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This is to certify that the
dissertation entitled

Two-Proton Intensity Interferometry For Impact-Parameter
Selected 36Ar + hSSc Collisions at E/A=80, 120 and 160 MeV

presented by

Damian Orest Handzy

has been accepted towards fulfillment
of the requirements for

Ph.D. degree in PhYSlCS

 

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Major professor

Date 2’4 July, 1995

 

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TO AVOID FINES Mum on or More duo duo.

DATE DUE DATE DUE DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

MSU loAn Minn-tho Action/Equal Opportunity Inflation
Wm:

TWO—PROTON INTENSITY INTERFEROMETRY FOR IMPACT-
PARAMETER SELECTED 35AR 4- 45Sc COLLISIONS AT
ElA=80, 120 AND 160 MEV

By

Damian Orest Handzy

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of PhYsics and Astronomy

1995

ABSTRACT

TWO-PROTON INTENSITY INTEREEROMETRY FOR IMPACT-PARAMETER SELECTED
36AR 4- 458c: COLLISIONS AT ElA=80, 120 AND 160 MEV

By
Damian Orest Handzy

Impact-parameter selected two—proton intensity interferometry is used to
study the space—time characteristics of emitting sources formed in medium-
energy heavy—ion collisions.

Building on a previous study for the same system at a lower energy, a
high-resolution 56-element Si-CsI(Tl) hodoscope was used to collect single- and

two—proton yields, for collisions of 36Ar + 45Sc at E / A=120 MeV and 160 MeV.
Coincident measurements of other charged particles emitted in the reaction were
made with the MSU 4:: Array, providing information about the impact-
parameter of the collision.

The Boltzmann-Uehling—Uhlenbeck (BUU) equation is used to predict the
emission of protons from the reaction zone created in heavy-ion collisions. The
Koonin—Pratt formalism is then used to calculate theoretical correlation functions
from the predicted single—particle phase space probability density.

Dependencies of predicted longitudinal and transverse correlation functions on

source velocity are examined for central and peripheral 3‘5Ar + 45Sc collisions at
E/A=80 MeV, and are compared to previously measured values. The usefulness
of the correlation function to distinguish exotically shaped sources, predicted by
microscopic transport models at this energy, is investigated.

Consistent with previous measurements, proton correlations are shown to
have larger peaks for more energetic protons, regardless of impact-parameter.

However, the measured correlations are shown to decrease as beam energy

increases from E/A=80 to 160 MeV, indicating that proton-emitting sources
formed in more energetic collisions appear to have larger space—time extents.
For central collisions at E/A=160 MeV, the correlation function shows no
dependence on the momentum of the proton pair, suggesting that the source
emits fast and slow protons on similar time scales.

The BUU theory is shown to over predict the magnitude of the measured
correlations for the reactions at E/A=120 and 160 MeV, possibly because of the
effects of proton emission from particle unstable resonance states not modeled

I

by the theory.

PeHi

iv

ACKNOWLEDGMENTS

I became interested in the study of the nucleus as an undergraduate student
working with Professors Fay Ajzenberg—Selove, David Balamuth and Keith
Griffieon at the University of Pennsylvania. My undergraduate thesis project

(on 6He) developed into a collaboration between David Balamuth’s group and
the A1200 group at the N SCL, which will continue with an experiment next
month. To the three of them I will be forever grateful for kindling in me a
passion for experimental physics.

At the N SCL, I have had the fortune of working with many outstanding
physicists. Most influential among them has been my advisor and friend, Claus-
Konrad Gelbke. I have learned far more from him than the methods of world-
class scientific research. I have been witness many times to his exceptional
organizational, interpersonal, and professional skills, and to the diligence and
integrity with which he approaches every situation. His influence on me, both
personally and professionally, will be everlasting. Thank you, Konrad.

During my graduate studies, I have also had the privilege of working with
and learning from other physicists who are ”neck deep” in correlation functions.
Wolfgang Bauer has worked closely with me on comparing predictions of his
BUU model with my experimental results, and has taken an active and
influential part in my education. Scott Pratt, whose door is always open, sat
patiently through my numerous questions, even though I'm sure he had just
answered them days before. Many thanks to you both.

I am also thankful to Gary Westfall, Skip Vander Molen, and the entire 41!:
Group for going through not one - but two - more hodopscope - 41:
experiments. My guidance committee deserves special recognition. To
Professors Pope and Mahanti I am thankful for their participation and input

during the first 3 years. To Professors Yuan and Duxbury, I am especially
grateful for serving on my committee on such short notice.

I also wish to thank a number of physicists from the NSCL and other
institutions. They are: David Bowman, Romualdo deSouza, Thomas
Glasmacher, Roy Lacey, Roy Lemmon, Bill Llope, Graham Peaslee, Carsten
Schwarz, M. Betty Tsang, and Gordon Wozniak. And of course, I (the cook), say
to Nicola Colonna (the chambermaid), ”gracia.”

I have been very fortunate to become close friends with a few fellow
graduate students who have made my years in Michigan especially enjoyable.
From over-boiling wort with Phil, Brian and Mathias, to gripe sessions over
Long Islands with Carlos and discussing everything from Wall Street to
Windows programming with David over cappuccino, the friendships I have
formed with them will last a lifetime. I have enjoyed our time together, and the
privilege of their company.

I am indebted to my friends Phil Zecher, Carlos Montoya, David McGrew,
Brian Young, Mathias Steiner, Sally Gaff, Iac Caggiano, Mike Fauerbach, Renan
. Fontus, Njema Frazier, Gene Gualtieri, Stefan Hannuschke, Wen-Chien Hsi,
Min-Iui Huang, John Kruse, Michael Lisa, Catherine Mader, Joelle Murray,
Raman Pfaff, Larry Phair, Chris (it’s 4n!) Powel, Easwar Ramakrishnan, and
Shigeru Yokoyama for sharing with me a small part of their lives. A professor
once told me that in graduate school you meet the most interesting people at the
worst time of their lives. He clearly did not know my friends, who have made
the past four years some of my best. I thank each of you.

I owe thanks to many NSCL employees for their roles in making my time
at the cyclotron lab enjoyable. They are: Kay Barber, Shari Conroy, Mike Ellis,
Laura Hernandez, Tina Keehn, Rilla McI-Iarris, Chris O'connor, Barb Pollack,
Reg Ronningen, Arlieta Sawyer, Greg Scoczlas, Dennis Swan, Chris Townsend,

and, of course, C-c—c—h-h-h—u—u-u—cw—c—k—k—k V—v—v—a—c—e—k.

During our trips back East, Renata and I were always met with relatives’
warm smiles, in amazement that we were able to make yet another family event.
To all of you, for always providing us a place to rest our heads and quench our
thirst, I am thankful.

From discussions about the height of the Saturn-V rocket (while crossing
the George Washington bridge) to multiple trips to the Hayden Planetarium and
the Museum of Natural History, my parents taught me at an early age that
science is both interesting and fun. Through their own lives, they have taught
me how to reach things not initially in my grasp, and how to enjoy the time in
between. Fl Bac mobmo i BaM unKyio.

As I conclude my graduate studies, my brother is still in the middle of his
own program in mathematics. Although we have not lived together in over 4
years, we continue to become closer, as each of us follows his own path. I thank
you, Nest, for reminding me to keep things in perspective as I struggled with the
demands of academic life, and for constantly reminding me of my ”war crimes.”
Now let me remind you of one thing: I love you, too.

Most of all, I thank Pet-ta, my wife. This thesis is dedicated to her because
of the tremendous support she has given me during my years of graduate
school. For everything you put up with, from late nights at the lab to family
event we didn’t make, I thank you. Now it’s time to move on to our next
adventure together, and it’s you turn to lead the way. Fl Tobi nyme meIo i
T866 nyxe mobmo.

vii

TABLE OF CONTENTS

List of Tables .............................................................................................................. x
List of Figures ............................................................................................................ xi
Chapter 1. Introduction .......................................................................................... 1
1.1 Medium Energy Heavy-Ion Collisions .................................................... 1
1.2 Intensity Interferometry ............................................................................ 2
1.3 The Two—Proton Correlation Function .................................................... 5
1.4 Motivation .................................................................................................. 8
Chapter 2. Experimental Details and Data Reduction .......................................... 12
2.1 Detector Elements ...................................................................................... 13
2.1.1 The 56-Element Hodoscope ............................................................. 13
2.1.2 The MSU 4t: Array ............................................................................ 16
2.2 Electronics .................................................................................................. 19
2.2.1 The 56—Element Hodoscope ............................................................. 21
2.2.2. The MSU 4n Array ........................................................................... 22
2.2.3 Master Electronics ............................................................................. 23
2.3 Calibrations ................................................................................................ 25
2.3.1 The 56—Element Hodoscope ............................................................. 25
2.3.2 The 4n Array 34
2.4 Impact Parameter Selection ....................................................................... 34
Chapter 3. The Two-Proton Correlation Function .......................................... . ..... 46
3.1 The Koonin-Pratt Formalism .................................................................... 46
3.2 Resolving the Space-Time Ambiguity of the Correlation Function ...... 49

viii

3.3 Calculating Correlation Functions from Model Predictions .................. 50

3.4 Constructing Experimental Correlation Functions ................................. 51
Chapter 4. Correlation Functions for 36Ar + 455C at E/A=80 MeV ...................... 57
4.1 Comparison of Angle—Integrated Correlation Functions ....................... 57
4.2 Longitudinal and Transverse Correlation Functions .............................. 60
4.3 Source Velocity Dependence of BUU-Predicted Correlation
Functions ..................................................................................................... 70
Chapter 5. Toroidal Density Distributions ............................................................. 74
5.1 Correlation Functions from Schematic Sources ....................................... 74
5.2 Correlation Functions from Model Predictions ....................................... 82
Chapter 6. 3‘5Ar + 458c Collisions at E/A=120 and 160 MeV ................................ 88
6.1 Single Proton Energy Spectra .................................................................... 89
6.2 Measured Proton-Proton Correlation Functions .................................... 92
6.3 Comparisons of BUU Predictions with Experimental Results ............... 100
6.3.1 Varying BUU Input Parameters ....................................................... 101
6.4 Importance of Particle Unstable Resonances ........................................... 104
Chapter 7. Summary and Conclusions ................................................................... 111
List of References ...................................................................................................... 113

ix

LIST OF TABLES

Table 2.1: Details of the three 36Ar + 455c experiments ........................................ 12
Table 2.2: Punch-In Energies for Ball Detectors .................................................... 17
Table 2.3: Punch-In Energies for Forward— and High-Rate— Array Detectors. 17

Table 2.4: Calibration Reactions .............................................................................. 29

Figure 1.1:

Figure 1.2:

Figure 2.1:

Figure 2.2:
Figure 2.3:

Figure 2.4:

Figure 2.5:

Figure 2.6:

Figure 2.7:

Figure 2.8:

Figure 2.9:

LIST OF FIGURES

Schematic diagram underlying the principle of
interferometry .................................................................................... 4

Spatial distribution of emitted protons from (a) a small source
of short lifetime, (b) a large source of short lifetime and (c) a
small source of long lifetime ............................................................. 7

Schematic diagram of the 56—element hodoscope coupled to the
MSU 41: Array .................................................................................... 14

Angular coverage of the 56—element hodoscope ............................ 15

A 32-faced truncated icosahedron, the MSU 41: Array
geometry ............................................................................................. 16

Schematic diagram of the front faces of the Forward Array (top)
and the High Rate Array (bottom) ................................................... 18

Electronics used by the 56—element hodoscope and the MSU 41:
Array in digitizing data .................................................................... 20

Master electronics timing diagram. The dashed lines indicate

the effect of the (approximately) 200 ns ”jitter” in the relative

time between 41: triggers and hodoscope triggers. The fast clear

is created about 1.5 us after the 41: trigger, and the fast clear

veto is issued immediately following a hodoscope trigger ........... 24

2—dimensional E vs. AE for a typical hodoscope detector.
Isotopes separate into hyperbolic bands on this logarithmic
scale ..................................................................................................... 26

2-dimensional total energy vs. PID plot showing linear
separation of isotopes. Again, logarithmic density scale is
used ..................................................................................................... 27

1—dimensional projections of PID using thin cuts in energy,
showing particle identification resolution as a function of
energy ................................................................................................. 28

Figure 2.10:

Figure 2.11:

Figure 2.12:

Figure 2.13:

Figure 2.14:

Figure 2.15:

Figure 2.16:

Figure 2.17:

Figure 2.18:

Energy spectra gated on detected particle type (p,d,t,3He,o:; top
to bottom) for our E/A=30 MeV calibration run ............................ 30

Linear calibration curves for a typical hodoscope” detector. The
dashed line in the deuteron panel is the average of the proton

and triton calibrations, which was used for those detectors in
which insufficient deuteron statistics prevented a direct

calibration ........................................................................................... 31

Relative timing of detectors both recording protons. The upper
panel shows the results before ”walk” correction. After the
correction (lower panel), the spectrum displays clear peaks
corresponding to different cyclotron beam bursts .......................... 33

Total transverse kinetic energy distributions measured in the 41:
Array under the three trigger conditions ”Ball,” ”Singles," and
”Coincidence.” ................................................................................... 36

Upper panels: total transverse kinetic energy spectrum for the
”Ball” trigger in our E/A=120 MeV reaction (left) and the
E/A=160 MeV reaction (right). Below each spectrum is the

corresponding reduced impact parameter scales b(Et) based on

these distributions .............................................................................. 38
Average reduced impact parameter distributions dP/db(Et )for
the three event triggers studied ........................................................ 39
The total transverse kinetic energy spectra, dP/dEt, measured

in the 41: Array (solid) is compared with the predictions of BUU
where neutrons are included (dashed line) and excluded (solid
line) ..................................................................................................... 41
Reduced impact parameter distributions dP/db(Et) for the

narrow cuts on b(NC) and b(ZY) indicated in the figure. The

upper and lower panels show dP/db(Et) for central and
rnidcentral cuts ................................................................................... 43

Upper panel: sharp cut in b(Et) used to define our central
events. Lower panel: distributions dP/db true of ”true”
reduced impact parameter corresponding to the cut on b(Et)

xii

Figure 3.1:

Figure 3.2:

Figure 4.1:

Figure 4.2:

Figure 4.3:

Figure 4.4:

(in upper panel) for our E/A=120 MeV results (solid line) and
our E/A=160 MeV results (dashed line) ...................................... 45

Correlation functions constructed with the mixed-events

technique (open circles), and with the singles technique (solid
circles). The suppression of the mixed-events constructed
correlation function is consistent with previous studies [Lisa 91,
93c] ...................................................................................................... 53

A theoretical correlation function constructed from the

2 _ 2
schematic source function g(r, p, I) ~ e('r/r°) e( p /2mT)8 (I)
using the coarse filter, which does not take into account energy
loss through the target, is shown in the solid line. The
correlation function calculated by using the exact filter and
correcting for energy loss through the target is shown in the
dashed line. The differences are smaller than the uncertainty in
our measurements ............................................................................. 56

Average height of the correlation function in the region as a
function of the total laboratory proton-pair momentum, Plab-

The right-hand axis gives the radius of a zero-lifetime spherical
source with a Gaussian density profile that produces a
correlation function of equal magnitude. The upper and lower
panels show results for central and peripheral events. Data are
shown by solid circles and impact parameter filtered BUU
predictions are shown in open circles. The BUU predictions of

strictly central collisions (5 = 0) are shown in solid diamonds.
The lines are drawn to guide the eye ............................................... 58

Distributions of the true reduced impact parameter, dP/dém ,
used for the theoretical calculations of our E/A=80 MeV

collisions. The solid line represents the probability distribution

for central collisions, the dashed line for peripheral collisions ..... 59

Solid and open points show longitudinal and transverse
correlation functions measured for central collisions at the
indicated momenta. The curves show the BUU predictions for

the idealized case of purely central collisions (b = 0 ). The cuts
on the angle \y are defined in the center-of—momentum frame of
the colliding system ........................................................................... 63

Solid and open points show longitudinal and transverse
correlation functions measured for central collisions at the
indicated momenta. The curves show the BUU employing the

xiii

Figure 4.5:

Figure 4.6:

Figure 4.7:

Figure 4.8:

Figure 4.9:

realistic impact parameter probability distribution shown in the
solid line of Figure 4.2. The cuts on the angle w are defined in
the center—of—momentum frame of the colliding system ............... 65

Schematic of our detector acceptance and momentum cuts.
Dashed lines represent the detector boundaries at
:9 lab = 30°—45°. solid and hatched areas represent the low— and

high— momentum cuts, Plab=400'600 and 700-1400 MeV/c,

respectively. The solid circles depict Fermi spheres of target
and projectile ...................................................................................... 66

Longitudinal (solid points and curves) and transverse (open
points and dashed curves) correlation functions constructed in
the laboratory (top) and projectile (bottom) rest frames. Left
and right panels show results for low— and high-momentum
cuts, respectively. Points show data selected by the centrality

cut b(Et ) s 0.36 , and the curves show BUU predictions for
B = o .................................................................................................... 68

Longitudinal (solid points and curves) and transverse (open
points and dashed curves) correlation functions constructed in
the laboratory (top) and projectile (bottom) rest frames. Left
and right panels show results for low— and high-momentum
cuts, respectively. Points show data selected by the centrality

cut b(Et ) S 0.36 , and the curves show BUU predictions

employing the realistic impact-parameter probability
distribution shown in the solid line in Figure 4.2 ........................... 69

Relative difference, (AR) / ( R) , between longitudinal and

transverse correlation functions as a function of the velocity 8",

of the rest frame in which the directional w cuts are defined.
Shown are BUU predictions for b=0 (solid circles), b=3 fm (open
diamonds) and b=6 fm (open circles). Results for low- and
high-momentum cuts are displayed in the upper and lower
panels, respectively ............................................................................ 71

Schematic representation of a moving source (shown here in the
center-of-momentum frame) and the directions probed when

the angle w is defined in various reference frames. The

direction defined by P (the longitudinal direction) is shown to
”rotate" as the velocity of the defining frame increases ................. 72

xiv

Figure 5.1:

Figure 5.2:

Figure 5.3:

Figure 5.4:

Figure 5.5:

Figure 5.6:

Figure 5.7:

Figure 5.8:

Figure 5.9:

Figure 6.1:

Residual system predicted by BUU transport calculations for

b=0 collisions of 36Ar + 455c at E /A=80 MeV at time t=10, so,

100, and 150 fm/c. The top panels show distributions viewed
along the beam axis; the bottom panels show distributions
projected onto a plane which contains the beam axis .................... 76

Geometry of a torus and the directions of applied cuts on
:1 used in all of our calculations ....................................................... 77

Comparisons of correlation functions calculated for a zero—
lifetime torus and disk, using the directional cuts defined in
Figure 5.2 ............................................................................................ 78

Left panels: density distribution of the torus at t=100 fm/c,

taken from Figure 5.1. Right panels: density distribution of a
”disk,” formed by uniformly filling in the hole of the torus. The
radius of the disk’s density distribution was chosen to produce

the same correlation-function peak height as that produced by

the torus .............................................................................................. 79

Two-proton correlation functions predicted for emission from a
torus assuming an exponential time dependence of mean
emission time ':=25 fm/c ................................................................... 81

The time dependence of proton emission, dP/dt , predicted for

b=0 collisions of 35Ar + 458c at E/A=80 MeV, showing that most
of the protons are emitted before t=100 fm / c, when the torus is
fully formed (see Figure 5.1) ............................................................. 83

Two-proton correlation functions predicted by BUU transport
calculations for b=0 36Ar + 455c collisions at E/A=80 MeV ........... 84

Residual system predicted at t=100 fm/c by BUU calculations
for 36Ar + 455c collisions E/A=80 MeV for the indicated
impact-parameters ............................................................................ 85

Two-proton correlation functions predicted by BUU transport

calculations for central 36Ar + 455c collisions at E/A=80 MeV.
The impact parameter distribution used for these calculations is
shown in Figure 4.2 ........................................................................... 87

Probability for nuclear reaction loss for protons incident on a 10
cm CsI(Tl) crystal ............................................................................... 90

XV

Figure 6.2:

Figure 6.3:

Figure 6.4:

' Figure 6.5:

Figure 6.6:

Figure 6.7:

Figure 6.8:

Laboratory frame proton energy spectra measured at 013b=31°
for central (b / bmax = 0 - 0.3) collisions of 35Ar + 458c at E/ A =
120 and 160 MeV (solid points) are compared with the

predictions of BUU (histograms). Relative normalization gives
equal areas for measured and predicted spectra for Eproton> 50

MeV. The arrows indicate average values of 51533,,"

corresponding to low and high momentum cuts used to analyze
the correlation function in this chapter ............................................ 91

Schematic of detector acceptance and center-of-mass
momentum cuts for our E/A=120 MeV reaction. Dashed lines
represent the detector boundaries at 13 lab = 30°-45°. Dotted
and hatched areas represent the low- and high—momentum
cuts, Pcm = 200 - 400 and 400—800 MeV/c, respectively. Solid

circles depict Fermi spheres of target and projectile ...................... 93

Schematic diagram similar to Figure 6.3, appropriate for our
E/A=160 MeV reaction ..................................................................... 94

Momentum-integrated two-proton correlation functions for our
E/A=120 MeV (top panel) and E/A=16O MeV (bottom panel )
reactions. Central events are indicated by solid points, while
peripheral events are shown in open points ................................... 96

Measured two-proton correlation functions, cut on total
proton—pair center—of-mass momentum, Pm, and on centrality

for our E/A=120 MeV reaction. Open and solid points

represent high- and low—momentum protons, cut on central
(upper panel) and peripheral (lower panel) events ........................ 97

Measured two—proton correlation functions, cut on total
proton-pair center-of-mass momentum, Pm, and on centrality

for our E/A=160 MeV reaction. Open and solid points

represent high- and low-momentum protons, cut on central
(upper panel) and peripheral (lower panel) events ........................ 98

The total proton-pair momentum dependence of the average
value of the two—proton correlation function

((1 + R) q=15_25 MeV , c ) in the peak region, q=15—25 MeV/c for
our E/A=120 MeV (top) and E/A=160 MeV (bottom) results.

Central events are shown in solid points, and peripheral events
in open points. The lines are drawn to guide the eye ................... 99

xvi

Figure 6.9:

Figure 6.10:

Figure 6.11:

Figure 6.12:

Figure 6.13:

Figure 6.14:

Two-proton correlation functions for central collisions of 36Ar +

455c at E/A = so MeV (top), 120 MeV (middle) and 160 MeV
(bottom). results with our low-momentum gates are shown in

solid points (data) and solid lines (theory), while high—

momentum gated results are shown in open points (data) and
dashed lines (theory) ......................................................................... 102

The average value of the correlation function,
(1+ R) 4:1 5_25 MeV / c , in the peak region is plotted versus total

proton momentum for our E/A=120 MeV (top) and E/A=160
MeV (bottom) reactions for central collisions. Data are shown
in solid points and BUU predictions in open points ...................... 103

The average peak height (at q = 15 - 25 MeV/c) of the two-

proton correlation function is shown versus the total proton-

pair momentum in the center-of-mass rest frame for central
collisions of 35Ar + “Sc at E/A=120 MeV. Data are shown as

solid points. BUU simulations were calculated for freeze-out
density pf = 1/ 8 (solid), 1/16 (hatched) and 1/ 32 (dotted).

Each band shows the range of BUU predictions for values of the
parameter (I (explained in the text) ranging from 0 to -0.4 ............ 105

Proton emission rates, dP/ dt, predicted by BUU calculations

from central 36Ar + 455C collisions at E/A=80 MeV (solid), 120
MeV (dashed) and 160 MeV (dotted) ............................................... 106

Two proton correlation functions (for low-momentum cut) for

central collisions of 36Ar + 45Sc at E/A=120 MeV. Data are

shown by points, and the results of ”delayed" BUU simulations

are shown in the hatched area, defined by t=160 fm/c and

f=30°/o to 50%. Details are given in the text. The insert depicts

the ambiguity between the parameters 1 and f ............................... 108

Two proton correlation functions (for low-momentum cut) for

central collisions of 36Ar + 455C at E/A=160 MeV. Data are

shown by points, and the results of ”delayed” BUU simulations

are shown in the hatched area, defined by t=160 fm/c and

f=20% to 30%. Details are given in the text. The insert depicts

the ambiguity between the parameters 1 and f ............................... 109

xvii

Chapter 1 - Introduction

1.1 Medium Energy Heavy-Ion Collisions

The study of heavy ion collisions at intermediate energies (E/AzZO—ZOO
MeV) is an active field of fundamental research aimed at understanding the
thermodynamic properties of strongly interacting quantum systems under
extreme conditions of pressure and temperature. The nuclear system under
study has between 20 and 200 constituents - too many to be understood in an
accurate few body formalism, and not enough to be treated in a truly statistical
manner. In addition, the system evolves over a very short time scale: on order
10'22 sec, 0r about 100 fm/ c. As a consequence, many phenomena may require
theoretical treatment by a quantum transport theory rather than allowing purely
statistical descriptions. For these reasons the study of nucleus-nucleus collisions
in this energy regime is interesting and very challenging, both experimentally
and theoretically.

The dynamical and statistical effects of nuclear collisions are both heavily
dependent upon the bombarding energy and the impact parameter. At low
energies (typically a few MeV / nucleon), the reaction depends on mean—field
effects and single body dissipation. Peripheral collisions can be described as
”quasi-elastic”, in which the target and projectile exit the reaction slightly
perturbed by the collisions, possibly in excited states. Alternatively, peripheral
collisions can also be classified as ”deeply inelastic”, in which case there is
substantial exchange of energy, angular momentum and even nucleons between
target and projectile. Central collisions result in a long-lived equilibrated
system which decays by statistical particle evaporation.

At much higher, relativistic, energies ( E/ A 2 1 GeV), nuclear reactions are
described in terms of individual nucleon-nucleon collisions. The geometrical
picture of the participant-spectator model is appropriate in which the hot
(participant) region is defined by the geometric overlap of projectile and target.

The deposited energy causes the system to "explode,” emitting constituent
nucleons as well as more exotic massive particles. Statistically models are
applicable to the participant zone, where thermal equilibrium is readily
achieved.

Between these two extremes is the intermediate—energy regime for which
both mean field effects and nucleon—nucleon collisions are important. Statistical
and dynamical processes compete as the system evolves. After an early emission
of fast nucleons, a semi—equilibrated system is predicted to remain which
continues to evolve through statistical decay, possibly accompanied by
expansion. Pre—equilibrium emission of particles is known to be important. At
low energies, the system undergoes a sequential binary decay, while at higher
energies, multifragment decays are known to occur on an increasingly shorter
time scale. Experimental evidence exists for Au 4» Au collisions at E/A=200 MeV
that the source expands as it emits fragments [Hsi 94, Jeon 94] and dynamical
model calculations have predicted the formation of exotically shaped sources for
central collisions in this energy domain [Baue 92a, Gros 92, Bord 85, Xu 93 and
Hand 94]. A liquid-gas phase transition is expected to occur in nuclear matter of
temperature 5—10 MeV and density 0.1 to 0.5 times normal nuclear density.

Experimental progress has been significant in recent years. Higher
resolution, lower threshold detector arrays have allowed more detailed
measurements of reaction products and unique detectors have permitted studies
in which nearly all emitted particles are detected, allowing a detailed

characterization of the collision process.

1.2 Intensity Interferometry

Interferometry has played an important role in physical and astronomical
measurements. The famous Young double-slit experiment, which used an
amplitude interferometer, was instrumental in establishing the wave theory of
light, and was crucial in the development of a unified electromagnetic theory.

Out of this theory came the notion of the aether, the medium which ”waved”
when light traveled. It was not until 1881 that Michelson built an amplitude
interferometer capable of disproving the existence of the aether that the
foundation was laid for the special theory of relativity. The Michelson
interferometer was used in this century to measure the angular diameters of the
red giant Betelgeuse and of many other stars [Swen 87]. The amplitude
interferometer had two difficulties in measuring angular sizes of stars
accurately. Better resolving power required increasing the size of the device,
which introduced mechanical instabilities. More importantly, atmospheric
turbulence imposes random relative phase shifts that destroy the interference
signal. Hanbury Brown and Twiss (HBT) proposed a conceptually different
interferometer in response to these shortcomings in 1954 [Hanb 54], and the
theory of the effect was refined in a series of papers [Hanb 57a, 57b].

The difference between intensity and amplitude interferometry can be
understood from Figure 1.1 [Boal 90]. The finite sized source emits particles
from points S1 and 52, which are later observed at points P1 and P2. In amplitude

interferometry, the points P1 and P2 could be slits through which the particles

pass. They would then interfere constructively or destructively at various points
on a screen located behind the points, producing the traditional fringe pattern.
Knowledge of the slit size and measuring fringe visibility could then provide
information regarding the source size and the degree of coherence of the light
emitted from points P1 and P2. In intensity interferometry, the two—particle
coincidence yield n12 and the single—particle yields n1 and n2 are measured. In
contrast to the Young experiment, the signal is maximal when the emission from

P1 and P2 is incoherent. From these measurements, a correlation function

C(p1,p2) is constructed through the relation:

C(PIIFZ)= 1+R(I31ri52)=(n—Sl.1-<2%5- (1-1)

Information about the source size can be extracted from the correlation function
in a two particle coincidence measurement.

The coincident yield, n12, is influenced by quantum statistics (for identical
particles) and by the interaction between the two detected particles. For
measurements of non-interacting identical bosons, e. g. photons and neutral
pions, the correlations arise due to the symmetrization of their relative wave
function: the presence of one boson in a particular momentum state will enhance
the probability that another will be found in the same state. The size of the
correlation depends upon the separation of the bosons upon emission. Hence,

the two-photon correlation function is sensitive to the source size.

    

P1

 

Source .

 

Figure 1.1 Schematic diagram underlying the principle of interferometry.

Although the technique of intensity interferometry was originally intended
for two-photon correlations, it was quickly applied to other particles, including
charged fermions and bosons. The first such application, by Goldhaber,
Goldhaber, Lee and Pais studied pion correlations in proton-antiproton
annihilation at about 1 GeV [Gold 59, Gold 60]. More recently, two-pion
correlation studies have been used extensively to study the shapes, sizes and
lifetimes of sources created in elementary particle and high energy nuclear
collisions [Boal 90]. This technique will be an integral part of the search for

signatures of the quark-gluon plasma at the Relativistic Heavy Ion Collider
(RHIC) in the early part of the next century.

The use of two—proton intensity interferometry was proposed by Koonin
[Koon 77] to investigate the space—time nature of medium energy heavy-ion
collisions. In contrast to the bosonic nature of photons and pions, protons
should exhibit an anti-correlation due to their quantum-statistical behavior as
fermions. The dominant contributions to the two—proton correlation, however,
are final-state interactions due to the strong nuclear attraction and long-range
Coulomb repulsion. Each of these three factors has definite consequences on the
shape of the two—proton correlation function, which was shown to be highly
sensitive to the space—time extent of the source at low relative momentum (less
than 50 MeV/ c). The two—proton correlation function has been used to study
medium energy heavy ion collisions over a wide range of bombarding energies
[Boal 90]. It may also provide information about the liquid—gas phase transition
in nuclear matter [Prat 87].

1.3 The Two—Proton Correlation Function

Two protons which are emitted from an excited nuclear system will repel
each other through the Coulomb force and attract one another through the
nuclear force. In addition to these final state interactions, quantum Fermi
statistics requires that the two-proton wavefunction be antisymmetric. These
three effects modify the two—proton distribution, and the magnitude of that
modification depends upon the relative separation of the protons upon emission.
Dividing the coincidence yield by the single particle yields largely divides out
the single particle phase space effects and isolates the correlation caused by the
final state interactions.

The attractive S—wave nuclear interaction leads to a pronounced maximum
in the correlation function at relative momentum q==20 MeV/c. This maximum

decreases for increasing source size and / or emission time scale. Coulomb

repulsion and Pauli antisymmetrization produce a minimum at q~0, which
becomes more pronounced for sources of smaller space-time extents.

Consider a source of radius r and lifetime I which emits protons of average
speed v. The initial spatial separation of the emitted protons, approximately
r + v: , is greater for longer lifetimes. The correlation function is sensitive to this
initial separation, which depends upon source radius and lifetime. Figure 1.2
shows the spatial distribution of protons emitted from a source. Larger radii
(panel b) or longer lifetimes (panel c) produce larger emission zones. Larger and
smaller emission zones can be discriminated by analyzing the peak height of the
two—proton correlation function; smaller emission zones have larger peaks. The
nuclear resonance (at q=20 MeV/ c) is spherically symmetric due to its S-wave
nature. Furthermore, Coulomb repulsion is dominated by the I = 0 partial wave
for spatial separations less than the two—proton Bohr radius of 58 fm. Therefore,
these final state interactions are sensitive only to the size (not shape) of the
emission zone; panels b and c cannot be distinguished from one another.

However, information regarding the shape of the distribution of emitted
protons may be extracted from quantum statistical effects. The Pauli
antisymmetrization requirement of the two-proton relative wavefunction is
directionally sensitive for values of relative momentum, q , and relative position,

'r‘ , which satisfy |r - q] .. n [Koon 77, Pratt 87, Awes 88]. In particular, correlation
functions are suppressed over a range qi S h/ri , for each Cartesian coordinate i.
Since the distribution of emitted particles (such as in panel c) is elongated in the
direction of total proton momentum, P = p1 + 132 , (measured in the system’s

center-of-momentum frame) this vector can be used to exploit the directional
sensitivity of the Pauli principle. Correlation functions constructed from proton

pairs whose relative momentum, q = %(j51 - p2 ), is oriented along the short

dimension of the distribution (transverse to P) will be suppressed when
compared with correlation functions constructed from pairs whose relative

.' '. . ' To detectors

_.__>

Figure 1.2. Spatial distribution of emitted protons from (a) a small source
of short lifetime, (b) a large source of short lifetime and (c) a small source of long
lifetime. -

momentum is oriental along the long dimension of the distribution (parallel to

P). Such cuts are performed, in practice, by analyzing the angle between

relative and total proton momenta, w = cos"1 (P - Ej/P q).

1.4 Motivation
The two—proton correlation function has been shown to have a dependence

on the magnitude of the total proton pair momentum, P = [P1 + p2] , for many

systems over a range of beam energies [Lync 83, Chen 87c, Poch 86, Poch 87,
Awes 88, Gong 90b, 90c, 91b, Lisa 91, 93a, 93b, Hand 94]. Higher momentum
protons have a larger correlation peak height, indicative of emission from a
smaller source. This dependence has been reproduced by calculation of the
Boltzmann-Uehling-Uhlenbeck (BUU) dynamic transport model [Gong 90c, 91b,
Baue 92a]. More recently, measurements of energy integrated (but impact
parameter gated) two—proton correlation functions have shown that central
collisions result in a larger source than peripheral collisions, consistent with a
geometric interpretation of the reaction zone [Lisa 91, 93a, 93c]. The BUU model
predicts a qualitatively similar trend. It was able to reproduce the detailed

momentum dependence of the correlation function for central collisions of

36Ar+"’53c at E/ A = 80 MeV, but it under predicted the peak height in more
peripheral collisions (especially for more energetic protons) [Lisa 93a, 93c].
Another important result from this analysis was the first experimental
observation of a difference between longitudinal (w s 0°) and transverse
(w s 90°) correlation functions consistent with a finite-lifetime effect [Lisa 93b,
93c], using the technique described in the previous section. Past experiments
had failed to identify such a directional dependence, possibly because a clear
characterization of emission sources had not been achieved. Comparisons of
these directional correlation functions with schematic sources were consistent
with a Gaussian source of radius 5 fm and lifetime 25 fm/ c. To gain further

insight into the predicted evolution of the source, detailed calculations of the
BUU theory were compared with the experimental data [Lisa 93b, 93c] as part of
this thesis. We investigated the dependence of the impact parameter and
momentum gated directional correlation function on the frame of reference in
which the total momentum is defined [Hand 94].

Serious discrepancies between predictions of the BUU model and

experimentally measured correlation functions for impact parameter inclusive

collisions of 40Ar+197Au at E/A=200 MeV were reported by Reference [Kund
93]. Improved agreement was obtained by calculations with the Quantum
Molecular Dynamics model (QMD), but these calculations failed to describe the
data at E/A=60 MeV where BUU calculations were successful in interpreting
impact-parameter inclusive data [Kund 93]. Furthermore, these data show a
vanishing minimum in the correlation function at q z 0, suggesting that the
experimental acceptance of the detecting device may have seriously affected the
data.

It was the goal of this thesis to investigate the details of proton emission in
heavy-ion reactions by measuring the two—proton correlation function’s

dependence on total momentum and impact parameter, and its directional

dependence for collisions of 36Ar+458c as a function of bombarding energy. We
were particularly interested in investigating the claims of Reference [Kund 93]
for a symmetric system with improved statistical accuracy capable of impact
parameter determination. We also investigated the utith of the correlation
function to extract a signature from exotically shaped sources predicted to form
in central heavy-ion collisions in this energy domain [Baue 92a]. We measured
coincidence yields at bombarding energies of E/A=120 and 160 MeV, using the
same high-resolution 56—element hodoscope array employed in the original
measurement at E/A=80 MeV [Lisa 93c].

10

1.5 Organization

Chapter 2 begins with a discussion of the details of the experimental setup,
triggering and digitizing electronics. We then discuss data reduction, including
particle identification and energy and time calibrations, including time walk
correction. Finally, we present our approach of determining the impact
parameter of each collisions from data taken with the 41: Array and the
construction of an impact parameter distribution appropriate for use with
dynamical models.

In chapter 3, we present the theoretical framework in which the two—proton
correlation function is calculated, and we present the results of illustrative
examples of the spac&time ambiguity of correlation functions and the use of
directional cuts. Finally, we discuss the construction of the correlation function
from measured data.

Chapter 4 presents the BUU transport model’s predictions of the directional

correlation function for collisions of 3“Ar + 455c at E/A=80 MeV. Sensitivities to
the motion of the emitting source are discussed for central and peripheral
collisions. Comparisons of these simulations with the data measured by M.A.
Lisa [Lisa 93a, 93b, 93c] are shown.

Chapter 5 deals with the BUU prediction of toroidally shaped sources [Baue
92a, Gros 92, Bord 85, Xu 93 and Hand 94]. We investigate the possibility of
extracting a signature of such an exotically shaped distributions using the two-
proton correlation function by presenting a step-by-step approach which
incorporates increasingly realistic assumptions.

In chapter 6 we discuss correlation functions for 36Ar + 458c at E/A=120
and 160 MeV. Single proton energy spectra are shown, followed by a discussion
of measured correlation functions and comparisons with BUU predictions. We
show the results of varying BUU input parameters. The importance of particle-
unstable fragment production in correlation measurements is discussed and
directional correlation functions are presented.

11

A summary and conclusions are given in chapter 7. Most of the results
presented in this dissertation have been submitted to refereed scientific journals
for publication [Hand 94, Hand 95a, Hand 95b].

Chapter 2 - Experimental Details and Data Reduction

This thesis project involves two experiments which continue the study of

36Ar+45$c collisions via two—proton intensity interferometry using the 56—
Element Hodoscope [Gong 88, 90a, 91b, 91c] coupled with the MSU 41: Array
[West 85, Winf 91, Lisa 93c]. Experimental details are summarized in Table 2.1

Table 2.1 Details of the three 36Ar + 45Sc experiments.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Experiment Number 88026 91034 93034
Experimenter M.A. Lisa D.O. Handzy D.O. Handzy
Beam Energy / Nucleon 80 MeV 120 MeV 160 MeV
Target Thickness 10 mg /cm2 40 rug/(:11:2 40 mg/cm2
Beam Intensity 3 x 108 particles/sec 3 x 107 particles/sec 5 x 107 particles/sec
Hours Approved 216 252 360
.Nflmber of H) 2.1 x 10" 1.2 x 10‘5 3.1 x lo6
corncldences recorded
41: Configuration Ball + Ball + Ball +
Forward Array Forward Array High Rate Array
Species Resolved in P ,d,t,3He,4He p,d,t,3He,4He p,d,t
Hodoscope
Dates of Run June 1991 November 1992 August 1994

 

In this chapter, we discuss the detectors, their processing electronics and

energy and time calibrations. Finally, we discuss impact parameter selection.

12

 

13

2.1 Detector Elements
A closely—packed array of 56 Si—CsI(Tl) telescopes [Gong 88,90a,91b,91c]

was used to detect the light charged particles p, d, t, 3He, and 4He. A1156 of the
telescopes, as well as the vacuum chamber used to mount the hodoscope onto
the 41: Array had been previously constructed for the theses of W.G. Gong [Gong
91c] and M.A. Lisa [Lisa 93c]. Figure 2.1 shows a schematic drawing of the
coupling of the two devices.

2.1.1 56—Element Hodoscope

Each hodoscope element consisted of a thin Si AE-detector backed by a
thick CsI(Tl) E—detector. We used Si surface barrier detectors of 300 and 400 um
thickness and active areas 450mm2. Each Si detector was attached to the
hodoscope’s mechanical support structure [Lisa 93c] using a brass mount, the
front face of which was covered by a thin (~10um) Ta foil to suppress signals
from electrons and photons. Typical bias voltages ranged from 100 to 200 volts,
and each raw AE signal was sent to a preamplifier (in vacuum) before being
processed by the electronics logic.

Behind the Si detectors were CsI(Tl) crystals [Gong 91c, Lisa 93c] of length
10 cm and diameter 4 cm. Their signals were read out with PIN diodes of active
area 2cm x 2cm attached to the back end of the crystals; preamplifiers for these
detectors were contained in the same casings as the crystals.

The telescopes, each of which subtended a solid angle of AQ=0.37 msr,
were packed with a nearest—neighbor spacing of A0=2.6°. Figure 2.2 shows polar
coordinates of the hodoscope array. Gain drifts, which were reduced by actively
cooling the hodoscope to about 20° C, were monitored by pulser input wst
signals for both the Si and CsI(Tl) detectors. The gains were found to be stable
to within 1% for both experiments.

l4

mmtmrmzmzd. IODOmOODm

Enema _zmmx:oz ...\.

ImOI>2_wZ

" do . ”u

um
>n mm>§

 

 

 

 

 

A

 

am
Mn
q>m0ma

nomigo S33.

o 2 m2...

 

Figure 2.1. Schematic diagram of the 56—element hodoscope coupled to the MSU

41: Array.

15

 

{SB—Element Hodoscope Acceptance

I

T I

 

d J

— -

@ddddddo
@@@@@®®®

@

o
oo

doeoode
oooooooo
decoded

@@90

@0

 

 

200 '-

190

_

0
. B

1

Amooumouv e

170 '-

160 '-

40 45
9 (degrees)

35

30

Figure 2.2. Angular coverage of the 56-element hodosc0pe.

16

2.1.2 The MSU 41: Array

The MSU 41: Array [West 85, Winf 91] is a closely—packed array of plastic
phoswhich detectors covering approximately 87% of 41:. The 41: Array is
arranged as a 32-faced truncated icosahedron built of 20 regular hexagonal faces
and 12 regular pentagonal faces, as shown schematically in Figure 2.3 Each of
the hexagonal (pentagonal) detectors is subdivided into 6 (5) light—charged-
particle detector telescopes. The 41: Array is capable of resolving nuclear species

of Zz1-18.

 

 

 

Figure 2.3. A 32-faced truncated icosahedron, the MSU 41: Array geometry.

The 41: Array is composed of 170 ”Ball” detectors, and 45 Forward Array
detectors. During the two years between the experiments, the 41: Array

augmented it’s Forward Array (FA) to be geometrically more efficient, by about
a factor of two. This improved device is called the High Rate Array (HRA). A
schematic diagram of the FA and the HRA is shown in Figure 2.4.

17

The 41: Array is made of 215 phoswhich telescopes, each of which is
composed of a thin ”fast” plastic scintillator optically coupled to a larger ”slow”
plastic scintillator. Pulse—shape discrimination of each telescope’s output signal
is used to separate the fast (rise timezlns, fall time-:33 ns) and slow (rise
time=25 ns; fall timez180 ns) components.

The minimum energy required to enter the ”slow” portion of the detector
(punch-in energy) depends on particle type, and these values are listed in Tables
2.2 and 2.3 for the Ball and the FA respectively. The HRA and the FA have
virtually identical punch-in energies.

Table 2.2 Punch-In Energies for Ball Detectors

Particle Punch-In Particle Punch-In

17 ' 140
24 214
28 287
70 380

 

Table 2.3 Punch-In Energies for Forward- and High—Rate Array Detectors

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Particle Punch-In Particle Punch-In Particle Type Punch-In
Type Energy (MeV) Type Energy (MeV) Energy (MeV)
p 12 C 259 A1 816

d 16 N 328 Si 927

t 19 O 403 P 1009

He 47 F 472 S 1124

Li 95 Ne 570 C1 1216

Be 146 Na 636 Ar 1403

B 195 Mg 737

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 2.4. Schematic diagram of the front faces of the Forward Array (top) and
the High Rate Array (bottom). '

19

2.2 Electronics

The purpose of these experiments was to measure two-proton correlations
as a function of impact parameter. The protons were measured in the hodoscope
while the impact parameter of each collision was determined from data taken
with the 41: Array. The electronics set—up of both of these experiments were
based on that used by Michael Lisa for his thesis experiment [Lisa 93c].

Because the 41: detectors had a much faster rise time than the hodoscope
detectors, the 41: Array triggered its own ADC’s. The digitized information was
read into the computer if the slower hodoscope satisfied its trigger condition.
Otherwise, the data was cleared in anticipation of the next event. The two
detector arrays had independent triggers, both of which had to fire to form a
valid event. Events in which the 41: Array fired, but the hodoscope did not, were
”fast cleared” before the computer recorded the event. Because of the much
greater geometrical acceptance of the 41: Array and because of the large polar
angle positioning of the hodoscope, events in which the hodoscope fired but the
41: Array did not were extremely rare. Because these events had no information
about the impact parameter, they were ignored in the subsequent off-line
analysis.

We used three different ”master” triggering conditions. The ”Coincidence”
trigger demanded that at least two hodoscope detectors fire. Data so taken was
used to construct the numerator of the correlation function. We used a second
”Singles” trigger in which a minimum of one hodoscope detector fired to have
data with which to construct the denominator of the correlation function.
Finally, we took short runs with the 41: Array as the trigger to construct an
impact parameter scale (see section 2.4).

Shown in Figure 2.5 is a schematic diagram of the electronics used for the
two detectors and how they work together.

20

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

33-14 1 i5
1 r a t
. " :2 El El _
: E E1
5! t g lflé
ET?
535.
[g] Elms [i]
f glint—1!: [9

 

 

 

 

 

 

 

 

 

Figure 2.5. Electronics used by the SG—element hodoscope and the MSU 41:
Array in digitizing data.

21

2.2.1 The 56—Element Hodoscope

The electronics logic for each individual telescope is schematically outlined
in the upper section of Figure 2.5. The shaping amplifiers provided two outputs:
a ’fast’ output (with integration time constant t=100 ns) which was used to
construct the trigger logic, and a ’slow’ output (with :z3 us) which was sent
directly to a peak-sensing Silena ADC. In order to detect high-energy protons
(which deposit too little energy to reliably trigger the Si detector discriminators),
two thresholds were used for the CsI(Tl) detectors. Signals passing the Si
threshold (typically ~200 keV) will be denoted AB; those passing the low CsI(Tl)
threshold (typically ~500 keV) will be denoted EL; and those passing the high

CsI(Tl) threshold (typically ~1—2 MeV) will be denoted EH. Because we were
interested in fast and slow protons, a valid event could have been formed in one
of two ways. Simultaneous AB and EL signals were considered valid, which

would include low to medium energy protons (which trigger both detector
discriminators). High energy protons would produce solitary EH signals (not
triggering the AE discriminator), and were also valid. A hodoscope event was
considered valid, then, if the following triggering condition was met:
(EL -AE)+(EL En)-

EL was the only signal present for all valid events; it therefore, was made
to determine the timing of each hodoscope event. Since the second signal to
arrive at an ”AND” gate determines the timing of the gate’s output, delaying the
EL signal, with a Delay-and—Gate—Generator (DGG), insured that it always
determined the timing of the hodoscope event. Since the timing of a
discriminator’s output depends upon the magnitude of the analog input pulse,
our timing technique had the advantage that this ”walk” or ”slewing” effect was
determined by the EL discriminator. By using a Leading-Edge Discriminator

(LED), we introduced a well-defined particle dependent slewing effect which was

later corrected for off-line.

22

The output of the final hodoscope ”AND” in Figure 2.5 was used for two
purposes: as an input into the overall ”hodoscope master," and for timing
relative to the RF cycle of the cyclotron. We used a LeCroy TFC—FERA system to
digitize the time information which requires (in order): an enable, a start and a
common stOp. The enable was generated by the hodoscope master, and the
individual detector starts were generated by the individual valid detector
signals. However, the hodoscope master could arrive as much as 200ns after the
first detector fires (due to walk). In order to insure that the starts arrived after
the enable, a rather unconventional delay was developed by M.A. Lisa [Lisa
93c]. The detector masters were re—discriminated with an ECL discriminator
(width: 500 ns), and sent along modified cables which physically flipped the
signals, the effect of which is to conjugate the logic. The trailing edge of the
(now inverted) signal is interpreted by the TFC (Time to FERA Converter) as a
leading edge, thus effectively delaying the start by the 500 ns "width." The
common stop to the TFC was provided by a discriminated RF signal in
coincidence with a hodoscope master.

During the analysis of the first experiment, we noticed that for certain
detectors, the particle identification lines in the E—AE matrices were
discontinuous; very low values of either variable were suddenly ”dropped” to
lower channel numbers than expected. A subsequent bench test with the Silena
ADC showed that this problem is reproduced when the gate width is larger than
about 2 us, because the ADC digitizes the first (not the largest) peak it encounters
within the gate. Using a wider gate allows ripples in the upward sloping signal
to be misinterpreted as the peak. In our second experiment, we set all Silena
gates to less than 1.5 us and did not observe this phenomenon.

2.2.2 The MSU 41: Array
Signals from the Ball and Forward Array photomultiplier tubes enter
"splitter boxes" designed especially for the 41: Array. These passive splitters

23

divide the analog signals into three identical signals known as ”fast,” ”slow,"
and "timing." Both the "fast” and ”slow” signals are appropriately delayed and
sent to FERA’s (Fast Encoding Readout ADC’s) where they undergo charge
integration and digitization.

The "timing” signal’s processing is more complicated. First, it is
discriminated in an Phillips Scientific 7106 discriminator. The individual
discriminator outputs are themselves delayed and sent to TFC start inputs.
These discriminators also have a ”sum” output which generates a voltage level
of ~50mV for each firing channel; these are used to determine the multiplicity of
each event. The 45 Forward Array (or High Rate Array) detector ”timing”
signals pass through three Phillips discriminators, the ”sum” outputs of which
are then linearly added in a ”summer" box, providing a measure of the Forward
Array multiplicity.

The 170 Ball detector ”timing” signals are similarly processed to measure
the ”Ball” multiplicity. Finally, the Ball sum signal and forward array sum
signal are added to measure the overall multiplicity. These three ”sum” signals
(Ball, FA and ”total”) are each sent to a maskable, CAMAC programmable CFD
(Constant Fraction Discriminator). By setting which of these three sum signals to
consider, through the discriminator’s masking capabilities, one forms the 41:
trigger by programming the discriminator’s threshold. In this way, one can
trigger on ”Ball," ”Forward Array,” or ”Total,” and one can set the minimum
multiplicity through the discriminator threshold.

2.2.3 Master Electronics

As described above, the usual trigger conditions required 2 hodoscope
detectors firing along with a minimum number of 41: detectors. The 41: Array
self-gated its own ADC’s and would fast-clear that digitized information if no

hodoscope master appeared. .
The vetoing of the triggers due to the computer—busy was affected by the

large difference in arrival times between the hodoscope master and the 41:

24

master. We used the original 41: computer busy to veto the 41: master signals,
and we used a stretched ”hodoscope busy" to veto hodoscope master signals.
For the higher energy experiment, we adopted a different method to veto the
master signals. In this last experiment, we used a delayed (not stretched)
hodoscope busy signal. Furthermore, we explicitly required a 41: master
(appropriately delayed) in coincidence with every hodoscope master before
issuing gates to the ADC’s. This change prevented the possibility of events
being written to tape in which hodoscope data was present without
corresponding information from the 41: Array. Figure 2.6 shows the timing of

the master electronics.

t-0 i=1 115 t=1.5 us

41: trigger

 

 

 

 

 

 

 

 

 

 

v———-—-—-il——————-+-———h————

 

l
l
I
I
hodoscope trigger P
l
l
I
I
I
I
j
|
I
l

fast clear
fast clear veto Ll
........ |
l I ......
not busy |
busy ------- I busy
not busy ;
hodoscope busy I l °°°°° § ' busy

 

 

Figure 2.6. Master electronics timing diagram. The dashed lines indicate the
effect of the (approximately) 200 ns ”jitter” in the relative time between 41:
triggers and hodoscope triggers. The fast clear is created about 1.5 us after the

41: trigger, and the fast clear veto is issued immediately following a hodoscope
trigger.

2.3 Calibrations
In this section, we discuss our methods for identifying particle species, and

calibrating the energy and time signals.

2.3.1 The 56—Element Hodoscope

Since the energy calibration of scintillating detectors depends upon the
nuclear species, one must first identify the detected particles before calibrating.
Figure 2.7 shows a plot of CsI(Tl) detector E-signal vs. Si detector AE-signal for
one of the hodoscope detectors in which the different nuclear species can be seen
as hyperbolas. Although clear bands are visible in this representation,
separation of protons from deuterons at high energy becomes difficult due to
limited channel resolution. We transform this E-AE map into a linearized E—PID
matrix which makes more efficient use of the two dimensional space. Particle
identification in areas of low resolution is then accomplished by linearly

extrapolating from regions of higher resolution. Figure 2.8 shows this function,

PID ~ (E+AE)1'80 — Elm, vs. total particle energy. Linear bands defining regions
in the E—PID space were then used to clearly identify the nuclear species: p,d,t,

3He and a. One—dimensional projections of these curves onto the PID axis for
various cuts in total energy are shown in Figure 2.9, where one sees that clean
separation of the nuclear species is achieved over a wide range of proton energy.
The Si detectors were calibrated using a highly linear capacitor and pulser
system, following the method described in [Gong 91c] and [Lisa 93c]. Si
calibrations were found to be linear, in agreement with these previous studies.
Calibration of the CsI(Tl) crystals was achieved by scattering alpha particles

off of a 10 mg/cm2 (CH2) n target at E/A=25,30 and 40 MeV into the hodoscope

detectors. Each beam was run for approximately 8 hours, resulting in the
reactions listed in Table 2.4 which provided calibrations accurate to about 1% for

all 5 nuclear species.

Raw Data E VS. AE
800 ITIIIIIFIIIIIIIIIIIITIllll

 

 
 
 
  

 

 

600 ; '~ —
A g, * -
.-—l i
(D -’ "
g __i _
Cl 1 .
CU F \
o ‘ ' ‘
V ' P 1 l
u ser . _
g _ i/ .
200 . 7
- ‘33:”; ‘ .f ’ . Helium Isotopes -
1:. . ‘ felté‘. :J‘ {fi§;:. :- :’..:._' . : _‘
n“ “ Hydrogen Isotopes ~

 

o 100 200 300 400 500
E (Channel)

Figure 2.7. 2—dimensional E vs. AE for a typical hodoscope detector. Isotopes
separate into hyperbolic bands on this logarithmic scale.

,I-F.

 

PID

27

Particle Identification
1200 l j I I F I l l T I I l I I I I l

 

‘
-

1000

800

600

400

200

lLIlllllLlllLLllLll'lLlll-lJJlJ

 

 

 

Oil 1 .100 200 300 400
E+AE (MeV)

Figure 2.8. 2—dimensional total energy vs. PID plot showing linear separation of
isotopes. Again, logarithmic density scale is used.

PID for various cuts on E

I I I T I l fi U I I I j I T T I

 

 

 

 

 

1000 p d .
t Ep=2s—42 MeV
100
Peak/Valley-wo
10
l . | . l l
l I r f l l
1000 p
d t E =60— 72 MeV
7d 100
F" Peak/Valleyaifil3
.: 1° i ll
>..
1000 p
' d
10 i Peak/Valleyfila E i
1000. i i
p
d ..
100 t E,=125—14o MeV :He '3
3He 2
1° Peak/ValleyflS I. -
r r n r L r r r l I r r n l l I I
250 500 750 1000

PID

Figure 2.9. l-dimensional projections of PID using thin cuts in energy, showing
particle identification resolution as a function of energy.

29

Table 2.4 Calibration Reactions

 

 

 

 

 

 

Particle Calibration State Scattered off in Residual Nucleus

Type Reaction Residual Nucleus: I“ (energy in MeV)

p p (a,p) a N / A: knockout reaction

d 12c (a,d) 14N 14N: 5* (8.96)

.. + .. ..

* 12C (out) 13N 13N: Jf (g.s.), 4} p} (3.55, 3.51), 32- (6.89)
3He no (a, 3He) 13c 13c: 4" (3.5.)
4He no (a,a')12C‘ 12c: 0+ (g.s.), 2+ (4.44)

 

 

 

 

 

Figure 2.10 shows the single particle energy spectra resulting from the
calibration run at E/A=30 MeV. The calibration curves were assumed to be
linear, based on previous work with these detectors [Gong 91c, Lisa 93c]. Losses
in the Ta foil and in the AE detector were accounted for, and in each case the
centroid of the scattering peak was used to determine the energy calibration
using two-body kinematics. Calibration curves for a representative detector are
shown in Figure 2.11. In the most backward detectors, in which deuteron peaks
could not be extracted, we averaged the proton and triton calibrations to provide
an approximate deuteron calibration. This method was compared with the true
calibration in more forward detectors, and Figure 2.11 shows that the
”approximate" calibration (dashed line in deuteron panel) is nearly
indistinguishable from the true calibration (solid line). Figure 2.11 also shows
the linearity of the Si detector calibration for this telescope.

Energy losses through the Ta foils and AB detectors were calculated using
the Littmark—Zieger tables [Litt 80], following Reference [Lisa 93c]. To test this
method, we also determined the energy loss through the Si detectors using their
known calibrations. For the lightest ions, the methods agreed to about 0.1%.

However, for 3He, there was substantial disagreement (about 2%). Therefore,
we recalibrated the entire device using the measured energy loss through the Si

30

 

_ _ O
Ea—124MeV, Blah—31.7
2000-"'I""I'”'I""I""-—

p(a.p)a

 

 

 

 

 

 

 

o 1 1 j l l l l 1 1 1 l l l l l 1 L
100 200 300 400 500 600
E (channels)

Figure 2.10. Energy spectra gated on detected particle type (p,d,t,3I-Ie,a; top to
bottom) for our E/A=30 MeV calibration run.

31

1000 0 1000 1000

 

 

50 Tfi U I I I I I I
1 t I .. l I
Ch-
I- di- all)-
D d. “P
125 ~~ —— ——
)- dt- cl-
)- cn- un-
u- an un-
- qu— un-
100 - —— —-
:- qtn u)-
. up an-
'- dl- all-
). un- u-u-
75 r- -- d n- t
L p dr- -(I-
u- an- db
l- u:- un-
- 1L 'l"
50 — — --
)- - un-
#- 1!- un-
)- q- cu-
:- di- qr-
25 )- ql- .1
I- II)- II
II I:
'l .

-
I.
p
p
_
h

 

20

150 _

125 :—

Energy (MeV)

: 15
100 :-
75 :— 10

507’

)-
llllllllllljlljlklllllllllllJllllljlllllLlllll

U ViriliriIl'rtilUlthIUUU

Si
Detector .

l l l l l L l l

25

 

 

lellllllllLlllLlllljjlIllll

 

 

 

rLllllLlllllllllllllllljlllll

O .1..l..4...ll
0 1000 1000 2000

 

Channel Number

Figure 2.11. Linear calibration curves for a typical hodoscope detector. The
dashed line in the deuteron panel is the average of the proton and triton
calibrations, which was used for those detectors in which insufficient deuteron
statistics prevented a direct calibration.

32

detectors and used the tabulated values to account only for the thin Ta foil. It
should be noted that this discrepancy did not affect the proton calibration of
Reference [Lisa 93c] to more than 0.05%, and that the two-proton correlation
function was unaffected within its uncertainty.

Any two—particle observable measured in experiments must take into
account ”real” and ”random” coincidences. In our work, misidentification of
random coincidences will lead to a suppression of the correlation function, and
therefore must be screened.

In both experiments, we recorded the time at which each hodoscope
detector fired, relative to the Radio-Frequency (RF) time of the cyclotron, Ti-

TRF. Calculating (T i'anl' (T j—TRF)= Ti-Tj then provides a measure of the relative
time between the firing of detectors i and j. This variable is plotted, after being
summed over all detector pairs recording two protons, in the upper panel of
Figure 2.12, where one sees a rather broad distribution due to the effects of
discriminator walk [Gong 91c, Lisa 93c]. By assuming a Gaussian shaped

amplifier output, V(t) = 13e(t-t°)2/1 , where E is the particle energy, t is the
observed detector firing time, t0 is the ”true” firing time, and ‘t is a particle-type
dependent rise time constant, one can easily correct for the energy-dependent
walk effect. Solving for to, one obtains: to = t "RAW where Eth is the
threshold level of the discriminator. After making this transformation, the
relative timing spectrum appears in the bottom panel of Figure 2.12, showing
distinct peaks corresponding to different cyclotron beam burst. We defined as
”real” events those which fell to within 2 )5 cyclotron RF bursts of the peak.
Proton pairs with relative times in the ”random” section of the spectrum were
used to correct for the small random background. Since four RF peaks were

included in the ”random” and five in the ”real” definition, we used the relation:

33

p-p Coincidence Relative Timing

 

 

 

104 IIIIIrTIIlIITITIIIYIIfiI IIII
103
102
Uncorrected '
1o1
2
l.|....ln...|a...l.. I .
.2 *j'l""l""|""l" I: '
>.. I
105 , <I—Rea}—> l
l I
104 I I
lRandom Randoml
I I
3
1° I I
I I
102 I I
I I
1 Corrected ~
1° 'w I ' I "I;
"lljllllllllLlllllllllJ4JIIJLIII

 

 

-300 -200 -100 0 100 200 300

Relative Time (nsec)

Figure 2.12. Relative timing of detectors both recording protons. The upper
panel shows the results before ”walk” correction. After the correction (lower
panel), the spectrum displays clear peaks corresponding to different cyclotron
beam bursts.

34

”True” = ”Real” - (5 / 4 ) x ”Random.” The random contribution amounted to

less than 2% of the coincidences for both experiments.

2.3.2 The 41; Array

Particle identification and calibration in the 41c Array was accomplished by
analyzing ”fast” vs. ”slow” spectra. The 41: Group has put together a standard
procedure which gain-matches all of their detectors in a systematic and routine
way [Wils 91]. Each fast-slow map is ”stretched” to fit into the standard
detector template. By matching the observed particle lines with predefined
”standard” positions, one identifies each species and calibrates simultaneously
[Li, T 93 and Cebr 90]. This calibration technique is accurate to approximately
10%.

2.4 Impact Parameter Selection
The impact parameter with which two nuclei collide cannot be determined
directly, but it can be roughly deduced from ”global” observables which are
related to the ”violence” of the collisions. Following the work of References
[Phai 92, 93b, Hand 94 and Lisa 93c], we studied three commonly used global
observables.
1) The charged particle multiplicity, NC, is the number of 41:
detectors that record a charged particle.
2) The total transverse kinetic energy is defined by:
NC
lat = E1 E, sin2(ei) (2.1)

where Bi and 9i are the kinetic energy and laboratory angle of
the ith measured particle by the 41: Array in an event. This

definition is appropriate at nonrelativistic energies since

35

Esin2(0) = pi /2m provides a measure for energy dissipation

into velocity components perpendicular to the beam axis.

3) The midrapidity charge, Zy, is defined by:

NC
zy =22.- -9(y.- —%y..,)-C~>(%y,m -y.-) (2.2)
'=1

where 9(x) is the standard Heaviside function and Ytarr Ypro
and yi are the rapidities of the target, projectile, and ith charged

particle detected by the 41: Array, in the center-of-momentum
system.

In order to reduce the effects of ”self-biasing,” particles detected in the 56-
element hodoscope were not included in the definitions of any of the global
observables used to construct the impact parameter.

Phair showed that E provides the best measured of centrality for collisions

of 4°Ar+197Au at E/A=50, 80 and 110 MeV [Phai 92, 93b], and Lisa showed the
same for collisions of 36Ar+45$c at E/A=80 MeV [Lisa 93c]. The total transverse
kinetic energy spectrum, dP/dEt , for collisions of 36Ar+455c at E/A=120 MeV is
shown in Figure 2.13 for three triggers: Ball (41: multiplicity 21) shown in solid
squares; Singles (hodoscope multiplicity 21) in open diamonds, and Coincidence
(hodoscope multiplicity 22) in solid circles. Since larger values of Et correspond

to more central collisions, Figure 2.13 shows that requiring particles to enter the
hodoscope preferentially selects more central collisions.
Following References [Phai 92, Phai 93b, Hand 94 and Lisa 93c], we

construct a reduced impact parameter scale by means of the geometric relation

 

b(X) = Jiégdx, (2.3)

36

Et Distributions for Three Triggers

 

 

   

 

 

 

I I I I I I I I I l I I I I I I I I T [ I I I I -
“AH“Sc; E/A=120 MeV 33
T
> —_
m 5
E :
If -
"U
\10"3 :- llf’ '1
e E 1":- E
"U : ..(, Z
.. Q _
I
_ .Ioo -
10—4 :— I Ball Trigger '2? -._:
E 0 Singles Trigger 5% I
- O Coincidence Trigger -
I— n + _
10-5 J 1 I l I l l 1 LJ 1 l I l I I l l
O 200 400 600 800 1000
E (MeV)

Figure 2.13. Total transverse kinetic energy distributions measured in the 4::
Array under the three trigger conditions ”Ball,” ”Singles,” and ”Coincidence."

37

where dP/dX is the normalized probability distribution for the global
observable X. If the variable X exhibits a strictly monotonic dependence upon
impact parameter, b, then b(X) = b(X)/bmax where bmax is the maximum impact
parameter for which the measured value of X assumes a nonzero value.
Furthermore, we assume that the detecting system has perfect efficiency
regardless of impact parameter, from which it follows that with an inclusive
(minimum bias) trigger, the probability of observing an event with impact
parameter b follows the geometric distribution: dP/ db~b.

Figure 2.14 illustrates the relationship between Et and b(Et) defined by

equation 2.3. The top panels show the transverse energy spectra dP/ dEt for the

36Ar+45$c reaction at E/A=120 MeV (left) and at E/A=160 MeV (right). The
lower panels show the corresponding reduced impact parameter scales, b(Et) ,
which range from 0 for the most central collisions to 1 for the most peripheral
collisions. The analysis of our data focuses on central collisions, defined
bme.) = 0.0 - 0.03.

Through the relationship bait) , any distribution dP/dEt can be

transformed into a reduced impact parameter distribution using

dP dP db
"7 = — — . 2.4
db dB: dEt ( )
Figure 2.15 presents the reduced impact parameter distribution corresponding to
the different experimental triggers for our E/A=120 MeV analysis (the E/A=160
MeV results are very similar). The solid line represents the minimum bias

trigger which, by construction, increases linearly with b. The dot—dashed and
dashed lines show the distributions for the single and two—proton inclusive

events in the 56-element hodoscope, respectively. Requiring one or more

38

E/A=120 MeV E/A=160 MeV

IIIIIIITjIII IIIIIIII IIII

 

Ball Trigger

dP/dEt (MeV-1)
5.
GO

 

 

 

 

 

10‘“4 ‘ ._
I: ‘

. --
10—5 : :
1.0 '1'“ "j
I ‘ :
0.8_ -I_ -E
5:: 0'6 r' 7? ?
(.o : :: :
0.4 .— —..— —.
0.2 I— -Ei- -I
0.0:....I..I..3....I.-L. 3

0 500 1000 0 500 1000
Et (MeV)

Figure 2.14. Upper panels: total transverse kinetic energy spectrum for the
”Ball" trigger in our E/A=120 MeV reaction (left) .and the E/A=160 MeV
reaction (right). Below each spectrum is the corresponding reduced impact

parameter scales b(Et) based on these distributions.

39

 

 

2.0 I I T I I I I T f Ti j I I l I I 1 fl I I I W I
1.5 — -
. .' / ’ \ \ .
. / .
I: / \ .
[3.3 / \
v b . f - \ -
.D .' '-

(1: 1.0 '- .- / °. \\ -
\ I- .‘ / .o d
On - l \

“U ' .' \ ‘
. -' I \ .
.' / , \L
- o: r ..o
0.5 F :/ -
- _.'/ — Bail trigger -
- .‘l - Hodoscope singles trigger °
- 4'7 ° - - Hodoscope coincidence trigger -

 

00rnrermrIIrrrrIJrrrIrrrr

'0.0 0.2 0-41 0.6 0.8 1.0
b(Et)

Figure 2.15. Average reduced impact parameter distributions dP/db(Et)for the
three event triggers studied.

40

protons in the hodoscope clearly biases the data to slightly more central
collisions.

In Figure 2.16 we compare the transverse kinetic energy spectrum to
predictions of the BUU transport model using a geometric weighting of impact
parameters. Since the BUU transport model only predicts the emission of
nucleons, Figure 2.16 shows the predicted values of E when neutrons are
included (dashed line) and not included (solid line) in the definition of E. The
obvious disagreement between observed and calculated Et spectra arises
primarily from the fact that the BUU model describes the time evolution of the
single-particle phase space distribution. As a consequence, the model cannot
reproduce observables sensitive to the emission of complex fragments
contributing to the sum in Equation 2.1. Similar difficulties are encountered for
other observables, such as NC and 2,. Since the calculations fail to reproduce the
observed Et spectrum, comparisons or Et-selected data to model predictions are
not straightforward. Clearly, the use of identical Et cuts on data and theoretical
predictions is inappropriate. 3

We attempt to construct a realistic impact parameter distribution sampled
by the data. For this purpose, we employ alternative reduced impact parameter
scales derived from the charged-particle multiplicity and midrapidity charge
(using X= NC and ZY, respectively, in Equation 2.3). By construction, a sharp cut
in the observable X corresponds to a sharp cut in b(X). In order to quantify the
values of impact parameter that are sampled by the data, one must consider the
effects of fluctuations in the relationship between the true impact parameter of a

collision and the global observable X. Following References [Phai 92, 93b, Hand
94 and Lisa 93c], we obtain an estimate of the scale of the fluctuations in the

relationship between the true reduced impact parameter btme and the

transverse kinetic energy Et by observing the effects of narrow cuts in b(NC) and

41

 

  
  
 

I I I I I I Ij I I TI I II I I I I I I I I I I I I-
__ 0 data
10 1 f?
— BUU (protons only) 3
- - BUU (protons and neutrons):
I I Z
> ' :
0) ..
2 ” r ‘
v '- \\ _
['3‘ —3 \i
g 10 E u' 5
0.. : :
'U _ '\ _
I— \ -
_ l‘ -I
I
10-4 :— l’ '—.:
'- IIl‘Il "3
I ,II I
_ Ill-ll
..I 1
IIIIIIIIllIIIILI

 

 

 

10—5 IIILLIJIIIIII
' 0 250 500 750 1000 1250

Et (MeV)

Figure 2.16. The total transverse kinetic energy spectra, dP/dEt, measured in the

41: Array (solid) is compared with the predictions of BUU where neutrons are
included (dashed line) and excluded (solid line).

42

b(Zy)on the distribution dP/db(Et). The upper panel of Figure 2.17 shows

dP/dbCEQfor narrow cuts in b(NC) (dotted line) and in 13(zy) (dashed line),
and for a double-cut on both variables (solid line) for our E / A=120 MeV

collisions. Similar distributions are shown in the lower panel for somewhat less

central cuts. The widths,0(b), 0f the double—cut distribution can be taken as
upper limits of the widths of the distributions of the true reduced impact

parameter for a given values of E.

To obtain a ”true” reduced impact parameter distribution dP/db true , it is

necessary to fold the effects of the finite widths into the impact parameter
distributions. As an ansatz, we take the following expression for the probability
distribution of true reduced impact parameters for events filtered by a sharp cut

on B ‘ Bmt):

A A A A 2
de;b ,. b-b ,. .
(warm: .223) sow-4..)

 

 

For a given cut, Efimn SEt SE9”, the true reduced impact parameter

distribution is given by:

at“: m db m at“:

b , dP i333 ) dP B

i I ”“de ( "“9 ) ( )J (2.6)
bmin

with 5min = 5(Etma") and 13m = 5(Efni"). By inserting the geometric
distribution dP/db = 213/ bfim , into Equation (2.6), we construct explicit impact

parameter distributions used to weight the impact parameters of the BUU
calculations. In our calculations, we assumed bmax=10 fm; an additional

43

 

fiIITIWIIIITfIIIIifiI—FIIIIfi
P

6 3- B(E,): B(X)=0.05—0.15

l I

l l l l I

x=m

dP/db(Et)

 

 

 

 

 

' BIB.)

Figure 2.17. Reduced impact parameter distributions dP/db(Et) for the narrow

cuts on b(Nc) and b(Zy) indicated in the figure. The upper and lower panels
show for central and midcentral cuts. '

requirement of proton emission in the direction of the hodoscope was imposed

when selecting phase—space points for the calculation of correlation functions.

For illustration, the upper panel of Figure 2.18 shows the sharp cut in bait) used

in the subsequent analysis of our data. The lower panel shows the corresponding
distributions, extracted via Equation 2.6 for our theoretical E/A=120 MeV
analysis (solid) and our theoretical E / A=160 MeV analysis (dashed).

45

 

 

 

 

2.0 I I I I I I I I I I I T I I I I I I I d
1.5 ~_-— -—
(.0 : 3
g 1.0 :- —_'
D-I - .
'5 I I
0.5 _— 'j
0.0 I \ '
. ’ \ :
I \ ~ .

o F I ‘
g 3 — , \ — E/A=120 MeV -—
4.: - ..
g - l’ ( — - E/A=160 MeV -
Q, . , \ -
"d - I \ q
1 h ' \ -
— \ —
. / \ .
n \ -
O h r r r I 1 r I I r 1\\ _.I _r J I r r r '-

.0.0 0.2 0.4 0.6 0.8 1.0

btrue I b

Figure 2.18. Upper panel: sharp cut in b(Et) used to define our central events.
Lower panel: distributions dP/db We of ”true” reduced impact parameter

corresponding to the cut on b(Et) (in upper panel) for our E/ A2120 MeV results
(dashed line) and our E/A=160 MeV results (solid line).

Chapter 3 - The Two-Proton Correlation Function

In this chapter we give a brief review of the formalism in which the two—
proton correlation function is calculated from the predicted single particle phase
space density. The sensitivity of the correlation function to the source size and
lifetime is discussed with the use of analytical source parametrizations. Finally,
we discuss the construction of the experimental correlation function from

measured data and the effects of target thickness.

3.1 The Koonin—Pratt Formalism

The theoretical framework known as the Koonin—Pratt formalism uses the
single particle emission probability and knowledge of the two—particle relative
wave function to calculate the correlation function. Since the derivation of the
general particle-particle correlation function is produced in detail elsewhere
[Koon 77, Prat 84, Prat 87, Boal 90, Gong 91a], we discuss only those results
which are pertinent to two-proton correlations.

The formalism requires knowledge of the probability of emitting a proton
at position ‘f with momentum p at time t. This information is contained in the
”source function” g('f,p, t), which can be predicted by theoretical models such as

the Boltzmann-Uehling—Uhlenbeck (BUU) dynamic transport model [Gong 90c,
91b, Baue 92a]. For demonstrative studies, schematic Monte-Carlo simulations
can produce analytical source functions; for example, Gaussian spatial
distributions with exponential lifetimes. In most of our analysis, the BUU model
is used to predict the theoretical source function g(‘f, p, t) . In Chapter 5,
however, we use schematic simulations to study whether of the correlation
functions can be used to identify exotically shaped sources such as tori and discs.

Once the source function is known, three assumptions are made in order to

calculate the correlation function. It is assumed that the protons are emitted

46

47

independently of one another, implying that all correlation between the two
protons are due to the final state interactions discussed in chapter 1: the nuclear
and Coulomb forces and the effects of the Pauli principle. However,
conservation laws restrict the independence of the particles [Lync 82]. Entrance
channel effects [Zhu 91] and correlations in the source itself [Bern 85, Alrn 93,
Kund 93] can also break this assumption.

It is also assumed that the two protons interact only with each other and not
with the emitting source nor with other emitted particles. For light particles well
above the Coulomb barrier this is a reasonable assumption because the Coulomb
field will not affect the relative coordinates of the emitted particles. For heavier
fragments, however, effects of the nuclear residue become important [Kim 91,
Kim 92, Fox 93, Bowm 93, Glas 93]. Even for light particles, especially near the
Coulomb barrier [Eraz 91, Gong 92], three—body calculations have shown a
sensitivity to the charge of the emitting source [Ledn 95, Mart 95].

The third and final assumption is that the source function g(‘r‘, p, t) is

approximately constant over regions of momentum of order q, the two-proton
relative momentum. This is reasonable since the momenta of the protons are
typically hundreds of MeV/c, while the correlation function’s region of interest
lies in the range q=0~50 MeV/c.

With these assumptions, References [Koon 77, Prat 84, 87, Boal 90, and
Gong 91a] have shown that the correlation function can be written:

1 + R(f’r 9) = C(P’ ii) = Y1 (2310; :32) =

 

_ _ _ _ 2

ld4fld4r2 gfilrlyrtl)'8621%tt2)' cP(?1 42 '35:;‘3 a‘1)l ( )

.. : .. , 3.1
ld4r1 8513741) ' ld41'2 8525732)

where q = é-(jil — 132) is the relative momentum in the rest frame of the proton

 

 

pair, P is the total momentum of the proton pair, and the relative wave function,

48

<p('r‘,2j), depends only on the relative phase space coordinates at the time of

emission of the second proton.

The relative wave function was calculated by modifying the full Coulomb
wave function [Mess 76] by the contribution from the nuclear interaction
between the protons. This contribution was calculated by solving the
Schrfidinger equation for the f = 0 and f = 1 partial waves with the Coulomb and
Reid soft-core potential [Reid {68]. The two—proton wave function was then
calculated as:

lq>(?.4‘)|2 =-,‘;I<p.(?.4‘)l2 +%|<p.(r.4)|2. (3.2)

where (p , ('r’,§) anqu , (7,5) are the singlet at triplet two—proton spatial wave

functions respectively, whose weights in Equation 3.2 are determined by
assuming that the proton spins are statistically distributed.
For noninteracting identical particles, the squared relative wave function

~.- _-.-2
e'rqie ' q

hasthe form: '(p 7,2} 2 ~

 

 

~ 121: cos(2'r‘ - 5), Where the upper (lower)

sign applies to bosons (fermions), due to the quantum mechanical (anti)
symmertrization requirement. The correlation function is then the Fourier
transform of the square of the source distribution [Kopy 72, Zajc 92]. For spin-
1 / 2 fermions, then, the correlation function reduces to 1 / 2 at q=0 , and returns to
1 with a characteristic width of qle /rx, for a source whose spatial extent in the

x-direction is about rx [Kopy 72]. A full measurement of the 3—dimensional

characteristics of the correlation function would, in principle, yield the size and
shape of the emitting source.

The Coulomb interaction between the protons produces a dip in the
correlation function that goes to zero as q goes to zero. As long as the spatial
extent of the source is well below the proton Bohr radius of 58 fm, this dip is
largely independent of the relative orientation of 'f and q . An unfortunate

aspect of the Coulomb interaction is that it lessens the number of available pairs

49

at small relative momentum, making it more difficult to determine the effects of
identical particle interference.

The strong interaction provides a very sensitive gauge of the size of the

source. An attractive 2He ”near—resonance” (with a phase shift of about 60°)
causes a bump in the correlation function at q z 20 MeV/c. The size of the bump
scales roughly as the proportion of proton pairs whose relative position is within

the size of the nearly bound state. The height to the bump, therefore, scales

approximately as R’3 . However, since this ”resonance" is S—wave, it provides

no directional information.

3.2 Resolving the Space-Time Ambiguity of the Correlation Function

Illustrative calculations of the two—proton correlation function have been
performed in detail in references [Gong 91c] and [Lisa 93c] for various source
parametrizations. Without reproducing those illustrations, we discuss some of
their relevant points.

Calculations of the correlation function with schematic sources of zero
lifetime are able to easily differentiate source sizes which differ by as little as 1/ 2
fm, provided the source radius is less than about 7 fm. For larger sources, the
sensitivity is diminished, but not removed. Calculations with schematic sources
of fixed size emitting protons of fixed momentum revealed sensitivity to the
lifetime of the source on order about 10 fm/c, provided the lifetime was less
than about 100 fm/ c. Similarly, the sensitivity was diminished, but not
removed, for longer lived sources. Calculations with fixed lifetime and fixed
source size showed that correlation function with cuts on the total momentum of
the proton pair were distinguishable provided that the momentum cuts differ by
about 200 MeV/c. It was shown that faster moving protons exhibited a smaller
peak at q=20 MeV/c than slower protons. This can be understood in terms of
the ’apparent’ source size. For a faster pair, the spatial separation at the time of

the second proton’s emission will be greater than for a slower pair, due to

50

kinematics. This faster pair, therefore, appearing to come from a larger source,
has a reduced peak in the correlation function.

Although the schematic calculations described above show sensitivity to the
radius (when lifetime is held fixed) and to the lifetime (when radius is held
fixed), they cannot differentiate a small source of large lifetime from a larger
source of shorter lifetime. To extract such information, it is necessary to exploit
the directional sensitivity of the quantum mechanical contributions to the
correlation function. This is accomplished, as described in chapter 1, by gating

on the angle between relative and total proton momentum, w = cos”1 GP ~ ql/Pq).

Care must be taken when defining this angle because although the relative
momentum is independent of reference frame (in the non-relativistic limit), the
total momentum depends upon the frame of reference. Since the phase-space
distribution of emitted protons is extended in the direction of P (defined in the
rest frame of the emitting source), the angle \[l should be constructed in the source
rest-frame [Gouj 91, Lisa 93b, Rebr 93]. In chapter 4, we discuss the effects of
defining the angle W in frames of reference moving with velocities ranging from

0 (laboratory frame velocity) to vpmj (projectile frame velocity) for both central
and peripheral collisions.

3.3 Calculating Correlation Functions from Model Predictions

The model (BUU, QMD, schematic calculation, etc.), is used to generate the
source function, g('r‘, p, t), from which the correlation function is then calculated
employing the Koonin-Pratt formalism described earlier. In practice, the model
produces a set of phase space points for different impact parameters and

reaction-plane orientations. In order to simulate a geometrical distribution of

impact parameters, Nb, the number of ensembles (or simulated events) with a

given impact parameter b, was set proportional to b. With n, the event number,

51

running from 1 to Nb and i, the particle number, running from 1 to Mn,b (the
multiplicity of the event), the correlation function was calculated as [Lisa 93c]:

1+R(q) =

2
NbNanlb Mngb p$1_ pmz p'"l-p'-n2
1b II 1_ II 2 1” lb
2.~—‘;22 2 2 (1— 6.,-6.,.,)6A[q— 2 (xi, " ,g'. 2

n1 II; I

 

 

 

 

 

 

C

 

NbiNbZM nrbI Marl,2 .
2 22 2 2 2 (1-6 ,- 6.1.,6r6b,)6.[q-—2——

b1 b211, n2 I

 

 

(3.3)
Here, the primed momenta are calculated in the center-of-momentum
frame of the proton pair and the double-primed coordinates are calculated in the

center-of-momentum frame at the time of emission of the second proton; (p is
the wave function of relative motion between the two protons; 6A(q) is the
”binning function” which is unity for [q] S %A and zero otherwise; and C is a
constant which sets R(q)=0 for q between 60 MeV/c and 80 MeV/c.
Construction of the correlation function gated on centrality involves setting Nb
proportional to the appropriate distribution dP/db true shown in the bottom
panel of Figure 2.18.

3.4 Constructing Experimental Correlation Functions
Experimentally, the two—proton correlation frmction, 1+R(q), is defined
through the relationship:

EY2(I'51:I'52)= ”(1+R(q))ZYback(i51:f’2)I (34)
where Y2 (p1 ,pz) is the measured coincidence yield for two protons with
momenta pzand p2, Yback(Per2) is the background yield, and N is a

normalization constant. The sums on both sides of Equation 3.4 are extended

52

over all combinations of detectors and proton energies corresponding to each q-
~ bin.

The background yield, Yback (filrfiz ), is typically constructed in one of two
ways. The first method, known as the ”singles” technique, defines the
background yield as the product of the single particle yields, measured with the
same trigger condition as the coincidence yield: Yback (131,132): 13651)- Y1 (p2)
[Zarb 81, Chit 85, 86a, Poch 86, Kyan 86, Chen 87a, 87b, Poch 87, Awes 88, Gong
90b, 90c, 91b, Kim 91, Zhu 91, Gouj 91, Lisa 93a, 93c, 93d].

The second method, referred to as the ”mixed—events” technique, generates
the background yield by ”mixing” particles from different coincidence events
[Kopy 74, Zajc 84, Gust 84, Dupi 88, Fox 88, DeYou 89, 90, Cebr 89, Rebr 90, 92,
Lisa 91, 93c]:

Yb...(61,162)=21{63(161 ‘Fl,i)'53(F2 -62,,-)+63(i62 -161,.)-63(161 -62,,)}.
(3.5)
Here, the indices i and j label the ith and jth recorded two-particle coincidence

events and p1,,- and sz denote the momenta of particles 1 and 2 recorded in

events i and j, respectively [Lisa 91,93c]. In our analysis, the index i ran over all
recorded coincidence events while the index j took on values from i to i+10.

The mixed-events and singles techniques have been shown to produce
similar correlation functions, with a slight suppression of the correlation
function when constructed with the mixed-events technique [Lisa 91, 93c]. We
present a comparison of correlation functions constructed with the mixed-events
and with the singles techniques in Figure 3.1, which shows that the differences
are small (about 3%), consistent with previous comparisons.

A mixed events analysis offers the advantage of simplicity since no
singles measurements are required to construct the correlation function.

However, the coincidence technique is known to suppress the final state

53

36Ar + 45Sc; E/A=16O MeV

 

 

 

 

l T I I j I l l l I T l l I
1.5 “- a
~ 6 6 0 singles -
O
r 8 0 mixed events -
8
.. 8 8 -
g 1.0 F 0 9 a 9 e O a 0—
+
H L- ._
L. -
» Pm=200—800 MeV/c —
» B=o.o—o.3 1
0.5 — A
.- l l l L l L l L l l l k l l l -‘
0 20 4O 60 80
q (MeV/e)

Figure 3.1. Correlation functions constructed with the mixed—events technique
(open circles), and with the singles technique (solid circles). The suppression of
the mixed-events constructed correlation function is consistent with previous
studies [Lisa 91, 93c]

54

correlations under study [Zajc 84]. We have adopted the singles technique in
our analysis of the correlation function for four reasons:

0 Consistency with previous analyses [Gong 89,90a,90b,90c,91a,91b,91c,92,
Lisa 91,93a,93b,93c, Kund 93].

0 Ease of interpreting theoretical predictions (the mixed events technique is
more difficult to model theoretically).

0 The singles and coincidence yields sample very similar ranges of impact
parameter (see Figure 2.15); hence the use of the singles yield to construct
the denominator should not distort the correlation function [Lisa 91,93c].

0 Unlike the coincidence method, the singles technique does not suppress

the correlations we attempt to measure (see Figure 3.1).

The correlation function’s sensitivity to small distortions in the coincidence
yield against the large background requires it to be rather insensitive to single-
particle phase space effects and to detector resolution. Reference [Lisa 93c]
shows that the resolution effects for our 56—element hodoscope are of minor

consequence to the correlation function, for cuts on both total proton

momentum, F , and on the angle, w, between relative and total momenta.

In the two experiments of this thesis, (36Ar+458c at E/A=120 and 160 MeV;
see Table 2.1), thick targets of area density 40 mg/cm2 were used to compensate
the lower beam intensities in order to collect reliable statistics. Uncertainties in
the outgoing proton energies (due to uncertainties in the ”location” of the
reaction within the thick target) may lead to distortions of the correlation
function.

Furthermore, theoretical comparisons are complicated by the use of the
software filter intended to mimic the acceptance of the 56—element hodoscope.
Because the use of a ”fine" filter (one which accepts particles into the exact polar
coordinates (0,¢) of each detector) would require prohibitively long calculations,
we have used a coarse filter which introduces a small distortion but allows

55

calculations to proceed approximately 10 times faster. The coarse filter first
accepts single protons that lie within the polar (6) coordinate range of the
hodoscope (6=30°—45°). After randomly picking pairs of protons (appropriately
weighted for relative angular acceptance), it screens these pairs to guarantee that
the individual protons are not closer than 2.6 degrees of one another (the
minimum spacing between detector elements).

The effects of both the target energy loss and the use of the coarse filter are
small. Figure 3.2 shows theoretical correlation functions constructed with the
coarse filter (and no target loss correction) in the solid line and with the fine
filter (taking into account target energy loss) in the dashed line. The source

2 _ 2
function for this comparison took on the form g(r, p, t) ~ e(-'/'°) e( p /2mT)6(t),

with ro=3.0 fm and T=20 MeV. The differences between the effects of the two

filters are smaller than the differences between using mixed-events and singles
techniques, and are within the uncertainty of our measurements, justifying the
use of the faster course filter in our calculations of theoretical correlation

functions.

56

 

     
  
 
    
 

 

 

 

'- I I I 1' j I I I I I I I I I I I I -
1.6 — __
I —-- coarse filter :
1.4 — fine filter ‘1
’3' ’ _‘
a; 1.2 _ _
+
r" ” a
1.0 —"
'- '1
- l
0.8 __ g(r,p,t)=e( r/r.) e( p /2mT)6(t) __
0.6 l l 1 l 4 l l l l L 1 l l l 1 l l l
0 20 4O 60 80

q (MeV/e)

Figure 3.2. A theoretical correlation function constructed from the schematic

2 _ 2
source function g(r, p,t) ~ e(-'/'°) e( p /2mT)8 (t) using the coarse filter, which
does not take into account energy loss through the target, is shown in the solid
line. The correlation function calculated by using the exact filter and correcting
for energy loss through the target is shown in the dashed line. The differences

are smaller than the uncertainty in our measurements.

Chapter 4 - Correlation Functions for 36Ar + 45Sc at EIA=80 MeV

This chapter presents a comparison of BUU predictions with two—proton
correlation function measured for impact-paramter gated collisions of 36Ar +
455c at E/A=8O MeV. We compare both angle integrated and directional
correlation functions for central and peripheral collisions and for high and low
momentum proton pairs. Since directional correlation functions depend upon
the frame of reference, comparisons are made in the laboratory, center-of-mass
and projectile rest frames. Finally, we present a detailed analysis of the BUU
predictions of directional correlation functions for different impact parameters,
proton momenta and frames of reference. The experimental results presented in
this chapter were taken from Reference [Lisa 93a, 93b, and 93c].

4.1 Comparison of Angle—Integrated Correlation Functions.

The main findings of Reference [Lisa 93a] are shown in Figure 4.1. The
solid points in the figure show the measured [Lisa 93a] total momentum
dependence of the average peak height, (1+ R) 4:1 5_25 M eV , c, of the two-proton

correlation function in the region around q=20 MeV / c. As described in chapters
1 and 3, for relatively small sources and short emission time scales, this quantity
is the primary indicator of the extent of the phase—space distribution of emitted
protons. The data are gated on central and peripheral collisions in the upper
and lower panels, defined by 5(E, ) S 0.36 and 505’, ) = 0.44 - 0.82 , respectively
(see chapter 2 and References [Lisa 93c, Hand 94]). Error bars indicate statistical
uncertainties as well as an estimate of the normalization uncertainty in the high-
q region. For orientation, the right hand axis gives the Gaussian radius of a
zero—lifetime spherical source that produces a correlation function with the same

value of (1+R) 4:15-25 MeV I c. Impact parameter filtered BUU predictions are

shown as open circles. The impact parameter probability distribution,

57

58

 

     
  

18” ‘lrfi'LflTjTHfll'H' 13'7
:+Data: b(Et)=O—O.36
. EUU- «4.0
1.6 —-O 'b selected
i-O-sz only /
I ‘45
{ 1.4—
> _
é : y 6.0
3 ~ 8 / .1
«'3 12_ / / 5.5 0
°‘ "'::|"""'"|""l""l""-37 E?
+— A o
6;: 1'8:-0-Data: b(Et)=O.44—O.82 V
i E . ~4.0
V 1.67
C “4.5
1.4”-
{Q‘s—e’¢_$65.o
1.2:— 6.5
.l....l.1..l....l...l

 

 

 

400 500 600 700 800 9001000
IDlab (MEV/C)

Figure 4.1 Average height of the correlation function in the region as a function
of the total laboratory proton—pair momentum, Plab- The right—hand axis gives

the radius of a zero-lifetime spherical source with a Gaussian density profile
that produces a correlation function of equal magnitude. The upper and lower
panels show results for central and peripheral events. Data are shown by solid
circles and impact parameter filtered BUU predictions are shown in open circles.
The BUU predictions of strictly central collisions (I; = 0) are shown in solid
diamonds. The lines are drawn to guide the eye.

59

dP/dlsm , used to weight these calculations was calculated in the same manner

as descried in section 2.4 (using equation 2.6) and is shown in Figure 4.2. For
central collisions (top panel in Figure 4.1), the agreement between experimental
and theoretical correlation functions is satisfactory, but for more peripheral
collisions (bottom panel), the BUU transport model substantially under predicts

the total momentum dependence of the correlation function.

 

dP/dbm.
\
I

 

 

 

Figure 4.2. Distributions of the true reduced impact parameter, dP/dgm , used

for the theoretical calculations of our E/A=80 MeV collisions. The solid line
represents the probability distribution for central collisions, the dashed line for
peripheral collisions.

The dynamics of strictly central (5 = 0) collisions cannot be investigated
experimentally because contributions from nonzero impact parameters cannot be

avoided. Nevertheless, it is interesting to investigate theoretical predictions for

60

this idealized case and to assess the significance of imperfect impact parameter

selection. For this purpose, we also present BUU predictions for collisions with

5 = 0 (shown by solid diamonds in the upper panel of Figure 4.1). As may be
expected (from the differences between central and peripheral predictions), the
removal of noncentral collisions from the calculations leads to an enhanced
momentum dependence of the two-proton correlation function. While the
agreement with data is somewhat worse than for the more realistically filtered
calculations, the qualitative observation of a strong momentum dependence of

two—proton correlation functions for central collisions is already well

reproduced in these simplified 5 = O calculations.

The success of the BUU model in predicting the strong momentum
dependence of the two—proton correlation functions observed in central
collisions (top panel of Figure 4.1) suggests that the BUU transport model
provides a reasonable description of the phase space density distribution of
nucleons emitted in collisions at small impact parameters. Predictions for more
peripheral collisions may be less reliable [Klak 93], because of a deficiency in
treating the nuclear surface.

In the following, we investigate data and theoretical predictions for central

collisions in more depth by exploring two-proton correlation functions with cuts

on the angle w between I.5 and q .

4.2 Longitudinal and Transverse Correlation Functions

”Angle integrated" correlation functions (those with no explicit cut on
W = cos'1(l-5 - q/Pq)) probe the volume of the phase-space distribution of emitted

particles with little sensitivity to its shape [Pratt 87, Gong 91a]. Without
independent knowledge of the size of the emitting system and the emission
mechanism (surface versus volume emission), ”angle-integrated” correlation
functions cannot discriminate between smaller sources of longer lifetime and

large sources of shorter lifetime.

61

This space—time ambiguity may be reduced by analyzing two-proton
correlation functions with cuts on \y [Koon 77, Pratt 87, Gong 91a, Awes 88, Gouj
91, Lisa 93b, Rebr 92]. Emission from a long-lived source leads to a phase space
distribution elongated in the direction of P , the total momentum of the proton
pair in the rest frame of the emitting source. The magnitude of this elongation is
of order P‘c/Zm where 1 is the average time interval between the emissions of the
detected particles. Two-proton correlation functions exhibit a directional
sensitivity primarily due to an increased Pauli suppression in the nonelongated
(transverse) direction. For very extended phase—space distributions, the
Coulomb interaction causes additional w dependencies [Gong 91a, Kim 92]. For
long—lived sources, transverse correlation functions (ELLIS) are therefore
suppressed at small q in comparison with longitudinal correlation functions

(i’llll-5 ). Since the total momentum, P , depends upon the rest frame of the source,
but the relative momentum, q , does not, the angle w, and hence the definition of

the longitudinal and transverse cuts, depends on the rest frame of the emitting
system. Care must therefore be taken to characterize the rest frame of the
emitting source [Gouj 91, Rebr 92, Lisa 93b].

Longitudinal and transverse correlation functions of low-energy proton
pairs (Plab=400'600 MeV/c) emitted in central 36Ar + 45Sc collisions were

published in Reference [Lisa 93b]. Significant differences were observed when
the longitudinal (u; long = 0°—50°) and transverse (w mm = 80°-90°) cuts were

defined in the 36Ar + 45Sc center of momentum frame. These differences were
largely washed out when the directional cuts were defined in the laboratory rest
frame. These observations could be reproduced by adopting a simple moving-
source parametrization simulating emission from a source of finite lifetime

1 = 20 - 40 fm I c and spherically symmetric Gaussian density profile
p(r) cc exp(—r2/r02) with r0 = 4.5 - 4.8 fm , moving with the center-of—momentum

frame of reference. Energy and angular distributions of the emitted protons

62

were selected by randomly sampling the experimental yields. As part of this
thesis project, we investigate whether the observed directional dependence of
the two-proton correlation function can be understood in terms of the phase-
space distribution predicted by the BUU transport model, including the impact
parameter and source velocity dependencies.

The solid and open points of Figure 4.3 show the longitudinal and
transverse correlation functions measured for central 36Ar + 45Sc collisions at
E/A=80 MeV, with the angle w defined in the center-of-momentum frame of the
colliding system [Lisa 93b]. The central cuts correspond to b(Et ) s 0.36. The top
and bottom panels of the figure show the results for low—momentum (Plab=400’
600 MeV/ c) and high-momentum (Plab=700—14OO MeV/c) protons, respectively.
For consistency with References [Lisa 93a, 93b, Hand 94], we define our cuts on
the magnitude of the total momentum defined in the laboratory frame, but we
will use difi’erent rest frames for the definition of the angle w. Significant
differences between longitudinal and transverse correlation functions are
observed for the emission of slow protons, Plab=400‘600 MeV/c, but not for the
emission of the faster protons, Plab=700-14OO MeV/c, likely reflecting decreasing
emission time scales for emitted protons of higher energy.

For orientation, the solid and dashed curves in Figure 4.3 show
longitudinal and transverse two—proton correlation functions predicted by BUU
calculations for the idealized case of b = 0. The calculations reproduce the
magnitude and the difference between longitudinal and transverse correlation
functions rather well, slightly over predicting the peak height at q~20 MeV/c for
the high—momentum protons, Plab=700—14OO MeV / c. This over prediction was
already visible in Figure 4.1. It is comparable in magnitude [Gong 91a] to the
theoretical uncertainty due to our choice of emission criteria tcut=150 fm/c and

p 5 pa /8. Calculations using tcut=200 fm/c predict slightly reduced correlation

63

 

 

 

 

 

 

.CT-.-,,....,,.-.T.
1.75 ~ Pmb = 400—600 MeV/c a
l
' Data: 0 O°<¢Qm<50° ‘
1-50f O so°<1pc,m_<9o° ‘
_ BUU b=O: — O°<¢cm<50°
1.25 _ so <¢c.m,<9o
1.00—
:5; .
0::
+ r
1.752
F
1.50-
125*
100*
L
0

Figure 4.3. Solid and open points show longitudinal and transverse correlation
functions measured for central collisions at the indicated momenta. The curves
show the BUU predictions for the idealized case of purely central collisions

(b = 0 ). The cuts on the angle \v are defined in the center-of-momentum frame
of the colliding system.

64

functions: they agree rather well with data for the high-momentum gate,
Plab=700'1400 MeV/c, but they under predict the maximum at q=20 MeV/c for

the low—momentum gate and they predict too large a split between the
directional correlation functions.

Figure 4.4 shows results for calculations which incorporate the effects due
to the finite resolution of the centrality filter using the impact—parameter

probability distribution, dP/ (15,,“ , shown in Figure 4.2. These more realistic
calculations reproduce the overall trends of the data rather well, with a slight
over prediction of the difference between longitudinal and transverse correlation
functions for the high—momentum gate, Plab=700'1400 MeV/ c (bottom panel).

In order to provide more insight into the transport model predictions, we
show in Figure 4.5 the acceptance of the hodoscope array in the p x vs. 1’: plane
for single-proton momentum cuts of p lab = Plab /2 . The dashed lines indicate
the angular acceptance of the hodoscope: 0 [ab = 30°-45° . The dotted and
hatched areas depict cuts corresponding to Phb=400—600 MeV/c and Plab=700‘
1400 MeV/c. For reference, the two solid circles depict the Fermi momentum
spheres of the projectile (centered at P2 /A = 395MeV / c) and target (centered at
P2 /A = 0 ), and the dashed circle depicts the region of final momenta accessible
by single nucleon-nucleon scattering processes, representative of a midrapidity
source (centered at pz /A =197MeVlc). The low-momentum cut Phb=400—600

MeV/ c selects protons emitted at large transverse angles with low energies with
respect to the center-of-momentum rest frame for projectile and target. This
kinematic region should be strongly populated by emission from the cooling
participant zone formed by geometrical overlap of projectile and target, and the
simple concept of emission from a source at rest in the center-of-momentum
frame of the projectile-target system may be well justified for central collisions.
In contrast, the high momentum cut, Plab=700'1400 MeV/c, selects fast protons

with velocities closer to the projectile than to the target velocity. In this

65

 

 

 

 

 

 

1.75r'v'rvrvv1vvvvyvfivvu
Plab = 400—600 MeV/C 4
Data: 0 0°<¢cm<50°
1-50 ” O eo°<1pm<90°
BUU b—seleeted: —- 0°<¢c_m.<50° ‘
so°<¢m<90° ‘
1.25—
1.00—
:5:
CE. » .
: 1'75 : : 6 t j 4 : t t % 4r t J. c j : : 4. t jlfi‘
1.50 — 0 / g _
r \h‘
125* _
l
> “ Q \ 2 4) .
1,00. 3 :w _, . 1
0 .
O
r r . . l r . . . l . r l r l . r . . l .
O 20 4O 60 80

q (MeV/c)

Figure 4.4. Solid and open points show longitudinal and transverse
correlation functions measured for central collisions at the indicated momenta.
The curves show the BUU employing the realistic impact parameter probability
distribution shown in the solid line of Figure 4.2. The cuts on the angle \v are
defined in the center-of-momentum frame of the colliding system.

-800 -600 -400 -200 O 200 400 600 800

 

   
    
    
  
  

    

 
       
        
  
        
    

 

BOOIIIIlIriqlrrrrlrrrr rflrltrvrlrrrrlrrrr 800
- \
'— \ \ -1
Z. \ . _‘ 600
600 _ \ «O'O‘x
\ o'o’o‘o’o‘ j
- I \2
_ 'o’o‘o‘o’o’o’o _
Q
_ (9.0.0.0...0. ‘
_ «one — 400
A 400 ‘....\‘
s - Wt?» -
r- ‘.‘
6 ~ ' l
v 200 -— — 200
u j' _
Q. r- _.
j. ..
l. ..
O O
-200 _ — -200
"lllllllllllllllllll rrrrerrrlrrlllrrrr"

 

 

 

 

—800 —600 —400 -200 0 200 400 600 800

p. (MeV/c)

Figure 4.5. Schematic of our detector acceptance and momentum cuts. Dashed
lines represent the detector boundaries at '0 lab = 30°—45° . Solid and hatched

areas represent the low— and high- momentum cuts, Plab=400'600 and 700-1400
MeV/c, respectively. The solid circles depict Fermi spheres of target and

projectile.

67

kinematic domain, contaminating emission from excited projectile spectator
matter is likely to occur, especially when contributions from noncentral
collisions exist. Here, the concept of emission from a single source at rest in the
center-of—momentum frame of projectile and target may become inappropriate.

Differences between longitudinal and transverse correlation functions
caused by lifetime expansion effects are best shown by defining the angle \v in
the rest frame of the emitting system [Lisa 93b]. In less well—defined situations,
other directional dependencies may exist which may not be revealed by our
choice of cuts on w. It is therefore instructive to explore angular cuts on w,
defined in different rest frames and compare them to predictions of the BUU
model.

The upper and lower panels of Figures 4.6 and 4.7 show longitudinal (solid
points) and transverse (open points) correlation functions with cuts on \y defined
in the laboratory and projectile frames, respectively. The right and left panels
show data for the low and high momentum cuts. In Figure 4.6, the data are
compared with BUU predictions for the idealized case of b = 0 . In Figure 4.7,
they are compared to the more realistic impact-parameter gated calculations.

For the low-momentum cut, Plab=400‘600 MeV/c, the differences between
longitudinal and transverse correlation functions become insignificant when the
cuts on \v are defined in the laboratory frame (top, left panels of Figures 4.6 and
4.7) and in the projectile rest frame (bottom, left panels). These trends are rather
well reproduced by the BUU calculations, using either the b = 0 (Figure 4.6) or
realistic impact parameter weights (Figure 4.7). For the present reaction, the
emission of low-energy protons at large angles (190m = 90°) appears to be
rather well described by the BUU calculations, with little sensitivity to
contributions from collisions at small, but nonzero, impact parameters.

For the high momentum cut, Plab=700-1400 MeV/c, no significant
difference between measured longitudinal and transverse correlation functions
is observed when the cuts on \v are defined in the laboratory frame (right, top

68

 

1.75 ;- 0°<v<50° :-
: - BO°<¢<90' 1»
1.50 ' Data: 0 0°<7<50° j;

0 00°<o¢<90°

1.25 E

100;

 

4

 

1+R(q)

 

  

- - l A
v I v .

l
' I

1

 

 

 

 

 

 

 

J
1'75 ’ Pm = 400-600 MeV/c Pm = 700‘1400 MeV/c3
1.50 E _
1.25 i
1.00 I
20 40 60 80
q (MeV/c)

Figure 4.6. Longitudinal (solid points and curves) and transverse (open points
and dashed curves) correlation functions constructed in the laboratory (top) and
projectile (bottom) rest frames. Left and right panels show results for low— and
high—momentum cuts, respectively. Points show data selected by the centrality

cut b(Et ) S 0.36, and the curves show BUU predictions for b = O .

1.75 f ..
: BUU b-selected:— 0°<¢<50°
. - 80°< <90“
1.50 7 I
I Data: 0 0°<¢<50°
’ 80°< (90°
1.25 f I
A 1.00 E
3" ,
m A L A l l A l
+ ’ I i I I ' ' I0 '
"‘ 1.75
1.50 C
1.25 ,
1.00 i

69

 

 

 

 

 

   
   
  
  
  

 

 

Figure 4.7. Longitudinal (solid points and curves) and transverse (open points
and dashed curves) correlation functions constructed in the laboratory (top) and
projectile (bottom) rest frames. Left and right panels show results for low— and
high-momentum cuts, respectively. Points show data selected by the centrality
cut b(Et ) S 0.36 , and the curves show BUU predictions employing the realistic

impact—parameter probability distribution shown in the solid line in Figure 4.2.

70

panels in Figures 4.6 and 4.7), but there is an indication for a small suppression
of the transverse correlation function in the projectile rest frame (right, bottom

panels). This difference, is, however, of marginal statistical significance. These

A

trends are reasonably well reproduced by the BUU calculations using b = 0
(Figure 4.6) which do, however, over predict the magnitude of the peak at q =6 20
MeV/c, as was already evident in Figures 4.1, 4.3 and 4.4. The BUU calculations
employing the realistic impact-parameter distribution (Figure 4.7) predict a
negligible difference between longitudinal and transverse correlation functions
in the laboratory rest frame, in agreement with the experimental findings. In the
projectile rest frame, however, these calculations predict a larger difference than
observed experimentally, possibly indicating that the calculations predict
somewhat too large emission from projectile spectator matter than is observed

experimentally.

4.3 Source Velocity Dependence of BUU-Predicted Correlation Functions

In order to gain additional insight into the rest—frame dependence of
longitudinal and transverse correlation functions predicted by BUU transport
calculations, we plot in Figure 4.8 the relative split, (AR) / (R) , between
longitudinal and transverse correlation functions calculated for specific impact
parameters, b=0, 3, and 6 fm, and for different rest frame velocities,
“I! = CB“, with respect to the laboratory frame. AR = R10” - Rm,” is the
difference between the longitudinal and transverse correlation functions
evaluated in a given rest frame, R is the angle integrated correlation function
which is independent of rest frame, and the ( ) denotes the average value over

the interval 15 Mev / c S q S 40 MeV / c. The top and bottom panels of Figure
4.8 show the values of (AR) / (R) predicted for the cuts Plab=400'600 MeV/c and
Plab=700‘1400 MeV/c, respectively.

71

 

    

 

 

 

 

 

0.20,.,....,.........l -l,‘

: Plab=400—600 MeV/c 2

0.15} _‘
0.10} (2K (2K _:
0.05} $1 ’ _:
0.00} _:

6} E 0b=01m 1
V -0.05} 01323 fm [f (1 _‘
\ ; Ob=6 frn B B 2 :
‘ c. . p0) ‘

6; —O-10 .[ #71 i i 1 1 I 1 1 1 1r? 1 1 1 1 J] 1 1 1 1 I] 4, 1‘
V 0.157 ¢ _‘
: / .

0.10} _;

g /f ~¢>\ 3

0.05} ago \¢\ #1
0.00} ;

L 1

—0.05} _:
—0.10:1 .1....1 ...BF-‘1‘.-..1...§Pfi°i. l

0.0 0.1 0.2 0.3 0,4
51

Figure 4.8. Relative difference, (AR) / ( R) , between longitudinal and transverse
correlation functions as a function of the velocity, 8‘", of the rest frame in which

the directional \y cuts are defined. Shown are BUU predictions for b=0 (solid
circles), b=3 fm (open diamonds) and b=6 fm (open circles). Results for low-
and high—momentum cuts are displayed in the upper and lower panels,

respectively.

72

For the low momentum cut, Plab=400"600 MeV/c, the predictions for
central collisions (b=0, solid circles) follows the trends of the data: the largest
value of (AR) / ( R) is predicted in the center-of-momentum of projectile and
target, and very small differences are predicted for longitudinal and transverse

cuts on \y defined in the target (laboratory) or projectile rest frames.

A qualitative interpretation of this observation was first given in Reference
[Lisa 93b]. As the velocity of the w—frame (BV) increases, the axis defined by P
”rotates" through the spatial distribution of emitted nucleons, as is portrayed
schematically in Figure 4.9. The relative difference between the longitudinal and
transverse correlation functions, (AR)/(R), will be maximal when the axis is
oriented along the distribution of emitted protons, which occurs when 8“,

assumes the value of the velocity of the rest frame of the emitting source.

 

Figure 4.9. Schematic representation of a moving source (shown here in the
center-of-momentum frame) and the directions probed when the angle \v is
defined in various reference frames. The direction defined by P (the
longitudinal direction) is shown to ”rotate" as the velocity of the defining frame
rncreases.

73

A very different behavior is predicted for a large (b=6 fm) impact
parameter (open circles in Figure 4.8) for the low—momentum cut. For these
glancing collisions, no observable differences are predicted in the center-of—
momentum frame. In contrast, significant differences are predicted when the 01
cuts are defined in the target and projectile rest frames, consistent with the
intuitive expectations that emission of midrapidity protons in peripheral
collisions is due to a superposition of emission from target and projectile like
sources.

For the high—momentum cut, Plab=700—1400 MeV/c, differences between

longitudinal and transverse correlation functions are predicted to be negligible
for central (b=0) collisions, independent of rest frame. These predictions follow
the trends of the data. Fast particle emission in central collisions appears to
occur on a fast time scale, and elongations of the phase-space distribution from
finite lifetime effects becomes negligible. For larger impact parameters,
however, (AR) / ( R) is predicted to become large for rest-frame velocities close to
the projectile velocity, indicating that fast, forward-emitted particles in
peripheral collisions are predicted to have substantially (if not predominant)
contributions from the decay of projectile residues. Note, however, that the BUU
predictions for energetic emission in peripheral collisions do not reproduce the
data — see Figure 4.1 and Reference [Lisa 93b]. Furthermore, for our peripheral
cut, no statistically significant differences between longitudinal and transverse
correlation functions were found in the target, projectile, nor in the center-of-
momentum rest frames [Lisa 93b, 93c]. These findings corroborate that the
details of proton emission in peripheral collisions are not well described by our
calculations.

Chapter 5 - Toroidal Density Distributions

This chapter addresses the practicality of using the two—proton correlation
function as a means of identifying toroidal density distributions that are
predicted to form in heavy—ion collisions. After defining directional cuts, we
explore the sensitivity of the correlation function to exotic shapes (disks and tori)
employing increasingly realistic assumptions, including the effects of proton

emission over extend time scales and impact parameter averaging.

5.1 Correlation Functions from Schematic Sources

Calculations with the BUU model predict that disk-shaped [More 92] or
toroidal [Baue 92a, Gros 92, Bord 85, Xu 93, Hand 94] configurations may be
produced in central heavy-ion collisions. Different observables have been
suggested to find a signature of a toroidal breakup [More 92, Baue 92a, Xu 93,
Phai 93a, Glas 93], but no experimental evidence for the formation of tori as yet
exists. The suggested observables were based on either intuitive arguments
[More 92, Baue 92a, Xu 93] or on schematic calculations [Phai 93a, Glas 93] in
which a multifragment disintegration of the torus was assumed. However, none
of these suggestions was substantiated by dynamical calculations capable of
exploring the space—time evolution of the reaction zone. A consistent dynamical
treatment is possible for nucleon emission and for the calculation of two—proton
correlation functions, using the Koonin-Pratt formalism and the actually phase—
space distributions predicted by the BUU theory.

Our analysis concentrates on simulations of central 36Ar + 455c collisions at
E / A=80 MeV, performed with very high statistics. As shown in the previous
chapter, measured impact-parameter—filtered correlation functions for this
reaction were reproduced successfully by BUU calculations. These same
calculations predict the formation of a toroidal density distributions [Hand 95a].

The data were, however, also reproduced by assuming emission from a spherical

74

75

source of finite lifetime [Lisa 93b]. Hence, the agreement with BUU calculations
could not be taken as evidence for the predicted toroidal shapes; see also [Hand
94].

Shown in Figure 5.1 is the calculated evolution of the residual system for
strictly central (b=0) 36Ar + 455c collisions at E/A=80 MeV. Panels from left to
right show density projections at times t=10, 50, 100 and 150 fm/c; top and
bottom panels show projections onto the (x,y) and (x,z) planes, respectively, with
the z—axis chosen parallel to the beam axis. The system is shown to evolve into a
toroidal configuration which is symmetric about the beam axis. In order to
isolate effects due to the spatial configuration of the emitting source, we make

stringent cuts on the directions of total and relative momenta, as shown in

Figure 5.2.
We first cut on the orientation of the total momentum, P , which is selected

to lie within 21:10° of the (x,y) plane. The relative momentum,ij , is then selected
(withini10°) along three directions: am, is parallel to P and thus sensitive to the
source dimension and its lifetime; ti beam and ti pup are perpendicular to P and
thus insensitive to lifetime effects. rim lies in the (x,y) plane and is thus
sensitive to the radius of the torus, while film,“ is parallel to the beam axis and is
thus sensitive to the thiclmess of the torus.

The solid and dashed curves in Figure 5.3 show two—proton correlation
functions calculated under the simplifying assumption of instantaneous emission
from the volume of the toroidal} density distribution at t=100 fm/ c, as
reproduced (from Figure 5.1) in the left hand panels of Figure 5.4. The momenta
for this simulation were chosen isotropic in the center-of-mass frame, with a flat
distribution for each Cartesian component ranging from -200 to 200 MeV/c.

Statistical uncertainties of the calculations are indicated by the error bars. The
correlation functions for ii pa, and Ejpeq, (not shown in the figure) are practically

76

 

1! (fm)

_5~

 

2 (fm)

 

 

 

 

 

 

 

Figure 5.1. Residual system predicted by BUU transport calculations for b=0
collisions of 36Ar + 455c at E/A=80 MeV at time t=10, 50, 100, and 150 fm/c. The
top panels show distributions viewed along the beam axis; the bottom panels
show distributions projected onto a plane which contains the beam axis.

77

Tori Coordinate Definitions
AZ

qbeam

 

 

Figure 5.2. Geometry of a torus and the directions of applied cuts on 5 used in
all of our calculations.

78

 

1.5 '-

1+R(q)

 

1.0 -

 

 

 

Figure 5.3. Comparisons of correlation functions calculated for a zero-lifetime
torus and disk, using the directional cuts defined in Figure 5.2.

79

   
    
  

 

 

.. ' 'c1
....l...rl.-..l.l.l
""I""I""I""

I‘ll

p

 

.4
1
<-

 

u’

 

 

v

o a

o 1

o 1

n 1
J— -4 5
'4» q

' u» 1

a.

.. l
—'n— —1 o
' o 4
. a»

' 0

ID
.-— -5
. I’

>

p

s

’....l...

_ .
. u.._..
.l ‘ l I I'l
AAA AAAA LLAA A‘LL AAAA AAA

-5 O 5
x (1m)

 

 

Figure 5.4. Left panels: density distribution of the torus at t=100 fm/c, taken
from Figure 5.1. Right panels: density distribution of a “disk,” formed by
uniformly filling in the hole of the torus. The radius of the disk’s‘ density
distribution was chosen to produce the same correlation-function peak height as
that produced by the torus.

80

degenerate, consistent with an instantaneous emission. The correlation function
for ti hm exhibits a strongly enhanced Pauli suppression, reflecting the shorter
dimension of the torus (see Figures 5.1 and 5.4). The open and solid points in
Figure 5.3 represent the results of calculations for instantaneous emission from a
disk—shaped distribution obtained by uniformly filling the ”hole" of the
doughnut. (The torus and ”disk” density distributions, which have
approximately equal volumes, are compared in Figure 5.4.) The calculations for
the two source geometries are very similar, indicating that our choice of
directional cuts is not sensitive to the ”hole” of the torus. It will be even more
difficult, if not impossible, to differentiate between a disk and a torus with
experimental correlation functions which average over the temporal evolution of
the reaction zone over a finite window of impact-parameters. However, one
may still hope to distinguish these ”flat” shapes from spherically symmetric
sources, even in a more realistic reaction scenario.

The degeneracy of R(qpe,p) and R(qpa,) is removed if emission occurs

with a fixed lifetime, as is shown in Figure 5.5. Specifically, we used the same
toroidal spatial distribution employed previously, with an added exponential
time dependence, dN/dt cc exp(-t/t), witht = 25fm / c. The choice of t was
made for consistency with Reference [Lisa 93b]. The magnitude of the peak of
the correlation function at q z 20 MeV/ c is reduced, and the correlation functions

are ordered approximately as R(q par) 2 R(&jperp ) 2 R(q beam ), reflecting

increased Pauli suppression due to smaller average particle separations along
the respective directions. ‘-
Figures 5.3 and 5.5 suggest that two—proton correlation functions are well-
suited to extract useful information about nonspherical sources of fixed
geometries. This simplified scenario (of emission from a non-evolving source)

serves only as an example. More realistic calculations will incorporate the

81

 

l

r- -— 'par .1
1.5 - - - 'perp' -

1+R(q)

 
     

2L0 .
' -r=25 fur/e"...

 

 

l..rll.

0‘ 20 40 60 00
q (MeV/c)

 

Figure 5.5. Two-proton correlation functions predicted for emission from a

torus assuming an exponential time dependence of mean emission time 1:25
fin/ c.

82

dynamics of proton emission, and will include contributions from larger impact-

parameter collisions, as described in the following section.

5.2 Correlation Functions from Model Predictions

Figure 5.6 shows that the BUU transport model predicts, for strictly central
collisions, that most of the protons are emitted before the torus is fully formed
(at t = 100 fm/c). This suggests that the differences between our directional cuts
will be reduced, when employing the phase—space density distribution predicted
by the BUU model, due to the substantial proton emission from the compact.
initial configurations (see Figure 5.1). Figure 5.7 shows that when employing the
same cuts as defined in Figure 5.2 (with no restrictions on the magnitude of the
total momentum) that the predicted differences are significantly reduced.
Although there is a clear difference between the correlation functions cut on ti pa,

and 5pm, , indicative of the source lifetime, the toroidal signature (the difference
between ripe“, and qbeam) is substantially diminished.

Further complications arise when unavoidable contributions of nonzero
impact parameters are taken into account. Figure 5.8 illustrates the strong
impact-parameter dependence of the shape and orientation of the residual
system calculated from the BUU theory at t=100 fm/c. Toroidal shapes are
shown to be predicted for small (nonzero) impact parameters, but their
symmetry axes are tilted away from the beam axis. For impact parameters
larger than about 3 fm, the residual system assumes stretched configurations
leading to a binary exit channel, possibly accompanied by neck emission [Mont
94].

The impact parameter distribution used for the calculation of the two—
proton correlation function for central collisions of 36Ar + 45Sc at E/A=80 MeV
was shown in the solid line of Figure 4.2. With our choice of bmax=10 fm, this

figure shows that the most likely impact-parameter for our central cut is about 3

83

 

b 03 m
I I I

dP/dt (arb. units)
to
I

 

 

 

 

Figure 5.6. The time dependence of proton emission, dP/dt , predicted for b=0

collisions of 36Ar + 455c at B/A=80 MeV, showing that most of the protons are
emitted before t=100 fm/c, when the torus is fully formed (see Figure 5.1).

 

1 +R(q)

 

 

 

 

Figure 5.7. Two-proton correlation functions predicted by BUU transport
calculations for b=0 36Ar + “Sc collisions at B/AaSO MeV.

85

 

 

 

     

 

I ‘ "'V"' U""U""""'F""“ " I "U""V""T' I '2
1» 1» 1
1» 1» 1
.. ~ b=3 f 1
db db d
1» 1» 1
1» 1» 1
1» 1» 1
1» 1» 1
A db db d
1» 1» 1
E 1» 1» 1
1» 1» 1
H 1» 1» 1
v db db d
1» 1» 1
1» 1»
h 1 1» I
1 1» 1
db db d
1» 1» 1
1» 1» 1
1» 1» 1
1» 1» 1
db db d
1» 1» 1
1» 1» 1
1» 1» 1
L. -1 1 ll 1“. A- -" Ar 1- “- r- 1 -‘
v I' 'U '8 ft '1' '0 " U' l' "0" 'U 'U 2
1» 1» 1
» 1» 1» 1
» 1» 1» 1
10 b db db d
» 1» 1» 1
» 1» 1» 1
» 1» 1» 1
» 1» 1» 1
5 b db db d
A » 1» 1» 1
» 1» 1» 1
» 1» » 1
E 1» I» 1
o b db db «-
H » 1» 1» 1
v » 1» 1» 1
» 1» 1» 1
» 1» 1» 1
N _5 b .. q»- q
» 1t 1» 1
» 1 1» 1
» 1» 1» 1
» 1» 1» 1
-10 b db db .1
» 1» 1» 1
» 1» 1»
» 1» 1»
’ 1 A 11-- 1 I n A " .11. n- -1-- 11 r “ -l-1- n ._---

 

 

 

 

 

-0-50 510 -10-50 510
x(frn)

-10—5 0 5 10

Figure 5.8. Residual system predicted at t=100 fm/c by BUU calculations for
“M + 4551c collisions B/A=80 MeV for the indicated impact-parameters.

86

fm, and that the distribution contains substantial contributions from larger
impact—parameters. A

Figure 5.9 illustrates the effects of impact-parameter averaging. The
correlation functions shown in the figure were calculated using the fixed-axis
directional cuts defined in Figure 5.2 and by employing the central-cut impact—
parameter distribution shown in Figure 4.2. For tilted tori (i.e., for collisions
with b > 0), the directional cut parallel to the beam axis no longer probes the

smallest dimension of the residual system. As a consequence, the correlation

function R(qbeam) is less suppressed, and the relative magnitudes of the three

correlation functions are different than in Figures 5.5 and 5.7. Furthermore,
dynamical correlations due to impact-parameter averaging [Gong 91b] become
important due to the non-negligible sideward directed flow, causing additional
distortions at larger values of relative momenta (q > 40 MeV/ c). The qualitative
features of impact-parameter averaged correlation functions are therefore
different from those for purely central collisions, making the extraction of a
signal of toroidal (or disk—shaped) density distributions difficult, if not
impossible. Nevertheless, BUU calculations predict correlation functions with
shapes distinct from those representing emission from spherical sources of finite
lifetime.

Unfortunately, the statistical accuracy of our experimental data (References
[Hand 94, Lisa 93a, 93b]) is not sufficient to allow such tests. Impact-parameter-
selected correlation functions of improved statistical accuracy and with
appropriately chosen directional cuts should, however, be able to test the
nontrivial space-time evolution predicted by microscopic transport calculations.
We believe, however, that such an experimental signature would be very
difficult to interpret in a model-independent way.

87

 

1.5 P

1 +R(q)

1.0 '—

 

 

 

 

Figure 5.9. Two-proton correlation functions predicted by BUU transport
calculations for central 35Ar + 455c collisions at E/A=80 MeV. The impact
parameter distribution used for these calculations is shown in Figure 4.2.

Chapter 6 - 35Ar + 45Sc collisions at EIA=120 and 160 MeV

The two-proton correlation function has proven to be a valuable tool in
studying nuclear reactions, and for testing the space-time evolution of nuclear
collision dynamics predicted by microscopic transport theories. They are
sensitive to source-size and lifetime effects [Lisa 93b]. For energies below a few
tens of MeV per nucleon, where long-lived evaporative emission is expected,
measured two-proton correlation functions were found to be consistent with
compound-nucleus model predictions [Gong 91b, Elma 93]. At high energies,
above 200 MeV per nucleon, nuclei should be vaporized and serni—classical
cascades should provide a valid description. For collisions at intermediate
energies, nuclei disintigrate by emitting a large number of light clusters and
intermediate mass fragments. Since this energy range represents the transition
from liquid—like to gas—like behavior, it may be the most interesting region to
study, but it is also the most difficult to model theoretically.

BUU calculations were successful in reproducing two—proton correlation
functions measured at energies below about 100 MeV per nucleon [Gong 91a,
91b, Lisa 93b, Hand 94, Poch 87, Kund 93]. In chapter 4 and References [Lisa,
93c], the BUU model was shown to reproduce detailed depenencies 0f the
measured two-proton correlation function on the total momentum of the proton
pair and on the orientation of the relative momentum remarkably well. The
emerging discrepencies for peripheral collisions were attributed to an
inadequate treatment of the nuclear surface [Lisa 93b, Hand 94], but not to a
fundamental limitation of the BUU formalism. Rather surprisingly, the model
failed to explain inclusive measurements for 4°Ar+197Au at E/ A=200 MeV
[Kund 93] where it should have been on even more firm theoretical grounds
than at the lower energy. Improved agreement was obtained by using two-
proton emission probabilities calculated with the Quantum Molecular Dynamics
model [Kund 93, Aich 91], but such calculations could not reproduce the

88

89

correlation functions measured at 60 MeV per nucleon [Kund 93, Poch 87], and
did not therefore resolve the problem. In addition, the correlation function
measured at E / A=200 MeV displayed an unusual insensitivity to the protons’
energy and a filling in of the minimum at relative momentum q =5 0, neither of
which were observed at lower energies.

Further insight can be expected from impact—parameter filtered data at
higher energy. Therefore, we measured two—proton coincidences for collisions
of 36Ar + 455c at E/A=120 and 160 MeV. In this chapter, we present
comparisons of measured correlation functions with predictions of the BUU
transport model and we explore the importance of particle—unstable resonances

in correlation studies.

6.1 Single Proton Energy Spectra

The measurement of light charged particles is complicated by the
possibility of reactions with nuclei within the detector. When this occurs, the
incident particle’s energy is not accurately measured because of a nonzero Q-
value and the emission of energetic neutral particles (neutrons and photons).
Following the methods of References [Gong 91 c, Kox 87], we calculated the
probability of nuclear reaction loss for protons entering a 10 cm long CsI(Tl)
crystal, shown in Figure 6.1 . A proton of 150 MeV incident energy has about a
20% chance of undergoing a nuclear reaction within the detector. The
correlation function is not affected by such reactions since the losses affect both
numerator and denominator. However, these losses must be corrected for in
singles spectra analyses, especially for reactions at higher beam energies, from
which many energetic protons are emitted.

Shown in Figure 6.2 are the single-proton energy spectra for our 3"’Ar 1-
455C collisions at E/A=120 and 160 MeV. The solid points points show the

energies of protons detected at (0 lab) = 31° and histograms show the predictions

 

0 5 f 7 V f] fir 7 j' Y I V T I f I T r ‘
o

b

h

b
p
o
01
b 1
»- .1

Percent Energy Loss

 

 

 

 

' 0 50 100 150 200

Figure 6.1. Probability for nuclear reaction loss for protons incident on a 10 cm
CsI(Tl) crystal.

91

 

   

 

 

 

 

 

I I I I T T I I I I I I I I l I I I I
i E/A=160 MeV
10_1 (X 2.5)—
. ..
0’ g. _
A 0 “
5 — i
D 5
s... _ ..
$3 I
C.‘
”U _ _
Lil
'5 _
\
0..
'U
10—2 “raaAr + 4580
E 1‘
_ J
l l l l I L l J l I l l l L I l l l l
O 50 100 150 200
Eproton (Mev)

Figure 6.2. Laboratory frame proton energy spectra measured at 013b=31° for
central (b/bmax = 0 - 0.3) collisions of 36Ar + 455c at E/ A = 120 and 160 MeV
(solid points) are compared with the predictions of BUU (histograms). Relative
normalization gives equal areas for measured and predicted spectra for
Eproton> 50 MeV. The arrows indicate average values of E 32,0" corresponding

to low and high momentum cuts used to analyze the correlation function in this
chapter.

92

of the BUU model. The BUU calculations over predict the low-energy proton
yields, but they reproduce the approximate shape of the high-energy tails. This
over prediction of proton yields at low energy has been observed before, and
attributed to the model’s inability to treat the formation of bound clusters [Gong
93]. These clusters should preferentially form in regions of phase—space where
nucleon population density is high, hence more protons should be ”lost” to

bound clusters at lower than at higher proton energy.

6.2 Measured Proton-Proton Correlation Functions

The correlation functions presented in this chapter are gated on centrality,
total proton center-of—mass momentum, and orientation of relative and total
proton momenta. In order to understand the effects of our detector upon the
correlation function, we show in Figures 6.3 and 6.4 the acceptance of the

hodoscope in the p x vs. p 2 plane for for single—proton center—of-mass
momentum cuts pm, = m /2, for our two energies, E/A=120 and 160 MeV,

respectively. These figures are similar to Figure 4.5 with the momentum cuts
now defined in the center—of—mass frame. The dashed lines indicate the angular
acceptance of the hodoscope 0 lab = 30°—45°. The dotted and hatched areas

correspond to our momentum cuts, Pcm = 200-400 and 400—800 MeV / c,

respectively. The two solid circles in each plot depict the Fermi momentum
spheres of projectile (centered at p z /A = 487 MeV / c and p2 /A = 568 MeV / c ,
respectively) and target (centered at pz /A = 0 ).

Similar to the lower energy reaction, the low momentum cut selects protons
emitted at large transverse angles with low energies with respect to the center-
of—mass rest frame. Again, the high-momentum cut selects protons with
velocities closer to the projectile than target velocity.

In Figure 6.5 we show measured momentum-integrated two-proton
correlation functions gated on centrality for our E/A=120 (top) and 160 (bottom)

MeV reactions. Consistent with References [Lisa 93a, 93c], our measurements

93

1066000 -750 -500 -250 O 250 500 750 10001000
\

\

 

\
\
\

750 \\ 750
a'Ar-I-wSc
,3 500 , E/A=120 Mev 500
> 11'.
0
5 250 250
6:
0 0
-250 -250

-1000 -750 -500 -250 0 250 500 750 1000
p: (MeV/c)

Figure 6.3. Schematic of detector acceptance and center-of-mass momentum
cuts for our E/A=120 MeV reaction. Dashed lines represent the detector
boundaries at 0“, = 30°-45°. Dotted and hatched areas represent the low- and
high-momentum cuts, Pcm = 200 -400 and 400-800 MeV/ c, respectively. Solid
circles depict Fermi spheres of target and projectile.

-7 .. ..
1 066 000 50 500 250 O 250 500 750 10001 000

 

\\ \
\ .
\
750 \\ 750
”Ar-t-“Sc
.3 500 .. E/A=160 MeV 500
\
E
v 250 250
a'.‘
0 0
—250 -250

—1000 -750 —500 -250 0 250 500 750 1000
p. (MeV/c)

Figure 6.4. Schematic diagram similar to Figure 63, appropriate for our
13/ A= 160 MeV reaction.

95

show a stronger correlation from peripheral than from central collisions, for both
bombarding energies. This observation is consistent with a geometric
interpretation of the reaction zone: the ”source” is formed by the physical
overlap of target and projectile, hence peripheral collisions lead to smaller
sources.

Figures 6.6 and 6.7 show the measured correlation functions cut on both
centrality and proton pair momentum, for our E/A=120 and 160 MeV reactions,
respectively. The correlation functions at E/A=120 MeV show a strong
dependence on total proton pair center-of—mass momentum, for both central
(top) and peripheral (bottom) collisions. Faster protons are shown to be more
strongly correlated, suggesting that they are emitted from a smaller source or on
shorter time scales than the slower protons. This trend is consistent with
previous impact-parameter gated measuremets [Lisa 93a, 93c] for the same
system at E/A=80 MeV. For our higher energy reaction (Figure 6.7), the
correlation function displays little dependence on proton-pair momentum for
central collisions (top panel), but does show a sensitivity for more glancing
collisions (bottom panel). The near—degeneracy of low- and high—momentum
correlation functions for central collisions may indicate that all protons are
emitted on similar time scales, and that there is no evaporative cooling of the
source. The momentum dependence at E/ A=160 MeV for both central and
peripheral collisions is smaller than that observed at E/A=120 MeV.

These trends are summarized in Figure 6.8, where the average value of the
correlation function in the peak region q=15—25 MeV/c, (1+ R) 4:15-25 Mev , c, is

plotted versus the total proton-pair center-of-mass momentum, Pm, for the two

centrality cuts for both the E/A=120 (top) and E/A=160 MeV (bottom) reactions.
Error bars include statistical uncertainties as well as an estimate of the
uncertainty due to normalization in the high—q region (q=60—80 MeV/ c). While
the measured results at E/A=120 MeV show a modest momentum dependence,

96

 

 

 

 

 

. 1.-....,.....
1.5 - A —
~ 0 o b=0.5—0.8 «
9 o a A
9 a o 5:00—03
»- . 11
1.0 — i e o o 0 e 9-
. .
° E/A=120 MeV
0.5 P$ Pcm=200—800 MeV/c ——
3 1 . . 1 .7 1 a 1 j 1
Q: . I I Iq
+
1—1 1.5"" O O '7
. . O . O .
o o
. ° 9 .
1.0 ~ 9 G 9 0 0 0 o o—
- E/A=160 MeV
0.5 — —
. a .
j ..
1 1
l . 1 l . 1 1 l 1 1 r l 1 1 1 l
0 20 40 60 80

q (MeV/C)

Figure 6.5 Momentum-integrated two-proton correlation functions for our
E/A=120 MeV (top panel) and E / A=160 MeV (bottom panel ) reactions. Central
events are indicated by solid points, while peripheral events are shown in open
points. .

97

 

_. . .I... 1' 1. 1 .-. T1

1.5— A -
’ Oo b=0.0—O.3

00:)

. 0.9. .

1.0— 0.9009

.0

' O Pcm=400-BOO MeV/c
0.5 ’_ o Pcm=200—400 MeV/c

 

 

 

 

3111,11 1111.111‘
0: .'* (5' 1 ‘ 1w '1.
+1.5” 0 A ~
H ’ O. O b=0.5—0.8
o 'oo
. . 1
1.o~— 0 99096.8991
.
0.5"+ —
* E/A=120 MeV
' 1 14 l 1 1 111 1 111 1 11‘
0 20 40 60 80
q(MeV/c)

Figure 6.6 Measured two—proton correlation functions, cut on total proton-pair
center—of-mass momentum, Pm, and on centrality for our E/A=120 MeV

reaction. Open and solid points represent high— and low-momentum protons,
cut on central (upper panel) and peripheral (lower panel) events.

98

 

 

 

 

 

fl I I I I
1.5 '— 0 A _1
. o e b=0 0—03 .
' 0
1 8 S 8
1.0 r o O O 8 O a G-
» + o Pcm=400—800 MeV/c
0.5 _ . O Pcm=200-400 MeV/c _
* 1 1 1 IA 1 1 l 1 1 1 J‘
I I I ’ ' I ' ' r I
E i 11 11 11
33 1.5 — . ' O 1 —
..1 + ' . b=0.5—O.8
, 1
r 6 ¢
1 O . . . .
1.0 r O O O a Q g g.
I 1
0.5; E/A=160 MeV _‘
’11111111111...1I
o 20 40 60 80
q (MeV/c)

Figure 6.7 Measured two-proton correlation functions, cut on total proton-pair
center-of—mass momentum, Pm, and on centrality for our E/A=160 MeV
reaction. Open and solid points represent high- and low—momentum protons,
cut on central (upper panel) and peripheral (lower panel) events.

99

 

  
 

 

 

 

 

 

1 fl I 7
C l
1.6 — -
E/A=120 MeV A, , 1
. / 1
1.4 — -
Q 1
‘3 I 1.
i 1 o b=0.5—O.8
3 1.2: o 13:00-03 i
E b t 1 : 1‘4 1 1 l 1 : : 1F : 4 .#d
/\ 1 ¢ 1
3 1 , , ’ ‘
E? l 6 r / ¢
+ I
v I
' i
1.4 — —
’ 1
‘ 1
1.2 + E/A=160 MeV —
b 1 1 1 l 1 1 1 J 1 1 1 I 1 1 4 ‘
200 300 400 500 600

P1111. (MeV/C)

Figure 6.8. The total proton-pair momentum dependence of the average value
of the two—proton correlation function ((1 + R) q=1 5_25 MeV I c) in the peak region,

q=15—25 MeV/c for our E/A=120 MeV (top) and E/A=160 MeV (bottom) results.
Central events are shown in solid points, and peripheral events in open points.
The lines are drawn to guide the eye.

100

the correlation functions at E/A=16O MeV show a weak dependence for

peripheral collisions, and no dependence for central collisions.

6.3 Comparisons of BUU Predictions with Experimental Results

The diminishing momentum dependence for higher beam energies was
established by Reference [Kund 93] for impact—parameter inclusive
measurements of 40Ar + 197Au collisions at E/A=200 MeV, which could not be
reproduced by BUU calculations. However, correlation functions are known to
be highly sensitive to impact-parameter (as shown in Figures 6.5-6.8), and a
more stringent test of the BUU model would include gating on centrality as well
as momentum. The BUU model is know to fail at lower energies for peripheral
collisions, possibly because of an inadequate treatment of the nuclear surface.

For this reason, and for better source characterization, we restrict ourselves to

central collisions, defiend by b = O - 0.3 , for both experiment and theory (see
section 2.3.2).

For this analysis, we used the BUU model of Bauer [Baue 86, Baue 87, Li
91a, Li 91b, Baue 92a] with a stiff equation of state (K=380 MeV), and the
nucleon-nucleon cross section was set to its value in free space. A proton was

considered ”emitted” if its local density was less than one—eighth that of normal
nuclear matter (po = 0.16 fm‘3), before a time t s ta“ = 150 fm I c (and if it did not
reenter a region of higher density before the calculation was terminated at #200
fm/ c). Some theoretical uncertainty exhists with respect to the choices of
10,5150 fm/c and freeze-out density, p f = %p,, which were consistent with

previous studies [Boal 90, Gong 90c, 91b, Lisa 93a, 93b, 93c].

While the BUU model reproduces the measured two—proton correlation
function rather well at E/A=80 MeV, it over predicts the correlation function for
the higher beam energies; see Figure 6.9. Measured (predicted) correlation
functions corresponding to the low-momentum cut are shown in solid points

(solid lines) and those corresponding to our high-momentum cut are shown in

101

open points (open lines), for the E/A=80 MeV (top panel), 120 MeV (middle
panel), and E/A=16O MeV (bottom panel), respectively, all for central collissions.
The data show less sensitivity to the bombarding energy than predicted by the
BUU calculations. At all three energies, the maximum of 1+R(q) at q z 20 MeV/c
is larger than for the Ar+Au reaction at E/A=200 MeV [Kund 93], and the

minimum at q z 0 is clearly observed for both cuts on Pam. (The approximate
momentum gates used in Reference [Kund 93] were Pcm z 160 — 340, 340 - 430,
and 430 — 570 MeV/c. No minimum was observed for the higher two gates).

Figure 6.10 shows the average value of the correlation function,
(1+R)q=15_25 MeV/c, in the peak region for measured (solid points) and BUU-

predicted (open points) events, as a function of the total momentum of the
proton-pair, for the central E/A=120 MeV (upper panel) and E/A=160 MeV
(lower panel) collisions. In both cases, the BUU model is shown to substantially
over predict the measured values, for all values of proton-pair center-of-mass

momentum.

6.3.1. Varying BUU Input Parameters

We have explored whether different BUU input parameters would lead to
better agreement with the data. Sensitivities to the stiffness of the equation of
state are small [Gong 91a]. However, the choice of freeze-out density (pf) could

affect the apparent source size. Also, BUU calculations were recently shown to
provide improved agreement with the balance energy in collective flow data if

102

 

 

 

 

      
    

 

 

 

1.5 f
L
1.0g
0.57
1.5—
3 1
0: 1.0—
+ .
1 E/A=120 MeV
0.5 f
’ k4111111§.11§
. / \ L 33:? Pcm=200-400 MeV/c
1.5 f Pcm=400-800 MeV/c
1.07
(if E/A=160 MeV
0.5 .— I.
L 1 1 l 1 1 1 l 1 1 1 l 1 1 1 I
O 20 4O 6O 80

q (MeV)

Figure 6.9. Two-proton correlation functions for central collisions of 36Ar + 455c
at E/ A = 80 MeV (top), 120 MeV (middle) and 160 MeV (bottom). results with
our low-momentum gates are shown in solid points (data) and solid lines
(theory), while high—momentum gated results are shown in open points (data)
and dashed lines (theory).

103

 

 

 

 

 

200 , 1 . 1 I 1 1 1 l 1 1 1 I
1.75% / O —
- (3. \ BUU/ , / e .
1.50 — \ 9’ —
0 Data
\
3125— —
2
$ E/A=120 MeV o b=0.0—O.3
W200 1 i i. : i i i } 4 i } i i 1
U 0
A / ’
0" I a /
E1 75 — , ’ -
: 9,1,9
V
1.50 f ~
1.25? i
: E/A=160 MeV 3
1.00" 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ml
200 300 400 500 600

P1111 (MeV/C)

Figure 6.10. The average value of the correlation function, (1 + R) q=15_25 MeV/c 1

in the peak region is plotted versus total proton momentum for our E/A=120
MeV (top) and E/A=160 MeV (bottom) reactions for central collisions. Data are
shown in solid points and BUU predictions in open points.

104

one allows for a density dependence of the nucleon-nucleon cross section (6 My)

[Klak 93]. By using a Taylor expansion for the density dependence of the in-
medium cross section, a W = (1+-a p / p0)o {W , best agreement with balance

energy data was obtained for on = -O.2 [Klak 93]. Therefore, we have performed
calculations for different choices of freeze—out density, pf = po/ 8, po/ 16 and

po/ 32, and for paramtera = O,-0.2,—O.4. Figure 6.11 shows the average peak
height of the correlation function at E/A=120 MeV as a function of the proton-
pair momentum for data and for BUU simulations. The solid, hatched and
dotted regions show predictions for pf = 1 / 8, 1 / 16, and 1 / 32 normal nuclear

matter density respectively, each for a between 0 and -O.4. These simulations
show little dependence on pf and on, except at lower total momentum where
smaller values of pf reduce the predicted peak height. However, no
commbination of of and a is able to reproduce the data, suggesting that the BUU
is deficient in its description of the emitted proton phase space distribution at
this energy.

The BUU predictions can be understood from the proton emission rates,
dP/dt, calculated for central 36Ar + 455c collisions, as shown in Figure 6.12. At
80 MeV per nucleon (solid line), the BUU calculations predict the existance of a
long—lived residue which slowly cools by particle emission [Lisa 93b, Hand 94].
In contrast, at E/A=120 MeV (dashed line) and 160 MeV (dotted line), a fast
flash of nucleon emission is predicted, without any residue. While this
drastically different scenario leaves no obvious signature in the single—particle
spectra shown earlier in this chapter, it results in enhanced two—proton

correlations, at variance with the data.

6.4 Importance of Particle Unstable Resonances
The lack of fragment formation in the current BUU formalism may be
responsible for its substantial over prediction of the correlation functions at

105

 

 
    
    
 

 

 

 

. . . . , . . . . , . . . . ,
1-8' - p1=1/a ‘
. BUU: m [2.31/16
Pr'1/32 l
{ «It
> I-
1'. 1.6
¥ .
3 I
9: ..-.':::- ------
+ 1-4 " 5232:3232? ''''' -
v ,. 1
B=o-o.3
1.2 ~ 1
3"AH“Se; E/A=120 MeV; <a>=3a° 1
1 1 1 1 l 1 1 4 1 l 1 1 1 1 l 1 1 1 1
200 300 400 500 600
P,,In (MeV/c)

Figure 6.11. The average peak height (at q = 15 - 25 MeV/c) of the two-proton
correlation function is shown versus the total proton-pair momentum in the
center-of-mass rest frame for central collisions of 36Ar + ”Sc at E/ A=120 MeV.
Data are shown as solid points. BUU simulations were calculated for freezeout
density pf = 1/ 8 (solid), 1/ 16 (hatched) and 1/ 32 (dotted). Each band shows the
range of BUU predictions for values of the parameter a (explained in the text)
ranging from 0 to -0.4.

106

 

_IllIIIIIIIIIIIIIIIITTIIIIIIIl

E/A=160 MeV
. — - E/A=120 MeV
/ 1 —— E/A=80 MeV

0.3

dP/dt. (arb. units)
0

l l l l l l I l I l l l l l l l

O
N
Ilrlllllllllllrl

 

 

 

 

0.0 ‘

O

25 5O 75 100 125 150
t (fm/c)

Figure 6.12. Proton emission rates, dP/dt, predicted by BUU calculations from

central 36Ar + 6Sc collisions at E/ A=80 MeV (solid), 120 MeV (dashed) and 160
MeV (dotted). '

107

E/A=120 and 160 MeV. At much higher energies (1.6 GeV), n+1t'correlation
functions were well reproduced by BUU calculations in which baryonic
resonances (such as A and N") played a large role [Plut 93]. At the energies
under consideration here, highly excited particle-unstable fragments may play an
important role in proton production by extending the time scale, I, over which
protons are emitted. We simulate the effects of proton emission from excited
states by taking a fraction of the protons (f), predicted to be emitted by the BUU
model, and delaying their emissions for a time t, statistically distributed

according to dP/dt = of”1 . By letting 1: range from 20 to 240 fm/c, we simulate
emission from resonances of approximately 0.8-10 MeV. The predicted spatial

coordinates and momenta are left unchanged. Figures 6.13 and 6.14 show
examples of simulations which reproduce the magnitudes of the experimental
correlation functions at E/A=120 and 160 MeV, respectively. For the E/A=120
MeV case, the plot shows that simulations with 1:160 fm/ c, and f ranging from
30% to 50% bracket the data, providing an excellent fit in the peak reagion of the
correlation function, which is the primary indicator of source size. For E/A=160
MeV (Figure 6.14), with 1:160 fm/c, the fraction of delayed protons needed to fit
the data lies between 20% and 30%. Somewhat surprisingly, fewer protons need
to be delayed at 160 MeV in order to agree with the data than at 120 MeV.
Figure 6.12 shows the BUU prediction that protons are emitted over a longer
time interval at 120 than at 160 MeV. Delaying a broader distribution (such as
the 120 MeV predictions) will have a smaller effect on the proton separation than
delaying a sharper distribution (such as the 160 MeV predictions). Therefore,
longer times (or more resonances) are required at 120 MeV. However, because
of the schematic nature of our calculations, this result should not be over
interpreted. The inserts of Figures 6.13 and 6.14 show the ranges of t and f
which give similarly good agreement with the data. To some extent, smaller
fractions of ”delayed” protons, f, can be compensated by longer delay times, t.

108

 

I I . I I I I I I I I I I r I I I

36Ar+wSc
Iii/A=120 MeV

1.5 —

 

1.0 '—

1 +R(q)

 

0.5
- i

 

 

 

 

- 40 60 80
q (MeV/c)

Figure 6.13. Two proton correlation functions (for low-momentum cut) for
central collisions of “Ar + 45Sc at E/A=120 MeV. Data are shown by points, and
the results of ”delayed" BUU simulations are shown in the hatched area, defined
byt=160fmlcandf=30°lot050%. Detailsaregiveninthetext. Theinsert
depicts the ambiguity between the parameters 1: and f.

109

 

 

 

 

 

 

 

fl ' l r'fil'T ' l ' r‘l
1'5— 0'2 3all1r+wSc
' T=160 fin/c
E/A=160 MeV
,3. 1
E 1.0-
+ .
v-l 1
0.5—
l §
0 20 40 60 80

Figure 6.14. Two proton correlation functions (for low-momentum cut) for
central collisions of 36Ar + 455c at E/ A=160 MeV. Data are shown by points, and
the results of ”delayed" BUU simulations are shown in the hatched area, defined
by t=160 fm/c and f=20% to 30%. Details are given in the text. The insert
depicts the ambiguity between the parameters 1 and f.

110

While the correlation functions are consistent with a delayed proton—
emission component, its magnitude is uncertain and not well determined by our
schematic simulation. In order to assess whether the range of parameters shown
in the inserts of Figures 6.13 and 6.14 is compatible with statistical expectations,
we have performed calculations with the statistical codes Firestreak [West 76,
Goss 78] and FREESCO [Fai 82, 83] to estimate sequential-decay contributions to

the proton yield. For a single source containing 81 nucleons, using p; = po/ 8 and
Tf = 8 MeV, the Firestreak model predicts that about 50% of emitted protons

come from resonancesm, while the FREESCO prediction is slightly less than
20°/o. The two predictions differ mainly because FREESCO incorporates more
resonances than Firestreak; for details, see References [West 76, Goss 78, and Féi
82, 83]. From the inserts of Figures 6.13 and 6.14, this range is qualitatively
consistent with values of 1 between 50 and 200 fm/c, or widths between 1 and 4

MeV, which are typical of light resonances such as 5Li" and on".

The successful reproduction of the experimental correlation function at
E/A=80 MeV by BUU calculations, without an ad hoc introduction of
resonances, may be due to the predicted formation of a heavy residue, which
emits protons over a similar time scale (see Figure 6.10). The observed
agreement of QMD calculations with impact-parameter averaged data at
E/A=200 MeV and the disagreement at lower energy [Kund 93] may be
coincidental because QMD does not properly contain the particle unstable

resonances found to be important.

Chapter 7 - Summary and Conclusions

The technique of two—proton intensity interferometry has been used to

study the space-time structure of the reaction zone created in 36Ar+458c
collisions at E/A=80, 120 and 160 MeV. We have shown how one constructs the
experimental correlation function from measured single- and coincident—proton
yields, and how one calculates the theoretical correlation function from the
single—particle phase space density using the Koonin—Pratt formalism.

The proton—proton correlation functions is sensitive to the spatio-temporal
characteristics of the emission zone due to the final state interactions between the.
protons. Specifically, the long-range Coulomb repulsion and short-range
nuclear attraction lead to a minimum at q z 0 and a peak at q z 20 MeV/c in the
correlation function, respectively. The size of the peak at q z 20 MeV/c is the
primary indicator of the space—time extent of the source: sources of smaller size
lead to more pronounced peaks. A directional dependence in the quantum
statistical suppression of the correlation function permits independent
extractions of source size and lifetime by gating on the orientation of relative and
total proton momenta. ,

We showed that a model based on the Boltzmann-Uehling—Uhlenbeck
(BUU) equation is able to reproduce the dependencies of experimental

correlation functions on total proton momenta and angle ‘V = cos '1 (F - q/Pq) for

central 36Ar+455c collisions at E/A=80 MeV remarkably well. Predictions for
peripheral collisions are less reliable.

The usefulness of the correlation function in identifying toroidally shaped
density distributions was examined. Although schematic studies show that the
correlation functions differ for toriodal and spherical sources, unavoidable
contributions from nonzero impact parameters and the dynamical effects

associated with emission of protons from various stages of the reaction severely

111

112

weaken the signal. Unless stringently tight cuts on centrality and orientation of
relative momenta can be achieved, there is little hope of extracting a toroidal
signature with the two—proton correlation function.

Measured correlation functions for the same system at E/A=120 and 160
MeV reveal a weaker dependence on the total momentum of the proton—pair
with increasing beam energy. This may indicate that at higher energies, all
protons are emitted on similar time scales, and that there is no significant cooling
of the source. The BUU model predicts that the peak height of the correlation
function rises with beam energy, at variance with the data. Furthermore, it
severely over predicts the magnitude of the correlation function at energies
above E/A=8O MeV, consistent with a previous study for a less symmetric
system.

Varying BUU input parameters, such as the freeze—out density and
magnitude of the density dependence of the nucleon-nucleon cross section, has
little effect on its predictions. It was shown that particle—unstable fragments,
such as 5Li", could play an important role in this energy regime, by extending
the time scale over which protons are emitted, thus reducing the magnitude of
the correlation function. Because such resonances are not modeled by BUU, we-
schematically incorporate their effects by delaying the predicted emission of
protons. Statistical calculations indicate that between 20% and 50% of all
protons are emitted from resonances at these energies, which we showed to be
consistent with emission from states of widths 1-4 MeV, typical of light
resonances. In order to fully assess the importance of such resonances, the
development of a truly quantum transport model, which incorporates the
production of fragments, is needed in this energy range.

LIST OF REFERENCES

[Aich 91]
[Alm 93]
[Awes 88]

[Baue 86]

[Baue 87]
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[Bern 85]

[Boal 86a]
[Boal 86b]
[Boal 90]

[Bord 85]

[Bowm 93]

[Cebr 89]

[Cebr 90]
[Chen 87a]

[Chen 87b]

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