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This is to certify that the
dissertation entitled
Impact Parameter-Gated Two-Proton Intensity Interferometry
in Intermediate Energy Heavy Ion Collisions

presented by

Michael Annan Lisa

has been accepted towards fulfillment
of the requirements for

Ph. D. degree in Physics

WW

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Date 2 September 1993

 

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MSU In An Affirm-(Iva Action/EM Opportunity Institution
WWI

Impact ParameteréGated ‘
Two-Proton Intensity Interferometry in
Intermediate Energy Heavy Ion Collisions

By

Michael Annan Lisa

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

1993

ABSTRACT

Impact Parameter-Gated Two-Proton Intensity Interferometry in
Intermediate Energy Heavy Ion Collisions

By

Michael Annan Lisa

Impact parameter gated two-proton intensity interferometry is employed
to probe the space-time structure of emitting sources created in intermediate
energy heavy ion collisions.

The Koonin-Pratt formalism relating the single-proton phase space
population density to the two-proton correlation function is discussed.
Schematic calculations illustrate the sensitivity of the correlation function to
source size and lifetime.

Single- and two-proton yields, used to construct the two-proton
correlation function, were measured with a high-resolution 56-element
hodoscope for the reaction 35Ar+455c at E/ A=80 MeV. Coincident measurement
of charged particle yields in a 41: detector array provided information about the
impact parameter of the collision. For both central and peripheral events, proton
correlations are larger (indicative of a smaller space-time extent for the source)

for proton pairs with larger total momentum. For central events, the total

momentum dependence is stronger than for peripheral events, indicating a
distinct reaction evolution for events with different impact parameter.

Methods for comparing impact parameter-gated data to theoretical
predictions are discussed, and predictions of a model based upon the Boltzmann-
Uehling-Uhlenbeck (BUU) equation are compared to experimental correlation
functions. For central collisions, the total momentum dependence of the
correlations is well reproduced. For peripheral collisions, this dependence is
underpredicted, suggesting that the model may be deficient in its description of
peripheral heavy ion collisions.

Differences between longitudinal and transverse correlation functions
allow independent extraction of source size and lifetime. Potential difficulties in
the search for such differences are discussed. For intermediate energy protons
emitted from central 35Ar+45Sc collisions at E/A=80 MeV, a source with a
Gaussian radius and lifetime of about 4.5 i 1.0 fm and 30 i 10 fm/c is indicated.
For the reverse kinematics reaction 129Xe+27Al at E/A=31 MeV, we observe
lifetimes of about 1700 fm/c and 1200 fm/ c for proton pairs with total
momentum P2480 MeV/c and P2580 MeV/c, respectively; this behaviour is

consistent with emission from a cooling system.

Dedicated, with love, to
my mother, Claire Marie Annan Lisa
and

my father, Charles Pasquale Lisa

iv

ACKNOWLEDGMENTS-

I have enjoyed extraordinarily good fortune in my brief scientific career so
far. During my graduate studies, many scientists and friends (non-exclusive sets)
have taught and supported me. I wish to thank a few of them here, for despite
claims of the title page, a thesis is rarely the work of one person.

Firstly, I would like to thank Dr. Claus-Konrad Gelbke. His support has
been crucial to me at every step, from my transition to MSU from Stony Brook, to
my thesis experiment, to our European experiments, to the writing of this
dissertation. The energy, integrity, and insight which he brings to his work leads
to my utmost respect, and will inspire me through my career. Without him, this
work would have been impossible.

I would also like to thank Dr. Pawel Danielewicz, Dr. S.D. Mahanti, Dr.
Bernard Pope, and Dr. Gary Westfall, who gave generously of their time to serve
on my thesis committee. Their cheerful and patient efforts make my degree
attainable.

As an undergraduate, I was introduced to scientific research through the
REU program at the University of Notre Dame and was fortunate to work with
two fine scientists, Dr. Gerald Jones and Dr. James Kolata, who impressed me
with their ability to work with developing physicists at all stages.

During my second of two years of graduate study at the State University
of New York at Stony Brook, I was privileged to work in the group of Dr. Peter
Braun-Munzinger and Dr. Johanna Stachel. Their friendly instruction style

combined with a strong grasp of important physical issues made for an

impressive introduction to the field of high energy nuclear physics. Particularly,
I wish to thank Dr. Stachel for serving on my committee at Stony Brook and Dr.
Braun-Munzinger for his kind recommendation as Ibegan my studies at MSU.

The enthusiasm and intelligence of the students and postdocs at Stony
Brook impressed me from the start. For infecting me with enthusiasm and for
their valuable friendship, I especially thank Mark Bellermann, Rene Bellwied,
Dave Coker, Jimmy Dee, Anindita Ghosh, Rajeev Gupta, Chris Kendsiora, Claus
Koestler, Dennis LaFosse, Steve Riemersma, Mark Widmer, and Christof
Wunderlich.

My training in experimental nuclear physics at Michigan State has been
quite enjoyable, due to skilled scientists at the lab who love their work. I was
privileged to learn the fine art of designing, optimizing, and debugging an
acquisition electronics system while working alongside Dr. David Bowman and
Dr. Skip Vander Molen. I am grateful to Dr. Betty Tsang for constantly
reminding me to take nothing for granted and for showing me that computer
control of an experiment can be done in a systematic manner. Thanks also to Dr.
Romualdo de Souza, who taught me not to lose sight of the physics when
conducting an experiment.

My work in two-proton interferometry builds largely on the foundation
laid by Dr. Wen Geng. His patient instruction in the theoretical and
experimental aspects of this field during the time that he was preparing to write
his dissertation is much appreciated. I am also indebted to Dr. Scott Pratt,
correlation guru, for many patient and insightful explanations of the subtleties of
interferometric physics and for the use of his correlation code. I was privileged
also to work with Dr. Bill Lynch, who can as easily discuss an involved

theoretical issue as a tricky technical one.

vi

I would also like to thank Dr. Wolfgang Bauer and Dr. Catherine Mader
for providing me with the results of their BUU calculations, and for fruitful
discussions. ' 1

The lab staff is the largest asset of the NSCL. In particular, I would like to
thank Jim Vincent for help with nuclear electronics problems, and Len Morris,
Renan Fontus, and Steve Bricker for help with the design and "field engineering"
aspects of the mechanical apparatus used in my experiment. Barb Pollack and
Ron Fox were there to help when I had computer problems, and Dennis Swan
always seemed to know some trick to deal with a detector problem I was having.
Finally, the entire operations group deserves my thanks for a steady, well-tended
beam during my experiment.

I am fortunate to count among my friends many fellow graduate students
and postdocs at the NSCL. I am indebted to the friendship of James Dinius,
Thomas Glasmacher, Gene Gualtieri, Damian Handzy, Stefan Hannuschke, Rick
Harkewicz, Wen Chien Hsi, Yeonduk Kim, Roy Lacey, Tong Li, Mike Mohar,
Raman Pfaff, Larry Phair, Don Sackett, Carsten Schwarz, Cornelius Williams, Bo
Zhang, and Fan Zhu. From them and with them, I learned a lot.

The greatest thanks must go to my family. My brother Greg and-my
sisters Shawna and Jennifer are a restless lot, and their energy and achievements
remind me never to get too lazy. Similarly, tales of the Irish Annans and
McCabes and the Italian Boninos and Lisas remind me that any of our
accomplishments stem from the sacrifices and efforts of those who have gone
before. Closest to me are my parents, whose own achievements serve as both
example and inspiration. Their loving support has meant everything to me.

Finally, the greatest gift I have received in Michigan has been to meet and
get to know Laura. Her love has supported me through trying times and has
made the future look like a pretty nice place to be.

vii _

Table of Contents

List of Tables ................................................................................................................. x
List of Figures ................................................................................................................ xi
Chapter 1 - Introduction .............................................................................................. 1
1.1 Intensity Interferometry .................................................................................. 1
1.2 Geometrical Measurements with Two-Proton Correlation
Functions ........................................................................................................... 5
1.3 Sources Created in Intermediate-Energy Heavy Ion Collisions ................ 7
1.4 Motivation for This Study ............................................................................... 10
1.5 Organization of This Work ............................................................................. 11
Chapter 2 - Experimental Details ............................................................................... 14
2.1 The 56-Element Hodoscope ............................................................................ 14
2.1.1 Detector Elements ................................................................................... 16
2.1.2 Mechanical Support ................................................................................ 19
2.2 The NSCL 41: Detector Array .......................................................................... 22
2.2.1 Detector Elements ................................................................................... 22
2.2.2 Mechanical Set-up ................................................................................... 28
2.3 Electronics .......................................................................................................... 28
2.3.1 56-Element Hodoscope ........................................................................... 29
2.3.2 The 41: Detector Array ........................................................ - ..................... 3 8
2.3.3 Master Circuitry ....................................................................................... 39
2.4 Nuclear System Studied .................................................................................. 41
Chapter 3 - Data Reduction ......................................................................................... 44
3.1 56-Element Hodoscope .................................................................................... 44
3.1.1 Particle Identification .............................................................................. 44
3.1.2 Silicon Detector Calibration ................................................................... 48
3.1.3 CsI(Tl) Detector Calibration ................................................................... 48
3.1.3 Slewing Correction .................................................................................. 52
3.2 41: Detector Array ............................................................................................ 59

viii

3.3 Physics Tapes .................................................................................................... 60

Chapter 4 - Impact Parameter Selection .................................. . ................................. 61
4. 1 Variables Used for Centrality Selection ........................................................ 61
4. 2 Construction of the Reduced Impact Parameter Scale ............ , .................... 68
4. 3 Comparison of Variables ................................................................................. 69
4. 4 Impact Parameter Distributions as Inputs to Models ................................. 70
4.4.1 The Method of Equivalent Cuts ............................................................ 76
4. 4. 2 The True Reduced Impact Parameter Distribution ............................ 81
Chapter 5 - Two-Proton Correlation Functions ....................................................... 88
5.1 Relation of the Correlation Function to the Spatio-Temporal
Structure of the Source .................................................................................... 88
5.1.1 Koonin-Pratt Formalism ......................................................................... 89
5.1.2 The Two-Proton Wavefunction ............................................................. 91
5.1.3 Illustrative Calculations .......................................................................... 94
5.2 Constructing a Correlation Function for Model Predictions ..................... 100
5.3 Experimental Construction of the Correlation Function ............................ 104
5.4 Effects of Finite Resolution ............................................................................. 113
Chapter 6 - Centrality-Cut Proton Data for 36Ar+455c at E/ A=80 MeV .............. 124
6.1 Singles Energy Spectra .................................................................................... 125
6.2 Measured Two-Proton Correlation Functions ............................................. 128
6.3 Comparison of Measured Correlations to BUU Predictions ...................... 134
Chapter 7 - Longitudinal and Transverse Correlation Functions ......................... 139
7.1 The Importance of Identifying the Source Frame ........................................ 140
7.2 Determination of the Source Lifetime in Central 35Ar+45Sc
Collisions at E/A=80 MeV .............................................................................. 143
7.3 Determination of the Source Lifetime in 129Xe+27Al Collisions at
E/A=31 MeV ..................................................................................................... 152
Chapter 8 - Summary and Conclusions .................................................................... 159
Appendix A - Raw and Physics Tape Formats ........................................................ 163
A.1 Raw Tape Format ............................................................................................ 163
A2 Physics Tape Format ....................................................................................... 164
List of References .............................. 168

ix

List of Tables '

Table 2.1. Time constants for fast and slow scintillating plastics .......................... 24
Table 2.3. Punch-in Energies for Forward Array detectors .................................... 25
Table 2.2. Punch-in Energies for Ball detectors ........................................................ 25
Table 3.1. Rise times used to correct for slewing effect for light particles ........... 58

List of Figures

Figure 1.1. The operation of the two types of interferometer is
schematically shown. In the upper panel, an amplitude (Michelson)
interferometer measures the single-body detection probability. The
lower panel shows an intensity (HBT) interferometer which measures
the two-particle yield. (From [Gong 91c]) ........................................................... 2

Figure 1.2. Spatial distributions of emitted protons originating from a
source with a small spatial and temporal extent (a), with a large spatial
extent (b), and with a large temporal extent (c). ................................................ 6

Figure 2.1. Schematic diagram showing the arrangement of the 56-
element hodoscope and the MSU 41: Array ........................................................ 15

Figure 2.2. Schematic anatomy of the CsI(Tl)-PIN diode assembly used as
the E detector in the light particle telescope ....................................................... 17

Figure 2. 3. Schematic of the preamplifier for the PIN diode used to read
out the CsI(Tl) detector. ......................................................................................... 20

Figure 2.4. Schematic of the front face of the 56-element hodoscope
structure. .................................................................................................................. 21

Figure 2.5. Angular coverage provided by the 56-element hodoscope. .............. 23

Figure 2.6. Schematic of phoswich detector operation. The signal from the
phototube is split and the light components from the "fast" and "slow"
plastic are separately integrated. ......................................................................... 26

xi

Figure 2.7. A "fast" vs. "slow" spectrum from a detector element from the
main ball. The punch-in line and neutral line frame signals from
identifiable particles. .............................................. - .............. . ................................. 2 7

Figure 2.8. A 32-faced truncated icosahedron, the underlying geometry of
the MSU 41: Array ................................................................................................... 29

Figure 2.9. Spatial arrangement of phoswich detectors in the forward
array. .............................................. , .......................................................................... 30

Figure 2.10. Electronics used to digitize data and trigger the computer .............. 32

Figure 2.11. Raw spectrum of signals of the Si and CsI(Tl) detectors is
shown for a typical AE-E telescope from the 56-element hodoscope.
Note the effect of the various discriminator thresholds in eliminating
coincident noise from the two components ........................................................ 34

Figure 2.12 Signal timing for the internal logic of a detector trigger. Solid
lines correspond to timing for large signals, while dashed lines indicate
timing for signals at threshold. The "EL" signal always determines the
overall timing. ......................................................................................................... 35

Figure 2.13. Signal timing for the determination of the time that a detector
element fired. .......................................................................................................... 36

Figure 2.14. Signal timing for the master circuit. .................................................... 40

Figure 2.15. Schematic illustrating, in a purely geometric picture, the
reduced sensitivity of global observables to the impact parameter for
very asymmetric systems. ..................................................................................... 43

Figure 3.1. Two-dimensional density plot of the PID function vs. the
particle energy. The z-axis has a logarithmic scale. ......................................... 46

Figure 3.2. One-dimensional projections of PID for thin cuts in particle
energy, showing particle identification resolution as a function of
energy ....................................................................................................................... 47

xii

Figure 3.3. Raw energy spectra measured in a typical detector for light
particles emitted when 120 MeV alpha particles bombarded a
polypropylene target. ............................................ 1 ............... , ................................. 50

Figure 3.4. Calibrated proton and alpha energy spectra from 120 MeV
a+CH2 reactions. The widths of the measured peaks are dominated by
kinematic broadening, indicating an intrinsic energy resolution on the
order of 1%. ................................. 51

Figure 3.5 Lines indicate calculated deuteron energies from a(12C,d)14N*
for Ea=120 and 100 MeV . Measured deuteron energies assuming a
deuteron energy calibration identical to the proton and triton
calibrations are shown as circles and x's, respectively. .................................... 53

Figure 3.6. Energy calibration curves for a typical CsI(Tl) detector from the
56-element hodoscope. .......................................................................................... 54

Figure 3.7. Relative timing spectrum for two particles recorded in
coincidence in the 56-element hodoscope. The upper and lower panels
show the spectrum before and after slewing correction, respectively.
Also shown are the relative timing regions defining a "real" and a
"random" coincidence in the correlation analysis .............................................. 55

Figure 3.8. The time when a detector fired relative to the cyclotron RF
signal vs. the energy is shown for protons measured in a typical AE-E
telescope. The raw spectrum (top panel) shows the effects of slewing
or time-walk. Slewing effects are absent in the corrected timing signal
(bottom panel) Different bands in both spectra reflect the RF structure
of the cyclotron beam .................... 56

Figure 4.1. Charged particle multiplicity distributions measured in the 41:

Array under 3 separate triggering conditions. Open circles, open
squares, and filled circles indicate dP/ ch for inclusive, hodoscope

singles, and hodoscope coincidence triggers, respectively. ............................. 63

Figure 4.2. Midrapidity charge distributions measured in the 41: Array are
shown for the same triggering conditions as for Figure 4.1 ............................. 65

xiii

Figure 4.3. Total transverse energy distributions measured in the 41: Array
for the same triggering conditions as for Figures 4.1 and 4.2. ......................... 66

Figure 4.4. Logarithmic two-dimensional probability distributions are
shown for pairs of global observables. The values of the three '
observables shown, the charged particle multiplicity (NC), the mid-
rapidity charge (Zy), and the transverse energy (Et) are highly
correlated. If one observable is a good measure of event centrality,
they all are. .............................................................................................................. 67

Figure 4.5. The inclusive transverse energy spectrum dP/dEt is shown in
the upper panel. The lower panel shows the average reduced impact

parameter scale ENE.) based on this distribution. ............................................. 71

Figure 4.6. Average reduced impact parameter distributions dP / (16031)
for the three event triggers studied. Curves in this figure correspond
to those shown in Figure 4.3. ................................................................................ 72

Figure 4.7. Average reduced impact parameter distributions (1? / C1605.)
for tight centrality cuts on 600, where X=Nc and Zy. A double-cut on
both observables does not narrow the distribution significantly for the
mid-central cut, indicating that the measured width is close to the scale
of the intrinsic fluctuations between impact parameter and transverse
energy ....................................................................................................................... 73

Figure 4.8. Average reduced impact parameter distributions dP / d6(Nc)
for tight centrality cuts on 1300, where X=Et and Zy. ..................................... 74

Figure 4.9. Average reduced impact parameter distributions dP / db(zy)
for tight centrality cuts on b(X), where X=Et and NC ...................................... 75

Figure 4.10. Total transverse energy spectrum dP/dEt measured under a
minimum bias trigger in the 41: Array is indicated by the solid line.
Also plotted is the prediction of the BUU model, after passing through
the detector acceptance of the 41: Array. Dashed and dot-dashed lines
indicate calculations in which the definition of E: (Equation 4.3)

includes contributions from all emitted nucleons, and from protons

xiv

only, respectively. Relative normalization gives equal area for all
spectra in the region Et2100 MeV ......................................................................... 78

' Figure 4.11. Plotted are the contributions to the total Et spectrum from

various ranges of impact parameter as predicted by the BUU when the
contribution of neutrons are included. The black curve represents the

impact parameter averaged spectrum, while the colored curves show

the spectrum for small impact parameter intervals. ......................................... 79

Figure 4.12. Experimentaland theoretical inclusive Et distributions are
shown in the upper and lower panels, respectively. "Equivalent" Et
cuts defining central and peripheral events are shown. ................................... 80

Figure 4.13. The top panel shows the total transverse energy distribution
dP/ dEt for events which have two protons in the hodoscope. The cuts
we define as "central" (high Elf) and "peripheral" (low Et) are indicated
by the shaded regions . Shown in the center panel are the average

reduced impact parameter distribution (11’ / (15(51) corresponding to

the total transverse energy distribution of the top panel. The bottom

panel shows the reduced impact parameter distributions dP/ db;-
corresponding to the centrality cuts shown in the upper two panels.

Solid lines indicate reduced impact parameter distributions for two-

proton coincidence events, while dot—dashed lines represent similar
distributions for singles proton events ................................................................ 82

Figure 4.14. The upper panel shows sharp average reduced impact
parameter cuts corresponding to the definition of central and
peripheral events are indicated for a geometric distribution of impact
parameter. The lower panel shows reduced impact parameter
distributions dP/ dbr corresponding to the centrality cuts indicated
above. ....................................................................................................................... 85

Figure 4.15. Comparison of experimentally reconstructed impact '
parameter distributions to those used in the BUU calculations. Solid
curves show the impact parameter distribution used to weight BUU
events (same as bottom panel of Figure 4.14). This distribution is
modified by the additional requirement of proton emission towards

XV

the hodoscope (dotted curves). The dashed curves show the

experimental reduced impact parameter distribution (see Equations 4.9

and 4.10) when a proton is detected in the hodoscope (same as dotted-
dashed curve in Figure 413) ................... 86

Figure 5.1. The square of the two-proton wavefunction as a function of the
relative proton separation in coordinate and momentum space. The
top and bottom panels show the wavefunction when q and r are
parallel and perpendicular, respectively. ........................................................... 92

Figure 5.2. Correlation functions corresponding to proton emission from
spherical sources with Gaussian density profile and negligible lifetime
(Equation 5.6) show a strong source size dependence for smaller
sources ...................................................................................................................... 95

Figure 5.3. The interplay between the momenta of emitted protons and the
timescale of emission is illustrated in the two-proton correlation
function. The upper panel shows the effect of increasing lifetime for a
source described by Equation 5.7, with the parameters indicated, for
proton pairs with a fixed total momentum. The bottom panel shows
the effect of increasing the total momentum P, when the lifetime is kept
fixed. ......................................................................................................................... 98

Figure 5.4. Longitudinal (solid lines) and transverse (dotted lines)
correlation functions are shown for schematic sources of various
lifetime parametrized according to Equation 5.7. Longitudinal cuts
correspond to a range w=0-50°. In the left- and right-hand panels,
transverse cuts correspond to ranges wfiO-90° and \v=60-90°,
respectively. Details are discussed in the text. ................................................... 101

Figure 5.5. Comparison of singles (circles) and pseudo-singles (diamonds)
energy spectra measured in [Gong 90c, Gong 91b] for the 14*N+27Al
reaction at E/A=75 MeV. Statistical errors are smaller than the size of
the symbols .............................................................................................................. 108

Figure 5.6. Comparisons of singles (circles) and pseudo-singles
(diamonds) angular distributions measured in [Gong 90c, Gong 91b]

for the 14N+27Al reaction at E / A=75 MeV. Statistical errors are smaller
than the size of the symbols. ................................................................................. 109

Figure 5.7. Two-proton correlation functions measured [Gong 90c] [Gong
91b] for the 14N+27Al reaction at E/A=75 MeV. Open and solid
symbols represent correlation functions constructed by the singles and
event-mixing techniques, respectively. Momentum cuts are indicated
in the figure. Statistical errors (larger than the size of the data points)
are shown only for open points. Solid points have errors of same
magnitude ................................................................................................................ 110

Figure 5.8. Energy-integrated two-proton correlation ftmctions measured
[Gong90c] [Gong 91b] for the 14N+27Al reaction at E / A=75 MeV.
Open and solid symbols represent correlation functions constructed by
the singles and event-mixing techniques, respectively. Top, center, and
bottom panels represent correlation functions extracted from arrays
consisting of 47, 23, and 7 detectors, respectively. ............................................ 112

Figure 5.9. The first and second moments of the distribution (qreal-qmeas)
are shown as a function of qmeas for representative cuts on P and 1'}.
Finite resolution effects are quantified by the second moment
(indicated by the diamonds), while the first moment (indicated by
circles) shows a very small systematic distortion of the value of q.
Details are discussed in the text. The distribution for low momentum
pairs with q transverse to P stops at q=60 MeV/c due to the finite solid
angular coverage of the hodoscope. .................................................................... 115

Figure 5.10. A uniform hit pattern over the faces of the detector elements
of the 56-element hodoscope was used to quantify finite resolution
effects on the correlation function for the 35Ar+45Sc measurement- .............. 119

Figure 5.11. A uniform hit pattern over the faces of the detector elements
of the (incomplete) 56-element hodoscope and 13-element hodoscope
was used to quantify finite resolution effects on the correlation
function for the 14N+27Al and 129Xe+27Al measurements ............................... 120

Figure 5.12. Calculated correlation functions for P=400-600 MeV/c.
Longitudinal and transverse correlation functions are defined in the

xvii

laboratory rest frame. Solid lines indicate the "true" or undistorted
correlation functions, while symbols indicate the correlation function
after accounting for the resolution of the 56-element hodoscope. .................. 121

Figure 5.13. Calculated correlation functions for P2 700 MeV/c.
Longitudinal and transverse correlation functions are defined in the
laboratory rest frame. Solid lines indicate the "true" or undistorted
correlation functions, while symbols indicate the correlation function
after accounting for the resolution of the 56-element hodoscope. .................. 122

Figure 5.14. Calculated correlation functions for P2 700 MeV/c.
Longitudinal and transverse correlation functions are defined in a
frame moving with [3:018 in the laboratory. Solid lines indicate the
"true" or undistorted correlation functions, while symbols indicate the
correlation function after accounting for the resolution of the 56-
element hodoscope. ................................................................................................ 123

Figure 6.1. Centrality-cut proton energy spectra measured with a singles
trigger at two extreme polar angles in the 56-element hodoscope.
Spectra for protons emitted from central events (as determined by the
total transverse energy measured in the event. See Figures 4.13 and
4.14.) are shown as solid symbols. Spectra corresponding to protons
emitted from peripheral events are represented as open symbols. ................ 126

Figure 6.2. Proton energy spectra selected by cuts on 8030 measured at
two extreme polar angles covered by hodoscope detectors are
compared to predictions of the BUU. Spectra for central and
peripheral events are shown in the upper and lower panels, .
respectively. Measured and predicted energy spectra are indicated by
symbols and histograms, respectively. Relative normalization gives
equal area for measured and predicted spectra at (mo-32° for
Epmmn250 MCV ....................................................................................................... 127

Figure 6.3. Energy-integrated two-proton correlation functions measured
in the 56-element hodoscope for the centrality cuts indicated in Figures
4.13 and 4.14. Central events (high Et) are indicated by solid points,

xviii

and peripheral events (low Et) by open points. Statistical errors are
smaller than the symbol size ................................................................................. 131

Figure 6.4. Measured two-proton correlation functions for a double cut on
the total momentum of the pair P = |p1 + p2 I and the impact
parameter. Centrality cuts are indicated in Figures 4.13 and 4.14. Open
and solid points indicate peripheral and central events, respectively.
The upper panel shows the correlation function for slow protons, 400
MeV/ c S P s 520 MeV/c. The lower panel shows the correlation
function for fast protons, P 2 880 MeV/c. Statistical errors are
indicated when they are larger than the symbol size. ...................................... 132

Figure 6.5. The total momentum dependence of the correlation function is
summarized by plotting the height of the correlation in the region 15
MeV/c s q S 25 MeV/c as a function of the total momentum P. For
reference, on the right-hand axis is indicated the radius of a zero-
lifetime spherical source of Gaussian density profile that would
produce a correlation of equal magnitude. Solid lines which connect
the points for central and peripheral events are drawn to guide the eye.
Error bars indicate statistical errors in the region 15 MeV/c s q s 25
MeV/c as well as uncertainties in the height due to uncertainties in
normalizing the data at large relative momentum ............................................ 133

Figure 6.6. Measured two-proton correlation functions for cuts in
centrality and total momentum are compared to BUU predictions.
Solid symbols represent data. Open symbols are BUU calculations.

The upper panels show the correlation function for slow protons (400

MeV/ c S P s 520 MeV/c). Lower panels correspond to faster protons

(P 2 880 MeV/c). Panels on the left- and right-hand sides, respectively,
correspond to central and peripheral events. Centrality cuts for the

data were based on Et, as indicated by Figure 4.14. Weighting for BUU
events was done according to the reduced impact parameter

distributions dP/ dbr indicated in the bottom panel of Figure 4.14.

Statistical uncertainties are indicated when they are larger than the size

of the points ............................................................................................................. 136

xix

Figure 6.7. The average height of the correlation function in the relative
momentum interval 15 MeV/c S q s 25 MeV/c, is plotted against the
total momentum of the pair P. For orientation, the right-hand axis
indicates the source radius of a zero- lifetime spherical source with a
Gaussian density profile that would produce a correlation of equal
magnitude. The upper panel displays the data and calculations for
central events, and the lower panel provides the comparison for
peripheral events. Data are indicated by solid circles, while BUU
predictions are indicated by the open symbols. The open circles
correspond to BUU results when the event selection is performed
according to the dP/ dbl» distributions shown in the bottom panel of

Figure 4.13. The open squares correspond to the selection of events via
the "equivalent-Et" method discussed in the text. ............................................ 137

Figure 7.1. Schematic illustration of phase space distributions at a time t =
70 fm/c, seen by a detector at Slab = 38°, for a spherical source of radius
r = 3.5 fm and lifetime 1: = 70 fm/c emitting protons of momentum 250
MeV/ c. Part a) Source at rest in the laboratory. Part b) Source moves
with vsource = 0.18c. In the distributions shown, the laboratory
velocities of the protons (vaab) are depicted by small arrows and the
directions perpendicular and parallel to vaab are depicted by large
double-headed arrows. In parts (a) and (b), vaab is kept constant, and
Vemit is different. Therefore, the elongations along Vemit are different- ........ 142

Figure 7.2. Measured longitudinal and transverse correlation functions for
protons emitted in central 3‘5Ar + 455c collisions at E/ A = 80 MeV. The
correlation functions are shown for proton pairs of total laboratory
momenta P=400-600 MeV/c detected at <013b>=38°. Longitudinal and
transverse correlation functions (solid and open points, respectively)
correspond to w = C05'1(q~P/qP) = 0° - 50° and 80° - 90°, respectively,
where P is defined in the rest frame of the presumed source. Upper
panel- vsource = 0.18c. Solid and dashed curves represent longitudinal
and transverse correlation functions predicted for emission from a
Gaussian source with to = 4.7 frn and t = 25 fm/c, moving with vsom-ce
= 0.18c. Lower panel- vsource = 0 .......................................................................... 146

XX

Figure 7.3. Contour diagram of xz/v (chi-squared per degree of freedom)
determined by comparing theoretical correlations functions to the data
shown in the upper panel of Figure 7.2. The fit was performed in the
peak region of the correlation function, q = 15-30 MeV/c. Details are
discussed in the text. .............................................................................................. 147

Figure 7.4. Longitudinal (solid symbols) and transverse (open symbols)
correlation functions constructed for protons emitted from peripheral
events with total pair momentum P=400~600 MeV/c. Cuts were
performed on the angle 1|! as constructed in the projectile, center-of-
momentum of the 35Ar+458c system, and the laboratory frame. .................... 148

Figure 7.5. Longitudinal (solid symbols) and transverse (open symbols)
correlation functions constructed for protons emitted from peripheral
events with total pair momentum P2700 MeV/c. Cuts were performed
on the angle w as constructed in the projectile, center-of-momentum of
the 36Ar+45Sc system, and the laboratory frame. .............................................. 149

Figure 7.6. Longitudinal (solid symbols) and transverse (open symbols)
correlation functions constructed for protons emitted from central
events with total pair momentum P2700 MeV/c. Cuts were performed
on the angle w as constructed in the projectile, center-of-momentum of
the 36Ar+45Sc system, and the laboratory frame. .............................................. 150

Figure 7.7. Longitudinal (filled circles) and transverse (open circles) two-
proton correlation functions for the reaction 129Xe + 27A] at E / A=31
MeV. Angular cuts in the upper panel were constructed in the
laboratory frame of reference, and in rest frames moving in the lab
with B = 0.2086 (center panel) and B=.4172 (bottom panel). The cut on
the total momentum of the proton pair was P 2 480 MeV/c. .......................... 153

Figure 7.8. Contour plots of xZ/v (chi-squared per degree of freedom)
evaluated by comparing measured longitudinal and transverse
correlation functions (over the range of 10 MeV/c S q S 40 MeV/c to
those predicted for emission from a schematic source wifll radius and
lifetime parameters R and 1:. The upper and lower panels compare
correlation functions constructed from pairs with P2480 MeV/c and

P2580 MeV/c, respectively. X's mark parameter sets used for
calculated correlation functions shown in Figure 7.9. ...................................... 157

Figure 7.9. Longitudinal (filled circles) and transverse (open circles) two-
proton correlation functions for the reaction 129Xe + 27A1 at E/ A=31
MeV, evaluated. in the center-of-momentum frame of the projectile and
target. Top and bottom panels show data for cuts on the total
momentum of the proton pair of P 2 480 MeV/ c and P 2 580 MeV/c,
respectively. Solid and dashed curves show calculations of
longitudinal and transverse correlation functions using the source
parametrization of Equation 7.3 with the parameters given in the
figure. ....................................................................................................................... 158

Figure A.1. The data format of the uncalibrated ”raw” tapes shows a
hierarchical structure. Data words are indicated by x’s. These are
grouped into physics events, FF buffers, and NSCL buffers. .......................... 166

Figure A2. The data format of calibrated ”physics" tapes follows a more
natural structure. Data for detector elements are grouped together. ............. 167

xxii

Chapter 1 - Introduction

1.1 Intensity Interferometry

The importance of interferometric measurements in the development of
modern physics can hardly be overstated. A crucial element in what was to
become a unified electromagnetic theory was revealed when Young's double slit
experiment firmly established the wave nature of light. Then, the existence of
waves in otherwise empty space naturally led to the question of 'what was
waving?’ and the concept of a universal medium through which massive bodies
move freely— the aether— was born. In 1881, Michelson built an interferometer
sensitive enough to disprove the existence of the aether, and the fundamental
basis for special relativity was laid. Finally, the essence of the third great
development of modern physics— quantum mechanics— is contained in an
interferometric experiment. Feynman states that truly quantum mechanical
effects can always be summarized by saying, 'You remember the case of the
experiment with the two holes? It's the same thing!’ [Feyn 65]. Indeed, in
Feynman's path integral formalism, any physical phenomenon may be viewed as
an interference pattern to which all possible paths contribute.

The cases mentioned above are all examples of amplitude interferometry.
In amplitude interferometry, schematically illustrated in the upper panel of
Figure 1.1, the waves emitted from the source 0 are split by two slits. The
probability amplitude at detection point D is the sum of the amplitudes
associated with the two paths O-Sl-D and 062-0 The dark fringe spacing may

2

 

Amplitude Interferometry] '

 

 

 

 

 

 

 

 

Slits ' Screen
D; or
E:
Source ' Detector
O <:
e s?
Intensity Interferometry ]
Detector 1

Source

6

 

I:

Detector 2

Figure 1.1. The operation of the two types of interferometer is
schematically shown. In the upper panel, an amplitude (Michelson)
interferometer measures the single-body detection probability. The lower
panel shows an intensity (HBT) interferometer which measures the two-
particle yield. (From [Gong 91c]).

be studied as a function of slit separation, and information on the angular size of
the source 0 may be extracted.

A Michelson interferometer may be based on this principle (employing a
beam splitter instead of a plate with two slits) to measure stellar sizes. In 1920,
the angular diameter of the red giant Betegeuse in the shoulder of Orion was
measured with such a technique [Swen 87]. However, measurement of
significantly smaller sources proved difficult with the Michelson interferometer.
Improving the resolving power meant increasing the baseline, and distances of
100 km or more involved more elaborate equipment [Hanb 54]. More
importantly, as the path lengths sample less and less common air volume,
atmospheric disturbances impose random relative phase shifts, destroying the
interference signal.

In response to this problem, Hanbury Brown and Twiss (HBT) proposed
an interferometer that operated on a different principle. The operation of their
intensity interferometer is schematically shown in the lower panel of Figure 1.1.
As opposed to the amplitude (Michelson) interferometer, which measures the
single-particle detection probability, the intensity (HBT) interferometer records
the two-particle probability. Information about the relative phase of wavefronts
impinging on Detectors 1 and 2 is lost (and is unnecessary), but the time-
structure of the measured probability at each detector is recorded. The two-
particle coincidence yield mg and the single-particle yields n1 and n2 are

measured. A correlation function C(p1,p2) is constructed according to:
C(sz) = 1+R(pl,P2) = <n12>/<n1><n2>- (M)

Source size information may be extracted from the two-particle correlation

function.

The correlation function depends only on the correlations present in the
coincidence yield. For a photon measurement, such correlations arise due to
their bosonic nature. The presence of one photon in a particular momentum state
increases the likelihood that another will be emitted into the same state. The
strength of such correlations depends on the average space-time separation of the
photon emission points— that is, the source size.

The use of intensity interferometry in subatomic physics followed a few
years later in a study of angular correlations of pions emitted from antiproton-
proton annihilation reactions at about 1 GeV. Goldhaber, Goldhaber, Lee, and
Pais (GGLP) [Gold 60] found that the statistical model described well the
correlations between pion pairs of unlike charge, but that distributions for like-
charge pions contained additional correlations. Assuming that these arose from
wavefunction symmetrization effects (or so-called Bose-Einstein correlations),
GGLP were able to reproduce the additional correlations with an assumed source
radius on the order of 5 fm. Today, pion correlations are used extensively as a
tool to probe the space-time extent of sources created in high energy nuclear
collisions [Boal 90] and are to be the subject of major experimental efforts in the
search for the quark-gluon plasma at the Relativistic Heavy Ion Collider (RHIC).

In 1977, Koonin [Kobn 77] proposed the use of proton correlations as a
means of studying intermediate-energy nuclear reactions. As fermions, protons
should show an anti-correlation in the two-particle yields, in contrast to photons
or pions. Dominating the correlations, however, are final-state interactions due
to the strong nuclear force and Coulomb repulsion. When correctly incorporated
into the theoretical formalism, these too may be used as probes into the space-
time extent of the source. Two-proton correlation functions have been studied

extensively in intermediate-energy heavy ion collisions (for references, see, e. g.

[Boal 90]) and may eventually pinpoint the elusive liquid-gas phase transition of
nuclear matter [Prat 87].
1.2 Geometrical Measurements with Two-Proton Correlation Functions

Two protons emitted from an excited region of nuclear matter will
mutually interact through the strong nuclear and Coulomb forces. These final-
state interactions, along with the quantum mechanical requirement of
wavefunction antisymmetrization, will alter the measured two-proton
distribution, compared to a scenario in which the protons are distinguishable and
do not interact. The magnitude of these effects depends on the phase space
separation of the two protons upon emission. By dividing the coincidence yield
by the single-particle yields, the correlation function measures the amount of
distortion in the final state, revealing information about the initial space-time
separation of the pair, while dividing out single-particle phase space effects.

An attractive S-wave component of the strong interaction leads to a
pronounced maximum in the correlation function at a relative momentum qz20
MeV/ c. Smaller emitting sources lead to more pronounced bumps. The
Coulomb repulsion and the Pauli antisyrrunetrization principle suppress the
correlation function at small q values. The amount of suppression increases with
decreasing source size. The interplay between these three effects, along with the
initial proton phase space distribution, determines the shape of the correlation
function.

Protons emitted with an average velocity v from a source with radius r
and an emission timescale characterized by 1:, will have an initial spatial
separation on the order of d= I r+v.‘: l . In Figure 1.2, the spatial distributions of
protons emitted toward a detector system to the right are shown. It is

qualitatively clear that an increase in the source radius (panel b) or in the

 

To Detector

 

 

. P. .:. 0.0 ’0 .' .20... 'V. .0. .: 0o “.0
Q. .g . . :0. . 0. .
. . . . . :g. C
C) ' : e o y . o . o‘
3. n a . {. . o. . . .
0. ~. I . .5 . . ’. . I
.Q . '5 o ‘ .o O so ‘ I. ‘ ° .

Figure 1.2. Spatial distributions of emitted protons originating from a
source with a small spatial and temporal extent (a), with a large spatial
extent (b), and with a large temporal extent (c).

emission timescale (panel c) leads to a more extended phase space distribution
than is shown in panel a.

The resonance in the strong interaction is S-wave in nature, and, for
spatial separations less than the Bohr radius of 58 fm, the long-range Coulomb
repulsion is dominated by the i =0 partial wave. Therefore, the magnitude of
these final-state interactions depends upon the average separation d only; the
distributions in panels b and c are not distinguishable through final-state
interactions.

However, information about the shape of the distribution may be
extracted through quantum mechanical effects. The antisymmetrization
requirement suppresses the correlation function over the range quI / Rx, where
qx and Rx are the component of the relative momentum and the spatial
separation of the phase space distribution in the x-direction, respectively.
Therefore, if the phase space distribution resembles that shown in panel b, a
correlation function constructed from proton pairs whose relative momentum is
oriented along the short dimension of the distribution will be suppressed as
compared to a correlation function constructed from pairs with the relative
momentum oriented along the long dimension. Through such cuts, both the size
and lifetime of the source may be extracted. .

1.3 Sources Created In Intermediate-Energy Heavy Ion Collisions

The study of heavy ion collisions is currently an active area of research,
pushing existing accelerators to the limits of their design and prompting the ‘
construction of a new machine able to accelerate heavy ions to energies not
currently attainable. The system under study is finite, quantum mechanical,
strongly interacting, and quickly evolving. Furthermore, the behaviour of the
system depends on variables that elude direct measurement. The complexity of
heavy ion collisions presents a true challenge to theorists and experimentalists

alike because both statistical and dynamical effects play a role in the evolution of
the collision, the relative importance of each beingdetermined by the
bombarding energy and impact parameter.

At low bombarding energies (E / A 510 MeV), the evolution of a heavy ion
collision is largely determined by mean field effects and one-body dissipation.
Central collisions are understood in terms of the creation and statistical decay of
a long-lived, equilibrated compound system which is formed through complete
fusion and which decays by particle evaporation Proton-emitting sources
created in such a scenario may be expected to have a spatial extent comparable to
that of the compound nucleus, and a lifetime on the order of hundres to
thousands of fm/c [Frie 83].

Heavy ion collisions that occur at relativistic energies (E/ A 2 1 GeV) are
largely understood in terms of individual nucleon-nucleon interactions. The
geometrical participant-spectator model applies, and the energy density is
greatest in the reaction zone, defined by the spatial overlap of the target and
projectile density distributions. The binding energies are negligible compared to
the energy deposited in the reaction zone, and the system explodes into its
constituents. A "source" in this case may be difficult to envision, and lifetimes
are expected to be on the order of the time required for a nucleon to traverse the
nuclear region— roughly 10 fin/c. .

Intermediate bombarding energies constitute the realm of the transition
between the statistical decay of an equilibrated compound system and the
dynamical explosion of nuclei into constituent components. Highly excited
composite systems may be created on very short timescales, allowing dynamical
and statistical processes to compete as the distinction between pre-equilibrium
and thermal emission from a "hot spot" blurs. After the early emission of high-
energy nucleons, an equilibrated hot residue is expected to remain, which then

undergoes statistical decay. At low excitation energies, the compound system
resembles a hot liquid drop which evaporates nucleons as it cools. At higher
excitation energies, thermal pressure may push the residue to the limits of
stability and cause it to expand to low density where disintegration into nucleons
and bound clusters may occur.

The transition to a mixture of nucleons and bound clusters is similar to a
liquid-gas phase transition. Although current heavy ion accelerators are capable
of creating nuclear systems hot enough to leave the liquid state, no definitive
experimental signature for the liquid-gas phase transition has been found in the
single-particle spectra. Kinetic energy "temperature" parameters vary smoothly
with projectile energy [West 82, Chit 86b], and no sudden "jump" in the fragment
multiplicities indicative of a phase transition is observed as a function of
bombarding energy or impact parameter me So 91, Ogil 91, Bowm 91, Peas 93,
Tsan 93].

A saturation in the temperature of a statistical system as the excitation
energy is increased may signal a phase transition. Indeed, temperatures
extracted by comparing relative yields of particle-unstable states emitted from
heavy ion collisions show surprisingly little sensitivity to projectile energy [Chen
87c]. (A small systematic dependence is, however, observed [Schw 93].) At
present, it is not yet certain whether the relative population of states is
determined by the temperature of the emitting system or whether it is
determined by the reaction dynamics [Fuch 90].

As discussed by Pratt and Tsang [Prat 87], an experimental determination
of the timescale of particle emission could give essential information about the
phase of the decaying nuclear system. The decay of nuclear matter in the liquid
phase would be characterized by lifetimes on the order of several thousand fm/c
[Frie 83, Prat 87], with longer lifetimes for less energetic particles, since the

10

temperature drops more slowly as the system cools [Boa186b]. A nuclear gas, on
the other hand, should exhibit emission timescales on the order of 100 fm/c or
less [Aich 85, Aich 86, Prat 87], as spontaneous expansion of the system follows
compression.
1.4 Motivation for This Study

The dependence of the correlation function on the total momentum of the
proton pair P: l p1+p2| has been studied previously for many systems [Lync 83,
Chen 87c, Poch 86, Poch 87, Awes 88, Gong 90b, Gong 90c, Gong 91b, Lisa 91].
Stronger correlations have been observed for pairs with larger total momentum,
qualitatively indicating a tendency towards sources of larger space-time extent as
the reaction evolves. Recently, this dependence was successfully reproduced by
a dynamical model based upon the Boltzmann-Uehling-Uhlenbeck (BUU)
equation [Gong 90c, Gong 91b, Baue 92], suggesting that the BUU may reliably
describe the evolution of the reaction zone. However, the P-dependence of the
correlation function predicted by the BUU was quite different for central and
peripheral collisions [Gong 91a], indicating a distinct reaction evolution for
central and peripheral collisions. In the past, such detailed predictions were
inaccessible to experimental tests, since high-statistics proton coincidence
measurements lacked experimental impact parameter filters. Those
measurements which did have impact parameter filters lacked sufficient statistics
to gate simultaneously on P. Experimental tests of the impact parameter and
total momentum dependence of predicted correlations may isolate those impact
parameters for which the BUU adequately describes the reaction evolution and
indicate areas in which further theoretical work is needed.

As discussed in Section 1.2, proton emission from a long-lived source
should be evident through cuts on the relative orientation of the total and relative

momenta of the proton pair, \|I=COS'1(P-q / P q). Comparison of the longitudinal

11

(wz0°) and transverse (w=90°) correlation functions should eliminate the
ambiguity between space and time effects and thus allow the determination of
I expansion and emission timescales in a relatively model-independent way.
Previous searches for a directional dependence in two-proton correlation
functions have produced null results [Zarb 81, Awes 88, Ardo 89, Gong 90b,
Gong 91b, Gouj 91, Rebr 92]. Recently, it was suggested [Rebr 92] that incorrect
assumptions about the rest frame of the emitting source due to incomplete
knowlege of the impact parameter could wash out the expected directional
dependence.

It was the aim of this thesis to measure the total momentum and
directional dependence of the two-proton correlation function for 35Ar+45Sc
collisions at E/ A=80 MeV as a function of impact parameter. This required a
high-resolution, high-statistics measurement of proton yields, with coincident
information on the violence of the collision from the yields of other charged
particles emitted in the reaction. Coincidence and single proton yields were
measured in a high-resolution detector array at <91ab>=38°- The MSU 41: Detector
Array, covering about 85% of the remaining solid angle, provided information
about the violence of the collision Two-proton correlation functions were
constructed from the proton yields and impact parameter selection was
performed with cuts on global observables measured in the 41: Array.

1.5 Organization of This Work

This work concentrates on a study of the nearly symmetric system
36Ar+45Sc at E/A=80 MeV. Full details of this study are presented here. Also
presented are some results from a re-analysis of data previously taken by Gong
et al. for the systems 14N+27Al at E/ A=75 MeV [Gong 90c, Gong 91b] and
129Xe+27Al at E/A=31 MeV [Gong 90b, Gong 91b]. Experimental details for

12

these data have been previously reported [Gong 91c]; we do not reproduce them
here.

Chapter 2 discusses the technical details of the Ar+Sc experiment,
including descriptions of the 56-element hodoscope and 41: Array detector
systems and electronic logic used. Chapter 3 presents the details of the data
reduction, including particle identification, time walk correction, and energy
calibration techniques.

Chapter 4 deals with experimental determination of the centrality of an
event. Impact parameter selection based on global observables is discussed, as
well as a method to experimentally determine the experimental resolution for
centrality determination. Also described are two methods for comparing
centrality-cut data to predictions of a dynamical model.

Chapter 5 discusses the two-proton correlation function. The theoretical
framework in which the correlation function is usually understood— the Koonin-
Pratt formalism- is presented, and specific application of the formalism to two-
proton data is discussed. Sensitivity of the correlation function to the space-time
structure of the source is illustrated by schematic calculations. Two commonly-
used methods for constructing the experimental correlation function are
discussed and compared for the system 1“Na-27A] at E/ A=75 MeV. Finally, a
discussion of the effects on the correlation function from finite experimental '
resolution is presented; the 36Ar+453c data measured with the 56-element
hodoscope is used as an example.

Chapter 6 presents the centrality-cut proton data measured for 3‘5Ar+458c
at E / A=80 MeV. Predictions of a Boltzmann-Uehling-Uhlenbeck transport
model are compared to the single-particle energy spectra and to the two-proton

correlation functions.

13

Chapter 7 discusses the use of longitudinal and transverse correlation
functions to independently determine the source size and lifetime for a short-
lived hot source created in central 36Ar+45Sc collisions at E/A=80 MeV, and for
an evaporative source created in 129Xe+27Al collisions at E/A=31 MeV. The
importance of proper source frame identification is discussed.

Finally, in Chapter 8, we present a summary and some brief conclusions.

Most of the results presented in this work have been submitted to refereed

journals for publication [Lisa 91, Lisa 93a, Lisa 93b, Lisa 93c, Lisa 93d].

Chapter 2 - Experimental Details

The experiment was performed at the National Superconducting
Cyclotron Laboratory at Michigan State University using particle beams
provided by the K1200 cyclotron. Two charged particle detector arrays were
combined to measure reaction products. A schematic diagram of the set-up is
shown in Figure 2.1. The energy and emission direction of light charged particles
were measured with high precision in a 56-element hodoscope, which was
centered at a laboratory angle of 38°. Information about the centrality or violence
of the collision was determined by measuring charged particles in the MSU 41:
Array [West 85], which surrounded the target. Event information from each
collision was digitized on an event-by-event basis and written to tape to be
analyzed off-line.

Here, we discuss the major subsystems of the experiment, concentrating
first on the detectors themselves, and then discussing the electronics. Finally, we
discuss our choice of nuclear system under study.

2.1 The 56-Element Hodoscope

Light particles (p,d,t,3He,a) were measured in a 56-element hodoscope,
comprised of a closely-packed array of 56 AE-E telescopes [Gong 91b]. The
mechanical structure of the hodoscope and 37 of the 56 detector elements had
already been constructed for the thesis experiment of WC. Gong [Gong 91b]. In
preparation for the measurements discussed in this work, we constructed the
remaining detector elements and the mechanical support/ vacuum chamber to
couple the hodoscope to the 41: Array.

14

15

MSU 41: Array

1'
i
E
i

 

Figure 21. Schematic diagram showing the arrangement of the 56-
element hodoscope and the MSU 41: Array.

16

2.1.1 Detector Elements

The AE component of the telescope consisted of a 300-um-thick planar
surface barrier detector with an active area of 450 mm2 (Ortec models TB-024-
450-300-S and TD-024-450-300-S). A positive voltage (typically ~ 100 V) across
the face of the detector allows the collection of free charge generated when
ionizing radiation (e.g. a charged particle) passes through the detector. The
signal is then passed to a preamplifier and sent out to the electronics system to be
discriminated and digitized.

The E component of the telescope consisted of a CsI(Tl) detector [Gong 88,
Gong 90a, Gong 91c]. A schematic of this detector is shown in Figure 2.2. The
CsI(Tl) crystal is a 10 cm long cylinder with a diameter of 38 mm. The crystals
used for the original telescopes were manufactured by BICRON corporation. For
the completion of the hodoscope, we purchased crystals of identical dimensions
from Hilger corporation of France. T

The crystals arrived unpolished. Following the "standard" crystal
processing developed in the construction of the original detectors, we sanded the
(vertical) sides of the crystal with fine sandpaper (#320) with strokes parallel to
the cylinder axis. The two end faces were first polished with fine sandpaper in a
circular motion; the crystal was held vertically over the sandpaper which sat on a
flat glass plate. Ettin alcohol was liberally applied to the sandpaper to help the
crystal glide smoothly, avoiding "jumps" which can deeply scratch the surface.
Next, a silk cloth with polishing compound (National Diagnostics ND-706) was
used to polish the surfaces, and finally, a tissue that had been very slightly
dampened with water was used. CsI is hygroscopic, so it is important to avoid
contact with water; however, a very small amount will "melt" a surface slightly,
greatly improving the polish. Although we eventually would roughen one end
of the crystal with #320 sandpaper, first polishing both ends allowed us to see

17

CsI(Tl) scintillator readout by PIN diode

 

 

 

 

 

white Teflon tape Teflon ring I Fixture
and Al foil A1 can Al adaptor Stand-off
CsI(Tl) crystal ; /A
u (1.5"121 x 4.0": ) . 7 ‘1 LE 0
collimator connectors

thinAlfoil Lucite RTV615 p diode.
(159m) light guide Silicon rubber (3(2sz) Preamplifier

Figure 2.2. Schematic anatomy of the CsI(Tl)-PIN diode assembly used
as the E detector in the light particle telesc0pe.

18

through the crystal. Most crystal interiors were uniformly clear. Two of the
crystals had small visible defects; these did not affect performance as determined
by resolution bench tests [Gong 90a]. When coupling the crystal to the light
guide with optical cement, bubbles of air can form which may interfere with the
detector performance. These were easily identified visually through the crystal.
When such defects were discovered, coupling was re-done.

The crystal was coupled at the back end to a clear Lucite light guide with
RTV 615 silicon rubber. It is relatively easy to couple two surfaces without air
bubbles using this compound, and good optical coupling as well as mechanical
stability is achieved. The sides of the light guide were sanded like the sides of
the crystal. Finally, to prevent light loss, the sides and back of the light guide
(where not coupled to the PIN diode) were painted with reflective paint.

A square PIN diode (Hamamatsu model 81790-02) of 2 cm X 2 cm active
area was coupled to the back end to the light guide. The photodiode has an
operating voltage of about 50 V and a leakage current of about 3 nA at room
temperature. Operation of the diode in vacuum can result in increased leakage
currents due to heating both from the diode and associated preamplifiers.
Therefore, it was necessary to actively cool the hodoscope in vacuum; details are
discussed below.

The crystal / light guide/ PIN diode assembly was next wrapped in Teflon
to further decrease light loss and nonuniformity along the detector edge. Long
strips of thick Teflon tape were laid overlapping along the axis of the crystal.
Next, thinner Teflon tape was wrapped around the assembly. Thin (15 um),
light-opaque aluminum foil covered the front face of the detector, and thick
aluminum foil was wrapped outside the Teflon tape.

The wrapped assembly was placed into the "front" portion of an
aluminum detector can, which provided mechanical rigidity. A copper ring in

19

the front of the can shielded the outer edges of the detector, giving a collimated
diameter of 2.5 cm. Teflon rings held the crystal assembly firmly in the front
portion of the can. i

The signal from the PIN diode was sent by short leads (3-4 cm long) to the
back portion of the can, where the preamplifier was mounted. This preamplifier,
shown schematically in Figure 2.3, was developed by Michael Maier at the NSCL.
The low power dissipation of this preamplifier (0.5 W) is nicely suited for
Operation in vacuum; a large array of these preamps can be handled by our
cooling system. The rise time of the preamp (1:; =0.2 us) is less than both of the
decay components of the CsI(Tl) (0.4 and 7 us). For an input impedance of 130
pF, corresponding to that of our PIN diodes, the full width at half maximum
(FWHM) noise of the preamplifier corresponds to a charge of 2000 electrons. For
a typical CsI(Tl) detector, this corresponds to a resolution of about 300 keV for 01
particles of 5.5 MeV. To monitor possible gain drifts (either in the preamplifier or
the amplifier that follows), a pulser test input was provided. In our experiment,
gains were found to drift by less than 1% over the week-long experiment. Since
both preamplifier gain and crystal light output depend on temperature, active

temperature stabilization was important to steady gains.
2.1.2 Mechanical Support

The 56 detector elements were mechanically mounted in the hodoscope
structure. A schematic of the front face of the structure is shown in Figure 2.4.
Silicon detectors are securely held in removable brass mounts on the front face of
the structure. Covering the front face of each mount is a thin (~10 um) Ta foil to
suppress unwanted signals from electrons and X-rays. Preamplifiers for these
detectors surrounded the main structure. They were in good thermal contact
with copper bars cooled by flowing alcohol to about 15° C.

20

 

 

 

 

 

 

 

 

 

‘3 BIAS
IDnF ‘ *IZV
. 1151.“:
1 !
PHOTO DIODE ‘* FL we”: 4 OUT

 

 

 

 

 

 

 

fi; *IZV

 

 

 

 

 

A TEST

Figure 2.3. Schematic of the preamplifier for the PIN diode used to read
out the CsI(Tl) detector.

21

 

 

 

 

 

2.4. Schematic of the front face of the 56-element hodoscope

Figure
structure.

22

Each CsI(Tl) detector is centered behind a Si detector. The back end fits
snugly into a hole in a square aluminum back plate, and the front end fits into a
recess in the back of the front plate. Cooling liquid flows through a copper tube
embedded in the back plate to remove heat from the CsI(Tl) preamplifiers.

Telescopes mounted in the hodoscope structure face the center of a sphere
with a radius of about 1 meter; this was well suited to our target-to-detector
separation of 105 cm.

The hodoscope structure was housed in a vacuum chamber attatched to a
hexagonal port of the 41: Array vacuum chamber, positioning the hodoscope at
an average polar angle of about 38°. Figure 2.5 shows the angular coverage
provided by the 56-element hodoscope.

2.2 The NSCL 41: Detector Array

The NSCL 41: Array [West 85] is a closely-packed array of plastic
phoswich detectors which covers approximately 84% of the solid angle
surrounding the target. Although many measurements at the NSCL are
performed using the "standard" 41: set-up, a primary consideration in the design
of the 41: Array was the flexibility to perform coincidence experiments with
other detector systems. This consideration led to the use of interchangeable
detector modules with nearly identical particle response, for ease of analysis. It
is customary to distinguish between two groups of detectors in the 41: Array, the
forward array and the main ball. Although in operation and response these

detectors are similar, their geometries differ.
2.2.1 Detector Elements

All detectors in the 41: Array are phoswich telescopes. A thin layer of
"fast" scintillating plastic is optically coupled to a larger piece of "slow"
scintillating plastic, which is then optically coupled to a phototube. The fast and

200

190

180

9!) (degrees)

170

160

5,6-element Hodoscope Acceptance

 

 

 

 

.@ (as i
L@ .@ (13..
7(5). @@@®j
. ® @ ..
’@.@@@@ ‘
' C53
r @ ® ® -
L .
@ ® ® @
. ® I
-@ ® ®
" @642 @0657
.@ (23$) <9.
1‘9 ® 3
30HH354 140‘ #45.

9 (degrees)

Figure 2.5. Angular coverage provided by the 56-element hodoscope.

24

slow plastic have different rise and fall times, as shown in Table 2.1. Figure 2.6
schematically shows the principle of operation. A charged particle entering the
detector through the front face loses energy in the fast plastic, causing a brief
flash of light . The particle then stops in the slow plastic and the light emitted

with the slower time scales is proportional to the remaining energy.

Table 2.1. Time constants for fast and slow scintillating plastics

 

 

 

 

 

 

 

BICRON plastic rise time (ns) fall time (ns)
BC-412 (fast) 1.0 3.3
BC-444 (slow) 19.5 179.7

 

The signal read from the photomultiplier tube is a linear sum of the fast
and slow components. Electronically, the components are approximately
separated by integrating the signal over different time windows, as shown in
Figure 2.6.

A typical detector response is shown in Figure 2.7, where a raw "fast" vs.
"slow" spectrum from a main ball element is shown. Particles of a given species
define a curve in the fast-slow space as a function of energy. Two features of the
spectrum deserve mention.

Particles with energy below the so-called punch-in energy stop in the fast
plastic and the signal has no slow component. The punch-iii energy varies with
particle type; values for the ball and forward array detectors are given in Tables
2.2 and 2.3. Although no light is generated in the slow plastic, residual light from
the fast plastic appears in the slow gate. In this case, the slow signal is
proportional to the fast signal, and the "punch-in line" is defined; see Figure 2.7.

A similar situation occurs for particles which do not lose energy in the fast
plastic, but lose energy in the slow plastic. Examples of such particles are

gamma rays, neutral particles, and cosmic rays that enter the detector from the

Table 2.2. Punch-in Energies for Ball detectors

 

 

Particle Punch-in Particle Punch-in
T1129 Energy (MeV) Type Enemy (MeV)
p 17 Li 140

d 24 Be 214

t 28 B 287

He 70 C 380

 

 

 

 

 

 

Table 2.3. Punch-in Energies for Forward Array detectors

 

 

 

 

 

 

Particle Punch-in Particle Punch-in Particle Punch-in
Type Energy (MeV) Type Energy (MeV) Type 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

 

 

 

 

26

slow plastic

mmpumm

. .'.'.‘. ............

   

.......

-2523: if; photomultiplier
tube

 
 
  

 

particle entering
detector

   

..........

 

m"—'O<

 

 

fast

gate slow gate

Figure 2.6. Schematic of phoswich detector operation. The signal from
the phototube is split and the light components from the "fast" and
"slow" plastic are separately integrated.

27

411 Module 48

 

250

200 — Punch-in Line —-

H

01

O
l
l

 

Fast (channels)
ES
:3

. -50

 

 

 

 

o 50 100 ‘ 150 200 250
Slow (channels) '

Figure 27. A "fast" vs. "slow" spectrum from a detector element from the
main ball. The punch-in line and neutral line frame signals from

identifiable particles.

28

side. The fast-slow curve of these particles defines the "neutral line," also shown

on Figure 2.7.
2.2.2 Mechanical Set-up

The basic geometry of the 411 Array is that of a 32-faced truncated
icosahedron, giving it the appearance of a soccer ball with twenty hexagonal and
twelve pentagonal faces, as shown in Figure 2.8. The beam enters through one
pentagonal face and exits through another. The entrance face is the only face not
covered by detectors, allowing space for target insertion. The exit face contains
the forward array, a set of 45 phoswich detectors with polar angles less than
about 18°. A forward array element has either a square or circular front face;
these shapes are not amenable for close-packing in a pentagonal geometry, so the
solid angle coverage is about 54% for the exit face. The arrangement of forward
array elements is shown in Figure 2.9.

The remaining faces are covered with ball modules. The pentagonal and
hexagonal modules are divided into five and six wedges, respectively. Each of
the 170 wedges is an independent phoswich detector.

2.3 Electronics

In this experiment, we measured two-proton correlation functions as a
function of impact parameter. The correlations were measured in the 56-element
hodoscope, while the impact parameter was determined by charged particle
measurements in the 41:] Array. All two-particle hits in the hodoscope were
recorded along with the corresponding information in the 41: Array.

Because of the much faster rise time of the 41: detector elements as compared to
those of the hodoscope, the 41: Array had to self-trigger its ADC's and the
digitized information had to be read into the computer, or it had to be cleared,

depending on whether the slower hodoscope satisfied the trigger condition.

29

 

 

 

 

\
c--- ”’ \
\
I I
I I
I l
’ I
A\\ 1’ ‘
O
\\ ’v ”o \
\
I I ‘\
I I
I I
| I
I
’I
I \
Y .
\
\
\ i
\ I

 

Figure 2.8. A 32-faced truncated icosahedron, the underlying geometry
of the MSU 4n Array.

30

Forward Array detector arrangement

@3880
063:9 @339:
0“.gé90 Bean; Pipe ”<2:
goo (a o a:

“0:3 33 32

 

 

 

 

 

Figure 2.9. Spatial arrangement of phoswich detectors in the forward
array. ‘

31

In this situation, two distinct trigger conditions operated in parallel. The
first condition is generated by the 41: to gate its ADC's. The second "master"

7 condition is generated by the hodoscope to start the readout of the event. Events
with a hodoscope trigger but without a 41: trigger contain no information about
the impact parameter for the event. Clearly, it is important that there be a 41:
trigger present every time there is a hodoscope (master) trigger, although the
reverse need not be true. Therefore, we set the 41: trigger condition at "minimum
bias" settings— usually at multiplicity 2 or 3 in the array. Due to the much
greater geometric coverage of the 41: Array as compared to the hodoscope and to
the fact that the hodoscope was positioned at a relatively large polar angle,
trigger rates in the 41: Array were much larger than the master (hodoscope)
trigger rates.

Three master trigger conditions were used in this experiment. For most of
the 5-day data-takingperiod, we ran with a coincidence trigger in the hodosc0pe.
At regular intervals, we took data under a singles trigger in the hodoscope to
have data with which to construct the denominator of the correlation function,
and also to be able to compare coincident and singles spectra. Finally, we took a
short run with the 41: Array as the master with the trigger condition set at
multiplicity 1. This last set of data was important in construction of an impact
parameter scale; see Chapter 4.

Figure 2.10 shows a schematic block diagram of the electronics setup used.
It is seen that the electronics for the two detector subsystems, the 56-element
hodoscope and the 41: Array, are largely independent. Below, we discuss the two

main branches of the electronics setup, and then how they work together.
2.3.1 SG-Element Hodoscope

Preamplifier signals from the Si and CsI(Tl) detectors were sent to shaping
amplifiers. The "slow" output of the amplifier, operated with a differentiating

32

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

/V
Emu n...” _ _ all... 3.3%
H u<
as... 18.-
o _ E. . on. .
as. 9? 9.0 So a... .8 In
3a a...
so
On 963 on.
.E O
a: 3:131
In:
n...”
Emu ct
<mwu ,
'10 sit
Cu .8
-E 203
3.)
:5
83950
so
582”
O!
In:
3a
a do. 063
(”mu ca.
IIIVE

 

Figure 2.10. Electronics used to digitize data and trigger the computer.

33

and integrating time constant t~3.5 us, was sent directly to a peak-sensing ADC,
Silena model 4418 / V. The "fast" outputs from the amplifiers, with a time
constant t~100 ns, were used to build the trigger logic. The logic for one AE-E
telescope is shown in the upper third of Figure 2.10. I

The fast output from the silicon detector amplifier was sent to a constant-
fraction discriminator (CFD) with a relatively low threshold. We will refer to the
logic signal generated by this component as "".AE The fast output from the
CsI(Tl) detector amplifier was passively split and sent to two leading edge
discriminators (LEDs). One LED was set with a very low threshold, so that it
fired somewhat in the noise; this logical signal we shall call EL. The EL signal
was then delayed and stretched by a delay and gate generator (DGG). The other
LED was set with a higher threshold; its signal we call EH.

The trigger for the 56-element hodoscope consisted of either one (singles
trigger) or two (coincidence trigger) detectors firing in an event. A telescope is
considered to have "fired" when it satisfies a logical condition EL*(EH+AE), where
EL, EH, and AE are considered as Boolean variables. A schematic indication of
the meaning of these conditions is seen in Figure 2.11, where a AE-E plot from a
typical detector element is shown, along with the threshold conditions EL, EH,
and AE. For real particles, a small signal in the CsI(Tl) detector (E component) is
accompanied by a large signal in the silicon detector (AE component). To reduce
experimental energy thresholds, we set the discriminator level for EL into the
noise. Therefore, any real particle in the telescope must at least fire EL. To
reduce the frequency of triggering on the noise, however, we required one of two
additional conditions to be satisfied. We set the AE threshold just above the
noise, so most particles deposit sufficient energy in the silicon detector to fire this
logical component; a EUAE trigger condition is sufficient for these particles.
However, high energy (180 MeV) protons leave a very small signal in our

Telescope 52 Raw ADC Spectrum

 

  

500—.~ I....,,s..,...fi,.fl.a

400)- —

* 1(pulser) ~

1" ‘ '

<11 300b —

C3 - .

C3 -

a .

.13 - -

3 - .

4 ' .

Z. «$51.13 2

100-;' an. _

I- .

- t -4
AE—">“. Bid

0 1 L 'e”: 1 1 1 [DI 1 1 1 l 1 1 1 1 l

 

 

     

 

of j 100 200 300 400 ~500

E1 E1! E (channels)

Figure 2.11. Raw spectrum of signals of the Si and CsI(Tl) detectors is
shown for a typical AE-E telescope from the 56-element hodoscope. Note
the effect of the various discriminator thresholds in eliminating
coincident noise from the two components.

35

l

l

I .

' 200 ns
I

I

I

 

 

 

 

51.
(disc) _LJ : 1
' l
L---1
I
EL 1 """" 1
(Doe) I I : I I
l u '
I

 

 

D

m

I

I

I

I
--.
—

 

If.
i"
2%

 

 

5?
it
_I

Figure 2.12 Signal tinting for the internal logic of a detector trigger.
Solid lines correspond to timing for large signals, while dashed lines
indicate timing for signals at threshold. The "EL" signal always
determines the overall timing.

36

 

 

 

 

 

 

 

 

 

 

 

 

200 ns
<——->
det A
det B i ,
I. TFC start
_ for dot A
det A I
tear
1 r
det B I I/
hodo , TFC

master enable

 

 

 

 

TFC stop veto
RF
* l I ; {I .|
Downscaled RF
to TFC stop [TFC stop

 

Figure 2.13. Signal timing for the determination of the time that a
detector element fired.

37

300-pm-thick silicon detectors (~250 keV) which may not fire our AE
discriminator. Therefore, we also accept a condition of EUEH.

The timing of the detector trigger signals is shown in Figure 2.12. All of
our discriminators show a time walk (or "slewing") effect; the timing of the
discriminated signal depends on the pulse height. The solid lines in the figure
show the timing for large pulses, while the dashed lines indicate the timing for
signals just above threshold. The overall timing of the detector trigger is
determined by the walk of EL; since EL is generated by a LED, this has the
advantage of exhibiting a well-defined time walk function which can be
corrected off-line (see Chapter 3).

Besides reading out AE and E signals through the Silena 4418 / v's, we also
read out the time that a detector fired relative to the RF cycle of the cyclotron.
Therefore, the "detector trigger" for a given element served two main purposes:
to go into the overall "hodoscope trigger", and to act as a time marker. We used
LeCroy TFC-FERA combinations to digitize this information. This system
requires three signals in time-order, an enable, a start, and a common stop.
Figure 2.13 shows the timing of these signals.

The enable signal was provided in our case by the master. The start
signals were generated by the individual "detector triggers. " However, the start
must arrive after the enable, and the master is determined by coincident detector
triggers. Due to time walk, the master may be generated as much as 200 ns after
the first detector fires. In this case, then, the individual starts should arrive about
500 ns after the detector triggers. Instead of creating this delay using cable delay
or DGGs, which would add considerable hardware to our electronics setup, we
used a somewhat unconventional technique. The detector triggers passed
through an Phillips 7106 ECL discriminator with an adjustable output signal
width We adjusted this width so that the "trailing edge" of the signal came 500

38

ns after the detector trigger. Then, since physical inversion of an ECL signal
conjugates the logical signal, we used modified cables that flipped the signal
from the ECL discriminators. Therefore, the trailing edge acts as a leading edge
to the TFC, and the start is delayed. The common stop to the TFC was generated
by the discriminated RF signal from the cyclotron. This signal, which had a
frequency of about 1 / (50 ns), was downscaled by a factor of 3 in order to

evaluate the timing resolution on-line.
2.3.2 The 41: Detector Array

Signals from the photomultiplier tubes of the ball and forward array
elements are sent to specially-made splitter boxes which passively divide the
signal into three components: "logic," "fast," and "slow."

The "fast" and "slow" signals are passed through a cable delay and into a
FERA for charge integration and digitization. Figure 2.6 shows a typical
phoswich signal and the FERA gate timing used to digitize the fast and slow
signal components.

The "logic" signal is simply the raw phototube signal passed to a Phillips
7106 discriminator. The individual discriminator outputs were then delayed and
sent to TFC starts for timing measurement.

The Phillips discriminator also has a "sum" output that generates a linear
signal with a magnitude‘of 50 mV for each firing channel. The sum output of a
discriminator, then, could be used to determine the hit multiplicity for 15
detectors. The 45 forward array elements passed to 3 discriminators, whose sum
outputs were summed in a summing module; the output of this module was then
the event multiplicity in the forward array. Similarly, the multiplicity from the
170 ball detectors was determined in a separate summing module. One output
from each of these summing modules was sent to a third summing module,

whose output was the device multiplicity. The outputs from each of the three

39

summing modules was sent to a maskable CFD. This device, a commercial
module produced by EG&G, can be programmed through the CAMAC system.
Any or all of the input channels can be masked off— allowing one to fire on ball
multiplicity, forward array multiplicity, or device multiplicity. The multiplicity
level is selected by setting the discriminator threshold.

The output of this module, then, is the "41: trigger." The 41: master is
generated by vetoing this signal with the Computer Busy signal. The 41: master
enables the TFCs, provides the TFC stops (after a delay), and sets the gates for
the time FERAs. The timing for the FERA gates for the E and AE phoswich signal
components are determined separately for the main ball and the forward array.
This is done by requiring a coincidence between the 41: master and a delayed
logical signal generated by the forward array and main ball separately. In this

way, timing jitter of the gates, which can degrade resolution, is reduced.
2.3.3 Master Circuitry

For the majority of the experiment, the hodoscope trigger condition
determined whether to record digitized information from both the 41: Array and
the hodoscope. As explained above, this required the use of a fast clear circuit to
allow the 41: Array to self-gate and store digitized signals in the FERAs for every
event, and then either erase those signals (fast clear) if the hodoscope trigger
condition was not met, or read them out via CAMAC if the hodoscope trigger
condition was satisfied. The signal timing for the master circuit is shown in
Figure 2.14.

If both the hodoscope and the 41: Array fired in an event, the hodoscope
trigger was generated about 1.1 us later, with a jitter of about 200 ns. The fast
clear signal is generated by delaying the 41: trigger by about 1.5 us. Dashed lines
in the figure indicate the effect of jitter. If the hodoscope triggered, then a fast

4O

 

 

 

 

 

 

 

 

 

400 ns
H
41: trigger _ 7
hodoscope U
trigger ' master f
: 5 LI
'. .1
Fast Clear
Fast Clear
Veto ......
f I
i..---] i I
Busy I I" - - -I
w | E l

 

 

Figure 2.14. Signal timing for the master circuit.

41

clear veto is generated to disable the fast clear, and a "computer start" is given to
the system.

When vetoing the triggers with the computer busy, it was appropriate to
account for the relatively large time delay between the 41: and hodoscope
triggers. Therefore, we employed the true computer busy to veto 41: masters and
the so-called busy' to veto hodoscope masters. Simple use of two fan-in/fan-outs
and an appropriate cable delay made a signal whose leading edge coincided with
that of the busy, but whose trailing edge lagged that of the busy by 800 ns. (In
the figure, the computer has become "live" (not busy) just before the 41: fires, and
then becomes busy again due to the new master.) The reason for such a signal
follows. If the computer is busy when a 41: trigger occurs, then no 41: gates will
be set, and the event should not be read. However, if the computer becomes live
again immediately after the 41: trigger, then use of the busy to veto the
hodoscope trigger would allow the event to be read. Vetoing with the busy’, on
the other hand, largely eliminated the readout of these useless events; if the busy
signal was set when the 41: trigger fired, then the busy' would be set when the
hodoscope fired, and the event would not be read out.

2.4 Nuclear System Studied

The system we chose to study with the set-up described above was.
36Ar+455c at E/A=80 MeV. The choice of this system was the result of many
considerations.

As discussed in Chapter 1, a high-statistics measurement of the two-
proton correlation function had been made by Gong [Gong91b, Gong91c] on the
relatively light system 14N+27Al at E/ A=75 MeV. Correlations for this system
showed a very strong dependence on the total momentum of the proton pair,
and BUU simulations were largely successful in reproducing this dependence.
The simulations also showed that the total momentum dependence was a strong

42

function of impact parameter; such predictions, of course, could not be tested by
Gong's data. We wished to perform our measurement in the energy region for
which the BUU is relatively well-understood and could explain impact-
parameter-averaged measurements.

We attempted to extract the impact parameter from global observables,
such as event multiplicity and transverse energy. In a naive participant-spectator
model, such global variables depend on the volume of overlap between the
projectile and target nuclei. As is shown schematically in Figure 215, this
overlap volume reaches a maximum when the smaller nucleus fits inside the
larger. For a very asymmetric system, then, global observables are expected to be
rather insensitive to impact parameter for central collisions. Our choice of a
nearly symmetric system was designed to maximize sensitivity to impact
parameter.

Correlation functions corresponding to emission from a very large system
are suppressed, and the effect of any cut, such as in impact parameter or total
momentum, is difficult to see. Therefore, we did not want to study a very large
system.

On the other hand, the search for impact parameter effects with a very
light system is difficult, since fluctuations in impact parameter may dominate the
centrality filter. Also, smaller systems are less sensitive to bulk-compressional
effects, which may be important in the difference between peripheral and central

collisions.

43

o.

POfiphCrSI

I

Mid-le

tree

Figure 2.15. Schematic illustrating, in a purely geometric picture, the
reduced sensitivity of global observables to the impact parameter for
very asymmetric systems.

Chapter 3 - Data Reduction

In the experiment, electronic signals from the detectors were digitized and
recorded on magnetic tape on an event-by-event basis. In this chapter, we
discuss our procedures for. obtaining particle identification, energy, and time
signals from the raw data. We concentrate first on the 56-element hodoscope,
and then on the 41: Array.

3.1 56-Element Hodoscope

One of the attractive features of the correlation function is its relative
insensitivity to complicated experimental distortions, such as position-dependent
energy thresholds, for example. However, one must still exercise care in the
treatment of correlation data. Contamination from deuterons in proton data, for
example, is not "corrected for" by the correlation function. Likewise, inclusion of
a large number of "random," or uncorrelated, pairs in the construction of the
correlation function will suppress true correlations.

Below, we discuss our procedure to determine particle identification,
energy, and relative time with precision. Although we discuss them separately,
we note that the extraction of each of these quantities is not independent. To
determine particle identification, for example, we used the proton energy
calibration, the extraction of which, in turn, required a rough proton
identification. '

3.1.1 Particle Identification

Particle lines can be identified in a two-dimensional map of the energy

deposited when punching through the silicon detector (AE) vs. the energy

45

deposited while stopping in the CsI(Tl) detector (E); see Figure 2.10. However,
the channel resolution required to cleanly separate particle lines and set gates in

i this map demands large amounts of computer memory. Therefore, we
transformed the AE-E map into a so-called PID-E map, where PID is a linearizing
function of AE and E [Shim 79].

The advantages of using a linearizing PID function are twofold. Firstly,
the two-dimensional channel space is more efficiently utilized, decreasing
"wasted" computer memory spent on unoccupied AE—E bins. Secondly, in
regions in which there is less resolution between particle lines, it is easier to
extrapolate the gate from regions with good resolution

We found that our PID-E maps were most linear with the parametrization:
PID = (E+AE)1°73 - E173. (3.1)

The PID-E map for a typical detector is shown in Figure 3.1 as a
logarithmic density plot. Particle species are identified by relatively straight
curves. An exception is the low-energy behavior of the helium isotopes, where
the nonlinearity of the PID function for nearly-stopped particles in the Silicon
detector is evident.

Two-dimensional gates on particle type were set in these maps, and clear
identification is achievable for 1H, 2H, 3H, 3He, and 4He. Particle identification
resolution is a function of energy. At low kinetic energies, where a considerable
signal is observed in the AE component, the distinction between particle types is
excellent. Separation is most difficult, however, for the lightest particles with
high (~ 100 MeV) kinetic energy. In Figure 3.2, we show one-dintensional PID
projections for various gates on E. The proton and deuteron peaks are absolutely
separated at low energy, while the peak-to-valley ratio is typically 7:1 in the

PID

 

300 I I I I I I T I I I I I I I r I I

200 —

 

100 _

 

 

 

 

Figure 3.1. Two-dimensional density plot of the PID function vs. the
particle energy. The z-axis has a logarithmic scale.

47

 

l I I

 

 

 

 

 

 

 

 

 

 

 

103 H E=25—30 MeV
102 A {I
101
Pidh # All I l 1 ll 1 l l l l 1
I T l I I I I I I I I I
103 Ease—55 MeV a
102 3He
101
I l I I I I I
10:3 E~105-110 MeV
102
1000 I l I ll 1 [12 011'] L

 

100
PID

Figure 3.2. One-dimensional projections of PID for thin cuts in particle
energy, showing particle identification resolution as a function of energy.

 

48

worst region, at a proton energy of about 100 MeV. Estimated deuteron

contamination in this energy region is of the order of 0.5%.
3.1.2 Silicon Detector Calibration

The AE components of our telescopes were relatively easy to calibrate
because of the well-known linearity of silicon detectors. A precision capacitor
and pulser system was calibrated with alpha particles of known energy from the

following radioactive isotopes stopping in one of our silicon detectors:

228Th Ea=5.423 MeV.
224Ra Ea=5.686 MeV.
22% Ea=6.288 MeV.
216Po Ea=6.779 MeV.
2123i Ea=8.784 MeV.
241Am Ea=5.486 MeV.

The pulser calibration resulted in a proportional relationship between voltage on
the capacitor (equivalent to charge injected into the preamplifier) and energy
deposited in the silicon detector. This allowed us to "dial in" any AE signal with
the pulser system.

Charge was injected into each silicon preamplifier with the pulser at
several settings AEI, and the corresponding peaks in the AE ADC spectrum at
channels Ni provided a calibration

AE(N) = b + m - N. (3.2)

for each detector.
3.1.3 CsI(Tl) Detector Calibration

A (CH2)n target of about 20m thickness was bombarded by a particles of
160, 120, and 100 MeV kinetic energy. The CsI(Tl) detectors were calibrated by

49

measuring monoenergetic particles emitted from the following elastic, inelastic,

and transfer reactions:

p(4He,p)4He (3.3)
12C(4He,d)14N" E"=8.96 MeV (3.4)
12C(4He,t)13N* E”=o.o, 3.51, 6.89 MeV (3.5)
12C(4He,3He)13c* E*=o.o, 3.86 MeV (3.6)
12C(4He,4He)12c* E*=o.o, 4.44 MeV (3.7)

For each detector angle and particle species, two-body kinematics were
used to calculate the kinetic energy associated with each peak in the energy
spectrum. Energy loss in the Ta foil and AE detector was accounted for. Because
of the finite solid angle of each AE-E telescope, the widths of calibration peaks are
greater than the intrinsic energy resolution of the detectors. In all cases, the
energy was calculated for a particle hitting the center of the detector, and the
peak centroid was extracted to establish the energy calibration.

Raw energy spectra of a typical detector for a calibration run with Ea=120
MeV are shown in Figure 3.3. Clear peaks for all light particle species are
observed. In Figure 3.4, we show the calibrated E+AE spectra for protons and
alpha particles. The widths of the calibration peaks in both cases are dominated
by kinematic broadening, as indicated by the energies of particles entering the
detector at the most forward and backward edges. The intrinsic energy
resolution of the detector is of the order of 1%.

Previous energy calibrations of the CsI(Tl) detectors with a larger set of
calibration points than was available to us have demonstrated a highly linear
response for light charged particles [Gong 91c]. Therefore, we assumed linearity

in our energy calibrations and observed no deviations from this assumption.

50

CH2(a,X) Ea=120 MeV

 

   

 

I T I I I I r‘fi‘rlr r T U 1 I V I I fl T rfi I I I I I

 

 

 

 

 

 

0 200 400 600 800 1000
E (channels)

Figure 33. Raw energy spectra measured in a typical detector for light
particles emitted when 120 MeV alpha particles bombarded a

polypropylene target.

 

 

 

Counts

600

a+CH2

Ea=120 MeV

51

<9>=30.73°

 

 

 

 

 

 

I I T I

f l I I I I I l I I I T

I I I I

 

     

 

 

 

 

 

 

400 - —
—><——
200 '_ FWHM=1.5 MeV _
M Ep(am)—Ep(9m)=1.5 MeV q
OW?“~#J‘IIIIEII ::I 4
600 L Ea=108.2 MeV _‘
400 - -
I
- F'WHM=1.7 MeV .
- <— «I
~ + Ea(6min)-Ea(9mu)1
200 f‘ =o.9 MeV 1
L
0'....1....1L... WULIWI
4O 60 BO 100 120 140 160
E+AE (MeV)

Figure 3.4. Calibrated proton and alpha energy spectra from 120 MeV
a+CH2 reactions. The widths of the measured peaks are dominated by
kinematic broadening, indicating an intrinsic energy resolution on the
order of 1%.

52

The determination of the excited states which were populated in the direct
reactions (3.4), (3.5), and (3.6) required interpolation. Proton and alpha
calibrations were straight-forward; only the well-known lowest excited state of
12C (E*=4.44 MeV) was excited. The previous detector calibrations by Gong
indicated that energy calibrations of the other light particles (d,t,3He) would lie
between the proton and alpha calibrations. With this assumption, it became clear
which states were populated. An example is shown in Figure 3.5. The energy of
a deuteron emitted from the direct reaction 12C(4He,d)14N* is calculated as a
function of lab angle, assuming the excitation of selected low lying states in 14N.
Also indicated is the measured deuteron energy in our detectors, using the
proton and triton calibrations as an approximation. With the knowledge that the
deuteron calibration lies between the proton and triton calibrations, it is clear
that the population of the E‘=8.96 MeV state in 14N is consistent with our
observations. Previous reports of the strong dominance of this state in this
reaction [Ajze 86, Harv 62] confirmed this assumption.

Calibration curves for a typical detector are shown in Figure 3.6. In the
most backward detectors, for which deuteron peaks were not available, we
averaged the proton and triton calibrations to provide the deuteron calibration.
This procedure was performed also for the forward detectors, and calibrations

were in good agreement with observed peaks.
3.1.3 Slewing Correction

The construction of a correlation function or any other two-particle
observable from experimental data requires the separation of real coincident
pairs from so-called "random" coincidences, which contain particles emitted from
two separate nuclear collisions. In our measurement, we recorded the time that
each hodoscope detector element fired relative to the (downscaled) RF time from
the cyclotron, Ti-TRF. In principle, then, if detector i and detectorj fire in an

53

 

105. T T r r l ' ' ' . I
100 3' Ea=120 MeV ;
95’ ‘
90:
85;

80

 

 

All
I

\
\K \\
0 Using proton calibration

Ed (MeV)

85 .— X Using triton calibration T
Ea= 100 MeV j

b
80;- 1
h d

75

B. j

70 g

E 8.96 M v E
65 . .

E>°<XX WXX mx 3“

- L . . 1 l I . ?‘X.
60

30 35 40
9(deg)

 

 

 

 

Figure 3.5 Lines indicate calculated deuteron energies from a(12C,d)14N’
for Ea=120 and 100 MeV . Measured deuteron energies assuming a
deuteron energy calibration identical to the proton and triton calibrations
are shown as circles and x's, respectively.

Energy (MeV)

54

 

150

I I I I

100

01
O

I I I I l I I I I I

 

  

 

150

100

I I I I I I I I I

50

I I I l

 

l J L J l l l

L

In /
«II- /

 

l l l l

   
   

l L l l

q

 

 

0 500 1000

1500 500

Channel

1000

1500

Figure 3.6. Energy calibration curves for a typical CsI(Tl) detector from

the 56-element hodoscope.

55

 

TlllIllllIlerlTlllIllllIrll

l 111

Uncorrected

H
0
4s
1 l lllllll

H
O
03
l 1 “MI

H
O

N
1 1 uqu

l

 

 

 

 

 

IIIIIII%%++':IIeI+IIIJIIIW
I I I I I I 3
Corrected I d
I 1
I
I ..
I ..
Random Real Random.
I
I
I
I
I
I
lLlLlllLLLLlIlLlanlll‘ll‘ll

 

—300 -200 -100 0 100 200 300
Relative Time (ns)

Figure 3.7. Relative timing spectrum for two particles recorded in
coincidence in the 56-element hodoscope. The upper and lower panels
show the spectrum before and after slewing correction, respectively.
Also shown are the relative timing regions defining a "real" and a
"random" coincidence in the correlation analysis.

 

700

600

500

TI'EW (ns)

IVIITlIWIIIII.TIITII1
\-°. .‘ I ..

   

400

IIII

~

300

 

L11111111111111111111'111111111

-

-

I.

' I I I
J J 1 1 1 1 1 l L L 1 1 1 1 1 i
I . I I r I I I I I T T I I I I r

I- .'

I-

I. .

i- R

500 — "

- -,- .e°:.'-~-. - .. .

L :. .9. \.'.4. u$:“. .L. ..'.‘.'“O l‘ . .I.!...‘ .

b . o-.'-o. ‘-
-1.""‘.1f'3‘.p't- . "'. . ._ . . _

'- ?a'ii'ffE'-i?‘dv\‘i'5tsz?4cfv“~’.'. ‘-'b.>:’-"v'v;"-.'= 'n-t‘shl'an'.‘ '

   

400 _..3-.- .4 , .

v »

 
  

c .
.41.

300

Tcor (ns)

-l.

200

100

1LlllllJlllllll’llillLllllllL

VIIIIIITIlIljrrIIII

  

 

l 1 4

   

0 50 100 150
Eprot (MeV)

Figure 3.8. The time when a detector fired relative to the cyclotron RF

signal vs. the energy is shown for protons measured in a typical AE-E
telescope. The raw spectrum (top panel) shows the effects of slewing or
time-walk. Slewing effects are absent in the corrected timing signal
(bottom panel) Different bands in both spectra reflect the RF structure of
the cyclotron beam.

 

 

57

event, the variable I (Ti-TRp)-(TI-TR1:) | = l TI-TI I should indicate whether the pair
constitutes a real or a random coincidence. The upper panel of Figure 3.7 is a
histogram of the variable Ti-TI, summed over all particle types and all detector
pairs (i,j). A broad smear in the relative timing spectrum is observed, and no
obvious distinction between real and random coincidences is visible. The cause
of such a smear becomes apparent when we examine a spectrum of the time vs.
the energy of the particle, shown for protons measured in a typical detector in
the top panel of Figure 3.8. There, it is seen that no unique relationship exists
between a detector's firing time and the RF signal. The relatively slow risetime of
the CsI(Tl) detector and the use of leading edge discriminators to construct the
logic signal for the hodoscope detectors gives rise to a large energy-dependent
"walk" or "slewing" effect in the timing signal.

In order to obtain a clean measure of the relative time signals between
coincident particles, the energy dependence of the time signal must be corrected
for. Moreover, since the slewing curve shows a dependence on particle type, this
correction must be performed separately for each particle type.

The detector timing signal was determined only by the EL signal, which
was generated by a leading edge discriminator (LED) (see Section 2.3.1). We
modeled the slewing effect by assuming a Gaussian-like leading edge from the
fast output of the shaping amplifier. If we may assume a proportional
relationship between the pulse height and the energy of the particle, then the fast
output as a function of time is

V(t) = m - E - e‘(*"*0)2/12 (3.8)

where t is a characteristic rise time which depends on the particle type, E is the
energy of the particle, m is the energy calibration constant, and to may be

considered the "true" firing time of the detector. The LED fires when the signal

58

reaches the threshold voltage Vth=m - Eu... Therefore, the slewing-corrected

firing time to may be written in terms of the recorded firing time t as

 

to = t - rJln(E) — math). (3.10)

A spectrum of to vs. E for protons in one detector is shown for protons in
the lower panel of Figure 3.8, where it is seen that the effect of slewing has been
removed. The threshold energies used in Equation (3.10) were determined for

each detector from the energy spectra.

Table 3.1. Rise times used to correct for slewing effect for light particles

 

 

 

 

 

 

Particle Type 1 (ns)
p 173.8
d 181.3
t 185.6
3He 111.6
a 94.5

 

 

 

 

It is important to adjust 1: for each particle type. The characteristic rise
times 1 were determined by trial and error and are given in Table 3.1. .

After correcting for slewing and adjusting the zero-time offset of each
detector relative to the others, relative timing spectra were constructed. The
bottom panel of Figure 3.7 shows such the spectrum when all detector pairs and
all particle species are summed over. The cyclotron RF structure of the Ar beam
is apparent through random coincidences between particles from different beam
bursts. Clearly, all real coincidences have both particles emitted in the same RF
burst. However, since our time resolution was not perfect, we included pairs
with a relative timing difference of up to two RF bursts in our definition of a

"real" coincidence.

59

Four representative "random" peaks are indicated in the figure. Particle
pairs in these peaks were used to correct the correlation function for the effects of
random coincidences (see Chapter 5). It is clear from the relative sizes of the real
and random peaks in Figure 3.7, however, that the effect of randoms will be
small. In the construction of coincidence spectra, we use the schematic

algorithm:

"True" = "Real" - (5 / 4) - "Random" (3.11)
3.2 41: Detector Array

Particle identification and energy calibration for the detector elements of
the 4n Array proceeded through what has become the "official" 41: algorithm
[Wils 91]. The fast-slow map is transformed through a number of operations
onto a standard pallet in which all detectors appear similar. Then, PID gates and
energy calibrations are performed.

In the first step, the neutral and punch-in lines (see Section'2.2.1 and
Figure 2.6) are identified for each detector. The position at which they intersect
corresponds to zero light output from both the fast and the slow plastic.
Therefore, this fast-slow value is defined as the origin of the new map by
subtracting its fast-slow value from every point; this corrects for offsets in the
FERA QDCs.

Next, the coupling between fast and slow signals that allow finite slope of
the punch-in and neutral lines was corrected for. For a given slow-value, the
fast-value of the neutral line was subtracted from every point. This operation
transformed the neutral line into the x-axis. In a similar fashion, the punch-in
line was transformed into the y-axis by subtracting the slow-value associated

with the punch-in line from every point according to its fast-value.

60

Multiplicative factors further transform the new fast and slow variables in
order to match the spectrum to a calibrated template [Cebr 90]. This template
contains gates for particle identification, and every point on the template has an
associated particle energy.

Therefore, six numbers are associated with each detector: the fast and slow
positions of the QDC offset point, the slopes of the punch-in and neutral lines,
and the multiplicative factors for the fast and slow variables. The "standard 4::
physics routine" uses these numbers to transform the fast and slow values for a
particle measured in a detector. The position of the transformed point relative to
the calibrated pallet indicates particle type and energy. Energy calibrations
obtained by this gain-matching procedure are estimated to be good to about 10%.
3.3 Physics Tapes

To decrease the processing time involved with physical analyses, we
created so-called "physics tapes," which contain recorded events in terms of the
calibrated data. Although in principle the use of physics tapes makes no
difference to a physical analysis, in practice the increase in speed encourages the
scientist to optimize the analysis and sample the data in more ways.

Simple histogramming of calibrated data from "raw," or unprocessed,
tapes required about 24 cpu hours per 2.2 GB (exabyte) tape on a VAX model 76.
Such a procedure also required almost all of the 32 MB of memory, rendering the
machine virtually useless for other purposes.

Such a procedure using our physics tapes required less than 2 cpu minutes
and less than 4 MB on the same machine. However, limited tape I/O speed
caused this process to take about 2 hours of real time, with a low cpu load.
Construction of correlation functions with impact parameter calculations did not
increase the real time of the analysis, only the cpu load.

The details of our raw and physics tape formats are given in Appendix A.

Chapter 4 - Impact Parameter Selection

The two-proton correlation function provides a measure of the space-time
extent of the proton source created in a nuclear collision. Dynamical calculations
with a Boltzmann-Uehling-Uhlenbeck model indicated that the source would
evolve differently in central and peripheral collisions, and that the correlation
function could reveal this difference [Gong 91a, ”Baue 92].

In order to investigate in detail the behaviour of the proton source
separately for central and peripheral collisions, cuts on global observables were
employed. Two-proton coincidences were measured in the hodoscope to
construct the correlation function. In order to avoid the effects of "self-cut"
biasing the impact parameter selected correlation functions, the "global"
obServables used were constructed from particles measured in the 41: Array only.

Below, we define the global observables used in this analysis. Next, we
discuss the average reduced impact parameter which provides a common scale
for the comparison of different global observables. Finally, we discuss two
possible methods to compare impact parameter cut data to theoretical
predictions.

4.1 Variables Used for Centrallty Selection

We studied three commonly used global observables, each of which are
assumed to provide a measure of the violence of the collision. To what degree
this assumption is true for each is examined in the next section. Here, we define

and present spectra of each observable for each of three trigger conditions (see

61

62

Section 2.3): inclusive (4:: multiplicity 2 1), singles (hodoscope multiplicity 2 1),
and coincidence (hodoscope multiplicity 2 2). ’

The charged particle multiplicity, NC, is defined as the number of 4n
detector elements that recorded a valid particle in an event. A "valid" particle is
defined as at least one charged particle in the detector; unidentified charged
particles that stopped in the fast plastic and double hits were counted as one
particle.

The charged particle multiplicity spectrum is shown for three different
event triggers in Figure 4.1. Open circles, squares, and filled circles show
dP/ ch for events measured with the inclusive, singles, and coincidence
triggers, respectively. Presumably, a larger multiplicity indicates a more central
event. Therefore, we see from Figure 4.1 that even without explicit centrality
cuts, the requirement of measuring two protons in the hodoscope already

preferentially selects more central events.

The second global observable we studied was the midrapidity charge, Zy,
defined as:

Nc
zy =Zzi-em —0.75ytarg>-e<o.75ypmj “Y0 . (4.1)
i=1
with
9 _ 1 if x > 0 (4 2)
(x) _ 0 if x S 0 °

In Equation 4.1, and ytarg, yme, and yi are respectively the rapidity of the
target, projectile, and ith charged particle in the center-of-momentum frame of
the system.

63

 

 

 

 

 

__ I T IfI I I I I l I I I I l I I I I I I I _
c .1
”Oo .
‘1 0 - QB _
10 g DEISOOB B E
: Do 00 B E
- 0 0o B _
-2 ID 006 I
10 E 08 —§
: 05 E
- 1
_ OB _
10"3 :— O Inclusive 03 ‘E’
E D Np(hodo)21 0. E
r 0 Np(hodo).2_2 Cl. -
10-4 E— oD.e 5
E QC] ..5
.. Cl -

10—5 1 1 1 1 I 1 1 1 1 I 1 1 1 1 I 1 1 1 1 I 1 '1 I? 1
0 5 10 15 20 25

NC

Figure 4.1. Charged particle multiplicity distributions measured in the 411
Array under 3 separate triggering conditions. Open circles, open
squares, and filled circles indicate dP/ ch for inclusive, hodoscope
singles, and hodoscope coincidence triggers, respectively.

64

Figure 4.2 shows the midrapidity charge distributions dP/ dZy for our
three triggering conditions. Qualitatively, a trend towards more central events
(as characterized by large values of Zy) is again observed as the trigger varies

from inclusive (open circles), to coincidence (filled circles).

Finally, centrality cuts were based on the total transverse energy of the

event, defined as:
N

13t = in -sin2 9i, (4.3)
i=1
where Bi and 91 are respectively the energy and laboratory angle of the 1th
particle measured in the event.

The spectra dP/dEt are shown in Figure 4.3 for our three trigger
conditions. Once again, the requirement of measuring two protons in the
hodoscope appears to bias our event distribution toward more violent collisions.

The distributions of all three of the global observables display similar
trends as the trigger condition varies. The assumption that a large value of any
one signals a central collision leads to the reasonable conclusion that measuring
two protons in our hodoscope at 38° in the lab frame biases the event sample
toward more central collisions.

If the value of one of our global observables indicates a central collision,
what is the likelihood that another will do likewise? This issue is addressed in
Figure 4.4, in which the correlations between Et, Zy, and NC are shown. On
average, the values of the other two global observables increase when the value
of any one increases. If any global observable is a measure of the centrality, they

all are.

65

 

 

 

 

 

 

 

 

_1111I1111I1111I11111111111_
IO .
_ O _
10"1 :DEQIBQQB —:
: O a :
_e o as -
- OO [1. -
.— O D. .1
I- O D. _
_2 0 1:10
10 : O C]. ——:
I E 0 :10 E
N - o -
"U _ OIj "
\ r- OD. ..
0.. -3 . [3.
T; 10 = O Incluswe 0 "E
5 00° 5
: 1:1 Np(hodo).2_1 OD. ._.
4 - 0 Np(hodo)22 01:1. -
10‘ - . ——_~
: O :
: D O :
: 0'3 o :
.. . _
10—5 1111I1111I1L11I.1L11I111Q$'1111
0 5 10 15 20 25 30
23'

Figure 4.2. Midrapidity charge distributions measured in the 4:: Array
are shown for the same triggering conditions as for Figure 4.1.

 

 

 

   

 

 

 

 

I I l I I I l I I 7 I l I I I I I I I _
10"2 —§
10_3 3- .3
C I
_. u 0. e
-4 , o s
10 _=_- O Incluswe CI: 5 —:'
E D N (hodo)-Z1 00 Ch ~. 3
.. P Q) CI: 0. _
- O Np(hodo)22 Cb CI: . -
10-5 E:— Q) 0%.. =
E 00 .~ 5
- 1 L l 1 l L L l i l l 1 l l l 1 J 9 13% I I!
0 100 200 300 400 500

E1:

Figure 4.3. Total transverse energy distributions measured in the 41:
Array for the same triggering conditions as for Figures 4.1 and 4.2.

67

  
 
  

lllllllllllllllll-

 

-1-
-1—

[III IIIIIIIII

llll

   

30

llllIlI
I

llLl

   

N.‘ 20

 

l 1

ft
Llllllllllllllll

10

  

 

0 5 10 15 20 0 200 400 600

Figure 4.4. Logarithmic two-dimensional probability distributions are
shown for pairs of global observables. The values of the three
observables shown, the charged particle multiplicity (NC), the mid-
rapidity charge (Zy), and the transverse energy (Et) are highly correlated.
If one observable is a good measure of event centrality, they all are.

68

4.2 Construction of the Reduced Impact Parameter Scale

In this and the following section, we follow closely the technique
described by Phair [Phai 92].

A uniform sc'ale indicating the event centrality may be based on any
global observable, which we generically label X. The validity of this scale as an
average impact parameter scale is based on three assumptions, which we list
below. The first two assumptions are quite feasible, while the last is only an
approximation for a non-ideal detector system.

We assume a strictly monotonic relationship between the average value of
X and the impact parameter of the event. Then the relationship between the
probability of measuring an event with impact parameter b and the probability

of measuring the observable with the value X may be written:

dP dP
—-db=—-dX. 4.4
db dX ( )

If we assume that the largest values of X are associated with the most

central collisions, we obtain:

,_ ”dP ,
-db —de—X7-dx. . (4.5)

 

Ib dP
0 db’
Finally, we assume that the detector system has unit efficiency, regardless
of impact parameter. Therefore, with an inclusive event trigger, the probability
of observing an event with impact parameter b follows the geometric distribution
dP/ db ~ b. Then, defining the average reduced impact parameter b, we have:

* _b(X)_ °°dP ,
b(X)_It:ax2_\/Ix37dx. (4.6)

 

69

Thus the centrality of an event may be quantified by the average reduced
impact parameter, which takes on a value of zero for the most central events and
unity for the most peripheral.

An example of such a scale based on the total transverse energy (X=Et) is
shown in Figure 4.5. The inclusive distribution dP/ dEt is shown in the upper
panel, while the bottom panel shows 6(Et), calculated through Equation 4.6.

Through such scales, distributions of global variables may be transformed
into average reduced impact parameter distributions through

dP _dP c113

d13(X) - d—X a? (4'7)

When cuts are made on X, extraction of the impact parameter distribution
becomes more involved. We discuss this in Section 4.4.2.

Figure 4.6 shows the distributions dP / db(Et) corresponding to our three
event triggers. These curves correspond to those shown in Figure 4.3, and
illustrate more quantitatively the shift toward centrality as we require the
observation of protons in the 56-element hodoscope.

4.3 Comparison of Variables

The average reduced impact parameter, discussed in the previous section,
is an approximate measure of the event centrality, based on the value of some .
global observable X. Clearly, however, there exist fluctuations in the relationship
between impact parameter and any global observable on an event-by event basis.
The relationship 6(X) describes only the average behaviour of this relationship.
In this section, we discuss how to estimate the magnitude of the fluctuations in
the relationship between the "true" reduced impact parameter br=b /bmax, and a
global observable X

The utility of such an exercise is twofold. It determines which global

observable provides the most precise measure of event centrality. Also, as we

7O

discuss in Section 4.4.2, the extracted widths may be convoluted with the average
reduced impact parameter distributions and used as input for dynamical models.

In Figure 4.7 the reduced impact parameter distribution dP/db(Et) is
shown for narrow cuts on b(X), where X=Nc (dashed lines) and Zy (dot-dashed
lines) for central and mid-central values of b. Also shown is the distribution
with a narrow double cut on both b(Nc) and 13(2),) (solid lines). All probability
distributions are normalized to unit area for ease of comparison.

For the midcentral cut b(X)=0.35-0.4 (bottom panel), the width of the
distribution dP/ d b(Et) is roughly the same for X=Nc and X=Zy. The fact that
this width does not significantly decrease when a double cut is performed
indicates that the extracted width (0 z 0.15) approaches the magnitude of the
intrinsic fluctuations between the impact parameter and Ft.

For narrow cuts centered at very small values of the average reduced
impact parameter (top panel), an edge effect-- due to the fact that 13 takes on only

positive values— distorts the distribution, moving its centroid to larger values of

A

b.

Distributions similar to those shown in Figure 4.7 are shown in Figures 4.8
and 4.9 for dP/db(Nc) and dP/db(Zy), respectively. Somewhat smaller values

of o(b) were extracted from the distributions of dP/db(Et), indicating that E is
the best measure of centrality. This is consistent with a similar study of other
nuclear systems [Phai 92].
4.4 Impact Parameter DIstrlbutlons as Inputs to Models

Dynamic models of nuclear collisions simulate the evolution of a nuclear
system as it undergoes a collision. This evolution can depend strongly on impact
parameter, and experimental cuts on global observables may be used to test the

details of theoretical predictions.

71

36Ar+458c; E/A=80MeV

_1IIITITTTIITIIIIIIIIITIII]Ill]

 

10

k

10‘2 '

10‘3
.10‘

10"5

dP/dEt (MeV‘l)

10‘6

‘W
I
All!

(18

 

(16

b(Et)

 

(l4

 

(12

ll'lllllllJllllllllJJlLU

 

 

IFIIIIIIT [Ill [[11

 

0.01111I1111l1111 .. llllUll-T
100 -200 300 400 500 600

Et (MeV)

C

Figure 4.5. The inclusive transverse energy spectrum dP/ dB: is shown in
the upper panel. The, lower panel shows the average reduced impact

parameter scale b(E,) based on this distribution.

72

 

   

 

 

 

2,5 I I f I I I I I r I I I I I l I I I I I I I I ‘
- a
- — 2 protons in hodoscope "c:
2 0 ;_ --- 1 proton in hodoscope 4313
- - - Minimum bias trigger , ’ - Cl:
C x ’ Z 8
/
- , .
“U .. -
\ - .,
IL ‘ ,
"d - .
1.0 — '—
: x :
- °\ _
:— .\. .—
0.5 r:
o O l J l l I l l l l l l L L l L l l l l l l l l
0 0 0.2 0.4 A 0 6 0 8 1 0
b

Figure 4.6. Average reduced impact parameter distributions dP/ d603,)
for the three event triggers studied. Curves in this figure correspond to
those shown in Figure 43.

 

TTITIIIII

 

 

 

dP/db(Et)

 

04

 

. A 0.6
b(Et)

 

900—86-HSN

llillilllllllJlllllllllL

 

0.8

-
db

llllJJlllllLlllllllllJJ

 

H 4
0

Figure 4.7. Average reduced impact parameter distributions dP / db(E.)

for tight centrality cuts on b(X), where X=Nc and Zy. A double-cut on
both observables does not narrow the distribution significantly for the
mid-central cut, indicating that the measured width is close to the scale
of the intrinsic fluctuations between impact'parameter and transverse

energy.

74

 

 

 

 

 

5 '- I I I I T I fi] I I: I I I I I r T T T _,
Z b(X) = 0.05-0.15 1
4 :— - ' -' X=Et ;
: — —- x=zy :
; —- Double cut I
3 - ./-~ -—
- ./ / ~ '\ \.\ j
2 _ -;/ \\\ ‘
.. j \x j
I \ :
\ —
’3 1 \\ \ \ _
5 'x \ \ :
.0 ~ ‘ ‘ i
13 0 - .
E : b(X) => o.35—o.4 :
-u 4 '_ .1
3 E— ‘3
I 1
2 :— "T.
I 1
1 .— i
O - .
0.0 0.2 0.4 0.6 0.8 1.0

Em.)

Figure 4.8. Average reduced impact parameter distributions dP / db(Nc)
for tight centrality cuts on b(X), where X=Et and Zy.

A

75

 

 

5 — I I T I I I filr I l: I I I I I I j I I
: b(X) = 0.05—0.15
4 7.. — - X=Nc
: — — ' X=Et
I —— Double cut
3 .—
: /” \
2 l— / \
I / \
- / \
A 1 h .’ \\
h "I \_
N \_\
Z 1 1 1 I 1 l L I 1 1 1 \\
U 0 - I I I I j I if I I AI I I
E C b(X) = o.35—o.4
“U 4 L.
3 E-
2 .— 12”.,
' // §\
:— ’7 o
1 L. / \.\‘x\
- .\
.— \\
t "\- .,
O l I I I L L I I I I I I I J I I I ~

 

llllIllllIllllIlllJIllll

 

 

0.0 0.2 - 04,. 0.6
b(Zy)

, lllllI'llllIllllIllllLllll

Figure 4.9. Average reduced impact parameter distributions dP I db(Zy)

for tight centrality cuts on b(X), where X=Et and NC.

76

Such tests, are complicated by the fact that the very quantity that describes
the initial state of the system- the impact parameter distribution- is not
measured directly. Therefore, before judging the predictive power of a model,
one should be as confident as possible that any test is free of the effects of impact
parameter selection.

Since an unambiguous method of comparing impact parameter selected
data to theoretical predictions does not exist, we discuss below two methods of
selecting similar ranges of centrality for data and model events. A transport
model based on the Boltzmann-Uehling-Uhlenbeck equation, to which we

compare our correlation results in Chapter 6, is used as an example.
4.4.1 The Method of Equivalent Cuts

We want to compare physical quantities constructed from experimental
and model events that represent similar regions of centrality. If we cut on Et in
our data, one obvious method of selecting model events would be to cut on the Et
of the event as predicted by the model. Such a procedure assumes that the model
correctly describes the relationship between Et and the impact parameter. We
discuss below a procedure to select model events based on a global observable
such as Et.

In Figure 4.10, we compare the total transverse energy spectrum dP/ dEt
measured for a minimum bias trigger with the impact-parameter-averaged
prediction of the BUU model. To generate dP/ dEt for the BUU, a geometric
impact parameter weighting was used (dP~b-db). For the experimental data, the
sum in Equation 4.3 runs over all charged particles and fragments detected in the
41: Array, while for the BUU results, the index is over all emitted nucleons, after
passing through the detector acceptance. The Et distribution predicted by the
BUU is shown when the sum in Equation 4.3 runs only over the protons (dot-

dashed curve), and when it runs over protons and neutrons (dashed curve).

77

Although free neutrons are not detected in our experiment, neutrons bound in
complex fragments do contribute to the experimental Et spectrum.

The BUU fails to reproduce the observed Et spectrum. As mentioned in
Chapter 1, the BUU is a theory that describes the time evolution of the single
particle phase space distribution function and contains no mechanism for
complex fragment formation. Therefore, distributions of global observables
sensitive to contributions from complex fragments, such as E, NC, or Zy, will not
be reproduced by the BUU.

From the inclusive Et distributions shown in Figure 4.10, it is clear that the
use of identical Et cuts on the experimental data and on the BUU events would be
rather meaningless. One may, however, define "equivalent" Et cuts for BUU
events and for measured events. We propose a procedure which assumes that,
on average, Et decreases monotonically with increasing impact parameter. The
validity of this assumption for the BUU model is illustrated in Figure 4.11, where
the contribution to the Et spectrum from various impact parameter ranges is
shown

Our method then characterizes both data and BUU events according to
where the E for the event falls in the respective Et distribution. For example, one
could assume that the top 10% of the experimental Et distribution is produced by
the same impact parameters as the top 10% of the theoretical Et distribution. In
practice, this implies that if dP/ dEt and dP'/ dEt are the total transverse energy

probability distributions for the BUU and for the data, respectively, then a cut in

the data at E corresponds to a cut in the BUU at Flt, where the relationship
between Et and fit is given by
dP’ dP

TdEh— = "dE'.—. 4.
IEt t dE; E: t dE; ( 3)

78

 

 

 

 

10—1 1 l _a
10-2 k — Data . —:
A - - BUU mth neutrons g
T ,\ -'- BUU without neutrons:
% —3 \ "~
3 10 \ g g \ .
1 °:\c\
E3 10-4 "\ \ \ \
E \ u
“U ’ \.‘ I. \/\o\
10‘5 III \ \
II. \. \l v‘
- \
I
10‘6 I
I l
I I L I I I I L I I ' I J I_I
O 200 400 600 800
Et (MeV)

Figure 4.10. Total transverse energy spectrum dP/dEt measured under a
minimum bias trigger in the 41: Array is indicated by the solid line. Also
plotted is the prediction of the BUU model, after passing through the
detector acceptance of the 41: Array. Dashed and dot-dashed lines
indicate calculations in which the definition ofE. (Equation 4.3) includes
contributions from all emitted nucleons, and from protons only,
respectively. Relative normalization gives equal area for all spectra in
the region 15ng MeV. '

dP/dEt (MeV—1)

 

79

BUU with neutrons

 

IIIIIIII I IIIIIIII I IllllIII

IIIlI

 

 

 

800

Figure 4.11. Plotted are the contributions to the total Et spectrum from
various ranges of impact parameter as predicted by the BUU when the
contribution of neutrons are included. The black curve represents the
impact parameter averaged spectrum, while the colored curves show the
spectrum for small impact parameter intervals.

80

dP/dEt (MeV‘l)

0 100 200 300 400 500

Figure 4.12. Experimental and theoretical inclusive Et distributions are
shown in the upper and lower panels, respectively. "Equivalent" E. cuts
defining central and peripheral events are shown.

 

600

81

Figure 4.12 shows "equivalent" cuts in the experimental and theoretical
inclusive Et distributions for central and peripheral events.

In summary, the procedure for comparing data to model predictions via
the methods of equivalent cuts is as follows. First, one generates an ensemble of
model events with a geometrical weighting according to impact parameter.

Then, experimental events selected according to the Et cut (E:0W ,Et‘igh) are

compared to model events selected from the ensemble according to

(E:°W,E:‘igh), where the equivalent cuts are determined by Equation 4.8.
4.4.2 The True Reduced Impact Parameter Distribution

A different event-weighting scheme makes use of the reduced impact
parameter scale developed in Section 4.2. In this section, we attempt to extract a
realistic "true" reduced impact parameter scale from the data which we may
input directly into a dynamical model.

Equation 4.6 defines a one-to-one relationship between the average
reduced impact parameter b and the transverse energy Et. Therefore, one may
associate a sharp region in b with any sharp region in Et defined by centrality
cuts. An example is shown in Figure 4.13, where the upper panel shows the
experimental transverse energy distribution when two protons are measured in
the hodoscope. The shaded regions indicate the values of Et that define our
"central" and "peripheral" cuts. The center panel shows the average reduced
impact parameter distribution that corresponds to this spectrum, calculated
through Equation 4.7. Sharp cuts in E correspond to sharp cuts in b, as
indicated by the shaded regions in the center panel.

However, it is clear from our discussions in Section 4.3 that while we may
postulate a one-to—one relationship between Et and the average value of the
impact parameter, event-by-event fluctuations will lead to a broad E; spectrum

even for a single impact parameter. Therefore, association of the b distributions

_1)
H
O

dP/dB dP/dEt (MeV

82

Et (MeV)
-2 0 100 200 300 400 500 60

O

LOO-SB-DSN

4 - 2 protons in hodoscope
-- 1 proton in hodoscope

dP/db,

 

000-5100)

.0 0.2 mm, bros 0.8 1.0

Figure 4.13. The top panel shows the total transverse energy distribution
dP/ dB. for events which have two protons in the hodoscope. The cuts
we define as "central" (high Et) and "peripheral" (low E9 are indicated by
the shaded regions . Shown in the center panel are the average reduced
impact parameter distribution at!" / d5(E,) corresponding to the total
transverse energy distribution of the top panel. The bottom panel shows
the reduced impact parameter distributions dP/ dbr corresponding to the
centrality cuts shown in the upper two panels. Solid lines indicate
reduced impact parameter distributions for two-proton coincidence
events, while dot-dashed lines represent similar distributions for singles
proton events.

83

shown in the center panel of Figure 4.13 with the impact parameter distribution
is unrealistic.

To obtain a true reduced impact parameter distribution dP/ db, , it is
necessary to fold the effects of the finite widths 0(8) into the impact parameter
distribution; we use the widths measured in Section 4.3 for the 0(8). As an
ansatz, we take the following expression for the probability that an event with
global observable corresponding to average reduced impact parameter b was in

fact the product of a collision with true reduced impact parameter b,:
dPB (br) ~

“ 2 “ 2
db b,.e-<br-b> W”) -e(1-br)~e(br). (4.9)
r

 

For a given cut Elm" s E, S Ef‘a", the reduced impact parameter
distribution can be obtained by incorporating the extracted widths 0(8) as
follows:

(I? _ 6(Effin)dfi,.dP(B')-dpfir(br)
dbr 15:“) db’ dbr '

 

 

(4.10)

. The effect of this smearing is illustrated in the bottom panel of Figure 4.13.
The solid curves show the distributions of the true reduced impact parameter b,-
deduced from the cuts shown in the upper two panels. These distributions
represent our best estimate of the distributions of impact parameter for which
two protons were measured in the hodosc0pe. For reference, the dot-dashed
curves in the bottom panel correspond to b, distributions calculated in a similar
fashion for proton singles data; a slightly enhanced probability for larger impact
parameters is noted in the singles data.

The solid curves in the bottom panel of Figure 4.13 correspond to impact
parameter distributions for "central" and "peripheral" events which had two
protons in the hodoscope. Hence, these impact parameter distributions have

84

been selected by the application of double gating conditions, and they may not
be used as direct input for model calculations. We assume that in nature,
reactions are caused by a geometric weighting of impact parameters, and our
two-proton correlation functions are constructed from a select ensemble which
produce two protons in the hodoscope detectors. To correctly simulate this
experimental scenario in the construction of input impact parameter distributions
for dynamical models, only the centrality cut on 8(Et) should be applied; the
requirement of particles being detected in the hodoscope is implicit in the
extraction of the correlation function from BUU events (see Chapters 5 and 6).

In order to construct impact parameter distributions that may be used as
input to dynamical calculations, we construct distributions dP/ dbr for our two
centrality cuts from the inclusive Et distribution, which corresponds to a
geometric impact parameter weighting. The distribution of true impact
parameters sampled by these two cuts may then be obtained by inserting the
geometric distribution, dP / d5 = 25 / 5,3,“, into Equation 4.10. The upper panel of
Figure 4.14 shows the sharp cuts in average reduced impact parameter
corresponding to our central and peripheral definitions. The solid curves in the
lower panel of Figure 4.14 show the distributions of true reduced impact
parameter deduced from these cuts. These curves may be used to weight events
generated by a dynamical model such as the BUU.

Not all events generated by the dynamical model will produce a proton in
the hodoscope acceptance. Thus, the impact parameter distributions
corresponding to the events that contribute to the correlation function will differ
from the "input" impact parameter distributions shown in the lower panel of
Figure 4.14. We investigate this effect in Figure 4.15. The solid curves in the
figure show the distributions of impact parameters used to weight our BUU
calculations. (The distribution representing the central cut is truncated at very

85

dP/db

dP/db,

 

Figure 4.14. The upper panel shows sharp average reduced impact
parameter cuts corresponding to the definition of central and peripheral
events are indicated for a geometric distribution of impact parameter.
The lower panel shows reduced impact parameter distributions dP/db,
corresponding to the centrality cuts indicated above.

86

 

dP/dbr

 

 

 

 

5 - I I I I I I I I I I I I I I I I I I I I I I I _
C Inclusive (input to BUU) :
4 :___ - - - - 1p in hodoscope (exp) __:
Z ---------- 1p in hodoscope (BUU) :
- +
_ \ _
— \ O. _.
3 _ \ fl
_ , \ _
- \ -' -— '_ _
- , \ // \Q _
2 L.— ’ \ .: l X _
.. \ ’ Ex _
- , \ : ’ '. \ _
— 1 ’ '-. \ -
1 .. .- \
1 L I :7 :_ \ _:
l. ’ -" '- \
I :7 '. \ ‘
l- , .7 .o. \ "
C’ .7." \C
0 ’1 1 L 1 A L 1 I 1 1 1 1 1 L L I L 131-“ .
0.0 0 2 0 4 0.6 0 8 1 0
br

Figure 4.15. Comparison of experimentally reconstructed impact
parameter distributions to those used in the BUU calculations. Solid
curves show the impact parameter distribution used to weight BUU
events (same as bottom panel of Figure 4.14). This distribution is
modified by the additional requirement of proton emission towards the
hodoscope (dotted curves). The dashed curves show the experimental
reduced impact parameter distribution (see Equations 4.9 and 4.10) when
a proton is detected in the hodoscope (same as dotted-dashed curve in
Figure 4.13).

87

low impact parameters, since we did not run calculations for very small impact
parameters for which the geometric cross section is small.) The dotted curves
show the corresponding distributions with the additional requirement of proton
emission into the direction of the hodoscope. For the peripheral cut, this
requirement produces a shift towards smaller impact parameters. The
distributions shown by the dotted curves may be compared to those extracted
from our data (dashed curves; they are identical to the dotted-dashed curves in
Fig. 6). The overall agreement between these two distributions illustrates the
quality of the impact parameter selection and reconstruction procedure used in

the present analysis.

To summarize this section, we have two possible methods of comparing
the data to a model that requires an impact parameter distribution as an input.
One may use "equivalent" Et-cuts on model and data events, or one may extract
reduced impact parameter distributions corresponding to a given cut in a global

observable. We consider each in Chapter 6. Finally, we note that the method of
"equivalent" Et cuts is tantamount to defining a relation b"(E,BUU) for BUU

events as we have done for the data, and to apply the same cuts on 515,3 U”) and

on 5(Efo).

Chapter 5 - Two-Proton Correlation Functions

In this chapter, we discuss general aspects of the two-proton correlation
function. We discuss what the correlation function can tell us about sources
created in nuclear collisions and present the formalism within which correlation
functions may be predicted from model results.

Then, we move onto experimental issues. We discuss the construction of
the so-called background correlation spectrum. We re-analyze data from a
previous study to determine the difference in the correlation function using two
popular methods of constructing the background. Finally, we discuss
experimental resolution effects. Detector granularity and finite resolution may
distort the correlation function. These effects must be accounted for in
theoretical correlation functions. For the acceptance of the 56-element
hodoscope, resolution effects are shown to be largely negligible in the context of
our analysis.

5.1 Relation of the Correlation Function to the Spatlo-Temporal Structure of
the Source

The Koonin-Pratt formalism provides the theoretical framework in which
the two-particle correlation functions is usually understood [Koon 77, Prat 84,
Prat 87, Boal 90, Gong 91a]. Below, we present this framework, and then discuss
the particular case of the two-proton correlation function. Finally, we discuss the
sensitivity of the correlation function to the space-time structure of the proton

source with illustrative calculations based upon the Koonin-Pratt formalism.

88

89

5.1.1 Koonin-Pratt Formalism

The Koonin-Pratt formalism relates the two-particlecorrelation function to
the single-particle phase-space population density. The formalism has been
derived elsewhere using basic assumptions [Koon 77, Prat 84, Prat 87, Gong 91a].
We do not reproduce the derivation here, but discuss the case of correlations
between identical particles.

The formalism begins with an emission function g(r,p,t) (often referred to
as the ”source function”), which represents the probability that a particle with
momentum p will be emitted at position r and time t. Then, the two-particle

correlation function may be written as

Y(p,,p;)
1+R(P. )=C(P. )=
q q up.) - up.)
- Idarldtldarzdtzg(r1rP/23t1)g(rzrP/29t2)|¢(qrr1 _ r2 — t{:12 ' I’lz
Id3r1dtlg(rl 9%.“) . Idsrzdtzguzi'lé’tz) .

 

(5.1)

 

 

In Equation 5.1, q is the momentum of relative motion in the rest frame of
the pair, and P is the total momentum of the pair. Y(p) and Y(p1,p2) are the
single- and two-particle yields. The wavefunction of relative motion for the
particle pair, ¢(q,r) depends only on the relative phase space coordinates at the
time of emission of the second particle.

Several assumptions enter into the derivation of Equation 5.1. Firstly, it is
assumed that the particles are emitted independently; the second particle never
knew that the first one left. This implies that all two-particle correlations are due
to final-state interactions. This is only an approximation since conservation laws
place constraints on the independence of the particles [Lync 82]. Furthermore,
entrance channel effects [Zhu 91] or correlations in the source itself [Bern 85, Alm

93, Kund 93] may destroy the independence hypothesis.

90

A second assumption is that the two particles interact only with each other
and not with the residue nucleus or with other emitted particles. For pairs of
[particles with the same charge-to-mass ratio emitted well above the Coulomb
barrier, it is reasonable to neglect the residue, since to first order the Coulomb
field of the residue does not affect the relative coordinates of the particles.
However, three-body Coulomb trajectory calculations indicate that for
correlations between heavy fragments [Kim 91, Kim 92, Fox 93, Bowm 93], or
even for identical light particles emitted close to the Coulomb barrier [Eraz 91,
Gong 92], interactions with the Coulomb field of the residue may be important.
Furthermore, many-body Coulomb trajectory calculations [Glas 93] indicate that
many-body interactions among emitted heavy fragments are of similar

magnitude as two-body effects.

These two assumptions allow a clean separation in the sequence of events.
The source, described by g(r,p,t), emits two particles independently. Therefore,
the two-particle emission probability may be written as a simple product of the

single-particle emission probabilities.
rI(P1432) = II(Pl) ~ 1"1092). (52)

Once the second particle is emitted, the interaction between the particles,
represented by the relative wavefunction ¢(q,r) is "switched on." This leads to a
correlation which destroys the factorability of the final probabilities:

Y(p1,pz) at Y(pi) - Y(p2)- (5.3)

The final important assumption entering into Equation 5.1 is that the
source function g(r,p,t) does not vary greatly over a momentum region of the

order of l q I . This assumption is appropriate when applied to intermediate

91

energy heavy ion collisions, since typical momenta (hundreds of MeV/c) are
much larger than the values of relative momentum (S 50 MeV/c) typically of
interest in correlation studies. I

With this assumption, the need for explicit integration over the initial
momenta in Equation 5.1 is eliminated, as the momentum dependence in g may
be represented by the average momentum of the pair. Also, by the conservation
of momentum, the final and initial average momentum are equal, making any
knowledge of the details of the evolution from initial to final momentum

unnecessary.
5.1.2 The Two-Proton Wavefunction

The behaviour of the correlation function for a given source configuration
depends on the particle pair used, since the two-particle relative wavefunction
gives rise to the correlations in Equation 5.1. Here, we discuss briefly the two-
proton wavefunction and the physical effects that influence the two-proton
correlation function.

In Figure 5.1, the square of the two-proton relative wave function, M) l 2, is
shown as a function of the relative separation I rl and relative momentum I q I
for the cases r | l q and r.Lq. The relative wavefunction 0 is determined by three
physical effects: the strong nucleon-nucleon interaction, the Coulomb repulsion,
and the quantum mechanical requirement of wavefunction antisymmetrization.

To calculate 0, the full Coulomb wavefunction [Mess 76] was modified by
the contribution from the strong interaction. This modification was determined
by solving the Schrtidinger equation for the 2 =0 and 2 =1 partial waves with the
Coulomb and the Reid soft-core potential [Reid 68]. Then, the two-proton
wavefunction is determined by

11mm)? =w1-|‘¢(q,r)|2+w3-|3¢<q,r>|2, (5.4)

92

Two—Proton Wavefunction (1:0,1)

 

 

 

 

 

 

Figure 5.1. The square of the two-proton wavefunction as a function of
the relative proton separation in coordinate and momentum space. The
top and bottom panels show the wavefunction when q and r are parallel
and perpendicular, respectively.

93

where 1ti)(q,r) and 3¢(q,r) are the singlet and triplet spatial two-proton
wavefunctions, respectively. We follow the usual assumption that the proton
spins are distributed statistically, so the singlet and triplet weighting factors are
set to w1=1 / 4 and W3=3 / 4. (See, however, [Zhu 91], for a study of 3He breakup,
where proton spin correlations from the entrance channel result in a
nonstatistical spin distribution of the final state.)

The squared wavefunction for identical non-interacting particles goes as
|¢(q,r) I2 ~ 1icos(2q-r) (where the upper sign applies for bosons and the lower to
fermions) due to the quantum mechanical (anti)symmetrization requirement. In
this case, the correlation function is just the Fourier transform of the square of the

source distribution [Kopy 72, Zajc 92]:

1 + R(q) = 11 |{;(q)|2 = H: je2‘4"dr ~ {jgm + Jfr)g(R - indR}, (5.5)

where the total momentum dependence has been suppressed for clarity. For
spin-1 / 2 fermions, identical particle interference suppresses the correlation
function to 1 / 2 at l q I =0. The correlation function returns to unity at a relative
momentum qx~1 / Rx, for a source whose extension in the x-direction is about Rx
[Kopy 72]. In principle, then, a measurement of the full 3-dimensional behaviour
of 1+R would reveal the size and shape of the emitting source.

The Coulomb repulsion between the two protons also suppresses the
correlation function at low relative momentum. The strength of the suppression
is greater for small sources. For source functions whose extent is much below the
proton Bohr radius of 58 fm, this suppression is largely independent of the
relative orientation between q and r. Therefore, the Coulomb effect obscures
shape information carried by the quantum interference by reducing the number

of proton pairs at low relative momentum. Worse, the weak directional

94

dependence that the Coulomb force does contribute to the correlation function
opposes that of the quantum interference effects [Gong 91a]. With good statistics
and tight directional cuts, however, shape information can be recovered, as we
discuss below and in Chapter 7.

A near-resonance (corresponding to a phase shift of about 60°) in the
strong interaction between the protons provides additional sensitivity to the size
of the source. The attractive "2He resonance" causes a bump in the correlation
function at q=20 MeV/c. Because the resonance is s-wave, no directional
information is carried by the size of this bump. However, since the strength of
the resonance goes roughly as the fraction of proton pairs whose relative position
is within this nearly-bound state, the height of the bump provides a good

measure of the volume of the system.
5.1.3 Illustrative Calculations

Typically, all three of the physical effects discussed above are important
for the correlation function. Here, we present correlation functions
corresponding to schematic source functions in order to illustrate the sensitivity
of the correlation function for various source sizes and lifetimes.

We parametrize the spatial dependence of the source with a spherical
function of Gaussian density profile [Koon 77]. In most comparisons with data,
the time dependence of the source function is neglected [Zarb 81, Lync 83, Gust
84, Chen 87a, Chen 87b, Poch 86, Poch 87, Fox 88, Awes 88, Cebr 89, Gong 91b,
Zhu 91, Hong 92, Xu 93, Kund 93]:

g(r.t.p>~8<t>-e“‘"”°”. (5.6)

Besides reducing the number of source parameters to one, a vanishing
source lifetime eliminates the I P l dependence of the correlation function In
Figure 5.2, correlation functions calculated according to Equation 5.1 with the

215

2.0

0.6

0.4

0.2

0.0

95

gas ,t)~6(t)°ee(l?|/r°)2

 

III

IIIIIIIIIIIIIIIIIIIII

fl
1
l/
Ildlli

IITITFITIIIII
.\

I I r I I I I I I I
""" r0=3 fm
"""" ro=5 fm
----- ro=7 fm-

ro=10 fIn

 

IIIIIIIIIIIIII

 

 

ul—
wi-
.1—
-1r-

 

 

 

  

‘b
qr-
-

 

 

 

-
cl
q
-|

 

 

 

E

L /_ .3
I l : :
I if 2
:- 1; -_
E I; ---. ----- ro=20 fm‘ 1
r"; - - - ro=50 fm -_j
E If E
:1 —_
:I :
.- I I 1 I L I I I I I I I I I I I I I I L I I I q
0 20 40 60 80 100

q (MeV)

Figure 5.2. Correlation functions corresponding to proton emission from
spherical sources with Gaussian density profile and negligible lifetime
(Equation 5.6) show a strong source size dependence for smaller sources.

96

zero-lifetime source parametrization (5.6). For source radius parameters to S 7
fm, the height of the bump at q = 20 MeV/c provides the best measure of the
spatial extent of the source. For larger sources, however, the strong interaction
and antisymmetrization effect become negligible [Gong 91a, Gong 91c], and
information on the source size is carried by the strength of the Coulomb
suppression at low relative momentum.

Although correlation functions similar to those shown in the lower panel of
Figure 5.2 have been measured [DeYo 89, DeYo 90, Elma91, Eraz 91, Gong 90c,
Gouj 91, Rebr 92], spatial sizes on the order of 50 fm are clearly unphysical for
nuclear sources. Finite lifetime effects will increase the apparent size of the
source if the parametrization (5.6) is used. In the Koonin-Pratt picture, the
interaction between a pair of particles begins when the second particle is emitted.
If 1: describes the average time delay between the emission of two particles with
velocity v, then the average particle separation will be on the order d = |Ar + v - 1|;
where Ar is the relative coordinate of the two emission points; i.e. l r I ~ro, In this
case, a source parametrization with Equation 5.6 will give an overly-large spatial
dimension. We now add an exponential time dependence to our source function.
To simplify the discussion and concentrate on the spatial aspects of the
correlation function sensitivity, we use a thermal energy distribution in the

source function:

2 2
g(r,t, p) ~ e“""’°’ -e‘”‘ -e“" ’2‘”. (5.7)

Here, m is the proton mass and T is the temperature, which we set arbitrarily to 8
MeV, since for tight gates on the total momentum P, the correlation function is
insensitive to the details of the energy spectrum.

97

Figure 5.3 illustrates the interplay between the momenta of emitted
protons and the lifetime of the source. In the upper panel, the lifetime parameter
1 is varied for a tight total momentum cut P=250-350 MeV/c. The bump in the
correlation function ,damps quickly as the lifetime is increased above about 100
fm/c. Correlation functions corresponding to long time delays between particle
emissions are quite similar to those corresponding to spatially large, zero-lifetime
sources shown in the bottom panel of Figure 5.2.

Displayed in the lower panel of Figure 5.3 are correlation functions for a
fixed lifetime as the total momentum of the pair P is varied. Since these curves
are degenerate for a zero-lifetime source, we see that correlations between
higher-momentum pairs are more sensitive to lifetime effects. Also, note that for
a fixed source size and lifetime, the correlation becomes more suppressed as the
total momentum of the pair is increased. This is opposite the behaviour observed
in correlation measurements [Lync 83, Chen 87a, Chen 87b, Poch 86, Poch 87,
Awes 88, Gong 91b, Zhu 91, Lisa 91, Hong 92, Lisa 93a, Lisa 93b], revealing that
protons of different energies are emitted from an evolving source, such as a
cooling compound nucleus or a hot expanding system [Prat 87]. Thus, the two-
proton correlation function may truly serve as a tool to probe the dynamics of a

heavy ion collision.

Such an optimistic statement, however, must be taken with a grain of salt.
For, although Equation 5.1 unambiguously associates a correlation function with
any given source function, inversion of the problem is more difficult. We have
seen in Figures 5.2 and 5.3 that the correlation function is affected similarly by an
increase in lifetime or in source size. Without further cuts, the correlation

function cannot distinguish space and time unambiguously.

 

98

 

2.0

1.5

 

 

H
O
IIII‘IIIIIIIIIIIIIIIIII

 

 

 

 

 

 

0.5
J L J l | l 1 l J l l L L l l L l 1 L l l l l l
O O : I I I I I I I I I I I I I T I I I I I I I I I I .
1.0 E- ~ ——----— au-Z‘
O 5 C. r°=3.5 frn -——-- ZOO-300 __-
° - = ---------- 300-400 ..
: 1' 100 fm/c _ _ _ _ 400-500 :
O o l l l J i l l 1¥ # l 1 l l l l L l l l l i l 1 .l 1
o 20 40 so so 100

q (MeV/C)

Figure 5.3. The interplay between the momenta of emitted protons and
the timescale of emission is illustrated in the two-proton correlation
function. The upper panel shows the effect of increasing lifetime for a
source described by Equation 5.7, with the parameters indicated, for
proton pairs with a fixed total momentum. The bottom panel shows the
effect of increasing the total momentum P, when the lifetime is kept
fixed.

99

The convention followed in the literature is to parametrize the source in
terms of a zero-lifetime spheres. If such a parametrization leads to an unphysical
source size, a finite lifetime is given to the source function. Then, the procedure
is to assume some reasonable source size and fit the correlation function by
varying the lifetime. In fact, in one study of 16O+197Au at E/ A=94 MeV,
correlation functions measured at forward angles are fit by assuming a negligible
lifetime and varying the radius [Chen 87c], while those measured at backward
angles assume a radius and vary the lifetime [Ardo 89].

For very large source sizes (r0 2 20 fm) or long lifetimes (r 2 1000 fm/ c)
[Gong 91a, Gong 92], the strong interaction and quantum interference effects
may safely be ignored in uncut correlation functions, and Coulomb trajectory
calculations [DeYo 89, DeYo 90, Elma 91, Eraz 91, Koro 91, Elma 93] are often
employed to extract lifetimes on the order of 1020-10“21 s (~300-3000 fm/c) for
evaporative sources. Other studies have varied both the size and the lifetime to
fit the shape of the correlation function for shorter-lived sources [Mach 92].

Accurate lifetime determinations of these types are difficult, since a large
change in source parameters leads to a small change in the correlation function.
Moreover, for long lifetimes, the correlation function changes only for very low
q, where statistics are low (due both to the correlations themselves and to phase
space considerations) and where experimental distortion effects are most

important; see Section 5.4 below.

Cuts on the direction of the relative momentum q with respect to the total
momentum P have been suggested as a means of independently determining the
source size and lifetime [Prat 87, Gong 91a]. Protons emitted from a long-lived
source will have a phase space distribution elongated along the direction of the
total momentum [Bert 89]; see Figure 1.2. The dimension of the source along the

100

direction of P will be on the order of IAr + v - 1|, while typical dimensions
perpendicular to P will be I Ar l ~r0. The suppression of the correlation function
due to the Pauli suppression will be stronger when q is oriented along the short
direction of the distribution. Therefore, the longitudinal correlation function
(q l l P) will be enhanced as compared to the transverse (q_LP). In principle,
then, fitting the magnitude of the correlations as well as the difference between
longitudinal and transverse correlation functions, would require varying both the
spatial and the temporal extent of the source.

Schematic calculations of correlation functions cut on
w = Cos‘l (P- q / |P| - |q|) are shown in Figure 5.4. The source function is
parametrized according to Equation 5.7, with the temperature parameter set to 8
MeV, the radius parameter set to 3.5 fm, and the total momentum of the pair kept
between 250 and 350 MeV/c. The longitudinal cut is defined as w=O-50°. In the
left- and right-hand panels, the transverse cut is defined as w=80—90° and w=60-
90°, respectively. w—cut correlation functions for the cuts shown in the right-hand
panels are sensitive to lifetime effects for 30 S ‘t S 300 fm/c [Gong 91a]. On the
other hand, tighter w cuts for the transverse correlation function extend the
sensitivity to larger lifetimes on the order of 1000 fm / c, raising the hope that w-
cut correlation functions may be used to measure lifetimes from evaporative
sources; see Chapter 7 for further discussion
5.2 Constructlng a Correlation Function for Model Prodlctlons

As discussed above, two-proton correlation functions measure the space-
time structure of emitting sources. However, the concept of a "source" is only a
mental crutch used to characterize a typical emission pattern. Like physical
reality, dynamical models produce individual collisions. The size and shape of
proton emitting regions created in each collision depends on the impact

parameter and the orientation of the reaction plane; event-by-event fluctuations

0.6

101

 

IIIIjIrIIIIIIIIIIII'I

‘r=10 fm/c

      

 

 

 

 

 

  

   

-r=1ooo fm/c I

q

 
 

 

  

    
   

 

r=1ooo fm/c I

 

 

; —¢= o-5o' ;_ —~p= 0-50°

_- ----¢=ao-90° 1 ----¢=so-90° —
-.-...l....l....l....l....-....l....l..4.l....l....
o 20 4o 60 so o 20 4o 60 so 100

q (MeV/e)

Figure 5.4. Longitudinal (solid lines) and transverse (dotted lines)
correlation functions are shown for schematic sources of various lifetime
parametrized according to Equation 5.7. Longitudinal cuts correspond to

a range v=0-50°. In the left- and right-hand panels, transverse cuts
correspond to ranges w=80-90° and w=60-90°, respectively. Details are
discussed in the text.

102

produce variations on top of these dependencies. Therefore, a modified version
of Equation 5.1 is used to test model predictions.

The model is used to generate a set of phase space points for different
values of impact parameter and reaction plane orientation. We denote the set of
phase space points as (x,p),’-'b, where b is the impact parameter vector. The event
number n runs from 1 to N5, where N], denotes the number of events with
impact parameter vector b. Finally, i is the particle number in the event (n,b);

i=1...Mn,b, where Mm}, is the multiplicity of event (n,b). Then, the correlation

 

 

 

function is calculated as
1 + R(q) =
M M , , , 2
le Nb ml) “2‘, lpisl P£2| n“1_»l‘|2.Ipi{;1 plnzl
2N— b2 2 2 Z (1 5ii5n1n2 ”A“! 2 ) (xib -ij , 2 )
Cb bnl n2 i j _
ij sz Mnlbl Man2 1111 1112 I

 

'Pl P
222 Z Z Z (1- 51i5n1025b1b2)54(q- 1’12 ibz)

b1 b2 n1 n2 1

(5.8)

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 of the pair at the time of emission of the second

particle; ¢ is the wavefunction of relative motion between the two protons; 8A(q)
is the "binning function" which is unity for |q| S -21—A and zero otherwise; C is a

constant adjusted such that R(q) vanishes for large q.

In the numerator of Equation 5.8, the summation is "coherent" in impact
parameter vector, while in the denominator an "incoherent" sum is performed.
This is analogous to the experimental situation. If large ranges in impact
parameter are sampled, this distinction between the two summations may lead to
so-called "dynamical" correlations [Gong 91b]. Tight centrality cuts in the data

103

and theory reduce the effects of averaging over the magnitude of the impact
parameter I b I .

In principle, tight centrality cuts do not remove all dynamical correlations,
since the denominator still averages over the orientation of the reaction plane.
Since proton emission is rather isotropic with respect to the reaction plane [Wils
90, Wils 91, Tsan 91], such effects are probably negligible for proton-proton
correlations. However, a recent study [Kamp 93] suggests that such effects may
be important for correlations between heavy fragments, when larger flow effects
exist.

The factor of (1 /Nb) in the numerator of Equation 5.8 simulates the impact
parameter weighting of the experimental correlation function. In the Koonin-
Pratt formalism, events with the same b are combined to generate the emission
function, and then the final state interaction is applied. Therefore, there is a
double sum in the numerator in Equation 5.8; this double sum causes the
contribution for a given b to go as N52. The factor of (1 / Nb) accounts for the fact
that in the data, the number of pairs contributing to the numerator for a given b
is equal to the number of events with that b. The denominator needs no such
correction factor, as the weighting from the double summation follows that
found in the data.

Construction of correlation functions with the "equivalent" Et technique
(see Section 4.4.1) is performed by setting N b proportional to I b I, consistent
with a geometric weighting of impact parameters. The sums in Equation 5.8 are
then required to include only those events that have a total transverse energy
which falls within the desired cut.

Construction of the correlation function according to the experimentally
determined impact parameter distributions (see Section 4.4.2) is performed by

104

setting Nb proportional to the distributions dP/ d I b l shown in the bottom panel
of Figure 4.13.
5.3 Experimental Construction of the Correlation Function

For collisions at fixed impact parameter, the correlation function 1+R(P,q)

is related to the single- and two-particle yields, Y(p) and Y(p1,p2) by

Y(PIQ) = Y(PpPz) = C'(1+R(PA))'Y(P1)'Y(P2) (59)

where p1 and p2 denote the momenta of the two detected particles, P=p1+p2 is

the total momentum of the proton pair, and q is the momentum of relative
motion in the center of momentum frame of the proton pair. The constant C can
be determined from the condition that R(P,q) =0 for sufficiently large relative
momenta for which modifications of the two-particle phase-space density due to
quantum statistics or final-state interactions become negligible.

Since the impact parameter of a subatomic collision cannot be measured
with precision, experimental determinations of two-particle correlation functions
involve averages over impact parameter. Furthermore, it is virtually impossible
to collect sufficient statistics to allow the determination of the full six-
dimensional dependence of the correlation function upon P and q. Hence,
implicit integrations are carried out in the constructions of experimental
correlation functions. For example, experimental correlation functions are

mostly defined according to the relation [Boal 90]:
ZY(p1.p,)= 5311+ R(C))- 2Y(p,.p,) (5.10)

In Equation 5.10, Y(pvh) is the ”background” yield, C is a normalization
constant which ensures proper normalization at large relative momenta, and C, is
the variable for which the explicit dependence of the correlation function is

evaluated (the most common and traditional choice is C=q). For each

105

experimental gating condition (representing implicit integrations over a number
of variables), the sums on both sides of Equation 5.10 are extended over all
energy and detector combinations corresponding to the given bins of C. The
experimental correlation function is defined in terms of the ratio of these two
sums. Comparisons with theoretical results must take this definition into
account (see also the discussions of the previous Section and the appendix of
[Gong 91b]).

Two different approaches are commonly used for the construction of the
background yield. In one approach (referred to as the ”singles technique”), the
background yield is taken as proportional to the product of the single particle
yields, measured with the same external trigger conditions as the true two-
particle coincidence yield [Zarb 81, Lync 83, Chit 85, Poch 86, Kyan 86, Chit 86a,
Chen 87a, Poch 87, Chen 87b, Awes 88, Gong 90b, Gong 90c, Kim 91, Gong 91b,
Zhu 91, Gouj 91, Lisa 93a, Lisa 93b, Lisa 93c, Lisa 93d]

Y(pimz) ~ Y(pl)-Y(pz). (5.11)

In the other approach (referred to as the ”event-mixing technique”), the
background yield is generated by mixing particle yields from different
coincidence events [Kopy 74, Zajc 84, Gust 84, Dupi 88, Fox 88, DeYo 89, Cebr 89,
Rebr 90, DeYo 90, Rebr 92]

Y(Pvpz) "' 2{83(Pi " P1,.)' 53(92 ‘ 92,...) + 83m: ’ pi...) ' 53(91 ‘ pa... )}- (5°12)

mum

Here, the indices n and m label the n-th and m-th recorded two-particle
coincidence events, while pm and 92m denote the momenta of particles 1 and 2
recorded in events n and m, respectively. In most analyses, the index n runs over
all recorded coincidence events and the index m is varied according to m=n+k,

with typically 0<k<1000.

106

The main advantage of the event-mixing technique is its simplicity, as no
singles measurements are necessary. In some situations, single- and two-particle
data could represent different averages of impact parameter. In such cases, the
use of the singles technique could lead to serious distortions of the correlation
function [Rebr 90] which could complicate comparisons with theoretical
predictions. Furthermore, less interesting correlations, resulting for example
from phase space constraints due to conservation laws [Kn0180, Lync 82, Lync
83, Tsan 84, Chit 86b, Baue 87, Baue 88], may be suppressed by using the event-
mixing technique. However, the event-mixing technique also attenuates the very
correlations one wishes to measure [Zajc 84]. The degree of attenuation depends
on the phase space acceptance of the experimental apparatus and on the
magnitude of the correlations. Quantitative analyses require careful Monte-Carlo
simulations. For the extraction of undistorted correlation functions, iterative
procedures have been developed [Zajc 84] (for an excellent discussion, see
Appendix B of [Mors 90]).

. For statistical emission processes in which the emission of a single particle
has negligible effect on further emissions, single and two-particle yields should
originate from similar regions of impact parameter. In such instances, the singles
technique appears to be the preferential choice since it avoids the attenuation of
the very correlations one wishes to measure. There are, however, scenarios
where the singles technique may become inappropriate. For example, emission
to extreme forward angles may have large contributions from breakup reactions
in which only one particle is emitted [Rebr 90]. In such instances, single and two-
particle detection will select different classes of collisions.

Two-proton correlations generated with the singles technique display a
strong dependence on the total energy (or momentum) of the emitted particle
pairs [Lync 83, Chen 87c, Poch 86, Poch 87, Awes 88, Gong 90b, Gong 90c, Gong

107

91b]. In a recent paper [Rebr 90] the issue was raised that this dependence might
be an artifact of the singles technique employed for the construction of the
experimental correlation function. We address this question and give a
quantitative comparison of two-proton correlation functions constructed by the
two techniques. For this purpose, we have re-analyzed the high-statistics data of
[Gong 90c] and [Gong 91b] taken for the 14N+27Al reaction at E/A=75 MeV.

In our event-mixing analysis, we have first projected out ”pseudo-singles”

spectra from the two—proton yields:

ir<p>=283<p-p.,.), (5.13)

where pm is the momentum of the i-th proton (i=1,2) detected in the n-th two-

proton coincidence event. These pseudo-singles yields were then inserted into
Equation 5.11 to generate the background yield. This procedure is equivalent to
summing Equation 5.12 over all indices n and (n+m), thus also allowing a
contamination of contributions from true coincidences (m=0). In our analysis,
this contamination is entirely negligible (<10'6). The background constructed via
Equations 5.11 and 5.13 provides the maximum statistical accuracy which can be
obtained with the event-mixing technique.

Useful insight can be gleaned by comparing true singles and pseudo
singles yields. In Figure 5.5, energy spectra are compared for three
representative regions of angle covered by the experimental apparatus of [Gong
90c] and [Gong 91b]. In Figure 5.6, angular distributions are compared for three
different energy intervals. Clearly, singles and pseudo-singles yields are very
similar.

Figure 5.7 gives a comparison of two-proton correlation functions

constructed with the two techniques, using the same momentum gates as

Proton Yield (arb. units)

NS

 

 

 

 

.60. . . i v T # f i %
00000000 i
0000
1000 r o o °o°31.5°< (a < 335° .3
t 26 < (9 < 28 o l
500 — °000°°°°<>oo° °e _.
000 0°C J
O o
o
18°< (9 < 20° °°e °o
_ ©°°°°°°©© ° 00 —
100E 9889 °°°° °°e 8a 3
_. 8 J
50_ 00° 00° .
i 0 singles . , 0° ,. 8
0 coincidence ": ii . 03:
oo
10 5' 9 l
5 t_ 14N +27“, E/A = 75 MeV °o ;
i . . . l . r . i l . . l .o l
O 50 100 150

E (MeV/c)

Figure 5.5. Comparison of singles (circles) and pseudo-singles
(diamonds) energy spectra measured in [Gong 90c, Gong 91b] for the
14N+27Al reaction at E/A=75 MeV. Statistical errors are smaller than the
size of the symbols.

109

 

 

 

 

. c 1 . . . . r . . . - T
. 9 14N +27A1, E/A = 75 MeV
t o o
g o 9 o
7; 8 o <>13:15-40 MeV
«u 0
E1000.- 8 e e g o
:3 r" 8 6 ° 0
. 5- 8 o

'8 700:- a
(U s s 8 E=50-80 MeV
v 3' 8 g 9
§ 500:- E=9o-120 MeV 8 ‘8
.... - e e 8
>-' :_ 8 8
c: 3 g
3 i 8 8
e BOOf Osingles 8
0.. ~ 0coincidence 8 8

200» . L l . . m . l i i . . 1 .

20 25 30
(9 (deg)

Figure 5.6. Comparisons of singles (circles) and pseudo-singles
(diamonds) angular distributions measured in [Gong 90c, Gong 91b] for
the 14N+27Al reaction at E/ A=75 MeV. Statistical errors are smaller than
the size of the symbols. ‘ ‘

1 + R(q)

110

 

 

 

 

I I ' I I ' I ' I I ' I r I I ' I I I I
2.0 — $3; a P=840-1230 MeV/c ~
+ ¢ (a o P=450— 780 MeV/c «
{i I? o P=270— 390 MeV/c
* ++ 00 ii 0 ° 0 Singles 4
1.5 L‘ 8°°99 t | 0 0 Event Mixing “
' e 9 $3
. it 9 .
L +2 o’ it ‘9 ‘3‘:
O ' ’9’, 969 ‘
. . é
1-0 *- 9 ”I""'giomtiiu'mil;Mg“
' ii
a I
_ I 14N +27Al, E/A = 75 MeV, eav= 25° ‘
05 L . . r l .4 . . i . . r 4 l . L J . 1 . . r L
O 20 4O 60 80 100
q (MeV/C)

Figure 5.7. Two-proton correlation functions measured [Gong 90c] [Gong
91b] for the 14N+27Al reaction at E/ A=75 MeV. Open and solid symbols
represent correlation functions constructed by the singles and event-
mixing techniques, respectively. Momentum cuts are indicated in the
figure. Statistical errors (larger than the size of the data points) are
shown only for open points. Solid points have errors of same magnitude.

111

displayed in Figure 14 of [Gong 91b]. Very similar results are obtained by the
two techniques. In particular, both techniques give very similar momentum
dependencies of the two-proton correlation functions. As expected, the event-
mixing technique gives correlation functions which are slightly attenuated in
comparison with the singles technique. The differences are small (typically
smaller than 5%) and only of marginal statistical significance.

We also explored whether the results were stable with respect to the
number of detectors employed in the experiment. In Figure 5.8, we show energy-
integrated two-proton correlation functions from data in which the entire
hodoscope (top panel), only 23 detectors (center panel), and only 7 detectors
(bottom panel) were incorporated into the analysis. (Integration over all
outgoing particle energies was performed to ensure good statistical accuracy for
the analysis employing only 7 detectors.) Again, the two techniques give rather
similar results. Here again, however, correlation functions obtained with the
event-mixing technique are slightly attenuated as compared to those constructed
with the singles technique, reflecting the presence of ”residual” correlations in
the background yields constructed from event-mixing analyses [Zajc 84].

In conclusion, the momentum (or energy) dependence of two-proton
correlation functions reported in references [Gong 90c] [Gong 91b] can be
considered as firmly established. A re—analysis of the data in terms of a
background constructed by event-mixing gives correlation functions which are

very similar (though slightly attenuated) to those published previously.

Differences between correlation functions constructed via the two
techniques must be checked on a case-by-Case basis. In our analysis of 35Ar+45Sc
at E/ A=80 MeV, the hodoscope was positioned at 38° in the laboratory. As we
discussed in Chapter 4, the region of centrality selected by measuring one proton

1 + R(q)

1.6
1.4
1.2
1.0
0.8
0.6
1.6
1.4
1.2
1.0
0.8
0.6
1.6
1.4
1.2
1.0
0.8
0.6

112

 

T I I Y I r I T T I Y Y I U I I Y I I I I Y I

     

 

 

 

 

 

 

 

47—element hodoscope

:— 6

g. s

g4 a

E‘ i 0 Singles —?

L g ' 0 Event Mixing _§

t...1....1....|....1.. :

L 1 r] u r v i I u r r r I r v v v I r r i i:

BIS-element hodoscope

r s s

L O :

: '1‘

E 0 w

E :

E- 0 —j

?....i....1 1...]...3

LH'I'H' '“.]-..r;

fix 7-element hodoscope 4

E 99m i

7 9 “$.30” I

t O O

:- le .

E

89 . 1 i 1 . . 1 . . . _
20 40 60 80 100

q (MeV/C)

Figure 5.8. Energy-integrated two-proton correlation functions measured
[Gong90c] [Gong 91b] for the 14N+27Al reaction at E/ A=75 MeV. Open
and solid symbols represent correlation functions constructed by the
singles and event-mixing techniques, respectively. Top, center, and
bottom panels represent correlation functions extracted from arrays
consisting of 47, 23, and 7 detectors, respectively.

113

in the hodoscope is very similar to that selected by measuring two. Therefore, as
expected, the correlation functions were largely insensitive to the technique used.

In Chapter 7, we re-analyze the data of [Gong 90b] and [Gong 91b] for the
reaction 129Xe+27A1 at E/ A=31 MeV. This reverse kinematics system was
viewed at 25° in the laboratory frame, where contributions from projectile decay
and from fusion residues exist. Therefore, it is somewhat more likely that one-
and two-particle events in the hodoscope represent different regions of centrality.
Correlation functions constructed with the singles technique differ from those
constructed according to the event-mixing technique. The degree to which the
event-mixing technique attenuates interesting correlations is proportional to the
strength of the correlations. For this system, the correlations are very weak.
Therefore, a difference between correlation functions constructed via the singles
and event-mixing techniques is attributed to different event class selection, and
we use the event-mixing method in our analysis.

5.4 Effects of Finite Resolution

The ability of the correlation function to extract relatively small effects
from a large background rests firmly on its insensitivity to single-body phase
space and detector acceptance effects. However, experimental uncertainties in
the determination of the binning variable (usually the relative momentum) will
distort the measured correlation function. Here, we briefly discuss how to .
account for this effect in calculated correlations, for the hodoscope set-up used in
our experiment. We find that finite resolution effects are of minor importance for
this case.

It is useful to distinguish the granularity of the detector system from the
resolution. A hodoscope with finite granularity and perfect resolution would
consist of infinitesimally small perfect detector elements with a finite spacing
between them. In realistic detector systems, finite solid angle coverage and

114

energy uncertainty lead to nonperfect resolution; q is not measured perfectly.
The correlation function itself is robust with respect to nonideal granularity, since
this is basically a single-particle phase space effect. However, the degree to which
the finite resolution of the detector distorts the correlation function depends on
the granularity. Therefore, calculations which quantify effects of finite resolution
should also account for detector granularity.

In Figure 5.9, we show the results of a Monte-Carlo calculation of the
resolution of the 56-element hodoscope for four representative cuts in P and
Wm, = Cos‘l (PM, - q / '1’qu . |q|). Diamonds and circles, respectively, indicate the
first and second moments of the quantity 8q=qreaj-qmas, where qmal represents
the "true" relative momentum between a pair of protons, and qmeas is the relative
momentum that the device measures, including all effects of finite granularity
and angular and energy resolution. The measured momentum spectrum was
averaged over for proper weighting.

The value of the second moment (A2q=(<(q,ea1-qmeas)2>)0-5) indicates the
magnitude of resolution effects. Its value is mostly independent of q, but
depends on the cut in P and Who The q-resolution of the hodoscope is
dominated by finite solid angle effects (even changing the energy resolution from
1% to 5% does not drastically alter Figure 5.9). Therefore, the resolution for
proton pairs for which the relative momentum is parallel to the face of the
hodoscope (mabz90°) is worse than for those pair whose individual momenta p1
and p2 are almost parallel (Wlab=0°). Also, precise measurement of q for more
energetic proton pairs (high total momentum P) is more diffith than for less
energetic pairs, since a small change in the direction of p1 or p2 leads to larger
changes in q. In all cases, the resolution is never worse than :i:2.5 MeV/c, and, as

we see below, is only important for the correlation function at very low q.

115

 

 

 

 

-TTI IIIIIIITIIIITT IIIIq-‘IIITITIIIrITTTlIIITTIITI
: P=700—14OO :: P=700-1400 _
:_ 1p: o-50° “C 1p=eo-90° gas
2 12 2
A ' " ‘
O - -i- q
\ .— Ji— 4
—- F'—
i; W_ Wee-00W
a I it 3
u HHiIHHIHHIHHIHH iiH'r'THi'IHHjimjm.
N )- ql- -
<1 ; P=400- eoo :: P=400- eoo :
‘2: f— w— o-50° —"— wee-90° _.
<1 : I: ..
I I: 2
1 :- ................. 69999999999" ‘1
inc ----------------- .1
0 —, . 1:22: ;: ; : : : : ; : ;; c 2 3 t ; : : f ; : a —-‘I
.- 9 -i
- -1
plLlllllllllllllllllllLll lillilllllllllllllLlllqu

 

 

 

0 20 40 60 80 0 20 40 60 80100

q (MeV/c)

Figure 5.9. The first and second moments of the distribution (qm-qmeas)

are shown as a function of qmeas for representative cuts on P and v.
Finite resolution effects are quantified by the second moment (indicated
by the diamonds), while the first moment (indicated by circles) shows a
very small systematic distortion of the value of q. Details are discussed
in the text. The distribution for low momentum pairs with q transverse
to P stops at q-60 MeV/c due to the finite solid angular coverage of the
hodoscope.

116

The distributions for the first moment of the distribution (A1q=<qrea1-
Qmeas>) deserve a comment. Naively, we might expect that this quantity should
vanish for all q, within statistics. However, A1q is slightly smaller than zero for
all cuts, especially at low q. This is basically a result of phase space
considerations: if y=x+8, where 8 is a small random vector, then, on average,

I y l > I x | . Therefore, in addition to the finite precision with which we measure q,
finite resolution effects also introduce a systematic distortion of the value of q.
This small distortion is significant only for very small values of q, and is well
within the overall uncertainty.

Next, we construct correlation functions corresponding to emission from a
schematic source function to determine the importance of the finite resolution.

Since resolution effects can depend on the shapes of the singles energy

spectra, we parametrize the source function according to
g(r,t,p) ~ e-(II'I/ro)2 . e-t/1: . Y(p), (5.14)

where Y(p) represents the experimentally-determined proton momentum yield.
Inthe data analysis, we assume that the proton hits the center of the detector,
and we store the energy spectra with a certain binning width (typically 25 keV).
These effects are retained in the Y(p) that enters into Equation 5.14, implicitly
accounting for the granularity of the detector.

A set of phase space points (x,p)i was generated by Monte-Carlo sampling
g(r,t,p), and correlation functions were calculated according to a simplified
(because we have no impact parameter averaging to deal with) version of

Equation 5.8:

 

i just

1+ R(‘LPJIO = ———£—26Aq *_ 8A? . 813V

i jati

~I ~I 2
, ,p-p
225M'5m5m'lflxi-xi; I2 1')I

 

(5.15)

117

Here, 51w 8 A1,, and 8A,, are binning functions similar to the function 5A(q) used

in Equation 5.8. As in Equation 5.8, double primes indicate that the spatial
separation is calculated at the time of emission of the second proton, and single
primes indicate that the relative momentum is calculated in the center of
momentum system of the proton pair. Finite resolution effects are accounted for
by evaluating the squared wavefunction at Iii: - iijl / 2, where i) is the proton
momentum after being smeared by detector resolution. So we bin the correlation
variable according to the binned momenta pi, but we weight the numerator and
denominator according to the smeared (or ”true”) momenta iii.

Our smearing procedure uses a Gaussian distribution in energy to
simulate energy resolution and assumes a uniform hit density over the face of the
detector to account for the finite solid angle coverage of each detector element.
(This assumes that the angular distribution is flat over the angular range covered
by a detector element— a good approximation for our small detectors.) Figures
5.10 and 5.11 show the hit pattern obtained by smearing the energy spectra
measured by us (35Ar+455c, E/ A=80 MeV) and from [Gong 91b] (129Xe+27Al,

E/ A=31 MeV) with our smearing routine.

Figures 5.12 and 5.13 show the calculated correlation functions for
representative cuts in P and \iflab- Correlation functions calculated with Equation
5.15 with the smearing turned ”off” (i=p) represent the undistorted correlations
(lines), while those calculated with the smeared momenta (symbols) show the
distortion effects of finite resolution. As expected and reported previously [Gong
91b], finite resolution effects are only important for the lowest q—bins; the
important peak region (q~10—40 MeV/c) is relatively free of such effects.
Furthermore, as expected from Figure 5.9, the effects are somewhat more

pronounced for correlatiOns cut on high total momentum of the pair P.

118

As a final note to this section, we point out that we have constructed
longitudinal and transverse correlation functions with cuts on the relative angle
\v between P and q as defined in the lab frame. Since P, as defined in the lab
frame, is normal to the hodoscope face, the relative momentum for pairs
contributing to the transverse correlation functions suffer maximum resolution
effects due to finite angular resolution. As we discuss in Chapter 7, it is often
more appropriate to cut on \y as defined in a different frame. In such a case, the
effect of finite angular resolution will be more equitably distributed among the
longitudinal and transverse correlation functions, since q will not always be
parallel to the hodoscope face for any given ill-cut. This is illustrated in Figure
5.14, where we show the effects of finite resolution on the correlation function
corresponding to a moving source (vsomce=0.18 c) of radius 3 fin and negligible
lifetime. The cut on the total momentum of the proton pair as calculated in the
laboratory is idendical to that used in Figure 5.13.

119

56-element Hodoscope Coverage

 

 

 

 

I ’ ' I ‘ I T ' i I l .
200— ~
5'??? I» . . s» :
- ,. , ’3‘ . an»
190— 43 . a: @—
: . ass ’ Q a. it? :
a.” “1% “3% is Q «
3180- #38 . w m ~
% .. a up i
170— $ “ 2'9 ‘3"
1% m
. w 5% ‘3 as set.
359
160- -
l . l . i L l L . i
so 35 4o _45

0 (degrees)

Figure 5.10. A uniform hit pattern over the faces of the detector elements

of the SG-element hodoscope was used to quantify finite resolution
effects on the correlation function for the 36Ar+455c measurement.

120

37— and ‘13—e1ement Hodoscopes Coverage

 

 

 

 

 

1 1 1 1 I l 1 1 I 1 1 1 l I I l T i
_ ~ _
~ 0 a un- -
- {D o o o _.
’a? - - ..
Q)
Q)
5.. — _
ii”
3 so —- —-~
'9- 1- -1
- - .1
b o '0 -
Q '
-
i" - - * - * -i
- O a. -
O O - a.
- - g ‘ _
_ o 1.... ‘2, "" _
#1 1 L I 1 1 L 1 I 1 4 1 1 I 1 1 1 1
15 20 25 30 35

0 (degrees)

Figure 5.11. A uniform hit pattern over the faces of the detector elements
of the (incomplete) 56-element hodoscope and 13-element hodoscope
was used to quantify finite resolution effects on the correlation function
for the 1“Ni-”Al and 129Xe+27Al measurements.

121

 

 

   

 

   
  

 

 

 

    
   

 

2.5- 1IIIIIIIIIIIIIIIIII-W'IIIITIIIII'IIIIIIIIIlIITq
. '— 5." 4:— P=400-600 MeV c3
2 O ; .. «p: o-90° 3; / 3
: " ~- — Undistorted -
1.5 " ._.-.:_ 1..
C " j: 0 Distorted :
1.0 : u .. ro=3 fm T=O —:
I- n 1- _
_ u #-
3 .3 ~- 1
m 0.5 _ .- .
+ "' “ di- '1
1-0 1- ": __ I q
2 e O _ 1‘: —I'_ "'.. .1}. . ——1
I ;; il’= -50° :1; 3 1 ¢=80—90° 3
1'5 '2. 7:". 1
1.0 _ If .3. q
i- :1. “.1 -1
l' e 3 p q
I... LLlLllLllllllLllLlJll g olllllllllllll llllllll

 

 

0.5
0 20 40 60 80 0 20 40 60 80 100
q (MeV/c)

Figure 5.12. Calculated correlation functions for P=400-600 MeV/ c.
longitudinal and transverse correlation functions are defined in the
laboratory rest frame. Solid lines indicate the "true" or undistorted
correlation functions, while symbols indicate the correlation function
after accounting for the resolution of the 56-element hodoscope.

122

 

 

     

  

 

 

 

2.5-IIIIIIIIIITIPfiIlIIrI-IIIWIIIIIlIIIIIIIIIl'IIII‘
2.0 _— I— P2700 MeV/c j
: : —- Undistorted :
1-5 .1— .— “i
3 : O Distorted j
1.0 _ h ro=3 frn T=O -j
A : .. :
3 '- ‘p -
m 0.5 _ 4L 4
+ .. .4... .‘ .1
I". "' . d: ; p 1
2.0 — ~ —— " _..
: :: 1fi= 0-50" I: " ¢=80-90° :
1- ': ;- .. W -
1.5 _ E 1:- __‘

1- ji-

1- i-

. a! "-
1.0 _ L.
l- a“
In1111JIlll|lllllllLllLlll‘l‘llllLllJllllJllllllll ,

 

 

0.5
o 20 40 so so 0 20 40 so so 100
q (MeV/C)

Figure 5.13. Calculated correlation functions for P2 700 MeV/c.
Longitudinal and transverse correlation functions are defined in the
laboratory rest frame. Solid lines indicate the "true" or undistorted
correlation functions, while symbols indicate the correlation function
after accounting for the resolution of the 56-element hodoscope.

2.5

2.0

1.5

1.0

   

 

123

 

 

 

 

. IIIIFIIIIIIIIIIIII-fii-IIIIIITIITIIIIIIIII IIII“l
.- db .1
- j P2700 MeV/c :
i— n , -"
- 1p: 0—90 1: —— Undistorted :
;_ 4L- 0 Distorted _-‘
- :- ro=3 fm T=0 :
: I vsource=0°18c :
L. g; . .12. 3.. J
- ~ ¢= O-50° :1: .. ¢=80—90° :
I '3' I 1: " I
)— v': “I" u': 4
.— :: dL- H '1
i- t. ‘1- _“

C .3

lLLLlIlLLIlllLllllLllJllllllljlllllllLJLllllllJJV

0.5 .
o 20 40 so so 0 20 40 so so 100
q (MeV/o)

Figure 5.14. Calculated correlation functions for P2 700 MeV/ c.
Longitudinal and transverse correlation functions are defined in a frame

moving with B=O.18 in the laboratory. Solid lines indicate the "true" or
undistorted correlation functions, while symbols indicate the correlation
function after accounting for the resolution of the 56-element hodoscope.

I Chapter 6 - Centrality-Cut Proton Data for 35Ar+45Sc at EIA=80
MeV; Comparison to BUU

Two proton correlation functions provide a sensitive probe of the space-
time extent of the emitting source created in a nuclear collision. The dependence
of the correlation function on the total momentum of the pair (P: I p1+p2 I)
provides valuable information on the evolution of the reaction zone, and may be
a method to determine emission and expansion time scales [Boal 86a, Boal 86b,
Prat 87, Boal 90, Gong 91a, Baue 92]. Previous measurements of two-proton
correlation functions have found stronger correlations at q=s20 MeV/ c for more
energetic protons [Lync 83, Poch 86, Chen 87a, Chen 87b, Poch 87, Awes 88, Gong
90, Gong 91b, Zhu 91, Lisa 91, Hong 92], indicating that energetic protons are
emitted from smaller sources and/ or with shorter characteristic timescales than
less energetic protons.

In previous studies [Gong 91a, Baue 92, Zhu 91, Gong 90c, Gong 91b],
twofproton correlation functions predicted by the Boltzmann-Uehling-Uhlenbeck
(BUU) transport model were seen to be in good agreement with the total
momentum dependence observed in the data. In these comparisons of measured
and predicted P-dependencies of the correlations, the data were taken without
explicit impact parameter selection, and BUU events were weighted
geometrically according to impact parameter and filtered through the detector
acceptance.

However, BUU calculations of the total momentum dependence of the
correlation function show drastically different behavior for central and

124

125

peripheral collisions [Gong 91a, Baue 92]. Comparison of these calculations to
impact parameter selected data represents a stringent test of the theory and
would be interesting. I

In this chapter, we present proton data selected by impact parameter
filters. We see that differences in the reaction zone evolution for central and
peripheral events do not express themselves in the single-proton energy spectra.
However, two-proton correlations reveal underlying differences in the source
geometry. We present the first study of the total momentum dependence of the
two-proton correlation function for different regions of centrality and the first
experimental test of the reaction zone evolution predicted by the BUU for impact
parameter-selected events. Such a double cut on the data requires higher
coincidence statistics than previously available; it provides unique information
about reaction zone dynamics in central and peripheral collisions.
6.1 Slngles Energy Spectra

Singles proton energy spectra measured in the hodoscope and gated on
the centrality conditions defined in Figures 4.13 and 4.14 are shown in Figure 6.1.
Filled and Open symbols indicate the energy spectra for central and peripheral
events, respectively, at the extreme polar angles covered by the hodoscope. The
normalization is independent of the polar angle. Clearly, no large difference
between central and peripheral events is evident from the singles spectra.

In Figure 6.2, energy spectra predicted by the BUU is compared with the
data. BUU events were centrality-selected according to the true reduced impact
parameter method (see Section 4.4.2). The relative normalization was chosen to
give equal areas for the distributions at 6=30-32° in the region Epmton250 MeV.
In agreement with the data, no large difference is seen in the singles spectra for

the centrality cuts used.

dP/dEdQ (a.u.)

126

36Ar+45Sc; E/A=80MeV

 

 

 

 

 

 

1 I lel.f.|.T I I I I I I I I l r I I I
- r- . _
10 : e . 9 . 9=3o°~32° :
_. | e . O _
:_' e o r
Q a . '—
_ 6 . _
_ 8 . .-
10—2___ 6=44°—46° 3 . ° . _
: O o g :
.. O n
r- . a —
.. O . ‘
_ O . O _
— O . O ..

0

10-35— 0 5::
E OCentral ‘ 0 . E
_ o ..
- OPeripheral -
_ o-
10'45— '"E
: l 1 l l l I l l l l l I l l I l l L 1 l -
O 50 100 150 200

Eproton (MeV)

Figure 6.1. Centrality-cut proton energy spectra measured with a singles
trigger at two extreme polar angles in the 56-element hodoscope. Spectra
for protons emitted from central events (as determined by the total
transverse energy measured in the event. See Figures 4.13 and 4.14.) are
shown as solid symbols. Spectra corresponding to protons emitted from
peripheral events are represented as open symbols.

dP/dEdfl (a.u.)

H

O

.1.
”ml—

 

127

 

 

 

 

 

 

 

 

 

 

 

 

E
— b(Et)=O-—O.36 o -
‘ o L
10's:- 00Data 0 .5
f_ O<>BUU . ._=.
'.,.J...,I..I...{"

Ill

    
   

 

 

l llllllll

 

 

 

10‘2—
E— A
: b(Et)=O.44—O.82

1045.. .0 Data
E. OOBUU
'1..il..L.l....l. ,
0 50 100 150 ” 200

Eproton (MeV)

Figure 6.2. Proton energy spectra selected by cuts on b (E0 measured at
two extreme polar angles covered by hodoscope detectors are compared
to predictions of the BUU. Spectra for central and peripheral events are
shown in the upper and lower panels, respectively. Measured and
predicted energy spectra are indicated by symbols and histograms,
respectively. Relative normalization gives equal area for measured and
predicted spectra at saw-32° for Epmmzso MeV.

128

At energies corresponding to proton emission below or around the
Coulomb barrier, the BUU overpredicts the proton yield, but reproduces the
spectra well for energies above the Coulomb barrier, in agreement with previous
comparisons which did not gate on centrality [Gong 91b]. The BUU's
overprediction of the low-energy proton yield has been attributed to the lack of a
mechanism for cluster formation [Gong 91b]. In a simple coalescence model
picture, clusters may be expected to form in regions of phase space where
nucleon population density is high. Thus, more protons will be lost to clusters at
low energies than at higher energies, where the phase space density is lower.

6.2 Measured Two-Proton Correlatlon Functlons

The two-proton correlation function is obtained by dividing the two-
proton coincident yield by a "background" yield which simulates the phase space
population of two non-interacting protons. This background yield is constructed
by treating as a pair, two protons measured in different singles events (see
Equation 5.11). In this way, the correlation function, usually evaluated as a
function of the relative momentum between the proton pair, measures primarily
the distortion effects due to the final-state interactions between the protons;
single-particle phase space effects largely divide out.

As a check, we have also constructed background yields via the event-
rnixing technique, in which the background yield is constructed by "mixing" two
protons from different coincidence events (see Equation 5.12). Correlation
functions constructed via the singles and event-mixing techniques are very
similar; differences are on the order of statistical uncertainties, with a slight
systematic damping in the correlations observed via the event-mixing technique.
This is consistent with previous studies (see Section 5.3). A large discrepancy
between results from the two techniques is not expected, since coincidence and
singles data sample very similar ranges of impact parameter (see Figure 4.6),

129

especially after gating on total transverse energy of the event (see bottom panel

of Figure 4.13).
Experimentally, the two-proton correlation function 1+R(q) is defined
through the relation
2Y,(p..p.)=N<1+R(q)>-2Y.(p.)v.(p,) (6.1)

where Y2(p1,p2) is the measured coincidence yield for two protons with
momenta p1 and p2, and Y1(p) is the measured singles yield for a proton with
momentum p. The summations are over events selected by the specified gates on
Et and on total momentum P = |p1 + p2 I . The correlation function is evaluated
as a function of the relative momentum of the proton pair q = l p1-p2 I / 2 as
measured in the center-of-momentum frame of the pair, and N is a normalization
constant set such that R(q) =0 for large q, where final state interactions are
believed to be negligible.

Previous studies [Gust 84, Kyan 86, Chen 87a, Chen 87b, Dupi 88] have
revealed differences between correlation functions constructed with central and
peripheral events, selected by cuts on some global observable. At relatively high
incident beam energies (E / A > 50 MeV), it has been observed [Gust 84, Kyan 86,
Dupi 88] that peripheral events show a more pronounced maximum in the two-
proton correlation function at q~20 MeV/ c than do central events. In this case, a
geometric parametrization indicates a smaller emitting source for peripheral
events, consistent with a geometric picture of the reaction zone. The opposite
trend, however, has been observed in the 14N + 197Au system at E/A=35 MeV.
In this latter experiment, measurements were performed at angles sufficiently
close to the grazing angle that gates on peripheral events selected l‘arge ‘
contributions from projectile decays. In this experiment, correlations for
peripheral collisions were attenuated as compared to those for central collisions,
and these data were interpreted in terms of the lifetime of excited projectile

130

residues and not in terms of a single geometric source size [Chen 87a, Chen 87b,
Lync 84].

In Figure 6.3, we display correlation functions corresponding to our
centrality cuts (indicated in Figures 4.13 and 4.14) and integrated over pair
momentum P. As in Reference [Dupi 88], our measurements of the two-proton
correlation function are performed at angles far back of the grazing angle where
contributions from projectile decays are small, and our results are qualitatively
consistent with the trends previously observed at higher beam energies.

In Figure 6.4, the two-proton correlation function is shown for cuts on the
total momentum of the proton pair. For the slowest protons (P=400-520 MeV/c),
the peripheral data show an enhanced correlation at qz20 MeV/c relative to the
central data, indicating a smaller implied source size for peripheral collisions.
For the fastest protons (P2880 MeV/c), however, the height of the bump at q=20
MeV/ c is similar if not higher for central as compared to peripheral cuts.

These trends are summarized in Figure 6.5, where the average value of the
correlation function in the region q=15-25 MeV/c, <1+R>15-25MeV/cI is plotted
against the total momentum of the proton pair for the two centrality cuts. Error
bars indicate statistical uncertainties as well as an estimate of the uncertainty due
to normalization in the high-q region. For orientation, the right scale gives the
radius parameters to calculated in the Koonin formalism [Koon 77] for zero-
lifetime spherical sources of Gaussian density profile, p(r) cc exp(-r2 /3), which
predict the values of <1 +R>15-25Mev ,C shown on the left.

Correlation functions from central and peripheral collisions display
distinct dependencies on the total momentum P of the proton pair. A decreasing
source size (or shorter emission time) for increasing P is indicated for both

regions of centrality, with a stronger dependence seen for central collisions. This

1+R(q)

131

36Ar+458c; E/A=80MeV; <6hb>=38°

 

 

 

 

16 — T I r I r I j I I I r r I I I f I f I I I I I I .4
_ _ a:
_ o P—integrated 453
l
_ o A _ L8
1.4 r:— O . . o 0 Central (b(Et)-O-O.36) J8
_ A qm
_ ' 0 o 0 Peripheral (b(E.)=o.44-o.82) .
1 2 :— 2 . O Q
' _ . O J
_ . a O _
_ . . :
Lo— 9 ..00090689308.8.0&
- 1
0.8 —- -—
.- 8 .1
0.6 - #-
1 1 1 1 I L 1 1 1 l 1 1 1 1 l 1 1 1 1 l 1 1 1 1 "‘
O 20 40 60 BO 100
q (MeV/c)

Figure 6.3. Energy-integrated two-proton correlation functions measured
in the 56-element hodoscope for the centrality cuts indicated in Figures
4.13 and 4.14. Central events (high E0 are indicated by solid points, and
peripheral events (low E0 by open points. Statistical errors are smaller
than the symbol size.

132

36Ar+453c; E/A=80MeV; <61ab>=38°

 

 

 

 

 

’- I T r I I I I 7 I I I I r I I I I 1 fl I r I I ‘5
2.0 - o b(Et)=O.44—O.82 ~13
- °b(Et)=O—O.36 1'8
- P=400—520 MeV/c «
A 1.5 - OOO _
o~ - _
V - o 0.90 q
a: ”’ .. .90 O -
H b O .90000 o O ‘
1'0 _— e .6080028999938;
_ Q
3 R i
1 1 1 1 l 1 1 L 1 l 1 1 1 l 1 1 1 1 l 1 1 1 1
0.5 .- IiT I I I I I I I I T r I TIw I I I 1 1 I y I I I d
2.0 - + _
3 W :
15 - i I. Pgaao MeV/c -
’5- ' ~ 0. .
E * O. 1
I h 090 ‘
1-0 _ 99.6 200009580.5
I .
0.5 L 1 1 1 l 1 1 1 1 l 1 1 #1 I 1 1 1 1 l 1 1 1 1
O 20 4O 60 80 . 100
q (MeV/e)

Figure 6.4. Measured two-proton correlation functions for a double cut
onthetotalmomentumofthepairP= |p1 +p2| andtheimpact
parameter. Centrality cuts are indicated in Figures 4.13 and 4.14. Open
and solid points indicate peripheral and central events, respectively. The
upper panel shows the correlation function for slow protons, 400 MeV/ c

S P s 520 MeV/c. The lower panel shows the correlation function for fast

protons, P 2 880 MeV/c. Statistical errors are indicated when they are
larger than the symbol size.

133

36.A.1~+“58c:; E/A=80MeV; <elab>=38°

 

    

I'I VII I I I I I I I I I I IITIIIIIIIIIII - E
1-8. __ A 3.7 Jo
— O <b>(Et)=O.44-O.82 J:
»- A O
_ O <b>(Et)=O—O.36
_ -4.0
§1 6 _ oh:
a "" A
1. — a
a; ..
+ _ ~45
V 1.4 r—
“ ~5.0
1.2 —- ‘55
”1111111111lhllllllJlllLlJllll

 

 

 

400 500 600 700 800 900 1000
P (MeV/c)

Figure 6.5. The total momentum dependence of the correlation function
is summarized by plotting the height of the correlation in the region 15
MeV/c S q S 25 MeV/c as a function of the total momentum P. For
reference, on the right-hand axis is indicated the radius of a zero-lifetime
spherical source of Gaussian density profile that would produce a
correlation of equal magnitude. Solid lines which connect the points for
central and peripheral events are drawn to guide the eye. Error bars

indicate statistical errors in the region 15 MeV/c S q S 25 MeV/c as well
as uncertainties in the height due to uncertainties in normalizing the data
at large relative momentum.

134

may indicate that the effect of source expansion is more prominent in central
collisions.
6.3 Comparison of Measured Correlations to BUU Predictlons
Interpretation of correlation functions is complicated by the lack of a one-
to-one relationship between space-time geometry of the source and the
correlation function [Prat 87]. However, any prediction of reaction zone
dynamics unambiguously gives rise to a correlation function and its dependence
on P and q [Prat 87]. Therefore, dynamical theories can be tested by comparing
the P-dependence of the predicted correlation function to that seen
experimentally. Here, we follow the formalism presented in Section 5.2 to

construct correlation functions from BUU phase space population predictions.

In this work, we use the BUU model of Bauer [Baue 86, Baue 87, Li 91a, Li
91b, Baue 92] with a stiff equation of state (K = 380 MeV) and the nucleon-
nucleon cross section set to its free value. The BUU transport equation was
solved via the test particle method [Bert 84], in which many ensembles (or
"events") for a given set of initial conditions are run simultaneously. To simulate
a geometric distribution of impact parameters, the number of ensembles at a
given impact parameter b was set to Nb = 130-b, where b has units of frn. Impact
parameters between 1 and 10 frn were run in 0.5 fm steps. In this way, one
obtains an ensemble of BUU events representing a geometric distribution of
impact parameters. "Central" and "peripheral" subsets of events are chosen from
this ensemble according to the methods described in Section 4.4 and below.

The phase-space population distribution was evolved by the BUU in steps
of 0.5 fm/c on a spatial grid of dimensions 81 x 81 x 161 fm3. An unambiguous
indication of when and where a particle is emitted is not provided by the BUU;

we employed emission criteria similar to those used previously [Gong 91a, Baue

135

92, Zhu 91, Gong 90c, Gong 91b]: a proton was considered "emitted" if it found
itself in a region of local matter density eight times less than that of normal
nuclear matter before a time tmax=150 fm / c. Following the formalism of Section
5.2, these sets of phase space points were used to construct correlation functions.

Correlation functions for BUU events selected according to impact
parameter (Section 4.4.2) were constructed with impact parameter weightings
given by the curves in the lower panel of Figure 4.14. They are compared to the
data in Figure 6.6. Good agreement between model predictions and observed
correlation functions as a function .of total pair momentum P = I p1 + p2| is seen
for central events. For peripheral events, the prediction for fast (high P) proton
pairs disagrees with the data.

The dependence of the correlation function on total momentum is again
summarized in Figure 6.7 where the height of the correlation function at q=20
MeV/ c is plotted against P, for central (upper panel) and peripheral (lower
panel) events. Predictions of the BUU model with the impact parameter
distributions indicated in Figure 4.14 are shown as open circles connected by
dashed lines (they are labeled "b-selected"). The total momentum dependence of
the correlation function for central events, shown in the upper panel, is seen to be
reproduced rather well. For peripheral events, the theoretical predictions
disagree significantly with measurement.

The heights of correlation functions constructed from sets of BUU events
which satisfy "equivalent-E1" cuts, as described in Section 4.4.1, are shown in
Figure 6.7 as the open squares (labeled "Ft-select "). The "equivalent-E1" impact
parameter selection procedure leads to predictions which are practically
equivalent to those obtained via the more elaborate b -selection procedure,

giving confidence to the stability of our results. The insensitivity of the

136

36Ar+453c; III/A: BOMeV <
b(E )= O- 036

91.b>= 38°
b(EI t)= 0-44 082

 

 

 

 

 

 

 

art—zs—nsn

20L. .1. .1“ .IHHITHE. wwwfifllflfllnfi
' : 0 Data 9:400-520 MeV/c 3; P=400-520 MeV/c ‘3
: 0 BUU :
1.5— —— .0.
; on it 0
. .00 1. 6
10:. o OW ' o
. _ e
A - +- O
3 : .0 I: o 1
mos IIIJI 111 T—IIIILfiIl Lrl LILLIL LlllLJlll 11411 1
:zoL I I I I Ufirl' 'Ifi'W’ ”II Y.
' ; 8. P2880 MeV/c it P2880 MeV/c ‘1
. g «I- M l
- 1r- 4
1.5 .— 33. 1:- i
: ¢ 0 1:
L. o m d’ ¢
1.0_ ”GM;-
1- 4; O
0.5b1 l l l 1 1 L] I [IL 1 l IILJ l I‘d-l l l l 1 L1 1 l 1 l
0 20 40 60 80 0 20 40 60 80 100
q (MeV/c)

Figure 6.6. Measured two-proton correlation functions for cuts in
centrality and total momentum are compared to BUU predictions. Solid
symbols represent data. Open symbols are BUU calculations. The upper

panels show the correlation function for slow protons (400 MeV/c S P s

520 MeV/c). Lower panels correspond to faster protons (P 2 880 MeV/ c).
Panels on the left- and right-hand sides, respectively, correspond to
central and peripheral events. Centrality cuts for the data were based on
I51, as indicated by Figure 4.14. Weighting for BUU events was done
according to the reduced impact parameter distributions dP/ db,-
indicated in the bottom panel of Figure 4.14. Statistical uncertainties are
indicated when they are larger than the size of the points.

137
:38 45 . _ . _ o
Ar+ SC, E/A—BOMeV, <61ab>—38

 

   
  
 
 

 

    

 

 

 

1.8 :LfTI Ij I I I IITIIIIIIIIrI IIIr 43.7 a
:+Data -4 0?;
1.8 —— 13”” 8
I--G- b selected
:-E+'Et selected _4.5
Q 1.4- :— /'/
i - - -5.0
5 '- I,’
N - I —1
J, 1.2---1:_=1:',’é 55 5‘
A :118'1111JHH'111111111'1111 c:
CE 1.8 .__.A l I l l I 13.7 B
._. - b(E,)=0.44-0 82 v
V I-
: -4.0
1.6 T
; . _ .45
1.4 — ’ ‘~.‘-%_/'
8 $=~<$—— _e”’ ~5.0
1.2 :— .55
:lllllLl 1 111111 I lllLLl l l l I l 1 Lil
400 500 600 700 800 900 1000
P (MeV/C)

Figure 6.7. The average height of the correlation function in the relative

momentum interval 15 MeV/c S q S 25 MeV/c, is plotted against the
total momentum of the pair P. For orientation, the right-hand axis
indicates the source radius of a zero- lifetime spherical source with a
Gaussian density profile that would produce a correlation of equal
magnitude. The upper panel displays the data and calculations for
central events, and the lower panel provides the comparison for
peripheral events. Data are indicated by solid circles, while BUU
predictions are indicated by the open symbols. The open circles
correspond to BUU results when the event selection is performed
according to the dP/ db, distributions shown in the bottom panel of
Figure 4.13. The open squares correspond to the selection of events via
the "equivalent-E1" method discussed in the text.

138

theoretical correlation functions to technical details of the impact parameter
weighting method is encouraging.

For central collisions, the BUU describes the total momentum dependence
of the correlation function quite well. It is then reasonable to assume that the
space-time evolution of the proton-emitting zone generated in central collisions
is also fairly well described by the theory. However, the theoretical calculations
fail to reproduce the total momentum dependence of the correlation function for
peripheral events. While some additional theoretical uncertainties exist, due to
ambiguities in the criteria for where and when a particle is emitted, the
discrepancies for peripheral collisions may indicate that the present model is
incomplete in its description of peripheral collisions and that it may be deficient

in its description of surface effects.

Chapter 7 - Longitudinal and Transverse Correlation Functions;
Finite Lifetime Effects ‘

As discussed in Section 5.1.3, the spatial and temporal extent of a proton-
emitting source may be independently determined through differences between
longitudinal and transverse correlation functions. However, previous studies
[Zarb 81, Awes 88, Gong 90c, Rebr 92] have failed to observe such differences. (A
weak signal of marginal statistical significance was reported in [Gouj 91].
However, in this study, longitudinal and transverse correlation functions were
normalized independently, an incorrect procedure, as discussed in [Gong 91b]
and below.) In this chapter, we discuss the main difficulty in extracting signals
of finite lifetime from the correlation function and present the first clearly
observed differences between longitudinal and transverse correlation functions
for two systems with very different emission time scales: a hot, rapidly evolving

nuclear zone and a slowly cooling compound nucleus.

The experimental two-proton correlation function is defined according to

N - ()
1 R = ._c2m;fl_, 7,1
+ (q) C N 1(q) ( )

where Nam-"C(q) represents the yield of coincident proton pairs with relative
momentum q. The background yield, Nback(q), is constructed using either the

singles or the event-mixing technique. Two-proton correlation functions were

139

140

constructed with cuts on the relative angle between the total momentum of the
pair and the momentum of relative motion, 11! = Cos"1 (|P ~ ql/IPI - |q|).

The normalization constant C in Equation 7.1, adjusted such that R(q)
vanishes for large q, should be determined independently of the angle w. This
may be understood through the following simple reasoning. At I q I =0, cuts on
the relative orientation of q and P become degenerate, and the normalizations for
longitudinal and transverse correlation functions are by definition identical.
Since the normalization constant is independent of l q l , this same constant must
apply for all q and w values. Thus, there is no a priori justification for separately
normalizing longitudinal and transverse correlation functions, and doing so may
produce misleading results [Gong 91b].

We should point out, however, that differences in the assymtotic
behaviour of the longitudinal and transverse correlation functions may arise
from other physical effects. As discussed in Section 5.2 and [Gong 91b], impact
parameter and reaction plane averaging may lead to "dynamical" azimuthal
correlations that distort the large-q behaviour of the correlations. Such effects
cannot be normalized away and must be accounted for by a reaction model or
else be considered part of the intrinsic uncertainty.

We are interested in the low-q behaviour of the correlation function,
where such distortion effects should be small. Indeed, as we see below, for the
systems we study, such effects are unimportant.

7.1 The Importance of Identifying the Source Frame

Proton emission from a long-lived source will lead to a phase space
distribution elongated in the direction of total momentum [Prat 87, Bert 89, Gong
91a]. For such distributions, the suppression of the correlation function at low q
due to the Pauli principle is less important for proton pairs whose relative

momentum is oriented along the direction of the total momentum, and cuts on w

141

will reveal finite lifetime effects in the correlation function. However, the relative
orientation of P and q is a function of rest frame (since P depends on the rest
frame, but q— at least in the nonrelativistic limit— does not). In order to see the
effects of finite lifetime, therefore, it is important that the angle 1|! is constructed
in the frame of the emitting source [Rebr 92, Lisa 93c, Lisa 93d].

The problem of identifying finite-lifetime effects is illustrated in Figure 7.1.
The figure depicts phase space distributions in the laboratory rest frame of
protons emitted with fixed velocity vaab towards the detector at 0131, = 38° for a
source at rest in the laboratory (part a) and for a source at rest in the center-of-
momentum frame of the system 36Ar+45Sc at E/ A=80 MeV (vsource = 0.18c, part
b). We assumed a spherical source of 7 fm diameter and 70 fm/c lifetime
emitting protons of momentum 250 MeV/ c in the laboratory.

For emission from a source at rest, the phase space distribution of particles
moving with a fixed velocity vpjab = Vemit towards the detector exhibits an
elongated shape [Prat 87, Bert 89, Gong 91b] oriented parallel to Vp’lab. A source
of lifetime 1 appears elongated in the direction of the proton momentum by an
incremental distance As z vemit - t = Vp,lab - 1:. Correlation functions for relative
momenta q J. vaab reflect a stronger Pauli-suppression, and hence a reduced.
maximum at q z 20 MeV/c, than those for q I l vpjab [Prat 87, Bert 89, Gong 91a,
Gong 91b]. .

Previous analyses [Zarb 81, Awes 88, Gong 90b, Gong 91b, Gouj 91]
compared the shapes of correlation functions selected by cuts on the relative
angle 111131, = C05‘1(q-P/ qP) between q and P = 1)] + p2 a 2mvp,1ab, where p1 and
p2 are the laboratory momenta of the two protons and q is the momentum of
relative motion. Such analyses are optimized to detect lifetime effects of sources
stationary in the laboratory system, but they can fail to detect lifetime effects for

moving sources. For the specific case illustrated in Figure 7.1b, the source

142

 

Figure 7.1. Schematic illustration of phase space distributions at a time t
= 70 fm/c, seen by a detector at 9111b = 38°, for a spherical source of radius
r = 35 fin and lifetime 1 = 70 fm/c emitting protons of momentum 250
MeV/ c. Part a) Source at rest in the laboratory. Part b) Source moves
with vm = 0.18c. In the distributions shown, the laboratory velocities
of the protons (VpJab) are depicted by small arrows and the directions

' and parallel to vaab are depicted by large double-headed
arrows. In parts (a) and (b), VP“, is kept constant, and vemit is different.
Therefore, the elongations along vemit are different.

143

dimensions parallel and perpendicular to vaab are very similar, and no
significant differences are expected for the corresponding longitudinal and
transverse correlation functions.

For a source of known velocity, the predicted lifetime effect is detected
most clearly if longitudinal and transverse correlation functions are selected by
cuts on the angle wsource = C05‘1(q'-P'/q'P'), where the primed quantities are
defined in the rest frame of the source. (In the rest frame of the source, the phase
space distribution is always elongated in the direction of vemit. Hence, in Figure
7.1b, the source dimensions should be compared in directions parallel and
perpendicular to Vemjt.) Such analyses can only be carried out for emission from
well characterized sources [Rebr 92, Lisa 93c, Lisa 93d]. In the following, we
corroborate these qualitative arguments by our experimental data.

7.2 Determination of the Source Lifetime In Central 35Ar+458c Collisions at
EIA=80 MeV

Cuts on 111 were applied to the centrality-gated two-proton correlation
functions measured for the 36Ar+455c system at E/ A=80 MeV in an attempt to
identify the effects of a finite lifetime. Since typical emission timescales may
differ for protons with different energies, cuts were also made on the total
momentum of the proton pair. Very good statistics is required to perform such
multi-dimensional cuts on the impact parameter, total momentum, and angle ‘I’
between P and q: in all, about 1.6 million "good" proton pairs were measured in

the present experiment.

In a symmetric system like the one under consideration, the source
velocity may take on many values, depending on the impact parameter. For very
central collisions, we assume stopping of projectile and target. The protons
would then be emitted from a hot source moving in the lab with the center-of-

momentum velocity of the system. For peripheral collisions, the linear

144

momentum transfer may be incomplete, and the source velocity may vary
significantly from event to event. Furthermore, many proton sources— for

example "target-like, projectile-like," and "participant" sources— with different
velocities and lifetimes may be created in a single peripheral collision. For
peripheral collisions, then, more sophisticated methods may be required to

extract the signatures of finite lifetime.

Figure 7.2 shows experimental longitudinal and transverse two-proton
correlation functions for central 36Ar + 455C collisions at E/ A = 80 MeV selected
by appropriate cuts on the total transverse energy detected in the 41: Array. In a
geometrical picture (see Section 4.2), the applied cuts correspond to reduced
impact parameters of b/bmax = 0 - 0.36. Longitudinal (solid points) and
transverse (open points) correlation functions were defined by cuts on the angle
‘1' = C05‘1(q-P/ qP) = 0°-50° and 80°-90°, respectively. (The normalization constant
C in Equation 7.1 is independent of 11!.) To maximize lifetime effects and reduce
contributions from the very early stages of the reaction, the coincident proton
pairs were selected by a low-momentum cut on the total laboratory momentum,
P = 400-600 MeV/ c. The top panel shows correlation functions for which the
angle \v was defined in the center-of-momentum frame of projectile and target (1|!
= 0130“,“); for central collisions of two nuclei of comparable mass, this rest frame
should be close to the rest frame of the emitting source. The bottom panel shows
correlation functions for which the angle 11! was defined in the laboratory frame
(\II = ‘l’lab)-

Consistent with the qualitative arguments presented in Figure 7.1, a clear
difference between longitudinal and transverse correlation functions is observed
for cuts on wwmce (top panel of Figure 7.2), but not for cuts on 111131, (bottom

panel of Figure 7.2). The clear suppression of the transverse correlation function

145

with respect to the longitudinal correlation function observed in the top panel in
Figure 7.2 is consistent with expectations for emission from a source of finite

' lifetime [Prat 87, Gong 91a, Gong 91b, Bert 89]. The solid and dashed curves in
the top panel of Fig. 2 depict calculations for emission from a spherical Gaussian
source of radius parameter to = 3.5 fm and lifetime I = 70 fm/c, see also Equation
7.2 below. The data in Figure 7.2 represent the first clear experimental evidence
of this predicted lifetime effect.

For a more quantitative analysis, we performed calculations assuming a
simple family of sources of lifetime 1 and spherically symmetric Gaussian density
profiles, moving with the center-of-momentum frame of reference. Energy and
angular distributions of the emitted protons were selected by randomly sampling
the experimental yield Y(p). Specifically, the single particle emission functions
[Gong 91a, Gong 91b] were parametrized as

g(r.p.t) «exm-rz/rf. -t/r)-Y<p). (7.21

In Equation 7.2, r, p, and t refer to the rest frame of the source. Phase-space
points generated in the rest frame of the source were Lorentz boosted into the
laboratory frame, and the two-proton correlation function was obtained by
convolution with the two-proton relative wavefunction, see Section 5.2 or [Prat
87, Gong 91a, Gong 91b] for details. '

Transverse and longitudinal correlation functions were calculated for the
range of parameters r0 = 2.5 - 6.0 fm and 1: = 0 - 150 fm/c. For each set of
parameters, the agreement between calculated and measured longitudinal and
transverse correlation functions was evaluated by determining the value of xz/v
(chi-squared per degree of freedom) in the peak region, q = 15 - 30 MeV/ c. A
contour plot of xZ/v as a function of r0 and ‘t is given in Figure 7.3. Good
agreement between calculations and data is obtained for a source with a

146

 

 

 

 

 

 

 

I I I I I I I I I I I I I I I I I I T I I I I
1-5 f P=400-600 MeV/c _
- Central events
1.0 '-

_ — o 0°<¢<50° -
E05MHIHHIHHIHHIHH
i 1.5 — -

: (I) Q 3 g

¢ 0 .

r

1.01- ¢.§é°°0°00'e.—
.- . .1
- O Vsource=o -
:- 0°<¢<50° .
o 80°<¢<90° .
0.5 L l l l I l l l_l l L l l l l_l l_ L _l 1 1 l l I
O 20 4O 60 80
q (MeV/C)

Figure 7.2. Measured longitudinal and transverse correlation functions
for protons emitted in central 36Ar + 455c collisions at E/ A = 80 MeV.
The correlation functions are shown for proton pairs of total laboratory
momenta P=400-600 MeV/c detected at <0hb>=38°. Longitudinal and
transverse correlation functions (solid and open points, respectively)
correspond to v = Cos'1(q-P/qP) = 0° - 50° and 80° - 90°, respectively,
where P is defined in the rest frame of the presumed source. Upper
panel- vsom = 0.18c. Solid and dashed curves represent longitudinal
and transverse correlation functions predicted for emission from a
Gaussian source with r0 = 4.7 fm and ‘t = 25 fm/c, moving with vsom =
0.18c. Lower panel- vsoume = 0.

I'0 (fm)

147

 

III

 

IIII

  
  
  

IIIIIIIII

  

-

—

"

 

 

Figure 7.3. Contour diagram of xz/v (chi-squared per degree of freedom)
determined by comparing theoretical correlations functions to the data

 

shown in the upper panel of Figure 72. The fit was performed in the
peak region of the correlation function, q = 15-30 MeV/c. Details are

discussedinthetext.

148

38Ar+""’Sc E/A=80 MeV; Peripheral Cut; P=400—600 MeV/c

 

 

 

 

 

 

 

2.0 - I I I I I I I I I 1 r I I I I I I r I I I I I I 4
3 00: 0-50° 3
1.5 -— _ _ o —-
: e 9 e 8 o 1p—80 90 j
I e
107 6590000000.?
: 0 8:.388 (pro?e¢tile frame) j
_ 111 l l I L I l l I l L l l l l l l l I 1 LI 1‘
0.5 - I I I r I I I I I I I I I I I I I I I I I I I I q
1.5 -_- 9 1
A - Q Q -
3 t 9 . - j
H : o B: 18 (c m. +me) j
- ll 1 l l I l l l l I l l 1 L I l l 1 LI 1 l l L-
0.5 _' l I I I I i r I I T I I I I I r T r I I r r rj
1.5 3— —3
: 0°09 _
I 90 g 2
107+ 0000000005?
'_' 8:0 (lab frame) 1
0 5 _ 1L] 1 1 LI 1 l l I l l l l I I l l l I l l l l q
0 20 40 60 BO 100

q (MeV/C)

Figure 7.4. Longitudinal (solid symbols) and transverse (open symbols)
correlation functions constructed for protons emitted from peripheral
events with total pair momentum P=400-600 MeV/c. Cuts were
performed on the angle v as constructed in the projectile, center-of-
momentum of the 35Ar+455c system, and the laboratory frame.

149

3°Ar+453c E/A=BO MeV; Peripheral Cut; P2700 MeV/c

1+R(q)

2.0

1.5

1.0

0.5

1.5

0.5

1.5

1.0

0.5

 

 

 

 

 

 

 

 

 

 

r- I I—r I I I Ff I I I I I Ifi F I I I I I I I -
E_ ¢ 8 . 0 1p: 0—50° 3
-_- o 0 o ¢=30—90°
- C
To 9999999900339
: . 3:.388 (projectile frame) 2
L I Ir L I I; I I I I I I I_I I i I I I I I I I I 4
.- I I I I I I I I I I I I I I I I I I I I I I .4
I a I
.— 8 9 0 ‘1
I i g o o g Ti
T . 9 i O Q . . O o f
: 5:18 (am. frame) 2
b IgI I I I LI I I I I I I I I I I I I I I I I I d
)— I I I I I I j I I I I I j r I I I I I I I I I II ‘
r .
C 8 l-
r ' J. fl
. . r
I 9 '3 I
7¢ 000909080004:
C 5:0 (lab frame) 3
L- I I I I I I I L I I I J I I I I I I I I I I I I
O 20 4O 60 BO 100
q (MeV/C)

Figure 75. Longitudinal (solid symbols) and transverse (open symbols)
correlation functions constructed for protons emitted from peripheral

events with total pair momentum P2700 MeV/ c. Cuts were performed
on the angle \y as constructed in the projectile, center-of-momentum of
the 35Ar+45Sc system, and the laboratory frame.

150

3"Ar+""58e E/A=80 MeV; Central Cut; P_2.'700 MeV/c

 

 

 

 

 

 

 

2.0 _. I I I I I I I I I I I I I I I I I I T I I I I _
5 ~__ 9 010: 0—50° _
1° : ¢ 99 0¢=80—90° :
L .
LOB-O 990000005964
: 5:. 388 (projectile frame) I
I. I I I LII I I II I II L I I I I LI_L Id
0.5 PI I I I II r I II T II I I I I IfrI I:
I 0 h
1.5 — —
A . as. 3
U“ - .4
v a . a
m 10r-Q 9 oeoaoae¢ 01>
+ l- q
v-I E 5:.18 (c. m. Jframe) ”Ti”? :
-J I I LI I I I d
0.5 r- I I I I I I I II ‘
- +
I 1
1.5 r $9 9 8 j
E II a" . ' 3 8i
for 9.090 33 3
F fi=0 (lab frame) :
0.5 I I I I LI I I I I I I I I I I I I LI I I I I“
O 20 4O 6O 80 100

q (MeV/C)

Figure 7.6. Longitudinal (solid symbols) and transverse (open symbols)
correlation functions constructed for protons emitted from central events
with total pair momentum P2700 MeV/c. Cuts were performed on the
angle w as constructed in the projectile, center-of-momentum of the
36Ar+455c system, and the laboratory frame.

151

Gaussin radius of about 4.5 fm and a lifetime of about 25-40 fm/c. It is gratifying
to note that these extracted emission timescales are consistent with those
predicted by BUU transport calculations [Li 91a, Li 91b], which are expected to
apply here.

Clean characterization of a relatively well-defined source frame appears
crucial. Figures 7.4 and 7.5 shows longitudinal and transverse correlation
functions constructed for peripheral collisions, where \v is defined in the
projectile frame, the center-of-momentum frame of the 35Ar + 455c system, and
the laboratory frame. Two-proton correlation functions selected by theperipheral
cut do not exhibit any differences between longitudinal and transverse
correlation functions, irrespective of the assumed rest frame. Indeed, for such
collisions no single unique source velocity may exist, and more sophisticated
analysis techniques may be necessary to exploit directional dependencies of two-
proton correlation functions.

More interestingly, even for central collisions no significant difference
between longitudinal and transverse correlation functions was observed for
protons selected by cuts on large total momenta, P 2 700 MeV/c. In Figure 7.6,
we show these correlation function assuming the same source frames as’
discussed above. The degeneracy of the longitudinal and transverse correlation
functions for this cut suggests that these very energetic protons are emitted on a
very fast time scale and / or at such an early stage of the reaction where the

concept of emission from a single source does not apply.

In conclusion to this section, we have measured the lifetime of the source
of intermediate-energy protons in central Ar+Sc collisions at E / A=80 MeV.
Consistent with dynamical. models, a lifetime of about 30 :l: 10 fm/c was

152

extracted. Such emission timescales are consistent with the creation of a
deconfined nuclear gas [Prat 87].

7.3 Determination of the Source Lifetime in 129Xe+27Al Collisions at
ElA=31 MeV

Extraction of finite lifetime effects in the measurement discussed in the
previous section was possible because central events could be selected, for which
the assumption of emission from a unique source may be reasonable. Previous
measurements of longitudinal and transverse correlation functions were
performed without impact parameter selection. Here, we discuss a re-analysis of
previously-measured high statistics two-proton coincidence data for the inverse
kinematics reaction 129Xe+27Al at E/A=31 MeV [Gong 90b, Gong 91b]. For this
reaction, ambiguities of the emitting source velocities should be relatively small;
the likely source velocities range from B = 0.209 (for complete fusion reactions) to
B = 0.251 (for emission from excited projectile fragments) [Gong 91b]. We will
show that clear differences between longitudinal and transverse correlation
functions exist when the appropriate angular cuts are made in the compound
nucleus rest frame. These differences become largely washed out when the
angular cuts are made in the laboratory rest frame as was done in Reference

[Gong 91b].

In the experiment of [Gong 91b], an 27A1 target of areal density 5.6
mg/cm2 was irradiated by a 129Xe beam at E/ A = 31 MeV from the K1200
cyclotron of the National Superconducting Cyclotron Laboratory. Light particles
were detected by two arrays of AE—E telescopes consisting of 300-400 urn-thick
silicon detectors backed by 10-cm-long CsI(Tl) or N aI(Tl) detectors. An array of
37 Si-CsI(Tl) telescopes was centered at polar and azimuthal lab angles of 0 = 25°
and 4) = 0°. Each telescope covered a solid angle of A!) = 0.37 msr with a nearest-
neighbor spacing of A0 = 2.6°. Centered at 0 = 25° and (I = 90° was an array of 13

1 +R(q)

153
129Xe+27AL E/A=31 MeV

 

I I I I r I I I I I I I I r I T I I I

 

 

 

 

 

 

 

 

- CD -
1.0 —" + Q Q Q 0 R g 5 m a . 0 a an 2

F- . -< I

4

- co

- 4) fi=0 (lab frame) 4 “I”

. o d o
08 __ . 11: 0’50 —i 2:

_ I I II I I I I I I I I I I fiI _‘

.. . .
1-0 — O (D a 8 a 8 a a 8 a 6 ._}

.— 0 CD q

- g 3:.2086 (c.m. frame) ‘
0.8 — "j

b 'r i i i I I i i i I i i I i I i i i i I

r; | | GI :
1.0 09.9.3... 0‘04

- ¢ fl=.4172 ‘
0.8 :— J

" l I 1 1 I l 1 l 1 I 1 1 1 I l 1' l "

0 20 40 60 . 80

q (MeV/C)

Figure 7.7. Longitudinal (filled circles) and transverse (open circles) two-
proton correlation functions for the reaction 129Xe + 27A] at E/A=31
MeV. Angular cuts in the upper panel were constructed in the
laboratory frame of reference, and in rest frames moving in the lab with B

= 0.2086 (center panel) and B=.4172 (bottom panel). The cut on the total
momentum of the proton pair was P 2 480 MeV/ c.

154

Si-NaI(Tl) telescopes, each covering an = 0.5 msr of solid angle with a nearest-
neighbor spacing of A9 = 4.4°. Proton energy resolution was on the order of 1%.
. More experimental details can be found in References [Gong 91b, Gong 91c].

In order to eliminate small differences in the shapes of the single- and two-
proton inclusive energy spectra, the background yield, Nbacflq) in Equation 7.1, is
constructed using the event-mixing technique (see Section 5.3). Differences in the
single— and two-proton inclusive energy spectra could, for example, arise from
different relative contributions from decays of projectile and fusion residues.

Also, in order to minimize the effects of proton interaction with the heavy
residue, which are not accounted for in the Koonin-Pratt formalism, we study
correlation functions only for particle emission above the Coulomb barrier of the
compound system.

As in the search for lifetime effects in the Ar+Sc system, it is important
that the angle \V is constructed in the frame of the emitting source. This is
illustrated inFigure 7.7. The individual panels of the figure show longitudinal (w
= 0° - 50°, filled symbols) and transverse (‘l’ = 80° - 90°, open symbols) two-
proton correlation functions selected by the cut on the total laboratory
momentum P2480 MeV/ c. For the top, center and bottom panels, the angle \V
was defined in the laboratory system (B = O), the center-of-momentum frame of
projectile and target (i.e. the rest frame of the compound nucleus, [3 = 0.209), and
a rest frame moving with twice the velocity of the center-of-momentum system
(B = 0.417). Consistent with the results of References [Gong 90b, Gong 91b], no
significant differences between longitudinal and transverse correlation functions
are observed when ‘I’ is defined in the laboratory rest frame. When the angle \y is
defined in the rest frame of the compound nucleus, however, significant
differences between longitudinal and transverse correlation functions emerge.

155

They disappear again when w is defined in a rest frame moving with twice the
speed of the compound nucleus. .

The identification of lifetime effects from differences between longitudinal
and transverse correlation functions can be difficult, if not impossible, for data
containing a broad distribution of source velocities. Fortunately, for the present
reaction, the range of source velocities is relatively narrow [Gong 91b], thus
allowing the study of lifetime effects without further event characterization. In
the following, we will use a simple parametrization of the source function to
illustrate the sensitivity of the present data to the average spatial dimension and
lifetime of the emitting system.

The Koonin-Pratt formalism (Section 5.1.1) allows the construction of the
two-proton correlation function from the single-particle emission function
g(r,t,p), which is a function of the space-time emission coordinates r and t, as well
as the momentum p [Prat 87, Gong 91a, Gong 91b] of the emitted particles. In
our schematic calculations, we parametrize the source in terms of surface
emission from a sphere of sharp radius R and with an exponential lifetime 1.’ [Kim
92].. Energy and angular distributions of the emitted protons were obtained by
sampling the experimental yield Y(p). In the rest frame of the source, the source
function was parametrized as

g(m. p) ~ «Sad—R)- 0(r- prfi- Y(p)-exp(—r/ r) . (7.3)

where 0(x) is the unit step function which vanishes for x<0, and 5(x) is the delta
function which vanishes for x¢0. The source was assumed to be at rest in the
center-of-momentum system, and the emitted particles were then boosted into
the laboratory rest frame (B = 0.2086). Longitudinal and transverse correlation

156

functions for a given R and 1: were constructed according to the Koonin-Pratt
formalism [Gong 91a].

For a given parameter set R and t, the agreement between measured and
calculated correlation functions was quantified by the value of xZ/v (chi-squared
per degree of freedom) evaluated in the region q=10-40 MeV/c, where the
measured difference between longitudinal and transverse correlation functions is
most pronounced and where distortions due to detector resolution are small
[Gong 91b]. Contour plots of xZ/v as a function of the parameters R and 1: are
shown in Figure 7.8 for the two cuts on the total momentum of the emitted
proton pairs, P2480 MeV/ c and P2580 MeV/c. The best agreement between
measured and calculated correlation functions is obtained for source radii of R =
3 - 4 fm; the extracted emission times are 1: = 17001: 200 fm/c and 1200: 200 fm/c
for the momentum cuts P 2 480 MeV/c and P 2 580 MeV/c, respectively.
(Uncertainties in extracted lifetimes are estimated from xZ/v values.) Calculated
longitudinal and transverse correlation functions giving the best fit to the data
are shown as solid and dashed curves in Figure 7.9.

The extracted source radius is smaller than that of the compound nucleus.
The reason for such a small source radius is not fully understood. It may reflect
an artifact of the present schematic source parametrization which neglects
anisotropies of emission resulting from angular momentum effects. It is
interesting, however, that the extracted average emission time scale decreases
slightly as the total momentum threshold is increased. Such a dependence is
expected [Prat 87, Gong 91a] for emission from a cooling compound nucleus.

157

12°Xe+27AL E/A=31 MeV

 

 

 

 

    
 

 

 

 

 

 

7 I I I fl r I I I I I I I I l I
6 F— . . P§480 MeV/(:3 %’
- ' - co
: :cio
5 '— ‘ 8
I : o
t —1
4 I"
: 1 1.5 -
3 :— ‘
E J .5 | 5
A 2 i i i ‘ ‘ u 'r i t l i i i i
8 : I | l :
b : P2580 MeV/c:
I I
5 :— ‘3
4 E. <2:
3 E— 1
2: 1 1 1 ‘ ' L m, 1 1 1 I 1 m L: .
0 1000 2000 3000 4000
1' (fm/c) '

Figure 7.8. Contour plots of xZ/v (chi-squared per degree of freedom)
evaluated by comparing measured longitudinal and transverse
correlation functions (over the range of 10 MeV/c s q s 40 MeV/c to
those predicted for emission from a schematic source with radius and

lifetime parameters R and 1:. The upper and lower panels compare
correlation functions constructed from pairs with P2480 MeV/c and
P2580 MeV/c, respectively. X's mark parameter sets used for calculated
correlation functions shown in Figure 7.9.

158

12"Xe+“"’.tu. E/A=31 MeV
I I I I I I I I j ‘l I I I r T I r I I

 

 
 
   
   

 

 

 

 

 

 

 

 

: J

K

1-1 7 R430 MeV/c 1 g

I jé

1.0 :— _, I

. : :§
0 9 ":— R=3.0 fm, r=1700 fm/c ‘1
o 8 E. . 1P: 0-50° j
' g — — - o 0:50-90" 3
A P / .
3 '- 4 l 1 1 I 1 1 1 1 I 1 1 4 +4 1 1 1 1 ‘
m 0 7 : I I I r I I I I I I r 1 I I I I fiI I j
i". 1 1 T P2560 MeV/c ‘1
1.0 :— I
3 1
0.9 — 1.
C cl
0.8 :— _:

0.7 P . .
0 20 40 60 80

q (MeV/C)

Figure 7.9. Longitudinal (filled circles) and transverse (open circles) two-
proton correlation functions for the reaction 129Xe + 27A1 at E/.A=31
MeV, evaluated in the center-of-momentum frame of the projectile and
target. Top and bottom panels show data for cuts on the total
momentum of the proton pair of P 2 480 MeV/c and P 2 580 MeV/c,
respectively. Solid and dashed curves show calculations of longitudinal
and transverse correlation functions using the source parametrization of
Equation 7.3 with the parameters given in the figure.

Chapter 8 - Summary and Conclusions

In this work, we have used the technique of two-proton intensity
interferometry to explore the space-time structure of emitting sources created in
intermediate energy heavy ion collisions. We have discussed the Koonin-Pratt
formalism, the theoretical framework which relates the two-proton correlation
function to the phase space population density. In the context of this formalism,
schematic calculations were presented that illustrated how the two-proton
correlation function may provide a "snapshot" of the proton configuration at the
time of proton emission.

The two-proton correlation function is sensitive to the spatial separation of
emitted protons due to final state interactions and quantum statistical effects. An
attractive S-wave resonance in the strong nuclear interaction leads to a I
pronounced bump in the correlation function at q=20 MeV/c. Sources with small
spatial and temporal extent produce larger bumps, while sources with large
spatial dimensions (r 2 10 fm) or long lifetimes (or Z 10 fm) lead to no bump at
all. The Coulomb repulsion produces a minimum at q=0, with smaller or
shorter-lived sources producing stronger minima.

-A directional dependence in the quantum statistical suppression of the
correlation function due to the Pauli principle allows independent extraction of
source size and lifetime. Appropriate cuts on the relative orientation between the
total and relative momentum of the proton pair define the longitudinal and

transverse correlation functions. The overall magnitude of the correlations

159

160

indicates the space-time extent of the source, while the difference between
longitudinal and transverse correlations reveals its lifetime. If the rest frame of
the emitting source is identified and if sufficiently tight cuts are performed,
lifetimes on the order of I ~ 30-3000 fm/c may be extracted. This range of
emission timescales cover those expected for proton emission from a hot nuclear
gas, as well as from cooling nuclear systems in the liquid state.

Single- and two-proton yields at <013b>=38° were measured with a high
resolution hodoscope in coincidence with charged particles detected in the MSU
41c Array. Two-proton correlation functions were constructed from the yields
measured in the hodoscope, and information about the impact parameter of the
event was obtained from the total transverse energy measured in the 41c Array.
Central and peripheral collisions were characterized by high and low values of
the total transverse energy, respectively. V

Correlation functions for both central and peripheral events exhibited
stronger bumps at q=20 MeV/ c when proton pairs with high total momentum
were selected. Statistical and dynamical models of heavy ion collisions associate
more energetic particles with the early stages of the collision. Using such
associations as a guide, we conclude that the proton-emitting source created in
the collisions increases in size and / or emission timescale as the reaction evolves.
The total momentum dependence was observed to be stronger for collisions with
high total transverse energy than for those with low total transverse energy,
indicating a significant difference in the evolution of central and peripheral
collisions. ‘

A dynamical model based on the Boltzmann-Uehling-Uhlenbeck (BUU)
transport equation was used to generate phase space distributions for protons
emitted from 35Ar+458c collisions at various impact parameters. Following the

Koonin-Pratt formalism, we calculated two-proton correlation functions for

161

central and peripheral events. The total momentum dependence of the
experimental correlations was well reproduced by the predicted correlation
functions for central events, indicating that the model describes the reaction
evolution well. For peripheral collisions, however, the total momentum
dependence was underpredicted by the model. This discrepancy may be due to
inadequacies in the model's description of nuclear surface effects.

The issue of impact parameter selection was examined in detail. In
particular, reduced impact parameter scales were constructed for three different
global observables, each of which presumably gives an indication of the impact
parameter. The impact parameter resolution associated with cuts on each of
these observables was approximately evaluated. Next, we developed two
different methods of realistically incorporating experimental impact parameter
cuts into theoretical calculations. In the construction of impact parameter
selected correlation functions for the BUU theory, the two methods gave almost
identical results, giving us confidence that the observed discrepancy between
experimental and theoretical correlations functions for peripheral events is not
due to incorrect impact parameter selection.

Next, we studied the use of longitudinal and transverse correlation
functions as a tool in a search for finite lifetime effects. We discussed the
importance of clear identification of the rest frame of the emitting source. Then,
we examined the longitudinal and transverse correlation functions for two
systems for which the source velocity is reasonably well-determined.

For central 35Ar+45Sc collisions, it is reasonable to assume that the proton-
emitting source moves with the center-of-momentum velocity of the system. We
found that if this source velocity was assumed, tight cuts on w = C03'1(qu/qP)
revealed the effect of a finite emission timescale on the order of 25-40 fm/c. Such

timescales are consistent with those predicted by the BUU model, and may

162

indicate the formation of a nuclear system in the gaseous state. Improper choice
of the assumed source rest frame obscured the lifetime effect. Correlation
functions for protons emitted from peripheral collisions, for which there may
exist no unique source frame, did not show a lifetime effect.

We also reanalyzed proton coincidence data for the reaction 129Xe+?-7Al at
E/A=31 MeV, for which the spread of possible source velocities is small.
Assuming that the source velocity was the center-of-momentum velocity of the
system, we observed differences between longitudinal and transverse correlation
functions consistent with emission timescales on the order of ~1500 fm/c, and
source radii on the order of 3 fm. Furthermore, more energetic protons were
associated with shorter emission timescales. These results are consistent with
statistical proton emission from a hot cooling residue, analogous to the

evaporation from a cooling liquid drop.

APPENDIX

Appendix A - Raw and Physics Tape Formats

Raw data taken on-line in our experiment followed the 4:: Array
formatting standard, which is itself a variant of the NSCL formatting standard.
Off-line, we created ”physics tapes,” which contained calibrated data in a
somewhat streamlined format.

Below, we briefly record the data formats of the raw and physics tapes for
completeness. Details of the raw tape format may be found in the 4n L’User’s
Guide [Winf 91].

A.1 Raw Tape Format

The 41: data acquisition system organizes data into ”buffers” of 4096 16-bit
words, in accordance with the N SCL standard. The first 16 words of each buffer
constitute a ”header” containing information about the buffer, such as the buffer
sequence number and number of valid words in the buffer.

The readout scheme of the 41c Array electronics imposes a unique ,
structure onto the raw data format. This is due to the use of ECLine readout of
ADCs and QDCs, as well as to the front end system used.

All valid data from an entire module (FERA or Silena ADC) is read out at
once in zero-suppressed mode via an ECLine front panel interface. The use of
FERA Drivers (LeCroy model #4301) to read out groups of modules mandated
the grouping of modules that used a common gate. Therefore, all FERA’s that
digitize Forward Array ”slow” signals constituted a group and were read out

sequentially. Other groups were then read out in sequence. Consequently, the

163

164

various elements of data concerning one detector- fast, slow, and time (or AE, E,
and time for a hodoscope detector)-- do not appear near each other in the raw
data buffer, and decoding of module header words is required to ”relink” the
data.

A second feature of the raw buffer in the 41: format arises due to the front
end system used, the so-called ”FERA Faucet System,” designed by Michael
Maier at the NSCL. In each event, data is loaded from the ECL data bus into a
512 kB memory array, which sits in a VME crate with the front end slave and
master CPU's. Before loading data from an event, a special word is loaded into
the memory to separate data from different events. When the memory becomes
nearly full, its contents are packaged by the slave cpu into small ”FERA Faucet
(FF) buffers,” each of which has its own 16-word header, as well as a two-word
trailer to mark the end of the FF buffer. These FF buffers are then inserted into
the 8 kB NSCL buffers. Although usually a NSCL buffer will contain only one FF
buffer, in principle it may contain several, each with its own header and trailer.
Therefore, the number of FF buffers is indicated in the NSCL buffer header.

The data format is indicated in Figure A.1. Events are separated by the
hexadecimal word ’FFFF’. Each module has a header, followed by one or more
data words. The crate and slot number of each module is encoded in each
module header, and the channel number is encoded in each data word. A simple
map links each data word with a detector signal component.

The buffer type, indicated by the second word in the buffer header,
follows the NSCL standard and is useful for uniform processing of control
information, such as begin-of-run markers.

A.2 Physics Tape Format

Calibrated data was written to tape in a simpler format designed to

maximize analysis efficiency. The buffer header follows the NSCL standard, and

165

41: event separators were employed. The more natural order of the data reduces
the need for cumbersome arrays necessary for data reorganization and
encourages the use of a simpler program architecture.

Our physics tape format is shown in Figure A.2. Each detector that fired
carries 3 or 5 words of data, depending on whether it was an element in the 41:

Array or the S6—element hodosc0pe.

166

 

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Figure A.1. The data format of the uncalibrated ”raw" tapes shows a
hierarchical structure. Data words are indicated by x’s. These are
grouped into physics events, FF buffers, and NSCL buffers.

2nd

1!!

oeperdorFEflA

event

FERA

(..... an. m for m FEM.......)

header (.. d.- mum FERA.)

'FFFF' heater

X

X

X

X

167

 

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Figure A.2. The data format of calibrated

”physics" tapes follows a more

natural structure. Data for detector elements are grouped together.

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