MEASUREMENT AND MODELING OF FRAGMENTS AND

NEUTRONS PRODUCED FROM PROJECTILE FRAGMENTATION

REACTIONS

By

Krystin Elizabeth Stiefel

A DISSERTATION

Submitted to

Michigan State University

in partial fulfillment of the requirements

for the degree of

Chemistry — Doctor of Philosophy

2018

ABSTRACT

MEASUREMENT AND MODELING OF FRAGMENTS AND NEUTRONS

PRODUCED FROM PROJECTILE FRAGMENTATION REACTIONS

By

Krystin Elizabeth Stiefel

Projectile fragmentation has been used to produce rare and short-lived nuclei for study at a

variety of isotope production beam facilities such as the National Superconducting Cyclotron

Laboratory (NSCL). Relatively few exclusive measurements of the final fragments from pro-

jectile fragmentation reactions have been made, even though these types of measurements

can provide more insight into the production process by comparing the measurements to

models that simulate collisions and reactions on nuclei. The present work examines ex-

clusive measurements of neutrons in coincidence with isotopically identified products. Two

intermediate-energy (55.5 MeV/u) projectile fragmentation beams of 30S and 40S nuclei were

produced and reacted with beryllium targets at the NSCL to produce a wide range of projec-

tile fragments. Resulting heavy residue fragments were measured with the Sweeper magnet

charged particle detectors and neutrons were detected in coincidence using the Modular Neu-

tron Array and Large-area multi-Institutional Scintillator Array (MoNA LISA) detectors. A

broad range of fragments was identified in each reaction for elements with Z = 6 − 11. To
explore the projectile fragmentation process, the present results were compared to predic-

tions from two different nuclear reaction models. The hit multiplicity distributions observed

in MoNA LISA for the summed elemental and individual isotopic products were compared

to the two models. The first calculational approach involved the Li`ege Intranuclear Cas-

cade (INCL++) model, a microscopic model and Monte Carlo based code that considers

the reaction as a two-step process with collisions between individual nucleons followed by a

de-excitation process of the intermediate and highly excited residue. The second approach

involved the Constrained Molecular Dynamics (CoMD) model, a more macroscopic quantum

mechanical model that follows the dynamical evolution of nuclear matter using the nuclear

equation of state with three options for the symmetry energy term. The output of the CoMD

model was coupled to GEMINI++ to de-excite any remaining hot fragments. Results from

both simulations were passed through the GEANT4 code to model the neutron response of

MoNA LISA to produce simulated hit multiplicity distributions that could be compared to

the experimental hit multiplicity distributions. The majority of identified fragments were

measured in coincidence with no neutron hits. Because the INCL++ model prediction better

matched the proportion of fragments with zero hits and the CoMD + GEMINI++ simula-

tions under-predicted the proportion of fragments produced with zero hits, INCL++ did an

overall better job at predicting the observed hit multiplicity proportions. However, INCL++

failed to generate enough events with higher hit counts in MoNA LISA. Furthermore, the

three CoMD + GEMINI++ symmetry energy options did not appear to produce noticeably

different hit multiplicity distributions and no constraint on the symmetry energy could be

made with this experimental data set. The distributions of precursor fragments in both the

INCL++ and CoMD models were also examined. Many of the precursor fragments contained

more nucleons than the projectile, indicating that both models predict that the projectile

picks up nucleons from the target during the initial encounter. This prediction differs from

common descriptions of these reactions.

To everyone who called this my “paper.”

iv

ACKNOWLEDGMENTS

I regret to announce- this is the end of my time at Michigan State University. I am going

now. But before I leave, I must acknowledge those who have helped me along the way.

Thanks to Dave Morrissey, my ever-patient and thoughtful advisor who has provided

calm and constructive guidance for this work. I appreciate that you took on my half-baked

project and stayed with me until the very end.

My committee members, Sean Liddick, Paul Mantica, and Michael Thoennessen, went

above and beyond their committee responsibilities. Sean, my second reader, was always

willing to provide some outside perspective, whether it was research-based or career-related.

Michael oversaw some of my most challenging transitions and problems throughout the years

and continuously offered his help. I would not be at Michigan State University if it was not

for Paul, who spoke about all the wonderful work being done at MSU while I was attending

Nuclear Chemistry Summer School.

A special thanks is due to Zach Kohley, who guided me in the first few years of graduate

school and proposed the initial design of this project with the MoNA Collaboration. It has

been an honor to work with the MoNA Collaboration, especially the local members (and for-

mer members) Thomas Baumann, Artemis Spyrou, Anthony Kuchera, Jesse Snyder, Jenna

Smith, Michael Jones, Thomas Redpath, Han Liu, Daniel Votaw, and Dayah Chrisman.

This material is based upon work supported by the Department of Energy National

Nuclear Security Administration through the Nuclear Science and Security Consortium and

by the National Science Foundation. Without funding, this dissertation would not have been

possible.

v

It takes a community to raise a scientist, and I am glad to have spent this time at the

National Superconducting Cyclotron Laboratory. You cannot find a more commendable

group of faculty, staff, and students than the ones found at the NSCL. In particular, I’d like

to thank my officemate Becky Lewis for putting up with all my muttering at my desk and

my former roommate Kalee Hammerton for putting up with all my muttering at our place.

There are some things you can’t share without ending up liking each other, and getting

through graduate school is one of them.

I would not be where I am without the unending encouragement from my family: my

parents, Dana and Kathy; my little sister, Breanna; my extended family- Dushanes, Masser-

ants, Morgans, Stiefels, Twynhams, and every other branch in my enormous family tree; and

my “GoodStief” family, Dianne, Geary, Lydia, and Brett Goodman. You have all pushed

and inspired me to keep moving forward, even when I felt like my odds of success were 3,720

to 1, and for that, I am eternally grateful.

And finally, to Richard Morgan, my husband, I’m afraid I don’t quite have enough words

to thank you for the loving and patient support you’ve given me through the years. You’ll

have to settle for just a few: Thanks for the adventure. Let’s go have a new one.

vi

TABLE OF CONTENTS

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

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

Chapter 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Projectile Fragmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Nuclear Equation of State . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Isospin Symmetry Energy . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . .
1.3 Dissertation Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2.1
1.2.2 Constraining Symmetry Energy with Heavy-Ion Collisions

2.1 Constrained Molecular Dynamics Model

Chapter 2 Modeling and Simulations . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
2.1.1 GEMINI++ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.2 Previous CoMD Comparison to MoNA LISA-Sweeper Data
. . . . .
. . . . . . . . . . . . . . . . . . . . . . . .
2.1.3 CoMD Model Parameters
2.1.4
. . . . . . . . . . . . . . . . . . . . . . .
2.2 Li`ege Intranuclear Cascade Model . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1
INCL++ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 ABLA07 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.3 Limitations of the INCL Model
. . . . . . . . . . . . . . . . . . . . .
INCL++ Model Parameters . . . . . . . . . . . . . . . . . . . . . . .
2.2.4
2.2.5
Isotope Production Output
. . . . . . . . . . . . . . . . . . . . . . .

Isotope Production Output

3.1.1
3.1.2

Chapter 3 Experimental Techniques . . . . . . . . . . . . . . . . . . . . . . .
3.1 Beam Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30S Beam Production . . . . . . . . . . . . . . . . . . . . . . . . . . .
40S Beam Production . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 A1900 and Target Scintillators . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Sweeper Magnet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Charged Particle Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 Blocker
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.2 Cathode Readout Drift Chambers . . . . . . . . . . . . . . . . . . . .
3.4.3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.4 Thin Timing Scintillator . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.5 Hodoscope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 MoNA LISA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6 Electronics and Data Acquisition . . . . . . . . . . . . . . . . . . . . . . . .

Ion Chamber

ix

xi

1
1
4
5
8
12

13
13
14
16
17
18
18
18
26
27
28
28

32
32
33
35
35
37
38
38
39
41
42
44
45
47

vii

Chapter 4 Data Analysis

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1 Calibrations and Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Charged Particle Calibrations and Corrections . . . . . . . . . . . . .
4.1.1.1 Cathode Readout Drift Chambers . . . . . . . . . . . . . . .
4.1.1.2
Ion Chamber . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1.3 Thin Timing Scintillator . . . . . . . . . . . . . . . . . . . .
4.1.1.4 Hodoscope
. . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1.5 A1900 and Target Timing Scintillators . . . . . . . . . . . .
4.1.2 MoNA LISA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.2.1 Charge Calibration . . . . . . . . . . . . . . . . . . . . . . .
4.1.2.2 Position Calibration . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . .
4.1.2.3 Time Calibrations
4.2 Event Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Beam Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.2 Event Quality Gates
. . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.3 Element Identification . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.4
Isotope Identification . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.5 MoNA LISA Multiplicity . . . . . . . . . . . . . . . . . . . . . . . . .

54
54
54
54
60
64
70
70
71
72
74
75
78
78
81
82
84
99
4.3 Experimental Simulation and Gates . . . . . . . . . . . . . . . . . . . . . . . 102
4.3.1 Fragment Gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.3.2 Neutron Gates
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

Chapter 5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.1 Hit Multiplicities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
. . . . . . . . . . . . . . . . . . . . . . 111
5.1.1 Hit Multiplicities by Element
5.1.2 Hit Multiplicities by Isotope . . . . . . . . . . . . . . . . . . . . . . . 117
5.2 Symmetry Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.3 Fragment Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.3.1
Isotope Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.3.2 Time Evolution in CoMD . . . . . . . . . . . . . . . . . . . . . . . . 132
5.3.3 Precursor Fragments in INCL++ . . . . . . . . . . . . . . . . . . . . 136

Chapter 6

Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . 142

APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
Appendix A Deriving the Nuclear Equation of State and the Symmetry Energy . 147
Appendix B CRDC Mask Positions . . . . . . . . . . . . . . . . . . . . . . . . . . 152
Appendix C Isotope Separation Parameters and Corrections . . . . . . . . . . . . 156
Appendix D χ2
v Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

viii

LIST OF TABLES

Table 4.1: Results for the CRDC mask calibration slopes and offsets for the 30S sec-
ondary beam. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Table 4.2: Results for the CRDC mask calibration slopes and offsets for the 40S sec-
ondary beam. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Table 4.3: Coefficients for the thin scintillator position corrections used in Equation
4.11. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

60

67

Table 4.4: Thin scintillator time offsets.

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

69

Table 4.5: Calculated timing offsets for the A1900 and target scintillators.

. . . . . .

71

Table 4.6: Global time offsets for each table of MoNA-LISA. . . . . . . . . . . . . . .

78

Table B.1: The XY coordinates for the holes and lines in the CRDC masks. The coor-
dinates from the diagram were converted to their corresponding locations
on CRDC1 and CRDC2.

. . . . . . . . . . . . . . . . . . . . . . . . . . . 155

Table C.1: Final correction factors used for isotope separation for fragments produced

from the 30S + 9Be reaction and the 1.51 Sweeper magnet setting.

. . . . 158

Table C.2: Final correction factors used for isotope separation for fragments produced

from the 30S + 9Be reaction and the 2.25 Sweeper magnet setting.

. . . . 159

Table C.3: Final correction factors used for isotope separation for fragments produced

from the 40S + 9Be reaction and the 2.01 Sweeper magnet setting.

. . . . 160

Table C.4: Final correction factors used for isotope separation for fragments produced

from the 40S + 9Be reaction and the 2.27 Sweeper magnet setting.

. . . . 161

Table D.1: Calculated χ2

v values for isotope hit multiplicity distributions produced by

the 30S + 9Be reaction and the 1.51 Sweeper magnet setting.

. . . . . . . 164

Table D.2: Calculated χ2

v values for isotope hit multiplicity distributions produced by

the 30S + 9Be reaction and the 2.25 Sweeper magnet setting.

. . . . . . . 165

Table D.3: Calculated χ2

v values for isotope hit multiplicity distributions produced by

the 40S + 9Be reaction and the 2.01 Sweeper magnet setting.

. . . . . . . 166

ix

Table D.4: Calculated χ2

v values for isotope hit multiplicity distributions produced by

the 40S + 9Be reaction and the 2.27 Sweeper magnet setting.

. . . . . . . 167

x

LIST OF FIGURES

Figure 1.1: Chart of the nuclei displaying the nuclear landscape. The black squares
indicate stable nuclei, the green squares indicate unstable nuclei that have
been observed, and the yellow “terra incognito” region indicates predicted
nuclei that have not yet been observed experimentally. The dashed lines
. . . . . . . .
follow nuclear magic numbers. Figure taken from Ref. [2].

Figure 1.2: Projectile fragmentation reaction depicted as a two-step abrasion-ablation
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

process.

Figure 1.3: The three different forms of the density dependence of the symmetry en-
. . . . .

ergy used within CoMD calculations. Figure adapted from [47].

Figure 1.4: Comparison of a selection of experimental constraints for symmetry en-
ergy parameters. SV is equivalent to Esym(ρ0) and L is equivalent to
Lsym. Filled bands indicate constraints from neutron skin thicknesses of
tin isotopes [49], dipole polarizability of 208Pb [51, 52], giant dipole res-
onances (GDR) [53], isotope diffusion in heavy-ion collisions (HIC) [54],
and energies of excitations to isobaric analog states (IAS) [55]. The filled
oval shows constraints from nuclear masses [56]. G and H refer to neutron
matter studies of Gandolfi et al. [57] and Hebeler et al. [58], respectively.
The hatched rectangle gives constraints from fitting astrophysical mass-
radius observations [43, 44]. The overlap region of the former constraints
is indicated by the enclosed white area. Figure taken from [38] with kind
permission of The European Physical Journal (EPJ).
. . . . . . . . . . .

Figure 1.5: Constraints on the slope (L) and magnitude (Esym(ρ0)) in Equation 1.6
of the symmetry energy from heavy-ion collision studies. The shaded
light gray and dark purple area are shown from [54] and [60], respectively.
Constraints at given values of Esym(ρ0) are shown from [61], [10], [36],
and [62] as the black lower-limit, solid green circle, open red circle, and
solid orange square, respectively. For clarity and to avoid overlap, the red
circle was offset by 0.1 MeV. Figure taken from [59] with kind permission
of The European Physical Journal (EPJ).
. . . . . . . . . . . . . . . . .

Figure 2.1:

Figure 2.2:

Inclusive isotope distributions from the stiff-2 CoMD + GEMINI++ sim-
. . . . . . . . . . . . . . . . . . . . . .
ulations for elements Z = 6 − 11.
Inclusive isotope distributions from the stiff-2 CoMD + GEMINI++ sim-
. . . . . . . . . . . . . . . . . . . . .
ulations for elements Z = 12 − 16.

2

3

8

10

11

19

20

xi

Figure 2.3:

Figure 2.4:

Figure 2.5:

Figure 2.6:

Figure 2.7:

Figure 2.8:

Inclusive isotope distributions from the stiff-1 CoMD + GEMINI++ sim-
. . . . . . . . . . . . . . . . . . . . . .
ulations for elements Z = 6 − 11.
Inclusive isotope distributions from the stiff-1 CoMD + GEMINI++ sim-
. . . . . . . . . . . . . . . . . . . . .
ulations for elements Z = 12 − 16.
Inclusive isotope distributions from the soft CoMD + GEMINI++ simu-
lations for elements Z = 6 − 11. . . . . . . . . . . . . . . . . . . . . . . .
Inclusive isotope distributions from the soft CoMD + GEMINI++ simu-
. . . . . . . . . . . . . . . . . . . . . .
lations for elements Z = 12 − 16.
Inclusive isotope distributions from INCL++ simulations for elements Z =
6 − 11. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Inclusive isotope distributions from INCL++ simulations for elements Z =
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12 − 16.

Figure 3.1: A section of the chart of the nuclei showing the isotopes of sulfur. Gray
boxes represent stable isotopes, while yellow, green, and blue boxes repre-
sent isotopes that decay by proton emission, electron capture or β+ decay,
or β− decay, respectively. The 30S and 40S isotopes highlighted in the red
boxes are the projectiles used in the experiment. . . . . . . . . . . . . . .

Figure 3.2: Schematic layout of the Coupled Cyclotron Facility and A1900 Fragment
. . . . . . . . . . . . . . . . . .

Separator. Figure reproduced from [93].

Figure 3.3: Concept mechanical drawing of the MoNA LISA-Sweeper experimental
equipment in the vault. Note that only some of the MoNA LISA bars are
displayed. Drawing provided by Craig Snow. . . . . . . . . . . . . . . . .

Figure 3.4: Schematic of the MoNA LISA-Sweeper experimental equipment in the
. . . . . . . . . . . . . . . . . . . . . . . . . .

vault. See text for details.

Figure 3.5: Layout schematic of the Sweeper magnet and charged particle detectors.
The solid black line indicates the path of the beam. The dashed line
represents the path of the unreacted beam if the Sweeper magnet was not
present.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 3.6: Schematic of a Cathode Readout Drift Chamber (CRDC). The Z-direction
has been expanded for detail, while the field shaping wires are omitted for
simplicity. Figure reproduced from [96].
. . . . . . . . . . . . . . . . . .

21

22

23

24

30

31

33

34

36

37

39

40

Figure 3.7: Schematic diagram of the ion chamber. Figure reproduced from [97].

. .

42

xii

Figure 3.8: Schematic drawing of the thin scintillator while looking in the direction
of the beam. Figure adapted from [95]. . . . . . . . . . . . . . . . . . . .

43

Figure 3.9: Schematic drawing of the Hodoscope CsI(Na) array. Figure taken from [97]. 44

Figure 3.10: Schematic drawing of a single plastic scintillator bar in MoNA LISA.

. .

45

Figure 3.11: Labels for each bar in MoNA LISA as seen from the sides of each table.
The front layers for the MoNA, LISA-1, and LISA-2 tables were A, J, and
N, respectively.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

Figure 3.12: Layout schematic of the Sweeper magnet and MoNA LISA detectors. The
solid black line indicates the path of the beam. The dashed line repre-
sents the path of the beam if the Sweeper magnet was not present. The
shaded blue regions and dotted lines represent the acceptance ranges of
the neutrons. Angles and distances from the target are given for the front
of the LISA-1 and LISA-2 tables.
. . . . . . . . . . . . . . . . . . . . . .

Figure 3.13: Schematic diagram of the MoNA LISA-Sweeper electronics. Start signals,
stop signals, and gates are indicated by green open arrows, red closed
diamonds, and blue solid arrows, respectively.
. . . . . . . . . . . . . . .

Figure 3.14: Abbreviated schematic diagram of the MoNA LISA electronics. Each
Level 1 module represents one layer or layer (bars 0-15). The nine Level 1
modules for each array (A-I for MoNA and J-R for LISA) feed into Level
2 modules. TDCs and QDCs are omitted for simplicity. . . . . . . . . . .

Figure 3.15: Example schematic diagram of determining valid events in the MoNA
LISA electronics. Two cases involving a Level 3 system trigger are shown
on the left. The first case includes two present CFD signals from one
bar. The second case includes at least one missing CFD signal. The right
example shows a case in which the two CFD signals are present but there
is no Level 3 system trigger. TDCs and QDCs are omitted for simplicity.

Figure 4.1: Pedestal subtraction in CRDC1 (top) and CRDC2 (bottom). The raw
charge prior to pedestal subtraction is shown in the left panels, while
the subtraction is shown in the right panels. Vertical lines indicate dead
channels in bad pads. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 4.2: Mechanical drawing of a tungsten mask used in the experimental configu-
ration. Each hole corresponded to a specific point in space and each line
corresponded to an X-position. Drawing made by J. Honke [102].
. . . .

48

49

50

52

56

58

xiii

Figure 4.3: Example data from CRDC2 mask run in which the holes and lines in
the mask were used to determine the slopes and offsets for the position
calibration. This mask run was performed with the 30S beam.
. . . . . .

Figure 4.4: Example data from a CRDC2 mask run in which the mask did not fully
insert. This mask run was performed with the 40S beam. . . . . . . . . .

Figure 4.5: Example of the Gaussian fitting procedure used for gain matching the
ion chamber. The reference pad, pad 9, is shown in the left, while the
charge collected from another example pad is shown on the right. The
red vertical line indicates the centroid of the reference pad. The examples
were taken from the centered run with the 40S beam.
. . . . . . . . . . .

Figure 4.6: Results of gain matching the pads of the ion chamber. Values of the raw
charge are displayed on the left, while calibrated charge values are shown
on the right. The white gaps for pads 1 and 8 indicate the results of poor
charge collection. These results were obtained with the 40S beam.
. . . .

Figure 4.7: Example data for the position dependence correction in the ion chamber
for pad 14. On the left is the raw signal as a function of X-position. The
right panel shows the position-corrected signal. Signals were taken from
the 30S beam. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 4.8: Example data for the position dependence correction in the ion chamber
for pad 14. On the left is the X-position corrected energy loss as a function
of Y-position. The right panel shows the position-corrected signal. Signals
were taken from the 30S beam.
. . . . . . . . . . . . . . . . . . . . . . .

Figure 4.9: Results of gain matching the four thin scintillator PMTs. Raw charge is
displayed on the left, while the gain-matched charge is shown on the right.
This gain matching was performed with the 40S beam.
. . . . . . . . . .

Figure 4.10: Correction of the position dependence in the thin scintillator. On the left
is raw charge signal as a function of the X-position. The right panel shows
the position-corrected signal. Signals were taken from the 30S beam.
. .

Figure 4.11: X-position-corrected signal versus the Y-position in the thin scintillator.
No noticeable dependence is observed. Signals were taken from the 30S
beam.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 4.12: An example of the QDC calibration procedure for MoNA LISA. The data
shown is from the left PMT from Layer K, Bar 8. On the left is the raw
charge from each QDC channel. The cosmic peak around 1000 is fit with
a Gaussian function (green). The right panel shows the calibrated charge
spectrum with the pedestal subtracted. . . . . . . . . . . . . . . . . . . .

59

60

62

63

64

65

66

67

68

73

xiv

Figure 4.13: An example of the TDC calibration procedure for MoNA LISA. The data
shown is the left PMT from Layer B, Bar 5. On the left is the “picket
fence” spectrum created by the pulser pulsing the system every 40 ns.
On the right is the relation between the TDC channel peaks and their
assigned time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 4.14: Example of the data for an X-position calibration for a bar in MoNA
LISA (Layer A, Bar 8). The time difference (tlef t − tright) between the
two PMTs is shown on the left, and the converted X-position spectrum is
shown on the right. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 4.15: Schematic representation of cosmic ray muon paths through MoNA LISA
bars. The schematic view on the left is a view from the front a table,
while the view on the right of is from looking at a side view of a table.
.

Figure 4.16: Calibrated time spectra for all detectors on each MoNA LISA table. The
smaller peaks on the left of each spectra are the gamma rays, while the
larger peaks are neutrons. The red vertical lines indicate the expected
time for the gamma rays to arrive in each set of bars. . . . . . . . . . . .

Figure 4.17: Beam component identification with flight time from the A1900 timing
scintillator to the target scintillator. The 30S beam and contaminants are
shown on the left, while the 40S beam peak is shown on the right.
. . . .

Figure 4.18: Beam component identification with flight time from the A1900 timing
scintillator to the target scintillator versus the energy loss in the ion cham-
ber. The 30S beam and contaminants are shown on the left, while the 40S
beam and contaminants are shown on the right.
. . . . . . . . . . . . . .

Figure 4.19: Beam component identification using the time-of-flight from the RF to
the A1900 scintillator. The 30S beam and contaminants are shown on the
left, while the 40S beam and contaminants are shown on the right. The
multiple peaking in the RF to A1900 time-of-flight is due to wraparound
of the RF. Beam gates were placed on the dominant packets encircled in
black.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 4.20: The σ of the gaussian fit as a function of total charge in the CRDCs for
the 40S + 9Be reaction and 2.01 Tm Sweeper setting. The quality gates
for each CRDC are drawn in black.
. . . . . . . . . . . . . . . . . . . . .

Figure 4.21: Total charge of CRDC1 plotted against the total charge of CRDC2 for
the 40S + 9Be reaction and 2.01 Tm Sweeper setting. The quality gate
(shown in black) excluded events that do not deposit charge in a linear
manner in both detectors.
. . . . . . . . . . . . . . . . . . . . . . . . . .

75

76

77

78

79

80

81

82

83

xv

Figure 4.22: Energy loss in the ion chamber versus the time-of-flight from the target
scintillator to the thin scintillator for reaction products from the 30S +
9Be reaction. The top panel contains products from the 1.51 Tm magnet
setting. The bottom panel contains products from the 2.25 Tm magnet
setting. Black lines indicate the locus of sulfur (Z = 16) products. . . . .

Figure 4.23: Energy loss in the ion chamber versus the time-of-flight from the target
scintillator to the thin scintillator for reaction products from the 40S +
9Be reaction. The top panel contains products from the 2.01 Tm magnet
setting. The bottom panel contains products from the 2.27 Tm magnet
setting. Black lines indicate the locus of sulfur (Z = 16) products. . . . .

85

86

Figure 4.24: Three-dimensional plots showing the correlation among the (dispersive)
X-position, X-angle, and time-of-flight for the oxygen isotopes from the
40S + 9Be reaction and the 2.01 Tm magnet setting. Isotope bands are
visible on the right edge in the Figure but merge towards the back left side. 88

Figure 4.25: Projection of the three-dimensional plot from Figure 4.24 onto the (dis-
persive) X-position and X-angle plane for the oxygen isotopes from the
40S + 9Be reaction and the 2.01 magnet setting. A solid black line is
drawn along the contours to demonstrate the quadratic function fit used
in the analysis.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 4.26: Emittance parameter from Equation 4.20 plotted against the time-of-flight
from the target to the thin scintillator for the oxygen isotopes from the
40S + 9Be reaction and the 2.01 magnet setting. A solid black line is
drawn along the contours to demonstrate the linear function fit.
. . . . .

Figure 4.27: The Z versus N/Z pattern for isotopes with Z = 6-16 and N/Z ≤ 2. Each
point is a separate isotope. Some of the isotopes shown do not exist. The
red diamond points represent isotopes in which N = Z. The two beams,
30S and 40S, are circled for reference. . . . . . . . . . . . . . . . . . . . .

Figure 4.28: Energy loss in the ion chamber for elements Z = 6−14 using the corrected
time-of-flight for oxygen isotope separation. The elements shown are from
the 30S + 9Be reaction and the 2.25 Tm Sweeper magnet setting. A
vertical line at N/Z = 1 is drawn for reference. This line corresponds to
the red diamond points in Figure 4.27. The 28Si fragments are circled in
red.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

90

92

93

Figure 4.29: One-dimensional distributions of carbon, nitrogen, oxygen, fluorine, neon,
and sodium isotopes produced from the 30S + 9Be reaction and the 1.51
Tm Sweeper magnet setting. Element and event quality gates were applied. 94

xvi

Figure 4.30: One-dimensional distributions of carbon, nitrogen, oxygen, fluorine, neon,
and sodium isotopes produced from the 30S + 9Be reaction and the 2.25
Tm Sweeper magnet setting. Element and event quality gates were applied. 95

Figure 4.31: One-dimensional distributions of carbon, nitrogen, oxygen, fluorine, neon,
and sodium isotopes produced from the 40S + 9Be reaction and the 2.01
Tm Sweeper magnet setting. Element and event quality gates were applied. 96

Figure 4.32: One-dimensional distributions of carbon, nitrogen, oxygen, fluorine, neon,
and sodium isotopes produced from the 40S + 9Be reaction and the 2.27
Tm Sweeper magnet setting. Element and event quality gates were applied. 97

Figure 4.33: One-dimensional distributions of magnesium and aluminum isotopes pro-
duced from the 30S + 9Be reaction and the 2.25 Tm Sweeper magnet
setting. Element and event quality gates were applied.
. . . . . . . . . .

99

Figure 4.34: Kinetic energy of neutrons detected in MoNA LISA for all fragments mea-

sured with the 40S + 9Be reaction and the 2.01 Tm Sweeper setting.

. . 100

Figure 4.35: Examples of the hit multiplicity distributions for all neon fragments from
the 30S + 9Be reaction with the 2.25 Tm Sweeper magnet setting (left)
and the 40S + 9Be reaction with the 2.27 Tm Sweeper magnet setting
(right). Distributions are presented in terms of a normalized probability.

101

Figure 4.36: Simulated hit interactions in MoNA LISA bars for neutrons from the 40S +
9Be reaction. The top panel shows interactions from the INCL++ model,
while the bottom panel (CoMD-2) shows interactions from the CoMD +
GEMINI++ for the second symmetry energy option (stiff-1). . . . . . . . 105

Figure 4.37: Simulated hit interaction angles in MoNA LISA bars for neutrons from the
40S + 9Be reaction. INCL++ and the second symmetry energy option
(stiff-1, labeled here as CoMD-2) for CoMD + GEMINI++ are shown.
The neutrons with angles of around -3 to 15 degrees were measured in
MoNA and LISA-1, while the neutrons with angles of around 24 to 53
degrees were measured in LISA-2. . . . . . . . . . . . . . . . . . . . . . . 106

Figure 5.1:

Isotopic fragments identified the experimental data. Red boxes indicate
isotopes only identified from the 30S + 9Be reaction, blue boxes indicate
isotopes only identified from the 40S + 9Be reaction, and purple dotted
boxes indicate isotopes identified from both reactions. For comparison,
the 30S and 40S incident beams are outlined in red and blue, respectively. 108

xvii

Figure 5.2: Hit multiplicity distributions for 16O fragments from the 30S + 9Be re-
action with the 1.51 Tm Sweeper magnet setting. Distributions are pre-
sented in terms of probability (see the text for details). Solid black squares
represent experimental data; open pink squares represent INCL++ sim-
ulations; and open blue diamonds, open up-pointing red triangles, and
open down-pointing green triangles represent CoMD + GEMINI++ sim-
ulations with the CoMD-1, CoMD-2, and CoMD-3 options, respectively.
For clarity and to avoid overlap, the INCL++, CoMD-1, CoMD-2, and
CoMD-3 points were offset in hit multiplicity by -0.1, 0.1, 0.15, and 0.2,
respectively. Points representing a hit multiplicity of one are highlighted
in shaded blue to demonstrate grouping.

. . . . . . . . . . . . . . . . . . 110

Figure 5.3: Hit multiplicity distribution for all carbon fragments produced in each
beam setting for the 30S + 9Be and 40S + 9Be reactions. See text and
Figure 5.2 for details. Sweeper acceptance gates were used, but no isotopic
gates were applied.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

Figure 5.4: Hit multiplicity distribution for all magnesium fragments produced in each
beam setting for the 30S + 9Be and 40S + 9Be reactions. See text and
Figure 5.2 for details. Sweeper acceptance gates were used, but no isotopic
gates were applied.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

Figure 5.5: Hit multiplicity distribution for all aluminum fragments produced in each
beam setting for the 30S + 9Be and 40S + 9Be reactions. See text and
Figure 5.2 for details. Sweeper acceptance gates were used, but no isotopic
gates were applied.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

Figure 5.6: Hit multiplicity distribution for all silicon fragments produced in each
beam setting for the 30S + 9Be and 40S + 9Be reactions. See text and
Figure 5.2 for details. Sweeper acceptance gates were used, but no isotopic
gates were applied.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

Figure 5.7: Hit multiplicity distribution for all phosphorus fragments produced in each
beam setting for the 30S + 9Be and 40S + 9Be reactions. See text and
Figure 5.2 for details. Sweeper acceptance gates were used, but no isotopic
gates were applied.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

Figure 5.8: Hit multiplicity distribution for carbon isotopes produced in each beam
setting for the 30S + 9Be and 40S + 9Be reactions. See the text and
Figure 5.2 for details of the display. . . . . . . . . . . . . . . . . . . . . . 118

Figure 5.9: Hit multiplicity distribution for nitrogen isotopes produced in each beam
setting for the 30S + 9Be and 40S + 9Be reactions. See the text and
Figure 5.2 for details of the display. . . . . . . . . . . . . . . . . . . . . . 119

xviii

Figure 5.10: Hit multiplicity distribution for oxygen isotopes produced in each beam
setting for the 30S + 9Be and 40S + 9Be reactions. See the text and
Figure 5.2 for details of the display. . . . . . . . . . . . . . . . . . . . . . 120

Figure 5.11: Hit multiplicity distribution for fluorine isotopes produced in each beam
setting for the 30S + 9Be and 40S + 9Be reactions. See the text and
Figure 5.2 for details of the display. . . . . . . . . . . . . . . . . . . . . . 121

Figure 5.12: Hit multiplicity distribution for neon isotopes produced in each beam
setting for the 30S + 9Be and 40S + 9Be reactions. See the text and
Figure 5.2 for details of the display. . . . . . . . . . . . . . . . . . . . . . 122

Figure 5.13: Hit multiplicity distribution for sodium isotopes produced in each beam
setting for the 30S + 9Be and 40S + 9Be reactions. See the text and
Figure 5.2 for details of the display. . . . . . . . . . . . . . . . . . . . . . 123

Figure 5.14: Isotope production distributions for the 30S + 9Be reaction and the 1.51
Tm magnet setting. Counts are given in terms of probability normalized
relative to the total number of observed fragments within the given ele-
ment. The CoMD + GEMINI++ model stiff-2, stiff-1, and soft options
are referred to as CoMD-1, CoMD-2, and CoMD-3, respectively. . . . . . 128

Figure 5.15: Isotope production distributions for the 30S + 9Be reaction and the 2.25
Tm magnet setting. Counts are given in terms of probability normalized
relative to the total number of observed fragments within the given ele-
ment. The CoMD + GEMINI++ model stiff-2, stiff-1, and soft options
are referred to as CoMD-1, CoMD-2, and CoMD-3, respectively. . . . . . 129

Figure 5.16: Isotope production distributions for the 40S + 9Be reaction and the 2.01
Tm magnet setting. Counts are given in terms of probability normalized
relative to the total number of observed fragments within the given ele-
ment. The CoMD + GEMINI++ model stiff-2, stiff-1, and soft options
are referred to as CoMD-1, CoMD-2, and CoMD-3, respectively. . . . . . 130

Figure 5.17: Isotope production distributions for the 40S + 9Be reaction and the 2.27
Tm magnet setting. Counts are given in terms of probability normalized
relative to the total number of observed fragments within the given ele-
ment. The CoMD + GEMINI++ model stiff-2, stiff-1, and soft options
are referred to as CoMD-1, CoMD-2, and CoMD-3, respectively. . . . . . 131

Figure 5.18: Contours in the N -Z plane of the fragment distribution showing the time
evolution of fragments in the 30S + 9Be reaction from the CoMD model
with the stiff-1 option. Only the heaviest fragments are shown. The boxes
indicate the N -Z of the beam. . . . . . . . . . . . . . . . . . . . . . . . . 133

xix

Figure 5.19: Contours in the N -Z plane of the fragment distribution showing the time
evolution of fragments in the 40S + 9Be reaction from the CoMD model
with the stiff-1 option. Only the heaviest fragments are shown. The boxes
indicate the N -Z of the beam. . . . . . . . . . . . . . . . . . . . . . . . . 134

Figure 5.20: Distribution of the intermediate precursor fragments in the N -Z plane
from INCL++ for final 13C fragments produced in each beam setting
for the 30S + 9Be and 40S + 9Be reactions. The intermediate precursor
fragments are represented by the solid boxes. Open squares representing
the beams (30S or 40S) and solid black circles representing the position of
13C are displayed for comparison. . . . . . . . . . . . . . . . . . . . . . . 137

Figure 5.21: Distribution of the intermediate precursor fragments in the N -Z plane
from INCL++ for final 16O fragments produced in each beam setting
for the 30S + 9Be and 40S + 9Be reactions. The intermediate precursor
fragments are represented by the solid boxes. Open squares representing
the beams (30S or 40S) and solid black circles representing the position of
16O are displayed for comparison. . . . . . . . . . . . . . . . . . . . . . . 138

Figure 5.22: Distribution of the intermediate precursor fragments in the N -Z plane
from INCL++ for final 23Na fragments produced in each beam setting
for the 30S + 9Be and 40S + 9Be reactions. The intermediate precursor
fragments are represented by the solid boxes. Open squares representing
the beams (30S or 40S) and solid black circles representing the position of
23Na are displayed for comparison.

. . . . . . . . . . . . . . . . . . . . . 139

Figure B.1: Schematic demonstrating the orientation of the raw signals TAC and grav-
ity (or pad number) in CRDC1 and CRDC2 and their transformed and
calibrated XY-positions.

. . . . . . . . . . . . . . . . . . . . . . . . . . . 153

Figure B.2: Mechanical drawing of a tungsten mask used in the experimental config-
uration. Each hole (1-26) and line (A and B) corresponded to a specific
point in space. Adapted from a drawing made by J. Honke [102].

. . . . 154

xx

Chapter 1

Introduction

The field of nuclear science is concerned with studying the nuclei of atoms, or the building

blocks that make up everything in the universe. The atomic nucleus is composed of protons

and neutrons, and the number of protons and neutrons within the nucleus defines the identity

of that nucleus as an isotope. The number of protons and neutrons can be used to organize

nuclei in a chart, as shown in Figure 1.1.

While more than 3000 different isotopes have been observed experimentally, over 7000

different isotopes are predicted to exist [1], indicating that there is much more work to be

done to fill the nuclear landscape. However, producing these rare isotopes is a challenge.

One of the important mechanisms to produce unknown isotopes is the subject of this work.

1.1 Projectile Fragmentation

Only a small percentage of known nuclei are stable. Most nuclei are unstable and decay, with

some of the more exotic species having half-lives of less than a second. Because these rare

nuclei exist for such a short time, rapid production, separation, and detection is necessary in

order to observe their properties. To produce rare nuclei for study, isotope production beam

facilities such as the National Superconducting Cyclotron Laboratory use a technique called

projectile fragmentation. This technique relies on reacting a high-energy beam of nuclei in

a relatively thin target.

1

Figure 1.1: Chart of the nuclei displaying the nuclear landscape. The black squares indicate
stable nuclei, the green squares indicate unstable nuclei that have been observed, and the
yellow “terra incognito” region indicates predicted nuclei that have not yet been observed
experimentally. The dashed lines follow nuclear magic numbers. Figure taken from Ref. [2].

Projectile fragmentation reactions have often been described as a two-step process origi-

nally referred to as the abrasion-ablation or fireball model [3–5]. Figure 1.2 shows a schematic

picture of the abrasion-ablation model. In the first step, the projectile rapidly collides and

passes by the target. The overlapping area is abraded or scraped away, creating a “fireball”

of participants that releases nucleons and/or fragments. The collision produces two excited

precursors, a projectile-like fragment and a target-like fragment. The excited fragments then

undergo ablation, a slower de-excitation process, by releasing nucleons, other light particles,

and energy. The end results are two “cold” nuclear residues, one moving very fast in the lab

frame and another nearly at rest.

Products from a projectile fragmentation reaction may be studied in inclusive or exclusive

experiments. In an inclusive measurement, only one product is measured from the collision.

2

projectile

projectile-like fragment

final fragment

γ

target

target-like
fragment

participants

nucleons

COLLISION

ABRASION

ABLATION

Figure 1.2: Projectile fragmentation reaction depicted as a two-step abrasion-ablation pro-
cess.

In an exclusive measurement, two or more products from one collision are measured in

coincidence. Consider the example of a projectile fragmentation reaction producing a final

fragment residue as well as several nucleons and gamma rays, as shown for example in

Figure 1.2. If only the residue fragment was measured, an inclusive measurement was made.

Measuring the residue fragments in coincidence with the gamma rays or neutrons would be

an example of an exclusive measurement.

If measured in an exclusive experiment, the final fragments can provide more insight

into the projectile fragmentation process than just measuring the production of individual

residues alone. The final fragments can be compared to models that simulate collisions and

reactions on nuclei. If the experimental data matches the model, then the model could be

considered a fair representation of the fragmentation process. A poor match between data

and simulation would indicate that the model has made incorrect assumptions or is missing

a piece of the puzzle. One goal of the present work is the use of exclusive measurements of

projectile fragmentations products to test nuclear reaction models.

3

1.2 Nuclear Equation of State

Some models of nuclear reactions treat the nuclei as fluid nuclear matter with a macro-

scopic equation of state for the total energy. For ground state nuclei, or cold nuclei at

saturation density (ρ0 = 0.16 fm3), the nuclear binding energy is described through the

Bethe-Weizs¨acker formula for a charged liquid drop model [6]. The nuclear binding energy

is then written in a semi-empirical mass formula as

BE(Z, N ) = aV A − aSA(2/3) − aC

Z2
A(1/3) − aA

(N − Z)2

A

± δpairing

(1.1)

where the binding energy, BE, is calculated with a volume term (aV ), a surface term (aS), a

Coulomb term (aC ), an asymmetry term (aA), and a pairing term (δpairing). The coefficients

are then determined by empirically fitting the equation to experimentally known masses [6].

While ground state nuclei are relatively well-described with the Bethe-Weizs¨acker for-

mula, determining the properties of nuclear matter with temperature, densities, and neutron-

to-proton ratios away from ground-state nuclei has been a challenge. The nuclear equation of

state is often used to describe how the energy per nucleon of infinite nuclear matter changes

as a function of the density (ρ) and isospin asymmetry (δ). Isospin asymmetry is often given

as

δ =

(ρn − ρp)
(ρn + ρp) ≈

(N − Z)

A

(1.2)

where ρn is the neutron density and ρp is the proton density of the matter. (As the difference

in proton mass and neutron mass is negligible, the approximation to terms of N , Z, and A

is valid.)

4

1.2.1

Isospin Symmetry Energy

The nuclear equation of state for infinite nuclear matter can be connected to the Bethe-

Weizs¨acker formula [6] for finite matter. (The connection is summarized below. Intermediate

steps are shown in Part 1 of Appendix A.) Consider the Bethe-Weizs¨acker formula for the

case of an infinitely large mass, A. As infinite matter has no surface, the surface term must

be zero. The Coulomb term is removed, as some infinite nuclear matter studies exclude

the term and including Coulomb forces makes the liquid drop unstable in this form [7–9].

(Similar derivations of the nuclear equation of state exclude the Coulomb term [10–15].)

The pairing term can also be excluded due to its small contribution in this case. Thus, the

Bethe-Weizs¨acker formula reduces to the following relation for matter with infinitely large

mass:

BE(Z, N )

A

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)A→∞

= aV − aA

(N − Z)2

A2

(1.3)

where the volume and asymmetry terms are the only remaining parts.

The asymmetry term can be further broken into volume and surface terms. As before,

the surface term for the asymmetry term can be excluded in infinite matter. The binding

energy per nucleon for infinitely large nuclear matter is then written as:

BE(Z, N )

A

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)A→∞

≈ aV − av

Aδ2

(1.4)

where av

A is the asymmetry-volume term. For symmetric nuclear matter (δ = 0), the second
Aδ2) becomes zero. Thus, the first term, aV , represents the

part of the above equation (av

bulk part of the nuclear equation of state, while the second term represents the isospin

5

asymmetric part. When discussing the nuclear equation of state, the above equation is often

written as E(ρ, δ) in terms of the density and isospin:

E(ρ, δ) = E(ρ, 0) + Esym(ρ)δ2 + O(δ4)

(1.5)

The first term in the expression, E(ρ, 0), represents the binding energy for symmetric

nuclear matter (N = Z) as a function of density and is independent of δ. The second term

contains the symmetry energy, Esym(ρ), or the difference in energy between pure neutron
matter (δ = 1) and symmetric nuclear matter (δ = 0). (The third term, O(δ4), is considered
negligible and often omitted in current symmetry energy work.) As nuclear matter becomes

more asymmetric, this second term becomes more important due to the δ2 dependence.

While the nuclear equation of state of symmetric nuclear matter is thought to be well-

understood and constrained by various measurements, the same is not true of asymmetric

matter [10, 16]. For example, the binding energies of ground state nuclei are well-known

and the binding energy of symmetric nuclear matter is well-defined at the nuclear saturation

density (ρ0 = 0.16 fm3) [17]. Furthermore, the nuclear equation of state symmetric term,

E(ρ, 0), is relatively well-defined from heavy-ion collision transverse flow measurements and

giant monopole resonances [18–23].

However, there are large discrepancies among the predictions for the isospin dependence

of the nuclear equation of state, in particular in the low and high density regions [10, 16, 17].

Different nucleon-nucleon interactions contain various parameters that are adjusted to fit

a variety of experimental data that have led to a variety of predictions. Thus, theoretical

models show inconsistencies in predicting the density dependence of the symmetry energy

[24, 25].

6

Determining the form of the symmetry energy and how it behaves in relation to a va-

riety of conditions such as density, temperature, and pressure remains a major objective

in understanding nuclear properties, astrophysical processes, and the fundamental nucleon-

nucleon interaction [26–37]. In particular, the properties of neutron-rich matter are clearly

dependent on the symmetry energy term. For example, neutron stars are thought to have

a very asymmetric proton-to-neutron ratio [38, 39]. Adjusting the symmetry energy term

has implications for predictions of the mass-to-radius ratios, density, and cooling timescale,

among other characteristics, of neutron stars [10, 29, 36, 40–42]. Furthermore, because the

characteristics of neutron stars are affected by the nuclear equation of state, the properties

of neutron stars can be used to constrain the symmetry energy term [43, 44].

While more sophisticated calculations of the equation of state have produced better

constraints of the symmetry energy [45, 46], it is important to continue to test these new

predictions of the symmetry energy and neutron matter by experimentally constraining the

form of the symmetry energy, establishing tighter and consistent constraints.

The form of the symmetry energy is often described as “soft” or “stiff” based on how the

symmetry energy evolves with density. Stiffer forms of the symmetry energy have a more

constant increase in symmetry energy as density increases, becoming increasingly larger

beyond the nuclear saturation density. Soft forms of the symmetry energy show larger

magnitude below the saturation density, with slower increase or even decrease beyond ρ0. A

comparison of three plausible forms of the symmetry energy is displayed in Figure 1.3 [47].

In order to quantitatively describe the form of the symmetry energy, the density depen-

dence of the symmetry energy form can be written as a Taylor expansion of the symmetry

energy:

7

Figure 1.3: The three different forms of the density dependence of the symmetry energy used
within CoMD calculations. Figure adapted from [47].

Esym(ρ) = Esym(ρ0) +

Lsym

3

(

ρ − ρ0

ρ0

) +

Ksym

18

(

ρ − ρ0

ρ0

)2 + ...

(1.6)

where Esym(ρ0), Lsym, and Ksym are the magnitude, slope, and curvature parameters,

respectively. (The expansion of the symmetry energy term to Equation 1.6 is shown in Part

2 of Appendix A.) These parameters provide quantitative values of the constraints, allowing

comparison of the derived constraints from different studies and models.

1.2.2 Constraining Symmetry Energy with Heavy-Ion Collisions

The density dependence of the symmetry energy has been constrained by the results from sev-

eral studies, including neutron skin measurements, giant dipole resonances, nuclear masses,

8

astrophysical mass-radius observations, and dipole polarizability studies [48–50]. A compar-

ison of a selection of constraints from diverse experimental studies can be seen in Figure

1.4 [38]. The magnitude of the symmetry energy form is relatively constrained compared to

the slope. As an example, the finite-range droplet model (FRDM) was fit using experimental

masses to find that Esym(ρ0) = 32.5 ± 0.5 MeV, a fairly restricted range (±2%), yet Lsym
had a much larger range of 70 ± 15 MeV (±21%) [50], indicating that it has been more
difficult to constrain the slope parameter relative to the magnitude.

Another method that can constrain the symmetry energy involves the use of heavy-ion

collisions (HICs), which probe nuclear matter at densities, temperatures, and pressures away

from those of ground state nuclei [36, 59]. Heavy-ion collisions produce fragments and other

light particles, which can be compared to models containing an adjustable symmetry energy

term. Several constraints obtained from experiments using heavy-ion collisions are shown in

Figure 1.5 [59] and the green HIC band in Figure 1.4 [38].

Heavy-ion collisions with radioactive ion beams may be a key to placing better constraints

on the form of symmetry energy. Compared to the limited possible stable beams, radioactive

ion beams can introduce larger asymmetries into the reaction, and thus a larger asymme-

try parameter, δ2. Thus, with these larger asymmetries comes an enhanced sensitivity to

symmetry energy.

Unconstrained parameters and differences in modeling the dynamics are the main uncer-

tainty in obtaining constraints from these reactions. There are two main classes of models

used in these evaluations: molecular dynamics models and mean-field models. In molecular

dynamics models, each nucleon is described as a Gaussian distribution to represent its quan-

tum wave packet with its characteristic nucleon properties. On the other hand, in mean-field

approaches, each nucleon is represented by a set of test particles distributed in a cell array.

9

Figure 1.4: Comparison of a selection of experimental constraints for symmetry energy
parameters. SV is equivalent to Esym(ρ0) and L is equivalent to Lsym. Filled bands indi-
cate constraints from neutron skin thicknesses of tin isotopes [49], dipole polarizability of
208Pb [51, 52], giant dipole resonances (GDR) [53], isotope diffusion in heavy-ion collisions
(HIC) [54], and energies of excitations to isobaric analog states (IAS) [55]. The filled oval
shows constraints from nuclear masses [56]. G and H refer to neutron matter studies of
Gandolfi et al. [57] and Hebeler et al. [58], respectively. The hatched rectangle gives con-
straints from fitting astrophysical mass-radius observations [43, 44]. The overlap region of
the former constraints is indicated by the enclosed white area. Figure taken from [38] with
kind permission of The European Physical Journal (EPJ).

10

Figure 1.5: Constraints on the slope (L) and magnitude (Esym(ρ0)) in Equation 1.6 of the
symmetry energy from heavy-ion collision studies. The shaded light gray and dark purple
area are shown from [54] and [60], respectively. Constraints at given values of Esym(ρ0) are
shown from [61], [10], [36], and [62] as the black lower-limit, solid green circle, open red circle,
and solid orange square, respectively. For clarity and to avoid overlap, the red circle was
offset by 0.1 MeV. Figure taken from [59] with kind permission of The European Physical
Journal (EPJ).

Nucleons and fragments are formed by phase-space coalescence, in which test particles within

an adjustable position range are grouped together.

The differences in the approaches of the models can produce different outcomes. For ex-

ample, Colonna et al. [63] compared multifragmentation reaction simulations from the anti-

symmetrized molecular dynamics (AMD) model [64] and the momentum-dependent stochas-

tic mean-field (SMF) model [65]. The study showed that the models gave different results in

that more free nucleons were produced in the SMF model compared to AMD model, while

light intermediate mass fragments (A = 5 − 15) were more abundant in the AMD model

11

compared to the SMF model. These differences were attributed to early clustering effects

and many-body correlations in the AMD model. The clustering grouped nucleons into light

intermediate mass fragments, reducing the amount of mass available for generating free nu-

cleons. The constraints made on the symmetry energy are thus model dependent, as the

reaction model chosen may affect predictions for the heavy-ion collisions, particularly for

particle emission.

1.3 Dissertation Overview

This experiment examined two different approaches to describe the fragment and neutron

distributions from projectile fragmentation reactions.

In one approach, the experimental

data set compared to the Li`ege Intranuclear Cascade (INCL++) model, a Monte Carlo

based code that first simulates the fast part of the reaction to create an excited remnant

that is then “cooled” through a statistical de-excitation code. The second approach involved

the Constrained Molecular Dynamics (CoMD) model, a dynamical model that contains three

versions of the symmetry energy term. The two models are described in Chapter 2.

The experimental data set was collected at the National Superconducting Cyclotron

Laboratory, where intermediate-energy (55.5 MeV/u) beams of 30S and 40S were reacted with

beryllium targets. Fast neutrons from the reaction were measured in coincidence with heavy

residue fragments using the MoNA LISA (Modular Neutron Array and Large-area multi-

Institutional Scintillator Array) detectors and Sweeper magnet charged particle detectors

[66–68]. Experiment techniques are described in Chapter 3, while analysis of the data is

outlined in Chapter 4. Results and comparisons of the two models with the experimental

data are discussed in Chapter 5, and final remarks and conclusions are given in Chapter 6.

12

Chapter 2

Modeling and Simulations

2.1 Constrained Molecular Dynamics Model

The Constrained Molecular Dynamics (CoMD) model [69,70] is a dynamical simulation used

to describe the interaction (collision) of two heavy-ions over a given length of time. The

length of time can be adjusted to encompass the full evolution of the collision. In CoMD,

each nucleon is described as a Gaussian wave-function to represent the quantum wave packet

with the distribution function:

fi(r, p) =

1

(2πσrσp)3 × exp" −

(r − hrii)2

2σ2
r

(p − hpii)2

2σ2
p

#

−

(2.1)

where hrii is the central position, hpii is the central momentum, σr is the width of the
position distribution, and σp is the width of the momentum distribution of the ith nucleon.

Nucleons were propagated according to the relevant equations of motion and a momentum-

independent Skyrme interaction. The symmetry term in the Skyrme interaction could be

changed, providing a probe to the form of the density dependence of the symmetry energy.

The CoMD model gives special care to the Pauli principle in respect to constraining the

equations of motion. In general quantum molecular dynamics models, a Pauli potential is

used to prevent nucleons of the same isospin and spin state from occupying the same phase-

space. This Pauli potential adds a repulsive force that does not exist in the nucleus. Within

13

the present CoMD calculations, the occupation density in phase space of each nucleon is

calculated every 1 fm/c time step. If an occupation density is found to be greater than one,

the nucleon is considered to be in violation of the Pauli principle. To correct the violation,

the momenta of the neighboring particles are randomly changed while the total momentum

and total kinetic energy of the subset are conserved. If the occupation density drops below

one after the change, the configuration is accepted. If the occupation density requirement

is not met, the nucleon-nucleon collision is rejected. This allows a faster simulation of

heavy-ion collisions compared to other molecular dynamics models. For example, in the

antisymmetrized molecular dynamics model (AMD) [64, 71], computation time scales with

A3 (where A is the number of nucleons). In CoMD, computation time scales with A2 [72].

The decreased computation time allows the simulation to be run for a longer time, enabling

the system to be followed as it dynamically evolves.

The CoMD model is an adjustable model in that it includes three different symmetry

energy parameterizations: stiff-2, stiff-1, and soft. Each parameter gives a distinct slope for

the form of the symmetry energy at a given magnitude. The variation of the equation of

state with these parameters was displayed in Figure 1.3 from Chapter 1.

2.1.1 GEMINI++

The CoMD model does not always produce the final products unless the code is run for

a very long time.

In the present case, the dynamical process was stopped at 2000 fm/c.

At this time, the system had, in general, stabilized with very little evolution in fragment

distributions, and fewer than 2% of fragments remained excited. (The fragment distributions

and their evolutions over time are discussed in Chapter 5.) A list of the excited prefragments

were passed to another code to calculate the final distributions by statistical decay.

14

GEMINI is a statistical decay and evaporation code used to de-excite hot nuclei [73, 74].

The excitation energy, angular momentum, charge, and mass of a hot fragment is used to

calculate its decay path in the nuclear landscape to the final, cold product. A Monte Carlo

method is used to calculate the most probable series of sequential binary decay steps from the

initial hot fragment. The Hauser-Feshbach formalism is used to calculate the decay widths of

the light charged particles (including n, p, d, t, 3He, α, 6He, and 6-8Li fragments), while the

fission channel is calculated with the Bohr-Wheeler formalism. The original GEMINI code

was converted to C++ and updated to improve the calculation of the width of the charge

and mass distributions from fission of heavy fragments. This revised code was renamed

GEMINI++ and was used in the present work.

GEMINI and GEMINI++ are often used as so-called “afterburner codes” that are coupled

to molecular dynamics models to decay any remaining hot fragments from the dynamical

model [72, 75–78]. Passing the leftover hot fragments from CoMD through GEMINI++

mimics a two-step reaction, with CoMD covering both the abrasion and initial ablation

period and GEMINI++ completing the ablation period leading to the final de-excited state.

However, it should be emphasized that CoMD attempts to reach a final de-excited state

without connection to an afterburner code. In the present case it was found that after a

few thousand fm/c, only a small percentage of fragments remain hot for GEMINI++ to de-

excite. For each of the three symmetry energy options within both simulated reactions, only

2% or fewer of the fragments were passed to GEMINI++. While the percent of fragments

affected was small, it was important to cool the fragments to their final states.

In using the GEMINI++ code as an evaporation code with CoMD simulations, some

assumptions had to be made. When the CoMD simulation is stopped at an arbitrary point,

hot fragments are likely deformed and at a density below normal nuclear density (ρ0 =

15

0.16 fm-3). The GEMINI++ code assumes that the hot fragment is at normal nuclear

density when calculating the de-excitation of the hot fragment. This difference in shape and

density could potentially affect the decay process. Furthermore, GEMINI++ decays each

of the hot fragments independently so final trajectories of the fragments are locked in and

no further external nuclear or Coulomb forces are considered. While these assumptions are

necessary in using GEMINI++ to cool remaining hot fragments, dynamical models connected

to GEMINI and GEMINI++ have produced results with reasonable agreement to a variety

of experimental observables [72, 75–79].

2.1.2 Previous CoMD Comparison to MoNA LISA-Sweeper Data

A similar analysis of CoMD predictions for prior MoNA LISA-Sweeper data from another

reaction system has been carried out. The original experiment [80,81] was designed to study

neutron-unbound states of 27F and 28F populated from direct nucleon knock-out reactions

from a 29Ne beam on a 9Be target. The experiment made exclusive measurements of final

residue fragments, neutrons, and gamma rays.

While 29Ne was the desired projectile in the proposed experiment, 87% of the beam

delivered to the experiment was actually 32Mg contaminant due to the difficulty of separating

the very weakly produced 29Ne ions. The more prevalent 32Mg + 9Be contaminant reaction

was simulated in the CoMD framework by Kohley et al. [60]. The 73 MeV/u reaction was run

in CoMD up to 1500 fm/c with no coupling to a de-excitation code. Their results indicated

that the N/Z of the final fragment residue may be sensitive to the form of the symmetry

energy.

While the neutron yield was partially explored in the work, there were a few limitations

in fully exploring neutron multiplicities. The experiment was designed to detect gamma

16

rays with the CAESAR (CAESium iodide ARray) detector [82]. The inclusion of CAESAR

forced the target to be set farther upstream of the normal target position typical for a MoNA

LISA-Sweeper configuration, resulting in a lower angular acceptance in the neutron detector

array. The present work did not include the CAESAR detector so that neutron distribu-

tions were more complete and could be compared to CoMD simulations. Furthermore, the

prior experiment required neutron coincidence events detected in MoNA LISA with charged

particle fragments measured in the Sweeper detectors. Thus, no fragments with zero hits in

MoNA LISA were examined. The present work, as described in Chapter 3, did not require

hits in MoNA LISA to measure charged fragments.

2.1.3 CoMD Model Parameters

In this analysis, CoMD simulations of the 30,40S + 9Be reactions were run at 55 MeV/u with

each of the three symmetry energy options.

A magnitude (Esym(ρ0)) of 32 MeV was chosen for this experiment due to its relatively

well-known constraint from [50] and [59]. The calculated slopes (Lsym) at the saturation

density for this magnitude are 105 MeV (stiff-2), 78 MeV (stiff-1), and 51 MeV (soft). These

values for slope and magnitude lie within the bounds of current potential constraints [38,59].

For comparison with the current set of experimental data, the CoMD heavy-ion collisions

were propagated to 2000 fm/c, allowing the system to evolve dynamically relatively far and

become relatively cool. Any hot fragments left at 2000 fm/c were further de-excited using

GEMINI++. The combined model will be referred to as CoMD + GEMINI++ in this work.

17

2.1.4

Isotope Production Output

The isotope production distributions taken directly from the CoMD + GEMINI++ sim-

ulations are shown in Figures 2.1-2.6. Figures 2.1 and 2.2 contain the distributions from

the stiff-2 option, Figures 2.3 and 2.4 contain the distributions from the stiff-1 option, and

Figures 2.5 and 2.6 contain the distributions from the soft option. The stiff-2 distributions

are skewed slightly more in neutron-rich fragments relative to the distributions from the soft

simulations. (Stiff-1 falls in the middle.) Below the saturation density, the soft symmetry

potential is more repulsive for neutrons than the stiff potential. Thus, the soft symmetry

energy option should result in an increase in neutron emission, which leads to a decreased

average N/Z of the residues. The stiffer option was less repulsive, resulting in increased

average N/Z of the residues.

It should be noted that the fragment distributions presented here represent the complete

inclusive distributions. Filtering the fragments and coincident neutrons through a set of

conditions which mimicked the treatment of the experimental data was necessary to compare

the simulated results to experimental results. The filtering procedure is described in Chapter

4 and filtered results of hit multiplicities will be presented in Chapter 5.

2.2 Li`ege Intranuclear Cascade Model

2.2.1

INCL++

The Li`ege Intranuclear Cascade (INCL) model is a Monte Carlo based code used to simulate

nucleon, pion, and light-ion induced reactions on heavy nuclei. The energy of these collisions

can range from a few tens of MeV to a few GeV. The latest release of the code, INCL4.6 [83],

18

3
10×

3
10×

s
t
n
u
o
C

s
t
n
u
o
C

s
t
n
u
o
C

4

3.5

3

2.5

2

1.5

1

0.5

0

7

6

5

4

3

2

1

0

16

14

12

10

8

6

4

2

0

30

S + 

40

S + 

9
Be

9
Be

9

10

11

12

13

15

16
14
Carbon A

17

18

19

20

21

3
10×

12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

Oxygen A

3
10×

16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

s
t
n
u
o
C

s
t
n
u
o
C

s
t
n
u
o
C

4.5

4

3.5

3

2.5

2

1.5

1

0.5

0

10 11

12 13

14 15

16 17 18

19 20

21 22

23 24

Nitrogen A

3
10×

10

8

6

4

2

0

22

20

18

16

14

12

10

8

6

4

2

0

15 16

17 18

19 20

21 22 23

24 25

26 27

28 29

Fluorine A

3
10×

19 20

21 22

23 24

25 26 27

28 29

30 31

32 33

Neon A

Sodium A

Figure 2.1: Inclusive isotope distributions from the stiff-2 CoMD + GEMINI++ simulations
for elements Z = 6 − 11.

19

3
10×

3
10×

s
t
n
u
o
C

s
t
n
u
o
C

50

40

30

20

10

0

120

100

80

60

40

20

0

21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39

Aluminum A

3
10×

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

Phosphorus A

s
t
n
u
o
C

s
t
n
u
o
C

30

25

20

15

10

5

0

90

80

70

60

50

40

30

20

10

0

s
t
n
u
o
C

120

100

80

60

40

20

0

30

40

Be9S + 
Be9S + 

19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

Magnesium A

3
10×

22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

Silicon A

3
10×

26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45

Sulfur A

Figure 2.2: Inclusive isotope distributions from the stiff-2 CoMD + GEMINI++ simulations
for elements Z = 12 − 16.

20

3
10×

3
10×

s
t
n
u
o
C

s
t
n
u
o
C

s
t
n
u
o
C

4

3.5

3

2.5

2

1.5

1

0.5

0

8

7

6

5

4

3

2

1

0

18

16

14

12

10

8

6

4

2

0

30

40

Be9S + 
Be9S + 

9

10

11

12

13

15

16
14
Carbon A

17

18

19

20

21

3
10×

12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

Oxygen A

3
10×

16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

s
t
n
u
o
C

s
t
n
u
o
C

s
t
n
u
o
C

6

5

4

3

2

1

0

12

10

8

6

4

2

0

24

22

20

18

16

14

12

10

8

6

4

2

0

10 11

12 13

14 15

16 17 18

19 20

21 22

23 24

Nitrogen A

3
10×

15 16

17 18

19 20

21 22 23

24 25

26 27

28 29

Fluorine A

3
10×

19 20

21 22

23 24

25 26 27

28 29

30 31

32 33

Neon A

Sodium A

Figure 2.3: Inclusive isotope distributions from the stiff-1 CoMD + GEMINI++ simulations
for elements Z = 6 − 11.

21

s
t
n
u
o
C

s
t
n
u
o
C

35

30

25

20

15

10

5

0

90

80

70

60

50

40

30

20

10

0

s
t
n
u
o
C

100

80

60

40

20

0

3
10×

3
10×

30

40

Be9S + 
Be9S + 

19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

Magnesium A

s
t
n
u
o
C

50

40

30

20

10

0

21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39

Aluminum A

3
10×

3
10×

100

s
t
n
u
o
C

80

60

40

20

0

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

Phosphorus A

22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

Silicon A

310×

26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45

Sulfur A

Figure 2.4: Inclusive isotope distributions from the stiff-1 CoMD + GEMINI++ simulations
for elements Z = 12 − 16.

22

s
t
n
u
o
C

s
t
n
u
o
C

s
t
n
u
o
C

3
10×

30

40

Be9S + 
Be9S + 

9

10

11

12

13

15

16
14
Carbon A

17

18

19

20

21

6

5

4

3

2

1

0

3
10×

10

8

6

4

2

0

18

16

14

12

10

8

6

4

2

0

12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

Oxygen A

3
10×

16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

s
t
n
u
o
C

s
t
n
u
o
C

s
t
n
u
o
C

8

7

6

5

4

3

2

1

0

14

12

10

8

6

4

2

0

25

20

15

10

5

0

3
10×

10 11

12 13

14 15

16 17 18

19 20

21 22

23 24

Nitrogen A

3
10×

15 16

17 18

19 20

21 22 23

24 25

26 27

28 29

Fluorine A

3
10×

19 20

21 22

23 24

25 26 27

28 29

30 31

32 33

Neon A

Sodium A

Figure 2.5: Inclusive isotope distributions from the soft CoMD + GEMINI++ simulations
for elements Z = 6 − 11.

23

3
10×

3
10×

s
t
n
u
o
C

s
t
n
u
o
C

50

40

30

20

10

0

80

70

60

50

40

30

20

10

0

21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39

Aluminum A

3
10×

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

Phosphorus A

s
t
n
u
o
C

s
t
n
u
o
C

s
t
n
u
o
C

35

30

25

20

15

10

5

0

80

70

60

50

40

30

20

10

0

30

40

Be9S + 
Be9S + 

19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

Magnesium A

3
10×

22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

Silicon A

3
10×

100

80

60

40

20

0

26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45

Sulfur A

Figure 2.6: Inclusive isotope distributions from the soft CoMD + GEMINI++ simulations
for elements Z = 12 − 16.

24

contains support for nucleon, pion, and composite projectiles. An object-oriented C++

version of the code, INCL++ [84, 85], is equivalent to the INCL4.6 release for nucleon and

pion induced reactions. Unlike INCL4.6, INCL++ contains support for light-ion projectiles

with masses up to 18. Note that the target in the present case is 9Be and falls within

the bounds of the supported projectile masses and can be used as the “projectile” in an

inverse-kinematics option available to INCL++ and described later in this section.

INCL++ has been described as a time-like intranuclear cascade model. In the first stage,

all nucleons are distributed appropriately in phase space. Target nucleons are given positions

and momenta at random in agreement with Saxon-Woods and Fermi sphere distributions,

respectively. The projectile particle is given the appropriate energy and an impact parameter

is assigned at random. Once the parameters are defined, the projectile moves toward the

target and initiates an avalanche of binary collisions within the target nucleus. Resulting

particles are individually tracked, with emission of nucleons, pions, and light clusters fol-

lowing. Light clusters are produced by a dynamical phase-space coalescence algorithm. In

cases of binary particle-particle collisions, nucleons are restricted by Pauli blocking. When

the fast particles escape the nuclei and the remnant nucleus shows signs of thermalization,

the cascade step stops.

Predictions from the standard version of the model, INCL4.2, have been tested against

a large body of experimental data from reactions in the range of 200 MeV/u to 2 GeV/u.

INCL4.2 was found to be in good agreement with experimental total reaction cross sections,

particle multiplicities, residue mass and charge distributions, residue recoil velocity distribu-

tions, and neutron, proton, pion, and composite double differential cross sections [86].

INCL4.6 (and, by extension, INCL++) incorporated several new features to improve the

standard version of INCL. New features introduced to INCL4.6 included cluster produc-

25

tion through a dynamical phase space coalescence model, Coulomb deflection for entering

and exiting charged particles, experimental threshold values for particle emission, and im-

proved treatment of Pauli blocking [83]. The new features were found to improve the model

predictions [83, 87, 88].

INCL++ simulates the intranuclear-cascade or fast part of the reaction. After this step,

the remaining nucleus, or cascade remnant, still holds excitation energy and may de-excite

through fission, multifragmentation, or the emission particles such as nucleons, light nuclei, or

photons. Coupling INCL++ to a de-excitation code is necessary to produce the distributions

of the final (observed) fragments and neutrons, similar to the CoMD code. The INCL

codes can interface with a few de-excitation models (including the previously described

GEMINI++), however the ABLA07 code [89] is the default option selected by the authors

of INCL++.

2.2.2 ABLA07

To de-excite the cascade remnant, information on the fragment is passed to a separate

statistical de-excitation code. In this case the INCL++ package includes the ABLA07 code.

ABLA07 is a semiclassical code that describes the de-excitation of a hot fragment through

comparison of the relative probabilities of simultaneous break-up, particle emission, and fis-

sion. Simultaneous break-up occurs when a hot fragment fractures into several fragments

due to thermal instability. Particle evaporation is based on the Weisskopf-Ewing formalism,

and the fission decay width is calculated by taking into account dynamical effects. An ana-

lytical approximation to the solution of the Fokker-Planck equation for the time dependent

fission width is included, allowing ABLA07 to become a more dynamical code as compared

to a purely statistical model. Earlier versions of the ABLA code focused on the emission

26

of neutrons, light charged particles (Z < 3), and gamma-rays, while ABLA07 included the

emission of intermediate-mass fragments (Z > 2) [89]. While ABLA07 was originally cre-

ated for describing the de-excitation stage of heavy-ion collisions and spallation reactions

at relativistic energies, it has been coupled to INCL++ to handle the de-excitation step in

nuclear reactions [83].

2.2.3 Limitations of the INCL Model

The INCL4.6 and INCL++ models have been well tested for projectiles up to mass 18,

target nuclei close to stability, and reaction energies from around 10 MeV/u to 20 GeV/u

[83, 84]. The upper limit of the projectile mass number is based on the validity of the

physical approximations and testing. Heavier projectiles may be simulated in the code, but

the outputs remain untested by the authors. To accommodate reactions with projectiles

above a mass of 18 with a light target, INCL++ comes with the ability to introduce an

inverse-kinematic transformation [84]. The initial set of variables are calculated in the direct-

kinematics frame, with the input projectile moving at the given velocity along the positive

Z-axis with the target at rest.

In the inverse-kinematic frame, the target moves at the

given velocity along the positive Z-axis with the projectile at rest. The calculation of the

interaction is then followed as normal and the reaction products are boosted back to the

laboratory frame at the end of the simulation.

While intranuclear-cascade style models are generally not used for reactions below 150

MeV/u, the INCL codes have shown reasonable agreement to experimental data down to a

few tens of MeV per nucleon [83]. Reactions that fall below a few MeV do not reproduce

inelastic events well. Low-energy reactions induced by neutral or negative particles have not

27

been tested. The upper energy limit for the INCL codes falls around 20 GeV/u and is not

relevant to the present work.

2.2.4

INCL++ Model Parameters

Because the projectiles for this experiment had masses above 18, INCL++ was run in the

inverse-kinematics mode for this analysis. The INCL++ collisions were run with the pro-

jectile 9Be and targets 30S and 40S. The fragments were then transformed into the inverse-

kinematics frame to give fragment output from the measured 55 MeV/u 30,40S + 9Be reac-

tions. The de-excitation of the cascade remnants was carried out by ABLA07, the default

recommended option in the INCL++ package. Unlike the CoMD model, INCL++ does not

contain adjustable symmetry energy parameters and thus has not been used in symmetry

energy research.

2.2.5

Isotope Production Output

The inclusive or total isotopic distributions of produced fragments from INCL++ are shown

in Figures 2.7 and 2.8. Similar to the previously described fragments from the CoMD +

GEMINI++ model, the fragment distributions presented here do not have an experimental

filter. Simulated INCL++ fragments and neutrons were filtered through an experimental

set-up as described in Chapter 4 to be compared to experimental data. Filtered results from

these calculations are also described in Chapter 5.

Note that the final fragment distributions are very different for the two projectiles, as

expected, except for the light fragments. The predictions from the two models are, in fact,

28

relatively similar, except that the INCL++ model has a large odd-even staggering for the

products from the 40S beam.

29

3
10×

3
10×

s
t
n
u
o
C

s
t

n
u
o
C

s
t
n
u
o
C

70

60

50

40

30

20

10

0

60

50

40

30

20

10

0

45

40

35

30

25

20

15

10

5

0

30

S + 

40

S + 

9

9

Be

Be

9

10

11

12

13

14

15

16

17

18

19

20

Carbon A

3
10×

13

14

15

16

17

18

19

20

21

22

23

24

Oxygen A

3
10×

17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

s
t
n
u
o
C

s
t

n
u
o
C

s
t
n
u
o
C

45

40

35

30

25

20

15

10

5

0

25

20

15

10

5

0

45

40

35

30

25

20

15

10

5

0

12

13

14

15

16

17

18

19

20

21

22

23

Nitrogen A

3
10×

17 18

19 20

21 22

23 24 25

26 27

28 29

30 31

Fluorine A

3
10×

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

Neon A

Sodium A

Figure 2.7: Inclusive isotope distributions from INCL++ simulations for elements Z = 6−11.

30

3
10×

3
10×

s
t
n
u
o
C

s
t
n
u
o
C

60

50

40

30

20

10

0

50

40

30

20

10

0

s
t
n
u
o
C

s
t
n
u
o
C

s
t
n
u
o
C

70

60

50

40

30

20

10

0

30

S + 

40

S + 

9

9

Be

Be

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

Magnesium A

3
10×

100

80

60

40

20

0

140

120

100

80

60

40

20

0

22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42

Silicon A

3
10×

26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45

Sulfur A

21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

Aluminum A

3
10×

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

Phosphorus A

Inclusive isotope distributions from INCL++ simulations for elements Z =

Figure 2.8:
12 − 16.

31

Chapter 3

Experimental Techniques

Two projectile fragmentation beams, 30S and 40S, were produced at the National Super-

conducting Cyclotron Laboratory for this work. The fragments and neutrons resulting from

collisions with a 9Be target placed in front of the Sweeper magnet were measured with the

MoNA LISA-Sweeper experimental set-up [66–68]. This system used a very large area of

plastic scintillators to detect neutrons in coincidence with a series of charged particle detec-

tors to characterize the fragments. The details of the experimental configuration are given

in this chapter. Calibrations, measurements, and data analysis are described in Chapter 4.

3.1 Beam Production

The experiment was performed at the National Superconducting Cyclotron Laboratory

(NSCL) at Michigan State University. Beams of 30S and 40S were used to provide proton-rich

and neutron-rich projectiles on either side of stability, as shown in Figure 3.1, thus allowing

a wide range of fragment isospin.

The 30S and 40S beams were produced by projectile fragmentation of stable beams [90]

that were accelerated by the Coupled Cyclotron Facility (CCF) [91]. The stable fragments

impinged on a 9Be target and resulting fragments were filtered by the A1900 Fragment

Separator [92]. A schematic diagram of the CCF and A1900 Fragment Separator is displayed

in Figure 3.2.

32

30Cl 31Cl 32Cl 33Cl 34Cl 35Cl 36Cl 37Cl 38Cl 39Cl 40Cl 41Cl 42Cl
29S 30S 31S 32S 33S 34S 35S 36S 37S 38S 39S 40S 41S
28P 29P 30P 31P 32P 33P 34P 35P 36P 37P 38P 39P 40P

Z

N

stable

proton emission

EC+, β+

β-

Figure 3.1: A section of the chart of the nuclei showing the isotopes of sulfur. Gray boxes
represent stable isotopes, while yellow, green, and blue boxes represent isotopes that decay
by proton emission, electron capture or β+ decay, or β− decay, respectively. The 30S and
40S isotopes highlighted in the red boxes are the projectiles used in the experiment.

3.1.1

30S Beam Production

To produce the 30S beam, a primary beam of 36Ar was ionized by an electron cyclotron

resonance (ECR) ion source. The resulting 36Ar7+ ions were injected into the K500 cyclotron

and accelerated to 13.06 MeV/u. The ions were sent to the K1200 cyclotron, where they

were passed through a stripper foil to remove the remaining electrons. The K1200 cyclotron

accelerated the fully ionized 36Ar18+ to 150 MeV/u.

Upon leaving the K1200 cyclotron, the 36Ar ions were impinged on a beryllium production

target with a thickness of 940 mg/cm2. Collisions in the target induced the projectile

fragmentation process and a wide variety of nuclei were produced. To filter out the desired

30S fragments, the produced nuclei were sent through the A1900 Fragment Separator, a 35-

meter long device which uses magnetic selection and energy loss in a solid wedge degrader to

select individual isotopes. The A1900 Fragment Separator is composed of four dipoles that

sort fragmentation products based on their momentum-to-charge ratio, or rigidity (Bρ =

p/q). Quadrupole magnets placed between the dipole segments refocused the beam.

33

Figure 3.2: Schematic layout of the Coupled Cyclotron Facility and A1900 Fragment Sepa-
rator. Figure reproduced from [93].

An aluminum wedge with a thickness of 855 mg/cm2 was placed between the second

and third dipoles to separate the isotopes with the same magnetic rigidity. As the different

elemental fragments passed through the wedge, they lost energy in proportion to Z2 [94].

Different element fragments with the same rigidity before the wedge had different rigidities

after passing through the wedge. This allowed a second stage of separation in the second

half of the separator.

Momentum slits were used to tune the acceptance of the A1900 Fragment Separator.

These slits were set to 0.5% momentum acceptance for this experiment. After passing

through the A1900 Fragment Separator, the secondary beam fragments were delivered to

the experimental vault. Only 32% of the secondary beam was composed of 30S fragments.

Major contaminants included 29P and 28Si. These contaminants were excluded from analysis

in a procedure described in Chapter 4. The 30S fragments were delivered to the experimental

vault at an energy of 55.5 MeV/u and a magnetic rigidity of 2.0403 Tm.

34

3.1.2

40S Beam Production

The 40S secondary beam was produced in a similar manner to the 30S secondary beam.

A 48Ca source was ionized to produce 48Ca8+ ions, which were accelerated in the K500

cyclotron to 12.28 MeV/u. Passing through the stripper foil again to remove the remaining

electrons, the ions were accelerated in the K1200 cyclotron to 140 MeV/u. The fully stripped

48Ca ions were then impinged on a beryllium production target with a thickness of 1151

mg/cm2.

Resulting fragments were sent through the A1900 fragment separator, where the dipole

magnets, quadrupole magnets, and an aluminum wedge with a thickness of 765 mg/cm2 were

used to filter out unwanted fragments and send the desired 40S fragments to the experimental

vault at energy of 55.5 MeV/u and a magnetic rigidity of 2.7201 Tm. The A1900 Fragment

Separator momentum slits were set to 0.5% momentum acceptance. While the 30S fragments

were accompanied by major contaminants, the 40S beam had a purity of around 98%.

3.2 A1900 and Target Scintillators

A timing scintillator made of 0.125 mm thick BC-404 plastic optically coupled to a photo-

multiplier tube (PMT) was placed at the end of the A1900 Fragment Separator to provide

a time-of-flight reference signal. After passing through the A1900 fragment scintillator, the

30S and 40S secondary beams were delivered to the experimental vault. A three-dimensional

concept drawing of of the experimental equipment is displayed in Figure 3.3 and a top-down

view is shown in Figure 3.4. Both figures show the locations of the target scintillator, reaction

target, sweeper magnet, and charged particle detectors.

35

Charged Particle Detectors

Sweeper Magnet

Reaction Target

MoNA LISA

Target Scintillator

Figure 3.3: Concept mechanical drawing of the MoNA LISA-Sweeper experimental equip-
ment in the vault. Note that only some of the MoNA LISA bars are displayed. Drawing
provided by Craig Snow.

Once the beams entered the experimental vault, the beams again passed through a thin

plastic timing scintillator referred to as the target scintillator to provide another time-of-

flight reference. The target scintillator, placed 1.15 meters before the target, was a 0.254

mm thick BC-404 plastic foil and coupled to a PMT. The secondary beams then entered the

target chamber, in which the 30S and 40S fragments were reacted in a 47 mg/cm2 beryllium

target.

36

Figure 3.4: Schematic of the MoNA LISA-Sweeper experimental equipment in the vault. See
text for details.

3.3 Sweeper Magnet

Charged particle fragments and neutrons from the secondary fragmentation reactions, as well

as unreacted beam particles, left the target chamber and entered the Sweeper magnet [68],

a large-gap dipole magnet with a bending angle of 43.3 degrees and a bend radius of 1

meter. The Sweeper magnet was designed to bend the trajectories of the unreacted ions

and charged reaction fragments away from the original beam axis into a series of charged

particle detectors. The neutrons, which were unaffected by the magnetic field produced by

the Sweeper, then passed through a 14 cm vertical gap and a 6.4 mm thick steel plate placed

at zero degrees before flying towards the MoNA LISA neutron detectors. The steel plate

was necessary to maintain vacuum in the magnet gap.

A Hall probe was used to measure the magnetic field strength, which had been previously

mapped [95]. During the production runs of this experiment, the magnetic rigidity of the

37

Sweeper MagnetReaction TargetTarget ScintillatorMoNALISA-1BeamCharged ParticleDetectorsLISA-2Sweeper magnet was set to bend the unreacted beam away from the center of the charged

particle detectors. This prevented the high beam rate from damaging the charged particle

detectors.

To accept a variety of fragments from each secondary beam reaction, two magnet settings

were chosen for each secondary beam. For the 30S + 9Be reaction, the Sweeper magnet was

set to 1.51 and 2.25 Tm. For the 40S + 9Be reaction, the Sweeper magnet was set to 2.01 and

2.27 Tm. Reaction products falling outside of the Sweeper acceptance, including products

with the same rigidities as the unreacted beam, were not measured. After passing through

the 47 mg/cm2 beryllium target, unreacted 30S and 40S beams had rigidities of 1.97 Tm and

2.65 Tm, respectively.

3.4 Charged Particle Detectors

After leaving the Sweeper magnet, the charged fragments entered a large vacuum box con-

taining a series of detectors. The positions and angles of each charged fragment were deter-

mined with two Cathode Readout Drift Chambers (CRDCs). Energy loss was measured in

an ionization chamber, and the arrival time was measured with a thin plastic timing scintil-

lator. Figure 3.5 contains a layout schematic of the charged particle detectors following the

Sweeper magnet.

3.4.1 Blocker

To prevent the unreacted beams from hitting the active area of the CRDCs, a tungsten

blocker with dimensions of 24.5 cm × 13 cm × 9.5 cm was inserted at the edge of the active
area of the detectors. The blocker was located at the exit of the Sweeper magnet and before

38

Beam

Reaction 

Target

Sweeper 
Magnet

Blocker

Ion Chamber

CRDCs

Thin Scintillator

Hodoscope

Figure 3.5: Layout schematic of the Sweeper magnet and charged particle detectors. The
solid black line indicates the path of the beam. The dashed line represents the path of the
unreacted beam if the Sweeper magnet was not present.

the first CRDC in the charged particle array box, as seen in Figure 3.5. The unreacted beam

was stopped in the blocker.

3.4.2 Cathode Readout Drift Chambers

Two CRDCs [96] were used in this experiment to measure the X and Y positions of the

fragments as they moved through the detector box. The first CRDC was placed immediately

after the blocker, 1.72 m downstream from the target. The second CRDC was located 1.55

m after the first CRDC. The full active area for each CRDC was 30 cm × 30 cm in the
XY-plane, though as mentioned previously, one side of the active area of the CRDCs was

covered using the tungsten blocker during production runs.

Each CRDC was filled with a gas mix of 20% isobutane and 80% CF4 at an operating

pressure of 40 Torr. The CRDCs acted similarly to two-dimensional ion chambers. Charged

particles passed through the detector and produced ionization pairs in the gas. A 1000 V

drift voltage was applied between a plate at the top of the detector and the Frisch grid at the

39

Charged Particle Path

–E

e- drift

z

x

y

Frisch Grid

Anode Wire

Cathode Pads

Figure 3.6: Schematic of a Cathode Readout Drift Chamber (CRDC). The Z-direction has
been expanded for detail, while the field shaping wires are omitted for simplicity. Figure
reproduced from [96].

bottom of the detector, creating an applied electric field and causing the ionization pairs to

drift apart. Field shaping wires were placed parallel to the X-direction along the two faces

of the CRDC. An anode wire ran below the Frisch grad and parallel to the X-axis. Along

the Frisch grid was a series of 116 aluminum cathode pads with a pad pitch of 2.54 mm. The

pads were segmented along the X-axis. A schematic diagram of a CRDC is shown in Figure

3.6.

As electrons drifted through the Frisch grid, the strong field caused by the anode wire

caused an avalanche of electrons. The X-position of a particle was determined from the

induced charge distribution upon the cathode pads. The charge was collected and was then

40

fit with a Gaussian function. The centroid of the Gaussian function was then taken to be

the X-position. The Y-position of a particle was determined by the relative drift time of the

electrons to the anode wire. The drift time was measured as the time difference between the

thin timing scintillator near the end of the charged particle array box and the signal from

the anode wire. Because there was no segmentation in the Z-direction and each CRDC had

a relatively small thickness (with an interaction length of about 8.6 cm through the center of

the detector), the Z-position was assumed to be the center of the each CRDC. The X and Y

positions of the fragment measured in each of the CRDCs was used to determine the angle

of its path.

3.4.3

Ion Chamber

Directly downstream of the CRDCs was the ion chamber [96], which was used to determine

the energy loss of the charged particles. The ion chamber had an active area of 40 cm × 40
cm, a length of 65 cm, and was filled with a gas mixture of 90% argon and 10% methane

at 300 Torr. The large windows were made of Kevlar filament, 12 µm PPTA, and epoxy.

The windows allowed the particles to enter and exit the detector with minimal energy loss.

The entrance window had an area of 30 cm × 30 cm to correspond to the active area of the
second CRDC. The exit window was larger, measuring 40 cm × 40 cm, to account for beam
divergence and multiple scattering. A schematic diagram of the ion chamber is shown in

Figure 3.7.

Like in the CRDCs, a charged particle passing through the gas of the ion chamber created

ionization pairs. The electrons drifted through the detector and passed through a Frisch grid.

The electrons were collected on 16 collection pads, which were segmented in the Z-direction

41

–E

Charged 
Particle Path

e- drift

y

z

Charge Collection Pads

x

Figure 3.7: Schematic diagram of the ion chamber. Figure reproduced from [97].

and extended parallel to the X-direction. The charge collected on all 16 pads was summed

and used to calculate the energy loss in the chamber.

3.4.4 Thin Timing Scintillator

A thin timing scintillator [96,97], located directly after the ion chamber, was used to measure

the time-of-flight of the particles and to trigger the data acquisition system. The scintillator

was made of EJ-204 plastic and measured 5 mm thick covering 55 cm × 55 cm in area. Two
light guides were attached to the top of the scintillator, while another two light guides were

attached to the bottom. Each light guide was optically coupled to a PMT. A schematic

drawing of the thin scintillator is show in Figure 3.8.

42

Left-Upper
PMT 0

Right-Upper
PMT 2

Scintillator

Light Guides

y

z

x

Left-Lower
PMT 1

Right-Lower
PMT 3

Figure 3.8: Schematic drawing of the thin scintillator while looking in the direction of the
beam. Figure adapted from [95].

As a charged particle passed through the thin scintillator, it deposited energy into the

organic plastic material. Organic scintillators like the thin scintillator material exploit π-

electron structures of the dopant molecules in the plastic. At room temperature, most of

the dopant molecules in the plastic were at the ground state. The deposited energy from

the charged particle excited π electrons. Some of the highly excited molecules de-excited to

the first excited singlet state through internal conversion. Any state with excess vibrational

energy quickly thermalized to reach equilibrium with its neighbor. The resulting excited

molecules at the first excited singlet state then de-excited to the ground state and emitted

43

Figure 3.9: Schematic drawing of the Hodoscope CsI(Na) array. Figure taken from [97].

scintillation light. This prompt fluorescence occured with a decay constant of 1.8 ns in

EJ-204 plastic. The light was then propagated to the light guides.

3.4.5 Hodoscope

The last component of the charged particle detector array was a CsI(Na) array referred to

as the hodoscope [97], which was designed to stop the charged particles and measure their

residual energy. The Sweeper hodoscope was similar to the hodoscope used with the S800

Spectrometer [98]. The hodoscope array was composed of twenty-five CsI(Na) crystals, each

with a dimension of 3.25 in × 3.25 in × 2.16 in. The 25 crystals were arranged in a five-by-
five array centered on the beam line. Each crystal was wrapped in 0.2 mm thick reflective

material and was optically coupled to a Hamamatsu PMT R1307 with magnetic shielding.

A mechanical drawing of the hodoscope is shown in Figure 3.9.

44

Figure 3.10: Schematic drawing of a single plastic scintillator bar in MoNA LISA.

The crystals were hygroscopic and easily damaged by moisture in the air. A four-sided

gas cover was fit to surround the crystals, and a gas inlet allowed the flow of dry air along

the face of the crystals when the chamber was not under vacuum. This cover was removed

prior to running the experiment.

3.5 MoNA LISA

The positions and times-of-flight of neutrons was measured with the Modular Neutron Array

(MoNA) and the Large-area multi-Institutional Scintillator Array (LISA) [66, 67, 99]. Both

MoNA and LISA are composed of 144 plastic scintillator bars measuring 200 cm × 10 cm
× 10 cm. MoNA bars were made of BC-408 plastic and LISA bars were composed of EJ-
200 plastic, equivalent plastics from Bicron and Eljen, respectively. Each bar was wrapped

in reflective material to reduce light lost and black plastic to prevent ambient light from

inducing signals. A schematic of a single bar is shown in Figure 3.10.

As uncharged neutrons passed through the bars, the neutrons scattered on the hydrogen

or carbon in the plastic, producing ionization and then scintillation light. Because the

neutron-to-hydrogen mass ratio is much larger than the neutron-to-carbon mass ratio, the

45

recoil is much greater when the neutron interacts with a hydrogen nucleus. Thus, most of the

scintillation light output came from neutron on hydrogen scattering reactions. The light was

propagated to each end of the bar, where it was collected by a PMT. MoNA used Photonis

XP2262/B PMTs and LISA used Hamamatsu R329-02 PMTs. The PMTs turned the light

into an electronic signal for recording.

Due to the modular nature of the array, the MoNA LISA bars may be configured in a

variety of settings. Bars were stacked 16-high, numbered 0 at the bottom and 15 at the

top. Each stack of 16 bars formed a vertical “wall” or layer. Example layers are shown in

the concept image in Figure 3.3. The assembled layers were elevated such that the layers

were centered on the beam axis, with eight bars falling above and eight bars falling below

the beam axis. Each layer was assigned a letter, A-R. For this experiment, the MoNA LISA

bars were split among three tables named MoNA, LISA-1, and LISA-2. The MoNA bars

were placed in a nine-layer arrangement (MoNA), while the LISA bars were divided into a

four-layer table (LISA-1) and five-layer table (LISA-2). Thus, the location of each bar was

defined by the bar’s table (MoNA, LISA-1, or LISA-2), layer (A-R), and stack-height (0-15).

Bar arrangements and labels for each table are shown in Figure 3.11.

The placement of the bars relative to the target is shown in Figure 3.12, where the blue

shaded region indicates the acceptance angles for neutron measurement from the target. The

center of the first layer of LISA-1 was placed 6.4 m from the target and offset 6 degrees from

the beam axis. The MoNA table was placed directly behind the LISA-1 table at a distance

of 7.6 m from the target. The LISA-2 table was placed at an angle of 38 degrees with the

center of the first layer 4.1 m from the target. The full array covered angles from about -3

degrees to 51 degrees.

46

r
e
b
m
u
N
 
r
a
B

15

14

13

12

11

10

9

8

7

6

5

4

3

2

1

0

෡

y

෡

z

MoNA

LISA-1

LISA-2

15

14

13

12

11

10

9

8

7

6

5

4

3

2

1

0

r
e
b
m
u
N
 
r
a
B

r
e
b
m
u
N
 
r
a
B

15

14

13

12

11

10

9

8

7

6

5

4

3

2

1

0

A B C D E F G H I

Layer

J K L M

Layer

N O P Q R

Layer

Figure 3.11: Labels for each bar in MoNA LISA as seen from the sides of each table. The
front layers for the MoNA, LISA-1, and LISA-2 tables were A, J, and N, respectively.

3.6 Electronics and Data Acquisition

The electronics and data acquisition (DAQ) systems have been described in previous work

[80, 95, 97, 100, 101]. A brief overview of the systems is presented here, and a condensed

schematic is displayed in Figure 3.13.

MoNA, LISA, and the Sweeper particle detectors were operated as three independent

DAQ subsystems. The subsystems were connected to a “Level 3” logic system that gener-

ated a system trigger and a “timestamp.” The system trigger was produced upon the arrival

of a signal measured in the left-upper PMT of the thin timing scintillator. The “timestamp”

was a 64-bit word generated by a “clock” in the Level 3 system. The “timestamp” was

simply an event tag and not a timestamp in the strict definition as the clock generating the

“timestamps” did not run continuously. When the system trigger was generated, the trigger

and “timestamp” were sent out to the three subsystems, which would then independently

47

Figure 3.12: Layout schematic of the Sweeper magnet and MoNA LISA detectors. The solid
black line indicates the path of the beam. The dashed line represents the path of the beam
if the Sweeper magnet was not present. The shaded blue regions and dotted lines represent
the acceptance ranges of the neutrons. Angles and distances from the target are given for
the front of the LISA-1 and LISA-2 tables.

record the data in their system. Once the data were collected, the three separate subsystem

data sets were merged offline by matching the event tag “timestamps.” The present experi-

ment was operated in a method in which the system trigger only required a valid signal from

the thin scintillator, allowing charged particle fragment events to be recorded regardless of

neutron signals from MoNA LISA.

The trigger logic was handled by programmable Xilinx Logic Modules (XLMs) grouped

into “levels.” One Level 1 module was assigned to each layer (i.e. 16 bars). The Level 1

modules collected the CFD (constant fraction discriminator) hit information for each layer

and passed the information on to the Level 2 modules. There were two Level 2 modules with

one assigned per array (MoNA and LISA). Figure 3.14 presents an abbreviated schematic of

how the Level 1 modules feed into the Level 2 modules for each array.

48

Sweeper MagnetReaction TargetBeamMoNALISA-1LISA-26°38°Sweeper
System

Sweeper Trigger

(PMT 0)

Level 3

Trigger

Gate

Thin Scintillator 

PMTs

CFD

TDC

ADC

Scintillator 

A1900
PMT 

Scintillator 

Target
PMT

CRDCs

Anodes
Pads

Ion Chamber  

Pads

Hodoscope
CsI(Na) 
PMTs 

CFD

TDC

CFD

FEE

TDC

ADC

ADC

XLM

ADC

ADC

Start Signal
Stop Signal
Gate

MoNA LISA
Systems

Level 2 
(Arrays) 

Level 1 
(Layers) 

TDC

QDC

CFD

MoNA Bar 

Layer A, Bar 0 Left 

TDC

QDC

CFD

MoNA Bar 

Layer A, Bar 0 Right 

...

TDC

QDC

CFD

LISA Bar 

Layer R, Bar 15 Right 

Figure 3.13: Schematic diagram of the MoNA LISA-Sweeper electronics. Start signals, stop
signals, and gates are indicated by green open arrows, red closed diamonds, and blue solid
arrows, respectively.

49

MoNA 
Level 2

.....

Layer A 
Level 1 

Layer I 
Level 1 

CFD

CFD

CFD

CFD

CFD

CFD

CFD

CFD

Bar 0
Left 

Bar 0
Right 

...

Bar 15
Left 

Bar 15
Right 

Bar 0
Left 

Bar 0
Right 

...

Bar 15
Left 

Bar 15
Right 

LISA 
Level 2

.....

Layer J 
Level 1 

Layer R 
Level 1 

CFD

CFD

CFD

CFD

CFD

CFD

CFD

CFD

Bar 0
Left 

Bar 0
Right 

...

Bar 15
Left 

Bar 15
Right 

Bar 0
Left 

Bar 0
Right 

...

Bar 15
Left 

Bar 15
Right 

Figure 3.14: Abbreviated schematic diagram of the MoNA LISA electronics. Each Level 1
module represents one layer or layer (bars 0-15). The nine Level 1 modules for each array
(A-I for MoNA and J-R for LISA) feed into Level 2 modules. TDCs and QDCs are omitted
for simplicity.

50

The Level 2 modules determined the validity of an event in MoNA or LISA and made a

decision about issuing a trigger based on the hit information. For a MoNA or LISA event

to be considered valid, the system required light detection and signals in the CFD channels

for both PMTs in a single bar. Level 3 held the clock for “timestamps” and handled the

coincidence trigger logic between MoNA LISA and the Sweeper. When there was a system

trigger, the Level 3 sent a “timestamp” to each of the subsystems and opened a coincidence

gate to allow for valid signals from MoNA LISA to be recorded and all signals were read

out and processed. If a valid signal was sent, each subsystem processed the event. If there

was no valid MoNA LISA signal, the coincidence gate closed and only information from the

Sweeper was recorded. If MoNA LISA had a valid signal but the Sweeper had not triggered,

MoNA LISA performed a fast clear. Figure 3.15 shows a schematic diagram with examples

for processing CFD signals for valid or invalid events.

The Sweeper data acquisition consisted of electronics for each of the CRDCs, ion chamber,

timing scintillators, and hodoscope. The outputs from the A1900 timing scintillator, target

scintillator, and four thin timing scintillator PMTs were each sent through a CFD and then

to a TDC (time-to-digital converter). All TDCs in the Sweeper subsystem were operated in

common start mode, in which the Level 3 trigger began the measurement and the timing

signals from each individual PMT stopped the measurement. The CRDC pad signals were

digitized with Front-End-Electronics (FEE) modules that sampled the pad pulse and sent

the pulse to an XLM [96]. The signals from the ion chamber pads were sent through a shaper

and then to an ADC (analog-to-digital converter). The hodoscope PMT signals were sent

through shaping amplifiers into ADCs.

MoNA and LISA electronics and DAQs were separate but identical. Each PMT had two

output signals. The anode signal was used for local triggering and timing. The anode was

51

Level 3
System Trigger

Event 

Processed 

MoNA  
Level 2 

Event Not 
Processed 

No System Trigger 

Fast 
Clear 
                    MoNA 
                     Level 2 

Layer A 
Level 1 

Layer B 
Level 1 

CFD

CFD

CFD

CFD

Bar 0
Bar 0
Left 
Right 
Both CFD 

Signals Present

Bar 0
Left 

Bar 0
Right 

Missing One or Both 

CFD Signals 

Layer A 
Level 1 

CFD

CFD

Bar 0
Bar 0
Left 
Right 
Both CFD 

Signals Present 

Figure 3.15: Example schematic diagram of determining valid events in the MoNA LISA
electronics. Two cases involving a Level 3 system trigger are shown on the left. The first
case includes two present CFD signals from one bar. The second case includes at least one
missing CFD signal. The right example shows a case in which the two CFD signals are
present but there is no Level 3 system trigger. TDCs and QDCs are omitted for simplicity.

52

sent to a CFD, which then sent outputs to a TDC and XLM. The TDCs were operated in

common stop mode, in which a signal from the detector triggered the start of measurement

and the Level 3 trigger ended the measurement. The dynode signal was used to measure the

charge collected in each PMT. The dynode signal was passed to a charge-to-digital converter

(QDC) for integration. Charge and timing information for each PMT was read out by the

DAQs from the QDCs and TDCs.

53

Chapter 4

Data Analysis

Signals from the MoNA LISA and Sweeper charged particle detectors were processed, cali-

brated, and corrected for non-linear responses. Once the calibrations were completed, events

were selected from the data set by isotope for analysis and comparison to the simulated

reactions.

4.1 Calibrations and Corrections

4.1.1 Charged Particle Calibrations and Corrections

The detectors used for identifying each charged particle included, in sequence within the

detector box, two Cathode Readout Drift Chambers (CRDCs), an ion chamber, a thin

scintillator, and a hodoscope, plus the upstream A1900 and target timing scintillators.

4.1.1.1 Cathode Readout Drift Chambers

The CRDCs provided measurements of the X and Y positions of the charged fragments.

As discussed in Chapter 3, the CRDC recorded the signals induced on an anode wire and

cathode pads for the horizontal position and the drift time for the vertical position [96].

To obtain the actual positions of the fragments, these signals from the CRDCs had to be

processed. When a charged particle was measured in one of the CRDCs, the charge read out

54

included the sum of the charge generated by its movement in addition to the pedestal, or the

small current observed on each pad with no beam presence. The pedestal was subtracted

pad by pad to determine the charge attributed directly to the detection of the particle.

The pedestals were obtained in a data collection run in which no beam was sent through

the Sweeper magnet. The pedestals shown in Figure 4.1 appear to have two components,

one at approximately channel 250 and another at approximately half of that. However,

the intensity of the lower component is less than 5%. The lower component resulted from

incomplete electronic signals. The larger value of the charge collected for each pad was then

fit with a Gaussian to determine the pedestal value. The pedestal subtraction is shown in

Figure 4.1.

Instead of integrating the entire charge collected, each pad in the CRDCs took a number

of samples, which were counted and summed to find an approximation of the total charge

deposited on the given pad. The total charge on each pad could then be calculated with the

following equation:

Qpad =

1
n

n

Xi=0

qi − qped

(4.1)

where Qpad was the total approximated charge deposited on the pad, n was the number of

samples, qi was the charge per sample, and qped was the pedestal charge.

Once the pedestal was determined and subtracted, the pads were gain matched to account

for differences in charge collection among the pads. To gain match the pads, a “sweep” run

was used.

In a sweep run, the unreacted beam was swept across the acceptance area of

the CRDCs, sequentially illuminating each pad. One pad in each CRDC was chosen as

a reference, and the remaining pads were mathematically matched to that reference. The

55

)
y
r
a
r
t
i
b
r
a
(
 
l

a
t
s
e
d
e
P

)
y
r
a
r
t
i
b
r
a
(
 
l
a
t
s
e
d
e
P

500

450

400

350

300

250

200

150

100

50

0
0

500

450

400

350

300

250

200

150

100

50

0
0

20

20

610×

3

2.5

2

1.5

1

0.5

0

310×
40

35

30

25

20

15

10

5

0

100

120

100

120

40
CRDC1 Pad Number

80

60

40
CRDC2 Pad Number

80

60

140

120

100

)
y
r
a
r
t
i
b
r
a
(
 

d
a
p
Q

80

60

40

20

0
0

20

40
CRDC1 Pad Number

60

80

140

120

100

)
y
r
a
r
t
i
b
r
a
(
 

d
a
p
Q

80

60

40

20

0
0

20

40
CRDC2 Pad Number

60

80

310×

25

20

15

10

5

0

310×
24

22

20

18

16

14

12

10

8

6

4

2

0

100

120

100

120

Figure 4.1: Pedestal subtraction in CRDC1 (top) and CRDC2 (bottom). The raw charge
prior to pedestal subtraction is shown in the left panels, while the subtraction is shown in
the right panels. Vertical lines indicate dead channels in bad pads.

total charge deposited (Qpad) could then be transformed into the gain-matched charge (Qcal)

through the following equation:

Qcal =

µref
µi × Qpad

(4.2)

56

where µref was the centroid of the charge on the reference pad and µi was the centroid of

the charge distribution on the given pad. For both CRDC1 and CRDC2, the reference pad

was taken to be a central pad, number 64.

Several of the pads displayed poor response and had to be removed from analysis. These

pads appeared to be overly sensitive to electronic noise, showing unusual charge collection

when compared to other pads, such as the dead channels seen in Figure 4.1. For CRDC1,

bad pads included 98, 100, 103, 105, 106, and 108. For CRDC2, the bad pads were 24, 73,

and 89.

The X-position of the charged particle fragment was determined by the charge distribu-

tion across the pads. After the total charge collected on each pad was calculated, a Gaussian

shape was fitted to each distribution of pad charge, with the centroid taken as the particle’s

position along the pads. The pad position was then converted to an X-position in mm in

the lab frame.

The Y-position of the charged particle fragment was determined by the drift time of the

electrons, specifically as the time difference between the charge collection on the anode wire

and the detection of the particle in the thin scintillator. The drift time was then converted

into a Y-position in mm in the lab frame.

Both the converted X and Y positions were determined through the calibration provided

by a “mask.” In a masked run, a tungsten mask was placed in front of the CRDC by a

hydraulic drive. Each mask had a known set of lines and holes to form the specific pattern

seen in Figure 4.2 [102]. The tungsten beam blocker was removed and the incoming beam

was defocused and swept across the area of the CRDC mask. Only the beam particles that

passed through a hole or line were detected in the CRDC. By knowing the position of the

hole or line in the mask, it was possible to transform the signals measured in the CRDC into

57

Figure 4.2: Mechanical drawing of a tungsten mask used in the experimental configuration.
Each hole corresponded to a specific point in space and each line corresponded to an X-
position. Drawing made by J. Honke [102].

a physical position. (The physical positions of the holes and lines and their corresponding

positions in each CRDC are given in Appendix B.) An example of data from a calibrated

mask run for CRDC2 is shown in Figure 4.3. Note that while the bottom of the CRDC

was not fully probed and not every hole was illuminated, enough points were measured to

determine the identities of the holes and lines and perform the calibration.

The offsets of the X and Y positions as well as the slope of the Y-position were determined

from the linear transformation of the signals from the mask runs and are given in Tables 4.1

and 4.2. The slope for the X-position calibration was determined by the physical pad pitch

of 2.54 mm. (Because of the orientations of the two CRDCs, the X-position slopes happened

to have opposite signs.) The pad numbers for CRDC1 increase in the positive X-direction

58

150

100

50

0

50−

)

m
m

(
 
n
o
i
t
i

s
o
P
Y

 

−

100

−

150
−

150

−

100

0

50−
50
X Position (mm)

100

150

Figure 4.3: Example data from CRDC2 mask run in which the holes and lines in the mask
were used to determine the slopes and offsets for the position calibration. This mask run
was performed with the 30S beam.

X Slope (mm/pad)

X Offset (mm)

Y Slope (mm/pad)

Y Offset (mm)

CRDC1
CRDC2

2.54
-2.54

-181.4
187.2

-0.1835
-0.1889

95.8
98.0

Table 4.1: Results for the CRDC mask calibration slopes and offsets for the 30S secondary
beam.

in the lab frame, while the pad numbers for CRDC2 increase in the negative X-direction in

the lab frame. The negative Y slopes are a result of the collecting electrons drifting upwards

to the anodes of each detector.

The hydraulic drive was unable to fully lift the CRDC2 mask at several points during

the experiment. This prevented the mask from covering the full active area of CRDC2, as

shown in Figure 4.4. There was no evidence of this occurring with the CRDC1 mask. Thus,

a “beam-down-center” run was used to find an additional offset in the Y-direction to place

59

X Slope (mm/pad)

X Offset (mm)

Y Slope (mm/pad)

Y Offset (mm)

CRDC1
CRDC2

2.54
-2.54

-178.3
187.6

-0.1910
-0.1932

95.0
92.9

Table 4.2: Results for the CRDC mask calibration slopes and offsets for the 40S secondary
beam.

150

100

50

0

50−

100

)
y
r
a
r
t
i
b
r
a
(
 
)

m
m

(
 
n
o
i
t
i

s
o
P
Y

 

−

−

150
−

150

−

100

50−
50
X Position (mm)

0

100

150

Figure 4.4: Example data from a CRDC2 mask run in which the mask did not fully insert.
This mask run was performed with the 40S beam.

the central path of the beam at zero on the Y-axis. Final CRDC slopes and offsets for the

30S and 40S secondary beams are given in Table 4.1 and Table 4.2, respectively.

4.1.1.2

Ion Chamber

The anode of the ion chamber was segmented into 16 strips along the beam path. The

16 pads (numbered 0 through 15) in the ion chamber each collected charge to determine

the energy loss of the particle. The pads were gain matched with the 30S and 40S beams

60

sent through the center of the ion chamber. The calibration procedure assumes that energy

loss is fairly uniform throughout the ion chamber. The energy loss in the ion chamber for

each beam was calculated by approximating the detector as 65 cm of pure argon gas. The

incoming 30S beam lost approximately 114 MeV in the first half of the ion chamber and 121

MeV in the second half of the ion chamber. The incoming 40S beam lost approximately 113

MeV in the first half of the ion chamber and 118 MeV in the second half of the ion chamber.

The charge on each pad was measured and the resulting distribution was fitted with a

Gaussian shape. Pad 9 was chosen as a reference pad for comparison to the others. The

centroid of the individual pad was compared to the centroid of the reference pad to determine

its slope. An example of the Gaussian fits for the reference pad and another pad are shown in

Figure 4.5. The calibrated charge on an individual pad was determined as in the calibration

of the CRDC pads:

qcal =

µref
µi × qraw

(4.3)

where µref was the centroid of the reference pad, µi was the centroid of the given pad, and

qraw was the raw charge collected on each pad.

Results of gain matching the pads in the ion chamber are shown in Figure 4.6. After

gain matching, the centroids of most pads lined up to the reference pad. The white gaps

in Figure 4.6 at pads 1 and 8 indicate that the pads were inefficient. Because of their low

resulting values, these pads were excluded from further analysis.

Because of unequal charge collection, the ion chamber showed a position dependence that

is a variation in charge collected. To correct the observed position dependence, a sweep run

was used to move the beam across the active area, illuminating all X-positions. Because

61

s
t
n
u
o
C

310×

18

16

14

12

10

8

6

4

2

0
800

1200
900
Pad 9 Charge (arbitrary)

1100

1000

310×

16

14

12

10

8

6

4

2

s
t
n
u
o
C

1300

0
800

1200
900
Pad 4 Charge (arbitrary)

1000

1100

1300

Figure 4.5: Example of the Gaussian fitting procedure used for gain matching the ion cham-
ber. The reference pad, pad 9, is shown in the left, while the charge collected from another
example pad is shown on the right. The red vertical line indicates the centroid of the reference
pad. The examples were taken from the centered run with the 40S beam.

the beam had the same kinetic energy despite the changing magnet setting, it deposited the

same amount of charge at each X-position. The calculated energy loss was plotted against

the X-position of the CRDC2 and fit with a polynomial. The dependence was corrected by

calculating the position-corrected value:

qx,pos =

(4.4)

qcal
i=0 aixi

P4

where qcal was the calibrated charge from Equation 4.3, x was the X-position, and ai repre-

sented the polynomial coefficients for the fit on the given pad. An example of the X-position

correction is shown in Figure 4.7.

A slight position-dependence was also observed in the Y-position. A similar method to

removing the X-position dependence was applied for removing the Y-position dependence to

find a final position-corrected value, qcorr:

62

2000

1800

1600

1400

1200

1000

800

600

400

200

)
y
r
a
r
t
i
b
r
a
(
 
e
g
r
a
h
C
w
a
R

 

0
0

2

4

8

6
10
Pad Number

12

14

16

2000

1800

1600

1400

1200

1000

800

600

400

200

)
y
r
a
r
t
i
b
r
a
(
 
e
g
r
a
h
C
d
e
h
c
t
a
m
n
i
a
G

 

0
0

2

4

8

6
10
Pad Number

12

14

16

410

310

210

10

1

310

210

10

1

Figure 4.6: Results of gain matching the pads of the ion chamber. Values of the raw charge
are displayed on the left, while calibrated charge values are shown on the right. The white
gaps for pads 1 and 8 indicate the results of poor charge collection. These results were
obtained with the 40S beam.

qcorr =

(4.5)

qx,pos
i=0 aiyi

P3

where qx,pos was the X-position corrected value, y was the Y-position, and ai represented

the polynomial coefficients from the fit for the given pad. An example of the Y-position

correction is shown in Figure 4.8.

Thus, once each pad had been gain matched and position-corrected, the total energy loss

in the ion chamber was calculated by summing the energy loss recorded on all good pads:

QT otal =

15

Xi=0

qcorr,i

(4.6)

Examples of the resulting distributions of QT otal, i.e. energy loss dE, are shown later in

this chapter.

63

1400

1200

1000

800

600

400

200

)
y
r
a
r
t
i
b
r
a
(
 

 

s
s
o
L
y
g
r
e
n
E

0
150−

100−

50−

0

50

100

150

CRDC2 X-Position (mm)

30

25

20

15

10

5

0

1400

1200

1000

800

600

400

200

)
y
r
a
r
t
i
b
r
a
(
 

 

s
s
o
L
y
g
r
e
n
E

0
150−

100−

50−

0

50

100

150

CRDC2 X-Position (mm)

30

25

20

15

10

5

0

Figure 4.7: Example data for the position dependence correction in the ion chamber for pad
14. On the left is the raw signal as a function of X-position. The right panel shows the
position-corrected signal. Signals were taken from the 30S beam.

4.1.1.3 Thin Timing Scintillator

The thin scintillator provided an important reference for the time-of-flight of the charged

particle fragments. The thin scintillator was located near the end of the beam line and was

connected to four photomultiplier tubes (PMTs), with two attached to the bottom and two

attached to the top. Each PMT was labeled with a number and its relative location on the

thin scintillator. The PMTs measured total charge collected and the time of collection.

Signals from each PMT were gain matched and combined to create calibrated time and

charge measurements. Similar to gain matching the ion chamber, a centered beam was sent

through the thin scintillator. A gate on position was placed to select events measured in

the center of the detector (-10 to 10 cm in the X-direction and -10 to 10 cm in the Y-

direction). The position on the thin scintillator was determined by projection from positions

of the CRDCs. The charge signals were fit with a Gaussian, and the slope was determined

by comparing the widths of each Gaussian fit to the reference width from PMT 3. The

centroids were used to calculate the offset required to place all centroids at the same charge

64

)
y
r
a
r
t
i
b
r
a
(
 

 

s
s
o
L
y
g
r
e
n
E

1200

1000

800

600

80−

60−

40−

20−

0

20

40

60

80

CRDC2 Y-Position (mm)

45

40

35

30

25

20

15

10

5

0

)
y
r
a
r
t
i
b
r
a
(
 

 

s
s
o
L
y
g
r
e
n
E

1200

1000

800

600

80−

60−

40−

20−

0

20

40

60

80

CRDC2 Y-Position (mm)

70

60

50

40

30

20

10

0

Figure 4.8: Example data for the position dependence correction in the ion chamber for pad
14. On the left is the X-position corrected energy loss as a function of Y-position. The right
panel shows the position-corrected signal. Signals were taken from the 30S beam.

position. The calibrated charge for the individual PMT could then be calculated with the

following equation:

qcal =

σref
σi × qraw + q0

(4.7)

where σref was the Gaussian width of the reference PMT 3, σi was the Gaussian width of

the given PMT, qraw was the raw charge, and q0 was the offset. Figure 4.9 shows the charge

collected on each individual PMT before and after gain matching.

Once all four PMTs were gain matched, their signals were combined to determine the

total deposited charge, qtotal:

qtop =

qLU + qRU

2

qbottom =

qLD + qRD

2

65

(4.8)

(4.9)

310×

310×

40

35

30

25

20

15

10

5

s
t
n
u
o
C

0
400

500

700

600
Raw Charge (arbitrary)

800

900 1000 1100 1200

PMT 0

PMT 1

PMT 2

PMT 3

35

30

25

20

15

10

5

s
t
n
u
o
C

0
400 450 500 550 600 650 700 750 800 850 900

Calibrated Charge (arbitrary)

Figure 4.9: Results of gain matching the four thin scintillator PMTs. Raw charge is displayed
on the left, while the gain-matched charge is shown on the right. This gain matching was
performed with the 40S beam.

qtotal = qq2

2

top + q2

bottom

(4.10)

In the above equations, qLU , qLD, qRU , and qRD represent gain-matched signals from

the left-upper (PMT 0), left-lower (PMT 1), right-upper (PMT 2), and right-lower (PMT

3) PMTs, respectively.

The observed light intensity from the four PMTs depended on where the fragments in-

teracted within the scintillator. If uncorrected, fragments with the same energy but different

flight paths would be read as having different energy losses. To correct for the position

dependence, the beam was swept horizontally across the active area of thin scintillator. The

combined energy loss was plotted against the X-position in the thin scintillator as projected

from the CRDCs and fit with a polynomial function to correct the position dependence. The

final corrected total charge, qcorr, was calculated with the following function:

66

Coefficient

a0
a1
a2
a3
a4

30S

359.486

-0.0266998
-1.33152 × 10-3
-4.32671 × 10-6
2.58745 × 10-8

40S

380.132

-0.0264493

-1.20136 × 10-3
-3.68012 × 10-6
2.29955 × 10-8

Table 4.3: Coefficients for the thin scintillator position corrections used in Equation 4.11.

)
y
r
a
r
t
i
b
r
a
(
 
e
g
r
a
h
C

440

420

400

380

360

340

320

300

280

260

200−

150−

100−

50−

0

50

100 150 200

X-Position (mm)

50

40

30

20

10

0

)
y
r
a
r
t
i
b
r
a
(
 
e
g
r
a
h
C

440

420

400

380

360

340

320

300

280

260

200−

150−

100−

50−

0

50

100 150 200

X-Position (mm)

50

40

30

20

10

0

Figure 4.10: Correction of the position dependence in the thin scintillator. On the left is raw
charge signal as a function of the X-position. The right panel shows the position-corrected
signal. Signals were taken from the 30S beam.

qcorr =

(4.11)

qtotal
i=0 aixi

P4

where qtotal was the total charge found in Equation 4.10, ai represented the polynomial co-

efficients, and x was the X-position. Figure 4.10 shows the energy loss in the thin scintillator

before and after the position correction for the 30S beam. There was no noticeable depen-

dence in the Y-direction when looking at the corrected charge as a function of Y-position,

as shown in Figure 4.11. The coefficients for the 30S and 40S beams are given in Table 4.3.

67

)
y
r
a
r
t
i
b
r
a
(
 
e
g
r
a
h
C

440

420

400

380

360

340

320

300

280

260

100−

50−

0

50

100

Y-Position (mm)

100

80

60

40

20

0

Figure 4.11: X-position-corrected signal versus the Y-position in the thin scintillator. No
noticeable dependence is observed. Signals were taken from the 30S beam.

Only an offset was used to calibrate the time signals from each PMT. Because the range

of the TDC was fixed, with a range of 400 ns and 4096 channels, the slope of the TDC was

assumed to be 0.1 ns/channel. A jitter was introduced into the TDC common stop signal by

the processing in the Field Programmable Gate Array (FPGA) in the Sweeper electronics.

While the jitter was around 20 ns, the exact jitter value varied by each event, requiring the

jitter to be eliminated event-by-event. Because the same stop signal was used for all TDCs,

the jitter was the same for each timing signal. The jitter can be eliminated by subtracting

one timing signal from remaining timing signals within an event. In this experiment, the

timing signal from the thin scintillator left-upper PMT was subtracted from the remaining

timing signals. To allow for different TDC ranges, this subtraction was done after the slopes

were applied.

68

Name

PMT No.

30S Offset (ns)

40S Offset (ns)

Left-Upper
Left-Lower
Right-Upper
Right Lower

0
1
2
3

42.37
-42.854
-53.604
-14.244

42.37
-38.437
-53.537
-13.877

Table 4.4: Thin scintillator time offsets.

The centered beam was also used with the thin scintillator to determine the time offsets

for each PMT. Fragments outside the center of the scintillator were excluded for determining

the time offsets. The flight-path distance from the target to the thin scintillator was 419.9

cm. The 30S and 40S beams had kinetic energies of 55.5 MeV/u. Thus, the calculated

time-of-flight for the unreacted beams to travel from the target to the thin scintillator was

42.4 ns. The reference PMT 0 was set to this time, and all thin scintillator PMT timing

signals were offset to match that value. Each offset was selected such that the centroid of

each PMT aligned with the signal from the reference time-of-flight of the unreacted beam.

The time calibration offsets for each PMT for each beam are listed in Table 4.4. Once the

individual offsets were determined, the final calibrated timing signal for the thin scintillator

was calculated from the average of each PMT:

tthin =

1
n

n

Xi=0

ti

(4.12)

where n was the number of PMTs that recorded for a given event and ti were the time signals

from each PMT.

69

4.1.1.4 Hodoscope

The final detector in the detector box was a CsI hodoscope made up from a 5 x 5 array of

crystals. The hodoscope was designed to measure the residual energy of the fragments. Prior

to this experiment, it was found that the hodoscope was malfunctioning. As detailed in [101],

a 20O beam was swept across the faces of the crystals in the center row of the hodoscope.

Each crystal in the middle row of the hodoscope displayed a non-uniform response, indicating

uneven light collection and poor signal quality.

Further inspection of the crystals showed that the teflon wrapping had stuck to the

crystals, possibly due to moisture condensing between the teflon and hydroscopic CsI(Na)

crystals. The teflon attachment to the crystals potentially allowed light leaks, affecting signal

quality in the hodoscope. Because the problems were not fixed prior to this experiment, the

hodoscope was not used for this analysis.

4.1.1.5 A1900 and Target Timing Scintillators

The time offsets for two other upstream timing scintillators were also determined for each

beam. The TDCs for both scintillators covered a range of 400 ns with 4096 channels, resulting

in a 0.1 ns/ch fixed slope. Once the TDC channel was converted to time, the FPGA jitter

was subtracted, as explained before.

To determine the offset, the secondary 30S and 40S beams were sent through the center

of the system and the time signals were recorded. Both beams had energies of 55.5 MeV/u

as determined by the magnetic rigidity in the A1900. The flight path from the A1900

scintillator to the target was 10.88 m, and the flight path from the target scintillator to the

target was 1.15 m. Thus, the time-of-flight for the beams to travel from the A1900 to the

target scintillator was 98.15 ns, while the time-of-flight from the target scintillator to the

70

Scintillator

30S Offset (ns)

40S Offset (ns)

A1900
Target

-108.5
65.0

-107.9
64.9

Table 4.5: Calculated timing offsets for the A1900 and target scintillators.

target was calculated to be 11.60 ns. The offsets were applied to adjust the time signals to

correspond to the calibrated time at the thin scintillator. The resulting timing offsets for

the A1900 and target scintillators are given in Table 4.5.

4.1.2 MoNA LISA

The MoNA LISA plastic scintillators were used to observe neutrons in coincidence with the

charged particles from the reaction. The array was made up from bars of BC-408 (MoNA)

and EJ-200 (LISA) scintillator that were 10 cm × 10 cm × 200 cm each with a photomultiplier
tube (PMT) on each long end. Each MoNA LISA PMT gave a raw signal for charge and

time based on the total light collected by the PMT and the arrival time of light. These

signals were calibrated to determine the deposited charge, position of interaction in the bar,

and time interaction in the bar using cosmic rays. Cosmic ray data collection was performed

prior to the experiment. Cosmic ray muons traveled at a velocity close to the speed of light,

depositing about 2.05 MeV per centimeter as they passed through the plastic bars [103].

The cosmic ray muons deposit approximately 20 MeVee of light into the bar, where 1 MeVee

is equivalent to the light deposited by an electron with 1 MeV of kinetic energy. The known

velocity of the muons was used to determine the relative timing of the bars.

71

4.1.2.1 Charge Calibration

Note that PMTs in general vary in sensitivity as a function of applied voltage. If one PMT

is more sensitive than another, flashes of light emitted at different positions along the bar

will not be detected with uniform efficiency. For example, if a weak light flash occurs near

a more sensitive PMT, that PMT has a better chance of detecting the light, while a less

sensitive PMT may not detect the light. Because both PMTs must detect light to accept

and record an event, this non-uniformity could skew the data towards one side of the given

bar. To minimize this issue and calibrate for charge signal, the PMTs were gain matched

by adjusting the applied voltages to put the cosmic ray muon peaks to roughly channel 900

out of 4096. The gain-matching process was iterated until cosmic ray muon peaks aligned

and voltage fluctuations in each individual PMT were below 10 V from the penultimate

gain-matching run to the final gain-matching run. In this experiment, the final voltages of

the PMTs ranged from around 1250 to 1930 V.

The charge signals were checked with a second set of cosmic data taken with gain-matched

PMTs. A pedestal, or a small offset, was observed in every QDC channel while the electronics

were running. The pedestal also caused every QDC channel to be read out for every charged

particle event even if the bars did not detect a valid interaction, resulting in significant

data collection dead time. To offset the pedestal so that channel 0 would correspond to 0

MeVee, a linear calibration transforming the channel number to MeVee was used to match

the pedestal to 0 MeVee and the cosmic muon peak to 20 MeVee. Thus, the QDC calibrations

were performed with the following relation:

qcal = mq × (qraw − qped)

(4.13)

72

510

410

310

210

10

s
t
n
u
o
C

1

0

500

1000

1500

2000

2500

3000

QDC Channel

410

310

s
t
n
u
o
C

210

10

1

0

10

20

30

40

50

60

70

Charge (MeVee)

Figure 4.12: An example of the QDC calibration procedure for MoNA LISA. The data shown
is from the left PMT from Layer K, Bar 8. On the left is the raw charge from each QDC
channel. The cosmic peak around 1000 is fit with a Gaussian function (green). The right
panel shows the calibrated charge spectrum with the pedestal subtracted.

where mq was the QDC slope in MeVee/channel, qraw was the raw QDC channel, and qped

was the pedestal channel. An example of this charge calibration procedure is shown in Figure

4.12.

Furthermore, to suppress the pedestal and decrease dead time, a hardware threshold was

applied slightly above the pedestal. The threshold (qthresh) was calculated with the following

equation:

qthresh =

qped
16

+ 2

(4.14)

The factor of 16 was used to convert the pedestal channel from a 12 bit scale to an 8 bit scale

as the pedestal and threshold were stored as 12 bit and 8 bit numbers, respectively. The

addition of two ensured that the threshold was set above the pedestal. With a threshold in

place, only channels with a signal greater than the threshold were included in the data. The

thresholds for this experiment ranged from 0.38 to 0.85 MeVee.

73

4.1.2.2 Position Calibration

The MoNA LISA bars were stacked to cover a wide range of neutron emission angles in the

fragment reference frame and thus had fixed spatial coordinates in two directions. The third

coordinate, or the point of interaction along the length of the bar, was found by measuring

the relative arrival time of the light pulse at each end of the bar. When an interaction

occurred within a bar, light traveled by total internal reflection to each end of the bar and

into each PMT. Because light travels at a constant speed in the plastic, the difference in

arrival times can be used to calculate the position of the interaction.

The TDC channel readout was converted into a time. Since the time difference between

the PMTs on a neutron detector was used to calculate the neutron position in space, the

time spectrum of each bar was calibrated individually. The slope was found with a “pulser,”

an Ortec NIM Time Calibrator module (model 462) that pulsed the system every 40 ns over

a TDC range of 350 ns. This created a “picket-fence” spectrum with a peak at every 40

ns interval, as shown for example in Figure 4.13. The first peak was assigned a time of 40

ns; the second peak was assigned a time of 80 ns; and so forth. The calculated slopes were

approximately 0.085 ns/channel.

Once the TDC slopes were determined, the time difference between the two PMTs in a

bar could be converted into a position along the bar. Cosmic rays interacted in the plastic

along the full length of the bar. Each bar was 200 cm long, with the center of the bar marked

as 0 cm and the ends defined at -100 cm and 100 cm. The edges of the time difference spectra

were adjusted so that the midpoint of the fall-off landed at the ends of the bar, and a linear

slope calculated from the two edges was used to find an offset to place the center of the

74

310×

80

70

60

50

40

30

20

10

s
t
n
u
o
C

0
0

500

1000 1500 2000 2500 3000 3500 4000

TDC Channel

)
s
n
(
 

 

i

e
m
T
y
r
a
r
t
i
b
r
A

300

250

200

150

100

50

0
0

500

1000

1500

2000

2500

3000

TDC Channel

Figure 4.13: An example of the TDC calibration procedure for MoNA LISA. The data shown
is the left PMT from Layer B, Bar 5. On the left is the “picket fence” spectrum created by
the pulser pulsing the system every 40 ns. On the right is the relation between the TDC
channel peaks and their assigned time.

bar at 0 cm within its own reference frame. Figure 4.14 shows an example of the position

calibration within a bar.

4.1.2.3 Time Calibrations

The time-of-flight of an interaction in a bar was calculated by averaging the two times

recorded from the two PMTs, but an offset was needed to place the time of the interaction

in the bar relative to the times of other interactions within the detector array. This offset

was again determined with cosmic ray muons that passed through the majority of the bars

in a layer and deposited about 20 MeVee in each bar. A schematic drawing showing an

example of a cosmic ray muon traveling through all bars within a single layer of MoNA

LISA is displayed in the left part of Figure 4.15. Since the cosmic ray muons travel with

a known velocity, it was possible to calculate the time, t, that it would take for a muon to

travel from one bar to another with the following equation:

75

1000

s
t
n
u
o
C

800

600

400

200

1800

1600

1400

1200

800

600

400

200

1000

s
t
n
u
o
C

0
20−

15−

10−

5−

0

5

10

15

20

0
150−

100−

Time Difference (ns)

50−
Bar X-Position (cm)

50

0

100

150

Figure 4.14: Example of the data for an X-position calibration for a bar in MoNA LISA
(Layer A, Bar 8). The time difference (tlef t − tright) between the two PMTs is shown on
the left, and the converted X-position spectrum is shown on the right.

t =

d
vµ

(4.15)

where d was the distance between the interactions and vµ = 29.8 cm/ns, the speed of the

muon. The difference between the expected travel time and the observed travel time was

the time offset.

There were three “tables” (labeled MoNA, LISA-1, and LISA-2) that were supporting

stacks of bars, as described in Chapter 3. Layers on the same table were linked together by

cosmic ray muons passing through the top bar of every layer on the table and the bottom

bar of the front layer. A schematic drawing showing example tracks of cosmic ray muons

traveling through the top bar in a layer and a bottom bar in the front layer is displayed in

the right part of Figure 4.15. The expected and observed times were used to calculate offsets

that were then added to each bar within the layer. The result was that each table of bars

was calibrated in time relative to all other bars on that table.

76

µ–

µ–

෡

y

z෡

෡

x

෡

y

෡x

෡

z

Figure 4.15: Schematic representation of cosmic ray muon paths through MoNA LISA bars.
The schematic view on the left is a view from the front a table, while the view on the right
of is from looking at a side view of a table.

Finally, each table had to be located in time relative to the interaction time in the target.

A thick target was inserted into the beam path, and when the beam collided with the thick

target, a variety of gamma rays and neutrons were produced. The known velocity of the

gamma rays, vγ = 29.98 cm/ns, was used in combination with the known physical positions

of the bars to determine the final time offset to align the gamma ray peaks by their calculated

times of flight. The final offsets for each table of bars were given in Table 4.6. Figure 4.16

shows the velocity spectra of all three tables with the gamma ray velocity peaks lined up

with their expected values.

77

s
t
n
u
o
C

450

400

350

300

250

200

150

100

50

0
0

20 40 60 80 100 120 140 160 180 200

MoNA Time of Flight (ns)

s
t
n
u
o
C

300

250

200

150

100

50

0
0

20 40 60 80 100 120 140 160 180 200

LISA-1 Time of Flight (ns)

140

120

100

s
t
n
u
o
C

80

60

40

20

0
0

20 40 60 80 100 120 140 160 180 200

LISA-2 Time of Flight (ns)

Figure 4.16: Calibrated time spectra for all detectors on each MoNA LISA table. The smaller
peaks on the left of each spectra are the gamma rays, while the larger peaks are neutrons.
The red vertical lines indicate the expected time for the gamma rays to arrive in each set of
bars.

MoNA Offset (ns) LISA-1 Offset (ns) LISA-2 Offset (ns)

469.92

410.61

409.65

Table 4.6: Global time offsets for each table of MoNA-LISA.

4.2 Event Selection

While this experiment was designed to measure as many fragments as possible, many collision

events had to be discarded from the final analysis. This section describes the process for

selecting appropriate, complete events for the analysis.

4.2.1 Beam Identification

The A1900 Fragment Separator filters out unwanted projectile fragments and attempts to

send only the desired fragments of interest to an experimental vault. However, the separation

technique trades purity against rate so that contamination by unwanted products occurs in

some reactions [92, 93]. Contaminants can be rejected in analysis by time-of-flight correla-

tions. Fragments with the same magnetic rigidity but with slightly different mass-to-charge

78

Figure 4.17: Beam component identification with flight time from the A1900 timing scintil-
lator to the target scintillator. The 30S beam and contaminants are shown on the left, while
the 40S beam peak is shown on the right.

ratios with different velocities will be accepted. In particular, the A1900 generally includes

isotones as contaminants due to the kinetics of the separation process at approximately 100

MeV/u [92, 93, 104]. Thus, the time-of-flight from the A1900 Fragment Separator timing

scintillator to the target scintillator can be used to identify the desired beam from the con-

taminants. Figure 4.17 shows time-of-flight spectra for both the 30S beam plus contaminants

and the 40S beam plus contaminants.

While the 40S beam was relatively pure with few contaminants, having a purity of 98%,

the 30S beam had a purity of only 32%. Major contaminants in the 30S beam were the two

isotones 29P and 28Si. These two contaminants arrived at different times when compared

to the desired 30S particles, making it relatively straightforward to exclude them from the

analysis. The energy loss of the unreacted fragments in the ion chamber was examined to

confirm that there were no significant contaminants with the same time-of-flights as the

79

Figure 4.18: Beam component identification with flight time from the A1900 timing scintil-
lator to the target scintillator versus the energy loss in the ion chamber. The 30S beam and
contaminants are shown on the left, while the 40S beam and contaminants are shown on the
right.

desired beams. Figure 4.18 shows the energy loss in the ion chamber versus the timing

spectra from Figure 4.17.

To further improve incoming beam selection, cuts using the time-of-flight from the K1200

cyclotron to the A1900 timing scintillator were also included in the analysis. This was done

by comparing the time difference between the cyclotron RF signal and the A1900 timing

scintillator and the time-of-flight from the A1900 timing scintillator to the target scintillator,

as displayed in Figure 4.19. The time signal from K1200 cyclotron was measured from the RF

main oscillator. The multiple peaks in the time-of-flight spectra for the RF to the A1900 was

due to wraparound of the RF [80]. The smaller peaks with relatively few events between the

larger major peaks were likely the result of inaccurate time measurements. These “echoes”

were excluded from the analysis.

80

Figure 4.19: Beam component identification using the time-of-flight from the RF to the
A1900 scintillator. The 30S beam and contaminants are shown on the left, while the 40S
beam and contaminants are shown on the right. The multiple peaking in the RF to A1900
time-of-flight is due to wraparound of the RF. Beam gates were placed on the dominant
packets encircled in black.

4.2.2 Event Quality Gates

Accurate position information from the CRDCs was found to be necessary to achieve identi-

fication of individual isotopes. Events in which the experimental system behaved abnormally

were removed by applying “event quality gates” to the produced charged fragments in the

data. In some events, the CRDCs failed to collect all charge deposited, resulting in abnor-

mally low signals. In other events, an unusually high charge was recorded, possibly due to

pile up. These events were identified by plotting the σ from the Gaussian fit for the X-

position against the total charge summed over all the pads, or the pad sum. CRDC quality

gates were constructed based on this relation, as shown in Figure 4.20. Furthermore, some

events displayed inconsistent charge deposit from one CRDC to the other. For example, an

event could deposit a high charge in CRDC1 and a low charge in CRDC2. To exclude these

81

Figure 4.20: The σ of the gaussian fit as a function of total charge in the CRDCs for the 40S
+ 9Be reaction and 2.01 Tm Sweeper setting. The quality gates for each CRDC are drawn
in black.

events that did not have a linear relation between the charge deposition between the two

detectors, another quality gate was created, as shown in Figure 4.21. These CRDC quality

gates excluded approximately 16% of events.

4.2.3 Element Identification

Two magnet settings of the Sweeper magnet were chosen for each secondary beam in order

to collect data on a variety of reaction products. For the 30S + 9Be reaction, the Sweeper

magnet was set to 1.51 Tm (115 A) and 2.25 Tm (174 A). For the 40S + 9Be reaction, the

Sweeper magnet was set to 2.01 Tm (157 A) and 2.27 Tm (180 A).

Element separation and identification was achieved by measuring the energy loss in the

ion chamber and correlating it with the time-of-flight. The Bethe-Bloch formula [5, 105]

shows that energy loss in the ion chamber is related to the charge number (Z) of the incident

particle and its velocity (β) in the following manner:

82

Figure 4.21: Total charge of CRDC1 plotted against the total charge of CRDC2 for the 40S
+ 9Be reaction and 2.01 Tm Sweeper setting. The quality gate (shown in black) excluded
events that do not deposit charge in a linear manner in both detectors.

dE
dx ∝

−

Z2
β2 × f (β)

(4.16)

where f (β) represents the other terms in the Bethe-Bloch formula.

(The full formula is

available in References [80, 105].)

Thus, element separation can be observed in a plot of the energy loss against a parameter

tied to the velocity. The experimental configuration provided a velocity parameter through

the time-of-flight from the target scintillator to the thin scintillator. Figures 4.22 and 4.23

show the element identification for products from the 30S + 9Be collisions and 40S + 9Be

collisions, respectively. Each band in these figures represents products from a different

element and the lines superimposed on the data indicate the sulfur (Z = 16) products. The

83

bands directly under the Z = 16 band contain phosphorus (Z = 15) products; bands under

phosphorus contain silicon (Z = 14) products; and so forth. Gates were placed around these

bands in order to make elemental cuts. Unfortunately, due to the hodoscope malfunction,

no comparison could be made between the energy loss measured in the ion chamber and the

total energy measured in the hodoscope.

4.2.4

Isotope Identification

Following the elemental identification of each event, the isotopic or mass number identifica-

tion was necessary for the data analysis. The magnetic rigidity (Bρ) of a charged particle is

based on its momentum to charge ratio:

Bρ =

p
q

=

mv
q

(4.17)

In the present reactions at relatively high energy, light isotopes of the same element have

the same number of protons and no electrons, and thus the same q. Isotopes with the same

Bρ and constant q could then be separated by their mass number A, as their masses are

inversely proportional to their velocity:

v =

Bρq
m ∝

Bρq
A

(4.18)

Furthermore, because the (non-relativistic) velocity is simply distance per unit of time,

the time-of-flight measured from the target to the detector and mass of a charged particle

are proportional. Thus, isotopes with a lower A would arrive sooner than isotopes with the

same Bρ and q but higher A.

84

Figure 4.22: Energy loss in the ion chamber versus the time-of-flight from the target scintilla-
tor to the thin scintillator for reaction products from the 30S + 9Be reaction. The top panel
contains products from the 1.51 Tm magnet setting. The bottom panel contains products
from the 2.25 Tm magnet setting. Black lines indicate the locus of sulfur (Z = 16) products.

85

Figure 4.23: Energy loss in the ion chamber versus the time-of-flight from the target scintilla-
tor to the thin scintillator for reaction products from the 40S + 9Be reaction. The top panel
contains products from the 2.01 Tm magnet setting. The bottom panel contains products
from the 2.27 Tm magnet setting. Black lines indicate the locus of sulfur (Z = 16) products.

86

However, charged particles produced by these nuclear reactions can have a large spread in

momentum. Variations in momenta lead to variations in the distance traveled in the present

case, as the track of a charged particle that passes through a dipole magnet is dependent on

its magnetic rigidity. Because these momenta distributions are so large and the track length

varies, the times of flight of the isotopes tend to overlap one another.

Thus, the measured path was used to separate isotopes. The magnetic rigidity and path

length traveled by the observed charged particles were correlated to their position and angle

with the position measurements in the CRDCs and the beam spot on the target. In standard

magnetic spectrometers, the fragments are highly focused in the horizontal or X-position and

isotopic separation is observed in a two-dimensional plot of plane angle versus the time-of-

flight [80]. However, this simple method of achieving isotopic separation was not possible

with the Sweeper magnet, which was a simple dipole magnet and did not contain focusing

elements such as those found in true magnetic spectrometers, such as the S800 spectrograph

at the NSCL [106]. The angle and position at the “focal plane” in the Sweeper are correlated

and entangle the X-position, X-angle, and time-of-flight. This correlation can be seen in a

three-dimensional plot of the three parameters. An example of such a three-dimensional plot

from the present work is shown in Figure 4.24.

For isotopic separation from the Sweeper magnet to be achieved, the full correlation

among the X-position, X-angle, and time-of-flight must be untangled. First, the three-

dimensional data for one element from one reaction was projected onto the X-position and

X-angle plane, as shown for example in Figure 4.25. Contours along this plot were fit with

a quadratic function:

f (x) = a2x2 + a1x + c

(4.19)

87

 

i

)
s
n
(
 
)
n
h
T
o
t
 
t
e
g
r
a
T
(
 
F
o
T

75

70

65

60

55

50

45

40

100

0

Dispersive Position (mm)

100−

100

( m r a d )

100−

0

D i s p e r s i v e   A n g l e  

Figure 4.24: Three-dimensional plots showing the correlation among the (dispersive) X-
position, X-angle, and time-of-flight for the oxygen isotopes from the 40S + 9Be reaction
and the 2.01 Tm magnet setting. Isotope bands are visible on the right edge in the Figure
but merge towards the back left side.

where x was the X-position, a2 and a1 were the coefficients from the quadratic fit, and c was

a constant offset. Varying c only moves the curve vertically along the angle axis.

A new linearized parameter describing the position and angle for a constant time-of-

flight was created. This parameter, referred to as an emittance (e(x, θx)), was written in the

following manner:

e(x, θx) = θx − (a2x2 + a1x)

(4.20)

where θx was the dispersive angle. Plotting the values of this emittance parameter against

the time-of-flight provided the degree of isotopic separation, shown by example in Figure

4.26, with each parallel band containing fragments from a single isotope.

88

150

100

50

0

)
d
a
r
m

(
 

X

 
-
 

l

e
g
n
A
e
v

 

i

s
r
e
p
s
−
D

i

50−

100

−

150
−

150

−

100

50−

0

50

100

150

Dispersive Position - X (mm)

63

62

61

60

59

58

57

56

55

Figure 4.25: Projection of the three-dimensional plot from Figure 4.24 onto the (dispersive)
X-position and X-angle plane for the oxygen isotopes from the 40S + 9Be reaction and the
2.01 magnet setting. A solid black line is drawn along the contours to demonstrate the
quadratic function fit used in the analysis.

The isotopic separation was finally obtained by applying a linear fit to one of the isotopic

bands and then projecting the data onto an axis called tcorr:

tcorr = m × e(x, θx) + ttarget→thin

(4.21)

where m was the slope of the line perpendicular to the linear fit and ttarget→thin was the
time-of-flight from the target to the thin scintillator. From another standpoint, one could

say that the time-of-flight was corrected by projecting the two-dimensional figure onto the

axis perpendicular to the linear fitted line.

89

100

50

0

50−

r
e
t
e
m
a
r
a
P
e
c
n
a
t
t
i

 

m
E

100−

52

54

58

64

66

56
ToF Target to Thin (ns)

62

60

16

14

12

10

8

6

4

2

0

68

70

Figure 4.26: Emittance parameter from Equation 4.20 plotted against the time-of-flight from
the target to the thin scintillator for the oxygen isotopes from the 40S + 9Be reaction and
the 2.01 magnet setting. A solid black line is drawn along the contours to demonstrate the
linear function fit.

Isotopic separation was further improved by plotting the corrected time-of-flight against

the dispersive X-position and X-angle as well as other parameters, including the non-dispersive

Y-position and Y-angle. Any observed correlations were then removed in a similar manner to

the corrected time-of-flight parameter from Equation 4.21. These corrections were generally

much smaller. The final time-of-flight separation parameter can be expressed in a general

format:

tf inal = ttarget→thin +X CiPi

(4.22)

90

where Ci were the correction coefficients and Pi were the parameters. The isotope separation

procedure was carried out for the data from each element in each magnet setting for both

sulfur beams. Correction coefficients and parameters for each element are summarized in

Appendix C. The projected isotopic distributions are shown later in this chapter.

Once the relative distributions were constructed, the relationship between energy loss

and the corrected time-of-flight was used to identify the isotopes. Energy loss is tied to Z, as

demonstrated in Figures 4.22 and 4.23, in which an element with a higher Z demonstrates

greater energy loss compared to an element with a lower Z.

Recall that the corrected time-of-flight is related to N/Z. The magnetic rigidity (Bρ) is

equal to momentum divided by charge (Equation 4.17). Thus, velocity (β) can be written

in terms of magnetic rigidity, charge (Z), and mass (m) in the present case because q = Z:

β =

Z(Bρ)

m

(4.23)

If the magnetic rigidity is constant, the velocity will be proportional to Z/A, meaning that

time-of-flight is proportional to A/Z. Thus, energy loss is comparable to Z and the corrected

time-of-flight is comparable to A/Z or N/Z.

Figure 4.27 displays the expected pattern for plotting Z against N/Z in the range of 6

≤ Z ≤ 16 and N/Z ≤ 2. Each point is the position of a single isotope. It is evident that
isotopes with the same N/Z (or same A/Z) fall in a vertical line, such as the red diamond

points representing isotopes in which N = Z (or A/Z = 2). This pattern was compared to

a plot of the energy loss versus corrected time-of-flight for each reaction in order to identify

the isotopes in the data. An example of a plot of energy loss versus a corrected time-of-

flight parameter is shown in Figure 4.28. While the final isotopic separation was performed

91

30S

40S

16

15

14

13

12

Z

11

10

9

8

7

6

0

0.25

0.5

0.75

1

N/Z

1.25

1.5

1.75

2

Figure 4.27: The Z versus N/Z pattern for isotopes with Z = 6-16 and N/Z ≤ 2. Each point
is a separate isotope. Some of the isotopes shown do not exist. The red diamond points
represent isotopes in which N = Z. The two beams, 30S and 40S, are circled for reference.

with the correction parameters for each element, the Z and A relative pattern and isotope

identification was found with the correction parameters for oxygen isotopes, which were

applied to every element to create this figure.

(In other words, if individual correction

parameters were applied by element, the isotopes would be shifted and out of sync with each

other.) The line indicating isotopes with the same “corrected” time-of-flight corresponds

to the red diamond points in Figure 4.27. Those isotopes that fall along the vertical line

have the same N/Z, and the remaining isotopes were identified by their relationship to the

vertical line.

The final one-dimensional projections for isotope separation for fragments from the 30S

+ 9Be collision in the 1.51 Tm and 2.25 Tm Sweeper settings are shown in Figure 4.29 and

92

Figure 4.28: Energy loss in the ion chamber for elements Z = 6 − 14 using the corrected
time-of-flight for oxygen isotope separation. The elements shown are from the 30S + 9Be
reaction and the 2.25 Tm Sweeper magnet setting. A vertical line at N/Z = 1 is drawn
for reference. This line corresponds to the red diamond points in Figure 4.27. The 28Si
fragments are circled in red.

Figure 4.30, respectively. Figure 4.31 and Figure 4.32 show the one-dimensional projections

for isotope separation from the 40S + 9Be collision in the 2.01 Tm and 2.27 Tm Sweeper

settings, respectively. Element gates from the energy loss in the ion chamber (as shown in

Figures 4.22 and 4.23) and event quality gates were applied. The four figures indicate that

separation of individual isotopes was only possible for carbon, nitrogen, oxygen, fluorine,

neon, and sodium isotopes. The separation was clearest for elements with lower proton

numbers, and the separation became more difficult with increasing proton number.

93

Figure 4.29: One-dimensional distributions of carbon, nitrogen, oxygen, fluorine, neon, and
sodium isotopes produced from the 30S + 9Be reaction and the 1.51 Tm Sweeper magnet
setting. Element and event quality gates were applied.

94

Figure 4.30: One-dimensional distributions of carbon, nitrogen, oxygen, fluorine, neon, and
sodium isotopes produced from the 30S + 9Be reaction and the 2.25 Tm Sweeper magnet
setting. Element and event quality gates were applied.

95

Figure 4.31: One-dimensional distributions of carbon, nitrogen, oxygen, fluorine, neon, and
sodium isotopes produced from the 40S + 9Be reaction and the 2.01 Tm Sweeper magnet
setting. Element and event quality gates were applied.

96

Figure 4.32: One-dimensional distributions of carbon, nitrogen, oxygen, fluorine, neon, and
sodium isotopes produced from the 40S + 9Be reaction and the 2.27 Tm Sweeper magnet
setting. Element and event quality gates were applied.

97

It is important to note that this experiment entered unexplored territory with the MoNA

LISA and Sweeper experimental set-up. Previous experiments have focused on studying

isotopes of the elements only up to magnesium (Z = 12) [80, 95, 97, 100, 101, 107–111]. The

30S and 40S (Z = 16) secondary beams used in this experiment were significantly higher in

Z than those in the past MoNA LISA and Sweeper experiments and isotopic separation with

Z = 12 − 16 had not been done before and was not possible here. Isotope separation results
from a previous study on fluorine, neon, and sodium isotopes produced from a 32Mg beam

on a beryllium target already suggested that separation was difficult [110]. From Figure

4.27, elements with higher Z contain more isotopes than elements with lower Z in a given

range of N/Z. For example, in the N/Z range from 0.75 to 1.25, carbon has three isotopes,

while phosphorus has seven isotopes. The example isotopic separation shown in Figure 4.28

indicates this compression of the isotopes in elements with higher Z. In particular, the 28Si

fragments appear to be overlapping the 29Si fragments in this case. Thus, the isotopes with

higher Z are more overlapped and difficult to separate. Figure 4.33 demonstrates the poor

one-dimensional separation for magnesium (Z = 12) and aluminum (Z = 13) isotopes from

Figure 4.28. This experiment pushed the limits on the current detector configuration and

the resolution was not sufficient to achieve isotope separation for the highest Z elements.

However, a broad range of fragments was identified in each reaction.

Furthermore, most previous Sweeper experiments characterized isotopes produced by

only one or a few nucleon removal or addition. Final fragments measured in the current

experiment were produced by removing many nucleons. For example, the 40S + 9Be reaction

produced 12C, a mass change of 28 units. This wide range was not standard for the MoNA

LISA and Sweeper configuration and would have to be improved for future experiments with

heavier beams.

98

Figure 4.33: One-dimensional distributions of magnesium and aluminum isotopes produced
from the 30S + 9Be reaction and the 2.25 Tm Sweeper magnet setting. Element and event
quality gates were applied.

4.2.5 MoNA LISA Multiplicity

Charged particles were measured in coincidence with neutron hits in the MoNA LISA de-

tectors. The kinetic energy spectrum of observed neutrons in MoNA LISA measured in

coincidence with fragments from the 40S + 9Be reaction and the 2.01 Tm Sweeper set-

ting is shown in Figure 4.34. The distribution peaks around beam energy, indicating that

most of the detected neutrons were generated the subsequent de-excitation process. Neu-

trons stripped from the projectile by the target would have a kinetic energy around half

the kinetic energy of the beam. The kinetic energy of a nucleon removed directly from the

projectile by the target would be equal to the energy of the center-of-mass. Because the

target is stationary, the stripped neutron would have half the kinetic energy.

As previously described in Chapter 3, an interaction in MoNA LISA is only considered

valid if there were good timing signals in the CFD channels for both PMTs within a single

bar. A valid interaction is referred to as a “hit” in MoNA LISA. The number of hits in

99

25000

20000

s
t
n
u
o
C

15000

10000

5000

0
0

20

40
Kinetic Energy (MeV)

60

80

100

120

Figure 4.34: Kinetic energy of neutrons detected in MoNA LISA for all fragments measured
with the 40S + 9Be reaction and the 2.01 Tm Sweeper setting.

MoNA LISA is called the hit multiplicity and its relation to the actual neutron multiplicity

have to be obtained from comparisons to detailed simulations.

The number of hits in MoNA LISA does not directly measure the true neutron multi-

plicity for an event due to neutrons scattering multiple times within the full array and dark

scattering. In dark scattering, neutrons scatter on carbon nuclei in the plastic scintillator

material. Due to the short range of the carbon recoil, insufficient scintillation light is pro-

duced for a valid measurement. Thus, the measured hit multiplicity is not the true neutron

multiplicity. Despite this, as the number of neutrons emitted increases, the number of valid

neutron events recorded also can be expected to increase, leading to a higher hit multiplicity.

The hit multiplicities were extracted for comparison to the simulations discussed in the

following section, Section 4.3. All of the extracted hit multiplicity distributions and the

comparisons to simulations are shown in Chapter 5, but two examples of these experimental

hit multiplicity distributions are shown in Figure 4.35. The distributions are shown as a

100

1

1−10

1

1−10

)
y
t
i
l
i

b
a
b
o
r
P

2−10

3−10

)
y
t
i
l
i

b
a
b
o
r
P

2−10

3−10

(
 

4−10

s
t
n
u
o
C

5−10

(
 

4−10

s
t
n
u
o
C

5−10

6−10
0

2

4
Hit Multiplicity

6

8

10

6−10
0

2

4
Hit Multiplicity

6

8

10

Figure 4.35: Examples of the hit multiplicity distributions for all neon fragments from the
30S + 9Be reaction with the 2.25 Tm Sweeper magnet setting (left) and the 40S + 9Be
reaction with the 2.27 Tm Sweeper magnet setting (right). Distributions are presented in
terms of a normalized probability.

normalized “probability” in which the distributions were normalized relative to the total

number of observed hits for each fragment measured in the magnet setting. Most charged

particles were measured with zero hits, or a hit multiplicity of zero, so the distributions are

shown on a log scale to better highlight differences among the distributions.

From the hit multiplicity distribution examples shown in Figure 4.35, it is evident that

the fragments produced from the 40S reaction were in coincidence with significantly higher

numbers of hits in MoNA LISA than the fragments produced from the 30S reaction. Because

30S contains ten less neutrons than 40S, there are fewer neutrons available for reactions.

Thus, fragments produced from the 40S collision could be more neutron rich as well as have

higher neutron multiplicities, leading to a higher number of valid hits in MoNA LISA.

101

4.3 Experimental Simulation and Gates

As discussed in Chapter 2, the two nuclear reactions were simulated using two different mod-

els: the Li`ege Intranuclear Cascade (INCL++) and the Constrained Molecular Dynamics

(CoMD) model.

INCL++ contained the ABLA07 code for the de-excitation of hot frag-

ments created from the fast stage of the reactions. For CoMD simulations, GEMINI++ was

used as an evaporation or “afterburner” code to decay remaining hot fragments from the dy-

namical model. Once the correlated distributions of fragments and neutrons from INCL++

and CoMD + GEMINI++ were obtained, the results were filtered through a model of the

detection efficiency developed in the GEANT4 code. The filtering is discussed next and the

results of the comparisons between the simulations and experimental data are presented in

Chapter 5.

4.3.1 Fragment Gates

To mimic the physical configuration of the detectors measuring the charged particles, frag-

ments from the CoMD + GEMINI++ and INCL++ simulations were filtered according to

the magnetic rigidity and angular acceptances of the experimental setup.

The Sweeper had an acceptance range of approximately ±10% in momentum. To deter-
mine the acceptance range for each beam, the unreacted beam was sent through the Sweeper

magnet in a series of regular momentum intervals. These step sweep runs were used to char-

acterize the beam as it moved horizontally across the faces of the charged particle detectors

as discussed above. Each step sweep run had an associated magnet setting and position in

the detectors at which the beam appeared. The limits on each side of the central magnet

102

setting could then be deduced and used to determine the actual magnetic rigidity acceptance

and active detector area.

The magnetic rigidities and trajectories exiting the target of the final simulated charged

particle fragments were determined based on the results of the CoMD + GEMINI++ and

INCL++ simulations. Gates were placed on the simulated data to accept or reject the

fragments based on whether the fragments would pass through the acceptance ranges of the

charged particle detectors to create sets of “accepted” events and the correlated neutrons.

4.3.2 Neutron Gates

The GEANT4 code is a well-known simulation package using Monte Carlo methods to model

the passage of particles through detectors [112, 113]. GEANT4 has been used to simulate

the interactions of the neutrons in MoNA LISA in the past [97,101,110,111] and also in this

work.

The neutron momentum vector in the laboratory frame and energy from the neutrons in

the accepted events were used to determine the neutron’s interaction in the plastic scintil-

lator material. Text files containing the neutron four-vector information were created from

INCL++ and each version of CoMD + GEMINI++ for each reaction. Each neutron was

tagged with an event number to connect it back to the collision event that produced it from

CoMD + GEMINI++ or INCL++. The text files were then passed to GEANT4.

In GEANT4, the neutrons were propagated through the Sweeper magnet steel because

MoNA LISA was configured to cover angles ranging from -3 degrees to 51 degrees relative

to the target. Interactions of the neutrons in the steel plate material in the Sweeper magnet

were incorporated into the GEANT4 package. The steel Sweeper material was approximated

with the neutron scattering cross section of iron [110].

103

The neutrons were then propagated forward towards the MoNA LISA bars. The location

of the bars relative to the target was matched to the physical location of the bars within

the experimental configuration. Figure 4.36 shows a top-down view of the simulated hit

interactions in the bars from INCL++ and the second symmetry option for CoMD + GEM-

INI++. The angles of the neutron interaction relative to the target are shown in Figure

4.37. As seen in both Figures 4.36 and 4.37, the angular distribution of neutrons from the

CoMD + GEMINI++ model was somewhat broader, with slightly more neutrons measured

in the LISA-2 table relative to the number of neutrons measured in the LISA-2 table for

INCL++. Neutron interaction cross sections in the hydrocarbon plastic bars were taken from

the MENATE R package [114], which includes neutron interaction cross sections for both

carbon and hydrogen for neutrons. The MENATE R package has been previously shown to

have better agreement with experimental data taken with MoNA LISA compared to simu-

lations run with the stock GEANT4 physics processes [115]. The GEANT4 code required

a threshold to mimic the QDC hardware threshold gate of the physical detectors. In this

analysis, only hits above 0.9 MeVee were counted towards hit multiplicity, so the simulated

threshold for the MoNA LISA PMTs was set to 0.9 MeVee.

The neutron interaction information output from GEANT4 was matched back with the

event number in CoMD + GEMINI++ or INCL++ to create a simulated hit multiplicity

for each single collision determined by counting the number of simulated valid neutron hits

within each event. The normalized probability distributions were then constructed for each

isotope and are shown in Chapter 5.

104

Figure 4.36: Simulated hit interactions in MoNA LISA bars for neutrons from the 40S +
9Be reaction. The top panel shows interactions from the INCL++ model, while the bottom
panel (CoMD-2) shows interactions from the CoMD + GEMINI++ for the second symmetry
energy option (stiff-1).

105

)
y
r
a
r
t
i
b
r
a
(
 
d
e
Y

l

i

0.035

0.03

0.025

0.02

0.015

0.01

0.005

0

INCL++

CoMD-2

0

10

20
30
 (degrees)
θ

40

50

Figure 4.37: Simulated hit interaction angles in MoNA LISA bars for neutrons from the 40S
+ 9Be reaction. INCL++ and the second symmetry energy option (stiff-1, labeled here as
CoMD-2) for CoMD + GEMINI++ are shown. The neutrons with angles of around -3 to
15 degrees were measured in MoNA and LISA-1, while the neutrons with angles of around
24 to 53 degrees were measured in LISA-2.

106

Chapter 5

Results and Discussion

After the calibrations were completed and fragment-neutron coincidence events were selected

from the data set for analysis, the experimental data set was then compared to the results of

the two reaction simulations. As previously discussed in Chapter 2, the two simulation data

sets involved in this work were the Li`ege Intranuclear Cascade model (INCL++) and the

combined Constrained Molecular Dynamics model and GEMINI++ (CoMD + GEMINI++).

The CoMD model contained three options for the symmetry energy: stiff-2, stiff-1, and soft,

which are referred to as CoMD-1, CoMD-2, and CoMD-3, respectively, in this work. As

described in Chapter 4, each simulation data set was passed through a GEANT4 model of

the neutron interactions with the detectors.

A summary of the isotopic fragments identified in the experimental data for each reaction

is given in Figure 5.1. Note that many other isotopes than those shown in the figure were

produced from the reaction, but not all produced fragments traveled through the acceptance

gates of the Sweeper magnet and the charged particle detectors. Furthermore, as discussed

previously in Chapter 4, no isotope identification was achieved for elements above sodium.

Thus, those element rows in the figure are left blank.

As Figure 5.1 shows, several isotopes were produced and identified in both the 30S +

9Be and 40S + 9Be reactions. These shared isotopes included symmetric nulcei (N = Z).

107

16

15

14

13

12

11

10

9

8

7

r
e
b
m
u
N
n
o
t
o
r
P

 

30S

40S

21Na 22Na 23Na 24Na 25Na 26Na

19Ne 20Ne 21Ne 22Ne 23Ne 24Ne

17F 18F 19F 20F 21F 22F

15O 16O 17O 18O 19O 20O

13N 14N 15N 16N 17N

Produced from 30S beam

Produced from 40S beam

Produced from 30S and 40S beams

6   11C 12C 13C 14C

5

6

7

8

9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Neutron Number

Figure 5.1: Isotopic fragments identified the experimental data. Red boxes indicate isotopes
only identified from the 30S + 9Be reaction, blue boxes indicate isotopes only identified
from the 40S + 9Be reaction, and purple dotted boxes indicate isotopes identified from both
reactions. For comparison, the 30S and 40S incident beams are outlined in red and blue,
respectively.

Additional fragments identified from the 40S beam were neutron-rich, while some fragments

identified from the 30S beam were proton-rich.

5.1 Hit Multiplicities

Hit multiplicity distributions were used to compare the experimental hits measured in MoNA

LISA in coincidence with fragments to the simulated hits from the reaction models. An

example of hit multiplicity distributions containing experimental data and simulated data

is shown in Figure 5.2. The example distribution shows the hit multiplicity distribution for

108

16O fragments from the 30S + 9Be reaction with the 1.51 Tm Sweeper magnet setting. The

other comparisons of the hit multiplicity distributions are shown later in this chapter.

The hit multiplicity distributions are shown in terms of “probability,” in which the dis-

tributions were normalized relative to the total number of observed hits for each fragment

measured in a given magnet setting. A hit multiplicity of zero indicated a fragment that did

not coincide with any measured hits in MoNA LISA; a hit multiplicity of one indicated a

fragment that coincided with one hit in MoNA LISA; and so forth. The probabilities of the

distributions were plotted on log scales to better display differences among hit multiplicities

as the hit multiplicity of zero dominates the distributions.

A consistent representation of the distributions is used in this chapter. Solid black squares

represent experimental data; open pink squares represent INCL++ simulations; and open

blue diamonds, open up-pointing red triangles, and open down-pointing green triangles rep-

resent CoMD + GEMINI++ simulations with the CoMD-1 (stiff-2), CoMD-2 (stiff-1), and

CoMD-3 (soft) options, respectively. The data points for the simulations were slightly offset

in hit multiplicity to prevent symbols from completely overlapping. The INCL++, CoMD-1,

CoMD-2, and CoMD-3 points were offset in the horizontal direction by -0.1, 0.1, 0.15, and

0.2, respectively, relative to the experimental data points.

The relative uncertainty or error was calculated with the following equation:

Ei =

Pi√Ci

(5.1)

where i was the hit multiplicity value (bin number), Pi was the probability value of the

multiplicity, and Ci was the number of counts in the multiplicity.

109

Figure 5.2: Hit multiplicity distributions for 16O fragments from the 30S + 9Be reaction with
the 1.51 Tm Sweeper magnet setting. Distributions are presented in terms of probability
(see the text for details). Solid black squares represent experimental data; open pink squares
represent INCL++ simulations; and open blue diamonds, open up-pointing red triangles,
and open down-pointing green triangles represent CoMD + GEMINI++ simulations with
the CoMD-1, CoMD-2, and CoMD-3 options, respectively. For clarity and to avoid overlap,
the INCL++, CoMD-1, CoMD-2, and CoMD-3 points were offset in hit multiplicity by -0.1,
0.1, 0.15, and 0.2, respectively. Points representing a hit multiplicity of one are highlighted
in shaded blue to demonstrate grouping.

110

5.1.1 Hit Multiplicities by Element

The hit multiplicity distributions were constructed for each element by summing over the

isotopes to have an overall sense of how the distributions behaved. Figure 5.3 shows the

hit multiplicities for all carbon fragments. No isotopic gates were applied to obtain the

data in this figure. Similar to Figure 4.35 in Chapter 4, these results demonstrate that

the fragments produced from the 40S reaction were, not surprisingly, in coincidence with

significantly higher numbers of hits in MoNA LISA than the fragments produced from the

30S reaction in the experimental data. While the difference between the two reactions is

clear in the experimental data, the simulations show a more subtle difference between the 30S

and 40S reactions. Furthermore, it is evident that the majority of experimentally detected

charged particles were measured with a hit multiplicity of zero. While INCL++ results

appear to replicate the high percentage of fragments with a hit multiplicity of zero, the

results of the CoMD + GEMINI++ simulations noticeably under-predict the percentage of

fragments with a hit multiplicity of zero. The hit multiplicity distributions for individual

carbon isotopes are further explored in Section 5.1.2.

Because no isotopic separation was achieved for elements above sodium, only the elemen-

tal hit multiplicity distributions could be extracted for comparison. Figures 5.4, 5.5, 5.6, and

5.7 show the experimental and simulated hit multiplicities for all magnesium, aluminum, sil-

icon, and phosphorus fragments. (While sulfur fragments were also detected, measured, and

simulated, comparisons are not shown here due to unreacted beam contamination in some

of the magnet settings.) No isotopic gates were applied to these element hit multiplicity

distributions.

111

Figure 5.3: Hit multiplicity distribution for all carbon fragments produced in each beam
setting for the 30S + 9Be and 40S + 9Be reactions. See text and Figure 5.2 for details.
Sweeper acceptance gates were used, but no isotopic gates were applied.

112

Figure 5.4: Hit multiplicity distribution for all magnesium fragments produced in each beam
setting for the 30S + 9Be and 40S + 9Be reactions. See text and Figure 5.2 for details.
Sweeper acceptance gates were used, but no isotopic gates were applied.

113

Figure 5.5: Hit multiplicity distribution for all aluminum fragments produced in each beam
setting for the 30S + 9Be and 40S + 9Be reactions. See text and Figure 5.2 for details.
Sweeper acceptance gates were used, but no isotopic gates were applied.

114

Figure 5.6: Hit multiplicity distribution for all silicon fragments produced in each beam
setting for the 30S + 9Be and 40S + 9Be reactions. See text and Figure 5.2 for details.
Sweeper acceptance gates were used, but no isotopic gates were applied.

115

Figure 5.7: Hit multiplicity distribution for all phosphorus fragments produced in each beam
setting for the 30S + 9Be and 40S + 9Be reactions. See text and Figure 5.2 for details.
Sweeper acceptance gates were used, but no isotopic gates were applied.

116

These comparisons provided a method to look at elemental fragments that could not be

identified by isotope. Overall a consistent pattern can be seen. It appears that for elements

with Z values closer to the beam, CoMD + GEMINI++ results begin to match more closely

with the data, particularly in results from the 30S reaction with the 2.25 Tm magnet setting.

This suggests that CoMD + GEMINI++ could be better at reproducing fragments that have

a smaller proton and mass loss from the projectile.

However, no isotopic comparisons can be made at these higher elements in the present

data set. Experiments designed to accomplish higher element isotopic separation may be

able to examine this effect more closely. This could be done with set-ups that already provide

better isotopic separation at higher proton numbers or with upgrades to the Sweeper magnet

and its associated charged particle detectors.

5.1.2 Hit Multiplicities by Isotope

Isotopic hit multiplicity distributions were generated by applying isotopic gates to the ex-

perimental data and simulations. Figures 5.8, 5.9, 5.10, 5.11, 5.12, and 5.13 show the hit

multiplicity distributions for carbon, nitrogen, oxygen, fluorine, neon, and sodium isotopic

fragments, respectively.

117

Figure 5.8: Hit multiplicity distribution for carbon isotopes produced in each beam setting for the 30S + 9Be and 40S + 9Be
reactions. See the text and Figure 5.2 for details of the display.

118

Figure 5.9: Hit multiplicity distribution for nitrogen isotopes produced in each beam setting for the 30S + 9Be and 40S + 9Be
reactions. See the text and Figure 5.2 for details of the display.

119

Figure 5.10: Hit multiplicity distribution for oxygen isotopes produced in each beam setting for the 30S + 9Be and 40S + 9Be
reactions. See the text and Figure 5.2 for details of the display.

120

Figure 5.11: Hit multiplicity distribution for fluorine isotopes produced in each beam setting for the 30S + 9Be and 40S + 9Be
reactions. See the text and Figure 5.2 for details of the display.

121

Figure 5.12: Hit multiplicity distribution for neon isotopes produced in each beam setting for the 30S + 9Be and 40S + 9Be
reactions. See the text and Figure 5.2 for details of the display.

122

Figure 5.13: Hit multiplicity distribution for sodium isotopes produced in each beam setting for the 30S + 9Be and 40S + 9Be
reactions. See the text and Figure 5.2 for details of the display.

123

Each of the six figures contains a panel for one isotope and they follow a standard format.

The top row contains fragments measured from the 30S + 9Be reaction and 1.51 Tm Sweeper

magnet setting, the second row contains fragments measured from the 30S + 9Be reaction

and 2.25 Tm Sweeper magnet setting, the third row contains fragments measured from the

40S + 9Be reaction and 2.01 Tm Sweeper magnet setting, and the bottom row contains

fragments measured from the 40S + 9Be reaction and 2.27 Tm Sweeper magnet setting. The

isotopes identified in common to these reactions are lined up in columns. The symbols for

the individual hit multiplicity distributions follow the description given in Figure 5.2.

These comparisons of the hit multiplicity distributions display several trends. When com-

paring isotopes produced from both reactions, such as 12C in Figure 5.8 or 15N in Figure 5.9,

it is apparent, again not surprisingly, that those fragments produced from the 30S reaction

are predicted to be in coincidence with fewer hits in MoNA LISA compared to fragments

produced from the 40S reaction. This was shown as well in Figure 4.35 from Chapter 4 and

Figure 5.3, though the previous figures did not involve any isotopic identification.

For isotopes produced from the 30S + 9Be reaction, each of the CoMD + GEMINI++

versions tended to under-predict the proportion of fragments measured with zero hits while

over-predicting the proportion of fragments measured with one or more hit. The INCL++ re-

sults, on the other hand, matched the experimental data points more closely. While INCL++

did fail to replicate the higher hit multiplicity values present in the data, it is important to

note that these higher multiplicity values contain a very small proportion of counts, some-

times representing fewer than 10 counts per bin.

Results from the 40S + 9Be reaction do not match the trends apparent in the 30S + 9Be

reaction. While each of the three CoMD + GEMINI++ versions still under-predicted the

proportion of fragments measured with zero hits, the overall hit multiplicity trend follows

124

the data more closely, with a more gentle slope downwards as multiplicity increases. The

INCL++ simulation does not appear to follow the trend in the experimental data. Instead,

INCL++ results display a much steeper slope in the hit multiplicity distributions due to

under-prediction of the number of fragments measured in coincidence with higher numbers

of hits.

For a quantitative comparison, a χ2 statistic was calculated with the following equation:

χ2 =

n

Xi=0

(P data

i

i

− P sim
i
)2

(Edata

)2

(5.2)

where i was the hit multiplicity value (bin number), P data

i

was the probability value of the

experimental data, P sim

i

was the probability value of the simulation set (INCL++, CoMD-1,

CoMD-2, or CoMD-3), and Edata

i

was the uncertainty in the data.

A reduced χ2

v was used to compare the models to the experimental data. The inclusion

of hit multiplicity values or bins with very few events could skew the comparison by placing

more importance on those few events. In order to minimize this effect, only hit multiplicity

values with more than five events were included. The reduced χ2

v was found with following

equation:

χ2
v =

χ2
ndf

(5.3)

where ndf was the number of degrees of freedom. As usual, if the given simulation matched

the data within the experimental uncertainties, the χ2

v value would be one. The larger the

χ2
v value, the worse the fit. These χ2

v values are shown in Appendix D.

In general, the χ2

v values demonstrate the features observed in the previous figures, but

are dominated by the lowest hit bins. Overall, the INCL++ simulations were closer in

125

predicting the hit multiplicity zero bin than the CoMD + GEMINI++ simulations. Because

the majority of identified fragments coincided with zero hits, the χ2

v values for the INCL++

predictions were lower (with typical χ2

v values in the tens and hundreds) than those of the

CoMD + GEMINI++ simulations (with typical χ2

v values in the hundreds and thousands).

The CoMD + GEMINI++ simulations greatly under-predicted the number of fragments

produced with zero hits, leading to larger χ2

v values.

The makers of the CoMD model acknowledged that the earliest iteration was over-

producing hydrogen and helium fragments from their test collisions (symmetric reactions

of 40Ca + 40Ca at 35 MeV/u and 197Au + 197Au at 35 MeV/u) [69]. The authors believed

that this was due to the occupation density limitation, which randomly changes the momenta

of the neighboring particles if the Pauli principle is violated and may be under-producing

intermediate-mass fragments through break-up or fission. While CoMD has been modified

since the earliest iteration, it appears that more attention is needed on neutron yields.

It is important to note that while INCL++ did a better job, it is still not a perfect

match.

In particular, INCL++ fails to generate enough events with higher hit counts in

MoNA LISA.

5.2 Symmetry Energy

The CoMD model provides a dynamical description of the evolution of the many-body sys-

tem. One of the advantages of the CoMD model is the inclusion of three different symmetry

energy parameterizations. Each parameter gives a distinct mathematical form of the sym-

metry energy. If the three options produce different results, the experimental data could be

126

compared to the simulations to determine which of the three versions fits the experimental

data best.

In this work, the three CoMD + GEMINI++ options do not appear to produce significant

differences in the hit multiplicity distributions as shown in Figures 5.3-5.13. Instead, the

CoMD + GEMINI++ points on these figures generally overlap. This indicates that neutron

multiplicity distributions from the intermediate-energy heavy-ion collisions are not a good

method for constraining the symmetry energy term in the nuclear equation of state. Thus,

no constraint on the symmetry energy was attempted using this experimental data set.

5.3 Fragment Production

5.3.1

Isotope Production

While this project focused on the exclusive measurement of neutrons measured in coinci-

dence with charged particle fragments, it was possible to examine the isotope production

distributions for the experimental data and simulations. The isotope production distribu-

tions for the for the 30S + 9Be reaction and 1.51 Tm and 2.25 Tm magnet settings are

shown in Figures 5.14 and 5.15, respectively. The isotope production distributions for the

for the 40S + 9Be reaction and 2.01 Tm and 2.27 Tm magnet settings are shown in Figures

5.16 and 5.17, respectively. For all four distributions, the counts were normalized relative to

the total number of observed fragments within the given element for that magnet setting.

The stiff-2, stiff-1, and soft options for CoMD + GEMINI++ are referred to as CoMD-1,

CoMD-2, and CoMD-3, respectively. The model distributions were filtered with fragment

gates as described in Section 4.3.

127

s
t
n
u
o
C

s
t
n
u
o
C

s
t
n
u
o
C

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

Experiment
INCL++
INCL++
Experiment
CoMD-1
CoMD-1
CoMD-2
CoMD-2
CoMD-3
CoMD-3

11

12

Carbon A

13

15

16

Oxygen A

17

19

20

Neon A

21

s
t
n
u
o
C

s
t
n
u
o
C

s
t
n
u
o
C

0.8

0.7

0.6

0.5

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0.3

0.2

0.1

0

0.8

0.7

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0.5

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0.3

0.2

0.1

0

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

13

14

Nitrogen A

15

17

18

19

Fluorine A

21

22

23

Sodium A

Figure 5.14: Isotope production distributions for the 30S + 9Be reaction and the 1.51 Tm
magnet setting. Counts are given in terms of probability normalized relative to the total
number of observed fragments within the given element. The CoMD + GEMINI++ model
stiff-2, stiff-1, and soft options are referred to as CoMD-1, CoMD-2, and CoMD-3, respec-
tively.

128

s
t
n
u
o
C

s
t
n
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o
C

s
t
n
u
o
C

0.9

0.8

0.7

0.6

0.5

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0.3

0.2

0.1

0

0.9

0.8

0.7

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0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

INCL++

Experiment

CoMD-1

CoMD-2

CoMD-3

11

12

Carbon A

13

15

16

Oxygen A

17

20

21

Neon A

22

s
t
n
u
o
C

s
t
n
u
o
C

s
t
n
u
o
C

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

14

15

Nitrogen A

18

19

20

Fluorine A

22

23

Sodium A

Figure 5.15: Isotope production distributions for the 30S + 9Be reaction and the 2.25 Tm
magnet setting. Counts are given in terms of probability normalized relative to the total
number of observed fragments within the given element. The CoMD + GEMINI++ model
stiff-2, stiff-1, and soft options are referred to as CoMD-1, CoMD-2, and CoMD-3, respec-
tively.

129

s
t
n
u
o
C

s
t
n
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o
C

s
t
n
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o
C

0.8

0.7

0.6

0.5

0.4

0.3

0.2

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0

0.8

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0.1

0

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

INCL++

Experiment

CoMD-1

CoMD-2

CoMD-3

12

13

14

Carbon A

16

17

18

Oxygen A

19

20

21

22

23

24

Neon A

s
t
n
u
o
C

s
t
n
u
o
C

s
t
n
u
o
C

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

14

15

16

Nitrogen A

17

18

19

20

21

Fluorine A

22

23

24

25

Sodium A

Figure 5.16: Isotope production distributions for the 40S + 9Be reaction and the 2.01 Tm
magnet setting. Counts are given in terms of probability normalized relative to the total
number of observed fragments within the given element. The CoMD + GEMINI++ model
stiff-2, stiff-1, and soft options are referred to as CoMD-1, CoMD-2, and CoMD-3, respec-
tively.

130

s
t
n
u
o
C

s
t
n
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o
C

s
t
n
u
o
C

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

INCL++

Experiment

CoMD-1

CoMD-2

CoMD-3

12

13

Carbon A

14

16

17

18

19

20

Oxygen A

21

22

23

24

Neon A

s
t
n
u
o
C

s
t
n
u
o
C

s
t
n
u
o
C

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

14

15

16

Nitrogen A

17

19

20

21

22

Fluorine A

23

24

25

26

Sodium A

Figure 5.17: Isotope production distributions for the 40S + 9Be reaction and the 2.27 Tm
magnet setting. Counts are given in terms of probability normalized relative to the total
number of observed fragments within the given element. The CoMD + GEMINI++ model
stiff-2, stiff-1, and soft options are referred to as CoMD-1, CoMD-2, and CoMD-3, respec-
tively.

131

The experimental data isotope distributions follow the one-dimensional distributions from

Figures 4.29-4.32 in Chapter 4. As a whole, the INCL++ simulations follow the trends of

the data, with a few notable suggestions. In the oxygen and fluorine isotope distributions

for the 30S + 9Be reaction and the 1.51 Tm magnet setting, INCL++ skews towards the

more neutron-deficient isotopes. For the 40S + 9Be reaction and the 2.01 Tm magnet

setting, INCL++ displayed a stronger preference for production of the even-even 12C and

16O isotopes. Each of the three CoMD + GEMINI++ options skewed neutron-rich in many

of the isotope distributions. The three CoMD + GEMINI++ options were relatively similar

to each other, though the stiffest option (CoMD-1) did skew to slightly higher mass (and

thus more neutron-rich) isotopes relative to the softest option (CoMD-3). This follows the

fully inclusive distributions shown in Figures 2.1-2.6 from Chapter 2.

5.3.2 Time Evolution in CoMD

In the present work, the CoMD heavy-ion collisions were propagated up to 2000 fm/c (about

7 × 10−21 s), allowing the system to evolve dynamically and become relatively cool. The
previously presented hit multiplicity distributions were taken from the reaction products

at 2000 fm/c from CoMD that were passed through GEMINI++ and then the neutrons

that were passed through GEANT4. However, output from the CoMD simulations included

information of precursor fragments at earlier time steps. This allowed examination of the

time evolution of fragments within the simulations.

Figures 5.18 and 5.19 show the time evolution of fragments in the 30S + 9Be and 40S +

9Be reactions, respectively, from the CoMD model with the stiff-1 option. The time steps

shown are 20, 40, 80, 200, 1000, and 2000 fm/c. (Other time steps were recorded but are not

represented in the figures since only an overview of the time evolution is important here.)

132

Figure 5.18: Contours in the N -Z plane of the fragment distribution showing the time
evolution of fragments in the 30S + 9Be reaction from the CoMD model with the stiff-1
option. Only the heaviest fragments are shown. The boxes indicate the N -Z of the beam.

133

Figure 5.19: Contours in the N -Z plane of the fragment distribution showing the time
evolution of fragments in the 40S + 9Be reaction from the CoMD model with the stiff-1
option. Only the heaviest fragments are shown. The boxes indicate the N -Z of the beam.

134

At 0 fm/c, the projectile (30S or 40S) approaches the target. The 20 fm/c panel shows

the early encounter with little change in the projectile. By 40 fm/c, the projectile has

interacted with the target, displaying some pick-up of protons and neutrons. In the 30S +

9Be reaction, two very distinct peaks appear, corresponding to fragments around 30S and

39Ca. (39Ca represents the full combination of the 30S projectile and 9Be target.) This

phenomenon is less pronounced in the 40S + 9Be reaction, but the effect is still visible. (The

full combination of the 40S projectile and 9Be target would produce 49Ca.) Sharp edges in

the top and right edges of the distributions indicate the physical limitation in the number

of protons and neutrons available when combining the projectile and target in the given

reaction system.

At 80 fm/c, the earlier precursor fragments have started to break up. The fragments at

this stage begin to reach downward in Z and N as the heaviest fragments shed mass. By

200 fm/c, a general form of the fragment distribution is apparent. At 1000 fm/c, the heavier

fragments shed more nucleons, creating more fragments with lower numbers of protons and

neutrons. While there is some change from 1000 to 2000 fm/c, the general form is preserved,

indicating that the evolution of fragments had stabilized.

In both CoMD reaction simulations, the final fragments are concentrated around the

original projectile fragments. The shape of this central concentration demonstrates the

direction in which fragments evolved. For example, fragments from the 40S reactions were

allowed to evolve more “horizontally” in the N -Z plane, losing neutrons while at higher

elements before drifting downwards as Z decreased. Fragments from the 30S evolved more

“diagonally” due to the ten fewer neutrons in the system and the proximity to the proton

drip line.

135

In the final time step panels, it is evident that the 30S + 9Be reaction is predicted to

produce more proton-rich (or neutron-deficient) isotopes relative to the 40S + 9Be reaction.

In the opposite manner, the 40S + 9Be reaction is predicted to produce more neutron-rich

isotopes relative to the 30S + 9Be reaction. While not all produced fragments were measured

in the experimental data, this qualitative trend from CoMD clearly matches the observed

isotopes that were identified from the experimental data set.

5.3.3 Precursor Fragments in INCL++

As described in Chapter 2, INCL++ uses two discrete steps to simulate the nuclear reaction

[83–85]. The first step represents the fast part of the reaction to create an excited remnant,

or intermediate precursor fragment. The remnant then “cools” through a statistical de-

excitation to its final state. In the present work, the statistical de-excitation code ABLA07

was used [89].

It was possible to track the precursor fragment channels that led directly to the final

de-excited fragments in such a simulation. Three isotopes were chosen to demonstrate this

tracking process. The intermediate precursor fragments that led to final 13C, 16O, and 23Na

fragments from INCL++ are shown in Figures 5.20, 5.21, and 5.22, respectively. These

fragments were chosen as examples because they were found in each magnet setting for both

beams. (For comparison to the CoMD model, the INCL++ precursor fragments would align

to CoMD precursor fragments early in the reaction, such as those around 40 fm/c.)

The magnet settings of the Sweeper magnet selected on a slice of fragments within a

given range of magnetic rigidity. Changing the magnet setting then changed what slice

of fragments was measured. The magnetic rigidity is equal to momentum to charge ratio

(Bρ = p/q = mv/q) of the fragments. When a specific isotope is selected, the proton

136

Figure 5.20: Distribution of the intermediate precursor fragments in the N -Z plane from
INCL++ for final 13C fragments produced in each beam setting for the 30S + 9Be and 40S
+ 9Be reactions. The intermediate precursor fragments are represented by the solid boxes.
Open squares representing the beams (30S or 40S) and solid black circles representing the
position of 13C are displayed for comparison.

137

Figure 5.21: Distribution of the intermediate precursor fragments in the N -Z plane from
INCL++ for final 16O fragments produced in each beam setting for the 30S + 9Be and 40S
+ 9Be reactions. The intermediate precursor fragments are represented by the solid boxes.
Open squares representing the beams (30S or 40S) and solid black circles representing the
position of 16O are displayed for comparison.

138

Figure 5.22: Distribution of the intermediate precursor fragments in the N -Z plane from
INCL++ for final 23Na fragments produced in each beam setting for the 30S + 9Be and 40S
+ 9Be reactions. The intermediate precursor fragments are represented by the solid boxes.
Open squares representing the beams (30S or 40S) and solid black circles representing the
position of 23Na are displayed for comparison.

139

number and mass remains constant. Thus, fragments of the same isotope that are measured

in different magnet settings must have different velocities.

In each of the three examples, the INCL++ precursor fragment distributions were found

to be quite similar for the two magnet settings for the 40S + 9Be reaction. Recall that the

two magnet settings chosen for the 40S reaction (2.01 and 2.27 Tm) were both below that of

the unreacted 40S beam (2.65 Tm). Because of the acceptance range in the Sweeper magnet,

these two settings overlapped and many of the fragments were measured in both settings.

This was seen in Figures 4.31 and 4.32 from Chapter 4, which showed only a small shift of the

isotopic distribution with higher numbers of neutrons in the higher magnet setting. On the

other hand, the INCL++ precursor fragment examples show a broader range of precursor

fragments for the lower magnet setting in the 40S reaction. This effect is most apparent in the

precursor fragments of 23Na shown in Figure 5.22. Thus, 23Na fragments that are observed

with higher velocities are predicted to be produced from a somewhat narrower distribution

of precursor fragments that are mainly heavier than the beam.

The INCL++ precursor fragment distributions for the two magnet settings for the 30S

+ 9Be reaction can also be seen to be somewhat different and follow a similar pattern.

For example, in Figure 5.22, the precursor fragments of 23Na appear to perhaps have two

separate concentrated regions of precursor fragments in the lower magnet setting. Meanwhile,

the precursor fragments for the higher magnet setting are more uniformly distributed around

a central spot at 35Cl. For reference, recall that the unreacted 30S beam had a rigidity of

1.97 Tm. The Sweeper magnet settings chosen for the 30S reaction fell at 1.51 Tm and 2.25

Tm, below and above the unreacted beam rigidity, respectively. Thus, different precursor

fragment distributions for one fragment are associated with different final velocities.

140

Interestingly, in every case, many of the precursor fragments contained more nucleons

than the given projectile, indicating nucleon pick-up or merging of the target and projectile.

This suggests that the beam did not simply interact with an inert target in the INCL++

and CoMD simulations.

It should be noted that some degree of nucleon pick-up was present in the experiment.

As Figure 4.23 in Chapter 4 showed, bands of the element chlorine were observed above the

sulfur element bands for the 40S + 9Be reaction. The proton in these chlorine fragments

would have to have come from the beryllium target, of course.

141

Chapter 6

Summary and Conclusions

Filling out the nuclear landscape by studying rare isotopes is a major objective in the field of

nuclear science. While a small percentage of known nuclei are stable, most of the landscape

is composed of unstable nuclei that decay. Some of the more exotic nuclei have half-lives of

less than a second, creating challenges in producing, separating, and detecting these nuclei in

a very short time frame. To produce these rare and short-lived nuclei for study, a technique

called projectile fragmentation has been used for a number of years at a variety of isotope

production beam facilities such as the National Superconducting Cyclotron Laboratory. This

reaction is also the foundation for the next generation of these facilities.

In projectile fragmentation reactions, a high-energy beam of large nuclei is impinged

in a relatively thin and low-Z target. Projectile fragmentation reactions have often been

described as a two-step process. In the first step, the projectile rapidly collides and passes

by the target and the overlapping area is rapidly removed. The collision produces two

excited precursors, a projectile-like fragment and a target-like fragment, which enter the

second, slower de-excitation process. The excited precursors release nucleons, other light

particles, and energy before becoming “cold” nuclear residues. These residues encompass

the broadest range of nuclei, which is the reason the reaction mechanism is so useful.

Exclusive measurements of the final fragments from projectile fragmentation reactions

can provide more insight into the process by comparing to models that simulate collisions and

142

reactions on nuclei, but relatively few such measurements have been made. The present work

used exclusive measurements of projectile fragmentations products to explore the predictions

from two very different nuclear reaction models.

This work was performed at the National Superconducting Cyclotron Laboratory (NSCL),

where two intermediate-energy (55.5 MeV/u) projectile fragmentation beams were produced

and delivered for reaction with beryllium targets. Beams of 30S and 40S nuclei were chosen to

provide proton-rich and neutron-rich projectiles, allowing a wide range of fragment isospin.

The resulting heavy residue fragments were measured with the Sweeper magnet charged par-

ticle detectors and neutrons were detected in coincidence using the Modular Neutron Array

and Large-area multi-Institutional Scintillator Array (MoNA LISA) detectors [66–68, 96].

Signals from the MoNA LISA and Sweeper charged particle detectors were processed,

calibrated, and corrected for non-linear responses. Once the calibrations were completed,

events were selected from the data set by isotope for analysis. This experiment was found

to exceed the limits of the current detector configuration. The resolution was not sufficient

to achieve isotope separation for the highest Z elements (Z = 12 − 17) observed in the
detectors. However, a broad range of fragments was identified in each reaction for elements

with Z = 6− 11. The hit multiplicity distributions observed in MoNA LISA for the summed
elemental and individual isotopic products were compared to simulations from two nuclear

reaction models.

The two fragmentation reactions were simulated with two very different models to de-

scribe the fragment and neutron distributions from projectile fragmentation reactions. In

one approach, the Li`ege Intranuclear Cascade (INCL++) model [83–85] uses a microscopic

model and Monte Carlo based code that considers the reaction to be a two-step process with

collisions between individual nucleons followed by a de-excitation process of the intermedi-

143

ate and highly excited residue. The de-excitation process was carried out by ABLA07, the

default option selected by the authors of the INCL++ code. In the second approach, the

Constrained Molecular Dynamics (CoMD) model [69, 70] uses a more quantum mechanical

and global approach that follows dynamical nuclear processes using the nuclear equation of

state with three options for the symmetry energy term. While most fragments were cool

by the end of the CoMD calculations, the GEMINI++ code [73, 74] was used as an “after-

burner” code with CoMD to decay the small percentage of remaining hot fragments. The

results from both models were passed through the GEANT4 code to model the neutron

detection efficiency of the MoNA LISA array.

The hit multiplicity distributions were compared to the experimental data hits measured

in MoNA LISA. The fragments produced from the 40S reaction were found to be in coin-

cidence with significantly higher numbers of hits in MoNA LISA compared to fragments

produced from the 30S reaction as expected, given that 40S has ten more neutrons than 30S,

leading to more neutrons available in the reaction. Furthermore, with more neutrons in the

reaction, more neutron-rich final fragments could be produced. Because 30S has relatively

few neutrons, the distribution of fragments is limited by the proton drip line.

The majority of identified fragments were measured with no neutron hits. The CoMD

+ GEMINI++ simulations greatly under-predicted the number of fragments produced with

zero hits. Because the INCL++ model was closer in predicting the hit multiplicity zero bin

than the CoMD + GEMINI++ simulations, INCL++ was found to do a better job overall at

predicting isotopic hit multiplicity distributions. The CoMD + GEMINI++ model was able

to produce events with higher hit multiplicities seen in the 40S data, unlike the INCL++

model.

144

The CoMD model has been an important part of symmetry energy research. Determin-

ing the form of the symmetry energy remains a major objective in understanding nuclear

properties, astrophysical processes, and the fundamental nucleon-nucleon interaction. While

the form has been constrained in a variety of methods, it is important to continue to test the

predictions by experimentally constraining the form of the symmetry energy and establishing

tighter and consistent constraints. In the present experiment, the hit multiplicity distribu-

tions were examined to determine if a constraint could be made. However, because the three

CoMD + GEMINI++ options do not appear to produce noticeably different hit multiplicity

distributions, no constraint on the symmetry energy was made from this experimental data

set.

The distributions of precursor fragments in both the INCL++ and CoMD models were

also examined in this work. Many of the precursor fragments contained more nucleons than

the projectile. The projectile was predicted to pick up nucleons from the target, something

that is different from common descriptions of these reactions. Experimental data confirmed

that nucleon pick-up occurred as chlorine fragments were detected from the 40S + 9Be

reaction. Thus, the beam did not simply fragment on an inert target.

145

APPENDICES

146

Appendix A

Deriving the Nuclear Equation of

State and the Symmetry Energy

This appendix describes the connection between the nuclear equation of state for infinite

nuclear matter and the Bethe-Weizs¨acker formula [6] for finite matter. Some of the derivation

below can be found in [14], but intermediate steps are included here for clarity.

Part 1: Nuclear Equation of State

To derive the nuclear equation of state, it is helpful to start with the Bethe-Weizs¨acker

formula written in the semi-empirical mass formula:

BE(Z, N ) = aV A − aSA(2/3) − aC

Z2
A(1/3) − aA

(N − Z)2

A

± δpairing

(A.1)

where the binding energy, BE, is calculated with a volume term (aV ), a surface term (aS), a

Coulomb term (aC ), an asymmetry term (aA), and a pairing term (δpairing). The coefficients

of these terms have been determined by fitting the equation to experimentally known masses

[6]. The formula can be rewritten to show binding energy per nucleon:

BE(Z, N )

A

= aV −

aS
A(1/3) − aC

Z2
A(4/3) − aA

(N − Z)2

A2

±

δpairing

A

(A.2)

147

The Bethe-Weizs¨acker formula is for finite matter. For “infinite matter,” A approaches

infinity. The surface term drops out, as the denominator leads to infinity and the numerator

remains the same, pushing the term to zero. (Furthermore, there would be no surface in

infinite matter.) The Coulomb term is removed per standard practice of defining the nuclear

equation of state with symmetry energy. Some infinite nuclear matter studies exclude the

term as including Coulomb forces makes the liquid drop unstable in this form. Finally, the

pairing term is dropped because of its negligible contribution. This leaves the volume term

and the asymmetry term.

BE(Z, N )

A

= aV − aA

(N − Z)2

A2

≈ aV − aAδ2

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)A→∞

In the above function, δ is the isospin asymmetry, which is given as

(A.3)

(A.4)

(A.5)

δ =

(ρn − ρp)
ρρn+ρp ≈

(N − Z)

A

where ρn is the neutron density and ρp is the proton density of the matter.

The asymmetry term can be broken into volume (av

A) and surface (as

A) terms.

aAδ2 =(cid:16)av

A +

as
A

A1/3(cid:17)δ2

Substituting the above relation into Equation A.3. giving the binding energy per nucleon

as the following:

BE(Z, N )

A

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)A→∞

≈ aV −(cid:16)av

A +

as
A

A1/3(cid:17)δ2

(A.6)

148

Similar to before, the surface asymmetry term is dropped. This leaves only a few terms

left in the equation:

BE(Z, N )

A

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)A→∞

≈ aV − av

Aδ2

(A.7)

Consider the above as two parts: the symmetric (N = Z) part and the asymmetric

(N 6= Z) part. For symmetric matter, δ = 0. If δ = 0 is substituted into Equation A.7, only
the volume term aV remains:

BE(Z, N )

A

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)A→∞

≈ aV − av

A(0)2 = aV = E(ρ, 0)

(A.8)

The volume terms (aV ) then represents the symmetric part of the nuclear equation of

state (E(ρ, 0)). Thus, av

Aδ2 becomes the asymmetric term (when N 6= Z).

Aδ2 = Esym(ρ)δ2
av

(A.9)

The symmetry energy term is written as Esym(ρ, δ). Thus the nuclear equation of state

is written in its common form.

BE(Z, N )

A

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)A→∞

≈ E(ρ, δ) = E(ρ, 0) + Esym(ρ)δ2

(A.10)

Part 2: The Symmetry Energy

The symmetry energy term is expanded using a Taylor expansion characterizing the depar-

ture of symmetry energy from its value at saturation density. The Taylor expansion generic

form [116] is given as the following:

149

f (x0) +

f′(x0)

1!

(x − x0) +

f′′(x0)

2!

(x − x0)2 + ...

The symmetry energy has been defined [14] in the following relation:

Esym(ρ) =

1
2

∂2

∂δ2 E(ρ, δ)(cid:12)(cid:12)(cid:12)δ=0

(A.11)

(A.12)

For the Taylor expansion formula, the (x − x0) value is based on the difference of the
density from the saturation density. The difference is often written as the following equation:

(x − x0) = z =

ρ − ρ0
3ρ0

(A.13)

where ρ is the density and ρ0 is the saturation density.

Due to the small contribution of the latter terms, only the first three terms in the expan-

sion are considered. Let J, K, and L represent magnitude, slope, and curvature, respectively.

The form of the symmetry energy can be given as:

Esym(ρ) = J + Lz +

Kz2

1
2

(A.14)

with the magnitude parameter, the slope parameter, and the curvature parameter written

as the below Equations A.15, A.16, and A.17, respectively.

J = Esym(ρ0)

Lz =

Kz2 =

1
2

d
dρ

Esym(ρ)(cid:16) ρ − ρ0
3ρ0 (cid:17) =
3ρ0 (cid:17)2
dρ2 Esym(ρ)(cid:16) ρ − ρ0

d2

150

Lsym

3 (cid:16) ρ − ρ0
ρ0 (cid:17)
ρ0 (cid:17)2
18 (cid:16) ρ − ρ0

Ksym

=

(A.15)

(A.16)

(A.17)

Adding the terms together gives the total expansion of the symmetry energy term in

magnitude, slope, and curvature.

Esym(ρ) = Esym(ρ0) +

Lsym

3 (cid:16) ρ − ρ0

ρ0 (cid:17) +

Ksym

ρ0 (cid:17)2
18 (cid:16) ρ − ρ0

(A.18)

151

Appendix B

CRDC Mask Positions

As Chapter 4 explained, to determine the XY-positions of particles in the CRDCs, the pad

position (labeled in the data as “gravity” or pad number) was converted to an X-position

in mm in the lab frame and the drift time of the electrons (TAC) was converted into a

Y-position in mm in the lab frame. Figure B.1 shows how the “raw” signals in each CRDC

appeared before they were transformed into the correct XY-positions.

Both the converted X and Y positions were determined through the calibration provided

by tungsten masks. The mask was placed in front of the CRDC by a hydraulic drives. Each

mask had a known set of lines and holes forming a specific pattern, as shown in Figure

B.2 [102]. The incoming beam was defocused and swept across the area of the CRDC mask,

and only the beam particles that passed through a hole or line were detected in the CRDC.

Because the positions of the holes and lines in the mask were know, it was possible to convert

the CRDC raw signals into physical positions.

Table B.1 gives the XY-positions of the holes and the X-positions of the lines in the

masks. The physical coordinates of the holes and lines in the mask were converted to the

corresponding coordinates for CRDC1 and CRDC2.

152

CRDC1

CRDC2

i

s
l
a
n
g
“
 
(cid:863)
w
a
R
(cid:862)

s
l
a
n
g
“

i

 
(cid:863)
d
e
t
a
r
b

i
l

a
C
(cid:862)

TAC

Y

Gravity
(Pad Number)

X

Figure B.1: Schematic demonstrating the orientation of the raw signals TAC and gravity (or
pad number) in CRDC1 and CRDC2 and their transformed and calibrated XY-positions.

153

Figure B.2: Mechanical drawing of a tungsten mask used in the experimental configuration.
Each hole (1-26) and line (A and B) corresponded to a specific point in space. Adapted from
a drawing made by J. Honke [102].

154

Diagram

CRCD1

CRDC2

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
A
B

0.25
0.25
0.25
0.25
14.75
14.75
7.50
7.50
4.55
4.55
4.55
4.55
9.27
9.27
9.27
9.27
9.67
10.06
10.45
7.50
7.50
7.50
7.50
7.50
7.50
7.50
5.99
11.90

Label X (in.) Y (in.) X (mm) Y (mm) X (mm) Y (mm) X (mm) Y (mm)
-166.85
-116.05
-65.25
-14.45
-14.45
-65.25
53.01
-46.99
33.00
22.99
13.01
3.00
33.00
22.99
13.01
3.00
3.00
3.00
3.00
-97.00
-72.01
-21.99
3.00
27.99
78.01
103.00

20.65
71.45
122.25
173.05
173.05
122.25
240.51
140.51
220.50
210.49
200.51
190.50
220.50
210.49
200.51
190.50
190.50
190.50
190.50
90.50
115.49
165.51
190.50
215.49
265.51
290.50

0.81
2.81
4.81
6.81
6.81
4.81
9.47
5.53
8.68
8.29
7.89
7.50
8.68
8.29
7.89
7.50
7.50
7.50
7.50
3.56
4.55
6.52
7.50
8.48
10.45
11.44

-181.15
-181.15
-181.15
-181.15
187.15
187.15
3.00
3.00
-72.03
-72.03
-72.03
-72.03
48.03
48.03
48.03
48.03
58.02
68.02
78.03
3.00
3.00
3.00
3.00
3.00
3.00
3.00
-35.28
114.71

-169.85
-119.05
-68.25
-17.45
-17.45
-68.25
50.01
-49.99
30.00
19.99
10.01
0.00
30.00
19.99
10.01
0.00
0.00
0.00
0.00

-100.00
-75.01
-24.99
0.00
24.99
75.01
100.00

-
-

-184.15
-184.15
-184.15
-184.15
184.15
184.15
0.00
0.00
-75.03
-75.03
-75.03
-75.03
45.03
45.03
45.03
45.03
55.02
65.02
75.03
0.00
0.00
0.00
0.00
0.00
0.00
0.00
-38.28
111.71

6.35
6.35
6.35
6.35
374.65
374.65
190.50
190.50
115.47
115.47
115.47
115.47
235.53
235.53
235.53
235.53
245.52
255.52
265.53
190.50
190.50
190.50
190.50
190.50
190.50
190.50
152.22
302.21

-
-

-
-

-
-

Table B.1: The XY coordinates for the holes and lines in the CRDC masks. The coordinates
from the diagram were converted to their corresponding locations on CRDC1 and CRDC2.

155

Appendix C

Isotope Separation Parameters and

Corrections

Chapter 4 describes the isotope separation procedure. The isotope separation procedure was

carried out for the data from each element in each magnet setting for both the 30S and 40S

beams. The corrections and parameters for each element are summarized in the following

tables. The numbers in the right columns were multiplied by the parameters in the far left

columns. The values were then summed to construct the final corrected time-of-flight in the

following equation:

tf inal = ttarget→thin +X CiPi

(C.1)

where Ci were the correction coefficients (the right columns) and Pi were the parameters (the

left column). Isotope separation was found by applying these corrections to the appropriate

element fragments.

Tables C.1 and C.2 provide the parameters and coefficients for fragments produced from

the 30S + 9Be reaction and the 1.51 Tm and 2.25 Tm Sweeper magnet settings, respectively.

Tables C.3 and C.4 provide the parameters and coefficients for fragments produced from the

40S + 9Be reaction and the 2.01 Tm and 2.27 Tm Sweeper magnet settings, respectively.

156

An example that of the calculation of Equation C.1 for the carbon fragments produced

from the 30S + 9Be reaction and the 1.51 Tm Sweeper magnet setting is shown below.

tcarbon
f inal =ttarget→thin + (−4.21014 × 10−1)x + (−3.51953 × 10−3)x2 + (1.12644)θx

x + (1.42990 × 10−1)y + (4.53751 × 10−3)y2

(C.2)

+ (4.29741 × 10−3)θ2
+ (−1.18388 × 10−3)θy + (−7.28025 × 10−3)θ2
+ (−1.70214 × 10−6)x2θ2

x + (9.25063)ttarget→thin

y + (4.13360 × 10−3)xθx

157

Carbon

Nitrogen

Oxygen

Fluorine

Neon

Sodium

-4.21014 × 10−1
-3.51953 × 10−3

-

1.12644

4.29741 × 10−3
1.42990 × 10−1
4.53751 × 10−3
-1.18388 × 10−3
-7.28025 × 10−3
4.13360 × 10−3
-1.70214 × 10−6

8.25063

-3.76333 × 10−1
-2.16416 × 10−3
-4.32900 × 10−5
9.77349 × 10−1
5.32203 × 10−4
1.29525 × 10−1
3.92893 × 10−3
4.29734 × 10−2
-1.52684 × 10−3
-2.15249 × 10−3
-1.77161 × 10−6
-4.09191 × 10−10

7.77247

-2.91627 × 10−1
-2.74075 × 10−3
-5.15389 × 10−5

1.04316

-2.91884 × 10−1
-2.82947 × 10−3
-4.22475 × 10−5

1.04248

-3.54995 × 10−1
-2.07933 × 10−3
-3.82195 × 10−5

1.05602

-3.02621 × 10−1
4.03140 × 10−4
-6.08871 × 10−5

1.07408

-

-

-

-

1.28582 × 10−1
3.76369 × 10−3
-1.16298 × 10−2
-5.00950 × 10−3
-1.39708 × 10−3
-2.82452 × 10−6
-7.69764 × 10−10

1.26766 × 10−1
3.79120 × 10−3
-1.12000 × 10−2

9.29421 × 10−2
3.40481 × 10−3
3.61747 × 10−2

-

-

1.17116 × 10−1
3.61254 × 10−3
1.36818 × 10−2

-

-2.22464 × 10−3
6.33964 × 10−7

7.80569 × 10−5
-6.89200 × 10−7

5.85587 × 10−4
-2.58906 × 10−7

-

-

-

6.45237

6.21714

6.90080

5.98684

x
x2
x3

θx
θ2
x

y
y2

θy
θ2
y

-

xθx
x2θ2
x
x3θ3
x
T oF∗
∆EIC
*T oFtarget→thin

-

-

-

1.54680 × 10−2

-2.44056 × 10−2

9.95481 × 10−3

Table C.1: Final correction factors used for isotope separation for fragments produced from the 30S + 9Be reaction and the
1.51 Sweeper magnet setting.

158

Carbon

Nitrogen

Oxygen

Fluorine

Neon

Sodium

1.002234

1.14073

-6.90426 × 10−1
-1.37997 × 10−3

-5.97753 × 10−1
-1.75521 × 10−3

-

-

9.75722 × 10−1
-2.14629 × 10−4
-9.31887 × 10−6
9.51435 × 10−2
4.01068 × 10−3
2.87760 × 10−2
1.96523 × 10−3
-8.01858 × 10−4

-

-

1.10371 × 10−3

-

1.00653 × 10−1
4.96386 × 10−3
3.58381 × 10−2

-

-7.91037 × 10−4

-

-

-6.85592 × 10−1
-2.73203 × 10−3
-1.16694 × 10−5

-6.57750 × 10−4
-5.89717 × 10−5
1.02590 × 10−1
4.23464 × 10−3
4.39263 × 10−2

-

-6.42464 × 10−1
-1.87090 × 10−3
-8.33033 × 10−6
9.37767 × 10−1
2.01998 × 10−3

-

1.24917 × 10−1
4.47153 × 10−3

-

-

-1.79488 × 10−3
-4.22700 × 10−7

-2.07703 × 10−3
-9.36356 × 10−7

-

-

-6.05760 × 10−1
-2.55234 × 10−3
-8.37146 × 10−6

1.014035

1.88481 × 10−3

-

1.29125 × 10−1
6.07290 × 10−3
2.06325 × 10−2
2.56012 × 10−3
-9.83607 × 10−4

-

-

x
x2
x3

θx
θ2
x
θ3
x

y
y2

θy
θ2
y

-6.31523 × 10−1
-2.28293 × 10−3

-

9.72445 × 10−1
1.83028 × 10−3

-

1.65249 × 10−1
5.15829 × 10−3
3.74497 × 10−2
1.67589 × 10−3

-

-

-

-

-

xθx
x2θ2
x
x3θ3
x
T oF∗
∆EIC
∆E2
IC
*T oFtarget→thin

3.42879 × 101

3.47553 × 101

2.66472 × 101

-

-

-

-

3.37634 × 101
-2.43065 × 10−1
3.33372 × 10−4

3.63524 × 101
3.58891 × 10−2

-

2.86306 × 101
-3.05655 × 10−1
-2.66471 × 10−4

Table C.2: Final correction factors used for isotope separation for fragments produced from the 30S + 9Be reaction and the
2.25 Sweeper magnet setting.

159

Carbon

-1.29621

Nitrogen

-1.21297

Oxygen

-1.19580

1.42778 × 10−3
-1.08579 × 10−3
8.18146 × 10−1
1.23215 × 10−1
1.24508 × 10−1
1.22695 × 10−1
1.25725 × 10−1
-1.72589 × 10−4
-9.38943 × 10−3
-4.63388 × 10−3
-1.74465 × 10−6

6.73078

1.23366

5.47301 × 10−3
-9.00213 × 10−4
8.75322 × 10−1
1.28244 × 10−2
-3.09585 × 10−4
1.51930 × 10−1
-5.61972 × 10−3
-2.67244 × 10−4
-1.10577 × 10−2
-5.20661 × 10−3
-2.13910 × 10−6

6.17817

1.03563

-2.82654 × 10−3

-1.56647 × 10−3

Fluorine

-8.72480 × 10−1
-2.57488 × 10−4
-9.98858 × 10−4
9.1623 × 10−1
1.27262 × 10−1
1.24630 × 10−1
1.14758 × 10−1
1.27029 × 10−1
-3.20693 × 10−4
-1.07260 × 10−2
-4.94628 × 10−3
-2.11771 × 10−6

8.33772

Neon

-1.14109

Sodium

-1.17800

1.41042 × 10−4
-9.86415 × 10−4
9.17048 × 10−1
1.27342 × 10−1
1.24635 × 10−1
1.14595 × 10−1
1.28281 × 10−1
-2.43192 × 10−4
-1.02837 × 10−2
-4.99835 × 10−3
-2.99474 × 10−6

7.45735

-1.19571 × 10−3
-1.01516 × 10−3
9.16177 × 10−1
1.26905 × 10−1
1.24589 × 10−1
1.07479 × 10−1
1.25565 × 10−1
-2.88624 × 10−4
-2.13894 × 10−2
-5.42562 × 10−3
-1.90380 × 10−6

7.65281

9.90138 × 10−1
-1.24955 × 10−3

8.26615 × 10−1
-8.64823 × 10−4

7.98638 × 10−1
-6.77071 × 10−4

x
x2
x3

θx
θ2
x
θ3
x

y
y2

θy
θ2
y

-3.89795 × 10−3
-7.14639 × 10−4
8.00033 × 10−1
1.22943 × 10−1
1.24519 × 10−1
1.83709 × 10−1
1.26645 × 10−1
-2.29830 × 10−4
-1.01731 × 10−2
-5.72727 × 10−3
-2.56692 × 10−6

6.64998

2.49880

xθx
x2θ2
x
T oF∗
∆EIC
∆E2
*T oFtarget→thin

IC -7.88068 × 10−3

Table C.3: Final correction factors used for isotope separation for fragments produced from the 40S + 9Be reaction and the
2.01 Sweeper magnet setting.

160

Carbon

-1.19777

Nitrogen

-1.26471

Oxygen

-1.18715

Fluorine

-1.33243

Neon

-1.35075

Sodium

-1.40206

x
x2
x3

θx
θ2
x
θ3
x

y
y2

θy
θ2
y

xθx
x2θ2
x
T oF∗
∆EIC
∆E2
IC
*T oFtarget→thin

-

-1.55306 × 10−2
-1.00078 × 10−3
8.84322 × 10−1
-4.82437 × 10−3
-3.09585 × 10−4
2.09501 × 10−1
-5.90194 × 10−3
-3.28330 × 10−4
-1.09471 × 10−2
-4.99835 × 10−3
-2.03215 × 10−6

7.57671

6.93957 × 10−2

5.47301 × 10−3
-9.58920 × 10−4
9.34056 × 10−1
-1.23330 × 10−2
-5.06380 × 10−4
4.53709 × 10−1
-6.21424 × 10−3
-1.91054 × 10−4
-1.21635 × 10−2
-5.15454 × 10−3
-2.78083 × 10−6

8.53880

1.28058

-2.97914 × 10−3

5.57342 × 10−3
-8.99616 × 10−4
9.19855 × 10−1
-1.00328 × 10−2
-4.82665 × 10−4
6.31530 × 10−1
-5.85912 × 10−3
-2.67334 × 10−4
-1.16106 × 10−2
-5.83140 × 10−3
-2.18188 × 10−6

6.24597

-9.30528 × 10−4
-8.98881 × 10−4
8.96637 × 10−1
-1.84343 × 10−2
-6.23105 × 10−4
1.89120 × 10−1
-5.68960 × 10−3
-2.66530 × 10−4
-1.06154 × 10−2
-5.46694 × 10−3
-2.26745 × 10−6

7.44467

-1.76240 × 10−3
-8.70187 × 10−4
9.57798 × 10−1
-9.92907 × 10−3
-5.01209 × 10−4
1.99088 × 10−1
-5.78834 × 10−3
-1.85992 × 10−4
-9.84135 × 10−3
-4.99835 × 10−3
-2.09632 × 10−6

7.09218

-3.76194 × 10−3
-9.24366 × 10−4
0.912594 × 10−1
-9.34532 × 10−3
-4.94422 × 10−4
3.71123 × 10−1
-6.28356 × 10−3
-2.87112 × 10−4
-1.13894 × 10−2
-4.89421 × 10−3
-1.71128 × 10−6

7.18210

9.21870 × 10−2

1.09720 × 10−1

-

-

9.22175 × 10−2
-1.56005 × 10−3

4.12172 × 10−1
-2.75234 × 10−4

Table C.4: Final correction factors used for isotope separation for fragments produced from the 40S + 9Be reaction and the
2.27 Sweeper magnet setting.

161

Appendix D

χ2

v Values

As described in Chapter 5, χ2

v values were calculated for a quantitative comparison to the

simulated models. As a brief reminder, the models used in this analysis were INCL++

and the three versions of CoMD + GEMINI++ (labeled here as CoMD-1, CoMD-2, and

CoMD-3).

First, χ2 values were calculated for each hit multiplicity value with the following equation:

χ2 =

n

Xi=0

(P data

i

i

− P sim
i
)2

(Edata

)2

(D.1)

where i was the hit multiplicity value (bin number), P data

i

was the probability value of the

experimental data, P sim

i

was the probability value of the simulation set, and Edata

i

was the

uncertainty in the data.

A reduced χ2

v was used to compare models with different numbers of parameters. The

inclusion of hit multiplicity values or bins with very few events could skew the comparison

by placing more importance on those few events. In order to minimize this effect, only hit

multiplicity values with more than five events were included. The reduced χ2

v was found

with the following equation:

χ2
v =

χ2
ndf

162

(D.2)

where ndf was the degrees of freedom (the number of bins subtract one).

If the given

simulation matched the data within the experimental uncertainties, the χ2

v value would be

one. The larger the χ2

v value, the worse the fit.

The χ2

v values are given in the following tables. Tables D.1 and D.2 provide the calculated

v Values for isotope hit multiplicity distributions produced by the 30S + 9Be reaction and
χ2

the 1.51 Tm and 2.25 Tm Sweeper magnet settings, respectively. Tables D.3 and D.4 provide

the calculated χ2

v values for isotope hit multiplicity distributions produced by the 40S + 9Be

reaction and the 2.01 Tm and 2.27 Tm Sweeper magnet settings, respectively. The average

given in the tables is the total of the χ2

v values of a simulation divided by the number of

isotopes in that setting.

163

Isotope COMD-1 COMD-2 COMD-3

INCL

11C
12C
13C
13N
14N
15N
15O
16O
17O
17F
18F
19F
19Ne
20Ne
21Ne
21Na
22Na
23Na

1898.47

1835.89

2252.20

39.10

5251.70

5962.74

5799.37

147.66

1154.09

1260.59

1142.56

23.39

955.09

1129.78

1266.93

33.57

3829.17

4276.70

3871.57

117.59

2158.96

2004.40

2214.18

128.27

5157.21

4794.39

4833.07

131.27

8545.54

8972.88

8329.43

256.99

1417.42

1392.30

1307.27

46.63

2050.41

2245.29

2295.02

87.18

2617.99

2931.60

2717.69

65.67

1498.47

1185.19

1148.44

44.30

1980.14

2295.78

2092.33

65.46

5571.36

6191.00

6031.88

293.89

2059.19

2322.77

2207.63

102.27

2258.28

2143.88

2226.45

97.75

5916.04

5171.33

4583.89

235.18

1612.18

1552.83

1730.95

44.62

Average

3107.32

3203.85

3113.94

108.93

Table D.1: Calculated χ2
30S + 9Be reaction and the 1.51 Sweeper magnet setting.

v values for isotope hit multiplicity distributions produced by the

164

Isotope COMD-1 COMD-2 COMD-3

INCL

11C
12C
13C
14N
15N
15O
16O
17O
18F
19F
20F
20Ne
21Ne
22Ne
22Na
23Na

564.48

607.85

692.62

6.08

7178.11

5655.27

4882.24

212.58

4662.11

4237.77

4214.36

182.72

6691.34

6311.20

9005.17

350.54

9573.68

6509.68

7525.17

414.74

2523.50

3284.80

3049.19

73.39

14570.80

16537.37

10277.38

792.90

5477.91

4705.06

4959.60

419.38

2827.75

2940.24

2569.97

400.14

4700.76

4363.91

3622.98

410.74

1298.89

1037.52

879.28

148.08

6978.30

6895.98

5139.13

1034.15

5619.90

5587.73

5472.99

1402.53

1702.93

1537.72

1473.48

491.18

1925.52

1752.52

2429.46

586.41

2822.84

2198.50

2642.74

1385.43

Average

4944.93

4635.19

4302.24

519.44

Table D.2: Calculated χ2
30S + 9Be reaction and the 2.25 Sweeper magnet setting.

v values for isotope hit multiplicity distributions produced by the

165

Isotope COMD-1 COMD-2 COMD-3

INCL

12C
13C
14C
14N
15N
16N
17N
16O
17O
18O
19O
18F
19F
20F
21F
20Ne
21Ne
22Ne
23Ne
24Ne
22Na
23Na
24Na
25Na

191.27

143.47

118.74

188.71

228.55

384.26

122.09

163.37

274.88

191.69

250.08

234.00

39.76

51.87

39.78

32.87

1098.27

865.57

771.21

163.00

190.14

120.40

167.40

72.16

83.06

77.42

340.15

572.82

358.37

40.76

27.83

93.62

380.88

363.63

411.01

108.21

829.47

543.53

428.19

111.24

218.29

176.09

186.32

16.47

144.97

108.54

35.99

44.98

338.46

599.31

543.64

152.97

545.30

691.20

796.32

156.19

416.27

371.60

430.89

101.43

231.54

167.62

304.36

120.87

603.47

562.76

690.89

213.42

1480.82

1054.62

1300.17

344.24

419.85

387.05

388.59

107.52

119.42

116.89

125.73

211.06

177.85

139.63

41.00

78.60

718.42

1006.32

975.37

388.00

1107.80

1006.23

1195.65

418.50

984.19

956.06

1044.62

263.00

Average

459.01

448.05

477.34

132.32

Table D.3: Calculated χ2
40S + 9Be reaction and the 2.01 Sweeper magnet setting.

v values for isotope hit multiplicity distributions produced by the

166

Isotope COMD-1 COMD-2 COMD-3

INCL

12C
13C
14C
14N
15N
16N
17N
16O
17O
18O
19O
20O
19F
20F
21F
22F
21Ne
22Ne
23Ne
24Ne
23Na
24Na
26Na
25Na

56.09

75.00

45.86

45.49

65.16

69.51

67.46

27.99

162.84

259.94

83.42

4.26

51.20

55.51

7.27

82.27

38.50

114.68

148.02

41.29

53.82

75.62

8.86

54.44

215.13

114.53

107.58

153.72

126.50

131.82

138.73

152.40

67.02

81.14

69.21

96.72

55.92

129.16

261.02

153.53

291.97

253.90

132.24

111.17

144.13

49.83

76.84

63.93

98.68

85.78

207.95

11.84

15.01

20.63

5.92

50.95

21.18

24.85

25.33

32.16

44.58

27.45

14.02

39.93

71.37

48.96

20.37

62.89

568.53

267.92

214.98

134.24

196.46

259.81

209.59

145.86

112.79

139.10

382.39

172.23

255.96

82.80

49.30

96.00

210.46

200.99

145.33

138.49

331.13

300.74

307.03

130.05

134.08

150.52

161.07

63.70

Average

148.48

135.64

138.05

51.33

Table D.4: Calculated χ2
40S + 9Be reaction and the 2.27 Sweeper magnet setting.

v values for isotope hit multiplicity distributions produced by the

167

BIBLIOGRAPHY

168

BIBLIOGRAPHY

[1] J. Erler, N. Birge, M. Kortelainen, W. Nazarewicz, E. Olsen, A. M. Perhac, and

M. Stoitsov. The limits of the nuclear landscape. Nature, 486:509512, Jun 2012.

[2] E. O’Donnell. Chart of nuclei. https://www.flickr.com/photos/nsclmedia/16247725170/

in/album-72157622074745816/, 2015. Accessed: 2018-05-30.

[3] J. D. Bowman, W. J. Swiatecki, and C. F. Tsang. Abrasion and ablation of heavy
ions. Technical Report LBL-2908, Lawrence Berkeley National Laboratory, July 1973.

[4] A. S. Goldhaber and H. H. Heckman. High energy interactions of nuclei. Annual

Review of Nuclear and Particle Science, 28(1):161–205, 1978.

[5] W. Loveland, D. J. Morrissey, and G. T. Seaborg. Modern Nuclear Chemistry. John

Wiley & Sons, Inc., Hoboken, NJ, 2006.

[6] H. A. Bethe and R. F. Bacher. Nuclear physics A. Stationary states of nuclei. Rev.

Mod. Phys., 8:82–229, Apr 1936.

[7] L. Satpathy and R. Nayak. Generalized Hugenholtz-Van Hove theorem and a new

mass relation for finite nuclei. Phys. Rev. Lett., 51:1243–1246, Oct 1983.

[8] R. D. Mattuck. A Guide to Feynman Diagrams in the Many-body Problem. Dover

Publications, 2nd edition, 1992.

[9] J. D. Walecka. Theoretical Nuclear and Subnuclear Physics. Imperial College Press

and World Scientific, Hackensack, NJ, 2nd edition, 2004.

[10] B. A. Li, L. W. Chen, and C. M. Ko. Recent progress and new challenges in isospin

physics with heavy-ion reactions. Physics Reports, 464(4):113 – 281, 2008.

[11] L. W. Chen, B. J. Cai, C. M. Ko, B. A. Li, C. Shen, and J. Xu. Higher-order effects on
the incompressibility of isospin asymmetric nuclear matter. Phys. Rev. C, 80:014322,
Jul 2009.

[12] P. Danielewicz and J. Lee. Symmetry energy I: Semi-infinite matter. Nuclear Physics

A, 818(1):36 – 96, 2009.

[13] G. Giuliani, H. Zheng, and A. Bonasera. The many facets of the (non-relativistic)
nuclear equation of state. Progress in Particle and Nuclear Physics, 76:116 – 164,
2014.

169

[14] S. Typel, H. H. Wolter, G. R¨opke, and D. Blaschke. Effects of the liquid-gas phase
transition and cluster formation on the symmetry energy. Eur. Phys. J. A, 50(2):17,
2014.

[15] R. Wang and L. W. Chen. Empirical information on nuclear matter fourth-order
symmetry energy from an extended nuclear mass formula. Physics Letters B, 773:62 –
67, 2017.

[16] B. A. Li and U. Schroder. Isospin Physics in Heavy-Ion Collisions at Intermediate

Energies. Nova Science Publishers, Inc., Huntington, NY, 2001.

[17] C. Fuchs and H. H. Wolter. Modelization of the EOS. The European Physical Journal

A - Hadrons and Nuclei, 30(1):5–21, Oct 2006.

[18] S. D. Gupta and G. D. Westfall. Probing dense nuclear matter in the laboratory.

Physics Today, 46:34, 1993.

[19] W. Reisdorf and H.G. Ritter. Collective flow in heavy-ion collisions. Annual Review

of Nuclear and Particle Science, 47(1):663–709, 1997.

[20] N. Herrmann, J. P. Wessels, and T. Wienold. Collective flow in heavy-ion collisions.

Annual Review of Nuclear and Particle Science, 49(1):581–632, 1999.

[21] D. J. Magestro, W. Bauer, and G. D. Westfall. Isolation of the nuclear compressibility

with the balance energy. Phys. Rev. C, 62:041603, Sep 2000.

[22] P. Danielewicz, R. Lacey, and W. G. Lynch. Determination of the equation of state of

dense matter. Science, 298(5598):1592–1596, 2002.

[23] S. Shlomo, V. M. Kolomietz, and G. Col`o. Deducing the nuclear-matter incompress-
ibility coefficient from data on isoscalar compression modes. The European Physical
Journal A - Hadrons and Nuclei, 30(1):23–30, Oct 2006.

[24] D. D. S. Coupland, W. G. Lynch, M. B. Tsang, P. Danielewicz, and Y. Zhang. Influence
of transport variables on isospin transport ratios. Phys. Rev. C, 84:054603, Nov 2011.

[25] M. B. Tsang, J. R. Stone, F. Camera, P. Danielewicz, S. Gandolfi, K. Hebeler, C. J.
Horowitz, Jenny Lee, W. G. Lynch, Z. Kohley, R. Lemmon, P. M¨oller, T. Murakami,
S. Riordan, X. Roca-Maza, F. Sammarruca, A. W. Steiner, I. Vida˜na, and S. J. Yen-
nello. Constraints on the symmetry energy and neutron skins from experiments and
theory. Phys. Rev. C, 86:015803, Jul 2012.

[26] R. J. Furnstahl. Neutron radii in mean-field models. Nuclear Physics A, 706(1):85 –

110, 2002.

170

[27] B. A. Li. High density behaviour of nuclear symmetry energy and high energy heavy-

ion collisions. Nuclear Physics A, 708(3):365 – 390, 2002.

[28] Bao-An Li. Probing the high density behavior of the nuclear symmetry energy with

high energy heavy-ion collisions. Phys. Rev. Lett., 88:192701, Apr 2002.

[29] J. M. Lattimer and M. Prakash. The physics of neutron stars. Science, 304(5670):536–

542, 2004.

[30] V. Baran, M. Colonna, V. Greco, and M. Di Toro. Reaction dynamics with exotic

nuclei. Physics Reports, 410(5):335 – 466, 2005.

[31] L. W. Chen, C. M. Ko, and B. A. Li. Determination of the stiffness of the nuclear

symmetry energy from isospin diffusion. Phys. Rev. Lett., 94:032701, Jan 2005.

[32] T. Kl¨ahn, D. Blaschke, S. Typel, E. N. E. van Dalen, A. Faessler, C. Fuchs, T. Gaitanos,
H. Grigorian, A. Ho, E. E. Kolomeitsev, M. C. Miller, G. R¨opke, J. Tr¨umper, D. N.
Voskresensky, F. Weber, and H. H. Wolter. Constraints on the high-density nuclear
equation of state from the phenomenology of compact stars and heavy-ion collisions.
Phys. Rev. C, 74:035802, Sep 2006.

[33] P. G. Krastev and B. A. Li. Constraining a possible time variation of the gravitational
constant G with terrestrial nuclear laboratory data. Phys. Rev. C, 76:055804, Nov
2007.

[34] A. W. Steiner, M. Prakash, J. M. Lattimer, and P. J. Ellis.

Isospin asymmetry in

nuclei and neutron stars. Physics Reports, 411(6):325 – 375, 2005.

[35] H.-Th. Janka, K. Langanke, A. Marek, G. Martnez-Pinedo, and B. Mller. Theory
of core-collapse supernovae. Physics Reports, 442(1):38 – 74, 2007. The Hans Bethe
Centennial Volume 1906-2006.

[36] D. V. Shetty, S. J. Yennello, and G. A. Souliotis. Density dependence of the sym-
metry energy and the nuclear equation of state: A dynamical and statistical model
perspective. Phys. Rev. C, 76:024606, Aug 2007.

[37] T. Fischer, M. Hempel, I. Sagert, Y. Suwa, and J. Schaffner-Bielich. Symmetry energy
impact in simulations of core-collapse supernovae. The European Physical Journal A,
50(2):46, Feb 2014.

[38] J. M. Lattimer and A.W. Steiner. Constraints on the symmetry energy using the
mass-radius relation of neutron stars. The European Physical Journal A, 50(2):40, Feb
2014.

171

[39] M. Duer, O. Hen, E. Piasetzky, H. Hakobyan, L. B. Weinstein, M. Braverman, E. O.
Cohen, D. Higinbotham, K. P. Adhikari, S. Adhikari, M. J. Amaryan, J. Arrington,
A. Ashkenazi, J. Ball, I. Balossino, L. Barion, M. Battaglieri, V. Batourine, A. Beck,
I. Bedlinskiy, A. S. Biselli, S. Boiarinov, W. J. Briscoe, W. K. Brooks, S. Buelt-
mann, D. Bulumulla, V. D. Burkert, F. Cao, D. S. Carman, A. Celentano, G. Charles,
T. Chetry, G. Ciullo, L. Clark, B. A. Clary, P. L. Cole, M. Contalbrigo, O. Cortes,
V. Crede, R. Cruz-Torres, A. D’Angelo, N. Dashyan, R. De Vita, E. De Sanctis, M. De-
furne, A. Deur, C. Djalali, G. Dodge, R. Dupre, H. Egiyan, A. El Alaoui, L. El Fassi,
P. Eugenio, R. Fersch, A. Filippi, T. A. Forest, G. Gavalian, Y. Ghandilyan, S. Gilad,
G. P. Gilfoyle, K. L. Giovanetti, F. X. Girod, E. Golovatch, R. W. Gothe, K. A. Grif-
foen, L. Guo, N. Harrison, M. Hattawy, F. Hauenstein, K. Hafidi, K. Hicks, M. Holtrop,
C. E. Hyde, Y. Ilieva, D. G. Ireland, B. S. Ishkhanov, E. L. Isupov, K. Joo, M. L.
Kabir, D. Keller, G. Khachatryan, M. Khachatryan, M. Khandaker, A. Kim, W. Kim,
A. Klein, F. J. Klein, I. Korover, S. E. Kuhn, L. Lanza, G. Laskaris, P. Lenisa, K. Liv-
ingston, I. J. D. MacGregor, C. Marchand, N. Markov, B. McKinnon, S. Mey-Tal Beck,
T. Mineeva, M. Mirazita, V. Mokeev, R. A. Montgomery, A. Movsisyan, C. Munoz-
Camacho, B. Mustapha, S. Nadeeshani, P. Nadel-Turonski, S. Niccolai, G. Niculescu,
M. Osipenko, A. I. Ostrovidov, M. Paolone, E. Pasyuk, M. Patsyuk, A. Papadopoulou,
K. Park, D. Payette, W. Phelps, O. Pogorelko, J. Poudel, J. W. Price, S. Procureur,
Y. Prok, D. Protopopescu, M. Ripani, A. Rizzo, G. Rosner, P. Rossi, F. Sabati´e,
A. Schmidt, C. Salgado, B. A. Schmookler, R. A. Schumacher, E. P. Segarra, Y. G.
Sharabian, G. D. Smith, D. Sokhan, N. Sparveris, S. Stepanyan, S. Strauch, M. Taiuti,
J. A. Tan, M. Ungaro, H. Voskanyan, E. Voutier, D. P. Watts, X. Wei, N. Zachar-
iou, J. Zhang, X. Zheng, Z. W. Zhao, and The CLAS Collaboration. Probing high-
momentum protons and neutrons in neutron-rich nuclei. Nature, 560(7720):617–621,
2018.

[40] J. M. Lattimer and M. Prakash. Neutron star structure and the equation of state. The

Astrophysical Journal, 550(1):426, 2001.

[41] J. Rikovska Stone, J. C. Miller, R. Koncewicz, P. D. Stevenson, and M. R. Strayer.
Nuclear matter and neutron-star properties calculated with the Skyrme interaction.
Phys. Rev. C, 68:034324, Sep 2003.

[42] B. A. Li and A. W. Steiner. Constraining the radii of neutron stars with terrestrial

nuclear laboratory data. Physics Letters B, 642(5):436 – 440, 2006.

[43] A. W. Steiner, J. M. Lattimer, and E. F. Brown. The equation of state from observed

masses and radii of neutron stars. The Astrophysical Journal, 722(1):33, 2010.

[44] A. W. Steiner, J. M. Lattimer, and E. F. Brown. The neutron star mass-radius relation
and the equation of state of dense matter. The Astrophysical Journal Letters, 765(1):L5,
2013.

172

[45] J. Erler, C. J. Horowitz, W. Nazarewicz, M. Rafalski, and P. G. Reinhard. Energy

density functional for nuclei and neutron stars. Phys. Rev. C, 87:044320, Apr 2013.

[46] I. Tews, T. Kr¨uger, K. Hebeler, and A. Schwenk. Neutron matter at next-to-next-to-
next-to-leading order in chiral effective field theory. Phys. Rev. Lett., 110:032504, Jan
2013.

[47] K. Stiefel, Z. Kohley, R. T. deSouza, S. Hudan, and K. Hammerton. Symmetry energy

dependence of long-timescale isospin transport. Phys. Rev. C, 90:061605, Dec 2014.

[48] T. Li, U. Garg, Y. Liu, R. Marks, B. K. Nayak, P. V. Madhusudhana Rao, M. Fujiwara,
H. Hashimoto, K. Kawase, K. Nakanishi, S. Okumura, M. Yosoi, M. Itoh, M. Ichikawa,
R. Matsuo, T. Terazono, M. Uchida, T. Kawabata, H. Akimune, Y. Iwao, T. Murakami,
H. Sakaguchi, S. Terashima, Y. Yasuda, J. Zenihiro, and M. N. Harakeh.
Isotopic
dependence of the giant monopole resonance in the even-A 112−124Sn isotopes and the
asymmetry term in nuclear incompressibility. Phys. Rev. Lett., 99:162503, Oct 2007.

[49] L. W. Chen, C. M. Ko, B. A. Li, and J. Xu. Density slope of the nuclear symmetry
energy from the neutron skin thickness of heavy nuclei. Phys. Rev. C, 82:024321, Aug
2010.

[50] P. M¨oller, W. D. Myers, H. Sagawa, and S. Yoshida. New finite-range droplet mass

model and equation-of-state parameters. Phys. Rev. Lett., 108:052501, Jan 2012.

[51] A. Tamii, I. Poltoratska, P. von Neumann-Cosel, Y. Fujita, T. Adachi, C. A. Bertu-
lani, J. Carter, M. Dozono, H. Fujita, K. Fujita, K. Hatanaka, D. Ishikawa, M. Itoh,
T. Kawabata, Y. Kalmykov, A. M. Krumbholz, E. Litvinova, H. Matsubara, K. Nakan-
ishi, R. Neveling, H. Okamura, H. J. Ong, B. ¨Ozel-Tashenov, V. Yu. Ponomarev,
A. Richter, B. Rubio, H. Sakaguchi, Y. Sakemi, Y. Sasamoto, Y. Shimbara, Y. Shimizu,
F. D. Smit, T. Suzuki, Y. Tameshige, J. Wambach, R. Yamada, M. Yosoi, and J. Zeni-
hiro. Complete electric dipole response and the neutron skin in 208Pb. Phys. Rev.
Lett., 107:062502, Aug 2011.

[52] X. Roca-Maza, M. Brenna, G. Col`o, M. Centelles, X. Vi˜nas, B. K. Agrawal, N. Paar,
D. Vretenar, and J. Piekarewicz. Electric dipole polarizability in 208Pb: Insights from
the droplet model. Phys. Rev. C, 88:024316, Aug 2013.

[53] L. Trippa, G. Col`o, and E. Vigezzi. Giant dipole resonance as a quantitative constraint

on the symmetry energy. Phys. Rev. C, 77:061304, Jun 2008.

[54] M. B. Tsang, Y. Zhang, P. Danielewicz, M. Famiano, Z. Li, W. G. Lynch, and A. W.
Steiner. Constraints on the density dependence of the symmetry energy. Phys. Rev.
Lett., 102:122701, Mar 2009.

173

[55] P. Danielewicz and J. Lee. Symmetry energy II: Isobaric analog states. Nuclear Physics

A, 922:1 – 70, 2014.

[56] M. Kortelainen, T. Lesinski, J. Mor´e, W. Nazarewicz, J. Sarich, N. Schunck, M. V.
Stoitsov, and S. Wild. Nuclear energy density optimization. Phys. Rev. C, 82:024313,
Aug 2010.

[57] S. Gandolfi, J. Carlson, and S. Reddy. Maximum mass and radius of neutron stars,

and the nuclear symmetry energy. Phys. Rev. C, 85:032801, Mar 2012.

[58] K. Hebeler, J. M. Lattimer, C. J. Pethick, and A. Schwenk. Constraints on neutron star
radii based on chiral effective field theory interactions. Phys. Rev. Lett., 105:161102,
Oct 2010.

[59] Z. Kohley and S. J. Yennello. Heavy-ion collisions: Direct and indirect probes of
the density and temperature dependence of Esym. The European Physical Journal A,
50(2):31, Feb 2014.

[60] Z. Kohley, G. Christian, T. Baumann, P. A. DeYoung, J. E. Finck, N. Frank, M. Jones,
J. K. Smith, J. Snyder, A. Spyrou, and M. Thoennessen. Exploiting neutron-rich
radioactive ion beams to constrain the symmetry energy. Phys. Rev. C, 88:041601,
Oct 2013.

[61] Z. Kohley, L. W. May, S. Wuenschel, A. Bonasera, K. Hagel, R. Tripathi, R. Wada,
G. A. Souliotis, D. V. Shetty, S. Galanopoulos, M. Mehlman, W. B. Smith, S. N.
Soisson, B. C. Stein, and S. J. Yennello. Investigation of transverse collective flow of
intermediate mass fragments. Phys. Rev. C, 82:064601, Dec 2010.

[62] P. Russotto, P. Z. Wu, M. Zoric, M. Chartier, Y. Leifels, R. C. Lemmon, Q. Li,
J.  Lukasik, A. Pagano, P. Pawowski, and W. Trautmann. Symmetry energy from
elliptic flow in 197Au+197Au. Physics Letters B, 697(5):471 – 476, 2011.

[63] M. Colonna, A. Ono, and J. Rizzo. Fragmentation paths in dynamical models. Phys.

Rev. C, 82:054613, Nov 2010.

[64] A. Ono. Antisymmetrized molecular dynamics with quantum branching processes for

collisions of heavy nuclei. Phys. Rev. C, 59:853–864, Feb 1999.

[65] J. Rizzo, M. Colonna, M. Di Toro, and V. Greco. Transport properties of isospin

effective mass splitting. Nuclear Physics A, 732:202 – 217, 2004.

[66] B. Luther, T. Baumann, M. Thoennessen, J. Brown, P. DeYoung, J. Finck, J. Hin-
nefeld, R. Howes, K. Kemper, P. Pancella, G. Peaslee, W. Rogers, and S. Tabor. Mona
the modular neutron array. Nuclear Instruments and Methods in Physics Research Sec-
tion A: Accelerators, Spectrometers, Detectors and Associated Equipment, 505(1):33 –

174

35, 2003. Proceedings of the tenth Symposium on Radiation Measurements and Ap-
plications.

[67] T. Baumann, J. Boike, J. Brown, M. Bullinger, J. P. Bychoswki, S. Clark, K. Daum,
P. A. DeYoung, J. V. Evans, J. Finck, N. Frank, A. Grant, J. Hinnefeld, G. W. Hitt,
R. H. Howes, B. Isselhardt, K. W. Kemper, J. Longacre, Y. Lu, B. Luther, S. T.
Marley, D. McCollum, E. McDonald, U. Onwuemene, P. V. Pancella, G. F. Peaslee,
W. A. Peters, M. Rajabali, J. Robertson, W. F. Rogers, S. L. Tabor, M. Thoennessen,
E. Tryggestad, R. E. Turner, P. J. VanWylen, and N. Walker. Construction of a
modular large-area neutron detector for the NSCL. Nuclear Instruments and Methods
in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated
Equipment, 543(2):517 – 527, 2005.

[68] M. D. Bird, S. J. Kenney, J. Toth, H. W. Weijers, J. C. DeKamp, M. Thoennessen, and
A. F. Zeller. System testing and installation of the NHMFL/NSCL sweeper magnet.
IEEE Transactions on Applied Superconductivity, 15(2):1252–1254, Jun 2005.

[69] M. Papa, T. Maruyama, and A. Bonasera. Constrained molecular dynamics approach

to fermionic systems. Phys. Rev. C, 64:024612, Jul 2001.

[70] M. Papa, G. Giuliani, and A. Bonasera. Constrained molecular dynamics II: An N-
body approach to nuclear systems. Journal of Computational Physics, 208(2):403 –
415, 2005.

[71] A. Ono and H. Horiuchi. Antisymmetrized molecular dynamics for heavy ion collisions.

Progress in Particle and Nuclear Physics, 53(2):501 – 581, 2004.

[72] Z. Kohley. Transverse Collective Flow and Emission Order of Mid-Rapidity Fragments

in Fermi Energy Heavy Ion Collisions. PhD thesis, Texas A&M University, 2010.

[73] R. J. Charity, M. A. McMahan, G. J. Wozniak, R. J. McDonald, L. G. Moretto, D. G.
Sarantites, L. G. Sobotka, G. Guarino, A. Pantaleo, L. Fiore, A. Gobbi, and K. D.
Hildenbrand. Systematics of complex fragment emission in niobium-induced reactions.
Nuclear Physics A, 483(2):371 – 405, 1988.

[74] R. J. Charity. In Joint ICTP-AIEA Advanced Workshop on Model Codes for Spallation

Reactions, Report INDC(NDS)-0530, IAEA, Vienna, February 2008.

[75] M. Mocko, M. B. Tsang, D. Lacroix, A. Ono, P. Danielewicz, W. G. Lynch, and R. J.
Charity. Transport model simulations of projectile fragmentation reactions at 140
MeV/nucleon. Phys. Rev. C, 78:024612, Aug 2008.

[76] G. Cardella, G. Giuliani, I. Lombardo, M. Papa, L. Acosta, C. Agodi, F. Amorini,
A. Anzalone, L. Auditore, I. Berceanu, S. Cavallaro, M. B. Chatterjee, E. De Filippo,
E. Geraci, L. Grassi, J. Han, E. La Guidara, D. Loria, G. Lanzalone, C. Maiolino,

175

T. Minniti, A. Pagano, S. Pirrone, G. Politi, F. Porto, F. Rizzo, P. Russotto, S. Santoro,
A. Trifir`o, M. Trimarchi, G. Verde, and M. Vigilante. Effects of neutron richness on
the behavior of nuclear systems at intermediate energies. Phys. Rev. C, 85:064609, Jun
2012.

[77] M. Papa, I. Berceanu, L. Acosta, F. Amorini, C. Agodi, A. Anzalone, L. Auditore,
G. Cardella, S. Cavallaro, M. B. Chatterjee, E. De Filippo, L. Francalanza, E. Geraci,
L. Grassi, B. Gnoffo, J. Han, E. La Guidara, G. Lanzalone, I. Lombardo, C. Maiolino,
T. Minniti, A. Pagano, E. V. Pagano, S. Pirrone, G. Politi, F. Porto, L. Quattrocchi,
F. Rizzo, E. Rosato, P. Russotto, A. Trifir`o, M. Trimarchi, G. Verde, and M. Vigilante.
Dipolar degrees of freedom and isospin equilibration processes in heavy ion collisions.
Phys. Rev. C, 91:041601, Apr 2015.

[78] C. Y. Qiao, H. L. Wei, C. W. Ma, Y. L. Zhang, and S. S. Wang. Isobaric yield ratio
difference between the 140A MeV 58Ni +9 Be and 64Ni + 9Be reactions studied by the
antisymmetric molecular dynamics model. Phys. Rev. C, 92:014612, Jul 2015.

[79] R. Wada, T. Keutgen, K. Hagel, Y. G. Ma, J. Wang, M. Murray, L. Qin, P. Smith,
J. B. Natowitz, R. Alfarro, J. Cibor, M. Cinausero, Y. El Masri, D. Fabris, E. Fioretto,
A. Keksis, S. Kowalski, M. Lunardon, A. Makeev, N. Marie, E. Martin, Z. Majka,
A. Martinez-Davalos, A. Menchaca-Rocha, G. Nebbia, G. Prete, V. Rizzi, A. Ruangma,
D. V. Shetty, G. Souliotis, P. Staszel, M. Veselsky, G. Viesti, E. M. Winchester, S. J.
Yennello, W. Zipper, and A. Ono. Reaction dynamics and multifragmentation in fermi
energy heavy ion reactions. Phys. Rev. C, 69:044610, Apr 2004.

[80] G. Christian. Spectroscopy of Neutron-Unbound Fluorine. PhD thesis, Michigan State

University, 2011.

[81] G. Christian, N. Frank, S. Ash, T. Baumann, P. A. DeYoung, J. E. Finck, A. Gade,
G. F. Grinyer, B. Luther, M. Mosby, S. Mosby, J. K. Smith, J. Snyder, A. Spyrou, M. J.
Strongman, M. Thoennessen, M. Warren, D. Weisshaar, and A. Wersal. Spectroscopy
of neutron-unbound 27,28F. Phys. Rev. C, 85:034327, Mar 2012.

[82] D. Weisshaar, A. Gade, T. Glasmacher, G. F. Grinyer, D. Bazin, P. Adrich, T. Baugher,
J. M. Cook, C. Aa. Diget, S. McDaniel, A. Ratkiewicz, K. P. Siwek, and K. A.
Walsh. CAESAR- A high-efficiency CsI(Na) scintillator array for in-beam γray spec-
troscopy with fast rare-isotope beams. Nuclear Instruments and Methods in Physics
Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment,
624(3):615 – 623, 2010.

[83] A. Boudard, J. Cugnon, J.-C. David, S. Leray, and D. Mancusi. New potentialities
of the Li`ege intranuclear cascade model for reactions induced by nucleons and light
charged particles. Phys. Rev. C, 87:014606, Jan 2013.

176

[84] S. Leray, D. Mancusi, P. Kaitaniemi, J.-C. David, A. Boudard, B. Braunn, and
J. Cugnon. Extension of the Lige Intra Nuclear Cascade model to light ion-induced
collisions for medical and space applications. Journal of Physics: Conference Series,
420(1):012065, 2013.

[85] D. Mancusi, A. Boudard, J. Cugnon, J.-C. David, P. Kaitaniemi, and S. Leray. Ex-
tension of the Li`ege intranuclear-cascade model to reactions induced by light nuclei.
Phys. Rev. C, 90:054602, Nov 2014.

[86] A. Boudard, J. Cugnon, S. Leray, and C. Volant. Intranuclear cascade model for a
comprehensive description of spallation reaction data. Phys. Rev. C, 66:044615, Oct
2002.

[87] J.-C. David, D. Filges, F. Gallmeier, M. Khandaker, A. Konobeyev, S. Leray, G. Mank,
A. Mengoni, R. Michel, N. Otuka, and Y. Yariv. Benchmark of spallation models.
Progress in NUCLEAR SCIENCE and TECHNOLOGY, 2:942–947, Oct 2011.

[88] J.-C. David. Spallation reactions: A successful interplay between modeling and appli-

cations. The European Physical Journal A, 51(6):68, Jun 2015.

[89] A. Keli´c, M. V. Ricciardi, and K.-H. Schmidt. In Joint ICTP-AIEA Advanced Work-
shop on Model Codes for Spallation Reactions, Report INDC(NDS)-0530, IAEA, Vi-
enna, February 2008.

[90] I. Tanihata. Radioactive beam facilities and their physics program. Nuclear Physics

A, 553:361 – 372, 1993.

[91] F. Marti, P. Miller, D. Poe, M. Steiner, J. Stetson, and X. Y. Wu. Commissioning of
the Coupled Cyclotron system at NSCL. AIP Conference Proceedings, 600(1):64–68,
2001.

[92] D. J. Morrissey, B. M. Sherrill, M. Steiner, A. Stolz, and I. Wiedenhoever. Commis-
sioning the A1900 projectile fragment separator. Nuclear Instruments and Methods in
Physics Research Section B: Beam Interactions with Materials and Atoms, 204:90 –
96, 2003. 14th International Conference on Electromagnetic Isotope Separators and
Techniques Related to their Applications.

[93] A. Stolz, T. Baumann, T. N. Ginter, D. J. Morrissey, M. Portillo, B. M. Sherrill,
M. Steiner, and J. W. Stetson. Production of rare isotope beams with the NSCL
fragment separator. Nuclear Instruments and Methods in Physics Research Section B:
Beam Interactions with Materials and Atoms, 241(1):858 – 861, 2005. The Application
of Accelerators in Research and Industry.

[94] K.-H. Schmidt, E. Hanelt, H. Geissel, G. Mnzenberg, and J. P. Dufour. The
momentum-loss achromat - A new method for the isotopical separation of relativis-

177

tic heavy ions. Nuclear Instruments and Methods in Physics Research Section A:
Accelerators, Spectrometers, Detectors and Associated Equipment, 260(2):287 – 303,
1987.

[95] N. Frank. Spectroscopy of Neutron Unbound States in Neutron Rich Oxygen Isotopes.

PhD thesis, Michigan State University, 2006.

[96] C. Freigang. Building the Sweeper magnet focal plane detector. Master’s thesis,

Michigan State University, 2001.

[97] J. K. Smith. Unbound States in the Lightest Island of Inversion: Neutron Decay
Measurements of 11Li, 10Li, and 12Be. PhD thesis, Michigan State University, 2014.

[98] K. Meierbachtol. Parallel and Perpendicular Momentum Distributions in Projectile

Fragmentation Reactions. PhD thesis, Michigan State University, 2012.

[99] T. Baumann, A. Spyrou, and M. Thoennessen. Nuclear structure experiments along

the neutron drip line. Reports on Progress in Physics, 75(3):036301, 2012.

[100] W. Peters. Study of Neutron Unbound States Using the Modular Neutron Array

(MoNA). PhD thesis, Michigan State University, 2007.

[101] M. D. Jones. Spectroscopy of Neutron Unbound States in 24O and 23N. PhD thesis,

Michigan State University, 2015.

[102] J. Honke. Sweeper Magnet Focal Plane Det.: CRDC Mask & Drive. National Super-

conducting Cyclotron Laboratory Drawing Number 7-SWF-4-6-D, Rev. A, 6 2001.

[103] J. M. Paul. The density effect and rate of energy loss in common plastic scintillators.

Nuclear Instruments and Methods, 96(1):51 – 59, 1971.

[104] A. Stolz, T. Baumann, N. H. Frank, T. N. Ginter, G. W. Hitt, E. Kwan, M. Mocko,
W. Peters, A. Schiller, C. S. Sumithrarachchi, and M. Thoennessen. First observation
of 60Ge and 64Se. Physics Letters B, 627(1):32 – 37, 2005.

[105] G. F. Knoll. Radiation Detection and Measurement. John Wiley & Sons, Inc., New

York, NY, 3rd edition, 2000.

[106] D. Bazin, J. A. Caggiano, B. M. Sherrill, J. Yurkon, and A. Zeller. The S800 spectro-
graph. Nuclear Instruments and Methods in Physics Research Section B: Beam Inter-
actions with Materials and Atoms, 204:629 – 633, 2003. 14th International Conference
on Electromagnetic Isotope Separators and Techniques Related to their Applications.

[107] C. Hoffman. Investigation of the Neutron-Rich Oxygen Isotopes at the Drop Line. PhD

thesis, Florida State University, 2009.

178

[108] M. Strongman. Neutron spectroscopy of 22N and the disappearance of the N=14 shell.

Master’s thesis, Michigan State University, 2011.

[109] S. Mosby. Spectroscopy of Neutron Unbound States in Neutron Rich Carbon. PhD

thesis, Michigan State University, 2012.

[110] M. Mosby. Measurement of Excitation Energy of Neutron-Rich Precursor Fragments.

PhD thesis, Michigan State University, 2013.

[111] J. Snyder. Spectroscopy of 15Be. PhD thesis, Michigan State University, 2014.

[112] S. Agostinelli, J. Allison, K. Amako, J. Apostolakis, H. Araujo, P. Arce, M. Asai,
D. Axen, S. Banerjee, G. Barrand, F. Behner, L. Bellagamba, J. Boudreau, L. Broglia,
A. Brunengo, H. Burkhardt, S. Chauvie, J. Chuma, R. Chytracek, G. Cooperman,
G. Cosmo, P. Degtyarenko, A. Dell’Acqua, G. Depaola, D. Dietrich, R. Enami, A. Fe-
liciello, C. Ferguson, H. Fesefeldt, G. Folger, F. Foppiano, A. Forti, S. Garelli, S. Giani,
R. Giannitrapani, D. Gibin, J. J. Gmez Cadenas, I. Gonzlez, G. Gracia Abril, G. Gree-
niaus, W. Greiner, V. Grichine, A. Grossheim, S. Guatelli, P. Gumplinger, R. Hamatsu,
K. Hashimoto, H. Hasui, A. Heikkinen, A. Howard, V. Ivanchenko, A. Johnson, F.W.
Jones, J. Kallenbach, N. Kanaya, M. Kawabata, Y. Kawabata, M. Kawaguti, S. Kelner,
P. Kent, A. Kimura, T. Kodama, R. Kokoulin, M. Kossov, H. Kurashige, E. Lamanna,
T. Lampn, V. Lara, V. Lefebure, F. Lei, M. Liendl, W. Lockman, F. Longo, S. Magni,
M. Maire, E. Medernach, K. Minamimoto, P. Mora de Freitas, Y. Morita, K. Murakami,
M. Nagamatu, R. Nartallo, P. Nieminen, T. Nishimura, K. Ohtsubo, M. Okamura,
S. O’Neale, Y. Oohata, K. Paech, J. Perl, A. Pfeiffer, M. G. Pia, F. Ranjard, A. Rybin,
S. Sadilov, E. Di Salvo, G. Santin, T. Sasaki, N. Savvas, Y. Sawada, S. Scherer, S. Sei,
V. Sirotenko, D. Smith, N. Starkov, H. Stoecker, J. Sulkimo, M. Takahata, S. Tanaka,
E. Tcherniaev, E. Safai Tehrani, M. Tropeano, P. Truscott, H. Uno, L. Urban, P. Ur-
ban, M. Verderi, A. Walkden, W. Wander, H. Weber, J. P. Wellisch, T. Wenaus, D. C.
Williams, D. Wright, T. Yamada, H. Yoshida, and D. Zschiesche. Geant4 a simulation
toolkit. Nuclear Instruments and Methods in Physics Research Section A: Accelerators,
Spectrometers, Detectors and Associated Equipment, 506(3):250 – 303, 2003.

[113] J. Allison, K. Amako, J. Apostolakis, H. Araujo, P. Arce Dubois, M. Asai, G. Barrand,
R. Capra, S. Chauvie, R. Chytracek, G. A. P. Cirrone, G. Cooperman, G. Cosmo,
G. Cuttone, G. G. Daquino, M. Donszelmann, M. Dressel, G. Folger, F. Foppiano,
J. Generowicz, V. Grichine, S. Guatelli, P. Gumplinger, A. Heikkinen, I. Hrivna-
cova, A. Howard, S. Incerti, V. Ivanchenko, T. Johnson, F. Jones, T. Koi, R. Kok-
oulin, M. Kossov, H. Kurashige, V. Lara, S. Larsson, F. Lei, O. Link, F. Longo,
M. Maire, A. Mantero, B. Mascialino, I. McLaren, P. Mendez Lorenzo, K. Minami-
moto, K. Murakami, P. Nieminen, L. Pandola, S. Parlati, L. Peralta, J. Perl, A. Pfeif-
fer, M. G. Pia, A. Ribon, P. Rodrigues, G. Russo, S. Sadilov, G. Santin, T. Sasaki,
D. Smith, N. Starkov, S. Tanaka, E. Tcherniaev, B. Tome, A. Trindade, P. Truscott,
L. Urban, M. Verderi, A. Walkden, J. P. Wellisch, D. C. Williams, D. Wright, and

179

H. Yoshida. Geant4 developments and applications. IEEE Transactions on Nuclear
Science, 53(1):270–278, Feb 2006.

[114] B. Roeder. In-beam validation of the recoil separator design concept for EURISOL.

EURISOL Design Study, Oct 2008. Accessed: 2018-05-30.

[115] Z. Kohley, E. Lunderberg, P.A. DeYoung, B. T. Roeder, T. Baumann, G. Christian,
S. Mosby, J. K. Smith, J. Snyder, A. Spyrou, and M. Thoennessen. Modeling in-
teractions of intermediate-energy neutrons in a plastic scintillator array with Geant4.
Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spec-
trometers, Detectors and Associated Equipment, 682:59 – 65, 2012.

[116] H. Anton, I. Bivens, and S. Davis. Calculus: Early Transcendentals. John Wiley &

Sons, Inc., Hoboken, NJ, 8th edition, 2005.

180