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ISOSPIN DYNAMICS AND THE ISOSPIN DEPENDENT EOS

presented by

TIANXIAO LIU

has been accepted towards fulfillment
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fl.

ISOSPIN DYNAMICS AND THE ISOSPIN DEPENDENT EOS

By

Tianxiao Liu

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

2005

ABSTRACT

ISOSPIN DYNAMICS AND THE ISOSPIN DEPENDENT EOS

By

Tianxiao Liu

Isotopic yields for light particles and intermediate mass fragments have been mea-
sured for 112Sn+1mSn, 112Sn+124Sn, 124Sn+1128n and 124Sn+124Sn with the Large
Area Silicon Strip Array (LASSA). LASSA consists of nine individual telescopes with
each telescope comprised of two Si strip silicon detector layers backed by four CsI(Tl)
crystals. LASSA provides high angular, energy and isotopic resolution over a large
detection area and energy range. The total solid angle covered by LASSA is about
540msr with polar angle 7°-58° and an angular resolution of :l:0.43°. The energy
range covered is E /A = 2.4-140 MeV for protons and E/A = 4.8-335 MeV for 16O.

Isotopic yields from central collisions were compared with predictions of stochastic
mean field calculations. These calculations predict a sensitivity of the isotopic dis-
tributions to the density dependence of the asymmetry term of the nuclear equation
of state. However, the secondary decay of the excited fragments modifies signifi-
cantly the isotopic distributions produced by the stochastic mean field model. The
predicted final isotOpe distributions after decay are narrower and more neutron-rich
than the experimental data and the sensitivity of the predicted yields to the density
dependence of the asymmetry term is reduced. The distributions calculated using the
stiff asymmetry term are more neutron-rich and are closer to the measured values.
However the less than satisfactory level of agreement between theory and experiment

precludes definitive statements about the density dependence of the asymmetry term

of the EOS.

Tianxiao Liu

Energy spectra and average energy of light particles and fragments from central
collisions were obtained and compared with statistic models calculations. We found
that neutron deficient isotopes are significantly more energetic than those of neutron
rich isotopes of the same element. This trend is well beyond what can be expected for
the bulk multi-fragmentation of an equilibrated system. It can be explained, however,
if many of these fragments are evaporated from the surface of the system while it is
expanding and cooling.

Measurements of particles and fragments from peripheral collisions were used to
study the dynamics in nuclear reactions. Isoscaling analyses imply that the quasi-
projectile and quasi-target in these collisions do not achieve isospin equilibrium, per-
mitting an assessment of the isospin transport rates. We find that comparisons be—
tween isospin sensitive experimental and theoretical observables, using suitably chosen
scaled ratios, permit investigation of the density dependence of the asymmetry term
of the nuclear equation of state. This observable appears to be one of those promis-
ing ones for providing much needed constraints on the asymmetry at sub-saturation

density.

ACKNOWLEDGEMENTS

First and foremost, I would like to express my most sincere thanks to my disser-
tation advisor, Dr. William G. Lynch. He is a unique combination of great intellects
and genuine caring for those persons for whom he has some responsibility. I was very
fortunate to be one of these people. To me, he is not only a world class advisor, but
also a dear brother and an influential mentor. He is a source of endless inspiration
even during the most stressful times. His guidance, support, vision and friendship
will be a long-lasting influence in my professional and personal life.

I would also like to thank Dr. Pawel Danielewicz, Dr. Chien-Peng Yuan, Dr.
Betty Tsang and Dr. Edward Brown for serving on my guidance committee. Special
thanks go to Dr. Betty Tsang for her enthusiasm. She is an inexhaustible source of
intellectual and creative energy. Her good sense in finding general physics laws to
explain complicated experimental data was an excellent lesson to me. I am extremely
impressed by her ability to get an intuitive feel of the most difficult ideas. Special
thanks also go to Dr. A. Wagner, Dr. H. S. Xu, Dr. H. F. Xi, Dr. G. Verde and Dr.
MJ van Goethem who helped me with software writing, experiment set-up and data
analysis.

My work on my dissertation and the research benefited from the successful collab-
oration of the groups at Michigan State University (MSU), Indiana University (IU)
and Washington University (WU). I would like to thank the MSU group of C. K.
Gelbke, W. P. Tan, X. D. Liu, W. G. Lynch, M. B. Tsang, A. Vander Molen, G.
Verde, A. Wagner, H. F. Xi, and H. S. Xu, the IU group of L. Beaulieu, B. Davin, Y.
Larochelle, T. Lefort, R. T. De Souza, R. Yanez and V. E. Viola and the WU group
of R. J. Charity and L. G. Sobotka.

A number of people have contributed to the work which culminates in this disser-

tation. Among them are: Drs. M. Colonna, M. Di toro, M. Zielinska-Pfabe, W. H.

iv

Woolter, W. A. Friedman, S. R. Souza, R. Donangelo, L. J. Shi and P. Danielewicz.
I gratefully acknowledge their assistance on model calculations.

This dissertation and research that it describes could not be accomplished had
it not been for the excellent support of the staff at the National Superconducting
Cyclotron Laboratory of Michigan State University. I owe my deepest thanks to
John Yurkon of the detector lab, Len Morris of the design group, Jim Vincent of the
electronics group, Andrew Vander Molen and Ron Fox of the computer group, who

helped with data acquisition.

 

CONTENTS

LIST OF TABLES viii

LIST OF FIGURES ix

1 Introduction

1.1
1.2

1
Background and Motivation ....................... 1
Theoretical Models ............................ 5
1.2.1 Boltzmann-Uehling-Uhlenbeck (BUU) Model .......... 5
1.2.2 Stochastic Mean Field (SMF) model .............. 8
1.2.3 Statistical Multifragmentation Model (SMM) Model and Macro-

canonical Approach ....................... 10
1.2.4 Expanding Emitting Source (EES) Model ........... 14
1.3 Studies Prior to This Dissertation .................... 16
1.4 Thesis Organization ............................ 18
2 Experimental Setup 20
2.1 Experimental Layout ........................... 20
2.2 LASSA Telescopes ............................ 24
2.3 The Miniball/Miniwall Array ...................... 33
2.4 The Ring Counter Foward Array .................... 41
2.5 Data Acquisition Electronics ....................... 47
3 Data Analysis 52
3.1 Data Reduction Algorithms ....................... 52
3.2 Data Analysis of LASSA ......................... 53
3.2.1 Overview ............................. 53
3.2.2 Energy Calibration of Si ..................... 54
3.2.3 Multiple-hit pattern Decoding .................. 59
3.2.4 Construction of PID function and Non-uniform Thickness Cor-
rection .............................. 61
3.2.5 CsI calibration .......................... 69
3.2.6 Background Subtraction ..................... 73
3.2.7 Efficiency Correction ....................... 76
3.3 Data Analysis of the Ring Counter ................... 80
3.3.1 Energy Calibration ........................ 84

vi

3.3.2 PID function ........................... 88

4 Experimental Data and Model Comparisons 90
4.1 Impact Parameter Selection ....................... 90
4.2 Isotope Multiplicities, Energy Spectra from Central Collisions and Model

Calculation ................................ 95
4.2.1 Experimental Data ........................ 95
4.2.2 Models for the Multiplicity Calculation ............. 104
4.2.3 Overall Behavior Predicted by the SMF Calculations ..... 110
4.2.4 Isoscaling Analyses ........................ 115

4.2.5 Sensitivity of the SMF Calculations to the Asymmetry Term . 119
4.2.6 Discussion of the Isotope Multiplicity and SMF Calculations . 123

4.2.7 Energy Spectra and Equilibrium Models Calculation ..... 126

4.3 Isospin Diffusion in the Peripheral Collisions and Model Calculation 135
4.3.1 Data from the Peripheral Collision ............... 135

4.3.2 BUU Calculations ......................... 144

5 Summary 148
LIST OF REFERENCES 152

vii

LIST OF TABLES

2.1

3.1

3.2

4.1

Table of the Miniball/Miniwall detectors used for the impact parameter

selection, their corresponding polar angles and solid angle coverage . 38
Table of the fitting parameter used in the energy calibration functions

of LASSA CsI(Tl) crystals ........................ 72
Table of fragmentation products used in the energy calibration of LASSA
CsI(Tl) crystals .............................. 74

Values for a and 5 obtained from fitting the isotope ratios R21 . . . . 103

viii

LIST OF FIGURES

1.1 The symmetry potential for neutrons and protons with different pa-
rameterization ...............................
1.2 Mean-field enhancement factor of the diffusion coefficient in symmetric
nuclear matter as a function of T at a fixed density ..........
1.3 Experimental isoscaling behavior exhibited by the central 124Sn + 124Sn
and 112Sn + 112Sn collisions . . . . . . . . . . . ............
1.4 The ratio of pre-equilibrium neutrons to protons as a function of nu-
cleon kinetic energy with both the Coulomb and symmetry potentials
1.5 Isoscaling parameter from central Sn+Sn collisions, experimental data
and calculations from the hybrid models ................

2.1 Major detectors used in the experiment, viewing in the downstream

direction of the beam ...........................
2.2 Mechanical Layout of the experiment, viewing from the side .....
2.3 The relative position of LASSA and the Ring Counter .........
2.4 Photo of a single LASSA telescope ...................
2.5 A schematic drawing of the LASSA detector ..............
2.6 A two dimensional histogram of the raw signal outputs of DE(first

silicon) vs. EF(second silicon) ......................
2.7 A two dimensional histogram of the raw output signals of DE+EF

(silicon) vs. E(CsI) ............................
2.8 LASSA geometric acceptance ......................
2.9 A schematic of the elctronics for the LASSA telescope .........

17

19

21
23
23
25
26

28
29

31
32

2.10 Schematic of phoswich assembly of individual Miniball/Miniball detector 34

2.11 Output of the phoswich of Miniball/Miniwall ..............
2.12 A two dimensional histograms of the raw signal from Slow vs. Fast

and PDT vs. SLOW ...........................
2.13 Schematic diagram of the electronics of the Miniball/Miniwall . . . .
2.14 A schematic drawing of the Ring Counter ................
2.15 A schematic figure of the side view of Ring Counter ..........
2.16 A two dimensional histogram of the raw signal from Pie side and C31
2.17 A block chart of the electronics of the Ring Counter ..........
2.18 Layout of the data acquisition ......................
2.19 Schematic drawing of the principle of the ECL readouts ........

ix

36

37
40
42
43
44
46
50
51

3.1

3.2
3.3
3.4
3.5
3.6
3.7

3.8
3.9
3.10
3.11
3.12
3.13

3.14

3.15
3.16
3.17
3.18
3.19
3.20

4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10

4.11

Major detectors used in the experiment, viewing in the downstream
direction of the beam ...........................
The PC-DAQ system used for the silicon calibration ..........
Calibration lines of energy(MeV) vs. Dial value(Volt) .........
ADC output vs. precise pulser dial value (Volt) ............
Comparison of isotope resolution before and after thickness correction
A histogram of counts vs. nominal DE energy and nominal EF energy
A histogram of nominal counts vs. nominal DE+EF energy and matched
CsI light output ..............................
1D PID histogram for particles stops in the second silicon detector . .
1D PID histogram for particles stops in the CsI crystals ........
Calibration curves of Carbon for LASSA CsI(Tl) ............
PID line of 3He shows the contamination of 4He ............
Background subtraction for the heavy fragments ............
The raw yield and the efficiency function for 4He produced in the pe-
ripheral collision for the 112Sn + 112Sn reaction .............
The corrected yields of 4He produced in the peripheral collisions for
different reactions .............................
LASSA 4He yield in peripheral collisions ................
Ring Counter analysis flow chart ....................
Ring Counter calibration for silicon detector(Pie side) .........
Ring Counter calibration for the CsI detector .............
The 2D PID lines of the Ring Counter .................
Ring Counter PID 1D ..........................

The probability distribution of the charged particle multiplicity and
impact parameter as a function of the charged particle multiplicity . .
Nc vs. szf .................................
LASSA 4He yield in central collisions ..................
LASSA 12C yield in central collisions ..................
Energy spectra of H, He, Li, Be from the central collisions of the reac-
tion 112Sn—l—112Sn .............................
Energy spectra of B, C, N, O from the central collisions of the reaction
112Sn+112Sn ................................
Isotope yields in central collisions ....................
Behavior of supper stiff and soft symmetry terms at low densities

Differential multiplicities for 124Sn+124Sn collisions and 112Sn+1128n
collisions as a function of A and Z ....................
Primary and secondary calculations of Carbon and asymmetries, as
well as the comparison with data ....................

55
56
58
59
63
65

66
67
68
71
77
78

81

82
83
85
86
87
88
89

93
94
97
98

100
102
106
111

114

Primary and secondary calculations of ratio of the yields from 124Sn+124Sn

and 112Sn+ImSn ..............................

117

4.12 Differential multiplicities for Carbon and Oxygen, data and calculation 120

4.13

Scaled function S (N) = R21 (N, Z) . 6‘”, data and calculation

122

4.14

4.15
4.16

4.17
4.18
4.19

4.20
4.21
4.22

11C and 12C energy spectra from the central collisions, data and calcu-

lations ................................... 128
Mean kinetic energies from central collisions, data and calculation . . 129
Mean kinetic energy and emission rates for 12C (11C) from the EES

calculation ................................. 133
LASSA 12C yield in central collisions .................. 137
Differential multiplicities for isotopes with 5 > 0.8 cut ......... 138
R21 using 124Sn + 124Sn collisions as reaction 2 and fitted values for

isoscaling parameter a .......................... 140
Rn from ln(Yz'eld(7Be)/Yz'eld(7Li)) ................... 142
Calculated R,- and comparison with data ................ 143
Calculated R,- and comparison with data ................ 146

xi

Chapter 1

Introduction

1.1 Background and Motivation

Heavy nucleus-nucleus collisions provide a unique way of exploring several interesting
topics, among which are the nuclear liquid-gas phase transition and nuclear equation
of state (EOS). The liquid-gas phase transition has been investigated for a long time.
It bears some relevance to the phenomenon of multi-fragmentation but its detailed
connection to that process is still open to questions.

The nuclear equation of state (EOS) describes the relation between pressure, den-
sity, temperature and isospin asymmetry for nuclear matter. Understanding EOS is
very crucial in astrophysical contexts. For example, the EOS governs the dynamics
of dense matter in supernovae [7]. It also influences strongly the internal structure,
radii, maximum masses and moments of inertia of neutron stars [59]. In addition,
the EOS influences the time scale for cooling of proto—neutron stars. In particular,
the density dependence of the asymmetry term of the EOS can determine whether
or not the direct Urca process can occur[57]. Finally, it is relevant to remember that
the EOS affects significantly the binding energy and rms radii of neutron-rich nuclei

[100] as well as the difference between neutron and proton radii in nuclei [17]. These

parameters may influence the formation of the heavy elements via the r-process.

Significant constraints have already been placed on the symmetric matter EOS at
high densities [31]. However, the constraints on isospin-asymmetry term of the EOS,
which describes the sensitivity of EOS to the difference between neutron and proton
densities, is poor. Better knowledge of this term is essential to understand both the
central density of neutron-rich isotopes [105] and the internal structure, radii and
moment of inertia for neutron stars [81, 58].

Theoretical studies have shown that the EOS of asymmetric nuclear matter can

be approximately expressed as [63]
E(p.5) =E(p,5=0)+5(p)52, (M)

where p = pn + )0p is baryon density, 6 = (pn - pp)Pn + pp) is the relative neutron
excess, and E(p, 6 = 0) is the energy per particle in symmetric nuclear matter. The
bulk symmetry energy is denoted by S (p) Its value at normal density, SOES(po), is
known to be in the range of 27-36 MeV [46]. Different density dependences of S (p)
have been found, depending on the nuclear forces used in the calculation. One simple

parameterization can be written as [85]
3
S(P) = (22/3 -1)(5E2~)[u2/3 — 1701)] + Sofia) , (12)

where E} is the Fermi energy and u E p/po is the reduced baryon density, and F(u)
denotes the potential energy contribution. To understand the role the density depen-

dence of F(u) has on various quantities, it has been approximated by the following

 

parameterizations:
F ( ) 2212
u =
1 1 + u’
F2(u) = u, (1.3)
F3(U) : u1/2,

which provide a range of reasonable density dependencies[63].

2

The symmetry potential for neutrons and protons due to the interaction compo-
nent of the symmetry energy can be obtained,

3(pF (1052)

V"(p)(p, 5) = ea
apub»)

asy

(1.4)

where an E [So —— (22/3 — 1)(g-E2«)] is the contribution of nuclear interactions to the
bulk symmetry energy at normal nuclear matter density.

Fig. 1.1 shows the symmetry potential as a function of u, using two forms of
F (u), i.e. F1 and F3 and So = 30MeV. The symmetry potential from F2, which is
not shown, always falls between those for symmetry potentials from F1 and F3.

From Fig. 1.1 we can see that the symmetry potential is quite sensitive to the form
of F (u), the neutron excess 6 and the baryon density p. In collisions of neutron-rich
nuclei at intermediate energies, both 6 and p can be appreciable in a large space-
time region where the isospin-dependent mean fields, which are opposite in sign for
neutrons and protons, are strong. This will affect differently the reaction dynamics of
neutrons and protons, leading to possible differences in their yields and energy spectra.
This suggested that it is possible to extract information on the asymmetry term of the
nuclear EOS in regions under laboratory controlled conditions. Although in heavy-
ion collisions at intermediate energies we certainly cannot reach very high-density
regions, we should be able to obtain information on the slope of the asymmetry term
below the saturation density. Qualitatively different effects arise in central collisions,
with bulk fragmentation, and peripheral collisions with neck fragmentation. In this
way we hope to put some experimental constraints on the effective interactions used

in astrophysical contexts [9].

 

100 T Y I I I T T‘TWI Y I, I I I I T T r I T T ‘1? I

 

 

 

 

 

 

 

C I 1
I l j
80 — /6'—0.8 _
r Neutron , :
r- , ..
l J
L 2
C 1
A ; j
> _ _
o i- -
E L
v i-
5‘
> i
i
.- a. -
. \ \
-40 :- g \\ “- \ \ -»
i ’ ' " f \ j
- Fl \ 5:0.8 J
-60 L 3 \ \ _
Z \\ (Hi:
-80 :_ Proton . _
’ 5—08
I \
-100 I . I I I 1 . . . r 1 . . .\ . J . . . . 1 . . I .
0.0 0.5 1.0 1.5 2.0 2.5
p/p0

Figure 1.1: The symmetry potential for neutrons and protons corresponding to the
two forms of F(u) and $0 = 30MeV with 6 = 0.2 and 0.8. Images in this dissertation
are presented in color.

 

1.2 Theoretical Models

The extraction of information about the EOS or the liquid-gas transition of nuclear
matter are usually done by comparing experimental data to a theory model calcula-
tion. In this dissertation, several models were used. Two of them are the Boltzmann-
Uehling-Uhlenbeck (BUU) and Stochastic Mean Field (SMF) models, which make
predictions for the reaction dynamics without any assumption about the degree of
thermal equilibrium achieved. The other two models, Expanding Emitting Source
(EES) and Statistical Multifragmentation Model (SMM), assume some degree of ther-

malization. In the following, we will discuss and contrast these different approaches.

1.2.1 Boltzmann-Uehling—Uhlenbeck (BUU) Model

The exact theoretical assumption of the collision of two complex nuclei at the incident
energies described in the dissertation can be in principle found in the many body
Schrddinger equation. This task, however, is fraught with difficulties, both formal
and technical, which motivate the introduction of approximate methods that focus
on specific aspects of the collisions. Here, we describe an approach formulated to
calculate observables that are related to the average dynamics of a collision. The
Boltzmann-Uehling-Uhlenbeck (BUU) [113, 74, 51] equation describes time evolution

of the single particle phase space distribution function f1 (p, r, t) and reads as follows:

(11.1)

at R'Vrfl—vv'val=Icolla (1-5)

where, [can accounts for collisions and can be written as
1
Icon = —/W0V12If1f2(1 — fl')(1— fz') — fl’f2'(1 — f1)(1— f2)l

X (2r)353(p1 + P2 - p’1 - p’z)d3p2d3p’1d3p’2 - (1-6)

Here, a is the differential scattering cross section for a certain change of momentum,

-'~'---c-_-A. H-“w g o

(p1,p2) -> (p’1,p’2), and v12 is the relative velocity for the colliding nucleons. Icon
differs from the classical collision by the Pauli-blocking factor (1 — f1(1:))(1 — f2(2I)).

In the BUU equation, V is the mean-field potential, which will be a function of
the local density. It can be parameterized as an arbitrary function of density, making
it possible to model a variety of equations of state. Typically, it can be written as
the sum of three terms:

vzw+m+nw, up

where VC, V" and Vasy represent the Coulomb potential, the isoscalar nuclear potential
and the symmetry energy, respectively.

With the right-hand side set to zero, Eq. 1.5 becomes the Vlasov equation, which
describes the evolution of a single-particle distribution under the influence of a self-
consistent mean field. Interpreting f1 (p, r, t) as the Wigner transform of the quantum
mechanical density matrix, the Vlasov equation can be derived from time-dependent
Hartree-Fock (TDHF) in the limit of smoothly varying potentials.

To solve BUU equation, given in Eq. 1.5, one often uses the so called Test
Particle method [5, 30], in which the continuous variables p and r are restricted to
a set of discrete values, and the phase-space density is represented in terms of a set

of 6-functions, or test particles,

f(p, r, t) = (33") X 6(p — p.(t))6<r — rm» , (1.8)
test I:

 

where NM, is the number of test-particles per nucleon and 6 is the Dirac delta func-

tion. In the test particle approach, f solves the BUU equation provided that rk and pk

are themselves the solutions of equations of motion for test particles in a self consistent

mean field. The local density can be obtained by integrating f over the momentum.

The changes in the density with time can be predicted by computer simulations, in

which the test-particles are required to follow the Hamilton’s equations:

mzfli=m
d_pr__6_H
dt— ark.

The spatial gradient in the lower equation can be calculated by taking the difference
of the mean field potential between the neighboring points divided by the lattice
spacing.

BUU equation can be used to calculate various transport quantities. For the
present work, one of the most interesting transport properties is the isospin diffusion
coefficient. For the binary fermion system which is composed of neutrons and protons,
the isospin current can be shown to depend on the isospin asymmetry and can be

written as [93]

Pa = pnvn - ppvp = —pDav.-6 . (1-10)
where, pp(p,,) and vp(vn) are the local nuclear matter density and collective velocity
of nucleons, 5 = (p7. — Pp)Pn + pp) is the relative neutron excess and D5 is the isospin
diffusion coefficient.

The diffusion of isospin is driven by isospin concentration, as described by the
particle flow equation Eq. 1.10. When self-consistent mean fields produced by the
particles depend on the concentration, then this dependence make an important con-
tribution to the diffusion. The diffusion coefficient D5 therefore depends on the sym-
metry energy in the equation of state (EOS) and on its density dependence.

In nuclear matter, the isospin asymmetry dependence for both the kinetic and
interaction energy per nucleon may be well approximated in the quadratic form of
isospin asymmetry, E335“) = ef;’:,[‘"‘)62, where 6:73;, and egg; are the kinetic and in-

teraction contributions to the asymmetry coefficient, respectively. In nuclear matter

7

at normal density,e km 12.6MeV and 6"“ 19i6MeV. The density dependence

esymO ~ symO ~

kin
sym

of the kinetic energy is given by e (p)- — ef‘J,‘,,0(p/po)2/3. The density dependence
of the interaction symmetry energy, however, is not well constrained experimentally.
In light of this uncertainty, one may parameterize this density dependence, calcu-
late experimental observables as a function of this density dependence and identify
measurable observables that can be used to constrain it. For example, one can try a
power-law dependence on the density, 1. e. e?“ ym(p)— — em‘ (p/po)”.

symO

One can define a mean-field amplification factor R = D5(ei’;fin)/D5(ewm "“ =0) for
the diffusion coefficient. Predicted values of R from BUU calculations as functions of
temperature are shown in the Fig. 1.2, where results for both the linear and quadratic
density dependence of 8:3;(1/ = 1 and 2) are shown. The quadratic dependence gives
higher amplification factors at p > p0, than the linear dependence, while the opposite
is true at p < p0. Thus one would expect the stiffer (u = 2) symmetry energy
term to have a large diffusion at high density and the soft (V = 1) symmetry energy
term to have a larger diffusion at lower density as shown by Fig. 1.2. In fact,
the difl’usion coefficient depends monotonously on the symmetry energy and is large
when the symmetry energy is large and small when the symmetry energy is small.
At low temperature and moderate to high densities the amplification is very strong
suggesting that the measurements sensitive to the diffusion can be used to probe the

asymmetry energy. The same calculation also predicts the diffusion coefficient to be

reduced by proton-neutron scattering by the residual interaction [93].

1.2.2 Stochastic Mean Field (SMF) model

The BUU equation provides an accurate description of the time dependence of the
one body distribution function. Accurate solutions of the BUU equation average

away fluctuations in the density that might lead to the formation of fragments in

 

 

\ — Linear
\ - - - - Quadratic

 

 

 

 

 

 

 

 

 

 

Figure 1.2: Mean-field enhancement factor of the diffusion coefficient in symmetric
nuclear matter, RED5(ei’;fin)/D5(e§';fin = 0), at a fixed density n, as a function of
temperature T. The solid and dashed lines, respectively, represent the factors for
the assumed linear and quadratic dependence of the interaction symmetry energy on
density. The lines from top to bottom are for densities p = 2po, p0, 0.5p0, and 0.1po,
respectively. At normal density the results for the two dependencies coincide. [93]

an individual collision. This is usually achieved by solving the BUU equations with
a large number of test particles per nucleon Nmt. Because the density fluctuations
that lead to fragment formation are suppressed in the BUU equation, the calculation
of fragment yields directly via the BUU model is not feasible. Therefore, alternative
Models, such as Stochastic Mean Field (SMF) model [23] has been developed to
address the density fluctuation. SMF, like BUU, describes the time evolution of the
collision using a self-consistent mean field. The application of the SMF model to the
unstable situations relies on the knowledge of the most important unstable modes,
which maybe difficult to identify in cases where the modes are not known apriori.
To account for the stochastic fluctuation in SMF model, two approaches are used.
In the first approach, a fluctuating term is added to the standard Boltzmann-Nordeim—
Vlasov (BNV) equation to account for the stochastic force, the strength of which
is adjusted to reproduce the growth of the most important unstable mode in the
system. In practical term, the stochastic force is a type of stochastic noise. The
second approach uses this fact by approximating it with numeric noise caused by

solving BUU with a small number of test particles (i. e. N¢m=50) per nucleon.

1.2.3 Statistical Multifragmentation Model (SMM) Model

and Macrocanonical Approach

Statistical models assume the existence of statistical (equilibrated) distributions of
states during the fragmentation. Equilibrium statistical theories provide one approach
to calculate the properties of such fragments. Such models can provide precise pre-
dictions. The accuracy of such prediction is directly related to the accuracy of the
equilibrium approximation itself.

The copious production of intermediate mass fragments (IMFs), with mass in-

termediate between that characteristic of light particle evaporation and fission, is a

10

distinguishing feature of intermediate-energy heavy-ion reactions [73]. The emission
of IMFs at low emission rate is a prediction of statistical evaporation theories [37, 69].
The observation of very high IMF multiplicities, however, lies outside of the realm of
prediction of evaporation theories [15, 37]. Such an observation suggests a different
mode of decay via a bulk disintegration [10, 37, 42].

Microcanonical Statistical Multifragmentation Model (SMM) uses a Monte Carlo
method to calculate physics observables in approximate microcanonical equilibrium
approach [102]. If one assumes that equilibrium is achieved for a source with excita-
tion energy E5, volume V, mass A0 and charge Z0 specified. Possible decay modes
are weighted by the entropy of that decay mode within this volume. All decay modes
that conserve the total mass, total charge and total energy are constructed.

The entropy of each decay mode is assumed to be additive and is obtained from
the entropy of the individual fragments. Both energies and entropies for individual

fragments can be calculated from the free energy of the individual fragments [14]

T2 T2 — T2
_ A 1 c 5/4A2/3
.0) + 8(——,.>

713+
(A — 2Z)2 + 32262
A 5 RAZ

mnAT
— T1n[g.znvo(—2;m—.)3” + T

FAz(T) = (—Wo -

[1 — (1 — n)_l/3] (1.11)

ln(NAzl)
NAZ ,

+25

 

where, W0 = 16MeV is the binding energy of nuclear matter; so is the inverse nuclear
level density (50 z 16MeV for the Fermi gas model); TC = 16MeV is the critical
temperature; R A z = 1.1714” 3 f m is the radius of a fragment corresponding to normal
nuclear matter density p0; m N is the nucleon mass; g},Z is the spin degeneracy factory;
K. is a model parameter, which is introduced since the individual fragments are as-
sumed to move freely in a volume of [Cl/0; V0 is the volume of the system corresponding
to normal nuclear matter density p0.

The first two terms in the r.h.s. are the contributions from volume and surface;

11

the third term is from symmetry energy; the fourth term is from Coulomb interaction
and the last two terms are from the translational motion. In this approximate mi-
crocanonical approach the internal degree of freedom of the fragments act somewhat
as a heat reservoir. The temperature T of the fragments are then determined by the

equation for overall energy conservation.

The equilibrium can also be more approximately studied within macrocanonical
frame work, which provides some insightful approximate expressions. Under the
Macrocanonical approach, the fragment yield for 2"" fragment in its k‘h state can be

written as [1]

1.3/2 n -31
16,],(14, Z) = VIA; (2.12-,1C +1)eNfl 1‘2” (:8 NT’Z e—TL , (1.12)

where, Y,,k(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 J”. is the ground state spin of a nucleus with
charge Z and mass A; pp and p" are the proton and neutron chemical potentials; A =
(/27rh2 / m; V is the free(unoccupied) volume of the system and T is the temperature.

Eq. 1.12 describes the primary fragment yields. The fragment yields from exper-
iment are the yields after secondary decay. To minimize the effects of the secondary
decay, in some cases, one can construct observables that are approximately insensitive
to the secondary decay. One can construct the ratio of the fragment yields for the
same isotope from two different reactions under similar conditions. By taking this
ratio in the macrocanonical approximation Eq. 1.12, one finds that the volume and
binding energy factor cancel out. If the feeding from the decay of particles of stable or
unstable states are similar in the two reactions, it may cancel approximately as well.

In this case, if we define R21 as the ratio of isotope yields from two different reactions

12

1 and 2, R21 = Y2(N, Z) / Y1(N, Z), the following isoscaling relationship [110] is found

to be quite robust in nuclear reactions

R21 or eanZ, (1.13)

Where: a : (Alum/T) = (#722 - Mud/T and .8 : (App/T) : (”p2 “ [Jpn/T- Fig-
1.3 illustrates the isoscaling property observed with the fragments produced in the
mid-rapidity region of 124Sn + 124Sn and 112Sn + 112Sn reactions [123]. Here, 124Sn

+ 124Sn is the reaction 2 and 112Sn + 112Sn is the reaction 1.

 

 

 

I“
o

 

9.0!"
OIQO

 

 

 

 

 

I“
o

 

Y(‘2‘Sn+ ‘z‘sm /Y(1128n+ 112Sn)

 

 

.09!"
cnszo

 

 

Figure 1.3: Experimental isoscaling behavior exhibited by the central 124Sn + 124Sn
and 11"’Sn + 112Sn collisions. The data are the nuclide yield ratios, R21(N, Z) from
the two reactions plotted as function of N (top panel) and Z (bottom panel)

13

a and 6 can be shown to directly relate to the neutron excess 6, i. e. 0 oc
(62 — 61)(1 — (61 + 62)/2) [111]. We are going to use a as an observable to study the
density dependence of the asymmetry term and explore the isospin diffusion in this

dissertation.

1.2.4 Expanding Emitting Source (EES) Model

The achievement of equilibrium assumed in Sec. 1.2.3 can be achieved and maintained
only if the emission of particles and their absorption by collisions with other particles
are in detailed balance. Statistical surface emission provides a way to calculate the
rates and to describe how an excited system might proceed towards equilibrium or
away from it. In this section, we discuss the Expanding Emitting Source (EES) [37, 38]
model which calculates the binary decay of excited nuclei via emission rate equations
within a statistical frame work. This formalism, which allows the description of
time dependent phenomena, integrates the contributions for each type of particle as
a function of time over the de-excitation process. The formulae that describe the
surface emission mechanism are similar to those developed by Weisskopf’s seminal
paper on compound nuclear decay [119]. They are also conceptually similar to the
ideas of Richardson’s thermionic emission.

Let us consider the decay of a compound system labeled G into a daughter B,
capable of further decay, and a particle labeled b,

C —> B + b
Let Nb be the number of emitted particles of type b. Then fi—fg expresses the rate

of emission (at kinetic energy E in to a given interval dE) of that type of particle.

The energy spectra can be obtained from the double differential quantity

«in- We,
dE " o dEdt

 

dt. (1.14)

14

The emission rate is obtained by an energy integeration

dNb_ °°d2Nb
cfi_odwt

 

dE , (1.15)
and the multiplicity by a further time integeration

°° dNb
N=/-—fi. 11
b 0 dt ( 6)

The rate of emission using the approach of Weisskopf [119] can be written as

dN 2s 1
m = (72;. >EMa.....le<Eg>/wc<Ea>i , (1.17)
where M, E, and 3 refer to the mass, kinetic energy and spin of the emitted particle b;
and w3(Eg) is the density of states for the daughter system B at excitation energy, E5,
which remains after particle b is emitted with kinetic energy E from the compound
system C, originally having excitation energy E5. The quantity 012+ B _) C is the cross
section for the formation of system C which we take to be geometrical.

For the basic EES calculation, initial temperature and excitation energy need
to be specified for the EES calculation. The evaporation is complicated by other
factors, i.e. the isospin of the excited system, the recoil momentum of the target,
the reduction of the Coulomb barrier which accompanies charge emission as well as
the range of temperatures traversed. A discussion can be found in [37]. The EES
model allows expansion of the nucleus subject to its internal pressure and predicts
a rapid emission of fragments once the density of the residue decreases below about

0.4p0. This suggests that the multifragmentation principally occurs after the system

expands to subnuclear density.

15

1.3 Studies Prior to This Dissertation

The isospin-dependent Boltzmann-Uehling—Uhlenbeck (BUU) transport model has
successfully explained several isospin-dependent phenomena in heavy-ion collisions
at intermediate energies [61]. In isospin-dependent Boltzmann-Uehling-Uhlenbeck
(BUU) transport model, the isospin dependence was included in the dynamics through
nucleon-nucleon collisions by using isospin-dependent cross sections and Pauli block-
ing factors, the symmetry potential V023”) (p, 6) and the Coulomb potential. The
isospin-dependent BUU model was used to calculate the ratio of yield of neutron and
protons in pre-equilibrium emission [63]. It is found that if one drops the Coulomb
and asymmetry term Vaggpr, 6) in the BUU model, the ratio of pre—equilibrium neu-
trons to protons is almost a constant as a function of nucleon kinetic energy. However,
when one includes the Coulomb and the asymmetric term of the EOS, the ratio of
pre—equilibrium neutrons to protons changes monotonously as a function of nucleon
kinetic energy. Fig. 1.4 has shown this calculation. One can then study the effects of
the symmetry energy S (p) since the Coulomb potential is well understood. Since the
symmetry potential Vajg") (p, 6) tends to make more unbound neutrons than unbound
protons, one expects that a stronger symmetry potential leads to a larger ratio of free
neutrons to protons. One can see from the figure that the effects of symmetry poten—
tial show up at higher energy for central collisions and at lower energy for peripheral
collisions.

A naive picture of nuclear reactions at intermediate energies undergoes three im-
portant stages: compression and expansion stage when the preequilibrium light par-
ticles are emitted, after which a prefragmetation source is formed; the disassembly
stage (evaporation) after the prefragmetation source reaches equilibrium; a secondary

decay process of various emitted fragments. While the dynamic model such as BUU

and SMF give a good description of preequilibrium emission during the compres-

16

 

 

 

      
 

 

 

 

      

4 Vvfi WW ,, v --
: i T f' i
: .ooo”.‘+ [I 1 1
3 .' '3ZSn-i-‘328n '1
’ i
2 ' 1
1 [ 4
4 -
; '24Sn+'24Sn
O. a; 1 '4
z : 1 ill
2 ; .‘?00’-'..**** + j ,
z: 3i i
"D f
1 ' .
4 ................... — -
] “28n+“28n —— F 1(U): * “2811-1-“2Sn
3,- l<b<5fm —~—F2(U)» 7_b512fm -
t --+ F3(U)i .
8: 1
1 :- .'.‘.O“O...‘." l

 

 

 

 

 

 

O 20 40 60 80 IODOAASO 40 ”60‘80 100
Ek,,,(MeV)

Figure 1.4: The ratio of pre-equilibrium neutrons to protons as a function of nucleon
kinetic energy for central (Left Panel) and peripheral (Right Panel) calculated from
h the Coulomb and symmetry potentials

the isospin-dependent BUU model with bot
[63].

17

sion and expansion stages, the statistical models give a better description of the later
stages as equilibrium is approximately achieved. At the end as hot fragments decay to
their ground states, nuclear structure data such as level densities and binding energies
are needed to describe the secondary decay process. It may be that the combination
of different models to describe the different stages of the reaction can give a better
description of the experiment data than can any of them individually. Hybrid models
in which the internal dynamics is described by a dynamic model and the later stage
by a statistical model have been attempted and used to test the asymmetry term
of EOS. Fig. 1.5 shows one of such calculations for the data obtained from central
Sn+Sn collisions. The data and calculations shown are based on the Eq. 1.13 and
uses 112Sn+“2Sn as reaction 1. In this figure, 6,, and [3,, are plotted as functions of
Ntot/Ztot- Where, ,6" and 13,, are just the e“ and 85 defined in Eq. 1.13, respectively;
NM and Ztot are the total numbers of neutrons and protons involved in reaction
2. The experimental data are shown as solid circles and squares while the open and
cross-hatched rectangles show corresponding hybrid calculations of R21 obtained with
the asy-stiff (left panel) and asy-soft (right panel) EOS’s respectively [101]. We can
see from this figure the final yields with the asy-stiff EOS (left) panel overlap the
data. In comparison, the calculations using the asy-soft EOS (right panel) show a
significantly weaker dependence on Ntot/Zm than do the data. In this dissertation,
we will use a different type of hybrid model to study the density dependence of the

asymmetry term in EOS.

1.4 Thesis Organization

In this dissertation, Chapter 2 describes the experimental setup, i.e. the devices
involved in the experiment of this dissertation: The Large Area Silicon-Strip/CSI de-
tector Array (LASSA), The Ring Counter and Miniball/Miniwall array. Chapter 3

18

describes the data analysis, i.e. the energy calibration for the detectors and extraction
of PID function. Chapter 4 provides the experiment results and various comparisons
with model calculations. Chapter 5 summarizes the thesis and provides the conclu-

sions.

 

‘ I V l’

lIsotope

1 .4 -
’ Asy— stiff

b

......,
Ratios

    
  
   

  

L\\\\‘ .‘ ‘ .“

 

 

1.0‘

   

\\\\\‘
_
A I l A I

[Isotone Ratios

0.6 i“"““““‘ --A-l--1.L....1
1.2 1.3 1.4 1.5 1.3 1.4 1.5

Ntot / Ztot

Figure 1.5: Both Panels: The solid circles and solid squares show values for p“, and
p", respectively; measured in central 112Sn+“2Sn, 112Sn+124Sn, and 124Sn+1248n col-
lisions at E / A =50MeV. Left panel: The open and cross-hatched rectangles show
corresponding hybrid calculations for R21 calculated from the primary and final frag-
ment 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 for R21 calculated from the primary and final fragment yields, respectively,
predicted by the hybrid calculations using the asy-soft EOS.

 

 

 

 

19

 

 

Chapter 2

Experimental Setup

2.1 Experimental Layout

The thesis experiment was performed at the National Superconducting Cyclotron
Laboratory at Michigan State University with beams from the K1200 accelerator.
112Sn+“2Sn, 124Sn+1128n, 112Sn+124Sn, 124Sn+m4Sn collisions were measured by us-
ing 50MeV per nucleon 112Sn and 124Sn beams and 5 mg/cm2 112Sn and 124Sn targets.
Three major devices were used in the experiment:

(1) Nine telescopes of the Large Area Silicon Strip Array (LASSA)

(2) The Miniball/Miniwall phoswich detector array.

(3) The annular silicon strip forward array (the Ring Counter)

Fig. 2.1 shows a photograph of these detectors used in this experiment. This
photograph is taken towards the downstream direction of the beam. In this picture,
one can see six of the nine LASSA telescopes. These are the large square telescopes
mounted above the beam axis. One can also see the Ring Counter, which appears
as a ring around the beam exit hole, just under the LASSA telescopes. The other
detectors in the picture are various elements of the Miniball array.

The LASSA detector Array is a large area, highly segmented silicon-strip/Csl de»

20

 

Figure 2.1: Major detectors used in the experiment, viewing in the downstream
direction of the beam

21

tector array. It provides excellent energy, angular and isotopic resolution for isotopes
with Z S 8. In this experiment, LASSA is used to measure the fragments produced
from central collisions as well as the fragments from the projectile—like source. The
Miniwall/Miniball multi-detector array is a moderate resolution detector array for de-
tecting charged particles. The Miniball detectors can identify elements with Z S 18
and isotopes for Z s 2. This array was employed to select impact parameters in this
experiment. The impact parameters were selected by gating on the multiplicity of
identified charged particles.

The Ring Counter forward array is also a highly segmented silicon-strip/CSI de-
tector array. It provides atomic-element resolution for fragments with 3 5 Z s 55
(Z is atomic number) as well as excellent angular position information. The Ring
Counter was used to detect projectile-like fragments (PLFs).

The LASSA detector array was constructed specifically for this experiment, while
Miniwall/Miniball multi-detector array and the Ring Counter forward array are exist-
ing devices available for use in this experiment. Details of the detectors are discussed
in the following sections.

Fig. 2.2 is a schematic drawing of the mechanical layout of this experiment.
LASSA, which was centered at 325°, subtended polar angles from 7° to 58°. The
Miniwall/Miniwall detector array covered polar angles from 695° to 150°. The geo-
metric acceptance of Miniball/Miniwall is about 76% of the 41r solid angle. The Ring
Counter, which is an annular shaped detector, was put at the most forward angle. It
was centered about the beam line and covered polar angles from 22° to 45°. The
relative placement of the LASSA and the Ring Counter can be viewed in the Fig. 2.2
and Fig. 2.3.

22

Lassa detector

Ringcounter

/

 

 

A _
U r Beam

    
  

Miniball

Figure 2.2: Mechanical Layout of the experiment, viewed from the side

I\

Figure 2.3: The relative position of LASSA and the Ring Counter

23

2.2 LASSA Telescopes

The Large Area Silicon—Strip/CSI detector Array, LASSA [32], consists of nine indi-
vidual telescopes, instrumented with strip silicon detectors. It provides high angular,
energy and wide-range isotopic resolution, a large detection area and a reasonable

energy range for the detected particles.

Structure of LASSA Detector
A photo of a single telescope is shown in Fig. 2.4. A schematic of a single telescope
is also shown in Fig. 2.5. There are three layers of detectors in each telescope. The
first two layers are highly segmented strip silicon detectors. These strip detectors
are backed by four CsI(Tl) detectors, which constitutes the third layer of detectors.
The first silicon detector, which we called the “DE” detector, is a 65 pm thick silicon
detector with 16 strips on one side of the detector; the total area of the detector is
50x50 mm2 with each strip 50mm in length and 3mm in width. The second silicon
detector, called “EF/ EB”, is a double-sided, 500nm thick silicon detector with 16
strips on each side, the direction of the strips on the front side are perpendicular to
the strips on the back side; the area of the EF/ EB detector is also 50x50mm2 with
the strips on each side being 3mm in width. For both the first and the second silicon
detectors, the gap between the strips is about 0.1mm. The third layer of detectors
consists of four CsI crystals. Each of these has an active area of 2.5x2.5 mm2 and
each is 60mm in thickness. The crystals are tapered such that front and back surfaces

subtend the same solid angle with respect to the target.

Principle of Operation
Each telescope in LASSA functions as an energy loss telescope. An energy loss tele-

scopes identifies the charge, mass, and energy of charged particles in the following

24

 

Figure 2.4: Photo of a single LASSA telescope. The Silicon detectors are not mounted
and one of the side covers is removed to show the CsI detectors.

25

 

 

 

 

 

 

 

 

 

Beam '

Figure 2.5: A schematic drawing of the LASSA detector

way. For particles passing through a detector, the loss energy is proportional to the
stopping power. The energy loss can be approximated by Bethe formula:

2
AE z kAZ

 

dcc, (2.1)

where da: is the detector thickness, k is a proportionality constant, A is the mass
number of the particle, and Z is the atomic number of the particle. One can easily
see that for a given dm, a plot of E verses AE will yield a family of a contours with
AE or 1 / E. Each line corresponds to an integer value of Z, and an integer values of
A.

If the various detectors generate signals proportional to the energy deposited by
the charged particle, then digitized signals can be used to identify the particles using
this energy loss relation. It is true for the detectors making up the LASSA telescope.
When a charged particle passes through a silicon detector in LASSA, for example, it
generates a signal that has a linear dependence of the energy it deposited. When a
charged particle stops in the CsI(Tl), it generates a measurable light output which is
roughly proportional to the energy deposited by the particle in the C31. Thus when

we plot energy loss in silicon vs. light output from CsI, we can expect to see the

26

contours of AE or 1/E, offset for each isotope according to the integer values of the
atomic number Z and mass number A. One can deduce the relationship between E,
AE and Z, A, E101 = AE + E empirically or analytically. In either case, we can
construct a PID function or PID value which uniquely identifies the Z and A of the
charged particle from the measured values of AE and E. In this thesis, we determine
the PID empirically.

The data for the LASSA telescope are analyzed as follows: If a charged-particle
passes through the first silicon and stops in the second silicon, then the signals from
the first and the second silicons are used to yield the angle, energy and PID informa-
tion. If the charged particle passes both silicons and stops in the C31, then the all the
signals from the silicon detectors and the CsI crystal are used to give out angle, energy
and PID (i.e. Z and A) information. By detecting the charged particle this way, we
can take advantage of the high angle and energy resolution of silicon detector as well
as large energy detecting range of the CsI(Tl) detector. Fig. 2.6 plots the measured
energy loss in the DE detector verses measured energy loss in the EF detector. In
Fig. 2.7, we plot the measured energy loss from the silicon detectors(with DE and EF
signals added together) verses that of the CsI crystals. Note that roughly hyperbolic
lines we see in both figures correspond to the various isotopes. In addition, on the
Fig.2.6, we see the isotope lines turn over at the lower right side and then continue
toward lower values of DE and EF. The particles corresponding to these lines “punch-
through” both of the silicon detectors, and stop in the CsI crystal. The energy of
the charged particle was obtained from the calibrated signals from the silicons and
CsI(Tl). The position information was given mainly by the signals from the EF and
EB strips of the second silicon detector.

LASSA provided isotopic identification for fragments with Z _<_ 8 and the ener-

gies of these fragments to better than 3% accuracy. For the energy calibration and

27

 

3500

5

§

5

 

DE (Raw ADC Output)
3
8

1000

..1 ___

    

I 1.

'-’~‘.’.

0 500 1000 1500 2000 2500 3000 3500 4000
EF (Raw ADC Output)

Figure 2.6: A two dimensional histogram of the raw signal outputs of DE (first
silicon) vs. BF (second silicon). In the figure, below the element labels, one can see
the hyperbolic lines for various isotopes of the elements.

28

§§§§§

DE+EF (Raw ADC Output)
§

 

 

 

 

. 4 , 4"“
-————--———--——-——————————__--—————-——-_—

500 1000 1500 2000 2500 3000 3500 4000

CsI (Raw ADC Output)

Figure 2.7: A two dimensional histogram of the raw output signals of DE+EF (silicon)
vs. E (CsI). In the figure, below the element labels, one can see the hyperbolic lines

for various isotopes of the elements.

multiple hit decoding of LASSA, details can be found in the next chapter.

Solid Angle Coverage

The LASSA telescope array was put at 20cm from the target and centered at a forward
angle of 325°. The 3mm width for the strips of the silicon detectors at a distance of
20mm corresponds to a width in angle of 086°. The position of the center of each
“pixel” (e.g. the intersection of the strips with perpendicular directions, EF and EB)
can be seen in Fig. 2.8. At the upper left corner, the relative placement of the 9
telescopes is shown. The given number corresponds to the labeling scheme for the
telescopes during the experiments. The whole array covers azimuthal angles ranging
from approximately 50° to approximately 150°, and polar angles from approximately

7° to approximately 58°.

Electronics of LASSA
High quality pre-amplifiers (PA) were used for LASSA to achieve high-energy res-

olution. Given the number of channels involved and the space constraints, all pre-
amplifiers were constructed using surface mount technology, with the exception of
the input field effect transistor (FET) and the load resistor. The pre-amplifiers of
C81, which amplified the signal of the photodiodes of the CsI crystals, were installed
inside the LASSA telescope; the silicon preamplifiers were mounted in separate boxes
adjacent to the LASSA array.

There are 432 Silicon channels in LASSA. A total of thirty 16-channel Analog-
to—digital Converter (ADCs) and nine Fast Encoding and Readout ADCs (FERAs)
modules (for the timing signals) were required to read out all the silicon signals from
LASSA. The large number of channels in LASSA required development of an inte-

grated shaper/ discriminator module that would allow easy computer control of the

30

60

50

40

30

20

10

 

l6

FT I I ‘

 

° 0'

03 co go
9/

'1-
'1'
x?

.0
.°‘

 

 

l

 

l L l l l l

///../

y—
y—
_—

........
................

1]l ll

1

 

0

20 40 60 3

31

0 100

120 140 160 180

(l)

Figure 2.8: A two dimensional graph of the solid angle coverage of LASSA in polar
angle 6 and azimuthal angle (15

 

 

LASSA electronics, including the gain of the energy signals and discriminator thresh-
olds. These special double-width Computer Automated Measurement and Control
(CAMAC) modules were developed by Washington University. Each module consists
of a slow shaper, a timing filter amplifier, a leading edge (LE) discriminator, and
a time-to-charge converter (TFC) for each of the 16 independent channels. These
special modules can be controlled by computer via a GPIB CAMAC interface. In
the experiment, a precision pulser (IU pulser) was used to track changes in electronic
gain. This pulser is stable to within 0.1% and can be controlled by a computer
via an RS-232 serial interface. A schematic of the LASSA electronics can be seen in

Fig. 2.9 [32].

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

R8232 WU Double-width Shaper
éMult 9
Control 30111 Disc.Or Time to 4
i I ' r. Fera
J z, : of 16 Chan. Fera .
r 4—:
P lser ; ON I I i
u ‘ v - :
[ LE Disc. Log” _;
[N] Inspect ‘
x16
; Timing filter Amp P11716411
Preamp Slow Shaper ADC
l Linear
DE/EF/ ’ Inspect
EB ..........................................................
of Silicon

 

 

 

 

I
,1 . r.‘

GPIB Control :32,

0

J

l

“ '
d

 

 

 

 

Pre- Quad PH7 1 641']
amp Shaper ADC

 

 

CsI(Tl)

 

 

 

 

 

 

 

 

 

 

Figure 2.9: A schematic of the elctronics for the LASSA telescope

32

2.3 The Miniball/Miniwall Array

Structure of Miniball/Miniwall

The Miniball detector array [96] is a 47r detector array. It consists of 11 independent
coaxial rings centered about the beam axis, with each ring at a specific polar angle.
A total of 188 fast-plastic CsI(Tl) phoswich can be mounted on these rings. Each
phoswich detector consists of a 8mg/cm2 (80pm) thick plastic scintillator foil and
a 2 cm thick CsI(Tl) crystal. The shape of the front face of the crystals are the
same within each ring but differ among rings, forming a spherical surface facing the
target. The front face of the phoswich assembly is covered by an aluminized mylar
foil (0.15mg/cm2 mylar and 0.02mg/cm2 aluminum). The crystal is coupled by a
light-guide to the photo-multiplier (PM) tube. The photomultiplier tube and the
cylindrical light guide are surrounded by a cylindrical p-metal shield. Fig. 2.10
shows a schematic drawing (the cylindrical p-metal shield is not shown).

Since the angular distribution of emitted particles in heavy-ion collisions is strongly
forward peaked [53], an additional Miniwall detector array [121] was constructed by
Washington University to cover the forward angle 19 S 25°. The Miniwall detector
array was constructed as an azimuthal group of the phoswich detectors similar to the
Miniball array. It consists of 6 rings, with a total of 128 detectors. Each of the Mini-
wall detectors was made with 3cm thick CsI(Tl) scintillator crystals and 8mg/cm2

thick scintillator foils.

Principle of Operation

The Miniball/Miniwall detector is based on the plastic-CsI(Tl) phoswich technique,
and detects charged particles in the following way: When charged particles pass
through the fast-plastic scintillator or the Tl—doped CsI crystal, visible light is gen-

erated. This light is collected and converted to amplified electric signals by the pho-

33

Fast scintillator Light guide

 

 

PMT

 

 

 

 

 

 

 

l

Al-mylar foil

 

 

 

 

\\\\\\\\\\\\\\\\\\\\\\\\\V

 

 

Optical cement

Figure 2.10: Schematic of phoswich assembly of individual detector elements.

34

tomultiplier. Because the light emitted by the scintillator materials have a variety of
decay time constants, it is useful to analyze respectively three different components in
the output signal of the photomultiplier, which we labeled: Fast, Slow and Tail. The
anode signal output from the photomultiplier tube and the three integrated regions
are shown in Fig. 2.11. These regions are integrated with FERA charge integrat-
ing QDCs within alternative applied gates that were triggered by the phoswich light
output signals of the detectors. The Fast component corresponds to the first 33ns
of the phoswich light output, which mainly originates in the thin fast-plastic scintil—
lator. Integrating the Fast component will yield a quantity roughly proportional to
energy loss 6E deposited in the fast-plastic scintillator by the charged particle. The
CsI(Tl) produces light that decays with a time dependence that is energy, charge
and mass dependent. We obtained energy and PID information from the Slow and
Tail components, corresponding to 150ns-540ns and 1.5ps-3ps after the beginning of
the signal. Integrating the Slow signal gives a quantity roughly proportional to the
energy E of the particle. Integrating the Tail component will yield a quantity which
depends strongly on the Z and A of the particle. The integrated Fast signal and
integrated Slow signal are combined to determine the charge of the detected particles
with Z _<_ 18. The integrated Slow signal and integrated Tail signal are combined
to give isotopic identification for the detected particles with Z S 4. The thresholds
to particle identifications are about Eth/A = 2 MeV for Z = 3, Eth/A = 3 MeV for
Z = 10, and Eth/A = 4 MeV for Z = 18 fragments. However, these thresholds are
not important to this dissertation since the Miniball/Miniwall were only used here
for providing the multiplicities of light particles and IMFs.

The particle identification resolution of one of Miniball detectors, located at the
polar angle of 90°, is shown in 2.12. To obtain the PID for isotopes from the Slow and

Tail signals, a special transformation was done to separate the PID lines even more,

35

resulting a new “transformed signal”, PDT [121]. A plot of counts versus Slow and
Fast signals is shown in the top panel and the corresponding plot of counts versus the

PDT and Slow signals is shown in the bottom panel.

Photomultiplier tube output signal

0 33ns l80ns 540ns 1.5113 3118

;/

     
            
     

 

Time

   

“-—‘—

\\\\\\\\\\\\\\\\\““
\\\

&\

  

“\\

\\\\\\

 
 

\.\\\\
\\\\\\

a.

 

 

 

Fast Slow Tail

Figure 2.11: The signal from the anode of phoswich detectors. The starting and
stoping times for the integrations of the three parts of the signal are shown.

36

 

Fast(Channel)

 

 

Slow (Channel)

 

 

 

 

O 50 100 150 200 250

PDT (Channel)

Figure 2.12: A two dimensional histograms of the raw signal from Slow vs. Fast and
PDT vs. SLOW

37

Solid Angle Coverage

In this experiment, the Miniball/Miniwall Array is used only for impact parameter
selection. A third of the Miniball/Miniwall detectors were taken out to allow insertion
of in LASSA telescopes. The ring involved and the corresponding solid angle covered
are list in Table 2.1. The Miniball/Miniwall detection array covered a solid angle

corresponding to about 76% of 47r.

 

 

 

Ring Number of Detectors 0(°) an (msr) A9 of Ring (msr)
2(W) 11 6.95 2.57 28.27
3(W) 12 10 2.59 31.08
4(W) l4 13 2.85 39.9
5(W) 12 16.625 5.56 66.72
6(W) 10 21.875 10.64 106.4
3’ 13 28 11.02 143.26
4 9 35.5 22.9 206.1
5 16 45 30.8 492.8
6 13 57.5 64.8 842.4
7 18 72.5 74 1332
8 16 90 113.3 1812.8
9 14 110 135.1 1891.4
10 12 130 128.3 1539.6
11 8 150 125.7 1005.6
Total 178 9538.33

 

 

 

 

 

 

 

Table 2.1: Table of the Miniball/Miniwall detectors used for the impact parameter
selection, their corresponding polar angles and solid angle coverage.

Electronics of Miniball/Miniwall
The electronics used to read out the detectors of Miniball and Miniwall are quite
similar. The electronics are divided into groups called banks. Each bank processes the
outputs of 16 detectors. There are four Fast Encoding and Readout ADCs (FERAs)
in each bank which processes the Fast, Slow, Tail and Time signals.

The signals from the anode of the Photomultiplier are first split by a passive

splitter into four signals. One signal is used for the logic circuit and the other three

38

signals are the Fast, Slow and Tail parts of the photomultiplier signals to be integrated
within FERAS. The four signals have relative amplitudes 1pm : I 310w : [Tail : [Tn-g m
0.82 : 0.04 : 0.04 : 0.1. The trigger signal is amplified and sent to a leading edge
discriminator which produces an Emitter Coupled Logics (ECL) logic signal. The
typical signal output from the detector and various integrating gates are shown in
Fig. 2.11. Individual gates are produced by the discriminator for gating the linear
gates which select the Fast region of the photomultiplier signals; while the gates for
integrating the Fast, Slow and Tail regions within the FERAs are produced by gate
generators that are triggered by the master trigger for each event. Since it takes
some time for the logic part of the electronics to generate the logic gates, Fast signal
is delayed by 150ns before the linear gate. The output of each of the linear gate is
then sent directly to its FERAs.

Since there are nearly two hundred Miniball/Miniwall detectors, it would take a
long time to read out the ADCs for every detector via CAMAC. Therefore, we need
the FERA Fast readout scheme, which read the data through the F ERA bus, with
zero suppression to achieve a faster readout. The readout details can be found in the
Data Acquisition Electronics section.

Fig. 2.13 shows the electronics block diagram for the Miniball array. In the figure,
LP denotes Light Pulser system. It was triggered at a rate of 1 Hz to generate visible
light that was injected by optical fiber into the light guide of the CsI(Tl) detectors

to monitor the stability of the gain during the experiment.

39

 
      
        
    
      
   
 
   
 
 

NIM/
ECL

 
    

  

l xl
FIFO Common
start

11

  

=>

Fast Slow Tail

   

Feta

gate

   
    

Logic
for Feras

  

::>

Figure 2.13: Schematic diagram of the electronics of the Miniball/Miniwall

40

2.4 The Ring Counter Foward Array

Structure of the Ring Counter

The Ring Counter Forward Array is an array of Silicon-CsI(Tl) telescopes. The sili-
con AE detector elements are segments of a large annular strip silicon detector, the
CsI(Tl) E detectors are elements of an annular array of 16 CsI(Tl) detectors. The
active area of the Ring Counter is defined by the ring-shaped 280 pm(67.05 mg/cm2)
thick silicon detector which forms the first element [34]. The inner diameter of the
detector is 47.88mm, while the outer diameter is 96.10mm. The front side, called the
Strip side, of the silicon detector is divided into four quadrants, and each of these
quadrant is subdivided into 16 strips, each at a constant polar angle. So there are
total 4x 16 2.8mm-wide strips. The back side, called Pie side, of the Ring Counter is
segmented into 16 azimuthal pie slices. The silicon detector is mounted directly on
a copper plate, which serves as a cooling bar for the silicon detector. Following the
silicon, there are 16, 2-cm thick, CsI(Tl) scintillators closely packed inside of a plastic
ring. The 16 CsI(Tl) crystals are aligned so that each CsI crystal is directly behind
one of the pie slices. Finally, a thin 5.0 mg/cm2 Sn—Pb (60% Sn, 40% Pb) foil was
placed in front of the array to absorb electrons and x-rays emitted from the target.
Fig. 2.14 is a schematics drawing showing the various components of Ring Counter.

Fig. 2.15 gives a side view of the Ring Counter.

Principle of Operation
The Ring Counter Foward Array identifies the charged particles using the same prin-
ciples as LASSA. It uses the signal from the silicon as AE, the signal from the
CsI(crystals) as E. Ordinarily, the signal from the pie side of the silicon detector is
used for AE, and the signals from the Strip side gives the polar angle and those from

the Pie side gives the anzimuthal angle. Fig. 2.16 shows the correlation between the

41

 

 

 

 

 

 

 

 

 

 

 

 

 

 

k

 

\\

Strip side of Si

 

Pie-sector side of Si

 

 

 

 

 

 

 

\,\

 

 

/

 

 

\l‘f

VK

CSI

”1

(T1) Crystals

Figure 2.14: A schematic drawing of the Ring Counter

42

AE

UV Lucite

 

 

 

 

 

 

l

CsI(Tl) Pin Diode

 

Annular Si

Sn-Pb foil

Figure 2.15: A schematic figure of the side view of Ring Counter

43

energy loss signals measured in the pie side of the silicon detector verses the light

output signals measured in the corresponding CsI(Tl) detector.

 

4000 -

DE (Raw ADC Output)
3 N 5.” "’ R
8 § 8 s s

 

 

§.

  

‘— 3 ”2000
L
3500 4000

0 ‘1 1 . 1 L I
0 500 1000 1500 20002500 3000

 

J.

CsI (Raw ADC Output)

Figure 2.16: A two dimensional histogram of the raw signal from Pie side and CsI

Electronics of Ring Counter
The pie side is the p-type side and the strip side is the n-type side. The diode de-
tector is reversely biased. Each strip or pie slice is read out independently. Similar

electronics were used to process the signals from the Ring Counter Forward Array as

44

for the LASSA telesc0pes. Lower gain LBL preamplifiers were used for Ring Counter
silicon signals, however, because the energy loss of the projectile-like fragments is
very large.

A schematic of the electronics is found in Fig. 2.17. The electronics used to
read out the 16 channels of the azimuthal pie-slice segments and the 64 channels of
the polar-angle strips are similar, the only differences are the polarity of the signals
and the additional requirement that the pie sector provides the timing information.
Although the preamplifiers of the signals from the 16 azimuthal pie sectors generate
both a Fast and a Slow outputs, we injected the Slow output into a Washington
University (WU) shaper-discriminator module, which contains a second stage slow-
shaping amplifier and a fast timing-filter amplifier. The fast timing—filter amplifier
output was used as the trigger to the experiment. The outputs of the fast amplifiers
of the 64 polar-arc segments and the 16 CsI were amplified in a second stage slow-
shaping amplifier in a single-width WU shaper. All the outputs from the slow-shaping
amplifiers for the strips, Pie’s and CsI’s were sent to the Phillips 7164B 4096 channel
peak-sensing ADCs. Only those signals above the low-threshold of the peak-sensing
ADC were digitized and read by the DAQ.

45

 

 

 

 

 

 

 

 

 

 

 

 

_—

 

 

 

Slow Shaping

_4

 

 

Peak sensing

 

 

 

 

 

 

 

 

 

 

 

 

Strip __ Preamplifier
Pie-Sector _S Preamplifier
CsI H Preamplifier

 

 

 

 

 

 

 

 

 

 

 

 

Amp ADC
Trig for
DAQ
Timing Filter
Amp __ DGG _,
Slow shaping Peak sensing
Amp '— ADC
Slow shaping Peak sensing
Amp b ADC

 

 

 

 

Figure 2.17: A block chart of the electronics of the Ring Counter

46

2.5 Data Acquisition Electronics

The Data Acquisition(DAQ) for this experiment was a big challenge. Strip-silicon
detectors usually require a lot of electronics. In this experiment, there are 9 x 16 DE
signals, 9 x 16 EF signals, 9 x 16 EB signals, 64 Strip signals, 16 Pie signals, for a
total of 512 energy signals from the strip detectors. In addition, there were 36 energy
signals from the CsI(Tl) detectors as well as 144 timing signals from the EB strips.
We also need to incorporate Fast, Slow, Tail and Time signals from the roughly 200
Miniball/Miniwall phoswich detectors, which corresponds to 800 electron channels.

To reduce the dead time and incorporate LASSA and the Ring Counter electronics
as well as Miniball/Miniwall electronics, all channels were read out via the F ERA
ECL readout scheme. The Miniball/Miniwall electronics used FERA 4300 ADCs
which could be read out more quickly via the ECL (Emitter Coupled Logic) port.
This is a double-bus readout scheme, with one bus for command and another for
ECL port data. A rate of up to 100nsec/word can be reached, comparing with the
typical CAMAC hardware cycle of 1 ,us. To make use of the ECL port readout, there
are some problems needed to be solved: First, Phillips ADC does not have an ECL
readout port like FERA. To solve this, an auxiliary CAMAC module, called 8832 is
used. This CAMAC module is designed by J. Toke of University of Rochester. 3832
works as a inter-media between Phillips ADC and Lecroy FERA Driver 4301. It acts
like a very fast CAMAC controller to the Phillips ADC, and outputs the data via
the ECL port like a standard Lecroy 4300 FERA. All the data from the FERAs and
8832 were sent via the ECL port to a FERA Driver 4301.

In more detail, the 8832 collects the data from Phillips ADC, handshakes with
FERA driver 4301, then puts the data to the data bus. Every converter executed
a hit-pattern—based sparse readout of up to 24 Phillips model 7164 ADCs via the

standard CAMAC protocol. This non-standard readout effectively moves the data

47

and control cords from the back plane of the CAMAC crate, to the front, ECL ports
of the 8832. The non-standard readout, which required no hardware modifications
to the 7164H ADCs, or the CAMAC crates, resulted in a 7.5-fold reduction in the
readout time of the LASSA electronics as compared to a standard CAMAC readout.

The old Miniball electronic design is tree-structured; one problem that needed to
be solved is that FERAs were not side by side as needed for a bus-structured readout.
Special cables and connectors are made to solve this problem. A schematic drawing
of the topology of the electronicsis shown in Fig. 2.18. The readout for the Miniwall
was analogous to the Miniball and the readout for the Ring Counter was analogous
to the LASSA, so the readout for the Miniwall and the Ring Counter were not shown
in this figure. In Fig. 2.18, the readout has a two-level scheme. The lower level is the
FERA modules and 8832 modules which are controlled by FERA drivers; the FERA
drivers, in turn, are controlled by a VME module Fera Faucet Maier (FFM).

In Fig. 2.18, those rectangles with thick black box lines are the CAMAC crates.
Those lines connecting the modules in the CAMAC crates denote the bus lines for
control signals and data. The signals from the Miniball detectors come from the
upper-right corner and go directly to the splitter. The trigger signals from the splitter
go through the discriminator and the multi-box to the main trigger control module
OCF 8000. The outputs from OCF 8000 are sent to FFM, different ADC and TFC
modules. The multi-box is used to determine the coincidence multiplicity of Miniball
trigger, which we set to 3 in this experiment. The Fast, Slow and Tail signals from
the splitter are sent directly to the corresponding FERAs in the CAMAC crates. The
signals from the LASSA are from the left side of Fig. 2.18. They first go to a double-
width WU shaper and discriminator/TFC module. The outputs of WU shaper are
sent to the Phillips ADCs while the outputs of TFCs are sent to FERAs for timing

signals. The data in those Phillips ADCs are collected and sent to FERA drivers by

48

S832.

The principle of ECL port readout can be understood from Fig. 2.19. For the
same reason as Fig. 2.18, only the parts for LASSA and the Miniball are shown.
Fig. 2.19, one sees a microprocessor, the “transputer”, developed by Inmos company,
was used as the front-end computer. CES High Speed Memory (HSM) 8170 is a
standard VME memory module. FFM integrates the CES 8170, FERA drivers and
ECL readout modules. The usual trigger used for this experiment was the observation
of a multiplicity in the Miniball/Miniwall > 3 or with the Ring Counter trigger,
downscaled by a factor of 25. Zero compression is used for all ADCs to suppress
all data words which are consistent with the experimental pedestal in the individual
ADC channels.

In upper part of Fig. 2.19, one can see that the electronics for the Miniball array
are basically the same as what is shown in Sec. 2.3. The main difference is in the
lower part, where one sees that the output signals from delay gate generator (DGG)
are sent to the gate input (GAI) of F ERA driver, instead of the gate input of FERA.
The LASSA elctronics, as described in Sec. 2.2 are incorporated with the Miniball
electronics. Also, the busy outputs from CBS 8170 and FFM are added to the busy

logic boxes.

49

 

ECL Readout “PM

MN? for ADC;

 

 

DE

 

Matti-box DISC 7

EF

EB

CsI

Time EF

:umuadunj

Miniball

 

Lassa

Figure 2.18: Layout of the data acquisition (FD means FERA Driver)

50

 

a

 
 
 
 

  

Miniball

 

ll

 
      
 

Driver
Terminator

  

Figure 2.19: Schematic drawing of the principle of the ECL readouts

51

Chapter 3

Data Analysis

3.1 Data Reduction Algorithms

As discussed in the last chapter, three types of devices were involved in the exper-
iment: LASSA, Ring Counter and Miniball/Miniwall detectors. Miniball/Miniwall
detectors are only used for impact parameter selection, so the energy calibration was
not done for Miniball/Miniwall detectors. However, a rough PID function was gener-
ated to separate LPs (light charged-particles with Z <3) and IMFs (heavier fragments
with Z Z 3). For LASSA, the calibration was done for isotopically resolved particles.
For the Ring Counter, the calibration was done for atomically resolved particles.
Since both LASSA and Ring Counter are composed of silicon-strip/CSI telescopes
array, the data analysis for these two devices are quite similar. This data analysis
is basically divided into two parts. One consists of the analysis of the signals in the
Silicon detectors while the other one is the corresponding analysis for the signals in
the CsI detector. For the Silicon detector part, the most difl‘icult issue is the multiple
hit correction. In multiple hit events, two or more particles hit the same telescope at
the same time. When a multiple hit happens, there will be more than one way to pair

up the signals from the vertical and the horizontal strips, the task is to determine the

52

correct pairing. For the CsI detector part, the main issue is the energy calibration.
Since the lightput out of CsI(Tl) depends on the mass number A, atomic number
Z and energy of the charged particles, different calibrations must be done for each
isotope (LASSA) or each element (Ring Counter). We will talk about details of the

data analysis of LASSA and the Ring Counter in the following sections.

3.2 Data Analysis of LASSA

3.2.1 Overview

The energy calibration and the generation of the PID function were the first two steps
of analysis for the data analysis of the LASSA telescopes.

The LASSA array was calibrated in three stages. The first stage involved cal-
ibration of the device in the running configuration with a precision pulser and a
charge terminator. In the second stage, a series of fragmentation beams was used and
each detector was repositioned to allow the beam to be put directly into the telescope.
Beams of 20 MeV/u 16O and 40 MeV / u 36Ar were accelerated by the K1200 cyclotron
at Michigan State University and used to bombard a production target at the exit
of the cyclotron. Reaction products from projectile fragmentation were subsequently
selected according to magnetic rigidity by the A1200 magnetic channel. These low
intensity secondary beams (~1300 p/s) were then used to directly scan the detector
face. The beams used are listed in Sec. 3.2.5. These beams were useful in assessing
the thickness non-uniformity of the 65 pm Si detector, as well as in calibrating both
of the Si detectors and the CsI(Tl) detector. Finally, proton recoils used for proton
calibration was measured by bombarding a Polyethylene target (CH2) with a 112Sn
beam.

Two sets of PID functions were generated for LASSA: A Si-Si PID function is

53

generated for particles stopping in the second silicon detector, a Si-CsI PID func-
tion is generated for particles stopping in the C81. To generate added 2D PID his-
tograms without losing resolution, one needs to do the correction for the thickness
non-uniformity of silicon detectors as well as the gain-matching of the CsI detectors.
The thickness non-uniformity correction is done for each “pixel” defined by the inter-
section of a given front and back strip in the 500pm detector. In order to properly
assign particles to their respective pixel, a multiple-hit decoding routine is needed to
determine which pixel the charged-particle hit. This routine will also give the posi-
tions and energies for each particle when more than one particle hit the same detector
at the same time.

After the energy calibration is done and the PID functions are generated, the full
information for each charged particles detected by LASSA telescopes can be obtained
and written to a “physics tape”. This information includes: the Particle ID, the total
kinetic energy and the emission angle.

In order to obtain the final yields and energy spectra, background from noise and
random coincidence should be subtracted. The LASSA array also has a limited solid
angle coverage. Thus, the detection efficiency should be calculated and corrections
for the efficiency should be made. The final yields are then obtained by applying
various cuts to these efficiency corrected data.

Fig. 3.1 gives an overview of the scheme of data reduction of LASSA. It also shows
the sequence following which the analysis is done. We will discuss the data analysis

details according to this sequence.

3.2.2 Energy Calibration of Si

For the range of charged particles we analyze here, the signal generated depends

linearly on the energy deposited by the charged particle, independent of the type of

54

 

Raw Data on Tapes

a?

\

 

 

 

\{iort Raw Data onjafisr/

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

[ DE Raw Output EF Raw Outpud [ EF Raw Output CsI Raw Output
1 . 1 1 r
Subtract Pedestal Subtract Pedestal Subtract Pedestal Subtract Pedestal
V Y '
DE Calibration® EF Calibration @ EF Calibration ®
1 f 1
fi T fi N F a
Dead Layer Dead Layer Dead Layer
[ Correction Correction J k Correction
6 1
Match EF & EB, Get rid of 1% difference
1 l
Multiple-hit Decoding, get hitting
position, final EF/EB Energy
i i
DE Non-unifonn EF & EB Non-unifonn C I G . M t h
thickness Correction thickness Correction 5 am a c
Silicon PID(PID1) CsI PID (PID2)
CsI Calibration®

 

 

 

@3, EF & EB Dead Layer CorrectiB
Physics Data(Energy, angle PID) o©

 

 

 

 

 

® Ortec Pulser +
Monte Carlo Simulation (Background Subtraction J charge terminator
Generate Efficiency ® Fragmentation Beam
F
unction Efficiency :Correction) And Proton Recoil

 

 

 

Final Yield, Energy, Angle )

 

Figure 3.1: Major detectors used in the experiment, viewing in the downstream
direction of the beam

55

the charged particle detected. Indeed, we expect an electron-hole pair to be collected
from the detector for each 3.1 eV of energy deposited. Based on these facts, we can
simply calibrate the energies of particles detected in the silicon. Specifically, we cross-
calibrate a pulser system by a source detected in the silicon detector at low energy.
Then we extrapolate the calibration to the whole energy range of Silicon with a linear
pulser system. This pulser system consists of two parts: an Ortec precision pulser
generator and a precision capacitor that is placed between the pulser and the cable
that connects the detector to is preamplifier. The Ortec precision pulser generates
precise pulser signals, with amplitude that is linearly dependent on its dial setting.
As the input to the preamp is a “virtual ground”, the capacitor value sets the scale
of the injected charge.

In our experiment, we used the following:

1) A PC based data acquisition system was used to calibrate pulser system relative
to known energies of 228Th and 241Am (1 sources. Fig. 3.2 is a schematic drawing of

this PC-DAQ set. The energy of the 0 particles from these 0 sources are relatively

 

 

a Source

228 Th 01' 24lAm

PC DAQ

 

 

 

 

 

 

I 7

J;
0 l" PA Shaper

Si detector

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Vacuum chamber

Figure 3.2: The PC-DAQ system used for the silicon calibration

low, comparing with the energy range of particles detected in the experiment. To

56

calibrate the pulser dial setting, we set the amplifier gain to put 8.784 MeV into
Channel 2220 of a 4k channel ADC. Then, we measured the pulse height spectrum
for the pulser for a variety of dial settings on the pulser.

2) This relation between the ADC spectrum and the pulser dial setting can be
expressed as:

ADC readout = a X (Dial + b1) + b0 (3.1)

where, a is a pulser attenuation factor, be is the channel number corresponding to
zero pulser height from the pulser, bl is a constant describes how the zero may change
with pulser setting.

3) Using 1) and 2) and several attenuation and dial settings we obtained the
relationship: Energy vs. Dial. Since the gain settings for DE, EF and EB are different,
calibration using different attenuation factors were needed when the dynamic energy
ranges for various detectors differ. Three different dynamic ranges for the pulser
calibration, 140MeV, 200MeV and 500MeV, were used to calibrate the DE, EB and
EF signals. Fig. 3.3 shows the relationship between the Dial and the Energy for the
DE, EF and EB pulser calibrations.

4) At the end of the experiment, we coupled the pulser capacitor to the cable
between the Silicon and Preamplifier by using a T—connector. Everything includ-
ing detectors and DAQ were maintained under the same conditions as during the
experiment.

5) Then we changed the dial of the pulser, to get a relationship: Energy vs. the
channel in the ADC, used in experiment, until the full response over the full energy
range of the detector is obtained. This full energy range depends on the gain settings
of preamplifier and shaper. Normally, the linearity is good for energies in the middle
of the energy range, but gets worse at low energies, when the ADC output is close to

zero channel or at high energies where the amplifier become non-linear. To address

57

this effect, more calibration points are done for ADC output below 100 and above
2500 channel in the 4095 channel Phillips ADC. Then we fitted the calibration with
a linear function for ADC output channels 0~100, a second linear function for ADC
output channels of 100~2500, and a polynomial of order 4 for ADC output channels
of 2500~4095. Between those fitting functions, a spline interpolation is used to make
a smooth connection. Fig. 3.4 shows the calibration line for a typical Silicon strip.
Step 4) and 5) of this procedure was repeated for all 432 strips of AE, EF and
EB one by one. After the calibration is done, the calibration is checked by plotting
the energy of EF vs. EB. If the energy calibration is correct, with no noise, the
energy from EF and EB should be exactly the same for each particle. However, we
had about a 1% difference in the energy between EF and EB after our calibration.
The reason for this is not clear. Possible reasons are: a difference in the efficiency
of charge collection between EF and EB or an systematic error in the attenuation

setting of the Ortec pulser. To correct for the discrepancy, we forced the energy from

Different Dynamic Range Settings for DE, EF and EB

 

 

 

  
    

 

 

 

 

 

 

16 . ._. . ._ _. _m_ ##### ,"‘ jfifih ,- -, _.. ....._. .,
14 :SOOMeV Dynamic Range l
y] = 49.70216x + 0.0285

12
e10 s
g . H W 1
E 8 il40MeV Dynamic Range ]
, I
g 6 y3= l.93486x + 0.0l483 ]
4 - - ~- —~~ »—- ~
[200 MeV Dynamic Range 7

2 y2= 20.7257lx + 0.01723
0

 

O

0.2 0.4 0.6 0.8
Pulser Dial (Volt)

Figure 3.3: Calibration lines of energy(MeV) vs. Dial value(Volt)

58

Energy Calibration for One Silicon Strip

_a
N

 

 

_ 4 3 2

10 y-a3x +b3x +C3X +d3x+e3
Ea
:: y=a2x+b2
36
G
'6
+34 Y=31X+b1
a

2

0

0 1000 2000 3000 4000

ADC Readout

Figure 3.4: ADC output vs. precise pulser dial value(Volt)

the EF channels to be equal to the energy from EB channels, which we took to be

more accurately matched to the a calibration points.

3.2.3 Multiple-hit pattern Decoding

Before one can get the PID function, some preparations need to be done. Most
important preparations are the Pixelation (Multiple-hit Decoding) and Non-uniform
Thickness Correction. In LASSA, strips of EF and strips of EB are perpendicular to
each other. The scattering angle of particle is determined from the X, Y coordinates
where the particle hits the telescope. Here, X and Y are basically determined by
which EF and EB strip is fired. (We use X to denote position information obtained
from EF and Y to denote position information obtained from EB.)

When only one particle hit the telescope, one can easily tell X and Y values from
the energies detected in the strips. However, when two or more particles hit the
telescope at the same time, several EF and EB strips will have data. For example,

when two particles hit one telescope at the same time, there will usually be four

59

signals X1, X2, Y1, Y2. Sometimes, there would be fewer if one of the strips in the
detector is not working properly or the two particles happen to hit the same strip.
If four signals are observed, one need to know whether the two particles hit (X 1, Y1)
and (X2, Y2), or in coordinates (X1, Y2) and (X2, Y1) instead. One needs also to
know the amplitude of the signals (corresponding to energy loss) induced by these two
particles. Most of the times, the position and energy information can be retrieved by
requiring the signals from EF and EB to be equal. But sometimes, this comparison
does not yield a unique answer. For example, if the signals (energy loss) induced by
the two particles happen to be the same then both parings (X1, Y1) and (X2, Y2),
or (X1, Y2) and (X2, Y1) are equally favored. However, when the signals from AE
and CsI are included in the paring, it is often possible to determine which paring is
the correct one. Thus in practice, various hitting patterns must be tested to find the
most probable one.

Occasionally, more than two particles hit the telescope at one time. We have not
found a reliable algorithm to analyze such events. Instead we reject them and count
them as lost.

When there are bad strips in a telesc0pe and particles hit these strips, the re-
construction fails and the information from that particle can not be retrieved. When
either position or energy information for a particle are irretrievable in a strip detector,
this particle is also lost. Other losses occur: when two or more particle go into the
same CsI crystal, then, the energy information is lost because the signals induced
by multiple particles superimpose on each other in the CsI and cannot be separated.
Also, particles are lost when they hit at the gaps between the C31, and no valid signals
for CsI can be constructed.

Another output of the Multiple-hit decoding program is a record of the fraction

of particles which are lost. This information is then used to constrain the efficiency

60

correction. More details about efficiency correction can be found in Sec. 3.2.7.

3.2.4 Construction of PID function and Non-uniform Thick-

ness Correction

When constructing the PID function, it is desirable to make it applicable to all two
dimensional (2D) AE vs. E histograms. However, when one does this, the isotopic
resolution of the 2D spectra becomes rather poor. This occurs, because the measured
values for AE and E depends on the position where the particle hits the telescope.
Part of this position dependence comes from the thickness variation of the silicon
detector across its surface. For AE, the thickness variation is of the order of 10
pm, causing a position dependent variation in the energy loss of the particles and
smearing the relationship between AE and E values. Only if the thickness variation
is of the order of 1%, can one calculate the PID without addressing this position
dependence. Second, when the 2D PID histogram is constructed by using silicon as
AE vs. CsI, it is easiest to use the uncalibrated output of the CsI detector. However,
the uncalibrated output from the CsI depends on the gain setting of the electronics,
so this signal will be crystal related.

To obtain a PID function that is applicable to all telescopes without losing reso-
lution, we should correct for these effects. For the silicon detector, we assumed that
the energy loss a particle experiences in passing through the detector is proportional
to the detector thickness, which is good for relatively small energy losses. Then we
determined the relative thickness of different AE by using particles of fixed energy
and measuring the energy losses in the silicon. This can be done if you have a beam of
defined energy or if you measure particles that stop in the second silicon. In the latter
case, you can gate on the total energy deposited in the silicon. Such gates applied to

01 particles were quite important for establishing the variation of AE thickness with

61

position.

In general, the thickness correction is done for each “pixel”, defined by the inter-
section of a given front and back strip in the 500 pm detector. In this fashion, we
obtained thickness corrections for both the 65pm silicon detector and the 500nm sili-
con detectors. (Note, the non-uniformity is less crucial for the 500pm silicon detector,
which is of the order of 1%). After the non-uniform thickness correction, we define
an effective “energy” so that particles with a given energy have the same “effective”
energy loss in the silicon detector regardless where it hits. Then we applied the PID
function to all telescopes.

The effect of making this non-uniform thickness correction is shown in Fig. 3.5.
On the Left panel of Fig. 3.5, the PID of Lithium and Carbon isotopes are obtained
without making the thickness correction. One can see that the resolution is bad. On
the right hand panel, the thickness correction is performed. A much better resolution
is achieved. A comparison of the left panel and right panel of Fig. 3.5 illustrates the
crucial importance of correcting for the thickness variation in the 65 pm detector in
achieving optimal particle identification. Details of the construction of the 1D PID
function will be discussed later in this section.

For the CsI detector, the gain match is done by matching 36 CsI crystals of
LASSA to one selected crystal, using Helium and Carbon calibration points from a
beam calibration run.

After the silicon detector calibrations, the non-uniform thickness correction for
the silicon detector and the gain matching of the CSI crystals. 2D AE vs. E PID
histograms were constructed. These PID histograms are now independent of the po-
sition where the particles hit the detectors. In practice, different measured quantities
were used for AE and E in order to construct the more accurate PID functions. For

example, the DE detector is used as AE and the value of EF is used as E for particles

62

 

500 — 7Li 7Li

Before Correction Li After Correction

 

0~6789 6789

12c 130 Before Correction 12c 13c Amer Correction

100

 

 

 

111213141516A 111213141516

Figure 3.5: Comparison of isotope resolution before and after thickness correction

that stopped in the second silicon. The low EF gain allowed higher Z fragments to
be properly identified. For p, d, t, 3He, 4He and 6He that stopped in the CsI crys-
tals, EB is used as AE and the matched raw output from CsI is used as E. The EB
signals, which used higher gain shaping amplifiers, improved the identification of the
isotopes of Hydrogen and Helium at the expense of dynamic range. The identification
of higher Z fragments required the use of the lower gain EF signal. For isotopes with
Z > 3 that stopped in the CsI crystals, DE+EF is used as AE and the matched
raw output from CsI is used as E. A typical DE—EF two-dimensional spectrum for a
telescope is shown in Fig. 3.6. In this figure, the measured energies in DE and EF
detectors are matched first, which we call them nominal energies, and then are used
to construct the 2D histogram. PID gates are drawn from this histogram. Compared

with the raw energy spectra shown in Fig. 2.6 in the previous chapter, CsI signals

63

are used to veto the punch through lines. A typical DE+EF-E(CsI) two-dimensional
spectrum for a telescope is shown in Fig. 3.7. The isotopes of oxygen can be clearly
distinguished in this figure. PID gates are drawn from this histogram too.

The 2D PID histogram is not convenient. To make various gates for different
isotopes as well as to estimate the background level, it is desirable to work on a 1D
histogram. One cannot simply project the 2D histogram to either X or Y since the
PID lines are curved in the 2D histogram. However, one can straighten the 2D PID
histogram by drawing curved line along each isotope and projecting the 2D histogram
according to these lines. Because the real isotope yields are not symmetric about these
lines, this projection can cause some problems in the normalization, making it difficult
to fit the curve of isotopic yield.

Fig.3.8 show the 1D PID histogram for particles stopped in the second silicon
detector. Fig. 3.9 show the 1D PID histogram for particles stopped in the CsI

crystals.

64

 

120

DE (Matched Energy in MeV)
ON
c:

 

 

.Hiliiiiliililiiiiliiiiliii
0 50 100 150 200 250 300
EF (Matched Energy in MeV)

1l1111
350 400

Figure 3.6: A histogram of counts vs. nominal DE energy and nominal EF energy

65

 

DE+EF (Matched Energy in MeV)

 

 

‘ ~-——'_
_. __ a, . _._____-::a-—---

00 500 1000 1500 2000 2500 3000 3500

CSI (Matched ADC Output)

Figure 3.7: A histogram of nominal counts vs. nominal DE+EF energy and matched
CsI light output. The isotopes of selected elements are indicated by labels that are
immediately above the corresponding data.

66

 

Counts(a.u.)

 

 

 

 

 

 

 

lljllll

llllllllllJllllllllllllllJlllLl

0 500 1000 1500 2000 2500 3000 3500

 

PID (a.u.)

Figure 3.8: 1D PID histogram for particles stops in the second silicon detector

67

Counts(a.u.)

 

 

j—d
O
O\

 

 

 

 

 

 

 

 

     

 

 

1 l r l

O 500 100015

I I 1 l

00 2000 2500

1 l 1 1
3000 3500 4000
PID (a.u.)

l

 

 

Figure 3.9: 1D PID histogram for particles stops in the CsI crystals

68

3.2.5 CsI calibration

The energy calibration of the CsI crystal is more difficult than that of Si detectors,
because the light output of CsI(Tl) crystals depend on the charge, mass and energy
of the detected particle. That is why one must construct the PID function before one
performs the energy calibration of a CsI crystal. The light output for a particle with a
certain charge and mass is a non-linear function of the energy loss for heavy ions and
low energies. To address this, a non-linear energy-to-light relation was established.
That depended both on the charge Z and mass A of the particle.

Many different empirical functions have been employed to construct energy cal-
ibrations of CsI(Tl) [47, 22, 72]. These parameterization typical are approximately
linear at high energies but become very non-linear for low energies. Some of these
parameterizations have both a charge and a mass dependence while others are simply
charge dependent. The best choice of calibration function depends on the T1 doping
of the crystals.

For the crystals used in LASSA in this experiment, four calibration functions
were used depending on the particle type [118]. These functions are similar to that
proposed by Y. Larochelle et al. [56]. Larochelle’s calibration function has the at-
tractive feature that the deposited energy E is expressed as an analytical function
of the light output, which is good for calibration purposes. It has an explicit mass
dependence, necessary for the calibration of heavy ions. We discuss the other details
in the following.

Following Larochelle, we parameterized the incident particle energy E for heavy
ions of Z>3 as a function of the light output L, the charge Z, and the mass A of the

particle, as follows:
E(L, A, Z) = aAZzL + b(1 + cAz2)L1-d~/7*Z . (3.2)
This expression has a linear part, that dominates at high energies and an exponential

69

part that dominates at low energies. In Eq. 3.2.5 (1, b, c, d are the fitting parameters
which can be determined by fitting experimental points. The parameters a, b, c depend
on the electronic gain and scintillation efficiency. The parameter a (affecting the linear
part) is important for small Z and high energies. The parameters b and c (weighting
the exponential part) describe the non-linearity at low energies and higher Z ’8. All
parameters in Eq. 3.2.5 are positive. The parameter d is responsible for the transition
from non-linearity to linearity in the energy response. Larochelle identifies L with the
slow component (7 [us time constant) of the light output. In our parameterization, L
is the total light output instead.

In Fig. 3.10 the fitting curves as well as the calibration points are shown for
Carbon. In Fig. 3.10, the solid and dot-dashed lines represent the best fit of Eq. 3.2.5
to the experimental energy calibration 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 11C isotopes does not pass through the high-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 calibration procedures adopted
elsewhere in the literature. This mass-dependent calibration curve appears to be a
more reasonable solution. Since several fragmentation beams would be required to
have the full calibration curve for each isotope, we adopt the mass-dependent ansatz
(closely related to the quenching effect) of Larochelle [56].

For light-charged particles with Z33, the parameterization described in Eq. 3.2.5
did not accurately describe the detected energies. Compared to the observation of
Larochelle et al. [56], a less pronounced isotopic effect was observed for light ions.

This may be the result of the increased concentration of the activator element, T1,

70

 

600

400

ECSI(MeV)

ZOO

 

 

 

l1.

O 1000 2000
Light(a.u.)

l l l 1

 

Figure 3.10: Calibration curves of Carbon for LASSA CsI(Tl)

in the CsI-crystals used in the present study compared to those studied in Ref. [56].
We find the mass dependence to be overestimated by the AZ 2 factor in Eq. 3.2.5,
and employ a modified function of Eq. 3.2.5 with a weaker dependence on A to fit
Z 53 particles. This will give us the following three modified calibration functions
for light particles:

For lithium (Z = 3) particles, we changed the first term of Eq. and used the

follows:

E(L, A, Z) = ax/AZzL + b(1 + cAZz)L1-d\/7‘Z . (3.3)

71

For helium (Z = 2) isotopes, the following calibration equation was used:

EXL,A,Z)==aL-+bA%1——d”). (34)

The variables a, b, c, and d in Eq. 3.2.5, 3.3 and 3.4 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 Eq. 3.2.5 - 3.4.

As we have only two calibration points for each isotope, p, d and t, we adopt the

simple linear function for Z = 1 particles.

E=aL+b, (am

where a, and b are fitting parameters. A linear CsI(Tl) response is consistent with
that observed for hydrogen isotopes by Handzy [45].

Table 3.1 show the fitting parameters for all the isotopes.

 

 

 

 

 

Z A a b c d

1 1 0.2010 -0.9587

1 2 0.1916 -l.214

1 3 0.1784 0.8212

2 0.1696 4.575 0.3380 -0.05772
3 0.01783 0.2456 0.09743 0.06358
4 0.0006680 0.4493 0.01015 0.02616
5 0.0006354 2.944 0.002940 0.03713
6 0.0004102 2.362 0.003566 0.02869
7 0.0003101 4.834 0.002213 0.02734
8 0.0002289 4.701 0.002042 0.02301

 

 

 

 

 

 

 

 

Table 3.1: Table of the fitting parameter used in the energy calibration functions of
LASSA CsI(Tl) crystals

The calibration functions used above have been used to calibrate 36 CsI (Tl) crys-
tals of LASSA. To provide the calibration points, fragmentation products ranging
from hydrogen to oxygen isotopes produced in fragmentation beams and direct a
and proton beam particles were used. The detectors were directly exposed to low-

intensity (1000 particles / s) beams of different isotopes and energies. These ions were

72

obtained by fragmenting 2160 MeV 36Ar and 960 MeV 160 primary beams from the
NSCL K1200 cyclotron in the A1200 fragment separator [90]. The main advantage of
this method is the availability of a large number of particles that could be detected
simultaneously (up to 52 isotopes were identified in the case of the 36Ar fragmen-
tation). Since particles are selected only by their magnetic rigidity (Bp=l.84l 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 FWHM 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 CsI crystals in the present work are listed in
Table 3.2. Hydrogen and helium isotopes were also calibrated by elastic scattering
of E /A=30 MeV p-4He molecular beams on a Au target and by 240 MeV direct 4He
beam particles.

Our fitting procedure resulted in a precision of the energy calibration better than

2% for isotopes from 4He to 200.

3.2.6 Background Subtraction

The spectrum shown in Fig. 3.6 is for all particles that do not have a CsI (Tl) signal;
i. e. particles that either stop in the 500nm Si or “punch-through” 500nm Si but
somehow escape the detection by the CsI(Tl) crystals behind the silicon detector. The
low intensity background haze visible in Fig. 3.6 between the element groups is due to
particles passing through inactive regions (edges, inter-strips, etc.) of the telescope,
resulting in inconsistent DE—EF combinations. Also visible in Fig. 3.6 is a faint band
of particles with a positive slope at the high energy event of the isotope lines between
He and C. Particles in this band “punch-through” the 500pm Si detector but are not
rejected because no signal is observed in the C31. These particles are present due to

a slight misalignment of the Si detectors relative to the CsI(Tl) crystals, or due to

73

 

 

16O fragmentation products E(MeV) 36Ar fragmentation products E(MeV)
p 77.17
(1 39.78 d 79.57
1: 26.72 t 53.75
3H6 105.00 3H6 210.00
4H6 79.99 4H6 160.00
6H6 53.64 6H8 107.90
6L1 119.90 6L1 240.00
7L1 103.10 7L1 206.80
8L1 90.40 8L1 181.60
7Be 182.20 7Be 363.40
9311 142.50 9311 285.60
loBe 128.40 loBe 257.90
lOB 199.90 1013 400.00
11B 182.10 11B 364.90
12B 335.40
11C 261.20 “C 521.60
12C 239.90 12C 480.00
13C 221.80 13C 444.40
”C 413.70
14N 279.90 14N 560.00
15N 524.00
l6N 492.40
150 340.80 150 680.70
160 640.00
170 603.70
180 571.30

 

 

 

 

 

 

Table 3.2: Table of fragmentation products used in the energy calibration of LASSA
CsI(Tl) crystals

the dead area between adjacent CsI(Tl) crystals (due to wrapping for the purpose
of optical isolation). One is able to eliminate this contribution, at the expense of
geometric efficiency, by gating selectively on the position in the Si detectors.

The background haze visible in Fig. 3.7 has the following origins. First, particles
passing through the inactive regions of the telescope give rise to spurious EF-E(CSI)
values. Also, coincidence summing of two or more particles detected within a sigle
CsI(Tl) crystal is particularly evident in the low EF region. One source of these double

hits may be that high energy light particles in combination with a second particle

74

in a single CsI(Tl) crystal distort the E(CsI) signal. A second source of coincidence
summing occurs when neutrons hit the CsI at the same time as a charged particle.
This produces signals which superimpose on the signals of the charged particles and
shift the PID lines. Higher order coincidence summing as well as noise from either
the detector or the electronics can shift the PID lines as well.

It is very hard to make correction for the signals causing the background, i.e. to
separate the noise from the real signals. A more practical way is to make an esti-
mation of the background level and correct for the background accordingly. There
are two types of corrections. One is the subtraction of background under the PID
curve of each isotOpe and the other is the correct for losses due to coincidence sum-
ming that remove particles from a given PID line and move them elsewhere. We will
discuss the background subtraction first. We studied the background and observed
that the background level depend on three factors. First, the coincidence summing
probabilities are detector related: charged particles have different coincident sum-
ming probabilities when they stop in the Silicon detector than when they stop in the
CsI crystals. Second, the background level is energy related: charged particles with
different energies have different background levels. Finally, the background level is
also isotope related: the background level is somewhat different for different isotopes.
For most of the isotopes, we estimated the background as follows. For central colli-
sions (the event selection will be discussed in the next chapter), we generated 1D PID
histograms for different detectors and in ten different energy bins, then background
levels were estimated for each elements. For peripheral collisions, since the statistics
is not high, we generated 1D PID histograms for different detectors, then background
levels were estimated for each elements. We treated 7Be, 3He, 11C, 15O separately,
because their background levels are not the same as other isotopes of the same charge

number. For example, 7Be has no neighboring isotopes, so the background level of

75

7Be was estimated separately. For 3He, 11C, 150, there were significant leakages into
these PID lines because their yields are weak compared with their neighboring 4He,
12C and 16O isotopes. Fig. 3.11 show the contamination of 3He by 4He. To obtain
the yields of 3He, 11C, 150, we only used the halves of the PID lines which are lo—
cated at the far sides of their neighboring isotopes, which are believed to be free of
contamination. Fig. 3.11 also shows that the PID line of 3He is asymmetric. The
reasons for that are: a) The PID algorithm does not requires isotope yields to be
strictly symmetric about the middles of PID lines. b) There is a contamination of
3He by 4He. Fig. 3.12 shows that the background levels for IMFs as estimated by

those fitted straight lines in the figure.

3.2.7 Efficiency Correction

Ideally, if the detectors have perfect 47r coverage and can detect charged particles with
any energies, there is no need for an efficiency correction. However, LASSA only has
a limited geometric coverage and can only be placed at certain angles. For example,
we can not detect those particles emitted in the direction very close to the beam line,
since we can not put the detectors too close to the beam line without damaging the
detector. We also cannot detect particles with energies that are below the detector
thresholds or above the dynamic range of the electronics. This type of efficiency was
easily noted but not easily corrected because we cannot definitely correct for data
loss at angles where the efficiency is zero. Even if the particles are emitted in the
directions covered by the LASSA array, they cannot be fully detected. The LASSA
array can only detect and identify charged particles that stop in either the second
silicon detector or the CsI crystal. The particles can not be detected and identified
if they stop in other places: such as the Sn-Pb foils that covered the telescopes, the

first silicon detector, the dead layers of the silicon detectors, the mylar foil that covers

76

2000

 

  
    
    
  

1800

1600

Counts (a.u.)

1400

1200

I I I I I I I I I I I I I I I I I I I

1000

I

800

I I T I I I I

600

I

400

I I I I l I I I

200

 

 

lllllllllllllllllllllllllllllllLLl114LilJlll

0 275 300 325 350 375 400 425 450
PID (a.u.)

Figure 3.11: PID line of 3He shows the contamination of 4He. It also shows that the
PID line of 3He is asymmetric.

 

O
N
UIfiii

77

é

 

 

 

 

Counts(a.u.)
S
O
Q

fl
‘
LIE——

 

csééggél

 

 

 

mammmimm

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

 

[C

     

 

 

l 1 11111 I 1 l l

1000 1500 2000 2500 3000 3500 4000 4500

 

PID (a.u.)

Figure 3.12: Background subtraction for the heavy fragments, the straight dashed
lines are the estimated background levels.

78

the crystals and the gap between Csi crystals. Multiple hits also cause losses in the
charged particle detection. At polar angles where the detection-efficiency is not zero,
the efficiency can be corrected for such effects.

In our analysis, efficiency functions were generated for each isotope and for differ-
ent reactions. We construct the efficiency functions for central collisions and periph-
eral collisions in different ways for different purposes of study. For central collisions,
we calculated 1D efficiency as a function of EC", within each degree of 6m. For pe-
ripheral collisions, we calculated 2D efficiency as a function of p 1 /A and 3],. The
eficiency functions are calculated as follows, isotopes were first generated from the
even phase space in the center of mass frame, and then the kinetic energy and an-
gle in the lab frame are calculated. If the isotope survives the double hits loss and
passes the detection threshold and was successfully detected by a good detector, then
a weight of l/N will be added to the corresponding point in either the EC". space or
the y, vs. p 1 /A space for this isotope. Where, N is the number of total particles
generated. The raw yields are then corrected by these efficiency functions.

On the top panel of Fig. 3.13, we show the raw yield of 4He produced in the
peripheral collisions of the 112Sn + 112Sn reaction as a function of vs. parallel relative
rapidity, called y, vs. the momentum per nucleon perpendicular to the beam direction,
called p 1 /A. The rapidity is defined by %ln%[—:' The parallel relative rapidity is
calculated by normalizing the rapidity parallel to the beam direction of 4He particles
by the beam rapidity. In the figure of the raw yield of 4He, we see that there is an
annular ring facing the origin with low counts. This discontinuity of the 4He was
caused by either 4He particles stop in the dead layer of the second silicon or in the
Mylar foils of C31. This discontinuity was also caused by 4He particles that stop in the
CsI, but cause a very low light output due to the non-linear response of the C31, which

is so low that we can not detect it. This loss in the efficiencies affects the yields for

79

central collisions at the order of 2%. We also see edges of enhanced yields for certain
polar angles 0. On the bottom panel of Fig. 3.13, we show the efficiency function,
with double hit effect taken into account, of 4He for the 112811 + 112Sn reaction. We
see the same enhanced angles of emission because we take the solid angle coverage
into account in the efficiency simulation. On the other hand, we make no attempt to
simulate the loss of counts due to the discontinuity where 4He punches into the CsI
because this is difficult to simulate accurately enough. In the first panel of Fig. 3.14,
we divided the measured yield by the efficiency function to obtain the corrected yields
vs. y, and p 1 /A for 4He produced in the peripheral collisions of the 112Sn + 112Sn
reaction. We see that the angle dependent of the efficiency is well corrected. In the
other 3 panels of Fig. 3.14, we show the corrected yields for all the other reactions.
We see from this figure that after correction, the yield is smoother comparing with
the raw yield. The “Coulomb ring”, which centered at 3;, ~ 0.95 is more prominent
in the corrected yield comparing with the raw yield. In Fig. 3.15, we reflected the
corrected yield of 4He with respect to the line of y, = 0.5 and the line of p 1 = 0, since
the yields should be symmetric with respect to these lines for symmetric reactions.
The corrected y, vs. p1 /A plot provides our best global picture for 4H8 emission in

this reaction.

3.3 Data Analysis of the Ring Counter

The Ring Counter has similar structure to the LASSA array. The data analysis for
the Ring Counter is basically the same as the LASSA array. One needs to do a
similar energy calibration and generate a Si-CsI PID function as what are done for
the LASSA array. We will discuss about the energy calibration and PID function
of the Ring Counter in detail in the following sections. Since the Silicon detectors

and the CsI crystals are not tightly mounted in a box as was LASSA, there could be

80

 

300_

   
  

 

ERaineld 4
250?
A :0.8<fi<1 He
0 L
S 200:
0 :
@1507
$ :
£21100:
50;—
300m
250 éEfficiency Function
A :0.8<6<1
Q L
3 200:
0 :
@150.—
s 5
£100:
50;—
0—111111.1...1...1...11111...1.

 

 

 

0 0.2 0.4 0.6 0.8 1 1.2 1.4
yr

Figure 3.13: The raw yield and the efficiency fimction for 4He produced in the pe-
ripheral collision for the 112Sn + 112Sn reaction

81

 

300 _

ECorrected Yield 4 ECorrected Yield 4
250 50.8 < b < 1 He 50.8 < 0 < 1 He
;1 12Sn+l 12Sn " ”2Sn+12.4Sn

I

N
O
O

p i/A(MeV/c)
5 5

       

IIIIIIIII'IIIIIIIIII

 

 

   

 

 

 

50 -
O _ J l 1 1 l l l l I l I l l l l l l l l 1 I l l l 1 l l l
ECorrected Yield 4 ECorrected Yield 4
A 250 50.8<6<1 He :_0.8<fi<1 He
g 200 Er124Sn +1:12Sn 21248 n +1248n
O : :
2 150 r 7
z I I
100 I- '-
21 . e
50 I— I-
0 : l I I 41—]; J L 1 l l l l l l : l l l J l 1 l l 1 l 1 l l l
0 0.5 l 0 0.5 1
yr yr

Figure 3.14: The corrected yields of 4He produced in the peripheral collisions for
different reactions.

 

l

200

: A 4
_08<b<1, , He

  

100 L

p i/A(MeV/c)
O

—100 ;

 

 

 

] l l

7.11111111411111111111111111. 1.11.111
-0.4-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4

 

Y1

Figure 3.15: Reflected yield of 4He produced in the peripheral collisions for the
112Sn + 112Sn reaction

83

a misalignment between the Silicon detectors and the CsI crystals. We found that
the misalignment effects were not the same for all of the 16 Pie sectors. The overall
effect of the misalignment caused a loss at the order of 7% of the total projectile-like
fragments detected. Fig. 3.16 gives the flow chart of the algorithm of the program

used for the Ring Counter analysis.

3.3.1 Energy Calibration

The calibration of the Ring Counter is almost the same as the LASSA array, except
that there is no proton recoil measurement, which is the last stage for LASSA cali-
bration. First, we used a different charge terminator to have a pulser calibration over
the dynamic range of 10GeV for the silicon calibration. Only the signals from the
pie side were used for energy. In the beam calibration stage, both silicon and CsI(Tl)
detectors in the Ring Counter were also calibrated with direct beams of 80MeV/ A
17O and 60MeV/ A 36Ar, and cocktail beams, consisting of a mixture of 60 MeV/ A
48Ti, 52Cr, 56Fe, 60N i, 64Zn, 68Zn, 12C beams. The response of the annular silicon-strip
DE detectors is a fairly linear function of the energy deposited and independent of
the species of ions. The Si energy calibration shown in Fig. 3.17, was achieved with
an accuracy of 1%. The CsI(Tl) detectors, however, display a non-linear response to
the energy deposition, which is mass and charge dependent [118]. For the CsI(Tl)
calibration, there was insufficient information to establish the calibration of the CsI
for each species. The energy calibration of CsI(Tl) is done by matching newest beam
calibration to the calibration curve obtained for the Ring Counter in a prior experi-
ment. We found that the prior calibration [34] appears to be very consistent and we
believe that it is very good. The calibration curve, shown in Fig. 3.18 which gives
the energy calibration of CsI(Tl) to an estimated accuracy of 3%. The total energy

calibration of AE + E is at an accuracy of about 3%.

84

 

C Begin D
I

[ Do Pie =1,16 ]
l

 

 

 

Multiple hit decoding
Test Strip

 

 

 

 

Strip Good?

 

 

 

Test CsI

 

 

CsI Good?

 

 

 

Calibrate Pie, Match CsI, Find Z

I
Test time signal for 20 ? Z

Time Signal Good?

Calibrate CsI
L

Generate PID, energy, angle

|

 

 

 

 

 

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Continue

1
Enddo Pie=l , 16

C Elnd D

Figure 3.16: Flow chart of the algorithm of the Ring Counter analysis

 

 

 

 

 

 

85

 

 

 

 

 

 

 

 

Rlng Counter P1e Callbratlon
9 800 _ XZ/ndf 0.3982 / 6
é) : Pl 2.075
~ P2 0.9885
3: 700 f
a :
O _
[f] 600 f
U h o
8 :
Cd _
:3 500 :
Tv‘ :
U 400 j
300 E—
200 :—
100 :_ 12C 36Ar 413—“ 52C1’ seFe 6ON1 6421’) 68Zn
L
0’ 1.111.111.1.11.1111...111.111.11.11...
0 100 200 300 400 500 600 700 800
Measured Energy(MeV)

 

 

 

Figure 3.17: The energy calibration of the silicon detector(Pie side) of the Ring
Counter

86

N

 

Ring Counter CsI Calibration

_.

h—I

11—11

1200
1000

Channel (Light Output)
4:. 3 oo
8 s s 5
I

ITIITIIIIT r1111

III

600 E
400 :—
200 f_ 36Ar 48T1 52Cr 56Fe 6ON1 64Zn 682m

 

 

E11111111111141111111111141111111411111
0 500 1000 1500 2000 2500 3000 3500 4000
E(Mev)

 

0

Figure 3.18: The energy calibration of the CsI detector of the Ring Counter

87

3.3.2 PID function

Iii-om the raw output of AE and E detectors, one can construct two-dimensional
matched AE and E histograms to identify the detected charged particles. Since the
Silicon detector of the Ring Counter is very thick (298 pm), the thickness correction
is not necessary. We simply used the calibrated emery deposited by the charged
particles in the Silicon detectors as AE, and the matched light output from the CsI
crystals as E. Fig. 3.19 shows the “stretched” PID lines of the Ring Counter. In this
figure, elements from Z = 2 to Z = 50 are clearly distinguished. Fig. 3.20 shows the
1D PID function of the Ring Counter.

 

 

 

 

 

”32500
3 i
942000:
1500
1000
500

O 4111111111111 1.144 1_1

0 200 400 600 800 1000

E(MeV)

Figure 3.19: The 2D PID lines of the Ring Counter

88

A, 6000

Counts(a. u

5000

 

I Ifj’ I I I I

~ F
4000 T 16

3000 :—

2000

1000 i

Am
Si ;

IE

I

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1111]

 

 

 

1111 111111 1 11 11

 

PHe
z 111111

1111

Sn

 

 

 

1

 

1111
fKX) 1000 1500 2000

Charge Identification(a. u.)

Figure 3.20: 1D PID function of the Ring Counter

89

2500

Chapter 4

Experimental Data and Model

Comparisons

4.1 Impact Parameter Selection

Classically, the impact parameter “b” of nucleus-nucleus collision is defined by the
distance between the straight-line trajectories of the centers of the two nuclei before
their interaction. In experiments, however, one can only use the experiment observ-
ables to get some information of impact parameter. Then one uses this information to
construct impact parameter filter which selects range of the impact parameter. Exam-
ples of such variables are the total charge particle multiplicity, NC, the total proton
multiplicity, Np, the total detected charge, Ztot[84], or the charge of projectile-like

fragment, Zn”. In the present work, we use the charged particle multiplicity [84, 53]

90

to construct the impact parameter filter for all of the experimental results. In a rough
picture, the charge particle multiplicity depends on the energy transferred from the
relative nuclei motion to internal degrees of freedom in the region of the overlap be-
tween projectile and target. More charged particles will be emitted in more central
collisions with a stronger overlap than in peripheral collisions where the overlap is
less. Based on the assumption that charged particle decreases monotonously with
impact parameter multiplicity, a reduced impact parameter or “b” can be calculated
from the charged particle multiplicity. The reduced impact parameter, first suggested

by Cavata et al. [19], can be written as:

b(Ne)

bmaz

E(Nc) =

 

= 1: 1001.11” 2/ [N 11: )P(Nc)]1/21 (4.1)

where NC is the charged particle multiplicity corresponding to b, P(Nc) is the rela-
tive frequency of events being detected with the charged particle multiplicity equals
to NC, and Nc(b,,m,) is the charged particle multiplicity corresponding to bum, the
minimum bias impact parameter. In this experiment, Nc(bm) = 4. This relation-
ship is rigorous only under two assumptions: First, the nucleus-nucleus cross section
can be well approximated by the geometrical cross section: ag=7rb2. This is good
for relativistic bombarding energies [19], but may not be strictly applicable to the
intermediate-ranged bombarding energy, like what it is in this experiment. Second,
there should be a monotonic correlation between multiplicity and impact parameter
without dispersion. This could not be well justified either. Considerable fluctuations

of the charged particle multiplicity can be expected even for collisions of well-defined

91

impact parameter. Nevertheless, we use this impact parameter to select the central-
ity of the nucleus-nucleus collision, and group the events to different bins, “central”,
“mid-central” and “peripheral” collisions. We then must be aware that these gates
may be somewhat empirical due to fluctuations in the charged particle multiplicities.

In our analysis, the charged particle multiplicity, NC, consists of all the charged
particles detected in the Miniball/Miniwall array and LASSA telescopes. Those de-
tected charged particles not only include the identified particles but also unidentified
particles. For Miniball/Miniwall array, such unidentified particles includes heavy frag-
ments that stop in the fast plastic and light particles that punch through the CsI(Tl)
crystals. For LASSA telescopes, the unidentified particles includes heavy particles
that stop in the first silicon, the light particles that punch through the CsI(Tl) and
the particles that hits at inter-CsI gaps.

In Fig. 4.1, the probability distribution of the charged particle multiplicity as well
as the impact parameters are shown for all four reactions. A threshold of NC = 4 is
chosen from the setting of the hardware trigger during the experiment. In Fig. 4.1,
we can see that the probability distribution of Nc is rather flat between Nc=4 and
Nc=25, then there is an exponentially decrease above Nc=25.

The impact parameter b is derived from b and bmaz. b is calculated from the
procedure described in Eq. 4.1 and bm is calculated by measuring the cross section
0 = «bin; for events with NC 2 4. We obtained values of bmaz =7.50fm for reac-
tion 112Sn+“28n; bmaz =7.20fm for reaction 112Sn+124Sn; bmaa: =7.38fm for reaction
124Sn+112Sn; bm =7.02fm for reaction 124Sn+12‘18n. While there is a difference in
the bum of the four reactions, it is within the systematic errors in the procedure. We
take bmaz =7.5fm for all four reactions. In the following discussions, central colli-
sion corresponds to the gate of 0 < b < 1.65 fm (0 < b < 0.2); peripheral collision

corresponds to the gate of 6fm < b < 7.5fm (0.8 < b < 1).

92

 

 

30‘ 40

vrv‘rv VV

 

0' 10 20 30"‘0' 1020

rv txrvv I TfiV f.

E ....oooooooooooo...... 1 .p..ooooooooooooo.... 1
. ‘ ‘ 0 ‘ ‘ .
D 0.01 I '5‘ a: ‘10, 10.01
’ 0‘. :5 °\. .
' . I: o l

112 112 . 124
Sn+ Sn -\ Sn+ Sn 4

5
% lE-3 \'\. 11- 112 \'\ 1113-3
1E-45 ‘11 \ 10134

v vvrvvv'l

 

 

\. \ i
b.
L 1 1 1 . 1 1 . 41 A A A L A 1 1 .
1- 4»
> 0000 4» O
o 000
0.0 0 0.0
O 4r 00
6 :- O‘O\O T'- 00 d 6
A ‘O 1’ 0‘0, l
v.0 P 0‘ 4
. O
0 1' 4
E 4 f Ob 4? \O‘O\o 4
W 1- \ '0' '1 4
b 0‘0 4b b\o 4
\ \ 4
_Q' ; 112 112 0.0 :1 112S 1248 50 ‘
1 Sn+ Sn 1., . n o .
r- O 41- 0‘ -4
. b 45 0‘ 4
b b L 0‘ 4
0'0 1» DU
'0
b
. . . . L J 11111 1 ..... 1 0900 1 ..... 1 ..... 1 ..... 1 OOQoom
''''' I V V V V V I r f '7‘, V V V V V I V V V V V I f r I
1 ....oooooooooo.... E .gooooooooooo... 3
. 0 f. .~ 4
P ‘. ... 4p . 5" 4
0 01 " 0 01
‘ O
U . :F .\. 4F ‘3 14 .
D \ : z

1
. 3. .
. \ ,. \ .

1E'3 E 124 112 '\ 1:? 124 °\. 1, 1E'3
b n 1’ 4
r
E

1.1:. \115-4

 

1: .
b4) 4
11 \1
44 e: :A 1: :¢ :: :‘ ‘cie44%47:‘ ‘: ‘ L‘ 4 #;,;44;A }44;¢ -1 #. .:*::—'
I) 4
00
> o
_ 000 I 000 I
o 00
> O 4} 4
o 0
6 I- 0‘0 di- 00 -1 6
, ‘ ‘O 4
A r 0.0.0 1: 0.0. ‘
”O 4» 00 4
O\ 4» ”O 4
(4.4 Al" 0‘0 1’ b‘ '41
‘0 4» 0‘ 4
‘-' b 0 Ob 1
_Q 1248 112S 5 1 124s 1248 ‘5‘ 1
4» o 4
2 n+ n ,1 , n+ n ,2
0‘ 4 O ‘
4
4

F

p

1 ‘0 .
1- O. 4
L

K

‘O O
'0 O
1 .OQOQnmn: 1 9000913

0”71‘0”2‘0”‘30 0102050 40
Nc Nc

Figure 4.1: The probability distribution of the charged particle multiplicity, NC, and
the derived impact parameter as a function of the charged particle multiplicity.

 

 

 

 

93

 

Zplf

 

 

 

0 llllllllLlilIllllllllllllllliJllll

0 5 10 15 20 25 30 35

 

Figure 4.2: The charge of project-like fragments, ZW, as a function of the charge
multiplicity Nc

94

Other observables can also be used to construct impact parameter filter. Since
there is a strong correlation between charged particle multiplicity and impact param-
eter. We can test the correlations between other observables and charged particle
multiplicity to check their correlations with impact parameter. We explored the cor-
relation between the charge of PLF(Project-Like Fragments), detected in the Ring
Counter, and charged particle multiplicity for example. In Fig. 4.2, we see that there
is a strong correlation between the Zpgf, the charge of PLF detected in the Ring
Counter, and NC. In the region of Zplf larger than 20, Z1,” decreases monotonously
with NC. This can be understood since the more charge left in the PLF, the less charge
left for the overlap regions of the reactions thus less charge particles were detected
by Miniball/Miniwall and LASSA. In the region of Zplf smaller than 20, Zplf smears
out, suggesting that the correlation does not exist anymore. We do not always detect
the PLF because it sometimes goes to smaller angles outside of our coverage. Thus,
we chose the reduced impact parameter constructed from Nc as the primary impact
parameter filter for this study and not the PLF charges. A two-dimensional cut on
two different impact parameter gates has not been used to place more restrictive gates
on the event selection, even though that might lead to greater precision in the impact

parameter selection for some studies.

4.2 Isotope Multiplicities, Energy Spectra from Cen-

tral Collisions and Model Calculation

4.2.1 Experimental Data

The differential multiplicities of different isotopes were obtained by first selecting the
central collision events with (3 < 0.2, as explained in the section 4.1, then within each

event selecting the fragments emitted at center of mass angles of 70° 3 0cm 3 110°.

95

The acceptance of LASSA and the impact of this acceptance cut on the data are
illustrated in Fig. 4.3. In this figure, we show the corrected yield in the center of
mass frame as a function of ECm and polar angle 06", for 4He produced in the central
collisions of the 112Sn + 112Sn reaction. The cut 70° 3 0cm 3 110° is indicated by
the dashed lines in the figure. At these angles, the coverage of the LASSA array is
excellent and the efficiency is well behaved. The only losses occurred for particles
emitted at very low energies E /A < 0.2 MeV in the center of mass, corresponding
to small laboratory angles of 61.1,, < 7°, as well as at high energies, corresponding
to laboratory angles of 010;, > 58°. These losses (total < 2%) were estimated by
fitting the energy spectra for each isotope and corrected by the efficiency function.
The acceptance of LASSA is also shown in Fig. 4.4. In this figure, the efficiency
corrected differential multiplicity dM/dydpi for 12C fragments is shown. The angle
cut is indicated by the dashed lines in the figure as well. The measured differential
multiplicities smoothly depend on the transverse momentum pi and rapidity y and are
centered about the rapidity of the center of mass, consistent with emission from the
participant source. They display no characteristic “Coulomb holes” near the projectile
rapidity, expected from the Coulomb repulsion of 12C fragments from projectile-like
residues moving with velocities somewhat less than that of the projectile.

The final differential multiplicities were obtained after background subtraction and
efficiency correction. Background level was approximated by fitting each element on
the 1D PID histogram with different energy bins cut, then subtract the background
corresponding to each energy bin. Totally ten bins were chosen according to the
statistics of the yields of the fragments. The efficiency correction was done for each
degree from 70° to 110° in the center of mass frame. Fig. 4.3 shows the 2D Emu-0cm
histogram for 4He after the background subtraction and efficiency correction. The 2D

Emu-06m histograms are then projected to 1D to generate the energy spectra and the

96

 

o 70° @110° 4

S
a
I I

 

 

00
§
1 I l l l

 

 

 

_ . m-__‘-__ _--lfl- -
0 20 40 60 80 100 120 140 160 180

(*9

cm

Figure 4.3: Corrected yield of ‘He produced in the central collisions for the 112Sn +
112Sn reaction

97

 

O
O

mSn+mSn

E B< 0.2
12C

  
 

p J_/A(MeV/c)
'5 171 N
95’

U]
Q
.,...

 

-150}
goo—0126' '012’014'016'618' '1' ”112
yr

Figure 4.4: Efficiency corrected differential multiplicity dM/dydpi for 12C fragments.
The cut 70° 3 0cm 5 110° is indicated by the dashed lines in the figure.

 

 

 

I
N

p-a
O

dM/(dEdQ)
3

ES
1.

1
{.11

10

 

1'1

111111111 W 11_11
.

1 11111111

 

71111

11111111111111111

I'l;'li:
i; ‘31‘
11 11.

11.

o Proton
1 Deutron
’ Triton f 1

 

 

O

25 50 75 100

 

 

(3111
O
‘5‘. AAAAAAAA
r. ..... 1‘1zsn+112Sn
g ;;.1»<0.2
I AA“ .‘A‘
2‘1: 1566‘ ::.!!!!6
l<_ 1~;
E— 1 - . °‘ ‘113'. ,
: 11111" via?
_ 4‘} .. 1‘
1‘ ‘0
,__ . § .4 1
'6L1 11 '
: ‘7 - I? I l
_ 8L1 1, ‘ 1
1? » L1 ‘ .
E 9 .
: A L1 1
1 1 1 l 1 l 1 1 1 1 1 l 1 1 1 l

 

 

0

50 100 150
Ecm(MeV)

 

 

10 -5 w“.
10

10_ 11:21}
10 ,1.

5111111

 

1 1 1 1 1 ‘1 1 1 l

 

050

100 150
Ecm(MeV)

 

A -
C.‘ 4 “
a 10 E' a 1!. llen+mSn
.0 11' =11, S<o.2
v _ J,
g -5 ~ ’31
L 11:1 .
‘10 E“ g11 ‘19;
:1 ‘ 1;:
- ~ 07Be'loBe1 ‘
10 §A9BCAHBC [1‘ i
:1 1 1 l 1 1 l 1 1 1 l l l l 1

 

 

 

050

100 150
Ecm(MeV)

Figure 4.5: Energy spectra of H, He, Li, Be from the central collisions of the reaction
112Sn+m’Sn. The energy spectra are obtained after the background subtraction and
efficiency correction.

99

 

 

A “““ A _4 _
Ci 4 0:10 _2333.I‘-
U10 !— " ‘-.HZSn+mSn ID _ ‘ .',|HZSn+“ ZSn
m : -.. m _5 _A ..... 1
"U _5 I 1 =_- 1b<02 "Q ~,' "1ng <02
{’10 _' 1‘1111’;! E10 ; 11 (1111'
2 ‘11 .1 -6 ~11“ 111 {11
U '6 a 1 “10 g A 1111 11
10 E_.10B ,‘1.B 111 : .HC 11 I ,
r 11 13 —7 12 1
7 _ A B A B 1 A C 1
' ~ 10 — 13C 15
10 E” - AMC 1?le
_1111111111111111 1111111111111111

 

 

 

 

 

 

10
50 100 150 0 50 100 150
Ecm(MeV) 4 Ecm(MeV)

O

 

 

A I‘. A 1.— ‘
$104 :_ .‘ dlo E‘ 111.“.
m :4 .".“H:SHEHESH a 5 :1 1.;1125n1én8n
_ - ‘ «.j ‘. <02 ' “1 . <02
310 511‘1‘11;7'9;; 310 E— . ' 11‘
2 1w 3 ”"1“”
:1 9 * , :1; ”1 ’ H
“U - —1 U - F 11 1 1
10 —':4N 1:1} 10 g.120 1 1
N C A O T
-7 ‘1 1 -7 : ,‘30 I50 1 1
10 — AanlsN l 10 A140 19 160 1

 

 

 

 

 

 

0 50 100150 0 50 100 150
Ecm(MeV) Ecm(MeV)

Figure 4.6: Energy spectra of B, C, N, 0 from the central collisions of the reaction
112Sn-i-ll'zSn. The energy spectra are obtained after the background subtraction and
efficiency correction.

100

differential multiplicity. Fig. 4.5 and Fig. 4.6 present the differential multiplicities for
the 35 isotopes we studied, from proton to 20O, as a function of energy. We see that
those energy spectra are very smooth within the energy and angular range selected.
We can also see that the loss of the low energy and high energy particles is almost
negligible.

Fig. 4.7 shows the measured average differential multiplicities of Li, Be, B, C, N
and O isotopes at 70° 3 0 _<_ 110° as a function of N — Z, corresponds to the neutron
excess of the nuclides. In this figure, the 112Sn+“28n and 124Sn+12‘18n data are shown
by the solid squares and circles, respectively. The 112Sn+mSn and 124Sn+ll2Sn are
essentially equal: they have been averaged and are shown by the open diamonds.
To show the isotopes in one figure, the yields of the B, C isotopes are offset by a
factor of 10 and the yields of the N and O isotopes are offset by a factor of 100 in
the figure. As expected, more neutron rich nuclides are produced by the neutron rich
system, 124Sn+1248n, while more proton rich nuclides are produced by the proton
rich system, 112Sn-l-ll'ZSn. The experimental results indicate that the multiplicities
of IMF’s are x 10 — 20% larger for the 124Sn-l—lMSn entrance channel than for the
112Sn-l-“l‘ZSn entrance channel, consistent with previous observations at an incident
energy of 40MeV per nucleon [54]. The peaks of the distributions are always located
at isotopes with N = Z + 1 for elements with odd values for Z, for all reactions. But
for elements with even values for Z, the peak of the distribution located at N > Z
for 124Sn+mSn and located at N = Z for 112Sn+”28n.

In general, the drop from the peak toward more proton-rich isotopes is rather
steep especially for elements with even values for Z. The main differences between
the isotope yields for the four different systems are observed in the tails of the iso-
tope distributions, where it is greater than a factor of 4 for 200. Larger differences

may be expected for even more exotic isotopes, but the background in the present

101

 

10_1:L,f...I....1,24...12.,-...,....,
~ —° Sn+ Sn; E/A=5OMeV

- ----- 0 mixed

___,. 1128n+1128n

 

 

 

 

 

Figure 4.7: Average differential multiplicities at 70°§ 60M 5 110° for Li, Be, B, C,
N and O isotopes as a function of neutron excess (N — Z) of the isotope. The solid
points drawn by solid lines to guide the eyes are data for the 124Sn+1248n system
with N /Z =l.48. The open circles are data for the lightest system 124Sn+12‘13n with
N / Z =1.24. The open diamonds are the averaged values from the two mixed systems,
124Sn+1128n and 112Sn+mSn. The dashed and dot-dashed lines are predictions from
Eq. 4.2. See text for more details.

102

 

measurement due to multiple hits in the LASSA telescopes does not allow for their
accurate determination.

For multi-fragmentation which can be described by statistical fragment emission
mechanisms, it was found that ratios of isotopic yields R21(N , Z )=Y2(N , Z ) / Y1(N , Z)
for a specific pair of reactions with different total isotopic composition, follow an

isoscaling relationship [123, 110, 112]

Y N,Z a
R21(N, z) = 7372—; = 0d ”+le . (4.2)

Here, C is an overall normalization factor, a and ,8 are the isoscaling parameters
for the chosen pair of reactions. This parameterization is discussed in greater detail
within the isosaling section below. If we adopt the convention that reaction 2 is more
neutron-rich than reaction 1, one expects a to be positive and ,8 to be negative. We
have adOpted that convention here and have fitted the ratios of the isotopic yields for
these four systems to extract the corresponding values for a and B. These values for

a and [3 are given in Table 4.1.

 

Reaction 2 Reaction 1 a fi

 

112Sn+124sn 112Sn+1128n 0.18:t0.01 -0.193L_0.01

124Sn+124Sn 112Sn+u2Sn 0.36i0.02 -0.39:t0.01

 

 

 

 

 

 

Table 4.1: Values for a and 3 obtained from fitting the isotope ratios R21.

Eq. 4.2, with only three parameters, C, a and [3 can be used to predict the
isotope yields of 112Sn + 1”Sn as well as the mixed system, 112Sn + 124Sn or 124Sn
+ 112Sn systems using the measured yields of one system. To illustrate how well this
parameterization relates the yields of these four systems, we take the yields of the

124Sn + 124Sn system as a reference and use those yields and the fitted values of a

103

 

and B to predict the yields for the other four systems. The dash and dot-dashed lines
in Fig. 4.7 are the calculated yields for 1123n + 112Sn and 112Sn + 124Sn, respectively.
These isoscaling parameters can, for this limited range of asymmetry, be described by
a linear dependence on either the initial N /Z or the asymmetry parameter, 6 = fig
of the reactions [112]. The excellent agreement between the predicted yields and the
data suggests that such scaling law extrapolations may have useful predictive power.
For example, we expect that these scaling predictions can be accurately extrapolated

to other mass-symmetric systems of A=200-250 nucleons at the same incident energy

per nucleon but with very different isospin asymmetry.

4.2.2 Models for the Multiplicity Calculation

Now we turn to the theoretical interpretation of the central collision data. To study
the density dependence of the asymmetry term of the EOS, we adopt the viewpoint of
the Stochastic Mean Field (SMF) approach described in refs. [2, 23]. In this approach,
the time evolution of the nuclear density is calculated by taking into account both the
average phase-space trajectory predicted by the Boltzmann-Northeim-Vlasov equa-
tion and the fluctuations of the individual collision trajectories about this average
that can be predicted by equations of the Boltzmann-Langevin type. Both the intro-
duction of fluctuating forces and the inclusion of numerical noise have alternatively
been utilized in modeling the fluctuations in practical solutions to such equations;
the simulations presented here utilize the latter technique.

The virtue of such a dynamical approach for. the study of isotopic effects lies
in its self-consistency. The flow of neutrons and protons is calculated under the
influence of Coulomb and asymmetry terms, which reflect self-consistently the motion
of these nucleons. Several different density dependences of the asymmetry term were

explored. Here, we present results for two different asymmetry terms. In both cases,

104

 

the asymmetry term is approximated by the form

Esym (p, 5) = S (p) 62 , (43)
where for the asymmetry term with a stronger density dependence

:a _P_ 2/3 200/Po)2
5“” (p0) ”Hm/p0) ' (4'4)

Here, p is the physical and p0 is the saturation density, a=13.4 MeV, b=19 MeV
[58, 64, 85]. In the following, we refer to this as the “super stiff” asymmetry term.
For the asymmetry term with the weaker density dependence,

S(p) = (4%)?” + 240.9p — 819.1,)2 , (4.5)
where a=12.7 MeV [23]. In the following, we refer to this as the “soft” asymmetry
term. In Fig. 4.8, it can be seen that the two expressions are nearly equal at satura-
tion density but differ at densities that are either much larger or smaller that p0. In
addition to the asymmetry term, the nuclear mean field has a Skyrme type isoscaler
mean field with a soft equation of state for symmetric matter characterized by an
incompressibility constant K 2201 MeV. The isoscaler mean field and the asymmetry
term of these equations of state are used for the construction of the initial ground
state and for the time evolution of the collision. The nucleon-nucleon collisions by
the residual interaction are calculated from an energy and angle dependent parame-
terization of the free nucleon-nucleon interactions and the isospin dependence of the
Pauli-blocking is considered during these collisions.

The calculation solves the transport equations by evolving test particles of finite
width. As mentioned above, we use a reduced number of test particles (50 test
particles per nucleon) in the present calculations to inject numerical noise into the
evolution. In test calculations, we alternatively employed the fluctuation mechanism

discussed in ref. [23], which involves damping the numerical noise by utilizing a large

105

A
O
_]

 

l—‘ [\3 [\D W ()3
U] :3 U1 O U1
I I I I I

  

3(9) (MeV)

—super stiff _
' " soft

)—l
O
I

   

 

 

CU!

0 0:2 0:4 0:6 03; 1 1.2
P/PO

Figure 4.8: The solid curve and dashed curves indicate the density dependencies for
the super stifl and soft asymmetry terms, respectively.

106

number of test particles and introducing explicitly physical noise according to thermal
fluctuations. It was checked that both methods lead to similar results. In contrast
to the Brownian one-body (BOB) method of ref. [44], these methods of inserting
fluctuations are well suited to reactions at finite impact parameter because they do
not presuppose knowledge of the most unstable modes.

When the system expands and reaches the spinodal instability (after about 1 10- 120
fm/c), the most unstable modes are amplified and initiate the formation of fragments
via spinodal decomposition. The evolution of the system is continued after spinodal
decomposition until freeze-out where the number of dynamically produced fragments
and their properties are finally determined. The system is decomposed into fragments
using essentially a coalescence mechanism in coordinate space; specifically, fragments
are defined by regions of density in the final distributions that are above a “cut-
off” density of 1/8po. By definition, the freeze-out time occurs when the average
calculated number of fragments saturates. This occurs about 260 fm/c after initial
contact of projectile and target nuclei in the present simulations. The excitation
energy of the fragments is determined by calculating the thermal excitation energy in
a local density approximation. The procedure is rather rough and will overestimate
the excitation energy particularly for light fragments.

Some of the important features of these calculations and of the prior BUU-SMM
[101] and EES [1 10] calculations can be understood simply by considering the influence
of the density dependence of the asymmetry term on the relative emission rates of
neutrons and protons. In all Sn+Sn collisions, the symmetry energy in the liquid
drop model is positive, i.e. repulsive. The interaction contribution to the symmetry
energy gives rise to a repulsive contribution to the mean field potential for neutrons
and an attractive contribution to the mean field potential for protons. The mean

field potential for an asymmetry term with stronger density dependence is larger at

107

high density and weaker at low density than that for an asymmetry term with weaker
density dependence.

It is the low-density behavior that dominates the predictions for the isoscaling
parameter. As the system expands and eventually multi-fragments, the prefragment
remains at subnuclear densities for a long time while it is emitting nucleons. The
asymmetry term with weaker density dependence around p0 increases the difference
between the neutron and proton emission rates leading to a more symmetric prefrag-
ment than is produced by calculations with the asymmetry term which has a stronger
density dependence.

The SMF calculations are interesting because they are free, in principle, of arbi-
trary assumptions about whether the fragments are formed at the surfaces or from the
bulk disintegration of the system. Comparisons between the fragmentation dynamics
for different asymmetry terms were reported in ref. [107]. The trends of these calcu-
lations are consistent with the prefragment isospin dependences discussed above. In
particular, fragments produced in calculations with an asymmetry term with strong
density dependence tend to be more neutron-rich than the fragments produced in
calculations with an asymmetry term with weak density dependence. In this respect,
these predictions are similar to the results of the BUU-SMM calculations of ref. [101]
and opposite to the results of the EES calculations of ref. [110].

However, the SMF fully dynamical formation of fragments should actually be
more sensitive than the hybrid BUU—SMM to the interplay of the EOS, i.e. to the
density dependence of the asymmetry term, with the fragmentation process. In the
hybrid BUU-SMM calculations, the EOS is entering only in the “pre-equilibrium” nu-
cleon emission described above. In the SMF approach, we have not only this isospin
eflect on fast particle emission but also the full dynamics of the isospin fractiona—

tion/distillation mechanism during the cluster formation. In a neutron-rich system,

108

this leads to a different N / Z “concentration” in the liquid phase (the fragments are
more symmetric) and in the gas phase (nucleons and light ions, are more neutron-
rich) [75, 3]. This effect is associated to the unstable behavior of dilute asymmetric
nuclear matter and so in this way we have the chance of testing the EOS also at very
low density.

Asymmetry terms with weaker density dependence around p0 must show a faster
increase at low densities and so a larger isospin fractionation / distillation during the
fragment formation [4]. Therefore in a fully dynamical picture of fragmentation events
a “sof ” behavior of the asymmetry term around saturation density will enhance
the formation of more symmetric fragments for two converging reasons: i) A larger
“preequilibrium” neutron emission rate as discussed before; ii) A stronger isospin
fractionation / distillation during the bulk disintegration. Opposite effects are of course
predicted for a rapidly increasing (“stiff”) asymmetry term around p0. In this sense
we can expect the SMF results to be more sensitive to the isospin dependences of the
EOS at sub-saturation density.

At freezeout, the fragments are highly excited. For simplicity, we assume that the
de—excitation of these fragments can be calculated as if the fragments are isolated. For
this de-excitation stage, we have tabulated the known masses, states, spins, isospins
and branching ratios for nuclei with Z 315. Where experimental information is
complete, it is used. Alternatively, empirical level density expressions are used for
the discrete levels. These discrete levels are matched to continuum level density
expressions as described in ref. [103]. The decay of primary fragments with Z 215
are calculated, following ref. [103], using known branching ratios, when available, and
using the Hauser-Feshbach formalism when the information is lacking. The decays of
heavier nuclei are calculated using the Gemini statistical decay code [103, 20].

While the SMF calculations predict the numbers and properties of the hot frag-

109

ments that are produced at breakup, the predictions for the relative abundances of
light clusters such as the isotopes with Z =1-2 that are emitted before the system
expands to sub-nuclear density are not very realistic. This prevents a precise model-
ing of complete events including their detection efficiency and means that the impact
parameter selection based on multiplicity cannot be imposed straightforwardly on
the calculated events as on the data. This and the considerable numerical effort it
requires have persuaded us to limit our comparisons to calculations composed of 600
events for each of the 112Sn+“28n and 124Sn-l-mSn reactions at a fixed impact pa-
rameter of b=2fm. We note, however, that the widths in the multiplicity distributions
at fixed impact parameter are large enough that a range of impact parameters may
contribute significantly to the experimental data. Future calculations are necessary

to assess quantitatively the importance of this impact parameter smearing.

4.2.3 Overall Behavior Predicted by the SMF Calculations

In Fig. 4.9, the solid circles and open squares in the left panel show the measured ele-
mental multiplicities for 23 Z S8 averaged over 70°§ 9cm $110° for 124Sn+mSn and
112Sn +1123n collisions, respectively. The right panels show the corresponding mea-
sured multiplicities as a function of the fragment mass. These averaged multiplicities
were obtained by summing the isotopic multiplicities for 2_<_ Z 38.

The dashed lines denote the corresponding distributions of hot primary fragments
calculated by the SMF model using the super stiff EOS. Due to the low total number
of events, we averaged these calculations over a slightly larger angular interval of
60° 5 0cm 3 120°. The solid lines show the multiplicities of cold fragments after
secondary decay. The statistical uncertainties in these calculations are shown in
the figure as vertical bars. The corresponding uncertainties in the data are smaller

than the data points. If the angular integration was performed over the entire solid

110

 

t V I V I f l I v V I V V l V V

124 124

Sn+ Sn E! . Data
_ . §§ O ---- Primary ,
E I . —— Secondary:

1r I

-b -
II

 

I- d.
. p {b
F i.
‘F
ii
I

 

 

.1:-

nt-

-P
p

 

dM/do (sr'l)
(3’):

.O
p—I
r1

 

0.01

 

 

 

 

Figure 4.9: Differential multiplicities for 124Sn+”"Sn collisions (upper panel) and
112Sn+“28n collisions (lower panel) as a function of the fragment charge (left panels)
and the fragment mass (right panels). The points are the data. The dashed and
solid lines are the calculated primary and final fragment differential multiplicities, re-
spectively. Statistical uncertainties are shown for the calculations; the corresponding
uncertainties in the data are smaller than the data points.

111

angle, the averaged calculated multiplicities are about 20-30% larger. This difference
reflects an anisotropy in the calculated primary angular distributions for the heavier
fragments. In the present calculations, however, we do not have the capability to
accurately calculate the modifications of the angular distribution due to secondary
decay so we presently cannot explore this issue more quantitatively. As we will show
later, this anisotropy has no impact on the shapes of the isotopic distributions for
Z =3-8.

In general, the calculated primary and secondary fragment multiplicities are smaller
than the measured values for the lighter fragments Z =3,4 and are somewhat closer
to the measured values for Z =6—8. The lighter fragments with Z < 4 are mainly
produced in secondary decay stage of the theoretical calculations; the primary yields
of these light fragments are much smaller relative to the final yields than are the
values for the heavier fragments. Because the fragment multiplicities and angular
distributions depend on impact parameter, the comparison shown in Fig. 4.9 may
be sensitive to the impact parameter ranges included in both calculation and data.
Future calculations over a wide range of impact parameters are needed to address
this issue. Concerning the greater discrepancy for Z =2-4 fragments, we have already
noted that the formation of light clusters in the dynamical stage before breakup is
not well described in BUU- and SMF-type simulations, because the unique structural
properties of these fragments are not therein well treated. (Treatments of the emission
of light clusters in coupled transport equations for nucleons and light clusters can be
found in refs. [109, 29] and in the framework of FMD [35] or AMD [78] simulations.)
On the other hand, there is a considerable emission of protons and neutrons during
this stage; the total emission and consequently the asymmetry of the remaining source
may still be realistic.

Now we turn to an examination of calculated isotopic yields. The upper left panel

112

of Fig. 4.10 shows the isotopes of carbon nuclei predicted by the SMF calculations
over the entire angular range for 124Sn+12‘18n (solid line) and 112Sn+m’Sn (dashed
line); the dotted-dashed and dotted lines show the corresponding calculations over
the 60° 3 0cm 3 120° gate. Not surprisingly, the more neutron rich 124Sn-l—liMSn
system preferentially produces the more neutron rich isotopes. The peak of the carbon
primary distribution for the 124Sn+12‘13n system occurs at about 15C while the peak
for 112Sn+1128n system occurs at lower mass, i.e. somewhere between 13C and 14C.
The differences between the angle gated and total primary yields are small, and
these difference translate into negligible differences in the shape of the isotopic yield
distribution after secondary decay; we therefore do not plot the gated data because
the two curves are indistinguishable when normalized to each other. As the statistics
of the present calculation make it difficult to perform comparisons to isotopic yields
with a 60° 5 6a,, 5 120° gate imposed on the calculation, the remaining calculated
multiplicities are integrated over the entire solid angle.

After sequential decays, one obtains the secondary distributions shown in the
lower left panel. No longer is there a noticeable difference between the peak locations
(at 12C in both systems); instead, the main differences are found in the shape of the
distribution, which is higher in the neutron-rich isotopes and lower in the neutron-
deficient isotopes for the 124Sn+”“Sn system than it is for the 112Sn+“2Sn system.
Such trends are also qualitatively observed in the experimental data shown for the
124Sn+1248n system by the solid circles and for the 11"’Sn+1128n system by the open
squares in the lower left panel. However, the experimental distributions are consider-
ably wider and more neutron rich than the model predictions. This trend is replicated
in the elemental distributions for all of the other measured elements.

Another way to quantify the differences in the isotope distributions is by the

asymmetry parameter 6 = (N — Z) / (N + Z). The average asymmetry < 6 > of the

113

 

. . v v g . r - I I I I F I I 0.30
Carbon primary
-- mSn + “2811 all e
1 " 124 124 if d 0'25
— Sn + Sn all 6

--- "23n+"23n 60°<e<120°

0-1 * mSn+mSn 60°<0<120° " ' 0-20
0.01 ~ ‘ 0.15
A lE-3 ~ 4 0.10

 

 

w-
-[u-
MF-

1
I

 

<g>

   

 

 

 

 

.. 0.15
]
0001- «~ ~O.10
1E-4 - D “2311+“st MIX ~~ « 0.05
OmSn+mSn data I ’ ‘,
lE-Slo“"‘1‘5"‘345mé°

A Z

Figure 4.10: Left panels: Calculated primary (upper panel), calculated final (lower
panel) and measured (lower panel) carbon isotopic yields for Sn+Sn collisions. Right
panels: Calculated primary (upper panel), calculated final (lower panel) and measured
(lower panel) mean isospin asymmetries as a function of the fragment charge for
Sn+Sn collisions. The lines and data points are further explained in the text.

114

isotopic distribution for each element is shown as a function of Z in the right panels
of Fig. 4.10. Following the same convention as in the left panels, the solid and dashed
lines show the average asymmetries for 124Sn+1248n and 112Sn+”28n collisions; the
upper and lower panels present results for the primary and secondary fragment dis-
tributions, respectively. The calculated differences between the two systems are more
pronounced prior to secondary decay than afterwards. The corresponding data, shown
for the 124Sn+124Sn system by the solid circles and for the 112Sn+uf"Sn system by the
open squares in the lower right panel, are larger and display a stronger dependence
on the asymmetry of the system than do the final calculated fragment yields after

secondary decay.

4.2.4 Isoscaling Analyses

A more sensitive way to compare isotopic distributions is to construct the isotopic
ratio R21(N, Z)=Y2(N, Z)/Y1(N, Z) from the isotope yields Y,-(N, Z) with neutron
number N and proton number Z from two different reactions. As discussed in the
experimental section, R21(N, Z) obeys a simple relationship R21(N, Z) = C e°N+BZ
where C is an overall normalization factor and a and ,8 are isoscaling parameters [101,
110, 123, 112] . Such an isoscaling relationship can be obtained in statistical theories
for two systems that are at the same temperature when they produce fragments.
Binding energy factors common to the yields for the fragments in each system are
cancelled by the ratio when the temperatures are equal, leaving terms related to
the chemical potentials or separation energies [110]. In grand canonical models of
multifragmentation, a = App/T and fl = Ayn/T for example, where Ann and Au...
are the differences in the chemical potentials for the neutrons and protons in the two
systems and T is the temperature [123, 112]. In some calculations [112], the values for

the isoscaling parameters extracted from equilibrium multifragmentation models are

115

“‘er «_—

similar before and after sequential decays, an observation that has been attributed to
a partial cancellation of secondary decay effects [123, 112].

While isoscaling can be expected for many statistical processes [101, 110, 123,
112] , the question of whether it can be expected for specific dynamical calculations
remains open. To investigate whether the SMF dynamical model displays isoscaling,
we construct the relative isotope ratios, R21, primary, using the primary fragments
produced in 124Sn+mSn collisions as reaction 2 (numerator) and in ll""Sn+11r"’Sn
collisions as reaction 1 (denominator).

The results are shown in the upper panel of Fig. 4.11. The error bars reflect the
statistical uncertainties. The predicted isotope ratios for these primary fragments
depend very strongly on the neutron number and follow trends that appear consistent
with isoscaling relationship defined by Eq. 4.2. The uncertainties are large reflecting
the low statistics of the simulations, but the strong dependence on neutron number
makes it possible to discern apparent isoscaling trends nonetheless. The lines are best
fits using Eq. 4.2 resulting in C = 0.96, a = 1.07 and B = —1.43. These values for
a are much larger than values observed in the experiment. The lower panel provides
the corresponding SMF predictions for the ratios, R21, final, of the yields of particle
stable nuclei after secondary decay. For comparison purposes, the ordinates of the
top and bottom panels are chosen to be the same; this demonstrates graphically
that the trends of the final isotope ratios are much flatter and the corresponding
isoscaling parameters (a = 0.286 and B = —0.288) are much smaller. Clearly, the
isoscaling parameters predicted by dynamic SMF calculations are strongly modified
by secondary data. This trend is very diflerent from some equilibrium statistical
models for multifragmentation where the isoscaling parameters have been predicted
to be insensitive to secondary decay [123, 112].

The isoscaling behavior of the dynamically produced fragments arises not from

116

 

 

20.0 :

e1.0'71N
. 10.0 [— BNV
primary

 
   

Z)

.oz-.m .01
01000
T |

F3
(\3
T

 

 

Y1“Sn+‘2‘Sn(N:Z)/ Y‘128n+“23n(N

.93

R21:
[_x
01 o o
K'

 

 

 

.0

Figure 4.11: Upper panel: R21 values obtained from the ratios of the primary isotopic
distributions for 124Sn+1248n collisions divided by those for 112Sn+112$n collisions.
Lower panel: Corresponding R21 values obtained from the ratios of the final isotopic
distributions. Each line in the two panels corresponds to ratios for a given element.
Elements with Z =2—8 (Z =2 1-8) are represented from left to right in the upper (lower)
panel. The lines are the result of fitting R21 with Eq. 4.2; the dependencies on
neutron number for the best fits are given in each panel.

117

 

thermal physics but rather from some special characteristics of the SMF primary
distributions predicted for these reactions. We find, for example, the SMF primary
isotopic and isotonic distributions can be roughly described by Gaussians, see Fig.

4.9. Isotopic distributions, for example, can be described by

(N — N(Z))2

Y(N,Z)=f(Z)€Xp _ 20%

 

, (4-6)

where N(Z) is the centroid of the distribution and 0% describes the width of the
distribution for each for each element of charge Z. This leads to an exponential

behavior of the ratio R21, since, neglecting quadratic terms in N,

In R21 = i, [37(2), — N(Z),] N . (4.7)
02

Note Eq. (4.7) requires the values for 0% to be approximately the same for both
reactions. We have observed this to be the case for our SMF calculations of Sn+Sn
collisions (to within the statistical accuracy ~10%). For the ratios for every element,
to be optimally described by the same parameter, the ratio [N (Z )2 — N(Z)1] /az
must be independent of Z. The statistics of the calculation do not allow a detailed
test of this assumption, but it does appear that this ratio increases somewhat with
Z, as Fig. 4.11 suggests. The primary distributions therefore do not respect the
isoscaling relationship as well the data do.

Now, at variance with the statistical fragmentation models, the secondary decays
substantially modify the isoscaling parameter. The width 0% decreases due to sec-
ondary decay and the difl'erence [N2 — N1] like ways decreases fractionally, but by a
larger amount. Moreover, the final shape is no longer Gaussian, but due to secondary
decay, it reflects the binding energy as a function of neutron excess more strongly (see
Fig. 4.10). These changes combine to decrease the isoscaling parameter as shown in

the lower panel of Fig. 4.11.

118

4.2.5 Sensitivity of the SMF Calculations to the Asymmetry

Term

The density dependence of the asymmetry term has a significant influence on the
relative emission rates of the neutrons and protons and, consequently, on the isospin
asymmetry of the hot fragments prior to secondary decay. As discussed previously,
an asymmetry term with weaker density dependence tends to remain more important
at lower densities, driving the fragments closer to isospin symmetry, than does an
asymmetry term with stronger density dependence. Consistent with this general
consideration, the calculated primary isotope distributions in 124Sn+124Sn collisions,
shown in Fig. 4.12 for carbon (upper left panel) and oxygen (upper right panel),
are more neutron-rich for the super stiff asymmetry term (solid line) than they are
for the soft asymmetry term (dashed line). A similar trend is also predicted for the
112Sn+m’Sn system, but is not shown in the interest of brevity.

A similar trend is observed in the corresponding final distributions that are ob-
tained after secondary decay and shown in the middle panel with the same convention
for the solid and dashed lines as in the upper panel. Both secondary distributions cal-
culated for super stiff and soft asymmetry terms, however, are significantly narrower
and more proton-rich than the experimental distributions shown by the closed circles
in the figure. (The lower panels, which display corresponding calculations when the
excitation energy is reduced by 50%, will be discussed in the next section.) Similar
trends are also observed for the 112Sn+1128n and for the other elements with 3$Z£8,
though we do not for brevity’s sake show those results.

In Fig. 4.13, we present the related dependence of the SMF predictions for the
isotope ratios R21 upon the density dependence of the asymmetry term. We take
advantage of the fact that the results in the Fig. 4.11 can be compactly displayed by

the scaled function S (N) = R21 (N, Z) -e"Bz which condenses the isotopic dependence

119

 

  

 

 

 

 

 

  

 

 

 

 

0.1- "'124"2'4""'10'1
1
Carbon S“ TL S“ Oxygen
0.01» anary ‘0.01
1E-3b «IE-3
.7" 0.01[ -0.01
H
U)
V
% 1E-3» -1E-3
% 11341 11134
C data
% + : jgfic : }* : i r 4
secondaryE/Z
0.01- 1- , «0.01
133. O, . 1 -lE-3
l: .;
1E-4- ' ‘lE-4
lE—S-A-A-eAA-AA 14‘113-5
10 15 15 20

Figure 4.12: Upper panel: Dependence of the primary distributions for carbon(left
panel) and oxygen (right panel) upon the density dependence of the asymmetry term.
Middle panel: Dependence of the final distributions for carbon (left panel) and oxygen
(right panel) upon the density dependence of the asymmetry term. The data are
also shown as the solid points. The various lines in the figure are described in the
text. The excitation energies for the fragments are taken directly from the SMF
calculations. Lower panel: The data are the same as in the middle panels. The curves
are the calculations obtained when the excitation energies of the primary fragments
are reduced by a factor of two.

120

 

for the various elements onto a single line [110]. The left panel in Fig. 4.13 shows
the results for the super stiff asymmetry term and the right panel shows the results
for the soft asymmetry term. In each panel, the values for S (N) obtained from the
primary distribution are shown by the symbols clustered about the solid lines, the
results obtained from the secondary distribution are shown by the symbols clustered
about the dashed lines and the results from the data are shown by the dotted-dashed
lines in each panel to provide a reference. Both the primary and secondary values for
S (N) have been fit by exponential functions to obtain corresponding values for the
scaling parameter and these values are given in the figure.

Generally, the primary distributions for both equations of state display a much
stronger dependence on neutron number than do the final isotopic distributions and
the data. However, the influence on the isoscaling parameter is statistically not very
significant. Indeed, as we pass from a “stiff” asymmetry term to a “soft” one, we
do have a stronger isospin fractionation / distillation, as already discussed before. The
centroid of the distribution, N2, decreases (see Fig. 4.12) but also the width 0% de-
creases; the decrease in width, however, is of the order of 10% and comparable to its
statistical uncertainty. The calculated final distributions display a weak sensitivity to
the density dependence of the asymmetry term; the values for (a=0.286) obtained for
the super stiff asymmetry term are larger than the values for (a=0.254) obtained for
the soft asymmetry term. The sensitivity to the asymmetry term is considerably less
than that reported for the EES model [110], and for the BUU-SMM hybrid calcula-
tions [101]. Unlike these latter two calculations, both super stiff and soft asymmetry
terms yield alpha values that are significantly lower than the value extrapolated from
the data (020.36). One should note, however, that the excitation energies of these
latter calculations could be more freely varied to achieve better agreement with the

experimental observations.

121

 

 

 

5,...-,....,g,....,...fi,.
10 E" ‘ ‘ Primary ;, “a? '3
E—final 1E 1
. 1. 52
data A3 ,3 ,. $ / .
A104 t" N0 ‘ 1'.- ng ’. 1
N 5 ° ’ 25 ' ’ a
3 5 super stlff (0g 1; sofi e93 :
a. . ’_ 1'
:4 i I,
a) 3 _._ _
,3 10 g
a?
l 2
z 10 ‘2
U) 1
,
101 _.
100

 

 

 

 

Figure 4.13: Dependence of the scaled function S(N) on the density dependence of
the asymmetry term. The left panel provides a comparison between values for S(N)
computed from the data (dashed line) and the calculated primary (points about solid
line) and final (points about dashed-dotted line) distributions obtained for the super
stiff asymmetry term. The right panel provides a comparison between values for S(N)
computed from the data and the calculated primary and final distributions obtained

for the soft asymmetry term.

122

4.2.6 Discussion of the Isotope Multiplicity and SMF Calcu-

lations

The calculated final isotopic distributions for both asymmetry terms differ from the
measured ones in three respects: (1) they are narrower; (2) they are more neutron
deficient; and (3) they show a weaker dependence on the isotopic asymmetry of the
total system. In these respects, the calculated results for the two different asymmetry
terms are more similar to each other than they are to the data. It is probably
premature at this stage to focus attention on the sensitivity of the predicted final
distributions to the asymmetry term. Instead, let us concentrate upon what may be
required to bring the final isotopic distributions into greater concordance with the
measurements.

The tendencies of the final isotopic distributions to be more neutron deficient
and to display a weaker dependence on the isotopic asymmetry of the system are
probably closely related. When the final isotopic distributions for both systems are
too neutron deficient, there is not very much room for a strong dependence on the
isotopic asymmetry of the system. The problem that the final isotopic distributions
are too neutron deficient can either be that the primary distributions were too neutron
deficient, that the secondary decay removed too many neutrons or that the freezeout
assumption is invalid.

First let us consider whether initial isotopic distributions may be too neutron
deficient. We note that the average isospin asymmetry of the initial distributions is
trivially related by charge and mass conservation to the average isospin asymmetry
of the nucleons emitted during the SMF calculations before the freezeout (t=260
fm/c) chosen for these calculations. Clearly, it is important that complimentary
measurements of the yields and energy spectra of light particles be performed to

provide guidance as to whether these missing neutrons are carried away primary

123

during fast pre-equilibrium emission or slower evaporative emission. However, the
present simulations neglect the emission of light clusters ((1, t, 3He, 4He, 6Li, 7Li, etc.)
during the dynamical evolution prior to the freezeout. Previous studies [29, 94, 92]
have noted that the neglect of the emission of 4He emission is particularly problematic
because it is abundant and because each 4He particle enhances the isospin asymmetry
of the remaining system by removing four nucleons without changing the neutron
excess. Indeed, it has been speculated that that 4He emission may have an influence
on the isospin asymmetry of the other clusters and fragments that is of the order
magnitude of the influence of the mean field [29, 94, 92]. The treatment of light
cluster emission clearly needs additional theoretical attention.

The number of neutrons removed by secondary decay depends primarily on the
fragment excitation energies and the relative branching ratios for neutron and charged
particle emission. There are significant uncertainties in the calculation of the exci-
tation energies of the fragments, which are related to the difficulty to establishing
precise their ground state binding energies. To explore the sensitivity of the results
to the excitation energy, we have reduced the excitation energy of each fragment by a
multiplicative factor f where 0.5 S f S 1 and recalculated the final fragment isotopic
distributions.

The solid and dashed lines in Fig. 4.12 for carbon fragments (lower left panel)
and oxygen fragments (lower right panel) show the calculated final distributions for
f =0.5 using super stiff and soft asymmetry terms, respectively. Clearly, it is possible
by reducing the excitation energy to shift the isotope distribution in the direction
of the more neutron-rich isotopes, so as to make the mean isospin asymmetry of the
calculated final and measured distributions to be the same. However, the widths of
the calculated final isotopic distributions will still be narrower than the measured

ones .

124

".

 

This discrepancy between the theoretical and experimental widths would be re—
duced if the theoretical primary distributions were wider in their excitation energy
distributions or their isotopic distributions or both. It is interesting that the primary
distributions of equilibrium statistical model calculations that reproduce the experi-
mental final distributions are much wider in excitation energy and neutron number
than those predicted by the SMF calculations [97, 112]. Future investigations will be
needed to address whether wider primary distributions in excitation energy or neu-
tron number can be attained in the SMF model by altering some of the underlying
model assumptions such as the manner in which they are defined at freezeout.

Increased widths may be achieved by performing calculations for a range of im-
pact parameters rather than the single impact parameter b=2fm presented here. We
note that the inclusion of larger impact parameter events may broaden the primary
distributions at mid-rapidity because it will require the inclusion of fragments emitted
from the neck joining projectile- and target-like residues. In ref. [4], it was shown
that SMF calculations predict such “neck” fragments to be more neutron rich because
the isospin fractionation / distillation effect is somewhat reduced in peripheral events,
leaving more neutrons in the fragments, and there can also be an overall neutron en-
hancement in the neck region for such events due to the neutron skins of the projectile
and target [4]. Moreover, the excitation energies of the primary fragments in more
peripheral collisions are also somewhat reduced. Experiments also suggest that neck
fragments in very-peripheral collisions are more neutron-rich [33], but there is little
detailed experimental information about the impact dependence at smaller impact
parameters. Combining more calculated peripheral events with the calculated central
collision events presented in this thesis may result in the broader isotopic distributions

that appear to be required by the data.

125

 

4.2.7 Energy Spectra and Equilibrium Models Calculation

The central events display many attributes consistent with bulk multifragmentation
following expansion and spinodal decomposition at densities of p S 1/3po [27, 70,
36]. The assumption of thermal equilibrium allows useful baseline predictions for the
behavior of complex systems, even when equilibrium may not be fully achieved. For
example, the yields of particles emitted in intermediate energy [27, 12], relativistic
[120, 60] and ultra-relativistic [77] nuclear collisions have been successfully compared
to equilibrium statistical ensembles. This success suggests the attainment of chemical
equilibrium, allowing interpretations [76, 11, 116] of such data in terms of phase
transitions in strongly interacting matter.

This interpretation assumes that systems must last long enough for the relevant
degrees of freedom to equilibrate, and the observables being described reflect that
equilibrium. Similar to the case for the early universe, observables related to these
degrees of freedom may reflect a freezeout time after which these observable cease
their evolution. Indeed, many observables in the central collisions of heavy ions have
been successfully described by equilibrium calculations for an expanding system with
an initial mass, charge, excitation energy and a freezeout density of p z 1 /6po — 1/3po
[27 , 60, 11]. Unlike the early universe, however, the small size of these systems may
allow particles to be emitted and decoupled from the surfaces of the system before the
rest of the system has its freezeout. In this section, we show that the energy spectra
of isotopically resolved fragments allow one to observe such effects and quantify the
breakdown of the freezeout approximation.

The expected properties for the energy sprectra of fragments from the disassembly
of thermalized freezeout configuration are straightforward. Because nuclear interac-
tions between particles are assumed to be negligible after freezeout, the observed

mean kinetic energies of particles < E. > reflect the thermal kinetic energy 3/ 2T,

126

 

 

the collective velocity Ucou at freezeout and the energy < ECO“; > gained due to
the accelerations of these particles by the Coulomb field of the remaining system

[70,50,48L
1

2Am~ < ”Sou > + < ECaul > 1 (4-8)

3

where Am N is the mass of the fragment with A nucleons and m N is the nucleon mass.
As the Coulomb energy depends nearly linearly on the fragment charge and for light
fragments, Z / 2, Eq. 4.8 suggests a roughly monotonic dependence of < E, > upon A;
this has been used previously to extract values for the collective expansion velocity ”can
after constraining < ECO“, > with assumptions about the breakup density [70, 50, 48].

Because ECO,“ depends on Z while the collective motion term depends on A,
Eq. 4.8 may permit experimental distinctions between the two terms. To minimize
sensitivity to the Coulomb effects, for example, one can compare the isotope mean
energy for each element. Such studies have been performed for light charge particles
Z S 2. These studies show mean energies for 3He that are higher than those for
4He, contrary to Eq. 4.8 [122, 115], suggesting the emission of 3He particles from the
surface of the emitting source rather than from an equilibrated freezeout. Since many
of the light charged particles such as 3He are strongly emitted before the fragments,
the question of whether fragments originate in a thermalized freezeout remains Open.
To answer this question, one must measure the isotope energy spectra for fragments
heavier than He.

In Fig. 4.14, we show the energy spectra for 11C (open circles) and 12C (closed
circles). The yield of 12C yield is nearly a factor of 10 higher than that for 11C due to
its higher binding energy. The peak in the energy spectrum occurs at higher energies
for 11C than for 12C and is broader. These two factors dictate a higher mean energy
for 11C than for 12C.

The measured mean energies of all the isotopes as a function of the mass number

127

 

 

 

/dEdO

—5
510

2—-———ISMM
O 0 data

 

 

[Llllll

l

 

 

10-6 1 1 1 1 I 1 1 1 1 I
0 50 100

M (MGV)

Figure 4.14: The solid and open points represent t e measured center of mass energy

spectra for 11C and 12C fragments, respectively. The solid lines represent the cor-
responding ISMM calculations. The dashed lines represent the corresponding EES

model calculations.

128

A are plotted in the left panel of Fig. 4.15. In this figure, the isotopes of each element
have the same symbols. The even Z (Z =2, 4, 6, 8) elements are represented by closed
symbols and the odd Z (Z=1, 3, 5, 7) elements by the open symbols. Generally,
< ECM > increases with A; however, the lightest isotopes in each element (3He,
6Li, 7Be, 10B, 11C, 13N) display a significantly larger mean energies than the neutron
rich isotopes. (Due to insufficient statistics, the uncertainties in the measured mean
energy of the neutron deficient oxygen isotope, 15O is very large and not usable.) This
striking trend is contrary to the behavior described by Eq. 4.8 for fragment emission

from a single equilibrated source.

 

 

llllllllilllllllll’T LITTITWTTIITIIAW’TT _IITTIYTIIIIIIIIIITI
60— 1‘c . 111:5: 14 1 — ISMM —
- ] N1
_ . . ‘ _ ‘1, __ ‘ _
_ 11‘ n‘_ B ‘1 ‘ _ <><3d<> _
5O — ' . ”9 — ) N
' L U ' 1 AK. ,9
/\ ' ' 5 " 6 . a L __ [213‘ ‘
z ‘ r“ If! * L1 ‘ 19
V :SHe :. . f? .- ._ _
~ ' - ‘. tiff) £6
- ll . a- ‘1), . a- I a
30 T i —._ o/ -_ GO _
F . ‘
[.. __ __ ~
~ . 13.1121 - Q/
20 . ne‘er” 112S11__ *— —
bLllllll|lllll|IlllI.‘_lll|lllllllllllllll ~1111l1_111l1111l1111

 

 

 

 

O 5 10 15 20 5 10 15 20 5 10 15 20
A

Figure 4.15: Left panel: experimental fragment mean kinetic energies. Middle panel:
mean kinetic energies calculated with the EES model. Right panel: mean kinetic
energies calculated with the ISSM model.

While Eq. 4.8 describes the trend expected for an equilibrated system of excited

129

 

fragments, sequential decay of the excited fragments emitted from a single equili-
brated source can change this trend. To estimate such effects, we use the improved
Statistical Multi-fragmentation Model (ISMM) of ref. [103] to model equilibrium
breakup configurations composed of excited fragments and light particles and calcu-
lated the sequential decay of the excited fragments with a Monte-Carlo Weisskopf
model [119].

To compare with experimental results, we calculated the energy spectra from the
decay of an initial source with Ao=168, Zo=75 and E‘/A= 6 MeV (roughly 75%
of the total mass, charge and energy of the combined projectile and target [106]).
Initially, we assign randomly a thermal velocity to each fragment and light particle
according to a Boltzmann distribution characterized by a breakup temperature of T:
5.24 MeV [103] and we position each particle or fragment randomly within a volume

of 8.3 fm. We add a collective velocity v = fivcdz,m to the thermal velocity, and

 

vary 1260“,”, = \/2(Ecou/A)/mN (i.e. Emu/A) to describe the mass dependence of
the experimental mean energies. We solve the classical equations of motion these
particles interacting via Coulomb forces.

The optimal choice of collective velocity depends on details of the placement of
fragments within the volume. If one excludes initial configuration that place any
part of a fragment outside R, the accepted configurations fragments will be more
concentrated in the interior than if one excludes configurations that place the center
of mass of any fragment outside R. In the former case, the fragments experience larger
Coulomb forces on average and the collective energy per nucleon needed to reproduce
the < E], > data, ECou/A z 0, is less than that for the latter case, ECou/A z 2MeV.
Regardless of the details of the spatial distribution of fragments at breakup, however,
these equilibrium calculations do not reproduce the measured enhancements of the

mean energies of the most neutron deficient fragments.

130

 

To demonstrate this, we show results for a calculation in which the centers of mass
of fragments are randomly distributed over a spherical volume of radius R. Predicted
11C and 12C spectra (normalized to the data) are shown as blue lines in Fig. 4.14.
The calculation reproduces the peak of the 12C spectrum better than it does the peak
of the 11C spectrum; neither the 11C nor the 12C calculation reproduces well the high-
energy tails of the corresponding experimental spectrum. The right panel in Fig.
4.15 shows that the predicted mean energies does not reproduce enhanced values
observed experimentally for the N < Z neutron deficient fragments. The slightly
depressed values for < E], > calculated for symmetric N = Z fragments reflect strong
secondary decay contributions to these N = Z nuclei from the secondary decay of
heavy particle unbound nuclei that have somewhat smaller initial velocities. The
strength of these secondary decay contributions reflect the Q-value for secondary
decay, which favor decays to well bound N = Z nuclei and suppresses decay to their
N < Z neighbors, but the calculated change in < E], > is very much smaller than that
observed experimentally. Changing the assumptions about the spatial distribution of
fragments or the collective velocity at breakup will not change this conclusion.

This failure and the aforementioned evidence for surface emission of helium iso-
topes suggest that fragments may also be emitted from the surfaces of the system
while it is expanding and cooling. Because the binding energies of the neutron defi-
cient isotopes are significantly less than those of their neutron rich neighbors, their
surface emission rates will decrease relative to their neutron rich neighbors as the
system cools. The Expanding Emitting Source (EES) model [39] provides a means to
test this scenario. First the model predicts a sequence of source conditions that reflect
the cooling of system, rather than a single one. Second it provides a self-consistent
calculation of unique sequences of emission rates for each isotope.

To illustrate these ideas, we have performed an EES calculation, which assumes

131

 

 

that particles can be radiated from the surface of the expanding system prior to
bulk disintegration. Unlike equilibrium models, which assume the system to have
already expanded, this time dependent model calculates the expansion and emission
of the system beginning at an earlier time as it expands through saturation density.
Specifically, the model assumes surface emission for p Z 0.4po, bulk emission for
p S 0.3po and a gradual transition from surface to the bulk emission for densities,
0.4po > p > 0.3p0. For our EES calculations, we take saturation density, E’/A=9.5
MeV, A0=224 and Z0=100 as the specific starting conditions.

As one example of the EES model results, we show the time evolution of 11C and
12C yields. At the time of emission, the primary fragment of each isotope acquires
an average kinetic energy dictated by the instantaneous Coulomb barrier, expansion
velocity and temperature of the expanding system. (The early surface emission con-
tributions increase the value of the Coulomb and collective contributions above those
obtained from the bulk emission alone.) Taking this time dependence into account,
we plot the < E], > values for 11C (blue dashed line) and 12C (red solid line) as a
function of the time of emission in the left panel of Fig. 4.16. Over the evolution of
the source the carbon isotopes are emitted with a range of kinetic energies but there
is very little difference between the values for the two carbon isotopes at any given
time. -

We next examine the emission rates as a function of time for the two isotopes.
This is shown in the lower right panel of Fig. 4.16 where the instantaneous rates for
each isotope are plotted and in the upper right panel of Fig. 4.16 where the ratio
of the rates is plotted. Here we find that the rate for the emission of 11C relative to
that of 12C changes drastically with time. The emission of the former is enhanced
at earlier time and the latter at later time. Within the EES model, the rates are

determined primarily by three factors: the spin degeneracy factors of each channel,

132

 

 

 

IIII[TIIIIIIIIIII lllllllllllllll

100

 

 

 

 

 

P l
l
A _ \ ,
> 1 X
m \ I”
2 r 1 [.1
V l,‘ ,1], Q
l "'1 _
A50 ‘1‘ I’ z
2 \ ’1 \
o _ 1..
1:1 ..__ 3
V r A
O
L \
1'15
F 3
_ p I]; \ v
+- // 4“ -
O1111l1111i1111111LII/1:"1l1111i1111l‘1‘1‘

 

0 50 100 1500 50 100 150

t (fm/c)

Figure 4.16: Left panel: Mean center of mass kinetic energies for 12C(HC) calculated
as a function of time with the EES model. Lower right panel: Emission rates for 12C
(“0) calculated as a function of time with the EES model. Upper right panel: Ratio
of the emission rate for 11C divided by the emission rate for 12C calculated with the
EES model.

133

 

 

 

free energy, exp((Nf,’;(T) + Z f;(T)) /T), and binding energy, erp(—Q/T). Here,
f;(p)(T) are the excited free energy per neutron (proton) of the source, and Q is the
Q-value associated with the emission. The degeneracy factors favor 11C. The values
for f "‘ are negative and hence the isotope with fewer nucleons is also favored by this
factor. The magnitude of f *, however, goes to zero like T2 so the relative advantage
for 11C arising from this factor deceases as the temperature goes down. The Q values
are greatly influenced by the respective binding energy factor, which strongly favors
the N = Z isotopes; this preference strengthens at reduced temperature. The net
effect is that the preference for the 11C at the highest temperatures shifts to 12C as the
temperature falls with the emission and expansion. We tested whether the binding
energy is the dominant consideration by forcing the binding energies for 11C and 12C
to be equal. In this test calculation, the relative emission rates for the two isotopes
changed little with time.

In the EES model, 11C is preferentially emitted earlier than 12C, and the shapes
of the energy spectra are consequently not the same. In Fig.4.16, we show the cal-
culated energy spectra for 11C and 12C (red dashed lines) by taking a convolution of
spectra each arising from the instantaneous conditions of the source. The EES model
reproduces the shift in the peak of the energy spectra, but somewhat underpredicts
the slope of the spectra for both isotopes. We note that it is necessary to multiply
the EES model predictions for both 11C and 12C by a factor of 0.3 to match them to
the data. We attribute this reduction to the fact that the emission of elements with
Z > 10 are not considered in these EES model calculations.

The total yield of 12C contains contributions from the neutron decay of excited
13C and the a decay of excited 16O. The yields of 11C are not affected significantly
by sequential decays. Integrating over the energy spectra, we find an average kinetic

energy of about 56.7 MeV for 11C and 45.2 MeV for 12C. The difference of about 11

134

 

 

MeV is in qualitative agreement with the data. This same scenario applies to the
other elements, each of which shows similar patterns for relative emission. We show
the EES calculations for the mean energies of all isotopes in the middle panel of Fig.
4.15. The EES model reproduces the basic trends of the data.

Besides the success of these schematic calculations, other considerations support
the hypothesis of an early surface emission of fragments prior to bulk emission from
an expanded system. We note that surface evaporation and fission are the basic decay
modes of equilibrated compound nuclei formed at lower incident energies. 'II'ansport
theory, which describes preequilibrium emission in collisions at Fermi energies and
above, predicts the abundant emission of nucleons and clusters as the system expands
to low density. The early emission of fragments should not be surprising. Indeed,
the preequilibrium emission of fragments have been previously reported in for proton
induced reactions [87], and similar isotopic effects were observed. Here we have shown
that such effects are relevant even for the multifragmentation process that have been
heretofore interpreted widely as an equilibrium process. Now, the task before us
is to understand how this early emission and other non-equilibrium effects impact
the utilization of multifragmentation as a probe of the liquid-gas phase transition of

nuclear matter.

4.3 Isospin Difl'usion in the Peripheral Collisions

and Model Calculation

4.3.1 Data from the Peripheral Collision

We used peripheral collisions for our isospin dynamics study. As experimental ob-

servables, we focus on features of isotopic yields Y,(N, Z) of particles measured for

cc '1)
2

reaction . Here, N and Z are the neutron and proton numbers for the detected

135

 

 

fragments. The diflerential multiplicities of different isotopes were obtained by first
selecting the peripheral collision events with f) > 0.8, then the background level was
approximated by fitting each element on the 1D PID histogram. Since the statistics
is not high enough to allow us to do the background level approximation with many
different energy bins, only two bins are used, for particles stop in the second silicon
detector or stop in the CsI crystals, respectively. The efficiency correction was done
by dividing the measured 2D yr-pi histogram by the corresponding efficiency function
to get a corrected 2D yr-pj histogram. A histogram of corrected yield of 12C after
reflection can be found in Fig. 4.17. A “Coulomb hole” can be seen clearly at the
rapidity region near that of the projectile. These 2D yr-p 1. histograms can be used
to get the isotopic yields in different rapidity bins.

To study the isospin diflusion, projectile rapidity fragments were selected by a ra-
pidity gate of y, = y/ybeam > 0.7, where y and ybeam are the rapidities of the analyzed
particle and the beam, respectively. The contributions from the neck fragments are
minimized in this rapidity region. Fig. 4.18 shows the measured average differential
multiplicities of Li, Be, B, C, N and O isotopes at y, > 0.7 which can be compared
with the corresponding data from central collisions. One can see that the yields from
112Sn+1243n and 124Sn + 112Sn are not the same, especially in the tails of the isotope
distributions. This property can be used to study the isospin diffusion.

We analyzed the ratios of isotopic yields R21(N, Z) = 13(N, Z) / Y1(N, Z) for a
specific pair of reactions with different total isotopic composition and found that they
follow the same isoscaling relationship as described in Eq. 4.2 [110]. It suggested this
may be a compact and accurate way to quantify the isotopic asymmetry effects caused
by isospin diffusion.

The left panel of Fig. 4.19 shows measured values for R21(N, Z) assuming 124Sn +

124Sn collisions as reaction 2 and 112Sn + 11"’Sn collisions as reaction 1 in Eq. 4.2. The

136

 

 

 

 

p i/A(MeV/c)
“ E a

-50
-100

-150

-2000 .

Figure 4.17: Efficiency corrected diflerential multiplicity dM/dydpi for 12C fragments
in peripheral collisions. A “Coulomb hole” can be clearly seen at 31, ~ 0.05 and

y, ~ 0.95.

UIO
GOO

 

0.8 < 0 <1 112Sn+“28n

12C

    

 

 

0.21461 10.204106018‘ ‘1' ”1.2
yr

137

400
350
300
250
200
150
100
50
0

-

 

 

 

1""""""""E/1;5'o'Mev'1
M. 124Sn+124$n 08 <’b< 1
[‘3 a _ <> 112Sn+121Sii 0'7 < Y/Ybeam
(f 4 \'\\ _ _ . 121Sn+ fsn .
10—2 0 \’:\ N :/ —
LI V\\\. O
C». Q
0 \‘9
/;°\\ 0
/ ' \\ ,.
[iv/1 \\.\\ ‘
B ?§»\\. XO.1
_4 — \ . \//
10 s 1
f../ ' \\ ‘
3 \ \g x001
I o\,
N Q» \
\ \ 9 (,1
\ §
10‘6 — "f
I l n l l l l

 

 

 

 

1i 2

Figure 4.18: Average differential multiplicities as a function of neutron excess (N — Z)
of the isotope for peripheral collisions, f) > 0.8, for Li, Be, B, C, N and O isotopes
with rapidity near that of the projectile. The solid circles are data for the 124Sn+12"Sn
system with N /Z 21.48. The open circles are data for the lightest system 112Sn+”2Sn
with N /Z=1.24. The solid diamonds are data for system 124Sn+“2Sn. The open
diamonds are data for system 1128n+mSn The solid, dashed and dot-dashed lines
are drawn to guide eyes.

138

 

fit to Eq. 4.2 are shown as the solid and dashed lines. Fits to the other systems (not
shown) are of similar quality. In the right panel of Fig. 4.19, we plot the best fit values
for the isoscaling parameter, 0, versus the overall isospin asymmetry of the colliding
system: 60 = (No — Z0) / (N0 + Z0), where No and Z0 are the corresponding total neu—
tron and proton numbers. The solid and open points denote data for 124Sn and 112Sn
projectiles, respectively. In general, the isoscaling parameter (1 increases with the
overall isospin asymmetry 60. Statistcal calculations, both from multifragmentation
and evaporation predict that the isoscaling parameter should depend approximately
linearly on the isospin asymmetry 60 of the entire system. If the linear correlation
between a and 60 [123] is assumed, the mixed systems at 60 = 0.153 should assume an
a value midway between that of 124Sn + 124Sn (60 = 0.193, a=0.57:l:0.02) and 112Sn +
112Sn (60 = 0.107, 0120). However, the measured value for the 124Sn projectile (solid
point) is much larger than the value for the 112Sn projectile (open point) at 60 = 0.153
indicating that the emitting source is not consistent with the achievement of isospin
equilibrium is not achieved in the asymmetric reaction systems [33]. The upper value
of a=0.42:l:0.02, obtained for 124Sn + 112Sn, represents an effective asymmetry of
about 0.17. This value corresponds to roughly half way from the projectile value of
0.193 to the “equilibrium” value of 0.153. The lower value of 0.16:1:0.02, obtained for
112Sn + 124Sn, has the same interpretation; except here the projectile is 112Sn and the
change in asymmetry is in the opposite direction. The consistency of these results
extracted from two independent measurements adds credibility to the approach.
Because there are no isospin differences between identical projectiles and targets,
we use symmetric 112Sn + 112Sn and 124Sn + 124Sn collisions in the following analyses
to establish diffusion-free baseline values for the measured and predicted observables.
The asymmetric 124Sn + 112Sn and 112Sn + 124Sn collisions, on the other hand, have

the large isospin differences needed to explore isospin diffusion. Following ref. [86],

139

 

 

 

 

 

 

0.6
”E
U)
51’.
fi+
C1
0)
,9); 0.4
3:
> C5
C1
to
S
_+_
1: 1 0.2
U) j»
R? .
:07:
;— b/bmax>0.8 i, ,’
: : /
05: Y/ybeam>0-7 <~1211128n+1128n — 0.0
: 1 L n 1 1 1 l 1 1 1 ‘ 1 1 1 1 l 1 1 1 1
10 15 0.10 0.15 0.20
A 50=(No‘zo)/(No+zo)
Figure 4.19: Left panel: Measured values for R21(N, Z) =

K241124(N, Z)/Y112+112(N, Z) (points) and fits with Eq. 4.2 (solid lines). The
solid line and points represent even Z =4,6,8 isotopes while the dash lines and open
points represent odd Z 23,5,6 isotopes. Right panel: Best fit values for a as function
of 60. The lines serve to guide the eye. The reactions are labeled next to the
data points. 60 = 0.107 and 0.194 for the symmetric systems of 124Sn+124Sn and
112Sn-l—“zSn respectively. The two mixed systems have the same 60 = 0.153. Solid
points denote 124Sn as the projectile and open points denote 112Sn as the projectile.

140

 

we define the isospin transport ratio, R, as

 

2113 - $124+124 '— 50112
+112
R,‘ = , (4.9)

$124+124 — $112+112

where :1: is an isospin sensitive observable. For the two symmetric systems 124Sn +
124Sn and 117‘Sn + 112Sn, R,- is automatically normalized to +1 and —1, respectively,
allowing quantitative comparison of the measured and predicted R.- values even if the
model calculations use isospin observables that are different from the experimental
ones. The only requirement is that both the experimental and theoretical observables
depend linearly upon the isospin asymmetry 60 of the emitting source. To represent
the experimental data, we chose the neutron isoscaling parameter (1 because the pro-
jectiles and targets for the asymmetric systems differ only in their neutron numbers.
Using a for :1: in Eq. 4.9, we obtain the isospin transport ratios of the two asymmetric
systems shown as the shaded bands in Fig. 4.21. The observed values, |R,(a)| z 0.5,
are consistent with previous discussion that the isospin asymmetry of the projectile
remnant is half way between that of the projectile and the “equilibration value”.

Changes in a may result from both isospin diffusion and preequilibrium emission.
In the absence of isospin diffusion, preequilibrium emission from the projectile should
be approximately equal for asymmetric 124Sn + 112Sn (“28n + 124Sn) collisions as for
symmetric 124Sn + 124Sn (llzsn + 112Sn) collisions. By focusing on diflerences between
mixed and symmetric systems, R,(a) largely removes the sensitivity to preequilibrium
emission and enhances the sensitivity to diffusion.

We can use other physical observables to construct R,. Since mirror nuclei have
similar secondary decay modes, we can reduce the effects of the secondary decay and
enhance the effects of diffusion if we take the ratio of the mirror nuclei. Fig. 4.20 shows
Rn, constructed from ln[Yz'eld(7Be)/Yz'eld(7Li)], as a function of relative rapidity in
the center of mass, (y — ycm)/ybeam. The Ry, from reaction 112Sn + 124Sn is reflected

about 0 to give an overall trend. Note, we use ln[Yz'eld(7Be)/Yield(7Li)] instead of

I41

the ratio Yield(7Be)/Yield(7Li) because ln[Yz'eld(7Be)/Yield(7Li)] cc ()1, — 11,,)/T
just as 0 oc Ayn/T. We expect this to give a corrected ratio R,- which is closer
to linear dependence on the isospin asymmetry 60 of the emitting source than does
Yield(7Be)/Yield(7Li). We see that R17 agrees with R,(a) for |(y—ycm)/ybeam| 2 0.2.

In addition, Rn give us the ability to explore isospin diffusion as a function of y and

P1-

 

1.5

l\ -
m ; 1n(Yield(7Be)/Yield(7Li))
1; os<9<1o
I r
0-5 :Iiiifiifiil::11:iiiiiliiiiiiiiiifiliffiIZlIifiiin
_ I:
0 l '
: .1
1 _.. _ I
............... I-~
—0.5 .' ---------------------------------------------------
1 i if: 1315“
: S + SI]
1 m1 ln 1 m L m 1 n n l 1

 

 

 

"LS-0.6 1 16.4 —0.2 0 10.21 0.4' 10.6

(y-ycmyybeam

Figure 4.20: R17, constructed using ln[Yz'eld(7Be)/Yz'eld(7Li)], as a function of (y —
yan)/ybeam-

142

 

1OLizfsprlfisajaqéiifgsjqn)________§
x .

p2 p 101/3 ‘” f

0 - 5 ;_ ' “MKJQZJQWYIYIYIZIYIYIQW'

   

 

 

 

0100- ....................................... e .............. _
o: 8 °
v i 5
"0'5. DATA
—1-0fris————-——————--—-—-—-----}

 

Figure 4.21: Average measured (shaded bars) and calculated (open points) values
for R,- . The labels on the calculated values represent the density dependence of
E8ym,,-,,t/A with decreasing stifl'ness from left to right.

143

 

 

4.3.2 BUU Calculations

We now explore the relationship between isospin diffusion, and the asymmetry term
of the EOS within the context of the Boltzmann-Uehling-Uhlenbeck (BUU) [6, 62]
formalism, which calculates the time evolution of the colliding system using a self—
consistent mean field. The isospin independent part of the mean field in these calcu-
lations is momentum independent and described by an incompressibility coemcient
of K =210 MeV [93]. The interaction component of asymmetry term in three sets of
calculations provides a contribution to the symmetry energy per nucleon of the form
Esym,,-m/A=C,ym(p/po)7 where Cay", is set to 12.125 MeV and 7:2, 7:1 and 7:1/3 .
Here, smaller values for 7 dictate weaker density dependence for anm,int- The fourth
set of calculations, referred to as SKM, uses an interaction asymmetry term providing
Esym,,-,,t/A =38.5 (p/po) — 21.0(p/p0)2 [2] and has the weakest density dependence.

Calculations were performed for the 124Sn + 124Sn, 124Sn + 112Sn, 112Sn + 124Sn
and 112Sn + 112Sn systems at an impact parameter of b=6 fm [93]. We employed
ensembles of 800 test particles per calculation and followed each calculation for an
elapsed time of 216 fm/c. At this late time, the projectile and residues can be cleanly
separated. Nonetheless, we require that all nucleons in the assigned regions to have
density less than 0.05po and velocities above half the beam velocity in the center of
mass to be consistent with the experimental gates. To reduce statistical fluctuations
in the results, we averaged 20 calculations for each system.

Using the average asymmetry of the pro jectile-like residue < 6 > in the calculation,
as the isospin observable, x, in Eq. 4.9, we plot predictions for R,(6) as a function
of time as bands in Fig. 4.22 for the stiffest asymmetry term, (p2 , top panel) and
the softest asymmetry term, (SKM, bottom panel). By construction, [2,-(6) describes
the evolution of isospin asymmetry for the projectiles (112Sn or 124Sn) in the mixed

reactions relative to that for the symmetric 124Sn + 124Sn (with R,- = 1) and 112Sn +

144

112Sn (R,- = —-1) systems. The widths of the bands reflect the statistical uncertainties
of the calculated values for R,(6). Initially, these predictions for R,(6) represent
the isospin of the projectile (R.(6) = 1 for 124Sn and 1%.-(6) = -—l for 112Sn) nuclei.
Subsequent isospin diffusion drives the R;(6) values towards zero. Even though pre-
equilibrium emission from the projectile remnants influences < 6 >, R;(6) is not
strongly modified because such pre-equilibrium emission is largely target independent
and therefore cancelled in R,(6) by construction.

The influence of the asymmetry term depends on its magnitude at sub-saturation
density [62, 93]. For the top panel, the energy is Ewmmt/A =Csym(p/po)2, which
decreases rapidly at low density and becomes very small, leading to little isospin
diffusion. For the bottom panel, E,ym,,-,,¢/A =38.5 (p/po) — 21.0(p/p0)2 remains
larger at low density, leading to stronger isospin diffusion and bringing the residues
to approximately the same isospin asymmetry. In both cases, the asymptotic value
for R,(6) is first reached at around 100 fm/c when the two residues separate and
cease exchanging nucleons. This timescale is comparable to the collision timescale

Tcou, which can be roughly (~20%) estimated by
Tcou z (4RN + d)/vbeam ‘3 80fm/c , (4.10)

where RN, vbeam and d are the nuclear radius, incident velocity, and the separation
d between the two nuclear surfaces at breakup as illustrated by the time evolution
images of the collisions for the 124Sn + 124Sn system in Fig. 4.22.

Assuming that the experimental isoscaling relationships, shown in Fig. 4.19, re-
flect particle emission from the projectile remnants after 100 fm/c and that such
emission can be described statistically, the calculated values for R,(6) may be easily
related to the measured ones. In doing so, we take advantage of the nearly linear re-
lationship between a and the 6 values that has been shown valid for evaporation and

for statistical multi—fragmentation of the remnants that emit the observed particles

145

 

 

 

-[o 11an Elab=50MeV, b=6.5fm ‘

1 l 1 l 1 1 I 1 l 1

0 20 40) 60 80 100120140
time(fm/c)

Figure 4.22: R,(6) from BUU calculations are plotted as a function of time for the
mixed systems. The symmetric systems are calibrated to +1 and —I automatically
by Eq. 4.9. The top and bottom panels show the calculated results with Em,.-,,¢/A
=C,,’,,,,(p/p0)2 and Eme-m/A =38.5(p/po)2 - 21.0(p/p0)2, respectively. The bands
above (below) the dashed line represent the system with 124Sn (llZSn) as the projec-
tile. The time evolution images of the collisions for the 124Sn + 124Sn systems are
superimposed on the upper panel suggesting that the projectile and targets separates
around 100 fm/c.

 

 

 

 

146

[112]:
02 + 01
2

 

aoc (62—61)(1— ), (4.11)

where 61 and 62 are the asymmetries for the two systems involved in the isoscaling

ratio. Inserting Eq. 4.11 into the expression for R;(oz) , one can show

(61' — 6112+112)(6i _ 61244-124)

Ri = Bi (5 _ O
(a) ( ) (5124+124 _ 5112“”) — (1 — (6124+124 + 5112+112)/2)

 

(4.12)

Since the second term is negligible (< 4%), we can obtain < 6 > from the flat
region at 216 fm/c, use it as :1: in Eq. 4.9 to obtain R,(6) values for these two
systems and we can compare them to the experimental values of R;(a). The calculated
values, shown as open points in Fig. 4.21 in the order of an increasing “softness”
from left to right indicate an increased isospin equilibrium for successively softer
asymmetry terms. The diamond shaped points indicate the R,(a) values obtained
as in the experiment by decaying the residues simulated in BUU to fragments using
the statistical model of ref. [112]. As expected, there is close agreement between the
diamonds and the corresponding open points according to Eq. 4.12.

Fig. 4.21 demonstrates the sensitivity of such data to the asymmetry term of
the EOS. The experimental R;(a) values are closest to the predicted R.-(6) values
derived from the theoretical predictions for < 6 > using the stiflest asymmetry term,
(p2, top panel) with Esym,,-,,t/A =C,ym (p/po)2. This conclusion depends, however, on
the assumption that the measured particles are produced after 100 fm/c when < 6 >
attains its asymptotic values. If the data include emission from earlier stages when 6 is
larger, a favorable comparison with calculations using softer asymmetry terms may be
possible. Relevant determinations of the emission timescales for the detected particles
are being explored [93] and may serve to make the present conclusions regarding the

density dependent asymmetry term more definitive.

147

Chapter 5

Summary

In this dissertation, we studied the asymmetry term in the EOS and isosipin diffusion
using heavy-ion collisions at the intermediate energy. Calculations have shown that
experiment observables such as the energy spectra, intermediate mass fragment (IMF)
multiplicities, isoscaling parameters and various nuclei yield ratios are sensitive to
the asymmetry term in EOS. We show for the first time in this dissertation that
isospin diffusion rates measured in our experiment can also provide constraints on
the symmetry term in the EOS.

We have measured the isotope energy and yields of Z=2-8 particles emitted in
four different Sn+Sn reactions with different isospin asymmetry at 50MeV/ A. To
make the measurement, nine telescopes of the Large Area Silicon-Strip/CsI detector
Array (LASSA) were developed and used in the experiment. The LASSA telescopes
provided a high isotopic, energy and angular resolution which are not available in the
previous experiments. Impact parameters were selected by the multiplicity of charged
particles, measured with LASSA and the Miniball/Miniwall phoswich array.

Techniques were developed to enable the subtraction of background in the ex-
perimental data and the correction for the experimental detection efficiency. The

background levels were mainly estimated at the elemental levels while for those weak

148

isotopes are estimated at the isotopic level. The efficiency correction was done us-
ing efficiency functions obtained from Monte Carlo simulations of the experiment.
These efficiency simulations take care of the multiple hits, the inefficiencies in the
solid angle and loss at low p 1- We examined the accuracy of the efficiency correction
by comparing forward angle data and backward angle data obtained in symmetry
collisions. From this comparison, we extract a systematic error bar for our data. For
the part which can not be corrected by efficiency function, i.e. the inconsistence of
the detection in silicon and C31, a systematic error bar is given.

The total charged particle multiplicity detected in Miniball/Miniwall and LASSA
was used for impact parameter determination. We selected central collision events,
corresponding to a reduced impact parameter of 6 S 0.2 and the top 4% of the
charged-particle multiplicity distribution. The impact parameter is estimated to be
less than 1.5 fm according to total cross section measurements. The peripheral colli-
sions are selected corresponding to a reduced impact parameter of 6 > 0.8 and bottom
40% of the charged-particle multiplicity distribution. The parameter is estimated to
be greater than 6.0 fm in this case.

We used the isotope distributions for Z =2-8 from the central collision to compare
with the results from dynamic model calculations which include fluctuations that give
rise to fragment production. The experimental data display a strong dependence on
the isospin asymmetry that can be accurately described by an isoscaling parameter-
ization. The theoretical calculation reproduces the yields for the heavier fragments
with Z =6—8, but underpredict the yields of the lighter ones which are not strongly
produced by the calculation as primary fragments. The calculated final isotopic dis—
tributions display isoscaling, but the calculated isotopic distributions are narrower,
more neutron deficient; and show a weaker dependence on the isotopic asymmetry

of the system than do the data. The density dependence of the asymmetry term

149

of the EOS has an effect on the calculated final isotopic distributions. The distri-
butions calculated using the asymmetry term with stronger density dependence are
more neutron—rich and are closer to the measured values. These trends are simi-
lar to prior results obtained for a BUU-SMM hybrid model, but different from the
trends for evaporated fragments predicted by EES rate equation calculations. The
present level of agreement between theory and experiment precludes definitive state-
ments about the density dependence of the asymmetry term of the EOS. A number
of theoretical issues, such as the pre—equilibrium of bound clusters, the calculations of
fragment excitation energies, and the importance of the extreme tails of the primary
fragment distributions, may influence the calculated results. Additional theoretical
work is required to explore these issues and to determine the role they may play in
the resolution of these discrepancies. Complimentary measurements of the yields and
energy spectra of light particles can help to determine whether the missing neutrons
in the calculations are carried away primarily by pre—equilibrium emission during the
compression-expansion stage or during the later evaporative decay of the hot frag-
ments.

We compared the isotope energy spectra from central collisions with the data.
We found that calculations from the bulk multi-fragmentation models, i.e. the im-
proved Statistical Multi-fragmentation Model(ISMM), can not reproduces well the
high-energy tails of the corresponding experimental spectra for heavy fragments like
11C and 12C. Neither can the bulk multi-fragmentation models reproduce enhanced
mean energy values observed experimentally for the N < Z neutron deficient frag-
ments. The Expanding Emitting Source(EES) model provides a better description of
the measured energy spectra of the IMF isotopes produced in multifragmentation and
mean energies of all isotopes. This success and other evidence for surface emissions

reveal the dynamics of the emission process. The more energetic neutron deficient

150

isotopes are consistent with the picture that they are emitted earlier from the surface
of the system while the source is expanding and cooling.

We have observed the influence of isospin diffusion with comparable diffusion and
collision timescales. Simulations using the BUU transport model predict that isospin
diffusion reflects driving forces arising from the asymmetry term of the EOS. The
comparisons of experimental data and results of calculation suggest better agreement
with stiff aymmetry term. This result agrees with what we have found by comparing
the isotope distributions from the central collisions with SMF calculations. However,
more stringent constraints on the emission timescales for the measured particles in the
BUU calculations are needed. The version of the BUU used in our comparison does
not have the momentum dependence of the mean field symmetry term. Recently
new calculations by other groups indicate that the momentum dependence of the
asymmetry needs to be carefully addressed. This issue as well as other issues such
as the dependence of isospin diffusion on the impact parameter and the influence
of the in—medium neutron-proton cross section by the rasidual interaction should be

explored in more detailed future work.

151

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