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This is to certify that the

dissertation entitled

TWO-PROTON AND TWO-NEUTRON CORRELATION FUNCTIONS FOR
INTERMEDIATE-ENERGY HEAVY-ION REACTIONS

presented by

Sally Janine Gaff

has been accepted towards fulﬁllment
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TWO-PROTON AND TWO-NEUTRON CORRELATION FUNCTIONS FOR
INTERMEDIATE-ENERGY HEAVY-ION REACTIONS
By

Sally Janine Gaff

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

1 997

ABSTRACT

TWO-PROTON AND TWO-NEUTRON CORRELATION FUNCTIONS FOR
INTERMEDIATE-ENERGY HEAVY-ION REACTIONS

By

Sally Janine Gaff

Two-particle intensity interferometry is used to study the space-time
characteristics of the reaction zone formed in intermediate-energy heavy ion reactions.
The results are compared to appropriate models.

Because of the recently reported discrepancies between BUU calculations and
measured two-proton correlation functions at beam energies E/A>100 MeV, a two-proton
correlation function has been measured for the reaction 16O + 197Au at E/A=200 MeV.
The measured total correlation function agrees with the BUU predictions, but the
dependence of the correlation function on the total momentum predicted by the BUU
model is larger than the data show.

In the evaporative regime, questions arise in interpreting charged particle
correlations because of the Coulomb ﬁeld of the heavy residue. Therefore, in this energy
region two-neutron correlation functions were measured. Two-neutron correlation
functions for the reaction 40Ar + 165H0 at ElA=25 MeV are presented and comparisons
made to models. Corrections and event selection are made for the coincident neutrons to
eliminate the effects of crosstalk and background scattering. These corrections are

compared to two independent simulations. The experimental results show a strong

correlation at q = 0, which is comparable to emission from the surface of a sphere with a
radius of 7 ﬁn and an exponential lifetime of 700 fm/c. An evaporative model with and
without preequilibrium corrections was compared to the data. The measured lifetime is

within a factor of two of the lifetime the evaporative model predicts. This comparison

provides a useful test of the statistical models used for ﬁssion lifetime measurements.

To the glory of God

iv

ACKNOWLEDGMENTS

I would ﬁrst like to thank my research advisor, Konrad Gelbke. He has provided
invaluable guidance and motivation. Every time I talked to him my work seemed more
valuable and more interesting.

I would also like to thank the members of my guidance committee: Aaron
Galonsky for many conversations about the Neutron Walls, Wolfgang Bauer for running
BUU calculations for me, and Bernard Pope and SD. Mahanti for their input and for
adding breadth to the committee.

Other professors at the Cyclotron have provided helpful information and ideas
about my research and what it means to be a research scientist. I would like to mention
Thomas Glasmacher, Betty Tsang, Bill Lynch, Ed Kashy, Sam Austin, Mike
Thoennessen, Dave Morrissey, and Paul Mantica. I especially want to thank Scott Pratt
for explaining the ﬁner points of correlation functions.

The people whose hard work made the two experiments included in this thesis
possible deserves special recognition. The two-proton experiment involved T.
Glasmacher, E. Gualtieri, D.O. Handzy, S. Hannuschke, R. Lacey, L. Martin, C.P.
Montoya, R. Pak, N. Stone, A.M. Vandermolen, G.D. Westfall, and J. Yee along with
theoretical help from W. Bauer, F. Dafﬁn, and K. Haglin. The two-neutron experiment

involved A. Galonsky, C.K. Gelbke, T. Glasmacher, M.J. Huang, J. Kruse, G.J. Kunde,

R. Lemmon, WJ. Lynch, 8. Pratt, M.B. Tsang, J. Wang, P. Zecher, F. Deak, A. Kiss, A.
Horvath, Z. Seres, K. Ieki, and Y. Iwata.

Many students at the Cyclotron have been good friends. I would like to thank my
ofﬁcemates who kept my life interesting: Mathias who was always interested in my
newest car adventures and was willing to help even though my car never broke in the
ordinary ways, Mike for his attempts to help me see how bad life really is and his ever
helpful critiques of my data, J ing for helping me with Neutron Wall questions, Don who
provided stories for my ﬁle, and also David for reminding me what it is like to start
graduate school. I would also like to thank Raman Pfaff, Chris Powell, Jac Caggiano, Jon
Kruse, Phil Zecher, Damain Handzy, Dave Brown, Gerd Kortemeyer, and Bettina
Huebner for various interesting conversations in the ofﬁce, at conferences and at the
Peanut Barrel. I would especially like to thank Njema Frazier for being a great roommate
and for being around when I was sick of talking about cars and football.

Other people at the Cyclotron were essential in my ﬁnishing this dissertation.
Gerd Kunde provided a lot of assistance in analysis and with PAW questions. Razvan
Popescu was also helpful in answering PAW questions and making the research group
meetings more interesting. I would like to thank Rich for help with technical questions
and organizing breakfasts. I am also indebted to Craig Snow and Christine O’Conner and
the rest of the staff at the Cycltron. I also appreciate being included as a LOL.

I would also like to thank those who helped with the actual preparation of my

thesis. This help includes the writing group, initiated by Betty Tsang, organized by

vi

Sharon Thomas, with participants Kim Kessler, Ray and Xi. I especially want to thank
Amy Hirshman for reading the whole manuscript and ﬁnding many grammar mistakes.

I want to thank my other friends who made my stay at Michigan State University
more fun, including friends from University Reformed Church and Graduate InterVarsity.
IfI started to list them all, I would need to add many more pages to these
acknowledgments. I thank them all.

My family deserves special thanks for their support. My Mom and Dad for
thinking graduate school is a worthwhile thing to do. And my sister for her conﬁdence in
me expressed in her frequent question, “ Did you fail out of grad school yet?” I also thank
her for taking a little longer to ﬁnish so my little sister would not graduate before me.

Last I would like to thank Misha for his input on both my thesis and my practice

talks. His friendship made my life much better during these last few stressful months.

vii

TABLE OF CONTENTS

LIST OF TABLES ............................................................................................................. xii
LIST OF FIGURES .......................................................................................................... xiv
Chapter 1 - Introduction ..................................................................................................... 1
l .1 Intermediate-Energy Heavy-Ion Collisions .......................................................... 1

1 .2 Particle Intensity Interferometry ........................................................................... 3

l .3 Motivation ............................................................................................................ 6

1 .3 .1 Two-Proton Intensity Interferometry ........................................................ 6

1 .3 .2 Two-Neutron Intensity Interferometry ...................................................... 7

Chapter 2 - Two-Proton Intensity Interferometry: 16O + '97Au at ElA=200 MeV ............ 10
2.1 Motivation ........................................................................................................... 10

2.2 Setup .................................................................................................................... 12

2.3 Results ................................................................................................................. 14

Chapter 3 - Two-Neutron Intensity Interferometry: Experimental Setup and Data

Reduction .......................................................................................................................... 21
3.1 Experimental Setup ............................................................................................. 22
3.1.1 Fragment Telescopes ................................................................................ 22

3.1.2 Neutron Walls ........................................................................................... 25

viii

3.1.3 Other Experimental Details ...................................................................... 28

3.2 Electronics ........................................................................................................... 28
3.2.1 Fragment Telescopes ................................................................................ 28
3.2.2 Neutron Walls ........................................................................................... 29
3.2.3 Trigger Electronics ................................................................................... 33

3.3 Data Reduction .................................................................................................... 34
3.3.1 Fragment Telescopes ................................................................................ 34
3.3.2 Neutron Walls ........................................................................................... 37

3.3.2.1 Neutron Energy ............................................................................. 37

3.3.2.2 Neutron Position ............................................................................ 40

3.3.2.3 Neutron Particle Identiﬁcation ...................................................... 46

3.3.2.4 Neutron Light ................................................................................ 50

3.3.3 Measured Background .............................................................................. 57

Chapter 4 - Two-Neutron Intensity Interferometry: Measured Correlation Functions ...... 63

4.1 Constructing a Correlation Function ................................................................... 63

4.2 Experimental Correlation Function ..................................................................... 67
4.2.1 Event Selection: Individual Neutrons ....................................................... 67
4.2.2 Event Selection: Crosstalk Correction ...................................................... 72
4.2.3 Background Corrections ........................................................................... 82
4.2.4 Singles vs. Mixed-Events Technique ....................................................... 86
4.2.5 Front-Wall Pairs ....................................................................................... 90

4.3 Longitudinal and Transverse Correlations .......................................................... 91

ix

Chapter 5 - Two-Neutron Intensity Interferometry: Model Simulations of Detector

Response ............................................................................................................................ 94
5.1 Description of Models ......................................................................................... 94
5.1.1 Neutron Interactions ................................................................................. 95

5.1.2 Neutron Initialization and Detector Response .......................................... 99

5.1.3 GEANT Simulation Code ....................................................................... 104

5.1.4 Wang Simulation Code ........................................................................... 105

5.2 Simulation Results ............................................................................................. 107
5.2.1 Efﬁciency ................................................................................................ 107

5.2.2 Crosstalk ................................................................................................. 107

5.2.3 Resolution of Relative Momenta ............................................................ 118

5.2.4 Correlation Function ............................................................................... 1 18

Chapter 6 - Two-Neutron Intensity Interferometry: Interpretation ................................. 129
6.1 Surface Emission Model ................................................................................... 129

6.2 Evaporative Model ............................................................................................ 130

6.3 Evaporation Model Plus Preequilibrium ........................................................... 140

6.4 Using Surface Emission with Two Components ............................................... 148

6.5 Preequilibrium and Time-Adjusted Evaporation Model ................................... 151

6.6 Summary and Outlook ....................................................................................... 151
Chapter 7 - Conclusion .................................................................................................... 155
Appendix - The Koonin-Pratt Formalism ...................................................................... 158
BIBLIOGRAPHY ........................................................................................................... 161

LIST OF TABLES

Table 4.1 Crosstalk cuts applied to the data. The difference in calculated and

measured scattering angles is gated on AC_ < cos0’-—cos0 < AC+; the
difference in the time between hits calculated and measured is gated by
AT_ < tn, — (t2 — t1) < AT+; the difference in energy deposited in the
second detector calculated and measured is gated on En, - Ep(Lz) > AB ............. 78
Table 5.1 Possible interactions between a neutron and the scintillator. For
illustration, cross sections for the reaction, on, are given for a 24 MeV neutron

[Wang 97]- 96

Table 5.2 Parameters used to model the light response. These values were

 

determined from a fit of the data given in [Verb 68] with the function L = a (l-

 

exp(b Eﬁ)+dE+f. 100
Table 5.3 Detector resolution for time, position, and light, used as input in models.
The resolution is assumed to be Gaussian with this standard deviation. ......... 103
Table 5.4 Crosstalk cuts applied for both models. The difference in calculated and

measured scattering angles is gated on AC_ < cos0’—cos0 < AC+; the

difference in the time between hits calculated and measured is gated by

AT_ < tn, — (t2 — t1) < AT+ ; the difference in energy deposited in the second

 

detector calculated and measured is gated on En, - Ep(Lz) > AE 114
Table 5.5 Statistics for two classes of events A. IPI < 250 MeV/c and B. I i”: > 250
MeV/c. The second column gives the number of events total, real, and crosstalk
events in each class for both GEANT and Wang models when no cuts have been
applied. The other columns show the ratio remaining in that category after the

cut has been applied. The cuts are defined in Table 5.4. 117

 

xii

LIST OF FIGURES

Figure 1.1. Comparison of amplitude and intensity interferometry. a) An amplitude
interferometer measures light intensity after the interference of two signals

from a source. b) An intensity interferometer measures single and coincident

 

yields from a single source. 4
Figure 2.1 Proton correlation function for ”Ar + 1”Au reaction at E/A = 200 MeV

[Kund 93] 11

 

Figure 2.2 Schematic drawing of the experimental setup: the 56-element hodoscope

attached to the MSU 41c array. 13

 

Figure 2.3 Energy distribution of protons measured in the laboratory rest frame at
e ..., = 30°. 45° for collisions of 16o + ”As at EIA = 200 MeV (solid points) and
predicted by BUU (solid line). To allow a comparison of shapes, the relative
normalization between experimental and theoretical energy spectra was chosen

to give equal areas for Eproton > 50 MeV. 15

 

Figure 2.4 Two-proton correlation functions for collisions of 1‘O 4» 19"Au at E/A =
200 MeV. The top panel shows the energy-integrated correlation function, and
the bottom panel shows correlation functions for proton pairs selected by the

indicated cuts on their total momentum 17

 

xiii

Figure 3.1 Arrangement of silicon telescopes around the target at angles of 30° and

45° and a distance of 15 cm. 23

 

Figure 3.2 Schematic drawing of the setup. The fragment telescopes were used for
event selection and the neutron detectors measured coincident neutrons. ........ 24

Figure 3.3 Picture of one detector from the Neutron Walls. 26

 

Figure 3.4 Mechanical drawing of one of the Neutron Walls. The glass cells are
protected by an aluminum cover, which is sealed to prevent light from reaching

the PMTs. 27

 

Figure 3.5 E versus AE showing the hardware cut. The counts in the lower left
corner have been reduced by the hardware trigger. Those remaining come from
events triggered by another Si detector. The lack of 8Be gives a useful reference

for isotope identiﬁcation. 30

 

Figure 3.6 Electronics for the Si telescopes. TPO = timing pick off; CFD = constant
fraction discriminator; AD811 is a peak-sensing analog-to-digital converter... 31
Figure 3.7 Schematic diagram of the trigger electronics from the experiment. The

elements marked D8 are downscalers. 35

 

Figure 3.8 PID for silicon telescopes. This plot of energy versus particle
identification shows the clear separation between isotopes. The gates are also

plotted. 36

 

Figure 3.9 Walk correction for the timing signal from the leading edge
discriminator. The top panel shows the time for the prompt gamma peak

measured by the Neutron Wall, Tn, versus the AE signal before correction. The

xiv

line is the fit used to correct for walk. The bottom panel is the same plot after

the walk correction 0t(E) has been applied. 39

 

Figure 3.10 Time spectra for the Neutron Walls after all correction, Ty - TOF + 500

ns. The bottom panel is an expansion of the region around the prompt gamma

peak. 41

 

Figure 3.11 Energy spectra for the Neutron Walls (front and back detectors). These

spectra are not efficiency corrected. 42

 

Figure 3.12 Position spectra for the Neutron Walls. A shadow bar creating a
shadow of 36 cm was placed in front of one section of these detectors, creating a
large suppression in the yield. The other structure in this spectra is the result of

electronic crosstalk. 44

 

Figure 3.13 The coincidence signal for one detector either in a front-back (top
panel) or front-front (bottom panel) pair divided by the singles yield for that

detector. 45

 

Figure 3.14 Neutron-gamma discrimination for the high energy neutrons in the
Neutron Walls. The plot of “Fast” versus “Attenuated Total” signal shows a

separation between neutrons and gammas. 47

 

Figure 3.15 Neutron-gamma discrimination for the low energy neutrons. “Total”
signal versus a particle identification, PID, chosen to straighten and expand the

spectra. See text for definition of PID 48

 

Figure 3.16 Timing spectra for the Neutron Walls. The top panel shows the time

spectrum for all hits. The middle panel is gated on neutrons identified by gates

on either the high or low energy discrimination plot. The bottom panel is gated

on gammas identiﬁed by either gate in the discrimination plots. 49

 

Figure 3.17 Light spectra showing the Compton edges used to calibrate the light
signal for the Neutron Walls. The sources and the energy of the Compton edge

are given in the figure. 51

 

Figure 3.18 Deﬁnition of x for the determination of the left and right light signals.52
Figure 3.19 Neutron energy versus the highest light yield. The solid line shows a ﬁt
to Verbinski data while the dashed line shows a linear ﬁt empirically chosen

from the data. 53

 

Figure 3.20 Light output versus energy of the neutrons. Neutrons that come directly
from the target cannot have light corresponding to a proton energy greater
than the neutron energy. The line shows the cut made to exclude background

neutrons. 55

 

Figure 3.21 Kinematics of neutron scattering. Variables in the laboratory frame are
shown in italics while variables in the center of mass frame have a standard

font. 56

 

Figure 3.22 Light yield at a given neutron energy. A ﬂat distribution from threshold
to maximum possible light is expected. The four panels represent different
gates on the energy of the neutrons. The gate applied is speciﬁed in the

panel. 58

 

xvi

Figure 3.23 Energy dependence of the ratio of background/total neutrons. The yield
from a shadow bar run is divided by a standard run and normalized by the

total beam current in these runs. 60

 

Figure 3.24 Background/total ratio for each detector in the Neutron Walls. ........... 61
Figure 4.1 Theoretical two-neutron correlation function. Surface emission from a
sphere of radius 7 fm with a lifetime of 700 fm/c. The solid line shows the
standard two-neutron correlation function. The dotted line shows the
correlation from only quantum statistics with no f’mal-state interactions. ....... 65
Figure 4.2 Spatial distribution of emitted particles of velocity v from three sources.
(a) Small source with a short lifetime (b) Large source with a short lifetime (c)

Small source with a long lifetime. 66

 

Figure 4.3 Deﬁnition of longitudinal and transverse directions. Transverse samples
the small dimension of the source while longitudinal samples the lifetime

dimension. 68

 

Figure 4.4 The position versus the geometric mean of the measured light for one

detector. A l MeVee threshold was applied to each PMT. 70

 

Figure 4.5 Elastic scattering kinematics, for a neutron scattering from a proton.
These formulas are used to calculate the energy and direction, Eat and 9’, of the

scattered neutron. 73

 

Figure 4.6 The variables involved in the crosstalk correction. Eat, Lat, tot are
calculated assuming elastic scattering, while the others are measured

quantities. 74

 

xvii

Figure 4.7 Crosstalk criteria. (a) Difference in angles calculated and measured. (b)
Difference in time between hits, calculated and measured. (c) Difference in the
energy of the scattered neutron and the energy corresponding to the light
measured in the second detector. The cuts are shown on the ﬁgure and deﬁned

in Table 4.1. 77

 

Figure 4.8 Crosstalk criteria constructed from mixed singles where no crosstalk is
present. Same variables as Figure 4.7. Cut 1 is shown in this case. .................... 79
Figure 4.9 Difference between measured and calculated angle versus time between
hits for real coincidences. The dashed lines show where the cuts for crosstalk
were made for Cut 1 and the dotted lines show Cut 2. The solid line is where

both cuts are the same. 80

 

Figure 4.10 The experimental correlation function with and without a crosstalk cut.
The neutron pairs have I P l < 250 MeV/c. The cuts are deﬁned in Table 4.1. 81
Figure 4.11 The correlation functions cut on total momentum of the neutron pair.

Crosstalk Cut 1 was applied to both spectra. 83

 

Figure 4.12 The correlation function corrected for background scattering (circles).

The uncorrected correlation function is shown for comparison (triangles).

(Both have IP I < 250 MeV/c and crosstalk Cut 1 applied.) 87

 

Figure 4.13 Correlation functions for front-front pairs (triangles) and front-back
pairs (circles). Both are corrected for background and have IPI < 250 MeV/c.
Cut 1 is applied to the front-back pairs and the cut described in the text applied

to the front-front pairs. - 92

 

xviii

Figure 4.14 Longitudinal and transverse two-neutron correlation functions with IPI

< 250 MeV/c. 93

 

Figure 5.1 GEANT total cross-sections for neutrons scattering on hydrogen and on

carbon as a function of the neutron energy. 97

 

Figure 5.2 Light response for electrons, protons, alphas, and carbons. These
response curves are ﬁts to the data from Verbinski [Verb 68, Wang 97]. The

light is measured in MeV electron equivalent, the light produced by a 1 MeV

electron. 98

 

Figure 5.3 Position of the interaction in the cell versus the measured light signal. A
light threshold of l MeVee at the left and right PMT is applied in both the data

and the simulation. 102

 

Figure 5.4 Efficiency predicted by Wang and GEANT simulations as a function of
neutron energy. The light threshold was set at 1 MeVee for left and right light

signals. - 108

 

Figure 5.5 Crosstalk criteria for simulations from the Wang model. The solid line is
for no crosstalk cut, while the dashed line shows the results after Cut 2 is
applied. (a) Difference in angles calculated and measured. (b) Difference in time
between hits, calculated and measured. (c) Difference in calculated energy and

energy of proton from light measured in the second detector 110

 

Figure 5.6 Same as Figure 5.5 but from the GEANT model. 111

 

xix

Figure 5.7 Same as Figure 5.5 for events that are crosstalk in the Wang model. The
solid line is events with no crosstalk cut, the dashed line is after Cut 2 is

applied, and the dotted line is after Cut 3. - 112

 

 

Figure 5.8 Same as Figure 5.7, but for the GEANT model 113
Figure 5.9 Momentum dependence of crosstalk. The top panel shows the real
coincidences from two different neutrons before and after Cut 2 is applied. The
bottom panel shows the fraction of crosstalk/total events before and after Cut 2

is applied. 116

 

Figure 5.10 Width of measured relative momentum for a given real momentum,

either 4 or 8 MeV/c, from the GEANT simulation. - 119

 

Figure 5.11 The total correlation function from the GEANT simulation. The line
shows the theoretical correlation function for a source with r = 7 ﬁn and 1: = 700

fm/c and the points show the results of the simulation after various crosstalk

cuts. I Pl < 250 MeV/c 122

 

Figure 5.12 The longitudinal and transverse correlation functions from the GEANT

simulation without crosstalk corrections. Longitudinal is selected by \v < 60°

 

and transverse is selected by w > 80°. lPl < 250 MeV/c- 124

Figure 5.13 Longitudinal and transverse correlation functions from the GEANT

simulation. The lines show the theory. The points represent the simulation after

crosstalk cuts. I Pl < 250 MeV/c. 125

 

Figure 5.14 Total correlation function for high momentum pairs. IPI > 250 MeV/c.

The cuts are described in Table 5.4. 126

 

Figure 5.15 Longitudinal and transverse cuts for the high-momentum pairs. IPI >

250 MeV/c. The cuts are described in Table 5.4. 128

 

Figure 6.1 Fit of the total, ﬁltered correlation function from a surface model, r = 7

fm, with different lifetimes. 131

 

Figure 6.2 Best lifetime from the surface model for a given radius- 132

 

Figure 6.3 Correlation functions with longitudinal (‘1’ = 0 - 60°) and transverse cuts
(‘1’ = 80 - 90°). The data (points) are compared to ﬁltered calculations for
surface emission from a sphere with a radius of 7 ﬁn and a lifetime of 700 fm/c

(lines) 133

 

Figure 6.4 Evaporation-model calculations. The solid line is a calculation with the
excitation energy of E*/A = 3.93 and a = A110. For the dashed line the excitation
energy is reduced to E*IA = 2.0. The dotted curve has the level density
parameter increased to a = A18 with the original excitation energy. ................ 135

Figure 6.5 The measured energy spectrum compared to predicted energy spectra for
two different excitation energies in the evaporation model [Frie 83]. The energy

is plotted in the laboratory frame. 136

 

Figure 6.6 The measured correlation function compared to the evaporation-model

predictions with two different excitation energies 138

 

Figure 6.7 Time spectra from the evaporation model compared to the time spectrum

for an exponential distribution with “c = 2000 fm/c. - 139

 

Figure 6.8 Two-component ﬁts of the energy spectra. The lines with the steepest

slopes correspond to the evaporation component; the lines with a more shallow

slope correspond to the preequilibrium component; the sum for both ﬁts lies
close to the data points. Fit 1: Tc = 4.9 MeV, Tl, = 17.0 MeV, and f, = 9%. Fit 2:

Tc = 3.0 MeV, Tp = 8.0 MeV, and fl, = 45%. 142

 

Figure 6.9 Correlation functions for various two-component ﬁts with the
evaporation-plus-preequilibrium model. Fit 1: f, = 34%, Fit 2: fl, = 25%, Fit 3:

fp=22%,Fit 4: fp=8% _ 143

 

Figure 6.10 Energy spectra predicted by BUU calculations compared to the

data. 145

 

Figure 6.11 Comparing the correlation function from the data with a two-
component model with preequilibrium fraction f, = 18%. The low-energy
component is from the evaporation model. The high-energy component has Tl,

= 12 MeV and 1,, = 0 or 50 fm/c (solid or dashed lines). 146

 

Figure 6.12 Yield of particles as a function of time from BUU calculations. The line
is the yield from an exponential distribution with a lifetime of 50 fm/c. .......... 147
Figure 6.13 The best ﬁt to the data for the compound-source lifetime, 1,, for various
values of fp, assuming I, = 0. The error bars represent the range of reasonable

ﬁts. 149

 

Figure 6.14 Two-component model with the time for both components sampled from
exponential time distributions. The preequilibrium component has either 1,, = 0
and 50 fm/c while the compound component has 1:c = 1000 fm/c and 1:c = 900

fm/c, respectively. The preequilibrium/total fraction is f, = 18%. ................... 150

xxii

Figure 6.15 Two-component model, with the low-energy component taken from the

evaporation model with each emission time scaled by a factor s. The

 

preequilibrium component has fp = 18% and 1,, = 0. 152

xxiii

Chapter 1 - Introduction

1 .1 Intermediate-Energy Heavy-Ion Collisions

Since the beginning of this century, understanding the atom, which is the basic
building block of matter, has been a topic of interest. In 1906, Rutherford began a series
of experiments which showed that most of the atomic mass is found in a nucleus which
ﬁlls only 10'15 of the atom’s volume. Since this experiment, other scientists have been
trying to understand the nucleus and the forces that hold it together.

With the advent of modern accelerators, it is now possible to study nuclear matter
in conditions, such as extreme temperatures and densities, that do not occur in ordinary
terrestrial settings but may exist in the cosmos, e.g., in supernovae explosions or neutron
stars. Current heavy-ion accelerators probe a broad range of nuclear properties including
high-spin states, high excitation energy, and unusual isospin conﬁgurations. By varying
the initial collision conditions, different aspects of the nuclear interactions can be
explored. As an example, with different bombarding energies collisions are produced
which are sensitive to different properties of the nucleus.

At low bombarding energies (E/A S 10 MeV), the collision is dominated by mean '

ﬁeld effects and one-body dissipation. The colliding nuclei can form a long-lived
equilibrated source which decays statistically over a long time (100,000 fm/c = 3x10 "9
sec). Often these reactions are modeled by assuming fusion of the two nuclei and total

equilibration of the energy among the nucleons. In this model, particles are then emitted

2
statistically over a long lifetime. Topics that can be studied at low energies include
nuclear rotation, particle evaporation, ﬁssion mechanisms and the viscosity of the
nucleus.

At relativistic energies (E/A Z 1 GeV), the individual nucleon-nucleon

interactions become dominant. These systems form and decay in a short period of time
(10 fm/c), and the nucleus-nucleus collision is often pictured with a ﬁreball model, where
the energy deposited in the overlap between projectile and target is equilibrated and
released in an explosive expansion. The particles are produced and emitted on a very
short time scale (less than a few tens of fm/c). One topic studied in the ultrarelativistic
energy range is under what conditions nucleons are merged to form a plasma of the
constituent quarks and gluons.

Another important region for understanding the reaction dynamics of nuclear
collisions is the intermediate energy region. In this region both the mean ﬁeld and the
individual nucleon-nucleon interactions determine the dynamics, so several processes can
occur. A heavy residue can be formed after emission of initial particles, or the composite
source can fragment into many small clusters. Statistical mechanics suggests a description
of nuclear matter as having various states: liquid at normal temperatures and pressures,
and gas when the nucleus is heated so much that it breaks up into individual nucleons.
This phase transition is currently being studied [Naya 92]. In general these studies try to
provide an understanding of the equation-of-state of nuclear matter, the relationship of

pressure, volume, and temperature.

3
At all energies, interpretation of nuclear reactions requires understanding the
dynamics, size, and time scale of the reaction. The focus in any one experiment is usually
on understanding one part of the reaction process by applying a certain technique. For
example, information about the space-time characteristics of the source can be obtained

by analyzing coincident particles using particle intensity interferometry.

1 .2 Particle Intensity Interferometry

The ﬁrst interferometer measured the fringe patterns produced when light from a
coherent source travels along two different paths and interferes, e.g., the two-slit
experiment. The interferometer is sensitive to the differences in phase between the two
signals when they reach a detector. In one early interferometry experiment, Michelson
tried to measure the earth’s motion relative to the ether with an improved interferometer,
and his null result provided evidence of wave propagation without the ether. Since that
interferometry measurement, many improvements and applications have been made for
interferometers.

One astronomy application uses an interferometer to measure star diameters [Hari
85]. The star emits light from points on its surface, as illustrated in Figure 1.1a. The
photons from points 81 and S2 pass through slits at P1 and P2, and interfere either
constructively or destructively at the detector. The visibility of an interference pattern, or

coherence, can be shown to depend on the angular diameter, 20, of the star [Stee 83]. If

the ﬁrst zero of the interference pattern is found, it can be translated into angular diameter

Amplitude Interferometry

I Screen I

 

 

 

 

Intensity Interferometry

    

b)

 

~I Detector 2 I
S2

Figure 1.1. Comparison of amplitude and intensity interferometry. a) An
amplitude interferometer measures light intensity after the interference of two
signals from a source. b) An intensity interferometer measures single and coincident

yields from a single source.

by the formula: 20: = 1.22 A, where s is the separation of the detectors, and 3. is the
s

wavelength. Only a few stars with large angular extent can be measured by this technique.
To increase the resolving power, signals must be selected by slits that are further apart.
As the slits are moved further apart, differences in the paths caused by mechanical
instability and atmospheric ﬂuctuations become important and can destroy the signal.
These limitations in amplitude interferometry inspired the development of a new
technique.

In the mid ﬁfties, Hanbury-Brown and Twiss introduced intensity interferometry
to overcome these limitations in measuring star diameters [Hanb 56]. Instead of
measuring fringes of light and darkness caused by relative phase differences, the
intensities of light at two points are compared. In Figure 1.1b intensity interferometry is
illustrated by a source emitting particles from S1 and 82. The yields at the detectors, 11.,

n2, and the coincidence yield, an, are measured, and the correlation function is

_. .. ("12)

constructed by the relation: C(pl, p2) = 1 + R(p1, p2) = m
111 n2

yields averaged over some time are compared for photons measured simultaneously and

. The coincident

for photons measured at different times. Because the photon’s wave function is
symmetrized, the coincidence yield is enhanced compared to that for photons with non-
overlapping wave-packets.

Within a few years, applications of intensity interferometry extended beyond

astronomy to nuclear physics. In nuclear physics, interferometry has been applied to

6
studying the size characteristics of the reaction region. While the astronomy experiments
detected photons, in nuclear physics many other particles can be detected, and correlation
functions can be constructed with all of these particles. With particles the interactions
include not only quantum statistics but also nuclear or Coulomb forces. The ﬁrst nuclear
results were published by Goldhaber, Goldhaber, Lee and Pais, who applied this
technique to study pion correlations in proton-antiproton annihilation at 1 GeV [Gold 59,
Gold 60]. Since that experiment, many other nuclear systems have been studied with
intensity interferometry. In this dissertation two proton and two neutron coincidence

measurements will be used to construct correlation functions.

1 .3 Motivation

1 .3 .1 Two-Proton Intensity Interferometry

Two-proton correlations have been studied extensively at intermediate energies
[Lync 83, Poch 86, Chen 87, Awes 88, Gong 90, 91a, 91b, Lisa 93, Hand 95b]. These
studies have shown that the correlation function depends on other reaction variables such
as the total energy of the proton pair and the impact parameter, or overlap, in the
collision. By comparing these results to simple geometric models, progress has been
made in understanding the evolution of the source. For example, changes in the
correlation function with total energy may be attributed to cooling effects from particle
emission, and changes with the impact parameter reﬂect different geometries of the

reaction zone [Lisa 93].

7
Because the actual time evolution is expected to be more intricate, experimental
results have been compared with transport models which include reaction dynamics.
Calculations with the Boltzmann-Uehling-Uhlenbeck (BUU) transport equations have
been rather successful in reproducing inclusive two-proton correlation functions at beam

energies below approximately E/A z 100 MeV with reactions such as 1"N + 27A] and 14N

+ 197Au at E/A = 75 MeV [Gong 90b, 91a, 91b]. At higher energies (200A MeV), Kunde
et al. found that the BUU model signiﬁcantly overpredicted the correlation function for
the 40Ar + 197Au reaction [Kund 93]. A similar over prediction in the proton correlation
function was seen for the reaction 36Ar + 45Sc at E/A = 120 and 160 MeV [Hand 95b,
95c], but not at E/A = 80 MeV [Lisa 93]

In order to further investigate correlation functions in this energy domain, the
reaction 160 + 197Au at E/A = 200 MeV was studied. The results of this investigation are

described in the second chapter of this thesis.

1 .3 .2 Two-Neutron Intensity Interferometry

To complement the information available from two-proton correlation functions,
two-neutron correlation functions can also be measured. Two-neutron correlation
functions are especially useful for avoiding ﬁnal-state Coulomb interactions with the
residual nucleus and the resulting distortion of the correlation function. These distortions
should be largest in the case of evaporation from a compound nucleus. An experimental
study of the reaction 129Xe + 27A1 at E/A = 31 MeV found that the direction-selected

correlation functions (directional cuts are discussed in Section 4.1) were best ﬁt using a

8
hard sphere of radius r = 3.5 ﬁn and an exponential lifetime 1: = 1300 fm/c [Lisa 94]. This

source size is much smaller than the geometric compound nucleus size (r = 1.2 A"3 = 6.5
fm). This discrepancy is difﬁcult to understand, unless one allows that the heavy residue
is distorting the outgoing particle trajectories and thus the correlation function. When
interpreting correlation functions, one usually assumes that the particles being studied
only interact with each other and not with the rest of the system. This assumption fails
when measuring charged particles, especially when they move slowly away from the
emitting heavy residue. The Coulomb forces may distort the proton correlation function
and thus must be considered when interpreting the two-proton correlation functions
[Gong 92]. Unfortunately, the three-body problem has not been solved exactly for this
quantum-mechanical system. Experimentally it is possible to eliminate this Coulomb
force by constructing a correlation function using neutrons.

Two-neutron correlation functions have not been extensively measured because of
the challenges involved in measuring coincident neutrons [Ghet 93, C010 95]. Neutrons
are scattered both by the detectors and by other materials in the experimental area. They
do not stop in a detector, so the same neutron can be measured in two detectors and make
a false coincidence. Also, many neutrons can be scattered into the detector after they have
interacted with other material, such as the walls of the vacuum chamber, or the concrete
ﬂoor or ceiling. These background neutrons can come in coincidence with a direct
neutron and distort the correlation function. Corrections for these effects must be made

before the correlation function can be interpreted.

9

To measure a two-neutron correlation function for a compound nucleus formed in
a low-energy reaction, the system 40Ar + 165H0 at E/A = 25 MeV was chosen. Lifetimes
for these low energy evaporation reactions are important, since neutron evaporation
“clocks" have been used to obtain information about the time-scale of nuclear ﬁssion
[Hind 84]. Two-neutron correlation functions can provide a calibration for these “clocks.”
This system was chosen because it has a high neutron multiplicity [Schw 94], and since
many neutrons are emitted from the system, the chance of measuring two within a limited
angular range is enhanced. Also, a similar system was measured to have a signiﬁcant
compound-nucleus cross section with most neutrons emitted after equilibrium had been
reached [Benr 93].

The second experiment discussed in this thesis involves neutron-neutron
correlation functions for the reaction 40Ar + 165H0 at E/A = 25 MeV. This experiment was
designed to measure the two-neutron correlation function which contains information
about the source size and lifetime of an evaporative collision. The results of this

experiment are described in Chapters 3-6

Chapter 2 - Two-Proton Intensity Interferometry: 16O +

197Au at EIA=200 MeV

2.1 Motivation

Calculations with the Boltzmann-Uehling-Uhlenbeck (BUU) transport equation
have been rather successful in reproducing inclusive two-proton correlation functions at

E/A S 100 MeV [Gong 90b, 91a, 91b, Lisa 93, Hand 95b]. At higher energies, the BUU

model was expected to be more appropriate. The model includes approximations,
representing the Pauli blocking, which should become more accurate at higher energies
where more states are available and the time scale becomes shorter. Therefore, it was
surprising that BUU calculations failed to reproduce the results from the 40Ar + 197Au
reaction at E/A = 200 MeV [Kund 93]. These results differed from lower-energy data in
at least two ways. First, the data showed an unusually small dependence of the measured
correlation function on the energy of the emitted protons, much in contrast to the strong
dependence observed at lower energies [Lync 83, Chen, 87, Awes 88, Gong 90b, 91b,

Lisa 93]. Second, the data did not show the expected minimum at q z 0 MeV/c (Figure

2.1). These rather surprising results left questions about the generality of the BUU failure
and a need to verify the anomalous shape of the observed correlation function.
In this chapter, the results of two-proton correlation measurements for 16O + 197Au

collisions at E/A = 200 MeV are presented [Gaff 95]. These measurements were

10

11

 

 

 

 

 

 

 

 

2 _ e".
a 0
. ‘
'- I
I
I
1.5 — .'
I
I
’— I
1 I
I
— .' I E....=160—290Mev
‘— l
+ ' l l l 1
ll 1 r l 1 r l r r i r
E05
(I
2 L—
15 -
‘I .—
. E,,,,,,=25—1OOMeV
oslllllllllllllllll

20 4O 6O 80
q (MeV/C)

Figure 2.1 Proton correlation function for ”Ar + l9"Au reaction at E/A = 200

MeV [Kund 93].

12

performed to explore whether the anomalous trends observed for the 40Ar + '97Au

reaction at E/A = 200 MeV [Kund 93] persist for a lighter projectile.

2.2 Setup

The experiment was performed at the National Superconducting Cyclotron
Laboratory using a beam of 16O ions at E/A = 200 MeV, the highest energy per nucleon
possible from the K1200 cyclotron. The '97Au target had an areal density of 60 mg/cmz,
and the typical beam intensity was 1.5x108 particles/sec. Coincident protons were
detected by a 56—element high-resolution hodoscope [Gong 88, Gong 90a] attached to the
MSU 4n: Array [West 85] at (0 lab ) = 38° (Figure 2.2). Each element of the hodoscope
consisted of a 300 or 400 rim-thick Si detector backed by a 10 cm—long CsI(Tl) detector,
allowing measurement of protons with energies between 10 and 200 MeV. The telescopes
each subtended a solid angle of A!) z 0.37 msr. The nearest-neighbor spacing between
telescopes was A6 = 26°, and the energy resolution for each telescope was about 1% for
50 MeV protons. Events were written on tape when they satisﬁed the trigger condition of
at least two detectors of the 56-element hodoscope and at least three detectors of the 41t-
Array ﬁring. This trigger eliminated the most peripheral collisions, but otherwise
accepted a broad range of impact parameters. Due to a major power outage, the
experiment had to be ended prematurely, and the statistics collected were insufﬁcient to
allow narrow cuts on impact parameter. Thus we can only present inclusive two-proton

correlation functions.

13

56-ELEMENT HODOSCOPE

 

 

 

 

TARGET INSERTION
MECHANISM
BEAM " 38° C.
_%>..-- T l ”
TARGET
FORWARD
4n DETECTOR ARRAY

Figure 2.2 Schematic drawing of the experimental setup: the 56-element

hodoscope attached to the MSU 4n array.

14

2.3 Results

The solid points in Figure 2.3 show the measured energy spectrum of protons

emitted at polar angles 61,], = 30° - 45°. The solid curve represents the corresponding

energy spectrum predicted by the BUU calculations [Baue 87, 92, Li 91a, 91b]. In these
calculations, a hard equation of state (K = 380 MeV) and an in-medium nucleon-nucleon
cross section set to its free value were used. The calculations were ﬁltered for the
experimental acceptance and energy thresholds. A proton was considered emitted if it was
located in a region of local mass density less than one-eighth normal nuclear density
[Gong 91a, 91b]. At energies near the Coulomb barrier BUU calculations overpredict the
proton yield. Such an overprediction of the proton yields at low energies has been noted
previously and attributed to the neglect of light cluster formation [Hand 95b, 95c, Gong
93]. Light clusters are expected to be predominantly formed in densely-populated regions
of phase space, thus reducing the ﬂux of free nucleons at low energies.

In order to investigate the inﬂuence of light cluster production on the calculated
two-proton correlation functions, the single-particle phase-space distribution of emitted
particles was subjected to a coalescence analysis. When adjusting the coalescence
parameters to the observed p/d/t ratio, the calculated singles yield of protons below 50
MeV was reduced by a factor of two relative to the high-energy protons in accordance
with naive expectations. Despite this change of energy distribution in the singles spectra,
the calculated two-particle correlation functions remained virtually unchanged. (This

effect was noted before [Gong 91a, Baue 92] and attributed to the division procedure

15

 

 
  
  
   

 

 

 

I‘r I I I V I T I I I T I I I I I I I I I

0.30— 16o + 197Au; E/A=200 MeV ——
”T 0.20— —BUU __
(I)
:::.’
C. 0 DATA
:1
9' 0.10 ..
L. —1
(6 -—
v ——
C _
—o _
m
to _
\
0..
—o 0.03 ——

o.oz—- __

1 l l l I l l 1 l I l 1 l l I l 1 l l I l
0 so 100 150 200
Eproton (MGV)

Figure 2.3 Energy distribution of protons measured in the laboratory rest
frame at e ..., = 30°. 45° for collisions of 16o + 197An at E/A = 200 MeV (solid points)
and predicted by BUU (solid line). To allow a comparison of shapes, the relative
normalization between experimental and theoretical energy spectra was chosen to

give equal areas for Em“... > 50 MeV.

16
employed to arrive at the correlation function as a ratio of the two—particle cross section
and the square of the singles cross section. Since coalescence changes only the density of
particles emitted but does not change the volume into which the particles are emitted, the
shape of the correlation function should not change.) Even after the coalescence step, the
predicted singles spectra were still somewhat steeper than the data. This is at least
partially due to the fact that the coalescence was applied to particles that had been

emitted, i.e. that came from a region of low density (p S 1/8), and the cluster formation

may be determined at somewhat higher densities. In light of the insensitivity discussed
above, the model parameters were not adjusted to produce a perfect ﬁt to the data. Instead
the singles spectrum shown is that produced by BUU and not corrected by a coalescence
step.

Two-proton correlation functions, 1 + R(q), are presented in Figure 2.4. The
experimental two-proton correlation function, shown by points, was deﬁned in terms of
the two-proton coincidence yield, Y2(p1, p2), and a background yield, Yback(p1, p2),
obtained by the event-mixing technique [Lisa 91].

Z Y2(P1.P2)= C11 + R(QHZ Yback(P1’ P2)-

Here q is the (invariant) magnitude of the relative momentum four-vector, given

lﬁl - M = uvrel. For a given experimental gating

NIH

non-relativistically as q =

condition, the sums of each side of the equation above extend over all proton energies and
detector combinations of the 56-element hodoscope corresponding to each q-bin. The

normalization constant C is determined by the requirement that R(q) vanish for large q,

17

 

        

 

 

 

 

 

1,8rwﬁ.l...yl..ral....l
I 16o + 197Au; E/A=200 Me «
1.6— _
P-Integrated
° 0
1.4- —— BUU __
0 DATA .
1.2—
L
1.0—
0.8— I
’c? f
V A l l l 1 A L l l A l L 1 l A
m 0,5 ....L....l..f.l....l
: , \ BUU, Data
1.6- / C] m — — a 135500—1220—
? ' \ o Pm=200—500 -
1.4* O
1.2"—
r
1.0—
0.3-
0.6
0

 

Figure 2.4 Two-proton correlation functions for collisions of 1“O + 197Au at
EIA = 200 MeV. The top panel shows the energy-integrated correlation function,
and the bottom panel shows correlation functions for proton pairs selected by the

indicated cuts on their total momentum.

18
where ﬁnal-state interactions between the emitted protons are negligible. Speciﬁcally,
both the experimental and theoretical correlation functions were normalized such that

(R(q)) = O for 60 MeV/c Sq S 80 MeV/c.

Theoretical two-proton correlation functions (shown by curves in Figure 2.4) were
calculated with the Koonin-Pratt formalism which relates the one-body phase—space
distribution with the correlation function [Koon 77, Prat 87, Gong 91a, 91b]. Speciﬁcally,
the theoretical correlation function was calculated by convoluting the one-body phase-
space distribution predicted by the BUU calculations described above with the two-proton
relative wave function [Gong 91a, 91b].

The upper panel of Figure 2.4 presents the energy-integrated correlation function
for all protons detected by the 56-element hodoscope with laboratory energy above 10

MeV and below 200 MeV. The lower panel shows correlation functions selected by two

different cuts on Ptot = lp, - p2| , the total momentum of the detected proton pair in the
center-of-momentum frame of projectile and target. The peak values, 1 + Rmax z 1.4 - 1.6,

of the measured correlation functions are larger than the values, 1 + Rm z 1.2, reported

for the 40Ar + 197Au reaction at FJA = 200 MeV [Kund 93]. Further, the magnitude of the
correlation function for the present reaction exhibits a clear dependence on total
momentum, PM, of the detected proton pair as compared to the negligible dependence on
proton energy observed for the Ar-induced reaction [Kund 93]. Within the resolution of
the present experiment, no evidence for a disappearance of the minimum at q = 0 was

found.

19
The BUU calculations reproduce the magnitude of the energy-integrated
correlation function (Figure 2.4 upper panel), but they predict too strong a dependence on
the total momentum, Pm. (Figure 2.4 lower panel). The calculations underpredict the
magnitude of the correlation functions measured for the low-momentum gate, 200 MeV/c

.<. PM 5 500 MeV/c, and they overpredict it for the high-momentum gate, 500 MeV/c S
Pm; S 1220 MeV/C.

A couple of differences may explain why the experimental correlation function for
the 16O + 197Au reaction at EIA = 200 MeV differs from the 40Ar + 197Au reaction with
the same energy per nucleon [Kund 93]. One difference is in the total energy in the
system. Since the oxygen is lighter, the total amount of energy deposited in the system is
a factor-of-two smaller, and the reaction might produce a residue not present with a
heavier projectile. Another difference in these experiments is the lack of an external
centrality ﬁlter in the [Kund 93] experiment.

In summary, proton energy spectra and two-proton correlation functions were

measured at (Bub) = 38° for 16O + 197Au collisions at E/A = 200 MeV. The single-particle

yield for the low-energy protons has a ﬂat slope that is not reproduced well by the BUU
transport model, even with a simple coalescence step added. The momentum—integrated
correlation function agrees well with the BUU calculations, and the momentum-gated
correlation functions show a dependence on the momentum of the proton pair in both
experiment and theory. However, the BUU theory overpredicts the magnitude of the

observed energy dependence. The correlation functions look signiﬁcantly different from

20
those for the ”Ar + 1”Au reaction at E/A = 200 MeV [Kund 93] in both total height and

momentum dependence.

Chapter 3 - Two-Neutron Intensity Interferometry:

Experimental Setup and Data Reduction

In the two-neutron intensity interferometry experiment, the intent was to measure
the size and lifetime of a compound source formed in the reaction 40Ar + 'GSHo at E/A =
25 MeV. During a collision many different types of reactions can occur. Thus, it is
important to select a speciﬁc type of reaction to simplify the interpretation of the
correlation functions. Nuclear reactions are often characterized by an impact parameter,
deﬁned as the distance between the centers of the particles perpendicular to the direction
of motion, before any interaction. Events with small impact parameters were selected in
this reaction. These “central events” are those where most of the beam energy is
transferred into internal energy. In these events a few preequilibrium particles may be
emitted and then a compound nucleus is expected to form.

The experimental setup was designed to select neutrons from a compound
nucleus, both by choosing central events and by minimizing the contributions from
preequilibrium neutrons. The detectors, used as a trigger, selected central events while
providing the zero time for measuring the neutron time-of-ﬂight. These silicon telescopes
triggered the electronics when a charged particle with Z > 2 was measured at 30° - 40° in
the laboratory, a condition shown to favor compound nucleus formation [Kim 89].
Coincident neutrons were measured in detectors placed at 90° in the center of mass. The

coincidence requirement biases the data toward more central collisions, and a neutron

detection angle of 90° was chosen to reduce the number of preequilibrium neutrons,

21

22
which are preferentially emitted in the forward direction. In this chapter the details of the

experimental setup, electronics, and data reduction are described.

3.1 Experimental Setup

3. 1.1 Fragment Telescopes

Charged particles, used to trigger the experiment, were measured with silicon AE-
E telescopes. Each telescope was constructed of two elements, a 75 um planar Si AE-

detector and a 5 mm Si(Li) E-detector. To improve the resolution of the Si(Li) detectors,

the detectors were cooled to 0° Celsius. Each telescope was covered by a collimator with

a diameter of 12.7 mm, assuring that only the active area of the detectors was exposed to
particles. Onto these collimators was glued a thin Ta foil to suppress electrons. Thirteen
of these telescopes were arranged on two circles around the beam axis with polar angles

of 30° and 45° (Figure 3.1), much larger than the grazing angle of 9°. The telescopes were

located 15 cm from the target. This distance was chosen to maximize the solid angle
coverage, while still providing access to the detectors and space for the mounting ring.
The whole assembly was placed inside a vacuum chamber. The chamber has a diameter
of 91.4 cm and walls made of steel that is only 3 mm thick, to minimize neutron
scattering. The detector mount was located so that it did not shadow any part of the

Neutron Walls (Figure 3.2).

23

      
 

 

 

 

 

 

I
0 “44's
f ,I/" '
. 44- '1]; '3 / ,
. ﬁll/(0“ ° °
4 ll ”t.
\"\ \IvI’Q
\zg ﬁ\ 1? /'.A‘
. Qasgs my)
~V-a
-
L— L_I H ..

 

 

 

 

 

 

 

 

 

Figure 3.1 Arrangement of silicon telescopes around the target at angles of

30° and 45° and a distance of 15 cm.

24

lm /
Neutron Walls
...... lag
“Fragment Detectors
Targetw
45°
Not drawn to scale

Figure 3.2 Schematic drawing of the setup. The fragment telescopes were

used for event selection and the neutron detectors measured coincident neutrons.

25

3.1.2 Neutron Walls

Neutrons were measured with the Neutron Walls, which were designed and built
at the National Superconducting Cyclotron Laboratory (N SCL) [Zech 97]. This detector
provides good angular coverage without much dead space. The Neutron Walls are made
up of glass cells that are two meters long with a height of 7.62 cm and a depth of 6.35 cm
(Figure 3.3). Each cell is ﬁlled with NE-213, a liquid scintillator commonly used to detect
neutrons. Each end of the rectangular shaped cells was tapered and sealed to a Pyrex
circle with a diameter 7.5 cm. On this circular surface a 7.5 cm diameter photomultiplier
tube (PMT) is attached with optical epoxy. Each wall is made up of 25 of these cells, of
which 24 were used in this experiment (Figure 3.4). The cells are placed as close together
as possible and the distance between the centers of adjacent cells is 8.55 cm. The cells are
mounted inside an aluminum box which provides mechanical stability and keeps ambient
light from reaching the detectors.

During this experiment, the Neutron Walls were positioned at 75° in the
laboratory, which corresponds to approximately 90° in the center-of-mass system. The

ﬁrst wall was four meters from the target, and the second wall was placed one meter
behind the ﬁrst (Figure 3.2). Unfortunately, the experiment was performed in an
experimental room where the concrete ﬂoor and ceiling are about one meter from the
detectors. Also, behind the second wall is a concrete wall about 1 meter away. This

surrounding concrete scatters neutrons into the detectors.

26

scintillator expansion
reservoir with Kynar
ﬁttings and Teﬂon tubing

photomultiplier tube

with ﬂ'metal shield \ﬁ
\ ruff“? :

ﬂI ‘ ' u131~ '5 ’

voltage divider

 

 

 

 

 

 

 

 

 
  

 

 

 

 

 

\ Pyrex cell with

NE-213 scintillator-

cross-sectional area
is 6.35x7.62 cm2

Figure 3.3 Picture of one detector from the Neutron Walls.

27

    
   
  

25 Cells

A1 Frame
\

Al Cover

Gas Manifold Blower

------------------

Figure 3.4 Mechanical drawing of one of the Neutron Walls. The glass cells
are protected by an aluminum cover, which is sealed to prevent light from reaching

the PMTs.

28

3.1.3 Other Experimental Details

The E/A = 25 MeV 40Ar beam was generated from the K1200 Cyclotron at the
NSCL. The beam, with an intensity of approximately 5x109 particles/sec, was delivered
to the target. The target was 165H0, which is naturally mono-isotopic and had an areal
density of 32 mg/cmz. The beam lost E/A = 2.6 MeV in the target [Hube 90], so the

interaction energy varied from EIA = 23.4 - 25.0 MeV.

3.2 Electronics

3.2.1 Fragment Telescopes

The two silicon detectors in the telescopes produce signals which can be used to
determine the charge, mass, and energy of a particle. The ﬁrst detector measures energy
loss for a speciﬁc distance, dex, while the thicker second detector measures the
remaining energy, E, of the stopped particle. These two quantities depend on the mass, M,

and charge, z, of the particle in the detector, and so a plot of dde, also referred to as AE,

versus B will show distinct bands for each isotope [Goul 64, Butl 70]. The Bethe-Block

describes the energy loss in the form

2
E = _B Ml ME),
dx E M

 

where B and C are constants depending on the detector characteristics [Engl 74].
In order to choose compound nuclei, events were selected where fragments were

measured with Z > 2 [Kim 89]. A ﬁrst cut for heavier fragments was done with hardware.

29

This cut requires the sum of E and AE signals, x = AE + OLE, to be above a threshold. The
threshold and weighting factor, on, were adjusted for each detector, so that most alphas

were excluded, but lithium was not (Figure 3.5). The silicon telescope electronics which

include this logic are shown in Figure 3.6. The attenuator allows or to be adjusted, and the

constant fraction discriminator (CFD) threshold adjusts the minimum x.
The fragment telescopes also provide the zero time for the neutron time-of-ﬂight.

A timing pick off (TPO) produces a timing signal from the AE signal, which has a faster

rise time than the E detector. The TPOs provide quick time signals, but they are prone to
oscillations and so require some attention in setup. This signal is delayed with cables by

250 ns and makes the stop for the Neutron Wall timing CFD.

3.2.2 Neutron Walls

Since neutrons do not ionize materials, the most common way to detect them is to
measure a recoiling nucleus, which has been scattered by the neutron. Mostly, the
Neutron Walls detect neutrons which scatter elastically on protons in a scintillator. These
protons excite and ionize the scintillator molecules. The scintillator produces light which
propagates by total internal reﬂection to the photomultiplier tubes (PMTs) where the light
signal is detected. If both PMT signals are larger than the threshold of 1 MeV electron
equivalent (MeVee) set in the constant fraction discriminators (CFDs), then the event will
be recorded.

In the Neutron Walls the phototubes provide two signals, one from the dynode

and one from the anode. The signal from the dynode is sent through a CFD and provides

30

140

120

g '.
J
l
1

AE (1 OaMev)

A
e
Jul-J .3. -.

l l I

 

 

0
0 20 40 60 80 100 120 140 160 180
Energy (MeV)
Figure 3.5 E versus AE showing the hardware cut. The counts in the
lower left comer have been reduced by the hardware trigger. Those remaining
come from events triggered by another Si detector. The lack of aBe gives a

useful reference for isotope identiﬁcation.

31

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

TPO __ preamp quad- fast
shaper
slow
Fast
Amp AD811
Leading Edge
Discriminator
silicon _
time - Analog CFD I logic
_ fan srgnal
fast
preamp __J quad- __ Attenuator _

 

shaper

 

 

 

 

 

 

 

 

 

slow
AD81 1

 

 

 

 

 

Figure 3.6 Electronics for the Si telescopes. TPO = timing pick off; CFD =
constant fraction discriminator; AD811 is a peak-sensing analog-to-digital

converter.

32
logic signals for the circuit. The charge from the anode signal is further processed to
provide a measure of the shape and intensity of the light signal.

The time for the neutron tirne-of-ﬂight is measured by a Time to FERA
Converter, TFC. This unit is started by the signal from the CFD and is stopped by the
time from the Si telescope, which has been delayed by 250 ns.

These neutron detectors can distinguish gamma rays from neutrons. When
ionizing radiation interacts with this organic scintillator, it produces a pulse of light
primarily with two components. One component is prompt (decaying in a few
nanoseconds) while the other is delayed (decaying over hundreds of nanoseconds).
Gamma rays are detected by Compton scattering on electrons. The electrons excite the
scintillator and produce light with a mostly prompt ﬂuorescence. On the other hand,
neutrons scatter on protons which ionize the scintillator and produce much of their light
as delayed ﬂuorescence. The light produces an electric pulse in the PMT, which has the
same shape as the light pulse. Because the shape of the electric pulse produced by
gammas and neutrons is different, it is possible to compare the shapes of the signals to
determine the source of the ionization. Traditionally the comparison was done by using
two analog-to-digital converters (ADCs). One ADC measured the total charge produced,
and the other measured some ﬁxed fraction of the pulse, for example the ﬁrst 30 ns.
Because this method requires separate gates for each ADC channel and this experiment
has almost one hundred channels, an alternate more cost-effective approach was chosen.

The Neutron Walls have a pulse shape discrimination circuit that was designed

and built for this detector. This circuit takes the anode pulse and produces four signals:

33
“Total”, “Attenuated Total”, “Fast”, and “Attenuated Fast”. “Fast” is proportional to the
charge in the ﬁrst part of the pulse. The attenuated outputs reduce the signals by
approximately a factor of four to give the detectors a wider range. A plot of “Fast” vs.
“Total” separates neutrons and gamma rays. During this experiment only enough
electronics were available to measure three of the possible signals, and so “Attenuated
Fast” was not recorded.

One potential source of error in this experiment is the detection of high energy
protons from the reaction in the neutron detectors. Protons produce a light signal exactly
like a neutron, and so protons cannot be excluded by pulse shape discrimination. The
vacuum chamber, with walls made of 3 mm-thick steel, has a large effect in stopping
protons. Using energy loss calculations based on Reference [Hube 90], a proton with
energy less than 40 MeV will be stopped by the steel in the chamber wall. Protons up to
57 MeV will be stopped before reaching the scintillator by the additional material present.
The protons must pass through 4.0 m of air, the 1/32 inch aluminum box and the 1/8 inch
glass cell. Very few protons are expected with energies higher than 57 MeV, thus the

effect of protons is not considered in the analysis.

3.2.3 Trigger Electronics

During the experiment three types of events were recorded. The ﬁrst type of event
was the coincidences with at least one hit in the Neutron Wall and a particle with Z > 2 in
a silicon telescope. The other two event types were downscaled telescope events and

downscaled Neutron Wall events. These were singles, where either the Neutron Wall or

34
the telescope ﬁred alone. Because these events greatly outnumber the coincidence events,
they were downscaled and only recorded one time out of 1000. A schematic of the trigger
electronics is shown in Figure 3.7. Because the processing time to make the master trigger

was over 1 us, the electronics required a Fast Clear for the Neutron Walls.

3.3 Data Reduction

3.3.1 Fragment Telescopes

The silicon telescopes must identify the isotopes and measure their energy. The

energy measured by the silicon detectors, both E and AE, was calibrated using an alpha-

particle source and a calibrated pulser. A 228Th source was used. It is especially
appropriate for the thin silicon detector since it provides an alpha particle with 8.78 MeV
which is close to the highest energy measured in the detector. The linearity of the
electronics for the full range of each detector was checked by using a pulser. An energy
calibration of 10% is adequate to ﬁnd the particle ﬂight time to within 0.2 ns, but the
actual energy calibration is much better (~2%).

Isotopes were identiﬁed for each detector after constructing a particle

identiﬁcation (PID) function: PID = (EAE)1/ 3 — (0.005 )AE. Figure 3.8 shows the
PID resolution for a typical detector. This PID makes setting gates on isot0pes easier
since instead of the strongly curved gates that would be needed in Figure 3.5, these gates
are approximately rectangles. The isotopes were identiﬁed based on ﬁxed points, such as

6Li as the lowest Z = 3 isotope and the absence of 8Be, and conﬁrmed by looking at

35

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Neutron Wall
Fast Clear "
NWATrigger j > AND _, "1’"
—>0 DS
TOF Start lln _
TOF Stop ‘
' AND b Computer
. ’ Start
Telescopes
ADC Gate ‘4 # 9 DS U
n
IT elescope Trigger ——5
AND W
Computer Busy

 

 

 

 

Figure 3.7 Schematic diagram of the trigger electronics from the experiment.

The elements marked D8 are downscalers.

36

Detector 12
400

350

 

 

0 50 100 150 200 250
Energy(MeV)

Figure 3.8 P11) for silicon telescopes. This plot of energy versus particle
identiﬁcation shows the clear separation between isotopes. The gates are also

plotted.

37
predictions for these detectors using an energy loss program. As expected, the energy

threshold increases as a function of mass.

3.3.2 Neutron Walls

For each phototube in the Neutron Walls, a time and light signals were recorded.
These Signals provide information about the energy, position, particle identiﬁcation, and

amount of light measured for each hit in the detector.

3.3.2.1 Neutron Energy

The neutron energy is determined from the time-of-ﬂight of a neutron between the
target and the detector. The time that is measured for each PMT corresponds to the time
between the silicon detector trigger and the PMT trigger. This measured time must be
corrected in order to ﬁnd the actual time-of-ﬂight.

The time Signals, TL and TR, are measured in the experiment by the TFCS which
start with the CFD signal for the left or right PMTS and Stop with the delayed silicon
telescope trigger. This raw time was calibrated with a time calibrator which gives start
and stop signals at given time intervals. The signals for the left and right times are then
averaged to remove the position dependence, Tn = (TL + TR)/2. TI, is proportional to the
time between a hit in the Neutron Walls and the silicon detector trigger. Because the
Neutron Wall starts the time, a smaller Tn corresponds to later times.

In order to ﬁnd the time-of-ﬂight, TOF, several corrections must be made to the
measured time. Speciﬁcally TOF is deﬁned as

TOF = - 'I‘n + T5; + ATC.

38
The measured time, T.., determines the relative time between various signals. In
order to ﬁnd an absolute time a reference time is used. The prompt gammas provide this

reference. In the case of prompt gammas

 

TOF = T7 = ./(d2 +x2 +y2) /c,
where dis the distance to the wall, 4 or 5 meters, x and y are the horizontal and vertical
positions of the hit, and c is the speed of light. For each combination of silicon telescope
and Neutron Wall cell, this reference point in the time spectra can be found and ATc
chosen appropriately. This correction takes into account differences in cable delays, and
electronic responses.

The time must also be corrected for the time a fragment takes to arrive at the
telescope and trigger the electronics. This correction, T3,, adds the ﬂight time for the
fragment and an energy dependent walk factor,

TSi = dSi/VSi + “(E)-
The velocity, vs, is calculated from the mass and energy of the isotope measured. The
fragment ﬂight time, dSi/VSi, is usually 25 us. The silicon time was measured with a
leading edge discriminator, so a walk correction is also needed, a(E).

The walk correction was determined empirically from the data. In a plot of the
energy signal measured in the AE detector versus Tn for prompt gamma rays, the

curvature caused by walk is evident (Figure 3.9 top). The maximum of the prompt

 

gamma peak is ﬁtted by a function oc(AE) = 1,] log(AE / AEth) , where Ed, is the energy

39

 

u:
c
c»
C

3;

(II
O
H
III IIIIIIIIIIIII'
i.

l‘rl'rllrlLr<.lllll‘-IlllI:II§IIIllllvlr'rli'llvvv
we w ,
a

Z

”17“,?
I
I

     
   
    

 

496
495

r,
R
i
s
IlllIIIlllllllllllllllllllIllllllllllll Illlllllllllll llllllllllllllllllll

 

 

 

u-"-1~'l:-”T'.:L~f:..”l...:l.I..l....l....l....l....
0 10 20 30 40 50 60 70 80 90 100
DE(MeV)

 

Figure 3.9 Walk correction for the timing signal from the leading edge
discriminator. The top panel shows the time for the prompt gamma peak measured
by the Neutron Wall, Tn, versus the AE signal before correction. The line is the ﬁt
used to correct for walk. The bottom panel is the same plot after the walk correction

or(E) has been applied.

40
threshold and 1: is adjusted for each detector. After correction the energy dependence of

the prompt gamma’s time is removed (Figure 3.9 bottom).
After these corrections, the time spectra shows the width of the gamma peak and
thus the resolution of the detector and electronics (Figure 3.10). The time is plotted as

time = 500 ns + TY-TOF.

For most detectors the full-width at half-maximum (FWHM) resolution is about 1.0 us.
One of the fan-in fan-outs (FIFOS) used to split the stop signal introduced additional
noise in the detector timing. This noise added a width of about 1 us to the resolution of
one third of the detectors. The gamma peak for these detectors measured about 1.4 ns
FWHM.

The neutron energy can be easily calculated from the TOF. Relativistically,

 

 

 

 

 

r \2
—"-mc2
En + me2 = m2c4+ C 2
I “(V-"J
K c I
where m is the mass of the neutron and vn = :SF. The distance to the hit is d“. The

energy spectra measured for single neutron events is shown in Figure 3.11. The energy
spectra for the front and back walls differ somewhat because the ﬁrst wall acts as a

scatterer for the neutrons arriving at the back wall.

3.3.2.2 Neutron Position

Besides providing energy information, the time signal also measures the position

of an interaction in the detector. If the neutron interacts close to one photomultiplier tube,

41

 

10‘: .. ...,.. ... ..e.., .. ... .. ... .. -,
103
10‘?
10 i

1 LLIJJ Pl 1 11 Ll4l 1 L1

. l . l . . . . 1 . . f
250 300 350 400 450 500 550

yield

 

1600 : ...,.........,..-.,.-.................
1400
1200
1000
800
600 g
400 g
200 a

:b ...... . . r 1 . l . -
0490 492.5 495 497.5 500 502.5 505 507.5 510

time(ns)

lrl

III

II'I'T'ITI'I

 

 

Figure 3.10 Time spectra for the Neutron Walls after all correction, T, - TOF

+ 500 us. The bottom panel is an expansion of the region around the prompt gamma

peak.

42

 

5
10 IIIIrIIIIIIIIIlIﬁIlII—IIIIIII'IIIIIIIIIIII—rI

— front wall

I llllll

l

--- back wall

10 4 “-

.3103
>~

I IIIII

 

I *IiTI'IIIIl

 

 

lllllllllllllllllllllllJlllllIlllllllJllllJl

5 10 15 20 25 30 35 40 45 50
energy (MeV)

Figure 3.11 Energy spectra for the Neutron Walls (front and back

detectors). These spectra are not efﬁciency corrected.

43
the light produced takes less time to reach that PMT and longer to reach the other side.
The difference in time between the two signals measures the position of the interaction,

tp(x) = tL - tn. The position in centimeters, x, was determined by the times at the edges of

the wall, x1, and xR, x = tp(x) 200cm . The edges of the distribution were

tp(xL) " tp(xR)

 

designated as the place where the number of counts fell to half of the full yield.

The position resolution can be seen by shadowing a part of the wall. A shadow bar
35 cm long that shadowed 36 cm on the wall was placed into the path of the neutrons
(Figure 3.12). These adjacent detectors measure the same size for the shadow bars, and
the resolution is better than 5 cm FWHM.

The position spectra also Show some structure, especially near the center of the
tube. This non-linear response was later shown to be the result of electronic crosstalk
between different channels within the CFD. A signal in one channel induces noise on the
neighboring channels, so when two Signals arrived at nearly the same time the crossover
time is shifted. Each detector was cabled so that the signal from the right and left PMT
were in adjacent channels. For one location in the detector, usually near the center of the

tube, the Signals arrive simultaneously at the CFD, At = 0 (Figure 3.12).

The correlation function should not be effected by this electronic crosstalk, Since
the effect for one detector changes the numerator and denominator in the same way. The
only way this electronic crosstalk could change the correlation function is if it caused a
correlation between different detectors. To investigate this correlation, the position is

plotted for one detector when another detector in the same (Figure 3.13, lower panel) or

 

250
200
150 j
100 i
50 E
250 .
200
g 150 E
51100 .
50 '
250
200
150 ; s
100 E- * -:
50 :' Atal-O 1

- 11111111111111 ...11 1
50 100 150 .200 250 300 350

posrtron (cm)

 

111 11

 

11l1

11111111111

 

 

 

 

Figure 3.12 Position Spectra for the Neutron Walls. A Shadow bar creating a
Shadow of 36 cm was placed in front of one section of these detectors, creating a
large suppression in the yield. The other structure in this spectra is the result of

electronic crosstalk.

45

 

 

 

004"'I"'l"'I"'l"'I“'l"'F"' IT
0'35 front-back
0.3
0.25
0.2 +
0.15! - -t- ~+d~+
i3 " +'F:l—-l- +‘l"
g 0.1
£0.05
g 0.5E :
0.45:- -:
E 0.4%. front-front +
o0.355 é
“3 '1—+ a

. = | I '
0.25 =- +T++++ ++
02}
0.15 E-
01}
0.05;

 

l I l l I 11111L111

 

 

100 120 140 160 180 200 220 240 260 280 300

Positon (cm)

Figure 3.13 The coincidence Signal for one detector either in a front-back

(top panel) or front-front (bottom panel) pair divided by the singles yield for that

detector.

46
opposite (upper panel) wall ﬁres. The spectra are divided by the singles spectra for the
same detector.

Coincidence __ 2 Y2 ([31. 92)
Singles 2 Y1(Pl)

 

 

where the numerator is summed over all {11 for one detector and all 152 for detectors in
the appropriate wall. The denominator is summed over all ['51 for the one detector. Within

the error bars there does not seem to be any correlation in these spectra. Detectors in
opposite walls Should not have any correlations since their signals are processed in

different CFDS.

3.3.2.3 Neutron Particle Identiﬁcation

The pulse shape discrimination in the neutron walls allows a separation of
neutrons and gammas. Two different scales were used for the particle identiﬁcation.
Figure 3.14 shows the “Fast” Signal versus the “Attenuated Total” signal, used for higher-
energy neutrons and Figure 3.15 Shows “Total” versus a PID function used for low-
energy neutrons. The PID function was determined empirically to expand the resolution
of the lower-energy part of the spectra and to straighten the results.

PID = 7.9898 F - 0.003413 F*T + 0.00188 T2 - 4.439 T + 1116,
where F is “Fast” and T is “Total.” By gating on these spectra it is possible to eliminate
most of the gamma rays. The timing spectra after the gammas are eliminated Show a

reduction of the height of the gamma peak from 2x104 to 1x103 counts (Figure 3.16).

47

 

IIIII‘Irl—I’IIIIiIIlIII'IfI'III'III'InI:II_

Detector 34 : i,

.1
.1

I
IL111|1L1LI1111|1111|111411111[1"111'

era“.
'V
v- '1
. ~ s.
A
.V‘
...;
1’10
'K'e.
.
- ~ ;
_ v w. or
1. .-.. 1..
. v ". v ’1.
‘1' '5'.
I."
c a. n
. nu . “
‘ ..
x '.’
- .
(“2‘ ”—
1 '3.
'd’ o-‘v ’1‘ on
° ‘5'
.
v
.vv‘.‘ ‘2»
...—V
r- ‘. -
U . A
.-

250

  
     

Fast light
‘é

Gommos- .3} .. - ‘

150

~ I
" u
.w A

I. .r- . _ . A v
j” _ - Neutrons

IIWI'ITITIITIﬁIrTI'II—IIIrIII

100

 

50

1‘ -\1

1.2—1.1“. 1 . 1'1 1 11. 1 111 -.-1l-1-:.'-.vl1'1"1‘r 1.13. l . 1111
00 20 40 60 80 100 120 140 160 180 "10

Attenuated light

 

 

Iﬁ I r I I I I
, r I
I I

 

 

 

Figure 3.14 Neutron-gamma discrimination for the high energy neutrons in
the Neutron Walls. The plot of “Fast” versus “Attenuated Total” Signal Shows a

separation between neutrons and gammas.

48

 

IIIIIIIIlIIlIIIlIlIleI'IIIIITIIIIIITT

' Detector 34 ‘

350 - _

' Gommos . - _» _ '

300-

PID

250 -

  

200 -

Neutrons

 

 

 

15011IlllIIIlLllllllllIlllllllllllllllI11

0 20 40 60 80 100 120 140 160 180
Total light

 

Figure 3.15 Neutron-gamma discrimination for the low energy neutrons.
“Total” signal versus a particle identiﬁcation, PID, chosen to straighten and expand

the spectra. See text for deﬁnition of PID.

49

 

   

|' 1'
104

103

102

Commas

10

L 1 1 1 l 1 141 1 I 1 1 1 1 I J 1 1 1 l 1 1 1 1 I 1 1 1 1

1
250 300 350 400 450 500 550
Time (ns)
Figure 3.16 Timing spectra for the Neutron Walls. The top panel shows the
time spectrum for all hits. The middle panel is gated on neutrons identiﬁed by gates
on either the high or low energy discrimination plot. The bottom panel is gated on

gammas identiﬁed by either gate in the discrimination plots.

50

3.3.2.4 Neutron Light

In order to correct for detector crosstalk (Section 4.2.2), it is important to relate
the light signal to the amount of energy deposited in the cell by a neutron. The light is
calibrated and then related to the proton energy necessary to produce this much light. The
proton energy is used to identify the crosstalk events.

The light intensity measured in each PMT by the “Attenuated Total” signal was
calibrated by looking at the Compton edge of various gamma sources: 12C(Ex = 4.44
Mev) in a PuBe, 228Th and 60Co (Figure 3.17). The Compton edges were found from the
light for events from a restricted range of positions in the Neutron Walls. The light is
calibrated in units compared to the light produced by a 1 MeV electron (1 MeV electron
equivalent, 1 MeVee) which is the most light a 1 MeV gamma ray can produce.

The total light signal is a position-independent combination of the calibrated left
and right Signals. The individual signals at a position x deﬁned from the left Side of the

cell can be represented by Lies, = L e -111 and Light = L e 4‘1“)”, where L is the total light
produced, (1 is the length of the cell and k is the attenuation length (Figure 3.18). A

position-independent signal can be constructed by taking the geometric mean of the two

 

light signals, L = JIM, * Lright .

From the total light, L, the proton energy can be reconstructed using a light
response curve, EP(L). Two light response curves were used. (Figure 3.19). The ﬁrst is

based on a ﬁt of the data from References [Verb 68, Ceci 79]. The second response curve

51

 

q

I rlijI I {III I I I rI Ij’I I I I II II

”Co 1.1 MeV

  

I'IfI

200

100_

1111'1111'111

 

 

   

400

,_, 300

0

s: 200
100

2”Th 2.4 MeV

  
   

 

PuBe 4.2 MeV
200 _-

100

    

111111111I11

 

 

  

    

1 1

1111111111111
200 300 400 500

Channels

111
0 100

Figure 3.17 Light spectra showing the Compton edges used to calibrate the
light signal for the Neutron Walls. The sources and the energy of the Compton edge

are given in the ﬁgure.

52

 

 

 

 

 

Figure 3.18 Deﬁnition of x for the determination of the left and right light

signals.

53

 

25 IIIIlIIIIIIIIIIIIIfI‘IijITﬁIIIIIII'II I

I I

22.5

[IIIIII
1'113111114

20_ x’

17.5

111'111

l—l
UI

Light yield (MeVee)
11.1
1— i"
e u:
IIIIIII—rqurf'IIIIlrIiIl’IIIIlrII
\

>1
u:

5

l
\

2.5

/
I

 

 

\
1111111111111111111l1111l_111111

11JLL411L114111111l1111ll11111111l1111

0 5 10 15 20 25 30 35 40
Energy (MeV)

 

0

Figure 3.19 Neutron energy versus the highest light yield. The solid line
shows a ﬁt to Verbinski data while the dashed line shows a linear ﬁt empirically

chosen from the data.

54
is a linear ﬁt to the maximum light as a function of energy. The second curve was chosen
to make symmetric cuts on the crosstalk variables described in Section 4.2.2.

The linear ﬁt was determined from the light intensity as a function of the neutron
energy (Figure 3.20). The maximum light for a given energy neutron can be seen as an
edge of the distribution, where a line is drawn in this ﬁgure. It is reasonable to use the
edge of this distribution to estimate the light response function, because the light
measured for a given neutron energy is expected to correspond to a ﬂat distribution of
proton energies up to a maximum of the total incident neutron energy. This expectation
can be seen from the following derivation. The solid angle in the center of mass of the
proton and neutron is

d!)c = 2n sin0c d0c,
where 0c is the scattering angle for the neutron in the center of mass. If one assumes that
the masses of the neutron and proton are the same, then the center of mass angle is twice

the laboratory angle, 0c = 201, (Figure 3.21). Then
d!)c = 81: sinOL cos0L d0L.
By scattering kinematics, the ﬁnal proton energy, Ep, and incident neutron energy, En, are

related by
Ep = En SinzeL.
(113‘, = 2 En C0891, 811191, (191,.

With this information, the yield can be found as a function of the proton energy:

55

1—1
Q

 

250

IIll'lllIlIlllllllIIIYITTlllll

225
200
175

Light Output (MeV ee)
u-

IIIlIll111l1IllIIlllllllllllllllllllllllI111

 

1111

 

 

 

.—

I l I I I l I I l l l l I l I I I I I I l l 1 I l l l l I

5 10 15 20 25
Energy (MeV)

 

G

A
c IIIIIIIIIjIIIIlIIIIIIIII—llllIlIIII'IIII'IIIT'IlII

OJ
6

Figure 3.20 Light output versus energy of the neutrons. Neutrons that come
directly from the target cannot have light corresponding to a proton energy greater
than the neutron energy. The line shows the cut made to exclude background

neutrons.

56

 

 

 

van

Figure 3.21 Kinematics of neutron scattering. Variables in the laboratory
frame are shown in italics while variables in the center of mass frame have a

standard font. Note: lvnl=lvn,l=lvpl=lvp’l

57

dN (mp: dN dQc,
dB (100

[Wm

 

 

 

P
P P

m

81f sin 9L COS OLdOL
2E“ sin 9L COS BLdOL

 

 

_dN;41
(jg-ICE“o

Since dN/d.(2c is a constant for s-wave scattering, the yield at a given neutron energy is

independent of the proton energy, as long as 13,, < E". This condition is required by energy
conservation. The light yield from the detector for certain selections of neutron energy is
shown in Figure 3.22. The detector resolution and detector inefﬁciency cause these

spectra to differ from the box shape expected.

3.3.3 Measured Background

Background neutrons can effect the correlation function, so it is important to
understand the amount and characteristics of the background. Some neutrons come from
the target, but scatter before they reach the detector. These background neutrons can
scatter from the walls of the vacuum chamber or from the walls and other material in the
experiment room.

In order to measure the magnitude of this background, a large shadow bar was
placed in the direct path from the target to the Neutron Walls. The shadow bar was

located just outside the vacuum chamber, 46 cm from the target. A pile of steel 40 cm

58

 

 

 

 

 

8 " .
000 3 5-6 MeV 5,- 9—1OMeV I400
7000 .- 3-
6000 '5 it '20”
5000 - -=':' '00“
4000 E ' 300
3000 =- 600
2000 :- 400
1000 E- 200
% 400 :4 5: :Hnnhh.
i ’ 16-17MeV 160
,: 140
gg . 120
loo
so
‘ .0
-.— 40
:‘E 20
Ad I l 0

 

 

 

 

0: _ ‘PI 11 lllllllll lllllll-
0 2.5 5 7.5 10 0 2. 5 7. 10
light (MeVee)
Figure 3.22 Light yield at a given neutron energy. A flat distribution from
threshold to maximum possible light is expected. The four panels represent different

gates on the energy of the neutrons. The gate applied is specified in the panel.

59
long and 35.5 cm x 35.5 cm was located so as to shadow the whole neutron wall. This
length was calculated in order to reduce the number of neutrons transmitted to a small
fraction [Remi 86]. The number of neutrons neither scattering nor interacting in a length

of material x can be determined from the relation

N = Noe-“GTX ,
where No is the original number of neutrons, n is the number of nuclei/cm3, and GT is the
neutron total cross section. An additional correction must be made for the fraction of
neutrons that scatter to small angles and so still reach the detector. This calculation can be

done more accurately by dividing the shadow bar into J sections with length x = III. The

fraction transmitted is
F = 111-1:103- + (1 — £90506. loT)),

here fj = Nj/NO, the fraction of neutrons making it through a short length of the shadow

bar.
do (0) .
A0 = GO—d—Znsmede,
el Ie=o d!)

where the integral is performed over all angles where the neutron will still reach the

detector, and 6e. is the neutron elastic cross section. The total and elastic cross sections

were taken from Reference [Hugh 58]. From these calculations it is possible to estimate
that about 2% of 20 MeV neutrons will arrive at the detector through this shadow bar.

By comparing the normalized spectra for the shadow bar run to a run without a
shadow bar it is possible to characterize the energy and detector dependence of the

inscattering. In Figure 3.23, the energy dependence of the background/total ratio for one

 

 

 

 

_....,T...,. ......,....,....

t Detector34
1+ _

.++

3 E

Q) _

$0.8 r i

f; . .

3 i

\06 _ -

E

3 + -

e . -

30.4 - -

U

U

m

0.2 ~ + -

h+++H +H++ ++ +++i1
1.111114. .11. 111
O0 5 10 15 210 25 30

Energy (MeV)

Figure 3.23 Energy dependence of the ratio of background/total neutrons.
The yield from a shadow bar run is divided by a standard run and normalized by

the total beam current in these runs.

61

 

0.4 I I I I j r I I I l I I I I I I I I I l I fI I

‘

 

 

 

 

 

 

 

 

G

E 3
0.35 .- 1
5 Front woll :
0.3 :- 1
0.25 E- -:
0.2 L'- i
: 1
20.15 E- d:
3 : 1
g, 0.1 E- -:‘
'2‘,0.05 E- i
o : 1
D '44t':44-':::t'::::'4r:::‘
“_ 0 E l l l l a
20.35 .- 1
o E Bock wall 5
1:, 0.3 T 1
o : :
90.25 E- -'.'
‘0— : :
0.2 :- '5
0.15 I- "I
0'1 E- bottom top .5
0.05 I- 3
: 1 J l I l l l l l l l l L l l I l l l l J l l ‘

0

5 10 15 20 25
detector number

Figure 3.24 Background/total ratio for each detector in the Neutron Walls.

62
detector is shown. For scattered neutrons, the energy is lowered and the ﬂight path is
lengthened by an unknown amount; both effects shift the scattered neutron events into the
low energy portion of the spectrum. Indeed, the yield below approximately 4 MeV is
mostly background. Figure 3.24 shows how the crosstalk percentage changes with
detector number. The detectors closer to the ﬂoor and ceiling have more crosstalk. There
was no noticeable dependence in the background on the horizontal position within a

detector.

Chapter 4 - Two-Neutron Intensity Interferometry:

Measured Correlation Functions

4.1 Constructing a Correlation Function

Many previous studies have looked at how correlation functions with identical
particles can be used to measure the emission zone of a nuclear reaction [Koon 77, Prat
87, Gong 91a, Gong 91b]. In this section the relevant details are reviewed.

Experimentally, the correlation function is constructed by histogramming events

as a function of the (invariant) magnitude of the relative momentum four-vector,

q = Hp, — p2| , and then dividing by a denominator constructed from uncorrelated

ZY2(P1,P2)

events. 1+ R(q) = N
X Yback (P1,P2)

 

, where Yback is the yield for uncorrelated

events, Y2 is the coincidence yield, and N is the normalization determined so that R(q) ->

O at large q [Boal 90 and references therein]. The sums are over all detector and
momentum combinations available in the experiment. The correlation function is deﬁned
as this ratio to eliminate effects of the single-particle phase space and the detector
geometry and efﬁciency. For this division to be effective, the numerator and denominator
must be treated in exactly the same way. In the correlation functions presented here, the

same cuts have been applied to numerator and denominator.

63

64

The shape of the correlation function for two neutrons reﬂects the quantum
statistics of neutrons and the ﬁnal-state interactions [Gong 91a, 91b, C010 95]. The basic
shape of the correlation is a combination of these two effects. If there were only quantum
statistics, the correlation would be suppressed at small q (Figure 4.1). (R(q) + 1 = 0.5 at q
= 0, but because the theory is binned in 2 MeV bins the correlation in this ﬁgure does not
reach 0.5.) However, the ﬁnal-state nuclear interaction is attractive, so the total
correlation is positive for relatively small emission zones.

Intensity interferometry is sensitive to the relative separation of the particles upon
emission, and thus the volume of the emission zone [Koon 77, Bert 89]. Consider a
source with a radius, r, and a lifetime, I, where the particles move away from the source
with an average speed v. The initial spatial separation, r + v1, depends on both the radius
and lifetime. If the source is small and the lifetime is short, the correlation will be strong
(Figure 4.2a). On the other hand, a weaker correlation could come either from a source
with a large size and short lifetime or from a small source with a long lifetime (Figure
4.2b and c).

In order to separate lifetime and size effects, it is possible to make a cut on the
direction of the relative momentum, q , compared to the total momentum, P , of the
particle pair. For dimensions on the nuclear scale, the attractive s-wave interaction gives
rise to a positive correlation. This nuclear effect is sensitive to the volume of the source
emitting particles. However, the effects of the Pauli exclusion principle depend on the
shape of the emission zone. The two-neutron wave function must be antisymmetrized,

and thus relative momentum states, Ap, will be suppressed over a range Ap = h / Ax

65

 

2 IIII'IIII'IIIIIITII'IIIIIIIII'IIII'IIIIIIIII'IITI

r=7 fm T=700 fm/c
— Totol

Quantum statistics

I I I I I I I l I I I I I I I L I I I

 

A AA-A
AAA AAAAA‘A AA A-

m""
.....
eeeeeeeeeee
e
00'
00‘
e...
e
.e
e
.e
e

lIIIlIﬁﬁilIIIlIII'IIIlIIT

0.8

I I
O

1

I l I I I l J I I

0 5 10 15 20 25 30 35 40 45 50
q(MeV/C)

 

 

 

Figure 4.1 Theoretical two-neutron correlation function. Surface emission
from a sphere of radius 7 fm with a lifetime of 700 fm/c. The solid line shows the
standard two-neutron correlation function. The dotted line shows the correlation

from only quantum statistics with no f’mal-state interactions.

a O

b O

c O

 

 

Figure 4.2 Spatial distribution of emitted particles of velocity v from three
sources. (a) Small source with a short lifetime (b) Large source with a short lifetime

(c) Small source with a long lifetime.

67
where Ax is the average spatial separation. The width of the suppression is primarily
sensitive to the spatial size in the direction of the relative momentum [Boa] 90, Prat 87,
Awes 88]. If the particles are emitted over a lifetime, the distribution of particles emitted
with velocity v z P/2m is elongated in the 13 direction by d z r + vt: (Figure 4.3) [Hand
95c]. Particles with their relative momenta perpendicular to the I5 direction sample a
smaller dimension (of the order of the source radius). At small relative momenta, these
particles experience a stronger anticorrelation from the antisymmetrization effects and
thus the transverse correlation function is reduced. The directional differences are seen by
comparing the two classes of events, longitudinal and transverse, based on the angle
‘I’ = cos"(P - q / Pq). This angle must be constructed in the reference frame of the

source emitting the particles, so that the directions are aligned appropriately with the

source [Lisa 93c].

4.2 Experimental Correlation Function

In constructing the correlation function, it is important to select the appropriate
coincidence data. Cuts must be made to choose the central events, to correct for crosstalk
between detectors, and to correct for scattering from the vault. Other cuts simplify

modeling of the detector.

4.2.1 Event Selection: Individual Neutrons

Because the event trigger recorded many events which did not satisfy the criteria

applied for construction of the correlation function, additional requirements were

68

To detector Transverse

P] P

2q
P2

Longitudinal

P
2q

 

P2

Figure 4.3 Definition of longitudinal and transverse directions. Transverse
samples the small dimension of the source while longitudinal samples the lifetime

dimension.

69
imposed in the analysis. Criteria include the charged-particle identiﬁcation, neutron
characteristics, and the multiplicity of neutrons.

One requirement was that the fragment telescope measures one identiﬁed isotope
with a charge greater than two. The hardware particle selection was not completely
effective and so the remaining light particles were excluded in the analysis. Particles that
could not be identiﬁed as speciﬁc isotopes were also excluded, since the particle mass is
necessary to correct the neutron time—of-ﬂight. Events with two or more fragments were
also excluded because the electronics did not unambiguously distinguish which particle
was ﬁrst, thus causing trouble in correcting the neutron time-of-flight.

No on-line selection was made for gamma-neutron discrimination so this
requirement was also added in software. Using the pulse-shape discrimination
characteristics of the Neutron Wall detectors, all hits were required to fall into one or both
of the neutron gates set for low-energy and high-energy neutrons. This requirement
eliminated most of the gamma events coming from the target (See section 3.3.2.3).

Each hit in the neutron detectors should give appropriate values for all measured
quantities. The light threshold was set slightly below 1 MeVee by the electronics, but
varied from detector to detector. A cut of 1 MeVee was applied to the light signal of each
PMT. This software cut makes the spectra easier to simulate. The effect of this cut on the
total light threshold as a function of position is shown in Figure 4.4. The position
measured was required to be within 1.05 meters of the center of the detector. Some events

appear to be outside of this range and probably come from double hits in the detector,

70

 

10_"T‘rr"'lI'I'l'll'lliﬁrtijt

ll IIIITII

light (MeV ee)
I I I—rrrI I I I

9
8
7
6
5
4
3
2
1

    

 

rIIIlrIIIlIIII

I l I I I l

O I PI I l I I I I l I I I I l I l I l

50 100 150 20 2501300 350°

position (cm)

Figure 4.4 The position versus the geometric mean of the measured light for

one detector. A 1 MeVee threshold was applied to each PMT.

7 1
where the ﬁrst hit triggered only one of the PMTS. The maximum energy of the neutron
was cut at 50 MeV to aid in simulation.

Several requirements were made of the neutron hits in order to eliminate
background scattering. As shown previously, the lowest-energy neutrons include many
events that are actually background neutrons (Figure 3.22), so a software threshold was
applied on the data at 5 MeV. This cut removes many of the background neutrons. A cut
on light measured versus energy of the neutron also provided a way to eliminate some
background events. Figure 3.20 shows the cut made to eliminate these background events.
The shadow—bar run indicates that about half of the background events were eliminated
by this cut.

Another selection requirement was the multiplicity of neutrons. For the
numerator, coincidence neutrons were required. Events with more than two neutrons were
not included because of the difﬁculty in making effective crosstalk rejections for three or
more hits. In the denominator, events were selected from those with one and only one
neutron measured (a gamma could be in coincidence). The standard deﬁnition of singles
includes all events with one or more particles, but in this case an ambiguity exists in how
to treat the coincidence events where one hit appears to come from crosstalk. Usually
selecting events based on detected multiplicity is a cut on the reaction multiplicity and
thus reaction type. Because the detection efﬁciency of the Neutron Walls is only about
10%, selection on detected multiplicity is not strongly related to the multiplicity of
neutrons produced in the event. 79% of the events where two neutrons are incident upon

the wall will appear as singles. Also, singles are at least 30 times more plentiful than

72
coincidences, so the difference between excluding coincidences and keeping them is
small. For one run there were 330,000 singles in the front and 150,000 in the back that
fulﬁll all individual neutron criteria. There were 7000 coincidences for the same run

where both neutrons fulﬁll all the individual neutron criteria.

4.2.2 Event Selection: Crosstalk Correction

In making neutron coincidence measurements, it is necessary to correct for
crosstalk. A signiﬁcant fraction of the measured coincidences comes from crosstalk,
where one neutron is detected and then scattering into another detector where it is
detected again.

Most neutrons are measured in the scintillator when they interact with a proton by
elastic scattering (Figure 4.5). Standard scattering kinematics allow calculation of the

energy and angle of the outgoing neutron, En- and 0’, if the incident neutron energy, E“,

and the scattered proton’s energy, Ep, are known. Since the neutron energy has been
calculated from the time-of-ﬂight and the proton energy can be found from the measured
light, it is possible to reconstruct with some accuracy where the scattered neutron should
go and with what energy.

For each multiplicity-two neutron event, several pieces of information are
available for each hit, including the time the neutron hit the detector, ti, the amount of
light detected, Li, the location of the hit relative to the target which gives the initial

velocity, v i , and the calculated energy of the neutron, E, = 1/2 mviz. This information can

be used to compare the measured event to the calculated crosstalk event (Figure 4.6).

73

En’=En'Ep

13“.=En cosze’

 

Figure 4.5 Elastic scattering kinematics, for a neutron scattering from a
proton. These formulas are used to calculate the energy and direction, Em and 0’, of

the scattered neutron.

74

 

 

Figure 4.6 The variables involved in the crosstalk correction. Ent, Lnt, tot are

calculated assuming elastic scattering, while the others are measured quantities.

75
Assuming the mass of the proton and neutron are equal, the calculated energy is

13,,» = E1 — Ep, where Ep = Ep(L1) is the energy from the light response. A linear

function (Figure 3.19) was used for the light response, so that the crosstalk variables were
symmetric around zero.

Various comparisons can be made between the calculated and measured values.
Three tests were used to ﬁnd events with a large probability of being crosstalk, suggested
by Reference [Wang 97].

1. The difference between the scattering angle calculated, cos 0’, for the

crosstalk neutron and the measured angle, cos 0 , is restricted to:

AC- < cos 0’— cos 0 < AC+.

V1 '(V2 - V1)

IVIHVZ - VII , while the

 

The measured scattering angle is deﬁned as cos 0 =

E I
calculated quantity is cos 0’ = ’13—“. The minimum and maximum difference in
l

angles that can be classiﬁed as a crosstalk event are deﬁned by AC- and AC.,
respectively.

2. The difference between the time needed for the scattered neutron to reach
the second detector, tn-, and the time between hits, t2 - t], is restricted to:

AT_ < tn» — (t2 — t]) < AT+.

. . d . . .
The calculated time 18 tnr = -v—" , where the distance to be traveled 18
n.

cl", = |'r'2 — fl] and the velocity relativistically is calculated as

76

 

2
’E . E - + 2mc , . .
V“: = c —“( n ) and m rs the neutron mass. The minimum and

 

(En. + mc2)2
maximum difference in time that are classiﬁed as crosstalk are AT- and AT..,
respectively.
3. The difference between the scattered neutron’s energy, E... and recoil
proton energy for the light measured in the second detector, E904), is restricted to
En~-Ep(I/2) > AB.
AB is the minimum proton energy possible for a crosstalk event.
If the neutron could be crosstalk by all three of these criteria, it is excluded from the
analysis. The differences in calculated and measured values, cos 0’ - cos 0, tut - (t2 - t1),

and Ent - Ep(Lz) show a strong correlation from crosstalk (Figure 4.7). The ﬁrst two
criteria peak at zero and the last criterion has a peak at positive values as expected for
crosstalk. Two different sets of gates (listed in Table 4.1) have been applied in this ﬁgure.

The crosstalk in true coincidences can be compared to the same variables when
coincidences are constructed by mixing singles events, which thus have no crosstalk
(Figure 4.8). The singles events do not show the sharp peak at zero for the cosine and
time criteria, and in the light criterion they do not show a second peak for positive
differences. (These peaks are the result of crosstalk.) A two-dimensional plot of the
differences in angles vs. the differences in times also shows that the peaks of these
criteria exhibit a signiﬁcant correlation (Figure 4.9) as is expected for crosstalk.

The crosstalk can also be seen in the correlation function. Figure 4.10 shows the

integrated correlation function with one neutron in the front wall and one in the back

77

 

IIITIiI'IIIIIIIlIII

    
 
 
 
  

Yield
H
e

N

   
  

: Cut2
l
IILIIiII IJIIIIIILLIIIjI IIIIJJIIII

-l :0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

cosO'O-cos
4
10 TWIIIIIIIIIIIlIIII IIIIIIIII

I
l l
l __ All events I
l l
l I
I

 

IIII'IIII'II—IIIIIII

 

 

2
.9
*102
....r....rri..i....1....|....r....1....r....Li...
-50 -40 -30 -20 -10 (0 $0 20 30 40 50
ty" tz-tg
104....,....,.......,.,....,..............
E
.2
)-

 

 

-20 -15 -10 -S 0 5 10 15 20
En' - EP(L2) (MeV)
Figure 4.7 Crosstalk criteria. (a) Difference in angles calculated and
measured. (b) Difference in time between hits, calculated and measured. (c)
Difference in the energy of the scattered neutron and the energy corresponding to

the light measured in the second detector. The cuts are shown on the figure and

deﬁned in Table 4.1.

78

Table 4.1 Crosstalk cuts applied to the data. The difference in calculated and

measured scattering angles is gated on AC_ < cosO’—cos0 < AC+ ; the difference in
the time between hits calculated and measured is gated by
AT_ < tn. -— (t2 — t1) < AT+; the difference in energy deposited in the second

detector calculated and measured is gated on En: - E904) > AE

Cut AC- AC. AT- AT+ AE

79

 

Yield

 

 

IIIIIIIIIILLIJI In. I n
1 -l -0.8 -0.6 -0.4 -0.92 0 0%) 0.4 0.6 0.8 1

 

Yield

 

 

Yield

  

_--—-- d

IIIIIIIIIIIIIIIII I
-20 -15 -10 5

- 0 5 .10....15....20
Em - 5.0-2) (MeV)

Figure 4.8 Crosstalk criteria constructed from mixed singles where no

crosstalk is present. Same variables as Figure 4.7. Cut 1 is shown in this case.

0.8
0.6
0.4

O 0.2

0

s O'— cos

30.2
-0.4
-0.6
-0.8

'1-50 .40 -30 -20 -10 0 10 20 30 40 50

80

 

        

 

       

 

 

TTI ITI I—rI I I I I l l I I I I l I l I l I I I I I I I I l I I I I I I I IT—[rTI

r- ' . ‘ -

L ' . ' - - .
r x

i-u : _ -I
.. P _y 1

L i "A ._ ~ ..... : g.

-. n :, . ,‘u
' I ‘- ,- ' r.—

5’1 a. “r Pie; -K ”....

H -_ - - u . g 10

H- . ‘ 'I

u_..' 1'- ‘.

4.3-l .. e

H"‘ . I

.: - . l-

.35..

9.7,! .

:51

,3? ..

-:'-.rr¢-

I 'J' «-

I'an.

".'-.'-'.

 

I.II'I
. 1.1....
I

 
    

Iﬁ'l’lllllll

.
' .
--_--q-_.
n
_ ll
.

 

 

 

llllllllllllll IIIIIIIIIIIIII IIIIlIIlIIIIlIllIII

 

  

tw- (112-t1)

Figure 4.9 Difference between measured and calculated angle versus time

between hits for real coincidences. The dashed lines show where the cuts for

crosstalk were made for Cut 1 and the dotted lines show Cut 2. The solid line is

where both cuts are the same.

81

 

 

-""l"'ll"'rIIIIIIITTTIIIIIIIIIIIlrrY‘
2.4;- -:
2.2L .-

: + o Nocut I

2 .I

. o Cutl I
1.8_ + Ct2 1
16:# + D u 1

+ ° . ‘
3 i + :
011.4. _¢_ + -‘
1

1.2_- it? +

 

 

 

 

 

 

0.8}

0.6:- -j
Lt.11...tittttltttiliiIilit.tlititltit
0 5 10 15 20 25 30 35 40

q (MeV/C)

Figure 4.10 The experimental correlation function with and without a
crosstalk cut. The neutron pairs have IPI < 250 MeV/c. The cuts are defined in

Table 4.1.

82
wall. The correlation function is plotted before and after the crosstalk cut is applied. The
crosstalk events produce an artiﬁcial peak at ~10 MeV/c. One advantage of using
detectors at two different distances is that the crosstalk does not peak at q = 0, where the
correlation function peaks.

The gates for crosstalk were adjusted by observing the effect on the crosstalk
variables, the correlation function, and in the simulations. The gates for the crosstalk must
be made so that most of the crosstalk is removed while retaining as many events as
possible. The two cuts, shown as examples, both eliminate most of the crosstalk.

The correlation functions in Figure 4.10 are constructed using Cut 1 and also

selecting events based on the total momentum of the pair, IPI < 250 MeV/c. By looking
at the correlation function from events above and below this cut, it is possible to see that
some crosstalk remains in the higher momentum pairs (Figure 4.11). The lack of statistics
for the higher momentum events does not allow further investigation with wider crosstalk
gates.

Unfortunately, not all of the crosstalk can be eliminated with this crosstalk
method. Some crosstalk neutrons scatter on carbon, or interact more than twice, so that
two-body scattering kinematics on hydrogen are not applicable. In the next chapter, the

effectiveness of these crosstalk cuts are investigated by comparison with a simulation.

4.2.3 Background Corrections

The experimental correlation function must be corrected for the background

scattered neutrons. Figure 3.24 shows that at least ten percent of the measured neutrons

83

 

2 IIIIfITITrIfIITIIIIIIII[IIII'IIII'IIII

 

1.8 - -
I e P < 250 MeV c .
t / .
1'6 T _ _ A P > 250 MeV/c '1
1.4 - -

a; .
’5 ' ‘
351.2: 1:: + -

 

 

 

 

L
1:- :H‘e‘g‘j‘: :‘r—z‘ k;

r .
0.8“ $

- J
0.6:- -'

hIIIIlILIIlI IIIIIIILLII IIIIIIIIIIJIIILIIIq

0 5 10 15 20 25 30 35 40

q (MeV/C)

Figure 4.11 The correlation functions cut on total momentum of the neutron

pair. Crosstalk Cut 1 was applied to both spectra.

34
are from background events, even after eliminating the lowest-energy events and cutting
on the light measured. These background neutrons add uncorrelated coincidences so that
the measured correlation will be smaller than the true correlation for events coming from
the target.
If the background events are assumed to be uncorrelated to the real neutrons, it is
possible to correct the correlation function for this suppression [Prat 96]. A corrected

correlation function can be deﬁned as

2 N031. 152) — on: Don route]. 52)).

1 + R’ =
(Q) 0‘2 13051, '152)(1 - b051, P2»

 

Here the functions N(f),, 5,) = T6651, oz) and ms“ 15,) = ?,(p,)?,(§,) are the
numerator and denominator as constructed for the standard correlation function. The
denominator in the standard correlation function is a distribution reﬂecting the geometry
and efﬁciency of the detectors without the correlations of the source. This uncorrelated
distribution should be similar to distributions of events that are added by the background
scattering. The function b('pl , ['52) is the fraction of times that an event measuring
neutrons with momenta p, and p2 would be a background event. This weighting is
applied on an event by event basis. The correction assumes the amount of scattered
background in a particular event depends on the detectors hit but is independent of the
energy and horizontal position of the neutrons. The opposite function is used to correct
the denominator of the corrected correlation function. Here the event is weighted based
on the fraction of good events for these detectors. One overall normalization constant is

available for the corrected correlation function, chosen as or.

85
Another way to perform this background correction is to subtract the background

from the singles yields before constructing the correlation function. The measured yield

of each type of particle, T , is considered to be the real yield, Y, plus a background yield,
Yb. Then the corrected correlation function, 1 + R”(q), can be written as the coincidence
yield minus corrections for events with one background and one real neutron and events

with two background neutrons.

1+ R" (q) =

2 (“(131, Pz) - [(?1(Pr) - Yih(Pr))Yzh (P2) + (371 (132) - Y2b(Pz))Yib (131)] - Ylb(Pl)Y2b(P2)) .
2 (T6031) - Ylb(Pl))(?l(P2) — Y2b(P2))

The normalization constant is A.

This method can be shown to correspond to the ﬁrst method if functions fl (1'51)

and f2 (15,) , the fraction of good neutrons in events with this momentum, are deﬁned by

Ylb(P1)
Y1(151)

_ Y2b (P2)

.. _. . The denominator of the R”
Y1 (P2)

f1(P1) = 1" and f2(132) =1

function becomes
f1(P1)Y1(P1)f2(i52)?1(P2) = 8(P1t P2)D(P1. 152).
where g(p1, p2) = l — b(p1, 132) represents the fraction of events that are good. The
correction for one background neutron, can be written as
“1 (mm (m1 — f2<§2))71(ﬁz) + <1 — ft<rt>)?t(in)f2<izﬁt032)]
= nominates» — f1(ﬁ1)f2(P2) + £2032) - f1(P1)f2(P2)]
= 13031, P2)[f1(i51) + f2052) - 2f1(P1)f2(P2)].

The second correction, for two background neutrons, can be written as:

86
[(1- fi(r>t))?1(ivt>(1— f2 032))?1 (§2)]
= Ytr'tst)?t(is2>[1— f1 (P1) — f2(P2) + f1(P1)f2(P2)]

= D(P1,P2)[1- f1(i51) - f2052) + f1(P1)f2(l52)]-

Combining these terms the numerator will be
2'. AN(P1, P2) - Z D(P1,i52)[1- f1(P1)f2(P2)].

Since [1 — f1(f),)f2 (p, )] = b(p1, p2) this form is the same as that given above with A =
lla- So both corrections, R’(q) and R”(q), will give the same result.

After applying this correction, the correlation function was enhanced by about
10% (Figure 4.12). This method provides an average correction for the correlation
function. The background ratios have not been corrected for the neutrons that will reach
the detector through the shadow bar or for changes in the background scattering resulting

from the shadow bars. The original plan for the experiment was to measure the
background spectra Y1 (p1)Y2b (oz) by using small shadow bars, to check the

assumption of uncorrelated background, but the statistics which could be collected in the

time available were too small.

4.2.4 Singles vs. Mixed-Events Technique

In previous studies two methods for creating the denominator have been used
[Zajc 84, Lisa 93b (p105)]. The ﬁrst is to mix neutrons from different coincidence events
and so create an uncorrelated function for the denominator [Troc 87, Fox 88, Cebr 89,
DeYo 89]. This technique requires only coincidence measurements, and it ensures that

the denominator and numerator both come from the same class of events. The second

87

 

2 I l l ' I ' l l l
1.8 0 background correction -
. A not corrected I
1.6 _ j

 

 

 

 

 

 

 

 

0.8-
0.6} -f
OrrrIénrrilldrrrllsrrrrzlorrrrzlsrrrr3lorrrr315rrrr4-O

q (MeV/c)

Figure 4.12 The correlation function corrected for background scattering

(circles). The uncorrected correlation function is shown for comparison (triangles).

(Both have IPI < 250 MeV/c and crosstalk Cut 1 applied.)

88
technique is to use singles events[Lync 83, Chit 85, 86, Poch 85a, 85b, Chen 87, Awes
88, Ghet 93]. Here particles from events where one particle is measured are combined to
construct the denominator.

For two-neutron correlation functions, an ambiguity exists in how to treat
crosstalk events in a mixed-event analysis. When an event has been identiﬁed as
crosstalk, what should happen to it? Both neutrons could be mixed with the rest, only the
“real” neutron included, or the whole event excluded. If the crosstalk is not removed,
neutrons that do not come from the target are included. However, if some events are
excluded before the events are mixed an inefﬁciency that depends on the momentum of
both particles is introduced.

The effect of this crosstalk cut on the correlation function can be seen by

considering how the correlation function is made. Inefﬁciencies in the detectors cause the
measured singles, T165), and coincidence yields, Y2 (15,, p2) , to differ from the true
yields. By constructing a ratio of correlated to uncorrelated events with the appropriate

conditions these differences will divide out. The usual case is when the coincidence yield

is measured with some efﬁciency factor, up] , “152) . The coincidence yield is then
i‘{2(P1rP2) = Y2(I31tP2)8(§1rP2).
In the mixed event technique pseudo-singles are constructed from the coincidence events.
?1 (151) = i 3"(2051, 15211132 = i Y2('151. P2)E(P1’ Pz)dP2 .
similarly

Yarn) = I To (131. P2)dP1 = I Your. P2)€(P1v P2)dP1 .

89

From these yields, the measured correlation function was constructed:

(3: Z?2(§1'ﬁ2) ._ 2Y2(Per2)€(Per2)

 

 

Z Y1(P1)?1(P2) _ Xi Y2(‘151.132)£(131. P2)dPl.1 Y2('151. P2)€(Plr P2>dP2 .
In order for this correlation function to be equal to the real correlation function,

= 2Y2(i51rP2)
2Y1(P1)Y1(P2)’

 

two assumptions must be true:
1. Efﬁciency depends only on the individual momenta.
8(Plr P2) = 8105082052)
2. Pseudo singles are proportional to real singles.

Y1(i52) °° iY2(I31»P2)81(i51)dP1

In the case of neutrons these assumptions cannot be made. The ﬁrst assumption is

especially a problem because the efﬁciency function is not separable. The crosstalk cuts

depend on the combination of "[51 and :32.

The singles technique allows the calculation of the real correlation function from

the measured yields. Corrections to the correlation function necessary to remove crosstalk

events are applied to both the numerator and denominator. With similar notation this can

be written as

C 2?2(§1.§2)g(§1.§2.?1.?2)

z Y1(i51)?1('152)g(§1.§2.'r'1.?2)

 

= ZY2(P19P2)31(P1)82(P2)8(P19P2,?1’f2) .
2 Y1 (P1)81(P1)Y1 ('152)82(P2)8(i51r P2. fl, ?2)

 

90
Since Y2 (151, p2) = CY1('p1)Y1(i52), if the correlation function, C, varies slowly over
this range, it can be factored out and the sums cancel.

Because the mixed-events technique is not valid in this case, a singles technique
was used. The problem of treating crosstalk remains if events with multiplicities greater
than one are included in the sample. So the denominator was constructed with n = 1
singles. The multiplicity-one events occur about a factor of 30 more often, so the effect of

neglecting the coincidence events should be small.

4.2.5 Front-Wall Pairs

The experimental correlation functions shown so far (Figure 4.10, Figure 4.11,
and Figure 4.12) were constructed from events with one neutron in each wall.
Coincidences were also measured when two neutrons were in the same wall. Two hits in
the front wall are more frequent than two hits in the back wall because of the larger solid
angle and because the neutron ﬂux is attenuated by the front wall.

Neutrons measured in the same wall have some disadvantages compared to
neutrons in different walls. Because of ﬁnite detector size, neutrons must hit different
detectors and so have at least some minimum relative momentum. Also, the crosstalk
correction for front-front pairs is less accurate because the angle between hits is not
measured. As a limiting case, the angles are calculated by assuming the ﬁrst hit occurs at
the front face of the detector, and the second hit occurs at the back of the detector. Then
the same crosstalk technique is applied for these pairs. The gates are less effective and

must be adjusted again for front-front pairs (AC+ = 0.7, AC- = -0.3, AT, = 10 ns, AT- = -10

91

ns, AB = -2.0 MeV). The biggest disadvantage for same-wall events is that the crosstalk

peaks at zero relative momentum just where the correlation function peaks.

The correlation function for front-front pairs is compared to front-back pairs in
Figure 4.13. The shape of the front-front correlation function is similar to that measured
for the front-back pairs. The difference in sensitivity to low q values can be seen in this

plot.

4.3 Longitudinal and Transverse Correlations

In order to distinguish between source size and lifetime, a cut can be made on the
direction of q relative to the total momentum of the pair in the source reference frame.
The source velocity is assumed to be the velocity of the center of mass of the system We =

0.045. After this source correction, the longitudinal, ‘l’ < 60°, and transverse, ‘1’ > 80°,

correlation functions are constructed. The cuts for front-back pairs are inconclusive but
are consistent with no separation between longitudinal and transverse directions (Figure

4.14).

92

 

I

2 lIITlIIIIIIIIIlIIII'IIII'IIII'IIII'IIII
I

1.8 - . front-back
A front-front

I I I I

 

I T

I

I"
a.
I

l"
N
l

I I I I I I I I L I l

 
  

+
it" -A-

 

0.8

0.6- -
r.rlnnrrlrrrrlrrrrlrrrrlrrrrlrrrrlrrrrd
0 5 10 15 20 25 30 35 40
q(MeV/c)

 

 

 

Figure 4.13 Correlation functions for front-front pairs (triangles) and front-

back pairs (circles). Both are corrected for background and have IPI < 250 MeV/c.
Cut 1 is applied to the front-back pairs and the cut described in the text applied to

the front-front pairs.

93

 

2 . .14...,....,....,........., ...,....
p

1.8 - -

. A Longitudinal .

' 0 Transverse C) ‘

1.6 " i f

1.4 — :

 

 

Rig)+1
N
—O—>—-
_o.
_.O_
O-——
...}...-
ID
> r

 

 

 

0.8 L f .2
0.6 _-

O 5 10 15 20 25 3O 35 40
q(MeV/c)

T I I

 

 

 

 

Figure 4.14 Longitudinal and transverse two-neutron correlation functions

with IPI < 250MeV/c.

Chapter 5 - Two-Neutron Intensity Interferometry:

Model Simulations of Detector Response

5.1 Description of Models

Constructing a correlation function for neutrons requires many corrections, as
described in the previous chapter. In order to assess how effective these corrections are,
and what distortions they might introduce into the results, a simulation which includes the
detector response is necessary. Simulations can be especially useful in studying crosstalk
and detector efﬁciency.

Two different simulation programs were used to model this experiment. One
program was developed at Michigan State University by Wang et a1. [Wang 97]
speciﬁcally to model crosstalk in the Neutron Walls. The other model, GEANT from the
CERN library software, is designed for general detector simulation and has been applied
to this speciﬁc detector. Both models are based on Monte Carlo principles. For a given
event some number of neutrons are initialized: each neutron is given an initial energy and
direction. The neutrons are propagated through the detector by small steps. At each step,
random numbers are used to sample probability distributions which describe the cross
sections for various neutron interactions. After all the steps are completed for all
neutrons, the result is compared to some criterion such as whether two detectors
registered a neutron interaction. If the event matches this criterion its properties are

recorded for further study.

94

95

5. 1.1 Neutron Interactions

A key element in the simulations is how the neutrons interact in the scintillator.
Neutrons can interact with the scintillator by several mechanisms, each with its own
partial cross section. For each interaction, the energy deposited must be converted to the
amount of light that would be measured by the photo-multiplier tubes, PMTS, and any
scattered neutrons must continue to be tracked to see if they interact again somewhere
else in the Neutron Walls.

The NE-213 scintillator is made up of hydrogen and carbon. The neutrons can
interact with these elements by one of six possible mechanisms, listed in Table 5.1 [Wang
97, Ceci 79]. The energy-dependent cross section for an interaction determines how often
that particular event occurs. For illustration, the cross sections for a 24 MeV neutron are
given in Table 5.1. Figure 5.1 shows the energy dependence of the neutron interaction
with hydrogen and carbon. The ﬁrst two mechanisms have the largest cross sections and
account for most interactions for low energy neutrons. The other mechanisms become
more probable at higher neutron energies.

When an interaction occurs in the scintillator, the amount of light produced must
be calculated. The simulation determines what kind of interaction has occurred, for
example, elastic scattering on a proton. Then the energy of the proton is dissipated in the
detector. Both simulations model this process by propagating the proton by small steps
until it stops in the scintillator. At each step the energy lost by the proton is converted into
light by a light response function. Each particle that excites the scintillator has its own

light response (Figure 5.2). The curves used in the simulation come from ﬁts to the data

96

Table 5.1 Possible interactions between a neutron and the scintillator. For

illustration, cross sections for the reaction, on, are given for a 24 MeV neutron

[Wang 97]-
INTERACTION OR (24 MeV)
1 n + p —) n + p 0.406 b
2 n+C —-> n + C 0.900 b
3 n+C —) n’ + C + 7(4.44 MeV) 0.104 b
4 n+C —) He + Be - 5.71 MeV 0.048 b
5 n+C —) n + 30L - 7.26 MeV 0.210 b

6 n+C—>p+B- 12.59 MeV 0.100b

97

 

 

 

10 Evttl IIlthtI1r1rllIIIIIIIIr1ﬁIT'IIII:
*‘ Hd '
_. yrogen .
1
A
(O
C
L
O
.0
V .
S 2
nnnnlnnnnlIn:nlnirnlLanJJnlnllnanLllnan’
:8 10 :UIIIIUIIIIIIIIIIIIrTIrIllIIIIIIIYIIIIII:
3 : :
a) - Carbon 1
CD .
0
L
O
1
10.1.‘lll""l'llnllllllnnn lllllllljllllll

 

 

 

 

.25
Energy (MeV)

10 15 20

Figure 5.1 GEANT total cross-sections for neutrons scattering on hydrogen

and on carbon as a function of the neutron energy.

30 35

40

98

 

d

lllllllllllL

25 Y—V'I'UUIIIUIITIUIIIIUUYU 'TII'IUUIIUV

22.5

electron

20

17 .5

IIITIIUlIﬁ'llU‘

proton

p—i
Ul

Olph

l-i
G

IIIIII'I‘THITUVVIIIII'IIIIIVI

Light yield (MeVee)
i3
u:

."
u:

lJllllLllllllllll'LlllLllLllll

carbon

 

 

5L

' ' ‘ I I
0 - ,,__._.__ 11111 IlJlllllll 1:11 1111

0 510 15 20 25 30 35
Energy (MeV)

.8

0

Figure 5.2 Light response for electrons, protons, alphas, and carbons. These
response curves are fits to the data from Verbinski [Verb 68, Wang 97]. The light is

measured in MeV electron equivalent, the light produced by a 1 MeV electron.

99

from Reference [Verb 68] of the form given in Table 5.2. The light deposited in the
detector is accumulated over all the steps until the particle stops. At the step when the
light exceeds the light threshold, Lu” the time and position of the interaction are recorded.

The probability of a neutron being detected by a certain mechanism is a
combination of the cross section and the light response for that mechanism. Since the
protons have the largest light response, and Mechanism #1 has a large cross section, the
low energy neutrons in this reaction are most likely to be detected by this mechanism.
Some of the interactions which produce alphas, Mechanism #5, exceed the experimental
light threshold of 1 MeVee. However, those interactions which have carbon in the ﬁnal
state, Mechanism #2, do not produce enough light to be detected. Neutrons that interact
by Mechanism #3 produce a gamma which can be eliminated by pulse-shape
discrimination.

After an interaction, any neutrons that remain continue to be tracked to see if they
interact again in any other detectors. These scattered neutrons can produce crosstalk

events.

5.1.2 Neutron Initialization and Detector Response

Besides the physics of the neutron interactions, other quantities dependent on this
experiment are also included in the simulations. The initial neutron energy spectra, the
detector resolution, and the light threshold must all be incorporated in a reasonable way.

The neutrons were initialized with values consistent with the measured results.

The energy of each initial neutron is chosen from a two-component exponential

100

Table 5.2 Parameters used to model the light response. These values were

determined from a fit of the data given in [Verb 68] with the function L = a (l-exp(b

E3) + d E + f.

particle a b c d f
proton ~5.6375 0.1 15 0.838 0.7875 0.015
alpha(>5 MeV) -6.l643 -0.001 2.2 0.01585 0.0
alpha(>5 MeV) -5.2688 0.0872 1.22 0.543 0.0

Carbon 0.0 0.0 0.0 0.017 0.0

101

distribution. This distribution was chosen so that the experimental singles spectrum for
the front wall is reproduced by the same spectrum from the simulation. The initial
neutron direction was chosen isotropically in the direction of the Neutron Walls.

Both simulations determine when a neutron is detected based on a light threshold.
This threshold is applied to both the left and right light signals. The total light, L, is
determined from the light response curve. Then the light measured at the left and right
PMT is calculated by: LL = Lexp(-x/k) and LR = Lexp(-(d-x)/X), where the position x is
deﬁned from the left side of the cell as in Figure 3.18. The attenuation length used is A =

226.8 cm. The value of I. affects the position dependence of the threshold (Figure 5.3).

Information is recorded for each detected neutron, including the original
momentum of the particle, the ﬁnal time and position of the hit, the total light, and if the
neutron has already interacted in another detector. The position and time of interaction are
deﬁned at the time when the total amount of light measured becomes greater than the
light threshold. The light is recorded after all interactions have been completed.

The models simulate detector resolution by adding a number sampled from a
Gaussian distribution to the measured signals. The widths of these distributions were
determined from the data and their values are listed in

. The time resolution was applied to the measured time of neutron detection. In the
experiment, the position resolution in x is related to the time resolution and the geometry

of the cell. In the simulations, all of these effects were combined into one width, 0,. The

y and 2 dimensions are constants assigned based on the cell where the interaction

occurred. The light resolution, 01,, is proportional to the number of photons detected in

102

 

'lllll'ﬁ'1liii'l1l'll

lo IT'IIIIIIIIVIIIIIIIII

   
  

  
  

 

simulation

Light (MeVee)

IIIIIIIIllllll'lll'llliilllllllll

   

llllllllllllllllllllllllllllllllllllll

l—YTI‘IIIIIIIUIIIIIIllllllllllllIIIITTIIIIIIIIllll
t’llllllllllllllllllllllll'lllllllllllll

 

'Ttlﬁllll

..
-.
.
..

_.
.1
..

 

 

 

:-
0 lllll'lllllllllllllll llllllllll'llllllllll

100 150 200 250 300 100 150 200 250 300

 

Position (cm)

Figure 5.3 Position of the interaction in the cell versus the measured light
signal. A light threshold of l MeVee at the left and right PMT is applied in both the

data and the simulation.

103

Table 5.3 Detector resolution for time, position, and light, used as input in

models. The resolution is assumed to be Gaussian with this standard deviation.

Standard deviation of Gaussian (0')

time resolution a} = 0.5 ns
position resolution (57‘ = 5 cm
L,
0'1. — LRLT —L- ,

light resolution

104

the PMT, N, which is proportional to JL. So for a measured light intensity, L, the light

resolution can be determined from the light resolution at threshold, R”, by the formula
CL = Li * RLT * 1/LT / Li , where by is the light threshold. This resolution is applied

to the left and right light signals. The light recorded is the geometric mean of the left and

right light signals after the resolution has been applied.

5.1.3 GEANT Simulation Code

One model used was GEANT, the Detector Description and Simulation Tool from
the CERN libraries. It was originally designed for high-energy experiments, but has been
modiﬁed to be applicable for lower-energy experiments [GEAN 93]. The simulation code
includes all of the routines for particle tracking. The user deﬁnes the detector geometry
and the initializes the incident particles and then the GEANT code propagates the
particles. At each step the user can access the information about the position and energy
of particles in the simulation. If the incident particle interacts with other particles,
information about those particles is also available. The user must deﬁne what results
should be recorded and in what format.

In order to use the appropriate cross sections for low-energy neutron interactions,
speciﬁc libraries must be used. The low—energy (E < 20 MeV) neutron cross sections are
sampled from the MICAP package (A Monte Carlo Ionization Chamber Analysis
Package). Cross sections for energies greater than 20 MeV are deﬁned by the FLUKA
routines. The total cross sections used for neutrons incident on protons and carbon are

shown in Figure 5 . 1.

105

The detector geometry must be deﬁned by the user. The cells are deﬁned as boxes
with dimensions the size of the cells, 7.62 cm x 6.35 cm x 200 cm and ﬁlled with a
material deﬁned by an atomic ratio of H/C = 1.212 and a density of 0.874 g/cm3. The
cells are located in a larger volume of air with a size of 300 cm x 300 cm x1200 cm. The
coordinate system is deﬁned so that the origin is the source of the particles, and the two
walls made of cells are located at z = 400 cm and 500 cm perpendicular to the z axis.

In this model many events were simulated. For each event either one or two
neutrons were initialized and the neutrons and any particles they interact with were
tracked by the code. When all particles have stopped or have escaped from the air-

volume, the event ends.

5.1.4 Wang Simulation Code

The other code we used was written speciﬁcally for the Neutron Walls [Wang 97].
This Monte Carlo simulation was based on a previous code [Ceci 79], but it includes the
most recent cross section data from the National Nuclear Data Center (NNDC).

The Wang code varies from the GEANT code in a couple of ways. First, it
includes the Pyrex cells in the calculations. Neutrons can interact with the Pyrex, but
these interactions do not produce any light. Another difference is that this code
speciﬁcally tracks the neutrons ﬁrst into the front wall and then into the back wall. This
algorithm does not consider the rare occurrence of backseattering from the back wall into

the front wall.

106

Several changes to this code were implemented in order to make it more
applicable to this experiment and easier to use. Technical changes included explicitly
deﬁning the variables in the FORTRAN code and modifying the output to write n-tuples
for analysis with the CERN library code, PAW. A different random number generator
was also used for sampling. Because of the large number of random numbers used in this
code and the large number of events simulated, a random number generator with a long
period was required.

A couple of implementation changes were also made. First, the light threshold for
neutron detection was applied to left and right signals. Originally the threshold was just
applied to the total light, and so did not simulate the position dependence of the threshold.
Also the deﬁnition of the recorded time was changed. The original code recorded the last
time any energy was deposited in a detector, while the modiﬁed code records the time
when the light in the detector exceeds the threshold. In many cases this is a small change,
but sometimes the proton takes a long time to lose the last bit of energy, and this extends
the measured time by a signiﬁcant amount.

This simulation was designed to model one or two initial neutrons. An event
consists of the neutrons being initialized, propagated through the ﬁrst wall, and then

propagated through the second wall.

107

5.2 Simulation Results

5.2.1 Efﬁciency

One useful result from these simulations is the efﬁciency of the detector as a
function of energy. In order to compare the measured energy spectra to model predictions,
this efﬁciency must be included. For both simulations the efﬁciency of the front wall was
calculated by dividing the measured energy spectra by the initial energy spectra (Figure
5.4). The two simulations show a similar dependence on energy. The efﬁciency depends
on the light threshold, which in this case was 1 MeVee for both the left and right light

signals.

5.2.2 Crosstalk

Both simulation codes model the neutron interactions which produce crosstalk.
Because the crosstalk in the simulation can be identiﬁed, it can be used to study how
much crosstalk is present, how well crosstalk elimination methods remove it, and how
these methods affect the distribution of real events.

The same method used to eliminate crosstalk in the data was also applied to the
simulation (Section 4.2.2). The calculated energy of the scattered neutron, En-, is
determined from the energy of the ﬁrst neutron, E1, and the proton energy, Ep. The
neutron energy E; is calculated from the interaction time and the proton energy for the

detected light, L1, from the Verbinski light curve, Ep = EP(L1). From these measured

108

 

 

 

 

”up...,...r,....,....,....,.........,....,..r.d
0'14.- e Wang ...
0.12:. A Geant _

* +1: -‘
0.1 ~ '.-e-:3'-'.‘. a
j 1' ""1-

6008' {5* '

C - _.

9 e - I’... - -

E, “13447

Lt! 1
0.06 - __ -

-:
0.04 - _
. l
0e02 _ _l
. :.- :
0 111l+LLJlllllllllllllllllLllIIIJLIJLLLLIIlllllljj

0 5 10 15 20 25 30 35 40 45 50
Energy (MeV)

Figure 5.4 Efficiency predicted by Wang and GEANT simulations as a
function of neutron energy. The light threshold was set at 1 MeVee for left and right

light signals.

109
quantities En- = ErEp is calculated. The same three criteria used with the data are used in
the simulation to ﬁnd the events with a large probability of being crosstalk:
1. The difference between the calculated scattering angle, cosO’, and the
measured scattering angle, cost), is within the range where crosstalk is expected :
AC_ < cos 6’— cos 0 < AC+.
2. The difference between the calculated time between hits, tn', and the
measured time between hits, (tz-tl), within a range close to zero:
AT_ < tn, — (t2 - t1) < AT+.
3. The difference between the calculated energy of the scattered neutron, En'.

and the minimum energy of the neutron which could deposit the measured light in

the second detector, Ep(Lz), indicates that this is a possible crosstalk event:

13,. spa/z) > AB.
Crosstalk events are identiﬁed as those where all three criteria are satisﬁed. These
variables from both models look qualitatively like the data (Figure 5.5 and Figure 5.6
compare to Figure 4.7). The crosstalk is evident in peaks centered at zero for the ﬁrst two
criteria, and at a positive value for the third criterion.

In the models, the crosstalk events can be distinguished from the true
coincidences. When the crosstalk criteria are plotted only for the events which are
crosstalk, the validity of these cuts can be seen (Figure 5.7 and Figure 5.8). For the
correlation functions, three different gates were applied to the simulations (Table 5.4).
Cuts 2 and 3 are shown in the ﬁgures of the crosstalk variables. Note that these cuts are

numbered differently than the cuts in Chapter 4.

110

 

  
 
 

    
   

 

 

103
:5 102
3: _ . All events
10 After Cut 2
1 Lill.1...i...1...I...|...1...LU111...
-l -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
cos O'- cos 0
103 b
E
a
s: 102
10
-50 -40 -30 -20 -10 0 10 20 30 40 50
to"- (tZ-ti)
...........l....,....l..-.,....,..
U
7‘."
)—

 

lklllllllllilllllllllllllllllllllllllji

-20 -15 -10 -5 0 5 10 15 20
En' - Eva-2) (MeV)

Figure 5.5 Crosstalk criteria for simulations from the Wang model. The solid
line is for no crosstalk cut, while the dashed line shows the results after Cut 2 is
applied. (a) Difference in angles calculated and measured. (b) Difference in time
between hits, calculated and measured. (c) Difference in calculated energy and

energy of proton from light measured in the second detector.

Yield

Yield

Yield

111

 

10 3
10 2

. V All events
10 —- - After Cut 2

      

l lllllllllJlllllllllllle
-l -0.8 -0.6 -0.4 -0.2 0 0.2 0.4
cosO’-cos@

..i..
0.6 0.8 l

 

IIITVUUIITTI'UTTTIIIIIIIIIU'UIIIIIUIUIIVFVIVIII

 

 

llllllllllllIllllllllllllllllllllllllllllllllllll
-50 -40 -30 -20 -10 0 10 20 30 40 50
tn'— (tZ-tl)

 

 

 

.26....” .10 5 0... 5...10....15....20
En' - Eva-2) (MeV)

Figure 5.6 Same as Figure 5.5 but from the GEANT model.

112

 

  

     

 

 

 

103 OIr ‘TT I
2 102
.2
>- 10 a _
1 ...I...| J znsiMiiu-Jd:ﬂiikn
-1 -0.s -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
cosO’-cosO
'U
1’
>-
-50 .40 -30 .20 .10 10 20 30 40 50
"'l " I
103 c
‘o 2
74'.) 10
>’

  

l

-15

-20 -10

 

Figure 5.7 Same as Figure 5.5 for events that are crosstalk in the Wang
model. The solid line is events with no crosstalk cut, the dashed line is after Cut 2 is

applied, and the dotted line is after Cut 3.

Yield

Yield

Yield

113

 

 

l . :
-l -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
cos 6 '- cos 0

    

.10 o 10 20 30 40 so
tn'- (tZ-tl)

VIII—[1"

 

     
   

Figure 5.8 Same as Figure 5.7, but for the GEANT model.

114

Table 5.4 Crosstalk cuts applied for both models. The difference in

calculated and measured scattering angles is gated on AC_ < cos0’—cos9 < AC+;
the difference in the time between hits calculated and measured is gated by
AT_ < tn, — (t2 — t1) < AT+; the difference in energy deposited in the second

detector calculated and measured is gated on En: - EP(L2) > AE

Cut AC- AC+ AT- AT+ AB
1 -O.1 0.1 -10 ns 10 ns -2 MeV
2 -O 3 0.3 -10 10 -2

l 15
The fraction of measured coincidences which are crosstalk depends strongly on
the total momentum of the neutron pair. The yield of real coincidences decreases with

increasing total momentum, lPl (Figure 5.9, top panel). At the same time the ratio of

crosstalk/total events increases (Figure 5.9, lower panel). Crosstalk events make up a
larger part of the coincidences at high momentum because higher-energy neutrons are
more likely to measured twice, i.e. to deposit light which will be above threshold in two

detectors. Even after applying a crosstalk cut, Cut 2, up to 30% of the remaining events at

high total momentum, l PI > 250 MeV/c, are crosstalk. The momentum dependence of the

crosstalk ratio explains why the experimental correlation function with | PI > 250 MeV/c

still shows a peak at 10 MeV/c: crosstalk remains in the numerator (Figure 4.11).

The momentum dependence of the events can be seen in more detail in Table 5.5.
The events are divided into two classes deﬁned by IPI < 250 MeV/c and IPI > 250

MeV/c. In each class, the number of events is given for a sample run with no crosstalk
cut. The number of total, real, and crosstalk events is given for both the Wang and
GEANT models. The three crosstalk cuts are applied and the fraction of events remaining
is listed for each type of event. As the cuts are made wider, the fraction of crosstalk
removed is larger, as is the fraction of the real events removed. The low-momentum
events start with ﬁve times less crosstalk. For the low-momentum pairs, the gates remove
more crosstalk. For example, Cut 2 removes about twice as much of the crosstalk from

the low—momentum pairs as from the high-momentum pairs.

Real Yield
...-i
c
N

amahmei—

ia'

Crosstalk/Total counts
9 e c a 9 c c c c
c i— N

116

 

 

 

— No Crosstalk Cut
-— - After Cut 2

' i
I” III
‘I ha”
0|]
L l.
A“;
'1rrL
LJ“

_ _, .-1r.'.'.‘|r....1....I....|....
200 225 250 2,75 300 325 350 375 400
lPl (MeV/c)

Figure 5.9 Momentum dependence of crosstalk. The top panel shows the real

coincidences from two different neutrons before and after Cut 2 is applied. The

bottom panel shows the fraction of crosstalk/total events before and after Cut 2 is

applied.

117

Table 5.5 Statistics for two classes of events A. IPI < 250 MeV/c and B. l Pl >

250 MeV/c. The second column gives the number of events total, real, and crosstalk
events in each class for both GEANT and Wang models when no cuts have been
applied. The other columns show the ratio remaining in that category after the cut

has been applied. The cuts are defined in Table 5.4.

A. I Pl < 250 MeV/c

 

 

No cuts Cut 1 Cut 2 Cut 3
# ratio ratio ratio remaining
remalmng remaining
GEANT all 11649 0.9 0.86 0.84
GEANT real 10506 0.980 0.95 0.93
GEANT crosstalk 1143 0.14 0.03 0.01
Wang all 20659 0.92 0.88 0.85
Wang real 18976 0.98 0.95 0.92
Wang crosstalk 1683 0.19 0.04 0.01

 

 

B. I i5: > 250 MeV/c

 

 

No cuts Cut 1 Cut 2 Cut 3
# ratio ratio ratio
remaining remaimng remaining

GEANT all 27152 0.65 0.57 0.53
GEANT real 15638 0.98 0.94 0.9
GEANT crosstalk 11514 0.20 0.06 0.02
Wang all 37171 0.66 0.58 0.53
Wang real 21847 0.98 0.94 0.89
Wang crosstalk 15324 0.20 0.07 0.02

 

 

118

5.2.3 Resolution of Relative Momenta

In constructing a correlation function it is important to understand the resolution
of the detector for measuring the relative momentum, q. This is difﬁcult to calculate
analytically, but the simulation provides a simple way to compare the real relative
momentum with the measured one. In the simulation two values of the relative
momentum were chosen to show the resolution at low relative momentum, q = 4 MeV/c

and 8 MeV/c. For these values, the ﬁrst neutron’s momentum, 151, was chosen by

sampling the experimental energy spectra and choosing a direction from an isotropic

distribution. Then the momentum of the second neutron, p52 , was constructed from
r; - "151, where the direction of G was chosen randomly. The resulting plot of measured q

shows a resolution of less than 2 MeV/c FWHM for q < 10 MeV/c (Figure 5.10).

5.2.4 Correlation Function

In order to investigate the effect of the crosstalk and its elimination on the
correlation function, the simulated detector response was combined with a theoretical
source to produce correlation functions. The correlation function was calculated from the
one-particle phase-space distribution using the Koonin-Pratt formalism (Appendix)
[Koon 77, Prat 87]. The one-particle phase-space distribution was determined from a
momentum distribution of the neutrons, their emission points and the time of emission.

The simulated correlation functions were produced by a two-step process. In the

ﬁrst step, the initial and measured momentum of the detected neutron events were

 

119

 

3500 TTTI'ITTIIITTTIIIIIlIIIIlIITI'IﬁI'IIT—rTIIIIlIIII

E

I I I I I I r

units)

H _ (arﬂtraryN

§ § § §

cjﬁIIIIIIjITIITTIIITTIIIIII
lljlllllllllllll

Measured Yield

§

 

 

Q

 

q (MeV/C)

Figure 5.10 Width of measured relative momentum for a given real

momentum, either 4 or 8 MeV/c, from the GEANT simulation.

120
recorded. In the second step, these events were combined with source points and a
correlation function constructed.

The ﬁrst step involves running these simulations for many events with one or two
initial neutrons. The numerator was constructed from events where two neutrons were
incident on the detector and two neutrons were measured. These measured coincidences
include both real and crosstalk events. The events used in the denominator had one
neutron incident on the detector and one neutron measured. For both types of events, the
original momentum, the measured time, and the measured position were recorded. The
measured values determine a measured momentum. The initial neutron energy for the
simulation was sampled from the efﬁciency-corrected energy spectrum from the
experiment.

Besides the momentum, it is also necessary to know the initial position and time
of the neutron to apply the Koonin-Pratt formalism. For the correlation functions shown
in this chapter, the source points were chosen from a surface model where the position
and time are independent of the momentum of the particles. For each recorded event, a
position was chosen from the surface of a sphere with a speciﬁed radius, r. The emission

time was chosen by sampling an exponential distribution with a lifetime I. More

complicated models can be used if a table or function is constructed to relate the emission
time and location to the initial momentum.

Given the original momentum and source points, it is possible to apply the
Koonin-Pratt formalism to construct a correlation function. For neutron pairs from the

target, this formalism calculates a weight corresponding to the correlation from

121
interactions between the neutrons. For crosstalk neutrons, there is no interaction and the
weight is set equal to one. The original momentum is used to calculate the weight for the
pair, but the measured momentum determines which q-bin is incremented with this
weight. The denominator is constructed by mixing singles and binning them by the
measured relative momentum, q. The resulting correlation function is normalized to one

for 20 MeV/c < q < 40 MeV/c.
The correlation function for front-back pairs with cuts on I Pl <250 MeV/c and E

> 5 MeV but without a crosstalk correction (Figure 5.11 open circles) is similar to the
correlation function for the data (Figure 4.10). The amplitude of the crosstalk peak in the
simulation is smaller than that in the data because of a different weighting of real to
crosstalk events. In the simulation, only events with two neutrons incident on the detector
are included in the numerator. For the data, the number of incident neutrons varies and
many events have one incident neutron. The number of crosstalk events in the data is
increased by crosstalk from these one neutron events.

The simulated correlation functions, after corrections, should reproduce
theoretical correlation functions without detector effects. These theoretical correlation
functions are also constructed using the Koonin-Pratt formalism. The correlations are
produced using the same source size and lifetime. The neutron energy is sampled from
the same experimental energy spectra and used directly in the correlation function, i.e. no
crosstalk, detector efﬁciency, or resolution effects. The theory calculates the numerator
and denominator at the same time. The numerator includes a weight to account for ﬁnal-

state interactions, while the denominator weights all pairs equally.

122

 

2 IIIIIIIIIIIIII'IIIIIIIIIIIIIIlIIIIlIIrI

 

 

0.8 _- -

0.6 '— -

l'lllllIlllllkLlllllLlllllIlllllllll

0‘” 5 10 15 20 25 30 35 40
Q(MeV/C)

 

 

 

Figure 5.11 The total correlation function from the GEANT simulation. The
line shows the theoretical correlation function for a source with r = 7 fur and 1: = 700
We and the points show the results of the simulation after various crosstalk cuts.

I Pl < 250 MeV/c.

123

The simulated correlation functions after crosstalk cuts are similar to the
theoretical correlations with no detector effects. The results of the correlation functions
after crosstalk cuts are also shown in Figure 5.11. All three crosstalk cuts give similar
results for the correlation function, showing that the correlation function is stable with
respect to this cut. The differences between the theory and simulation for the lowest
relative momentum bins are due to the ﬁnite resolution in the detector.

For the data, directional cuts were made on the correlation functions, and these

cuts can also be made for the simulated correlation function. As in the data, the angle
‘1’ = cos’1 (P - q / Pq) is calculated in the source rest frame (Section 4.1). Longitudinal
events are deﬁned as ‘I’ < 60°, while transverse events have ‘I’ > 80°. The effects of

crosstalk, obvious in comparison to the theory, appear primarily in the longitudinal
direction (Figure 5.12). These directional correlation functions must be constructed
carefully since a cut that does not remove enough crosstalk can artiﬁcially show a
separation between longitudinal and transverse correlation functions. With the crosstalk
corrections Cut 2 or Cut 3, the correlation function reproduces the theory (Figure 5.13). In
the case of Cut 1, the simulated longitudinal correlation functions is higher than the same
correlation function for the other two cuts, i.e. some crosstalk remains. The directional
cuts are more sensitive to crosstalk than the total correlation function, but with enough
crosstalk removed, they seem to reproduce the strength of the correlation expected from
the theory.

The simulation also shows that the correlation function is not as stable for the

high-momentum neutron pairs. In Figure 5.14 the total correlation function is plotted for

 

124

 

 

 

2 I I I I l I I I I I I I I l I I I I I I l I l I I l I

b -A- I
I- -l
. J

1e8 ’- _‘_ q
l- . o q
.+ ‘ Longitudinal .
. - 9 Transverse *

1.6 - ‘4' A
i- “I‘ —' Theory .
b -A-

1.4 ‘
v- I' + 4
+ -A-

9 -l
V 1.2 - ‘
O: ..‘_ -l
l b h g -1

:- -l

0.8 b ‘

:- cl

. cl

0.6 ' '-
III I I I I I I I I I I I I I I I I I I I I_LLLI I14 I LI I I

 

 

 

0"5 10 15 20 25 30 35 4o
C1(MeV/C)

Figure 5.12 The longitudinal and transverse correlation functions from the

GEANT simulation without crosstalk corrections. Longitudinal is selected by ‘l’ <

60° and transverse is selected by w > 80°. IPI < 250 MeV/c.

125

 

2 IIjTIIIIIIITj’I'IIIIIIIIIIIIiIITTIﬁIIIII
h

 

 

 

 

1'8 _- 0 Cut 1 -

El Cut 2 .

' A Cut 3 '

1'6 . — Theory 2
1.4 _
2; ,3 Longitudinal O-50° :
E; 1.2 _ , -
1 . - . ‘ ' j::..;’.-e-;-i-L‘+- —

0.8 h. Transverse 80-90° .:
0.6 - -

 

 

 

0” '5‘ ' I '10 ”15””20 2530' ”35' ' 40
q (MeV/c)
Figure 5.13 Longitudinal and transverse correlation functions from the

GEANT simulation. The lines show the theory. The points represent the simulation

after crosstalk cuts. l Pl < 250 MeV/c.

126

 

     

 

 

2 1—r I I I I I I I I—T I I I l I I I I I I I I I I I IiI‘IT I I I I l I I r!
.- -O-O- -l
1.8 - 0 Cut 1 -
. E] Cut 2 .
1 6 L '9' -O- A Cut 3 '
° . — Input correlation .
-o- ‘
1.4 - -
i -0- .
U . .
FE 1.2 _ .
D 0 I
* '°'-o- .
1 1 .~ . _
0.8 - -
0.6 - -
I I L I I I I I I I L I I I I I I I I I L LLI I I I I I I I I I I I I I I I‘

 

 

 

0 5 10 15 20 25 30 35 40
q(MeV/C)

Figure 5.14 Total correlation function for high momentum pairs. IPI > 250

MeV/c. The cuts are described in Table 5.4.

127
IF! >250 MeV/c. Cut 1 is no longer enough to remove crosstalk, and the correlation

functions for the other cuts show more variation. The longitudinal and transverse cuts

show a similar effect in Figure 5.15. Here the simulation does not match the theory as

well as it did in the low total momentum case.

128

 

   

 

 

 

 

 

2 - l l I l t I ﬂ l l 1 t l

1.8 - -

C u Cut 2 .

L A Cut 3 j

1'6 . 3 - Theory .

P i

.. a _

'- l.4 a: ‘

i l- \ ﬁ ..

g 1.2 .. ii 3 '1 ' a
ﬁg :é: .‘ .- Longitudinal O-50°

1 -I 4539-76, A Z T? "2' 19W“

_j 13:6: ' ' ' " .

0.8 f Transverse 80-90° ‘

0.6 - -

. 1[11111111111null-11.1111.Innnnlnnnn

 

0'” 5 10 15 20 25 3o 35 40
.Q(MeV/C)

Figure 5.15 Longitudinal and transverse cuts for the high-momentum pairs.

I Pl > 250 MeV/c. The cuts are described in Table 5.4.

Chapter 6 - Two-Neutron Intensity Interferometry:

Interpretation

Chapter 4 shows how the correlation function for the system 40Ar + I65Ho at EIA
= 25 MeV was constructed and corrected for background effects. In order to interpret the
experimental correlation function in terms of a source size and lifetime, it must be
compared to a model. In this chapter, several models are used in order to understand how
the neutron emission time from the correlation function compares to the predictions of an

evaporative model.

6.1 Surface Emission Model

One simple model to compare to the correlation function is a source emitting
particles with an exponential lifetime. Speciﬁcally, surface emission was used. Particles
are emitted from the surface of a sphere of radius, r, with times chosen according to an

exponential distribution exp(-t/t) described by a lifetime 1:. The energy for each neutron is

chosen by sampling the efﬁciency-corrected singles spectrum from the experiment. The
energy selected for a neutron is assumed to be completely independent of its emission
time.

The single-particle phase-space distribution produced by this model was ﬁltered,
made into a correlation function by the Koonin-Pratt formalism (Appendix) [Koon 77,
Prat 87], and compared to data. These calculations were ﬁltered using the simulations

described in the previous chapter. A radius of 7 fm was chosen to correspond to total

129

130
fusion of the reacting nuclei (r = 1.2 Am, A = 205). The correlation functions predicted
for various lifetimes were compared to the data (Figure 6.1). The best ﬁt for this radius

was a lifetime of 700 i 200 fm/c. Since the angle-integrated correlation function does not

distinguish size from lifetime, a variety of lifetimes and radii can produce a good ﬁt
(Figure 6.2). The points represent the best ﬁt for that radius, while the error bars indicate
the range of reasonable ﬁts.

The directional cuts on the correlation functions were expected to distinguish the
space and time dimensions. For a lifetime of 700 fm/c, a separation between longitudinal
and transverse correlation functions is predicted by the simulations, but the data do not
show a clear separation for the directional cuts (Figure 6.3). The explanation of the
inconsistency between the data and the model may be that the model is too simplistic in

describing the source.

6.2 Evaporative Model

In order to investigate the emission time expected, a more physical evaporative
model was used. This model should provide a description of neutron emission from the
compound nucleus formed in the reaction 40Ar + 165H0 with E/A = 25 MeV. Speciﬁcally,
a code which implements the statistical formalism for particle emission from Reference
[Frie 83] was used. This model predicts the position, time, and energy for particles
emitted from an excited source, speciﬁed by the initial number of protons and neutrons,

the excitation energy, and the energy-level density.

131

 

 

 

 

   

 

 

 

2i..e,....,.........,....,....,....,.....
. \ 9 Data :
1.8 - \ " - T=4OO fm/c j
' \ T=600fm/c .
' '. \ — 1:700 fm/c _‘
1'6.” \ ““T=800fm/c .
~__ \ """" T=1OOOfm/c
1.4L \ ‘ 4
i .- ...:
o. \
E12?
1 -- - -
0.8- j
0.65 -‘
b11141!lllllllLiJllLJlllllllllllllllllll.
0 5 10 15 20 25 30 35 40

q (MeV/C)

Figure 6.1 Fit of the total, filtered correlation function from a surface model,

r = 7 fm, with different lifetimes.

132

 

1800 :-
1600

IIIIr'II

E
S

 

 

Lifetima(fméc)
§ 2
I . . . I . .
—.—
$

on
8
.rrﬁﬁr...

 

IIT'I

 

 

 

3 4 5 6 7 8 9
Radius (fm)

Figure 6.2 Best lifetime from the surface model for a given radius.

-
0nannlnnln[11111111allianlennnlnnnn

 

133

 

 

 

 

 

 

 

 

 

2 I I [I II I II I I T I I I I I
ie' 1
1.8 - ~
r """" ‘ Longitudinal .
' ' '. — ' 0 Transverse {} ‘
1.6 - -
104 III" \ -
'- - \ . .-
i I- \ u
o- . - .
E12 _- - ._ _ i? 4
i- I! . . - .. 'l
l- \ - e
1 ‘ -L:d>' " -D . I -1: \I \A
i- \ ' " - - - - a— '" /d
. . . .
0.8 L ' ' -
- 1
0.6 - '-
- I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I

 

0 5 10 15 20 25 30 35 40
q (MeV/c)

Figure 6.3 Correlation functions with longitudinal (‘P = 0 - 60°) and

transverse cuts (‘1’ = 80 - 90°). The data (points) are compared to filtered

calculations for surface emission from a sphere with a radius of 7 fur and a lifetime

of 700 fmlc (lines).

134

The parameters for the statistical model were chosen as follows: First, the mass
and charge of the compound nucleus were chosen to be the size of the total system, i.e.,
the effects of preequilibrium emission were assumed to have a small effect on the number
of nucleons available to the compound source. Second, initial excitation energy and level-
density parameter were determined. Since increasing the level density or decreasing the
excitation energy changes the shape of the energy spectrum and correlation function in a
similar way (Figure 6.4), the level-density parameter was ﬁxed at a = A/ 10 and the
excitation energy was adjusted.

The excitation energy is the amount of energy the compound system has initially.
It determines the slope of the energy spectrum. The excitation energy calculated assuming
total fusion was E*lA = 3.93 MeV. When the predicted energy spectrum was compared to
the energy spectrum in the laboratory frame (Figure 6.5), the predicted energy spectrum
was too shallow. (The model results were shifted into the laboratory frame by using the
center-of-mass velocity of 0.045 c.) To ﬁt the lower-energy part of the measured energy
spectrum, the excitation energy was reduced to E*lA = 2.0 MeV.

Several factors might lower the excitation energy but do not completely account
for the factor-of-two reduction. First, perhaps the energy-level density was not chosen
properly. A larger level density would produce a steeper energy spectrum, but for this
heavy system a = N10 is already quite large. Another explanation might be incomplete
fusion. For reactions at this energy, previous studies suggest that about 80% of the energy

should be transferred to the compound nucleus with the remaining energy carried away by

135

 

    
 

 

2 I I I I I I I I I l I I I I l I T I I I I I IITI I I I l I I I I I I I I I ﬂ

l- .

. — E/A=3.93 MeV a=A/10 .

1.8 - d

. - - E/A=2.0 MeV a=A/10 1

1'6 FF E/A=3.93 MeV a=A/8 -.

l" u

' i

1.4 - -

v— i- "l

+ " d

s r. 1
1 — N - — — ~ — ﬂ -

0.8 - _

l. .

r- u

0.6 - _

J I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I

 

 

 

0 5 10 15 20 25 30 35 40
q (MeV/C)

Figure 6.4 Evaporation-model calculations. The solid line is a calculation
with the excitation energy of E*IA = 3.93 and a = A110. For the dashed line the

excitation energy is reduced to E*IA = 2.0. The dotted curve has the level density

parameter increased to a = AI8 with the original excitation energy.

136

 

6
10 -IIIIIIIIIl—IIII'IIII'IIII'IIII'IIII'TIIIIIIII'IIII:
; Front woll ;
i 3 . Efficiency corrected 2

  
  

-- -- Evop E*/A=3.93 o=A/1O
—— Evop E*/A=2.0 o=A/1O ‘

105

I IIIIIII

I IIIIIII

l

j

13

'E

3

Z‘ \

8 4 o. \

.3 10 E" ... \ ‘7‘:

s E ...; 3

V L- ’0‘. .

U h ‘

713 ”9+ +

'- ' \ §

>‘ 3 \+#++
10 m.

I IIIIIII

I
I

 

 

IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIJJIIIIJJIIIJ—I

0 5 10 152025 30 35 404550
energy (MeV)

 

Figure 6.5 The measured energy spectrum compared to predicted energy
spectra for two different excitation energies in the evaporation model [Frie 83]. The

energy is plotted in the laboratory frame.

137
light preequilibrium particles [Viol 82, Nato 86]. Incomplete fusion could reduce the
excitation energy by 20%.

This model, like the surface model, can be used to predict the correlation function.
Here the results from the unﬁltered theory are shown. (The simulation and theory results
were already shown to be very similar in Section 5.2.4.) Figure 6.6 shows the comparison
of this model with the data. For the excitation energy that ﬁts the data, the calculated
correlation function is much smaller than the measured one indicating that the model
predicts too low an emission rate; in other words too large a lifetime of the source. The
discrepancy between predicted and observed correlation functions cannot be attributed to
the uncertainties in the initial excitation energy of the emitting source. Even for the
extreme assumption of complete fusion with no preequilibrium emission the predicted
lifetime is too long.

The correlation functions calculated from the evaporation model can be compared
to the correlation functions produced by the surface model described in the last section.
The lifetime parameters for the surface model that would make a correlation function of
similar strength quantify the lifetime predicted by the evaporation model. For the lower
excitation energy, which ﬁts the energy spectrum, the lifetime determined by this
comparison is 2000 fmlc for a radius of 7 fm. The correlation function at the higher
excitation energy is comparable to a lifetime of 1200 fmlc. The strength of the correlation
functions are compared between these two models because the dissimilarity of the shapes

of the time spectra makes direct comparison difﬁcult (Figure 6.7).

138

 

2 . 1 rI I I I I I l I I I I ' I I I I l I III I l I Iﬁ I I I I I I [j I I I
p q
+ J
. C Date .

1.8 -
- - E/A=3.9 MeV o=A/1O

 

1-6 _' — E/A=2.0 MeV o=A/1O

 

 

 

 

IIIIIJIIIIIIIIJIIIIIILIJLJIILIIIIIII

0'” 5 10 15 20 25 30 35 4o
Q(MeV/c)

 

Figure 6.6 The measured correlation function compared to the evaporation-

model predictions with two different excitation energies.

139

 

IIIIrIIIIIIIII'IIIrITTIIIIIII'IIIITIIIIIIIIIIIIII

b d

10 4 — Evaporation model

7' = 2000 fm/c

IIIIII

yield (arbitrary units)

in
G
U
rrrl

 

 

 

....1....1....1....1....I....1....1.....'r.-..'..I....
0 10002000300040005000600070008000900010000

time (fm/c)

 

Figure 6.7 Time spectra from the evaporation model compared to the time

spectrum for an exponential distribution with t = 2000 fmlc.

140
The evaporative model is predicting longer lifetimes than are observed
experimentally. One possible explanation is that the neutrons are not all coming from a

compound source. Even though the neutrons were detected at 90°, it is possible that some

of the neutrons are coming from preequilibrium emission, thus the measured time is a
combination of a fast neutron component and the evaporation lifetime. This possibility is

investigated in the next two sections.

6.3 Evaporation Model Plus Preequilibrium

In order to explore the effects of preequilibrium, a fast component was added to
the evaporation model. The resulting hybrid model is made up of two independent
sources of neutrons, each with its own temperature (energy-spectrum slope parameter)
and lifetime. The experimental energy spectrum is ﬁtted by two exponential distributions
with temperatures Tp and Tc. The distribution with the lower temperature, Tc, corresponds
to a compound source, while the distribution with the higher temperature, Tp, corresponds
to preequilibrium emission. The ratio of neutrons of each type is energy dependent. In
order to quantify the fraction of preequilibrium/total neutrons, fp, the energy spectra are

compared for energies 5 - 20 MeV.

20MeV
N

f = SMCV
P 20MeV (

-E/T
e pdE

e‘E’Tc + Ne'E’TP )dE
SMeV

The program to make correlation functions was modiﬁed in order to sample

neutrons from these two distributions with the appropriate weights. The excitation energy

141

for the evaporation model was chosen to reproduce Tc. Its emission time and position are
determined by the evaporation model. The energy for the preequilibrium particles is
sampled from an exponential distribution with temperature, Tp. The emission points are
sampled from the surface of a sphere with r = 7 ﬁn, and the emission time was chosen, for
simplicity, as IP = 0.

Allowing for some systematic uncertainties, the decomposition of the energy
spectrum into two components is not unique. By adjusting Tc and Tp, a range of

weighting factors can be chosen (Figure 6.8). A ﬁt with fp = 9% has a similar x2 value to

a ﬁt with fp = 45%. Correlation functions were constructed for various weights with the
appropriate temperatures (Figure 6.9). Assuming that the evaporation model is accurately
determining the emission time of the compound source, the resulting correlation functions
imply that about 25% of the neutrons may come from preequilibrium emission.

Using IP = 0 provides a lower limit for the amount of preequilibrium necessary to

reproduce the data. Since the preequilibrium time may be longer a larger fraction of
preequilibrium would be necessary to reproduce the experimental correlation function. In
the extreme case of all the neutrons coming from preequilibrium, this model would be

identical to the surface model in Section 6.1 and a lifetime of IP = 700 fmlc would

reproduce the correlation function.

In order to determine a more realistic value for the fraction of preequilibrium
neutrons, calculations based on the Boltzman-Uehling-Uhlenbeck (BUU) equations [Baue
87, Li 91a, Baue 92] were used to predict the temperature for preequilibrium emission.

The model was run for 200 fm/c, but the slope of the energy spectrum is fairly constant

142

 

10 6 .
: i
, e Corrected Data .
‘ — Fit 1 ‘
. \ .
‘:. \

A 5 , ..... Fit 2

.910 .— . 1

'E Z I

3 . ..

b I- .

g - .

:45" r-

L.

3 " .

E 4

0

31° :' ':

 

 

0 5 10 152025 30 35 40 45 50
energy (MeV)

Figure 6.8 Two-component fits of the energy spectra. The lines with the
steepest slopes correspond to the evaporation component; the lines with a more
shallow slope correspond to the preequilibrium component; the sum for both fits lies
close to the data points. Fit 1: Tc = 4.9 MeV, Tl, = 17.0 MeV, and fl, = 9%. Fit 2: Tc =

3.0 MeV, T9 = 8.0 MeV, and r, = 45%.

143

 

 

 

 

 

 

0....5 . .10....15....20. ..25....30....35...40
q(MeV/C)

Figure 6.9 Correlation functions for various two-component fits with the
evaporation-plus-preequilibrium model. Fit 1: fl, = 34%, Fit 2: f9 = 25%, Fit 3: fl, =

22%, Fit 4: r, = 8%.

144

over this time, so the choice of cutoff is not important. The BUU results, with a
geometrical average over impact parameters b = 1 - 5 fm, have an energy spectrum with a
slope parameter corresponding to a temperature of about 12 MeV (Figure 6.10). When the
temperature, Tp, is determined, Tc and fp are determined by the slope of the energy
spectrum. Using the preequilibrium temperature from the BUU calculations, the fraction
of preequilibrium neutrons fp = 18%. With 1,, = 0, the correlation function was
constructed for this combination of sources (Figure 6.11). This correlation function is not
as strong as that in the data, so a more realistic fraction of preequilibrium cannot bring the
evaporation model into agreement with the data.

The preequilibrium neutron emission is not really instantaneous. To use a more

realistic value for tp, the time distribution of neutrons from the BUU calculations was

used as a reference. The neutron yield over time can be approximated by an exponential
lifetime with IP = 50 fmlc (Figure 6.12). If the emission time of the preequilibrium
component is chosen from this exponential distribution, the resulting correlation is
somewhat reduced from the instantaneous value (Figure 6.11).

With either instantaneous emission or a lifetime ID = 50 fmlc for the
preequilibrium component, the measured correlation function is still larger than that from
the evaporation model with 18% of the neutrons from the preequilibrium component.

Even with this hybrid approach the time scale in the evaporation model is too short.

145

 

p—i
G
a
l
O
I

e e Data 1

’. — BUU energy

a
u:
l

O
a
I

yield (arbitrary units)

 

 

y—L
c
W
l
I

 

 

JIIIJ IIIIIIII IIIILIIIIIIIILJIIIIIIIIIII IIIIIIII

0 5 1'0 15 20 25 30 35 40 45 50
laboratory energy (MeV)

 

Figure 6.10 Energy spectra predicted by BUU calculations compared to the

data.

146

 

III'IIII'IIII'III

 

0 Data
1.8 - -
t — Evap+T,= Ofm/c :
r I
1'6: — - Evap + T, = 50 fm/c
_ t

 

 

 

 

1."
0.8}
0.6; -
.111llllllllJlllllllllllllelllllllIllll
0 5 10 15 20 25 30 35 4o
C1(MeV/C)

Figure 6.11 Comparing the correlation function from the data with a two-
component model with preequilibrium fraction fl, = 18%. The low-energy
component is from the evaporation model. The high-energy component has T, = 12

MeV and 1,, = 0 or 50 fmlc (solid or dashed lines).

147

 

q
.l....l

IITIII

 

yield (arbitrary wits)
l

750 f I
; . BUU yield

500 _ — 7' = 50 fm/c -

250 - -

 

 

#1 llllllllllllllllllllllllllllllll
20 40 6 80 100 120 140 160 180
time(fm/c)

.1.“
200

Figure 6.12 Yield of particles as a function of time from BUU calculations.

The line is the yield from an exponential distribution with a lifetime of 50 fmlc.

148

6.4 Using Surface Emission with Two Components

In the previous section, the compound-source component of the model was ﬁxed
by a statistical evaporation model. Because the evaporation model predicts a lifetime
which is too long to match the data, it is instructive to treat this lifetime tc as a free
parameter. Again a two-component model is used, with temperatures, Tc and Tp, and
weighting, fp, selected to match the energy spectra. The two components have emission
points from the surface of a sphere of radius 7 fm. The emission times for these

components are sampled from two exponential distributions with lifetimes 1,, and re.

By varying 1,; as a function of fp, the best ﬁts to the correlation function can be
found (Figure 6.13). These ﬁts were done with IP = O, instantaneous emission of the
preequilibrium particles. The point plotted is the best ﬁt as determined by a x2 test for the
low-q region of the correlation functions, and the error bars include the range of
reasonable ﬁts. In this scenario it is possible to ﬁt the correlation function for
preequilibrium components up to about 30%. For larger contributions, the 1,, = 0
preequilibrium component alone makes the correlation stronger than the data. At 18%,
the fraction of preequilibrium that the BUU calculations predict, the best ﬁt is with a
lifetime of 17,; = 1000 fmlc.

For the fraction of preequilibrium predicted by the BUU calculations, the model
was also run with the preequilibrium time sampled from an exponential distribution with
a lifetime, 1,, = 50 fmlc (Figure 6.14). The longer lifetime for preequilibrium reduces the

compound nucleus lifetime by about 100 fmlc.

149

 

IfTIﬁ'IITIIIjjfﬁIIIIIIIIIIIIIIIIII

H
G
A
l
l 1111i

he
6
M
_.__‘
_._
IIIIII

Compound source lifetime (fm/c)

IIIIIIIlLIIIiIIIIHIIIIILIJIIJLJLJI

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Fraction preequilibrium

 

 

 

Figure 6.13 The best fit to the data for the compound-source lifetime, 1., for
various values of f9, assuming 1,, = 0. The error bars represent the range of

reasonable fits.

150

 

 

 

2 LI—r I I I r I I I I I I I I l I I l I I I I I I l I I I I I I I I I I I I 1
. 0 Data .
1.8 - ..
_ — 1',= 1000 fm/c+1',= D fm/c .
1.6 P‘ —l
. ‘\ - - r,= 900 fm/c+1',= 50 fm/c .
L I
F \ .
104 - \ .-
+ ’ .
far ' .1
E 1.2 - ..
1 - + ‘ _
- + .
0.8 - _
b q
0.6 I a
I L I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I Ii I

 

 

 

0 5 10 15 20 25 30 35 40
q (MeV/c)
Figure 6.14 Two-component model with the time for both components
sampled from exponential time distributions. The preequilibrium component has
either 1:, = 0 and 50 fmlc while the compound component has 1:c = 1000 fmlc and “cc =

900 fmlc, respectively. The preequilibrium/total fraction is f, = 18%.

151

6.5 Preequilibrium and Time-Adjusted Evaporation Model

Another way to determine by how much the evaporation lifetime needs to be
reduced is to adjust the emission time by scaling the time for each event, te. The new
time, t = s te, is used as the compound-source component in the hybrid model described in
Section 6.3. This technique retains a scaled correlation between time and energy for
particle emission. Since the evaporation model predicts a different shape for the time
distribution than a simple exponential distribution (Figure 6.7), this correlation must be
different in the two models.

Again the two-component model is used with the low-energy component sampled
from the evaporation model and IP = O and fp = 18%. Using a scaling factor 5 = 0.4 brings
the results into the best agreement with the data (Figure 6.15). With a longer

preequilibrium time, 1p, the scaling factor would have to be even smaller. The

evaporation time seems to be a factor-of-two too long.

6.6 Summary and Outlook

Several observations result from this experiment measuring the two-neutron
correlation function for 40Ar + 165H0 at EIA = 25 MeV. The data show a strong peak at q
= O in the correlation function, but no clear separation of longitudinal and transverse
correlation functions.

The measured total correlation function can be described by a source of radius 7
ﬁn and an exponential lifetime of 700 fmlc. To compare this lifetime to the expectations

for a compound nucleus, a statistical evaporative model was used. This model predicts a

 

152

 

 
 

 

2 IIIIIIIIlIIIIIiIﬁIlIIIIlII‘TT—II I IIII IIIJ

4

O DCltO i

1.8 L, -

s=0.3 .

1'6 — s=O.4 -.

o -4

\ . . .

\ .0. -- 3:007 '-

v— \ .' .

+ \ a

G \\ .
\

E" 1.2 .

0.8

P
&

ITI I I I I I I l I I III I1. I I I‘rI I I Ii
/

lllllIJll‘

0.6

 

 

IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII

5 10 15 20 25 30 35
q(MeV/c)

 

6
h
6

Figure 6.15 Two-component model, with the low-energy component taken
from the evaporation model with each emission time scaled by a factor s. The

preequilibrium component has fp = 18% and t, = 0.

153
correlation function that would be consistent with a surface-model parameterization of
2000 fmlc with a radius of 7 fm.
How preequilibrium emission changes the apparent lifetime of the correlation
function was investigated by various hybrid models. Using the experimental energy
spectrum as an additional constraint, it is found that the addition of 25% preequilibrium

with 1,, = 0 can enhance the correlation function from the evaporation model so that it ﬁts

the data. Longer lifetimes for the preequilibrium component require a larger fraction.
Using the Boltzman-Uehling—Uhlenbeck model to predict the temperature of the
preequilibrium component implies that 18% of the neutrons come from preequilibrium.
With this ratio for instantaneous preequilibrium neutrons, the correlation function can be
reproduced by a surface emission model with a lifetime of 1000 fmlc. If the emission
time in the evaporation model is scaled by a factor of 0.4, it can also reproduce the
experimental correlation function with 18% of the neutrons from instantaneous
preequilibrium emission. All of these models suggest that the evaporation model predicts
a lifetime that is too long by about a factor-of-two compared to the data

The lack of separation for longitudinal and transverse correlation functions is
puzzling, as the lifetime from the total correlation function seems long enough that all the
models predict a separation of longitudinal and transverse directional cuts. This lack of
separation could be the result of a mixture of sources due to poor impact-parameter
selection.

Future two-neutron intensity interferometery measurements could be improved in

several ways. First, an experiment should be run at a lower beam energy where fewer or

154
no preequilibrium neutrons would be produced. Also it would be beneﬁcial to restrict the
class of events further by measuring a heavy residue in coincidence with the two
neutrons. These two improvements should produce a cleaner source determination,
perhaps resulting in a measurable difference between the angle-selected correlation
functions. A technical improvement would be to perform the measurement in a larger
room where there would be fewer background neutrons. Better statistics would also allow
a more thorough analysis. To obtain better statistics, more solid-angle coverage would be
necessary for either the trigger detector or the neutron detectors or both. As this
experiment already ran for almost two weeks, just running longer would be a less
attractive option. With these improvements two-neutron intensity interferometry could

make a stronger statement about the lifetime of the compound source.

Chapter 7 - Conclusion

In this thesis, intensity interferometry was used to measure the size and lifetime of
the emission zone in two intermediate-energy heavy-ion reactions. One reaction used
two-proton intensity interferometry to investigate 16O + 197Au collisions at EIA = 200
MeV. The other reaction employed two-neutron intensity interferometry to measure 40Ar
+ 165H0 collisions at FJA = 25 MeV.

Two-proton correlation functions were measured for 1‘50 + '97Au collisions at E/A
= 200 MeV. When compared with BUU calculations [Baue 87, 92, Li 91a, 91b] the
momentum-integrated correlation function agrees well with the data, but the dependence
of the correlation function on the momentum of the particles is overpredicted by the
theory. Several correlation functions measured in this energy region [Kund 93, Hand 95b]
show a source that is larger or longer lived than the BUU model predicts. The BUU
model takes into account the dynamics of the reaction but is missing some physics that
would account for the longer emission times seen in the data. Protons from long-lived
resonances are not included in this model and may account for at least part of the
difference between theory and experiment [Hand 95b].

Two-neutron intensity interferometry was used to measure the reaction 40Ar +
165Ho at EIA = 25 MeV. Near-central collisions were selected and a correlation function
constructed from coincident neutrons. The coincident neutrons were corrected for

crosstalk and background scattering, and the crosstalk corrections were tested using two

independent simulation codes. The neutrons used to construct the correlation function

155

 

156

were required to have energies greater than 5 MeV, in order to reduce background
scattering. The simulations for I Fl < 250 MeV/c showed that the correlation function is

not distorted by this crosstalk elimination method. However, the higher total-momentum
pairs have a signiﬁcant amount of crosstalk remaining.

For this beam energy, EIA = 25 MeV, incomplete fusion is expected with the
formation of a compound nucleus. The results are compared to an evaporation model. For
this reaction, the extracted lifetime is shorter than the model predicts. Emission from the
surface of a 7 fm sphere with an exponential time distribution ﬁts the data for a lifetime

of 700 :l: 200 fmlc. The correlation function from the evaporation model [Frie 83] is

compared to the same surface model. The correlation function for the evaporation model,
after accounting for preequilibrium emission as predicted by BUU, is comparable to the

correlation function from this model with an exponential lifetime of 2000 :1: 500 fm/c.

The neutron data have larger uncertainties than the proton data and a measurement with
more statistics and a stricter event selection would improve the experimental results. The
data from this experiment are consistent with a lifetime about a factor-of—two shorter than
the evaporation model predicts. This evaporation model is quite basic, so a more
elaborate model might explain some of this difference. This evaporation model was
chosen because it gives a reasonable approximation of evaporation without a large
number of free parameters.

Checking the time scale in evaporative models has relevance to the interpretation
of lifetimes from ﬁssion reactions. Many experiments have measured ﬁssion lifetimes as

a way to understand dissipation in the nuclear medium. At least three different methods

157

have been used to measure the ﬁssion lifetime. One method is to measure the pre- and
post-scission neutron multiplicities. The emission of each neutron requires a ﬁnite time
and so the number of pre-ﬁssion neutrons is assumed to be related to the lifetime of the
compound nucleus before ﬁssion [Hind 84]. For excitation energies from 45-90 MeV,
ﬁssion lifetimes from 2 - 5x10"20 8 = 6000-15,000 fm/c have been extracted using various
models [Hind 85, Siwe 95, Andr 95]. The second method of measuring ﬁssion lifetimes
is to measure the gamma multiplicity from giant dipole resonances [Paul 94].
Experiments using this technique for a source with an excitation energy of 64 MeV,
predict time scales of 2.9x10'19 5 (87,000 fmlc) [Buts 91]. The third method uses crystal
blocking and predicts a lifetime greater than 10'19 s = 30,000 fmlc for excitation energies
less than 250 MeV where the fusion time becomes longer with decreasing excitation
energy [Morj 97].

Because the ﬁrst two methods rely on statistical models to extract a lifetime from
the measured quantities, using two-neutron intensity interferometery in this energy regime
to conﬁrm the time scale of these models is especially useful. The results presented in this
thesis indicate that the evaporation model [Frie 83] is accurate to about a factor-of-two.
Discrepancies in the ﬁssion lifetimes differ by factors of ten, which are not accounted for
by uncertainties in the evaporation model.

The original question of how the heavy residue distorts the two-proton correlation
function still remains. It would be interesting to compare two-proton and two-neutron
intensity interferometry methods in this evaporative region. Unfortunately, no two-proton

measurements are available for the 40Ar + 165H0 system.

APPENDIX

Appendix - The Koonin-Pratt Formalism

Most theories modeling nuclear reactions predict the one-particle phase-space
distribution. In order to transform these into a correlation function, the Koonin-Pratt
formalism is applied. This formalism has been derived elsewhere [Koon 77, Prat 84, Prat
87, Gong 91a]. Here the basic form and assumptions relevant to two-proton and two-
neutron correlation functions are reviewed [Lisa 93b].

The basic formula relates the single-particle source function g(?, 'p, t) to the
correlation function, 1 + R(P, ii). The source function represents the probability that a
particle of momentum 'p will be emitted at position 'r’, and time t. Then the correlation
function can be written as a function of P , the total momentum of the pair and q , the
invariant relative momentum:

1+ 1203.61) = ——pr" p3)
I’(131)P(132)

ld3f1dt1d3fzdtzgfﬁ. 1% . t1)g(?2. % . t2)l¢(fi. f1 - f2 - ((t1 42%!!!) - 13)]
l d3fldtlg61’ 1% , tr) ‘ [d3fzdt2gﬁ2.% .12)

 

158

159

In this equation P(pl , p2) and P(p) are the probabilities of measuring two particles and
a single particle with the given momenta The wave function ¢(q', f) depends only on the

relative phase-space coordinates at the time of emission of the second particle“

Several assumptions enter into deriving this form for the theoretical correlation
function. First, the particles are assumed to be emitted independently. All correlations are
assumed to result from the ﬁnal-state interaction and not from the actual emission
process. This assumption is not completely valid since at least conservation laws link the
emission of particles [Lync 82]. Other entrance-channel effects [Zhu 91] or correlations
in the source [Bern 85, Alrn 93, Kund 93] may also destroy the independence hypothesis.

The second assumption for this derivation is that the two particles interact only
with each other not with the residue or other emitted particles. For neutrons this is
especially appropriate, since no Coulomb interactions apply.

These assumptions allow the source-emission phase-space to be factored into the

product of the single-particle source functions g(f, 13, t). The relative wave function,
¢(q, 'r’) , creates the correlation which destroys the factorability of the ﬁnal probabilities:
P651, P2) *5 P(P1)P(P2)-
The third assumption is that the source function does not vary greatly over a

momentum region of orderl ql. This is appropriate for these heavy-ion collisions where

the particle momenta are typically hundreds of MeV/c while the relative momenta of

interest are less than 50 MeV/c.

160

The two-particle relative wave function, m, 'r') , determines the theoretical

correlation function. For applications of this theory in this thesis, the wave function is

determined numerically by solving the Schrodinger equation for the t = 0 and 1 = 1 partial

waves with the Coulomb and Reid soft-core potential [Reid 68]. In the case of neutrons
the charge for the Coulomb potential is set to zero. The full wave function is then

determined by
- - 2 1 - - 2 3 - s 2
|¢(q.r)| = -|¢s(q.r)| +—|¢.(q,r)|
4 4
where ¢S(q, f) and ¢,(q, f) are the singlet and triplet two-neutron spatial wave

functions, respectively. The weighting of this equation is determined by assuming a

statistical distribution of spins.

 

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