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3 1293 00908 1997 -

 

 

 

 

 

 

 

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This is to certify that the
dissertation entitled
Two-proton Intensity Interferrmetxy
in Intermediate Energy Heavy-ion Collisions

presented by

Wen Guang Gong

has been accepted towards fulfillment
of the requirements for

Ph.D. degree in Physics

 

Zé’fi. *‘iflfibflé’aév

 

Major professor

(Dr. Claus—Konrad Gelbke)
Date October 21, 1991

 

MS U i: an Aflinmm'w Action/Equal Opportunity Institution 0- 12771

 

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MSU Is An Affirmative Action/Equal Opportunity Institution
cmmd

 

 

 

 

 

 

 

 

TWO-PROTON INTENSITY
INTERFEROMETRY IN INTERMEDIATE
ENERGY HEAVY-ION COLLISIONS

By

Wen Guang Gong

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

1991

C)

/,D/

654‘

ABSTRACT

TWO-PROTON INTENSITY INTERFEROMETRY IN
INTERMEDIATE ENERGY HEAVY-ION COLLISIONS

By

Wen Guang Gong

Two-proton intensity interferometry has been studied both experimentally
and theoretically in order to probe the space-time evolution of heavy-ion nuclear
collisions at intermediate energy.

An approximate relation is derived which allows the calculation of the two-
proton correlation function for any reaction model capable of predicting the single-
particle phase-space distribution or the Wigner function in the exit channel. The
sensitivity of two-proton correlation functions to source radii and lifetimes is
illustrated by calculations with simple parametrizations. More realistic
calculations are presented for two different regimes of emission time scales: slow
particle evaporation from equilibrated compound nuclei, as predicted from the
Weisskopf formula, and fast non-equilibrium particle emission in intermediate
energy nucleus-nucleus collisions, as predicted from the Boltzmann-Uehling-
Uhlenbeck transport equation.

Two-proton correlation functions have been measured at elab z 25 ° for the
"inverse kinematics" reaction 129Xe + 27A1 at E/ A = 31 MeV, and for the nearly
symmetric reaction 129Xe + 122Sn at E/ A = 31 MeV, for the "forward kinematics"
reactions 14’N + 27A] and 14N + 197Au at E/ A = 75 MeV. For the reactions at 31 MeV

per nucleon, the two-proton correlation functions do not exhibit maxima at q = 20

ii

MeV/c, but only minima at q = 0) MeV/c. These correlations indicate emission on
a slow time scale. They can be reproduced by calculations based on the Weisskopf
formula for evaporative emission from fully equilibrated compound nuclei. For
the reactions at 75 MeV per nucleon, the correlation functions exhibit pronounced
maxima at relative momenta, q = 20 MeV/c, and minima at q = 0 MeV/c. These
correlations indicate emission from fast, non-equilibrium processes. They are
analyzed in terms of standard Gaussian source parametrizations and compared to
microscopic simulations performed with the Boltzmann-Uehling-Uhlenbeck
equation. For all reactions, the measured longitudinal and transverse correlation
functions are very similar, in agreement with theoretical predictions.

Two-proton correlation functions predicted by the BUU model for the
reaction 14N + 27A] at E/ A = 75 MeV reveal large sensitivity to the magnitude of
the in-medium nucleon-nucleon cross sections and little sensitivity to the
compressibility of nuclear matter. Moreover, they exhibit strong dependence on
the total momenta of the emitted proton pairs when the collision impact

parameter is selected.

iii

To
my father, Yong-quart Gong,
and

my mother, Gnu-fang Wen.

iv

ACKNOWLEDGEMENTS

I would like to express my deepest gratitude to Dr. Claus-Konrad Gelbke,
first of all. This thesis would be simply impossible without his supervision,
encouragement and support. Over past years, he has taught me what
experimental nuclear reaction physics is about and how to become a good
researcher, I have enjoyed the most working with him. His enthusiasm for
scientific research, his insight into a problem, his understanding of people, his
efficiency of working are held in my greatest respect. His influence on my career
will be long-lasting.

I should thank Dr. Wolfgang Bauer, Dr. George F. Bertsch, Dr. William G.
Lynch, Dr. Bernard Pope, and Dr. Michael F. Thorpe for serving kindly on my
degree committee and giving generously cn'tical advice and helpful suggestions.
Without their work and time, it would be impossible to get my degree. In
particular, Dr. Lynch has given plenty of advice and ideas from which I benefitted
a lot, he helped my thesis experiment to make it a success. Dr. Bauer generously
accepted to be in my oral committe during Dr. Lynch’s absence. Dr. Bauer was an
essential collaborator to perform the BUU calculations. Dr. Bertsch kindly
recommended me twice to attend the Summer School in Nuclear Physics at Gull
Lake (1989) and at Santa Cruz (1990) where I learned a lot of important nuclear
physics and made many new friends.

I greatly appreciate the essential contributions from all of my collaborators.
Dr. Giacomo Poggi helped me develop the CsI(Tl) detector. It was fun working

with him who is nice and humorous. I especially appreciated his invitation and

V

hospitality for a recent visit to the IN PM in beautiful Florence. Dr. Betty Tsang
helped greatly during the experiment and taught me a lot about SMAUG. Dr. Josef
Pochodzalla generously sent me many FORTRAN codes which were useful for
calibrating CsI(Tl) detectors. Extremely productive was the collaboration with Dr.
Scott Pratt who formulated the theoretical basis for calculating two-proton
correlation functions. It was very fruitful and enjoyable to work with Drs. Dave
Sanderson, Tetsuya Murakami, Nelson Carlin, Remualdo de Souza, Hongming
Xu, Ziping Chen, and Yeongduk Kim at Michigan State University and Drs. Kris
Kwiatkowski, Vic Viola, Sherry Yennello, R. Planeta, and Doug Fields from
Indiana University who had helped me at various stages.

I am very grateful to Prof. J.S. Kovacs for attracting me through the CUSPEA
program to study in the Physics Department and work at the National
Superconducting Cyclotron Laboratory. I sincerely thank all the fellow students,
staff and faculty members of the Physics Department and the Cyclotron
Laboratory for the‘ quality education and the excellent support that I have
received. Particularly, I acknowledge a number of individuals who provided me
with useful assistance from time to time: Jerry N olen, Peter Miller and the whole

operator crew who provided the first 4 GeV 129x923+

beam for my experiment;
John Yurkon and Dennis Swan at the Detector Lab.; Michael Maier and Jim Vicent
at the Electronics Shop; Richard Au, Ron Fox and Babara Pollack at the Computer
Dept.; Len Morris, Phil Fighter, and Steve Hickson at the Mechanical Design and
Machine Shop; Rilla McHarris, Jackie Bartlett and Stephanie Holland for graphics
and paper works. I greatly appreciate the letter of recommendation from Director
Sam Austin. Furthermore, I am indebted to the friendship with David Bowman,
Ziping Chen, Yuming Chen, Yibing Fan, Mingzhuan Gong, Rich Harkewicz,
Yeongduk Kim, Yongsheng Li, Mike Lisa, Tapan N ayak, Graham Peaslee, Larry

Phair, Bo Pi, Hongming Xu, Fan Zhu during my stay at MSU.

The last but not the least thanks should go to my dear family: my father and
mother, my brother, my sister and brother-in-law for their love, understanding
and support throughout my life. It is they who always value my education at the
highest priority and encourage me to go forward. I am fortunate enough to meet
Ms. Yiwen Wang who becomes my dearest friend. Her understanding and love

has given me new strength to move on. She deserves my heartfelt thanks.

vii

TABLE OF CONTENTS

LIST OF TABLES .................................. xi
LIST OF FIGURES .................................. xii
CHAPTER 1. INTRODUCTION .......................... 1
1.1 Intensity Interferometry ........................... 1
1.2 Intermediate Energy Heavy-Ion Collisions ................. 5
1.3 Two-proton Intensity Interferometry .................... 7
1.4 Motivation ~ ................................... 8
1.5 Organization .................................. 11
CHAPTER 2. THE CSI(TL) DETECTOR ...................... 13
2.1 General Considerations ............................ 13
2.2 Description of the CsI(Tl) Detector. ..................... 18
2.3 Resolution Test of Individual Detectors .................. 22
2.4 Pulse Shape Discrimination ......................... 37
2.5 Quality Tests of CsI('I'l) Scintillators ..................... 46
CHAPTER 3. EXPERIMENTAL DETAILS ..................... 53
3.1 Mechanical Setup ............................... 53
3.2 Electronics ................................... 58
3.3 Reactions .................................... 61

viii

CHAPTER4. DATA REDUCTION ......................... 63

4.1 Particle Identification ............................. 63
4.2 Energy Calibration .............................. 65
4.2.1 Two-body Reaction Kinematics .................... 65

4.2.2 Energy-loss Calculations ........................ 72

4.3 Time-walk Corrections ............................ 81
4.4 Corrections due to Nuclear Reaction Loss ................. 86
CHAPTER 5. SINGLE-PROTON ENERGY SPECTRA .............. 90
5.1 Inclusive Energy Spectra ........................... 90

5.2 Comparison with BUU Calculations .................... 97
CHAPTER 6. TWO-PROTON CORRELATION FUNCTION .......... 99
6.1 Theoretical Formalism ............................ 99
6.2 Illustrative Calculations and Discussions ................. 111
6.2.1 Spherical Sources of Negligible Lifetime ............... 113

6.2.2 Spherical Sources of Finite Lifetime .................. 120

6.3 Comparison with Classical Trajectory Calculation ............ 125

CHAPTER 7. TWO-PROTON CORRELATION FOR

THE GAUSSIAN SOURCE ................. 127
7.1 Angle- and Energy-integrated Correlation Functions ........... 128
7.2 Dependence on Total Momentum ...................... 130
7.3 Effect of Instrumental Resolution ...................... 139

CHAPTER 8. TWO-PROTON CORRELATION FOR
EVAPORATIVE EMISSION ................. 141
8.1 Statistical Evaporation Model ........................ 141

8.2 Model Calculations . . . ' ........................... 144
8.3 Angle-integrated Correlation Functions .................. 153
8.4 Longitudinal vs. Transverse Correlation Functions ............ 157

CHAPTER 9. TWO-PROTON CORRELATION FOR

N ON-EQUILIBRIUM EMISSION ............. 159
9.1 BUU Transport Equation for Collision Dynamics ............. 159
9.2 Impact Parameter Averaging ........................ 165
9.3 Predicted Two-proton Correlation Functions ............... 167
9.4 Dependence on Total Momentum ...................... 170
9.5 Longitudinal vs. Transverse Correlation Functions ............ 178

9.6 Dependence on the Nuclear Equation of State and in-Medium

N ucleon-N ucleon Cross Section ...................... 186

9.7 Dependence on Impact Parameter ...................... 191
CHAPTER 10. SUMMARY AND CONCLUSIONS ................ 196
LIST OF REFERENCES ...................... 200

LIST OF TABLES

Table 2.1: Properties of N aI(Tl) and CsI(Tl) scintillators. ............. 14

Table 4.1: Fit parameters for energy calibrations of various particles in a
typical N aI(Tl)/ CsI(Tl) detector (N 04/ C10) using the
equation: E (MeV) = A + BON (E) + C-N(E) 2 . ............. 69

Table 4.2: Differential light output, dL/dE, of CsI('I'l) scintillator v.s.
differential energy loss, dE/ dx (MeV/ mg/ cm 2 ). ........... 71

Table 5.1: Fit parameters used for the description of the inclusive single-
proton cross sections shown in Figure 5.1. The spectra
for the l ‘ N and i 2 ° Xe-induced reactions were fitted by
Equations (5.1) and (5.2), respectively. ................. 92

LIST OF FIGURES

Figure 1 .1: Schematic diagram of (1) an amplitude interferometer (upper
part) and (2) an intensity interferometer (lower part). ........ 3

Figure 1 .2: Illustration of source functions for emission from short-
lived (upper part) and long-lived (lower part) nuclear
systems. The dots indicate the locations of protons of
a given momentum after the last proton has been
emitted. .................................. 9

Figure 2 .1: Measured energy spectrum of y-rays from a 6 ° Co source with a

small volume CsI(Tl) scintillator (1x2x4 cm 3 ) read out

by a Hamamatsu PIN diode (51790: 1x1 cm 2 ), where a

commercial charge-sensitive preamplifier (Canberra 2003)

was used. ................................. 16
Figure 2 .2: Measured energy spectra of y-rays from a ‘ 3 7 Cs source (upper

part) and a 6 ° Co source (lower part) with a small cubic

CsI(Tl) crystal (2x2x2 cm ’ ) read out by a Hamamatsu PIN

diode (2x2 cm 2 ), where a homemade charge-sensitive

preamplifier was used. ......................... 17

Figure 2 .3: Anatomic diagram of the CsI(Tl) detector. .............. 19

Figure 2 .4: Schematics of the preamplifier used for the readout of the
PIN diode. The preamplifier is mounted on a circuit
board of 38 mm diameter. ........................ 21

Figure 2 .5: Crystal-fixed frame of reference for y-ray scanning tests. ..... 24

Figure 2 .6: Left-right asymmetric response of crystal #1 for collirnated
y-rays of 662 keV, protons of 56.8 MeV, and a-particles
of 89.1 MeV. Shown is the relative shift of the peak
centroid as a function of the X-coordinate of the
collimator, keeping Y=0. The data are normalized to the
peak position measured for X=Y=O. .................. 26

xii

Figure 2 .7: Position sensitive response of crystal #2 for collimated 7-
rays. The open points show measurements as a function
of the Y-coordinate of the collimator keeping X=0; the
solid points show the dependence on the X-coordinate
keeping Y=0. Part a) and b) show measurements using a
square PIN diode and a photomultiplier with circular
photocathode, respectively. ....................... 28

Figure 2 .8: Position sensitive response of crystal #3 for collimated 'y-
rays. The open points show measurements as a function
of the Y-coordinate of the collimator keeping X=0; the
solid points show the dependence on the X-coordinate
keeping Y=0. Part (a): Standard sanding treatment of
the reflecting surfaces; part (b): original detector
with polished front face. ......................... 30

Figure 2 .9: Percentage shift of the AE, E, and summed AE+E signals as a
function of the X-coordinate of the particle
collimator, keeping Y=0. (Crystal #3) ................. 32

Figure 2.10: Energy spectra of protons and (It-particles obtained at 9:20 °
for reactions induced by 96 MeV a-particles on a CH 1
target. The detector collimator was 3 mm in diameter.
(Crystal #3) ................................ 33

Figure 2.11: Energy spectrum of a-particles detected at 9:20 ° for the 0t+Au
reaction at 96 MeV. A collimator of 20 mm diameter
was used. (Crystal #3) .......................... 34

Figure 2.12: Energy calibration for crystal #3. ................... 35

Figure 2.13: Position sensitive response of crystal #3 as detected by
collimated protons of 178 MeV (solid points) entering
the front face (Z=0) of the crystal with the PIN diode
mounted at the rear end (Z=112 mm) and by collimated y-
rays (open points) entering the rear face (Z=100 mm) of
the crystal with the PIN diode mounted at the front end
(Z=-12 mm). Part (a) shows measurements as a function
of the X-coordinate of the collimator keeping Y=0; part
(b) shows measurements as a function of the Y-
coordinate of the collimator keeping X=0. .............. 36

Figure 2.14: Particle identification spectrum obtained with the AE-E

xiii

technique and employing the linearization function PID

= (E+AE)1°8- E18. The interval of energy integration

was 45-70 MeV. To eliminate inhomogeneities of the AE-
silicon detector, a collimator of 3 mm diameter was
used. .................................... 39

Figure 2.15: Particle identification spectrum obtained with the AE-E
technique and employing the linearization function PID

= (E+AE)1'8- E18. The interval of energy integration

was 45-70 MeV. A collimator of 20 mm diameter was used.
Identification spectra are shown for the count rates of

10 ’ cps (shaded histogram) and 1.4x10 ‘ cps (upper

histogram). ................................ 40

Figure 2.16: Particle identification spectra obtained with the pulse shape
discrimination technique. The interval of energy
integration was 45-70 MeV. A collimator of 20 mm
diameter was used. Identification spectra are shown for
the count rates of 10 ’ cps (shaded histogram) and
1.4x10‘ cps (upper histogram). .................... 42

Figure 2.17: Particle identification efficiency as a function of count
rate in the CsI detector obtained with the pulse shape
discrimination technique. The slow pulse shaping was
performed using a semi-gaussian bipolar filter of
shaping time constants of 3 us (upper part) and 2 us
(lower part). The interval of energy integration was
45-70 MeV. A collimator of 20 mm diameter was used. ....... 44

Figure 2.18: Relative timing between the CsI(Tl) E and the silicon AE
detectors using constant fraction discriminators. The
interval of energy integration was 45-70 MeV. ............ 45

Figure 2.19: Maximum variations of scintillation response observed for
collimated y-rays entering the detector front face at
various points located on a circle of 1 cm radius and
centered at the detector axis. The shaded histogram
shows the measurements for detectors produced with the
new growth technique; the unshaded histogram shows
measurements for detectors fabricated with the old
technique. ................................. 49

xiv

Figure 2.20: Energy spectra measured with a CsI(Tl) detector produced with

the new technique. ...........................

Figure 2.21: The stability of pulse-height of a CsI('Tl) detector. ........

Figure 3 .1: Schematic view of the experimental setup with two hodoscopes

being used. ...............................

Figure 3 .2: Schematic view of the front face of the 56 element hodoscope.
Figure 3 .3: Schematic view of the front face of the 13 element hodoscope.

Figure 3 .4: Polar coordinates of 50 AE-E telescopes used in the

experiment. ...............................

Figure 3 .5: Block diagram of the electronic circuits and trigger logic. . . . .

Figure 3 .6: Diagram for processing the logic signals in a Si-NaI(Tl) or

Si-CsI(Tl) telescope. ..........................

Figure 4 .1: Two-dimensional plots of PID v.s. E for a Si-NaI(Tl)
telescope (left panel) and for a Si-CsI(Tl) telescope

(right panel) whereon particle gates were set. ...........

Figure 4 .2: Energy calibration of protons, deuterons, tritons, ’ He and
‘ He for a CsI(Tl) detector (C10) obtained by two-body
reaction kinematics. The curves are explained in the

text. ...................................

Figure 4 .3: Energy calibration of protons for a NaI(T1) detector (N04)
obtained by two-body reaction kinematics. The curves

are explained in the text. .......................

Figure 4 .4: Two-dimensional map of AE (the energy loss in Si. detector)
v.5. E (the energy loss in CsI(Tl) detector) which was
used for the off-line energy calibrations of all

identified particles in a CsI(Tl) detector (C10). ...........

Figure 4.5: Two-dimensional map of AE (the energy loss in Si. detector)
v.s. E (the energy loss in N aI(T l) detector) which was
used for the off-line energy calibrations of all

identified particles in a N aI(Tl) detector (N04). ...........

XV

.50

.51

.54

.55

.56

.57

.59

.60

.64

.67

.68

.73

.74

Figure 4.6: Energy calibration of protons, deuterons, tritons, 6 He, 6 He,
and 6 He for a CsI(Tl) detector (C10) obtained by both
two-body reaction kinematics and energy-loss
calculations. The curves are explained in the text. .......... 76

Figure 4 .7: Energy calibration of 6 Li, 6 Li, 6 Li, 6 Be, 6 Be, 6 6 Be, and B
for a CsI('Il) detector (C10) obtained by energy-loss
calculations. The curves are linear fits to the data
points. ................................... 77

Figure 4 .8: Comparison of energy calibrations for particles such as
proton, 6 He, 6 Li, 6 Be, and B measured in a CsI(Tl)
detector (C10). The curves are the corresponding fits. ....... 78

Figure 4 .9: Energy calibration of protons, deuterons, tritons, 6 He, 6 He,
6 He, 6 Li, and 6 Li for a N aI(Tl) detector (N04) obtained
by energy-loss calculations. The curves are
corresponding fits to the data points. ................. 79

Figure 4.10: Comparison of energy calibrations for particles such as
proton, 6 He, and 6 Li measured in a N aI(Tl) detector
(N04). The curves are the corresponding fits. ............. 80

Figure 4.11: Two-dimensional plots of (T - TRF) v.s. E for a Si-NaI(Tl)

telescope (left panel) and for a Si-CsI(Tl) telescope
(right panel) where necessary time-walk corrections
were made. ................................ 82

Figure 4.12: Relative timing spectra between any two telescopes in the
75MeV/nucl. 6 6 N induced reactions on 6 6 Al (left part)
and on 6 6 6 Au (right part). Timing gates for real and
random coincident events are indicated by vertical
lines. ................................... ‘ . 83

Figure 4.13: Relative timing spectra between any two telescopes in the
31MeV/ nucl. 6 6 6 Xe induced reactions on 6 6 Al (left part)
and on 6 6 6 Sn (right part). Timing gates for real and
random coincident events are indicated by vertical
lines. .................................... 84

Figure 4.14: The probabilities of nuclear reaction loss for light-charged
particles detected by a 10cm long N aI(Tl) or CsI(Tl)

xvi

scintillator. ................................ 88

Figure 5 .1: Inclusive proton cross sections measured, at elab=18 ° and

33°,forthereactions 6 6 6Xe+6 6Aland 6 6 6Xe+6 6 6Sn at

E/A=31 MeV (left hand panels) and the reactions

‘ 6N+ ’ 6A1 and ‘ 6N+6 ° ’ Au at E/A=75 MeV (right hand

panels). .................................. 91

Figure 5 .2: Decomposed three moving source fits to inclusive proton
energy spectra measured, at elab=18 ° and 33 ° , for 75

MeV/nucl. 6 6N induced reactions on 6 6 Al (left two
panels) and on 6 6 6 Au (right two panels). .............. 93

Figure 5 .3: Decomposed two moving source fits to inclusive proton energy
spectra measured, at 01ab=18 ° and 33 ° , for 31 MeV/nucl.

Xe induced reactions on 6 6 Al (left two panels) and
I 1 2

on Sn (right two panels). ...................... 94

[29

Figure 5 .4: Single-proton cross sections calculated with the BUU theory
(open points) are compared to experimental cross
sections (solid points) for the reactions 6 6 N + 6 6 Al (top
panel) and 6 6N+6 6 6Au (bottom panel) at E/A=75 MeV.
Circular and diamond shaped symbols indicate laboratory
angles of 18 ° and 33 ° , respectively. .................. 98

Figure 6 .1: Two-proton correlation functions calculated for Gaussian
sources of negligible lifetime with representative
radius parameters, r 0 =2.5, 5, 10, 20 fm. The solid
curves show the result of the full calculations; the
dotted curves show calculations for which the nuclear
interaction is neglected; the dashed curves shown
calculations for which the nuclear interaction and the
Pauli principle are neglected. ...................... 113

Figure 6 .2: Two-proton correlation functions calculated for Gaussian
sources of negligible lifetime with representative
radius parameters, r 0 =2.5, 5, 10, 20 fm. Assuming the
two-proton relative wavefunction to be:
I 4) I 6 =w80 l 6 q; l 6 +wt- l 6 d) l 6 , the solid, dotdashed and dashed

curves show calculations for (1) ws=1 and wt=0, (2)

xvii

ws=1 / 4 and wt=3/ 4, (3) ws=0 and wt=1, respectively. ....... 114

Figure 6 .3: Two-proton correlation functions calculated for sources of
negligible lifetime assuming Gaussian (solid lines) and
sharp sphere (dotted lines) density distributions. The
radius parameters, r o and Rs’ are indicated. ............. 116

Figure 6 .4: Relation between radius parameters, r o and R5’ of Gaussian

and sharp sphere density distributions for which
equivalent two-proton correlation functions are
obtained in the limit of negligible lifetime. Crosses
indicate results of numerical calculations, the solid
line represents a linear fit, and the dotted curve
shows the relation RS=/(5 / 2) r 0 used in the literature

[Boal 90]. ................................. 117

Figure 6 .5: Two-proton correlation functions predicted for emission from
spherical sources of radius RS=5 fm, decaying

isotropically with fixed life-time, Equation (6.28).
The top and bottom panels depict the dependence on
life-time, T, and total momentum, PC m , respectively. ...... 119

Figure 6 .6: Longitudinal and transverse correlation functions calculated
for emission from sources decaying with constant
lifetimes, Equation (6.28). The parameters used in
these calculations are indicated in the figure. ............. 121

Figure 6 .7: Two—proton correlation functions for a schematic source,
Equation (6.29), representing thermal surface emission
with fixed life-times, I=200fm/c (upper part),
I =500fm/c (middle part), and I =1000fm/ c (lower part).
The parameters are indicated in the figure. Different
symbols and curves are explained in the text. ............ 125

Figure 7 .1: Comparison of energy integrated two-proton correlation
functions measured for the reactions 6 6 N + 6 6 Al and
6 6 N + 6 6 6 Au at E / A=75 MeV (top panels) and the reactions
6 6 6Xe+6 6A1 and 6 6 6Xe+6 6 6Sn at E/A=31 MeV (bottom
panels). The solid curves represent correlation

xviii

functions predicted for Gaussian sources of negligible
lifetime with the indicated radius parameters, r o . .......... 129

Figure 7 .2: Two-proton correlation functions measured for the reactions
66 6Xe+ 6 6A1 and 6 6 6Xe+6 6 6Sn at E/A=31 MeV. The gates on
the total momenta, P, of the coincident proton pairs
are indicated; solid and open points represent center-
of-mass energies below and above the compound nucleus
Coulomb barriers. ............................ 132

Figure 7 .3: Two-proton correlation functions measured for the reaction
6 6 N + 6 6 A1 at E/A=75 MeV. The gates placed on the total
momenta, P, of the coincident particle pairs are
indicated. The solid curves represent calculations for
Gaussian sources of negligible lifetime assuming a
dependence of the radius parameter, r o , on total
momentum, P, as shown by the open points in Figure 7.5
and folded with the response of the experimental
apparatus. ................................. 133

Figure 7.4: Two-proton correlation functions measured for the reaction
6 6 N + 6 _6 6 Au at E/A=75 MeV. The gates placed on the total
momenta, P, of the coincident particle pair are
indicated. The solid curves represent calculations for
Gaussian sources of negligible lifetime assuming a
dependence of the radius parameter, r o , on total
momentum as shown by the solid points in Figure 7.5 and
folded with the response of the experimental apparatus. . . . . 134

Figure 7 .5: Radius parameters, r o , for Gaussian sources of negligible
lifetime extracted from two-proton correlation
functions gated by different total momenta, P, of the
coincident particle pairs for 6 6 N induced reactions on
6 6 Al and 6 6 6 Au at E/A=75 MeV. The error bars indicate

estimated systematic errors. ...................... 136
Figure 7.6: Systematics of Gaussian source radii, r o (v/vbeam), extracted

from various asymmetric reactions. The points and

curves are explained in the text. .................... 138

Figure 7.7: Monte-Carlo simulation for the response of the experimental
apparatus. The curves show the undistorted correlation

xix

functions assumed for the indicated gates on total

momentum, P. The points represent the calculated

response of the apparatus after taking the energy and

angular resolution effects into account. The error bars

show the statistical accuracy of the Monte-Carlo

calculations. ................................ 140

Figure 8 .1: Temporal evolution of particle emission from equilibrated
6 6 6 Ho nuclei of different initial temperatures, T (top
panel), and for different total momenta, PC m (bottom

panel), of the emitted twooproton pairs. ................ 145

Figure 8 .2: Dependence of two-proton correlation function on the total
momentum, PC m (bottom panel), and on the initial

temperature, T (top panel), of emitted proton pairs
calculated for the decay of equilibrated 6 6 6 Ho nuclei. ..... 146

Figure 8 .3: Sensitivity of calculated two-proton correlation functions to
different assumptions on the level density parameter,
a=E"/ AT 6 , for the decay of 6 6 6 Ho compound nuclei of
initial excitation energy E*/A=6 MeV. ................ 148

Figure 8 .4: Sensitivity of calculated two-proton correlation functions to
different assumptions on the level density parameter,
a=E"‘/ AT 6 , for the decay of 6 6 6 Ho compound nuclei of
initial temperature T=10 MeV. ..................... 149

Figure 8 .5: Longitudinal and transverse correlation functions predicted
for evaporative emission from 6 6 6 Ho compound nuclei of
initial temperature, T=10 MeV. Different panels show
predictions for different total momenta, PC m , of the

emitted proton pairs: PC m =500 MeV/c (top panel),
P =400 MeV/c (center panel), P =300 MeV/c (bottom
c.m. c.m.
panel). ................................... 151
Figure 8.6: Two-proton correlation functions measured for the 6 6 6 Xe+ 6 6 Al
(part a) and 6 6 6Xe+6 6 6Sn (parts b,c) reactions at
E/A=31 MeV for the indicated gates on the total

momenta, P, of the two-proton pairs. The curves
represent calculations for evaporative sources at rest

XX

in the center-of-mass frame of reference; the
parameters are indicated on the figure. ................ 153

Figure 8 .7: Two-proton correlation functions measured for the 6 6 6 Xe+ 6 6 Al
(part a) and 6 6 6 Xe+-6 6 6Sn (part b) reactions at E/A=31
MeV for the indicated gates on the total momenta, P, of
the two-proton pairs. The curves represent calculations
for evaporative sources at rest in the projectile frame
of reference; the parameters are indicated on the
figure. ................................... 155

Figure 8 .8: Longitudinal (‘I’=0 ° -40 ° or ‘I’=140 ° -180 ° ) and transverse (‘I’=60 ° -
120 °) two-proton correlation functions measured for the
reaction 6 6 6 Xe+ 6 6 Al (top panel) and 6 6 6 Xe+ 6 6 6 5n (bottom
panel) at E/A=31 MeV. ......................... 157

Figure 9 .1: Dependence of correlation functions predicted by BUU
calculations on the emission time intervals, Ate, and

the emission densities, pe. The values of individual

parameter choices and selected total momenta of the

proton pairs are given in the figure. In these

calculations, the in-medium cross section was

approximated by the experimental free nucleon-nucleon

cross section, and the stiff equation of state was

used. .................................... 169

Figure 9 .2: Two-proton correlation functions, measured for the reaction
6 6 N + 6 6 A1 at E/A=75 MeV, are compared with correlation
functions predicted with the BUU theory. The gates
placed on the total momenta, P, of the coincident
proton pair are indicated. ........................ 171

Figure 9 .3: Radius parameters r o (P) for Gaussian sources of negligible
lifetime extracted from two-proton correlation
functions gated by different total momenta P. Solid and
open circles represent experimental and theoretical
correlation functions, respectively. ................... 172

Figure 9 .4: Two-proton correlation functions, measured for the reaction
6 6 N + 6 6 6 Au at E/A=75 MeV, are compared with correlation
functions predicted with the BUU theory. The gates
placed on the total momenta, P, of the coincident

xxi

proton pair. are indicated. ........................ 173

Figure 9 .5: Comparison of two-proton correlation functions predicted for
different source geometries: the solid points represent
the results of BUU calculations averaged over the
momentum range P=450~780 MeV/c. The curves show
emission from sources of negligible lifetime: the solid
and dashed curves are obtained, respectively, for a
spherical Gaussian source of radius parameter, r 0 =4.5
fm, and a source consisting of two sharp spheres of
radius, Rs=5 fin, and separated by the distance, d=20

fm. ..................................... 175

Figure 9 .6: N ucleon density distributions in the reaction plane
calculated from the BUU equation for 6 6 N + 6 6 Al
collisions at E/ A=75 MeV and for an impact parameter of
b=2 fm. Different panels depict the distributions at
different times, t. ............................. 176

Figure 9 .7: Spatial distributions of emitted nucleons in the reaction
plane calculated from the BUU equation for 6 6 N + 6 6 Al
collisions at E/ A=75 MeV and for an impact parameter of
b=2 fm. Different panels depict the distributions at
different times, t. ............................. 177

Figure 9 .8: Longitudinal (‘I’=0 ° -40 ° or ‘I’=140 ° -180 ° ) and transverse (‘I’=60 ° -
120 °) two-proton correlation functions measured for the
6 6 N + 6 6 Al reaction at E/A=75 MeV. In the left hand
panels, longitudinal and transverse correlation
functions were normalized independently; in the right
hand panels, the normalizations were determined from
the ‘Y-integrated data. .......................... 180

Figure 9 .9: Longitudinal (‘f'=0 ° -40 ° or ‘I-‘=140 ° -180 ° ) and transverse (‘I’=60 ° -
120 °) two-proton correlation functions measured for the
6 6 N + 6 6 6 Au reaction at E/A=75 MeV. In the left hand
panels, longitudinal and transverse correlation
functions were normalized independently; in the right
hand panels, the normalizations were determined from
the ‘P-integrated data. .......................... 181

Figure 9.10: Longitudinal (‘I’=0 ° -40 ° or ‘l’=140 ° -180 ° ) and transverse (‘I’=60 ° -

xxii

120 °) two-proton correlation functions predicted by BUU

calculations. The left and right hand panels show

calculations for the reactions 6 6 N + 6 6 Al and 6 6 N + 6 6 6 Au,
respectively. The momentum cuts are indicated in the

individual panels. ............................ 184

Figure 9.11: Longitudinal, and transverse correlation functions predicted
by BUU calculations for 6 6 N + 6 6 Al collisions at E/A=75
MeV for proton pairs for the indicated total momenta,
P. ...................................... 185

Figure 9.12: Sensitivity of two-proton correlation functions to the
nuclear equation of state (E05) and the in-medium
nucleon-nucleon cross section. Different panels show
results calculated for different total momenta, P, of
the proton pairs. ............................. 188

Figure 9.13: Two-proton correlation functions, measured for the reaction
6 6 N + 6 6 A] at E/A=75 MeV, are compared with correlation
functions predicted with the BUU theory assuming two
different values of the in-medium nucleon-nucleon cross
section. The total mometum gates are indicated. ........... 190

Figure 9.14: Two-proton correlation functions predicted by BUU
calculations for 6 6 N + 6 6 Al collisions at E/A=75 MeV for
the indicated impact parameters and total momenta of
the emitted proton pairs. ........................ 192

Figure 9.15: Momentum dependence of the heights of the maxima of two-
proton correlation functions predicted by BUU
calculations for 6 6 N + 6 6 Al collisions at different
impact parameters. Lines connect points corresponding
to a given impact parameter to guide the eye. ............ 193

xxiii

CHAPTER 1. INTRODUCTION

1.1 Intensity Interferometry

Interferometric measurements are of great importance and wide application
in physics and astronomy [Mich 03, Hari 85, Swen 87, Boal 90]. In the early 19th
century, the interference study by Young’s double-slit experiment contributed to
the establishment of wave theory of light, which seemed to settle down the
N ewtonian-Huyghensian controversy over the ultimate nature of light (vibratory
wave or corpuscular particle) [Swen 87]. Several decades later, Michelson
perfected the technique of interferometry to perform the famous ’aether drift’
experiment in 1881 [Mich 03, Mich 27]. The null results obtained by him led to the
rejection of the ’aether’ concept and laid the foundation for the special theory of
relativity. Two of the very important applications of Michelson’s interferometer
(an amplitude interferometer) were to measure the length of the Pt-Ir bar (which
was then the international standard of meter) in terms of the wavelength of light
in 1896 [Mich 03] and to measure the diameters of Betelgeuse and six other stars in
1920 [Hari 85].

A conceptually different interferometer (an intensity interferometer) was
proposed by Hanbury Brown and Twiss in 1954 [Hanb 54]. To measure the
angular size of stars (on the order of 10.4 arc second), the use of an amplitude
interferometer had two difficulties. One was mechanical stability associated with a
long base-line, the other was atmospheric turbulence which introduced random

phase variation in the light-path. However, the intensity interferometer

1

2

enabled Hanbury Brown to overcome those difficulties and measure angular sizes
of 32 stars accurately [Hanb 74].

Figure 1.1 shows the schematics of two types of interferometers: the
amplitude interferometer and the intensity interferometer [Boal 90]. In an
amplitude interferometer, particles or waves from the source are split by double
slits (or beam splitter in Michelson’s interferometer) into two paths. They interfere
constructively or destructively at the detector or on the screen to give rise to fringe
patterns. The measurement of fringe visibility and slit seperation can provide
information on the angular size of source. In an intensity interferometer, two

particles emitted from the source are detected in coincidence by two detectors. A

correlation function C(p6 1,1362) or R(p61,p62) is generated as follows:

<n12 >

<n1> <n2>

 

C(f; 1,1362) = 1 + R6132) = (1.1)

Here n12 is the number of counts where particles are detected in coincidence by

both detectors and ni(i=1,2) are the numbers of counts where particles are detected

individually by each detector. Information on the source size can therefore be
extracted from such correlation functions in a two-particle coincidence
experiment.

While amplitude interferometry involves the one-particle probability,
intensity interferometry measures the two-particle joint probability. As photons
are indistinguishable bosons, the symmetrization of the two-photon wave function
results in the two-photon correlations [Paul 86]. After its original application in
astronomy [Hanb 74], two-photon intensity interferometry has recently been used
[Frib 85] to study the process of parametric down-conversion [Burn 70] in which

pump photons incident on a non-linear dielectric fission into two highly

 

LAmplitude Interferometry

 

 
 

 

 

 

Slits Screen
D3
S gl:
ource ' Detector
O
52
Intensity Interferometry I
Detector 1

Source

6

 

Q

Detector 2

Figure 1 .1: Schematic diagram of (I) an amplitude interferometer (upper part)
and (2) an intensity interferometer (lower part).

4

correlated lower frequency signal and idler photons. The time interval between
the signal and idler photons were accurately measured to be about 100fs [I-Iong
87]. Further experiments demonstrated the non-classical and non-local behaviors
of photons [Cu 89, Cu 90]. Based upon two-particle interferometry technique, new
experiments were proposed to test the fundamental question of completeness of
quantum mechanics [Fran 89, Horn 89].

The study of pion correlations in proton-anti-proton annihilation by
Golderhaber et. al [Gold 59, Gold 60] marked the first application of intensity
interferometry using pions in subatomic physics. Since the origin of pion
correlations was understood due to Bose-Einstein statistics, two-pion intensity
interferometry was mostly referred to as Bose-Einstein correlation. Following
extensive theoretical work [Shur 73, Cocc 74, Kopy 74, Yano 78, Gyul 79], two-pion
intensity interferometry has become a valuable tool to study various properties
such as chaoticity (the degree of incoherence), radius, lifetime and shape of the
pion emitting source in elementary particle collisions as well as nuclear collisions
[Boal 90]. Promisingly, it may be able to probe the formation of quark-gluon
plasma which is a new state of matter to be created in the ultra-relativistic heavy-
ion collisions [Bert 89].

Two-proton intensity interferometry was put forward by Koonin in order to
probe the space-time structure of the collision dynamics in medium-to-high
energy nuclear reactions [Koon 77]. Since protons are fermions, the anti-
symmetrization of two-proton wavefunctions results in an anti-correlation.
However, two-proton correlations are mainly dominated by correlations resulting
from final state interactions including strong and Coulomb interactions. It was
shown that the shape of two-proton correlation functions at small relative
momenta (less than 50MeV/c) is very sensitive to the source size. Two—proton

correlations have subsequently been measured to investigate the space-time

5

characteristics of the emitting source in heavy-ion induced reactions at a wide
range of energies [Boal 90].
1.2 Intermediate Energy Heavy-Ion Collisions

Heavy-ion collisions at intermediate energy (E/A=20-200MeV) has become
an active area of research using heavy—ion accelerators [Gelb 87, Bord 90, Greg 86].
The goal of research is to understand the thermodynamical properties of nuclear
matter under extreme conditions and the nuclear dynamics leading to thermal
equilibration. It is a difficult and challenging task because of the unique
complexity involved with the finite nuclear system which interacts strongly and
evolves on a very short time scale.

The dynamics and thermodynamics of nucleus-nucleus collisions are largely
governed by the bombarding energy and the impact parameter. At low energy of
a few MeV per nucleon, heavy-ion collisions are dominated by the nuclear mean
field and one.body dissipation. Peripheral collisions are characterized by quasi-
elastic and deeply inelastic collision. Central collisions lead to complete fusion of
projectile and target, and the process can be well described by the formation of
compound nucleus and its decay by statistical evaporation of particles and y-rays
[Birk 83].

Relativistic heavy-ion collisions at high energy (E/ A 2 0.5 GeV) are
dominated by individual nucleon-nucleon collisions as the nucleon mean-free
path becomes short and comparable to nucleonic size. The geometrical concept in
the participant-spectator model proves to be valid. Thermal equilibrium may be
rapidly obtained within the volume of participant nucleons to which statistical
models may be applicable [DasG 81, Cser 86, Stoc 86, Stiic 86].

Heavy-ion collisions at intermediate energy represent the transition region in
which both nuclear mean field and nucleon-nucleon collsions are important. The
incomplete fusion model combines mechanisms of both low energy and high

energy collisions. The partially fused system can be highly excited. As the

6

emission of particles prior to the] attainment of full statistical equilibrium of the
composite system is observed to be important, it remains an open question to
know accurately how much excitation energy is deposited into the composite
system. The knowledge of the excitation energy is essential to determine the decay
of the system in a statistical approach. At low excitation energy, the excited system
undergoes binary sequential decays. As the excitation energy increases, multi-
fragment emissions become important. At even higher excitation energy, the
system may approach its limit of stability and explode into its constituents. A
nuclear liquid-gas phase transition of the excited system is expected to occur.
Therefore, systematic study of the final products of collisions can provide
information on the properties of hot nuclei of temperature T=0-10MeV and
density of approximately 0.1 to 1.5 times normal nuclear density.

Significant progress has been made recently in both experimental
measurements and theoretical studies to understand intermediate energy heavy-
ion collisions. More exclusive experiments were carried out to measure most of the
final reaction products including y rays, neutrons, light charged particles,
intermediate mass fragments and fission fragments using high efficiency detector
arrays and/ or 41: detectors [Tsan 89, Kim 89, Pias 91, Bowm 91, DeSo 91, Soho 91].
A complete characterization of the final states of the collsion may be possible
experimentally. Microscopic models have been based upon computer simulations
of the transport equation of nucleons under the influence of nuclear mean field,
nucleon-nucleon collision, and phase-space blocking due to Pauli principle [Bert
88]. They provided useful insights into the dynamics leading to statistical
thermalization. To extract any information on the thermodynamical property of
nuclear matter (e.g. the nuclear equation of state) from the measurement of
reaction products, we need a good understanding of both the dynamical and the
statistical aspects of the collision and develop and test microscopic models capable

of describing the space-time evolution of the reaction.

7

We have exploited the two-proton correlation function as an observable to
test space-time geometries of intermediate energy heavy-ion collisions predicted
by dynamic models such as the Boltzmann-Uehling-Uhlenbeck model, as it was
originally proposed in Reference [Koon 77].

1.3 Two-proton Intensity Interferometry

Two protons, emitted at small relative momenta from an excited nuclear
system, carry information about the space-time characteristics of the emitting
source since the relative two-proton wave function reflects the interplay of the
mutual Coulomb and nuclear interactions and the exclusion due to Pauli principle
[Boal 90, Koon 77, Prat 87, Gong 91a]. The attractive S-wave nuclear interaction
leads to a pronounced maximum in the two-proton correlation function at relative
momentum, q==20 MeV/c, when the average distance upon emission is of the
order of 10 frn or less. The long-range Coulomb interaction and the Pauli exclusion
principle give rise to a minimum at q=0 MeV/c. The detailed shape of two-proton
correlation functions has to be calculated by incorporating correctly all the
physical ingredients.

The average distance between the two protons upon emission depends on
the spatial dimension and the lifetime of the emitting system. Consider two
protons with an average velocity, v, emitted from a static source of radius, r, and
lifetime, 1:. After emission, the separation between the two protons is r+v1. For
the decay of equilibrated compound nuclei with temperatures below 5 MeV,
estimated emission times are larger than several hundred fm/c [Frie 83]. As a
consequence, the average distance between emitted protons is much larger than
the size of the emitting nucleus and the effects of the Coulomb interaction and the
Pauli principle should dominate. On the other hand, non-equilibrium proton
emission in intermediate energy heavy-ion collisions is calculated to proceed on

much shorter time scales [Aich 85, Cass 88] and average proton separations may

8

reflect the spatial dimension of the emitting system rather than the emission rate.
Here, the nuclear interaction should be prominent.

Figure 1.2 illustrates the apparent sources expected for emission from short-
lived and long-lived nuclear systems [Bert 89]. For correlations at small relative
momenta, the detected protons have nearly the same final momenta directed
towards the detection system. The dots in the figure illustrate the locations of
protons moving towards the detector with a given momentum, 3. Protons emitted
from a short-lived source (upper part) occupy a small region of space, but protons
emitted from a long-lived source (lower part) occupy a large and elongated region
of space [Prat 87, Awes 88, Bert 89]. The direction of elongation is along the
direction of B. In general, reduced correlations are expected for emission from long-
lived sources due to the larger apparent source size. Moreover, for an elongated
source, the Pauli anti-correlation should be less in the longitudinal (elongated)
direction than in the transverse (non-elongated) direction due to the directional
dependence of anti-symmetrization effects [Koon 77, Prat 87, Awes 88] which are
important for Ian—'6 I zh, where If and 176 denote the relative momentum and position
vectors upon emission. The longitudinal correlation function (for which the
relative momentum,q6 #:1136132), is parallel to total momentum, P6 = 31 +32) of
a long-lived source may therefore be enhanced as compared to the transverse
correlation function (for which :16 is perpendicular to P6), unless the apparent source

region becomes so large that sensitivity to anti-symmetrization effects is lost.

1.4 Motivation

In most measurements of two-proton correlation functions, implicit
summations over the relative angle, ‘I’=cos6l(P6°q6/Pq), between relative and total
momenta of the proton pair were performed. While such measurements did not
explore the shape of the source function, they did corroborate the qualitative
expectations based upon the lifetime arguments outlined above. Two-proton

 

MSU-90-067
FAST EMISSION
/
REACTION 662
DETECTOR

o 0.... ...:o..°TO
D

°..16..:DETECTOR

 

 

Figure 1.2: Illustration of source functions for emission from short-lived (upper
part) and long-lived (lower part) nuclear systems. The dots indicate the locations
of protons of a given momentum after the last proton has been emitted.

10

correlation functions measured in kinematic regions dominated by evaporation
from equilibrated reaction residues [DeYo 89, Ardo 89, DeYo 90, Gong 90b] exhibit
a minimum at q=0 MeV/ c, but no maximum at qz20 MeV/ c. The shapes of these
correlation functions could only be described by assuming emission from long-
lived compound nuclei or, alternatively, from short-lived systems of unphysically
large dimensions. In contrast, two-proton correlation functions measured in
kinematic regions dominated by fast non-equilibrium emissions exhibit a clear
maidmum at q~20 MeV/c [Zarb 81, Lync 83, Gust 84, Poch 86, Chen 87a, Chen
87b, Poch 87, Fox 88, Awes 88, Cebr 89, Queb 89, Gong 90b, Gong 90c] which
becomes more pronounced with increasing kinetic energy of the emitted protons
[Lync 83, Poch 86, Chen 87b, Poch 87, Awes 88, Gong 90b, Gong 90c]. The shapes
of these correlation functions could be described in terms of short-lived sources
with dimensions comparable to those of the respective compound nuclei; emission
from systems with even smaller dimensions was required for the description of
correlation functions measured for the most energetic protons [Lync 83, Poch 86,
Chen 87b, Gong 91b].

More recently, longitudinal and transverse correlation functions were
measured, both for non-equilibrium [Awes 88] as well as equilibrium emissions
[Ardo 89, Gong 90b]. None of these investigations found definitive evidence for
elongated source shapes. For the case of equilibrium emission, these findings were
shown to be consistent with theoretical correlation functions predicted by the
Weisskopf formula for evaporation from equilibrated compound nuclei [Ardo 89,
Gong 90b].

In order to elucidate similarities and differences of two-proton correlation
functions for equilibrium and non-equilibrium emission processes, we performed

measurements at elab == 25° for 6 6N induced reactions on 6 6 Al and 6 6 6Au at

E/A = 75 MeV and for 6 6 6 Xe induced reactions on 6 6 Al and 6 6 6Sn at E/A = 31

MeV. When light projectiles impinge on heavy target nuclei, emission at forward

11

angles is dominated by non-equilibrium processes and emission at backward
angles is dominated by equilibrium processes [Poch 87, Queb 89, Ardo 89]. Taking
advantage of this angular dependence, we studied non-equilibrium emission in
"forward kinematics" for the reactions 6 6N + 6 6A1 and 6 6N + 6 6 6Au at E/A =
75 MeV and equilibrium emission in "inverse kinematics“ for the reaction 6 6 6 Xe
+ 6 6 Al and for the nearly-symmetric reaction 6 6 6 Xe + 6 6 6Sn at E/ A = 31 MeV.
The measurements were performed with identical detector geometries, energy
calibrations and energy thresholds.

Two-proton correlation functions have been calculated by applying the
Wigner function formalism [Koon 77, Prat 87, Gong 91a] to one-body phase-space
distribution predicted by the Boltzmann-Uehling-Uhlenbeck transport equation as
well as the Weisskopf formula. Comparisons between the measured and
calculated two-proton correlation functions provide tests of the space-time
structure of specific reaction models. N on-equilibrium emission predicted with
good numerical accuracy 6 6N + 6 6A1 reaction at E/ A = 75 MeV. Particle
emission rates predicted by the Weisskopf formula were tested for the 6 6 6 Xe
induced reactions at E/ A = 31 MeV.

1.5 Organization

In Chapter 2, we describe various tests and performances of the CsI(Tl)
detetctor developed for this experiment. Details of the experimental setup are
outlined in Chapter 3. Subsequently, Chapter 4 explains the data analysis
procedures for particle identification, energy calibration, time-walk correction and
detector efficiency correction.

In Chapter 5, the inclusive single-proton cross sections are shown. They are
fitted in terms of moving-source parametrizations and compared to BUU model
predictions for pre—equilibrium emissions in reactions 6 6N + 6 6A1 and 6 6N +

‘ ° ’ Au at E/A =75 MeV.

12

In Chapter 6, we present a brief derivation of the Wigner function formalism
which relates the two-particle correlation function to the single particle Wigner
function. The sensitivity of two-proton correlation functions to source radii and
emission time scales is illustrated by performing calculations for a number of
simple source parametrizations. Approximations underlying the Wigner function
formalism are examined in comparison with the classical trajectory calculations.

The measured two-proton correlation functions are presented in Chapter 7
and analyzed in terms of Gaussian sources of negligible lifetime to allow
comparisons with previous measurements.

Chapter 8 gives a brief review of the Weisskopf formula used for the
calculation of particle evaporation from equilibrated compound nuclei, together
with some numerical results. The measured two-proton correlation functions for
equilibrium emissions are compared to predictions of the Weisskopf evaporation
model.

In Chapter 9,- we begin with a brief review of the basic assumptions
underlying the derivation of the Boltzmann-Uehling—Uhlenbeck transport
equation, and then discuss the impact parameter averaging procedure and a few
numerical calculations. The measured two-proton correlation functions for non-
equilibrium emissions are compared to predictions of the BUU model. We further
explore the sensitivity of predicted two-proton correlation functions to the
equation of state of nuclear matter, in-medium nucleon-nucleon cross section and
the impact parameter of the collision.

A summary and conclusions are finally given in Chapter 10. Most results of
this thesis were published in referred journals [Gong 88, Gong 90a, Gong 90b,
Gong 90c, Gong 91a, Gong 91b, Gong 91c].

CHAPTER 2. THE CSI(TI..) DETECTOR

2.1 General Considerations

The study of intermediate energy heavy-ion reactions often requires the
detection of light charged particles (p, d, t, on, ...) emitted with energies ranging
from the exit channel Coulomb barriers up to several hundred MeV. For moderate
resolution requirements, plastic scintillators are adequate. Better resolution can be
obtained with inorganic scintillators such as N aI(Tl), for which energy resolutions
of the order of one percent have been achieved for light charged particles of about
100 MeV energy [Poch 87].

Particle detectors using NaI(Tl) scintillators have a number of disadvantages.
The crystals are hygrosc0pic and must be hermetically sealed, typically with a thin
entrance window (z 811m Havar). Small pinholes in the entrance window can
lead to deteriorations of the detectors with time when they are stored in air. The
use of thicker entrance windows leads to higher detection thresholds and a loss of
resolution. The detectors usually employ photomultipliers for photon detection. In
general, photomultipliers exhibit rather poor long-term stability. Although it is
possible to monitor gain drifts with accuracies of the order 1%, the procedures
used are rather time consuming and cumbersome [Chit 86b, Poch 87].

The use of CsI(T1) scintillators promises to overcome some of these
difficulties [Knol 89, Gras 85a]. The crystals have superior thermal and mechanical
properties (see Table 2.1). Since they are only slightly hygroscopic, they do not
have to be hermetically sealed. Furthermore, their spectral response is well

matched to that of silicon photodiodes [Gras 85a] which exhibit excellent

13

14

Table 2.1: Properties of N aI(T1) and CsI(Tl) scintillators.

 

Scintillator I CsIITl) | NaI(Tl) I Units

Maximum Emission | I I

 

 

 

 

 

 

 

 

 

 

Wavelength, I. I 540 I 415 I run
max
Refractive Index @k l 1.80 | 1.85 I
max
Pulse 10-90% Rise Time I 4.0 I 0.5 | us
Decay Constant, Tf/ TS I0.4-0.7/ 7. I 0.23 I us
Total Light Yield I 52000 I 38000 I PhotoxLS/ MeV
(dE/dx) . I 5.6 I 4.85 l MeV/ cm
min
Radiation Length, X n I 1.86 I 2.59 l cm
Hardronic Interaction I | I
Length, 7» I 36.4 I 41.3 I cm
Specific Gravity. p I 4.51 l 3.67 I gm/ cm 6
Melting Point, Tm I 621 I 651 I 6 C
Hygroscgaicity l slkLhtly I yes I

 

Thermal and Mechanical I I l

Resistance lexcellent I poor |

15
long-term stability (see Figure 2.21 below) when read out with good quality

charge sensitive preamplifiers [Bluc 86]. A number of investigations using small
volume crystals have pointed out these generally attractive features [Vies 86, Meij
87, Kreu 87, Gras 85a, Gras 85b]. In fact, energy resolutions of the order of 1% were
reported for a-particles of 100 MeV energy [Vies 86]. However, the use of CsI(Tl)
scintillator does have obvious disadvantages due to rather slow time response of
the scintillator. The detector usually has poor time resolution and count-rate
tolerance.

At the beginning of our detector development, we tested a small volume
CsI(Tl) scintillator (1x2x4cm6) read out by a PIN diode of 1x1cm6 active area
using a commercial charge-sensitive preamplifier (Canberra 2003). Standard
electronics was set up with a amplifier (Tennelec TC241) and a Multi-Channel-
Analyzer (Canberra MCA-85). Figure 2.1 shows the measured energy spectrum of
'y-rays from a 6 0Co radioactive source. The FWHM resolutions are 6.0%, 5.6%,
and 4.4% for E7: 1.173 MeV, 1.332 MeV, and 2.505 MeV, respectively. The inset
magnifies the pile—up peak originated from two lower energy 'y-rays detected in
the crystal in coincidence. We built another detector by using a cubic CsI(Tl)
crystal (2x2x2cm 6), a larger PIN diode of 2x2cm 6 active area, and a homemade
charge-sensitive preamplifier. The y-ray energy spectra measured with this

I 3

detector is shown in Figure 2.2 for a 6 Cs source (E Y= 0.662 MeV) (upper part)
and for a 6 6 Co source (E Y: 1.173, 1.332 MeV) (lower part). Indicated in the figure
are the FWHM energy resolutions, which are very comparable to the energy
resolutions of the y-rays measured by a standard N aI(Tl) detector read out by a
photomultiplier tube [Gras 85a]. Assuming that the total resolution (0 to t) consists

of the intrinsic resolution (0. ) and the electronic noise (6 . ), we obtained that
mtr noise

 

_ 2 _ 2 .
Ointr - 6/(Otot6 (onoise6 . In the upper part of Figure 2.2, for an example, Otot

= 59 keV , 0 . = 46 keV , and then 0. = 37 keV. The intrinsic detector
noise mtr

resolution is somewhat less than the electronic noise level of the pre

16

MSU-87-243
1XZx4cm6 CsI(Tl)+Sl790+Can2003

 

 

 

 

 

 

    

 

 

 

 

I ' v I . I j

_ 71: 6.0% 666Co .

2x104 7 I 6

I 7.2: 5.6% ;
m .6 I T
...)
CI r .
:1 . 4
O I .
0 1x106 L 400 6

I-

} II 200 6

O 6 . r L . l . . . L L . . 6 . l . .
o 200 400 600 800

Channel Numbers

Figure 2.1 : Measured energy spectrum of 'y-rays from a 6 6C0 source with a
small volume CsI(Tl) scintillator (1x2x4 cm6) read out by a Hamamatsu PIN
diode (51790: 1X1 cm 6), where a commercial charge-sensitive preamplifier
(Canberra 2003) was used.

17

 

 

 

 

 

 

 

 

MSU-87—242
2x2x2cm3 CsI(T1)+PD400

20000 ,. a r, . '. ., . .- ., . .. e ,. a
15000 [— 636Cs: 8.9% —2
; Pulser: 1
10000 ; FWHM=46keV 7
5000 E €
.5:: - j
g 0 . . : I :~+ l .4

o I 1: 5.4% 60
15000:— 6 C0-
: 72: 5.1% 6
10000 7 .1
5000 3 €
0 6 . . r . l . . L . l . . l - a . I ._n6

0 200 400 600 800

Channel Numbers

Figure 2.2 : Measured energy spectra of y-rays from a 6 6 6C5 source (upper

part) and a 6 6C0 source (lower part) with a small cubic CsI('I1) crystal (2x2x2
cm 6) read out by a Hamamatsu PIN diode (2x2 cm 6), where a homemade charge-
sensitive preamplifier was used.

18

-amplifier. Reduction ,of the electronic noise in the preamplifier should lead to
even better detector resolution.

Though small volume CsI(Tl) crystals have demonstrated excellent detection
resolution, our objective is to use rather large volume cylindrical crystals of typical
dimensions of 38 mm diameter and 100 mm length for a 56 element detector array.
We have performed detailed resolution tests for a number of large volume CsI(Tl)
scintillators read out by a 2x2cm 6 PIN diode. Resolutions of 1% have been
achieved for protons of 50 MeV. However, in general, the resolution was found to
be limited by local nonuniformities causing variations of the scintillation efficiency

of several percent.

2.2 Description of the CsI(Tl) Detector.

Our CsI('Tl) crystals were manufactured by BICRON corporation. They are
cylinders of identical physical dimensions (diameter = 38 mm, length = 100 mm),
which can stop, for example, protons up to energies of about 190 MeV and alpha
particles up to energies of about 780 MeV.

The structure of the CsI('Tl) detector is depicted in Figure 2.3. We used a clear
lucite light guide of 12 mm length connected to the rear flat surface of the CsI(Tl)
crystal which had been polished with a tissue moistened with alcohol. Good
optical coupling with sufficient mechanical rigidity was obtained using RTV 615
silicon rubber for the interfaces between the CsI(Tl) crystal and the light guide,
and between the light guide and the photon detectors. Most measurements were
performed using a square shaped PIN diode of 2x2cm 6 active area for photon
detection. In several instances, the diode was replaced by a photomultiplier with a
circular photocathode of 38 mm diameter to verify that our results were not an
artefact of the square geometry of the active area of the PIN diode.

The detailed response of the detector assembly depends on the treatment of

the scintillator surface. Some of our test results will be discussed in the next

19

CsI(Tl) scintillator readout by PIN diode

 

 

 

 

white Teflon tape T600“ ring Fixture
and Al foil Al can Al adaptor Stand-off
CsI(Tl) crystal + ..//
u (l.5"¢ X4.0"I ) - Wu LEMO
collimator // connectors
' ' ' RTV615 -
thm Alforl Lucrte P diode Preamplifier

(15pm) light guide Silicon rubber (2X20m2)

Figure 2.3 : Anatomic diagram of the CsI(Tl) detector.

20

section. Our standard treatment consists in sanding with fine paper (#3204400) all
the outer surfaces of the assembly (i.e. crystal and light guide). The sides are
sanded by movements parallel to the cylinder axis; the front surface is sanded by
circular motions. Teflon tape is wrapped around the sides of the detector
assembly; the front face of the crystal is covered by an aluminum foil of 15 um
thickness. A light intransparent aluminum foil is wrapped around the layers of
teflon tape. The detector is then inserted into an aluminum can which provides
mechanical protection and the housing of the preamplifier.

Our 2x2cm 6 PIN diode is a prototype supplied by Hamamatsu corporation.
It has a leakage current of about 5 nA at room temperature and at an operating
bias of 60 V. Because of slight heating, operation in vacuum resulted in an increase
of leakage current. In our detector array, cooling lines were installed to remove
excessive heat and maintain the detectors at constant temperature (about 15 ° C). It
was measured [Gras 85a] that the temperature dependence of the light yield of
CsI('Tl) scintillator read out by photodiode shows a flat maximum around 3040 ° C
and has a temperature coefficient of 0.3%/ °C at 20°C. The diode has a
capacitance of 135 pF at full bias. Its area-to-capacitance ratio is better by about a
factor of two as compared to the standard 1x1cm 6 PIN diodes such as the
Hamamatsu 51723 and 51790 diodes. For low level signals one can expect an
improved signal-to-noise ratio. However, the actual improvement does not track
the area-to-capacitance ratio because the use of a larger diode of area comparable
to that of the light guide results in a non-negligible reduction of the reflecting
surface of the light guide (which is wrapped with teflon tape). The signals
measured with the 2x2cm 6 PIN diode were about a factor of 2 larger than those
measured with 1x1cm 6 PIN diodes.

The photodiode was furnished without a protecting window at the front
face. We, therefore, applied an optically clear epoxy (EPO—TEK 301-2, by Epoxy
Technology Inc.) to the surface of the PIN diode to protect the silicon wafer and

AAA

21

 

PHOTO DIODE ‘r

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

. : eras
10 F 666666 6 10636 H 6 6666
n $130M 220 + u
I’ :Eiaa film: I
I 2.5V . . * lSuF
150?
2N3<I|4 W ~ fzuaaas ""5"“: ,
NJ903 ' 0005'“ {, lOOnF
n a \n -
r . . - our
10'“: Haseezlfl r
I I
IOOnF I I
"lpF lOOnF lOOnF
. GND
4.7K ilfiuF (I:
I I I
51 ‘ ,Iisur , 150?
Jsasd: ”PF l
10' r—'
1 F ' Ié’BET-‘I 6 ‘12"
p ‘fi
= TEST IN

 

Figure 2.4: Schematics of the preamplifier used for the readout of the PIN diode.
The preamplifier is mounted on a circuit board of 38 mm diameter.

22

allow repeated optical couplings with different detectors. To provide the
necessary mechanical stability, the diode was mounted on a substrate made of
Boron Nitride. The diodes were operated for months without measurable
deterioration of its performance.

A low cost preamplifier of good resolution and low power dissipation was
designed by MR. Maier at the NSCL and its schematics is shown in Figure 2.4. Its
risetime of 6r: 0.2 Its is smaller than the fast decay component of the fluorescent
light of CsI(Tl) (about 0.4 us). The power dissipation of 0.5 watt is sufficiently low
to allow Operation of the preamplifier in vacuum without overheating the diode.
The preamplifier noise at full-width-half-maximum (FWHM) corresponds to a
charge of 2000 electrons using an input capacitance of 130 pF and a shaping time
constant of 1.4 us for the subsequent amplification with a spectroscopy amplifier.
For our CsI(Tl) detector assembly, this noise figure corresponds to a resolution of
about 300 keV for a-particles of 5.5 MeV, the exact value depending on the light
output of individual crystals.

2.3 Resolution Test of Individual Detectors

In this section, we describe a series of tests for a number of crystals
concerning their resolution for energetic light particles and the position sensitivity
of the detector response. For these measurements a AE surface barrier detector of
400 um thickness and 450 mm 6 active area (ORTEC TB-27-450400) was placed in
front of the CsI detecor. In order to unambiguously identify intrinsic properties of
individual crystals each crystal was provided with a well defined frame of
reference, which was kept fixed with respect to the individual crystal throughout
our measurements. Our choice of reference frame is depicted in Figure 2.5.

For the purpose of book-keeping the crystals were labelled with numbers.
We present representative results obtained for detectors #1 and #3 during test runs
with energetic light particles and from bench tests with collimated radioactive

sources.

23

For the bench test, we used a 6 6 6

Cs 'y-ray source which was collimated by a
3 mm diameter lead collimator of about 6 cm thickness. The collimated y-ray
source was mounted on a movable platform. The position of the collimator was
determined with an accuracy of better than 0.05 mm. During these measurements
the position of the CsI-detector was kept fixed. The measurements were
performed with semi-gaussian pulse shaping using a peaking time of 6tts for the 6
main amplifier. Typically, the FWHM resolution for the 662 keV line of 6 6 6 Cs
was of the order of 25%; it is affected by photon statistics and electronic noise.
Therefore, these measurements had to be performed with good statistical
accuracy. For an accurate determination of the peak position, it was necessary to
subtract the background due to noise and Compton scattering. Because of the
excellent stability of the PIN diode, peak shifts of less than 0.2 % could be detected

by measuring the 6 6 6

Cs peak with high statistical accuracy (typically larger than
2x10 6 counts in the photopeak).

Most measurements of detector responses were performed with a beam of 96
MeV OI-particles delivered by the K500 cyclotron of the NSCL. For these
measurements, the CsI(Tl) crystals were mounted as the stopping detectors of AE-
E telescopes, the AE detector being a 400 pm thick silicon surface barrier detector
with an active area of 450 mm 6. Gold and polycarbon, (CH 2 6n’ targets of 2 and 1
mg/ cm 6 thickness were used to obtain alpha particles and protons of known
energy. The scattered particles were collimated by a 12.5 mm thick block of copper
with a circular hole of 3 mm diameter. For these measurements, the position of the
collimator was kept fixed in the laboratory and the telescope was moved with
respect to the collimator exposing different points of the detector entrance
window to scattered particles of fixed energy. The distance between the collimator
and the target was 40 cm; the kinematic smearing for recoil protons was less than

1% in all cases. The absolute resolution of the AE—silicon detector (550 keV) is
considerably better than that of the CsI(Tl) detector.

24

MSU-87-I79

 

 

 

 

Figure 2.5 : Crystal-fixed frame of reference for y-ray scanning tests.

25

One can, therefore, largely eliminate the effects of energy loss variations in the AE-
detector due to straggling or detector inhomogeneities by adding the AE and E
signals. By this procedure one can determine the intrinsic resolution of the CsI(Tl)
detector with good accuracy.

Tests were also performed at the Indiana University Cyclotron Facility. A
direct, low intensity beam of 200 MeV protons was degraded by passing it
through an aluminum absorber and then collirnating it with 50 mm thick Cu
collimators with circular holes of 3 or 25 mm diameter, respectively. For these
collimator sizes, outscattering of the particles is of minor importance. In these
measurements no silicon AE-detector was used.

Results for Crystal #1

This crystal was of clear transparent appearance with a few tiny imper-
fections in the bulk material. For the results presented here, crystal #1 was
packaged following the procedure outlined in Section 2.2 with the single exception
that the detector surfaces were not sanded by us. The crystal had been ordered
from the factory with its front and rear faces polished and the cylinder mantle
unpolished.

Figure 2.6 shows a scan of the detector response obtained by moving the
collimated radiation source (scattered protons and a-particles as well as y-rays)
parallel to the X-axis of the crystal entrance window. Plotted is the percentage
shift of the peak location measured at position X as compared to the peak position
measured at the center, =0. The exact magnitude of the measured shift depends
on the nature of the detected radiation. This can be expected from the different
penetration depths of y-rays, protons and OI—particles. For all three measurements,
the detector exhibits a response which is distinctly left-right asymmetric with
respect to the cylinder axis, X=0. For the measurements with the collimated
6 6 6 Cs source, the signal to (electronic) noise ratio is considerably worse than for

measurements with energetic particles. As a consequence, the experimental

26

 

 

 

 

 

2 MSU-87-I89
i D6646 1 a 7 (.662MeV) 1
C o p (56.8MeV)
1 _ 0 a (89.1MeV) _
¢.-.-._¢_Ib\.’\
A ' T9.
1,3: I
3 07 _.—¢ 6
E .\$ ..... ”(3....
m
-1; .;
I
L .
_2 . . . l l . . . . I . . . L l . . r .
-20 -10 0 10 20

X (mm)

Figure 2.6 : Left-right asymmetric response of crystal #1 for collimated y-rays of
662 keV, protons of 56.8 MeV, and (at-particles of 89.1 MeV. Shown is the relative
shift of the peak centroid as a function of the X-coordinate of the collimator,
keeping Y=0. The data are normalized to the peak position measured for X: =0.

27

uncertainties for the y-ray measurements are larger. Due to Compton scattering
and variations of the penetration depths the effective volume in which scintillation
is produced is larger for y-rays than for particles. The larger sampling volume
might slightly reduce the sensitivity of tests with 'y-rays. However, it is important
that the approximate magnitude of the asymmetry can be well established with a
simple bench test using y-rays. This allows testing of individual crystals without
having to take recourse to expensive accelerator time.

With a collimator of 3 mm diameter, the detector exhibits an excellent energy
resolution of the order of 1% for a-particles of 94 MeV. Because of the strong
position dependence of the detector response, the energy resolution is
considerably worse if a larger collimator is used. With a collimator of 20 mm
diameter a double humped peak structure was observed with a FWHM resolution
of about 3%.

Different treatments of the front face of the crystal #1 were investigated. The
front surface was polished, sanded, covered with aluminum foil, aluminized
mylar, or teflon tape. Although the detailed detector response is affected by these
treatments, we were unable to eliminate the underlying left-right asymmetry. The
effects of surface treatments will be illustrated in some more detail for a different
crystal, #3, which exhibited a less asymmetric response at one end of the crystal.

It was verified that the asymmetric response of the detector was not due to
the square shape of the photon detector. As an example, Figure 2.7 shows
measurements with a collimated y-ray source for a different crystal, #2, performed
using a PIN diode and a photomultiplier tube as photon detectors. Both
measurements give very similar results: the response of this particular crystal is
relatively flat for a scan along the X-axis and shows a pronounced asymmetry for

a scan along the Y-axis.

28

 

 

 

MSU-87-I88
3 _ . . . . I . T - I x I . e
: Det.# 2 . S=X /+
2_— a)PD Readout o S=Y 4),” ‘1
I / I
1 ,- ‘1
A 0} -'
be » I
V "
+3 3 _ : I I I I e I 1r I I a I I ‘. I I : .L ‘
3‘: I b)PMT Readout 4, 3
..CI 2 I. - / J
m » ,/ .

 

 

 

02- ¢ / -

I ,,¢———- ’ I

_1 . .¢: . I . l . . I . . . HI . . . . 6

—20 —10 0 10 20
S (mm)

Figure 2.7 : Position sensitive response of crystal #2 for collimated y-rays. The
open points show measurements as a function of the Y-coordinate of the
collimator keeping X=0; the solid points show the dependence on the X-coordinate
keeping Y=0. Part a) and b) show measurements using a square PIN diode and a
photomultiplier with circular photocathode, respectively.

29

Results for Crystal #3
This crystal was of slightly opaque appearance. As compared to crystal #1, it

had a reduced light output. Initial tests revealed that the response of the detector
was nearly symmetric with respect to the cylinder axis. This detector was then
chosen for rather detailed investigations concerning the treatment of surfaces as
well as its resolution for energetic light particles, including protons of 178 MeV
energy.

Figure 2.8 shows the response to collimated y-rays for the original
(untreated) crystal as it arrived from the factory and for the crystal with the front
surface and the cylinder mantle sanded. For the untreated detector (with a
polished front surface) the light collection efficiency decreases as the radiation
source is removed from the cylinder axis, see Figure 2.8b. A considerably more
uniform response is obtained when the crystal is sanded, see Figure 2.8a. The
major effect of sanding the surfaces of the detector is the removal of a reflection
symmetric position sensitivity along the X and Y directions, which could be
caused by variations of the effective solid angle for light collection due to total
internal reflection at the side surfaces.

We have investigated whether sanding of the front surface introduces
significant dead layers which could affect the energy resolution for low energy
charged particles. Within experimental uncertainties, the energy resolution for
collimated (IL-particles of 5.5 MeV energy was found to be the same for polished
and sanded entrance windows. In addition, the signal amplitudes were very
similar for the two different surface treatments. Therefore, we conclude that no
major damages are introduced by also sanding the front surface of the detector.
An optimally uniform response is obtained for diffusely reflecting detector
surfaces. It should be noted that different treatments of the cylinder mantle can
significantly alter the light collection efficiency as a function of the Z-coordinate.

Such a dependence is of minor concern for the detection of charged particles

30

MSU-BT-IBG

 

E bet.# 3
1 Ea)Surface Sanded

 

 

Shift (7.)

 

 

 

___3 . . L . l . r . . I . . . . l . . . .
—20 —10 O 10 20

S (mm)

Figure 2.8 : Position sensitive response of crystal #3 for collimated y-rays. The
open points show measurements as a function of the Y-coordinate of the
collimator keeping X=0; the solid points show the dependence on the X-coordinate

keeping Y=0. Part (a): Standard sanding treatment of the reflecting surfaces; part
(b): original detector with polished front face.

31

entering through the front window: it merely introduces nonlinearities in the
energy calibration of the detector without significantly affecting the energy
resolution. Quite generally, scintillation detectors have nonlinear energy
calibrations due to the dependence of the differential light output on ionization
density [Poch 87, Vies 86].

Figure 2.9 shows the response of the elements of the AE-E telescope to a-
particles of 92 MeV restricted by a 3 mm collimator moved along the X-axis of the
crystal. The upper part of the figure shows the response of the AE-detector which
shows clear evidence of a convex shape with an inhomogeneity of the order of 8%.
The center part of the figure shows the complementary response of the CsI
detector. After summation of the AE and E signals, the total response of the
telescope is obtained to be flat within 0.5%.

Figure 2.10 shows energy spectra of protons and (at-particles emitted at
6=20° from a polycarbon target (CH 2 )n irradiated with 96 MeV a-particles. In
order to reduce kinematical broadening of the elastic scattering peaks a collimator
of 3 mm diameter was used. The energy resolution (FWHM) of the telescope is
1.0% for 55 MeV protons and 0.8% for 92 MeV a-particles. Due to the rather
uniform response of crystal #3, the detector resolution is of comparable quality for
larger collimators. For 95 MeV a-particles (obtained from the elastic scattering on
Au), a resolution of 0.9% was measured with a collimator of 20 mm diameter, see
Figure 2.11. Figure 2.12 shows the energy calibration of the detector. Good
linearity is observed over the range of energies measured. Consistent with
previous measurements [Vies 86], a-particles exhibit a significant pulse height
defect as compared to protons.

The tests described so far probe only a small fraction of the crystal. In order
to test the response to more penetrating radiation, we performed measurements
with protons of 178 MeV. Using a collimator of 3 mm in diameter a resolution

(FWHM) of 1.4% was obtained. When a 25 mm diameter collimator was used, the

32

 

 

 

 

 

 

 

MSU-87-l90
10
AE
5 — _
O "' _
..5 — __
-10 l l l
A E
N 1 — _
v
q...
.--c
.-C‘. -1- _
U)
_2 ‘ l l l
E + AE
1 '— a
O .— k 4 fi.\.\. ._
-1— ..
-2 l l l
-10 -5 0 5 10

X (mm)

Figure 2.9 : Percentage shift of the AB, E, and summed AE+E signals as a function
of the X-coordinate of the particle collimator, keeping Y=0. (Crystal #3)

Counts

33

MSU-87-l87

 

2000

1500 '-

1000 —

500 '-

 

Page.

Ep = 54.7 MeV

a+ CH2 , Em = 96 MeV
Deon. = 2.7 mm

—> +FWHM 550keV (1.07.)

 

_A_

 

 

 

 

 

 

 

600 _ Ea = 92 MeV .5
400 — —
—-~> «FWHM
200 __ 730keV(0.8%)_
40 60 80

E + AE (MeV)

120

Figure 2.10: Energy spectra of protons and a-particles obtained at 9=20° for
reactions induced by 96 MeV (at-particles on a CH 2 target. The detector collimator
was 3 mm in diameter. (Crystal #3)

 

 

 

 

 

 

700 MSU-87—l84
O(+ Au , Em: 96 MeV
600— Dcoll. = 19.6 mm “
500 _ E0, = 95 MeV _
m 400 - —
4..)
E.
:3
O 300 — ‘ —
o
FWHM —> <—
200 — 860 keV (0.9%) -—
100 _
0 . I
70 80 90 100 110

E + AE (MeV)

Figure 2.11 : Energy spectrum of a-particles detected at 9:20 ° for the 0t+Au
reaction at 96 MeV. A collimator of 20 mm diameter was used. (Crystal #3)

35

 

 

 

 

MSU-87-l80
2000 I I I I
CsI + Pin Diode
1500— _
La
0)
.o
E
:1
z 1000—— __
'03
5::
53 p
rd _ —_
o 500
o l l I l
0 20 4O 60 80 100
E (MeV)

Figure 2.12: Energy calibration for crystal #3.

36

 

. . , . , . . I . fl . . {Msp—a'z—iez
; Det.# 3 . p (178MeV) 1
1 r a)S=X o 7 (0.662MeV)

I

L 1 L l l A l l A

 

Shift (7.)

 

 

 

20

 

Figure 2.13: Position sensitive response of crystal #3 as detected by collimated
protons of 178 MeV (solid points) entering the front face (Z=0) of the crystal with
the PIN diode mounted at the rear end (Z=112 mm) and by collimated 'y-rays
(open points) entering the rear face (Z=100 mm) of the crystal with the PIN diode
mounted at the front end (Z=-12 mm). Part (a) shows measurements as a function
of the X—coordinate of the collimator keeping Y=0; part (b) shows measurements as
a function of the Y-coordinate of the collimator keeping X=0.

37

energy resolution was 1.7%. Scans of the detector response along the X and Y
directions revealed non-trivial variations of the peak position as a function of
position, see solid points in Figure 2.13. Reduced position sensitivities were
measured for protons degraded to lower energy (=146 MeV). In order to verify
that the position sensitivity of the detector was associated with the properties of
the crystal itself and not an artefact of our detector assembly, we removed the light
guide and PIN diode from the crystal and attached it at to the former entrance
window (Z=0 mm coordinate). After applying our standard surface treatment and
packaging, we measured the response of the crystal by injecting collimated y-rays
through the rear window of the crystal (Z=100 mm coordinate). The
measurements are shown by the open points in Figure 2.13. We observe a
pronounced asymmetry in the detector response which tracks the qualitative
features observed for 178 MeV protons. Of course there are quantitative
differences since the two tests sample different (though partly overlapping)
regions of the crystal. However, they do confirm that the variations in the
scintillation efficiencies are due to inhomogeneities in the rear part of the crystal.
In summary, our results indicate that the resolution of large volume CsI(Tl)
crystals is mainly limited by local inhomogeneities of the crystal response (most
probably due to non-uniform doping of T1 concentrations). Improved fabrication

techniques might overcome these limitations.

2.4 Pulse Shape Discrimination

In this section, we describe the timing characteristics of the detector and
compare light particle identification spectra obtained with the AE-E technique
using a 400 pm thick transmission surface barrier detector of 450 mm 2 active area
(ORTEC TB-27-450-400) with those obtained by pulse shape discrimination
technique. Special attention will be paid to the degradation of particle

identification by pulse shape discrimination technique at higher count rates.

38
Particle identification spectra obtained with the AIS-E technique are shown in
Figures 2.14 and 2.15. For these measurements the preamplified PIN diode signals
were shaped with a standard semi-Gaussian bipolar filter using a peaking time of
3 us for the unipolar output. (This corresponds to a time constant of 1.4 ps.) The
particle identification spectra were linearized by using a standard particle
identification (PID) function:

PID °c (E+AE)1'8- E1°8 . (2.1)

The PID spectra were then integrated over an interval of 45-70 MeV for the
energies of the detected particles. Figure 2.14 shows the quality of particle
identification which can be obtained with planar AE detectors. In these
measurements inhomogeneities of the AE detector were reduced to a negligible
level by using a collimator of 3 mm diameter in front of the telescope. Between
protons and deuterons a peak-to-valley ratio of about 100 can be obtained; the
separation between 3 He and ‘ He is even better. When a larger collimator is used
the inhomogeneities of our AE-detector are not negligible and the particle
identification deteriorates somewhat, see Figure 2.15. In this figure, particle
identification spectra are shown for two different count rates, 10 3 and 1.4x10‘
counts per second (cps), using a collimator of 20 mm diameter. (In our
measurements sizeable contributions to the count rate of the CsI(Tl) detector are
due to y-rays and neutrons.) At count rates of the order of 103-10‘ cps, the
particle identification is of good quality, it degrades slightly with increasing count
rate.

It has been known for many years that the temporal decay of the CsI(Tl) light
output depends on the ionization density of the detected particles [Stor 58]. The

fluorescent light has two major decay time constants. In a good approximation the

39

 

 

 

 

 

 

 

MSU-87-|83
_ I I I I .
104 E‘ AE—E Identification .1
: p Deon. = 2.7 mm , E 3 45—70 MeV :
E3 3 ‘ d
z 10 " a “‘
Z i t 3 I
.35 I He
8 2 b i
to 10 b ‘5
E
z
:> I
O 1
O 10 5- 1
100 I I I I
0 20 4O 60 80 100

(E + AE)” -— Ia"8 (arb. u.)

Figure 2.14: Particle identification spectrum obtained Wh tI-Ig AE-E technique
and employing the linearization function PID = (E-I-AE) ' - E ' . The interval of
energy integration was 45-70 MeV. To eliminate inhomogeneities of the AE-silicon
detector, a collimator of 3 mm diameter was used.

4O

 

MSU-87-IBI
I I I I
104 _ AE-E Identification
I; ‘3 t Es45~70Mev ‘
t a j
1.4x 10“ cps I
103 — ,/ _.

 

 

 

COUNTS/CHANNEL

 

 

 

 

 

 

Figure 2.15: Particle identification spectrum obtained whth tiig AE-E technique
and employing the linearization function PID = (E+AE) ' - E ' . The interval of
energy integration was 45-70 MeV. A collimator of 20 mm diameter was used.
Identification spectra are shown for the count rates of 10 ’ cps (shaded histogram)

and 1.4x10 ‘ cps (upper histogram).

41

decay of the scintillation pulse can be expressed as [Stor 58, Alar 86]:
L(t)=LS exp(-t/IS) +Lfexp(-t/If) . (2.2)
While the slow time constant, Is: 7 Its, is independent of the ionization
density generated by the detected particle, both the detailed value of the fast time

constant, I = 0.4-0.7 us, as well as the relative intensity, L f/ Ls’ are known to

exhibit suclfi a dependence. It is, therefore, possible to discriminate between
different charged particles via pulse shape discrimination techniques [Kreu 87,
Alar 86, Bigg 61].

Figure 2.16 shows particle identification spectra obtained via pulse shape
discrimination for two count rates, 103 and 1.4x10‘ cps. For these data, we
measured the time difference, AT, between the time derived with a constant
fraction discriminator from a fast signal of the PIN diode, shaped with 400 ns
differentiation and 100 ns integration constants, and the zero-cross-over time of a
slow doubly differentiated signal shaped with a semi-gaussian bipolar filter
(ORTEC 572) using a time constant of 3 us. (A theoretical analysis, using
measured time constants for different particle species and the noise characteristics
of the filter network, predicts an optimum pulse shape discrimination for time
constants between 2-3 us, assuming that pile-up effects are negligible.) At low
count rates, the pulse shape discrimination technique yields good separation
between hydrogen isotopes and somewhat marginal separation between 3 He and
‘ He. However, the quality of particle separation deteriorates rather rapidly with
increasing count rate. This effect can be understood as due to the sensitivity of the
zero-cross-over of the slow signal to pile-up. The AE-E technique is clearly
superior, especially at higher count rates. Nevertheless, pulse shape
discrimination can provide useful results in low count rate applications
encountered, for example, in detector arrays of high granularity in which each

module subtends only a small solid angle. In addition, it can be used to

42

 

 

 

 

5 MSU-87-l92
1° _ I I I I a
E Pulse Shape Identification g
104 :_ E g 45—70 MeV a
a E “H83 t d P
2 ~ H I I I -
Z 103 =— _
< E E
I 3 1.4x104cps E
U " -
\ ' ..
z E 7 7 7 7 a
D ' 7 7 7 7 3cps :
o - 7, 7 7 7 .
E I 7 77 :IIIWI
100 M I flE//% : : fl/fll ‘
0 20 40 100

A T (arb. Ou.)

Figure 2.16: Particle identification spectra obtained with the pulse shape
discrimination technique. The interval of energy integration was 45-70 MeV. A
collimator of 20 mm diameter was used. Identification spectra are shown for the
count rates of 10 3 cps (shaded histogram) and 1.4x10 ‘ cps (upper histogram).

43

discriminate between. different “hydrogen isotopes of very high energy in
applications where dynamic range considerations dictate the use of thin AE
detectors less suitable for the separation of protons, deuterons, and tritons.

Since the particle identification efficiency of the AE-E technique is close to 1,
one can determine the efficiency of the pulse shape discrimination (PSD)
technique by setting gates corresponding to various particles in the PID-E matrix
and then sorting the gated data into the AT-E matrix. The efficiency of the PSD

technique can then be defined in terms of the ratio

epsd= NPSD/NPID , (2.3)

where NPID denotes the total number of counts sorted for a given gate in the PID-E

matrix and NPSD is the number of events which fall into the appropriate region in

the AT-E matrix.

We have measured the dependence of .9de as a function of count rate in the
CsI(Tl) detector #3 using time constants of 2 and 3 Its in a semi-gaussian filter
network. The results obtained for p, d, t, and a-particles are shown in Figure 2.17.
For both shaping times, the pulse-shape discrimination efficiency decrease
strongly as a function of count rate. At count rates of the order of 10 J cps, epsdz 95-
100%, the exact value depending on shaping constant and particle type. At count

rates of the order of 10 ‘ cps, e z 70-80%, i.e. between 20% to 30% of the particles

are misidentified. Such rrusidijicfifications can pose a serious handicap for high
resolution coincidence measurements since the number of correctly identified
particle pairs depends quadratically on spsd: For epsd= 75%, only 56% of the
coincident particle pairs will be identified correctly.

Some improvements of the count rate dependence of the pulse shape
discrimination efficiency might be obtained by using pulse shaping by double

delay-line differentiation because of its faster baseline recovery characteristics.

100

90

80

7O

80

7O

6O

50

Figure 2.17: Particle identification efficiency as a function of count rate in the CsI
detector obtained with the pulse shape discrimination technique. The slow pulse
shaping was performed using a semi-gaussian bipolar filter of shaping time
constants of 3 Its (upper part) and 2 us (lower part). The interval of energy

 

 

 

 

 

msu-e7—204
“-8 O O T = 3 ,us -
__ - 45—70 MeV_
I
O
a
_ I I _
II
I I I
g Q T = 2 ILLS
- I B ._
I
_ B +
_ I _
(3:1) Elzt
‘ on d II a ‘
I I I
O 5 10 15 20

Counts per second ( ><103 )

integration was 45-70 MeV. A collimator of 20 mm diameter was used.

COUNTS /CHANNEL

800

600 I 20 ns —>|
/ (X3)
400 FWHM—- <—
3 7 us
—> <— FWHM

200 0( p g 13 ns

0 _ I_ I I_ l_ I _ _

50 60 70 80 90 100 110

45

MSU-87-193

 

 

E '5’ 45—70 MeV

 

 

 

TCsI “ TAE (HS)

120

Figure 2.18: Relative timing between the CsI(Tl) E and the silicon AE detectors
using constant fraction discriminators. The interval of energy integration was 45.

70 MeV.

46

However, one cannot expect to match the excellent count rate dependence
achieved for the AE-E technique, since the amount of time during which the
baseline is occupied will not change considerably. The AE-E technique is less
sensitive to count rate since the time constant in the main shaping amplifier for the
AE signal can be kept small; it could even be reduced with respect to the value of
1.4 us adopted for our measurements since electronic noise is not the limiting
factor.

The rather poor count rate characteristic of the pulse shape discrimination
technique is intimately connected to the fact that even the faster of the two
components of the fluorescent light of CsI(Tl) is sufficiently slow to cause severe
pile-up problems. Another consequence of the slow CsI(Tl) light response is the
quality of the timing that one can obtain from such a detector. With mono-
energetic protons of 178 MeV, time resolutions of the order of 4 ns were obtained
with the constant fraction technique. This value is close to the optimum value
which can be expected from our detector: at lower energies, the signal to noise
ratio becomes worse and the time resolution will deteriorate. Further problems
arise from the existence of different decay constants for different particle species
which cause additional walk problems when one uses charge integrating
preamplifiers. To illustrate this problem, Figure 2.18 shows the time difference
derived from the signals of the CsI(Tl) E and the silicon AE detectors using

constant fraction timing techniques. The particle dependent walk is clearly visible.
2.5 Quality Tests of CsI(Tl) Scintillators

The resolution of large volume CsI(Tl) detectors was found to be limited by
local crystal nonuniformities which caused significant variations of the light
output efficiency. The magnitude of the local variations of light output efficiency
can vary significantly between individual crystals. It can be assessed with an
inexpensive bench test in which the position dependent response to collimated 7-

rays is measured (see Section 2.3). In order to make detectors of satisfactory

47

quality for a 56 element hodoscope, we have tested a large number of CsI(Tl)
scintillators manufactured by BICRON corporation. Based upon extensive
measurements with collimated y-ray scanning, we compared crystals grown by a
previous technique to crystals grown with a newly developed process showing
considerably better uniformity of scintillation efficiency.

All crystals tested were cylinders of 38 mm diameter and 102 mm length.
They are packed according to the structure depicted in Figure 2.3. The response of

137

all detectors was tested with 662 keV y-rays from a Cs source collimated by a
3 mm diameter lead collimator of about 6 cm thickness. The collimated y-rays
entered through the flat front face of the CsI(Tl) crystals in a direction parallel to
the cylinder axis. As a quantitative measure for the non-uniformities of
scintillation response, we evaluated the maximum shift of the centroid of the
photopeak as the collimator moved along a circle of 1 cm radius centered at the
axis of the cylindrical CsI(Tl) crystals. The measured shifts of the photopeak were
normalized to the response for irradiation at the center of the circular entrance
window.

Techniques used for the growth of CsI(Tl) (and other alkali halides) on a
commercial scale are not documented in the open literature, but are based on
years of research and development in material purification and growth furnace
design. Techniques described in Reference [Birk 64] make it possible to grow good
quality crystals of small size and on a small scale. The growth of large volume
crystals (2100 I) on a monthly schedule requires a detailed understanding of the
interplay between purity of starting material, growth furnace design, and growth
rates.

The two different sets of crystals described here represent an old and a new
growth process. BICRON began growing CsI(Tl) in 1985 using the old process.
The resulting crystals were acceptable for Compton suppression shields and

phoswich [NaI(Tl)/CsI(T1)] applications. Under close examination with a finely

48

collimated y-ray beam these crystals were found to have pulse height variations of
as much as 6% over distances of the order of 1 cm. Our test results for a large
sample of such crystals are shown by the unshaded histogram in Figure 2.19.
These scintillators exhibit a broad distribution in scintillation uniformity.

Analysis of the crystals from the old growth process revealed severe internal
stress and strain. Fluorescent emission under strong photo excitation did not show
wide variations. An immediate correction was to further customize the
Stockbarger furnace and to reduce the diameter of the crystals from 19 inches to 6
inches to improve the control of thermal gradients in the crucible full of molten
material. Thermal gradients at the liquid/ solid interface must be large and the
vertical plane within the furnace at which crystal growth occurs must be constant.

The performance of crystals grown with the new technique is encouraging.
They exhibit much better scintillation uniformities, see the shaded histogram in
Figure 2.19. Attempts to go to larger, more useful diameters have led to new
purification processes and additional furnace changes. Yields of high quality
crystal material are low. In Figure 2.19, 37 crystals measured with "Shift[%]" less
than or equal to 1% were accepted and used for our purpose.

There is a close relation between scintillation uniformity and energy
resolution when photon statistics or electronic noise cease to be limiting factors.
When used for the detection of energetic light charged particles, the majority of
crystals grown with the old technique will yield detectors with only modest
energy resolution. Crystals grown with the new technique exhibit exceptional
energy resolutions for energetic light particles, see Figure 2.20 for illustration. The
energy spectra shown in the figure were obtained by irradiating a polycarbonate
foil with (II-particles of 160 MeV incident energy. For these measurements, a AE-E
detector telescope was used which consisted of a 300 pm thick planar silicon AE-

detector and a CsI(Tl) E-detector grown with the new technique. The sharp peaks

49

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

MSU-89-095
Cl 1 I T I I l I I I I l I I l l I I I l l l_
10— —:
C I
.22. : "
as e— —"
...) ..

U) .- -I
t 7///// :
U *' 7 ‘
e—; —
H—I - / -
O .. 7 _
I... ‘ /// / - ‘
cu ' 7 “
n 4:- 7 “
8 . 7 :
:1 - 7 -
z 2- 7 t
_ g ‘.
.. g .4
.. / ..I
0L1 I7 I l L I l 4 l I I I I I I I I/LJ I I 1‘

o 1 2 3 7’ >4

Shift [7.]

Figure 2.19: Maximum variations of scintillation response observed for collimated
'y-rays entering the detector front face at various points located on a circle of 1 cm
radius and centered at the detector axis. The shaded histogram shows the
measurements for detectors produced with the new growth technique; the
unshaded histogram shows measurements for detectors fabricated with the old

technique.

50

MSU-89-096
60 . EQZI'IGQMGV’ ®laIb:20-I7° I
50‘ 5 0.57:
40-
30.‘ 12C((X,d) 14N
20-

13‘ I W :53...) I I I I I L I
250 - (wan/2') *1
200 ' 0.6%
150 .-
100 r
50 -

 

 

 

12C(a,t)13N

Counts

V2 1/2‘(¢.s.)

 

 

 

' I O‘kgm.) L A
3000 ‘ 0.6%

2000 - 12C(0(,0I')12C {

2+
1000 '
: .- . A
A A A M

110 120 130 140 150 160
Elab [MeV]

 

 

 

Figure 2.20: Energy spectra measured with a CsI(Tl) detector produced with the
new technique.

51

MSU-BB-OB?
Stability of CsI(Tl) Detector

r I t 7 j V l I f V t l I I
. . . . . .
. . C . . .
0 . . . . .

.0
m

 

.0
O
I

it
++

++ Ms +++t W “2* .
+
+5 ++++

I
o
m
1"-
f.—

..
--1
.4
..
q

C
O
O
O
I
I
I
I
I
I
I

—

J
.4

Shift (7.)
I l
p o
05 4}
f r I
l

I
O
(D

1

 

 

‘1 . .5. PSI . . :l .31 ‘2‘ . ‘24
150 200

O 50 100
Time [hrs.]

 

Figure 2.21: The stability of pulse-height of a CsI(Tl) detector.

52

resulting from the ‘ 2C((x,d), ‘ 2C(a,t) and ‘ 2(Kan) reactions indicate excellent
detector resolutions of 0.5-0.6% (FWHM). For this particular CsI('Tl) crystal, a
maximum peak shift of 0.6% was measured with our y-ray scanning test. The
measured electronic noise was about 300 keV.

Finally, to illustrate the degree of pulse-height stability, Figure 2.21 gives an
example for one typical CsI('Tl) detector. Except for the initial phase-in period of a
few hours during which the temperature of the detector and its assciated
electronics had to stabilize, the pulse height remained constant within 0.5% up to 8
days. The gain stability of our actual CsI(Tl) detectors was measured to be of
comparable quality during the whole experiment for up to one month.

In summary, our investigations have shown that improved fabrication
techniques have allowed the production of large volume CsI('Tl) detectors with
excellent energy resolution and excellent long-term stability when photodiode

readout is used.

CHAPTER 3. EXPERIMENTAL DETAILS

3.1 Mechanical Setup

The experiment was performed in the 92 inch scattering chamber of the
National Superconducting Cyclotron Laboratory of Michigan State University
using beams from the K1200 cyclotron. Light particles were detected with AE-E
telescopes consisting of silicon Ali-detectors and CsI(Tl) or N aI(Tl) E-detectors.

The detectors were arranged in two seperate hodosc0pes. The geometrical
placement of these hodoscopes is indicated in the schematic drawing shown in
Figure 3.1.

One hodosc0pe with 37 Si-CsI(Tl) telescopes was centered at 0=25° and
¢=0° (where 0 and (1: denote the polar and azimuthal angles with respect to the
beam axis) and at a distance of 105 cm from the target. Its front face geometry is
shown in Figure 3.2. (Due to the low yield of CsI(Tl) crystals grown by the new
technique [Gong 90a] at BICRON Corp., we were not able to fully implement the
56 element hodosc0pe at the time of this experiment.) Each telescope of this
hodosc0pe subtended a solid angle of AQ=0.37 msr and consisted of a 300 um
thick planar surface barrier detector of 450 mm2 active area and a cylindrical
CsI(Tl) scintillator (length: 10cm, diameter= 4cm) read out by a 400 mm 2 PIN
diode [Gong 88]. The nearest neighbor spacing between adjacent detectors was
A6=2.6°. The CsI(Tl) detector array was kept at a constant temperature around
15°C degree with an ethanol refrigerator and had excellent gain stability (better

than 1% over a time period of one month).

53

Experimental ~ B
l3 Element earn
setup Hodoscope‘ Out

56 Element
H \cope

   

 

Figure 3 .1: Schematic view of the experimental setup with two hodoscopes being
used.

55

 

 

 

 

Figure 3 .2: Schematic view of the front face of the 56 element hodoscope.

56

 

 

 

 

 

 

 

Figure 3 .3: Schematic view of the front face of the 13 element hodoscope.

57

 

 

 

 

msu-so-oez
. I f . . I . . .
_ Q j
100 - O O O 4
, O 1
- O O O 4
. O O O O .
75 f' O T
A 50 - ‘
no .. q
0 . .
'd
'6' o ‘
25 - ‘
. O O .
. O O .
_ o O o .
O O O
* O O o O ‘
0 - O O O o -
b O O O O O ‘
O o C .
. O o o *
. O o O '1
-25 "' . . 1 A . . . 1 . . . . I . L “
20 25 30 35
0 (deg)

Figure 3.4: Polar coordinates of 50 AE-E telescopes used in the experiment.

58

The other hodosc0pe with 13 Si-N aI(T1) telescopes was centered at 6=25°
and ¢=90° and at a distance of 60 cm from the target. Its front face geometry is
shown in Figure 3.3. Each telescope of this array subtended a solid angle of
AQ=0.5 msr and consisted of a 400 pm thick Si surface barrier detector of 200
mm2 active area and a cylindrical NaI(Tl) scintillator (length: 10cm, diameter:
4cm) read out by a photomultiplier tube. The nearest neighbor spacing between
adjacent detectors was A9=4.4°. Gain drifts of the individual photomultiplier
tubes were measured by a light pulser system as well as by changes in the location
of the particle identification lines in the AE-E matrix [Poch 87]. These gain drifts
were determined in the off-line analysis and corrected with an overall accuracy of
better than 2%.

Figure 3.4 shows the polar angles covered by individual telescopes in the the
0-¢ plane.

3.2 Electronics

The principle for the electronic processing of all detector signals is sketched
in Figure 3.5. The analog signals from either Si solid-state detector or
CsI(T l)/ N aI(T l) scintillation detector were first amplified by charge-sensitive pre-
amplifiers (PA). They were further shaped and amplified by amplifiers (A). The
slow outputs of the amplifiers were digitized by peak-sensing ADCs and read out
and recorded later on magnetic tape as the energy information. The fast outputs of
the amplifiers were converted to logic signals by either Constant-Fraction-
Discriminators (CFD) or Leading-Edge-Discriminators (LED). Figure 3.6 depicts
the sequences in processing NIM logic signals for a Si-NaI(Tl) telescope (upper
part) and a Si-CsI(Tl) telescope (lower part), respectively. In order to detect high
energy protons which deposit little energy in the transmission Si detectors, two

thresholds of lower energy, E , and higher energy, EH’ were set for the fast E-

L
outputs from NaI(T1)/CsI(T1) detectors. The valid telescope event was then

defined as: (AE’SiOEL + EL.EH) for a Si-CsI(Tl) telescope and EL0(Al-ZiSi + EH) for a

PIN

59

 

 

 

 

Note:

PA Preamplifier

A ; Amplifier

CFD -— Constant Fraction Discriminator
LED Leading Edge Discriminator
FIFO —— Fan-In-Fan-Out

QDGG Quad Delay Gate Generator
DGG -— Dual Gate Generator
NVW — time-delay cable (not resistor)

 

 

 

 

 

 

 

 

 

 

Flt-MASTER

Single Event
Coinc.Event D
Pulse: Event —>

 

 

 

 

 

MASTER

 

 

 

 

 

l---—0 to ADCSuobe(w=6ps)
—[—D to TDCStIt

——>toCOMPUTERSm

 

 

‘ CFD 1 ou‘e’e’
m b «as: Em
to ADC 2

 

 

 

Figure 3.5: Block diagram of the electronic circuits and trigger logic.

6O

Si-NaI(Tl) Tel.

gall

.KDm

 

 

r-‘-n

IIIII

 

ll

 

 

(AE+EH)

 

EL*(AE+EH)

 

 

 

 

 

4.». II
T.
.m.
I
s
1 c
..m l I
file ....... --l"-'v'l """"

 

 

 

 

 

(EL*(AE+EH))’

 

 

 

 

 

 

 

 

I-"I-""l-'l

1"---

 

(AE’II EL+EL’ *EH)

Figure 3 .6: Diagram for processing the logic signals in a Si-NaI(Tl) or Si-CsI(Tl)

telescope.

61

Si-NaI(T1) telescope, where the primes indicated the delay-and-stretch operations
on the corresponding signals. Advantage was taken of faster timing responses of
Si detector.” in the Si-CsICI‘l) telescope and NaI(Tl) detector in the Si-NaI(Tl)
telescope. The timing information of a valid telescope event was therefore
i or EL(CsI) (when AE

determined by AE i was not present) in the Si-CsI(Tl)

telescope or by ELfNaI) in the Si-NaI(Tl) teliscope. For protons, EL was about 5
MeV and EH about 30 MeV.

The logic signal of a valid telescope event was fanned out to (1) the
coincidence multiplicity unit to register as a coincidence event when at least two
telescopes fired within the coincidence timing window (around 400 ns); (2) the
multiple-OR module to register as a single event after a certain factor of down-
scaling; (3) Scalers and TDC Stop to register event rate and timing after
appropriate delays.

Electronic pulser signals were provided for all pre-amplifiers in addition to
the light pulser system for photomultiplier tubes. They were very useful for
testing electronic modules and trigger logic. Gains shifts were monitored by the
pulser events and corrected for in the off-line analysis wherever necessary.

Coincidence and down-scaled singles data were taken simultaneously,
together with pulser events taken at much lower rate. A VETO circuit was
incorporated to prevent ADC conversions during computer busy period. The
MASTER logic thus defined provided trigger signals for TDC Start, ADC Strobe,

and Computer Start. BIT registers were used to keep track of what type of event

occurred and which detector fired.
3.3 Reactions

Four nuclear reactions with heavy ion beams were measured in the
experiment. For reactions induced by ‘ 2 ° Xe at E/A=31 MeV, we used 2 7 Al and
l 2 2

Sn targets with areal densities of 5.6 and 5.3 mg/cmz, respectively. For
reactions induced by ' ‘N at E/A=75 MeV, we used 2 7Al and 1 9 7Au targets

62

with areal densities of 15.0 and 15.9 mg/ cm 2, respectively. Typical beam
intensities on target were approximately 5x10 ° ‘ ‘N(5+) ions per second and
1x10' ' .il’Xe(23+) ions per second. The beam spots on target had elongated
shapes of typically 1-2 mm width and 2-3 mm height.

For the purpose of energy calibrations of individual detectors, a (CH 2 )n
target (25.4ttm thick) was used to scatter a-particles of 160, 116, and 90 MeV
incident energy after passing through Al degraders with thickness of 0, 880, and
1250 mg/ cm 2 , respectively. Resulting from two-body reaction kinematics, mono-
energetic light particles such as protons, deuterons, tritons, 3 He, and ‘ He were
measured at various laboratory angles.

During the experiment, raw data were sampled periodically and diagnostic
spectra were monitored on line to insure that all detectors were functioning well

and good quality data were recorded on magnetic tape.

CHAPTER 4. DATA REDUCTION

In this chapter, we explain the offline data-reduction procedures used for
identifying particle, calibrating energy, correcting time-walk, and correcting
detector efficiency.

4.1 Particle Identification

Using the AE-E telescope, charged particles can be clearly identified [Goul
75] on the two-dimensional map of AE (the energy loss in the thin transmission
detector) v.s. E (the energy loss in the stopping detector). However, we did not set
the particle gates in the AE-E map because excessively large computer memory is
required for sufficient channel resolution. Instead, we used a linearization
function [Shim 79] to transform the AE-E map into a PID-E map where PID is the
value of the corresponding linearization function.

For illustration, Figure 4.1 shows typical PID-E plots for a Si-NaI(Tl)
telescope in the left panel and for a Si-CsI(Tl) telescope in the right panel. The PID
function for a Si-NaI(Tl) telescope was defined as:

PID = 100 '[log(b°AE) + (b-1)°10g(E+c0AE) - 0.7]. (4.1)
where c=0.5, b=1.5-0.18 OAE/ t, and t=400um (the Si detector thickness). The PID
function for a Si-CsI(Tl) telescope was similarly defined as:

PID = 80 0[log(b°AE) + (b-1)°log(E+c0AE) - 2.5]. (4.2)
where c=0.5, b=1.825-0.18 OAE/ t, and t=300um (the Si detector thickness). To set
particle gates unambiguously for the isotopes of Li and Be in the CsI(Tl) detector,

we magnified their existing regions by using another PID function which was

63

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cos—3:3 m

 

 

 

.3 mo :2

 

      
  

 

   

         

; . ...2.‘ (Nichols. .'.vli
.. 3... 1 (J. ...
I. i. x ...t..«\r\. ; .l. ...V,...‘..~ ...v.aA‘u..A‘ r ..
\v
. ."‘l1 v I
.I/ .1.
..t ..x.. lita‘.
‘
1,... {marquis .
'5'. ...) ..D.) a)“.
.2 . .4. t
u .. Hitllwriéu .
u .u .

 

 

 

 

CIIcl

65

defined as:

PID = 220 °[log(b-AE) + (b-1)0log(E+c0AE) - 5.5]. (4.3)
where c=0.5, b=1.65-0.18 OAE/ t, and t=300ttm (the Si detector thickness).

In Figure 4.1, it is clear that p, d, t, 3 He, ‘ He, 6 He, and 6 Li, and 7 Li, are
identified in the Si-NaI(Tl) telescope and p, d, t, 3 He, ‘ He, 6 He, 6 Li, 7 Li, ' Li,
9 Li, 7 Be, ° Be, ‘ ° Be, and B are identified in the Si-CsI(Tl) telescope.

4.2 Energy Calibration

Because of their high linearity, the AE-Silicon detectors were simply
calibrated by a precision electronic pulsers and by means of 5.5 MeV (1 particles
from a 2 ‘ lAm source. Once calibrated by the 5.5 MeV 0: particles, the pulser
would inject known amounts of charge through a calibrated capacitor into a Si
preamplifier. By setting the pulser at several amplitudes ABi’ we obtained the
corresponding peaks Ni(AE) on the AE-ADC spectrum. A linear fit by the
equation:

AEi = a °Ni(AE) + b, (4.4)
provided the desired energy calibration coefficients (a, b) for a Si-detector.

Energy calibrations for NaI(T1) or CsI(Tl) scintillator detectors were more
involved. Previous energy calibrations for NaI(Tl) [Chen 88] were performed by
directing secondary charged particles of known rigidities into each telescope. This
method allowed the simultaneous calibrations for different particles of precisely-
known incident energies. Unfortunately, this technique could not be used in the
present setup as a magnetic beam analysis setup was not available. Alternative

energy-calibration methods had to be employed for our detectors.
4.2.1 Two-body Reaction Kinematics
A (CH 2 )n target of 25.4um thickness was bombarded by (it-particles of 160,

116, and 90 MeV incident energy after passing through the Al degraders with
thickness of O, 880, and 1250 mg/ cm 2 during the calibration run. Calibration

66

points could be obtained in moSt telescopes by detecting mono-energetic light-

charged particles (p,d,t, 3 He, ‘ He) from scatterings or direct reactions such as:

p(‘He,p)‘He, (4.5)
p(‘He,d) ’ He, (4.6)
2 2 C( ‘ He,d) ‘ ‘ N, (4.7)

‘ 2 C( ‘ He,t) ‘ 2 N, (4.8)
p( ‘ He, 3 He)d, (4.9)

‘ 2 C( ‘ He, ’ He) 2 2c, (4.10)
2 2C(‘He,‘He)‘ 2C. (4.11)

Other beams or targets were also used to supply a few extra calibration points for
p and ‘ He in the following scatterings:

p( 2 °Ne,p) 2 °Ne, (4.12)

‘ 2 ’Aut‘He,‘He)‘ 2 ’Au. (4.13)

Typical energy spectra of light-charged particles were already shown in
Figures 2.10, 2.11, and 2.20. Recoil protons in Equation (4.5) or recoil deuterons in
Equation (4.6) had broader peak widths than the real detector resolution because
of kinematical smearing in detectors of finite angular acceptance. In all cases, the
peak centroids were extracted to establish energy calibrations. Meanwhile, the
corresponding energy values of all peaks were calculated by two-body kinematics.
Typical energy calibrations derived from this technique are shown in Figure 4.2
for a CsI(Tl) detector and in Figure 4.3 for a N aI(T1) detector.

For the energy calibration of protons in the N aI(Tl) detector (N04), a
quadratic polynomial fit was performed [Chen 88], indicated by solid line in
Figure 4.3. Included in the fit was one special point on the X-axis, which was the

ADC channel offset (N ) determined from pulser calibrations. Since no data

offset
were available at higher energy, a linear extrapolation was used for E > 60MeV as

indicated by the dashed line (obtained by a linear fit to all available

67

 

CsI(Tl) Energy Calibration (C10)

  
    
  

 

 

 

 

 

: p i; d 1 s
. .. . I
100 r - 3
. . N
i I
50 j J
A
b
a 158 )— f I '—' l I l L:
13: t I
100 - j
50 r J
0’..grl....l.L..I.L..J...._L‘ ... Lkl..4.l....l....l‘
0 200 400 600 800 1000 200 400 600 800 1000

Channel

Figure 4.2 : Energy calibration of protons, deuterons, tritons, 3 He and ‘ He for a
CsI(Tl) detector (C10) obtained by two-body reaction kinematics. The curves are
explained in the text.

68

NaI(Tl) Energy Calibration (N04)
. . 1 . . . I . . . . .

 

 

 

 

150 I ‘ 5
<,=
—-quadratic fit ,I’ 2 t?
----1inear fit 2%
' proton (data) //
100'- -
A
:>
a)
2
v
m .1
50" _
r ,’
O',’...1...1..11...
0 200 400 600 800
Channel

Figure 4.3 : Energy calibration of protons for a N aI(Tl) detector (N 04) obtained by
two-body reaction kinematics. The curves are explained in the text.

69

Table 4.1: Fit parameters for energy calibrations of various particles in a typical
NaI(T1)/ CsI(Tl) detector (N04/ C10) using the equation: E (MeV) = A + B-N(E) +
C0N(E)z .

 

 

 

 

 

 

 

 

 

 

Isotope l A l B I C | Comment
p,d,t I -1.575 I 0.1721 IO.2158e-4 lEs60MeV,NaI(Tl)
p.d,t I -3.941 I 0.1868 I 0.0 I E>60Me\L NaI(TD
2He I 0.629 I 0.199 I 0.0 I NaI(Tl)

2He I -0.258 I 0.168 I 0.0 I CsICl'l)

‘He I 3.677 I 0.193 I 0.0 I NaI(Tl)

‘He I 1.548 I 0.166 I 0.0 I CsI(Tl)

‘He I 4.524 I 0.199 I 0.0 I NaI(Tl)

‘He I 2.643 I 0.167 I 0.0 I CsI(Tl)

‘Li,’L1 I 13.507 I 0.214 I 0.0 I NaI(Tl)

“URL: I 8.523 I 0.178 I 0.0 I CsI(Tl)

2Lier I 11.891 I 0.177 I 0.0 I CsI(Tl)

“’Be I 20.505 I 0.181 I 0.0 I CsI(Tl)

°Be,‘ °Be I 19.705 I 0.195 I 0.0 I CsI('I‘l)

B I 33.181 I 0.198 I 0.0 I CsI(Tl)

 

70

data). The fitting coefficients are listed in Table 4.1 for this detector.

Energy calibrations of p,d,t, 3 He and 3 He are shown in the upper left and
right panel 3‘and in the lower left and right panel of Figure 4.2, respectively, for the
CsI('l'l) detector (C10). The points and crosses were measured data points, and the
curves were calculated responses for p,d,t , 3 He and 3 He together with a line fit
for 3 He and 3 He only.

The scintillation response to charged particles can be described [Birk 64] by a
relation between dL/dx, the differential light output, and dE/dx, the stopping

power of incident particles. Commonly referred to is the empirical Birk’s formula:

S°dE/dx

dL/ dx"'1 + kB-dE/dx ’

(4.14)
where S is the scintillation efficiency, k is the relative quenching probability, and
BOdE/dx is the density of damaged molecules along the track of the penetrating
particle. The functional values of dL/dE v.5. dE/dx are listed [Birk 64, Pink 89] in
Table 4.2, which were used for calculating light outputs for p,d,t, 3 He and 3 He in
CsI('Tl) detectors.

For a particle of given energy E, the light output L(E) can then be calculated

by the following integral:
R(IE)
1.6)... $35.35- .. (4.15)
0

where dE/dx was the stopping power in the layer between x and x+dx and R(E)
was the stopping range, both of which were calculated from Ziegler’s parametri-
zations [Zieg 77]. L(E) was then associated with the ADC channel N(E) by the
linear relationship:

N(E) = NOffset + n -L(E). (4.16)

Table 4.2: Differential light output, dL/dE, of CsI(Tl) scintillator v.s. differential

energy lossidE/dx (MeV/ mg/ cm 3 ).

71

 

 

 

 

dE/dx | 0.001 | 0.002 |0.003 I 0.004 | 0.005 | 0.006 I 0.007 | 0.008
dL/dE I 0.980 I 0.980 I0.985 I 1.000 | 1.010 | 1.018 I 1.030 I 1.042
dE/dx | 0.009 I 0.010 I0.020 I 0.030 | 0.040 I 0.050 | 0.060 I 0.070
dL/dE I 1.057 I 1.071 I1.198 | 1.255 | 1.241 I 1.201 I 1.175 | 1.141
dE/dx | 0.080 I 0.090 I0.100 | 0.200 | 0.300 | 0.400 | 0.500 I 0.600
dL/dE | 1.108 | 1.085 I1.057 I 0.849 I 0.717 | 0.613 I 0.540 I 0.500
dE/dx | 0.700 I 0.800 I0.900 I 1.000 I 2.000 |10.000 |20.000 |100.00
dL/dE I 0.465 I 0.435 l0.415 I 0.400 | 0.300 | 0.170 I 0.140 | 0.100

 

72

Here, 11 is the normalization constant which is the same for p, d, and t and slightly
different for 3 He and 3 He.

Figure.L 4.2 compares the calculated responses to the measured data. The
calculations have satisfactorily reproduced the energy calibration data for p, d, t
and therefore provide the final energy calibrations for these particles. However,
the calculations, indicated by dotted and dashed curves, respectively for 3 He and
3 He, did not go through the data points at all energies and showed larger non-
linearity at low energies. The discrepancy might be related to the fact that fast
scintillation component of CsI(Tl) is known to depend on dE/dx. In our analysis,
we had adopted the simple linear fit to the measured 3 He and ‘He calibration
points. This is justified by the quality of the fits.

4.2.2 Energy-loss Calculations

One of the advantages of using a AE-E telescope to measure charged
particles is that some energy calibrations can be accomplished in the off-line
analysis when calibration beams of well known energy are not available. With this
technique, one uses the known AB calibration and AE detector thickness to deduce
the energy deposited in the CsI(Tl)/NaI(Tl) detector.

Figures 4.4 and 4.5 show the centroids of particle identification lines in the
AE-E maps for representative Si-CsI(Tl) and Si-NaI(Tl) telescopes, respectively. For
a specific particle, each point represents N (E), the centroid position of the E-ADC
channel numbers corresponding to N (AE), a narrow bin of AE-ADC channels. A
function can be defined by:

N (E) = f(N(AE)). (4.17)

For a charged particle (Z,A) of given incident energy (E), we calculated [Zieg
77, Zieg 85, Hube 90] the energy losses: AB in Si detector of known thickness and E
in NaI(Tl)/CsI(Tl) detector. The energy calibration of N(E) v.5. E was. then
established through Equations (4.4) and (4.17). Using this procedure, we were able

73

 

3000..-.l....l....,-.

Si—CsI(Tl) Telescope (C10)
2500

N
O
O
0

AE (channel)
3
O
O

H
O
O
O

500

 

+—E A

Particles:

Identified 3

1

A .l l L A L I l A A A

 

 

 

1000 1500
E (channel)

2000

961-16-0871

2500

Figure 4.4 : Two-dimensional map of AE (the energy loss in Si. detector) v.s. E
(the energy loss in CsI(Tl) detector) which was used for the off-line energy

calibrations of all identified particles in a CsI(Tl) detector (C10).

74

 

 

    

 

 

 

L _
4000 .u . . 1 . . . . I . . . . I . . . . I . . . . g
. 13- Si— aI(Tl) Telescop (N04) Identified 1 I2
Particles: 2‘2
. 1 I
.. 7“ N
3000 '11 _ 3
2 .He 3
a) 2 3
He
g ~ 8He
.1:: 2000 t _
O ' d
v
\. P
Lil
<1 _ ‘:§\ 3
1000 ""‘~‘=”.:--~..
. “re-“ta —
o o-.;;- .‘I:': :2wa J
H . . . . . . . . ' " w e...
0 . . 1 .‘1 .'.-'.-."'1 .'-. .". 1 . . . . I L . . .
0 500 1000 1500 2000 2500

E (channel)

Figure 4.5 : Two-dimensional map of AE (the energy loss in Si. detector) v.5. E
(the energy loss in N aI(Tl) detector) which was used for the off-line energy
calibrations of all identified particles in a N aI(T1) detector (N04).

75

to calibrate almost all particles for the energy ranges of interest. Typical examples
are given in Figures 4.6-8 for the CsI(Tl) detector (C10) and in Figures 49-10 the
NaI(T1) detéctor (N04).

In Figure 4.6, the solid points are the same data as shown in Figure 4.2,
which were obtained from two-body reaction measurements, while the open
points and crosses are derived from energy-loss calculations. After adjusting the
thickness of the Si-detector within 5%, we achieved remarkable agreement
between the two calibration procedures for protons, deuterons, and tritons. It
should be pointed out that the two-body reaction technique mainly provided data
points at high energies, and the energy-loss calculation technique supplied most
data points at low energies. Data points at very high energy given by the second
technique were less reliable since the numerical interpolation or extrapolation had
large uncertainty. As seen from Figure 4.4 and 4.5, the AE-E curves become flat at
large N (E), this technique therefore loses its sensitivity. For the Helium particles,
we adopted the simple linear fits which could represent all data rather well.

Energy calibrations of the Lithium, Beryllium, and Boron particles were
obtained by the energy-loss calculations only, as shown in Figure 4.7. For
simplicity, linear fits were used. The fitting coefficients are listed in Table 4.1. In
Figure 4.8, energy calibrations of p, ‘ He, 7 Li, 7 Be, and B are compared for the
CsI(Tl) detector (C10). The non-linear behavior of proton response with respect to
other particles was not well understood and was observed for almost all detectors.
It might be related to light collection efficiency for particles stopping at different
ranges.

In the same way, we obtained the energy calibrations of p,d,t, 3 He, ‘ He,
° He, 6 Li, and 7 Li for the N aI detector (N04) as shown in Figures 49-10. We had
used the same proton energy calibration as discussed in Section 4.2.1 for deuterons

and tritons since they showed very similar responses. Energy calibrations for ’ He,

76

 

 

 

 

 

 

 

5»
....C?S.I(_T.1) Energy Calibration (C10)
0 .c:
P :: d ‘4,
100- _"_ ‘1.
I ‘1 73
L J
507 -- ..
A .
%
5158'- . ' ' ' L-
ca : t
100_- __
I' .
50: ..
O .1.1..1 1 ...1. - 1 .. 1 .1

800 1000 200 400 600 SODA-1000
Channel

0 200 400 600

Figure 4.6 : Energy calibration of protons, deuterons, tritons, 3 He, ‘ He, and
6He for a CsI(Tl) detector (C10) obtained by both two-body reaction kinematics
(solid points) and energy-loss calculations (open points). The curves are explained

in the text.

"

,CS,I(T.1) Energy Calibration (C10)

 

250 E
200 i
150 i
100 i

50

$61-16—DEH

 

250 i
200 i
150 §
100 i

50 i

E (MeV)

 

250 _
200 i
150 Q
100 i

50 I

O I . . . L L - - L . 1 . . . I
0 500 1000 1500

Channel

 

 

 

 

Figure 4.7 : Energy calibration of 6Li, 7Li, 8Li, 7Be, 9Be, ‘ °Be, and B for a
CsI(Tl) detector (C10) obtained by energy-loss calculations. The curves are linear
fits to the data points. See text for explanations.

78

 

 

 

 

s.
CsI(Tl) Energy Calibration (C10)
/ * 1; . g
2507 ;/fl 71;". I]: 4,
- 7/ fiX.-:«' .T
..x , . s
. . . 01
200 — /7 I .x' _
. I x .
: ’ X 1'
- fi/ #5 ,
A / ,
> 150 _ f [y -
Q) ’ / /
2 - ,6 lo ’9’
V ' / / ‘5'
Ed ’ / ,’.." E’ o ___
' -////x " "I ° “‘- Be
_ // //.X I x ........... 7111
50 r/i//.__.x' .0 ------- 4H6 -
/ , 3‘ .
( .irI 00 P
0 .- , . . . . . _ . , _ . . .
0 500 1000 1500
Channel

Figure 4.8 : Comparison of energy calibrations for particles such as proton, ‘ He,
7Li, 7Be, and B measured in a CsI(Tl) detector (C10). The curves are the

corresponding fits. See text for explanations.

E (MeV)

150 _

100 j

150
100

200E
150 g
100;

79

 

50:

 

50 I

. NaIr(Tl)

Energy Calibration (N 04)

r‘vv r T

.0

. . . . I
OBI-IB-DSN

"CPU
1

X4-

 

 

 

 

; i L L A L A A l l A L

L 4 L l L

 

400 600
Channel

200

Figure 4.9. : Energy calibration of protons, deuterons, tritons, ’ He, ‘He, 6He,
6 Li, and 7 Li for a N aI(Tl) detector (N04) obtained by energy-loss calculations. The
curves are corresponding fits to the data points. See text for explanations.

80

'0

fiNTaI‘(Tl)v Energy Calibration (NO4)V

 

 

 

 

 

K
m
c.
4.
200 T
' §
150
’>‘
G)
2 .
V100
[:3 .
5O
0'..L.1...1-.-1..-1..L
O 200 400 600 800 1000
Channel

Figure 4.10: Comparison of energy calibrations for particles such as proton, ‘ He,
and 7Li measured in a NaI(Tl) detector (N04). The curves are the corresponding
fits. See text for explanations.

81

‘ He, 6He, 6Li, and 7 Li particles were once again fitted by straight lines. The
fitting parameters are again listed in Table 4.1.

The azcwracy of this calibration technique is mainly limited to (1) the
accuracy of the energy-loss calculation; (2) the accuracy of thickness measurement
for any foil or detection media; (3) the accuracy of energy calibration for the Si
detector; (4) the accuracy of numerical interpolation or extrapolation in using
Equation (4.17).

4.3 Time-walk Corrections

In two-particle coincidence measurements, it is essential to seperate real
coincidence events coming from one single projectile-target interaction from
random coincidence events due to two or more projectile-target interactions.
Random event contaminations are corrected by constructing the relative time
spectrum between any two telescopes and by setting appropriate time gates.

It is well known that the time response of NaI(Tl) scintillators is fast
(td=0.23p.s, Tr=0.5tts) and that of CsI(Tl) scintillators is very slow (1: d=1.Ot.ts,
1:41.15) (see Table 2.1). While constant-fraction-discriminators (CFD) were used in
the timing circuits for the N aI(Tl) detectors, it was not economical and practical to
use CFDs for the CsI(Tl) detectors in this experiment. Instead, we employed
simple leading-edge-discriminators (LED) (see Figure 3.5). Consequently, a large
time-walk became inevitable for the CsI(Tl) detectors in contrast to the negligible
time-walk for the NaI(Tl) detectors as shown respectively in the right and left
panel of Figure 4.1]. Here the time difference between a detector and the down-
scaled RF signal (T-TRF) is plotted against the stopping energy signal (E). Each band
represents one beam burst. To simplify the analysis for the CsI(Tl) detector, we
straightened the curved response of CsI('Tl) detector.

After making this kind of walk correction for the CsI(Tl) detectors and
aligning the zero-time with each other among all detectors, the relative time
spectra were constructed. They are displayed in Figures 4.12 for 75MeV/nucl.

82

 

NaI Tel. CsI Tel.

 

 

  

 

 

A:

E (channel)

 

Figure 4.11: Two-dimensional plots of ( T - TRF) v.5. E for a Si-NaltTl) telescope

(left panel) and for a spa-1m) telescope (right panel) where necessary time-walk
corrections were made.

I,“

N .+, .27441.E,/A.=75¥8.V. . . .‘.‘N.+, .‘°f’4u. 30575qu .

1

106 Ir CsI—CSI CsI—CsI 1

 

r

79! -IB-flSN

   

 

 

 

  

 

 

 

 

 

103 :.:‘::::E.L:4::::;:I::.‘v
106 I NaI—CsI NaI—CsI 1
3 105 ’ l l ‘
G F I 1
'3 4 I l
U 10 r l l
10: ’ . :l : : I ::: c : I : t :5 : I : : L
10 g-
5 5 Nal— a1
.0 . t
:r g s . I I 1
: : l I l
104 r : s s I I 1 1
. 3 : l I 1 .
3 I E = ‘
.' ° 1 . . . 1 . . P- - A - 1 . . . a; A‘
-400 -200 0 200 -400 -200 0 200 400
t1 — t2 (ns)

Figure 4.12: Relative timing spectra between any two telescopes in the
7SMeV/nucl. ‘ ‘N induced reactions on 2 7Al (left part) and on 1 ° 7Au (right
part). Timing gates for real and random coincident events are indicated by vertical

lines.

.(fi

c‘.2°.X.e.+. .ZTAL EAAéBINc-N . .‘.“’?X.e.+ .‘izvsnuE/PPJWY
CsI-Csl 081-081

H
0
0|
f'fi

SUI-IO-OSR

H
O
00
r"""l""‘l

 

Counts

H H

o o

to as
"""1"""l""'l"""r" "

   
 

 

NaI-NaI

-400 -200 0 200 -400 -200 O 200 400
t'1 "' t'2 (ns)

..n
O
(D
""1 ""‘l """1 """l " A

     

 

 

Figure 4.13: Relative timing spectra between any two telescopes in the
31MeV/nucl. ' 2 °Xe induced reactions on 2 7Al (left part) and on ‘ 2 2Sn (right
part). Timing gates for real and random coincident events are indicated by vertical
lines.

85

191

‘ ‘N induced reaction on 2 7 Al (left parts) and on Au (right parts) and in

Figure 4.13 for 31MeV/nucl. ‘ 2 ° Xe induced reaction on 2 7 Al (left parts) and on

1 1 2 Sn (right parts), respectively. Three categories of the relative time spectrum
were defined as (1) between two CsI('l'l) detectors (upper panel); (2) between
N aI(Tl) and CsI(Tl) detector (middle panel); (3) between two NaI(Tl) detectors
(lower panel). For each reaction, the relative time spectra of different category had
very similar peak structures except that the NaI(Tl) detector showed better time
resolutions than the CsI(Tl) detectors. The highest peak in the center contains real
coincidence events and also a random contamination, while the side peaks of
about equal height arise from random coincidence events.

)

To be on the safe side, we have defined the "real" coincidence events (Nreal

by the events locating between two dotted lines and the random coincident events

(N ran dom ) by the events lying between the dotted and the dashed lines on both left
and right hand sides. Assuming that the number of random peaks contained in

Nreal is mreal and Nrandom included mrandom random peaks, the

corrected real coincidence events are:

Nreal = Nreal - Nrandom.mreal lmrandom' (4'18)

One can estimate from Figure 4.12 and 4.13 that the ratio of N over
random

Nreal was about 5% for ‘ ‘N induced reactions and about 0.3% for ’ 2 °Xe

induced reactions. After random correction, the numbers of real two-proton

coincidence events, Nre =2.25x10°, 1.94x10‘, 1.13x10‘, and 0.46XIO‘ for the

al
a 7 27 29 2
reactions”N+‘° Au, HN+27Al, 129Xe+ Al,andl Xe+” Sn,

respectively.

86

4.4 Corrections due. to Nuclear Reaction Loss

Energetic light-charged particle can undergo nuclear reactions with target
nuclei in a detector. When such a nuclear reaction occurs, the particle’s energy is
generally not measured correctly due to non-vanishing Q-value and the emission
of neutrons and y-rays. This poses a small inefficiency problem for the detector. In
this section, we discuss corrections for detection losses due to nuclear reactions of
light charged-particles entering CsI(Tl) / N aI(T1) scintillators.

The total reaction cross section for heavy-ion collisions in the intermediate

energy range can be parametrized as [Kox 87]:

B

2 c
OR = “Rim ' (1' E
cm

), (4.19)
where EC m is the c.m. energy and BC is the Coulomb barrier of the projectile-target
system. They are given by:
1 / 3 1 / 3

4.

0A /(A +A) and B =z Zez/[r (A A )1, (4.20)
t p t c p t c p

E .=E t

cm lab
with rc=1.3fm, Zp(zt) and Ap(At) are the atomic and mass number of projectile
(target) nuclei. The interaction radius, Rint’ can be divided into volume and surface
terms as:

= R + R (4.21)

int vol surf °

The volume radius is given by

R = r0(A1/3 + A1/3), (4.22)
vol p t

and the surface radius is parametrized as

87

A1/3.A1/3
D t

R =ro(a -

‘surf; - 1 /3+A1/3 ' C) + D’ (423)

Ap t
where r o = 1.1 fm, a = 1.85, D = SZP(At-22t)/(APAI) due to neutron skin excess,
and c is empirically parametrized by the formula [Town 88]:

c = 191-16.0 0exp(-0.7274°E0°3493)'COS(O.0849'EO'5904), ' (4.24)

with E = Elab/Ap being the laboratory kinetic energy of projectile in units of
MeV/ nucleon.

We first obtain the stopping range (S in units of mgr/cmz) for a particle
incident on CsI(Tl)/ N aI(Tl) detectors by using Ziegler’ s parametrization [Zieg 77].
The detector depth is then devided into many thin slices within the particle’s
stopping range. In the i-th slice of detector (Ax=S/N), we use the thin-target

approximation to calculate the nuclear reaction rates by:
Ri(Cs) = n0(Cs)° Ax(Cs) . (rigs), (4.25)
R.(I) = n (no Ax(I) . o‘ (I), (4.26)
1 0 R

where n0(Cs) = 6.022x10-7/A(Cs), n00) = 6.022x10-7/A(I), both in units of
[cmz/mgr/mb], and Ax(Cs) = Ax 0 A(Cs)/[A(Cs) + A(I)], Ax(I) = Ax 0
A(I)/ [A(Cs) + A(I)].

Finally, the total probability of nuclear reaction loss is obtained by

N

Rtot = l - i131 [l-Ri(Cs)]°[1-Ri(l)]. (4.27)

 

0.54,_....Ifi...I.s..I...4
S _ NaI(Tl) Detect r ‘
0.3 " ..dy/fu ~~ t “
I P _.-'/’
0.2}
0.1 ‘

     

l L l

 

r

l
tor

 

0.0 I

CsI(Tl) Detec

Nuclear Reaction Loss

 

 

 

 

Figure 4.14: The probabilities of nuclear reaction loss for light-charged particles
detected by a 10cm long N aI(Tl) or CsI(Tl) scintillator.

89

Figure 4.14 shows the calculated probability of nuclear reaction loss for light-
charged particles (p, d, t, 3 He, ‘ He, 6 He) entering a 10cm long NaI(Tl) detector
(upper part; and a 10cm long CsI(Tl) detector (lower part). Typically, a proton of
150MeV energy has about 20% probability of nuclear reaction inside the detector.
At the energies of interest, the probability of nuclear reaction is smaller for the
Helium particles and negligible for heavier fragments. By comparison, the
probability of nuclear reaction is slightly larger in N aI(Tl) detector than in CsI(Tl)
detector since NaI(Tl) has less stopping power and the particles have a longer

range than in CsI('Tl).

5.

CHAPTER 5. SINGLE-PROTON ENERGY SPECTRA

5.1 Inclusive Energy Spectra

Examples of inclusive energy spectra for protons detected at the extreme

angles covered by our detector array, 9 = 18 ° and 33 ° , are shown in Figure 5.1.

lab
Spectra measured for reactions induced by ' 2 ° Xe and ‘ ‘ N projectiles are shown
in the left and right panels, respectively. In order to gain qualitative insight and
allow comparisons with other data, we have fit these cross sections with the
moving source parametrisations. We have to caution, however, that the extracted
parameters are not uniquely determined by our data and must not be over-
interpreted since our measurements covered only a small range of angles.

For the ' ‘N-induced reactions, we have chosen a simple three-source
parametrization, representing isotropic Maxwellian contributions from a target-

like source, a projectile-like source, and an intermediate velocity nonequilibrium

source:

 

3 _—
= Z N.~/E—U Oexp{-[E-U +E.-2~/E.(E-U )Ocose]/T.}. (5.1)
. 1 c c 1 1 c 1

(19 dB
i=1

Here, Ni and Ti are relative normalization and kinetic temperature parameters,
respectively. The energy E. is the kinetic energy of a particle co-moving with the i-
th source, Ei= % mczfiiz. The Coulomb energy, Uc’ corrects for the Coulomb
repulsion from heavy reaction residues assumed at rest in the laboratory rest
frame.

The solid curves displayed in the right hand panels of Figure 5.1 show fits

9O

91

 

 
   

MSU—90—198

*"'l""l""l"" ..I....I.r..I...rI

106 g. E/A=31MeV _ E/A=75MeV _.

105 I. -.
. -----

104 r I ' . "=

103 s- '1.

:°18°

dzo/deE (,ub/sr-MeV)
8
N

   

 

 

 

: o O .1.-
g 33 t1... v.
197Au(1“N,p) :
105 .
104
103
102 ..L.1.-.-1..¢ 111‘-
0 50 100 150 0 50 100 150 200

E (MeV)

Figure 5 .1: Inclusive proton cross sections measured, at 61 =18 ° and 33 ° , for the
reactions ‘ 2 °Xe+ 2 7Al and ‘ 2 °Xe+‘ 2 2Sn at E/A=31 eV (left hand panels)
and the reactions ‘ ‘N+2 7Al and ‘ ‘N+‘ ° 7Au at E/A=75 MeV (right hand

panels).

92

Table 5.1: Fit parameters used for the description of the inclusive single- proton

cross sections shown in Figure 5.1. The spectra for the

reactions were fitted by Equations (5.1) and (5.2), respectively.

The normalization constants, N i’ are given in units of [mb/ (erMeV

I ‘N and 1 2 °Xe-induced

3”'11,U,A.
C C

and Ti are all in units of [MeV]. Also given are the velocities, Bpro and Bcn’ of the

projectile and the compound nucleus, respectively.

 

 

 

 

 

Reaction prrol Bcn l Uc I Ac '1 I Ni l Bi I Ti

“N+”A1 10.40 10.14 11.72 1- 11 I 4.67 10.371 I 6.04
I I I I - 12 1 1.62 1 0.200 1 19.59
1 I 1 1- 13 I 3.98 10.073 I 4.62
1 1 I 1 1 I I 1

“N+‘ ° ’Au 10.40 I0.026|8.57 1- 11 1 8.52 10.376 1 5.88
I 1 | 1 - 12 1 4.55 1 0.200 I 18.10
| 1 1 1- 13 122.81 10.025 1 3.84
I I I 1 1 1 I I

12°x(-2+“AI 10.26 10.21 15.18 12.0 l1 152.40 10.234 | 4.07
1 1 1 1 12 1 3.91 10.213 1 8.94
1 I 1 I 1 I 1 1

‘2 °Xe+2 7A1 10.26 10.21 17.25 12.0 11 136.57 10.213 1 6.0
1 I 1 1 1 1 1 1

‘2 "Xe+12 251110.26 10.13 14.23 12.0 11 166.05 10.241 I 4.25
1 I l I 12 1 7.13 I 0.130 1 13.92
1 I I 1 l I I I

‘2 °Xe+‘ ’ 251110.26 10.13 19.88 12.0 11 158.0 10.130 1 9.13

 

93

 

.... MSU-90-196

106 27A1(14N1p) 197.411(‘“N .p)
5 E/A=75MeV E/A=75MeV
10 1

 

dzo/dOdE (,ub/sr-MeV)

 

q

 

101111 -11.L
0 50 100 150 200 50 100 150 200

E (MeV)

 

Figure 5 .2: Decomposed three moving source fits to inclusive proton energy
spectra measured, at 9 a =18° and 33 °, for 75 MeV/nucl. ' ‘ N induced reactions
on 2 7 Al (left two panelsyand on ' ° 7 Au (right two panels).

(126/de1: (,ub/sr-MeV)
s
N

94

MSU-90-l95

 

er"'771""1"' ‘
27Al(129X8,p) 1228n(129xe,p)_;

E/A=31MeV E /A=31Mev

 

 

 

‘ \\ . 33° "
5 .

o.\ .

°. \ 1

 

 

 

Figure 5.3: Decomposed two moving source fits to inclusive proton energy
spectra measured, at 9 =18° and 33°, for 31 MeV/nucl. ‘ 2 °Xe induced

. a
reactions on 1 7 Al (left two panels) and on

122

Sn (right two panels).

95

obtained with this parametrization; the parameters are listed in Table 5.1. To
illustrate the relative importance of contributions from various moving sources,
the fits were decomposed into contributions from target- and projectile-like source
and intermediate velocity source, indicated by dot-dashed, dotted, and dashed
lines, respectively, in Figures 5.2 and 5.3. The spectra can be rather well described
by assuming emission from target- and projectile-like sources with temperature
parameters of about 4 to 6 MeV and by including a non-equilibrium component
described, as before [Chen 87b, Poch 87, Gelb 87] in terms of an intermediate
velocity source characterized by a high kinetic temperature parameter, T==18-20
MeV. Particularly for the ‘ ‘N + ‘ ° 7Au reaction, the fits indicate significant
evaporative contributions from a target-like source to the low-energy portion of
the spectrum.

For the Xe—induced reactions, fusion-like and projectile-like residues have
large velocities in the laboratory rest frame, and emission from these two sources
is strongly forward focussed. In comparison, contributions from target-like
residues are of minor importance at our detection angles. Furthermore, non-
equilibrium emission may be expected to be small. Therefore, we adopted a two-
source parametrization representing emission from projectile- and fusion-like
sources. For these sources, the simple Coulomb correction adopted in Equation
(5.1) is inappropriate since the heavy reaction residues have large velocities with
respect to the laboratory rest frame. Furthermore, the measurements include
energies which lie below the projectile and compound nucleus Coulomb barriers.
Hence, the sharp truncation of the energy spectra for sub-barrier energies is
inappropriate. For these reasons, we adopted a parametrization similar to that

used in Reference [Fie186]:

 

 

 

2 2 U exp[-(U-UC)2/2A2]

d O N. ldU C JE(1-U/E .)e
1 cm

-(Ecm i-U)/'I'i
d9 dE A /2 1t ,1 ’
1= 1 U c

. (5.2)

96

Here, Ecm,i = E + Ei- Zfiecose and Bi = %m c2 B? ; the integration limits were
chosen as Ul =max(0,Uc-5Ac) and U2=min(Ecm,i’Uc+5Ac)' In Equation (5.2), the
Coulomb field is assumed to be stationary in the rest frame of the emitting source,
and an average is performed over an ensemble of Coulomb barriers using
Gaussian weighting factors.

The fitted spectra are shown as solid lines in the left hand panels of Figure
5.1. The spectra can be rather well described by assuming evaporative emission
from a fusion-like source and a projectile-like source. The decomposition of source
contributions is presented in Figure 5.3, where dashed lines indicate fusion-like
sources and dotted lines indicate projectile-like sources. (For the l 2 ° Xe + 2 7 Al
reaction, fusion-like sources have velocities very similar to the projectile velocity,
[3 z 0.26. For the ‘ 2 ° Xe + ‘ 2 2 Sn reaction, fusion-like sources have
approximately half the beam velocity.) The fits shown by the solid curves indicate
strong contributions from decays of excited projectile residues. The inclusion of a
projectile-like source largely improves the fits at lower energies. At these energies,
the calculations are sensitive to details of the parametrization of the ensemble of
Coulomb barriers. Again, it must be stressed that the extracted source parameters
are not uniquely determined because of the small angular range covered by our
detector array. In order to illustrate some of the existing uncertainties, we have
also described the tail of the energy spectra (E240 MeV) by assuming emission
from a single source moving with the velocity of the compound nucleus. These
calculations are shown by the dashed curves in the left hand panels of Figure 5.1;
the parameters are listed in Table 5.1.

It should be kept in mind that the moving source parametrization does not
offer a unique interpretation of the single-particle inclusive cross sections. The
adopted parametrizations should, therefore, be viewed with a "grain of salt".
Nevertheless, the calculations indicate possible contributions from a number of

different sources which cannot be disentangled without ambiguity.

97

5.2 Comparison with BUU Calculations

With respect to our future discussion, it is instructive and useful to compare
the single-proton cross sections measured for the ’ ‘N-induced reactions with
those predicted by BUU calculations [Baue 87]. Comparisons of the BUU theory to
the experimental two-proton correlation functions will be presented later in
Chapter 9.

In Figure 5.4, the solid points represent the measured single-proton cross
sections for the reactions 1 ‘N + 2 7Al (upper panel) and ‘ ‘N + ‘ ° 7Au
(lower panel); cross sections predicted by BUU calculations are shown by open
points. For the emission of energetic protons, the cross sections predicted by the
BUU calculations are in rather good agreement with the data. However, at lower
energies, E s 70 MeV, the predicted cross sections are larger than the measured
ones. At least part of this discrepancy may be attributed to the fact that the present
calculations do not incorporate cluster formation and emission. The formation and
emission of clusters is expected to be particularly important when the phase space
density is high, i.e. at low kinetic energies. In these regions of phase space, the flux
of emitted nucleons will appear, in part, in the form of bound clusters. On the
basis of these qualitative arguments, one can expect that proton cross sections
predicted by BUU calculations should be larger than the experimental cross
sections for free protons; the effect should be most pronounced for protons of low

energies.

98

 

 

 

 

 

MSU-90-202
I . . . . r . . . . I . . a . 1
105 _ 27A1(1“N,p), E/A=75MeV .
- 000 .00.... 0 18°
; 104 ~— .....08¢°° 'oo°°°o ‘3
g E 0.09.0.9... .292: O . ...°°:¢¢ ..
. 9.3 o. .
s... i ' 9 EXP. 05¢ .0 32¢ ‘
.. t t i _=
£ 10‘3 o o BUU 33° +2§++; .03 oi Tlri
..O
3 _ : s : .L g : if
m 5 L 0,, o 19".em(”1\1,p) E/A= 75MeV
PC 10 E 8003 000000
c E 000 .2220 o 4
'd ' ’. .M 0 .83 so. 0 18
\ ’ .300099022.o o ...0
(‘1t: 104 - ’0‘ 2292 33% "
'° ‘ ’¢ Wm -
3 ' ° EXP 33o §o.i + i i ‘
1° W n
_ n . . . L . . . . l . . . . 1 . 7 .
0 50 100 150 200

E (MeV)

Figure 5 .4: Single-proton cross sections calculated with the BUU theory (open
points) are compared to experimental cross sections (solid points) for the reactions
”N-t-2 7A1 (top panel) and l‘N+‘ 9 7Au (bottom panel) at E/A=75 MeV.
Circular and diamond shaped symbols indicate laboratory angles of 18° and 33 ° ,
respectively.

CHAPTER 6. TWO-PROTON CORRELATION FUNCTION

In this chapter, we give a brief derivation of the general formalism which
allows the calculation of two-proton correlation functions from the knowledge of
the single-particle phase-space density. The sensitivity of two-proton correlation
functions to source radii and lifetimes is illustrated by means of simple analytical
source parametrizations.

6.1 Theoretical Formalism

A number of formalisms have been published [Boal 90] which derive two-
particle correlation functions from the knowledge of the emission function, g(p),x),
i.e. the probability of emitting a particle with momentum I; from space-time point
x=(?,t). The derived expressions differ only in minor details and the predicted
results are similar. Here, we derive an expression for the correlation function in
most general terms assuming complete knowledge of all two-particle quantum
mechanical matrix elements. We then introduce and justify approximations which
allow practical calculations. For simplicity, we will restrict ourselves to the
important case of correlations between two identical particles.

In the following, we will use four-vector notation to keep our formulae
compact and manageable. However, our formalism is not relativistically covariant.

Our final expression for the two-particle correlation function is identical to

the expressions given in References [Koon 77, Prat 87]:
-} ->
—) -) —) -» n( P ‘1’ P2)
R(P,q)+1 = C(P,q) = _) a =
II( p 1 )l'I( p 2)

 

99

100

Id4x d4x g( 1372,): )g(1'>’/2,x Huff, E’ 4? -(t -t )13’/2m)l2
_ 1 2 1 2 1 2 1 2 (61)
_ 4 _’ 4 _I’ o c
Id xlg(P/2,x1)ld x2g< P/2,x2)

Here, 1161,32) and 116%) denote the two-particle and single-particle emission
probabilities, l5) and 3 are the total and relative momenta, I; =31 + 32 and {13:61-
32M; respectively, and o is the relative wave function.

This expression was given in Reference [Koon 77] without derivation. It was
derived in Reference [Prat 87] by using the sudden approximation for which the
particles are assumed to be on shell in their final state and the mutual interaction
is switched on suddenly. Equation (6.1) was then derived assuming the thermal
wavelength is much smaller than the size of the system. These approximations are
not very well justified. Here, we start from the full quantum-mechanical
expression for two-particle emission and show how, under most circumstances,
we can justify a few approximations which lead to Equation (6.1). Results obtained
in many other formalisms are similar. For instance, in the literature on two-pion
interferometry the function g(R/2,x) is often replaced by something similar, for

-> —+ 1 /2 . —§ .
example, [ g(p1,x)-g(p2,x) ] . As long as the relative momentum l q I IS

much smaller than a characteristic momentum, I; I, of g(3,x) (such as If; I z JET,
where T is the temperature of the emitting system), there is little difference
between the formalisms.

The complete matrix element for the creation of the n-body final state
includes all information necessary for the calculation of correlation functions. Here
we will derive an approximation in which the correlation function will only
depend on properties of the one-body emission probability which can be extracted
from the one-body matrix elements.

We start out by first considering the probability for creating two particles

with final momenta 3] and 32 from sources 1 and 2 at space-time points x1 and x2:

101

-+ -9 4. 4 -) —> 2
l'I(p1,p2) = I i d xld x2M1(xl)M2(x2)U(x1,x2.p1,p2)| . (6.2)

Here, U(x1,x2:31,32) is the evolution operator for particles created at x1 and x2 which
—) -)

end up in the asymptotic momentum states p1 and p2. The matrix element Mi(xi)

creates the particle at xi and the remaining collision products into a state which

henceforth does not interact with the particle. In Equation (6.2), we have assumed

that the particles are emitted independently. This allows the factorization of the '

matrix element for two-body emission into the product of single-body elements.

By squaring the matrix elements and transforming to the new coordinates xi

(mean) and 5xi (relative), we obtain

-’ -’ 4 4 4 4
”(P1,P2)= id xld 5x1d xzd 5x251(x1,5x1)82(x2,8x2)o .

“1131432041, ,5x1, x2,5x2 ). (6.3)

where 4
f -> —)
W31’32(x1,,5x1,,x2,5x 2)= U (x1+5x12/2,x +5x 2/2:p1,p2)0
U(x1-5x1 /2,x2-5x2/2:p 1,p 2) , (6.4)

and

_ ‘l'
51(x1,5x1) - M1 (x1+5x1/2)M1(x1-5x1/2) . (6.5)

The expression becomes physically more transparent when the dependences on
5xi are replaced by dependences on the momenta, ki’ through four-dimensional

Wigner transforms. As a result, we obtain:

—) -)
H(p1,p2)- - id4x ld4x 24d k d 4k2§1(x1 ,k1)§2(x2, ,1<2)wI-;1 32(x1,,k1,,x2,,k2) (6.6)

with the Wigner transforms g and W defined by:

102

S 1i(x,1<i)= ld4 5xSi(x. I'l‘,5x)e46xi 1, (6.7)
and
W31 32(x1, ,k1 ,x2,k2)- =
id4 5x1d4 5x W-) -> (x ,5x ,x2,5x2)e-48x1.k1-46x2.k2. (6.8)
2 p1,p2 1’1

For non-interacting distinguishable particles, the time evolution operators,

U, in Equation (6.2) become simple exponentials, and we obtain from Equation

(6. 4) the relation W-+ -+ =exp(i5x1-p +i5x p ). In this case, we can see from
p1, p 2 1 2 2
4
Equation(6. 8) that W31 3(2x1,k 1,x ’2k2)— 54(p1-k1)5 (pz-kz), asexpected. Non-

interacting distinguishable particles retain their four-momentum after the
emission.

The functions §(x,k) are the quantum mechanical analogues of the emission
probability for particles with four-momentum k from space-time point x. This can
be seen from the following argument. Performing the same steps as above, but
now for the single particle distributions, and making use of the relation

W3(x,k)=54(p-k), we obtain the result:

N(E) = i d4xd4k §(x,k)54(p-k) . (6.9)
From here on, we will imply the on-shell condition p ° = 13(3) when referring to the
0th component of the asymptotic momentum. Equation (6.9) shows that the

emission probability is given by:

g(3,x) =§(x,p>. (6.10)

103

k ), we assume that the

In order to obtain an expression for W-> -* (x1,k 1 ,x2, 2,

pl’pz
first emitted particle propagates freely for a time ( t2 - t1 ) before it interacts with

the second particle which is created at t With this assumption, the evolution

2.
operator,

,-2->.-)—+

_,_, -53 3 44-1k-x-1k0x
U(k1,1, k2,t2 :p1,p2)-(21t) id xld x2U(x1,x2.p1,p2)e 1 1 2 2, (6.11)

for momentum states, Ei’ evolving into the true scattering states, 31’ can be written
as

~ -) —> -) —> -) a —> -) . .

U(k 1,t1,k 2,t2.p 1,p 2) = <D(k 1,k 2.13 1,p 2) 0exp[1E12t2+1E1(t1-t2)] . (6.12)

Here E1245 the total kinetic energy of the proton pair, and <l> is the projection of the

total wave function on the plane wave states :1 and R42.

If the emissions of the two particles are not far apart in time, the assumption
of time-ordered emission entering Equation (6.12) becomes questionable due to
the uncertainty principle.

One can now calculate W in terms of the wave function projection (D.

W31 32(x1, ”k1,,x2 ,k2)=

UT_+

idfit 63 8k 12d8t 63 6k 2U (k k+5k 1,1/21 +51 /2,1< 2+151< 2,2/2t +5: 21/2p1,p2)

-U(k 1-5k 1/2,t1-5t1/2,k 2-8k 2/2,t2-5t2/2:p 1,132)
3 3 3 3 o o
_ I d 5k1d Skzd Rld R25(k1-E1)5[k2-(E12 E 12m (12’1,k2,R ,Rzzp1,p2)

. -> -) —) --) . —) -) -)
e-15k10 (x1+k1(t2—t1)/m-R1)-15k2 (x 2-R2). (6.13)

104

Here, the symbol ki° is the O-th component of the four vector ki' To arrive at the
second part of Equation (6.13), we performed Fourier transforms and made the
linear approximation for the energy, 13(12’+512) z E0?) + 81? 01-2/ m. The function f2
introduced into Equation (6.13) is the Wigner decomposition of the total two-

particle density matrix:
—) -1 -) -) -) —) 3 3 . —> —> . -+ -)
f2(k1,k2,R1,R2.p1,p2) - I d 5121a 5122621301512 1R1-16k2R2)

*—> -> -) —1 —§ —> —> -> -) -> -9 —)
(D (k1+5k1/2,l<2+5k2/2:p1,p2)¢(k1-5k1/2,k2-5k2/2:p1,p2)
3 3 . —> —) . -—> -)
= 1d 5x1d 8x2exp(-15x1k1-15x2k2)
*4 -—> —-> —> -—> —) —> —, -—) —> -) —)
(I) (R1+5x 1/2,R2+5x 2/2:p1,p2)<l>(R1-5x 1/2,Rz-5x 2/2:p 1,132) . (6.14)

We insert Equation. (6.13) into the expression for the two-particle probability,
Equation (6.6), perform the integration over kio and obtain
3 3 3

-+ -* 3 3 3 4 4 ~ -’ ~ -)
U(p1,p2)— ld Skld 5k2d Rld de kld kzd xld x2$(x1,E1,k1)S(x2,E12-E1,k2)

'512’ "12’<u)/ 12’ '5E’("12’)
-)-)-)—)-) - 0 - - - O -
. e1 1("1+121 m 1)1 2 "2 2 (6.15)

If we had used a different "reasonable" formula for Equation (6.12) or kept more
terms in the expansion of E(l_<)+61_<’), we would have obtained the same formula as

Equation (6.15) except that the partition of energy into the source at x1 and the

source at x2 would have been different. These details are not important since they

will be absorbed into the approximation of the next paragraph.
In order to make the formalism tractable, that is depend only on 30?, E,

(21+? 2) / 2), we now make the assumption that the product of the matrix elements

105

31032 depends weakly on the partition of the four-momentum. In order for the
particles to interact, x1 and x2 must be very close together. The function 5 should
then have the same momentum dependence at both points. If the momentum
dependence is thermal, then the product of the Boltzmann distributions has no
dependence on k1 ~k2. Even for arbitrary momentum dependence, the product 31 0 32
has, to first order, no dependence on relative momentum. If we do not make the
approximation of weak dependence on the partition of four-momentum, the
formalism is intractable, unless we know quantum details of the emission matrix
§(k,x) for k ° ¢E(l_<)). This assumption allows us to integrate over d35ki and d3Ri and
express the two-particle probability as:

3

—> -> 3 4 4 ~ -> —> ~ -) —>
U(p1,p2)= I a 1216 12221 x161 x25(x1,E12/2,(k1+k2)/2)S(x ,1312/2,(121+122)/2)

f(l2’12"E’( )/ —’-_’_’) (616)
2 1’ 2"‘1+ 1 t2"‘1 m"‘2‘191’1’2 ' '
Since the total momentum of the pair is conserved during the evolution towards

—) —> —) -> —)
their asymptoticmomentum states, we can substitutek +k 2 =p 1 + p 2: =P .We obtain:

1
-> -+ 3 3 4 4 ~ ~
U(p1,p2) - Id kld 122d xld x25(x1,P/2)S(x2,P/2)

—>—)-—>
f(k,k,x

12’(11)/ "-"") (617)
2 1 2 1+ 1 2'1 m"‘2'pl’p2' ‘

The emission function g is evaluated at the four-momentum P=(E(F),13)). In

general, the O-th component of this four-vector is not equal to E . However, as

12
long as 3132 is small, this difference can be neglected. This approximation allows us
now to replace the function 3 (x,P/ 2) with the single particle emission probability
g(I_”/2,x), see Equation (6.10). We can now calculate the two-particle probability in

terms of single particle probabilities. It is prudent to make the same

106

approximations in the expression for the single- particle emission probabilities.
Then the correlation function for non-interacting particles will remain at unity
even when 3132 is not small.

Dividing the two-particle emission probability from Equation (6.17) by the
single particle emission probabilities thus yields for the two-particle correlation

function:

C(fila) =

4 4 3 3 .. -+ ->
Id x 21 22 d 12 d k2g(P/2,x )g(P/2, 22)fg(121,12’2,2’ +12’l(12 111/61,2 2.151412)
3

-+ 4 3
ld4x1d klg( P/2, x1)f1(k11:p1)ld xzd kzg(P/2, x2)f1(k2, 2’ 2: p2)
(6.18)
Using the definition of the Wigner decompositions, Equation (6.14), and the
fact that the two-particle wave function can be factored into the center-of-mass

wave function multiplied by the relative wave function,
-) —) —9 —) . -) -) -> -) —) -) -) —)
<I>(x 1,x 2:p1,p2) = exp [-1(p1+p2)(x 1+x 2)/2] q) [(pl-p2)/2,x l-x 2l , (6.19)

one finds for the integrals over the Wigner functions f1 and f2:

3 -)—>->
Id kif1(ki,xi.p) _ 1, (6.20)
Id312213121(12’,12’,"p" 3-31)ld(k +2312)d(k1-—2—k-22)1(12’,12’,:"2’2’")
22 1’ 2”“1”‘2p 1’p 2 ‘ 1’ 2’x1’xzpl’pz

k -k
= I 53021242 )53( 1 ——22)53(f>’-12’-1212’2)

I535rexp1152’ -(i2’1-12’2)/2]¢*(§’,x+5r/2)¢(fi’,2’ -22’ -52’/2)

107

->—>->

= l¢(q,x2-x 1)Iz. (6.21)

Inserting these relations into Equation (6.18) now yields our final result which was
already given in Equation (6.1).

The central assumption underlying our derivation is the approximation
§(x,E,l_<’) == §(x,E,(l-<+1 +1.: 2)/ 2). This approximation becomes exact when the emission
function §(x,k) is broad, i.e. when its characteristic momentum is much larger than
the relative momentum or the momentum spread of the resonance. In
intermediate energy heavy-ion collisions this is an appropriate assumption since
the characteristic momenta are of the order of magnitude of a few hundred MeV/ c
and the relative momenta of interest are smaller than 50 MeV/c. (The important
momentum components for the 2 He "resonance" lie below 100 MeV/c.) If we keep
the momentum dependence in Equation (6.15), expanding k: about 13/ 2, we can
show that this dependence cancels out in first order. Thus we expect the
formalism to give very close to the correct answer, unless the characteristic
momentum of the emission becomes quite small. Our derivation does not rely on
the assmnption that the system be semi-classical and that the thermal wavelength
be much shorter than the size of bound states or even the size of the emitting
system. Our result is valid as long as the product of 31 0 32 depends only on 12,1 +122 and
not 131 1:2. For systems which sample many states, such as a thermal system, this is a
good assumption. Since usually there is no knowledge of the matrix elements as a
function of k ° , these approximations are the best we can do. Luckily, since we are
interested in correlations for small relative momenta and since the characteristic
momenta of the protons are sufficiently high, these are excellent approximations.

The correlation function C(33) depends only on the final relative positions of

all the particles with momentum 1372. To make this more clear, we write Equation

(6.1) as:

108

 

C(iJ’,21’)= I431 1:132? Huff? )12, (6.22)
-+ —> -) -)
wherer =r 1- 2is therelativecoordinateoftheemittedparticles.Thefunctioangk )is
defined by:
_, Id3R1(f>’/2,R’+F’/2, 1))1(R’/2,12’-F’/2,1>)
FI-;(r ) = 2 (6.23)

IId31'1(13’/2,F",1>)1

where I? = i 5 +?2) is the center-of-mass coordinate of the two particles. The Wigner

1
function ((331)) is the phase-space distribution of particles of momentum 3 at

_,
position r at some time, t), after both particles have been emitted:

t
>
1(3,F’,1>)= I211 g(3,F’-3(1>-1)/m,1) . (6.24)

-00

For a given momentum 13’, the correlation has three degrees of freedom, 21’, which are
a function of F1316"). The most we can hope to extract from the correlation function is
EEG"), the normalized probability of two protons with the same momentum 13V 2
being separated by R We have shown in the derivation above that the calculation
of Fifi?) requires only the knowledge of the single-particle phase-space
distributions or the emission probability g; ,1? ,t).
6.2 Illustrative Calculations and Discussions

In this section, we will discuss the physical information contained in two-
proton correlation functions, such as the source size and lifetime, and point out
characteristic signatures of slowly-cooling and explosive sources, respectively. The
function 1736'-> ), Equation (6.23), contains significant information about the dynamics
of the collision. For instance, a long lived source will lead to an extended
separation when 17’ is parallel to the velocity 13/ 2m. Thus in addition to the spatial

extent of F136?) one gains insight into the lifetime of the source [Prat 87].

109

The E dependence yields insight into the dynamics of the collision. Both the
size and the lifetime can depend on the total momentum. Cooling [Prat 87, Erie 83,
Boal 86] is signified by increasingly large lifetimes for particles with smaller
energies. We discuss this later in the section on the compound nucleus. Collective
explosive flow is signaled by short lifetimes and shrinking apparent source
dimensions for increasing energy [Prat 87]. There should be a transition in reaction
mechanisms, from evaporative to exploding, at excitation energies near the
nuclear binding energy. This is the dynamical equivalent of the liquid-gas phase
transition. At low excitation, nuclei slowly evaporate particles, cooling like a hot
liquid drop. At sufficiently high excitation, nuclei explode and expand like a gas
[Prat 87].

The relative wave function, ¢(q’,17’), is influenced by three different effects:
identical particle interference, short range hadronic interaction, and the Coulomb
repulsion of the protons. Additionally, it is affected by the weighting factors for
the partition of various spin states. We briefly discuss how these effects contribute
to the correlation function and how the resulting correlations can be used to
determine F307,).

To calculate the relative wave function numerically, we solved the
Schrodinger equation for the 1. =0 and 1 =1 partial waves with Coulomb and the
Reid soft-core potential [Reid 68]. We used the full Coulomb wave function [Mess
76], tied-121?), and added the modification 5w; ,1? ) which is the contribution to the
relative wave function from the first two partial waves minus the contribution
which would have occurred if the strong interaction were absent. The two-proton

relative wave function, (116;), is then obtained by

1116,1312 =wS-1‘¢(21’,E’)I‘ +wt0| 3216,2311, (6.25)

110

where ’ 0G,?) and ’ $6,?) are the singlet and triplet ’ He spatial wave functions,
respectively and. ws and wt are the spin weighting factors for the singlet and triplet
2 He, respectively. In most cases, two-proton spins are assumed to be statistically
distributed so that ws=1/ 4 and wt=3/ 4. This assumption is followed in our
analysis.

For identical non-interacting particles, the squared wave function has the
form: 16(3, 1?) Izoc (1icos(230? )) .In thatcase, the inverse Fourier transform of the
correlation function would yield the complete three-dimensional function F13€ ). For
spin-half particles, the correlation function is reduced to one half at 131:0 [Kopy
72] and returns to unity with a width of qx=1/Rx. Experiments that gate on the
direction of the relative momentum can then determine all three spatial
dimensions of F369).

Coulomb interactions yield a dip in the correlation function which goes to
zero as 21’ approaches zero. As long as the characteristic dimensions of Fifi?) are
much smaller than the two-proton Bohr radius of 58 fm, the shape of the dip does
not depend on the shape or size of the source but only on the Gamov factor. For
larger sources, the Coulomb interaction, too, provides information about both the
source size and, to a weak degree, on the shape. The Coulomb dip in the
correlation function at I (Y I =0 will diminish for large sources. The most unfortunate
aspect of the Coulomb interaction is that the Coulomb dip lessens the number of
available pairs at small relative momentum, making it more difficult to see the
effects of identical particle interference.

Strong interactions provide excellent gauges of the size of smaller sources,
R310 fm. The 2He "resonance" appears as a bump in the two-proton correlation
function at q= 20 MeV/c. (Strictly speaking, the 2 He "resonance" is not a
resonance, since the phase shift does not increase by 90 °, but only by about 60 ° .)

The size of the bump is proportional to the percentage of pairs whose relative

111

position is within the size of the nearly bound state. Thus the height goes roughly
as R's. This provides a very sensitive test of the size, but not of the shape.

As was recently pointed out [Zhu 91], two-proton spin distribution could be
distributed non-statistically in fast break—up reactions of light projectile nuclei. In
such cases, the spin weighting factors can be different from ws=1 / 4, wt=3/ 4 and
the shape of correlation function will be modified accordingly.

For typical source sizes of about 5 fm, all the above effects are important.
Choosing a distribution F1341?) that fits the correlation function requires more than
just the appropriate size as measured by a single parameter. One must have F30?)
correct for large 17 in order to fit the correlation function at low 1? in the Coulomb
dip and one must have F134?) correct at small 1") in order to fit the height of the
correlation function. If the source is not so large that the Coulomb dip erases the
effect of identical particle interference, the shape must be chosen correctly as well
to fit the correlation function for different directions of 3.111115 all the physical
characteristics of F36") can be tested with correlation measurements provided that
all effects due to quantum statistics and final state interactions are properly taken
into account. In the following subsections we illustrate some these qualitative

expectations by calculations performed for simple analytical emission sources.

6.2.1 Spherical Sources of Negligible Lifetime

The relative importance of anti-symmetrization and the nuclear and
Coulomb interactions depends on the size of the emitting system. In order to
provide a quantitative comparison of these effects, we have calculated two-proton

correlation functions for Gaussian sources of negligible lifetime,
—) -> 2 2
g(p,r It) = p(r)8(t‘t o) = p o epo‘ /1' o )8(t‘t o ) I (626)

in which the nuclear interaction and the Pauli principle were turned off

successively. These calculations are compared in Figure 6.1 for a number of

112

representative radius parameters (to = 2.5, S, 10, 20 fm) . The solid curves show
the results of the full calculations which include the nuclear and Coulomb
interactions and the Pauli exclusion principle. The dotted lines show calculations
for which the nuclear interaction has been turned off; these calculations still
include quantum effects of the two-proton Coulomb interaction and the anti-
symmetrization of the relative wave function. The dashed curves represent
calculations for which both the nuclear interaction and the anti-symmetrization of
the relative wave function have been turned off. (These latter calculations differ
from classical Coulomb trajectory calculations [DeYo 89, DeYo 90, Elma 91] since
the Coulomb repulsion between the two protons is treated quantum mechanically.
For large source dimensions or emission from long-lived sources, this difference
should be of minor importance.) For radius parameters, r o g 10 frn, both the anti-
symmetrization of the relative wave function and the nuclear interaction have
important effects on the detailed shape of the calculated correlation functions. For
much larger source dimensions, r o z 20 fm, the Coulomb interaction dominates and
the neglect of the Pauli principle and the nuclear interaction can be justified [DeYo
89, DeYo 90, Elma 91].

Furthermore, we have calculated two-proton correlation functions for
Gaussian source of negligible lifetime with three different assumptions about the

spin weighting factors (ws,wt) in Equation (6.25) in order to assess the relative

113

MSU-90-087

 

f . . . 1 . .
complete calculation

 

°°°°°°°°° no nuclear interaction ‘

- - - - no nuclear interaction“:
no antisymmetrization .

   

 

1 + R(q)

 

vvvv

ro=20fm ‘

 

 

 

 

 

50 0 50 100

Figure 6 .1: Two-proton correlation functions calculated for Gaussian sources of
negligible lifetime with representative radius parameters, r 0 =2.5, 5, 10, 20 frn. The
solid curves show the result of the full calculations; the dotted curves show
calculations for which the nuclear interaction is neglected; the dashed curves
shown calculations for which the nuclear interaction and the Pauli principle are
neglected.

   
 
  

114

MSU-Ql-IBO

 

I
singlet 2He -
‘ °°°°° weighted total ’
" " '- ‘ triplet 2He

 

 

 

 

 

 

 

 

 

100

Figure 6 .2: Two-proton correlation functions calculated for Gaussian sources of
negligible lifetime with representative radius parameters, r 0 =2. 5, 5, 10, 20 fm.
Assuming the two-proton relative wavefunction to be. I11)! 2 ws-I ’¢| '

wt. | ’ (j) I 2 ,the solid, dotdashed and dashed curves show calculations for (1) ws=
1 and wt: 0, (2)ws= 1/4 and wt= 3/4, (3)ws= Oand Wt: 1, respectively.

115

importance of singlet and triplet spin states. In Figure 6.2, solid and dashed curves
represent the calculated two-proton correlation functions for the singlet (S = 0:
ws= 1, wt= 0) and triplet (S = 1: ws= 0, wt= 1) 2 He spin states, respectively. The
dot-dashed curves are the averaged calculations using ws= 1/ 4 and wt = 3/ 4. Four
representative radius parameters (r0 = 2.5, 5, 10, 20 fm) are used to see the
dependence of the spin-state effect on the emission-source sizes. For radius
parameters, ro s5 fm, the singlet two-protons have very strong correlations
around q = 20 MeV/c and the triplet two-protons show large anti-correlations.
These are consistent with the respective proton-proton potentials [Reid 68]. At
larger radii, the singlet two-proton correlations become weak and the peak shifted
to q less than 20 MeV/c, and the anti-correlations for triplet two-protons are
smaller, too. Overall, the correlation peak at q=20MeV/c is most dominated by the
singlet spin state.

Most analyses of fast-particle-emission processes [Zarb 81, Lync 83, Gust 84,
Chen 87a, Chen 87b, Poch 86, Poch 87, Fox 88, Awes 88, Cebr 89] have used
sources of negligible lifetime with spherically symmetric Gaussian density
distributions, Equation (6.26). For spherically symmetric sources, the shape of the
two-proton correlation function is rather insensitive to details of the density
profile. To illustrate this point, we compare the shapes of two-proton correlation
functions calculated for sources of negligible lifetime with Gaussian, Equation

(6.26), and uniform sharp-sphere density distributions,

g(3,2’,t)=p 69(r)9(RS-r)5(t-t6 ). (6.27)

In Equation (6.27), Rs is the sharp sphere radius and E-Xx) is the unit step function
which vanishes for negative arguments. The solid and dotted curves in Figure 6.3
represent calculations performed with Eqs. (6.26) and (6.27), respectively. In these

calculations, the radius parameters have been adjusted to match the magnitudes

116

 

 

 

 

 

 

 

2 . 5 1 1 1 1 1 1 1 MSUI- 90402
)- l .1
2.0 3— _:
C I
3 1.5 :— _—
n: _'_' -
+ _ _
1—1 1.0 _— AAAAAA %
0.5 -— _L'
0.0 ‘

0 100

 

q (MeV/c)

Figure 6 .3: Two-proton correlation functions calculated for sources of negligible
lifetime assuming Gaussian (solid lines) and sharp sphere (dotted lines) density
distributions. The radius parameters, r o and Rs’ are indicated.

117

 

   
   
 
  

 

 

 

’ MSU-90-l03
10 L“I I I I I I I I I l I I I I I I I I I I I I I I I I I I I I.:.I I I r1 TL;
E R8: V5/2 oro \ ...g' E
a 3- —3
”11‘ :
'4—1 : S
m I :
D: 4 E... J
E \ R.=1.493-ro+o.0475 3
2 :— _3
E ,. ° 3
O 7' I I I I I II I I I I I I I J I I I I L I I I I I I I I I I I I I I I I I l-I

r0 (fm)

Figure 6.4: Relation between radius parameters, r o and R , of Gaussian and sharp
sphere density distributions for which equivalent two-profon correlation functions
are obtained in the limit of negligible lifetime. Crosses indicate results of
numerical calculations, the solid line represents a linear fit, and the dotted curve

 

shows the relation Rs=~/(5/ 2) r a used in the literature [Boal 90].

118

of the maxima at q=20 MeV. The correlation functions calculated for these rather
different density profiles are virtually identical in shape.
Previously, Gaussian source parameters, r0, have been translated [Boal 90]

into equivalent sharp-sphere radii, Rs’ by employing the approximate relation, Rs=

 

/(5/ 2) to. This relation can be justified by equating the rms radii of the two
density distributions. Equivalent radii, r o and Rs’ of the two density profiles can
also be defined by the requirement that the calculated correlation functions have
maxima of identical height. Such equivalent radii are indicated by X-shaped
symbols in Figure 6.4. These equivalent radii can be rather well described by the

relation Rs: 1.493ro +0.0475 fm; this relation is represented by the solid curve in

 

Figure 6.4. For comparison, the analytically derived relation, Rs= /(5/ 2) r0, is

indicated by the dotted curve; it is a surprisingly good approximation.
6.2.2 Spherical Sources of Finite Lifetime

In order to illustrate the sensitivity of the shapes of two-proton correlation
functions to the lifetime of the emitting system, we have performed calculations
under the simplifying assumption that particles are emitted from a spherical
volume of radius Rs according to a simple exponential law,

2
-p /2mT-t/ T , (6.28)

gar-1t)“ P 0 9(r)O(RS-r)E-)(t)pe

where T and 1' denote the (constant) source temperature and lifetime, respectively.

Calculations for two-proton correlation functions, integrated over all relative
orientations between I; and 21’, are shown in Figure 6.5. The parameters used in these
calculations are given in the figure. The upper panel of the figure illustrates the

sensitivity to the lifetime of the emitting system for proton pairs of total

119

 

 

 

 

 
 
 

 

 

 

MSU-QO-IO?’
I 1 1 y 1 -
2.0 f ,v‘.’ -
: .o ‘2. R,=5fm, T=6MeV
1.5 ’1
1.0 E-
3 0.5 f j
0: : 4" '. """"" $88 Pc.m.=300MeV/c :
+ 0 0 " ............. 1000 -
2-1 : “““ 3000 1
1.5 . t ‘ I 2 § 2 IL : 1 .r
i 4.1.7.3 R,=5fm, T=6MeV
1.0 f " ' ______-__-_-__._______4jl
E Pc.m.(Mev/c) :
0.5 ; ----- 200 7
’ ............. 300 } Tzloofm/C
E —— 400
0.0 - ' 4 1 - . - 1 .
o 20 40 60 80 100
q (MeV/c)

Figure 6 .:5 quproton correlation functions predicted for emission from
spherical sources of radius RS =5 fm, decaying isotropically with fixed life—time,
Equation (6. 28). The top and bottom panels depict the dependence on life-time,1:
and total momentum, Pc.m’ respectively.

120

momentum, Pc.m.= 300 MeV/c, measured in the rest frame of the emitting source.
The calculated two-proton correlation functions exhibit considerable sensitivity to
lifetimes of the order of 30-3000 fm/c. For much shorter lifetimes, the shape of the
correlation function becomes dominated by the spatial dimension of the emitting
system; for much longer lifetimes the correlations disappear. The lower panel of
Figure 6.5 illustrates how the correlation function depends on the total momentum
of the emitted proton pairs for the case of a fixed lifetime, I=100 fm/c. The
calculated correlations become more pronounced for smaller total momenta, i.e.
for particles emitted with lower kinetic energies. This dependence can be
understood in terms of the spatial extent of the Wigner distribution. The
longitudinal dimension of the apparent source is of the order of (Pc.m./2mp)‘t,
where m denotes the proton mass. For fixed lifetime, I, this quantity increases for
larger vallues of Pam. and the correlation function becomes attenuated. Such a
momentum dependence stands in contrast to experimental observations [Lync 83,
Chen 87a, Chen 87b,. Poch 86, Poch 87, Awes 88, Gong 91b] and is opposite to that
calculated for emission from compound nuclei for which the effects of cooling
produce a strong momentum dependence of the effective decay times (see also the
discussion in Chapter 8).

Figure 6.6 illustrates the dependence of the two-proton correlation function
on the angle, ‘1’ = cos.1 ( 1333/1321 ) , between the relative and total momentum vectors
of the two-proton pair. As was done in the experimental analysis [Gong 91b], we
define longitudinal and transverse correlation functions by the cuts lcos‘f'l 20.77
(‘l’=0 ° -40 ° or 140 ° -180 ° ) and l cos‘I’ I s 0.5 (‘I’=60 ° -120 ° ), respectively. Different
panels of Figure 6.6 show longitudinal and transverse correlation functions
calculated for different lifetimes 1:0, 10, 30, 100, 300, 1000 fm/c. In these
calculations, the total momentum of the proton pair was kept constant at Pc.m.=300

MeV/c. For very short lifetimes, 1310 fm/c, the apparent source is essentially

121

 

 

  
 

  
  
  

 

  

  

 

 

 

 

 

 

 

MSU-QO-IOB
2.0 ' I l ' I .. I * I ' I ' I r _
: T=0fm/c j; T=10fm/c
C ‘_ R,=51m 1
1,5 .. T=6MeV 1
. II P6.m.=300MeV/C ;
1.0 I {f- ...................
’ —-—¢=(o-4o.14o-1ao)° ‘
05 -L ........ 1":(60-120)‘, _
A 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0‘ 5 I I ' I ' I b I ' I ' I ' I '
v 1.5 -- ~
0:“. ; T=30fm/c ; T=100fm/c j
+ : :
"‘ 1-0 .................... f
0.5 y _i-
1.0 ,
f T=300fm/c f T=1000frn/c .
0.5 ’ V j
0 20 40 60 80 0 20 40 60 80 100
q (MeV/c)

Figure 6 .6: Longitudinal and transverse correlation functions calculated for
emission from sources decaying with constant lifetimes, Equation (6.28). The
parameters used in these calculations are indicated in the figure.

122

spherical in shape and life-time effects are negligible for the calculation of the two-
proton correlation function. For an intermediate time-window, I=30-300 fm/c,
longitudinal and transverse correlation functions are sensitive to the lifetime of the
emitting system. For larger lifetimes, 121000 fm/c, this sensitivity is essentially lost
as the average separation between emitted particles becomes so large that anti-
symmetrization effects become negligible. In fact, for extremely large lifetimes the
effect is reversed because Coulomb induced correlations contain a weak amount of
directional information. The Coulomb force is parallel to the relative displacement
of the protons; therefore, the Coulomb hole in the correlation function will be
strongest when the relative momentum is parallel to the longest dimension of the
pair’s separation. For long-lived sources this is the longitudinal direction.
Comparisons of longitudinal and transverse correlation functions yield the
strongest sensitivity to the lifetime of the emitting system on the order of 30-300
fm/c. By more judicious choices of the gates on Pc.m.’ one may stretch the
sensitivity of such measurements beyond these rough boundaries.
6.3 Comparison with Classical Trajectory Calculation

Two-proton correlation functions have been used to study emission time
scales of evaporative emission processes [DeYo 89, Ardo 89, DeYo 90, Gong 90b,
Elma 91, Gong 91b]. Two different techniques have been used to calculate two-
proton correlation functions. One technique is based upon the Wigner function
formalism outlined in Section 6.1, and the other employs classical trajectory
calculations [DeYo 89, DeYo 90, Elma 91]. As it is not clear whether and under
which conditions the two approaches yield similar (or identical) results, we have
performed numerical calculations for a simplified source function in order to
make quantitative comparisons between the two approaches and explore the
validity of the approximations underlying different theoretical treatments.

Similar to Equation (6.28), we used a simple source function corresponding

to thermal emission from the surface of a sharp sphere of radius R:

123

A A A A - - ..
361:2) °~= (r 0p)9(r 0p)5(r—RS)O(t)9(E-VS)(E-Vs)e (E VS”T t/ I . (6.29)
A " . -) -+ 2 2 .
Here, r and p are un1t vectors parallel to r and p; E=p / 2m; VS = ZSe /RS 15 the
Coulomb barrier. Calculations were performed with the parameters: ZS = 78, R5 =

8 fm, T = 4 MeV, and 1: = 200, 500, 1000 fm/c. We have chosen a relatively low
temperature parameter to emphasize the emission of low-energy protons
pertinent to evaporative emission from hot compound nuclei.

In the Wigner function formalism, the two-particle correlation function
depends only on the final relative positions of all the particles with momentum
P / 2. It is derived under the assumptions that the final-state interaction between the
two protons dominates and that final-state interactions with the remaining system
are negligible, that the single-particle phase-space distribution varies only slowly
over particle momenta P/Ziq, and that the correlation functions are determined
by the two-body density of states as corrected by the interactions between the two
particles.

For a long-lived source, we assume that the Coulomb force is the only
interaction between two protons. After integrating over all relative orientations
between I” and If, we obtained the classical approximation to Equation (6.22) as:
[Kim 91a, Kim 91b]

1+R(P,q) = I d3r F1316") [ l-mez/qzr] ”2 .

(6.30)
Classical trajectory calculations similar to References [Kim 91a, Kim 91b]

were performed for proton emissions. The final-state Coulomb interaction with

the emitting system was incorporated (ZS = 78), or neglected (ZS = 0). Recoil effects

were taken into account (the mass of the emitting system was taken as M = 197 u).

S
The emission function in the rest frame of the emitting system, Equation 6.29, was

sampled by Monte-Carlo techniques. For trajectory calculations in which the

124

Coulomb interaction with the emitting system was taken into account (ZS = 78),

the Coulomb parameter in Equation (6.29) was set to V5 = O. For calculations in

which the Coulomb interaction with the emitting nucleus was turned off, the

S = 78e2/RS was used in Equation (6.29) to ensure close

similarity of the asymptotic kinetic energy spectra for all calculations. Upon

Coulomb parameter V

emission, the particle trajectories were calculated by integrating N ewton’s
equations, and the asymptotic particle momenta were stored as coincidence events
and as single-particle spectra. These simulated events were treated in the same
manner as the measured data to generate two-proton correlation functions
according to Equation (7.1).

Correlation functions calculated with various approximations are compared
in Figure 6.7 for emission times of T=200fm/c (upper part), SOOfm/c (middle
part), and IOOOfm/c (lower part). The solid curves represent the correlation
functions calculated with the Wigner function formalism by including nuclear and
Coulomb interactions and anti-symmetrization of wave-function. The dotted
curves represent the same calculations except that the nuclear interaction was
turned off. Comparison between these two calculations indicates that the nuclear
interaction is important for life-times TS 500 fm/ c.

The dashed curves in Figure 6.7 represent calculations with the Wigner
function formalism in which the two protons are assumed to be spinless and
distinguishable particles experiencing only Coulomb interaction. The relative
wave function is given by the (non-antisymmetrized) Coulomb scattering wave-
function. The dot-dashed curves show the results obtained with Equation (6.30).
For the lifetimes considered, the classical approximation provides an excellent

approximation for the calculations of angle integrated correlation functions

125

 

 
  
      

  

 

 

 

* I r I r r ' ' r 1*
1.0 ' ° ° °
0.9—
3‘2 T=200fm/c
0.6 0 trajectory (Z,=0)
0'5 0 trajectory (Z,=78)
1.0 "8° _ . .
’3 0 9 55°
3:“ 0.8 o T=500fm/c
+ 0.7 nuclear,Coulomb,spin
O 6 . '''''''' nuclear-off
H ° - - - nuclear—off. spinless
0'5 :.::::;.::fl:s:1?t{:;::}.
1.0 _.,_.--—- —, . ' .“
0.9 .. 0 '
g: ‘ T=IOOOfm/c
0.6 . R,=arm,'r=4Mev
0-5¥¥..1....1....1....1
O 10 20 3O 40

q (MeV)

Figure 6 .7: Two-proton correlation functions for a schematic source, Equation
(6.29), representing thermal surface emission with fixed life-times, I=200fm/c
(upper part), T=500fm/c (middle part), and T=1000fm/c (lower part). The
parameters are indicated in the figure. Different symbols and curves are explained
in the text.

126

arising from Coulomb final-state interactions between distinguishable particles,
even if they are as light as protons. (Note, however, that anti-symmetrization
effects can only be safely neglected for 121000 fm).

The open points in Figure 6.7 show the results of classical trajectory
calculations incorporating the Coulomb interaction with the emitting system
(ZS=78). For comparison, the solid points show results obtained by turning off the
Coulomb interaction with the emitting source (25:0). (These latter calculations are
not necessarily identical with results obtained from Equation (6.30) as they do not
rely on the approximation, implicit in the Wigner function formalism, of using
only the one-body phase-space density at the center-of-mass velocity of the two-
proton pair.) The differences between the two trajectory calculations are small. For
short emission time scales (1:200 fm/c), however, they are not negligible. (This
conclusion depends on the shape of the energy spectrum: Coulomb distortions in
the field of the heavy reaction residue decrease for the emission of more energetic
protons.) For large emission time scales, I=1000 frn/c, all calculations agree to a
good degree of accuracy.

In summary, our investigation suggests that the neglect of the two-proton
nuclear interaction and the anti-symmetrization of the two-proton wave-function
leads to comparable or larger inaccuracies than the neglect of the Coulomb
interaction with the emitting heavy residue. For large emission times 121000 fm/c,
pertinent for the decay of low-temperature compound nuclei, the classical
formula, Equation (6.30), provides an excellent approximation for the calculation
of angle integrated two-proton correlation functions and gives virtually identical

results to the classical trajectory calculations.

CHAPTER 7. TWO-PROTON CORRELATION FOR THE
GAUSSIAN SOURCE

The experimental two-particle correlation functions, R(q), are presented as a

function of relative momentum, q, using the definition:
—> -+ —) —§
2Y12(p 1,p 2) = C12(1+R(q)) 0 2Y1(p 1)Y2(p2) . (7.1)

Here, 31 and 32 are the laboratory momenta of particles 1 and 2; q =% | 31 - 32 l is the
relative momentum of the particle pair; Y1 261,32) is the coincidence yield; and
qu-i 1) and Yip-)2) are the single particle yields. For each experimental gating
condition, the sums on both sides of Equation (7.1) are extended over all energy
and detector combinations corresponding to the given bins of q. The
normalization constant, C12, is determined by the requirement that R(q)=0 for large
relative momenta.

Theoretical correlation functions are calculated with Equations (622-24). In
most previous analyses, g(p),i-'),t) was parametrized in terms of a simple Gaussian

source of negligible lifetime,

30372.?» = p . 5(t)exp {-r 2 /r0 ’(Pn . (7.2)
where the radius parameter was allowed to be momentum dependent. To allow
comparisons with the previous analyses and to systematize our data, we will

adopt this parametrization for the calculations presented in this Chapter.

127

128

7.1 Angle- and Energy-integrated Correlation Functions

Two-proton correlation functions corresponding to sums over all detectors
and all proton energies above the applied software energy threshold of 10 MeV
are compared in Figure 7.1. The correlation functions measured for the reactions
1 ‘N + 2 7A1 and ‘ ‘N + ‘ ° 7Au exhibit pronounced maxima at relative
momenta q z 20 MeV/c, see top panels of Figure 7.1. Small but significant
differences exist at small relative momenta where the minimum at q a 0 MeV/c is

‘ 9 7Au target. The correlation

more pronounced for the 2 7 Al than for the
functions measured for the ‘ 2 ° Xe-induced reactions do not exhibit a pronounced
maximum at q z 20 MeV/c, but only a minimum at q z 0 MeV/c, see bottom
panels of Figure 7.1. The correlation functions measured for the reactions ' 2 ° Xe
+ 2 7Al and ‘ 2 °Xe + ‘ 2 2Sn have very similar shapes. On the other hand, the

1 2 ° Xe induced reactions are

correlation functions measured for HN and
strikingly different, corroborating previously observed differences between
correlation functions measured for equilibrium and nonequilibrium emission
processes in slightly different reactions [DeYo 89, Queb 89, Ardo 89, DeYo 90]. For
orientation, the solid lines show theoretical correlation functions predicted for
Gaussian sources of negligible lifetime. For the reactions induced by ‘ ‘N and
l 2 °Xe, radius parameters of r0 = 4.4 and 70 fm were used, respectively. These
source parameters should be compared to the equivalent radius parameters for Al

and Au nuclei, r0 (A1) = 2.5 fm and r0 (Au) 2 4.4 fm , which are obtained from
tabulated [Brow 84] r.m.s. charge radii using the approximate relation r o = v/(7/3)

rrms' For the ‘ ‘N-induced reactions a source radius ofro =4.4 fm is not
necessarily unreasonable. However, it is astonishing that this radius parameter
exhibits no obvious dependence on the size of the target nucleus. A purely
geometrical interpretation of the correlation function is, therefore, in doubt. For

the large source parameter, r o z 70 fm, used to describe the correlation functions

129

 

    
 
   
     
    

 

  
 
  

     

 

 

 

MSU-90-063
. , . , . . .
1.5 - -
L _ro=4.4fm _ro=4.4fm '
1.0 r i'
C 27Al(“N,pp) ° 1°7Au(“N,pp) :
"a o 5 '_’ E/A=75MeV E/A=75MeV _‘
53’ ° .+ .
: :j : : t . } r r
+ . .
1.5 _ —
"" . —-ro=70fm —-ro=70fm «
L .
10 _ ..oooooo...
L. .
; 27Al(12°Xe,pp) 1228n(129xe.pp) ;
0.5L E/A=31MeV E/A=31MeV __
r . . . . l . i . . 7 . . . . l . . j ml
0 50 100 50 100
q (MeV/C)

Figure 7.1: Comparison of energy integrated two-proton correlation functions
measured for the reactions ' ‘N+ 2 7 Al and ‘ ‘N+‘ ° 7 Au at E/A=75 MeV (top
panels) and the reactions ‘ 2 °Xe+2 7Al and ‘ 2 °Xe+‘ 2 2Sn at E/A=31 MeV
(bottom panels). The solid curves represent correlation functions predicted for
Gaussian sources of negligible lifetime with the indicated radius parameters, r o.

130

129

measured for the Xe induced reactions, a purely geometrical interpretation is

clearly unphysical and the lifetime of the emitting system must play a major role.

7.2 Dependence on Total Momentum

Two-proton correlation functions are known to exhibit strong dependences
on the energy of the emitted particles [Lync 83, Chen 87a, Chen 87b, Poch 86, Poch
87, Awes 88,Gong 90b, Gong 90c], or, equivalently, on the total momentum}: = 31 +
1322, of the coincident proton pair. Figure 7.2 shows two-proton correlation functions
for two representative momentum gates, measured for the reactions 2 2 9Xe +
2 2A1 (upper panel) and 2 2 °Xe + 2 2 2Sn (lower panel). The momentum gates
represented by the solid and open points correspond to protons emitted with
kinetic energies below and above the compound nucleus Coulomb barriers,
respectively. For the 2 2 °Xe + 2 2A1 reaction, the two momentum gates, P=480-
570 and 660-750 MeV/c, select protons with kinetic energies of Ec.m.z 5-10 and 15-
23 MeV, respectively, in the center-of-mass frame of reference (i.e. the rest frame of
the compound nucleus). For the 2 2 °Xe + 2 2 2Sn reaction, the two momentum
gates, P2270640 and 540-660 MeV/c, select protons with kinetic energies of Ecm.”
1-15 and 15-27 MeV, respectively. As can be expected from qualitative time scale
arguments, sub-barrier emission results in a reduction of the minimum at qu
MeV/ c. Furthermore, correlation functions at sub-barrier energies can suffer
enhanced attenuations and/ or distortions from sequential decays of primary
fragments emitted in particle unbound states [Frie 83] and from deflections in the
Coulomb field of the heavy reaction residue. Because of these additional
complications, we will refrain from a more detailed analysis of two-proton
correlation functions for protons emitted with sub-barrier energies. Calculations of
two.proton correlation functions for evaporative processes will be presented in
Chapter 8.

l

The correlation functions measured for the ‘ N-induced reactions exhibit a

more pronounced dependence on the total momentum of the detected proton

131

pairs. Figures 7.3 and 7.4 show correlation functions for representative ranges of
the total momentum, P, for the reactions 2 4N + 2 7Al and 2 ‘N + 2 2 2Au,
respectively. COnsistent with previous measurements, the maximum at q z 20
MeV/c becomes more pronounced for larger total momenta, i.e. for the emission
of more energetic particles. For the lowest momentum gate, P=270-390 MeV/ c, the
correlation functions are distinctly different for the two targets. For the 2 2 Al

2 ° 2 Au target, on

target, a clear maximum at q = 20 MeV/c is measured. For the
the other hand, this maximum is barely visible and the shape of the correlation
function resembles that measured for evaporative processes.

The solid curves in Figures 7.3 and 7.4 show correlation functions calculated
for Gaussian sources of negligible lifetime, Equation (7.2), using momentum
dependent source parameters shown in Figure 7.5. In these calculations,
appropriate averages over total momentum were performed, and the resolution of
the hodosc0pe was taken into account. The overall trends of the data are well
described. However, the shapes of the measured correlation functions are not
reproduced in all details. The peaks of the calculated correlation functions are
slightly narrower than the peaks of the measured correlation functions. The
disagreement is particularly evident in the region around q = 30-40 MeV/c. In
addition, for the 2 2N + 2 2 7Au reaction, the adopted parametrization fails to
reproduce the exact shape of the minimum at q z 0 MeV/c for the low momentum
gate, P = 270-390 MeV/c.

In order to provide a simple description of the momentum dependence of
the two-proton correlation functions measured for the 2 2 N-induced reactions, we
have constructed experimental correlation functions for a number of narrow gates

placed on the total momentum, P. Each such correlation function was

characterized in terms of a Gaussian source, Equation (7.2), by requiring that

132

MSU-90-O74

 

. . f . , . . . .
1-2 " 27141(229Xe,pp), E/A=31MeV, 6"=25° "

1.0 — “,;;w?¢g;;a;aaoz; was...”

08 _. § . P=480—57OMeV/c _
° P=660-75OMeV/c

 

 

 

 

‘2: 12 ” 122Sn(2222Xe,pp),E/A=31MeV, e,,=25° ‘
*4 ¢
1.0 — + 4:25:33 WWW‘W
' 22¢
08 _+ r} . P=270—54OMeV/c .
l o P=540—660MeV/c
[- 4, 1
o 50 100

q (MeV/C)

Figure 7.2: Two-proton correlation functions measured for the reactions
22 °Xe+2 2A1 and 22 °Xe+2 2 2Sn at E/A=31 MeV. The gates on the total
momenta, P, of the coincident proton pairs are indicated; solid and open points
represent center-of-mass energies below and above the compound nucleus
Coulomb barriers.

1 + R(q)

Figur

2.0

1.5

1.0

0.5

133

MSU-90-067

 

 

‘
\

 

l l l I l l

27.A.1(“‘N,pp), E/A=75MeV j
®.v=25° 4

o P=840-1230MeV/c .
0 P=450— 780MeV/c ‘
u P=27o— 390MeV/c —

 

 

e 7.3:

¢¢ ¢ .
xii": .1-:"-_ n!!! :¢ é
.. .- ¢ . 3+? _
. . I L , . . 2
50 100
q (MeV/C)

Two-proton correlation functions measured for the reaction

2 4N<I~2 7A1 at E/A=75 MeV. The gates placed on the total momenta, P, of the
coincident particle pairs are indicated. The solid curves represent calculations for
Gaussian sources of negligible lifetime assuming a dependence of the radius
parameter, r o , on total momentum, P, as shown by the open points in Figure 7.5
and folded with the response of the experimental apparatus.

1 + R(q)

Figure 7 .4:

2.0

   

134

MSU-SO-OGB
l

 

 

l I I l I

2222Au(2‘2N,pp), E/A=75MeV—_

av: o ..
0 P=840-123OMeV/c '
0 P=450- 780MeV/c *
P=27o- 390MeV/c —

 

 

Two-proton correlation functions measured for the reaction

2 ‘N +2 2 2 Au at E/A=75 MeV. The gates placed on the total momenta, P, of the
coincident particle pair are indicated. The solid curves represent calculations for
Gaussian sources of negligible lifetime assuming a dependence of the radius
parameter, r o , on total momentum as shown by the solid points in Figure 7.5 and
folded with the response of the experimental apparatus.

135

the correlation function calculated for this source could reproduce the height of
the maximum of the measured correlation function. The dependence of the
extracted radius parameters on the total momentum of the detected particle pair,
r o (P), is shown in Figure 7.5. The error bars indicate estimated systematic errors.
For the 2 4N + 2 2 2Au reaction, the adopted parametrization fails to reproduce
the shape of the minimum at q z 0 MeV/c for the low momentum gates, P<350
MeV/c. Here, the assumed monotonic dependence of the radius parameter on
total momentum appears to be inadequate, and one may have to mix in additional
contributions from much larger sources or, alternatively, from long-lived
evaporative processes to fit the shape of the minimum at q == 0 MeV/c. Because of
the inherent ambiguities of such an approach, we did not pursue this possibility.
Instead, we have indicated these possible contributions by open-ended error bars.

We wish to comment that the radius parameter, r o (P), cannot be interpreted
at face value since the lifetime effect must be present for emissions of low energy
protons. Even though correlation functions are calculated without lifetime, this
approach is used only to summarize and display the full momentum dependence
and some subtle target dependence of two-proton correlation functions.

While the average correlation functions measured for the 2 2N + 2 2 Al and
2 2N + 2 2 2 Au reaction are very similar (see Figure 7.1), significant differences
surface when one explores the dependence on the total momentum of the proton
pairs. Such more subtle differences, already apparent in the raw data shown in
Figures 7.3 and 7.4, are clearly revealed in Figure 7.5. For the 2 2N + 2 2 2Au
reaction, the extracted source dimensions exhibit a nearly monotonic increase with
decreasing total momentum of the detected proton pair. For the 2 2N + 2 2Al
reaction, on the other hand, the extracted source dimensions are rather constant

over the range of P=400-750 MeV/c. The extracted source dimensions are

136

 

tl'VUVUIU

° 222AU(22N.pp) '5
° 222A1(2‘2N.pp)
e:— + E/A=75MeV

i
.2+++ .
¢¢¢¢3$+¢¢$22
3 ¢ '5

3i- 2 0

q

L l l 1 l L l l l l l I l 1 l L _k

200 400 600 800 1000 1200
P (MeV/c)

'TU'U'U

 

llljllllll

IUT'UU
lllllllllll

I'UW' I'

I'o (fm)
01
[

IUIII'U'
l

T

J. '1

 

 

 

Figure 7 .5: Radius parameters, r o , for Gaussian sources of negligible lifetime
extracted from two-proton correlation functions gated by different total momenta,
P, of the coincident particle pairs for 2 2 N induced reactions on 2 2 Al and 2 2 2 Au
at E/A=75 MeV. The error bars indicate estimated systematic errors.

137

comparable for the two targets at high total momenta, P2 800 MeV/c, indicating
that very energetic particles are emitted by comparable processes. At low total
momenta, P5500 MeV/c, the extracted source dimensions are considerably larger

1

for the reactions on 2 2 Au than for the reactions on 2 2 Al, most likely indicating

larger contributions from slow evaporation processes for the 2 2 2

Au target.

As more two-proton correlation measurements become available, interesting
systematic trends emerge. Figure 7.6 shows the radius parameters for Gaussian
source of negligible lifetime extracted from 2 He + Ag(natural) at E/ A = 67 MeV
[Zhu91], 2 2Ar+ 22 2Au at E/A = 60 MeV [Poch 87], and 2 2N +22 2Au at E/A
= 75 MeV (this work, see Figure 7.5), as a function of v/vbeam’ the ratio of v, the

mean velocity of the two-proton pairs, to v , the beam velocity. The solid

curve is an interpolation to the data obtainedbfoarmthe 2 2 N + 2 2 2 Au reaction. The
dashed and dotdashed lines are obtained by scaling the solid line with
multiplicative factors of (3/14)2/3 and (40/ 14)“3 for the reactions 2He + Ag
and 2 2Ar + 2 2 2_Au, respectively. The good agreement between extrapolated
and measured source radii at v/vbeam 2 0.7 for both 2He and 2 0Arinduced
reactions suggests that the spatial extent of the emitting region for energetic
protons scales with the radius of the light projectile impinging on a heavy target.
For lower energy protons emitted from an evaporative source on a long time scale,
the extracted radius parameters may not represent the source size but rather
reflect the lifetime of the emitting system (see Section 6.2.2 and Chapter 8). This
qualitative argument is in accordance with the observation that the scaling of the
measured source radii with the radius of the projectile does not apply at v/vbeam <
0.7.

It is interesting to note that similar source-size scaling with the projectile
radius has been observed from two-pion intensity interferometry in relativistic

nucleus-nucleus collisions [Bart 86].

138

 

 

 

 

 

 

MSU-91-201

12'rl"’*r*fi*‘r'
[:1 1922Au(2'22Ar,pp),E/A=60MeV -
10; o 222Au(22N,pp), E/A='75MeV _'
. <> mtAg("He,pp), E/A=67MeV .
B "' _J
A - -
..a - 4
v j- a
o 6 ._ _l
S-c L .
)- d
4 ‘- ._
2 ._ —

L L l L L L L l L . . L l L

05 1.0 1 5
V/Vbeam

Figure 7.6: Systematics of Gaussian source radii, r0 (v/v ), extracted from

. . . . ea .
various asymmetric reactions. The points and curves are exp amed in the text.

139

7.3 Effect of Instrumental Resolution

In order to evaluate instrumental distortions of the measured correlation
functions, we have simulated the response of our experimental apparatus by
taking its known energy and angular resolution into account. The solid curves in
Figure 7.7 show the undistorted correlation functions, calculated for Gaussian
sources with the indicated radius parameters. The points are results of Monte
Carlo calculations in which the angular and energy resolutions of the experimental
apparatus are taken into account. In these calculations, the coincidence yield was

taken as
Y12(p1,p 2) = (1+R(q) ) Y1(p1)Y2(p2) , (7.3)

where the singles yields, Y1(p21) and Y1 (1322), were taken from the single particle yields
measured for the 2 2N+2 2Al reaction and R(q) was calculated by assuming a
Gaussian source of negligible lifetime, Equation (7.2), with the radius parameters
given in the figure. Both singles and coincidence distributions were smeared by
the energy and angular resolution of the experimental apparatus and sorted in the
same way as the experimental data, using three representative momentum gates.
The simulated correlation functions are in close agreement with the original,
undistorted correlation function. Except at very small relative momenta, line
shape distortions caused by the resolution of the experimental apparatus are
negligible. Note, in particular, the absence of visible distortions in the region of q
z 30-40 MeV/c where the Gaussian-source fits deviate from the experimental

correlation functions (see also Figures 7.3 and 7.4).

1 + R(q)

Figure 7 .7:

2.0

0.5

140

MSU-90-066

 

 

 

I I I I I I I I I

1

Monte Carlo simulation of
experimental resolution

0 P=870-900MeV/c, ro=3.6frn; ~
. P=750-780MeV/c, ro=4.2fm; 2
a P=330—360MeV/c, ro=5.2fm. -

 

l l l I I 1 l l l I

 

50 100
q (MeV/c)

Monte-Carlo simulation for the response of the experimental

apparatus. The curves show the undistorted correlation functions assumed for the
indicated gates on total momentum, P. The points represent the calculated
response of the apparatus after taking the energy and angular resolution effects
into account. The error bars show the statistical accuracy of the MonteoCarlo

calculations.

CHAPTER 8. TWO-PROTON CORRELATION FOR
EVAPORATIVE EMISSION

Two-proton correlation functions for particle evaporation from long-lived
compound nuclei can be calculated by using the Wigner-function formalism [Prat
87, Gong 91a], Equations (6.22-24). We have used the statistical model of Reference
[Frie 83] to construct Wigner functions for evaporative emission from equilibrated

compound nuclei.

8.1 Statistical Evaporation Model

In this model, the average particle emission is calculated from a generalized
Weisskopf formula and cooling of the compound nucleus is calculated from the
average mass and energy emission rates [Frie 83]. Sub-barrier emission is not
incorporated because of the use of the sharp cut-off approximation for the inverse
cross sections. For simplicity, the level density is approximated by that of an ideal
Fermi gas at the density of normal nuclear matter.

The generalized Weisskopf formula gives the probability per unit time and
energy intervals of emitting a particle, b, with energy, E, at time, t, from a

compound nucleus, c:

2 2
—d Nb = (25 +1)——222222R22(E-v )C-XE-V )expl2E+ZbF2T2p222+NbF2TIPV22Bb
dEdt b nzfls b b 'r

 

]. (8.1)

Here, Sb, mb, Zb’ Nb’ and E denote the spin, mass, charge, neutron number,

141

142

and energy of the emitted particle; T is the temperature of the compound nucleus;
F(T,pn) and F(T,pv) are the Helmholtz free energies per particle for protons and
neutrons, respectively, and EXx) is the unit step function. The free energies were
calculated by assuming that the nucleons behaved like an ideal Fermi gas with a

density of p = 0.145 fm23. The height of the Coulomb barrier, V was taken as the

bl
Coulomb energy between the daughter nucleus and the emitted fragment when
they are separated by the absorption radius, Rb. For simplicity, the absorption

radius was calculated as:

1/3]

%=0 r [(A C-l/Ab) +Ab] ;r0= 1 .2 fm. (8.2)

Here, AC and Ab denote the mass numbers of the parent nucleus, c, and the
emitted particle, b, respectively. The binding energy, Bb, IS the difference in masses
of the parent nucleus and that of the daughter nucleus and the emitted particle.
The masses of parent and daughter nuclei were calculated from a liquid-drop
formula:

M(A,Z)=Zmpc2+(A-Z)mnc2- [ 14.1A-13A2/ 3-0595221; 2/3-19(A-ZZ)2/A] (8.3)

The spatial distribution of emission points was chosen to be uniform in the
two coordinates transverse to the velocity of the emitted particles; the third
coordinate was chosen such that the emission point corresponds to the surface of
the sphere of radius Rb. For particles with energies very near the Coulomb barrier,
this is not a particularly satisfying choice. These particles undergo a significant
change in their trajectory due to the Coulomb field of the compound nucleus.
Distortions of two-proton correlation functions by deflections of the emitted
protons in the Coulomb field of the daughter nucleus are manifestations of long
range three-body effects. Such effects are not incorporated in the formalism

presented in Chapter 6. We also neglected effects due to angular momentum

143
coupling which, for rapidly rotating compound nuclei, could modify the extracted
radii by up to 20% [Koon 89].

If the spatial separation between emitted protons is much larger than the two-
proton Bohr radius of 58 frn, one can neglect the identical-particle interference of
the emitted protons and their strong mutual interaction, and also the quantum
nature of their mutual Coulomb repulsion. Under these conditions, one can
calculate correlation functions from the classical Coulomb trajectories, where the
electric field of both protons and that of the compound nucleus are taken into
account [DeYo 89]. For compound nuclei with excitation energies below about 2.0
MeV/ nucleon, emission time scales are sufficiently large that the above conditions
are satisfied; classical trajectory calculations should provide a good approximation
(see the discussions in Section 6.3).

For increasingly high excitation energies, the time scales for emission become
shorter. As the protons are emitted closer to one another, the quantum nature of
the Coulomb interaction, identical particle statistics, and the strong interaction
become important in that order. Fortunately, Coulomb deflections in the field of
the compound nucleus become less important for particles emitted with kinetic
energies significantly above the Coulomb barrier. For collisions with sufficient
energy to dissolve the nuclei, E/AzSO MeV, the neglect of three-body effects for the
two-proton relative wave function should provide a good approximation. At these
energies, the effects of the strong interaction and identical particle interactions
dominate. These effects can only be calculated by a full quantum treatment of the

relative wave function.

144

8.2 Model Calculations

It is instructive to explore the sensitivity of the calculated correlation
functions to various parameters and momentum cuts. The following illustrative
calculations are performed for narrow ranges of the total momenta, Pc.m.’ of the
emitted particle pairs. (The momenta, Pc.m.’ are defined with respect to the
compound nucleus rest frame.)

The shape of the two-proton correlation function depends on the time-scale
governing the emission of the detected particles. For particle emission from
equilibrated compound nuclei, this time scale depends on the level density and,
therefore, on the initial temperature. Because of cooling via particle emission, the
time scale also depends on the energy of the emitted particles. In Figures 8.1 and
8.2, such dependences are illustrated for the decay of highly excited 2 2 2H0
nuclei. Figure 8.1 shows the calculated time dependence of the respective emission
processes, i.e. the relative probability per unit time for the emission of the
specified protons. The predicted two-proton correlation functions are shown in
Figure 8.2. The top panels in the two figures present calculations for initial
temperatures T = 5, 10, and 20 MeV, keeping the total momentum of the two-
proton pair fixed at 22cm. = 400:10 MeV/ c. (This momentum bin selects protons of
kinetic energy Ec.m./ A z 21 MeV in the compound nucleus rest frame.) At low
temperatures, T 5 5 MeV, the decay times are large and the predicted correlation
functions exhibit only a minimum at q z 0 MeV/ c. With increasing temperature,
the decay times decrease and the minimum at q = O MeV/c becomes more
pronounced. For very hot nuclear systems, T Z 20 MeV, the calculated emission
time scales become so short that the two-proton nuclear interaction becomes
significant and the maximum in the correlation function at q z 20 MeV/c emerges.
The bottom panels of Figures 8.1 and 8.2 show calculations for different total
momenta, Pc.m.’ and fixed initial temperature, T = 10 MeV. Because of cooling of the

compound nucleus, particles of higher energy are emitted on faster

145

 

 

MSU-SO-HS
100 22 I 2 l ' I r I '
.j\ A=156. Z=67. Pm=400MeV/c
. \
10_1 .3 \ \ \
x \ 'r(ueV) _
., \~c\ (r/a
10'2 2 ‘ ~ ~ .. c ‘ _750
-3 ............ 10

1° ............. 070
A ......
d 10‘4
3 222
:3 100
\
2: 1
1: 10—

10"2 2%.. ‘--—

10"3

10‘4

 

 

() 1000 2000 2 3000 2 4000 2 5000
t (fm/c)

Figure 8.1: Temporal evolution of particle emission from equilibrated 2 2 2H0
nuclei of different initial temperatures, T (top panel), and for different total
momenta, PC m (bottom panel), of the emitted two-proton pairs.

146

 

MSU-SO-IIG

 

 

 

 

 

 

1.0 -
0.8 —
0.6 — , ----- 5 A=156 -
. ............. 10 2:67
’3. 0.4 - ___—- 20 Pcm=400MeV/c -
m 2 2 . l L l l
+ 0.2 _ ' I I I i I
H 1.0 -
0.8 -
0.6 —- ----- 300 A=156 —
. ............. 400 2:67
0.4 — — 500 T=10MeV -
02 L l L l L l l
0 20 40 6 80

0
q (MeV/c)

100

Figure 8.2: Dependence of two-proton correlation function on the total
momentum, P (bottom panel), and on the initial temperature, T (top panel), of

emitted proton 'pairs calculated for the decay of equilibrated

156

Ho nuclei.

147

time-scales than particles of lower energy. As a consequence, the two-proton
correlation functions reflect smaller apparent source dimensions for the emission
of particles of increasing energy. At lower temperatures, qualitatively similar
dependences exist, but their effects on the two-proton correlation functions are
less visible due to the larger emission times.

For fixed excitation energy, E‘, the decay rate of a compound nucleus
depends on the relation between excitation energy and temperature since higher
temperatures lead to higher emission rates and, therefore, to smaller apparent
source dimensions. Figure 8.3 illustrates the sensitivity of the calculated two-
proton correlation functions to the assumed relation between excitation energy

and temperature for the decay of l 5 6

Ho compound nuclei of fixed initial
excitation energy, E / A=6.0 MeV. In these calculations, the level density was taken
as that of an ideal Fermi-gas of the indicated density p. For an ideal Fermi gas, the

relation between excitation energy and temperature is given to first order by:

 

'I'2 = E / a . (8.4)
The level density parameter, a, depends on the Fermi energy, 81::
1:2 A 2/ 3 -1
a = z 0.065A( p o / p ) MeV . (8.5)
4 8F

/3

In Equation (8.5), we have used the relation 2 z 38(p / p o )2 MeV, where p o =

F
0.17 fm-3 denotes the density of normal nuclear matter. For reference, the level
density parameters are also indicated in Figure 8.3. The calculations indicate only
a moderate sensitivity to the level density parameter. As expected, higher
temperatures produce shorter emission times and, hence, more pronounced

correlations.

Particle emission rates depend largely upon the temperature of the

148

MSU-SO-l I8

I I I I I I I I I

 

1.: .7. 3.“ OT. "0“ 0?. :10?- .7. 3‘: .1.- RAW

€*=6.0MeV, A: 156, 2:67

5:; Pc.m.=4OOMeV/c _

+ a(MeV") p(fm'°) T(MeV)

._. ------ A/B 0.0626 3.0 -
.............. A/lz 0,115 9.3

A/16 0.177 10.5 -

 

 

l I I l l I l l

50 100
q (MeV/C)

 

 

Figure 8 .3: Sensitivity of calculated two-proton correlation functions to different
assumptions on the level density parameter, a=E*/ AT 2 , for the decay of ' 5 6 Ho
compound nuclei of initial excitation energy EV A=6 MeV.

149

 

 

 

 

 

MSU-90-ll7
I I I I I I I I I
1.0 —
A L , T=10MeV, A=156, Z=67
U —
E 0.8 _ Pc.m.—4OOMeV/c _
+ a(MeV‘1) p(rm'3) e'(MeV)
"' - , ------ A/B * 0.0626 8.5 .
:5 .............. A/12 0.115 6.8
0.6 — .': —-——— A/16 0.177 5.5 —
,.
I
I, I I I I I J I I
0 50 100
q (MeV/c)

Figure 8 .4: Sensitivity of calculated two-proton correlation functions to different
assumptions on the level density parameter, a=E"/ AT 2 , for the decay of ' 5 6 Ho

compound nuclei of initial temperature T=10 MeV.

150

compound nucleus. Yet, the initial temperature does not specify the decay
characteristics unambiguously, because different level density parameters
represent different heat capacities of the decaying nuclear system. Figure 8.4
shows the sensitivity of the calculated two-proton correlation functions to the level
density parameter for l 5 6 Ho compound nuclei of fixed initial temperature, T =
10 MeV. The predicted correlations are slightly more attenuated for larger level
density parameters, i.e. for increasing values of the heat capacity of the compound
nucleus. However, the sensitivity is not very pronounced; it is even less significant
at lower temperatures.

Since the predicted emission times depend strongly on the momenta of the
emitted particles, the differences between longitudinal and transverse correlation
functions can be expected to be momentum-dependent. Low energy particles may
be emitted on such long time-scales that sensitivity to anti-symmetrization effects
is lost. As emission time scales decrease for more energetic emissions, differences
between longitudinal and transverse correlation functions may become more
significant for particle pairs of higher total momenta. These qualitative
expectations are corroborated by the calculations shown in Figure 8.5. The solid
and dashed curves show longitudinal and transverse correlation functions
predicted for narrow gates on the total momentum, Pc.m.= 300, 400, 500 MeV/c, of
the particle pair with respect to the center-of-mass frame of reference (i.e. the rest
frame of the compound nucleus). Significant differences between longitudinal and
transverse correlation functions are predicted for large momenta, P=500 MeV/c
(top panel). These differences are reduced for smaller momenta, Pc.m.=400 MeV/c
(center panel). For even smaller momenta, Pc.m.=300 MeV/c, longitudinal and
transverse correlation functions differ only in the detailed shape of the minimum
at q=0 MeV/c (bottom panel) which is more pronounced for the longitudinal than

for the transverse correlation function.

151

 

 

 

 

 

 

 

 

 

 

 

 

MSU-90-H4
. I . I . I .
1.0~ - ——————— --
O.8 - I’ Pc.m.=5OOMeV/c '-
.. I ——¢=(o_4o,14o-1ao)° T=10Hev -
A=158
0.6 " " " 7450-120)’ 2:67 ‘
_ l f l i I ‘ 4
1.0 " -"
A b q
3
D: 0.8 - Pcm=4OOMeV/c '-
+ - —-1[v=(0-40,140-180)° T=10MeV -
A=156
"'* 0.6 - - - 1I'=(60-120)" =67 '-
L I L L L
_ I r l : .
1.0 "'
0.8 — , Pm=600MeV/c -—
’ ——v=(0-40.140-160)° T=10HeV -
A=156
0.6 " "' " 7450420)" Z=67 "
L 1 L 1 L 1 L
0 20 4O 60 80
q (MeV/c)

Figure 8.5: Longitudinal and transverse correlation functions predicted for
evaporative emission from ‘ 5 6H0 compound nuclei of initial temperature, T: 10
MeV. Different panels show predictions for different total momenta, PC ,of the

emitted proton pairs: P cm =500 MeV / c(top panel) PC c..m =400 MeV/ c(centmer panel),
PC m = 300 MeV/c (bottom panel).

152

8.3 Angle-integrated Correlation Functions

12°Xe+

Figure 8.6 shows two-proton correlations measured for the reaction
2 7A1 at E/A = 81 MeV (part a) and for the reaction ‘ 2 °Xe + ‘2 2Sn at E/A = 31
MeV (part b,c). The gates placed on the total momenta are indicated in the figure.
In Section 8.2, model calculations assuming evaporation from compound nuclei
have exhibited sensitivity to total momenta of the two-proton pair, the initial
temperature of the compund nucleus, and the level density parameters. The
shapes of two-proton correlation functions at q s 30 MeV/c are largely
determined by the average emission time scales. However, those calculations were
performed in the rest frame of the emitting source.

In order to compare the calculations to our data, we have transformed the
single-proton phase-space distributions from the source rest frame to the
laboratory frame. The calculations were folded with the resolution of the
experimental apparatus and averaged over the appropriate momentum bins using
the experimental proton yields as relative weights.

Since the inclusive cross sections are consistent with substantial
contributions from evaporative emission from excited projectile residues, we
calculate correlation functions for two extreme cases, (1) emission from a source at
rest in the compound nucleus rest frame and (ii) emission from a source at rest in
the projectile rest frame.

Firstly, we used the compound nucleus values for the mass, A, and charge,
Z, but treated the temperature as a free parameter to illustrate the sensitivity of the
calculated emission rates to the initial temperature of the emitting system. The
level density parameter is chosen to be a=A/16, corresponding to a Fermi gas of
density of 0.145 fm-3. In Figure 8.6, the curves represent calculations for three
initial temperatures. Good agreement between calculations and data is obtained
for initial temperatures of about 7—10 MeV. For complete fusion of ‘ 1 ° Xe + 2 7 Al

and ‘ 2 °Xe + ‘ 2 2Sn, initial temperatures of 8.2 and 10.3 MeV, respectively,

153
MSU-90-ll2

r I ' l r I .
Ca) ”N(“fieppx E/A=31MeV,
- 9“,:25"

 

1.0

 

. P8 600-750 MeV/c
o. - - - TIGHOV
' —— r-mv } "15°-

0.5 247 -

....... T- lonev
I L I L I l'

: l . I v I i
jb) 122s::1(“““xt=.,pp), E/A=31MeV, ;
6,,=25° I

 

    

I ' P- 540-660 MeV/c

""" T-5IIaV } A-251,

' . -—— T-lOIleV
o .5 . 1 ...... T- 15|lev 2'1!“ r

- f I ' I ‘ I i
:c) 1228n(lz°Xe,pp), E/A=31MeV, I
C safes:+

L

 

 
  

 

   

P- 060-750 MeV/c
--- 'r-snev } A3251.

 
 

— T-IOIIeV '
I ...... T'I lslnov 2' t04 r

0 20 40 60 80
q (MeV/c)

0.5 ~

 

 

 

Figure 8.6: Two-proton correlation functions measured for the ‘ 2 °Xe+2 7Al
(part a) and l 2 9 Xe+l 2 2 Sn (parts b,c) reactions at E/A=31 MeV for the indicated
gates on the total momenta, P, of the two-proton pairs. The curves represent
calculations for evaporative sources at rest in the center-of-mass frame of
reference; the parameters are indicated on the figure.

154

are calculated if one assumes the level density of an ideal Fermi gas of normal
nuclear matter density; the more common relation, T2=(8 MeV) 0E‘/ A, gives values
of 5.8 and 7.3 MeV. However, the equilibrated emitting systems should have
temperatures which are somewhat lower than those calculated for compound
nuclei formed in complete fusion reactions [Poch 87, Wada 89, Jian 89, Xu 89] since
some energy is carried away by pre-equilibrium emission.

In the second case, we choose the equilibrated projectile to be the proton
emitting source moving at the initial beam velocity. We used the projectile values
for the mass, A, and charge, 2, and treated the temperature as a free parameter.
In Figure 8.7, we compare our data with calculations at T=5,10,15 MeV. For
projectile decays, the lower momentum bin, P=540~660 MeV/c, largely
corresponds to subbarrier emission. Therefore, we only present calculations for
the higher momentum bin, P=660-750 MeV/ c. Reasonable agreement with the
data is obtained for temperatures of about 10-15 MeV.

Fits to the correlation functions require higher temperatures when one
assumes emission from projectile-like sources rather than emission from fusion-
like sources. This is related to the fact that average emission times become shorter
for increasing temperature and for increasing emission energy with respect to the
rest frame of the decaying nucleus [Frie 83]. In our detection geometry and for our
laboratory-momentum gates, the emitted particles have lower kinetic energies in
the projectile rest frame than in the compound nucleus rest frame. For fixed
temperature, the correlations are therefore attenuated if one assumes emission
from the rest frame of the projectile as compared to emission from the rest frame
of the compound nucleus. To reproduce the experimental correlation function, one
must choose a higher temperature for the projectile-like source than for the fusion-
like source.

Temperatures which provide the best description of the experimental

155

 

 

 

 

 

 

 

‘

 

 

MSU-SO-l94
27Al(129Xe,pp), E/A=31MeV, 6.,=25°
T
1.0 '
’ P= 660-750 MeV/c
’ .. ........... T=15uev - ‘
’o“ O 5 P r=10uev } 2:39, _
+ 122r 129l ' I t I f
H ’ Sn( Xe,pp), E/A=31MeV, Gu=25°
1.0 '
P= 660-750 MeV/c ‘
’ . ........... T=15uev
‘l T=10uev “129' ‘
0.5 - ____ T=5MeV z=54 -
O 20 40 60 80
q (MeV/c)

Figure 8.7: Two-proton correlation functions measured for the ' 2 ”Xe+2 7Al
(part a) and ‘ 2 "Xe-I-l 2 2Sn (part b) reactions at E/A=31 MeV for the indicated
gates on the total momenta, P, of the two-proton pairs. The curves represent

calculations for evaporative sources at rest in the projectile frame of reference; the
parameters are indicated on the figure.

156

correlation functions (see Figures 8.6 and 8.7) may be unrealistically high.
However, it may also be unrealistic to assume pure equilibrium emission. Some
contributions from pre-equilibrium emission would decrease the average lifetime
of the emitting system and produce stronger correlations. Within the present
equilibrium model, stronger correlations can be produced by raising the
temperature of the source (see Figure 8.2). Clearly, some quantitative uncertainties
about the exact nature of the emitting system remain. In order to have a better
understanding of those uncertainties, it is desirable to employ a more
sophisticated statistical model of compound nucleus. However, the qualitative
interpretation of the measured two-proton correlation functions as predominantly
caused by slow evaporative emission is not affected by our incomplete knowledge
of the mass, temperature, and velocity of the decaying nucleus or by small
contributions from pre-equilibrium emission processes.
8.4 Longitudinal v.s. Transverse Correlation Functions

We have explored the dependence of the two-particle correlation function on
the angle)? = cos.1 ( I; Oq/Pq ) , between the relative and total momentum vectors of
the proton pairs to search for clues on the source lifetime and shape [Koon 77, Prat
87, Awes 88, Ardo 89]. As was illustrated in Figure 1.2 and discussed in References
[Prat 87, Awes 88, Bert 89, Gong 91a], emission from a long-lived system produces
phase-space distributions elongated in the longitudinal direction. Because of the
reduced Pauli anti-correlation in this direction, the longitudinal correlation
function (‘on ° or 180 °) of a long-lived source may be enhanced compared to the
transverse correlation function (‘I’z90°), unless the average particle separations
become so large that sensitivity to anti-symmetrization effects is lost.

Figure 8.8 shows longitudinal and transverse two-proton correlation

129

functions measured for the reactions ‘ 2 ° Xe + 2 7 A1 (top panel) and the Xe +

122

Sn (bottom panel). The longitudinal correlation functions, shown by solid

points, were evaluated for the gate lcos‘I’ I20.77 (corresponding to the

l

157

 

 

 

 

 

 

 

 

 

 

MSU-SO-lll
' I . I . I r
1.2 _ 2'2Al(2"29Xe.pp). E/A=31MeV, 0,“:25"
1.0 -
O 8 _ '9’ P=460-750MeV/c
' ov=(0-40.140—160)° Egg ‘
9 ' ....... o ¢=(60—120)° Z;67
E: 0'6 1;2 122 2 2 2 l 2 I
+ 1.2 .. Sn( 9Xe.pp). E/A=31MeV, 0av=25° _
‘_.|
1.0 -
O 8 _ P=270—750MeV/c
+¢ ov=(0-4o,140-160)° §=éguev ‘
' '3' ....... o ¢=(60-120)° 2;].01
0.6 I l - I . I . .
0 20 40 60 80
q (MeV/c)

Figure 8.8: Longitudinal (‘I’=0°-40° or ‘I’=140°-180°) and transverse (‘I’=60°-
120°) two-proton correlation functions measured for the reaction ' 2 °Xe+ 2 2 Al
(top panel) and ‘ 2 °Xe+2 2 2 Sn (bottom panel) at E/A=31 MeV.

158

angular cuts of ‘I‘ =0°-40° or 140°-180° ). The transverse correlation functions,

I.
shown by open points, were evaluated for the gate lcos‘I’tl 50.5 (corresponding to
the angular cut-20f ‘I’t=60°-120°). For improved statistical accuracy, the gates on
the total momenta of the proton pairs were made wider than in Figure 8.6; the
values are indicated in the figure. No statistically significant difference between
longitudinal and transverse correlation functions is visible. However, this result
does not contradict theoretical expectations for evaporation from long-lived
compound nuclei. The solid and dotted curves in Figure 8.8 show longitudinal
and transverse correlation functions calculated for evaporative emission using the
parameters indicated in the figure. The calculations were averaged over the
appropriate momentum bins and folded with the resolution of the experimental
apparatus. The predicted differences between transverse and longitudinal
correlation functions are of the order of a few percent and, therefore, below the
statistical sensitivity of the present experiment.

It was shown in Figure 8.5 that differences in the shapes of longitudinal and
transverse correlation functions exhibit a strong dependence on the total momenta
of the proton pairs. For emission from equilibrated ‘ ° 2 Ho compound nuclei,
significant differences between longitudinal and transverse correlation functions
are mainly predicted for large momenta, but not for small momenta
corresponding to emission close to the barrier. Since evaporative cross sections
decrease exponentially as a function of increasing kinetic energy of the emitted
particles, integrals over wide momentum gates have predominant contributions
from lower momenta for which the differences between longitudinal and
transverse correlation functions are predicted to be small and difficult to detect.
Unfortunately, the statistical accuracy of our experiment was insufficient for a

more detailed exploration of longitudinal and transverse correlation functions at

higher total momenta of the emitted proton pairs.

CHAPTER 9. TWO-PROTON CORRELATION FOR
NON-EQUILIBRIUM EMISSION

For intermediate energy nucleus-nucleus collisions, particle emission already
sets in at the early, non-equilibrated stages of the reaction for which purely
statistical treatments are clearly inappropriate. These early stages of the reaction
can be treated in terms of semi-classical models based upon the Boltzmann-
Uehling-Uhlenbeck equation [Bert 84, Bert 88] which describes the space-time
evolution of the one-body phase-space distribution function. The theory
incorporates mean-field effects, nucleon-nucleon collisions, and the Pauli-
exclusion principle in the semiclassical approximation; Coulomb effects are
included. Within the Wigner-function formalism, knowledge of the one-body
phase-space distribution function is sufficient for the characterization of the size
and lifetime of the reaction zone formed in the nuclear collision and for the
calculation of the two-proton correlation function at small relative momenta.

9.1 BUU Transport Equation for Collision Dynamics

In this section, we briefly review the derivation of the BUU transport
equation which provides the basis for microscopic calculations for nonequilibrium
particle emission in intermediate energy nucleus-nucleus collisions. We start with
the Schriidinger equation of the N -particle system,

i 8 t‘I’ = HP. (9.1)

Here, H is the N-particle Hamiltonian and ‘I’ is the N-particle wave-function.

159

160

From ‘I’, one constructs the N -particle density,

= W , (9.2)

which leads to the von—Neumann equation of motion:

(N) (N)

iatp =[H,p 1. (9.3)

We introduce the reduced density matrices via

p(n)(r1r 1;) I? F), )=
n n
3 3 3 (N) -) -> .—)’ -—)'
(N—n )! _ld3 rn +1...d rNd (“find r’Np (r 1,...,r N,r 1,...,r N). (9,4)

Inserting Equation (9.4) into Equation (9.3), leads to the Bogoliubov-Born-Green-
Kirkwood-Yvon (BBGKY) hierarchy of the reduced density matrices. This

hierarchy links the time derivative of p (n) to p (n+1). The lowest two member’ s of the

BBGKY hierarchy are:

13 p(l)(r1,r_21)= @5072 -Vf,>p(2)(? :7 1
1 1

+ld r (v(r r2)-v(r1r2))p ( 1, 2; ’1, 2) (9.5)
and
. (2)-”>554, _ L2 V2 2(21-»->,->,—>,
222:” (21”2’21”22"2m {2 ( 1'.-V1') ('1’22’21'22)

161

(2)—1-1—1—9
If 1'

-§ -I (—)I 4’) ( . ’ I)
+ (v(r1,r2)-vr1,r2 )p 1, 2,r 1, 2
2 Id3 (44%(4’4) (3)(—) —) —) —1’—),->) (96)
+1-1 r3(vri,r3 vri,r3 )p r1,r2,r3,r1,r2,r3 . .

A closed solution of the equations of this hierarchy is only possible, if one

truncates at a level n and with it neglects (n+1)-body correlations. Truncating at

(2)

n=1, neglecting two-body correlations, and approximating p as an

antisymmetrized product of one-body densities, one obtains the TDHF equations:

iatpU) = 111.9(2)] , (9.7)

where h is the single particle Hamiltonian,
-')—I V? + 5361;”)! d3r2p(2’(?;?2)v(?,?2)

- pm(?,?')v(?,?') . (9.8)
Performing a Wigner-transformation gives the Vlasov equation:

.9

-)-> 2 . —)-) - -> . —>-) =
atf(p,r,t)+m Vrf(p,r,t) 6:06) ‘V’Pf(p,r,t) 0. (9.9)

In Equation (9.9), U(:) is the mean field or Hartree potential,

162

—) 3 -> -) -+ —)
U(r ) = I d r2v(r ,r 2>90 .r 2), (9.10)

and {(3521) is the Wigner-transform of the single-particle density matrix,

(12(12- ip 0s

Nlml

_)
I'

Nlml

((3511) = I d36 p + )e (9.11)

I

In the derivation of Equation (9.9), the semiclassical approximation has been used.
This approximation is valid since typical wavelengths (232.5 fm for p=500
MeV/c) are shorter than the size of the regions with the mean field (typically of
the order of 8 fm diameter). The TDHF and Vlasov approximations to the many-
body problem are pure one-body theories in mean field approximation in which
all multi-particle correlations are neglected.

A truncation of the BBGKY hierarchy which includes two-body correlations,
but neglects three- and higher-particle correlations leads to the Boltzmann-

Uehling-Uhlenbeck (BUU) equation:

—) -) E . -) —) - —-) . —) —-> _
atap; ,t) +m V’rflp; ,t) Vruu) Vpap; ,t) _

 

1 3,33,_122,2 -)-+,,—>_(1(_5
3 2 2d qld qzd (1222(2115p +q2‘11 "12 2)) 53 (92212 ql'q2)do
21: m
A—p —) A—) —) "-—)—) A—9 —)
x{f (q’1,r ,t)f (q’2,r ,t) (1-f (p,r ,t)) (1-f (q2,r ,t))
A_, A_,
-f(p,r,qt)f(q2, r,t)(1-f(q’1,r,t))(1-f(q’2,r t,))} (9.12)

163

A
In Equation (9.12), f (3,31) is the phase-space density averaged over one phase-space

cell. The left-hand side of the equation is the Vlasov term describing the temporal

change of the one-body Wigner-function, ((1323:), due to the interaction of the
nucleons with the mean field. The right-hand side is the collision integral which
represents the effects of the correlations due to two-body collisions on the one-
body Wigner-function. Equation (9.12) was first obtained by N ordheirn [Nord 28]
as a quantum mechanical extension of the Boltzmann equation which incorporates
Ferrnion statistics.

Equation (9.12) is solved by using the pseudo-particle method [Wong 82]. In

this method one compares the left-hand side of the equation to the complete
-) ->
differential of f(p,r ,t):
a

-) —)
d— -) —) _ -) -> dr . -> —-) d . -) —)
map; ,t) -—afif(p,r ,1) +—dt v’rap; ,t) + _de v’pf(p,r ,t). (9.13)

From Equations (9.12) and (9.13), one obtains a set of six coupled first-order

equations for every occupied phase-space point:

d'i P i

a? = "11"“, (9.14)
dpi -) a ->

(it =Ii(p)- atom. (9.15)

l

_,
Here, Ii(p) is the change in momentum pi due to nucleon-nucleon collisions

(i=1 ,2,3). The differential equations can be interpreted as the classical Hamiltonian

164

equations of motion for a pseudo-particle. Since the total occupied phase space is
proportional to the number of nucleons, it is convenient to specify the total
number of phase space points per nucleon as a measure for the numerical
precision with which Equation (9.12) is solved. In the present calculations, we
have used up to 700 pseudo-particles per nucleon, resulting in a total of up to
172,200 coupled first-order differential equations which were solved
simultaneously. The average phase-space occupancies used to determine the
effects of the Pauli principle in the collision integral were determined from the

coordinates and momenta of the pseudo-particles. The cell size for averaging was
1 3
chosen to be 4(h/21t) .

We solve the BUU equation by numerical methods which are similar to the
ones introduced by Reference [Bert 84], see also Reference [Bert 88]. The major

new numerical technique used in our present calculations is the treatment of the

Pauli exclusion principle. By explicitly storing ? (352,1) on a six-dimensional lattice in
every time-step, we were able to greatly speed up the computer program without
relaxing the accuracy of the treatment of the Pauli-exclusion principle [Baue 90].

It seems at first sight surprising that it could be used to calculate two-particle
correlation functions since BUU is basically a theory describing the time evolution
of the one-body phase-space distribution function. However, within the Wigner-
function formalism outlined in Chapter 6, the knowledge of the one-body phase-
space distribution function is sufficient for the calculation of the two-proton
correlation function at small relative momenta and the characterization of the size
and lifetime of the reaction zone formed in the nuclear collision.

Correlations between coincident particles do not only arise from quantum
statistics and / or final state interactions, but also from a number of dynamical and
kinematical effects. Previously, it was shown [Knol 80, Lync 82, Lync 83, Tsan 84a,
Tsan 84b, Chit 86a, Baue 87, Baue 88, Tsan 90, Ardo 90] that the main features of

165

two-proton coincidences at large angles are explained by the effects due to total
momentum conservation, finite particle number, and/ or collective motion in the
reaction plane 2without requiring the detailed information about two-particle
correlations. For example, the detection of a single particle will shift the total
momentum of all remaining particles; for small systems this effect can lead to
significant correlations at large relative momenta. If the single-particle distribution
is azimuthally anisotropic, the detection of one particle can filter out a non-
isotropic distribution of reaction-plane orientations [Tsan 84a, Tsan 84b, Chit 86a,
Tsan 90, Wils 90, Ardo 90] causing non-isotropic azimuthal correlation functions.
In the next Section, we will explain in detail the procedure for averaging over
impact parameters and point out the approximations used for calculating two-

proton correlation functions at small relative momenta.

9.2 Impact Parameter Averaging

For semi-classical reaction models, the impact-parameter averaged

correlation function, consistent with Equation (7.1), can be written in the form:

{

_,_, _, jbdbd¢{fl( b. 0. £34412) m b, (1)/:12)- Ef)cf( b, 6,3,an
C(P,q)=N(P) r 1-+ _9 [ qu . (9.16)
debd¢{l'l(b, (p EP+q) }debd¢{11(b,¢, EP-q)}

 

Here, b denotes the impact parameter; 0 denotes the azimuthal orientation of the
reaction plane; 1109,03) is the probability of emitting a particle with momentum f;
for events characterized by b and 0; Cf(b,¢,P2,q2) is the correlation function due to
final state interactions and/ or quantum statistics for given b and It); N(P) is a
suitably chosen normalization constant which makes the normalization at large
relative momenta consistent with the experimental data. We can write this

expression in the form:

166

erdbd¢{II( b ”0 2P+q)l'I(b, 0 ‘P-q)Cf (b ,0 P,,q)}

C(P,q)=N(P) Cf(P ,q)°Cd (P ,q){ {
debd¢{rl(b,¢, EP)II(b,¢i:'P)Cf( b.¢.P.q>}

 

 

 

Jfbdbd¢{l'l(b,¢,2 lI-’2)l'l(b,‘lh 1132)}
debd¢{l'l( b ,0, :P+q)l'l(b, (I), ‘P-q)}
with
f 19 1.» f -+ -+
{_H debd¢{l1(b, ¢.2P)Il(b,¢, 2P) C (b. 9.29)}
C (P,q) = r fi , (9.18)

debd¢{I'I(b. 0, if“ b. 0, 2:13)}

 

 

cd(1‘3’,c'f)= { 1—9 A f 1_)_) (9.19)
debd¢{l'l( b, 0,2P+q) }debd¢{l‘l(b, q), EP-q)}

Here, C263) is the correlation function due to final state interactions and/ or
quantum statistics, renormalized to unity for large relative momenta for which
Cf(b,¢,P-2,q2) = 1, and Cd(P £12) is the "dynamical correlation function" which describes
correlations caused by averaging over b and 0.

In Equation (9.17), the terms in curly brackets can be neglected to a good
approximation. (If the correlation function Cf(b,¢,P2,q2) is independent of b and 0,
the terms in the curly brackets cancel exactly.) For two-proton correlation
functions, the correlation function Cf(b,¢,P2,q2) is non-trivial only for small relative
momenta, q530 MeV/ c, for which one may approximate l'Iz(b,¢,P2/2) z
I'I(b, 0,1; / 2+3 ) 0 l'I(b, ¢,P/ 2-3). This approximation reduces the two factors in the curly
brackets to unity. At larger relative momenta, Cf(b,¢,P2,q2) z 1 and the denominators
and enumerators of the two terms in the curly brackets cancel cross-wise. It

should, therefore, be reasonable to use the approximation:

167

c0323) =N(f>’)-cf(13’,fi’)-cd(13’,3). (9.20)

With Equation (9.20), one can incorporate dynamical correlations in a fairly
straightforward fashion. (Note, however, that other physical processes, not
considered in our calculations, may also affect the correlations at small total
momenta, e.g. distortions in the Coulomb field of the heavy reaction residue or
feeding from the decay of particle unbound states.)

In our measurements, distortions due to dynamical correlations may be
present in some of the correlation functions extracted for low total momenta. For
example, there is evidence for shape distortions in the correlation function for the
2 ‘N + 2 ° 2 Au reaction at small total momenta, see Figure 7.4. More significant
are the difficulties encountered in the normalization of transverse and
longitudinal correlation functions for the 2 ‘N-induced reactions at low total
momenta discussed below (see Figures 9.8 and 9.9). However, the present BUU
calculations do not provide an accurate description of the cross sections for such
low-energy emissions, see Figure 5.4. Furthermore, dynamical correlations test
different aspects of the model than correlations due to final state interactions.
Therefore, we have decided to neglect them in our calculations and used Equation
(9.18) for the calculation of the impact parameter averaged correlation functions.
For comparisons with experimental data, the calculated correlation functions were
renormalized at larger relative momenta, q z 60 - 100 MeV/c, to make them

consistent with the normalization conventions adopted in our data analysis.

9.3 Predicted Two-proton Correlation Functions

In our standard BUU calculations, we used a stiff equation of state and
energy-dependent free nucleon-nucleon cross sections. (For the present reactions,
the calculations exhibit little sensitivity to the stiffness of the equation of state.)
For the reactions l‘N + 2 2Al and HN + 2° 2Au, the correlation

functions were calculated from the phase-space points obtained from a total

168

of 5250 and 4500 computational events, respectively, with impact parameters
distributed according to their geometrical weights.

The Wigner functions of emitted particles were constructed from nucleons
emitted during a time interval of Ate=140 fm/c following initial contact of the
colliding nuclei; the time t> was taken at the end of the time interval Ate. N ucleons
were considered as emitted when, during this time interval, the surrounding
density fell below pe = p o / 8 and when subsequent interaction with the mean field
did not cause recapture into regions of higher density. This test for recapture was
continued over a time interval of At = 180 fm/c after contact. The finite size of our
lattice did not allow us to explore much larger emission times. However, the
consideration of much larger emission times would not necessarily lead to more
reliable results since, in our present approximation, the nuclei are not stable over
long time scales and the BUU calculations become inaccurate due to spurious
decays.

In our calculations of the correlation functions, appropriate averages over
impact parameter, orientation of the reaction plane, and the indicated gates on the
total momentum and the angle, ‘1’, between the relative and total momentum of

the two protons were taken into account.

While our particular choice of the parameters Ate and pe is reasonable, it
involves a certain degree of arbitrariness. The sensitivity of the calculations to
different choices of the emission time interval, Ate’ and the freeze-out density, pe,
is illustrated in Figure 9.1. Larger emission time intervals reduce the height of the
maximum of the correlation function at q s 20 MeV/c due to an increase of the
average emission time. On the other hand, smaller freeze-out densities lead to a
slight increase in the height of the maximum of the correlation function. This can

be understood as follows: lower emission densities select subsets of particles

27 14 MSU-SI-OSB
BUU: A1( N,pp), E/A=75MeV, 6,,=26°
L --- At =140fm/c, p.=po/16 T 2 1
———At =1401'm/c. p,=po/8 " 1
L ........ At =1601m/c. p.=p°/3 L
1.5 . “ ,\\ P=500MeV/c 1
P=300MeV/c 1 .
A 1.0 -
3 .
D:
+
H
1.5 '-
1.0 b
0
Figure 9.1 : Dependence of correlation functions predicted by BUU calculations

on the emission time intervals, At , and the emission densities, p . The values of
individual parameter choices andeselected total momenta of the proton pairs are
given in the figure. In these calculations, the in-medium cross section was
approximated by the experimental free nucleon-nucleon cross section, and the stiff

169

 

 

   

  

 

 

 

 

 

 

 

 

equation of state was used.

 

170

considered as emitted for higher freeze-out densities by eliminating late emissions
(i.e. particles which have not yet reached the lower densities) and thus selecting
particles which had left the higher density regime at earlier times. The
corresponding reduction of the average temporal separation between emitted
particles leads to enhanced correlations. Typically, different reasonable choices of
Ate and pe introduce uncertainties of the order of 5-10% into the magnitude of the
predicted correlation functions. However, in some instances these uncertainties

can be larger.
9.4 Dependence on Total Momentum

Two-proton correlation functions measured for the 2 2N + 2 2 Al and 2 ‘N +
2 2 2Au reactions at E/ A = 75 MeV are compared in Figures 9.2 and 9.4 to
calculations using density distributions predicted by the BUU equation. We made
cuts on the total laboratory momenta, P, and no selection on the angle, ‘1’, between
the total and relative momentum vectors for the emitted proton pairs. By means of
extracting Gaussian radius parameters, ro (P), we present a more detailed
comparison between measured and calculated two-proton correlation functions
for 2 2 N + 2 2 A1 at E/ A = 75 MeV in Figure 9.3. The errors represent estimates of
accuracy in extracting the radius parameters from correlation functions.

Overall, two-proton correlation functions predicted by the BUU theory (solid
curves in Figures 9.2 and 9.4, open points in Figure 9.3) are in rather good
agreement with the measured correlation functions (points in Figures 9.2 and 9.4,
solid points in Figure 9.3). It is particularly gratifying that the calculations can
qualitatively reproduce the observed strong dependence of the correlation
functions on the total momentum of the emitted proton pairs. For the low-

momentum gate of the 2 4N + 2 ° 2 Au reaction, the maximum of the calculated

correlation function is larger than that of the experimental correlation function,

171

 

' I l I I I

2'0 .— 221, t} 27M(“N,pp), E/A=75MeV
1)

e,,,=25°

o P=840-1230MeV/c -
. P=450- 780MeV/c -
D P=270- 390MeV/c —

 
   

 

 

 

:10F““:I=.. . . .¢ . ' "
0 ' - _-
a=0ml (BUU) -
50 100
q (MeV/c)
Figure 9.2 : Two-proton correlation functions, measured for the reaction

2 ‘N-I- 2 7A1 at E/A=75 MeV, are compared with correlation functions predicted
with the BUU theory. The gates placed on the total momenta, P, of the coincident
proton pair are indicated.

172

MSU-90-I92
7 b T T f I r I r l’ f T U I I I T I I

27Al(“'N.pp), E/A=75MeV, 0“=25°
0 Experiment
: 0 Theory (BUU)

53. +2)

 

I”? 9 .

_ +++$++I¢++++ 3

4E- . ¢ 4* _:
I +¢ I
(1

V'l'I'U
L

 

 

P
I.
L L l l l L l l l l l I J l

200 400 600 800 2 10002
P (MeV/c)

 

Figure 9.3 : Radius parameters r o (P) for Gaussian sources of negligible lifetime
extracted from two-proton correlation functions gated by different total momenta
P. Solid and open circles represent experimental and theoretical correlation
functions, respectively.

173

 

  
    

 

 

 

 

MSU-91-005
I I I I I I I I I
2.0 b ._
. 32¢ 197Au(1“N.pp). E/A='75MeV ,
- 6,525"
" 0 P=840—1230MeV/c -
~ . P=450- 780MeV/c -
A 1.6 L a P=270— 390MeV/c -
3 " .
fr. ' -
+ - -
H .2 "' I o -
1.0 -- 221‘: :' .3" - -
.. 1:) ' d
' 12.1. a-am (BUU) -
0.5 L I I I i I I L I
0 50 100
q (MeV/c)
Figure 9.4 : Two-proton correlation functions, measured for the reaction

2 2 N + 2 2 2 Au at E/A=75 MeV, are compared with correlation functions predicted
with the BUU theory. The gates placed on the total momenta, P, of the coincident
proton pair are indicated.

174

see Figure 9.4. This discrepancy is not surprising since the emission of low-energy

protons is expected to have significant contributions from slow evaporative
processes which are not incorporated into our calculations. In fact, the existence of
a strong evaporative component in the low-energy portion of the proton spectrum
for the 2 2 N + 2 2 2 Au reaction was already inferred from the shape of the
minimum of the correlation function at qg 15 MeV/ c (see Figure 7.4) as well as from
the shape of the single proton spectra (see Section 5.2). The inclusion of
evaporative processes would lead to more extended Wigner functions and, hence,
to more attenuated correlation functions.

A close comparison of Figures 7.3 and 7.4 with Figures 9.2 and 9.4 reveals
that correlations functions calculated from the BUU theory provide an improved
description of the shape of the experimental correlation functions in the region of
q=30-40 MeV/ c as compared to those calculated for spherical Gaussian sources.

Figure 9.5 compares the detailed shapes of two-proton correlation functions
predicted for different space-time geometries. The solid points show correlation
functions predicted from the BUU equation, averaged over the indicated range of
total momenta, P, of the proton pairs. These calculations are in excellent
agreement with data (see figure 9.2). The solid curve shows results obtained for
instantaneous emission from a Gaussian source, Equation (7.1), with radius
parameter r 0 =45 fm. The two-proton correlation function calculated for the
Gaussian source, exhibits a narrower maximum than that calculated for the more
realistic density distribution obtained by means of the BUU equation. This
difference in shape can be attributed to the fact that sources predicted by the BUU
equation are non-spherical. To illustrate the sensitivity of the shape of two-proton
correlation functions to the source geometry, we show the correlation function
predicted for emission from a source consisting of two sharp spheres of negligible

lifetime. Both spheres were assumed to have radii of RS = 5 fin, and the centers

175

 

 

 

 

 

 

MSU-90-l89
' I I“ I ' I I I ' I
1.5 -
Q
0 BUU: P=450-780MeV/c
A
0" L
v
0:: 1.0 — -
+
H
One Gaussian Source:
2' r0=4.5fm 1
Two Sharp Sphere Sources:
0.5 - ---- R,=5fm, d=20fm -*
L l L I L l L I L l
0 20 40 60 80 100

q (MeV/c)

Figure 9.5 : Comparison of two-proton correlation functions predicted for
different source geometries: the solid points represent the results of BUU
calculations averaged over the momentum range P=450-780 MeV/c. The curves
show emission from sources of negligible lifetime: the solid and dashed curves are
obtained, respectively, for a spherical Gaussian source of radius parameter, r 0 =4.5
frn, and a source consisting of two sharp spheres of radius, R =5 fm, and separated
. s
by the d15tance, d=20 fm.

176

     

1'0 "III:

0° "some

 

0 "IS! .
. IO III: 119 "73 "III:

 

'0‘ I - 30’s- I:

     

10° N ‘9 Ian

  

1' '05 Im/c

 

 

Figure 9.6 : Nucleon density distributions in the reaction plane calculated from
the BUU equation for 2 2N + 2 2 Al collisions at E/A=75 MeV and for an impact
parameter of b=2 fm. Different panels depict the distributions at different times, t.

177

 

 

30 30
20 20

1O 10

   

-1O —10

I‘IIIIIIIIIIIITIIIIII
_TIIIIIIIIII1IFIIIIIIIII

lllllLLLLLIllILLlLIIIlI

—20 -10 O 10 20

 

 

 

 

IljlllllLIlllLiLLiLlLll

-10 O 10 20

 

 

I
ha
CD

t=30 fm/c t=45fm/c

 

 

30 30

20

20

  

Io 10

'°:::' . -10

O . O .
1 1.1,1.1,1 1 l 1 1 1 1 1 1'1 1 1 l 1 1 1 1 l 1_l 1 1 1,1 l 1'1 1,1 1 111 1 1 l 1 1 1 1 l 1

-20 -10 O 10 20 -20 -1O 0 10 20

 

—1O

IIIWIIIIIIIIIrIIIIIIIII

ijTIWIIIIIIITrIIIIIIIl—r

 

 

 

 

 

 

t=60 fm/c t=75fm/c

Figure 9.7 : Spatial distributions of emitted nucleons in the reaction plane
calculated from the BUU equation for 2 2 N + 2 2 Al collisions at E/A=75 MeV and
for an impact parameter of b=2 fm. Different panels depict the distributions at
different times, t.

178

of the two spheres were assumed to be separated by a distance of d=20 fm and
aligned along the beam direction. The correlation function for this two-source
distribution is depicted by the dashed curve in Figure 9.5; it has a wider maximum
than the single Gaussian source distribution and is rather similar in shape to that
predicted from the BUU calculations. (Remember: two—proton correlation
functions obtained for single spherical sources of sharp sphere and Gaussian
density profiles are virtually indistinguishable in shape; see also the discussion of
Figures 6.3 and 6.4.)

Figures 9.6 and 9.7 depict the space-time evolution of the collision process as
predicted by the BUU equation. Figure 9.6 shows the time evolution of the
nucleon density in the reaction plane for a 2 2 N + 2 2 Al collision at E/A=75 MeV
and an impact parameter of 2 fm. Different panels of the figure represent
snapshots taken at the indicated times after contact of the colliding nuclei. The
calculations predict that the two colliding nuclei essentially survive the collision
and separate into two hot nuclear objects which may then decay on longer time
scales for which BUU calculations cannot make accurate predictions. More
relevant for the calculations of two-proton correlation functions is the density
distribution of the emitted nucleons in the reaction plane, shown in Figure 9.7 for
selected times after the initial contact of the colliding partners. The distribution of
emitted nucleons clearly undergoes an evolution from a near-spherical source at
early times to a two-source distribution at larger times. This two-source
distribution at larger times may explain the similarity of the correlation functions
obtained for a two-sphere distribution with that predicted by the BUU

calculations.
9.5 Longitudinal vs. Transverse Correlation Functions

To explore the shape of the emission sources in 2 2 N induced reactions, we
have compared longitudinal and transverse correlation functions following similar

discussions in Section 8.4. Figures 9.8 and 9.9 show longitudinal and transverse

179

correlation functions measured for the 2 2N + 2 2Al and 2 2N + 2 2 2Au
reactions at E/ A = 75 MeV, respectively. As before, ‘I’ = cos-1(P2-q2/Pq),
longitudinal (solid points) and transverse (open points) correlation functions were

evaluated for the gates Icos‘I’ l 20.77 and lcos‘I’tl s 0.5, respectively. The upper

and lower panels of the figure: show data for different gates on the total momenta
of the emitted particle pairs, P=270~420 MeV/c and P: 420-780 MeV/c for the
‘ 2N + 2 ’ Al reaction, and F: 270450 MeV/c and P= 450-760 MeV/c for the
2 2 N + 2 2 2 Au reaction. For each gate, left and right hand panels show results
obtained with different normalization conventions. The right hand panels depict
longitudinal and transverse correlation functions normalized with a single

normalization constant, C1 which was determined, for each gate on P, by

2!
normalizing the angle integrated correlation function, R O(q), by the condition,

erq R0(q) = 0 , (9.21)
Aq

where Aq = 60-100 MeV/c. With this normalization, longitudinal and transverse
correlation functions gated by low total momenta (P=270-420 and 270-450 MeV/c,
top right-hand panels of Figures 9.8 and 9.9) attain distinctly different values for
larger relative momenta, qz 40 MeV/c. For higher total momenta (P=420-780 and
450-780 MeV/c, bottom right-hand panels of Figures 9.8 and 9.9) differences at
large relative momenta are less significant.

At small relative momenta, residual dynamical correlations are expected to
be small and the use of a single normalization constant should be justified. With
this presumption, we do not find statistically significant differences between
longitudinal and transverse correlation functions at small relative momenta, qg 30

MeV/c. This experimental result is in agreement with that of Reference [Awes 88]

180

MSU-SO- O72

'27Al(22N.pp), E/A=75MeV, 0“=25°

 

1.5 -

; #2222222
; I3
2 1°
L?

t

P=270- -420MeV/c
o v=(60- 120)°

. v=(0-40,140- 180)°
L L L I L L L L

22W¢ll22¢2

#2

33W

IW" 133323222212?)

P= 270-420MeV/c
°1I=(60—120)°

ov=(0-40.140-160)°
: L L I L : L

.

1

[5

 

.¢:

 

P=420-76011eV/c

 

 

P=420-780MeV/c

 

 

0 ¢ o¢=(60-120)° ¢ o¢=(60-120)° I
-5 0 V=(0-40.140—180)° l 0 V=(O-40,140-180)° "
L L L L l L L . L 7 L L L L 1 L L L L
0 50 0 50 100
q (MeV/c)
Figure 9.8 : Longitudinal (‘I’=0°-40° or ‘I’=140°-180°) and transverse (‘I’=60°-

120°) two-proton correlation functions measured for the 2 2N+ 2 2 Al reaction at
E/A=75 MeV. In the left hand panels, longitudinal and transverse correlation
functions were normalized independently; in the right hand panels, the
normalizations were determined from the ‘P—integrated data.

181

MSU-90- 073

 

19"Au(“N.pp), E/A=75MeV, 6,,=25°

 
    
  

1.5 .- .L
: w
1 WWW “M
. ¢
1-0 721* ”2222‘ ° it 22222222222 2%
;¢¢° P=270-450ueV/c ¢¢ 13:270- -450neV/c
: °v=(60-120)° °v=(60-120)° .
0'5 T '19-(0-40.140-160)° ov=(0-40,140-160)° 'j

 

l A A A A A A A I A A
l V f v I r I’ I I V V '

 

 

 

 

_ O
_ P=450-780MeV/c P=450-780MeV/c
. °+ o¢=(60-120)° o¢=(60-120)° .
0.5 f -v=(0-40.140-160)° w=(0-40.140-160)° j
L L L 1 L L . L 7 L L L L L L L .
0 50 0 50 100

q (MeV/c)

Figure 9.9 : Longitudinal (‘I’=0°-40° or ‘I’=140°-180°) and transverse (‘I‘=60°-
120 °) two-proton correlation functions measured for the 2 2 N + 2 2 2 Au reaction at
E/A=75 MeV. In the left hand panels, longitudinal and transverse correlation
functions were normalized independently; in the right hand panels, the
normalizations were determined from the lP-integrated data.

182

for which the same (angle independent) normalization convention had been
adopted.

We have checked by Monte-Carlo calculations that the different asymptotic
values assumed by longitudinal and transverse correlation functions are not due
to trivial effects of detector acceptance or resolution. These differences cannot be
understood in terms of the present model for intensity interferometry. They might,
however, be related to dynamical correlations caused by impact parameter
averaging effects when the single-particle distributions exhibit significant
azimuthal asymmetries. It is possible that some dynamical correlations are caused
by impact-parameter averaging (see Section 9.2). However, we feel that the
present BUU calculations may not model such effects to a sufficient degree of
accuracy since the theory does not reproduce the low-energy portion of the energy
spectrum (see Figure 5.4). One should, therefore, not expect to reproduce
dynamical correlations at the required level of accuracy of a few percent.

The left hand panels in Figures 9.8 and 9.9 show longitudinal and transverse
correlation functions normalized independently over the relative-momentum

interval, Aq = 60-100 MeV/ c, by separately enforcing the conditions,
(
J dq RL T(q) = 0 , (9.22)
Aq '

for the longitudinal and transverse correlation functions RL(q) and Rr(q). With this
renormalization, the longitudinal correlation functions gated by the low total
momentum cuts (P = 270-420 and 270-450 MeV/c, top left-hand panels of Figures
9.8 and 9.9) exhibit larger maxima than the transverse correlation functions,
qualitatively consistent with an elongated source or a source of finite lifetime. For
the 2 2 N + 2 2 Al reaction, this difference disappears for higher total momenta (P =
420-780 MeV/c, bottom left-hand panel of Figure 9.8). For the 2 2N + 2 2 2Au

reaction, the transverse correlation functiOn for the higher momentum gate (P =

183

450-780 MeV/c, bottom left-hand panel of Figure 9.9) exhibits a larger maximum
than the longitudinal correlation function, consistent with an oblate source.

We should caution, that independent normalizations of longitudinal and
transverse correlation functions cannot be justified a priori. Therefore, the
correlation functions shown in the left hand panels of Figures 9.8 and 9.9 should
not be misconstrued as experimental evidence for deformed source shapes. The
different normalizations adopted for the construction of the correlation functions
shown in the right and left hand panels of the two figures are only used to
illustrate the uncertainties within which differences and similarities between
longitudinal and transverse correlations are established experimentally.

Figure 9.10 shows theoretical predictions for longitudinal and transverse
correlation functions for the 2 2N + 2 2Al and 2 2N + 2 2 2Au reactions. These
calculations employed the same cuts on P and ‘I’ which were used in the data
analysis. Differences predicted for longitudinal and transverse correlation
functions are small. They are of the order of the statistical uncertainty of our
measurements, but considerably smaller than the systematic normalization
uncertainties illustrated in Figures 9.8 and 9.9.

In order to present the detailed sensitivities to various cuts on .P and ‘I’,
Figure 9.11 shows longitudinal and transverse correlation functions predicted by

BUU calculations . We have adopted the angular cuts, ‘1’ = 0 ° -40 ° and ‘I’t = 60 °-

1
90°, for the calculation of longitudinal and transverse correlation functions,
respectively. For the transverse correlation functions, we define the in-plane and
out-of-plane directions by constraints on the azimuthal angle, 0, of the relative
momentum vector, £12, in a coordinate system with z-axis parallel to the total

momentum vector, P, of the proton pair. Defining ¢=0° as the plane spanned

184

 

 

 

   

 

 

 

   

 

 

 

 

v v v v I v v v v v v I MSU-9'-OO4
1.5 _ A1(22N,pp), E/A=75MeV L- 1""'LIIu(“N.pp). E/A=75MeV 1
- e,,=26° : 0,,=26° I
1.0 - ——
; P=270-420MeV/c ;; P=270—450MeV/c 4
,3 05; —— ¢=(0-40,140-160)° L. — ¢=(o-40.140-160)°L
BE . r --- y=(60—120)° ;- --- ¢=(60-120)° -:
L L L L l L L L L L L L L I L 2
I . . . . L . . L I . 2 t %
+ . 0 4
1.5 b '1'- .L
v-I L .L L
r .
1.0 f -
: P=420-780MeV/c I P=450-780MeV/c ‘
05; — ¢=(o-40.140-160)° .. -— v=(0-4o.14o—160)°.
- L --- v=(60-120)° ‘f --— ¢=(60-120)° 1
L L L L I L L L L L L L L l L L L L
0 50 0 50 100

q (MeV/C)

Figure 9.10: Longitudinal (‘I’=0°-40° or ‘I’=140°-180°) and transverse (‘I’=60°-
120°) two-proton correlation functions predicted by BUU calculations. The left
and right hand panels show calculations for the reactions 2 2N+2 2Al and
2 2N +2 2 2 Au, respectively. The momentum cuts are indicated in the individual
panels.

r

185

. sup: 27Al(“"N,pp), E/A=75MeV, e,,=25°

MSU-Sl-OTO

 

1.5

V I T—T‘f

A A
1 V

-—-1on¢. Ai-40'Aofi-180‘
- - 'trans.(ln-plane) Air-30' “
- ° -- trans.(out-plano) A¢-30' --

 

P=500MeV/c

r

 

1 + R(q)

 

 

 

 

 

100

Figure 9.11: Longitudinal, and transverse correlation functions predicted by BUU

calculations for ‘ ‘N+2 7Al collisions at E/A=75 MeV for proton pairs for the
indicated total momenta, P.

186

by the beam direction and lg, the in-plane transverse correlation function
corresponds to the cut |¢-n1c| 530° (n=0,1,2); the out-of-plane transverse
correlation function corresponds to the cut I ¢-(2n+1)1r/ 2| 530° (n=0,1). Different
panels of the figure show the results for the indicated cuts on the total momenta of
the proton pairs. In general, the out-of-plane transverse correlation functions are
smaller than the in-plane transverse correlation functions since they are less
affected by finite lifetime effects and collective velocity components. At low
momenta, the longitudinal correlation function is more pronounced than the
transverse correlation functions, possibly due to slow emission times; at higher
momenta, longitudinal and in-plane transverse correlation functions are
comparable in magnitude.

Overall, correlation functions are insensitive to various cuts on angles.
Without an accurate understanding of the distortions caused by dynamical
correlations it appears futile to extract information on the shape of the phase -
space distributions of emitted protons from differences between longitudinal and

transverse correlation functions.

9.6 Dependence on the Nuclear Equation of State and in-Medium

Nucleon-Nucleon Cross Section
The two major ingredients entering into Equation (9.12) are the mean field
potential, U(?), and the nucleon-nucleon cross section, dO/dfl. In principle, one
should be able to derive both from a fundamental nucleon-nucleon interaction, as
has been done in some G-matrix calculations. In this paper, however, we proceed
differently and use the conventional density-dependent Skyrme-type

parametrization,

U(p) =A(p/p .) +B(p/p L)“, (9.23)

 

187

where the parameters A and B are determined by the nuclear matter binding
energy and the saturation density of nuclear matter at p=p o (0.17fm.3). A choice of
0:2 results in A=-124 MeV and B=70.5 MeV and a nuclear compressibility of K =
380 MeV. This set of parameters is referred to as the "stiff" equation of state. The
"soft" equation of state, with K=200 MeV, corresponds to the parameter set,
o=7/ 6, A=-356 MeV, and B=303 MeV. The simple parametrization, Equation
(9.23), is only chosen to investigate the possible sensitivity of our calculations to
the value of the nuclear compressibility.

We approximate the in—medium nucleon-nucleon cross section, d0/ d9, by
the energy-dependent free nucleon-nucleon cross section, doNN/dfl, parametrized
from experimental data. Since the exact value of the in-medium nucleon- nucleon
cross section has attracted some recent attention, we also vary this input by

multiplying the experimental d N/d£2 by different factors ranging from 0 to 1.

o

For our numerical examplefjve calculate two-proton correlation functions for
protons emitted at the laboratory angles Blah-:25 ° 19° for ‘ ‘ N + 2 7 Al collisions
at E/ A = 75 MeV. Unless stated differently, we will use the stiff equation of state
and in-medium cross sections equal to the experimental free nucleon-nucleon
cross sections.

Figure 9.12 shows correlation functions calculated from the Wigner functions
predicted by the BUU equation using various assumptions on the in-medium
nucleon-nucleon cross section and the stiffness of the equation of state. Individual
panels of the figure show correlation functions calculated for different values of
the total laboratory momenta, P, of the proton pairs.

The solid and dotted curves show correlation functions predicted for the stiff
and soft equations of state, using do/dfl = dO'NN/dfl. These two calculations are
very similar, indicating little sensitivity of the two-proton correlation functions to

the stiffness of the equation of state. The lack of sensitivity of the present

calculations to the stiffness of the equation of state may be due to the small size of

188

MSU- 9l-067

EUIl: ”Al(“fN,pp), E/A=75MeV, eu=25°

 

  

 

 

      
   

 

 

 

 

 

2.0
i —- a-O ..
-.-. ,_ f\
' - - . 3.3333:] stiff nos ., I \
r /_\ — 0-0“ L; I \
l / \\ ...... no... soft 208 I a \ l
1.5 - , _ __ m \
l .I- . \ 0 I. \\. Pa _
. ’1 ’ 3K. \ P-aooneV/c .. I ”"9; \ 5°°"°V/°
. I! . I X \
' ° I §§\\
’3' 1.0 - _ I . \\ A
V ' I
‘3' 2.0 I : -
+ - _ q
H A\ , 1
la, -
1 5 ' ’ ’ "\ “ .
. _ P-7ooueV/c f P-eooueV/c
r ..
h r
1.0 - '_ ......
b I ‘ l .............. 1‘
0 50 o 50 100

Figure 9.12: Sensitivity of two-proton correlation functions to the nuclear
equation of state (E05) and the in—medium nucleon-nucleon cross section.
Different panels show results calculated for different total momenta, P, of the
proton pairs.

189

projectile and target and the relatively moderate bombarding energy. One can try
to explore it at higher energy for heavier system.

The solid, dashed, dot-dashed, and dot-dot-dashed curves represent
calculations with the stiff equation of state performed with the assumption that
the in—medium nucleon-nucleon cross-section is equal to 1.0, 0.8, 0.5, and 0.0 times
the free nucleon-nucleon cross section. For Pg 700 MeV/c, the predicted correlation
functions become more pronounced for decreasing values of do/dfl indicating
that increased nucleon-nucleon sacttering leads to slower emission time scales and
the space-time characteristics of the emitting system is more sensitive to the
magnitude of the in-medium nucleon-nucleon cross section than to the stiffness of
the equation of state.

To see how strongly the sensitivity to in-medium nucleon-nucleon cross
section is constrainted by our measurement, we have compared data to
calculations assuming one-half of the free nucleon-nucleon cross sections (see
dotted curves in Figure 9.13) as the in-medium nucleon-nucleon cross sections. For
the low and intermediate momentum gates, the calculated correlation functions
exhibit enhanced maxima when the in-medium nucleon-nucleon cross sections are
reduced. For the high momentum gate, the sensitivity is lost. The agreement with
the data is significantly worse for the calculations using the reduced in-medium
nucleon-nucleon cross sections.

For the l ‘ N + ’ ° 7 Au reaction, dependences on the stiffness of the equation
of state and the magnitude of the in-medium nucleon-nucleon cross sections were
not explored because calculations for this heavier system would have required
large additional amounts of computer time.

Future work should also study the effects of a momentum-dependent mean
field [Gale 87, Aich 87, Welk 88, Grei 90]. Such effects are expected to be important
at higher beam energies (E/Azl GeV). At these energies, it has been shown [Grei

90] that the effects of the momentum-dependence of the mean field on the

190

 

2.0 _-

  
  
 
  

1+R(q)

 

Figure 9.13:

MSU-9l-003

r l I I l I 1— U j
_ ¢¢+ 27A1(”N,pp), E/A=75MeV __
. _° '°-...¢¢ eav=25° ,

0 P=840—1230MeV/c -
. P=450- 780MeV/c -
n P=270- 390MeV/c —

 

 

 

:‘r’l’PT-ngnfi .. . n flél ” .
'- HH¢ .- I ”a
.............. -05 '1
°" ' ”m} BUU -
=a'nu .
50 100

Two-proton correlation functions, measured for the reaction

‘ ‘N+ 2 7 A1 at E/A=75 MeV, are compared with correlation functions predicted
with the BUU theory assuming two different values of the in-medium nucleon-
nucleon cross section. The total mometum gates are indicated.

 

I91

collective nuclear matter flow can be approximated by a momentum-independent
mean field with a suitably changed compressibility. The same effect has been
found at intermediate beam energies (E/ A = 50-150 MeV) where it was found that
the disappearance of nuclear collective flow could be reproduced by using a
momentum-dependent mean field with a compressibility of 210 MeV [Krof 89] or,
alternatively, a momentum-independent mean field with a slighly higher
compressibility of 240 MeV [Ogil 90]. The investigations of Reference [Ogil 90]
corroborate our present finding that fast particle emission in intermediate energy
nucleus-nucleus collisions (E/ A a 50-150 MeV) depends sensitively on the
magnitude of the in-medium nucleon-nucleon cross sections, but only very little

on the nuclear compressibility.

9.7 Dependence on Impact Parameter

Details of the Wigner function must depend on the impact parameter of the
collision. The predicted dependence is illustrated in Figure 9.14; the various cuts
on the total momentum of the proton pairs are indicated for the individual panels
of the figure. In order to summarize the predicted trends, Figure 9.15 shows the
heights of the maxima of the calculated correlation functions as a function of P for
different ranges of impact parameters. For orientation, the equivalent Gaussian
source radii are labeled on the right side of the figure.

For small impact parameters, b=0.5-2.S fm, the predicted correlation
functions increase in magnitude as a function of increasing total momentum of the
emitted protons, consistent with shorter time scales for the emission of more
energetic protons.

For peripheral collisions, b=5-7 fm, the calculated correlation functions are
weakest for proton pairs with very high, P=800 MeV/c, or very low momenta,
P5300 MeV/c; they are largest at intermediate momenta, PzSOO MeV. This
correlation pattern may be understood in terms of emission from fairly well

defined projectile and target-like sources. Proton pairs of low and high momenta

192
usu-st-oss

 

BUU: 27AM14N,pp),E/A=75MeV,®av=25°
C -—-b=0 5—2.5 fm I L i
2.0 _' """" b=3 0-4-5 fm ':' I \\ P=5OOMeV/c -
.-—-b=5 o-7.o fm \ I

‘
I

j-L
0‘!
...l..

I
u

l

I O
1 .-
l:'
'5
'5
1:
If

I

 

    

 

 

 

 

 

 

 

Figure 9.14: Two-proton correlation functions predicted by BUU calculations for
‘ 4N+2 7Al collisions at E/A=75 MeV for the indicated impact parameters and

total momenta of the emitted proton pairs.

 

1 +R(q=20MeV/c)

193

 

 

 

 

 

 

 

 

MSU-Sl-OGG
-fi.,...,..-,.,-r-
3 0 ;BUU: 27Al(14N.pp),E/A=75MeV,®"=25°
' - I--—- b=0.5-2.5(fm) _ 2 5
I o ---------- b=3.0—4.5(fm) '
» o----- b=5.0-7.0(fm) :-
2'5 _' o b=0.5-7.0(fm) ,/
f — 3.0 A
I E
- A L L...
2.0 ~ .0’ ,’ ‘6
. . U" I _3 5 L.
. —4.0
L.

1'5. -4.5
--5.0
~5.5

1 O L L L l L L L i L L L 1 L L L L L

200 400 600 800 1000
P (MeV/c)

Figure 9.15: Momentum dependence of the heights of the maxima of two-proton
correlation functions predicted by BUU calculations for ‘ ‘N+ 2 7 Al collisions at

different impact parameters. Lines connect points corresponding to a given impact
parameter to guide the eye.

194

correspond to low-energy emissions in the rest frames of target and projectile-like
sources, respectively. More energetic emissions from these sources are selected by
intermediate momenta. For these emissions the correlation functions are expected
to be enhanced because of the reduced size of the participant zone and/ or because
of shorter emission time scales. However, nucleon-nucleon collisions appear to
play only a minor role since the the maximum of the correlation function at P=500
MeV/ c remains when the in-medium nucleon-nucleon cross section is set to zero.

For intermediate impact parameters, bz3-4.5 fm, the predicted momentum
dependence is weak, most likely because of overlapping contributions from
participant and spectator regions.

For total momenta, Pg 700 MeV/c, more pronounced correlations are
observed for peripheral than for central collisions - in accordance with a simple
geometric interpretation of the size of the reaction zone. However, for protons
emitted with velocities higher than the beam velocity, P> Zpbeam z 760 MeV/c, the
maxima of the predicted correlation functions are larger for central than for
peripheral collisions. Apparently, high energy protons from central collisions are
emitted on a very fast time scale. It may be speculated that such fast emission
processes are related to Fermi-jets [Bond 80], i.e. nucleons accelerated by the action
of the mean field and emitted without significant nucleon-nucleon collisions.
However, because of the semi-classical nucleonic momentum distribution used in
our calculations, the predicted correlations may not be reliable for the highest
momenta, P>1000 MeV/c, and should be viewed with caution. In any case, the
calculations clearly indicate that one should be able to extract a wealth of
information about the space-time evolution of the reaction zone by detailed
investigations of the momentum and impact parameter dependence of two-proton
correlation functions. Previous measurements of two-particle correlation functions
did not determine the simultaneous dependence on impact parameter and

momentum of the emitted particle pair and thus averaged over valuable

 

 

195

information. New measurements capable of determining such dependences are
clearly desirable. The experimental effort to explore these effects is currently
getting under way.

In summary, the measurement of two-proton correlation functions have
provided a new sensitive observable to test the space-time geometries of nuclear
collisions. With the technique of two-proton intensity interferometry, one can

hope to gain more insight into nuclear reaction dynamics.

 

CHAPTER 10. SUNIMARY AND CONCLUSIONS

Heavy-ion collisions at intermediate energy provide an unique opportunity
to study the nuclear collision dynamics and thermodynamical properties of highly
excited nuclei. In this thesis, we have used the technique of two-proton intensity
interferometry to probe the space-time evolution of nuclear collisions.

Considerable efforts were made to develop CsI(Tl) detectors read out by PIN
diodes. The detector resolution is limited by local non-uniformities of the CsI(Tl)
scintillator. After selecting CsI(Tl) scintillators of good uniformity based upon a y-
ray scanning technique, we built a 56 element multi-detector array using Si AE-
detectors and CsI(Tl) E-detectors. The detectors show high gain stability and good
energy resolution for the detection of light charged particles.

We have measured two-proton correlation functions for emission processes
governed by different emission time scales. Slow evaporative processes were
studied at E/ A = 31 MeV for the near-symmetric reaction 1 2 ° Xe + l 1 2 Sn and
for the "inverse kinematics" reaction ‘ 2 °Xe + 2 7Al. Fast non-equilibrium

l

processes were investigated, in "forward kinematics", for ‘ N induced reactions

on 2 7Al and

19‘]

Au at E/ A = 75 MeV. Two-proton correlation functions
measured for these qualitatively different reaction mechanisms exhibit significant
differences in shape. These differences are well understood in terms of the
different emission time scales governing equilibrium and non-equilibrium

processes.

I96

 

 

 

197

We have used the Wigner-function formalism which allows the calculation
of two-proton correlation functions for any reaction model capable of predicting
the one-body phase-space distribution of the emitted particles. We demonstrated
the sensitivity of two-proton correlation functions to the space-time geometry of
the emitting system via calculations for schematic emission sources. Results
obtained with the Wigner-function formalism were compared with classical
trajectory calculations for long-lived evaporative sources.

Two-proton correlation functions measured for evaporative processes in
1 2 °Xe induced reactions at E/ A = 31 MeV do not exhibit maxima at q =3 20
MeV/c, but only minima at q z 0 MeV/c. Particle emission from equilibrated
compound nuclei typically proceeds on such long time scales that anti-
symmetrization and nuclear interaction between the two emitted particles play a
relatively minor role. The correlation functions are dominated by final state
Coulomb interactions. The measured correlation functions can be rather well
understood by applying the Wigner-function formalism to one-body phase-space
distributions predicted by statistical model calculations based on the Weisskopf
formula. For these reactions, rather small emission rates lead to large spatial
separations between emitted particles and a loss of memory of the size of the
emitting nucleus. The predicted correlation functions exhibit only moderate
sensitivity to detailed properties of the decaying nucleus.

Two-proton correlation functions measured for non-equilibrium emission in
1 ‘ N induced reactions at E/ A = 75 MeV exhibit pronounced maxima at relative
momenta q -- 20 MeV/c and minima at q z 0 MeV/c. The maximum at q z 20
MeV/c is caused by the attractive singlet S-wave interaction between the two
emitted protons. The minimum at qu MeV/c reflects the combined effects of the
Coulomb repulsion and the exclusion due to Pauli principle. For these reactions,
the emission time scales are sufficiently short that the final phase-space

distributions of the emitted particles are of nuclear or smaller dimensions. Once

 

198

emission times are of the order of a few hundred fm/c or less, anti-
symmetrization effects, nuclear and Coulomb interactions must all be
incorporated into the calculations. The measured correlation functions can be
rather well understood by applying the Wigner function formalism to one-body
phase-space distributions predicted by the Boltzmann-Uehling—Uhlenbeck
transport equation. It is particularly gratifying that the theory can reproduce the
observed strong dependence of the experimental correlation functions on the total
momentum of the coincident proton pairs.

For all cases investigated, longitudinal and transverse correlation functions
were found to be very similar. Our data could not provide definitive evidence for
elongated source shapes expected from simple lifetime arguments. The
experimental observations are, however, consistent with more detailed
calculations for which the predicted differences between longitudinal and
transverse correlation functions were too small to be detected by the present
experiment. _

The calculated two-proton correlation functions predicted by the BUU model
for the reaction ’ ‘ N + 2 7 A1 at E/ A = 75MeV exhibit significant sensitivity to
details of the space-time evolution of the reaction zone. They indicate large
sensitivity to the magnitude of the in-medium nucleon-nucleon cross sections
used in the BUU calculations. Considerable sensitivity is predicted for
measurements which explore the dependence of the two-proton correlation
function on the collision impact parameter and on the total momenta of the
emitted proton pairs. However, there is very little sensitivity to the compressibility
of nuclear matter.

Exclusive measurements of two-proton correlation functions with selection
on impact parameters can be pursued by combining the 56-element hodosc0pe
with the 4K detector array. Better characterization of the collision geometry will

enable us to make more stringent tests on the dynamical models for heavy-ion

 

199

collisions. It is desirable to extend future studies of two-proton intensity
interferometry to higher beam energies (E/ A z 1 GeV) and to heavier projectile-
target combinations. Such studies may provide additional insight into the collision
dynamics and compressibility of nuclear matter, which could not be gained in an

unambiguous fashion from observables tested so far.

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