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This is to certify that the
dissertation entitled

RARE ISOTOPE PRODUCTION

presented by

MICHAL MOCKO

has been accepted towards fulfillment
of the requirements for the

Ph. D. degree in Physics

 

 

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September 21, 2006

 

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DATE DUE DATE DUE DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2/05 p:/CIRCIDateDue.indd-p.1

 

RARE ISOTOPE PRODUCTION
By

Michal Mocko

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
RARE ISOTOPE PRODUCTION
By

Michal Mocko

Projectile fragmentation is one of the prominent methods of rare isotope beam
production. A series of projectile fragmentation experiments have been performed us—
ing 40Ca, 48Ca, 58Ni, and 64Ni primary beams at 140 MeV/ u and 86Kr at 64 MeV/u.
Two targets: 9Be and 181Ta were used in the present experiments in order to inves-
tigate the target and projectile dependence of the fragmentation production cross
sections. The study resulted in 1740 measured cross sections created in reactions of
10 systems.

The ﬁtted parameters of the momentum distributions are compared to the sys-
tematics and parameterizations derived from the high energy projectile or target frag-
mentation reactions. Extracted width parameter corresponding to the “pure” frag—
mentation component of the momentum distribution is in good agreement with the
limiting fragmentation models, and consistent with the relativistic energy projectile
fragmentation experiments. The fragmentation cross sections are reproduced well by
the empirical parameterization EPAX, while discrepancies in tails of the isotopic dis-
tributions and for elements close to the neutron-rich projectiles were observed.

Fragmentation cross section target dependence is more complex than the trend
suggested by the limiting fragmentation framework. Nevertheless the rather small
enhancement of the fragmentation cross section when using 181Ta target does not
translate to the production yields because of relatively small atom density in tanta-
lum versus beryllium material. The projectile dependence of the fragmentation cross
section resembles isoscaling for light fragments but is more complicated for heavier

fragments.

Calculations by three different theoretical models are presented. A macroscopic
geometrical Abrasion-Ablation model results in excellent reproduction of the cross
section distribution, with only two ﬁtted parameters for each reaction system. A
macroscopic-microscopic Heavy Ion Phase Space Exploration (HIPSE) model yields
very good description of the fragment velocity distributions for all reaction systems
involving 9Be target, but fails in the case of 181Ta target showing the limits of its ap—
plicability. A more SOphisticated microscopic Antisymmetrized Molecular Dynamics
(AMD) calculation can only be applied to reaction systems containing 9Be target.
The AMD does not reproduce the fragment velocity distributions. However, its de-
scription of the fragmentation cross sections is similar in quality as the HIPSE model

comparison.

to my wife Veronika

iv

ACKNOWLEDGMENTS

I would like to start by thanking my Ph.D. supervisor Professor M. Betty Tsang
for providing the opportunity to do my dissertation research at the NSCL. I was
fortunate enough to join a new project on projectile fragmentation that she had
started only a short time before I came to MSU. I really appreciate her determination
and persistence in whatever she does: starting with the experiments and ending with
the data analysis. Her ability to get things “organized” and “done” is admirable. I
have learned a lot about doing data analysis by working with her. I would like to
thank her for the continuing support and guidance during my stay at MSU.

Being a part of the HiRA group headed by Professor W. G. Lynch really broadened
my experience, by giving me the ability to help to work on development of a new
detector system like the High Resolution Array. Even though I worked on a different
project than the rest of the HiRA, they never let me feel isolated. It has been my
privilege to work with a group of people like them. Over the years the group had
its share of postdocs coming through, and all of them are people I enjoyed working
and interacting with. Any question or problem related to my data analysis or issues
with PAW were always answered by Giuseppe Verde. I enjoyed having Marc-Jan van
Goethem as a friend and colleague, and being able to discuss anything from the most
compelling physics questions to chatting over a cup of good coffee. Michael Famiano
is the most hard working person I have met in my life. Despite his very busy schedule
he got me started with ob ject-oriented data analysis using the ROOT framework, and
he always found time to address the endless problems I experienced in the beginning.

All of this probably would not be possible without my very good friend and “coffee
buddy” Mark Wallace. He has been greatly helpful since my joining of the HiRA
group. He always found time to help or explain things I could not grasp. I really

enjoyed having him at RIKEN for one of our projectile fragmentation experiments.

Life would have been much tougher without our regular 10 o’clock coffee.

I am thankful to Ingo Wiedenhover for allowing me to work with him and intro-
ducing me to the A1900 fragment separator during the busy commissioning period.
I am grateful for the Opportunity to work with Andreas Stolz. He helped me to un-
derstand the A1900 and has always been there when I needed help, whether it be the
analysis and understanding the data acquisition or any kind of physics question. He
even squeezed the proofreading of this manuscript to his already very busy schedule!
I would also like to thank other members of the A1900 group at the NSCL: Thomas
Baumann, Thomas Ginter, Mauricio Portillo and Mathias Steiner.

I am indebted to Prof. Hiroyoshi Sakurai and Nori Aoi for their invaluable help
and providing very hospitable environment for me and my colleagues during our frag-
mentation experiment at RIPS fragment separator at RIKEN. I want to acknowledge
all postdocs, graduate students and staff scientists working at Radioactive Isotope
Physics Laboratory at RIKEN for their help and support during my stay in Wako—
shi.

I am grateful to Mikhail and Lyudmila Andronenko and Hui Hua for their help
during the data analysis. I really appreciate the help of Sergei Lukyanov, who carefully
checked every single momentum distribution (out of 1740) presented in this disserta-
tion. Oleg Tarasov’s help was indispensable, especially during the ﬁrst fragmentation
experiments and I greatly acknowledge his help during my studies.

I would like to acknowledge also other members of the HiRA group like Andrew
Rogers, Franck Delaunay, Jenny Lee, Wanapeng Tan for their support and help dur-
ing my graduate studies. I acknowledge the help of fellow graduate students Jeremy
Armstrong and Ivan Brida with reading the manuscript of this dissertation.

The dissertation would not be complete without the extensive calculations I have
done in order to understand the experimental data. I enjoyed interacting and working
with many theorists, and this experience deﬁnitely deepened and broadened my un-

derstanding of the physics of nuclear reactions. I hereby want to express my thanks

and gratitude to Akira Ono for his help with the AMD code, Denis Lacroix for in-
troducing me to his HIPSE model, Bob Charity for many useful discusions and help
using GEMINI, and Pawel Danielewicz for his support and fruitful discussions.
Last, but not least I want to thank my parents Miroslav and Anna Mocko for
always being there for me and for their continuous support and help. I am grateful

to my wife Veronika for her endless love, support and belief in me.

vii

Contents

1 Introduction 1
1.1 Rare isotope beam production ...................... 6
1.2 Projectile fragmentation reactions .................... 8
1.3 Organization of dissertation ....................... 11

2 Experimental setup 12
2.1 Method of measurement ......................... 12
2.2 Particle identification ........................... 15
2.3 Experimental setup for the N SCL experiments ............. 16

2.3.1 Primary beam and reaction targets ............... 19
2.3.2 A1900 Fragment separator .................... 20
2.3.3 Detectors at the A1900 fragment separator ........... 23
2.3.4 Magnetic rigidity settings .................... 30
2.3.5 Data acquisition system ..................... 31
2.4 Experimental setup for RIKEN experiment ............... 34
2.4.1 Primary beam and reaction targets ............... 34
2.4.2 RIPS fragment separator ..................... 35
2.4.3 Detectors at the RIPS fragment separator ........... 36
2.4.4 Magnetic rigidity settings .................... 39
2.4.5 Data acquisition system ..................... 4O

3 Data analysis 42
3.1 Primary beam charge state distributions ................ 42
3.2 Calibration of beam intensity monitors ................. 44

3.2.1 NSCL experiments ........................ 44
3.2.2 RIKEN experiment ........................ 48
3.3 Particle identiﬁcation ........................... 49
3.3.1 NSCL experiments ........................ 49
3.3.2 RIKEN experiment ........................ 51
3.4 Cross section analysis ........................... 53
3.4.1 Differential cross sections ..................... 53
3.4.2 Momentum distribution ﬁtting procedure ............ 54
3.4.3 Evaluation of fragmentation production cross section ..... 55
3.4.4 Transmission correction evaluation ............... 59
3.5 Error analysis ............................... 69

viii

4 Experimental results
4.1 Momentum distributions .........................

4.2
4.3

4.1.1
4.1.2

Widths of the momentum distributions .............
Centroids of the momentum distributions ............

Fragmentation production cross sections ................
Cross section comparisons ........................

4.3.1
4.3.2
4.3.3
4.3.4
4.3.5

EPAX parameterization .....................
Comparison to EPAX ......................
Comparison to other data ....................
Target dependence ........................
Projectile dependence ......................

5 Comparison to models
5.1 Reaction models ..............................

5.2
5.3
5.4

5.5
5.6

5.1.1
5.1.2
5.1.3

Abrasion Ablation model .....................
Heavy Ion Phase Space Exploration ...............
Antisymmetrized Molecular Dynamics .............

Primary fragment distributions .....................
Excitation energy .............................
Evaporation codes ............................

5.4.1
5.4.2

LisFus evaporation code .....................
Statistical evaporation code GEMINI ..............

Cross section distributions ........................
Velocity distributions ...........................

6 Summary and conclusions

A Fitting results

Results for 40Ca projectile ........................
Results for 48Ca projectile ........................
Results for 58Ni projectile ........................
Results for 64Ni projectile ........................
Results for 86Kr projectile ........................

A.1
A2
A3
A4
A5

Bibliography

ix

70
7O
72
77
84
86
86
91
92
110
114

119
120
120
123
130
137
139
144
144
147
150
164

171

178
178
185
195
205
217

224

List of Figures

1.1 Nuclear landscape. ............................ 2
1.2 Schematic representation of an ISOL (left) and an in-flight Projectile
Fragmentation (right) technique to produce rare isotope beams. . . . 7

1.3 Projectile fragmentation reaction in a two step Abrasion-Ablation model. 10

2.1 Layout of the experimental areas at the NSCL. ............ 17
2.2 A schematic of the Coupled Cyclotron Facility at the N SCL. ..... 17
2.3 Calculated yields of different isotope species for the Coupled Cyclotron

Facility. .................................. 18
2.4 Detailed view of the A1900 fragment separator. ............ 22
2.5 Simulated fragment distributions. .................... 22
2.6 Simulation of N = Z setting for 58Ni+9Be reaction using LISE++. . . 26
2.7 Principle of operation of an avalanche counter ............. 27
2.8 Parallel Plate Avalanche Counter ..................... 27
2.9 Detector setups in the A1900 used for our fragmentation experiments. 28
2.10 Photographs of the two beam intensity monitoring devices. ...... 29
2.11 Magnetic rigidity settings for Ca primary beams. ........... 31
2.12 Magnetic rigidity settings for Ni primary beams ............. 32
2.13 The electronic diagram used for the A1900 fragmentation experiments. 33
2.14 Layout of RIKEN Accelerator Research Facility. ............ 35
2.15 RIPS setup used for the fragmentation experiment. .......... 37
2.16 Beam intensity monitor installed at the target position ......... 38
2.17 Magnetic rigidity coverage for the 86Kr runs .............. 39
2.18 Block schematic of the electronic circuit used for the 86Kr fragmenta-

tion experiment at RIKEN. ....................... 41

3.1 Primary beam charge state distributions for 58"“‘Ni and 86Kr primary
beams plus 9Be and 181Ta targets ..................... 43
3.2 Faraday cups Z001 and 2014 in the extraction beam line and the target
box, shown relative to the K1200 cyclotron and the A1900 fragment

separator. ................................. 45
3.3 NaI(Tl) beam monitor calibration for all primary beams. ....... 46
3.4 Ban beam monitor calibration for primary beams mentioned in the

text. .................................... 47
3.5 Uncalibrated and calibrated particle identiﬁcation spectrum. ..... 50
3.6 Uncalibrated particle identiﬁcation spectrum of fragments created in

reaction 86Kr+181Ta ........................... 51

3.7
3.8
3.9
3.10
3.11
3.12
3.13
3.14
3.15

3.16
3.17

4.1
4.2

4.3
4.4

4.5
4.6
4.7

4.8
4.9
4.10
4.11
4.12
4.13

4.14

Particle identiﬁcation spectrum for a 2.07 Tm magnetic rigidity setting
using the 86Kr primary beam. ......................
Momentum distributions of 50V and 50Mn created in fragmentation of
64Ni on 9Be target. ............................
Systematics of the centroids and variances of the momentum distribu-
tions for the reaction 64Ni+9Be. .....................
Coordinate system used in ion optical calculations of transmission cor-
rection ....................................
Angular acceptance of the A1900 and RIPS fragment separators as
calculated by M OCADI in (,6 versus 9 plane. ..............
Comparison of angular transmission calculated using LISE++ and
MOCADI ..................................
Calculations of the primary beam emittance ellipse for all beams de-
livered by the K1200 cyclotron in stand alone mode during the 19903.
Angular transmission correction as a function of fragment mass number
for the A1900 and RIPS fragment separators. .............
Angular distributions in the target plane for 44Ca and 59Co fragments
from the 64Ni+QBe reaction. .......................
Final transmission correction, 5, for 40"‘8Ca primary beams .......
Final transmission correction, 5, for 58"54Ni and 86Kr primary beams. .

Examples of the momentum distributions. ...............
Width of the right side of the momentum distribution 03 for the 40"‘E‘Ca
and 58"“‘Ni primary beams and the two reaction targets Be and Ta.
Width of the left side of the momentum distribution 0L for the 40"‘8Ca
and 58"“‘Ni primary beams and the two reaction targets Be and Ta.
Width of the left 0L and right on sides of the momentum distribution
for the 86Kr primary beam on 9Be and 181Ta reaction targets.
Relative deviations from the projectile velocity (vp/vp — 1) for frag-
ments with complete momentum distributions identiﬁed in the frag-
mentation of 40"“‘Ca isotopes on 9Be and 181Ta targets. ........
Relative deviations from the projectile velocity (Up/12p — 1) for all frag-
ments with complete momentum distributions identiﬁed in the frag-
mentation of 58NM isotopes on 9Be and 181Ta targets. ........
Relative deviations from the projectile velocity (vp/vp — 1) for all frag-
ments with complete momentum distributions identiﬁed in the frag-
mentation of 86Kr primary beam on 9Be and 181Ta targets. ......

Nuclides identiﬁed in the fragmentation of 40Ca projectile on two targets.
Nuclides identiﬁed in the fragmentation of 480a projectile on two targets.
Nuclides identiﬁed in the fragmentation of 58Ni projectile on two targets.
Nuclides identiﬁed in the fragmentation of 64Ni projectile on two targets.
Nuclides identiﬁed in the fragmentation of 86Kr projectile on two targets.

Isotopic distributions for elements 5 S Z S 20 produced in the 40Ca+9Be
reactions at 140 MeV/ u. .........................

reactions at 140 MeV/ u. .........................

xi

53

56

57

59

61

62

63

64

66

67
68

71

76

78

79

81

82

83
95
96
97
98
99

100
Isotopic distributions for elements 5 3 Z S 20 produced in the 4°Ca+181Ta
101

4.15 Isotopic distributions for elements 5 S Z S 20 produced in the 48Ca+9Be

reactions at 140 MeV/u. .........................

102

4.16 Isotopic distributions for elements 5 S Z S 20 produced in the 48Ca+181Ta

reactions at 140 MeV/ u. .........................
4.17 Isotopic distributions for elements 5 S Z S 28 produced in the 58Ni+9Be
reactions at 140 MeV/u. .........................

103

104

4.18 Isotopic distributions for elements 5 S Z S 28 produced in the 58Ni+181Ta

reactions at 140 MeV/u. .........................
4.19 Isotopic distributions for elements 5 S Z S 28 produced in the 64Ni+9Be
reactions at 140 MeV/ u. .........................

105

106

4.20 Isotopic distributions for elements 5 S Z S 28 produced in the 64Ni+181Ta

reactions at 140 MeV/ u. ......................... 107
4.21 Isotopic distributions for elements 25 S Z S 36 produced in the

86Kr+9Be reactions at 64 MeV/ u. .................... 108
4.22 Isotopic distributions for elements 25 S Z S 36 produced in the

86Kr+181Ta reactions at 64 MeV/ u. ................... 109
4.23 Target ratios of the fragmentation cross sections, aTa(A, Z) /aBe(A, Z), -

of fragments 8 S Z S 18 for two projectiles 40Ca and 48Ca. ...... 112
4.24 Target ratios of the fragmentation cross sections 0T8(A, Z) /aBe(A, Z)

of fragments 10 S Z S 26 for two projectiles 58Ni and 64N i. ...... 112
4.25 Target ratios of the fragmentation cross sections 0T3(A, Z) /aBe(A, Z)

of fragments 25 S Z S 36 for 86Kr projectile ............... 113
4.26 Ratios of cross sections 043(N, Z) /040(N, Z) of fragments created in

4"3Ca and 40Ca reactions on 9Be (top panel) and 181Ta (bottom panel)

targets .................................... 116
4.27 Ratios of cross sections 064(N, Z) /0‘58(N, Z) of fragments created in

64Ni and 58Ni reactions on 9Be (top panel) and 181Ta (bottom panel)

targets .................................... 117
5.1 N ucleus-nucleus potential VATAP as a function of the relative distance. 125
5.2 A schematic representation of the quantum branching for multichannel

reactions [115]. .............................. 133

5.3 Examples of the time evolution of the 86Kr+9Be collisions at 64 MeV/u.136

5.4 Primary fragment isotopic distributions for 40"‘8Ca+9Be reaction system. 138
5.5 Primary fragment isotopic distributions for 58"54Ni+9Be reaction system. 139
5.6 Excitation energy per nucleon, E*/A, as a function of the mass num-

ber of the primary fragments in different models for reaction systems

involving 181Ta. .............................. 141
5.7 Excitation energy per nucleon, E*/A, as a function of the mass num-

ber of the primary fragments in different models for the 48Ca+181Ta

reaction system ............................... 142
5.8 Excitation energy per nucleon, E*, as a function of the mass number

of the primary fragment different models for 86Kr beam and 9Be and

181Ta targets. ............................... 143
5.9 Two Abrasion-Ablation calculations compared for 40"‘E‘Ca projectiles

with 9Be target ............................... 146

xii

5.10

5.11

5.12

5.13

5.14

5.15

5.16

5.17

5.18

5.19

5.20

5.21

5.22

5.23

5.24

5.25

Influence of the level density parameterization on the ﬁnal isotope dis-
tributions in 40Ca+9Be reaction ......................
Inﬂuence of the level density parameterization on the ﬁnal isotope dis-
tributions in 48Ca+9Be reaction ......................
Fragmentation cross sections of 40Ca+9Be compared to theoretical mod-
els. .....................................
Fragmentation cross sections of 48Ca+9Be compared to theoretical mod—
els. .....................................
Fragmentation cross sections of 58Ni+QBe compared to theoretical mod-
els. .....................................
Fragmentation cross sections of 64 N i+9Be compared to theoretical mod-
els. .....................................
Fragmentation cross sections of 86Kr+9Be compared to theoretical mod-
els. .....................................
Fragmentation cross section isotopic distributions of 40Ca+181Ta reac-
tion are compared to theoretical models. ................
Fragmentation cross sections of 48Ca+181Ta compared to theoretical
models. ..................................
Fragmentation cross sections of 58Ni+181Ta compared to theoretical
models. ..................................
Fragmentation cross sections of 64Ni+181Ta compared to theoretical
models. ..................................
Fragmentation cross sections of 86Kr+181Ta compared to theoretical
models. ..................................
Deviation from the projectile velocity (Up/Up — 1) in percent for frag-
ments in the 40"“‘Ca and 58MN i projectile and 9Be target collisions.

Deviation from the projectile velocity (’UF/‘Up — 1) in percent for frag-
ments in the 48Ca projectile on 181Ta target collisions ..........
Deviation from the projectile velocity (vp/vp — 1) in percent for frag-
ments in the 86Kr projectile and 9Be, 181Ta target collisions. .....
Results of the HIPSE calculations in terms of the excitation energy
per nucleon, E*/A, (left) and the fragment velocity (right) for the
483Ca+181Ta reaction system ........................

IMAGES IN THIS DISSERTATION ARE PRESENTED IN COLOR

xiii

148

149

153

154

155

156

157

158

159

160

161

162

165

166

166

168

List of Tables

2.1 List of primary beams used in our experiments. ............ 19
2.2 List of targets used in the series of experiments at the NSCL. . . . . 21
2.3 Fundamental parameters of the A1900 fragment separator [55]. . . . . 21
2.4 The magnetic rigidity, Bp, values of N = Z settings, used for particle

ID calibration ................................ 24
2.5 Primary beam energy measured and target thickness measurement in

the 86Kr experiment. ........................... 35
2.6 Fundamental parameters of the RIPS fragment separator [35]. . . . . 36
3.1 Values of 00 and 0'0 parameters in Equation (3.5) used in the angular

transmission correction calculations for different primary beam target

combinations. ............................... 69
4.1 Goldhaber reduced width parameters for all reaction systems. . . . . 74
4.2 Number of fragments and pick-up (including exchange) cross sections

measured for all reaction systems ..................... 85
4.3 Best ﬁt values of the isoscaling parameters for all investigated systems. 118
5.1 Best ﬁt values for the K and S parameters of the excitation energy in

the Abrasion-Ablation model. ...................... 122
5.2 Values of HIPSE adjustable parameters as determined by Lacroix [111]. 129
A.1 Fitting results for the reaction system 40Ca+9Be. ........... 178
A2 Fitting results for the reaction system 40Ca+181Ta. .......... 181
A3 Fitting results for the reaction system 48Ca+9Be. ........... 185
A4 Fitting results for the reaction system 48Ca+181Ta. .......... 190
A5 Fitting results for the reaction system 58Ni+9Be ............. 195
A6 Fitting results for the reaction system 58Ni+181Ta ............ 199
A.7 Fitting results for the reaction system 64Ni+9Be ............. 205
A8 Fitting results for the reaction system 64Ni+181Ta ............ 210
A9 Fitting results for the reaction system 86Kr+9Be. ........... 217
A.10 Fitting results for the reaction system 86Kr+181Ta ............ 221

xiv

Chapter 1

Introduction

The nucleus is the core of the atom. Containing more than 99.9% of the atom’s mass,
the nucleus deﬁnes its center and inﬂuences the chemical properties of the atom
through its electrical charge. A nucleus is composed of Z protons and N neutrons
which determine its mass number A = N + Z. Since the protons and neutrons have
similar mass and behave similarly in the realm of the strong interaction they are
called nucleons. They themselves have a sub-structure, quarks and gluons, with the
gluons holding the quarks together in a tightly bound entity [1—3]. This results in the
attractive force between nucleons as the residue of the quark and gluon interactions [4].

Nuclei come in a large variety of combinations of protons and neutrons. However,
due to fundamental laws of nature, which are still being investigated, not all com-
binations are possible. Figure 1.1 illustrates the landscape of those nuclei that we
presently think might exist in a map spanned by the number of protons on one axis
and the number of neutrons on the other. This landscape shows several thousands
of nuclei that are expected to be bound by the strong force. Of these, slightly fewer
than 300 isotopes make up the assortment of the 82 stable elements (marked as black
squares in Figure 1.1) that exist in nature. When displayed in the representation of a
nuclear landscape, these stable isotopes lie along a slightly curved line, called the line

of stability (or the valley of stability) [6]. There are thousands of unstable nuclides

 

Nuclear Landscape

  
   
   
  

‘
7

less than 300 stable ]

proton number Z

 

terra incognita
known nuclei

A
V

2 8 neutron number N

 

 

 

Figure 1.1: Nuclear landscape —— nuclei shown in proton versus neutron number rep-
resentation [5].

which are bound against the release of protons and neutrons, but they are subject to
a [7] and/or 6 [8] decay. Some of the unstable nuclei have very long half lives and
can be found on Earth. Others are man-made, while thousands more have not been
discovered. The yellow region in Figure 1.1 indicates short lived nuclei that have been
produced in laboratories. By adding either protons or neutrons one moves away from
the valley of stability, ﬁnally reaching what is referred to as the drip lines [9] where
the nuclear binding ends because the forces between the neutrons and protons are no
longer strong enough to hold these particles together.

Many thousands of radioactive nuclei with very small or very large N / Z ratios are
yet to be explored. In the nuclear landscape they form the term incognita indicated in
green [10]. The proton drip line has been reached experimentally for all odd Z nuclei

up to Z = 91 [11], because the long range Coulomb repulsion among the protons

prohibits formation of extremely proton-rich nuclides. In contrast, the neutron drip
line is considerably further away from the valley of stability [11], hence harder to
approach. Except for the lightest nuclei where it has been reached experimentally,
the neutron drip line is estimated on the basis of nuclear models, therefore it is very
uncertain due to the extrapolations involved.

The red lines in Figure 1.1 show the “magic numbers” known around the valley of
stability [12]. However, since the structure of nuclei is expected to change signiﬁcantly
as the drip lines are approached, we do not know how nuclear shell structure evolves
at the extreme nuclear asymmetries.

For decades, nuclear physics experiments have been performed with reactions be-
tween beams and targets of stable nuclei found in nature. Basic properties of the
atomic nuclei have been discovered and many fundamental models developed (e. g.,
Weizsacker mass formula [13], shell model [6], etc.). In order to understand nuclear
matter, we need to explore and study not only the properties of stable nuclei, but
also those close to the limits of nuclear existence. These investigations of unstable
nuclei pose a great experimental challenge. Such studies at the extremes will provide
important insights into the structure of nuclei, the dynamics of nuclear reactions,
the underlying symmetries of nature, and the nucleus as a fundamental many-body
system governed by the strong interaction. With recent developments in heavy-ion
accelerators and rare isotope beam production many new surprising phenomena have
been observed in unstable nuclei (e. g., neutron halo [14], neutron and proton skins
of nuclei far from stability [15,16], large deformations of neutron rich isotopes [17],
etc.).

Further study of the unexplored regions of the nuclear landscape will provide
answers to many important questions outlined in the “Scientiﬁc opportunities with

Fast Fragmentation Beams from RIA” [18]:

o What is the origin of the elements in the cosmos?

0 Where are the limits of nuclear existence?
0 What are the properties of nuclei with extreme N /Z ratios?

0 What are the properties of bulk neutron-rich matter under extreme conditions

of temperature and density?

0 How must existing theoretical models be changed to describe the properties of

rare isotopes?

o How are the prOperties of rare isotopes related to the basic nucleon-nucleon

interaction?

Projectile fragmentation reactions serve as an important tool for producing the
rare isotope beams used in a wide variety of experiments looking for answers to the
above mentioned questions. The presented dissertation explores the fragmentation
reaction mechanisms, to better understand and optimize the production of rare iso-
topes. It allows experimenters to better estimate and optimize the yields of the rare
ion beams available for their studies.

Understanding fragmentation reactions is not only important for the fundamental
science experiments investigating new phenomena with exotic isotope species, but is
also needed in many applications. Three concrete applications of fragmentation reac-
tions will be mentioned here brieﬂy: space radiation, heavy-ion therapy and nuclear
waste transmutation.

Space radiation consists of many components, from which the high energy and
high-charge ions constitute about 1%. High energy highly charged ions possess signif-
icantly higher ionizing power this means they have a greater potential for radiation-
induced damage. Since nuclear fragmentation is an important reaction mechanism for
these relativistic heavy ions, better understanding and modeling of these processes is
very important [19]. Realistic simulations of the radiation damage to equipment and

humans are essential for the success of any space missions that last an extensive period

of time. Realizing that there is a lack of experimental data, NASA joined efforts with
the Brookhaven National Laboratory in building the NASA Space Radiation Labora-
tory (NSRL) in Brookhaven, New York [20]. The new facility will study the reactions
of heavy ions in various biological materials (tissues, cells or DNA in solution). It also
evaluates industrial samples for their suitability in spacecraft shielding.

Heavy-ion therapy is a new emerging technique to treat tumors resistant to
other conventional treatment methods. Close to the end of a heavy-ion track, there is
a well localized energy deposition by the ion. This maximum in the stopping power
is called the Bragg peak. When positioned properly, the peak concentrates the dam-
age to the tumor tissue and minimizes the radiobiological effects to the surrounding
tissue [21]. Fragmentation of the heavy-ion beams used to treat tumors may produce
lighter isotopic species that may penetrate deeper into the healthy tissue. The en-
ergy deposited by the beam and the produced ions causes a severe damage to both
helices of the DNA of a cell nucleus, also referred to as “double strand breaks” [22].
Therefore, energy and mass of the treatment means used are constrained by the num-
ber of the fragment species created in the treatment tissue. Better understanding of
the fragmentation reaction mechanism improves calculations of the radiation dam-
age inﬂicted by the fragments of the treatment heavy-ion beam. A pilot program of
treatment using relativistic ions of 12C was started in 1997 at GSI in Darmstadt,
Germany [23]. Based on the results of this project, a new cancer therapy center [24]
is under construction in Heidelberg, Germany.

Nuclear waste is predominantly comprised of used fuel discharged from oper-
ating reactors. Almost all issues related to risks arising from long-term disposal of
spent nuclear fuel are attributable to about 1% of its content [25]. This small fraction
consists of transuranic elements and long-lived isotopes of iodine and technetium.
'I‘ransmutation [26] of these isotopes by a proton beam can decrease the toxic nature
of the spent fuel below that of natural uranium ore, reducing the challenges associ-

ated with long-term storage. In order to Optimize the nuclear waste transmutation

it is very important to study and understand the spallation (target fragmentation)
reaction mechanism. Realizing the difficulties connected with long-term storage of
the nuclear waste materials, large collaborations such as the Advanced Fuel Cycle
Initiative at Los Alamos National Laboratory [27] have been formed to identify the
relevant cross sections and thereby address the scientiﬁc questions concerning the
nuclear waste transmutation as a solution of the nuclear waste problem.

These three examples of nuclear fragmentation applications are by no means ex-
haustive; they only illustrate the broad impact of the fragmentation reaction research
in human society. This dissertation, however, focuses mainly on the study of cross
sections and the reaction mechanism of projectile fragmentation reactions in basic

science.

1.1 Rare isotope beam production

The two most prominent techniques to produce rare isotope beams are Isotope Separa-
tion On-Line (ISOL) and in-ﬂight Projectile Fragmentation (PF) [28]. The underlying
principle of the production of the exotic nuclei is to transform the stable nuclei into
unstable species using nuclear reactions.

In an ISOL facility the radioactive nuclei are produced essentially at rest in a
thick target by bombardment with particles from a driver accelerator. After ionization
and mass separation, the nuclei of rare isotopes are accelerated in a post-accelerator
(see the left panel of Figure 1.2). A wide range of primary beams including thermal
neutrons [29], high energy protons [30], intermediate energy heavy ions [31] can be
used as projectiles. For the thick target construction high Z materials are used such
as tantalum or uranium. Similarly a range of different types of post-accelerators are
being used including cyclotrons [32], linacs [33], and tandem accelerators [34].

Post-accelerators — optimized for high quality beams — provide easy energy

variability, high energy precision and small emittances. These characteristics are re-

Isotope Separation On-Line Projectile Fragmentation
_ isobar/isotope .. . .

  
    

IL...
p

    
 

 

 

    
   

 

 

 

 

 

 

 

 

se ration
Pa fragment
] If .- ~, separator
production _7 e
tar et “
driver thiCk. hot target 9094' tn 9
acce'eratOF a°°° 9'3 0 heavy ion radioactive beam
d' ct' b accelerator
ra Ioa we earn
Experiment
Experiment

 

 

 

 

Figure 1.2: Schematic representation of an ISOL (left) and an in-ﬂight Projectile
Fragmentation (right) technique to produce rare isotope beams.

quired by many experiments in nuclear structure and nuclear astrophysics. Another
advantage of ISOL facilities is the high luminosity that can be achieved through
the combination of a thick production target and a very intense primary beam. The
time delay resulting from the stopping, ionization and extraction of the radioactive
fragments from the production target and forming a secondary ion beam limits the
production of particularly short-lived rare isotope beams. The extraction of the rare
isotope species from the target material is done using chemical methods which make
the whole process charge, Z, dependent. Modern day facilities that use this production
technique include: the REX-ISOLDE facility [33] at CERN in Geneva, Switzerland
and the ISAC facility [30] at TRIUMF in Vancouver, Canada.

In an in—ﬂight PF facility, as illustrated in the right panel of Figure 1.2, an en-
ergetic heavy-ion beam is fragmented when it interacts with a (production) target.
The desired reaction products are subsequently transported to an experiment after
mass, charge and momentum selection in a fragment separator. In—ﬂight ﬁssion of
very heavy beams, and also charge exchange and transfer reactions, have been used
as an alternative to projectile fragmentation. The high energy that the fragments au-
tomatically carry from the primary beam in this production method, eliminates the
need for post-acceleration. The in-ﬂight production also means that the experiments

with radioactive fragments as a secondary beam can be done promptly. Because they

are only delayed by the ﬂight time through the separator and beam line system, typi-
cally < 1 us, production of very short-lived exotic nuclides is possible. Contemporary
facilities using the PF method of rare isotope beam production include: Riken Accele-
rator Research Facility (RARF) [35] at RIKEN in Wako—shi, Japan and the Coupled
Cyclotron Facility (CCF) [36] at NSCL in East Lansing, MI.

These two techniques of rare isotope beam production are, in many respects,
complementary. Different classes of experiments impose very different requirements
on rare isotope beams in terms of velocity, momentum and angular spread and pu-
rity, allowing experimenters to choose the most suitable secondary beam production

method possible.

1.2 Projectile fragmentation reactions

Nuclear reaction [37] is the single most important technique to probe the fundamen-
tal properties of nuclear matter. Nuclear collisions are usually classiﬁed according to
the impact parameter and the incident energy of the projectile. We associate periph-
eral collisions with large impact parameters and central collisions with small impact
parameters. In terms of the kinetic energy the collisions are usually categorized as
low (< 20 MeV/u), intermediate (20—200 MeV/u) and high energy (>200 MeV/u).
Projectile fragmentation reactions are peripheral collisions of heavy ions at interme-
diate to high energies, when the projectile breaks up — fragments into one or more
heavy ions — on impact with the target nucleus. More central collisions of heavy
ions in intermediate to high energy regime, also referred to as multifragmentation
reactions [38], occur when the interaction region of the reaction system shatters into
many smaller pieces (i. e. clusters).

From the experimental viewpoint, two classes of fragmentation reactions are recog-
nized: target fragmentation, more commonly referred to as spallation, and projectile

fragmentation, as described above. These two reaction categories are identical in the

center of mass frame of reference, allowing one to use the same theoretical treatment
and model description. However, the experimental techniques used in their study and
the historical evolution were quite different.

Target fragmentation reactions have been investigated for more than four decades
and large number of data sets are available. Many of the fragmentation models, pa-
rameterizations and systematics are based on the target spallation experimental data.
Projectile fragmentation reactions were experimentally inaccessible until the 19703
when high-energy heavy-ion beams became available [39,40]. The ﬁrst pioneering pro-
jectile fragmentation experiments at relativistic energies with 40Ar and 48Ca beams
were carried out at Bevalac at Lawrence Berkeley Laboratory (LBL) [41,42]. These
experiments showed the potential of this method to produce rare isotope beams, and
many fragment separators have been constructed (e. g., A1200 (Analysis 1200) at
MSU [43], F RS (FRagment Separator) at GSI [44], RIPS (RIken Projectile fragment
Separator) at RIKEN [35]) utilizing this technique.

The ﬁrst attempts to describe the projectile fragmentation reactions were done at
LBL. In 1973 Bowman, Swiatecki and Tsang introduced the Abrasion-Ablation (AA)
reaction model [45]. Which forms the basis of our understanding of the fragmentation
reactions. This simple model approximates the reaction in two stages; Figure 1.3
illustrates the two very distinct processes. The nuclei are treated as perfect spheres
colliding on classical parallel trajectories separated by the impact parameter, b. Any
alteration from the straight path is neglected in the model because it assumes that
the projectile is moving with relativistic velocity. At the end of the abrasion stage the
overlap (participant) region of two spherical nuclei is removed. In its simplest form,
no treatment of the participant region is necessary to explain the data using the
AA model. The abrasion stage ends with a deformed and highly excited prefragment
which decays in the second ablation stage by evaporating light clusters, nucleons and
gamma radiation. The second step of the AA model is much slower (z 10‘16-10“18

8, depending on the excitation energy) as compared to the abrasion step (a: 10‘23 s).

Since its introduction in 1973, many different versions of the AA model have been
developed. Many of these AA models differ in the second ablation stage. There is
one common feature of all the AA models: the inability to predict the excitation
energy of the prefragment. Most of the models rely on various parameterizations of
the excitation energy and in many cases it is taken as a free parameter. The details

of the AA model used in this dissertation are explained in Section 5.1.1.

 

 

  

 

projectile prefragment
e
O
*
e
E . v
0 t
O 0
target
Abrasion Ablation (deexcitation)

Figure 1.3: Projectile fragmentation reaction in a two step Abrasion-Ablation model.

While there are many puzzling aspects to the fragmentation phenomenon, it
does display some simplifying characteristics at high incident energies. For exam-
ple, many experimental observables in peripheral collisions at high incident energies
( >200 MeV/u), such as charge or multiplicity distributions, vary little with incident
energy and target mass. This “limiting fragmentation” behavior forms the basis for
the EPAX parameterization used to calculate the secondary beam rates in many ra-
dioactive beam facilities and even the rates predicted for the next generation rare
isotope facility [46]. This parameterization assumes that the isotopic distributions
and their dependence on the isospin of the projectile and target are consistent with
limiting fragmentation. However, EPAX derives its results from a careful empirical
ﬁt to a limited data set of production cross sections measured under a wide variety of
experimental conditions. As the parameterization is not based upon a speciﬁc theory
for projectile fragmentation, EPAX is better at interpolating between measured data

points taken under similar conditions than at predicting the production of an isotope

10

further away from the valley of stability. The original EPAX parameterization [47]
included only the target spallation data; it was revisited in 2000 [48] and improved
by including, the contemporary projectile fragmentation data available.

The threshold energy for the limiting fragmentation is not uniquely deﬁned and
fragmentation data in the intermediate energy range are rather scarce. Very little data
exist to examine the detailed dependence of isotopic distributions on target and beam
in the intermediate energy regime. As Siimmerer et al. [48] pointed out: “It would
be interesting, however, to compare EPAX also to cross sections obtained at lower
incident energies once high-quality data become available”. The present study in this
dissertation provides high-quality and comprehensive projectile fragmentation data
at intermediate energy available from the Coupled Cyclotron Facility at the National

Superconducting Cyclotron Laboratory at Michigan State University.

1.3 Organization of dissertation

The dissertation is organized in the following way. Chapter 2 introduces the exper-
imental method used for the fragmentation measurements. The description of the
fragment separators, the A1900 at the NSCL and RIPS at RIKEN, along with ex-
perimental details of the measurements are also included. Chapter 3 describes the
principal steps taken in the analysis of the experimental data, emphasis is given to
the discussion of the fragmentation production cross section extraction. In Chapter 4
the experimental data are presented, starting with the discussion of the momentum
distributions and ending with the cross sections of fragments and nucleon pick-up
reactions measured in our experiments. Three different theoretical models of frag-
mentation reactions are introduced in Chapter 5, along with the comparisons to the
experimental data. Conclusions of the dissertation are summarized in Chapter 6. The
measured experimental cross sections of fragments and nucleon pick-up products are

listed in the Appendix.

11

Chapter 2

Experimental setup

The present experiments were carried out at two laboratories -— NSCL and RIKEN.
A series of four fragmentation experiments with 40"‘8Ca and 58MM projectiles at 140
MeV/ u using the A1900 fragment separator were carried out at the NSCL. The frag-
mentation experiment using the 86Kr primary beam at 64 MeV/ u was done at RIKEN
using the RIPS fragment separator. In this chapter a description of the method of
measurement is followed by introduction of the particle identiﬁcation techniques com-
mon to all experiments. Since the details of the experimental setup differ, the frag—
ment separators and the detector setups for the NSCL and RIKEN measurements are

discussed separately.

2.1 Method of measurement

One of our goals is to perform comprehensive measurements of fragmentation produc-
tion cross sections produced in the reactions of 5 different projectiles in intermediate
energy regime. The emitted fragments were collected and identiﬁed with a fragment
separator. In order to obtain the ﬁnal production cross sections, one must understand
how to correctly reconstruct the fragment distributions in angular and momentum

coordinate spaces. The angular distributions are not directly measured in our exper-

12

iments and are taken as an efficiency correction from simulations and are referred to
as the transmission correction in Chapter 3. The momentum distributions of frag-
ments are measured in the magnetic ﬁeld of the fragment separator. Because of the
limited momentum acceptance of current fragment separators (z :t(2—3)% around
the central momentum) the momentum distributions must be measured by multiple
magnetic settings of the fragment separator. In doing so, one can use two methods of
obtaining the momentum distributions running the fragment separator with its full
momentum acceptance or running with a narrow momentum acceptance.

The full momentum acceptance of the fragment separator is generally used in pro-
duction mode to maximize the production yield of the rare isotope beams. We chose
the narrow momentum acceptance mode for our experiments since the main objec-
tives of our experiments were to measure comprehensive sets of fragmentation cross
sections minimizing the associated uncertainties. The following issues have been con-
sidered in choosing the narrow momentum acceptance mode: primary beam charge
states, transmission characteristics, particle identiﬁcation, number of magnetic set-
ting, optimum use of the beam intensity and the electronic dead time.

The energetic heavy ions can pick-up, exchange or loose electrons when they tra-
verse a foil of a given material composition. These complex interactions depend on
the heavy-ion energy and charge of both projectile and target. The heavy ions then
emerge from the foil in different charge states —— differing by number of electrons they
possess. Fragment production rate is approximately 103—104 lower than the primary
beam intensity or its charge states. If the full momentum acceptance of the fragment
separator is used, regions of the magnetic rigidity that accepts the charge states must
be blocked Off. This large difference precludes the measurements of fragments and the
primary beam (or its charge states) in one magnetic rigidity, Bp, setting. If narrow
momentum acceptance is used we can map the Bp space between the charge states
of the primary beam resulting in a better Optimization of the beam intensity and a

reduction of the data acquisition dead time.

13

In order to measure the absolute fragmentation cross sections, we need to under-
stand the transmission characteristics of the fragment separator. These characteristics
express the efﬁciency of our measurement. With the full momentum acceptance we
are also ﬁlling the spatial acceptance of the fragment separator. Optical abberations
start to play more important role for particles traveling farther away from the optical
axis Of the separator. On the other hand if we restrict ourselves to a narrow momen-
tum acceptance we are limited to particles traveling very close to the optical axis
of the separator and therefore minimizing the effects of the Optical abberations. The
experimental measurement of the transmission is very difﬁcult because it requires ac-
curately knowing the cross sections. The transmission is determined by an ion-optical
calculation. Transmission for particles within the narrow momentum acceptance of
the fragment separator can be simulated more accurately. This consideration favors
narrow momentum acceptance mode.

With wide momentum acceptance, particle identiﬁcation requires a measurement
of the magnetic rigidity at the dispersive image plane, to correct for the different
particle trajectories. In narrow momentum acceptance mode no additional position
sensitive detectors are required at the dispersive image plane, because the particles are
bound to very similar trajectories and the momentum is determined by the magnetic
rigidity setting of the fragment separator. This consideration again favors narrow
momentum acceptance mode.

One disadvantage of narrow momentum acceptance is that it requires more mag—
netic settings of the fragment separator compared to the wide momentum acceptance
to cover the same range in Bp. But with narrow momentum acceptance we gain other
advantages: measurement between the charge states of the primary beam, easier par-
ticle identiﬁcation, and better utilization of the beam time (data acquisition dead

time characteristics).

14

2.2 Particle identiﬁcation

Particle identiﬁcation was obtained by using the Bp—ToF -AE-TK E method [44] on
an event-by-event basis. The magnetic rigidity, Bp, was given by the magnetic setting
of the fragment separator. Fragment time of ﬂight, ToF, energy loss , AE, and total
kinetic energy, TK E , were measured by charged-particle detectors. The determination
of these four quantities Bp, ToF, AE, TK E deﬁnes unambiguously the momentum,
p, mass and charge numbers A and Z along with the charge state, Q, for every ion.

The magnetic rigidity, Bp, of a charged particle, can be expressed as a function
of the A/Q ratio:

Bp = ﬂvuc x (2-1)

a,
where c is the speed of light in vacuum, B = v/c is the velocity of the charged particle,
7 = 1/ W is the relativistic factor and u is the atomic mass unit.

The time of ﬂight, T 0F , of a charged particle through a distance, L, is given by

its velocity, 6, as follows

L
ToF = 53' (2.2)

The Bethe-Bloch formula [49] connects the differential energy losses, —dE/d:r, of
a charged particle, with nuclear charge, Z, and velocity, 6, in a material with the
mean ionization potential, I:

dE 47re4n 2 2mec2 62 2
'E’W'Z'lln( I 'l—wl‘ﬁl’ (2‘3)

 

where e is elementary charge, and n is the electron density of the material. Hence,

the energy loss of a charged particle in a thin detector can be expressed as:

AE = K1Z2 x [1 — $111 (IX—2%)] , (2.4)

 

where K1 and K2 are parameters containing all medium speciﬁc and universal con-

15

stants.
The total kinetic energy, TK E , of a charged particle is given by its mass number,

A, and the relativistic factor 7:
TKE = (’y — 1) X Auc2. (2.5)

This gives one uniquely the mass, A, provided that one knows the velocity, ,6.

The A/Q ratio for a given charged particle with the magnetic rigidity, Bp, is
obtained by expressing the 6 from Equation (2.2) and inserting it into Equation
(2.1). The energy loss, AE, in a thin layer of matter, is proportional to the square of
nuclear charge (Equation (2.4)), providing the resolution of nuclear charge, Z. Finally,
the relation between the total kinetic energy, TK E, and particle mass, A, (Equation
(2.5)) is utilized to calculate the charge state, Q, using the A / Q ratio from Equation
(2.1).

The general Bp—ToF -AE—TK E identiﬁcation technique can be reduced to the
Bp—ToF-AE method, when Q = Z for all charged particles of interest (i. e. all ions
of interest are fully stripped of electrons). In this special case the determination of
the total kinetic energy, TK E, is not necessary, because the measurement of the
magnetic rigidity, Bp, yields the A/Z ratio (Equation (2.1)). The nuclear charge, Z,
is calculated from Equation (2.4) based on the energy loss AE, while the particle

velocity, 6, is determined from the time of ﬂight, ToF, in Equation (2.2).

2.3 Experimental setup for the NSCL experiments

In 1999—2001 the National Superconducting Cyclotron Laboratory(NSCL) underwent
a major upgrade [36]. The two superconducting cyclotrons (K500 and K1200) had
previously been used to accelerate heavy-ion beams individually. In the new Coupled

Cyclotron Facility (CCF) [36, 51,52], the two cyclotrons are coupled so the facility

16

ECR K500 N2 N3 N4 N15 N6

 

 

 

 

SRF Clean room

Cryogenic plant

 

Figure 2.1: Layout Of the experimental areas at the NSCL. The Coupled Cyclotron Fa-
cility consisting of the K500 and K1200 cyclotrons and the A1900 fragment separator
is shown with respect to the experimental vaults N2—N6 and Sl—S3.

 

Figure 2.2: A schematic of the Coupled Cyclotron Facility at the NSCL [50]. RT-ECR
and SC-ECR are Room Temperature and Superconducting ECR ion sources. The
K500 and K1200 cyclotrons are connected by the coupling line. The A1900 fragment
separator is enclosed in the dashed-line area.

is able to deliver higher beam intensities at higher energies. Figure 2.1 shows the
layout of the experimental building at the NSCL housing the CCF. The CCF, as
shown in more detailed view in Figure 2.2, consists of the K500 cyclotron, K1200
cyclotron and the A1900 fragment separator. The main function of the CCF is to
produce and deliver rare-isotope beams (RIB) to any of the 8 experimental vaults

(labeled 81—83 and N2—N6 in Figure 2.1). The CCF is able to accelerate all stable ion

17

beams, from hydrogen to uranium, prepared by either the Room Temperature [53]
or Superconducting [54] Electron Cyclotron Resonance ion sources, also referred to
as RT-ECR and SC-ECR in Figure 2.2. To date, 20 primary beams from oxygen to
bismuth have been developed.

The CCF uses the projectile fragmentation technique to produce secondary beams
of exotic isotopes. Since 2001 more than 330 secondary beams have been delivered to
various experiments. To produce a rare isotope beams, ﬁrst a stable beam (such as
400a, 48Ca, etc.) is accelerated in the K500 cyclotron to energies of approximately 10—
20 MeV/u (z 0.2 c) (Figure 2.2). As the beam is injected into the K1200 cyclotron,
it passes through a carbon stripper foil increasing its charge state substantially to
maximize its energy in the ﬁnal stage of acceleration. For ions with mass number
A S 136 the maximum energy attained in the K1200 is approximately 120—140 MeV/u

(z 0.5 c). The accelerated heavy-ion (primary) beam strikes a production target,

 
  
   

Yield #/8

I>1OZ
l> 10
I> 10
I> .01_5
l> 10

Proton Number
I
I
01
O

Neutron Number

Figure 2.3: Calculated yields of different isotope species for the Coupled Cyclotron
Facility at the NSCL—MSU [36]. Nuclides are plotted in proton versus neutron number
representation. The colors represent the yield in number of particles per second in
logarithmic scale.

18

creating various isotopic species lighter (fragmentation) or slightly heavier (nucleon
pick-up) than the primary beam nucleus. The A1900 fragment separator works as an
“isotopic ﬁlter” —— separating and transporting speciﬁc rare isotope(s) to different
beam lines for experiments [50]. Figure 2.3 displays calculated rates (by EPAX [48])
for different rare isotopes produced by the CCF [36] in the proton versus neutron

number (nuclear chart) representation.

2.3.1 Primary beam and reaction targets

In the study carried out at the N SCL we used four primary beams: 40"‘8Ca and 58"MN i.
Basic characteristics of these beams, like charge state and energy, are listed in Table
2.1. The primary beam intensity was controlled by a series of meshes (attenuators),
allowing experimenters to attenuate primary beam intensity from a factor of 1 to
106, with a step size of approximately a factor of 3. Beam spot in the target plane
was approximately 2 x 2 mm2 (determined using a CCD camera looking at a viewing
plate covered with phosphorous material, scintillating upon interaction with charged
particles).

Table 2.1: List of all primary beams used in our experiments. Nominal energies and

charge states of the primary beams are given at the exit of ECR, K500 and K1200
respectively. Energy in the A1900 is determined by measuring the Bp.

 

 

 

Ion ECR Charge state E (MeV/ u) Charge state E (MeV/ u) E (MeV/ u)
K500 K500 K1200 Nominal A1900
40Ca8+ 8+ 12.36 19+ 140.00 140.81
“Ca“ 8+ 12.23 19+ 140.00 141.96
58Ni11+ 11+ 12.35 27+ 140.00 140.96
64Ni11+ 11+ 12.35 27+ 140.00 141.23

 

 

 

 

 

 

 

 

One of the goals of our experiments was to investigate the target dependence of the
fragment yields. In our measurements we used two target materials — Be (beryllium)
and Ta (tantalum). Due to its relatively large nuclear number density, beryllium is a

commonly used material for production targets at fragmentation facilities. Beryllium

19

forms a mechanically stable solid material from which a sturdy self-supporting target
can be easily manufactured, with relatively good heat transfer and radiation hardness.
The tantalum target was chosen to investigate the dependence of the fragmentation
cross section and the production yield dependence on the nuclear charge and neutron-
proton asymmetry of the target material.

Table 2.2 lists all beam-target combinations used in our fragmentation experi-
ments at the NSCL along with energy of the primary beam before, in the middle, and
after the target material. All combinations of two targets, 9Be, and 181Ta and four pri-
mary beams, 40' 48Ca, and 58"“‘Ni, were used in our fragmentation experiments carried
out at the NSCL. The 9Be target thickness was chosen as a reasonable compromise
between maximizing the yield of the Observed fragments and minimizing the effects
of the energy loss, angular and energy straggling on the ﬁnal fragment momentum
distributions. In order to ensure these effects were comparable for both targets, the
thickness of the 181Ta target was chosen such that the energy losses Of all primary
beams were similar to the 9Be target (z 4—9 MeV/ u). The data could then be taken
with both targets using the same magnetic settings, thus, minimizing the number of

settings required in the experiments.

2.3.2 A1900 Fragment separator

Since its commissioning in 2001 the A1900 fragment separator have been used rou-
tinely to separate and deliver rare isotOpe beams to experiments at the NSCL. In
our experiments the A1900 was used to collect and identify the emitted fragments.
Figure 2.2 depicts the A1900 fragment separator as part of the CCF at the NSCL.
A more detailed view of the A1900 is shown in Figure 2.4. The A1900 consists of 4
superconducting dipoles D1—D4, with a radius of 3 m and a bending angle of 45°.
The A1900 uses 24 quadrupoles grouped in 8 cryostats.

The A1900 fragment separator was constructed with a considerably larger (10x)

angular acceptance than its predecessor the A1200 device [43]. Angular and mo-

20

Table 2.2: List of targets used in the series of experiments at the NSCL. Measured
thickness is determined by measuring energy losses of the primary beam in the target
material. Energy of the primary beam before, in the middle and after the target
material are also given.

 

 

 

 

 

 

 

 

 

 

 

 

Beam Target Measured Energy
material thickness before middle after
target target target
(mg/cm2) (MeV/u) (MeV/u) (MeV/u)
Be 103 138.41 135.98
40
C“ Ta 221 ”0'81 137.48 134.11
Be 105 139.93 137.89
48
ca Ta 228 ”1'96 139.12 136.24
Be 104 137.68 134.35
58 -
N1 Ta 226 ”0'96 136.35 131.64
. Be 105 138.24 135.20
64
N1 Ta 225 14123 137.08 132.85

 

 

Table 2.3: Fundamental parameters of the A1900 fragment separator [55].

 

 

 

[ Parameter [ A1900 ]
Max. rigidity (Tm) 6
Solid angle (msr) 8
Momentum acceptance (%) 5
Dispersion (mm/%) 59.5
Resolving power 2915

 

 

 

 

mentum acceptances determine the overall efﬁciency for collecting the fragmentation
products. While we used the narrow momentum acceptance mode in our measure—
ments, the large momentum acceptance is usually utilized in the production mode,
when one wants to transmit a large fraction of the desired fragment to an experiment.
Fundamental parameters characterizing the A1900 fragment separator are listed in
Table 2.3.

The principle of operation of the A1900 fragment separator in the production
mode is illustrated below. The exotic fragments are produced by the fragmentation of
the primary beam in a production target and have the initial beam velocity (left panel

of Figure 2.5). The mixture of unreacted primary and secondary ions is bent by the D1

21

 

Figure 2.4: Detailed View of the A1900 fragment separator. Bending dipoles are labeled
D1—D4. The ﬁve focal planes are labeled as target, image 1, image 2, image 3 and
focal plane. Eight cryostats each housing 3 quadrupoles one is also indicated.

 

.117

50Co

pro ons

 

neutrons

, i‘
-4 . aneutrons

 

 

 

 

 

Figure 2.5: Distribution of fragments in the chart of nuclides (proton versus neutron
number) at three positions along the axis of the A1900 fragment separator: target,
dispersive image, focal plane, respectively from left to right. Simulation is provided
for the production of 50Ca in fragmentation of 58Ni on a 9Be target, with momentum
acceptance, 1%, target thickness, 300 mg/cmz, and degrader thickness, 200 mg/cm2.
Simulation was done using LISE++ [56].

and D2 dipoles to select a single magnetic rigidity, Bp, using a momentum slit (middle
panel of Figure 2.5). Isotopic selection is completed by passing the ions through an
energy-degrading “wedge”, placed after the momentum slit in the intermediate image
plane (image 2 in Figure 2.4). Ions enter the degrader with a single Bp but may have
different charges, Q, and atomic mass numbers, A, and exit with different momenta
that depend on Q and A. A second dispersive beam line then provides, in most
cases, isotopic separation (right panel of Figure 2.5). The nature and the thickness
of the production target and the energy degrader, as well as the sizes of momentum
apertures, are parameters that are adjusted to control the secondary beam intensity
and purity [50].

We used the A1900 fragment separator in stand alone mode with narrow mo-

22

mentum acceptance (0.2% in dp/ p). Particle identiﬁcation was achieved using the
detectors mounted in the focal plane detector box. We tried to minimize all factors
that introduce uncertainties to our measurement. There was no degrader material
used in the intermediate image plane in order to diminish the transmission uncertain-
ties. (The wedge was used only during the measurement of 50Ca produced in 58Ni+9Be

reaction, as discussed in Section 4.3.2.)

2.3.3 Detectors at the A1900 fragment separator
Particle identiﬁcation detectors

It is a good approximation to assume that all fragments produced in our experiments
at 140 MeV/u are predominantly fully stripped of electrons (Q = Z). (From the
charge state distribution measurement for the primary beams we expect less than 2%
contribution from hydrogen-like charge states for all projectile-target combinations
except nickel projectiles on a tantalum target where it is less than 5%. ) This means we
can use the simpler particle identiﬁcation method, Bp—ToF-AE, discussed in Section
2.2. In this case the measurement of three quantities, the magnetic rigidity, Bp, time
of ﬂight, ToF, and energy loss, AE, suffice to identify the various species. Since we
used the narrow momentum acceptance mode (0.2% in dp/ p), the Bp values are given
by the magnetic ﬁeld settings of the fragment separator.

For the time of ﬂight (T 0F in Equation (2.2)) detector we used a plastic scintillator
with thickness of 100 mm and an area, 150 x 100 mm”, placed at the focal plane as
the last detector. The radio frequency (RF) pulse of the cyclotron is used as a time
reference. The plastic scintillator (SCIN) provided an event trigger and the TOF start
signal. The time of ﬂight was stopped by the RF pulse. The time structure of the
primary beam produced by the CCF spans approximately 2—4 ns due to multi—turn
extraction in the K1200 cyclotron. Nevertheless, we are able to resolve the mass for

all fragments in our experiments. The length of the time of ﬂight path (the distance

23

from target to the focal plane position along the fragment separator axis) was taken

35.48 m.

Table 2.4: The magnetic rigidity, Bp, values of N = Z settings, used for particle ID
calibration for every primary beam discussed in the text.

 

 

Primary beam B p(Tm)
40Ca 3.12
4”Ga 3.20
58Ni 3.60
64Ni 3.30

 

 

 

For energy loss measurement (AE in Equation (2.4)) we used a 500 pm thick
silicon PIN (Positive Intrinsic Negative) detector. The active area Of the PIN detector
is 50 x 50 mm2.

A PID spectrum is obtained by plotting the energy loss, AE, versus the time of
ﬂight, TOF. Figure 2.6 displays a simulated PID spectrum for the 58Ni+9Be reaction
using LISE++ [56]. The individual fragments are displayed as well separated groups
of events forming characteristic bands in horizontal and vertical directions. All nu-
clides with a constant neutron excess, N — Z, form vertical bands in Figure 2.6. The
slightly tilted horizontal bands correspond to isotopes of individual elements. The
leftmost vertical band corresponds to fragments with N = Z, the neighboring band
corresponding to N — Z = 1, etc. (The time of ﬂight was simulated such that the
more neutron-rich fragments are located to the right of the N — Z = 0 band.) By
recognizing these characteristic features of the PID spectrum we can easily identify
nuclides with different N — Z. Identiﬁcation of the element bands is done by locating
“the hole”, which corresponds to particle unbound 8Be nucleus, as shown in the mag-
niﬁed region of the PID spectrum in Figure 2.6 allows one to identify all fragments.
For reference, the isotope of 30P and the band corresponding to isotopes of calcium
(Z = 20) are labeled in Figure 2.6. The magnetic rigidity setting for a given reaction
allowing the observation of the N — Z = 0 band is also referred to as the “N = Z

setting”. The N = Z magnetic rigidity settings of the A1900 fragment separator used

24

in the PID calibrations for all primary beams are listed in Table 2.4.

The AE and TOF detectors are calibrated using the fragments from the N = Z
setting, to the calculated values of TOF and AE using the LISE++ code [56]. First
the AE and T 0F values are calculated using LISE++ for the fragments identiﬁed
in the N = Z setting. Then the channel values of AE and TOF are extracted from
the experimental spectrum and the coefﬁcients of a linear calibration from channels
to MeV and ns are Obtained for AE and TOF, respectively. Next, the coefﬁcients,
K1 and K2, of Equation (2.4) are ﬁtted to reproduce the nuclear charge numbers of
all identiﬁed fragments in the N = Z setting. After this calibration the Equation
(2.1), (2.2) and (2.4) are used to calculate the momentum, nuclear charge and mass
numbers for all fragments in our analysis. The PID calibration was done for each

projectile separately.

Position sensitive detectors

The detection of the position of particles traveling through the fragment separator is
important for diagnostic purposes (measurement of the positions and angles ensures
that the magnetic rigidity tune corresponds to the ion optical calculation). In the
case of wide/ full momentum acceptance, position measurement gives the momentum
of fragments. A total of four position sensitive Parallel Plate Avalanche Counters
(PPAC) detectors is installed in the intermediate image (2) and the focal plane (2)
along the fragment separator axis.

The principle of the PPAC detectors is described here. A charged particle ionizes
the gas in the active volume of the detector. The electrons freed in the ionization
process are multiplied in a cascade (avalanche), by ionizing surrounding gas molecules,
in high electric ﬁeld in the active volume as shown in Figure 2.7.

PPACs used in the A1900 fragment separator have segmented cathodes, providing
the position sensitivity. Figure 2.8 shows two different (front and side) views of a

PPAC detector. Each detector consists of two separate chambers sharing one anode

25

 

 

 

   
  
 
 

 

 
 

 

 

 

 

 

 

I I I I l I I I I l m W
r ,. a; , .
_ . g 100— It . 1 . he
‘ O ' > , III *
I O . l I § ' b '
I ' . . g * 53 5o— 2 . ‘ ‘—
" _ ”3 4.
_N-Z—1 . C. _ , at. ‘
300 O . 9 .. . j
’\ _ -.. . z . ‘1 r? > a: BB;
. , ‘ ,/ .. ..
% i: . ._ ‘ . ‘ 0 II m: l L I L L A I 1 .
g . Q 235 240 245 250 255
x, - TOF (ns)
_. I .E
_ O " .» e "‘ l
I ] ' "’ ' . ~ 2:20 -
i- . . 4:. "'
O ‘_ *_ III
4. .— 1 "' -
100 —L 6.? I: 1 ', . .. _
_ : - ‘ «up. .. _
_ {:1 I l I 1-]-l ['14 + I r1 I I'I I I I I I‘d
930 240 250 260 270 280
TOF (ns)

Figure 2.6: The PID spectrum of a simulated of N = Z setting for 58Ni+9Be reaction
at 140 MeV/u using LISE++ [56] for the A1900 fragment separator. Parameters of
the simulation: Bp=3.6 Tm, 0.2% momentum acceptance.

26

Ionizing
particle \ Anode

E :\° ° Gas

 

3. Cathode

Figure 2.7: Principle of Operation of an avalanche counter. The charged particle ionizes
the gas in sensitive volume of the detector between the anode and cathode. The
electrons freed in this process are multiplied in an avalanche, creating an electric
signal in the cathode.

foil. Segmented cathodes have evaporated strips connected to the readout electronics.
Two different designs of PPAC detectors are used in the A1900, referred to as I2

PPACs and PP PPACs.

III-I-I-I-I-I-II-I-
:19: x 1;: c.1pz;,gmxw~ 335,12:ij ﬂavzg-g

    

 

 

 

 

Figure 2. 8: The front (left panel) )and side (right panel) view of a Parallel Plate
Avalanche Counter (PPAC). 1 — cathode plane with horizontal strips, 2 ~ common
anode plane, 3 — cathode plane with vertical strips

A pair of I2 PPAC detectors is installed at the dispersive image of the A1900
on a retractable platform. Each detector has an active area of 400 x 100 mm2 corre—

sponding to 160 x 32 strips. Every strip is individually read out using the Front End

27

 

 

Figure 2.9: Detector setups in the A1900 used for our fragmentation experiments.
Beam intensity monitors (NaI(Tl) and Ban) shown in the target area (TA), momen-
tum slit at intermediate image (12) and the particle identiﬁcation setup (PIN and
Scin) at the focal plane (PP). The position sensitive PPACO and PPACl were used

to track the fragment trajectories.

Electronics (FEE) boards. This arrangement of the read out allows experimenters to
use the detectors up to the particle rates of 1 MHz.

The ion Optics of the A1900 fragment separator is set up such that the particles
with different momentum have different spatial positions in the horizontal plane.
Hence, the measurement of position at the intermediate image is equivalent to the
measurement of momentum. In our experiments the I2 PPACs were used only for the
measurements of momentum of the primary beam and its charge states (see analysis
of the charge state distributions in Section 3.1). These detectors were removed from
the beam during our fragment cross section measurements, because they were not
needed in narrow momentum acceptance mode.

Each of the two focal plane PPAC (FP-PPAC) detectors have an active area of
100 x 100 mm2 with resistive read-outs. The strips are interconnected by a resistor
chain and read out by two channels on both ends. Hence one PPAC detector provides
signals in 5 channels: left, right, up, down and anode. These detectors were used for the
tracking all fragments during our fragmentation experiments (Figure 2.9). rIracking
of the fragments at two different positions along the fragment separator axis at the
FP (before and after the PIN detector) allowed us to interpolate the paths to ensure

that all fragments were hitting the active area of the PIN detector (Figure 2.9).

28

I“
I

1.2; '11—?
.' in I '

A1900 ‘-
triplet J

l2.
_ . i. in"
[‘2' I :1,
iv a ,_..._
chamber
Figure 2.10: Photographs of the two beam intensity monitoring devices. Left panel:
four NaI(Tl) detectors attached to a ladder; Right panel: Ban detector in a lead
fortress (used to shield the detector from the background gamma radiation).

 

Primary beam intensity monitors

For overall normalization of fragment yields, precise continuous measurement of pri-
mary beam intensity is required. In order to calculate absolute fragmentation cross
sections we needed to know the absolute primary beam intensity. Intensities of the
primary beam ranged from 106 to 1010 particles per second (pps) during our produc—
tion runs. The absolute beam intensity is measured by a Faraday Cup (FC) which
stops the primary beam, at the CCF. As the direct measurement by a charged—particle
detector is not possible, because of the high rates of the primary beam used in our
experiments (> 106 pps), we used an indirect method to determine the beam intensity
by monitoring the radiation created by beam-target interactions.

The ﬂux of light particles and gamma radiation emitted in the interactions Of the
primary beam with the production target is proportional to the beam intensity. This
feature provides means to monitor the primary beam intensity continuously. Two
detector systems of light particles and gamma radiation were used in order to cover
the primary beam intensity range used in our experiments (approximately 4 orders
of magnitude). The ﬁrst detector conﬁguration was an array of four N aI(Tl) crystals;
the second one was a single Ban crystal coupled to a photomultiplier tube (Figure

2.9).

29

The N aI(T l) array consists of 4 crystals mounted on a ladder which is attached to
the outside of the target vacuum chamber (left panel of Figure 2.10) approximately
30 cm from the target at an angle of approximately 45°. The bias voltages and dis—
criminator levels were set up to detect the high energy charged particles emitted in
the reaction of the primary beam and target.

The BaF2 detector was shielded from the surrounding gamma radiation in the
target area by a lead fortress. It was placed on the ﬂoor behind the target chamber
(right panel of Figure 2.10) approximately 130 cm from the target at an angle of
approximately 72°. The bias voltage and the discriminator level were set up to detect
the gamma radiation created in the beam-target interactions.

Calibration of the absolute beam intensity monitor was performed using a Fara-
day Cup. The FC was placed in the extraction channel of the K1200 cyclotron (Z001)
in the case of 58Ni fragmentation, and the PC was placed in the target box of the
A1900 fragment separator for 40"“3Ca and “Ni primary beams. For the NaI(Tl) array
the background count due to radiation produced by activated target area has been
subtracted. This rate was determined by simply measuring the ambient radiation in
the target area with no beam on target. Since the beam intensity monitoring tech-
nique relies on nuclear reactions of the projectile and target nuclei, it is necessary to
perform the beam intensity monitor calibration for each beam-target combination.
For every beam-target combination we performed 3 beam monitor calibrations (at
the beginning, middle and end of an experiment). With the three calibration mea-
surements we conﬁrmed that changes (e.g., due to activation of the target area, beam

spot position) during our experiment were negligible.

2.3.4 Magnetic rigidity settings

Measurement of the momentum distributions for a wide range of fragments was carried
out by systematically scanning across the magnetic rigidity settings of the A1900

fragment separator. Figure 2.11 and 2.12 show magnetic rigidity, Bp, settings used

30

for all the primary beams with the 9Be and 181Ta targets. The individual points in
Figure 2.11 and 2.12 represent the magnetic rigidity settings of the A1900 fragment
separator. The horizontal bars display the momentum acceptance for individual Bp
settings. For 40Ca we took measurements between 3.2—4.2 Tm in 33 steps and for 48Ca
we scanned region between 3.2—5.1 Tm in 50 settings. For 58Ni and “Ni beams we
took data between 3.25—4.3 Tm in 26 steps and 3.3—4.5 Tm in 34 settings, respectively.

The gaps in the Bp coverage correspond to the primary beam and its charge states
rigidity (Figure 2.11 and 2.12). The primary beam intensity was optimized at each
magnetic rigidity such that the counting rate of the PIN detector was approximately
700—900 counts per second. The detection efﬁciency of the PIN detector is close to

100% when the counting rates are below 1 kHZ.

2.3.5 Data acquisition system

Figure 2.13 shows a schematic diagram Of the electronic circuit used in the A1900

fragmentation experiments. The trigger was produced by the focal plane scintillator

 

T j T I r I ﬁr I I I I I l I I I '

 

 

 

 

 

 

 

' r
BeL—uuuuunnuu HHHHHHHHHHHNHHHHHHHHHHHH1
Ta—uuuuuunu HHHHHHHHHHHHHHHHHHHHHHHHH—
J L 1 1 .4 I l a l l l 1 L a l
3.2 3.4 3.6 3.8 4 4.2
Bp(Trn)
1'...l.....1.'...'.uerthiurvir..l......
Be—IIIIIIIIIIIIIIIIIIIIIIIIIIIIIII linen-ulnllnnlu III—II
Ta—Ielulnll IIIIIIIIIIIIIII IIIIIIIIIIII III-luau II —I
.L.;.l.;.l;..llamleggnnl...l...l...lL4
3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 5
Bp(Tm)

Figure 2.11: Magnetic rigidity settings for Ca primary beams. Top panel for 40Ca and
bottom panel for 48Ca. The error bars in horizontal direction indicate the 0.2% in
dp/ p momentum acceptance in the Bp space.

31

 

Bel-I'D! H H H H H H H H HH H HHH HHH H H H H H —]

 

k P L +4 I l I I 1

. I . 1 . . 1
3.2 3.4 3.6 3.8 4 4.2
39 (Tm)

 

 

 

' I T I I I If I '

lIII'IIfo‘rI
Bel—IIIIIHIIIIIHHIIIIHHHHHHHHHHHHH HHHHHHHH-A

TaI—IHIHHIIIIIHHHHHIINIIHHNN HHNHHHHHH—I

 

 

I 4 J 14 L m l I I I l I I I I I I j . I I I l

3.4 3.6 3.8 4 4.2 4.4 .
BP (Tm)

 

Figure 2.12: Magnetic rigidity settings for Ni primary beams. Top panel for 58Ni and
bottom panel for “Ni. The error bars in horizontal direction indicate the 0.2% in
dp/p momentum acceptance in the Bp space.

signals (labeled as FP SCI N&S in Figure 2.13). The time of ﬂight was measured by
the F P SCI with respect to the Cyclotron Radio Frequency (RF).

The focal plane PPAC (Parallel Plate Avalanche Counter) detectors with standard
readout provide signals in 5 channels (up, down, left, right and anode signals). All
channels from the two PPAC detectors are denoted by using thick line in Figure 2.13.

Timing and analog signals were digitized using standard CAMAC (Computer Au-
tomated Measurement And Control) Analog to Digital Converters (ADC), Charge to
Digital Converters (QDC), and Time to Digital Converters (TDC). The CAMAC bus
was interfaced through the Versa Module Eurocard (VME) to CAMAC Branch Driver
(CES CBD8210) and controlled by a personal computer (PC) running the standard
NSCL data acquisition system NSCLDAQ [57] in the Linux Operating system. The
data were collected and distributed by the SpectroDaq server [57]. The NSCLDAQ al-
lows users to run multiple simultaneous online analyses using the NSCL SpecTcl [58],
plus archive the data on the network ﬁlesystem. The ﬁnal back-up of the experimental

data was stored in a 100 GB Digital Linear Tape (DLT).

32

 

[ CFD L, > Trigger

FPSCI —~~—b~—@§E \ QDC

N&S

 

 

 

 

 

»—] Gate Generator ]

 

 

 

 

 

 

 

> Scaler
ADC
CYC. {E— TAC -

 

 

RF

PIN —[E—i Shaper

FP
PPACl —[®—J Shaper > ADC
PPAC2 Q

[— cm
NaI —[E—L%aper |—[CFD]— Z ——>Sca1er

BaF 41 Shaper [—[ CFD 1r +>Scaler

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

— 5+5 signals — 4 signals

Figure 2.13: Schematic diagram of the electronic diagram used for the A1900 frag-
mentation experiments.

33

2.4 Experimental setup for RIKEN experiment

Projectile fragmentation experiment using 86Kr beam was performed at RIKEN (The
Institute of Physical & Chemical Research) Accelerator Research Facility (RARF) in
Wako-shi, Japan. The layout of the RARF facility at RIKEN is shown in Figure 2.14.
The beam of 86Kr was accelerated in the Ring Cyclotron (K540 in Figure 2.14) and
delivered to experimental area D (Figure 2.14). The fragments were analyzed and
identiﬁed using the RIKEN Projectile-fragment Separator (RIPS).

Figure 2.14 shows the RIPS fragment separator located across two adjacent rooms
D and E6. More detailed view of the RIPS is shown in Figure 2.15. Primary produc-
tion targets were located in the target position of RIPS (Figure 2.15). Projectile-like
fragments produced in interactions of the primary beam with the target nuclei were
collected in the RIPS fragment separator and transported to doubly achromatic focal
plane F2 (Figure 2.15). Detector setups at focal planes F2 and F3 allow us to identify
all fragments reaching F2. In order to cover the full momentum space we changed the

magnetic settings of the two dipole magnets D1 and D2.

2.4.1 Primary beam and reaction targets

Ions of 86Kr30+ were accelerated in the K540 ring cyclotron at RIKEN. The primary
beam energy, determined to be 63.77 MeV/ u, was measured by a direct measurement
of the magnetic rigidity using the RIPS. The beam intensity, which was adjusted by
a beam attenuator at the exit of the Electron Cyclotron Resonance (ECR) ion source
varied from 106 to 1011 particles per second (pps), during our experiment. Beam spot
size on the target was approximately 1 mm in diameter.

For the fragmentation experiments with the 86Kr primary beam, two target mate-
rials 9Be and 181Ta were used. Energy of the primary beam before, in the middle, and
after the target is listed in Table 2.5, along with the measured thickness of the target

materials from energy loss consideration. The target thickness was chosen as a reason-

34

able compromise between minimizing the distortion of the momentum distributions

and maximizing the yields Of reaction products.

Table 2.5: Primary beam energy measured and target thickness measurement in the
86Kr experiment. Energy before, in the middle and after the target is also listed.

 

 

 

Target Thickness measured Energy
before middle after
(mg/cmz) (MeV/u) (MeV/ u) (MeV/ u)
9Be 96 63.77 57.90 51.60
181Ta 153 63.77 58.02 51.89

 

 

 

 

 

2.4.2 RIPS fragment separator

The fundamental parameters characterizing the RIPS fragment separator are listed

in Table 2.6. Since its commission in the beginning of the 19903, it has been routinely

 

Figure 2.14: Layout of RIKEN Accelerator Research Facility (RARF) [35]. The K540
cyclotron is shown with respect to the experimental areas labeled as E1—E6. The
RIPS fragment separator is located across the two adjacent rooms D and E6.

35

used to deliver a variety of radioactive beams for experiments. Conﬁguration of the
RIPS fragment separator in the RARF facility is shown in Figure 2.14. It consists
of two 45° dipole magnets, twelve quadrupoles (Q1—Q12), and four sextupoles (SXI—
SX4). The ﬁrst section with a conﬁguration of Q-Q-Q-SX-D-SX-Q gives a dispersive
focus at F 1 allowing measurement of the magnetic rigidity of the particles. The second
section with a conﬁguration Of Q-Q-SX-D—SX-Q-Q-Q compensates the dispersion of
the ﬁrst section and gives a doubly achromatic focus at F2. Quadrupole triplet of
the last section gives the third focus at F3, where special experimental setups can be
installed [35].

For our fragmentation experiment the main particle identiﬁcation detection setup

was placed in F3 focal plane (Figure 2.15).

Table 2.6: Fundamental parameters of the RIPS fragment separator [35].

 

 

 

[Parameter | RIPS ]
Max. rigidity (Tm) 5.76
Solid angle (msr) 5
Momentum acceptance (%) 6
Dispersion (mm/%) 24
Resolving power 1600

 

 

 

 

2.4.3 Detectors at the RIPS fragment separator

Charged-particle detectors installed at the RIPS fragment separator were used for
particle identiﬁcation and primary beam intensity monitoring. Figure 2.15 shows the
RIPS fragment separator with all the detector setups used in the 86Kr fragmentation

experiment in ovals next to focal planes F0, F2 and F3.

Particle identiﬁcation detectors

In section 2.2, we introduced the particle identiﬁcation method used. The charge state

distributions of 86Kr primary beam at 64 MeV/u are much broader as compared to

36

 
   
 
 

Primary Beam

86Kr :DFB-gﬁﬁqﬁ“?

D1
Production Target
(Be, Ta)

        
  

[953-] = 0.2%

’ Momentum Slit
I

 

  
 

Beam intensity
monitor

      
   
 
      
 

Fragments

F2 plastic (TOF start)

Fragments

F3 plastic (TOF stop)

Si stack
dE+E 1+E2+E3+E4

Figure 2.15: RIPS setup used for the fragmentation experiment [59]. Beam intensity
monitor MOMOTA was placed at F0, particle identiﬁcation setup was placed at F2

and F3 focal planes. Momentum slit at F1 deﬁned the momentum acceptance.

primary beams measured at the NSCL. Hence, a signiﬁcant contamination also from
the charge states of individual fragments is expected (more than 10% in hydrogen-like
charge state for the primary beam after passing through the 9Be target). To identify
the momentum, mass and nuclear charge numbers along with the charge state of each

fragment, the full Bp—ToF -AE—TK E particle identiﬁcation technique must be used

(as explained in Section 2.2).

The magnetic rigidity, Bp is given by selecting different magnetic settings of the

37

fragment separator. The time of ﬂight, ToF, is measured between two 500 pm—thick
plastic scintillators on the ﬂight path of 6 m mounted at the F2 and F3 planes, respec-
tively. This is displayed in the ovals next to F2 and F3 focal planes in Figure 2.15. A
stack of ﬁve silicon detectors was used to measure energy loss, AE, and total kinetic
energy, TK E. The silicon stack is comprised of one 300 pm-thick detector and four
500 [rm-thick detectors. The stack was installed at the very end of the RIPS beamline,
as shown in the oval next to F3 in Figure 2.15. Energy 1033 simulations showed that all

particles heavier than argon will be stopped in this arrangement of silicon detectors.

Primary beam intensity monitoring

The primary beam intensity is an essential quantity to calculate the absolute frag-
mentation cross sections. We monitored the primary beam intensity using a telescope
called MOMOTA. It consists of three plastic scintillators and detects the light par-
ticles produced in nuclear reactions in the production target. We required triple co-
incidence rates to reduce the detector background. Figure 2.16 shows a schematic

drawing of the beam monitor telescope at the target position.

MOMOTA

:1

Primary
beam

 

 

Production
target

 

Figure 2.16: A schematic view of the MOMOTA beam intensity monitor with respect
to the target and the downstream Faraday Cup (F C).

As MOMOTA beam monitor was located further downstream from the target
position as the scattered particles off the FC inﬂuenced the MOMOTA reading during

the beam intensity calibration. Unlike the N SCL experiments, we could not use the

38

Faraday Cup (F C) to calibrate the MOMOTA beam monitor. Instead, the absolute
calibration of the MOMOTA telescope was provided by the direct rate measurement of
primary beam charge state 8°Kr33+ and 86Kr31+ for 9Be and 181Ta targets, respectively,

at the F2 focal plane by the plastic scintillator (detailed discussion in Section 3.2.2).

2.4.4 Magnetic rigidity settings

Measurement of the momentum distributions for a wide range of fragments was carried
out by changing the magnetic rigidity settings of the RIPS fragment separator. Figure
2.17 shows the magnetic rigidity settings used for 9Be and 181Ta targets. Settings for
the 9Be target covered the range in Bp from 1.79 to 2.93 Tm in 45 steps. Settings
for the 181Ta target covered much smaller range in Bp from 1.79 to 2.35 Trn in 29
steps, because the charge state distribution Of 86Kr at 64 MeV/ u is much broader

than in the case of the 9Be target, resulting in signiﬁcant contamination of the PID

 

 

 

 

spectrum.

. I . . 4 , . . . I . . . , . . . , . . r r .
Be—IIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIII II III II II II—
TaI—IIIIIIIIIII III-IIIIIIIIIIIII -—]

1 L; 1 1 l 1 1 1 l 1 1 1 l 1 1 1 l 1 L 1 l 1

1.8 2 2.2 2.4 2.6 2.8

39 (Tm)

Figure 2.17: Magnetic rigidity coverage for the 8°Kr runs. The error bars in horizontal
direction indicate the 0.2% in dp/ p momentum acceptance in the Bp space.

The step size we used for the overall scanning was 0.03 Tm. To avoid the charge
states Of the primary beam from coming to the focal plane detectors (at F2 and F3) the
corresponding Bp settings were not taken, forming the gaps in the magnetic rigidity,
Bp coverage in Figure 2.17. At each magnetic rigidity setting we Optimized the beam
intensity such that the counting rate of the silicon stack detector was 5 1, 000 counts

per second.

39

2.4.5 Data acquisition system

Figure 2.18 shows a schematic diagram of the electronic circuit of 86Kr fragmentation
experiment. The event trigger was generated by the F2 plastic scintillator. The tim-
ing and analog signals were digitized by standard CAMAC (Computer Automated
Measurement and Control) Analog to Digital Converters (ADC), Charge to Digital
Converters (QDC), and Time to Digital Converters (TDC).

The data were collected using the standard data acquisition software at RIKEN
[60]. The online data analysis has been done using the ANAPAW [61], a tailored
version of the Physics Analysis Workstation (PAW) [62] for the online analysis. All
the data were archived to 120m Digital Audio Tapes (DAT).

40

‘n—--—--——-——---—--——---—------—‘

 

 

 

 

 

 

 

 

 

 

 

E4

 

Delay
PPAC —[>« m [m
corners

[ Delay QDC
{gim- ~—[>—‘H

E Ii-——-———> Trigger

5 > Sealer

i Delay QDC
F3 Scint. E

TDC

L&R .

E > Scaler

Bit

I > register
dB, E1, p i rs-h-ae
E2, E3, :

 

 

E > Trigger
PPAC Im‘rm m... we
anode , -
; Sealer

 

 

    

Scaler

Okuno %:—E & ——> Scaler

 

 

 

 

 

 

 

 

 

 

 

 

 

\

--—-_----—--———--—-—----_—-----o

Figure 2.18: Block schematic of the electronic circuit used for the 86Kr fragmentation
experiment at RIKEN.

41

Chapter 3

Data analysis

The primary beam charge state measurements are presented in the beginning of the
chapter. It is followed by description of the analysis procedures for the beam inten-
sity calibration, particle identiﬁcation, differential cross section evaluation, angular
transmission correction and momentum distribution. The method of fragmentation
production cross section calculation and the uncertainty evaluation are discussed at

the end of the chapter.

3.1 Primary beam charge state distributions

In our experiment, we want to avoid detecting charge states of the primary beam
—— as discussed in Section 2.1. Therefore, it is important to know the positions and
widths of these distributions in the magnetic rigidity space. Different charge states,
Q, of one ion with mass number, A, traveling at velocity, 6, are spatially separated in
a magnetic ﬁeld according to Equation (2.1). In our case the two neighboring charge
states were spatially separated in the dispersive plane of the fragment separator due
to different bending radii in the magnetic ﬁeld.

The charge states of the primary beam were measured by tuning the fragment

separator to a Bp between the magnetic rigidities of two neighboring charge states,

42

allowing us to Observe their relative Bp distributions. For each measured beam-target
combination the results were normalized with respect to each other and the probabil-
ities for different charge states were calculated. A series of the primary beam charge
state distribution measurements have been done for 58'“Ni and 86Kr, for which two

neighboring charge state distributions ﬁt within the acceptance of the fragment sep-

 

 

    

 

 

 

 

 

 

 

 

 

 

 

arator.
I I I 1 _ I f T I m . I 4
102;- uBe target a 10’? 036 target 1
g 10;? ........... ATa target 1. § 10:; 4T3 target 1
s 15 5 g ’5 i
‘- E i t: I; I
"Suﬁ: A a €10? - E3
“§ 5 58Ni : “@102; a
3.104;: a: a.
103:- 0 1 10-35? ‘ E
. I 31045111-. .1
0 1 2 0 2 4 6
Z-Q Z-Q
: l l I l 2
10’E DBe target ‘3’
’8 10_ ATa target Q
s . f
E I. 14
lgm’E 1,
'8 -2 ‘
£10 3
103: a
. A 2
.4
10 E- l l l l E
0 1 2 3
Z-Q

Figure 3.1: Primary beam charge state probability distributions for 58'“Ni and 86Kr
primary beams plus 9Be (open squares) and 181Ta (Open triangles) targets plotted as
a function of number of electrons Z — Q. Solid and dashed lines show calculation by
GLOBAL parameterization [63] as implemented in LISE++ [56] for 9Be and 181Ta
target, respectively.

43

The separation of two neighboring charge states in Bp for ions of Ca is larger (2 5
% in dp/ p) than the momentum acceptance of the A1900 fragment separator (z 5 %
in dp/ p) so the charge state distributions were not measured for 40"‘8Ca beams.

The charge state distributions measured for 58’“Ni and 86Kr beams at 140 MeV/ u
and 64 MeV/ u, respectively, are presented in Figure 3.1 in terms of probability as a
function of number of electrons, Z — Q. The distributions obtained on 9Be and 181Ta
target are presented as open squares and triangles, respectively. The experimental
data are compared to a leading parameterization of the charge state distribution ——
GLOBAL [63], shown as solid (9Be) and dashed (181Ta) lines. The target thicknesses
used in the experiments were much larger than the equilibration thickness so all the
GLOBAL calculations were done for the equilibration thickness of the target mate-
rial. The charge state probability distributions are wider for 181Ta for all projectiles
shown in Figure 3.1. All probability distributions are reproduced rather well in spite
of the fact the parameterization GLOBAL, was derived for Z > 28 ions. Some dis-
crepancies are observed in the case of 58Ni and 86Kr projectiles. The GLOBAL code
has no isotope dependences; the calculated distributions for 58Ni and “Ni are iden-
tical. However, the experimental data show broader probability distributions in the
case of the 58Ni projectile (especially for 9Be). The experimental data and GLOBAL
predictions for 8°Kr33+ differ by a factor of 10, while the experimental data for 181Ta

target are reproduced well up to Z — Q = 6.

3.2 Calibration of beam intensity monitors

3.2.1 N SCL experiments

Precise knowledge of the primary beam intensity is essential in determination of the
absolute fragmentation production cross sections. In the fragmentation experiments
carried out at the NSCL we used two primary beam intensity monitors referred to as

N aI(Tl) and Ban (Section 2.3.3). Since the experiments were carried out at different

44

times over the course of almost two and half years there are small differences in
the experimental conﬁguration and analysis procedures including the primary beam
intensity monitoring.

The primary beam intensity monitors were calibrated with respect to the Fara-
day Cup (FC) beam current measurement. For the 58Ni fragmentation experiment
only one FC was available (Z001); for the other systems (40'48Ca and “Ni), another
FC was located in the target box (Z014). Figure 3.2 shows the Z001 FC (left) in
the extraction channel and Z014 FC (right) in the target box, in front of the A1900
fragment separator (right). Since both F C’s are installed upstream from the target
position (which is Z015) it is not possible to measure the beam current and monitor
the counts of the secondary particles from the target simultaneously. We assume the
beam intensity variations over a short period of time (2—3 minutes) are negligible.
The beam current is taken as an average from multiple instantaneous measurements

5 seconds apart in duration of up to approximately 1 minute. Immediately after the

 

 

 

 

 

Figure 3.2: Faraday cups Z001 and Z014 in the extraction beam line and the target
box, shown relative to the K1200 cyclotron and the A1900 fragment separator. The
Z001 and 2014 Faraday Cups are magniﬁed by a factor of 4 in the presented drawing.

45

 

 

 
 
  

 
  

 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 
   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

.8 $01le I I FIVIWI V I lllllil l I I In"! I V IlYlI‘] I I II .1th Y I I 11"" I l I IIlll' l I l IIIII'I T l I FFFYTI I
10 5 Ca /( a 10“; Ca ’ E
Z 0 Be target / : : 0 Be target 1
" -9- 4 " 10'9=- 4
31 10 A Ta target a S 5 A Ta target / a
Q. E q 1:. : / 1
3 510-10 r ;
@1040 :- 1, >.
: 1 I
4 4s
"S . . -u _
L k 10 1:
12m“- e 3
e 1042? a
10-[2 l:- 3 1 ': ~
E1 U l 1 1111111 1 l lllllll l 1 1111111 I l lllllll l l 1 i-lllll L 1 1 Ill ll 1 1 1 llllll l l l llllll l l 1 111111 1“
10 102 103 104 1 10 102 103 104 105
NaI(Tl) count rate (cps) NaI(Tl) count rate (cps)
.- 5'8' VUHWI U I I'll"! I I IUITII'I Y I IIIHI' I I IIIIII' 1 IE _I& I I I IUIYII I T T111111 I I I an'r U U ITITTII .1
: Ni 3 10:8 Ni ‘2
: 0 Be target f/ : 4 0 Be target 4
A I I A ' ‘
3, 10'95 A Ta target “a S, 109: A Ta target “a
s 5 a s E i
o 1 o ~ .
>10 0 .4 5 4,1010 / 4.
g 5 s
x 10-“ E— 3 B I 1
a : c: -11 .— 5
L1. -12 4 LL. [0 : g 1:
10 F E:
-13; :_ I l [0‘12 : 5: a
10 h l l llllﬂlV I I 111111] 1 l lllﬂll l l llllll‘ L 1 1111111 1 1E“ : 111 l l I 111111 1 1 l Lllul 1 l l llllll l l 1 111111 :
1 10 102 103 104 105 10 102 10’ 104 105
NaI(Tl) count rate (cps) NaI(Tl) count rate (cps)

Figure 3.3: NaI(Tl) beam monitor calibration for all primary beams. Beam current
measured by the Faraday Cup is plotted as a function of the NaI(Tl) background
subtracted counting rate. Calibration data for 9Be target are shown as red Open
squares and 181Ta target as blue open triangles. 181Ta target FC reading is offset by a
factor of 0.1 in PC current reading for clarity. Solid lines through the calibration data
show the ﬁt by a second order polynomial in logarithmic representation. Horizontal
arrows indicate the range of the NaI(Tl) counting rate used in our fragmentation
experiments for both 9Be and 181Ta targets.

46

 

 

I I I IIUIII I l IIIIIII I I Y Illlll _‘ III I I I Ulllll I l IIUIUII I 1 {VIII
—I

§4°Ca E“Ni 2/ 3
10,7: 0 Be target __ 107; 0 Be target 1,
A Ta target A Ta target / ]
_ . ,4! 4

 

/

L a

N
Q.
‘0
l I
11111

Faraday cup (A)
N N
O o
‘0 Go
I l
x \
1]
N Faraday cup (A)
Q S
'5 co
l ‘ l

 

 

 

‘

 

 

 

 

 

 

 

 

 

 

10-10 5— El

1 LJLJJLLLL l I 111111! L llllllll" [.11 I

1111 I I 1111111 I 1 ll I I
10 102 103 104 10 102 103
Ban count rate (cps) BaFZ count rate (cps)

lllll

 

 

Figure 3.4: BaF2 beam monitor calibration for primary beams mentioned in the text.
Beam current measured by the Faraday Cup is plotted as a function Of the Ban
background subtracted counting rate. Calibration data for 9Be target are shown as
red open squares and 181Ta target as blue open triangles. 181Ta target FC reading
is offset by a factor of 0.1 in F C current reading for clarity. Solid lines through the
calibration data show the ﬁt by a second order polynomial in logarithmic represen-
tation. Horizontal arrows indicate the range of the BaF2 counting rate used in our
fragmentation experiments for both 9Be and 181Ta targets.

beam current measurement is taken the FC is retracted from the primary beam and
the counts of the beam intensity monitors are recorded in sealers for 1—2 minutes.
By repeating this procedure and varying the beam intensity by a series of meshes
(attenuators), we obtained a dependence of the primary beam current and the beam
monitor count rate. Since the beam monitors used are sensitive to the reactions of
beam with target the calibration depends on the target material. The beam monitor
calibrations were measured three times during each of the 4 fragmentation experi-
ments (the beginning, middle and end). No time dependence of the calibration was
Observed during our experiments. For each set Of beam intensity calibration a mea-
surement of the background radiation was performed by stopping the primary beam
at the exit of the ECR ion source.

Figure 3.3 shows all the N aI(Tl) monitor calibrations for 4(”480a and 58’“N i beams,

in terms of correlation of the FC current reading with the background subtracted

47

count rate of the N aI(Tl) monitor. The calibrations with 9Be target are shown as
red squares and with 181Ta as blue triangles. The ﬁt by a second order polynomial in
logarithmic representation is displayed as red (QBe) and blue (181Ta) solid line. The
horizontal arrows indicate the range of applicability of our primary beam intensity
calibration with the NaI(Tl) monitor. The calibration data are described by our fit
function within 7—10% over the range of beam intensity we measured. In the case
of the more neutron-rich projectiles 48Ca and 64Ni, when the beam current exceeded
beyond 10“7 A, we used the BaF2 monitor. The lower part was calibrated using the
NaI(Tl) monitor and the upper part using the Ban monitor. Figure 3.4 displays the
calibration curves of the BaF2 beam intensity monitor.

The calibration curves shown in Figure 3.3 and 3.4 are used in the analysis to

determine the total number of beam particles traversing the target in each run.

3.2.2 RIKEN experiment

Primary beam intensity monitoring for 86Kr fragmentation experiment was provided
by the MOMOTA telescope placed at forward angle outside the target chamber as
described in Section 2.4.3.

The FC was located after the target. This arrangement, in principle, allowed us
to simultaneously measure the beam current (FC) and the secondary particle count
rate (MOMOTA). Unfortunately, the primary beam reacts with the FC material and
the back-scattered radiation affects the count rate of the MOMOTA monitor. This
introduced an uncertainty upward of 15-20%. Thus we had to use a different method
of beam intensity monitor calibration.

For the 86Kr+gBe system we measured the absolute beam intensity by counting
86Kr33+ ions directly using the F2 plastic scintillator. From the measured charge state
distribution as shown in Figure 3.1, the 86Kr33+ charge state is 0.028% of the 86Kr

primary beam. During a 16 minute interval we recorded 1,054,792 events of 86Kr33+

in the F2 scintillator, 10,607 counts were recorded in the MOMOTA telescope at the

48

same time. Background measurement yielded a background count rate of 0.008 cps in
the F2 scintillator and 0 cps in the MOMOTA telescope. The background subtracted
count rate of the MOMOTA telescope is directly proportional to the primary beam
intensity.

In the case of 181Ta we measured the absolute beam intensity by counting 86Kr31+

ions directly in the F2 plastic scintillator. The 86Kr31+

charge state probability is
0.016% of the primary beam 86Kr. During a 4 minute measurement we recorded
214,525 events of 86Kr31+ in the F2 scintillator and 479 counts were recorded in the

MOMOTA beam intensity monitor.

3.3 Particle identiﬁcation

The general particle identiﬁcation method utilized in our fragmentation experiments
is described in detail in Section 2.2. Since there are underlying differences in detector
setups and details in the method of measurement of particle identiﬁcation observables,
we describe the particle identiﬁcation analysis of the NSCL and RIKEN experiments

separately.

3.3.1 N SCL experiments

The particle identiﬁcation detector setup was located in the focal plane box of the
A1900 fragment separator as shown in Figure 2.9. As discussed in Section 2.3.3 we
used the Bp—ToF-AE particle identiﬁcation technique for all beam-target combina-
tions measured at the NSCL, because we have Q = Z for all fragments of interest.
This simpliﬁes the whole particle identiﬁcation analysis because now by plotting the
energy loss, AB, in the PIN detector versus the time of ﬂight, ToF, for a ﬁxed Bp,
we can separate all fragments traveling through the fragment separator. A typical
raw experimental particle identification pattern is shown in the left panel of Figure

3.5. The example provided is for the 48Ca+9Be reaction at Bp = 3.2 Tm. We can

49

 

 

 

 

 

 

 

 

 

' ' I\l-z'='2' ' ' ' '* 20." ..'..,'. I I 7
- d - '"u‘ ”L. N...
4 “ N - M ﬁll an -
- i ' < . a, - I“; l h" .4.
A _ 2:16 ,_ " .g a 915- '43-. - ﬁg . -e~- _
$2000 \ - ﬂ. * 1 ‘ (U . a. an
C \1- " .-' ‘ c - _. .. ~F .
c ,. * a. o - an. E .. .
«s .. *5 .. ta * r~ g: ..... -
s .. 34. .. 9w: :; 2,... :.-.
Emce— .— .4, _ a» — g l '2.- _~,. g.
.,..--> a '. :2 5L _ -. —
. *‘I: . E889 —_’-~ I ‘
- l l l
0 1 500 2000 2500 3000 -1 1 2

ToF (channels)

0
Neutron excess N-Z
Figure 3.5: Uncalibrated (left panel) and calibrated (right panel) experimental particle

identiﬁcation (PID) spectra. The PID plots are shown for the 48Ca+QBe reaction at
Bp = 3.2 Tm.

see a clear separation of individual groups of events corresponding to different frag-
ments. By recognizing the typical features of the PID spectrum and locating the hole
corresponding to particle unbound 8Be nucleus, as described in Section 2.3.3, we can
identify all nuclides in the uncalibrated PID spectrum shown in the left panel of Fig-
ure 3.5. (The time of flight measurement was set up such that the more neutron—rich
fragments are located to the left of the N — Z = 0 band.)

The values of AE and TOF in channels, were calibrated (Section 2.3.3) to the
calculated energy loss and time of ﬂight using the LISE++ code [56]. Then, param-
eters K1 and K2 in Equation (2.4) were ﬁtted to the calculated values of energy loss
and time of flight to reproduce observed nuclear charges, Z. Mass numbers were cal—
culated using Equation (2.1). By performing the calibration for one Bp setting for
each primary beam we can extract a set of calibration coefﬁcients which allowed us
to present all particle identiﬁcation spectra in nuclear charge versus neutron excess
(N — Z) as shown in the right panel of Figure 3.5.

Fragment yields were extracted from calibrated particle identiﬁcation spectra (ex-

ample shown in the right panel of Figure 3.5) by counting events corresponding to

50

 

         
 
    

 

 

 

3000c- r
f a
* E’
A — g
1,2000— 5’
“c’ a
C _
(U
c _ ' 'b
0 '11111-.___-....-.-_.
V _ 700 750
UJ ToF(channeB)
<1 _
1000—
_ / .i' “
_ lgzo '
1 1 1 l 1 i
200 800

400 600
ToF (channels)
Figure 3.6: Uncalibrated particle identiﬁcation spectrum of fragments created in re-

action 86Kr+181Ta at Bp=2.07 Tm. The inset shows zoomed area around the missing
8Be.

individual isotopes. This procedure was applied to all magnetic rigidity settings and

all beam-target combinations.

3.3.2 RIKEN experiment

At 64 MeV/ u signiﬁcant fractions of 86Kr are not fully stripped of electrons after

traversing the target. The charge state distribution of 86Kr showed that approxi-

51

mately 10% of the primary beam intensity of 86Kr is in 86Kr35+ charge state after
passing through the 9Be target. The fraction is even larger in the case of 181Ta target
because the charge state distribution for 86Kr primary beam is broader as discussed
in Section 3.1. In this case we had to use the general Bp-ToF-AE—TK E particle
identiﬁcation method. This technique allowed us to identify the momentum, p, mass
number, A, nuclear charge, Z, and charge state, Q, for every fragment of interest by
measuring four quantities the magnetic rigidity, Bp, time of ﬂight, ToF, energy loss,
AE, and total kinetic energy, TK E.

Basic particle identiﬁcation plot and calibration procedure is the same as in the
case of the N SCL experiments. Figure 3.6 shows an uncalibrated particle identification
plot, AE versus ToF, for a 2.07 Tm magnetic rigidity setting using 86Kr+181Ta
reaction. The inset shows the zoomed area around the 8Be hole. (The time of ﬂight
measurement was set up such that the more neutron-rich fragments are located to the
right of the N — Z = 0 band.) From that reference point we can identify all isotopes
in the PID spectrum. The T 0F and AE detector calibrations were done following
the same procedure used for the NSCL reaction systems. In the case of 86Kr beam
isotopes of all elements in the analysis (Z 2 25) come in multiple charge states.
The measurement of the total kinetic energy, T K E , must be used to provide clean
separation.

The analysis procedure is demonstrated in Figure 3.7 for the 86Kr+9Be reaction
run with 2.07 Tm magnetic rigidity setting. The left panel of the Figure 3.7 shows
the energy loss, AE, versus time of ﬂight, ToF, PID spectrum with gates around
elements with Z = 28, 31 and 34. The right panel of Figure 3.7 plots the charge state,
Q versus A/ Q ratio plane for fragments with Z = 28, 31, and 34 gates from bottom
to top, respectively. We see isotope charge states clearly separated for all elements
showed. The counts of individual isotopes in different charge states for the differential
cross section calculation were extracted from the charge state, Q, versus A /Q ratio

plots.

52

8
8

A O

1’ CD

a) -o-I
(U

c

g ‘63
a)

.1:

8 9’
(u

'él . 5

2000 ~ "I“

 

 

 

 

 

 

H. ' n '1 1

 

- ' 2h2 A
Ratio NO

70 750 800
Sl'oF (channels)

Figure 3.7: Particle identiﬁcation spectrum for a 2.07 Tm magnetic rigidity setting for
the 86Kr+9Be reaction. Left panel shows the PID with three gates around elements
with Z = 28, 31 and 34. Right panel shows projections to charge state, Q, versus
A/ Q ratio plane of events within the corresponding nuclear charge gates Z = 28, 31,
34 from bottom to top, respectively.

3.4 Cross section analysis

3.4.1 Differential cross sections

Yield for a given fragment and the momentum acceptance of a magnetic rigidity
setting are combined together with primary beam intensity determined by the cali-
bration described in Section 3.2 to obtain the differential cross section, da/dp, using

the following formula:

do N(A, Z) - M,

——A,Z =—,
( ) I‘t'NA'dt'Ap'TLIVE

dp (3.1)

53

where
N (A, Z) Number of events for a fragment (A, Z),

I Primary beam intensity during the run (s‘l),
t Time duration of the run (8),

N A Avogadro’s number (mol‘l),

Mt Atomic number of target material (gmol‘l),
d, Target thickness (gem—2),

Ap Momentum acceptance (0.2% in dp/p),
TLIVE Live time correction.

By plotting the differential cross section, da/dp, as a function of momentum, p,
for a given fragment a momentum distribution is obtained. The area of the momen-
tum distribution, in this representation, is the fragment production cross section.
The momentum distributions are discussed in more details in the following chapter.
Here we focus our attention to the transmission correction factor evaluation and the

momentum distribution ﬁtting procedure.

3.4.2 Momentum distribution ﬁtting procedure

Momentum distributions of fragmentation products provide insight into the fragmen-
tation reaction mechanism. Momentum distributions of fragments created in rela-
tivistic heavy-ion fragmentation are well described by a Gaussian function [64]. It
has been shown that the momentum distributions created in the intermediate energy
(20—200 MeV/ u) fragmentation are asymmetric [65]. We can separate the momentum
distribution into two half Gaussians joined at maximum, p0, the high momentum
side created by a “pure” fragmentation mechanism characterized by GR and the low
momentum side created by a “mixed” reaction mechanism described by UL [59]:

da S - exp (-(p - ref/(20 )) for P S P0:

f.
_ _ (3.2)
dp S - exp (-(p - IMP/(20%)) for P > P0,

54

where S is the normalization parameter.

The experimental width parameters at, and 03, were extracted from the asym-
metric experimental momentum distributions using ﬁts with a modiﬁed Gaussian
function of Equation (3.2). Such procedure, however, results in correlations among
the three parameters pg, 01,, and JR characterizing the shape of the fragment mo-
mentum distributions. A minute change in the centroid, 190, will result in different
division of the total width of the momentum distribution between the UL and 03.
These complications, however, do not affect the determination of the fragmentation

production cross sections.

3.4.3 Evaluation of fragmentation production cross section

The fragmentation production cross section is obtained by integrating the area of the
momentum distribution and divided by the transmission coefficient, 5. This trans-
mission correction factor, 5, expresses the efficiency correction of our measurement is
addressed in more details in the following section. The momentum distributions for
fragments in our analysis were obtained by a systematic scan of the magnetic rigidity
space as described in Sections 2.3.4 and 2.4.4 for the A1900 and RIPS fragment sep-
arator experiments. Most of the experimental momentum differential cross sections
extend over 1—4 orders of magnitude (typical examples shown in Figure 4.1). Since
the range in the Bp space covered in our experiments is limited, there are fragments
in our data with parts of the momentum distribution (either low or high momentum
side) not measured. Figure 3.8 shows examples of complete and incomplete momen-
tum distributions of 50V and 50Mn fragments, respectively, produced in 64Ni-l-S’Be
fragmentation reactions.

The complete momentum distributions were ﬁtted by a modiﬁed Gaussian func—
tion of Equation (3.2) by minimization of X2 method. Most of the experimental mo-
mentum distributions were described extremely well over almost three orders of mag-

nitude (e. g., see 50V in Figure 3.8). Overall approximately 60—80% (depending on

55

 

 

 

 

 

 

 

 

E , f . - r a ﬁ , ......... , ......
2: 104.- a
10 g” ‘ :
’7: ‘ ”79105:— 1
133103? ‘ .03 E
E (D : : E 0) ‘

2104 210'“? a
V [5— ‘E V E 5
b'8 E i b|g ’ 50 i
U 105: 50V —: D 107;F Mn 1

C l 8;. 1
106—111 IIIIIIIII 1 IIIII 10:1111 IIIIIIIII 1IIIIL+‘
24000 26000 24000 26000

p|| (MeV/c) pll (MeV/c)

Figure 3.8: Momentum distributions of 50V and 50Mn created in fragmentation of 64Ni
on 9Be target. Shown fragments demonstrate a typical complete (50V) and incomplete
(5°Mn) momentum distributions. Experimental data are depicted by open squares and
the ﬁt by modiﬁed Gaussian function of Equation (3.2) is shown as a solid line.

the reaction system) of the fragments have complete momentum distributions.

The above described ﬁtting procedure was not suitable for the distributions with
very few experimental points (< 5), or for distributions with missing left or right
sides. In order to extract fragmentation production cross sections for these fragments
and to minimize the uncertainties we took advantage of the larger set of completely
measured momentum distributions. This set of data provided a data base for empirical
systematics of the three parameters of the modiﬁed Gaussian function (Equation
(3.2)): the matching point 190, the variances of the left, 01,, and the right, 03, sides. The
analysis procedure for extraction of the production fragmentation cross sections for
the fragments with incompletely measured momentum distributions is demonstrated
in the case of fragmentation of 64Ni on a 9Be target.

In the top left panel of Figure 3.9, the ﬁtting parameter p0 is plotted as a function
of fragment mass number. All p0 determined by ﬁtting of the experimental data
with the modiﬁed Gaussian function were within 1% of the polynomial systematics
shown as a solid line. This parameterization assumes that pa for a given mass number

are identical, but the experimental data show deviations. In order to get an even

56

 

 

 

 

 

 

 

 

 

‘li ' l j l l I

30000? _ 25800} -:
a . l G 3 1
S _ S 25700 f j
§ 20000 e — g : :
Q0 I (1025600 j j
r - _ _
. 1 1 1 . 1 . 1 . 1 ‘ : 1 1 I 4 41 1 1 ‘

10 20 30 40 50 60 2 4 6 8 1o

Fragment mass A Neutron excess N-Z

 

 

 

 

 

1.
20

Asa-Pr

 

0R (MeV/c)

 

 

l

 

 

C)
O

1
20

Ala-A:

‘40

Figure 3.9: Systematics of the centroids and variances of the momentum distributions
for the reaction 64Ni+9Be. Upper row shows the centroids, p0; the variances, UL and
03, are shown in the bottom row.

57

more precise prescription for the centroids we ﬁtted individual isobar chains as a
function of neutron excess to get a “local” parameterization. The top right panel
of Figure 3.9 shows a linear ﬁt for the centroids of isobars A = 50. In this local
systematics for the centroids it was shown that the experimental data were within
0.5% of the parameterization line. The variances 0L and on are modeled by the

Goldhaber model [64]

UL 2: 0’0 AP—l , (3 3)
a A(Ap—A) °

where Ap and A are atomic mass of the projectile and fragment respectively and
06‘, 062 are used as ﬁtting parameters. Best ﬁts to the experimentally determined 0;,
and 03 are shown in bottom panels of. Figure 3.9. To describe the experimental data
06’ = 140 and 06* = 94 MeV/c were used. The incomplete momentum distributions
were ﬁtted with Equation (3.2) by having the centroid p0 and variances 0L and an to
range over values centered at values determined by the parameterization. The size of
the interval over which the parameters were allowed to vary was :|:1% for the centroid
p0 and :l:13% for the variances 0L and 03.

Figure 3.8 shows examples of complete and incomplete momentum distributions
(open squares) along with the ﬁtted modiﬁed Gaussian function of Equation (3.2)

(solid line) for fragments 50V and 50Mn respectively.

All fragmentation cross sections Ufrag were calculated using

Ufrag=S'\/Z2T:'(0'L+0’R), (3.4)

where S, 0L and 03 are parameters of the modiﬁed Gaussian function (3.2). The calcu-
lation of fragmentation production cross section, afrag, in Equation (3.4) is equivalent
to integrating Equation (3.2) over all momenta, p.

In the case of 86Kr primary beam the total fragment yield is distributed over

different charge states. For 86Kr+9Be reaction system we analyzed fragments with

58

Z — Q = 0 charge states. Since the measured charge state distribution for the pri-
mary beam is reproduced well for the 86Kr+gBe reaction, we use the GLOBAL code
to correct the ﬁnal cross sections for all fragments in the analysis. The calculated
corrections vary between 1—9% for isotopes of 25 S Z S 36 elements.

For the 86Kr+181Ta reaction we analyzed the three most abundant charge states
(Z -— Q = 0, 1, 2) in order to harvest most of the total cross section. The sum of these
three charge states for the 86Kr beam on 181Ta target is reproduced very well by the
GLOBAL calculation. The corrections for fragment cross sections in the 86Kr+181Ta
analysis, calculated by GLOBAL, vary between 0.1—3% for isotopes of 25 S Z S 36

elements.

3.4.4 Transmission correction evaluation

The transmission correction, 5, is essentially a correction factor for the experimen-
tal acceptance. It accounts for ﬁnite spatial and angular acceptance and transport
eﬂiciency of the fragment separator used in the experimental measurement. The dif-
ﬁculties in the evaluation of this correction arise from our inability to measure the
essential ingredients. The transmission correction can be factored into momentum
(longitudinal direction) and angular (transverse direction) transmission. The coor—
dinate system used in our calculations and in further discussion is shown in Figure
3.10, where ‘z’ is in beam direction and ‘x’, ‘y’ are in plane perpendicular to the beam

direction.

 

Figure 3.10: Coordinate system used in ion optical calculations of transmission cor-
rection. Where ‘Z’ is in the primary beam direction and ‘X’ and ‘Y’ are in the per-
pendicular plane.

59

In Section 2.1 we discussed that the narrow momentum acceptance implies sim-
pler evaluation of the transmission. To evaluate the momentum acceptance (0.2% in
dp/p) effect we used a universal Monte Carlo code for the transport of heavy ions
through matter within ion-optical systems MOCADI [66]. All MOCADI simulations
were carried out in the ﬁrst order. The momentum acceptance was deﬁned as a ratio
of the number of particles transported to the focal plane of the fragment separator
and the number of particles within i0.1% of the central momentum in the target
plane. This ratio is calculated only for particle transmitted in angular space with
100% probability, in order to ensure the separation of the angular and momentum
transmission corrections. The momentum acceptance in our deﬁnition is independent
of the magnetic rigidity setting and the fragment mass and charge. The momentum
transmission in our case expresses the “effectiveness” of the momentum cut made by
the slit system at the intermediate image of the fragment separator. The calculated
momentum transmission is 962t2% and 98:1:2% for the A1900 and RIPS fragment sep-
arators, respectively. The uncertainties in the momentum slit opening and angular
cut contribute to the ﬁnal error of the momentum transmission. Calculated angular
acceptance areas, used in the evaluation of the momentum transmission, of the two
fragment separators are presented in Figure 3.11. Calculations were carried out using
the primary beam (“Ni and 86Kr for the A1900 and RIPS respectively) with unreal-
istically large uniform angular distribution (i80 mrad) in both the 0 (‘x’ direction)
and o (‘y’ direction) angles. Simulated plots in Figure 3.11 show the projection of
particles transported to the focal plane of the fragment separator to the 45 versus 0
plane at the target position, thus deﬁning the angular acceptance of the A1900 (left
panel of Figure 3.11) and RIPS (right panel of Figure 3.11) fragment separators in
our MOCADI simulations.

For evaluation of the angular transmission we used LISE++ [56]. This code uses
analytical methods to calculate the transport of particles through the fragment sepa-

rator, which enable it to execute the calculations much faster than Monte Carlo based

60

 

 

50

IjﬁTj—TrIIII
-1

(p (mrad)

50

I I rj I

 

 

 

 

 

 

‘ 100

 

Figure 3.11: Angular acceptance of the A1900 (left panel) and RIPS (right panel)
fragment separators as calculated by MOCADI in 43 versus 0 plane. Dashed-dotted
line denotes the input angular distribution (:l:80 mrad) for the MOCADI calculation.

MOCADI. A benchmark calculation was performed for the 58Ni+QBe system using
identical parameterization for the angular distributions of fragments and the primary
beam emittance. In Figure 3.12 we show the calculated angular transmission curves
using LISE++ and MOCADI codes. Since the overall agreement is better than 4%,
we use LISE++ for all angular transmission calculations.

The angular transmission was deﬁned as a portion of the angular distribution, for
a given fragment, transported to the focal plane of the fragment separator. Results
of the angular transmission calculations depend on the primary beam emittance and
the fragment angular distribution assumptions. Since the experimental data for these
two components are scarce, we had to rely on parameterizations and estimates.

The emittance of the primary beam is given by its spatial and angular distribu-
tions. The primary beam spot size in the target plane is approximately 2 x 2mm2,
which gives us an estimate of the spatial component. The direct measurement of the
angular distribution of the primary beam in the target plane is not available. We
did get an estimate of the primary beam angular distribution for the 58Ni primary

beam by measuring the position and angular distributions at the intermediate image

61

(I2) of the A1900 fragment separator. In order to reproduce the measured distribu-
tions using MOCADI Gaussian proﬁles of the primary beam with variances in spatial
ox 2 ay 2 1mm and angular 0'9 2 0,5 = 7mrad coordinates were used.

Another estimate of the primary beam emittance can be inferred from the calcu-
lations for all beams delivered by the K1200 cyclotron in stand alone mode during
the 1990S as shown in Figure 3.13. Most of the beam emittances fall within approx-
imately 147r - mm - mrad in both ‘x’ and ‘y’ directions. If we assume the normalized
emittance of the K1200 and the CCF is the same we can take the K1200 stand alone
mode emittance as an upper limit of the CCF primary beam emittance. This result
is consistent with the above mentioned conclusions of MOCADI calculation.

In the case of the RIKEN fragmentation experiment we had to rely solely on
ion optical calculations [68]. The primary beam spot size in the target plane for the
RIKEN experiment was approximately 1 x 1mm2. The ion optics calculations suggest
that the emittance of the primary beam is roughly half of that at the CCF, resulting
in 0'9 = 043 = 3mrad [68].

 

 

 

 

 

C
.9
*6
“5
0
0 " .1
0-4 ,- —LISE++ 4
o. ' ----- MOCADI 1
l 1 1 l L d
20 40

Fragment mass number A

Figure 3.12: Comparison of angular transmission calculated using the LISE++ and
MOCADI codes. Parameterization of Equation (3.5) for the variance of the transverse
momentum distribution is used in both cases with 00 = 100 MeV/c and Up = 200

MeV/c. Primary beam emittance parameters: a_.c = 0,, = 1 mm and 09 = 0,), = 7
mrad.

62

on

O)

IIIILLIAII

A
I

 

theta (mrad)
phi (mrad)
‘3. .. 1?. . . .

I
N
llllll

 

 

 

 

 

 

 

Figure 3.13: Calculations of the primary beam emittance ellipses in 0 versus a: phase
space (left panel) and d) versus y phase space (right panel) for all primary beams
delivered by the K1200 cyclotron in stand alone mode during the 19903 [67].

63

 

 

I I I I I 1 I I TI 1 l T ﬂ T I I I I I—l j I
—1 —

     
      
 

 

 

 

 

 

 

f j - 86Kr+Be
08* ~ '
c c l—
.9 : ‘ .9 °"’.
*6 I ‘6'
m a)
t0.6— _ t
o - 006 _
0 c.9453 mrad 0 ‘ i, . ........ *°e¢=1mrad .
0 4 __ ce¢=7 mrad j g7 __ cs9 1,=3 mrad
' _ . ..... cew=10mrad . ~ ..... °e¢=7 mrad
~ —..09¢=15 mrad + 0.4K _..c”=10 mrad -
“11111111111111.1‘ .IIL1IIIIIIPII1II‘
20 30 40 50 60 70 80
Fragment mass number A Fragment mass number A

Figure 3.14: Angular transmission correction as a function of fragment mass number
for the A1900 (48Ca+9Be) and RIPS (86Kr+9Be) fragment separators. The calcula-
tions were done for different values of variances of the Gaussian proﬁle of the primary
beam angular distributions, aw. All calculations used Equation (3.5) to evaluate the
variance of the transverse momentum distribution 01 with 00 = 100 MeV/c, (ID = 200
MeV/c.

Due to uncertainties associated with the determination of the primary beam an-
gular distributions we carried out a series of angular correction calculations using the
LISE++ code. The primary beam angular distributions were simulated with Gaussian
proﬁles, in all cases, varying the variances, “M: in ‘x’ and ‘y’ directions respectively.
The LISE++ calculations in Figure 3.14 for the A1900 fragment separator were done
using the 48Ca+gBe reaction with 09",, = 3, 7, 10 and 15 mrad, and for the RIPS
fragment separator using the 86Kr+9Be reaction with a“, = l, 3, 7 and 10 mrad. We
see that the uncertainty of the 09,4, is within i3 mrad of the above estimated values
of 7 and 3 mrad for the A1900 and RIPS, respectively, resulting in the angular trans-
mission correction uncertainty at the level of approximately 3—4%. For all angular
transmission calculations we used 0'9 = ad, = 7mrad, 0x = a3, = 1mm for the NSCL
primary beams and 09 = 04, = 3mrad, 0x 2 0y = 0.75 mm for the RIKEN primary
beam as explained above.

Experimentally the longitudinal and the transverse components of momentum

64

distributions after fragmentation are equal (within 10%) in relativistic energy regime
[69]. However, this is not the case at intermediate energies [70,71]. Since there are no
published experimental data available for the angular distributions of the fragments
measured in our fragmentation experiments, we relied on the parameterization [70]

of the variance of the transverse momentum distributions, 0 13

Amp —— A) A(A — 1)
2 = 2 2
“i ”0 Ap —— 1 + ”DAP(AP -— 1)

 

(3.5)

where A p and A are mass numbers of the projectile and fragment, respectively. The
ﬁrst term in Equation (3.5) comes from the Goldhaber model for the width of a
longitudinal momentum distribution and the second term results from the deﬂection
of the projectile by the target nucleus [71]. 00 can be determined by our measurement
of the longitudinal momentum distributions and an, the orbital dispersion, is taken
from Ref. [70], where it is shown to describe the fragmentation data of 16O at 92.5
and 117.5 MeV/ u. From the published data [70] we estimated the orbital dispersion
parameter an to be 185 i 15 MeV/c for the NSCL experiments and 225 :l: 25 MeV/c
for the RIKEN experiment.

We were able to conﬁrm the parameterization of Equation (3.5) with the fragment
angular distributions measured in a separate experiment at the N SCL using the S800
spectrograph [73]. Figure 3.15 shows the angular distributions for 44Ca and 59Co
fragments produced in the 64Ni+QBe reaction at 95 MeV/u. The angular distributions
were reconstructed with the inverse mapping technique using the S800 spectrograph
for the reaction 64Ni+9Be [72]. Based on the measured angular distributions in the ‘x’
direction, we calculated the variance of the transverse angular distribution al(0) =
33 :l: 2 and 16 :l: 1 mrad for 44Ca and 59Co fragments, respectively. These transverse
angular distributions translate to transverse momentum widths a; = 482 i 29 and
319 :i: 20 MeV/c for 44Ca and 59Co, while the parameterization (Equation (3.5))
predicts ai = 446 :l: 12 and 301 :l: 12 MeV/c. There is a good agreement between the

65

 

 

 

 

 

 

 

 

 

1oq'rlr 1 1OOA-ﬁ‘ﬁﬁTT'l""l“—
* 44C ‘ . 59
. a Co
A A ~ I
=5 =2 L
53 £3
a) i a)
*" _ _ ~500— -
S 50 S
0 ~ 0 f
o o
I I L I I I I I l I I I I l I I l I I m l I
0 -100 0 100 0 -100 0 100
0 (mrad) 9 (mrad)

Figure 3.15: Angular distributions in the target plane for 4"‘Ca and 59Co fragments
from the 64Ni+9Be reaction, measured at the NSCL using the S800 spectrograph from
Ref. [72]. See text for details.

measured width and predictions from Equation (3.5).

As was discussed in the previous sections, the experimental momentum distri-
bution for our fragmentation data are described by a modiﬁed Gaussian function of
Equation (3.2) characterized by two width parameters 0;, and 03. For the purposes of
the angular transmission calculations we approximated the momentum distributions
with equivalent Gaussian shape. The actual ao parameter was determined by ﬁtting
(01, + 03) / 2 from the experimental data. Values of co and 01) parameters used in our
transmission correction calculations are listed in Table 3.1.

Final transmission corrections were obtained as a product of the momentum and
the angular transmission corrections. The calculated transmission correction, s, is
plotted as a function of fragment mass number in Figures 3.16 and 3.17 for all reac-
tion systems. Figure 3.16 displays the mass dependence of the transmission correction
for the 40Ca (t0p panels), 48Ca (bottom panels) beams with 9Be (left panels), 181Ta
(right panels) targets. The transmission correction for the 58Ni (top panels), 6“Ni
(middle panels), and 86Kr (bottom panels) beams with 9Be (left panels), 181Ta (right

panels) targets are showed in Figure 3.17. The shaded region denotes the estimated

66

 

ion

t

ISSIOn COFTGC
o
'00

.0
a:

9
.5

.0

Transm
N
‘ ‘ ' It...

 

,,r:.; I
. . I

1.
1-4...
1 l I I I

l
1

A; ......
11:15] '

14
ills
.C:
llnggLJ_1_lIl

 

l l 1 I l 1 1 1 I L I l l

 

l
20 30 4o
Fragment mass number A

 

1 r r r I r 771' I l i I I I j—ﬁ I’m
_ ,‘3‘12‘17? _,
C r~~ '
_ 48 9 ...::;;~;»~ .
o a+ e
o— . .Lxﬁ‘I-U‘
+- * size” ‘
o 3 — LL —
.3367”
(D ' _ «
h '- WW ‘1
o a”
.255?
0 6~ - —
C - .rfsféii‘
)- ‘.:‘el:i" *
o I n:
o- "‘ 41%;?” -—1
a) e"
I. ”gr-4 .4
J '3
-- 0 4— a e
' i111?
f as ‘
U) *' riff”! ‘
C l- .25
‘ ‘ '1
cu a a
h 0.2 if?
F “
_,
)- '4
1 1 1 l 1 1 1 1 l 1 1 1 1 L L a

 

 

 

20 30 4o
Fragment mass number A

 

.0
on

.o .o
O)

, a

.0

Transmission correction
N
. . r w,

 

 

-1

 

[A L l l I l 1 1 l

 

30
Fragment mass number

I.
40
A

 

 

1 l l ' l ’ _.-:—::-:-:-'
C. I _._.-:-:':'::::::. lllll
g :“Ca-i-‘a’Ta ,-
0 _ _.
G) 0.8_ : d
t _ -
C 0.6: .. “
.9 - .
-- 0.41- a? a
E r .5" 1
(D r j
C L ~ 1
g 0.2:? a

_II.1II..111111111:

    

 

 

20 30 40
Fragment mass number A

Figure 3.16: Final transmission correction, c, as a function of fragment mass number
for 40"“BCa primary beams with 9Be and 181Ta obtained for 0.2% momentum accep-

tance.

67

 

.0
on

.0
O:
l
5%,

.0
.p
1
,

x'r'
:fr’

Transmission correction
.0
AL

 

1 l I I l l I I I I I

_ .' "1T 1111

1 I l l +

|
IIL+1J_14_1111

 

.T

 

o 'ILJLI

20 30 40

50

Fragment mass number A

 

 

1 I l T l l I I T» ‘ 1:???
."I:15?i:.:~::“_.-,-,II. 4
8 I “Ni 936 i
._ _ + a
g 0.82 g _1
L- P ‘l .,
h ’- _;Cv.;+' 4
o j
F iii... 4
U 0 6* . ﬂ
C .
T w ‘4
.9 . g 4
~ift‘v“
o— 0.4—- 9:325" a
E ” .515 A
ﬁr;
U) - $.43 ‘
C F :1,
as i .5 A
h 0
1— ' 5 4
i
I , l l l

 

 

20

40
Fragment mass number A

60

 

.0

b

"1.
”it

.°
N
rrth‘

L l L I

 

r r v r t r 1 T T r l I m
l l l

c b 86 9 ,.v:?'. ””1" I I

_Q . Kr+ Be —

8 0 8V 5” _]

- F -5
g I. “$.53” -<
— .4” -t
5

8 f 55‘ I

0.6” ”3‘5"; 1

c ._ d1? .1

.9 r if ‘

.vz-‘i”

(‘2 I 15‘”?- 4
(I)
C
(O
p

. l

1 l I I 1

I14

 

 

1 1 I I l I l l I
o I
70

80

Fragment mass number A

 

d

‘I

r-

.o .o .o
O) on
r 1 T I T IT

Transmission correction
0
ix:

 

L l

k
- 5’Ni+’°’Ta

T—I—II—II—I—TIIIITIIIIF'

 

.....

IIII

 

liIllllIllllIJlllILll

 

O

20 30 40 50
Fragment mass number A

 

.°
p 1

Transmission correction
O
in

 

   
 

 

P l l

 

o L I l l l I

1
4O 60

 

I

A
I
-q

n

I.

Transmission correction

 

 

I I I I I I I I I I

Kr-I-1 "Ta

 

 

60 7O 80

Fragment mass number A

Figure 3.17: Final transmission correction, e, as a function of fragment mass number
for 5854Ni and 86Kr primary beams with 9Be and 181Ta targets obtained for 0.2%

momentum acceptance.

68

Table 3.1: Values of 00 and 00 parameters in Equation (3.5) used in the angular
transmission correction calculations for different primary beam target combinations.

 

 

 

 

 

 

 

Beam Target 00 OD
(MeV/c) (MeV/c)
“oCa jBe 106 :l: 4 185 :l: 15
181Ta 101 j: 5 185 :l: 15
48Ca 9Be 109 :t 5 185 :l: 15
181Ta 106 :l: 4 185 :t 15
58Ni 7"Be 113 :l: 3 185 :l: 15
181Ta 110 i 3 185 :t 15
64Ni ”Be 117 :l: 5 185 :t 15
181Ta 114 :t 3 185 :l: 15
86Kr 9Be 147 :l: 5 225 :l: 25
181Ta 153 i 5 225 :l: 25

 

 

 

 

 

uncertainty in the transmission correction factors which are dominated by the un-
certainty of the co parameter for light fragments and the uncertainty of momentum

transmission for heavy fragments.

3.5 Error analysis

The uncertainties in the ﬁnal fragmentation cross section are calculated based on the
statistical uncertainty, the beam intensity calibration (7—10%), the error calculated
by the ﬁtting procedure and the transmission uncertainty (2—8%). Uncertainty of
the target thickness determination and momentum slit opening were estimated to be
smaller than 2% and hence they are neglected in our error analysis. Other uncertain-
ties like reaction losses in detectors and secondary reactions in the target material
are smaller than 1% so their contributions are also neglected. For the fragments mea-
sured with incomplete momentum distributions, a systematic error stemming from
the extrapolation of the parameterization of pg, 0;, and 0’}; were included in addition

to the above mentioned errors.

69

Chapter 4

Experimental results

This chapter presents the experimental data extracted from the four projectile frag-
mentation experiments carried out at the NSCL and one at RIKEN. The fundamental
parameters, such as the shape, maximum, and width of the momentum distributions
of fragments are presented. The production fragmentation cross sections are com-
pared to the empirical parameterization of the fragmentation cross sections and to
published experimental data measured at higher incident energies. The target and
projectile dependence of the measured fragmentation cross sections is discussed at

the end of the chapter.

4.1 Momentum distributions

More quantitative discussion of the properties of the fragment momentum distribu-
tions is provided in this section. The obtained fragment momentum distributions are
compared to high energy systematics.

The basic properties of the momentum distributions of the target or projectile
fragmentation products were the subject of many studies [65, 74—76]. In general, the
momentum distributions created in the fragmentation of relativistic projectiles are

very well represented by Gaussian functions centered at velocities near that of the

70

 

 

 

 

 

 

 

 

 

 

 

 

10'2E
10:3
o o
93 nSWﬁ
£10“i Em :
v E MO4E
Sl‘1 " 8|Cl :
U10-5E U _
i 106: .
10": .5
104%" l l' “g
I: : 10-2
A1025 j A
o : o
g 5 133103
E c0103: 5 E a:
2 i 2
o. 4’ . 0.0.4
Slum a 8'0
_ : 1o~5
10.5? Ea

 

 

 

 

 

 

 

Figure 4.1: Examples of the momentum distributions. Distributions of 33P are shown
for the 40"’E’Ca+9Be and 49V for the 58"5“Ni+9Be systems. The solid line shows the
modiﬁed Gaussian ﬁt of Equation (4.1), and the dashed line shows the Gaussian ﬁt
to the high momentum side of the distribution.

projectile [75]. Many models and parameterizations have been developed to describe
the variance (width) and the maximum of the Gaussian-shaped momentum distribu-
tions [64,77].

On the other hand, the fragment momentum distributions at intermediate ener-
gies are asymmetric [65]; they can no longer be approximated with a Gaussian func-
tion. Different functions have been used to reproduce the asymmetric momentum

distributions: Gaussian functions with a different cutoff [76], polynomial exponential

71

functions [65], or Gaussian functions with exponential tails [74]. Our experimental
data are very well described by a modiﬁed Gaussian functions of Equation (4.1). To
demonstrate the asymmetry of the measured momentum distributions, we show the
momentum distributions of 33P from the fragmentation of 40"‘8Ca on a 9Be target and
49V from the 58"’4Ni+9Be reaction in Figure 4.1. The solid lines in Figure 4.1 show

the ﬁts using the following modiﬁed Gaussian function:

do _ S - exp (-(p - POP/(20%)) for p S 100,

Zp‘ _ (4.1)

S - exp (-(P — 1902/9030) for P > no,

where S is the normalization factor, p0 is the centroid, 0L and 03 are width parameters
of the “left” and “right” halves of the momentum distribution. The modiﬁed Gaussian
function describes the experimental data over 3—4 orders of magnitude extremely well.
On the contrary, the ﬁt to the high momentum side of the momentum distribution
by a symmetric Gaussian function (dashed curve) shows a large deﬁcit in the low

momentum part of the distribution.

4.1.1 Widths of the momentum distributions

At high bombarding energy (> 200 MeV/ u), the momentum distributions were found
to be independent of the bombarding energy and the target. The observed widths of
the experimental momentum distributions showed a parabolic dependence on frag-
ment mass number. The Goldhaber model [64] developed in 1974 considers the pro jec-
tile nucleus, as being composed of independent nucleons moving freely in a spherical
potential well. The only correlations assumed in the model are those arising from
momentum conservation. The momentum distribution of the nucleons results in to-
tal momentum zero at any time in the rest frame of the projectile nucleus. When a
certain number of nucleons are removed suddenly from the projectile, the remaining

residual fragment recoils in the opposite direction with the same momentum magni-

72

tude. Furthermore, assuming the isotropy of the internal momenta of the nucleons in
the projectile, the recoil momentum distribution of the fragment, projected on a given
axis, has a Gaussian shape. Goldhaber showed that the width 0 of this distribution

is related to the the masses of fragment, A, and projectile, A p [64]:

Amp — A)

AP _1 , (4.2)

0:00

where the 00 parameter, also called reduced width, is the root-mean—squared momen-

tum of the individual nucleons, < pf >:

 

(4.3)

Assuming the nucleons are distributed in the projectile according to the Fermi gas

model we have:

3
< 1)? >2 gpf’ermit (4'4)

where ppermi is the Fermi momentum. By inserting this equality into Equation (4.3),
we get:

2
03 = 3%5. (4.5)

Hence, the Goldhaber model relates the Fermi motion of the nucleons to the reduced
widths of the fragment momentum distributions in the projectile fragmentation reac-
tions.

The Fermi momentum, ppermi, can be experimentally obtained from quasielastic
electron scattering. Moniz et al. [78] measured values of the Fermi momentum for tar-
get materials varying from 11Li to 208Pb. By interpolating their results, we estimated
the pperm, to be approximately 250 MeV/c for 40"‘8Ca and 260 MeV/c for 58MN i and
86Kr nuclei. Calculated reduced width, 00, parameters are listed in Table 4.1 along
with the values determined from our experimental data.

In 1989 Morrissey [77], assuming that the momentum distribution is a convolution

73

Table 4.1: Goldhaber reduced width parameter, 06‘, for the right side and 06‘, for the
left side of the experimental momentum distributions values listed for all investigated
reaction systems. Last column shows the reduced width parameter of the Goldhaber
model calculated based on the Fermi momentum from Ref. [78].

 

 

 

 

 

 

 

Beam Target 05 of Fermi mom.
material material (MeV/c) (MeV/c) (MeV/c)

40 Ca gBe 125:1:10 851:8 112
181Ta ll7:I:9 84:}:7 112
48Ca 9Be 134i9 893:6 112
181Ta 129:}:9 88i7 112
58M gBe 133i10 9755 116
1"lTa 129i10 95i6 116
“Ni 9Be 1403:? 94:l:5 116
lBlTa. 136i7 93i4 116
86m jBe 175:1:11 12118 116
l81Ta 181:1:9 119i10 116

 

 

 

 

 

 

 

of the primary (fast) process and the subsequent sequential decay, and introduced a
modiﬁed systematics for widths of the momentum distribution based on compilation
of high energy fragmentation data (projectile and target). The momentum width in

his systematics is given by:

a = const- Ap — A, (4.6)

where the constant is a parameter of the systematics, usually taken as 85—100 MeV/c.
In the case of projectile fragmentation, it describes only the projectile-like fragments,
while light fragment widths are generally overestimated, which can be already seen
in the original paper.

Asymmetric momentum distributions in the intermediate energy regime are ex-
plained by existence of different, competing reaction mechanisms. Apart from the
“pure” fragmentation component, which completely dominates at relativistic ener-
gies, both the low momentum tail typical for more dissipative processes [79] and the

broadening associated with nucleon pick-up reactions in the prefragment formation

74

phase [80] will inﬂuence the shape of the ﬁnal momentum distribution. There are
two width parameters 0'}; and (IL in the description of our experimental momentum
distributions for the left and right halves of the momentum distribution. Guided by
the previous works [59,80,81] we assume the right side of the momentum distribution,
03, is completely dominated by the “pure” fragmentation reaction mechanism and
the left side, 01,, has more signiﬁcant contributions from different reaction processes
so it is not expected to behave according to the fragmentation parameterizations.
The extracted width parameters of the right side of the momentum distribution,
03, are shown as a function of fragment mass loss with respect to that of the pro-
jectile, Ap — A, in Figure 4.2 for 40*""Ca, 58""‘Ni projectiles and in the top panels of
Figure 4.4 for the 86Kr beam. The experimental data do not follow monotonically-
increasing trend with the number of removed nucleons from the projectile as suggested
by the Morrissey systematics plotted as dashed curves in Figure 4.2, with const = 85
MeV/c. The disagreement with the experimental data is not surprising as this sys-
tematics based mainly on target residues from light-ion induced reactions was not
meant for extrapolation towards large mass differences between the projectile and
reaction products (Ap — A > Ap / 2). The solid line shows the ﬁt by Goldhaber for-
mula (Equation (4.2)), where the reduced width, 00, is taken as a ﬁtting parameter.
The values of the Goldhaber reduced width parameter, 062, for the right side of the
experimental momentum distributions, are listed in Table 4.1. For a given projectile
the of,” does not depend on the target material, but increases slightly with the mass
of the projectile. The values of the of extracted in our fragmentation experiment
should be compared to the values calculated from the Fermi momenta measured with
quasielastic electron scattering, listed also in Table 4.1. We see that the experimental
values from this study are lower than the Fermi momentum values for all investigated
beam-target combinations. This is in line with other intermediate and relativistic en-
ergy fragmentation experiments when similar discrepancies have been reported [75].

To explain the difference, Bertsch [82] in his model used Pauli correlations. Weber et

75

 

 

 

 

 

 

 

   
 

      

 

 

 

 

 

— I T. 1" I I ﬁ'fF—‘F ' I I
400 — .25 __
: ' O o o '9¢ . . .I... . O o .
r '6 O. 0 O 98-.«1: GOO . O 01
- Q {6. 0;.’£:.. '0'.‘ --- 9 .. -l
3°07 0 .2?" ' ° °o 1
: 43'. 000 _g°:;= ' :
200 L :03 21;? _‘
I ~-' 40 9 c 48 9 i
’5 3 -- Ca+ Be ~» Ca+ Be 3
S 100 .. .5 1
a: .' ~
2 I . I
v __ 'x o, S
m 400 7 .0" o'A é @ d
b : '0' Q) 0 .0 (30...; j
I. 0' o O 0.554.: O 1
300 _ O f O .' _:l' I" O ..
'_' O :7?" O :
. , 00 0:; " j
200 f O 333’ I,
- 4O 18 48 18
- 0 Ca+ ’Ta } Ca+ ’Ta 3
100 : t‘ -
L I I I l I I I I I I I I I l u 1 1 1 1 I 1 1 1 1 I 1 14 L I 1 1_1 1"
O 30
I
I ‘ ‘ ,. é o .1
.. «’9‘- . f; 0
400 .“'- _>‘4.(“.°D. “ d
A . " _
(\3 200
> I
d) t
2
v
(I
b
400 -
200 ‘
0

Figure 4.2: Width of the right side of the momentum distribution 03 for the 40"’E’Ca
and 58""‘N i primary beams on 9Be and 181Ta targets, plotted as a function of number
of removed nucleons, Ap — A. The solid line shows the ﬁt by Goldhaber formula
with reduced width, of, listed in Table 4.1 and the dashed line depicts the Morrissey
systematics with const=85 MeV/c.

76

al. [75], alternatively, attributed the discrepancy to the mass loss in the evaporation
stage which is not included in the Goldhaber model.

The values of the of parameters for fragments created in fragmentation of 86Kr at
64 MeV/ u are larger (z 120 MeV/c) than the ones obtained for the NSCL reaction
systems (% 90 MeV/c). However, no correlation with target material used in the
projectile fragmentation reactions at different beam energies (64 and 140 MeV/ u) has
been observed, which is consistent with the conclusions of other similar investigations
[59].

The width parameters of the left side of the momentum distribution, 01,, are shown
as a function of the mass loss of the fragment with respect to that of the projectile,
Ap — A, in Figure 4.3 for the 40"“‘Ca, 58"‘4Ni projectiles and in the bottom panels of
Figure 4.4 for the 86Kr beam. The solid line through the data points represents a ﬁt
by the Goldhaber formula (Equation (4.2)), however this ﬁt is just formal, because we
do not expect any fragmentation formula to apply for the left side of the momentum
distribution, 01, (as discussed above). Nevertheless, it is clear that the experimental
widths, 01,, can be roughly described by the Goldhaber model using larger values for

the reduced width parameter 03‘, as listed in Table 4.1.

4.1.2 Centroids of the momentum distributions

Fragment velocity after the reaction is a rather important experimental observable
bearing information about the ﬁrst stage of the reaction — abrasion. The observed
momentum distributions peak near the beam velocity. Small deviations in fragment
mean velocities, Up, from the projectile velocity, ’01), can be seen in a simple picture as
the friction phenomenon during nuclear reaction. Nuclear bonds are broken during the

nucleon removal process, causing the projectile to slow down. In a model introduced

77

 

 

 

 

 

 

 

 

 

 

 

 

 

600—- —— o —
L o I Cb.
_ O .J ¢¢ 00 o
O
400— 00630 —- -
I I o
r _
G 200— 40Ca+9Be -- 48Ca+9Be -
\ - .
a . r
2600—4 'rkkr'A‘Akﬁj Tf+1jltr .,::‘ﬁd
b" b 0
O
" __ 9830(1) ¢ (:3 q
400 _ Q30 ‘
200 -- 48Ca+"3'Ta j
1 l I 1 " I l I l l 4
0 1o 20 30 1o 20 30
AP'AF
P I l J v I r f '7 r
600- it
_ 00 -. q, 4
: “”295 QDOCSJQO :: f) ‘
400— j- ‘
a r 58Ni+9t3e . 64Ni+9I3e ,
S 200 - ,
g e § : f s % . g g
I} ‘ _: :
r é q.)
I o 35’
r- o r
-— 64Ni+“"Ta
Id 14L 1 I l
20 4o

 

AP'AF
Figure 4.3: Width of the left side of the momentum distribution 0;, for the 40"‘8Ca

and 58MM primary beams on 9Be and 181Ta targets. The solid line shows the fit by
Goldhaber formula with reduced width, 06’, listed in Table 4.1.

78

 

    

400

(IR (MeV/c)

200

llirrlliilt

 

.4

un-

‘-

I1
I"4J1iil;1114‘
' I "I 'l' I

O’L (MeV/c)

 

 

 

 

Figure 4.4: Width of the left 0;, and right 0’}; sides of the momentum distribution for
the 86Kr primary beam on 9Be and 181Ta reaction targets. The solid line shows the ﬁt

by the Goldhaber formula with the reduced width listed in Table 4.1 and the dashed
line depicts the Morrissey systematics with const=85 MeV/c.

79

by Borrel et al. [83] the fragment velocity deviation, vp/vp, is expressed as:

 

 

It depends on the mass of the projectile Ap and fragment A plus the energy of the
beam Ep in units of MeV/ u. The underlying idea is that the energy cost is 8 MeV
per removed nucleon.

Other models describe the change of velocity in terms of the momentum trans-
fer. A number of models have been proposed to treat the small momentum transfer
from the bombarding particle. They all assume a two-step mechanism and treat the
result as a quasi two—body system. Although differing in details of the initial interac-
tion, Cumming et al. [84] realized they all predict the same functional form for the

momentum transfer in the projectile reference frame p“ as

PM = A—fz' (1+k(1-/32)”2) (4.8)

where 8 is the pro jectile’s velocity, AET is the energy transferred to the prefragment,
and the parameter It sets the rate at which p” decreases as ,8 increases. Different
- authors arrived at different values of the AET and 1: parameters studying various
systems. Kaufman [85] compiled high energy fragmentation data (0.4—2.1 GeV/u)
and described the data with parameters k = O and AET = 13 MeV/AA. On the
other hand, Morrissey analyzed heavy residues from target spallation reactions [77]
and obtained k = 1 and AET = 8 MeV/AA.

Experimental fragment velocities, Up, are calculated by translating the po param-
eter of the modiﬁed Gaussian ﬁtting function (Equation (4.1)) for every fragment in
our analysis into the velocity space. In order to make the cross system comparisons
clear, we plot them as relative deviations from the projectile velocity (Up/Up — 1) in

%, in Figure 4.5, 4.6 and 4.7 for the 40"180a, 58"MM and 86Kr beams, respectively.

80

 

 

   
   

    
  

 

 

 

 

 

 

 

 

 

 
  

 
  

 

 

 

 

 

 

Iﬁ I I r I I I I T I f I I I I I I l I I I I r I l I I I I I
0 ———————————— —] o————————————
.. .] L.
o O
- O . -
A O A
.\° - ¢ 0 < s ~ 0
v v 0000 000
£1 ' o 1 g: 0 ° 9 f
‘9. O O
> ’ ‘ > ’ ° 40Ca+""'l'a *
- ~ - “Barrel ]
4— --—-Kaufman — -4r --°-Kaufman J
- , — Morrissey . ~ — Morrissey ]
L 1 1:! L l l I I 1 L4 1 I l l 1 L141] 14 l L l J_1 1 1
1o 20 30 4o 10 20 30 4o
Fragment A Fragment A
T T I I I r I Iﬁ I I I I I I IT I I I I I I I I I I I I I I I ﬁI I I I Iii I
o ————————————— 0————~—*——————ﬁ
. . . 4
.. . I. .J
A O A
o\° [ ¢ (f ¢<I> o\° - 0 é
‘- -2— — 1- -2— ~
. o , - é
>‘L ' (pf ' ] >‘L ' “3% i
:a - Mali ~ I :1 -<> ,go New.
* “Barrel ~ -
-4~ ""Kaufman — -4F “Kaufman -
- — Morrissey . — Morrissey
l l l l 1 J I 1 l 1 l l l l l l l l 1 l l l l l l l l 1 1+1 P
10 20 30 4o 50 1o 20 30 4o 50
Fragment A Fragment A

Figure 4.5: Relative deviations from the projectile velocity (vp/vp — 1) for all frag-
ments with complete momentum distributions identiﬁed in the fragmentation of
40"“‘Ca isotopes on 9Be and 181Ta targets plotted as a function of fragment mass
number, A. Parameterizations of Borrel, Kaufman and Morrissey are shown as dot—
ted, dashed and solid lines respectively.

81

 

 

       

 
  

 
  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I I T— I— I T I I I I I I I Iﬁ—I —I—I I I I I I I I I I T I I I rj I I I I I T I I
o _____________ 0 _____________
’0‘ _ ’o‘ _
E 90¢? o E o
‘TCL -2 0 o i To. -269 200%6 r
> ' ‘ > "
:0. _ _ :0- Q o
- - Borrel ‘ ~ ‘
-4- ""Kaufman -‘ -4- ----Kaufman -
- — Morrissey - [ —Morrissey «
l 1 1 1 1 l 1 1 1 L l 1 1 1 L l 1 1 1 1 J + 1 1 1 l 1 L 1 L
20 30 4o 50 60 20 30 4o 50 60
Fragment A Fragment A
I I I I I I I I I I I I I I I I I I I I T I I I I I I I I I I I I I I I I I I T I I I I I
o ___________ o ___________
4’41, " J
2\: ’- ¢¢¢ é - o .00 ..
'7 “”88? '7 fiffoféé is r
i * at ‘P <9 ‘
> - > ' ° “NH“"I’a ‘
- ~ ...... Borrel ]
4F -----Kaufman - -4L ----Kaufman a
— — Morrissey - — Morrissey
L1 J 1 1 l 1 1 111 1 1 1 1 1 L1 1 1 1 1 1:1 1 1 1 l 1 1 1 l l 1 1 1 1 I l 1
20 30 4o 50 6O 20 30 4o 50 60
Fragment A Fragment A

Figure 4.6: Relative deviations from the projectile velocity (Up/Up — l) for all frag-
ments with complete momentum distributions identiﬁed in the fragmentation of
58"5“Ni isotopes on 9Be and 181Ta targets plotted as a function of fragment mass num-
ber, A. Parameterizations of Borrel, Kaufman and Morrissey are shown as dotted,
dashed and solid lines respectively.

82

 

 

   

     

 
 

 

 

 

 

 

 

_[ o _____________
- , .
.. .] h-

-2— ~ -2— —
A [- -« A - .4
o\0 . . °\° .

‘T 4* -— '7 -4- —
>°’ _ >°' _
:a. - ‘cL .
«3- — > -6~ —
~ - Borrel 4 ~ ”- Borrel
- , Kaufman - Kaufman
-8 ” ' — Morrissey ‘ -8 r ' — Morrissey ‘
.- I 1 1 1 I l 1 1_1 L 1 141_L J_L 1 4 1" ’- I l l 1 1 I 1 1 J_L AI_;1 1 1 I 1 1 _
50 60 7o 80 so 60 7o 80
Fragment A Fragment A

Figure 4.7: Relative deviations from the projectile velocity (vp/vp —— 1) for all frag-
ments with complete momentum distributions identiﬁed in the fragmentation of 86Kr
primary beam on 9Be and 181Ta targets plotted as a function of fragment mass num-
ber, A. Parameterizations of Borrel, Kaufman and Morrissey are shown as dotted,
dashed and solid lines respectively.

The experimental data are shown as open circles and three calculations by models of
Borrel, Kaufman and Morrissey are shown as dotted, dashed, and solid lines, respec-
tively. The experimental fragment velocity deviations exhibit a considerable scatter
making the comparisons with the above discussed parameterizations difficult. The
studied projectile fragmentation data show no dependence of the fragment velocities
on the projectile or the target material (within the scatter of the experimental data).
The overall proﬁle of the relative experimental velocity is similar for all the NSCL
reaction systems (Figure 4.5 and 4.6) in that the velocities gradually decrease with
the fragment mass number and saturate around -2%. All presented models predict a
further decrease of the velocity with the mass number which is not experimentally
observed. A clear deviation from the experimental data starts around 10 removed
nucleons for the 40"“BCa beams and around 15 removed nucleons for the 58"MM beams.
The Borrel parameterization predicts overall larger fragment velocities as compared
to the systematics of Kaufman and Morrissey (see Figure 4.5 and 4.6). The Kauf-

man and Morrissey systematics, on the other hand, produce similar predictions for

83

velocities of projectile-like fragments. However, it must be noted that all presented
parameterizations fail to reproduce the overall shape of the data.

For the 86Kr reaction systems, only the fragments with A > 50 have been ana-
lyzed, so we only see an overall trend of the fragment velocities as a function of the
fragment mass number, A in the vicinity of the projectile (Figure 4.7). In the case of
86Kr+9Be reaction the experimental data are scattered much more than the data of
the NSCL reaction systems. This effect is most probably caused by a combination of
much lower beam energy (64 MeV/ u) and rather thick target (z 100 mg/cm2). The
Borrel systematics predicts larger fragment velocities than the data for most of the
fragments (left panel of Figure 4.7). The Kaufman and Morrissey calculations inter—
sect the experimental data points close to the projectile, but deviate signiﬁcantly for
fragments with mass numbers 55—60. In the case of the 86Kr+181Ta reaction (right
panel of Figure 4.7) the number of complete momentum distributions extracted is
much lower than for the 86Kr+9Be system, making the conclusions more difficult.
Due to broader charge state distributions of the 86Kr beam on 181Ta target a smaller
range in the magnetic rigidity was covered for the 86Kr+181Ta reaction. The momen-
tum distributions of fragments in the vicinity of the projectile were not measured
(or measured completely). This limitation does not allow us to compare the experi-
mental fragment velocities in the vicinity of the projectile to the parameterizations.
The experimental velocities for lighter fragments A a: 52—65 are, again, larger than

predicted by Kuafman and Morrissey parameterizations.

4.2 Fragmentation production cross sections

Fragmentation production cross sections were extracted according to the analysis
procedure described in detail in Section 3.4.2. This section summarizes the results
of the reaction cross section analysis for all studied systems. The vast majority of

measured nuclides are classiﬁed as fragmentation reaction products. However, nucleon

84

exchange and pick-up isotopes have been also identiﬁed in our study. Table 4.2 lists the

Table 4.2: Number of fragments and pick-up (including exchange) cross sections mea-
sured for all reaction systems.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Beam Target Number of
fragments ] pick-up

9Be 100 11

4003' 181Ta 101 15

9Be 176 26

4803 181Ta 167 32

58 . 9Be 184 12

N1 181Ta 179 10

64 . 9Be 240 3

N1 ISITa 232 2

9Be 180 0

861“ T8%. 70 0
| Total: | 1629 ] 111 J

 

number of measured fragmentation products and pick-up products (including nucleon
exchange). The presented table should be understood as a summary of the nuclides
identiﬁed in our study. It is by no means an exact summary of reaction mechanism for
the various isotopes measured in our data. Because the experimental data were taken
in separate experiments, scanning different parts of the magnetic rigidity spectrum,
Table 4.2 only lists the number of cross sections we extracted.

Figure 4.8—4.12 present an overall view of the fragment cross sections in the style
of the nuclear chart for all investigated reaction systems. The range of the measured
cross sections spans over 8 orders of magnitude, from nb to hundreds of mb. Figure 4.8
and 4.9 present the isotope cross sections measured in the fragmentation of 40Ca and
48Ca, Figure 4.10 and 4.11 display the fragment cross sections of 58Ni and 64Ni and
Figure 4.12 shows the 86Kr beam fragmentation product cross sections. For reference,
the projectile is represented by the symbol of box with a cross inside and the stable
isotopes are highlighted with the black framed boxes. The number of nuclides and their
position in the chart of nuclides (Figure 4.8—4.12) using 9Be and 181Ta target is very

similar for all investigated systems except for the 86Kr projectile where measurements

85

of fragments with 181Ta target were incomplete.

Even though more neutron-rich fragments are expected to be produced by the
neutron-rich projectile 48Ca, the number of measured isotopes in 48Ca (202) fragmen-
tation is almost twice that of 400a (111). On the other hand, for the 58Ni and 64Ni
projectiles the cross section of 196 and 243 isotopes were measured, respectively. For
the reactions of the 86Kr beam, we did not have nucleon pick-up isotopes with full
momentum distributions identiﬁed because of the rather limited range in magnetic

rigidity covered.

4.3 Cross section comparisons

Before a more quantitative presentation of the measured fragmentation cross sections,
we introduce the empirical parameterization of fragmentation cross sections (EPAX).
Then the measured cross sections are compared to EPAX and previously published

experimental data (where available).

4.3.1 EPAX parameterization

A large amounts of early spallation and subsequent projectile fragmentation data have
stimulated an interest to understand the underlying dependences and correlations and
to systematize them. By studying the numerous cross sections measured in different
experiments in the relativistic energy regime, two very important facts have been

learned [47]:

1. Fragment isotopic distributions become approximately energy independent above

beam energy z 200 MeV/ u;

2. Target spallation cross sections for fragments close to the target nucleus pro-
duced by proton or heavy ion beams differ only by a constant factor close to

the ratio of the total reaction cross section.

86

The terms “limiting fragmentation” and “factorization” [86] have been introduced
to refer to the latter two observations. They formed the basis of many empirical
parameterizations valid for speciﬁc projectile and target combinations [87]. In 1990
Siimmerer et. at [47] combined the efforts of many different groups developing em-
pirical parameterizations into a universal empirical parameterization of cross sections
(EPAX) version 1. This version has been reﬁned and improved with inclusion of rel-
ativistic projectile fragmentation data available in 2000 resulting in EPAX version
2 [48].

In the EPAX2 parameterization [48] the fragmentation cross section of a fragment
with mass, A, and nuclear charge, Z, created from projectile (AP, 2,) colliding with

a target (At, 2,) is given by
0(A, Z) = YAn exp (—R|Z,,,.ob — Z|Un<P>). (4.9)

The ﬁrst term YA describes the sum of the isobaric cross sections with mass number,
A, and the second term exp (—R|Zp,ob - Z [Un<P>) is the so called charge dispersion,
i. e. the distribution of the elemental cross sections around maximum value, meb,
for a given mass. The shape of the charge distribution is controlled by the width
parameter, R, and the exponents, UIn and Up, describing, the neutron-rich (n) and
proton-rich (p) side, respectively. Where the neutron-rich fragments are deﬁned with
meb —— Z > 0 all others are considered proton-rich. The factor n = W normalizes
the integral of the charge dispersion to unity.

The mass yield, YA, is parameterized as an exponential function of the number of

removed nucleons, AP — A,

YA = SP exp [—P(A,, — A)]. (4.10)

87

S is the overall scaling factor which accounts for the peripheral nature of the frag-

mentation reaction and proportional to the sum of the projectile and the target radii:
S: 82(A],/3+A,1/3+Sl). (4.11)

With SI = —2.38 and $2 = 0.270 being EPAX2 parameters. The slope of the expo-
nential function in Equation (4.10), P, is taken as a function of the projectile mass,

Ap, with P1 = —2.584 and P2 = —7.57 x 10"3 as EPAX parameters:
P = exp (PgAp + P1). (4.12)

The charge dispersion in Equation (4.9) is described by three parameters R, Zprob,
and Un(p)- These parameters are strongly correlated [48] and are very difficult to obtain
by a ﬁtting technique. To account for the asymmetric nature of the shape of isobaric
distributions, the exponents, UI, and Up, for the neutron-rich and proton-rich sides are
different. The maximum of the isobar distribution, meb, lies in the valley of stability

and it is parameterized as:
2....(A) = 277A) + A, (4.13)

where Z3(A) is approximated by a smooth function of the mass number, A:

A
2‘3“) ‘ 1.98 + 0.0155A2/3’ (4'14)

 

and the A parameter is found to be a linear function of the fragment mass, A, for

heavy fragments and a quadratic function of A for lower masses:

A3A2 ifA < A4,
A = (4.15)

A2A+A1 ifA 2 A4.

88

Where A1 = —1.09, A2 = 3.05 x 10”, A3 = 2.14 x 10‘“, and A4 = 71.35 are
EPAX2 parameters. Similarly, the width parameter, R, of the charge distribution is
parameterized as a function of the fragment mass, A, with R1 = 0.885 and R2 =

—9.82 x 10“3 EPAX2 parameters:

R = exp (R2A + R1). (4.16)

This description is sufficient to predict the cross sections of fragments located close
to the line of stability and far from the projectile nucleus, also referred to as the
“residue corridor”. For fragments with masses close to the projectile, corrections to

the parameters A, R, and YA are introduced, according to the following equations:

A = A [1+ d1(A/A,, — d2)2] , (4.17)
R = R [1 + r1(A/A,, — m2] , (4.18)
YA = YA [1 + y1(A/AP “ 3M2] - (4-19)

The corrections to A, R, and YA parameters are applied only for fragments with
mass number, A, fulﬁlling (A/Ap — d2) > 0, (A/Ap — m) > 0, and (A/Ap — 312) >
0, respectively. With EPAX2 parameters d1 = —25.0, d2 = 0.80, 1‘; = 20.0, r2 =
0.82, and y1 = 200.0, y2 = 0.90, characterizing the corrections to the A, R, and
YA parameters. A ﬁnal correction is applied in the case of projectile nuclei far from
the line of ,B-stability, ZB(A,,). In this case it has been shown that the fragment
distributions keep some memory of the A /Z ratio of the projectile nucleus resulting

in a correction to the maximum, meb, of the charge distribution:

zpmbm) = 23m) + A + Am, (420)

89

where Am is expressed separately for neutron-rich ((Zp — Zg(Ap)) < 0) and proton-

rich ((Zp — Z3(Ap)) > 0) projectiles:

Am : (Zp — Zg(Ap)) [711(A/Ap)2 + n2(A/Ap)4] for neutron rich, (4.21)

(2,, — Zg(A,,)) exp [p1 + p2(A/Ap)] for proton rich,
where m = 0.40, 712 = 0.60 and p1 = —10.25, [)2 = 10.10 are parameters of EPAX2.

The EPAX2 parameterization altogether contains 24 parameters (SI, 52, P1, P2,
R1, R2, A1, A2, A3, A4, Un, U1, U2, U3, m, 712,121,122, d1, d2, 7'1, 7'2, yl, and y2), many
of which are strongly inter-correlated. These parameters have been determined by a
ﬁt of the fragmentation data of 40Ar [88], 48Ca [42], 58Ni [89], 8f’Kr [75], 129Xe [90],
and 208Pb [91] projectiles measured at 600, 212, 650, 500, 790, and 1000 MeV/u,
respectively.

The main goal of the EPAX2 parameterization is to obtain a smooth analytic de-
scription of the fragmentation data within a factor of two [48]. The empirical param-
eterization assumes the limiting fragmentation regime when the fragmentation cross
sections are no longer beam energy and target dependent. Since it is based on analytic
formula, the cross section calculations are fast. Because of its simplicity, the EPAX2
parameterization has been implemented in many codes (MOCADI [66], LISE++ [56]).
It is not only used to calculate the yields of rare isotopes for the existing (CCF-
NSCL [52], GSI [44], RIKEN [35]), but also fragment yields for the next-generation
radioactive beam facilities (RIA [92], FAIR [93], RI beam factory at RIKEN [94]). The
EPAX2 is also available at http: / /www-aix.gsi .de/ 'sue/epax/epax_v2.htm1.

Since the EPAX2 parameterization is reproducing an “average” behavior of the
fragmentation cross sections for a variety of beam—target combinations, it produces
reliable results while interpolating between experimentally measured data points. The
extrapolation to very exotic (neutron-rich or proton-rich) fragments may be unreli-
able, because the slopes of the isotopic distributions are only measured in the vicinity

of their maxima. A minute change of the slope parameter may cause differences in

90

orders of magnitude in the extreme tails of the fragment isotopic distributions. Hence,
the EPAX2 should be used with extreme caution when calculating yields for fragments

very far from the experimentally measured regions.

4.3.2 Comparison to EPAX

The experimentally determined fragmentation production cross sections for the 10
reaction systems are plotted in Figure 4.13 (4°Ca+9Be), Figure 4.14 (4°Ca+181Ta),
Figure 4.15 (48Ca+gBe), Figure 4.16 (“8Ca+181Ta), Figure 4.17 (58Ni+9Be), Fig-
ure 4.18 (58Ni+181Ta), Figure 4.19 (64Ni+gBe), Figure 4.20 (64Ni+181Ta), Figure 4.21
(86Kr+9Be) and Figure 4.22 (86Kr+181Ta). Each panel represents isotope cross section
data (ﬁlled squares) for one element, plotted as a function of neutron excess, N — Z, of
each isotope. The nucleon pick-up and exchange cross sections are shown as ﬁlled tri-
angles. All comparisons shown in Figure 4.13—4.22 were calculated using EPAX2 [48].
EPAX does not predict pick-up cross sections and the description of the light frag-
ments (A < Ap / 2) is generally not as good as the predictions for fragments closer to
the projectile, because light fragments may be produced in more central collisions in
other reaction mechanism such as multifragmentation. The maxima of very light ele-
ments with Z < 9 (for the NSCL reaction systems), which may be produced in more
central reactions, do not agree with EPAX calculation. In general, isotope distribu-
tions from EPAX are wider than the measured ones, resulting in over-predictions for
the very neutron-rich and proton-rich fragments for all investigated reaction systems
measured at 140 and 64 MeV/u. This behavior is especially pronounced in the case
of 40Ca projectile.

For all investigated systems, we also observed differences between the EPAX cal-
culated maximum of the isotopic distribution and the experimental data for elements .
close to the projectile. The differences between the EPAX maximum and the exper—
imental data are noticeable for almost all projectiles measured at 140 MeV/u but

are really pronounced for fragments of 86Kr (Ebenm = 64 MeV/u), which suggests

91

stronger sensitivity to the incident energy. This systematic discrepancy between the
intermediate energy fragmentation data and EPAX parameterization has been re-
ported by Siimmerer [97]. The Fermi spheres of the target and projectile nuclei have
larger overlap at intermediate energies than at relativistic energies. There may be
increasing contributions to the prefragments with charge numbers greater than that
of the projectile from the transfer-type reactions. Subsequent decay of these primary
fragments feeds the neutron-deﬁcient isotopes close to the projectile.

The production of 50Ca in the reaction of 58Ni+9Be represents the lowest cross sec-
tion measured from among the NSCL reaction systems. To achieve the measurement
of such low cross section (0.41 :t 0.16 nb), we used a thick 9Be target (578 mg/cm2).
In addition, an Al degrader (240 mg/cmz) was placed at the dispersive image (image
2 in Figure 2.4) position to further separate the fragments with the same A/ Q ratios.
The wedge is especially effective in deﬂecting the light fragments from reaching the
focal plane detector setup, thus reducing the counting rate of the PIN detector, allow-
ing us to use the maximum beam intensity. All other cross sections (including those
in RIKEN) were measured without the use of the degrader. The lowest cross section
(15 :f: 7 pb) of the study was measured for a fragment 79Cu produced in 86Kr+9Be

reactions at 64 MeV/u.

4.3.3 Comparison to other data

In addition to our 40Ca+9Be data, Figure 4.13 also shows the fragmentation data of
40Ca on a hydrogen target at 356 MeV/u [95]. The latter data are shown as open
squares. Compared to our data, the isotope distributions obtained from the hydrogen
target tend to be narrower than our data and the cross-sections are smaller. The
narrower isotopic distributions generally indicate lower excitation energy reﬂecting
lower center of mass energy for the system with hydrogen target.

Our fragmentation data of 48Ca on 9Be target are compared to the data measured

at 212 MeV/u at Berkeley [42] shown as open squares in Figure 4.15. There are

92

differences between the two experimental data sets. Even though the fragmentation
data of Ref. [42] were included in extracting the EPAX parameters, the deviation of
this data set from the overall EPAX ﬁt (by a factor of 3) was noted in the original
paper [48]. On the other hand, our fragmentation data seem to agree pretty well with
EPAX predictions except for a small shift of the maximum for elements (S—Ca) close
to the projectile (discussed at the end of the previous section).

More recently, cross sections of 46"‘F’Ca, 44’45'46'47K, 41’42'44145Ar and 39"‘0'41’42Cl iso-
topes have been measured in the projectile fragmentation of 48Ca on a deuterium
target at 104 MeV/ u [96]. These data are shown in Figure 4.15 as open triangles. As
mentioned in Ref. [96], the cross sections for fragments with few nucleons removed
obtained with the 2H target are higher than our data with the 9Be target as expected
from EPAX. In the case of Ca and K isotOpes, they are even higher.

The experimental cross sections for the reaction of 58Ni+9Be are plotted along with
the relativistic energy fragmentation of 58Ni on a 9Be target at 650 MeV/ u by Blank
et al. [89] measured at GSI shown as open squares in Figure 4.17. The experiment of
Blank et al. focused on the proton-rich side of the fragment spectrum for fragments of
elements 21 g Z s 28. Where there are overlapping data points, an interesting trend
is observed. Where there are overlapping data points, an interesting trend is observed.
The fragment cross-sections from the GSI experiment are consistently higher by 70%
than our data for the proton-rich (Z > N) isotopes. In the slightly neutron-rich the
GSI data are slightly lower by 30%. Unfortunately there are not enough overlaps
between the two sets of data to determine if the widths of the distributions are wider
or narrower.

In Figure 4.21 our cross section data are compared to an early experiment with
86Kr projectile and 9Be target at 650 MeV/u by Weber et al. [75], which were used
to ﬁt the EPAX parameterization [48]. For light fragmets the latter data show wider
isotope distributions as compared to our fragmentation data analysis. The cross sec-

tions on top of the isotopic distributions for Co—Zn elements agree rather well with

93

the measurements done at very different energies. The isotopic distributions measured V
at 650 MeV/ u generally appear wider than our measurement at 64 MeV/ u. It must
be noted that there is considerable scatter (Figure 4.21) in Weber et al. data, making

detailed comparison of the distributions width very difﬁcult.

94

Protons

      

E projectile
5 7 D stable \

Neutrons,

Figure 4.8: Nuclides identiﬁed in the fragmentation of 40Ca projectile on two targets.
There are 111 and 116 isotopes shown for 9Be and 181Ta targets respectively.

 

95

Protons .

 

E prolectlle

U stable

 

\/

 

 

Neutrons

Figure 4.9: Nuclides identiﬁed in the fragmentation of 48Ca projectile on two targets.
There are 202 and 200 isotopes shown for 9Be and 181Ta targets respectively.

96

Protons \

 

a projectile

El stable

 

 

Neutrons '

Figure 4.10: N uclides identiﬁed in the fragmentation of 58Ni projectile on two targets.
There are 197 and 189 isotopes shown for 9Be and 181Ta targets respectively.

97

\
I

Protons

      

“Ni+'8'TO

E projectile

D stable

\

Neutrons ﬂ

 

Figure 4.11: N uclides identiﬁed in the fragmentation of 64Ni projectile on two targets.
There are 246 and 234 isotopes shown for 9Be and 181Ta targets respectively.

98

 

Protons

   
   
 

a projectile

El stable

86Kr+ ””Ta

Mn

27 29 31

 

Neutrons '

Figure 4.12: Nuclides identiﬁed in the fragmentation of 86Kr projectile on two targets.
There are 180 and 70 fragments shown for 9Be and 181Ta targets respectively.

99

 

 

 

 

 

 

 

 

 

 

 

 

 

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Neutron excess N-Z

Figure 4.13: Isotopic distributions of measured cross sections plotted as a function
of neutron excess for 5 5 Z s 20 elements detected in the 40Ca+9Be reaction at
140 MeV/u. The experimental fragmentation data are shown as ﬁlled squares, cross
sections of pick-up reactions are depicted with ﬁlled triangles, the solid line shows the
calculation by EPAX and open squares show 40Ca+1H data at 356 MeV/u [95].

100

 

 

 

 

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Neutron excess N-Z

Figure 4.14: Isotopic distributions of measured cross sections plotted as a function
of neutron excess for 5 3 Z s 20 elements detected in the 40Ca+181Ta reaction at
140 MeV/u. The experimental fragmentation data are shown as ﬁlled squares, cross

sections of pick-up reactions are depicted with ﬁlled triangles and the solid line shows
the calculation by EPAX.

101

 

   

 

 

 

 

 

 

 

 

 

 

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O 5 10 O 5 10 O 5 10 O 5 10
Neutron excess N-Z

Figure 4.15: Isotopic distributions of measured cross sections plotted as a function
of neutron excess for 5 3 Z S 20 elements detected in the 48Ca+QBe reaction at
140 MeV/ u. The experimental fragmentation data are shown as ﬁlled squares, cross
sections of pick-up reactions are depicted with ﬁlled triangles, the solid line shows the
calculation by EPAX and open squares show 48Ca+9Be data at 212 MeV/u [42]. Open
triangles depict the fragmentation data of 48Ca on a 2H target at 104 MeV/u [96].

102

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Figure 4.16: Isotopic distributions of measured cross sections plotted as a function
of neutron excess for 5 3 Z S 20 elements detected in the 48Ca+181Ta reaction at
140 MeV/u. The experimental fragmentation data are shown as ﬁlled squares, cross
sections of pick-up reactions are depicted with ﬁlled triangles and the solid line shows
the calculation by EPAX.

103

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Figure 4.17: Isotopic distributions of measured cross sections plotted as a function
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sections of pick-up reactions are depicted with ﬁlled triangles, the solid line shows the
calculation by EPAX and open squares show 58Ni+QBe data at 650 MeV/u [89].

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o 5100 5100 5100 510
Neutron excess N-Z

Figure 4.18: Isotopic distributions of measured croSs sections plotted as a function
of neutron excess for 5 3 Z s 28 elements detected in the 58Ni+181Ta reaction at
140 MeV/ u. The experimental fragmentation are data shown as ﬁlled squares, cross
sections of pick-up reactions are depicted with ﬁlled triangles and the solid line shows
the calculation by EPAX.

 

 

 

 

 

 

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Neutron excess N-Z
Figure 4.19: Isotopic distributions of measured cross sections plotted as a function
of neutron excess for 5 3 Z S 28 elements detected in the 64Ni+9Be reaction at
140 MeV/ u. The experimental fragmentation data are shown as ﬁlled squares, cross

sections of pick-up reactions are depicted with ﬁlled triangles and the solid line shows
the calculation by EPAX.

 

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Neutron excess N-Z

Figure 4.20: Isotopic distributions of measured cross sections plotted as a function
of neutron excess for 5 3 Z S 28 elements detected in the 64Ni+181Ta reaction at
140 MeV/u. The experimental fragmentation data are shown as ﬁlled squares and the
solid line shows the calculation by EPAX.

107

 

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Neutron excess N-Z

Figure 4.21: Isotopic distributions of measured cross sections plotted as a function of
neutron excess for 25 3 Z S 36 elements detected in the 86Kr+9Be reaction at '64
MeV/u. The experimental fragmentation data are shown as ﬁlled squares, the solid
line shows the calculation by EPAX and open squares show 86Kr+gBe fragmentation
data at 500 MeV/u [75], the nucleon pick-up cross sections from this data set are
shown as open triangles.

l 14—L

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Neutron excess N-Z

Figure 4.22: Isotopic distributions of measured cross sections plotted as a function
of neutron excess for 25 3 Z S 36 elements detected in the 86Kr+181Ta reaction at
64 MeV/ u. The experimental fragmentation data are shown as ﬁlled squares and the
EPAX calculation is shown as a solid line.

109

4.3.4 Target dependence

In order to explore the target dependence of the fragmentation cross sections for a
given projectile, we constructed target isotOpe ratios, Rtgt(A, Z), for isotopes with
mass, A, and proton number, Z. R¢9t(A, Z) E aTa(A, Z)/aBe(A, Z), where 0T8(A, Z)
and 038(A, Z) are the fragmentation cross sections measured for reactions with 181Ta
and 9Be target, respectively. The RAMA, Z) ratios were calculated for all projectiles
in our study. By convention, fragment cross sections created in interactions with the
more neutron-rich target (181Ta) are used as numerators.

Rtgt(A, Z) for elements 8 5 Z 3 18 are plotted as a function of the fragment mass
number, A, in Figure 4.23 for 40Ca (left panel) and 48Ca (right panel) beam. Figure
4.24 shows the Rtgt(A, Z) as a function of fragment mass number, A, for elements
10 _<_ Z s 26 for 58Ni (left panel) and 64Ni (right panel) beam. The target isotope
ratios, Rtgt(A, Z) for elements 25 3 Z S 36 for the 86Kr primary beam are plotted as
a function of fragment mass number, A, in Figure 4.25.

The isotopes with even proton number, Z, are plotted with ﬁlled symbols and the
isotopes with odd Z are plotted with open symbols in Figure 4.23—4.25. The lines
connecting the isotopes of one element are drawn to guide the eye. Only ratios with
errors smaller than 20% are plotted; for clarity we do not show the error bars of the
Rt9t(Ai Z)-

In the geometrical limit, the target isotope ratios, Rtgt(A,Z), are given by the
differences of the total reaction cross sections which are proportional to the sum of

radii squared [98]. This leads to the expression:

(Ayn/1w

Rm: 13 13 2’
(AP/ +AB/e)

(4.22)

 

where Ap is the mass number of the projectile, ATE and Age are mass numbers

of 181Ta and 9Be. The horizontal dotted lines in Figure 4.23—4.25 indicate the Rtgt

110

values calculated by Equation (4.22). The target isotope ratios calculated using EPAX
parameterization are shown as horizontal dashed lines in Figure 4.23-4.25. EPAX
systematics assumes only peripheral interactions parameterizing the cross sections as
a linear function of the sum of projectile and target radii. The experimental data,
however, exhibit a complex dependence of the target isotope ratios as a function of
the fragment mass number, A, for all investigated projectiles.

In the case of the 40Ca primary beam, left panel of Figure 4.23, we can see the U-
shaped curves for different elements suggesting enhanced production of neutron-rich
as well as neutron-deficient isotopes of a given element with 181Ta target. This is a
direct consequence of the wider isotope distributions for reactions with the 181Ta tar-
get compared to the ones with the 9Be target. In the case of the 48Ca projectile, right
panel of Figure 4.23, the U-shaped curves are not as pronounced and smooth as those
seen in the left panel of Figure 4.23 for the 40Ca beam. The magnitude of the effect is
small but larger than EPAX and geometrical limit predictions for light neutron-rich
isotopes. Such differences may be interpreted as an effect of isospin transfer from the
target to the projectile. The relatively steep increase of the production of neutron-rich
fragments in the fragmentation of 40Ca (Z /A = 0.50) on a 181Ta (Z /A = 0.40) target
may be due to the relatively large difference of the isospin asymmetry, whereas in the
case of 480a (Z /A = 0.42) projectile the asymmetry is similar to that of the 181Ta
target.

For the 58Ni projectile, left panel of Figure 4.24, the enhanced production of
neutron-rich as well as neutron-deficient isotopes is not as pronounced as in the case
of 400a, but the overall increase of the fragmentation cross sections with decreasing
fragment number can be observed. The more neutron-rich 64N i projectile, right panel
of Figure 4.24, exhibits less pronounced U-shaped curves. A clear enhancement, in-
versely proportional to the fragment mass number, A, of the production cross section
using the 181Ta target, can be observed for light elements (10 S Z 5 16). Again, this

behavior may be interpreted as an effect of the isospin transfer from the target to

111

 

 

 

 

 

 

 

 

 

 

 

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Mass number A Mass number A

Figure 4.23: Target ratios of the fragmentation cross sections U'ra(A, Z )/aBe(A, Z),
of fragments 8 3 Z S 18 for two projectiles 40Ca (left panel) and 48Ca (right panel).
The horizontal dashed and dotted lines indicate the ratio calculated by the EPAX
formula and Equation (4.22), respectively.

 

 

 

 

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20" "30 '140‘ 50“ "60
Mass number A Mass number A

Figure 4.24: Target ratios of the fragmentation cross sections 0T8(A, Z) /oBe(A, Z) of
fragments 10 3 Z s 26 for two projectiles 58Ni (left panel) and 64Ni (right panel).
The horizontal dashed and dotted lines indicate the ratio calculated by the EPAX
formula and Equation (4.22), respectively.

the projectile. For the 58Ni (Z /A = 0.48) projectile, however, the isospin asymmetry
is larger than in the case of the 40Ca beam, which may account for the less pro-
nounced enhancement of cross sections for neutron-rich fragments. In the case of the
64Ni (Z /A = 0.44) beam, we do not observe strong enhancement of the production
cross sections for the neutron-rich fragments because of a rather similar asymmetry
between the target and the projectile.

The overall trend of the target isotope ratios, R¢gt(A, Z), for the 86Kr projectile

measured at 64 MeV/u is similar to the projectiles measured at 140 MeV/ u, but with

112

 

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oTa(A,Z)/oBe(A,Z)

 

 

I I TTITI
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g
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60 70
Mass number A

Figure 4.25: Target ratios of the fragmentation cross sections 0T8(A, Z) /0’Be(A, Z) of
fragments 25 3 Z s 36 for 86Kr projectile. The horizontal dashed and dotted lines
indicate the ratio calculated by the EPAX formula and Equation (4.22), respectively.

steeper overall SIOpe. The experimental cross section data set for the 86Kr+181Ta
reaction is much smaller than the one for the 86Kr+QBe reaction, not allowing us to
investigate the enhancement of the production cross section of neutron-rich isotopes
for individual elements.

There are many differences in the shape of the target isotope ratios for different
projectiles. One common feature is that the overall slope of the target isotope ratios,
Rtgt(A, Z), seems to get steeper with increasing mass of the projectile.

Unfortunately, the observed small enhancement of the fragmentation cross sections
using 181Ta target is offset by the large difference in atomic mass of the two target
materials (approximately a factor of 20). The 181Ta target must be more than 20
times thicker in order to have the same number of target nucleons per cm2 as 9Be.
Phrthermore, effects like the energy loss and angular straggling must be considered
when using thick targets for production of rare isotopes. In the case of the 86Kr
projectile at 64 MeV/u a relatively broad charge state distribution for 181Ta target
makes it undesirable to us. However, if the rising trend of the Rtgt for the 86Kr primary
beam in Figure 4.25 continues for fragments of lighter elements, it may overcome the
above mentioned handicaps of the 181Ta target material. Unfortunately, our limited

set of the fragmentation cross sections does not allow us to make this conclusion.

113

4.3.5 Projectile dependence

The ratios of the fragmentation cross sections for a given fragment with neutron,
N, and proton number, Z, Rpm-(N, Z), from different projectiles were calculated to
address the projectile dependence. Rpm-(N, Z) E 043(N, Z) /a40(N, Z) in the case of
the Ca primary beams and Rpm-(N, Z) E 064(N, Z) /058(N, Z) for the Ni projectiles.
By convention, fragment cross sections from the more neutron-rich projectile (“Ca
or 64Ni) are used as numerators.

Figure 4.26 displays the ratios, RpmﬁN, Z), for the reactions of the 48Ca and 40Ca
with 9Be and 181Ta targets. The Rpm-(N, Z) ratios calculated for even and odd Z
elements are shown as ﬁlled and open symbols, respectively. Similarly, Figure 4.27
shows the Rpm-(N, Z), for the reactions of the 64N i and 58N i primary beams with 9Be
and 181Ta targets. Ratios obtained with the 9Be and 181Ta targets are plotted in the
top and bottom panels, respectively, of Figure 4.26 and 4.27 as a function of fragment
neutron number, N.

Peripheral projectile fragmentation reactions are mostly responsible for produc-
tion of fragments with mass A > A p / 2. Most likely, lighter fragments are produced by
more central collisions with different reaction mechanisms including multifragmenta-
tion. Indeed, light fragments (Z S 9 and Z S 12 for Ca and Ni systems, respectively)

exhibit the linear isoscaling behavior as observed in multifragmentation [99]
Rpmj(N, Z) = R21(N, Z) = Cexp(aN + BZ), (4.23)

where the isoscaling ratio R21(N, Z) is factored into two fugacity terms a and ,8,
which contain the differences of the chemical potentials for neutrons and protons of
the two reaction systems. C is a normalization factor of the isoscaling ratio. The lines
in Figure 4.26 and 4.27 for elements Z S 9 and Z S 12, respectively, are the best
ﬁts of the data to Equation (4.23). The best ﬁt values of a and B parameters for all

investigated reaction systems are listed in Table 4.3.

114

For heaviest fragments (Z 2 15 and Z 2 22 for Ca and Ni systems, respectively),
longer chains of isotopes were measured. The data no longer exhibit the linear isoscal-
ing behavior of Equation (4.23). Instead, the data are better represented by a function

which contains second order terms in neutron, N, and proton number, Z,

BMW-(N, Z) = Cexp(aN+alN2+ﬁZ+ﬁlZ2), (4.24)

where C is the normalization factor, a, m, ,8, and 61 are parameters of the proposed
function. The curves in Figure 4.26 for elements 15 S Z S 19 are the best ﬁts of the
data to Equation (4.24). In the case of the Ni reaction systems we ﬁtted the Rpm,-
ratios for elements 22 S Z S 26 by function of Equation (4.24).

It may seem that this second order function may be in contradiction with the
ﬁndings of the deeply inelastic scattering of Kr isotopes on Ni isotopes. However,
close examinations of previous data [100] show that the observed isoscaling is more
similar to Equation (4.24) than to Equation (4.23). Furthermore, closer examination
of the 10 S Z S 14 and 13 S Z S 21 regions for the Ca and Ni reaction systems,
respectively, revealed that they cannot be described by a single set of parameters of
Equation (4.23) or (4.24). Instead we noticed a gradual change of the a, a1, 6, and
61 parameters with every isotope chain of 10 S Z S 14 and 13 S Z S 21 elements
for Ca and Ni systems, respectively. This observation suggests that the isotopes of
these elements contain contributions from both projectile fragmentation as well as
multifragmentation. Unlike multifragmentation which can be assumed to be statistical
and thermal the large prefragment residues are most likely formed in non-equilibrium
processes and there is no a priori reason to assume that the fragments would observe
isoscaling. Impact parameter event selection detector in future experiments should

allow better distinction between the above described processes.

115

 

   

 

 

 

 

 

 

 

 

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Figure 4.26: Ratios of cross sections 043(N, Z)/a40(N, Z) of fragments created in
48Ca and 40Ca reactions on 9Be (top panel) and 181Ta (bottom panel) targets. The
projectile ratios for even and odd Z elements are denoted by ﬁlled and open symbols,
respectively. Solid and dashed lines show the ﬁt by Equation (4.23) for 5 S Z S 9.
Curved solid and dashed lines denote ﬁts by Equation (4.24) for 15 S Z S 19.

116

5..

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Neutron number N

Figure 4.27: Ratios of cross sections 064(N, Z) /058(N, Z) of fragments created in
64Ni and 58Ni reactions on 9Be (top panel) and 181Ta (bottom panel) targets. The
projectile ratios for even and odd Z elements are denoted by ﬁlled and open symbols,
respectively. Solid and dashed lines show the ﬁt by Equation (4.23) for 8 S Z S 12.
Curved solid and dashed lines denote ﬁts by Equation (4.24) for 22 S Z S 26.

117

Table 4.3: Best ﬁt values of the a, 011, ,8, and 31 parameters of the isoscaling function
of Equation (4.23) and the second order function of Equation (4.24) for Ca and Ni
systems. me and me. values indicate the range of elements for which the best ﬁt
values were obtained.

 

 

 

 

 

 

Projectiles Target Range

material me Zmax a 01 ﬁ 61
988 5 9 0.8714 0.0 —0.9971 0.0

Ca 15 19 — 1.2764 0.0776 1.8367 —0.1116
181Ta 5 9 0.7586 0.0 —0.8706 0.0

12 19 — 1.5350 0.0850 2.5532 -0.1340
9Be 8 12 0.5503 0.0 —0.6142 0.0

Ni 22 26 — 1.8047 0.0514 1.4966 —0.0556
181Ta 8 12 0.3961 0.0 —0.4670 0.0

22 26 — 1.4904 0.0444 1.4062 —0.0533

 

 

 

 

 

 

118

Chapter 5

Comparison to models

In order to get a physical understanding of the underlying fragmentation processes
and the reaction mechanism, three reaction models, varying in complexity and so-
phistication, were used to understand the results of our experimental measurements.
Calculations by the macroscopic Abrasion-Ablation (AA) model, the macroscopic-
microscopic Heavy Ion Phase Space Exploration (HIPSE) model and the sophisti-
cated fully microscopic Antisymmetrized Molecular Dynamics (AMD) model were
compared to the experimental data of this study. The AA model used in the present
work is a modiﬁed version of the geometrical AA model implemented in LISE++ [56].
In all described theoretical models the nuclear reactions are simulated in two stages:
primary (fast nuclear reaction dynamics) and secondary (slow decay of excited pri-
mary fragments).

First the three reaction models are discussed, introducing the main assumptions
and the theoretical concepts. Then the primary fragments generated by these models
are compared with each other. The discussion continues with the description of evap-
oration codes used to de—excite the residues obtained by the reaction models. At the
end of the chapter the calculated properties of the ﬁnal fragments are compared to

the experimental data.

119

5. 1 Reaction models

5.1.1 Abrasion Ablation model

In the fully macroscopic Abrasion-Ablation (AA) model, the projectile and target
nuclei are assumed to be spherical. The Coulomb deﬂection of the projectile trajectory
is neglected. In the case of a peripheral reaction, at a given impact parameter, b, the
projectile and target nucleons which lie in the overlap region (participant nucleons)
are removed from the original nuclei, while the remainder nucleons do not participate
(spectator nucleons). This ﬁrst stage of the reaction, called abrasion, is very fast; the
timescale is given by the relative velocity of projectile and target (see Figure 1.3).
The AA model does not describe the ultimate fate of the participant nucleons, since
they are not necessary to model the fragmentation of projectile or target nuclei.
Analytic formulas relating the impact parameter, b, to the number of removed
nucleons, AA, have been derived [101]. Unfortunately the expression of impact pa-
rameter, b, as a function of number of removed nucleons must be obtained numerically.
The cross section, ame(A), for producing a primary fragment with mass, A, from a

projectile, A p, is calculated from the obtained b(AA) function:
0mm) = 7r { [b(AA — 0.5)]2 — [b(AA + 0.5)]2}. (5.1)

Assuming there are no proton-neutron correlations in the reaction system, the proba-
bility to form a residue with a given mass number, A, and nuclear charge, Z, is given
by a hypergeometric distribution. Hence the total cross section, 0pm(A, Z), to form a

primary fragment (A, Z) is

0161A, 2) = [(AZZ) (IQ/($451)] 0mm). (5.2)

Where AP, Zp, and Np are mass, charge and neutron numbers of the projectile and AA,

120

AZ, and AN are numbers of nucleons, protons, and neutrons, respectively, removed
in the interaction. The projectile and target dependence is contained in the b(AA)
dependence.

The highly excited and deformed spectators de-excite in the second stage, called
ablation, by evaporating light clusters and nucleons (see Figure 1.3). This evaporation
stage of the reaction is taken to be slower (z 10‘21—-10‘16 3) compared to the abrasion
step (z 10‘22—10‘23 s) and results in the formation of the de—excited nuclei detected
in the experiments. In the original AA model [45] the excitation energy of the primary
fragments is determined from the surface energy (typical value z 1MeV/fm2). This
value of the excitation energy, however, must be regarded as a lower limit, because
the energy of the primary fragment can be altered by other processes (such as transfer
of energetic participant nucleons into the projectile prefragment).

To increase the excitation energy, many different mechanisms have been proposed,
such as scattered nucleons from the participant zone to the spectators [102] or in-
clusion of the frictional energy in the target-projectile interaction [103]. Gaimard
et al. [104] calculated the excitation energy based on vacancies created in the Fermi
distribution of the nucleons in the spectator nucleus. This simple statistical “hole-
energy” model yields an average excitation energy of 13.3 MeV per hole (abraded
nucleon). Another investigation by Schmidt et al. [105] resulted in an average ex-
citation energy of 27 MeV per abraded nucleon for fragments of heavy projectiles
(A > 100) in the most peripheral collisions. The above mentioned estimates of the
mean excitation energy per nucleon illustrate the level of uncertainties in determina-
tion of the excitation energy in the AA model.

In our calculations we used a modiﬁed version of geometrical Abrasion-Ablation
model that has been implemented in the LISE++ code [56]. The mean excitation
energy, E*, of the primary fragment is expressed as a linear function of the number
of abraded nucleons, AA:

E* = K . AA. (5.3)

121

Table 5.1: Best ﬁt values for the K and S parameters of the excitation energy in
the Abrasion-Ablation model for all investigated reaction systems. Uncertainties es-
timated from the minima in the two-dimensional K versus S space.

 

 

 

 

 

 

 

Beam Target K S
(MeV/u) (MeV/ﬂ)
400a 9Be 1112 621:1
181Ta 9 2t 1 4 :l: 1
480a 9Be 1112 6i1
181Ta. 11 :i: 2 6 :l: 1
58Ni 9Be 13 j: 2 7 :l: 1
181Ta 13 i 2 8 :l: 1
“Ni 9Be 13 i 2 7 i: 1
181Ta 15 i 2 8 ﬂ: 1
86Kr 9Be 19 :l: 2 10 :l: 2
181Ta 18 ﬂ: 2 10 d: 2

 

 

 

 

 

 

The ﬂuctuations around this mean value are described by a Gaussian distribution

with a variance, 031, parameterized as:
0E: : S ’ V AA. (5.4)

The K and S are taken as ﬁtting parameters of the model. The excitation energy
is assigned to prefragments according to Equations (5.3) and (5.4). These excited
primary fragments decay as described in the analytical LisFus evaporation code [106].

The K and S parameters are determined by ﬁtting the measured cross sections
of projectile-like fragments (A > Ap/ 2) for a given reaction system. The obtained
best ﬁt values of the K and S parameters for all our reaction systems are listed in
Table 5.1. This approach has been applied in the case of the 78Kr+9Be reactions at
140 MeV/ u by Stolz et al. [107]. The measurement focused on proton—rich fragments
in Ge—Se region and obtained K = 12 MeV/ AA and S = 3 MeV/m.

The obtained values for the K and S parameters in our analysis as listed in Table
5.1, do not depend on the target for all investigated projectiles within the quoted

uncertainties. Both Ni projectiles seem to require a slightly larger values for the K

122

parameter than the Ca beams, but the differences are still within the large uncer—
tainties. The 86Kr reactions, however, require much steeper slopes of the excitation
energy as a function of the number of removed nucleons in order to describe the
fragmentation data.

The LISE++ implementation of the AA model [104] provides fast calculations over
a wide dynamic range of ﬁnal fragment cross sections. With only two parameters, the
excitation energy and its ﬂuctuations, the predictions reproduce the experimental
data extremely well as will be discussed in Section 5.5. Currently it requires ﬁtting of
principal parameters to the experimental data for each reaction, its predictive power is
rather limited. However, it may be used to extrapolate the production cross sections,
in the case of one reaction system, for very exotic fragment species when a systematic

set of fragmentation cross sections becomes available.

5.1.2 Heavy Ion Phase Space Exploration

The Heavy Ion Phase Space Exploration (HIPSE) model has been implemented to
bridge the gap between the statistical models, which reduce the description of the
reaction to a few important parameters, and fully microscopic models which require
a large amount of CPU time to describe the motion of individual nucleons. Based
on a macroscopic-microscopic “phenomenology,” it accounts for both dynamical and
statistical aspects of nuclear collisions.

The nuclear reaction, as described by HIPSE [108], can be separated into three
stages: approach of projectile and target nuclei, partition formation and the cluster
propagation phase (ending with an in-flight statistical decay). In the entrance channel
at a given beam energy, Eg, a classical two—body dynamics of the center of masses of
the target and the projectile nuclei is assumed. For projectile and target nuclei with

mass numbers, Ap and AT, and positions, rp and rT, respectively, the dynamical

123

evolution associated with the following Hamiltonian is considered:

—AL—-EB = p—2 ‘1' VA A (er — I’pl) (5.5)
AT +1413 211. T P ,

where VATAP is the nucleus-nucleus potential, p is the relative momentum and ,u =
me p / (mT+m p) is the reduced mass with mp and mT being the projectile and target
mass, respectively. The proximity potential [109] is used to express the nucleus-nucleus
potential VATAP at large distances 7" > (RT + Rp), with RT and Rp radii of the target
and the projectile nuclides, respectively. This potential is unambiguously deﬁned when
the two nuclei are well separated. For small relative distances 7' S (RT + Rp), a simple
third order polynomial extrapolation is used, assuming continuity of the derivative of

the potential at each point. The value at r = 0 is expressed as:
VAT/119(0) = Gal/£310 = 0) (5-6)

where the potential hardness parameter, aa, is an adjustable parameter of the model,
and Vﬂ'ﬁ: (r = 0) is the energy of the system assuming that the two nuclei overlap
completely in the frozen density approximation. This energy corresponds a pm’om’ to
the maximal value of VATAP (0), leading to 01,, S 1. Figure 5.1 shows a dependence of
VATAP on relative distance between target and projectile nuclei for different values of
the an parameter. The frozen density or “sudden” approximation assumes that the
internal degrees of freedom do not have time to reorganize themselves and the system
has a strong memory of the initial conditions. At energies close to the fusion, barrier,
the 010 parameter represents a measure of the degree of reorganization of the internal
degrees of freedom during the reaction. Since the model does not treat the internal
reorganization of nucleons explicitly, an is taken as a free parameter, depending only
on the beam energy, representing the absence of knowledge of the nucleus-nucleus

potential at large overlaps.

The trajectories of the projectile and the target nuclides are determined by the

124

 

 

 

 

 

O LALlLLAA[LPAAIAAAAILJLALLJAL

10 20 30
Relative distance (fm)

Figure 5.1: Nucleus-nucleus potential VATAP as a function of the relative distance for
the 129Xe+1208n system with (1,, = 0, 0.01, and 0.2 [108].

Hamiltonian in Equation (5.5). At the minimum distance of approach, the two re-
acting nuclei overlap according to the impact parameter and the potential hardness
parameter, aa. The frozen density approximation is applied again, explicitly, to sam-
ple the positions and the momenta of the nucleons in the center of mass of each of
the reaction partners. The semiclassical Thomas-Fermi theory is used to get realistic
ground state density distributions for each of the reaction partners. The nucleons
are sampled according to a Metropolis algorithm where the Pauli principle is taken
into account in each nucleus by requiring that Ar, - Apr 2 h, where Ar.r and Ap,
are the relative position and momentum of two nucleons with the same isospin, T.
This procedure ensures a uniform arrangement of the nucleons in each of the reaction
partners. At the end of the ﬁrst stage of the reaction positions (2:, y, z),- and momenta
(p1, py, 132),- for a set of AT + Ap nucleons are obtained.

Before the clusters are deﬁned, the overlap region is determined based on the

following consideration: a nucleon initially in the target at position r,-, is assumed to

125

be in the overlap region if Ir,- — rpl S Rp, where rp is the position of the projectile
in the center of mass frame, and Rp is the radius of the projectile. For the nucleons
initially in the projectile analogous criteria are required Ir,- — rTI S RT, where r7» is
the position of the target in the center of mass frame, and RT is the radius of the
target. This approach corresponds to the so-called “participant-spectator” picture.
However, experimentally it is observed that the quasi-projectile fragments have a
slightly reduced kinetic energy with respect to that of the initial projectile. This
effect is taken into account by introducing the exchange of particles between the two
partners during the reaction. In the HIPSE model it is introduced ad hoc by assuming
that a fraction, 9:”, of the nucleons coming initially from the target (projectile) are
transferred to the projectile (target). It is expected that the number of exchanged
nucleons decreases with the beam energy, and thus mt, depends only on the initial
kinetic energy.

A simpliﬁed procedure is applied to treat nucleon-nucleon collisions inside the
overlap region characterized by a mass number, Awer. Only a fraction of the nucleons
in the participant region experiences a collision. Thus the number of collisions is
deﬁned by NCO“ = Am, - 1:60“. The fraction of the nucleon-nucleon collisions, mm",
is the third free parameter of the HIPSE model. It is expected to depend only on
the energy of the beam. A two-body nucleon-nucleon collision is simulated only in
the momentum space. The ﬁnal spatial positions of the two nucleons are randomly
distributed inside a sphere with radius

Rcoll : 12A1/3 + really (57)

0087‘

where parameter rwu is taken to be 4 fm which reproduces the low energy multifrag—
mentation data [30]. It muSt be noted that due to the Pauli exclusion principle not
all the volume of the sphere with radius, R60", is accessible.

After nucleon-nucleon collisions take place, the clusters are deﬁned according to

126

a straight-forward coalescence algorithm. First, one of the nucleons is chosen at ran-
dom, constituting a starting point from which the fragment is built. Then, another
nucleon, 2', from the participant region is chosen. Since the exact conditions when a
nucleon in the nuclear medium can be absorbed by a cluster are not known, a simple
phenomenological criterion is used. In particular, the nucleon with relative position,
1",, and relative momentum, 19,-, with respect to fragment is assumed to be absorbed
by it if

2 V
—p‘— + a“ < 0, (5.8)

2m 1 + exp [4”? ]

 

where m is the nucleon mass and the parameters d; and th correspond to limits in
position, 7', and momentum, p, space, respectively. The distance, d f, is expressed as
Rf + rent, where Rf is the radius of the fragment and rent is 7 fm for all nucleons
undergoing a collision and 2.5 fm for all others. Using cht = —p2/2m, pcut=500
MeV/c was obtained and the diffuseness parameter, a, has been ﬁxed to be 0.6 fm.
If the phase space condition of Equation (5.8) is fulﬁlled the nucleon is absorbed by
the fragment and a new cluster is formed provided that it exists in the mass table. If
the above mentioned requirements are not fulﬁlled the simulated event is discarded
as unphysical.

The whole procedure is repeated until all nucleons in the participant region are
exhausted. The position, momentum, mass and charge of the clusters are then updated
at each step. If there is more than one possibility for aggregation at a given step, the
nucleon is absorbed by one of the fragments at random.

Once all the residues are deﬁned with mass number, position, momentum and
angular momentum (A,, R.,-, P,-, L,), deduced from the nucleon properties in every
cluster, a clock is started corresponding to t = 0 fm/c for the forthcoming dynamics.

The fragments are propagated according to the Hamiltonian

 

Pr?
H: E; m, +ZVA.A,(IR.- —R.-I)- (5.9)

i<j

127

In order to ensure the consistency of the calculation, the same nucleus-nucleus poten-
tial of Equation (5.5) is used as in the entrance channel. The dynamical propagation
of the clusters is stopped at t = 150 fm/c, when a rather signiﬁcant reorganization in
the phase space may have occurred. The relative energy of any of the combinations
of two fragments is calculated, and if it is lower than the fusion barrier, the corre-
sponding clusters fuse and the properties of the fused systems are calculated. It is
then assumed that the thermalization occurs on a time scale of tens of fm/c, which is
not described on the microscopic level in the model, the excitation energy is obtained

from the overall energy balance of the reaction in the Center of Mass System (CMS):

E0 = Q + BK + Epot + E,“ + Erota (5.10)

where E K and EM are, respectively, the sum of the kinetic and potential energies of all
primary fragments, Emt is total rotational energy and Q is the energy balance between
the entrance channel and the ﬁnal distribution of residues. Finally, E“ represents the
total excitation energy available for all the primary fragments. The excitation energy

is distributed among the residues according to their mass number:

 

(5.11)

where A,- and A are the mass numbers of the fragment and the whole system, re-
spectively. At the end of the third phase of the reaction in the HIPSE model, a
distribution of residues deﬁned by their mass number, charge, position, momentum,
angular momentum and excitation energy (A,, Z,-, R, P,-, L,-, E:) is deﬁned. The
secondary decays on the set of primary fragments are performed by the statistical
sequential evaporation code GEMINI [110].

The HIPSE model has three freely adjustable parameters: an, 33m and 3:60", which
depend only on the incident beam energy. Their energy dependence and the resulting

values for beam energy, EB = 10, 25, 50 and 80 MeV/u are listed in Table 5.2 [111].

128

Table 5.2: Values of HIPSE adjustable parameters as determined by Lacroix [111] for
energy 10, 25, 50 and 80 MeV/ u. Extrapolated values to 140 MeV/u and interpolated
values for 64 MeV/ u used in our calculations are also shown.

 

 

 

 

Beam energy
(MeV/ 11) 0,, art, (Econ
10 —0. 10 0.60 0.00
25 0.10 0.45 0.02
50 0.20 0.30 0.05
80 0.25 0.25 0.10
140 0.55 0.09 0.18
64 0.22 0.27 0.06

 

 

 

 

 

 

The parameter values used in our calculations for 86Kr at 64 MeV/ u were interpolated
using a second order polynomial function and the results are listed also in Table
5.2. For the 40"‘8Ca and 58MM projectiles at 140 MeV/ u, we used the extrapolation
by second order polynomial (an and mm") and the exponential decay function (ctr)
resulting in values listed in Table 5.2.

The results of the HIPSE calculations at 140 and 64 MeV/ u for 40"‘BCa, 58"MN i and
86Kr projectiles on Be and Ta targets, are insensitive to the choice of a,, parameter
because the energy available in the center of mass frame is always much larger than the
value Vf;j’ﬁ,(r = 0) in Equation (5.6). The nucleon exchange rate, xtr, and fraction
of nucleon-nucleon collisions, 32w“, parameters, on the other hand, set the overall
distribution of the excitation energy of the residues.

Since the HIPSE model requires three adjustable parameters whose values are
beam energy dependent, its predictive power is considerably larger as compared to
that of the geometrical AA model. It has been applied to the multifragmentation data
from central collisions of 129Xe+""‘tSn measured at 32, 39, 45 and 50 MeV/ u [112],
58Ni+58Ni measured at 32 to 90 MeV/u [113]. Until this study the HIPSE model
had not been applied extensively to the peripheral collisions at beam energy higher
than 100 MeV/u [114]. This investigation provided a test of the HIPSE model, which

uncovered some unexpected deﬁciencies of the model, as will be discussed in Section

129

5.3 and 5.6.

5.1.3 Antisymmetrized Molecular Dynamics

From the many microscopic models available today to simulate the nuclear collisions,
the Antisymmetrized Molecular Dynamics (AMD) model was used to compare to our
experimental projectile fragmentation data. As the most sophisticated model used in
our study it describes the nuclear reaction at the microscopic level of interactions of
individual nucleons. The AMD model has a potential to mimic all reaction processes
involved in the complex heavy-ion collisions. This investigation also represents one
of the ﬁrst attempts to use the AMD model in a study of projectile fragmentation
reactions at 140 MeV/u [115].

The AMD model developed by A. Ono and H. Horiuchi [116] is, an antisym-
metrized version of a molecular dynamics model. The original version of AMD [117]
contains some quantum features, like the use of the wave functions and a correct
treatment of antisymmetrization, plus the effect of nucleon-nucleon collisions. The
version of the AMD used in this work has been improved by incorporating the advan-
tages of other microscopic models such as the time-dependent Hartree—Fock (TDHF)
theory and the Boltzman-Uehling—Uhlenbeck (BUU) equation [116]. Implementation
of a mean ﬁeld theory in a nuclear reaction model usually results in the improvement
of the single particle description [118,119], but the fragment formation description
gets worse [116]. The authors of AMD attempt to overcome this conceptual difﬁculty
by incorporating quantum branching [120].

AMD is a special case of the time-dependent variational theory [121]. The AMD
wave function is given by a Slater determinant of Gaussian wave packets for individual

nucleons [116]:

(1‘1 - ° -rA|‘1>(Z)> = det [<pz.(rj)xa.(j)l, (5-12)

130

where the wave functions in spatial coordinates are expressed as:

(11,02) = (27:1)3/4 exp {—u (r — 7%)2 + ézi’} (5.13)

and X0, is the spin and isospin part of the wave function. The many-body state, |<I>(Z)),
is parameterized by a set of complex variables Z E {Z1, - -- ,ZA}, with A being the
total number of nucleons in the system. The width parameter, u, is treated as a
constant parameter in AMD, while the complex centroid, Z, is treated as a dynamical
variable. The real part and the imaginary part of Z correspond to the centroids of the
position and the momentum respectively [116], therefore the V parameter deﬁnes the
balance of the uncertainties of the position and the momentum, having a minimum
given by the Heisenberg uncertainty principle. In the AMD calculations V = 0.16
fm‘2 was chosen in order to reproduce the ground state energy of the light nuclei.

An effective two-body nuclear interaction is utilized in the AMD Hamiltonian:

2
i

A
H = 2; 2pm + Zvij, (5.14)

i<j

 

where m is the nucleon mass and 11,-, is the effective two-body force. For our calcula-
tions a modiﬁed Gogny-type force (GOGNY-AS [122]) has been used with parameters
listed in Ref. [116].

The time evolution of the centroids, Z, which parameterize the many-body wave

function [@(Z)), is determined by the time-dependent variational principle

 

6 / dt<<I>(Z)lih% — H|<I>(Z))

<<I><Z>I<I>Iz>> 2 0’ (5'15)

from which the classical Hamiltonian equation of motion for Z is derived. The equa-
tion of motion in AMD can be regarded as a generalization of the time-dependent
cluster model (e. g., [123]). The frictional cooling method, derived from the equation

of motion [116], is utilized to calculate the ground states of colliding nuclei.

131

The AMD wave function of Equation (5.12), as a single Slater determinant, does
not contain any two-body correlations, except the implicit correlations given by the
mean ﬁeld and the antisymmetrization. In order to describe nuclear collision, corre-
lation beyond mean ﬁeld is included in a more explicit way by considering nucleon-
nucleon scattering in the dynamically evolving system. The two-nucleon scattering
was introduced in order to improve description of the (multi) fragmentation reactions
in the intermediate energy regime. It is simulated by changing the momenta of two
nucleons when their centroids approach in spatial coordinates. The implementation
of the nucleon-nucleon scattering in AMD is non-trivial because the wave packet cen-
troids, Z, do not have a physical meaning due to the antisymmetrization procedure.
In order to overcome this difﬁculty, physical coordinates W E {W1, - -- ,WA} are
introduced, which have a physical meaning as the centroids of the nucleon wave pack-
ets [116]. The transformation from Z coordinates to W space is chosen such that it
respects the Pauli exclusion principle (i. e., the inverse transformation from W to Z
exists only if the Pauli principle is obeyed).

A typical two-nucleon scattering event is calculated as follows. Any two nucleons,
1' and j, in the system are scattered with a probability based on the differential cross
section, daNN/dQ, of nucleon-nucleon collisions. The cross section, aNN, depends on
the relative kinetic energy of the nucleons, the isospin and the environment surround-
ing the two scattering nucleons. Then, at every time step in solving the equation of
motion for AMD, it is examined whether each pair of nucleons should make a stochas-
tic collision based on the real part, R,- and Rj, of the physical coordinates, W, and
the probability of the scattering is determined by the overlap of the Gaussian wave
packets of the two nucleons and their relative velocity. Once the two nucleons are
allowed to collide based on the probability described above, their physical momenta,
P,- and P j, are changed in the following way. First, the unit vector, n, representing the
scattering angle is chosen randomly based on the differential cross section, daNN/dﬂ.

The spatial positions, R,- and R,, of the nucleons are not changed and the ﬁrst guess

132

.rv‘Ta‘ 4a A 1

1;; ,9 e» .3 Initial State

\
/\\\ //\]/<1\ b\

c1]~,.:)3 +02 6:) . +03 {a}. +...

:5" . xv "y 0
eUD .

Figure 5.2: A schematic representation of the quantum branching for multichannel
reactions [115].

0-0-

 

of the ﬁnal momenta in the ﬁnal state, W’, is calculated:

1
P; = EPCM + Pr’ern, (5-16)
I 1 I

where POM = P, + P, and Pr’e, is the relative momentum (1/2)|P, — le- Next,
the ﬁnal state W’ is transformed to Z’ coordinates. If the corresponding Z’ solution
does not exist, it means the calculated ﬁnal state is Pauli blocked and the collision is
discarded. If the corresponding Z’ solution exists, the total energy, E’, of the state,
|<I>(Z’)), is calculated. Then energy conservation between the initial and the ﬁnal

states is required. If not fulﬁlled the next guess of the relative momentum P,’;, is

taken from
PII2 I2
—"~’—‘ — Q = E — E’. (5.18)
m m

The procedure is iterated until energy conservation is satisﬁed.

It is necessary to go beyond ﬂuctuations introduced via nucleon—nucleon collisions

133

in order to better describe the multifragmentation data. Although a single Slater de-
terminant is sufﬁcient for the initial two nuclei and the fragments in each reaction
channel, it is far from sufﬁcient for the superposition of the ﬁnal channel wave func-
tions (see Figure 5.2). In such cases, one can expect that by solving the deterministic
time evolution, one gets a ﬁnal state which resembles one of the possible channels. In
other cases, however, the obtained ﬁnal state does not look like any of the channels.
By introducing the quantum branching to AMD, the many-body state wave function,
[\Il(t)), which is evolving from the initial to the ﬁnal state according to the Shr6dinger

equation, is expressed as an ensemble of AMD wave functions, |<I>(Z)):

[\Il(t))(\II(t) z lggglggilw(Z, t)dt, (5.19)

 

where the right hand side is the statistical density operator of the ensemble and
w(Z, t) represents the weight of the state, |<I>(Z)), at time, t. The approximation in
Equation (5.19) neglects the off-diagonal terms which are expected to have negligible
contributions [116]. Although different channels may interfere, it is impossible to
perform the numerical calculations by keeping all of the interference. It should be
noted that the time evolution is solved independently for each channel. Namely, the
state is decomposed into independent branches of channels, which is called quantum
branching.

The ground states of 40"‘BCa, 58"“‘Ni, and 86Kr projectiles and 9Be target were
prepared by applying the frictional cooling method to the AMD wave function [115].
The time evolution of the system in the AMD model is governed by the equation of
motion, together with the stochastic nucleon-nucleon scattering. In order to compare
the results of the AMD calculation to our inclusive experimental data, it is necessary
to perform many collisions with different impact parameters, b. The range of impact
parameters chosen for our AMD simulations was 0 S b S 14 fm. The system dyna-

mical evolution was followed up to time, t = 150 fm/c, when the produced clusters

134

of nucleons are spatially well separated. Then a simple coalescence algorithm applied
to the physical positions of the centroids of the Gaussian wave packets can be used
to deﬁne the primary fragments. The cut-off radius was chosen to be rout = 5 fm,
meaning that if the centroids of two Gaussian wave packets are closer than 5 fm, they
belong to one cluster. After applying this algorithm, all primary fragments 2' are de-
ﬁned with mass and charge number, momentum, angular momentum and total kinetic
energy (A,, Z,, P,~, L,-, E,). The excitation energy of a given residue is then deﬁned
with respect to the experimental binding energy taken from the Audi-Wapstra atomic
mass evaluation [124]. These residues were then de-excited using the statistical decay
code GEMINI [110].

Since the AMD calculation is extremely CPU—intensive, all calculations have been
carried out at the High Performance Computing Center (HPC) at Michigan State Uni-
versity using a 64-processor cluster [125] running under OS Linux. For the 40Ca+9Be
reaction one event requires approximately 6 CPU minutes. The computing time scales
as A3, where A is the total number of nucleons in the reaction. For the 40”‘8Ca, 58MM,
and 86Kr projectiles on 9Be target we gathered approximately 20,000 events / reaction.
The AMD calculations have not been performed for the 181Ta target, because the large
number of nucleons, A > 200, results in an impractically long execution time to gather
enough statistics.

A time evolution of four sample events (in center of mass frame of reference)
with different impact parameters is shown in Figure 5.3 in the case of 86Kr+9Be
reaction. The nuclear collision is represented by the nuclear density projection onto
the reaction plane (labeled as ‘Y’ versus ‘Z’ in the bottom left pane of Figure 5.3).
Each row of snapshots in Figure 5.3 represents the state of the reaction, simulated
by the AMD model, at different time, t = 0, 30, 60, 90, 120, and 150 fm/c. The
snapshots are grouped in four vertical columns representing events with four different
impact parameters, b = 0.57, 4.14, 7.47, and 10.97 fm, from left to right. The area of

each snapshot shown in Figure 5.3 is 40x40 fm2. The projectile 86Kr (large density

135

b = 0.57 fm b = 4.14 fm b = 7.47 fm b = 10.97 fm

, ...j—r.ﬁ Y.#Y,v+..r17.
< [

 

"V—YV‘IiYYV—VVY—VYYiT

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 5.3: Examples of the time evolution of the density projected onto the reaction
plane Y versus Z, where the projectile (from left) collides with the target (from right)
nucleus along the Z axis. Snapshots from t = 0 tot = 150 fm/c for 86Kr+9Be collisions
at 64 MeV/ u are shown in vertical columns for impact parameters b = 0.57, 4.14,
7 .47 and 10.97 fm. The area shown for each time step corresponds to 40x40 fmz.

136

contours) and the target 9Be (small density contours) nuclei are well separated for
all impact parameters at t = 0 fm (ﬁrst row in Figure 5.3). In the case of central
collisions (2 leftmost columns) the target nucleus is absorbed by the projectile with
few prompt nucleons emitted. For more peripheral reaction, b = 7.47 fm (third column
from left), we see the abrasion of the projectile nucleus by the target. For extremely
large impact parameter, b = 10.97 fm, (rightmost column) we observe no overlap
of the projectile and target nuclei. Events of all impact parameters are summed up
and normalized to the maximum impact parameter in order to be compared to the

inclusive experimental data.

5.2 Primary fragment distributions

In order to directly compare the results of all three calculations (AA, HIPSE, and
AMD) one must examine the properties of the primary fragments before sequential
decays occur. Excitation energy of the residues will be discussed in Section 5.3. Here
we compare the isotopic distributions of the primary fragments.

Figure 5.4 and 5.5 display the primary fragment distributions obtained in three
different models for 40"“3Ca and 58MN i primary beams on 9Be target. The AA, HIPSE,
and AMD model calculations are shown as dotted, solid, and dashed lines, for 13 S
Z S 20 isotopes for the Ca beams and 21 S Z S 28 isotopes for Ni beams, respec-
tively. In the case of projectile-like primary fragments of (up to approximately 10
nucleons removed), all three models predict very similar isotopic distributions. The
Abrasion-Ablation model (dotted lines) calculates the primary residue distributions
according to geometrical assumptions (Equation (5.2)), resulting in smooth and broad
distributions, calculated over an extremely wide range of isotopes. The HIPSE and
AMD models based on Monte Carlo techniques, on the other hand, create fragments
over a limited range of cross sections proportional to the number of simulated events.

The disagreements of the three reaction models grow with the number of removed

137

 

.5
_Lo

 

Cross section (mb)
as. 32. <2.

A
C:
d

 

 

 

 

 

 

AAAl‘JAAlLA‘I

 

 

 
 

 

0 5 -5 0
Neutron excess N-Z
1o: , 1: .~...'.3--‘-«-...
.. ‘ 1 '.
E ‘”
c 7:3" .
$10,: Al .. SI
48 9 r
g; [3: Ca+ Be
m .
9
o ,
10'1

 

 

 

 

 

 

 

   

o 5 10 o 5 1o
Neutron excess N-Z

Figure 5.4: Primary fragment isotopic distributions for 40Ca+QBe (top) and 48Ca+-9Be
(bottom) reaction system plotted as a function of neutron excess, N —Z. Dotted, solid
and dashed lines show calculations by AA, HIPSE and AMD models, respectively.

nucleons for all investigated reaction systems. Overall the isotopic distributions of
primary fragments in the HIPSE calculation are wider than the ones obtained by
the AMD model. Furthermore, clear trends of deviations between the HIPSE and
AMD models are observed for the lightest primary fragments shown in Figure 5.4
and 5.5, when the AMD predicts systematically larger cross sections than the HIPSE
calculation.

One should note that unlike the HIPSE or AMD models which follow the dynamics
up to t = 150 fm/c, the AA model does not describe the dynamical evolution of the

fragments in the collision. Since the primary fragments are created in an instantaneous

138

_A
0

Cross section (mb)
5 a. a.

—L
Q
d
c

_L
0

Cross section (mb)
8

d
0.

Figure 5.5: Primary fragment isotopic distributions for 58Ni+9Be (top) and 64Ni+9Be
(bottom) reaction systems plotted as a function of neutron excess, N -Z. Dotted, solid
and dashed lines show calculations by AA, HIPSE and AMD models, respectively.

geometrical abrasion the prefragments from the AA model could be formed at much

earlier times.

d

d

5‘33.

 

 

 

 

 

P.
b

 

 

 

 

 

 

 

 

-1 i- .U
0
-‘ Co
.-
l- 0..
A I l A A l A A A A

.1

1AA

 

 

 

0 1o 0
Neutron excess N-Z

5.3 . Excitation energy

The excitation energy is one of the principal parameters describing the nuclear colli-
sions, but it is very difﬁcult to measure experimentally. These diﬂiculties stem from
the technical issues during experiments like: incomplete measurement of all particles

in an experiment or problems associated with detection of neutrons, one of the most

139

10

 

 

dominant evaporation/de—excitation channels. In the case of our experimental data,
we did not detect the entire spectrum of emitted particles, rather only the nuclides
with velocity close to that of the projectile were measured. The mean excitation en-
ergy must be determined from the theoretical calculations by comparing measured
observables.

In the case of the Abrasion-Ablation model the excitation energy and its ﬂuctua-
tions are parameterized with respect to the number of removed nucleons by Equations
(5.3) and (5.4), respectively. The excitation energy of primary fragments used in our
AA calculations is determined by ﬁtting the K and 5' parameters to our experimental
cross section data (Section 5.1.1).

The dynamical calculations used in our analysis (HIPSE and AMD), on the other
hand, deﬁne all the residues with their excitation energies. The mean excitation energy
for a given prefragment mass number, A;, was calculated as an average over all events
resulting in primary fragments with A,- for both the HIPSE and AMD models.

The average excitation energy per nucleon for prefragments produced in the frag-
mentation of 40"‘8Ca and 58"“‘Ni primary beams on 9Be target is shown in Figure 5.6
as a function of the primary fragment mass number. The mean excitation energy used
in the AA calculations is plotted as a dashed line. The mean excitation energy for a
given prefragment mass number, A5, calculated by the HIPSE and AMD models is
shown by the black and cyan-shaded curves in Figure 5.6. The shaded regions rep-
resent the variance of the excitation energy distribution in all simulated events. In
addition to these three models, we show a calculation by a single-particle, microscopic
Boltzman-Uehling-Uhlenbeck (BUU) equation [126] (solid line).

The excitation energy per nucleon, E*/A, used in the AA model calculations
(dashed line) has a monotonically increasing trend determined by the K parameter of
Equation (5.3) and the number of removed nucleons from the projectile, AA. It results
in extremely large E*/A for lighter primary fragments (> 10 MeV), but the excitation

energy for primary fragments in the vicinity of the projectile is lower than predicted

140

 

 

     
  

   
  
  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6 IV I I I l f 6
9 4 " 9 4 ‘
a: i a)
E i E
$ - 4°Ca+gBe S - 4'3Ca+9Be ~
4: 4: x
m z—IHIPSE ._ UJ 2—IHIPSE “*~':" » ----- —
- ---AA - ---AA ‘
'IAMD 'IAMD
- —BUU ‘ - -BUU ‘
i . i . l . i . I l . . . . ‘ . G n n l u 1 i 1 l i ‘7
90 25 30 35 4o 30 4o 50
Residue mass A Residue mass A
6Ll""‘l""'l"" ""'IT\"I""
i ‘
J
J
’>‘ 4 ‘ 9
0) (D
E. . E
g 4E58Ni+gBe . g
Lu 24IHIPSE a UJ
-x--AA T
“-AMD <
~ —BUU . ‘
GLAAAJIIAIAIIIIJ“I 044.1 Ill Alli .‘u
30 4o 50 60 4o 50 so
Residue mass A Residue mass A

Figure 5.6: The mean excitation energy per nucleon, E*/A, as a function of the mass
number of the primary fragments in different models. HIPSE and AMD calculations
are shown for reactions of 40“Ca and 58'64Ni projectiles on 9Be target. The mean
excitation energy used in our Abrasion-Ablation calculations is shown as a dashed
line.

by all other models (HIPSE, BUU, AMD). In the case of dynamical models (HIPSE
and AMD) we notice a rather sharp rise of the E*/A with the number of removed
nucleons up to approximately 4—5 MeV, when it remains roughly constant. The E*/A
calculated in the BUU model exhibits very similar trend as the HIPSE results for
all reaction systems shown in Figure 5.6. The HIPSE and AMD model calculations
differ mainly for residues close to the projectile (z 0—10 abraded nucleons). The AMD

calculation produces systematically higher excitation energy for this region of primary

141

 

E*/A (MeV)
I a, I I

A

 

 

 

 

-

C

r- I

F' .‘ "

F .Q‘ q
1 L 1 1 I 1 l I “a

. l i
30 4o 50
Residue mass A

 

Figure 5.7: The mean excitation energy per nucleon, EVA, as a function of the mass
number of the primary fragments in different models. HIPSE calculation is shown for
the 48Ca+181Ta reaction system. The mean excitation energy used in our Abrasion-
Ablation calculations is shown as a dashed line.

fragments.

Plotting the E*/A as a function of the primary fragment mass number inevitably
mixed events with different impact parameters. The deterministic BUU calculations
result in the mean mass and charge numbers of the primary fragment for a given
impact parameter. This procedure makes the direct comparison with the AA, HIPSE
and AMD models difﬁcult. Nevertheless, we conclude that the BUU result in terms
of average excitation energy per nucleon is similar to that of the HIPSE model for all
reaction systems shown in Figure 5.6.

Figure 5.7 shows the mean excitation energy of primary fragments, created in
the 48Ca+181Ta reaction, as a function of their mass number. Results of the HIPSE
calculations are shown along with the excitation energy used in the AA model. Opti—
mum excitation energy determined using the AA calculation in the case of 181Ta and
9&5 target is the same. On the other hand, the E*/A proﬁle obtained by the HIPSE
calculation is very different for the two target materials. In the case of 181Ta target,

the excitation energy saturates at much higher values (le MeV/u), not shown in

142

 

 

    
  

 

 

 

 

 

 

   

 

 

5...,....,.ﬁ.ﬁrr. 6a..,....,....,....,..
. “KHQBe
.HIPSE .
...AA .
g 4‘. ...EAMD _, g ...... _
E —BUU ‘ 2
.s .s
LIJ2 ...- LU
0.xiiii..i....‘~. o..ilin..1....i...:‘i‘...n
75 80 85 7o 75 80 85
Residue massA Residue massA

Figure 5.8: The mean excitation energy per nucleon, E*/A, as a function of the mass
number of the primary fragment different models for 86Kr beam and 9Be and 181Ta
target materials. The HIPSE calculations are shown in black, the AMD results are
presented in cyan and the mean excitation energy used in our AA model calculation
is shown as a dashed line.

the plot because of poor statistics for the intermediate mass range fragments. The
HIPSE calculation suggests a rather big difference between the excitation energy of
primary fragments when using 9Be and 181Ta targets for 400a and 58MM projectiles
as well.

The mean excitation energy per nucleon calculated for 86Kr projectile is compared
to the values used by the AA model in Figure 5.8. Results for the 86Kr+9Be are
shown in the left and 86Kr+181Ta in the right panel. The E*/A used for the AA
calculation (dashed line) is very similar (K = 18 and 19 MeV/AA, respectively)
for both targets (9Be and 181Ta). Such K values are larger than those obtained in
the lighter projectile systems (K x 11—15 MeV/AA). On the other hand, the E*/A
predicted by the dynamical models is lower for 86Kr reaction systems in Figure 5.8
than that of the NSCL reaction systems due to lower beam energy (64 MeV/ u). The
HIPSE simulation of 86Kr+9Be reaction produces very low excitation energy (z 2
MeV) as compared to the AMD result (z 3 MeV). The E*/A calculated using the

HIPSE model in the case of 86Kr+181Ta reaction is higher than the assumptions

143

used in the AA model calculations (especially for the heaviest prefragments). The
saturation excitation energy per nucleon is approximately 6 MeV, which is lower
than in the case of the N SCL reaction systems.

The HIPSE model predicts rather different excitation energy for reactions of all
investigated projectiles with 9Be and 181Ta targets, which is in strict contrast with
the AA model where we do not observe any differences in the excitation energy
necessary to reproduce the ﬁnal cross section distributions. Unfortunately, the AMD
calculations could not be carried out for the reaction systems involving 181Ta target,

because of excessive CPU—time required.

5.4 Evaporation codes

The calculated primary fragment distributions (Section 5.2) cannot be compared
directly to the experimental data, because the experimental observation of fragment
is performed after hundreds of nanoseconds —— orders of magnitude later than the
prompt step, simulated by the nuclear reaction models (geometrical-cut Abrasion,
HIPSE, and AMD). The (hot) primary fragments prepared by these three nuclear
reaction models must be de-excited before comparing to the experimental data.

In our study, two different evaporation codes were used to calculate the ﬁnal states.
The analytic LisFus [106] code was used in conjunction with the AA model, and the
GEMINI [110] code was applied to primary fragments generated by the HIPSE and
AMD calculations.

5.4.1 LisFus evaporation code

LisFus [106] is a fusion-evaporation code developed in the framework of LISE++
[56]. The evaporation stage of the LisFus is treated in a macroscopic way on the
basis of a master equation which leads to diffusion equations as proposed by Campi

and Hiifner [127]. For each i-th point of excitation energy, E,, distribution, P, of

144

parent nucleus, the LisFus model calculates the probabilities, Wk, for 8 possible decay
channels (n, 2n, p, 2p, d, t, 3He, a) and a daughter excitation energy distribution
function, Dk(E.-). With the above mentioned deﬁnitions the i-th segment of the parent

excitation function is expressed as:

[E P(E)dE = Z Wk(E,-)Dk(E,). (5.20)

Ei k=1

Masses of all parent and daughter nuclei are taken from Audi-Wapstra atomic mass
evaluation [124]. Level density, p(E*), is calculated by the phenomenological level

density systematics based on the Fermi-gas model [128]:

p(E*) = 113ﬁa-1/4(E* — A)-5/4 exp [2,/a(E* — m], (5.21)

where a is the level density parameter and E* is the excitation energy of the nucleus
with mass number, A. A is the pairing energy correction expressed as A = x527,
where X = 0, 1, or 2, for odd-odd, odd-even or even-even nuclei, respectively. The

level density parameter, a, is expressed as:

1 _ e—«Eum

a(E*,Z, N) = (aA+ﬁA2/3) - 1+6W(Z, N) E, _ A ,

 

(5.22)

where 6W(Z, N) is a shell correction in the liquid drop model and a, ,6 and '7 are
parameters taken to be 0.111, 0.107, and 0.46, respectively [128].

The analytical approach of this code allows one to calculate the cross sections of
nuclei far from stability. Currently, the LisFus code [106] does not take into account
the contributions of the angular momentum.

To demonstrate the importance of the evaporation code we show two Abrasion-
Ablation calculations in Figure 5.9 for pro jectile-like fragments created in 40Ca+9Be
and 48Ca+9Be reactions. The experimental fragmentation and pick-up cross sections

are shown as open squares and triangles, respectively. The Abrasion-Ablation model

145

 

_A
0.
M

 

Cross section (mb)
3 <3. 9,.

_L
q
—L

    

' .-' A
10-35:-.14-..i.‘:;E.L1..l.l.i*-'::..J....I.:15..1....1.
o 5 o 5 o 5 o 5
Neutron excess N-Z

 

 

 

 

 

   

 

 

 

 

 

 

 

 

 

 

102? " 1 1

I 1. I1

1. 1

A 1 q

.o 4. .

E ‘1- i

v _, .

C . .
o ..
I": ‘ ..
0 ' 1:
m ‘-
m 1?
o '1.“
~10“2 t - "
0 1:3 Al
rs' -
- .- A

105 *- Cl . .
AlllllullLlAllll 1 11‘ 1.
0 5 10 0 5 10 0 5 10

Neutron excess N-Z

Figure 5.9: Measured isotopic cross section (open squares) distributions for 40Carl-989
(top panel) and 48Ca+"Be (bottom panel) for 12 3 Z S 19 elements are compared
to two Abrasion-Ablation calculations with the same excitation energy assumptions.
Solid line shows a calculation by the ABRABLA code [104] and a dotted line displays
LISE++ AA calculation. The average excitation energy is determined by Equation
(5.3) with K = 27 MeV/ u in both calculations.

146

calculation by the ABRABLA [104] code is shown as a solid line. The average excita-
tion energy in this code is calculated with SIOpe parameter of Equation (5.3), K = 27
MeV/AA which reproduces the high energy projectile fragmentation cross sections
very well [129]. This calculation is compared to LISE++ AA code with K = 27
MeV/AA shown as a dotted line in Figure 5.9. We observe a fairly good reproduction
of the experimental cross sections (especially in the peak of isotopic distributions) for
the ABRABLA calculation (solid lines) for both reaction systems in Figure 5.9. The
description of the isotopic distributions in the case of the 40Ca+9Be reaction system
by LISE++ AA calculation with changed excitation energy assumptions (K = 27
MeV/ u) is also reasonable. However, in the case of the 48Ca+gBe reaction system
the LISE++ AA model shows large deﬁciencies for the more neutron-rich fragments
(N — Z > 5) of 12 S Z s 17 elements. The proton—rich (N — Z < 0) fragments, on
the other hand, are described rather well. It must be noted that the Abrasion stage of
the two models is identical and the evaporation code is based on the same principles
(Campi and Hiifner [127]). The discrepancy between the two AA models is clearly
pronounced in the case of neutron-rich projectiles (“Ca and 64Ni), while calculated
isotopic distributions for the projectiles closer to the line of stability (“Ca and 58Ni)
are comparable. Based on our previous discussion we conclude that the differences
in the ﬁnal cross section distributions between the LISE++ AA and HIPSE (AMD)

models are caused by the very different decay codes used.

5.4.2 Statistical evaporation code GEMINI

The GEMINI code calculates the decay of a primary fragment by sequential binary
decays. All possible binary decays from light-particle emission to symmetric division
are considered. Monte Carlo technique is employed to follow all decay chains until
the resulting products are unable to undergo further decay. For the purposes of the
sequential decay calculations the excited primary fragments generated by the HIPSE

and AMD calculations are taken as the compound nucleus ([130] input of the GEM-

147

INI code. Every primary fragment is decayed as a separate event in the GEMINI

calculation.

102.

 

_ul
0.
...A

 

.53

Cross section (mb)

—L
0.
d

 

 

 

 

 

 

 

5 l o
Neutron excess N-Z

Figure 5.10: Inﬂuence of the level density parameterization on the ﬁnal isotope dis—
tributions in 40Ca+9Be reaction. Open squares show the experimental fragmentation
data and open triangles depict the pick-up reaction products. The AMD primary frag—
ment distributions are decayed using GEMINI with level density parameter a = A/ 8
MeV“1 (dashed line) and a = A/ 12 MeV‘1 (solid line).

For the evaporation of particles lighter than a-particle, the Hauser—Feshbach [131]
formalism is applied. For binary divisions corresponding to emission of heavier frag-
ments, the decay width is calculated using the formalism of Morretto [132]. The liquid
drop model with shell corrections [133] was used to calculate the masses of all parent
and daughter nuclei in the calculation. For all level densities, the Fermi gas [134]

expression is assumed in the GEMINI calculation:

 

2 3/2 1.
h. J ﬁexp (Ric—LET), (5.23)

p(E*, J) = (N +1) [5 ‘13 5*

where J, I , and E" are the spin, moment of inertia and excitation energy of the
residual nucleus or saddle—point conﬁguration, respectively.

The ﬁnal isotope distributions are sensitive to the level density parameter, a.

148

 

.3?»

A
o.

d
‘

 

Cross section (mb)
EE.

 

—L
Q
d

 

K

. . (A ..
o 5 1o 0 5 10 o 5 1o
Neutron excess N-Z

 

 

 

 

 

 

Figure 5.11: Inﬂuence of the level density parameterization on the ﬁnal isotope dis-
tributions in 48Ca+9Be reaction. Open squares show the experimental fragmentation
data and open triangles depict the pick-up reaction products. The AMD primary frag-
ment distributions are decayed using GEMINI with level density parameter a = A/ 8
MeV‘1 (dashed line) and a = A/ 12 MeV‘1 (solid line).

To demonstrate this effect, two different primary fragment decay calculations for
400a and 480a beams and 9Be target are shown as solid and dashed lines in Figure
5.10 and 5.11. The only difference between the two calculations is the level density
parameterization. The solid lines are calculated using a = A/ 12 MeV‘1 and the
dashed lines depict calculations using a = A/8 MeV—1. In the case of 40Ca+9Be
reaction we see the two calculations with different level density parameter assumptions
are nearly identical in the peaks of the isotopic distributions shown in Figure 5.10.
Differences can be observed only in the tails of the distributions. On the contrary,
calculations with different level density parameter assumptions for the 48Ca+9Be
system, shown in Figure 5.11, differ much more. We clearly see a shift of the peak
of the isotopic distributions for elements shown in Figure 5.11 towards the more
neutron-rich isotopes in the case of a = A / 12 MeV‘1 as compared to a = A/ 8 MeV“1
calculation. The same conclusions are reached in the case of the 58Ni projectile, where

no shift of the isotopic distributions is observed. For the 64Ni beam, shifts towards

149

the neutron-rich isotopes are clearly visible.
In the following discussions we choose a. = A / 10 MeV‘1 for all GEMINI sequential
decay calculations as a reasonable compromise between the two investigated cases

(a = A/8 and A/12 MeV—1).

5.5 Cross section distributions

The fragmentation production cross section is by far the most important experimen-
tal observable in the present experiments. In this section we are going to compare
the experimentally determined reaction cross sections with the ﬁnal fragment cross
sections calculated by the AA, HIPSE and AMD reaction models in conjunction with
the LisFus and GEMINI evaporation codes.

Figures 5.12—«5.16 present the comparisons of the measured isotope cross section
distributions with 40"180a, 58MM and 86Kr primary beams and the 9Be target. Exper-
imental fragmentation cross sections and pick—up (nucleon exchange) cross sections
are shown as open squares and triangles, respectively. The calculations are shown
with dotted, solid and dashed lines for AA, HIPSE, and AMD models respectively in
Figure 5.12—5.16.

The Abrasion-Ablation model in LISE++ reproduces the experimental fragmen-
tation cross sections extremely well, which is not surprising for the projectile-like
fragments, because the excitation energy dependence on the number of abraded nu-
cleons and its fluctuations have been adjusted for each reaction system individually.
The cross sections calculated by the AA model for fragments lighter than half cf the
mass of the projectile describe the experimental data very well too. This feature is
not expected because the AA model based on the geometrical cut of the projectile by
the target nucleus, has clear deﬁciencies when describing more central collisions, and
the excitation energy used for the intermediate mass prefragments (z A p / 2) is rather

large (> 10 MeV/ u). The simple geometrical AA model is very good at reproducing

150

the tails of the isotope distributions —— this is especially clear in the case of 480a and
64N i projectiles, where the experimental data extend to very neutron-rich isotOpes.

The HIPSE and AMD are stochastic calculations, so the ﬁnal cross section dy-
namical range depends on the number of simulated events, which was approximately
100,000 and 20,000 for the HIPSE and AMD models, respectively. Hence we are re—
stricted to compare the ﬁnal cross sections only in the peak region of the isotope
distribution. For reference the total number of events detected for 64Ni+QBe reaction
system was of the order of 107!

The isotope cross section distributions calculated using the HIPSE model are
generally wider than the experimental ones for all investigated projectiles on 9Be
target (Figure 5.12—5.16). This effect is especially pronounced in the case of 86Kr
primary beam (Figure 5.16), when the excitation energy is clearly not sufficient to
decay the residues on the neutron-rich side of the isotope distributions, resulting in
cross section deﬁcit in the peak regions (Ga—Se). Overall the peaks of the isotope
distributions are described well for reactions of 40"“3Ca, 58"MM and 9Be by the HIPSE
calculations. The overestimation of the experimental cross section in the tails of the
isotope distributions may be caused by large ﬂuctuations of the excitation energy in
the HIPSE model. Unfortunately, the calculated cross sections of proton or neutron-
rich fragments are very important when considering the predictive power of a model.

The experimental fragmentation cross sections of 40Ca, 58Ni, and 86Kr primary
beams (Figure 5.12, 5.14, and 5.16) are described rather well in the AMD model. The
AMD model has larger deﬁciencies when describing the cross section distributions
of fragments created in the reactions of more neutron-rich projectiles 480a and 64N i
plotted in Figure 5.13 and 5.15. These problems may be caused by problems of AMD
describing the ground states of these projectiles [115]. The successful application of
the AMD for the 400a and 58N i projectiles is important, because the calculations were
done without adjustments of the parameters. This ﬁnding is encouraging for the de-

velopment of the AMD model to describe projectile fragmentation phenomena. AMD

151

has been known for its successful applications in more central, multifragmentation
reactions [117].

Measured fragment isotope cross section distributions with 40ABCa, 58’“Ni and
86Kr primary beams and 181Ta target are presented as open squares in Figure 5.17—
5.21, along with the nucleon pick-up and exchange cross sections depicted by open
triangles. The AA and HIPSE calculations are shown as dotted and solid lines, re-
spectively.

The Abrasion-Ablation calculation reproduces the experimental fragmentation
cross sections extremely well, as a consequence of the adjustment of the mean ex-
citation energy and its ﬂuctuations. We notice a surprisingly good description of
the isotope distributions even for the lightest elements (8—0) for 40"‘8Ca and 58Ni
projectiles. It should be noted that the light elements (B—Ne) for the “Ni primary
beam (Figure 5.20) are described very poorly, as expected in a simple geometrical
AA model.

The isotope cross section distributions calculated by the HIPSE model are overall
broader than the experimental ones for both reaction systems involving 9Be and 181Ta
target. The experimental cross sections for the lightest elements (8—0) are reproduced
well for all projectile measured at the NSCL (Figure 5.17—5.20). The intermediate
mass fragment (3 Ap/2) cross sections in HIPSE, on the other hand, show rather
large deﬁcits (almost a factor of 10) with respect to the experimental data. This might
be a direct consequence of the large excitation energy produced by the HIPSE model
for the intermediate mass residues (as shown in Figure 5.7). The overall description of
the isotope distributions gradually improves for higher Z isotopes. The projectile—like
fragment cross sections are reproduced rather well by HIPSE, especially in the case
of the 48Ca and “Ni projectiles (Figure 5.18 and 5.20). Cross sections of fragments
near the peak of the isotope distributions for all measured elements (Mn—Kr) in the
86Kr+181Ta reaction are reproduced very well using the HIPSE model calculations

(see Figure 5.21). However, the experimental distributions measured in the case of

152

 

 

  
   

—L-‘
0?;

—L

 

Cross section (mb)

10‘2

 

 

 

 

 

 

A
-1 1 41

l J 4 A .l 1 L I. A A I I I I I I I I l l I

0....5..0...5 o 5 o 5
Neutron excess N-Z

 

Figure 5.12: Fragmentation (open squares) and nucleon pick-up (Open triangles) cross
section isotopic distributions of 40Ca+QBe reaction are compared to the calculations
by HIPSE (solid line), AMD (dashed line) and Abrasion-Ablation (dotted line) mod-
els.

86Kr+181Ta system contain only approximately 6 isotopes per element versus 12—16
for the 86Kr+9Be reaction. The dynamic range in cross section is only a factor of 10
versus many orders of magnitude (3—9) in the case of the 86Kr+gBe reaction. It is
therefore impossible to draw conclusions about the tails of the isotope distributions.
Nevertheless, the HIPSE calculation describes the cross sections of fragments in the
peak of isotope distributions much better for the 86Kr+131Ta than the 86Kr+9Be

reaction.

153

 

 

10‘2 .

 

 

 

 

 

 

 

 

 

 

3
E
c 4’.
.910
1‘6 .
Q) ~‘
(I) 10:
(D __
(D 1)
Q :-

10“?

102?

.2 5
‘° . 3.9
@......n....n...’£.l....ll...|..A‘:.i....1....1...’.§l....1....i..c
0 5 10 0 5 10 0 5 10 5 10

Neutron excess N-Z

Figure 5.13: Fragmentation (open squares) and nucleon pick-up (open triangles) cross
section isotopic distributions of 4(’3Ca-i-9Be reaction are compared to the calculations
by HIPSE (solid line), AMD (dashed line) and Abrasion-Ablation (dotted line) mod-
els.

154

 

102.

 

 

     

 

p
1b . 1b 1
- l I l .2 l l .‘ A l .. l . A l 4__ Al,
I I I' p ' I I ' ' I I ‘ I I q
’ 1

b
F 1

Cross section (mb)

c
c
o"
.0
o‘ 1
.0
c
A A A A A A A A
'U 'I'"‘." I I 'I '5 'l 'I V U ' I
I " 1 '-
o...
o 0 .......
Cl...
0 5.,
1:; o .
o o
u 1!
’ 0 '
1v . 1p ..
U
1 .0 ....
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0 5 1 0 0 5 1 0 0 5 1 0
Neutron excess N-Z
Figure 5.14: Fragmentation (open squares) and nucleon pick-up (open triangles) cross

section isotopic distributions of 58Ni+9Be reaction are compared to the calculations by
HIPSE (solid line), AMD (dashed line) and Abrasion-Ablation (dotted line) models.

 

 

 

 

 

155

Cross sectio
0.

 

 

 

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Neutron excess N-Z

Figure 5.15: Fragmentation (open squares) and nucleon pick—up (open triangles) cross
section isotOpic distributions of “ Ni+9Be reaction are compared to the calculations by
HIPSE (solid line), AMD (dashed line) and Abrasion—Ablation (dotted line) models.

156

 

 

 

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Neutron excess N-Z

5

10 15 20

Figure 5.16: Fragmentation (open squares) cross section isotopic distributions of
86Kr+QBe reaction are compared to the calculations by HIPSE (solid line), AMD
(dashed line) and Abrasion-Ablation (dotted line) models.

157

 

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Neutron excess N-Z

Figure 5.17: Hagmentation (open squares) and nucleon pick-up (open triangles) cross

sections of 40Ca-l—lmTa are compared to the calculations by HIPSE (solid line) and
Abrasion-Ablation (dotted line) models.

158

 

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Neutron excess N-Z

Figure 5.18: Fragmentation (open squares) and nucleon pick-up (open triangles) cross
section isotopic distributions of 48Ca+181Ta reaction are compared to the calculations
by HIPSE (solid line) and Abrasion-Ablation (dotted line) models.

159

 

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Neutron excess N-Z

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5

10

Figure 5.19: Fragmentation (open squares) and nucleon pick-up (open triangles) cross
section isotopic distributions of 58Ni+181Ta reaction are compared to the calculations
by HIPSE (solid line) and Abrasion-Ablation (dotted line) models.

160

 

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Neutron excess N-Z

Figure 5.20: Fragmentation (open squares) cross section isotopic distributions of

64Ni+181Ta reaction are compared to the calculations by HIPSE (solid line) and
Abrasion-Ablation (dotted line) models.

161

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Neutron excess N-Z

l l

 

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Figure 5.21: Fragmentation (open squares) cross section isotopic distributions of .
8"sKr-t-lslTa reaction are compared to the calculations by HIPSE (solid line) and
Abrasion-Ablation (dotted line) models.

162

It must be noted that the overall shape and the peak position of the isotope dis-
tributions are not only inﬂuenced by the simulation of the fast part of the reaction ——
done by the geometrical cut of projectile by the target nucleus in AA, or dynamical
evolution in HIPSE and AMD models, but also by the statistical cascade decay code,
LisFus in the case of AA and GEMINI for HIPSE and AMD calculations. The results
of the HIPSE and AMD calculations can be compared directly, because identical se-
quential decay code (GEMINI) has been used. The fragment cross sections calculated

by the AA model, on the other hand, used a different decay code (LisFus).

163

5.6 Velocity distributions

The fragment velocity is one of the experimental observables sensitive to the dynamics
of the nuclear collision. It allows one to disentangle (to a certain degree) the fast
reaction dynamics from the long in-ﬂight decay of the excited primary fragments.
The ﬁnal fragment velocity distributions contain a signiﬁcant imprint of the nuclear
collision as described by a particular nuclear reaction model.

The Abrasion—Ablation model used in our calculations does not treat the dyna-
mical evolution of the abrasion stage, it produces only one relevant observable — the
fragmentation production cross section. More sophisticated models like the HIPSE
and AMD, on the other hand, describe the dynamics of the nuclear collision, and can
predict more observables (e. g., excitation energy, momentum), to be compared to
experimental data.

The experimental fragment velocity is deﬁned as the velocity of the weighted
mean of the momentum distribution for every fragment measured with complete
momentum distribution. The fragment velocity has been extracted from the HIPSE
and AMD calculations by taking the mean of the calculated velocity distribution for
every fragment after decay using GEMINI with more than 100 events simulated.

The ﬁnal fragment velocity deviations with respect to the projectile, in percent,
are presented in Figure 5.22 and 5.24 for 40*‘18Ca, 58MM and 86Kr primary beams
on 9Be target, respectively. The experimental fragment velocities are shown as open
circles, the HIPSE and AMD calculation results are displayed as ﬁlled triangles and
squares, respectively. The fragment velocity results from the fully microscopic BUU
[126] calculation are also available for 9Be target for all NSCL experiments (Figure
5.22) and are shown as solid lines.

The HIPSE calculation reproduces the velocities for all 40ABGa and 58MM primary
beams and 9Be target, as can be seen in Figure 5.22. It must be noted that the

experimental velocity deviations are slightly lower for lighter fragments for 48Ca,

164

 

 

        

 

 

 

 

 

 

l I T I I I I I I I I I I I I I W r Iﬁ
. “Ca+9
O
_ O ‘
A
O 0 0A ‘A
u- . .
O A A
_ QR ‘ ' .
t . "-
‘ .
I l l l l I l l 1.1 I l l l 1 I I l l I‘l I l l l I I I i l I l l l l
10 20 so 40 1o 20 so 40 so
Fragment A Fragment A

 

 

I I I I I l I I r I I I I I I I I I I I l I I I I I I I I l I I I I l I I

 

 

 

 

 

 

 

J 1 l l 44 1 .L *I l l 1 L I l L 1 1 1 l l l I 1 ‘4 I l l l l I 1 1
20 so 40 so so 40 so so
Fragment A Fragment A

Figure 5.22: Deviation from the projectile velocity (vp/vp— 1) in percent for fragments
in the 40’480a and 58""“Ni projectile on 9Be target collisions shown as open circles.
Filled triangles and squares depict the HIPSE and AMD calculations, respectively.
BUU calculation is also shown as a solid line.

165

 

vF/vP-1 (%)

 

 

 

‘ -4
I I l l I I l l l l 4 l l l 1 l J l l I I
1 0 20 40 50

 

so
Fragment A

Figure 5.23: Deviation from the projectile velocity (Up/vp— 1) in percent for fragments

in the “Ca projectile and 181Ta target collisions shown as Open circles. Filled triangles
depict the HIPSE calculation.

 

 

 

 

 

 

 

 

 

 

I l l l I I I I I T V I I I I I I I l l I I I I I I
0.._._____._._ _.___ o._..____..__ _____ ._—
: 86Kr : 86Kr+“’*ra at???
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A -
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b I l l l l I l l 1 1 l 1 .4 Pl I | I | |M Iél l l I l l l l I 1 1 1 T
50 so 70 so so so 70 so
Fragment A Fragment A

Figure 5.24: Deviation from the projectile velocity (Up/vp— 1) in percent for fragments
in the 86Kr. projectile and 9Be, 181Ta target collisions shown as open circles. Filled

triangles depict the HIPSE calculation and ﬁlled squares show the AMD simulation
results.

166

58Ni, and 64N i beams (Figure 5.22). It must be emphasized that the free parameters
(see Section 5.1.2) of the HIPSE model have not been adjusted to reproduce this
observable. The velocities predicted by the AMD model are lower and have smaller
spread, probably due to insufﬁcient fluctuations in the AMD theory. The BUU model
velocity predictions are located between the HIPSE and AMD calculations. The BUU
is a deterministic model, such that it predicts average values of observables for a
given impact parameter, hence one must be careful when comparing the velocity as
a function of the fragment mass.

In the case of the 86Kr+9Be reaction, the ﬁnal fragment velocity deviations with
respect to the projectiles shown in Figure 5.24, we see a rather good reproduction of
the fragment velocities by the HIPSE model and very steep decline of the velocities
calculated by the AMD. The fragment velocity comparison prefers the HIPSE model,
which in turn results in rather low excitation energies. These calculated excitation
energies are not sufficient to decay the neutron-rich primary fragments resulting in
unsatisfactory fragmentation cross section values.

For the reactions involving the 181Ta target material, there are no AMD and BUU
calculations available, hence Figure 5.23 presents the experimental velocities for the
48Ca+181Ta reaction system compared to the HIPSE model only. We can see the
HIPSE model is no longer able to describe the fragment velocities as in the case
of 98e target material presented in Figure 5.22. This is another indication that the
excitation energy for 181Ta systems is overestimated (see Section 5.3).

For the nuclear collisions involving the target nuclei of 181Ta, the HIPSE calculated
fragment velocities show a clear deviation from the experimental data. The origin of
the deﬁciencies for all reactions involving 181Ta target is not fully understood. A
series of benchmark calculations were carried out to understand the underestimation
of the fragment velocities. Alterations of the three adjustable parameters do, Jim
and 2:60" did not show signiﬁcant changes in the velocity distributions. The excitation

energy, on the other hand, varied signiﬁcantly as expected when changing the nucleon

167

 

 

  
 
   

 

 

 

 

 

 

 

1G. I I I I I r I T r y d
I 48Ca+mTa :
' 'HIPSE :
9 ~ .
‘D 6"“ " “W‘IHIPSE1 -1
E : :
<2 “ -
F 4‘ ~~
2:“ "i

o'- llllllllllj.‘ _IlfllIA‘IleﬁgllllJl

10 20 3O 4O 50
30 4O 50
Residue mass A Fragment A

Figure 5.25: Results of the HIPSE calculations in terms of the excitation energy per
nucleon, E*/A, (left) and the fragment velocity (right) for the 48Ca+181Ta reaction
system. The calculation with changed distribution of the excitation energy among
the ﬁnal fragments is labeled as HIPSEl. The deviations from the projectile velocity
(vp/vp — 1) for fragments in the 86Kr-i-mlTa collisions, shown as open circles, are
compared to the HIPSE (ﬁlled triangles) and the HIPSEl (green triangles).

exchange parameter, :13".

For the calculations using the 9Be target the maximum number of nucleons able
to collide and exchange with the quasi-projectile is 9. The 181Ta nucleus offers much
larger pool of available nucleons for the interactions with the quasi-projectile. The
rather small number of nucleons of the 9Be target nucleus is reﬂected in the saturation
of the mean excitation energy per nucleon as shown in Figure 5.6, while in the case of
181Ta target the mean excitation energy saturates at much larger values (Figure 5.7),
when using the same set of HIPSE parameters. By comparing two proﬁles of the mean
excitation energy for the 9Be and 181Ta target, respectively, we see a very different
evolution as a function of the fragment mass number (Figure 5.6 versus 5.7).

Beside the change of input parameters, several tests have been done, by direct
modiﬁcations of the HIPSE code to improve the description of the fragment velocity.
One of the ways to improve the description of the velocities is to increase the exci-

tation energy, E*, of the projectile-like primary fragments for our reaction systems

168

with 181Ta target. In order to achieve this change, the algorithm of total excitation
energy distribution among the prefragments was modiﬁed. In the new algorithm the
total excitation energy, obtained from the energy balance of Equation (5.10), is equally
distributed among all the residues. This new approach assigns relatively larger excita-
tion energy (per nucleon) to lighter projectile-like fragments as compared to (much)
heavier target-like fragments. The results of this changed energy distribution algo-
rithm (labeled HIPSEl) are shown in Figure 5.25 for the 48Ca+181Ta reaction. The
left panel of Figure 5.25 displays the excitation energy per nucleon as a function of
fragment mass number obtained by the modiﬁed calculation (green) compared to the
previously shown result (black). The right panel of Figure 5.25 shows the results of
the HIPSEl with changed excitation energy (green ﬁlled triangles) compared to the
experimental data (open circles) and the previously shown HIPSE calculation (black
ﬁlled triangles). It is clear that the excitation energy per nucleon for the heaviest
projectile-like primary fragments has increased. But in terms of fragment velocities,
the change of the calculation of the excitation energy is not sufﬁcient to bring a signif-
icant increase of the calculated velocities. In the extreme case, when all the excitation
energy is deposited in the quasi-projectile, the fragment velocities get closer to the
experimental data. However, even this increase of the excitation energy is not suf—
ﬁcient to reproduce the experimental fragment velocities. Not to mention that such
unequal sharing of the E'* among the primary fragments is very hard to justify.
However, these tests illustrate that the HIPSE calculation may become consistent
with the experimental data if larger amount of the excitation energy is deposited
in the quasi-projectile compared to the one originally produced by the model. The
experimental fragment velocity distributions prefer the overall shape of the mean
excitation energy per nucleon as a function of the fragment mass to be similar to that
of the reactions with 9Be target (Figure 5.6). Such an increase of the E* can only be
obtained by additional dissipative processes for very peripheral collisions with heavy

targets, which does not seem to be treated properly in the current version of the

169

HIPSE model.

170

Chapter 6

Summary and conclusions

An extensive study of projectile fragmentation reactions is presented in this disser-
tation. A series of systematic measurements have been undertaken to extend the
available data sets for projectile fragmentation at intermediate energies. The experi-
mental work has been done at two laboratories. Experiments with primary beams of
40Ca, “Ca, 58Ni, and 64Ni at 140 MeV/u were done at the NSCL using the A1900
fragment separator. The experiment using 86Kr primary beam at 64 MeV/u was done
at RIKEN using the RIPS fragment separator. Two targets 9Be and 181Ta, were used
in experiments with all primary beams in order to investigate the target dependence
of the fragmentation production cross sections. The experiments with a total of 10
reaction systems resulted in 1740 measured cross sections and momentum distribu-
tions.

The fragment momentum distributions were asymmetric unlike those measured
in projectile fragmentation reactions at high energy (>> 200 MeV/u). The ﬁtting of
the experimental parallel momentum distributions was done using a modiﬁed Gaus-
sian function consisting of two half Gaussians in Equation (3.2). We found that the
right side of the momentum distribution described by the Width parameter, 03 (high
momentum), is reproduced well by the Goldhaber systematics derived for high en-

ergy fragmentation phenomena. The reduced width parameters, of, of the Goldhaber

171

model are lower than those calculated from the measured Fermi momenta for all N SCL
reaction systems. This deﬁcit is consistent with other intermediate and high energy
fragmentation experiments. of deduced from the experimental momentum distribu-
tions of 86Kr fragments are larger than the values obtained for the NSCL reaction
systems. This difference may be caused by the lower energy of the projectile (64
MeV/u).

The left side of the experimental momentum distributions, described by the width
parameter 0;, (low momentum), also follows the behavior of the simple Goldhaber
model, but requires higher values for the reduced width, 05, for all the N SCL reaction
systems. The 86Kr fragmentation experiment at 64 MeV/ u results in even larger 0L
values and we noticed that the experimental data of 01, began to deviate from the
Goldhaber description. This was especially pronounced in 86Kr+9Be reaction. The
data for the 86Kr+181Ta reaction are not so conclusive because of the small number
of fragments with completely measured momentum distributions. The left side (low
momentum) of the momentum distribution likely includes contributions from different
reaction mechanisms (more important at lower energy like 64 MeV/u) which are
clearly outside of the Goldhaber model assumptions.

The fragment velocities, deﬁned by the maximum of the momentum distribution,
p0 parameter of the ﬁtting function of Equation (3.2), were compared to momentum
transfer systematics and parameterizations of Borrel, Kaufman and Morrissey. These
parameterizations have their foundation in target spallation. All the above parame-
terizations provide similar predictions in terms of fragment velocity for the heaviest
projectile-like fragments, close to the experimental data. They all fail to describe
the overall shape of the experimental velocity distributions for all investigated sys-
tems. This fact is not surprising as all of the above mentioned parameterizations were
derived and ﬁtted only for the projectile or target-like fragments.

The experimental cross sections have been obtained by integration of the parallel

momentum distribution ﬁtting function of Equation (3.2) over the whole interval of

172

parallel momentum. All extracted fragmentation cross sections were compared to the
empirical code EPAX [48]. This parameterization is based on the limiting fragmenta-
tion assumptions (Section 4.3.1), when the fragmentation production cross sections
do not depend on primary beam energy and very weakly on the target material.
The observed cross sections for the projectile—like fragments are in agreement with
the EPAX formula for all NSCL reaction systems, however, some discrepancies re-
main. A clear shift of the peak of the isotope distributions between the EPAX values
and the experimental data was observed for the fragments of neutron-rich projectiles
(48Ca, “Ni and 86Kr) for elements in the vicinity of the projectile. This effect can
be attributed to the memory parameter in the EPAX. It is much more pronounced
for the fragmentation of 86Kr primary beam, which might indicate a stronger depen—
dence of the memory parameter on the incident energy. The enhanced production of
neutron-deﬁcient fragments for elements close to the neutron-rich projectiles is prob-
ably caused by transfer-type reactions, which play a more dominant role at lower
beam energies [97]. The experimental isotope distributions tend to be narrower than
the EPAX calculated distributions. This discrepancy is clearly pronounced for the
intermediate mass fragments (z Ap / 2) in the case of the 40Ca beam, and a smaller
effect is still visible for the rest of the reaction systems.

The target dependence of fragmentation cross section has been investigated by
calculating the target cross section ratios, Rtgt. It is shown that the Rtgt dependence on
the fragment mass number is much more complicated than suggested by the limiting
fragmentation framework (EPAX). A clear enhancement of cross sections for neutron-
rich as well as neutron-deﬁcient fragments was observed for the 40Ca projectile when
using a 181Ta versus a 9Be target. However, the effect is small, around a factor of 3—4.
Other investigated projectiles did not show the cross section enhancement for very
neutron-rich fragments. An overall cross section enhancement trend was noticed for
lighter fragments in the case of all investigated projectiles when using 181Ta target.

Unfortunately, the observed enhancement of fragmentation production cross section

173

does not translate into yields (Section 4.3.4).

Projectile dependence of the fragmentation cross section was addressed by calcu-
lating the projectile cross section ratios, Rpmj, for the Ca and Ni projectiles in our
study. These ratios for isotopes of the lightest elements are equivalent to isoscaling
ratios studied in central multifragmentation reactions. For isotopes of heavier ele-
ments the ratios for isotopes of one element formed a slightly curved line. A modiﬁed
isoscaling-like function with second order terms has been used to reproduce the trends
for individual elements. Study of Bp,-03' allows one to see the difference between two
dominating reaction mechanisms — projectile fragmentation (heaviest fragments with
curved Rpm, trends) and multifragmentation (lightest fragments with straight Rpm,-
trends).

In order to get a better understanding of the physical processes involved in the
studied reactions, calculations using different theoretical models were carried out.
The geometrical Abrasion-Ablation, the macroscopic-microscopic Heavy Ion Phase
Space Exploration (HIPSE) and the microscopic Antisymmetrized Molecular Dynam-
ics (AMD) models have been used in our studies. The above mentioned calculations
were done for all investigated reaction systems, but the AMD calculations were car-
ried out only for the ﬁve reaction systems using 9Be target, because of unrealistically
long execution time for systems involving 181Ta.

A revised version of a simple geometrical Abrasion-Ablation model in conjunction
with the LisFus evaporation code [106] as implemented in LISE++ [56] has been
used. After adjusting the excitation energy as a function of the number of removed
nucleons and its ﬂuctuations for each system independently we get an extremely
good description of the experimental fragmentation cross sections. The excitation
energies required to reproduce the cross section distributions for all NSCL reaction
systems are x 11—13 MeV per removed nucleon, indicating a very slow rising trend
with the mass of the projectile in the studied beam mass range of 40 < A < 64.

For the 86Kr primary beam a little higher excitation energy of 18 MeV per removed

174

nucleon is required. There is no target dependence of the excitation energy observed
for all the investigated reaction systems. Small differences in terms of the excitation
energy for individual projectiles are well within the errors of its determination. Even
though the simple linear dependence of the excitation energy produces extremely
large excitation energy for lighter primary fragments, which is unphysical, we observe
very good reproduction of the isotope distributions for the lightest elements (except
for “Ni+181Ta reaction). Since the AA model assumes that the fragment velocity is
equivalent to that of the projectile and it requires adjustment of the excitation energy
with experimental data, its predictive power is severely limited. More sophisticated
nuclear reaction models are needed to get a clearer picture of the excitation energy
and the fragment velocity.

The HIPSE and AMD models have been used to get more insight into the reaction
mechanisms involved in the investigated nuclear collisions by treating the collisions
dynamically. Aside from the fragment cross sections we compared the primary frag-
ment excitation energies and the ﬁnal fragment velocities. For all reaction systems
involving the 9Be target material, we noticed the excitation energy for the primary
fragments close to the projectile determined by the dynamical models (HIPSE and
AMD) was higher than the one required by the AA model. For lighter prefragments,
however, the dynamical models saturate around 4—5 MeV/ u while the AA model uses
large values of the excitation energy (>10 MeV/ u). For all reaction systems involving
the 181Ta target, the excitation energy obtained from the HIPSE model was qualita-
tively and quantitatively equal to the one used in the AA calculation (Figure 5.7).
The excitation energy in the HIPSE model, however, saturates at approximately 10
MeV/u for all primary beams measured at the NSCL and at approximately 6 MeV/u
for the 86Kr+181Ta reaction.

The excitation energy of the primary fragments obtained from the HIPSE and
AMD models for all reaction systems involving 9Be target showed rather similar

proﬁles close to the fully microscopic BUU calculations. Calculations by the HIPSE

175

model for all reaction systems containing 181Ta target, on the other hand, produced
very low excitation energies for primary fragments close to the projectile and very
large values for the lighter fragments (Figure 5.7).

The comparisons of the relative fragment velocity to the calculated values strictly
preferred the HIPSE model for all reaction systems containing 9Be target. The AMD
fragment velocities plotted as a function of the fragment mass number are lower and
the overall trend is much steeper than the experimental data exhibit. The HIPSE
model not only reproduced the overall trend of the fragment velocity as a function
of the fragment mass, but also more or less successfully described the spread of the
experimental values. On the contrary, for the reaction systems containing 181Ta target
the HIPSE model was not able to reproduce the fragment velocities. Our study with
the HIPSE model showed that the fragment velocity description was improved slightly
when we increased the excitation energy for the heaviest primary fragments. Most
probably the description of peripheral nuclear collisions with heavy targets like 181Ta
at 140 MeV/ u is beyond the scope of the current HIPSE model.

The failure to describe the fragment velocities and probably the excitation ener-
gies by the HIPSE, resulted in a rather poor description of the fragmentation cross
section for reaction systems that include a 181Ta target. The experimental isotope
cross section distributions involving a 9Be target, on the other hand, were described
fairly well by the results of both dynamical calculations (HIPSE and AMD). It should
be noted that calculated cross section isotOpe distributions in the HIPSE model were
slightly broader than the experimental ones. It would suggest that the excitation en—
ergy was not large enough to decay the neutron and proton-rich primary fragments.
This conclusion is consistent with the AMD calculated isotope distributions which
appeared to be narrower, keeping in mind that we used the same statistical cascade
evaporation code GEMINI for the two calculations.

It must be noted that the calculated ﬁnal cross section distributions are the results,

not only of the primary (fast) step of the nuclear reaction (modeled by HIPSE or

176

AMD), but also of the secondary (slow) step modeled by the statistical evaporation
code (GEMINI). Hence, if we want to put the dynamical nuclear collision calculations
to a more stringent test, it is imperative to fully understand the de—excitation part of

the nuclear collision — the evaporation process.

177

 

Appendix A

Fitting results

The ﬁtting results of the experimental momentum distributions of the reaction prod-
ucts identiﬁed in projectile fragmentation reactions using all primary beam and target
combinations are presented in this appendix. Values of po, 0;, and 03 parameters of
the ﬁtting function of Equation (3.2), plus the value of the ﬁnal production cross
section, 0pm,, are given for all measured nuclides in reactions of 40Ca, “Ca, 58Ni, “Ni
and 86Kr beams with 9Be and 181Ta targets. Values of pg, 0;, and an quoted without
errors were extrapolated according to the procedure described in Section 3.4.3. All
entries in italics are products of nucleon exchange or pick-up reactions.

A.1 Results for 400a projectile

Table A.1: Fitting results for the reaction system

40Ca+9Be.

 

[ A I Z [ p0 (MeV/c) [ 0L (MeV/c) I03 (MeV/c) I

Uprod (mb)

l

 

 

 

 

 

 

 

 

 

 

10 5 5232 371 247 7.52E+00 :t 6.43E—01
11 5 5719 :l: 2 462 :l: 4 244 :l: 1 1.24E+01 :l: 8.19E—01
12 5 6148 476 275 9.03E—01 d: 1.54E—01
11 6 5692 363 258 5.10E+00 ﬂ: 8.27E—01
12 6 6228 i 2 436 :t 13 250 :l: 1 2.39E+01 i 1.61E+00
13 6 6728 :l: 2 454 :l: 3 256 :t 1 1.20E+01 :t 7.51E—01
14 6 7118 d: 6 405 :l: 9 353 :l: 5 2.66E+00 :l: 1.66E—01
15 6 7662 371 311 1.23E-01 :t 1.73E—02
l3 7 6740 423 346 1.43E+00 i 2.29E-01
l4 7 7234 j: 10 441 :l: 17 280 i 4 1.21E+01 :t 8.00E—01
15 7 7692 :l: 2 385 i 1 274 :l: 1 1.74E+01 :l: 1.02E+00
16 7 8123 :l: 4 376 :l: 2 _ 362 :l: 9 1.14E+00 :l: 1.12E—01
l7 7 8632 436 365 2.70E—01 :t 2.93E-02
15 8 7757 371 301 3.70E+00 :l: 9.27E—01
16 8 8248 :l: 8 436 :l: 13 283 :l: 3 2.43E+01 d: 1.47E+00
17 8 8717 :t 6 407 :t 4 295 :l: 2 9.23E+00 :l: 5.16E—01

 

 

178

continued on next page

 

Table A.1: continued from previous page

 

A I Z I p0 (MeV/c) I 0L (MeV/c) I an (MeV/c) I

”prod (mb)

l

 

 

 

 

 

 

 

 

 

 

 

 

18 8 9152 :l: 8 395 :l: 4 351 :l: 3 2.96E+00 :l: 1.60E—01
19 8 9674 i 9 471 j: 14 369 :l: 14 3.40E—01 :l: 1.94E—02
20 8 10340 384 273 5.50E—02 :l: 9.36E—03
17 9 8812 513 296 1.57E+00 :l: 2.59E—01
18 9 9264 :l: 11 452 :l: 14 311 :t 5 7.34E+00 :l: 4.23E-01
19 9 9721 :l: 6 411 :l: 3 313 :l: 2 8.98E+00 :l: 4.66E—01
20 9 10254 :l: 6 512 :l: 4 323 :l: 2 3.39E+00 :l: 1.69E—01
21 9 10700 :1: 5 467 :l: 9 369 :l: 4 7.57E—01 :l: 3.71E—02
22 9 11185 434 368 6.30E—02 :l: 7.51E—03
23 9 11875 513 318 8.06E—03 :t 3.42E—03
18 10 9300 449 313 1.25E—01 :l: 3.95E—02
19 10 9833 518 305 1.90E+00 :l: 2.74E—01
20 10 10272 :i: 9 421 i 10 315 :l: 4 1.49E+01 :l: 7.74E—01
21 10 10746 :t 5 411 :l: 3 320 :l: 2 1.64E+01 i 7.79E—01
22 10 11287 :l: 5 474 :l: 3 318 :l: 2 8.05E+00 :l: 3.65E—01
23 10 11727 :t 10 449 :l: 10 358 :l: 5 7.82E—01 i 3.56E—02
24 10 12267 485 362 1.12E—01 :l: 1.42E—02
20 11 10485 406 289 9.44E—02 :l: 1.75E—02
21 11 10840 518 305 1.67E+00 :l: 2.23E—01
22 11 11287 :1: 10 424 :l: 9 331 :l: 4 1.20E+01 :l: 5.73E—01
23 11 11802 :1: 5 433 i 3 313 :l: 2 2.17E+01 :l: 9.37E—01
24 11 12310 i 5 466 :l: 3 318 :l: 2 6.62E+00 d: 2.72E—01
25 11 12807 :1: 7 466 :l: 7 333 :l: 3 1.55E+00 :l: 6.19E—02
26 11 13304 :t 7 465 :l: 10 353 :l: 7 1.36E-01 :l: 5.50E—03
27 11 13840 458 256 1.22E—02 d: 1.96E-03
22 12 11391 449 326 1.76E—01 :l: 5.21E-02
23 12 11878 514 314 3.17E+00 2t 3.81E-01
24 12 12346 :l: 8 424 :l: 7 314 :l: 3 2.75E+01 :l: 1.17E+00
25 12 12863 :1: 5 438 i 3 304 i 2 2.89E+01 :l: 1.12E+00
26 12 13385 :t 5 456 :l: 3 302 :l: 2 1.53E+01 :l: 5.56E—01
27 12 13873 :l: 7 456 i 7 315 :l: 2 1.73E+00 :l: 6.21E—02
28 12 14370 :l: 11 451 :l: 11 328 :l: 5 2.25E—01 :l: 8.17E—03
29 12 14880 427 325 9.28E-03 :t 9.58E—04
30 12 15439 449 278 9.42E-04 :l: 2.71E—04
24 13 12381 442 362 1.43E—01 :l: 2.86E—02
25 13 12914 426 304 1.80E+00 :l: 1.52E—01
26 13 13404 :t 9 427 :l: 6 304 :l: 4 1.71E+01 :l: 6.52E—01
27 13 13936 d: 5 438 :l: 2 290 :l: 2 3.85E+01 :l: 1.31E+00
28 13 14430 :l: 5 439 :l: 3 294 :l: 2 1.30E+01 :l: 4.14E—01
29 13 14949 :t 6 440 :t 6 293 j: 2 2.90E+00 :t 8.99E-02
30 13 15472 :1: 9 441 :l: 10 293 :t 3 2.25E—01 :t 7.22E—03
31 13 15938 :l: 20 424 :l: 19 312 :l: 15 2.31E—02 :l: 1.01E—03

 

 

179

continued on next page

Table A.1: continued from previous page

 

LA I Z I p0 (MeV/c) I 0L (MeV/c) I O'R (MeV/c) I

O'pmd (mb)

 

 

 

 

 

 

 

 

 

 

 

 

32 13 16405 415 296 8.50E—O4 1 1.60E—04
26 14 13432 431 306 2.11E—01 1 4.53E—02
27 14 13976 442 294 3.71E+00 1 1913—01
23 14 14434 1 7 403 1 4 231 1 3 4.29E+01 1 1.41E+00
29 14 15031 1 5 423 1 2 267 1 2 4.88E+01 1 1.44E+00
30 14 15554 1 4 424 1 3 253 1 1 3.16E+01 1 3.60E—01
31 14 16026 1 6 403 :1 5 271 1 2 3.36E+00 1 9.00E—02
32 14 16565 1 7 407 1 7 262 1 3 3.45E—01 1 9.64E—03
33 14 17036 1 16 373 1 17 273 1 3 1.4lE—02 1 6.18E—04
34 14 17542 332 252 7.79E—04 1 1003—04
23 15 14550 401 279 12313—01 :1 2703-02
29 15 15103 1 14 426 1 14 247 1 3 2.06E+00 1 331153—02
30 15 15556 1 7 393 1 3 269 1 3 19213101 1 5.53E—01
31 15 16127 1 4 413 1 2 241 1 2 4.56E+01 1 1.15E+00
32 15 16536 1 3 330 1 1 266 1 1 2.25E+01 1 5.14E—01
33 15 17117 1 6 372 1 5 243 1 2 4.83E+00 1 1.25E—01
34 15 17577 1 5 317 1 1 259 1 2 3.48E—01 1 8.46E—03
35 15 13115 325 244 2403—02 1 3.25E—03
36 15 13639 1 17 311 1 9 222 1 3 6.02E—04 1 5103-05
30 16 15594 449 233 2.07E—01 1 3.57E—02
31 16 16169 1 5 393 1 6 223 1 5 4.00E+00 1 1.20133—01
32 16 16671 1 5 373 1 2 234 1 2 3.72E+01 1 9.28E—01
33 16 17222 1 4 394 1 2 214 1 2 5.95E+01 1 1.48E+00
34 16 17633 1 3 309 1 1 213 1 1 4.85E+01 1 1.12E+00
35 16 13193 1 5 330 1 4 216 1 2 5.54E+OO 1 1.46E—01
36 16 13753 1 4 311 1 16 191 1 2 6.39E—01 1 2.62E—02
37 16 19212 1 5 273 1 2 195 1 1 1433—02 1 4.64E—04
33 16 19753 1 31 226 1 12 161 1 5 13113—04 1 3493-05
32 17 16705 413 234 1.53E—01 1 2.63E—02
33 17 17256 1 4 342 1 3 203 1 2 2.233100 1 59313—02
34 17 17742 1 2 354 1 2 213 1 1 2.26E+01 1 5.66E—01
35 17 13293 1 2 343 1 1 133 1 1 5.81E+01 1 1.40E+00
36 17 13772 1 5 311 1 6 193 1 2 2.99E+01 1 8.42E—01
37 17 19309 1 3 273 1 1 169 1 1 7.38E+00 1 1303—01
33 17 197771 2 226 1 1 161 1 1 2533—01 1 6.54E—03
39 17 20271 1 25 336 1 59 2771 12 34116-03 1 3. 7911—04
34 13 17772 293 204 2.96E—Ol 1 1433-02
35 13 13323 1 3 313 1 2 131 1 3 5.84E+OO 1 1.55E—01
36 13 13361 1 3 311 1 1 167 1 2 4.33E+01 1 1.19E+00
37 13 19347 1 3 273 1 1 164 1 1 8.22E+01 1 2.11E+00
33 13 19395 1 3 226 1 1 135 1 1 5.71E+01 1 1.47E+00
39 13 20301 1 1 191 1 1 141 1 1 2.403900 1 6.01E—02

 

 

180

continued on next page

Table A.1: continued from previous page

 

I A I Z I p0 (MeV/c) I 01, (MeV/c) I UR (MeV/c) I

Uprod (mb) J

 

 

 

 

 

 

 

 

 

 

40 13 209971 10 2171 2 170 1 6 20919—02 1 9023—01
36 19 13359 1 12 274 1 10 165 1 11 2.62E—01 1 1.20E-02
37 19 19423 1 2 273 1 1 144 1 1 5.49E+00 1 13213—01
33 19 19901 1 4 226 1 1 144 1 2 4.36E+01 :1 1.50E+00
39 19 20433 1 1 162 1 1 35 1 1 9.56E+01 1 2.42E+00
10 19 207371 12 1371 6 1371 5 2.31E+00 1 11313—01
11 19 21463 1 7 2971 2 173 1 4 1173-01 1 14413—02
12 19 21772 1 73 333 1 17 223 1 32 5133—03 1 5933-04
33 20 19959 1 3 221 1 2 119 1 4 1.29E+00 1 4.13E—02
39 20 20470 162 115 4.00E+01 1 6.53E+OO
11 21 21262 1 6 134 1 4 165 1 1 2.10E—01 1 33913—02

 

Table A2: Fitting results for the reaction system

40Ca+181Ta.

 

Z Ipo (MeV/c) I 0;, (MeV/c) I OH (MeV/c) I

O‘pmd (mb)

 

 

 

 

 

 

 

 

 

 

 

10 5 5198 :l: 6 343 :l: 1 249 :l: 3 2.52E+01 :l: 2.03E+00
11 5 5623 :l: 3 365 :t 1 257 :l: 1 4.43E+01 :l: 3.47E+00
12 5 6107 :t 7 429 :l: 11 305 :l: 1 4.29E+00 :l: 3.37E—01
13 5 6662 357 287 6.06E—01 :l: 6.10E—02
11 6 5651 440 243 2.25E+01 :l: 2.60E+00
12 6 6167 344 261 6.13E+01 d: 5.52E+00
13 6 6648 :l: 3 391 :l: 1 275 :l: 1 3.57E+01 :1: 2.66E+00
14 6 7178 :t 1 363 :l: 1 284 i: 2 7.12E+00 i- 5.18E—01
15 6 7688 :t 7 391 :l: 1 293 :l: 6 4.83E—01 :l: 3.51E—02
16 6 8215 373 252 6.85E—02 :l: 1.11E—02
14 7 7160 374 309 2.97E+01 :l: 2.38E+00
15 7 7712 :l: 5 430 :t 4 272 :l: 1 4.08E+01 :l: 2.90E+00
16 7 8133 :l: 8 397 :t 6 326 :l: 4 3.35E+00 :l: 2.33E—01
17 7 8597 :l: 12 415 :l: 9 356 :l: 6 9.51E—01 :l: 6.48E-02
18 7 9132 379 346 6.77E—02 :l: 1.02E—02
19 7 9593 381 257 1.14E—02 :l: 2.09E—03
15 8 7756 391 264 1.07E+01 :l: 2.13E+00
16 8 8195 :l: 4 351 :t 1 293 i 2 5.08E+01 :l: 3.49E+00
17 8 8710 :l: 3 407 :l: 1 291 :l: 1 2.01E+01 :l: 1.33E+00
18 8 9135 i 6 379 :l: 3 339 :l: 3 7.25E+00 :l: 4.66E-01
19 8 9670 :t 6 456 :t 8 333 :l: 11 9.29E—01 :t 5.99E-02
20 8 10188 :t 9 403 i 1 331 i 9 1.47E—01 :l: 9.30E—03
21 8 10675 439 348 1.25E—02 :l: 2.05E—03

 

181

continued on next page

 

 

Table A2: continued from previous page

 

LA LZ I p0 (MeV/c) J 0;, (MeV/c) I an (MeV/c) I

Uprod (mb)

l

 

 

 

 

 

 

 

 

 

 

 

 

17 9 8774 340 304 3.24E+00 :l: 5.25E—01
18 9 9203 :l: 9 391 :l: 12 333 :l: 4 1.48E+01 :l: 9.98E—01
19 9 9720 :l: 6 417 :l: 4 309 i 2 1.77E+01 :l: 1.10E+00
20 9 10198 :1: 4 446 i 1 327 :t 1 7.68E+00 :l: 4.54E—01
21 9 10684 :t 9 451 :l: 9 353 d: 6 1.84E+00 :l: 1.07E—01
22 9 11177 :1: 17 454 :l: 14 363 :1: 12 1.82E-01 i 1.08E—02
23 9 11720 :1: 14 399 :l: 4 329 :l: 3 2.22E-02 :l: 1.45E—03
19 10 9840 477 285 4.31E+00 :l: 1.15E+00
20 10 10256 :1: 8 402 :l: 9 317 :l: 3 2.69E+01 :l: 1.65E+00
21 10 10769 :l: 5 423 d: 3 307 :l: 2 2.92E+01 :l: 1.66E+00
22 10 11247 :1: 3 429 :l: 1 317 :l: 1 1.54E+01 :l: 8.29E—01
23 10 11740 :1: 9 453 :l: 8 335 :l: 4 1.64E+00 :l: 8.67E—02
24 10 12251 d: 12 471 :l: 12 336 :l: 9 2.64E—01 :l: 1.40E—02
25 10 12787 515 326 1.64E—02 :l: 1.43E—03
26 10 13200 492 245 1.71E—03 :l: 5.57E—04
21 11 10827 372 303 3.03E+00 :l: 4.80E—01
22 11 11283 :1: 9 410 :t 9 327 :l: 4 2.09E+01 :l: 1.18E+00
23 11 11806 :1: 5 435 :l: 3 307 :l: 2 3.56E+01 :l: 1.84E+00
24 11 12280 :t 4 440 :l: l 321 i 1 1.14E+01 :l: 5.51E—01
25 11 12791 :1: 5 435 :l: 1 321 :L- 2 2.87E+00 :t 1.31E—01
26 11 13299 d: 14 467 :l: 13 334 :t 9 2.84E—01 :l: 1.37E—02
27 11 13772 450 347 3.35E—02 :1: 2.03E—03
23 12 11876 392 301 5.52E+00 :l: 4.05E—01
24 12 12348 :1: 7 412 :l: 7 307 :l: 3 4.29E+01 :l: 2.16E+00
25 12 12868 :1: 5 438 :l: 3 296 :l: 2 4.25E+01 :l: 1.96E+00
26 12 13347 :t: 3 421 :l: 1 304 :t 1 2.34E+01 :i: 9.95E—01
27 12 13840 :i: 4 417 :l: 1 310 :h 2 2.85E+00 :l: 1.14E—01
28 12 14356 :L- 6 442 :l: 4 314 :l: 3 4.17E—01 :l: 1.58E—02
29 12 14834 439 350 2.26E—02 :l: 2.22E—03
30 12 15334 446 301 2.60E—03 :l: 5.58E—04
25 13 12930 395 295 3.11E+00 :l: 1.72E—01
26 13 13402 :1: 8 417 :l: 6 299 :l: 4 2.50E+01 :l: 1.12E+00
27 13 13922 :1: 3 431 d: 1 287 :l: 1 5.27E+01 :l: 2.09E+00
28 13 14394 :1: 3 411 :l: 1 295 :l: 2 1.80E+01 :l: 6.59E—01
29 13 14901 :1: 4 405 :l: 1 296 i 2 4.34E+00 :l: 1.46E—01
30 13 15409 :1: 6 406 :l: 3 298 :l: 3 3.87E—01 :l: 1.23E—02
31 13 15912 :1: 19 429 :l: 17 309 :l: 13 4.60E-02 :l: 2.05E—03
32 13 16430 412 279 2.43E—03 :l: 4.66E—04
27 14 13970 . 385 286 5.91E+00 :t 2.98E—01
28 14 14471 :t 7 394 :l: 4 278 :l: 3 5.67E+01 :t 2.18E+00
29 14 14993 :1: 2 413 :l: 1 269 :l: 1 6.25E+01 :l: 2.10E+00
30 14 15476 :1: 3 374 :t 1 269 d: 1 4.11E+01 :l: 1.25E+00
31 14 15975 :1: 4 381 :l: 1 274 :l: 2 4.62E+00 :l: 1.28E—01

 

 

182

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Table A.2: continued from previous page

 

I A I Z Ipo (MeV/c) T01, (MeV/c) FIR (MeV/c) I

owed (mb)

 

 

 

 

 

 

 

 

 

 

 

 

32 14 16496 1 5 330 1 19 277 1 2 53513—01 1 2.06E—02
33 14 17025 1 6 392 1 4 265 1 1 279153—02 1 8.67E—04
34 14 17431 301 254 2.01E-O3 1 3.76E—04
23 15 14574 349 259 2.59E—01 1 5473-02
29 15 15072 1 13 363 1 25 251 1 3 3.09E+00 1 1.72E—01
30 15 15537 1 7 333 1 4 263 1 3 2.45E+01 1 8.04E—01
31 15 16043 1 3 335 1 1 253 1 1 5.73E+01 1 1.53E+00
32 15 16534 1 3 353 1 1 254 1 2 2.61E+01 1 6.38E—01
33 15 17047 1 4 351 1 1 249 1 2 6.16E+OO 1 144133—01
34 15 17543 1 5 353 1 3 256 1 3 50913—01 1 1.29E—02
35 15 13034 1 14 351 1 14 233 1 7 4.68E-02 1 1.92E—o3
36 15 13535 1 34 359 1 31 245 1 22 1.84E—03 1 1.61E—04
30 16 15604 330 261 3.58E-01 1 747153—02
31 16 16123 1 5 354 1 5 237 1 4 5.73E+OO 1 1.80E—01
32 16 16646 1 5 366 1 3 226 1 2 4.50E+01 1 1.2OE+OO
33 16 17122 1 3 350 1 1 229 1 1 7.33E+01 1 1.72E+00
34 16 17636 1 3 310 1 1 216 1 1 5.25E+01 1 1.23E+00
35 16 13121 1 4 323 1 1 223 1 2 6.50E+00 1 1.57E—01
36 16 13670 1 4 303 1 4 216 1 3 8.52E—01 1 2.25E—02
37 16 19033 1 17 294 1 16 239 1 7 32413—02 1 15313—03
33 16 19950 1 3 215 1 3 119 1 24 11213—03 1 2363-04
32 17 16695 310 220 2.77E—01 1 39313—02
33 17 17223 1 4 319 1 4 197 1 7 3.08E+00 1 9.61E—o2
34 17 17712 1 2 343 1 2 214 1 1 2.65E+01 1 6.77E—01
35 17 13193 1 3 309 1 1 193 1 1 6.76E+01 1 1.63E+00
36 17 13690 1 3 239 1 1 193 1 1 3.26E+01 1 7.85E—01
37 17 19215 1 3 260 1 1 179 1 1 8.05E+OO 1 2.00E-01
33 17 19665 1 3 254 1 7 224 1 4 3.61E—01 1 1233—02
39 17 203571 12 263 1 1 206 1 6 22213—02 1 34319—04
40 17 20365 1 62 209 1 43 244 1 33 3453-04 1 1423—04
34 13 17771 259 132 5.53E—01 1 3.03E—02
35 13 13290 1 3 297 1 4 173 1 1 8.21E+00 1 2.23E—01
36 13 13736 1 1 239 1 1 173 1 1 5.18E+01 1 1.19E+00
37 13 19277 1 3 261 1 1 163 1 1 8.75E+01 1 2.19E+00
33 13 19314 1 1 223 1 1 134 1 1 5.35E+01 1 1.36E+OO
39 13 20243 1 2 233 1 3 146 1 1 2.93E+00 1 3353—02
40 13 20943 1 5 133 1 1 150 1 3 2.26E—01 1 6.78E—03
41 13 21419 1 6 190 1 3 141 1 3 1073—02 1 5.13E—04
36 19 13339 1 6 231 1 6 149 1 4 5.14E—01 1 192151—02
37 19 19345 1 2 245 1 2 161 1 1 7.28E+00 1 2.07E—01
33 19 19343 1 1 243 1 1 146 1 1 3.75E+01 1 900133—01
39 19 20331 1 2 126 1 1 93 1 1 1.18E+02 1 3.46E+00
40 19 20692 1 3 133 1 4 136 1 5 4.11E+00 1 1.56E—01

 

 

183

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Table A.2: continued from previous page

 

I A I Z I p0 (MeV/c) I 0;, (MeV/c)J O’R (MeV/c) I

”prod (mb)

 

 

 

 

 

 

 

 

 

 

41 19 214973: 2 148 :l: 1 109 :1: 1 5.69E—01 :l: 1.64E—02
42 19 22000 :1: 10 301 :l: 2 180 :t 3 3.66E—02 :1: 1.64E—03
38 20 19897 :1: 2 185 i 2 113 j: 1 2.34E+00 :l: 7.35E—02
39 20 20416 i 2 114 :i: 1 73 :t 1 6.54E+01 :l: 1.96E+00
41 20 2139421: 4 210 i 1 155 :l: 51 2.17E+00 :l: 3.10E-01
40 21 20818 :t 3 110 :l: 1 76 :t 2 2.14E—01 i 9.04E-03
41 21 21259 :1: 11 158 :l: 9 158 :l: 4 2.22E—01 :l: 4.74E—02

 

184

 

A.2 Results for 480a projectile

Table A.3: Fitting results for the reaction system

 

 

 

 

 

 

 

 

48Ca+9Be.

I A TZ I p0 (MeV/c) I01, (MeV/C)J UR (MeV/c) I owed (mb) I
11 5 5617 :t 10 516 i 3 349 :1: 2 1.45E+01 :1": 9.95E-01
12 5 6249 i 6 532 :1: 15 266 :1: 1 2.51E-1-00 :t 1.77E—01
13 5 6786 480 273 7.75E-01 :1: 7.01E-02
14 5 7106 460 378 4.01E-02 i 3.43E—03
12 6 6069 393 344 8.31E+00 :1: 1.30E+00
13 6 6667 :1: 18 482 :E 17 298 :1: 6 8.58E+00 :1: 6.27E-01
14 6 7095 i 19 440 :1: 14 376 i 7 4.40E+00 :1: 3.08E-01
15 6 7811 :t 21 324 :1: 18 318 :1: 7 4.73E-01 :1: 3.67E-02
16 6 8333 :1: 11 494 :1: 29 290 :1: 8 1.77E—01 :1: 1.34E—02
17 6 8739 434 398 1.41E-02 :1: 1.45E—03
14 7 7213 558 279 3.53E-1-00 :1: 6.57E—01
15 7 7602 :1: 22 479 :t 20 352 :1: 9 1.16E+01 :E 8.40E—01
16 7 8212 i 7 428 :1: 1 329 :t 3 1.96E+00 :1: 1.24E—01
17 7 8825 i 15 556 :1: 13 328 :1: 4 1.10E+00 i 7.36E-02
18 7 9381 615 339 2.31E—01 :1: 3.47E-02
19 7 9862 555 356 6.45E—02 d: 7.56E—03
20 7 10518 606 356 6.22E-03 :1: 1.12E—03
16 8 8139 438 332 6.86E+00 :1: 9.44E—01
17 8 8629 :t 28 474 :1: 21 352 :1: 13 4.93E+00 :1: 3.55E—01
18 8 9154 :1: 20 486 d: 10 369 :1: 8 4.14E+00 i 2.63E—01
19 8 9683 :1: 18 453 :1: 8 390 :1: 6 1.15E+00 i 7.03E—02
20 8 10265 461 378 4.23E—01 :1: 7.19E—02
21 8 10920 :1: 34 516 :1: 43 328 :1: 15 5.95E—02 :1: 4.91E—03
22 8 11440 614 390 1.10E-02 :1: 1.79E—03
23 8 11825 614 361 3.04E—04 :1: 65813-05
17 9 8669 511 398 1.56E—01 i 2.98E—02
18 9 9237 567 307 1.56E-1-00 :1: 1.69E-01
19 9 9667 i 28 499 :1: 21 358 :1: 13 5.02E+00 :1: 3.48E-01
20 9 10206 :1: 24 504 :1: 11 359 :1: 11 4.55E-1-00 :t 2.81E-01
21 9 10742 :1: 8 450 :1: 1 376 :1: 4 2.31E-1-00 :1: 1.31E-01
22 9 11341 :1: 18 517 j: 14 364 j: 5 5.37E—01 :1: 3.31E-02
23 9 11878 :1: 25 491 :1: 41 367 d: 9 1.56E—01 :1: 1.26E-02
24 9 12540 621 312 1.69E—02 :1: 2.43E—03
25 9 12935 496 368 1.66E-03 :1: 2.83E—04
26 9 13496 452 360 1.29E-04 :1: 5.10E-05
19 10 9792 444 301 1.16E—01 :1: 3.35E-02
20 10 10282 595 298 3.19E-1-00 :1: 4.17E-01
21 10 10722 :1: 27 517 :1: 20 337 :1: 11 6.82E+00 :1: 45213—0]

 

 

 

 

 

 

 

 

185

continued on next page

Table A.3: continued from previous page

 

IA I Z Ipo(MeV/c)

I 0'1, (MeV/c) I 0'}; (MeV/c) I

”prod (mb) 1

 

 

 

 

 

 

 

 

 

 

 

22 10 11193 d: 23 486 :l: 9 376 :1: 10 7.78E+00 :t 4.53E—01
23 10 11755 :1: 21 499 :t 7 370 d: 9 2.39E+00 :1: 1.35E—01
24 10 12328 :1: 20 534 i 13 381 i 6 9.09E—01 :1: 5.38E—O2
25 10 12888 :1: 31 495 :1: 41 372 i 11 1.44E-01 :1: 1.17E—02
26 10 13509 :1: 29 452 d: 36 336 i: 10 3.15E—02 :1: 2.44E—03
27 10 14087 572 314 1.42E-03 :l: 1.91E—04
28 10 14590 553 364 5.88E—04 :1: 7.81E—05
29 10 15262 575 354 3.94E—05 :1: 1.32E-05
21 11 10719 609 373 1.24E—01 d: 1.83E—02
22 11 11267 587 335 2.10E+00 i 3.67E—01
23 11 11747 :1: 26 513 :l: 18 340 :l: 10 9.40E+00 :1: 5.95E—01
24 11 12225 :1: 23 490 :t 9 368 :l: 10 7.18E+00 :t 4.01E-01
25 11 12818 :1: 21 515 :t 7 356 :l: 9 4.56E+00 :t 2.45E—01
26 11 13368 :1: 21 518 j: 11 364 :1: 8 1.23E+00 :1: 6.89E—02
27 11 13947 :1: 27 519 :1: 20 358 :l: 9 3.82E—01 :1: 2.33E—02
28 11 14531 :1: 31 483 :l: 43 345 j: 10 5.96E—02 :1: 5.22E—03
29 11 15111 596 335 1.57E—02 :1: 1.87E-03
30 11 15560 532 412 1.84E—03 :l: 1.83E—04
31 11 16113 535 294 3.05E—04 :1: 5.07E—05
23 12 11719 614 389 1.75E—01 :1: 3.09E—02
24 12 12310 578 318 4.87E+00 :1: 4.80E—01
25 12 12782 :1: 24 514 :1: 16 338 :1: 9 9.74E+00 :1: 5.88E—01
26 12 13275 i 21 493 :1: 8 358 d: 8 1.26E+01 :1: 6.62E—01
27 12 13873 :1: 20 524 :l: 7 342 d: 9 5.16E+00 :1: 2.67E—01
28 12 14455 i 23 536 :t 10 340 :1: 10 2.26E+00 :t 1.20E—01
29 12 14984 i 29 517 i 18 356 :1: 10 3.84E—01 :l: 2.23E—02
30 12 15668 :t 28 572 :1: 31 304 :1: 9 1.37E—01 :l: 9.59E-03
31 12 16194 633 317 2.30E—02 :l: 2.61E—03
32 12 16741 :1: 34 585 i 54 320 :l: 14 6.20E—03 :l: 5.07E—04
33 12 17173 478 333 4.98E—04 :1: 8.35E—05
34 12 17704 558 279 8.06E—05 :l: 1.42E—05
25 13 12719 454 410 7.75E—02 d: 1.65E—02
26 13 13282 :1: 37 540 :1: 50 354 :1: 13 2.21E+00 d: 1.92E—01
27 13 13822 :1: 23 512 :1: 15 334 :l: 8 1.28E+01 :l: 7.33E—01
28 13 14353 :1: 20 515 :l: 8 341 :l: 8 1.17E+01 :l: 5.98E—01
29 13 14931 :1: 19 530 :l: 7 340 :t 8 8.63E+00 :1: 4.23E—01
30 13 15543 :1: 22 543 :1: 10 316 :t 9 2.72E+00 :1: 1.36E—01
31 13 16034 :1: 21 497 :1: 12 341 :1: 7 1.14E+00 :t 5.74E—02
32 13 16669 :1: 30 516 :1: 22 318 :1: 10 2.24E—01 :l: 1.35E—02 .
33 13 17224 :1: 26 484 :1: 29 313 :1: 8 8.89E—02 :1: 6.36E—03
34 13 17731 :1: 30 491 :1: 41 337 :l: 10 1.28E—02 i 9.53E—04
35 13 18339 :1: 32 539 :1: 35 310 i 14 3.78E—03 :l: 2.51E—04
36 13 18847 :1: 46 496 :1: 42 360 i 8 3.18E—04 :l: 3.96E—05

 

186

continued on next page

 

Table A.3: continued from previous page

 

 

 

 

 

 

LA I Z IPo (MeV/c) I 0L (MeV/c) I 0'}; (MeV/c) I 0pm; (mb) I
37 13 19412 420 304 7.33E—05 :1: 1.92E—05
38 13 19797 369 250 2.64E-06 :1: 9.27E—07
26 14 13139 532 372 3.50E—03 :1: 1.73E-03
27 14 13775 609 370 1.31E—01 :1: 1.81E—02
28 14 14318 511 344 5.00E+00 :t 6.27E-01
29 14 14855 :1: 24 497 :1: 13 333 :t 9 1.27E+01 :1: 7.00E—01
30 14 15424 i 18 513 :1: 7 323 :1: 6 2.02E+01 :1: 9.82E-01
31 14 16013 :t 18 529 :1: 7 318 :1: 7 1.02E+01 :1: 4.88E—01
32 14 16628 :1: 21 536 :1: 9 296 :1: 8 4.98E-1-00 :1: 2.45E—01
33 14 17128 :1: 23 486 :1: 13 318 :1: 9 1.37E+00 :1: 7.28E-02
34 14 17769 :1: 22 520 :1: 16 287 :1: 7 5.67E—01 :1: 3.06E—02
35 14 18307 :1: 25 449 :1: 31 292 :t 8 1.19E—01 :t 9.81E—03
36 14 18839 :t 23 443 :1: 25 296 :1: 7 4.28E—02 :1: 2.90E—03
37 14 19424 :t 31 502 :1: 32 287 :1: 11 6.19E—03 :1: 4.20E—04
38 14 19970 :1: 24 463 :1: 24 284 :1: 8 1.57E—03 :t 9.83E—05
39 14 20513 406 240 1.18E—04 :1: 1.63E—05
40 14 21010 458 310 2.79E—05 :1:: 4.06E—06
41 14 21671 371 255 1.84E—06 :1: 8.25E—07
28 15 14418 527 303 1.08E—03 :1: 4.07E-04
29 15 14781 395 361 3.89E—02 :1: 1.14E—02
30 15 15272 :1: 37 449 :t 29 397 :1: 15 1.17E-1-00 :t 8.11E—02
31 15 15921 d: 22 498 :1: 12 322 d: 8 1.13E+01 :1: 5.93E—01
32 15 16491 :1: 18 515 :1: 7 309 :1: 6 1.60E+01 :1: 7.64E—01
33 15 17090 i 17 524 :1: 6 297 :1: 7 1.51E+01 :1: 6.90E—01
34 15 17686 :t 19 526 :1: 9 291 :1: 7 6.90E+00 :1: 3.31E—01
35 15 18243 :t 20 488 :1: 10 286 :1: 8 3.79E-1-00 :1: 1.87E—01
36 15 18852 :1: 20 486 :1: 13 266 :1: 7 1.19E+00 :1: 6.11E—02
37 15 19433 i 19 477 :1: 14 258 i 6 5.04E—01 :1: 2.76E-02
38 15 19947 :1: 22 451 :1: 20 271 :1: 7 1.34E—01 :1: 9.08E—03
39 15 20478 :1: 21 421 :1: 20 273 :1: 7 4.11E—02 :1: 2.43E-03
40 15 21016 :1: 23 425 :1: 18 278 :1: 9 6.12E-03 :1: 3.45E—04
41 15 21583 :1: 23 400 :1: 17 256 :1: 7 1.25E-03 :1: 7.36E—05
42 15 22130 :1: 27 395 :1: 27 252 :1: 10 1.19E—04 :t 1.08E—05
43 15 22659 375 254 1.26E—05 :1: 2.21E—06
30 16 15246 517 402 7.85E—04 :1: 3.02E—04
31 16 15772 469 390 4.95E-02 :1: 1.53E-02
32 16 16335 :1: 33 439 :1: 24 377 :1: l3 1.88E+00 :1: 1.21E-01
33 16 16960 :1: 22 484 :1: 10 320 :1: 8 9.31E+00 :1: 4.74E-01
34 16 17551 :1: 17 500 :1: 7 304 :1: 6 2.08E-1-01 :1: 9.87E—01
35 16 18170 :1: 17 517 :1: 6 281 :1: 6 1.69E+01 :1: 7.71E-01
36 16 18757 :1: 18 506 i 8 270 :1: 7 1.16E+01 :1: 5.47E—01
37 16 19328 :1: 20 474 :1: 9 267 :t 7 5.62E-1-00 :1: 2.79E—01
38 16 19874 :1: 21 436 :1: 11 270 :1: 8 2.96E+00 d: 1.57E—01

 

 

 

 

 

 

 

 

187

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Table A.3: continued from prcvimrs page

 

 

 

 

 

 

 

 

 

 

 

A Z p0 (MeV/c) O’L (MeV/c) UR (MeV/c) awed (mb)
39 16 20443 :t 20 412 d: 11 272 d: 6 1.08E+00 :1: 5.37E—02
40 16 21015 :1: 20 403 :1: 15 254 :1: 6 48913—0] :1: 3.07E—02
41 16 21571 :1: 21 385 :1: 17 246 :1: 7 1.04E—01 :t 6.78E—03
42 16 22132 :1: 19 379 :1: 15 233 :l: 6 2.76E—02 :t 1.56E—03
43 16 22663 :1: 14 375 :t 9 228 :1: 7 2.78E—03 :t 1.43E—04
44 16 23227 :1: 22 333 d: 16 210 :1: 8 5.14E—04 i 3.43E—05
45 16 23590 :t 18 297:1: 53 201 d: 34 1.42E—05 i 2.97E—06
32 17 16326 504 392 4.04E—04 :t 1.53E—04
33 17 16898 484 334 1.30E—02 :1: 2.50E—03
34 17 17375 :1: 15 442 :t 17 378 :1: 8 5.91E-01 :t 3.52E-02
35 17 18060 :1: 20 486 :1: 9 288 :1: 7 7.85E+00 :1: 3.88E—01
36 17 18620 d: 17 493 :1: 7 288 :1: 6 1.65E+01 :1: 7.91E—01
37 17 19238 d: 18 497 :1: 7 271 :1: 7 2.14E+01 :t 1.04E+00
38 17 19804 :1: 19 478 :1: 8 266 :1: 7 1.39E+01 :1: 7.33E—01
39 17 20376 :1: 20 441 :t 9 259 :1: 7 1.10E+01 :1: 5.70E—01
40 17 20904 :1: 18 402 :1: 9 273 :1: 7 5.98E+00 :t 33813—01
41 17 21498 :1: 18 380 :t 10 252 :1: 6 3.78E+00 :1: 20113—0]
42 17 22091 :1: 21 367 :t 11 237 :1: 6 1.54E+00 :1: 9.02E—02
43 17 22659 :1: 17 347 :t 11 221 :1: 5 6.53E—01 :E 4.36E—02
44 17 23164 d: 19 311 :1: 12 228 :1: 7 1.56E—01 :t 9.20E—03
45 17 23752 :t 10 297 :t 1 183 :1: 4 3.43E—02 :t 1.70E—03
46 17 24183 :t .9 245 :1: 1 166:1: 1 9.16E—04 :1: 4.79E—05
47 17 2471.9 :1: 32 307 :1: 77 227:1: 60 2.95E—05 :t 6.25E—06
73—5 18 17872 390 343 1.50E—02 :1: 2.28E-03
36 18 18471 :1: 31 425 :1: 17 340 :1: 14 8.15E—01 :1: 5.03E-02
37 18 19110 :1: 21 469 :1: 9 284 :t 8 5.57E+00 :1: 2.88E-01
38 18 19711 :1: 17 474 :1: 7 263 :t 7 1.68E+01 :1: 89513—0]
39 18 20311 :1: 18 474 :1: 7 249 :1: 7 2.13E-1-01 :1: 1.13E+00
40 18 20907 :1: 18 452 :1: 8 234 :1: 7 2.07E+01 :t 1.18E+00
41 18 21432 :1: 19 395 :1: 8 250 :1: 7 1.71E+01 :1: 9.62E—01
42 18 22026 :1: 19 370 :1: 8 231 :t 8 1.42E+01 :1: 8.77E—01
43 18 22549 :1: 19 322 :1: 9 247 i 7 8.61E+00 d: 5.03E—01
44 18 23170 :1: 24 309 :1: 10 210 :1: 10 5.60E+00 d: 3.81E—01
45 18 23719 :1: 4 297 :1: 6 201 :1: 1 2.30E+00 :1: 1.01E—01
46 18 24299 :1: 10 245 :1: 3 147 :t 4 1.08E+00 :1: 6.61E-02
47 18 24668 :t 5 203 :t 1 150 :t 4 1.45E—02 d: 3.40E—03
48 18 25164 :1: 14 280 i 5 185 :1: 7 1.06E—03 :1: 8.83E—05
4.9 18 25736 :1: 23 239 :1: 50 1.90 :t 15 3.01E—04 :1: 5.42E—05
37 19 19015 499 302 4.51E—03 d: 1.21E—03
38 19 19493 :1: 34 407 i 17 345 :1: 20 2.28E—01 :t 1.54E—02
39 19 20191 i 19 448 d: 8 262 :t 8 3.83E+00 :1: 2.12E—01
40 19 20782 :1: 18 451 d: 7 241 :1: 7 1.15E+01 :t 6.80E—01
41 19 21372 :1: 18 431 :1: 7 230 :t 7 2.01E+01 :1: 1.15E+00

 

188

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Table A.3: continued from previous page

 

I A I Z 1P0 (MeV/c) LIL (MeV/c) I 03 (MeV/cﬂ

0pm,} (mb)

 

 

 

 

 

 

 

 

 

 

42 19 21953 1 16 402 1 7 220 1 6 2.31E+01 1 1.41E+00
43 19 22513 1 17 356 1 3 213 1 6 2.82E+01 1 1.88E+00
44 19 23092 1 24 323 1 9 211 1 12 2.46E+01 1 2.07E+00
45 19 23644 1 7 276 1 5 201 1 6 2.61E+01 1 1.78E+00
46 19 24245 1 4 245 1 1 166 1 1 2.19E+01 1 9.44E—01
47 19 24775 175 119 1.91E+01 1 430131-00
43 19 25230 1 20 293 1 11 203 1 5 3.501?— 01 1 3.55E- 02
49 19 25405 1 31 121 1 25 121 1 25 1595—02 1 1.02E—02
50 19 26222 1 7 135 1 3 140 1 3 11319—02 1 8.88E—04
39 20 20170 433 212 3.29E—03 1 5.61E—O4
40 20 20659 1 23 400 1 13 276 1 15 2.56E—01 1 1.74E—02
41 20 21294 1 13 424 1 3 223 1 3 2.4OE+OO 1 1496—01
42 20 21376 1 9 406 1 4 205 1 5 9.32E+00 1 5.28E—01
43 20 22433 1 10 375 1 3 204 1 6 1.67E+01 1 1.09E+00
44 20 23013 1 12 340 1 3 192 1 7 2.39E+01 1 1.98E+00
45 20 23599 1 9 297 1 4 176 1 5 3.06E+01 1 2.04E+oo
46 20 24131 1 10 245 1 1 157 1 5 4.57E+01 1 3.23E+00
47 20 24777 1 10 175 1 1 105 1 4 7.59E+01 1 6.11E+00
49 20 25336 1 24 290 1 11 209 1 10 7.68E—01 1 7. 713—02
50 20 263671 44 192 1 27 166 1 15 14013—02 1 2135—03
51 20 26969 1 19 230 1 46 173 1 42 1023—03 1 1.32E—04
41 21 21233 1 67 320 1 91 293 1 73 3.9919— 04 1 10913—04
42 21 21664 1 37 369 1 19 270 1 59 3673—02 1 4.56E—03
43 21 22315 1 12 375 1 3 205 1 3 53219—01 1 3. 7016—02
44 21 22392 1 11 340 1 2 131 1 6 2.58E+00 1 1.68E—01
45 21 23450 1 13 2971 1 169 1 7 5.53E+00 1 4393—01
46 21 24016 1 5 245 1 1 166 1 4 6.33E+00 1 3.15E—01
47 21 24619 1 6 175 1 1 119 1 1 3.33E+00 1 4.76E—01
43 21 25442 1 17 331 1 11 131 1 6 5.26E+00 1 5.68E—01
49 21 25995 1 40 396 1 20 435 1 79 1.14E+00 1 21719—01
44 22 227071 23 340 1 3 230 1 9 5413—03 143719—04
45 22 232471 15 2971 2 201 1 2 3.56E—02 1 2133-03
46 22 23343 1 10 345 1 13 255 1 3 1.05E—01 1 6.43E—03
47 22 24406 1 13 393 1 3 294 1 13 1035—01 1 6.53E—03
43 22 25051 1 17 473 1 10 350 1 4 71313—02 14412—03

 

 

189

Table A4: Fitting results for the reaction system

48Ca+181Ta.

 

A I Z I p0 (MeV/c) I01, (MeV/c) I 03 (MeV/c) I

”prod (mb)

1

 

 

 

 

 

 

 

11 5 5591 :l: 13 450 :l: 4 319 :l: 6 7.47E+01 i 4.18E+00
12 5 6212 :l: 1 482 :1: 14 263 :l: 1 1.48E+01 :1: 8.19E—01
13 5 6645 :t 13 365 :1: 19 319 :l: 7 3.59E+00 :1: 2.10E—01
14 5 7124 445 370 2.11E—01 2t 2.48E—02
12 6 6168 499 298 4.75E+01 :t 7.93E+00
13 6 6644 i 21 438 :1: 34 319 d: 7 4.28E+01 :1: 3.17E+00
14 6 7175 :1: 19 457 :l: 14 328 :1: 6 2.11E+01 :l: 1.22E+00
15 6 7674 i 16 366 :l: 13 345 :l: 5 2.38E+00 :l: 1.44E—01
16 6 8277 :1: 18 525 :t 5 306 :1: 11 7.57E—01 d: 3.69E—02
17 6 8645 393 390 5.99E—02 :l: 7.25E—03
14 7 7177 408 285 1.56E+01 :l: 1.56E+00
15 7 7579 :1: 22 432 i 19 362 :1: 10 4.75E+01 :1: 3.08E+00
16 7 8118 :1: 16 376 :l: 6 348 :l: 5 8.80E+00 :1: 4.64E—01
17 7 8623 :l: 13 458 :1: 12 383 :1: 4 4.51E+00 :1: 2.49E—01
18 7 9192 i 15 398 :l: 23 377 j: 5 8.64E—01 :l: 5.04E—02
19 7 9828 579 366 2.45E—01 :l: 1.76E-02
20 7 10254 488 402 2.07E—02 :t 2.86E—03
16 8 8083 :1: 21 388 d: 26 358 :1: 10 2.59E+01 :l: 1.79E+00
17 8 8669 :1: 26 472 :l: 20 334 :1: 12 1.83E+01 :1: 1.19E+00
18 8 9199 i 18 499 :1: 10 345 :1: 7 1.54E+01 :1: 8.37E—01
19 8 9708 i 11 504 :l: 7 367 :l: 4 4.36E+00 :1: 2.12E-01
20 8 10181 :1: 14 405 :1: 19 384 :1: 5 1.48E+00 :1: 8.27E-02
21 8 10832 i: 19 468 :t 8 345 :t 8 1.91E—01 :1: 8.66E—03
22 8 11360 538 394 3.52E—02 :1: 3.46E—03
23 8 11735 532 389 1.17E—03 :l: 1.80E—04
17 9 8882 202 147 3.54E—01 :1: 1.10E-01
18 9 9132 399 371 5.61E+00 :1: 6.66E-01
19 9 9703 :l: 26 492 :1: 42 343 :1: 11 1.71E+01 :l: 1.33E+00
20 9 10238 :1: 24 515 :1: 11 342 :l: 11 1.54E+01 :l: 8.39E—01
21 9 10722 i: 17 502 :l: 11 371 :l: 5 7.79E+00 :1: 4.00E—01
22 9 11265 :1: 18 500 :l: 12 365 :1: 7 1.73E+00 :1: 9.44E-02
23 9 11820 :1: 20 451 :l: 31 357 :l: 6 4.94E—01 :1: 3.25E—02
24 9 12419 :1: 20 493 d: 5 351 :l: 10 4.79E—02 :l: 1.99E—03
25 9 12974 549 303 5.37E—03 :l: 7.51E—04
19 10 9853 369 278 5.22E—01 :l: 1.09E—01
20 10 10213 446 339 9.65E+00 :1: 1.10E+00
21 10 10749 :1: 13 493 :1: 14 334 :t 7 2.11E+01 :l: 1.15E+00
22 10 11269 :t 24 507 :l: 10 340 :l: 10 2.29E+01 :1: 1.21E+00
23 10 11828 i 22 522 :1: 9 331 :1: 9 7.13E+00 :1: 3.56E—01
24 10 12321 :1: 13 516 d: 4 367 :1: 5 2.67E+00 :l: 1.27E—01
25 10 12940 :1: 19 427 :1: 5 316 :1: 7 4.13E—01 i 2.10E—02

 

 

 

 

 

 

 

190

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Table A4: continued from previous page

 

 

 

 

 

 

 

 

A I Z I p0 (MeV/c) I 0;, (MeV/c) I an (MeV/c) I 0pm; (mb) I
26 10 13414 :1: 19 462 :1: 11 354 :1: 7 9.17E-02 :1: 4.21E—03
27 10 13942 552 394 4.82E-03 :1: 5.79E-04
28 10 14502 493 314 1.74E—03 d: 1.96E—04
21 11 10801 462 364 4.06E—01 :1: 8.69E-02
22 11 11266 474 345 6.15E+00 :1: 4.43E—01
23 11 11816 :1: 24 518 :1: 18 326 :1: 9 2.53E-1-01 :1: 1.54E+00
24 11 12340 :1: 23 530 :1: 10 332 :1: 9 1.87E-1-01 :1: 95913—0]
25 11 12893 :t 20 547 :1: 8 341 :1: 8 1.16E+01 :1: 5.53E—01
26 11 13382 :1: 13 518 d: 6 353 d: 6 3.22E+00 :t 1.50E—01
27 11 13933 :1: 15 471 :1: 3 337 :1: 6 1.04E-1-00 :1: 4.72E—02
28 11 14480 rt 17 443 :t 20 340 :1: 6 1.54E—01 :1: 8.32E—03
29 11 15008 :1: 20 477 :1: 7 359 :1: 8 3.73E—02 :1: 1.42E—03
30 11 15583 :t 26 545 :1: 24 356 :1: 22 5.25E—03 :1: 2.88E-04
31 11 16015 462 390 1.03E—03 :1: 1.93E—04
23 12 11762 497 397 5.92E—01 :1: 9.72E—02
24 12 12296 486 344 1.21E+01 :1: 9.82E—01
25 12 12838 :1: 13 495 :1: 5 328 :1: 6 2.42E-1-01 :1: 1.19E-1-00
26 12 13390 :1: 21 523 :1: 14 325 d: 8 2.87E-1-01 :1: 1.47E-1-00
27 12 13940 :1: 21 535 :1: 8 317 :1: 9 1.20E-1-01 :1: 5.67E—01
28 12 14503 :1: 23 533 :1: 11 307 :1: 9 5.41E+00 :1: 2.66E—01
29 12 14985 :1: 15 471 :1: 3 335 :t 6 9.66E-01 :1: 4.50E—02
30 12 15578 :1: 18 424 :t 3 312 :1: 7 3.41E-01 :1: 1.61E-02
31 12 16032 :1: 29 430 :t 40 351 :1: 10 5.34E-02 :1: 4.13E-03
32 12 16614 :1: 18 482 :1: 17 354 :1: 10 1.54E—02 :1: 6.25E—04
33 12 17161 516 378 1.61E—03 :1: 2.20E—04
34 12 17422 449 274 1.82E-04 :1: 4.22E-05
25 13 12784 500 390 2.59E-01 :1: 6.31E-02
26 13 13343 507 357 5.26E-1-00 :1: 4.30E—01
27 13 13881 :1: 12 495 :1: 5 324 :1: 6 2.81E-1-01 :1: 1.38E-1-00
28 13 14423 :1: 9 519 :1: 2 327 :t 4 2.48E+01 :1: 1.14E+00
29 13 14983 :1: 8 535 :1: 5 327 :t 3 1.82E+01 :1: 8.14E—01
30 13 15505 :1: 13 496 :1: 2 326 :1: 6 6.09E-1-00 :1: 2.62E—01
31 13 16047 :1: 14 476 :1: 3 322 :1: 6 2.66E+00 :1: 1.21E-01
32 13 16604 :1: 16 441 :t 3 315 :1: 6 5.73E—01 :1: 2.54E-02
33 13 17145 :1: 15 418 :t 5 319 :1: 6 2.00E-01 :1: 8.46E-03
34 13 17622 :1: 26 431 :1: 39 353 :1: 10 2.97E—02 :t 1.94E—03
35 13 18245 :1: 16 470 :1: 7 331 :1: 10 9.18E—03 :1: 3.46E—04
36 13 18806 :1: 29 413 :1: 13 297 :1: 13 9.41E-04 2t 5.51E—05
37 13 19150 346 253 1.58E—04 :1: 4.34E—05
27 14 13902 552 333 3.31E—01 :1: 6.30E—02
28 14 14392 487 342 1.03E-1-01 :1: 7.64E-01
29 14 14940 :1: 12 488 d: 3 313 :l: 6 2.58E+01 d: 1.24E-1-00
30 14 15484 :1: 9 509 :1: 2 317 :1: 5 3.84E-l-01 :1: 1.70E+00

 

 

 

 

 

 

 

191

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Table A4: continued from previous page

 

 

 

 

 

 

IA I Z I p0 (MeV/c) I 0;, (MeV/c)I O’R (MeV/c) r up,“ (mb) I
31 14 16037 :1: 5 515 :1: 3 309 :1: 2 2.03E-1-01 :1: 7.15E-01
32 14 16601 i 24 502 :t 11 310 :1: 11 1.03E-1-01 :1: 5.14E—01
33 14 17144 :1: 14 481 :t 4 309 :1: 8 2.88E+00 :1: 1.35E—01
34 14 17656 d: 13 426 :1: 3 306 :1: 5 1.30E+00 :1: 5.72E-02
35 14 18219 :t 16 395 :1: 8 300 :1: 6 2.56E—01 :1: 1.27E—02
36 14 18704 :1: 14 376 :1: 14 315 :1: 6 9.48E—02 :1: 4.17E-03
37 14 19276 i 18 447 :1: 29 326 :1: 10 1.47E-02 :1: 7.65E—04
38 14 19894 :1: 16 410 :1: 11 273 i: 11 38613—03 :1: 1.79E-04
39 14 20382 447 243 4.43E—04 :1: 7.66E—05
40 14 20924 403 298 9.01E—05 :1: 2.20E—05
29 15 14874 433 361 1.00E-01 d: 2.33E—02
30 15 15410 :1: 42 480 :1: 33 401 :1: 16 2.40E+00 :1: 1.64E—01
31 15 16005 :1: 21 496 :t 43 316 :1: 8 2.04E-1-01 i 1.54E+00
32 15 16535 :t 9 502 :1: 2 314 i 5 2.92E+01 :1: 1.27E+00
33 15 17093 :1: 5 509 :1: 2 306 :1: 13 2.82E+01 i 1.05E+00
34 15 17624 :1: 12 468 :t 2 308 :1: 6 1.36E+01 :1:: 5.85E—01
35 15 18210 :1: 12 459 :1: 1 289 :1: 6 7.63E+00 :1: 3.43E-01
36 15 18759 :t 13 421 :1: 2 284 :t 6 2.49E+00 :1: 1.17E—01
37 15 19309 :1: 14 385 :1: 1 277 :1: 5 1.08E-1-00 :1: 4.81E—02
38 15 19833 :1: 12 378 :1: 6 284 :1: 5 2.88E-01 :1: 1.23E-02
39 15 20359 :1: 12 388 :1:: 10 290 :1: 5 8.44E—02 :1: 3.04E-03
40 15 20942 :t 13 400 :1: 2 277 :1: 7 1.36E-02 i 4.80E—04
41 15 21484 i 28 377 :1: 21 263 :1: 15 30813-03 :1: 1.88E—04
42 15 22004 365 269 3.98E—04 :1: 7.33E-05
43 15 22261 261 216 2.54E—05 :1: 1.07E-05
31 16 15900 453 384 1.07E-01 :1: 2.19E—02
32 16 16476 :1: 35 460 :1: 25 365 :t 13 3.42E+00 :1: 2.13E—01
33 16 17041 :t 25 483 :1: 11 316 :1: 11 1.56E+01 :1: 9.32E—01
34 16 17594 :1: 9 487 :1: 1 305 :1: 5 3.58E-1-01 :1: 1.63E-1-00
35 16 18152 :1: 9 489 :1: 1 296 :1: 5 3.05E+01 :1: 1.31E+00
36 16 18686 :1: 11 439 i 1 294 :1: 5 2.29E+01 :1: 9.95E-01
37 16 19280 :1: 11 453 :1: 2 287 :1: 6 1.04E-1-01 :1: 4.73E—01
38 16 19850 i 13 409 :1: 2 259 :1: 6 6.08E+00 :1: 3.16E—01
39 16 20369 :1: 22 402 :t 17 279 i 8 2.13E-1-00 :1: 1.29E—01
40 16 20932 :1: 12 336 :1: 2 254 :1: 4 9.91E—01 :t 4.42E—02
41 16 21444 :1: 10 347 :1: 11 257 :1: 4 2.09E—01 2t 85813-03
42 16 22009 :1: 9 343 :1: 2 241 :1: 5 5.14E—02 :1: 1.63E-03
43 16 22524 :1: 12 357 :1: 5 245 :1: 7 6.36E-03 :1: 2.70E—04
44 16 23086 :1: 11 308 :1: 7 225 :1: 7 1.30E-03 :1: 6.32E—05
45 16 23555 :1: 120 353 :1: 76 2.92 :1: 9.9 9.64E— 05 :E 2. 7913—05
33 17 17007 500 364 2.59E—02 :1: 6.38E-03
34 17 17576 506 370 1.06E+00 :1: 5.00E-02
35 17 18122 :1: 9 479 :t 7 293 d: 12 1.27E+01 :1: 7.23E—01

 

 

 

 

 

 

 

 

192

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Table A4: continued from previous page

 

IA I Z Ipo(MeV/c)

IO’L (MGV/C) I On (MeV/c) I

aprod (mb)

 

 

 

 

 

 

 

 

 

 

36 17 13639 1 10 472 1 1 294 1 5 2.76E+01 1 1.29E+00
37 17 19205 1 9 463 1 1 232 1 4 3.88E+01 1 1.73E+00
33 17 19746 1 11 421 1 2 233 1 5 2.64E+01 1 1.18E+00
39 17 20360 1 5 432 1 4 259 1 12 1.91E+01 1 7.76E—01
4o 17 20901 1 11 396 1 3 262 1 5 1.14E+01 1 5.69E—01
41 17 21453 1 11 370 1 3 250 1 6 7.08E+00 1 3.97E—01
42 17 22014 1 11 351 1 31 242 1 4 2.77E+00 1 1.99E-01
43 17 22556 1 3 296 1 1 216 1 3 1.31E+00 1 5.47E—02
44 17 23073 1 9 301 1 3 220 1 4 2.63E—01 1 8.60E—03
45 17 23645 1 3 231 1 1 132 1 4 5303—02 1 191153—03
46 17 240371 13 320 1 11 2171 9 33315—03 1 2.05E—04
47 17 24473 1 41 330 1 3 233 1 39 6.03E—04 1 5535-05
35 13 13066 427 311 2.39E—02 1 4.54E—03
36 13 13643 472 353 1.34E+00 1 3.14E—02
37 13 19165 1 9 461 1 7 290 1 4 8.64E+00 1 4.89E—01
33 13 19725 1 5 457 1 2 274 1 4 2.61E+01 1 9.80E—01
39 13 20271 1 9 443 1 2 269 1 5 3.68E+Ol 1 1.74E+00
40 13 20322 1 10 337 1 2 260 1 5 3.89E+01 1 1.83E+00
41 13 21412 1 10 393 1 2 250 1 5 2.93E+01 1 1.52E+00
42 13 22011 1 3 372 1 6 226 1 3 2.46E+01 1 1.43E+00
43 13 22530 1 19 326 1 3 233 1 3 1.55E+01 1 1.07E+OO
44 13 23100 1 17 313 1 3 222 1 6 9.31E+00 1 6.30E—01
45 13 23633 1 4 269 1 2 197 1 1 4.07E+00 1 1.83E—01
46 13 24202 1 6 222 1 1 133 1 2 1.63E+00 1 6.56E—02
47 13 24545 1 13 236 1 1 1771 6 4503—02 1 17419—03
43 13 25046 1 23 315 1 14 201 1 12 13917—02 1 7473—04
49 13 25614 1 5 220 1 6 130 1 1 4555—03 1 1.96E—04
50 13 26050 1 46 331 1 32 249 1 75 1.3013—04 1 31713—05
37 19 19093 394 342 7.48E—03 1 1.18E—03
33 19 19614 424 331 3.53E—01 1 5.06E—02
39 19 20210 1 9 435 1 7 231 1 2 5.49E+00 1 3.35E—01
40 19 20779 1 5 442 1 3 265 1 3 1.61E+01 1 6.35E—01
41 19 21216 1 15 351 1 1 232 1 6 3.51E+01 1 1.97E+00
42 19 21339 1 9 359 1 3 237 1 4 4.14E+01 1 2.08E+00
43 19 22472 1 9 339 1 1 216 1 4 5.00E+01 1 2.66E+00
44 19 23065 1 17 330 1 3 212 1 7 3.35E+01 1 2.77E+00
45 19 23630 1 17 295 1 3 224 1 6 4.21E+01 1 3.30E+00
46 19 24174 1 7 272 1 5 134 1 4 2.89E+01 1 2.09E+00
47 19 24757 1 5 159 1 1 116 1 1 4.25E+01 1 2.23E+00
43 19 25020 1 6 245 1 6 131 1 1 9373-01 1 5.86E—02
49 19 25512 1 9 203 1 7 173 1 4 22315—01 1 1035—02
50 19 26113 1 5 162 1 1 120 1 2 1565—01 1 6.28E—03
51 19 26273 1 40 225 1 21 204 1 13 1.36E—03 1 13513—04

 

193

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Table A4: continued from previous page

 

I A I Z I p0 (MeV/c) I 0;, (MeV/c) I 03 (MeV/c) I

(Tm-0d (mb)

l

 

 

 

 

 

 

 

 

 

 

52 19 27043 1 15 231 1 52 2071 73 5.66E—04 1 1.11E—04
53 19 27734 1 24 196 1 19 162 1 56 4.47E—04 1 1135—04
40 20 20737 400 269 3.69E—01 1 2303-02
41 20 21294 1 13 416 1 3 235 1 6 3.52E+00 1 23113—01
42 20 21320 1 9 403 1 3 257 1 1 9.42E+00 1 8.17E—01
43 20 22343 1 16 301 1 1 232 1 6 2.90E+01 1 1.64E+00
44 20 22969 1 10 304 1 1 197 1 5 4.67E+01 1 2.93E+00
45 20 23516 1 10 273 1 1 204 1 5 4.93E+01 1 3.58E+00
46 20 24174 1 11 269 1 2 202 1 6 7.64E+01 1 6.79E+00
47 20 24362 1 6 270 1 6 199 1 1 1.34E+02 1 9.62E+00
51 20 26590 1 22 165 1 26 142 1 7 1.04E+00 1 15319—01
52 20 27496 1 14 133 1 12 126 1 7 4996-03 1 4255—04
42 21 21724 1 22 375 1 17 239 1 16 4.99E—02 1 3.96E—03
43 21 22331 1 15 336 1 11 299 1 5 3315-01 1 6.45E—02
44 21 22343 1 20 336 1 12 273 1 5 1.87E+00 1 21313—01
45 21 23377 1 17 246 1 41 201 1 3 6.43E+00 1 30413—01
46 21 23390 1 40 276 1 17 243 1 19 4.3034001 9.36E—01
47 21 24363 1 3 204 1 4 269 1 7 7.93E+00 1 7143— 01
43 21 25164 1 12 250 1 3 190 1 5 3.23E+00 1 2103—01
49 21 25669 1 51 329 1 29 231 1 96 4.10E— 01 1 7.7311— 02
50 21 25905 1 1 410 1 39 409 1 42 2.56E—02 1 6.21E—03
44 22 22694 1 131 3741 142 342 1 125 6.05E—03 1 13717—03
45 22 23299 1 74 354 1 30 393 1 34 6. 01E— 02 1 2. 3613— 02
46 22 23322 1 26 393 1 22 314 1 13 9295-02 1 1133-02
47 22 24433 1 59 296 1 50 234 1 45 1.45E— 01 1 2.93E—02
43 22 24995 1 107 375 1 47 426 1 99 90113—02 1 1395—02
49 22 25466 1 16 209 1 43 276 1 13 3536-01 1 5473-02
50 22 262071 21 279 1 10 206 1 10 4395— 02 1 4255—03
51 22 26743 1 17 123 1 21 93 1 3 4043‘— 02 1 6.54E—03

 

 

194

 

 

 

 

 

 

 

 

 

 

A.3 Results for 58Ni projectile
Table A5: Fitting results for the reaction system
58Ni+9Be.
I A I Z Ipo (MeV/c) IaL (MeV/c) I 0’}; (MeV/c) I 0pm,; (mb)
10 5 5450 496 263 4.11E+00 1 5.11E—01
11 5 5723 514 377 6.28E+00 1 1.82E+OO
12 5 5396 393 233 3.23E—01 1 13011—01
12 6 6326 531 233 7.64E+00 1 52213—01
13 6 6745 547 401 4.85E+00 1 5.40E—01
14 6 7033 457 304 9.00E—01 1 1.45E—01
13 7 6957 131 130 1.51E—01 1 5.68E—02
14 7 7313 493 334 3.05E+00 1 4.07E—01
15 7 7673 1 21 472 1 17 403 1 12 5.77E+00 1 3.26E—01
16 7 3203 501 317 4.01E—01 1 5.64E-02
17 7 3467 441 330 1.00E—01 1 1.64E—o2
16 3 3305 536 330 5.85E+00 1 1.1713100
17 3 3696 1 13 457 1 14 333 1 3 2.35E+00 1 1.56E—01
13 3 9233 594 333 9.79E—01 1 9.02E—02
19 8 9614 570 392 1.31E—01 i 3.33E—02
17 9 3373 507 402 24213—01 1 54511—02
13 9 9263 607 402 1.51E+OO 1 1.70E—01
19 9 9730 1 17 490 1 13 391 1 7 2.54E+00 1 1.31E—01
20 9 10269 1 12 461 1 4 411 1 14 9.66E—01 1 44313—02
21 9 10616 593 341 2.86E—01 1 5.44E—02
22 9 11213 471 405 3.66E—02 1 79413—03
19 10 9356 455 426 2.52E—01 1 3.67E—02
20 10 10263 624 330 2.35E+00 1 1.67E—01
21 10 10752 1 14 499 1 10 393 1 5 4.07E+00 1 1.88E—01
22 10 11307 1 27 576 1 15 403 1 15 2.06E+00 1 1.07E—01
23 10 11690 474 371 2.69E—01 1 3.06E—02
24 10 11379 307 226 2.60E—02 1 775133—03
21 11 10303 526 419 2.09E—01 1 3.67E-02
22 11 11246 637 420 2.10E+00 1 2.76E—01
23 11 11755 1 13 503 1 9 397 1 5 5.21E+00 1 2.25E—01
24 11 12214 1 22 526 1 13 443 1 9 1.95E+00 1 93713—02
25 11 12676 503 476 5.93E—01 1 42313-02
26 11 13114 574 353 6.50E—02 :l: 1.22E—02
27 11 13662 434 417 91413—03 1 3.69E—03
23 12 11783 642 470 4.06E-01 :1: 5.34E-02
24 12 12216 574 436 4.36E+00 i 2.96E—01
25 12 12772 :1: 14 540 :1: 10 403 :1: 5 6.15E+00 :t 2.54E—01
26 12 13252 :1: 16 531 :1: 9 427 :1: 6 3.99E-l-00 :1: 1.81E—01

 

 

 

 

 

 

 

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Table A5: continued from previous page

 

 

 

 

 

 

 

 

 

A I Z Ipo (MeV/dial, (MeV/c) I 03 (MeV/c) L 0pm; (mb) I
27 12 13697 :1: 45 524 :1: 25 470 i 25 6.91E—01 :1: 3.96E—02
28 12 14156 559 355 1.23E-01 :t 1.46E—02
25 13 12849 650 405 1.99E—01 :E 1.93E-02
26 13 13240 562 416 2.41E+00 :1: 1.47E—01
27 13 13802 :t 15 571 :1: 10 400 :t 5 8.34E-1-00 :1: 3.27E-01
28 13 14230 :1: 13 514 :1: 8 432 :1: 4 4.01E+00 :1: 1.68E-01
29 13 14730 :1: 13 485 :1: 3 454 :1: 8 1.19E-1-00 i 4.31E—02
30 13 15180 500 480 1.51E-01 :t 1.87E-02
31 13 15633 499 417 2.25E—02 :1: 4.93E—03
27 14 13757 484 465 3.22E—01 :1: 5.86E—02
28 14 14274 618 429 6.08E+00 :1: 3.92E-01
29 14 14781 :t 20 566 :1: 11 414 :1: 6 9.45E+00 :1: 3.70E—01
30 14 15250 :t 10 498 i 5 420 :1: 3 9.79E+00 :t 3.58E—01
31 14 15727 :1: 25 504 :1: 11 458 :1: 9 1.60E-1-00 d: 7.52E-02
32 14 16218 482 465 2.78E—01 :1: 1.85E—02
33 14 16694 547 425 2.19E—02 :1: 6.85E-03
34 14 16934 480 412 1.94E—03 :1: 4.49E—04
28 15 14375 570 480 9.21E—03 :1: 2.24E—03
29 15 14739 485 477 1.39E-01 :1: 2.16E—02
30 15 15284 599 440 2.09E+00 :1: 3.58E—01
31 15 15826 :1: 20 578 d: 11 402 :1: 7 9.02E-1-00 :1: 34013-01
32 15 16287 :1: 11 507 :1: 5 417 :1: 4 7.72E-1-00 :1: 2.76E—01
33 15 16840 :1: 24 546 :1: 13 421 :1: 7 2.37E+00 :1: 1.25E—01
34 15 17315 :1: 34 589 :1: 34 430 :1: 21 4.03E-01 :1: 2.08E-02
35 15 17742 575 387 6.20E—02 :1: 5.50E—03
31 16 15767 484 475 2.28E—01 i 4.94E-02
32 16 16279 549 449 3.65E-1-00 :1: 1.82E—01
33 16 16821 :1: 22 556 :1: 10 420 :1: 8 1.03E-1-01 2E 3.86E—01
34 16 17302 :1: 10 492 :1: 4 420 :t 3 1.36E-1-01 :1: 4.58E—01
35 16 17852 :1: 19 539 :t 10 419 :1: 5 3.53E+00 :t 1.67E—01
36 16 18326 :1: 28 531 d: 16 445 :1: 14 7.99E-01 :1: 3.20E-02
37 16 18855 646 393 9.71E—02 :1: 2.55E-02
38 16 19448 596 397 1.36E-02 :1: 3.58E—03
39 16 19294 455 392 4.82E-04 :1: 1.50E-04
32 17 16139 567 478 7.92E—03 :1: 7.48E-03
33 17 16908 480 403 7.25E—02 :h 1.49E—02
34 17 17266 i 30 508 :1: 32 468 :1: 16 1.75E-1-00 :t 8.10E-02
35 17 17881 :1: 22 560 :1: 10 402 :1: 9 9.94E+00 :1: 3.63E-01
36 17 18339 :1: 13 505 :1: 5 416 :1: 4 1.19E+01 :1: 4.08E-01
37 17 18913 :1: 15 548 i 8 405 :1: 4 5.32E+00 :1: 2.36E-01
38 17 19367 :1: 27 516 :1: 15 432 :1: 10 1.02E+00 :1: 4.53E-02
39 17 19883 595 428 2.11E—01 :1: 1.27E—02
40 17 20377 539 386 24913-02 :1: 3.17E—03

 

 

 

 

 

 

 

196

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I A I Z I p0 (MeV/c) I 0L (MeV/c) I an (MeV/c) T

0pm; (mb)

 

 

 

 

 

 

 

 

 

 

 

 

41 17 20702 441 380 2.20E—O3 :1: 6.82E—04
42 17 21432 565 444 2.13E—04 :1: 1.47E—05
34 18 17141 478 473 9.22E—03 :l: 7.31E—03
35 18 17819 597 475 1.77E—01 :1: 3.74E—O2
36 18 18367 :1: 24 546 :1: 24 421 :l: 13 2.94E+00 :1: 1.13E—01
37 18 18918 :1: 22 555 :1: 9 403 i 9 1.05E+01 :l: 3.77E-01
38 18 19424 :1: 13 517 :1: 4 394 :l: 4 1.69E+01 :t 5.83E—01
39 18 19974 :1: 14 543 :1: 7 389 :l: 3 7.72E+00 :1: 3.48E—01
40 18 20495 :1: 22 548 :l: 14 395 :1: 7 1.77E+00 :1: 7.89E—02
41 18 20893 :1: 30 498 :t 28 448 :l: 20 3.24E—01 :1: 1.46E—02
42 18 21443 529 317 4.00E—02 :l: 5.57E—03
43 18 22107 574 366 4.38E—03 :t 1.42E—03
44 18 22471 527 446 3.45E-04 :1: 3.52E—05
45 18 23133 538 538 1.96E—05 :l: 2.55E—06
37 19 19006 631 348 7.68E—02 :L— 1.38E—02
38 19 19408 :1: 26 540 :l: 23 432 :t 15 1.63E+00 :l: 6.24E—02
39 19 19955 :1: 21 536 :1: 8 406 :t 11 1.02E+01 :t 3.84E-01
40 19 20439 :1: 16 510 i 5 397 :1: 6 1.57E+01 :1: 6.39E—01
41 19 21029 :1: 5 540 :1: 7 379 i 2 9.64E+00 :1: 2.97E—01
42 19 21539 :1: 12 521 :1: 2 385 :1: 3 2.68E+00 :1: 8.80E—02
43 19 22006 :t 21 530 :1: 23 411 :1: 11 6.10E—01 :l: 2.32E—02
44 19 22559 558 327 7.19E—02 i 7.82E—03
45 19 23068 453 385 8.23E—03 :1: 7.19E—04
46 19 23566 496 403 5.57E—04 :l: 9.69E—05
47 19 24072 427 373 2.76E—05 :1: 3.71E—06
38 20 19598 542 359 3.71E—03 :1: 1.10E—03
39 20 19957 551 422 1.35E—01 :1: 3.08E—02
40 20 20472 :1: 21 512 :l: 16 408 :l: 12 2.71E+00 :1: 8.97E—02
41 20 21026 d: 21 531 :1: 7 391 :1: 11 1.21E+01 :t 4.71E—01
42 20 21542 i 15 510 :1: 4 375 i 5 2.12E+01 :1: 8.66E—01
43 20 22083 :t 13 522 :l: 6 372 :l: 4 1.49E+01 :1: 6.72E—01
44 20 22617 :1: 12 478 :1: 1 363 :1: 3 5.54E+00 :1: 1.99E—01
45 20 23103 :1: 14 500 :1: 5 378 :1: 6 8.32E—01 :1: 2.29E—02
46 20 23664 543 340 1.15E—01 :l: 8.31E—03
47 20 24175 :1: 9 415 :t 2 352 :1: 3 8.57E—03 :l: 2.16E—04
48 20 24678 :1: 11 440 :t 9 362 :l: 6 5.52E—04 :t 1.63E—05
49 20 25172 i 21 458 d: 29 338 :1: 19 2.03E—05 :t 2.77E-06
50 20 23863 176 176 3.76E—07 :t 1.47E—07
41 21 21022 594 437 4.28E—02 :1: 6.55E—03
42 21 21512 :t 22 505 :l: 15 405 i 14 1.39E+00 :t 4.91E-02
43 21 22103 :1: 23 521 :t 7 370 :1: 13 1.02E+01 2t 4.48E-01
44 21 22652 :1: 16 519 :1: 5 356 :1: 6 2.05E+01 :1: 8.86E-01
45 21 23189 i 15 520 :1: 7 345 :t 4 1.89E+01 :l: 9.63E—01

 

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Table A5: continued from previous page

 

LA I ZIPo (MeV/c) 1 0L (MeV/c)I0‘R (MeV/c) I

0pm,; (mb)

1

 

 

 

 

 

 

 

 

 

 

 

46 21 23668 :t 10 465 :1: 1 355 :t 2 6.50E+00 :t 2.30E-01
47 21 24212 :1: 11 443 :1: 2 344 :l: 4 1.34E+00 :l: 3.67E—02
48 21 24766 :t 16 521 :1: 12 318 i 14 1.61E-01 :1: 5.56E—03
49 21 25285 :t 8 377 :l: 1 319 :1: 3 1.33E—02 2t 3.40E—04
50 21 25744 :1: 11 400 :1: 10 335 :1: 6 5.97E—04 i 1.79E—05
51 21 26294 470 366 2.73E—05 :1: 4.18E—06
43 22 22081 487 409 1.50E—01 :1: 1.51E—02
44 22 22612 :t 20 483 :t 11 380 :l: 11 2.20E+00 :l: 7.06E—02
45 22 23213 :1: 9 510 :1: 4 336 :l: 4 1.17E+01 :l: 3.49E—01
46 22 23763 :1: 7 507 :t 4 326 :l: 1 2.83E+01 :1: 9.87E—01
47 22 24293 :t 8 499 :l: 3 315 :t 3 2.65E+01 :l: 1.18E+00
48 22 24766 :1: 8 428 :l: 1 327 :1: 2 1.23E+01 :1: 4.38E—01
49 22 25330 :1: 10 394 :l: 1 311 :l: 3 1.74E+00 :1: 4.95E—02
50 22 25888 :t 10 449 :l: 6 296 :1: 7 2.64E—01 :t 7.39E—03
51 22 26363 :t 8 355 :t 1 297 i 3 1.35E—02 :l: 3.65E—04
52 22 26864 :1: 10 352 :l: 4 298 :t 4 8.14E-04 :l: 2.27E—05
53 22 27377 d: 30 378 :l: 16 286 :l: 20 2.20E— 05 :1: 1.58E— 06
44 23 22463 488 411 8.10E—03 :t 2.34E-03
45 23 23129 440 383 1.01E—01 :l: 7.82E—03
46 23 23688 :t 20 467 :l: 9 366 :l: 11 1.56E+00 :1: 5.13E—02
47 23 24296 :1: 9 483 :l: 3 320 :1: 3 1.05E+01 :1: 6.06E—01
48 23 24844 :1: 9 474 :l: 1 299 :1: 5 2.71E+01 :l: 1.32E+00
49 23 25379 :1: 9 466 :l: 2 294 :1: 4 3.22E+01 :l: 1.59E+00
50 23 25835 :1: 8 408 :t 2 310 :l: 2 1.40E+01 :1: 5.24E—01
51 23 26450 i 9 350 :1: 1 277 :l: 3 2.55E+00 :l: 7.59E—02
52 23 26942 :t 8 393 :h 2 279 :1: 4 2.56E—01 :l: 6.59E—03
53 23 27494 356 238 l.91E-—-02 :l: 2.69E—03
54 23 27964 :1: 10 294 :l: 50 24.9 :1: 5 6. 7119—04 :1: 6. 75E—05
55 23 28400 :1: 39 325 2t 65 249 :1: 44 1.13E—05 :1: 1.85E—06
46 24 23783 393 288 6.19E-03 :l: 1.19E—03
47 24 24222 :1: 31 427 :l: 29 382 :1: 23 1.60E—01 :l: 9.32E—03
48 24 24841 :t 20 450 :l: 8 316 :l: 12 2.02E+00 :1: 7.39E—02
49 24 25411 :1: 8 460 :t 3 303 :1: 2 1.23E+01 :l: 1.18E+00
50 24 25943 :1: 10 433 :1: 1 277 :1: 6 3.45E+01 :1: 2.94E+00
51 24 26455 :1: 10 402 d: 1 267 i 4 4.07E+01 :1: 2.19E+00
52 24 26952 :1: 7 361 :l: 3 274 i 2 1.98E+01 :l: 8.28E—01
53 24 27515 :t 11 338 :1: 1 257 :1: 3 3.09E+00 :1: 1.10E—01
54 24 28066 d: 6 327 :l: 1 236 :l: 2 3.69E—01 :l: 9.34E—03
55 24 28582 :1: 7 291 :t 2 213 :l: 1 1.28E— 02 :l: 8.53E— 04
48 25 24773 496 363 6.20E—03 :1: 1.10E—03
49 25 25298 :t 33 386 :l: 23 385 :1: 26 9.56E—02 :1: 5.71E—03
50 25 25955 :1: 11 425 :1: 4 279 :l: 6 1.54E+00 :1: 4.74E—-02
51 25 26502 i 8 413 i 4 273 :1: 2 1.17E+01 :t 4.04E—01

 

 

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I A I Z I p0 (MeV/c) I 0;, (MeV/cu UR (MeV/c) L 0pm,; (mb) I
52 25 27014 :1: 9 382 :1: 1 257 :1: 6 3.55E-1-01 :1: 3.12E+00
53 25 27566 :1: 11 360 :1: 1 228 :1: 4 4.84E+01 :1: 2.91E+00
54 25 28062 i 13 332 :1: 5 237 :1: 5 1.88E+01 :t 1.33E-1-00
55 25 28619 :1: 9 289 :1: 2 219 :1: 3 4.36E+00 i 1.68E—01
56 25 29112 :1: 16 264:1: 14 197:1: 7 2.15E—01 :1: 1.01E—02
57 25 29600 :1: 25 290 :1: 9 235 i 9 1.90E—03 :1: 13813-04
50 26 25914 334 286 5.31E—03 :1: 1.55E—03
51 26 26484 :1: 28 378 :1: 14 302 :1: 19 1.31E—01 :1: 65513—03
52 26 27098 d: 8 380 :1: 4 250 :1: 2 2.05E+00 :t 6.55E—02
53 26 27621 :1: 8 374 :1: 4 251 :1: 2 1.36E-1-01 :1: 1.46E+00
54 26 28102 :t 14 318 :1: 1 217 :1: 7 4.70E+01 :1: 4.92E+00
55 26 28651 i 12 298 :t 1 216 :1: 5 6.16E+01 i 4.86E-1-00
56 26 29191 :1: 12 239 :t 1 184 d: 4 3.23E-1-01 :1: 2.33E-1-00
57 26 29595 1 16 249 1 13 224 1 4 1.60E+00 1 12913-01
58 26 29996 :1: 12 237 :1: 2 312 d: 9 2.91E-02 :1: 2.67E—03
52 27 26994 323 285 48613-03 :1: 1.71E—03
53 27 27548 d: 22 320 d: 11 313 :1: 21 1.01E—01 :1: 5.62E-—-03
54 27 28202 :t 21 324 :1: 8 218 :1: 12 1.82E+00 :1: 1.01E-01
55 27 28694 :1: 5 278 :1: 1 204 :1: 2 1.65E-1-01 :1: 1.24E-1-00
56 27 29228 :1: 17 267 i 7 206 :1: 6 5.15E-1-01 i 4.57E-1-00
57 27 29808 174 146 6.67E-l-01 :1: 1.08E-1-01
58 27 30264 :t 19 219 :t 1 166 :1: 9 8.45E+00 :h 1.92E+00
59 27 30920 :1: 22 161 :1: 35 161 :1: 4 2.58E—01 :1: 6.27E—02
54 28 28117 251 243 5.60E-03 :1: 6.27E—04
55 28 28770 291 157 1.79E-01 :1: 2.20E-02
56 28 29407 i 8 290 i 3 198 :1: 9 3.21E+00 :1: 1.48E—01
57 28 29873 191 117 4.55E-1-01 :1: 6.72E+00
56 29 29244 1 1 239 1 11 140 1 15 1.36E—03 1 31213-04
57 29 29708 :t 25 216 :1: 2 216 :1: 2 1.41E—02 :1: 4.79E-03

Table A6: Fitting results for the reaction system
58Ni+181Ta.

I A I ZJ p0 (MeV/c) I 01, (MeV/c) I 0'}; (MeV/c) I am“; (mb) I
10 5 5246 373 353 2.63E+01 1 4.57E+00
11 5 5604 1 6 333 1 16 352 1 3 3.43E+01 1 4.64E+00
12 5 5933 344 230 3.17E+OO 1 3.55E—01
12 6 6104 459 354 4.33E+01 1 2.54E+00
13 6 6646 1 4 442 1 1 325 1 1 2.51E+01 1 1.23E+00
14 6 7091 1 6 432 1 19 400 1 1 7.48E+00 1 3943-01

 

 

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A I Z I p0 (MeV/c) I 01, (MeV/c) I03 (MeV/c) I

aprod (mb)

 

 

 

 

 

 

 

 

 

 

 

 

 

14 7 7121 500 385 1.78E+01 :1: 2.83E+00
15 7 7719 :1: 2 498 :t 4 303 :1: 1 2.57E+01 :t 1.21E+00
16 7 8174 :l: 9 366 :t 7 305 :l: 3 2.10E+00 :1: 1.01E—01
17 7 8573 452 333 8.24E—01 :l: 4.04E—02
15 8 7727 438 343 4.67E+00 :1: 6.76E—01
16 8 8124 474 367 2.56E+01 :l: 1.88E+00
17 8 8599 :l: 9 408 :1: 7 382 :1: 3 1.09E+01 :1: 5.12E—01
18 8 9212 :1: 8 418 i 4 331 d: 3 4.24E+00 :l: 1.90E—01
19 8 9547 :l: 12 391 i 17 384 i 6 7.09E-01 :1: 3.50E—02
20 8 10101 404 331 1.03E—01 :1: 1.02E—02
18 9 9151 533 408 6.80E+00 :l: 4.97E—01
19 9 9636 3‘: 9 450 :l: 8 385 :l: 3 9.20E+00 :l: 4.11E—01
20 9 10210 d: 9 399 :l: 4 364 :l: 2 4.03E+00 :1: 1.75E—01
21 9 10639 :1: 8 510 :t 18 392 :1: 4 1.42E+00 :1: 6.42E—02
22 9 11048 :1: 6 437 :1: 5 434 :l: 1 1.67E—01 :l: 6.60E—O3
20 10 10155 552 405 1.06E+01 :1: 9.64E—01
21 10 10649 :1: 8 452 :1: 7 394 d: 2 1.27E+01 :t 5.29E—01
22 10 11210 :1: 4 504 :1: 1 377 i 1 7.87E+00 :1: 3.04E—01
23 10 11688 i 8 515 d: 7 398 :t 3 1.15E+00 :1: 4.53E—02
24 10 12080 d: 19 493 :1: 28 468 :l: 10 2.54E—01 :l: 1.27E—02
25 10 12614 500 463 1.92E-02 :l: 9.20E—04
22 11 11179 543 417 7.17E+00 :l: 5.17E-01
23 11 11638 d: 8 450 :1: 7 415 :1: 3 1.45E+01 d: 5.66E—01
24 11 12175 i 8 530 :l: 6 410 :l: 2 5.97E+00 :l: 2.32E—01
25 11 12714 i 9 498 :1: 8 407 :t 3 1.81E+00 i: 6.94E—02
26 11 13124 :1: 15 479 :l: 23 461 :1: 6 2.64E-01 i 1.19E—02
27 11 13699 :1: 8 574 :1: 9 467 :l: 10 4.66E—02 :1: 1.69E—03
28 11 14169 575 467 3.98E-03 :l: 5.88E—04
24 12 12194 540 414 1.31E+01 :1: 8.14E—01
25 12 12651 :1: 9 476 :l: 7 424 :1: 3 1.51E+01 :l: 5.60E—01
26 12 13200 :1: 7 521 :l: 5 410 :l: 2 1.01E+01 :1: 3.61E—01
27 12 13700 :1: 10 534 :l: 11 421 :1: 3 1.88E+00 :t 7.11E—02
28 12 14187 i 12 566 i 40 444 :l: 4 4.08E—01 :l: 2.19E-02
29 12 14734 :1: 7 575 :t 7 468 :1: 7 3.05E—02 :l: 1.07E—03
30 12 15254 575 405 5.93E—03 :1: 3.81E—O4
31 12 15521 574 467 5.08E—04 :l: 8.98E—05
25 13 12814 421 417 5.25E—01 :l: 7.24E—02
26 13 13209 548 425 6.64E+00 i 4.40E—01
27 13 13712 :1: 10 534 :l: 8 412 :l: 4 1.80E+01 :1: 6.33E—01
28 13 14195 :1: 8 517 :t 5 421 :l: 3 8.30E+00 :1: 2.80E—01
29 13 14724 :1: 10 533 :1: 9 421 :1: 3 2.87E+00 :1: 1.01E—01
30 13 15197 :1: 12 530 :1: 21 444 :l: 4 4.01E—01 :t 1.65E—02
31 13 15732 :1: 15 608 :1: 10 464 :l: 6 7.79E—02 :1: 2.70E-03

 

 

200

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LA I Z I p0 (MeV/c) I 0;, (MeV/c) I 03 (MeV/c) I 0pm; (mb) ]
32 13 16278 :1: 29 572 :1: 10 453 :t 25 6.56E-03 :1: 2.71E-04
28 14 14243 542 416 1.31E+01 i 6.84E-01
29 14 14740 :1: 10 543 :1: 8 415 :1: 4 1.87E-1-01 :1: 6.16E—01
30 14 15207 :1: 7 499 :1: 4 425 :t 2 1.64E+01 :1: 4.98E—01
31 14 15740 :t 10 517 :1: 7 419 :t 3 3.18E-1-00 :1: 1.04E—01
32 14 16242 :t 12 546 i 17 431 :1: 4 6.19E—01 :1: 2.28E—02
33 14 16724 :1: 15 570 :1: 32 459 i 5 5.93E—02 :1: 2.79E-03
34 14 17292 :1: 26 566 :1: 11 455 :1: 14 9.44E—03 i 33813—04
35 14 17710 521 432 7.05E—04 :1: 8.39E—05
29 15 14861 466 370 3.19E—01 :1: 4.00E—02
30 15 15267 535 434 4.27E-1-00 :t 2.47E—01
31 15 15770 :1: 10 549 :1: 8 418 :1: 4 1.60E-1-01 :1: 4.96E—01
32 15 16235 :1: 8 506 :1: 4 427 :1: 3 1.17E+01 d: 3.39E—01
33 15 16784 :1: 9 538 :1: 6 415 It 3 4.47E-1-00 :1: 1.33E—01
34 15 17248 :1: 13 512 :t 14 431 :t 4 7.83E—01 :1: 2.69E—02
35 15 17751 :t 14 579 :t 32 445 :t 5 1.49E—01 :1: 6.81E-03
36 15 18294 :1: 25 601 :1: 16 458 :1: 10 1.42E-02 :1: 5.55E—04
37 15 18901 553 414 2.10E-03 :1: 1.21E—04
38 15 19261 531 445 2.12E—04 :1: 4.29E-05
31 16 15885 495 393 5.04E-01 :1: 8.28E-02
32 16 16331 563 407 6.83E-1-00 :1: 2.89E-01
33 16 16833 :1: 10 556 :h 8 399 :1: 4 1.71E+01 :1: 4.92E—01
34 16 17298 :1: 7 506 :1: 4 411 :1: 3 1.90E-1-01 :1: 4.95E-01
35 16 17808 :t 9 535 :t 5 415 :1: 3 5.69E-1-00 :1: 1.52E—01
36 16 18330 :t 11 535 :t 11 410 :1: 3 1.37E-1-00 :t 4.07E—02
37 16 18793 :1: 12 556 :1: 17 431 :1: 4 1.84E—01 :1: 6.15E—03
38 16 19324 :1: 18 570 :1: 8 442 :1: 6 2.74E-02 :1: 9.67E-04
39 16 19921 :1: 29 568 :1: 24 423 :t 16 2.71E-03 :1: 1.21E—04
40 16 20348 532 433 3.66E—04 :1: 4.35E-05
33 17 16850 570 445 1.82E-01 :1: 2.61E—02
34 17 17351 538 416 3.10E+00 :1: 1.64E—01
35 17 17867 :1: 10 552 :E 7 399 :1: 5 1.48E+01 :E 4.00E—01
36 17 18335 :1: 8 515 :1: 4 408 :1: 3 1.55E-1-01 :1: 3.77E—01
37 17 18837 :1: 9 530 i 5 414 :1: 3 7.71E+00 :1: 1.89E-01
38 17 19364 :1: 8 539 :t 6 408 i 3 1.61E-1-00 :t 4.14E—02
39 17 19852 :1: 11 540 i 24 416 :t 4 35613—0] :1: 1.31E—02
40 17 20386 :1: 13 538 i 5 416 :1: 5 5.05E—02 :1: 1.60E-03
41 17 20917 i 21 557 :E 33 426 i 9 7.15E—03 :1: 33513—04
42 17 21466 i 19 514 :1: 18 418 :1: 12 5.67E—04 :1: 2.89E-05
35 18 17940 478 373 3.08E-01 :t 6.72E—02
36 18 18402 :1: 18 524 :1: 21 396 :t 11 4.66E+00 :1: 1.70E-01
37 18 18919 :1: 10 546 :1: 7 391 :1: 5 1.49E+01 :1: 3.73E—01
38 18 19404 :1: 7 516 i 3 392 :1: 3 2.13E-1-01 :1: 4.93E—01

 

 

 

 

 

 

 

 

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A I Z I p0 (MeV/c) I 0;, (MeV/c) I UR (MeV/c) I aprod (mb) I
39 18 19899 :1: 8 530 :1: 5 399 :1: 3 1.01E-1-01 :t 2.44E—01
40 18 20420 :t 7 529 :1: 2 398 :1: 3 2.67E+00 :1: 6.43E—02
41 18 20905 :1: 10 518 :1: 5 399 :1: 3 5.03E—01 :1: 1.34E—02
42 18 21400 :1: 19 536 :1: 36 414 :t 7 8.09E-02 :1: 4.15E-03
43 18 21973 i 24 532 :1: 14 407 :1: 9 7.37E—03 :1: 3.17E—04
44 18 22508 :1: 15 492 :1: 21 400 :1: 18 69613—04 :t 3.90E—05
37 19 18883 576 455 1.50E-01 :1: 2.37E-02
38 19 19455 :1: 22 543 :1: 23 403 :1: 13 2.47E+00 :1: 9.51E—02
39 19 19960 :1: 11 532 :1: 6 390 :1: 6 1.37E+01 :1: 3.48E—01
40 19 20444 :1: 7 513 :1: 3 389 :1: 3 1.97E-1-01 :1: 4.61E—01
41 19 20927 :1: 5 504 :1: 1 394 i 2 1.20E+01 :1: 2.74E—01
42 19 21456 :1: 10 506 :1: 7 395 :t 4 3.59E-1-00 :1: 9.61E—02
43 19 21985 :1: 8 493 :l: 3 377 :1: 3 8.85E—01 :1: 2.23E-02
44 19 22445 :1: 14 527 :1: 9 402 :1: 5 1.30E—01 i 3.92E—03
45 19 23038 :1: 17 485 :t 8 374 :1: 7 1.35E-02 :1: 5.45E—04
46 19 23531 508 389 1.02E-03 :1: 1.30E—04
47 19 24138 451 367 5.89E—05 :1: 1.13E—05
39 20 20035 540 390 2.33E-01 :1: 3.22E—02
40 20 20517 i 17 505 :1: 15 375 :1: 10 3.88E-1-00 :1: 1.27E-01
41 20 21030 :1: 10 527 :1: 5 368 :1: 5 1.56E+01 :1: 3.86E—01
42 20 21529 :1: 7 511 :1: 3 365 :1: 3 2.50E-1-01 :1: 5.85E-01
43 20 22051 :1: 7 521 :1: 4 360 :t 3 1.80E+01 :1: 4.27E-01
44 20 22545 :t 8 495 :1: 6 368 :t 3 6.69E-1-00 :1: 1.71E—01
45 20 23039 :1: 8 483 :1: 7 365 :1: 3 1.17E+00 :t 3.06E—02
46 20 23504 :1: 12 500 :1: 40 380 :1: 4 1.87E—01 :1: 1.01E—02
47 20 24080 :1: 18 475 :t 8 362 :1: 8 1.42E—02 :1: 6.22E-04
48 20 24656 463 346 8.98E—04 :1: 1.51E-04
49 20 25064 416 339 4.05E-05 :t 1.06E-05
41 21 21088 504 365 6.67E—02 :t 1.30E—02
42 21 21562 d: 19 500 :1: 15 382 :1: 11 2.04E-1-00 :1: 6.97E-02
43 21 22079 :1: 10 508 :1: 5 362 :1: 6 1.24E+01 :1: 3.15E—01
44 21 22597 :1: 7 503 :1: 3 354 :1: 3 2.40E-1-01 :1: 5.70E-01
45 21 23115 :1: 7 498 :1: 4 346 :1: 3 2.21E-1-01 :1: 5.30E-01
46 21 23612 :1: 5 479 :1: 2 349 :1: 3 7.22E+00 :1: 1.70E—01
47 21 24135 :1: 7 469 :1: 4 343 :1: 3 1.82E+00 :1: 4.58E—02
48 21 24574 :1: 13 437 :1: 17 350 :t 4 2.54E—01 :1: 9.63E-03
49 21 25195 :1: 21 452 :1: 3 323 :1: 9 2.04E-02 :1: 8.20E-04
50 21 25716 :1: 26 397 :1: 16 323 :1: 9 8.49E—04 :1: 9.74E—05
51 21 26210 375 305 5.27E-05 :1: 1.33E-05
43 22 22149 407 354 2.34E—01 :1: 2.66E-02
44 22 22650 :1: 15 480 :1: 11 360 :1: 9 2.94E+00 :1: 8.99E-02
45 22 23171 :1: 9 493 :1: 4 337 :1: 5 1.36E-1-01 :t 3.40E—01
46 22 23707 i 7 489 :1: 3 319 :1: 3 3.07E-1-01 :1: 7.41E—01

 

 

 

 

 

 

 

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LA I Z I p0 (MeV/c) I 01, (MeV/c) I 03 (MeV/c)I 0pm; (mb) I
47 22 24196 :1: 4 474 :t 1 320 :1: 2 2.78E+01 :1: 6.38E—01
48 22 24708 :1: 5 439 :1: 1 315 :1: 2 1.23E+01 :1: 2.85E—01
49 22 25217 i 6 431 i 2 315 :1: 3 2.40E+00 :1: 5.88E—02
50 22 25703 :1: 11 394 :1: 12 314 :1: 4 4.02E-01 :1: 1.40E—02
51 22 26130 :1: 23 465 i 18 345 :1: 9 2.30E-02 :1: 1.11E—03
52 22 26809 350 285 9.38E—04 :1: 1.33E—04
45 23 23251 463 313 1.62E-01 :t 1.52E—02
46 23 23720 :t 17 469 :t 10 338 :1: 10 2.04E+00 :1: 6.52E-02
47 23 24259 :1: 8 478 :1: 4 314 :1: 5 1.18E+01 :1: 3.00E—01
48 23 24782 :1: 3 467 :t 2 300 :h 2 2.80E+01 :1: 6.59E-01
49 23 25279 :1: 4 444 :1: 2 297 :1: 2 3.19E-1-01 :1: 7.43E—01
50 23 25776 :1: 4 409 :1: 1 294 :1: 2 1.37E+01 :1: 3.21E—01
51 23 26315 :t 5 391 :t 1 281 :1: 3 3.37E+00 :1: 8.27E-02
52 23 26769 :1: 12 362 :1: 13 275 :1: 5 4.40E—01 d: 1.70E—02
53 23 27233 :t 21 399 :1: 5 293 :1: 8 3.07E—02 :1: 1.26E-03
54 23 27750 :1: 20 373 :1: 89 276 :1: 63 1.25E— 03 :1: 3.22E—04
47 24 24277 410 334 2.49E-01 :1: 2.12E—02
48 24 24843 i 17 450 :1: 8 296 :1: 9 2.49E-1-00 :1: 7.69E—02
49 24 25349 :1: 7 445 :1: 4 290 :t 4 1.28E-1-01 :1: 3.27E—01
50 24 25872 :1: 3 421 :t 1 271 d: 2 3.43E+01 d: 8.12E-01
51 24 26361 :1: 3 397 :1: 1 267 :1: 2 3.91E+01 i 9.16E-01
52 24 26903 :1: 4 360 :1: 1 249 :t 2 2.00E-I-01 :1: 4.81E-01
53 24 27393 i 8 359 :1: 6 263 :1: 3 3.74E+00 :1: 1.06E—01
54 24 27928 :1: 5 310 :t 1 214 :1: 3 7.17E-01 :1: 1.95E—02
55 24 28253 :1: 11 255 :1: 7 207 :t 1 2.63E— 02 :1: 2.08E—03
49 25 25372 407 325 1.36E-01 :1: 1.12E—02
50 25 25942 :1: 7 428 :1: 4 269 :1: 10 1.90E+00 :1: 5.76E-02
51 25 26437 :1: 7 403 :t 4 259 :1: 4 1.18E+01 i 3.18E-01
52 25 26948 :1: 3 375 :1: 1 243 :t 2 3.37E-1-01 :1: 8.13E—01
53 25 27457 :1: 6 349 :1: 3 235 :1: 3 4.59E+01 :1: 1.16E+00
54 25 27979 :1: 3 323 :1: 3 220 i 2 1.83E+01 :1: 4.69E—01
55 25 28469 d: 7 299 :1: 5 239 :t 3 4.71E-1-00 :1: 1.38E—01
56 25 29029 1 3 245 1 1 131 1 6 6.60E— 01 1 2.46E—02
57 25 29231 :1: 11 155 :1: 45 201 :1: 7 3.30E-02 :t 6.16E—03
51 26 26463 352 267 2.11E—01 :1: 1.32E-02
52 26 27060 :1: 6 387 :1: 4 239 :1: 4 2.41E-1-00 :1: 6.69E-02
53 26 27526 :E 6 343 :1: 4 238 :1: 4 1.37E-1-01 :1: 3.79E—01
54 26 28054 :1: 3 311 :1: 1 210 :1: 2 4.50E+01 i 1.18E-1-00
55 26 28565 i 11 298 :t 4 181 :t 8 5.73E-1-01 :1: 1.91E-1-00
56 26 29099 :1: 2 253 :1: 1 175 i 1 3.01E-1-01 :b 7.56E-01
57 26 29603 1 9 304 1 31 212 1 4 2.92E+00 1 4.64E— 01
58 26 30188 :1: 4 240 :1: 4 198 :1: 29 9.51E—01 :1: 1.44E—01
59 26 30410 :1: 18 200 i 40 198 :1: 28 2.30E—02 :1: 3.31E—03

 

 

 

 

 

 

 

 

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Table A6: continued from previous page

 

I A I Z I p0 (MeV/c) F11, (MeV/c) I an (MeV/c) I

Uprod (mb)

1

 

 

 

 

 

 

 

 

53 27 27575 1 23 339 1 15 235 1 16 143133—01 1 8.62E—03
54 27 23157 1 14 339 1 6 212 1 11 2.17E+00 1 33213—02
55 27 23623 1 4 232 1 4 133 1 2 1.46E+01 1 4.73E—01
56 27 29090 1 2 210 1 1 171 1 1 5.06E+01 1 1.22E+OO
57 27 29706 150 101 9.83E+01 1 2.27E+01
53 27 29953 1 6 136 1 3 249 1 1 6.91E+00 1 7445-01
59 27 30796 1 2 154 1 24 134 1 2 3.46E+00 1 3055—01
55 23 23669 232 163 3.04E—01 1 22313—02
56 23 29194 1 4 210 1 9 162 1 3 4.42E+00 1 1.86E—01
57 23 29639 150 122 2.47E+01 1 2.85E+00
60 23 31055 1 7 151 1 47 201 1 3 3.94E+00 1 6.38E—01

 

 

204

 

 

 

 

 

 

 

 

A.4 Results for “Ni projectile
Table A.7: Fitting results for the reaction system
“Ni+9Be.
I A I ZTpo (MeV/c) I 0;, (MeV/c) I 0‘}; (MeV/c) I aprod (mb)
10 5 5455 477 304 2.20E+00 :1: 49813-0]
11 5 6079 214 155 3.68E+00 d: 3.66E—01
12 5 6396 542 387 9.99E—01 :1: 1.13E—01
13 5 6449 413 399 1.58E—01 :l: 5.82E—02
12 6 6413 542 286 4.37E+00 :1: 8.93E—01
13 6 6785 486 295 5.85E+00 :1: 1.32E+00
14 6 7366 :1: 12 550 i 12 343 :l: 6 1.64E-1-00 :1: 1.43E—01
15 6 7765 588 420 1.69E—01 :1: 7.94E—02
16 6 8104 444 373 2.52E—02 :1: 7.69E-03
14 7 7434 574 355 1.60E+00 :1: 2.58E-01
15 7 7816 435 310 3.67E+00 :t 1.06E+00
16 7 8397 444 330 6.07E-01 :l: 6.04E—02
17 7 8826 613 438 3.40E—01 :1: 3.18E—02
18 7 9223 624 446 4.19E—02 :1: 6.75E—03
19 7 9484 469 394 5.73E—03 :1: 1.64E—03
l6 8 8369 575 322 2.87E+00 :1: 5.45E—01
17 8 8706 540 403 1.89E+00 :t 1.72E—01
18 8 9509 401 314 9.27E—01 :1: 1.16E-01
19 8 9931 634 371 3.06E—01 :1: 3.63E—02
20 8 10328 643 459 8.67E—02 d: 1.39E-02
21 8 10679 652 465 8.17E—03 2t 1.43E—03
22 8 11041 487 409 7.21E—04 :1: 2.27E-04
17 9 8908 533 324 6.4lE—02 i 1.67E—02
18 9 9358 505 376 5.78E—01 :1: 1.17E—01
19 9 9720 608 439 1.86E-1-00 :1: 3.52E—01
20 9 10379 :1: 25 643 :t 2 369 :1: 7 1.46E+00 :1: 1.07E—01
21 9 10872 :1: 27 540 d: 4 394 :1: 10 6.23E-01 :1: 4.66E—02
22 9 11263 :1: 14 510 :1: 37 471 :1: 4 1.14E—01 :1: 9.35E-03
23 9 11838 666 476 2.19E—02 :1: 4.70E—03
24 9 12159 497 480 1.28E—03 :1: 2.73E—04
25 9 12569 501 420 9.26E-05 :1: 6.79E—05
19 10 9975 469 453 7.50E—02 :1: 1.66E-02
20 10 10315 603 400 1.20E+00 :1: 2.21E—01
21 10 10762 i 34 652 :1: 31 423 :1: 13 2.45E+00 :1: 2.00E—01
22 10 I 11384 :1: 22 659 :t 3 382 :1: 6 2.30E+00 :1: 1.62E—01
23 10 11862 :t 21 666 :1: 3 412 :1: 7 5.95E-01 i 4.20E—02
24 10 12334 :1: 22 634 :1: 32 442 :1: 6 1.96E-01 d: 1.60E—02
25 10 12841 596 484 1.93E—02 :1: 3.68E—03

 

 

 

 

 

 

 

205

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Table A.7: continued from previous page

 

A I Z I p0 (MeV/c) I 0;, (MeV/c) I an (MeV/c) I

a prod (mb)

1

 

 

 

 

 

 

 

 

 

 

 

 

26 10 13304 682 487 3.03E—03 :l: 5.13E—04
27 10 13886 685 426 1.08E—04 :1: 3.23E—05
22 11 11371 659 394 8.07E—01 :1: 1.12E—01
23 11 11781 666 424 3.22E+00 :1: 4.52E—01
24 11 12299 :1: 18 633 :l: 16 419 :1: 3 1.98E+00 :t 1.48E—01
25 11 12905 :t 18 677 :1: 3 401 :t 5 1.09E+00 :1: 7.23E—02
26 11 13356 :1: 18 618 :l: 27 436 :1: 3 2.32E—01 :t 1.79E—02
27 11 13861 685 474 5.67E—02 :1: 7.69E—03
28 11 14386 600 492 5.10E—03 :l: 9.35E—04
29 11 14866 691 493 8.98E—04 i 1.71E—04
30 11 15648 641 430 1.17E—04 :l: 3.05E—05
24 12 12315 598 430 1.79E+00 :1: 3.40E—01
25 12 12787 677 429 3.41E+00 i 5.74E—01
26 12 13345 i 41 633 :l: 29 411 :1: 11 3.37E+00 :1: 2.78E-01
27 12 13926 :1: 31 655 :l: 22 407 :l: 8 1.12E+00 :1: 8.58E—02
28 12 14415 i 32 647 :1: 27 427 :1: 9 3.84E—01 :1: 3.00E—02
29 12 14940 691 435 4.90E—02 d: 7.71E—03
30 12 15416 693 478 1.22E—02 i 1.87E—03
31 12 15891 694 495 1.24E—03 :l: 2.38E—04
32 12 16430 669 431 2.14E—04 :t 5.84E—05
26 13 13352 681 429 9.27E—01 :1: 1.71E—01
27 13 13791 649 429 4.50E+00 :1: 7.61E—01
28 13 14193 :t 49 542 :l: 28 480 :1: 17 3.37E+00 :t 2.67E—01
29 13 14936 i 36 660 :1: 19 411 :l: 10 1.90E+00 :h 1.38E—01
30 13 15458 :1: 30 660 :1: 16 416 :1: 8 4.56E—01 :1: 3.21E—02
31 13 15958 691 434 1.43E—01 :1: 2.74E—02
32 13 16404 694 478 1.93E—02 :l: 2.74E—03
33 13 16944 694 495 4.29E—03 :1: 6.50E—04
34 13 17463 658 495 3.60E—04 :l: 5.95E—05
28 14 14329 637 428 2.15E+00 :1: 3.90E—01
29 14 14771 595 437 4.84E+00 :t 6.94E—01
30 14 15263 :1: 43 586 i: 29 452 :t 14 6.34E+00 :1: 5.04E—01
31 14 15884 i 38 622 :l: 19 427 d: 10 2.28E+00 :l: 1.64E—01
32 14 16486 :1: 28 660 :1: 14 413 j: 7 7.82E—01 :1: 5.29E—02
33 14 16907 i 39 570 :1: 39 454 :t 10 1.33E—01 :l: 1.18E—02
34 14 17489 693 446 3.53E—02 :l: 6.34E-03
35 14 18014 691 459 3.38E-03 :1: 4.95E—04
36 14 18497 689 492 7.65E—04 :1: 1.23E—04
37 14 19290 637 426 9.03E—05 :l: 2.98E—05
3O 15 15302 607 450 5.81E—01 :1: 1.05E—01
31 15 15815 :1: 16 613 :1: 45 430 :1: 8 4.63E+00 :l: 4.11E—01
32 15 16249 :t 53 582 :1: 34 478 :l: 22 5.30E+00 d: 4.36E—01
33 15 16861 :t 38 598 :1: 17 444 i 11 3.51E+00 :1: 2.42E—01

 

 

206

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Table A.7: continued from previous page

 

I A I Z I p0 (MeV/c) I 0;, (MeV/c) I 0’}; (MeV/QT

Uprod (mb)

1

 

 

 

 

 

 

 

 

 

 

 

34 15 17483 :1: 33 643 :l: 14 420 :1: 9 1.08E+00 :1: 7.09E—02
35 15 18033 :1: 35 666 d: 35 420 :1: 9 3.47E—01 :t 2.86E—02
36 15 18468 689 455 6.81E—02 :l: 1.26E—02
37 15 19030 617 460 1.35E—02 i 1.85E—03
38 15 19526 682 487 2.11E—03 :l: 3.56E—04
39 15 20050 677 484 3.31E—04 :1: 5.55E—05
32 16 16379 694 419 1.12E+00 :l: 1.97E—01
33 16 16811 :1: 33 590 :t 42 440 :l: 7 4.64E+00 :1: 4.00E—01
34 16 17275 :t 48 569 :1: 30 465 i 18 8.32E+00 :1: 6.63E—01
35 16 17795 :t 41 565 :t 17 475 :1: 14 4.49E+00 :t 3.02E—01
36 16 18490 :t 34 626 d: 14 423 i 10 1.92E+00 :i: 1.25E—01
37 16 19055 :1: 34 668 i 26 420 :1: 9 4.96E—01 2t 3.68E—02
38 16 19524 :1: 43 572 :1: 38 436 :l: 11 1.38E—01 :t 1.20E-02
39 16 20056 677 449 2.64E—02 :1: 3.44E—03
40 16 20579 584 469 4.43E-03 :1: 7.24E-04
41 16 21095 625 476 4.39E—04 :1: 7.86E-05
42 16 21728 659 471 6.28E—05 ﬂ: 1.44E—05
34 17 17330 616 455 4.35E—01 :1: 6.67E—02
35 17 17786 d: 36 541 :1: 37 465 :l: 7 4.44E+00 :l: 3.75E—01
36 17 18305 :1: 50 573 d: 31 462 :t 19 7.47E+00 :t 5.94E—01
37 17 18811 :1: 38 554 :l: 15 473 :t 14 6.46E+00 :1: 4.19E—01
38 17 19468 :1: 35 615 :1: 13 445 :t 11 2.49E+00 :l: 1.57E—01
39 17 20118 :1: 37 666 :1: 24 405 :1: 10 9.82E—01 :l: 7.13E—02
40 17 20554 :1: 29 615 :1: 30 432 :t 3 3.02E—01 :1: 2.14E—02
41 17 21036 i 31 534 :l: 9 450 :1: 9 7.79E—02 :l: 4.78E—03
42 17 21574 :1: 44 553 :1: 15 460 :l: 15 1.13E—02 :l: 7.56E—04
43 17 22117 618 500 1.64E—03 :1: 2.70E—04
44 17 22643 643 459 1.63E—04 :1: 2.73E—05
45 17 23411 469 394 1.92E-05 :1: 8.69E—06
36 18 18325 689 478 7.23E—01 :l: 1.14E—01
37 18 18900 632 431 3.95E+00 :1: 6.44E—01
38 18 19380 :t 46 582 :1: 29 446 i 16 9.43E+00 :1: 7.46E—01
39 18 19849 :1: 38 549 :1: 14 472 :l: 15 7.96E+00 :l: 5.29E—01
40 18 20492 :1: 36 607 :t 13 441 i 13 4.05E+00 :1: 2.55E—01
41 18 21110 :1: 36 618 i 19 409 :t 10 1.56E+00 :l: 1.05E—01
42 18 21597 :1: 7 594 :t 24 417 :t 3 5.37E—01 :l: 3.69E—02
43 18 22135 :1: 43 537 :t 36 421 :t 11 1.12E—01 :t 9.61E—03
44 18 22720 643 406 2.15E—02 :l: 2.82E—03
45 18 23168 634 453 2.50E-03 :t 3.03E—04
46 18 23761 574 446 2.32E—04 :l: 3.10E—05
47 18 24289 613 438 1.45E—05 :1: 2.46E—06
38 19 19287 682 472 3.12E—01 :1: 5.21E—02
39 19 19831 :1: 39 558 :l: 40 476 :l: 9 3.49E+00 :l: 2.99E—01

 

 

207

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Table A.7: continued from previous page

 

IA I Z Ipo(MeV/c)

I 0L (MeV/c) I 0’}; (MeV/c) I

”prod (mb)

1

 

 

 

 

 

 

 

 

 

 

40 19 20470 i 48 611 :1: 29 423 :1: 17 8.50E+00 :1: 6.64E—01
41 19 20977 :1: 38 587 :1: 15 430 :1: 15 9.19E+00 d: 6.19E—01
42 19 21561 :1: 35 615 :1: 12 424 :l: 12 5.58E+00 :l: 3.51E—01
43 19 22188 :1: 37 635 :l: 20 396 :1: 11 2.80E+00 :1: 1.92E—01
44 19 22673 :t 28 607 :1: 19 406 :1: 3 9.51E—01 :1: 6.55E—02
45 19 23271 :1: 39 611 :1: 36 387 :t 10 2.50E—01 :1: 2.16E—02
46 19 23769 624 397 4.69E—02 :l: 6.22E—03
47 19 24301 613 409 6.26E—03 :l: 9.02E—04
48 19 24806 601 429 5.53E—04 :1: 6.43E-05
49 19 25336 451 420 5.34E—05 :1: 8.62E—06
41 20 20900 561 438 3.72E+00 d: 6.05E-01
42 20 21464 :1: 51 579 :t 37 422 :l: 16 9.88E+00 :1: 8.47E—01
43 20 21994 :1: 40 581 :l: 19 429 :l: 14 1.12E+01 d: 7.90E—01
44 20 22599 :1: 38 603 :1: 16 412 :t 13 7.46E+00 :1: 4.93E—01
45 20 23220 :L- 39 626 i 19 401 :l: 13 2.97E+00 :1: 2.11E—01
46 20 23729 :1: 6 540 :1: 16 387 :t 3 1.22E+00 :1: 8.10E-02
47 20 24330 :1: 23 613 :1: 17 375 :l: 7 3.11E—01 :1: 1.82E—02
48 20 24872 :1: 34 601 :1: 27 373 :1: 9 5.48E-02 i 3.97E—03
49 20 25433 :t 31 588 :1: 34 372 :l: 9 6.64E—03 :l: 5.24E-04
50 20 25959 518 386 6.76E—04 :l: 1.08E—04
51 20 26481 558 399 6.14E—05 :1: 1.17E—05
52 20 27125 542 387 5.01E—06 i 9.38E—07
42 21 21400 659 451 2.15E—01 :1: 6.74E—02
43 21 21964 573 421 2.69E+00 :1: 5.35E—01
44 21 22474 :1: 59 545 :l: 37 434 :1: 22 8.85E+00 :l: 7.71E—01
45 21 23059 :1: 40 572 :1: 18 416 i 14 1.36E+01 :1: 9.61E-01
46 21 23642 :1: 38 589 :l: 15 411 :l: 14 9.05E+00 :1: 6.09E—01
47 21 24292 :1: 40 607 :t 17 384 :1: 15 4.73E+00 :1: 3.35E—01
48 21 24866 :1: 51 562 :l: 32 371 :l: 18 1.71E+00 2t 1.48E—01
49 21 25478 :1: 39 571 :1: 33 338 :l: 10 5.42E-01 i 4.94E—02
50 21 25957 574 354 9.60E—02 :1: 1.61E—02
51 21 26543 :1: 42 558 i 25 343 :1: 12 1.79E-02 :1: 1.40E—03
52 21 27038 :1: 29 425 d: 16 352 :1: 11 2.33E—03 :t 1.66E—04
53 21 27576 485 374 3.29E—04 :l: 4.64E—05
54 21 28066 556 403 2.75E—05 :l: 4.72E—06
44 22 22446 643 459 2.96E—01 :l: 6.74E—02
45 22 23044 :1: 46 565 :t 50 396 :1: 7 2.56E+00 :1: 2.45E—01
46 22 23576 d: 51 544 :l: 31 400 :1: 17 1.07E+01 :1: 9.02E—01
47 22 24150 :1: 41 564 :1: 17 .390 :l: 15 1.58E+01 :1: 1.14E+00
48 22 24758 :1: 33 582 :1: 13 370 :l: 11 1.33E+01 :1: 9.01E—01
49 22 25335 :1: 46 554 :1: 25 371 i 16 6.51E+00 :t 5.21E-01
50 22 25904 :1: 10 521 d: 17 358 :t 5 2.86E+00 :1: 2.20E—01
51 22 26528 :t 27 558 :l: 20 324 i 9 8.28E-01 :1: 5.60E—02

 

 

208

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Table A.7: continued from previous page

 

A I Z Ipo (MeV/c)

IO'L (MeV/c) I03 (MeV/CLI

”prod (mb) J

 

 

 

 

 

52 22 27029 d: 46 515 :1: 37 341 :l: 4 2.26E—01 2t 2.05E—02
53 22 27626 i 30 524 :l: 14 312 :l: 8 4.33E—02 :l: 3.02E—03
54 22 28184 i: 31 406 :l: 9 304 d: 9 7.34E—03 :l: 5.26E—04
55 22 28652 :1: 16 437 :1: 23 324 :t 8 9.21E-04 :1: 7.04E—05
56 22 29220 478 331 1.28E-04 :1: 1.23E—05
46 23 23505 461 440 1.34E—01 :1: 3.42E—02
47 23 24078 :1: 45 514 :1: 39 401 :1: 8 1.92E+00 :l: 1.81E—01
48 23 24672 d: 44 536 :t 26 368 :t 14 8.84E+00 :l: 7.38E—01
49 23 25263 :1: 36 553 :1: 15 353 :t 12 1.72E+01 :t 1.28E+00
50 23 25842 d: 30 563 :t 11 352 :1: 10 1.54E+01 :l: 1.08E+00
51 23 26446 :1: 39 532 :l: 21 338 :l: 12 9.57E+00 :1: 7.80E—01
52 23 27036 d: 44 522 :l: 28 329 :l: 14 3.71E+00 :l: 3.44E—01
53 23 27624 :t 38 509 i 23 303 :l: 10 1.74E+00 :1: 1.51E-01
54 23 28129 :t 28 396 :l: 8 301 :1: 7 4.77E-01 :1: 3.40E—02
55 23 28654 i: 26 417 :1: 5 304 :1: 7 1.49E—01 :l: 9.78E—03
56 23 29319 :1: 38 459 :l: 26 266 :t 11 2.15E—02 :1: 1.87E—03
57 23 29785 i 25 393 :l: 15 287 :1: 9 4.69E-03 :l: 3.54E—04
58 23 30268 390 295 4.77E—04 :l: 7.88E—05
59 23 30846 434 290 6.59E—05 :1: 1.43E—05
48 24 24700 601 324 1.47E—01 :1: 3.16E—02
49 24 25187 i 50 520 :l: 33 356 :l: 7 1.65E+00 :l: 1.52E—01
50 24 25755 :1: 45 510 :t 23 355 :l: 16 8.91E+00 :1: 7.42E-01
51 24 26380 i 34 540 :t 13 323 :t 12 1.83E+01 :1: 1.38E+00
52 24 26953 :1: 16 542 :l: 10 321 :1: 8 1.97E+01 :1: 1.33E+00
53 24 27547 :1: 39 505 :1: 19 304 :1: 12 1.21E+01 :l: 9.97E—01
54 24 28092 :t 20 446 :l: 11 307 :1: 7 7.21E+00 d: 5.11E—01
55 24 28657 :1: 51 445 :l: 24 305 :l: 21 2.89E+00 :l: 2.68E—01
56 24 29235 :1: 27 416 :t 21 282 :l: 12 1.24E+00 :l: 1.06E—01
57 24 29749 :1: 27 389 :l: 7 277 :t 8 3.47E—01 :1: 2.52E—02
58 24 30322 :1: 28 377 :1: 7 265 :l: 9 8.41E—02 :1: 6.31E—03
59 24 30888 :1: 29 344 :1: 24 248 :l: 12 1.13E—02 i 1.10E—03
60 24 31414 :t 27 333 :1: 7 237 :l: 8 1.87E—03 :l: 1.45E—04
50 25 25657 574 410 7.77E—02 :1: 1.66E—02
51 25 26230 :1: 67 459 i 45 360 :1: 27 1.17E+00 :1: 1.34E—01
52 25 26826 :1: 46 482 :1: 20 336 :1: 20 7.12E+00 :1: 6.30E—01
53 25 27500 :t 29 521 :1: 10 290 :l: 10 1.77E+01 :1: 1.37E+00
54 25 27999 :1: 13 481 :l: 9 308 :l: 13 2.12E+01 :l: 1.61E+00
55 25 28622 :1: 18 451 :l: 11 281 :1: 8 1.84E+01 :l: 1.39E+00
56 25 29168 :1: 22 410 :l: 4 282 :l: 8 1.02E+01 :l: 7.72E—01
57 25 29728 :t 40 392 d: 10 292 :1: 8 6.30E+00 i 6.02E—01
58 25 30333 d: 30 377 :1: 9 251 :t 13 2.80E+00 :1: 2.47E—01
59 25 30886 :1: 22 310 :1: 4 223 :l: 5 1.30E+00 :1: 9.83E-02
60 25 31448 :1: 29 305 :l: 76 223 :l: 12 2.53E—01 i 4.34E—02

 

 

 

 

 

 

209

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Table A.7: continued from previous page

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I A I Z I p0 (MeV/c) I01, (MeV/c) I 0’}; (MeV/c) I aprod (mb) I
61 25 31989 i 32 297 :1: 4 200 :t 9 6.87E-02 :1: 6.17E—03
62 25 32519 i 53 210 i 47 164 :1: 14 1.94E—03 i 6.08E—04
52 26 26783 485 343 5.53E—02 :1: 1.21E-02
53 26 27354 :1: 55 445 :1: 34 311 :1: 21 8.52E-—-01 :1: 9.12E—02
54 26 28020 :t 38 481 :1: 7 271 :1: 5 5.89E+00 :1: 4.97E—01
55 26 28552 i 8 467 i 8 280 :1: 8 1.59E+01 :1: 8.94E—01
56 26 29128 :1: 16 441 :1: 14 271 :1: 5 2.51E+01 :1: 2.55E+00
57 26 29709 :1: 14 417 :1: 10 265 :1: 10 2.10E+01 :1: 1.87E+00
58 26 30304 :1: 21 347 :1: 20 235 :1: 9 1.96E+01 :1: 1.85E+00
59 26 30852 :t 24 333 :t 6 228 :1: 8 1.21E+01 :1: 1.06E-l-00
60 26 31476 :1: 25 331 :t 14 204 :1: 13 8.22E+00 :1: 8.53E—01
61 26 31991 :1: 22 275 :1: 12 191 :1: 8 3.68E+00 :1: 3.69E-01
62 26 32575 :1: 27 246 i 3 151 :1: 8 1.56E+00 :1: 1.59E-01
63 26 33040 i 8 172 :1: 44 123 :1: 20 6.09E—02 :1: 1.41E-02
54 27 27816 385 354 2.42E—02 :t 6.91E-03
55 27 28464 :1: 24 433 :1: 16 290 :1: 17 3.94E—01 d: 4.56E—02
56 27 29021 :1: 17 421 :1: 10 280 :1: 8 3.31E+00 2t 3.05E—01
57 27 29634 :1: 14 396 :1: 11 245 :1: 11 1.38E+01 :1: 1.32E-1-00
58 27 30204 :t 24 387 2t 15 265 :1: 8 2.00E-1-01 :1: 2.38E+00
59 27 30777 i 17 313 :1: 12 220 :1: 6 3.25E+01 :1: 2.83E-1-00
60 27 31394 :1: 40 297 :1: 10 199 :1: 5 2.77E+01 2E 3.24E-l-00
61 27 31975 :t 21 288 :1: 17 197 d: 13 3.10E-1-01 i 6.47E-1-00
62 27 32549 :1: 14 271 :1: 17 175 :t 6 2.07E+01 :1: 2.88E-1-00
63 27 33123 172 123 3.20E+01 :1: 7.37E-1-00
56 28 28984 385 273 8.97E—03 :1: 1.60E—03
57 28 29584 380 280 1.87E—01 i 4.93E-02
58 28 30136 :1: 14 381 :1: 11 245 d: 3 1.45E-1-00 :1: 1.64E—01
59 28 30704 :1: 19 357 :1: 10 252 :1: 9 6.88E+00 :1: 6.80E—01
60 28 31316 :1: 12 298 :1: 11 189 :1: 7 2.01E+01 :1: 2.32E+00
61 28 31922 :1: 15 293 :1: 9 162 :1: 6 2.70E-1-01 :1: 3.95E+00
62 28 32497 :1: 19 271 :1: 15 190 :1: 14 2.94E+01 :1: 4.13E+00
63 28 33069 131 119 7.19E+01 :1: 2.97E+01
60 29 31152 :1: 22 268 d: 6 179 :1: 15 32913—0] :1: 4.96E—02

Table A8: Fitting results for the reaction system
“Ni+181Ta. ' ‘

I A I Z I p0 (MeV/c) I 01, (MeV/c) I 0'}; (MeV/c) I 0pm; (mb) I
11 5 5543 449 275 6.60E+01 1 1.66E+01
12 5 6037 516 335 1.09E+01 1 2.33E+00

 

 

 

 

 

 

 

 

210

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Table A8: continued from previous page

 

LA I Z Ipo (MeV/c) I01, (MeV/c) I an (MeV/c) I

Uprod (mb)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

13 5 6665 395 293 1.66E+00 :1: 3.10E—01
14 5 6572 404 301 1.48E—01 :t 3.49E-02
12 6 6208 405 315 3.78E+01 :l: 7.18E+00
13 6 6675 :l: 15 422 :t 5 317 :1: 4 3.50E+01 :1: 2.05E+00
14 6 7070 :l: 35 487 i 22 379 :t 16 1.55E+01 :1: 1.07E+00
15 6 7526 480 393 1.31E+00 :t 2.23E—01
14 7 7188 477 457 1.59E+01 d: 1.96E+00
15 7 7694 :1: 19 453 :t 9 335 :1: 6 3.73E+01 :1: 2.22E+00
16 7 8132 i 23 573 :1: 9 380 i 8 6.30E+00 i: 3.66E—01
17 7 8669 :1: 18 432 :t 3 380 :l: 17 2.10E+00 :1: 1.42E—01
18 7 9108 440 408 2.76E—01 :1: 3.85E—02
16 8 8233 573 352 2.38E+01 :1: 3.09E+00
17 8 8664 :1: 23 499 :l: 17 376 :1: 8 1.41E+01 :1: 8.73E—01
18 8 9173 :1: 18 595 :t 8 380 :l: 5 1.07E+01 :1: 5.89E—01
19 8 9620 i 24 620 :l: 23 422 :t 12 2.35E+00 :1: 1.48E—01
20 8 10098 599 457 5.71E—01 :t 1.09E—01
21 8 10569 459 463 4.83E—02 i 8.58E—03
18 9 9202 595 443 5.32E+00 :l: 7.86E—01
19 9 9680 :1: 25 523 :1: 20 396 :1: 9 1.21E+01 :l: 7.60E—01
20 9 10175 :1: 21 526 :1: 2 393 :l: 6 9.77E+00 i 5.28E—01
21 9 10669 :1: 24 621 :1: 7 414 :1: 9 4.25E+00 :1: 2.34E—01
22 9 11128 :1: 27 464 d: 21 454 :1: 19 6.04E—01 :l: 4.18E-02
23 9 11589 469 473 1.14E—01 i 2.18E—02
20 10 10156 453 448 7.34E+00 :l: 1.26E+00
21 10 10721 d: 19 596 i 44 399 :1: 4 1.47E+01 :1: 1.14E+00
22 10 11253 :1: 30 584 :l: 26 393 :1: 8 1.29E+01 :t 9.23E-01
23 10 11710 :t 16 635 :1: 15 417 :1: 5 3.25E+00 :1: 1.76E-01
24 10 12172 :1: 24 634 :1: 18 441 i 5 1.00E+00 :l: 6.31E—02
25 10 12632 524 481 9.50E—02 :1: 1.27E-02
26 10 13126 650 484 1.62E—02 :1: 2.65E—03
21 11 10797 540 463 3.93E—01 :l: 1.21E—01
22 11 11272 628 404 5.03E+00 :l: 6.96E—01
23 11 11733 611 414 1.65E+01 :1: 2.93E+00
24 11 12239 d: 33 606 j: 30 413 :1: 9 1.02E+01 i 7.27E—01
25 11 12766 :1: 16 645 :1: 7 409 :t 5 5.24E+00 :l: 2.67E—01
26 11 13245 :1: 21 650 :t 32 432 :l: 8 1.02E+00 :1: 6.31E—02
27 11 13671 492 476 2.18E—01 :1: 3.43E—02
28 11 14176 656 489 2.57E—02 i 3.29E—03
29 11 14720 654 491 4.45E—03 :t 6.30E—04
24 12 12256 618 421 9.17E+00 :1: 1.43E+00
25 12 12759 612 412 1.51E+01 :t 2.48E+00
26 12 13251 :1: 35 561 :1: 24 417 :1: 10 1.49E+01 :1: 1.01E+00
27 12 13817 i 17 653 :l: 27 408 :1: 6 4.67E+00 :l: 2.68E—01

 

211

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Table A8: continued from previous page

 

A I Z I p0 (MeV/c) I 01, (MeV/c) I an (MeV/c) I

Uprod (mb)

1

 

 

 

 

 

 

28 12 14260 :1: 31 646 :l: 22 437 :1: 9 1.55E+00 :l: 1.00E—01
29 12 14711 :1: 49 568 :l: 56 462 :t 16 1.89E—01 :t 1.72E—02
30 12 15237 i: 30 598 i 58 484 :1: 16 4.41E—02 :1: 3.92E—03
31 12 15721 661 493 5.23E—03 :t 8.52E—04
32 12 16226 627 429 9.50E—04 i 1.73E—04
26 13 13238 494 450 3.70E+00 :1: 7.95E—01
27 13 13725 533 441 1.68E+01 :1: 2.16E+00
28 13 14259 :1: 38 544 :l: 24 423 d: 11 1.25E+01 :1: 8.55E—01
29 13 14827 :1: 32 613 :1: 19 416 :1: 9 6.72E+00 :l: 4.41E—01
30 13 15310 :1: 32 646 :l: 18 428 d: 10 1.59E+00 :l: 9.83E—02
31 13 15811 :1: 41 626 :t 40 443 :l: 12 4.43E—01 :l: 3.66E—02
32 13 16258 520 473 5.68E—02 :1: 8.24E—03
33 13 16786 :1: 30 661 :1: 29 493 i: 23 1.52E—02 :l: 9.96E—04
34 13 17304 590 492 1.47E—03 :t 2.49E—04
27 14 13768 568 477 3.12E—01 :l: 5.31E—02
28 14 14322 656 413 7.86E+00 :l: 9.27E—01
29 14 14804 600 421 1.56E+01 :1: 2.48E+00
30 14 15300 i 40 584 :1: 28 423 :1: 13 1.99E+01 :l: 1.42E+00
31 14 15830 :t 34 605 :1: 18 424 :l: 10 7.23E+00 :l: 4.55E—01
32 14 16412 :1: 30 643 :t 17 404 i 8 2.30E+00 :l: 1.52E—01
33 14 16822 :1: 44 592 :l: 35 444 :l: 12 3.91E-01 :l: 3.26E—02
34 14 17306 600 462 1.05E—01 :1: 1.48E—02
35 14 17790 658 491 1.16E—02 :l: 1.48E—03
36 14 18344 656 489 2.66E—03 :t 4.56E—04
37 14 18797 483 360 2.75E—04 :1: 6.68E—05
29 15 14816 573 490 9.39E—02 :1: 1.56E—02
30 15 15298 594 451 1.93E+00 :l: 3.28E—01
31 15 15819 592 431 1.29E+01 :1: 1.47E+00
32 15 16336 :1: 48 602 :l: 32 424 :1: 17 1.47E+01 :1: 1.11E+00
33 15 16840 :1: 33 569 :1: 16 425 :1: 10 9.93E+00 i 5.96E—01
34 15 17413 :1: 28 653 i 13 419 j: 9 2.87E+00 :h 1.72E-01
35 15 17886 d: 37 599 :1: 23 431 :l: 10 9.38E—01 :t 6.33E—02
36 15 18367 :1: 49 562 :1: 42 441 :1: 14 1.69E—01 :1: 1.47E—02
37 15 18886 653 458 3.99E—02 :1: 4.66E—03
38 15 19395 597 484 6.04E—03 :l: 9.95E—04
39 15 19940 590 414 1.07E—03 :1: 1.87E-04
32 16 16368 661 419 3.09E+00 d: 4.79E—01
33 16 16843 585 432 1.14E+01 :l: 1.34E+00
34 16 17384 :1: 43 603 :1: 30 419 :t 14 1.98E+01 :l: 1.47E+00
35 16 17869 :1: 36 578 :i: 15 423 :h 9 1.10E+01 d: 5.72E—01
36 16 18443 :t 34 623 :l: 15 412 :1: 11 4.66E+00 :1: 2.94E—01
37 16 18954 :1: 37 613 d: 25 412 :1: 10 1.23E+00 :1: 8.70E—02
38 16 19428 :1: 48 633 i 38 435 :l: 14 3.45E—01 :l: 2.95E—02

 

 

 

 

 

 

 

212

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Table A8: continued from previous page

 

I A I Z Ipo (MeV/c) I 0,, (MeV/CU 0'3 (MeV/c) I

0pm; (mb)

1

 

 

 

 

 

 

 

 

 

 

 

39 16 19921 645 455 6.50E—02 i 1.05E—02
40 16 20402 521 477 1.22E—02 :l: 2.25E-03
41 16 20922 635 473 1.59E—03 :1: 2.03E—04
42 16 21315 628 468 2.75E—04 d: 6.19E—05
34 17 17402 660 427 1.15E+00 :l: 2.00E—01
35 17 17873 :1: 20 569 :t 39 433 :l: 7 9.67E+00 :1: 7.92E—01
36 17 18396 d: 51 595 :l: 32 426 :1: 20 1.61E+01 :l: 1.24E+00
37 17 18913 :1: 35 576 :l: 15 421 :1: 12 1.42E+01 :1: 8.59E—01
38 17 19474 i 37 627 :1: 15 416 :t 14 5.56E+00 :1: 3.54E-01
39 17 19989 :1: 39 623 :1: 23 419 :1: 13 2.20E+00 :l: 1.49E—01
40 17 20483 :1: 29 613 :1: 19 414 :l: 4 6.63E-01 :l: 4.22E—02
41 17 21003 d: 36 566 i 30 416 i 13 1.54E—01 :1: 1.59E—02
42 17 21435 :1: 50 524 :1: 52 453 :l: 15 2.46E—02 :l: 2.48E—03
43 17 22026 621 463 3.53E—03 :t 4.60E—04
44 17 22550 453 457 3.04E—04 :l: 5.39E-05
35 18 17885 573 491 5.18E—02 :1: 1.36E—02
36 18 18441 656 402 1.67E+00 :l: 3.21E—01
37 18 18936 613 416 8.21E+00 :l: 1.35E+00
38 18 19473 :1: 46 610 :l: 30 410 :t 16 1.86E+01 :1: 1.40E+00
39 18 19922 :1: 40 566 :t 15 436 :l: 17 1.56E+01 :l: 1.01E+00
40 18 20563 :1: 32 635 :1: 14 388 :1: 10 8.07E+00 :1: 5.12E—01
41 18 21016 :t 36 592 :l: 17 414 :l: 12 3.23E+00 :1: 2.06E—01
42 18 21547 :1: 44 589 :t 27 406 :1: 14 1.01E+00 :1: 7.48E—02
43 18 22093 :1: 50 602 i 41 398 :1: 15 2.17E—01 :1: 1.90E—02
44 18 22516 556 435 4.25E—02 :1: 6.49E-03
45 18 23004 604 450 5.87E—03 :1: 9.91E—04
46 18 23731 595 399 5.67E—04 :l: 9.19E—05
39 19 19972 603 418 6.97E+00 :t 1.11E+00
40 19 20508 :1: 48 606 :1: 30 406 :l: 18 1.63E+01 :1: 1.26E+00
41 19 21021 :1: 36 581 :l: 14 400 :1: 13 1.76E+01 :l: 1.11E+00
42 19 21555 :1: 35 598 :1: 13 410 :l: 13 1.05E+01 :t 6.72E—01
43 19 22093 :1: 36 593 :t 16 396 :1: 12 5.33E+00 :1: 3.45E—01
44 19 22609 :1: 39 589 i 21 395 :1: 12 1.68E+00 d: 1.16E—01
45 19 23114 :1: 52 514 :1: 41 398 :l: 15 4.29E—01 :l: 4.12E—02
46 19 23617 595 406 8.28E—02 :1: 1.17E—02
47 19 24157 :1: 37 584 :l: 34 409 :1: 5 1.20E—02 :1: 8.78E-04
48 19 24729 573 404 1.04E—03 :1: 1.47E—04
49 19 25292 546 418 1.45E—04 i 2.69E-05
40 20 20429 473 439 9.18E—01 :l: 1.31E—01
41 20 21004 579 418 6.91E+00 :t 7.90E—01
42 20 21573 :1: 44 599 :l: 27 389 :1: 15 1.82E+01 d: 1.39E+00
43 20 22093 d: 38 589 :l: 15 386 :t 15 2.17E+01 d: 1.46E+00
44 20 22624 :t 30 589 :l: 11 380 :t 10 1.55E+01 :1: 9.76E—01

 

 

213

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Table A8: continued from previous page

 

I A41 Z IPo (MeV/c)

I 0;, (MeV/c) I 03 (MeV/c) I

Uprod (mb)

1

 

 

 

 

 

 

 

 

 

 

45 20 23135 :1: 39 596 :1: 18 385 :l: 12 6.46E+00 :1: 4.44E-01
46 20 23692 :1: 37 583 :t 20 375 :1: 12 2.27E+00 :l: 1.65E—01
47 20 24209 :t 29 584 :1: 30 382 :l: 11 5.10E—01 :l: 3.67E—02
48 20 24791 573 358 9.38E—02 :1: 1.57E—02
49 20 25251 560 380 1.37E—02 :t 2.29E—03
50 20 25847 547 350 1.57E—03 :1: 2.49E-04
51 20 26324 434 368 1.59E—04 j: 2.45E—05
42 21 21426 464 468 3.75E—01 :l: 7.29E—02
43 21 22028 i: 73 562 :t 15 426 :1: 26 4.63E+00 :1: 5.02E—01
44 21 22589 :1: 48 578 :t 26 401 :t 19 1.54E+01 :1: 1.21E+00
45 21 23162 i 37 588 :1: 14 374 :1: 15 2.46E+01 :1: 1.70E+00
46 21 23686 :1: 17 595 :1: 9 375 :1: 10 1.75E+01 :1: 1.11E+00
47 21 24182 :t 42 561 :1: 18 379 j: 14 9.69E+00 :1: 7.15E—01
48 21 24714 :1: 25 573 :l: 11 376 :1: 10 3.33E+00 :1: 2.40E—01
49 21 25293 :t 47 548 :t 32 358 i 14 9.71E—01 d: 8.93E-02
50 21 25840 480 357 1.43E—01 :1: 2.52E-02
51 21 26367 :t 56 492 :1: 42 350 :1: 17 3.13E—02 :1: 3.26E—03
52 21 26888 464 352 4.47E—03 d: 7.64E—04
53 21 27376 i 47 499 :1: 25 357 :t 17 7.67E—04 :l: 6.19E—05
54 21 27891 387 358 7.58E—05 :l: 1.69E—05
44 22 22547 472 400 4.53E—01 :l: 1.00E—01
45 22 23080 544 413 4.14E+00 i 5.03E—01
46 22 23663 :1: 49 565 :1: 24 380 :1: 21 1.70E+01 :l: 1.37E+00
47 22 24239 :1: 17 584 :l: 10 356 :1: 8 2.67E+01 :l: 1.64E+00
48 22 24767 :t 16 573 :t 19 349 :l: 7 2.45E+01 d: 1.58E+00
49 22 25242 :1: 34 542 :t 15 362 :t 11 1.27E+01 :1: 9.01E—01
50 22 25820 i 41 532 :1: 18 344 :l: 13 5.06E+00 :1: 3.71E—01
51 22 26372 :t 42 522 :1: 19 338 :1: 4 1.25E+00 :1: 9.67E—02
52 22 26903 :1: 33 454 :t 10 330 :l: 11 3.46E-01 :l: 2.41E—02
53 22 27432 :t 37 471 d: 27 329 :1: 6 7.04E—02 :t 6.25E-03
54 22 27973 i 36 430 d: 39 324 :t 13 1.36E—02 :1: 1.26E—03
55 22 28450 460 343 1.96E—03 :l: 2.02E—04
56 22 29068 404 293 2.64E—04 2t 5.40E-05
46 23 23526 488 443 2.18E—01 :l: 4.47E—02
47 23 24186 564 388 2.78E+00 :t 4.81E—01
48 23 24758 :1: 43 563 :l: 21 353 :l: 16 1.28E+01 :l: 1.02E+00
49 23 25307 :1: 18 560 :1: 5 337 :1: 9 2.67E+01 d: 1.85E+00
50 23 25812 d: 35 544 :t 16 340 :1: 13 2.63E+01 :1: 1.98E+00
51 23 26315 :1: 21 532 i 14 358 :1: 14 1.65E+01 :1: 1.26E+00
52 23 26893 :1: 30 544 :l: 25 340 :l: 16 6.26E-l—00 :1: 5.14E—01
53 23 27465 :1: 22 499 :t 10 313 :1: 8 2.57E+00 :l: 1.71E-01
54 23 27982 i 37 449 :1: 24 320 :l: 16 6.67E—01 :t 5.55E—02
55 23 28576 :1: 29 425 :1: 21 292 :t 14 1.86E—01 :l: 1.60E—02

 

 

214

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Table A8: continued from previous page

 

A I Z I p0 (MeV/c)T0L (MeV/c) I 0'}; (MeV/c) I

0pm; (mb)

 

 

 

 

 

 

56 23 29052 :1: 35 420 :1: 13 299 :1: 13 3.56E—02 :l: 2.83E—03
57 23 29582 413 300 8.15E-03 :1: 1.29E—03
58 23 30141 386 287 7.43E—04 :l: 1.36E-04
59 23 30673 355 265 1.43E—04 i 1.35E—05
48 24 24690 533 366 1.89E—01 :l: 4.08E-02
49 24 25240 :t 64 511 :1: 40 345 :1: 26 2.15E+00 :1: 2.22E—01
50 24 25804 i 39 526 i 18 337 j: 14 1.13E+01 :1: 9.02E—01
51 24 26348 :1: 19 532 d: 4 335 :t 12 2.35E+01 :l: 1.81E+00
52 24 26889 :1: 14 530 i 10 332 :1: 3 2.64E+01 :1: 2.03E+00
53 24 27423 :1: 22 499 :1: 8 316 :l: 10 1.87E+01 :1: 1.43E+00
54 24 27994 :1: 26 471 :l: 8 297 i 10 1.10E+01 :1: 9.64E-01
55 24 28502 :1: 23 446 i 5 298 :t 8 4.36E+00 :l: 3.31E-01
56 24 29108 :t 32 395 j: 34 278 :1: 14 1.48E+00 :l: 1.44E—01
57 24 29619 :1: 32 405 i 10 278 :l: 13 4.28E—01 :l: 3.56E—02
58 24 30143 :1: 15 386 j: 11 287 :l: 16 1.14E—01 :t 1.24E—02
59 24 30662 382 272 1.79E—02 :1: 3.78E—03
60 24 31242 306 243 2.33E—03 :l: 5.44E—04
50 25 25729 473 343 9.02E—02 :1: 1.89E—02
51 25 26304 :1: 54 488 i 32 330 :l: 21 1.43E+00 :1: 1.41E-01
52 25 26872 :t 40 504 :h 16 316 j: 15 8.53E+00 d: 7.10E—01
53 25 27444 :1: 17 499 :1: 3 293 :1: 8 2.36E+01 :1: 1.72E+00
54 25 27977 :1: 8 494 d: 15 297 :1: 5 2.81E+01 :l: 1.57E+00
55 25 28514 :1: 14 465 :1: 10 292 :t 5 2.54E-l-01 i 2.04E+00
56 25 29069 :1: 23 477 :1: 22 306 :1: 15 1.39E+01 :l: 1.36E+00
57 25 29651 i: 23 415 :t 4 253 :l: 9 8.95E+00 i 7.17E—01
58 25 30201 :1: 29 386 d: 18 249 :t 15 3.20E+00 :1: 3.08E—01
59 25 30730 :t 29 348 :t 9 237 :1: 11 1.36E+00 :l: 1.22E—01
60 25 31261 :t 27 335 :l: 6 223 :t 10 3.17E—01 :1: 2.79E—02
61 25 31812 280 208 7.78E—02 :1: 1.23E-02
52 26 26780 414 350 6.61E—02 :l: 1.34E—02
53 26 27358 :t 63 452 :1: 29 331 :l: 32 1.01E+00 :1: 1.09E—01
54 26 27968 :1: 17 475 j: 11 290 :l: 9 6.86E+00 :1: 5.40E—01
55 26 28492 :1: 14 471 :1: 10 297 :l: 3 1.75E+01 :t 1.59E+00
56 26 29030 :1: 12 463 :l: 10 285 :t 1 2.76E+01 :1: 2.60E+00
57 26 29571 d: 17 446 :t 12 299 :t 5 2.64E+01 :l: 2.31E+00
58 26 30149 d: 23 386 :1: 3 250 :l: 13 2.65E+01 :t 2.58E+00
59 26 30717 d: 8 393 :l: 15 236 :l: 3 1.34E+01 :1: 9.19E—01
60 26 31319 :1: 11 336 :1: 16 202 :l: 19 9.14E+00 :1: 7.89E—01
61 26 31839 i 24 289 :1: 10 194 :1: 10 3.30E+00 :l: 3.32E—01
62 26 32386 :1: 13 230 :1: 2 171 :l: 4 1.50E+00 :1: 1.26E—01
54 27 27816 389 358 2.77E—02 :1: 4.90E-03
55 27 28496 :1: 56 440 :t 11 264 :l: 9 4.95E—01 :1: 4.73E—02
56 27 29018 :1: 14 443 :1: 11 273 :1: 4 3.46E+00 d: 3.32E—01

 

 

 

 

 

 

 

215

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Table A8: continued from previous page

 

 

 

 

 

I A I Z I p0 (MeV/c) I Q, (MeV/c) I 03 (MeV/c) I 0pm, (mb) I
57 27 29589 :1: 18 413 d: 2 242 :1: 8 1.61E+01 :1: 1.35E-1-00
58 27 30122 :1: 22 386 i 4 244 :1: 12 2.87E-1-01 :1: 2.88E-1-00
59 27 30677 :1: 27 355 :t 1 221 :1: 10 4.18E+01 d: 4.42E-1-00
60 27 31246 :1: 23 320 :t 2 201 :1: 7 4.00E+01 :1: 3.48E+00
61 27 31824 280 208 3.76E+01 :1: 4.65E+00
62 27 32380 230 171 2.49E+01 :1: 3.26E-1-00
63 27 32961 164 100 2.62E+01 :1: 5.33E+00
56 28 29038 437 267 1.15E-02 :t 2.40E—03
57 28 29538 373 251 1.96E—01 :t 4.88E—02
58 28 30104 371 247 1.93E-1-00 :1: 3.28E—01
59 28 30641 :1: 13 367 :t 13 227 :t 2 7.13E-1-00 :1: 8.93E—01
60 28 31164 d: 22 353 :1: 12 289 :1: 9 1.78E+01 :1: 1.86E+00
61 28 31672 :1: 38 324 i 25 234 :1: 16 2.05E+01 :1: 2.84E-1-00
62 28 32406 :1: 7 170 :1: 4 127 d: 10 5.94E+01 :1: 5.85E+00
63 28 32943 :1: 3 121 :1: 5 90 :1: 3 2.32E-1-02 :t 2.26E+01
59 29 30526 :1: 18 294 :1: 54 217d: 50 3.20E—02 :1: 8. 75E—03
60 29 31093 1 33 310 1 20 254 1 71 29013—01 1 5.46E— 02

 

 

 

 

 

 

 

 

216

A.5 Results for 86Kr projectile

Table A9: Fitting results for the reaction system

 

 

 

 

 

 

86Kr+9Be.

I A I Z I p0 (MeV/c) I 0L (MeV/c) I 0’}; (MeV/c) I 0pm; (mb) I
52 25 16606 996 629 6.42E—01 :1: 8.53E—02
53 25 16831 :1: 32 884 :1: 20 698 :t 18 2.11E+00 i 1.42E-01
54 25 17239 i 17 857 :1: 11 528 :1: 3 3.23E+00 It 2.09E—01
55 25 17670 :1: 46 847 :1: 15 508 :1: 23 3.18E-1-00 :t 2.18E—01
56 25 17928 :t 56 801 :1: 17 609 :t 46 1.66E-1-00 :1: 1.34E—01
57 25 18331 :1: 63 803 :1: 20 550 It 48 9.05E—01 d: 7.44E—02
58 25 18729 981 468 3.57E—01 :1: 1.01E—01
59 25 19096 726 627 1.13E—01 :1: 5.19E-02
60 25 19001 718 459 2.49E—02 d: 6.49E—03
53 26 16505 880 653 5.49E—02 :1: 1.95E-02
54 26 17140 917 682 6.71E—01 :t 1.40E-01
55 26 17529 :1: 62 878 :1: 28 552 :t 33 2.49E-1-00 :1: 1.85E—01
56 26 17854 :1: 49 830 :1: 18 551 :t 24 4.97E+00 :1: 3.46E—01
57 26 18213 :1: 45 814 :1: 15 541 :1: 23 4.40E+00 :1: 3.05E—01
58 26 18560 :1: 45 786 d: 14 551 :1: 24 3.13E+00 :1: 2.19E—01
59 26 18818 :1: 53 744 :1: 17 611 :1: 32 1.57E-1-00 d: 1.17E-01
60 26 19259 :1: 78 804 :1: 56 585 :1: 43 7.04E—01 :1: 6.46E-02
61 26 19588 797 614 2.38E-01 i 4.37E-02
62 26 20374 356 295 4.06E—02 :1: 6.09E—03
63 26 20383 692 442 1.65E—02 :1: 3.52E—03
64 26 20745 682 473 4.36E—03 :1: 2.02E—03
65 26 20957 790 429 7.63E-04 :1: 5.37E—04
55 27 17410 951 480 4.54E—02 :1: 1.09E—02
56 27 17712 858 644 5.71E—01 :1: 1.27E—01
57 27 18119 :1: 49 828 :1: 23 530 :1: 23 2.99E-1-00 :1: 2.22E—01
58 27 18450 :1: 44 803 :1: 17 523 :1: 21 5.94E+00 :1: 4.17E—01
59 27 18824 :1: 37 792 :t 13 512 :E 15 7.20E+00 :t 4.83E-01
60 27 19142 :1: 43 752 :t 14 537 :t 19 4.80E-1-00 :1: 3.28E—01
61 27 19516 :t 44 742 :t 14 535 :t 22 2.96E+00 :1: 2.06E-01
62 27 19843 :t 49 717 :1: 20 569 d: 25 1.32E-1-00 :1: 9.64E-02
63 27 20231 i 45 764 :1: 45 598 :1: 31 5.42E—01 :1: 4.59E—02
64 27 20490 :t 80 725 :1: 35 585 i 33 1.69E—01 :1: 1.45E—02
65 27 21143 909 581 4.74E—02 :1: 1.31E—02
66 27 21512 660 483 1.00E—02 :1: 3.65E—03
67 27 21854 684 415 2.96E—03 :1: 1.43E—03
68 27 22126 748 406 5.05E—04 :1: 2.79E-04
69 27 22485 623 398 8.90E—05 :1: 4.39E—05
57 28 17907 847 472 2.77E—02 :1: 5.26E—03
58 28 18288 806 633 4.49E—01 :1: 69613—02

 

 

 

 

 

 

 

 

217

continued on next page

Table A9: continued from previous page

 

 

 

 

 

I A I Z Ipo (MeV/c) I 01, (MeV/CH UR (MeV/c) I aprod (mb) I
59 28 18696 i 58 794 :1: 24 538 :1: 29 2.74E+00 i 2.10E—01
60 28 19006 :1: 44 761 :1: 16 545 :t 23 7.62E+00 i 5.49E—01
61 28 19407 :1: 38 761 :1: 13 516 :1: 15 9.85E+00 d: 6.68E-01
62 28 19746 :t 39 730 :1: 14 528 d: 14 8.69E+00 i 5.82E—01
63 28 20105 :1: 41 717 :1: 14 541 :1: 21 5.10E-1-00 :1: 3.59E—01
64 28 20513 :t 45 728 :1: 20 520 :1: 18 2.53E-1-00 :1: 1.77E-01
65 28 20800 :1: 75 714 :1: 46 591 :1: 39 1.06E-1-00 :t 9.08E—02
66 28 21255 :1: 42 670 :1: 13 537 :1: 18 3.43E-01 :1: 2.53E—02
67 28 21894 797 415 9.34E—02 :t 2.11E—02
68 28 21899 636 550 2.50E—02 :1: 6.50E—03
69 28 22793 842 445 4.75E—03 :1: 1.81E-03
70 28 22940 608 389 1.65E—03 :1: 6.80E-04
71 28 23470 593 379 1.88E—04 :1: 7.82E—05
72 28 23777 781 499 4.01E—05 :1: 2.20E-05
60 29 18936 804 579 3.13E—01 :1: 4.28E-02
61 29 19201 :1: 52 743 :1: 21 559 :1: 30 2.26E-1-00 :t 1.79E—01
62 29 19585 :1: 42 739 :1: 16 542 :t 23 7.35E-1-00 :1: 5.37E—01
63 29 19978 :1: 39 738 d: 13 541 :1: 19 1.24E+01 :1: 8.67E—01
64 29 20437 :1: 37 747 :1: 12 489 :1: 15 1.16E-1—01 :1: 7.86E—01
65 29 20740 :1: 38 704 i 13 519 :1: 16 8.67E-1-00 :1: 5.81E—01
66 29 21144 :1: 42 713 :1: 19 505 :1: 16 4.68E-1-00 i 3.24E—01
67 29 21457 d: 50 668 :1: 26 542 :1: 20 2.30E+00 d: 1.72E—01
68 29 22036 :1: 62 815 :1: 43 494 :1: 22 7.90E—01 :t 6.53E—02
69 29 22347 i 49 651 :1: 15 515 :1: 19 2.60E-01 i 1.98E—02
70 29 22944 823 452 8.28E—02 d: 1.68E-02
71 29 23518 803 397 1.77E-02 :1: 5.58E-03
72 29 23848 577 369 3.91E—03 :1: 1.46E-03
73 29 24238 560 358 1.00E—03 :1: 3.05E—04
74 29 24501 542 419 2.02E—04 :1: 5.91E-05
75 29 24870 522 334 2.56E—05 :1: 1.07E—05
76 29 25252 501 320 4.18E—06 d: 1.39E-06
77 29 25619 479 306 4.60E—07 i 1.57E-07
78 29 25900 475 315 8.79E—08 :1: 2.36E-08
79 29 26064 427 273 1.47E-08 :1: 7.23E-09
61 30 19134 780 614 1.74E—02 :1: 4.37E—03
62 30 19499 765 567 2.81E-01 i 6.74E—02
63 30 19858 :1: 23 731 d: 14 442 :1: 14 2.23E-1-00 :1: 1.63E—01
64 30 20212 :1: 41 718 :1: 15 530 d: 19 9.03E+00 :1: 6.59E-01
65 30 20620 :t 36 721 :1: 12 503 :1: 16 1.65E+01 d: 1.14E+00
66 30 21033 :1: 36 712 :1: 12 483 :1: 13 1.88E+01 :1: 1.25E-1-00
67 30 21374 :1: 38 689 :1: 12 501 i 14 1.34E+01 i 8.87E—01
68 30 21725 :1: 49 671 :1: 20 531 d: 19 7.76E-1-00 :1: 5.34E—01
69 30 22233 :1: 45 700 :1: 19 474 :1: 16 3.83E+00 :L' 2.68E-01

 

 

 

 

 

 

 

 

218

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Table A9: continued from previous page

 

A I Z I p0 (MeV/c) I01. (MeV/c)I OR (MeV/c) I

Uprod (mb)

l

 

 

 

 

70 30 22623 :1: 66 711 :1: 34 510 j: 28 1.56E+00 :l: 1.28E-01
71 30 22970 3: 35 628 :1: 8 494 :t 13 6.21E—01 i 4.31E—02
72 30 23559 :1: 46 526 :l: 4 408 :l: 13 2.14E—01 :l: 1.58E—02
73 30 24177 594 358 5.02E—02 d: 1.55E—02
74 30 24522 733 383 1.65E—02 :1; 6.40E—03
75 30 24959 522 334 2.39E-03 2t 6.24E—04
76 30 25295 501 320 8.77E—04 :l: 3.10E—04
77 30 25540 479 328 1.68E—04 i 7.05E—05
78 30 25866 614 304 3.61E—05 :l: 8.83E—06
79 30 26147 531 337 5.52E—06 :l: 1.47E—06
80 30 26418 398 344 1.06E—06 j: 3.52E—07
63 31 19694 692 525 9.28E—03 i 2.89E—03
64 31 20048 719 581 1.68E—01 :l: 4.28E—02
65 31 20470 :1: 58 722 :l: 22 523 :t 43 1.64E+00 d: 1.41E—01
66 31 20840 :t 38 708 :l: 14 478 :t 19 7.50E+00 2t 5.54E—01
67 31 21233 :1: 37 700 j: 13 499 :l: 16 1.75E+01 i 1.22E+00
68 31 21633 :1: 35 691 d: 12 482 :1: 13 2.22E+01 :1: 1.48E+00
69 31 22030 i 40 679 i 13 485 :1: 17 1.96E+01 :1: 1.32E+00
70 31 22413 3: 59 654 :l: 9 486 :1: 5 1.32E+01 :1: 7.73E—01
71 31 22788 :1: 25 647 :1: 6 483 :t 4 6.98E+00 d: 4.28E—01
72 31 23378 d: 51 730 d: 27 430 :1: 20 3.17E+00 :l: 2.34E-01
73 31 23769 i: 27 694 :l: 23 484 i 24 1.53E+00 :l: 1.18E—01
74 31 24093 :1: 40 600 :l: 4 453 :l: 14 4.76E—01 :1: 3.54E—02
75 31 24642 :1: 45 502 :t 5 388 :t 14 1.61E—01 :1: 1.26E—02
76 31 25179 401 335 4.87E—02 :t 6.55E—03
77 31 25625 338 288 1.50E-02 :1: 2.02E—03
78 31 25974 303 280 3.49E-03 d: 1.00E—03
79 31 26152 282 332 9.06E—04 :1: 5.30E—04
80 31 26427 479 344 2.21E-04 :l: 9.77E—05
81 31 26841 366 284 2.38E—05 :t 5.43E—06
65 32 20413 754 527 8.12E—03 :1: 1.84E—03
66 32 20687 707 571 1.34E—01 :l: 3.35E—02
67 32 21154 :1: 55 718 :l: 21 484 :l: 37 1.47E+00 :1: 1.22E-01
68 32 21568 d: 38 716 :1: 14 413 :l: 16 7.62E+00 :1: 5.42E—01
69 32 21882 :1: 35 688 i 12 456 :1: 17 1.90E+01 :1: 1.36E+00
70 32 22208 :1: 16 659 i 8 507 :l: 4 2.71E+01 :1: 1.78E+00
71 32 22612 :1: 16 646 d: 8 499 :t 12 2.48E+01 d: 1.59E+00
72 32 23087 :1: 10 634 i 5 468 i 23 2.03E+01 :l: 1.19E+00
73 32 23498 d: 16 638 i: 9 488 :l: 5 1.12E+01 :1: 7.00E—01
74 32 23953 :1: 11 585 :l: 2 432 :1: 6 6.95E+00 :1: 3.85E—01
75 32 24400 :1: 13 607 :1: 11 449 :l: 11 3.08E+00 :t 1.78E—01
76 32 24807 :1: 13 584 :l: 5 432 :h 3 1.04E+00 :1: 6.19E—02
77 32 25518 :1: 13 639 :1: 27 306 j: 3 4.47E-01 :l: 4.25E—02

 

 

 

 

 

 

 

219

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Table A9: continued from previous page

 

IA I Z I p0 (MeV/c) IaL (MeV/QT O’R (MeV/c)I

Uprod (mb)

1

 

 

 

 

 

 

 

 

 

 

78 32 25923 324 276 1.52E-01 :1: 2.08E-02
79 32 26222 352 282 6.16E—02 ﬂ: 8.07E—03
80 32 26614 398 273 2.00E-02 :t 5.94E-03
81 32 26961 276 231 3.78E—03 :t 9.06E—04
82 32 27249 349 223 1.24E—03 :t 2.27E—04
68 33 21336 694 550 7.90E—02 :t 1.39E—02
69 33 21746 :1: 52 695 :t 19 449 i: 33 8.70E—01 :1: 7.15E—02
70 33 22045 :1: 19 656 d: 10 517 :t 5 5.23E+00 :1: 3.57E—01
71 33 22446 i 18 648 i 9 509 :1: 4 1.64E+01 :t 1.11E+00
72 33 22719 :1: 3 604 :t 5 454 i 2 2.34E+01 :1: 1.61E+00
73 33 23293 :1: 13 632 :1: 7 480 :1: 2 3.07E+01 :l: 1.92E+00
74 33 23720 :t 17 596 :t 3 444 :l: 8 2.51E+01 :t 1.56E+00
75 33 24302 :t 34 643 :1: 12 387 :t 11 1.69E+01 :1: 1.09E+00
76 33 24725 d: 22 632 :t 11 433 :l: 15 9.52E+00 :t 6.17E-01
77 33 25336 i 20 647 :l: 19 309 :t 8 6.69E+00 :t 4.56E—01
78 33 25708 i 30 614 :l: 10 371 i 21 2.79E+00 :l: 2.37E-01
79 33 26186 :1: 13 574 :l: 16 273 :l: 3 1.62E+00 :1: 1.46E—01
80 33 26585 d: 15 518 d: 18 255 :l: 8 7.10E-01 :t 7.22E—02
81 33 26917 :1: 9 469 :1: 20 234 :1: 2 2.80E—01 :1: 2.98E-02
82 33 27006 327 356 9.64E—02 :1: 2.88E—02
83 33 27474 218 249 3.33E—02 2t 7.87E—03
70 34 21932 667 526 5.19E—02 :l: 9.66E—03
71 34 22336 :1: 29 656 i 19 518 :1: 11 6.22E—01 :1: 4.66E—02
72 34 22723 :1: 21 641 :1: 12 503 :1: 11 4.30E+00 :t 3.05E—01
73 34 23110 d: 19 633 d: 9 500 :1: 3 1.39E+01 :l: 9.55E—01
74 34 23648 :1: 39 651 :l: 12 417 :t 21 2.92E+01 :t 2.12E+00
75 34 23962 :1: 18 616 d: 9 489 :1: 4 3.19E+01 :1: 2.06E+00
76 34 24500 :1: 27 608 :l: 12 433 :l: 18 3.23E+01 :l: 2.17E+00
77 34 24965 :1: 20 603 :1: 9 414 :l: 80 2.27E+01 :1: 2.29E+00
78 34 25554 :1: 22 614 :l: 6 330 i 16 1.81E+01 :1: 1.34E+00
79 34 25954 :1: 17 578 :i: 4 370 :t 59 1.13E+01 :l: 9.92E—01
80 34 26421 :1: 23 539 :l: 4 274 :l: 10 6.62E+00 :l: 5.36E-01
81 34 26900 495 234 4.65E+00 :l: 8.77E—01
82 34 27083 347 257 2.46E+00 :1: 5.88E—01
83 34 27468 375 286 1.12E+00 :l: 3.22E-01
84 34 27933 319 181 7.53E—01 d: 2.04E—01
72 35 22687 689 381 2.71E—02 :t 8.00E—03
73 35 23093 678 465 3.50E—01 :l: 9.94E—02
74 35 23363 :1: 22 625 :1: 12 497 j: 5 2.35E+00 :l: 1.67E—01
75 35 23922 :1: 41 648 :l: 14 384 i 24 1.04E+01 :1: 8.16E-01
76 35 24330 :1: 38 631 :1: 12 385 :l: 22 2.23E+01 :l: 1.65E+00
77 35 24844 :1: 38 640 :t 11 351 i 21 3.18E+01 :1: 2.30E+00
78 35 25269 :1: 43 599 :l: 15 341 :1: 22 3.24E+01 i 2.44E+00

 

 

220

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Table A9: continued from previous page

 

I A I Z FPO (MeV/c) I31, (MeV/dim; (MeV/c) I

0pm,; (mb)

1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

79 35 25732 d: 19 578 :1: 9 356 i 27 3.09E+01 :l: 2.63E+00
80 35 26163 :t 10 539 :i: 8 255 :1: 6 2.16E+01 :t 1.27E+00
81 35 26646 i: 15 495 :l: 2 316 :l: 12 2.17E+01 :l: 1.46E+00
74 36 23276 651 346 9.44E—03 :1: 3.57E—03
75 36 23742 639 410 1.79E—01 :1: 5.95E—02
76 36 24256 i 20 651 j: 13 320 d: 18 1.42E+00 :1: 1.09E—01
77 36 24643 i: 41 629 :l: 14 347 i 26 6.15E+00 d: 4.89E—01
78 36 25058 :t 34 603 :l: 11 355 :l: 17 1.58E+01 :l: 1.14E+00
79 36 25511 :1: 50 571 :1: 20 336 :t 27 2.55E+01 :t 2.13E+00
80 36 25948 :t 21 539 j: 2 329 j: 17 3.10E+01 :t 2.60E+00
81 36 26468 542 455 3.45E+01 :l: 4.61E+00
82 36 26885 445 285 3.81E+01 :1: 4.25E+00
Table A.10: Fitting results for the reaction system
86Kr+181Ta.

I A Z I p0 (MeV/c) I 0L (MeV/c) I03 (MeV/c) I aprod (mb) I
52 25 16640 :1: 30 816 :l: 16 509 :l: 20 1.12E+01 :1: 7.85E—01
53 25 16805 :i: 6 763 i 16 587 :1: 10 2.65E+01 :1: 1.88E+00
54 25 17272 i 21 852 :1: 18 535 :l: 4 3.15E+01 :1: 2.17E+00
55 25 17738 i 26 905 i 16 484 :1: 16 2.53E+01 i 1.93E+00
56 25 18142 d: 56 952 :l: 10 480 :l: 35 1.29E+01 :1: 9.01E—01
57 25 18378 924 652 5.98E+00 j: 8.07E—01
54 26 17271 i 29 797 :l: 19 518 :t 19 8.77E+00 i 6.45E—01
55 26 17462 i 6 748 :1: 14 569 d: 9 2.27E+01 :1: 1.56E+00
56 26 17957 :1: 19 838 :1: 16 518 i 5 3.24E+01 :l: 2.16E+00
57 26 18263 :t 26 858 d: 17 587 :l: 6 2.29E+01 :l: 1.54E+00
58 26 18732 911 673 1.68E+01 :1: 1.43E+00
59 26 18979 908 621 7.69E+00 :l: 1.81E+00
56 27 17963 :1: 31 814 i 30 519 j: 25 5.70E+00 :1: 5.46E—01
57 27 18304 i 12 804 i 9 483 :t 32 1.76E+01 i: 1.14E+00
58 27 18619 :t 11 823 :1: 20 494 j: 3 2.78E+01 :1: 1.70E+00
59 27 18927 i 22 840 i 15 564 :1: 4 2.68E+01 :l: 1.74E+00
60 27 19430 904 463 1.75E+01 :l: 1.59E+00
61 27 19628 887 598 1.31E+01 :l: 4.39E+00
58 28 18639 i 32 811 j: 32 511 d: 37 4.23E+00 d: 4.80E—01
59 28 18960 :1: 22 815 :l: 20 502 :l: 12 1.21E+01 :1: 8.42E—01
60 28 19269 i: 17 811 :l: 15 499 d: 3 2.45E+01 :l: 1.60E+00
61 28 19581 :1: 19 818 d: 14 536 i 3 2.59E+01 :l: 1.67E+00
62 28 20015 :1: 33 879 :l: 17 630 :l: 9 2.05E+01 :1: 1.30E+00

 

 

 

 

 

 

 

 

221

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Table A.10: continued from previous page

 

_—

A I Z IPo (MeV/c) I 0;, (MeV/c) I 0’}; (MeV/c) I

0pm; (mb)

l

 

 

 

 

 

 

 

 

 

63 28 20278 867 560 1.09E+01 :t 3.06E+00
61 29 19590 :t 25 780 :l: 21 503 :1: 11 8.51E+00 i 5.68E—01
62 29 19907 d: 20 805 :t 18 510 :1: 4 1.88E+01 j: 1.31E+00
63 29 20255 :t 21 815 :l: 15 541 :t 5 2.45E+01 :l: 1.55E+00
64 29 20784 i 19 884 :l: 13 440 :1: 8 2.11E+01 :l: 1.35E+00
65 29 21220 936 702 1.90E+01 i 5.91E+00
66 29 21368 802 527 1.26E+01 :l: 2.40E+00
63 30 20254 :1: 73 757 d: 36 488 :1: 38 7.81E+00 i 6.21E—01
64 30 20599 i 26 767 :l: 20 510 i 9 1.74E+01 :l: 1.06E+00
65 30 20954 :1: 19 783 i 16 494 :h 6 2.29E+01 d: 1.39E+00
66 30 21279 :1: 21 782 j: 15 520 :1: 5 2.33E+01 i 1.62E+00
67 30 21724 :1: 39 831 :l: 20 601 i 17 1.71E+01 :1: 1.08E+00
68 30 21994 :1: 22 759 :t 17 481 :1: 3 1.05E+01 :l: 8.36E—01
66 31 21220 :1: 77 744 :1: 36 488 :1: 38 1.02E+01 :1: 1.33E+00
67 31 21592 :t 23 747 i 16 496 :l: 14 2.06E+01 j: 1.61E+00
68 31 21952 d: 24 766 :t 15 517 :t 7 2.04E+01 :1: 1.18E+00
69 31 22339 :1: 29 772 :1: 18 537 :l: 9 1.81E+01 :l: 1.21E+00
7O 31 22845 801 597 1.31E+01 :1: 1.51E+00
71 31 23130 :1: 28 747 :1: 24 459 :1: 7 9.61E+00 :t 1.71E+00
68 32 21903 :1: 42 715 i 27 501 :1: 34 8.53E+00 :1: 1.02E+00
69 32 22190 i: 21 707 :l: 17 437 :t 4 1.79E+01 d: 2.06E+00
70 32 22657 :1: 21 749 :1: 17 469 :1: 5 2.19E+01 :1: 1.49E+00
71 32 23060 i 37 751 :t 20 527 :l: 12 1.86E+01 :1: 1.67E+00
72 32 23260 :1: 46 706 i: 22 525 :t 26 1.34E+01 :1: 1.00E+00
73 32 23867 772 361 9.18E+00 i 1.68E+00
70 33 22544 675 419 5.60E+00 :1: 8.25E—01
71 33 22977 682 518 1.30E+01 :t 1.60E+00
72 33 23343 i 46 690 :t 28 495 :1: 47 1.78E+01 :1: 1.97E+00
73 33 23843 735 554 2.02E+01 i 3.33E+00
74 33 24570 843 843 2.36E+01 :1: 5.28E+00
75 33 24412 :t 31 613 :t 18 337 :t 19 1.13E+01 :l: 1.16E+00
71 34 23035 818 450 1.54E+00 :l: 2.45E—01
72 34 23256 650 460 4.05E+00 :l: 6.32E—01
73 34 23688 681 479 9.19E+00 :1: 1.40E+00
74 34 24100 687 495 1.53E+01 d: 2.53E+00
75 34 24535 723 541 1.96E+01 d: 3.63E+00
76 34 24901 714 536 1.88E+01 d: 3.48E+00
77 34 25391 660 309 1.49E+01 :t 3.69E+00
74 35 23927 699 443 2.22E+00 :t 3.86E—01
75 35 24316 :1: 35 648 i 33 423 :1: 49 5.40E+00 :1: 4.88E—01
76 35 24699 i 43 639 :1: 25 460 i 44 9.72E+00 :t 7.16E—01
77 35 25260 697 521 1.38E+01 :1: 1.53E+00

 

 

 

 

 

 

 

222

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Table A.10: continued from previous page

 

I AJ Z I p0 (MeV/c) I Q, (MeV/c) I (73 (MeV/c) I

0pm; (mb)

1

 

 

 

 

 

 

 

 

78 35 25610 673 500 1.72E+01 :1: 2.15E+00
77 36 25017 629 403 3.06E+00 :1: 4.15E—01
78 36 25600 665 500 6.43E+00 :l: 7.13E—01
79 36 25895 640 480 9.28E+00 :1: 2.12E+00
80 36 26241 509 367 1.22E+01 i 2.48E+00

 

 

223

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