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ANGULAR CORRELATIONS NEAR THE FERMI ENERGY

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Daniel Fox

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ANGULAR CORRELATIONS NEAR THE FERMI ENERGY

By

Daniel Fox

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

1987

ABSTRACT

ANGULAR CORRELATIONS NEAR THE FERMI ENERGY

BY

Daniel Fox

Angular correlations between light particles have been studied to
probe the extent to which a thermally equilibrated system is formed in
heavy ion collisions near the Fermi energy. The results of two
experiments are reported.

Light particle correlations have been measured at energies (35-50
HeV/nucleon) near the fermi energy. Light particle spectra and large
angle correlations were measured for ‘40 and 50 MeV/nucleon C+C, Ag and
Au. The single particle inclusive energy spectra are well fit by a three
moving source parameterization. The correlations were done with tag
detectors at G=ZS° and “5° with ¢=O°, 90° and 180°. Coincident particles
were detected at G:15°-150°. The decay of particle unstable states was
found to contribute noticeably to the same side in-plane correlations.
The two-particle correlations were found to be dominated by momentum
conservation effects.

Light particle correlations at small relative momentum were measured
for 25 and 35 MeV/nucleon N+Ag and single particle inclusive spectra were
measured for 35 MeV/nucleon N+Ag at 0:35°, 145°, 60° and 80°. Source
radii were extracted from the two-particle correlation functions and were
found to be consistent with previous measurements using two-particle
correlations and the coalescence model. The populations of particle
unstable states were extracted and used together with the measured single

particle inclusive spectra and previously measured Y and neutron emitting

states to determine the temperature of the emitting system using the
quantum statistical model. The temperature extracted from the relative
populations of states, using the quantum statistical model, was finnui to
be 4.833 MeV which is substantially lower than the 114 MeV temperature

extracted from the slopes of the kinetic energy spectra.

ACKNOWLEDGMENTS

I would like to thank my advisor Gary Westfall for his help and
guidance over the last three years which has helped to make this work
possible. I would also like to thank Dan Cebra, Jeff Karn, Cam Parks,
Ashok Pradhan, Bob Tickle, Skip VanderMolen, Hans van der Plicht and Ken
Wilson for their help during the experiments. I would also like to thank
my other fellow graduate students, past and present, for their help,
support and friendship over the last four years. Finally I would like to
thank the staff of the National Superconductiong Cyclotron Laboratory for
providing the facilities and beam time without which this work could not

have been completed.

TABLE OF CONTENTS

Chapter
LIST OF TABLES ...................................................... viii
LIST OF FIGURES ..................................................... ix
I Introduction .................................................... l
A. Large Angle Correlations .................................. 2
B. Nuclear Temperature Measurements .......................... A
C. Contents And Organization Of The Thesis ................... 7
II Experimental Setup .............................................. 9
A. Unstable Resonance Experiment. ............................ 9
B. Singles Measurement For The Unstable Resonance Experiment. 17
C. Large Angle Correlation Experiment. ....................... 20
III Data Analysis and Reduction ..................................... 25
A. Phoswich Energy Calibrations. ............................. 25
1. Unstable Resonance Experiment. ........................ 25
2. Large Angle Correlation Experiment. ................... 29
B. Silicon Detector Calibration. ............................. 33

vi

C. Reaction Loss Corrections. ................................ 33
D. MWPC Efficiency. .......................................... 37
IV Large Angle Correlation Experiment Data And Discussion .......... A2
A. Inclusive Energy Spectra .................................. A2
1. Light Particle Spectra ................................ A2

2. Moving Source Parameterization. ....................... A2

3. Heavy Fragment Spectra ................................ A8

B. Two Particle In-Plane Correlations. ....................... 53
1. Two Particle Correlations, 45° Tag .................... 53

2. Momentum Conservation Model. .......................... 61

3. Two Particle Correlations, 25° Tag .................... 66

C. Out—of-plane Correlations ................................. 75
1. Out-of-plane Correlation Functions .................... 75

2. In-plane To Out-of-plane Ratios ....................... 79

V Unstable Resonance Experiment Data And Discussion ............... 89
A. Two Particle Correlation Functions ........................ 89
1. 35 MeV/nucleon N+Ag ................................... 90

2. 25 MeV/nucleon N+Ag ................................... 102

B. Source Sizes .............................................. 102
C. Bound State Spectra ............ . .......................... 109
D. Populations 0f Particle Unstable States ................... 113

E. Quantum Statistical Model And Extraction 0f Nuclear
Temperature ................................................... 115

F. Final State Interactions Vs. Emission 0f Particle Unstable

vii

VI Summary And Conclusions .........................................

APPENDIX

Momentum Conservation Model .....................................

LIST OF REFERENCES

OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO

125

Table
II-1
II-2

II-3

II-u

III-l

IV-1

IV-2

IV-3

Iv-u

V-5

LIST OF TABLES

Plastic Scintillator Properties ........................... 10
Phoswich Low And High Energy Cuts ......................... 13
Telescope Energy Ranges For Unstable Resonance Singles
Measurement ............................................... 18
Telescope Position And Solid Angles For The Large Angle
Correlation Experiment .................................... 22

Values 0f a2 And b For 2:2-6 ............................. 32

2
Energy And Angle Ranges Used For Moving Source Fits And

The Coulomb Shifts For Ag And Au Data ..................... A9
Moving Source Parameters .................................. 50
Moving Source Parameters For Li And 7Be ................... 5A
Energy Ranges For Correlations ............................ 6O
Extracted Source Radii From Two Particle Correlations ..... 108
Moving Source Parameters For 35 MeV/nucleon N+Ag .......... 112

Extracted Cross Sections 0f Particle Unstable States
(mb/sr) ................................................... 11H
Excited State Ratios For 35 MeV/nuc. N+Ag ................. 117

Key To Particle Unstable And Gamma States In Figure V-15 .. 119

viii

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

III-l

III-2

III-3

III-N

III-5

III-6

III-7

IV-1

IV-2

LIST OF FIGURES

Phoswich signals and ADC gates. .....................
Phoswich array used in the unstable resonance
experiment. .........................................
Electronics diagram for the unstable resonance
experiment.

Fraction of 'Be nuclei that identify as ’Li. ........
Chamber setup for the large angle correlation
experiment.

Electronics diagram for the large angle
correlation experiment. .............................
AE-E spectra from the unstable resonance

experiment. .........................................
Example of a phoswich calibration for the

unstable resonance experiment. ......................
Example of a phoswich calibration for the

large angle correlation experiment. .................
Comparison of the two different a

calibrations.

MHPC wire efficiency as a function of mass,
charge and energy for the wire in front of one
phoswich. ...........................................
Calculated efficiency for MHPC and phoswich

array for p-t, d-a and a-a particle pairs for

.. O
OMWPC'us '
Light particle energy spectra for A0 MeV/nuc.

C+C. The lines are the results of moving
source fits.

Light particle energy spectra for 50 MeV/nuc.
C+C. The lines are the results of moving
source fits.

ix

X

Figure IV-3 Light particle energy spectra for A0 MeV/nuc.
C+Ag. The lines are the results of moving
source fits. ........................................ A5

Figure IV-A Light particle energy spectra for A0 MeV/nuc.
C+Au. The lines are the results of moving
source fits. ........................................ A6

Figure IV-S Heavy fragment energy spectra for A0 MeV/nuc.
C+C (top) and 50 MeV/nuc. C+C (bottom). The
lines are the results of moving source fits. ........ 51

Figure IV-6 Heavy fragment energy spectra for A0 MeV/nuc.
C+Ag (top) and C+Au (bottom). The lines are
the results of moving source fits. . ................. 52

Figure IV-7 Two-proton correlation function for which one
proton is detected at 0=-A5°. The lines are
described in the text. .............................. 55

Figure IV-8 Proton-deuteron correlation function for which the
deuteron is detected at 0:-A5°. The lines are
described in the text. .............................. 56

Figure IV-9 Two-deuteron correlation function for which one
deuteron is detected at 0=-A5°. The lines are
described in the text. .............................. 57

Figure IV-10 Deuteron-alpha correlation function for which the
alpha is detected at 0=—A5°. The lines are
described in the text. .............................. 58

Figure IV-11 Alpha-alpha correlation function for which one
alpha is detected at 0:-A5°. The lines are
described in the text. .............................. 59

Figure IV-12 Contribution from the three sources to the
deuteron singles cross section (a) and to the
two-deuteron coincidence cross section for
=-A5° (b) and for 0t =-25° for A0
Mgegnuc. C+Au. ......... mg ............................ 63

Figure IV-13 Contribution from the three sources to the
deuteron singles cross section (a) and to the
two- deuteron coincidence cross section for
=-AS° (b) and for 0ta g=-25° for A0
Mgegnuc. C+C. ....................................... 65

Figure IV-1A Two-proton correlation function for which one
proton is detected at Oz-25°. The lines are
described in the text. .............................. 67

Figure IV-15 Proton-deuteron correlation function for which the
deuteron is detected at 0=-25°. The lines are
described in the text. .............................. 68

Figure

Figure

Figure

Figure
Figure

Figure

Figure

Figure

Figure

Figure

Figure

IV-16

IV-l7

IV-18

IV-19

IV-20

IV-21

IV-22

IV-23

IV-2A

IV-25

IV-26

xi

Two-deuteron correlation function for which one
deuteron is detected at O=-25°. The lines are
described in the text. ..............................
Deuteron-alpha correlation function for which the
alpha is detected at O=-25°. The lines are

described in the text. ..............................
Alpha-alpha correlation function for which one

alpha is detected at O=-25°. The lines are

described in the text. ..............................
Lithium-proton correlations for which the lithium

is detected at 0:-25°. ..............................
7Be-proton correlations for which the 7Be is

detected at O=-25°. .................................
Two-proton out-of-plane correlation function

for O,=25° (triangles and dashed lines) and

0,:A5° (squares and solid lines). ...................
Two-deuteron out-of—plane correlation function

for O,=25° (triangles and dashed lines) and

O,=A5° (squares and solid lines). ...................
Two-alpha out-of—plane correlation function

for 0,:25° (triangles and dashed lines) and

O,=A5° (squares and solid lines). ...................
Ratio of in-plane to out-of-plane correlations

for two protons for A0 MeV/nucleon C+C

(squares and solid lines), C+Ag (circles and

dash dot lines), C+Au (diamonds and dashed

lines) and 50 MeV/nucleon C+C (triangles and

dotted lines). One proton is detected at

O,=A5°. .............................................

Ratio of in-plane to out-of—plane correlations

for two deuterons for A0 MeV/nucleon C+C

(squares and solid lines), C+Ag (circles and

dash dot lines), C+Au (diamonds and dashed

lines) and 50 MeV/nucleon C+C (triangles and

dotted lines). One deuteron is detected at

O,=A5°. .............................................

Ratio of in-plane to out-of-plane correlations
for two alphas for A0 MeV/nucleon C+C (squares
and solid lines), C+Ag (circles and dash dot
lines), C+Au (diamonds and dashed lines) and
50 MeV/nucleon C+C (triangles and dotted
lines). One alpha is detected at O,=A5°.

xii

Figure IV-27 Ratio of in-plane to out-of—plane correlations
for two protons for A0 MeV/nucleon C+C
(squares and solid lines), C+Ag (circles and
dash dot lines), C+Au (diamonds and dashed
lines) and 50 MeV/nucleon C+C (triangles and
dotted lines). One proton is detected at
O,=25°. ............................................. 8A

Figure IV-28 Ratio of in-plane to out-of-plane correlations
for two deuterons for A0 MeV/nucleon C+C
(squares and solid lines), C+Ag (circles and
dash dot lines), C+Au (diamonds and dashed
lines) and 50 MeV/nucleon C+C (triangles and
dotted lines). One deuteron is detected at
O,=25°. ............................................. 85

Figure IV-29 Ratio of in-plane to out-of-plane correlations
for two alphas for A0 MeV/nucleon C+C (squares
and solid lines), C+Ag (circles and dash dot
lines), C+Au (diamonds and dashed lines) and
50 MeV/nucleon C+C (triangles and dotted
lines). One alpha is detected at O.=25°. ........... 86

Figure IV-30 Ratio of in-plane to out-of-plane correlations
for two protons using the BUU model
(histogram) [Ba 87] compared to the data

(circles). .......................................... 88
Figure V-1 Two-proton correlation function for 35

MeV/nuc. N+Ag. The lines are described in the

text. ............................ . .................. 91
Figure V-2 Proton-deuteron correlation function for 35

MeV/nuc. N+Ag. The lines are described in the

text. .. ............................................. 92
Figure V-3 Proton-triton correlation function for 35

MeV/nuc. N+Ag. The lines are described in the

text. ............................................... 9A
Figure V-A Proton-alpha correlation function for 35

MeV/nuc. N+Ag. The lines are described in the

text. ..... . ......................................... 95
Figure V-5 Two-deuteron correlation function for 35

MeV/nuc. N+Ag. The lines are described in the

text. ....... ........ . ..... ..... ..................... 97
Figure V-6 Deuteron-triton correlation function for 35

MeV/nuc. N+Ag. The lines are described in the

teXt. cocoa-00000000000009...eeee oooooooooooooooooooo 98
Figure V-7 Deuteron-alpha correlation function for 35

MeV/nuc. N+Ag. The lines are described in the
text. .. ............................................. 99

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

V-9

V-lO

V-ll

V-12

V-13

xiii

Two-triton correlation function for 35
MeV/nuc. N+Ag. The lines are described in the
text. ............................................... 100

Triton-alpha correlation function for 35
MeV/nuc. N+Ag. The lines are described in the
text. ............................................... 101

Two-alpha correlation function for 35 MeV/nuc.
N+Ag. The lines are described in the text. ........ 103

Two-proton, proton-deuteron, proton-triton and
proton-alpha correlation functions for 25

MeV/nuc. N+Ag at 0=A5°. The lines are

calculations for final state interactions for

sources of radius r:A (dashed dot), 5 (dashed)

and 8 fm (solid) for nuclear and coulomb

interactions and r=8 fm (dotted line) for

coulomb only interactions. The dashed line in

c.) if for 9 fm. ......... . .......................... 10A

Two-deuteron, deuteron-triton and deuteron-

alpha correlation functions for 25 MeV/nuc.

N+Ag at O:A5°. The lines are calculations for

final state interactions for sources of radius

r=8 (solid) and 9 fm (dashed) for nuclear and

coulomb interactions and r=8 fm (dotted line)

for coulomb only interactions. ...................... 105

Two-triton, triton-alpha and two-alpha

correlation functions for 25 MeV/nuc. N+Ag at

O=A5°. The lines are calculations for final

state interactions for sources of radius r:A

(dashed dot), 5 (dashed) and 8 fm (solid) for

nuclear and coulomb interactions and r=8 fm

(dotted line) for coulomb only interactions.

The dashed dot line in c.) if for r=6fm. ............ 106

He, Li, Be and B energy spectra for 35
MeV/nuc. N+Ag. The lines are moving source
fits described in the text. ......................... 111

Quantum statistical model calculation of the

production of the measured states for T=A.8

and 1A.0 MeV and p=0.18p,. The excited states

and particle unbound ground states shown in

the bottom half of the figure are identified

in Table V-5. ....................................... 118

Quantum statistical calculation of the feeding

to the observed states for a source

temperature of T=A.8 MeV and freeze—out

density of p=0.18po. ................................. 120

Chapter I

Introduction

The extent to which thermalization occurs in intermediate and
relativistic energy heavy ion nuclear physics has been the focus of many
experiments in the last decade. At relativistic energies single particle
inclusive spectra of light particles were found to be thermal in
appearence, leading to the introduction of the fireball model [We 76, Go
77]. The basic idea behind the fireball model is that the region of
overlap between the projectile and target forms a hot source moving at a
velocity slightly less than half the beam velocity. The source expands
emitting particles with Maxwell-Boltzmann type energy distributions. In

3He,uHe) were found to be well

addition the spectra of light nuclei (d,t,
described by the coalescence model [Cu 76, Go 77]. In the coalescence
model composite fragments are formed when nucleons are emitted close
together in phase space.

At intermediate energies, 20<E eam<200 MeV/nucleon, single particle

b
inclusive spectra were measured for light and intermediate mass particles
and found again to be well described by a Maxwell-Boltzmann type energy
distribution [We 82, Ja 83, We 8A]. Additional evidence for a thermal
source was found in the behavior of the d/p and ”He/p ratios [We 82]
which were found to be in general agreement with calculations using the
fireball model [We 76, Go 77]. The intermediate mass particles were
shown to come from the same source as the light nuclei [Ja 83]. Also the
coalescence model, which had previously been applied at relativistic
energies [Cu 76, Go 77] and at lower energies [Aw 82] to light nuclei,

was applied to both light and intermediate mass fragments [Ja 85]. The

extracted coalescence and source radii showed little variation with

2

fragment mass indicating a common origin for the observed particles.
Particles detected in coincidence with intermediate rapidity fragments
and projectile like fragments were also found to come the same source [Ha
85]. Particles in coincidence with projectile like fragments very close
to the beam mass were found, however, to have a slower source velocity by
about 30%, but the source temperature was found to be the same as the
temperature extracted from the singles spectra.

While single particle inclusive kinetic energy spectra of light arm!
intermediate mass fragments have suggested emission of all particles from
the same thermally equilibrated intermediate velocity source some
evidence of non-thermal processes has been observed. At relativistic
energies quasi-elastic p-p scattering has been seen [Na 79, Ta 80, Ta
81]. Other two-paritcle correlations have suggested that momentum
conservation plays a key role in intermediate energy reactions [Ly 82, Ts
8A, Kr 85, Ch 86b]. More recently experiments have sought to measure the
temperature of the intermediate source by measuring the relative
populations of various nuclear states [Mo 8A, Mo 85, Mo 86, Xu 86, Ch
86a, Ch 87, P0 86b, Bl 87b]. The temperatures extracted ffiwnn the
populations of nuclear states has been shown to be considerably lower

than the temperatures extracted from the kinetic energy spectra.

A. Large Angle Correlations

Two-particle large-angle correlations have been used to study the
extent to which thermalization occurs at relativistic [Na 79, Ta 80, Ta
81] and at intermediate [Ly 82, Ts 8A, Kr 85, Ch 86b] energies. The
first experiments [Na 79, Ta 80, Ta 81] measured the ratio of in-plane to

out-of-plane correlations. Tag detectors were placed at (O,¢)=(A0°,t90°)

3
and (A0°, 180°). Coincidences were measured between these tag detectors
and a spectrometer at <I>=0° that covered angles from 0=10° to 50°. The

ratio of the number of in-plane (ttag=180 ) to out-of-plane (Otag:t90 )

correlations was measured. For a completely thermalized system this
ratio should be one. For 800 MeV/nucleon C+C an enhancement of about 70%
was observed in the ratio of in-plane to out-of—plane correlations at
angles and energies corresponding to quasi-elastic p-p scattering. For
Ar+KCl a smaller enhancement of about 30% was observed. For C+Pb the
ratio of in-plane to out-of-plane correlations was found to be about 0.9
independent of angle. This was interpreted to be due to shadowing by the
target.

In a similar experiment [Kr 85a, Kr 85b] the ratio of in-plane to
out-of-plane correlations was measured at 85 MeV/nucleon for C+C, Al, Cu
and C+Au [Kr 85a, Kr 85b]. As at 800 MeV/nucleon the C+C system showed
an enhancement in the ratio of in-plane to out-of-plane correlations.
This enhancement was attributed not to quasi-elastic scattering but
rather to emission from a recoiling thermal source. In addition the
enhancement was found to increase with increasing mass of the detected
particle pair.

At lower energies momentum conservation has been shown to be an
important factor in two particle large angle correlations for light
systems while heavy systems have been described using a rotating ideal
gas [Ly 82, Ts 8A, Ch 86b]. Azimuthal correlations of light particles in
25 MeV/nucleon O+Au reactions have been shown to have a strong preference
for emission in the same plane [Ts 8A]. The data were well described by
a calculation in which the particles were assumed to be emitted from a

rotating moving source. For lighter systems (0+C) both in-plane and

u
azimuthal correlations are dominated by momentum conservation effects
which lead to a strong enhancement of planar emission on opposite sides
of the beam [Ch 86b].

In recent results for 60 MeV/nucleon Ar+Au reactions the two
particle correlations were found to be much more isotropic [Ar 85]. 'The
correlation functions for two proton correlations vary by only about 10%
as a function of O and t compared to variations of 50% for 25 MeV/nucleon
O+Au [Ts 8A, Ch 86b]. Even two alpha correlations showed much less
angular dependence at 60 MeV/nucleon, varying by 50% as a function of
angle compared to a factor of five variation for two triton correlations
at 25 MeV/nucleon. One interesting feature in the 60 MeV/nucleon data is
that the azimuthal correlations have their minimum at ¢=A5° and not ¢=90°
as is seen at 25 MeV/nucleon. The more isotropic behavior at 60
MeV/ruuileon is most likely due to the larger size of the emitting source

which makes it easier to balance the momentum of the emitted particle.
B. Nuclear Temperature Measurements

A few years ago Morrissey et al. [Mo 8A, Mo 85] tested the idea of
thermal equilibrium by measuring the populations of excited states of
light nuclei that decay by Y-ray emission. For a thermalized system the
ratio of the populations of the excited state to the population of the

ground state is given by

2] +1 -E /kT
eX ex

R=2j +1 e , (I-l)
gs

 

5

where jgs and jex are the spins of the ground and excited states,
reespectively, Eex is the excitation energy of the excited state, and kT

is the temperature. Morrissey et al. measured the populations of the

6 7

low-lying, Y-ray emitting states of Li, 7

Li, 8Li and Be emitted from

the reaction 35 MeV/nucleon mNa-Ag. The kinetic energy spectra of the

6L1, 7L1, 8Li and 7

Be ground states indicated a temperature of about 10
MeV. The populations of the Y-ray emitting states, however, were found
to correspond to a source temperature of less than 1 MeV.

Morrissey et al. checked their method of measuring the nuclear

temperature by studying the system 1uN+12C at low bombarding energies

(87.5-350 MeV) [Mo 86]. The Y-ray emitting states of 7Be and 10B emitted

from the 26Al compound nucleus were measured. The temperatures extracted
from the excited state populations were found to be constant down to
about 112 MeV at which point the temperature began to decrease. Between
87.5 and 112 MeV the measured temperature agreed with the expected
temperature from the compound nucleus when the rotational energy of the
compound nucleus is taken into account.

Two possible explanations for the surprisingly low temperatures
extracted from the Y-ray emitting states have been proposed. In one
explanation nuclei emitted in particle bound excited states experience
final state interactions with neutrons resulting in their deexcitation
[Bo 8A]. As a result the number of observed excited nuclei is
considerably less than those originally emitted and hence a lower
temperature is extracted. The second explanation is based on the quantum
statistical model [St 63, Ha 86, Ha 87]. In this model the temperature

measured by the populations of Y-ray emitting states is lower than the

6

true temperature of the emitting states because of the sequential decay
of heavier particle unbound states. Sequential decay primarily feeds the
ground states, thus lowering the measured temperature [Ha 87]. Using the
quantum statistical model Xu et al. [Xu 86] showed that the populations
of Y-ray emitting states from the reaction S+Ag at 22.3 MeV/nucleon was
consistent with a temperature of T>A MeV. Without using the quantum
statistical model the extracted temperature was between 0.5 and 2.A MeV
depending on wich excited state was used to extract the temperature.

The method of Morrissey et al. has more recently been extended to
particle unstable states of light nuclei [Ch 86a, Ch 87, P0 85a, Po 85c,
Po 86b]. The use of particle unstable states has the advantage of
allowing the measurement of higher temperatures than most of the Y-ray
emitting states due to the higher excitation energies of most of the
particle unstable states compared to those of the Y-ray emitting states.

For the system 35 MeV/nucleon N+Au [Ch 86a] temperatures of between 3 and

9 MeV were extracted from the particle unstable states of 6L1 neglecting

the effects of feeding from higher lying states. The higher temperatures

were obtained for higher total energy of the emitted 6L1 parent nuclei.

5Li and 6Li in coincidence with

Temperatures extracted for states in
fission fragments were found to be about 5 MeV [Ch 87] independent of the
folding angle of the two fission fragments. For 60 MeV/nucleon Ar+Au [Po
86b] a temperature of about 5.5 MeV was obtained using the quantum
statistical model to fit the populations of a large number of states. In
the Ar+Au experiment a large array of plastic scintillator was used to

provide multiplicity information. Although the two particle correlations

varied considerably as a function of multiplicity, the extracted

7
temperature showed no significant variation as a function of multiplicity
[P0 87].

More recent experiments have examined particle unstable states tfluat
decay by neutron emission. The system 35 MeV/nucleon N+Ag was studied

looking at neutron emitting states in 7L1, 8L1, 10Be and 12B [Bl 87a, Bl

87b]. These states were found to yield higher temperatures than the Y-

ray states of the same system. The temperature extracted from the
a
Li7 A56 state, when corrected for feeding of the ground state, was found

to 2.8:0.3 MeV.
The use of'particle unstable states to measure nuclear temperatures
is not without problems. The effects of feeding from higher states is

still important and may change the apparent temperature by a factor of

a
two or more. Even some of the particle unbound states, such as Be3 OA’

can be heavily fed [Ha 87]. In addition the extraction of the
populations of the excited states may be complicated by final state

interactions of randomly emitted particles.
C. Contents and Organization of the Thesis

In order to probe the extent to which a thermally equilibrated
source is created in heavy ion reactions around the Fermi energy we have
performed two experiments. In the first experiment single particle
inclusive kinetic energy spectra and two-particle large angle
correlations of light nuclei were studied for A0 and 50 MeV/nucleon C
induced reactions on C, Ag and Au targets. The measured single particle:

inclusive energy spectra are fit using a three moving source fit. A

8
momentum conservation model is used to determine the extent to which
energy and momentum conservation effects the measured two-particle
correlations.

In the second experiment the populations of particle unstable states
were measured for 25 and 35 MeV/nucleon N induced reactions on a Ag
target using two-particle correlations at small relative momentum. The
populations of particle unstable states are extracted from the two-
particle correlations. The quantum statistical model is used to extract
the source temperature for 35 MeV/nucleon N+Ag from the populations of
the particle bound and unbound states measured in the present work
combined with the data of Morrissey and Bloch for the Y and neutron
(unitting states. In addition information about the space-time extent of
the emitting system is extracted from the two-particle correlation
functions.

The rest of this thesis is arranged as follows. The experimental
setups are described in Chapter II. The detector calibrations, reaction
loss correction, and efficiency calculation for a multi-wire proportional
counter used in the unstable resonance experiment are discussed in
Chapter III. The results of the large angle correlation and unstable
resonance experiments are presented and discussed in Chapters IV and V
respectively. Chapter VI contains a summary of the results and

conclusions.

Chapter II

Experimental Setup

Both experiments described in this dissertation were performed in
the 1.5 meter scattering chamber at the National Superconducting
Cyclotron Lab (NSCL) at Michigan State using beams from the K500
Cyclotron. The chamber and electronics setups will be discussed in this

chapter. The detector calibrations are described in Chapter III.
A. Unstable Resonance Experiment.

Two types of detectors were used in the unstable resonance
experiment. The energy of the observed light particles (p,d,t,3He,uHe)
were measured using sixteen fast-slow plastic phoswich telescopes.
Position information was obtained by the use of a three plane multi-wire
proportional counter (MWPC). The phoswiches will be discussed first.

Particle identification and energy measurement was achieved using an
array of sixteen fast-slow phoswich telescopes. Each telescope consisted
of a 1.6 mm thick fast plastic (BC-A12) AE detector optically coupled to
a 127 m long slow plastic (BC-AAA) E detector. The properties of the
plastic scintillators are given in Table II-1. The signals from each
telescope were read using a single Amperex 2202 ten stage
photomultiplier. The photomultiplier has a maximum response for light
with wavelength A=A25330 nm which is the same as the wavelength of
maximum emission of the plastics given in Table II-1. In order to get
both AE and E from the single signal the signal was split using a 500
splitter and sent to two different charge integrating ADCs. Figure II-1

shows what the signals from the AE and E detectors look like. The pure

10
Table II-l

Plastic Scintillator Properties

Fast Plastic Slow Plastic

 

(BC-A12) (BC-AAA)
Rise Time (ns) 1.0 19.5
Decay Time (ns) 3.3 179.7
Pulse Width (FWHM ns) A.2 171.9
Wavelength Of Maximum Emission (nm) A3A A28
Light Output, 1 Anthracene 60 A1

(From Bicron Corporation data sheets [Bi 85].)

11

MSIJ-I 87 - 057

Pure AE Signal Pure E Signal
0 20 50 IOOnS' O 200 400 600 ns
I I l I 1 1 1 l 1 I l I
53 MeV/nucleon (1 Signal

20 ICC 200 300 400 ns
1 1 l I 1

AE E

Figure II-1 Phoswich signals and ADC gates.

12

AE signal comes from a very low energy particle which stops in the fast
plastic AE. The pure E signal comes from a neutron or gamma which passes
through the AE without interacting and stops in the slow plastic E
detector. The signal shown at the bottom of Figure II-1 is that of a 53
MeV/nucleon a from the large angle correlation experiment calibration.
Also shown in the bottom of Figure II-1 are the ADC gates used to
separate the AE and E signals.

3

The lower and upper energy cuts of the phoswiches for p,d,t, He and

“He are given in Table II-2. The energy resolution of the phoswiches was
measured as part of the calibration of the phoswiches for the large angle
correlation experiment. For a 106 MeV deuteron, which loses 2 MeV in the
AE detector and 10A MeV in the E detector, the resolution was determined
to be 391 in the AE and 2.81 in the E. Resolution of the proton band
from the neutrons and gammas is usually lost around 100 MeV. Due to
problems with the photomultiplier bases the resolution of the helium
isotopes was poor in the unstable resonance experiment. In this
experiment the gains on the phoswiches were set such that the high
energy cuts ranged from 75 to 1A0 MeV for 2:1 and from 135 to 250 MeV far
2:2. In addition a previously unknown delay of about 25 ns in the ADCs
resulted in most of the phoswiches having higher low energy cuts than
those listed in Table II-2, for 2:1 particles the low energy cuts ranged
from 15 to 25 MeV and for 2:2 from A5 to 65 MeV.

The phoswiches were designed to fit, when stacked as shown in Figure
II-2, behind the MWPC and to be placed as far from the target as possible
so as to provide the greatest possible angular resolution. In this
experiment the phoswiches had an opening angle of 5.7° each and the

entire array had a solid angle of 165 msr.

13
Table II-2

Phoswich Low And High Energy Cuts

 

Particle Low Energy High Energy
Cut (MeV) Cut (MeV)
p 11.9 135
d 16.1 182
c 19.2 ‘ 218
3He A1.7 A78
14

He A7.A 5A1

Msu.5,_°27

 

Figure II-2 Phoswich array used in the unstable resonance experiment.

15

To improve further the angular resolution of the array a multi-wire
proportional counter (MWPC) [Ha 8A] was placed in front of the
phoswiches. The MWPC contained three wire planes, two at right angles
with respect to each other and the third at A5° with respect to the first
two planes in order to resolve possible ambiguities in events with more
than one particle. The wire spacing of this counter is 2.5 mm, but the
wires are paired together giving an effective spacing of 5 nm to provide
angular resolution of 0.7°. The relative momentum for two 60 MeV alphas
with an opening angle of 0.7° is 5 MeV/c. The MWPC was filled with a
mixture of Argon-Ethane (501/501) gas at 500 torr and had an operating

voltage of V =-3000V.

Oath
Beams of 25 and 35 MeV/nucleon 1“N were used on a target of 1.6

mg/cm2 Ag. The beam energies were determined by the cyclotron settings.
Data were taken with the MWPC at central angles of O=35°, A5°, 60° and

80° for the 35 MeV/nucleon 1A

N beam and at 0=A5° for the 25 MeV/nucleon
beam.

The electronics diagram for the unstable resonance experiment is
shown in Figure II-3. Use of the new LeCroy FERA modules (Fast Encoding
And Readout ADCs) enabled us to do some preliminary gating before writing
an event to tape [Va 85]. Using software gates, we were able to
eliminate from each event particles that stopped in the AE detector, for
which particle identification was not possible. In addition only those
events in which two or more of the detectors had valid particles were
written to tape.

The data acquisition for both experiments was controlled by a VAX-

750 connected to two MC68010 CPUs residing on a VME bus [Va 85]. The

first MC68010 acquires data from the CAMAC crate containing the ADCs and

 

_.4-way

16

 

 

 

 

 

 

Splitter

 

Figure II-3

AE

CFO

 

 

 

 

 

 

 

 

 

 

MSU-G7-026
TDC Stop
iii»: ”9'" ,
m Lwe J——— Btt Gate
All 16 __ Multi l N ADC Gates
Detectors[: L09“? ): AND F0" L_._.TDC Common Star
>
Unit Scale: L/Lsco'" Out Hit Detector
Scaler Trigger

 

 

 

Electronics diagram for the unstable resonance experiment.

17
TDCs until its 8 Kbyte buffer is full. Control of the CAMAC crate is
then passed to the second CPU while the first CPU transfers its buffer to
the VAX. Sealers are read every 10 seconds and sent to the VAX by an
LSI-11/23 which resides in a second CAMAC crate along with the scalar
modules. The LSI also controls the starting and stopping of runs.

In both experiments the beam was monitored in a shielded faraday cup
located two meters beyond the scattering chamber. The current was
integrated by a BIC Current Integrator and was recorded using a CAMAC
scaler module. In order to avoid previous problems [Ha 8A] with the
current integrator double firing, the current integrator's output was run
through a variable width gate generator with the width adjusted until the
multiple firing was no longer observed. The average intensity of the

beam was 2 particle namps.
B. Singles Measurement For The Unstable Resonance Experiment.

The singles energy spectra for the ground states of the He, Ini, and
Be isotopes studied in the unstable resonance experiment were measured in
a separate experiment using a single two element Si stack. The stack
consisted of a A00 an AB detector and a 5 mm E detector. Data were taken
at the same angles, 0:35°, A5°, 60° and 80°, that were measured in the
unstable resonance measurement. The energy range measured for each
isotope is given in Table II-3. In order to reduce the contaminatitni<3f

7

the Li cross section from decaying 8Be nuclei [BI 86] the Si stack was

positioned so as to have a solid angle of 1.5A msr. The fraction of 88c

7

that identify as Li as a function of energy is shown in Figure II-A.

18
Table II-3
Telescope Energy Ranges For Unstable Resonance

Singles Measurement

 

Particle Low Energy High Energy
Cut (MeV) Cut (MeV)

3H8 26 111

“He 29 126

6He 35 151

6L1 5a 237

7L1 59 25u

7Be 80 350

93a 92 392

10

Be 95 A11

19

 

 

 

 

 

(3.55 1 1 r 1 1 11 7’] T’T 11 1*T’1 1 I 11 1 T T {iii 1 1

:3 r ‘1
b‘ '- ‘—
0: _.
m 0.4m-

"C

Q) "1
$3 .4
and

c: 0.3 __
a) _
“U

II-i —
as ..
an 0.2 _.
H—t —
O

:3

o —i
vt—fl

*5 0.1 -1
<6 -1
L.

12:. -1

C) C) .J L 11,l 1 11 1 1 11,4 L 1144 1 1 l1 1.i 1l 1 1 11

0 50 100 150 200 250 300
8Be Energy (MeV)

Figure II-A Fraction of 6Be nuclei that identify as ’Li.

20

Even for this small solid angle the fraction of 8Be that were identified

as 7L1 is between 10 and 35% depending on the energy of the decaying 88e.

C. Large Angle Correlation Experiment.

The experimental setup for the large angle correlation experiment is
shown in Figure II-5 and the positions and solid angles of the detectors
are given in Table II-A. All thirteen telescopes used in this experiment
were fast-slow plastic phoswiches of the type used in the unstable
resonance experiment. Two pairs of telescopes were used as tag counters
and nine telescopes were used as array counters. One paIJ‘13f tag
telescopes was placed at O:A5° with ¢:180° and 90°. The second pair of
tag telescopes was placed at 0=25° with the same azimuthal angles as the
first pair. The tag telescopes at ¢:180° are known as in-plane tag
telescopes and the pair at ¢:90° are called the out-of-plane tag
telescopes. The nine array telescopes were placed at angles ranging from
O:15° to O=1SO° with ¢=0°. Copper collimators were placed in front of
all thirteen telescopes. The collimators were thick enough to stop 160
MeV protons. During the experiment the array detector at 0:35° did not
function properly and is not included in the analysis.

The electronics for the large angle correlation experiment are shown
in Figure II-6. Three types of events were taken, singles, coincidences
involving at least one tag telescope and one arnay telescope, and
coincidences involving any two telescopes. Singles were taken only
during separate singles runs. Low coincidence rates (250-1000
coincidences/second) allowed the taking of all coincidences and not just

of coincidences involving a tag telescope and an array telescope.

21

Out—of—plane
Tags

 

 

 

 

 

 

 

 

 

Figure II-5 Chamber setup for the large angle correlation experiment.

22

Table II-A

Telescope Positions And Solid Angles For

The Large Angle Correlation Experiment

 

Telescope O t Solid
(Deg.) (Deg.) Angle (msr)

In-Plane Tag 1 25 180 1.22
In-Plane Tag 2 A5 180 2.A1
Out-Of—Plane Tag 1 25 90 1.22
Out-Of-Plane Tag 2 A5 90 2.A1
Array Telescope 1 15 0 1.02
Array Telescope 2 25 0 1.22
Array Telescope 3 35 0 1.02
Array Telescope A A5 0 2.A1
Array Telescope 5 55 0 1.02
Array Telescope 6 70 0 2.A1
Array Telescope 7 85 0 2.A1
Array Telescope 8 100 0 2.A1
Array Telescope 9 150 0 2.A1

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2A

The targets used in the experiment were a 26 mg/cngraphite target,

A.O mg/cm2 Ag, and 5.5 mg/cm2 Au. Beams of A0 and 50 MeV/nucleon 12 C

were used. The A0 MeV/nucleon beam was used on all four targets while the
50 MeV/nucleon beam was used only on the graphite target. The beam
energies were determined from the cyclotron settings. The average beam
intensity on the graphite target was 0.5 particle namps, on the other

targets the average intensity was 3 particle namps.

 

Chapter III

Data Analysis And Reduction

In both experiments the data were written to tape in event by event
mode for offline analysis at a later time. In the offline analysis two
dimensional AE-E plots were used to set gates around the different
particles. These gates were used in sorting the data into singles and
coincident spectra. In this chapter the energy calibrations of the
detectors will be discussed, as well as reaction loss corrections for the
plastic scintillators and the efficiency of the MWPC used in the unstable

resonance experiment.
A. Phoswich Energy Calibrations.
1. Unstable Resonance Experiment.

The phoswich telescopes used in the unstable resonance experiment
were calibrated using beams of 25 MeV/nucleon deuterons and alphas on

targets of CD and Ag. The phoswich array was placed at central angles

2
hanging from O:0° to 75°. Calibration points were obtained from elastic
and inelastic scattering.

As part of the calibrations, corrections had to be made for two
different problems. The first correction was required because the AE
gate unavoidably samples part of the E signal causing the spectra to curl
up at large energy as shown in Figure III-1. In this experiment the
problem was compounded by a then unknown delay in the FERA gates

mentioned in Chapter II. To correct for these problems the AE value used

to calculate the energy of a particle was taken to be the difference

25

26

 

A

O

00-!

4.3

U)

as

II-O

CL

4-)

{G .

9— He

a We
0" d
.- - P

 

 

 

 

E (slow plastic)

Figure III-1 AE-E spectra from the unstable resonance experiment.

27
between the ADC value and the center of the neutron and gamma line at the
same E value. This is done because the neutrons and gammas are most
likely to interact in the E detector and not in the AE detector, hence
the AE signal is zero for these events.

The second problem that had to be corrected for was a small
(iifference in tinung, and hence in the gates, between singles and
coincidence events that was discovered during the experiment. After
about 36 hours of operation the MWPC failed. In order to calibrate the
data taken up to that point a full set of calibration data was taken at
that time. Upon playing back these data it was found that the
calibration data which were taken in singles mode had a small shift in
timing compared to the coincidence data. To allow us to correct for this
timing problem we took two sets of deuteron calibration data at the end
of the experiment. The first set was taken in the same singles mode as
the first deuteron calibration. The second set was taken in1a simulated
coincidence mode by putting two wires for each telescope onto the
multiplicity logic unit and requiring a two fold coincidence. By
comparing the two sets of deuteron calibrations a multiplicative factor
was obtained that related the two settings and was used to correct the
first set of calibrations which was taken only in a singles mode.

An example of the phoswich calibrations is shown in Figure III-2.
The calibration points were found to be linear. The deuteron calibration
shown is from the second calibration and was taken in the simulated
coincidence mode. The alpha calibration is for the same phoswich but is
from the first set of calibrations. The larger positive intercepts are
due to the timing problem involving the ADC gates that was discussed in

Chapter II.

28

100........

 

 

 

 

 

 

  
   

     

E AE calibre on j
75 :' ‘2
I- d
I 3
50 t.- 2
F I
25 E- '3
_. E 3
g l 1 1 1 1
:1 O 5 10
(U
3 500 b I I I 7 I I I I I r I I r-‘r' tf' I I I trIr' I I l rfiTT‘
0 ; E calibration ;; E calibration .
400 :- for deuteron :: for alphas j
- J. i
300: t :
200: : €
I I: 1
100: I :
I- 4- -1
0.11 InilliirlllL-ilin llLllLllJllLle

 

 

 

 

O 20 4O 60 BO 25 5O 75 100
Energy (MeV)

Figure III-2 Example of a phoswich calibration for the unstable
resonance experiment.

29

2. Large Angle Correlation Experiment.

In the large angle correlation experiment the phoswich telescopes
were calibrated using beams of 53 MeV/nucleon deuterons and alphas
incident on varying thickness aluminum degraders. The degraders had a
small hole in the center to allow part of’the undegraded beam to pass
through in addition to four different thicknesses to provide.a series CM?
calibration points ranging from approximately 1/5 of the total beam
energy up to the full beam energy in equal steps. Using the degraders
enabled us to get calibration points that spanned a wide range of
energies. As mentioned in the previous section a correction had to be
made to the AEs because the fast AE gate samples part of the slow E
signal.

Examples of the energy calibrations are shown in Figure III-3 for
one of the phoswiches. The calibrations were found to be linear with
intercepts very close to zero. Because the gains on the phoswiches
varied, fewer than the five points the degraders were designed to provide
were obtained for all counters. For those that had only one or two
points the calibrations were forced to go through zero. The punch-in
energies of the light particles were then checked and small adjustments
(1-A MeV) made in the intercepts were made to force agreement with range-

3 A

energy calculations for the punch-in energies for p, d, t, He and He.

In the large angle correlation experiment the phoswich gains were
set such that we were able to observe fragments with 2:1-6 in the forward
detectors. It is well known that the light output of plastic

scintillators is nonlinear as a function of 2 [Ba 67, Ne 61, Bu 76, Be

76]. We obtained approximate calibrations for particles with 2:3-6 by

3O

 

ITTrr'Yl'TTjIr

AE calibration t

200
150
100

50

 

 

IU‘U'IUIUIIIIIIII‘UF

1111111111111111111

 

 

     

'8 1 1 1 1 I 1 1 1 1 l 1 1 1 1 l 1 1 1 1
g o 5 10 15 20
£0
a 1 000 p Y I I I U fi I l f r ‘- I I I V r I Y 1 I I T I' I U I T r 1
0 - E calibration ~~ E calibration
. .. 1
800 :for deuterons xfor alphas :
r- w- -1
.- qr- .1
600: t 1
1- -u- q
400C i 1
1- dr- .1
2001 A E
O 1 1 11 l 1 1 1 l 1 1 1 1 1 1 1 1 l 1 1 1 1 1 L1 1 1 l 1 1 1 ‘

 

 

 

 

O 40 80 120 50 100 150 200
Energy (MeV)

Figure III-3 Example of a phoswich calibration for the large angle
correlation experiment.

31
using the AE punch-through points and comparing the measured AE signal to
the expected AE value assuming that the deuteron calibrations could be
applied linearly to higher 23. The calibration was assumed to be of the

form

E : ma c-b (III-1)

where E is the energy deposited in the detector, 111 is the slope of the

calibration for deuterons, c is the ADC value, and b is an intercept

Z

chosen to provide the proper low energy out for the given element. a2 is

a correction factor that accounts for the nonlinearity in response of the

plastic scintillator and is defined as

 

(III-2)

where Cz and Cd are the punch through channels for a particle of charge

2, and for a deuteron. Ez and Ed are the punch through energies for a

particle of charge 2, and for a deuteron. The values for a2 were

determined for every telescope for D1, when a given 2 was seen in more
than one telescope the value used in the calibration was the average of

all telescopes. The values for (12 and b2 are given in Table III-1.

As a check on the accuracy of this method ”He spectra were
generated using both this method and the a calibration obtained from the

calibration beam. A comparison of the two spectra is shown in Figure

32

 

Table III-1
Values of a2 and b2 for 2:2-6

Particle oz bZ
(MeV)

”He .659 10
Lithium .502 20
713a .1166 30
Boron .372 A5

Carbon .33A 65

33
III-A. The spectra generated by the a calibration are shown as lines
while the data points are for the spectra generated by the calibration
obtained from the punch through points. As is seen in Figure III-A, the
two calibrations yield almost identical spectra. Another indication that
the punch—through calibration is reasonably good is that moving source
fits, which are shown in Chapter IV, to the Li and Be spectra yielded
source velocities similar to those of the light particles. The excellent
agreement between the two calibrations means that in future experiments
it will not be necessary to calibrate the phoswiches separately for
deuterons and alphas as has been done in the past. Instead the
calibration need be done for one element and then extended to all other

observed elements using the method outlined above.

B. Silicon Detector Calibration.

The silicon detectors used in the singles measurement fkn~ the
unstable resonance experiment were calibrated using calibrated pulsers.
The calibrations obtained using the pulsers were checked by comparing the
punch-in and punch—through energies for the observed isotopes given by
the calibrations to those calculated using the code DONNA [Me 81]. The

silicon detector calibrations are shown in Figure III-5.

C. Reaction Loss Corrections.

Nucleons traveling through plastic scintillators lose energy not
only through interactions with the atomic electrons, but also through
nuclear reactions in the scintillators. The interaction with the atomic

electrons produces the full energy peak in the scintillator. The nuclear

3A

 

 

 

 

 

 

 

2 4O MeV/nucleon C+C
11(3 1 1 1 1 1 1 1 1 1’ 1 1 1 1 1’1 1 1 1 1 {’1 1 1 ‘r
1
.——.lo
’2.‘
a:
=> 0
EEQIICD
‘\\
A: -—1
310
C: 135°
1:3 —2
.g 10
\ 1
° 9
“2:, -3
10 55° 4“ b— +‘
1 145°
10“ L1] 111 111 [L 111 111 1L1 111
0 50 100 150 200 250

Energy (MeV)

Figure III-A Comparison of the two different a calibrations.

35

 

 

 

 

 

4096 L. I r I I I I I I ‘1 I I I l I I I l

{E calibration :AE calibration j
_. 3072 - -1- a
CD I I I
g I- d). d
CU 2048 : :- j
.c.‘ r + .
o 1024 _- j: j

i I j

0 1 1 1 1 L 1 1 1 1 1 1 1 l 1 1 1 l1 1
O 250 500 40 80 120

Energy (MeV)

Figure III-5 Silicon detector calibration.

36
reactions, however, produce reaction products such as neutrons, gammas
and alphas, which, because of the nonlinearity of the scintillator,
produces less light in the scintillator than the original particle would
have produced. The reaction leads to the loss of the particle from the
full energy peak, thus making particle identification impossible using
AE-E plots.

We have used the calculations of Hasselquist [Ha 8A] for the
reactnmilosszniplastic scintillator. The calculation is patterned
after that of’Measday and Richard-Serre [Me 69]. In the calculation the
range of the particle is divided up into small steps and the reaction
cross section for each step is calculated. The fraction of particles

lost due to interactions is then given by

i i
f:1-e (III-3)

where 111 is the number of atoms/cm‘2 in the ith step. The calculated

reaction cross section, 01 is given by

oianR2(1-Vc/E)(1+(K/E)A) (111-u)
where R is the interaction radius given by

R:rO(A11/3 + A 1/3-1) (III-5)

2

with r0:1.2 fm. The coulomb potential, Vc, is given by

37

2
V0-2122e /R. (III-6)

N, x and A are adjustable parameters determined by fitting to data given
in Measday [Me 69]. The energy spectra are corrected by dividing by 1-f
which is the fraction of particles that do not interact in the
scintillator. For 100 MeV protons 101 of the incident protons are lost

due to nuclear interactions in the scintillators.

D. MWPC Efficiency.

As mentioned in Chapter II a multi-wire proportional counter (MWPC)
was used to obtain more precise position resolution in the unstable
resonance experiment. In order to measure the populations of particle
unstable states the efficiency of the MWPC must be calculated. In this
section the procedure used to calculate the overall MWPC efficiency will
be outlined and all the factors that influence the efficiency will be
discussed.

The MWPC efficiency was calculated using a Monte Carlo simulation.
For each trial the parent nucleus is constrained to hit somewhere on the
phoswich array with its energy and angle determined using the energy and
angular distributions for the ground state of the same isotope, or the
closest stable isotope for those cases where the ground state is unbound.
The nucleus is then allowed to decay in it's rest frame, with the two
decay products having a relative momentum tip. The decay products are
then transformed into the lab system with a random rotation about the
trajectory of the parent nucleus. The position in the phoswich array

that each decay product hits, if it hits the array, is then found. If

38

both particles hit the array, and if they hit two different phoswiches,
then their energies are compared to the individual energy cuts of the
phoswiches they hit. Next the wire efficiency as a function of energy
for the particles involved is checked. Finally a check is made for the
possibility that the particles scatter out of the scintillator or are
lost due to a nuclear reaction in the scintillator. The fraction of
trials for a given Ap in which both particles survive all of these checks
is the efficiency for detection at that Ap.

The wires of the MWPC have efficiencies that vary as a function of
the particle's energy, mass, and charge. In Figure III-6 the efficiency
of the MWPC wires for particles that were detected in one of the
phoswiches behind the MWPC is shown. The lines are simple linear fits to
the curves using one or two intersecting lines. For all wires the
efficiency for the helium isotopes shows no variation as a function of
energy. The efficiency for the hydrogen isotopes has a strong variation
as a function of energy and mass. The efficiency of the wires was
determined for each MWPC angle setting and was incorporated into the
efficiency calculation using linear fits like the ones shown in Figure
III-6.

A charged particle passing through a medium will experience many
small elastic scatterings which might scatter the particle out of the
detector. In the unstable resonance experiment the phoswiches were
uncollimated, and hence it was possible for particles to be close enough
to the edge of the phoswich to be scattered out of the detector. This
correction can be especially important for those particle pairs with very
small momentum differences Ap, because their small opening angles greatly
increase the probability that one or both of the particles will hit near

the edge of a phoswich. The scattering angle due to multiple scattering

39

35 MeV/ nucleon N+Ag

 

   

 

 

 

 

 

 

 

 

 

 

1.O._1 1 1 1 1 1 1 1 1 M1++¥ g h 1—

_ +""*"TiITTTT+11I 1W—

._ H; d

0.8 — _

_ t _

30.6 1— d —
.23. : :
0° ._ —1
S 0.4 _ p _
0.2 — I
().C) — 1 1 1 1 I L 1 1 1 I 1 1 1 1 I 1 1 1 1 fl
0 50 100 150 200

Energy (MeV)

Figure III-6 MWPC wire efficiency as a function of mass, charge and
energy for the wire in front of one phoswich.

A0

is assumed to have a Gaussian distribution peaked about 00 given by [Pa

8A]

0 a ——°—z.nc/E7ER [1+l log,o(L/LR)] [1+ ] (radians). (III-7)

___AE1__.
0 p8 1 9 (t+mo)As

where p, B and Zinc are the momentum, velocity and charge of the incident

particle and L/L is the thickness in radiation lengths of the scattering

R
medium. The scatter out calculation was incorporated into the efficiency
calculation.

The reaction loss correction discussed in the previous section of
this chapter was also used in calculating the efficiency.

The calculated efficiency for the detection of coincident p-t, d-a
and a-a pairs are shown in the top part of Figure III-7 for the case when
the MWPC array was at A5°. In the bottom part of Figure III-7 the
average opening angles as a function of Ap are shown. The efficiency
peaks when the average opening angle is about the same as the opening
angle of the phoswiches. For smaller values of Ap the chief loss of
efficiency comes from both particles hitting the same phoswich. At
larger values of Ap the chance of one of the particles hitting outside

the array increases, causing the efficiency to decrease.

A1

 

Y Y r T r I l T I F Y r r I I T T r V Y I U

Efficiency

11111111111111

(.0
O

N
O
111111111111111111‘

Angle (deg)

1.;
O

111111111111111‘111

 

 

 

 

O 11111111[1111111111111111111

0 50 100 150 200 250 300
Ap (MeV/c)

Figure III-7 Calculated efficiency for MWPC and phoswich array for p-t,
d-a and a-a particle pairs for OwaczA5°.

Chapter IV

Large Angle Correlation Experiment Data And Discussion.

In thiscflmpter the results of the large angle correlation
experiment are presented and discussed. The inclusive energy spectra
will be discussed first, and then the two-particle correlations will be

presented.

A. Inclusive Energy Spectra

1. Light Particle Spectra

3He, and ”He for A0

The inclusive energy spectra for p, d, t,
MeV/nucleon C+C, Ag, Au and 50 MeV/nucleon C+C are shown in Figures IV-1-
A, for 0:15°, 25°, A5°, 55°, 70°, 85°, 100°, and 150°. The spectra have
been corrected for reaction losses in the phoswiches, and the errors
shown are statistical. The spectra are smooth, decreasing monotonically
with increasing angle and decaying exponentially at high energies,
suggesting a thermal origin for the observed particles. The spectra show
signs of emission from a fast projectile-like source at the forward

angles, Oz15° and 25°. At low energies there is evidence of emission

from a slow target-like source.
2. Moving Source Parameterization.

We have fitted the data with a triple moving source
parameterization, which assumes the presence of a projectile-like source,

a target-like source, and an intermediate velocity source. 1UJ.three

112

A3

 

C+C 4O MeV/nucl.

 

1 1111.11 11 . 111 1111111 i\t1Lk111'1&111

 

1
.1
4
.4
1
1
.1
1

C)
U
:3
m
.0-
II
(D

I

d‘2 a/dEdQ [mb/(MeV sr)]
ES
8
81
S
8
8’1
"3
O

(1' O: H
/ , I
.uuflnlmdm 11111111 111111 111111

 

o 50 100 150 200 0 so 100 150 200
Energy (MeV)

Figure IV-1 Light particle energy spectra for A0 MeV/nuc. C+C. The
lines are the results of moving source fits.

AA

C+C 50 MeV/nucl.

  
  
 
 
  
  

 

1

10
0

10

-l

10
10 _
10 -

-4

10

2

3

 

 

 

 

d‘2 U/dEdO [rub/(MeV sr)]

 

 

 

o 50 100 150 200 o 50 100 150 200
Energy (MeV)

Figure IV-2 Light particle energy spectra for 50 MeV/nuc. C+C. The
lines are the results of moving source fits.

A5

C+Ag 4O MeV/nucl.

 

  

 

 

 

VIIIIIIIIYY I YIr IIIIVITYI'IUVIYI 1
l ' t i
10 g
o 5
10 = .. 1
-1- O 0‘
10 1 .. 1
-2:
10 I 0" ”O
-35
10 g T
.4':'
1111111

 

O 25 50 75100 O 25 50 75100 O 25 50 75100

 

d2 a/dEdQ [Inb/(MeV sr)]

 

 

o 50 100 150 200 o 50 100 150 200
Energy (MeV)

Figure IV-3 Light particle energy spectra for A0 MeV/nuc. C+Ag. The
lines are the results of moving source fits.

U6

C+Au 4O MeV/nucl.
IIITI'YTYIYTTY'VYVYIT YTTTIYTVYIVT'YT|.T_'3

 

 

 

 

 

d2 o/dEdQ [mb/(MeV sr)]

 

 

o 50 100 150 200 o 50 100 150 200
Energy (MeV)

Figure IV-H Light particle energy spectra for 40 MeV/nuc. C+Au. The
lines are the results of moving source fits.

M7
sources are assumed to move in the beam direction. While it is true that
at 40-50 MeV/nucleon the three sources are not well separated, the energy
spectra show definite signs of each of the three sources. The energy
spectra for each source are assumed to be described by a relativistic
Maxwell-Boltzmann energy distribution. In the rest frame of each SMMJPCG

the distribution is given by

d2o o e-E'/1
= 0 (IV—1)

p'2dp'd0' unm3 2(I/m)2K1(m/I)+(I/m)KO(m/I)

where p' and E' are the momentum and total energy of a particle in the
source rest frame, and K0 and K1 are modified Bessel functions of the
second kind, and 00 and T are the production cross section and source

temperature for the given source. The energy spectra in the lab are then

 

given by
9:)— : pE'dZO
dEdQ p'2dp'd0' (IV-2)
where
E' : Y(E-chosOlab), (IV-3)

p and E are the momentum and total energy of a particle in the lab frame,
Olab is the lab angle, and B is the velocity of the source in the lab.
For each source 8, 00, and T are fit using those angles and energies that

are dominated by the source being fit. The angles and energy ranges fit

48
for each source are listed in Table IV-1. Also listed in Table IV-1 are
the Coulomb shifts that were applied to the Ag and Au data prior to
fitting. The fits are shown in Figures IV-l-u as lines and the extracted
parameters are listed in Table IV-2. Typical uncertainties are 5-20% for

i0.002-0.007 for B and t0.2-O.5 MeV for T. The target-like source
3

00’
was not fitted for the He spectra for the Ag and Au targets. Due to the
lack of helium spectra backward of Oz85° the target-like source for the
helium isotopes had to be fit using the low energy part of the spectra at
more forward angles. In addition the fit for the intermediate source
included the 45° spectra for the helium isotopes. The extracted
parameters for the helium isotopes have higher velocities and
temperatures for the target-like and intermediate sources than do the
hydrogen isotopes. For the projectile-like source the helium isotopes
have the same velocities as the hydrogen isotopes, but higher
temperatures. These differences between the extracted parameters might
be due to the different angular and energy ranges used in the fits.

The fits are excellent except for some differences at forward angles
for the carbon target. These differences may be caused by the small size

of the sources for the C+C system. The intermediate source size with the

largest weight (A2nbdb) is twelve nucleons.

3. Heavy Fragment Spectra

The gains on the most forward phoswiches, O:1‘3°, 25° and 115°, were
set such that particles between Li and C were also seen. The spectra for
these particles are shown in Figures IV—5,6. Both the Li arm! 7Be
decrease monotonically with increasing angle. The lines are the results

of a single moving source fit to the Li and 7Be spectra, the extracted

A9

Table IV-1

Energy and Angle Ranges Used for Moving Source Fits and the Coulomb

 

Shifts for Ag and Au Data

 

 

Particle Target Source
Ag Au Target-like Intermediate Projectile-like
V0 V0 0 E O E O E
(MeV) (MeV) (MeV) (MeV) (MeV)
p A 5 loo-150° 10-50 55-85° HO-125 15-25° 40-125
d A 5 100-150° 10-50 55-85° 50-125 15-25° uo-150
A S 85—150° 10-50 55—85° 50- 75 15-25° 50-100
3He 5 1o 45- 70° uo—ao u5-70° 80-200 15-25° 80-250
"He 5 10 us- 85° A0-6O u5-55° 100-200 15—25° 80-250

50

 

 

 

 

Table IV—2
Moving Source Parameters
Target like source Intermediate source Projectile like source
Cross Velocity Temp. Cross Velocity Temp. Cross Velocity Temp.
Section Section Section
Particle o B I o B I o B I
(m8) (MeV) (m8) (MeV) (m8) (MeV)
A0 MeV/nucleon C+C
p 620 0.016 A.6 A72 0.1A6 11.A 311 0.259 A.9
d A70 0.027 A.3 138 0.12A 11.1 215 0.23A 5.5
5 A60 0.037 3.8 56 0.12A 10.3 7A 0.201 5.7
”He 280 0.06A 5.5 62 0.152 11.8 95 0.251 9.6
He 970 0.071 5.3 260 0.1A9 9.7 757 0.2A0 7.0
50 MeV/nucleon C+C
p 600 0.022 5.3 A60 0.17A 1A.0 361 0.288 5.2
d 380 0.026 A.8 158 0.1A5 13.0 239 0.267 6.6
g 205 0.0A7 5.5 99 0.153 8.8 51 0.237 5.5
”He 350 0.053 6.2 60 0.18A 12.7 90 0.291 9.9
He A70 0.080 6.8 111 0.165 11.7 652 0.271 7.9
A0 MeV/nucleon C+Ag
p 8200 0.015 3.5 1910 0.129 11.9 760 0.257 6.1
d 1800 0.019 A.2 920 0.123 11.9 A11 0.251 7.8
5 800 0.020 A.2 A20 0.113 11.7 157 0.210 6.2
“He 2A1 0.179 15.A 66 0.278 11.A
He 16A0 0.063 7.0 808 0.163 16.0 737 0.255 7.0
A0 MeV/nucleon C+Au
p 8700 0.012 3.6 2A50 0.123 12.2 670 0.263 6.7
d 2000 0.015 3.8 1020 0.118 12.2 A1A 0.256 7.5
g 1100 0.031 A.3 A80 0.115 13.3 15A 0.208 5.3
”He 162 0.168 17.1 88 0.267 8.7
He 1250 0.069 7.7 569 0.15A 18.A 866 0.253 6.6

51

C+C 40 MeV/ nucleon

 

 

 

 

 

F_W f
A '1

Lo :

m : 1
=2 . .
v ‘
'8 -. .1
H10 !- . 1
55‘ . i 1
Eg 1° F 1
\\~ 3
e: a 1
“3o 10 .

O 100 200 300 400 O 100 200 300 400 O 100 200 300 400
Energy (MeV)

r-H '

’1? C+C so MeV/nucleon

m

>

O

3

<10 '4

A:

8

H

-I
c: 10
5
-s

>10

N O 100 200 300 400 O 100 200 300 ‘00 O 100 200 300 4100
1:

Energy (MeV)

Figure IV-S _ Heavy fragment energy spectra for A0 MeV/nuc. C+C (top) and
50 MeV/nuc. C+C (bottom). The lines are the results of
moving source fits.

)1

ya
C)

pa
(3

pa

dz a/dEdfl [rub/(MeV 31'
o .

pa
(3

pa
C)

p;

d2 a/dEdO [Ebb/(116V 311]

pa
C)

I
p

I1
V

 

b
-3
1

52

C+Ag 40 MeV/ nucleon

7Be

 

111111111ln.1111111 111111111111111111111111

a zoo 200 300 400 o 100 zoo 300 400 o 100 zoo aoo coo
Energy (MeV)

C+Au 4O MeV/nucleon

I
u

C)

Figure IV-6

 

 

100 200 300 coo o 100 zoo 300 400 o 100 zoo 300 400
Energy (MeV)

Heavy fragment energy spectra for A0 MeV/nuc. C+Ag (top)
and C+Au (bottom). The lines are the results of moving
source fits.

53
source parameters are listed in Table IV-3 along with the Coulomb :HIifts
that were applied prior to fitting the data. The fits were carried out
for E>200 MeV. The fits indicate that the particles are coming ownn the
projectile-like source. The extracted parameters for Li and 7Be are
similar to those of the projectile-like source for the helium isotopes

9,10

listed in Table IV-2. The Be, B and C spectra are shown only for

15°. The 9’mBe, B and C spectra are peaked at energies near the beam

energy indicating that they are coming from the projectile.
B. Two-Particle In-Plane Correlations.
1. Two-Particle Correlations, A5° Tag

The two-particle correlations are shown in Figures IV-7-11 for p-p,
p-d, d-d, d-uHe, and “He-“He. The correlations are shown in terms of the
two-particle correlation cross section, 012, divided by the singles cross
sections, 01 and 02. In each case one of the particles is detected at
0=-A5° and the second particle is detected between 0=-170° and +170°. 11
negative angle for the second particle means the two particles were
observed on the same side of the beam, while a positive angle indicates
emission on opposite sides of the beam. For the nonidentical particle
cases (p-d, d-uHe), the case for which the heavier particle was detected
at 0=-A5° is shown. The correlations have been integrated over the
energy ranges given in Table IV-A.

In general the correlations for the C+C systems show a broad maximum
at positive angles which indicates a preference for emission of particles

to opposite sides of the beahL The magnitude of the peak increases as

the mass of the observed particles increase and is slightly larger at 50

5A

 

Table IV—3
Moving Source Parameters For Li and 7Be
Particle V Cross Velocity Temperature
Section
00 B I
(MeV) (mb) (MeV)
A0 MeV/nucleon C+C
Li 0 75:2 0.2A910.001 8.10:0.12
7Be 0 25:1 o.2uu:o.002 8.18:0.1u
50 MeV/nucleon C+C
Li 0 5916 0.28310.001 10.A5:0.16
7Be 0 23:1 0.280:0.002 8.81:0.19
A0 MeV/nucleon C+Ag
Li 12 82:2 0.2AA:0.002 12.86:0.20
7Be 16 19:1 o.229:o.oo1 12.20:o.21
A0 MeV/nucleon C+Au
Li 15 80:2 0.2A310.002 10.55:0.15
7Be 20 15:1 o.238:o.oo3 9.85:0.29

55

91 ="45° PP

 

      

 

  

    

 

 

 

 

1.8 I 1 I l r I I I I .1 I 1' I I I I I I I I I I 1 I 1
I , q- .
1 5 ' 5 '1 " "
' C " -- I
I III-
2 I- . .' .—
1 1- : 41- +
I- ~0- q
A 0.9 - x -_ _
T P .1- 1, ‘1 I, x"
e 0'6? , ~ ‘1
. ’1 \“‘- "I '1
0.3 -~ A
N -40 MeV/nucl. C+C .. 50 MeV/nucl. C+C «
b P- 1 L 1 I 1 1 1 l 1 1 1 l 1 1 1 +- 1 1 1 J 1 g 1 1 1 1 L L 1 1 1
00 I I 1 I I I T fififi ] f I I I I I 1 r I I l I I I T I I I
d .- ql- «
Q - 41' 1
:- fib -1
N O 6 "' . 41- _1
6' ' " q
I . I .
I
004' - ———————— \\ .’I’ .1,-
.— \ .’ qr “ .I
1- ‘\\. l,’ 4-. ......... “\. .9, ‘ ‘ - - .4
p- \‘I «1- -4
0.2 - T Q” a
P _ fi
.40 MeV/nucl. C+Ag «- 40 MeV/nucl. C+Au 1
O O P 1 1 1 l 1 1 1 1 L 1 1 L 1 1 1 “ 1 1 1 L 1 L 141 1 11 1 1 1

-180 -90 O 90 180 -90 0 90 180
@2 (degrees)

Figure IV-7 ‘Two-proton correlation function for which one proton is
detected at ez-A5°. The lines are described in the text.

56

__ __ o
(9, — 45 pd
2.8 I I I l I I I I I I I I I I I I I I [TTI I I I7 I I I I

:40 MeV/nucl. C+C 50 MeV/nucl. C+C

 

            

 

 
 
 

 

 

 

P

I: .
1- " dr- .1
I \ qI- -
2.0 E ,' ‘\ :1: .2.
I- -11- '1
D "b d
1.6 r —- _
D d d
-‘ L2: :5 .
I . q. ,4
1- db H
a O 8 :+ .. <5" " I :1
1- “1- ‘\ ,” ‘\ ’I 4
'>\ "P ‘-" '1
cu 0'4 1’ 1: 1
- ‘P d
n «I .4

COOIIHIHHHHH'HHHHHHH.
‘ 1- 11:: -<
b , ._ I
\ r- 4- 1
—1 —<
93(3.€5;: . 1
b _ _ 1
D ‘4
0.4 - a
- -1
1- ---‘ \- I <
1- \~’ .- .4
()Ja — _ _
-4O MeV/nucl. C+Ag «~40 MeV/nucl. C+Au 1

b db

d d

O O 1 1 1 l 1 1 1 l 1 1 1 l L 1 1 1 1 1 l 1 1 1 l 1 L 1 1 L4 1

 

—180 —90 O 90 180 —90 O 90 180

@2 (degrees)

Figure IV-8 Proton-deuteron correlation function for which the deuteron

is detected at 0=-A5°. The lines are described in the
text.

57

     
  

 

   
 

 

 

2-8 P ' r ' T ' I ' fry I I I I I If I I fiI T r 1
2 4 E40 MeV/nucl. _~
E C+C ,’ ‘ :
2313 : '+ ~\ 2
1.6} 3
A : I
14 1.23 : ,\_
:3 0 8 : *1
0.4 : 1
N - ~~~~~~ :
b0.0”11%%11+111%1%‘111111111h11h1
d b db -l
b . .1 .
\ 1- 4- 4
01(3.€51- __ I
d h db '1
b - dr- ¢ + + -<
()34 ‘ 7r ‘
. V ., .
0.2 " '1'- .1
- .1.- .1
r40 MeV/nucl. C+Ag --40 MeV/nucl. C+Au ~

 

 

 

00111111111111111 41111111111111

—180 -90 O 90 180 —90 O 90 180

 

@2 (degrees)

Figure IV-9 Two-deuteron correlation function for which one deuteron is
detected at 0=-A5°. The lines are described in the text.

58

1 @1 =_450 dd
10 j I I I I T I I I I I I I IT

~40 MeV/nucl. C+C
. .

 

TIII
\

r

 
   

    

I IIIIIII

III/‘E” C +C ‘1—
. 50 MeV/nucl.

LL11111111L11L14
IIffTIIIIIIIIf

E40 MeV/nucl. C+Au

 

11111

I IIIIII'
11411111

 
 

I
I
1

I \
I
10 111111—111111111 #1111111111L111V

 

 

 

 

-180 -90 0 90 180 -90 0 90 180

02 (degrees)

Figure IV-10 Deuteron-alpha correlation function for which the alpha i}:
detected at 0=-A5°. The lines are described in the text.

59

@1 =—45° aa

 

       
   

   
  

 

       

 

10.0’. TIIIrI[7:FTjIII-'bfiyrr‘I‘r'2,
I 40 f E I 50 3
7 5; MeV/nucl. f '1 i MeV/nucl. :
. C C+C E 1 1r C+C
L : E I
A 5.0 i 5 ‘3 .2:
—~ I u;- z I:
I r- I ‘ 4.-
3 2 5 :- ' “:7
I I
N '- r-
b 0.0 ' ‘
b“ E40 MeV/nucl'.
\ +C A + s
N r + g ;
; + - ,:
0.5 _.
.- ’- '1-
L \
00 1 1 1 L1 1 1 L1 1 1 1 1 1 1

 

 

 

—180 -90 O 90 ‘ 180 —90 O 90 180
@2 (degrees)

Figure IV-11 Alpha-alpha correlation function for which one alpha is
detected at 0=-u5°. The lines are described in the text.

60
Table IV-N

Energy Ranges For Correlations

 

Particle Energy
(MeV)
p 12- 80
16- 80

“He u7-135

61
MeV/nucleon than at “0 MeV/nucleon. For negative angles the correlations
are almost flat as a function of the angle of the second particle. The
p-p and d-uHe correlations are exceptions to this trend because they show
peaks at 0=-55°. These peaks come from the decay of particle unstable
states in 2He and 6L1. The C+Ag and C+Au systems have different
systematic behavior. The correlation function is in general almost
symetric about 0=0° with the same side being only slightly preferred.
The p-p peak coming from 2He break up in the light C+C systems is
considerably weaker in the heavier systems. Both the d-uHe and ”He-”He
correlations show strong peaks coming from the decay of particle unstable
states of 6L1 and 88e. The “He-“He peak was not observed in the lighter
systems because it is difficult to form a 8Be given that the average
intermediate velocity source contains only 12 nucleons. The particle
pairs that are not shown all exhibit the same characteristics that are

seen in Figures IV-7-11. The p-t, p-uHe, and t-uHe correlations all have

peaks at 0=-55° which comes from the decay of particle unstable states.
2. Momentum Conservation Model.

Energy and momentum conservation effects have been used to explain
the observed correlations for small systems at lower energies [Ly 82, Ts
818, Ch 86b]. In order to explore the extent to which our data is
affected by conservation laws we have carried out calculations
incorporating these effects. The calculation [Ly 82, Ha 8”] assumes
emission of two particles from a source of size A. After the first
particle is emitted the source recoils, re-equilibrates, and then emits
the second particle. The calculation is repeated with the second

particle being emitted first. The two cases are then averaged to produce

62

the final coincidence cross section. The entire calculation is
integrated over impact parameter with each impact parameter having a
weight dw=2nb0db0A where b is the impact parameter and A is the source
size which comes from a fireball model [We 76, Go 77] calculation. The
calculation is normalized by the total reaction cross section. The
parameters listed in Table IV-2 for the intermediate source were used to
describe the emitting source. For greater detail on the momentum
conservation calculation see the Appendix.

The results of the momentum conservation calculations are shown in
Figures IV-7-11 as solid lines. The calculations have been renormalized
to the data at 0=+45° and +55°. The calculations do a good Job of
reproducing the general trends in the data but miss some of the details.
In the lighter systems the same side correlations are basically flat,
except for those cases that have contributions from the decay of
resonances, while the calculation predicts a broad minimum at about
0=-80°. The heavier systems have a "V" shaped dip around 0=0°, but the
calculations show no such dip.

Until now we have assumed that the observed two-particle correlations
are coming only from the intermediate source although from the inclusive
spectra shown in Figures IV-1-A it is evident that a substantial fraction
of the observed particles come from sources other than the intermediate
source. In Figure IV-12a the contribution to the deuteron singles cross
section from each of the three observed sources is shown for C+Au system.
For small angles the projectile source contributes heavily to the singles
cross section while for large angles the target like source dominates the
cross section. Even for small angles for which one would expect the
projectile-like source to dominate, the target-like source contributes

about ten percent of the total cross section. Figure IV-12b shows the

63

 

 

 

3
31° 3.) hb'Mévyifii'ci.
\ 2 '
"010
E
C:: 1
.610
\ ‘
b o .
"010 L1111L‘1411111L1L1L»

o 60 120 180'
0 (degrees)

5 40 MeV/nucl. C+Au-*dd

 

-180 —90 o 90 180 -90 o 90 180
@2 (degrees)

Figure IV-12 Contribution from the three sources to the deuteron singles
cross section (a) and to the two deuteron coincidence cross

° -_ o =_ o
Segfiion for Gtag' A5 (b) and for Gtag 25 for no MeV/nuc.

614

contribution to the two-deuteron coincidence cross section from each of
the three sources according to momentum conservation calculations using
the parameters for each of the sources given in Table IV-2. From Figure
IV-12b we can see that adding the contributions of the three sources
together will increase the coincidence cross section around 0:0° by less
than 10% while in Figure IV-12a we see that the addition of the
projectile and target-like sources more than doubles the singles cross
sections near 0:0°. This summation will lead to the dip around 0° which
is observed in the data.

For the lighter systems the effect of the other two sources is
somewhat different than for the heavier systems. Figure IV-13a shows the
contribution to the deuteron singles cross section for each of the three
sources for '40 MeV/nucleon C+C. At forward angles the projectile-like
source is far more dominant than it is for C+Au. The projectile-like
source is also the leading contributor to the two-deuteron coincidence
cross section at the most forward angles as seen in Figure IV-13b for
correlations with 91:45". The target-like source contribution to the

two-deuteron cross section is peaked at about 0 =+50°.

2

The dashed lines in Figures IV-7-11 are the results of three-source
momentum conservation calculations. These calculations take into account
only correlations between two particles coming from the same source.
Unlike the single source calculation discussed earlier no normalization
has been applied to the three source calculation. The calculation now
produces a "V" shape for small 0 for the heavier systems. Also where the
single source calculation had a broad minimum for same side correlations,
the three source calculation shows a maximum at about 0=-55°. This

maximum is almost as large as the opposite side maximum for the heavier

systems. The most notable disagreement between the data and the three

65

 

 

3
351° ' 3.1; ‘4'0' newaha
\\ . C+C+d
JD
8
v
C.‘
"U
\\\
b
U 1~1~¢_1_

 

180
0 (degrees)

40 MeV/nucl. C+C-Kid

 

—180 -90 o 90 180 -90 o 90 180
@2 (degrees)

Figure IV-13 Contribution from the three sources to the deuteron singles
cross section (a) and to the two deuteron coincidence cross

section for 0 :-45° (b) and for 0 =-25° for NO MeV/nuc.
C+C. tag tag

66
source calculation is the under prediction of the coincident cross
section for large 0. This disagreement is probably due to correlations
between in which one particle comes from the intermediate velocity source
sun: the other particle comes from the target-like source. Correlations
involving particles from different sources are not included in the three-

source momentum conservation calculation.
3. Two-Particle Correlations, 25° Tag

The two-particle correlations triggered on a light particle at 0:25°
are shown in Figures IV-1H-18 for p-p, p-d, d-d, d-uHe and “He-“He. The
correlations have the same general features as the correlations triggered
on the A5° tag telescopes. The light C+C systems show a strong
enhancement of the opposite side, while the heavier C+Ag and C+Au systems
are synmetric about 0=0°. Strong peaks are seen in the heavy systems on

the same side of the beam for the d-uHe and “He-”He correlations coming

from the decay of particle unstable states in 6L1 and 8Be. There is also
some evidence in all four systems of a enhancement at 0=-15° in the p-p
correlations and in the C+C systems for the d-uHe correlations.

“The lines in Figures.IV-1H-18 are the results of three source
momentum conservation calculations. The calculations do not reproduce
the data for 01z-25° as well as they do for 01z-AS°. Most noticeably the
calculations predict a strong peak at small positive angles for p-p
correlations and for d-d correlations in the C+C systems and for p-p
correlations in the heavy systems. The general trends are, however,
reproduced in most cases with the best agreement is for d-d correlations

in the heavy systems and for “He-“He correlations in the C+C systems.

67

(’31 =-25° PP

105 TTI'IUI'YUrIIrI TITrYrrIfrrIVYr

 

1111111111

IIIIIIIIII

   

       

111111111L1411
III

TfII'IjIIIIIII

1D

 

L 11
Y—UIII

 

 

 

 

 

0.1 - 4O MeV/nuc C+Ag #- 4O MeV/nuc C+Au ‘
"’ Jr- .4
0.0 L #4 L ‘ l Li L 1 l l L 1 1 1 L L l 1 1 1_1 1 1 1 L 1 1 L

-180 -90 O 90 180 —90 O 90 180

92 (degrees)

Figure IV-Hl Two-proton correlation function for which one proton is
detected at 0=-25°. The lines are described in the text.

68

@1 =_25° pd.

 

 
 

 
 

    

 

 

 

 

 

 

2.4 P fit Y I if T T r ‘r r I f r T d fi'r F I V r T [V r f 7" V V J
I I» I
:- ~11- .4
h 1h 1
C' 2'? 1
:- co- 4
1- un- ..
I- d1- 4
u- q- q
I— q— .1
r- d:- d
b db 1
1- 4- -4
D '1- d
1- db ..
P .. 1
D d. d
1- 1. d
N p- m- d
D d

500 +:::;:IHH:;:P
d :- q- -1
£ 0 5 r- -r- -‘
1- GL -(
b :- db . .5 . d
I— —:

r 1
)- -1- d
)— —1— .1

E T
0.1 - 40 MeV/nuc C+Ag 4- 40 MeV/nuc C+Au d
p I- d

O O L 1 1 1 1 1 1 1 1 1 1 1 1_1 1 1 1 1 1 1 L 1 1 4 1 L L; 1 1
O

-180 —90 0 ‘90 180 -90 0 90 180
@2 (degrees)
Figure IV-15 Proton-deuteron correlation function for which the deuteron

is detected at 0=-25°. The lines are described in the
text.

69

@1 =—25° dd

 

   
 

  

 

2.8 I I I I I I I I I r I I I I I ., r I r I I I I I I I I I I I

2.4:»- 40 MeV/nuc C+C 5:- 50 MeV/nuc C+C 3

2.0 E. g 3+ *# 1

. I: " ‘

L6} 1} + I 5

:‘szg g; + I +5

in o a _-+ l":-'. 55- " ' :

‘~—/ . Ii 2: 1

0.4 1 " E

N : x :

zoo driH}§#H%#H"}%%::H::H%1”

b 1
\

N _
v-i
b

o 4’0 MeV/nuc C+A3 '- 40 MeV./nuc C+Au -

.0 1 1 1 1 1 1 I 1 1 1 I 1 1 1 1 1 1 l 1 1 1 l 1 1 1 l 1 1 1

 

 

 

 

-180 -90 O 90 180 -90 O 90 180
02 (degrees)

Figure IV-16 Two-deuteron correlation function for which one deuteron is
detected at 0=-25°. The lines are described in the text.

70

@1 =_25O da

 

  

      

 

 

 

 

5.0 I I T I I I I I I 1 T I I I I -. I I I I I I I I I I I I I I I J
I 2‘: 50 MeV/nuc 1
I- .1. fi
4.0 _- 40 MeV/nuc C+C :- C+C + j
I- qp u
I- qt- ‘J -1
. 1* 4. 'I 1
I- + JI- III
)I '0'- all
’ ‘2313 t __.+ II II +1-
7 . :: :
. «L .
n I- «1- u
v '- '-"'" “
N '- ..
b I I I I 1r I rI I I 1 1I I 1 TL
p b -1
6" ~ - 40 MeV/nuc C+Au ~
N )- _. .,
d
b L r- . .1
v- 1- d
u- b c:
I. L J
1- . . ~ . d
T 1' i:
)- - ..
0.0 1 1 1 1 1 1 1 1 1 1 1 l 1 1 1 I. 1 1 1 l 1 1 1 l 1 1 1 1 1 1 1

 

-180 -90 0 90 180 -90 O 90 180
02 (degrees)

Figure IV—17 Deuteron-alpha correlation function for which the alpha is
detected at 0:-25°. The lines are described in the text.

71

@1 =_25O aa

 

       

     

 

 

   

10.0 r I I I T I I I I T I r r r r I I I I I FT T I I I
" J
C 2: 1' :
7 5 _-_ 4o MeV/nuc + J; so MeV/nuc+ J
’ I C+C I: C+C 3
I 1: I
5.0 " 4- .
A " . .-
"' ' " :
I " 1
e 2 5 3 j
o: 1 j 1 j
b 0.0» :.++‘r .%.L+
d p- 4- .1
b d)- +
\ ' -- 40 MeV/nuc C+Au «
N 0.6 " -n- _
v-I _ .. .
b _ .. .
. ‘1' ..
0.4 '- q)— .1
. +- d
I I .
0.2 )- ..- -
h- “ ‘
. +
0.0 1 1 1 l 1 1 1 1 1 1 1 1 14 1 1 1 1 L4 L1 I 1 1 1 l 1 1 1

 

 

 

 

-180 -90 0 90 180 —90 O 90 180
02 (degrees)

Figure IV-18 Alpha-alpha correlation function for which one alpha is
detected at 0=-25°. The lines are described in the text.

72

The peak at small positive angles for C+C systems comes from the very
large contribution to the two-particle correlation function coming form
the projectile-like source. This contribution is shown fin~the two-
deuteron correlations:U1ngure IV-13c. For the C+Au system the
projectile-like source contribution at the most forward angles is still
less than that of the intermediate source as shown in Figure IV-12c. For
both the C+C and C+Au systems the importance of the target-like source is
considerably less for 01=-25° than it is for 02:45". The contribution
to the two-deuteron coincidence cross section coming from the
intermediate source is very similar for the two tag angles.

In addition to light particle-light particle correlations,
correlations between light particles and Li and 7Be were measured using
the 25° tag telescopes. In these correlations the heavy fragment, Li or

7Be, is detected at 0=25°.

7

Figures IV-19,20 show the p-Li and p- Be correlation functions. Once

again the light systems show a very strong preference for emission to

opposite sides of the beam. It should be noted however, that the peak for

positive angles is considerably smaller than the peak in the ”He-“He

correlations. This is an exception to the general trend of the peak

increasing with increasing mass of the detected particle pair and is

7

probably due to it being more likely that the proton and the Li or Be

come from different sources. For the C+Ag system there is a large peak
in the p-Li correlation function between 0=-15° and -55°, this peak comes
from the decay of particle unstable states in 7Be and 88e that decay by
proton emission. For p-7Be the C+Ag system looks very much like the C+C
systems in that it shows a strong preference for emission to opposite
sides of the beam. The C+Au system is basically flat as a function of

7

angle for both p-Li and p- Be.

5.0

4.0

3.0

(1)" )

H
O

.0
o

012 /01 02
.0
03

0.2

0.0

73

o, =—25° Li p

 

 

 

bIIII.lIT[lIITIIl-‘IITTTITTrITrII ..
E 40 MeV/nuc C+C 5} 50 MeV/nuc +++ 5
E, .+ SE C+C * + 2
: '1' f 3
. ++ I
+- q- I -
e _- + 3E . - a
— - .- q
: I :: III I :
: 1' EI' . 1: + ++"
1 1 I 1 1 1 1 1 1 1 l 1 1 1 1 1 1 1 1 1 1 1 1 1 1 l 1 1 1

.. I Ifi I I I l I I I I I I I I . I I I I I I I I r I r I I I T‘
E +1 1: 40 MeV/nuc C+Au :
: O + : ¢
.‘+ +++ "+11 +1? '1' u + 1
.- . .. + . . +* + + 1
- db

- 40 MeV/nuc C+Ag --+ ++ -
I I l

1 1 1 l 1 1 L1 1 1 1 I 1 L1 1 1 1 J 1 1 1 l 1 1 1 1 1 1 1

 

 

 

—180 -90 0 90 180

—90 0 90 180

02 (degrees)

Figure IV-19 Lithium-proton correlations for which the lithium is

detected at 0:-25°.

7H

@1 =_250 7 86 p

 

 

 

 

 

 

 

 

600 I I I I I I I I rrIj r fl I I T r I I I I I I I rfr
1r } d
I « .
4.0 - 40 MeV/nuc C+C -- 50 MeV/nuc+ -
- ~- C+C .
A _ -r + .
T ao- ++ l I
a _ ‘++ ._ .
\_/ 4. " ‘
. + J' J I ++ ' .
m . r-
b<101 .?:%%%+:::4 :::%

6 k i .
\ 0 6 - 40 MeV/nuc C+Ag «L 40 MeV/nuc C+Au ~
01 . "" 1 ..
g ; I 1
. + .4. .I
03: ++++++¢ jf+ ++++ ++ +:
.+ :; H + ‘

0.0 1 1 1 l 14 J l 1 1 1 l 1 1 1 1 1 1 j 1 1 1 l 1 1 l_i 1 1 1

--180 -90 0 90 180 -90 0 90 180
02 (degrees)

Figure IV-ZO ”Be-proton correlations for which the 7Be is detected at
0=-25°.

75

C. Out-of-plane Correlations

In addition to the in-plane correlations described in the previous
section, two-particle out-of-plane correlations were also measured.
Correlations were measured between the out-of—plane tag telescopes
(¢=90°) and the nine in-plane detectors with ¢=0°. The correlations will
first be presented in terms of the two-particle correlation function used
in the previous section. Then the ratio of in-plane to out-of-pLane

correlations will be discussed.
1. Out-of-plane Correlation Functions

The two-particle out-of-plane correlation functions for p-p, (L41 and
”He-“He are shown Figures IV-21-23. The lines are the results of three
source momentum conservation calculations. One particle is detected at
<I>=90° with 0:25° (triangles and dashed lines) or 185° (squares and solid
lines). The second particle is detected at <l>=0° with 1S°SGS150°. The
correlations are integrated over the same energy ranges as the in-plane
correlatnmuL The correlation functions show very little difference
between the different systems. The p-p and d-d correlations rise slowly
from 02=1S° to ”5° and then level off. The ”He-“He correlation functions
show a slight decrease with increasing angle.

For the p-p correlations the momentum conservation calculations do an
lexcellent.job of reproducing the general trend and magnitude of the data
for the heavy systems. In the C+C system, the trend but not the
magnitude of the correlations in reproduced for O1z-45°. For G1z-25° the

calculation has a large maximum at forward angles where the data has a

minimum.

76

 

   

 
    

 

 

 

     
 

 

 

 

 

 

1 8 I I I I I I I I I I I I I I I I I I I I I
1.5 _~ 1 -'
1.2 - 11x. 3
. I 2: ‘~ ¢ :
A 0.9 - -- -
T i I ‘
,0 0.6 - -- ~~~~~~~ I
V " ~~~~~ :_ ----------- I
0.3 - ‘ -- '3
b" :40 MeV/nucl. C+C : 50 MeV/nucl. C+C -
dcxo_. 1:: :: ::: i:
b . I:
\ . ..
$3 Cl13 : '-
b _ I
T.--_ I ..
'- --“ u 1'-
l. “' .8 1:4! ~~~~~~~ :1 «—
0.2 - "“—~
:40 MeV/nucl. C+Ag «4o MeV/nucl. C+Au 3
0.0 1 1 l 1 1 l 1 1 l 1 1 I. 1 1 l 1 1 I 1 1 l 1 1

90 135 180 45 90 135 180

@2 (degrees)

Two-proton out-of-plane correlation function for 0.=25°
(triangles and dashed lines) and 0.:u5° (squares and solid

lines).

Figure IV-21

77

<11 =90° <12 =0° dd

 

 

 

 

 

     

 

 

   

2.1
I I I I I I I I I I I ‘ I I I I r I I I I I q
.L d
1.8 L -r- + i 4
l-ESC. 3: II ‘:+ l+ 1
"~ + I I A I
1.2 :‘1 . __ “ _I
A : .‘\' A4 I 2:. i
T 0.9 1— “ * # ar— :1
' ‘ 'C .
L -: ‘-
03:. 1’ q
a ~40 MeV/nucl. C+C ~50 MeV/nucl. C+C «
boo-14111L1LL11d-1PI1L11111L.
. I I I I I 1 I I r I I 17' I I I r I j t I I
H ,. 11-
b . ..
“\~ . ..
oz . - +-
_ o 6 _ 1.
b L ..
I- db
()u4': , '+ --
’ I I ' x I
x‘ ‘ ‘ ‘4~-__-1~
0.2

1 1 1 l
I I I
1 1' 4 l

rTI'II

0 MeV/nucl. C+Ag 40 MeV/nucl. C+Au

001111111LL11 11111L11L1L

0 45 90 135 180 45 90 135 180

 

 

 

 

02 (degrees)

Figure IV-22 Two-deuteron out-of—plane correlation function for 6,:25°
(triangles and dashed lines) and G.=HS° (squares and solid
lines).

78

 

 

 

 

 

 

   
 
 

 

 
 
 

I I I I I T :I I I I I I
I- ' d
+ i .
#50 MeV/nucl. C+C 4
cl I 4
1t ‘ 'I' .1
-11- .+ i A
41- ’ .1
fl ’ '"J\ I, 'i
| " "" ‘I ’7’ '1
0'0 .. x,” J_ \‘ I, 4
003 '- "- \\ 1’ ‘1
I- d- «I
N " " .1

b on .31: :%: =%: : : rl h%% :+: %%
6" :Me nucl. C+Ag «40 MeV/nucl. C+Au ~
\ 0 4 i "“1. II —1
95 ' “'1 ,‘I .1

b x x

0.3 "P: ’I -(
.4" I, q

  
 

0.2 - ..
~22 a+ ‘ .I ‘
0 1 - ‘A * ‘u t -

 

 

 

 

0 45 90 135 180 45 90 135 180

02 (degrees)

Figure IV-23 Two-alpha out-of-plane correlation function for G.=25°
(triangles and dashed lines) and G,=u5° (squares and solid
lines).

79

For the d-d correlations the calculation once again gets the trend
but not the magnitude of the C+C systems for 01z-45° and has a maximum at
forward angles for 02=-25°. In the heavier systems the general trend is
reproduced by the calculations for both angles when one remembers that
for large angles correlations between deuterons emitted from the target
source and those emitted from either the projectile or intermediate
sources are not included in the calculation. These correlations between
deuterons from different sources will raise the calculation at backward
angels where the singles cross sections are dominated by the target like-
source.

For the “He-“He correlations the calculations do a very poor Job of
explaining the data. For the heavy systems the magnitude of the
correlation function is greatly over estimated although the shape is
reasonably similar to the data. For the C+C systems the the calculations

do a very poor job of explaining the data.
2. In-plane To Out-of-plane Ratios

The ratio of in-plane to out-of-plane correlations for the tag

detectors at O1 is defined as

012(0, <l> =180°,202,¢=0°)/o1(01,¢1:180°)

R(0): .. 0 I
2 012(01’;1=9O° gezgézzoo )/01(01,¢1-90 )

(IV-l4)

where 012 is the two-particle coincidence cross section for in-plane

(¢1=1BO°) or out-of-plane (¢1=90°) and o is the singles cross section

1
for the in-plane tag (¢1=180°) or the out-of—plane tag (¢1=9O°). The

80

ratio of in—plane to out-of—plane correlations are shown in Figures IV—
2M-26 for the ”5° tag telescopes and Figures IV-27-29 for the 25°
telescopes. Tflmzlines are the results of three source momentum
conservation calculations. The data show very little difference between
the two C+C systems and between the two heavier systems (C+Ag and C+Au).
The ratio of in-plane to out-of—plane correlations triggered on the 25°
tag telescopes is generally slightly lower than the ratio triggered on
the 15° tag telescopes for the same particle pair and system.

For the 45° tag telescopes the momentum conservation calculations ck)
an excellent job of reproducing both the shape and magnitude for the
heavy systems. For the C+C systems the momentum conservation
calculations over predict the in-plane out-of—plane rathm. The over
prediction of the ratio for the C+C systems is probably due to the
greater importance of correlations between particles from different
sources in the light systems than in the heavy systems. Correlations
between particles coming from different sources would lower the ratio of
in-plane to out-of-plane correlations.

The momentum conservation calculation does not do as good of a Job
explaining the in-plane to out-of—plane ratio triggered on the 25° tag
telescopes. Most of the calculations show two peaks in them, one at very
forward angles and the second near 90°. While the structure is not well
reproduced the magnitude of the in-plane to out-of—plane ratio is well
reproduced by the momentum conservation calculation fkn~‘the heavy
systems. Ckuy'for the p-p correlations is the magnitude of the in-plane
to out-of—plane ratio well reproduced for the C+C systems. For txyth the
d-d and ”He-“He correlations the data is much lower than the momentum

conservation calculation.

81

 

 

 

 

 

e1 =45° PP

1-8IIIITIIIIWIIIIIIIIIIIIIIIIIIIIITI
m — [m —1
a - / -
F-O — I" ""
fin. 1.€3 r- /’ ., -—
«a __ / _.
dls : -1
° 1.4— A _
° — -1
4.3
m — + _
E 2 " ’ . _j
.5 - ,x'“¢""f.ff’.'f:§3-':‘~‘~ ;_ ~¥.-. —
“a Iffi‘ ------- ~ - -- - :‘111112111—‘2 ..... '1

1 O -- "MW“—
0 —1
:5. _
<5 -— .—
m _ -1

0.8 11111l11l11L1111111111111111111L11

O 30 60 90 120 150 180

03 (degrees)

Figure IV- 2‘4 Ratio of in- -plane to out-of- -plane correlations for two-
protons for NO MeV/nucleon C+C (squares and solid lines),
C+Ag (circles and dash dot lines), C+Au (diamonds and
dashed lines) and 50 MeV/nucleon C+C (triangles and dotted
lines). One proton is detected at G.=45°.

82

 

 

 

 

 

 

®1=45° dd
3~0_IIIIIIIIIIIIHIIIII IIIIIIIIIIIIII_A
32-8: 1:
CU r- I:
I-d —
‘? 23u4»:7 —-
322: 1
<3 ' :: ::
" ‘1
.8 2.0: T j
3 18: ‘L l
m ‘ — j
1-O : —
r|:"1.6_-_- :
I: — + :
'” 1H45; + , “1+ '2
H _ :7.._ ‘~.‘ -1
0 :: I I¢+w ‘M ‘:.\‘ _
o 1 .2 : ’.:';”"’ + ,‘.‘~::: .... T:
01-. .Ca’ “~::-- -—
”£3 E” t ""“'- ---.:
0:: 1.0__ Q I
11111111111111111“
0.81””“1'1'l111 ,

0 3O 60 90 120 150
02 (degrees)

H
CD
C)

Figure IV-25 Ratio of in-plane to out-of—plane correlations for two-
deuterons for 40 MeV/nucleon C+C (squares and solid lines),
C+Ag (circles and dash dot lines), C+Au (diamonds and
dashed lines) and 50 MeV/nucleon C+C (triangles and dotted
lines). One deuteron is detected at O,=u5°.

101)

.‘1
01

Ratio of in—plane to out-of—plane
m 01
b1 (3

.0
o

83

 

 

 

 

 

 

 

3O 6O 90
02 (degrees)

Figure IV-26 Ratio of in-plane to out-of—plane correlations for two-

alphas for N0 MeV/nucleon C+C (squares and solid lines),
C+Ag (circles and dash dot lines), C+Au (diamonds and
dashed lines) and 50 MeV/nucleon C+C (triangles and dotted
lines). One alpha is detected at 0.:H5°.

Ratio of in—plane to out—of-plane

81:250 pp
1-8 11111IIIIIIIIIIIIIIIITIIIIIIIIIIWI
I -1
_ I A
1.6— ,1, _.
1.4%; + ._
.' ". 4
[.5 f x /?/A “ 1
_0: '3‘ x / \ _.
_:' '~. \ ._
~/" $“‘~~.§;i- _§ .............. ~ -
. . my; ----- g -
1.0—3 ~-....,_I
0.811111]11111111111111111L11111l11111‘
O 30 60 90 120 150 180
02 (degrees)

8“

 

 

 

 

 

Figure IV-27 Ratio of in-plane to out-of-plane correlations for two-

protons for 180 MeV/nucleon C+C (squares and solid lines),
C+Ag (circles and dash dot lines), C+Au (diamonds and
dashed lines) and SO MeV/nucleon C+C (triangles and dotted
lines). One proton is detected at O,=25°.

8‘5

9, =25° dd

IIIITIIIIIIIIIIIIIIIIIIIIIITIIIIIII

.03
o

 

1

50.10
05C!)

N!“
we

 

 

 

 

06‘
0+
*.
\“ I.
I
/
l

/
11Y11111111111111111111111111111111111111

Ratio of in—plane to out—of—plane

IIILIT IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII

L

 

 

11111111111111111111111111111 11111

30 60 90 120 150 180
02 (degrees)

 

OHHHHHN
momeoaooo

0

Figure IV-28 Ratio of in-plane to out-of-plane correlations for two-
deuterons for “O MeV/nucleon C+C (squares and solid lines),
C+Ag (circles and dash dot lines), C+Au (diamonds and
dashed lines) and SO MeV/nucleon C+C (triangles and dotted
lines). One deuteron is detected at O,=25°.

86

 

10.0

>1
01

Ratio of in-plane to out-of—plane
go 01
01 o

 

 

.0
o

 

 

Figure IV-29 Ratio of in-plane to out-of-plane correlations for two-
- alphas for 140 MeV/nucleon C+C (squares and solid lines),
C+Ag (circles and dash dot lines), C+Au (diamonds and
dashed lines) and 50 MeV/nucleon C+C (triangles and dotted

lines). One alpha is detected at O,=25°.

87

Recently Bauer [Ba 87] has studied two-proton correlations using the
Boltzmann-Uehling-Uhlenbeck (BUU) equation modified to conserve momentum.
In the BUU approach the reaction being studied is simulated using 100
parallel ensembles of’test particles. The test particles move in a mean
field hHJfll collisions treated using a cascade approach with Pauli
blocking. The results, shown in Figure IV-3O for NO MeV/nucleon C+C»pp,
reproduce the overall trend of‘ the data. Bauer's BUU calculations have
also reasonably reproduced the in-plane to out-of-plane ratio for 85
MeV/nucleon C+C [Kr 85] and the azimuthal correlations for 25 MeV/nucleon

O+C [Ch 86b].

88

 

.4
——4
1
q
-—1
——<
.4
.4
.1

93
o
o

A 1

12C+12C—>2p+X
Ebeam. = 40 MeV/nucl .4

j. A

rey
1—6—1

 

a
£3
1—€}*
k—{3——1
a l

 

 

H

~J

CH
IIIIIITTIIUIIIIIIfIIfiYI

 

 

 

c(45°,92.1ao°) / c(‘459,02,9o°)
2;
CD

 

 

1.25
1.00 -——--——--—------1
1
l l . Ll L,. l, l
20 4O 60 80
92 (deg)

Figure IV-3O Ratio of in-plane to out-of—plane correlations for two-

protons using the BUU model (histogram) [Ba 87] compared to
the data (circles).

Chapter V

Unstable Resonance Experiment Results And Discussion

In this chapter the results of the unstable resonance experiment
will be presented and discussed. First the two-particle correlation
functions will be shown. Next the source sizes which were extracted from
the correlations will be presented. The populations of the bound and
unbound states will then be extracted, and the temperature of the source

will be determined from the quantum statistical model.

A. Two-Particle Correlation Functions

In this section the two-particle correlation functions at small
relative momentum will be presented. The correlation function, R(Ap)+1,

is defined as

R(Ap)+1 : No12(Ap)/0;2(Ap), (V-1)

where 012 is the number of particle pairs with relative momentum Ap, 01.2
is the number of randomized particle pairs with relative momentum Ap and
N is a normalization constant picked such that the correlation function
for large Ap is one. The random particle pairs are created by keeping
the last five of each particle type and calculating their momentum

relative to the current particle pair. Thus for each actual correlation

ten random correlations are generated.

89

9O

1. 35 MeV/nucleon N+Ag

The two-particle correlations are shown in Figures V-1-10. The
lines in each figure are the results of calculations by Boal and
Shillcock [Bo 86] for final state interactions between the emitted
particles for various source radii coming from Coulomb and nuclear
interactions between the two-particles. For some of the particle pairs
(p-t and d-t) the calculation has been carried out using the Coulomb
interaction only. The calculations have been smeared to account for the
experimental resolution of the MWPC and telescope array. Both the
calculations and the smearing will be discussed in the next section.

The two-proton correlation function at small relative momentum is
shown in Figure V-1. The correlation function has a broad maximum at
Ap=20 MeV/c. The peak height decreases with increasing angle. This peak
has been described previously in terms of both emission of 2Me [Be 85]
and in-flight final state interactions between two randomly emitted
protons [Ko 77]. In Koonin's [Ko 77] description the correlation
function depends on the space-time extent of the emitting system and is
often used to extract the source size of the emitting system. The
extraction of source sizes from the correlation functions will be
discussed later in greater detail. The correlation function also shows
an anti-correlation for small Ap caused by Coulomb repulsion between the
two protons. Calculations of the correlation function as a function of
the size of the emitting source are shown in Figure V-1 for r:‘4.0 (solid
line), “.5 (dotted line), 5.0 (dot-dashed line) and 7.0 fm (dashed line).

The proton-deuteron correlation function is shown in Figure V-2.
The correlation function rises smoothly until it reaches one and then

stays flat at one. The calculated correlation function for source radii

 

91

35 MeV/nuc N+Ag->pp

itifrlill Ilirlrit

 

   

I I

I

O

s

s

‘s

I
. t.

O
-

a.) o=35° b.) o=45°

111111111

.0
- 01

R(Ap) + 1
f" E"
O U]
E

.0
U}
IlUlrI’lU
0
v
G)
ll
- O) :
- O
0
1'
D.
v
‘7’
m
0
0
11111111

 

" I l
“{b.
'I
Q

 

 

 

o 25 50 '75 25 so '75
Ap (MeV/c)

Figure V-1 Two- -proton correlation function for 35 MeV/nuc. N+Ag. The
lines are described in the text.

92

35 MeV/nuc N+Ag-*pd

1.5 IrIIVIIIIIIIT ITIIITIITrIITI

 

 

 

 

 

 

11111111

 

 

 

0 50 100 150 50 1010 150
Ap (MeV/c)

Figure V-2 Proton-deuteron correlation function for 35 MeV/nuc. N+Ag.
The lines are described in the text.

93
of r=7.0 (dashed line), 8.0 (dot-dashed line) and 9.0 fm (solid line) are
shown in Figure V-2.

The proton-triton correlation function is shown in Figure V-3. The
correlation function has a series of three unresolved peaks corresponding
to the first three excited states (Eexz20.1, 21.1 and 22.1 MeV) in “He
which decay via proton or neutron emission. The solid line is the
calculated correlation function for a source of radius 7.0 fm.

The proton-alpha correlation function is shown in Figure 1141. The
correlation function has a broad maximum at Ap=70 MeV/c which corresponds
to the particle unstable 5L1 ground state. The 5L1 ground state is not
resolved from the first excited state (Eexz7.5 MeV, P=5 MeV). The
calculated correlation function for source radii of r:N.O (dashed line),
5.0 (dot-dashed line) and 7.0 fin (solid line) including both nuclear and
Coulomb interactions are shown in Figure v-11. The shift in the location
of the peak in the correlation function is at least in part due to the
effects of the Coulomb force of the emitting source [Po 85b]. The proton
and alpha coming from the decay of the ground state of'SLi are
accelerated differently by the Coulomb field of the emitting source due
to their different charge to mass ratios. In this experiment the protons
had a higher low energy out in MeV/nucleon than the alphas, it has been
previously shown [Po 85b] that for correlations for which the proton has
a higher velocity than the alpha that the peak in the correlation
function is shifted towards larger Ap. The peak location could also be
shifted some due to experimental resolutions and calibrations. The
chatted line is the calculated correlation for a Coulomb only interaction

with a source radius of r=7.0 fm.

9H

35 MeV/nuc N+Ag-*pt

 

 

 

 

1.5 rvvlvvttjvtrrlvuri ttrtrtfi1r[vvtw—[IVTV’
20.1 22.1
$2&.1‘
1.0
" '1
D d
- 1
_. ° _ o
0.5 - a.) @—35 b.) @—45 -
1- .1
fl p .1
+ - a
A 1- q
n‘ 1 5 111 111 1 111L11 1L1111 llllllLLllJlllLlll
< . vvvri Urt‘TYrTrFUTT YVYY'UI’VYr'VV—rrT—ri
v
9:: ‘
Q‘, -
O .1
1.0

0.5 c.) o=ao° d.) o=ao°

Jilllkll l AJAILJI

C).()"‘ .. .1,L...L. ...... .. .. 1, . .7
0 50 100150200 50 100150200
Ap (MeV/c)

Figure V-3 Proton-triton correlation function for 35 MeV/nuc. N+Ag.
The lines are described in the text.

 

95

2 0 35 MeV/nuc. N+Ag->pa

 

   

 

IFYITWTIIIIIIIIITIi
1‘ I‘
-1 "‘
I \
1 ' ~
I \
151 ' ‘
Q 'l. \
1 1", +
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10 ’ “
O .1

 

 

 

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V
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ll
DJ
01
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v
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||
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01
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v-O
+ "

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an. 1111111111111111111 1111111111111114111
< . TYUVIYUTTIrrT'FITYI'I YUFrIITIITFYTIU' I’
a: .. ..

I \ 1

LllLlll

   
  
    

 

 

1.0
0.5

   

 

 

I I IL! 111 LJJJ

 

c.) ®=60°

L

00 AllLllllllljlllLLllilLllLlLLLLLl—lLJLJLLI
.

0 50 100 150 200 50 100 150 200
Ap (MeV/c)

Figure V-ll Proton-alpha correlation function for 35 MeV/nuc. N+Ag.
The lines are described in the text.

96
The two-deuteron correlation function is shown in Figure V—S. Like
the proton-deuteron correlation function the two-deuteron correlation
function rises smoothly to one. The calculated correlation function is
shown for sources of radius 7.0 (solid line) and 8.0 (dashed line).
The deuteron-triton correlation function is shown in Figure V-6.
The correlation function has two peaks corresponding to the second and

5

third excited states (Eex=16.76 and 19.8) in He.

The deuteron-alpha correlation function is shown in Figure V-7. The
large peak at Apzuo MeV/c comes from the decay of the first excited state
of 6Li (Eex=2'186 MeV). Two other states are observed around Ap=90-100
MeV/c, these are the third and fifth excited states (Eex=ll.31 and 5.65
MeV) of 6Li. The calculated correlation function for a source of radius
7.0 fm is shown in Figure V-7 with the nuclear interaction included
(solid line) and with only the Coulomb interaction (dashed line).

The two-triton correlation function is shown in Figure V-8. Like the
proton-deuteron and two-deuteron correlation functions the two-triton
correlation function rises smoothly to one. The calculated correlation
function is shown in Figure V-8 for sources of radius 6.0 (dot-dashed
line), 7.0 (solid line) and 8.0 fm (dashed line).

The triton-alpha correlation function is shown in Figure V-9. The
large peak at Ap=70 MeV/c comes from the decay of the second excited

7

state (Eex:u.630 MeV) of Li. Lying at higher Ap are the next two states

7

of Li (Eexz6.68 and 7.14597 MeV). The bump at Ap=30 MeV/c has been

attributed by Pochadzalla et. al [Po 86b] to the 2—step decays such as

8.” 5

L1 + t+ Li -> t+a+n.

    
    

Figure V-S

97

35 MeV/nuc N+Ag—>dd

 

TIIIIIIIIITIIIFTTII IIIIIIfifiIIIIIIIIIfT

a.) o=35° I} b.) o=45° +

    

1 I I T

 

 

111111

 

 

[11141111111111
IrTTIrIIIIITTTIY

CL

v

(9
II

CD

CD

+0
41’227

 

 

 

 

 

 

 

 

 

 

l
I-
\
-n:
+
' 'fEEatZZ
t‘ 1
1
111111111

 

O 50 100 150 200 50 100 150 200

Ap (MeV/c)

Two deuteron correlation function for 35 MeV/nuc. N+Ag.
The lines are described in the text.

1.5

0.5

00':

98

35 MeV/nuc N+Ag-:dt

 

’rTTrl r1 I

[j

 

a.) ®=B5°

c.) ®=60°

11141

UIU'

 

 

 

 

 

 

 

111L111]

 

0 50 100150200 50 100150200

Figure V-6

Ap (MeV/c)

Deuteron-triton correlation function for 35 MeV/nuc. N+Ag.
The lines are described in the text.

99

 

35 MeV/hue N+Ag eda

  
  
 

     

1. a) o=35° {- ,b.) 9:43 5
I a, .I '9 I
’. .. I _

3. 1L‘ :1 1' '1 ‘-
1. :q .4. lg; _
_ . l . -

2. t: P :: '.Z_
; r1-

 
  

 

        

bbboobboooo
r‘1j1‘J

 

.—4 1 .........
+- ' .
a "

’- 1 11 ‘ »
<1 '3 1' ' ~
v __ 1 I
‘3‘ 4. i- . .:

7' - -
1' .1
:3 t " ;
° t a I
- .1 -
2 E I ~‘
", 1' ’ ‘_‘
1 1 'l.__._.....‘
" '1 1.113 :
O F:’ 111111'1111 1111T1111 :
O 50 100 150 200 50 100 150 200
1p (MeV/c)
Figure V-7 Deuteron-alpha correlation function for 35 MeV/nuc. N+Ag.

The lines are described in the text.

100

35 MeV/nuc N+Ag—vet

1.5 TIIrTIrrtrrirI’rTIrlrrIIIIfiFT

 

 

T I I T
l

 

 

\c‘
0‘.
v
(9
II
#5
CH
0
11111111

 

 

 

 

11111111111111
I ftfirrfl Fr;
+ .
1". I 1 __y
0’ ‘

 

 

 

 

\
111111

 

0.01111411-11411- 1W1
o 50 100 150 50 100 150

Ap (MeV/c)

Figure V-8 Two-triton correlation function for 35 MeV/nuc. N+Ag. The
lines are described in the text.

101

35 MeV/nuc N+Ag->ta

 

     

 

’ITIITIIIIIIIIIrITr IIWIFIIIIIIIITliir’
1
I
1
25 :
. : -
: M
I
f x r I
I
1.5 a

   

 

 

 

 

 

 

 

0 50 100 150 200 50 100 150 200
Ap (MeV/c)

Figure V-9 Triton-alpha correlation function for 35 MeV/nuc. N+Ag.
The lines are described in the text.

102
The two—alpha correlation function is shown in Figure V-10. The
peak at Ap=18 MeV/c comes from the decay of the particle unstable ground

state of 88e. The broad peak at Ap=110 MeV/c corresponds to the first

excited state (Eexz3.0u) of 88c. The peak at ApzltO MeV/c is the
!
combination of the decay of the 9Be2 ”3 state which decays by emitting an

5L1 which in turn decays to a proton and an alpha [Po

alpha leaving a
86b], and the ghost state of the 88c ground state [Ba 62, Be 71, Be 81].
The calculated correlation function is shown if Figure V-‘IO for a source

of radius 7.0 fm including the nuclear interaction (solid line) and for a

Coulomb only interaction (dashed line).
2. 25 MeV/nucleon N+Ag

The two-particle correlation functions for 25 MeV/nucleon N+Ag are
shown in Figures V-11-13. The gross features of the correlations are the

same as the 35 MeV data.
B. Source Sizes

fTwo-particle correlations have been used for some time to determine
the size of the emitting source. In astronomy the Hanbury-Brown Twist
interferometer is used to measure the size of stars by looking at two-
photon correlations [Ha 56]. Two-pion correlations have been used in
laigh energy [£2 77] and nuclear physics [Be 83, Za 8“] to measure the
space-time extent of the interaction region. In nuclear physics two-

proton correlations have been used to extract information about the

51)

£10

{30

21)

R(Ap)+1

m3

35 MeV/nuc N+Ag—max

 

IrrrIIrrrTrIlrITIIIrflrr IIrTIIIrIIIIrI[FIIrrIrr

 

 
 

EL°Be o o 3
:4, a.) (9:35 b.) (9:45 :
Ii 1
3* i a
3: ¢ 1

‘ -1
f 1
- . 4
C I j

v'-'
o“
a

   

lllllllllllllllliLlllL

 

 

50 100 150 200 250

 

 

 

 

O 50 100 150 200 250

Figure v-10

Ap (MeV/c)

Two-alpha correlation function for 35 MeV/nuc. N+Ag. The
lines are described in the text.

2.0
1.5
1.0

0.5

H
+
’5.

3 1.5
a:

0.0

Figure V-11

1014

25 MeV/nuc N+Ag 45°

 

 

I’I

IIIl'T

 

   

 

 

r I I I I I I I I I I I I I I I I l T T I I I I I I I I I 1 I I
i- .1 q
C ‘ o". ‘
I- :1 "I \‘ 1
P a ‘ I' ‘1 '"
- I1 - ' ’1’ " ~

. \ d I. I ‘ \\
b ‘ \

I ‘_ “ ' I - \‘Q‘

un- I

   

 

 

 

 

 

JILIJLL

 

1111111111111171111111111111111111

 

50 100 150 50 100 150
Ap (MeV/c)

Two-proton, proton-deuteron, proton-triton and proton-alpha
correlation functions for 25 MeV/nuc. N+Ag at 0:45°. The
lines are calculations for final state interactions for
sources of radius rzu (dashed dot). 5 (dashed) and 8 fm
(solid) for nuclear and coulomb interactions and r=8 fin
(dotted line) for coulomb only interactions. The dashed
line in c.) if for 9 fm.

105

1 5 25 MeV/nuc N+Ag 45°

 

1 I r I
+—

  

1.0

 

 

  
   

 

 

b.) dt

IIIrjilIl
11111111

1111111111111111111

50 100 150 200

 

 

 

R(Ap)+1

 

 

 

 

O 50 100 150 200

Ap (MeV/c)

Figure V-12 Two-deuteron, deuteron-triton and deuteron-alpha
correlation functions for 25 MeV/nuc. N+Ag at 9=fl5°. The
lines are calculations for final state interactions for
sources of radius r=8 (solid) and 9 fm (dashed) for nuclear
and coulomb interactions and r=8 fm (dotted line) for
coulomb only interactions.

106

25 MeV/nuc N+Ag 45°

 

4.0
3.0
2.0

l
l
1
I
l
l
I
p
i
I

I

1.0

 

 

5. '1IIFIIIIIIIIIIrIIrI' IIY[If'IT]rFrI1111111111.1
I
' d
1+ 1
d
q
.1
.‘5 J
. I
I ' . '.
.I 1',
rt: , ‘ J _ ::Il‘i
p + l

IIII

R(Ap)+1

1.5

IIrI

a.) ta aa

+

50 100 150 200 250

 

1.0

firrrv

0.5

 

11111111111111111111117
’ IIII "I
I Id
+ - ~
d

+11,“ 0.) tt ‘
.y ‘

.1

 

 

OO111L111111111111141
.

0

Figure V-13

50 100 150 200

Ap (MeV/c)

Two-triton, triton-alpha and two-alpha correlation
functions for 25 MeV/nuc. N+Ag at 0=H5°. The lines are
calculations for final state interactions for sources of
radius r=fl (dashed dot). 5 (dashed) and 8 fm (solid) for
nuclear and coulomb interactions and r=8 fm (dotted line)
for coulomb only interactions. The dashed dot line in c.)
if for r=6fm.

107

space-time extent of the interaction region by many authors [Za 81, Ly
83, Cu 84, Ba 86] using the description of Koonin [Ko 77]. Recently two-
particle nuclear interferometry has been extended to other particle pairs
by Boal and Shillcock [Bo 86]. Chitwood et al. [Ch 86a] and Pochodazalla
et al. [Po 86a, Po 86b] have used the calculations of Boal and Shillcock
to extract source sizes from several particle pairs. In most cases the
finite lifetime of the emitting source is neglected. The effect of
neglecting the finite lifetime of the emitting source is to increase the
extracted source radius.

The source sizes have been extracted for p-p, p-d, p-a, d-d, d-a,
t-t and (MI correlations by using the calculations of Boal and Shillcock
smeared for the experimental resolution. The smearing was done by taking
the calculated correlation function and multiplying by the random
correlations, 032(Ap), to produce 0

0 was then smeared assuming a

12' 12
Gaussian shaped distribution of width 6p. For the correlations with
resonances the width, 6p, was adjusted so that the FWHM of the smeared
correlation function was the same as the FWHM of the peak in the data.
For the correlations without resonances the width was adjusted to give a
shape that looked as similar to the data as possible. The smeared
calculations are shown in Figures V-1-13 as lines.

The source radii were determined by chi square minimization at each
angle for each particle pair. The extracted radii, given in Table V-1,
are in general agreement with previous results [Ch 86a, Po 86a, Po 86b] .
The extracted radii show little variation with angle or beam energy. For
the p-a correlations the source radius was extracted by looking at the

peak height only due to the large shift in the location of the peak. Due

to the poor resolution for the d-a correlations the source radius was

108

Table V-1

Extracted Source Radii From Two-Particle Correlations

35 MeV/nuc. N+Ag

25 MeV/nuc. N+Ag

 

Correlation 35° u5° 60° 80° Average 45°

pp n.32o.3 4.520.3 4.520.3 H.820. n.52o.2 u.02o.3
pd 9.320.8 9.120.? 9.620.8 10.220. 9.520.u 10.220.9
pa 5.020.3 H.620.3 U.620.3 u.u20. n.720.2 U.5:O.3
dd 6.720.8 7.320.? 8.820.9 7.621. 7.52o.u 8.820.9
da 5.220.5 H.920.“ 5.0:O.5 H.920. 5.0:O.2 5.0:O.5
tt 6.621.1 5.920.8 6.u21.2 7.122. 6.320.5 5.621.0
aa u.720.3 4.220.3 u.320.5 u.uzo.2 3.720.u

 

109
extracted by integrating both the calculated correlation function and the
data over the range of lip dominated by the first peak. The radius
extracted from the d-a correlations are larger by about 2 fm than those
obtained previously [Ch 86a, Po 86b]. This difference is probably due to
the resolution in the present experiment.

One interesting feature of the extracted radii is that for those
correlations involving particle pairs that do not have resonances, (p-d,
d-d and t-t), the size of the emitting source is larger than for those
pairs that have resonances. This pattern has been seen before [Ch 86a,
Po 86a, Po 86b]. Chitwood et al. [Ch 86a] also found that the source
radii extracted from the first peak in the d-a correlations was smaller
by about 0.5 fm from the radii extracted from the second peak. The
calculations by Boal [Bo 86] assume that all the two-particle
correlations come from particles that are randomly emitted and then have
in-flight final state interactions. For randomly emitted particles
experiencing in-flight final state interactions one would expect that a
single source size would describe all the states. If, however, some of
the observed correlations come instead from the decay of emitted particle
unstable nuclei, then the extracted source radius would be smaller than
the true radius and since the different states are populated according to
the temperature of the system, it is possible that different states will
yield different apparent source radii.

It has been suggested that the different extracted source radii may
result from a sequential freeze-out [Na 82, Bo 86]. In the case of
sequential freeze-out different particles may be used to probe different
stages of the interaction.

The source radii extracted using two-particle correlations at small

relative momentum can also be compared to the radius extracted using the

 

 

110
coalescence model [Le 79, Sa 81, Ja 85]. For 92 and 137 MeV/nucleon
Ar+Ca, lhi reactions [Ja 85] the source radius extracted using the
coalescence model for particles of mass 2-1N was r = 3.7 fm. This radius
is slightly smaller than the radius extracted from pp and aa correlations
and much smaller than the radius extracted from pd, dd and tt
correlations (6-10 fm). It should be remembered that the extraction of
the source radius from the two-particle correlations was done neglecting;
the lifetime of the emitting system. If one includes the finite lifetime

of the emitting system then the measured source radius will decrease.
C. Bound State Spectra

The single particle inclusive energy spectra for He, Li, Be and B
are shown in Figure V-H-I. The lines are the result of single moving
source fits to Ozu5°, 60° and 80°. The extracted source parameters are
listed in Table V-2 along with the Coulomb shifts used in the fits. The
fits indicate that the intermediate velocity source dominates the cross
sections for G=N5°-80°. The 35° spectra show signs of enhancement due to
contributions from the projectile-like source. The extent of
contamination of the 7L1 spectra by the decay of'BBe, discussed in
Chapter II, is estimated to be “.3 mb of 8Be identified as 7Li per 50 mb

of 88e. Based on the cross sections for the stable Be isotopes this

means that about 2-4 mb of the 7

8Be.

Li cross section comes from the decay of

 

111

35 MeV/ nucleon N+Ag

 

 

 

 

1° .IIIITTfIIIIIr VIIIIITIrIIII rTIIlIIIT[IITI
3 4»
1 He He
1()
0
1C)
-1
1C)
-.
1C)
-8
1C)
10-‘ 1111111111111 LLJLIILLllllLlillllllLllll .
O 50 1 OO O 50 1 00 O 50 1 00 1 50
0

 

1° rIII1IIIIIIrII IIIIITTIITII
-
p

IE) I ' O 141

 

 

 

dz a/dEdfl [tub/(Nev sr)]

 

  
  

 

.
--3
10 g
b
-O
10 lllllL 11 11 11111111
0 100 200 O 100 200 300
-3
1° ‘ IIIIIIIIIIII If! I 11111;!1117
: .110 .
-3' BC B
10 g
E
.-a'
10 g; *
_. +
10 g If
-5” + . ‘
10 - ~

 

O 100 200 300 O 100 200 300 O 100 200 300 400
Energy (MeV)

Figure v-1u He, Li, Be and B energy spectra for 35 MeV/nuc. N+Ag. The
lines are moving source fits discribed in the text.

112

Table V-2

Moving Source Parameters For 35 MeV/nucleon N+Ag

 

Particle VC Cross Velocity Temperature
Section
(MeV) (mb) (MeV)
3He 8 2232 15 .16220.011 12.u20.5
”He 8 20902100 .09520.003 12.120.u
6He 8 552 5 .07220.oou 1u.220.5
6Li 12 662 u .11620.003 15.120.3
7L1 12 1192 7 .10820.00u 1u.620.3
7Be 16 352 3 .12820.003 1u.620.3
9’1OBe 16 no: 3 .11520.002 15.220.3
B 20 1u02 50 .12220.00u 13.u20.6

113

D. Populations Of Particle Unstable States

In order to extract the populations of particle unstable states ther
contribution to the two-particle correlation functions coming from
randomly emitted particles must be subtracted. The populations of
particle unstable states were found by integrating the two-particle
coincidence cross section 012(Ap) over those values of Ap that correspond

to the state of interest. For each Ap step the cross section is given by

o=(o,2(Ap)-o;2(Ap)[R(Ap)+1I/N)/e(Ap). (V-2)

where 012(Ap), 032(Ap), and N are the same as before (sect. A). R(Ap)+1
is the theoretical correlation function for Coulomb only final state
interactions for a source radius R, and e(Ap) is the efficiency of the
WPC-phoswich array calculated by the procedure outlined in Chapter III,
section D. The cross sections were extracted using source radii of R=7
and 8 fm for the theoretical correlation functitui, R(Ap)+1. 'The
extracted cross sections are given in Table V-3 are for a source size of
R=7 fin with errors including the uncertainty arising from the use of a
cdifferent source size, a 10% uncertainty for the efficiency calculation,
and statistics.

The extraction of the populations of the particle unstable states
was done assuming that randomly emitted particles have Coulomb only final
state interactions. Two-particle correlations coming from randomly
emitted particles which interact through nuclear forces to produce
resonant states are indistiguishable from emitted particle unstable
states. Possible corrections for these random correlations will be

discussed in section F.

 

 

114

Table V-3

Extracted Cross Sections of Particle Unstable States (mb/sr)

 

35 MeV/nucleon 25 MeV/nuc.
State(s) 35° u5° 60° 80° u5°
u e20,21’22 3.520.6 1.1120.16 0.2u20.0u 0.0u320.007 0.5520.08
5H°:6.76 0.3920.18 0.1u20.08 0.0u220.015 0.00820.003 0.02120.010
She:9 8 0.7520.2u 0.uu20.08 0.08120.018 0.01020.003 0.00620.008
5 183,7.5 11.5210.6 2.720.3 0.u220.05 0.03120.005 1.u220.19
6 i; 186 11.3210.u 2.720.3 0.u620.06 0.03120.00u 1.6520.21
6L1; 31,5.65 1.5620.29 0.3u20.06 0.05820.010 0.002720.0008 0.0820.03
7Li: 63 7.220.9 1.0520.13 0.15120.019 0.007u20.0012 0.6620.08
7Li;.68’7.46 3.220.5 0.uu20.06 0.0u320.008 0.001120.0005 0.2020.04
° ' 3.7720.03 0.39920.00u 0.050320.0011 0.u320.06

e3.01:

115

E. Quantum Statistical Model And Extraction Of Nuclear Temperature

Previous work has established the importance of feeding of light
nuclei by the decay of heavier particle unbound states [Xu 86, P0 86b, Ha
87]. In order to extract information such as temperature from the
relative populations of states, the amount of feeding to each individual
state needs to be measured or calculated. Using the quantum statistical
model of Hahn and St'o'cker [Ha 86] the extent of feeding to each of the
observed states may be estimated. In the quantum statistical model the
initial population of a state is determined from its chemical potential,
statistical weight and the temperature. Approximately “0 stable and 500
unstable nuclear levels up to mass 20 are included in the calculation.
After the initial populations are determined the excited states are
allowed to decay using the known branching ratios.

We have extracted the nuclear temperature for the 35 MeV/nucleon
system by x2 minimization using the quantum statistical model. The
neutron to proton ratio calculated using the fireball model [We 76, Go
77] was found to be 19/17 for the most probable impact parameter. The
break-up density was fixed at p:O.18pO which is the density obtained from
a break-up radius of 7 fm and a source of 36 nucleons. The temperature
was extracted using the populations of the particle bound states measured
in the singles run and the particle unstable states extracted from the
two-particle correlations. In addition the data of Morrissey [Mo 81%, Mo
85] and Bloch [BI 86, Bl 87] for Y and neutron emitting states for the
same system were also included. For the particle bound states the ratio
of the cross section each for isotope extracted from the moving source
fit to the 6Li ground state cross section was compared to the ratio

calculated by the quantum statistical model. For the particle unstable

116
states the cross sections given in Table V-3 were integrated from 0:115.o
to 80°, which corresponds to O:35° to 90° due to the 20° opening angle of
the MWPC—phoswich array. The 35° data have been excluded because they
contain a component coming from the prOJectile like source. The ratio of
these integrated cross sections were then taken to the cross section of
either the ground state of the appropriate isotope or to 6L1 ground state
integrated over the same angles and energies using the moving source
parameters. These ratios are given in Table v-u, the quoted errors
include a 10% uncertainty in the absolute normalizations for the particle

unstable states and the particle bound states. Using these parameters

+2.8
-2.u M

calculated ratios for temperatures of T=l1.8 and 111 MeV are compared to

the temperature was determined to be ”.8 eV. In Figure V-15 the
the data. The moving source fits to the bound state kinetic energy
spectra indicate a source temperature of about 14 MeV. The 1“ MeV
calculation shown in Figure V-15 clearly does a very poor Job of fitting
the data. The extracted temperature of 11.8 MeV agrees with the
temperature extracted at higher energies [Po 86b] using the quantum
statistical model.

Because the particle bound states were not measured at 25
MeV/nucleon, it was not possible to extract the temperature in the same
way as for the 35 MeV/nucleon data. Instead the ratio of each state to
the 6Li;.186 state was taken. These ratios were found to be similar to
the same ratios for the 35 MeV/nucleon data, hence it can be assumed that
the temperature is the same at 25 and 35 MeV/nucleon.

To illustrate the importance of feeding Figure V-16 shows the
calculated feeding for several states. In general over half the observed
bound state spectra come from the decay of heavier states. Two of the

a a
particle unstable states (7LiO N76 and 88e3 on) are heavily fed, but most

 

 

117
Table V—M

Excited State Ratios For 35 MeV/nucO. N+Ag

 

State Ratio To Ratio
“He;0.1,21.1,22.1 “Hegs °'°13:g.8gg
5He16.76 6L1gs O'O1S:g.803
SHe19.8 6L1gs .048:g:g?g
5L1 6Li .77+°'35
85,5 gs -0.25
6L12.186 6L1gs °6O:g.§0
6L1£315.65 6L1gs .071j8:33$
7L13.65 Ligs .20:g:82
7Li;.68,7.1156 Ligs '078:g:83;
BBe;.OU Ligs '17:8.82

118

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3 35 MeV/nucleon N+Ag QSM p=.18po
10
, ........ I _ T=4.8 MeV
10 -------- T=14.0 MeV
o I
10
10’1
O
13' 3He ‘He
(6
at: 1
10 abodefghijklmnopq
o ----
10
‘-1
10
10"
-3
10

 

 

 

Figure V-15 Quantum statistical model calculation of the production of
the measured states for T=l1.8 and 111.0 MeV and p=O.18p.,.
The excited states and particle unbound ground states shown
in the bottom half of the figure are identified in table V-
5.

119
Table V-S

Key To Particle Unstable And Gamma States In Figure V-15

 

Letter State Decay Reference
u u
a He20.1,21.1,22.1 p‘“
5 I
b “616.7 d-t
5 fl
0 ”€19.7 d-t
5
d Ligs,7.5 °'°
6 .*
e L12.186 d-a
r 6 " 8n
L13.6 Y [Mo ]
6L.“
3 lu.35,5.65 d‘“
h 7L-' 86
1.u76 Y [31 1
7 I
i Liu.65 t-a
7L.*
3 16.68.7.1456 t‘“
7 .* 6 .
k L17.N56 n- Li [Bl 87a]
1 7B ' 86
e N30 Y [Bl ]
8 .*
m L1 98 Y [Mo 85]
8 .* .
n L12O255 n-7L1 [Bl 87a]
8 ‘l
o 883.04 a-a
10 * 9
p Be7.371 n- Be [Bl 87b]
12 * 11

q 83.388 n- B [81 87b]

120

35 MeV/nucleon N+Ag QSM p=.18po

 

100
80
60
40
20

3He 4I-Ie°He °Li 7L1 7Be 9Be

__I_—_,___.

 

 

 

 

 

 

 

7. Feeding

30
25
20
15
10

5

0

 

IIIIIIIIIIIIITIIIIIIITITIII'IIII I I I l I I III I I [1 IIfi Ir

 

 

 

111111111111111111?111111111111111 11 1 1 11 1111111 1 111 11

 

 

 

 

 

 

Figure V-16 Quantum statistical calculation of the feeding to
the observed states for a source temperature of T=l1.8 MeV
and freeze-out density of p=0.18po.

121
(if the particle unstable states are affected very little by feeding from
higher states.

The difference between the temperature extracted from the kinetic
energy spectra and the population of the states may be related to time of
formation of the complex fragments. The kinetic energy spectra of the
protons and neutrons in the intermediate source are fixed in the early
hot stage of the interaction. If complex fragments are formed later as
the source is cooling, then the distributions of the states will reflect
the later cooler temperature. The kinetic energy spectra of the complex
fragments reflect are determined by the spectra of the protons and
neutrons that coalesced to form the fragment.

Similarly it has been pointed out that while the average energy pmn~
particle stays constant during the expansion of the source, the relative
abundances of the states change as the source expands and cools [Inn 86].
Thus the kinetic energy spectra will reflect the temperature of the early
hot stage of the interaction, while the populations of different states
will reflect the temperature at the time of freeze-out. If this is the
case, then freeze-out appears to occur at a temperature of around 5 MeV

for interactions in the 35-60 MeV/nucleon range.

F. Final State Interactions Vs. Emission Of Particle Unstable Nuclei

One of the most important questions that arises in the measurement
of the nuclear temperature through the measurement of particle unstable
states is the question of at what point does one consider a particle to
have been formed and emitted. In particular, is the emission of particle
unstable nuclei the same as in-flight final state interactions or is it a
totaly different process? If the latter is the case, then one is only

interested in those two-particle correlations that come from the decay of

122

emitted unstable nuclei when extracting the temperature of the emitting
source. The problem then arises of how to eliminate the in-flight final
state interactions from the observed two-particle correlations. This
could be accomplished if one could determine a unique source size for all
particles. One would then be able to use the calculated correlation
function for the nuclear and Coulomb final state interactnnusin the
extraction of the populations of the particle unstable states.

The problenlof'determining whether final state interactions are the
same as emission of particle unstable nuclei can be examined by looking
at the trends shown in the two-particle correlations. As mentioned
earlier correlations involving two particles that exhibit resonances have
a consistently smaller extracted source radii then those particle pairs
with no resonances. It has been suggested that differences in the time
of freeze-out for different particles may be responsible for these
different radii [B0 86]. On the other hand if all particles come ownn 3
source that is the same size and if the emission of particle unstable
nuclei and final state interactions are different processes, then
correlations involving particles with resonances will be enhanced and a
smaller source radii will be extracted for these correlations. In
addition the difference between the extracted source radii for the two
peaks in the d-a correlations [Ch 86a] could then be understood 1J1 terms
of a greater enhancement of the lower lying state leading to a smaller
apparent radius for it.

It is also interesting to look at the behavior of two-proton
correlations as a function of the size of the target. 'Two-jnwaton
correlations at small relative~momentum have been studied for 25

16 12 2

MeV/nucleon O + C, 7A1 [Be 85] and 197

Au [Ly 83]. The peak of the

correlation function increases with increasing target size. Using

123
Koonin's model [Ko 77] this leads to a smaller extracted source radii as

one increases the target size. If, on the other hand, one interprets the
data in terms of the decay of 2He then the data may be explained in terms

of the 2He cross section increasing with increasing source size.

The two-particle correlations have been observed to be stronger for
higher two-particle total energy [Ch 86a, Po 86a, Po 86b], leading to a
smaller extracted source radius. It has been suggested that this may
mean that higher energy particles tend to come from smaller sources [Po
86b]. On the other hand, in the case of correlations with resonances
this might be due to there being a greater cross section for the emission
of the article unstable nuclei at a high energy than the emission of two
lighter nuclei of equal energy.

It is known from the large angle correlation presented in Chapter IV
that the decay of particle unstable nuclei contribute heavily to the two-
particle correlation function at relatively large angles (10°-25°). For
the system nearest 35 MeV/nucleon N+Ag, 110 MeV/nucleon C+Ag, the large
angle correlations shown in figure IV-11 indicate that there are about
twice as many a-a coincidences on the same side of the beam as on
opposite sides of the beam. In p-d and d-d correlations it was shown

that ‘there is a slight favoring of the opposite side over the same side

for this system. The excess clearly must come from the emission of 8Be
and not from two alphas that have final state interactions.
Using Boal's [Bo 86] calculations of the final state interaction

including both the nuclear and Coulomb potentials for a radius of 7.0 fin

the cross section for the first excited state of 8Be was extracted in the
same fashion as described in section D. The cross section was found to
be 050720.093 mb compared to 0.69020.097 mb obtained from the Coulomb

only calculation. In addition to being smaller by about 251 the cross

12”
section is now much more sensitive to the source radius that is used.
Unfortunatedy the full nuclear and Coulomb calculation exists for only a
limited number of the two-particle correlations that have resonances in
them. In addition in the present experiment the experimental resolution
has good enough to perform the nuclear plus Coulomb background
subtraction for the p-p and a-a cases only. The effect of subtracting a
background that assumes a component coming from in-flight final state
interactions would be to lower the extracted cross sections of particle

unstable states and hence reduce the extracted temperatures.

Chapter VI

Summary and Conclusions

Single particle inclusive kinetic energy spectra and two-particle
large-angle correlation functions have been measured for 40 and 50
MeV/nucleon C induced reactions. The light particle energy spectra have
been well fdJ;Ivith a triple moving source parameterization assuming the
emission of particles from three distinct moving sources all moving in
the beam direction. One source, the projectile—like source, has a
velocity of about 85% of the incident beam velocity and a temperature of
about “-5 MeV. The second source, the target-like source, has a velocity
of about 5-10% of the incident beam velocity with a temperature of 3.5-
4.5 MeV. The spectra are dominated by the third source, the intermediate
velocity source, which has a velocity intermediate between the projectile
and target velocities and a temperature of 10-15 MeV. Heavier fragments
(Li-C) were measured at forward angles, 0:15°-45°. .These particles were
found to come primarily from the projectile-like source.

Two-particle in-plane correlations were measured with tag detectors
at 0=-25° and 45°. For both tag angles the C+C systems showed a very
strong preference for emission of two particles to opposite sides of the
beam. The enhancement in opposite side emission increases as the total
mass of the observed particles increases. Very little difference was
observed between the two beam energies, 110 and 50 MeV/nucleon. For the
heavier systems, C+Ag and C+Au, the correlations were more symmetric
about the beam axis with opposite side correlations only slightly
enhanced over same side correlations. For all four systems the decay cM‘
particle unstable light nuclei contributed heavily to the two-particle

large-angle correlations. For an opening angle of 10° , enhancements in

125

126
. . 11 11 11
the correlation function are observed for p-p, p-t, p- He, d- He, t- He

and “He-“He. The contribution to the correlation function for particle

unstable resonances can be as high as 80% (d-uHe for no MeV/nucleon C+C).
Out of plane correlations were also measured with tag telescopes at O=25°
and 145° at 0:90°. The correlations were found to be nearly flat with a
slight decrease at more forward angles. The ratio of in-plane to out-of-
plane correlations shows an enhancement in the emission of particles in
the same plane. As in the in-plane correlations, the enhancement is
largest for the light systems and for higher total mass of the observed
particles.

In order to determine the extent to which conservation of momentum
effects the measured correlations, momentum conservation calculations
were carried out. Using a momentum conservation calculation
incorporating emission from all three sources the general trend of the
in-plane 45° tag correlations were reproduced. The overall agreement was
not as good for the correlations with the 25° tag, perhaps due to the
increased importance of correlations involving particles from different
sources. For the out-of-plane correlations the agreement between the
momentum conservation calculations and the data was good only for the
heavier systems. The momentum conservation calculation did a reasonably
good Job of explaining the ratio of in-plane to out-of-plane correlations
for the heavy systems. For the light systems the calculation
consistently over predicted the ratio of in-plane to out-of—plane
correlations, this may in part be due to correlations between two
particles from different sources.

A close packed array of sixteen phoswich telescopes positioned
behind a multi-wire proportional counter was used to measure two-particle

correlations at small relative momentum for the reaction 35 MeV/nucleon

 

 

127
N+Ag. The correlations were measured at central angles of O:35°, 115°,
60° and 80°. The correlations have been used to extract information
about the space-time extent and temperature of the intermediate velocity
source. Source radii extracted from two-particle correlations are in
good agreement with the radii extracted by previous authors. The radii
are smallest for particle pairs with resonances, pp, pa, do and cm: (“.5-
5.0 fin), and largest for nonresonant pairs, pd, dd and tt (6.3-9.5 fin).
Little difference was found in the extracted source radii as a function
of angle and beam energy. The radii are somewhat larger than those
extracted using the coalescence’model due, in part, to neglecting the
lifetime of the emitting source. If a nonzero source lifetime is used
then smaller source radii are obtained.

One important question that has come up is when is a particle
considered to have been emitted from the source? In particular are
particle unstable nuclei emitted from the source or are all observed
resonances the results of in-flight final state interactions? 11‘
particle unstable nuclei are emitted in addition to other particles
experiencing in-flight final state interactions, then the observed two-
particle correlations are the results of a combination of the two
processes. If this is the case, then radii extracted from particle pairs
with resonances will be smaller than radii extracted from nonresonant
pairs because the decay of particle unstable light nuclei will enhance
the correlation functions which leads to a larger extracted radius. If,
on the other hand, all observed resonances are the results of in-flight
final state interactions then the differences in the extracted radii may
indicate different emission times for different particles or perhaps that

different particle pairs come from different types of collisions.

128

The difference between the source radii extracted using particle
pairs with resonances and those without resonances has also been
attributed to different freeze-out times for different particles. Those
particles that freeze-out earlier will yield a smaller source radius than
particles that freeze-out at a later stage of the interaction. In this
view information about the source at different times may be obtained by
looking at different particles.

Using the populations of bound and unbound states measured in the

present experiment and the Y and neutron emitting states measured by

+2.8
-2.‘1

been extracted using the quantum statistical model of Hahn and Stécker in

Morrissey et al. and Bloch et al. a source temperature of 9.8 MeV has
order to correct for feeding from higher lying states to the observed
states. Overall the quantum statistical model does an excellent Job of
fitting the populations of all the measured states at the same time. The
use of the quantum statistical model to take into account feeding from
higher lying states has eliminated the previous descrepency in the
temperature between measurements using particle bound Y-ray emitting
states and measurements using particle unbound states. The temperature
was extracted using the quantum statistical model of Hahn and Stécker in
order to correct for feeding from higher lying states to the observed
states. The temperature is in good agreement with the temperature
extracted from particle unbound states for 35 MeV/nucleon N+Au and 60
MeV/nucleon Ar+Au. The temperature extracted from the population of
states is about 10 MeV lower than the temperature extracted from the
kinetic energy spectra. The reason for this disagreement may be that the
two temperature measurements represent the temperature of the emitting
source at different times. The kinetic energy spectra are fixed early in

the interaction while the populations of the states change as the source

129
expands and cools. Thus the temperature measured from the populations of
the states represents the temperature at the time of freeze-out. If this
is the case, then the freeze-out temperature does not change measurably
for beam energies between 35 and 60 MeV/nucleon.

In conclusion it has been shown that two-particle large angle
correlations are consistent with emission from a thermally equilibrated
system. While the measured two-particle correlations are not isotropic“
as emission from a thermally equilibrated source would require, the
effects of momentum conservation requirements have been shown to account.
for most of the deviation from an isotropic distribution. As the number
of nucleons in the emitting source increases the correlation functions
become more isotropic.

The use of the quantum statistical model has resolved the
«discrepency between temperature measurements using the populations of Y-
ray emitting states and those using particle unstable states. The
temperature extracted from the populations of nuclear states is still
considerably lower than the temperature extracted from the kinetic energy
spectra. Further work is needed to eliminate the contribution of final
state interactions to the populations of the particle unstable states.
Theoritical calculations are needed for the nuclear final state
interactions of several particle pairs and either a theoritical or

experimental value for the freeze out radius is needed.

APPENDIX

Appendix

Momentum Conservation Model

The momentum conservation model used in Chapter IV is based
treatment of Lynch et al. [Ly 82]. The model assumes that particles are
emitted isotropically in the rest frame of a moving source of size A
nucleons with a kinetic energy spectra given by Eqs. (IV-1) and (IV-2).

The emission of the first particle of mass m with lab momentum Euab at

1

lab angles 91, d> changes both the temperature and momentum of the

1

emitting source. In the rest frame of the moving source the Lorentz

transformations for the first particle give

p1x:Y(p11ab°°391'BE1lab)’ (A-1)
p1yzp1labsin61cos¢1, (A-2)
p1z=p1labsin01sin¢1, (A-3)
where

Y=(1-32)"/2 (A-N)

and B: the velocity of the moving source in the lab frame. The x-
component is parallel to the beam direction. The cross section in the
moving source rest frame is given by
2 -E /T
e

1 = 01 1 , (A-S)

3
p1dp1d01 lHum1 K(m1/T)

130

131

where
K(m1/I)=2(T/m1)2K1(m1/T)+(T/m1)K0(m1/T), (A-6)

K0 and K1 are modified Bessel functions of the second kind, 001 is the
cross section for particle 1 from the moving source fits of Chapter IV,
and 1 is the source temperature from the moving source fit. The cross

section in the lab frame is given by

 

 

2 -E /T
d °1iab = pllabE1°O e 1 . (A-7)
3
dE1labdfl1lab u"m1 K(“‘1”)

The source velocity in the lab frame is now given by

. B-p1x/m'
Bx = W , (A-8)
B; : -p1y/mé , (A-9)
B; = -p1z/mé , (A-10)
where m; = AmO-m1 (m0: 931.5 MeV). The excitation energies before, E2X ,

and after, E;, the emission of the first particle are related by
E' = E -(e -m )-(6 )2/2m' (A-11)
x x 1 1 1 s '

The initial excitation energy is given by the empirical relation

132

Ex: A12/(8 MeV), (A-12)
and the temperature of the recoiling source may be expressed as

1' = (E;-8 MeV/A')1/2. (A-13)
'The source may now emit particle 2 with mass m2, lab momentum p21ab

' O
at lab angles GZlab’ ¢21ab' The second particle s momentum transformed

in to the source rest frame is given by

cos92-B'E ), (A-1u)

_ !
p2x’Y (p21ab x Zlab

sinezcosé -m B , (A-15)

p2y‘p21ab 2 2 y

p22=p213bsin02cos¢2-m282. (A-16)
The source rest frame and lab frame cross section for the emission of
particle 2 are given by Eqs. (A-5) and (A-7) by changing subscript 1 to 2
and T to I'.

The two-particle coincidence cross section for the emission of
particle 1 followed by the emission of particle 2 is proportional to the
product

dzo

. » 1lab(p1’B’T’m1) d 021ab(p2’B ’1 'mz)
P(p1,p2) = dB

. (A-17)
1labd01lab dE21abd921ab

 

Experimentally it is impossible to tell which particle is emitted first,
hence we also define 9(52,61) in the same manner as 9611,62) using Eq.

(A-17) by exchanging the subscripts 1 and 2 and by replacing 1' and B' by

133
1".and B". The new parameters 1" and 8" are defined in the same
manner as I' and B' using Eqs. (A-8) through (A—13). The two-particle
coincidence cross section is now given by

duo

dE1d01dEzd02 ‘ Co[P(p1'p2)*P(Pz’p1)]: (A-18)

 

where CO is a normalization constant taken to be

CO : ORA/2 , (A-19)

with oR is the total reaction cross section given by

2
0R _ 10nr (1-VC/Ecm) (A-20)
where
_ 1/3 1/3
r'1'2(AproJ+Atarg'1)’ (A-21)
VC=1.HH ZtargzproJ/r’ (A-22)
E:cm:TbeamAproJAtarg/(Atargmproj)’ (A'23)
where A and A are the number of nucleons in the projectile and
proJ targ
target, respectively, zproJ and ztarg are the number of protons in the

projectile and target, respectively, and T is the total beam energy

beam
in MeV.

13A
To compare the momentum conservation model to the data Eq. (A-18) is
integrated over E1 and E2 using the energy ranges listed in Table 1v-u in
3 MeV steps. The values of 001, 002, T and B are taken from Table IV-2
for the appropriate source. The calculation is also integrated over the
source size A. Each source is given a weight dW=2nbodb-A, where b is the
impact parameter and A is the number of nucleons from a fireball model

calculation [We 76, Go 77].

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