PROTON-PROTON CORRELATION FUNCTIONS AS A PROBE TO REACTION
DYNAMICS
By
Micha A. Kilburn

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

ABSTRACT
PROTON-PROTON CORRELATION FUNCTIONS AS A PROBE TO
REACTION DYNAMICS
By
Micha A. Kilburn
In an experiment at NSCL, proton-proton (p-p) correlation functions were measured
in 40 Ca+40 Ca and 48 Ca+48 Ca reactions, both at E/A = 80 MeV. The High Resolution
Array (HiRA) detected light particles with excellent energy (≤200 keV) and angular (≈0.2◦ )
resolution. The MSU 4π Array covered 77% of the total 4π solid angle and was used to
determine the impact parameter for collisions using transverse energy (Et ) as the relevant
observable.
Two-particle correlation functions are employed in this work to measure the space-time
extent of the source. A transport model previously predicted that the p-p correlation functions would be sensitive to the density dependence of the symmetry energy, while other work
had already shown the p-p correlation functions to be sensitive to nucleon-nucleon (NN )
in-medium cross sections.
More detailed calculations performed in this dissertation indicate that that sensitivity to
the symmetry energy is subtle. Much less subtle is the dependence of the p-p correlation
functions on the laboratory angle of the total momentum vector of the two protons. At
forward angles, where the correlation function is sensitive to the projectile spectator, the
measured correlation functions appear consistent with sources that are very extended in
space-time. The space-time extent of these sources exceed the predictions of BUU transport calculations, which are the main tool for probing the symmetry energy via correlation
functions. At backward angles, where it is sensitive to the expanding participant source,

the observed sources are more compact; there the trends can be reproduced by the BUU
calculations.
At the most forward angles, we note that the qualitative trends of the correlation function
with angle and energy run counter to the qualitative trends of smaller sources for particles
with higher momentum typically reported by published work in this incident energy domain. While we observe this latter trend at backward angles, the momentum dependence
in the source size observed at forward angles is comparatively weak and trends in the opposite direction, with the most energetic protons displaying the weakest correlation functions.
These energetic protons are closer to the expected velocity for projectile spectator remnants,
suggesting their origins in the decay of these remnants.
Further analysis of the correlation functions with gates on rapidity and transverse momentum allowed a clean exclusion of projectile decay. After excluding this kinematic domain,
it was possible to obtain data that can be compared to a BUU transport model. This model,
however, predicted a weak sensitivity to the density dependence of the symmetry energy that
is too small to be experimentally probed. Consistent with prior work, we find a strong sensitivity to the NN in-medium cross section reduction as well as a strong previously unobserved
sensitivity to the production of light clusters. Comparisons between the BUU calculations
establish the sensitivity of data to these transport quantities as well as the range of values
for these transport quantities that may be consistent with the present measurements.

To the women behind the scenes: Debbie, Kim and Shari

iv

ACKNOWLEDGMENT

This dissertation and the work within could not have been completed without the help
of many advisers, colleagues and friends. To not acknowledge them at the beginning would
be a crime on par with excluding the bibliography at the end, for without them there would
be nothing to read in between.
I would like to thank my adviser, Bill Lynch, for his guidance, knowledgeable mind,
and faith in me. When we had a similar vision, the course was clear. When our views
differed, the result was clarity in my own thoughts and conceptions. In addition, I thank
Pawel Danielewicz for advising me as a theorist for two years. He was understanding when
I switched to an experimental group, and continued to provide guidance as a committee
member. I also thank the rest of the members on my guidance committee: Betty Tsang,
for her “get it done” attitude and useful suggestions; Piotr Piecuch for providing another
theoretical eye; and Norman Birge, for advising me as a TA and asking probing questions.
The analysis for this thesis work was a joint effort between myself and two postdocs:
Vlad and Daniela. Without their efforts, the experiment would not have ran, and it certainly
would have taken me longer to graduate. Without their friendship, it would have felt like an
eternity. I will never forget Vlad’s persistence and Daniela’s patience during the late nights
working with them before, throughout, and after the experiment.
The analysis would have taken much longer to complete if Zibi had not joined our group
as a postdoc, after the Henzl’s departure, and helped with the finishing touches. His work
on extracting source sizes from the correlations was pivotal and his advice was most useful.
He was diligent and thoughtful while we worked on a paper together, and he is also a mean
v

poker player.
I must thank the rest of the HiRA family, the other siblings, uncles, and cousins in
addition to the parents and older siblings mentioned above. The cooperation of the entire
family is necessary in any successful HiRA experiment. I am grateful for my big brothers:
Mark and Andy. I thank Mark for being my mentor, getting me on the Art’s softball team,
and convincing me to make the switch from theory to join HiRA. I thank Andy for being
my officemate and patiently answering my numerous questions about HiRA, ROOT and
C++ in general. I want to thank my little brothers, Dan and Mike for keeping it real
and fun, my other siblings Ali and Jenny for all their help, and cousin Clemens for all
of his hard work and attention to meticulous details. A special thanks to Dan for all of
those morning conversations and his helpful descriptions. I am immensely appreciative to
the stepson, Brent, for his friendship and help with BUU. Hats off to my uncles: Sergei for
rolling the calcium target and teaching me words in Russian, Bob who taught me more about
programming in a couple of hours than I had learned in months, and Lee for reminding me
why to use “Do not use” tags. Uncle Mike was also pivotal in the experimental setup. His
work ethic and humor are to be admired. Most thanks to my godfather, Giuseppe, for his
questions and ideas pertaining to BUU, correlation functions and the imaging technique.
I also want to give a shout out to members of the 4π family: Skip Vander Molen for
his extensive help with the experimental setup and his masterful control of Freddie; Gary
Westfall for answering all of my questions during analysis about what they did many moons
ago; and Dan Brown for his cheerful outlook and friendship during the experimental setup.
Although only vaguely related to the family, Scott Pratt has also been very patient and
helpful, providing advice and codes to help with the analysis.
vi

I am indebted to various staff around the lab. I thank Renan Fontus for his extensive
help with the laser system analysis, and always being friendly. I thank Len Morris for all of
his patience and work designing parts for the experiment. I thank Dave Sanderson and Andy
Thulin for teaching me everything I wanted to know about vacuum systems, and more. I
thank John Yurkon for his extensive knowledge, and help with targets and flanges. I thank
Hang Lui for his help getting running on the HPCC, where most of the simulations were
performed. My most sincere thanks go to Shari Conroy. She not only helped me with travel
arrangements, and kept me caffeinated, but she was a friend, and hooked me up with hockey
tickets even after she retired.
Other professors at the lab also made my stay here more welcoming. Filomena made me
feel at home immediately, and always asked how things were going when we passed in the
hallways. Remco and I also chatted often, usually outside, but he has opened some doors
that I didn’t know could be opened. His friendship has also been invaluable. Thomas was
only here briefly, but I won’t forget how he rooted for me during my last subject exam. The
first time I met Michael, he stole the basketball from me during the annual REU/faculty
game. Throughout my time here, he’s helped keep me on track by providing a voice of reason
when it was needed most. His humor hasn’t hurt either.
Throughout my years at the lab, I became more and more interested in explaining nuclear
science to the public. After all, it’s their money funding us, and they should know what they
are paying for. For that I thank Geoff Koch for exposing me to the publicity side of science,
and getting me involved in numerous projects which gave me a voice. Perhaps one of my
greatest mentors during grad school came from the most unlikely of places: the outreach
coordinator, Zach Constan. He arrived at the lab two years after me, shortly after I had
vii

began giving tours of the lab. He completely revamped the tour program and added many
other outreach programs. His enthusiasm was contagious and I somehow caught the outreach
bug. His ideas are infectious and his advice has been curing. I am glad to be following in
his footsteps.
During my time here, I wasn’t just a part of the NSCL, I was also a part of the Physics
department. I will look back on my time with the recruiting committee with fondness, and
thank them all for treating me as an equal colleague. I also thank Wolfgang for listening
when we asked him to, and letting me take over the grad section of the website. My most
sincere thanks go to Kim and Debbie, for providing answers, advice, and their friendship.
I wouldn’t have made it through graduate school without the camaraderie of the original
posse: Dan, Jon, Josh V, and Josh S. Those late night homework sessions would have been
all-nighters without their help and those first few years would have been disastrous without
their companionship. There are many more friends and family I should thank for their
support over the last seven years. They are too many to name, but they are not forgotten.
The most pivotal years for me were the first and the last. While the posse got me through
the first, Samantha got me through the last. I thank her for all of her support and motivation
in getting through the final push. She was the light at the end of the tunnel.

viii

PREFACE

In nuclear science, we build large detectors to measure tiny particles with the goal of
understanding giant astronomical objects. To “understand” we must solve a puzzle that has
had us scratching our heads since the beginning of humanity: Where did we, and everything
around us, come from? Carl Sagan wrote “We are made of star stuff,” but a puzzle remains:
How did the stars make all of the elements? People in nuclear science, particle physics, and
astrophysics each work on a different piece of the puzzle, and over time, the solution to it
is starting to take shape. In this way, nuclear scientists are like detectives, piecing together
bits of information to unravel the mystery of existence.
The tiny particles that we study are atomic nuclei comprised of nucleons (protons and
neutrons) at the center of the atom. These nuclei have sizes that are smaller than that
of atomic physics and yet larger than that of high energy physics studies. Because nuclei
are too small to be seen with the most powerful microscope, we must develop not only
the macroscopic detectors needed to study these submicroscopic objects, but we must also
be creative in how we “measure” properties of nuclei. We can’t put a single nucleus on a
bathroom scale, but we can deduce its mass. We can’t look at a single nucleus, but we
can determine its shape. There is not a physical ruler small enough to measure the size
of a nucleus, however, we can not only determine its matter radius, but also the radius of
the charge distribution (due largely to protons). For nuclei with many more neutrons than
protons, nuclear scientists can also detect a neutron skin or a halo neutron around the outer
edges of the nucleus if they exist. In this way, nuclear scientists are like biologists, dissecting
the details of how nuclei are built and how their internal dynamics work.
ix

The properties mentioned thus far only deal with the structure of the nucleus. However,
there are questions concerning what can take place during nuclear collisions in general, and
complex central collisions in particular. What fragments are created after a central collision?
How many neutrons will pass from a neutron-rich nucleus to a neutron-poor nucleus during
a collision? If we regard nuclei as droplets of nuclear matter, what is the nuclear equation of
state? Complicating this task is the fact that just as you can’t measure size with a ruler, you
can’t put a probe in the middle of the reaction. Instead, we detect nuclei produced from a
nuclear collision far away from the actual collision, and long after the collision is over. From
these measurements, we then infer what happened in the past. In this way, nuclear scientists
are like archaeologists, reconstructing the collision dynamics from the events that occurred
in the past.
In both nuclear structure and nuclear reaction studies, the data often is not enough on
its own to give a complete representation: It must be compared to a theoretical model. In
nuclear structure, one might compare data to predictions from shell model theory, while
in the case of reactions, one might compare data to transport models that simulate nuclear
collisions. The knowledge gained from both structure and reactions can be used as input into
astrophysical models that simulate processes such as stellar nucleosynthesis or supernovae
explosions, resulting in predictions for elemental abundances. In this way, nuclear scientists
are like architects, designing a construct from abstract ideas.
Nuclear science is all of this and more. While this dissertation focuses on only one small
piece of the puzzle, it’s an important piece necessary for tying together what we learn from
nuclear structure, nuclear reactions, and nuclear astrophysics. As you read on, remember
that this isn’t the beginning or the end, but merely a step in the right direction.

x

TABLE OF CONTENTS

List of Tables

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xv

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Introduction to p-p Correlation Functions . . . . . . . . . . . . . .
1.3 Introduction to the BUU Transport Model . . . . . . . . . . . . . .
1.3.1 Defining the Symmetry Energy Parametrization . . . . . . .
1.3.2 Defining the Nucleon-Nucleon Cross Section Parametrization
1.4 Organization of Thesis . . . . . . . . . . . . . . . . . . . . . . . . .

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

1
2
5
12
14
17
18

2 Experimental Setup . . . . . . . .
2.1 Reaction Systems . . . . . . . . .
2.2 Experimental Layout . . . . . . .
2.3 The High Resolution Array, HiRA
2.4 4π Detector Array . . . . . . . .
2.5 Target Details . . . . . . . . . . .
2.6 Electronics . . . . . . . . . . . . .
2.6.1 HiRA . . . . . . . . . . .
2.6.2 4π Array . . . . . . . . . .
2.6.3 Trigger Logic . . . . . . .

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

19
20
21
24
29
32
35
35
38
41

3 Data Analysis . . . . . . . . . . . . . . . . . .
3.1 HiRA . . . . . . . . . . . . . . . . . . . . .
3.1.1 Silicon Calibration . . . . . . . . . .
3.1.1.1 Readout Order Corrections
3.1.2 CsI Crystal Calibration . . . . . . . .
3.1.2.1 Linearization . . . . . . . .
3.1.2.2 Particle Identification (PID)
3.1.2.3 Offset Extraction . . . . . .
3.1.2.4 Gain Extraction . . . . . .
3.1.2.5 Calibrating A>1 Particles .
3.2 4π Array . . . . . . . . . . . . . . . . . . . .

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

43
44
44
46
49
50
52
53
57
65
66

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

xi

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

3.2.1 Phoswich Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Impact Parameter Selection . . . . . . . . . . . . . . . . . . . . . . .
Quantification of Calcium Target Oxidation . . . . . . . . . . . . . . . . . .

66
73
77

4 Exploration of BUU Physics Inputs . . . . . . . . . . . . . . . . . . . . . .
4.1 Description of Parameter Space for Inputs . . . . . . . . . . . . . . . . . . .
4.2 Size of Computational Region and Reaction Duration . . . . . . . . . . . . .
4.3 Influence of Number of Test Particles . . . . . . . . . . . . . . . . . . . . . .
4.4 Sensitivity of Observables to Impact Parameter . . . . . . . . . . . . . . . .
4.5 Sensitivity of Observables to Cluster Production and Momentum Dependence
4.6 Effects of Symmetry Energy . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.7 Comparing Free and In-Medium Cross Sections . . . . . . . . . . . . . . . .
4.7.1 Exploring the Density Dependent In-Medium Cross Section . . . . . .
4.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82
82
84
97
102
112
119
129
138
141

5 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1 Imaging and Gaussian Fits . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.1 Theory Adaptation . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 p-p Correlation Functions . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Correlation Functions Selected by Laboratory Angle and Momentum .
5.3.1 Comparison to BUU Transport Theory . . . . . . . . . . . . . .
5.4 Correlation Functions Selected by Rapidity and Transverse Momentum
5.4.1 Comparison to BUU Transport Theory . . . . . . . . . . . . . .
5.5 Three Particle Correlations . . . . . . . . . . . . . . . . . . . . . . . . .

.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.

144
145
148
150
152
162
165
172
179

6 Summary . . . . . . . . . . . . . . . . . . . . . . . . .
6.1 Conclusions from BUU Transport Simulations . . .
6.2 Summary of Experimental Results . . . . . . . . . .
6.2.1 Dependencies in the Laboratory Frame . . .
6.2.2 Dependencies on Transverse Momentum and
6.3 Outlook . . . . . . . . . . . . . . . . . . . . . . . .

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

185
186
187
187
188
189

3.3

Appendix A Laser Measurements
Appendix B ROOT Analysis

. . . . .
. . . . .
. . . . .
. . . . .
Rapidity
. . . . .

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

. . . . . . . . . . . . . . . . . . . . . . . . . 192

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

Appendix C Calculation of r1/2 and Error . . . . . . . . . . . . . . . . . . . . 202
References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

xii

LIST OF TABLES

2.1

Biases applied to E and DE detectors and corresponding sample leakage currents. 26

2.2

Mean polar angles of the ball phoswiches. . . . . . . . . . . . . . . . . . . . .

30

2.3

Thicknesses and composition of the phoswiches in the ball and FA. Rise and
fall times are also given, although the signal from the photomultiplier tube
for the thin scintillator will have much long rise and fall times. . . . . . . . .

30

2.4

Mean angles of the FA phoswiches. . . . . . . . . . . . . . . . . . . . . . . .

32

2.5

List of targets used with their respective thickness. . . . . . . . . . . . . . .

32

3.1

Energy punch-ins for ball phoswiches [1]. . . . . . . . . . . . . . . . . . . . .

67

3.2

Energy punch-ins for Forward Array phoswiches. . . . . . . . . . . . . . . . .

71

4.1

Results of particle conservation for central 40 Ca+40 Ca reactions. BG = Big
Grid without clusters, NG = Normal Grid without clusters, BGC = Big Grid
with Clusters, and NGC = Normal Grid with Clusters. . . . . . . . . . . . .

90

r1/2 values for each combination of grid size and cluster production for both
reaction systems and two transverse momentum cuts. BG = Big Grid without
clusters, NG = Normal Grid without clusters, BGC = Big Grid with Clusters,
and NGC = Normal Grid with Clusters. . . . . . . . . . . . . . . . . . . . .

96

4.2

4.3

r1/2 values for each value of test particle number for low and high pT and
both reaction systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

4.4

r1/2 values for Dep-On,Dep-Off,Indep-On, and Indep-Off for both reaction
systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
xiii

4.5

r1/2 values for 3 choices of cross section for low and high total momentum
cuts and both reaction systems. . . . . . . . . . . . . . . . . . . . . . . . . . 138

5.1

The normalization of the reconstructed source distribution according to Eq. 5.2
and the χ2 /ndf of the reconstructed correlation function obtained from imaging method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

5.2

Normalization and fit results to the experimental correlation functions assuming Gaussian source distribution. . . . . . . . . . . . . . . . . . . . . . . . . 159

5.3

Given are the fractions of fast emission protons from both the imaging technique and from a Gaussian fit routine. . . . . . . . . . . . . . . . . . . . . . 161

5.4

Comparison of systems, angular and momentum dependence of r1/2 for imaging, the Gaussian fitting procedure, and BUU. . . . . . . . . . . . . . . . . . 162

5.5

Fit results to the experimental correlation functions assuming Gaussian source
distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

5.6

Comparison of system size, angular and momentum dependence of r1/2 obtained from reconstructed source distribution with imaging method, Gaussian
fitting procedure and BUU transport model simulations. . . . . . . . . . . . 171

xiv

LIST OF FIGURES

1.1

1.2

1.3

Before and after cartoon of a heavy-ion collision. The black arrows represent
momentum vectors for two particles which are strongly correlated. For interpretation of the references to color in this and all other figures, the reader is
referred to the electronic version of this dissertation. . . . . . . . . . . . . . .

7

Same cartoon as above, however, here the black arrows represent momentum
vectors for two particles which are weakly correlated. . . . . . . . . . . . . .

7

A p-p correlation function constructed from a Gaussian source with r1/2 = 6
fm. Important features are labeled. . . . . . . . . . . . . . . . . . . . . . . .

9

1.4

FWHM of a p-p correlation as a function of r1/2 of a Gaussian source function. 11

1.5

Different density dependencies of the symmetry energy. . . . . . . . . . . . .

16

2.1

Layout of the CCF, from above, in early fall of 2006. . . . . . . . . . . . . .

20

2.2

Drawing of the HiRA Array mounted on a modified 4π flange. . . . . . . . .

22

2.3

Photograph of the HiRA Array with special lighting. . . . . . . . . . . . . .

23

2.4

Photograph of the HiRA Array inside the 4π Array. . . . . . . . . . . . . . .

23

2.5

Drawing of HiRA telescope components [2]. . . . . . . . . . . . . . . . . . .

25

2.6

The HiRA array, labeled with telescope numbers, as viewed from the target.
In this diagram the beam travels from left to right so that telescopes 4 and
17 are the most forward. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

2.7

Angular coverage of the entire HiRA array. . . . . . . . . . . . . . . . . . . .

27

2.8

Actual angular coverage of the HiRA array. . . . . . . . . . . . . . . . . . . .

28

xv

2.9

Schematic diagram of the forward array. Also shown is the lab reference for
the azimuthal angle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

2.10 Photograph of target carousel which housed all targets used. Pictured, clockwise, are an old scintillator target at 5 o’clock, an old broken gold target at 8
o’clock, and a new empty oblong frame at 11 o’clock. . . . . . . . . . . . . .

34

2.11 Photograph of chipboard with quarter for size comparison. . . . . . . . . . .

36

2.12 Photograph of HiRA motherboard with six chipboards installed. . . . . . . .

36

2.13 Diagram of the ASIC electronics logic. [2, 3] . . . . . . . . . . . . . . . . . .

37

2.14 Diagram of the HiRA electronics logic. [2, 4] . . . . . . . . . . . . . . . . . .

38

2.15 Idealized signal from a phoswich detector. [5] . . . . . . . . . . . . . . . . . .

39

2.16 Diagram of the 4π electronics logic, part (a). . . . . . . . . . . . . . . . . . .

40

2.17 Diagram of the 4π electronics logic, part(b). . . . . . . . . . . . . . . . . . .

40

2.18 Diagram of the coincidence trigger logic. [6] . . . . . . . . . . . . . . . . . .

42

3.1

Decay chain for 228 Th. Energies of prominent α particles are given in red. .

45

3.2

Example of a raw α spectra for a single strip. . . . . . . . . . . . . . . . . .

45

3.3

Example of a calibrated α spectra for all EF strips in one detector. . . . . .

46

3.4

Pulser ramp for Telescope 0, crystal 0. The red triangles are the peaks as
found by ROOT [7]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

50

A 3rd order polynomial was used to fit the pulser data due to slight nonlinearities in the electronics. Shown here are the pulser data and fit for one
crystal in telescope 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

The difference between the input voltage and fit voltage is plotted as a function
of channel number. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

Calibrated PID for all 40 Ca+40 Ca data. . . . . . . . . . . . . . . . . . . . .

54

3.5

3.6

3.7

xvi

3.8

Calibrated PID for all 48 Ca+48 Ca data. . . . . . . . . . . . . . . . . . . . .

55

3.9

Calibrated EF energy vs raw CsI energy. . . . . . . . . . . . . . . . . . . . .

56

3.10 PID for 40 Ca+CH2 at E/A = 25 MeV. . . . . . . . . . . . . . . . . . . . . .

58

3.11 The proton energies plotted as a function of θ in the laboratory frame. . . .

59

3.12 Energy as a function of θ (laboratory) for elastically scattered protons. . . .

60

3.13 Cross sections for the 3- state of 40 Ca as a function of θ in the center of mass
system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

3.14 Cross sections for the 2+ state of 40 Ca as a function of θ in the center of mass
system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

3.15 Energy as a function of laboratory angle for inelastically scattered protons
from the 3- excited state of 40 Ca. . . . . . . . . . . . . . . . . . . . . . . . .

62

3.16 Energy as a function of laboratory angle for inelastically scattered protons
from the 2+ excited state of 40 Ca. . . . . . . . . . . . . . . . . . . . . . . .

62

3.17 Energy loss through relevant foils as a function of proton energy. . . . . . . .

63

3.18 Projection of proton PID for selection of the punch through channel. . . . .

64

3.19 Calibration for crystal 0 of telescope 0. . . . . . . . . . . . . . . . . . . . . .

65

3.20 Raw PID plots from a ball phoswich (top) and a forward array phoswich
(bottom). Axes are in channels. . . . . . . . . . . . . . . . . . . . . . . . . .

68

3.21 Raw PID from a ball phoswich with important calibration features labeled. .

69

3.22 Unfolded (disentangled) PID from a ball phoswich (top) and a forward array
phoswich (bottom). Axes are in arbitrary units. . . . . . . . . . . . . . . . .

70

3.23 Calibrated PID from a ball phoswich (top) and a forward array phoswich
(bottom). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

72

xvii

3.24 The relationship between the total multiplicity in the 4π array and the multiplicity in the ball elements of the array is shown in the upper panel. The
lower panel shows the mean total multiplicity as a function of ball multiplicity. 75
3.25 Reduced impact parameter as a function of multiplicity. . . . . . . . . . . . .

76

3.26 Reduced impact parameter as a function of transverse energy. . . . . . . . .

77

3.27 Total multiplicity as a function of transverse energy. . . . . . . . . . . . . . .

78

3.28 Oxidation of natural Ca target as a function of time. . . . . . . . . . . . . .

78

3.29 Comparison of reactions on Calcium, for early and late data, with reactions
on mylar for both Ca beams. . . . . . . . . . . . . . . . . . . . . . . . . . . .

80

3.30 Comparison of reactions on Ca, over time, with reactions on mylar for both
Ca beams. Details are given in the text. . . . . . . . . . . . . . . . . . . . .

81

XZ plane projection of central 40 Ca+40 Ca reaction at 395 fm/c without
cluster production. The masses of the two large residues were calculated. The
sum of the residue masses is displayed in Table 4.1. . . . . . . . . . . . . . .

85

XZ plane projection of central 40 Ca+40 Ca reaction at 495 fm/c without
cluster production for a larger computational grid size. The masses of the two
large residues were calculated. The sum of the residue masses is displayed in
Table 4.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

86

XZ plane projection of central 40 Ca+40 Ca reaction at 395 fm/c with cluster
production. The masses of the five largest residues were calculated. The sum
of the residue masses is displayed in Table 4.1. . . . . . . . . . . . . . . . . .

87

4.1

4.2

4.3

4.4

XZ plane projection of central 40 Ca+40 Ca reaction at 495 fm/c with cluster
production for a larger computational grid size. The masses of the four largest
residues were calculated. The sum of the residue masses is displayed in Table 4.1. 88

4.5

Change in average single particle energy as a function of time, illustrating
conservation of energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

Effect of using a time cut on source functions for 40 Ca+40 Ca without cluster
production. r1/2 sizes for the sources are given in Table 4.2. . . . . . . . . .

91

4.6

xviii

Effect of using a time cut on source functions for 40 Ca+40 Ca with cluster
production. The legend is the same as the previous figure. r1/2 sizes for the
sources are given in Table 4.2. . . . . . . . . . . . . . . . . . . . . . . . . . .

92

Effect of using a time cut on source functions for 48 Ca+48 Ca without cluster
production. r1/2 sizes for the sources are given in Table 4.2. . . . . . . . . .

93

Effect of using a time cut on source functions for 48 Ca+48 Ca with cluster
production. The legend is the same as the previous figure. r1/2 sizes for the
sources are given in Table 4.2. . . . . . . . . . . . . . . . . . . . . . . . . . .

94

4.10 Snapshots of the 40 Ca+40 Ca reaction in the XZ plane at 300 fm/c. These
are thin slices in the Y plane centered about y = 0. The top panel is for a
reaction with 200 test particles, the middle panel is for 800 test particles and
the bottom panel is for 1600 test particles. . . . . . . . . . . . . . . . . . . .

99

4.7

4.8

4.9

4.11 Snapshots of the 40 Ca+40 Ca reaction in the XZ plane at 300 fm/c. Thin
slices in the Y plane centered about y = 0. All panels are for 200 test
particles, with different random number seeds. . . . . . . . . . . . . . . . . . 100
4.12 Lab energy spectra for protons emitted from 40 Ca+40 Ca reaction for three
different values of test particle number. . . . . . . . . . . . . . . . . . . . . . 101
4.13 A comparison of the shape of source functions for different numbers of test
particles used. The upper quadrants are for low pT for 40 Ca+40 Ca. The
bottom quadrants are for high pT for the same reaction system. . . . . . . . 103
4.14 A comparison of the shape of source functions for different numbers of test
particles used. The upper quadrants are for low pT for 48 Ca+48 Ca (right).
The bottom quadrants are for high pT for the same reaction system. . . . . 104
4.15 Transverse energy as a function of impact parameter for 40 Ca+40 Ca reaction
(top) and 48 Ca+48 Ca reaction (bottom) for the screened cross section reduction.106
4.16 Transverse energy as a function of impact parameter for 40 Ca+40 Ca reaction
(top) and 48 Ca+48 Ca reaction (bottom) for the Rostock cross section reduction.107
4.17 Transverse energy as a function of impact parameter for 40 Ca+40 Ca reaction
(top) and 48 Ca+48 Ca reaction (bottom) for the free cross sections. . . . . . 108
xix

4.18 r1/2 of source functions resulting from screened cross section reductions as a
function of impact parameter. The upper quadrants are for a low transverse
momentum cut <150 MeV/c for 40 Ca+40 Ca (left) and 48 Ca+48 Ca (right).
The bottom quadrants are for a high transverse momentum cut >150 MeV/c
for the same reaction systems. . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.19 r1/2 of source functions resulting from Rostock cross section reductions as a
function of impact parameter. The upper quadrants are for a low transverse
momentum cut <150 MeV/c for 40 Ca+40 Ca (left) and 48 Ca+48 Ca (right).
The bottom quadrants are for a high transverse momentum cut >150 MeV/c
for the same reaction systems. . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.20 r1/2 of source functions resulting from free cross sections as a function of
impact parameter. The upper quadrants are for a low transverse momentum
cut <150 MeV/c for 40 Ca+40 Ca (left) and 48 Ca+48 Ca (right). The bottom
quadrants are for a high transverse momentum cut >150 MeV/c for the same
reaction systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.21 Density snapshots in the XZ plane summed over all Y. Top panels show
calculations for momentum dependent mean fields. Bottom panels are for
momentum independent mean fields. Left panels are with cluster production
on, and right panels are without cluster production. All reactions are for
40 Ca+40 Ca. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.22 Change in average single particle excitation as a function of time, illustrating
conservation of energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
4.23 Laboratory kinetic energy spectra for protons emitted between 18-58◦ for
Dep-On,Dep-Off,Indep-On, and Indep-Off for the 48 Ca+48 Ca reaction. . . . 118
4.24 Source functions for low pT (top) and high pT (bottom) gates for the 40 Ca+40 Ca
reaction system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
4.25 Source functions for low pT (top) and high pT (bottom) gates for the 48 Ca+48 Ca
reaction system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
4.26 Two nucleon correlation functions for a stiff and soft density dependence of the
symmetry energy. The left panels are for low total momentum <300 MeV/c
while the right panels are for high total momentum >500 MeV/c from Ref. [8].123
xx

4.27 p-p correlation function for higher total momentum pairs from Ref. [8] renormalized to match peak heights (by Verde for E03045 proposal) so that the
widths can be compared. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
4.28 Average emission rate for protons with different symmetry energies for the
48 Ca+48 Ca reaction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
4.29 Average emission rate for protons emitted between 18-58◦ with different symmetry energies for the 48 Ca+48 Ca reaction. . . . . . . . . . . . . . . . . . . 126
4.30 Average emission time for protons and neutrons for γ=2 and γ=1/3 for the
48 Ca+48 Ca reaction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
4.31 Average emission times of nucleons as a function of momentum [8]. . . . . . . 128
4.32 r1/2 values, obtained from calculations using the code of Danielewicz, as a
function of average total momentum of the proton pair for γ=2 and γ=1/3
for the 40 Ca+40 Ca reaction. . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
4.33 r1/2 values as a function of average total momentum of the proton pair for
γ=2 and γ=1/3 for the 48 Ca+48 Ca reaction. . . . . . . . . . . . . . . . . . 131
4.34 Correlation functions for stiff and soft symmetry energy. The upper quadrants
are for a low total momentum cut P<300 MeV/c for 40 Ca+40 Ca (left) and
48 Ca+48 Ca (right). The bottom quadrants are for a high total momentum
cut P>500 MeV/c for the same reaction systems. . . . . . . . . . . . . . . . 132
4.35 In-medium cross section reductions for the screened (η=0.7) and Rostock
parametrizations for 48 Ca+48 Ca reaction. The left panel is for nn/pp collisions, and the right panel is for np collisions. . . . . . . . . . . . . . . . . . . 134
4.36 Laboratory kinetic energy spectra for the free cross sections and reduced Rostock and screened (η=0.7) cross sections. Only protons emitted between
18-58◦ are included. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
4.37 Source functions for the free cross sections and reduced Rostock and screened
(η=0.7) cross sections. Only protons emitted between 18-58◦ are included.
The 40 Ca+40 Ca reaction is shown for low total momentum 500-640 MeV/c
(top) and high total momentum 740-900 MeV/c (bottom). . . . . . . . . . . 136
xxi

4.38 Source functions for the free cross sections and reduced Rostock and screened
(η=0.7) cross sections. Only protons emitted between 18-58◦ are included.
The 48 Ca+48 Ca reaction is shown for low total momentum 500-640 MeV/c
(top) and high total momentum 740-900 MeV/c (bottom). . . . . . . . . . . 137

4.39 Plot of r1/2 values for select values of η for both np and nn/pp collisions.
When one type of collision cross section is varied, the other η is fixed to 0.7.
Only protons between 18-58◦ in θ were used in calculating the source. . . . . 139

4.40 2D plot of r1/2 values for all values of η for both np and NN collisions.
Only protons between 18-58◦ were used in calculating the source. The upper
quadrants are for low total momentum 500-640 MeV/c for 40 Ca+40 Ca (left)
and 48 Ca+48 Ca (right). The bottom quadrants are for high total momentum
740-900 MeV/c for the same reaction systems. . . . . . . . . . . . . . . . . . 140

5.1

p-p correlation functions for both reaction systems at midrapidity,
-0.05<ycms <0.05. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

5.2

Total laboratory momentum of proton pairs as a function of its angle. . . . . 153

5.3

Experimental correlation functions from 40 Ca+40 Ca (left) and 48 Ca+48 Ca
(right). The upper panels include protons with low total momentum (500-640
MeV/c) while the lower panels represent protons with a high total momentum
(740-900 MeV/c). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

5.4

Experimental correlation functions from 40 Ca+40 Ca (left) and 48 Ca+48 Ca
(right). The upper panels include protons with low total momentum (500-640
MeV/c) while the lower panels represent protons with a high total momentum
(740-900 MeV/c). The green dotted lines represent the results of the fit assuming a Gaussian source distribution. The purple solid lines are reconstructed
correlation functions from imaging. . . . . . . . . . . . . . . . . . . . . . . . 156

5.5

Comparison of the imaging fits to the Gaussian fits of p-p correlation functions
for 40 Ca+40 Ca (left) and 48 Ca+48 Ca (right). The upper panels include
protons with low total momentum (500-640 MeV/c) while the lower panels
represent protons with a high total momentum (740-900 MeV/c). The green
lines are Gaussian source distributions while the purple filled areas are source
functions from imaging. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
xxii

5.6

r1/2 as a function of average total momentum for both reaction systems and
all three angular gates. The size of sources from data using the imaging
technique is given by blue triangles while that from the Gaussian fit is shown
as red squares. Source sizes from BUU are shown as black circles. . . . . . . 164

5.7

Total transverse momentum as a function of pair center of mass rapidity. . . 165

5.8

Experimental correlation functions from 40 Ca+40 Ca (left) and 48 Ca+48 Ca
(right). The top panels include protons with low pair rapidity while the middle
panels include protons with intermediate pair rapidity and the lowest panels
represent protons with a high pair rapidity. . . . . . . . . . . . . . . . . . . . 166

5.9

Experimental correlation functions from 40 Ca+40 Ca (left) and 48 Ca+48 Ca
(right). The top panels show correlations for proton pairs in the lowest rapidity
bin. The middle panels show correlations for proton pairs in an intermediate
rapidity bin. The bottom panels show correlations for proton pairs in the
highest rapidity bin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

5.10 Experimental source functions from 40 Ca+40 Ca (left) and 48 Ca+48 Ca (right)
for both the imaging and Gaussian fit methods. . . . . . . . . . . . . . . . . 169
5.11 r1/2 values as a function of average total transverse momentum for low rapidity (top), intermediate rapidity (middle) and high rapidity (bottom). Results
using a stiff symmetry energy are shown by red filled circles while those using
a soft symmetry energy are shown with black open squares. . . . . . . . . . . 173
5.12 r1/2 values as a function of average total transverse momentum for low rapidity (top), intermediate rapidity (middle) and high rapidity (bottom). Results
using free cross sections are shown as red filled circles while those using a
screened in-medium cross section reduction are shown as black open squares. 174
5.13 r1/2 values as a function of average total transverse momentum for low rapidity (top), intermediate rapidity (middle) and high rapidity (bottom). Results
with cluster production are given by red filled circles, results without cluster
production are given by black open squares. . . . . . . . . . . . . . . . . . . 176
5.14 r1/2 values as a function of average total transverse momentum for low rapidity (top), intermediate rapidity (middle) and high rapidity (bottom). Results from calculations using a momentum dependent mean field potential
are shown with red filled circles, while those using a momentum independent
potential are shown with black open squares. . . . . . . . . . . . . . . . . . . 177
xxiii

5.15 Source r1/2 values from 40 Ca+40 Ca (left) and 48 Ca+48 Ca (right) for data
using the Gaussian fit and 4 parametrizations of BUU. . . . . . . . . . . . . 178
5.16 3-particle correlation for 6 Be→ p+p+α. Known states are labeled by red
arrows. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
5.17 3-particle correlation for 9 B→ p+α + α. Known states (in MeV) are labeled
by red arrows. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
5.18 3-particle correlation for 12 C→ α + α + α. Known states are labeled by red
arrows. States (in MeV) are listed in ascending order. . . . . . . . . . . . . . 183
5.19 Dalitz plot for 12 C→ α + α + α and 12 C→ 8 Be+α. . . . . . . . . . . . . . . 184
A.1 The black dots are corners of the detectors as measured by LBAS. The red
dots are corners of the detectors as given by the design of the mechanical setup
for the array. The discrepancy shows the need for such a laser measurement.
Units are in inches. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
A.2 A photograph of LBAS with components labeled. . . . . . . . . . . . . . . . 194
A.3 An example of an edge scanned with LBAS. . . . . . . . . . . . . . . . . . . 195
A.4 The red lines are the mechanical drawing of the target frame used to hold a
target. The green dots are from the LBAS scans of a target while it was inside
the chamber. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
A.5 The red lines are the mechanical drawing of the reference block used to link
different measurement reference frames together. The green dots are from
LBAS scans of the block. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
A.6 The LBAS scans with mechanical drawings are shown on top while a photograph of the actual setup is shown below. The target can be seen on the right
hand side of the figure, while the reference block can be seen on the left hand
side of the figure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
B.1 Tree structure for creating 2 particle correlation functions. . . . . . . . . . . 201

xxiv

Chapter 1
Introduction
All matter is made of atoms, each of which is comprised of a nucleus and electrons. The
nucleus, the core of the atom, consists of nucleons: Neutrons(n) and protons(p). Nuclear
science, at the interface between chemistry and physics, is the study of this nucleus. The
nucleus is bound together by nuclear forces. Its binding energy can be described by a
complicated relationship involving its mass number and charge. The mass number, denoted
as A = N + Z, is the number of nucleons, where Z denotes the charge number and represents
the number of protons, and the difference between mass number and charge number is
denoted by N, which represents the number of neutrons. One term in the parametrization
of the nuclear contribution to the binding energy depends upon the asymmetry, N − Z,
between the number of neutrons and protons. This asymmetry term plays an important
role not only in nuclear science, but also in nuclear astrophysics, for when this difference
between neutron and proton number is extrapolated to the extrenum, pure neutron matter
is obtained, similar to that found in the interior of a neutron star. One original goal of this
dissertation work was to study nuclei with an excess of neutrons to extract parameters that
1

affect the properties of neutron stars.
This introductory chapter will first provide a historical motivation, listing previous studies
this dissertation is built upon in section 1.1. Next, the main experimental observable, protonproton (p-p) correlation functions are introduced in section 1.2, with an explanation of how
they are measured, a description of their features, and an explanation of their relation
to the space-time extent of the emitting source during a nuclear collision. In section 1.3,
background information on the transport simulation used in this work is provided, and two
of the physics inputs, the density dependence of the symmetry energy and nucleon-nucleon
(NN ) in-medium cross sections, are explained. Finally, the organization of the remainder of
the dissertation is outlined.

1.1

Motivation

The three main areas of nuclear physics: structure, reactions, and astrophysics, are sometimes viewed as discrete entities with only the atomic nucleus to tie them together. One
common thread between them is the equation of state (EOS) of nuclear matter. An equation of state is a relationship between pressure, temperature, and density. One well known
equation of state is the ideal gas law

pV = nRT.

(1.1)

The ideal gas law can be used to describe nuclei, but only at very high temperatures and at a
low phase space density where the Pauli principle and nuclear interactions can be neglected.
In reality, a different EOS is required for many of the important systems that are at a higher
2

density where NN interactions and the Pauli exclusion principle are important.
The EOS is important for describing the bulk and thermodynamic properties of macroscopic nuclear systems [9]. Consequently, aspects of the EOS are essential to the description
of astrophysical phenomena such as the evolution of the universe after the Big Bang [10],
supernovae [11], and the structure of a neutron star [12]. For the latter it is useful to distinguish between the symmetric matter EOS, which is the limiting form of the EOS that
applies to nuclei and nuclear systems with N ≈ Z, and the symmetry energy, which can be
thought of as a correction term to the symmetric matter EOS for nuclei and nuclear systems
with N − Z > 0.
Since the turn of the millennium, it has appeared that heavy-ion collisions had provided
some constraints on the isospin symmetric matter EOS [13], however, the symmetry energy
is still widely unconstrained. Attention has been turned towards constraining the symmetry
energy with the increasing availability of neutron rich radioactive beams with large asymmetries in N-Z.
The symmetry energy is important for neutron rich nuclei, and the structure of the nucleus is governed by the symmetry energy in many aspects. While the most direct connection
is through the binding energy (see subsection 1.3.1), the symmetry energy also influences
other aspects such as neutron skin thickness [14], pygmy dipole resonance [15, 16], giant
monopole resonance [17], and isobaric analog states [18].
Many properties of neutron stars are governed by the symmetry energy, including their
maximum mass [19], and the relation between the mass and radius [20]. While it is known
that a neutron star cools by neutrino emission, it is unknown whether it occurs via the direct
or modified Urca process (named after the Cassino da Urca in Rio de Janeiro). The direct
3

Urca process may be possible. Whether this occurs or not depends in part on the density
dependence of the symmetry energy [12, 21].
The dynamics of nuclear reactions have been predicted to be sensitive to the symmetry energy, and many observables have been developed to explore this relationship. Based
on theoretical calculations, for example, one may expect that two-particle correlation functions [8], single and double ratios of free neutrons and protons [22], isoscaling [23], isotopic
distributions [24], and isospin diffusion [25] among others [26, 27] should be sensitive to the
symmetry energy. An overview of some of the current constraints on the symmetry energy
can be found in Ref. [28].
In order to constrain the symmetry energy from a heavy-ion collision (HIC), a comparison
to a theoretical model is needed. Hydrodynamic models were one of the first used, but
they assumed a zero mean free path for nucleons inside the nuclear medium. In its place,
theorists have developed transport theories for use at intermediate energies, many of which
can be understood from the point of the semiclassical Thomas Fermi (TF) model. The TF
model, applied first to the description of atoms, describes nucleons as a semiclassical Fermi
gas interacting by a self consistent mean field. Like the TF model, the BUU (BoltzmannUehling-Uhlenbeck) equation models the nucleus as a semiclassical distribution of nucleons
interacting by a self consistent mean field. In addition to the mean field potential, NN
collisions via the residual interaction are calculated in a collision term that enforces the
Pauli exclusion principle by incorporating Pauli blocking factors, discussed in section 1.3.
Ambiguities arose in the transport calculations due the treatment of the mean field potential.
There may be different physical origins for momentum dependence in the mean field such
as momentum dependence of the NN effective interaction, similar to that of the free-space
4

NN scattering matrix. In addition, the fact that nucleons are identical particles, leads to
momentum dependencies from the Fock or exchange term. The inclusion of a mean field
potential modifies the EOS from the Fermi gas EOS that would result in the limit of a
vanishing mean field potential.
Current BUU transport models used to study nuclear reactions have many unconstrained
physics inputs, only one of which is the symmetry energy. In order to constrain the symmetry
energy using any observable from heavy-ion collisions, one must also constrain other inputs
such as the NN in-medium cross section (described in Subsection 1.3.2). It turns out that
the p-p correlation function is not as sensitive to the density dependence of the symmetry
energy as it is to the in-medium cross section. In addition, there are other features of
reaction dynamics in transport models that also affect p-p correlation functions such as
momentum dependence of the mean field potential and cluster production, which will be
addressed in section 4.5. While constraining the density dependence of the symmetry energy
was the original motivation for this work, this motivation has evolved to focus on the other
unconstrained inputs in transport theory because the theoretical predictions have evolved.

1.2

Introduction to p-p Correlation Functions

For uncorrelated events, P (A(1), B(2)) = P (A(1)) · P (B(2)), that is the probability of an
observable taking a value of A(1) for particle 1 and B(2) for particle 2 in an event can be
described by the product of observing A(1) for particle 1 times the probability of observing
B(2) for particle 2 independently. If the observables are correlated, this is no longer true.
Correlation functions are one way to describe how the correlated two particle distributions
differ from the product of the uncorrelated single particle distributions. These difference
5

can arise from final state interactions (FSI) between the two particles during a HIC and,
in the case of identical Fermions, from the anti-symmetrization required for the relative
wavefunctions. Due to FSI and identical particle effects, correlation functions reflect the size
of the source from which the particles are emitted. For pairs of photons, the symmetry of
the relative wavefunction of two photons can be used to measure the radii of stars, while
for pairs of protons, the combination of anti-symmetrization of the relative wavefunction
and FSI can be used to measure the space-time extent of the emitting nuclear source. The
source is the region of the nuclear reaction which emits particles. Not all particles emitted
from a HIC are correlated however. A simple illustration of how correlations can be large or
small depending on the source of the nucleons is provided in Figs. 1.1 and 1.2 in the context
of the participant-spectator model. In the participant spectator model, when a nucleus
strikes another nucleus, the overlap region along the beam axis is called the participant
zone. The part of the projectile which does not overlap with the target is “sheared off” and
is called the projectile spectator. Likewise, the part of the target which does not overlap is
called the target spectator zone. Figs. 1.1 and 1.2 provide crude representations of particles
which may or may not be correlated. Fig. 1.1 gives an example of momentum vectors for
emitted particles which are likely to be strongly correlated while Fig. 1.2 is an example of
emitted particles which are likely to be weakly correlated. While the participant-spectator
description is not as precise at intermediate energies as it is at relativistic energies, it remains,
nevertheless, qualitatively correct and provides a useful terminology for describing the various
sources of particles produced in a collision.

In this dissertation, correlations are measured via intensity interferometry. Such correlations are sometimes called HBT correlations, named after Hanbury-Brown and Twiss [29].
6

Figure 1.1: Before and after cartoon of a heavy-ion collision. The black arrows represent
momentum vectors for two particles which are strongly correlated. For interpretation of the
references to color in this and all other figures, the reader is referred to the electronic version
of this dissertation.

Figure 1.2: Same cartoon as above, however, here the black arrows represent momentum
vectors for two particles which are weakly correlated.

7

The HBT correlation technique was invented to measure the size of radio sources in the galaxies Cygnus and Cassiopeia, and then used to measure the angular radius of the star Sirius.
In these early studies, the relevant correlations were between two photons and the main effect was caused by the fact that the two photons are bosons and their relative wavefunction
must be symmetric. Since then, HBT has also been used to measure femtoscale sources by
measuring correlations between pairs of pions or pairs of protons [30]. This method has been
subsequently used successfully to study the size of the source for a wide range of particle
types and colliding nuclei at a wide range of collision energies. [31, 32, 33, 34, 35].

In this dissertation, the primary concern is of p-p correlations, C(q). Experimentally, this
observable is calculated by

C(q) = 1 + R(q) ≡ ℵ

Y12 (p1 , p2 )
Y1 (p1 )Y2 (p2 )

(1.2)

the ratio of correlated to uncorrelated events, where the relative momentum, q, is calculated
in the center of mass of the proton pair and is defined by

1
q = (p1 − p2 ).
2

(1.3)

In Eq. 1.2, C(q) = 1 for particles which are uncorrelated. Consequently, R(q) describes the
degree of correlation, and should be 0, in the absence of collective effects, at large q where the
effect of final state interactions and identical particle effects are vanishing small. In Eq. 1.2,
the numerator, Y1,2 is the yield of events where two protons are detected with p1 and p2
respectively. The denominator mixes protons with momenta p1 and p2 from different events
with the same impact parameter and calculates q as if the two protons were emitted from
8

Figure 1.3: A p-p correlation function constructed from a Gaussian source with r1/2 = 6
fm. Important features are labeled.

the same event. Both numerator and denominator must be summed over the same range of
measured particle momenta and over the same range of impact parameters.
An example of a p-p correlation function, as a function of relative momentum, is shown in
Fig. 1.3. The peak at 20 MeV/c is due to the attractive 1 S0 potential and the anticorrelation
at low relative momentum is due to Coulomb repulsion and Pauli blocking. It is clear at
high relative momentum, there is no correlation between proton pairs.
The peak of the correlation function includes contributions from both early, fast emission
protons, and late, slower protons from secondary decays and evaporation. Many transport
models, such as BUU, do not adequately account for protons emitted from secondary decays
and evaporation and thus comparing the height of the correlation between experiment and
theory is difficult. The shape, ie. width, of the correlation function is a better indicator
for comparison. Experimental correlation functions are often “imaged” to produce source
9

functions which are compared to source functions from transport theory.
Imaging is a term that refers to the process of extracting a source function from a correlation function. Correlation functions are related to the space-time extent of the source by
the angle-averaged Koonin-Pratt [36, 37] equation

C(q) = 1 + 4π

K(q, r)S(r)r2 dr

(1.4)

where the kernel, K, given by
K(q, r) = |φq (r)|2 − 1

(1.5)

is the relative two proton wavefunction squared minus one. The relative wavefunction describes the propagation of two protons separated by distance, r, in the pair’s center of mass
frame, traveling to a detector at infinity with asymptotic relative momentum q. The source
function, S(r), is the probability of two protons being separated by a distance, r, at the time
the second proton is emitted.
Source functions for two protons typically peak at r = 0 fm and often have a Gaussianlike shape. For a purely Gaussian source, the half width half maximum, r1/2 , has a negative
linear correlation with the full width half maximum (FWHM) of the correlation function [38],
as shown in Fig. 1.4. That is, the wider the correlation function, the smaller the source size.

To interpret p-p correlation functions from the data, it is instructive to compare them
to transport models that simulate nuclear reactions in order to learn how the experimental
observables can be related to reaction dynamics and to theoretical quantities such as the
EOS, the momentum dependence of the mean field and the NN in-medium cross sections.
10

Figure 1.4: FWHM of a p-p correlation as a function of r1/2 of a Gaussian source function.

11

1.3

Introduction to the BUU Transport Model

The Boltzmann-Uehling-Uhlenbeck (BUU) transport equation was developed and applied to
nuclear collisions in Ref. [39] where it was initially applied to relativistic heavy ion collisions.
A useful review of transport models, including the BUU equation can be found in Ref. [40].
While there are many programs that solve the BUU equation in use currently, we have
performed most of our calculations using the BUU transport model developed by Danielewicz
and collaborators [41, 42, 43]. The following description adopts the notation of this version.
BUU transport models simulate heavy-ion collisions by self-consistently solving the modified
Boltzmann equation

∂f ∂ p ∂f ∂ p ∂f
+
−
=
∂t
∂p ∂r
∂r ∂p

dp2

dσ
((1
dΩ v12
dΩ

f1 )(1

f2 )f1 f2 − (1

f1 )(1

f2 )f1 f2 )
(1.6)

where Uehling and Uhlenbeck introduced the quantum statistical terms [44] involving f1
and f2 that are found in the collision integral on the right hand side of the equation. The
terms on the left hand side of Eq. 1.6 describe the changes in Wigner transform of the one
particle density matrix, f, due to the motion of particles in the mean field, while the terms
on the right hand side describe changes in f due to collisions between particles, scattering
into or out of various momentum states, and include Pauli-blocking [43]. The minus signs
are for fermions and the plus signs are for bosons on the right hand side. The mean field
potential is created by all other nucleons and

is the single particle energy.

When the right hand side of the equation is set to 0, the Boltzmann equation reduces
to the Vlasov equation which describes the evolution of a single particle under the influence
of the mean field potential. By Liouville’s theorem, the phase space density will not change
12

as a result of the mean field potential, but the inclusion of the statistical collision term on
the right hand side can cause a modification of the phase-space density and a corresponding
violation of the Pauli principal. The blocking factors f1 and f2 on the right hand side
prevent that.
While Eq. 1.6 is derived as a semi-classical limit of the quantum mechanical Hartree-Fock
equation, it can be solved classically by a test particle approach. To accurately determine the
mean field and Pauli-blocking probability, many test particles are needed for each nucleon
in the heavy-ion collision [39, 45]. The nuclei on their own have too few particles to create a
stable mean field to calculate the Pauli blocking factors. In the case of a static phase space
density, the test particles evolve through the left hand side of the Boltzmann equation by
Hamilton’s equations
˙
p = − rH

(1.7)

and
˙
r=

pH

(1.8)

following the lattice method of Lenk and Pandharipande [46]. The stability of the transport
code with respect to test particle number is explored in section 4.3.
The colliding nuclei are initialized using Thomas-Fermi equations which find the density
configuration of nucleons which results in the lowest ground state energy for the nucleus.
The surfaces of the nuclei are nearly touching at time = 0 fm/c. The test particles collide
on a computational grid of variable size, and the dependence on this size is investigated
in section 4.2. The Coulomb force affects charged particles at a larger range than can be
accounted for in this model, so when particles leave the grid, they are boosted to infinity
where the Coulomb force becomes negligible. Any free particles remaining on the grid at the
13

end of the simulation are also boosted to infinity.
In this work, only particles emitted from reactions are important, so it is important to
define what constitutes emission within the BUU transport code. First, “bound” and “free”
must be defined. Physically, particles are considered free when they are decoupled from the
mean field and no longer collide. At each time step, particles are checked to see whether or
not they are bound. If the local density is greater than ρ0 /15, where ρ0 = 0.16 fm−3 is the
saturation density, the particle is bound. If the density is less than ρ0 /15, the particle is
temporarily boosted to the rest frame of the projectile. If the particle’s energy in this rest
frame is less than the rest mass minus the binding energy, it is bound. If not, it is boosted
to the rest frame of the target to see if it is bound within that residue by the same energy
criteria. If it is not bound in either residue, it is marked as unbound. A collision resets this
process. If the particle is unbound when it exits the computational grid, or at the end of the
simulation, it is considered emitted. If it emitted, its position, momentum, and density at
the time of last collision are written to a file to calculate source functions.
The source function, and thus p-p correlation function, can depend on many inputs in the
BUU transport model. Two inputs, the symmetry energy and NN in-medium cross section
reduction, are introduced in detail below.

1.3.1

Defining the Symmetry Energy Parametrization

One of the original goals of this dissertation work was to constrain the density dependence
of the symmetry energy. The symmetry energy is the difference in binding energy between
pure neutron matter, A=N, and symmetric matter, Z =N. A simple parametrization of the
14

binding energy of a nucleus is given by the Bethe-Weizaecker formula
a
Z(Z − 1)
(N − Z)2
EB = av A − as A2/3 − ac
+ δ0 (A, Z) P
− aA
A
A1/3
A1/2

(1.9)

in the liquid drop model. This binding energy assumes a density independent form of the
asymmetry term. In this parametrization, the first term is the volume term, proportional to
the volume of the nucleus, and based on the strong nuclear force. The second term is the
surface term, also based on the strong force, however, similar to surface tension in liquids.
The third term is due to Coulomb force, taking into account electrostatic repulsion between
protons. The fourth term is the asymmetry term, (also called the Pauli energy) which shows
that when the number of neutrons is much greater than the number of protons, the energy is
different than it would be for a system with the same number of nucleons but equal neutrons
and protons. The last term is the pairing term, where δ0 = 1 for Z,N both even, δ0 = −1
for Z,N both odd, and δ0 = 0 for A odd.
A phenomenological parametrization of the energy as a function of density and isospin
can be written as follows

E/A(ρ, δ) = E/A(ρ, 0) + (Ekin (ρ/ρ0 )2/3 + Eint (ρ/ρ0 )γ )δ 2

(1.10)

where δ = (ρn − ρp )/ρ. This is an expansion about isospin where only even terms are
kept due to charge exchange parity. Fourth order and higher terms are small enough to be
negligible. The first term, where δ = 0, is the symmetric matter EOS and has been fairly well
constrained over the past few decades. The second term, is a correction for the asymmetry.
The constant, Ekin , which governs the size of the kinetic contribution is on average about
15

Figure 1.5: Different density dependencies of the symmetry energy.

12.5 MeV/A for a free fermi gas [47] and the constant for the interaction contribution, Eint ,
was chosen to be 17.6 MeV/A so that results are consistent with Ref. [26]. Values of the
exponent for the density dependence, γ, explored were 1/3, 0.4, 0.7, 1, and 2 as shown in
Fig. 1.5 where the density dependence of the symmetry energy for pure neutron matter is
shown. Smaller values of γ < 1 have a weakened density dependence which is called soft.
Larger values with γ > 1 have a stronger density dependence and are referred to as stiff or
hard.
16

1.3.2

Defining the Nucleon-Nucleon Cross Section Parametrization

One can obtain information about the symmetry energy from heavy-ion collisions, but these
collisions are influenced by other transport quantities as well. One of these quantities is the
nucleon-nucleon cross section, σN N . The baryon scattering cross section may be modified
in the nuclear medium, but this modification is poorly constrained [8,48,49]. Three different
σN N parametrizations were explored for this work. The first were free cross sections, that is,
with no in-medium modification, determined by fitting to tables of data. The second, referred
to as Rostock, was an energy dependent in-medium reduction, which is a parametrization of
that given by Brueckner-Hartree-Fock microscopic calculations [50], described by

σ(ρ) = σf ree exp − 0.6

ρ
1
.
ρ0 1 + (KEcm /150M eV )2

(1.11)

The last was a strongly density dependent in-medium reduction [43], referred to as screened,
and described by
ση (ρ) = σ0 tanh[σf ree /σ0 ]

(1.12)

where σ0 = ηρ−2/3 . The screened in-medium cross section reduction ensures that the
geometric cross section radius does not exceed the interparticle distance. This is similar to
requiring that the mean free path is not much less than one mono-layer of nucleons at normal
nuclear density.
17

1.4

Organization of Thesis

The remainder of the thesis is organized as follows: Chapter 2 characterizes the experimental
setup, both mechanical and electronic. A detailed description of the targets is also given.
Then, chapter 3 chronicles the calibration procedures for all detectors and describes how
events are selected based on the centrality of the reaction. It also describes how target oxidation is quantified. Next, chapter 4 explores many inputs of the BUU transport simulation
code such as momentum dependence of the mean field potential, light cluster production,
and NN in-medium cross section reduction. Chapter 5 provides experimental data in the
form of two and three-particle correlation functions. There, the dependence of p-p correlation functions on laboratory momentum, both in angle and magnitude is examined, as is
the dependence on transverse momentum and center of mass rapidity. This chapter also
includes corresponding source functions, and comparisons to BUU theory. Finally, chapter
6 summarizes the results and conclusions of this work. It also provides an outlook for the
future. Appendices include a description of the precise laser measurements of detector positions, a description of the analysis packaged used, and an explanation of how to quantify
the space-time extent of the source.

18

Chapter 2
Experimental Setup
The “4π+HiRA” experiment (NSCL-PAC number 03045 [51]) was performed in the fall of
2006 using stable beams from the National Superconducting Cyclotron Laboratory (NSCL)
Coupled Cyclotron Facility (CCF) at Michigan State University. The experiment was named
after the two detection systems, the soccer ball shaped NSCL 4π Array [52] and the High
Resolution Array (HiRA) [53], used in the experiment. The experiment was located in
the N2 vault, as shown in Fig. 2.1, which shows the layout of the experimental area at
NSCL in 2006. Immediately after the experiment, the NSCL experimental area underwent
major reconfiguration and the N2 and N3 vaults were combined into one vault. The 4π was
decommissioned and its frame was placed in the Biomedical and Physical Sciences building
on MSU campus as an exhibit, marking the end of an era of experiments with this device.
This chapter begins with a brief description of the reaction systems studied. It continues
with a description of the purpose of each of the two detector arrays and a description of their
layout. Next, the components of HiRA are explored in detail. This is followed by specifics
of the 4π detector array. Then, a description of the targets is given. Finally, descriptions of
19

Figure 2.1: Layout of the CCF, from above, in early fall of 2006.

the electronics for HiRA, the 4π, and the trigger logic are provided.

2.1

Reaction Systems

40 Ca+40 Ca, 48 Ca+48 Ca, and 48 Ca+40 Ca reaction systems were studied at E/A = 80
MeV. This dissertation will focus on the two symmetric reactions for constructing correlation
functions. While rare, unstable, neutron-rich beams would have been ideal for the studies
in question, stable beams were employed to achieve the statistics needed for p-p correlation
functions.
In addition, the 40 Ca+CH2 at E/A = 25 MeV reaction was used. The hydrogen in the
polyethylene (plastic) target was scattered into HiRA for calibration of the CsI crystals, as
explained in section 3.1.2.
Both beams of 40 Ca and 48 Ca at E/A = 80 MeV impinged on a mylar foil target to
monitor the calcium targets for oxidation, which is discussed more in section 3.3.
20

2.2

Experimental Layout

Two detector arrays were used in this dissertation experiment: The 4π detector array, which
is comprised of the “ball” (shaped like a soccer ball) and a forward array1 (FA), and a High
Resolution Array (HiRA) [54]. The 4π array is a truncated icosahedron with 20 hexagonal
modules, 10 pentagonal modules, and 2 pentagonal faces that serve as the beam entrance and
exit. Each module in the ball houses 6 (hexagonal) or 5 (pentagonal) logarithmic detectors
to be described in section 2.4. In addition, the forward array contains 45 phoswich detectors
and is mounted on the exit pentagon. The 4π was used to determine the impact parameter
using transverse energy. Nuclei with Z≤3 could be identified in the ball and nuclei with Z≤7
could be identified in the forward array. Individual isotopes could not be resolved in the 4π
array.
HiRA, described in greater detail in section 2.3, consists of a variable number of telescopes. Each telescope is comprised of a single sided silicon strip detector, a double sided
silicon strip detector and four CsI(TI) crystals. HiRA was used to identify Z≤3 isotopes
with good energy resolution for correlation functions. The excellent angular resolution of
HiRA allows us to identify pairs of nuclei with a relative angle of less than 1◦ .
Seventeen telescopes of HiRA, arranged in five towers, were mounted on a custom made
vacuum flange, shown in Figs. 2.2 and 2.3 to replace one hexagonal module of the 4π detector.
The towers of telescopes were mounted on rails to easily separate them during assembly. The
rectangular holes in the flange, seen in Fig. 2.2, were necessary for electrical feedthroughs.
There were also water feedthroughs to allow for cooling of the detectors. The sides of each
telescope canisters were mechanically modified to allow for close packing. The telescopes
1 Referred to as the high rate array (HRA) in references [1, 5, 52].
21

Figure 2.2: Drawing of the HiRA Array mounted on a modified 4π flange.

were angled so that the center of the front face of each telescope pointed to a common point
60 cm away, which was roughly the target position. From laser position measurements (see
App. A), it was determined that the middle of the thick silicon detectors of the HiRA array
were at an average distance of 61.9 cm from the target.

Fig. 2.3 shows HiRA just before insertion into the 4π and Fig. 2.4 shows most of the
detector system with HiRA inserted. On the left of Fig. 2.4 is the side of the HiRA array
and just left of center one can see the reflective faces of part of the forward array. The rest
of the detectors are elements of the 4π ball. In this photograph, the beam comes in from
the right and the forward array is downstream. Two more hexagonal modules were removed
from the 4π for access to HiRA and the installation of a cryogenic vacuum pump.
22

Figure 2.3: Photograph of the HiRA Array with special lighting.

Figure 2.4: Photograph of the HiRA Array inside the 4π Array.

23

2.3

The High Resolution Array, HiRA

Each HiRA telescope, a schematic of which is shown in Fig. 2.5, is equipped with a thin
65 µm silicon detector, DE, with 32 vertical strips on the front and a thick 500 µm double
sided strip detector, E, with 32 vertical strips on the front and 32 horizontal strips on the
back. The E detector will be referred to as EF for the energy detected by the front strip
and EB for the energy detected by the back strip. The double sided strips on the E detector
provide 1024 pixels, each 1.95 mm by 1.95 mm for an angular width of ∼ 0.18◦ at 62 cm.
The silicon detectors are backed by four CsI crystals which are 39 mm in length on average.
An excellent description of the HiRA telescopes is given by Rogers and Wallace [2, 3, 54].
Particles which stop in the E can be identified using DE vs E energy plots. Particles which
stop in the CsI crystals can be identified using E vs CsI energy plots. In this dissertation,
particles stopping in the E detectors are not analyzed.
Seventeen telescopes, arranged as in Fig. 2.6 and labeled by their number, were used in
the experiment. Telescopes 4 and 17 did not have the DE installed since most protons at
forward angles would have high enough energy to punch through the E detector into the CsI
crystals. Telescopes 2, 7 and 9 have been excluded from this analysis due to malfunctioning
electronics.
The HiRA array spanned roughly 18◦ -57◦ theta in the lab. The optimal angular coverage
for the array is shown in Fig. 2.7.

In reality, most detectors had some EF and EB strips

which were not working throughout the experiment. The actual HiRA coverage can be seen
in Fig. 2.8. Many channels of the chip electronics did not work properly. Also, the middle
strips from each silicon detector are removed from analysis because particles hitting these
strips will often pass in between two CsI crystals.
24

Figure 2.5: Drawing of HiRA telescope components [2].

During the experiment, all components of the telescopes had bias applied to them. All
CsI crystals were biased to 80 V. The EBs were biased to 100 V. The biases for DEs and EFs
are given in Table 2.1. The net bias across the 1.5 mm thick double sided silicon strip detector
is the difference between the EF and EB bias values. Also in the table are sample leakage
currents, which drifted during the experiment. These were monitored regularly during the
experiment to ensure that there was no discharge on the silicon surfaces, or breakdown of
the detectors. The telescopes are listed from top to bottom, and from forward to backward
angles from the beam axis.
25

Figure 2.6: The HiRA array, labeled with telescope numbers, as viewed from the target. In
this diagram the beam travels from left to right so that telescopes 4 and 17 are the most
forward.
Tele #
4
17
0
1
3
5
6
8
10
11
12
13
16
19

EF Bias [V]
-200
-200
-190
-250
-315
-240
-340
-360
-150
-250
-340
-310
-160
-210

E Current [µA]
1.64
1.90
0.70
1.60
1.34
2.38
1.78
1.80
1.58
1.36
1.72
1.48
1.76
1.04

DE Bias [V]
-9
-8
-7
-9
-7
-11
-7
-8
-8
-7
-7
-8
-7
-6

DE Current [µA]
0.16
0.06
0.108
0.27
0.171
0.11
0.21
0.057
0.033
0.015
0.038
0.27
0.03
0.081

Table 2.1: Biases applied to E and DE detectors and corresponding sample leakage currents.
26

Figure 2.7: Angular coverage of the entire HiRA array.

27

Figure 2.8: Actual angular coverage of the HiRA array.

28

2.4

4π Detector Array

The 4π Detector Array was a logarithmic detector system covering 3/4 of the total 4π solid
angle. It was designed in late 1982 to detect all charged particles from intermediate energy
collisions of nuclei. The detector system consisted of multi-wire proportional counters, Bragg
curve counters, and fast and slow phoswiches. In addition there was a zero degree detector
that was removed prior to this work. Only the phoswiches were used in this experiment.
The ball has an inside diameter of 70" , and is 101" long, resulting in a volume of 3400
liters3 . RTV was used to seal the aluminum pieces of the framework, nicknamed the “soccer
ball” for its shape. 145 of rubber O-rings were used to seal the modules and feedthrough
plates [52].
Before this experiment, the 4π had sat dormant for a number of years. One of the first
tasks was to make the chamber vacuum tight. After two weeks of leak testing, fixing leaks,
and the installation of a cryogenics pump, a reasonable vacuum could be attained. During
the experiment, the vacuum averaged 5x10−5 Torr and broke the 10−6 Torr mark just hours
before the end of beam time. [55].
Each hexagonal module in the 4π contained six phoswiches and each pentagonal module
contained five phoswiches, whose front and rear surfaces were triangular in shape. The polar
angle, with respect to the beam line, of the midpoint of each module in the array is given in
Table 2.2 [52]. Each hexagonal module subtended a solid angle of 6 x 65.96 msr and each
pentagonal module subtended a solid angle of 5 x 49.92 msr [52]. Module 5 was removed for
the insertion of HiRA, module 15 was removed to have access to HiRA, and module 29 was
removed to install a cryogenics pump to help obtain a better vacuum. After removing these
modules, the ball covered 9.37 sr. The FA covered 280 msr giving a total of 9.65 steradians
29

Mod
1-5
6-10
11-15
16-20
21-25
26-30

A
θ (◦ )
23.1
54.7
64.9
86.5
105.4
128.3

B
θ (◦ )
32.3
54.7
72.4
93.5
112.7
134.0

C
θ (◦ )
46.0
67.3
86.5
107.6
125.3
147.7

D
θ (◦ )
51.7
74.6
93.5
115.1
125.3
156.9

E
θ (◦ )
46.0
67.3
86.5
107.6
112.7
147.7

F
θ (◦ )
32.3
72.4
93.5
134.0

Table 2.2: Mean polar angles of the ball phoswiches.

Element
Ball Fast ∆E
Ball Slow E
FA Fast ∆E
FA Slow E

Thickness [mm]
3
250
1.7
194

Bicron plastic
BC-412
BC-444
NE-110
NE-115

Rise Time [ns]
1
20
1.1
8

Fall Time [ns]
3.3
180
3.3
320

Table 2.3: Thicknesses and composition of the phoswiches in the ball and FA. Rise and
fall times are also given, although the signal from the photomultiplier tube for the thin
scintillator will have much long rise and fall times.

or roughly 77% of total 4π solid angle.
Each phoswich was comprised of a thin wafer of fast plastic scintillator to detect the
initial rate of energy loss as it entered the 4π module, and a thick slow plastic scintillator to
stop the charged particle and detect the total energy deposited. Some specifications of the
phoswiches in the ball and FA are given in Table 2.3 [52].
The forward array was comprised of 45 closely packed phoswich detectors, surrounding
the exit beam line as shown in Fig. 2.9. Also shown in the figure is the relationship to
the first ring of modules [5] and the HiRA array.

Although the θ angle was defined with

respect to the beam axis, the direction of φ is somewhat arbitrary. In Fig. 2.9 you can see
the reference for φ, with 0◦ defined as the center of FA element 1. The polar and azimuthal
angle for the center of each FA element is given in Table 2.4 [5].
30

Figure 2.9: Schematic diagram of the forward array. Also shown is the lab reference for the
azimuthal angle.

31

Detector
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15

θ (◦ ) φ (◦ )
5.4
0.0
5.4 324.0
5.4 288.0
5.4 252.0
5.4 216.0
5.4 180.0
5.4 144.0
5.4 108.0
5.4
72.0
5.4
36.0
10.6
6.0
9.6 342.0
10.6 318.0
10.6 294.0
9.6 270.0

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

θ (◦ )
10.6
10.6
9.6
10.6
10.6
9.6
10.6
10.6
9.6
1-.6
15.9
14.3
14.3
15.9
15.9

φ (◦ )
246.0
222.0
198.0
174.0
150.0
126.0
102.0
78.0
54.0
30.0
9.0
351.0
333.0
315.0
297.0

Detector
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45

θ (◦ )
14.3
14.3
15.9
15.9
14.3
14.3
15.9
15.9
14.3
14.3
15.9
15.9
14.3
14.3
15.9

φ (◦ )
279.0
261.0
243.0
225.0
207.0
189.0
171.0
153.0
135.0
117.0
99.0
81.0
63.0
45.0
27.0

Table 2.4: Mean angles of the FA phoswiches.

2.5

Target Details

A nuclear reaction occurs when a projectile nucleus strikes a nucleus in a target, which is
a thin metal foil. The composition and thicknesses of targets used in this experiment are
summarized in Table 2.5.

Calcium targets were used for the nuclear reactions of interest.
Target
40 Ca
48 Ca
CH2
mylar
scintillator
Natural Ni
empty frame

Thickness [mg/cm2 ]
2.2
5.1
2.95
0.9
N/A
4.6
N/A

Table 2.5: List of targets used with their respective thickness.
A polyethylene target was used to elastically scatter protons for CsI calibration described
in section 3.1.2. A mylar target was used to monitor the oxidation of the 40 Ca and 48 Ca
targets, described in section 3.3. A scintillator target was used for initial beam tuning. A
32

natural nickle target was used for “target in” measurements and debugging procedures, while
an empty frame was used for “target out” measurements to assess the amount of beam hitting
the target frame. During beam tuning, the “target in” to “target out” ratio was minimized
to reduce the amount of beam on frame. This was especially important for the E/A = 25
MeV 40 Ca beam which was degraded significantly from E/A = 140 MeV and had a relatively
large beam spot.

The calcium targets were rolled at the NSCL in a glove box filled with argon because
calcium is highly reactive with oxygen. Most adhesives either require oxygen to cure or
contain oxygen in them, therefore vacuum grease was used to “glue” the calcium targets to
the aluminum frames. All frames used in the experiment were oblong shaped and rotated
37.5◦ so that the beam impinged on a circular target while the targets actually faced the
center of the HiRA array. This had a dual advantage: The operators were able to tune on a
circular target and the difference in energy loss of particles leaving the target was minimized.
Had the target been perpendicular to the beam, it would have significantly degraded the
energy resolution of the unbound resonances detected with the HiRA detectors.

The frames were connected to a base using carbon fiber rods, which are lightweight
enough that the target would not sag appreciably, and yet flexible enough to be navigated
by the target driver, “Freddy”, into the cylindrical carousel shown in Fig. 2.10. “Freddy
Krueger” was the nickname “fondly” given to the system of stepping motors and machinery
used to move a target, by its base, from the carousel to the center of the ball. Freddy earned
its nickname for its predilection of shredding targets and for its longevity.
33

Figure 2.10: Photograph of target carousel which housed all targets used. Pictured, clockwise, are an old scintillator target at 5 o’clock, an old broken gold target at 8 o’clock, and a
new empty oblong frame at 11 o’clock.

34

2.6
2.6.1

Electronics
HiRA

The electronics for the HiRA silicon detectors were housed outside of the vacuum chamber
due to lack of space inside the chamber. There are 100 channels per telescope that need to
be processed. All of these signals were routed through a vacuum tight flange using 32 pin
individually shielded ribbon cables. Each detector required three cables, one for the DE,
one for the EF, and one for the EB. Outside of the chamber, each cable was connected to a
chipboard housing 2 Application Specific Integrated Circuit (ASIC) chips, where each chip
can process 16 channels. A photograph of a chipboard with a prototype chip is shown in
Fig. 2.11 with a quarter to provide a scale. In the case of the DE detectors, due to the
high capacitance of the thin silicon, external pre-amplifiers were used to reduce noise on the
onboard pre-amplifiers. The E detectors are thicker, and not as sensitive to electronic noise,
thus the onboard pre-amplifiers were sufficient.
A maximum of 15 chipboards in principal can be connected to a single motherboard,
shown in Fig. 2.12. Each motherboard distributes power for the chipboards, distributes
high voltage for the detectors, and reads out all signals. In the version at the time of this
dissertation experiment, each motherboard could only supply current to ten chipboards. The
copper bars seen in Fig. 2.12 were necessary for cooling the chips which operate at ∼ 30◦ C.
Six motherboards were used in this experiment, two for DEs and four for Es.
The signal from the silicon is processed in the ASIC chip, which contains the preamplifier,
shaper and discriminator circuitry, shown in Fig. 2.13 for a single channel.

The chip

itself provides several inspection channels for the user to inspect the preamplifier, shaper
35

Figure 2.11: Photograph of chipboard with quarter for size comparison.

Figure 2.12: Photograph of HiRA motherboard with six chipboards installed.

36

Figure 2.13: Diagram of the ASIC electronics logic. [2, 3]

and discriminator outputs. The settings of the ASIC chip, such as discriminator threshold
and preamplifier gain, are controlled using an external program and set through an XLM
(universal logic module). In the ASIC, the detector signal is split into time (not used in this
experiment) and energy branches, which enter the Flash ADC (FADC) on a pair of doublelemo cables. This reduces the number of electronic modules needed, as well as number of
cables, considering the high number of channels in HiRA.
The schematics of the HiRA silicon electronics is shown at the top of Fig. 2.14. The data
were read out from the motherboard using an XLM. To start the XLM data acquisition, the
XLM needed to be triggered. The XLM trigger in this experiment was generated based on
signals in CsI exceeding multiplicity of one or two (HiRA trigger) in coincidence with a 4π
signal exceeding a multiplicity of four.
When triggered, the XLM read in channel information from a motherboard and then
sent a clock signal to the FADC to digitize the energy each time it had a trigger. The XLM
generated a complete signal once it was finished acquiring data from the motherboard. In
the case of HiRA singles, the complete signal was used as a computer trigger. During regular
37

Figure 2.14: Diagram of the HiRA electronics logic. [2, 4]
data taking, it was the HiRA trigger in coincidence with the 4π that triggered the computer.
For the CsI crystals, shown at the bottom of Fig. 2.14, the pulses were processed in a
preamplifier located inside of the HiRA telescope can. The signal was then routed into a
CAMAC Pico System shaper/discriminator module before being routed to the ADC (CAEN
V785). The sum output from Pico shaper/discriminator was used to generate total sum of
all CsI signals in order to generate a multiplicity based HiRA trigger. This signal was sent
in coincidence with 4π to generate XLM trigger and other important logic signals.

2.6.2

4π Array

An idealized current pulse from a phoswich detector can be seen in Fig. 2.15 where the solid
line is the total signal, the dashed line is the fast component and the dotted line is the slow
component. The ∆E and E gates are shown to separate the fast and slow components. It
38

Figure 2.15: Idealized signal from a phoswich detector. [5]
can be seen that the signals overlap, and each is contaminated by the other. The method
of disentangling them is necessary for generating accurate calibrations and is described in
subsection 3.2.1.
There were two independent and fairly similar electronic branches for the FA and ball
as shown in Figs. 2.16 and 2.17. The phoswich, photomultiplier tube (PMT), PMT base
and voltage divider are housed inside of the vacuum chamber of the 4π. The PMTs serve
to amplify the light created by the scintillator plastic. The bias for the PMT and the signal
from it are transmitted over a single SHV cable for each detector. This cable is connected to
a splitter where the signal is separated from the high voltage and split into fast (∆E), slow
(E), and time components.
The ∆E subbranch entered through a 100 ns passive delay before entering a V862 Caen
QDC, which was used in a common gated mode by supplying a logical yes level to the
individual gate inputs. Each ∆E had its own corresponding common gate corresponding
to an OR of the discriminators for the signal input to the QDC. The E subbranch entered
39

Figure 2.16: Diagram of the 4π electronics logic, part (a).

Figure 2.17: Diagram of the 4π electronics logic, part(b).

40

through a 100 ns passive delay before entering a V792 Caen QDC which was used in common
gate mode. Both of these modules were fast cleared if there were no valid HiRA data in
coincidence with the 4π.
The time subbranch is processed in a Phillips 7106 discriminator. The sum output of
the discriminator was used to enter summers to generate total, ball or FA+ball sum signal.
Each Phillips discriminator preamplifier output corresponds to ∼50mV per channel fired.
The total sum of the FA+ball can thus be used to generate multiplicity based 4π trigger
signal. In this experiment a multiplicity at least four in the FA+ball was required to reject the
most peripheral events. The sum signal was processed in a constant fraction discriminator
to the master live signal, which was the 4π trigger signal, also used to generate the slow E
gate. To generate time information, time signals from the Phillips discriminator are also sent
to V1190 TDC as individual start signals. Stop for the TDC is provided by the 4π master
live plus HiRA coincidence. This was a substitute for a fast clear since this module did not
have one.

2.6.3

Trigger Logic

For the purpose of taking regular data, data events were only recorded if there was a HiRA
event in coincidence with an event in the 4π. The coincidence trigger logic is shown in
Fig. 2.18. For the data acquisition to trigger, a multiplicity greater than one (or two) in the
CsI crystals was required, fulfilled by setting a threshold on sum of CsI discriminator signals.
This signal entered a coincidence with 4π master, which was selected based on FA+ball
multiplicity above four. The lower portion of Fig. 2.18 shows the busy logic, which was used
to generate the computer trigger. If a HiRA and 4π coincidence signal was generated and
41

Figure 2.18: Diagram of the coincidence trigger logic. [6]
the computer was ready to acquire the data, the computer trigger signal was issued. CsI
multiplicity two events were recorded to construct the numerator of the correlation functions
while multiplicity one events were recorded to construct the denominator of the correlation
functions.
A variety of other triggers were also needed and employed. An FA singles trigger was used
for target in, target out measurements. An FA+ball singles trigger was used for debugging
and minimum bias runs used for impact parameter selection. A HiRA singles trigger was
used for debugging HiRA and setting thresholds. For the CsI calibration runs the data was
triggered by multiplicity one in the CsI crystals, while vetoing events with any data in the
4π.

42

Chapter 3

Data Analysis

Detectors do not immediately provide the energy of particles detected. They provide a signal which is digitized, processed by various electronics modules, and encoded before being
written to a computer file. In this thesis experiment, one hundred hours of experimental data corresponds roughly to one hundred gigabytes of raw computer data. This data
must be transformed, processed, and condensed into meaningful physics quantities. Detailed
information about the nature of this processing can be found in Appendix B.

This chapter explains preliminary analysis, including the method with which the raw
electronic channels were converted into energies for each detector component. HiRA calibrations are described first, beginning with a discussion of silicon strip calibrations and finishing
with the CsI crystal calibrations. The 4π section is next, starting with the calibration of the
phoswiches and ending with a detailed description of the selection of central events. The final
section in this chapter quantifies any oxidation of the calcium targets during the experiment.
43

3.1

HiRA

Each component of HiRA, (DE, EF/EB, and CsI) was calibrated separately. However, the
same basic procedures were followed for both silicon detectors and will only be described once.
Any differences in the calibrations for the silicons will be highlighted. The CsI calibration
will then be described in depth.

3.1.1

Silicon Calibration

There are two stages in calibrating a silicon strip detector. The first step is to test the
linearity of the electronics with a pulser, and the second step is to calibrate each strip with
an α source. It was discovered that an additional correction step was needed, and this is
discussed in Subsection 3.1.1.1.
A precision BNC PB-5 pulser was used to send specific voltages to the test input of the
Si detectors, mimicking a real signal from a particle. A pulser ramp was performed, with
21 steps at 0.25 V intervals. There were no significant non-linearities found on any of the
channels.
The second stage of silicon calibration involves exposing all strips to an α source. At the
end of the experiment, a 228 Th α source was employed to calibrate the DEs and the Es on
the two most forward telescopes, 4 and 17, which had no DE installed. The α particles stop
in the DE, however, such that the DEs had to be removed to calibrate the Es at a later date
with the same 228 Th α source. Fig. 3.1 shows the decay chain of 228 Th with prominent
energies for the α source listed in red. These five energies are used for calibration purposes.
Fig 3.2 shows an example of an α energy spectrum for one single strip in an E detector. The
five most prominent peaks are those used for calibration purposes.
44

Figure 3.1: Decay chain for 228 Th. Energies of prominent α particles are given in red.

Figure 3.2: Example of a raw α spectra for a single strip.

45

Figure 3.3: Example of a calibrated α spectra for all EF strips in one detector.
A variety of energy losses must be taken into account when using an α source to calibrate
silicon strips. The α particles lose energy going through a 50 µg/cm2 thick gold window seal
on the source itself. They then lose energy going through 1.9 µm thick aluminized mylar
foil covering each detector. Finally, the α particles lose energy going through an inactive
layer of the silicon surface called the dead layer. After these energy losses are taken into
account, all strips are calibrated with a linear extrapolation between channel numbers and
known energies. The result is a calibrated spectrum such as that shown in Fig. 3.3.

3.1.1.1

Readout Order Corrections

This experiment used the first version of the Application Specific Integrated Circuit (ASIC)
electronics developed by collaborators at Washington University in St. Louis. It was noticed
during this experiment that the channel number corresponding to a particle with a given
energy is dependent upon the order in which the electronics reads out that channel, hence
the name “readout order problem.” This experiment ran in sparse mode which means only
46

channels with a signal above threshold are read out. Channels are read out in a specific
order each event. In a given motherboard, chipboards are read out from bottom to top. The
chipboards for EBs are always lower than those for EFs, so they are always read out first.
The “readout order problem” results from digital-analog cross-talk. The digital signals
from control logic of the ASIC during readout add to the amplitudes of the analog signals
that are present and sent out from the switched capacitor array to the ADC. The cross-talk
actually occurs at the ASIC level, so it could only be removed with a redesign of the ASIC,
which was completed after this experiment.
Suppose only one particle enters a HiRA detector for a given event. If the particle
deposits 5 MeV in an EB strip, its shaped signal would be digitized in the ADC to provide
data in a channel value of 667 in accordance with the approximate energy calibration of
7.5 keV/channel. Now suppose instead, another particle deposits 40 MeV in another EB
strip. With the same conversion, a channel of 5333 would be expected. However, such an
energetic particle may also induce cross talk on neighboring strips in the same EB detector.
Although the cross-talk signal may be small, the electronics for these other strips will often
be triggered by these cross-talk signals. Most likely, one of these other strips will also be
read out before the strip that measures the 40 MeV particle. Thus, there may be a number
of strips with data in the EB chipboard electronics for this event. The value recorded by the
data acquisition for each strip will depend both on the charge collected and also the order
that this strip is read out during the event. For example, if this strip with the 40 MeV signal
is read out later, and is not the first channel read out, it may not register channel 5333, but
another value such as 5365. The shift depends on the precise order of the readout. There is
a specific shift in channel if the 40 MeV signal is the second channel read out, another if it
47

is the third read out and so on.
Each face of each detector is connected to two ASICs. Chip 0 reads out the even strips,
and chip 1 reads out the odd strips. Chip 0 is read out before chip 1. Depending on how
large the signal is, up to 2 strips on each side may be affected, which means the signal, which
contains the useful data, may be read out as 1st, 2nd, 3rd, or even 4th strip due to the cross
talk between neighboring channels. In addition, there can be real particles in the detector
and in the ASIC, doubling the number of nonzero readout channels. The actual digitized
channel number recorded is affected by this order, and the readout order shift must be
removed during the analysis process. The average correction for the order on the chipboard
is listed below:
• 1st channel read out = no shift
• 2nd channel read out = 22-24 channel shift ∼ 180 keV
• 3rd channel read out = 11-13 channel shift ∼ 80 keV
• 4th channel read out = 8-10 channel shift ∼ 70 keV
The shift in the channels due to the readout order on the chipboard is independent of the
size of the original signal. It only depends on the order in which that signal is read out.
There is another readout order correction. Up to 8 chipboards with 2 chips each are
housed on one motherboard. The order in which the chipboard is readout also affects
the corresponding channel number. This correction is only 2-3 channels, however, at 7.5
keV/channel, correcting for this can improve the resolution by 10-20% [56]. The motherboard and chipboard corrections just sum together, although they affect the channel number
in opposite directions.
48

The DEs were not affected by the readout order problem because they used external preamplifiers. This confirms that the cross-talk problem involves coupling the digitized control
signals on the ASIC on to the analog signals on the CSA (preamplifier) of the ASIC. In
order to properly calibrate the EF and EB using alpha spectra, one must correct for the
order in which each strip is read out on the chipboard and within the motherboard. This
can affect the calibration by 0.5%. It is important to calibrate the EF and EB as accurately
as possible since their energies are used in the calibration of the CsI crystals described in
the next section.

3.1.2

CsI Crystal Calibration

The photodiodes produce an electronic signal which is linearly proportional to the light
produced by the particle detected in the scintillator. This signal is amplified in the electronics
and digitized by the Analog to Digital Converter (ADC). This may result in small nonlinearities in the relationship between the signal produced by the photodiode and the channel
number calculated by the ADC. There are four possible steps in calibrating proton energies
in a single CsI(TI) crystal: Linearize the electronic channels, identify the particles, calculate
the offset, and calculate the gain. As a first step, these non-linearities are checked and
corrected for by using a precision electronic pulser. Depending on the charge and mass of
the detected particle, the light output from the crystal can be a non-linear function of the
energy deposited by particle in the CsI scintillator. Light output non-linearities have been
previously investigated [57]. The results of prior investigations have been used to correct the
non-linearities of A > 1 particle energies. Such non-linearities are small for protons, but can
be much larger for low energy heavy particles.
49

Figure 3.4: Pulser ramp for Telescope 0, crystal 0. The red triangles are the peaks as found
by ROOT [7].
3.1.2.1

Linearization

To determine the non-linearity, a pulser ramp was used. Electronic pulses with evenly spaced
voltages with known values were sent into the test input of the CsI pre-amplifiers. Because
some pulser amplitudes may be too low or too high to be within the channel range of the
ADC, the middle pulse, with half the maximum voltage value, was recorded for twice as
long, giving twice as many pulses so that it can be easily seen and identified, as in Fig. 3.4.
The middle pulse is then used to identify the voltage values for all channel numbers.
After plotting voltage vs channel number, as shown in Fig. 3.5, each crystal was fit
with a 3rd order polynomial in the range of interest. This polynomial function allowed us
to convert from non-linear channel numbers (up to ≈ 3500) to linearized voltages.

To

ensure the accuracy of the fit, the difference between the input voltage was compared to the
reconstructed voltage from the fit with a 3rd order polynomial. This difference is plotted as
a function of channel number in Fig. 3.6 for crystal 0 from telescope 0. All differences are
50

Figure 3.5: A 3rd order polynomial was used to fit the pulser data due to slight non-linearities
in the electronics. Shown here are the pulser data and fit for one crystal in telescope 0.

51

Figure 3.6: The difference between the input voltage and fit voltage is plotted as a function
of channel number.
smaller than 2 mV. The second stage in the calibration procedure, particle identification is
discussed below.

3.1.2.2

Particle Identification (PID)

In order to calibrate the CsI detectors, each species of particle must be accurately identified.
Particles can be separated based on their charge and mass by taking advantage of the Bethe
Bloch formula

−

4π
nZ 2
e2 2
2me c2 β 2
dE
=
·
·
· ln
− β2
dx
4π 0
me c2 β 2
I · (1 − β 2 )
52

(3.1)

which shows that the energy lost by a particle traversing through a material is proportional
to the square of the charge of the incident particle. Other factors in the formula are β, the
velocity scaled by the speed of light, n, the electron density of the material, and I, the mean
excitation potential of the material. Approximating the kinetic energy by 1 mc2 β 2 , one
2
2
can show that the Bethe Bloch equation can be approximately reduced to dE ∝ mz f (v)
E
dx
where f(v) is a slowly varying function of velocity. Thus, when the energy lost in the silicon is
plotted against the energy deposited in the CsI, the data are organized into bands, of which
each corresponds to a different isotope. Examples of these particle identification bands
can be seen in Fig. 3.7 and 3.8 which are from the collisions 40 Ca+40 Ca and 48 Ca+48 Ca
respectively.
The bottom three bands are the hydrogen isotopes, starting with protons on the bottom,
deuterons in the middle, and tritons on top. The next two bands are the abundant helium
isotopes, helions and alphas. While 5 He is absent, there is a faint 6 He line above the alpha
line. 6 Li and 7 Li can then be seen at the top of the PID. Shown here are fully calibrated
PID for all EF strips and CsI crystals in all telescopes. In order to calibrate a CsI crystal,
PID must be generated for each crystal separately.

3.1.2.3

Offset Extraction

The second step in the calibration procedure is to extract the offset. Ordinarily, this can be
done by running a pulser calibration and finding the channel number corresponding to zero
pulse height in the pulser. However, in this experiment, the pulser chopper system had a
shift in the zero offset which was undetermined. Therefore we used the energy loss in the
telescope to deduce the offset. Partially calibrated PID plots from any reaction system can
be used for this purpose. Once the Es are calibrated, it is known how much energy each
53

Figure 3.7: Calibrated PID for all 40 Ca+40 Ca data.

54

Figure 3.8: Calibrated PID for all 48 Ca+48 Ca data.

55

Figure 3.9: Calibrated EF energy vs raw CsI energy.
proton lost in the thick silicon. By plotting calibrated energy from the EF as a function of
the raw CsI channels for each crystal, PID are obtained similar to that shown in Fig. 3.9.
If the exact thickness of the silicon detectors are known, the energy deposited in the CsI
can be calculated for each proton from the energy lost in the EF, using a stopping power
program such as LISE++1 . One such point could be used to deduce the offset. However,
the thicknesses of the silicon strip detectors are not known precisely enough to determine
the offset from just one point on the E vs CsI plot at low energies. In fact, the thickness is
known to only ≈ ±50 µm. However, this is precise enough to extract the offset using several
points on a E vs CsI curve for protons such as that shown in Fig. 3.9.
1 www.nscl.msu.edu/lise
56

The following procedure was used to extract the offset for the CsI calibration. A few
points along the proton PID line, between 6-10 MeV in EF energy, were used to calculate the
offset. For each point, the energy lost in the EF was known. By assuming nominal thickness
as provided by the manufacturer, the energy deposited in the CsI could be calculated. For
each point, the digitized but uncalibrated energy signal in the CsI, in channel number,
was known. This channel number was converted into a “pulser” voltage obtained by the
linearization procedure. This corrected for the non-linearity of the electronics. The energies
were then plotted against values in volts for the corresponding signal in the CsI detector.
This correlation was used to extrapolate the y-intercept for each crystal.

3.1.2.4

Gain Extraction

To extract the gain in the calibration, one could just use the channel value at which protons
punch through the CsI crystal. This energy, Epunchthrough = 114 MeV can be accurately
calculated by energy loss programs using the known thickness of the CsI crystal. However,
the average error in this punch through data point would alter the calibration significantly.
A better calibration can be obtained by elastically scattering protons into the detectors.
This was achieved by degrading a beam of 40 Ca to E = 25 MeV and impinging it on a
plastic target. Two of the reactions: The elastic and inelastic scattering of 40 Ca nuclei on
the Hydrogen in the polyethylene target

40 Ca + p → 40 Ca + p

(3.2)

40 Ca + p → 40 Ca∗ + p

(3.3)

and

57

Figure 3.10: PID for 40 Ca+CH2 at E/A = 25 MeV.
can be used to calibrate the detector using the beam energy and two-body kinematic relations. The angle of the 40 Ca does not deviate significantly from 0 and should not enter any
4π detectors. If we detect a recoil proton in HiRA, no other particles should enter any 4π
detectors. To reduce background, we require that only one recoil proton is observed in our
proton recoil calibration data.
From conservation of energy and momentum, protons scattered from the reaction 40 Ca+CH2
will have known energies depending on the proton scattering angle, θ, with respect to the
beam-line. Protons scattered from the plastic target can be seen clearly at the high energy
end of the PID line in Fig. 3.10. Inelastically scattered protons can also be seen in the figure,
however, they cannot always be cleanly separated from protons emitted from other reactions.

In order to cleanly separate not only elastically scattered protons, but also inelastically
scattered protons, only protons were selected from the PID and their energy was plotted
58

Figure 3.11: The proton energies plotted as a function of θ in the laboratory frame.
as a function of θ as shown in Fig. 3.11. Protons with the highest energy at each angle
are elastically scattered. Protons that were inelastically scattered have slightly less energy
because some energy was transferred to the 40 Ca to excite it. Protons with the lowest
energies were the result of reactions such as 40 Ca+12 C that were not completely removed
by the requirement of zero charged particle multiplicity in the 4π.
After the relevant protons were selected, both the elastic and inelastic groups were divided
into 4 angular bins for each crystal. The elastic protons in each bin were assigned an energy
based on angle (θ) from Fig. 3.12, which shows energy as a function of angle, minus any
energy losses in foils.
The inelastically scattered protons result from 40 Ca being in a 3- or 2+ excited state.
These states are close in energy and could not be distinguished from one another. Therefore,
the energy assigned to a proton was a weighted average of the energies expected from these
two states. It was weighted by the expected contribution from each cross section for the
59

Figure 3.12: Energy as a function of θ (laboratory) for elastically scattered protons.

given angle. The cross sections are shown in Figs. 3.13 and 3.14

The energy distributions

for each inelastically scattered state are shown in Figs. 3.15 and 3.16.
Energy losses through various foils were taken into account to determine the actual energy
of a recoil proton entering a CsI crystal. Due to the trajectory, each proton actually traveled
through a slightly different amount of material. The effective thickness of each material was
calculated from the angle of the proton, and then the energy loss in Fig. 3.17 was multiplied
by this effective thickness for a total energy loss due to the material. Energy losses were
calculated sequentially. First, the proton must travel through the second half of the target,
assuming that the reaction occurs in the center of the target. Then, the protons travels
through Sn-Pb foil. Next, it travels through the mylar foil. Finally, it travels through the
dE before entering the thicker silicon.
It was mentioned earlier that the punch through could be utilized to calculate the gain for
each crystal. The crystals were 39 mm long on average resulting in a punch through energy
of ∼114 MeV. Protons punching through a crystal could have slightly more or less energy
60

Figure 3.13: Cross sections for the 3- state of 40 Ca as a function of θ in the center of mass
system.

Figure 3.14: Cross sections for the 2+ state of 40 Ca as a function of θ in the center of mass
system.

61

Figure 3.15: Energy as a function of laboratory angle for inelastically scattered protons from
the 3- excited state of 40 Ca.

Figure 3.16: Energy as a function of laboratory angle for inelastically scattered protons from
the 2+ excited state of 40 Ca.

62

Figure 3.17: Energy loss through relevant foils as a function of proton energy.

63

Figure 3.18: Projection of proton PID for selection of the punch through channel.

due to the angle it went through the crystal, and any scattering that took place. SRIM2
calculations showed a shoulder in the punch through similar to that seen in Fig. 3.18. The
channel value that corresponds to the punch through energy is that at the half-max of the
shoulder.

The last step in the proton calibration is combining all of the information from linearization and the offset to get the gain. This is shown in Fig. 3.19. The offset has been subtracted
from all points, so the lowest point is (0,0). The middle points are from elastic and inelastic
scattered protons. The highest energy point is the punch through energy.

2 http://www.srim.org/
64

Figure 3.19: Calibration for crystal 0 of telescope 0.

3.1.2.5

Calibrating A>1 Particles

The initial calibration calculates the energy of each particle as if it were a proton, referred
to as Ep . Different species of particles interact with the CsI crystal differently, producing
different amounts of light for the same energy deposited. Thus, obtaining the correct energy
for each isotope requires its own energy calibration. For Z=1 particles, there is a linear
relationship to the proton calibration such that

E = aEp + b

(3.4)

while the energy calibration for Z=2 particles can be approximated by an exponential
term [57]
E = aEp + bAc (1 − edEp ).
65

(3.5)

In both of these, Ep is the energy that would be assigned if the particle was a proton and
a,b,c, and d are fitting parameters. To determine the fitting parameters, LISE++ was used
to calculate the energy deposited in the CsI crystal as determined by the energy loss in the
E detector. The punch throughs were also used when available as a data point in the fit.

3.2

4π Array

The 4π array provided a support structure for HiRA, a vacuum chamber, and, more importantly, it was also used as a detector to measure the positions and energies of all emitted
nuclei which do not enter HiRA. This is essential to determine the impact parameter of each
collision, described in Subsec: 3.2.2, which is needed to accurately compare the data to theoretical transport models. In order to calculate impact parameters, the phoswich detectors
must be calibrated.

3.2.1

Phoswich Calibration

The fast and slow plastic scintillators have different decay times as was shown in Table 2.3
allowing the signals to be separated. In the following, we designate DE as the signal from
the fast plastic and E as the signal from the slow plastic. As in the case of HiRA, DE vs
E plots can be used to identify particles due to dE/dx being approximately proportional to
AZ 2 . Particles which stop in the fast plastic deposit no energy in the slow plastic, but the
slow gate still contains a little of the fast signal, resulting in a “punch-in” line with a finite
slope. Some particles such as cosmic rays and neutrons deposit energy in the slow plastic
without going through the fast. These particles fall in the neutral line which has a slight
slope due to the fast signal containing some of the slow rising signal. These lines are shown
66

in Fig. 3.21. The first task in calibrating the phoswiches is to disentangle the fast and the
slow signals so that the punch-in line has an infinite slope and the “neutral line” has a slope
of 0. The intersection of the lines corresponds to 0 energy, and any difference from 0 is due
to offsets in electronics. The electronic offsets are extracted and subtracted to place this
intersection at 0.
Calibrating each detector is a three step process. First, PID plots are generated by
plotting the fast plastic signals versus the slow plastic signals from the raw uncalibrated
data as shown in Fig. 3.20 for one ball phoswich and one FA phoswich.

The slope for

both the neutral and the punch-in lines are found as well as the intersection point between
the two lines as shown in Fig. 3.21.

Using Eqs. 3.6 and 3.7, the dE and E signals are

disentangled from one another where Mn is the slope of the neutral line, Mp is the slope of
the punchthrough line and X0 and Y0 are the coordinates of the intersection point.

F ast = (dE − Y0 ) − (E − X0 ) ∗ Mn

(3.6)

Slow = (E − X0 ) − (dE − Y0 )/Mp

(3.7)

Once the fast and slow components are disentangled, as shown in Fig. 3.22, the punch-ins
listed in Table 3.1 can be utilized to calibrate the phoswiches.

Particle Type
p
d
t
He
Li

Punch-in Energy [MeV]
17
24
28
70
140

Particle Type
Be
B
C
N
O

For the ball phoswiches, the

Punch-in Energy [MeV]
214
300
380
483
593

Table 3.1: Energy punch-ins for ball phoswiches [1].
67

Figure 3.20: Raw PID plots from a ball phoswich (top) and a forward array phoswich
(bottom). Axes are in channels.

68

Figure 3.21: Raw PID from a ball phoswich with important calibration features labeled.

69

Figure 3.22: Unfolded (disentangled) PID from a ball phoswich (top) and a forward array
phoswich (bottom). Axes are in arbitrary units.

70

light response function of the fast plastic is described by Eq. 3.8 while the response of the
slow plastic is described by Eq. 3.9 where Ci is a gain factor specific to each detector [58].
√
Li = Ci dE

(3.8)

E 1.4
Li = C i
Z 0.8 A0.4

(3.9)

The gain factor for the slow plastic, Ci , (for both the ball and the FA) was found by using
the energy loss in the fast plastic and LISE++ as in the case of the CsI crystals.
The Forward Array is calibrated similarly with the punch-in energies given in Table 3.2.
For the fast phoswich, the energy is still given by the square of the light output, but the

Particle Type
p
d
t
He
Li
Be
B

Punch-in Energy [MeV]
13
17
20
50
99
152
212

Particle Type
C
N
O
F
Ne
Na
Mg

Punch-in Energy [MeV]
269
341
419
515
591
687
767

Table 3.2: Energy punch-ins for Forward Array phoswiches.

slow phoswiches have a slightly different calibration given by Eq. 3.10 [5].
E 1.35
Li = Ci
Z 0.772 A0.386
The calibrated spectra for both a ball and a FA detector can be seen in Fig. 3.23.
71

(3.10)

Figure 3.23: Calibrated PID from a ball phoswich (top) and a forward array phoswich
(bottom).

72

3.2.2

Impact Parameter Selection

The impact parameter of a nuclear reaction is defined as the distance between straight line
trajectories of the centers of the nuclei. If the two nuclei hit head on, the impact parameter
is 0 fm and the reaction is classified as central. At the other end of the spectrum, if the nuclei
barely graze each other, the reaction is characterized as peripheral. These limits correspond
to zero cross section, so ranges of impact parameters must be combined to obtain central
collision data. In this dissertation, the impact parameter range of 0 ≤ b ≤ 4 fm is defined
as central for 40 Ca+40 Ca and 48 Ca+48 Ca reactions.
BUU transport simulations show that pp correlation functions are related to different
sources depending on the impact parameter, or equivalently, the overlap region of the nuclear
reaction. In the simulation, impact parameter is an input variable. In order to compare
between theory and experimental data, it is important to be able to select ranges of impact
parameters in the data. This work focuses on central reactions because the correlation
functions were predicted to be sensitive to the density dependence of the symmetry energy
at low impact parameters.
The 4π array was used to extract an experimental impact parameter for each event.
Different observables can be used to select ranges of impact parameter in nuclear reactions.
Two commonly used observables are charged particle multiplicity, Nc , [24,59] and transverse
energy, Et [60]. Nc is the number of charged particles detected by the 4π array for a given
event. If two particles hit the same detector, they are counted as one. Et , is defined by
Nc
Et =

Ei sin2 θi

i=1
73

(3.11)

where θ is the laboratory angle between the particle’s trajectory and the beam axis and E
is the measured energy in the laboratory frame.
If one assumes a strictly monotonic relationship between impact parameter and an observable such as Nc or Et , labeled X, the reduced impact parameter is calculated by
∞
P (X)
b(X)
ˆ
=
b(X) =
bmax

X
∞

(3.12)
P (X)

X(bmax )
following Ref. [59]. Two other assumptions are that X increases as the reaction becomes
more central and the fluctuation of the parameter x at a fixed value of b is small compared
to the change of x with b. This assumption is only approximately valid and will be addressed
in section 5.1.1. In Eq. 3.12, P(X) refers to the probability distribution of observable X, ˆ
b(X)
is the reduced impact parameter and bmax is the maximum impact parameter consistent
with the experimental setup. bmax is calculated by

1/3
1/3
bmax = R1 + R2 = 1.25A1 + 1.25A2

(3.13)

where A is the mass.
For 40 Ca+40 Ca, bmax = 8.55 fm while for 48 Ca+48 Ca, bmax = 9.09 fm. For an impact
parameter of b = 4 fm, and the sake of simplicity, a reduced impact parameter of ˆ = 0.5
b
was used for both.
The impact parameter that can be probed in an experimental device such as the 4π
detector ranges from 0 up to the largest impact parameter contained in the minimum bias
74

Figure 3.24: The relationship between the total multiplicity in the 4π array and the multiplicity in the ball elements of the array is shown in the upper panel. The lower panel shows
the mean total multiplicity as a function of ball multiplicity.

trigger. The minimum bias trigger required a multiplicity of at least two on the 4π array.
During the data analysis it was realized that the FA was not triggered during the minimum
bias run. Thus the procedure of establishing a relationship between the multiplicity or Et
and impact parameter was first performed using the Ball detectors. Then, a regular data run,
in which both the Ball and FA detectors were working, was used to relate the ball multiplicity
to the total multiplicity. This total multiplicity was related to the impact parameter selected.
The correlation between ball multiplicity and total multiplicity can be seen in Fig. 3.24.
75

Figure 3.25: Reduced impact parameter as a function of multiplicity.

The total multiplicity was extrapolated from this correlation to what it would have been
if the FA was working using a second order polynomial. The same procedure was employed
for Et which lead to the expression

2
3
T otalEt = 10.970 + 1.088 · BallEt − 4.076e−4 · BallEt + 4.976e−7 · BallEt .

(3.14)

Once Et and Nc distributions are known from the minimum bias data, a reduced impact
parameter, ˆ can be calculated for each using Eq. 3.12.
b,
The reduced impact parameter for 40 Ca+40 Ca as functions of Nc and Et are shown in
Figs. 3.25 and 3.26 respectively.

It can be seen that for a centrality condition of ˆ < 0.5
b

this corresponds to Nc ≥ 8 or Et ≥ 150 MeV. There was no usable minimum bias data for
48 Ca+48 Ca, therefore, the same conditions to define centrality are used for both reaction
systems. This is a reasonable assumption since Et is dependent upon charged particles and
both reaction systems have the same number of charged particles.
76

Figure 3.26: Reduced impact parameter as a function of transverse energy.
In Fig. 3.27 it can be seen that Nc and Et are correlated, so that if one observable
is a good requirement for impact parameter, they both are. However, Nc saturates as Et
becomes large. Phair showed [60] that Et was a slightly better indicator of centrality than
Nc . While Nc was used for some preliminary analysis, Et was used to select central events
for the purpose of this thesis.
There can be a variety of reasons why the multiplicity in the 4π array may change over
time. Two explanations are changing thresholds on the detector elements, and oxidation of
the Ca target. The latter is explored below.

3.3

Quantification of Calcium Target Oxidation

Calcium is highly reactive with oxygen, forming CaO. The percentage of oxygen on a natural
calcium target exposed to air is shown in Fig. 3.28 where the percentage was calculated from
the increase in weight of the target over time.

Fig. 3.28 shows that a target left in air

for three hours is nearly 40% oxygen. There was a concern that the targets may oxidize
77

Figure 3.27: Total multiplicity as a function of transverse energy.

Figure 3.28: Oxidation of natural Ca target as a function of time.

78

during the experiment since the vacuum in the 4π chamber was only in the range of 10−5
Torr. Targets were visually inspected before putting them in the target carousel and upon
removing them from the carousel. A calcium target will be shiny but an oxidized target will
appear duller. The 40 Ca target actually appeared more shiny after 226 hours in vacuum,
and the 48 Ca target did not change in appearance after 63 hours in vacuum.
To monitor target oxidation, the reactions 40 Ca+mylar and 48 Ca+mylar were studied
to observe the effects of the beam reacting with oxygen or carbon instead of calcium. Mylar,
(polyethylene terephthalate or PET) has the chemical formula C10 H8 O4 . Aluminized mylar
was chosen to represent an oxygen target due to high carbon and oxygen content, wide
availability, and solid form. The thickness of the aluminum layer on the mylar foils was
approximately 2-3 µg/cm2 which is small compared to the 200 µg/cm2 thickness of the
mylar foils, so reactions on the aluminum layer can be neglected.
When calcium reacts with mylar, it is reacting with either carbon, oxygen or hydrogen,
all of which have fewer protons than calcium. This results in a lower charged particle
multiplicity distribution that peaks at a lower multiplicity for a mylar target than for a
calcium target. The average multiplicity in the 4π array for the mylar target is about five,
and it does not extend to as high multiplicities as the multiplicity spectra measured on the
Ca targets, as shown in Fig. 3.29. If a calcium target is oxidizing, its multiplicity should
decrease significantly over time. Fig. 3.29 also shows 4π multiplicity for 40 Ca+40 Ca (top)
and 48 Ca+48 Ca (bottom) reactions from the early data and later data. There is a slight
decrease in 4π multiplicity over time for both reaction systems. However, the decrease is
slight for the 48 Ca system. The larger decrease observed in the 40 Ca system mainly reflects
changes in the trigger that were introduced early in the experiment which lead to a small
79

Figure 3.29: Comparison of reactions on Calcium, for early and late data, with reactions on
mylar for both Ca beams.

80

Figure 3.30: Comparison of reactions on Ca, over time, with reactions on mylar for both Ca
beams. Details are given in the text.
loss of efficiency for charged particle detection.
To be more quantitative, we take the difference between the late and earlier runs and show
that with the late run multiplicity in Fig. 3.30. We see that for both targets, there is a few
percent increase in this difference at low multiplicity. If we normalize the mylar spectrum to
the difference in the calcium spectra at low multiplicity, we see that the correction at higher
multiplicities (M>8) consistent with our central collision trigger is negligible.

81

Chapter 4
Exploration of BUU Physics Inputs
In this chapter, a Boltzmann-Uehling-Uhlenbeck (BUU) transport simulation model, developed by Danielewicz and collaborators [41, 42, 43], and introduced in section 1.3, is used to
predict the effects of the density dependence of the symmetry energy on source functions for
central reactions of 40 Ca+40 Ca and 48 Ca+48 Ca at E/A = 80 MeV. The size of the source
is the primary quantity explored, while the shape is not ignored. Appendix C provides details on how the size, r1/2 , is calculated from sources. In addition to the source size, many
observables are explored including lab energy spectra for protons, particle distributions, and
average emission time as a function of momentum. The first section outlines the rest of the
chapter.

4.1

Description of Parameter Space for Inputs

There are many inputs that can be changed within the BUU transport code, some are basically computational while others are physical. Clearly, observables should be stable against
changes in computational inputs, including the size of the computational grid, duration of
82

reaction, and the number of test particles used. With the exception of section 4.2, the computational grid size was ±30 fm in the x and y directions and ±45 fm in the z direction
(beam axis) with a cell size of 0.92 fm in all directions. A time step of .5 fm/c was used with
1000 steps so that each collision took place over 500 fm/c. Either 800 or 1600 test particles
per nucleon were used for calculations in this dissertation, although the effect of using only
200 test particles is explored in section 4.3. In addition, each calculation was performed with
20 independent runs with a different random number seed to enhance statistics.

The effects of impact parameter, momentum dependence of the mean field potential, and
composite production on observables are also explored. Except in section 4.4, all simulations
used an impact parameter of b=1.4 fm. This represents an idealization of what is achievable
experimentally. Using a single impact parameter, however, has the advantage of simplicity
when comparing the effects of one set of transport inputs to another. In addition, the effects
of the symmetry energy are expected to be largest at small impact parameters. Thus, the
trends at b=1.4 fm represent close to the maximum sensitivity to the density dependence of
the symmetry energy that one might expect. Calculations presented in this chapter reveal
that much larger effects are predicted for the NN in-medium cross section than for the density
dependence of the symmetry energy. The combined effects of momentum dependence and
cluster production are explored in section 4.5. Averaging over impact parameters is discussed
later in Ch. 5. This chapter ends with a summary of the predicted sensitivity of the source
sizes to various transport inputs.
83

4.2

Size of Computational Region and Reaction Duration

Two of the first inputs of a numerical simulation that need to be chosen are the computational grid size and the duration of evolution of the reaction. For most calculations, the
computational grid size was ±30 fm in X and Y and ±45 fm in the Z direction (beam axis)
with a cell size of 0.92 fm in all directions. The calculations ran to 500 fm/c and it was
noticed that residues begin to go off of the grid at different times depending on composite
production and momentum dependence of the mean field potential. This could make the
calculation inaccurate because the mean field goes to zero when the residues go off the grid,
influencing some of the experimental observables.
To examine the effect of the grid size, a momentum independent mean field potential and
the Rostock in-medium cross section were employed. The grid size was increased to ±45 fm
in the X and Y directions and ±90 fm in the Z direction. Figs. 4.1, 4.2, 4.3, and 4.4 show
snapshots of the 40 Ca+40 Ca reaction either at 395 fm/c for the regular grid, or at 495 fm/c
for the larger grid. They are all XZ projections of density, summed over all density in the
Y dimension. In Figs. 4.1 and 4.2, there is no cluster production, with the latter having a
larger grid size. In Figs. 4.3 and 4.4 there is cluster production, with the latter having a
larger grid size. The method of cluster production is described in section 4.5.
In the case of the normal sized grid space, Figs. 4.1 and 4.3, the residues start to exit the
grid at about 400 fm/c. While in the case of the large grid, Figs. 4.2 and 4.4, the residues
remain on the grid until the end of the simulation at 500 fm/c. The figures also show that
cluster production has a large effect on the dynamics of the reaction; this is explored further
in section 4.5.
For each reaction, energy and baryon number should be conserved. To evaluate the energy
84

Figure 4.1: XZ plane projection of central 40 Ca+40 Ca reaction at 395 fm/c without cluster
production. The masses of the two large residues were calculated. The sum of the residue
masses is displayed in Table 4.1.

85

Figure 4.2: XZ plane projection of central 40 Ca+40 Ca reaction at 495 fm/c without cluster
production for a larger computational grid size. The masses of the two large residues were
calculated. The sum of the residue masses is displayed in Table 4.1.

conservation, the BUU code outputs the energy per baryon at specified time intervals. For a
40 Ca+40 Ca reaction at E/A = 80 MeV, the energy per nucleon of the projectile and target
in the center of mass is 20 MeV. After taking the binding energy (≈ 8 MeV) into account,
the average energy per nucleon is 12 MeV. Fig. 4.5 shows the absolute energy (top) and the
change in the average energy per nucleon (bottom) as a function of time, where all values
in the bottom plot are with respect to the energy at time=0 fm/c. It is clear that energy
is conserved in all cases within approximately E/A = 0.2 MeV. There is a decrease in the
case of the normal grid size without clusters, shortly after 300 fm/c, which occurs when the
residues exit the grid. That decrease is of the order of 300 keV per nucleon.
Table 4.1 shows how the mass of the system is distributed among the emitted particles.
Regardless of cluster production and computational grid size, particle number is conserved.
86

Figure 4.3: XZ plane projection of central 40 Ca+40 Ca reaction at 395 fm/c with cluster
production. The masses of the five largest residues were calculated. The sum of the residue
masses is displayed in Table 4.1.

87

Figure 4.4: XZ plane projection of central 40 Ca+40 Ca reaction at 495 fm/c with cluster
production for a larger computational grid size. The masses of the four largest residues were
calculated. The sum of the residue masses is displayed in Table 4.1.

88

Figure 4.5: Change in average single particle energy as a function of time, illustrating conservation of energy.

89

It should be noted that the residues are calculated based on one transport run, while the
emitted particles are averaged over twenty runs. This explains why the total mass does not
sum up to exactly 80.
particle
p
n
d
3He
t
mass of residues
total nucleons

BG
30.9
30.5
0
0
0
18.9
80.3

NG
30.9
30.5
0
0
0
18.8
80.2

BGC
11.4
11.4
5.1
6.8
6.7
7.1
80.6

NGC
11.3
11.3
5.1
6.8
6.8
6.2
79.8

Table 4.1: Results of particle conservation for central 40 Ca+40 Ca reactions. BG = Big Grid
without clusters, NG = Normal Grid without clusters, BGC = Big Grid with Clusters, and
NGC = Normal Grid with Clusters.

The source function is the probability of two protons being separated by some distance, r,
at the time the second proton is emitted. As a measure of the space-time extent, the source
function can depend on the time at which the source is evaluated. Selecting protons emitted
before 250 fm/c excludes protons emitted late in the collision. This late emission would
increase the size of the source. Thus, the source will be larger when all protons emitted at
later times are included.
In general, source functions are calculated by evaluating the separation distance, r, between pairs of protons emitted in BUU simulations at the time when the second of these
protons are emitted. Figs. 4.6, 4.7, 4.8, and 4.9 compare sources calculated for two selections
of proton transverse momentum, with and without cluster production.

They show source

functions for high (circles) and low (squares) transverse momentum cuts for four cases: large
grid no clusters (upper left), large grid with clusters (upper right), normal grid no clusters
(lower left), and normal grid with clusters (lower right). The sources for protons emitted
90

Figure 4.6: Effect of using a time cut on source functions for 40 Ca+40 Ca without cluster
production. r1/2 sizes for the sources are given in Table 4.2.

91

Figure 4.7: Effect of using a time cut on source functions for 40 Ca+40 Ca with cluster
production. The legend is the same as the previous figure. r1/2 sizes for the sources are
given in Table 4.2.

92

Figure 4.8: Effect of using a time cut on source functions for 48 Ca+48 Ca without cluster
production. r1/2 sizes for the sources are given in Table 4.2.

93

Figure 4.9: Effect of using a time cut on source functions for 48 Ca+48 Ca with cluster
production. The legend is the same as the previous figure. r1/2 sizes for the sources are
given in Table 4.2.

94

with t<250 fm/c are in red open symbols and sources for all protons emitted with t<500
fm/c are in black closed symbols. Figs. 4.6 and 4.7 show the results for the 40 Ca+40 Ca reaction and Figs. 4.8 and 4.9 show the results for the 48 Ca+48 Ca reaction. The figures show
that the source becomes somewhat larger with increased time. The effect is small for high
pT protons, but stronger for low pT protons which is consistent with the notion that slower
protons are often emitted later in time from a cooling, expanding source. The sensitivity to
grid size in these figures is very small.
More quantitative information can be obtained by constructing better measures of the
space-time extent of these sources. One measure of the space-time extent of the source
is the radius at which the source function decreases to 1/2 of its maximum value, r1/2 ,
described in Appendix C. The r1/2 size for sources in Figs. 4.6, 4.7, 4.8, and 4.9 are given
in Table 4.2. For calculations without clusters, 4-5 protons after emitted after 250 fm/c,
whereas only 1 proton is emitted after 250 fm/c for calculations with clusters. The source
size does not change significantly with changes in the size of the grid. Thus, the normal grid
size is adequate for the present purposes.
The total transverse momentum cut has a large effect on the source size, with higher
total transverse momentum having a smaller source. This is consistent with the notion that
faster protons come from the pre-equilibrium source, early in the reaction, when it is smaller.
This larger size for low energy protons can be a reflection of both the physical size of the
source, which is expanding with time, and the lifetime for emission, which is longer for lower
energy protons. The production of clusters also influences the size of the source, making it
larger. If two protons are close together in space after emission, they are more likely to be
in a region with a larger density in phase space, making it more likely that a proton will
95

cluster with other nucleons to form a mass two or three cluster. By depleting the highest
density regions, it reduces the number of pairs of protons with a small distance between
them and effectively increases the size of the source. Cluster production reduces the relative
phase space density. The projection of relative phase space density into coordinate space is
the source distribution.

40 Ca+40 Ca
pT <150 MeV/c
t<250 fm/c
pT >150 MeV/c
t<250 fm/c
pT <150 MeV/c
t<500 fm/c
pT >150 MeV/c
t<500 fm/c
48 Ca+48 Ca
pT <150 MeV/c
t<250 fm/c
pT >150 MeV/c
t<250 fm/c
pT <150 MeV/c
t<500 fm/c
pT >150 MeV/c
t<500 fm/c

BG
r[fm]

NG
r[fm]

BGC
r[fm]

NGC
r[fm]

6.39±0.20

6.27±0.25

6.96±0.18

7.12±0.21

4.27±0.25

4.25±0.25

4.84±0.23

4.64±0.21

7.23±0.25

6.94±0.20

7.30±0.24

7.35±0.23

4.30±0.23

4.25±0.25

4.86±0.23

4.65±0.22

6.86±0.21

6.77±0.24

7.73±0.24

7.70±0.23

4.59±0.19

4.63±0.20

5.03±0.23

5.27±0.29

8.03±0.19

7.82±0.23

8.13±0.20

8.02±0.20

4.65±0.22

4.64±0.21

5.09±0.21

5.29±0.30

Table 4.2: r1/2 values for each combination of grid size and cluster production for both
reaction systems and two transverse momentum cuts. BG = Big Grid without clusters, NG
= Normal Grid without clusters, BGC = Big Grid with Clusters, and NGC = Normal Grid
with Clusters.

By comparing calculations with the big grid without clusters (BG) to calculations with the
normal grid without clusters (NG) and calculations with the big grid with clusters (BGC) to
calculations with the normal grid with clusters (NGC), one can see that the normal grid size
of 30 by 30 by 45 fm3 is sufficient for our purposes. The source size does not change, within
uncertainty, if the grid is made larger. The source gets smaller if only protons emitted before
96

250 fm/c are included because protons emitted at later times are excluded which increases
the space-time extent of the source, however, all protons should be included in the source
regardless of emission time because emission time is not independently measured. We can
only make selections on measurable observables.

4.3

Influence of Number of Test Particles

In BUU calculations, a specified number of test particles are created for each nucleon in the
reaction in order to better simulate the continuous phase space distribution corresponding
to the Wigner transform of the one body distribution function. These test particles collide
with each other and can be emitted from the source. It is important that the calculations
are stable with changes in test particle number. Too few test particles per nucleon can cause
too large fluctuations in the mean field potential and too large fluctuations in the Wigner
transform which is used to calculate the Pauli blocking in the collision term. There is no hard
limit on the minimum value of test particles used, but the number of test particles should
be greater than about 200 to achieve stability in the mean field and phase space distribution
for the Danielewicz version of BUU [61]. Even with 200 test particles per nucleon, there are
additional fluctuations not seen with higher numbers of test particles such as 800 or 1600,
which is demonstrated below. Calculations with a momentum dependent, soft isoscalar
mean field, but without cluster production, were performed to explore the dependence on
test particle number. The calculations all used the Rostock in-medium cross sections and a
density dependent symmetry energy exponent term, γ=0.7. Calculations shown elsewhere in
this dissertation employed 800 or 1600 test particles per nucleon. Here, the calculation with
200 test particles is examined to explore the computational stability against test particle
97

number.
In Fig. 4.10 snapshots of the 40 Ca+40 Ca reaction are compared for 3 values of test
particle (tp) number, 200, 800, and 1600. The size of the large residue is unchanged when
scaled by the number of test particles. Fig. 4.10 shows that the residues are farther apart
for 1600 and 800 tp than for 200 tp. In the case of 200 tp, the centers of the residues are
separated by about 67 fm. In the case of 800 and 1600 tp, the centers of the residues are
separated by about 71 fm. The case of 200 tp was examined with 4 different random seeds
to probe whether this difference in separation distance was real, or a result of fluctuations.
The velocities of residues , however, becomes more sensitive to fluctuations when the
calculation evolves with fewer test particles. Fig. 4.11 shows how the residue velocities can
fluctuate, for a small number of test particles, from collision to collision in the simulation.
Different collisions were simulated with different random number seeds, and the variation in
the separation of the nuclei at 300 fm/c is consistent with the variation with test particle
number. For example, in the case of 200 tp with 4 different random seeds, the separation of
the centers of the residues varies from 68-74 fm. This emphasizes the importance of averaging
observables over many runs.
Fig. 4.12 shows kinetic energy for protons emitted between 18-58◦ in θ with respect to
the beam axis. The lab energy spectra is nearly identical for 800 or 1600 tp. In the case
of 200 tp, there are fewer protons emitted with low energy, and more protons emitted with
energies between 40-125 MeV. The low energy regime is dominated by long-lived emission
from the residues. Its relative absence for the calculation with 200 tp suggests that the
source might be smaller when fewer test particles are used.
In the case of 800 or 1600 tp, the source has a nice smooth shape as seen in Figs. 4.13
98

Figure 4.10: Snapshots of the 40 Ca+40 Ca reaction in the XZ plane at 300 fm/c. These are
thin slices in the Y plane centered about y = 0. The top panel is for a reaction with 200
test particles, the middle panel is for 800 test particles and the bottom panel is for 1600 test
particles.

99

Figure 4.11: Snapshots of the 40 Ca+40 Ca reaction in the XZ plane at 300 fm/c. Thin slices
in the Y plane centered about y = 0. All panels are for 200 test particles, with different
random number seeds.

100

Figure 4.12: Lab energy spectra for protons emitted from 40 Ca+40 Ca reaction for three
different values of test particle number.

101

and 4.14 and the sources are similar in shape. The source function resulting from protons in
the 200 tp case has a somehwat larger tail at large r, especially with high pT , and a slight
peak at low pT . It is not clear how significant the peak is, as the statistics at low r are
small.
Table 4.3 shows r1/2 values for simulations with different numbers of test particles. While
there may be some differences in dynamics with small test particle number, the differences
in r1/2 are statistically insignificant.

40 Ca+40 Ca
pT <150 MeV/c r1/2
pT >150 MeV/c r1/2
48 Ca+48 Ca
pT <150 MeV/c r1/2
pT >150 MeV/c r1/2

200 tp
r[fm]
5.59±0.55

800 tp
r[fm]
6.10±0.30

1600 tp
r[fm]
6.32±0.25

3.84±0.29

3.95±0.19

4.05±0.19

5.99±0.83

6.66±0.30

6.97±0.21

4.57±0.38

4.26±0.25

4.26±0.25

Table 4.3: r1/2 values for each value of test particle number for low and high pT and both
reaction systems.

4.4

Sensitivity of Observables to Impact Parameter

This section explores the effects of impact parameter on transverse energy and source size.
For the source in these calculations, emitted protons were only included if the laboratory θ
was between 18-58◦ , which corresponds roughly to the acceptance of HiRA. For all values
of b explored here, 800 test particles were used with momentum dependent interactions and
no cluster production. A somewhat soft symmetry energy exponent of γ=0.7 was used.
All three cross sections reduction schemes are explored in this section: free, Rostock, and
screened with η=0.7 for all collisions.
102

Figure 4.13: A comparison of the shape of source functions for different numbers of test
particles used. The upper quadrants are for low pT for 40 Ca+40 Ca. The bottom quadrants
are for high pT for the same reaction system.

103

Figure 4.14: A comparison of the shape of source functions for different numbers of test
particles used. The upper quadrants are for low pT for 48 Ca+48 Ca (right). The bottom
quadrants are for high pT for the same reaction system.

104

The total transverse energy, Et , defined by Eq. 3.11, has been shown to be a good candidate for determining the impact parameter [60]. Figs. 4.15, 4.17 and 4.16 show calculated
values for Et as a function of impact parameter for screened, Rostock, and free in-medium
NN cross sections respectively. The transverse energy is quite different depending on which
NN cross sections are used with larger values for free cross sections than reduced in-medium
cross sections.

In all cases, however, transverse energy is a monotonic function with

respect to impact parameter.
Source functions were constructed as a function of impact parameters for this set of
simulations. Figs. 4.18, 4.19, and 4.20 show the impact parameter dependence on source
size for screened, Rostock, and free cross sections, respectively. The source radii generally
decrease with impact parameter. The size of the source becomes increasingly dependent on
impact parameter at larger values of b.

Attention must be paid to impact parameter

selection when precision comparisons are made between theory and data.
The impact parameter dependence can be understood qualitatively within the participant
spectator model. In this picture, most of these protons originate from the participant zone
offered by the geometric overlap of the projectile and target nuclei. The collision creates
a participant source that expands to the freeze-out density, ρf , where the mean free path
1
ρσ becomes approximately equal to the volume of the expanding system. At this point, the
protons are emitted. The observed source radius, R ∝ (N/ρf )1/3 usually depends on the
number of participant nucleons and on the freeze-out density.
In the participant spectator model, the value of r1/2 at b = 0 fm reflects the freeze-out
density when all nucleons are in the participant source. From the value of r1/2 at b = 0 fm in
Figs. 4.18 and 4.20 one can estimate that the freeze-out density is a factor of (4.5/7)3 = 0.26
105

Figure 4.15: Transverse energy as a function of impact parameter for 40 Ca+40 Ca reaction
(top) and 48 Ca+48 Ca reaction (bottom) for the screened cross section reduction.

106

Figure 4.16: Transverse energy as a function of impact parameter for 40 Ca+40 Ca reaction
(top) and 48 Ca+48 Ca reaction (bottom) for the Rostock cross section reduction.

107

Figure 4.17: Transverse energy as a function of impact parameter for 40 Ca+40 Ca reaction
(top) and 48 Ca+48 Ca reaction (bottom) for the free cross sections.

108

Figure 4.18: r1/2 of source functions resulting from screened cross section reductions as a
function of impact parameter. The upper quadrants are for a low transverse momentum cut
<150 MeV/c for 40 Ca+40 Ca (left) and 48 Ca+48 Ca (right). The bottom quadrants are for
a high transverse momentum cut >150 MeV/c for the same reaction systems.

109

Figure 4.19: r1/2 of source functions resulting from Rostock cross section reductions as a
function of impact parameter. The upper quadrants are for a low transverse momentum cut
<150 MeV/c for 40 Ca+40 Ca (left) and 48 Ca+48 Ca (right). The bottom quadrants are for
a high transverse momentum cut >150 MeV/c for the same reaction systems.

110

Figure 4.20: r1/2 of source functions resulting from free cross sections as a function of impact
parameter. The upper quadrants are for a low transverse momentum cut <150 MeV/c for
40 Ca+40 Ca (left) and 48 Ca+48 Ca (right). The bottom quadrants are for a high transverse
momentum cut >150 MeV/c for the same reaction systems.

111

smaller for the free cross sections than for the screened cross section reduction. This is
not surprising given that the freeze-out varies as σ −3/2 . The decrease of the source size
with impact parameter varies as the cube root of the number of nucleons contained in the
participant volume, which decreases monotonically with impact parameter and vanishes at
large impact parameter where the nuclei do not interact.
Figs. 4.18, 4.19, and 4.20 also show that the source is systematically larger for the neutron rich system for both momentum cuts. The source is also larger for proton pairs at
low transverse momenta than at high transverse momenta in both reaction systems. The
sensitivity to cross section is mainly for low momentum pairs.

4.5

Sensitivity of Observables to Cluster Production and
Momentum Dependence

In heavy-ion reactions, clusters and fragments are produced in addition to free nucleons. An
option was added to this BUU simulation code [41,42] such that cluster production could be
enabled, producing clusters with A ≤3. When this option is enabled, deuterons are created
by three nucleon collisions, and tritons and helions are created by four nucleon collisions,
where the extra particle is needed in order to conserve 4-momentum in the formation of
the cluster. The formation of clusters is a process which is essentially an inverse to the
break-up of an incident cluster that collides with the nucleus. A deuteron is formed if two
nucleons collide within an area described by the NN cross section while a third nucleon is
within a certain phase-space volume centered about the center of mass of the first two. For
the formation of helions (3 He) and tritons, two nucleons must be within the approximate
112

phase-space volume surrounding the two colliding nucleons. NNd and dd channels are not
included in the formation of A=3 clusters.
The mean field potential is attractive when the nucleon has a low relative momentum with
respect to the mean field, and repulsive when the nucleon has a high relative momentum.
This is due to the different length scales probed in the two cases, giving rise to a momentum
dependence of the mean field potential. Microscopically, one can expect a momentum dependence if the NN force originates from the exchange of virtual particles, or when the exchange
potential in Hartree-Fock is important. To first order in p2 , this momentum dependence can
be approximated by replacing the mass in the kinetic energy term with an effective mass for
the nucleons. Thus the momentum dependence is sometimes described as an effective mass
correction. In the BUU model, the momentum dependence is modeled beyond first order in
p2 . Fig. 4.5 showed that using test particles leads to fluctuations in the net energy of the
system with energy rising over time due to the diffusive process. This is a larger effect when
a momentum dependent mean field is employed because driving terms in both of Hamilton’s
equations fluctuate instead of just one, as in the case of momentum independent mean field.
The exact momentum dependence is not fully understood, therefore, calculations were
performed to determine the sensitivity to the momentum dependence. For the momentum
dependent (MD) mean field calculation, a soft isoscaler equation of state (EOS) was used
with an incompressibility K =210 MeV, whereas in the momentum independent (MI) case,
a stiff isoscalar EOS was used with K =380 MeV. The incompressibility K, given by Eq. 4.1,
is the curvature of the isoscalar mean field at ρ0 and isospin=0 [47].
∂2 E
K = 9ρ2
0 ∂ρ2 A |ρ=ρ0 ,δ=0,T =0
113

(4.1)

The momentum dependencies were chosen to have a non-relativistic effective mass of 0.7
mN . Variations in the momentum dependence of the mean field for asymmetric matter
were limited to these two options because these two options predict elliptic and transverse
flow observables that are consistent with experimental data [62, 45]. However, these two
optionsmay not make similar predictions at low energies.
Certain effects of producing A=2 [41] and A=3 clusters [42] in this BUU code have
been previously reviewed. It should be noted that inelastic rates for cluster production and
breakup are not implemented, which may be important. This section illustrates the sensitivity of the calculations to the combined effects of including or excluding cluster production
and the momentum dependence of the mean field. In these calculations, the Rostock inmedium cross section reductions and the density dependent power law dependence γ=0.7
for the symmetry energy are utilized. The reaction dynamics, energy conservation, single
particle energy spectra and source functions for four cases are explored: Momentum independent interactions with clusters off (Indep-Off), momentum dependent with clusters off
(Dep-Off), momentum independent with clusters on (Indep-On), and momentum dependent
with clusters on (Dep-On).
Before quantifying differences between these four cases, it is instructive to look at the
dynamics qualitatively. Fig. 4.21 shows that cluster production has a strong impact on the
reaction dynamics.

Without clusters, there are two large A ≈ 14 residues. With cluster

production there is more fragmentation with smaller residues A ≈ 6. Momentum dependence
has a small influence without cluster production on these figures, but a much stronger effect
when clusters are produced. Based on the density snapshots, Dep-Off and Indep-Off should
be very similar in single and multi-particle observables while Dep-On and Indep-On should
114

Figure 4.21: Density snapshots in the XZ plane summed over all Y. Top panels show calculations for momentum dependent mean fields. Bottom panels are for momentum independent
mean fields. Left panels are with cluster production on, and right panels are without cluster
production. All reactions are for 40 Ca+40 Ca.

115

be somewhat different and the difference between Dep-Off and Dep-On should be even larger.
The energy conservation is considered in each of the four cases. Fig. 4.22 shows the change
in single particle energy as a function of time in the reaction.

Momentum independent

reactions tend to conserve energy better, but the energy per nucleon in the momentum
dependent cases is still conserved to within 1 MeV which is similar to what one expects for
this transport code [61]. The valley at early times corresponds to the point at which the nuclei
are passing through each other, showing transparency. At this point it is difficult to calculate
∆E precisely. The drop in energy per nucleon at late times, t>300 fm/c, corresponds to a
large fragment or residue leaving the computational grid.
As energy is released in creating composite particles, emitted protons not involved in
cluster production should get a boost in kinetic energy when cluster production is enabled,
which is shown in Fig. 4.23.

With clustering on, the number of remaining protons in the

interacting system is reduced, and those that remain have a higher average kinetic energy
than with clustering off. Also, momentum dependence seems to enhance kinetic energy
slightly in the case of clustering. One can see that clustering has a much larger effect in the
kinetic energy spectra than making the mean field potential momentum dependent.
With cluster production turned on, protons close together in space are likely to be “eaten
up” by clusters, reducing the number of nucleons close together and reducing the magnitude
of the source function at small r. This effectively increases the size of the source. The
shapes of the sources are shown in Figs. 4.24 and 4.25. For protons with low transverse
momentum, momentum dependence and clustering has a small effect on the shape of the
source, with cluster production having a larger effect than momentum dependence. For the
high transverse momentum selection, both momentum dependence and cluster production
116

Figure 4.22: Change in average single particle excitation as a function of time, illustrating
conservation of energy.

117

Figure 4.23: Laboratory kinetic energy spectra for protons emitted between 18-58◦ for DepOn,Dep-Off,Indep-On, and Indep-Off for the 48 Ca+48 Ca reaction.

118

have a significant effect on the shape of the source.
Table 4.4 quantifies the differences in shape by presenting the size of each source. In all
cases, enabling cluster production systematically increases the size of the source as expected,
although the sources are not always significantly larger. At low transverse momentum, the
momentum dependent mean field leads to a smaller source size, but at larger transverse
momentum, no clear trend can be seen. Also of interest is that in each case, the neutron
rich system has a larger size than the symmetric system, which is expected if the freeze-out
density remains roughly constant.
Dep-Off

Dep-On

Indep-Off

Indep-On

6.36±0.23
4.01±0.18

6.96±0.22
4.81±0.23

6.94±0.20
4.25±0.25

7.35±0.23
4.65±0.22

6.97±0.20
4.28±0.24

7.54±0.38
5.29±0.24

7.82±0.23
4.64±0.21

8.02±0.20
5.29±0.30

40 Ca+40 Ca
pT <150 MeV/c
pT >150 MeV/c
48 Ca+48 Ca
pT <150 MeV/c
pT >150 MeV/c

Table 4.4: r1/2 values for Dep-On,Dep-Off,Indep-On, and Indep-Off for both reaction systems.

4.6

Effects of Symmetry Energy

The symmetry energy has been predicted to have a strong effect on proton emission rate,
average emission time as a function of momentum, and p-p correlation functions [8] calculated
using the IBUU transport code for the 52 Ca+48 Ca reaction at E/A = 80 MeV at b = 0 fm
impact parameter. A principal motivation for this dissertation was a prediction using the
IBUU model that a stiffer symmetry energy would result in a wider correlation function for
high total momentum proton pairs, while no difference in source size would be expected for
low momentum proton pairs, as seen in Figs. 4.26 and 4.27. Fig. 4.27 is the p-p correlation
119

Figure 4.24: Source functions for low pT (top) and high pT (bottom) gates for the
40 Ca+40 Ca reaction system.

120

Figure 4.25: Source functions for low pT (top) and high pT (bottom) gates for the
48 Ca+48 Ca reaction system.

121

function from Fig. 4.26 for high total momentum.
This section explores those observables using a different BUU transport code, developed
at MSU by Pawel Danielewicz and collaborators [41], which has been used previously to
compare sources to experimental data [48, 38]. Free cross sections and momentum independent interactions were employed to be consistent with Ref. [8]. This section also explores
both 40 Ca+40 Ca and 48 Ca+48 Ca reactions at E/A = 80 MeV for central collisions (b=1.4
fm). The results should be consistent with those in Ref. [8], as the inputs are similar.
Fig. 4.28 shows average emission rates for protons emitted over all angles from the neutron
rich 48 Ca+48 Ca reaction for soft (γ=1/3) and stiff (γ=2) symmetry energies. It is clear that
the stiffer symmetry energy enhances early pre-equilibrium proton emission while in the case
of a soft symmetry energy proton emission is delayed. This is consistent with the higher
pressures present in the case of the stiff symmetry energy, forcing protons to be emitted
early with higher momenta. Fig. 4.29 explores the effect of an angular cut between 18-58◦ ,
in the laboratory frame, on the emission rate. Although there are fewer statistics in this
angular region, the trends do not change.
Next, Fig. 4.30 shows the average time of emission as a function of momentum. Emission
is slightly delayed for the soft symmetry energy as is to be expected.

What is striking, is

that the difference for protons with different γ values is very subtle, in contrast to previous
results with IBUU [8], shown in Fig. 4.31. The trends are also different. In BUU, neutrons
are always emitted before protons, on average, regardless of the γ value. In IBUU, for low
energy protons, the stiff symmetry energy delays neutron emission such that protons are
emitted earlier, on average.
There is a difference in emission times for different density dependencies of the symmetry
122

Figure 4.26: Two nucleon correlation functions for a stiff and soft density dependence of the
symmetry energy. The left panels are for low total momentum <300 MeV/c while the right
panels are for high total momentum >500 MeV/c from Ref. [8].

123

Figure 4.27: p-p correlation function for higher total momentum pairs from Ref. [8] renormalized to match peak heights (by Verde for E03045 proposal) so that the widths can be
compared.

124

Figure 4.28: Average emission rate for protons with different symmetry energies for the
48 Ca+48 Ca reaction.

125

Figure 4.29: Average emission rate for protons emitted between 18-58◦ with different symmetry energies for the 48 Ca+48 Ca reaction.

126

Figure 4.30: Average emission time for protons and neutrons for γ=2 and γ=1/3 for the
48 Ca+48 Ca reaction.

127

Figure 4.31: Average emission times of nucleons as a function of momentum [8].

128

energy, and this should be reflected in the size of the source. Figs. 4.32 and 4.33 examine the
r1/2 size of the source as a function of total momentum of the proton pair for 40 Ca+40 Ca
and 48 Ca+48 Ca reactions respectively.

The gates on total momentum of the pair were

0-300 MeV/c, 500-640 MeV/c, and 740-900 MeV/c. It is clear that the sources are the same
size, within error, regardless of momentum cut. It is curious that the strongest difference
between source sizes, although not significantly different, occurs at low total momentum.
To be complete, correlation functions were made from the sources for the same momentum
gates used in [8] P<300 MeV/c and P>500 MeV/c for both the symmetric and neutron rich
reactions. The p-p correlation functions are shown in Fig. 4.34. There is no discernible
difference in magnitude or shape.
Within statistical uncertainties, there is little sensitivity to symmetry energy for source
sizes. For all combinations of momentum dependence, clustering, and in-medium cross sections explored in this dissertation, no combinations showed a sensitivity to the symmetry
energy. For the remainder of the discussion of BUU, γ =0.7 was used to explore the effects
of other parameters in BUU. This value is consistent with recent results [26].

4.7

Comparing Free and In-Medium Cross Sections

The NN cross section is thought to be reduced in nuclear medium, but the exact nature of
this reduction is unknown. This section examines the differences between free cross section
(no reduction), Rostock (Eq. 1.11) and the screened (Eq. 1.12) NN in-medium cross section
reduction. For the screened, a variety of values for η are investigated. The Rostock originated
from a parametrization of microscopic calculations, while the screened is geometrical in
nature. This section also explores the reduction factor, kinetic energy spectra and sources
129

Figure 4.32: r1/2 values, obtained from calculations using the code of Danielewicz, as a function of average total momentum of the proton pair for γ=2 and γ=1/3 for the 40 Ca+40 Ca
reaction.

130

Figure 4.33: r1/2 values as a function of average total momentum of the proton pair for γ=2
and γ=1/3 for the 48 Ca+48 Ca reaction.

131

Figure 4.34: Correlation functions for stiff and soft symmetry energy. The upper quadrants
are for a low total momentum cut P<300 MeV/c for 40 Ca+40 Ca (left) and 48 Ca+48 Ca
(right). The bottom quadrants are for a high total momentum cut P>500 MeV/c for the
same reaction systems.

132

for the various cross sections. For all cross sections, momentum dependent interactions are
employed with no cluster production and 1600 test particles.
The Rostock and screened in-medium reductions reduce the free cross sections by a factor,
which varies from collision to collision. In the case of the screened NN in-medium cross
section, it acts to limit the mean free path to be greater than approximately one nuclear
mono-layer in the limit of no Pauli blocking. In effect, the larger the free cross section,
the more the screened parametrization reduces the free cross section, by construction. The
distribution of the values for the reduction factors of a NN collision are plotted in Fig. 4.35
for 48 Ca+48 Ca. Collisions per test particle are plotted for nn, np, and pp collisions. In
this figure, a cross section reduction factor of 0.4 means that the calculated cross section
is 0.4σf while a cross section reduction factor of 0.9 means the calculated cross section is
0.9σf where σf is the NN cross section in free space.
On average, the reduction in cross section assumed by the Rostock formula is less than
that of the screened cross section formula, that is the Rostock cross sections are closer to
free cross sections. The Rostock reduction also has the same shape distribution for all types
of collisions, although there are more np collisions than nn or pp collisions, which is to
be expected. In regards to the screened reduction, it is clear that np cross sections are
reduced more than nn or pp cross sections due to the np cross section being larger than
the nn and pp cross section. The screened cross section are generally quite reduced with
respect to the Rostock and the free cross sections. The Rostock results in no collisions with
reduction factors less than 0.4 while the screened cross section results in very few collisions
with reduction factors greater than 0.9.
The more collisions a proton encounters, the greater chance it has of being emitted. This
133

Figure 4.35: In-medium cross section reductions for the screened (η=0.7) and Rostock
parametrizations for 48 Ca+48 Ca reaction. The left panel is for nn/pp collisions, and the
right panel is for np collisions.

134

Figure 4.36: Laboratory kinetic energy spectra for the free cross sections and reduced Rostock
and screened (η=0.7) cross sections. Only protons emitted between 18-58◦ are included.

is evident in Fig. 4.36 which shows that the free cross section emits the most protons, and
Rostock, having a small reduction, emits more protons than the screened cross section which
reduces the cross section more. For free cross sections, the energy spectra falls exponentially,
while for reduced cross sections, there is a slight bump in the spectra starting around 50
MeV. Fewer lower energy protons are emitted with reduced cross sections.
As the cross section is decreased, the source gets smaller, which is shown qualitatively in
Figs. 4.18, 4.19, 4.20, in Figs. 4.37 and 4.38 and in Table 4.5. It is, however, more evident
for low total momentum than high total momentum selections, which is consistent with the
kinetic energy spectra.

As you decrease the cross section, you emit fewer low energy
135

Figure 4.37: Source functions for the free cross sections and reduced Rostock and screened
(η=0.7) cross sections. Only protons emitted between 18-58◦ are included. The 40 Ca+40 Ca
reaction is shown for low total momentum 500-640 MeV/c (top) and high total momentum
740-900 MeV/c (bottom).

136

Figure 4.38: Source functions for the free cross sections and reduced Rostock and screened
(η=0.7) cross sections. Only protons emitted between 18-58◦ are included. The 48 Ca+48 Ca
reaction is shown for low total momentum 500-640 MeV/c (top) and high total momentum
740-900 MeV/c (bottom).

137

free [fm]

Rostock [fm]

screened [fm]

low P r1/2
high P r1/2
48 Ca+48 Ca

5.12±0.20

4.77±0.25

3.84±0.22

3.90±0.20

3.98±0.18

3.60±0.19

low P r1/2
high P r1/2

5.56±0.21

5.16±0.22

4.13±0.20

4.23±0.25

4.24±0.25

3.80±0.23

40 Ca+40 Ca

Table 4.5: r1/2 values for 3 choices of cross section for low and high total momentum cuts
and both reaction systems.
protons, protons which most likely contribute to the tail of the source at large r. As you
remove protons from the tail of the source, you proportionally increase the relative number
of protons in the peak, reducing the overall size of the source.

4.7.1

Exploring the Density Dependent In-Medium Cross Section

Thus far, the screened in-medium reduction has been set with η=0.7. Although the size
of the source is not sensitive to the symmetry energy, perhaps it is sensitive to the inmedium reduction. This section explores all values of η from 0.5 to 0.9 in 0.1 increments for
both nn/pp and np collisions. Momentum dependent interactions were used with no cluster
production. The symmetry energy exponent, γ, was set to 0.7.
Fig. 4.39 examines the effect of varying either the np or the nn/pp cross section. When
the nn/pp cross section is varied, the source sizes for both 40 Ca+40 Ca and 48 Ca+48 Ca
show little sensitivity to η. When the np cross section is varied, the source size for both
systems shows a stronger sensitivity to η, with 48 Ca+48 Ca showing the strongest effect.
Fig. 4.40 explores the effect of varying both η’s at the same time. The np cross section
is varied along the y-axis and the nn/pp cross section is varied along the x -axis. The weight
138

Figure 4.39: Plot of r1/2 values for select values of η for both np and nn/pp collisions. When
one type of collision cross section is varied, the other η is fixed to 0.7. Only protons between
18-58◦ in θ were used in calculating the source.

139

Figure 4.40: 2D plot of r1/2 values for all values of η for both np and NN collisions. Only
protons between 18-58◦ were used in calculating the source. The upper quadrants are for low
total momentum 500-640 MeV/c for 40 Ca+40 Ca (left) and 48 Ca+48 Ca (right). The bottom
quadrants are for high total momentum 740-900 MeV/c for the same reaction systems.

140

of each cell is the r1/2 value of the source. In the case of high momentum (bottom panels)
the source varies very little with cross section. However, for low momentum (top panels) the
source varies more significantly. Most important are the diagonal lines of nearly equal r1/2
values for low momentum for both reaction systems. For a given source size, there are many
combinations of np and nn/pp cross sections that give the same result.
We note that the sensitivity to the np cross section would be much greater for the Rostock
style parametrization of the in-medium cross section. Thus it would be of interest to redo
these calculations for the Rostock style cross section or for another one without such extreme
reductions in the in-medium np cross sections.

4.8

Conclusions

This chapter explored many parameters of the BUU transport simulation code. The collision
grid used was of sufficient size even though large fragments or residues may go off the
computational grid near the end of the reaction. Increasing the size of the grid did not
affect any of the observables. Because residues go off the grid and energy is not conserved
as well when this happens, the effect of a time cut was explored. Using a time cut removes
a significant amount of low energy protons emitted during late stages when no clusters are
produced.
Using the Rostock NN in-medium cross section, the effect of the number of test particles
was investigated. 200 test particles appear to be too few, as the reaction dynamics are
different than when more test particles are used. The code is stable against changes from
800 to 1600 test particles. As long as at least 800 test particles are used, and the collision
evolves to 500 fm/c, the results are stable against changes in these computational parameters.
141

Next, the influence of impact parameter on energy spectra and the size of the source
was explored. Both were found to be sensitive to impact parameter, however for b<3, the
sensitivity to b is relatively small. When comparing to experiment, mixed central impact
parameters should be weighted appropriately.
Cluster production had a strong effect on the reaction dynamics of the collision while
momentum dependence of the mean field had a weaker effect. Cluster production increases
the available kinetic energy to protons and increases the size of the source. Because the
source is a probability distribution, removing pairs which are close together during emission
increases the overall size of the source. When cluster production is enabled, protons that are
close to each other are likely to be “eaten up” by cluster formation, resulting in a larger source
size. Without cluster production, the source size is underestimated. The most reasonable
physical options appear to be to calculate cluster production and to include momentum
dependent mean fields.
Although the symmetry energy shows an effect on proton emission rates, the source size
was relatively insensitive to symmetry energy for all gates on momentum and all combinations
of cross section, cluster production, and momentum dependence. This is in contrast to
previous results using IBUU [8].
The screened in-medium reduction reduces the cross section more than the Rostock and
its reduction is considerably greater for the np than for the nn or pp cross sections. Both
kinetic energy spectra and source size were dependent upon the cross section reduction. As
the cross section is decreased, the number of protons emitted with low kinetic energy is
decreased, removing them from the tail of the source and thus decreasing the size of the
source. The reduced cross section also decreases the chance of two neighboring protons to
142

collide before being emitted, resulting in more proton pairs with smaller r in the source
distribution.
The source size might be used to constrain η in the screened in-medium cross section.
However, a given source size can be reproduced by many combinations of nn/pp and np cross
section reductions. Source sizes can constrain the relationship between these reductions but
can not fix the individual values for the reductions in σnp , σnn and σpp .

143

Chapter 5
Experimental Results
Nuclear reactions occur over very short timescales on the order of 10−21 seconds. For
the most part, experimental observables only sample the final stages of the reaction and
theoretical models must be used to access the information about the initial stage of the
collision as well as the dynamics occurring during the reaction. One exception is intensity
interferometry [29, 30], introduced in section 1.2, and known for its sensitivity to the spacetime extent of the source from which particles are emitted.
In this chapter, we extract information about the sources for particle emission by fitting the experimental correlation functions with the Koonin-Pratt formula for two different
parametrizations of the source. The first involves approximating the source by a Gaussian
function of the separation between the two protons at the time of emission. The second
involves a more general source function that can be varied to get an "image" of the source.
We begin this chapter by introducing these two methods. Then we discuss how to calculate
a comparable source using transport theory. We then present the data, extract the sources,
and compare measured source sizes to those calculated via the BUU transport model. All
144

p-p correlation functions are from central events, but different gates on rapidities, transverse
momenta and laboratory momenta and angle are explored. The chapter ends with a brief
presentation of three-particle correlation functions.

5.1

Imaging and Gaussian Fits

We want to study how shape of the source function for proton emission depends on constraints on quantities such as the centrality of the collisions or the momenta of the emitted
protons or both. Recall that the definition of the source function is the probability distribution for the distance between two protons at the time of emission of the second proton. This
source function reflects an ensemble average over nucleus-nucleus collisions subject to the
aforementioned constraints. From the central limit theorem, one might expect that a reasonable approximation to the ensemble averaged source for particle emission would be Gaussian
in form. From the seminal work of Koonin [36], Gaussian sources have been frequently
assumed.
If one assumes a Gaussian two particle source profile, given by

S(r) =

λ
√ G 3
(2 πRG )

2
− r2
4RG
e
,

(5.1)

one can obtain a correlation function using the Koonin-Pratt formula (Eq. 1.4). Then the
parameters of this Gaussian can be obtained by fitting the experimental correlation function
by varying the parameters until the χ2 per degree of freedom (ndf ) is minimized.
There are three free parameters in this fitting procedure; an overall normalization of the
correlation function not reflected in the parametrization of the Gaussian source, the λG
145

parameter, and the size of the source RG , which are parameters of the Gaussian source
function. Correlation functions are often normalized such that at large values of q the
correlation is nearly 1. This makes the overall normalization nearly unity.

The correlation function has some limits to its sensitivity, particularly when emission
involves a superposition of short-lived and long-lived sources [38]. When the λG parameter
is equal to one, in Eq. 5.2, it assumes that all protons used to construct the experimental
correlation function are described by the source. However, if it is Gaussian, it typically
neglects very long timescale emissions, which would contribute to the total source whose
integral must be unity. In general, however, protons are emitted on both short and long
time scales, often described by two different sources, whose combined integral must be unity.
In the rest frame of the source, fast protons are often emitted dynamically from collisions
over short time scales, while slow protons are often emitted from evaporation and secondary
decays which occur over longer time scales. These latter protons still influence the height of
the correlation function even though they are uncorrelated with protons emitted earlier from
the same collision [38]. They do this by contributing to the uncorrelated background without
making a comparable contribution to the correlated peak. The width of the correlation
function maximum is mainly sensitive to the fast emitted protons, but the maximum is
reduced by the factor λG in the presence of a significant contribution from the long timescale emission. In this case, Eq. 5.2 has a general form that reflects other sources of protons.

4π

S(r)r2 dr = λ

(5.2)

Then, the λ parameter can be related to the fraction of short time scale emitted protons
146

according to [38]
λ = f 2.

(5.3)

The fraction of long time scale emitted protons can be written as 1 − f .

Previous work [38] has offered the half width half maximum (r1/2 ) of the source function
as one way to characterize its size. The relation between r1/2 and the size of the Gaussian
source distribution (Eq. 5.1) is given by
√
r1/2 = 2 ln2RG .

(5.4)

In this dissertation, r1/2 is used to quantify sizes of sources.

It is possible to relax the assumption of a Gaussian source, provided the correlation function maximum is sufficiently large and measured with sufficiently high statistical accuracy.
In the "imaging" approach of Ref. [63, 64, 65], the source function inserted in the KooninPratt formula (Eq. 1.4) is not assumed to be Gaussian. Instead, it is parametrized to allow a
more general shape, which is then adjusted to optimally reproduce the correlation function
data.

Thus, there is a fitting procedure that connects terms in the expansion of the source, indexed by j, to measurements of the correlations function, Ci , measured at relative momenta,
qi . Specifically, the source functions S(r) are expanded in a superposition of polynomial
splines,
S(r) =

Sj Bj (r).
j
147

(5.5)

This converts the Koonin-Pratt equation into a matrix equation

Ci = 1 + Ri =

Kij Sj ,

(5.6)

j
where the terms Kij can be obtained from

Kij =

4πr2 drK(qi , r)Bj (r).

(5.7)

As in the case of the Gaussian source parametrization, the unknown coefficients Sj in the
matrix equation above are obtained minimizing the value of the χ2 between the experimental
and theoretical correlation functions. In the following, we used six 3rd order splines to
construct the sources.
When comparing the experimental data to BUU calculations, one can either compare
experimental and theoretical correlation functions or experimental and theoretical sources.
In practice, it is easier to compare the sources extracted from fits to the experimental data
to the theoretical sources provided by the BUU calculations. Since the BUU transport code
only describes the protons emitted on a short time scale, it does not describe the later
evaporative emission. Thus, the BUU source will need to be normalized so that its integral
over r is equal to λ.

5.1.1

Theory Adaptation

Any experimental setup can, in principal, bias the data extracted from it. To avoid this, an
experimental filter can be applied to the theoretical results to select only those particles which
could have been experimentally measured during the experiment. This includes imposing
148

energy thresholds and energy maxima on the protons from the simulations. For example,
only protons within a certain range are able to stop in the CsI crystal without punching
through completely. A more restrictive cut imposed was the requirement that the proton was
emitted at angles such that it would enter a working EF/EB pixel in HiRA and yet not slip
between two CsI crystals. Multiple simulations were performed with different random seeds
to increase the proton statistics after applying the filter. To allow an arbitrary orientation
between the HiRA detectors and the total angular momentum vector of each event, the
protons in the event were randomly rotated 100 times before applying the HiRA experimental
filter, ensuring that all orientations of the reaction plane were sampled.
Most of the BUU simulations in the preceding chapter were done at an impact parameter
of b = 1.414 fm. However, the centrality of the data is determined by Et > 150 MeV which
corresponds approximately to b < 4 fm. The transverse energy decreases monotonically
with impact parameter on the average. Event by event, however, the impact parameter
fluctuates. This fluctuation was studied by Michael Lisa, who analyzed correlation functions
for the 36 Ar+45 Sc system at E/A=80 MeV [66] and showed that event by event fluctuations
are approximately Gaussian in distribution with a standard deviation of about 1.2 fm.
Depending on the cross section employed, the source size changes differently as a function
of b. The impact parameter dependence of the source radii was shown in Figs. 4.18, 4.19,
and 4.20 in the preceding chapter for several assumptions about the NN in-medium cross
sections. This dependence is approximately described by a second order polynomial,

r1/2 (b)th = r1/2 (b0 )th + a1 · (b − b0 ) + a2 · (b − b0 )2

(5.8)

Neglecting Et fluctuations as fixed impact parameter, one needs simply to integrate r1/2 (b)th
149

over the impact parameter interval 0 ≤ b ≤ 4 fm. This is the main effect, and it reduces the
values of r1/2 (b)th by about 5%. Impact parameter fluctuations typically reduce r1/2 (b)th
by an additional 1% compared to what one calculates without taking them into account. In
the following, we consider only the effects of the impact parameter gate and not those of
impact parameter fluctuations when making comparisons between BUU calculations and the
experimental data for average quantities such as r1/2 (b)th . We combine the sources at
different impact parameters within the gate, using the correct impact parameter weighting.

5.2

p-p Correlation Functions

An experimental correlation function, introduced in section 1.2, is constructed by taking a
ratio of yields, as was defined by Eq. 1.2. For two particle correlation functions, such as
p-p these yields are a function of the relative momentum of the pair in the center of mass
of the pair. The numerator is constructed by pairing particles from the same event. In this
thesis, the event-mixing method [33] is used to construct the denominator. Particles from
different events are paired as if they were from the same event, and the relative momentum
is calculated. Any selections on data, such as centrality, are applied to the numerator and
denominator equally. To obtain reasonable error bars, the denominator usually had 15 times
more statistics than the numerator. When the two yields are divided, the result is a twoparticle correlation function.
For this dissertation, p-p correlations were measured over a large angular range, wide
kinematic range, and with high statistics. As a proof of concept, correlation functions were
constructed for 40 Ca+40 Ca and 48 Ca+48 Ca at midrapidity, as shown in Fig. 5.1.

The

isospin symmetric system is shown in red, while the neutron-rich system is shown in blue. At
150

Figure 5.1: p-p correlation functions for both reaction systems at midrapidity,
-0.05<ycms <0.05.

151

first glance, the widths of the correlation functions appear to be very similar, although the
width of the correlation function for 40 Ca+40 Ca is slightly wider than that for 48 Ca+48 Ca.
This corresponds to a larger source for the latter system. This could be a result of the simple
fact that the 48 Ca+48 Ca system is geometrically larger to begin with. Another suggested
reason is that the extra neutrons in the 48 Ca+48 Ca system act to delay or block protons
from being emitted. Further study is needed to decide which of the two explanations is more
accurate.

5.3

Correlation Functions Selected by Laboratory Angle
and Momentum

Experimental measurements at intermediate energies are typically performed with devices
of limited solid angle. Selection by laboratory angle and momentum is potentially more
easily understood than selection by center of mass angle and momentum or by rapidity
and transverse momentum because the experimental efficiency is completely straightforward in the laboratory frame. Many analyses of laboratory correlation functions have been
performed [48, 34, 67], which typically exhibit larger and broader proton-proton correlation
functions for higher momentum proton pairs consistent with these protons being emitted
earlier before the source has had time to expand. Lower momentum protons typically exhibit weaker and narrower correlation functions consistent from the emission of lower energy
protons at later times from a cooling, expanding source. Such trends were well documented
for the 36 Ar+45 Sc system at E/A=80 MeV [48,67] by measurements, Gaussian and imaging
analyses from a device with an angular acceptance that was more limited than HiRA utilized
152

Figure 5.2: Total laboratory momentum of proton pairs as a function of its angle.
in this dissertation.
This section examines the total momentum dependence in the laboratory frame, both
in magnitude and angle, on the source size. The phase space covered by HiRA in the total
momentum as a function of the laboratory angle is shown in Fig. 5.2. To explore the angular
dependence with precision and sensitivity, the lowest and highest third of momentum bins
(500-640 MeV/c and 740-900 MeV/c) were subdivided into three angular bins (18 − 26◦ ,
26 − 33◦ , and 33 − 58◦ ) with comparable statistics. The correlation functions measured in
the experiment for each of these bins are shown in Fig. 5.3. The left panels present results
from 40 Ca+40 Ca and the right panels from 48 Ca+48 Ca collisions. The upper panels are
for protons with low total momentum of the pair in the laboratory frame and the lower
panels are for protons with high total momentum. The correlation functions for the most
backward angles (33 − 58◦ ) in the laboratory frame are represented by black stars, results
for intermediate angles (26 − 33◦ ) are plotted with red triangles, and results for the forward
153

Figure 5.3: Experimental correlation functions from 40 Ca+40 Ca (left) and 48 Ca+48 Ca
(right). The upper panels include protons with low total momentum (500-640 MeV/c) while
the lower panels represent protons with a high total momentum (740-900 MeV/c).

154

angles (18 − 26◦ ) are shown as blue circles. For all, only events with an Et > 150 MeV were
used, selecting more central events. This corresponds to a reduced impact parameter ˆ = 0.5
b
corresponding approximately to b < 4 fm.
Fig. 5.3 shows that at backward and intermediate angle selections, higher momentum
protons correspond to a more pronounced correlation function, consistent with prior measurements. However, the trend of the smaller source for high momentum protons is not
generally true. Indeed it is false at forward angles of 18 − 26◦ . There the correlation functions for faster protons with total momentum 640 ≤ PT ot ≤ 740 MeV/c are less pronounced,
corresponding to a source that is more extended in space-time. This gate corresponds to
proton kinetic energies of 63 ≤ E/A ≤ 80 MeV, which encompasses much of the expected
energy window for nucleons evaporated from an expanding and fragmenting projectile spectator source. The lower momentum gate of 500-640 MeV/c, which corresponds to proton
kinetic energies of 22 ≤ E/A ≤ 43 MeV, contains a mixture of emission from the participant and projectile spectator source, but probably misses most of the contribution from
the evaporative emission of excited projectile-like remnants. Based on these observations,
we conclude that proton emission from expanding and evaporating projectile remnants is
the dominant factor leading to large space-time extent of the source of protons in the high
momentum gate for both reactions.
In Fig. 5.4, the data are shown with reconstructed correlation functions from imaging
and Gaussian methods.

To obtain quantitative information about the emitting source

from the correlation functions the imaging technique is employed to construct the source
function. Such source distributions from the experimental correlation functions are presented
in Fig. 5.5. Also shown are the Gaussian sources which best fit the data.
155

Each panel of

Figure 5.4: Experimental correlation functions from 40 Ca+40 Ca (left) and 48 Ca+48 Ca
(right). The upper panels include protons with low total momentum (500-640 MeV/c) while
the lower panels represent protons with a high total momentum (740-900 MeV/c). The green
dotted lines represent the results of the fit assuming a Gaussian source distribution. The
purple solid lines are reconstructed correlation functions from imaging.

156

Figure 5.5: Comparison of the imaging fits to the Gaussian fits of p-p correlation functions
for 40 Ca+40 Ca (left) and 48 Ca+48 Ca (right). The upper panels include protons with low
total momentum (500-640 MeV/c) while the lower panels represent protons with a high total
momentum (740-900 MeV/c). The green lines are Gaussian source distributions while the
purple filled areas are source functions from imaging.

157

system
40 Ca+40 Ca

P [MeV/c]

angle θ [◦ ]

[500,640]

33 − 58

40 Ca+40 Ca

[500,640]

26 − 33

40 Ca+40 Ca

[740,900]

33 − 58

40 Ca+40 Ca

[740,900]

26 − 33

48 Ca+48 Ca

[500,640]

33 − 58

48 Ca+48 Ca

[500,640]

26 − 33

48 Ca+48 Ca

[740,900]

33 − 58

48 Ca+48 Ca

[740,900]

26 − 33

λI
0.93+0.13
−0.11
0.85+0.14
−0.13
0.69+0.19
−0.12
0.52+0.17
−0.10
0.84+0.17
−0.14
0.81+0.16
−0.12
0.60+0.16
−0.11
0.58+0.22
−0.13

χ2 /ndf
1.48
1.18
1.06
1.58
1.41
1.17
1.51
1.15

Table 5.1: The normalization of the reconstructed source distribution according to Eq. 5.2
and the χ2 /ndf of the reconstructed correlation function obtained from imaging method.
this figure corresponds to the same panel in Fig. 5.4 and the color convention is also the
same. The imaging technique was able to construct source distributions for backward and
intermediate angles but failed for forward angles.
It is a feature of the imaging technique that it cannot construct the source distribution
if the correlation effect at 20 MeV/c is small or is not seen at all (for source sizes larger
than 5-6 fm). Larger statistical errors in the correlation function make it even harder for the
imaging method to converge.
The fit quality and the λI from Eq. 5.2 for imaging, are given in Table 5.1. In general, the
imaging method can provide many solutions, source distributions S(r), that will give smaller
value of χ2 /ndf than found in Table 5.1. However, solutions with S(r) < 0 are not physical,
so they were excluded from the analysis. The errors on λI parameter were calculated under
condition that S(r) ≥ 0. The imaging method and Gaussian fit method should give roughly
the same results for λ since this parameter is related to the fraction of fast emission protons,
and this fraction should be independent of the method used to extract it.
The results from the Gaussian fits are presented in Table 5.2.
158

The systematic errors

system

P [MeV/c]

angle θ [◦ ]

40 Ca+40 Ca

[500,640]

33 − 58

40 Ca+40 Ca

[500,640]

26 − 33

40 Ca+40 Ca

[500,640]

18 − 26

40 Ca+40 Ca

[740,900]

33 − 58

40 Ca+40 Ca

[740,900]

26 − 33

40 Ca+40 Ca

[740,900]

18 − 26

48 Ca+48 Ca

[500,640]

33 − 58

48 Ca+48 Ca

[500,640]

26 − 33

48 Ca+48 Ca

[500,640]

18 − 26

48 Ca+48 Ca

[740,900]

33 − 58

48 Ca+48 Ca

[740,900]

26 − 33

48 Ca+48 Ca

[740,900]

18 − 26

λG
+0.03(0.03)
0.86
−0.03(0.01)
+0.03(0.01)
0.84
−0.03(0.01)
+0.03(0.01)
0.84
−0.03(0.01)
+0.06(0.05)
0.61
−0.06(0.02)
+0.06(0.06)
0.48
−0.06(0.01)
+0.04(0.01)
0.54
−0.04(0.01)
+0.05(0.02)
0.81
−0.05(0.01)
+0.05(0.01)
0.80
−0.05(0.01)
+0.05(0.01)
0.77
−0.05(0.01)
+0.09(0.03)
0.59
−0.09(0.02)
+0.07(0.01)
0.59
−0.07(0.02)
+0.35(0.01)
0.64
−0.08(0.01)

χ2 /ndf
1.48
1.05
2.12
0.88
1.28
1.52
1.08
1.08
0.97
1.27
1.47
1.04

Table 5.2: Normalization and fit results to the experimental correlation functions assuming
Gaussian source distribution.

159

are printed within parentheses next to statistical errors and were estimated by varying the
fit range and excluding the first data point of the correlation function from the fit.
Fig. 5.4 shows that the correlation functions reconstructed from imaging and obtained
from the Gaussian fit are very similar, and both match the data well. Fig. 5.5 shows that the
sources are also consistent between the imaging technique and the Gaussian fitting procedure.
Thus, it may not be so problematic that imaging fails at forward angles, because the Gaussian
fitting procedure can be used in this region. From both the shape of the source distributions
shown on the Fig. 5.5 and the quality of the reconstructed correlation functions, one can
draw the conclusion that the correlation functions at backward angles are for the most part
consistent with assumption that the source function has a Gaussian shape.
From λG and λI ,the fraction of fast emission protons can be calculated using Eq 5.3.
The results from this are presented in Table 5.3. We note the consistency between the fast
fractions extracted by both the imaging and Gaussian fits and that both fast fractions are
well above 50%. The fast fractions are similar for both reactions. These values for the fast
fraction are much higher than those extracted from correlation functions for slow velocity
protons in mass asymmetric 14 N+197 Au collisions at E/A=75 MeV [38], where values as
low as f =0.3 were reported. These values are a little sensitive to the range in r over which
the source is evaluated, but the difference also suggests that conventional evaporation may
be less relevant for these symmetric collisions as for mass asymmetric collisions at about
the same incident energy/nucleon. How to reconcile this with the large radii observed at
forward angles in the present work is not clear. To resolve this, it may be useful to analyze
directional correlations for the present systems in the forward angular domain [68]. Such
directional correlations can distinguish between large spatial sizes, and long lifetimes.
160

system
40 Ca+40 Ca

P [MeV/c]

P [MeV/c]

angle θ [◦ ]

[500,640]

584

33 − 58

40 Ca+40 Ca

[500,640]

588

26 − 33

40 Ca+40 Ca

[500,640]

594

18 − 26

40 Ca+40 Ca

[740,900]

794

33 − 58

40 Ca+40 Ca

[740,900]

798

26 − 33

40 Ca+40 Ca

[740,900]

802

18 − 26

48 Ca+48 Ca

[500,640]

583

33 − 58

48 Ca+48 Ca

[500,640]

587

26 − 33

48 Ca+48 Ca

[500,640]

594

18 − 26

48 Ca+48 Ca

[740,900]

799

33 − 58

48 Ca+48 Ca

[740,900]

804

26 − 33

48 Ca+48 Ca

[740,900]

805

18 − 26

fG
0.93+0.02
−0.02
0.92+0.02
−0.02
0.92+0.02
−0.02
0.78+0.04
−0.04
+0.05
0.69−0.05
0.73+0.03
−0.03
+0.03
0.90−0.03
0.89+0.03
−0.03
+0.03
0.88−0.03
0.77+0.06
−0.06
0.77+0.05
−0.05
0.80+0.22
−0.05

fI
0.96+0.07
−0.06
0.92+0.08
−0.07
–
0.83+0.11
−0.08
0.72+0.12
−0.07
–
0.92+0.10
−0.08
0.90+0.09
−0.07
–
+0.10
0.77−0.07
0.76+0.14
−0.09
–

Table 5.3: Given are the fractions of fast emission protons from both the imaging technique
and from a Gaussian fit routine.

As mentioned before, the imaging method failed to reproduce the source distribution for
the most forward angles where the size of the source is large and the correlation effect is
not as strong. This emphasizes the need to find a consistent method of determining the
size of the source so the results can be compared to each other. The r1/2 for each source
is presented in Table 5.4 for both reaction systems and both pair momentum ranges in the
laboratory frame. Also listed are results from BUU, described in the next section.
Except for three out of eight selections on momentum and angle where both imaging
and the Gaussian fits converge, the resulting values for r1/2 are statistically consistent. At
the largest angles and the low momentum cut, the imaging fit provides an anomalously
small value for r1/2 . Detailed comparison of the correlation function fits in Fig. 5.4 show
that the main difference between the two fits occurs at relative momentum near 40 MeV/c.
This small difference in the fits could be an interplay between the background and spline
161

System

P [MeV/c]

Angle [◦ ]

40 Ca+40 Ca

[500,640]

33-58

40 Ca+40 Ca

[500,640]

26-33

40 Ca+40 Ca

[500,640]

18-26

40 Ca+40 Ca

[740,900]

33-58

40 Ca+40 Ca

[740,900]

26-33

40 Ca+40 Ca

[740,900]

18-26

48 Ca+48 Ca

[500,640]

33-58

48 Ca+48 Ca

[500,640]

26-33

48 Ca+48 Ca

[500,640]

18-26

48 Ca+48 Ca

[740,900]

33-58

48 Ca+48 Ca

[740,900]

26-33

48 Ca+48 Ca

[740,900]

18-26

Gaussian fit
5.20+0.07
−0.09
6.46+0.10
−0.10
8.11+0.22
−0.24
4.20+0.15
−0.17
4.85+0.22
−0.25
8.99+0.55
−0.42
5.62+0.12
−0.12
6.85+0.19
−0.17
8.74+0.55
−0.42
4.75+0.22
−0.24
5.90+0.22
−0.24
16.37+8.46
−4.16

r1/2 [fm]
imaging
4.49 ± 0.15+0.08
−0.36
6.85 ± 0.33+0.07
−0.14

BUU
5.04 ± 0.10
5.91 ± 0.09

−

6.68 ± 0.10

4.06 ± 0.13+0.10
−0.27
4.71 ± 0.32+0.08
−0.12

4.14 ± 0.09

−

5.33 ± 0.10

4.94 ± 0.21+0.05
−0.33
8.35 ± 0.43+0.23
−0.30
−
4.69 ± 0.33+0.19
−0.10
5.85 ± 0.56+0.14
−0.18

5.51 ± 0.11

−

5.80 ± 0.10

4.66 ± 0.09

6.58 ± 0.09
7.31 ± 0.11
4.50 ± 0.10
4.93 ± 0.14

Table 5.4: Comparison of systems, angular and momentum dependence of r1/2 for imaging,
the Gaussian fitting procedure, and BUU.
parametrization. The sharp decrease in the radius at small momentum, appears to be
inconsistent with the general trend. From that perspective the Gaussian fits appear to
be more reasonable.

5.3.1

Comparison to BUU Transport Theory

40 Ca+40 Ca and 48 Ca+48 Ca collisions at E/A = 80 MeV were simulated using the BUU
transport model. The production of A ≤3 clusters [41,42] were included, which was found to
increase the size of the source, as shown in section 4.5. The Rostock parametrization of NN
in-medium cross section reduction was employed [43]. Momentum dependence in the mean
field potential was also included, as was a soft equation of state with an incompressibility
of K=210 MeV. The density dependence of the symmetry energy was chosen to be γ = 0.7,
162

which is in agreement with Ref. [26].

From the information provided by the model, source functions were constructed for the
same momentum and angular bins in the laboratory as used in the experimental analysis.
Then, the quantity r1/2 was calculated from the source distribution for protons with the energy and angle gates that are consistent with the acceptance of HiRA during the experiment.
This allows the source distributions and the values of r1/2 for BUU and the experimental
data to be directly compared. Figure 5.6 shows the latter comparison for both Gaussian fits
and the imaging technique.

BUU with the specific transport inputs given earlier in this section qualitatively reproduces the measured values for r1/2 at backward and intermediate angles for both pair
momentum ranges. It also qualitatively reproduces forward angle source sizes for low momentum but underpredicts the sizes at high momentum. In the laboratory frame, these
high momentum protons are moving at the velocity of the projectile-like fragment which is
evaporating and fragmenting. BUU does not incorporate evaporation or secondary decay
processes into the model so it is not completely unexpected that the source size is smaller
than the data for this kinematic region. In the next sections, we present correlation functions
gated on rapidity and transverse momentum. Then we vary the transport inputs to see for
which values they reproduce the measured values better and which do not.
163

Figure 5.6: r1/2 as a function of average total momentum for both reaction systems and
all three angular gates. The size of sources from data using the imaging technique is given
by blue triangles while that from the Gaussian fit is shown as red squares. Source sizes from
BUU are shown as black circles.

164

Figure 5.7: Total transverse momentum as a function of pair center of mass rapidity.

5.4

Correlation Functions Selected by Rapidity and Transverse Momentum

This section explores dependencies on total transverse momentum, an invariant with respect
to the frame of reference, and the pair rapidity in the center of mass reference frame. The
phase space in transverse momentum as a function of rapidity as covered by HiRA is shown
in Fig. 5.7.
In terms of the velocity, βz,cm , which is the component of the proton velocity in the
center of mass frame parallel to the beam divided by the speed of light, the center of mass
rapidity can be defined as

y = ln((1 + βz,cm )/(1 − βz,cm ))/2.
165

(5.9)

Figure 5.8: Experimental correlation functions from 40 Ca+40 Ca (left) and 48 Ca+48 Ca
(right). The top panels include protons with low pair rapidity while the middle panels
include protons with intermediate pair rapidity and the lowest panels represent protons with
a high pair rapidity.

Rapidity is additive under Lorentz transformation and reduces to βz,cm in the non-relativistic
limit. All rapidities, y, in this dissertation are in the center of mass frame.
This available phase space was divided into six regions with similar statistics. Fig. 5.8
examines the total transverse momentum dependence (

350MeV/c) of correlation functions

for three rapidity selections and both reaction systems. The left panels present results from
166

40 Ca+40 Ca and the right panels from 48 Ca+48 Ca collisions. For all, an E > 150 MeV
t
was again used to select more central events. To make a correspondence with the preceding
sections, we note that selecting events based on center of mass rapidity, y given by Eq. 5.9,
is somewhat similar to selecting events on center of mass angle, with y = 0 being similar to
θ = 90◦ .
However, dual gates on rapidity and transverse momentum can select source more precisely than gates on angle and total momentum. Low transverse momentum and high rapidity particles are moving slowly in the rest frame of the projectile remnants. Low transverse
momentum and low rapidity is more selective of particles moving in the rest frame of the
participant source. The emission to high transverse momentum is characteristic of short
time-scale pre-equilibrium emission.
Fig. 5.8 shows that the highest gates on y leads to the least pronounced correlation
functions. However, the lowest and intermediate selections of y give similar correlation
functions in most of the cases. In all cases, increasing the total transverse momentum
results in a more pronounced peak in the correlation function independent of y.
The correlation functions were fit by both the imaging technique and by assuming a
Gaussian source. The reconstructed correlation functions from both methods are shown in
Fig. 5.9 with the data. The agreement is good except in the case of low rapidity and low
transverse momentum for each reaction system where the data does not peak at 20 MeV/c.
This shift in the peak could be due to collective effects not accounted for by the two-particle
formalism. Note that the peak for the low transverse momentum and high rapidity selections
is nonexistent. This correlation function could not be imaged as in two cases in the previous
section.
167

Figure 5.9: Experimental correlation functions from 40 Ca+40 Ca (left) and 48 Ca+48 Ca
(right). The top panels show correlations for proton pairs in the lowest rapidity bin. The
middle panels show correlations for proton pairs in an intermediate rapidity bin. The bottom
panels show correlations for proton pairs in the highest rapidity bin.

168

Figure 5.10: Experimental source functions from 40 Ca+40 Ca (left) and 48 Ca+48 Ca (right)
for both the imaging and Gaussian fit methods.

Fig. 5.10 shows the imaged and Gaussian sources for the same set of data in the same
panel structure. The top panels show sources for proton pairs in the lowest rapidity bin. The
middle panels show sources for proton pairs in an intermediate rapidity bin. The bottom
panels show sources for proton pairs in the highest rapidity bin. Both methods provide
similar sources for all rapidity selections, except the case that could not be imaged.
The results for the Gaussian fit method are shown in Table 5.5. The systematic errors
169

system

PT [MeV/c]

Y

40 Ca+40 Ca

294

0.04

40 Ca+40 Ca

289

0.10

40 Ca+40 Ca

300

0.16

40 Ca+40 Ca

417

0.04

40 Ca+40 Ca

426

0.10

40 Ca+40 Ca

409

0.16

48 Ca+48 Ca

295

0.04

48 Ca+48 Ca

291

0.09

48 Ca+48 Ca

303

0.16

48 Ca+48 Ca

419

0.04

48 Ca+48 Ca

429

0.09

48 Ca+48 Ca

413

0.16

RG [fm]
+0.04(0.03)
3.79
−0.04(0.07)
+0.06(0.05)
4.01
−0.06(0.02)
+0.11(0.01)
4.70
−0.10(0.03)
+0.04(0.02)
2.99
−0.04(0.01)
+0.06(0.08)
2.91
−0.06(0.01)
+0.10(0.12)
2.82
−0.11(0.08)
+0.06(0.08)
3.88
−0.06(0.01)
+0.11(0.04)
4.18
−0.10(0.01)
+1.39(0.05)
6.16
−0.50(0.03)
+0.07(0.04)
3.12
−0.07(0.02)
+0.10(0.01)
3.10
−0.11(0.06)
+0.12(0.01)
3.31
−0.13(0.14)

λG
+0.02(0.01)
0.88
−0.02(0.01)
+0.06(0.01)
0.78
−0.05(0.01)
+0.03(0.01)
0.59
−0.03(0.01)
+0.04(0.01)
0.89
−0.03(0.01)
+0.05(0.03)
0.78
−0.05(0.01)
+0.05(0.04)
0.47
−0.05(0.03)
+0.04(0.01)
0.83
−0.04(0.01)
+0.05(0.01)
0.67
−0.05(0.01)
+0.04(0.01)
0.52
−0.04(0.01)
+0.05(0.01)
0.78
−0.05(0.01)
+0.07(0.01)
0.65
−0.07(0.02)
+0.07(0.01)
0.49
−0.06(0.04)

χ2 /ndf
3.48
1.72
1.01
1.57
1.20
1.07
2.50
1.32
1.08
1.06
1.92
1.21

Table 5.5: Fit results to the experimental correlation functions assuming Gaussian source
distribution.

are printed within parentheses next to statistical errors and were estimated by varying the
fit range and excluding the first data point of the correlation function from the fit.

The source r1/2 sizes for both imaging and the Gaussian fit method are shown in Table 5.6. Also shown are sizes from BUU. For this comparison, the mean field potential was
momentum dependent, γ = 0.7, clusters were produced, and the Rostock in-medium cross
section reductions were used.
170

System

PT [MeV/c]

r1/2 [fm]

Y

40 Ca+40 Ca

[150,350]

[-0.07,0.07]

40 Ca+40 Ca

[150,350]

[ 0.07,0.12]

40 Ca+40 Ca

[150,350]

[ 0.12,0.26]

40 Ca+40 Ca

[350,700]

[-0.07,0.07]

40 Ca+40 Ca

[350,700]

[ 0.07,0.12]

40 Ca+40 Ca

[350,700]

[ 0.12,0.26]

48 Ca+48 Ca

[150,350]

[-0.07,0.07]

48 Ca+48 Ca

[150,350]

[ 0.07,0.12]

48 Ca+48 Ca

[150,350]

[ 0.12,0.26]

48 Ca+48 Ca

[350,700]

[-0.07,0.07]

48 Ca+48 Ca

[350,700]

[ 0.07,0.12]

48 Ca+48 Ca

[350,700]

[ 0.12,0.26]

Gaussian fit
+0.07(0.05)
6.31
−0.07(0.12)
+0.10(0.09)
6.68
−0.10(0.04)
+0.19(0.02)
7.83
−0.17(0.05)
+0.07(0.04)
4.98
−0.07(0.02)
+0.10(0.14)
4.85
−0.10(0.02)
+0.17(0.20)
4.70
−0.19(0.14)
+0.10(0.14)
6.46
−0.10(0.02)
+0.19(0.07)
6.96
−0.17(0.02)
+2.32(0.09)
10.26
−0.84(0.05)
+0.12(0.07)
5.20
−0.12(0.04)
+0.17(0.02)
5.16
−0.19(0.10)
+0.20(0.17)
5.51
−0.22(0.24)

imaging

BUU

5.65 ± 0.39+0.23
−0.47

5.86 ± 0.08

6.63 ± 0.54+0.33
−0.45

6.00 ± 0.10

7.80 ± 0.58+0.21
−0.43

5.52 ± 0.10

4.70 ± 0.13+0.21
−0.32

4.64 ± 0.09

4.51 ± 0.15+0.10
−0.08

4.42 ± 0.09

4.93 ± 0.24+0.12
−0.28

4.53 ± 0.08

5.55 ± 0.28+0.12
−0.19

6.49 ± 0.09

6.78 ± 0.47+0.04
−0.32

6.52 ± 0.11

N/A

6.02 ± 0.10

5.04 ± 0.21+0.11
−0.26

5.23 ± 0.09

4.77 ± 0.15+0.18
−0.29

4.92 ± 0.09

5.55 ± 0.52+0.22
−0.39

4.79 ± 0.13

Table 5.6: Comparison of system size, angular and momentum dependence of r1/2 obtained
from reconstructed source distribution with imaging method, Gaussian fitting procedure and
BUU transport model simulations.

171

5.4.1

Comparison to BUU Transport Theory

Chapter 4 examined source dependencies on a variety of transport inputs. It is worthwhile
to revisit the BUU source dependencies when experimental filters are taken into account. In
this subsection, BUU is briefly re-examined. For this exploration, the mean field potential is
momentum dependent, clusters are being produced, γ = 0.7, and the Rostock NN in-medium
cross section reduction is employed, unless otherwise stated.
Fig. 5.11 shows the effect of changing the symmetry energy on the source size for the
two reaction systems.

In the case of the N=Z symmetric reaction system, the symmetry

energy has little effect, which is to be expected. In the neutron-rich reaction system, the
stiffer symmetry energy systematically results in a larger source size. The increase is not
significant at low transverse momentum, but is significant at higher transverse momentum.
The trend is opposite of that predicted by the IBUU model [8] and Fig. 4.33, neither of which
included cluster production. This difference may not be experimentally distinguishable. In
addition, there are other unconstrained inputs which affect the source size, as we saw in
Ch. 4, and is explored again below.
Fig. 5.12 shows that free cross sections lead to a larger source [48] while screened inmedium cross section reductions [43] result in much smaller sources. The energy dependent
(Rostock) NN in-medium cross section reduction [43] is between free and screened, lying
closer to free in terms of reduction and source size.

The effect of in-medium cross section

on source size is stronger for lower transverse momentum proton pairs than higher transverse
momentum pairs. In addition, the effect is much stronger at lower rapidity. At the highest
rapidity, the effect is subtle.
When colliding nuclei using BUU, one has the option of producing clusters. Fig. 5.13
172

Figure 5.11: r1/2 values as a function of average total transverse momentum for low rapidity (top), intermediate rapidity (middle) and high rapidity (bottom). Results using a stiff
symmetry energy are shown by red filled circles while those using a soft symmetry energy
are shown with black open squares.

173

Figure 5.12: r1/2 values as a function of average total transverse momentum for low rapidity
(top), intermediate rapidity (middle) and high rapidity (bottom). Results using free cross
sections are shown as red filled circles while those using a screened in-medium cross section
reduction are shown as black open squares.

174

shows the effect of A ≤3 cluster [41, 42] production. Including clusters in the simulation
increases the size of the source significantly and systematically.

The effect of cluster

production on source size is slightly stronger for lower momentum proton pairs and is mostly
independent of rapidity.
Including momentum dependence with a soft EOS (K=210 MeV) gives rise to a slightly
different size source, as shown in Fig. 5.14, than using a momentum independent mean field
potential with a stiff EOS (K=380 MeV) although the two parametrizations lead to similar
elliptic flow observables [45] for high beam energies. The effect of momentum dependence
of the mean field is most noticeable for higher transverse momentum proton pairs at the
highest rapidity, but is not experimentally distinguishable.
Depending on the choice of momentum dependence in the mean field, cluster production,
and cross section reduction, one could imagine that different conclusions could be made about
the symmetry energy by comparing p-p correlation functions between theory and data. This
dissertation makes no attempt to constrain the symmetry energy while these other inputs
are still somewhat unconstrained.
Now that the dependence of various inputs on source size has been re-explored, Fig. 5.15
shows a comparison between data for the three in-medium NN cross sections, one without
cluster production. For this comparison, the mean field potential is momentum dependent
with a soft EOS and γ = 0.7.
The agreement between theory and data is reasonable for the free and Rostock cross
sections with cluster production. Turning cluster production off with the Rostock cross
section underpredicts the size of the source. This also occurs when using the screened inmedium cross section reduction. It is clear that cluster production and larger in-medium
175

Figure 5.13: r1/2 values as a function of average total transverse momentum for low rapidity
(top), intermediate rapidity (middle) and high rapidity (bottom). Results with cluster production are given by red filled circles, results without cluster production are given by black
open squares.

176

Figure 5.14: r1/2 values as a function of average total transverse momentum for low rapidity
(top), intermediate rapidity (middle) and high rapidity (bottom). Results from calculations
using a momentum dependent mean field potential are shown with red filled circles, while
those using a momentum independent potential are shown with black open squares.

177

Figure 5.15: Source r1/2 values from 40 Ca+40 Ca (left) and 48 Ca+48 Ca (right) for data
using the Gaussian fit and 4 parametrizations of BUU.

178

cross sections are needed to explain the p-p correlation functions from the data. It is not
clear whether free cross sections or the Rostock in-medium reduction is in better agreement
with the data.
Protons emitted with higher transverse momentum always come from a smaller source
than those emitted with lower transverse momentum. At high rapidity, protons are being
emitted from an evaporating, fragmenting projectile like source. Such emission occurs over
longer timescales resulting in a large source. BUU does not incorporate secondary decays
or evaporation so this can be a reason why it fails to reproduce the trend seen in the data.
It should be noted that the disagreement between theory and data at high rapidity is not
due to the experimental resolution of the detectors, as the correlation functions from data
are narrower and not broader than the theory. The correlations are also narrower than what
they would be if the cross section was smaller.

5.5

Three Particle Correlations

Just as p-p correlation functions were constructed in the previous sections, 3-particle correlations can also be constructed from heavy-ion reactions which probe the excited state of
the parent nucleus. When constructing p-p correlation functions, the ratio of the yields is
plotted as a function of relative momentum in the pair’s center of mass frame. In the case
of 3-particle correlations, the ratio of yields is plotted as a function of relative kinetic energy

Erel = T 1 + T 2 + T 3
179

(5.10)

where T is the kinetic energy of each particle in the center of mass of the system of three
particles. The mass excess for the decay is subtracted from the relative kinetic energy to
determine the state in the parent nucleus. For this section, there were no selections on
centrality of the collisions. The normalization of the correlation function is arbitrary. While
these have not been analyzed in detail, four 3-particle correlations are described in this
section.
Fig. 5.16 shows the resonant 0+ ground state and 2+ state from the breakup of 6 Be. The
ground state peak is more pronounced despite the spin degeneracy of the excited state. This
is probably due to the increase in the background under the 2+ peak. The background can
be seen below and between the peaks. It decreases to zero at small relative energy reflecting
the Coulomb repulsion between the 6 Be decay products. The background in these multiparticle correlations increase with relative momentum more strongly with each additional
particle. This is a phase space effect. The peak appears as a single level, multiplied by the
2J+1 spin degeneracy factor. It appears at the energy of the excited state; its detection
efficiency modulates the number of detector events in the peak. The background has the
same efficiency factor, so it divides away in the correlation function. The natural strength
in the uncorrelated background scales with the semi-classical phase space density. As a
function of energy, it has an additional phase space factor of E 3/2 for each addition particle
in the multi-particle final state. As the relative energy is increased, relative contribution of
the resonance decays become steadily smaller. The width in the excited state 1.67 MeV is
broad. Its appearance is similar to that observed by Charity et al. [69]. More information
on the three-body decay of 6 Be can be found in Refs. [70, 71].
Fig. 5.17 shows the ground state and 2 excited states of 9 B that undergo 3-body breakup
180

Figure 5.16: 3-particle correlation for 6 Be→ p+p+α. Known states are labeled by red
arrows.

181

Figure 5.17: 3-particle correlation for 9 B→ p+α + α. Known states (in MeV) are labeled by
red arrows.

into a proton and 2 α particles. The ground state is quite pronounced, while the two excited
states are close together so that they appear to form one peak. More information on the
low lying states of 9 B can be found in Ref. [72]. For a discussion regarding the difference
between 9 B→ p+α+α and 9 B→ p+8 Be channels see Ref. [71].
The Hoyle state was predicted [73], as an excited state in carbon. This state was measured
a few years later [74] and thought to make possible a thermal resonance in the Salpeter
182

Figure 5.18: 3-particle correlation for 12 C→ α + α + α. Known states are labeled by red
arrows. States (in MeV) are listed in ascending order.

process, to take place in red giant stars. Fig. 5.18 shows the 0+ Hoyle state at the lowest
lying energy, along with many states at higher relative kinetic energies. The second lowest
peak corresponds to a 2+ resonance at 9.7 MeV [75, 76].
A phase space Dalitz plot [77] is also included for the triple α breakup of 12 C. Fir. 5.19
shows that nearly all of the 12 C breakup goes through the 8 Be channel. Significant 3-body
breakup would be noticeable in the center of the Dalitz plot, where T1=T2=T3. Instead,
183

Figure 5.19: Dalitz plot for 12 C→ α + α + α and 12 C→ 8 Be+α.
there are three “hot spots” which each correspond to one α particle having a higher kinetic
energy than the other two, which have nearly equal kinetic energies. It may be possible to
extract the three branch ratio for the ground state decay from this Dalitz plot, but additional
work is needed to accomplish this.

184

Chapter 6
Summary
The experiment for this dissertation (the final experiment for the 4π Array) was part of
the HiRA group’s campaign to constrain the density dependence of the symmetry energy.
Although it was found that p-p correlation functions are sensitive to changes in γ with a
certain choice of inputs and gates on transverse momentum, no constraints can be placed on
γ. There are other unconstrained inputs in the BUU transport theory that are competing
effects when comparing- source sizes. Constraining the density dependence of the symmetry energy via transport codes is a multidimensional issue in that it requires a number of
constraints on different inputs.
Dependencies of p-p correlations were studied for central 40 Ca+40 Ca and 48 Ca+48 Ca
nuclear reactions at E/A = 80 MeV. Measurements were performed with the HiRA detector
complemented by the 4π Array at NSCL. In this chapter we first summarize the conclusions
from a systematic study of the BUU transport simulations. This is followed by a discussion
of the data and a summary of the comparison between theory and data for both center of
mass and laboratory selections on data. The chapter will end with a discussion of possible
185

future work to be completed.

6.1

Conclusions from BUU Transport Simulations

Many inputs of the BUU transport simulation code were explored including those which
affect the numerical aspects of a transport code. BUU was found to be stable with the
choice of grid size and test particle number. The main observable, source size, was stable
against changes in these computational inputs.
Many calculations were performed to explore the sensitivity of the calculated source radii
to theoretical quantities. These quantities include the density dependence of the symmetry
energy, in-medium NN cross sections, the production of clusters, and the momentum dependence of the mean field potential. The momentum dependence of the mean field had little
effect on the size, although momentum independent mean fields produced slightly larger
sources in most cases. The density dependence of the symmetry energy had only a slight
effect on the source size. Reducing the NN in-medium cross section reduces the size of the
source because it increases the free mean path of the protons and increase the freeze-out density. Cluster production increased the size of the source which is consistent with the picture
that protons close together in space prefer to be in clusters, and are taken out of the source.
Since clusters are produced in nuclear collisions in this energy domain, this suggests that
cluster production needs to be considered more carefully in reaction dynamics calculations.
Efforts should be made to include and improve the description of cluster production, as it
has a significant effect on source size.
Thus one finds that there are significant sensitivities to both the in-medium cross sections and the cluster production. Consequently, p-p correlation functions, and thus source
186

sizes, cannot constrain any transport property such as the in-medium cross sections, without making some choice about the implementation of cluster production mechanism. It a
multidimensional project. Other observables with different sensitivities to these quantities,
such as in-medium cross sections, need to be sought out and explored. Then the various constraints can be combined to provide significant constraints on quantities such as the density
dependence of the symmetry energy. More theoretical work is needed in this area.

6.2

Summary of Experimental Results

At midrapidity, p-p correlation functions show that the 48 Ca+48 Ca reaction systems results in a larger source size than the 40 Ca+40 Ca reaction system. Part of this is due to
48 Ca+48 Ca having a larger geometric size. The observed effect is somewhat larger than
what one would get by scaling up by A1/3 . If extra neutrons act to delay proton emission,
through the large np cross section or the density dependent symmetry energy, until the
source has expanded, the effect is relatively small.

6.2.1

Dependencies in the Laboratory Frame

When selecting on laboratory observables in central reactions, the trend of the smaller source
for high momentum protons is not generally true. For backward and intermediate angle
selections, higher momentum protons correspond to a more pronounced correlation function,
however for forward angle selections, the high momentum protons come from an evaporating,
fragmenting projectile spectator source. This source is much larger due to the longer time
scales. The trends are the same in both 40 Ca+40 Ca and 48 Ca+48 Ca reaction systems.
A strong angular dependence within p-p correlation functions was found, reflecting the
187

different space-time extent of the source selected. Sources observed at backward angles, in
the laboratory frame, reflect the participant zone of the reaction, while much larger sources
are seen at forward angles which reflect the evaporating, fragmenting projectile-like residue.
A decrease of the source size was observed with increasing momentum of the proton pair
emitted at backward and intermediate angles while an opposite trend is seen at forward
angles. In all cases, the 48 Ca+48 Ca reaction system results in larger sources than the
40 Ca+40 Ca reaction system. It was also found that BUU transport calculations reproduce
the data well at backward and intermediate angles for low momentum protons, but failed to
reproduce the data at forward angles. This probably reflects the lack of a suitable description
of fragmentations, evaporation and secondary decays in the BUU model.

6.2.2

Dependencies on Transverse Momentum and Rapidity

When selecting on center of mass and frame invariant observables in central reactions, we find
that the protons with largest rapidities exhibit the least pronounced correlation functions.
In contrast, the protons measured at the lowest and intermediate rapidities exhibit larger
correlation functions of similar magnitude. In all cases, correlations at larger total transverse
momenta exhibit more pronounced peaks in the correlation function. These correlations
at large transverse momenta are relatively independent of rapidity. In BUU, this trend
can be approximately reproduced, indicating that the space-time extents of early emitted
protons at large transverse momenta may allow quantitative comparisons between theory
and experiment.
Both cluster production and larger in-medium cross sections as parametrized by the
Rostock formula, are needed in BUU transport simulations to reproduce the source sizes seen
188

in the data. BUU with a momentum dependent mean field potential, cluster production, γ =
0.7, and the Rostock in-medium cross section reduction (or free cross sections) reproduced
most of the trends seen in the data. This is true for both the neutron deficient 40 Ca+40 Ca
and the neutron-rich 48 Ca+48 Ca system, indicating no need for unusual reductions of the
np cross section in the medium. The only place where these comparisons fail is in the
low transverse momentum and high rapidity domain, which corresponds to the projectile
like spectator source. This source is evaporating and fragmenting which results in protons
being emitted over longer time scales and a much larger source. BUU does not incorporate
evaporation or secondary decays into the transport of particles. Again, BUU should only be
compared to data in kinematic regions where the model adequately describes the relevant
physical processes.

6.3

Outlook

A specific set of experiments could distinguish whether the 48 Ca+48 Ca system results in a
larger source than the 40 Ca+40 Ca system because of initial geometric size, or whether it is
isospin dependent. An example of two reactions that could be compared are 96 Ru+96 Ru
and 96 Zr+96 Zr. While 96 Ru and 96 Zr have similar geometric sizes, the latter is neutronrich, while the former is neutron-poor. A comparison of source sizes from these two reactions
would examine the role of isospin in p-p correlation functions.
The most obvious work to be continued in the future is to continue to constrain the density
dependence of the symmetry energy by using other experimental observables. Experiments
have already been performed as part of the HiRA campaign to constrain γ using n/p ratios
and isospin diffusion. In addition, as stated earlier, more work is needed to find observables
189

which isolate inputs in reaction dynamics.
On the theoretical front, transport models in general need to incorporate cluster production, especially alpha particles. The inclusion of evaporation and secondary decays would
alleviate discrepancies between theoretical models and experimental data where these processes are important.

190

Appendices

191

Appendix A
Laser Measurements
The Laser Based Alignment System [78] (LBAS) provides accurate positions of detectors
and targets. It is a non-contact, high precision, detector alignment tool. LBAS was essential
in measuring the location of the target and the detectors in this dissertation experiment.
The target would move if touched, so a non-contact method was needed to measure it. Also,
the detector faces are too sensitive to touch, therefore a non-contact measurement was again
needed. The detector array sagged a little under the weight of the detectors and thus the
design specifications were not precise enough to use in the analysis. The position of each pixel
in the detector needs to be known to within 0.25 mm for the necessary angular resolution.
The discrepancy between the location of the detectors from the mechanical design and the
actual location of the detectors can be seen in Fig. A.1. The measuring device also needed
to be small enough to fit inside the 4π array to measure the location of the target. LBAS
fit all of these requirements.
LBAS is composed of 3 main devices, two OWIS rotary stages [79] and an Acuity Laser
Displacement Sensor [80]. The rotary stages provide the theta and phi angles while the laser
192

Figure A.1: The black dots are corners of the detectors as measured by LBAS. The red dots
are corners of the detectors as given by the design of the mechanical setup for the array. The
discrepancy shows the need for such a laser measurement. Units are in inches.

193

Figure A.2: A photograph of LBAS with components labeled.

gives the distance within a range of 25.4-40.6 cm. Light diffusely reflecting off of surfaces is
focused by a lens in the sensor onto a CCD which focuses the position of the image. Since
the lens is located above the beam, the intensity maxima in the CCD shifts upwards if the
object is closer to the sensor and downwards if it is located farther away. A triangulation
method is used to compute the distance to the measured object. A photograph of LBAS
can be seen in Fig. A.2.

The displacements of the rotary stages and the laser itself are

taken into account in the final calculations of positions. The distance between the object and
LBAS can be measured to within 60 µm. For a 0.010◦ step in angle, the position resolution
is ∼0.2 mm which exceeds the specifications of 0.25 mm.
Software was developed to control the laser and scan edges of objects with specified step
194

Figure A.3: An example of an edge scanned with LBAS.

sizes. An example of a scan of the edge of a detector is shown in Fig. A.3. The output from
LBAS is distance, theta (horizontal angle), and phi (vertical angle), although the angles are
modified from normal spherical coordinates with the z-axis along the laser beam. Under
normal operation, the conversion from LBAS angles to spherical angles is given by:.

θspherical = 90◦ + θLBAS

(A.1)

φspherical = −φLBAS .

(A.2)

and

Some examples of the LBAS scans are shown in Figs. A.4 and A.5. The green dots are
195

Figure A.4: The red lines are the mechanical drawing of the target frame used to hold a
target. The green dots are from the LBAS scans of a target while it was inside the chamber.
directly from the LBAS scans and the red outline is the mechanical drawing of the target
frame and reference block respectively.

The block and target together, along with a

photograph of the actual positions can be seen in Fig. A.6. LBAS measured:
• The target
• The edges of all HiRA detectors once the array was extracted from the 4π
• The reference block
The reference block was used to link together measurements of positions made by two separate lasers, LBAS and a contact based laser system owned by NSCL.
The NSCL laser was a contact based measuring device. The laser locked on to the position
of a ball that touched the surface of the object being measured. By moving the ball around
a surface at least three times, the plane of the surface could be determined. The NSCL laser
measured:
196

Figure A.5: The red lines are the mechanical drawing of the reference block used to link
different measurement reference frames together. The green dots are from LBAS scans of
the block.

Figure A.6: The LBAS scans with mechanical drawings are shown on top while a photograph
of the actual setup is shown below. The target can be seen on the right hand side of the
figure, while the reference block can be seen on the left hand side of the figure.

197

• 3 reference points within the vault to connect the vault to the rest of the lab
• 3 reference points on the last quadrupole in the beam line before the 4π
• The front and back of the 4π to define the beam axis
• The reference block
• Reachable sides of HiRA
The last two items were the most important for linking together different measurements.
The target was measured by LBAS in its place, but HiRA was measured with LBAS outside
of the vacuum chamber. To reference the position of HiRA during the experiment, the NSCL
laser was necessary, even though it only measured a few sides of one HiRA detector. The
reference block was needed to link together the target, the beam line, and HiRA into one
reference frame, since it was measured by both lasers. From the measurement of the corners
of each HiRA detector, the location of each pixel could be calculated from the precision
design of HiRA. From the location of the target, assuming the beam was centered on target,
the angle of each particle emitted from the reaction could be calculated.

198

Appendix B
ROOT Analysis
For the analysis of this experimental data (and some theoretical simulations), the ROOT [7]
analysis software package was utilized. First, the hexadecimal data buffers from the data
acquisition software were unpacked into electronic module and channel values. This was
then mapped onto detector components. With that information, the ROOT tree structure
was employed to make a tree for each data file, with branches for each individual detector
component. For example, within HiRA, the DEs, EFs, EBs, and CsI crystals each had their
own branch which were arrays of 32 (4) for the silicon strips (CsI crystals). The 4π array
was also divided into branches, one for each slow and fast scintillator in the phoswich, with
additional branches for the FA elements. Initially, the values assigned to each branch was
filled with energies in raw channels. After calibrating, a new tree was made with calibrated
energies. Total transverse energy from the 4π array was added to this tree.
Each particle was recorded in a particular DE, EF, EB, and CsI for a given telescope.
First, the energy loss in each detector was summed to provide a total energy for each particle.
Then, from the detector information coupled with position measurements from LBAS, each
199

EF/EB pixel could be assigned a position in θ-φ space. From PID plots, each particle could
be identified and labeled with a number corresponding to a certain isotope. For each event,
the number of particles recorded in HiRA for that event and the total transverse energy
from the 4π were written to a text file. Then the isotope label, energy, and position of each
particle in HiRA was written to the same text file. Each data run was transcribed into a
text file in this manner.
Physics ROOT trees were then made from these text files for the purpose of making two
particle correlation functions. In this tree structure, each “event” was comprised of two particles entering the HiRA detector array. From these particles, the relative momentum for each
pair in their center of mass was calculated (Qrel) along with center of mass rapidity (CMSrapid), individual momentum (pLAB/pCMS), total momentum (PpairLAB/PpairCMS),
transverse momentum (pT), energy (LAB/CMS) and angle relative to the beam axis (thetaLAB/CMS), azimuthal angle (phiLAB/CMS), telescope number (tscope) in the laboratory
(LAB) and center of mass (CMS) frames as shown in Fig. B.1. Here each branch is a different observable for the event.

The pairangleLAB was the angle between the pair of

particles, while the pairthetaLAB/CMS was the angle of the total momentum vector of the
pair. The total transverse energy (Etrans) and 4π multiplicity (multi4pi) for each event
were also recorded for selection on impact parameter. In the tree, 1 and 2 refer to particle
number. For identical particles, the number was randomized so that the distribution for the
particle 1 branch was identical to the corresponding distribution for the particle 2 branch.
For different particles, the one with the smallest A (or Z ) is particle 1. Any observables
without a number are for the pair.
Identical trees were made for the numerator files and the denominator files, although
200

Figure B.1: Tree structure for creating 2 particle correlation functions.
different event files were used. In the case of the numerator, data taken with a minimum CsI
multiplicity of two were used, and protons were mixed from the same event. In the case of
the denominator, data files with a minimum CsI multiplicity of one were used, and protons
were mixed from different events. Similar trees were made from the text files for constructing
3 particle correlation functions. In this last case, the trees included relative kinetic energy
instead of relative momentum.

201

Appendix C
Calculation of r1/2 and Error
When source functions are calculated using an ancillary code to BUU, they are normalized
such that the volume integral is 1 (Eq. C.1) which means the height depends on the source
at very large separation distance, r.
∞
4π

S(r)r2 dr = 1

(C.1)

0
It is important that the half width half maximum (or r1/2 ) of the source, and its associated
error, are calculated in a consistent manner for all sources. Different methods for finding the
maximum of the source were explored. The first bin usually has a large error associated with
it and was excluded. One method was to regard the height of the second bin (r =0.75 fm) as
the maximum. This method was eventually rejected because the second bin was sometimes
not consistent with a smooth continuous source due to poor statistics. To avoid discrepancies
in the height of the source function, it was thought to use the rms of the source, assume a
√
Gaussian distribution, and then the r1/2 is simply rms · 2ln2. However, the rms is very
sensitive to the maximum radius in the summation, and does not converge.
202

The best method was to fit bins 2-5 (0.5<r <2.5 fm) with a + bx2 and use the y-intercept
as the maximum of the source. This averages out any inconsistencies in a single point and
forces the derivative of the source to be 0 at r = 0. The error in the maximum is the error
of the fit scaled by

χ2 /2.

Once the maximum, ymax , is found, calculating the r1/2 is elementary. The bins with
contents just above and below the half maximum give points (x1 ,y1 ) and (x2 ,y2 ) respectively
where x is r and y is S(r). A linear extrapolation is used to find the x value corresponding
to y = ymax /2 as shown by

r1

/2

=(

x − x1
ym
− y1 )( 2
) + x1 .
2
y2 − y1

(C.2)

The error in the r1/2 is found by propagation of errors
∂r1

δr1

/2

=

∂r1
∂r1
∂r1
∂r1
/2
/2
/2
/2
/2
2+(
2+(
2+(
2+(
(
δym )
δx1 )
δx2 )
δy1 )
δ )2 .
∂ym
∂x1
∂x2
∂y1
∂y2 y2
(C.3)

The error in the height, δym , was discussed above, while δy is the error of the source function
for those points below and above the maximum, and δx =0.1 fm (half of the bin size). The
partial derivatives are given in Eqs. C.4, C.5, C.6, C.7,and C.8, where dy = y2 -y1 . The
contributions to the error from the error in the source at y1 and y2 are smaller than the
contributions from the other three variables.
∂r1

/2
x − x1
= 2
∂ymax
2dy
203

(C.4)

∂r1

/2
−x2 + x1 (ymax − 2y1 )(x2 − x1 )
=
+
∂y1
dy
2dy 2

(C.5)

∂r1

(ymax − 2y1 )(x2 − x1 )
/2
=
∂y2
−2dy 2

(C.6)

∂r1

/2
−ymax + 2y1
=
+1
∂x1
2dy

(C.7)

∂r1

/2
ymax − 2y1
=
∂x2
2dy

204

(C.8)

REFERENCES

205

References
[1] W. K. Wilson. Azimuthal Distributions in Intermediate Energy Heavy-Ion Collisions.
PhD thesis, Michigan State University, 1991.
[2] A. M. Rogers. Study of 69 Br Ground State Proton Emission and Effects on the
rp-Process 68 Se Waiting Point. PhD thesis, Michigan State University, 2009.
[3] M. S. Wallace. Experimental and Theoretical Challenges in Understanding the
rp-Process on Accreting Neutron Stars. PhD thesis, Michigan State University, 2005.
[4] J. Lee. Survey of Neutron Spectroscopic Factors and Asymmetry Dependence of
Neutron Correlations in Transfer Reactions. PhD thesis, Michigan State University,
2010.
[5] R. Pak. Collective Flow in Intermediate Energy Heavy-Ion Collisions. PhD thesis,
Michigan State University, 1996.
[6] D. Henzlova, 2010. Private communication.
[7] Cern. http://root.cern.ch, 2011.
[8] L. W. Chen, V. Greco, C. M. Ko, and B. A. Li. Isospin effects on two-nucleon
correlation functions in heavy-ion collisions at intermediate energies. Phys. Rev. C,
68:014605, 2003. http://dx.doi.org/10.1103/PhysRevC.68.014605.
[9] R. B. Wiringa, V. Fiks, and A. Fabrocini. Equation of state for dense nucleon matter.
Phys. Rev. C, 38(2):1010–1037, Aug 1988.
http://dx.doi.org/10.1103/PhysRevC.38.1010.
[10] D. Boyanovsky. Phase transitions in the early and the present universe:From the big
bang to heavy ion collisions. 2001. hep-ph/0102120.
[11] Matthias Liebendörfer, Anthony Mezzacappa, Friedrich-Karl Thielemann,
O. E. Bronson Messer, W. Raphael Hix, and Stephen W. Bruenn. Probing the
gravitational well: No supernova explosion in spherical symmetry with general
relativistic Boltzmann neutrino transport. Phys. Rev. D, 63(10):103004, Apr 2001.
http://dx.doi.org/10.1103/PhysRevD.63.103004.
[12] J. M. Lattimer and M. Prakash. The physics of neutron stars. Science, 304:536, 2004.
http://dx.doi.org/10.1126/science.1090720.
[13] P. Danielewicz, R. Lacey, and W. G. Lynch. Determination of the equation of state of
dense matter. Science, 298:1592, 2002.
http://dx.doi.org/10.1126/science.1078070.
[14] B. A. Brown. Neutron radii in nuclei and the neutron equation of state. Phys. Rev.
Lett., 85:5296, 2000. http://dx.doi.org/10.1103/PhysRevLett.85.5296.
206

[15] A. Klimkiewicz, N. Paar, P. Adrich, M. Fallot, K. Boretzky, T. Aumann,
D. Cortina-Gil, U. Datta Pramanik, Th. W. Elze, H. Emling, H. Geissel,
M. Hellström, K. L. Jones, J. V. Kratz, R. Kulessa, C. Nociforo, R. Palit, H. Simon,
G. Surówka, K. Sümmerer, D. Vretenar, and W. Waluś. Nuclear symmetry energy and
neutron skins derived from pygmy dipole resonances. Phys. Rev. C, 76:051603, 2007.
http://dx.doi.org/10.1103/PhysRevC.76.051603.
[16] Andrea Carbone, Gianluca Colò, Angela Bracco, Li-Gang Cao, Pier Francesco
Bortignon, Franco Camera, and Oliver Wieland. Constraints on the symmetry energy
and neutron skins from pygmy resonances in 68 Ni and 132 Sn. Phys. Rev. C,
81:041301, 2010. http://dx.doi.org/10.1103/PhysRevC.81.041301.
[17] U. Garg, T. Li, S. Okumura, H. Akimune, M. Fujiwara, M.N. Harakeh, H. Hashimoto,
M. Itoh, Y. Iwao, T. Kawabata, K. Kawase, Y. Liu, R. Marks, T. Murakami,
K. Nakanishi, B.K. Nayak, P.V. Madhusudhana Rao, H. Sakaguchi, Y. Terashima,
M. Uchida, Y. Yasuda, M. Yosoi, and J. Zenihiro. The giant monopole resonance in
the Sn isotopes: Why is tin so fluffy? Nucl. Phys. A, 788:36 – 43, 2007. Proceedings
of the 2nd International Conference on Collective Motion in Nuclei under Extreme
Conditions - COMEX 2 http://www.sciencedirect.com/science/article/
B6TVB-4MV74YN-G/2/43398042e042f02c27aaf964d6aa56ba.
[18] P. Danielewicz and J. Lee. Symmetry energy I: Semi-infinite matter. Nucl. Phys. A,
818:36, 2009. http://dx.doi.org/10.1016/j.nuclphysa.2008.11.007.
[19] M. Prakash, T. L. Ainsworth, and J. M. Lattimer. Equation of state and the
maximum mass of neutron stars. Phys. Rev. Lett., 61:2518, 1988.
http://dx.doi.org/10.1103/PhysRevLett.61.2518.
[20] J. M. Lattimer and M. Prakash. Neutron star structure and the equation of state. The
Astrophysical Journal, 550:426, 2001. http://dx.doi.org/10.1086/319702.
[21] J. M. Lattimer, C. J. Pethick, M. Prakash, and Pawel Haensel. Direct URCA process
in neutron stars. Phys. Rev. Lett., 66:2701, 1991.
http://dx.doi.org/10.1103/PhysRevLett.66.2701.
[22] M. A. Faminao, T. Liu, W. G. Lynch, M. Mocko, A. M. Rogers, M. B. Tsang, M. S.
Wallace, R. J. Charity, K. Komarov, D. G. Sarantites, L. G. Sobotka, and G. Verde.
Neutron and proton transverse emission ratio measurements and the density
dependence of the asymmetry term of the nuclear equation of state. Phys. Rev. Lett.,
97:052701, 2006. http://dx.doi.org/10.1103/PhysRevLett.97.052701.
[23] M.B. Tsang, C.K. Gelbke, X.D. Liu, W.G. Lynch, W.P. Tan, G. Verde, H.S. Xu,
W. A. Friedman, R. Donangelo, S. R. Souza, C.B. Das, S. Das Gupta, , and
D. Zhabinsky. Isoscaling in statistical models. Phys. Rev. C, 64:054615, 2001.
http://dx.doi.org/10.1103/PhysRevC.64.054615.
[24] T. Liu. Isospin Dynamics and the Isospin Dependent EOS. PhD thesis, Michigan
State University, 2005.
207

[25] M. B. Tsang, T. X. Liu, L. Shi, P. Danielewicz, C. K. Gelbke, X. D. Liu, W. G. Lynch,
W. P. Tan, G. Verde, A. Wagner, H. S. Xu, W. A. Friedman, L. Beaulieu, B. Davin,
R. T. de Souza, Y. Larochelle, T. Lefort, R. Yanez, V. E. Viola, R. J. Charity, and
L. G. Sobotka. Isospin diffusion and the nuclear symmetry energy in heavy ion
reactions. Phys. Rev. Lett., 92:062701, 2004.
http://dx.doi.org/10.1103/PhysRevLett.92.062701.
[26] 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, 2009. http://dx.doi.org/10.1103/PhysRevLett.102.122701.
[27] B. A. Li, L. W. Chen, and C. M. Ko. Recent progress and new challenges in isospin
physics with heavy-ion reactions. Phys. Rep., 464:113, 2008.
http://www.sciencedirect.com/science/article/B6TVP-4SDGR1C-1/2/
0e1d94465e37ec9f9e3342b9120435d7.
[28] M.B. Tsang, Z. Chajecki, D. Coupland, P. Danielewicz, F. Famiano, R. Hodges,
M. Kilburn, F. Lu, W.G. Lynch, J. Winkelbauer, M. Youngs, and YingXun Zhang.
Constraints on the density dependence of the symmetry energy from heavy ion
collisions. Progress in Particle and Nuclear Physics, 2011.
[29] R. Hanbury Brown and R. Q. Twiss. A test of a new type of stellar interferometer on
sirius. Nature, 178:1046, 1990. http://dx.doi.org/10.1038/1781046a0.
[30] D. H. Boal, C. K. Gelbke, and B. K. Jennings. Intensity interferometry in subatomic
physics. Rev. Mod. Phys., 62:553, 1990.
http://dx.doi.org/10.1103/RevModPhys.62.553.
[31] S. J. Gaff, C. K. Gelbke, W. Bauer, F. C. Daffin, T. Glasmacher, E. Gualtieri,
K. Haglin, D. O. Handzy, S. Hannuschke, M. J. Huang, G. J. Kunde, R. Lacey, W. G.
Lynch, L. Martin, C. P. Montoya, R. Pak, S. Pratt, N. Stone, M. B. Tsang, A. M.
Vander Molen, G. D. Westfall, and J. Yee. Two-proton correlations for 16 O + 197 Au
collisions at E/A=200 MeV. Phys. Rev. C, 52:2782, 1995.
http://dx.doi.org/10.1103/PhysRevC.52.2782.
[32] D. O. Handzy, M. A. Lisa, C. K. Gelbke, W. Bauer, F. C. Daffin, P. Decowski, W. G.
Gong, E. Gualtieri, S. Hannuschke, R. Lacey, T. Li, W. G. Lynch, C. M. Mader, G. F.
Peaslee, T. Reposeur, S. Pratt, A. M. Vander Molen, G. D. Westfall, J. Yee, and S. J.
Yennello. Two-proton correlation functions for 36 Ar+45 Sc at E/A=80 MeV. Phys.
Rev. C, 50:858, 1994. http://dx.doi.org/10.1103/PhysRevC.50.858.
[33] M.A. Lisa, W.G. Gong, C.K. Gelbke, and W.G. Lynch. Event-mixing analysis of
two-proton correlation functions. Phys. Rev. C, 44:2865, 1991.
http://dx.doi.org/10.1103/PhysRevC.44.2865.
[34] F. Zhu, W. G. Lynch, T. Murakami, C. K. Gelbke, Y. D. Kim, T. K. Nayak, R. Pelak,
M. B. Tsang, H. M. Xu, W. G. Gong, W. Bauer, K. Kwiatkowski, R. Planeta, S. Rose,
208

Jr. V. E. Viola, L. W. Woo, S. Yennello, and J. Zhang. Light particle correlations for
the 3 He+Ag reaction at 200 MeV. Phys. Rev. C, 44:R582, 1991.
http://dx.doi.org/10.1103/PhysRevC.44.R582.
[35] Z. Chen, C. K. Gelbke, W. G. Gong, Y. D. Kim, W. G. Lynch, M. R. Maier,
J. Pochodzalla, M. B. Tsang, F. Saint-Laurent, D. Ardouin, H. Delagrange,
H. Doubre, J. Kasagi, A. Kyanowski, A. Péghaire, J. Péter, E. Rosato, G. Bizard,
F. Lefèbvres, B. Tamain, J. Québert, and Y. P. Viyogi. Inclusive two-particle
correlations for 16 O induced reactions on 197 Au at E/A=94 MeV. Phys. Rev. C,
36:2297, 1987. http://dx.doi.org/10.1103/PhysRevC.36.2297.
[36] S. E. Koonin. Proton pictures of high-energy nuclear collisions. Phys. Lett. B, 70:43,
1977. http://www.sciencedirect.com/science/article/B6TVN-47284FF-43/2/
f60f4d817f619c15ad5692892a47493c.
[37] S. Pratt, T. Csörgő, and J. Zimányi. Detailed predictions for two-pion correlations in
ultrarelativistic heavy-ion collisions. Phys. Rev. C, 42:2646, 1990.
http://dx.doi.org/10.1103/PhysRevC.42.2646.
[38] G. Verde, D. A. Brown, P. Danielewicz, C. K. Gelbke, W. G. Lynch, and M. B. Tsang.
Imaging sources with fast and slow emission components. Phys. Rev. C, 65:054609,
2002. http://dx.doi.org/10.1103/PhysRevC.65.054609.
[39] G. F. Bertsch, H. Kruse, and S. D. Gupta. Boltzman equation for heavy ion collisions.
Phys. Rev. C, 29:673, 1984. http://dx.doi.org/10.1103/PhysRevC.29.673.
[40] G. F. Bertsch and S. Das Gupta. A guide to microscopic models for intermediate
energy heavy ion collisions. Phys. Rep., 160:189 – 233, 1988.
http://dx.doi.org/10.1016/0370-1573(88)90170-6.
[41] P. Danielewicz and G. F. Bertsch. Production of deuterons and pions in a transport
model of energetic heavy-ion reactions. Nucl. Phys. A, 533:712, 1991.
http://dx.doi.org/10.1016/0375-9474(91)90541-D.
[42] P. Danielewicz. Formation of composites emitted at large angles in intermediate and
high energy reactions. Nucl. Phys. A, 545:21c, 1992.
http://dx.doi.org/10.1016/0375-9474(92)90443-N.
[43] P. Danielewicz. Hadronic transport models. Acta. Phys. Pol. B, 33:45, 2002.
http://arXiv:nucl-th/0201032v1.
[44] E. A. Uehling and G. E. Uhlenbeck. Transport Phenomena in Einstein-Bose and
Fermi-Dirac Gases. I. Phys. Rev., 43:552, 1933.
http://dx.doi.org/10.1103/PhysRev.43.552.
[45] P. Danielewicz. Determination of the mean-field momentum-dependence using elliptic
flow. Nucl. Phys. A, 673:375, 2000.
http://dx.doi.org/10.1016/S0375-9474(00)00083-X.
209

[46] R. J. Lenk and V. R. Pandharipande. Nuclear mean field dynamics in the lattice
Hamiltonian Vlasov method. Phys. Rev. C, 39:2242, 1989.
http://dx.doi.org/10.1103/PhysRevC.39.2242.
[47] L. Shi. Transport Phenomena in Heavy-Ion Reactions. PhD thesis, Michigan State
University, 2003.
[48] G. Verde, P. Danielewicz, W. G. Lynch, D. A. Brown, C. K. Gelbke, and M. B. Tsang.
Probing transport theories via two-proton source imaging. Phys. Rev. C, 67:034606,
2003. http://dx.doi.org/10.1103/PhysRevC.67.034606.
[49] B. A. Li and L. W. Chen. Nucleon-nucleon cross sections in neutron-rich matter and
isospin transport in heavy-ion reactions at intermidiate energies. Phys. Rev. C,
72:064611, 2005. http://dx.doi.org/10.1103/PhysRevC.72.064611.
[50] H.-J. Schulze, A. Schnell, G. Röpke, and U. Lombardo. Nucleon-nucleon cross sections
in nuclear matter. Phys. Rev. C, 55(6):3006–3014, 1997.
http://dx.doi.org/10.1103/PhysRevC.55.3006.
[51] NSCL. http://www.nscl.msu.edu/experiments#PAC27, 2003.
[52] A.M. Vander Molen, J.S. Winfield, and N.T.B. Stone. L’user’s guide to the MSU 4π
Array. 3rd Edition, 1996.
[53] HiRA. http://groups.nscl.msu.edu/hira/, 2011.
[54] M. S. Wallace, M. A. Famiano, M. J. van Goethem, A. M. Rogers, W. G. Lynch,
J. Clifford, F. Delaunay, J. Lee, S. Labostov, M. Mocko, , L. Morris, A. Moroni, B. E.
Nett, D. J. Oostdyk, R. Krishnasamy, M. B. Tsang, R. T. de Souza, S. Hudan, L. G.
Sobotka, R. J. Charity, J. Elson, and G. L. Engel. The high resolution array (HiRA)
for rare isotope beam experiments. Nucl. Instr. and Meth. A, 583:302–311, 2007.
http://dx.doi.org/10.1016/j.nima.2007.08.248.
[55] M. A. Kilburn, 2006. Private communication.
[56] V. Henzl, 2009. Private communication.
[57] W. Tan. Probing the Freezeout Mechanisms and Isospin Effects in Multifragmentation.
PhD thesis, Michigan State University, 2002.
[58] D. A. Cebra. Charged-Particle Correlations from Intermediate Energy Nuclear
Reactions. PhD thesis, Michigan State University, 1990.
[59] C. Cavata, M. Demoulins, J. Gosset, M. C. Lemaire, D. L’Hôte, J. Poitou, and
O. Valette. Determination of the impact parameter in relativistic nucleus-nucleus
collsions. Phys. Rev. C, 42:1760, 1990.
http://dx.doi.org/10.1103/PhysRevC.42.1760.
210

[60] L. Phair, D. R. Bowman, C. K. Gelbke, W. G. Gong, Y. D. Kim, M. A. Lisa, W. G.
Lynch, G. F. Peaslee, R. T. de Souza, M. B. Tsang, and F. Zhu. Impact-parameter
filters for 36 Ar+197 Au collisions at E/A = 50, 80 and 110 MeV. Nucl. Phys. A,
548:489, 1992. http://dx.doi.org/10.1016/0375-9474(92)90697-I.
[61] P. Danielewicz, 2009. Private communication.
[62] P. Danielewicz, R. A. Lacey, P.-B. Gossiaux, C. Pinkenburg, P. Chung, J. M.
Alexander, and R. L. McGrath. Disappearance of elliptic flow: A new probe for the
nuclear equation of state. Phys. Rev. Lett., 81:2438, 1998.
http://dx.doi.org/10.1103/PhysRevLett.81.2438.
[63] D. A. Brown and P. Danielewicz. Imaging of sources in heavy-ion reactions. Phys.
Lett. B, 398:252, 1997. http://www.sciencedirect.com/science/article/
B6TVN-3SPGXP4-72/2/edac150984f33fa44917287b2afec651.
[64] D. A. Brown and P. Danielewicz. Optimized discretization of sources imaged in
heavy-ion reactions. Phys. Rev. C, 57:2474, 1998.
http://dx.doi.org/10.1103/PhysRevC.57.2474.
[65] D. A. Brown and P. Danielewicz. Observing non-gaussian sources in heavy-ion
reactions. Phys. Rev. C, 64:014902, 2001.
http://dx.doi.org/10.1103/PhysRevC.64.014902.
[66] M. A. Lisa. Impact Parameter-Gated Two-Proton Intensity Interferometry in
Intermediate Energy Heavy-Ion Collisions. PhD thesis, Michigan State University,
1993.
[67] M. A. Lisa, C. K. Gelbke, W. Bauer, P. Decowski, W. G. Gong, E. Gualtieri,
S. Hannuschke, R. Lacey, T. Li, W. G. Lynch, C. M. Mader, G. F. Peaslee,
T. Reposeur, A. M. Vander Molen, G. D. Westfall, J. Yee, and S. J. Yennello.
Impact-parameter-selected two-proton intensity interferometry for 36 Ar+45 Sc at
E/A=80 MeV. Phys. Rev. Lett., 70(24):3709, 1993.
http://dx.doi.org/10.1103/PhysRevLett.70.3709.
[68] T. Glasmacher, L. Phair, D. R. Bowman, C. K. Gelbke, W. G. Gong, Y. D. Kim,
M. A. Lisa, W. G. Lynch, G. F. Peaslee, R. T. de Souza, M. B. Tsang, and F. Zhu.
Two-fragment correlation functions with directional cuts for central 36 Ar+197 Au
collisions at E/A=50 MeV. Phys. Rev. C, 50:952, 1994.
http://dx.doi.org/10.1103/PhysRevC.50.952.
[69] R.J. Charity and et al. Investigations of three, four, and five-particle exit channels of
levels in light nuclei created using a 9 C beam. 2011.
http://arxiv.org/abs/1105.1144v1.
[70] L. V. Grigorenko, T. D. Wiser, K. Mercurio, R. J. Charity, R. Shane, L. G. Sobotka,
J. M. Elson, A. H. Wuosmaa, A. Banu, M. McCleskey, L. Trache, R. E. Tribble, and
211

M. V. Zhukov. Three-body decay of 6 Be. Phys. Rev. C, 80(3):034602, Sep 2009.
http://dx.doi.org/10.1103/PhysRevC.80.034602.
[71] R. J. Charity, T. D. Wiser, K. Mercurio, R. Shane, L. G. Sobotka, A. H. Wuosmaa,
A. Banu, L. Trache, and R. E. Tribble. Continuum spectroscopy with a 10 C beam:
Cluster structure and three-body decay. Phys. Rev. C, 80:024306, 2009.
http://dx.doi.org/10.1103/PhysRevC.80.024306.
[72] R. Sherr and H. T. Fortune. Low-lying levels of 9 B. Phys. Rev. C, 70(5):054312, 2004.
http://dx.doi.org/10.1103/PhysRevC.70.054312.
[73] F. Hoyle. On Nuclear Reactions Occuring in Very Hot STARS.I. the Synthesis of
Elements from Carbon to Nickel. Astrophysical Journal Supplement, 1:121, 1954.
http://adsabs.harvard.edu/abs/1954ApJS....1..121H.
[74] C. W. Cook, W. A. Fowler, C. C. Lauritsen, and T. Lauritsen. 12 B, 12 C, and the red
giants. Phys. Rev., 107(2):508–515, Jul 1957.
http://dx.doi.org/10.1103/PhysRev.107.508.
[75] M. Freer, H. Fujita, Z. Buthelezi, J. Carter, R. W. Fearick, S. V. Förtsch, R. Neveling,
S. M. Perez, P. Papka, F. D. Smit, J. A. Swartz, and I. Usman. 2+ excitation of the
12 C hoyle state. Phys. Rev. C, 80:041303, Oct 2009.
http://dx.doi.org/10.1103/PhysRevC.80.041303.
[76] T Munoz-Britton, M Freer, N I Ashwood, T A D Brown, W N Catford, N Curtis, S P
Fox, B R Fulton, C W Harlin, A M Laird, P Mumby-Croft, A St J Murphy, P Papka,
D L Price, K Vaughan, D L Watson, and D C Weisser. Search for the 2 + excitation
of the Hoyle state in 12 C using the 12 C(12 C,3α)12 C reaction. J. Phys. G,
37(10):105104, 2010. http://stacks.iop.org/0954-3899/37/i=10/a=105104.
[77] R. H. Dalitz. Decay of τ mesons of known charge. Phys. Rev., 94(4):1046–1051, May
1954. http://dx.doi.org/10.1103/PhysRev.94.1046.
[78] A. M. Rogers et al. in preparation.
[79] OWIS. http://www.owis-staufen.de, 2011.
[80] Acuity Laser. http://www.acuitylaser.com/, 2011.

212