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THESIS
3.
1“ UC+

This is to certify that the
dissertation entitled

PRODUCTION OF NUCLEI NEAR THE NEUTRON DRIP-LINE BY
PROJECTILE FRAGMENTATION

presented by

Elaine Kwan

has been accepted towards fulfillment
of the requirements for the

Doctoral degree in physics and astronomy

 

 

 

 

 

 

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PLACE IN RETURN BOX to remove this checkout from your record.
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6/07 p:/C|RC/DateDue.indd-p.1

PRODUCTION OF NUCLEI NEAR THE NEUTRON
DRIP-LIN E BY PROJECTILE FRAGMENTATION

By

Elaine Kwan

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

2006

ABSTRACT

PRODUCTION OF NUCLEI NEAR THE NEUTRON DRIP-LINE BY
PROJECTILE FRAGMENTATION

By

Elaine Kwan

One of the most fundamental questions in nuclear physics is related to the existence
of nuclei. Nuclei at the limits of the neutron (or proton) drip—line can no longer bind
additional neutrons (or protons) and may have very different physical and structural
properties compared to nuclei near stability. The neutron drip-line is experimentally
known up to oxygen (Z=8). The discovery of 31F extended the boundaries of existence
in this region by an additional six neutrons. This large increase in the number of
bound neutrons with the addition of a single proton is not seen anywhere else in the
chart of nuclides. Many theoretical models make different predictions of the exact
locations of the neutron drip-line in this region. Most of these models are unreliable
in their precision because they predict 31F to be unbound. Those that do predict 31F
to be bound suggest that this nucleus is located at the drip-line. It has yet be shown
experimental that 31F is the last bound fluorine isotope.

Two experiments were performed using the coupled cyclotron facility at the Na-
tional Superconducting Cyclotron Laboratory to produce nuclei near the neutron
drip-line of Z = 8. Nuclei were produced from the fragmentation of an 40Ar projectile
with a 9Be, a "atNi, and a 181Ta target and a 480a beam with a 181Ta target. The
production yields from each reaction were measured to determine the effect of the
neutron excess of the target at intermediate energies and in an attempt to determine
the location of the drip-line above oxygen. The cross sections of the most neutron-rich
nuclei at intermediate energies of ~ 127 MeV/ nucleon were found to be enhanced by

the target’s neutron excess. The momentum distributions and momentum transfers

of the produced neutron—rich nuclei were measured and compared to theoretical pre-
dictions. Simulations from an intranuclear cascade code developed for slightly higher
bombarding energies (E/ A > 200 MeV/ nucleons) and a deep inelastic transfer code
developed for low energies (i.e. energies where the De Broglie wavelength >> distance
between nucleons) have been compared with experimental results from the present
work in an attempt to gain an understanding of reaction mechanism. In addition, the
cross sections were compared with the predictions from a semi-empirical parameteri-
zation formula developed for high-energy reactions and to experimental results from
RJKEN. The cross sections were found to be energy independent for the projectile
energies ranging from 90-130 MeV/ nucleon. The widths of the momentum distribu-
tions were consistent with widths resulting from the statistical emission of a single
cluster and the fragment velocities were similar to the velocities resulting from low

energy transfer mechanism.

To

my family

iv

ACKNOWLEDGMENTS

I would like to thank my advisor professor Dave Morrissey for his guidance and
patience as I learned my profession. I appreciate the opportunity that he gave to me
to participate in other types of experiments. I would also like to express my sincere
appreciation to the A1900 group, especially Torn Ginter, Mathias Steiner, Andreas
Stolz, for all their help, their hard work to insuring proper delivery of the beam and
maintaining the A1900 and their willingness to take time out of their busy schedule
to answer questions. It was because of the caliber of people that works at the NSCL,
that made it an enjoyable and fulfilling experience. I also like to thank the faculty
and staff at the N SCL for their willingness to help and to answer any questions that
I had.

I am grateful for professor Paul Mantica for his suggestions and advise about
careers in nuclear science and the members of my committee, professors Tim Beers,
Filomena N unes, Wayne Repko and Brad Sherrill, for their guidance and sugges-
tions. In addition, I would like to thank George Souliotis for his assistance with the
DIT+Gemini simulations.

It has been fun teasing the chemistry graduate students, Debbie Davies, Sean
Liddick, Bryan Tomlin, and Chandana Sumithrarachchi over the years. They were
great friends and make my experiences here very enlighting and interesting even
though they are chemist. If it was not for the aid of Chandana “no one can pronounce
my name properly”, Debbie and our post-doc Leo Weissman and their willingness
take a shift every day that my experiment ran, I would not of been able to complete
part of my dissertation. I would also like to thank Chandana for our discussions on
everything from beta-delayed neutrons to basic principles, and my lunch companion
Susan Musser for her friendship and conservations.

I would also like to thank my undergraduate professor Aruna Nadasen. He was the

one that gave me my first real glimpse of current research at this facility and based

on his advice that I chose to attend this university. I am grateful for the Physics
and Astronomy Department for supporting me in this long journey, especially for
the graduate secretary Debbie Simmons and Professor Manhanti for their help with
the physics graduation requirements. I have been fortunate to meet many interesting
peOple through the department, such as Carlos Maidana. I am so thankful that I
have met him and for his love, support, and encouragement during these last couple
of months.

Finally, I would like to thank my parents, Ming and Sue, and my brother Radley
for allowing me to chase after my dreams. Without them, I would of been unable to

complete this journey.

Contents

Introduction 1
Fragmentation Models 6
2.1 Empirical Parameterization (EPAX) .................... 10
2.2 Abrasion-Ablation Model .......................... 13
2.3 Internuclear Cascade (INC) Model ..................... 18
2.4 Deep Inelastic Transfer (DIT) Model .................... 22
2.5 De-excitation and evaporation of prefragments .............. 27
Nuclei Near the Neutron Drip-Line 31
3.1 Motivation ................................... 31
3.2 Experimental Setup .............................. 37
3.3 Electronic System ............................... 41
3.4 Particle Identification ............................. 45
Discussion and Experimental Results 60
4.1 Energy Loss .................................. 60
4.2 Parallel Momentum Widths ......................... 66
4.3 Parallel Momentum Transfer ........................ 69
4.4 cross sections .................................. 76
Summary 89
A Input Parameters for ISABEL and GEMINI 93
Derivations 96
B.1 Calculating uncertainties in the cross-sections .............. 96
Data from the Fragmenation of 40Ar 98
Cl Fragments produced from the fragmentation of 40Ar +9Be ....... 98
C.1.1 Cross Section ............................. 98
C12 Parallel Momentum Widths ..................... 100
C13 Parallel Momentum Transfer .................... 102
C.1.4 Asymmetric factor .......................... 104

C2 Fragments produced from the fragmentation of 40Ar +"atNi ...... 105
C21 Cross Section ............................. 105
C.2.2 Parallel Momemtum Width ..................... 107

vii

C.2.3 Parallel Momentum Transfer .................... 109

C24 Asymmetric factor .......................... 111

C3 Hagments produced from the fragmentation of 40Ar +181Ta ...... 112
C31 Cross Section ............................. 112

C32 Parallel Momentum Widths ..................... 113

C33 Parallel Momentum Transfer .................... 115

C34 Asymmetric factor .......................... 118

D Data from the Fragmenation 48Ca with 181'I‘a 120
DJ Fragments produced from the fragmentation of 48Ca +181Ta ...... 120
D.l.l Cross Section ............................. 120

D.l.2 Parallel Momentum Widths ..................... 122

D.l.3 Parallel Momentum 'Il‘ansfer .................... 123

D.l.4 Asymmetric factor .......................... 125
Bibliography 128

viii

List of Figures

1.1 Chart of the N uclide ............................. 2
1.2 Cross sections of nuclei produced by projectile fragmentation of 48Ca

with a 181Ta target ............................... 3
2.1 The N-N cross sections as a function of projectile energy ........ 7
2.2 EPAX comparison ............................... 13
2.3 Overview of the Projectile Hagmentation Process ............ 16
2.4 Nuclear Density ................................ 19

2.5 Excitation may of prefragments from the reaction of 40Ar with 12C 20

2.6 Comparison of prefragment distributions from the ablation—abrasion

model in LISE and ISABEL ......................... 22
2.7 Comparison of the prefragment excitation energies from ISABEL for

the reaction of 40Ar with three N / Z targets ................ 23
2.8 Cross section comparison ........................... 30
3.1 The neutron single particle energy levels .................. 32
3.2 Disappearance of magic numbers ...................... 33
3.3 Chart of the Nuclides: Z520 ......................... 34
3.4 Nuclei along the neutron drip-line ..................... 35
3.5 One- and two neutron separation energies of fluorine .......... 36
3.6 Schematics of the A1900 fragment separator ............... 39
3.7 Schematic diagram of the focal plane detectors .............. 40
3.8 Angular acceptance of fragments at the focal plane ........... 41
3.9 Schematic electronics diagram of the A1900 ................ 42
3.10 Walk due to the CFD ............................. 43

ix

3.11 CFD correction ................................ 44

3.12 A PID of Neutron-Rich Nuclei ....................... 45
3.13 Energy resolution of the Si PIN detectors ................. 47
3.14 Mass resolution ................................ 49
3.15. Angular acceptance .............................. 51
3.16 Angular acceptance: Mocadi simulations .................. ' 52
3.17 Efficiency of the PPAC ............................ 53
3.18 Current calibration .............................. 54

3.19 Yield distribution of 20N produced from the reaction of 40Ar with 9Be 55

3.20 Momentum distribution of the fluorine isotopes from the reaction of

40m with 913e, WM, and 181Ta ...................... 56
4.1 Secondary reactions .............................. 59
4.2 Broading of parallel momentum width ................... 62
4.3 Determination of the target constant .................... 63
4.4 Experimental reduced widths ........................ 65
4.5 Parallel momentum widths (high momentum side) ............ 67
4.6 Momentum transfer of fragments produced in the reaction of 238U with

Ti and Pb ................................... 69
4.7 Average momentum transfer ......................... 71
4.8 Momentum transfer comparisons ...................... 73
4.9 Average fragment velocities ......................... 74
4.10 Cross sections from the reaction of 40Ar with 9Be ............ 76
4.11 Cross sections from the reaction of 40Ar with "“tNi ........... 77
4.12 Cross sections from the reaction of 40Ar with 181Ta ........... 78
4.13 Cross sections from the reaction of 48Ca with 181Ta. .......... 79

4.14 Cross section comparisons of nuclei produced in the reaction of 40Ar
with 93e and "“‘Ni .............................. 80

4.15 Cross section comparisons of nuclei produced in the reaction of 40Ar
and 4‘BCa with 181Ta ............................. 81

4.16 Nuclear binding energy. ............................ 82

4.17 Effect of neutron excess on cross sections ................. 83
4.18 Particle Identification of 31F ......................... 84
4.19 Cross sections for producing 31F ...................... 84

Images in this dissertation are presented in color

xi

List of Tables

2.1
3.1
3.2
3.3
3.4
3.5
4.1
4.2
4.3
4.4
4.5
4.6
A.1
A.2
C.1
C2
C3
G4
C5
C6
C8
C9

Summary of the reaction models ...................... 9
Theoretical Neutron Separation Energies of Neutron-rich Fluorine . . . 35
Effective thickness of targets ......................... 37
Resolution of Si PIN detectors ....................... 47
Coefficients for the angular transmission .................. 51
Corrections to beam current at low intensities .............. 55
Coefficients of the range ........................... 61
Target constants ................................ 62
Contributions due to energy straggling in the momentum width of 22O 64

Reduced width ................................. 68
Longitudinal momentum parameters .................... 68
The slopes of the average parallel momentum transferred ........ 70
ISABEL Input Parameters .......................... 90
GEMINI Input Parameters ......................... 91
Reaction Cross Sections of 40Ar+QBe for N eutron-Rich Nuclei ..... 94
Parallel momentum width from the reaction of 40Ar + 98c. ...... 96
Parallel momentum transferred from the reaction of 40Ar + 9Be. . . . 98
Asymmetric factor. .............................. 100
Reaction Cross Sections of 40Ar+"“tNi for Neutron-Rich Nuclei . . . . 101
Parallel momentum width from the reaction of 40Ar + "“tNi. ..... 103
Asymmetric factor. .............................. 107
Reaction Cross Sections of 40Ar+181Ta for Neutron-Rich Nuclei . . . . 108

xii

C.10 Parallel momentum widths from the reaction of 40Ar + 181Ta ...... 110

CH Parallel momentum transferred from the reaction of 40Ar + 181Ta. . . 111
C.12 Asymmetric factor. .............................. 114
C.13 Angular transmission of fragments ..................... 115
DJ Reaction Cross Sections of 48Ca+181Ta for Neutron-Rich Nuclei . . . . 116
D2 Parallel momentum widths from the reaction of 48Ca + 181Ta. . . . . 118
D3 Parallel momentum transferred from the reaction of 48Ca + 181Ta. . . 119
DA Asymmetric factor. .............................. 121
D5 Angular transmission of fragments ..................... 123

xiii

Chapter 1

Introduction

The completion of the coupled cyclotron facility (CCF) in 2001 permitted the National
Superconducting Cyclotron Laboratory (NSCL) to accelerate stable nuclei to energies
well above 100 MeV/ nucleon at high intensities [1]. As a result, more exotic regions of
the chart of nuclide may be explored than in the past. Nuclei in these exotic regions
typically have very short half-lives, on the order of a few hundred milliseconds or less,
and they can have significantly different structural properties than those nuclei located
near the valley of fl-stability. As shown in Figure 1.1, presently about 2500 nuclei (dark
gray squares), including the ~270 stable isotopes (black squares) that are present in
nature, have been observed out of the perhaps ~7500 nuclei (light gray squares) that
are predicted to be particle bound. Nuclei located near the limits of stability are
extremely useful in testing nuclear theories due to the higher sensitivity to the input
parameters of the nuclear potential [2]. Extrapolations of nuclear properties of nuclei
near fl-stability to predict the location of the drip-line have been undependable [3,4].
Unique features such as halo structures and nuclear deformations that lead to shell
quenching have been observed in some of these nuclei near the drip-line [5].
Radioactive isotopes have been produced by a variety of different techniques such
as Coulomb dissociation, direct reactions, fission, fusion, and other processes. To ob-

serve short-lived nuclei such as those located at drip-lines, a technique known as

 

 

protons

N=82

 

 

80

 

A=100

 

 

 

60

 

 

 

40

 

Z=28

 

20-

 

 

 

 

 

 

 

 

 

0 20 40 60 80 100 120 140
neutrons

Figure 1.1: The terra incognita or unexplored region of the chart of nuclide as pre—
dicted by Tachibana is shown in light gray, the valley of stability is depicted by the
black squares, and the observed nuclei are shown in dark gray. The figure is from
reference [3].
projectile fragmentation was developed to quickly produce, separate and identify the
nuclei of interest in-flight before they are able to decay. Projectile fragmentation has
been widely used to produce nuclei along the proton and neutron drip-lines (regions
where an additional nucleon can no longer be bound because the proton or neu-
tron separation energy approaches zero). A large number of nuclei lighter than the
projectile (or target in the case of target fragmentation) are produced in a pure frag—
mentation process. Shown in Figure 1.2 are the nuclei expected to be produced by
the fragmentation of 48 Ca with 181Ta and their distributions made by the empiri-
cal formula EPAX [6, 7] (see section 2.1) in LISE [8]. The fragments created in the
fragmentation process may be used as secondary beams to produce very exotic nuclei
that may be important in astrophysical processes or in the study of nuclear structure
and reaction properties.

The CCF located on the campus of Michigan State University produces radioac-

tive ions by such a method. This technique has been proven useful in observing the

20

      

lTTrj'I'IVIII

Proton number
8

 

0 5 10 15 20 25
Neutron number

Figure 1.2: The predicted cross sections from EPAX of nuclei (in mb) produced by
projectile fragmentation of 48Ca with a 181Ta target.

shortest-lived particles near the drip-lines for the first time [3,9,10,11]. Facilities such
as Rikagaku Kenkyusho (RIKEN) in Japan, Gesellschaft fiir Schwerionenforschung
mbH (GSI) in Darmstadt, Germany, Grand Accelerateur National d’Ions Lourds
(GANIL) in Caen, France and the NSCL in the United States are at the forefront of
producing these rare and exotic nuclei. The use of synchrotron facilities such as the
Heavy Ion Synchrotron (SIS) at GSI allows projectiles to be accelerated to very high
energies (~ 90% the speed of light) [12]. Current cyclotrons are unable to produce
such high—energy projectiles due to the large radii required in the magnets, yet they
remain useful because of their ability to produce continuous beams at higher intensi-
ties than synchrotrons [13]. Measurements of the systematic trends in the observables
from fragmentation products (i.e. the momentum width, the momentum centroid, and
the production yields) at a number of bombarding energies are necessary in order to
improve on the accuracy of predicting the intensity of fragments away from fl-stability

and towards the drip-lines. The intensities of these fragments will determine the fea-

sibility of observing rare particles near the drip—lines in a reasonable amount of time
at the existing facilities.

Early work done in the late 1970’s at Lawrence Berkley Laboratory (LBL) demon-
strated the usefulness of projectile fragmentation in creating radioactive nuclei [14].
The fragmentation process creates radioactive nuclei through peripheral collisions of
a projectile nucleus with a target nucleus (impact parameter (b) > R1 - R2, where
R1 and R2 are the radii of the interacting nuclei). The fragments are emitted in a
narrow forward facing cone with velocities approximately equal to those of the pro jec-
tiles. Devices placed at zero degrees in the reaction plane such as fragment separators
were developed to capitalize on this property of the fragments and can be used to
separate and isolate specific groups of nuclei for further study. Central collisions (b
< R1 - R2) at intermediate and high energies lead to the annihilation of the nucleus
through multi-fragmentation. These violent collisions tend to have large multiplicities
and typically occur only about 10% of the time for A1 /A2 2 0.2. Hagments produced
in this type of collision are thought to be emitted isotropically in space in the moving
reference frame [15]. Devices such as the 41r arrays are well suited to study such re-
actions while fragment separators such as the A1900 and RIPS are more appropriate
in studying nuclei produced in peripheral collisions. The rest of this dissertation will
be restricted to examining the fragments produced in peripheral collisions.

The probability of a projectile nucleus to collide with a target with an areal density

(n) can be described using Beers’ law:
1. : e—nax. (1.1)

Given this dependence, it is easy to see that the intensity of the projectile (I) will be
attenuated from its initial value (Io) as it travels a distance x through the absorbing
material or target. The attenuation coefficient (no) of the absorber determines the

likelihood with which the projectile will collide with a target nucleus. In most cases,

the projectile will not interact with a target nucleus by the short-range nuclear force,
but will always interact with the target atoms via the long range Coulomb force. This
Coulomb interaction will cause the projectile to deposit part of its kinetic energy in
the target as it travels through the absorbing material. Thus, the final products from
a nuclear interaction will also have atomic energy-losses. This effect on the observed
momentum distributions will be discussed in detail in Chapter 4.

The reaction mechanism that describes the fragmentation of a nucleus has been
studied in detail by many authors using various models such as an incoherent droplet
model [16], the abrasion ablation model [17, 18], the internuclear cascade model [19,
20,21], and a deep inelastic transfer model [22,23]. The latter three models will
be presented in details in the following chapter. The experimental objectives and
motivations along with a description of the experiments conducted at the N SCL will
be presented in Chapter 3 with the experimental results presented in Chapter 4 and
summarized in Appendices C and D. An overview of several theoretical models that
attempt to explain the mechanism(s) that produces the observables of the projectile
fragmentation process will also be discussed in Chapter 4, along with comparisons
to the internuclear cascade and deep inelastic transfer models and other experiments
will be made. Finally in Chapter 5, conclusions from the present work will be drawn

and summarized.

Chapter 2

Fragmentation Models

The products of the projectile fragmentation process can be used to determine the
existence of rare nuclei along the drip-lines or they can be used to create short-lived
radioactive beams for a variety of other reactions. The intensity of these secondary
beams is crucial in determining the feasibility in studying properties of rare radioac-
tive nuclei. Ftom Equation 1.1, it can be seen that the production rate is related
to the reaction cross section (a). The cross sections of heavy-ion reactions are of-
ten approximated using a empirical formula EPAX [6, 7] in simulation programs like
INTENSITY [24] and LISE [8] where the user seeks to quickly calculate the ex-
pected production yields. The disadvantage of using a parameterization of the yields
is that it gives no insight into the mechanism that produces the observed nuclei. More
computer-intensive codes that typically involve Monte Carlo calculations have been
developed in an attempt to understand the reaction process that produces the ob-
served fragments [25,26]. Characteristics of the “prefragment” distributions predicted
by these codes are not directly measurable by experimental means but are important
in the determination of the final fragments. These properties can only be tested in-
directly through the observables of the final fragments. Thus, it is important to test
the validity of these codes against experimental data.

The reaction mechanism that produces the observed fragments is known to change

with the interaction time and particle momenta and hence is energy dependent [29]. At
high energies, where the projectile’s de Broglie wavelength (AD) is much smaller than
the average distance between nucleons in the target ((1), the nucleons in the projectile
can interact with those in the target through nucleon-nucleon interactions. Gaussian
momentum distributions of the reaction products, whose widths are governed by
the Fermi motion of the projectile nucleons, are observed because of the statistical
nature of the process. Asymmetries in the distributions occur at lower projectile
energies where the diffusion process starts to become important. At low projectile
energies (i.e. A D >> (1), the projectile interacts with the whole target nucleus and
at Coulomb barrier energies can form a compound nucleus in which nucleon-nucleon
interactions are suppressed due to strong Pauli blocking [29]. The time of interaction

for these low-energy reactions can be as much as 106 times longer than that for

 

[ I fIIrrnl :1lllnu' ]

 

i

10 11 fluid :1 viliutl

 

 

 

10 so too 500 1000
En. (MeV)

Figure 2.1: The free neutron-proton (n-p), neutron-neutron (n—n) and proton-proton
(p—p) cross sections are shown as a function of projectile energy. The figure is from
reference [27]. The region between the two vertical lines are the projectile energies of
primary beams produced at the NSCL, see [28]. The arrow indicates the bombarding
energies of the two projectiles used in the current work.

reactions at high energies. Central collisions at these energies result in fusion of the
nuclei while deep inelastic transfers dominate in collisions that are more peripheral.
The reaction mechanisms at the two energy extremes have been studied for many
years [30,31]. More complete studies of fragmentation at the intermediate energy
domain (A D ~ d) are needed to determine whether the known reaction mechanisms
relevant at high and low projectile energies persist. At these energies, it is well known
that the free nucleon-nucleon cross sections are reduced (see Figure 2.1), the transfer
mechanism that leads to nuclei that are heavier than the projectile is suppressed and
the momentum distributions are more symmetric. Studies at this energy range near
the Fermi energy (E F ~ from 20 to 50 MeV/ nucleon [32] in the center of mass) are
important to better understand how the reaction mechanisms transitions from one to
the other. The current CCF can accelerate stable beams, such as 78’86Kr and 58"MN i,
to the intermediate energy regime. Four projectile fragmentation reactions from beams
produced at the CCF were used to test the predictions of the internuclear cascade
model (valid at relativistic energies) and a deep inelastic transfer model (valid at
non-relativistic low projectile energies) to determine whether these mechanisms can
describe the distribution of products at intermediate bombarding energies (~ 130
MeV/ nucleon). These models along with the abrasion-ablation model, and EPAX
and will be discussed further in the following subsections. The four models discussed

above are summarized in Table 2.1.

2.1 Empirical Parameterization (EPAX)

As mentioned above, EPAX is an empirical formulation fitted to known cross sections
of heavy-ion reactions in the limiting fragmentation regime. Currently there are two
versions of EPAX available, EPAX 1.0 [6] and EPAX 2.15 [7]. The parameterization
obtained for the first version of EPAX (EPAX 1.0) relied primarily on proton-induced

spallation cross sections. Modifications were necessary to be able to describe more

Table 2.1: Comparison of the four reaction models discussed in the text.

 

 

 

 

 

 

 

 

 

 

 

EPAX [6, 7] Abrasion-Abrasion
type parameterization type macroscopic
number of - number of 2
stages stages
energy relativistic energy relativistic
regime (limiting fragmentation) regime (A << (1)
interaction - interaction geometricli2,
excitation - geometric + frictionz,
energy diabatic3
excitation AS-E31’4,
energy Esurf+mFSI (EFSI>2’5,
constant-AA3
INC [19,20, 25] DIT [23]
type microscopic type microscopic
number of 2 number of 2
stages stages
energy relativistic energy non-relativistic
regime (A << d) regime (A >> (1)
interaction nucleon-nucleon interaction nucleon transfer
excitation ZEparticle + thole excitation fPAEpdt + fPAEhdt
energy energy

 

 

1From reference [17].

 

2From reference [18].
3From reference [33].
4A8 = 41rR-(l + P - (1 - F)2/3), where F and P are functions of the impact parameter and nuclear

radii (see reference [34]). E, is the nuclear surface energy coefficient. This is a lower limit to the

excitation energy.
5E,,,,, is the extra surface energy, mm“ is the number of final state interactions and <Ep31 > is

the average energy deposited due to friction.

complete and recent data from relativistic heavy-ion reactions produced by projectile
fragmentation. The more recent work showed that the cross section distributions were
more neutron-rich and broader than the previous measurements of lighter nuclei. The
parameterizations made in EPAX are valid in the case where the reaction yields are
energy independent (i.e. in the regime of “limiting fragmentation”) and does not
describe products produced from fission or nucleon pickup.

For proton induced reactions at energies well above the Fermi energy, Rudstam [35]
suggested some time ago that the fragment yield a(Z,A) for a projectile of mass Ap
and charge Zp incident on a AtZt target in the case of projectile fragmentation can
be written as

0(Z, A) = Y(A)-0‘(Zp,.ob — Z) (2.1)

where the first term in Equation 2.1 is the isobaric mass yield (total isobaric cross
section). The second term describes the charge dispersion centered about the most
probable charge number (meb) for a given mass number (A) and is written in terms
of an exponential

—R|zp,.Ob—Z|U .

0(meb — Z) = ne (2.2)

Siimmerer et al. found the simplicity of Equation 2.1 convenient and with small
modifications to Equation 2.2, data from a variety of relativistic reactions could be

reproduced. They parameterized the isobaric mass yield as

1 + , A _ 2 A 2
Y(A) = aRP(Ap)e-P(Ap)'(Ap-A) ”1 (I? ”2) 7‘3 Y2 (2.3)
1 otherwise
where the slope of the exponential is defined as
Ppr) = epzAp+P1 (2.4)

10

and the scaling factor in barns is written as
0,, = a2.(A],/3 + Atl/3 + 01). (2.5)

Siimmerer et a1. redefined the second term for the charge dispersion in Rudstam’s

formula (2.1) by the following Equation

_ _ U
(7(me — Z) = n-e Rlzfl + A + Am Zl (2.6)
were the 6 stable charge (Z [3) is approximated by the liquid drop expression
zfl — A (2.7)

 

_ 1.98 + 0.0.155A2/3

and the difference between the experimental values of 2,, and Z fl is parameterized as

AA+A A 2A
A=AA 2 1 4 (2.8)

A3112 otherwise

where A A is defined as

1 + d - A — d 2 A 2 r
AA = 1 (74-1; 2) z; 2 (2.9)
1 A near fl-stability.
The three parameters n, R, and U in Equation 2.6 are defined by
normalization term 11 = Wif— (2-10)

11

1+r1A -( A —r2)4 A Zrz
width parameter R = eR2A+R1 p I; Z;

l A near fl-stability
(2.11)
and
U + U A + U A2 roton-rich e onent
U = 1 2 3 p Xp (2-12)
Un neutron-rich exponent

are used to control the shape of the distributions. The value of meb was found to lie on
the neutron-rich side of the valley of stability and in the case of target fragmentation
meb was dependent on the neutron / proton excess of the target relative to fi-stability.
For targets close to B stability, meb was found to depend only on the fragment mass
while neutron-rich and neutron-deficient fragments were found to retain some memory

of the neutron/ proton excess of the target. The parameter
Am = (z — zfl)epl+(92A)/Ap (2.13)

is necessary in Equation 2.6 to account for this “memory effect”. The resulting set
of twenty parameters y1, y2, d1, d2, P1, P2, 01, 02, R1, R2, r1, r2, U1, U2, U3, p1,
p2, A1, A2 and A3 in Equations 2.3-2.13 were determined by fitting the available
experimental data and their fitted values can be found in reference [7]. The values of
the constants obtained from target fragmentation are also valid for predicting frag-
ment cross sections produced by projectile fragmentation because the same laws must
govern both reactions. No effects from the use of different projectiles were included
into the parameterization aside from the cross section normalization in Equation 2.5.

The parameterization obtained in the second version of EPAX (EPAX 2.15) has
been found to be a good approximation for calculating cross sections of many nuclei
near and far from fl-stability. For example, a comparison of experimental cross sections

from the reactions of 48Ca with Be (panels a and b) and 86Kr with Be (panels 0

12

 

OO

O

crosssectlon(b)
d d. .5 d. d
O O

a. a. 1. a. .t,

    

 

28

 

 

70 75 so as 55 so as 70 75
fragment mass A fragment mass A

Figure 2.2: Comparison of the cross sections from EPAX with data from the reactions
of 480a with 9Be to produce a) sulfur and b) fluorine isotopes and 86Kr with 98a to
produce c) Selenium and d) Nickel. The dashed curves are predictions from the first
version of EPAX and the solid curves are from the second version. The figure is from
reference [7] and the data are from references [36,37].

and d) with the two versions of EPAX are shown in Figure 2.2. The data from the
fragmentation of 480a and 86Kr were obtained from references [36] and [37]. The 86Kr
data appears to be better described by the second version of EPAX (solid curve),
while the original version (dashed curve) describes the 480a data better. The cross
sections predicted by EPAX 2.15 will be compared with data from this work in order
to establish the validity of the EPAX predictions for nuclei near the neutron drip-
line and to determine whether EPAX can reproduce the cross sections of fragments

produced in projectile fragmentation on different targets.

13

2.2 Abrasion-Ablation Model

One of the earliest models that attempted to describe fragmentation of heavy-ion
beams at high bombarding energies, known as the geometric abrasion-ablation model,
was developed in 1973 at LBL by Bowman et a1. [17]. The geometric abrasion-ablation
model is a macroscopic model applied to heavy-ion reactions at relativistic energies (E
> 200 MeV/ nucleon). Bowman et al. speculated that at extremely high energies, the
interacting nucleons have such large momenta that they continue along straight lines
during the nucleon-nucleon collisions. Residues resulting from peripheral collisions are
thus emitted in the forward direction with velocities near that of the projectile. The
geometric abrasion-ablation model elaborated on a concept first proposed by Serber
in 1947. Serber suggested that the process for high energy proton collisions occurs
in two distinct stages: a quick abrasion stage and a slower ablation stage [38]. The
interactions between the projectile and target nucleons during the abrasion stage will
typically last on the order of 10‘23 8 during which time the nucleons in the region of
interaction (the participants) are removed or “abraded” from the projectile leaving
the remaining nucleons in the projectile and target (the spectators or prefragment)
in an excited state. The collisions between two sharp well defined spheres of the
projectile and target nuclei with radii roA1/3 (to ~ 1.2 fm) will gouge out concave
cylindrical surfaces from each sphere exposing larger surface areas. The number of
nucleons removed during this process is dependant on the impact parameter b(A) and
the nuclear radii. Geometric formulas that determine the number of removed nucleons
are available, see reference [15] for an example. Ptom the integration of the volume
of overlap between a cylinder and a sphere, it can be shown that the cross section of
the residual mass (ar(A)) can be represented as a function a function of b(A :l: 0.5)
only [17, 34]:

0(A) = 1r . b2(A + 0.5) — b2(A — 0.5). (2.14)

The neutron-to—proton ratio (N / Z) remains undetermined from geometric consid-

14

erations alone. Other assumptions must be made to determine the number of each
constituent in the prefragment. These assumptions range from having a fixed or cor-
related proton-to-neutron ratio (i.e. N p F/ Z p p = N / Z) to an uncorrelated or statis-
tical probability of removing a nucleon [34]. In the statistical model, the probability
P(APF,ZPF) of obtaining a prefragment with N F p neutrons and Z PF protons from

a AZ N projectile can be described using the hypergeometric distribution:

Z N

Z -— pr N — NPF
P(APFa ZPF) = (2-15)
A

 

A- NPF - ZPF

where the terms within the parenthesis represents binominal coefficients. The distri-
bution function described in Equation 2.15 has no correlations and tends to be very
broad. An alternative method of determining the neutron-to—proton ratio proposed
by Morrissey et al. [34] yields a much narrower distribution. They suggested that
fluctuations in the neutron-to-proton ratio are due to zero-point vibrations of the
giant dipole resonance (GDR) that can arise due to the collective motion of the nu-
cleons. The distribution of nucleons from the GDR about the primary charge Z can

be described by a Gaussian of the form:

_ (Z-AA-(figs/111019))2
P(A, Z) = —L_ 202 , (2.16)

‘ /27rAAaZ e

where a Z is a width parameter that depends on the relative motion of the nucleons in
the giant dipole resonace. The width of this distribution can be obtained by determin-
ing the expectation value of the displacement of the neutrons relative to the protons
using harmonic oscillator wave functions. Even though these two distributions are

very different, they can lead to similar final products through statistical de-excitation

15

(ablation) [39].

After nucleons are removed by the abrasion process, the nucleons remaining in
the prefragment are left in an excited state. The excited prefragment will de-excite
through the emission of nucleons (protons and neutrons), light particles such as a
particles, and 7-rays. This secondary stage can last more than 106 times longer than
the abrasion process and “washes out” the primary distribution. A schematic overview

of the abrasion-ablation process is depicted in Figure 2.3.

P ectile WW
roj Participants Prefragment 2:93;”

7'; \.—./’ r V;

—9 ~ VPF
I \ —'° ——9
k I |\b
‘ ' Nucleon

/' '\ 0

Figure 2.3: An illustration of the projectile fragmentation process.

The number of nucleons emitted during the ablation stage is strongly dependent
on the excitation energy (E*) of the prefragment. In Bowman et al.’s geometrical pic-
ture, the excitation energy arises from the excess surface area of the deformed nucleus
with a concave cylindrical surface gouged out compared to a sphere of equal volume.
The cross section distributions calculated from the excitation energy obtained from
this method were found to be too broad. Modifications were necessary to increase the
magnitude of the excitation energy. Additions to the excitation energy can arise from
interactions between the spectators and participants near the boundary of overlap
during the time of collision. For example, these interactions will result in the transfer
of kinetic energy to the spectators through scattering [18]. The addition of the fric-
tional spectator interaction to the excitation energy has been shown to give a better
description of the final cross sections.

Gaimard and Schmidt suggested an alternative approach to calculate the excita-

16

tion energy based on the probability that a nucleon will be in the participant zone
during the projectile-target interaction [33]. This approach is known as the diabatic
model. Nucleons within the nucleus are assumed to occupy single-particle energy lev-
els within a nuclear potential that is approximated using the liquid dr0p1et model.
Nucleon-nucleon interactions will remove some nucleons from their energy levels dur-
ing the collision creating holes within the potential well. An equal probability of
removing a nucleon from any energy level is assumed. The total excitation energy
of the prefragment after all interactions have occurred is then determined by the
summation of the energies of these single particle holes relative to the Fermi Energy.

The geometric abrasion-ablation model (including friction) of Wilson et a1 [40]
and the hole excitation model of Gaimard and Schmidt have been implemented in
the simulation program LISE [8] in order to calculate the excitation energies of the
prefragments. The input parameters used in LISE to determine the excitation energy
can be constrained by comparing to a second model, the internuclear cascade (INC).

The INC will be described in detail in the following section.

2.3 Internuclear Cascade (INC) Model

The nucleon-nucleon collisions during the fragmentation process at relativistic and
intermediate energies have been described in a microscopic framework using inter-
nuclear cascade models. These classical microscopic models use few free parameters.
The basic concept of all INC codes is that the incoming nucleons of the projectile
undergo a series of nucleon-nucleon collisions within the target and projectile until
they escape or fall below the binding energy of the nucleus. There are several versions
of INC that differ on the treatment of nucleon-nucleon interactions. For the purposes
of this dissertation, only the assumptions made by the two codes VEGAS [25] and
ISABEL [19,20] will be discussed. The INC codes called VEGAS (proton induced
reactions) and ISABEL (generalized VEGAS code) assume time-dependent two-body

17

collisions between bound or cascading nucleons with other cascading nucleons and
nucleons within the Fermi seas. The Fermi energy of the it“ type of nucleon (i.e. pro—

ton or neutron) in each projectile and target nuclei is given by the usual expression:

’12
En = 5,;(27r2m)2/3. (2.17)

where p,- is the nuclear density of the nucleon of mass m. Collisions of nucleons within
the two Fermi seas are also considered. The only restriction made in ISABEL is that
the cascading nucleons may not sequentially collide with the same cascading nucleon
without first interacting with another nucleon.

The projectile and target are assumed to move along classical trajectories in the
center of mass frame until the two nuclei collide with each other allowing nucleons to
interact. The nuclear densities of the projectile and target potentials are calculated
in ISABEL using 16 step-functions (histograms) that approximates a folded-Yukawa-
sharp-cutoff distribution with cutoff radius R3 = 1.18 Al/ 3 (dashed curve), see Fig-
ure 2.4. The older code VEGAS uses one of three models to calculate the nuclear
densities: a constant density, a trapezoidal distribution or a simple step function. The
nucleons within the target and projectile nuclei are assumed to be cold degenerate
Fermi gases trapped within their perspective potential wells. All quantum mechanical
effects except for the Pauli exclusion principle are neglected. These classical approx-
imations are valid in the regime where AD << (1.

The cascading nucleons generally traverse a small distance within the nuclear
medium before interacting with another nucleon. The mean free path between col-
lisions is determined using the free nucleon-nucleon and nucleon-pion cross sections.

Pions (1r) are created and absorbed as the result of the nucleon-nucleon (N—N) inter-

18

actions. The 1r modes of interactions can be described using the A resonance:

N+N 2.- A (2.18)

A S 1r+N. (2.19)

The cascading nucleons will deplete the Fermi sea of its collision partner through the
creation of holes with a volume of l / p in the density distributions. It is assumed that
no other interactions can occur at the site of these holes. The incoming and cascading
nucleons are tracked in small intervals of time until they escape the region of overlap,
the volume defined by the nuclei (for nucleons outside the overlapping region) or until
their total energy falls below the proton or neutron cutoff energy, where the nucleon

will become captured. The cutoff energies of the nucleons are defined as

mtg?" = EF + 2 < BE > (2.20)
E + 2 < BE >
Eggm = max F (2.21)

EF-i- < BE > +ECoul

 

0.16 -
0.14 t
0.12 i
6‘ 0.10 l
g 0.08 l
Q006l
0.04 '-

0.02 ~40
0.00 4 ,

0

 

 

 

 

 

radius (fm)

Figure 2.4: The density of 40Ar as a function of the nuclear radius is shown. The
histogram is the estimation made in ISABEL to approximate a folded-Yukawa-sharp—
cutoff distribution (dashed curve).

19

 

 

 

 

 

 

 

 

 

 

 

l IALJIA

 

 

LAJlAAAlAlAlAA

 

 

 

 

 

 

 

 

 

P(E*)l50 MeV (%IMeV)

 

 

 

100500 300‘ o 100 300
2. AA=6
L
1r
0L
012112611 “““
E(MeV)

Figure 2.5: The probabilities to remove the first six nucleons from a 40Ar projectile
are shown. The open circles are the excitation energies calculated in reference [39]
and the solid squares are the calculated excitation energy from a LINUX version of
ISABEL used in the present work. The two data sets were used to determine whether
the Linux version of ISABEL is comparable with an older version of ISABEL.

where <BE> represents the average binding energy, ECO“, is the Coulomb energy
and E F is the Fermi energy of the nucleus. All captured nucleons and holes in the
density distribution contribute to the final excitation energy of the prefragment. The
contributions from these terms are written as

N hole N particl e

E*= Z Ehole+ Z Eparticle (2'22)
'=0 '=0

where the hole and particle energies, Eparticle and Ehole: respectively, are measured
with respect to the Fermi energies.

To test a recent LINUX version of ISABEL [19,20], a comparison of the exci-
tation energies of the prefragments for removing zero to six nucleons from a 213
MeV/ nucleon 40Ar projectile was made to an older version found in the literature [39].

The open circles in Figure 2.5 were taken from reference [39] and normalized by the

20

total number of counts and the filled squares are from the version used in this work.
The probabilities were determined from the number of counts in 50 MeV increments
(the data points including statistical error bars are positioned at the center of each of
the 50 MeV increments). Both versions of ISABEL predict approximately the same
excitation energy distributions and show that the prefragment becomes very excited
as the number of removed nucleons increases.

A comparison of the predicted prefragment distributions for removing the first
12 nucleons from a 48Ca projectile from ISABEL (filled squares) using 30,000 pri-
mary events and the abrasion—ablation model in LISE (unfilled circles) are shown in
Figure 2.6. The error bars associated with the ISABEL calculations are statistical.
Note that the ablation-abrasion model does not allow for masses heavier than the
projectile or transfer of nucleons, hence the predictions are only shown up to Z =
20. The abrasion-ablation model predicts similar distributions to those from ISABEL
for small mass losses (A f > 41) and broader distributions of prefragments for larger
mass losses. This variation is a direct consequence of the difference in excitation en-
ergy. ISABEL predicts narrower mass distributions, but the excitation energies of the
prefragments are much larger (open squares) than the energies from the abrasion-
ablation model (lines = 13.3 MeV/ AA). Even so, the prefragment distributions of
ISABEL and the abrasion-ablation model can lead to similar cross sections of the
final fragments.

The excitation energy of the prefragments in the LISE version of the abrasion-
ablation model is a variable parameter, while the energies predicted by ISABEL are
not. The excitation energies of the prefragments from ISABEL do depend on the
target. This dependence on the reaction targets can be seen in Figure 2.7 for the
reaction of a 127 MeV/nucleon 40Ar with a 9Be target (triangles), a "atNi target
(squares), and a 181Ta target (circles). The excitation energies shown in the eight
panels increase with the target mass. The fate of the prefragment will depend largely

on the amount of excitation energy it has received during the cascading process. The

21

 

 

’ .05 1021A=46 10.8

 

 

 

 

 

 

 

 

  

 

 

 

 

 

 

  
  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

10.3 1o°1 ,. {0-4
1
- 0.010
10.6
03310
9°10 141618202224 '
0'6101A342 . ° 51 2
3 30-3 i 50.6
(E, ‘ 101: ’2 ",1,
.510 14 161820222490 12141561842022 00?)
810‘: ' ' .06 1A“? ° ° ° 96%
on 3 ° 0: ;
g 10": :0.810 E .03
810"!- ...........
315.17 re 21 @90101214 16 18 20 22 °

 

 

 

 

 

 

 

 

 

_s
O

A-39 , ;A=38 . . - . 0.6
' . 10.8100; .
. «[04 ° 0'4

w
1012141618 20 22001011 13151719 21°0

 

 

 

 

 

 

    

 

 

 

 

 

 

 

 

 

 

 

E :37. o 0 0 ° ° €1.5 E, A=3 $0.8
10"; ° . 11.01001 . 1

: o I ' 40.4
16:!“ I ’.5005 9 j
.. _. 1 t 1 ° .

............ o_0102.r....-...0.0

1012141618 20
roton Number

Figure 2.6: The distribution of prefragments from ISABEL (squares) and the ablation-
abrasion model in LISE (unfilled circles) produced in the reaction of 480a with 181Ta.
The average excitation energies of the prefragments from ISABEL (unfilled squares)
and ablation-abrasion (horizontal lines) that lead to the predicted distributions are
also shown. The error bars are statistical and the cross sections are in mb.

22

 

 

 

 

 

 

 

 

 

 

P(E*)/50MeV (%IMeV)

 

 

 

 

 

0.0 0.0

 

 

if K
' ' 3'1.
. - 1‘3..- .
0 200 400 200 400
15* (MeV)

Figure 2.7: The eight panels shows the excitation energies predicted by ISABEL for
removing the first seven nucleons from a 40Ar projectile. The open triangles are the
energy probabilities from the reaction with a 9Be target, the open squares are from
a ““‘Ni target, and the open circles are from 181Ta target.

excited prefragments de-excite by emission of nucleons and 7-rays in the same manner
as in the ablation-abrasion model. The cross sections and velocities of neutron-rich
fragments after de-excitation were calculated using the internuclear cascade code
ISABEL coupled to a statistical de-excitation code GEMINI [41] and compared with
data produced from projectile fragmentation of 40Ar with 9Be, 40Ar with "atNi, 40Ar
with 181Ta, and 480a with 181Ta at an intermediate energy of 127 MeV/ nucleon (the

energy of the projectile at the center of the target).

2.4 Deep Inelastic Transfer (DIT) Model

At low energies, it is widely known that nucleon transfer mechanisms are the dominant
processes that lead to the production of radioactive nuclei. The transfer mechanism is

expected to subside as the kinetic energy of projectile increases due to the increasing

23

difference between the projectile and target nucleon momenta. The disappearance of
transfer mechanism signatures such as the suppression of the yields of masses heavier
that of the projectile (in the case of projectile fragmentation) has been thought to be
an indication of the onset of the fragmentation mechanism, while others have argued
that these nuclei tend to be highly excited and the suppression of their yields may be
the result of evaporative processes that occur to de-excite the nucleus thus reducing
the number of observed heavy nuclei [23]. The presence of the transfer mechanisms
may result in the dissipation of energy in heavy-ion reactions, which can occur well
past the Fermi energy. Many theoretical models that use successive nucleon transfer to
excite the nucleus have been produced in attempts to explain results of reactions near
the Fermi energy. One such model was developed by Tassan-Got and Stephan [23]. An
outline of their model will be given below. The assumptions made by Tassan-Got and
Stéphan are similar to those in early work by Samaddar et a1. [42] with the addition
of angular momentum and a friction force. The nucleons in this model are assumed to
move in classical trajectories. As one nucleus approaches the other nucleus, 3 window
in the potential barrier opens for stochastic transfer of nucleons (excluding clusters).

Pauli blocking reduces the transfer probability to
P = fT-nd - (1 - no) . (1)1150. (2.23)

The first term (T) is the barrier penetrability including nuclear and Coulomb inter-
actions and is calculated using the Hill-Wheeler formula for a parabolic barrier. The
barrier penetrability determines the probability of nucleons to tunnel across the po-
tential barrier. The second and third terms (nd-(l- na)) determines the occupational
probability of the donor nucleus (nd) and acceptor nucleus (na) for a given system in

equilibrium. This term is given by

1

ni = 1 + 13:3qu — 6m r 3111/ I )/T,:] (224)

 

24

where

i = donor or acceptor
S,- = spin of the nucleus
l,- = nucleon’s angular momentum with respect to the nucleus i

I,- = the moment of inertia.

Only the components of the angular momentum that are perpendicular to the reaction
plane are expected to increase on average with each nucleon transferred and are
considered. The final term (<I>) in Equation 2.23 is defined as the one way local phase-

space flux and is equal to

<I>d5cr = 2h-3vxd372’pdpd0, pg; 2 0 (2-25)

where p and 0 are coordinates used to determine the position of the transferred
nucleon with respect to the center of the region of overlap, and Va; is the velocity in
the direction of the other nucleus and perpendicular to the projectile trajectory and
p is momentum of the nucleus. Each trapped nucleon (i.e. particle) or transferred
nucleon (i.e. hole) of mass m makes a particle and hole contribution to the excitation

energy equal to

I
AB], = 13(1) F + ”r1202 — EF — AU, particle excitation energy (2.26)
2
mo
AEh 2 EF — -TF-, hole excitation energy (2.27)

where v F is the transferred nucleon’s intrinsic Fermi velocity in the donor nucleus,
Vrel is the relative velocity, E F, is the Fermi energy of the accepting nucleus after
receiving the transferred nucleon and AU is the potential energy of the composite
system. Nucleons will continue to be transferred from one nucleus to the other until

the two nuclei move apart from each other after some interaction time At has elapsed,

25

after which time the exchanged nucleons become trapped. The total excitation energy
of the nucleus of mass M after all nucleon transfers has occurred is given by the
expression:

E* = fin/151,041+ [PAEhdt (2.28)

where P is the transfer probability defined in Equation 2.23.

The product nuclei with the excitation energy from the Monte Carlo DIT code by
Tassan-Got have recently been coupled to the statistical de-excitation code GEMINI
in order to compare cross sections of fragments from the reaction of a 25 MeV/ nucleon
86Kr beam in a 64Ni target, a 112Sn target and a 124Sn target [43]. The overall
agreement of the calculated cross sections with the data at these low bombarding
energies indicates the general validity of this approach to produce neutron-rich nuclei.
The authors note that the enhancement in the production of the most neutron-rich
nuclei with neutron-rich targets is not reproduced by the calculations. They suggest
the peripheral collision that creates these nuclei restricts the nucleon exchange to the
surface of the neutron-rich target (i.e. the neutron skin) resulting in a preferential
exchange of neutrons. This detail related to the neutron skin is lacking from the
present model. The cross sections and fragment velocities from the DIT+GEMINI
codes will be calculated at an intermediate energy of 140 MeV/ nucleon and compared

with data from the present work. The results will be described in Chapter 4.

2.5 De-excitation and evaporation of prefragments

Once an excited prefragment is created, the nucleus will decay towards a more stable
and less excited nucleus. The probability for an excited A0 Zo nucleus with spin Jo
to decay to a nucleus with mass A1, charge 21, and spin J1 can be described by the

Breit—Wigner resonance formula [44]:

_ 2.13 + 1 12
(2.11 +1)(2J2 + 1) (E — ER)2 + 12/2

 

(2.29)

26

where J R and E R are the spin and energy of the resonance state and the width (A)
is defined by Fermi’s Golden rule [13]:

2 I
A = 7:5[1/fil2pwf). (2.30)

Equation 2.30 depends on the density of the final states p(E'f) within an energy

interval dEf and the transition probability
[14,? = (fdmpfv’wg? (2.31)

depends the initial and final wavefunctions (\II; and \Ilf, respectively) and on a small
perturbation (V’) to the nuclear potential that allows the transition to occur.

A variety of evaporation codes are available such as CASCADE [45], PACE [46],
and GEMINI [41] that attempt to solve Equation 2.30 analytically. The solution
to Equation 2.30 depends on how the transition probability and density of states are
defined. Intensive studies have been made to determine the density of states of nuclear
matter [47,48]. Bethe demonstrated that the level density of energy independent single
particle states of a Fermi gas can be described by an exponential of the form [49]:

p(U) = fiaém. (2.32)
An energy shift (A = x172) to the excitation energy (U) was found necessary in

Equation 2.32 in order to account for the pairing effect [50]. For a Fermi gas, the level

density parameter (a) may be approximated by the semi-empirical function

(2.33)

a:

A
m

with m taken as a variable. This assumption is valid for a large volume homogeneous

gas where the effects near the surface can be ignored (i.e. regions where the particle

27

density is constant) [51]. Higher order effects due to surface energy and shell effects
may be included into Equation 2.33 which may have a large influence in the light-
est nuclei where there exist relatively few states compared to heavy nuclei. For the
purpose of this work, the code GEMINI was used to de-excite the prefragments and
higher order effects to the density parameter were neglected.

GEMINI is a statistical de-excitation code developed to track the sequential binary
decays of compound nuclei. The decay width (I‘(Zo, A0, JOIZI, A1, J1; 22, A2, J2) = M)
from Equation 2.30 has been modified to include the angular momenta so that the

decay width for the evaporation process for light nuclei can be written as

 

+11
2J + 1 J” U2+E
1‘(Zo, A01JolzlvA11 J1; Z21A2, J2) = 22 [0 fdépz(U1,Jl)TI(€)
"Po
1: (Jo-J1) ( )
2.34
where the transmission coefficient is defined as
h2z(z+1)
0 e < E +
11(6) = “m“ M (2.35)

1 otherwise

and R is the absorptive radius, the 2.] + l arises from the degeneracy in the angular
momentum and the subscripts 0, 1, and 2 indicate the pr0perties of the initial and final
two nuclei. For light particles, the nucleus is assumed to always decay if the difference
between the final kinetic energy (E f) and the energy of the Coulomb barrier (U) is
greater or equal to< —-2. Here, R 18 the same absorptive radius for a classical system
and p is the reduced mass. The level density for a Fermi gas can be rewritten to

include the nuclear spin such that:

p(U, J)— _ (2J+1)(— 1:)3/21—l—ggze 2W (2.36)

where I is the residual nuclei’s moment of inertia. A similar Equation for the decay

width of heavier nuclei (A > 12) can be written by replacing the energy of the Coulomb

28

barrier with the thermal energy of the saddle point and assuming full transmission
over the barrier.

The prefragments calculated by ISABEL produced in the reaction of 40Ar with
12C have been deexcited by GEMINI to determine if the final fragment distribu-
tions can be correctly predicted. Figure 2.8 shows a comparison of the cross sections
from reference [39] (symbols) and ISABEL+GEMINI (histograms). The magnitude
and widths of the distributions for the oxygen, fluorine, and neon isotopes in pan-
els (a), (b), and (c), respectively, are well reproduced by ISABEL+GEMINI except
for the lightest oxygen isotopes near the proton drip—line. The similar distributions
suggest that the two codes of ISABEL and GEMINI may be suitable for describing
the production of light neutron-rich nuclei. Thus, the prefragment distributions from
ISABEL and DIT have been coupled to the de-excitation code GEMINI to deter-
mine if the cross sections from this work can be reproduced and if either codes can
give information about the reaction mechanism for light nuclei produced in projectile

fragmentation.

29

 

 

 

 

 

 

 

 

 

 

 

a e + .
.' __ : 1

.0 -1: F [ E I
510 5F [1 10% .
.510-2Er ......... j 10-2t. . . . 1
g 13151719 21 17 19 21 23
a)
m .1 * 1 N91
3101 1 "‘1
o 1 1.

10 [F T

10'1 [7

‘1
10'

1'8 ‘ 2'0 ‘2'2 ‘ 2141246
A

Figure 2.8: A comparison of cross sections for the oxygen, fluorine, and neon iso—
topes in panels (a), (b), and (c), respectively, produced from the fragmentation of
40Ar with 12C. The experimental data (symbols) is from reference [39] and the

dashed histograms are the predicted cross sections from simulations made by IS-
ABEL+GEMINI.

30

Chapter 3

Nuclei Near the Neutron Drip-Line

3.1 Motivation

The establishment of the limits of stability is one of the major objectives in nuclear
physics. The determination of the locations of the drip-lines is a means to test the
validity of current nuclear structure theories. Fundamental properties obtained from
the study of nuclei near fl-stability are expected to change as nuclei become more
neutron-rich. For example, possible “tailing” of the nuclear wave function due to the
weak binding of the last nucleon(s) allows halo structures to exist [52]. In addition,
the levels structure of nucleons in a nucleus described by three dimensional harmonic
oscillator wavefunctions, which work well for stable nuclei, may no longer be valid. The
spin-orbit interaction, orbital angular momentum (l), and the spin angular momentum
of nucleons play a significant role in determining the location of a specific energy level
and thus influences the locations of the so called “magic numbers”. For example,
states with parallel coupling of angular momenta lie significantly below the energies
of states with antiparallel coupling. The 1f7/2 level is so much lower than the other
35w harmonic oscillator levels that it opens a gap in the level spacing at the N = 28
magic number, see Figure 3.1 (a). It has been suggested that this spin-orbit interaction

will decrease in strength near the drip-lines [53]. As another example, unusual filling

31

patterns allow strong attractive nucleon-nucleon interactions between the protons in
the j = l + 1/2 level and neutrons in the j = l - 1 / 2. This can result in the shifting
of the energy of the levels and the so called quenching of the known magic numbers
as shown in Figure 3.1 (a), and the creation of “new” magic numbers such as the one
putatively shown in Figure 3.1 (b). Evidence for the changing of the nucleon number’s
magicity can be seen in the one-neutron separation energies of isospin chains. The
neutron separation energies of ten total isospin chains of nuclei near the neutron
drip line are shown in Figure 3.2 as a function of neutron number to determine the
locations of possible magic numbers. The separation energies were obtained from the
2003 Atomic Mass Evaluation [54]. For the smallest isospins, the magic numbers 8
and 20 are suggested by the sudden drops in the one-neutron separation energies.
As the isospin increases, the disappearance of the magic numbers 8 and 20 and the

appearance of a new magic number N = 16 become evident.

5O 1

1 9/2 2 ,2
2 1/2 ‘2sz
2a figs—e 17“
173/2 16 ,2
—— 2s 23
mg,”2 id's/g

 

_ l

 

-G | |

(a) (b)

Figure 3.1: Schematic representation of the neutron single particle energy levels for
a) nuclei near fi-stability and b) a scenario for nuclei away from fi-stability.

Much attention has been focused in the region of the drip-line around the predicted
N = 20 shell closure because of the apparent disappearance of this magic number for
neutron-rich nuclei [56]. An N = 20 shell closure would mean that 280 should be

doubly magic (i.e. Z and N are both magic numbers) and should be particle bound.

32

 

 

 

 

 

 

 

 

12 14 .
1 '°
10 12 a “rz=o
’ 1

8 10 [T =1
% ’ E a 1%”
g 6 g 12:3
c ' c 6 =
(I) 4 (D 4 16 T2 4

2_ 2 fl '

°é”'1b‘“‘1‘5‘”2'd”‘2‘5”'ao °o

51015202530
N N

(a) odd N and even Z (b) odd N and odd Z

Figure 3.2: One-neutron separation energies from the 2003 Atomic Mass Evalua-
tion [54] for T; < 5.
Experiments performed at the NSCL [57,58] and other facilities [59, 60] failed to
observe 28O and not even 260 (only even-even oxygen nuclei are expected to be
bound near the drip-line due to the added stability from the pairing of like nucleons).
The pairing of like nucleons adds stability to a nucleus and thus causes the observed
odd-even staggering in the chart of nuclides [61]. Thus, 25370 are unlikely to be
particle bound and have yet to be observed [62,59]. The non-observation of these
nuclei suggest that 24O is located along the neutron drip-line of the oxygen isotopes,
see Figure 3.3.

The location of the neutron drip-line is of particular interest because it provides
a definitive test of the limit of the binding of neutrons to protons. In the late 19903,
Sakurai et al. [10] explored the production of nuclei in the region around the known
neutron drip-lines of Z 5 12, see Figure 3.4 and the 31F nucleus was observed for
the first time. The rapid change in the number of bound neutrons, N = 22 for 31F
compared to N = 16 for 24O, with the addition of one proton into the sd shell is
surprising and it may suggest the onset of deformation around the N = 20 magic
number for nuclei in the vicinity of the neutron drip-line [4, 10]. Utsuno et al. [63]

argued that if N = 20 were to remain a magic number, the unbound 26O and 280

33

 

 
 
 
 
  
  

 

 

 

 

 

m
C A=40
O
.- 20 l
5. — measured
- - - calculated A: 30
TUYY
a calc.uncerlaln
A: 20
10
stable
bound
unbound-measured
unbound—Ilmrl

 

 

 

 

I .
0 10 2030

40
neutrons
Figure 3.3: The chart of nuclides along the neutron-drip line for Z320. The figure is
taken from reference [3]. The solid black line is the experimentally determined loca-
tion of most neutron—rich nuclei of the isotopic chain. The dashed lines are possible
locations of drip-line nuclei for a fixed N based on the uncertainties in the 2003 atomic
mass evaluation (2003AME). The dashed boxes are regions where it is unclear whether »
nuclei may or may not be bounded based on the uncertainties in the 2003AME pre-
dictions. The gray boarders labeled TUYY for the authors are the predicted location
of the drip—line by Tachibana. et a1. [55]

isotopes would also require 29F to be particle unbound. 29F was first observed in
1989 at GANIL [4], thus N = 20 is unlikely to be a magic number far from stability.
Utsuno et al. suggest that an addition of a proton to the 240 core allows the unbound
neutrons in the 261280 1d3/2 orbital to be loosely bound in the fluorine isotopes and
the additional binding due to the two valence neutron holes in 27F is lost in 29F and
results in 29F being unbound. Only through the narrowing of the neutron effective
shell gap can 29F be bound. Models such as the finite range droplet model (FRDM) by
Méiller et a1. [64] and a Hartree—Fock model using a MSK7 interaction by Brown [65]
predict 31F to be the last particle bound fluorine isotope along the neutron drip-line,

see Table 3.1 and Figure 3.5. Both models predict that 32F to be unbound against

one- and two-neutron emissions and 33F to be unbound against two-neutron emission.

34

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

4 suqhb 27 FewJoN [51chsz 34,435 F7
N Na Na Na
3 4 6 7 a 1 2
N :NerHbNdh NJ NeFNa E45
20 21 22 23 27 29 31
F F F F 2F 25F 26F F F F
19 2o 21 22 23 24
O O O O O O
18 17 1a 19 2o 21 22 23
14 15 13 17 13 19 20 22
c c C C C c C c
12 13 14 15 17 19
B 8 8'8 8 a

 

 

 

 

 

 

 

 

 

 

Figure 3.4: The known neutron-rich nuclei from Be to Na are shown.

Table 3.1: Predicted one- and two-neutron separation energies (in MeV) of 31F, 32F,
and 33F.

 

FRDM :64] HF + MSk7 [65]
nuclei $171. 8an 8171 821,

31F 3.84 2.46 195(12) 0.78(12)
32F -1.76 2.08 -2.2(16) 0.01(23)
33F -057 -2.32 0.75(12) -1.52(20)

 

 

 

 

 

 

 

 

The establishment of the locations of the drip-lines is of such great interest that
in 2003, the Nuclear Science Advisory Committee (N SAC) released a report that sug-
gested as one of the milestones in the nuclear physics program should be to determine
the locations of the neutron drip-line up to the sodium isotopes (Z = 11) by 2007 [66].
Currently, the only method to produce such exotic nuclei and identify the exact loca-
tion of the drip-lines is projectile fragmentation. The location of the neutron drip-line
is only experimentally known up to the oxygen isotopes [3]. As part of the present
work, two experiments were performed at the NSCL to measure the production of
neutron-rich nuclei by projectile fragmentation from fl-stability to the neutron drip-
lines near the oxygen and fluorine isotopes and to try to determine the location of

the drip-line above oxygen. Experimental results will be presented in Chapter 4.

35

 

 

 

 

 

4L 4 l l l J 1 1

1'6 17 18 19 20 21 22 2 24

 

 

     

 

 

 

10
’ \
g 5_ \.
a L
g 0
(I) .
_5 1 l 1 l 1 1 I

 

116 1‘7 18 19 20 21 22 23 2
Neutron Number

Figure 3.5: The one and two neutron separation energies of the fluorine isotopes
near the neutron drip-line. The open squares and triangles are the predicted one
and two neutron separation energy predicted by M611er and Nix [64] and Brown [65],
respectively, and filled circles are data from reference [54]. Values below zero (dashed
horizontal line) indicate that the nucleus should be unbound.

3.2 Experimental Setup

Radioactive nuclei were produced at the NSCL from the fragmentation of 40Ar and
48Ca beams. In the first experiment, the 40Ar18+ beam was attenuated by 1 / 3 the
source intensity to about 3.5 electrical nanoAmps (enA)(~ 109 particles per second)
and accelerated to 140 MeV/ nucleon (fl ~ 0.5 c) using the coupled cyclotrons before
fragmenting in three different targets of Beryllium, Nickel, and Tantalum. The target
thicknesses were chosen to maximize the production of 31F and to yield similar energy
losses for the fragments of interest. This allowed the same fragments produced in each
target to be transmitted through the separator without changing the rigidity settings.
The effective thickness of the targets summarized in Table 3.2 were determined from
the initial and final energies of the 40Ar projectile based on the magnetic rigidity
used to center the beam at the midpoint of the fragment separator using the Image
2 viewer. A maximum % mm deviation of the beam’s centroid to the center of the

Image 2 viewer was estimated from photographs of the centered beam. This variation

36

Table 3.2: The effective thickness of the targets used in the present work.

 

 

 

 

 

 

 

 

beam manufacturer’s effective beam energy
target projectile energy thickness thickness1 at center1
(MeV/ nucleon) (mg/cmz) (mg/cmz) (MeV/ nucleon)
9Be 40Ar 141 658 668(39) 127.8(12)
"“‘Ni 40.41 141 767 775(42) 127.8(15)
181Ta 40A: 141 1041 1086(53) 127(2)
181111 4803 142 1181 1168(57) 127(3)

 

 

ICalculated using equation 4.3.

was included in the uncertainties listed in Table 3.2. Three target materials were
chosen to fragment the 40Ar projectile in order to investigate the influence of the
neutron-excess of the target on the N / Z ratio of fragments. In the high-energy limit
of the projectile fragmentation mechanism, the target N / Z ratio does not play any
role in the determination of the neutron excess of the products (see for example the
EPAX parameterization of the yields). However, at energies near the Coulomb bar-
rier, the target and projectile can rapidly equilibrate their N / Z ratio due to their long
interaction time. Large N / Z targets are thought to increase the yields of neutron-rich
fragments at intermediate energies due to the reabsorption of nucleons from the par-
ticipant zone or nucleon-nucleon exchange during the target-projectile collision [29]
and the yield is expected to only depend on the targets neutron excess and be inde-
pendent of the size of the target [67]. The effect of the neutron-excess of the target
on the production yields may be determined by studying the extent of products from
the interaction of the 40Ar beam with each of the three targets. Results from the first
experiment were used to select the reaction target for the higher intensity production
experiment. In the second experiment, a ~ 160 enA 48Ca19+ beam was accelerated
to 140 MeV/ nucleon and fragmented with a 181Ta target to produce neutron-rich
nuclei in the same region as the first experiment. Momentum distributions were again
measured and an attempt was made to determine the location of the neutron drip-line
above oxygen.

Nuclei produced in the four reactions were separated in-flight based on their mass-

37

to—charge ratio (A/ Q) using the A1900 fragment separator [68]. The A1900 is a third
generation separator consisting of fifty six superconducting multipole magnets (twenty
four quadrupoles, sixteen hexapoles and sixteen octupoles) and four dipoles magnets
that can be Operated at a maximum rigidity of six Tm [68]. The four dipole magnets
are used to select fragments produced in the target-projectile reaction based on the
ion’s magnetic rigidity (Bp) and the other the magnets were used to focus the beam
and correct for aberrations. In the present work, the A1900 was operated as a zero-
degree spectrometer. Ions with the correct momentum-to—charge ratio and angular
acceptance were transmitted ~ 35 meter downstream to an achromatic focal plane and
identified. The A1900 fragment separator was designed to have an angular acceptance
of 8 msr. The magnetic field strengths of the dipoles within the A1900 were set to the
LISE [8] predicted peaks of the momentum distributions of the fluorine isotopes in
the ranges from 22F to 31F (3.628 s Bp g 5.228 Tm) and 25F to 32F (4.042 _<_ Bp g
5.703 Tm) for the first and second experiments, respectively. Apertures located at
Image 2 and Image 3 limited the momentum acceptance of the separator to Ap/ p =
:l:0.5%, see Figure 3.6. These apertures were opened at the highest magnetic fields
to allow particles to be transmitted through the full acceptance Ap/p = i2.5% of
the separator in order to search for the production of the weakest channels: 31F and
32F. Particles were tracked and identified using various detectors located at the focal
plane. A schematic diagram of the focal plane detectors and their location are shown
in Figure 3.7. Five 5 x 5 cm2 Si PIN detectors in the dE telescope with thickness
of 980, 966, 1001, 988, and 992 pm, and a 10 cm thick plastic scintillator with an
active area of 15 x 10 cm2 were used to measure the energy loss and kinetic energy
in order to identify the transmitted particles. A pair of position sensitive parallel
plate avalanche counters (PPAC’s) with a 10 x 10 cm2 active area and a 28 mg/cm2
plastic BC—400 scintillator located at Image 2 were used to track the positions of the
ions. The positions measured by the PPAC’s located at the front and the back of

the focal plane box were previously calibrated with a mask. The two positions of the

38

fragments measured by the PPAC’s were used to determine the angular acceptance
at the focal plane. Shown in fFigure 3.8 is the horizontal and vertical spatial and
angular distributions of fragments produced from the fragmentation of 40Ar with 9Be
at the focal plane of the A1900, left and right panels, respectively. The transmitted
fragments in both experiments were found to be ~ i60 (1:15 mm) and ~ 3:40 mrads
(:l:15 mm) centered about 0° in the horizontal (9) and vertical (¢) planes, respectively.
Four inch thick copper slits located at the focal plane opened to :l:2.5 cm about the
central beam axis were used during the second experiment to prevent ions from going
around the Si dE telescope and triggering the larger focal plane scintillator. This
reduced the probability of particles from hitting the edges of the Si telescope where
partial charge collection can occur thus increasing the CPU’s dead time. No wedges
were used at Image 2 in order to avoid any complications that may occur during the
study of the projectile fragmentation process. Thus, the Image 2 detector was not
used in the first experiment at the 1% momentum acceptance. At full momentum

acceptance, the scintillator was inserted to measure the positions of the ions at Image

2. The insertion of the thin Image 2 scintillator resulted in a small wedge effect

8

      
  

8.5

mm

«1.:

.Eg

i=2"...-

W08

1. Hi

' I

I I
5").
75 o
385%
oE"°o
s. d)
QBQE

Focal Plane

Figure 3.6: Schematic diagram of the A1900 during the two experiments. Fragments
travel from the entrance of the spectrometer (left) to the focal plane (right). The
small rectangles indicate the positions of the 24 focusing quadrupoles.

39

 

 

 

 

 

10x100m210.2 x16.7cm2 5x5cm2 10x10cm2 15x10x10cm3

 

PPAC CU S'its AE detector PPAC Scintallator
1 1 1 1 1=

distance from
target (cm) 3479 3501 3506 3534 3552

Figure 3.7: The focal plane detectors used during the two experiments. The x—axis is
not drawn to scale.

(i.e. small shift in the rigidities of the fragments), but allowed the fragments to be
corrected for their different flight paths. The scintillator was left in during the second
experiment so that no additional changes to the system were necessary except for the

variation to the momentum acceptance.

3.3 Electronic System

The standard A1900 electronic setup was used during the two experiments, see Fig-
ure 3.9. Fast preamplifiers were used to match the impedance of the detectors to
the shaping amplifiers before being recorded for further processing by other elec-
tronic modules. Analog—to-digital converters (ADC’s) and charge—to—digital converters
(QDC’s) read the energy signals from the PPAC’s, Si detectors, and charge signals

from the two scintillators. The time-of—flight (TOF) of the particles through the full ~

40

2/m_l_

 

50. .

~30

 

  
   

 

 

_50 I u
—50 -30 ~10 10
mm
Y VS X

 

 

Figure 3.8: The fragment profiles in position and angular phase space at the focal
plane of the A1900, left and right panels, respectively.
35 m of the separator (distance from the target to focal plane) and the second half ~
17.5 m (Image 2 to the focal plane) were measured by four time-to-analog converters
(TAC’s). The 966 am Si PIN detector and the 10 cm plastic scintillator each provided
start signals for the two TAC’s, while the radio frequency (RF) of the cyclotrons and
the Image 2 plastic scintillator provided the stop signals for the TAC. In this way, four
redundant measurements of the TOP were recorded for each event. The path length
of through the entire separator resulted in a ~ 250 ns flight path for the transmitted
fragments. This longer path length increased the resolution in the time of flight spec—
tra allowing for mass separation at the largest momentum acceptances of the A1900,
but allowed the slowest and fastest fragments to overlap in time—of—flight relative to
the cyclotron rf. The RF freq of the coupled cyclotrons (23.1844 and 23.08667 MHz
for the 40Ar and 48Ca beams, respectively) produced a beam packet every 43 ms. The
analysis of the TOF will be discussed in section 3.4. The timing logic pulse from the
constant fraction discriminators (CFD’s) had a small walk at low pulse heights, visi—

ble in the TOP distribution of Figure 3.10. Fragments with the same charge-to-mass

41

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Master Gate _St_a__ TAC 3 r- ADC
Cydoéron OOMHz 8 Stop
R 01 . 17.
30 Stop
Master Gate ————1
TAC 4 _ ADC
o
a Fast CDF E Sealer
Am- —-—ooc
QDC gate
0066 000 gate
0
- Fast GDP 1.1.- —Seeler
FP N. Amp. _QDC “- TAC‘ 3““
—FP N. Scln. trigger g TAC 2 3m
nd DISC. l-l- .
2 31 TAC 3 Start
$199121, Fast
AE f Am .
Pm Sha r
Am - pe ”bSeeler
4&4!» a...
o 9006 000 Strobe
, ll.
W £1118 Shaper r TDC A --ITDC StartMmte
—94 ADC Bit t—LocGa Stop Common]
FP Pre cor: cc “‘7' ”Ga
E 1" Sha r 1S-caler
a... p.
Stan_ TAC 2 L.
AD
12 3. Fast 900 o 3.6,, C
3010- Amo. CDF I”: -—Scaler

 

 

 

 

QDGG

 

 

 

 

 

 

 

 

F0

 

 

12 N. * Fast CDF
‘ Soln. Amp._QDC

Sesler

 

 

 

 

 

 

 

 

Start TAC 1 —ADC

 

Figure 3.9: Schematic electronics diagram of the A1900 during the two experiments.
Acronyms: ADC - analog-to-digital converter, Amp—amplifier, CFD - constant frac-
tion discriminator, Coin. Reg. - coincidence register, disc - discriminator, FIFO - fan
in fan out, LFO - Logic fan out, TAC - time-to—analog converter, TDC - time to digital
converter, QDC - charge—to—digital converter, QDGG - quad delay gate generator

42

 

 

3041‘
2541‘
2041‘

1541"'

AE (arb units)

1041”

      

 

 

0. ‘
I .l O C . .
. C C
- I . ’ . .9 I Q. .
a I. o
. . Iz' . - -
.. I... . e an .5 P .‘ e
e ' . . .-

41481 681 881' 1081 1281

   

 

1481 1681 1881 2081 228
time of flight (arb units)

 

Figure 3.10: The particle identification of fragments the reaction of 40Ar with 93c
at a Bp setting of 3.6279 Tm. The walk introduced by the CFD is illustrated in the
curvature of the N/ Z = 5/ 2 line at small AE values.

ratio should have the same TOF through the separator. Thus, the N/ Z = 5/2 line
(dashed line) should not have any curvature in its time-of—flight. A quadratic function
in ln(dE/dx) (where dE/dx is the energy loss) was applied to the TOP in order to
correct for this effect, see Figure 3.11. This correction factor was found to hold for all
the rigidity settings used in the current work. Information from the ADC’s, QDC’s
and TAC’s was written to disk and stored on tape for later analysis. A veto signal
present when the CPU was busy limited the data acquisition (DAQ) system in the

number of collected events written to file.

3.4 Particle Identification

The silicon PIN detectors were used to measure the dE/dx of the fragments as they
traverse through each of the five detectors. The pulse heights from the ADC’s were

converted to an energy loss assuming a linear function. The expected energy losses

43

 

 

 

 

o _
E -20 [-
2
c -40 f
_2
9, -60 -
"5
<1 '80 P y = 1.02 ln(x)3- 28.0 ln(x)2
_100 _ '1' 242.0 ln(x) - 668.1

 

4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0
ln(AE) (channel #)

Figure 3.11: Correction to the walk in the tof spectrum introduced by the CFD. The
time difference of the N / Z = 5/ 2 line is relative to the centroid of 408.

of the fragments used in the calibration were determined using ATIMA 1.2 with LS-
theory [69, 70,71] in LISE. All of the observed nuclei penetrated into the 10 cm thick
focal plane (FP) plastic scintillator due to the large ranges of these light neutron-rich
fragments (31F has a range of ~ 21 mm in Si under these conditions). Silicon detectors
of approximately 1 mm were chosen because large surface area Si detectors are not
generally available with thickesses greater than ~1 1mm at present. By comparing the
dE/dx measured in a PIN detector with the total kinetic energy signal from the FP
scintillator, the charge of the fragments could be determined. Only one charge state
of the fragments was observed and this charge state was found to be fully stripped
based on the magnetic rigidities and the energy losses.

The proton number (Z) and the mass-to-charge ratio (A/ Q) of the ions were
calculated on an event-by-event basis using multiple TOF’s for the ~35 m (timing
difference calculated from the RF timing and the second Si detector) and the ~17.5 m
path lengths (timing difference measured from RF and FF scintillator), dE/dx’s, and
the magnetic rigidities. The particle rates were measured using the TOF’s from the
RF signals due to the better overall resolution from the longer path length and the

44

«44441
wbflswq'»

 

 

#NQ

Figure 3.12: The particle identification of neutron-rich nuclei that are transmitted to

the focal plane of the A1900 from the reaction of 480a + 181Ta with Bp3,4 = 4.4835
Tm and Ap/p = 10.5%.

45

resolvability of masses at full acceptance. At the lowest rigidities (~ 3.6 Tm), the time
difference between the fastest and slowest fragments produced by the fragmentation of
a single beam packet was larger than the flight time of the fastest fragments produced
by the fragmentation of the next beam packet. The additional TOF signals from the
Image 2 scintillator and the kinetic energy allowed the fastest and slowest fragments in
the RF spectra to be resolved. One structure in the particle id spectrum was obtained
from the RF by subtracting or adding the time of one RF cycle to the second structure.
Time was added or subtracted to the RF TOF depending on which side that the RF
signal was relative to the first PID structure in the Image 2 TOF vs RF TOF spectra.
The average atomic number and TOF from the last four Si PIN detectors were used
to construct a particle identification (PID) plot such as the one shown in Figure 3.12
in order to determine the fragment intensities. The N/ Z = 5/ 2 and 7/ 3 lines in the
PID plots were used to identify the isotopes in both experiments. The holes in the
PID plots due to the unbound nuclei of 8113Be provided additional conformation
of the particle’s identification. The first detector was used in the calculation of the
kinetic energies and to construct software gates to clean the PID but not used in the
identification due to the poor resolution of the detector caused by radiation damage.
The resolution of the Si detectors can be seen in the widths of the energy loss for the
5600 MeV 40Ar beam shown in Figure 3.13. The detector resolution summarized in
Table 3.3 were determined from the full width at half maximum and the centroid of
each Si detector. The resolution from an 8.78 MeV alpha produced from a 232U source
is also listed as a comparison. The resolution of the first detector is approximately
two times worse than that of the other detectors. A shim was placed under the Si
detectors to lower the detectors by 1 mm in an attempt to avoid the most damaged
area on the first detector.

The atomic numbers were obtained from the dE/dx’s measured by each of the Si

detectors using the Bethe-Bloch formula for emery loss in material by heavy ions.

46

 

 

 

 

 

 

 

    

 

 

 

 

 

 

  
  

 

 

 

100 _ C Detector #2 15 Detector #3
1 1500: .
E E 1000-
500- 10001 _
g 500E- 500;”
2 . : _
m “a . - ’- 1.... .44 1.111 1.1
‘9 275 325 375 150325 350 375 400 350 375 400 42
. D .
g 1 000: etector #4 _ Detector #5
8 I 1000}
500} 500:

 

     

 

 

 

375

400 ‘ 1425

 

 

450 400 425 A 450 475‘
Energy (MeV)

Figure 3.13: The emery deposited in the five Si detectors by a 5600 MeV (Bp =
3.93898 Tm) 40Ar beam. The spectra were fitted with a Gaussian (blue curve) to de—

termine the emery resolution displayed in Table 3.3. The thicknesses of the detectors
are given in Table 3.3.

Table 3.3: The measured emery resolution of the silicon stack detectors along with
the FWHM and the < AE>1 of the emery loss by 40Ar are given.

 

 

 

 

 

 

 

Detector thickness <AE> FWHM Resolution Resolution 1
no. (11m) (MeV) (MeV) ('70) (%)
1 980 337.63(0.05) 7.21(0.13) 5.10(0.04) 2.2
2 966 357.24(0.02) 7.78(0.05) 2.178(0.15) 1.3
3 1001 390.36(0.04) 10.55(0.10) 2.70(0.02) 1.3
4 988 409.24(0.04) 11.93(0.10) 2.92(0.03) 1.3
5 992 441.76(0.02) 9.69(0.06) 2.193(0.13) 1.3

 

 

 

1Resolution calculated from a 8.78 MeV alpha produced from the decay of a 232U

source.

The Bethe-Bloch formula can be written in terms of the proton number (Z) as:

 

 

Z = m\/dE/dx
[32

-(109(112)- 02) + b

(3.1)

where the ionization potential (I) of the stopping material (Si) is ~ 0.17 MeV. The

slope m and the offset b are calibration constants determined for each detector. Nu-

47

clear magnetic resonance (NMR) probes were used to monitor the magnetic field
strength (B) in each of the four dipoles and to determine the magnetic rigidity of the
transmitted fragments. From the Lorentz force equation, it can be shown that A/ Q
is related to the magnetic field strength B (in Telsa) and the particles momentum P

(in MeV/c) via the equation:

A P 68p

5 = 1165162 = 11001 (3'2)

 

where p is the radius of curvature of the fragments given in meters, e is the charge
of an electron, c is the speed of light, 11 is the atomic mass unit, 0 is the relativistic
velocity, and 7 is the Lorentz factor. The dispersion of the ions due to the variation in
momenta causes the fragments to follow different flight paths through the separator.
The separator is designed such that a dispersion (Ax) at Image 2 is related to the

spread in the magnetic rigidities (ABp) via the relation

11:: A2: Bp— mm Bp

ABp _ WABp' (33)

Using this relationship, the true fragment rigidities were determined. The horizontal
position x of each event was determined using the timing difference between the
photomultiplier tubes of the Image 2 scintillator and hence the true rigidity is related

to the central rigidity is the along the beam axis Bpo by the following equation:

Bp = Bpo(1 — Sg—fnfi). (3.4)
‘7‘

0

Using the corrected rigidities in the mass-to—charge ratio allowed for separation be-
tween the adjacemt masses at the full momentum acceptance of the separator and
provided a particle resolution of A/ Q of ~ 0.5% and Z of ~ 1%. For example, Fig-
ure 3.14 shows the resolution of beryllium isotopes produced in the reaction of 40Ar

with 9Be.

48

 

450
400
350
300
8 250
2 200
a
8 150

100

 

 

50

 

 

 

2.0 2.1 2.2 2.3 2.4 2.5 2.6
Mass to Charge

Figure 3.14: The mass resolution of Beryllium isotopes produce in the reaction of
40Ar witthe using a rigidity setting of 3.6279Tm.

Software gates were implemented to insure that each event passed through the
Si telescope by requiring the fragments had valid angular and spacial distributions
that rejected ions with partially deposited charge. The events that fulfilled these
requirements were used to determine the fragment yields at each rigidity setting. The
yield of the observed fragment (Y F) in the spectrometer can be approximated using

the standard formula for thin targets:

(3.5)

 

Y _ N f a.)
F ‘ teTIbApeppac aoxaay'

where Ib is the primary beam intensity, N / t is the number of particles per unit time, or
is the fractional live time of the CPU and Ap is the momentum spread. The angular
acceptances of the fragments (f Kai—”5715) were determined to first order using the
convolution of a Gaussian and exponential for the parallel-momentum distributions
in the simulation program LISE, the solid curves in panels (a) (”Ar + 9Be), (b) (40Ar
+ "atNi), (0) (”Ar + 181Ta), and (d) (48Ca + 181Ta) in Figure 3.15. The angular
transmission was found to depend mostly on the mass number of the fragment and

not the fragment charge. The fragment’s angular transmission of the fragments has

49

been fitted with a fourth order polynomial of the form y = P0 +P1x + P2X2 + P3x3
+ P4X4 (solid curve). The coefficients from the fits are summarized in Table 3.4.
Simulations using a Monte Carlo code Mocadi [72] were also done in order to
include higher order effects to the angular acceptance of the separator. Mocadi is a
program that was deve10ped to simulate the transport of heavy ions through mat-
ter in optical systems and it calculates the trajectories of ions through an optical
system using third order optics. Fragments were assumed to be produced from the
fragmentation of an elliptical 40Ar beam that is confined to x2 + y2 50.01 in x-y
coordinate space and a2 + b2 g 25 in a—b angular phase space, where x and y are
in cm and a and b are in mrads. The angular transmission of fragments was deter-
mined from the number of events within a momentum acceptance of 0.02% from the
central rigidity that reached the focal plane relative to the number of events at the
target position. Figure 3.16 shows a comparison of the predicted angular acceptance
for fragments produced in the reaction of 40Ar with (a)9Be, (b)”atNi, and (c)181Ta
from LISE (filled squares) and Mocadi (first order calculations are the open circles
and third order simulations are the filled triangles). The error bars shown in the figure
are statistical. The first order simulations from Mocadi using a 9Be reaction target
tend to be smaller (up to ~ 65(29)% less for the lightest fragments) than the angu-
lar acceptance from LISE. Third order corrections to the optical transmission were
also calculated in Mocadi for a more realistic simulation of the transmission. Large
deviations in the angular transmission appears for the heaviest fragments (A p > 25)
produced in the reaction of 40Ar with 181Ta. The transmission of these fragments is
~ 40% regardless of the mass. This behavior has not been seen in simulations using
a thinner 181Ta target. The angular distribution of the fragments at the focal plane
from Mocadi using the thickness of the 181Ta target in this work (A03; ~ 135 mrad
and A01, ~ 21:40 mrad) were found to be smaller than the measured distributions. The
same angular distribution was found at the focal plane regardless of the initial beam

profile. The angular transmission from LISE was used to correct the differential cross

50

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.0 1.0
(a) [11»
0.8 0.8;
1-
0.6 06‘;
A04 _ 0.4;-
a: .
5 0-2 0.2:-
.6 .namaIJJJLLLanal-analsaaalsasslannnlnnu
"E 0"’0 5 10 15 20 25 30 35 4o 0"’0 5 10 15 20 25 30 35 40
m 1.0 1.0
c 1 1
e :(C) :(d)
0.6} 0-63
0-4} 0.4}
0-29 0.2}
[L I....1.‘J‘lllllllLllllllklll..|l.., :nanl‘nnnlnnlAlanaAIAAAAlnlnAlAA1A1L11‘.AIAAIAA
0.00 5 10 15 20 25 30 35 40 0""0 5 10 15 20 25 30 35 40 45

A

Figure 3.15: The angular acceptance through the A1900 for fragments produced in
40Ar with (a)9Be, (b) "atN i, and (0) 181Ta and (d) 48Ca with 181Ta. The acceptances
are calculated to first order using the simulation program LISE.

Table 3.4: The coefficients for the angular transmission determined from the fits to
Figure 3.15.

 

 

 

 

 

 

 

 

reaction P0 P1 P2 P3 P4
40Ar+9Be 3.1512 4.4512 3.4513 9.5515 8.5E—7
40Ar+thi 2.6E—2 1.1512 1.6E-3 5.5515 5.6E—7
40Ar+ 181Ta 5.2E—3 1.1512 1.9513 4.45.5 8.5E—7
48Ca+181Ta 6.3E—3 1.1512 1.3513 4.0515 3.7517

 

 

sections due the inability of the Mocadi simulations to correctly predict the angular
phase space at the focal plane. Corrections to the optical matrices are currently being
done to include higher order effects and to correct for optical abrasions.

The particle rates (N/ t) were determined from the number of events (N) in the
Z vs. A/ Q plots. Losses to the particle yields due to PPAC efficiencies were also
included into Equation 3.5 since the PPAC’s were the only detectors that did not

have a 100% efficiency. Their efliciency was calibrated by determining the number of

51

 

 

 

 

 

 

 

 

 

 

 

 

.1
091(3) Jig-(b) "[11]
' 1 iii! * 1"!' 111*
l 1 III ' Iii
1 I l ' I 111
A06_ ;1[[ 06- .431 1111
2.3 L .1 ' 1,1 ' " [l][[i
503 1311* 1 ' ." ifjii1
3,) ' .g' .51 0.3- .-' [1[1Hii
'5' - ii "11:1 ‘ 1" 11 1
m '.g!. 5 _ A E 5 "'
c 00 115‘ f
2 1.1.1. .ot.‘1.1.1.1.1.
I: 10 20 30 40 10 15 20 25 30 35 40
E
309-(c) llll'l
1: II3
< I ll .
I"'
0.6” "|!:. I
F I. I 1 [Ii 1
.n' [ lliéiiflixif égflég
03-_,-' 1,1 1 it 1 ..
1' I] 111% f
1* ‘
0 1 . 1 . 1 . 1 . 1 . 1 .
10 15 20 25 30 35 40
Mass

Figure 3.16: Mocadi simulations for the angular transmission of the A1900 separator

for the reaction of 40Ar with (a)9Be, (b)"atNi, and (c)181Ta. The angular transmission
from Mocadi (the open circles are simulations using first order calculations and the
filled triangles are simulations using third order corrections) and LISE (filled squares)
are shown as a function of mass number.

counts detected in the Si detectors with and without using software gates from the
PPAC’s. The efficiency (appac) was found to depend on the emery and charge of the
ions. In the case of the nitrogen isotopes, the efficiency fell to about 55%. Figure 3.17
shows the efficiency of a PPAC as a function of the emery loss as determined by the

Born-Bethe equation:

dE cAZ2

7% = TKE (3'6)

 

where c is a constant and TKE is the total kinetic emery of the AZ ion. The open
squares and triangles are the fragment efficiencies at the lowest and highest primary

beam intensities (Ib’s), respectively, and the open stars are the efficiencies at beam

52

intensity of ~ 1.7 enA. No rate dependence on the efficiency was found. The data was

fitted with a Hill function of the form:

1.,(Az2 /TKE)"

= 3.7
5“” AC k"+(AZ2/TKE)" ( )

 

where k, n, and 60 are constants determined from the fit.

A BaF2 detector located the corner of the target box that was sensitive to various
charged particles produced in the target was used to monitor the relative primary
beam intensity (lb) as a function of time. The rates measured by the BaF2 for each
projectile-target combination were calibrated against the ion current measured by
the faraday cup closest to the target box, see Figure 3.18. Attenuators consisting
of four fine wire meshes located in the injection beam line before the cyclotrons
with attenuation factors of 3, 10, 100, and 1000 were used in combination to limit
the rates on the dE telescope to ~ 1500 particles/second. The linear calibrations
in Figure 3.18 are only valid at the highest beam intensities. At the lowest beam

intensities (i.e. the attenuation factor > 30000), the rates measured by the BaF2

 

39311151102501
0 Ib~65 epA

a lb~1.7enA
A Ib~57 enA

I

 

I I IIUFIII

 

 

2 °-‘ : i
n. _. ,
.3 . t
: fife
1- E, §
0.01 a 4’
A
E / y = 0.967*x3°77/(0.3833-"+x3-")
EL/fi 1 l l l l l l I I 1 L

 

1

.°
—\

AZZITKE

Figure 3.17: Efliciency of the focal plane PPAC’s for three beam intensities. The open
squares are at the lowest intensity, the open triangles are at the highest intensity, and
the open stars at an intensity midway between the highest and lowest intensities.

53

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.5 [_40AH-980 2.0 [MAN-"31M
E 1.5%
1.0 :- :
; 1.0 :-
0'5 T 0.53—
: 1= 1.507‘10‘3x+0.0187 I l=2.154*10'3x+0.0435
A 0.0 . . J 1 0.0 . . .
g 0 500 1000 0 500 1000
0)
_ 1.5 [MON-Flaws 8 F43Ca+181Ta
1.0 :- 6T
r 4
t 1
0.5 1_- . 2 P .
0 0 : ll=1.58§*10'3x‘+.0368 : 0 ' ‘ [4:500:11311 L09” .
' 0 500 100 0 500 1000 1500 2000

Banlsec

Figure 3.18: Calibration of the primary beam intensities for the four experiments used
in the present work.
detector were approximately equivalent to the rates due to background radiation (~
2 x 10‘8 pps), thus the linear calibration shown in Figure 3.18 was not valid. The
beam currents at these lowest intensities were approxirmated by dividing the rates
from the BaF2 detector (i.e. back ground current) by a constant that is dependent on
the attenuator used in the beam line. The values of the constants were determined
by the effect the attenuators should have on the beam current. For example, it was
found that changing from a 30k attenuator to a 10k attenuator increased the beam
current by 3.5 times, thus the beam current from the cyclotron using a 30k attenuator
was scaled down by a factor of 3.5 relative to the beam current from the cyclotron
using a 10k attenuator, see Table 3.5. These corrections were only necessary in the
first experiment where the 30k, 100k, and 300k attenuators were used.

The momentum distributions of the transmitted fragments were determined from
the measurement as a function of Bp. Figure 3.19 shows a typical momentum distri-

bution for the fragments observed in this work. The uncertainties in the differential

54

Table 3.5: Corrections to beam current for low intensities. The beam intensities listed
below are relative to the beam current from the cyclotron using a 10k attenuator.

target 30k 100k 300k

9Be 3.3(1.3) 8.2(1.1) 30.4(1.7)
"“‘Ni 3.2(0.2) 8.9(0.2) -
181Ta 3.5(0.2) 12.4(3.1) 43.6(2.0)

 

 

 

 

 

 

 

 

momentum distributions include statistical uncertainties as well as systematic errors
due to the target thickness, primary beam intensities and fragment transmission. An
asymmetric Gaussian (solid curve) was found to describe the distributions well. The
parameters from the fit were used to extract information about the reaction mecha-
nism. The fit and parameters will be described in the next chapter.

The isotopic yields of fragments produced by projectile fragmentation were de-
termined using Equation 3.5. The isotOpic yields of the fluorine isotopes produced
from the fragmentation of a 40Ar projectile used in first experiment in 9Be (black
triangles), ”atNi (blue squares), and 181Ta (red circles) are shown in Figure 3.20.
The fragment yields also include the angular acceptance of the A1900. The points
are connected by a solid line to guide the eye. The fluorine fragments were found
to have a Gaussian momentum distribution with an asymmetric tail on the low mo—

mentum side of the distribution. The momentum distributions of the curren work

 

  

 

 

 

 

I y = 3.91104 Epox - 9.87)2I(.642{1 - 0.30(x - 9.67m]
_ 8: {o x z 9.67
A 1E-3 ;- 1 x < 9.87
'1': E
\ L
3 .
9, 1
.0
§1E-4 5'
a, t
2 I
b
1, .
“Ar + “Bo -‘°N
1E—5 1 f 1 1 1 4 I 1 1 1 A
9 10 11
P (GeV/c)

Figure 3.19: The yield of 20N produced from the reaction of 127 MeV/ nucleon 40Ar
with 9Be. The solid curve is an asymmetric Gaussian fitted to the data.

55

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   

----A---- 939 target

yields/Ap (ppslpnA/(MeV/c))

an natNi target
9.115290110191113

 

+ 989 target
+ "atNi target
P (GeV/c) + 181Ta target

 

Figure 3.20: A comparison of the momentum distribution of the fluorine isotopes from
the reaction of 40Ar with 9Be (triangles), "atN i (squares), and 181Ta (circles). The

open and filled symbols are the momentum distributions prediced by LISE++ using
a convolution momentum distribution and the current work, respectively . The lost
of particles due to the angular acceptance through the separator is also included.

(filled symbols) are compared with simulations made in LISE++ version 7.4.75 [8]
using a convolution model which assumes a Gaussian momentum distribution with
an exponential tail on the low momentum side [73] (open symbols). The curves are
colored by the reaction target and have the same color code as the experimental
data points. The approximate shapes and centroids are reproduced by LISE near
the peaks of the distributions, but large deviations appear on the high momentum
side. The shapes and positions of the momentum distributions provide important

clues into the mechanism that creates these fragments. Various fitting techniques

that range from a double Gaussian [74,75] to a variable cutoff percentages on the low

56

momentum side [76] have been used in the intermediate energy regime in attempts
to extract information about the reaction mechanism. For the purpose of this work,

the momentum distributions were fitted with an asymmetric Gaussian of the form:

 

—(x — 30)?
y = yoEzp (3-8)
2 60. 220—23
20 (1 + 20
where the asymmetry factor is defined as:
0 x > centroid
6a 2 _ x0 ( ) (3.9)

a x<xo

and a is the width of the Gaussian on the high momentum side. An example of
equation 3.8 fitted to data is shown in Figure 3.19. The creation of fragments at
different locations within the target results in different energy losses of the projectile
and fragment that broadens the observed momentum widths. The effects of this
energy straggling must be separated from the measured widths in order to gain an

understanding in the reaction mechanism and will be discussed in the next section.

57

Chapter 4

Discussion and Experimental

Results

4.1 Energy Loss

To produce lighter nuclei than the projectile, the projectile must first interact with
a nucleus in the target with some thickness t. The type of interaction the charged
particle undergoes will depend on the impact parameter (b) [77]. If the impact
parameter is small enough (i.e. b g R p + Rt), the projectile will be fragmented into
, smaller nuclei. These nuclei may also interact with the target to produce other nuclei.
The probability of fragments to undergo a secondary interaction can be determined

from Beer’s Law, see Equation 1.1. Using the geometric cross sections

the probability for fragments to interact somewhere in the second half of the target
in the four reactions measured in this work were estimated, see Figure 4.1. At most,
4% of the initial fragments, in the case of 40Ar + 9Be, are lost due to multiple

interactions. The cross sections listed in the Appendice C and D were adjusted to

58

 

 

 

 

 

 

 

 

 

 

  
   

0.7 .
0.6 .
0.5 ’
0.4

410) 1.2
3.5
3 l
2-5‘ 0.8:
F. 2 ’
33,15] 0.!»
5 l - - . n - - c 0.4.
E 510152025303540 51‘01‘52‘02‘53‘05540
g 0.9 (c)
E 0.8]

 

 

1‘0 2o 3o 30
A

Figure 4.1: The percentage of nuclei that interact twice with the (a)9Be, (b)”atN i and
(c)181Ta targets are shown as a function of' fragment mass number.

 

include this effect.

Electromagnetic interactions of heavy—ions with material will result in the lost in
kinetic energy by bremsstrhlung radiation, excitation, or by ionization [78]. The ac-
celeration of the ions (~ Zthez/Mp) due to the Coulomb repulsion with the absorber
nuclei will cause the ions to radiate energy by electromagnetic radiation. These in-
elastic collisions are known as bremsstrahlung [79]. This process is nearly negligible
for heavy ions due to the M52 dependence of the energy lost.

If the ion passes close enough to an absorber nucleus, it can interact with highest
lying orbital electrons in the absorber nucleus and excite the electrons to higher lying
shells. This process is known as excitation. If the energy transferred to the orbiting
electrons is larger, electrons attached to the absorbing nucleus can break free and
the atom will be ionized. The amount of energy lost by the projectile due to the
ionization of the target is dependent on the charge and bombarding energies. In

addition to losing kinetic energy, these interactions will deflect the Zp projectile by

59

an angle dependant on its impact parameter:

Zthez
/ z z 2
b41rco 1 + (fifg-P

where Ea is defined as the permittivity constant and Zte is the charge of a target

0 = 1r — 2Arccos (4.2)

nucleus. The interactions of electrons with matter due to ionization have been
studied since the early 1930’s [80,81] and has since been extended to charged heavy-v
ions in matter. The energy straggling due to the ionization matter has implemented
in many codes such as ATIMA [69,70] and SRIM [82] to predict the characteristics
of the interaction such as the energy straggling and stopping-rangas. The average
energy loss of ions for the purpose of this work was calculated based on the work of
Hubert et al. [83].

Dufour et al. suggested that the convolution of the reaction mechanism and the
AB in the momentum widths can be unfolded by calculating one-half the difference
in the energy between those fragments. formed at the front and those formed at the
back of target [84]. Hubert et al. showed that the energy straggling of an AZ ion
with energy per nucleon E can be determined from its range in matter (R) and can

be parameterized by the function:
A

The constants k, C and 7 depend on the stopping material. The values for these
coefficients for the three target materials used in the present work were determined
from fits to the ranges of 19F in reference [83]. The values of the coefficients found
are summarized in Table 4.1. Horn Equation 4.3, it can be shown that the energy of
a fragment formed at a depth 6 can be expressed as:

6 t—d

E(6) = Ep(6)(1 - R(AP’ 21,, EP) - CAP " R(Af, Zf,Ep) — 04!),

(4.4)

 

60

Table 4.1: The coefficients of the range for 9Be, WNi, and 181Ta obtained from fits
to the ranges of 19F. The data is from reference. [83].

 

target k 7 c

9Be 5.4(0.2) 1.640(0.007) 330.1)
"mm 7.4(o.3) 1.611(0.007) -3.4(1.1)
181'ra 11.0(0.4) 1.595(0.006) -3.6(1.4)

 

 

 

 

 

 

 

and the energy difference of the fragments with initial energy E f can be written as

iii-”’1:

d9:

2 2
flfi

1
7-1'
k'yEf

 

(4.5)

 

 

The differential energy loss of ions results in a broading of the momentum widths
and a shift to the centroid. Dufour et al. [84] suggested that formation of fragments
at different locations within the target will increase the parallel momentum widths.
The contributions to the predicted width of 220 (solid blue curve) due to the energy
straggling of the fragment produced at the front (dashed black curve), middle (dotted
red curve), and back (dotted dashed green curve) of a 668 mg/cm2 9Be target on the
parallel momentum width can be seen in Figure 4.2. For an infinitely thin target, the
variance, 0N: in the momentum distribution is a direct consequence of the statistical
nature of the fragmentation process, but the measured widths, Utotal(E), in terms of

energy will be altered by an amount:
”Emma” = 031E + 012V(E), (4.6)

where the first term is broading due to the energ straggling of the ions. For thin
targets, ad E is approximately zero, but for thick targets, the different energy loss of
fragments formed at the front and back of the target will alter the parallel momentum

widths by an amount

22 22
odE’ = Ctarg (1T: - 2%) (4.7)

61

 

 

dc/dp (mb/(MeV/c))

 

 

40Al‘ “1336022
-10 4 4 . 1 . g m 1 .
10 7000 8000 9000 10000 11000 12000
P"(MeV/c)

Figure 4.2: The LISE predicted parallel momentum widths for 220 produced in the
fragmentation of 40Ar with 9Be at the front (dashed black curve), middle (dotted red
curve) and the back (dotted-dashed green curve) of the target. The solid black curve
is the total predicted width.

where Ctarg is a constant that is dependent on the target material and thickness, and
Zp,f and Ap,f are the proton number and atomic mass number of the projectile and
fragment, respectively. The constant for the four reaction targets used in the current

2 22
work were determined from linear fits of the LISE predicted “dB/(fig — 2%) vs the

average fragment energy, see Figure 4.3. The values of the constants are summarized

 

 

in Table 4.2.
Table 4.2: The fitted target constants from Figure 4.3.
40Ar+9Be 40Ar+natNi 40Ar+181Ta 4801a+181v1~a
Ctarg 1.245(6) 1.209(5) 1.263(5) 1.415(6)

 

 

 

 

 

 

 

The broading due to the fragments energy straggling in the target can have a
large effect on the measured momentum widths. The contributions due to the energy
straggling from each term can been seen in the example in Table 4.3. The energy
straggling of the fragment increased the parallel momentum widths by 143% in the
case of 220 produced by the fragmentation of 40Ar with a 668mg/cm2 9Be target.

Equation 4.4 with 6 = t / 2 was used to determine the contribution to the momentum

62

 

 

 

 

 

 

 

 

 

 

2.5 2.5
2.01 (a) 2,0; M
1.5:- !“ Q Q; i i 1.51. . H § l
E 10W'h’w— 1OWH§ ___..
r. ' : - 1
g 0.5 0.5]
{ttmiaoa‘éoba‘iota‘smg‘ 1%002000300040005000
N 2.5 2.:
if. “f “’l 2.0] (d)
N .
N 1.5L H i f if
E . ,- “0.....“IQfiWQIi 1'5;_—J'1r.£“fi" x". 1‘
'0 1.0” . 1.0_
f6 051 0.5»
.0 L‘ t . x4 1 14 .

 

 

 

 

 

 

x4.‘.14..1.xa o. 1A,...ALJ...
1000 2000 3000 4000 5000 10002000300040005000
Ecentmid(Mev)

2 22
Figure 4.3: The predicted widths due to energy straggling, adE: per %% — If for

fragments produced in the fragmentation of 40Ar with (a)9Be, (b) ”atNi, and 181Ta,
and (d) 48 Ca with 181Ta are plotted as a function of the fragments average energy at
the back of the target. The horizontal dotted lines are the fitted values for the target

constant.

shift due to energy lossas in the target. The corrected momentum widths and cen-
troids were analyzed and compared with theoretical predictions in an attempt to give

insight into the reaction mechanism and non-observables.

4.2 Parallel Momentum Widths

The statistical process that creatas the fragments in the fast disintegration process of
heavy ions was shown to have a Gaussian momentum distribution [85]. The paral-
lel momentum width (0“) of this distribution can give information on how nucleons
are removed from the nucleus during the target-projectile interaction. In the early
1970’s, Feshbach and Huang attempted to understand the widths of these momentum
distributions produced from the sudden emission of nucleons from a nucleus in terms

of an “incoherent droplet model” [16]. Nucleons in this model were treated as Fermi

63

Table 4.3: Contributions to the momentum widths of 220 produced in the reaction
of 40Ar with 9Be due to the energy straggling.

 

 

Lab flame Projectile E‘ame
Ep (MeV/ nucleon) 141 0
0 total [P 1 (MeV/ 0) 428(9) '
a total [El (MeV) 203 (7) ‘
adE(MeV)1 l44.2(0.6) -
aN[E] (MeV) 142(10) -
aN[P] (MeV/c) 311(19) 276(23)

 

 

 

 

 

1The reduced width calculated from Equation 4.7.

gases with spherically symmetric momentum distributions. They suggested that the
momentum width was constant (0 ~ mwc) regardless of the fragment mass. Later ex-
perimental evidence indicated that the momentum width doas vary with the fragment
mass. Goldhaber showed that the conservation of momentum in the rest frame of
the projectile for the statistical process that leads to the emission of a random cluster
of AA nucleons from an AZ projectile would naturally lead to a parallel momentum

width of
2 _ AA(A — A1003

0“ A _ 1 (4.8)

 

where the reduced width (00) is related to the Fermi momentum (P F) via the rela-
tionship [86]:
0% = P—F- (4.9)

Using the Fermi momentum of ~ 260 - 265 MeV/c from nuclei with nuclear density
of p ~ 0.17 fm—3 from quasielastic electron scattering data, a reduced width of 00 ~
117 MeV/c is found [87]. Experimentally, 0'0 was found significantly smaller (0'0 ~ 90
MeV/c) than the reduced width predicted using the Fermi momentum for relativistic
bombarding energies. Goldhaber suggested that the difference may be the result of
lower P F at the nuclear surface for very light or very fragments and that medium mass
fragments are likely to have larger than average nucleon momenta. Other authors at-

tribute the lower 00 value to nuclear and quantum effects such Coulomb repulsion and

64

Pauli exclusion that are not taken into account in Goldhaber’s simple model [88,89].
It has also been suggested that value of the reduced width predicted using the Fermi
momentum should be attributed the prefragments distribution [37]. The reduced
width has been measured for a variety of reactions at different bombarding energies.
Figure 4.4 shows a log-log plot of selected values of 00 found in literature (unfilled
symbols) [90] - [91] for a wide range of projectile energies. These widths appear to
depend strongly on the bombarding energy of the projectile. At low energies (E/ A <
30 MeV/ nucleon), the values of the reduce widths decrease rapidly. At these energies,
the fragmentation process is no longer the dominate method in producing the nuclei
and the transfer process that creates these nuclei can result in the smaller value for
the reduced width. At higher energies (E/ A > 85 MeV/ nucleon), the experimental
values of the reduced widths appear to be approximately constant. This variation of

the reduced width with the bombarding energies is not consistent with Goldhaber’s

 

 

 

 

140

130 - B

120 -

110 - ]

133E % 4 0

n

:8 80:- A 00 Q 9
E 70;- o
a 60E-
o I
o 503 V

40 E V

30 P'I'V'IV 1 4 1:11:11 1 n llllLll 1

10 100 1000
EIA (MeV/nucleon)

Figure 4.4: The reduced widths of selected reactions found in literature is plot-
ted a function of the bombarding energy. The reduced widths are measured from
the following reactions: 197Au(9Be,7Li) [90](o), 197Au(QBe,160) [92](A),160(9Be,X),
160(CH2,X). 16002090, l60(27AI.X), 160(208Pb,X), l2C(913e.X). ”C(CHzX),
120(120,X), 120(27A1,X), and 120(208Pb,X) [931(0), 40Ar(68Zn,X) [941(u),
22Ne(93Nb,X) [951(21), 12C(9Be,X) and 130(9Be,X) [96](O),84Kr(197Au,X) [971(0),
120(120,7Be) [98] (<1), 86K187 A1,X) [99](>), 208Pb(160,120) [91](v), present work (I).

65

model. It has been suggested that decreasing values of the observed width at low
energies may be the result of distortions due to the Coulomb force that alter the
fragment velocities [88]. At high energiae, the Coulomb distortions become insignifi-
cant and thus should not affect the reduced widths. The value of the reduced width
may be altered if the nucleons were removed over the total interaction time instead
of simultaneously. Morrissey suggrsted that the sequential evaporation of A p - A F

nucleons would lead to a width equal to [100]:

0'" ZUON/AP—AF (4.10)

where do is taken to be equal to 85 MeV/c.

The momentum widths in the projectile frame for the high momentum side of
the Gaussian distribution for fragments produced in the reactions of 40Ar in a)9Be,
“atNi, and 181Ta and b) 48Ca in 181Ta are shown in Figure 4.5 and are listed in
Table 4.4. The widths were corrected for energy straggling using Equations 4.6
' and refenergylosswidth. The predicted widths from the sequential evaporation of
nucleons assuming a reduced width of 85 MeV/c are plotted using dmhed curves. The
behavior of the momentum widths for large mass losses from the current experiment
(filled symbols) is not reproduced by the sequential evaporation of nucleons model,
but the momentum widths in all four cases are more consistent with the parabolic
distribution predicted from the simultaneous emission nucleons and the simulated
widths from ISABEL+GEMINI (open circles). The parallel momenta were fitted
using Equation 4.8 (solid curves) to determine the value of the reduced width. The
dotted curvae represent the standard deviation values of the reduced width. The
predicted widths from ISABEL+GEMINI agree with the data within the uncertainties
of the parallel momentum widths from the current work. The values of the reduced
widths from the current work are listed in Table 4.4 and were found to lie in a

range between 70 - 105 MeV/ nucleon. These values are independent of the target and

66

 

 

l" “.1

O
- 1. ‘k 1’ ~\
v' f Y . -q

.3? n; '3} “if; 8“.

 

 

 

 

 

 

 

 

 

 

 

0 O
05101520253035400510152025303540

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0
0 510152025303540 0 51015202530354045
MassLoss

Figure 4.5: The parallel momentum widths from the reaction of 40Ar in (a)9Be,
(b)”“tNi, and (0)181Ta and ((1) 48Ca in 181Ta. The dashed curves are the predicted
momentum widths from a sequential emission of nucleons model using 00 = 85 MeV/c
and the solid and the dotted curves are the fitted widths 10, respectively, assuming
a single emission of particles. The unfilled circles are the simulated widths from IS-
ABEL+GEMINI.

projectile and are consistent with the literature values found for high bombarding

energies, see the filled squares in Figure 4.4.

4.3 Parallel Momentum Transfer

The collision between the projectile and target nuclei results in a transfer of momen-
tum to the nucleons. If the nucleons gain enough energy, they may break free of their
potential well and escape, otherwise, the nucleons will remain bound to the nucleus
dissipating their energl through collisions with other nucleons [38]. This interaction

will alter the mean nuclear velocities of the prefragments. The changes in the ve-

67

Table 4.4: The reduced widths deduced from the present work.

 

 

 

 

 

00(MeV/c) 00(MeV/c) 00(MeV/c) 00(MeV/c)
(40Ar+9Be) (40Ar+"“tNi) (40Ar+181Ta) (48Ca+181Ta)
sequential 85 85 85 85
emission
simultaneous 86(15) 85(20) 86(16) 87(13)
emission
P F (MeV/c) 193(34) 190(44) 192(36) 194(30)

 

 

 

 

 

 

locities that may arise from the breaking of the nuclear bonds to remove nucleons
during the fragmentation process has been attributed to a friction phenomenon [101].
N ucleons receive a ”kick” or transfer of momentum q in the projectile rest frame dur-
ing the projectile—target interaction. These nucleons may aecape from the potential
well of the nucleus with momentum q’. Abul-magd et al. [101] suggested that the
momentum gained in the transverse direction by a fragment is proportional to the
momentum transferred. On average, there is no net contribution to the perpendicular
component of the observed momentum distribution due to the equal probability of
the projectile to interact on either side of the target nucleus (i.e. -(R1 + R2) < b <

R1+R2). The parallel momentum transfer has been described by the equation:

< p” >= $311 +k<1 401/21. (4.11)

where AE is the energy transferred to the prefragment and the constant k (=6m/mp)
is the rate at which the <p“ > changes with the projectile velocity (fi) [102], see
Table 4.5. The value of k varies for the different models. For proton-induced reactions,
a value of k equals one is predicted from a collective tube model (CTM), while a k value

of zero is obtained from a single fast neutron model (SFNM). It was also shown that

Table 4.5: A summary of the valuae for the parameters in Equation 4.11.

 

 

 

 

 

 

theory k AE/c mp

(MeV / c / nucleon)
SN FM 0 13 all nucleons
CTM 1 8 interacting nucleons

 

 

68

 

0.3
0.2

I I T I 1 j

0.1

 

 

0.0

 

 

Mean velocity / (cm/ns)

 

I
l’m l i 1 . l

0 50 100 150 200 250
Mass number

 

Figure 4.6: The velocity distribution of fragments from the fission-fragmentation

reaction of 238v with Pb (black squares) and 238v with Ti (unfilled circles) at 1
GeV/ nucleon. The figure is from reference [103].

the longitudinal momentum transferred by target fragmentation (<P”’>) is related
to the average velocity of the fragment (< 6" >) in the projectile’s rest frame can be
written as:

M

I
< P" )2 mtarg < ,3“ > m . (4.12)

where 37/(7 + 1) is a kinematic factor which depends on the incident velocity and the
Lorentz factor of the projectile in the lab frame. A literature survey of < P", > avail-
able at the time showed an empirical relationship between the momentum transferred

and the mass loss such that

< P”, >= mAA (4.13)

where m was found to be approximately 8 MeV/ c / nucleon, the average binding energy
per nucleon for heavy nuclei. This linear trend has been termed the Morrissey
systematics. At the time, large scattering in the momentum transfer were evident
for large mass losses. Later work at GSI, showed large deviations from the Morrissey
systematics for AA < A/ 2, see Figure 4.6 [103]. The velocities of the fragments with
AA < A/ 2 showed the opposite trend than the heavier fragments (i.e. fragments were

slowing down instead of speeding up). This behavior is thought to be the result

69

Table 4.6: The slopes of the average parallel momentum transferred for fragment with
masses greater than half the onectile’s mass.

 

 

40 A1. 4803
Target Projectile Projectile
9Be 5.9(0.4) . -

"“tNi 5.0(0.4) -
181Te 6.3(0.5) 4.9(0.4)

 

 

 

 

 

of nuclear forces from the participant blast [104]. Friction between the nucleons
works to slow the projectile nucleus while the interaction of the participants with
the spectator nucleons will accelerate the prefragment. More nucleons can interact
at small impact parameters and the transfer of momentum will be larger than the
momentum lost through friction, thus the fragments will appear to be reaccelerated.
The momentum transferred from the fragmentation of 40Ar with a)gBe, b)'“‘tNi, and
181Ta and d)1810a with 181Ta from the current work are shown in Figure 4.7. The
velocities have been corrected for the energy straggling using Equations 4.3 and 4.4
with 6 = t / 2. The linear trend predicted by Morrissey systematics is evident for
A p < A F < A p/2, but slopes smaller than the 8 MeV/c/ nucleon (solid lines) were
found. The values of the fitted slopes (dotted lines) are summarized in Table 4.6.
The small variations between the slopes indicate that there is a small dependence
on the target and projectile combination for fragment masses greater than half the
projectile mass. Notani et al. found that only the light fragments produced in the
reaction of 40Ar with 9Be are reaccelerated in the same manner as the data observed
at GSI. They suggested that the rate at which the fragments are reaccelerated varies
with the target—projectile due to the different impact parameters [105]. They found
that the light fragments created in the reaction of 40Ar with 9Be were reaccelerated at
a greater rate than the same fragments produced in the reaction of 40A: with 181Ta.
This effect is not clear in the current data due to the large scatter of the data.

The average isobaric velocities have been compared to the predicted velocities at

the high and low projectile energies in an attempt to gain a better understanding of

70

 

 

300 _

 

 

 

 

 

 

 

 

 

 

 

 

 

 

200: l
200_ : ] [. l ,.
f; 100 100:1] [[1 ]] [[1,]
S . g 1 l] l
a.) 0, L: ' II l 'll]
x? A 00‘ 10 20 30 4o
- = . (d)
o. 300- 300-
Y I ;
200} """""""""" 200} ]
100:- """" 100} ]] ]] ]]]]l
00 if ‘10”‘A2'0‘U‘50H 00H10m2l0m310m410
AA

Figure 4.7: The average momentum transfer for A { < Ap/ 2 in the projectile frame
8

for the reactions of 40Ar with a)9Be, b)"°tNi, and 1Ta and d) 181Ca with 181Ta is
fitted with a liner function (dashed line). A slope of 8 MeV/c/ nucleon (solid line) is
also shown for reference. The slopes of the lines are summarized in Table 4.6.

the reaction mechanism that creates these fragments. The velocities at the high and
low bombarding energy were calculated from the Monte Carlo codes ISABEL [19,
20]+GEMINI [41] (unfilled circles) and deep inelastic transfer (DIT) [23]+GEMINI
(unfilled triangles) and compared with the weighted average isobaric velocity in the
lab frame from the current experiment (filled squares), see Figure 4.9. The dot-
ted lines represent the primary beam velocities at the center of the target. Thirty
thousand primary events (prefragments) were used in the INC calculation with the
ISABEL code [19,20]. The excited prefragments were then deexcited with the GEM-
IN I code. Ten statistical decays were preformed for each primary event to determine
the final fragment distributions. The velocities of the fragments predicted by the

ISABEL+GEMIN I calculations were found to decrease linearly as mass was removed

71

from the projectile. The rate at which the fragment velocities decrease is proportional
to the rate predicted by Morrissey systematics using a slope of 8 MeV/ c/ nucleon
(solid line) while the rate at which the fragment velocities decrease due to the nu-
cleon transfer process is small in the low may limit. The simulations made by the
DIT+GEMINI code were calculated using a 140 MeV/ nucleon primary beam using
10,000 prefragment events. The velocities were Lorentz boosted into the projectile
frame at the center of the target in order to correct for the energy straggling of the
primary beams through half of the thicknesses of the targets. For light neutron-rich
nuclei at intermediate energies, the corrected DIT+GEMINI velocities are similar to
the observed velocities from this work. The overall agreement of the velocities with
the DIT+GEMINI may be due to the small range of impact parameters necessary to
create the fragments. Thus, a small change in the impact parameter can change the
collision from a peripheral one to a central one. The agreement of the velocities with
low energy reaction mechanism may indicate that the deep inelastic transfer process
persists at an intermediate energy of ~ 130 MeV / nucleon.

The momentum transfers may be sensitive to the target thickness. The energy
losses for the thick targets used in the current work tend to be large and the devia-
tion from the Morrissey systematics may be due to corrections for energy straggling.
The velocities should be measured for thin targets where the energy losses of the
fragments are small to determine whether or not this is the case. The velocities were
also compared with work by N otani et al. (open circles) [74] and Mocko et al. (open
triangle) [75] to determine if the velocities are sensitive to the energy loss corrections,
see Figure 4.8. The trend of the momentum transfers from the current work from
the reaction of 40Ar with 9Be and 181Ta agrees with the trend measured by Notani
et al. for much thinner targets (94.6(O.1) mg/cm2 9Be and l7.0(0.1) mg/cm2 181Ta)
(panels (a) and (c) in Figure 4.8, respectively). The momentum transfers of fragments
produced from the reaction of 48Ca with 181Ta are shown in panel (d). A slope of

5.38(0.07) MeV/c/ nucleon and intercept of 26.3(0.06) for the momentum transfer of

72

 

300 (a)

   
  

N
O
O

_L
O
O

 

 

 

>(MeV/c)

"PM

 

 

 

o 0 .l . i . . . .
O 5 1O 15 20 25 30 35 40 0 10 20 30 40
AA

Figure 4.8: The momentum transfers of the fragments from this work are compared
to the works by Notani et al. (open circles) [74] and Mocko et al. (open triangle) [75].
The reaction partners in each panel are the same as in Figure 4.7.

fragments with mass between A p/ 23 A f g A p was found from the data of Mocko
et al. using a 228 mg/cm2 Ta target. This slope agrees with the value found in the
current work (see Table 4.6). In addition to comparing the velocities, one has to
compare other observables with predictions from the simulations to determine if the
deep inelastic tranfer mechanism is the process that creates these nuclei. The third

observable one can measure directly is the reaction cross sections of the fragments.

4.4 Cross sections

The cross sections are an important quantity that describes the possibility for a

particular reaction to occur. This probability is dependent on the bombarding energy

73

 

 

a
0.48- . , ( )0.48 0”
........... .. ..+V ‘. +‘. ‘ A134 0

Us M43411 Q

1545605644 “3”” 643.79

.0
A
O)

 

 

 

 

 

 

 

lab frame>
p
i

<13

.0

.1;
1m

 

Ii 1 sit?
foA]AhAAAAAAAMéXAU“f" 995:4;4464

9
J;
a:

 

 

 

 

 

 

0.44 I . I . l . l 0.44

 

 

Figure 4.9: The average fragment velocities are plotted as a function of their mass
number. Morrissey systematics (solid lines), the projectile bombarding velocities
(dashed lines) and the velocities from ISABEL+GEMINI (unfilled circles) and
DIT+GEMINI (unfilled triangles) are also shown. The reaction partners in each
panel are the same as in Figure 4.7.

and spatial distribution. The reaction cross sections for fragments produced in the
four reactions measured in this work were calculated using Equation 1.1 and by the
numerical integration of Equation 3.8. The uncertainties in the cross sections were
obtained from the propagation of errors and with the use of Leibniz rule in a similar
fashion as outlined in reference [106] (see Appendix B for derivation). The cross
sections from the reaction of 140Ar with 9Be, "atNi, and 181Ta and 48Ca with 181Ta
are listed in Appendices C and D and are plotted in Figures 4.10, 4.11, 4.12, and 4.13,
respectively, using filled squares. Comparisons with data measured by N otanti et al. at

RIKEN for the fragmentation of 40Ar with 9Be and 181Ta at 90—94 MeV/ nucleon [74]
(filled stars), Mocko et al. at the NSCL using the A1900 at 140 MeV/nucleon [75]

74

(filled hexagon), EPAX [6, 7] (solid lines) and to simulations from DIT+GEMINI
and ISABEL+GEMINI (open triangles and circles, respectively) are also shown. A
statistical limit of ~1/10 mb were obtained in the simulations for 106 events. Smaller
cross sections may be obtained by increasing the number of prefragments generated,
but larger CPU times are required. Forexample, approximately three months would
be required to calculate a cross section of 10“7 mb. Thus, only cross sections for
the nuclei near stability are shown. The rates of nuclei near the peaks of the cross
sections distributions were not measured during this experiment in order to focus on
the production of the most neutron-rich elements.

The log of the of the ratio of the cross sections from simulations and RIKEN with
the current work are plotted in Figures 4.14 and 4.15. Smaller values for the fragment
cross section from the reaction of 48Ca with 181Ta were found in the work Mocko et
al. (approximately one and a half times smaller). Small deviations can be contributed
to the neglecting of secondary interactions with the target and detector (up to 1%)
and the transmission uncertainties (up to 2%) in the analysis of Mocko et al. work.
The experimental results from RIKEN and the results from the current work for the
fragmentation of 40Ar with 9Be in Figure 4.14 (a) and 181Ta Figure 4.15 (c) show no
dependence of the cross section on the bombarding energies and the two simulations
and EPAX are able to predict the correct magnitude for the cross sections for nuclei
lighter than the projectile. EPAX under predicts the cross sections for masses heavier
than the projectile and ISABEL tends to over predict the cross sections by a factor
of three on average for all but one reaction.

The overestimation by ISABEL may be due to the smaller binding energies of
these light fragments, see Figure 4.16. Larger excitation energies maybe obtained by
lowering the cutoff energies and thus shifting the distributions towards stability. The
variation in the binding energy with atomic mass is not taken into account in the
current version of ISABEL.

The effect of the target neutron excess on the N/ Z of the fragments can be mea-

75

 

 

 

 

   

 

 

 

 

C

'3E1 . 1 i -51.1.11
10 10141818

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   
   

 

 

 

 

 

 

 

 

     

 

    

10 .
., 10'1
- §
10 10'3
10 10'5
10 1 1 g '71F. 1 i 1 i 1
.1°_19 21 23 1° 21 23 25 27
10” . 10° 3
10'2 -2
104 10 ,
106 10.4 :
10-71N.e1.1. 10‘81N1a1.111 '61Mgi1eii
24 26 28 30 26 28 30 32 1° 28 30 32
f; . 10 P
10. EPAX
Si 8 gggLfiGEMlfll

 

 

 

 

 

 

’ i 1 i 1 . -5_1 i 1 . 1 1
1° 31 33 35 1° 33 35 3710 35 37
Mass Number

Figure 4.10: Comparison of the cross sections of fragments produced from the reaction

of 40Ar with 9Be. The solid lines are the predicted values from EPAX 2.15, and the
filled squares and stars are the cross sections measured in this work and from RIKEN,
respectively, and the unfilled circles are predictions made by ISABEL+GEMINI.

76

 

 

 

 

 

 

 

 

 

 

 

 

 

  
 

 

 

10-50. 1 m m...
19 21 23
10°. .
10'1 8

10-2
10'3
Ne. . .1021“: . ,
24 26 283010 252830
Si

 

 

 

 

    
      
 

       

 

 

 

 

 

 

 

 

 

 

 

 

 

 

10-
- I A1900
10 A EEQXGEMINI .
_ e I§ABFL+§EMINI _2
1° 33 35 37 1° 35 37

Mass Number

Figure 4.11: The cross sections of fragments produced from the reaction of 40Ar with
”“tNi. The unfilled triangles are the simulated cross sections from the DIT+GEMINI
codes. The other symbols have the same meaning as in Figure 4.10.

77

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  
  
  
  

10'3
1o-5

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Ne ' =
10'? 2’4 ‘ 2'8 ‘ 2‘8 ‘ 3'0 10728 ' 2'8 ‘ 3'0 ‘ 3'210' 29 ‘ 3'1 ‘3'3
10°: 100 e P
10-1; 10'1 .9 A1900
10-2; . 10'2 7.52%... a
1° 31 L313 4 35 10633 1 315 ‘ 10-3 3'88”?”6558'

Mass Number

Figure 4.12: The cross sections of fragments produced from the reaction of 40Ar with
181Ta. The symbols are the same as in Figures 4.10 and 4.11.

78

 

 

 

 

  

10'6

10".

 

 

 

 

 

 

 

 

 

 

6iAI

l

 

 

34 36 38

 

l 1 10’5hl

 

 

 

 

. 10"5

 

 

10"2'0 ’ 212 ‘ 24

 

10'1 8

10'3
Na

 

‘ 10-55'

 

10' 2'8 ' 30

L32

 

.2E
103;.
104;—

 

 

 

'5 h I 1
10 37 39
Mass Number

 

104;

10-3E

 

 

 

 

 

1o-1
10'3

    

F
10' 2'3 7215 ’ 27 29

 

 

 

104:3

1—
I—
I—
I—
I—
I-
-
p—
1-
I.
1—
.-

 

 

 

 

30 ' 3'2 ’ 34

I A1900

EPAX

A DIT+GEMINI

o ISABEL+GEMINI

 

Figure 4.13: The cross sections of fragments produced from the reaction of 48Ca with

181Ta. The filled hexagon are cross sections from the work of Mocko et al. [75] and
the other symbols are the same as in Figures 4.10 and 4.11.

79

1.0

 

 

.°

 

 

 

 

 

 

 

 

 

$0.5: $0.
‘3200:» 22:14.4 {L17 ' '2; Efiiii'] ”(0
Ti 1‘? 11:," t3 il‘f] E0.
ef'oe ,0 g,
3-1 ............... .8-0.
10152025303540 152025303540
A A

 

+1.:

;IM 11. ] [,] [[1]]. *ii] ]]1]**1

 

 

 

 

“9‘" ISAPACE+GEMINII “111900)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

II 5
_1. .1. .1. --44. ..1 ---
10 15 20 25 30 35 40
A
a) 40Ar+9Be
1 _ A L
b _. n- _ ._!I l=:::fili:"=.' g f {
Ag- b orfi'.-':.:-i' " fl' 2 1 i
'15 fin- ] I
b( t i z [I 5 I I . I
31 50:. ii-mii'li 1
81'3“ 9 ~ 'd- -i"’-- '
. . e » i [1' [1*
§ - a I l i '
__5: . $-15 l
.10AW‘20‘ ........ 3o. ....... .40. _ 1020-n-3040
A A
$29 1
\‘L [ ]
"5,1: l 1
, I
9 : I 'I' .i¥!fili. £.' {liii'i
do. 1 L 1 Ji
2 _ l
5.4 b 1 . 1 . 1 .
g: 10 20 30 40
— A
)40Ar+"“tNI

Figure 4.14: The logarithmic ratios of the cross sections of nuclei produced in the

reaction of 40Ar with 8) 9Be and b) natNi predicted by EPAX, DIT/GEMINI, and
RIKEN relative to data obtained in this work.

80

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

’3 °5‘":"’::i::"'-’i:"i'='“=35‘=‘i"5‘;¢" 3 '--------* ardent-4141....-.
Dz 2E . . \b; .
- _ --1. ’
if E 5 i I l
3314'- g ’
_3’ E .9? -
_e’ 1 ......... 1 ......... J ,,,,,,,,, .T 3 '3 1 ......... 1 ......... 1 .........
10 20 30 40 1° 20 30 40
’3 i * 3° i l
.. *0 ‘ i 4’. 0.51 i ]
b< . g: i 1]; l: _ I i ‘3? EDGE ll;':i.;1¥l¥.ég ,
E 1}! 1 ii iii? 1 30.5; ’ ] M‘lfhi“ i ]
t:E i 10E ]
‘31 1 b3“
—_4 116........216........31o. ........ v-1.5 .16........20....-....3o.........40
A — A
c)40Ar +181Ta
A 1. A :
§ P l i 1 g 1’0?
DE 0.5 ,1] 1!] i if i i l 3. 0.5%
30 if I! x! ii 5 If i] g 00:
at, I § '!X§ '1' 1 $ 0 E
.305 {”11 ,__-0.5:-
2 ' t AAAAAAAAA 1,? AAAAAAAA , AAAAAAAAAAAAAAAAA 313-10
10 20 A30 40 3’ 10
A "as
$1.5 g1 i
< .
\bi 1.0[31 ] g :
5 : 1 i i b - [ ]
l“0.51 1’! [l \-0 .I 131 r
9 : i 1 il ] 1] ] E [ ] ]}
t7110’ llil l‘l : lif]*]l]ln]l*ll]
% . I 1 [I ‘ l g
6'4) 1: ................... 1.. * .-1. b _1 Wm; ..................
‘6, M10 20 30 40 ‘0,“ 10 20 30 40
.2 A _o A

d)48Ca + 181Ta

Figure 4.15: The logarithmic ratios of the cross sections of nuclei produced in the

reaction of c)40Ar and d) 480a with 181'ra predicted by EPAX, DIT+GEMINI, NSCL
and RIKEN relative to data obtained in this work.

81

 

 

 

dNO#0|O)NQO

O
E- F "0"

 

A 1 A A

‘50 “1éo‘“‘1éd"‘2m'5"‘250

Binding Energy (MeV/nucleon)

0

Atomic Mass

Figure 4.16: The nuclear binding energies per nucleon. The data is from refer-
ence [107].

sured by a comparison of the fragment cross sections produced in the different targets.
The cross sections of the fragments produced from the reaction of 40Ar with ”Ni
and 181 Ta are compared to those produced in 9Be Figure 4.17. The ratios of the cross
sections have been normalized to the geometric ratios calculated from Equation 4. l in
order to remove the size dependence and are plotted as a function of the fragment ’3
neutron excess relative to fl-stability. At intermediate bombarding energies of E / A
~ 130 MeV / nucleon, nuclei near fl—stability have no dependence on the target’s neu—
tron excess while the yields of most neutron-rich nuclei appear to enhanced. This
enhancement suggested that neutron-rich targets are more likely to produce rare iso-
tOpee along the neutron-drip—line, thus a 181Ta target was chosen for the search for
the heaviest fluorine isotope in the second measurement.

In the second experiment, an attempt was made to determine the location of the
drip-line above oxygen. The nuclei observed along the drip-line are shown in the
particle identification plot, Figure 4.18. The heaviest nuclei in the isotopic chain are
labeled. Eight 31F ions were observed from the fragmentation of a ~ 8.5 pnA 48Ca
projectile with a 181Ta target during two days with the A1900 optimized for 31F and

82

 

 

 

 

 

 

4.‘ 45
4. 4.
a 3. 8:69-33.
23. \ 3.0
\b
8 32.5 abfz.
“b 2. bmz.
bat { f i ‘I ‘31.
”-1. f b 1
bzo- livflhfiia‘kl { o. H
0' .o 0.2.;4A046403 1.0 0o ‘MLzloi4‘o:e‘o:al1:o
N/Z- NB/ZBO

Figure 4.17: The ratios of the cross sections of fragments produced in the fragmen-
tation of 40Ar with "at Ni and 181Ta are compared with the fragment cross sections
from the reaction 40Ar in 9Be.

32F (five and three particles for the 31F and 32F setting, respectively). The presence
of the N = 2Z line and the absence of 133e, 16,188, 210 and 2830F allowed the un-
ambiguous identification of 31F. N 0 new nuclei were observed. The differential cross
sections were fitted with a Gaussian function using the characteristics determined
from the fits to the other nuclei. The measured cross sections from GANIL [ll](open
triangle), RIKEN [10,105] (Open circles) and the NSCL (present work given by the
filled square) are plotted in Figure 4.19(b).

The cross section of ‘ 31F produced in the reaction of 40Ar with 181Ta (Fig-
ure 4.18(a)) and the the predicted cross sections made by EPAX (dashed lines) for the
fragmentation of 40Ar and 480a with 181Ta (dashed lines) are also shown for refer-
ence. No uncertainties were reported in reference [11], statistical errors were assumed
for the cross section. The current experiment and the values obtained from RIKEN
and GANIL indicate that 31F has a cross section of about 1 pb independent of the
projectile energy. This small cross section demonstrates the difficulty of producing
and observing very neutron-rich nuclei even for light elements and this is why the

drip-line is only known for the lightest elements.

83

 

l

11
10

I
1'?

A 01 O) N on CD
I

 

 

 

 

Figure 4.18: Eight 31F ions were observed at two A1900 Bp settings optimized for

the transmission of 31F and 32F. The heaviest isotopes observed in the chart of the
nuclides are labeled up to fluorine. Twenty-six isotopes were observed in these runs.

 

 

 

 

 

0.3
302 RIKEN (a)
e ' .' l
b 0.1 - l
. Epax 2.15
00--1-r-1--n-1--.-q-r-i-a-t-r-i-L-i-s-t-r-J-
50 60 7o 80 90 100110120130140
E/A (MeV/nucleon)
2.5 ............... Epax2.15 ............... L b).
32.0;-
3 1'5: RIKEN NSCL
*3 1.0;- A I}
0.5;-
0.0:GANH14.1.1.|4L.1L|.1

 

 

 

50 60 7o 80 90 10011012013014
E/A (MeV/nucleon)

Figure 4.19: The cross sections of 31F produced in the reactions of (a)40Ar with 181Ta

and (b)4SCa with 181Ta measured at GANIL (open circle), RIKEN (open triangles),
and the NSCL (the present work, filled square). The dashed lines are the predicted
cross sections made by EPAX 2.15.

84

Chapter 5

Summary

Two experiments were performed at the NSCL using the A1900 fragment separator
to produce fragments near the neutron drip—line. The establishment of the limits of
stability is important in the understanding of isospin asymmetries and how the shell
structure changes as one moves away from fl stability. The yields of nuclei produced
in the fragmentation of 40Ar with 9Be, "“tNi, and 181Ta and 480a with 181Ta were
measured using the A1900 fragment separator. An asymmetric Gaussian (Gaussian
+ asymmetric tail on the low momentum side) was found to describe the momentum
distributions of fragments and the asymmetric factor was found to be independent
of the fragment mass, see Appendices C and D. The parallel momentum widths,
parallel momentum transfers, and fragmentation cross sections were extracted from
the parameters of the asymmetric Gaussian and the observables from the current
work were compared with an intranuclear cascade code ISABEL and a deep inelastic
transfer code (DIT) both coupled to a deexcitation code GEMINI in an attempt to
understand the reaction mechanism.

The cross sections for 255 isotopes (201 from the fragmentation of 40A: and 54
from the fragmentation of 48Ca) were measured in the current work. The parallel mo-
mentum widths of light neutron—rich nuclei produced from projectile fragmentation at

intermediate energies are consistent with the predicted momentum widths from the si-

85

multaneous emission of nucleons and the momentum widths from ISABEL+GEMINI.
The reduced widths from the present work were found to be ~ 85 MeV/c (Fermi mo-
mentum ~ 190 MeV/c) independent of the target and projectile combination. This
value agrees with literature values for higher bombarding energies. The momentum
transfer due to the breaking of the nuclear bonds during the emission of the cluster
has been observed to follow the linear trend predicted by Morrissey systematics for
fragments with AA < Ap/ 2, but with smaller slopes. This behavior was also observed
in the dissertation work of Notani et al. for the fragmentation of 40Ar with much
thinner 9Be and 181Ta targets. No conclusions about the reacceleration phenomena
of light fragments (A F < A p / 2) observed at GSI could be made due to the large scat-
ter of the data. The average fragment velocities were also compared with simulations
valid at the high and low energy regimes, ISABEL+GEMINI and DIT+GEMINI, re-
spectively. The predicted velocities of fragments from ISABEL+GEMINI were found
to have velocities similar to the velocities predicted by Morrissey systematics, but
disagreedwith the average velocities of the light fragments measured in the current
work. The velocities were found to be similar to the velocities predicted by a deep
inelastic transfer code (DIT) code. The velocities were also compared with fragments
produced from the fragmentation of 40Ar and 480a with thin targets to determine
if the thick targets used in the present work distorted the transfer mechanism. Simi-
lar trends from data taken at RIKEN and the N SCL indicated that the momentum
transfer of fragments produced in the current work may be different than the trend
seen for heavier projectiles (i.e. the Morrissey systematics). This effect may be due
to the small impact parameters required to create these fragments.

The high predictive power of the two Monte Carlo codes DIT+GEMINI and IS-
ABEL+GEMIN I for the cross sections of the neutron deficient nuclei in the present
experiment has been demonstrated (i.e. the simulations are able to predict the cross
sections to within a factor of ten). The excitation energies of the prefragements from

ISABEL were found to depend on the target and the number of nucleons abraded

86

from the projectile and become significantly larger than the 8 MeV/ nucleon excitation
energy used in LISE. The different excitation energies and distributions of the pre-
fragment predicted ISABEL and LISE were found to lead to very similar predictions
of the cross sections. This indicates that the cross sections of the final fragments are
largely dependent on the excitation energy, prefragment distribution, and the decay
widths of the products. The similar cross sections and fragment velocities from the
DIT+GEMINI codes with the present work indicates that the low energy mecha-
nism is most likely the dominant factor in creating light neutron-rich nuclei near the
drip-line.

The existence of 31 F was reconfirmed and no new nuclei along the neutron drip-
line were observed. A cross section of ~ 1 pb from the fragmentation of 480a with
181Tn for 31F has been measured at RIKEN, GANIL, and the NSCL independent of
projectile energy. No conclusions about the existence of 32i33F could be drawn under
the running conditions of the current work. The current work extents the measured
cross sections to the most neutron rich nuclei along the drip-line and can be used
to improve on the extrapolation for the cross sections heavier neutron-rich nuclei.
The cross sections of nuclei produced from the reaction of 40Ar with 9Be and 181Ta
were. observed to be independent of the bombarding energy in the energy range of 60
- 130 MeV/ nucleon. This suggests that the “limiting fragmentation” energy regime
has been reached and that the EPAX formulation should be valid. The empirical
formula in EPAX has been shown to be able to predict the cross sections of all nuclei
produced by fragmentation (i.e. A f g A 19) including the most neutron-rich nuclei in
this work to within a factor of ten. It has been found that EPAX overpredicts the
yields of nuclei near the drip-line by a factor of two in the case of 31F.

The cross sections of nuclei produced from the fragmentation of 40Ar were com-
pared with each other to determine the effect of the N/ Z ratio of the target on the
fragmentation yields. There was a strong indication that the target’s neutron-excess

does enhance the yields of the most neutron-rich nuclei and that the target with larger

87

neutron—excess has a larger effect. This effect becomes important when searching for
new nuclei where time constraints are an issue. More work in this region is necessary
to determine the exact locations of the drip-line. The cross sections in this work
may be incorporated into EPAX to improve on the reliability of the predictions near
the neutron-drip line. A recent upgrade in the 48Ca beam intensity (~ 15 times the
intensities used in the present work) would allow the search for new particle bound
nuclei in this region. With higher beam intensities, new challenges arise, such as
managing the heat load on the production target. One must be able to circumvent

these challenges in order to extend the known limits of existence.

88

Appendix A

Input Parameters for ISABEL and
GEMINI

The following tables are sample input files used in ISABEL (Table A.l) and GEMINI
(Table A2) codes for the reaction of 48C3. + 181Ta at ~ 127 MeV/ nucleon. Similar

input files were used for the other three reactions with the 40Ar projectile.

89

Table A.1: Input parameters used in ISABEL.

projectile type
1059.3 incident energy (MeV/ n) + 931.5
181.0 A target
73.0 Z target
8.02 binding energy per nucleon in target
10000 number of cascades
8.666 binding energy per nucleon in projectile
0 cutofa(l)
0 cutofa(2)
0 cutofa(3)
0 cutofa(4)
0 cutofa(10)
0 cutofa(ll)
73 model used
0 vpion
0 isonsw
0 mprint (print option)
20. timef
3 ntdel
3.0 rcas
48.0 A projectile
20.0 Z projectile
-99 yp (impact parameter)
3 jkey (option for evaporation file)
1 option for Pauli principle IPAULI
1 option to restrict distance NP
2 type of restriction of distance
1.1 gapl
~2.5 1.0 0.6 0.6 0.6 0.6 0.6 1.0 dx(1-8)
—2.5 1.0 0.9 0.6 0.3 0.2 0.2 0.2 0.3 0.3 0.4 0.4 0.4 0.4 0.4 0.4 0.4 dxp(l-16)
7 iden (option radial density)
0. vap (amplitude of potential anti-pr)
0 nopot
0 kaskew
0. gamiso
6000000 nzman (variable to stop program, in 1 / 100 8)
5564350 irand

90

Table A.2: Input parameters used in GEMINI.

**************** Olltpllt choices ************************

.false. ,diagnostics, true=events typed out

.false. ,time_flag=.true. emission times written out on event file
.false. ,J_flag

.FALSE. ,ex_fiag

.true. ,I_angle=switch to calculate angles,

.false. ,true=qua.ntum treatment of angles, false=semi—classical
**************** fission and IMF parameters ************************

2 ,imf_option,l=symmetric fission,2=all asymmetries,0=no imf
0. ,t.delay=fission delay time in 10E—21 seconds

0. ,sig_delay = first moment of delay as function of eta

.TRUE. ,sharp.delay, gamma(t) = 0 (T) or gamma._0*t/t_delay(F)
5 ,Z_imf.min, minimum imf charge considered,

0.0 ,Kramers factor (only for IMF_option=1)

.FALSE. ,LESTONE

1. ,b.scale

**************** evaporation + gamma emissionuuuunununuuu
.true. ,tl_iwbc, true=tl’s from IWBC model,False=sharp cut-off
.FALSE. ,k_sum,

.false. ,polarization

2 ,exotiandex

-1.0 ,ratio

2 ,mass.option,0=liquid drop,l=expimental,2=shell fadeout

50.0 ,E2 strength in Weiskofp units (default=50)

0.1 ,ELstrength

le—5 ,threshold

*************** level density *zoom*************************************
0 ,aden_type

8.5 ,aden_0=level density constant (k=A/a)

1.00 ,a_scale

91

Appendix B

Derivations

B.1 Calculating uncertainties in the cross-sections

Numerical integration of the asymmetric Gaussian

 

= E: [— (“’40P 1 (B 1)
y y" 1" a2(1 +(x0 —a:)6/a) '
where
5 = 0 x 2 x0 (32)
l x < x0 '

has been used to determine the cross-sections listed in Appendices C and D. The errors
associated with function B.1 were determined using Leibnitz Theorem in a similar
manner as outlined in reference [106]. Leibnitz Theorem states that the derivative of

an integral can be written as

7%]: f(=v,y)dy = Nam-33“ m)—-- + [m 6——f(:’ mdy (B3)

92

where f(x,y) is an function of x and y. Thus, it can be shown that the uncertainty in

equation B.1 may be calculated by the following function

= /(-‘§’-02(yo»2 + (%o2(xo)>2 + (3—3 02m»? +(‘;— —§j2a-2(b>)

+2i’lo(yo>(—o (mo) +§ —§o(a> + —o<b»

6y 631

+26 %don—yda) +:— ya(b)) + 2— a—-(a)a(b) (BA)

The partial derivatives of the asymmetric Gaussian are given by:

y _ (x — $0)2
$6 _ Empl— a2 (1 + L “3’:$02)] (8'5)

 

A b(x—xo)2 2 + 2(z—xo)
6y “3(“b #ij ) “2(l+b _x‘fxo )
E = (B6)
0 Exp[ (:l:—1:0)2

02(1+ba](—x+zo))

 

 

A _ b(sc—xo)2(—a:+:co)2 + K7301?
ESE = “4 (”K—#3321) 030+ $0 $0 ) (3.7)

60 (x—xo)2

E17402 (1+b(-x:-xo)) l

 

 

6_y _ A (a: — x0)2 (—a: + x0)

6b — (x—xo)2 3 bS—x-Hm) 2
Ea:p[a2 (1+b;_$:_xol ) ]a (1 + a )

(B.8)

 

93

Appendix C

- Data from the Fragmenation of

40Ar

The following tables list the isotropic cross sections in mb for the reaction of a~ 128
MeV/nucleon 40Ar projectile with 98c (Table 0.1), "“tNi (Table 0.5), and 181Ta
(Table 09). Only momentum distributions that could be fitted are listed below. The

uncertainties quoted for the two simulations are statistical.

C.1 Fragments produced from the fragmentation
of 40Ar +9Be

C.1.1 Cross Section

Table 01: Reaction Cross Sections of 40Ar+9Be.

 

 

 

 

 

 

 

 

z A a EPAX ISABEL+GEMINI
4 10 4.09(l.69)E+0 1.23E+00 3.48(0.12)E+0
4 11 2.01(1.62)E—1 2.2013101 3.92(O.41)E—1
4 12 3.72(3.99)E—2 2.62E—02 3.48(1.23)E-2
5 12 1.79(2.31)E+0 1.62E+00 4.28(0.14)E+0
5 13 6.70(4.93)E—1 3.19301 1.23(0.07)E+O
5 14 3.23(3.48)E—2 4.1912102 1.35(O.24)E—1
5 15 7.76(9.98)E-3 3.91E—03 3.48(1.23)E—2

 

94

Table C.l — continued from previous page

 

 

 

 

z A 0‘ EPAX ISABEL+GEMINI
6 14 3.26(4.25)E+0 2.163+00 7.49(O.18)E+0
6 15 4.88(3.66)E—l 4.68E—01 4.18(0.43)31
6 16 1.45(1.00)31 6.81302 3.57(0.39)31
6 17 1.01(7.23)32 7.09303 1.74(0.87)32
6 18 1.72(2.22)33 5.53304 8.70(6.15)33
6 19 2.59(0.41)35 3.34305 -(-)

7 16 9.36(1.28)31 2.89E+00 3.29(O.12)E+0
7 17 7.87(8.06)E—1 6.93301 l.67(0.09)E+0
7 18 1.39(1.03)31 1.12301 2.22(0.31)31
7 19 3.45(2.51)32 1.30302 7.83(1.85)32
7 20 2.31(1.65)33 1.14303 8.70(6.15)33
7 21 2.87(3.68)34 7.78305 -(-)

7 22 6.09(1.04)E—6 4.25306 -(-)

8 19 6.54(8.55)31 1.03E+00 1.44(0.08)E+O
8 20 2.48(3.17)31 1.86301 9.31(0.64)31
8 21 3.12(2.38)32 2.42302 7.83(1.85)32
8 22 4.26(3.55)33 2.38303 2.61(1.07)32
8 23 1.19(1.02)34 1.84304 1.04(1.04)33
8 24 7.72(1.09)36 1.14305 —(-)

9 21 1.05(1.47)3+0 1.543+00 3.46(0.12)E+0
9 22 2.71(3.96)31 3.11301 7.40(0.57)31
9 23 6.88(5.16)32 4.54302 2.65(0.34)31
9 24 5.96(4.54)33 5.03303 2.18(0.97)32
9 25 5.10(4.96)34 4.39304 -(_)

9 26 3.05(4.01)35 3.10305 -(-)

9 27 1.80(2.80)36 1.81306 -(-)

10 24 4.32(6.64)31 5.21301 l.44(0.08)E+0
10 25 5.22(4.85)32 8.55302 l.83(0.28)E—1
10 26 1.06(8.24)32 1.07302 6.96(1.74)32
10 27 3.85(6.15)34 1.06303 -(-)

10 28 1.07(1.28)34 8.49305 -(-)

10 29 3.58(6.46)36 5.69306 -(-)

10 30 2.93(0.64)37 3.24307 -(-)

11 26 5.00(6.85)E—l 8.73301 l.32(0.08)E+0
11 27 1.38(1.91)31 1.61301 5.83(0.50)31
11 28 1.71(1.52)32 2.28302 6.96(1.74)32
11 29 3.57(3.26)33 2.56303 1.74(0.87)32
11 30 2.56(2.57)34 2.35304 -(-)

11 31 3.46(4.77)35 1.80305 -(-)

11 32 1.05(2.19)36 1.18306 -(-)

11 33 8.86(3.67)38 6.88308 -(-)

12 28 6.69(1.14)31 1.46E+00 3.43(0.12)E+0
12 29 1.18(2.24)31 3.05301 2.74(0.35)31
12 30 3.57(4.78)32 4.89302 1.52(0.26)31

 

 

 

 

95

 

 

Table 0.1 - continued from previous page

 

 

 

z A a EPAX ISABEL+GEMINI
12 31 3.39(4.44)33 6.24303 4.35(4.35)33
12 32 7.73(7.84)34 6.53304 3.11(1.79)33
12 33 3.19(4.72)35 5.86305 -(-)

13 31 3.12(6.20)31 5.73301 1.37(0.08)E+0
13 32 5.31(7.85)32 1.05301 2.44(0.33)31
13 33 1.39(1.33)32 1.55302 6.53(1.69)32
13 34 1.36(1.90)33 1.97303 8.70(6.15)33
13 35 1.97(2.39)34 2.10304 —(-)

13 36 6.03(1.35)36 2.10306 -(-)

14 33 3.25(5.89)31 1.093+00 l.68(0.09)E+0
14 34 1.11(8.49)31 2.37301 8.49(0.61)31
14 35 1.05(1.38)32 4.21302 7.40(1.79)32
14 36 2.61(2.78)33 5.62303 --(-)

14 37 7.12(1.24)35 5.62305 -(-)

15 35 1.06(1.91)E+0 2.203+00 5.95(0.16)E+0
15 36 1.69(3.35)31 5.50301 1.21(0.07)E+0
15 37 6.60(1.67)32 1.04301 2.57(0.33)31
15 38 3.74(5.44)33 1.04303 -(-)

 

 

 

 

 

 

C.1.2 Parallel Momentum Widths

Table 02: Parallel momentum widths from the reaction

 

 

 

 

 

 

 

 

 

 

 

of 40Ar + 9Be.
Lab Projectile

flame flame

z A a<P> «(13) ads 10N(E) lam?) 0MP)
(MeV/c) (MeV) MeV MeV (MeV/c) (MeV/c)

4 10 306(43) 144(16) 80.9(0.4) 119(20) 253(112) 223(128)
4 11 292(6) 137(10) 91.0(0.4) 102(13) 218(18) 193(22)
4 12 322(16) 146( 10) 101.1(0.5) 106(14) 233(24) 208(29)
5 12 311(4) 147(9) 89.9(0.4) 1 16(12) 246(12) 217(15)
5 13 355(7) 166(10) 99.9(0.5) 132(12) 283(17) 251(20)
5 l4 382(18) 177(10) 110.0(0.5) 139(13) 300(23) 266(27)
5 15 431(11) 197(10) 120.1(0.6) 156(12) 342(10) 304(13)
6 14 316(3) 148(8) 96.3(0.4) 113(11) 241(10) 213(13)
6 15 342(4) 160(8) 106.4(0.5) 119(11) 255(13) 225(16)
6 16 400(7) 185(9) 116.5(0.5) 144(11) 311(13) 276(16)
6 17 416(7) 192(9) 126.6(0.6) 144(12) 313(13) 277(17)

 

96

 

Table 0.2 — continued from previous page

 

 

 

 

Z A 0(P) 0(E) 043 1057(3) 10.570”) 0117(1))

(MeV/c) (MeV) MeV MeV (MeV/c) (MeV/c)
6 18 478(10) 220(9) 136.7(0.6) 172(12) 375(8) 333(11)
6 19 389(30) 179(8) 146.7(07) 102(13) 222(19) 197(23)
7 16 345(7) 161(8) 100.3(05) 126(10) 270(15) 238(17)
7 17 362(7) 169(8) 110.4(05) 127(11) 274(19) 243(22)
7 18 394(7) 182(8) 120.5(0.6) 137(10) 296(13) 262(16)
7 19 395(4) 183(7) 130.6(0.6) 129(10) 277(10) 245(12)
7 20 450(7) 207(8) l40.6(0.6) 152(11) 330(12) 293(15)
7 21 515(11) 236(8) 150.7(07) 181(11) 396(9) 352(11)
7 22 437(38) 201(7) 160.8(0.7) 120(12) 261(17) 232(20)
8 19 366(3) 169(7) 111.9(05) 127(9) 275(8) 244(10)
8 20 398(3) 183(7) 122.0(0.6) 136(9) 297(8) 264(10)
8 21 430(8) 197(7) 132.0(0.6) 147(10) 320(13) 284(15)
8 22 428(9) 197(7). 142.1(07) 137(10) 297(13) 264(15)
8 23 521(23) 239(8) 152.2(07) 184(11) 401(16) 357(19)
8 24 497(33) 227(8) 162.3(0.7) 159(11) 348(22) 310(26)
9 21 451(26) 207(8) 110.9(05) 175(9) 381(16) 338(19)
9 22 367(7) 170(6) 121.0(0.6) 119(9) 258(19) 228(22)
9 23 386(4) 178(6) 131.1(0.6) 121(9) 261(9) 232(11)
9 24 400(5) 184(6) 14l.l(0.6) 119(9) 257(9) 228(11)
9 25 419(10) 193(6) 151.2(0.7) 121(10) 261(13) 232(15)
9 26 480(11) 220(7) 161.3(0.7) 150(10) 326(13) 290(16)
9 27 499(26) 228(7) 171.4(0.8) 151(11) 330(25) 293(29)
10 24 421(14) 193(6) 117.5(05) 153(8) 334(8) 297(10)
10 25 377(6) 174(6) 127.6(0.6) 118(8) 255(12) 227(14)
10 26 397(5) 183(5) 137.6(0.6) 120(8) 261(8) 232(11)
10 27 457(13) 209(6) 147.7(07) 147(9) 323(18) 287(22)
10 28 468(20) 215(6) 157.8(0.7) 146(9) 317(18) 282(21)
10 29 545(44) 248(8) 167.9(0.8) 182(11) 400(42) 357(48)
10 30 455(40) 210(7) l78.0(0.8) 112(13) 242(40) 215(46)
11 26 368(5) 168(5) 111.5(05) 126(7) 276(7) 245(8)
11 27 380(7) 174(5) 121.6(06) 125(7) 272(11) 241(13)
11 28 381(6) 175(5) 131.7(0.6) 115(8) 251(10) 223(12)
11 29 417(9) 191(5) 141.8(0.7) 128(8) 279(11) 248(13)
11 30 404(10) 186(5) 151.8(07) 107(9) 232(11) 206(14)
11 31 394(15) 182(5) 161.9(07) 83(11) 180(15) 159(19)
11 32 419(39) 192(6) 172.0(0.8) 85(13) 186(29) 166(33)
12 28 389(19) 177(5) 103.1(05) 144(7) 317(20) 282(23)
12 29 353(17) 162(5) 113.1(05) 116(7) 252(20) 224(24)
12 30 416(16) 190(5) l23.2(0.6) 145(7) 317(20) 282(23)
12 31 398(7) 182(5) 133.3(0.6) 123(7) 270(8) 241(9)
12 32 370(8) 170(4) 143.4(07) 91(8) 199(10) 177(12)
12 33 376(13) 173(4) 153.5(07) 79(10) 172(13) 153(15)
13 31 398(18) 181(5) 102.2(05) 149(5) 328(8) 292(9)

 

 

 

 

 

 

 

97

 

Table 0.2 — continued from previous page

 

 

 

 

 

 

 

 

 

 

Z A 0(P) 0(E) 083 l0N(E) 101%") 0MP)
(MeV/c) (MeV) MeV MeV (MeV/c) (MeV/c)
13 32 325(10) 149(4) 112.3(05) 98(6) 214(14) 190(16)
13 33 330(8) 152(4) 122.3(0.6) 89(6) 195(8) 173(10)
13 34 392(19) 179(5) 132.4(0.6) 121(7) 264(16) 235(18)
13 35 359(13) 165(4) 142.5(07) 82(8) 179(11) 159(13)
13 36 422(36) 192(5) 152.6(0.7) 116(9) 256(28) 228(33)
14 33 329(9) 150(4) 88.7(0.4) 120(4) 265(8) 236(10)
14 34 322(50) 147(6) 98.8(0.5) 109(9) 238(65) 212(74)
14 35 280(11) 129(3) 108.9(0.5) 68(6) 149(12) 132(14)
14 36 322(10) 148(3) 119.0(0.5) 87(6) 191(8) 169(9)
14 37 356(25) 162(4) 129.1(0.6) 98(7) 216(17) 192(19)
15 35 281(10) 128(3) 72.8(0.3) 105(4) 231(7) 206(8)
15 36 313(7) 142(3) 82.9(0.4) 116(4) 255(4) 227(5)
15 37 244(12) 112(3) 93.0(0.4) 63(5) 137(16) 121(19)

 

 

C.1.3 Parallel Momentum 'Iransfer

Table 0.3: Parallel momentum transferred from the re-
action of 40Ar + 9Be. All units are in MeV.

 

 

 

 

Lab Frame Projectile flame
Z A <P> 1<P°°"' > 1<Pc°"" > l<P’>
4 10 4952(93) 5010(69) -31(84) -31(85)
4 11 5434(16) 5492(26) -51(41) -47(37)
4 12 5690(22) 5753(31) -266(47) -224(40)
5 12 5983(12) 6072(22) 16(39) 13(33)
5 13 6384(18) 6475(27) -73(45) -57(35)
5 14 6836(26) 6929(33) -118(51) -85(37)
5 15 7188(12) 7285(31) -249(52) -168(35)
6 14 6929(11) 7059(23) —4(44) -3(32)
6 15 7395(16) 7525(27) -37(48) -25(33)
6 16 7764(16) 7899(29) -152(52) -96(33)
6 17 8251(18) 8386(32) -l68(56) -100(34)
6 18 8661(11) 8798(33) -249(59) -140(33)
6 19 9146(28) 9283(41) -266(67) -142(35)
7 16 7871(19) 8049(28) -20(51) -13(32)
7 17 8325(26) 8505(33) -63(57) -37(34)

 

 

 

 

 

 

 

 

1 Widths are corrected for the broading due to energy straggling.

98

Table 0.3 — continued from previous page

 

 

 

 

 

 

 

 

 

Lab flame Projectile flame
Z A <P> 1<Pc°rr > 1<P°°" > 1<P’>
7 18 8751( 19) 8933(31) -130(57) -73(32)
7 19 9275( 15) 9456(32) -114(59) -61(32)
7 20 9672(19) 9857(35) -205(64) -104(32)
7 21 10076(14) 10264(37) -292(67) -141(32)
7 22 10601(29) 10788(43) -275(74) -126(34)
8 19 9228(12) 9465(29) -106(58) -56(31)
8 20 9651(13) 9892(31) ~175(61) -88(31)
8 21 10107(21) 10350(36) -216(66) -104(32)
8 22 10636(22) 10876(38) —196(70) -90(32)
8 23 11060(29) 11303(42) -265(75) -117(33)
8 24 11479(41) 11725(50) -338(83) -143(35)
9 21 10111(27) 10415(39) -158(68) -76(33)
9 22 10694(33) 10993(43) -93(73) -43(34)
9 23 11144(16) 11446(36) -139(70) -61(31)
9 24 11619(17) 11921(38) -165(73) -69(31)
9 25 12127(25) 12429(42) -162(79) -66(32)
9 26 12502(26) 12809(44) -272(82) ~106(32)
9 27 12966(53) 13274(59) ~307(96) -115(36)
10 24 11515(14) 11892(38) -190(74) -80(31)
10 25 12085(24) 12458(42) ~136(78) -55(32)
10 26 12528(17) 12903(41) -188(80) -73(31)
10 27 12909(39) 13291 (52) -291(90) -109(34)
10 28 13476(40) 13853(53) -240(93) -87(33)
10 29 13792(94) 14178(90) -400(127) -139(44)
10 30 14555(95) 14927(91) -183(129) -62(44)
11 26 12484(14) 12939(42) -157(80) -61(31)
11 27 12981(24) 13435(46) -164(85) ~61(32)
11 28 13488(22) 13941(46) -163(87) -59(31)
11 29 13917(24) 14373(48) -226(91) -79(32)
11 30 14482(26) 14934(50) -177 (94) -60(32)
11 31 15006(38) 15457(56) -161( 100) —53(33)
11 32 15379(72) 15835(78) -272(120) -86(38)
11 33 15928( 100) 16381(98) -235(140) —72(43)
12 28 13348(44) 13894(60) -204(98) -74(36)
12 29 13967(47) 14503(62) -112(102) -39(35)
12 30 14361(47) 14904(63) -203(104) —69(35)
12 31 14812(18) 15358(51) —248(96) -81(32)
12 32 15433(24) 15970(53) —153(100) ~48(32)
12 33 15877(33) 16417(57) -203(105) -62(32)
13 31 14760(19) 15401(55) -210(99) -68(33)
13 32 15409(36) 16037(61) -94(106) -30(34)
13 33 15882(21) 16512(56) ~120(104) —37 (32)
13 34 16292(42) 16928(66) -198(113) -59(34)

 

99

 

Table 0.3 -— continued from previous page

 

 

 

 

 

 

 

 

 

Lab flame Projectile flame
Z A <P> 1<P°°"‘ > 1<P¢°" > l<P’>
13 35 l6833(31) l7465(61) -169(112) -49(32)
13 36 17125(79) l7771(90) -344(l38) -97(39)
14 33 15701(21) 16443(62) -18l(108) -55(33)
14 34 16268073) l7003(l71) -132(2l6) -39(64)
14 35 . l6870(33) 17596(66) -54(1 15) -l6(33)
14 36 17273(22) 18006(63) -137(115) -39(32)
14 37 17615(47) 18360(75) -270(127) -74(35)
15 35 l6674(19) 17520(68) -12l(ll7) -35(34)
15 36 17089(ll) 17942(68) -l93(118) -54(33)
15 37 17785(48) 18620(80) -4l(l30) -ll(36)
15 38 18221(21) 19060(69) -97(123) -26(33)

 

C.1.4 Asymmetric factor

Table 0.4: The asymmetric term to the parallel momen-
tum widths from the reaction of 40Ar + 9Be.

 

 

 

 

 

 

 

 

Z A a Z A a

4 10 -(-) 10 25 364(90)
4 11 618( 147) 10 26 265(81)
4 12 117(152) 10 27 771(606)
5 12 -(-) 10 28 220(294)
5 13 178(131) 10 29 -(-)

5 l4 523(222) 10 30 -(-)

5 15 -(-) 11 26 -(-)

6 14 -(-) ll 27 -(-)

6 15 354(91) 11 28 371(114)
6 16 126(60) ll 29 519(134)
6 17 270(91) ll 30 413(152)
6 18 -(-) 11 31 683(317)
6 19 -(—) 11 32 -(-)

7 16 -(-) 12 28 -(-)

7 17 296(155) 12 29 566(224)
7 18 215(77) 12 30 116(246)
7 19 648(107) 12 31 -(-)

7 20 192(81) 12 32 563(151)

 

 

 

1 Corrected for the broading due to energy straggling.

100

 

C.2

Table C.4 — continued from previous page

 

 

 

 

 

 

 

 

Z A a Z A a

7 21 -(-) 12 33 -(-)

7 22 -(-) 13 31 -(-)

8 19 -(-) 13 32 447(111)
8 20 -(-) 13 33 297(78)
8 21 169(94) 13 34 454(367)
8 22 457(119) 13 35 362(162)
8 23 134(178) 13 36 -(-)

8 24 -(-) 14 33 -(-)

9 21 -(-) 14 34 217(404)
9 22 448(163) 14 35 336(107)
9 23 293(64) 14 36 212(101)
9 24 436(88) 14 37 -(-)

9 25 639(190) 15 35 -(-)

9 26 -(-) 15 36 -(-)

9 27 -(-) 15 37 414(203)
10 24 -(-)

 

 

of 40Ar +"atNi

C.2.1 Cross Section

flagments produced from the fragmentation

Table 05: Reaction Cross Sections of 40Ar+"“tNi.

 

 

 

 

 

 

 

 

z A a EPAX DIT+GEMINI ISBEL+GEMINI
4 10 5.66(0.67)3+0 1.943+00 l.57(0.02)E+01 l.96(0.04)E+1
4 (11 3.76(o.45)31 3.48301 4.26(0.25)301 1.92(O.12)E+0
4 12 9.87(1.78)32 4.14302 2.12(0.18)E—01 3.16(0.48)31
5 12 2.94(0.35)3+0 2.56E+00 2.60(0.06)E+00 1.24(0.03)E+1
5 13 l.08(0.06)E+0 5.04301 7.91(0.34)301 3.21(0.15)E+0
5 14 5.46(0.44)32 6.62302 4.20(0.79)302 3.31(0.49)31
5 15 1.66(0.20)32 6.19303 2.25(0.58)302 1.22(0.30)31
6 14 4.79(0.57)3+0 3.413+00 1.03(0.01)3+01 1.65(0.03)3+1
6 15 6.74(0.45)31 7.40301 4.13(0.25)301 1.18(0.09)E+0
6 16 1.84(0.11)31 1.08301 5.28(0.28)301 6.83(0.70)31
6 17 1.42(0.10)32 1.12302 7.50(3.40)303 5.75(2.03)32
6 18 3.88(0.59)33 8.74304 6.00(3.00)303 7.19(7.19)33
7 17 1.10(0.13)3+0 1.103+00 1.28(0.04)E+00 3.53(0.16)E+0

 

101

 

Table 0.5 — continued from previous page

 

 

 

 

 

z A a EPAX DIT+GEMINI ISABEL+GEMINI
7 18 1.92(0.12)31 1.77301 1.34(0.14)301 5.11(0.61)E—1
7 19 4.76(0.27)32 2.06302 2.85(0.65)302 7.91(2.39)32
7 20 3.41(0.40)33 1.80303 1.50(1.50)303 1.44(1.02)32
7 21 5.45(0.90)34 1.23304 -(-) 7.19(7.19)33
8 19 8.91(1.06)31 1.63E+00 1.17(0.04)E+00 1.98(0.12)E+0
8 20 3.09(0.20)31 2.94301 7.64(0.34)301 1.13(0.09)E+0
8 21. 3.65(0.21)E—2 3.83302 1.80(0.52)302 8.63(2.49)32
8 22 5.15(0.35)33 3.77303 1.20(0.42)302 5.03(1.90)32
8 23 1.45(0.18)34 2.91304 -(-) 1.44(1.02)32
8 24 9.42(2.72)36 1.80305 -(-) -(-)

9 21 1.83(0.22)E+0 2,443+00 1.68(0.05)E+00 4.41(0.18)3+0
9 22 3.78(0.36)31 4.92301 2.13(0.18)301 8.77(0.79)31
9 23 8.46(0.50)E—2 7.18302 6.75(1.01)302 2.30(0.41)31
9 24 6.71(0.52)33 7.95303 9.00(3.70)303 5.03(1.90)32
9 25 6.64(0.67)34 6.94304 -(-) 1.44(1.02)32
9 26 4.10(0.53)35 4.90305 -(-) ()

9 27 2.58(0.45)36 2.87306 -(-) -(-)

10 24 5.38(0.65)31 8.23301 1.12(0.04)E+00 1.80(0.11)E+0
10 251 6.32(o.44)32 1.35301 5.40(0.90)302 8.63(2.49)E—2
10 26 1.16(0.08)32 1.69302 1.65(0.50)302 5.03(1.90)32
10 27 4.64(0.69)E-4 1.67303 3.00(2.10)303 -(-)

10 28 1.33(0.22)34 1.34304 4.50(2.60)303 -(-)

10 29 5.70(1.04)E—6 8.99306 -(—) ()

10 30 8.50(1.90)37 5.12307 () ()

11 26 5.57(0.70)31 1.38E+00 4.08(0.25)301 9.64(0.83)31
11 27 1.53(0.15)31 2.55301 1.04(1.32)301 4.24(0.55)31
11 28 1.77(0.12)32 3.61E—02 4.50(2.60)303 7.19(7.19)33
11 29 3.49(0.37)33 4.05303 6.00(3.00)E-03 -(-)

11 30 3.10(0.37)34 3.71304 -(-) ()

11 31 3.81(0.54)E—5 2.84305 () -(-)

12 28 9.45(1.22)31 2.303+00 l.62(0.05)E+0 2.45(0.13)3+0
12 29 1.92(0.24)31 4.82301 2.99(0.21)301 1.37(0.31)31
12 30 5.28(0.63)32 7.73302 3.02(0.21)301 8.63(2.49)E—2
12 31 4.62(2.33)33 9.86303 2.70(2.70)303 7.19(7.19)33
12 32 7.85(0.82)34 1.03303 -(-) -(-)

12 33 3.24(0.51)35 9.27305 -(-) -(-)

12 34 4.92(1.22)36 7.49306 -(-) -(—)

12 35 2.10(1.72)37 7.49308 -(-) -(-)

13 31 4.10(0.82)31 9.07301 9.26(0.37)E—01 6.40(0.68)31
13 32 5.66(0.67)32 1.65301 1.35(0.45)302 1.37(0.31)31
13 33 1.55(0.13)32 2.45302 6.00(3.00)303 4.31(1.76)32
13 34 1.51(0.18)33 3.11303 () -(-)

13 35 2.20(0.34)34 3.32304 () ()

13 36 1.15(0.26)35 3.32306 -(-) -(-)

 

 

 

 

102

 

Table 0.5 - continued from previous page

 

 

 

 

 

 

 

 

z A a EPAX DIT+GEMINI ISABEL+GEMINI
14 33 3.99(0.52)31 1.723+00 3.27(0.22)301 7.69(0.74)31
14 34 1.34(0.14)31 3.74301 6.00(0.95)302 2.80(0.45)E—l
14 35 1.38(0.18)32 6.66302 -(-) 7.19(7.19)33
14 36 2.83(0.40)33 8.89303 () -(-)

14 37 1.98(0.36)34 8.89305 () ()

14 38 8.37(2.49)36 1.78307 —(—) ()

15 35 1.39(0.19)3+0 3.47E+00 3.41(0.23)30 3.59(0.16)E+0
15 36 2.54(0.46)31 8.69301 2.10(0.56)302 4.31(0.56)31
15 37 5.37(1.14)32 1.64301 6.60(0.99)302 4.31(1.76)32
16 41 2.11(0.32)33 1.97308 -(-) -(-)

 

C.2.2 Parallel Momemtum Width

Table 0.6: Parallel momentum width from the reaction

 

of 40Ar + "atNi.

 

 

 

 

 

 

 

 

 

 

 

Lab Projectile

flame flame

Z A 0(P) 0(E) 043 105703) 10MP) 0310’)
(MeV/c) (MeV) MeV MeV (MeV/c) (MeV/c)

4 10 324(4) 151(11) 78.6(0.3) 128(13) 277(16) 245(20)
4 11 409(9) 188(13) 88.4(0.4) 165(15) 361(19) 321(23)
4 12 384(41) 177(12) 98.2(0.4) 148(15) 320(34) 283(40)
5 12 333(4) 156(10) 87.3(0.3) 129(12) 276(13) 244(16)
5 13 344(4) 160(9) 97.1(0.4) 128(12) 274(13) 243(16)
5 14 386(10) 179(10) 106.9(0.4) 144(13) 310(19) 274(23)
5 15 516(18) 238(12) 116.7(0.5) 208(14) 450(15) 400(19)
6 14 327(3) 153(8) 93.6(0.4) 121(10) 259(10) 229(13)
6 15 343(4) 160(8) 103.4(04) 122(11) 262(12) 231(15)
6 16 392(6) 181(9) 113.2(04) 142(11) 307(12) 272(15)
6 17 445(11) 206(9) 123.0(05) 165(12) 357(15) 316(19)
6 18 522(47) 243(11) l32.8(0.5) 204(13) 438(19) 387(23)
7 17 376(3) 174(8) 107.2(04) 136(10) 295(8) 262(11)
7 18 405(7) 187(8) 117.0(05) 146(10) 316(12) 281(15)
7 19 416(5) 192(8) 126.8(0.5) 144(10) 312(11) 277(13)
7 20 490(7) 225(8) 136.6(0.5) 179(11) 390(8) 346(11)
7 21 578(51) 268(10) 146.4(0.6) 225(12) 484(21) 429(25)
8 19 370(4) 171(7) 108.7(0.4) 132(9) 285(8) 253(10)
8 20 373(4) 173(7) 118.5(0.5) 125(9) 271(10) 240(13)

 

103

 

Table 0.6 — continued from previous page

 

 

 

 

 

 

 

 

 

 

Z A 0(P) 0(E) 083 10MB) 10M?) 0MP)

(MeV/c) (MeV) MeV MeV (MeV/c) (MeV/c)
8 21 418(6) 192(7) 128.3(0.5) 143(9) 311(10) 277(13)
8 22 407(7) 188(7) 138.1(0.5) 127(10) 276(11) 245(14)
8 23 492(12) 226(8) 147.9(0.6) 170(10) 372(15) 330(18)
8 24 559(136) 256(10) 157.7(0.6) 202(13) 440(46) 391(52)
9 21 471(17) 215(8) 107.7(04) 186(9) 408(11) 363(13)
9 I 22 372(8) 172(6) 117.5(0.5) 125(9) 271(16) 241(18)
9 23 393(5) 181(6) 127.3(0.5) 129(8) 279(9) 248(11)
9 24 422(11) 194(6) 137.1(0.5) 138(9) 299(13) 265(16)
9 25 439(20) 202(7) l46.9(0.6) 139(10) 302(18) 268(21)
9 26 501(18) 230(7) 156.7(0.6) 168(10) 366(19) 325(22)
9 27 556(50) 257(8) 166.5(0.7) 196(11) 424(32) 376(37)
10 24 378(4) 173(5) 114.1(05) 130(7) 284(6) 253(8)
10 25 372(6) 171(5) 123.9(05) 118(8) 256(10) 228(12)
10 26 400(7) 184(5) 133.7(05) 127(8) 275(10) 245(12)
10 27 418(16) 192(6) 143.5(0.6) 128(9) 279(20) 247(24)
10 28 429(31) 198(6) l53.3(0.6) 125(10) 271(25) 241(29)
10 29 475(43) 218(7) 163.1(0.6) 144(11) 314(32) 280(37)
10 30 532(90) 242(7) 172.9(0.7) 169(10) 371(32) 331(37)
11 26 398(8) 182(5) 108.3(0.4) 146(7) 320(9) 285(10)
11 27 339(8) 157(5) 118.1(0.5) 103(7) 223(13) 198(15)
11 28 374(6) 172(5) 127.9(05) 115(7) 250(8) 222(10)
11 29 438(19) 201(6) 137.7(0.5) 146(8) 319(18) 284(20)
11 30 392(14) 181(5) 147.5(0.6) 104(9) 226(14) 201(17)
11 31 396(23) 183(5) 157.3(0.6) 93(10) 202(14) 179(17)
12 28 397(12) 180(5) 100.1(0.4) 150(6) 330(14) 294(16)
12 29 308(5) 141(4) 109.9(0.4) 89(6) 193(4) 172(6)
12 30 322(8) 147(4) 119.7(0.5) 86(6) 188(5) 167(7)
12 31 402(64) 183(7) 129.5(0.5) 130(9) 285(50) 254(57)
12 32 365(12) 168(4) 139.3(0.6) 94(8) 204(13) 181(15)
12 33 388(22) 179(5) 149.1(0.6) 99(8) 215(16) 191(19)
12 34 424(69) 194(6) 158.9(0.6) 111(11) 243(47) 216(54)
12 ’35 406(169) 186(9) 168.7(0.7) 78(22) 170(105) 151(120)
13 31 328(27) 150(4) 99.3(o.4) 113(6) 247(21) 219(24)
13 32 314(10) 144(4) 109.1(0.4) 94(6) 205(13) 182(15)
13 33 336(9) 154(4) 118.9(0.5) 98(6) 214(8) 191(9)
13 34 324(11) 149(4) 128.6(0.5) 75(7) 163(12) 145(14)
13 35 434(34) 199(5) 138.4(0.5) 143(6) 312(12) 277(14)
13 36 356(36) 163(4) l48.2(0.6) 68(10) 148(20) 132(23)
14 33 324(5) 147(3) 86.2(0.3) 119(4) 262(6) 234(7)
14 34 289(7) 132(3) 96.0(0.4) 91(4) 199(7) 177(8)
14 35 361(14) 165(4) 105.8(04) 126(5) 277(8) 246(10)
14 36 285(13) 131(3) 115.6(0.5) 61(7) 133(12) 119(14)
14 37 369(36) 167(4) 125.4(05) 110(7) 243(26) 217(30)

 

104

 

 

Table 0.6 — continued from previous page

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Z A 0(P) 0(3) 033 10N(E) 1057(1)) 0MP)
(MeV/c) (MeV) MeV MeV (MeV/c) (MeV/c)
15 35 267(10) 121(3) 70.7(0.3) 99(3) 217(4) 193(5)
15 36 249(13) 114(3) 80.5(0.3) 80(4) 176(11) 156(13)
15 37 217(13) 100(3) 90.3(0.4) 42(6) 92(15) 82(18)
16 41 247(16) 111(2) 92.0(0.4) 63(4) 139(9) 124(10)
C.2.3 Parallel Momentum Transfer
Lab flame Projectile flame
Z A <P> 1<Pc°" > 1<Pc°" > 1<P’>
4 10 4883(13) 4942(29) -89(41) -90(41)
4 11 5298(18) 5359(34) -166(47) -153(43)
4 12 5860(45) 5920(44) -116(58) -98(49)
5 12 5899(12) 5990(30) -54(44) -46(37)
5 13 6336(10) 6430(34) -112(49) -87(38)
5 14 6755(23) 6850(40) -185(56) -134(41)
5 15 7279(18) 7373(43) -169(60) -114(41)
6 14 6857(16) 6989(32) -63(49) -45(36)
6 15 7291(12) 7426(35) -122(53) -83(36)
6 16 7764(11) 7900(38) ..150(57) -95(36)
6 17 8254(17) 8389(42) -162(63) -97(37)
6 18 8842(35) 8974(50) -91(71) —51(40)
7 17 8240(12) 8424(36) -132(57) -78(34)
7 18 8726(12) 8910(39) -148(61) -83(34)
7 19 9157(10) 9343(42) -211(65) -113(35)
7 20 9655(12) 9841(45) -217(70) -110(35)
7 21 10238(37) 10420(54) -150(79) -72(38)
8 19 9183(16) 9423(38) -140(62) -75(33)
8 20 9654(12) 9895(40) -168(65) -85(33)
8 21 10159(12) 10399(43) -169(69) -82(33)
8 22 10610(14) 10852(46) -214(74) -99(34)
8 23 11064(27) 11307(52) -258(80) -113(35)
8 24 11530(86) 11774(78) -291(106) -123(45)
9 21 10022(18) 10332(41) -228(68) -110(33)
9 22 10600(12) 10905(42) -167(70) —77(32)
9 23 11084(13) 11389(44) -185(74) -82(32)

 

 

 

 

 

 

 

 

 

‘ Widths are corrected for the broading due to energy straggling.

105

 

Table 0.7 — continued from previous page

 

 

 

 

 

 

 

 

 

Lab flame Projectile flame
Z A <P> 1<Pc°rr > 1<Pco" > l<P’>
9 24 l 1605(14) 11909(47) -172(78) -73(33)
9 25 12103(21) 12407(51) -178(82) -72(33)
9 26 12498(38) 12806(59) -270(91) -105(35)
9 27 13131 (70) 13432(74) -163(106) -61(40)
10 24 11526(15) 11904(45) -176(76) ~74(32)
10 25 12025(15) 12402(47) -182(79) -74(32)
10 26 12509(15) 12886(49) ~199(83) -78(32)
10 27 13013(38) 13390(58) -200(92) -75(34)
10 28 13583(31) 13955(57) -146(93) -53(33)
10 29 13914(74) 14295(78) -292(113) -102(39)
10 30 14256(74) 14643(80) -429(116) -145(39)
11 26 12452(21) 12910(50) -178(83) -69(32)
11 27 l2993(15) 13447(50) -150(85) -56(32)
11 28 l3473(16) 13928(52) -l70(88) -61(32)
11 29 13940(19) 14396(54) -202(92) -71(32)
11 30 14472(26) 14926(58) -180(96) -6l(33)
ll 31 15064(34) 15458(62) —155(101) -51(33)
12 28 13301(1) 13852(51) -237(87) -86(32)
12 29 13961(10) 14500(52) -111(90) -39(31)
12 30 14365(12) 14909(55) -194(94) -66(32)
12 31 14868(17) 15411(57) -196(97) -64(32)
12 32 15421(24) 15960(60) -157(101) -50(32)
12 33 l6005(54) 16538(73) -92(113) -28(35)
12 34 16270(125) 16817(116) -291(156) -87(46)
12 35 16754(287) 17302(232) -308(283) -89(82)
13 31 14833(l4) 15470(57) -145(98) -47(32)
13 32 15378(20) 16010(60) -113(101) -36(32)
13 33 l5847(18) 16481(61) -l42(104) -44(32)
13 34 16308(20) 16944(63) -l79(107) —53(32)
13 35 16867(33) 17499(68) -134(1 12) -39(32)
13 36 17292(56) 17928(78) -201(123) -57(35)
14 33 15684(14) 16429(63) -188(105) -58(32)
14 34 16303(l5) 17037(64) -97(107) -29(32)
14 35 16738(23) 17476(67) -154(112) -45(32)
14 36 17286(18) 18020(67) -120(1 13) -34(32)
14 37 17493(74) 18249(91) —362(137) -99(37)
14 38 18170(79) 1891l(95) -223( 141) -60(38)
15 35 16641 (12) 17493(68) -140(113) -40(33)
15 36 l7189(22) 18035(71) -106(117) -30(33)
15 37 17760(34) 18599(75) -54(121) -15(33)
16 41 19317(27) 20303(82) -330(135) -81(33)

 

106

 

C .2.4 Asymmetric factor

Table C8: The asymmetric term to the parallel momen-

tum widths from the reaction of 40Ar + ”atNi.

 

 

 

 

 

 

 

 

Z A a Z A a
4 10 ~(-) 10 28 532.09(455.14)
4 11 -(-) 10 29 -(-)
4 12 505.75(282.47) 10 30 -(-)
5 12 -(-) 11 26 -(-)
5 13 403.75(96.57) 11 27 796.77(142.15)
5 14 582.96(169.55) 11 28 148.32(92.23)
5 15 ~(-) 11 29 - -
6 . 14 -(-) 11 30 516.75(293.24)
6 15 688.53(135.19) 11 31 -(-)
6 16 194.16(85.41) 12 28 -(-)
6 17 369.6(113.79) 12 29 -(-)
6 18 213.1(171.79) 12 30 -(-)
7 17 -(-) 12 31 -(-)
7 18 226.15(85.85) 12 32 552.09(171.56)
7 l9 314.86(75.72) 12 33 -(-)
7 20 -(-) 12 34 -(—)
7 21 118.71(185.44) 12 35 -(-)
8 19 -(-) 13 31 612.39(240.02)
8 20 471(87.12) 13 32 437.36(119.24)
8 21 1 18.48(84. 1) 13 33 306.36(103.15)
8 22 397.08(102.22) 13 34 577.97(214.4)
8 23 -(-) 13 35 -(-)
8 24 -(-) 13 36 -(-)

1 9 21 -(-) 14 33 -(-)
9 22 645.98(174.91) 14 34 352.12(74.52)
9 23 257.37(72.39) 14 35 -(-)
9 24 340.99(138.46) 14 36 489.36(284.97)
9 25 541 .44(231.07) 14 37 -(-)
9 26 -(-) 14 38 -(-)
9 27 -(-) 15 35 -(-)
10 24 -(-) 15 36 515.66(93.43)
10 25 401.22(95.37) 15 37 700.16(202.09)
10 26 328.75(122.56) 16 41 -(-)
10 27 433.03(360.93)

 

 

1 Corrected for the broading due to energy straggling.

107

 

C.3 flagments produced from the fragmentation

of 40Ar +181Ta

C.3.1 Cross Section

Table 0.9: Reaction Cross Sections of 40Ar+181Ta.

 

 

 

 

 

 

 

 

z A a(mb) EPAX DIT+GEMINI ISABEL+GEMINI
4 10 0.97(0.10)3+1 2.64E+00 2.37(0.03)E+01 1.13(0.02)E+l
4 11 6.20(0.71)31 4.72301 7.78(0.46)301 1.08(0.07)E+0
4 12 1.92(0.54)31 5.62302 3.38(0.3)301 1.39(0.23)31
5 12 4.89(0.53)E+0 3.483+00 4.13(0.11)3+00 8.58(0.19)E+0
5 13 2.03(0.16)E+0 6.84301 1.27(0.59)E+00 1.90(o.09)3+0
5 14 1.04(0.10)31 8.99302 7.02(1.38)E—02 1.98(0.28)31
5 15 3.13(0.67)32 8.40303 2.16(0.76)302 4.36(1.32)32
6 14 9.89(l.21)E+0 4.633+00 1.52(0.2)E+01 1.29(0.02)3+1
6 15 1.26(O.15)E+0 1.003+00 6.26(0.41)301 7.78(0.56)31
6 16 3.38(0.37)E-1 1.46301 9.261(0.5)30 3.85(0.39)31
6 17 2.82(0.22)E—2 1.52302 2.7(0.85)302 2.38(0.97)E—2
6 18 7.42(0.98)33 1.19303 1.62(0.66)302 3.97(3.97)33
6 19 1.75(0.25)34 7.16305 () ()

7 17 1.91(O.21)E+0 1.49E+00 2(0.07)E+00 2.6l(0.lO)E+0
7 18 3.69(0.27)31 2.41301 1.76(0.22)301 3.41(0.37)31
7 19 9.15(0.67)32 2.80302 7.83(1.45)302 6.74(1.64)32
7 20 7.84(0.68)33 2.45303 -(-) -(-)

7 21 1.17(0.14)33 1.67304 -(-) -(-)

7 22 3.63(0.53)E—5 9.12306 () ()

8 19 1.63(O.19)E+0 2.22E+00 2(0.07)E+00 l.69(0.08)E+0
8 20 5.91(0.68)31 4.00301 1.33(0.06)E+00 7.82(0.56)31
8 21 6.84(0.51)32 5.20302 4.59(1.11)3o2 4.36(1.32)32
8 22 1.06(0.09)32 5.12303 1.89(0.71)302 3.17(1.12)32
8 23 3.09(0.31)34 3.94304 () 3.97(3.97)33
8 24 2.79(0.40)34 2.45305 - - -(-)

9 21 3.82(0.41)E+0 3.323+00 2.56(0.82)E+00 3.53(0.12)E+0
9 22 7.12(0.80)31 6.67301 3.97(0.33)301 5.16(0.45)31
9. 23 1.61(0.12)31 9.74302 1.27(0.19)301 1.75(0.26)31
9 24 1.36(0.11)32 1.08302 5.4(3.8)303 7.93(5.61)33
9 25 1.39(0.14)33 9.42304 () 7.93(5.61)33
9 26 1.19(0.17)34 6.65305 -(-) —(-)

9 27 8.91(1.47)36 3.89306 -(—) ()

10 24 9.46(1.09)31 1.12E+00 1.77(0.07)E+00 1.17(0.07)E+0
10 25 1.18(0.10)E-1 1.83301 8.91(1.55)302 6.35(1.59)32
10 26 2.33(0.19)32 2.30302 1.62(0.66)302 1.19(0.69)32
10 27 8.32(0.99)34 2.27303 -(-) 3.97(3.97)33

 

108

 

Table 0.9 - continued from previous page

 

 

 

 

 

 

 

 

z A a(mb) EPAX DIT+GEMINI ISABEL+GEMINI
10 28 3.18(0.68)34 1.82304 () ()

10 29 1.10(0.20)35 1.22305 -(-) -(-)

10 30 1.67(0.46)36 6.94307 () -(-)

11 26 1.18(0.13)E+0 1.87E+00 6.99(4.42)301 7.22(0.54)31
11 27 3.10(0.30)31 3.46301 1.78(0.22)301 1.94(0.28)31
11 28 3.55(0.30)32 4.90302 1.62(0.66)302 1.19(0.69)32
11 29 7.18(0.68)33 5.50303 8.1(4.7)303 -(-)

11 30 6.97(0.84)E—4 5.04304 () ()

11 31 1.21(0.27)34 3.86305 () -(-)

12 29 2.56(0.26)31 6.54301 4.56(0.35)301 6.74(1.64)32
12 30 7.09(0.65)32 1.05301 5.08(0.37)301 4.36(1.32)32
12 31 7.79(0.80)33 1.34302 2.7(2.7)303 -(-)

12 32 1.64(0.19)33 1.40303 () ()

12 33 l.26(0.19)E—4 1.26304 () ()

13 31 6.81(0.81)31 1.233+00 9.61(0.51)301 3.61(0.38)31
13 32 1.03(0.09)31 2.24301 2.16(0.76)302 2.78(1.05)32
13 33 2.74(0.25)32 3.32302 () 3.97(3.97)33
13 34 3.13(0.45)33 4.22303 () -(-)

13 35 5.87(0.77)34 4.51304 () -(-)

13 36 3.34(0.53)35 4.51306 () -(-)

14 33 7.68(0.81)31 2.333+00 4.37(0.34)301 4.17(0.41)31
14 34 2.13(0.20)31 5.08301 7.56(1.43)E-02 1.15(0.21)31
14 35 1.92(0.28)32 9.04302 () 3.97(3.97)33
14 36 4.24(0.79)33 1.21302 () ()

15 36 3.55(0.40)31 1.18E+00 1.13(0.18)301 1.83(0.27)31
15 37 1.04(0.10)31 2.23301 3.24(0.94)302 1.03(0.20)31
15 38 4.93(0.74)33 2.23303 -(-) 3.97(3.97)33
16 38 1.49(0.18)E+0 3.343+00 l.54(0.06)E+00 1.05(0.06)E+0
16 41 5.98(0.82)33 2.67308 -(-) ()

 

C.3.2 Parallel Momentum Widths

109

 

Table C. 10: Parallel momentum widths from the reaction

of 40Ar + 181Ta.

 

 

 

 

 

 

 

Lab Projectile

flame flame

Z A 0(P) 0(E) 0.13 10N(E) 1057(1)) 0570’)
(MeV/c) (MeV) MeV MeV (MeV/c) (MeV/c)

4 10 292(8) 137(11) 82.1(0.3) 109(14) 234(21) 207(26)
4 11 384(14) 176(12) 92.3(0.4) 149(14) 327(17) 291(21)
4 12 349(50) 160(13) 102.6(0.4) 123(17) 268(69) 238(79)
5 12 336(4) 156(10) 91.2(0.4) 127(12) 273(15) 242(18)
5 13 340(5) 158(9) 101.4(04) 121(12) 260(13) 230(16)
5 14 372(9) 172(10) 111.7(05) 130(13) 282(19) 250(23)
5 15 379(41) 175(10) 121.9(0.5) 125(14) 272(40) 241(47)
6 14 398(13) 182(11) 97.8(0.4) 154(13) 335(32) 298(37)
6 15 385(23) 178(9) 108.0(0.4) 142(12) 307(19) 272(23)
6 16 423(13) 194(9) 118.2(0.5) 154(11) 336(10) 299(12)
6 17 419(8) 193(9) 128.5(0.5) 144(12) 313(15) 278(18)
6 18 405(25) 188(8) l38.7(0.6) 127(12) 273(20) 242(23)
6 19 432(31) 197(8) 148.9(O.6) 129(13) 283(19) 252(23)
7 17 420(11) 193(9) 112.0(05) 157(11) 342(15) 304(19)
7 18 387(5) 178(8) 122.3(0.5) 130(10) 282(10) 250(13)
7 19 425(6) 195(8) 132.5(0.6) 144(11) 313(11) 278(14)
7 20 428(8) 198(8) 142.7(0.6) 137(11) 296(14) 263(17)
7 21 445(24) 206(8) 153.0(0.6) 138(12) 298(19) 264(22)
7 22 407(23) 185(7) 163.2(0.7) 88(14) 193(12) 172(15)
8 19 413(12) 189(8) 113.6(0.5) 151(10) 330(21) 294(25)
8 20 427(21) 196(8) 123.8(O.5) 152(10) 331(20) 294(24)
8 21 403(5) 185(7) 134.0(0.6) 128(10) 279(9) 248(12)
8 22 436(7) 200(7) 144.3(0.6) 138(10) 301(12) 268(14)
8 23 446(13) 205(7) 154.5(0.6) 135(11) 294(19) 261(23)
8 24 449(25) 207(7) 164.7(0.7) 125(12) 272(18) 241(21)
9 21 460(12) 208(7) 112.5(05) 175(9) 387(9) 346(11)
9 22 450(17) 205(8) 122.8(0.5) 164(9) 360(18) 321(21)
9 23 377(4) 174(6) 133.0(0.6) 111(9) 242(7) 215(10)
9 24 420(7) 192(6) l43.2(0.6) 128(9) 280(10) 249(12)
9 25 426(10) 195(6) 153.5(0.6) 121(10) 263(13) 234(16)
9 26 481(19) 220(7) 163.7(0.7) 147(11) 321(27) 286(31)
9 27 474(36) 218(7) 173.9(07) 132(12) 287(28) 255(33)
10 24 427(15) 194(7) 119.2(05) 153(8) 337(16) 300(19)
10 25 374(6) 172(5) 129.5(0.5) 113(8) 245(9) 218(12)
10 26 374(6) 172(5) 139.7(0.6) 100(9) 218(8) 194(11)
10 27 416(11) 190(6) 149.9(0.6) 117(9) 256(14) 228(16)
10 28 407(21) 187(6) 160.2(0.7) 97(11) 211(22) 188(25)
10 29 439(44) 201(7) 170.4(0.7) 106(13) 232(42) 207(48)
10 30 484(71) 222(8) 180.6(0.8) 129(14) 281(55) 250(63)

 

 

 

 

 

110

 

Table C.10 — continued from previous page

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Z A 0(P) 0(E) ”(1E 10MB) 10N(P) 0N(P)
(MeV/c) (MeV) MeV MeV (MeV/c) (MeV/c)
11 26 443(11) 200(6) 113.2(0.5) 165(7) 365(5) 326(7)
11 27 404(10) 184(5) 123.4(0.5) 137(7) 299(11) 267(13)
11 28 373(7) 171(5) 133.6(0.6) 106(8) 232(8) 206(10)
11 29 386(8) 177(5) 143.9(0.6) 103(8) 224(11) 199(13)
ll 30 412(10) 188(5) 154.1(0.6) 108(9) 236(10) 210(12)
11 31 433(37) 199(6) 164.3(0.7) 111(11) 243(29) 216(33)
12 29 381(11) 173(5) 114.8(0.5) 130(6) 285(11) 254(13)
12 30 349(8) 160(4) 125.1(0.5) 100(7) 218(9) 194(11)
12 31 361(10) 166(4) 135.3(0.6) 95(7) 208(10) 185(12)
12 32 368(12) 168(4) 145.5(0.6) 85(9) 185(13) 165(15)
12 33 394(18) 179(5) 155.7(0.6) 89(9) 196(15) 174(18)
13 31 365(13) 166(4) 103.7(0.4) 130(6) 285(13) 254(15)
13 32 343(7) 156(4) 113.9(0.5) 107(6) 235(7) 209(9)
13 33 332(8) 152(4) 124.2(0.5) 88(6) 191(7) 170(9)
13 34 341(15) 156(4) 134.4(0.6) 79(8) 173(13) 154(16)
13 35 449(19) 204(5) 144.6(0.6) 143(7) 316(13) 282(16)
13 36 355(35) 161(4) 154.9(0.6) 42(16) 94(23) 84(28)
14 33 367(7) 166(4) 90.1(0.4) 140(4) 309(2) 275(3)
14 34 304(7) 139(3) 100.3(0.4) 96(5) 210(6) 187(7)
14 35 333(25) . 152(4) 110.5(0.5) 104(6) 229(15) 204(17)
14 36 316(28) 144(4) 120.8(0.5) 79(7) 173(15) 154(18)
15 36 266(9) 121(3) 84.1(0.3) 87(4) 191(9) 170(10)
15 37 277(6) 126(3) 94.4(0.4) 83(4) 183(6) 163(7)
15 38 299(25) 136(3) 104.6(0.4) 87(5) 190(11) 170(13)
16 38 230(7) 104(2) 65.4(0.3) 81(3) 179(7) 159(8)
16 41 282(20) 126(3) 96.1(0.4) 82(4) 183(12) 164(13)
C.3.3 Parallel Momentum Transfer
Table 0.11: Parallel momentum transferred from the re-
action of 40Ar + 181Ta. All units are in MeV.
Lab flame Projectile flame
z A <P> 1<Pc°" > 1<Pwrr > 1<P’>
4 10 4936(18) 4996(39) -34(50) -35(51)
4 11 5270(15) 5334(44) -181(56) —167(51)

 

 

 

 

 

 

 

 

 

1 Widths are corrected for the broading due to energy straggling.

111

 

Table C.11 — continued from previous page

 

 

 

 

Lab flame Projectile flame
Z A <P> 1<PC°" > 1<P°°" > l<P’>
4 12 5663(57) 5729(52) -277(65) —234(55)
5 12 5873(14) 5968(42) -65(55) -55(46)
5 13 6349(14) 6444(47) —90(61) -70(47)
5 14 6787(22) 6884(52) -146(67) -105(49)
5 15 7264(48) 7361(59) -169(76) -114(51)
6 14 6715(35) 6856(47) -171(62) —123(45)
6 15 7300(23) 7438(50) -101(66) -68(45)
6 16 7695(12) 7836(54) -194(71) -123(45)
6 17 8234(20) 8373(59) -165(77) -98(46)
6 18 8766(28) 8905(64) -140(84) -79(47)
6 19 9079(28) 9222(69) -304(90) -162(48)
7 17 8175(21) 8366(52) —171(70) —102(42)
7 18 8710(14) 8899(56) —145(75) -81(42)
7 19 9147(16) 9338(60) -202(81) -107(43)
7 20 9699(23) 9888(65) -161(87) -82(44)
7 21 10195(31) 10383(70) -168(93) -81(45)
7 22 10482(20) 10678(74) -354(98) -162(45)
8 19 9091(31) 9342(56) -198(77) -106(41)
8 20 9618(32) 9867(60) -180(81) -91(41)
8 21 10117(15) 10365(62) -185(85) -89(41)
8 22 10570(20) 10820(66) -227(90) -104(42)
8 23 11100(35) 11348(72) —205(97) —90(43)
8 24 11605(34) 11853(76) 205(102) -86(43)
9 21 9908(14) 10233(57) -302(80) -145(38)
9 22 10505(31) 10823(62) -225(86) -103(40)
9 23 11110(14) 11422(63) —141(89) -62(39)
9 24 11515(19) 11831(68) —224(94) —94(40)
9 25 12029(26) 12344(72) -216(100) -87(40)
9 26 12424(54) 12743(80) —308(109) —120(42)
9 27 13034(60) 13348(85) -219(115) —82(43)
10 24 11419(29) 11813(64) —240(91) -101(38)
10 25 12044(18) 12430(66) —140(93) -57(38)
10 26 12532(17) 12918(69) -153(98) -60(38)
10 27 12928(29) 13318(74) -245(104) -92(39)
10 28 13524(48) 13909(81) -167(112) -60(40)
10 29 13871(94) 14263(96) -300(128) -105(44)
10 30 14404(130) 14794(110) —275(143) -93(48)
11 26 12293(11) 12773(65) -281(94) -109(37)
11 27 12890(24) 13363(69) —205(99) -77(37)
11 28 1346208) 13929(72) —150(103) —54(37)
11 29 13923(25) 14391(76) -186(108) —65(38)
11 30 14352(24) 14825(79) -248(112) -84(38)
11 31 14921(70) 15388(91) -195(125) -64(4l)

 

 

 

 

 

 

112

 

Table 0.11 — continued from previous page

 

 

 

 

 

 

 

 

 

Lab flame Projectile flame
Z A <P> 1<PC°" > l<Pc""'> l<P’>
12 29 13814(24) 14376(72) -199(104) -69(36)
12 30 14403(22) 14959(74) -l30(108) -44(36)
12 31 14890(25) 15446(78) -145(112) -47(37)
12 32 15347(32) 15904(82) -184(ll8) -58(37)
12 33 15749(40) 16312(87) -268(124) -82(38)
13 31 14717(32) 15379(77) -203(111) -66(36)
13 32 15254(18) 15912(77) -178(113) -56(36)
13 33 15835(19) 16486(80) -116(ll7) -35(36)
13 34 16283(35) 16937(85) -161(123) -48(37)
13 35 16593(36) 17259(89) -322(128) -93(37)
13 36 17009(62) 17680(98) -394(138) -111(39)
14 33 15614(5) 16384(78) -205(115) -63(35)
14 34 1620305) 16964(80) -138(118) -41(35)
14 35 16732(41) 17489(87) -118(l26) -34(36)
14 36 17210(43) 17968(90) -140(130) -39(36)
15 36 17134(25) 18006(85) -107(l26) -30(35)
15 37 17620(16) 18492(87) -122(128) -33(35)
15 38 18068(3) 18944(92) -168(134) -45(36)
16 38 18042(19) 19035(89) -88(l32) -23(35)
16 41 19197(36) 20219(100) -376(147) -93(36)

 

 

1 Corrected for the broading due to energy straggling.

113

 

C.3.4 Asymmetric factor

Table C .12: The asymmetric term to the parallel momen-

tum widths from the reaction of 40Ar + 181Ta.

 

A

a

Z

A

a

 

5sococcococooooooooommszxrsrxl-q‘lcacaczcaczczmcnmcnhas.1:-N

 

 

10
11
12
12
13
14
15
14
15
16
17
18
19
17
18
19
20
21
22
19
20
21
22
23
24
21
22
23
24
25
26
27
24
25

 

609.46(234.91)

-(-)
1186.08(693.7)

-(-)
577.63(117.67)
658.15(191.9)
788.19(305.11)

-(-)

757.4(249.9)

-(-)
392.27(104.25)
669.35(163.74)

-(-)

-(-)
427.05(88.61)
276.16(81.89)
734.53(158.51)
531.58(167.27)

-(-)

-(-)
456.32(267.28)
300.94(73.73)
359.74(128.79)

290.1(226.3)

-(-)

-(-)
332.65(434.98)
530.87(72.64)
128.81(86.19)
519.03(187.15)
467.84(450.33)

-(-)

-(-)
570.05(108.59)

 

10
10
10
10
10
11
11
11
11
ll
11
12
12
12
12
12
13
13
13
13
13
13
14
14
14
14
15
15
15
16
16
15
16

 

26
27
28
29
30
26
27
28
29
30
31
29
30
31
32
33
31
32
33
34
35
36
33
34
35
36
36
37
38
38
41
37
41

 

523.33(95.48)
-(-)
839.03(846.29)

426.47(231.32)
478.96(91.95)
3250405737)

-(-)
722.16(586.97)
386.93(265.73)
720.43(126.34)
542.68(127.55)
367.31(187.68)

-(-)

-(-)
337.75(108.03)
529.42(83.78)
604.33(279.51)

292.28(93.83)
341.01(191.36)
200.69(382.35)
562.19(169.6)
205(78.43)

700.16(202.09)
-(-)

 

114

 

Table 0.13: The angular transmission of fragments in
percent as predicted by LISE.

 

 

 

target target target target target target
z A 9Be M‘Ni 181'ra z A 9Be Wm 18115
4 10 0.14 0.25 0.25 10 28 0.73 0.73 0.73
4 11 0.19 0.27 0.27 10 29 0.74 0.73 0.72
4 12 0.21 0.31 0.31 10 30 0.76 0.75 0.75
5 12 0.21 0.31 0.30 11 26 0.68 0.71 0.70
5 13 0.24 0.33 0.33 11 27 0.72 0.74 0.73
5 14 0.27 0.37 0.37 11 28 0.74 0.75 0.73
5 15 0.28 0.40 0.40 11 29 0.76 0.76 0.75
6 14 0.27 0.36 0.36 11 30 0.77 0.78 0.76
6 15 0.28 0.40 0.40 11 31 0.79 0.79 0.78
6 16 0.31 0.42 0.42 11 32 0.79 - -
6 17 0.35 0.46 0.45 11 33 0.82 - -
6 18 0.42 0.49 0.49 12 28 0.75 0.75 -
6 19 0.45 - 0.50 12 29 0.77 0.78 0.76
7 16 0.35 - - 12 30 0.80 0.80 0.78
7 17 0.35 0.45 0.45 12 31 0.80 0.81 0.79
7 18 0.41 0.49 0.48 12 32 0.79 0.82 0.80
7 19 0.44 0.52 0.51 12 33 0.82 0.82 0.81
7 20 0.47 0.53 0.53 12 34 - 0.83 -
7 21 0.51 0.56 0.55 12 35 - 0.84 -
7 22 0.54 - 0.57 13 31 0.81 0.83 0.81
8 19 0.45 0.52 0.51 13 32 0.79 0.83 0.82
8 20 0.48 0.54 0.54 13 33 0.84 0.83 0.82
8 21 0.51 0.57 0.57 13 34 0.84 0.83 0.83
8 22 0.55 0.60 0.60 13 35 0.87 0.85 0.85
8 23 0.58 0.62 0.61 13 36 0.87 0.86 0.85
8 24 0.62 0.74 0.61 14 33 0.85 0.85 0.84
9 21 0.52 0.57 0.56 14 34 0.84 0.86 0.85
9 22 0.55 0.59 0.59 14 35 0.87 0.86 0.85
9 23 0.57 0.63 0.62 14 36 0.89 0.89 0.87
9 24 0.61 0.65 0.65 14 37 0.96 0.89 0.87
9 25 0.63 0.67 0.66 14 38 - 0.91 0.87
9 26 0.67 0.67 0.68 15 35 0.90 0.89 0.89
9 27 0.69 0.69 0.69 15 36 0.89 0.89 0.92
10 24 0.61 0.66 0.66 15 37 0.91 0.91 -
10 25 0.66 0.69 0.68 15 38 0.92 - -
10 26 0.68 0.70 0.69 16 41 - 0.93 0.91
10 27 0.71 0.72 0.70

 

 

 

 

 

 

 

 

 

 

 

115

 

Appendix D

Data from the Fragmenation 48Ca

with 181Ta

The following tables list the isotropic cross sections in mb for the reaction of a~ 128
MeV/ nucleon 480a projectile with 181Ta (Table 0.9). Only momentum distributions
that could be fitted are listed below. The uncertainties quoted for the two simulations

are statistical.

D.1 Fragments produced from the fragmentation
of 48Ca +181Ta

D.l.l Cross Section

Table D1: Reaction Cross Sections of 48Ca+181Ta.

 

 

z A a EPAX DIT+GEMINI ISABEL+GEMINI
4 10 3.32(0.38)E+0 2.313+00 1.48(0.02)E+01 2.12(0.03)E+1
4 11 4.36(o.50)31 5.38301 7.62(0.39)301 2.07(0.10)3+0
4 12 2.28(0.29)31 8.80302 4.50(o.30)301 4.32(0.47)31
5 12 1.99(0.23)3+0 3.04E+00 2.83(0.08)E+00 1.55(0.03)E+1
5 13 1.55(0.17)E+O 7.76301 1.20(0.05)E+00 4.30(O.15)E+0
5 14 1.18(0.11)31 1.40301 1.16(0.15)301 5.09(0.51)31
5 15 5.46(0.54)32 1.90302 4.20(0.92)302 1.93(0.31)31

 

 

 

 

 

 

 

 

116

Table D.1 — continued from previous page

 

 

 

z A a EPAX DIT+GEMINI ISABEL+GEMIN:I_
5 17 4.37(0.57)34 1.70304 () ()

6 15 7.23(0.79)31 1,143+00 6.74(0.37)301 1.57(o.09)3+0
6 16 3.84(0.31)E—1 2.28301 1.13(0.05)3+00 1.11(0.08)E+0
6 17 4.62(0.38)E—2. 3.45302 3.20(o.80)302 6.10(1.76)32
6 18 1.60(0.14)32 4.09303 1.40(0.53)302 2.03(1.02)32
6 19 5.13(0.54)34 3.92304 () ()

6 20 7.10(0.81)E—5 3.13305 —(—) ()

7 18 3.79(0.41)31 3.77301 1.64(0.06)E+00 7.53(0.62)31
7 19 1.52(0.12)31 6.38302 3.04(0.25)301 1.88(0.31)E-l
7 20 1.76(0.16)32 8.52303 1.04(0.14)301 ()
7 21 3.27(0.28)33 9.26304 8.00(4.00)E-03 ()

7 22 1.42(0.15)34 8.41305 4.00(2.80)303 ()

7 23 5.56(0.75)36 6.53306 —(-) -(-)

8 20 5.04(0.56)31 6.29301 1.57(0.06)E+00 2.12(0.10)3+0
8 21 1.13(0.10)31 1.20301 1.18(0.15)301 2.49(o.36)31
8 22 2.36(0.20)32 1.81302 1.16(0.15)301 5.60(1.69)32
8 23 1.12(0.12)33 2.23303 2.00303(2.00) 5.09(5.09)33
8 24 1.05(0.15)34 2.31304 () -(—)

9 23 2.21(0.24)31 2.28301 2.38(0.22)301 5.34(0.52)31
9 24 3.12(0.29)32 3.89302 2.60(0.72)302 9.16(2.16)32
9 25 4.63(0.57)33 5.47303 2.00(2.00)303 -(-)

9 26 5.42(0.94)34 6.50304 4.00(2.80)303 -(-)

9 27 3.22(0.48)35 6.66305 () -(-)

9 29 1.97(0.52)37 4.85307 -(-) -(-)

10 25 1.45(1.07)31 4.36301 3.50(0.27)301 3.82(0.44)31
10 26 5.15(0.58)32 8.48302 1.86(0.19)301 1.63(0.29)31
10 27 1.19(0.65)33 1.36302 6.00(3.50)303 1.53(0.88)32
10 28 1.38(0.20)E-3 1.86E-03 () 5.09(5.09)33
10 29 7.00(1.10)35 2.20304 () —(—)
10 30 1.08(0.20)35 2.30305 () -(-)

11 28' 6.46(0.74)E-2 1.86301 8.60(1.31)302 1.07(0.23)31
11 29 2.19(0.25)32 3.43302 1.80(0.60)302 1.53(0.88)32
11 30 3.63(0.45)33 5.39303 2.00(2.00)303 -(-)

11 31 7.29(l.08)E—4 7.37304 -(-) ()

1.1 32 5.71(0.97)35 8.96305 () -(-)

12 30 1.10(0.13)31 4.10301 4.94(0.31)301 2.39(0.35)31
12 31 2.55(0.30)32 8.69302 3.60(0.85)302 2.54(1.14)32
12 32 9.56(1.12)33 1.58302 2.80(0.75)302 2.54(1.14)32
12 33 8.77(1.14)34 2.51303 () -(-)

12 34 1.48(0.23)34 3.55304 () ()

13 34 1.48(0.18)E-2 4.64302 4.00(2.80)303 1.02(o.72)32
13 35 5.81(0.60)33 8.58303 () 1.53(0.88)E-2
13 36 6.02(0.84)34 1.42303 () -(-)

13 37 1.51(0.16)34 2.15304 () ()

 

 

 

 

 

 

 

117

Table D.1 - continued from previous page

 

 

 

 

 

 

 

 

z A a EPAX DIT+GEMINI ISABEL+GEMINI
13 38 1.08(0.29)35 3.01305 -(-) -(-)

14 37 7.71(o.96)33 2.94302 8.00(4.00)303 1.02(0.72)32
14 40 5.94(1.23)35 1.77304 () ()

 

D.l.2 Parallel Momentum Widths

Table D2: Parallel momentum widths from the reaction

of 48Ca. + 181Ta. All units are in MeV/c.

 

 

 

 

 

 

 

 

 

 

 

Lab Projectile

flame flame

Z A 0(1’) 0(E) 0413 103703) 10MP) 0117(1))
(MeV/c) (MeV) MeV MeV (MeV/c) (MeV/c)

4 10 239(4) 112(9) 82.1(0.3) 76(13) 162(13) 143(17)
4 11 268(7) 124(9) 92.3(0.4) 82(14) 178(22) 158(26)
4 12 302(12) 138(9) 102.6(0.4) 92(14) 201(27) 179(32)
5 12 307(5) 146(9) 91.2(0.4) 114(12) 240(13) 211(16)
5 13 351(2) 163(9) 101.4(04) 128(12) 275(9) 244(12)
5 14 405(7) 187(10) 111.7(0.5) 150(13) 325(18) 288(22)
5 15 415(11) 191(10) 121.9(0.5) 147(13) 319(19) 284(23)
5 17 394(23) 177(8) 142.4(0.6) 104(14) 233(24) 208(28)
6 15 359(4) 167(8) 108.0(0.4) 128(11) 274(9) 242(12)
6 16 398(4) 184(9) 118.2(0.5) 141(11) 305(11) 271(14)
6 17 428(4) 197(9) 128.5(0.5) 149(12) 324(12) 288(15)
6 18 456(9) 209(9) 138.7(0.6) 156(12) 341(15) 303(18)
6 19 440(10) 201(8) 148.9(O.6) 135(12) 296(17) 263(21)
6 20 484(17) 219(8) 159.2(07) 150(12) 332(15) 297(18)
7 18 406(3) 187(8) 122.3(0.5) 142(10) 307(8) 273(10)
7 19 440(4) 201(8) 132.5(0.6) 151(11) 331(10) 294(13)
7 20 454(4) 209(8) 142.7(0.6) 152(11) 331(11) 294(14)
7 21 479(5) 218(8) 153.0(0.6) 155(11) 341(11) 304(14)
7 22 471(10) 216(8) 163.2(0.7) 142(12) 309(18) 275(22)
7 23 581(29) 263(10) 173.4(07) 198(13) 437(37) 390(42)
8 20 398(4) 183(7) 123.8(0.5) 135(9) 294(8) 261(10)
8 21 432(6) 198(7) 134.0(0.6) 146(10) 318(13) 283(16)
8 22 442(6) 203(7) 144.3(0.6) 142(10) 311(12) 276(14)
8 23 482(7) 220(8) 154.5(0.6) 157(11) 344(16) 306(19)
8 24 503(12) 230(8) 164.7(0.7) 161(12) 352(25) 313(29)
9 23 436(5) 199(6) 133.0(0.6) 148(9) 324(7) 288(9)

 

118

 

 

Table D2 - continued from previous page

 

 

 

 

 

 

 

 

 

 

Z A 0(P) 0(15"?) ”dB l"N(E) l0MP) °N(P)
(MeV/c) (MeV) MeV MeV (MeV/c) (MeV/c)
9 24 442(8) 202(7) 143.2(0.6) 143(9) 312(13) 277(16)
9 25 446(10) 205(7) 153.5(0.6) 135(10) 295(16) 262(19)
9 26 501(16) 229(7) 163.7(0.7) 159(11) 350(24) 312(28)
9 27 511(13) 234(7) 173.9(07) 156(11) 341(25) 303(29)
10 25 418(81) 191(11) 129.5(0.5) 140(15) 307(116) 273(133)
10 26 452(6) 206(6) 139.7(0.6) 151(8) 332(9) 295(11)
10 27 468(19) 213(7) 149.9(0.6) 151(9) 333(21) 296(25)
10 28 490(21) 224(7) 160.2(O.7) 156(10) 342(24) 304(28)
10 29 503(22) 229(7) 170.4(0.7) 153(10) 336(27) 300(31)
10 30 476(20) 219(7) 180.6(0.8) 124(12) 269(30) 239(35)
11 28 436(7) 198(5) 133.6(0.6) 146(7) 322(8) 287(9)
11 29 460(8) 209(6) 143.9(0.6) 151(8) 334(10) 298(12)
11 30 471(15) 214(6) 154.1(0.6) 149(8) 328(17) 292(20)
11 31 463(22) 212(6) 164.3(0.7) 134(9) 293(22) 260(25)
11 32 524(29) 238(7) 174.6(07) 161(10) 356(34) 317(39)
12 30 409(7) 186(5) 125.1(0.5) 138(6) 304(6) 270(8)
12 31 433(9) 196(5) 135.3(O.6) 143(7) 314(9) 280(11)
12 32 473(10) 214(5) 145.5(0.6) 157(7) 347(11) 309(13)
12 33 497(19) 226(6) 155.7(0.6) 163(8) 360(17) 321(20)
12 34 456(24) 208(5) 166.0(0.7) 125(9) 274(19) 244(22)
13 34 440(12) 199(5) 134.4(0.6) 147(6) 325(11) 290(13)
13 35 535(2) 241(5) 144.6(0.6) 193(6) 428(1) 382(3)
13 36 544(29) 246(6) 154.9(0.6) 191(7) 423(20) 378(23)
13 37 472(6) 215(4) 165.1(0.7) 138(7) 303(3) 270(5)
13 38 491(54) 224(6) 175.3(07) 139(10) 306(45) 272(52)
14 37 422(14) 191(4) 131.0(05) 139(6) 307(9) 274(11)
14 40 417(40) 190(4) 161.7(0.7) 100(8) 220(21) 196(24)

 

D.l.3 Parallel Momentum Transfer

Table D.3: Parallel momentum transferred from the re-
action of 480a + 181Ta. All units are in MeV.

 

 

 

 

 

 

 

Lab flame Projectile flame
Z A <P> 1<Pc°" > 1<P°°"" > l<P’>
4 10 4922(11) 4986(40) -28(52) -34(63)

 

 

 

 

 

1 Widths are corrected for the broading due to energy straggling.

119

 

Table D.3 — continued from previous page

 

 

 

 

 

 

 

 

 

Lab flame Projectile flame
Z A <P> 1<P°°"‘ > 1<Pc°"' > 1<P’>
4 11 5343(20) 5409(45) -98(59) -107(65)
4 12 5724(26) 5793(51) -202(65) -204(66)
5 12 6023(13) 6120(43) 87(58) 88(58)
5 13 6356(10) 6458(48) -57(64) -53(59)
5 14 6771(20) 6876(53) -132(71) -1l4(61)
5 15 7238(23) 7343(59) -163(77) -131(62)
5 17 7939(30) 8051(69) -425(90) -3o2(64)
6 15 7352(11) 7498(51) -25(69) -20(56)
6 16 7748(15) 7898(56) -115(75) -87(57)
6 17 8215(16) 8365(60) -146(81) -104(58)
6 18 8646(21) 8798(66) . -207(88) 439(59)
6 19 9085(26) 9238(71) -261(94) -166(60)
6 20 9457(22) 9614(75) -373(100) -226(61)
7 18 8702(11) 8905(58) -112(80) -75(53)
7 19 9112(15) 9319(62) -190(86) -121(55)
7 20 9640(18) 9845(67) -169(92) -102(55)
7 21 10006(18) 10215(72) -285(98) -164(56)
7 22 10573(32) 10779(77) -230(104) -126(57)
7 23 10873(64) 11086(85) -402(113) -212(60)
8 20 9672(12) 9936(60) -88(84) -53(51)
8 21 10085(22) 10353(65) -163(91) -94(52)
8 22 10574(20) 10842(69) -174(96) -96(53)
8 23 11024(28) 11293(74) -219(102) -115(54)
8 24 11512(46) 11781(80) -231(110) -116(55)
9 23 11010(13) 11350(66) -169(95) -89(50)
9 24 11493(25) 11833(71) -186(101) -94(51)
9 25 12022(31) 12360(76) -163(107) -79(52)
9 26 12404(48) 12747(82) -264(114) -123(53)
9 27 12918(53) 13260(87) -255(120) -114(54)
9 29 14024(77) 14359(99) -170(134) -7l(56)
10 25 11964(227) 12382(147) -l43(181) -69(88)
10 26 12367(17) 12790(73) -226(105) -105(49)
10 27 12865(45) 13287(38) -230(113) -103(51)
10 28 13379(52) 13800(85) -221(119) -95(52)
10 29 13835(60) 14257(90) -260(126) -109(52)
10 30 14486(72) 14899(46) -136(l33) -55(53)
11 28 13324(16) 13834(75) -191(110) -83(48)
11 29 13760(21) 14273(79) -246(116) -103(48)
11 3o 14287(39) 14797(85) -226(122) -91(49)
11 31 14853(53) 15358(91) -174(129) -68(50)
11 32 15180(83) 15695(101) -320(140) —121(53)
12 30 14286(15) 14890(78) -144(116) -58(47)
12 31 14711(22) 15320(82) -208(121) -81(47)

 

120

 

Table D.3 — continued from previous page

 

 

 

 

 

 

 

 

 

Lab flame Projectile flame
Z A <P> 1<P°°" > 1<P°°" > 1<P’>
12 32 15112(27) 15726(86) -292(127) -110(48)
12 33 15655(44) 16266(92) -259(133) -95(49)
12 34 16242(49) l6845(96) -190(138) -68(49)
13 34 16075(29) l6792(89) -237(132) -84(47)
13 35 16486(2) 17208(91) -313(135) -108(47)
13 36 16993(55) 17713(100) -310( 145) -104(49)
13 37 17651(77) 18359(46) -183(143) -60(47)
13 38 18125(133) 18834(128) -206(175) -66(56)
14 37 17464(26) 18296(95) -239(142) -78(46)
14 40 19119(64) 19935(110) «121(160) -36(48)

 

D.l.4 Asymmetric factor

Table D4: The asymmetric term to the parallel momen-
tum widths from the reaction of 48Ca + 181Ta.

 

 

 

 

 

z A a z A a
4 10 -(-) 10 27 -(-)

4 11 692.48(212.85) 10 28 462.74(369.91)
4 12 833.17(285.72) 10 29 ()

5 12 () 10 30 -(-)

5 13 () 11 28 -(-)

5 14 273.31(121.46) 11 29 -(-)

5 15 644.26(152.54) 11 30 603.33(221.82)
5 17 659.97(259.02) 11 31 474.98(273.25)
6 15 -(-) 11 32 -(-)

6 16 229.75(76.76) 12 30 -(-)

6 17 413.53(95.63) 12 31 -(-)

6 18 461.84(112.15) 12 32 -(-)

6 19 425.42(183.43) 12 33 -(-)

6 20 -(—) 12 34 ()

7 18 -(-) 13 34 -(-)

7 19 108.03(58.66) 13 35 -(-)

7 20 542.68(14268) 13 36 -(-)

7 21 243.73(100.53) 13 37 -(-)

7 22 419.63(227.11) 13 38 ()

 

 

 

 

 

 

 

1Corrected for the broading due to energy straggling.

121

 

Table D.4 — continued from previous page

 

 

 

 

 

 

 

 

z A a z A a

7 23 () 14 37 -(-)

8 20 -(- 14 40 -(-)

8 21 252.5(10524) 13 36 -(-)

8 22 301.6(87.66) 14 33 -(-)

8 23 426.79(234.15) 14 34 292.28(93.83)
8 24 354.74(347.35) 14 35 341.01(191.36)
9 23 -(-) 14 36 200.69(382.35)
9 24 267.42(109.99) 15 36 562.19(169.6)
9 25 751.19(315.18) 15 37 205(78.43)

9 26 632.87(454.19) 15 38 -(-)

9 27 () 16 38 -(-)

9 29 () 16 41 ()

9 31 -(—) 15 37 700.16(202.09)
10 25 -(-) 16 41 ()

10 26 ()

 

122

 

Table D5: The angular transmission of fragments in per-
cent as predicted by LISE.

 

 

Z A Transmission Z A Transmission
4 10 0.22 9 26 0.62
4 11 0.25 9 27 0.65
4 12 0.27 9 29 0.68
5 12 0.27 9 31 0.68
5 13 0.29 10 25 0.60
5 14 0.33 10 26 0.63
5 15 0.36 10 27 0.65
5 17 0.41 10 28 0.68
6 15 0.35 10 29 0.70
6 16 0.38 10 30 0.71
6 17 0.41 11 28 0.68
6 18 0.43 11 29 0.70
6 19 0.46 11 30 0.71
6 20 0.49 11 31 0.72
7 18 0.43 11 32 0.72
7 19 0.46 12 30 0.71
7 20 0.49 12 31 0.73
7 21 0.51 12 32 0.75
7 22 0.53 12 33 0.77
7 23 0.55 12 34 0.78
8 20 0.48 13 34 0.78
8 21 0.51 13 35 0.80
8 22 0.53 13 36 0.81
8 23 0.55 13 37 0.82
8 24 0.58 13 38 0.82
9 23 0.55 14 37 0.83
9 24 0.58 14 40 0.85
9 25 0.60

 

 

 

 

 

 

 

 

 

123

 

Bibliography

[1] F. Marti, P. Miller, D. Poe, M. Steiner, J. Stetson, and X.Y. Wu. In P. Lucas
and S. Webber, editors, PA 02001 Proceedings of the 2001 Particle Accelerator
Conference, page 2557. (IEEE Operations Center, 2001), 2001. 1

[2] Z. Dlouhy, J. C. Angéligue, R. Anne, G. Auger, D. Baiborodin, C. Borcea,
E. Caurier, A. Gillibert, S. Grevy, D. Guillemaud-Mueller, A. S. Lalleman,
M. Lewitowicz, S. M. Lukyanov, W. Mittig, J. Mrézek, A. C. Mueller,
F. Nowacki, F. de Oliveira, N. Orr, R. D. Page, Yu. E. Penionzhkevich,
F. Pougheon, A. T. Reed, Z. Ben, D. Ridikas, P. Roussel-Chomaz, M. G.
Saint-Laurent, H. Sakurai, F. Sarazin, H. Savajols, O. Sorlin, O. Tarasov,
A. de Vismes, and J. Winfield. Nucl. Phys. A, 702:189, 2002. 1

[3] M. Theoennessen. Rep. Prog. Phys, 67:1187, 2004. and reference therein. 1,
2, 3, 34, 35

[4] D. Guillemaud-Mueller, Yu.E. Penionzhkevich, R. Anne, A.G. Artukh,
D. Bazin, V. Borrel, C. Détraz, D. Guerrea, B.A. Gvozdev, H.C. Jacmart,
D.X. Jiang, A.M. Kalinin, V.V. Kamanin, V.B. Kutner, M. Lewitowicz, S.M.
Lukyanov, A.C. Mueller, N. Hoai Chau, F. Pougheon, A. Pichard, M.G. Saint-
Laurent, and W.D. Schmidt-Ott. Z. Phys. A, 332:189, 1989. 1, 33, 34

[5] D. Bazin, W. Benenson, B.A. Brown, J. Brown, B. Davids, M. Fauerbach, P.G.
Hansen, P. Mantica, D.J. Morrissey, C.F. Powell, B.M. Sherrill, and M. Steiner.
Probing the halo structure of 19’17’15c and 14b. Phys. Rev. C, 57, 1998. 1

[6] K. Siimmerer, W. Briichle, D.J. Morrissey, M. Schiidel, B. Szweryn, and
Y. Weifan. Phys. Rev. C, 42:2546, 1990. 2, 6, 8, 9, 75

[7] K. Siimmerer and B. Blank. Phys. Rev. C, 61:034607, 2000. 2, 6, 8, 9, 12, 13,
75

[8] D. Bazin, O.B. Tarasov, M. Lewitowicz, and O. Sorlin. Nucl. Inst. and Meth.
A, 482:307, 2001. 2, 6, 17, 38, 56

[9] M. Hencheck, R.N. Boyd, M. Hellstréim, D.J. Morrissey, M.J. Balbes, F.R.
Chloupek, M. Fauerbach, C.A. Mitchell, R Pfaff, C.F. Powell, G. Raimann,
B.M. Sherrill, M. Steiner, J. Vandegriff, and SJ. Yennello. Phys. Rev. C',
50:2219, 1994. 3

124

[10] H. Sakurai, S.M. Lukyanov, M. Notani, N. Aoi, D. Beaumel, N. Fukuda, M. Hi-
rai, E. Ideguchi, N. imai, M. Ishihara, H. Iwasaki, T. Kubo, K. Kusaka, H. Ku-
magai, T. Nakamura, H. Ogawa, Yu.E. Penionzhkevich, T. Teranishi, Y.X.
Watanabe, K. Yoneda, and A. Yoshida. Phys. Lett. B, 448:180, 1999. 3,
33, 83

[11] S.M. Lukyanov, Yu.E. Penionzhkevich, R. Astabatyan, S. Lovastov, Yu.
Sobolev, D. Guillemaud—Mueller, G. Faivre, F. Ibrahim, A.C. Mueller,
F. Pougheon, O. Perrru, O. Sorlin, I. Matea, R. Anne, C. Cauvin, R. Hue,
G. Oliveira, Santos, D. Verney, Z. Dlouhy, J. Mrazek, D. Baiborodin, F. Ne-
goita, C. Borcea, and A. Buta. J. Phys. C, 28:L41, 2002. 3, 83

[12] H. Geissel, P. Armbruster, K.H. Behr, A. Brnle, K. Burkard, M. Chen, H. Fol-
ger, B. Franczak, H. Keller, 0. Klepper, B. Langenbeck, F. Nickel, E. Pfeng,
M. Pfiitzner, E. Roeckl, K. Rykaczewski, I. Schall, D. Schardt, C. Scheiden-
berger, K.-H. Schmidt, A. Schriiter, T. Schwab, K. Smmerer, M. Weber, and
G. anenberg. Nucl. Instr. and Meth. B, 70:286, 1992. 3

[13] KS. Krane. Introductory Nuclear Physics. John Wiley 85 Sons, Inc., 1987. 3,
27

[14] Y.P. Viyogi, T.J.M. Symons, P. Doll, D.E. Greiner andH.H. Heckman, D.L.
Hendrie, P.J. Lindstrom, J. Mahoney, D.K. Scott, K. Van Bibber, G.D. Westfall,
H. Wienman, H.J. Crawford, C. McFarland, and CK. Celbke. Phys. Rev. Lett.,
42:33,1979. 4

[15] J. Gosset, H.H. Gutbrod, W.G. Meyer, A..M Poskanzer, A. Sandoval, R. Stock,
and GD. Westfall. Phys. Rev. C, 16:629, 1977. 4, 14

[16] H. Feshbach and K. Huang. Phys. Lett. B, 47:300, 1973. 5, 63

[17] J .D. Bowman, W.J. Swiatecki, and CE Tsang. Lawrence Berkeley Laboratory
' Report No. LBL-2908, 1973 (unpublished). 5, 9, 14

[18] LR Oliveira, R. Donangelo, and JD. Rasmussen. Phys. Rev. C, 19:826, 1979.
5, 9, 16

[19] Y. Yariv and Z. Fraenkel. Phys. Rev. C, 20:2227, 1979. 5, 9, 17, 20, 71
[20] Y. Yariv and Z. Fraenkel. Phys. Rev. C', 24:488, 1981. 5, 9, 17, 20, 71
[21] J. Cugnon. Nucl. Phys. A, 387:1910, 1982. 5

[22] M. Lefort and C Ngo. Ann. Phys. (Paris), 3:5, 1978. 5

[23] L. Tassan-Got and C. Stépan. Nucl. Phys. A, 524:121, 1991. 5, 9, 24, 71

[24] J.A. Winger, B.M. Sherrill, and DJ. Morrissey. Nucl. Instr. and Meth. B,
70:380,1992. 6

125

[25] K. Chen, Z. flaenkel, G. fliedlander, J.R Grover, J.M. Miller, and Y. Shi-
mamoto. Phys. Rev., 166:949, 1968. 6, 9, 17

[26] C. Wong and K. Van Bibber. Phys. Rev. C, 25:2990, 1982. 6
[27] BC. Harvey. Nucl. Phys. A, 444:498, 1985. 7

[28] http : //www . nscl .msu. edu/aud/exp/propexp/beamlist . php. 7
[29] D. Guerreau. Nucl. Phys. A, 447 :37c, 1985. 7, 37

[30] CD. Westfall, P.J. Johansen, A.M. Poskanzer, W.G. Meyer, H.H. Gutbrod,
A. Sandoval, and R. Stock. Phys. Rev. Lett., 37:1202, 1976. 8

[31] W.U. Schriider and J.R. Huizenga. Annu. Rev. Nucl. Part. Sci, 27:465, 1977.
8

[32] B. Grabez. Phys. Rev. c, 64:057601, 2001. 8
[33] J.-J Gaimard and K.-H. Schmidt. Nucl. Phys. A, 531:709, 1991. 9, 17

[34] DJ. Morrissey, W.R. Marsh, R.J. Otto, W. Loveland, and GT. Seaborg. Phys.
Rev. C, 18:1267, 1978. 9, 14, 15

[35] G. Rudstam. Z. Naturforsch, 21:1027, 1966. 10

[36] G. D. Westfall, T.J.M. Symons, D.E. Greiner, H.H. Heckman, PlJ. Lindstrom,
J. Mahoney, A.C. Shotter, and D.K. Scott. Phys. Rev. Lett., 43:1859, 1973. 13

[37] M. Weber, C.Donzaud, J .P. Dufour, H.Geissel, A. Grewe, D. Guillemaud-
Mueller, H. Keller, M. Lewitowicz, A. Magel, A.C. Mueller, G. Miizenberg,
F. Nickel, M. Pfiitzner, A. Piechaczek, M. Pravikoff, E. Roeckl, K. Rykaczewski,
M.G. Saint-Laurent, I. Shall, C. Stéphan, K. Siimmerer, L. Tassan-Got, D.J.
Vieira, and B. Voss. Nucl. Phys. A, 578:659, 1994. 13, 65

[38] R. Serber. Phys. Rev., 72:1114, 1947. 14, 67

[39] DJ. Morrissey, L.F. Oliveira, J .O. Rasmussen, G.T. Seaborg, Y. Yariv, and
Z. Fraenkel. Phys. Rev. Lett., 43:1139, 1979. 16, 20, 29, 30

[40] J.W. Wilson, L.W. Towsend, and RF. Badavi. Nucl. Instr. and Meth. B, 182225,
1987. 17

[41] R. Charity. Nucl. Phys. A, 483:391, 1988. 23, 27, 71
[42] SK. Samaddar, J.N. De, and K. Krishan. Phys. Rev. C, 31:1053, 1985. 24

[43] G.A. Souliotis, M. Veselsky, G. Chubarian, L. 'IIache, A. Keksis, E. Martin,
D.V. Shetty, and S.J. Yennello. Phys. Rev. Lett., 91:022701, 2003. 26

[44] H. C. Ohanian. Principles of Quantum Mechanics. Prentice Hall, 1990. 26
[45] F. Piihlhofer. Nucl. Phys. A, 280:267, 1977. 27

126

 

[46] A. Gavron. Phys. Rev. C, 212230, 1980. 27
[47] Charles L. Critchfield and Sophie Oleksa. Phys. Rev., 822243, 1951. 27

[48] S. I. Al-Quraishi, S. M. Grimes, T. N. Massey, and D. A. leer. Phys. Rev. C,
63065803, 2001. 27

[49] H. A. Bethe. Phys. Rev., 50:332, 1936. 27

[50] A. S. Iljinov, M. V. MebelN. Bianchi, E. De Sanctis, C. Guaraldo, V. Lucherini,
V. Muccifora, E. Polli, A. R. Rcolon, and P. Rossi. Nucl. Phys. A, 543:517,
1992. 27

[51] J. Toke and W. J. Swiatecki. Nucl. Phys. A, 3742141, 1981. 28
[52] PG. Hansen and BM. Sherrill. Nucl. Phys. A, 6932133, 2001. 31

[53] G.A. Lalazissis, D. Vretenar, W. Paschl, and P. Ring. Phys. Lett. B, 379:13,
1996. 31

[54] G.Audi, A.H.Wapstra, and C.Thibault. Nucl. Phys. A, 729:337, 2003. 32, 33,
36

[55] T. Tachibana, M. Uno, and M. Yamada S. Yamada. At. Data and Nucl. Data
Tables, 392251, 1988. 34

[56] BA Brown and W.A. Richter. Phys. Rev. C, 722057301, 2005. 32

[57] M. Fauerbach. Study of Neutron Rich Nuclei. PhD thesis, Michigan State
University, 1997. 33

[58] A. Schiller, T. Baumann, J. Dietrich, S. Kaiser, W. Peters, and M. Thoennessen.
Phys. Rev. C, 722037601, 2005. 33

[59] H. Sakurai, S.M. Lukyanov, M. Notani, N. Aoi, D. Beaumel, N. Fukuda, M. Hi-
rai, E. Ideguchi, N. imai, M. Ishihara, H. Iwasaki, T. Kubo, K. Kusaka, H. Ku-
magai, T. Nakamura, H. Ogawa, Yu.E. Penionzhkevich, T. Teranishi, Y.X.
Watanabe, K. Yoneda, and A. Yoshida. In B.M. Sherrill, D.J. Morrissey, and
C. N. Davids, editors, ENAM98:E':1:otic Nuclei and Atomic Masses, pages 233-
238. The American Institute of Physics, 1998. 33

[60] O Tarasov, R Allatt, J .C. Angelique, R. Anne, C. Borcea, Z. Dlouhy, C. Don-
zaud, S.Grévy, D. Guillemaud-Mueller, M. lewitowicz, S. Lukyanov, A.C.
Mueller, F. Nowacki, Yu. Oganessian, N.A. Orr, A.N. Ostrowski, R.D. Page,
Yu. Penionzhkevich, F. Pougheon, A. Reed, M.G. Saint-Laurent, W. Schwab,
E. Sokol, O. Sorlin, W. Trinder, and J .S. Winfield. Phys. Lett. B, 409264, 1997.
33

[61] D. Bazin, B.A. Brown, J. Brown, M. Fauerbach, M. Hellstriim, S.E. Hirzebruch,
J.H. Kelley, R.A. Kryger, D.J. Morrissey, R. Pfaff, C.F. Powell, B.M. Sherrill,
and M. Thoennessen. Phys. Rev. Lett., 74:18, 1995. 33

127

 

[62] M. Langevin, E. Quiniou, M. Bernas, J. Galin, J. C. Jacmart, F. Naulin,
F. PougheonR. Anne, C. Dtraz, D. Guerreau, D. Guillemaud-Mueller, and A. C.
Mueller. Phys. Lett. B, 150271, 1985. 33

[63] Y. Utsuno, T. Otsuka, T. Mizusaki, and M. Honma. Phys. Rev. C,
64:011301(R),2001. 33

[64] P. Méiller, J.R. Nix, and K. L. Kratz. At. Data and Nucl. Data Tables, 662131,
1997. 34, 35, 36

[65] BA. Brown. Prog. Part. Nucl. Phys, 472517, 2001. 34, 35, 36

[66] Subcommittee on Performance Measures. Report submitted by the subcommit-

tee on performance measures. http: Hm. sc .doe.gov/np/nsac/nsac .html,
2003. 35

[67] V. Borrel, B. Gatty, D. Guerrau, J. Galin, and D. Jacquet. Z. Phys. A, 324,
1986. 37

[68] DJ. Morrissey, B.M. Sherrill, M. Steiner, A. Stolz, and I. Wiedenhover. Nucl.
Instr. and Meth. B, 204290, 2003. 38

[69] C. Scheidenberger and H. Geissel. Nucl. Instr. and Meth. B, 135225, 1998. 44,
60

[70] H. Geissel and C. Scheidenberger. Nucl. Instr. and Meth. B, 136:114, 1998.
44, 60

[71] http://www-linux.gsi.de/“weick/atima/. 44
[72] Nucl. Instr. and Meth. B, 1262284, 1997. 50
[73] O. Tarasov. Nucl. Phys. A, 7342536, 2004. 56

[74] M. Notani. Projectile Fragmentation Reactions and Production of Nuclei near
the Neutron Drip-line. PhD thesis, University of Tokyo, 2000. 56, 72, 73, 74

[75] M. Mocko. private communications. 56, 72, 73, 74, 79

[76] R. Pfaff. Projectile Fragmentation of Krypton Isotopes at Intermediate Energies.
PhD thesis, Michigan State University, 1996. 57

[77] GE Knoll. Radiation Detection and Measurement. John Wiley & Sons, Inc.,
third edition edition, 2000. 58

[78] N. Tsoulfanidis. Measurement and Detection of Radiation. Taylor and Francis,
second edition, 1995. 59

[79] RD. Evens. The Atomic Nucleus. Robert E. Krieger Publishing Co., 1982. 59
[80] HA. Bethe. Ann. Physik, 5:325, 1930. 60

128

[81] F. Bloch. Ann. Physik, 772287, 1932. 60

[82] J. Ziegler, J. Biersack, and U. Littmark. In The Stopping and Ranges of Ions
in Matter. Pergamon Press, New York, 1985. 60

[83] F. Hubert, R. Bimbot, and H. Gauvin. At. Data and Nucl. Data Tables, 4621,
1990. 60, 61

[84] JP Dufour, R. Del Moral, H. Emmermann, F. Hubert, D. Jean, C. Poinot,
' M.S. Pravikoff, A. Fleury, H. Delagrange, and K.-H. Schmidt. Nucl. Instr. and
Meth. A, 2482267, 1986. 60, 61

[85] H.H. Heckman et al. In G. Tibell, editor, High Energy Physics and Nuclear
Structure, Proceedings of the Fifth International Conference on High-Energy
Physics and Nuclear Structure, pages LBL-2052 (unpublished). North—Holland,
Amsterdam / American Elsevier, New York, 1973. 63

[86] A. S. Goldhaber. Phys. Lett. B, 532306, 1974. 64

[87] E.J. Moniz, I. Sick, R.R. Whitney, J.R. Ficenec, R.D. Kephart, and W. P.
Trower. Phys. Rev. Lett., 262445, 1971. 64

[88] W.A. Friedman. Phys. Rev. C, 27 :569, 1983. 65, 66

[89] G. Bertsch. Phys. Rev. Lett., 462472, 1981. 65

[90] M.J.Murphy and RC. Stokstad. Phys. Rev. C, 282428, 1983. 65
[91] BC. Harvey. Phys. Rev. Lett.,472454,1981. 65

[92] C. Egelhaaf, G. Bohlen, H. Fuchs, A. Gamp, H. Homeyer, and H. Kluge. Phys.
Rev. Lett., 46:813, 1981. 65

[93] DE. Greiner, P.J. Lindstrom, H.H. Heckman, B. Cork, and ES. Bieser. Phys.
Rev. Lett., 35:152, 1975. 65

[94] F. Rami, J.P. Coffin, G. Guillaume, B. Heusch, P. Wagner, A. Fahli, and
P. Fintz. Nucl. Phys. A, 4442325, 1985. 65

[95] C. Vincent-Donnet, M. Stern, E. Gerlic, M.T. Magda, B. Chambon,
A. Chevarier, N. Chevarier, B. Drain, C. Pastor, A. Giorni, D. Heuer, A. Lleres,
and J.B. Viano. J. Phys. C, 222397, 1996. 65

[96] G. Fares, R. Bimbot, S. Hachem, M. Mirea, R. Anne, T. Benfoughal, C. Cabot,
F. Clapier, P. Delbourgo—Salvador, T. Ethvignot, A. Lefebvre, M. Lewitow-
icz, P. Roussel-Chomaz, M.G. Saint-Laurent, J.E. Sauvestre, J.L. Sida, and
Yang Yong Feng. Eur. Phys. J. A, 192105, 2002. 65

[97] C. Stéphan, L. Tassan-Got, D. Bachelier, C.O. Bacri, R. Rimbot, B. Borderie,
J.L. Boyard, F. Clapier, C. Donzaud, T. Hennino, M.F. Rivet, P. Roussel,
D. Bazin, C. Grunber, D. Disdier, and B. Lott. Phys. Lett. B, 26226, 1991.
65

129

 

[98] J. Mougey, R. Ost, M. Buenerd, A. J. Cole, C. Guet, D. Lebrun, J. M. Loiseaux,
P. Martin, M. Maurel, E. Monnand, H. Nifenecker, P. Perrin, J. Pinston, C. Ris-
tori, P. de Saintignon, F. Schussler, L. Carlén, B. Jakobsson A. Oskarsson, I. Ot-
terlund, B. Schroder H. A. Gustafsson, T. Johansson, H. Ryde, J. P. Bondorf,
O. B. Nielsen, and G. Tibell. Phys. Lett. B, 105225, 1981. 65

[99] R. Pfaff, D.J. Morrissey, M. Fauerbach, M. Hellstriim, J .H. Kelley, R.A. Kryger,
B.M. Sherrill, M. Steiner, J.S. Winfield, J.A. Winger, S.J. Yennello, and BM.
Young. Phys. Rev. C, 5121348, 1995. 65

[100] DJ. Morrissey. Phys. Rev. C, 392460, 1989. 66
[101] A. Abul-magd, J. Hiifner, and B. Schiirmann. Phys. Lett. B, 602327, 1976. 68

[102] J. B. Cumming, P. E. Haustein, and H. C. Hseuh. Phys. Rev. C, 2422162, 1981.
68

[103] M.V. Ricciardi, P. Armbruster, J. Benlliure, M. Bernas, A. Boudard, S. Cza-
jkowski, T. Enqvist, A. Kelié, S. Leray, R. Legrain, B. Mustapha, J. Pereira,
F. Rejmund, K.—H. Schmidt, C. Stéphan, L. Tassan-Got, C. Volant, and O. Yor-
danov. Phys. Rev. C, 732041607, 2006. 69

[104] L. Shi, P. Danielewicz, and R. Lacey. Phys. Rev. C, 642034601, 2001. 70

[105] M. Notani, H. Sakurai, N. Aor, Y. Yanagisawa, A. Saito, N. Imai, T. Gomi,
M. Miura, S. Michimasa, H. Iwasaki, N. Eukada, M. Ishihar, T. Kubo,
S. Kubono, H.Kumagai, S.M. Lukyanov, T. Motobayashi, T.K. Onishi,
Yu.E. Penionzhkevich, S. Shimoura, T. Teranishi, K. Ue, V. Ugryumov, and
A. Yoshida. Phys. Lett. B, 542249, 2002. 70, 83

[106] Genie 2000 Customization Tools Manual, 2000. 74, 92
[107] G. Audi and A.H. Wapstra. Nucl. Phys. A, 56521, 1993. 82

130

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