another way,

Γi = 2π|Ai ( )|2 = γ ( )Si .

(2.14)

The single-particle decay width can be approximated with the calculated width for a
neutral particle in a square well [134]. This width has a slight mass dependence, so the
decay for mass A to mass A − 1 will be slightly greater than that for the decay of A − 1 to
A − 2. In this work, however, that difference is small, so the energy dependence (in γ ) is
separable from the index dependence (in Si ). Equation 2.14 can be used then to simplify
2.13:
−1
AT ( 1 , 2 ) = √
4π

γ ( 1 )γ ( 2 )S1 S2 [2S + E −
[S + 1 − 2i Γ2 ( 2 )][S +

i
2 (Γ2 ( 1 ) + Γ2 ( 2 ))]
i
2 − 2 Γ2 ( 1 )]

Finally, substitute this into Equation 2.11 and write the differential in terms of the
relative energy Er :
2S + E − 2i (Γ2 ( 1 ) + Γ2 ( 2 ))
dΓ(E)
1
=
γ ( 1 )γ ( 2 )S1 S2
dEr
8π
[S + 1 − 2i Γ2 ( 2 )][S + 2 − 2i Γ2 ( 1 )]

2

This defines the relative energy of the two neutrons for a sequential decay.

36

(2.15)

2.2.3

Dineutron decay

Like the sequential decay theory, the dineutron decay theory is also based on work previously
published by A. Volya [132, 133]. The amplitude for the decay is:

AT ( K , I ) =

A1 ( K )A2 ( I )
i
I − (E2 − 2 Γ2 ( I ))

Instead of 1 and 2 for the energies of the neutrons, let K be the kinetic energy of the
dineutron and I be its intrinsic energy. The initial state energy, E1 is the sum of those
energies: E1 = K + I . A1 and A2 are the amplitudes of the emission of the dineutron (the
first step) and the separation of the dineutron (second step). Putting this directly in the
Fermi Golden Rule (Equation 2.11) produces

dΓ
1
Γ1 ( K )Γ2 ( I )
=
δ(E1 − K − I )
.
d Kd I
2π
( I − E2 )2 + Γ22 ( I )/4

(2.16)

Assume that both the decay of the dineutron from the fragment and the decay of the dineutron itself can be parametrized as an s-wave decay. The decay width of a neutral

= 0

particle in a square well derived by Bohr and Mottelson [134] is used then to define Γ1 and
Γ2 :
2 2
Γ1 ( K ) =
k
mD R K
2 2
Γ2 ( I ) =
k
µr I

(2.17)
(2.18)

where mD is the mass of the dineutron (2m), µ is the reduced mass (m/2), R is the channel
radius for the dineutron separating from the A − 2 daughter fragment defined by R =

37

r0 [(A − 2)1/3 + 21/3 ], r is the channel radius for the breakup of the dineutron, defined by
R = r[11/3 + 11/3 ], and r0 is the scaling factor that parametrizes that radius, here set to 1.4
fm.
The value for the “intermediate” state enegy E2 is not as obvious as in the sequential
decay. As the intrinsic energy I goes to zero though (or in this case gets very small relative
to E2 ), the denominator must be proportional to 1/as + ikI where as is the s-wave neutronneutron scattering length to satisfy the effective range approximation. Associated with this
scattering length is an energy, 0 , defined as:
2
0=

2µa2s

.

To make the denominator proportional, the width Γ2 ( I ) should be equal to −2kI as E2 ,
which defines the “intermediate” state energy E2 as:
2

E2 = −

2as
=− 0
.
µr0 as
r0

Using the definition for 0 and the following definitions for I and K ,
2k2
K
=
K
2mD
2k2
I
I =
2µ

38

Equations 2.18 and 2.17 can be re-written as

|as | √
Γ1 ( K ) = 2
0 K
R
|as | √
Γ2 ( I ) = 4
0 I
r

(2.19)
(2.20)

Substituting Equations 2.19 and 2.20 into Equation 2.16 produces:

dΓ(E1 )
1
=
dI
π

(E1 − 1 ) 1
r

1 + 2asI
0

2
0

+ 1

r
R

which defines the distribution of the intrinsic energy of the dineutron.

39

Chapter 3
Experimental techniques

3.1

Beam production

The experiment was performed at the National Superconducting Cyclotron Laboratory
(NSCL) in July and August of 2012. A 13 B beam was provided by the Coupled Cyclotron
Facility (CCF) [135] and the A1900 Fragment Separator [136] using the process of projectile
fragmentation [137]. A schematic of the CCF and A1900 is shown in Figure 3.1. Ions of
18 O3+

were injected into the K500 cyclotron, where they were accelerated to an energy of

10.91 MeV/u. They then traveled to the K1200 cyclotron where they immediately passed
through a stripper foil that stripped off the rest of their electrons. These fully ionized nuclei
were accelerated through the K1200 and emerged fully stripped at an energy of 120 MeV/u.
After the K1200 cyclotron, the oxygen beam impinged on a Be production target (2491
mg/cm2 ), creating a wide variety of fragments. All these fragments entered the A1900
Fragment Separator, which was set to the magnetic rigidity of the desired 13 B secondary
beam to filter out most other fragments. A 300 mg/cm2 aluminum wedge was included
at the dispersive midplane of the A1900 to separate nuclei of different charge that entered
the wedge with the same magnetic rigidity. Energy loss through the aluminum wedge is
proportional to Z 2 where Z is the charge of the nucleus. Different elements that entered
the wedge with the same magnetic rigidity left with different rigidities, allowing them to be
filtered out by the magnets after the wedge. The 13 B beam was delivered to the experimental
40

13 B

(71 MeV/u)

K500
or

(10.91 MeV/u)

on

s

18 O3+

t
ara

tr

ep

Cy

clo

tS
en

m

pl

ed

rag
F
0

Co
u

90

A1

Aluminum wedge
(300 mg/cm2 )
K1200

Be production target
(2491 mg/cm2 )
18 O8+

(120 MeV/u)

Figure 3.1: Schematic of the Coupled Cyclotron Facility and A1900 Fragment Separator
with the production setting for this experiment.

41

vault with an energy of 71 MeV/u, corresponding to a magnetic rigidity of 3.2271 Tm.
In addition to the 13 B beam, the CCF and A1900 also provided a 12 B beam. The primary
beam, target, and degrader were all the same in the production of the 12 B beam, but the
magnet settings were different. The 12 B beam arrived at the experimental vault with an
energy of 71 MeV/u and a magnetic rigidity of 2.97720 Tm.

3.2

Beam detector and target

As it left the A1900, the beam passed through a timing scintillator (named the A1900 scintillator) that was optically coupled to a photomultiplier tube (PMT). Ten meters downstream,
upon entering the experimental area (shown in Figure 3.2), the beam hit a thin plastic timing
scintillator, called the target scintillator. The 0.254 mm thick target scintillator was made of
BC-404 and optically coupled to one PMT. At a location 1.04 m downstream of the timing
scintillator, the beam impinged upon a 51.37 mg/cm2 beryllium target. Such a thin target
was able to be used because of the high beam rate, which was approximately 800,000 Hz on
target.

3.3

Sweeper magnet

Unreacted beam, charged reaction fragments and neutrons leaving the target entered the
Sweeper magnet [138]. This large-gap dipole magnet had a bending angle of 43.3 degrees
and a radius of 1 meter. The Sweeper magnet bent the trajectories of the unreacted beam
and charged reaction fragments into a suite of charged particle detectors while the neutrons,
unaffected by the magnetic field, continued straight through a vertical gap of 14 cm into the
neutron detectors MoNA and LISA, detailed later in Section 3.5. The maximum magnetic
42

MoNA-LISA

x
z
y

Be target
Ion chamber
CsI(Na) array
x
Sweeper
CRDCs
z
magnet Blocker Thin scintillator y

Target scintillator
r
lato
l
i
t
in
0 sc
0
0
9
190
A1
A
rom
mf
a
e
B

Figure 3.2: Schematic of the experimental area.
rigidity for the Sweeper magnet is 4 Tm. For reactions with the 13 B beam, the magnetic
rigidity of the Sweeper was set to 3.76 Tm. For reactions with the 12 B beam, it was set to
3.71 Tm. Both settings were set to center the 9 Li fragments in the detectors and minimize
the amount of unreacted beam interacting in CRDC1.

3.4

Charged particle detectors

After the Sweeper magnet was a large vacuum box containing a suite of detectors for the
charged fragments. The positions of the particles was measured with Cathode Readout Drift
Counters (CRDCs), the energy loss was measured in an ion chamber, the time of flight was
measured with a plastic timing scintillator, and the total remaining energy was measured
with a CsI(Na) scintillator array.

43

3.4.1

Blocker

A 25.4 cm x 13 cm x 9.5 mm tungsten blocker was located at the entrance to the Sweeper
detector box. In this experiment it sat just off the entrance of the beamline into the box,
not blocking the active area of any detector. Most of the unreacted beam was bent into the
blocker.

3.4.2

Cathode Readout Drift Counters

Two CRDCs were positioned along the beam line. The first CRDC was positioned immediately after the blocker (1.72 m from the target) and the second was located 1.55 m
downstream from the first. The active area of each CRDC was 30 cm x 30 cm in the
xy-plane. Each detector was filled with a gas mix of 20% isobutane and 80% CF4 at an
operating pressure of 300 Torr.
Each CRDC operated similarly to a two-dimensional ion chamber optimized for position.
A schematic of the operation is shown in Figure 3.3. The passage of charged particles
through the detector created ionization pairs in the gas, which drifted apart due to an
applied electric field. In this experiment, the field was created by the application of a 1000 V
drift voltage between a plate at the top of the detector and the Frisch grid near the bottom
of the detector. Field shaping wires were placed at regular intervals on the two faces of
the detector and parallel to the x-direction. Above the Frisch grid was a charged anode
wire (parallel to the x-axis) and a series of 116 aluminum cathode pads with a pitch of 2.54
mm in width, segmented along the x-axis. Once the electrons drifted through the Frisch
grid, they felt the strong field created by the anode wire. This field caused an avalanche of
electrons, increasing the number available for detection. The y-position of the particle was

44

Cathode Pads

Anode Wire
e− Drift
Frisch Grid

Gas Ionization
-E
Charged Fragment Track

y
z
x

Figure 3.3: Schematic of CRDC operation modified from [139] where the z-direction has
been expanded for detail. The field shaping wires mentioned in the text are not included for
visibility and simplicity.

45

y
z
x

-E
Gas Ionization

e− Drift

ds

o

cti

e
arg
h
C

le
col

a
np

Figure 3.4: Schematic of ion chamber.
determined by the drift time of the electron, measured as the time difference between the
thin timing scintillator near the end of the beam line and the detection of charge along the
anode wire. The x-position of particle was determined with induced cathode readout [140].
In this method, a charge distribution is induced upon the cathode pads by the avalanche
electrons. The peak of this distribution gives the x-position of the particle.

3.4.3

Ion chamber

A schematic of the ion chamber is shown in Figure 3.4. The active area of the ion chamber
was 40 cm x 40 cm and its length was 65 cm. It was filled with P-10 gas (90% argon and 10%

46

methane) at 300 Torr. Windows made of Kevlar filament, 12 µm PPTA (an aramid plastic),
and epoxy allowed the particles to pass in and out of the detector with little energy loss.
The upstream window had an area of 30 cm x 30 cm to match the geometric acceptance of
the second CRDC, while the downstream window had an area of 40 cm x 40 cm, to allow for
the spread of the beam. A large plate at the top of the ion chamber and 16 charge collection
pads at the bottom were biased to create a drift voltage of 10 V. As charged particles passed
through, they created ion pairs. The electrons drifted down and were collected on the 16
collection pads which discretized the z-direction and extended along the x-direction. The
charge collected on all 16 pads was used as a measurement of the energy loss in the ion
chamber.

3.4.4

Timing scintillator

A thin timing scintillator sat immediately after the ion chamber to serve as the charged
fragment trigger as well as to measure the time of flight of the particles. The 55 cm x 55
cm x 5 mm scintillator was made of EJ-204, an equivalent to BC-404, both types of plastic
from Eljen and Bicron, respectively. Two trapezoidal light guides were attached to the top
edge of the scintillator and another two were attached on the bottom. Each light guide was
optically coupled to the thin scintillator and a PMT as shown in Figure 3.5.
Organic scintillators such as this one take advantage of the π-electron structure of the
dopant molecules within the plastic. The electronic energy levels consist of singlet states
and vibrational states which are associated with those singlet states. The spacing between
vibrational states that correspond to a singlet state is much smaller than the spacing between
singlet states. At room temperature, due to thermalization, most of the molecules are in
the ground state. As a charged particle passes through, kinetic energy from the passage
47

PMT 0

PMT 2
y
x

z

light guides
scintillator

PMT 1

PMT 3

Figure 3.5: A schematic of the thin scintillator, as seen from upstream of the detector.
of the particle is absorbed by the molecules, exciting them. Higher excitations are quickly
de-excited to the first excited singlet state through internal conversion. Any vibrational
energy is thermalized and most of the affected molecules are now in the first excited singlet
state. This state then decays and emits scintillation light to any of the states associated
with the ground singlet state. This light propagates to the light guides on either end of the
detector. For a timing scintillator, the decay constant associated with this decay should be
small - for EJ-204 this constant is 1.8 ns.

3.4.5

CsI(Na) array

At the end of the beamline, charged fragments were stopped in a CsI(Na) array. A mechanical
design of the array is shown in Figure 3.6. The array consists of twenty-five 3.25” x 3.25”
x 2.16” CsI(Na) crystals. There were five assemblies, each of which consisted of one row of
five crystals. When these assemblies were stacked, it produced a 5x5 array. Since the array
was centered on the beamline, the cross-sectional area seen by the beam was 16.25” x 16.25”

48

Figure 3.6: A mechanical design drawing of the CsI(Na) array.
or 41 cm x 41 cm. For optical separation and to maximize internal reflection, each crystal
was wrapped in reflective material 0.2 mm thick. Each crystal was optically coupled on its
downstream side to a Hamamatsu PMT R1307 with magnetic shielding. Each PMT was
powered by a voltage divider which also provided a single signal output.
Each assembly was fitted into a stainless steel frame which was bolted to a stainless steel
support structure. This structure sat on the rails of the Sweeper detector box, enabling
the CsI(Na) array to move along the full length of the box. The crystals themselves were
hygroscopic and easily damaged by moisture in the air. To deal with this, a four sided gas
cover surrounded the crystals. When the thin scintillator was pushed up against the cover, it
created a five sided box. A gas inlet allowed the flow of dry air along the face of the crystals
without requiring the dry air to fill a large space. This air flowed over the face of the crystals
49

PMTs
light guides

aluminum foil

plastic scintillator
black plastic
Figure 3.7: A schematic of a MoNA LISA bar.
whenever the detector box was not under vacuum.

3.5

MoNA-LISA

Both the Modular Neutron Array (MoNA) and the Large-area multi-Institutional Scintillator
Array (LISA) consisted of 144 2 m x 10 cm x 10 cm plastic scintillator bars. The MoNA bars
were BC-408 and the LISA bars were EJ-200. Each bar (shown in Figure 3.7) was wrapped
in reflective material to reduce light lost as well as black plastic to prevent ambient light
from entering the detector. When a nuclear interaction occured between the neutron and
the hydrogen or carbon atoms in the plastic, the kinetic energy of that hydrogen or carbon
atom was converted to prompt fluorescence in the same manner as described in Section 3.4.4.
Because the neutron to hydrogen mass ratio is so much larger than the neutron to carbon
mass ratio and the recoil is therefore much greater, most of the light output came from

50

neutron on hydrogen reactions. The emitted light was funneled to the ends of the bars via
total internal reflection and reflection off the foil wrapping. A shielded PMT at each end
of the bar collected the light and turned it into an electronic signal. MoNA used Photonis
XP2262/B PMTs and LISA used Hamamatsu R329-02 PMTs.

3.6

Electronics and Data Acquisition

The electronics and data acquisition (DAQ) systems have been described in detail in [139,
141,142], but a short overview is provided here and an abbreviated schematic of the systems
is shown in Figure 3.8.
MoNA, LISA, and Sweeper each operated as independent data acquisition subsystems
connected by a “Level 3” system that generated a system trigger and “timestamp” that was
relayed to each system. The system trigger in this experiment was generated solely based on
the arrival of a signal in the left upper (0) PMT of the thin scintillator. The “timestamp”
was a unique 64-bit word generated by a clock in the “Level 3” system. Upon the generation
of a system trigger, that trigger and the “timestamp” were relayed to the three subsystems,
which proceeded to read out the data independently.
MoNA and LISA had independent but identical electronics set-ups. Each PMT had two
outputs, an anode and a dynode. The negative signal from the anode was used for a local
trigger and for timing. The positive signal from the dynode was used to measure the charge
collected in each PMT. The timing and triggering branch signal went from the anode to a
constant fraction discriminator (CFD). Two outputs from the CFD went to a time-to-digital
converter (TDC) and to a Xilinx Logic Module (XLM) for the subsystem trigger logic.
The TDC modules were operated in common stop mode, where a signal from the detector

51

Sweeper trigger

Level 3 XLM
system trigger gate

Sweeper system
CFD

MoNA-LISA systems
TDC
Level 2 XLM

thin PMTs
ADC

Level 1 XLM
CFD

TDC

TDC

target PMT
ADC
A1900 PMT

CFD

CRDC anode
CRDC pads
ion chamber pads

QDC

CFD

TDC

MoNA-LISA PMTs

ADC
FEE

XLM

stop signal
start signal
gate

ADC

module accessed by DAQ
detector signal

CsI(Na) PMTs

ADC

Figure 3.8: Schematic of the MoNA-LISA-Sweeper electronics set-up.

52

triggered the start of measurement and the System 3 trigger ended the measurement. The
charge branch signal went from the dynode through a passive inverter to a charge-to-digital
converter (QDC) module. This process was duplicated for all 288 PMTs in each subsystem.
Charge and timing information for each PMT was read out by the DAQ computer directly
from the QDCs and TDCs.
The electronics set-up for the Sweeper included electronics for each of the three timing
scintillators, CRDCs, ion chamber and CsI(Na) array. The A1900 timing scintillator signal
was sent to a time-to-amplitude converter (TAC). The output from the TAC was then sent to
an amplitude-to-digital converter (ADC). The output from the PMT attached to the target
scintillator as well as the output from the four PMTs attached to the downstream thin
scintillator were sent through a CFD into a TDC. All TDCs in the Sweeper DAQ operated
in common start mode, where the Level 3 trigger began the measurement and the signal
from each individual PMT stopped the measurement. The pad signals from the CRDCs
were digitized by Front-End-Electronics (FEE) modules which sampled the pad pulse and
sent the information into an XLM for readout. The signal from each pad in the ion chamber
was sent through a shaper and into an ADC. The output from the four PMTs attached to
the thin scintillator were split to allow for the time measurement described previously and
a charge measurement by delivering the signals directly to a QDC. The signals from each
PMT in the CsI(Na) array were sent through a shaping amplifier and into an ADC. Each
time the system was triggered, all ADCs, TDCs, QDCs, and XLMs were read out by the
Sweeper DAQ computer.

53

3.7

Invariant mass spectroscopy

Neutron unbound states typically have decay widths on the order of 10-1000 keV with mean
lifetimes ranging from 7×10−20 to 7×10−22 seconds. At a beam velocity of approximately
11 cm/ns, these nuclei travel 7000-70 fm before decaying. In comparison, the radius of the
9 Be

target nucleus is approximately 2.5 fm and the target was 278 µm thick. The short

time scale makes direct measurement impossible. Invariant mass spectroscopy was used to
reconstruct the energy released in the decay, also known as the decay energy.
Assume particle A decays into m daughter particles. Momentum and energy must be
conserved and the cleanest way to do this is by utilizing four-vectors. We equate the fourµ

vector of the original particle, PA , with the sum of the four-vectors of the decay products,
µ

Pi and then can calculate the invariant mass MA of the original particle
m
µ
PA

=

µ

Pi

i=1
m m
µ

µ

(PA )2 =
2
MA

=

Pi Pjµ
i=1 j=1
m
Mi2
i=1

m−1

(Ei Ej − 2pi · pj )

+2

i=1 j=i+1

m

m−1

Mi2

MA =

m

i=1

m

(Ei Ej − 2pi · pj )

+2

i=1 j=i+1

where Mi,j are the masses of the decay products and Ei,j and pi,j are the energies and
three-momenta of the decay products.
The decay energy is defined as the difference in mass between the initial and final prod-

54

ucts, so we subtract the final product masses to get the equation for the decay energy:

m

m−1

Mi2

Edecay =
i=1

m

m

(Ei Ej − pi · pj ) −

+2

i=1 j=i+1

Mi

(3.1)

i=1

For a single neutron decay, this simplifies to:

Edecay =

Mf2 + Mn2 + 2Ef En − 2pf · pn − Mf − Mn .

For a two-neutron decay, Equation 3.1 reduces to:

Edecay =

MT2 + 2(ET2 − p2T ) − Mf − 2Mn .

where
MT2 = Mf2 + 2Mn2
ET2 = Ef En1 + Ef En2 + En1 En2
and
p2T = pf · pn1 + pf · pn2 + pn1 · pn2 .
It is important to note that when there are more than two decay products, the invariant
mass and decay energy are accurate calculations regardless of decay mechanism (sequential,
democratic, etc.).

55

m1

m1
X12
m2
m2

x1

m3

Y123
m3

x2
x3

Figure 3.9: Three possible coordinate definitions for a three-body system.

3.8

Particle correlation techniques

The mode of decay is primarily illuminated in this work by observing correlations between
the particles through Jacobi and Dalitz plots.

3.8.1

Jacobi coordinates

To construct the Jacobi coordinates, we begin with a three body system, as illustrated on
the left side of Figure 3.9. Each mass mn has a coordinate vector xn . By eliminating the
center of mass motion, the system can be redefined with only two relative coordinate vectors
Xij and Y ijk (shown on the right side of Figure 3.9). The first Jacobi coordinate, X is the
vector between two particles. The second vector, Y , begins at the third particle and ends at
the center of mass of the system defined by the first two particles. Mathematically,

Xij = xj − xi

Yijk =

mi xi + mj xj
− xk
mi + mj

where i, j, and k are the indices of particles. An example of this in the general three-body
case is shown on the right side of Figure 3.9.

56

ky

−kx

θk

kx

X

kx

−ky
Y

Y

X

θk

ky
−kx

−ky
T system

Y system

Figure 3.10: T and Y systems and associated Jacobi coordinates for fragment+n + n system.
The rotation of particles through the three indices gives us three different coordinate
systems: X12 and Y123 , X23 and Y231 , and X31 and Y312 . For three-body systems with two
identical particles, these three systems, shown in Figure 3.10, are called the T system and
the Y systems. The momentum vectors associated with X and Y are:

kx =

ky =

mj ki − mi kj
mi + mj

mk (ki + kj ) − (mi + mj )kk
mi + mj + mk

where ki is the momentum vector of particle i.
The specific Jacobi variables used in this analysis are the cosine of the angle θ between
the vectors kx and ky and the ratio of the energies (Ex /ET ) of the two-body system defined
by X and the total three body system:

Ex =

(mi + mj )kx2
2mi mj

cos θk =

57

kx · ky
kx ky

These variables can be used in one and two-dimensional plots for analysis purposes. Although each system (T and Y) fully defines the correlations between the particles, sometimes
it is easier to see specific correlations in one or the other. Some example Jacobi plots for
phase-space, sequential, and dineutron decays calculated using the models described in Section 2.2 are shown in Figure 3.11. For the T-system, the energy ratio shows how the energy
is split between the decay of the neutron pair and the decay between that two-body system
and the core. For a dineutron decay, this would show the energy ratio between the dineutron
scattering state and the total decay energy. A large total decay energy would skew the system
toward the “zero” end of the energy ratio spectrum, while a small decay energy would skew
the plot toward the “one” end of the spectrum. The energy ratio, Ex /ET , in the Y-system
is ideal for showing how the energy is shared between the neutrons. Unequal sharing of the
energy is indicative of a sequential decay through a narrow state that is much closer to the
initial(final) state than it is to the final(initial) state. In Figure 3.11, the equally shared
sequential decay line has shoulders which indicate unequal sharing of the energy between
the neutrons. These shoulders are correlated with a long, high-energy tail in the three-body
decay energy distribution. Though the intermediate central energy is half the initial state’s
central energy, neutrons decaying from that high-energy tail are more likely to share the total
energy unequally. The cos(θ) plot for the T-system should, by definition, be symmetric and
can also discriminate between types of sequential decays. The cos(θ) plot for the Y-system
shows how correlated the neutrons are in space: if the angle is large (and cos(θ) ∼ −1), the
system has a dineutron-like structure and if the angle is small (and cos(θ) ∼ 1), the system
has a cigar-like structure, with the neutrons on either side of the core.

58

0

0.2

0.4
0.6
0.8
Ex /ET - T system

1

0

0.2

0.4
0.6
0.8
Ex /ET - Y system

1

Phase space
Dineutron
Sequential, even
Sequential, uneven

-1

-0.5
0
0.5
cos θ - T system

1

-1

-0.5
0
0.5
cos θ - Y system

1

Figure 3.11: Example Jacobi plots for a phase space decay (black), a dineutron decay (red),
and a sequential decay (blue). The left column shows Jacobi coordinates from the T-system
and the right column shows Jacobi coordinates from the Y-system. The energy ratios are
shown on the top row and the angular distributions are shown in the bottom row. All
distributions are without any acceptances or resolutions applied. In the cos θ-T system plot,
the dineutron and phase space lines overlap.

59

3.8.2

Dalitz plots

Dalitz plots are two-dimensional graphs commonly used in high-energy physics to study
correlations in three-body decays. Introduced by R.H. Dalitz in the early 1950s [35, 36],
their primary feature is that they are uniformly filled for a simple phase-space decay. Any
correlations or resonances between the three bodies will show up as an enhancement within
a uniformly filled section.
The plots utilize the fact that two coordinates define the kinematics of a three-body
system (as in the Jacobi systems). When Dalitz introduced the plots, he chose two variables
constructed from kinetic energies of the particles [35]. Another option is to plot pairs of
squared invariant masses against each other. If mass A decays to masses i, j, and k, the axes
of the Dalitz plot would be m2ij and m2jk . This approach is more commonly used now than
the kinetic energy coordinates used by Dalitz.
The boundary of the uniformly filled area is defined by the kinematics and decay energy
of the system. In Dalitz’s original paper, the boundary was circular [35]. When invariant
masses are used, the shape more often resembles a rounded, skewed triangle. Different total
decay energies lead to different boundaries for the Dalitz plots. When a wide decay energy
range is considered, though, these different boundaries reduces the efficacy of the plot. To
restore the uniformity of the filled area, Marqu´es and collaborators introduced a normalized
invariant mass [37]:
m2ij

=

2 − (m + m )2
Mij
i
j

(mi + mj + Ed )2 − (mi + mj )2

2 is the squared invariant
where mi,j,k are the masses of the individual decay products, Mi,j

mass of the two-body system i, j, and Ed is the decay energy. This normalization creates
two coordinates that range from 0 to 1 and which create a circular boundary when plotted

60

1

0.8

0.8

0.6

0.6

2
Mcn

2
Mcn

1

0.4

0.4

0.2

0.2

0

0

0.2

0.4
0.6
2
Mnn

0.8

0

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0.2

0.4
0.6
2
Mnn

0.8

1

0

0.2

0.4
0.6
2
Mnn

0.8

1

2
Mcn

2
Mcn

1

0

0

0.2

0.4
0.6
2
Mnn

0.8

0

1

Figure 3.12: Example Dalitz plots for a phase space decay (upper left), a dineutron decay
(upper right), and two sequential decays (bottom row). The sequential decay in the bottom
left panel shows a decay through an intermediate state approximately halfway between the
initial and final states, while the sequential decay shown in the bottom right has a more
asymmetric energy distribution. All distributions are without any acceptances or resolutions
applied.

61

against each other.
Just like the Jacobi plots, the Dalitz plots can be used in two-dimensional form or
one-dimensional form, where the one-dimensional plots are simply projections of the twodimensional plot. Some example two-dimensional plots are shown in Figure 3.12. The
physical intuition and understanding that can be gained from these plots is similar to that
from the Jacobi plots. A dineutron-like correlation between the neutrons will produce an enhancement near m2nn = 0 and m2cn ≈ 0.5 that will steadily decrease as m2nn → 1. The decay
through a narrow resonance in an intermediate system will produce a pair of enhancements
in m2cn that will show up as bands in the two-dimensional plot. The location and widths of
those bands will be indicative of the energies released in the two sequential decays and their
widths.

62

Chapter 4
Data Analysis
This chapter discusses the steps taken to transform electronic signals to a set of meaningful
data spectra and the simulation used to interpret those spectra.

4.1

Calibrations and corrections

The electronic signals recorded in the data stream were extracted, calibrated and corrected
before they were used in calculations of any sort. This section details the calibration of the
individual detectors.

4.1.1

Charged particle calibrations and corrections

4.1.1.1

CRDCs

The signals from the CRDCs went through a significant amount of processing on their way
to producing a position spectrum. The raw signals included samples of charge found on each
pad, the total charge collected on the anode wire, and the drift time of the electron. The
drift time was measured as the time difference between the collection of charge on the anode
and the arrival of the particle downstream in the thin scintillator. The samples on each pad
were combined to give a raw total charge for each pad and the distribution of charge over
the pads was fit to give a pad position of the particle’s interaction location. Finally, this pad
position was converted to an x-position and the drift time was converted to a y-position.
63

104
102
Counts

Counts

103
102

10

10
1

1
0

200 400 600 800 1000
CRDC2 pad 64 raw charge (arb)

0

200 400 600 800 1000
CRDC2 pad 24 raw charge (arb)

Figure 4.1: Examples of total charge spectra for two pads in CRDC2. Pad 64 (left) is an
example of a good total charge spectrum and pad 24 (right) is an example of a bad total
charge spectrum.
Certain pads in the CRDCs displayed abnormal behavior. For a good position spectrum,
the behavior of the pads should be as uniform as possible across the entire detector. Pads with
abnormal charge collection, identifiable by comparing raw charge spectra, were eliminated
from further calibrations, corrections, and calculations. Examples of charge spectra from
good and bad pads are shown in Figure 4.1. Only one pad, pad 69, was identified as bad in
CRDC1, while three, pads 24, 73, and 89, were identified as bad in CRDC2.
Instead of integrating the entire charge collection, each pad in the CRDCs took 0-4
samples in a Riemann sum approximation to the full integral (Figure 4.2). The samples
were counted and then summed as an approximation of the total charge deposited on that
pad.
In this experiment, the number of samples collected varied regularly as a function of
y-position for both CRDCs (Figure 4.3). More specifically, for events near evenly spaced

64

Pad charge

Pad charge

Time

Time

Figure 4.2: Example charge distribution on CRDC pads as a function of time. A full charge
integration measurement would integrate the shaded area on the left. The sampling method
adds the values of a discrete number of points, as shown on the right.

150

150

100

105

50

104

0

103

-50

102

-100

10

-150

1

2
3
4
Number of samples

CRDC2 y-position (mm)

CRDC1 y-position (mm)

106

1

100

105

50

104

0

103

-50

102

-100

10

-150

1

2
3
4
Number of samples

1

Figure 4.3: CRDC y-position versus number of samples collected for each pad (left: CRDC1,
right: CRDC2). Both plots are shown with the z-axis on a log scale. Each CRDC collects
fewer samples at certain regularly spaced intervals along the y-direction, visible in these plots
as enhancements at lower numbers of samples and vacancies at the spectra for four samples.

65

y-positions (approximately 65 mm spacing), the number of samples collected for each pad
was less than four. This spacing was the same as the y-spacing of the electric field shaping
wires along the faces of the detector. To correct for this, the total charge collected on each
pad was normalized by the number of samples collected by that pad for a given event. This
allowed accurate comparison of the total charge collected in each CRDC, known as the pad
sum.
After the raw signal from each pad had been normalized to the number of samples, the
next step was to subtract the pedestal, the small current that always ran through the cathode
pads of the CRDCs. When a charged particle traveled through the detector, the charge read
out by the electronics was the sum of the pedestal and the charge generated by the particle’s
passage.
To measure the pedestal value of each CRDC pad, the charge in each pad was measured
when no particles were passing through. The distribution of charge collected on the pads
was normalized to the number of samples and then fit with a Gaussian lineshape to extract
the centroid - the pedestal value.
Event by event and pad by pad, the total charge on each pad was determined with the
following equation:
Qpad =

nsamples

1
nsamples

qi − qped
i=0

where Qpad was the total charge deposited on the pad, nsamples was the number of samples
on that pad for that event, qi were the individual samples collected and qped was the measured
pedestal charge.
Once the total charge collected on each pad was calculated, the charge distribution across
all the pads was used to find the pad position. A Gaussian lineshape was fit to each distri-

66

Charge collected (arb)

500
400
300
200
100
0

15

20

25

30

35 40 45
CRDC pad

50

55

60

Figure 4.4: An example of a charge distribution fit for CRDC2. The centroid of the Gaussian
fit, shown here by the vertical line, would be used as the position of the particle in this
detector.
bution, as shown in Figure 4.4. The central value of the fit was taken as the position of the
particle relative to the set of pads.
The final step was to convert this pad position to an x-position in the lab frame and
convert the drift time to a y-position. Both of these conversions were performed at the same
time by utilizing the tungsten masks in front of each detector and a 17 N beam. The masks
had a known pattern of holes and lines. When the beam passed through each hole, that
known location was associated with a pad position and drift time. With enough of these
holes and lines illuminated, linear transformations from pad position to x-position and drift
time to y-position were calculated. An example of the data collected for this calibration is
shown in Figure 4.5.
A schematic of the masks and their relations to the lab frame, beam direction, and pad
numbers is shown in Figure 4.6. The magnitude of the slope of the x-position linear fit was

67

CRDC2 y-position (mm)

150
100
50
0
-50
-100
-150
-150 -100 -50
0
50 100
CRDC2 x-position (mm)

150

Figure 4.5: Left: mechanical design drawing of the CRDC mask. Right: Example of CRDC
mask data from run 157.

CRDC1
y
x
CRDC2
z

bea

m

115
0

r
umbe
pad n

12
127

umbe
pad n

r

Figure 4.6: A schematic of the CRDC masks and their relation to the lab coordinate frame,
the beam, and the pads in each detector.

68

Table 4.1: CRDC1 mask calibration slopes and offsets from all CRDC1 mask runs. Because
there were so few holes in the y-direction (typically only 3), errors were unreasonably small.
Slopes and offsets have been truncated at the same significance as their CRDC2 counterparts
(Table 4.2).
CRDC1 mask run
25
96
156
240
average

x slope (mm/pad) x offset (mm) y slope (mm/ns)
2.54
-178.0 ± 0.2
-0.19
2.54
-178.7 ± 0.2
-0.19
2.54
-178.5 ± 0.1
-0.19
2.54
-178.7 ± 0.2
-0.19
2.54
-178.49 ± 0.08
-0.19

y offset (mm)
101
99
101
101
101

Table 4.2: CRDC2 mask calibration slopes and offsets from all CRDC2 mask runs.
CRDC2 mask run
26
97
157
241
average

x slope (mm/pad)
-2.54
-2.54
-2.54
-2.54
-2.54

x offset (mm) y slope (mm/ns)
186.8 ± 0.2
-0.19 ± 0.01
186.2 ± 0.2
-0.19 ± 0.01
186.7 ± 0.1
-0.19 ± 0.02
187.0 ± 0.2
-0.19 ± 0.02
186.68 ± 0.08
-0.19 ± 0.01

y offset (mm)
113 ± 2
112 ± 3
112 ± 3
112 ± 3
112 ± 1

defined by the pitch of the pads - 1/10”. The sign of the slope depended on the orientation
of the pad electronics - CRDC1 had a positive slope, while CRDC2 had a negative slope.
Since the collected electrons drifted upward to the anode of the detector, the sign of the
y-position slope was negative. An additional offset was included in the y-direction to place
the central path of the beam along the y = 0 axis. The offsets for the x-position fit were
determined by the mask.
The masks were located 45 mm in front of the detector [141]. Because of this z-offset,
particles going through the same hole at different angles could pass through the detector
at different positions. This effect was corrected by iterating the calibration process. After
performing position calibrations for both CRDCs (and assuming no z-offset), the angle of
the particle was calculated and the position calibrations were performed again, this time
accounting for the z-offset. This process was repeated until the slopes and offsets converged.

69

Mask data was collected approximately every 48 hours to account for possible drifts in
the detectors. The final slopes and offsets from each pair of calibration runs are shown in
Tables 4.1 and 4.2. Since the spread was fairly small and exhibited no drift in positions for
the production data, an average was used for each slope and offset (included in Tables 4.1
and 4.2).

4.1.1.2

Ion chamber

The ion chamber was segmented in 16 pads along the beam direction, each of which registered
a charge collected for each event. While the charge collected in the ion chamber can be used
almost without any calibrations, the data can be cleaned up significantly by removing bad
pads, gain-matching the remaining pads, and applying position and time corrections.
Pad 1, the second pad from the front, was removed from further calibrations and calculations because it registered almost a factor of 5 less charge than any other pad during the
experiment.
Approximating the ion chamber as 65 cm of argon gas (instead of the P-10 mixture it
was), the incoming 13 B beam should have lost approximately 36 MeV in the first half of the
detector and 37 MeV in the second half of the detector. This indicates that the beam did
not slow down significantly as it passed through the detector and that each pad should have
collected the same amount of charge.
To gain match the ion chamber pads, the beam was sent down the center of the ion
chamber and the charge on each pad was measured. One pad, pad 8, was chosen as the ideal
pad to which all others would be gain matched. Each charge distribution was fitted with a
Gaussian (see Figure 4.7). Those centroids, qpad , were then compared to calculate a slope,

70

×103

30

30

25

25

20
15

20
15

10

10

5

5

0

0

×103

35

Counts

Counts

35

0

20 40 60 80 100 120 140
Ion chamber pad 4 charge (arb)

0

20 40 60 80 100 120 140
Ion chamber pad 4 charge (arb)

Figure 4.7: An example of fitting used in gain-matching the ion chamber. Left: points show
raw charge signal from pad 4, thick solid line is Gaussian fit to data, and thin vertical line
indicates centroid of “ideal” pad 8. Right: points show gain-matched charge signal from pad
4 and thin vertical line indicates centroid of “ideal” pad 8.
Table 4.3: Ion chamber position and time correction coefficient values.
Position correction
Time correction
Coefficient
Value
Coefficient
Value
a0
129.728
b0
8.93895
a1
-0.01787
b1
119.669
a2
-0.00034
b2
0.460586
mpad , for each pad:
mpad =

qref
qpad

(4.1)

where qref was the centroid of pad 8.
For each event, the calibrated charge on each pad was calculated by multiplying the
charge collected on each pad by that pad’s slope to produce a calibrated charge value. The
total energy loss through the detector, referred to as the ion chamber sum, was calculated on
an event-by-event basis by summing together the calibrated charge values of all good pads.

71

160

1.2

120

140

100

80

80

60

60

40

40
20

1
Energy loss (arb)

Energy loss (arb)

120

100

20

0
0
-150 -100 -50 0 50 100 150
CRDC2 x-position (mm)

100

0.8

80

0.6

60

0.4

40

0.2

20

0
0
-150 -100 -50 0 50 100 150
CRDC2 x-position (mm)

Figure 4.8: Position dependence of the ion chamber. The left panel shows the summed signal
from the ion chamber as a function of CRDC2 x-position when using a cocktail beam. The
right panel shows the position-corrected summed signal from the ion chamber.
The ion chamber exhibited a slight position dependence in the x-direction. In the data
shown in Figure 4.8, a cocktail beam with many different fragments was swept across the
entire face of the ion chamber. The position dependence here was likely caused by inefficient
charge collection. Inefficient charge collection was much more apparent in the cases of a
high Z fragment when more ionization was created and the difference was greater than the
resolution of the detector. To correct this position dependence, the highest Z component of
Figure 4.8 (nitrogen) was fit with a second-degree polynomial with coefficients (a0 , a1 , and
a2 ) and each summed signal, Qic,sum , was divided by the value of that polynomial at that
x-position to produce the position-corrected value, Qic,poscorr :

Qic,poscorr =

Qic,sum
a0 + a1 x + a2 x 2

(4.2)

The fitted coefficients are shown in Table 4.3 and the results are shown on the right side of
72

Figure 4.8.
A slow drift of the signal in the ion chamber over time was noticed during the production
runs (Figure 4.9). This drift did not positively correlate with drift of signals in any other
detector. The position-corrected energy loss was observed to have the following functional
form:
b0
+ b2
t + b1

(4.3)

where t was the run number. The coefficients b0 , b1 , and b2 were all coefficients to be
fitted. The time dependence was removed by dividing Qic,poscorr by the fit to create a new
parameter, Qic,corr :
Qic,corr =

Qic,poscorr
.
b0 /(t + b1 ) + b2

(4.4)

The fitted coefficients for the drift in the ion chamber are shown in Table 4.3 and the
results are shown on the right side of Figure 4.9.

4.1.1.3

Thin scintillator

The thin scintillator was located near the end of the beam line and had four PMTs, attached
as in Figure 3.5. The four PMTs were labeled by a number and a description of their location
(Table 4.4). The signal from each PMT was split to measure both total charge collected and
the time of collection. Those raw signals were combined for a measurement of the total
energy loss in the detector and the time of interaction in the detector. As with the ion
chamber, inhomogeneities in position and time were corrected.
The time and charge signals in each of the thin scintillator’s four PMTs had an inherent
position dependence due to the attenuation through the plastic. In order to construct correct
time and charge measurements, the information from all four PMTs was gain-matched and
73

Energy loss (arb)

0.8

×103

0.7

14

0.6

12

0.5

10

0.4

8

0.3

6

0.2

4

0.1

2

0

20

40

60

80

100

120 140
Run number

160

180

200

220

×103

1.4

12

1.2
Energy loss (arb)

0

10

1
0.8

8

0.6

6

0.4

4

0.2

2

0

20

40

60

80

100

120 140
Run number

160

180

200

220

0

Figure 4.9: Time dependence of the summed ion chamber signal. The top panel shows
the summed, position-corrected ion chamber signal as a function of run number. The red
line is the fitted function 4.3 using parameters from Table 4.3. The bottom panel shows
the summed, position-corrected, and time-corrected ion chamber signal as a function of run
number.

74

Table 4.4: Thin scintillator locations and gain-matching slopes and offsets.
Location
Number
Left upper
0
Left lower
1
Right upper
2
Right lower
3

Slope
1
1
1.22494
1.68587

Offset
Time offset (ns)
0
37
a
349.819 44.8542 ,11.7117b ,10.8104c
12.0517
-8.46805
-155.151
25.2282

a runs 1-31
b runs 32-75
c runs 76-245

combined. To gain-match the four thin scintillator PMTs, the 13 B beam was sent down the
center of the detector. Events at the very center of the detector were selected because the
light from these events traveled similar distances to each PMT, so the signals arriving at the
PMT would have been equal in magnitude and travel time.
The charge signals from each PMT were fit with a Gaussian lineshape. The left lower,
right upper, and right lower signals were all gain-matched to the signal from the left upper
PMT. For the linear gain-matching, the slope was determined by the ratios of the widths of
the Gaussian fits while offsets were chosen to put all of the centroids at the same location.
Slopes and offsets from the gain-matching are shown in Table 4.4. An absolute measurement
of energy loss was not required, so calibrations to convert from QDC channels to total MeV
were not needed.
Once all four PMTs were gain-matched, their signals were combined to provide a measurement of the total charge (e) deposited in the detector:

q + qRU
etop = LU
2
q
+ qRD
ebottom = LD
2

75

Table 4.5: Thin scintillator position and time correction coefficient values.
Position
Coefficient
a0
a1
a2
a3
a4
a5
a6

correction
Time correction
Value
Coefficient
Value
1053.34
b0
-6.064
0.235699
b1
88.6789
-0.00802787
b2
0.861741
-2.7786e-05
2.30353e-07
3.19837e-10
-2.0901e-12

e=

e2top + e2bottom
2

where qLU,RU,LD,RD are the gain-matched signals from the left upper, right upper, left
lower, and right lower PMTs, respectively.
The total charge from the thin scintillator exhibited both position and time dependence.
The corrections for both were similar to those for the ion chamber described in Section
4.1.1.2, except the position dependence was fit with a sixth-order polynomial:

eposcorr =

e
6
i
i=0 ai x

.

(4.5)

Figure 4.10 shows the energy loss in the thin scintillator before and after the position
correction. The time drift in the total energy signal was corrected using Equation 4.3 and
before and after spectra are shown in Figure 4.11. Correction parameters for both are shown
in Table 4.5.
To calibrate the time signals from each PMT, only an offset was used, since a slope of
0.1 ns/channel was fixed by the range of the TDC. A 20 ns jitter was included in the timing
by the use of a Field Programmable Gate Array (FPGA). All TDC channels received a

76

1.6

1.6

120

1.2

1.4
100

1

80

0.8

60

0.6

40

0.4
0.2

Energy loss (arb)

Energy loss (arb)

1.4

20

100

1.2
1

80

0.8

60

0.6

40

0.4
0.2

0
0
-150 -100 -50 0 50 100 150
CRDC2 x-position (mm)

120

20

0
0
-150 -100 -50 0 50 100 150
CRDC2 x-position (mm)

Figure 4.10: Position dependence of the charge signal from the thin scintillator. The left
panel shows the total signal from the thin scintillator as a function of CRDC2 x-position when
using a cocktail beam. The right panel shows the position-corrected summed signal from
the thin scintillator. The correction was chosen so that the isotopes of interest (lithium and
beryllium) had no position dependence. The remaining position dependence was primarily
for the heavier isotopes, nitrogen and carbon.

77

×103

1

14
0.9
Energy loss (arb)

12
0.8

10

0.7

8

0.6

6

0.5

4

0.4

2

0.3

20

40

60

80

100

120 140
Run number

160

180

200

220

×103
14

1.2
1.1
Energy loss (arb)

0

12

1
10
0.9
8

0.8

6

0.7
0.6

4

0.5

2

0.4

20

40

60

80

100

120 140
Run number

160

180

200

220

0

Figure 4.11: Time dependence of the total thin scintillator charge signal. The top panel
shows the summed, position-corrected thin scintillator charge signal as a function of run
number. The red line is the fitted function 4.3 using parameters from Table 4.5. The
bottom panel shows the summed, position-corrected, and time-corrected charge signal as a
function of run number.

78

start command from the individual timing detectors and the stop signal was generated by
the FPGA. The amount of jitter varied on an event-by-event basis but since the same stop
signal was used for all TDCs, the jitter was the same for each timing signal within an event.
To eliminate the jitter, one timing signal (the thin left upper PMT) was subtracted from all
other timing signals to create a time spectrum for that detector. This subtraction was done
after the slopes were applied to allow for different TDC ranges.
After the jitter was subtracted, the time calibration was performed using the same event
selection for the charge calibration. The timing signals from the other three thin scintillator
PMTs were fit with Gaussians. The timing offset was calculated to align the mean values
of all the Gaussian fits. Time calibration offsets for the thin PMTs are shown in Table 4.4.
The calibrated timing signals from each PMT were then averaged to form one timing signal
from the device:
t + tLD + tRU + tRD
tthin = LU
4

(4.6)

where tthin is the final timing signal and tLU,RU,LD,RD are the timing signals from the
individual PMTs.
Faulty electronics caused some signals to drop out or shift during the experiment. During
the experiment the signal from the left lower PMT jumped twice. The first time was due to
a loss of signal in the middle of run 14 and was not fixed until after run 31. The second was
a shift that was correlated with fixing a double peaking problem in the target scintillator
immediately after run 75. Different offsets, shown in Table 4.4, were applied for each of these
sections. In addition, for a short time during the beginning of the experiment, that same
time signal disappeared. For those events, the time signals from the three remaining PMTs
are averaged. This slightly worsens the timing resolution but allows the inclusion of many

79

Table 4.6: Calibration parameters for the timing signals from the A1900 and target scintillators.
Parameter
Value
A1900 slope
0.1 ns/channel
A1900 offset
-134.57 ns
Target slope
0.1 ns/channel
Target offset (run number< 72)
−40.4 ns
Target offset (run number≥ 72)
−24.1 ns
c0
4.698 ns
c1
128.9 channels
c2
−1.5 ns
more events.

4.1.1.4

Other timing scintillators

Two other timing scintillators were used, one immediately after the A1900 (the A1900 scintillator) and another approximately one meter upstream from the target (the target scintillator). The target scintillator had one attached PMT whose signal was split to record both
the charge and time. The A1900 scintillator was located approximately 11 m upstream from
the target and the signal from its one PMT recorded only the time of interaction in the
scintillator.
Since an absolute value of energy loss through the scintillator was not necessary, the
charge signals were always displayed as raw QDC channel values. The basic time calibrations
for both the target and A1900 scintillators were performed in the same way and used a
calibration similar to that for the thin scintillator PMTs. First, the TDC channel value was
multiplied by a slope of 0.1 ns/ch to convert it to nanoseconds. Second, the FPGA jitter was
removed by subtracting the signal from the left upper PMT of the thin scintillator. Third,
an offset was applied so that the time difference between scintillators reflected the calculated
speed of an unreacted beam sent through the system. Calibration values are shown in Table
80

Counts

20
18
16
14
12
10
8
6
4
2
0
11

×103

Counts

×103

12

13

14 15 16
Time (ns)

17

18

45
40
35
30
25
20
15
10
5
0
-12 -11 -10

-9 -8 -7
Time (ns)

-6

-5

Figure 4.12: Timing signal from the target scintillator before (left) and after (right) the walk
correction.
4.6.
For events with low charge deposition in the target scintillator, a shift in timing (a
“walk”) was observed (see Figure 4.12). To fix this and improve the timing resolution of the
detector the charge dependence of the timing signal was fit with an exponential function and
corrected. The final correction took the form:

ttarg. = tns − c0 e−q/c1 − c2 ,

(4.7)

where ttarg. was the walk-corrected time, tns was the calibrated but uncorrected time, q was
the charge collected by the PMT, and c0 , c1 , and c2 were correction parameters. The final
calibration parameters for the target calibration are shown in Table 4.6 and the result is
shown in Figure 4.12.
In the first few days of the experiment, the target scintillator exhibited a double peaking
as shown by the black line in Figure 4.13. This was attributed to faulty electronics and fixed
81

×103
18
16
14
Counts

12
10
8
6
4
2
0
-30

-25

-20
-15
-10
-5
Target scintillator time (ns)

0

Figure 4.13: Target scintillator timing spectra from run 41 (black), taken while the target
exhibited double peaking, and from run 90 (blue), taken after the double peaking was fixed.
approximately two days into the experiment. Due to the change in electronics required to
fix the double-peaking, the target timing signal required different offsets before and after
the change. Values before and after the change are shown in Table 4.6. The double peaking
had a minimal effect on the beam identification, but if used for the particle identification,
would significantly worsen the resolution. To avoid this, the time of flight from the A1900
scintillator to the thin was used as the timing signal for the particle identification.

4.1.1.5

CsI(Na) array

The CsI(Na) array was used sparingly in this analysis because particle identification was
achieved through a corrected time of flight, so extensive calibration of the detector was not
required. A peak-sensing ADC was used to read out the charge collected in each PMT. More
extensive calibrations were carried out by collaborators at Augustana College but were not
used in the current analysis.

82

4.1.2

MoNA-LISA calibrations and corrections

Raw output from each MoNA-LISA PMT (576 total) included a charge measurement that
reflected the total light collected by the PMT and a time measurement of the light’s arrival
at the PMT. A series of calibrations converted these signals to a calibrated deposited charge
measurement, a time of interaction in the bar, and a position of interaction along the length
of the bar.
Cosmic ray data taken before the experiment was used for most MoNA-LISA calibrations.
Muons from cosmic rays deposit roughly 2 MeV/cm of energy in each detector [143] as they
pass through the detectors at roughly the speed of light [144]. This will deposit approximately
20 MeV electron equivalent of light into the bar. One MeV electron equivalent (MeVee) is
equivalent to the light deposited by a 1 MeV electron.

4.1.2.1

Charge calibration

The first step in charge calibrations for MoNA-LISA was to roughly gain-match the PMTs
by changing the applied high voltage. Each high voltage was adjusted until the cosmic-ray
peak from each PMT showed up in roughly the same channel. Final high voltage values
ranged from 1298.5 to 1951.5 V.
When the detectors were off and the electronics were on, a small current ran through
every QDC channel. When the detectors were on and collecting data, this small current,
called a pedestal, had two consequences. The first was that it offset the true charge collected
so that channel 0 did not correspond to zero energy. The second was that it caused every
QDC channel to be read out for every triggered event, regardless of how many bars registered
a real interaction. This deluge of data to be read-out would cause significant dead time.
The solution to the first problem was to apply a linear calibration to translate the channel
83

105

104

10

104

103

0

1000

1500

102

Counts

20

Counts

105

103
102

10

10

1
0

500 1000 1500 2000 2500 3000
Layer A, bar 15 charge (channels)

1

0

5 10 15 20 25 30 35 40
Layer A, bar 15 charge (MeVee)

Figure 4.14: An example of MoNA-LISA’s charge calibration procedure. The raw charge
from each QDC channel is plotted (left). The cosmic peak around channel 1200 is fit with
a red Gaussian line. The inset shows channels 500 to 2000 with a linear scale on the y-axis.
Those values are used to construct a linear fit whose slope and offset are applied to produce
a calibrated charge signal in MeVee (right).
number to a light measurement where the offset value was the same as the peak of the
pedestal. Using the pedestal and muon peak as known values (0 and 20 MeVee, respectively),
a linear fit for each PMT was found. An example of this procedure is shown in Figure 4.14.
The solution to the second problem was to apply a hardware threshold so that channels
without a signal greater than the threshold were not included in the data readout. The
thresholds, which should be above most of the pedestal peak, not at its peak, can only be
set in multiples of 16 channels. In this experiment, a mistake in the code for calculating
the thresholds set the thresholds higher than they needed to be; the thresholds for this
experiment ranged from 0.4 to 0.9 MeVee.

84

3
30 ×10

40 ns

25
Counts

20
15
10
5
0

0

1000
2000
3000
TDC time (channels)

4000

Figure 4.15: An example of the raw time spectrum during MoNA-LISA’s TDC calibration
procedure. A pulser pulses the system every 40 ns, creating a spectrum with peaks at
intervals that correspond to 40 ns.
4.1.2.2

Position calibration

Two of the spatial coordinates (roughly y and z for the array at zero degrees from the
incoming beam) of each interaction were given by the discretization of the bars. The third
spatial dimension, the measurement of the interaction along the bar, was given by the time
difference between the light’s arrival at the PMTs on either end of the detector (see Figure
3.7).
The first step in this process was to convert the channel readout of the TDCs to a
time. An absolute time was not possible, but a slope was applied to convert channels to
nanoseconds. This slope was roughly given by the set range of the TDC (350 ns over 4096
channels is approximately 0.085 ns/channel), but since time resolution was essential, the
TDC calibration slope was also measured with a pulser that pulsed the system every 40 ns,
generating a pulse structure as seen in Figure 4.15. The peaks were fitted and the difference
between various peaks was used to generate a linear fit to convert channel difference to
time difference. In MoNA, those slopes centered around 0.083 ns/channel with a full range
85

25
MoNA
Number of PMTs

20

LISA

15
10
5
0
0.08

0.082

0.084 0.086 0.088 0.09
TDC slope (ns/channel)

0.092

0.094

Figure 4.16: TDC slope values for MoNA and LISA.
between 0.08 and 0.09 ns/channel (Figure 4.16). In LISA, those slopes centered around 0.089
ns/channel with a full range between 0.081 and 0.092 ns/channel (Figure 4.16).
With the PMT times calibrated, the next step was to convert the time difference between
the two PMTs to a position along the bar. Cosmic-ray data was used to illuminate the entire
length of the bar uniformly. An example of the raw data is shown in Figure 4.17. The edges
of the step-like distribution are the edges of the bar, designated as -100 and 100 cm. To
set the position calibration, a linear slope was calculated from those two points and then an
offset was used to place the center of the bar at 0 cm.

4.1.2.3

Time calibrations

The measurement of a time difference between the interaction at the target and the interaction of the neutron in MoNA-LISA was crucial to the operation of the device. While the
position along the bar was determined by the time difference between the two PMTs, the
time of the interaction was determined by an average of the two times and then an offset to
86

300
250

200

200
Counts

Counts

250

150

150
100

100

50

50

0
-20 -15 -10 -5 0 5 10 15 20
Layer H, bar 8 time difference (ns)

0
-150 -100 -50
0
50 100 150
Layer H, bar 8 x-position (mm)

Figure 4.17: X-position calibration example. The final x-position spectrum (right) for one
bar was calculated based on the time difference between the two PMTs (left).
appropriately place the time of the interaction within the time of all the other interactions
in the detector.
The known velocity of the cosmic-ray muons, v, was utilized to calibrate the timing of
each bar relative to the bottom bar of its vertical layer. Only events that triggered a majority
of the bars in the layer and deposited approximately 20 MeV in each bar were used. Then,
for each bar, a time difference, ∆ta0 , and path distance, ∆da0 , between the interaction point
in that bar (a) and the bottom bar (0) were calculated. The value of

∆ta0 − ∆da0 /v

(4.8)

was plotted, which should be centered at zero if the time offsets for bar a are correct. If the
Gaussian distribution was not centered at zero, an offset was added to the time calibration of
bar a so that it was. At this point, the velocity of the muon, when calculated using the time
and position of interactions within a single vertical layer, should be measured accurately.
87

∆tD15,A0 − ∆dD15,A0 /v
∆tC15,A0 − ∆dC15,A0 /v
∆tB15,A0 − ∆dB15,A0 /v

∆t15,0 − ∆d15,0 /v

∆t10,0 − ∆d10,0 /v
Figure 4.18: Schematics showing the differences used to calibrate the time measurement in
MoNA-LISA. The view on the left is from the front of the array and the schematic on the
right shows the side view of the array, looking head-on at the PMTs.
The left diagram of Figure 4.18 shows these interactions and the differences used.
Once each vertical layer was “tied together in time”, the next step was to link layers on
the same table (right diagram of Figure 4.18). Specifically, the time of each vertical layer
needed to be calibrated relative to the time of the bottom bar in the front layer. The same
process was used, but this time utilizing events that had interacted in the bottom bar of the
front layer as well as in the top bar of other layers. From this, each vertical layer received
an additional time offset that was applied to all its bars.
At this point, each table of bars (1 table for MoNA and 2 for LISA) are time calibrated
relative to all other bars on that table. The final step was to place each table appropriately
in time relative to interactions in the target. This required data from the experiment where
a thick target was placed in the target position. The 13 B collided with the thick target,
creating many gamma rays. Based on the locations of the bars relative to the target, a final
offset was applied to all bars in each table such that the velocity of the gamma rays was

88

Table 4.7: Final time offsets for each table of MoNA-LISA, before and after the electronics
change.
Time period
MoNA offset (ns)
Before run 76
433.83
Run 76 and after
429.103

LISA at 0◦ offset (ns)
411.295
406.392

LISA at 30◦ offset (ns)
409.901
405.25

600
400
Counts

200

200

Counts

Counts

400

300
200

100

100
00

50
100
150
MoNA ToF (ns)

00

50
100
150
LISA at 0◦ ToF (ns)

00

50
100
150
LISA at 30◦ ToF (ns)

Figure 4.19: Calibrated timing spectra for each table. Impinging the beam on a thick target
produced many γ-rays and neutrons. Vertical lines indicating the expected arrival of γ-rays
and neutrons at the front of that table. The left panel shows the time-of-flight to the MoNA
table, the center panel shows the time-of-flight to the first LISA table, and the right panel
shows the time-of flight to the second LISA table.
29.98 cm/ns. Two different sets of offsets were applied because the electronics that created
the start signal for MoNA-LISA changed part of the way through the experiment. Both sets
of final offsets are shown in Table 4.7. As a final check, Figure 4.19 shows the timing spectra
for all three tables as well as vertical lines to indicate the expected arrival times of γ-rays
and neutrons from the thick target.

4.2

Event selection

Once the information from each detector was calibrated, information from different detectors
was combined to make a specific measurement. Essential to each specific measurement is
the selection of data events with the necessary information. This section details how almost
89

dE (arb)

2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0

14 C

102

13 B

10

1
75

80
85
90
Time-of-flight (ns)

95

100

Figure 4.20: Beam component identification using energy loss through the ion chamber
(y-axis), and time-of-flight from the A1900 scintillator to the target scintillator (x-axis).
200 hours of data was reduced to roughly 3 hours of useful events.

4.2.1

Beam identification

Event selection began by identifying the 13 B component of the incoming beam. Components of the beam were identified by removing the target and allowing the beam to enter the
Sweeper detector box. The magnetic rigidity of the beam as it entered the vault was 3.2271
Tm. At this rigidity, the velocity of the desired 13 B beam was 11.1 cm/ns. A 14 C contaminant traveled the distance between the A1900 and target scintillators 8 ns faster than the
13 B

and was identifiable in a plot of energy loss in the ion chamber against the time-of-flight

between the two detectors (Figure 4.20). Separation of the 13 B from the 14 C contaminant
was achieved with a time-of-flight gate between 85 and 92 ns, as shown in Figure 4.20.
A similar process was repeated to identify and isolate the 12 B beam, which arrived in the
vault with a rigidity of 2.9772 Tm. A 13 C contaminant was clearly visible in the energy loss

90

vs. time-of-flight spectrum. The velocity of the 12 B beam was also 11.1 cm/ns, so the same
time-of-flight was used to separate the 12 B beam from the contaminant as was used for the
13 B

beam.

4.2.2

Event quality gates

The detectors did not always behave in a uniform and expected manner. Events in which the
experimental system behaved abnormally could lack information from a detector or contain
incorrect information. The gates described in this section were used to remove those events
from the data stream.
The incoming beam rate was approximately 800 kHz, slightly less than one beam particle
every microsecond. Within the approximately 400 ns range of the TDCs then, there was
a possibility that another particle could go through the beamline, cause scintillation, and
record a time. In this case separate detectors would have measured the time for different
beam particles, resulting in a different time-of-flight than expected.
The frequency of the cyclotron was 21.8259 MHz, so these events with errorneous time
measurements should show up as smaller “echos”, separated by 46 ns, as seen in Figure 4.21.
This figure shows the time difference between the A1900 scintillator, 11 m upstream from
the target, and the thin scintillator near the end of the beamline. The echos were eliminated
by a time-of-flight gate as indicated by the vertical lines in Figure 4.21.
In other events, the position information from the CRDC was not considered reliable.
The y-position was measured by the time difference between the particle interacting in the
thin scintillator and the collection of the charge carrier at the anode of the CRDC. In some
events, the charge carrier wasn’t collected at the anode and the y-position measurement was
unable to be made.
91

106
105

Counts

104
103
102
10
1
10−1
-200

-100

0

100

200

300

400

Time-of-flight, A1900 to thin (ns)

106
105
104
103
102
10
1

105
104
Counts

Counts

Figure 4.21: Time difference between the A1900 and thin scintillators, showing “echos” from
events in which the A1900 scintillator recorded the time of a different particle than the thin
scintillator. The vertical lines indicate the limits of the gate used to eliminate these events.

103
102
10
1

0
1000
CRDC1 TAC (channels)

0
10
20
CRDC1 pad sum (arb)

Figure 4.22: Event quality gates applied to the CRDC1 parameters TAC (left) and pad sum
(right). Red vertical lines with arrows indicate the requirements for an event to pass through
the gate.

92

105

104

104
Counts

105
Counts

Counts

106
105
104
103
102
10
1

103

103

102
10

102
10

1

0
1000
CRDC2 TAC (channels)

1
0
10
20
CRDC2 pad sum (arb)

0

50
100
CRDC2 χ2

Figure 4.23: Event quality gates applied to the CRDC2 parameters TAC (left), pad sum
(center), and χ2 (right). Thick red vertical lines indicate the requirements for an event to
pass through the gate.
The x-position measurement was also compromised in some events. To calculate the
position accurately, charge needs to be induced on multiple pads. Some events showed abnormally low total charge across the pads. In addition, some events showed an abnormally
high charge collection, possibly in connection to multiple particles passing through the detector. These events were eliminated by gating on the total charge summed over all the pads
(the pad sum). Those gates are shown in Figures 4.22 and 4.23.
When the x-position was calculated by fitting the charge distribution across the pads
with a Gaussian, several variables were stored, including the centroid, the width, and the
χ2 value for the fit. A limit on the maximum χ2 value removed events with bad x-position
fitting and is shown in Figure 4.23.

4.2.3

Isotope identification

Many products were expected from the 13 B+9 Be reaction. Primarily, isotopes with a Z and
A less than or equal to that of the beam were created. The magnetic rigidity of the Sweeper
(3.76 Tm for 13 B) restricted the isotopes successfully traversing the magnet to those with
93

Corrected ion chamber energy loss (arb)

1.4
1.2
B
1
0.8
Be
0.6
Li

0.4
He
0.2
0.2

0.4

0.6

0.8

1

1.2

Corrected thin energy loss (arb)
Figure 4.24: Element identification.
magnetic rigidities within an acceptance window. For the Sweeper, this was roughly the
rigidity of the magnet plus or minus 8%. For this experiment, nuclides within that range
included 14,15 B, 10,11,12 Be, 7,8,9 Li, and 6 He.
Element identification was achieved by measuring the energy losses through the ion
chamber and the thin scintillator. Although elemental separation can be seen in the individual PMTs of the thin scintillator as well as the individual pads of the ion chamber,
two-dimensional gates on the elements produced a cleaner identification (see Figure 4.24).
The next step was isotopic separation. Ideally, a dipole magnet should separate the
isotopes by magnetic rigidity. In practice however, the distributions are broad enough that
they overlap one another and make isotope separation more difficult. The Sweeper magnet
convolutes x-position, x-angle, and time-of-flight. This convolution must be untangled for
94

Time-of-flight, A1900 to thin (ns)

150
145
140
135
130
150

125
100

120
150

50
100
CR 50
DC
2 x- 0
ang
le ( -50
mra -100
d)

0
-50

-p

-100 C2 x
D
-150 -150 CR

iti
os

on

)

(m

m

Figure 4.25: Three-dimensional plot showing lithium isotope identification by plotting
x-position, x-angle, and time-of-flight.
clean isotope identification. A detailed description of the process can be found in [139], but
an overview as well as the final results are presented here.
The convolution can be seen using a three-dimensional plot of x-position, x-angle, and
time-of-flight (Figure 4.25) gated on one element, in this case, lithium. The goal was to
find a combination of parameters to correct the time-of-flight for variances in x-position and
x-angle. To do this, this three-dimensional plot was projected down to the x-position-x-angle
plane and the contours guided the convolution of x-position and x-angle into a new parameter. When this new parameter was plotted against time-of-flight, three two-dimensional
95

×103
2.5

8 Li
9 Li

Counts

2
1.5
7 Li

1
0.5
0
5

×103
11 Be

Counts

4
10 Be

3
2

12 Be

1
0 2

2.2

2.4

2.6
2.8
Fragment A/Z

3

3.2

3.4

Figure 4.26: Lithium (top) and beryllium (bottom) particle identification from the 13 B beam
and in coincidence with an interaction in MoNA-LISA.
groups of events were visible - isotope separation. The final goal was isotope separation from
one single time-of-flight parameter, so the time-of-flight was corrected for this convoluted
x-position-x-angle parameter.
The large gap in the magnet caused a dependency on the y-position and y-angle. The
time-of-flight was corrected for both of these parameters as well as the energy loss through
the ion chamber in a similar fashion to correcting for the x-position-x-angle convolution.
Parameters and correction factors used to construct the final time-of-flight parameter are

96

Table 4.8: List of correction factors used to produce the lithium isotope separation parameter.
The numbers in the right column were multiplied by the parameter on the left, summed, and
added to the time-of-flight from the A1900 to the thin to produce a corrected time-of-flight
parameter for the lithium isotopes from the 13 B beam. Both position parameters, x and y,
were calculated 12.59 cm downstream of CRDC1.
Parameter
x
x2
y
y2

Correction factor
−1.14 × 10−1
−3.6 × 10−4
5.29126 × 10−3
6.90292 × 10−4

Parameter
θx
θx2
θx3
∆Eic.corr

Correction factor
1.8 × 10−1
1.95932 × 10−4
−8.54275 × 10−6
13.0485

Table 4.9: List of correction factors used to produce the beryllium isotope separation parameter. The numbers in the right column were multiplied by the parameter on the left, summed,
and added to the time-of-flight from the A1900 to the thin to produce a corrected time-offlight parameter for the lithium isotopes from the 13 B beam. Both position parameters, x
and y, were calculated 12.59 cm downstream of CRDC1.
Parameter
x
x2
y
y2
ToFA1900−thin

Correction factor
−6.5 × 10−1
−2 × 10−3
5.29126 × 10−3
2.8 × 10−3
2.55

Parameter
θx
θx2
θx3
∆Eic.poscorr

Correction factor
1.00191851
1.95932 × 10−3
−8.547275 × 10−6
60

shown in Table 4.8. To create the final separation parameter,
P

T oFcorr = T oFA1900−thin +

ci p i
i=0

where ci and pi are the coefficients and correction factors noted in Table 4.8, T oFcorr is the
corrected time of flight, and T oFA1900−thin is the time of flight from the A1900 to the thin
scintillator.
This time-of-flight correction for lithium could be used the separate the beryllium isotopes
as well, but the separation was better when a separate set of corrections were used. Those
parameters are shown in Table 4.9. Separation of the lithium and beryllium isotopes in
coincidence with an interaction in MoNA-LISA is shown in Figure 4.26. The 12 Be peak
97

Counts

180
160
140
120
100
80
60
40
20
0

8 Li
9 Li
7 Li

Counts

250

10 Be

200
150
100

11 Be

50
0 2

2.2

2.4

2.6
2.8
Fragment A/Z

3

3.2

3.4

Figure 4.27: Lithium (top) and beryllium (bottom) particle identification from the 12 B beam
and in coincidence with an interaction in MoNA-LISA.
is small because creating 12 Be from the beam requires all the neutrons to remain in the
fragment. The same plot for isotopes created from the 12 B beam is shown in Figure 4.27. In
this figure, 11 Be is the fragment created by removing a proton and leaving all the neutrons
in the nucleus, so it is smaller than the 10 Be peak.

4.2.4

Neutron interaction selection

Though MoNA-LISA was designed to detect neutrons, it also detected some background
radiation. This background consisted primarily of muons from cosmic rays and gamma rays
from the surrounding environment or from reactions in the target. The neutron interactions

98

needed to be separated from these interactions.
The raw neutron time-of-flight spectrum in coincidence with 9 Li is shown in the top panel
of Figure 4.28. The bottom panel shows the charge versus time spectrum for the interactions
in MoNA-LISA. For each event, there could be multiple interactions in MoNA-LISA; these
histograms only show the first interaction in MoNA-LISA, sorted based on interaction time.
The one-dimensional histogram has three main features: a sharp peak around 36 ns, a wider
peak around 70 ns, and a large background component. The two-dimensional histogram
shows that the peak around 70 ns deposited a lot of charge, the peak around 36 ns deposited
a small amount of charge, and the background also deposited a small amount of charge.
Not shown in the two-dimensional spectrum are all the first interactions in MoNA-LISA
that generated a time measurement but whose total charge deposited was below the QDC
threshold. These events constituted the majority of the background component and the
sharp peak around 36 ns.
Cosmic ray background interactions would have deposited 20 MeVee of energy in the
detector and appear as a horizontal band in the two-dimensional spectrum. Such a band is
not visible, therefore any background from cosmic rays was negligible.
The shape of the background in the top panel Figure 4.28 was due to three factors: the
staggering of the time ranges for the three sections of MoNA-LISA, the time ordering of the
interactions, and the high beam rate.
The first interaction in MoNA-LISA could occur anywhere within the detector, so Figure
4.28 contains interactions from MoNA and both tables of LISA. The time window of each
table, however, was slightly different. This softened the sharp edges expected of the background. Separating this spectrum into the individual spectra from each Table (Figure 4.29)
shows that the perceived bump around 15 ns is the beginning of the background collected
99

×103
50

Counts

40
30
20
10
0
-50

0

50
100
150
200
MoNA-LISA time-of-flight (ns)

250

300

-50

0

50
100
150
200
MoNA-LISA time-of-flight (ns)

250

300

Deposited energy (MeVee)

70
60
50
40
30
20
10
0

Figure 4.28: Top: Raw neutron time-of-flight spectrum. Bottom: Charge deposited vs.
time-of-flight for the first interaction in MoNA-LISA.

100

22

×103
Data
MoNA
LISA at 0 degrees
LISA at 30 degrees

20
18
16
Counts

14
12
10
8
6
4
2
0
-50

0

50

100

150

200

250

300

MoNA LISA time-of-flight (ns)
Figure 4.29: A breakdown of the time-of-flight spectrum of Figure 4.28 by sections (tables)
of the detector.
from the MoNA table. In addition, the γ-ray peaks and neutron peaks from each table
become sharper.
The time-ordering process also skewed the time-of-flight spectrum, particularly the background. If the background was evenly distributed across time, then for events in which
multiple interactions are registered, this spectrum only shows the time of the first interaction, which is guaranteed to be before the time of the second interaction. This is illustrated
in Figure 4.30. The top panel breaks down the events with two interactions in the first table of LISA. The distributions for the first and second interactions are noticeably different,
with the first interaction spectrum (the one that would be included in Figure 4.28) skewed
toward smaller times-of-flight. The bottom panel shows the distribution of events with one
interaction compared with the summed “two interaction” spectrum from the top panel.
The background before the γ-ray peak in Figure 4.30 is flat, while the background after
101

×103
10

Multiplicity=2 interactions
First interaction
Second interaction

Counts

8
6
4
2
0

-50

0

×103

50
100
150
200
◦
LISA at 0 time-of-flight (ns)

300

Multiplicity=1 interactions

10

Multiplicity=2 interactions

8
Counts

250

6
4
2
0

-50

0

50
100
150
200
◦
LISA at 0 time-of-flight (ns)

250

300

Figure 4.30: Top: Time-of-flight breakdown of events with two (and only two) interactions
in the section of LISA at 0 degrees. The blue line shows the first interaction spectrum, while
the green line shows the spectrum of the second interaction. The black line is the sum of
those two components. Bottom: Comparison of the time-of-flight spectrum for events with
one interaction (red line) and the summed spectrum from above (black line). The red line
has been normalized to the flat background before the gamma peak of the black line.

102

Counts

3
20 ×10
18
16
14
12
10
8
6
4
2
0
10

20

30
40
50
60
MoNA-LISA time-of-flight (ns)

70

Figure 4.31: The γ-ray peak in MoNA-LISA during the 13 B production runs.
it is not. The remaining gentle downward slope is likely due to the high rate of the beam and
the abnormally large γ-ray peak (the γ-ray peak is often one-tenth the size of the neutron
peak). The TDCs registered only the first interaction in each bar, so any background events
occurring in the same bar after another are immediately discarded. The effect of this is to
remove events predominantly from the higher end of the time-of-flight spectrum.
A zoomed-in spectrum of the peak at 36 ns is shown in Figure 4.31. The shape and charge
deposition (mostly below threshold) are consistent with γ-ray interactions from a point
source, but the time-of-flight is not - γ-rays from the target should arrive at approximately
26 ns. Separating the γ-ray peak into the constituent tables (Figure 4.32), shows that each
table has a unique time difference between the expected arrival time and and the measured
arrival time of the γ-rays (10.9 ns for MoNA, 10.7 ns for LISA at 0◦ , and 13.0 ns for
LISA at 30◦ ). Assuming the source of these γ-rays was in the experimental area and that
the difference in time-of-flight can be correlated with the distance from that source, these
differences can be used to see where in the vault this source could be.
103

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8

1
0 20
30
40
50
MoNA ToF (ns)

6

6

Counts

2

Counts

Counts

8

×103

4

4

2

2

0 20
30
40
50
◦
LISA at 0 ToF (ns)

0 20
30
40
50
◦
LISA at 30 ToF (ns)

Figure 4.32: The arrival time of the large γ-ray peak, as observed by MoNA (left), the LISA
table at 0 degrees (center), and the LISA table at 30 degrees (right). Vertical lines indicate
expected arrival time of γ-rays from the target area.
The expected time difference of gamma arrival at two different detectors was calculated
for every point in the vault. The measured time difference was subtracted from this calculated
time difference. This calculation was done for each pair of detectors and is shown in Figure
4.33. The source can be located anywhere along the line where this quantity is zero, deliniated
by the curved black lines.
Overlaying all of these zero-lines on the vault layout diagram (shown in Figure 4.34) can
give some indication of where the gamma source could be located. In this case, it appears the
γ-rays are coming from the negative x side of the Sweeper detector box, near the tungsten
blocker. This is close to where the unreacted beam was dumped, so it seems likely that this
large γ-ray peak was coming from unreacted beam being dumped on the tungsten blocker
or the wall of the Sweeper detector box. The expected rate of two incoming beam particles
arriving at the same time is small, but not negligible.
Accounting for the shape of the background is not enough - the neutrons still needed to
be separated. The γ-ray peak was easily separated from the neutron peak by time-of-flight,
but the continuous background underneath was not. It was separated, however, by requiring
104

600

20
15

400

10
200

5

0

0
-5

-200

-10
-400
-600

-15
0

500

1000

600

-20
20
15

400

10
200

5

0

0
-5

-200

-10
-400
-600

-15
0

500

1000

-20

Figure 4.33: Difference of time differences between two of the three detectors. The top panel
shows the comparison between LISA at 0◦ and LISA at 30◦ , and the bottom panel shows the
comparison between MoNA and LISA at 30◦ . Black lines indicate the possible locations of
a point gamma source to produce the measured time difference between the two detectors.

105

Figure 4.34: The two zero-lines from Figure 4.33 overlaid on the vault layout. The intersection of these lines indicates the likely location of a point gamma source. Within the
error of the calculation, and given the layout of the experimental vault, a gamma source was
identified inside the Sweeper box.

106

a certain amount of charge to be deposited in the bars. Since the QDC thresholds were
set abnormally high for this experiment, the lowest requirement of this type that could be
made was to simply have a charge measurement for the event. Including this requirement
effectively eliminated the background.
One concern in requiring a charge measurement was the possibility that while the first
interaction might not have had a valid charge and thus not be included as a neutron event,
the second (or third or fourth) interaction might qualify. In this case, the second interaction
should be considered the first, the third the second, and so on. This becomes especially
important when looking for the interactions of multiple neutrons. In an attempt to count
the neutron interactions more accurately, the interaction algorithim was modified. Whereas
previously, interactions needed only a valid time signal to pass the calibration and be incorporated into the data, they now required valid time and charge signals. The raw data still
included all of the information, but the calibrated neutron time-of-flight spectrum changed.
The final time-of-flight spectrum, shown in Figure 4.35, eliminated the background and
enhanced the statistics by mining the later interactions for interactions with valid charge
signals.

4.3

Inverse tracking

All the position measurements were performed after the fragments had been bent by the
Sweeper magnet. In order to retrieve the momentum and position information of the particles
at the target, they needed to be tracked through the magnet. This reconstruction of particle
trajectories before the magnet is fully described in [145], but a summary is presented here.
Ion-optical matrices are used to transform the coordinates of a particle before a magnet

107

2.2

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2
1.8
1.6
Counts

1.4
1.2
1
0.8
0.6
0.4
0.2
0
-50

0

50

100

150

200

250

300

MoNA LISA time-of-flight (ns)
Figure 4.35: Final time-of-flight spectrum for MoNA-LISA. This spectrum requires that each
interaction not only have a valid charge but also a valid time.
to the coordinates of the particle after it traverses the magnet. Specifically, the Sweeper
ion-optical matrix relates the set of variables (x, θx , y, θy , δ)(T ) at the target to the variables
(x, θx , y, θy , ∆L)(D) at the detector. In this case, x is the dispersive position, y is the
non-dispersive position, and the θ variables are ratios of transverse momentum in that (x or
y) direction over the total momentum:

sin θ(x,y) = p(x,y) /p0 .

The variable ∆L is the length difference between the path traveled by the particle and the
central path. The variable δ is the relative energy deviation:

δ=

E − E0
E0

108

where E is the energy of the particle and E0 is the energy of a particle following the central
trajectory through the magnet. The value of E0 can be determined by the magnetic field of
the magnet.
The forward matrix Mf is defined such that:


(D)

 x 
 
 
 θx 
 
 
 
 y 
 
 
 
 θy 
 
 
∆L

 (T )
x
 
 
θx 
 
 
 
= Mf  y 
 
 
 
 θy 
 
 
δ

This matrix is useful given the incoming particle information, but that only happens
during a simulation. For the data, this matrix needs to be transformed to one that takes
in the measured variables and gives back the unknown variables at the target- an inverse
(D)
(D)
mapping. There were measurements of x(D) ,θx , y (D) , and θy and the needed variables
(T )
(T )
were x(T ) , θx , y (T ) , θy , and δ.

One way to fix this situation of four input variables and five output variables is to measure
the incoming beam distribution with tracking CRDCs before the target. Then x(T ) and y (T )
are available and the reconstruction is overdetermined. In this experiment though, the use
of tracking detectors was precluded by the high incoming beam rate.
In this case, the solution was to assume that every particle arrived at the target with
x(T ) = 0. If the beam was centered at x = 0 (or close to it), this assumption will slightly
worsen the resolution of the measurement, but should not shift it. If the beam were not
centered, this would result in a reconstructed energy and x-angle that were also shifted

109

from the input values [145]. A series of transformations then produce a matrix Mi , which
transforms the measured variables to the variables needed to reconstruct the four-momentum
of the fragment at the target:






(T )
θx 

x(D)











 (D) 
(T
)
y 
 θx 








 (T ) 
 (D) 
θy  = Mi y 










 (D) 
 ∆L 
 θy 








δ
x(T )

4.4

Modeling and simulation

The final step in the analysis was to create a simulation, compare it to the data, and use it
to extract values of decay energy and width. An in-house Monte Carlo simulation took into
account the incoming beam profile, reaction, and subsequent decay as well as the geometric
acceptances and intrinsic resolutions of the detectors. In addition, the simulation was able to
reproduce cuts made on the data for a one-to-one comparison between data and simulation.
Various kinds of data were used as input to determine the parameters of the simulation.
A flow chart is shown in Figure 4.36. During the experiment, the strength of the Sweeper
magnet was measured with a Hall probe. Those values were used to create the ion-optical
maps. Several runs of data were designed to help calibrate the simulation. Runs where the
beam was sent down the central trajectory of the Sweeper magnet (“beam down center” runs)
were used to fix the incoming beam profile. Next, the current running through the Sweeper
magnet was changed to sweep the beam in discrete steps across the active area of the detectors
(“step sweeps”). This data was used to fix the beam energy. The fragment distributions are

110

Data input

Simulation
parameters

beam down center

beam profile

step sweeps

beam energy

fragment
distributions

reaction
parameters

neutron data

decay
parameters

Figure 4.36: A flow chart of the simulation process and the various data that are used as
input.
primarily dominated by the reaction kinematics, so reproducing those distributions in the
simulation sets the reaction parameters. Finally, the combination of fragment and neutron
data was used to extract the decay parameters such as energy, width, and decay type.
The motion of the charged particles was simulated using an in-house simulation titled
st mona. Acceptances were applied by eliminating particles that did not hit the active
area of the detectors and resolutions were applied by smearing the signals with Gaussians
representative of the resolution. The interactions of the neutrons with MoNA-LISA was
modeled with geant4 [146,147] and the custom neutron interaction model menate r [148].

4.4.1

Incoming beam parameters

The incoming beam parameters of the simulation were set by duplicating the distributions
from sending the unreacted beam through the detectors. In this data set, the beryllium

111

CRDC1 y-position (mm)

100

25

×103

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Counts

Counts

10

50

8

0
-50

Counts

100

×103

×103
5
Counts

CRDC2 y-position (mm)

8
7
20
50
6
5
15
0
4
10
3
-50
2
5
1
-100
0
0
-100 -50
0
50 100 -100 -50
0
50 100 -100 -50
0
50 100
CRDC1 x-position (mm)
CRDC1 x-position (mm)
CRDC1 y-position (mm)

6
4
2

4
3
2
1

-100
0
0
-100 -50
0
50 100 -100 -50
0
50 100 -100 -50
0
50 100
CRDC2 x-position (mm)
CRDC2 x-position (mm)
CRDC2 y-position (mm)

Figure 4.37: Incoming beam distributions in CRDC1 (top row) and CRDC2 (bottom row).
The left column shows the two-dimensional positional profile, the middle column shows the
x-projection, and the right column shows the y-projection.
target was retracted and an attenuated beam was sent into the Sweeper magnet, which was
set to direct the beam along the central trajectory of all detectors.
In this case, the goal was to duplicate the position and angular distributions at both
CRDC detectors, CRDC1 and CRDC2. A first look at the beam profile in position space
(Figure 4.37) shows that these were not simple Gaussian distributions. It looked like there
were three components to this beam: a horizontal peak around 30 mm in the y-position of
CRDC2, a square background and a slanted peak around -20 mm in the y-position.
Many different parameters were included in the duplication of the beam profile. A summary of them is shown in Table 4.10 and the final result is compared to the data in Figures
4.38 and 4.39. An initial estimate of the beam energy and spread was taken from the infor112

Table 4.10: List of simulation parameters used to recreate the incoming 13 B beam profile.
Part 1 recreated the horizontal peak around 30 mm in the y-position of CRDC2, Part 2
recreated the square background, and Part 3 recreated the slanted peak around -20 mm in
the y-position.
Simulation flag
cx
dx
ctx
dtx
cy
dy
fy
cty
dty
fty
sxtx
sxy
scale

Part 1 Part 2
-0.001 -0.001
0.008 0.0075
-0.004 -0.004
0.005 0.002
0.005
0.00
0.00
0.007
20
0.0055 0.00
0.0015 0.0085
20
-0.82
-0.82
20
100

Part 3
0.00
0.004
0.001
0.002
-0.012
0.0005
0.00
0.009
-0.82
0.5
7

Significance
x-distribution centroid
x-distribution width
θx -distribution centroid
θx -distribution width
y-distribution centroid
y-distribution width
y-distribution shape factor
θy -distribution centroid
θy -distribution width
θy -distribution shape factor
Rotation angle between x and θx distributions
Rotation angle between x and y distributions
Relative intensities of the components

mation from the A1900 group. Initial x, y, θx , and θy distributions were created and then
tweaked. Most distributions were modeled as Gaussian and so were defined by a centroid
and a width.
The exception to this were the y and θy distributions for the second component, the
square background. These distributions were defined with a generalized normal distribution:

β

Ae−(|x−µ|/α)

where µ, α, and β are parameters that can be varied. In this case, µ varies the location
of the centroid, α scales the width of the distribution, and β is a shape parameter. When
β = 2, this is a normal distribution with centroid µ and width α. In the limit that β = ∞,
this is a uniform density from µ − α to µ + α.
A correlation between x and θx was necessary to reproduce the beam profile. The corre-

113

×103

10

×103

×103
20

20

10

15

Counts

Counts

Counts

15

5

10
5

5

0
0
0
-100 -50
0
50 100 -100 -50
0
50 100 -100 -50
0
50 100
CRDC1 x-position (mm)
CRDC2 x-position (mm)
CRDC x-angle (mrad)
×103

×103

×103

8
20
4
15

Counts

4

Counts

Counts

6

10

2

2

5

0
0
0
-100 -50
0
50 100 -100 -50
0
50 100 -100 -50
0
50 100
CRDC1 y-position (mm)
CRDC2 y-position (mm)
CRDC y-angle (mrad)

Figure 4.38: Simulated (blue) and data (black) position and angular spectra for the incoming
13 B beam profile. Top row: CRDC1 x-position, CRDC2 x-position, and x-angle. Bottom
row: CRDC1 y-position, CRDC2 y-position, y-angle.

114

100

180
160
140

50

120
100

0

80
60
-50

40
20

-100
-100

-50
0
50
100
CRDC2 x-position (mm)

0

400
350

CRDC2 y-position (mm)

CRDC2 y-position (mm)

100

50

300
250

0

200
150

-50

100
50

-100
-100

-50
0
50
100
CRDC2 x-position (mm)

0

Figure 4.39: CRDC2 position spectra from data (left) and simulation (right) for the incoming
13 B beam.
lation was imposed on an event-by-event basis. A random value was picked for each of the
two independent distributions and then a rotation was imposed upon that coordinate pair.
The same process was used to reproduce the 12 B beam. Those beam parameters are
given in Table 4.11 and a comparison between data and simulation is shown in Figures 4.40
and 4.41.

4.4.2

Beam energy

The energy of the incoming beam was determined by the magnetic rigidity of the last magnet
before the target and the energy spread was set by momentum slits located upstream of the
experimental setup. The use of the slits created an energy distribution that was similar to
a step function, so the simulated beam energy distribution was defined by a centroid energy
and a width which was the full width of a step function distribution.
The momentum slits were set at 0.2% dp/p, which corresponded to a 0.4% spread in
115

×103

×103

×103

30
10

Counts

Counts

Counts

20

20

5

10

10

0
0
0
-100 -50
0
50 100 -100 -50
0
50 100 -100 -50
0
50 100
CRDC1 x-position (mm)
CRDC2 x-position (mm)
CRDC x-angle (mrad)
×103

×103

Counts

4

20

4
Counts

Counts

6

×103

15
10

2

2

5

0
0
0
-100 -50
0
50 100 -100 -50
0
50 100 -100 -50
0
50 100
CRDC1 y-position (mm)
CRDC2 y-position (mm)
CRDC y-angle (mrad)

Figure 4.40: Simulated (blue) and data (black) position and angular spectra for the incoming
12 B beam profile. Top row: CRDC1 x-position, CRDC2 x-position, and x-angle. Bottom
row: CRDC1 y-position, CRDC2 y-position, y-angle.

116

Table 4.11: List of simulation parameters used to recreate the incoming 12 B beam profile.
Part 1 recreated the horizontal peak around 30 mm in the y-position of CRDC2, Part 2
recreated the square background, and Part 3 recreated the slanted peak around -20 mm in
the y-position.
Part 1 Part 2
0.00
0.00
0.0055 0.0055
-0.003 -0.003
0.002 0.002
0.004
0.00
0.005 0.007
50
0.004
0.00
0.0015 0.0085
20
-0.82
-0.82
15
50

Part 3
0.00
0.004
0.003
0.002
-0.012
0.0005
0.00
0.0009
-0.82
0.25
20

CRDC2 y-position (mm)

100

50

0

-50

-100
-100

-50
0
50
100
CRDC2 x-position (mm)

Significance
x-distribution centroid
x-distribution width
θx -distribution centroid
θx -distribution width
y-distribution centroid
y-distribution width
y-distribution shape factor
θy -distribution centroid
θy -distribution width
θy -distribution shape factor
Rotation angle between x and θx distributions
Rotation angle between x and y distributions
Relative intensities of the components

200
180
160
140
120
100
80
60
40
20
0

100
CRDC2 y-position (mm)

Simulation flag
cx
dx
ctx
dtx
cy
dy
fy
cty
dty
fty
sxtx
sxy
scale

600
500

50

400
0

300
200

-50
100
-100
-100

-50
0
50
100
CRDC2 x-position (mm)

0

Figure 4.41: CRDC2 position spectra from data (left) and simulation (right) for the incoming
12 B beam.

117

energy. For the 13 B runs, the quadrupole magnet immediately before the magnet was set
at 3.2271 Tm (71.38 MeV/u). For the 12 B runs, this was set at 2.9772 Tm (71.33 MeV/u).
The centroid energy was tuned from these nominal values by comparing simulation to data
sets in which the Sweeper was set to different magnetic rigidities, which moves the centroid
of the beam in the dispersive (x) direction. These “step sweep” data sets (so called because
the beam was swept across the detectors in discrete steps) were also used to confirm that the
beam profile does not change as a function of the optics imposed by the Sweeper magnet.
The final beam energy used in the simulation was 71 MeV/u.

4.4.3

Reaction parameters

The 1p- and 2p-knockout reactions were simulated by removing the nucleons from the beam
and giving the unbound system a momentum “kick”. The momentum kick was parametrized
based on the Goldhaber model [149] for parallel momentum distributions and the model
proposed by Van Bibber [150] for perpendicular momentum distributions. In these models,
the parallel and perpendicular momentum kicks are Gaussian in shape and defined by widths,
σ for the parallel distribution and σ⊥ for the perpendicular distribution. These widths
were fixed by the observed position and angle distributions of the fragments and are given
in Table 4.12. In addition, a friction term was included to reproduce the low-momentum
tail observed in the data (Figure 4.43) and commonly attributed to dissipative interactions
within the target [151]. This friction term was created by multiplying every negative parallel
momentum kick by a constant factor. In this case, the factor 2 was chosen to reproduce the
tail observed in the data. A comparison of simulation and data for the 9 Li fragments from
the 13 B beam is shown in Figure 4.42.

118

Table 4.12: Widths of the parallel and perpendicular kicks used to reproduce the kicks of
the knockout reaction.
Beam Fragment σ (MeV/c) σ⊥ (MeV/c)
13 B
9 Li
183
244
13 B
11 Be
90
180
13 B
10 Be
122
244
12 B
9 Li
121
243

300

Counts

200

400

150
Counts

Counts

300

100

200

100

50

0

0

-100
0
100
CRDC1 x-position (mm)

100
0

-100
0
100
CRDC2 x-position (mm)

-100
0
100
CRDC x-angle (mrad)
×103

Counts

Counts

600

600
Counts

800

400

400
200
0

-100
0
100
CRDC1 y-position (mm)

200
0

-100
0
100
CRDC2 y-position (mm)

2

1

0

-100
0
100
CRDC y-angle (mrad)

Figure 4.42: Simulated (blue) and data (black) position and angular spectra for the 9 Li fragments from the 13 B beam. Top row: CRDC1 x-position, CRDC2 x-position, and x-angle.
Bottom row: CRDC1 y-position, CRDC2 y-position, y-angle. See Section 4.4.5 for information about the cut on CRDC2 y-position.

119

1400
1200

Counts

1000
800
600
400
200
0
2

2.5

3

3.5

4

4.5

5

Total momentum (GeV/c)
Figure 4.43: Total momentum of the 9 Li fragments in the data (black line) and simulation
(blue line).

120

4.4.4

Other parameters

The target area density of 51.37 mg/cm2 was fixed by measuring the thickness of the target
(0.278 mm) with a pair of calipers and converting to area density, assuming natural beryllium composition. The magnetic field strength used to create the field maps for the Sweeper
magnet was fixed by measurements by a Hall probe inserted in the field of the magnet.
The geometric acceptances of the detectors were also included. Those were based on measurements made before and during the experiment. Resolutions for MoNA-LISA included a
Gaussian resolutions for time measurements and position measurements along the length of
the bar as well as discretization in the directions perpendicular to the length of the bar [142].

4.4.5

Additional cuts

For a consistent comparison of data to simulation, additional cuts had to be made to both
the simulation and the data. As an example, intrinsic limits on the data due to the finite
physical size of the detectors had to be explicitly imposed upon the simulations. Imposed
detector limits included the x and y limits imposed by the windows in the front and back
of the ion chamber, as well as the CRDCs. The simulated light threshold for PMTs in
MoNA-LISA was raised to account for the high QDC thresholds (Section 4.2.4).
Other cuts were applied to both the simulation and the data. A “notch” of missing data
was observed in the lower half of CRDC2. It was determined that this was an effect of the
detector and not representative of the actual fragment distributions. To eliminate any bias
due to this notch, both the data and simulation were restricted to events above the beam
axis (y > 0). Due to the known non-uniformity in the y-direction of the dipole field within
the Sweeper magnet, events were also required to be within 20 mm of the dispersive plane

121

150

100

100
CRDC1 x-angle

CRDC1 x-angle

150

50
0
-50

-100

50
0
-50

-100

-150
-150 -100

-50
0
50 100
CRDC1 x-position

150

-150
-150 -100

-50
0
50 100
CRDC1 x-position

150

Figure 4.44: CRDC1 x-angle versus CRDC1 x-position for a beam sweep run with no target
(left) and for production (right). The red polygon indicates the “sweep” gate.
when they reached CRDC1. Finally, events that were bent by the Sweeper magnet in a
regular fashion had a strong correlation between x-position and x-angle after the magnet
(Figure 4.44). A cut was applied (the “sweep” gate) to select only those events.
This experiment was not designed to measure the decay of 12 Be, so the 11 Be fragment was
off-center. The x-position distribution of 11 Be fragments should be centered off the −x side
of CRDC2. The experimental spectrum, in contrast with the simulated spectrum in Figure
4.45, shows the x-position distribution covering the entire detector. This contamination was
reduced by only analyzing 11 Be fragments arriving at a CRDC2 x-position of less than -50
mm.

4.4.6

Decay models

Once the fragment distributions were reproduced, the final step was to use the data, specifically the decay energy plots, to pin down the decay parameters such as energy, width,
122

×103
16

Data

14

Simulation

Counts

12
10
8
6
4
2
0
-100

0

100

CRDC2 x-position (mm)
Figure 4.45: Full x-position spectrum for 11 Be, data (black line) and simulation (blue line).
scattering length, and decay mode.

4.4.6.1

Single neutron decay

The decay energy for a single neutron decay was modeled as an energy-dependent BreitWigner, as described in Section 2.1.1:

σ (E; E0 , Γ0 ) = A

Γ (E; E0 )
.
[E0 − E + ∆(E; Γ0 )]2 + 14 [Γ (E; Γ0 )]2

When using this model, the fitting parameters were the centroid energy, E0 , the width, Γ0 ,
the angular momentum of the neutron released, , and the relative intensity of the lineshape.
In this analysis, this lineshape was used to model the decays of 12 Be, 11 Be, and 10 Li.

123

4.4.6.2

Two neutron decay

The decays of 12 Be and 11 Li were modeled as two neutron decays to 10 Be and 9 Li, respectively. The decay of 12 Be was only modeled as a phase-space decay (Section 2.2.1), while
the 11 Li was modeled as a phase-space decay, a sequential decay, and as a dineutron decay.
In the two-neutron 12 Be, the energy was fixed by minimizing the one-neutron decay energy
(from the 12 Be→11 Be+n decay) and then converting that to a two-neutron decay energy
(for the 12 Be→10 Be+2n decay). The width was fixed with a single-particle decay width
calculation from Bohr and Mottelson [134]. For the 11 Li decay, the total decay energy, total
decay width, intermediate decay energy, second decay width, and the relative intensity were
all free fitting parameters.
The two-neutron phase space decays utilized the TGenPhaseSpace class as implemented
in ROOT [152]. The four-vector of the decaying particle and the mass distribution of the
decay products were input and the four-vectors of the three decay products were output.
The sequential decay used the relative energy distribution as defined in Section 2.2.2 to set
the energies of the two decays and then isotropically distributed the two decays. In the
sequential decay, the energies and widths of the two unbound states as well as the angular
momentum of each decay were the input used to create the lineshape. In a similar fashion,
the dineutron decay simulation utilized the energy distribution defined in Section 2.2.3 to
define the decay energies that separated the two neutrons from the charged fragment and
the two neutrons. The two “decays” were isotropically distributed. The inputs for each
dineutron lineshape were the central decay energy and the total two-neutron decay width.

124

4.5

Fitting and error analysis

Simulated spectra were compared to experimental spectra using the least-squares fitting
method, commonly called a χ2 analysis. For binned data and simulations, the following test
statistic was constructed:
n

S=

exp

(yi

− yisim (α))2
σi2

i=1
exp

where n is the number of bins, yi

(4.9)

is the experimental number of events in bin i, yisim (α)

is the simulated number of events in bin i as a function of the parameter set α, and σi is the
exp

error of yi

exp

, which is taken to be the square root of yi

. This test statistic was used for

minimization of parameters and error estimation. This test statistic is commonly referred
to as χ2 , so for consistency, that is how it will be referenced throughout the document.
For each comparison of data to simulation, the parameter set α was varied until χ2 was at
a minimum. The least squares method assumes a Gaussian distribution of events within each
bin, which cannot be assumed when the number of events in a bin is small. The inclusion
of bins with very few events could also skew the minimization to place more importance on
those events. To reduce this effect, only bins with more than 5 events were included in any
fitting procedure. It is also possible to use this statistic to test many histograms at once, by
simply expanding the sum to include bins from different histograms.
This test statistic also provides us with a way to estimate the statistical error in the
parameter minimization. The most common way to determine a 1σ statistical uncertainty
is by taking the values of αj at which χ2 = χ2min + 1. This assumes that the χ2 vs. αj
plot is parabolic in shape. Although the data is not always the perfect linear situation that
generated this definition for the 1σ limit, for consistency with comparison to other data, this
method has been used here to estimate the statistical uncertainty. The minimum of each χ2
125

vs. αj plot was fit with a parabola and limits corresponding to χ2 + 1 were extracted.
When multiple parameters were simultaneously minimized (say, α0 and α1 ), then each
point in the χ2 plot was constructed by fixing α0 and varying all other parameters (α1 , α2 ,
...) until a minimum was found. This was repeated for multiple values of α0 .
During the analysis, it was necessary to fit the same data set with different models. In
this case, a reduced test statistic was used to compare models with different numbers of
parameters. A reduced test statistic was constructed by dividing the original statistic, χ2 ,
by the number of degrees of freedom. Here, the number of degrees of freedom was defined as
the number of bins participating in the fit less the number of parameters in the set α. This
comparison of reduced χ2 was used to select a preferred model.
In addition to the statistical uncertainty in the measured values, the systematic error was
established by varying potential sources of error within reasonable limits and re-fitting that
new spectrum. Previous experience had shown that most sources of systematic error were
primarily associated with the neutron measurements, so the following sources of experimental
error were considered: MoNA-LISA global time offsets and measurements of MoNA-LISA
detector positions. In addition, three values set in the simulation were also varied to fix a
systematic error: the x-position resolution, the time-of-flight resolution, and the threshold
set for light collection at the PMTs.
Some systematic error was also identified from fitting the entire χ2 curve and not just
choosing the parameter set with the lowest χ2 . All three of the sources of systematic error in
the simulation produced the same additional error for the 10 Li fit even though the histograms
simulated with some error looked almost identical to those with no added error. The same
systematic error measurement was then generated by fitting a simulation with itself and
determined to be due to the curve fitting, rather than either resolution or the threshold.
126

This error is identified in the systematic error tables as “Error from fitting.”

127

Chapter 5
Results and Discussion

5.1

11

Li

The three-body decay energy spectrum for the 11 Li system is shown in Figure 5.1. This
spectrum uses the first two interactions in MoNA-LISA that pass a time-of-flight requirement
and a charge threshold, as detailed in Section 4.2.4. These requirements, however, do not
ensure that those first two interactions come from two different neutrons. Figure 5.1 has
contamination from neutron scattering cross-talk: events in which a single neutron interacts
in the detector, scatters, and then interacts again, creating two interaction sites from one
neutron.
To eliminate this fake two-neutron signal and enhance the real two-neutron spectrum, a
pair of requirements, known as causality cuts were applied to the data. The first requirement
was placed on the velocity of a hypothetical neutron scattering from the first interaction site
to the second (v12 ). This velocity was compared to the velocity of the neutron that traveled
from the target to the first interaction site (v01 ). If v12 < v01 , then it is possible that the first
two interactions were caused by the same neutron interacting twice. The first requirement,
then, is that v12 > v01 . The second requirement was placed on the distance between the two
interaction points. The probability that a second interaction was caused by the same neutron
as the first decreases sharply with the distance between the interactions. In addition, the
discretization in the y and z directions produces an uncertainty in the absolute position of
128

300

Counts

250
200
150
100
50
0

0

0.5

1

1.5
2
2.5
3
3.5
4
Decay energy, 3-body (MeV)

4.5

5

Figure 5.1: Decay energy for the 9 Li+n+n system. No causality cuts are applied.
∼10 cm, which corresponds to a large uncertainty in v12 . In contrast, the distribution of
distances between interactions from two different neutrons is relatively flat. Figure 5.2 shows
the number of real and fake two-neutron events that pass the causality cuts as a function of
the distance used in the causality cut for a phase space decay from a 1.5 MeV state in 11 Li
(Γ = 2.0). The cut was set at 50 cm for a combination of efficiency and purity. Though the
efficacy of these causality cuts changes as a function of energy and decay mode, the events
that make it through these causality cuts are approximately 90% real two-neutron events.
These causality cuts were applied to the three-body decay energy plot shown in Figure
5.1 and the resulting histogram is shown in Figure 5.3. The broad peak around 700 keV
appears to be consistent with previous measurements, though the width appears larger than
the measurements from Refs. [55, 62–64, 66, 67]. There is a suggestion of a state around 2.2
MeV decay energy, but the statistics are low.
Figure 5.4 shows the Jacobi plots for the 11 Li data below 1.6 MeV decay energy and with
causality cuts applied. The most obvious feature is a peak near -1 of the Y-system cos θ
129

6000
5000

Counts

4000
3000
2000
1000
0

20

40
60
80
Distance of cut (cm)

100

Figure 5.2: Number of real two neutron events (green, top line), and fake two neutron events
(red, bottom line) that pass the causality cut as a function of the magnitude of the distance
cut, based on a simulation of a phase space decay from 11 Li.

50

Counts

40
30
20
10
0

0

0.5

1

1.5
2
2.5
3
3.5
4
Decay energy, 3-body (MeV)

4.5

5

Figure 5.3: Decay energy for the 9 Li+n+n system. Causality cuts are applied.

130

spectrum, indicating directional correlation of the two neutrons. This is supported by the
asymmetric T-system energy ratio in the upper left panel. The lack of sharp peaks in either
the Y-system energy ratio or the T-system angle suggest that the decay does not proceed
sequentially through a narrow state in 10 Li.
To go beyond these initial qualitative conclusions requires comparison with the simulation. Figure 5.5 shows the potential decay paths through which an excited 11 Li nucleus could
decay. It could decay sequentially through states in 10 Li (red, blue, green, and magenta),
or it could decay in a three-body decay directly to 9 Li by emission of correlated neutrons
(dineutron model, orange) or uncorrelated emission of neutrons (phase space or democratic
model, cyan).
The qualitative analysis suggested that a dineutron model will fit best and that a sequential decay through a narrow state will not fit the Jacobi plots. There are four potential
sequential decays through 10 Li. According to Bocharev and Charity [31, 32], the energy
difference between the initial and intermediate states needs to be greater than the width of
that intermediate state for a sequential decay to be possible. Recent work from Egorova et
al. set a more stringent limit with the observation of a “suppression of the sequential decay
mechanism in favor of three-body democratic dynamics” at initial energies below twice the
intermediate energy plus the intermediate width [47]. A decay through a low-lying

=1

state in 10 Li at approximately 200 keV with a width of approximately 100 keV fits the description of a sequential decay and would produce sharp peaks in the Jacobi spectrum that
are not seen in the data. This decay (represented by the blue line in Figure 5.5) is not
considered. The lower-lying

= 0 ground state has a much larger width and remains in

consideration.
The simulation should also include potential single neutron decays from direct population
131

60

60

50

50

40
30

40
30

20

20

10

10

0

Counts

Counts

70

0

0.2

0.4
0.6
0.8
Ex /ET - T system

0

1

70

70

60

60

50

50
Counts

Counts

70

40
30

10

10
-0.5
0
0.5
cos θ - T system

1

1

-1

-0.5
0
0.5
cos θ - Y system

1

30
20

-1

0.2
0.4
0.6
0.8
Ex /ET - Y system

40

20

0

0

0

Figure 5.4: Jacobi plots for the 9 Li+n+n system. Causality cuts are applied, as well as a
requirement that the total three-body decay energy be less than 1.6 MeV.

132

2p

1

2

1

1

Ph

as

0

0
11

es

1

Energy MeV

3

1

pa
ce

Dineutron
10

Li

1

1

Li n

0
9

Li 2n

Figure 5.5: Possible decay paths from an ≈1 MeV state in 11 Li populated using a 2p knockout
from the 13 B beam. The red, blue, green, and magenta decay paths go through intermediate
states in 10 Li, while the orange and cyan decay paths explicitly bypass those states in favor
of correlated (dineutron) or uncorrelated (phase space) two-neutron emission.
(via 2p1n-knockout) of 10 Li. Those potential decays are shown in Figure 5.6. The gray dotted
line indicates population of a higher-lying state in 10 Li and subsequent decay to the ground
state of 9 Li.
In addition to the two spectra seen in Figures 5.1 and 5.3, these two-neutron decay models
were fit to two two-body decay energy spectra: one constructed with all events containing
both a 9 Li fragment and an interaction in MoNA-LISA and another constructed with events
containing a 9 Li fragment and only one interaction in MoNA-LISA. This second two-body
decay energy spectrum enhances the signal due to one-neutron decay. All four decay energy
spectra are shown together in Figure 5.7.
A fit was performed for each two-neutron decay path from 11 Li. Each fit included a
single decay path from 11 Li and all five one-neutron decay paths from 10 Li. The energies

133

2p1n

2

1
1

Energy MeV

3

1
1

1

0
0
11

10

Li

Li n

9

Li 2n

Figure 5.6: Possible decay paths from states in 10 Li populated using a 2p1n knockout from
the 13 B beam. The gray path indicates population of and decay from a much higher-lying
state.
and widths of the states were allowed to vary, as were the relative initial populations. In
all the fits, the first three decay energy spectra were dominated by contributions from the
one-neutron decays from 10 Li, while the fourth, causality-gated spectrum was dominated by
the two-neutron decay from 11 Li. The ratio of events between the different decay plots was
best reproduced with a threshold of 1.0 MeVee for MoNA-LISA’s PMTs, rather than the
initial setting of 0.6 MeVee. This generated an overall better fit as defined by χ2 and did
not influence the energy minimization.
Assuming a single decay path from a single unbound state in 11 Li allows another simplification. Decay through the highest-lying

= 1 state in 10 Li (shown as the magenta decay

in Figure 5.6) is energetically unfavorable, but a decay through the tail of the state could
occur if the state were wide enough. Regardless, given the other potential decay paths, it is
unlikely to be the sole decay path and therefore has been removed from the analysis.

134

Counts
0.5
1
1.5
2
2.5
Decay energy, 2-body (MeV)

3

300
250
200
150
100
50
0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Decay energy, 3-body (MeV)

700
600
500
400
300
200
100
00

0.5
1
1.5
2
2.5
Decay energy, 2-body (MeV)

3

50
40
Counts

Counts

Counts

1200
1000
800
600
400
200
00

30
20
10
0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Decay energy, 3-body (MeV)

Figure 5.7: Decay energy plots used when fitting simulation to data. The upper left panel
shows the two-body decay energy in coincidence with 9 Li and one neutron. The upper
right panel shows the same two-body decay energy spectrum, but this time requiring only
one interaction in MoNA-LISA. The lower left panel shows the three-body decay energy
spectrum, as shown in Figure 5.1. The lower right panel shows the same three-body decay
energy spectrum but with causality cuts, as shown in Figure 5.3.

135

350

350
Sequential, low

=1
300

Phase space

ron

ut
ine

250

χ2

χ2

300

250

D
200

200
Sequential,
600

=0

800
1000 1200
11 Li energy (keV)

500

1000
1500
11 Li width (keV)

2000

Figure 5.8: χ2 minimization plots for the 11 Li energy (left) and width (right), for many
different decay paths.
Two χ2 minimization plots for the energy and width of the 11 Li state, broken down by
decay path, are shown in Figure 5.8. In general, the minimization pushed the width to
unphysical values, so the width was required to be less than or equal to twice the central
energy value. The minimized energies and widths for each two-neutron decay model are
shown in Table 5.1, along with statistical errors. Three of the models select consistent
energies at 840±70, 860±80, and 890±70 keV, while the phase space model minimizes at
a lower energy of 750±70. The phase space decay path, while a good simple model for
reference is considered an unphysical model. A phase space decay assumes no interactions
between the decay products and there is a known neutron-neutron interaction. Eliminating
the phase space result from consideration then shows that the measurements of the decay
energy and width are insensitive to decay mode.
The model with the best fit to the decay energy spectra is a sequential decay through
the

= 0 ground state of 10 Li, as suggested by Simon et al. [55]. The best fits from all four
136

Table 5.1: Energies and widths for best fits of each two-neutron decay model. Error bars
are only statistical.
Type of 2n decay
Sequential, low = 1
Sequential, = 0
Phase space
Dineutron

Counts
0.5
1
1.5
2
2.5
Decay energy, 2-body (MeV)

3

300
250
200
150
100
50
0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Decay energy, 3-body (MeV)

700
600
500
400
300
200
100
00

50
Counts

Counts

Counts

1200
1000
800
600
400
200
00

Decay energy (keV) Width (keV)
860±80
>1300
890±70
>1400
750±70
>1200
840±70
>1400

0.5
1
1.5
2
2.5
Decay energy, 2-body (MeV)
Sequential

=1

40

Sequential

=0

30

Phase space

20

Dineutron

3

10
0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Decay energy, 3-body (MeV)

Figure 5.9: The best fits to the decay energy spectra from Figure 5.7 with all four decay
models. The upper left panel shows the two-body decay energy in coincidence with 9 Li
and one neutron. The upper right panel shows the same two-body decay energy spectrum,
but this time requiring only one interaction in MoNA-LISA. The lower left panel shows the
three-body decay energy spectrum. The lower right panel shows the same three-body decay
energy spectrum but with causality cuts.

137

Table 5.2: Energies and widths used for the best dineutron fit shown in Figures 5.9 and
5.10. Decay through a ∼200 keV = 1 state, shown in Figure 5.5, was minimized at zero
contribution to the total spectrum.
State
11 Li*
10 Li
10 Li
10 Li
10 Li

Decay energy (keV) Width (keV) Percent population
Color
840
1600
9%
orange
= 0 ground state
50
42%
red
= 1 excited state
650
1000
18%
green
= 1 excited state
1000
250
8%
magenta
= 1 excited state
2250
1400
23%
gray

Table 5.3: Sources of systematic error in the 11 Li “dineutron” decay. Sources of error in the
data are in the top section and sources of error in the simulation are in the second section.
Here, MoNA-LISA is abbreviated with ML.

Source
ML global time offsets
ML detector positions
ML x-position resolution
ML time resolution
ML PMT threshold
Error from fitting
Total systematic
Statistical
Total error

Associated error
Variance
Energy (keV) Width (keV)
±0.2 ns
20
200
±2 cm
20
100
3±1cm
10
0.18±0.05 ns
10
1.0+0.2
100
−0.4 MeVee
200
30
300
70
200
80
400

decay paths are shown in Figure 5.9. Although each fit is comprised of a single 2n decay
and five 1n decays, only the summed spectra are shown for simplicity. The Jacobi spectra
for each of these best fits is shown in Figure 5.10. As expected, the dineutron model fits the
Jacobi spectra best. The dineutron lineshapes shown in Figures 5.9 and 5.10 were composed
of one two neutron component from Figure 5.5 and all one neutron components from Figure
5.6. Those full lineshapes broken down into 2n and 1n decay path components and displayed
against the decay energy spectra in Figure 5.11 and against the Jacobi spectra in Figure
5.13. Dalitz plots for both data and simulation are compared in Figure 5.14.
The addition of systematic error (detailed in Table 5.3) increases the error margins so

138

60

60

50

50

40
30

40
30

20

20

10

10

0

Counts

Counts

70

0

0.2

0.4
0.6
0.8
Ex /ET - T system

0

1

70

70

60

60

50

50
Counts

Counts

70

40
30

10

10
-0.5
0
0.5
cos θ - T system

1

1

Sequential = 1
Sequential = 0
Phase space
Dineutron

30
20

-1

0.2
0.4
0.6
0.8
Ex /ET - Y system

40

20

0

0

0

-1

-0.5
0
0.5
cos θ - Y system

1

Figure 5.10: The Jacobi coordinate spectra (as shown in Figure 5.4) compared to the same
spectra from the best fits of the four decay models.

139

Counts
0.5
1
1.5
2
2.5
Decay energy, 2-body (MeV)

3

300
250
200
150
100
50
0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Decay energy, 3-body (MeV)

700
600
500
400
300
200
100
00

50
40
Counts

Counts

Counts

1200
1000
800
600
400
200
00

30
20

0.5
1
1.5
2
2.5
Decay energy, 2-body (MeV)

3

Sum
2n dineutron
1n = 1
1n = 0
1n = 1
1n = 1

10
0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Decay energy, 3-body (MeV)

Figure 5.11: The best fit to the decay energy spectra of Figure 5.7 with the dineutron model.
Different colored curves show the different components as labeled in Figures 5.5 and 5.6. The
black line shows the summed total. The upper left panel shows the two-body decay energy
in coincidence with 9 Li and one neutron. The upper right panel shows the same two-body
decay energy spectrum, but this time requiring only one interaction in MoNA-LISA. The
lower left panel shows the three-body decay energy spectrum. The lower right panel shows
the same three-body decay energy spectrum but with causality cuts.

140

Kobayashi (1993) [67]
Sackett (1993) [63]
Bohlen (1995) [68]
Korsheninnikov (1996) [61]
Korsheninnikov (1997) [62]
Zinser (1997) [64]
Gornov (1998) [65]
Simon (2007) [55]
Gurov (2010) [66]
0

1

2

3
4
Excitation energy (MeV)

5

6

Figure 5.12: Previous measurements of 11 Li with excitation energies less than 3.5 MeV. The
result from this work is shown by the box.
the final values are a decay energy of 840±80 keV (corresponding to an excitation energy of
1210±80 keV) and a width greater than 1200 keV. This measurement is compared to previous
results in Figure 5.12. It is consistent with 6 of the 8 measurements under 1.5 MeV. Two
of those six measurements are from experiments dominated by Coulomb excitation [63, 64].
Coulomb excitation was concluded to be a direct excitation to the continuum, not a resonant
excitation [64, 77], so the two Coulomb excitation measurements are measurements of peak
values of an excitation function rather than measurements of a resonant state.
The double charge exchange reaction from Kobayashi et al. is expected to primarily
populate positive parity states via the E1 mechanism, whereas a two-proton knockout from
the 3/2− ground state of a 13 B beam would favor the population of negative parity states. In
a Faddeev calculation of unbound states in 11 Li, Garrido et al. predicted thirteen states of

141

varying parity below 1.6 MeV excitation energy [82], suggesting it is reasonable to populate
opposite parity states near the same energy. The width of the unbound state seems large
compared to previous results and the calculations by Garrido et al.. Given the many potential
states in that region and the general lack of understanding of the unbound structure, there
is also the possibility that multiple states were populated.
The population percentages in Table 5.2 show that approximately 10% of the events came
from decay of an excited state of 11 Li while 90% came from unbound states in 10 Li directly
populated via 2p1n knockout. A two proton knockout from 13 B in a 0s21/2 0p33/2 configuration
has ten different possible knockout pairs. Knocking out the paired 0p3/2 protons is most likely
to populate the ground state and wouldn’t show up in coincidence with 9 Li. Knocking out
any of the deeply bound protons in the 0s1/2 orbital is likely to populate a high energy state
in 11 Li. At least one of the neutrons emitted from the decay will have a high kinetic energy
and be less likely to interact with MoNA-LISA. This would make the observed event appear
much more like a 10 Li decay. Two of the remaining nine possible knockout combinations
(subtracting the one likely to go to the ground state) only knock out protons from the 0p3/2
orbital and would be available for populating low-lying excited states in 11 Li. The inclusion
of real 10 Li decays from a 2p1n knockout make the measured 10% a reasonable number.
The dineutron model used here parametrizes the interaction between the two neutrons
as an

= 0 decay with a −18.7 fm scattering length, a value that is based on the neutron-

neutron scattering length measured by Gonzalez Trotter et al. [153]. The use of the −18.7 fm
scattering length is consistent with a democratic or phase-space decay modified by a finalstate interaction between the two neutrons. It is also important to note that both the data
and the model indicate a directional correlation, a correlation in momentum-space. Neither
necessarily implies a spatial correlation of the neutrons within the nucleus.
142

70

60

60

50

50
Counts

Counts

70

40
30

30

20

20

10

10

0

0

0.2

0.4
0.6
0.8
Ex /ET - T system

Sum
2n dineutron
1n = 1

70
60

1n
1n
1n

0

1

=0
=1
=1

0.2
0.4
0.6
0.8
Ex /ET - Y system

1

-1

-0.5
0
0.5
cos θ - Y system

1

60
Counts

50

40
30

40
30

20

20

10

10

0

0

70

50
Counts

40

-1

-0.5
0
0.5
cos θ - T system

1

0

Figure 5.13: The Jacobi coordinate spectra (as shown in Figure 5.4) compared to the same
spectra from the best fit from the dineutron model. Different colored curves show the
different components as labeled in Figures 5.5 and 5.6. The black line shows the summed
total.

143

The cos θ Jacobi spectrum in the Y-system suggests a slight additional correlation between the two neutrons may be evident in the data (Figure 5.13). As a test, the scattering
length was changed to −100 fm, increasing the directional correlation between the two neutrons. A comparison with the data and the −18.7 fm fit is shown in Figure 5.15. The
simulation with increased correlation visually appears to be a better fit than the −18.7 fm
simulation.
The two sequential decays (11 Li→9 Li+2 n →9 Li+2n) used in the dineutron model of the
simulation is a simplification of the physical process. A true calculation should track the
neutrons behavior from inside the nucleus all the way through the decay. The better fit
of the −100 fm scattering length in this model suggests that the data require an additional
correlation beyond the final state interaction parametrized by the −18.7 fm scattering length.
This correlation must come from some interaction inside the nucleus, but any discussion of
the source or behavior of this correlation based on this data and model would be pure
speculation.
There was no gamma-ray detection used in this experiment, so though we assume decay
to the ground state of 9 Li, it is possible that the decay was to an excited state in 9 Li. A
previous proton knockout from Chartier et al., however, found that a single proton knockout
from 11 Be to unbound 10 Li decayed to the first excited state of 9 Li only 7% of the time.
This, combined with comparison to previous results, makes decay to the excited states of
9 Li

unlikely.

144

Counts

60
40
20
01
0.8
2
Mcn

0.6
0.4
0.2
0

0

0.2

0.4 0.6
2
Mnn

0.8

10

20 40 60
Counts

0

0.2

0.4 0.6
2
Mnn

0.8

10

20 40 60
Counts

Counts

60
40
20
01
0.8
2
Mcn

0.6
0.4
0.2
0

Figure 5.14: Dalitz plots for 11 Li. The top set is the data and the bottom set is the best fit
simulation using the dineutron model. The data only uses events with a three-body decay
energy less than 1.6 MeV that also passed the causality cuts.
145

70

60

60

50

50
Counts

Counts

70

40
30

30

20

20

10

10

0

0

70

0.2

0.4
0.6
0.8
Ex /ET - T system

0

1

0.2
0.4
0.6
0.8
Ex /ET - Y system

1

-1

-0.5
0
0.5
cos θ - Y system

1

60

as = −100 fm

50
Counts

50
40
30

40
30

20

20

10

10

0

0

70

as = −18.7 fm

60
Counts

40

-1

-0.5
0
0.5
cos θ - T system

1

0

Figure 5.15: Two different dineutron models in comparison to the experimental data. The
solid line simulation used a scattering length of -18.7 fm while the dashed line simulation
used a scattering length of -100 fm, implying stronger correlation.

146

Counts

100
90
80
70
60
50
40
30
20
10
0
0

0.5

1
1.5
2
Decay energy, 2-body (MeV)

2.5

3

Figure 5.16: Decay energy in coincidence with 9 Li and one neutron, populated by two-proton
knockout from a 12 B beam.

5.2

10

Li

Like 11 Li, 10 Li was populated from a two-proton knockout, this time from a 12 B beam. The
two-body decay energy in coincidence with 9 Li is shown in Figure 5.16. A peak is visible
around 500 keV and its full-width at half-maximum is approximately 800 keV. There is also
some evidence of a shoulder around 1 MeV.
The two-proton knockout reaction is expected to selectively populate positive parity
states in 10 Li. The 1+ ground state of the 12 B beam has its three valence protons located
primarily in the p3/2 orbital. The odd proton couples with the odd neutron in the p1/2
orbital. The two protons should principally be knocked out as a pair, which will selectively
populate positive parity states. Any low-lying states will then decay by an

= 1 neutron

to the 3/2− ground state of 9 Li. This allows for the identification of any low-lying
states without significant background and without the broad
obscuring the identification.
147

=1

= 0 ground state component

100
90
80

Counts

70
60
50
40
30
20
10
0

0

0.5

0

0.5

1
1.5
2
Decay energy, 2-body (MeV)

2.5

3

2.5

3

Residual

20
10
0
-10
1

1.5

2

Decay energy, 2-body (MeV)
Figure 5.17: A fit of a single = 1 decay energy lineshape to the data from Figure 5.16 (top)
and the residual (bottom).

148

90
80
70

Counts

60
50
40
30
20
10
0
0

0.5

1

1.5

2

2.5

3

Decay energy, 2-body (MeV)
Figure 5.18: A fit of three

= 1 decay energy lineshapes to the data from Figure 5.16.

The peak around 500 keV is consistent with the previous measurements shown in Figure
1.11. Most of those measurements, however, measured a width of <500 keV, in contrast
with the wide peak seen in Figure 5.16. Using those measurements as a reference, the
decay energy spectrum was fit with a single = 1 decay lineshape whose central energy was
constrained between 450 and 650 keV. When the width of this resonance was unconstrained,
the minimization procedure pushed the width to unphysical values. To avoid this, the width
was required to be less than or equal to twice the central energy. With this requirement
applied, the best fit of a single = 1 decay to the decay energy spectrum is shown in Figure
5.17.
The residual shown in the lower panel of Figure 5.17 indicates that the fit underpredicts
the lowest portion of the spectrum (<300 keV) and the shoulder area around 1 MeV. Including a single additional resonance could not fit both of those areas, so a fit with three

149

Table 5.4: Best fit parameters as used in Figure 5.18. Quantities with error bars were
measured. Other parameters, such as relative contribution and the width, are reported
without error bars, simply for reference.
Color
Energy (MeV)
Blue
0.11±0.04
Green
0.5±0.1
Magenta
1.1±0.1

10

χ2

20
χ2

χ2

15

5

10

5
0

Percent population
13%
55%
32%

30

20

10

Width (MeV)
0.2
0.8
1.0

50 100 150 200
Blue energy (keV)

0300 400 500 600 700
Green energy (keV)

0

1000
1200
Magenta energy (keV)

Figure 5.19: χ2 plots that show the minimization of the central energies of the blue, green,
and magenta lineshapes in Figure 5.18. Dashed lines indicate statistical error and solid lines
indicate the total error margin.
= 1 decays was performed and is shown in Figure 5.18. The best fit utilized resonances at
110±50 keV (blue), 500±100 keV (green), and 1100±100 keV (magenta). The widths and
relative populations of the fit shown in Figure 5.18 are shown in Table 5.4. The details of
the systematic error are shown in Table 5.5.
The χ2 plots for the lower two resonances are shown in Figure 5.19 and the values measured in this work have been superimposed on a summary of previous measurements in
Figure 5.20. The resonance measured at 0.5 MeV is consistent with 10 previous measurements, whereas the resonance at 0.11 MeV is consistent only with the measurement from
Santi et al.. Near that energy Young et al. fit a low-lying resonance with an = 0 lineshape
but “could be a p wave or an s wave” [94] and Zinser et al. fit another resonance with an

150

Wilcox (1975) [92]
Young (1994) [94]
Kobayashi (1997) [96]
Zinser (1997) [64]
Gornov (1998)a [97]
Bohlen (1999) [99]
Caggiano (1999) [98]
Santi (2003)a [101]
Simon (2004) [102]
Jeppesen (2006) [54]
Simon (2007) [55]
Aksyutina (2008) [56]
Gurov (2011) [103]
Chernyshev (2013) [72]
0

0.5

1
1.5
Decay energy (MeV)

2

2.5

Figure 5.20: Previous measurements of unbound, = 1 states in 10 Li (same as Figure 1.11)
with the limits of the new measurements overlaid in blue, green, and magenta lines. Measurements reported without error have arrows for error bars. Two different measurements in
Chernyshev’s 2013 paper produced two different measurements of 0.7±0.05 and 0.78±0.15
MeV, here the weighted average is used [72].
a Both Gornov’s 1998 paper [97] and Santi’s 2003 paper [101] found two possible fits for the
data. Details are in Table 1.2 and footnotes within.

151

Table 5.5: Sources of systematic error in the 10 Li decay. Sources of error in the data are
in the top section. The sources of error from the simulation tabulated in Table 5.3 (ML
position resolution, ML time resolution, and ML PMT threshold) were all dominated by the
error from the fitting. Here, MoNA-LISA is abbreviated as ML.

Source
ML global time offsets
ML detector positions
Error from fitting
Total systematic
Statistical
Total error

Associated error in energy
Variance Blue (keV) Green (keV) Magenta (keV)
±0.2 ns
20
10
10
±2 cm
20
80
50
10
20
30
80
50
30
60
100
40
100
100

= 1 lineshape but stated that it was probably an

= 0 resonance, due to the slope of the

lowest three data points [64].
The two lowest-lying positive parity states in 10 Li are likely the 1+ , 2+ doublet from the
coupling of the single 0p3/2 proton with a neutron in the 0p1/2 state. Within the doublet,
the strong Nordheim rule [154] predicts the 1+ state to have a lower energy than the 2+
state. This assignment is consistent with that of Bohlen et al. [99], but the energy measured
here is approximately 0.1 MeV lower than in that work. Similarly, the two lowest-lying
negative parity states in 10 Li are expected to be 2− and 1− from the coupling of the 0p3/2
proton with the 1s1/2 neutron. The relative populations of these two states, shown in Table
5.4, is inconsistent with this assignment. The ground state of 12 B is 1+ and two-proton
knockout from there is expected to populate the 1+ state with a higher cross-section than
the 2+ state. This measurement was not the primary focus of the experiment and therefore
received limited beam time. More experimental data might resolve these two peaks and clear
up any uncertainty.
Given those spin assignments, the 0p1/2 single particle energy can be calculated using

152

4
1/2+

E − Ep1/2 (MeV)

3

Ep1/2
Es1/2

2

1−
2−

1
0
-1

1/2−

2+

2+
1+

1/2−
1/2+

1/2−
1+

1/2+

-2
9 He

10 Li

11 Be

12 B

13 C

Figure 5.21: Calculations of 0p1/2 (red) and 1s1/2 single-particle levels of 10 Li, 11 Be, 12 B,
and 13 C. Energies of single-particle levels and relevent energy eigenstates are given relative
to the calculated 0p1/2 single particle energy level. Dashed line shows the extrapolation of
the linear fit to the 1s1/2 single-particle levels. 9 He levels are from Ref. [155], 10 Li levels are
from this work, 11 Be and 12 B levels are from Ref. [156], and 13 C levels are from Ref. [157].

153

the weighted average described by Talmi and Unna [15]:

Ep1/2 =

J (2J

+ 1)EJ
J (2J + 1)

(5.1)

where J is summed over the states that results from pairing a 0p1/2 neutron with a 0p3/2
proton. The same sum can be made for Es1/2 . Talmi and Unna postulated that “the
change in interaction energy of a p1/2 , or s1/2 , neutron when two p3/2 protons are removed
is twice the change due to the removal of one p3/2 proton” [15]. Their paper extrapolated
that linearity from measurements of 12 B and 13 C to 11 Be and predicted that the 1s1/2 level
would be 0.21 MeV below the 0p1/2 level, only 0.11 MeV away from the measured 0.32
MeV level difference. When that same linearity is extrapolated out to 10 Li (Figure 5.21), it
predicts a difference of 2 MeV between the 0p1/2 and 1s1/2 single-particle energies. Based
on the measurements in this thesis, Ep1/2 = 0.4 ± 0.1. Given the current understanding that
at least one negative parity state lies below 0.05 MeV decay energy, this suggests that the
second state would need to be at least 4 MeV lower than the currently known

= 0 ground

state. This extrapolation would make 10 Li bound, which contradicts experimental evidence.
Since the difference between the 0p1/2 and 1s1/2 orbitals would be greater in 9 He (where
current measurements have it at approximately 1 MeV), the simple linear model presented
by Talmi and Una is not appropriate for calculating the position of the 1s1/2 single-particle
energy.
A Faddeev calculation by Thompson and Zhukov calculated the relationship between the
11 Li

binding energy and the 0p1/2 single-particle energy in 10 Li [80] as a function of the

1s1/2 scattering length. Figure 1 from that paper has been redrawn here in Figure 5.22. The
recent measurement of 11 Li at TITAN has placed stringent limits on the binding energy of

154

0.2
vs = vp
-11 fm
-18 fm
-27 fm
-44 fm

11 Li

binding energy (MeV)

0.1
0
-0.1
-0.2
-0.3
-0.4
-0.5

0.15

0.2

0.25

10 Li

0.3

0.35

0.4

0.45

0.5

p1/2 resonance energy (MeV)

Figure 5.22: Calculations by Thompson and Zhukov show the relationship between the 11 Li
binding energy and the 10 Li 0p1/2 single-particle energy as a function of the 1s1/2 scattering
length [80]. The line at -0.369 MeV shows the limits imposed by the measurements of Smith
et al. [51] and this thesis.

155

11 Li

(369.15±0.65 keV) [51]. That measurement and the weighted average of the two

=1

states measured here constrain the box (that looks more like a line) shown in Figure 5.22.
The location of the box indicates that the 1s1/2 virtual pole needs to be more negative than
−44 fm, which means it needs to be lower than 10 keV. To keep the 1s1/2 weighted average
below 10 keV requires that the highest-lying state of this negative parity doublet be below 27
keV. The currently known position of the = 0 ground state is below 50 keV. If both the 2−
and the 1− states are below 50 keV, it is unlikely they will be distinguished experimentally.

5.3

12

Be

Unbound 12 Be was populated via a single-proton knockout from the 13 B beam. The twobody decay energy spectrum is shown in Figure 5.23. The spectrum is dominated by a
Breit-Wigner-like peak around 1.2 MeV. Comparison with Figure 1.12 suggests that a state
here is consistent with the energy of a previously measured state at 4580±5 keV excitation
energy [114].
Given the single-particle levels available to the valence neutrons in 12 Be, we expect the
emitted neutron to have

= 0, 1, 2. The best fit with a single

= 1 resonance is shown in

the left panel of Figure 5.24. A small background (< 3% population) was required to fit
the small shoulder at low energies. The Maxwellian background used has a shape similar to
the “decay energy” of a small amount of 12 Be contamination. The best-fit shown uses an
energy of 1243±20 keV and a width of 634±50 keV. The χ2 minimization curves for each of
these parameters are shown in Figure 5.25 and a summary of the systematic error analysis
is shown in Table 5.6. Neither an

= 0 or

= 2 lineshape (shown in Figure 5.26) fit the

data equally well.

156

700
600
Counts

500
400
300
200
100
0

0

0.5

1
1.5
2
Decay energy, 2-body (MeV)

2.5

3

700
600
500
400
300
200
100
00

Counts

Counts

Figure 5.23: Decay energy of the 11 Be+n system after population of 12 Be via a two-proton
knockout from a 13 B beam.

0.5
1
1.5
2
2.5
Decay energy, 2-body (MeV)

3

700
600
500
400
300
200
100
00

0.5
1
1.5
2
2.5
Decay energy, 2-body (MeV)

3

Figure 5.24: Different simulated lineshapes fitted or compared to the 11 Be+n decay energy
lineshape shown in Figure 5.23. Left panel: data is fit with a single = 1 resonant lineshape
and a background. Right panel: data is compared to two lineshapes assuming total decay
from a previously measured state at 4580 keV in 12 Be to the excited 1/2− state in 11 Be (left
lineshape) and to the 1/2+ ground state (right lineshape).

157

200
150

100

χ2

χ2

150

50

0
1.2

100

50

1.22
1.24
1.26
Energy (MeV)

1.28

0
0.4

0.5

0.6
0.7
Width (MeV)

0.8

Figure 5.25: χ2 plots for the energy (left) and width (right) of the single = 1 decay fit to
the data in the upper left panel of Figure 5.24. Dashed vertical lines show the statistical 1σ
limits and the solid vertical lines indicate the additional systematic error.

Table 5.6: Sources of systematic error in the 12 Be decay. Sources of error in the data are in
the top section and sources of error in the simulation are in the second section.

Source
MoNA-LISA global time offsets
MoNA-LISA detector positions
MoNA-LISA x-position resolution
MoNA-LISA time resolution
MoNA-LISA PMT threshold
Error from fitting
Total systematic
Statistical
Total error

Associated error
Variance
Energy (keV) Width (keV)
±0.2 ns
10
20
±2 cm
15
40
3±1cm
5
13
0.18±0.05 ns
2
12
+0.4
0.6−0.15 MeVee
2
6
2
4
20
50
5
25
20
50

158

Table 5.7: Parameters for potential decays from positive and negative parity states in 12 Be,
using Sn =3169±16 keV [117].
Positive Negative
Decay energy (keV) 1243±20 1243±20
Multipolarity
=1
=1
Final state E* (keV)
320
0
π
−
Final state J
1/2
1/2+
E*−Sn
1563±20 1243±20
E*
4732±26 4412±26
The two bound states in 11 Be allow for the possibility of decay to either the 1/2+ ground
state or the 1/2− excited state at 320 keV. Since no gamma detection was available for this
experiment, there is no way to distinguish these two decays. The unbound state could be
at 4412±26 keV or at 4732±26 keV excitation energy, depending on if it decayed to the
ground or first excited state, respectively. If the previously measured 4580±5 keV state were
to decay to these states, it would decay with an energy of 1411 or 1091 keV, on either side
of the measured 1243±20 keV decay energy. To show the significance of this difference, two
= 1 lineshapes representing those two potential decay energies are shown in the right panel
of Figure 5.24.
These two potentially open decay channels also suggest that an unbound state in 12 Be
could decay to both bound states of 11 Be. The parity of the unbound state can be determined
by the multipolarity of the transition ( = 1) and the parity of the final state. Relevant
information for the two possible decays are shown in Table 5.7.
Though a single
An

= 0, 2 decay does not fit the data, an admixture of decays is possible.

= 0, 2 decay, which is of opposite multipolarity, would decay to the state left unpop-

ulated by the

= 1 decay. Assuming a negative (positive) parity for the unbound state,

the additional lineshape would have a central energy 320 keV lower (higher) than the

=1

decay. The fit of these two lineshapes to the data would push the central energy of the = 1
159

20
18
16
14
12
10
8
6
4
2
00

3

20
18
16
14
12
10
8
6
4
2
00

Counts
3

700
600
500
400
300
200
100
00

0.5
1
1.5
2
2.5
Decay energy, 2-body (MeV)
χ2 = 213

χ2 / bin

Counts
χ2 / bin

700
600
500
400
300
200
100
00

0.5

1
1.5
2
2.5
Decay energy (MeV)

0.5
1
1.5
2
2.5
Decay energy, 2-body (MeV)

3

χ2 = 184

0.5

1
1.5
2
2.5
Decay energy (MeV)

3

Figure 5.26: Fits of the 12 Be decay energy spectrum with an = 0 (left) or = 2 (right)
lineshape. The fits are shown on the top row and the bin-by-bin contributions to the χ2
value are shown in the bottom row.

160

lineshape higher (lower) than 1243 keV, which would in turn push the excitation energy of
4412 (4732) keV higher (lower). The addition of a second open decay channel would keep the
excitation energy within in the limits established by the

= 1 single channel decay. With

this consideration in mind, the excitation energy measured here is between 4400 and 4800
keV.
Any state within this excitation energy range is also unbound to two-neutron decay and
would be expected to release decay energies ranging from 738 to 1058 keV. The three-body
decay energy spectrum in coincidence with 10 Be fragments is shown in the lower left panel
of Figure 5.27. A peak is visible around 500 keV and perhaps another one around 1250 keV,
but there appears to be a dip in the spectrum between 739 and 1058 keV. When the causality
cut is applied (shown in the lower right panel of Figure 5.27), very few events remain and
no structure is visible. This decimation of counts suggests that very little of this unbound
state in 12 Be decays by two neutron emission.
To put a more quantitative upper limit on the branching ratio to the two-neutron decay
channel, simulations were fit to the two-body decay energy histogram, the two-body decay
energy gated on only one interaction histogram, and the three-body decay energy histogram,
all shown in Figure 5.27. The only histogram in Figure 5.27 not used in the fit was the
causality-gated spectrum in the lower right. In addition to the two-neutron decay simulation,
the fit included one-neutron decays from unbound states in 11 Be. These decays were taken
from previous measurements of neutron decays after neutron removal reactions from 12 Be
[109, 115] and included decays from the 5/2+ state and the first 3/2− state in 11 Be to the
ground state in 10 Be (decay energies of 1277 keV and 2690 keV, respectively) and the decay
of the second 3/2− state to the first excited state of 10 Be (with a decay energy of 80 keV).
In all cases, the energies and widths were fixed and only the relative intensities of each
161

300

200

Counts

Counts

250

150
100
50
0

0.5
1
1.5
2
2.5
Decay energy, 2-body (MeV)

3

60

6

50

5

40

4

Counts

Counts

0

160
140
120
100
80
60
40
20
0

30

2

10

1
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Decay energy, 3-body (MeV)

0.5
1
1.5
2
2.5
Decay energy, 2-body (MeV)

3

3

20

0

0

0

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Decay energy, 3-body (MeV)

Figure 5.27: Various decay energy spectra in coincidence with 10 Be fragments. Upper left
panel: two-body decay energy. Upper right panel: two-body decay energy spectrum, requiring only one interaction per event in MoNA-LISA. Lower left panel: three-body decay
energy spectrum. Lower right panel: three-body decay energy spectrum with causality cuts
applied.
component were varied.
The result of this fit is shown in Figure 5.28. The two-body decay energy spectra (top
row) and the three-body ungated decay energy spectrum (bottom left panel) are dominated
by signals from one-neutron decays, but a small proportion of the two-neutron decay (blue
dashed line) still contributes. Combining this with the previous simulation results in Figure 5.24, the branching ratio to the two-neutron decay energy channel has a less than 5%

162

300

200

Counts

Counts

250

150
100
50
0

0.5
1
1.5
2
2.5
Decay energy, 2-body (MeV)

3

60

6

50

5

40

4

Counts

Counts

0

160
140
120
100
80
60
40
20
0

30

2

10

1
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Decay energy, 3-body (MeV)

0.5
1
1.5
2
2.5
Decay energy, 2-body (MeV)

3

3

20

0

0

0

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Decay energy, 3-body (MeV)

Figure 5.28: The data from Figure 5.27, fitted with simulated spectra. The green, red, and
purple lineshape show single-neutron decays from 11 Be as used by Peters et al. [115]. The
blue line shows a two-neutron decay. The energies and widths of each decay were fixed,
leaving only the relative amplitudes to vary in the fitting. All spectra except the lower right
were included in the fitting.
branching ratio from the unbound state in 12 Be.
Although the range of possible excitation energies includes a 4580 keV decay, the total
width of the state measured by Fortune et al. was 107±17 keV [114], significantly smaller
than the presently observed full-width at half-maximum (approximately 1 MeV) or the
partial widths deduced from simulation. Even the inclusion of an empirical enhancement
factor of 1.6 (suggest by Fortune in Ref. [158]) cannot reconcile this discrepancy. It is likely
then, that this is a wider state with a similar energy that has not previously been measured.
163

Due to the ongoing discussion between Garrido and Fortune described in Section 1.6.2,
the spins and parities of states in this regions are also of interest. The inability to fit a single
= 0, 2 decay to the data indicates that the decay must contain the emission of an

=1

neutron. Since both of the potential final states in 11 Be have a spin of 1/2, this restricts the
spin of this state to 0, 1, or 2. The restriction can be slightly strengthened by consideration
of the reaction mechanism. The ground state of the 13 B beam is 3/2− . States of 0+ and
2+ can be populated by the knockout of a p3/2 proton, 1− and 2− states can be populated
by the knockout of a s1/2 proton, and the knockout of a proton from the p1/2 orbital can
populate states with spin-parities of 1+ and 2+ . This leaves the 0− state which can only
be populated by coupling a 3/2+ hole to the 3/2− ground state. This proton would have to
come from the 0d3/2 orbital, which is very unlikely.
We can place a loose restriction on the parity by observing that the branching ratio
to the two-neutron decay channel is very low. A recent paper by Garrido et al. reported
calculations of branching ratios for different unbound states in 12 Be [122]. The positive parity
states all showed significant branching to the two-neutron decay channel and the negative
parity states showed significantly less. This calculation suggests then, that this new unbound
state is likely of negative parity.
A continuum shell model calculation (CSM) [132] is compared to the experimental spectrum in Figure 5.29. The state measured here is shown by the grey box where the width
indicates the uncertainty in the state’s location. There are many predicted states in this
energy range of the CSM calculation. Specifically, there is a 1− state slightly lower in energy
and a 2− state slightly higher in energy. The calculated 1− state likely corresponds to the
bound 1− state at just under 3 MeV excitation energy but the 2− state is close.
The measurement of this new state doesn’t contradict the previous (t, p) measurements.
164

Data

CSM

6

1
3
2
2

5

0 ,4 ,0
3 ,2

Energy MeV

7

3
2
1

4
3
2

Sn
1
0
2

2
0

1
0

0

12

0

Be

12

Be

Figure 5.29: Level diagram of measured states in 12 Be from Figure 1.12 compared to a continuum shell model calculation of states in 12 Be [132]. The state presented here is represented
by the grey box. The width of the box represents the error of the energy measurements.
Fortune et al. stated that ”Below 6 MeV excitation energy, our data allow us to set an upper
limit of 30 µb/sr cross section for any possible missing narrow state of 12 Be” [114]. The state
measured in this work, however, with a width of 634 keV, would not have been identifiable
in that data set. This measurement is supported by a 11 Be(d, p) reaction reported in the
PhD thesis of Johansen [159]. In that work, a neutron unbound state at an excitation energy
of ≈ 4500 keV was observed. It was concluded to be the same state measured by Fortune et
al., but the width of the measurement (> 200 keV) was greater than that measured in the
(t, p) experiment.
Finally, this state could resolve the ongoing discussion between Fortune [160] and Garrido
et al. [123], by allowing the spin and parity of the narrow state to stand at 3− while providing
another state in that energy region that could be identified with the 1− state calculated by
Garrido et al.

165

Chapter 6
Summary and Conclusions
In this dissertation, measurements of three neutron-rich isotopes, 11 Li, 10 Li, and 12 Be have
been reported. All three lie in the lightest known island of inversion on the chart of nuclides.
In addition, 11 Li is a two-neutron halo nucleus whose excited states all decay by two-neutron
emission to 9 Li. As one of the lighter Borromean nuclei, 11 Li is a testing ground for threebody models that will likely be used along the neutron drip-line at higher masses. Crucial
to understanding 11 Li is accurately measuring the states of 10 Li, one of its subsystems. The
unbound state measured in 12 Be is located near an unbound state whose spin and parity
have been the subject of recent debate.
Previous measurements of unbound states in 11 Li suggested at least one state around
1000 keV excitation energy was present. The unbound state measured in this experiment
had an excitation energy of 1210±80 keV and a width greater than 1200 keV. Beyond the
measurement of the energy and width of this state, Jacobi coordinate spectra and Dalitz
plots were used to investigate correlations between the three decay products in order to
gain insight into the decay mechanism. Jacobi coordinates constructed from the momentum
vectors of the two neutrons and the daughter fragment showed a correlation between the
neutrons that was stronger than that expected from a final state interaction alone.
The population of 10 Li from 12 B was expected to populate primarily positive parity
states, which allowed for the measurement of low-lying positive parity states without interference from the negative parity ground state. The decay energy spectrum showed a broad
166

peak around 500 keV decay energy. While the peak energy was consistent with previous
measurements, the width was not. The decay energy spectrum was unable to be fit with a
single peak and required two additional peaks to fit the spectrum and be consistent with
previous measurements. A positive parity state below the 500 keV state was hinted at in
previous papers but measurements have been plagued by background and interference from
the ground state. The measurement of two low-lying

= 1 states at 110±40 and 500±100

keV decay energy are assumed to be the 1+ /2+ doublet from the coupling of the 0p3/2 proton
to the 0p1/2 neutron. Comparison of these states with a Faddeev calculation suggests that
both states of the 2− /1− doublet are below 50 keV and thus unlikely to be distinguished
experimentally. A third

= 1 state was also measured at 1100±100 keV.

The unbound state populated in 12 Be was best fit with an

= 1 decay at a decay

energy of 1243±20 keV. The two potential final states in 11 Be as well as the potential for an
= 0, 2 decay admixture complicated the assignment of the decay energy to an excitation
energy in 12 Be. The range of potential excitation energies ranges from 4400 to 4800 keV.
One state had previously been measured in this excitation energy range, at 4580±5 keV,
but a difference in widths (107±17 keV for the 4580 keV state and 634±50 for the state
measured here) is evidence that the two states are not the same. Spin selectivity from the
production mechanism and

= 1 best fit restricts the spin and parity of the unbound state.

Although every excitation energy possible is above the two-neutron separation energy, no
significant branching through the 2n decay channel to 10 Be is seen in the data. A limit of 5%
branching ratio was placed on the 2n decay channel. Most two-neutron decays are expected
to be emitted from the same orbital, so a negative-parity assignment for the unbound state
would likely produce a small branching ratio to the 2n channel. The most likely spin-parity
assignments for the unbound state are 1− or 2− .
167

These results suggest a few other studies that could aid in understanding the complex
relationships in three-body systems like those described here. A higher-statistics experiment
studying the unbound structure of 10 Li that was able to distinguish the two low-lying

=1

states would be helpful in confirming the single-particle energy of the 0p1/2 level. This
would then be used to estimate the single-particle energy of the 1s1/2 orbital, a value which
seems unlikely to be directly measurable. The more data that is gathered on 10 Li, the more
accurate the calculations and predictions of 11 Li will be. Reconciling the assortment of
measurements of unbound states in 10,11 Li would assist in understanding which components
of a three-body system calculation are most important. With regards to the 12 Be system in
particular, repeating this experiment with gamma-ray detection would allow the separation
of multiple 1n decay channels and the assignment of a parity of the state as well as an
excitation energy. It would also be useful to repeat Fortune’s (t, p) reaction with neutron
detection. Observation of the state’s decay would assist in assignment of spin and parity
and would also provide another way of distinguishing the two states.

168

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