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3 LIBRARY
7 0'7; MiChigan State
University

 

 

 

This is to certify that the
dissertation entitled

PROBING THE FREEZEOUT MECHANISMS AND ISOSPIN
EFFECTS IN MULTIFRAGMENTATION

presented by

WANPENG TAN

has been accepted towards fulfillment
of the requirements for the

Ph.D. degree in Physics

 

 

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Date

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6/01 c:/CIRC/DateDue.p65-p.15

PROBING THE FREEZEOUT MECHANISMS
AND ISOSPIN EFFECTS IN
MULTIFRAGMENTATION

By

Wanpeng Tan

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

2002

ABSTRACT

PROBING THE FREEZEOUT MECHANISMS AND ISOSPIN
EFFECTS IN MULTIFRAGMENTATION

Bv

‘/

“’an pen g Tan

IV'Iultifragmentation processes have been studied by measuring 129Xe+197Au col-
lisions at 50A MeV with a large area silicon—strip/Csl detector array (LASSA). The
LASSA was designed and constructed to provide excellent energy, angular and iso-
tope resolution for the detection of charged particles. Each of nine LASSA telescopes
consists of two silicon strip detectors (65/1111 and 500mm thick, respectively) and four
CsI(Tl) crystals. A non-uniformity in light output of the CsI(Tl) crystals better than
100 was obtained via the crystal selection and quality control procedure. Isotopi-
cally resolved particles from hydrogen to oxygen were detected in polar angles of

12" < 6 < 62° with an angular resolution of $0.43".

Experimental correlation functions for particles detected in LASSA are analyzed
using the Koonin-Pratt formalism and by making an assumption of thermal equilib-
rium. A method of probing the breakup density is demonstrated and the results from
the equilibrium approach are compared with those from the Koonin-Pratt formalism
for light particle correlatitms. The extracted values of the free volume are about 2.5
(10% uncertainty) times the volume of the total system at saturation density, which

corresponds to a freezeout density of 1/5-1/3 times the nuclear saturation density.

A new technique of spin determination from particle correlation functions is pro-

posed utilizing the same equilibrium assumption. A few examples of correlation

functions are studied to demonstrate the promising sensitivity of this new teclmique.
In particular, the spin of the first excited state of 8I3 at 0.774 MeV is determined as

J21 via the p—7Be correlation function.

To better understand the isospin effects and to further confront calculations
with experimental data. an empirically modified statistical multifragmentation model
(SMM) was developed. This model treats the properties and decay of nuclear excited
states Self-consistently and calculates the secondary decay using the empirical infor-
mation about the excited states. The importance of using empirical binding energies
and experimental level densities is shown in detailed comparisons. Nuclear ther-
mornetry is studied and con‘ipared with data to demonstrate that the corrections for

secondary decay can be modelled accurately within this approach.

From the isotopic composition of particles emitted during an energetic nucleus-
nucleus collision, we can test the isospin dependence of the nuclear equation of state
(EOS). Hybrid calculations were performed using isospin dependent BUU transport
model and the improved SMM. Comparisons between measured isotopic yield ratios
and theoretical predictions in the equilibrium limit are used to assess the sensitivity
to the density dependence of the asymmetry term of the EOS. This analysis suggests
preference for a stiff density dependence and indicates that such comparisons may

provide an opportunity to constrain this important property of nuclear matter.

To
my wife - LII/wt Zhai
and

my parents - G'uoyang Tan 8; Fengqm Wang.

iv

ACKNOWLEDGEMENTS

First of all, I would like to express my deepest gratitude to my thesis adviser, Dr.
William G. Lynch. His invaluable support and guidance are indispensable for the
completion of my thesis. In the past years, I have learnet‘l nmch from his deep under-
standing of nuclear physics on both experimental and theoretical aspects, his attitude
and patience of doing physics research without omitting any details, and his strong
leadership and enthusiastic stamina of completing research projects successfully. I
would also like to thank Dr. S. Billinge, Dr. P. Danielewicz, Dr. C. Schmidt, and Dr.

B. Sherrill for reading my dissertation and also serving on my guidance committee.

I am deeply indebted to the LASSA collaboration. W'ithout the productive collab-
oration of the groups at I\-Iichigan State University (MSU), Indiana University (IU)
and W’ashington University (WU), the LASSA project would not have been completed
in such a great. success. I would like to thank the MSU group of CK. Gelbke, T.X.
Liu, X.D. Liu, WC. Lynch, I\I.B. Tsang, A. Vander Molen, G. Verde, A. Wagner,
H.F. Xi and HS. Xu, the IU group of L. Beaulieu, B. Davin, Y. Larochelle, T. Lefort,
RT. de Souza, R. Yanez and V.E. Viola, and the WU group of R.J. Charity and LC.

Sobotka.

I am also very grateful to my collaljmrators on model calculations. I own my
sincere thanks to Dr. Sergio Souza and Dr. Raul Donangelo from Brazil for their
great help on SMM calculations. Dr. Bao—An Li from Arkansas State University
offered valuable discussions and BUU calculations for the collaboration on studying
the isospin dependence of the equation of state. I would also greatly appreciate the
help of Dr. Bob Charity from Washington University for incorporating his GEMINI
model into our statistical calculations. In the later stage of my thesis, Dr. G. Verde

kindly offered a great amount of help on the imaging analysis and model simulations

V

of collective effects and I owe great thanks to him.

Special thanks go to Dr. Betty Tsang who has been very helpful throughout my
thesis experiment and later data analysis. I have also benefited nmch from various
discussions with Dr. Pawel Dai'iielewicz, Dr. Konrad Gelbke and Dr. Scott Pratt.
I want to thank Dr. Aaron Calonsky, Dr. Pawel Danielewicz and Dr. Betty Tsang
for their kindness of providing me letters of references. I will cherish the friendship
and various helps during my stay at NSCL from l\v‘Iichael Famiano, Marc-Jan van
Goethem, Paul Hosmer, Patrick Lofy, Michal Mocho. Lijun Shi, Richard Shomin,

Mark Wallace, besides people 111eutioned above.

I would like to acknowledge the excellent support of the staff at the National
Superconducting Cyclotron Laboratory of Michigan State University. I appreciate
the help from John Yurkon of the detector lab, Len Morris of the design group, Jim
Vincent of electronics group, Ron Fox of the computer group and Mathias Steiner of

the operations group.

Last but not the least, I would thank the love and patience of my wife and my

parents, and the understanding and support of my sisters and brothers.

Contents

LISI OF TABLES
LIST OF FIGURES

1 Introduction

1.1 Freeze-out Conditions of l\Iultifragmentation ..............
1.2 Fragmentation and Isospin Effects ....................
1.3 Organization of the Thesis ........................

2 Experimental Setup

2.1 Mechanical Setup .............................
2.2 Electronic Scheme .............................
2.3 Miniball/Miniwall .............................

3 LASSA Telescopes

3.1 Silicon Detectors .............................
3.1.1 Speciﬁcations ...........................
3.1.2 Energy Calibration ........................
3.2 CsI(Tl) Detectors .............................
3.2.1 Pre—selecting and Scanning CsI(Tl) Crystals ..........
3.2.2 Testing Wrapping Materials ...................
3.2.3 Other Effects and Assembly ...................
3.2.4 Energy Calibration ........................

4 Data Reduction and Analysis

4.1 Overview of the Analysis .........................
4.2 Pixelation Technique ...........................
4.3 CsI Crosstalk ...............................
4.4 Uniformity Correction ..........................

vii

xii

10

12
12
14

18
20
20
22
23
24
28
31
34

39
39
41
44

4.5 Isotope Resolutions and PID ....................... 52

5 Particle Correlations 55
5.1 Selecting the Impact Parameters ..................... 56
5.2 Two Particle Correlations and the Koonin-Pratt Formalism ...... 59
5.3 Inﬂuence of Collective Motion ...................... 69
5.4 Equilibrium Correlation Functions .................... 84
5.5 Interpretations of Correlation Functions Using the Equilibrium Corre-

lation Approximation ........................... 88
5.6 Spin Determination of Particle Unstable States ............ 103
5.7 Multiple Particle Correlations ...................... 114

6 Statistical Multifragmentation Model with Empirical ModiﬁcationleO

6.1 Microcanonical Statistical Multifragmentation Model (SMM) ..... 121
6.1.1 Underlying Formalism ...................... 121
6.1.2 Temperature Distributions .................... 125
6.1.3 Effects of Temperature Variations ................ 130
6.1.4 Chemical Potentials ........................ 135

6.2 Empirically Improved Model ....................... 140
6.2.1 Ground State Energies ...................... 140
6.2.2 Internal free energy ........................ 144
6.2.3 Empirical Sequential Decay ................... 155

6.3 Model Predictions and Comparisons
158
6.3.1 Caloric Curve ........................... 158
6.3.2 Elemental and Mass Distributions ................ 162
6.3.3 Isotope Thermometry ....................... 167

7 Isospin Dependence of the EOS 177

7.1 Density Dependence of Asymmetric EOS ................ 178

7.2 Hybrid Model Calculations ........................ 181

7.3 Isotopic Composition and Isospin Dependence ............. 184
7.3.1 Relative Fme n/ p Densities and Mirror Nuclei Ratios ..... 186
7.3.2 Mirror Nuclei Ratios ....................... 190

7.4 Remarks .................................. 190

8 Summary 193

viii

LIST OF REFERENCES 200

ix

List of Tables

2.1

3.1

5.1

U"
N)

5.3

5.4

5.5

List of the original number of detectors in a ring of I\Iinil)all/.\Iiniwall,
the solid angle of a detector in that ring. and the numbers of mounted
and removed detectors in that ring, respectively. ...........

List of fragmentation 1’)ro(.lucts used in the energy calibration of the

LASSA CsI(T 1) crystals. .........................

List of quantities are shown for the p—p correlatimis (Figs. 5.3-5.6) for
inclusive and three different c.m. energy gates. The A that relevant
to the fraction of protons contributed to the ﬁtted source and the
rms radius rms of the two proton source are extracted from the ﬁtted
source distribution. The X values are calculated from the integral of
the source functions for r l 7 fm in Eq. 5.5. The density p/po is
estimated by taking into account collective effects in Eq. 5.9 ......

List of quantities are shown for the (LG correlations (Figs. 5.12-5.13)
for inclusive data and three different c.m. energy gates. The A that rel-
evant to the fraction of the involved particles contributed to the ﬁtted
source and the rms radius rm, of the two proton source are extracted
from the ﬁtted source distribution. The density p/po is estimated by
taking into account collective effects in Eq. 5.9. ............

Relevant spectroscopic information of 6Li, 7Li. 7Be and 8Be which is
adopted from ref. [98] is listed for the correlation functions discussed
in this section ................................

List of quantities are shown for the d-(i, t-(t', 3He-oz and 0-0 corre—
lations (Figs. 5.16-5.19). The values of Vf/VU are obtained from the
corresponding experimental correlation functions. Price is the correction
factor in Eq. 5.26 calculated from secondary decay contributions in the
SMM. The density p/po is estimated in Eq. 5.28 using the secondary
decay correction factor Fdec from the SMM while p/p0(KP) is obtained
using Fdec(K P) (see text). The corresponding pf/po and pf/ p0(K P)
are calculated in Eq. 5.29 without considering excluded volumes. . . .

Spectroscopic information of IOB" —+ a+6Li adopted from ref. [98].
The state at the 6.56 MeV is assigned tentatively with J=4 and only
alpha decay is conﬁrmed. The last three states contribute very little
to the ﬁtting. For details see text .....................

17

35

66

78

93

102

112

6.1

6.2

7.1

Best ﬁt parameters to the LD;\»I formula. of ref. [105]—[106]. All the
values are given in MeV, except for k which is unitless. ........

List of isotopic thermometers with AB > 10 MeV. The left column
shows the IMF thermometers involving isotopes of 3 3 Z S 8. The
right column lists the He t.hermometers involving the isotope pair of

SJHQ

The ﬁrst two columns provide the N / Z ratio and number of nucleons
in the prefragments produced in the calculations for an elapsed time
of 100 fm/c and density cutoff of po/ 8. The next two columns provide
corresponding information for the same cutoff density but a shorter
elapsed time of 80 fin/c. All calculations were performed at an impact
parameter of 1 fm. ............................

xi

 

170

List of Figures

2.1

2.2
2.3

2.4

3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10

4.1
4.2

4.3

4.4
4.5
4.6

Schematic drawing of l\-‘Ii11ibtrll/Miniwall + LASSA setup in the vertical
plane. ...................................

Miniball/h'finiwall + LASSA setup housed in the 92” chan‘iber. . . . .

Miniball/L‘Iiniwall + LASSA detector setup viewed inside from the
beam direction. ..............................

Schematic diagram of electronics setup for LASSA. ..........

One of the nine identical LASSA telescopes. ..............
Picture of one 500 nm thick double-sided silicon strip detector. . . . .
Pulser calibration for three dynamic ranges. ..............
Energy calibration for a typical silicon strip. ..............
CSI crystal scanning setup with alpha source. .............
Uniformity results of accepted and rejected CsI crystals. .......
Comparisons of different wrapping materials used for CsI crystals. . .
CsI preampliﬁers and their housing ....................

CSI packing in one telescope ........................

Calibration curves for 11C, 12C, 13C and 14C for the CsI (Tl) crystals
obtained using direct fragmentation beams listed in Table 3.1. The
curves are the best ﬁt according to Eq. 3.1. ..............

Flow chart of data analysis for the LASSA ................

EB-EF ﬁne-tuning calibration for pixelation. On the left panel dis-
crepancy between EB and EF calibrated from the precision pulser is
shown. After the EB-EF ﬂattening routine is applied, the good agree-
ment between EB and EF is shown on the right panel. ........

layout of CsI crystals within one telescope and typical crosstalk shown
for light leakage from CsI X to C51 Y in Telescope 2. Similar results
for CsI Y ——i CsI X are not shown. Units are in raw channel numbers.

Non-uniformity of one typical 65 pm thick silicon detector. ......
Non-uniformity of one typical 500 pm thick silicon detector. .....

PID lines of Li and C isotopes before and after uniformity correction.

xii

37

40

45
47
48
49

4.7 Isotopically resolved PID lines from H to O are shmvn for 1_)articles
stopped in Silicon Detectors ....................... 50

4.8 Isotopically resolved PID lines from H to O are shown for particles
stopped in CsI crystals .......................... 51

4.9 One-dimensional PID lines from H to O are shown for particles stopped
in Si detectors ............................... 52

4.10 One—dimensional PID lines from H to O are shown for particles stopped
in C31 crystals ............................... 53

4.11 Mass resolution is illustrated from one-dimensional PID plots of Car—
bon isotopes. Left panel is for Carbons stopped in the second layer of
Silicon detectors. Right panel is for Carbons stopped in the CsI crystals. 54

CI!
)—‘

reduced impact parameter as function of total charged particle multi—
plicity. ................................... 57

5.2 Differential multiplicites of hydrogen and helium isotopes at angles of
17°, 27°, 37°, 470 and 57". ........................ 58

5.3 Inclusive p—p correlation function in central 129Xe-l—lg-fAu collisions at
E/ A = 50 MeV is shown. The black points are experimental data and
the grey line is the best ﬁt. by the imaging technique. See table 5.1 for
speciﬁc parameters ............................. 61

5.4 The p-p correlation functions are shown for three c.m. energy gates of
0 < Em, < 15 MeV, 15 < Em < 30 MeV, and ECm > 30 MeV. For
the middle and high energy gates, the image technique is used. The
simple Gaussian source parametrization in Eq. 5.6 is applied for the
low energy gate. See table 5.1 for speciﬁc parameters .......... 62

5.5 The imaged source function is shown by inverting the pp correlation
in Fig. 5.3. See table 5.1 for the extracted source radius. ....... 63

5.6 The source functions are shown for the p—p correlation functions in Fig.
5.4 for three c.m. energy gates of 0 < Em, < 15 MeV, 15 < Em < 30
MeV, and Em > 30 MeV. For the middle and high energy gates, the
image technique is used. The simple Gaussian source parametrization
in Eq. 5.6 is applied for the low energy gate. See table 5.1 for extracted
source radii. ................................ 65

5.7 The inclusive d—a correlation function for the central collision gate is
shown as a function of relative momentum by the black points. The
solid line is the ﬁt by the simple Gaussian source parametrization in
Eq. 5.6 without corrections for collective motion. The blown-up in the
top right shows the poor quality of the ﬁt to the second peak of the
d—a correlation. .............................. 68

5.8 A source with only thermal motion (open arrows) is shown on the left.
The collective velocity ﬁeld (solid arrows) is drawn on the right and
results in a grey area of the source where emission into the right half
plane is improbable. ........................... 69

xiii

5.9

5.10

5.11

5.12

5.14

5.15

5.16

The source reduction factor, fwu. obtained from the ratio of the rms
radius to the system radius for the two particle sources with collective
motion, are shown by the solid line as a function of the mass number.
The ratio without collective motion is denoted by the dashed line. The
solid line is calculated for a uniform spherical system with a radius of
11 fm while the three symbols denote the corresponczling calculations
for a system with a radius of 7 fm. ...................

The solid line shows a simulation of the resonant distribution as a
function of relative energy for the d-alpha correlation. The dashed line
denotes the simulation for non-resonant background while the dotted
line demonstrates the relative energy distribution for mixed event back—
ground. All the distributions are normalized to one for comparisons. .

The ratios of the resonant yield over the mixed event distribution for
the d-a correlation are shown in three panels for maximum, 80% and
zero of the collective motion, respectively. The corresponding values
of Te” are extracted from the ﬁts (dashed lines). ...........

The inclusive d-a correlation function for the central collision gate is
shown as a function of relative momentum by the black points. The
solid line is the best ﬁt by the simple Gaussian source parametrization
in Eq. 5.6 with Teff corrections for collective motion. The blown-up
in the top right shows the good reproduction of the second peak of the
d-a correlation after Te" corrections. The extracted source radius is
To = 3.3 fm (see Table 5.2 for the rms value). The dashed and dotted
lines are the ﬁts assuming that the source radius is 1‘0 2 2.0 ﬁn and
r0 = 5.0 fm, respectively. .........................

The d-a correlation functions are shown for three c.m. energy gates
of 0 < Ed,cm < 20 MeV and 0 < Ema" < 25 MeV, 20 < Edm, < 40
MeV and 25 < Emcm < 45 MeV, and Ed”, > 40 MeV and Emcm > 45
MeV. Fits by the simple Gaussian source parametrization in Eq. 5.6
are shown by the solid lines after Teff corrections for collective motion.
See table 5.1 for extracted source radii and /\ values. .........

The ratios of the resonant yield over the mixed event distribution for
the p—p correlation are shown in three panels for maximum, 50% and
zero of the collective motion, respectively. The corresponding values
of T8” are extracted from the ﬁts (dashed lines). ...........

The experimental d—a correlation function is shown. The smooth solid
line is the empirical background used for the analysis. The dashed and
dotted lines are the Coulomb correlation calculated from Eq. 5.21 for
sharp sphere radii of R=12.4 and 15.6 fm, respectively. ........

The left panel shows the experimental d—a correlation function. The
solid, dashed and dotted lines are the different backgrounds used for
the estimation of uncertainty. The right panel exhibits the ﬁt of the
ﬁrst resonance peak after subtracting the background (solid line in the
left panel) ..................................

xiv

77

79

80

89

92

5.17

5.18

5.19

5.21

5.22

5.23

5.24

5.25

5.26

5.27

5.28
5.29

The experimental t—a correlation function is shown in the left panel.
The solid, dashed and dotted lines are the different backgrounds used
for the estimation of uncertainty. The right panel exhibits the ﬁt of
the ﬁrst resonance peak after subtracting the background (solid line in
the left panel). ..............................

The left panel shows the experimental 3He-Oz correlation function. The
solid, dashed and dotted lines are the different backgrounds used for
the estimation of uncertainty. The right panel exhibits the ﬁt of the
ﬁrst resonance peak after subtracting the background (solid line in the
left panel) .................................

The experimental a-a correlation function is shown in the left panel.
The solid, dashed and dotted lines are the different backgrounds used
for the estimation of uncertainty. The right panel exhibits the ﬁt of
the ﬁrst resonance peak after subtracting the background (solid line in
the left panel). ..............................

The densities of the resonance states from d—a, t-a, 3Han and a-
a correlations are ﬁtted after subtracting the backgrounds carefully
selected by the solid lines in Figures 5.16-5.19 ..............

The proton—7Li correlation function is plotted. The dashed line is the
selected background. See text for details .................

The density proﬁle of the resonances of 8Be is shown in the p-7Li cor-
relation after subtracting the background selected in Figure 5.21. . . .

The best ﬁt (solid line) is performed for the p—7Li correlation by vary-
ing the background and the spin value of the 17.64 MeV state. The
dashed line is the ﬁtted background. Two calculations are shown as
the dotted lines assuming that the spin of the 17.64 MeV state is 0 and
2, respectively and keeping the other parameters the same. ......

The p—7Be correlation function is ﬁtted by the solid line assuming only
two states at 0.774 MeV and 2.32 MeV. The dashed line is the ﬁtted
background. ................................

The p—7Be correlation function is ﬁtted by the solid line assuming the
existence of an additional state at 1.4 MeV. The dashed line is the
ﬁtted background ..............................

The a-6Li correlation function is ﬁtted by taking into account all the
spectroscopic information shown in Table 5.5 except for the 6.56 MeV
state. By varying the spin of the 6.56 MeV state from 1 to 4 and
assuming the decay branching ratio is 100%, one obtains the dashed,
solid, dotted and dot-dashed ﬁtting lines, respectively. If a branching
ratio of 55% is assumed, the solid line represents a ﬁt of J =4.

The a-a—a correlation function is shown. Resonances from the excited
states of 12C are labelled with the ﬁrst peak seen more clearly in the
inner upright panel .............................

The p—p—oz correlation function is shown. ................

The p—a-a correlation function is plotted. The resonances from the
decay of 9B are shown. ..........................

XV

94

95

97

104

106

107

109

110

111

116
117

118

5.30 The d-a-a correlation function is shown. The resonance states of 10B

6.1

6.2

6.3

6.4

' 6.5

6.6

6.7

6.8

that contribute this decay are listed. See (‘letailed discussions in text.

The points denote distributions of temperatures calculated with the
SMM approach for the decay of a ”2511 nucleus at three different
excitation energies. The lines denote gaussian ﬁts to the calculated
distributions. ...............................

The points denote temperature distributions calculated with the SMM
approach for the different isotopes considered in the carbon thermome-
ter for an excitation energy of E5 / A = 6MeV. The lines denote gaussian
ﬁts to the calculated distributions. ...................

Comparisons of various primary temperatures Tat/C, T1,.” p and "mm

from the SIV’IIVI and 72:3} from the analytical calculation in the grand
canonical limit. For details see the text. One point is missing for
7103‘: with or = 0.8‘MeV because the calculated value for p for the

correction term in Eq.(6.17) becomes negative at EJ/A = 3MeV, i.e.
the expansion breaks down in this case ..................

The solid squares and circles denote the free proton and neutron yields,
respectively, calculated via the SMIVI approach. The solid and dashed
lines denote ﬁts to the calculated yields following Eq. (6.23). .....

The squares, circles and triangles denote neutron chemical potentials
derived from Eq. (6.24) using SMM predictions for Carbon and Lithium
isotopic yields at various initial excitation energies for the decay of the
nucleus 1128a. The stars and the dot-dashed line denote approximate
values calculated from Eq.(6.28) for T20 and 4.58 MeV, respectively.
The error bars denote the statistical errors in the calculation, which in
many cases are too small to be observed in the ﬁgure ..........

Difference between the total binding energies predicted by the LDM
and those recomended in ref. [104]. Plot A corresponds to the param-
eter set adopted in standard SIV'IM [103], whereas Plot B is obtained
using the parameters presented in this work. ..............

Total binding energies for different nuclei. The full lines correspond to
the corrected LDM formula, whereas the symbols represent the exper-
imental data of ref. [104]. The dashed lines correspond to the predic-
tions given by Eq. (6.30). For details see text. .............

Internal free energies for A = 20 (upper panel) and A = 200 (lower
panel). The standard SMM expression [Eq. (6.34)] is represented by
the full line whereas the dashed lines stand for the results obtained
with the Taylor expansion [Eq. (6.36)]. The Free energy calculated

119

127

129

134

137

142

143

through the level density given by Eq. (6.38) is depicted by the symbols. 147

xvi

6.9

6.10

6.11

6.12

6.13

6.14

6.15

6.16

6.17

6.18

Level densities as a function of excitation energy for 2”Ne and '31P. Two
energy ranges are plotted to show the behaviors of level densities at
both low and high energy ends. The density of experimentally known
levels is shown as bars in the low energy region. The dashed lines are
the extrapolations of the empirical values according to Eq. 6.43. The
dotted lines are the level density (Eq. 6.38) parametrized from the
standard SMM. The solid lines are the level density adopted in this
work (Eqs. 6.49-6.52). ..........................

Comparison between F *(E ) calculated through Eqs. (6.35) and (6.49)-
(6.52), symbols, and the approximation given by Eq. (6.53), full line.
To illustrate the influence of quantum effects at low temperatures, the
dashed line represents the free energy used in standard Sl\-’IM calcula-
tions Eq. (6.34). For details see text. ..................

Best ﬁt values of T0 for different nuclei (symbols). The dashed line
corresponding to Eq. 6.54 is used for Z > 15. .............

Caloric curves are shown for calculations of the system of A=168 and
2:75 at ﬁxed breakup density and multiplicity-dependent density. The
dotted lines are calculated from the standard Sl\-'IM. The dashed lines
are calculated as empirical binding energies are taken into account.
The solid lines are obtained from the improved model with empirical
modiﬁcations of both binding energies and free energies. .......

Pressure curves due to kinetic motion and Coulomb interaction (see Eq.
6.58) are plotted for the system of A=168 and 2:75 at ﬁxed breakup
density and multiplicity-dependent density. The dotted lines are cal—
culated from the standard SMM while the improved SMM presents the
solid lines ..................................

Average breakup multiplicities are shown for the system of A=168 and
2:75 at ﬁxed breakup density and multiplicity-dependent density. The
dotted lines are calculated from the standard SMM while the improved
SMM presents the solid lines. ......................

Dependences of temperature on excitation energy and breakup density
are shown for the system of A=168 and Z=75. Calculations as function
of excitation energy at ﬁxed density of 1 / 6 normal density are shown
as solid circles. Calculations as function of density at ﬁxed excitation
energy are shown as open squares .....................

Mass and charge distributions are shown for the system of A2186 and
Z=75. The dashed lines are the calculations from the standard SMM.
The solid lines are calculated using the improved model. .......

Final mass and charge distributions after applying the empirical sec-
ondary decay procedure discussed in Sect. 6.2.3. The dashed lines are
calculated from the primary results of the standard SMM while the
solid lines are from the improved model. ................

Final mass and charge distributions from the present model (solid lines)
and the Botvina version (dashed lines) are shown. For reference, some
measured data is plotted as solid circles. ................

xvii

155

160

161

162

163

164

166

6.19

6.20

6.21

6.22

6.23

6.24

6.25

7.1

7.2

Primary isotopic distributions are shown for Be, C. O and Ne nuclei.
The dashed lines correspond to the calculations of the Botvina code
while the solid lines represent the results of the improved model. . . .

Isotopic distributions are shown for isotopes from Li to O. Experimen~
tal data is shown as the solid circles. The dashed lines denote the
Botvina calculations and the solid lines are the ﬁnal distributions after
decaying the hot primary fragment via the empirical secondary decay
procedure discussed in Sect. 6.2.3 .....................

Isotopic temperatures are extracted from 18 IMF thermometers (see
table 6.2) with 3 3 Z S 8 and AB > 10 MeV. Experimental data is
shown as the solid circles. The open squares are the calculations from
the improved model. For reference, the primary temperature calculated
from the present model is shown as the dashed line. ..........

Isotopic temperatures are extracted from 18 IMF thermometers (see
table 6.2) with 3 S Z S 8 and AB > 10 MeV. Experimental data is
shown as the solid circles. The open triangles are the calculations from
the Botvina model. For reference, the primary temperature calculated
from the present model is shown as the dashed line. ..........

Isotopic temperatures are extracted from 12 light thermometers (see
table 6.2) satisfying AB > 10 MeV and involving the isotope pair
of 3“‘He. Experimental data is shown as the solid circles. The open
circles are the calculations from the present model without corrections
of nonequilibrium emissions. For reference, the primary temperature
calculated from the present model is shown as the dashed line. . . . .

Isotopic temperatures are extracted from 12 light thermometers (see
table 6.2) satisfying AB > 10 MeV and involving the isotope pair
of 3"1He. Experimental data is shown as the solid circles. The open
squares are the calculations from the present model with corrections
of nonequilibrium emissions of 3He. For reference, the primary tem—
perature calculated from the present model is shown as the dashed
line. ....................................

Isotopic temperatures are extracted from 12 light thermometers (see
table 6.2) satisfying AB > 10 MeV and involving the isotope pair of
3"lHe. Experimental data is shown as the solid circles. The open circles
are the calculations from the Botvina code. For reference, the primary
temperature calculated from the present model is shown as the dashed
line. ....................................

The symmetry potential for neutrons and protons is shown for two
different density dependences of asymmetry term: asy—stiff F1 and asy-
soft F3 (see Eqs. (7.4)) ...........................

Isotopic distributions from Li to O are shown for central collisions of
124Sn-l-mSn. The full circles are experimental data while the solid
(dashed) lines denote the ﬁnal (primary) calculations from the hybrid
model using the density dependence asy-stiff F1 (Eq. 7.4) for the asym-
metry term of the EOS. .........................

168

169

171

172

174

175

176

180

7.3

7.4

7.5

7.6

Isotopic distributions from Li to O are shown for central collisions of
1“SIM—124811. The full circles are experimental data while the solid
(dashed) lines denote the ﬁnal (primary) calculations from the hybrid
model using the density dependence asy-soft F3 (Eq. 7.4) for the asym-
metry term of the EOS.

. . . - 0. x ‘) 9 ‘ f ‘
Relative isotope ratios, R21, of two reactions l"‘SIHJASn and ll“SnJrIUSn

are shown as a function of neutron number. The upper panel presents
the primary calculations using the hybrid model while the ﬁnal isotope
ratios after secondary decay are plotted in the lower panel. The lines
denote the best ﬁts through the symbols with the same slope.

Both panels: The solid circles and solid squares show values for [3,,
and {3", respectively; measured in central “28114—112811, 112Sn+124$n
and 124Sn+12ilSn collisions at E/A=50 MeV. Left panel: the open
and cross-hatched rectangles show corresponding hybrid calculations
for R21 calculated from the primary and ﬁnal fragment yields, respec-
tively, predicted by the hybrid calculations using the Asy-stiff EOS.
Right panel: the open and cross-hatched rectangles show correspond—
ing hybrid calculations for B21 calculated from the primary and ﬁnal
fragment yields, respectively. predicted by the hybrid calculations us-

ing the Asy—soft EOS.

The solid and open points in the upper and lower panels show the mir-
ror nuclei ratios measured for 124Sn+mSn and 112Sn+UZSn collisions,
respectively. Left panels: The open and cross-hatched rectangles show
corresponding hybrid calculations of the mirror nuclei ratios calculated
from the primary and ﬁnal fragment yields, respectively, predicted by
the hybrid calculations using the Asy-stiff EOS. Right panel: The open
and cross-hatched rectangles show corresponding hybrid calculations of
the mirror nuclei ratios calculated from the primary and ﬁnal fragment
yields, respectively, predicted by the hybrid calculations using the Asy-
soft EOS.

xix

185

187

189

191

Chapter 1

Introduction

Nuclear matter is a strtmgly—interacting Fermi liquid [1, 2] at low temperature and is
expected to undergo a. phase transititm to a nucleonic gas within a mixed phase region
bounded by a critical temperature of order 15 MeV [3]-[6]. This liquid-gas phase
transition is one of two bulk phase transitions in strongly interacting matter. The
other is the transition between hadronic matter and the quark—gluon plasma, which

is being studied at the Relativistic Heavy Ion Collider (RHIC) and elsewhere [7, 8].

The properties of mixed phase of the liquid-gas phase transition have been calcu-
lated in a number of equilibrium theories [9, 10, 11]. In general, these theories predict
the equilibrium distribution will be characterized by a mixture of light particles with
Z s 2 and intermediate mass fragments (IMF’s) with 3 3 Z s 30. Such fragments are
rarely emitted in the decay of compound nuclei at low excitation energies E*/A g 2
MeV, but become emitted with increasing multiplicity as the excitation energy is in-
creased signiﬁcantly beyond that approximate threshold [12, 13, 14]. These excitation
energies can be easily achieved at the NSCL as well as at other intermediate energy

heavy ion facilities.

Experimentally there are basically three ways to generate n'iultifragmentation.

One is by central collisions of two heavy ions of comparable masses at incident en-

ergies of 40-100 MeV per nucleon. This is the type of experiment perforn’ied in this
thesis. Dynamical calculatitms indicate that such collisions lead to dilute systems
in central collisions at energies greater than E/A=35 MeV [15, 16]. From a purely
dynamical point of view, multiple fragment production from the bulk system is very
likely in such collisions when the density drops below and the system cross into the
region of adiabatic instability, density ﬂuctuations grow exponentially and spinodal
decomposition n‘iay occur [17]. In addition to central collisions, one can also investi-
gate multifragrnentation in peripheral collisions (large impact parameters) of heavy
ions at incident energies of a few hundred MeV per nucleon. Such studies have been
pursued by groups at GSI and Berkeley [18, 19, 20]. The third approach is by light
ion induced collisions with much higher bombarding energies (> 1 GeV per nucleon)
[21]. All these reactions can form sources with excitation energies that exceed the
threshold predicted for statistical multifragmentation and multifragmentation has

been observed for the highly excited systems produced in all three reaction scenarios.

Studies of nmltifragmentation invoked the application of statistical techniques to
nuclear systems with a ﬁnite number of particles. Such applications have a long tra-
dition in nuclear physics that. stated with the description of highly excited compound
nuclei. The properties of nuclei or nuclear matter have been well studied at their low-
excited states, where the nuclear shell and collective models have been successfully
used to predict the behaviors of these states. The beta, gamma, alpha and ﬁssion
decay modes of these states have been well studied. Formalisms exist to calculate
these decay rates, and to provide information about the parent and daughter nuclear
states. As the system becomes more excited, the decay properties of individual states
become more difﬁcult to isolate. Then the average properties of groups of states or
levels become the quantity that is measured. The natural choices for model descrip-

tions of such average properties become the statistical formalisms by VVeisskopf or

2

Hauser and Feshbach[22. 23] and such models describe well the emission of nucleons

or light clusters.

In such models, the concept of nuclear temperature is introduced as a parameter
that describes the ensemble average over the various collisions with slightly different
incident energies, which contribute to the measured data. The validity of a thermal
approximation for such finite systems is sometimes questioned by scientists that are
more familiar with the thermal properties of much larger systems. In the nuclear
case thern‘ial approximations are justiﬁed because of the large number of initial states
that one is averagii‘ig over when one collides projectile and target nuclei under well
controlled conditions. To illustrate this point. consider the formation of a compound
nucleus with 150 nucleons at. an excitation energy of 2 MeV per nucleon (E* = 300
MeV). Like all Fermionic systems at moderately low temperature, the excitation
energy of such a system in contact with a. thermal bath at temperature T is given
by a quadratic relationship: E*/A = Tz/co. where T is the temperature in units of
MeV and 60 is a parameter which for nuclei is of order 8 MeV. It is straightforward
exercise in thermodynaruics to show that implies that the number of states per unit

energy p(E*) is given by

p(E*) oc exp (2.4(/E*/ (8A)) z exp(200). (1.1)

Using the K1200 cyclotron. one can study such a system using a beam with a
typical energy resolution of about 0.3% (AE* = 0.9.1! cV). Taking a standard nuclear
level density formula [24] one can estimate that there are about 1083 levels in this
energy window. Even the population of a small fraction of these states in the entrance
channel leads to a vast ensemble of quantum states over which one is averaging. One
therefore does not need the existence of an external thermal reservoir to justify the

thermal limit.

At the higher excitation energies characteristic of 1'1mltifragnmutation processes,
the thermal limit is therefore justiﬁed provided the systems reach a state where the
approximation of local thermal equilibrium may be applied. Et‘ptilibrium models
simply assume that equilibrium is achieved at a given freezeout time and characterized
by a freezeout density (probably 0.1-0.4,00) and temperature (typically 4-7 MeV) or
excitation energy (typically E*/A 2 3 MeV). Phase transition arguments may be
relevant when equilibrated system is dilute. \Vhether such dilute systems can be
described as an equilibrated phase mixture of fragments (liquid) and light particles
(gas) is one of the important questions that need to be addressed [13, 14]. Although
some problems related to the ﬁnite size effects have been studied [25, 26], the question
of how the system goes out of equilibrium and equilibrium observable are modiﬁed
as fragments and nucleons decouple from each other and prepagate to the detectors

is usually not addressed.

For the interpretation of such fragment production in terms of a bulk disintegra-
tion, it was important to assess how quickly the system disintegrates and exclude
the possibility of a slow sequential series of binary ﬁssion-like decays, each of which
increases the fragment multiplicity by one. Two observations rule out such decays
occurring over the long time scales normally associated with ﬁssion: 1) Correlations
between pairs of fragments emitted in such breakups reveal an anticorrelation in the
fragment emission at low fragment-fragment relative energies due to their mutual
Coulomb repulsion [27]. Strong anticorrelations are observed between every pair of
fragments which implies short breakup timescales that are inconsistent with a slow
sequence of binary decays, but within the range expected for a bulk multifragment
disintegration [28]. 2) A11 approximately linear mass dependence of the mean frag-
ment energies in the center of mass has been observed indicating a collective expansion

that at incident energies of E / A=50 MeV and above exceeds values consistent with

4

a purely Coulomb induced expansion. Both observations imply lneakup time scales

of the order of 100 fm/c or less.

Temperatures have been obtained for such events from measurements of fragment
isotopic yields [29, 30. 3'1, 32] and excited state populations [33, 34, 35]. These mea-
surements are consistent with there being a plateau of the nuclear caloric curve at
temperatures of about T N 5 MeV and E*/.4 a: 3-10 MeV [29]. In equilibrium sce-
narios, such a plateau is theorr-étically expected [36] for multifragmentation process,
reﬂecting the latent heat for transforn‘iing the nuclear liquid to the nucleonic vapor.
Scaling laws have also been observed in the fragment elemental [37, 38] and isotopic
[39] distributions. The former observation has been loosely interpreted by many au-
thors using Fisher liquid drop theory[40] to extract critical parameters and the critical
temperature for nuclear multifragmentation [38]. The ol,)servation of isotopic scaling
laws, discussed in this thesis, provides a powerful simpliﬁcation of the dependence of

the fragment isotopic. distributions on the overall isospin of the system [41, 42].

Despite these interesting developments, there are many missing elements in the
interpretation of fragmentation observables in terms of phase transitions. The early
stages of the reaction during which energy is deposited and may or may not be equi-
librated are not described by such models. Thus, practitioners have the freedom
to adjust the thermal energy (or temperature) and the breakup density to improve
the agreement with data. This has (.lelayed the determination of whether equilibrium
bulk fragmentation models are justiﬁed and should be used instead of time dependent
rate equation [43] or dynamical fragmentation [44] approaches. There have also been
very few real tests of equilibrium multifragmentation models other than comparisons
to fragment charge distributions and their related multiplicities [45]. Comparisons
to energy spectra within equilibrium models are not very quantitative because such
require the manual inclusion of collective motion induced by the earlier dynamical

5

evolution. More and better tests are possible if one can compare equilibrium calcu-
lations to isotopically resolved data, excited state populations and other observables.
The predictions for such observables. however, require an accurate descriptirm of the
secondary decay of the excited fragments that had not been available prior to this

thesis work [26, 46].

Even though most theoretical studies of multifragmentation have involved equi-
librium models. it is relevant to consider the alternative approaches. More generally.
theoretical models for the multifragmentation process can be divided into two cate-
gories: static and time dependent. Examples of static models include multi—particle
phase space models. such as the Statistical Multifragmentation Model (SMM) [36. 9]
and the Berlin multifragmentation model [47]. which can incorporate speciﬁc nuclear
properties more directly. Accordingly, a semi—microcanoiiical version of SMM [36]
that incorporates detailed miclear structure information relevant to the population
and secondary decay of the excited fragments was (levelt')1‘)ed [26. 46] and utilized as
part of this thesis work. Additional static models include percolation [48] and lattice
gas [49] approaches; these approach have the virtue of providing relatively simple
schematic algorithms suitable for the exploration of critical phenomena in ﬁnite sys—

tems.

Statistical rate equatitms, which allow the descrij‘)tion of time dependent. phe-
nomena within a statistical frz‘unework, have long been the standard aj")proach for
calculating the decay of equilibrated compound nuclei and also provide an alternative
description of nmltifragmentation [43]. An extension of these approaches to the case
of fragment emission from an expanding residue has been developed by Friedman [10].
This Expanding Emitting Source (EES) model predicts a rapid emission of fragments
once the density of the residue decreases below about 0.4/20, and is consistent with
many basic multifragment observables [50, 51]. In its present form, however, the

6

description of multifragmentation is still rather schematic, leaving out many speciﬁc
nuclear properties that inﬂuence fragu’ieut isotopic distributions and excited state

populations.

More complete dynamical armroaches. such as the Boltzn'iann-Uehling—Uhlenbeck
transport model (BUU)[44], solve equations of motion that involve a self-consistent
mean ﬁeld. Solutions of the BUU equation provide the Wigner transform of the one
body density matrix. Hybrid calculations that use the BUU equation to deﬁne the
input excitation energies and source size for various statistical models can then be
explored (e.g. [42]). The calculation of fragment yield directly via the BUU model
without the hybrid approach is not feasible, however, because density ﬂuctuations
that lead to fragment formation are suppressed in the BUU equation. Therefore,
alternative formalisms, such as the Stochastic Mean Field model [52] and Antisym-
metrized Molecular Dynamics model (AMD)[53] have been developed to address this
deficiency. These approaches, however, have not progressed to the point that accurate
predictions of isotopic observables are currently possible. In the absence of a better
description of dynamical fragment prmluction, we therefore test the equilibrium as-
sumptions via comparisons of data to equilibrium calculations or hybrid BUU-SMM
calculations, using the improved SMM we developed, which can provide quantitative

comparisons to fragmentation observables.

1.1 Freeze-out Conditions of Multifragmentation

Every equilibrium multifragmentation approach requires external input that. speciﬁes
two independent intensive parameters of the equilibrated system at freezeout. These
two parameters could be the excitation energy per nucleon and temperature. Excita-

tion energy be somewhat constrained by n'ieasurements are sensitive to the thermal

energy of the system. Examples of such n'ieasurements are the energies and mass
distributions of particles emitted from projectile- or target-like residues in peripheral
heavy ion collisions [29]. Such information is not readily available in central colli—
sions. Isotope temperatm'es may be extracted from measurements of double ratios
of isotopic yields [29, 30, 31, 32] of IMF ’s using the Albergo formula[54] and excited
state temperatures may be obtained from measurements of the relative populations of
excited states of intern’iediate mass fragments [33, 34, 35]. Systematic measurements
indicate that the values of T = 53:05 MeV may be expected in central nucleus-mtcleus

collisions ten‘iperatures at incident energies of about. E / A=50 MeV.

The relationship between the excitation energy and the temperature is only weakly
density dependent. Both excitation energy and temperature dependent observables
are sensitive to preequilibrium emission prior to freezeout. Thus, it would be ex—
tremely useful to have an independent measure of the density or volume of the sys-
tem, even in systems for which n'ieasurements of both quantities can be attempted.
Compared to the large numbers of ten'ijj)erature measurements that have been per-
formed, there are comparatively few measurements aimed at detern'iining the density.
The main attempts have involved classical analyses of fragment-fragment correlation

functions [55, 56] and a recent attempt involving light particles [57].

In this thesis, we explore alternative density determinations, which utilize two
particle correlation functions involving light 1.)a.rticles and much lighter fragments.
This work is more closely related to the intensity interferometry techniques, applied
originally to the problem of measuring stellar radii by Hanbury-Brown and Twiss [58],
adapted to strongly interacting particles by Koonin and others [59, 60] and utilized
extensively in heavy ion reactions over a wide range of energies [61, 62]. Recently,
Verde et al. [63] have revealed the importance of describing the detailed shape of
such correlation functions; this work shows that the source size is related mainly

8

to the width of the correlation peak instead of the height [63]. Verde et al. also
indicate how one can assess the degree to which the height of correlation functions
are diminished by secondary decay effects and determine the importance of secondary

decay processes [63].

Fruitful results are obtained in this dissertation by applying these ideas to mul-
tifragmentation events. In addition, such particle correlation functions are extended
to much heavier particles by utilizing thermal correlation function techniques. These
thermal correlation function techniques allow the extraction of additional information
relevant to the freezeout assumption and suggests a new technique for spin determi-

nation of particle unstable states using correlation functions.

1.2 Fragmentation and Isospin Effects

The study of isospin effects in heavy ion reactions is expected to be a major theme for
research at the CCF and later at RIA. Recently, im’estigations reveal the existence of
isoscaling laws that govern the dependence of fragment isospin distributions on the
total isospin of the system [41, 39]. This isoscaling behavior is manifested in multi—
fragmentation, compound nuclear decay and in strongly damped heavy ion collisions
[64]. The isoscaling parameters extracted for nmltifragmeritation processes reveal
the occurrence of isospin fractionation [41] whereby the gas phase manifested by the
properties of the primary light particles prior to secondary decay is more enriched in

neutrons than the liquid phase represented by the fragments.

The degree of fractionation appears to be sensitive to the density dependence of
the asymmetry term of the nuclear equation of state (EOS). The nuclear EOS is a
key property of nuclear matter that. is very relevant to supernova explosions and to

the structure and stability of neutron stars [65, 66, 67]. Experiments have succeeded

in constraining the EOS for symmetric matter at a variety of densities. The nuclear
n'ionopole and isoscalar dipole resonances[68], for example, have been performed to
sample the curvature of the symmetric matter EOS near the saturaticm density ,0”.
The collective ﬂow measurements[69] place constraints on the symmt—ttric matter EOS

at densities as high as 4 -— 5/20.

The properties of very neutron rich systems such as neutron stars, however, can
be dominated by the asymmetry term. Unfortunately, the difﬁculties of ﬁnding ob—
servables sensitive only to the isospin effects and of creating very asymn'ietric matter
under laboratory—cont.rolled conditions have left the density dependence of the asym—
metry term largely tmconstrained. 111 this dissertation, we develop an improved SMM
model and use it to investigate the sensitivity of the isoscaling parameters to the
asymmetry term of the EOS [42]. These investigatitms suggest that the availability
of high-intensity radioactive beams at the newly—upgracled coupled cyclotron facility
(CCF) of Michigan State University and possibly at the proposed rare isotope accel-
erator (RIA), will make the exploration of the isospin dependence of the EOS one of

the more lIlt(-‘.I'(‘.Stlllg issues to be explored in coming years.

1.3 Organization of the Thesis

The follmving chapters are organized as follows. In Chapter 2 experimental setup
for 129Xe+m7Au at 50A MeV is described. Detailed speciﬁcations of the LASSA
telescoj'x-s, the preparation of CsI scintillation detectors and energy calil')rations of
the LASSA telescopes are described in Chapter 3. In Chapter 4 the data reduction
and analysis procedures are described in detail for the LASSA array. Two and three
particle correlations obtained in this experiments are described in Chapter 5. There,

a. new technique for spin determination from particle correlations is proposed and a

10

 

probe to freeze-out density of lgireakup is discussed. In Chapter 6 an modiﬁed statis-
tical multifragmentation model (SMM) which incorporates self-consistelitly nuclear
structure information and calculates self—consist.ently the secondary decay of the ex-
cited fragments is described and predictions of the model and comparisons to data
are discussed. In Chapter 7 possible constraints on the isospin (l(,>pe11cle11ce of the
nuclear equation of state are presented by use of the model described in Chapter 6.

Chapter 8 summarizes the ﬁndings of this dissertation.

11

Chapter 2

Experimental Setup

The experin‘ient of l29Xe+m7Au at 50A MeV was performed at the National Super—
conducting Cyclotron Laboratory (NSCL) with the K1200 accelerator. To measure
the fragmentation of heavy ion collisions with particles emitted from all directions,
a large detection system with nearly full solid angle coverage is required. In this
experiment, the Miniball/hIiniwall array[70] was utilized to cover most of the solid
angle. However, the. Mi1iiball/Miniwall array alone can not fulﬁll the demanding
measurements for this study due to its relatively coarse granularity and limited mass
resolution. Therefore, a large area. silicon-strip/Csl detector array (LASSA)[71, 72]
was constructed to provide excellent energy, angular and isotope resolution for charged
particles. This experiment was one of the four consecutive experiments conducted in
the ﬁrst campaign with the complete nine LASSA telescopes. More details about the

LASSA will be discussed in the next chapter.

2. 1 Mechanical Setup

The 92” scattering chamber at the NSCL was used to house the wli(‘)le com ﬂex
(1)
detector system (Miniball/l\=‘Iiniwall + LASSA) in 'acuum. The 129Xe beam with an

intensity of ~ 108 particles per second was produced from the K1200 cyclotron to

12

bombard the 3 mg/cm2 thick Au target.

 

 

 

Figure 2.1: Schematic drawing of Miniball/h’liniwall + LASSA setup in the vertical
plane.

A schematic drawing of the detector setup in the vertical plane is shown in Figure
2.1. The 3 mg/cm2 thick Au target was situated at the center of the Miniball array.
The Miniwall array covered the forward angles 0 g 25" and provided a better granu-
larity over this angular domain. Some of the forward elements of Miniball/Miniwall
array were removed to insert the LASSA, which was centered at 35° and at a distance
of 20 cm away from the target. The geometric acceptance of the combined apparatus

was about 80% of 47r.

To reduce the noise level, the preampliﬁer system for the LASSA was mounted
along with the detectors in the vacuum chamber. Photo 2.2 shows the sideview
of the detector setup with these preampliﬁers mounted above the array. A cooling

system was run inside the chamber to remove the heat generated by the detectors and

13

 

Figure 2.2: Miniball/Miniwall + LASSA setup housed in the 92" chamber.

preampliﬁers. A temperature monitoring system was also installed with temperature
sensors attached to critical elements and read out by a FERA module to ensure that

the system remained near room temperature during the experiment.

Figure 2.3 shows a. photograph viewing from the beam direction. This picture
allows one an inside view of the detector setup after the experiments were ﬁnished.
Some Miniball detectors in the upper and backward quadrant were removed in order

to Show all nine LASSA telescopes in the photo.

2.2 Electronic Scheme

The LASSA detectors required the development of a high density electronic system ca-
pable of processing the signals. Special CAMAC modules that contained 16 channels

of shaping ampliﬁers and 16 channels of discriminators were developed by Washing-

14

 

Figure 2.3: Miniball/Miniwall + LASSA detector setup viewed inside from the beam
direction.

ton university and used for the LASSA readout. A schematic of our electronic setup

for processing the LASSA is shown in Figure 2.4.

2.3 Miniball/Miniwall

The original Miniball is a portable 47r phoswich detector array for the detection of
charged particles. The array of 187 phoswich detectors covers the angular range
9° 3 91“], 3 160° with a solid angle of 89% of 47r. Each phoswich detector consists of
a 0.08 mm (8mg/cm2) thick plastic scintillator foil backed by a two cm thick CsI(T1)

crystal.
Particles which penetrate the plastic scintillator foil and stop in the CsI(T1) crys-

tal are identiﬁed by atomic number up to Z=18. Pulse shape discrimination in the

15

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Trigger Fast ECL
R8232C Readout
Interface ’ D' T' /A\
Control 3 18C am to
OR FERA g\> FERA
P 1
user L.E. Disc 1: . Lin FIFO
nspection
: ‘ L"'Scope
Pre-Amp 3‘ T.F. Amp E
I 3 PH7164H
? ADC
Dt t ‘ Slow Shaper 3 .
6 CC or 1 ..... , ....... , ................................... {“Specn-on Lm FIFO

 

 

 

 

/ L“’ Scope

 

 

 

\
GPIB Bus for Slow Control >
/

Figure 2.4: Schematic diagram of electronics setup for LASSA.

Csl(Tl) crystals provides isotopic resolution for H and He isotopes as well. Approx-
imate energy thresholds are of the order of 1 MeV for the detection of a charged
particle ”hit” in the plastic scintillator. The thresholds to particle identiﬁcations
are about E,h/A=2 MeV for Z=3, Eth/Az3 MeV for Z=10, and EU,/A=4 MeV for
Z=18 fragments. However, these thresholds are not very relevant to this thesis work
because the h'liniball/h'liniwall array were used only for providing the multiplicity
of hits. The hit multiplicity was used to gate centrality of collisions as discussed in
Chapter 4. More details about the detector can be found in the reference by de Souza

et al., Nucl. Instr. and Meth. A295, 109 (1990).

To increase the granularity of the l\r’liniball in the forward direction, the Wash-
ington University group (L. Sobotka and R. Charity) constructed a forward array of

16

128 fast plastic - CSI phoswich detectors that mates exactly to the MSU Miniball at
25". Similar to the Miniball detectors. each I\=Iiniwall phoswich detector consists of a

~ 8mg / cm2 plastic foil and a three centimeter long CsI(Tl) crystal.

In the experiment 130 Miniball phoswich detectors and 80 Miniwall detectors
were mounted while the other MB/ MW detectors were removed to make space for
LASSA (for details see Table 2.1). The total solid angle coverage of the detectors
of the combined MB/ MW apparatus that were used in the experiment corresponds
to about 80% of 47r. To reduce the counting rates of electrons and gamma rays, all

Miniball/Miniwall detectors were covered by a layer of 4mg/cm2 Sn—Pb foil.

 

 

MB / MW Ring Detectors AQ(msr) Mounted Removed
MB3 28 11.02 15 13
M134 24 22.9 11 13
M85 24 30.8 20 4
MB6 20 64.8 14 6
MB7 20 74.0 19 1
MB8 17 113.3 17 0
MBQ 14 135.1 14 0
MBIO 12 128.3 12 0
MEN 8 125.7 8 0
MW2 16 2.57 16 0
MW3 22 2.59 17 5
MW4 26 2.85 19 7
MW5 24 5.56 15 9
MVVB 24 10.64 13 11

 

 

 

 

 

 

 

Table 2.1: List of the original number of detectors in a ring of Miniball/Miniwall,
the solid angle of a detector in that ring, and the numbers of mounted and removed
detectors in that ring, respectively.

17

Chapter 3

LASSA Telescopes

Since the initial measurements that demonstrated the phenomenon of multifragmen-
tation, there has been a growing need to explore this phenomenon with higher reso-
lution devices. h-Ieasurements of isotopic distributions, excited state populations and
correlation functions are much needed to extract the relevant temperatures and den-
sities of such processes. However, existing charged particle detection arrays such as
the Miniball/Miniwall[70] lack the energy, angular and isotopic resolutions for such

studies.

The Large Area Silicon—Strip/Csl detector Array (LASSA)[71, 72] was designed
and constructed to fulﬁll the highly demanding requirements of such studies. The
LASSA consists of nine identical telescopes, each of which is composed of two silicon-
strip detectors of 48 channels and four CsI crystals as shown in Figure 3.1 for one
telescope. The whole array was centered at the forward angle of 35° in the experiment
setup and positioned 20 cm from the target. The 3 mm pitch of the silicon-strip
detectors corresponds to an angular resolution of :l:0.43° for this setup. The total
array covered a solid angle of about 540 msr with the polar angle 6 ranging from 12°
to 62° and the azimuthal angle (5) ranging from 24° to 156°. Figure 2.1 indicates the

distances and angular coverage of LASSA.

18

 

Figure 3.1: One of the nine identical LASSA telescopes.

19

3. 1 Silicon Detectors

Silicon detectors are widely used in nuclear experiments because of their excellent
energy resolution and linear response for charged particles. Extremely thin (~ several
tens of am) to relatively thick (~ 1mm) silicon strip detectors of large areas have been
manufactured to provide position sensitivity for the detection of charged particles.
Both layers of silicon strip detectors used in the LASSA are variants of the Micron

design~W[73] and have an active area of about 5x5 cm2 ( Figure 3.2 ).

/ Hi

_l
am

 

Figure 3.2: Picture of one 500 pm thick double-sided silicon strip detector.

3.1.1 Speciﬁcations

In each LASSA telescope, the front (labelled here as 7’ DE”) silicon detector is about 65
pm thick and 50x50 mm2 in area; it has 16 strips on the front side that are 3 mm wide

each and has a single electrode (no strips) on the back. The second layer of silicon is

20

about 500 pm thick and is also 50x50 mm'“) in area. It is double-sided: Each side is
divided into 16 strips with 3 mm pitch. For convenience, we label 16 strips of the front
side by ”EF” and the strips on the back side, which are. perpendicular to the front
strips, by ”EB”. For both types of silicon detectors, there is a 0.1 mm wide interstrip
region without electrodes and its effects will be discussed in the data analysis later.
The thicknesses of the front layer of silicon can vary up to 10% and the back layer
can vary by up to 2%. These variations will affect the particle identifications in ways

that will be elucidated in the next chapter.

The closely packed design required the development of a highly flexible ﬁat printed
circuit board cable connecting the silicon strip detectors with the pre-ampliﬁer hous-
ings. These cables were wire—bonded directly to the silicon wafer (see Figure 3.2).
All 432 preampliﬁers were housed in 9 boxes, each one housing 16 preampliﬁers for
the DE, 16 preampliﬁers for the EF and 16 preampliﬁers for the EB of one telescope.
Each one of the preampliﬁers is driven by :l:12V supply voltage and has a power
consumption of approximately 300 111W. The gains of the DE preampliﬁers are 15

mV/MeV and the gains of the EF and EB preampliﬁers are 5 mV/MeV.

The silicon detectors need to be fully depleted to achieve the best energy reso-
lution. Interestingly we found that the full depletion voltage listed by Micron was
typically a factor of two smaller. The DE and EF strips were biased in the experiment

to be fully depleted.

The stopping energy for alpha particles is 10 MeV in the ﬁrst layer of silicon and
36 MeV in the second layer after passing through the ﬁrst. layer. Because any particle
stopped in the ﬁrst layer of silicon can not. be identiﬁed by this device through the
AER identiﬁcation technique, the stopping energy of the ﬁrst silicon sets the lower

threshold for the corresponding particle we can analyze. Software thresholds were set

21

on the analysis of the experimental data, which were a little higher than the physical
stopping energies. For example. the low energy cutoff for alpha particles is about 10

MeV for the LASSA.

3.1.2 Energy Calibration

One advantagemis feature of silicon ('letectors is their linear and largely particle—
independent energy response. \Vhile ﬁssion fragments have in general, rather different
energy calibrations than alpha particles due to space charge effects near the stopped
fragments, only one energy calibration should be needed in principle for any species
of charged particles with relatively low charge number Z s 8. As few as two points
should be able to deﬁne the energy calibration curve for such species. However, due
to the nonlinearity of electronics system including preampliﬁers, shapers and ADCS
at. low and high energies, the calibrations of both low and high energy particles must

be treated more carefully than those of particles at the center of the energy spectrum.

An Ortec precision pulser generator was used to calibrate the silicon detectors.
The pulser has a group of attenuation switches to calibrate detectors with different
dynamic ranges. Three different dynamic ranges of 140 MeV, 200 MeV, and 500 MeV
corresponding to three different combinations of pulser attenuation switch settings
were used to calibrate LASSA silicon detectors. An absolute calibration was obtained
from the measurements of 2‘”Am and 2”Th alpha sources for these three settings.
Figure 3.3 shows the linear calibration relations of energy in MeV versus pulser dial

value in Volt for the three dynamic ranges.

Right after the data—taking experiment was ﬁnished, all the electronics setup was
retained the same except that the pulser calibration system was plugged in each input
channel of the silicon detector preamps while the detectors were still attached. Then

the one-by-one pulser calibration was carried out for all 432 silicon channels. For each

22

161-“ -———--
y = 49.702x + 0.0286

  
 
    

If y = 20.726x + 0.0172

12- /

y =13.935x + 0.0148

 

Energy (MeV)
03
\

   

O 140 MeV Range
I 200 MeV Range
A 500 MeV Range
— Linear (140 MeV Range)
Linear (200 MeV Range)
--- Linear (500 MeV Range)

0 I I I r I
0 0.2 0.4 0.6 0.8 1 1.2

Dial Value (Volt)

 

 

 

 

 

 

.1“--- _. __ T. _ _

 

 

 

Figure 3.3: Pulser calibration for three dynamic. ranges.

channel, more calibration points were applied to the low energies and high energies
to measure the possible nonlinearities on those regions as shown in Figure 3.4. In
the low and middle energy regions labelled as 1 and 2, respectively, in Figure 3.4,
linear ﬁtting procedures were employed while in the high energy end (labelled as 3) a
4th order polynomial ﬁt was adopted. Spline interpolaticm was used to join the three

pieces together smoothly.

3.2 CsI(Tl) Detectors

Csl(Tl) scintillation crystals are cost effective for detecting charged particles with
energies of E/Az30—200 MeV. Less expensive than silicon detectors, less hygroscopic
than Nal crystals, and easily machined into different. shapes, CsI(Tl) detectors have
been used to stop higher energy particles in many AE-E type detection arrays with

large solid angle coverage.

23

 

.3
O

,: 4 3 2
) 03x +b3x +c3x +d3jc+e3

 

fill

 

-.'l‘

V M Spline ﬁt
I

 

 

 

 

 

= PM + bra

i 1 1 l l

Pulser Dial Value (Volt)
0 A N 00 £> 01 O) \l 03 CD
it

0 500 1000 1500 2000 2500 3000 3500 4000
Channel Number

F io'ure 3.4: Energy calilnation for a tv )ical silicon stri ).
b C). v

The Csl crystals had the original dimensions of 3.5x3.5x6.0 c1113. The ﬁnal shape
of Csl crystals is deﬁned by tapering the two outer sides and making the front surface
reduced to dimensions of 2.5x2.5 cm2 while keeping the dimensions of the back surface
the same. Four crystals are grouped together and placed behind the two layers of
silicon detectors. With a thickness of 6 cm, all CsI(Tl) can stop alpha particles with

energies up to 580 MeV and protons with energies up to 145 MeV.

3.2.1 Pre-selecting and Scanning CsI(Tl) Crystals

To ensure that. Csl crystals in LASSA have good energy resolution. only the crystals
with light output uniformity better than 100 were adopted. A careful alpha—scanning

procedure was applied to control the quality selection.

24

L Amp MCA COMPUTER

 

  

CsI Box
with Preamp

Vacuum chamber

 

Sliding Scales

Figure 3.5: Csl crystal scanning setup with alpha source.

25

The CsI(Tl) crystals uianufacturetl by Scionix[74] were originally rectangular in
shape with dimmisions of 3.5x3.5x6.0 curl. The front square surface was polished and
the other sides were sanded when obtained commercially. After a careful inspection
for visual cracks or imperfections, the back square surface was sanded down and
polished. It was then optically coupled to a clear acrylic light guide that was in
turn optically connected to a 2x2 cm2 photo—diode. The front surface of the crystal
was covered with an aluminized mylar foil and the other sides were wrapped with
two layers of teflon tape to ensure uniform light collection. Details of discussion on
wrapping materials will be shown in the next section. The whole detector set was
contained in a small aluminum box along with a charge sensitive preampliﬁer attached

in the back of the box to maximize the signal to noise ratio.

The 5.486 MeV alpha. particles from a collimated 1 uCi Q'HAm alpha source were
used to scan the crystals. Figure 3.5 shows the scanning setup inside a vacuum
chamber. A two dimensional sliding scale supported the alpha source and the crystal
box on two perpendicularly crossing arms, res1')ectively, where the scanned surface
position of the crystal was measured by the scale. For each crystal nine regions that
uniformly divided the front surface were in turn irradiated by the alpha source as
shown in Figure 3.5. The ﬁrst point was scanned again after ﬁnishing all nine points
to make sure that the scanning system was stable and not affected by temperature
changes and other drifting effects. The alpha spectra corresponding to the centers
of the nine sub-squares were recordI—xl with a. multi—channel analyzer equipped with a.
peak sensing ADC and then stored in computer for offline analysis as shown in the

electronics scheme of Figure. 3.5.

26

 

-0.03

+0.03

+0.14

 

-0.09

-0.01

+0.05

 

 

0.0 0.11

 

-0.04

 

+0.05

 

 

 

 

Crystal 652

 

 

 

 

 

 

 

 

 

   

-1.0
+1.0
+0.95 -0.05 -0.92
+0.88 -0.01 -0.87
0.0 +0.80 +0.14 091
Crystal 291
-1.0

Figure 3.6: Uniformity results of accepted and rejected CsI crystals.

27

The different grey levels in Figure 3.6 represent the percentage deviations of the
nine alpha peaks from the median value. The speciﬁc numbers of these deviations
are shown in the corresponding table next. to the shaded crystal surface. Only the
crystals with light output varyii‘ig within i051: were accepted. such as the one shown
in the top panel with deviations ranging from -0.11% to 0.05% of the mean. On the
other hand, crystals with light output uniformity beyond i0.5% were rejected and
returned to the manufacturer. The bottom panel demonstrates one of the rejected
samples, which clearly shows the existence of a light output gradient that is caused

by the non-uniformity of T1 doping inside the crystal.
3.2.2 Testing Wrapping Materials

A reﬂective entrance foil is needed to cover the front surface of a Csl crystal for
optimal light collection efficiency that improves the energy resolution. In order to
minimize the dead layer loss due to the Csl front foil, a very thin layer of aluminized
mylar foil (0.15 mg/cm2 mylar + 0.02 ing/cm2 Al) was applied to the C81 front
surface. As for the other sides of the crystals. opaque materials are needed for diffuse
light reflection. This side wrapljnng is very critical not only for maximizing the light
collection efﬁciency but also for minimizing the possible light crosstalk between four

bundled crystals within one telescope.

To achieve the best resolution and the thinnest wrapping, we tested different
materials for wrapping up crystals. Two different materials, 0.1 mm thick white
teﬂon tape and 0.14 mm thick white cellulose nitrate membrane with pore size of 0.2

nm are shown in ﬁg 3.7 for different numbers of wrapping layers.

28

 

900 WWHW

m Ot SOUI'CG

If
0.

I

l

l
O

”5800: /
13700 ;—

 

....... O
_ ........... 9""
CL. : /®

600 T ________ we”

. per

r ,

bt+®f l++k+l+s++laﬁ
500 _ q I OTeflonITape I

t

- 9 Cellulose Membrane

.\7
Q
' l
0

FWHM(%)
0

O
I I Y
\
l
\
\
\
\
\
\
\
\
\

 

 

 

Number of Layers

Figure 3.7: Comparisons of different wrapping materials used for Csl crystals.

29

Using the same 2"”Am alpha source as used in the scanning 1.)rocedure, we found
that the light output, i.e., the peak channel of alpha spectrum, increases as more
layers are wrapped around the crystal for both materials. This is shown in the upper
panel of Figure 3.7. The improvement of the percentage of energy resolution as shown
in the lower panel is directly related to the enhanced light output since the FW’HM
of the alpha peaks dictated by the electronic noise is constant at about 40 channels
(corresponding to 250 keV for the largest light output). When the number of layers
is large enough, for example, two for cellulose membrane and five for teﬂon tape,
however, the collected light output saturates and nothing can be gained by increasing

the thickness further.

The overall performance of cellulose nitrate membrane is much better than that of
teﬂon tape. \Vhen the same number of layers are used, the cellulose nitrate membrane
shows about 40 percent increase of light collection over the teﬂon tape, making it
certainly the choice. Since four crystals are grouped together behind the two silicon
detectors in each telescope, a. large gap between CsI(Tl) crystals leads to a loss in solid
angle. This is also argues for the cellulose membrane. Based on the saturation shown
in Fig. 3.7, two layers of the cellulose membrane is the optimal choice. In the ﬁnal
wrapping, each crystal was wrapped with two layers of cellulose nitrate Inen‘ibrane
around the two outer sides and one. layer around the two inner sides which touches
two other adjacent crystals. Counting both shared layers of the adjacent crystals,
each crystal had effectively two layers of cellulose nitrate membrane on all four sides.
This was enough for measuring the light collection and resolution but not, enough to

remove all cross—talk as we discovered later.

30

3.2.3 Other Effects and Assembly

The light. output of Csl crystals is also affected by temperature changes. During the
preselection scanning, we also tested the temperature effect which typically gives a 1%
decrease in light output for every three degrees centigrade increase at room tempera-
ture. In Xe+Au experiment, a cooling plate was attached right behind each telescope
to stabilize the temperature within one degree centigrade at room temperature. So
the variation of temperature would not be the limiting factor of energy resolution of

the Csl detectors.

Optical clear front and back surfaces are essential for collecting the light and moni-
toring the gluing process. A step by step sanding and polishing procedure was applied
by using 400 grit sanding paper and 30 ,um, 15 pm, 5 am, and 1 am polishing paper
with the aid of pure ethanol (any water-containing resolvent could easily damage the
Csl crystals). The final step with silk cloth and polishing compound would make the
surfaces crystal clear. In addition, four other sides were slightly sanded with 400 grit
sanding paper with the sanding grooves parallel to the long axis of each crystal for a

better uniformity and light diffusion.

For each crystal, a 1x3.5x3.5 cm3 acrylic light guide was glued to the back surface
with optical epoxy BC600[75] and a photodiode with active area of 2x2 cm2 was glued
with clear silicone rubber compound RTV615[76]. The gluing process was monitored
to ensure that no bubbles were left in the layer of glue. To prevent the light leak and
crosstalk between adjacent crystals, the outer sides of light guide and the photodiode
were painted with a reflective white paint BC620[75]. Four crystals were bundled in

the back of silicon detectors for stopping high energy particles.

Csl preamplifiers are attached right behind the crystals within the telescope box.

Figure 3.8 shows the set of four Csl preamps and their housing which is divided

31

 

Figure 3.8: CsI preampliﬁers and their housing.

and shielded with copper into four small cells to eliminate the crosstalk between the
preamps. The closely packed four crystal setup in one telescope box is shown in
Figure 3.9. The two layers of 5x5 cm2 silicon detectors and their frame sit on top of
the box in the ﬁnal assembly with the ﬂexible signal cables running inside the metal
box next to the wall. The silicon preamps are placed in separate boxes because there

are 48 silicon channels per telescope.

32

,\.i.um. walnut—u. but.»

 

Figure 3.9: Csl packing in one telescope.

33

3.2.4 Energy Calibration

The fluorescent light emitted by the CsI(Tl) scintillator has two major components
of a fast (500 ns) and a slow (7 ,us) decay time ccmstants. Both components have
a relationship of light output and energy that is mass and charge dependent. This
property has been exploited to provide mass identification for light ions using pulse-
shape discrimination. This pulse-shape discrimination capability is not needed in the
AE-E type LASSA telescopes where silicon detectors are used as AE detectors and
Csl crystals as the stopping detectors. However, the mass dependence of pulse shape

remains important because of its influence on the energy calibration.

At low energy, the light output of a Csl crystal shows a non—linear response to the
deposited energy, especially for heavy ions. It also depends on the T1 doping of CsI

crystals.

To determine the energy calibration for different ions, the detectors were directly
exposed to low intensity (~ 103 particles per second) beams of different isotopes and
energies. These ions were obtained by fragmenting 2160 MeV 3“Ar and 960 MeV 160
primary beams from the NSCL K1200 cyclotron in the A1200 fragment separator.
The main advantage of this method is the a failability of a large number species of
particles that could be detected simultaneously (up to 52 isotopes were identiﬁed
in the case of the 36Ar fragmentation). Since particles are selected only by their
magnetic rigidity (8,0 = 1.841 Tm for the 36Ar beam and Bp = 1.295 Tm for the 16O
beam) one obtains a broad range of different isotopes and energies. The FVVHM of
the momentum widths for these particles were selected to be 0.5%. The atomic and
mass numbers as well as energies of the particles used to calibrate the Csl crystals
in the present work are listed in Table 3.1. Hydrogen and helium isotopes were also

calibrated by elastic scattering of E/A=3O MeV p-4He molecular beams on a Au

34

target and by 240 MeV direct 4He beam particles.

 

 

16O fragmentation E (MeV) 36Ar fragmentation E (MeV)
p 77.17
(I 39.78 (1 79.57
t 26.72 t 53.75
3He 105.00 3He 210.00
4He 79.99 4He 160.00
6He 53.64 6He 107.90
6Li 119.90 6Li 240.00
7Li 103.10 7Li 206.80
8Li 90.40 8Li 181.60
7Be 182.20 7Be 363.40
9Be 142.50 9Be 285.60
10Be 128.40 lOBe 257.90
10B 199.90 10B 400.00
11B 182.10 11B 364.90
128 335.40
11C 261.20 11C 521.60
12C 239.90 12C 480.00
13C 221.80 13C 444.40
1“C 413.70
14N 279.90 14N 560.00
15N 524.00
16N 492.40
150 340.80 150 680.70
16C) 640.00
170 603.70
180 571.30

 

 

 

 

 

 

Table 3.1: List of fragmentation products used in the energy calibration of the LASSA
CsI(Tl) crystals.

The energy calibration for each isotope was done following the mass and charge
dependence of the light output described in ref. [77], which in turn was based on
previous studies of the light emission of Csl-crystals and on semi-empirical model
proposed by Birks [78]. In this approach, the incident particle energy E is parameter-

ized as a function of the light output L, the charge Z, and the mass A of the particle,

35

‘dS
E(L, A, Z) = aAZ‘ZL + b(1+ (-AZ'~’)L1—"V’EZ (3.1)

where a, b, c and d are the ﬁtting parameters with values greater than zero. This
expression describes a linear term, dominating at high energies and a non—linear term
dominating at low energies. In Fig. 3.10, the solid and dashed lines represent the best
ﬁt of Eq. 3.1 to the experin'iental energy calibraticm data corresponding to different
carbon isotopes (A=11-14). The need for a mass dependence can be demonstrated by
examining the light output of the higher energy carbon isotopes. At high energy, the
light response is expected to be linear. Both the 11C points should lie in the linear
domain. However, a straight line joining the two llC isotopes does not pass through
the l'1igh-energy 12C, 13C, and 14C isotopes. A curve going through all points for
the 11‘14C would lead to a very large and unreasonable curvature compared to other
Csl(Tl) calibrations adopted elsewhere in the literature. Instead we adopt another
solution which assumes a mass dependent calibratitm (closely related to the quenching
effect)[77]. To conﬁrm this mass dependent ansatz that could allow the constructioin
of the full calibration curve for each isotope. several fragn'ientation beams of different

incident energies would be required.

For light charged particles with 233, the parameterization described in Eq. 3.1
did not accurately describe the detected energies. Compared to the observation of
Ref. [77], a less pronounced isotopic effect was observed for light ions. This may be
the result of the increased concentration of the activator element, T1, in the LASSA
Csl crystals compared to those studied in ref. [77]. We ﬁnd that the AZ 2 factor in
Eq. 3.1 overestimates the mass dependence. we therefore employ a modiﬁed function
of Eq. 3.1 with a weaker dependence on A for the particles with 233. The expression
is modiﬁed for each element. For Lithium (Z=3) particles, we change the ﬁrst term

36

 

600 , , ... - . . ,

 

l ' , ’4;

_ g f //..

_ I/ /.~/////

_ 12 C (7:). ,// ’

_ [3” / 1 N
f~4oo—- 14 cvy/ C a
l?) P C /'/‘* /

(Jig/I 1:) i
\2/ {27?1/ )C
200 — 12 / —
: ‘C/Tl
/ C

’ I

_/ 113 ,s.

I ‘\ l l l

 

 

O 1 "i I i l i i i 1
O 1000 2000

Light(a.u.)

Figure 3.10: Calil‘n‘ation curves for 11C, 12C, 13C and MC for the Csl (Tl) crystals
obtained using direct fragmentation beams listed in Table 3.1. The curves are the
best ﬁt according to Eq. 3.1.

of Eq. 3.1 and used,
E(L. A, Z) = (MEL + b(1+ (:A22)L1~dm (3.2)
For Helium (2:2) isotopes, we use
E(L, He) = 0L + bAC(1— a”) (3.3)

The variables a, b, c, and d in Eq. 3.1-3.3 are ﬁt parameters. There are sufﬁcient
data to reproduce with good accuracy the light-output response for all the isotopes
of the same element using Equations 3.1-3.3. Our ﬁtting procedure resulted in a

precision of the energy calibration better than 2% for isotopes from He to O.

37

As we have only limited calibration points for p, d and t. two calibration points

from each isotoi‘w, we adopt the simple linear function for 221 particles.

E(L,H) = 0L + b (3.4)

where a. and b are ﬁt 1')arameters. A linear CsI(Tl) response is consistent with that

observed for hydrogen isotopes by Handzy [79].

38

Chapter 4

Data Reduction and Analysis

4.1 Overview of the Analysis

The detection system is very complex with 432 silicon and 36 Csl channels from
LASSA plus hundreds of detectors from Miniball/l\-'Ii1‘1iwall. To reduce the complexity
in the analysis, we decided to use the MB/MW array to determine the charged particle
multiplicity as the impact parameter ﬁlter so that detailed energy calibration of the
MB / MW system could be avoided for this experiment. N evertl'ieless, we did separate
the light particles (LP, ZS?) from the heavy fragments (223) in the MB/ MW in

order to obtain separately the IMF multiplicity in addition to the total multiplicity.

Detailed calibrations and analysis on the LASSA, 011 the other hand, were con-
ducted for isotopically resolved charged particles. Figure 4.1 shows the flow chart of
data analysis for the LASSA. After the raw data from the ADC was decoded and all
the zero’s and pulser events were suppressed, presorted data tapes were generated for
further analysis. The whole analysis procedure depends on the pixelation subroutine
which associates speciﬁc x and y strips (BF and EB) with speciﬁc particles to locate
where they hit. This procedure also allows one to associate the particle with speciﬁc
data in the proper DE strip and the Csl crystal. The silicon energy calibration, one

of the two main analysis tasks, was done using the information obtained with the

39

LASSA Calibration Flow Chart

 

 

 

 

 

 

Raw Data Tapes:

 

 

[ Pedestal offset?
for Csl

 

 

 

[ Csl crosstalk ]
corrections

 

 

 

[Gain-Match for Csl]

 

 

Presort Tapes I

 

 

 

 

7
Csl PID] /

\

 

 

 

Final PID I

\

 

 

Pedestal Offset for Si

 

 

 

 

[ Si Pulser Calib (DE, EF, EB) ’

 

 

 

V

 

 

Flatten EB-EF]

 

 

 

 

[PﬁelatiogbLow channel calib

 

\

[Si uniformity]

correction

 

 

v
Si PID

 

 

 

 

 

 

 

 

 

EB associatiog

 

 

 

._/

 

DE, E energy, angle

 

 

 

 

 

l

[Csl Energy Calib]

 

  

\

Physics Tapes

[ (Energy, angle, PID, marker) ]

 

Figure 4.1: Flow chart of data analysis for the LASSA.

40

 

precision pulser (for more details see the previous chapter). The energy calibration
of Csl crystals, on the other hand. was conducted using the inforn'iation from a series
of beam calibration runs and from measurements of proton recoils during the clam-
taking runs. After making the corrections for the thickness variations in the silicon
detectors, particle identification (PID) gates were constructed. In the ﬁnal physics
tapes, the particle ID value, the total kinetic energy, and the emission angle for each
particle were recorded as well as a marker to label the circumstances when the parti-
cle \, ’as identiﬁed. In the following sections, several aspects of this analysis procedure

are discussed in greater detail.

4.2 Pixelation Technique

One of the most important advantages of Silicon-strip detectors is their position
sensitivity. Since the second layer of Silicon in the LASSA array is double—sided, we
can use this to obtain (x,y) pixelwise position resolution that tells us which pixel(3x3
mn12) a particle strikes. This is simple, in principle, if only one single particle hits
the silicon in one telescope during one event. When more than one particle hits the
silicon, i.e., in multiple-hit cases, however, one must use care in pairing up the signals
from the vertical and horizontal silicon strips in order to obtain the correct position
information for each particle. This pairing is done in the ”pixelation procedure”

described below.

\Vhen there are two particles or more hitting into one telescope, there can be
ambiguities in assigning the position of each particle, which need to be addressed.
First, one need to determine whether the particles can be identiﬁed or not. For
example, if two charged particles punch through two layers of silicon detectors and

stop in the same Csl crystal, then it is not feasible to obtain the correct particle ID

41

and energies for these particles. Second, the signals from DE, EF, EB and C‘sI need

to be associated properly with speciﬁc particles for the analyses to be correct.

When a particle is stopped in the ﬁrst layer of single-sided silicon detector its
proper ID (mass and charge) and position can not be determined; such particles
have to be disregarded. For the other cases (eg. particles stopped in the second
silicon or the C31), signals from the second double-sided silicon detector can provide
the best choice for the determination of particle multiplicity and position since the
perpendicular EF and EB strips granulate the detector into ﬁne pixels and have
basically the same energy signal for the same particle when the calibrations for both
EF and EB are accurate. We use this to pair up the signals of EF and EB and identify

the particles.

 

 

      

 

 

5.0
9 2-5 7.:
(l)
2.
LL 0
”a w:
53 -2.5
_5' 7. ‘ I L l . l 1 l
0 40 80 I20 40 80 120

EB (MeV) EB (MeV)

Figure 4.2: EB-EF ﬁne-tuning calibration for pixelation. On the left panel discrep-
ancy between EB and EF calibrated from the precision pulser is shown. After the
EB-EF flattening routine is applied. the good agreement between EB and EF is shown
on the right panel.

In order to make the best use of the pairing process. the energies of EF and EB
should be calibrated in the same way. However, the independent silicon calibration of

EF and EB strips can only offer a precision of 1% which results in larger deviations

42

of EF from EB signals at higher energies and consccpiently a difficulty in resolving
two hits with similar energies. A EF-EB flattening routine was developed to enforce

that the energy signals from both EF and EB for the same particle are the same.

Figure 4.2 shows how the spread of EB-EF difference changes before and after
the flattening routine. On the left panel the difference between EF and EB signals
before the flattening procedure is plotted. One can see one intense, nearly horizontal
contour, a line that slopes downward, and a grey background. The grey background
results when the EB and EF signals are from different 1')articles. The crooked, nearly
11(1)rizontal line corresponds to the correlation between EB and EF when both are
calibrated properly with the alpha-sources and pulser as described in Sect. 3.1.2.
The steeper line corresponds to the case that particles pass through a EB or EF
strip that has problems with its ralibration. After performing this procedure, one
straightens out the correlatitm between EF and EB as shown in the right panel of
Figure 4.2 and recovers these ill—calibrated silicon strips. A gate of $0.5 MeV (shown
by the dashed lines in Figure 4.2) is applied to select the correct pairing of EF and

EB signals.

For signals that can not be paired up with any EB or EF strip within the gate of
:t().5 MeV, we need to distinguish cases where two I.)articles hit in the same silicon
strip (double—hit, which occurs about 3% of the time) and cases where the charge
of one particle is split between two neighboring strips (split-hit, which occurs about
5% of the time). Speciﬁcally speaking, if the magnitude of one EB signal is close to
the sum of two neighboring EF signals, then with a high probability this case can
be identified as a single particle hitting into the 0.1mm wide interstrip gap with its
signal split into two adjacent channels; if those two EF channels are separated, then
one knows that one has the double-hit situation where two particles have gone into
the same EB strip.

43

Due to limited dynamic range of the electronics the signals of some particles,
esl')ecially heavy particles, are saturated in a small fraction (< 1%) of all the signals,
which need to be excluded in the data analysis since the pairing technique and energy
determination become obscure for these particles. Such is the case for the particles
hitting into strips that give no output signals because either their electronics has

failed or the detectors or the wire bounds to the strips are had.

4.3 CsI Crosstalk

During this experiment with the LASSA device, it was discovered that a small frac-
tion (typically < 2%) of the light emitted in one crystal could leak into an adjacent
crystal. After this experiment, a thin layer of alumiuized mylar (0.15 ing/cm2 mylar
+ 0.02 ing/cm2 aluminum) was inserted between adjacent. crystals. This additional
foil decoupled the detectors completely. Since this decoupling was achieved after
this dissertation experiment was performed, the present analysis must deal with and

remove the effects of the light leak as discussed below.

Four Csl crystals are closely packed together within one telescope as indicated by
the sketch at the top of Figure 4.3. Figure 4.3 also shows the correlations between
the light output observed in pairs of CsI crystals within one typical telescope (the
reversed crosstalk is similar and not shown). Note that the Y scale is blown up for
showing the light output correlation between CsI X and Y when a signal is observed
in Csl X. It is clearly seen that the crosstalk is about the level of 1-2% between two
directly neighboring crystals and basically vanishes between diagonal crystals. In the
cases of two diagonal crystals (two middle plots in the ﬁgure), small slightly tilted
lines close to Y axis are just the correlation of two crosstalk signals when one of their

common neighbors has been ﬁred.

44

 

Csl2 Csl4

 

Csl] Csl3

 

 

 

 

 

 
    
    

Csl] —>Csl2 Csl] —>Csl3

Csl] —>Csl4 Csl2—>Csl3

   

 

Figure 4.3: layout of CsI crystals within one telescope and typical crosstalk shown
for light leakage from 031 X to Csl Y in Telescope 2. Similar results for CsI Y —> GSI
X are not shown. Units are in raw channel numbers.

45

During the experiment. it was determined that a linear crosstalk correction could
be applied. Therefore the decoupling foil was not inserted between Csl crystals in
an effort to conserve. data taking time. This linear assumption is demonstrated by
Figure 4.3. In order to correct the light output in each crystal a conversion matrix is

constructml as follows:

L, 2 142+ 2((10-‘(10-(112— L?J)) — 2((11,(L3 — L3») (4.1)

J?“ J¢i
where the L' is the channel number of apparent light output; L is the channel number
of reconstructed light. output after correcting the crosstalk between CsI crystals; (1, is
deﬁned as the crosstalk matrix; Q is the gain factor due to the gain differences between
crystals; L0 is the channel number of the offset. The second term is the light gain
from other crystals while the last term stands for the light loss to adjacent crystals.
The matrix elements of a and L0 can be easily obtained from the slope and offset

parameters of the crosstalk lines in Figure 4.3.

After the reconstruction of Csl light output, calibration of individual Csl crystals

were carried out according to the procedure described in Sect.3.4

4.4 Uniformity Correction

Before starting the particle identiﬁcation (PID) through the DE-E technique, one
has to correct the non-uniformity of Silicon thicknesses. As mentioned in Sect. 3.1.1
Large area thin silicon detectors have a disadvantage of non—uniform thickness up
to 10%. Different energy losses are in general recorded for identical particles of the
same energy passing through different pixels. To correct for this thickness variation,
it becomes essential to apply the pixelation procedure described in Sect. 4.2 and use

it to map the thickness variation.

46

 

 

 

 

 

 

 

Figure 4.4: Non-uniformity of one typical 65 pm thick silicon detector.

47

 

 

 

 

 

 

 

 

 

 

 

m:_m> cmoE 9 mmmcxoé @259

Figure 4.5: Non-uniformity of one typical 500 pm thick silicon detector.

48

500

 

Li Before Correction Li After Correction

250 *-

 

 

 

 

 

 

12
c:
:3 O
8 , 6 7 8 9 6 7 8 9 10
i C Before Correction 0 After Correction
l
IOU—
l
0 l l l l l l l l 4 i
ll 12 13 14 15 lb ll l2 I3 14 15 It:
A

Figure 4.6: PID lines of Li and C isotopes before and after uniformity correction.

The relative thickness of silicon detectors was determined from the direct beam
calibration experiment ( which is also part of calibration for CSI crystals, see Sect.
3.2.4). The direct beam with ﬁxed high energy was applied to punch through the
silicon detectors. The correction matrix can then be constructed from the measured
energy loss in each pixel. Assuming a linear relation between energy loss and thickness
the thickness variation can be obtained according to the energy loss in each pixel.
Figure 4.4 shows an example of the non-uniformity of the 65 um silicon detectors.
On the other hand, the 500 um silicon detectors have much better uniformity as
illustrated in Figure 4.5, suggesting that the absolute thickness variations of both
types of detectors are comparable and the uniformity scales merely with the overall

thickness.

49

 

 

 

 

DE with thickness correction (MeV)

 

 

EB (MeV)

Figure 4.7: Isotopically resolved PID lines from H to O are shown for particles stopped
in Silicon Detectors

After both layers of the silicon detectors have their thickness variation determined,
the effects of thickness variations on particle identiﬁcation (PID) can be removed
so that PID gates can be imposed independent of hitting position. As showed in
Figure 4.6 the PID lines of Lithium and Carbon isotopes are blurred before uniformity
correction applied because the same particles don’t fall in the same PID line when
they go through different paths. On the right panel well deﬁned sharp PID lines are

achieved, indicating the importance of making an accurate uniformity correction.

50

 

EB with Thickness correction (MeV)

 

40 " "o 7““? 24
Ecsi (MeV for 4H9)

Figure 4.8: Isotopically resolved PID lines from H to O are shown for particles stopped
in C31 crystals

51

 

 

 

 

E He Particles stopped in Si
10‘1 =l-l
Li
Be 3
3
U 10 C N
a l o
102 U
10
|
, | | l l l l

 

 

 

500 1000 1500 2000 2500 3000
PID (Arbitrary Units)

Figure 4.9: One—dimensional PID lines from H to O are shown for particles stopped
in Si detectors

4.5 Isotope Resolutions and PID

Two sets of PIDs can be constructed for charged particles stopped either in the second
layer of silicon or in the Csl crystal. Figure 4.7 shows the DEcorr (energy loss in DE
with uniformity correction) vs. E (energy loss in Si2) for particles stopped in Si2.
And similar plot in Figure 4.8 is shown for particles stopped in Csl. Different PID
lines are well resolved in both plots for nuclei up to Oxygen and resolution could be
achieved for heavier particles if the data were not limited by statistics and detector

dynamic range.

However, these curved lines are not very convenient for identifying particles and
further analysis. One solution is to straighten them out and make one-dimensional
PID plot for easy-handling. To preserve the density of particle distribution on the

DE—E plots, the ﬂattened PID lines are drawn according to the intervals normalized

52

 

H He Particles stopped in Csl
105 g)
5W Li
H)4 g] Be B C
i N
% i03r
S O
l
102 l
l
to ;

 

l
1iii i ll 1

500 l 000 l 500 2000 2500 3000
PID (Arbitrary Units)

Figure 4.10: One—dimensional PID lines from H to O are shown for particles stopped
in Csl crystals

along the diagonal direction of the DE-E plots. Simple one-dimensional PID spectrum
can therefore be extracted as in Figures 4.9 and 4.10. Speciﬁcally for carbon isotopes
the mass resolution can be seen in Figure 4.11 for carbons stopped in Silicon detectors

(left panel) and in Csl crystals (right panel).

53

500
400
*3 300 »
3
O
O
200
100

0M1 l l

 

l

 

 

 

 

 

 

 

 

 

 

10 12 14 16
Mass

‘l

l

l

 

 

  

 

 

 

 

A .

10 12 14 16
Mass

   

1 250

1000

750

500

250

Figure 4.11: Mass resolution is illustrated from one-(liniensional PID plots of Carbon
isotopes. Left panel is for Carbons stopped in the second layer of Silicon detectors.
Right panel is for Carbons stopped in the Csl crystals.

54

Chapter 5

Particle Correlations

Historically intensity interferometry via particle correlations was ﬁrst studied in as-
trophysics. In the 1950’s, Hanbury—Brown and Twiss applied this technique (now
commonly called HBT) for measuring the diameters of distant astronomical objects
by examining photon correlations[58]. This idea has later been generalized to cor-
relations in nuclear physics involving various types of particles[61, 62]. The early
examples such as pion-pio1'1[80] and proton—proton[59] correlations involve identical
bosons and fermions, respectively. These have been widely used for studying the
properties of the sources of particles emitted in heavy ion reactions. Subsequently,
non-identical particle correlations such as d-alpha correlations and correlations in-
volving heavier fragments (up to Carbon) have also been studied; these studies have
provided insight regarding the freeze-out conditions for multifragmentation processes
after which interactions effectively cease [34, 81, 82]. Among such studies are the
determinations of the populations of excited states of emitted fragments [83]. Con-
siderable knowledge about the temperatures at the time of fragment emission has
been gained and the hypotheses of thermal equilibrium for the emission mechanism

has been tested in these studies[34].

In this chapter, we investigate the correlations of various charged particles emit-

55

ted during central 129Xe+m7Au collisions at E/ A = 50 MeV. The. data presented
here were measured with the LASSA array. We begin by discussing how to select
the impact parameters included in our analyses. Then we introduce the basic corre—
lation function obser tables. The correlation functions are interpreted via within the
Koonin-Pratt formalism [59, (50] and within the assumption of thermal equilibrium.
After discussing some of the strengths and weaknesses of the two approaches and
their interrelationship, we explore the data for central l29Xe+197Au collisions more
fully within the equilibrium approach. A determination of the breakup density for
these multifragmentation events is attempted and the imcertainties of this determi-
nation are addressed. A technique for extracting spin information of unbound states
from correlation functions is also presented, and applied to the determination of the
spin of the 0.774 MeV excited state of 8B (which is J21) and to some other nu-
clei as well. Three particle (:(iirrelations are also shown and discussed for the cases
of deuteron-alpl1a—alplia, 1')rot.(_)n—alpl121—alpha, alpha-alpha—alpha, and proton-proton-

alpha correlations.
5.1 Selecting the Impact Parameters

To select the events of central collisions, we utilized charged particle multiplicity
detected in both Miniball/l\rIi1'iiwall and LASSA as the impact 1;)arameter ﬁlter. This
can be achieved from a monotonic relation between total charged particle multiplicity

NC and the reduced impact parameter b[84],

r, _ Kawdivci/(Nc) ”‘2

i3: ._ .
bmar [39¢ d‘NrCY ( ‘IVC )

 

 

(5.1)

where Y(NC) is the number of events associated with charge particle multiplicity
NC. bnm is the mean impact parameter for the collisions with NC = 3. Figure 5.1

shows the above relation for this experiment, where b = 1 cor'res1')(_)nds to the glancing

56

 

 

 

 

 

Figure 5.1: reduced impact parameter as function of total charged particle multiplic-
ity.

collisions and b = 0 represents the head—011 collisions. In the following discussions,
we will choose the gate of 0 < b < 0.3 for the selection of central collisions, which

amounts to about 7% of total statistics.

Before going on to extract source information in the correlation functions, we show
the single particle energy spectra. Five panels in Figure 5.2 present the differential
multiplicities for p, d, t, 3He and 4He isotopes at angles of 17", 27°, 37", 47" and 57".
These nearly equally spaced angular distributions are at angles where the detection
efﬁciencies are large. The spectra at forward angles are more energetic, reflecting
that emission occurs from a frame (i.e. the center of mass) that is moving rapidly
with respect to the laboratory frame. Within the energy and angular range of the
correlation analysis described below, the detection coverage and efficiency is smoothly

well behaved.

57

dM/dE/dQ

dM/dE/dQ

 

 

 

 

 

 

   
 

 

 

 

 

    
 

 

 

 

 

 

 

 

 

 

 

 

l '3 V. L
X‘ ~ -2
-2 10 i .
1O . ’
o
C" t 17
U -3
17° % 10 T 270
-3 27° ,3, l
10 r
. 370 . Deutron 37°
Proton .. 470 10 -41. 470
57° r 570
-4 t
10 l r l l 4 1 l
0 25 50 75 100 125 15 0 100 150 200
Elab (MeV) Emb (MeV)
.2 i l
10 L , -3
i; ' ‘0 r
i L
-3 ’ L
10 g g"
E Q 10 '4 :70
i : 7
.4’ S t ‘V J 2
10 f U 370
: ) 47°
. -5 o
-5 _ 57
10 r 10 :
0 so 100 150 ”200 250 o 50 100 150 200 250 300
Elab (MeV) Em (MeV)
-2i i : 17°
10 r
3 .. 37° ° .
g 10 r i “ 47°
m : . .3 ‘
E ‘ 5 ‘
2 l ‘He
'0 -4
10 g
L
-5
10
O 50 100 150 200 250 300

Ebb (MeV)

Figure 5.2: Differential multiplicites of hydrogen and helium isotopes at angles of 17°
27°, 37°, 47° and 57°.

58

3

5.2 Two Particle Correlations and the Koonin-Pratt
Formalism

Experimentally the two particle correlation function may be defined as follows,

grams.) = C<1+ thiizizmmmii (5.2)

where Ylg is the two particle coincidence yield of a given pair of particles with their
individual mmnenta $2 and 7?.2 respectively, and the Y,(?,) are the single particle
yields for the two particles measured under the same impact parameter selection
but not in the same event. The summations on both sides of the equation run over
pairs of momenta $2 and 7?2 corresponding the same bin in relative momentum q.
The correlation function describes how the correlation between coincidence particles
measured in the same event differs from the underlying correlation dictated largely
by phase space and modelled by mixing the single particle distributions of particles
from two different events (the so—called event-mixed yield). The correlation constant
C is typically chosen to ensure that R(q) = 0 at large relative energies where the
correlations due to final state interactions and quantum statistics can be neglected.
If the yields are normalized to be the appropriate differential multiplicities, C will be

of order unity.

In the case of proton-proton and pion-pion correlations, most correlation function
analyses have compared data to the Koonin-Pratt equation [59, 60]. If the summation
in Eq. (5.2) does not involve strong constraints on the emission angles of particles
1 and 2, the appropriate comparison is to the angle-averaged Koonin-Pratt equation

[59, 60],

C(q) 5 1+ R(q) = 1+ 47r/d7'7‘2K(q,-7')S(r), (5.3)

59

where the source function S (7) is defined as the probability distribution for emit-
ting a pair of particles with relative distance 7‘ at the time when the second particle
is emitted; it should be normalized to unity if all the emission components are in—
cluded. The angle—averaged kernel, K(q, r), is obtained from the radial part of the

ant.isyn'm'ietrized two—proton relative wave function as follows [59, 60],
Ice, 7) = are»? — 1 (5.4)

where the wave function <I>q(r) describes the propagation of the pair from a separation

r out to the detector at infinity, where relative mmnentum q is reached.

Correlation functions have been analyzed using Eq. (5.3) for a variety of cor-
relations involving hydrogen and helium isotopes [85, 61, 62]. Figure 5.3 shows the
proton-proton correlation function measured in central 129Xe+w7Au collisions at E / A
= 50 MeV. The p—p correlation function, as a function of the relative momentum, is
measured over angles of 12° 3 6 _<_ 62" covered by LASSA without c.m. energy cut.
The maximum at relative momentum q 2 20 MeV/c is caused by the strongly attrac-
tive singlet S-wave proton—proton interaction; the minimum at q o: 0 MeV/c arises
from the interplay between antisymn'ietrization and the long range repulsive Coulomb

interaction [59] .

The p-p correlation function with different c.m. energy cuts is also analyzed as
shown in Fig. 5.4. The bottom panel shows the p—p correlation function for the same
angular gate and with a gate which selects low energy protons with 0 < Em, < 15
MeV. The middle panel demonstrates the corresponding correlation function with a
gate of 15 < Em < 30 MeV and the top panel presents the p-p correlation function
with a gate of Em, > 30 MeV. Consistent with previous oliiservations, the correlation
function is clearly more enhanced for protons with higher energies, corresponding to

the emission of more energetic protons from sources that are smaller or shorter-lived

60

 

 

 

 

1.4 . 0 Exp Data
[:l Imaging

1.2 ,

lyk“
g 1 r. 5g!“ KW‘_‘M.D;_M;_.
0.8 r
0.6 ..
O 20 4O 60 80 100
qre,(MeV/c)

Figure 5.3: Inclusive p—p correlation function in central 129Xe+197Au collisions at E / A
= 50 MeV is shown. The black points are experimental data and the grey line is the
best fit by the imaging technique. See table 5.1 for specific parameters.

(or both) than the sources that emit lower energy protons.

In studying the properties of the two proton emission sources, the simple assump-
tion of a single Gaussian source [85], with unit normalization, has been used but
could not reproduce all the features of the correlation functions. Here, we use the
less model-dependent imaging technique of refs. [86, 87, 88, 63] where p—p correla-
tion functions have been analyzed by numerically inverting the correlation function
in Eq. (5.3) to obtain the source. Such an inversion is not completely straightforward
because one has to take into account experimental factors (limited statistics, finite

widths of momentum bins, etc.) and the intrinsic resolution of the kernel K(q,r).

61

 

 

 

 

1'5 0 Exp Data
4r" Imaging
1177"" ‘Iit‘m...
1 ,___. ¥ ‘ W..+-A-+- 4
Ecm > 30 MeV
l l l l
1'5 0 Exp Data
Imaging
g 1 _ frﬁhiw‘g’t—o- HW-o—mﬁ— -—O-- —-O--—-—-0-
O 5 15 < Ecm < 30 MeV
1. 5 l l l l I I
' 0 Exp Data
— Gaussian
1 _ o
l O < Ecm < 15 MeV
O 5 l l L l

 

 

 

0 20 4O 6O 80 1 00
are, (MeV/c)

Figure 5.4: The p-p correlation functions are shown for three c.m. energy gates of
0 < Em < 15 MeV, 15 < Em < 30 MeV, and EC", > 30 MeV. For the middle
and high energy gates, the image technique is used. The simple Gaussian source
parametrization in Eq. 5.6 is applied for the low energy gate. See table 5.1 for
specific parameters.

62

Small fluctuations in the data, even if well within statistical or systen‘mtic errors. can
lead to large changes in the imaged source function. On the other hand, successful

inversion of the correlation function has the virtue of being model independent.

The numerical inversion of the correlation in Eq. (5.3) was achieved via. the
optimization algorithm of refs. [87, 88, 63]. Using the imaging technique, we analyzed
the data shown in Fig. 5.3 by decomposing the source in a superposition of six b—spline

polynomials of the 3rd order over the interval 0 g 7' g 20 fm [87, 88, 63].

 

 

 

 

-3
x10
0.2
0.15 7 Imaging
'E
E 0.1
(71’
0.05
O 1 l i l l L r 72' l
0 2 4 6 8 10 12 14

ram)

Figure 5.5: The imaged source function is shown by inverting the p—p correlation in
Fig. 5.3. See table 5.1 for the extracted source radius.

The thick gray line in Fig. 5.3 is the best fit of the imaging approach to the
experimental data (black points) of the two proton correlation. The imaged source

function is shown by the thick grey line in Fig. 5.5 where the width of the line

63

indicates the uncertainty in the inversion. Since. the correlatirm function is difficult
to measure at low relative momentum (say. (1 < 10 MeV/c), the imaging technique
could not reliably produce the tail of the source at large radii (say, 7' > 13 fm) where
the dominant sources are delayed emissions of protons, such as from secondary decay.
Thus the source is only speciﬁed for 7' < 13 fm. Better data of the correlation function
at small relative momentm‘n could provide better estimation 011 the source at larger
radii where the source shape is mainly affected by the slow emissions of protons. The
extracted rms radius of the source is rm, 2 7.97 fm for fast protons that contribute
to r < 13 f m (see Table 5.1). However, the fraction of protons from these slow sources
can be obtained via the integration of the imaged source [63]. Due to the omission
of those long—lived components in the imaging of the source, the integration of the

source is no longer unity [63]
f2 E .\ = / S('r)d3r, (5.5)

where f is the fraction of protons emitted from fast sources while l-f stands for the
fraction of protons emitted from slow sources like secondary decay which can not be
imaged in the long tail of the source profile. By integrating out the source shown in
Fig. 5.5, we obtain f=0.48, which indicates that the emitted protons with a slow time
scale (or in other words, 7* > 13 fm in the source profile) amount to about half of all

the protons.

The two proton emission sources are also analyzed for the p—p correlation function
with different c.m. energy gates in Fig. 5.4. For the middle and high energy gates, the
extracted source functions are shown in the middle and top panels of Fig. 5.6 using the
imaging technique. Unfortunately, the higher energy gate has no data at low relative
momentum and the data at lowest point in relative momentum is measured with poor
statistics. Thus the inversion provides no reliable information at r > 7 frn. We have

64

x10

 

0.6
0.4
0.2

Imaging
Ecm > 30 MeV

1 ii. i "1' . ' l

 

S(r) (fm'3)

0.4 —
0.2

Imaging
15 < Ecm < 30 MeV

 

0.4 ~
0.2 _

 

Gaussian
0 < Ecm < 15 MeV

l l l l

 

Figure 5.6: The source functions are shown for the p-p correlation functions in Fig.
5.4 for three our. energy gates of 0 < Em, < 15 MeV, 15 < EC", < 30 MeV, and
EC", > 30 MeV. For the middle and high energy gates, the image technique is used.
The simple Gaussian source parametrization in Eq. 5.6 is applied for the low energy

r (tm)

gate. See table 5.1 for extracted source radii.

4 6 81012

 

 

 

 

p-p Inclusive 0 < Em" < 15 15 < EM,m < 30 Elm" > 30
Correlation MeV MeV MeV
/\ 0.22 0.93 0.50 0.14
X 0.11 0.22 0.16 0.14
r.,.,,,,, 7.97 11.26 10.95 4.69
p/[),, 0.52 0.18 0.20 2.52

 

 

 

 

 

 

Table 5.1: List of quantities are shown for the pp correlations (Figs. 5.3-5.6) for
inclusive and three different c.m. energy gates. The A that relevant to the fraction
of protons contributed to the ﬁtted source and the rms radius rm, of the two proton
source are extracted from the fitted source distribution. The X values are calculated
from the integral of the source functions for r i 7 fm in Eq. 5.5. The density p/po is
estimated by taking into account collective effects in Eq. 5.9.

calculated rm, by integrating the source for the highest energy gate over 1" < 7 fm
and the source for the medium energy gate over 7" < 13 fm and obtained sources
sizes of rm, :2 10.95, 4.69 fm and corresponding fast. proton fractions A = 050,014
for the middle and high energy bins, respectively (see Table 5.1). The corresponding
reconstructed correlation functions are shown by the grey lines in the middle and top
panels of Fig. 5.4. Because the correlation data for the high energy gate (top panel
in Fig. 5.4) has no points below q < 10 MeV/c one cannot say what are the source
contributions at 7‘ > 7 fm. If the source function obtained from a better measurement
that includes data below q < 10 MeV/c reveals a tail on the source function at larger
radii r > 7 fm similar to that for the middle energy gate, the extracted /\ value for
the high energy gate will be much higher. On the other hand, if the tail of the source
function for the middle energy gate are cut off, its A value will be much lower, which

is shown by column X in Table 5.1 for the integration of the source functions in Eq.

5.5 for r < 7 fm.

As for the lowest energy gate, however, we have not succeeded in stably imaging
the data because the data do not conform to the shape expected for a correlation

function of the Koonin-Pratt type. Instead we have fitted the correlation function

66

data with a non—unity Gaussian source of the form,

/\ 7‘2
‘5"1 :~"" ' . 1‘4 —_ 1 [7.7
(r) (2,),,,.,.8u<p 2,3 . (a6)

where r0 is the source radius paran'ieter and A corrects for the fact that some of the
protons are emitted over long time scales. The Gaussian source function fit is shown
in the lowest panel of Fig. 5.6 and the corresponding correlation function is shown by
the solid line in the bottom panel of Fig. 5.4. The quality of the ﬁt is poor, especially
at. low relative momentum where there is an enhancement in the yield whose origin
we do not understand. Whether this is some effect caused by secondary decay of some
heavier isotope, we cannot say, but we have analyzed the correlation function under

that assumption by ﬁtting only the data at q >20 MeV/c.

In general, the extracted sources for lower energy protons display larger source
radii, which is consistent with emission from a source that has expanded and may be
disassembling over a time frame that is somewhat longer than is the characteristic of
the higher energy protons. With the uncertainties in the present correlation functions,
we can not image the sources for these long-lived decays. We do, however, have
indirect sensitivity to those decays from the fraction of fast protons f = A extracted
from the integral of the source in Eq. 5.5, which indicates there is a signiﬁcant

contribution from secondary decay to the proton emission.

Before relating the proton source functions to the density of the multifragment-
ing source, we investigate d-alpha correlations, which can also be addressed by the
more conventional correlation function techniques involving the Koonin-Pratt equa-
tion. At present, d-alpha correlations have not been successfully inverted with the
imaging technique. Instead, we illustrate the ﬁt here with the simple Gaussian source

parametrization described previously.

67

 

 

 

 

 

 

3
09 F4..__.J_._‘_ __l*-_l_-4
7 so 90 100
[I
+ 2 -
,_
1 a
O
. 0 Exp Data
. . ' —- Gaussian w/o correction
i l l 4 l l l

 

 

0
0 20 40 60 80 100120140
qrel(MeV/c)

Figure 5.7: The inclusive d—a correlation function for the central collision gate is
shown as a function of relative momentum by the black points. The solid line is the
ﬁt by the simple Gaussian source parametrization in Eq. 5.6 without corrections for
collective motion. The blown-up in the top right shows the poor quality of the ﬁt to
the second peak of the d-a correlation.

Figure 5.7 shows the inclusive d-alpha correlation function as a function of relative
momentum. The solid line in Fig. 5.7 shows the best ﬁt to the experimental data with
Gaussian source without making any correction for collective motion. The Coulomb
suppression of the correlation function at small relative momentum (Ire! < 30 MeV/c
can not be fit with one single Gaussian source assumption like the case of the p—p
correlation due to the existence of slow sources like secondary decay. Therefore the
ﬁt is performed at qrer > 30 MeV/c and describe the magnitude and shape of the ﬁrst

excited state of 6Li at 2.186 MeV well. However, the magnitude of the second peak at

68

4.31 MeV (see the blown-up in Fig. 5.7) is overpredicted and its shape is incorrect, a
problem that has been noted by previous authors. Before interpreting this correlation
data further, some discussion of the possible effects due to collective motion is needed
because collective motion may be responsible for this difficulty. After this discussion,

we will return to the interpretation of both d-a and p—p correlation functions.

5.3 Influence of Collective Motion

Collective motion is characterized by an average velocity ﬁeld that assumes values
that are deﬁned by time and by the location within the colliding system. Examples
of collective motion include the collective radial expansion of an excited system under
the influence of pressure due to compression and to the repulsive Coulomb interaction,
or collective ”rotation” induced by a collision that is at non-zero impact parameter.
Another form of collective flow, directed transverse flow [89], is small at E/A=50 MeV
due to the balancing of repulsive and attractive momentum transfers at the ”balance

7

energy’ and will not be discussed further here.

 

Figure 5.8: A source with only thermal motion (open arrows) is shown on the left.
The collective velocity ﬁeld (solid arrows) is drawn on the right and results in a grey
area of the source where emission into the right half plane is improbable.

Collective motion has several impacts on the behavior of correlation functions.

First, it effectively decreases the ”source size”, i.e., size of the region that can con-

69

tribute to the emission at the measured angles and energies. The origin of this effect.
is simply illustrated in Fig. 5.8. O11 the left side, we consider the scenario where
there is no collective velocity and on the right side, we include a collective velocity
ﬁeld. In this drawing, the collective velocity ﬁeld is drawn by the solid arrows and the
random, i.e., thermal velocities are drawn in open arrows. The grey area indicates
the region of the source where emission into the right half plane is improbable. In
the long mean free path scenario, the source without collective motion corresponds
to the total volume of the system. With collective motion, however, emission to the
right is precluded in the grey region where the collective velocity has a component
to the left that exceeds the typical random velocity, which if thermal in origin would

. _ ‘)
decrease With mass as 1) oc (m) 1/ “.

To estimate the influence of collective motion on the source sizes, we simulate the
interplay between the collective and random source velocities as follows. We assume
a spherical source of A,,,,,,.(_.,. = 0.8(Ap,.0jm,rC + Amget) = 260 nucleons, of which 106
are protons and that the source moves with the velocity of the center of mass. Within

this spherical source, the collective velocity ﬁeld at 7? is given by

r ‘ 1

+ ’l’tJnaxQ ® 1 (5-7)

$201K?) : Unmax

:Ulrl
:Ulri

where vmm is the radial velocity of the surface expansion of the system at freeze-
out, vtmax is the tangential collective velocity at the surface, R is the radius of the
expanded source, and CC} deﬁnes the direction of the rotational velocity ﬁeld. The
orientation of (I) is approximately perpendicular to the beam axis, but the azimuthal
orientation with respect to LASSA is unknown. Therefore, the azimuthal orientation
is varied randomly assuming the relative azimuthal orientation between LASSA and
(D is isotropic.

We consider ﬁrst the fraction of the source that can contribute to particle emission

70

into LASSA. Emission was simulated via Monte Carlo for the interplay of this collec-
tive velocity ﬁeld with random or thermal motion [90]. In this simulation, the thermal
velocity was calculated by assuming m, = /3T/m for simplicity where m is the mass
of the particle and the temperature T -_— 4 MeV is roughly ('tonsistent with isotope and
excited state temperatures in this bombarding energy domain. we choose a radial
expansion velocity consistent with a radial kinetic energy of expansion of 2 MeV (and
it is also affected by the additional Coulomb energy gained by continuing to acceler-
ate the particles by the Coulomb ﬁeld of the source). It is rather approximate but
consistent with the radial expansion velocities reported in the literature. In addition,
we choose an tangential velocity of chum = 0.16 c, which corresponds to a surface
tangential velocity consistent with the initial velocities of the projectile and target.
We use the above collective flow 1.)arameters for the ” maximum” flow discussed below

and reduce them to see the dependence on the collective motion.

To demonstrate the mass dependence of the source size decreased by the collective
motion, two particle sources from p-p, d-d, up to 20;\le-2U1\Te are studied assuming 80%
in velocity (or 64% in energy) of the ”maximum" collective motion discussed above. In
essence, the rms radius for a two partice source is calculated as 7‘3,” = ((72 — 791?),
where 7’, is the location of the i” particle at the time of emission and the average
is over particles that make it into the LASSA array. If the two particles are emitted
from the same source, the rms radius for the two particle source is \/2 times the
rms radius for a single particle source. The velocity ﬁeld in Eq. 5.7 is constructed
so as to make the ratio of the rms radii to the overall system radius invariant with
respect to the system size. The ratio fwu, of the rms radius with collective motion
divided by the rms radius without collective motion for the two particle sources, is

plotted as the solid line in Fig. 5.9 as a function of the mass number while the ratio

without collective motion is denoted by the dashed line. The solid line was obtained

71

 

1.25 "

0.75 *-

fcoIl

0.5 '-

0.25 *‘

 

 

l l l

0 5 10 15 20
Mass Number

 

C

Figure 5.9: The source reduction factor, f coir, obtained from the ratio of the rms radius
to the system radius for the two particle sources with collective motion, are shown by
the solid line as a function of the mass number. The ratio without collective motion is
denoted by the dashed line. The solid line is calculated for a uniform spherical system
with a radius of 11 fm while the three symbols denote the corresponding calculations
for a system with a radius of 7 fm.

from calculations for a uniform spherical system with a radius of 11 fm. The three

symbols denote the corresponding calculations for a system with a radius of 7 fm;

this demonstrates the independence of this ratio with respect to the system size.

The corresponding reduction in the fraction of the participating source, fgou, es-
sentially dictates the fraction of the participating source, i.e. the mass fraction,
independent of the actual source volume. It depends on 1.3mm, Laymax, and the ran-
dom velocity v”, = y/3T/ m, but not on the source radius R. Clearly, the collective
velocity ﬁeld decreases the mass of the source signiﬁcantly, for example, by about a
factor of 2 for the p-p correlation. In Fig. 5.9 no cuts on relative momentum are

applied since we want to Show the mass dependence of the source reduction factor.

72

In reality, we need to obtain the ff

,,(

,,, factor for a gate of relative momentum where
we study the correlz-ition function. The corresponding reduction of the d-a source
is also calculated for the purposes of density extraction from the d-a correlation to
be discussed later in this section. This calculation yields a mass reduction factor of

about 5 for a relative momentum gate of 0-150 MeV/c.

In addition, the collective motion causes the particles to be more frequently emit-
ted in the plane perpendicular to of). This leads to a preference for the relative
azimuthal angle of the two particles to be 0° or 180” [91] and also modiﬁes the shapes
and magnitudes of broad correlation function peaks. In such cases, the two particles
are emitted from nearly the same point and experience nearly the same collective
velocity ﬁeld. On the other hand, the mixed event background includes particles that
are from two different events in which the collective velocity ﬁelds are completely dif-
ferent. Thus the relative velocity distribution for the resonant contribution is dictated
essentially by thermal motion and consequently narrower than the relative velocity
distribution for the non-resonant and mixed event backgrounds, for which the relative

velocity reflects differences in the collective velocities as well.

Fig. 5.10 shows a simulation of that effect for the d-alpha correlation. The solid
line shows the relative energy distril'nition between two particles that lie within a
distance of 2 fm from each other as they would be if they were both emitted from a
particle unbound fragment formed at freezeout. This relative energy distribution is
essentially consistent with an exponential decline of the form oc \/E,-€[ exp (—-E,.,_,1/T)
given by the thermal motion. The dashed line shows the corresponding relative energy
distril‘mticm for pairs of particles that are chosen randomly throughout the system as
they would be for the non-resonant background. Here the relative energy (.listribution
is much wider consistent with signiﬁcant contributions from the difference between the

collective velocity ﬁeld at the two emission points. The dotted curve in the same ﬁgure

73

 

Yield

 

 

 

 

ll 11 l in!
0 5101520253035

Erel (MeV)

Figure 5.10: The solid line shows a simulation of the resonant distribution as a func-
tion of relative energy for the d—alpha correlation. The dashed line denotes the sim—
ulation for non-resonant background while the dotted line demonstrates the relative
energy distribution for mixed event background. All the distributions are normalized
to one for comparisons.

shows the additional broadening that occurs in the event mixing in the denominator
of the correlation function where the two particles originate from different events
with different orientations of the reaction plane, i.e., different azimuthal orientations
for (2’. For simplicity, we could describe the distribution of relative energies here by

CC Ere! exp (—Erel/Tnii;r)-

The different broadening in these distributions has important consequences for the
correlation function, where the two distributions are essentially divided by each other.

The division of the resonant yield by the mixed event distribution is illustrated in the

74

  
   
 
   

 

 

      

 

 

 

10 100% collective motion
Teff = 6 MeV
1
-1
1 0 g
l J l l l l l
10 80% collective motion
E Tait = 7 MeV
t i
g -1 E
>7 10
r l l l l l l
10 E no collective motion
1 ; - - -
-1 E
1 0 E‘
f l

 

0 5101520253035
Ere,(MeV)

Figure 5.11: The ratios of the resonant yield over the mixed event distribution for
the d-a correlation are shown in three panels for maximum, 80% and zero of the

collective motion, respectively. The corresponding values of T6” are extracted from
the ﬁts (dashed lines).

75

three panels of Fig. 5.11 for maximum. 80% and zero of the collective motion discussed
above, respectively. The dashed lines in the ﬁgure are the best ﬁts of an exponential
function o< exp(—E,.,.,/T,,ff), which have values of T...” = 6,7 MeV for maximum
and 8000 of collective fiow, res1.)ectively; the line is fiat in the bottom panel without
collective motion. This simulation indicates that one should expect the resonant
peaks in the correlation function to decline exponentially with the relative energy
or relative momentum according to ac exp(—Er€r/T + Era/Tina) = exp(—E,er/Teff).
The non-resonant background is also expected to decline somewhat, but to a much

reduced extent.

Taking the simulations in Fig. 5.11 into account, we return to the d—a corre-
lation in Fig. 5.7. Consistent with the simulations, we ﬁnd that the non-resonant
background is relatively well represented in the original calculations. However, the
resonant yield can be much better described by multiplying the resonant contribution
by a factor ex1)(—E,.,.1/'T,,ff) with Te” 2 6.5 MeV. This revised correlation function
is given by the solid line in Fig. 5.12, which ﬁts the second peak of the d-or correla-
tion function at gm 2 84 MeV/c (the blown-up) much better than that in Fig. 5.7.
The extracted source radius is To = 3.3 ﬁn and the rms value and A can be seen in
Table 5.2. Using this empirical correction, we have explored the sensitivity of the d-a
correlation function in general for the extraction of information about the size of the
emitting source using a Gaussian source of the form discussed in Eq. 5.6. Consistent
with conclusions recently obtained for d-a correlations on other systems [92], we ﬁnd
that the width of the second peak in the correlation function at qrei = 84 MeV/c
is very sensitive to the source size. This is illustrated in Fig. 5.12 where two other
ﬁts are shown by the dashed and dotted lines with the assumption of source radii
of To = 2.0 ﬁn and 5.0 frn, respectively. This situation is similar to that. for the p—p

imaging analysis where the source radius information is primarily obtained from the

76

 

 

 

 

 

 

  

 

 

 

 

1.1 r
1
3 _
0.9_ l __ I_.____L l .4
70 80 90 100
CC
+ 2 r
T.
; 0 Exp Data
. . —— Gaussian w/ correction
. (Te,,=6.5MeV)
[ I l l L i l

 

O .
0 20 40 60 80 100120140
are, (MeV/c)

Figure 5.12: The inclusive d-a correlation function for the central collision gate is
shown as a function of relative momentum by the black points. The solid line is the
best ﬁt by the simple Gaussian source parametrization in Eq. 5.6 with Te” corrections
for collective motion. The blown-up in the top right shows the good reproduction of
the second peak of the d—a correlation after Te” corrections. The extracted source
radius is r0 = 3.3 fm (see Table 5.2 for the rms value). The dashed and dotted
lines are the ﬁts assuming that the source radius is To = 2.0 ﬁn and r0 = 5.0 fm,
respectively.

77

width of the p—p correlation function, not its height[63].

 

 

 

 

 

 

d-a Inclusive 0 < Edm, < 20 20 < Eacm < 40 detn, > 40
Correlation 0 < E01,", < 25 25 < Ema” < 45 Ema" > 45
MeV MeV MeV
A 0.11 0.14 0.13 0.13
rm, 5.72 7.35 5.58 3.01
p/po 0.54 0.37 0.59 3.74

 

 

 

Table 5.2: List of quantities are shown for the d-(r correlations (Figs. 5.12-5.13) for
inclusive data and three different c.m. energy gates. The A that relevant to the
fraction of the involved particles contributed to the ﬁtted source and the rms radius
rm, of the two proton source are extracted from the ﬁtted source distribution. The
density p/po is estimated by taking into account collective effects in Eq. 5.9.
Different energy gates for the d-alpha correlation are also studied as shown in Fig.
5.13. By varying the parameter Teff from collective effects, we can ﬁt the correlation
functions for the high, middle, and low energy gates, especially for the second peak at
q = 84 MeV/c which is blown up in the top right. corner of the corresponding panel.
While the ﬁrst peak can be ﬁtted somewhat better by adjusting Teff in this simple
one Gaussian source parametrization and the Coulomb suppression at low relative
momentum can be ﬁtted better by adding a tail to the gaussian source, we will defer

further discussion of the ﬁrst peak until the next. section and devote our attentions

here to achieving the best ﬁt to the second peak.

The collective motion effects are more evident for the lower energy gate where the
Ten value is lower than for the highest energy gate where the effects virtually vanish
and no Teff correction is needed. The extracted rms radii and A values for the three
energy gates are in Table 5.2. Same as the case in the p-p correlation, the sources
extracted from the d—a correlation functions have smaller radii for the higher energy

gates.

In principle, sources extracted by inverting p—p correlation functions may also be

influenced by collective motion. Fig. 5.14 shows the corresponding results from simu-

78

 

 

 

 

 

 

 

 

lac,cm > 40MeV
4 ”Emu" > 45MeV
(Teﬂzoo)
O P‘ l l n} i l l
20 < Edm < 40MeV 1.1 '
4 25 < Eum < 45MeV 1 A
[E ne,=6.6MeV) L _ J“ J _.
‘— 2 L-
O
O "P ". l l l l 1 i
0 < Ed.cm < 20MeV 1.1 i
4 #0 ( Emcm < 25MeV 1 W
=4.6MeV
(Ten ) . 0.9] m l 1 l
2 n 70 80 90 100
.0
0 "f i L l l l i

 

0 20 40 6 80 100120140
qre,(MeV/c)

Figure 5.13: The d-a correlation fimctions are shown for three our. energy gates
of 0 < Edy", < 20 MeV and 0 < EM", < 25 MeV, 20 < Ed,cm < 40 MeV and
25 < Ema", < 45 MeV, and Edm, > 40 MeV and Emcm > 45 MeV. Fits by the simple
Gaussian source parametrization in Eq. 5.6 are shown by the solid lines after Tc”
corrections for collective motion. See table 5.1 for extracted source radii and A values.

79

 

    

 

 

 

 

 

 

 

 

 

 

10 100% collective motion
. T9,, = 8.7 MeV
1 r
-1
1 0
h l L l l
10 6:: 50% collective motion
'5 : A T9,, = 29 MeV
: i " M
g -1 E
>-5 1 O
l l 1 l
10 S no collective motion
1 r v
-1 E
1 0 g
7 1 l l l
0 5 1 0 1 5 20

Erel (MeV)

Figure 5.14: The ratios of the resonant yield over the mixed event distribution for
the p—p correlation are shown in three panels for maximum, 50% and zero of the
collective motion, respectively. The corresponding values of Teff are extracted from
the ﬁts (dashed lines).

80

lations of the infliu—mce of collective motion on the p-p correlation functions assuming
three different sets of collective flow parameters. The corresponding values of the
effective temperature T,“ are much larger than the values extracted from the do
correlation, which indicates this effect is much smaller for the p-p correlation. In
addition, the peak of the p-p correlation function falls within the very low energy
region of EM < 0.5 MeV, unlike the case of the second peak in the do correlation
function which is at EM 2 2.836 MeV. Therefore, the (,listortion effects from Te];
should be much smaller for the p—p correlation than those for the d-a correlation. In

the interest of brevity, we neglect them in the following analyses.

Using these analyses of the p—p and d-a correlations, we attempt to place some
constraints on the densities of the system at the time of particle emission. Because
we are using source radii determined from the widths of the p—p correlation function
and the width of the second peak in the d—a' correlation, we are insensitive to how
collective motion reduces the fraction of deuterons, alphas or protons that come to the
LASSA and contribute to the mixed event background to the correlation function. On
the other hand, we are sensitive to how the imaged region may be effectivr—ély shrunk
by the collective motion and we must make some correction for that effect. For
these estimates, we use the factor f2,” determined earlier using T,” = 7 MeV, which
is close to the value obtained for most inclusive correlation functions, to determine
the fraction of the system that is contributing to these emissions. Take the p—p
correlation function shown in Fig. 5.3, for example, the rms source radius rm, =
7.97. The simulation in Fig. 5.9 suggests that the p—p correlation function, actually
originates from a fraction of the source. This fraction is given by f3,“ where f mu = 0.8.
We assume that the mass contained in the region sampled by the pp correlation
function to be f2,” - A,,,...f,.ag,,,e,,t :2 (0.8)3 - 1419”,",ng where Apnqmgmem is the mass

of the system after the end of pre-equilibrium emission. For this estimate. we take

81

Aprefmgment a: 260 corresponding to 80% of the total system.

To obtain an averaged density. we need to find the volume lr’_,.(,,,,.a. of the breakup

source measured by the two 1;)article source function. It. is impm‘tant to note that the
rms radius of the two particle source function is obtained by adding the rms radii of
single particle source functions in quadrature, i.e.. rm, 2 \/21'1. For an equivalent
sharp uniform spherical source with a radius of R, the rms radius of the two proton

source has a relation of rm, 2 ,/(6/5)R and consequently we can obtain the source

 

 

 

volume,
4 4 3
1/source 2 §R3 : i: ' rrms - (5.8)
Therefore, the density can be written as,
P/PO = 30" ' A’ (5.9)

7r 5 . i
‘3—[ afr'ms] 100

where the saturation (ilensity [)0 = 0.16 nucleons/fin3 is assumed. Densities obtained
for the all energy gates for the p—p correlations are. given in Table 5.1 by using Eq.
5.9. The extracted value of the density for the low energy gate is not as reliable as
those obtained by the imaging technique since the Gaussian source approach does not
reproduce the correlation for the low energy gate in Fig 5.4. However, one does see
a trend that the system is at a higher density when higher energy protons are being
used to construct the correlaticm. The density value from the high energy gate is so
high (/)/[)0 = 2.52) that it seems the correlation either ’sees’ the very early source
(compressed and before being expanded) since fast protons tend to emit earlier, or it.
just ’sees’ a smaller fraction of the source. The latter case would apply if the source
in the early stage is limited to the surface due to the short mean free path in the

dense interior.

82

For the d-a correlation, we l'l‘dV’C a different collective correction factor fmu = 0.58,
which is obtained by Simulating d—a correlation sources as discussed previously at
low relative momentum where we ﬁt the data. Values for the density obtained from
the d—a correlation functions are given in Table 5.2 by using Eq. 5.9. Since the
simple Gaussian source paranretrization is used in this case, it can not reproduce all
the features of the d-a correlation fimctions and the presumed Gaussian shape for
the sources may be rather schematic. Consequently. the extracted source radii and
densities may not. be as reliable, as those obtained from the p-p correlation functions
by the imaging technique. After taking into account the effects of collective motion
properly, however, we obtain similar values of density (0.52 and 0.54, respectively)
from both the inclusive p-p and do correlaticm functions. Moreover, a similar trend of
the energy dependence of the density is observed. which indicates that higher energy
particles are emitted at denser regions or earlier times. we note that it can be argued
that more accurate values for the density could be obtained if fa,” were calculated
for each of these energy gates via simulations tuned to reproduce the values for T6”
observed for each gate. Since lower values of Toff will result in smaller fwu, the values

for density in this case woulcl be somewhat. lower.

Clearly, there are large uncertainties in this approach. These uncertainties stem
from several different sources. First concerns whether the corrections for reduction of
the source sizes from the Monte Carlo simulations are of the correct magnitude. We
believe that while they are qualitatixr'ely correct, they could be off in their magnitudes
somewhat and we are considering ways to try to estimate this uncertainty more quan-
titatively. Secondly, we have some concerns about the assumption of an inﬁnite mean
free path that this density estimate employs. Clearly, this assumption is more correct
if the estimated density is small. When it is large, however, transport theoretical cal-

culations indicate a. sensitivity of two particle correlation fimctions to the iii-medium

83

cross section. Thus the estimated values for density for the p-p correlation functions
in Table 5.1 and the d-a correlation function in Table 5.2 are somewhat uncertain
on this account and if there are corrections needed, it. would be in the direction of

reducing the density.

5.4 Equilibrium Correlation Functions

One of the factors limiting the extension of the Koonin—Pratt equation to heavier
particles is the care needed to construct the Kernel K(q,r). Essentially, one must
search for a set of attractix-e nuclear potentials that can reproduce the experimental
phase shifts. Right now, we have only the necessary 1‘)otentials for the p—p and d-a
correlation functions. To rapidly extend the correlation function to heavier particles
and to facilitate the comparison to statistical models, we develop here a. f(,)rn'ialism

for calculating the correlation function within equilibrium theory.

The starting point for this development is the coi’isideration of elements needed
for the equilibrium description. First, one needs to have a con'ipact method for
incorporating both the long range Coulomb and short range nuclear interactions.
Second, one must address the volume that is occupied by other particles. We choose
to address the second issue by invoking the excluded volume approximation. This
essentially amounts to counting as particles only those that are isolated, a procedure

that is consistent with most equilibrium multifragmentation approaches [9, 10].

Equilibrium correlation function expressions are derived by considering how the
two particle phase space is modiﬁed by interactions. For simrﬂicity, we consider the
simpliﬁed geometry wherein the pair of spinless particles with charges 21 and Zg is
with its center of mass at the center of a volume V. To calculate how the phase space

of relative motion is modiﬁed by the Coulomb interaction, we follow semi-classical

84

theory which states that the phase space density is given as a function of the relative

spatial separation 7’ and relative momentum 77 by

(In. _ 1 (L, 10)
JWWWL—W’ 0~

where TKL is the local momentum given in terms of q, the reduced mass it 2: Ml Alg/(il[1+

ill-3) and the Coulomb potential by

 

2/I.Zl 2202

T

‘11. = (.12 (5.11)

Reexpressing Eq. 5.10 in terms of the momentum 7 at. large distances where the

Coulomb interaction can be neglected, we have

 

(In (In, (19% 1 2 [Z 2 e73
.. .nz—e LHL;%< an)
(1371!qu (Pg [1“ H]-

 

m lCi'oulomb :

If the above equation is integrated over a volume V and divided by the corresponding
integral of the relative phase space density of two free particles, (In/(1371" = V/h3, an

expression for the Coulomb correlation function 1 + Ram, may be obtained as follows,

 

2/1-212282

5.13
"I, < )

1 .
1 + R0011] ((1) = V f‘ (137‘ 1

The extension of this Coulomb correlation scenario to include the inﬂuence of
short ranged nuclear interactions can be accommodated using a formalism for calcu-
lating the second virial coefﬁcient[93]. One begins by imagining that two interacting
particles are placed in a spherical container centered about their common center of
mass. Assume that the appropriate boundary condition is for the wavefunction of
relative motion to vanish at the container walls. In the asymptotic region, the radial
wavefunction is of the form

sin [qr/h — 7; ln (2qr/h) — 57V? + 5t((1)l
o<

(17‘

 

Kw an)

85

The lfioundary condition therefore requires

qR/fz. — I]lll (‘ZqR/h) — [7r/‘2 + (55((1) 2 77m,

A
v.31
p—A
C,“

v

r r) , . . .
where 17 = Z1Z2c“/hv is the Coulomb parameter. Separating the phase shift into

0 o I u a
Coulomb m- and strong interaction (5; components, the density of states 1s

a : (2(+1)d—"i
dq

dq (5.16)

(I

= %:(2€+ 1) {7})? + % [nln(2qR/ﬁ.) +O'g(Q)]} + —71;Z(2€+ 1) {$51}

In Eq. 5.16, the ﬁrst term represents the density of states for the pure Coulomb
problem and the second term is the density of states due to the strong interacticm.
Since the ﬁrst term is difﬁcult and unwieldy to use, one can use the seu’iiclassical
expression in Eq. 5.13 or some similar shape for the Coulomb density of states. We
will use the second term for the strong interaction effects. Taking the spin of the

particles and resonances into account, the two particle phase space of relative motion

 

 

 

 

becomes
(in 25 1 23 1 V . 2 .Z Z‘ :22
,"13 = ( ‘+ )l 2+ l—f (137*1———”‘2‘ (5.17)
(1'3 q h~‘ V . v 1712
+47r1q2J :(2€ )dq ’

where V f and V are the free (unoccupied) and total (including occupied) volumes
of the system, respectively. Given this relaticmship, the correlation function as a

function of relative momentum becomes

1+ R((1) : 1+ RCioul(q )+ Rim-C(q) (518)

_ v/‘d3__,\/1_2;_.___1Z1Z2e2
_ rq2

86

 

I13 do"
_. + .H.’

+ .. . . , , 2/ 1 .
‘7r2q2f‘f'(2.51+1)(252 + 1) ;( (Iq

 

 

and as a. function of relative energy Em, becomes

 

 

1 + R (Erel) : 1 + R(‘()111(Erel) + Rnuc (Erel) (519)
1 . 212-262
v /v f ' 7-E,.,.,
11.3 , dd,”

 

 

2f+1)

+ , ' f 1 , .
47M», - (251 + 1) (252 + 1) lz.\/2;1E.e, g; “Err!

It is useful to reflect on how the equilibrium correlation function both resembles
and differs from the correlation functions calculated via the Koonin-Pratt equation.
\Vhile the general case of a ﬁnite emission time lies outside the equilibrium cm‘relation
formalism, there is a close connection in the limit of instantaneous emission between
ec‘niilibriun'i and Koonin-Pratt predictions for the second term in Eq. 5.18, stemming
from the influence of strong interactions. This term in the equilibrium correlation
pictui‘e remesents static iiimliﬁcations in the phase space distributions due to reso-
nances and other strong interaction effects. In this respect, the resonances can be
regarded as ”pie—existing". ()n the other hand, the correlations within the Koonin—
Pratt formalism have been examined in the "prompt” or instantaneous emission limit
for a. source that is much larger than intrinsic ”size” of the resonance itself by Boal

et al., and can be expressed by [94, 95]

3 . ~'
It do,”

. 2z+1 ,
4W2(12-(281+1)(252+1);( ) dq

 

1?"... (q) = S (0) (5.20)

for the strong interaction correlation function. If the source is taken to be uniformly
spherical, the 5(0) 2 UV}, and the equilibrium and Koonin—Pratt forn‘ialisms yield
identical expressions. Thus, the assumption of instantaneous emission implies that the
resonance structures predicted by the Koonin—Pratt formalism are those of unstable

particles that are ”pro-existing” at breakup.

87

Unlike the general situation for correlation functions calculated within the Koonin-
Pratt formalism (see section 5.2), the width of the resonance structure within this
large source approximation is given by the line shape of the resonance state. The
source size inforn'iation is therefore only contained in the magnitude of the correlation
structure through its dependence on S (0) = 1 / Vf. For states that are much narrower
than the intrinsic resolution of the detection apparatus, such as the case for the
2.186 MeV resonance in the d-a correlation, this does not. lead to a signiﬁcant loss of
inforn'iation. Comparisons between analyses performed in equilibrium and Koonin-
Pratt limits by Jennings et al. indicate that the correlation function predictions for
the magnitude of this resonance peak by both approaches are about the same for

gaussian sources with tyrﬁcal source radii [95].

5.5 Interpretations of Correlation Functions Using
the Equilibrium Correlation Approximation

To extend correlation function analyses to heavier particles that are more identiﬁed
with multifragmentation and the liquid-gas phase transition, we can at the present
time only apply the equilibrium correlation approximation. As mentioned in the pre-
vious section, this approximation is equivalent in many respects to the Koonin-Pratt
formalism. However, it does limit one to examine only the magnitude of the two par-
ticle correlation function peaks and there remains a. sensitivity to the secondary decay
corrections that cannot be determined from the technique itself, unlike the case for
the imaging procedure. This sensitivity means that one must employ statistical mod-
els to estimate the secondary decay and make the necessary corrections as discussed

below.

We begin our discussion of equilibrium correlation functions by returning to the

d-a correlation function, as shown again in ﬁgure 5.15 as a function of relative energy.

88

 

2.186 MeV 3+
4 d-alpha correlation
3 e .
[I
+
'— 4.31 MeV 2"
2 a

 

 

 

 

Ere, (MeV)

Figure 5.15: The experimental d-a correlation function is shown. The smooth solid
line is the empirical background used for the analysis. The dashed and dotted lines
are the Coulomb correlation calculated from Eq. 5.21 for sharp sphere radii of R2124
and 15.6 fm, respectively.

Here one can see clearly the peaks from the resonance states of 6Li: the 3+ state at
2.186 MeV and the 2+ state at 4.31 MeV. Also shown by the dashed and dotted lines

in the figure are Coulomb correlation functions calculated for a spherical volume of

radius R as follows,

 

1 . Z Z 1’2
1+ RCoul (Emil) : V /V dif1‘\/1 — TIEZ: (521)

_ [1 7‘min:| ”2 1 1 Tmin 3 (Tum)?
_ R 4 R. 8 R

 

 

 

 

89

 
 

I?

7 min

 
    

—1

 

 

3 (rmin > 31 . +
8 R rmin

where r,,,_,,, = Z1 ch’Q/Erd and we have used the spherical source radii of R12877
and 11.05 fm (corresponding ap1_)r(‘)ximately to the two-particle source radii of R =
ﬂR1=12x~1 and 15.0 fm) consistent. with breakup at densities of l/Gpo and 1/3/)(),
respectively, if half of the 200 nucleon source can be ”seen” by the correlation due to

collective effects (see sect. 5.3).

While the general trend of the predicted Coulomb correlation functions are simi-
lar to that (solid line) experimentally observed, the calculated correlation functions
underpredict. the measured one at very low relative energies and there are some differ-
ences in the overall shape. The extra measured yield in the data at very low relative
energies i'nay reflect long lived secondary decays that are not modelled by the equi-
librium Coulomb correlation function. Other (.lifferences, however, may reflect the
fact that the long ranged behavior of the Coulomb interaction makes it difﬁcult to
('listinguisl‘i two body from multi-body Coulomb effects. For example, a third frag-
ment in between the two measured fragments would repel them in opposite directions,
widening the correlation function minimum. Thus we have decided not to insist upon
ﬁtting the measured correlation function with Eq. 5.21, but simply ﬁt an empirical

l‘)ackground function (solid line) to the data, instead.

We therefore extract information about the source volume by considering what is
required to ﬁt the resonance peaks with the nuclear correlation function in Eq. 5.19.
This fit, however, requires the application of the empirical correction for collective
motion described in Section 5.3. When this correction is applied and the derivative
of the nuclear phase shift is given in a Breit-W’igner form [96]

d6}, ~ r,/2
dq "‘ (Em, — E:)2 + r3/4

 

(3.13.), (5.22)

90

the nuclear correlation function in Eq. 5.19 l‘)ecomes,

3
1?,.1,C(E,..-z) z . 1 . ’1 (ad/'1‘. n (,
(251+ I)(252 + I) 477 /f[l\/2,11.E,-€]

 

en
[\3
00
v

r,/2

 

>< :lr—Zczj. + 1) (BR)

For resonance states, the 1.)opulation will not be affected by collective motion
and if equilibrium is achieved, it should populate according to the breakup thermal
temperature Tmﬂml. If there is no collective motion, the relative energy spectrum
for the mixed event background in the denon'iinator of correlation function will also
be described by the same temperature Tmmnaz. On the other hand, due. to collective
motion of the hot source, the relative energy of the two particles in the event-mixing
case tends to be higher because the particles originate from different. places with
different collective velocities and because the reaction plane may also be different
from one event to another. Therefore, the effective temperature of the two particles

from different events is characterized by a. much higher value Tm”. This results in an

extra. factor in the resonance correlation Rmm where

1 1 1

chf 71thermal T001]

 

 

(5.24)

we find that d-a correlations are well described by assuming, T0” = 7 MeV (see
sect. 5.3) which could arise if the breakup temperature T,;,,_.,.,m,1 = 4 MeV and col-
lective ” temperature” is Ta,” = 9.3 MeV. Such values are typical of the temperatures
observed for isotopic and excited state temperatures (Tina-ml : 4 MeV) and for the
slopes of energy spectra (T6011 2 9.3 MeV) [30, 97]. If one is interested mainly in
resonance states near the threshold, i.e., Ere, _<_ 2 MeV, then the uncertainty of the
correlation Rune caused by the uncertainty of Te]; (i.e., varying between 5-10 MeV)
is less than 10%.

91

d-alpha correlation d-alpha correlation

 

 

 

 

 

 

 

 

 

 

 

2.186 MeV 3* 4 f 2.186 MeV 3"
4 .
3 - v,/v0=2.5
3
CC 5
3 CE 2 +-
4.31 MeV 2+
2 r
M»m..---_r -1 - 1 r
awn.— ._...._
1 I i o l l I l I A [H
4 6 8 1O 0 0.2 0.4 0.6 0.8 1 1.2 1.4
Erel (MeV) EmI (MeV)

Figure 5.16: The left panel shows the experimental d-a correlation function. The
solid, dashed and dotted lines are the different backgrounds used for the estimation
of uncertainty. The right panel exhibits the ﬁt of the ﬁrst resonance peak after
subtracting the background (solid line in the left panel).

Another imcertainty factor is the Coulomb correlation shape. We have used
particle-correlations in other channels that display fewer resonances to assess the
shape of the Coulomb correlation for interesting cases. For example, the p-d, and
t—t correlations have no resonances. Either may be used to estimate the background
of the p-t correlation. When we pay concentration on strong peaks, we also signiﬁ-
cantly reduce the sensitivity to the Coulomb correlation uncertainties. Nevertheless,
this remains a signiﬁcant potential source of uncertainty, which we must assess by

considering other shapes for the Coulomb correlation.

In the left panel of Figure 5.16 for the d-a correlation, we show three different
backgrounds (solid, dashed, dotted lines) that we have considered in order to assess
the uncertainty in the peak. In the right panel, we Show the nuclear correlation of
the ﬁrst excited state of 6Li after subtracting the background correlation denoted

by the solid line in the left panel. The remaining correlation function can be di-

92

Table 5.3-
adopted frt ,
tion.

rectly comp
this spectru
Table 5.3. A

parameter. t

 

tion. We. fin

lfflb : 29]

nuclear sat in
certainty in ‘

this places 0

Z < 3 to see

The trito

States of 7Li I

7.")

3 at. 9.67
(Solid, (1

étSher

in the right y

extract the l;

 

 

J7r E” (MeV) Fm“, (MeV) Channel BR. (%)
6L1 3+ 2.186 0.024 d—o 100
2+ 4.31 1.7 d-o' 97
7Li 7/2- 4.63 0.003 t-a 100
.5/2— 6.68 0.88 t-a 100
5/2- 7.46 0.089 t-a 18
7/2- 9.67 0.4 t-a 40(fit)
3/2- 0.85 1.2 t—o 40(fit)
7Be 7/2’ 4.57 0.175 3He-a 100
5/2- 6.73 1.2 3He—a 100
8Be 0+ gs. 6.8eV 01-04 100
2+ 3.04 1.5 a-a 100

 

 

 

 

 

 

 

 

Table 5.3: Relevant spectroscopic information of 6Li, 7Li, 7Be and 8B0 which is
adopted from ref. [98] is listed for the correlation functions discussed in this sec-
tion.

rectly compared to the nuclear correlation Rnu.c(Erel). We have done so by ﬁtting
this spectrum with Eq. 5.23 and using the known structural information, given in
Table 5.3. Applying Eq. 5.23 without any further considerations, we have only one
parameter, the free volume Vf, that can be varied to reproduce the correlation func-
tion. We ﬁnd that the correlation is well described with free volume of the breakup
Vf/VO = 2.50 d: 0.27, where V0 is the volume of the total system (326 nucleons) at
nuclear saturation density. Uncertainties in the background constitute the main un-
certainty in the extracted volume. Before discussing the constraints in density that
this places on the breakup, we examine other correlations between fragments with

Z < 3 to see whether any consistent trends emerge.

The triton—alpha correlation function is shown in Figure 5.17 with the resonance
states of 7Li (see Table 5.3): 7/2‘ at 4.63 MeV , 5/2‘ at 6.68 MeV, 5/2‘ at 7.46 MeV,
7/2“ at 9.67 MeV and 3/2‘ at 9.85 MeV. In the left panel, different backgrounds
(solid, dashed and dotted lines) are plotted for estimating the uncertainties. As shown
in the right panel, the very pronounced ﬁrst resonance above the threshold is used to

extract the breakup volume and this gives a value Vf/VO = 2.55 :i: 0.28 that is very
93

t-alpha correlation t-alpha correlation

 

 

 

 

 

 

 

 

 

 

2.5
1
2 c 4 63 MeV 7/2‘ V,/V0 = 2.55
6 68 MeV 5/2' 2
l | 7.46 MeV 5/2' 0'8 4.63 MeV 72‘
1 5 - a, I 9.67 MeV 7/2‘
[I ' 3' 1 [9.85 MeV 312' 8 0.6 . ,
+ ; ‘ ~
‘— 7’, "\ now-v . l m
w foMm ‘ ‘-‘M:n-'/.W--v L ‘- .
0.4 2
0.2 .
f
' lllllﬂmttftw “WWW
O I I l J— l l l L l 1
O 2 4 6 8 1O 0 0.5 1 1.5 2 .5 3 3.5 4
Erel (MeV) Erel (MeV)

Figure 5.17: The experimental t-o correlation function is shown in the left panel. The
solid, dashed and dotted lines are the different backgrounds used for the estimation
of uncertainty. The right panel exhibits the ﬁt of the ﬁrst resonance peak after
subtracting the background (solid line in the left panel).

similar to the one obtained from the d-a correlation function.

Similarly the resonance states of 4.57 MeV 7/2‘ and 6.73 MeV 5/2‘ (see Table
5.3) of the mirror nucleus 7Be are depicted in the 3He—alpha correlation in Figure 5.18.
The 5 / 2‘ state at 7.21 MeV is not shown due to a large decay branching ratio of about
97% through the p—GLi channel. The rise close to the threshold most likely comes from
the contaminant in the 3He PID from alpha particles and what one sees is the ground
state of 8Be from the alpha—alpha correlation (see below). Nevertheless, it does not
detract from our efforts to extract the free volume, which yielded Vf/VO = 2.22 :t 0.36

from the ﬁt to the resonant state at 4.57 MeV.

In the case of the alpl'ia-alpl'ia correlation as shown in Figure 5.19, the ground
state 0+ of 8Be and its ﬁrst excited state 2+ at 3.04 MeV are illustrated (see Table
5.3). In addition, the peak at about Ere, : 0.6 MeV is mainly from the decay of the

2.43 MeV state of 9Be [81]. In this example the identical particle effect is observed.

94

1+R

 

Figure 5.18
Solid. daslu
0f uncertai
SlIl')ira(‘llll“

’l

 

 

 

1+R

3He-alpha correlation

 

 

 

 

 

 

0.6 *
4.57 MeV 7/2’
1.5 ~ 6.73 MeV 5/2'
x it I 0.4 ‘
n:
0.2
0 l l l 0
O 2 4 6 10

Erel (MeV)

3He-alpha correlation

 

 

4.57 MeV 7/2'

v,/vo = 2.22

l + H +
[Lilli ll 1"] i

Hfﬁl‘l

 

 

0

1 2 3
E... (MeV)

Figure 5.18: The left panel shows the experimental 3He-a correlation function. The
solid, dashed and dotted lines are the different backgrounds used for the estimation

of uncertainty.

subtracting the background (solid line in the left panel)

1+R

alpha-alpha correlation

 

 

 

 

 

, 25
9.5. 0
2.5 --
20
2 r
l 98e'(2,43MeV) w 15
1 .5 fl 3.04 MeV 2’ (ES
10
5
0 1 I l
0 2 4 6 1O
Ere, (MeV)

alpha-alpha correlation

The right panel exhibits the ﬁt of the ﬁrst resonance peak after

 

 

9.5. 0‘

+

V'IV0 = 2.61

 

+ J l J

 

 

Ere, (MeV)

0
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Figure 5.19: The experimental o-a correlation function is shown in the left panel. The
solid, dashed and dotted lines are the different backgrounds used for the estimation

of uncertainty.

subtracting the background (solid line in the left panel).

95

The right panel exhibits the ﬁt of the ﬁrst resonance peak after

 

That is. th.
consequent .
By taking .
hunt the 3n

In all t.
this study.
temperatui
test t0 see i
I0 make tl
by the pro]
rcsenance ]

given belov

 

Performing
”I Fig- 5.21

excitation
essent ial a:

with T6” :

over this ‘
0f the ble-

lif I/l‘e : D

-

That is, the phase space of the two identical particles is reduced by a factor of 2 and
consequently the resonance correlation Rm“, becomes twice as large as in Eq. 5.23.
By taking into account this effect, one obtains a free volume of Vf/Vo = 2.61 i 0.03

from the ground state of unstable {Be as shown in the right panel of Figure 5.19.

In all the above correlations, only the first most pronounced peak is ﬁtted for
this study, which is sensible in reducing the sensitivity to the uncertainty of the
temperature factor and the background. Nevertheless, it is an interesting consistency
test to see if we can ﬁt all the peaks we observed. In ﬁgure 5.20, we explore this issue.
To make these higher peaks more visible, we divide the nuclear correlation function
by the prefactors in Eq. 5.23. This leaves only the free volume and the density of
resonance states coming from the derivative of the strong interaction phase shifts
given below

3 ‘1
1 . h (Era/W (5.25)
(2J1 + 1)(2]2 + I) 47TVO/LV (ZIMETCJ

1 1 r./2

= —— 2.1. 1 . .
vf/v0 7r 2,] + )(E,.,_., — E:)2 + r3/4

 

 

Rnuc(Erel) '

 

(13.11.) E t;,/V,,),.,.,(E,.e,).

Performing this operation on the correlation function data provides those data points
in Fig. 5.20. The solid lines are the sum of the resonance line shapes calculated using
the parameters in Table 5.3. To make a good representation of these peaks at higher
excitation energies, a careful normalization of the Coulomb correlation background is
essential as is the correction due to collective motion corresponding to the exponential
with T6,, = 7 MeV. In other words, ﬁtting over the domain of these higher resonance
states is useful to constrain the background and the parameter Teff. After ﬁtting
over this extended excitation energy range, the numbers obtained for free volume
of the breakup are changed very slightly: ’f/VO = 2.63 for the d—alpha correlation;

Vf/VO = 2.79 for the t-alpha correlation; V f / V0 2 2.44 for the 3He-alpha correlation;

96

d-alpha correlation

 

       

 

 

  

VO/Vi pres(Erel)

 

 

 

 
 
  

2.186 MeV 3+
30 r
q V/V0=2.63
we
v3 20 -~
Cf
3°
10 r 4.31 MeV 2”
O ...-"' 1 1 1 L 1H
0 0.5115 2 2.5 3 3.5 4
Ere, (MeV)
3He-alpha correlation
125 P 4.57MBIV7/2‘
10 ‘
u} V/Vo=2.44
g 7.5
Q.
:0 5 L. 6.73MeV5/2'
2.5 1. i” .
. ,, if
0 ++x+ﬁ++§+ H’ 1 1

 

  

 

 

0 1 2 3 4 5

Erel (MeV)

VO/Vf pres(Erel)

t-alpha correlation

 

 

 

 

 

 

 

 

 

15 -
4.63 MeV 7/2'
V/vo=2.79
10 l
6.68 MeV 5/2‘
l7.46MeV5/2'
9.67 MeV7/2'
9.85 MeV3/2’
5 ~ I
l
I! l J i
O 1 2 3 4 5 6 7 8
E (MeV)
alpha-alpha correlation
20 g.s.0‘
15 , V/Vo=2.54
10 ~
5 h °Be'(2.43MeV) 3'04 MW 2‘
‘ 1
0 V i 1 '
0 1 2 3 4 5
Ere, (MeV)

Figure 5.20: The densities of the resonance states from d-a, t-a, 3He—cr and CH}
correlations are ﬁtted after subtracting the backgrounds carefully selected by the

solid lines in Figures 5.16-5.19.

Vf/VO = 2.54 for the alpha-alpha correlation. The free volume measurements are

given in Table 5.4.

The cm‘isistency between the 1')reli1'ninary numbers given for the free volume in
the ”VI/V0” column in Table 5.4 is surprising. However, further estimates on the
breakup density need the consideration of corrections from collective motion and
from the secondary decay of heavier particle unstable isotopes that can produce the
particles in the correlation as well as nuclei in the resonance states we observe. Here we
attempt a correction for such effects. To consider the effects of collective motion, we
need to revisit the discussion in Section 5.3 on collective motion where the reduction
of the source size due to the interplay of collective and thermal motion is discussed.
The problem we need to solve is more complex than the problem we needed to solve
in order to determine the density from the p—p correlation function and from the
second peak in the d-a correlation function. In those latter cases, the shape of the
resonance was sufﬁcient to determine the volume of the source while one needed the
simulation to determine the mass in the source. Here we need to consider how the
collective motion influences the magnitudes of both the resonant, non-resonant and
mixed event yields. (The previous analysis showed that the latter two are inﬂuenced
similarly. )

Generally speaking, rearranging Eq. (5.2) yields the correlation function 1 + R(q)
expressed as a ratio of 5 dimensional integral over the coincidence yield divided by a
corresponding 5 dimensional integral over the mixed event yield product. If we try a
Monte Carlo simulation as in Section 5.3 of these yields, we break down the spatial
origins of the contributions to the momentum distributions. This means that starting
from these phase space distributions at breakup, we need to do an 11 dimension inte-
gral: 3 spatial and 3 momentum integrations for each particle minus one integration
corresponding to the non-integrated dependence on E,.._.,. Collective motion builds

98

correlations between the emission points and momenta of the emitted particles that
effectively reduce the fraction of the source contributing to the correlation function.
This reduction factor, however, is depem'lent on EN, as the exponential dependence

in Fig. 5.11 demonstrates.

Fig. 5.11, however, suggest. a simple way to deal with collective motion. Let us
assume that the data are consistent. with T.” = 7 MeV. Thus, for the source volume
of this simulation and this specific collective velocity ﬁeld, the resonant correlation
function should be proportional to the middle panel. Without collective motion, the
same source volume leads to the lower panel. The ratio of the middle panel divided
by the lower panel is what one would expect for the ratio of correlation function from
a source of the same dimension with collective motion divided by the correlation
function without it. This ratio is independent of the actual volume of the source.
Like the volume reduction factors in Fig. 5.9, it depends 011 vhmax, otmax, and the
random velocity 1’11; = y/3T/m, but not on the source radius R. Thus the inverse of
this ratio frat,” provides a way to correct for the collective motion effects and obtains
what the correlation function should be without collective motion. Fig. 5.11 shows

what the correction factor frat“, should be for the d-a correlation function in ﬁgure

5.15.

In applying this c<:>rrection factor, we simply need to extract the value for the cor—
rection factor ﬂan-(,(Eml) at the energy of the resonance in question. This dictates the
factor by which the correlation functions would have been smaller without collective
motion. This smaller correlation function would have resulted in the extraction of a
new free volume that would be larger by a factor of fm,,-O(E,.el)’1. The production
of heavier particle unstable nuclei and their decay into the particles included in the
correlation function or into the observed resonance peak also alter the correlation
function by increasing the yields of the involved nuclei. It is the correlation function

99

of primary fragments before decay which has the simple dependence on the emission
volume predicted by Eq. 5.19. Preliminary calculations of secondary decay suggest
that the shape of the spectrum after decay is not very different from the shape before
decay. Under this assumption, the primary correlation functions before decay differ
from the corresponding ones after decay by a multiplicative factor Elem where in the

case of the d-a correlation

(5.26)

where the factor DRd reflects the multiplicative factor by which the yield for deuterons
is enhanced by secondary decay, and the other factors in Fdec reﬂect the application
of the same considerations to the yields of 0’s and to the resonance in 6Li as well.
Secondary decay correction factors were obtained by calculating the equilibrated mul—
tifragment decay of a system consisting of 260 nucleons and 106 protons at an exci-
tation energy of E*/A =4 MeV using the SMM model described in the next chapter
of this dissertation. The secondary (‘lecay factors for the correlation functions shown

in Figs. 5.16—5.19 are given in the third column of Table 5.4.

we note that these secondary decay corrections are rather large. In particular, a
value for Fdec of 6.09 is obtained for the d-a correlation, which should be compared
to the reciprocal of the value for A = 0.11 that was obtained by ﬁtting the shape for
the second peak of the inclusive d—a correlation in Section 5.3. This suggests that
the SMM predicts about the same (differ by 49%) secondary decay contributions as
the Koonin-Pratt approach. To get a feeling for what a 49% correction implies, we
increase all the Fdec factors in the third column of Table 5.4 by 49%. These modiﬁed
values of Fdec(KP) are also shown in Table 5.4. These modiﬁed values should be
viewed with caution beyond some indication of how sensitive the densities are to the
secondary decay corrections.

100

Based 011 the above discussions. a correlation function Rnucpri without collective
motion before secondary decay can be obtained by an‘pr, ~ Rum - E1“ . fmm, (EM).
Since the factor exp(—E,.(,1 [11,”) consistent with Fig. 5.11 has already been taken into
account in Eq. 5.23 in extracting the free volumes for the correlation functions shown
in Figs 5.16-5.19, the normalization of the exponential factor frauoﬂ :—: f,.(,,,-0(En.1 = 0)
should be directed out to correct the free volumes for the remaining effects of collective
motion. Therefore an estimate of the expanded free volume after corrections for

secondary decay and collective motion might. be given by

 

Vf
V .source 1: _~ 527
f chc ' fratio.0 ( )
and the freezeout density p/po by
p
P/Pn = +—’—— 15.28)
Pf + 100
where
A we runner) 0-8fra io.f ' Flier:
mm, = —,’— = ’ ’ (5.29)

fgsourcept) L'f/f’f)

where the correction factor fmhoy) = 0.15 is obtained from Fig. 5.11 for the d—a
correlation and for simplicity it is used for calculating the densities for the correlations
shown in Figs. 5.16—5.19 assuming they all have the similar collective correction

factors. We note that they all have similar effective temperature of T8” = 7 MeV.

Values for the density, in units of saturation density p0 are calculated from Eqs.
5.28-5.29 and given in Table 5.4, indicating that the freezeout density of the system is
about 1/5-1/3p0. For comparisons, the density values of p/ p0(K P) are also shown by
applying the source reduction factor f2,” = 0.2 obtained for the Koonin-Pratt analyses
instead of the factor ﬂan-0,0 discussed here, and using the modiﬁed secondary decay

factor EMU/(P). If we omit the contributions of the excluded volume as in the case of

101

the Koonin—Pratt approach and apply the same secondary decay and collective motion
corrections as the results from the Koonin-Pratt approach. the pf/pn(KP) should be
comparable to the density obtained from the Koonin-Pratt approach in Sect. 5.3.
For the do correlation, indeed, the value of the density (pf/pUUX'P) = 0.57 in Table
5.4) extracted in the equilibrium limit are similar to the value (p/po = 0.54 in Table
5.2) extracted from the Koonin-Pratt analysis. Such a conclusion is in agreement
with the work by Jennings et al. [95], which equate the thermal and Koonin—Pratt

approaches for Gaussian sources under the assumptions of instantaneous emission

and no corrections from secondary decay and excluded volume.

The density extracted from the equilibrium approach has similar large uncertain-
ties as in the Koonin-Pratt analysis, which have been discussed in the end of Sect.
5.3. Especially the estimates of the correlation enhancement factor f,.at,,,(E,.€,) due to
collective motion are in need of better accuracy. On the other hand, the secondary
decay process signiﬁcantly modiﬁes the extracted free volume in the opposite direc-
tion. The resulting correction factor Fder: also need to be studied more carefully by

accurately checking the modelling of secondary decay by the statistical model.

 

 

Correlation Vf/VO Em, E16,.(KP) pf/po p/po pf/pn (KP) p/p0(KP)
d-o' 2.50 :t 0.27 6.09 9.09 0.29 0.23 0.57 0.37
t—o' 2.55 :1: 0.28 7.09 10.6 0.33 0.25 0.66 0.40
3He-(r 2.22 d: 0.36 8.45 12.6 0.46 0.31 0.90 0.47
0-0' 2.61 :l: 0.03 4.76 7.11 0.22 0.18 0.43 0.30

 

 

 

 

 

 

 

 

 

 

Table 5.4: List of quantities are shown for the d-a, t-a, 3He—a and 01-0 correlations
(Figs. 5.16—5.19). The values of Vf/VO are obtained from the corresponding experi-
mental correlation functions. chc is the correction factor in Eq. 5.26 calculated from
secondary decay contributions in the SI\«"Il\-~'I. The density p/po is estimated in Eq. 5.28
using the secondary decay correction factor Fdec from the SMM while p/p0(KP) is
obtained using Fdec(K P) (see text). The corresponding pf/po and pf / p0(K P) are
calculated in Eq. 5.29 without considering excluded volumes.

Besides the density extraction, it is interesting to see via the above multi-peak

102

ﬁttings that. unknmvn spin of one resonziu‘ice can possibly be obtained if the structural
information is well known for the other states. In the following section we will discuss

how to use this technique to determine spins of particle unstable nuclei.

5.6 Spin Determination of Particle Unstable States

One of the striking observations of this work is the degree to which the equilibrium
assumption can describe the relative magnitudes of the various peaks in the corre-
lation function. This is illustrated by the good agreement between the calculated
and measured values of the nuclear correlation function in Fig. 5.20. Noting the
dependence of the correlation function on the spin of the state (see Eq. 5.23), we
now explore whether this can be used to determine the spin of a nuclear state, if that
spin is unknown. By armlyzing density spectra. of resonance states like those shown
in Fig. 5.20, we try to determine the unknown values of spins and even unveil new

structures of nuclei.

To ensure the validity of applying this technique on studying the structures of
unbound nuclei, one simple case is examined in the following. In Figure 5.21, the
proton—7L1 correlation function is shown with the resonance states of excited 8Be that
are close to the tl'iresl‘iold: 17.64 MeV 1+, 18.15 MeV 1+, 18.91 MeV 2", 19.07 MeV
3+ and 19.24 MeV 3+ [98]. The branching ratios of these states through the proton
decay channel are as follows [98], the 17.64 MeV state is 100%, the 18.15 MeV state
is 100% where 96% goes to the ground state of 7Li and 4% to the ﬁrst 1/2' excited
state of 7Li at 0.478 MeV, the 18.91 MeV state is 86%, the 19.07 MeV state is 100%

and the 19.24 MeV state is 50%.

The reason why we choose the p—7Li correlation as an example is the structural

information of 8Be is quite complete and these pronounced resonance states are very

103

p-7Li correlation

 

17.64MeV 1+

f

1.5

18.15MeV cm
*1
1

18.91MeV 2'
19.07MeV 3+
19.24MeV 3+

+9311,
1’
+H’
1 + + H ““3
+ [#1 i “:1?“ 1’ +++++ff++f+m + + ” + + + + + + + +
1 — {1+1 ____________ {ft Efﬁmt’t +1.”; #1- #41511; ‘bﬁtﬂ- *- Mew
Y'- f I I

0.5

 

 

O l l l l
2 3

Erel (MeV)

Figure 5.21: The proton-7Li correlation function is plotted. The dashed line is the
selected background. See text for details.

 

104

close to the threshold, which can help reduce the sensitivity to the background se-
lection. For simplicity, we will parameterize the background correlation function as

follows,
1 + R59 2 1 — exp[—(ET€1/EC)”] (5.30)

which vanishes at zero relative energy and reaches unity at large relative energy. By
comparing with the proton-“Li and proton-8M correlations, the background of the
proton-7Li correlation function (the solid line in Figure 5.21) is selected with parame-
ters EC 2 0.16 MeV and a = 0.5 since these correlations should have similar Coulomb
interaction. After subtracting this background, the density of resonances is plotted
in Figure 5.22. The dotted lines represent the individual resonance states and the
solid line is the corwoluted ﬁt by applying the known structural information of these
resonances. The ﬁt is good everywhere except at the high energy end where the con-
tributions from higher lying states are not included due to incomplete spectroscopic

information.

To further test. the reliability of the background selection and the sensitivity to
the spin factor, a ﬁt to the overall correlation function is carried out by varying
the parameters in the background expression (Eq. 5.30) and the spin value of the
17.64 MeV state. In Figure 5.23, the best ﬁt (solid line) yields a background shown
as the dashed line with parameters of EC 2 0.152 MeV and a = 0.547 which are
very similar to what we have assumed above and a spin value of 1.06 i 0.1 which is
consistent with the experimental value. To illustrate the sensitivity of this technique
to the spin determination, calculated correlations (dotted lines) are also shown in
Figure 5.23 assuming that the spin of the 17.64 l\x‘leV state is 0 and 2, respectively
and meanwhile keeping the other parameters the same. The signiﬁcant separation of
J = 0, 2 calculations from the J = 1 ﬁt shows a good sensitivity to spin determination.

105

p-7Li correlation

 

18.91MeV 2'
1 * 19.07MeV 3+
19.24MeV 3+
17.64MeV 1+ [ [ [
”7:, 0.75 ” [ [

18.15MeV1+
0.5 T [1H 1‘]

const preS(E

 

D
O
.l

 

 

 

Figure 5.22: The density proﬁle of the resonances of 8Be is shown in the p—7Li corre-
lation after subtracting the background selected in Figure 5.21.

106

p-7Li correlation

 

17.64MeV 1+

18.91Mevz;
.; 19.07MeV 3+
15 _ 53 19.24MeV3
- 18.15MeV? [ I [

 

 

O
- +
. o -------------------------- Jbtt nt‘]

 

 

 

0 l l 1 l
0 0.5 1 1.5 2

Erel (MeV)

Figure 5.23: The best ﬁt (solid line) is performed for the p-7Li correlation by varying
the background and the spin value of the 17.64 MeV state. The dashed line is the
ﬁtted background. Two calculations are shown as the dotted lines assuming that the
spin of the 17.64 MeV state is 0 and 2, respectively and keeping the other parameters

the same.

107

 

As (1111‘
smnhdma
dwwnfor
obvious [1
state at 2
stzitc 1111s
.\le\" stat
5Li. a 1+
iMXC}Pt[

tlu?inter(

 

functhn1,
a 8D111 \"dl

Houe‘
humthuf
buiknov
Infigun
sphivahi
lhuﬂissl
neonanc
Syﬁenir
ﬁnnihul
Welnay
SDinoft
C‘dSeS’ th
of other

experhiu

As one can see from the above example. this technique can provide access to the
spin information of resonances. 111 Figure 5.24. the proton-TBe correlation function is
shown for an example with an unknown spin to be determined. In this (’tase the two
obvious peaks correspond to the ﬁrst excited state of 8B at 0.774 MeV and the 3+
state at. 2.32 MeV, respectively [98]. Interestingly the spin value for the 0.774 MeV
state has not been measured although it is 1,)elieved to be the 1+ analog of the 17.64
MeV state in 8Be [98, 99]. Also from the corresponding state in the mirror nucleus
8Li, a 1+ assignment would be expected for this state. However, experimental efforts
have yet. to justify that statement. If we assume that only these two states exist in
the interesting region, the solid line in Figure 5.24 is the best ﬁt to the correlation
function, which ﬁts a background (dashed line) along with the resonances and yields

a spin value of J 1 = 0.98i 0.29. The result conﬁrms the hint from the mirror nucleus.

However, the ﬁt does not present all the features of the experimental correlation
function. Especially a small bump at about 1.4 MeV seems to exist in between the
two known states. The calculations in ref. [100] predict a 1+ state of 8B at 1.4 MeV.
In Figure 5.25, the three resonances including the one at 1.4 MeV are ﬁtted. A similar
spin value J1 = 0.95 i033 of the ﬁrst state is extracted. But the background (dashed
line) is sluiillower than that of the previous ﬁt to accommodate the additional 1.4 MeV
resonance. If we can reduce the background by 111easuring a reaction with a small
system or constrain more strongly the background contribution to the correlation
function in some way and have better statistics and resolution in the correlation,
we may determine further if there indeed is a state at 1.4 MeV. Nevertheless the
Spin of the ﬁrst state at 0.774 MeV is conﬁrmed to be one in either case. In both
cases, the underestimated tail of the 2.32 MeV state seems to indicate the existence
of other nearby states at slightly higher energy which have not been identiﬁed yet in

experiments. However, we don’t have the resolution to distinguish these states.

108

p-7Be correlation

 

 

 

 

spin unkown
1.5 —- 0.774MeV
2.32MeV3”
+
- ++
CE 1 F ‘ + _______ .Jf .............. Jr—Hr
i V ,,,,, +~ ‘
0.5 c ff
0 ' L J I 1
0 0.5 1 1.5 2 2.5 3
Erel (MeV)

Figure 5.24: The p—7Be correlation function is ﬁtted by the solid line assuming only
two states at 0.774 MeV and 2.32 MeV. The dashed line is the ﬁtted background.

109

P-7Be correlation

 

 

 

 

Spin UnkOWn
1.4Mev 1+
1 ‘ ++++.

1 f— k
a: + " _____________ w
+ I .................
‘— + j ,,,,,,,

0.5 7- "I,
0 . 1 1 I I

0 0.5 1 1.5 2 2.5 3
Ere! (MGV)

Figure 5.25: The p—7Be correlation function is ﬁtted by the solid line assuming the
existence of an additional state at 1.4 MeV. The dashed line is the ﬁtted background.

110

01-6Li correlation
B.R.=100% (B.R.=55°/o)

 

[ 3 ...--..,/J =4
'.'-. ' .............. x/J =3
. ------- . J=2(J=4)
J=1

P O ..
‘ O
. .§ ‘I. o
I
~ ' . ‘.. \
[_. .4 Q '
~~ ..C ‘-

d

b ~~ ..
~
~ . 0..
~~ o
1.4 T F‘s .'
s '0'
C
1- “
.

 

 

2.2 2.4

 

 

1+R
(.0

 

 

 

 

 

 

O 0.5 1 1.5 2 2.5 3
Ere,(MeV)

Figure 5.26: The a-SLi correlation function is ﬁtted by taking into account all the
spectroscopic information shown in Table 5.5 except for the 6.56 MeV state. By
varying the spin of the 6.56 MeV state from 1 to 4 and assuming the decay branching
ratio is 100%, one obtains the dashed, solid, dotted and dot—dashed ﬁtting lines,
respectively. If a branching ratio of 55% is assumed, the solid line represents a ﬁt of
J=4.

111

In the following case. the c1-6Li correlation is shown with ample resonance states
forming peaks near the decay threshold. The structural information of 10B resonances
is listed in Table 5.5 where we notice that the spin of 6.56 MeV state is not well
measured (J24 is tentatively suggested [98]). Figure 5.26 shows the ﬁt with the
known structural information except for the state at 6.56 MeV. The spin J =1 of the
7.002 MeV state and its B.R. of 30% (see the d-a-a correlation in sect. 5.7) are
used in the ﬁtting, which have little effects on determining the spin and B.R. of the
6.56 MeV state. In this ﬁt we vary the spin value of this state from 1 up to 4 while
assuming that 100% of this state decays to the ground state of 6Li. As we can see,
it seems J =2 gives the best ﬁt while the cases of J21,3 are still possible due to poor
resolution. On the other hand, if J = 4 is assumed for the 6.56 MeV state, then a

branching ratio of this state decaying to alpha—6U channel can be obtained as about

 

 

50%.

.17r 13* (MeV) Fem, (keV) B.R. (%)
3+ 4.774 8.4x10-3 100
2‘ 5.1103 0.98 100
2+ 5.1639 1.76x10‘3 13
1+ 5.180 110 100
2+ 5.9195 6 100
4+ 6.025 0.05 100
3‘ 6.1272 2.36 97

(4)"? 6.560 25.1 100‘?
1“ 6.873 120 38

(1,2) + 7.002 100 small
2" 7.43 100 <30
2+ 7.478 74 <35

 

 

 

 

 

 

Table 5.5: Spectroscopic information of 108* ——> a+6Li adopted from ref. [98]. The
state at the 6.56 MeV is assigned tentatively with J=4 and only alpha decay is
conﬁrmed. The last three states contribute very little to the ﬁtting. For details see
text.

An extra 3+ state is predicted at about 6 MeV in ref. [101]. If this additional

3+ state exists and all states are populated with their full statistical weights, we

112

must. have overestimated that background in the ﬁt above. If instead, one lowers
the background to accon‘miodate the 3+ state, the ﬁtted spin value for the 6.56 MeV
statecould be a little higher. But it still can not account for a spin of J=4 if we
assume that the branching ratio to the a-GLi channel is 100%. In the next section,
we see some weak evidence of a resonance at Em, : 0.6 MeV in the deuteron—cr-a
correlation indicates that some of the 6.56 MeV state decays via either the d-cr-oz
three—particle channel or the d-8Be two-particle channel. If it accounts for half of the
decay of the 6.56 MeV state, then the best ﬁt (solid line) in Figure 5.26 favors a spin
of J = 4. We don’t have enough statistics, however, to determine the branching ratio
for the decay of the 6.56 MeV state to this channel, leaving the determination of spin
for the 6.56 MeV state open at the present time. Further studies 011 the branching

ratios are needed to clarify this issue.

To further apply this technique on probing nuclear structures, better statistics
and resolution are important to distinguish overla.11)ping resonances. Better statistical
accuracy and lower backgrounds can help reduce the uncertainty in the ﬁt. Lower
backgrounds can be attained by using a smaller system. However, the agreement with
equilibrium correlation functions could be worse because there may be stronger non-
equilibrium contributions. Resonance peaks far from the threshold are suppressed
signiﬁcantly due to the energy dependent suppressing factor in Rnuc which comes
from two particle phase space and collective effects. Therefore the states close to the
decay threshold are easier to study by this technique. As far as the secondary decay
effects concerned, proton—rich nuclei are preferred in this technique. In addition, all
the decay branching ratios, if not known, for the low-lying resonances of proton—rich
nuclei can in principle be determined by studying all relevant particle correlations.
In this case we don’t need to deal with neutron decay which can be omitted for

proton-rich nuclei. It will be an advantage when only charged particles are detected

113

in experiments.

5.7 Multiple Particle Correlations

Similar to two particle correlation functions, correlation of three or more particles can
be constructed in heavy ion reactions. W hile we have not analyzed them in detail, it
is nevertheless interesting for future studies to examine them. Choosing some of the

stronger correlation functions, we discuss some of their properties in this section.

In Figure 5.27 the three-alpha correlation function is shown. The resonance peaks
of 12C [102] are labelled in the plot. In two particle correlations one sees a weak energy
dependent factor 1/\/E,.el in Eq. 5.23 which stems from the two-body phase space.
In contrast to two particle correlations, three body phase Space yields a more strong
and singular energy dependent factor in three-particle correlation functions. For a
three body decay, this factor can be as strong as l/Efd and consequently increases
the height of resonance states well below 1 MeV enormously and suppresses the peaks
at higher energies. Due to this singular factor the ﬁrst peak is extremely pronounced
corresponding to the 0+ state of 12C at 7.654 MeV. The second peak showing the
3‘ state at 9.64 MeV is much reduced compared to the ﬁrst peak even though it
has a larger spin degeneracy factor. The states of 0+ at. 10.3 MeV and 1“ at 10.844
MeV are less evident because they are much broader and have low spins. A broad
pronounced group is observed consisting of ﬁve states of 2+ at 11.16 MeV, 2‘ at
11.828 MeV, 1+ at 12.71 MeV, 2" at 13.352 MeV and 4+ at 14.083 MeV, which can
not be distinguished from each other due to poor resolution. Beyond the broad 2+

state at 15.44 MeV the proton decay channel becomes dominant.

111 Figure 5.28 the proton-proton-alpha correlation is plotted depicting the reso-

nances of the 0+ ground state and the 2+ state at 1.67 MeV from 6Be [98]. Figure

114

5.29 shows the proton—a11.111a—al})lia correlation function from the decay of 9B. The
ﬁrst resonance of the 3/2" ground state of 9B is so pronounced because of the singular
three-body phase space factor (liscusst—ét’l above. And the 5/2‘ state at 2.361 MeV is
also pronounced while there is a weak evidence showing the existence of a broad state

at about 1.6 MeV [98].

In Figure 5.30 the deuteron-alpha—alpha correlation function shows the resonance
peaks from the decay of 1”B in the range of 6-8 MeV [98]. Although poor statistics,
one can see a suggestion of a resonance from the 6.56 MeV state to this decay channel,
which is just. above the detection threshold. This might indicate that the branching
ratio for the o-GLi channel from the 6.56 MeV state is less than 100%. Thus there
is a possibility that the spin of this state could be larger than the magnitude of the
peak in the (if-6L1 channel might indicate. The second peak in Fig. 5.30 could be the
decay of the 1‘ state at 6.873 MeV and the (1, 2)+ state at 7.002 MeV. A shoulder
on the spectrum corresponds to the decay from the 2~ state at 7.43 MeV and the 1‘L
state at 7.67 MeV and there is a pronounced peak showing the decay of the 2+ state

at. 8.07 MeV.

As well as identifying resonance states of particle unstable nuclei, we can gain
access to other valuable information in correlatitm functions. which is difﬁcult to
be obtained elsewhere. The three-body Coulomb correlation can be examined in
these three particle correlation functions. Although the resonance states far from
the threshold are difﬁcult to access due to the large phase space suppressing factor
1/E3d, this factor also makes the peaks very close to the threshold astoundingly
pronounced, which may be useful for further studies on the properties of sources and

so on. However, more needs to be done 011 the formalism for three body correlations

before this can be a quantitative tool.

alpha-alpha-alpha correlation

 

 

 

 

 

 

 

 

 

12
C 100 ~
. l 7.654lvleV
1116Mev
2.5 7’ ‘1828MCV
12.71MeV -
13.352Mev 50 _
14.083MeV
—L I l-
” [ [1+
1 +111 .
0 ......... tL +4)
m 2. [ [l] +++MW++++*+*+ m 0 0.5
+ 1.5 [ [ [#11 f +23 ”,4;
‘— 1 1 f 1 m, 1 Wffwowm 4
[ H [1 1'1 ++ fwﬂW’”’*’“’°-M’Jw
1 _ [[1] [_J 15.44MeV
.. [ 10.3Mev
10.844lv1eV

 

0.5 r

T7654Mev

O l l l l l l l
O 2 4 6 8 101214

Erel (MeV)

Figure 5.27: The a-oz-a correlation function is shown. Resonances from the excited
states of 12C are labelled with the ﬁrst peak seen more clearly in the inner upright
panel.

 

 

 

116

1+R

proton-proton-alpha correlation

 

1.5

1 _

0.5 c

 

]

 

 

11*

l

0+

1“+

F 9.3.

1.67MeV 2+

+++,,++++++
H++++++ when: w.»

+

1

6Be

4* + «4+ n *
9 + 0".“ *N“”§ H, ”0.. I”O.+N«

L

 

 

Figure 5.28: The p-p—a- correlation function is shown.

2 4 6
Erel (MeV)

117

8

1O

12

p-4He-4He correlation
QB 9.8. (3/2)’

 

I I t

60

I
(.—

2-5 W 9.8. 40

Ur'TUrTI’I

(5/2)' 20
2 “’ 2.361MeV 03.11.}?

 

 

 

 

1+R
01

 

 

 

0 1111111
012345678

Erel (MeV)

Figure 5.29: The p—(I-a correlation function is plotted. The resonances from the decay
of 9B are shown.

 

118

d-oc-oc correlation

 

 

 

 

 

 

 

3
CD CD (41316.56 MeV
® 1'at6.873 MeV
2.5 *- l®® @ ® (1,2)+ at 7.002 MeV
I] @ @ 2‘" at 7.43 MeV
@ © (1*) at 7.67 MeV
2 ~ I <6) 2* at 8.07 MeV
++
qI" 1 5 T [+1 ++++++++W++ ++ +
‘— [ + + ++ +++W++N+++++ﬁw
1 _ ii
0.5 _
O l 1 1 l
O 2 4 6 8 10

Erel (MeV)

Figure 5.30: The d—a-a correlation function is shown. The resonance states of 10B
that contribute this decay are listed. See detailed discussions in text.

119

Chapter 6

Statistical Multifragmentation
Model with Empirical
Modiﬁcations

The statistical multifragmentation model (SMM)[36] has been successful in describing
the thermmlynanrical properties of multifragmeiitation processes, especially for corn-
parisons to measured fragment multiplicities, mass and charge distributions and so
forth[19, 21]. Further studies on the degree of thermalization of such reactions involve
accurately constructing thermometric observables such as isotopic thermometers[54,
31]. Isospin effects in multifragn1entation are also being explored for the study of
isospin fractionation[41], isotope scaling[39] and even the isospin dependence of the
nuclear equation of state[42]. Such studies, however, have been rendered less conclu-
sive because of the inaccurate modelling of the later stages of the breakup in current
models where detailed nuclear structural infornu-ition are critical to accurately model
the secondary decay process. In this chapter, an empirical statistical multifragmen-
tation model incorporated with an empirical secondary decay procedure is described
by taking into account experimental information such as binding energies and level
densities as much as possible. By means of this improved model, nuclear thermometry

and isospin effects are studied and compared with experimental data.

120

6.1 Microcanonical Statistical Multifragmentation
Model (SMM)

6.1.1 Underlying Formalism

In the SMM, it is assumed that an excited, expanded and equililn‘ated source is
formed after the most violent stages of a heavy ion reactitm and it then decays simul-
taneously and statistically. The Inicrocanonical version of the SMM uses the Monte
Carlo method and averages oliiservzitbles with the statistical weight over fragmentation
1nodes[3(5]. A multifragment decay mode is defined in the SMM approach as a specific
set of emitted fragments and light particles. For simplicity, each fragmentation mode
in the SMM approach is weighted according to the entropy of the mode. This entropy
is approximated by analytical expressions rather than by an event by event sampling
of the phase. space as in ref. [11]. These approxinlations rely upon that fact that the
domiimnt contribution to this entropy comes from internal phase space of fragments
which plays the role of a heat bath within the. SMM approach just as an excited

residue plays the role of a heat bath within compound inn-lear decay theory [43].

The characteristics of this source, such as its thermal excitation energy E *, density,
mass and atomic numlmrs, A0 and 20, respectively, are taken for granted in the model.
For a given fragmentation mode, mass and charge conservation is strictly imposed:

A0 = Z NAZ A and Z0 2 : NAZ Z (6.1)

{AZ} {AZ}
where NAZ is the multiplicity of a fragment with mass number A and Charge number
Z. By making a Wigner Seitz approximation to the Coulomb energy, energy conser-

vation within the SMM approach leads to the expression [36]

 

, , Z2 v ”3
Eg‘ + O 2 (1C—% (V?) + Z IVAZEAZ (6.2)
A0 {AZ}

121

where E85 is ground state energy of the fragmenting source. The first term on the
right hand side stands for the Coulomb energy of a homogeneous sphere of matter
containing the total charge ZO and mass A0 at a (‘lensity p 2 p0 (Hy/V) where if) is the
normal volui'ne at saturation density [)0 and V is the, breakup volume occupied by the
system. The second term is the sum of the energy of each individual fragment in this
decay mode. Enz is the kinetic plus internal energy for each of these fragments. It is
related to the temperature by assuming all fragments are at a common temperature

as follows,

EM = 3T + 13pm + E552 — Big (63)
where the internal excitation energy of the fragments, 32(T), may be approximated
by an extension of the. semi—ei'npirical mass formula to finite temperatures [36], and
the extra Coulomb energy of the fragment in the fragmentation volume, Effz, may
be calculated within the VVigner-Seitz approximation (see below). B AZ stands for
the ground state binding energy for the fragn‘ient. Eqs. 6.2 and 6.3 result from an
average of the niicrocanonical expression for energy conservation over the phase space

corresponding to the specific fragmentation mode.

By applying the energy conservation relationship in Eqs. (6.2-6.3) one obtains
a ten’iperature T that describes the internal excitation and translational energies of
fragments within a given fragn'ientation mode. Even though the overall system is
assumed to be in equilibrium at a fixed excitation energy E5, different decay modes
have different Coulomb, binding, and translational energies and, consequently, dif—
ferent excitation energies of the emitted fragments. Consistency with Eqs. (6.2-6.3)
therefore requires that the temperature T of the fragments varies from one decay mode
to another, reflecting the differences between the Coulomb, binding and translational
energies of the various fragmentation modes.

122

Labelling the fra mentation mode N43 with the index, m, a )hvsical observable
b ..
0, such as the yield of a fragment or the temperature, can be expressed by a weighted

average over decay motles as,

:77] L1If"? ()171

O = 6.4

< > Zn: ”km- ( )
where the statistical weight associated with the mode m,

l’Vm : exp 2 NAZSAZfT) (6.5)

{AZ}
may be found by expressing the entropy of the fragments, SAZ, through the thermo-

dynamical relation with free energy,

8F
8T ’

 

S = — (6-6)

The corresponding free energy associated with each fragment can be written as,

2‘2 / 1/3
FAZ(T) : —BAZ — €1ch (V0)
+ BMW MT), (67)

where the kinetic term corresponds to:

771,, AT] 3/2
2nh2

+ T111 (1V4Zl) /]V/lZ . (6.8)

Ffzm = —-Tln g,.,zvf[

where V, = V — V0 is the free volume, m" represents the nucleon mass, and 942 is
the spin degeneracy factor. Empirical values at ground state are used for A < 5 since
one assumes that these nuclei have no excited states, except 4He. In all the other
cases, gAZ = 1 because we assume that this effect is, to some extent, already taken

into account by the level density used in the model. The Coulomb term in Eq. (6.7)

123

is associated with the remaining corrections in the \Vigner-Seitz approximation. The
excitation of the intrinsic degrees of freedom is taken into account by F§Z(T) and

therefore it is zero for the light particles as just mentioned.

To calculate the properties of the multifragment (‘nnission from the excited source,
one should sum the contributions of all the fragmentation modes consistent with mass,
charge and energy conservation imposed by Eqs. (GU—(6.2). However, this would be
extremely time consuming owing to the huge number of possible fragmentation modes
[103]. Therefore, the present. approach samples the more probable modes via a Monte
Carlo calculation. This is carefully discussed in ref. [103]; we note in passing that the
l\lonte Carlo procedure introduces a bias since not all the mass and charge partitions
enter with the same weight. Therefore a)", must be modified to correct for this bias
[103].

In the standard SMM formalism, simple paramet.rizations [36] of the ground state
energy and internal free energy are adopted for convenience. In order to have accurate
calculations, however, more careful treatments related to the nuclear structures are
needed, especially for the secondary decay process. Before we introduce an empirically
modified SMM with an consistent empirical secondary decay procedure in section 6.2,

issues related to primary temperatures need to be addressed in the following.

Temperatures have been extracted from the isotopic yields of heavy ion collisions
using the isotope thermometry method proposed by Albergo et al. [54]. The idea of
the method is to obtain temperature from the double ratios of the yields of four suit-

ably chosen isotopes, (A1, 21), (.41 + 1, 21 ), (142,22), (A2 + 1, 2;) via the relation[54],

AB
T - = ————— 6.9
150 ln(aAY) ( )

124

where

Y(AlaZI)/Y('41+1’Zl)
A}, : ‘ (.10
l/(flg, ZQ)/}r(A2 +122)! () )

AB = B(.4,, 21) — B(A1 + 1.21)

 

 

_B(.42.Z2) + B(.42 +1.22). (6.11)
and
a 2 (2122-42 + 1) (2.121,..n+-1 +1) .42 (A, +1) 3/2 (612)
(2'121v/11+1)(2J22..42+1 +1) Al (42 +1) ‘ -

Here Y(A, Z) is the yield of a given fragment with mass A and charge Z; B (A, Z) is
the binding energy of this fragment; and JZA is the ground state spin of a nucleus with
charge Z and mass A. However, this isotope thermometry method is derived within
the context of the grand canonical ensemble which could be doubtful as compared
with experiments which are probably closer to microcanonical limit. In the following
sections, the validity of the Albergo formula (Eq. 6.9) will be tested within this

approximate microcanonical SMIV’I.

6.1.2 Temperature Distributions

The SMM procedure expressed in Eqs.(6.2-6.4) leads to a distrilmtion of the temper-
atures of the fragmenting system for a given excitation energy in the same sense that
the temperature of the daughter nucleus in compound nuclear decay theory varies as
a function of the Coulomb barrier and separation energy of each decay channel. The
points in Fig. (6.1) denote the temperature distributions for the fragn‘ientation of an
excited “2872. nucleus at three different excitation energies obtained with the SMM.
These distributions are well ﬁtted by gaussian functions, shown by the curves in the
figure, with variances 0:}. that are fairly independent of the energy, O'T z 0.4 MeV,

in the range 3 MeV S Eg/A S 10 MeV. At each excitation energy, we average over

125

 

- —c+ EJA=4Mev
_.. EJA=6Mev

0.12 r

.0
o .0
m _s

W) (MeV‘)
.0
8

0.04

0.02 *

 

 

 

Figure 6.1: The points denote distributions of temperatures calculated with the SMM
approach for the decay of a 112571 nucleus at three different excitation energies. The
lines denote gaussian fits to the calculated distributions.

all of the fragmentation modes and define this average value as the ”approximate

microcanonical” temperature TM 1C.

Since each of the isotopes employed in the thermometer has a specific mass, charge
and binding energy, the application of conservation laws sets a constraint on the values
available to the remainder of the system. Because of this ﬁnite size effect, the tem-
perature distribution obtained when a. speciﬁc isotope is present is slightly different
from the one obtained when all fragmentation modes are considered. In particular,
a small difference (3 0.1 MeV) is observed between the average temperatures for the
various isotopes; this is illustrated in Fig. (6.2) for carbon isotopes from the fragmen-

tation of a ”2572. nucleus at E} /A = 6 MeV. Even though the average temperatures

126

 

0.12

.0

o

m
,

 

 

 

l l l l

o- -
3.5 4 4.5 5 5.5 6 6.5 7

T (MeV)

Figure 6.2: The points denote temperature distributions calculated with the SMM
approach for the different isotopes considered in the carbon thermometer for an ex-
citation energy of Ea/A = 6MeV. The lines denote gaussian ﬁts to the calculated
distriljmtions.

are different reﬂecting the different binding energies of the three isotopes, all these
distributions are gaussians with nearly the same variances. we can extract another
temperature T, M p by averaging over fragmei‘itation modes which contain an Interme-
diate Mass Fragment (IMF) with 3 _<_ Z S 10. It’s interesting to note that TM”; can
exceed TIMF at low energies by as much as 0.2 MeV, in part because it takes more
energy to emit an IMF than to emit an equivalent mass in the form of alpha particles,

leaving less energy for thermal excitation.

The basic idea contained in Eq. (6.9) was derived under the assumption that the
primary yields are well represented by the grand canonical approximation at a single

breakup temperature; the double ratio was invoked to cancel out the contribution to

127

the yields coming from the neutron and proton chemical potentials. In the Sh‘lhx‘l,
however, the temperature varies from one fragmentation mode to another and the
chemical potentials, which appear within the grand canonical formalism as Lagrange
multipliers that conserve charge and mass, are not explicitly invoked. Thus, we can
not presume the validity of the Albergo’s formula ( Eq. 6.9) in the SMM and must

test its validity instead.

We begin with a test of the validity of Eq. (6.9) when one employs the primary

yields. The primary yield for the ground state can be related to the total yield by

N912 = 1 AZ -yf§}€X1)lF.§z('T)/’Tl (6-13)

for a given fragmentation mode. Following the procedure described in the previous
section, we will use this expression and Eq. (6.4) to obtain the average g.s. yield
distribution (N512). This, in turn, can be used in Eq. (6.9) to extract isotopic

temperatures as follows,

 

 

r93 1 ms
<AAl.Zl>/<‘\.‘11+I.Zl> , AB
= C 9X1) -. (6.14)
ANS JVQS T‘sm m
AZZQ / 1 A‘2+1,Z‘2 1.30

In previous SMM calculations, experimental binding energies and spin degeneracy
factors 93,} were used for light nuclei with A < 5. Liquid drop binding energies
and spin degeneracy factors of unity were used for A Z 5. In this work, we will
retain these conventions on spin degeneracy factors so as to be consistent with prior

calculations, but we will use empirical binding energies for all nuclei.

In Fig. (6.3), the isotopic temperatures Tg’gm for the carbon thermometer (Z1 =
Z2 = 6, A1 = 11,A2 = 12) are plotted as the stars for the nuiltifragmentation of
a ”2371 source at excitation energies Eg/A = 3 — 10 MeV. For comparisons, the

corresponding Time and TIMF for the same system are also shown in Fig. (6.3) as

128

the dashed and solid lines. respectively. While supporting the concept of isotopic
thermometry, the good agreement. between Tum and T,‘;’(;”" is smnewhat surprisind,
given the strong dependence of the Boltzmann factor on temperature for large AB
and the width of the temperature distribution shown in Fig. (6.1). As shown in the
following section, it occurs in part due to a large cancellation involving the Boltzmann
factor and the temperature (le1,)endences of the effective chemical rmtentials. Fig.
(6.3) also reveals that fairly precise information about TM”: and somewhat less precise
information about TM“; is provided by the primary yields. This suggests that given
a precise relationship between primary to the ﬁnal yields, it would be possible to

determine the breakup temperature from the measured yields.

 

T (MeV)
A

 

 

 

l l

2 4 6 8 10

 

O

Eg/A (MeV)

Figure 6.3: Comparisons of various primary temperatures TM [(3, TMW and Trim"!

from the SMM and ng from the analytical calculation in the grand canonical limit.
For details see the text. One point is missing for T3"! with 0T 2 0.8MeV because

the calculated value for p for the correction term in Eq.(6.17) becomes negative at
a/A = 3MeV, i.e. the expansion breaks down in this case.

129

6.1.3 Effects of Temperature Variations

The surprising consistency between TIMF and T512“ in Fig. (6.3) suggests that the
corrections to the grand canonical prediction for the isotope temperatures are small,
and one may utilize this approach to understand why the temperature variations have
so little influence on the results. Taking this tact, we assume that the isotopic distri-
butions are well a1.)proximated for each fragmentation mode by the grand canonical
limit, use this limit, to gain insight. into the finite size effects and at the same time,
investigate the accuracy of this approximation. We take this approach to consider

ﬁrst the influence of the temperature variations and later the consequences of the

ﬁnite size on the effective chemical p(_)te1'itials.

Considering the influence of the ten’iperature variations in this approximation,
we average the grand canonical approximation over the temperature distribution in
Fig. (6.1). If the approximation works, the expressions that result from this average
should be appropriate for the consideration of the effects of temperature distributions
arising from other effects and within other equililn‘ium models of multifragmentation
as well. Taking this approach, the yield of a particular isotope i in the framework
of Albergo’s method [54], when averaged over all possible fragmentation modes, be-

COIDCSI

00 Am ,- T
(14>: V/o mew—f?

9X1) [(Zi HPF (T) + Ni HNF (T) + Bi)/Tl ((5-15)

where f (T) is the temperature distribution, V represents the free volume of the

system, AT = ‘/27rh2/mT, m is the nucleon mass and app (imp) stands for the

130

chemical potential associated with free protons (neutrons) at temperature T. The

internal partition function of the fragment i is given by:

 

AE1] (6.16)

<.('T>=>:g{exp[— T

where AEj is the excitation energy of the state j with respect to the ground state

and 9'} stands for the spin degeneracy factor of this excited state.

Assuming that f (T) is a gaussian centered at (T) and with width or << (T) (see
Fig. 6.1 ), one may expand 1 / T, T3/2, and the chemical potentials. By considering

only fragments observed in the ground state, i.e. (,(T) = 9?, we obtain that

g? v A?” com

(Yes) )‘3

I

 

- x g “PF (<T>) Z,- + [1.3m ((T)) N,-
e‘ p [<T> + (T)

1 q? (617)
m exp 4]) . .

 

where A, E y/27rh2 / m. In the above expression, the corrections to the grand canon-

. 0 0 I O 2 I
ical relationship are prov1ded by the correction factor ﬁ - exp [3;] WlllCh depends
on assumed width of the temperature distribution, the binding energy of the i-th
fragment, the neutron and proton chemical potentials and their derivatives through

the parameters p and q. These two parameters are deﬁned by

 

, — 1+ £2 Z-a +N-a +§—Bi (618)
p _ 2 (T) 1’ PF 1‘ IVF 4 (T) .
(7T 3 Bi
: Zi 3 Ni 31 _ _ 1

131

where

l-‘PF ( (Tl) 1 ~

an» = ,.1.’p,.((r)) — _<_?»— — 5“”"“ ((T)) (T) (6.19)
(31’1“ = .U-‘IPF (f(T)) * HP—ZTTgm

(1...: (.12..F<<T>>— L—QTQQ— — gt... <<T>> <T>

. _ ’ _ I’NF (<T>)

I‘ijF — “NF (<T>) (T)—

The isotopic temperature can be extracted from the above corrected yields. Re-
placing Y(A, Z) in Eq. (6.9) by the right hand side of Eq. (6.17), one cancels out the

spin and mass dependent term C and then obtains:

C(A1,Z1)/G(‘41+1,~Zi)
C(A'Za Z2)/G(A2 +1.22) 3

 

(6.20)

280

exp [AB / TC”, ] =

where

 

 

1 qz (6 21)
-— - ex — . .
y/2p p 4p

In the above double ratio the terms involving the chemical potentials evaluated at
the average temperature cancel; however, terms in the correction factor involving the

derivatives of the chemical potentials remain.

Quantitative estimates of the correction factor require one to obtain estimates for

the effective chemical potentials and their derivatives with respect to temperature.

132

The proton and neutron chemical potentials at temperature T may be calculated from

the free proton and neutron multiplicities via. the expression:

/\f{}.’) .‘ T
QPFV

/\3.Y. T

#AWKYU == Tdog ._L_A££_l
gNFV

where 91),.‘(g]\:1.~) represents the spin degeneracy factor of the proto1'1(neutron). For
the calculations presented in this work, it has proven advantageous and reasonably
accurate to approximate the yields YpF(T) and YNF(T) over a modest range in tem-

perature by power law expressions in the ten’iperature. In this approximation,

Ypp(T) = CppTTPF, (6.23)

},"'vl" (T) Z (HV’L‘T’NF

For the decay of “2572. nuclei at tenmeratures ranging over 4 S T S 7MeV , Ypp
and YNF are well described by "fpp = 4.5 and my): 2 1.0 (epp = 1.33 x 10"4 and
ch = 0.267) according to the Sh-Ih-‘l; comparisons of this parameterization to yields
calculated with the SMM model are shown Fig. (6.4). These values depend on the
density, which has been chosen to be one third that of the saturation density of nuclear

matter. Larger values of the free nucleon yields are obtained at lower density.

Using this approximation, the explicit forms of the correction factors in Eqs.
(6.17)-(6.19) become 20p}? 2 [ipp : (7,7,: — 55) = 3 311C120'NF : ,«3NF. : (amp _ g) =

—%. We note that the correction factor to the temperature T33 in Eq. (6.20) depends

133

 

 

 

 

-O— neutron yield
1.75 r -I- proton yield
1.5 r
1.25 r
2
.9 1 r
>-
_ /
0.75 .
0.5 r y/d
0.25 r '0';
0 1 l_ 9.1—- ’1 1 1 l
0 1 2 3 4 5 6 7 8

T (MeV)

Figure 6.4: The solid squares and circles denote the free proton and neutron yields,
respectively, calculated via the SMM approach. The solid and dashed lines denote
ﬁts to the calculated yields following Eq. (6.23).

on the power law exponents 7;)1:(')'Np) and not on the absolute values of the pro-

ton(neutron) yields.

Even though Eq. (6.15) has an exponent that appears to be strongly temperature
dependent, there is a strong cancelation between the contributions from the chemical
potentials and binding energy factors in the expressions for p and q. As a result, the
correction factor is of order unity. Values in the range of 31);}; -exp [54%] m 1 -— 2 are
obtained, for example, in the decay of “QSn nuclei at temperatures in the range of

4 _<_ 'T g 7AIeV.

The isoto ic tem eratures Tall calculated from E . 6.20 for carbon thermometer
P p q

180

are shown in Fig. (6.3) in comparisons with temperatures TM [(7, TIMF and T-s’m"

ISO

134

derived from the Sh-‘IM in the previous session. The very good agreement between
7122’ , 'T,:’;"" and TIM}: indicates that the corrections to the isotopic temperatures
associated with these temperature variations are small, although the yields can change

by as much as a factor of two. This comparative insensitivity arises because the

isotopic thermometers depend logaritln’nically on the yields.

This insensitivity cle1,)e.nds on the nature and magnitude of the temperature vari-
ation. The corrections to the isotopic temperatures will be somewhat larger in other
contexts or other models where the temperature variations are larger. The limited
precision with which systen‘is may be selected experimentally may also have a sim-
ilar inﬂuence because the excitation energy and temperature varies experimentally
from collision to collision due to variations in the impact parameter or in the energy
removed by preequilibrium particle emission. The inﬂuence of this temperature vari-
ation, which may exceed the variation in temperature caused by the averaging over
decay modes, can also be estimated via techniques outlined in the present section. To
illustrate how one can estimate the possible corrections due to an imprecision in the
excitation energy definition, the circles in Fig. (6.3) show calculations using Eq. (
6.20) for carbon thermometer assuming a width of OT z 0.8 MeV for the temperature
distribution, which is twice as large as that predicted in Figs. (6.1) and (6.2). This
width is not based upon a dynamical calculation; it is only to illustrate that larger

isotopic temperatures can result if the excitation energy is poorly deﬁned.
6.1.4 Chemical Potentials

The rand canonical limit has a reat advantage of rovidine a sim )le anal rtic ex-
0 D
pression for the isotopic yields from which other useful expressions can be derived.
However the conce )t of uniform chemical otentials is not a rediction of micro—
7

canonical or canonical models and must be investigated to determine its applicability

135

to ﬁnite systems. We do this by trying to comparing the grand canonical expres-
sion for the isotopic yields to the 1:)redictions of approximately n1icrocanonical SMM
calculations. We start by assuming that these isotopic distributions can be calcu-
lated within the grand canonical am'n‘oximation and then test this assumption as
follows.Using a pair of adjacent isotopes, we invert the grand canonical expression
for the isotopic yields of two adjacent. isotopes to obtain an equation for the (ﬂeet-rue

neutron chemical potential:

 

98 A 3/2
mm“) = Tlogi 3.1” ( )
QAHZ “4+1
YE” .-
exp ((8.42 — 8.4.12) /T) 74] (6.24)
AZ

where gTZ, BAZ and Vi]; are the ground state spin degeneracy, the binding energy
and the ground state primary yield for a fragment with (A,Z), respectively. If the Y3}
taken to be the ground state yields predicted by the SMM, pf,” (A, Z) becomes an
”effective SMM” chemical potential. By performing SMM calculations, we ﬁnd the
temperature- and isotopic- dependences of the effective neutron chemical potentials

given in Fig. (6.5) for Carbon and Lithium isotopes from the decay of a “2811 nucleus

at excitation energies of g/A = 3, 6, 9 AIeV.

These effective chemical potentials are essentially the same for the Carbon and
Lithium isotope chains. This insensitivity to element number is consistent with the
concept of a chemical potential and offers support for the use of the grand canonical
expression to describe isotopic distributions. There is a dependence on the neutron
number of the isotope, however, that lies outside of the grand canonical approxima-
tion. This variation in the neutron chemical potential basically comes as a result of

mass, charge and energy conservation for a ﬁnite-size system. We can understand

136

 

  

 

 

 

3|! Binary at T=0MeV
----- Binary at T=4.58MeV
-2 -« — m E5/A=3MeV 0 Li Eg/A=3Mev
9" - — — fit E'/A=6MeV .
x ,,,,, ﬁtEglA=9Mev I CEO/A=3Mev
-4 r 3" x A Li E3/A=6MeV
i x v CE3/A=5Mev
°‘. x . .
A -6 ~ x 0 LI Eo/A=9Mev
E C] CEa/A=9M8V
Z -8 ~ i i ' *6
c ~35
3.
-10 _
a \'
-12 »— f --8.. f
-14 _ @a
-16 ‘ J i A 1
-4 -2 0 2 4 6 8
N-Z

Figure 6.5: The squares, circles and triangles denote neutron chemical potentials
derived from Eq. (6.24) using SMM predictions for Carbon and Lithium isotopic
yields at various initial excitation energies for the decay of the nucleus “25m The
stars and the dot-dashed line denote approximate values calculated from Eq.(6.28)
for T=O and 4.58 MeV, respectively. The error bars denote the statistical errors in
the calculation, which in many cases are too small to be observed in the ﬁgure.

the inﬂuence of these conservation laws most easily at low excitation energies, where
the two largest fragments in the final state are the IMF (Carbon or Lithium in this
case) and a heavy residue which contains most of the remaining charge and mass.
We estimate the inﬂuence of conservation laws at low excitation energy qualitatively
by considering binary decay conﬁgurations. Assuming that a parent nucleus (A0, Z0)

decays into a light fragment (A,Z) and a heavy residue (A0 — A, Z0 — Z) , we can

approximate the yield of fragment (A,Z) in its ground state by

137

liz QC {’08 (A Z) If (do — A, Zn — Z) 7’12“ (625)

a pie - exp [8* (A. — A. Zn - Z >1
A ' (do — A) 3/2 1
A0 Xi

 

where p‘” = (15,7482, [2* and S * are the density of states for the light nucleus in its ground

state level, the density of states and entropy of the heavy residue in its excited state,

.xi.(.4..——.4 ) 3 ’

. _ /‘~ _x .
respectively, The other factor, ,0” 131. z [ 40 ] AT", is the thermal average of the

state density of relative motion.

Replacing the yields in Eq.(6.24) with Eq.(6.25) and assuming A << A0, one ﬁnds
that the effective chemical potentiail depends on the difference in residue entropies,
S" (An — A — 1, Z) — 8* (A0 — A, Z0 — Z). Using an expansion for small changes in
the nuclear entropy from ref. [43], this difference can be expressed in terms of the
difference of binding energies.

8*(A0—A— 1.7) ——S*(A0-—-A,Z0——Z)
1‘ —(B.1(,—A.z.,—Z — BAo—A—1.Zn—Z)/T

—(B,.(Z — BA+1Z)/T + f*/T (6.26)

plus a term depending on the free excitation energy per nucleon, f * = E * /A0—TS /A0.

This difference in binding energies is further related to the neutron separation energy:
Sn(AO — A. Z0 - Z) = BAo—AZo—Z — BAO~A—I.Zn—Z- ((3-27)
One consequently obtains the following expression for the effective chemical potential:

[1." = —8n(A0 — A, Z0 — Z) + f*. (6.28)

138

where the reduced free excitation energy has been approximated by its low energy

limit,

T2
ff: = —g, 60 Z 81l[€l/. (6.29)

For the decay 112Sn——>12C+X , the chemical potential at T = 0, i.e., —s,,(A0—A, Z0—
Z), is plotted as the stars in Fig.(6.5); the binding energies for these calculations
were calculated using the liquid-drop parametrization in ref.[103]. The reduced free
energy f * gives a reasonable estimate for the trend with excitation energy. The dot-
dashed line in Fig.(6.5) gives the chemical potential predicted from Eq. (6.28) for

(j/A = 3 M 6V (T = 4.58ﬂ-IeV). The predicted trend is close to that predicted by
the SMM model (solid circles and squares) but has a somewhat stronger dependence

011 N — Z.

In general, the slope of the effective neutron chemical potential is getting slightly
ﬂatter as the excitation energy or temperature increases. If we consider that the
system undergoes a multiple fragment decay at higher temperatures, it is clear that
approximating the entropy of the remaining system by that of a residue of compa-
rable mass becomes rather inaccurate. The constraints imposed on the total system
by the isospin asynunetry of one observed fragment. should, in that case, be less sig-
niﬁcant. While there is a mass dependence to the effective chemical potential that
is inconsistent with the grand canonical approach, it is useful to note that the mass
dependence of the chemical potential (for these systen'is of more than 100 nucleons) is
small if one is mainly concerned with nuclei near the valley of stability. If one cancels
the chemical potential effects by constructing double ratios like that of the Albergo

formula, the consequence of such ﬁnite size effects becomes negligible indeed.

139

6.2 Empirically Improved Model

6.2.1 Ground State Energies

Since the predicted primary yields of excited fragments is exponentially related to
their binding energies, it is natural to assume that accurate values for the ground
state masses for the observed fragments are needed. Recently, however, it has been
observed that. the masses of nuclei far from the valley of stability can also inﬂuence the
predicted observables through their inﬂuence on the primary distribution[26]. The
conclusion of such studies was that that Liquid Drop Mass (LDM) parametrizations
used in most current SMM codes [103] are insufﬁciently accurate for the prediction

of isotopic distril-)utions.

To address this problem, we use a more accurate description of the masses. In
1.)articular, we use the recommended binding energies values from ref. [104] when avail-
able. Although the sampling of the most probable decay modes discards too exotic
fragments, which would contribute with a vanishing statistical weight, it is still nec-
essary to know some binding energies whose empirical values are not reported in the
literature. Therefore, Souza, Tsang and Danielewicz [105] modify the parametrization

of the LDM formula given in ref. [106]:

.- A—2Z2
3312”" = (a-ux4—a..A2/") 1—k(T)
22 _ 22
“GEM/“A ””0171" (6.30)

where a, and as are the coefficients of volume and surface contributions to the binding

energy and k is the factor of asymmetry modiﬁcation. The coefficient (5A2 corresponds

140

to the usual pairing term:

+6P,,,,.,-,,g, N and Z even
(54,12 2 0, .4 Odd (6.31)

-.

—()p,,,-,.,;,,g, N and Z odd

where N and Z are the numbers of neutrons and protons, respectively. This formula
is as simple as the others adopted in similar statistical multifragmentation models.
However, it is superior to those since it includes corrections to the surface term due
to asymmetry, which are usually neglected. Furthermore, the extra Coulomb term
(i.dZ2/A, also disregarded in the models, takes into account corrections of the Coulomb

energy associated with the diffuseness of the nuclear surface.

Although the parameter set given in ref. [106] predicts binding energies which are
in very good agreement with the recommended values of ref. [104], we used those
data to improve the ﬁt, considering all nuclei in the table with A Z 5. The param-
eters corresponding to the best fit of the experimental data are listed in table 6.1.
To illustrate the in'iprovement in the model, the top panel of Figure 6.6 shows the
difference between the calculated binding energies from the parametrization of the
LDM of ref. [103] used in most current SMM codes and the empirical values (labelled
as A). The middle panel shows the corresponding comparison between the calculated
binding energies using Eq. (6.30) with the improved parameters and the empirical val-
ues (labelled as B). One should note that the total binding energies are used, rather
than the binding energy per nucleon. This improved agreement suggests that the

predictions for unmeasured masses will also be improved.

Despite the improvement in the overall mass predictions, there can be discon-
tinuity between the extrapolated (dashed lines) and empirical values (symbols) as
illustrated in Fig. 6.7. To improve the matching between the binding energies of

the known masses and the ones predicted by our mass formula, we compute average

141

 

 

ABE (MeV)

 

 

 

 

-10 J l I _.

 

O 50 100 150 200 250
Mass Number

Figure 6.6: Difference between the total binding energies predicted by the LDM and
those recomended in ref. [104]. Plot A corresponds to the parameter set adopted in
standard SMM [103], whereas Plot B is obtained using the parameters presented in
this work.

Table 6.1: Best ﬁt parameters to the LDM formula of ref. [105]-[106]. All the values
are given in MeV, except for k which is unitless.

 

av as k ac (Spairing ad
15.6658 18.9952 1.77441 0.720531 10.8567 1.74859

 

 

 

 

 

 

 

 

 

142

5 1O 15 1O 15 20

 

 

 

 

 

 

 

120 k l l l l l ﬁ
100 ‘ 250
A 80 N=15 — 200
E
2 60
V 4 . 150
ES 0
0)
5 ~ 1400
01800 P
E
E 750 ~ ~~ 1350
“0 N=1OO
700 1300
650 ~

 

30 4O 50 60 80 100
Charge Number

Figure 6.7: Total binding energies for different nuclei. The full lines correspond to
the corrected LDM formula, whereas the symbols represent the experimental data of
ref. [104]. The dashed lines correspond to the predictions given by Eq. (6.30). For
details see text.

shifts of the LDM formula from the empirical values and use these shifts to correct
the values in Eq. (6.30). For an isotone that has a lower charge than its isotonic
partners in the compilation of ref. [104] we use the three lightest isotones with the
same value of N in the compilation to compute the shift. Similarly for an isotone that
has a higher charge than its isotonic partners in the compilation of ref. [104] we use
the three heaviest isotones with the same value of N in the compilation to compute

the shift. This shift is then subtracted from the prediction of the LDM formula:
3:1er,): BLD'U __ AN , (6.32)

where

223:1(BA [DZ 1I__ 8.411(2), (6.33)

143

.00an

Bﬁ“‘§is the corresl‘mnding value from the compilation of ref. [104]. Two shifts are
therefore computed for each value of N. This procedure minimizes the systematic
deviations from the recommended values, as is illustrated for four cases in fig. 6.7
with the solid lines denoting the corrected values. Although we use B313?!" whenever
it is available, the solid lines in this figure also include the corrections to the LDM in
the mass region where empirical values are known. In this case, the corrected values
are calculated through an expression similar to Eqs. (6.32)-(6.33), but one takes 2
neighbors to the left and to the right to calculate the average shift. For comparison,
we also show in this picture the uncorrected values of the LDM formula, which are
represented by the dashed lines. The overall behavior of the corrections shows that
the discontimiity between the empirical and extrapolated values is removed. The
difference of these values and the empirical values is also shown in the bottom panel

(labelled as C) of Fig. 6.6. However, we stress the fact that we only use these

corrected values if the empirical information is not available.

6.2.2 Internal free energy

In this work, we have modified the SMM so as to allow accurate predictions of isotopic
properties, but have limited the extent of these modifications in an effort to retain as
many of the predictions of the original theory. In particular, we have retained the high
temperature properties of the fragment free energies, F12, which are paran‘ieterized

here and in the original SMM as:

. T2 _ T2 5/4 T2
* r : A2/3‘l- __Q.__.____ _ _ __ .
.4.A(T) I30 T}; + T2 1 A 60 , (6 34)

where [30 = 18.0 MeV, 60 = 16.0 MeV, and TC 2 18.0 MeV. This expression holds
only for temperatures smaller than TC. For T < TC, this expression has the some of

the expected qualitative behavior of a Fermi liquid: it depends quadradically on T at

144

low temperatures. and falls to zero at the critical temperature Tc,- where the surface
tension vanishes. We do not calculate decays at higher temperatures than that and
tl1(j‘r(:>fore do not concern ourselves here with the properties of the system at T 2 TC.
For 3 MeV < T < 10 MeV, where multifragmentation is important, however, other
expressions for F 3.2(T) with very different thermal properties are conceivable and

should be explored.

Instead we turn our attention to the fact that most fragments at T > 2 MeV are
particle unstable and will sequentially decay after freezeout. This decay is sensitive
to nuclear structure properties of the excited fragments such as their nuclear levels,
binding energies, spins, parities and decay branching ratios. The ﬁrst three of these
quantities also influence the free energies; this can be calculated via the fragment
internal partition functions. There is a self-consistency requirement in the freeze-out
a1‘11_)roximation which dictates that the states from which these fragments decay after
freezeout should be consistent with the Helmholtz free energies used in calculating

the primary yields of the hot fragments at freeze-out.

In order to discuss this self-consistency requirement, we must consider the den-
sity of states /)_,m,es(E) and its mathematieal relationship with the Helmholtz free

excitation energy F *(T):
w. '30 V
e“ /T = [0 dEe—E/Tpstates(E), (6.35)

where the integral is over the excitation energy E of the nucleus. Here we have, for
simplicity, neglected the complications of a degenerate ground state. In the original
papers on the Sh'lh‘l, the level densities corresponding to the Sh’lh’l were not stipu-
lated. We now consider what is required of the density of states to achieve the high
temperature behavior for F *(T) given by Eq. (6.34). Then we will address the general

issue of making the level densities consistent with empirical information and how that

145

impacts the free energies. Finally, we will discuss specific details of the incorporation

of the empirical information into the level density expressions.
High temperature behavior

First we investigate what forms of level densities may be consistent with the free
energies in Eq. (6.34). We note that the functional dependence of F *(T) used in
Eq. (6.35) makes its analytical inversion difficult at high temperatures. It may be
more reasonable to find a smooth real functional form for p_._,mt_es(E) that reproduces
the numerical values for F *(T) at high temperatures than it would be to perform an
inverse Laplace transformation of F *(T) in the complex plane. We note that a Taylor
expansion of F *(T) up to 2—nd order in 1 / T yields the Fermi gas expression:

1 (119/111 .11 _—
PFG.states(E) = 5W eRP (2705'1111115) , (6.36)

where a511,.” is the coefficient of the 2—nd order term of the expansion:

A + 5 3 A2/3
(I, . = — —l[ - .

 

(6.37)

However, this expression is unsatisfactory at high temperatures, as is illustrated in
Fig. 6.8 when the free energies obtained from Eq. (6.36) (dashed lines) are compared
with the standard SIV‘IM free energies in Eq. (6.34) (solid lines). Instead, we take
Eq. (6.36) as a starting point and obtain a useful analytic expression by multiplying
pstatapdE) by an ad hoc energy dependent term to obtain free energy values in

numerical agreement with Eq. (6.34):

)3/2

p5'r1ljll,states(E) = pFG,state(E) €_bs‘”“(as“‘”E . ((5-38)

where b51111! is given by:

146

 

 

 

 

 

 

-100
-200
E -300
v -400
5;
a; 0
1%
(D -1000
(D
LE -2000
-3000
-4000
O 5 1O 15
T (MeV)

Figure 6.8: Internal free energies for A = 20 (upper panel) and A = 200 (lower panel).
The standard SMM expression [Eq. (6.34)] is represented by the full line whereas the
dashed lines stand for the results obtained with the Taylor expansion [Eq. (6.36)].
The Free energy calculated through the level density given by Eq. (6.38) is depicted
by the synrlmls.

 

[15:11:11 2 0.07A_T, (6.39)
182(1+ A ) (640)
T = . . .
4500

The free energies obtained through Eqs. (6.35) and (6.38) are also (_lisplayed in Fig. 6.8
as symbols for two different mass regions. This simple 1‘)ara1neterization is fairly

accurate at temperatures in the range of interest, i.e. T g 10 MeV.

147

Empirical Level densities at low excitation energies for Z315

Several factors motivate the efforts to develop an accurate treatment for the level
densities at low excitation energy for Z 3 15. The first factor is that multifragmen-
tation experimental data are available for lighter fragments in this mass range. The
second is that empirical nuclear structure information[107] is also available for such
light nuclei. A comparable treatment of the level density for the heavier fragments
would be interesting, but necessary structure infm‘matitm is frequently incomplete or
entirely missing. Fortunately, the absence of an accurate treatment for the secondary
decay of the heavier nuclei does not negatively impact. the calculation of the ﬁnal
yields of the lighter nuclei so one can proceed towards accurate predictions at the

present time.

At lower excitation energies, it is customary to discuss the density of levels plate),
rather than the density of states because this deﬁnition is more useful experimentally
when the spins of speciﬁc levels are not accurate known. l\Iathematically, the density
of states is related to the densities of levels for individual spin values p1,.,.,..1_,.( E, J) by:

pstates(E) : Z (2] + 1) plevels(E-. J) (641)
.1

While the spacings between energy levels in a given nucleus generally decrease smoothly
with excitation energy, one often as a practical matter decomposes the empirical level
density p,.,,,,,,,e,.,.13( E, J ) into two expressions that apply in two different approximate
excitation energy domains: (1) one (labelled as ppyl,.,,,.1,,( E, .I)) containing discrete well
separated states at low excitation energies and (2) another (labelled as pCvJWCME, J ))
containing a continuum of overlapping states at higher excitation energies. For Z315,

empirical level ii‘ifor1na.tion[107] is applied as much as possible to the low-lying discrete

148

level density, wherever the experimental level scheme seems complete,
/)D.lr:1rr_‘ls(E1 I) :— 2: (NE, “' E), (642)
1

where the summation runs over the excitation energies E,- corresponding to states
of spin J. Examples for ”We and 31F are shown as bars in Fig. 6.9. For higher
excitation energies, a good approximation to the continuum level density has been
obtained by ref. [108] by combining Fermi liquid theory, a simple spin dependence
and experimental kni‘1wledge. The relevant expressions (shown as dashed lines in Fig.

6.9) for E > EC are [100],

pC,Icvels(E7 I) = [)('(E)f(.]. 0') (6.43)

where

ex12 a E—Eo
”((E) ll 1/ ( )l

 

 

: lgﬁalf'le - Eels/“0‘ ((1.44)

(2.] + 1) exp —(,] +1/2)‘2/202 . r

f(J, 0) Z [20‘2 l’ (0.41))
02 Z 0'0888 “(5 — EMA”. (6.46)

and the level density parameter a = A/ 8. E”, J, A and Z are the excitation energy,
spin, mass and charge numbers of the fragment. E0 is determined by matching the

total high-lying level density to the total low-lying level density as follows,

EC E.~
/ dEfdeaz.....(E,.1)= / dE/dem...i.(E..1). (6.47)
F O

V

40

The comparison in Eq. (6.47) is between the total level densities summed over
spin. This is done primarily to reduce the sensitivity in the matching to uncertainties

in the spin assignments for some of the discrete states. By adjusting the parameter

149

 

 

 

 
  
   
 
    
   

 

 

 

    

 

 

m :2
I. j 10
10 1°
F.) 10 4
m ‘21
E j 24
: '9 18 22
lg E
Q E
10 ‘4
- EXP E
‘0 2 '22:: 22:81:31“: 10 6
- —' SMM-MSU :j
l L l L -:
20 40 O 100 200 300
E (MeV)

Figure 6.9: Level densities as a function of excitation energy for ”We and 31F. Two
energy ranges are plotted to show the behaviors of level densities at both low and
high energy ends. The density of experimentally known levels is shown as bars in the
low energy region. The dashed lines are the extrapolations of the empirical values
according to Eq. 6.43. The dotted lines are the level density (Eq. 6.38) parametrized
from the standard SMM. The solid lines are the level density adopted in this work
(Eqs. 649-652).

150

E0, the total level density for cm’itinuum states can be made to connect smoothly to

the total level density for low-lying states at E < EC and Z < 12.

For the case of Z Z 12, low-lying states are not well identified experimentally and
a continuum approximation to the discrete level density [109] was used by modifying

the empirical interpolation formula of Ref. [108] to include a spin dependence:

1
90.16.181.45 J) = T exp[(E — Ell/T1]
1

(2.] + l) exp[—(J +1/2)2/203]
21(211 + 1) exp[—(J,- + 1/2ll‘2/2‘761la

 

(6.48)

for E 3 EC, where the spin cutoff 1‘)a.rameter 08 = 0.0888‘/(1(EC — E0)A2/3. For
Z Z 12, the values of EC 2 EC(A, Z) were taken from Ref. [108] as well as parameters
T1 = T1(A, Z) and E1 : E1(A, Z) and in this case, the approximate level density
(Eq.6.48) was used to complement the empirical level density for low-lying states. The
connection point EC to high-lying states, for Z < 12, was chosen to be the maximum
excitation energy up to which the information concerning the number and locations
of discrete states appears to be complete so that the empirical level density (Eq. 6.42)

was solely applied for low—lying states.
Matching low and high excitation energy behavior

Now, we turn to the requirement of self-consistency between the expression for F*(T)
and the level density relevant to secondary decay. In general, secondary decay is
more sensitive to nuclear structure quantities such as the excitation energies, spins,
etc. as the system decays towards the ground state. At low excitation energies, it is
more accurate using empirical level densities in place of the expression in Eq. (6.34),
which does not even depend on Z. As the excitation energy is increased, however,

the continuum level density becomes very large, little sensitivity to nuclear structure

151

details remains and a simpler expression like Eq. (6.34) may suffice.

In the following. we take p_g,\,,\,ﬂ,t,,,e,,(E) to be the behavior of the state density
at high energies and match it to the continuum part of the empirical state densities
at lower excitation energies. The net result is a. set of level density and state density
expressions that span the range of excitation energies relevant to multifragmentation
phenomena. For E* < EC, one uses the expression for the discrete, low-lying state

density

p(E, J) = pD(E*,J). (6.49)

For EC < E < E(. + AE, the new level density is an interpolation involving the contin-

uum expression relevant at low excitation energies between [)(vgstam and psmjuﬁmm,

, * E“ — EC
,1}; .J) =-- ME .J><1— TE )
E“ — E,

+ps.11111(E*1J)—AE—1 (6-50)

where AE 2 2.5.4 MeV provides a smooth transition from pC to [15,111]. The SMM
level density (shown as dotted lines in Fig. 6.9) can be incorporated with a similar

spin dependence as in Eq. (6.43),

ps.11.11(E*1J) = ps.11.11(E*)f(J~0)- (6-51)

For E* > EC + AE, the new density simply becomes the same as the SiV’IlV’I level

(lCIISitY 105.1111].

P(E*1J) = MAM/NE: Jl- ((5-52)

In Fig. 6.9, the empirically modiﬁed level density described in Eqs. (649-652) is

plotted as solid lines for 20N e and 31F.

152

This procedure uses the empirical information for excitation energies E < EC, a
linear interpolation for E, < E < E, + AE, and [25,1],.\,‘St,,,..,,(E) at higher values of
the excitatimi energy. The level density pg in Eq. (6.43) can be used as an proper
extension to the l(,)w-lying level density [2D in Eqs. (6.42) and (6.48) and a bridge
for matching to the SMM level density at continuum. Such a matching procedure
provides a state density that is emrnrically based at low excitation energies but become
progressively more uncertain as the excitation energy is increased above E* z EC.
This uncertainty in the thermal properties of nuclei at such high excitation energies
is not a question of ﬁnding an appropriate interpolation, but is, in fact, a fundamental
issue that must be resolved by comparisons to experimental data. For example, other
expressions can be proposed for the level density at E“ > EC and comparisons of
experimental data to SMM predictions of sensitive multifragment observables can be

used to constrain the level densities at high excitation energies.

Free energies F*(T), which reflect contributions from the discrete excited states
are obtained by inserting this parametrization for pst,,,es(E) into Eq. (6.35), and
performing a numerical integration. To facilitate the insertion of these free energies

into the SMM algorithm, we parameterize F*(T) by:
1

F*(T) = 3*td(T) 1_ 1+ exp[(T — T0)/ATl ,

 

(6.53)

where 8*td(T) stands for the standard SMlV‘I internal free energy, Eq. (6.34). The
parameters To and AT are adjusted to reproduce the numerical calculation of F *(T)
provided by Eqs. (6.35) and (6.49)-(6.52) for T g 10 MeV. In these ﬁts, a value for
AT 2 1.0 MeV is used for most nuclei (The exceptions are mainly very light nuclei),
while T0 is varied freely. The accuracy of the ﬁt is illustrated in Fig. 6.10, which
compares the exact values of F *(T) (symbols) to the approximation given by Eq.

(6.53) (solid line), for a 2ONe nucleus. The dashed line in this figure represents the free

153

 

 

 

 

A -50 ~ ”Ne
>
(D
3
53
a; -100 ~
C
Lu
(1)
9
LL
-150 1
_200 l l 1 1
0 2 4 6 8 10
T(MeV)

Figure 6.10: Conmarison between F*(E) calculated through Eqs. (6.35) and (6.49)-
(6.52), symbols, and the approximation given by Eq. (6.53), full line. To illustrate
the influence of quantum effects at low temperatures, the dashed line represents the
free energy used in standard Sl\r"ll\/I calculations Eq. (6.34). For details see text.

energy used in standard 8111M calculations in which the experimental discrete levels
are neglected. The matching procedure allows the discrete excited states to dominate

the low ten'iperature behavior, while the high temperature behavior remains similar

to that of the original Sl\'Il\»’I consistent with the goals stated above.

Because the empirical level densities vary from nucleus to nucleus, the parameters
To and AT used to obtain F*(T) must be ﬁtted for each nucleus. Fits of the same
quality as that. for QONe are achieved for all the light nuclei (Z S 15). These ﬁtted
values of T0 are shown as symbols in Fig. 6.11. We do not perform such ﬁts for
Z > 15, because the level density information there is less complete. We nevertheless
extrapolate the main trend of the parameters to heavy nuclei, for which detailed

experimental information on discrete excited states is not available in order to prevent

154

 

10 ~

 

 

 

I J. 1111L1_L L 1 IlLllil

1 10 102
A

 

Figure 6.11: Best ﬁt values of T0 for different nuclei (symbols). The dashed line
corresponding to Eq. 6.54 is used for Z > 15.

spurious discontinuities in the equilibrium primary yields. As mentioned above, there
seems to be a very weak dependence on AT and, therefore, we assume AT 2 1.0 MeV
for Z > 15. In spite of the uncertainty in extra.1')olating To, the dashed line in Fig.

6.11 shows that
T0 = 2264‘”8 MeV (Z > 15) (6.54)

describes the trend for the lower masses and we adopted it at for the higher masses

as well.
6.2.3 Empirical Sequential Decay

The typical time scale for a multifragmentation process is less than about 100 fm/c.

The ﬁnal production of isotopes is modiﬁed greatly by secondary decay from the hot

155

primary fragments at a much longer time scale. However, only the contaminated
ﬁnal yields can be directly 111easured in this type of heavy ion reactions. Therefm‘e,
an accurate secondary decay procedure is indispensable to assess the contributions
from secondary decay and deduce the information of the primary hot system from

ex1'1erimental data.

A11 einrnrical sequential decay procedure was developed to decay the hot fragments
produced in the primary SMM code. The strategy is to include as much experimental
information as possible for a precise calculation of secondary decay and to make
secondary decay and primary breakup procedures intrinsically consistent, such as
using the same level density and empirical information. Basically the sequential
decay procedure consists of two parts. One is to decay particles with Z315 through
a large empirical table. including all the. states of nuclei with known information such
as binding energy, spin, isospin, parity and decay brancl‘iing ratios. The other part is

to use the Gemini code[110] for particles outside the empirical table (usually Z>15).

Decay table for Z315

From the inmlmnentation of the level density (Eqs. ( 649-652)), we can construct
a ’table’ containing properly sampled levels associated with properties like spins,
isospins, parities and so on. For excitation energies E < EC and Z 3 15, each of
the entries in the table corresponds to one of the tabulated empirical levels. When
the information 011 the level is complete, it is used. For known levels with incom-
plete spectroscopic information, values for the spin, isospin, and parity were chosen
randomly as follows: spins of 0—4 (1/ 2-9/ 2) were assumed with equal probability for
even-A (odd-A) nuclei, parities were assumed to be odd or even with equal proba-
bility, and isospins were assumed to be the same as the isospin of the ground state.

This simple assumption turns out to be sufﬁcient since in effect most of spectroscopic

156

information is known for these low-lying states.

For unknown low-lying states and high—lying states, the levels were sampled ac—
cording to the sopl‘iisticated level density algorithm discussed in the previous sec-
tion at (’liscrete excitation energy intervals of 1 MeV for E“ < 15 MeV, 2 MeV for
15 < E‘ < 30 MeV, and 3 MeV for E" > 30 MeV in order to reduce the computer
memm'y requirements. The results of these calculations do not appear to be sensitive
to these binning widths. A cutoff energy of Eaton/A = 5 MeV was introduced cor-
responding to a mean lifetime of the continuum states at the cutoff energy about 125
fm / c. For simplicity, parities of these states were chosen to be positive and negative
with equal probability and isospins were taken to be equal to the isospin of the ground

state of the same nucleus.

Sequential decay algorithm

Before sequential decay starts, hot fragments from primary breakup need to be pop-
ulated over the sampled levels in the prepared table according to the temperature.
For the ith level of a given nucleus (A,Z) with its energy E: and spin J,, the initial
population is,

(2.], + 1) ex1_)(—Ef/T)p(Ef, J,)
242$ + 1) €XI)(-E£"/T)ﬂ( £3111)

 

xznmz) ma)

where Y0 is the primary yield of nucleus (A,Z) and T is the temperature associated

with the intrinsic excitation of the fragmenting system at breakup.

Finally all the fragments will decay to their ﬁnal states from top to bottom
throughout the table. Eight decay branches of n, 211, p, 2p, (1, t, 3He and alpha were
included for particle unstable decays of nuclei with Z315. The decays via gamma rays
were taken into account for calculations of the ﬁnal particle stable yields. If known,

tabulated branching ratios were used to describe the decay of particle unstable states.

157

Where such information was not available, the branching ratios were calculated from

the Hauser- F eshbach formula[23],

F( C.
— = —+, (6.56)
F Zd Ga

where

Cf'd : (1(1181d3163llplp3)2
|Jd+Jp| IJ +J| I
p 1 1 c e —1
X Z Z ”"3“ )Tz(E) (6.57)
le‘jd-JellzlJp—Jl

 

for a given decay channel (1 (or a given state of the daughter fragment). JP, Jd, and
J8 are the spins of the parent, claugl'iter and emitted nuclei; J and l are the spin and
orbital angular momentum of the decay channel; T[(E) is the transmission coefficient
for the lth partial wave. The factor [1 + ﬁpwd7r€(—1)’] / 2 enforces parity conservation
and cliqjiends 011 the parities 7r 2 i1 of the parent, daughter and emitted nuclei.
The Clebsch—Gordon coefficient involving 1p, Id, and [6, the isospins of the parent,
daughter and emitted nuclei, likewise allows one to take isospin conservation into

account.

For decays from empirical discrete states and l 3 20, the transmission coefﬁcients
were interpolated from a set of calculated optical model transmission coefficients;

otherwise a parameterization described in Ref. [109] was applied.

6.3 Model Predictions and Comparisons

6.3. 1 Caloric Curve

Before presenting predictions for isotope distributions and other observables for which

the present theoretical developments were undertaken, we examine predictions of the

158

present. model for the caloric curve and the primary fragment multiplicities, both
of which displayed features in prior SMM calculations that are characteristic of low
density phase transition. For example, SMM calculations predict an enhanced heat
capacity for multifragnientating systems reflecting the latent heat for transforming
nuclear fragments (Fermi liquid) into nucleonic gas. Fig. 6.12 shows the caloric curve,
i.e. the dependence of the mean fragmentation temperature (Tm) on excitation energy,
for a system with A02168 and Z0275. In both panels, the dotted lines indicate the
relationships predicted by the orginal SI\~‘II\-I, the solid lines denote the corresponding
predicticms of the Sl\ll\l with all the modiﬁcations discussed in this paper and the
dashed lines present the results provided by an SMM calculation that uses the new
binding energies of Eqs. (630-633) and the old parameterization of ref. [36] for the
Helmholtz free energies. These latter calculations allow one to assess the impact of

the changes in the binding energies and free energies independently.

The two panels provide the caloric curves COI‘I‘CSI)OI1(1111g to two different c011—
straints on the density. In the lower panel, a multiplicity-dependent breakup density
[36] is assumed, corresponding to a ﬁxed interfragment Spacing at breakup; this leads
to a pronounced plateau in the caloric curve for all three calculations. By taking into
account kinetic motion and Coulomb interaction, we have calculated the pressure
using the relationship

M - T Z3 62

p:
v, + 6RV’

 

(6.58)

where P is the pressure, M is the total multiplicity, Vf is the free breakup volume and
V is the total volume. The pressure corresponding to these multiplicity-dependent
breakup densities is plotted in the lower panel of Fig. 6.13. The corresponding pri—
mary fragment multiplicities are shown in the lower panel of Fig. 6.14. Consistent

with the conclusions of ref. [111], we ﬁnd the requirement of approximately constant

159

 

 

 

 

 

 

 

8 e Fixed density at p/p0 = 1/6
6
4
A 2 --
i
2 O l l l l l l
*- 8
6 .
4
2 _
0 L l 1 I l 4
O 2 4 6 8 1O 12
E*/A (MeV)

Figure 6.12: Caloric curves are shown for calculations of the system of A=168 and
2:75 at ﬁxed breakup density and multiplicity-dependent density. The dotted lines
are calculated from the standard SMM. The dashed lines are calculated as empirical
binding energies are taken into account. The solid lines are obtained from the im-
proved model with empirical modiﬁcations of both binding energies and free energies.
interfragment. spacing corresponds to nearly constant breakup pressures. 111 the up-
per panel of Fig. 6.12, we show the corresponding caloric curves calculated at ﬁxed
breakup density p/po = 1/6; these show a steeper dependence on excitation energy
and the small maximum displayed in the lower panel at excitation energies of about

3 MeV disappears. The corresponding pressures at constant density, shown in the

upper panel of Fig. 6.13, increase strongly with excitation energy.

These ﬁgures reveal the trends of three models to be similar. In general, the
temperatures in the plateau region at E*/A = 3 — 8 MeV in the lower panel are

larger for the calculations using the improved free excitation energies. Calculations

160

 

0.1

 

 

  

 

 

 

 

008 Fixed density at p/p0=1/6
0.06
0.04 1’
“E 0.02
a 0 1 I 1 1 1 1
E 0.3 . .. .
o_ Multnphcnty-dependent densnty
0.2 ~
0.1 3 ............
O m l ' m l 1
0 2 4 6 8 1O 12

E*/A (MeV)

Figure 6.13: Pressure curves due to kinetic motion and Coulomb interaction (see Eq.
6.58) are plotted for the system of A=168 and 2275 at ﬁxed breakup density and
n1ultiplicity-dependent density. The dotted lines are calculated from the standard
SMM while the improved SMM presents the solid lines.

with the i1111')roved free excitation energies require lower mean total excitation energies
to achieve the same temperature than do calculations with the original free excitation
energies . This lowers the latent heat for the transformation from excited fragments to
nucleon gas and lowers the temperature at which the transition occurs. The influence
of the improved binding energies on the caloric curve is less obvious, but this change
seems to be largely responsible for the differences between the original SMM and the

ﬁnal improved model at E*/A = 6 MeV.

Discussions of the nuclear caloric curve usually focus 011 the excitation energy
dependence of the temperature and ignore the density dependence. To illustrate

that the phase diagram is two dimensional and a density dependence does exist, we

161

 

4O

 

 

 

 

 

 

30 1 Fixed denSIty ammo: 1/6 ............
20
a 10»
32
TQ- O 1 1 1 1 1 l
“‘5
2 30 —
20 e
10 -
O -. 1 1 1 L 1

E*/A (MeV)

Figure 6.14: Average breakup multiplicities are shown for the system of A=168 and
2:75 at ﬁxed breakup density and multiplicity-dependent density. The dotted lines
are calculated from the standard SMM while the improved SMM presents the solid
lines.

constrast in Fig. 6.15 the density dependence (upper scale) of the temperature at
a ﬁxed excitation energy of E*/A=6 MeV (open squares) to the excitation energy
dependence (lower scale) of the temperature at a ﬁxed density of p/Po = 1/6 (solid
circles). Both the excitation energy and the density dependences of the caloric curve
are clearly important. It is therefore relevant to ﬁnd and measure observables that

constrain signiﬁcantly the freezeout density.

6.3.2 Elemental and Mass Distributions

Calculations of the mass distribution (left panel) and charge distribution (right panel)

for excited primary fragments are shown in Fig. 6.16 for a system with A0 = 186

162

p/pO

 

 

 

 

 

10 1
8-.Mﬁ~. 3A.. ...3
---E1 T(p/p0) at fixed PM = 6MeV

7 .- _. T(E*/A) at fixed p/p0=1/6 -~ 7

9 6 ” '16
(D
E

l— 5 a ‘ 5

1:1"

4 3 ‘ 4

3 L ! 1 l L 3

0 2 4 6 8 1O 12

E*/A (MeV)

Figure 6.15: Dependences of temperature on excitation energy and breakup density
are shown for the system of A=168 and Z=75. Calculations as function of excitation
energy at ﬁxed density of 1 / 6 normal density are shown as solid circles. Calculations
as function of density at ﬁxed excitatimi energy are shown as open squares.

163

 

Standard

Empirical

A‘_1 i

1 l L LLLLli ‘4

Yield

1 L4 1 A LlLl

 

 

L

 

‘JLL

 

lLLlLlLLLLJl

4

,4

a

1 l AlLLJiLllll'Ll‘lllLJlJJl

20 40
A Z

O

 

 

Figure (3.16: Mass and charge distributions are shown for the system of A=186 and

Z=75. The dashed lines are the calculations from the standard SMM. The solid lines
are calculated using the improved model.

164

and Z0 2 75 at E*/A : 6 MeV. The dashed lines denote the predictions using the
original SMM and the solid lines denote the predictions using the improved SMM.
The improved SMM calculations for the I_irimary distributions fluctuate about the
smooth distrilmtions of the original SMM for Z < ‘20 and A < (50 and then fall
l‘mlow the original SMM at higher mass and charge. This trend of reduced yields
at higher masses and charges is related to the tendency shown in Fig. 6.6 for the
binding energies in the original SMM to consistently exceed the empirical values at
Z > 20 and A > 60. Because conservation of mass and charge dictates that an
increase in the yields of heavier fragments must be compensated by a ('lecrease in
the yields of the lighter ones, one does not see a comparable over-prediction of the
primary yields of the lighter fragments by the original SMM. Besides the decrease in
the yields of heavier fragments, there is a tendency of the improved SMM predictions
for the lighter fragments to fluctuate about the smoother predictions for the original
SMM. These fluctuations are related to the influence of shell and pairing effects on
the ground state masses. These fluctuations have no significant impact on the final

yields after secondary decay as discussed below.

The corresponding final mass (left panel) and charge (right panel) distributions
after secondary decay are shown in Fig. 6.17. The solid lines denote the predictions
using the improved SMM. To explore how significant are. the fluctuations in the pri—
mary distributions due to the influence of shell and pairing effects on the ground
state masses, we have decayed the primary fragments from the standard SMM via
the same empirical secondary decay procedure discussed in Sect. 6.2.3. The final
mass and charge distributions of the standard SMM are shown as the dashed lines in
Fig. 6.17. Minimal discrepancies are seen in low mass and charge regime i1‘1di(‘:ating
that the secondary decay mechanism washes out the fluctuations in the primary dis-

trilmtions due to the influence of shell and pairing effects on the ground state masses.

165

 

Yield

1111i

O
‘1' 111

 

 

 

 

 

Wﬁq

111111!liLlllllllllljlllllll l I l

0 25 50 75010 20 3O 40
A Z

 

Figure 6.17: Final mass and charge distributions after applying the empirical sec-
ondary decay procedure discussed in Sect. 6.2.3. The dashed lines are calculated
from the primary results of the standard SMlV‘I while the solid lines are from the
improved model.

Meanwl'iile, significant differences on heavy fragments remain.

Secondary decay corrections to the standard SMlVI have been implemented by
Botvina et al. in ref. [112], and the latter code has provided the bulk of the corn-
parisons to experimental data prior to the development of the present model. In the
Botvina code [112, 9], the secondary decay is calculated by the W'eisskopf or Fermi
breakup formalisms and the final ground state masses of nuclei with A > 4 are taken
from a liquid drop mass formula. In the following, we compare the predictions of the
present model for the final yields to the corresponding predictions of the Botvina code
[112, 9] and to the experimental data for central 124Sn+1248n collisions at E/A=50
MeV of refs. [41, 113]. The predictions of the Botvina code for the same system are

shown by the dashed lines and the results from the present model by the solid lines

166

Yield

 

 

 

_ LLLJLlJlllLLJLiLllLII

l

 

llllllillli

 

111111111

 

'
Illlllllllllflllllllllillllll

 

O

25 50 75 0

A

10 20 30 40
Z

Figure 6.18: Final mass and charge distributions from the present model (solid lines)
and the Botvina version (dashed lines) are shown. For reference, some measured data
is plotted as solid circles.

in Fig. 6.18. For reference, the solid points show the corresponding experimental
data. No special attempt has been made to optimize the parameters of the calcula-
tions to achieve the best representation of the data. Clearly the present model and
the Botvina code differ significantly in their predictions for the elemental and mass
distributions, especially for the heavy fragments, and more important comparisons of

isotopic composition are shown in the next section.

6.3.3 Isotope Thermometry

In Fig. 6.19, the primary isotopic distributions for four elements emitted are shown
for a system with A02186 and 20:75 at E* / A = 6 MeV. The solid lines show pre-
dictions for the present model and the dashed lines show predictions of the Botvina

code [112, 9]. The two calculations produce primary isotopic distributions that are

167

 

 

 

 

Yield

 

 

 

 

.' 1 l l I. l l 1

-5 O 5 10 O 5 10
N-Z

 

Figure 6.19: Primary isotopic distributions are shown for Be, C, O and Ne nuclei.
The dashed lines correspond to the calculations of the Botvina code while the solid
lines represent the results of the improved model.

considerably broader and more neutron rich than corresponding distributions after
secondary decay shown in Fig. 6.20. The final distributions for the present model
(solid histograms) are also broader and more neutron—rich than the corresponding dis-
tributions predicted by the Botvina code(dashed histograms); the major differences
between the two calculations are found primarily in the predictions of the neutron
rich isotopes, where the fall—off for the Botvina code is more precipitous than it is
for the present code. The differences in the yields for isotopes on the proton rich
side are not as large.The measured isotopic distributions denoted by the solid points
are broader than those predicted by the Botvina code and more neutron rich than
either calculation; however, the parameters of the two codes were not optimized to

reproduce the data.

Isotope thermometers have been utilized as the primary probes for extracting the

168

W—

 

 

 

 

dM/dQ

 

 

 

 

 

 

N-Z

Figure 6.20: Isotopic distributions are shown for isotopes from Li to 0. Experimental
data is shown as the solid circles. The dashed lines denote the Botvina calculations
and the solid lines are the ﬁnal distributions after decaying the hot primary fragment
via the empirical secondary decay procedure discussed in Sect. 6.2.3.

 

169

 

 

Table 6.2: List of isotopic thermometers with AB > 10 MeV. The left column shows
the IMF thermometers involving isotopes of 3 *5 Z S 8. The right column lists the
He thermometers involving the isotope pair of 3*4He.

 

 

 

 

IMF-thermometers AB (1% eV) a He-thermometers AB (M eV) a
61713/1th 11.472 5.898 2’3H/3’4He 14.321 1.591
7~8Li/11~12C 16.690 5.361 (”Li/“He 13.328 2.183
8.9Li/11,12C 14.658 3.351 7v8Li/3'4He 18.546 1.984

Q'IOBe/II'IQC 11.910 1.028 8’9Li/3viHe 16.514 1.240
'1‘~12B/1‘"2C 15.352 3.000 9'10Be/3’4He 13.766 0.380
7.81.1 #5460 13.631 2.773 ”HE/“He 17.208 1.110
12,133/11120 13.844 5.278 ‘2‘13B/3'4He 15.700 1.953
121130/11'12C 13.776 7.917 l2*13C/3'4He 15.632 2.930
8,9111 /15»160 11.599 1.733 l3440/314er 12.401 0.726
l3“C/H’IQC 10.545 1.962 15'IGN/3'4He 18.089 3.578
15’16N /11»12C 16.233 9.669 16»1"'O/3~4He 16.434 8.536
11.1213/15460 12.293 1.551 17’180/3'4He 12.534 0.236
16v170/“912e 14.578 23.069
12Jim/15.160 10.785 2.729
lame/15,160 10.717 4.094
17~180/11~12C 10.678 0.637
15’16N/15'160 13.174 5.000
16,170 /15»160 11.519 11.930

 

170

 

 

 

 

6 1
3‘ ......................................................... r
a)
Z 4
8 g ,
f— -' 5% I.
DD 5! ..
2" DC]

 

 

l l l l l l J

0
30 35 4O 45 50 55 60 65 7O
A1+A2+A3+A4

 

Figure 6.21: Isotopic temperatures are extracted from 18 IMF thermometers (see
table 6.2) with 3 3 Z S 8 and AB > 10 MeV. Experimental data is shown as the
solid circles. The open squares are the calculations from the improved model. For
reference, the primary temperature calculated from the present model is shown as the
dashed line.

caloric curve of the nuclear liquid-gas phase transition. Since these observables are
(:(.)11st.ructed from the isotopic distributions, they share the sensitivity to structure ef-
fects in the secondary decay discussed above. In the isotopic thermometer technique,
the temperature is extracted from a set of four isotopes produced in multifragment
breakups [54] as shown in Eqs. (6.9-6.12) of section 6.1.1. Although this isotopic
thermometry method is derived within the context of the grand canonical ensemble,
it is still valid in the microcanonical ensemble as discussed in section 6.1.2. It has
been applied to a wide variety of reactions where it has been regarded as an effective
or ” apparent” temperature that may differ somewhat from the true temperature T

because of the extra yield in the ground state due to feeding from secondary decay.

171

 

 

 

 

 

 

67 1
5 A A
g 4 A
8 5 .3
l-_ ’l 6. l.
IA ‘3 .
25- A.
A A
A A
A
l l l l l l

 

O l
30 35 404 505 606570
A1+A2+A3+A4

Figure 6.22: Isotopic temperatures are extracted from 18 IMF thermometers (see
table 6.2) with 3 S Z S 8 and AB > 10 MeV. Experimental data is shown as the
solid circles. The open triangles are the calculations from the Botvina model. For
reference, the primary temperature calculated from the present model is shown as the
dashed line.

The relationsl‘iip between T130 and T can be calculated within an appropriate statis-
tical model for the fragmentation process if one exists. In general, one should choose
a set of four isotopes with large AB if one want to extract reliable values of apparent

ten'iperature Tm.

To studying the corrections due to secondary decay, measured and calculated tem-
peratures are extracted from the double ratios of suitably chosen isotopes. In Fig.
6.21, 18 IMF thern‘iometers are plotted with the requirement of 3 S Z 3 8 and
AB > 10 MeV (see details in Table 6.2) in order to limit the effects of nonequilib-
rium emissions and reduce the temperature ﬂuctuations. As one can see in Fig. 6.21,

the calculated isotopic temperatures (open squares) from the in‘iproved model agree

172

 

 

 

well with the c()rresponding experinnmtal values (solid points). The two tlmrrmnne-
ters involving 1”Be and 18O are significantly higher than the others due to structural
effects[31], which are well reproduced by the improved SMM since we have already in-
corporated the experimental structural information in secondary decay. All the other
16 IMF thernn)meters give a much lower temperature than the primary temperature
(dashed line) calculated in the model. which shows the effect of the additional feed—
ing to the ground states due to secondary decay. The calculations give somewhat
smaller values for T150 than the data, which may indicate that the secondary decay
corrections are overestimated or that the internal temperature or density are not op-
timal. In addition, these 16 thermometers follow a decreasing trend as a. function
of the total mass number of the four involved isotopes, indicating that the heavier
fragments are more strongly influenced by secondary decay. The open triangles in
Fig. 6.22 represent the corresponding predictions of the Botvina Code of ref. [112, 9]
where the secondary decay of hot primary fragments is calculated without regard to
the detailed structure of the discrete levels in the excited nuclei. The neglect of this
structural information is probably responsible for the poor level of agreement. with

the experimental isotopic temperatures for the heavier isotopes.

On the other hand, light isotopic thermometers are also studied to show the
possible effects from the radiative emission of light particles prior to the multifragment
breakup. In Fig. 6.23, He thermometers involving the pair of 3"‘He with AB >
10 MeV (see details in Table 6.2) are shown. Although the calculated temperatures
with the present model (open squares) are systematically lower than the experimental
values, they seem to track the measured trends. And indeed a better reproduction
of the experimental values can be obtained by assuming that 2/ 3 of the measured
3He yield is of a pie—equilibrium origin such as surface emission from the expanding

system prior to breakup. Adding such a contribution to the predicted yields results in

173

 

V‘Q
..

 

 

 

15—

9
m 10 -- 1
3
.3
F.
i o
5 _,___-Q ..... i "‘4 ................ _
o o .
O O O o O o o ’ ’
O O

 

 

I l l l l

10 15 20 25 30 35 4O 45
A1+A2+A3+A4

 

Figure 6.23: Isotopic temperatures are extracted from 12 light thermometers (see
table 6.2) satisfying AB > 10 MeV and involving the isotope pair of 3"lHe. Experi-
mental data is shown as the solid circles. The open circles are the calculations from
the present model without corrections of nonequilibrium emissions. For reference, the
primary temperature calculated from the present model is shown as the dashed line.
predictions given by the open squares in Fig 6.24 which are in good agreement with
the experimental data. (solid circles). In contrast to the improved SMM, the isotopic
temperatures calculated in the Botvina code are compared to the same measured
values for these He thermometers in Fig 6.25. The calculations from the Botvina
code don't follow the experimental trends well although they do better replicate
the ten‘iperatures of the H/ He and Li/ He thermcm‘ieters without corrections. The
isotopic: yields relevant to these thermometers reflect the secondary decay of heavier
fragments that is calculated for A > 16 using the VVeisskopf model and for A g 16

via a ”Fermi breakup” model; both models neglect the detailed structure of the

lighter nuclei that is included in the present model. If this structure information were

174

 

 

 

 

15 —
’>‘
an 10 - i
3
I_fe

3 .
5 ‘“ "I """"""" g "; """"" 5 g a """""""""
a 0
10 15 20 25 30 35 40 45

A1 +A2+A3+A4

Figure 6.24: Isotopic temperatures are extracted from 12 light thermometers (see
table 6.2) satisfying AB > 10 MeV and involving the isotope pair of 3"‘He. Experi-
mental data is shown as the solid circles. The open squares are the calculations from
the present model with corrections of 11(')11equilibrium emissions of 3He. For reference,
the primary temperature calculated from the present model is shown as the dashed
line.

included in the Botvina Code, it is likely that the predictions shown in Fig. 6.25
would be changed; the agreement shown for the H / He and Li / He thernn‘nneters may

be somewhat fortuitous.

175

 

 

 

15—

 

 

 

9
a: 10 -
g 1
FE?
A
w-- _--------..‘-A ............... . ................ -
5 t 6g
A X:
A
A
10 15 20 25 30 35 4O 45
A1+A2+A3+A4

Figure 6.25: Isotopic temperatures are extracted from 12 light thermometers (see
table 6.2) satisfying AB > 10 MeV and involving the isotope pair of 3"’He. Experi-
mental data is shown as the solid circles. The open circles are the calculations from
the Botvina code. For reference, the primary temperature calculated from the present
model is shown as the dashed line.

176

 

 

Chapter 7

Isospin Dependence of the EOS

The nuclear equation of state (EOS) has been one of the main topics on studies of
nuclear matter recently. To better understand astrophysics phenomena such as super-
novae [65] and neutron stars [66, 67] as well as nuclear physics, a better understanding
of the nuclear EOS is in need. For example, maximum mass of neutron stars [114] and
explosion mechanisms in core-collapsed supernovae [67] require accurate knowledge
of the EOS. Under laboratory-controlled conditions, the EOS has been investigated
by colliding nuclei and measuring compression sensitive observables. The nuclear
monopole and isoscalar dipole resonances, for example, sample the curvature of the
EOS near the saturation density pg [68]. Measurements of the collective flow of par-
ticles emitted from the dense and compressed matter formed at relativistic incident
energies can sample the EOS at densities as high as 4m, [69]. In both types of ex-
periment, investigations have primarily focused upon terms in the EOS that describe
symmetric matter (equal numbers of protons and neutrons), leaving the asymmetry
term that reflects the difference between neutron and proton densities largely unex-
plored [115]. For very asymmetric matter, however, details of this asynnnetry term
are critically important. For example, the asymmetry term dominates the pressure
within neutron stars at densities of p 3 2,00, determines certain aspects of neutron
star structure, and modifies proto—neutron star cooling rates [66, 67].

177

In the following sections hybrid model calculations are performed with the isospin
dependent BUU model [116, 117] and the microcanonical SMM model incorporated
with an empirical secondary decay procedure [26]. The isotopic distrilmtions of final
fragments are shown to compare the. model calculations with the experimental data.
Two observables of relative free 11 / p densities and mirror nuclei ratios are discussed
to elucidate the sensitivity to the density dependence of the asyi‘m'netric terms in the

EXDS.
7 .1 Density Dependence of Asymmetric EOS

Various studies have shown that the mean energy per nucleon ((0, (5) in nuclear matter
at density p and isospin asymmetry parameter 6=(p,.-pp)/(pn+pp) can be approxi-

mated by a parabolic function
()(p, 6) = e(/), 0) + S(p)(52 (7.1)

where (’(p, 0) provides the EOS of symmetric matter, and S(p) is the synnnetry en—
ergy [66, 67, 115]. Different functional forms for S (p) have been proposed [114], all
consistent with constraints on S ( p0) from nuclear mass measurements. Some theoret-
ical studies have explored the influence of the density dependence of S(p) on nuclear

reaction dynamics [114]—[119].

Calculations of energetic nucleus-nucleus collisions [116]- [119] reveal that the rel—
ative emission of neutrons and protons during the early non-equilibrium stages has a
robust sensitivity to the density dependence of S (p). In general, pre-equilibriun'i neu-
tron emission increases relative to pre—equilibrium proton emission when the density
dependence of 5(7)) is made weaker, e.g.“softer”. Enhanced pre-equilibrium neutron
emission reduces the neutron-to-proton ratio in the dense region that remains behind

[116,118]
178

A!

 

 

 

Central collisions of complex nuclei of comparable mass provide the principal
means to produce and study nuclear matter at densities either signiﬁcantly above
or below the saturation value. In near central Sn+Sn collisions at an incident en-
ergy of E/A=5() MeV, for example, matter is compressed to densities of about 1.5
[)0 before expanding and disassmnbling into 6—7 fragments with charges of 3S Zﬁ 30
plus assorted light particles. Detailed analyses imply that such multifragment dis-
assemblies occur at an overall density of p z [)0/6-p0/ 3 and over a time interval of
about 7' $304 00 f772/(: [120]-[125]. Essentially all initial isotopic compositions are
determined by the prrmerties of the system during this narrow time frame when the
density is signiﬁcantly less than p0. This implies that fragment isotopic distrilmtions
may have a signiﬁcant. sensitivity to the density dependence of S (p). One can also
enhance the sensitivity to the asymmetry term S(p) - 62 by varying the N/Z of the

initial system.

Following the reference By M. Prakash et al [114] we write down the synnnetry

energy,
S(/)) 2 (22/3 — 1)gE2~[(/.2/3 — F('u.)] + SOF(u) (7.2)

where E? is the Fermi energy at saturation, u E p/po is the reduced nucleon density,
So E S (/)0) is the symmetry energy at normal nuclear density and F (n) represents
the potential contributions to the symmetry energy with F (1) E 1. The mean-ﬁeld
potentials for neutrons and protons due to the symmetry energy can be deﬁned via

F (u) as follows,

- 0 F1162
14:22:47). A) = 33—31“)- (7.3)
‘ dl’nm
where ea E [SO — (22/3 — 1)—:E[’ . In this chapter two speciﬁc forms of F (u) are

investigated [114, 117],
179

 

 

 

 

100 ....r.r.T,....,r,.,
h Asy-StiffFl

   

 

50 '_ Asy-soft F3]
- , -—- "' 7' 1

9

Q)

2 0 ’/ Neutron‘
>. .\ Proton.
In
a .

>
-50 ~ 1

 

 

 

b u
100 J A L l A A l A l 1 A L A l A I I l
-

P’Po

Figure 7.1: The symmetry potential for neutrons and protons is shown for two dif-
ferent density dependences of asymmetry term: asy-stiff F1 and asy—soft F3 (see
Eqs. (7.4)).

180

 

21.12
F1(U.) 2 1+ “I, (7.4)

F3(u) : 111/2.

 

In Figure 7.1, the symmetry potentials for the two parametrizations are shown for
given 6 = 0.9. In the following, the stronger dependence F1 will be labelled as asy-stiff
and the weaker dependence 173 as asy-soft. As one can see from Figure 7.1, the asy-
stiff term gives lower neutron potential and higher proton potential at low density,
which means more. protons and less neutrons are emitted at the preequilibrium stage
leaving the hot ”prefragment” more neutron rich, and vice versa for the asy-soft term.
In the next section, calculations were performed in an effort to distinguish the two

different asymmetry terms.

7 .2 Hybrid Model Calculations

As discussed previously, the isospin asymmetries of the excited systems prior to mul—
tifragment. breakup are sensitive to the density dependence of the asymmetry term of
the EOS [116]— [118]. The “prefragment” is reduced in size relative to the total system
by preequilibrium emission when it disintegrates into the ﬁnal fragments. Both the
Stochastic Mean Field (SMF) [52] and the Boltzmann—Uehling-Uhlenbeck (BUU) [44]
f(_)rmalisms, which describe the time evolution of the collision using a self—consistent
mean ﬁeld (with and without ﬂuctuations, respectively), predict. preequilibrium emis-
sion that is increasingly neutroll—deficient. and corresponding prefragments that are
more neutron-rich for symmetry terms 8 (p) that have a stiffer density (le1:)e11(.lence
[116, 126]. These two formalisms are essentially identical during the early stages of
the collision when the densities exceed po/ 2 and fluctuations in the mean field are

negligible.

181

The mechanism for the disintegration of the prefragment into the observed frag-
ments with 33 Z3 30 is an issue that is not settled but, instead, is evolving consider-
ably as new measurements and models become ave-rilable. Dynamical multifragmen-
tation models [121, 127] have been used with some success, as have statistical models
either with fragment emission probabilities determined from the rates for evaporative
surface emission [10] or from the yields assuming thermal equilibrium [9, 11]. Here,
we examine the isotopic effects in the latter limit, which assumes that thermal equi-
librium is achieved at breakup. Such calculations have provided surprisingly accurate
predictions for the fragmentation of pro jectile- and target-like residues in peripheral
and mid—impact parameter heavy ion collisions at incident energies Elm", / A > 200
MeV [128, 129], central heavy ion collisions at Eben," / A S 50 MeV [55, 45] and in
light ion induced collisions at Ebeam > 4 GeV [130], after some accounting is made
for preequilibrium light particle emission. Con'iparisons of experimental data to such
approaches provide an assessment of the importance of non-equilibrium phenomena;
accordingly, more difficulties in such appr(')aches are encountered in central heavy ion
collisions at Ebmm / A > 50 MeV, reflecting the decreased time available for equilibra-

tion [45, 131].

To examine the isospin dependence of the EOS, hybrid model calculations were
carried out for central collisions (b=1fm) of 112Sn+llQSn, 112Sn+1248n and 12“Sn+mSn
at E/Az50 MeV. Speciﬁcally, we solved the isospin dependent BUU equation to ob-
tain 11)redictions for the dynamical emission of light particles during the compression
and expansion stages of the collision. Then, we calculate the multifragment. disinte-
gration of the denser portions of the system via the Statistical Multifragmentation
Model (SMM) as discussed in the previous chapter [26, 36]. In the ﬁrst step of the
hybrid calculations described here, the mean ﬁeld for synmietric nuclear matter in

the BUU calculations was chosen to have a stiff EOS (K = 386 MeV) [132]. Calcu-

182

lations were 1')erforme(’l with the two different expressions for the asymmetry term,
“asy-stiff” and "asy-soft", corresl‘mnding to F1({)/p0) and E3([)/p0) , respectively, as
shown in Figure 7.1. Using these mean ﬁelds, BUU calculations were followed through
the initial comI’u'ession and subsequent expansion for an elapsed time of 100 fm/c at
which point the central density decreased to a value of about p0/6. The regions with
densities p > p” / 8 were then isolated and their decay was calculated with the SMM.
Table 7.1: The ﬁrst two columns provide the N / Z ratio and number of nucleons in
the prefragments produced in the calculations for an elapsed time of 100 fm/c and
density cutoff of po / 8. The next two columns provide corresponding information for

the same cutoff density but a shorter elapsed time of 80 fm/c. All calculations were
performed at an impact parameter of 1 fm.

 

reaction t=100 fm/c, pc=p0/8 t.=80 fin/c. pc=P0/8

 

 

 

 

 

 

 

 

 

 

 

 

 

 

asy-soft asy-stiff asy—soft asy—stiff
N/Z A N/z A N/Z A N/Z A
msn+msn 1.16 153 1.27 152 1.17 165 1.27 105
“‘~’sn+l24sn 1.19 161 1.36 162 1.22 174 1.36 175
131811+124Sn 1.23 172 1.44 173 1.27 183 1.45 185

 

 

The \ / Z ratio and the nucleon number A of these fragn'ienting systems (“prefrag—
ments”) are given in two leftmost columns in Table 7.1. To illustrate the sensitivity
of prefragment size and asynunetry to the elapsed time and density cutoff, values for
N / Z and A are also given in Table 7.1 for an elapsed time of 80 fm/c. Calculations
have shown that. the N/Z ratio is not sensitive to the density cutoff [116]. While
A is sensitive to these parameters, the N/ Z ratio is I‘Cl‘dthCly insensitive; to within
3% , values of N/Z of 1.27 (1.16), 1.36 (1.19) and 1.44 (1.23) are obtained for the
source asynnnetry of asy-stiff (asy-soft) calculations for 112811+”28n, 112Sn+mSn and
l2"’SII+12“Sn collisions independent of matching condition. The excitation energy per
nucleon of the prefragment depends strongly on the matching condition; however,
this quantity is presently difficult to calculate accurately. A range of values for the

183

.11

 

 

Asy-s’riff F] 124Sn+i248n

 

 

 

I IIIITI

 

I I 1111"]

 

Yield
8

I I IIIIIII

I TIIIITI'
I

 

 

 

-2
1O —
,._ »-
P- i-
P F'-
- >-
D- FI-
»— h-
1.4 ,__
)— D-t
L‘- C
p- h-
1— b-
y. b
-
b— u—
1 1.— .___
I: :
>- P
..
.-
2>_ ’-
-
1— h—
.—
.—
I— h.
— l—
l- b
3 h
-
>- u-

 

 

 

 

Figure 7.2: Isotopic distributions from Li to O are shown for central collisions of
124Sn+124Sn The full circles are experimental data while the solid (dashed) lines
denote the ﬁnal (primary) calculations from the hybrid model using the density de-
pendence asy-stiff F1 (Eq. 7.4) for the asymmetry term of the EOS.

excitation energy per nucleon of E*/ A = 4—6 MeV was therefore assumed in the sub-

sequent SMM calculations to estimate the range of possible values consistent with

the present approach.

7 .3 Isotopic Composition and Isospin Dependence

Isotopic distributions calculated with the asy-stiff and asy-soft EOS’s are compared

with those measured for central collisions of 124Sn+1248n at 50A MeV [41, 113]. Ac-

184

 

Asy-soﬁ F3 124Sn+1243n

E—

"B

'- ‘

>- "

_ e,,
' \

 

—\
II III[
C

   

     
  
 
 

_\

I

A
TI 1 IT
..~

I

I

~
..~
.-

I
N

 

A
r TITTTII] HUI-I]

Yield
8

 

A
O
I II]

o
‘u

 

A
TTII.III[ TIITTIITIPIII

_x
O
"i

 

 

 

 

 

Figure 7 .3: Isotopic distributions from Li to O are shown for central collisions of
124Sn+124Sn. The full circles are experimental data while the solid (dashed) lines
denote the ﬁnal (primary) calculations from the hybrid model using the density de-
pendence asy-soft F3 (Eq. 7.4) for the asymmetry term of the EOS.

185

 

curate calculations for isotopic yields from the multifragment decay of the excited
prefragment within the Sh’lh’l approach require a careful accounting of the structure
and branching ratios of the excited fragments. Using an SMM code (see the previous
chapter) that carefully addresses such effects, the isotopic distributions in Figs. 7.2

and 7.3 were calculated for the prefragment source parameters in Table 7.1.

In Figure 7.2, the ﬁnal isotopic distributions (solid lines) for the asy-stiff EOS
agree well with the experimental data while in Figure 7.3 the ﬁnal distributions (solid
lines) for the asy—soft EOS are relatively narrower. From the comparisons of isotopic
distributions the asy-stiff density (.iependence of asymmetry term of the EOS is likely
favored. However, in both ﬁgures, the primary isotopic distributions (dashed lines)
are enormously modiﬁed by secondary decay, which could easily overshadow the sig-
nals of the density dependence of asymmetry energy. In the following, we will show
observables that are not as sensitive to secondary decay to distinguish the different

isospin dependences of the EOS.

7.3.1 Relative Free 11 / p Densities and Mirror Nuclei Ratios

Unfortunately, the observed isotopic distributions are also inﬂuenced by secondary de—
cay, making it very important to identify observables that are insensitive to sequential
decay. Statistical calculations have identiﬁed certain ratios of isotopic multiplicities
as being robust with respect to the secondary decay [41, 39]. For example, the ratio
of the n'ulltiplicities R21 (N,, Z,-) = 1112(Ni, Z,)/1’l[1(N,-, Z,) of an isotope with neutron
number N,- and proton number Z, from two reactions 1 and 2 is relatively insensitive
to the distortions from sequential decay. For multifragmentation, compound nuclear

evaporation, and selected strongly damped collisions, such ratios as functions of N,-

186

 

 

 

0.0 2.5 N 5.0 7.5 10.0
’,-+.,...-,........,‘

ESMMIMSU (Primal?)

9’

o

x
l

 

 

_—

 

 

P 4
’1 I 1 A 1 l . - . 1 l . I I . l I I . l‘

0.0 2.5 5.0 7.5 10.0

 

Figure 7.4: Relative isotope ratios, R21, of two reactions 124311+1243n and 112Sn+1128n
are shown as a function of neutron number. The upper panel presents the primary
calculations using the hybrid model while the ﬁnal isotope ratios after secondary decay
are plotted in the lower panel. The lines denote the best ﬁts through the symbols
with the same slope.

and Z, have been experin‘ientally shown [64] to satisfy a power law relationship:
R21(.’V,,Z,) = M-2(N,, Z,)/]l1'1(1’V,-, Z,)=C (5,.)2" (5,.)“9 (7.5)

where 6,, and 6,, are empirical parameters that have the interpretation, in the grand
canonical approximation, of being the ratios of the free proton and free neutron

densities in the two systems, 5,, = pp-g/ppi ; [3n = Pn2/ pill [41].

Before we use this technique to test the density dependence of the asymmetry
term, the relative isotope ratios calculated from the model are shown in Fig. 7.4 to
verify if the calculations have the same behavior as the experimental data. In the

upper panel of Fig. 7.4, the ratios of primary isotope yields are plotted as symbols

187

..|.

'1-

 

which lie along the dashed and solid Z lines with the same slope. Clearly one see that
the calculations are in agreement with the parametrization in Eq. (7.5) and similar
to the experimental data in ref. [41]. More importantly, in the lower panel of Fig.
7.4, the slopes of these ratios after secondary decay are very similar to the prin'iary
calculations. Therefore, this secondary decay insensitive observable is used below for

probing the isospin dependence.

The solid circles and squares in Fig. 7.5 show values for pp and pn, respectively,
obtained from fragments with 33 Z 1 S8 detected in central 112Sn+112Sn, 112Sn+mSn
and 124SI‘1+124Sn collisions at E / A=50 MeV [41]. The 112811+“28n reaction was la-
beled as 1 in Eq. 7.5; the different data points correspond to the three choices for
reaction 2 and are plotted in both left and right panels as a function of Nun/2101. where
NM and Ztot are the total numbers of neutrons and protons involved in reaction 2.
To indicate the sensitivity of these ratios to the secondary decay of heavier parti-
cle unstable nuclei, the open rectangles indicate the ratios obtained from the yields
of primary fragments and the cross-hatched rectangles indicate the ratios obtained
from the yields of the ﬁnal fragments after secondary decay. The vertical height of
each rectangle reflects the range of values for each quantity as the assumed excitation

energy is varied over the range of E* / A = 4-6 MeV.

The left and right panels in Fig. 7.5 provide values calculated for prefragments
obtained with the asy-stiff and asy-soft EOS’S, respectively. In both panels, it can
be seen that the ratios calculated from the primary yields (open rectangles) and
those calculated from the secondary yields (cross-hatched rectangles) are similar,
indicating that values for R21(N,Z) are relatively insensitive to secondary decay. With
the exception of the value of < [)p > for the l248114424811 reaction, Nmt/Ztot = 1.48,
the ratios calculated from the ﬁnal yields with the asy—stiff EOS (left panel) overlap
the data. In comparison, the calculations using the asy-soft EOS (right panels) show

188

_‘1

 

 

   
   
 

    

 

 

 

 

 

 

 

>-+..,-1.TT.,.-w..-.,..-a,....,q
ISOtOpe Ratios Jr Efinal Uprimary

1.4 ~ —— _ _
. Asy—stiff Asy—soft 1
/\ 1 1
< c: .. ,
3 1 2 — 4— ~ ~
1
1.0 *- —*- 5
~ 1

A 1
<5 0 8 — ~— —
V J. ,

1.

t
Isotone Ratios

O6 1 1 1 1 l 1 1 1 1 l 1 1 1 1 1 1 1 1 l . . 1 1 1 1 1 1 1 l
1.2 1.3 1.4 1.5 1.3 1.4 1.5

Ntot/Ztot

Figure 7.5: Both panels: The solid circles and solid squares show values for [)p and ﬁn,
respectively; measured in central 112Sn-l-112Sn, 112SIH-mSn and 124Sn+1248n collisions
at E / A250 MeV. Left panel: the open and cross-hatched rectangles show correspond—
ing hybrid calculations for R21 calculated from the primary and ﬁnal fragment yields,
respectively, predicted by the hybrid calculations using the Asy—stiff EOS. Right panel:
the open and cross-hatched rectangles show corresponding hybrid calculations for R21
calculated from the primary and ﬁnal fragment yields, respectively, predicted by the
hybrid calculations using the Asy-soft EOS.

189

a signiﬁcantly weaker dependence on Nmt/th than do the data.
7 .3.2 Mirror Nuclei Ratios

One can also reduce the influence of secondary decay by taking ratios of the multi—
plicities of mirror nuclei 111(N,,Z,~)/11*[(Z,~, N,) measured in a single reaction [41, 39],

but the reduction of secondary decay effects may be less effective in this case.

The solid and open points in Fig. 7.6 show the experimental values for the mirror
nuclei ratios constructed from the multiplicities of 7Li, 7Be, 11Band 11C fragments
[41]. The upper and lower panels are for 124Sn+mSn and 112Sn+1128n collisions,

respectively.

The left and right panels in Fig. 7.6 provide values for the mirror nuclei ratios
calculated with the asy-stiff and asy-soft EOS’S, respectively. For these ratios, the
sensitivity to the density dependence of the symmetry energy and to the secondary
decay corrections are more signiﬁcant. Ratios of mirror nuclei calculated with the
asy—stiff EOS exceed those calculated with the asy-soft EOS by about a factor of two

and overlap with the experimental values for three of the four ratios measured.

7 .4 Remarks

In the present simpliﬁed approach, the sensitivity of isotope and the mirror nuclei
ratios to the asymmetry term arises from the different (N / Z) ratios of the prefragments
that are predicted by BUU calculations. There is little sensitivity to the total mass
of the prefragment, but additional sensitivity to its excitation energy per nucleon.
Within the present model dependent analysis, this uncertainty in excitation energy

is the limiting factor that prevents a more quantitative constraint on S ([1).
Light cluster emission during the early compression and expansion stages of the

190

2:.

 

 

 

NCO
0CD
\

l
\
\
1—0—1

[_A
0
TI 1
\\\V
1
\
1
1
1
1
1
O
1
\
1
1
1
1
1
1
l

 

 

.\
s\\‘
-
RV

 

e
9

v Ilvvvv I
AlLlAlAl Ill]

 

 

Asy—stiff Asy—soft

1 AL I L l A I L L L l l l l l l L 1 L 1

 

7
g, 3? Sn+ S ‘
+4 1 LL41 1 1 l :gL‘
>38E— l I I T U:
N. EllZSn+112S 3: :
_. a) \ -—
220 m m
‘2’ i K =
1 ”—4
103 Iii/.H' '11; ’//,A—O E
3.
2.

O)

8 10 6 8 10

Mass

Figure 7.6: The solid and open points in the upper and lower panels Show the mir-
ror nuclei ratios measured for 124Sn+mSn and 112Sn+1128n collisions, respectively.
Left panels: The open and cross-hatched rectangles show corresponding hybrid cal-
culations of the mirror nuclei ratios calculated from the primary and ﬁnal fragment
yields, respectively, predicted by the hybrid calculations using the Asy-stiff EOS.
Right panel: The open and cross-hatched rectangles Show corresponding hybrid cal-
culations of the mirror nuclei ratios calculated from the primary and ﬁnal fragment
yields, respectively, predicted by the hybrid calculations using the Asy-soft E08.

191

 

 

 

collision can inﬂuence the N / Z ratio and excitation energy of the prefragment. In-
corporating the emission of light particles up to A=4 within transport model calcula—
tions will help address this issue [133, 134]. While the present hybrid model approach
demonstrates a sensitivity of the isotopic fragment yields to the asymmetry term of
the EOS, the detailed nature of this sensitivity is model dependent. For example,
the hyl‘n‘id model predicts that an asy-stiff EOS leads to fragments that are more
neutron-rich than those produced when the EOS is asy—soft. On the other hand,
recent calculations with the Expanding Evaporating Source (EES) model, which as-
sumes the fragments originate from surface emission and not froiu the equilibrium
decay of the residue, predict the opposite trend [(54]. It is therefore highly desirable
to explore the connection between the fragment isotopic distributions and the EOS
within other statistical and dynamical fragment production models currently in use
and under development. These long—term goals require significant future theoretical

efforts.

192

Chapter 8

Summary

Multifragmentation is main reaction mechanism for central nucleus-nucleus collisions
at incident energies in excess of E /A = 3.5.lIeV. The assumption of local thermal
equililn'ium is often invoked by model descriptions of this process. In such descrip-
tions, copious fragment. production occurs when the system undergoes a low density
phase transition conceptually similar to the liquid-gas phase transition of nuclear
matter. The fragment observables in such models are assumed to reflect the temper-
atures and densities at freezeout, after which the interacting system decouples and

fragments propagate to the detectors.

In order to study this freezeout stage, a Large Area Silicon-Strip/CSI detector
Array (LASSA) was devel(')pe(:l. The LASSA consists of nine identical telesmpes,
each of which is composed of two silicon—strip detectors and four CsI(Tl) crystals and
provides excellent energy, angular and mass resolution for the detection of charged
particles. Using LASSA and the Miniball/h‘liniwall 47r fragment detection array,
central collisions of 129Xe+197Au at 50A MeV were measured with high energy and

angular resolution.

These high quality measurements allowed the study of correlation functions for

particles emitted from the central 129)(eer’Au collisions. An imaging technique was

193

used to analyze p—p C(in‘relation functions by numerically inverting the correlation
function in the Koonin-Pratt formalism to obtain the source function. Both the in-
clusive and em. energy gated correlation functions were studied by this technique and
the energy dependence of the source distribution was observed. The main observed

trend was that sources of less energetic protons are more extended.

The d-a correlations were also studied in the Koonin-Pratt formalism, using a
simple. Gaussian source parametrization instead of the imaging approach which has
not yet provided reliable source information for d-a correlations. However, the d-a
correlation functions could not be fitted well, and especially the second resonance
peak at excitation energy 4.31 MeV was overpredicted and displayed an incorrect
shape. Simulations indicated this could be the effect of collective motion and that
the phenomenon could be modelled by an effective temperature correction. The origin
of this effect is that the mixed event yield has a broader relative energy distribution
than the resonant yield when a significant collective ﬂow occurs. After taking into
account this effect, we fitted the d-a correlation functions with the Gaussian source
parametrization. We found that the d-a source displayed an energy dependence that

was similar to that observed for the p-p correlati(_)ns.

This collective motion also led to a reduction in the detectable source due to the
competition between the collective velocity field and the thermal or random velocity
of the particles. The size reduction factor of the sources probed by the two particle
correlations in LASSA is more significant for l‘ieavier particles. After taking this
source reduction into account, we calculated the breakup density for both p-p and
d-a correlations. As the energy of the particles is reduced, the extracted values of the
density become smaller, indicating that the lower energy particles are emitted later

after the system is expanded.

194

An equilibrium a1*_)pr(‘)ach is developed for permitting the extension of these tech-
niques to heavier particles for which the kernel of the conventional Koonin—Pratt
approach has not yet been calculated. With this equilibrium assumption and taking
collective effects into account, we fitted the d-a. t-(l’, 3He-a and a-a (.torrelations and
extracted the detected free volume of the total source. After corrections for secondary
decay and collective motion, we calculated the ralues of the density which are around
1/5-1/3/20. When the collective and secondary decay effects are applied in the same
way for both the equilibrium and Koonin-Pratt approaches, the extracted density val-
ues for the d-a correlation function are very similar for the two approaches. However,
the uncertainties of this 'a.1‘)proa.cl'1 are still somewhat difficult to presently estimate.
Additional inf(.)r1nation about the secondary decay and collective motion corrections
to these data are needed to improve our understanding of the uncertainties in the

extracted breakup density.

The equilibrium approach was also used to determine the spins of particle unstable
states. The sensitivity to spin determination of this procedure is illustrated in the
p-7Li correlation function where three groups of resonances are ﬁtted. The spin of
the first excited state of 8B at 0.774 MeV has not been measured although the spin
of the analog states of 8Be and 8Li indicates a 1‘L assignment for this state. By ﬁtting
the p—7Be correlation function, the spin value of this 0.774 MeV state is determined
to be 0.98 i 0.29 if there is no other state between the two known 0.774 MeV and
2.32 MeV states, or 0.95 :l: 0.33 if a 1+ state at 1.4 MeV is considered. In either of
the cases, we conﬁrm the spin of the 0.774 MeV state of 8B is one. We also analyze
the a-GLi correlation function in the similar way where more resonances of 10B are
involved in the ﬁtting. Though the resolution and statistics are limited, it seems that
a spin of J = 2 is favored for the 6.56 MeV state if the branching ratio of this state

decaying through the a-GLi channel is 100%. If this state also decays to the d-d-a

195

Channel, the spin could be higher. Some strength near threshold is observed in this
channel, but we currently cannot say whether it is sufficient to influence this spin
assignment. Besides the d—a—a correlation, other stronger three particle correlations
(a-a-a, p-p—a, and p-a-a) are also shown; some exhibit huge enhancements for the
peaks near the threshold due to the small three body phase space near threshold.

Such high sensitivity near threshold may prove interesting for future studies.

In order to test the thermal properties of multifragmenting systems, we develop
an improved Statistical Multifragmentation Model (SMM). In this improved SIVIIVI,
we incorporate experimental values of binding energies if possible or otherwise use the
smoothly extrapolated values from an improved LDM if such experimental informa—
tion is not availal‘ne. The free energies used for calculating the primary populations
of the fragments are also modiﬁed by taking into account the experimental level den-
sities at low energy. To apply a consistent calculation, we model the secondary decay
process with the same empirically modiﬁed level density scheme, which adopts all
the experimental levels up to where the information seems complete and thereafter
is smoothly interpolated from the empirical extension to the SMM limit of the level
density at high energy. For all the experimentally known states, we also adopted
the experimental values of excitation energies, spins, isospins, parities and branching
ratios if possible. Before sequential decay starts, hot fragments from the primary
breakup are populated according to the primary temperature over the sampled levels
in the constructed ’table’. In the end, all the fragments will decay to their final stable
states from top to bottom throughout the ’table’ by use of the known branching ratios

or the values calculated from the Hauser—Feshbach formula.

Before comparing this model to experimental data, we examined the caloric curves
calculated from the improved SMM to check that they show similar behavior as those
obtained from the standard (original) SMM. We point out that the caloric curve also

196

shows an important density dependence. which manifests the significance of finding
and measuring observables that can constrain the freezeout density. In cmnparisons of
the charge and mass distrilmtions, the standard SMM overpredicts the yields of heavy
fragments due to the fact that the binding energies calculated in the standard SMM for
heavy fragments C(‘msistently exceed the empirical values used in the improved SMM.
We have shown that the Albergo formula for extracting the isotopic temperatures
from the double ratios of the isotopic yields is valid in this semi-microcanonical SMM

even though the formula itself is derived from the grand canonical limit.

Secondary decay corrections are calculated with this model to provide corrections
to the correlation functions and enable the determination of the breakup density.
In addition, the ﬁnal isotopic temperatures are extracted by this approach from a
set of IMF thermometers with 3 3 Z 3’ 8 and AB > 10 MeV to limit. the effects
of non—equilibrium emission and reduce the calculation fluctuations. We ol')tained a
very good agreement between the qualitative trends of the experimental data and
the calculated values using the improved SMM, thanks to the incorporated empiri-
cal nuclear structure information in the model, especially in the later stages of the
breakup, even though we did not try to adjust the input parameters of the model
to optimize the agreement between experiment and theory. In contrast, calculations
from the Botvina’s version of the SMM which neglects the experimental structure de-
tails can not reproduce the experimental data. However, these ﬁnal temperatures for
the IMF’s, either from the data or from this improved model, are significantly lower
than the primary values calculated in the model, which shows that these observables
are modiﬁed extensively by secondary decay. Furthermore, these IMF thermome-
ters follow a decreasing trend as a function of the total mass number of the four
involved isotopes, indicating that the heavier fragments are more strongly influenced

by secondary decay.

197

The accuracy of this statistical approach allowed us to explore the sensitivity of
the fragment isotope distributions to the isospin dynamics of the initial stage and
to the isospin dependence of the nuclear equation of state (EOS). Hybrid model
calculations are performed with the improved SMM and an isospin dependent BUU
model for exploring the isospin dependence of the EOS. Two forms (one stiffer, the
other softer) of the density dependence of the asymmetric terms in the EOS are tested
for central collisions of 11“2811+112Sn, 112Sn+124Sn, and 124811+124Sn at E/A=50 MeV.
Since the observed isotopic distrilmtions are enormously modiﬁed by secondary decay,
we need to identify observables insensitive to sequential decay in order to distinguish
the different density dependences of the asymmetry term. The scaling parameters (a,
13), which factorize the isotope ratios from two reactions into a simple parametrization,
R21(N, Z) = C exp(aN +32), and are discovered in studying the isoscaling and
isospin fractionation phenmnena in multifragmentation, are found very robust with
respect to secondary decay. By comparing the Sn+Sn data to the calculations with
the hybrid model for the isoscaling parameters, the stiffer density dependence of the
EOS is favored. Another test on the mirror nuclei ratios of 7'Li/7Be and 11B / 11C also
shows a better agreement l‘)etween the data and the calculations with the stiffer isospin
dependence though the reduction of secondary decay effects may be less effective in
this case.

While the present hybrid model approach demonstrates a sensitivity of the isotopic
yields to the asymmetry term of the EOS, the detailed nature of this sensitivity
is model dependent. For example, calculations with the EES model, which has a
different emission mechanism from the SMM, predict the opposite trend. Therefore,
it is highly desirable to explore this sensitivity of isotopic yields to the asymmetry
terms of the EOS within other statistical and dynamical models. Experimental efforts

are also in great need to produce larger isospin signals with more asymmetric systems,

198

which is becoming the focus of nuclear reaction studies with the available or proposed

high intensity radioactive beam facilities.

After these extensive explorations of the multifragmentation process, it is some-
what surprising that we ﬁnd few aspects that are well outside of the realm of de-
scriptitm by equilibrium models. To be certain there are problems. For example, the
freezeout density definitely depends on the energy of the emitted particle, consist-
ing with the lower energy particles being emitted after the system has cooled and
expanded to low density. On the other hand, the populations of excited states and
isotopic distributions are remarkably well described by the equilibrium approach. The
picture that appears to be emerging is the one where particles are emitted as a func-
tion of time from the bulk of the expanding system. \Vhile more time to digest these

results is needed, this picture does not appear to correspond to any existing model.

199

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