N = 7 SHELL EVOLUTION AT AND BEYOND THE NEUTRON

DRIPLINE

By

Daniel Gregory Jenks Votaw

A DISSERTATION

Submitted to

Michigan State University

in partial fulfillment of the requirements

for the degree of

Physics - Doctor of Philosophy

2019

ABSTRACT

N = 7 SHELL EVOLUTION AT AND BEYOND THE NEUTRON DRIPLINE

By

Daniel Gregory Jenks Votaw

This dissertation reports the results of a search for the controversial parity inversion in

the unbound N = 7 isotones 9He and 10Li. Proton removal reactions from two different

radioactive ion beams (11Be, Elab = 44 MeV/u and 12B, Elab = 45 MeV/u) on a beryl-

lium target were used to directly and selectively populate unbound resonances of a given

neutron configuration in the nuclides of interest. Neutrons and fragments were measured in

coincidence using the MoNA-LISA-Sweeper setup, and invariant mass spectroscopy or the

time-of-flight method were used to determine the decay energy or relative velocity of the

neutron decays. Results on multiple unbound resonances in the 9He and 10Li systems will

be presented, as well as a discussion of ground state level inversion.

ACKNOWLEDGMENTS

My time at MSU has been very rewarding, and I owe many thanks to all of the people

who helped make it that way. First, I want to thank my advisor, Michael Thoennessen, for

always providing guidance and mentorship, even after leaving MSU and starting a new job

in a different state. His commitment to mentoring his students is unparalleled, and it was

truly an honor to be one of them. I would also like to thank the members of my guidance

committee: Daniel Bazin, Scott Bogner, Laura Chomiuk, and Wade Fisher, whose feedback

and suggestions over the years have been very helpful. I am very grateful for their time and

attention throughout my time here.

All of the members of the MoNA Collaboration also contributed greatly to my time at

MSU. The Collaboration has always been welcoming and supportive, and I will greatly miss

being a part of it.

I would like to specifically point out the collaborators who regularly

attended my weekly analysis meetings: Thomas Baumann, Paul DeYoung, and Anthony

Kuchera. Their consistent, vigilant guidance is what got me through a major DAQ/analysis

software upgrade right before my thesis experiment, all the way through years of data analy-

sis, and finally to the results which will be presented in this dissertation. I would also like to

specifically acknowledge Paul Gueye for moving from Virginia to Michigan in order to help

maintain a strong local MoNA presence at the NSCL after Michael’s move. And I would

also like to thank Paul for opening doors for collaboation with Jefferson Lab, and for his

seemingly infinite supply of “crazy” ideas. To the rest of the (present and former) MoNA

PIs with whom I’ve have the pleasure of working, Jim Brown, Joe Finck, Nathan Frank,

Jerry Hinnefeld, Bryan Luther, Warren Rogers, Artemis Spyrou, and Sharon Stephenson, I

offer my thanks to all of you as well.

iii

I also must acknowledge the MoNA graduate students with whom I’ve worked over the

years. Each and every one of them has positively influenced by graduate school experience

in some way. Michael Jones, Krystin Stiefel, Han Liu, Thomas Redpath, Dayah Chrisman,

and Hank Thurston, thank you all, and I wish you all the best of luck in your careers.

A few NSCL staff members who made great contributions to setting up my thesis ex-

periment deserve recognition: Ron Fox, Jorge Pereira, Jeromy Tompkins, and Daniel Bazin.

Without them, we may not have had functioning detectors and data acquisition software.

I would also like to thank Sean Liddick and the Nuclear Science and Security Consortium

for the opportunities to attend many conferences, workshops, and summer schools related

to nuclear nonproliferation, and the chance to spend a summer at Los Alamos National

Laboratory. Aaron Couture, Shea Mosby, Chris Prokop, and the rest of the P-27 group at

LANL deserve thanks for the great summer I had out in New Mexico.

Finally, I would like to thank my family and friends, who always supported my choice to

move to Michigan and pursue a Ph.D. in physics. None of them have ever shown me a shred

of doubt in my decision. I look forward to seeing what the next chapter brings.

iv

TABLE OF CONTENTS

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

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

Chapter 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 The nuclear landscape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 The nuclear shell model
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Shell evolution and inversion . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Previous experiments with the N = 7 isotones . . . . . . . . . . . . . . . . .

Chapter 2 Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . .
2.1
s-wave ((cid:96) = 0) lineshape . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Energy-dependent Breit-Wigner ((cid:96) > 0) lineshape . . . . . . . . . . . . . . .

Chapter 3 Experimental Techniques . . . . . . . . . . . . . . . . . . . . . . .
3.1 Primary beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Secondary beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 N2 vault . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1
Sweeper magnet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.2 Charged particle detectors . . . . . . . . . . . . . . . . . . . . . . . .
3.3.2.1 Timing scintillators . . . . . . . . . . . . . . . . . . . . . . .
3.3.2.2 Cathode Readout Drift Chambers . . . . . . . . . . . . . . .
3.3.2.3
Ion chamber . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.3 Neutron detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.4 Electronics and DAQ . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.4.1 Hardware . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.4.2
Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Invariant mass spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4
3.5 Time-of-Flight method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.1 Detector calibrations and corrections

Chapter 4 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . .
4.1.1 Charged particle timing scintillators . . . . . . . . . . . . . . . . . . .
4.1.2 CRDCs
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Ionization chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.3
4.1.4 MoNA-LISA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.4.1 QDC calibration . . . . . . . . . . . . . . . . . . . . . . . .
4.1.4.2 TDC calibration . . . . . . . . . . . . . . . . . . . . . . . .
4.1.4.3
x-position calibration . . . . . . . . . . . . . . . . . . . . . .
4.1.4.4 Tmean calibration . . . . . . . . . . . . . . . . . . . . . . .
4.2 Event selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
1
5
7
9

11
11
13

17
17
18
19
20
22
22
23
24
25
28
28
31
35
37

39
39
39
43
48
51
52
54
55
56
58

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4.2.1 Beam gate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.2 CRDC quality gates
. . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.3 Charged fragments . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.4 Neutrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Fragment tracking
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 Beam parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.2 Reaction parameters
. . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.3 Decay parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Parameter estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.1 Binned likelihood parameter estimation . . . . . . . . . . . . . . . . .
Simulated annealing . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.2

Chapter 5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . .
12B(-3p)9He . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11Be(-2p)9He . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12B(-2p)10Li . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11Be(-1p)10Li
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.1
5.2
5.3
5.4

58
59
60
64
65
69
71
73
75
76
76
79

82
82
85
92
94

Chapter 6

Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . .

96

APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

98

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

vi

LIST OF TABLES

Table 3.1: Estimated secondary beam parameters.

. . . . . . . . . . . . . . . . . . .

20

Table 4.1: Unreacted secondary beam velocities.

. . . . . . . . . . . . . . . . . . . .

40

Table 4.2: Charged particle timing detector offsets. . . . . . . . . . . . . . . . . . . .

41

Table 4.3: CRDC mask calibration parameters.

. . . . . . . . . . . . . . . . . . . . .

47

Table 4.4: The common inputs to COSY used for all settings in this analysis.

. . . .

68

Table 4.5: Sweeper current, magnetic field, and rigidities for various settings used in
this work. The Hall probe is physically located near the edge of the magnet,
so the Hall probe value is not indicative of the central field strength, which
is usually on the order of 1 Tesla. . . . . . . . . . . . . . . . . . . . . . . .

69

Table 4.6: The simulated beam parameters used in this analysis.

. . . . . . . . . . .

71

Table 4.7: The simulated reaction parameters.

. . . . . . . . . . . . . . . . . . . . .

75

Table 5.1: Statistical and systematic uncertainties in the fitting of the resonance pa-
rameters for 12B(-3p)9He. The values for the E1 and E2 parameters are for
two-sided intervals, and the values for the Γ1 and Γ2 parameters correspond
to one-sided deviations from the likelihood maximum.
. . . . . . . . . . .

Table 5.2: Statistical and systematic uncertainties in the fitting of the scattering
length for 11Be(-2p)9He. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Table 5.3: 10Li resonance parameters from the Ph.D. dissertation of J. K. Smith [1].

84

92

93

vii

LIST OF FIGURES

Figure 1.1: The Chart of Nuclides, from NNDC [2]. The colors indicate ground state
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

half-lives.

Figure 1.2: The first few single-particle orbitals in the nuclear mean field potential,
from Wikipedia [3]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 1.3: The chart of nuclides, with the observed regions of inversion highlighted,
. . . . . . . . . . . . . . . . . .

from the dissertation of J. K. Smith [1].

2

6

8

Figure 2.1: Normalized s-wave lineshapes for various values of the scattering length
(a) and binding energy (εb) parameters. The effective range parameter
was fixed at 4 fm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

Figure 3.1: The Coupled Cyclotron Facility and the A1900 fragment separator.

. . .

18

Figure 3.2: The N2 vault experimental layout.

. . . . . . . . . . . . . . . . . . . . .

20

Figure 3.3: A schematic drawing of the Sweeper magnet, from Reference [4]. Both the
superconducting Sweeper coils (horizontal) and normal-conducting trim
coils (vertical) are shown in the diagram on the right. . . . . . . . . . . .

Figure 3.4: A schematic drawing of the thin scintillator. Adopted and modified from
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Reference [5].

21

23

Figure 3.5: A schematic drawing of a CRDC, from Reference [6].

. . . . . . . . . . .

24

Figure 3.6: A schematic drawing of the ion chamber, from Reference [1]. . . . . . . .

25

Figure 3.7: A schematic drawing of a single MoNA-LISA bar, from Reference [1].

. .

26

Figure 3.8: The layout of the tables, layers, and bars of MoNA-LISA.

. . . . . . . .

27

Figure 3.9: A diagram of the DAQ hardware. Data sources are shown in green, signal
processing modules in yellow, Level 1, 2, and 3 logic in grey. Green arrows
represent START signals, red arrows represent STOP signals, and blue
arrows represent GATE signals. The cyan arrow is the system trigger
from the Thin Left Up. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 3.10: A diagram of the DAQ software. Data sources are shown in green, data
sinks in red, ring buffers in yellow, and event builders in blue. The arrows
indicate the direction of the data flow.
. . . . . . . . . . . . . . . . . . .

30

32

viii

Figure 3.11: The contents of a PHYSICS EVENT ring item in the Sweeper ring buffer,
adapted from the Sweeper DAQ Wiki [7]. Ring items like this are con-
sumed both by a Sweeper-only SpecTcl program, and the master event
builder, which will combine this item with the corresponding ring items
from MoNA and LISA. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

Figure 3.12: A schematic representation of the 2-body decay 9He → 8He + n.

. . . .

35

Figure 4.1: The first stage of time calibration of the target scintillator. The left panel
shows the raw TDC spectrum. The right panel shows the spectrum after
applying the slope, but before subtracting the jitter, and applying the
offset. These data were taken from a target-out run where the Sweeper
magnet was set to center the 11Be beam (Run 2005).
. . . . . . . . . . .

Figure 4.2: The second stage of time calibration of the target scintillator. The left
panel shows the TDC spectrum after applying the slope and jitter subtrac-
tion, but before applying the offset. The right panel shows the spectrum
after applying the offset. These data were taken from a target-out run
where the Sweeper magnet was set to center the 11Be beam (Run 2005).

Figure 4.3: The pedestal distributions for the CRDCs, taken offline at the beginning
of the experiment (Run 2001). The left panel shows CRDC1 and the right
panel shows CRDC2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 4.4: The gain-matching of the CRDC2 pads. The left panel shows the pad
summary before pedestal subtraction and gain-matching, and the right
panel shows the pad summary after. The same procedure was applied to
both CRDCs, and the reference pad for both was chosen to be Pad 75.
The data come from a chain of Run 2026 (centered 12B), and Runs 2032
- 2039 (sweeping 12B).
. . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 4.5: The mask calibration of CRDC2. The left panel shows the uncalibrated
spectrum, and the right panel shows the calibrated spectrum. Due to the
physical orientation of the detector, the calibration involves a horizontal
and vertical reflection. The data are taken from Run 2083, where the
target was removed, the 12B beam was centered, and the CRDC2 mask
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
was inserted.

Figure 4.6: The energy loss spectra of each good pad overlaid. The left plot shows all
of the raw energy loss spectra, and the right panel shows all of the gain-
matched energy loss spectra. The largest two peaks in the spectra were
fit to Gaussian functions, and their centroids were linearly transformed to
align with the reference pad (marked in red). These data were taken from
a target-out run where the Sweeper magnet was set to center the 11Be
beam (Run 2005).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42

43

45

46

48

49

ix

Figure 4.7: The horizontal position correction of the ion chamber energy loss. The
left panel shows the uncorrected spectrum, and the right panel shows the
corrected spectrum, with reduced position-dependence. The data come
from a chain of Run 2026 (centered 12B), and Runs 2032 - 2039 (sweeping
12B). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 4.8: The QDC calibration for one PMT in LISA. The left spectrum is un-
calibrated, and the right spectrum is calibrated. After calibration, the
pedestal is removed, and the features which remain are a sharp peak at
low charge due to background γ-rays, and a broad cosmic muon peak. . .

Figure 4.9: The TDC calibration for one PMT in LISA. The left spectrum is uncali-
brated, and the right spectrum is calibrated. These data are taken from a
time calibrator run, where the “picket fence” spacing is set to 40 ns. This
is used to determine the TDC slope for each PMT in the array. The offsets
are determined later, using the method described in subsequent sections.

Figure 4.10: The x-position calibration for one bar in LISA. The left spectrum is un-
calibrated, and the right spectrum is calibrated. These data are taken
from a cosmic muon run. The vertical lines in the calibrated spectrum
represent the physical ends of the bars. Some small number of counts fall
outside these lines due to the stochastic nature of the detector response.

Figure 4.11: 11Be secondary beam identification gate. A 1D cut requiring the ToF to lie
between the red vertical lines is used to select events where the incoming
particle is 11Be. The main contaminant in this case is 14C, which is easily
removed by applying this gate. The same process is applied to the 12B
beam, even though it is almost pure.
. . . . . . . . . . . . . . . . . . . .

Figure 4.12: The CRDC quality gates used for the 11Be beam. The same gates were
used for the 12B beam. The left panel shows CRDC1, and the right panel
shows CRDC2. The events above both the horizontal and vertical lines
are accepted as good events, and the rest (within the cross-hatched area)
are removed from the analysis.
. . . . . . . . . . . . . . . . . . . . . . .

Figure 4.13: Element identification for the 11Be beam. The helium element gate is
shown in red. A similar gate can be made for other elements, and the
same procedure is followed for the 12B beam.
. . . . . . . . . . . . . . .

Figure 4.14: 1D helium isotope identification for the 11Be beam. . . . . . . . . . . . .

51

53

54

56

59

60

62

64

x

Figure 4.15: Neutron selection for the 11Be beam. The same cuts are made on the 12B
data. The corrected ToF is the ToF, position-corrected as if the interaction
had occurred in the front and center bar of the array. Events which lie
above the horizontal line and between the vertical lines are identified as
neutron events. All others (within the cross-hatched area) are presumed
to be cosmic muons or photons, and are removed from the analysis.
. . .

Figure 4.16: Comparison of the CRDC1 position distributions between data and simu-
lation for the 11Be beam. The data were taken in a target-out run, where
the Sweeper magnet was set to center the unreacted beam (Run 2005).
The points are data and the curves are simulation. The same comparison
was done for the 12B (Run 2026). . . . . . . . . . . . . . . . . . . . . . .

Figure 4.17: Comparison of the CRDC angle distributions between data and simulation
for the 11Be beam. The data were taken in a target-out run, where the
Sweeper magnet was set to center the unreacted beam (Run 2005). The
points are data and the curves are simulation. The same comparison was
done for the 12B (Run 2026).
. . . . . . . . . . . . . . . . . . . . . . . .

Figure 4.18: Comparison of the CRDC2 position distributions between data and simu-
lation for the 11Be beam. The data were taken in a target-out run, where
the Sweeper magnet was set to center the unreacted beam (Run 2005).
The points are data and the curves are simulation. The same comparison
was done for the 12B (Run 2026). . . . . . . . . . . . . . . . . . . . . . .

Figure 4.19: Comparison of the CRDC1 x and fragment kinetic energy distributions
between data and simulation for the 11Be beam. The data were taken
from a chain of target-in runs. The points are data and the curves are
simulation. The same comparison was done independently for the 12B
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
beam.

Figure 5.1: The decay energy spectrum from the 12B(-3p)9He reaction. . . . . . . . .

Figure 5.2: The likelihood maximization for the first resonance energy from the 12B(-
3p)9He reaction. The points are the calculated likelihoods, and the black
curve is a cubic spline interpolation. The red vertical lines correspond to
. . . . . . . . . . . . . . . . . . . . . . . . . .
the 1σ (stat.) error band.

Figure 5.3: The 11Be(-2p)9He decay energy at the global maximum of the coarse-
grained search. The simulated parameter values are a = −2 fm, Θ = 2.5
. . . . . . . . . . . . . . . . . . .
MeV, and a relative scale factor of 0.5.

Figure 5.4: A 1D projection of the log-likelihood near the global maximum of the
coarse-grained search. The points are the calculated log-likelihoods, and
the curve is a cubic spline interpolation to guide the eye.
. . . . . . . . .

65

72

72

73

75

83

83

86

87

xi

Figure 5.5: The 11Be(-2p)9He decay energy at the maximum in the restricted pa-
rameter space. The points are data and the curves are simulations. The
simulated parameter values are a = −7 fm, Θ = 1.5 MeV, and a relative
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
scale factor of 1.5.

Figure 5.6: A 1D projection of the log-likelihood near the maximum of the coarse-
grained search, when the parameter space has been restricted to large
scale factors. The points are the calculated log-likelihoods, and the curve
is a cubic spline interpolation to guide the eye. . . . . . . . . . . . . . . .
Figure 5.7: The 1D log-likelihood of fine-grained search around a = −2 fm. The
points are the calculated log-likelihoods, and the curve is a parabolic fit.
Figure 5.8: The 1D log-likelihood of fine-grained search around a = −7 fm. The
points are the calculated log-likelihoods, and the curve is a parabolic fit.

Figure 5.9: The relative velocity spectrum from the 12B(-2p)10Li reaction. The points
are data from this work, and the curves are the simulations of the reso-
nances from the previous work. The three resonances are shown individ-
ually in the red, green, and blue curves. The black curve is the combined
simulation.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 5.10: The 11Be(-1p)10Li decay energy at the global maximum of the coarse-
grained search. The simulated parameter values are a = −1 fm, Θ = 1.0
. . . . . . . . . . . . . . . . . . .
MeV, and a relative scale factor of 0.5.

Figure 5.11: The 11Be(-1p)10Li decay energy at the maximum of the restricted, coarse-
grained search. The simulated parameter values are a = −7 fm, Θ = 1.0
. . . . . . . . . . . . . . . . . . .
MeV, and a relative scale factor of 1.5.

88

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90

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93

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Chapter 1

Introduction

1.1 The nuclear landscape

The atomic nucleus is a system of one or more nucleons (protons and neutrons), bound

by the strong force. Each particular species of atomic nucleus, called a “nuclide”, is char-

acterized by its atomic number (the number of protons, Z), its number of neutrons (N ),

and its mass number (the number of nucleons, A). The mass number is related to Z and

N via A = Z + N , so only two of these quantities are required to uniquely specify a nu-

clide. The nuclear landscape is the set of all known nuclides, plotted in the (N, Z)-plane,

as shown in Figure 1.1. A nuclear system in general may contain various discrete bound

states, quasi-discrete resonant states, and a continuum of scattering states. Each nuclear

bound or resonant state is characterized by some set of quantum numbers, including excita-

tion energy, angular momentum, parity, and isospin. In addition to these quantum numbers,

there are a number of other observable quantities of interest, such as the half-life/energy

width, electomagnetic multipole moments, decay modes, and their branching fractions. The

study of these properties of bound and resonant nuclear states constitutes the field of nuclear

structure physics.

Not only are the properties of individual nuclear levels of interest, but the systematic

trends across the nuclear landscape are of utmost importance to forming a comprehensive

1

understanding of nuclear structure. Nuclei are in general complicated, strongly-interacting

quantum many-body systems, so in most cases, there is no simple rule that dictates how any

one of these properties will evolve as a function of Z and N .

Figure 1.1: The Chart of Nuclides, from NNDC [2]. The colors indicate ground state half-
lives.

Across the nuclear landscape, observed half-lives of unstable nuclides can vary from
many times the current age of the universe, down to ∼ 10−22 seconds, for particle-unbound
nuclides. At the extreme limit of large N/Z, there is a sharp jump in half-lives from ∼ 1
ms to ∼ 10−18 ms where the neutron separation energies (the “energy cost” for removing

one neutron) become negative. The longer half-lives are for particle-bound nuclides, with

positive separation energies, which tend to decay by β-emission, whereas the shorter half-

lives are for particle-unbound nuclides, with negative separation energies, which decay by

2

nucleon emission. This transitional region is known as the neutron “dripline”, where the

neutron separation energies become negative [8]. The sharp difference in half-lives across

the neutron dripline is due to the relative strengths of the weak and strong forces, at the

energy scales relevant to nuclear structure. The “weakness” of the weak force means that β-

decay half-lives tend to be very long compared to the characteristic timescale of the “motion”
of nucleons within a nucleus (∼ 10−22 sec).

The overarching goal of nuclear structure theory is to be able to predict all of the prop-

erties of any given nuclear state within a concise, unified framework. At present, there are

many different theoretical approaches to nuclear structure, at various levels of “granularity”.

At the most fundamental level, nuclear structure is dominated by strong interactions,

and can be described using quantum chromodynamics (QCD) (with relatively small correc-

tions due to the electroweak interactions). Because QCD is non-perturbative at low energies,

the study of nuclei from QCD must be done on a discretized spacetime lattice, within the

formalism of lattice QCD (LQCD). The fundamental degrees of freedom in an LQCD calcu-

lation are the quark fields at each lattice point, and the “gauge links” between points, which

would represent the gluon fields in the continuum theory. To study a nucleus from LQCD,

interpolating operators composed of quark fields coupled to the desired quantum numbers

are constructed, and their vacuum expectation values are evaluated using the Feynman path

integral formalism in Euclidean spacetime. In the limit of large Euclidean time, the expec-

tation value decays exponentially with the energy of the ground state, which is simply the

mass of the nucleus of interest. These so-called “baryon spectroscopy” calculations allow

for the discrete nuclear energy levels to be predicted from the most fundamental description

of the strong force currently available. However the ability to model nuclei from LQCD is

currently limited by computational cost, so only the lightest nuclei are currently accessible

3

with this method [9].

At the expense of a “less fundamental” approach, gains have been made in the computa-

tional ease of modeling a nucleus. Rather than choosing quarks and gluons as the fundamen-

tal degrees of freedom from which to build nuclei, effective field theories (EFTs) of hadrons

can be used instead. In this case, mesons and baryons are treated as fundamental particles,

and the nucleon-nucleon interaction is modeled diagrammatically as a meson exchange force

[10, 11]. These EFTs can be constructed to respect the same physical symmetries as the un-

derlying theory of QCD, and nuclei can be built “ab initio” from the nucleon-nucleon forces

derived from the EFT [12]. These approaches are ultimately limited by computational cost

as well, but can presently reach mass numbers significantly higher than LQCD can.

Rather than exactly solving a quantum A-body problem with complicated ab initio in-

teractions, which may be computationally intractable for sufficiently high A, so-called “con-

figuration interaction” (CI) methods can be used. CI methods will be discussed in detail in

the next section.

For the heaviest known nuclides, even CI methods can’t be used due to the computational

cost, and density functional theory (DFT) must be used instead. DFT methods have been

used for many years in atomic and condensed matter physics [13, 14], and have since been

applied to atomic nuclei as well [15]. In these methods, rather than considering individual

particle degrees of freedom, continuous densities and currents are used, and the equations

for the nuclear density and the interactions are solved self-consistently. This method allows

for determination of ground and excited state energies, and therefore calculations of binding

energies.

The combination of these methods allows nuclear structure theorists to study nuclei
from individual nucleons, all the way up to the superheavy nuclides (Z (cid:38) 104) [16], and even

4

make predictions about the yet unobserved “hyperheavy” (Z (cid:38) 126) nuclides [17]. However

rather than dividing the nuclear landscape into regions of applicability of multiple distinct

theoretical frameworks, a unified approach which is computationally tractable over the entire

landscape, while simultaneously maintaining a close connection to the fundamental degrees

of freedom, is desired. This would arguably be the “Holy Grail” of nuclear structure theory.

1.2 The nuclear shell model

The nuclear shell model is a framework with which the time-independent Schr¨odinger equa-

tion is solved for a bound system of A non-relativistic nucleons by direct diagonalization of a

many-body Hamiltonian. The matrix elements for a given Hamiltonian are often determined

phenomenologically for a certain mass range, and the eigenvalues and eigenstates of a given

shell model Hamiltonian can then be used to predict physical properties of a given nuclide

within its range of applicability. The nuclear shell model is an example of a CI method,

mentioned in the previous section.

The motivation for developing the nuclear shell model was the experimentally observed

sudden jumps in quantities like binding energies and nucleon separation energies near certain

“magic” numbers of nucleons (2, 8, 20, 28, 50, 82, etc.). Previous models, like Gamow,

Bethe, and Weizs¨acker’s liquid drop model [18, 19], were unable to explain this behavior,

which bears similarity to the behavior of atoms with closed electron shells (noble gases).

If nucleons within a nucleus are arranged in shells similar to those of atomic electrons, it

could explain the observed behavior. The problem then became finding an appropriate set

of interactions to reproduce the experimental magic numbers.

In the late 1940’s and early 1950’s, it was determined by Wigner, Goeppert-Mayer,

5

Figure 1.2: The first few single-particle orbitals in the nuclear mean field potential, from
Wikipedia [3].

Jensen, and Haxel that a Woods-Saxon plus spin-orbit potential would produce the correct

experimental magic numbers between single-particle nucleon orbitals [20, 21], a discovery

that would go on to earn the 1963 Nobel Prize in physics. An energy level diagram for the

single-particle orbitals in this potential is shown in Figure 1.2. The physical intepretation

is that a valence nucleon primarily feels a “mean-field” potential from all of the nucleons

in the core, in addition to “residual” interactions with each of the other valence nucleons.

Then in order to calculate observable quantities from this model, all that is necessary are

the appropriate parameters for the mean-field potential, and a model for the residual inter-

actions. Many-body wavefunctions can be constructed in the basis of Slater determinants

6

of single-particle orbitals in the mean-field potential, and corrections to the wavefunctions

and energies can be calculated taking into account the residual interactions. Because the

residual interactions are not zero in general, the mean-field potential does not encompass

all of the physical interactions within the nucleus, and Slater determinants of single-particle

orbitals (also called “configurations”) are not truly eigenstates of the full nuclear Hamil-

tonian. So nuclear states with definite energies are in general linear combinations of shell

model configurations. This is known as “configuration mixing”.

Clearly this method should be expected to work best near magic numbers, where the

proton and neutron valence shells are fully-occupied, and there are no residual interactions.

As particles or holes are added, and the system moves away from closed-shell configurations,

the residual interactions become more important, and the mean-field description begins to

break down. Near a closed shell the nuclear Hamiltonian would be expected to be almost

diagonal in the basis of shell model configurations, but away from closed shells this will not

be the case. Away from closed shells, interesting collective phenomena like nuclear vibrations

and rotations emerge. Excited states will not necessarily consist of a single valence nucleon

promoted into a higher single-particle orbital, but instead many nucleons can participate

in the excitation together. A topic that is currently of great interest in nuclear structure

physics is the evolution of shell structure away from stable and near-stable nuclides. This

topic will be discussed in the next section.

1.3 Shell evolution and inversion

The shell model has been remarkably successful in predicting the structure of nuclei near

closed shells however in recent decades, emphasis in nuclear structure physics has moved

7

toward nuclei away from closed shells. It has been shown that the structure of nuclei evolves

in interesting and nontrivial ways, due to a combination of effects. For example, the energies

of single-particle configurations can evolve along an isotopic or isotonic chain, and even

rearrange in order. So in nuclei far from stability, the levels predicted by considering mean-

field configurations may be in the wrong order. This phenomenon is called “shell inversion”,

because the mean-field levels are “inverted” relative to the true level scheme. The primary

effects driving this structural evolution appear to be many-body forces [22], the tensor force

[23], and close to the driplines, continuum effects [24]. Regions of the chart of nuclides where

inversion has been observed (depicted in Figure 1.3) are known as “islands of inversion” [25].

The lightest island of inversion contains some of the N = 7 isotones, which are the subject

of this dissertation.

Figure 1.3: The chart of nuclides, with the observed regions of inversion highlighted, from
the dissertation of J. K. Smith [1].

8

1.4 Previous experiments with the N = 7 isotones

There has been a great deal of discussion in previous works about the structural evolution

of single-particle neutron orbitals along the N = 7 isotone. Given the proximity to the

“light island of inversion” and the well-known case of the 2n-halo nucleus 11Li, it’s a topic

of great interest in the study of the structure of light nuclei. It’s been suggested that the

presence or absence of inversion in 10Li may shed light on the formation of the 2n halo in

11Li [26, 27, 28, 29, 30]. It has also been indicated that presence or absence of inversion in

9He could put constraints on a low-lying “double halo”-like ground state of 10He [31, 32].

In a mean-field approach, the first six neutrons of an N = 7 nucleus completely fill

the ν(1s1/2) and ν(1p3/2) orbitals, leaving the single valence neutron to occupy a ν(1p1/2)

orbital. However it is known that the lowest-Z particle-bound isotone, 11Be, has an inverted

shell structure due a strong ν(2s1/2) occupation, evident from its well-known neutron halo.

Calculations have suggested that this inverted neutron structure could be shared by the more

proton-deficient, neutron-unbound isotones, 9He and 10Li [33, 34, 32], but there is still some

theoretical disagreement [35].

Previous experiments have yielded contradictory results about the low-lying structure of

9He [36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47]. Early experimental works [36, 37, 38, 39]

relied on concurrent nuclear structure theory to make a spin and parity assignment of 1
2

−

to

the observed ground state resonance. In the work of L. Chen, et al. [40], it was first suggested
that a low-lying s-wave virtual state had been observed with scattering length as < −10

fm. However this experiment was not able to simultaneously constrain other resonances to

confirm parity inversion. Subsequent experiments using transfer reactions to populate 9He

directly [42, 43, 46], or studying isobaric analog states in 9Li [41, 47] have not painted a

9

consistent picture of the structure of 9He. Concise summaries of the experimental results as

of 2013 and 2015 are given by T. Al Kalanee, et al. [46] and H. T. Forture [48], respectively.

10Li has also had a long and controversial experimental history [49, 50, 51, 52, 53, 54, 55,

56, 57, 39, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67]. Some experiments have provided evidence

for a low-lying s-wave neutron resonance in 10Li [49, 50, 51, 53, 54, 57, 39, 59, 60, 62, 63],

while others have not [52, 55, 56, 58, 61, 64, 65, 66, 67]. For both 9He and 10Li, many of

the previous experimental works suffer from poor statistics and high signal-to-background

ratios, making the results and conclusions somewhat uncertain.

10

Chapter 2

Theoretical Background

2.1

s-wave ((cid:96) = 0) lineshape

This section will provide a simplistic derivation of the lineshape for decays via the emission

of an s-wave neutron. The treatment here will follow what was done in Refs. [57, 68]. We

are interested in the transition matrix element between a loosely-bound neutron state from a

neutron-halo beam, to an s-wave continuum state, upon the sudden removal of some number

of protons. The initial core-neutron state will be modeled by a spatial wavefunction of the

form:

Ψ0( (cid:126)R, (cid:126)r) = f ( (cid:126)R)

e−kr
r

(2.1)

where (cid:126)R is the coordinate vector of the core and (cid:126)r is the coordinate vector of the loosely-

bound neutron relative to the core. The parameter k ≡ √

2µεb, where µ is the reduced mass

of the core-neutron system and εb is the neutron separation energy of the beam nuclide.

In the final state, the neutron wavefunction will take the form of an (cid:96) = 0 spherical wave

of the form:

ψp((cid:126)r) ∼ sin(pr + δ)

pr

(2.2)

11

where p is the relative momentum of the neutron and the fragment in the final state
( = 1), and δ is the s-wave phase shift. The matrix element that we are interesting in

caclulating is then:

(cid:90)

M ( (cid:126)R, p) =

d3r ψ∗

p((cid:126)r)Ψ0( (cid:126)R, (cid:126)r)

(2.3)

Substituting in the forms of the wavefunctions and carrying out the integration, the

matrix element becomes:

M ∼ f ( (cid:126)R)

cos δ − k
k2 + p2

p sin δ

We now convert this into an energy-differential cross section:

∝ p|M|2 ∝

dσ
dE

p

(k2 + p2)2

(cid:104)

(cid:105)2

cos δ − k
p

sin δ

(2.4)

(2.5)

Finally, we parametrize the phase shift in terms of the effective range approximation:

p cot δ ≈ − 1
a

+

1
2

r0p2

(2.6)

The previous two equations determine the s-wave lineshape used for this analysis. There

are two unknown parameters: the scattering length (a) and the effective range (r0). However

it can be shown that the shape of the distribution is very insensitive to the value of the
effective range over a physically reasonable range (∼ 2 − 8 fm), so this parameter is usually

fixed to an arbitrary value within that range [68]. For this analysis, an effective range of 3

fm was used. This leaves only the scattering length as an undetermined parameter, so it was

used as the single fitting parameter for s-wave decays in this analysis.

12

This lineshape in the limit where a → 0, corresponding to zero final state interactions

between the continuum neutron and the fragment, will be referred to in this work as a

“correlated background”. Examples of the s-wave lineshape are plotted in Figure 2.1.

Figure 2.1: Normalized s-wave lineshapes for various values of the scattering length (a) and
binding energy (εb) parameters. The effective range parameter was fixed at 4 fm.

2.2 Energy-dependent Breit-Wigner ((cid:96) > 0) lineshape

This section will provide a derivation of the lineshape for decays via the emission of a

neutron with (cid:96) > 0, using R-matrix theory, following the treatment in Reference [69]. Only

the minimum amount of R-matrix theory needed to derive the result will be covered. For a

more comprehensive treatment, the reader should consult the textbook by Thompson and

Nunes [69], or the review article by Lane and Thomas [70]. In this work, we are interested in

decays via neutron emission immediately following direct proton removal reactions. Within

the R-matrix formalism, the neutron emission can be treated as inelastic scattering. For a

13

 [MeV]decayE01234567Counts [normalized]00.20.40.60.811.21.4 = 500 keVbεa = 0 fm,  = 5 MeVbεa = 0 fm,  = 500 keVbεa = (cid:173)10 fm,  = 5 MeVbεa = (cid:173)10 fm, s(cid:173)wave lineshapestransition from some entrance channel α to some exit channel β, the R-matrix can be written

as a sum over energy poles λ:

(cid:88)

λ

γλαγλβ
ελ − E

Rβα =

(2.7)

where E is the bombarding energy and ελ is the energy associated with the pole λ.

Then the transition S-matrix element can be written in terms of the R-matrix as:

S = t1/2 H− − aR(H(cid:48)− − ηH−)
H+ − aR(H(cid:48)+ − ηH+)

(2.8)

where tα ≡ 2/2µα, H± are diagonal matrices with the Hankel functions on the diagonals,

and η is the logarithmic derivative of the R-matrix evaluated at the channel radius a. The

Hankel function of a given partial wave is given by a linear combination of regular and
irregular Coulomb functions, H±

(cid:96) = G(cid:96) ± iF(cid:96), and the Coulomb functions are the solutions

to the time-independent radial Sch¨odinger equation with a repulsive Coulomb potential.

We now define the penetrability factor:

and the L matrix:

and the Ω matrix:

Pα ≡ kαa

F 2
α + G2
α

Lα ≡ H(cid:48)+

α
H+
α

− η

Ωα ≡ eiφα, tan φα ≡ − Fα
Gα

14

(2.9)

(2.10)

(2.11)

With a similarity transformation to put the S-matrix into symmetric form, it can be

written as:

˜S ≡ v1/2Sv−1/2 = Ω

(cid:104)
1 + 2iP1/2(cid:16)

1 − aRL

(cid:17)−1

RP1/2(cid:105)

Ω

(2.12)

where v is a diagonal matrix with elements vα = kα/µα.

Assuming there are only two relevant channels, the elements of the transformed S-matrix

are given explicitly by:

(cid:34)

˜Sβα = eiφα

δβα +

2iP

1/2
α γαγβP

1/2
β

(ελ − E)(1 − aR11L1 − aR22L2)

We now define the partial widths:

and the total resonance width:

Γα ≡ 2γ2

αPα

Γ ≡(cid:88)

α

Γα

The expression for the S-matrix elements simplifies to:

(cid:34)
δβα − iΓ

1/2
α Γ

1/2
β

E − Er + iΓ/2

(cid:35)

iφβ

e

˜Sβα = eiφα

where the total resonance energy Er is defined by:

Er ≡ ελ − γ2

1 − γ2

1S0

2S0
2

15

(cid:35)

iφβ
e

(2.13)

(2.14)

(2.15)

(2.16)

(2.17)

From elementary multi-channel reaction theory, the relationship between the cross section

and the S-matrix element is given by:

σβα =

π
k2
α

2J + 1

(2Ip + 1)(2It + 1)

| ˜Sβα|2

(2.18)

where the subscripts “p” and “t” refer to “projectile” and “target”, respectively. Explic-

itly plugging in the S-matrix element gives:

(cid:32)

(cid:33)(cid:34)

(cid:35)

ΓαΓβ

(E − Er)2 + Γ2/4

(2.19)

σβα =

π
k2
α

2J + 1

(2Ip + 1)(2It + 1)

This is the familiar functional form of an isolated Breit-Wigner resonance, however the

quantity Er is not exactly equal to the pole energy ελ. Instead, it’s the pole energy, shifted

due to the couplings between channels. This dependence can instead be absorbed into

an “energy-dependent width”, and the cross section can be written explicitly in terms of

the pole energy. This form was chosen to illustrate the similarity to a standard Breit-

Wigner lineshape. The name of this lineshape is an “energy-dependent Breit-Wigner”, or

“asymmetric Breit-Wigner” distribution. The final form, in terms of the pole energy and

total width is:

σ(cid:96)(E; ελ, Γ(cid:96)) ∝

(cid:16)

(cid:17)2
ελ − E + ∆(cid:96)(E; ελ, Γ(cid:96))

Γ(cid:96)

+ 1

4Γ2
(cid:96)

(2.20)

This is the lineshape used to fit decays with (cid:96) > 0 in this analysis. The free parameters

are the pole energy and the total width.

16

Chapter 3

Experimental Techniques

This chapter provides an overview of the experimental setup in the N2 vault of the National

Superconducting Cyclotron Laboratory (NSCL) at Michigan State University (MSU).

3.1 Primary beam

The first step in producing the two secondary beams of interest is to produce a primary beam

for in-flight projectile fragmentation. The primary beam of 120 MeV/u 18O is provided by

the CCF [71]. Figure 3.1 depicts everything necessary to produce a fast radioactive ion

beam for delivery to the experimental area. The stable primary beam particles originate

in an electron-cyclotron resonance (ECR) ion source, where electron cyclotron motion in a

plasma is resonantly excited using microwaves. This causes heating of the plasma, and an

increased degree of ionization. The 18O ions are then extracted from the ion source and

delivered to the K500 cyclotron for the first stage of acceleration. After being accelerated
to ∼ 10 MeV/u in the K500 cyclotron, the primary beam is transported and injected into

the K1200 cyclotron for the second stage of acceleration. There is a stripper foil inside the

K1200 that increases the charge state of the primary beam upon its injection into the K1200

to aid with acceleration. After the second stage of acceleration, with a final kinetic energy

of 120 MeV/u, the primary beam is directed onto a beryllium production target. Projectile

fragmentation of a beam particle of mass number Ap will produce every bound nucleus with

17

A < Ap according to some stastical distribution. The task now becomes separating out the

fragments of interest in order to deliver a relatively pure secondary beam to the experimental

area.

For the 11Be secondary beam, the production target had a thickness of 2938 mg/cm2,

and for the 12B secondary beam, the production target had a thickness of 3523 mg/cm2.

Figure 3.1: The Coupled Cyclotron Facility and the A1900 fragment separator.

3.2 Secondary beams

In order to separate out a desired species for a secondary beam, the A1900 fragment separator

[72, 73] is used. The A1900 is a series of magnets consisting for four main dipoles and eight

quadrupole triplets. There are three intermediate images where separation of unwanted

fragments occurs, and finally an achromatic focus in the A1900 focal plane, where the beam

is then directed into one of the various experimental areas. The main dipoles are tuned to

the magnetic rigidity (Bρ = p/q) of the secondary fragment of interest. The dipole magnets

introduce a dispersion to the beam (a correlation between the deviation from central rigidty

and horizontal position). At Image 1, the further off-rigidity a particle is, the further it

is from the beam axis. A set of horizontal slits with controllable width are then used to

18

physically remove particles that are sufficiently off from the desired Bρ. After Image 1, the

beam enters the second dipole and is directed toward a wedge at Image 2, which provides

differential energy loss according to Z2. The third dipole disperses and directs the beam to

Image 3, where a second set of slits is used to remove more off-rigidity particles from the

beam. Finally the fourth dipole exactly cancels the remaining dispersion and the filtered

secondary beam is brought to an achromatic focus in the A1900 focal plane, where there are

various detectors and diagnostics available. The quadrupole triplets throughout the A1900

are used for beam focusing in both the horizontal and vertical directions. By the end of

the A1900 separator, the 11Be beam was 85% pure, with the main contaminant being 14C
(∼ 15%), and the 12B beam was ∼ 99% pure.

3.3 N2 vault

This section provides an overview of the experimental setup witin the N2 vault (Fig. 3.2).

Properties of the secondary beams upon delivery to the experimental vault are summarized

in Table 3.1. Once the beam is delivered to the vault, it impinges onto a reaction target in

which the unbound nuclides of interest are produced. The reaction target used with both

secondary beams was a 188 mg/cm2 of beryllium.

Following the reactions in the target, the fragments enter the Sweeper magnet (dis-

cussed in Section 3.3.1, where their trajectories are bent horizonally into a vacuum chamber

containing multiple charged particle detectors. The vacuum chamber contains two Cathode-

Readout Drift Chambers (CRDCs) [74], an ionization chamber, a thin BC-404 scintillator,

and a CsI(Na) hodoscope. The CsI(Na) hodoscope was not used for this analysis, but the

rest of the charged particle detectors will be described in detail in Section 3.3.2.

19

Figure 3.2: The N2 vault experimental layout.

Table 3.1: Estimated secondary beam parameters.

Beam Energy [MeV/u] Rate [pps] Purity [%]
11Be
12B

2 × 105
4 × 105

44
45

85
99

The prompt neutrons emitted from the unbound nuclide travel, unaffected by the Sweeper
magnet, across a ∼ 6 meter flight path to MoNA-LISA, an array of 272 (10 cm) × (10 cm)
× (2 m) BC-408/EJ-200 scintillator bars, with a photomultiplier tube (PMT) on each end

[75]. The neutron detectors will be described in detail in Section 3.3.3.

With the combination of the charged particle and neutron detector arrays, the full four-

momentum vector of both the fragment and neutron can be determined for each valid event.

3.3.1 Sweeper magnet

The Sweeper magnet is a large-gap dipole superconducting electromagnet used to sweep

charged reaction products and unreacted beam out of the path of the neutrons. The vertical

gap has a full height of 14 cm in order to accomodate the neutron cone between the target

location and the front face of MoNA-LISA. The Sweeper has a maximum magnetic rigidity

20

of 4 Tm, with a bending radius of about 1 meter, and a bending angle of 43.3°. The magnetic

field inside the Sweeper gap is monitored using a Hall probe, and the details of the shape of

the magnetic field have been characterized in previous works [76, 77, 78]. The Sweeper field

is controlled by directly setting the current that flows through the superconducting coils.

To center the trajectories of the 11Be and 12B secondary beams, the Sweeper current

was set to 211 Amps (Bρ = 2.639 Tm) and 173 Amps (Bρ = 2.320 Tm), respectively. To

center the 8He decay products in the production setting, the current was set to 355 Amps

(Bρ = 3.776 Tm).

Figure 3.3: A schematic drawing of the Sweeper magnet, from Reference [4]. Both the
superconducting Sweeper coils (horizontal) and normal-conducting trim coils (vertical) are
shown in the diagram on the right.

The side of the Sweeper magnet is also fitted with two normal-conducting trim coils, to

help cancel the fringe field outside the magnet. It has been shown in previous works that

these coils have negligible effect on the field within the gap. The trim coil current was set to

200 Amps during this experiment. A schematic illustration of the Sweeper magnet is shown

21

in Figure 3.3.

3.3.2 Charged particle detectors

There are various charged particle detectors positioned along the path of the secondary beam

and fragments whose signals are used in the analysis, beginning at the A1900 extended focal

plane (XFP), and ending with a suite of detectors in the vacuum chamber just downstream

of the Sweeper magnet. The A1900/XFP scintillator, target scintillator, and the “thin”

scintillator are all used for timing (the target and thin scintillators provide charge signals

as well, but they are not used in this analysis). The CRDCs provide transverse positions

(x and y) at two different locations in z (separated by about 1.55 meters), allowing for

the determination of transverse angles. The ionization chamber detects energy loss as the

fragments traverse its gaseous active volume. There is also a thick CsI(Na) scintillating

hodoscope at the end of the Sweeper vacuum chamber for total energy deposition, but it

was not needed for this analysis. Each detector system will be described in detail in the

following sub-sections.

3.3.2.1 Timing scintillators

Various plastic scintillators are used in this analysis to extract timing information from

charged ions. Specifically, there are the A1900 extended focal place (XFP), target, and thin

scintillators. All of these detectors are organic scintillators optically coupled to photomulti-

plier tubes (PMTs). The XFP and target scintillators are made of BC-404 with thicknesses
of 1000 µm and 420 µm, respectively. The thin scintillator is a 55 cm × 55 cm × 5 mm

block of EJ-204 with four PMTs attached to it, one on each corner, coupled to the scintillator

via light guides. The PMTs are named “Left Up”, “Right Up”, “Right Down”, and “Left

22

Down”, corresponding to their locations. The thin scintillator PMTs are biased to 1260 V,

1290 V, 1380 V, and 1260 V, respectively, and the target scintillator PMT is biased to 1500

V. A diagram of the thin scintillator is given in Figure 3.4.

Figure 3.4: A schematic drawing of the thin scintillator. Adopted and modified from Refer-
ence [5].

3.3.2.2 Cathode Readout Drift Chambers

This work involved two Cathode Readout Drift Chamber (CRDC) detectors, separated by

1.55 meters. CRDCs are gas-filled detectors used for tracking of charged fragments. The
active area of each detector is 30 cm × 30 cm, and the active gas is a mixture of 20%

isobutane and 80% CF4, at a pressure of 40 Torr. Inside each CRDC, a drift voltage of 1000

V is applied. When an ion traverses the detector, it creates some ionization in the active gas.

The freed electrons then drift along the direction opposite the drift electric field, towards a

23

Frisch grid. Beyond the Frisch grid is a horizontal anode wire, biased to 1150 V, which serves

to create an electron avalanche. Parallel to the anode wire, there is a segmented plane of 2.54

mm cathode pads. The charge distribution induced on the pads will provide information

about the horizontal (x) coordinate of the incident ion, and the drift time of the electrons

to the anode will profive information about the vertical (y) coordinate of the incident ion.

A more detailed description of the CRDC detectors can be found in Refs. [74, 6], and a

schematic drawing is shown in Figure 3.5.

Figure 3.5: A schematic drawing of a CRDC, from Reference [6].

3.3.2.3 Ion chamber

The ion chamber is similar to the CRDCs, but it is used for particle identification (PID)

rather than tracking. The ion chamber is segmented into 16 pads in the longitudinal (z)

direction. The active gas is P10, with an operating pressure of 500 Torr. The drift voltage is

24

set to 1000 V, and the anode voltage is set to 200 V. Similarly to the CRDCs, ionization in

the active gas leads freed electrons inducing a charge on the pads. The pad charge signals are

pre-amplified and processed through peak-sensing analog-to-digital converter (ADC) circuits.

The combined signal amplitude from all of the pads is proportional to the energy loss of the

incident ion as it passes through the active gas. A diagram of the ion chamber is shown in

Figure 3.6.

Figure 3.6: A schematic drawing of the ion chamber, from Reference [1].

3.3.3 Neutron detectors

The neutron detectors used in this analysis were the bars of the MoNA (Modular Neutron Ar-

ray) and LISA (Large-area multi-Institutional Scintillator Array) arrays [75]. The combined

MoNA-LISA array consists of 272 plastic scintillator bars (BC-408 for MoNA and EJ-200 for
LISA). Each bar has dimensions of 10 cm × 10 cm × 2 m, and an optically-coupled PMT

on each end (Photonis XP2262/B for MoNA and Hamamatsu R329-02 for LISA). The bars

25

are wrapped with reflector, and then with black tape in order to enhance light transmission

inside the bar, and prevent external light from leaking in and corrupting the output signals.

A schematic of a bar is shown in Figure 3.7.

Figure 3.7: A schematic drawing of a single MoNA-LISA bar, from Reference [1].

In this experiment, the bars were stacked into 17 “layers”, each 16 bars high. All of the
detectors were aligned with their centers at 0◦ from the beamline. Layers A through H of

MoNA were stacked on a single table (Layer I was located at Los Alamos National Laboratory

for detector characterization experiments). LISA was segmented into two separate tables,

consisting of Layers J - M (LISA I) and Layers N - R (LISA II), respectively. In order of

increasing z distance from the target position, the tables were ordered: LISA II, LISA I,

MoNA. The layout of the bars and layers of MoNA-LISA is shown in Figure 3.8.

26

Figure 3.8: The layout of the tables, layers, and bars of MoNA-LISA.

Each PMT has three feedthroughs: an HV supply input, an anode output, and a dynode

output. The anode signal is used for timing pickoff, and the dynode signal is used for

measuring light output in the scintillator. The important physics comes with the timing

signals, while the light output signal is optionally used to enhance the neutron event selection.

The segmentation of the array allows for localization of neutron hits in both the vertical

(y) and depth (z) directions. In order to localize the hits along the length of the bars (x),

the difference between the timing signals of the PMTs on each end is used. The average of

the two timing signals, measured relative to the common stop (the target scintillator) gives

the time-of-flight of the neutron from the target to the hit in the neutron array. Both the

3D position and the time-of-flight for each neutron hit are recorded, so this fully determines

the neutron four-momentum, assuming it originated from the target position.

27

LISAIINOP0123456789101112131415BarQRLISAIJK0123456789101112131415LMABC0123456789101112131415MoNADEFGHLayerˆzˆyˆx3.3.4 Electronics and DAQ

3.3.4.1 Hardware

The MoNA-LISA-Sweeper DAQ hardware has been thoroughly explained in previous disser-

tations (Refs.

[76, 79, 80, 1, 6, 81, 5]), but the main points will be highlighted here. The

three subsystems (MoNA, LISA, and Sweeper) each have their own independent acquisition

hardware, connected by the “Level 3” logic system, which generates a system trigger that

is sent to each of the individual subsystems and a 64-bit timestamp unique to each event.

In this experiment, the system trigger comes from the Thin Left Up PMT. The logic for

this experimental setup is handled by field-programmable gate arrays (FPGAs) in Xilinx

Logic Modules (XLMs). The logic hardware is separated into three “Levels”. Levels 1 and 2

are used to determine whether a valid event (valid time signals from both PMTs of a given

bar) was detected in MoNA-LISA. Level 3 contains a clock which is counting whenever the

system is not busy, and the coincidence logic between MoNA-LISA and Sweeper.

This experiment was run in Sweeper singles mode, meaning that the whole system is

triggered by the Sweeper, and read out whether or not there is a coincident event in MoNA-

LISA. When Level 3 receives a signal from the Thin Left Up detector, a coincidence gate is

opened, and the system waits for a valid signal from MoNA or LISA. Because the Sweeper

provides the trigger, MoNA-LISA will never be read out unless there is a coincidence with

Sweeper. So if MoNA-LISA receives a signal without Sweeper (for example due to a cosmic

muon), it will cause Level 3 to go busy, preventing a system trigger from the Sweeper, and

leading to a fast clear of MoNA-LISA.

Each PMT in the MoNA-LISA array outputs two signals for a valid event: an anode signal

and a dynode signal. The anode is used for timing, while the dynode is used to measure the

28

amount of light deposited on the photocathode. The anode signals are processed through

constant fraction discriminators (CFDs) before being sent to time-to-digital converter (TDC)

circuits, as well as the XLMs for trigger logic. The dynode signals are sent directly into

charge-to-digital converter (QDC) circuits.

The suite of Sweeper detectors includes three plastic timing scintillators (the thin scintil-

lator, the target scintillator, and the A1900 XFP scintillator), two CRDCs, and the ionization

chamber. All of the timing signals from the plastic scintillators are processed through CFDs

and sent to TDCs. In addition, the timing signal from the Left Up PMT of the thin scintilla-

tor is sent to Level 3 to serve as the system trigger. The charge signals from each pad of the

ionization chamber are sent through shaping amplifiers and then to an analog-to-digital con-

verter (ADC). The charge signals from each CRDC pad are processed by frontend electronics

(FEE) housed on each CRDC detector, before being sent to XLMs.

A diagram of the DAQ hardware in its entirety is shown in Figure 3.9.

29

Figure 3.9: A diagram of the DAQ hardware. Data sources are shown in green, signal
processing modules in yellow, Level 1, 2, and 3 logic in grey. Green arrows represent START
signals, red arrows represent STOP signals, and blue arrows represent GATE signals. The
cyan arrow is the system trigger from the Thin Left Up.

30

3.3.4.2 Software

Each of the three subsystems have their own readout programs using NSCLDAQ 11.2

[82]. Each readout program runs on the DAQ computer dedicated to each device (sp-

daq40, spdaq42, and spdaq34, for MoNA, LISA, and Sweeper, respectively), and each has

its own corresponding single-producer, multi-consumer ring buffer (named “mona”, “lisa”,

and “sweeper”). (The “sweeper” ring consists of event-built data from both the VME and

CAMAC portions of the Sweeper hardware, as shown in Figure 3.11.) A ring buffer is a

data structure with a fixed size and modulo addressing, and two pointers called put and get.

The ring buffer is defined to be empty if (put == get) is true, and full if (++put == get)

is true. When a data producer wishes to store data in the ring buffer, it puts that data at

the location of the put pointer, and the pointer is incremented. When a consumer wishes to

take data from the ring buffer, it reads from the location of the get pointer, and the pointer

is incremented. Thus, the ring buffer operates as a first-in-first-out queue in which data are

enqueued by one producer and dequeued by some number of consumers. A more detailed

discussion can be found in the NSCLDAQ documentation [82].

31

Figure 3.10: A diagram of the DAQ software. Data sources are shown in green, data sinks
in red, ring buffers in yellow, and event builders in blue. The arrows indicate the direction
of the data flow.

When all of the subsytems are being operated in tandem, as is the case during an ex-

periment, there is a “master” readout program which enslaves the readouts of each of the

subsystems, and a master ring buffer (named “built”). The master readout program handles

event building, where the ring items of corresponding physics events from each subsystem

32

are combined using the unique 64-bit timestamp carried with each ring item. Finally, a

consumer can grab the event-built ring items from the “built” ring and write them to a data

file. The binary data file then just needs to be unpacked offline and saved into raw ROOT

files, which can then be calibrated and analyzed.

33

Figure 3.11: The contents of a PHYSICS EVENT ring item in the Sweeper ring buffer,
adapted from the Sweeper DAQ Wiki [7]. Ring items like this are consumed both by a
Sweeper-only SpecTcl program, and the master event builder, which will combine this item
with the corresponding ring items from MoNA and LISA.

34

The data producers in this system are the detectors: MoNA, LISA, and Sweeper. The

Sweeper system is further subdivded into VMUSB and CCUSB controlers for the VME and

CAMAC portions, respectively. The data consumers are various SpecTcl programs, and

eventually the event files which are written to disk. (SpecTcl is a tool for online viewing and

analysis of the incoming data [83].) A diagram of the DAQ software is shown in Figure 3.10.

3.4

Invariant mass spectroscopy

In this section, the relativistic kinematics of a 2-body decay of the form A → B + C will be

considered, and an expression for the decay energy will be derived from conservation of four-

momentum. An example of this kind of decay relevant to this work is shown in Figure 3.12.

The decay energy is simply the amount of energy released by the decay in the center-of-mass

frame. Units with c = 1 will be used.

Figure 3.12: A schematic representation of the 2-body decay 9He → 8He + n.

Conservation of energy and momentum requires that the total four-momentum of the

initial state be equal to the sum of the four-momenta of all particles in the final state.

Therefore,

pµ
A = pµ

B + pµ

C

35

(3.1)

(pµ

A)2 = (pµ

B + pµ

C )2

(3.2)

Squares of four-momenta are Lorentz-invariant, so they can be evaluated in any frame of

reference. Defining the decay energy to be the rest energy of the initial particle, minus the

sum of the rest energies of the final particles, the quantity on the left side will be evaluated

in the center-of-mass frame:

(pµ

A)2 = (Ed + mB + mC )2

The quantity on the right side of the previous equation factors into three terms:

(pµ

B + pµ

C )2 = (pµ

B)2 + (pµ

C )2 + 2(pB)µ(pC )µ

(3.3)

(3.4)

The squared terms are trivial to evaluate in the rest frame of each particle, and the cross

term is evaluated in the laboratory frame.

(pµ

B + pµ

C )2 = m2

B + m2

C + 2(EBEC − (cid:126)pB · (cid:126)pC )

Combining these results, and solving for the decay energy, the result is:

(cid:113)

Ed =

m2

B + m2

C + 2(EBEC − (cid:126)pB · (cid:126)pC ) − mB − mC

(3.5)

(3.6)

Now the decay energy is specified for an arbitrary 2-body decay, in terms of the masses,

total energies, and three-momenta of the two particles in the final state. This equation

is symmetric between particles B and C. For the purposes of this dissertation, “B” will

36

represent a neutron, and “C” will represent the daughter of the decay, referred to as the

“fragment”.

3.5 Time-of-Flight method

Invariant mass spectroscopy works very well for cases where the fragment of interest moves

approximately along the design trajectory through the Sweeper magnet. However, if one

wishes to analyze fragments with a magnetic rigidity significantly different than what the

Sweeper magnet is set to, the small-angle approximation may be violated, and the application

of a COSY optics matrix can give unreliable results. Instead of analyzing a decay energy,

which relies on knowledge of the fragment angles at the target position, one can analyze a

somewhat less robust observable, the “relative velocity”:

vrel ≡ |(cid:126)vn| − |(cid:126)vf|

(3.7)

To calculate this quantity, the neutrons are treated in the exact same way that they are

in the invariant mass method. However the fragment angles are no longer necessary, and the

fragment kinetic energy is simply calculated from the time-of-flight between the target and

the thin scintillators, without involving the COSY optics matrix. The flight path length of

the fragment is not known on an event-by-event basis, so a single value is used, representing

the arc length of the path of the centroid of the magnetic rigidity distribution for the desired

fragment. In order to ensure that this approximation is valid, a tight gate is made around the

center of the neutron x versus y distribution in order to restrict the analysis to cases where

the fragment and neutron are emitted in the forward/backward direction in the center-of-

mass frame. The path length can be used as a tunable parameter (within uncertainties) to

37

align the centroid of the relative velocity distribution with zero. The important information

in the relative velocity distribution is carried in the shape and the width rather than the

location of the centroid, which should be close to zero.

Because the relative velocity requires less information than the decay energy, it can be

calculated even in cases where the fragment is not centered in the Sweeper magnet. But

the fact that it contains less information also means that it is a less sensitive observable

for fitting. To summarize: a decay energy should be used when possible, but when a decay

energy is not accessible, a relative velocity can be used instead.

38

Chapter 4

Data Analysis

4.1 Detector calibrations and corrections

This section details the calibrations and corrections performed on all of the various detectors

used in this analysis.

4.1.1 Charged particle timing scintillators

Signals from the various charged particle timing detectors are used (after constant-fraction

discrimination) to START and STOP time-to-digital converter (TDC) circuits. In this ex-

perimental setup, the TDCs are operated in common start mode, meaning that all TDC

channels are started based on a common signal from Level 3, and the TDC STOP is pro-

vided by each individual detector. So the timing signal for each detector is measured relative

to the common start. However, only time differences are physically relevant for this analysis,

so the overall timing offset for each detector is arbitrary. In this analysis, the convention

will be that t = 0 should represent the passage of the secondary beam through the target

position. Offsets for each timing detector will be set to enforce this condition.

The TDC slopes, on the other hand, are not arbitrary. They represent a conversion

factor between TDC channels and physical times, measured in nanoseconds. All of the

charged particle timing signals are processed through different channels in the same Mesytec

39

Table 4.1: Unreacted secondary beam velocities.

Beam: Speed after A1900 XFP [cm/ns]: Speed after target scintillator [cm/ns]:
11Be
12B

8.83
8.90

8.92
9.01

TDC module, and the slopes are fixed by the hardware to be 0.0625 ns/channel.

The constant-fraction discrimination helps to mitigate timing walk, however the raw

TDC spectra are still distorted by jitter, introduced by the Field Programmable Gate Arrays

(FPGAs) controling the Sweeper logic. The FPGA jitter varies on an event-by-event basis,

but it is common to all of the charged particle timing detectors. In order to correct for this

effect, a jitter subtraction is done using the Thin Left Up time signal. Subtracting the Thin

Left Up signal from all of the others before applying the individual offsets helps to cancel

the common FPGA jitter.

Taking the above value of the slope, the raw time spectrum from each detector in a

target-out run can be converted into partially calibrated, jitter-subtracted time spectra in

nanoseconds, but without any offset. Then, using the known distances between each detector

and the target position, and the known speed of the secondary beam based on the A1900

rigidity acceptance, an offset can be applied such that the peak in each time spectrum can

be placed at the appropriate physical time. Even though there is no reaction target present

during these runs, the beam will still lose a small amount of energy as it passes through

the scintillator detectors. However it is straightforward to calculate these energy losses

in LISE++, and use the appropriate beam velocity when calibrating each detector. The

relevant velocities are shown in Table 4.1. Because there are two different secondary beams

used in this analysis, with different velocities, the timing detector offsets are determined

independently for each secondary beam. The results are shown in Table 4.2.

40

Table 4.2: Charged particle timing detector offsets.

Detector:
Thin Left Up
Thin Left Down
Thin Right Up
Thin Right Down
Target scintillator
A1900 scintillator

z position [cm]:

11Be TDC offset [ns]:

12B TDC offset [ns]:

405
405
405
405
-107
-1088

45.9
47.2
48.4
50.1
14.6
-571.2

45.6
46.9
48.0
49.7
12.3
-571.5

The offsets are all very similar between the two different calibrations, except for the

target scintillator, where the difference in offsets is about 2 ns. This difference is due to

the fact that the bias voltage applied to the PMT on this detector was changed between

the calibration runs. This difference does not affect any other quantities in the analysis, as

the time-of-flight is only used for particle identification (PID). As long as PID can still be

achieved, even the absolute value of the time-of-flight can be treated as arbitrary.

The target and A1900 scintillators are standalone detectors, but the “thin scintillator”

represents the combination of the four quadrants of the detector (Left Up, Left Down, Right

Up, and Right Down), each of which has its own PMT and signal processing circuits. After

each of the quadrants is calibrated in the way described above, they are combined in software

into a single quantity by taking the arithmetic mean of their calibrated times for each event.

An example of the charged particle timing scintillator calibration is shown in Figures 4.1

and 4.2.

41

Figure 4.1: The first stage of time calibration of the target scintillator. The left panel
shows the raw TDC spectrum. The right panel shows the spectrum after applying the slope,
but before subtracting the jitter, and applying the offset. These data were taken from a
target-out run where the Sweeper magnet was set to center the 11Be beam (Run 2005).

42

Raw time [TDC channel]05001000150020002500300035004000Counts110210310RawTime [ns] (arbitrary offset)050100150200250Counts110210310With slope appliedFigure 4.2: The second stage of time calibration of the target scintillator. The left panel
shows the TDC spectrum after applying the slope and jitter subtraction, but before applying
the offset. The right panel shows the spectrum after applying the offset. These data were
taken from a target-out run where the Sweeper magnet was set to center the 11Be beam
(Run 2005).

4.1.2 CRDCs

The CRDCs are two gas-filled detectors used to measure the positions and angles of the

fragments after they travel through the Sweeper magnet. Each one is segmented horizontally

into 128 pads (the first 12 are not present in the physical detectors, so there are really only

116 active pads). The charge induced on each pad is read out for each event and processed

through an analog-to-digital converter (ADC) circuit. By fitting the charge distribution as

a function of pad number, the physical position of each event in the horizontal (x) direction

can be inferred. And by measuring the time for the the freed electrons to drift vertically

through the detector, a vertical (y) position is measured as well. The drift time for each

43

Time [ns] (arbitrary offset)35−30−25−20−15−Counts110210310410510Jitter(cid:173)subtractedCalibrated time [ns]25−20−15−10−5−0Counts110210310410510Offset appliedevent is recorded using a time-to-amplitude converter (TAC) circuit. Therefore each CRDC

provides an (x,y) coordinate pair for each event, and the combination of the two CRDC (x,y)

positions determines the angles of each fragment relative to the beam axis in the horizontal

and vertical directions. This section discusses the calibrations performed on the CRDC

detectors in this work.

Noise in the signal processing electronics creates a “pedestal” signal which shows up in

the energy loss spectrum of each CRDC pad. Before any further calibrations are done, an

offset is applied to subtract this pedestal from the spectrum of each pad. These offsets are

determined by running the acquisition code offline while the CRDCs are turned on. Only

the pedestal signal will show up in the pad charge spectra. Then each pad spectrum can be

fit to a Gaussian function, and the centroid of that Gaussian fit is the pedestal value for that

pad. In all subsequent uses of the CRDCs, those pedestal values will be subtracted from the

raw data in each pad, in each CRDC.

The CRDC pedestal distributions can be seen in Figure 4.3.

44

Figure 4.3: The pedestal distributions for the CRDCs, taken offline at the beginning of the
experiment (Run 2001). The left panel shows CRDC1 and the right panel shows CRDC2.

Once the pedestals are removed from each CRDC pad, the pads must be gain-matched

to each other in order to reconstruct the x-position correctly. The CRDC pad gain matching

is very similar to the ion chamber pad gain matching, described in detail in the next section.

Any pads that are not functioning properly are declared “bad pads”, and removed from the

analysis. In theory the removal of a few isolated pads does not hinder the determination

of the x-position, as long as a Gaussian fit to the charge distribution as a function of pad

number can still be performed. The gain-matching is performed with a chain of data runs in

which the unreacted 12B beam is swept across the vacuum chamber by varying the Sweeper

current.

It can be helpful to apply an energy loss gate on the ion chamber in order to

isolate a single element and get the cleanest possible pad summary spectra. The results of

gain-matching CRDC2 are shown in Figure 4.4. In the raw spectrum, four distinct bands

are visible. This is because the pad charge distribution is time-sampled four times, and

the results are averaged in order to obtain the raw charge induced on each pad. The three

45

Pad number020406080100120Energy loss [arb.]020406080100120140160180200110210310410CRDC1Pad number020406080100120020406080100120140160180200110210310410CRDC2“shadow” bands below the main peak correspond to cases where one, two, or three samples

for that pad are missing.

Figure 4.4: The gain-matching of the CRDC2 pads. The left panel shows the pad summary
before pedestal subtraction and gain-matching, and the right panel shows the pad summary
after. The same procedure was applied to both CRDCs, and the reference pad for both was
chosen to be Pad 75. The data come from a chain of Run 2026 (centered 12B), and Runs
2032 - 2039 (sweeping 12B).

After the gain matching is done, the position calibrations can be performed. The position

calibrations are done using “mask runs”, where a tungsten mask (0.25 inch thick) with holes

drilled at specific, known locations is placed in front of each CRDC. Highly-charged heavy

ions will be stopped within the thickness of the tungsten, but pass freely through the holes.

This leads to an image of the mask holes being projected onto each CRDC, and then the

coordinates of each hole can be mapped via a linear function to the physical values. Some

low-Z ions may be able to penetrate through the entire thickness of the tungsten and degrade

the quality of the mask calibration data, but these events can be removed in software by

placing a gate on something with high energy loss in the ion chamber, isolating particles

46

Pad number020406080100120Energy loss [arb.]01002003004005006007008009001000110210310410RawPad number02040608010012001002003004005006007008009001000110210310410510Gain(cid:173)matchedTable 4.3: CRDC mask calibration parameters.

Detector: x slope [mm/pad]: x offset [mm]:
CRDC1
CRDC2

-186.103
186.055

2.54
-2.54

y slope [mm/channel]:

y offset [mm]:

-0.0761
-0.0767

106.040
98.610

of high Z. Even though the physical origins of the horizontal and vertical positions signals

(fitting of pad charge distributions, and a TAC time spectrum, respectively) are different,

the mask calibration works in the exact same way for both: a linear calibration function is

used to map the raw signal onto physical positions in millimeters. The slope of the x-position

is fixed by the geometry of the pads, but the x-offset, y-slope, and y-offset are determined

by linear fits to the locations of the holes. For this experiment, the CRDC1 mask run was

Run 2028 and the CRDC2 mask run was Run 2083. Both runs were taken with the target

out and the 12B beam centered in the Sweeper magnet. The results of the CRDC2 mask

run are shown in Figure 4.5, and the calibration parameters used in this analysis are shown

in Table 4.3.

47

Figure 4.5: The mask calibration of CRDC2. The left panel shows the uncalibrated spectrum,
and the right panel shows the calibrated spectrum. Due to the physical orientation of the
detector, the calibration involves a horizontal and vertical reflection. The data are taken
from Run 2083, where the target was removed, the 12B beam was centered, and the CRDC2
mask was inserted.

4.1.3 Ionization chamber

The ion chamber is a gas-filled detector which is used to measure the energy loss of charged

ions which pass through its active volume. The detector is segmented into 16 pads along

the beam axis, and the charge induced on each pad is read out for each event. Each of the

signals is processed through an ADC circuit. For this analysis, four of the ion chamber pads

(Pads 0, 4, 7, and 15) were dysfunctional, and were removed from the analysis. The energy

loss value used in the analysis comes from the summed signals on each of the good pads, and

in order to improve the energy resolution of the detector, the pads must be gain-matched

to each other. This is done in software by applying a slope and offset to each good pad

in order to align peaks in a calibration spectrum to those of an arbitrary reference pad.

48

x(cid:173)position [pad number]020406080100120y(cid:173)position [TAC units]05001000150020002500Rawx(cid:173)position [mm]150−100−50−050100150y(cid:173)position [mm]150−100−50−050100150CalibratedThe calibration data were taken with the target out and the secondary beam centered in the

Sweeper magnet. Because the beam is not pure, and particles of different Z will lose different

amounts of energy in the active gas, there will be multiple peaks in the energy loss spectrum

of any given pad. These peaks can be fit to Gaussian functions, and their centroids can be

used to determine the coefficients of the linear function necessary to align each pad with the

reference pad. The result of the ion chamber gain-matching is a linear function which maps

the energy loss in raw ADC channels to an energy loss in arbitrary, physical units. Because

the ion chamber energy loss is only used for particle identification, the numerical value in

absolute units is not relevant. The pad spectra before and after gain-matching are shown in

Figure 4.6.

Figure 4.6: The energy loss spectra of each good pad overlaid. The left plot shows all of
the raw energy loss spectra, and the right panel shows all of the gain-matched energy loss
spectra. The largest two peaks in the spectra were fit to Gaussian functions, and their
centroids were linearly transformed to align with the reference pad (marked in red). These
data were taken from a target-out run where the Sweeper magnet was set to center the 11Be
beam (Run 2005).

After the ion chamber is gain-matched, no further calibration is necessary, and the energy

49

Energy loss [ADC channel]050100150200250300350400450500Counts110210310410RawEnergy loss [arb.]050100150200250300350400450500Counts110210310410Gain(cid:173)matchedloss is simply proportional to the sum of the signals in each of the gain-matched pads. How-

ever there are some unwanted dependencies that can be corrected in order to achieve better

energy resolution, and therefore cleaner particle identification. The ionization and charge

collection inside the ion chamber are correlated with both the horizontal (x) and vertical (y)

positions of the incident particle (horizontal moreso than vertical). These dependencies can

be corrected one-by-one by fitting the correlation to some polynomial function, and dividing

the summed energy loss parameter by that polynomial. Formulaically:

∆Ex =

∆E(cid:80)Nx

i=1 αixi

(4.1)

where ∆E is the uncorrected (gain-matched) energy loss signal, and ∆Ex is the energy

loss after correction for x-dependence. Then to correct the y-dependence, simply repeat the

procedure using the x-corrected energy loss:

∆Exy =

∆Ex(cid:80)Ny

i=1 βiyi

(4.2)

Nx and Ny are the chosen orders of the polynomial fit functions (both were set to 3 for

this analysis), and αi and βi are the fit parameters. The x and y values used for the position

corrections are the x and y positions of the particle measured by CRDC2. Now the energy

loss signal for the ion chamber, still in arbitrary units, can be taken to be ∆Exy, which

has reduced dependencies on the horizonal and vertical positions. This makes for a cleaner

particle identification plot, which makes event selection easier.

The position corrections are done using a chain of target-out runs with the 12B secondary

beam, where the Sweeper current is varied in order to sweep the unreacted beam horizontally
across the CRDC detectors. The sweep runs allow access to regions of large |x|, where the

50

position-dependence is strongest. The horizontal position correction is demonstrated in

Figure 4.7. The vertical correction is done in the same way, except the dependence is much

weaker. The position dependences in this analysis are very weak, so these corrections are

not particularly important. This concludes all of the calibrations and corrections needed for

the ion chamber.

Figure 4.7: The horizontal position correction of the ion chamber energy loss. The left panel
shows the uncorrected spectrum, and the right panel shows the corrected spectrum, with
reduced position-dependence. The data come from a chain of Run 2026 (centered 12B), and
Runs 2032 - 2039 (sweeping 12B).

4.1.4 MoNA-LISA

Each bar of MoNA-LISA has two PMTs, and each PMT provides a time signal and a light

output (“charge”) signal. The charge signal is processed through a charge-to-digital converter

(QDC), and the time signal is processed through a time-to-digital converter (TDC) (after

discrimination). Both the time and the charge spectra are calibrated, and then there are

additional calibrations applied for the horizontal (x) position, for tying the timings of each

51

Dispersive position [mm]150−100−50−050100150Energy loss [arb.]100120140160180200220240260280300−10110UncorrectedDispersive position [mm]150−100−50−050100150Energy loss [arb.]0.40.60.811.21.41.6−10110Correctedbar to the other bars within the same table, and finally, the overall timing of each table

relative to the common stop.

Most of the calibrations are done offline using cosmic muons, but some are done using

a time calibrator module, or online using bremsstrahlung photons. The cosmic muons are

ultrarelativistic, so their speed can be approximated as c. Before any calibrations are per-

formed, each of the PMTs needs to be gain-matched to the others. This is done offline,

before the experiment begins. The procedure is to take cosmic ray data, and iteratively

change the bias voltage on each PMT until the cosmic peak in the charge spectrum aligns

with a fixed distance from the QDC pedestal. The final voltages for each PMT typically lie

within the range of 1300-1950 V. Once the gain-matching is complete, the QDC calibrations

are performed. The MoNA-LISA calibrations are performed with the array operating in

standalone mode, so no coincidence with Sweeper detectors is required.

4.1.4.1 QDC calibration

The QDC calibration consists of determining a threshold, which is used by the data acquisi-

tion system to ignore the QDC pedestal, and a linear calibration to set the cosmic muon peak

to about 20 MeVee (MeV electron equivalent, the amount of light produced by an electron

with a kinetic energy of 1 MeV). This value comes from the average energy loss of a cosmic

muon in the detector bars (from Reference [84]), and the thickness of each bar. If qp is the

location of the pedestal in QDC channels, the QDC threshold is calculated as:

Q0 =

qp
16

+ 2

(4.3)

The factor of 16 is used to convert the channel number from 12 bits to 8 bits, and the

52

2 is simply to place the threshold slightly above the pedestal value. The linear calibration

function which converts the raw charge in QDC channels (qr) into a calibrated charge in

MeVee (qc) is then:

qc = m(qr − qp)

(4.4)

Figure 4.8: The QDC calibration for one PMT in LISA. The left spectrum is uncalibrated,
and the right spectrum is calibrated. After calibration, the pedestal is removed, and the
features which remain are a sharp peak at low charge due to background γ-rays, and a broad
cosmic muon peak.

The determination of the calibration parameters is automated using Gaussian fits to find

the locations of the pedestal and the cosmic peak for each PMT. The resulting calibrated

spectrum should have the pedestal subtracted, and the cosmic peak aligned with 20 MeVee.

And example of the QDC calibration for one PMT is shown in Figure 4.8.

53

Charge [QDC channels]050010001500200025003000Counts110210310410510RawCharge [MeVee]0102030405060708090100Counts110210310410510Calibrated4.1.4.2 TDC calibration

The first step in calibrating the neuton time-of-flight spectra is to determine the TDC slopes

for each individual PMT in the array. This is done using a time calibrator module, which pro-

duces a START signal, and a STOP signal randomly at discrete intervals after the START.

This produces a “picket fence” in the MoNA-LISA time spectra, and the spacing of the picket

fence is tunable in the hardware of the time calibrator. For this analysis, the picket fence

spacing was chosen to be 40 ns. Using the known spacing, the slope of a linear function can

be determined in order to convert the raw peak locations in TDC channels to physical times

in nanoseconds. This calibration only determines the slope, so there is still an overall offset

which will be determined in subsequent sections. An example of a TDC slope calibration is

shown in Figure 4.9.

Figure 4.9: The TDC calibration for one PMT in LISA. The left spectrum is uncalibrated,
and the right spectrum is calibrated. These data are taken from a time calibrator run, where
the “picket fence” spacing is set to 40 ns. This is used to determine the TDC slope for
each PMT in the array. The offsets are determined later, using the method described in
subsequent sections.

54

Time [TDC channels]05001000150020002500300035004000Counts110210310RawTime [ns]100150200250300350400450500Counts110210310Calibrated4.1.4.3

x-position calibration

In order to determine the x-position of each hit in the array, the time difference between

the signals in the left and right PMTs is used. Because light propagates through the bars

at constant speed, this time difference should be proportional to the x-position where the

light originated from. In order to determine the slopes and offsets of the linear calibration

function for each bar, cosmic muons are used to illuminate the bars uniformly in position.

A histogram of the time differences for each bar should then have a rectangular shape.

The edges of the rectangle can be fit to sigmoid functions, and a slope and offset can be
determined to map the edges of the bars to their physical distances (±100 cm). In terms

of the TDC-calibrated left and right PMT times (tl and tr, respectively) the calibration

function has the form:

x = m(tl − tr) + b

(4.5)

This calibration determines the parameters m and b. An example is shown in Figure

4.10.

55

Figure 4.10: The x-position calibration for one bar in LISA. The left spectrum is uncali-
brated, and the right spectrum is calibrated. These data are taken from a cosmic muon run.
The vertical lines in the calibrated spectrum represent the physical ends of the bars. Some
small number of counts fall outside these lines due to the stochastic nature of the detector
response.

However, this only determines the position of the hit relative to the center of the bar, it

does not contain any information about the absolute locations of the bars in the experimental

vault. In order to account for the locations of each bar relative to the beam axis, there is

an additional offset for the locations of the bars in beamline coordinates. The bar locations

were determined from laser measurements.

4.1.4.4 Tmean calibration

The TDC calibrations discussed above set the slopes for each TDC, but there is still an

overall timing offset which must be determined for each bar. The calibrated time-of-flight

for a given bar is calculated as the average of the time-of-flights registered by each PMT.

The overall timing offset is applied to this averaged time. The timing offset can be split into

56

Time difference [ns]30−20−10−0102030Counts110210310Rawx [cm]200−150−100−50−050100150200Counts110210310Calibratedtwo parts, where each bar has some offset relative to a reference bar (typically the top bar

in the first layer of each table), and then each table has an offset with respect to the target.

The first part, the relative offset between bars in a given table, is determined using cosmic

muons. The idea behind this calibration is simply to select high-muliplicity events, where

a single muon interacts with many bars on the same table. Then, using the differences in

the calibrated positions of the hits, and the known speed of the muon (assumed to be c),

an absolute timing offset can be applied to each bar relative to the reference bar (shown in

Equation 4.6). This is done separately for each of the three tables, and then all that remains

is to place each table on an absolute time scale relative to the target scintillator.

|(cid:126)rj − (cid:126)ri|

c

tij =

(4.6)

Finally, the second part of the overall timing offset is determined online using bremsstrahlung

gamma rays. When the beam passes through the target, bremsstrahlung x-rays are produced,

and some of them will deposit their energy into MoNA-LISA. Using the calibrated position

of the x-ray hit in MoNA-LISA, and the known speed of the photons, the mean time-of-flight

for each table can be placed into an absolute time scale. This calibration assumes that the

photons originate from the target position. One should be aware that there will also be

bremsstrahlung from the unreacted beam hitting the walls of the Sweeper vacuum chamber,

but these are not usable for the time-of-flight calibration, because they originate from some

unknown position, not at the target location. Once each bar and each table has been placed

on the same absolute time scale, the MoNA-LISA calibrations are complete.

57

4.2 Event selection

In order to reconstruct a decay energy spectrum of a given unbound nuclide, we need to

isolate the events in which the decay products of that unbound nuclide are detected in co-

incidence. Since the trigger logic for the entire system is based on the Sweeper detectors, a

coincidence with a charged fragment is required for any event in MoNA-LISA. But coinci-

dence alone does not guarantee that the detected event is an event of interest. Impurities

in the secondary beam, reactions other than the reaction of interest in the target, and ran-

dom coincidences between the Sweeper and MoNA-LISA systems will all result in spurious

events in the decay energy spectrum, if not properly removed. The following subsections will

describe the process of removing these spurious events from the analysis.

4.2.1 Beam gate

Secondary beams coming from the A1900 separator are not pure. The first step in event

selection is to isolate events coming from the desired secondary beam, removing events

coming from the undesired “contaminant” beams. This is done using a software cut (the

“beam gate”) based on the time-of-flight between the A1900 XFP scintillator and the target

scintillator. For this analysis, making a 1D gate on this spectrum is sufficient to greatly

suppress contaminants, as shown in Figure 4.11. For secondary beams with a higher amount

of contamination, and/or smaller separation between species, additional requirements may

need to be employed.

58

Figure 4.11: 11Be secondary beam identification gate. A 1D cut requiring the ToF to lie
between the red vertical lines is used to select events where the incoming particle is 11Be.
The main contaminant in this case is 14C, which is easily removed by applying this gate.
The same process is applied to the 12B beam, even though it is almost pure.

4.2.2 CRDC quality gates

To make good identification of fragments of interest, and correctly reconstruct their kinetic

energies at the target position, good measurements positions and angles at the CRDCs are

needed. However the CRDC detectors sometimes give erroneous results due to incomplete

charge collection. In order to remove these events from the analysis, “CRDC quality gates”

are made to select events in which both CRDCs functioned properly. The CRDC quality

gates used in this analysis are shown in Figure 4.12. Events with very low pad sum (the

59

Time of flight [ns]9095100105110115120Counts050100150200250300350310×C14Be11sum of the charge collected on each pad) and/or abnormally small fit σ may give unreliable

results for the position, and are removed from the analysis using these gates.

Figure 4.12: The CRDC quality gates used for the 11Be beam. The same gates were used
for the 12B beam. The left panel shows CRDC1, and the right panel shows CRDC2. The
events above both the horizontal and vertical lines are accepted as good events, and the rest
(within the cross-hatched area) are removed from the analysis.

4.2.3 Charged fragments

Once the desired seconday beam has been selected using the beam gate, there are still

potentially multiple different reaction channels that the beam particles can undergo when

they interact with the target nuclei. One particular reaction channel contains the unbound

nuclide of interest, and the rest do not, so they must be removed from the analysis. To select

the appropriate reaction channel, the selection is made on the charged fragments detected

by the Sweeper detectors.

The process of selecting the fragments of interest is referred to as “particle identifica-

tion” (PID). PID is a two-step process in which the detected fragments are separated by

60

Pad sum [arb.]0100020003000400050006000700080009000 [number of pads]σFit 012345678110210310CRDC1Pad sum [arb.]0100020003000400050006000700080009000 [number of pads]σFit 012345678110210310CRDC2Z (element identification), and then within a given element, they are separated by A (iso-

tope identification). Once individual isotopes of the desired elements have been separated,

software gates can be made to select events with a given species of fragment in the final

state.

Element identification is done using the energy loss of the charged fragments as they

pass through the ionization chamber detector. From the well-known Bethe-Bloche formula

for the energy loss of non-relativistic heavy ions, the stopping power is related to the Z of

the fully-stripped ion by:

− dE
dx

= Z2f (v)

(4.7)

where f (v) is some function that depends on the speed of the ion, but not its charge

[85]. The speed of the ion is related in the obvious way to its time-of-flight (ToF). So a 2D

histogram of the ion chamber energy loss versus ToF between the target and thin scintillators

will show bands of constant Z2. Then, an element can be selected in software by drawing

a 2D gate around the corresponding band in the 2D histogram. An example is shown in

Figure 4.13.

61

Figure 4.13: Element identification for the 11Be beam. The helium element gate is shown in
red. A similar gate can be made for other elements, and the same procedure is followed for
the 12B beam.

After element identification is complete, all that remains for PID is separating out the

isotopes of the desired element. The Sweeper magnet produces dispersion of the fragments

based on magnetic rigidiy (Bρ = p/q). The Sweeper magnet current was set such that

particles with the rigidity of 8He should travel along the design trajectory, and given that

the neighboring isotopes of helium are unbound, isotope separation in this case mainly just

entails separating 8He from 6He. This can be done by simply plotting the dispersive phase

space coordinates (x, θx) of the fragments at the CRDCs. The plot should show two main

groups, one corresponding to 8He, and one corresponding to 6He (with some contamination

62

Time of flight [ns]50556065707580859095100Energy loss [arb.]050100150200250300110210310HHeLiBedue to leakage of other particles into the helium Z gate). If the Sweeper current is indeed

set to center 8He, then the group which crosses (x, θx) = (0, 0) will be 8He, and the other

will be 6He. In order to improve separation, one can include the correlations of the phase

space coordinates with the ToF. For simplicity, rather than looking at the full 3D correlation

between these three variables, one can define a “corrected angle” which is some nonlinear

(typically quadratic) combination of the x and θx coordinates, and then plot that quantity

against the ToF from the target to thin scintillators. At this point, a 2D gate can be drawn

around the group of interest, or if desired, the distribution can be rotated and projected onto

an axis in order to make 1D isotope ID gates (see Figure 4.14). If the isotope separation is

not very clean, making 1D gates is less arbitrary than just drawing a 2D gate around each

group. However for this experiment where the fragments are light and with some of the

neighboring isotopes being unbound, the different isotopes are very well-separated, and it is

straightforward to omit the last step and simply draw 2D isotope ID gates.

63

Figure 4.14: 1D helium isotope identification for the 11Be beam.

4.2.4 Neutrons

Once the charged particle event selection is complete, a MoNA-LISA spectrum in coincidence

with the desired fragment can be made. This spectrum will in general contain neutrons

which are in true coincidence with the fragment, and unwanted accidental coincidences with

photons or muons. In order to remove the spurious photon and muon events from the decay

energy spectrum, one last event selection is made on the MoNA-LISA data in order to isolate

neutrons. The “neutron gate” used in this analysis is a combination of two 1D cuts based on

the MoNA-LISA time and light output spectra. The neutrons should appear in the MoNA-

64

A/Z22.533.544.55Counts0100020003000400050006000He6He8LISA velocity spectrum as a peak around the beam velocity, and a 1D cut can be made

around that peak (equivalent to a 1D cut in time-of-flight). If further suppression of the

background is necessary, an additional 1D cut can be made to remove events of very low

light output. The neutron gate used in this analysis is shown in Figure 4.15.

Figure 4.15: Neutron selection for the 11Be beam. The same cuts are made on the 12B data.
The corrected ToF is the ToF, position-corrected as if the interaction had occurred in the
front and center bar of the array. Events which lie above the horizontal line and between
the vertical lines are identified as neutron events. All others (within the cross-hatched area)
are presumed to be cosmic muons or photons, and are removed from the analysis.

4.3 Fragment tracking

Because the neutrons are not significantly scattered or slowed in the air, measuring the 3D

position vector and time-of-flight of the neutrons between the target position and MoNA-

LISA determines its four-momentum vector immediately after the decay. The fragments, on

the other hand, are measured after having their paths bent through the Sweeper magnet. So

the measured fragment positions and angles at the CRDCs need to be mapped to the angles

65

Corrected ToF [ns]050100150200250300Light output [MeVee]0102030405060110210Neutron gateand energy before entering the Sweeper magnet. This mapping is done using a nonlinear

ion-optics calculation with the program COSY INFINITY. The details of COSY INFINITY

will not be explained here, but the interested reader should consult Reference [86].

The purpose of an ion-optics calculation is to relate the phase space coordinates of the

beam in one part of the beamline to the phase space coordinates of the beam in another

part of the beamline. Instead of working with the standard (x, px, y, py, z, pz) phase space

coordinates, COSY INFINITY uses a more convenient set of coordinates, labeled by (cid:126)Z =

(x, a, y, b, (cid:96), δ), where

a ≡ px
pz

= tan θx ≈ θx

b ≡ py
pz

= tan θy ≈ θy

(cid:96) ≡ − v0γ0
1 + γ0

(t − t0)

δ ≡ K − K0

K0

≈ 2

∆p
p

(4.8)

(4.9)

(4.10)

(4.11)

The subscript 0 denotes the reference particle, which travels exactly along the design

trajectory, with exactly the design energy. In words, a and b represent angles transverse to

the beam direction, (cid:96) is a “time-of-flight-like” variable which is not relevant to this analysis,

and δ is the normalized deviation from the kinetic energy of the reference particle. For a

more detailed discussion of this choice of coordinates, the interested reader is directed to

Reference [87].

66

With these coordinates established, the equations of motion for a charged particle in the

electric and magnetic fields of a given beamline element can be solved for the elements of a
matrix transformation M (s(cid:48)|s) such that:

(cid:126)Z(s(cid:48)) = M (s(cid:48)|s) (cid:126)Z(s)

(4.12)

This linear transformation can be extended to higher orders, where nonlinear terms are

included as well, and M is no longer a square matrix. COSY INFINITY uses differential

algebraic methods to solve these equations of motion to arbitrary order for the elements of

the transfer matrix. For this particular case, a matrix is needed which relates the phase

space coordinates at the target position (denoted “t”) to the phase space coordinates at the

CRDC1 detector (denoted “d”):

(cid:126)Zd = M (d|t) (cid:126)Zt

(4.13)

Because the measured quantities are those at the CRDCs, this equation must be inverted:

(cid:126)Zt = M−1(d|t) (cid:126)Zd

(4.14)

For this analysis, the optics calculation was carried out to third order, and the elements

of M were provided by COSY INFINITY.

Not all 6 phase space dimensions are relevant for this analysis, so following the procedure

in Reference [77], the dimensionality can be reduced, and the transfer matrix can be “partially

inverted” to yield a set of equations of the form:

67

Table 4.4: The common inputs to COSY used for all settings in this analysis.

Parameter:
order
angle
pos
drift

Significance:
Order of the optics calculation
Incoming beamline angle

Value:
3
-0.3◦
0.50 m Conversion between COSY coordinates and field map coordinates
1.5791 m Arc length of the design trajectory through the field map





at

yt

bt

(cid:96)t

δ





xd

ad

yd

bd

xt

= MP I (d|t)

(4.15)

The outputs relevant to the decay energy are at, bt, and δ, and the measured inputs are

xd, yd, ad, and bd. The horizontal position at the target (xt) is not a measured quantity, so

it must be assumed that the incoming beam distribution is a Dirac delta function at x = 0.

This is an approximation, and it contributes to the total systematic uncertainty on the final

result.

The inputs required to generate the partial inverse matrix are the properties of the beam

particle (A, q, and reference Bρ), and geometric properties of the beamline, such as its

incoming angle, and the arc length of the design trajectory through the Sweeper field map.

The value of the inputs common to all settings in this experiment are displayed in Table 4.4.

A text file containing By(x, z) in the midplane of the Sweeper magnet is also supplied. It

can be shown by Fourier-transforming Amp`ere’s law that the magnetic field in the midplane

fully determines By everywhere inside the gap. The fringe field extends slightly outside the

Sweeper gap, and is modeled using Enge functions. It is assumed due to the design of the

68

Table 4.5: Sweeper current, magnetic field, and rigidities for various settings used in this
work. The Hall probe is physically located near the edge of the magnet, so the Hall probe
value is not indicative of the central field strength, which is usually on the order of 1 Tesla.

Setting:
Centered 11Be
Centered 12B
Centered 8He

Current [A]: Hall probe [kG]: Bρ [Tm] for 43.3◦ bending:

211
173
355

-7.307
-6.179
-10.70

2.60
2.26
3.81

magnet that Bx and Bz are negligible.

These midplane text files are generated using the magnetic field strength measured by

a Hall probe inside the Sweeper gap. The Sweeper field was characterized using a 3D Hall

probe scan at various currents, so using the measured Hall probe values, one can interpolate

between the measured 2D field values to determine the midplane map at any given value of

the Hall probe field. The relationship between the Sweeper current setting, the measured

magnetic field strength at the Hall probe location, and the design trajectory rigidity needed
to produce a bending angle of 43.3◦ is shown in Table 4.5.

4.4 Simulation

Once the quantity of interest (i.e., the decay energy or relative velocity) is calculated in

data, it must be compared with simulations in order to fit a decay model to the exper-

imental data. Monte Carlo simulations including the incoming beam parameters, target

energy loss, knockout reaction kinematics, neutron emission decays, propagation of the neu-

trons and fragments into the detectors, and detector resolutions/acceptances are used for

the comparison. The simulation code is a combination of an in-house fragment simulation

package called ST MONA, and a GEANT4 [88, 89] simulation for the neutrons. The neutron

physics package used in the GEANT4 portion of the code is MENATE R.

69

The transverse phase space coordinates of the incoming beam are modeled in ST MONA

by Gaussian distributions with tunable means and standard deviations. The kinetic energy

distribution is modeled using a uniform distribution, with the center and width fixed by the

A1900 rigidity and slit settings. The beam is then propagated through the target, including

energy losses and straggling, until a randomly-chosen position where the reaction occurs.

The dynamics of the knockout reactions are modeled within the Glauber formalism using

a longitudinal momentum kick from the model of A. S. Goldhaber [90] and a transverse

momentum kick from the model of K. Van Bibber, et al.

[91]. Following the reaction,

a decay energy is randomly generated according to some input distribution, and the two-

body decay is simulated using the TGenPhaseSpace class in ROOT. The fragments are then

propagated to the end of the target, where the COSY forward matrix is used to propagate

the fragments to CRDC1, and a drift is applied to reach CRDC2. Realistic spreading based

on the resolution of the CRDC detectors is applied to the pseudodata, and then the COSY

inverse matrix is applied to reconstruct the fragment angles and energies at the target, in

order to treat the simulated pseudodata identically to the real data. The simulated neutron

energies and momenta are written to a file which is used as input to the GEANT4 part of the

code, which propagates the neutrons from the target position into the MoNA-LISA array.

The GEANT4 simulation contains a realistic model of the detector response, including timing

and position resolutions, and a calculation of the light output using Birks’ model [92]. The

simulation package has been extensively benchmarked and characterized in previous works

(Refs. [76, 79, 80, 1, 6, 5]).

The main tunable parameters in the simulation can be divided into three classes: incom-

ing beam parameters, reaction parameters, and decay parameters. Each of these quantities

must be systematically set in order to force the simulation to best reproduce the data. Each

70

Table 4.6: The simulated beam parameters used in this analysis.

43.27

43.94

Parameter: Value for 11Be: Value for 12B: Significance:
eBeam
beamA
beamZ
targA
targZ
dEbeam
bSpotCx
bSpotCtx
bSpotCy
bSpotCty
bSpotDx
bSpotDy
bSpotDtx
bSpotDty
bSpotSxtx
bSpotSyty

Beam energy [MeV/u]
Beam A
Beam Z
Target A
Target Z
Fractional width of energy distribution
Beam spot x centroid [m]
Beam spot θx centroid [rad]
Beam spot y centroid [m]
Beam spot θy centroid [rad]
Beam spot x width (σ) [m]
Beam spot y width (σ) [m]
Beam spot θx width (σ) [rad]
Beam spot θy width (σ) [rad]
Beam spot x − θx correlation [(m rad)1/2]
Beam spot y − θy correlation [(m rad)1/2]

0.01
-0.003
-0.011
-0.005
0.001
0.0093
0.0070
0.0093
0.0044
-0.0012
-0.0008

12
5
9
4

11
4
9
4

0.01
-0.005
-0.010
-0.001
0.000
0.0095
0.0070
0.0100
0.0041
0.0000
0.0013

set of parameters will be covered in the following subsections.

4.4.1 Beam parameters

The incoming beam parameters can be fixed directly to measured values (i.e.

the beam

energy and A1900 momentum acceptance), or via some kind of minimization procedure. In

order to set the parameters of the Gaussian distributions modeling the transverse phase

space coordinates of the incoming beam, the measured CRDC distributions from a target-

out run can be used. When there is no target in the beamline, and the Sweeper current is

set to center the unreacted secondary beam, the beam propagates into the CRDCs without

any reactions or energy losses. The incoming beam parameter space can then be scanned in

order to best reproduce the CRDC data. The results of this procedure, for the 11Be beam,

are shown in Figures 4.16, 4.17, and 4.18, and the parameter values are displayed in Table

4.6.

71

Figure 4.16: Comparison of the CRDC1 position distributions between data and simulation
for the 11Be beam. The data were taken in a target-out run, where the Sweeper magnet
was set to center the unreacted beam (Run 2005). The points are data and the curves are
simulation. The same comparison was done for the 12B (Run 2026).

Figure 4.17: Comparison of the CRDC angle distributions between data and simulation for
the 11Be beam. The data were taken in a target-out run, where the Sweeper magnet was set
to center the unreacted beam (Run 2005). The points are data and the curves are simulation.
The same comparison was done for the 12B (Run 2026).

72

x [mm]150−100−50−050100150Counts051015202530354045310×CRDC1 dispersive positiony [mm]150−100−50−050100150Counts01020304050310×CRDC1 non(cid:173)dispersive position [mrad]xθ150−100−50−050100150Counts0102030405060310×CRDC dispersive angle [mrad]yθ150−100−50−050100150Counts0102030405060708090310×CRDC non(cid:173)dispersive angleFigure 4.18: Comparison of the CRDC2 position distributions between data and simulation
for the 11Be beam. The data were taken in a target-out run, where the Sweeper magnet
was set to center the unreacted beam (Run 2005). The points are data and the curves are
simulation. The same comparison was done for the 12B (Run 2026).

4.4.2 Reaction parameters

Once the incoming beam parameters have been set based on target-out runs, target-in runs

can be used to fix the reaction parameters. The presence of the target causes both energy

losses and reactions to occur. Because the fragment of interest has a low Z, energy losses
are not very significant (∼ 4 MeV for 8He over half the target thickness), but they are easy

to correct for with energy loss calculations. More significant, however, are the energy and

momentum kicks provided to the reaction products by the knockout reactions.

As mentioned previously, the knockout reactions are modeled using the Glauber model,

which includes transverse and longitudinal momentum kicks as discussed in Refs.

[90, 91].

However these models were developed for high-energy fragmentation reactions, while the

energies relevant to this analysis are somewhat lower. So in order to ensure that the reaction

73

x [mm]150−100−50−050100150Counts051015202530310×CRDC2 dispersive positiony [mm]150−100−50−050100150Counts05101520253035310×CRDC2 non(cid:173)dispersive positionmodel is best reproducing the physics of these reactions, additional scale factors are used to

tune the strength of the transverse and longitudinal momentum kicks.

The three parameters used in this work are a transverse scale factor, a longitudinal scale

factor, and a velocity scale factor. The momentum kicks from the Glauber model are both

in the form of Gaussian distributions with zero mean, and some standard deviations (the

functional forms are given in Refs. [90, 91]). The transverse and longitudinal scale factors

are just scaling the widths of these Gaussian distributions. The velocity scale factor is a

scaling applied directly to the fragment velocity, and can be used to represent an energy kick

given to the fragment by the knockout reaction.

The values of these parameters are chosen in order to simultaneously minimize the dif-

ferences between the fragment kinetic energy and fragment dispersive position distributions

between data and simulation for target-in runs. The Kullback-Leibler divergence (Equation

4.16) is used to quantify the difference between the histogrammed data (q) and pseudodata

(p). The summation runs over histogram bins.

D(p||q) =

(cid:88)

i

(cid:16)pi

(cid:17)

qi

pi ln

(4.16)

The method is simply a brute force search over a predefined parameter space to find the

values which simultaneously minimize the K-L divergences for the fragment kinetic energy

and fragment x. The velocity shift parameter space searched was the interval [1.00, 1.04],

in steps of 0.01, and the Glauber scale parameter spaces searched were [0.0, 1.5] in steps of

0.5. The minimization process was performed separately for each beam, as these parameters

relate to the specific reaction which is occurring, and the reactions of interest are different

for the two different secondary beams. The results are shown in Figure 4.19 and Table 4.7.

74

Table 4.7: The simulated reaction parameters.

Beam: Longitudinal scale: Transverse scale: Velocity scale:
11Be
12B

1.02
1.02

1.0
1.0

0.0
0.5

The plots in Figure 4.19 are on a logarithmic scale to emphasize the disagreements in the

tails.

In spite of these disagreements, the shapes and locations of both peaks are fairly

well-reproduced in simulation, given the simplicity of the reaction model.

Figure 4.19: Comparison of the CRDC1 x and fragment kinetic energy distributions between
data and simulation for the 11Be beam. The data were taken from a chain of target-in
runs. The points are data and the curves are simulation. The same comparison was done
independently for the 12B beam.

4.4.3 Decay parameters

Once the beam and reaction parameters are fixed, all that remains is to find the parameters

of the neutron decay lineshape(s). The decay lineshape functions are discussed in detail in

Chapter 2.

75

x [mm]150−100−50−050100150Scaled counts110210CRDC1 dispersive positionKE [MeV/u]020406080100120Scaled counts110210310Fragment kinetic energyIn this work, all s-wave decays are parametrized by a scattering length, denoted by a. All

decays with (cid:96) > 0 are parametrized by a resonance energy, denoted by E, and a resonance

width, denoted by Γ. Backgrounds are modeled phenomenologically using a Maxwellian

function:

f (E; Θ) ∝

√

E exp

(cid:16) − E/Θ
(cid:17)

(4.17)

The Maxwellian function has a single free parameter, Θ. The Maxwellian function is a

good model for the background in a MoNA-LISA-Sweeper experiment because it produces a

decay energy spectrum resembling coincidences between correlated fragments and neutrons

when convolved with the experimental response function. This model has been used in many
previous MoNA works, and typical values of the parameter Θ range between ∼ 1 − 5 MeV.

In addition to these “intrinsic” parameters, there are scale factor parameters whenever

more than one component is included in the fit. If the fit contains N components, then there
are N − 1 free scale factor parameters, with the overall scale of the summed spectrum being

used to normalize simulation to data.

The values of the decay parameters are determined using standard statistical techniques

for parameter estimation, to be discussed in the next section.

4.5 Parameter estimation

4.5.1 Binned likelihood parameter estimation

This section will describe the use of likelihood maximization for parameter estimation, fol-

lowing the treatment in the data analysis textbook by G. Cowan [93].

76

An experimental data set can be thought of as a sequence of random values drawn from

some underlying probability density function (PDF). Then, one can assume some functional

form of the PDF, and try to estimate the values of its parameters from the experimental

data. One method for doing so is known as maximum likelihood parameter estimation. The

idea behind maximum likelihood is very simple: just find the values of the parameters of

some assumed PDF such that the probability of observing the experimental data given those

parameters is maximal. Mathematically, the goal is to find:

ˆθ ≡ argmax
(cid:126)θ

L((cid:126)θ) = argmax

(cid:126)θ

fjoint((cid:126)x|(cid:126)θ)

(4.18)

The quantity L((cid:126)θ) is known as the “likelihood function”. fjoint((cid:126)x|(cid:126)θ) is the joint PDF for
observing the data (cid:126)x, given the parameters (cid:126)θ. For independent and identically-distributed

data, the joint PDF is simply a product of PDFs for the individual components of the

data vector. For computational ease, it is typical to instead maximize the logarithm of the

likelihood function, since the logarithm is a monotonic function.

In this approach, the values of the estimators of the parameters can be determined

by finding the values of (cid:126)θ that maximize log L((cid:126)θ), and the statistical uncertainties and

correlations of the estimators can be estimated (assuming the estimators are unbiased) using:

(4.19)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:126)θ=ˆθ
in a histogram, where the number of counts in the bins (ni,(cid:80)N

ij = −∂2 log L((cid:126)θ)
ˆV −1
∂θi∂θj

For purposes of this work, we will be concerned with the summation running over N bins
i=1 ni ≡ ntot) can be treated
as independent variables with means νi = νi((cid:126)θ), with a joint probability density given by a

multinomial distribution:

77

fjoint((cid:126)n, (cid:126)ν) = ntot!

N(cid:89)

i=1

(cid:16) νi

(cid:17)ni

ntot

1
ni!

The logarithm of this joint PDF is then equal to:

log fjoint = log(ntot!) +

N(cid:88)

(cid:104)

i=1

ni log νi((cid:126)θ) − log(ni!) + ni log ntot

(4.20)

(4.21)

(cid:105)

The terms which do not depend on the values of (cid:126)θ are irrelevant to the maximization, so

they can be dropped, and what remains is:

N(cid:88)

i=1

log L((cid:126)θ) =

ni log νi((cid:126)θ)

(4.22)

In this analysis, this log-likelihood function is maximized with respect to (cid:126)θ. The numerical

algorithms used to maximize the log L in this work were either brute force in low-dimensional

parameter spaces, or simulated annealing in high-dimensional spaces (D > 5). Brute force

maximization is straightforward, and will not be explained further in this work. Simulated

annealing will be discussed in the following section.

Finally, it is useful to consider the special case of a likelihood function of only one

parameter. Near the maximum, the log-likelihood can be Taylor-expanded to give:

log L(θ) = log L(ˆθ) − (θ − ˆθ)2
2ˆσ2
ˆθ

(4.23)

The constant term is simply log Lmax, and the linear term is zero at a maximum. Equiv-

alently, this equation says:

log L(ˆθ ± ˆσˆθ

) = log Lmax − 1
2

78

(4.24)

This gives way the graphical method of quantifying statistical uncertainties. The 1σ

statistical uncertainty of the estimator ˆθ is given by the points where the log-likelihood

decreases from its maximum value by 1/2.

In the case of more than one parameter, a profile likelihood method was used to determine

statistical confidence intervals and limits. In this method, the likelihood as a function of D
parameters is first maximized with respect to D − 1 of its parameters, and then the 1D

“profile likelihood” is scanned in the final dimension. The 1σ interval is then given by the

locus of points near the maximum such that:

(cid:16)

2∆

(cid:17)

log L(θ)

< QD−1(α)

(4.25)

where QD−1(α) is the quantile of the chi-squared distribution for a significance α, with
D − 1 degrees of freedom. Finding the interval simply entails using the graphical method,

but with 1/2 replaced by QD−1(α)/2.

4.5.2 Simulated annealing

In order to minimize the log L function in the 7-dimensional parameter space for the 12B(-

3p)9He reaction, a stochastic hill climbing (SHC) algorithm coupled with simulated annealing

(SA) was used. The algorithm was developed because the 7D space was too large to do a

brute force search in a reasonable amount of time. The SHC allows maxima to be found

without having to iterate over the entire space, and the SA prevents the maximizer from

getting trapped in local maxima. With an appropriately chosen “cooling schedule” and

hyperparameters of the algorithm, it can be made to converge to the true global maximum

with a probability arbitrarily close to 1.

79

The SHC algorithm employed for this analysis was simply a random Gaussian step, sam-

pled from an uncorrelated 7-dimensional Gaussian distribution with mean (cid:126)0, and covariance

matrix 2 ˆI (in the discretized parameter space). The initial point is sampled from a uniform

distribution over the entire space. Then in each iteration, the subsequent point is chosen

based on this Gaussian step distribution. If a value outside the specified volume of parameter

space is chosen, that value is simply resampled until a valid value is obtained. With each

step, an acceptance/rejection condition is applied, and the acceptance/rejection condition

changes as a function of iteration number according to the cooling schedule of the SA.

Defining ∆(log L) ≡ log Lcurrent − log Lproposed, the acceptance/rejection condition is as

follows:

• if ∆(log L) < 0, accept the proposed point

• if ∆(log L) ≥ 0, accept the proposed point with probability e−∆(log L)/T , otherwise,

reject it

where T is the “temperature” of the SA algorithm. T is initialized to some pre-defined

value (in this case, 5), and then it evolves as a function of Monte Carlo time according to

the cooling schedule. The temperature is meant to start with a high value, which allows

the maximizer to frequently jump in the direction of a worse log L. This is what allows the

algorithm to escape from local maxima, and eventually converge to the global maximum. A

linear cooling schedule was chosen for this analysis, of the form Tnew = αTold, where α = 0.9.

The criterion for the temperature to change was taken to be the number of consecutive
rejections, Nrej, reaching a value of Nstays ≡ 5. In other words, if the maximizer decides

to stay at the same point for five propositions in a row, the temperature decreases, and the

algorithm continues. This process repeats until some predefined minimum temperature (in

80

this case, 10−6) is reached. As the temperature decreases in Monte Carlo time, it becomes

more and more unlikely for a worse log L to be accepted, which forces to algorithm to stay

near the maximum. However the fact that the steps are Gaussian means that there is a

nonzero probability in any given sampling to propose any other point in the parameter

space, no matter how far away it is. So even if the maximizer is somewhere on a large local

maximum, it has the chance to escape.

In order to test convergence, random restarts were applied. This simply entails running

the algorithm multiple times with random initial conditions, and verifying that it converges

to the same maximum each time. The user can perform as many random restarts as they

deem necessary to be convinced that they have found the true global maximum. Once the

algorithm is known to be converging, the user can play with the values of the hyperparameters

(initial temperature, minimum temperature, cooling rate, cooling criterion, Gaussian step

size, etc.) in order to try to shorten the runtime of the algorithm. The number of iterations

until convergence will vary each time the algorithm runs, but for this particular case, the
total number of iterations before convergence tends to be on the order of ∼ 1% of the total

volume of the parameter space. So the global optimum of the 7D parameter space can be

reached by iterating over a small fraction of the entire space. This allows the optimum values

to be found much faster than with the brute force method. The name “simulated annealing”

comes from the analogy of this procedure with annealing in materials science. Pseudocode

demonstrating the entire combined algorithm (SHC+SA) is shown in the Appendix .

81

Chapter 5

Results and Discussion

5.1

12B(-3p)9He

Figure 5.1 shows the invariant mass results for this reaction, which was expected to prefer-

entially populate p-wave resonances. The shape of the experimental spectrum is not well-

reproduced by any single input distribution, so it was fit with a combination of two p-wave

Breit-Wigner resonances and a Maxwellian background. In this case, the likelihood maxi-

mization has seven variable parameters: the energy, width, and scale factor of each resonance
(3×2 = 6), plus the Θ parameter of the Maxwellian background. The log-likelihood between

data and simulation was maximized in this 7D parameter space using the simulated anneal-

ing algorithm described in Section 4.5.2. The scale factors for the two BW distributions, and

the Maxwellian Θ parameter were treated as nuisance parameters in the analysis, but best

fit parameters and 1σ uncertainties were obtained for the resonance energies, and 1σ limits

were obtained for the decay widths of the two BW distributions. The statistical uncertainty

quantification for each parameter was done by fixing the values of the other parameters to

their best estimates, and scanning the 1D profile of the log-likelihood function, using the

“graphical method” described in Section 4.5.1. This is demonstrated for the first resonance

energy in Figure 5.2. The systematic uncertainties were calculated by varying parameters

of the analysis over a physically reasonable range and looking at the shift in the location of

82

the likelihood maximum, following the procedure in the dissertation of M. D. Jones [6]. A

breakdown of the uncertainties is shown in Table 5.1.

Figure 5.1: The decay energy spectrum from the 12B(-3p)9He reaction.

Figure 5.2: The likelihood maximization for the first resonance energy from the 12B(-3p)9He
reaction. The points are the calculated likelihoods, and the black curve is a cubic spline
interpolation. The red vertical lines correspond to the 1σ (stat.) error band.

83

 [MeV]decayE00.511.522.533.544.55Counts051015202530DataTotal simulated spectrumMaxwellian backgroundFirst BW resonanceSecond BW resonanceHe decay energy9B((cid:173)3p)12First resonance energy [MeV]00.511.522.5log L660670680690700710720730log L 1D projectionsTable 5.1: Statistical and systematic uncertainties in the fitting of the resonance parameters
for 12B(-3p)9He. The values for the E1 and E2 parameters are for two-sided intervals,
and the values for the Γ1 and Γ2 parameters correspond to one-sided deviations from the
likelihood maximum.

Fit values [MeV]:

1.1

2.3

3.1

2.8

Contribution:

Variation:

E1 [keV]: Γ1 [keV]: E2 [keV]: Γ2 [keV]:

Statistical

Target energy loss

Global Tmean offset

Simulated θx mean

CRDC1 x offset

CRDC2 x offset

COSY “drift”

–

±0.222 MeV

±0.2 ns
±15 mrad
±1 mm
±1 mm
±10 cm

Total systematic

–

+700−400

1.7

0.03

22

20

21

37

52

-1000

18.5

43.2

30

13

31

98

118

+1000−800

81.8

143.9

20

46

3

118

209

-1100

21.4

74.2

13

45

24

174

238

The results of the fit suggest that there is a relatively low-lying p-wave resonance at

E = 1.1+0.7−0.4 MeV (Γ > 1.3 MeV), and a higher-lying resonance at E = 3.1+1.0−0.8 MeV

(Γ > 1.7 MeV). Due to the high energy and width of the higher-lying resonance, the p-wave

and d -wave lineshapes cannot be strongly distinguished. A p-wave lineshape was assumed

for the analysis, although a d -wave lineshape would fit the data similarly well. From both

theoretical and previous experimental level schemes, no resonances with J > 5

2 are expected

to contribute within the range of decay energies that this experiment was sensitive to. The

use of a p or d lineshape for the higher resonance does not affect the values of the other

fitted parameters in a statistically significant way. The fit was also performed with the lower

resonance replaced by a d -wave, but the scale factor for the first resonance was pushed to

84

zero. Therefore, the lower resonance can be definitively assigned an (cid:96) of 1. The value of the

Maxwellian parameter at the global likelihood maximum is Θ = 3.5 MeV.

The presence of a p-wave resonance at about 1 MeV above the 8He ground state is in

close agreement with multiple previous experiments [36, 37, 38, 39, 41, 42, 44, 45, 46]. There

are large variations in the experimental level schemes of the multipolarities and resonance

energies of higher excited states, although many agree that J = 3
levels exist in the ∼ 2 − 5 MeV range [36, 37, 38, 39, 41, 43, 46].

2 and/or J = 5

2 excited

5.2

11Be(-2p)9He

The 11Be(-2p)9He data are where ground state inversion of 9He should be evident if it is

present. The presence of an s-wave resonance with a large, negative scattering length would

signal that 9He has a significant contribution from an s neutron in the ground state. On

the other hand, a scattering length close to zero would indicate weak final state interactions

between the neutron and 8He core, and suggest that there is no low-lying s state in 9He. If
the energy of the resonance is assumed to be E ≈ 2
2µa2 , then for an s-wave resonance to
have an energy less than 1 MeV, the scattering length must satistfy |a| (cid:47) 5 fm. In the limit
of a → 0, the s-wave simply becomes a “correlated background”, where the halo of the 11Be

beam is broken up by the reaction, with no final state interactions between the neutron and

fragment.

The comparison between experiment and simulations for this analysis were done as a

maximum likelihood fit to the combination of the decay energy and relative velocity spectra.

The simulations included two components: an s-wave, and a Maxwellian background, for a

total of three paramaters: the s-wave scattering length, the Maxwellian Θ, and the relative

85

scale factor. The scale factor is applied to the Maxwellian spectra before adding them to the

s-wave spectra, so the scale factor represents the fractional contribution of the Maxwellian

background. Because the parameter space has sufficiently low dimension, the fit was done

using a brute force maximization. The first step is a coarse-grained fit to find the approximate

global maximum, and then a more fine-grained fit around the maximum is performed to get

accurate estimates of the uncertainties.

The coarse-grained search was performed over the regions a ∈ (−16, 0] fm (step size of 1
fm), Θ ∈ (0, 6] MeV (step size of 0.5 MeV), and scale factor in (0, 5] (step size of 0.5). The

resulting decay energy spectrum is shown in Figure 5.3, and a 1D profile likelihood is shown

in Figure 5.4.

Figure 5.3: The 11Be(-2p)9He decay energy at the global maximum of the coarse-grained
search. The simulated parameter values are a = −2 fm, Θ = 2.5 MeV, and a relative scale
factor of 0.5.

86

 [MeV]decayE00.511.522.53Counts050100150200250300DataTotal simulations(cid:173)waveMaxwellianFigure 5.4: A 1D projection of the log-likelihood near the global maximum of the coarse-
grained search. The points are the calculated log-likelihoods, and the curve is a cubic spline
interpolation to guide the eye.

However, this result seems to show an abnormally small fractional contribution from

background when compared with the previous measurements using the same technique, in-

cluding that of L. Chen, et al. [40], which used the same exact reaction to populate 9He. If

the search space is limited to large values of the scale factor parameter, another maximum is
found with a = −7 fm, Θ = 1.5 MeV, and scale factor of 1.5, shown in Figures 5.5 and 5.6.

Qualitatively, this represents a combination of s-wave and Maxwellian contributions similar

in height, while in the previous case, the spectrum is dominated by the s-wave. The values

of the scattering length parameter at these two likelihood maxima are markedly different,

and have contradictory implications about whether or not inversion is present in the ground
state of 9He. The case with a = −2 fm would imply a non-observation of inversion, while
the case with a = −7 fm would imply that 9He is inverted in the ground state. The latter

case seems to be more consistent with the observation made by Chen, et al., where it was

87

Scattering length [fm]5−4−3−2−1−0log L4813048140481504816048170481804819048200T = 2.5 MeV, scale = 0.5concluded that 9He has an s-wave ground state.

Figure 5.5: The 11Be(-2p)9He decay energy at the maximum in the restricted parameter
space. The points are data and the curves are simulations. The simulated parameter values
are a = −7 fm, Θ = 1.5 MeV, and a relative scale factor of 1.5.

Figure 5.6: A 1D projection of the log-likelihood near the maximum of the coarse-grained
search, when the parameter space has been restricted to large scale factors. The points are
the calculated log-likelihoods, and the curve is a cubic spline interpolation to guide the eye.

88

 [MeV]decayE00.511.522.53Counts050100150200250300DataTotal simulations(cid:173)waveMaxwellianScattering length [fm]10−9−8−7−6−5−4−3−log L48140481504816048170481804819048200T = 1.5 MeV, scale = 1.5In the analysis by Chen, et al., the background was modeled using event mixing rather

than a Maxwellian function. Event mixing assumes that the neutron and fragment are

uncorrelated, and in random coincidence, and a background lineshape was generated by ran-

domly selecting fragments and neutrons from different events, and calculating their relative

velocities. However this background model is unrealistic, because any reaction or decay pro-

cess which produces neutrons and fragments in coincidence will have kinematic correlations

between the fragment and neutron. Therefore, the Maxwellian function is a more realis-

tic background model for both this work and the work of Chen, et al.. Additionally, the

Maxwellian Θ parameter and the scale factor parameter are correlated, because changing

the value of Θ can push particles in and out of the geometric acceptance of the detectors,

and therefore the scale factor has to be varied accordingly in order to achieve a good fit.

Therefore, using this improved background model allows for access to a region of parameter

space which was not searched by Chen, et al., and this explains why the current analysis can

produce seemingly ambiguous results while the Chen analysis did not.

On the other hand, a scattering length close to zero seems to be consistent with the

conclusion of Uberseder, et al. [47], which states that if an s state exists, it must lie at several

MeV above the neutron separation energy. Given the ambiguity of the current result, no

definitive statement can be made about the ground state inversion of 9He, but we can begin

to understand why previous analyses have reached seemingly contradictory conclusions.

The fine-grained fit near the a = −2 fm maximum was performed over the range a ∈
[−3,−1] fm in steps of 0.1 fm. The values and step sizes of the Θ and scale factor parameters
were kept fixed. The global likelihood maximum was found at a = −2.1 ± 0.6 (stat.) ± 0.2

(syst.) fm, Θ = 3.0 MeV, and scale factor 0.5. The value of the Maxwellian Θ is slightly

different between the fine- and coarse-grained fits, but the difference is only one unit in the

89

discretization of Θ. The 1D profile likelihood near this maximum is shown in Figure 5.7.

Figure 5.7: The 1D log-likelihood of fine-grained search around a = −2 fm. The points are
the calculated log-likelihoods, and the curve is a parabolic fit.

For the a = −7 fm maximum, the fit range for the scattering length was [−8,−6] fm,

in steps of 0.1 fm, with the values of Θ and the scale factor fixed to 1.5 MeV, and 1.5,
respectively. The likelihood maximum on this interval was found at a = −6.8 ± 1.2 (stat.)
± 0.7 (syst.) fm. The 1D profile likelihood for this fit is shown in Figure 5.8.

90

Scattering length [fm]3.5−3−2.5−2−1.5−1−0.5−log L48150481604817048180481904820048210T = 3.0 MeV, scale = 0.5Figure 5.8: The 1D log-likelihood of fine-grained search around a = −7 fm. The points are
the calculated log-likelihoods, and the curve is a parabolic fit.

The values of the log-likelihood at the −2 fm and −7 fm maxima are 48200.0 and 48199.8,

respectively, so the difference between them is less than the 1σ deviation calculated from the
quantile of the chi-squared distribution (QD−1(α)/2 ≈ 1.15). A breakdown of the statistical

and systematic uncertainties for both maxima is shown in Table 5.2.

91

Scattering length [fm]8.5−8−7.5−7−6.5−6−log L4817548180481854819048195482004820548210T = 1.5 MeV, scale = 1.5Table 5.2: Statistical and systematic uncertainties in the fitting of the scattering length for
11Be(-2p)9He.

Contribution:

Statistical

Target energy loss

Global Tmean offset

Simulated θx mean

CRDC1 x offset

CRDC2 x offset

COSY “drift”

Total systematic

Total (rounded to 0.1 fm)

Variation:

a = −6.8 fm: a = −2.1 fm:

–

±0.222 MeV

±0.2 ns
±15 mrad
±1 mm
±1 mm
±10 cm

–

–

1.2 fm

0.1 fm

0.1 fm

0.6 fm

0.2 fm

0.1 fm

0.2 fm

0.7 fm

1.4 fm

0.6 fm

0.02 fm

0.04 fm

0.16 fm

0.09 fm

0.02 fm

0.12 fm

0.22 fm

0.6 fm

5.3

12B(-2p)10Li

Resonances in 10Li have been populated using a 12B beam and Be target, albeit with a

different beam energy, in a previous MoNA experiment [1]. In that work, invariant mass

spectroscopy was used to fit a decay energy spectrum to a combination of three p-wave

resonances. The results of that fit are shown in Table 5.3. These resonances can then serve

as a benchmark for the current analysis, which aims to reconstruct 10Li from a 11Be beam

using the Time-of-Flight method (see Section 3.5).

92

Table 5.3: 10Li resonance parameters from the Ph.D. dissertation of J. K. Smith [1].

Multipolarity: Resonance energy [keV]: Width [keV]: Population [%]:

(cid:96) = 1

(cid:96) = 1

(cid:96) = 1

110 ± 40
500 ± 100
1100 ± 100

200

800

1000

13

55

32

Without any fitting or tunable resonance parameters, the resonances from the previous

work can be simulated and overlaid with the data from the current work. The result is shown

in Figure 5.9.

Figure 5.9: The relative velocity spectrum from the 12B(-2p)10Li reaction. The points are
data from this work, and the curves are the simulations of the resonances from the previous
work. The three resonances are shown individually in the red, green, and blue curves. The
black curve is the combined simulation.

This demonstrates that the Time-of-Flight method correctly reconstructs previously-

known states in 10Li.

93

 [cm/ns]f (cid:173) vnv3−2−1−0123Counts0510152025Li relative velocity10B((cid:173)2p)125.4

11Be(-1p)10Li

Producing 10Li using a neutron-halo beam should preferentially populate states with an s

neutron. If such a state with a large scattering length (in absolute value) is observed in the

relative velocity spectrum, it would indicate ground state inversion of 10Li. In this work,

the relative velocity spectrum was produced using the ToF method (see Section 3.5). The

fit was performed in a similar way to the fit of the 11Be(-2p)9He reaction, the likelihood was

maximized over the same 3D parameter space, where the three parameters are the s-wave

scattering length, the Maxwellian Θ, and the relative scale factor.

Similar to the 11Be(-2p)9He case, in a coarse-grained fit, two distinct maxima appear,

depending on the size of the scale factor parameter. The global maximum occurs at a
scattering length of −1 fm, a Θ of 1.0 MeV, and a scale factor of 0.5, as pictured in Figure

5.10.

Figure 5.10: The 11Be(-1p)10Li decay energy at the global maximum of the coarse-grained
search. The simulated parameter values are a = −1 fm, Θ = 1.0 MeV, and a relative scale
factor of 0.5.

94

 [cm/ns]f (cid:173) vnv3−2−1−0123Counts020406080100120DataTotal simulations(cid:173)waveMaxwellianHowever, if the scale factor parameter is again assumed to take on more reasonable values,
there is a local maximum at a scattering length of −7 fm, a Θ of 1.0 MeV, and a scale factor

of 1.5, pictured in Figure 5.11.

Figure 5.11: The 11Be(-1p)10Li decay energy at the maximum of the restricted, coarse-
grained search. The simulated parameter values are a = −7 fm, Θ = 1.0 MeV, and a
relative scale factor of 1.5.

These parameters are qualitatively more consistent with the results of Kryger, et al.

[51], where the background was also very wide, and the s-wave scattering length was large
in magnitude. However the maximum with the −1 fm scattering length is consistent with

previous works which did not observe ground state inversion.

After performing a fine-grained search around both maxima, following the same procedure
as with the 11Be(-2p)9He data, the final values of the scattering lengths are a = −7.0 ± 2.4
fm and a = −1.4 ± 1.0 fm.

95

 [cm/ns]f (cid:173) vnv3−2−1−0123Counts020406080100120DataTotal simulations(cid:173)waveMaxwellianChapter 6

Summary and Conclusions

In this dissertation, unbound states in 9He and 10Li have been reported. 11Be and 12B sec-

ondary beams were provided by the NSCL Coupled Cyclotron Facility, and proton removal

reactions were performed using a beryllium target in order to populate the unbound states.

The neutron-emission decays of these unbound states were experimentally reconstructed us-

ing the MoNA-LISA-Sweeper setup, and the decay energy or relative velocity was compared

with theoretical models via Monte Carlo simulations.

Invariant mass spectroscopy was used to reconstruct decay energies for 9He from both

beams. From the 12B beam, two levels in 9He were assigned resonance energies, widths,

and (cid:96) values. From the 11Be beam, the puzzle of the possible low-lying s-wave ground

state remains unsolved. Due to an ambiguity in the background model, the data from this

work could either be consistent with previous results which have supported or opposed the

presence of ground state inversion in 9He. While there is no definitive answer to the question

yet, this gives some insight into why there are apparently contradictory results in previous

experimental analyses.

Additionally, the Time-of-Flight (ToF) method was used to reconstruct relative velocities

for 10Li from both beams. The ToF method applied to this data set was able to correctly

reproduce previously-observed levels in 10Li from the 12B beam. Once the method was

verified, it was applied to the data from the 11Be beam. Similar to the case of 9He from the

96

11Be beam, the 10Li relative velocity spectrum can be fit in multiple ways. Like 9He, 10Li

also has a controversial experimental history, where some have claimed to observe ground

state level inversion and others have not.

To resolve the persisting ambiguity about the possible inversion of 9He and 10Li, more

experiments are needed, using various combinations of reaction mechanism, beam energy,

and target, in order to more systematically understand how these factors contribute to

the observed spectra. Additionally, improved reaction theory which includes the detailed

structure of the beam and target nuclei would make experimental data easier to intepret.

97

APPENDIX

98

Appendix

Simulated annealing

Initialize temperature

Select first point randomly (uniform over all allowed parameter space)

Calculate log L at current point

while T > Tmin do

while Nrej < Nstays do

Gaussian step

Calculate log L at proposed point
Accept/reject with acceptance probability P (A) = min{1, e−∆(log L)/T}

if Accept then
Reset Nrej

else

Increment Nrej

end

end

Reduce temperature according to Tnew = αTold

end

Print result

Random restart (as many as desired)

99

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