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This is to certify that the

dissertation entitled

Study of Projectile Fragmentation Reactions
at Intermediate Energies

presented by

George A. Souliotis

has been accepted towards fulfillment
of the requirements for

Ph. D Chemical Physics

degree in

 

 

Date 1/27/92

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STUDY OF PROJECTILE
FRAGMENTATION REACTIONS AT
INTERMEDIATE ENERGIES

By

George A. Souliotis
A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the Degree of

DOCTOR OF PHILOSOPHY

Department of Chemistry

1992

ABSTRACT

STUDY OF PROJECTILE FRAGMENTATION REACTIONS AT
INTERMEDIATE ENERGIES

George A. Souliotis

The experimental measurements of the present work represent a first attempt
towards obtaining systematic data of projectile fragment distributions near 0 degrees
produced by intermediate-energy beams delivered by the K1200 cyclotron at the
N SCL. The measurements were performed with two different devices operated as
0-degree spectrometers: the K1200 interim beamline and the A1200 mass separator.
Momentum distributions and limited angular distributions of pro jectile-like fragments
(PLFs) from a 1‘N (75 MeV/ u) beam with 2"'Al and 181Ta targets were measured using
the interim K1200 beamline, but the results were of limited quality. In a first series
of measurements with the A1200, the 0-degree momentum distributions of PLFs
from several beams [2°Ne (85 MeV/u), 1"’N (70 MeV/u) and 18O (70 MeV/u)] were
measured, but only a limited number of complete momentum spectra were obtained.
In a second series, a detailed measurement of neutron-rich PLFs from the reaction 180
(80 MeV/ u) + 2"Al was successfully performed. Accurate measurements of 0-degree

momentum distributions and relative yields of PLFs were obtained and analysed.

Analysis of the centroids of the momentum-per—nucleon (velocity) distributions

of the fragments indicated that, along with the fragmentation (break-up) channels,

i

direct stripping and/or pick-up channels contribute significantly to the production of
the fragments. Interestingly, the mean velocities of nucleon pick-up products were
considerably lower than that of the beam, consistent with momentum conservation
and the additional assumption that the nucleon(s) picked-up from the target have
preferentially momenta (on average) equal to the nuclear Fermi momentum oriented
in the direction of the projectile motion. The widths of the momentum distributions
of both nucleon-removal and nucleon pick-up products were in agreement with the
independent-particle model of Goldhaber and the cluster stripping model of Friedman.
In the case of pick-up products, the momentum distributions of both the projectile
part and the picked-up nucleon(s) had to be taken into account. Calculations of
yields with the abrasion-ablation model and the intranuclear cascade model showed
that both these high-energy models were not able to describe the trend of the data.
However, calculations with Friedman’s model provided a good description of the ex-
perimental yields, implying that a break-up (and/or pick-up) mechanism involving
low excitation energy is responsible for the production of PLFs from the present light

projectile.

ii

To my Parents

Antonios and Kyriaki

iii

ACKNOWLEDGEMENTS

I wish to express my sincere thanks and deepest appreciation to my advisor, Prof.
Dave Morrissey, for his guidance, support and encouragement throughout the cource
of my dissertation work. His advice, suggestions and continuous interest in my work

and progress have been crucial for the completion of the present thesis.

I am grateful to Dr. Brad Sherrill for his contribution, help and suggestions during
the experiments and data analysis of the present work. Sincere thanks go to Dr. Dave
Mikolas for his important help, guidance and friendly discussions during the initial

stages of my work at the N SCL.

I would like to acknowledge Dr. Jeff Winger, Dr. Nigel Orr, Prof. Jerry Nolen
and all the people that helped in the various phases of the experiments. John Yurkon
and Denis Swan deserve thanks for their assistance in detector testing and operation
and Orilla McHarris for her assistance in the preparation of various figures of my

thesis work.

It has been an important experience to work with the excellent support of the
staff and faculty of the N SCL, which I acknowledge very much. My dissertation

commitee members, Profs. Dave Morrissey, Jerry N olen, Walt Benenson and Kathy

iv

Hunt deserve recognition for their support and guidance during my graduate career.

It is my pleasure to thank Prof. Aristides Mavridis of the University of Athens
for his scientific inspiration and support during my undergraduate years in Athens.
Special thanks go to Eleni Archontaki for her friendship and assistance during my

first year at MSU.

I would like to express my thankfulness and gratefulness to my family, especially
to my parents who have played a major role in my education with their devotion,
support and encouragement throughout the years. Finally, I am grateful to Eleni
not only for her friendship, thoughtfulness and encouragement during my graduate

studies, but for giving inspiration to my life as well.

Contents

LIST OF TABLES
LIST OF FIGURES
l Introduction-Motivation

2 Projectile Fragmentation: Overview-Models

I Introduction ................................
II Isotopic Distributions ...........................
III Momentum Distributions .........................

3 Operation of the K1200 Interim Beamline as a Spectrometer

I Introduction ................................
II Description of the spectrometer and the detectors ...........
III Beam optics calculations .........................
IV Electronics .................................
V The experiment .............................
VI Data analysis and results ........................

vi

ix

34

35

40

43

46

4 Projectile fiagmentation Measurements with the A1200 Mass Sep-

arator 56

I Introduction—Description of the A1200 ................ 56

II Experiment and results .......................... 6O

5 Results and Discussion 73
I Introduction ................................ 73

II Study of the centroids of the momentum distributions ......... 74

A N ucleon removal products .................... 74

B Nucleon pick-up products .................... 79

III Study of the widths of the momentum distributions .......... 83

A Calculations of the widths with Goldhaber’s model ...... 85

B Calculations of the widths with Friedman’s model ....... 88

IV Study of Relative Yields ......................... 90

6 Summary and Conclusions 97
APPENDICES 102
A Derivation of Equations of hagmentation Models 102
I Goldhaber’s Model ............................ 102

II Friedman’s Model ............................ 104

B Supplement on Calculations for the Interim K1200 Beamline Exper-

iment 109

vii

I Beam Optics concepts .......................... 109

II Derivation of the expressions for the momentum and the reaction angle113

LIST OF REFERENCES 118

viii

List of Tables

2.1

3.1

3.2

4.1

5.1

Comparison of experimental reduced widths from the fragmentation
of relativistic projectiles with theoretical values calculated with Gold-

haber’s model ................................

Characteristics of the interim beamline spectrometer and its elements.

Beam-optical characteristics of the interim beamline spectrometer. . .

Characteristics of the operation modes of the A1200. .........

Examples of break-up patterns of 130 projectiles ............

ix

27

37

41

88

List of Figures

2.1

2.2

2.3

2.4

2.5

2.6

3.1

3.2

3.3

3.4

An abrasion-ablation picture of a peripheral collision ..........
The total nucleon-nucleon scattering cross sections. ..........

A comparison of the calculated production cross sections (using the
abrasion-ablation model and the INC model) with the data from the

fragmentation of 4oAr(213 MeV/u). ...................

A comparison of experimental isotopic yields for the reaction of ‘°Ar(44

MeV/u) projectiles with 58Ni and 19"Au targets. ............

The parallel-momentum distribution of 10Be fragments from the reac-

tion of 12C projectiles at 2.1 GeV/u on a Be target. ..........

The total-momentum dispersion of projectile fragments from the reac-

tion “Kr (200 MeV/u) + 19"An ......................

Schematic layout of the interim beamline spectrometer .........

Schematic diagram of the MWPC and the delay-line position measure-

ment .....................................

Schematic diagram of the electronics for the interim beamlime experi-

ment .....................................

Calibration of the radius of curvature versus front-counter position.

9

l3

15

18

22

23

36

39

44

47

3.5

3.6

4.1

4.2

4.3

4.4

4.5

4.6

4.7

4.8

4.9

Coordinate systems used in the calculations of momentum and reaction

angle. ................................... 48

Momentum and angular distributions of several pro jectile-like isotopes

produced by a 1“N beam at 75 MeV/u on 27A1 and 181Ta targets. . . . 54
Schematic layout of the A1200 beam-analysis device. ......... 58
The principle of resistive charge division for position measurements. . 61

Schematic diagram of the electronics for the fragmentation experiments

with the A1200 ............................... 63

Momentum/nucleon distributions of PLFs produced by a ”Ne (85

MeV/u) beam on 181Ta and 9Be targets. ................ 66

Momentum/ nucleon distributions of PLFs produced by a 15N (70 MeV/u)

beam on a 181Ta. target. ......................... 67

Momentum / nucleon distributions of PLFs produced by a 18O (70 MeV / u)
beam on a 181Ta target. ......................... 68
Rigidity vs Image #2 position calibration for the 18O (80 MeV/u) +

2"Al experiment. ............................. 69

An example of 3. AE vs A/q histogram from the 18O (80 MeV/u) +

"Al experiment. ............................. 70

Momentum / nucleon distributions of PLFs produced by a uBO (80 MeV/u)

beam on an 2"Al target. ......................... 71

4.10 Momentum / nucleon distributions of PLFs produced by a 18O (80 MeV/u)

beam on an 2"Al target. ......................... 72

xi

5.1

5.2

5.3

5.4

5.5

5.6

5.7

B]

Variation of the centroids of the parallel momentum/ nucleon distribu-

tions of nucleon-removal products from the reaction 1‘30 (80 MeV/ u)

+ 2"Al ....................................

Variation of the centroids of the parallel momentum/ nucleon distribu-
tions of nucleon pick-up products from the reaction 180 (80 MeV/u)

+ 27Al ....................................

Variation of the centroids of the momentum] nucleon distributions of
few-nucleon pick-up products from the reactions studied in the present

work .....................................

Variation of the widths of the parallel momentum distributions of

projectile-like fragments from the reaction 180 (80 MeV/u) + 2"Al.

Relative yields of projectile-like fragments from the reaction 180 (80
MeV/u) + "Al ...............................

Plot of the function W = ln(7,,,:cg/S) versus 202:0 for the yields of

projectile-like fragments from the reaction 180 (80 MeV/ u) + ”Al.

Comparison of the relative yields of PLFs from the reaction 180 (80

MeV/u) + 2"A1 with calculations using Friedman’s model. ......

Coordinate systems used in general beam optics calculations ......

xii

75

80

84

86

91

94

95

111

’-

Chapter 1

Introduction—Motivation

Projectile fragmentation is a specific nuclear reaction that takes place when heavy
projectile nuclei (i.e. heavier than ‘He) undergo peripheral collisions with target
nuclei, mainly at intermediate and high bombarding energies. A large number of
isotopes can be produced during the fragmentation of a heavy projectile. These
fragments (pro jectile-like fragments-PLFs) emerge in a narrow cone in the forward
direction with velocities near that of the projectile. Among the different PLFs, various
exotic (radioactive) nuclei can be produced. These exotic nuclei can be separated
and used as secondary (radioactive) beams, allowing the possibility of a new class of
experiments to study nuclear structure and nuclear reaction mechanisms. For these
reasons, projectile fragmentation has become a very important and powerful tool to
produce and study exotic nuclei. The technique of converting a primary beam into a
secondary radioactive beam by projectile fragmentation has been succesfully used at
several laboratories: LBL [Alon 84], GANIL [Anne 87] and RIKEN [Kubo 90]; also,
fragment separators are in the final construction stages for other laboratories (e.g.
GSI [Mfinz 90]). At the National Superconducting Cyclotron Laboratoty (NSCL),

this technique is realized with the A1200 beam analysis device [Sher 90].

In order to relate projectile fragmentation with other nuclear reaction phenomena,

it is convenient to use a simple classification scheme based on two parameters of the

1

reaction: the relative kinetic energy of the pro jectile—target system and the impact

parameter of the collision.

Nuclear reactions can be divided in three categories according to the bombarding
energy (usually expressed in MeV/ nucleon: MeV/ u): low energy reactions (below 20
MeV/u), intermediate (or medium) energy reactions (between 20 and 200 MeV/u)
and high (or relativistic) energy reactions (above 200 MeV/ u). In the low energy
region, the projectile kinetic energy is near the Coulomb barrier of the pro jectile—
target system (E/ A z 5 MeV). In the relativistic energy region, the projectile kinetic
energy is comparable to its rest mass (E/ A z 931 MeV). These latter energies are
large compared to typical nuclear energy scales, like nucleon separation energies or
the Fermi energy, which characterize the nuclear ground state. The changes between
the different energy regions are gradual and the numerical values mentioned here are

somewhat arbitrary.

Depending on the impact parameter of the collision, which is a measure of the

degree of proximity of the reaction partners, the collisions can be divided as follows

[e.g. Gelb 78, Frie 87]:

1. W: elastic Coulomb scattering (Rutherford scattering) or in-

elastic Coulomb scattering (Coulomb excitation or dissociation).

2. Peripheral collisions: at low energy: direct nucleon transfer and/or incomplete

 

fusion; at intermediate energy: projectile and target fragmentation and / or nu-

cleon transfer; at high energy: projectile and target fragmentation.

3. Mom: at low energy: fusion, compound nucleus reactions; at inter-
mediate and high energy: shattering of the projectile and the target into light

fragments and nucleons.

In this framework, projectile and target fragmentation reactions are the reactions
that occur when the nuclei just begin to overlap and have relatively large velocities.
From a physical standpoint, projectile and target fragmentation are the same process
in complementary rest frames. However, the experimental techniques used in their

study and their historical evolution have been quite different.

For more than three decades, a wealth of data has been accumulated for the pro-
duction of fragmentation products from proton-nucleus collisions using high-energy
proton beams. The target fragments produced by these interactions, usually called
spallation products, were studied by radiochemical methods, and extensive systemat-
ics have been established [Gold 78, Hufn 85, Morr 89]. More than a decade ago, with
the availability of relativistic energy heavy-ion beams (up to a few GeV/ u), the field
of high-energy projectile fragmentation was opened up. The main advantage of study-
ing projectile rather than target fragmentation is that, while fragments originating
from target break-up are slow-moving (and may be stopped in the target material),
fragments from the projectile are fast-moving and can be easily studied by magnetic

spectrometers.

A number of systematic measurements of production cross sections and momen-
tum distributions of PLFs from light projectiles exist at high energies [e.g. Heck 72,
Grei 75, Viyo 79]. A rather complete parametrization of the variations has been es-
tablished and, several disparate models have been successful in describing the obser-
vations [e.g. Gold 74, Oliv 79, Morr 79, Frie 83, Siimm 90a]. In general, these models
are based on the assumption of a two step process consisting of a fast collision step
and a slow deexcitation one. On the other hand, at low energies (below 20 MeV / u),
where heavy-ion beams have been available for about two decades, projectile-like
products have also been observed, but the fragmentation mechanism is not operat-

ing, because of the low relative velocity of the nuclei [Bond 71, Gelb 78, McVo 80,

Egel 81, Murp 83, Home 84]. At such low energies, direct transfer, deep-inelastic and
incomplete fusion reactions can take place during peripheral collisions of the projectile

with the target nucleus.

Intermediate energy heavy-ion beams have become available only during the last
few years and a few projectile fragmentation measurements have been performed in
this energy region [e.g. Guer 83, Blum 86, Borr 86, Bazi 90]. However, the data
obtained so far are not adequate to provide a systematic parametrization and a clear
understanding of the transition region. Apart from characteristics similar to those
observed at high energies, the data exhibit new features that may yield important
information on the reaction mechanism of peripheral collisions (e.g. orbital deflection
[Bibb 79, Silk 88]). Also, along with “pure” fragmentation channels (where nucleons
are only removed from the projectile), nucleon transfer channels have been observed
(where one or more nucleons have been picked-up by the projectile [e.g. Borr 86,
Soul 91]). In addition, the only estimates of yields and momentum distributions are
presently based on extrapolations of the high energy parametrizations for lack of a
better method. For example, at the N SCL, estimates of production rates of secondary
radioactive beams of PLFs separated by the A1200 are performed with the computer
code “IN TEN SITY” [Wing 91], which is based on the parametrizations of Siimmerer

and Morrissey [Siimm 90] of high-energy fragmentation reactions.

Therefore, in order to understand the reaction mechanisms operating in peripheral
collisions at intermediate energies and obtain parametrizations of PLF distributions,
additional refined measurements are necessary, mainly near 0" (since the pro jectile-
like isotopes are produced in a narrow cone about 0°). Such systematic data and
parametrizations, apart from a nuclear reaction standpoint, will improve our predic-
tive power to choose the proper energy and beam-target combinations to produce

radioactive beams. Measurements of reaction products at 0° are difficult in general,

because of the presence of the primary beam and require the use of magnetic spec-

trometers.

In this work, as a first attempt towards obtaining systematic data on intermediate-
energy fragmentation, the results from a series of O—degree measurements of pro jectile—
like fragments from several beams produced by the K1200 Cyclotron at the NSCL
are presented. The characteristics of the distributions of the PLFs are described and

analysed with several theoretical models.

In chapter 2, a more detailed discussion of projectile fragmentation reactions is
presented along with a description of models previously developed to interpret the
observations. For two of the models, namely Goldhaber’s model and Friedman’s

model, a derivation of the relevant equations is presented in appendix A.

In chapters 3 and 4, our measurements of projectile fragmentation distributions
at the NSCL are described in detail. In chapter 3, the preliminary measurements
performed with the interim K1200 beamline operated as a spectrometer are presented
(a discussion of Beam Optics concepts and a derivation of the equations developed
for the analysis of the data are given in appendix B). In chapter 4, the measurements
with the recently commissioned A1200 beam analysis device (operated in a O-degree

spectrometer mode) are presented.

In chapter 5, the systematic variation of the centroids and the widths of the mo-
mentum distributions and the relative yields of the observed pro jectile-like fragments
(both fragmentation and transfer products) are presented and compared with calcu-

lations performed using extended versions of the models presented in chapter 2.

Finally, in chapter 6, conclusions from the present study of intermediate—energy
fragmentation reactions are summarized and areas for further investigation are men-

tioned.

Chapter 2

Projectile Fragmentation:
Overview—Models

I Introduction

In this chapter a detailed description of the projectile fragmentation process is pre-
sented and various models used to interpret the experimental results are discussed.
In section II, the isotopic distributions of PLFs are discussed. Two high-energy mod-
els, the abrasion-ablation model and the intranuclear cascade model are described
and their predictions are compared to high-energy fragmentation data. In addition,
the features of the isotopic distributions of PLFs at lower energies are presented. In
section III, the momentum distributions of PLFs are discussed. The characteristics
of the momentum distributions at high-energies are first presented and two models
(Goldhaber’s model and Friedman’s model) are described. Also, the features of the
momentum distributions at lower energies are discussed, and various mechanisms

involved in the production of PLFs at these energies are mentioned.

II Isotopic Distributions

A large variety of fragments is produced by the fragmentation of a medium or heavy

projectile. As an example, approximately 70 isotopes were identified between 2:8

6

and 16 from the fragmentation of “Ar at 213 MeV/ nucleon [Viyo 79]. In general, it
has been found that for a given element, the variation of the cross section a( Z, A) with
respect to A, called the isotopic distribution, is approximately Gaussian in shape; for
elements close to the projectile, the centroids of the distributions are close to the
N / Z ratio of the projectile, whereas for lighter elements the centroids correspond
to isotopes closer to the valley of fl-stability. Very little systematic data exist for
pro jectile-fragmentation of heavy beams (A>40) and here one has to rely on radio—
chemical measurements of target fragments [e.g. Siimrn 90a]. The general features of
the distributions of fragments from these nuclei seem to be dominated by the effects
of statistical deexcitation of primary fragments. Two basic models have been used to
interpret the features of the isotopic distributions, one based on a macroscopic and
the other on a microscopic description of the collision. These models are outlined

below.
Abrasion—ablation model

The abrasion-ablation model [Bowm 73, Morr 78, Oliv 79] is a simple macro-
scopic, geometric model used to describe the isotopic distributions of target-like and
pro jectile-like fragments from high-energy collisions. According to this model, pro jec-
tile and target are taken to be spherical and the Coulomb deflection of the projectile
trajectory is neglected (the deflection is very small because of the high relative veloc-
ity). For a peripheral collision with a given impact parameter, projectile and target
nucleons which lie in the geometric overlap region interact strongly (participant nu-
cleons), and are removed from the initial nuclei, whereas the other nucleons do not
interact (spectator nucleons). This first step of the process, called abrasion, is fast
because of the large relative velocity of projectile and target. The remaining specta-
tor pieces, called primary fragments, are excited (it should be noted that no specific

assumption concerning the fate of the removed parts of the projectile and the target

is necessary to describe projectile or target fragmentation). In the original model,
the excitation energy of the primary fragments was assumed to originate from surface
tension: the abraded primary fragment, having a “bite” on it, has an odd shape with
a surface that is larger than that of a spherical shape of the same volume; the sur-
face energy is given by the product of this “extra” surface times the nuclear surface
tension coefficient (with a typical value of ~ 1.0 MeV/fm’) [Bowm 73]. Apart from
the excitation energy due to surface tension, which could be considered as a lower
limit, additional excitation energy can be deposited to the primary fragments by
nucleon—nucleon collisions in the participant region; some nucleons, scattered out of
this region, are directed towards the projectile and the target spectators, depositing

part of their energy in the spectators by further nucleon—nucleon collisions [Oliv 79].

From the geometric character of the abrasion, the number of nucleons removed
from the projectile ap(b) can be calculated as a function of the impact parameter
b of the collision [e.g. see Morr 78]: ap(b) is proportional to the overlap volume
of the projectile (with mass Ap) and the target (with mass AT), (see Fig. 2.1).
From the function ap(b), the cross section 0(App) for the production of all primary
fragments with mass App can be calculated. In order to calculate the cross section
0(pr,App) of a given primary isotope [0(pr,App-) is a fraction of the primary
mass yield 0(App)], a model for the relative number of removed protons and neutrons
from the nucleus has to be introduced. Several models concerning proton-neutron

correlations will be described in a moment.

The excited primary fragments are then assumed to undergo deexcitation by sta-
tistical emission (evaporation) of nucleons (or light fragments). This evaporation
step, called ablation, is slow compared to the abrasion and leads to the formation
of the final fragments. In order to calculate the yields from the ablation step, stan-

dard evaporation codes have been used [e.g. Blan 76, Tass 91], which are based on

PROJECTILE
PRIMARY
FRAGMENT

     
 

FINAL
- -- PROJECTILE
PARTICIPANT
PROJECTILE NUCLEONS FRAGMENT

TARGET

Figure 2.1: An abrasion—ablation picture of a peripheral collision. The collision be-
tween the projectile and the target shears off a part from both the projectile and the
target. The excited primary fragments (spectators) undergo deexcitation by statisti-
cal emission of particles to produce the final (observed) fragments.

10

the statistical theory of the compound nucleus. Whether the highly excited primary
fragments relax into a thermally equilibrated state implied by the compound nucleus
model is open to debate. Calculations indicate that it is at least plausible and in
fact likely [Harp 71]. It is the nature of the decay process that an ensemble of even
a single primary fragment will generate numerous final fragments. Consequently, in
order to calculate the production cross section of a given final fragment, all primary

fragments leading to it have to be taken into account.

Fluctuations in the number of protons (or neutrons) in the primary fragment (or
equivalently, in the abraded part of the projectile) must also be considered. Three
different models describing these fluctuations in terms of the proton-neutron corre-

lations in the projectile nucleus can be found in the literature:

Complete proton—neutron correlation. Under this assumption, the nucleus is taken

 

to be “frozen” and therefore the proton-to-neutron ratio of the abraded part of the
projectile is exactly equal to that of the projectile. Only one primary fragment nucleus
corresponds to this situation, with cross section equal to 0(App). This assumption

has found limited application [West 76].

No proton-neutron correlation. Under this assumption, every abraded nucleon

 

has a probability Zp/Ap to be a proton and N p /Ap to be a neutron. The probability
‘P(Zp,Np) that the abraded part (with mass A3) of the projectile has 23 protons

and N R neutrons is given by a hypergeometric distribution:

Z N Z N
p(zR,Nn) = R R = PP PP (2-1)
Ap AP
AR APF
where ZR = Zp - pr, NR = Np - Npp and A3 = Ap — App are the number of

protons, neutrons and nucleons, respectively, removed from the projectile and the

11

parentheses denote combinatorials, e.g. (3:) = Zp!/[(Zp — ZR)!ZR!]. The cross
section of a given primary fragment is the product of the probability 'P(ZR, N 3) given
by eq. (2.1) times the primary mass yield 0(App). In general, this assumption can
only describe the data when coupled to very high excitation energies of the primary

fragments [Oliv 79], because it predicts wide primary isotopic distributions [Viyo 79].

Proton-neutron correlation arising from GDR vibrations. According to this model,

 

the mean value of the ratio of the abraded neutrons to protons is equal to that of
the projectile, and fluctuations in the number of abraded projectile protons (and neu-
trons) arise from giant dipole vibrations of the nucleus. The giant dipole resonance
is a collective excitation mode of the nucleus in which, upon absorption of electric
dipole radiation, the protons vibrate collectively against the neutrons. According to
a macroscopic description given by Goldhaber and Teller [Gold 48], the protons and
the neutrons behave like two separate rigid but interpenetrating density distributions
and the resulting resonance consists of a harmonic displacement of these spherical
distributions with respect to each other. Myers et al. [Myer 77] have presented a de-
scription of the GDR mode in terms of the macroscopic droplet model of the nucleus.
This formulation was used by Morrissey et al. [Morr 78], to describe the proton—
neutron correlations during the abrasion step of fragmentation reactions. According
to this treatment, the production cross section 0(pr, App) of a primary fragment is

related to the mass yield 0(APF) by a Gaussian distribution:

 

1 _(zgr-2Fr,o)’
0(pr,App) = c 3 I 0(App) (2.2)
«2W0;
where the centroid ano of the primary charge distribution is determined by the
projectile Zp/Ap ratio: prp = App-if. Notice that this value for the centroid prp
is the same as the prediction of the complete correlation model for Z pp. The width 02

of the charge distribution is calculated from eqs. 15—17 of [Morr 78]. The predictions

12

of this model for the widths of the isotopic distributions lie, as expected, between the
previous two limiting cases. The model has been successful in describing the isotopic
distributions of the projectile fragmentation products from the reaction ‘°Ar(213
MeV/ u) + C [Viyo 79]. However, comparisons with limited target fragmentation
data from heavy targets [Morr 80] and also recent projectile fragmentation data from

heavy beams (A >40) [Stép 91] have been less satisfactory.
Intranuclear cascade model

The intranuclear cascade (INC) model is a microscopic model in which the pro-
jectile and target are considered to be made up of (nearly) independent nucleons
confined in spherical potential wells. The production of projectile fragments also oc-
curs in a two-step process. During the first (fast) step, the motion of the potential
wells past one another leads to individual nucleon-nucleon collisions. The result of
these collisions is the formation“ of excited primary fragments. In the second (slow)
step, the primary products deexcite by statistical evaporation, leading to the forma-
tion of the final fragments. Thus, the intranucler cascade model includes exactly the
same second step as the abrasion—ablation model, but importantly, it differs in the

description of the first step, namely the production of primary fragments.

From a calculational point of view, the INC model is implemented as a Monte
Carlo simulation of simultaneous nucleon-nucleon (N—N) scatterings. The principal
physical information required for an INC calculation is the free N —N scattering cross
section [Chen 68]. This cross section, which is usually taken from experimental data,
is energy and isospin dependent, as illustated in Figure 2.2 [Harv 85]. We notice
that, at intermediate energies, the proton-neutron scattering cross section (0,") is
approximately three times larger than the proton-proton or the neutron-neutron
scattering cross sections (0,, and an“, respectively). However, at high energies (>500

MeV/u), 03,.,, becomes approximately equal to 0,, or an“.

13

 

I IIITIII T IIIIIIIT

I

0' (mb)

 

 

 

 

10 I 1 1 LILIHL l llillLlL
10 w 100 500 1000
E‘(MeV)

Figure 2.2: The total nucleon—nucleon scattering cross sections [Harv 85]. At in-
termediate energies, 0,, is ~3 times larger than 0,, or 0,”; this difference almost

disappears at high energies ( >500 MeV/ u).

14

At the beginning of the calculation, both projectile and target nuclei are considered
as noninteracting Fermi spheres; during the collision step, random N—N scattering is
allowed to occur in the overlap region which removes nucleons from the projectile and
the target, creating excited primary fragments. The excitation energy of these frag-
ments originates from the “holes” which cascade nucleons leave in the volume of the
projectile or target remnant or from nucleons trapped in the remnant. Subsequently,

the excited primary products deexcite by statistical evaporation.

One of the most well known INC codes is the “VEGAS” code, originally developed
by Chen et al. for proton-nucleus collisions [Chen 68] and later generalized to rela-
tivistic heavy-ion collisions by Yariv and Fraenkel [Yari 79]. This latter code, called
“ISABEL”, takes into account, along with the Fermi motion of the nucleons inside
the colliding nuclei, the effects of the Pauli principle on the nucleon motion and the

nucleon-nucleon scattering.

The “ISABEL” code has been used to describe projectile fragmentation data
[Morr 79] for the reaction ”Ar(213 MeV/u) + C [Viyo 79], along with a correlated
abrasion-ablation model. The predictions of the INC model for the primary frag-
ments are different from those of the abrasion-ablation model. Specifically, the range
of mass removed from the projectile (and the target) is lower, the excitation energy
of the primary fragments is higher and the primary isotopic distributions are wider
than those predicted by the abrasion—ablation model. However, after the deexcitation
step, the final fragment distributions are rather similar, with noticeable differences
only for fragments very close to the projectile. In Figure 2.3, the measured isotopic
distributions of O, F and Ne isotopes from the reaction ”Ar(213 MeV/u) + 12C are
shown, along with the predictions of both the INC model and the abrasion-ablation
model [Morr 79]. We notice that both models agree with the trend of the data equally

well and cannot be differentiated by these data. N 0 measurements for this reaction

15

 

5-
I

 

 

s——.—-0
I

 

20/dZ deb/ZuniI/A unit)
a
O
I

F Iuorins Neon
-' —d
1: '0 '-

 

 

 

 

 

 

IL; L l I L I

J J I l 11
I2 I4 I6 I8 20 " l6 IO 20 22 24" IS 20 22 24 26
Moss number, A

 

Figure 2.3: A comparison of the calculated production cross sections (using the
abrasion—ablation model and the intranuclear cascade model) with the measured data
from the fragmentation of 40Ar(213 MeV/u). Solid curve: abrasion-ablation model,
histograms: INC model, solid points: data [Morr 79].

16

are available for near-projectile fragments, where the predictions of the two models

are slightly different.

Up to the present, relatively few detailed measurements on production cross sec-
tions of fragments from relativistic projectiles exist in the literature. Apart from
the measurements of 40Ar residues, careful measurements of fragments from the rela-
tivistic light ions 1‘N, 12C and 160 have been performed [Heck 72, Grei 75]. Also
another important measurement has been performed on the elemental cross sec-
tions (charge distributions) of fragmentation products of relativistic 58Fe projectiles
[West 79]. Very recently, a detailed measurement of cross sections and momentum
distributions of projectile fragments from the reaction 84Kr(200 MeV/ u) + 19"Au
has been performed [Stép 91], along with calculations with the correlated abrasion—
ablation model [Morr 79] and the intranuclear cascade model [Yari 79]. For this heavy
projectile, the abrasion—ablation model gave a poor description of the isotopic distri-
butions of fragments near the projectile. This was attributed to an underestimation
of the excitation energy of the primary fragments (taken, as usual, to be the excess
surface energy of the abraded projectile). However, calculations with the INC model
were found to be in good agreement with the measured isotopic distributions. There-
fore, the magnitude of the excitation energy of the primary products predicted by the
INC model was consistent with the data. In addition to the isotopic distributions,
the INC model gave good predictions of the centroids and the widths of the fragment

momentum distributions (see also section III).

An interesting characteristic of high energy projectile fragmentation is that the
relative yields of projectile fragments are independent of the target [e.g. see 0180 81,
0130 83]. This means that, for a given projectile, the variation of fragment cross
sections with target mass is given only by a single factor. This property of the

cross sections, referred to as “factorization”, reflects the geometrical character of

17

high energy peripheral collisions.

Some of the features of the distributions of high-energy fragmentation cross sec-
tions are also observed in the intermediate energy region, along with new ones. A
limited number of measurements of isotopic distributions have been made in this
energy region; e.g. reactions with 4oAr projectiles at 44 and 27 MeV/u on vari-
ous targets [Guer 83, Rami 84, Rami 85, Blum 86, Borr 86] and also reactions with
86Kr(44 MeV/u) projectiles on various targets [Bazi 90]. The isotopic distributions
of fragments far in mass from the projectile are similar to those observed at high
energies, approximately having a Gaussian shape and a width which is independent
of the target. However, the isotopic distributions of near-projectile fragments are

slightly shifted towards the neutron rich side when neutron-rich targets are used (see

Figure 2.4).

It has been pointed out by Borrel et al. [Borr 86] that the target effect is related
only to the neutron excess of the target and not to its size: in the reactions of 4"Ar(44
MeV/u) with the neutron rich targets 6“Ni and 1”Rh (both having the same N /Z
ratio, but different size), the relative yields of the observed near-projectile fragments
were the same. This effect can be qualitatively understood, if we assume that the re-
moval of nucleons from the projectile occurs only through individual nucleon-nucleon
collisions with the target and, in addition, we take into account the energy depen-
dence of the N—N scattering cross sections (see Fig. 2.2). In the intermediate energy
region, we expect that protons from the projectile are more likely than neutrons to
be scattered from target nucleons, if a neutron rich target is used. Therefore, the
projectile fragments tend to be proton-deflicient, that is, neutron-rich. At higher
energies (>500 MeV / u) however, where the nucleon-nucleon cross sections are all
similar, the N /Z ratio of the target will have no effect to the isotopic distributions of

the fragments.

 

A

   
  
   

1 l IIIIIII
I I llllllI

l llluul

In I nnn

 

5

I I Illll"
I A A nuul

Relative Yield

l l ' Inn]
A A Allllll

Ii'b
T
N
co

0
0‘
N
‘3‘

II

   

 

 

 

 

IO 5 :3
Z . 2

F ‘ —
pd \ at

P ‘ “ 4

I

I r; Z-l5 o ‘. ‘:
I o :

I- .

b u

\ ‘

° .

I I l L I 1 L I l I 1

 

3032 3436 3234363834 363840

Fragment mass (A,)

Figure 2.4: A comparison of experimental isotopic yields for the reactions of ‘°Ar(44
MeV/u) projectiles with 58Ni target (open circles) and 19"Au target (closed circles)
[Guer 83]. Near projectile elements, as S and Cl, exhibit wider isotopic distributions
on their neutron rich sides for the more neutron rich 19I’Au target than for the ”Ni
target.

19

Another new feature of intermediate energy “fragmentation” is the production of
PLFs resulting from a net nucleon transfer (pick-up or exchange) between the pro jec-
tile and the target (see also Fig. 2.4). Consequently, in this energy regime a nucleon

transfer mechanism is present, along with the “pure” fragmentation mechanism.

At low energies, where a substantial number of measurements has been performed
[e.g. Gelb 78, Egel 83], the isotopic distributions of PLFs reflect the characteristics
of a composite nuclear system formed by a partial fusion of the projectile with the
target [Bond 71 , Home 84]. It has been shown that the relative isotopic yields of PLFs
produced at low energies depend exponentially on the ground-state-to-ground-state
Q-value, Q”, of the two-body rearrangement reaction leading to a given fragment
[Bond 71]. Thus, the reaction mechanism of peripheral collisions at low energies is
completely different from that at high energies. At low energies (<20 MeV/u), the
relative pro jectile—target velocities are smaller than the mean velocity of nucleons
inside the nuclei (the mean nucleon velocities due to Fermi motion are ~ 0.2a) and
therefore, the collision times are longer than the transit time for nuclear dimensions.
Because of the long collision times, a partially equilibrated dinuclear system can
be formed. N ucleon exchange between projectile and target can occur during the
interaction stage. Thus, the average N /Z of the PLFs can be seriously affected by
the N / Z of the target. Such a behaviour is not observed at high energies, because of
the high relative velocities (e.g. ~v 2 0.90 for E /A z 2.0 GeV) and, consequently, the

sudden “clean-cut” character of the collision.

At intermediate energies, however, the reaction mechanism exhibits some features
of both high and low energy mechanisms. At these energies, direct nucleon transfer
and/or deep-inelastic (partial equilibrium) mechanisms compete with the “sudden”

fragmentation mechanism.

It should be pointed out that a very important question, concerning the production

20

of projectile-like fragments is the magnitude and the source of excitation energy im-
parted to the primary spectator nuclei (primary fragments). Predictions of this energy
are model dependent and still open to debate. The very broad momentum distribu-
tions and the relative proton—excess of TLFs from heavy targets [Morr 80, Siimm 90a]
and PLFs from heavy projectiles (A >40) [Bazi 90, Stép 91] imply that the excitation
energies of the primary fragments are high and rather long deexcitation chains lead
to the observed fragment distributions. As already mentioned, it has been shown
that the INC model correctly accounts for the excitation energy of the primary
products [Stép 91]. On the other hand, measurements of PLF distributions from
lighter projectiles at intermediate energies [Gelb 78, Guer 83, Blum 86, Borr 86] seem
to indicate that the excitation energy of the projectile primary fragments remains
surprisingly low, consistent with a rather “cold” break-up (or pick-up) mechanism

[Home 84, Harv 85].

Valuable information on the excitation energy spectrum of primary fragments can
also be obtained from exclusive measurements. There are only a few measurements
of this type in the literature. Measurements of charged particles in coincidence with
PLFs for the systems 20Ne + 19"An at 11 and 17 MeV/ u [Stok 84] and 130(16 MeV/u)
+ CsI [Murp 83a] have shown very low charged particle multiplicities in coincidence
with PLFs, implying very low primary fragment excitation energies at low bombard-
ing energies. Also, recent neutron multiplicity measurements on the reaction ”Ar +
19"Au at 27.2 MeV/u [Morj 88] have shown that very large excitation energies are
found in the target-like fragments. In addition, a comparison of the average neu-
tron multiplicities from reactions with Ar and Kr projectiles at 35 MeV/u on various
targets [Gali 88] have shown that the Kr projectile induces 50% higher neutron mul-
tiplicities, implying that more excited primary fragments are formed when using a

heavier projectile.

21

III Momentum Distributions

In addition to the isotopic distribution, another characteristic of projectile-like frag-
ments that contains important information on the reaction mechanism is the momen-
tum distribution. At high energies, systematic measurements of momentum distribu-
tions have been performed for the fragmentation products of relativistic 12C and 16O
projectiles [Grei 75] and for the products of ‘°Ar(2l3 MeV/ u) [Viyo 79] and 8“Kr(200
MeV/u) [Stép 91] projectiles, whose isotopic distributions were discussed in the pre-
vious section. As an example, the momentum distribution of 1"Be isotopes from the

fragmentation of 12C at 2.1 GeV/u is shown in Fig. 2.5a.

In general, the momentum distributions are well represented by Gaussian functions
centered at velocities near (slightly lower than) that of the projectile, and have widths
following a parabolic dependence on the fragment mass: 02 or Ap(Ap — Ap), where
Ap and Ap are the mass numbers of the projectile and fragment respectively (e.g.
see Fig. 2.5b). Deviations from this behaviour have been observed in fragmentation
studies of heavy projectiles [Stép 91] and heavy targets [Morr 89]: the momentum
distributions of fragments far in mass from the parent nuclei are wider than predicted

by the parabolic parametrization (see also Fig. 2.6).

, The longitudinal (or parallel) momentum distributions were found to be indepen-
dent of the target and the bombarding energy. Transverse momentum distrubutions
were also measured for 12C and 16O fragmentation and their widths were found to be

equal to the widths of the corresponding parallel distributions within 10%.

A small downshift in velocity (relative to the beam velocity) of the centroids of
the fragment momentum distributions has been observed. This downshift increases
with the number of nucleons removed from the projectile and has been interpreted

as a “friction” phenomenon: in order to remove bound nucleons from the projectile,

22

 

I

—1
c1
—1
——(
-—(
q
I—
I

Be

T

at? L
g. [l I‘I‘I\,\ :

g‘émld3 :‘i J \ -:'

5
I

mb/(MeV/cII

 

 

 

 

 

 

 

' Fitted I
‘ Gaussian '
IO"IIILIILIL 'fi""""”r'I"
-400 -200 O 200 4(1) "0 PROJECTILE
I: (MeV/c) ]
200'- ] 4 ‘
(a) I I )1
’8 I v 4“; 7T 1
a 2 2” 1 Nil]
3:50— : L f -
but I. a i l", I
E ’1
I00" .7 d
{s
50 “
L 4 l I
O 5 IO

 

FRAGMENT MASS (AMU)

(b)

Figure 2.5: a) The parallel-momentum distribution of 1“Be fragments from the reac-
tion of a 12C projectile at 2.1 GeV/u on a Be target. The distribution is expressed
in the projectile rest frame. A Gaussian function is fitted to the data points. The
average momentum and the width of the fitted distribution are indicated [Grei 75].
b) Dependence of the parallel-momentum distribution widths on the fragment mass.
The data are from the fragmentation of 16O at 2.1 GeV/u [Grei 75]. The plotted
symbols indicate the atomic number Z of the fragments. The parabola represents the
best fit to the data and has a reduced width 00 =1 83 :I: 1 MeV/c.

23

1000

 

 

 

I 7r T I I I ,
b inugedW-Z '(s
. 1 a. o"
a 2 o
O 3 o i """""""""""""
I a 4 ~33? ' I I
’8 5 I J
v I
g .99"
o “'3’
a 500 I .w‘ "
a -'.'
V
0.! I “3?: -------- Goldhahsr (sawmill/c)
3y ---- Experimental systematics
fig' — [intranuclear cascades
II .93.;
,5" “ n + ”An zoo MeV/u
‘0 a... " 0 0. I
o 0' 1 I I 4 l I l

 

0 l 2 3 4 5 0 7 0

VAA...

Figure 2.6: The total-momentum dispersion P,,,,. of projectile fragments as a function
of the square root of mass loss (AA) from the reaction I”Kr (200 MeV/u) + l9"Au
[Stép 91]. (P,.,m = \/30", under the assumption of isotropy, see text.) Two model cal-
culations are shown: Goldhaber’s model (thin dotted line) and intranuclear cascade
model (solid line). Fragments far in mass from the projectile (AA > 25) have momen-
tum dispersions larger than predicted by Goldhaber’s model. For these fragments,
the considerable fluctuations in the INC calculation are essentially statistical.

24

energy has to be consumed. This energy is assumed to come from the projectile kinetic
energy and consequently, the resulting fragment has, on average, lower velocity than
that of the initial projectile. In order to calculate the fragment average velocity
according to this “friction” concept, a picture of the way the nucleons are stripped
from the projectile has to be introduced. For this purpose, two different assumptions
are discussed below. First, the nucleons are assumed to be stripped from the projectile
one after the other [with an average of EM 3 8 MeV (for medium and heavy nuclei)
required for each removal]. Second, the nucleons are assumed to be removed from the
projectile simultaneously as a single (hound) cluster R, thus creating the observed
fragment F. The energy required in this case is equal to the separation energy
of the projectile into the clusters F and R. Both these simple assumptions give
predictions in agreement with the trend of the high energy fragmentation data and
correspond, respectively, to an independent-particle description of the fragmenting
projectile nucleus, and to a cluster structure description (see below the discussion of

Goldhaber’s model and Friedman’s model, respectively).

In order to interpret the origin and the variation of the widths of the PLF momen-
tum distributions, two simple models, Goldhaber’s statistical model and Friedman’s

peripheral model, each based on a different description of the projectile nucleus, are

discussed below.
Goldhaber’s Model

This model [Gold 74] is an independent-particle description of the projectile nu-
cleus: the nucleus is considered as a spherical potential well in which the nucleons
move independently. It is assumed that the only correlation between the nucleons
arises from momentum conservation; the nucleons inside the projectile have some dis-
tribution of momenta that, at any time, gives a total momentum of zero in the rest

frame of the projectile. Consequently, if one part of the projectile is removed by a

25

rapid interaction with the target, it carries away its relative momentum and, because
of momentum conservation, the remaining fragment recoils with an opposite momen-
tum. Assuming isotropy of the internal momenta of the nucleons in the projectile,
the recoil momentum distribution of the fragments, projected on a given coordinate
axis, is a Gaussian distribution. The width of the distribution can be related to the

fragment mass, if either of the following two assumptions is made:

Sudden fragmentation hypothesis: a part is suddenly removed from the projectile

 

by the interaction with the target. This assumption suggests a process similar to the
first step (abrasion) of the abrasion-ablation model (but here an ablation step is
missing). It has been shown by Goldhaber that conservation of momentum leads to

the following relation for the width:

Aer-Ar)
3_ 2
0 —0o Ap-l

 

(2.3)

where the parameter 0.), called the reduced width, is related to the nucleon momenta

in the projectile by the expression:

N
”U
:3»
v

 

00:

(2.4)

where (p3,) denotes the average-square momentum of the nucleon motion.

Assuming a Fermi-gas distribution of the nucleon momenta in the projectile, we

have:

(193.) = grim..- (2-5)

where zip-"m,- is the Fermi momentum. Inserting this expression of (p3,) into eq. (2.4),
the reduced width is related to the Fermi momentum of the nucleon motion according

to:

N

 

pFerms' (2.6)

a“
n

26

Consequently, according to the sudden fragmentation hypothesis, the observed Gaus-
sian distributions are directly related to the Fermi motion of the nucleons in the

projectile.

The nucleon Fermi momentum ppflm; can be obtained experimentally by quasielas-
tic electron scattering [Moni 71]. It has shown by Moniz et al. that ppm,“- increases
from the lighter to the heavier nuclei up to nickel (A = 58), where a value ppemg c: 260
MeV/ c was obtained. This increase of ppflmg with mass is expected because ppflm;

is related to the nuclear matter density p = A/ V according to the equation [see e.g.

Path 72]:

<.%)""h (3)” ' m

where h is Plank’s constant and it has been assumed that N = Z = A/2 and also that
ppnm, is the same for protons and neutrons. The nuclear density is known to increase
with mass until it becomes roughly constant for heavier nuclei; this behaviour is a
reflection of “saturation” of the nuclear forces [e.g. see Blat 52]. Expression (2.7)
provides another way to obtain ppemgz from measurements of the nuclear matter
density (e.g. via elastic electron scattering). For heavy nuclei, a typical value 0pr is
~ 0.17 nucleons/fm"; with this value, eq. (2.7) gives ppm, 2 270 MeV/c. This result

is in good agreement with the values obtained from quasielastic electron scattering

data.

In Table 2.1, the experimentally obtained values of the reduced width 0’0 from
the fragmentation of relativistic 12C, 16O and ‘°Ar projectiles are listed. Also, the
corresponding “theoretical” values of 00 are calculated using eq. (2.6), where the

values of ppm.- from Moniz et al. [Moni 71] are used.

From this table, we notice that the measured reduced widths increase with increas-

ing projectile mass (as the nuclear Fermi momentum does) and, interestingly, they

27

Table 2.1: Comparison of the experimentally obtained reduced widths from the frag-
mentation of relativistic projectiles [Grei 75, Viyo 79] with the theoretical values ob-
tained from eq. (2.6), where for the Fermi momentum ppemg, the values reported by

Moniz et al. [Moni 71] have been used.

 

 

Projectile E (M e V/ u) 00,“, 00,1.“ pram.-
HC 2100 81 :I: 2 99 221
16O 2100 83 :1: l 103 230
4oAr 213 94 :I: 5 112 251

 

 

 

are always smaller than those calculated using the Fermi momentum values. This dis-
crepancy can be understood if we recall that Goldhaber’s independent-particle model
ignores all correlations between nucleons, beyond the purely kinematical ones. It has
been shown by Bertsch [Bert 81] that the Pauli exclusion principle between identical
nucleons reduces the widths of the observed momentum distributions. A qualitative
interpretation of this result is that a large anticorrelation between the momenta of
two identical nucleons is expected when they are found close together in coordinate

space.

Thermodynamic equilibrium hypothesis: in this second fragmentation picture, the
projectile, after interaction with the target, is assumed to be in a highly excited state
at thermodynamic equilibrium. The excited projectile decays into two fragments by
a statistical break-up process. This process is similar to the second step (ablation)
of the abrasion-ablation model, but here the the abrasion step is missing. In this
case, conservation of momentum leads to the same relation between the width and
the fragment mass, eq. (2.3), but the reduced width is now related to the excitation

temperature T of the projectile at the moment of its statistical decay:

Ap—l
AP

 

03 = mnkT (2.8)

where m, is the nucleon mass and k is the Boltzmann constant. Furthermore, the

28

excitation energy of the nuclear Fermi gas is related to its temperature with the

following equation:
E" = aucT)2 (2.9)

where a is a level density parameter, empirically expressed as: a = Ap/ (8 MeV).

Thus, the excitation energy per nucleon of the projectile can be expressed as:

_E‘__("_T):
-Ap—8MeV

(2.10)

Inserting a typical value of 00 t: 90 MeV/c into eq. (2.8), we get IcT c: 8 MeV,
which gives, according to eq. (2.10), e“ z 8 MeV. This value of the excitation energy
per nucleon is comparable to the average binding energy per nucleon (~ 8 MeV) of
the projectile, implying that the projectile has some probability to disintegrate into
nucleons. Thus, taking into account the experimental fact that bound fragments are
abuntantly produced in a fragmentation reaction, the formation of a highly excited

projectile in thermodynamic equilibrium should be considered as an extreme case.

It should be pointed out that in both fragmentation models, the variation of
the widths of the momentum distributions with mass loss is a result of momentum
conservation. Between these two extreme cases, these is a third possibility, that of
a long evaporation chain, which also leads to the same result. Consequently, the
widths of the momentum distributions cannot distinguish between a fast nucleon-
removal process and a slower equilibration process, therefore they are insensitive to
the speed of the fragmentation process. In order to obtain information about the
course of the fragmentation reaction, the isotopic distributions of PLFs along with

correlations among PLFs and light particles should be investigated in detail.

Goldhaber’s model has been successful in describing the experimental parallel mo-
mentum distributions also at lower energies (e.g. fragmentation of ”Ar at 44 and 27

MeV/u [Blum 86, Borr 86]). However, the behaviour of the transverse momentum

29

distributions of fragments from intermediate energy projectiles is more complicated.
Measurements on the fragmentation of 16O projectiles in the 100 MeV / nucleon region
have provided evidence that the isotropy observed at high energies is not present at
these energies. The transverse momentum distributions of the observed fragments
are wider than the parallel distributions. This effect is attributed to orbital deflec-
tion of the projectile by the combined Coulomb-nuclear field of the target before
fragmentation takes place [Bibb 79, Silk 88, Brad 88]. It has been shown by Bibber
et al. [Bibb 79] that including orbital deflection to Goldhaber’s model leads to an

expression for the width of the transverse momentum distribution of the form:

Ap(Ap - AF) Ap(AF - l)
2 _ 2 2
0’ —-0’0 Ap-l +01 Ap(Ap—l)

 

 

(2.11)

where 0’0 is the reduced width due to the intrinsic nucleon motion and 0: = iIPIuI
is the variance of the transverse momentum p.41, of the projectile due to orbital
deflection at the moment of fragmentation. Typical values of the parameters 00 and

01, obtained from the 160 data, were 00 2: 80 MeV/c and 01 2: 200 MeV/c [Bibb 79].

Another important feature of the momentum distributions of PLFs produced at
intermediate and low energies is the existence of a long tail on the low-energy side
of the peak, which has been attributed to the presence of dissipative mechanisms
[Blum 86, Borr 86]. When applying Goldhaber’s model to describe these data, the

tail is excluded from the Gaussian fit.

A compilation of the best fit values of the reduced width parameter 0'0 obtained
from the analysis, within the framework of Goldhaber’s model, of existing fragmen-
tation data at high and low energies has been reported by Murphy and Stokstad
[Murp 83, Stok 85]. For energies down to ~40 MeV/ u, a. rather constant behaviour
of 00 with energy has been observed, indicating the essential validity of Goldhaber’s

model. However, at lower energies, a decrease of the reduced width 00 (and also of the

30

“unreduced” width 0) with energy has been observed. This has been attributed to the
dominance of a direct nucleon-transfer mechanism over the “sudden” fragmentation
mechanism. Using a local-momentum plane-wave Born approximation (LM-PWBA)
model, McVoy and N emes [McVo 80] showed that such a narrower width is expected
for the transfer component of the momentum distribution than for the fragmenta-
tion component. In a transfer reaction, the particle is transfered to a definite final
state due to momentum and angular momentum conservation; this imposes strong
constraints on the available phase space of the exit channel of the reaction and thus
leads to a narrower momentum distribution of the observed fragment. Such strong

constraints do not exist in a fragmentation reaction.

It should be noted that Goldhaber’s model uses only the average (statistical)
properties of the nucleons inside the projectile. The detailed nuclear structure of the
projectile is completely ignored. More recently, a simple quantum mechanical model
has been proposed by Friedman [Frie 83] in which, in addition to the average prop.
erties of the projectile nucleons, the nuclear structure is considered in a simple way
in the calculation of the momentum distribution widths and the yields of projectile

fragments. This model is discussed below.
fiiedman’s Model

This model considers the probability that a part R to be removed from the pro-
jectile, preexists as a (single) cluster in the projectile before fragmentation. This
cluster is stripped from the projectile upon interaction with the target and a final
fragment F remains from the projectile and is observed. In this picture, in contrast
to Goldhaber’s model, the nucleons removed from the projectile are not independent,
but they are strongly correlated in space to form a cluster (with given binding energy)

near the surface of the projectile.

31

The relative motion of the cluster R with respect to the fragment F inside the
projectile is described by a radial wave function ¢p_p(r) which, near the nuclear

surface, can be written as:

 

e-ar
$17-30*) CC r (2.12)
where a is the wave-number of the relative motion:
1
01 ==‘ E 211E, (2.13)

where p is the reduced mass of the system F—R, p = Ap(Ap - Ap)/Ap, and E. is its

separation energy.

The parallel momentum distribution of the observed fragment, determined by the
Fourier transform of the cluster wave function ¢p-p(r), is a Gaussian distribution
(centered at a velocity equal to that of the projectile, as in Goldhaber’s model) with

a width 0 given by:
02 = 11’ 3— (1 + -1—) (2.14)

where so = 1.2A];(3 is the radius of the fragment. If the Coulomb interaction between
the fragment F and the removed part R is taken into account in the wave function

¢p_p(r), the width has been shown to be:

2- 2_°‘_. .1.)
0 —h 2z0( l+y+ azo . (2.15)

where y = (be/zo)/E. is the ratio of the Coulomb energy of the system F -R at
the “touching” point (at r = so) to the separation energy 13.; b6 is the Coulomb
parameter of the system F—R, 66 = s’ZpZR. Equation (2.15) shows that the Coulomb
interaction between the fragment and the removed part results in a small increase of
the width of the parallel momentum distribution of the fragment. This effect is not

present in the simpler independent-particle model.

32

According to Friedman’s model, the production cross section or the yield 7p of a

given fragment (Zp, Ap) from the projectile (Zp, Ap) can be written as:

e-2a205

7F 0C 5(ZR.AR) m- (2-16)

where the factor S (Zp, AR) expresses the relative probability that the removed part
R of the projectile contains ZR protons and N p = A3 - ZR neutrons and is expressed
by the hypergeometric distribution, eq. (2.1); the second factor of this equation is the
square modulus (evaluated at r = so) of a normalized radial wave function ¢p_p(r).
The wave function is assumed to be constant up to a radius Rc = 1:0(1 — b) and
then exponentially decrease according to eq. (2.12) at larger distances. The “cut-off”
parameter b for a given projectile can be graphically determined as follows. Taking
the logarithms of both sides of eq. (2.16), we have:

W a In fl— : —b(2crzo) — ln(1 - b) + const (2.17)
5(ZR. As)

For a set of fragmentation yields from a given projectile, the function W should vary

linearly with 202:0 with a slope equal to the parameter b.

It should also be pointed out that the predictions of the model are strongly affected
by the way the nucleons are assumed to be stripped from the projectile: as a single
cluster (the simplest assumption) or, as more than one cluster or, as separate nucleons.

Each of these choices has a different separation energy and thus results in a different

value of the width and the yield.

A drawback of this model is that it gives only relative yields of fragments; no
absolute cross sections can be predicted due to the complete neglect of the impact
parameter (or the angular momentum) of the collision. A detailed microscopic cal-
culation which takes into account the impact parameter along with the “Friedman”
probability for the removal of a cluster from the projectile has been reported [Harv 85]

and has provided yield predictions in agreement with experimental data.

33

In general, Friedman’s model is in good agreement with the experimental data,
as is Goldhaber’s model, but it gives better results for fragments very close to the
projectile, especially in cases where the cluster structure of the projectile may play
an important role in the fragmentation process (e.g. stripping of an a particle from
an 1"O or 20Ne projectile [Frie 83, Home 84]). A derivation of the equations for the
momentum-distribution widths and the yields according to Friedman’s model is given

in appendix A.

Finally, apart from the simple models of Goldhaber and Friedman, the character-
istics of the momentum distributions can be calculated with the INC model [e.g. see
Stép 91]. Calculations with the models described in this chapter and comparisons
with the distributions of pro jectile-like fragments obtained from the measurements of

the present work (chapters 3 and 4) are presented in chapter 5.

Chapter 3

Operation of the K1200 Interim
Beamline as a Spectrometer

I Introduction

The work presented in this chapter represents the first attempt to measure fragmen-
tation products at 0 degrees using intermediate energy beams delivered by the K1200
cyclotron. Soon after the commissioning of the K1200 cyclotron, a short interim
beamline [Sher 87] was installed outside the cyclotron in order to do nuclear research
while the final beamlines and experimental devices were under construction. During
this time, we designed and performed an experiment in which the interim beamline

was operated as a O-degree spectrometer [Soul 91].

In this chapter, after a description of the beamline, its characteristics and the
detector system used, the measurements performed with this system are described.
In addition, the procedures developed for the analysis of the data are presented along
with several representative results of projectile fragment distributions obtained from
these data. In appendix B, we include a discussion of relevant Beam Optics concepts

and a derivation of the equations developed for the present study.

34

35

II Description of the spectrometer and the de-
tectors

The interim K1200 beamline consisted of four superconducting magnets (fig. 3.1),
which were: a quadrupole singlet, a quadrupole doublet, a 3:16" bending dipole mag-
net and a quadrupole doublet. The details of the construction of these magnets
have been described elsewhere [Ze1185]. For the present study, we operated the
two quadrupole doublets and the bending magnet as a O-degree spectrometer (the
quadrupole singlet was turned off). The characteristics of the magnetic elements of

the spectrometer are summarized in Table 3.1.

Projectile fragments, produced in a fragmentation target placed 40 cm after the
exit of the K1200 cyclotron, traveled through the device and were focused and de-
tected inside a 92-inch diameter cylindrical scattering chamber. The detection system
for the present work was mounted on the movable radial arm of the chamber and con-
sisted of two position counters and a AE—E plastic scintillator telescope. The first
position counter was placed near the entrance to the 92-inch scattering chamber and
the second counter 1.8 In further downstream. Both position counters were low pres-
sure multiwire proportional counters (MWPC’s) of Breskin type [Bres 82]. From
these detectors (see Fig. 3.2), a position signal is obtained by a delay-line readout
of the electric charge induced on two layers of interleaved cathode strips. The time
difference between the signals from the two ends of the delay line can be measured
and is proportional to position along the counter. The position measurement from
the front counter contained information on the rigidity, whereas the difference be-
tween the position measurements of the two counters contained information on the
scattering angle of a particle. Section VI describes in detail how the rigidity and the

scattering angle were obtained from the measured position and angle.

36

  
  
     
   
 
  
 

/ / / / / / /
mrsnm vauu

 

 

ouaosusors
/ruousunno« oodem
, unszr

/ OUA oaurou
/ oousrrr
I oumnuvoL:

   

 

( smchr

\ AE-E

. PLASTIC

Q‘- s——-‘———-‘ scmrlLuroe
,r s W o 1 2.

mos moo

cvcwrnon ] 7 / / / / / / /

 

Figure 3.1: Schematic layout of the interim beamline spectrometer.

37

Table 3.1: Characteristics of the interim beamline spectrometer and its elements.

 

 

 

 

 

 

Front MWPC—back MWPC

Quadrupoles

Effective length 43.0 cm
Pole-tip radius 5.08 cm
Max. magnetic field at pole tip 15 kG
Electric current at max. field 14 A
Distance between quads in a doublet 16.7 cm
Dipole

Mean deflection angle 16.4°
Mean bending radius 3.08 m
Entrance face angle 0.0°
Exit face angle 16.4° '
Effective length of central trajectory 0.882 m
Pole-gap width 5.0 cm
Maximum magnetic field 18 kG
Electric current at max. field 100 A
Distances between elements

Target— first doublet entrance 1.187 m
First doublet exit—dipole entrance 1.677 m
Dipole exit—second doublet entrance 0.852 m
Second doublet exit—front MWPC 2.662 m

1.840 m

 

38

The AE—E telescope, mounted behind the back position detector, consisted of
a thin (3 mm) fast plastic scintillator for energy loss measurement and a thick (5
cm) plastic scintillator as a stopping detector. Each scintillator was viewed by a
photomultiplier tube (PMT) on each end (left or right). A AE parameter was cal-
culated event—by-event by taking the square root of the product (geometric mean) of
the left and right PMT signals of the AE scintillator. Due to light attenuation by
self-absorption in the plastic scintillator, each PMT signal is exponentially dependent
on the position of a particle along the scintillator. The product of the two signals is
position independent and its square root proportional to AE (whereas the logarithm
of the ratio of these signals is linearly related to position). The atomic number Z of
the particles was obtained from the AE parameter, since according to Bethe equation

see e.g. Kran 87 : AB 0: E, where v is the velocity of the particles.
”2

The back MWPC had a low efficiency, so its position information was not used
in the analysis of the data. Instead, a back position parameter was constructed
from the logarithmic ratio of the two PMT signals of the AE scintillator. This
position measurement was found to be consistent with the position measurement
provided by the back MWPC for highly ionizing particles and was combined with
the front position parameter to provide a final angle parameter for the particles. The
position resolution of both MWPC’s was better than 1 mm; the position resolution
of the back position parameter Obtained by the AE scintillator was 1.0 cm and the
angle resolution obtained by the combination of this position parameter and the front
position parameter was 6 mr. The fact that the angle measurement had to rely on
the less accurate position values from the plastic scintillator had a significant effect

on the overall momentum resolution (see section III).

Time of flight (TOF) was measured relative to the K1200 radiofrequency (RF)

cycle. A start timing signal was obtained from the AE scintillator. A combination of

LEFT
DELAY

CATHODES

SIGNAL

 

~<I-I

 

 

 

 

 

 

39

 

 

 

 

 

 

 

 

 

 

 

 

>""l—I

 

 

 

 

 

 

i i I STRIPS
* WIRES

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I II I III II "’ i' MODE
1| [[7 . NSIGNAL
, I V
' I || | .1
, . 1 RIGHT DELAY
iiu’IiiIiiiM SIGNAL
DELAY LINE *
TAC
EXTRA c T
c DELAY SUE 1: p
D STOP
001 ADC

Figure 3.2: Schematic diagram of the MWPC and the delay-line position measure-
ment. The anode of the detector consists of a series of closely and evenly spaced wires.
Each of the two cathods consists of a thin plastic foil with evenly spaced aluminum
strips deposited on it. The strips of both cathods are interleaved and alternately con-
nected to a common delay line. The electron avalanche created by an ion entering the
gas volume of the detector is collected by the nearest anode wires, whereas inducing
a positive charge on the corresponding cathode strips. The time difference between
the delay-line signals created by this induced charge is measured with a TAC and is
linearly related to the position of the ion along the detector (for electronic module
acronyms see Fig. 3.3).

40

TOF and position provided an A / q (mass-over-charge) parameter, which was com-
bined with the AE parameter in a two-dimensional histogram to provide particle
identification. Due to multiple turn extraction from and the phase acceptance of the
cyclotron, only limited TOF resolution (approximately 4 us) was obtained, which

hindered the isotope identification for Z’s higher than 6.

III Beam optics calculations

Beam optics calculations (up to third order), performed with the program TRANS-
PORT [Care 88], showed that a moderate acceptance solution was possible for this
simple spectrometer, i.e. 35 mt horizontal and 10 mr vertical angular acceptances and
8% momentum acceptance. The optical characteristics of the spectrometer, the first
order TRANSPORT matrix elements and the important second and third order ma-
trix elements at the front position counter are summarized in Table 3.2. The TRAN S-
PORT beam optics notation and units have been used in Table 3.2 and throughout
this study. An overview of relevant Beam Optics concepts is given in appendix B.
At the end of Table 3.2, the field values of the magnets corresponding to setting the

spectrometer for 14N” at 75 MeV/u [which has a momentum-over-charge ratio p / q

of 0.762 (GeV/c)/e] are given.

For the present system, there are no first order correlations between the coordi-
nates of the particle motion on the horizontal (bending) :2 plane and the vertical
(non-bending) yz plane. However, such correlations arise in second and higher orders.
The TRANSPORT calculation showed that the second and third order correlations
of horizontal and vertical coordinates are negligible for the relatively small vertical
angular acceptance of 10 mr; in all cases, the contributions to the position and angle

resolution are less than 1 mm and 1 mr, respectively, and these contributions were

41

Table 3.2: Beam-optical characteristics of the interim beamline spectrometer.

 

Important parameters

Horizontal magnification

Dispersion at the front MWPC

Horiz. beam spot at the target (full-width)

 

(zlx) = —0.63
(:rl6) = 0.65 cm/%
An; = 6 mm

Max. horiz. angular acceptance A00 = 35 mr
Max. vertical angular acceptance A¢o = 10 mr
Max. solid angle A0 = 0.35 msr

 

First order TRANSPORT matrix elements

 

a: 0 3] (II I 5
—0.631 19 -0.07302 0.00000 0.00000 0.00000 0.64914
18.66188 0.57452 0.00000 0.00000 0.00000 0.61110

0.00000 0.00000 1.60686 0.80937 0.00000 0.00000
0.00000 0.00000 1.53734 1.39669 0.00000 0.00000
1.24998 0.04176 0.00000 0.00000 1.00000 —0.01198
0.00000 0.00000 0.00000 0.00000 0.00000 1.00000

 

Second and third order matrix elements

 

(zlzd) = -1.18 x 10’1
(.1106) = —4.53 x 10'3
(2w?) = -5.8 x 10‘“5
(zld’) = 1.02 x 10'3
(3106’) = 2.78 X 10'”

(0|:r6) = —6.65 x 10'1
(0|06) = -5.07 x 10‘2
(0'92) = -2.3 x 10’”
(0|6’) = 1.58 x 10'2
(0|06’) = 1.28 x 10"3

 

Magnetic fields (in KC) for i‘IN at 75 Meru

 

First quad 6.204 Third quad 1.701
Second quad -3.667 Forth quad -0.041
Dipole 8.244

 

42

neglected in the present study. Hence, only the correlations of the coordinates in
the bending plane were taken into account when extracting the momentum and the
scattering angle of a particle from the measured position and angle. Simple alge-
braic formulas and a calculation procedure were developed for this purpose and are

described in section VI.

From the first order TRANSPORT matrix, we see that a relatively small value
of the horizontal magnification (zlz) is achieved, but a considerable (:rIO) term is
present at the front counter, which introduces a first order correlation of final position
with initial scattering angle. Consequently, a first order horizontal focus was only
approximately created at the front detector. The actual focal plane [where (:rl0) is

zero] lies in between the two detectors.

In any spectrometer with relatively large momentum and angular acceptance,
there will be large second order aberrations [mainly due to the (1106) term]. These
can be corrected with sextupoles or by focal plane reconstruction. In the present
study, we chose the latter method. Consequences of this choice are that it is not
necessary to have a first order focus at a position sensitive detector, but an accurate

angular measurement is required.

Generally, it is desirable for the contributions of the second and higher order
matrix elements at the focal plane to be small fractions (i.e. less than 0.1) of the
displacement due to the dispersion. When one position counter is used at the exit of
a spectrometer, it is necessary that this detector be placed at the position of the focal
plane. However, if two position detectors are used (thus final position and angle of
the particles can be measured, as in the present system), it is not necessary to place
one of the detectors along the focal plane. In such a case, the position of the focal
plane can, in principle, be reconstructed offline, provided that the angle measurement

has sufficient accuracy.

43

For the present system, where reaction angle and momentum reconstruction are
performed, an experimental uncertainty in the the reaction angle of approximately
15 mr was obtained from the elastic scattering data. This value is consistent with a
conservative estimate of 22 mr obtained from eq. (B.29), which takes into account the
contribution from the final angle uncertainty and the beam-spot size (approximately
6 mm). Similarly, the experimental momentum resolution of approximately 2.0% can
be mainly attributed to the reaction angle uncertainty combined with the presence
of the (a:|0) term. An estimate of the momentum resolution of approximately 2.0%,
based on the beam optics, can be obtained from eqs. (B.27) and (B.28). However,
the momentum resolution of the system with only the front counter position, without
scattering angle reconstruction, would be worse. Specifically, at the full horizontal
angular acceptance of A00=35 mr. and momentum acceptance of A6 = 8% of the
spectrometer, the effects of the (zl0) and (3.106) terms to the image width would

result in a momentum resolution of approximately 4.5%.

IV Electronics

A schematic diagram of the electronic modules used in the interim beamline exper-
iment is shown in Fig. 3.3. The basic characteristics are discussed below. Each
MWPC was connected to a TAC measuring the time difference between a START
signal from the right end of the MWPC cathode delay-line and a STOP signal (addi-
tionally delayed) from the left end. Time-of-flight versus the RF was also measured
by another TAC with a START signal from the AE scintillator and a STOP signal
from the cyclotron RF. The signals from the TAC’s were sent to ADC’s. The pulse
height signals of both ends of the AE and E scintillators were sent to QDC’s and
the corresponding timing signals were AN D’ed. The event trigger (master gate) was

defined by the coincidence of the timing signals from the scintillators (indicating that

FRONT LP-MWPC (Identical for BACK LP-MWPC)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

moor : 7 FEASCALER
A f. D TDC
"m — CALER

RIGHT 1 r

DELAY r A r not START

‘ E L 0 ‘ TDC

LEFT T I}: TJCNIR

DELAY r A F mrm 5109
A L. D TDC

 

 

 

 

 

 

 

    
    
 
 
  

AC3 START

STROBE
cpu nor susv smosz
51m (couuou)
STOP (common)
STROBE

'a some
1'.
1

__I’ACJ STOP

 

 

CYCLOTRON
RF

 

 

 

 

 

Figure 3.3: Schematic diagram of the electronics for the interim beamlime experi-
ment. Electronic modules: TFA = timing filter amplifier, FA = fast amplifier, CFD
= constant fraction discriminator, TAC = time-to-analog converter, TDC = time-to-
digital converter, ADC- = analog-to-digital converter, QDC =charge—to-digital con-
verter, GDG = gate and delay generator.

45

a particle passed through the system) and the NOT-BUSY signal from the front-
end computer. For diagnostic purposes, timing signals were recorded with TDC’s
for both MWPC’s, both scintillators and for the RF. The TDC’s were set-up in the
“COMMON START” mode: a COMMON START signal was sent to them from the
event trigger and the different timing signals were sent as STOP signals. A total of
17 signals were recorded by the front-end computer for each event trigger. The data
acquisition software and hardware system used to read the ADC, QDC and TDC
modules and record the data to tape was the standard N SCL data acquisition system

[Fox 89].

V The experiment

As a test of the beamline spectrometer, we measured the products from the reaction of
a 1‘N beam at 75 MeV/u with Aland Ta targets with thicknesses of 8 and 10 mg/cm’,
respectively. Data were collected for approximately 10 field settings in steps of 5%,
covering the A / q region of 2.0 to 3.0. The magnetic field of the dipole was measured
with an N MR magnetometer. For primary beam charge collection, an aluminum bar
was placed inside the dipole and was connected to a current integrator. The position
and angle calibration of the detectors was performed in two independent ways, giving
consistent results: first, the radial arm (on which the detecting system was mounted)
was rotated by a known angle and the change in position of the 0° elastically scattered
beam particles was measured, and second, a slotted copper mask was inserted between

the two counters and was irradiated with a dispersed primary beam.

46

VI Data analysis and results

In the present 0-degree spectrometer, the momentum and the scattering angle of a
particle leaving the target are related to the measured position and angle at the front
position counter. The calculations and the procedure that were developed to extract

these quantities from the data are described below.

We can write the magnetic rigidity Bp of a particle moving in the magnetic field
of a dipole as [Care 87]:

Bp = 3.3356 g (3.1)

where B is the magnetic field induction (in kG), p is the radius (in cm) of the trajec-
tory of the particle in the dipole field, p is the momentum of the particle (in MeV/c)
and q is its charge (in units of elementary charge). In the present experiment, the
position of the undeflected beam particles at several charge states (“NH and uN“)
obtained at several different magnetic field settings, directly calibrated the curvature
radius p versus position (fig. 3.4) corresponding to a scattering angle 01, with respect
to the spectrometer optical axis at the target position (fig. 3.5a). This angle, made by
the beam emerging from the accelerator, was found to be 0;, .5. 1°. We had no control
over this angle, since the spectrometer was fixed and there were no bending elements
after the cyclotron and before the target. However, it was exploited to increase the

observed reaction-angle range to 2°.

The curvature radius—position calibration, shown in Fig. 3.4, can be well de-

scribed by a linear function:

PM) = A0(95) - 41(00er (32)

where pr is the position (in channel number) of a particle at the front position

counter. Two additional calibration points, obtained in a subsequent experiment with

47

 

 

 

 

I I I r I I r I I I I I f r I I-I
_ \ \ . 14N6+ ..
A v-Ov
' \ 14 7+ “
E x a
0 320 " \ N —
v . \ 20 3+ I
\
a . \ . Ne 1
:3 \O
4" " \ -1
I \
g 300 -- \ \ I. _
.. ' w ‘
o - \ .
\
U) r- s
.E’. . X .
'U
«1 280 —- 3R: —
III \
L \ '
l l l I I I l L l l l l i I l J ‘
200 400 600 800

Position (Channel No)

Figure 3.4: Calibration of the radius of curvarute versus front-counter position. The
error-bars on the points have widths of 20. The widths of the peaks of the elastically
scattered beam particles which arrive near the center of the front detector are mainly
due to the magnification term (zlx). The growth of the peak widths with increasing
distance from the center of the detector is attributed to the momentum dependent
term (142:6). The dashed straight line is a least-squares fit to the data points.

48

I
XFPTxrs ‘
(fir. fiaFP
a, T T T T T
BEAM )9" 9»

DIRECTION ["o a

, 4—" =
/
00 z 0 7-

 

 

 

 

 

 

(a) (b)

Figure 3.5: Coordinate systems used in the calculations of momentum and reaction
angkx 4

a) Initial coordinate system (at the target position). The beam strikes the target with
an offset so and an angle 05. Reaction products emerge from the target at angles 00
relative to the spectrometer axis (z-axis). The interesting nuclear reaction angle is
0,- = 00 — 05.

b) Final coordinate system (at the position of the front counter). In this system, a and
0, respectively, are the position and angle coordinates of the 0° elastically scattered
beam particles of one run that was chosen as reference; 2:, 0 are the final position
and angle coordinates, respectively, of a particle that entered the spectrometer with
initial coordinates so, 00 and momentum deviation 6; the quantity —£L¢£’!2’- is the
position of the particle relative to the reference position a and Opp is the angle of the
particle relative to the reference angle 6.

49

a 20Ne beam at 65 MeV/ u, agree very well with the nitrogen data. So, for a particle
emitted at an angle 01, relative to the spectrometer axis, eqs. (3.1) and (3.2) relate its
momentum to the position on the front counter. But for angles 00 different from 01,,
the correlation of front counter position with angle 00, produced in the spectrometer,
renders eq. (3.2) inadequate. A 00 versus front counter position calibration would be
neccessary, which we were not able to obtain in the present experiment. There are
at least two ways to perform this calibration; first, by bending the beam before the
target with an inflector magnet system and second, by inserting a slotted aperture at
a certain distance after the target. In the present work, the calibration was obtained
from a TRANSPORT calculation, which should be adequate, as the fields of the
magnets and their positions along the beamline are well known. In order to take into
account the angle dependence of the curvature radius, a relation similar to eq. (3.2)
was derived in which the coefficients A0 and A1 were expressed as functions of the

angle 00 in first order. Specifically we can write, similar to eq. (3.2):

10(00) = Ao(9o) - 141(90)er (3-3)
where now

Ao(90) = A009») - Ko(9o - 95) (3-4)

A1(00) = A1(05) - K1(00 -— 01,) (3.5)

where the coefficients K0 and K1 are related to the calculated first, second and third
order matrix elements with the following expressions (these equations are derived in

appendix B):

_ po 3 (3W);
K° ‘ 100(z16)'(( '0’ + (210 W) (3'6)

50

 

 

K‘ T loopco'pp I335); (3’7)
where:

(3)05) ..-= (3)03) + (sIO62)6 (3.8)

(3(5) .—. (2|6) + (11(23):. (3.9)

30,, = g“: + a _ (3):)30. (3.10)

In these expressions we have: 6 = 1009192 the momentum deviation (in %) of the
particle from the momentum of the central ray, po the radius of curvature (in cm)
of the central ray inside the dipole magnet, so the s-coordinate of the beam-spot at
the target position (fig. 3.5a), Cpp the position calibration coefficient of the front
position counter (in channels/ cm), X", the position (in channel number) of the 0°
elastically scattered beam particles of one run that was chosen as reference, and a
the final position (in cm) of the reference beam particles relative to the optical axis
of the spectrometer (fig. 3.5b). Terms including so were taken to be zero, because

the value of so was not known (but was thought to be small).

One can see that eqs. (3.4) and (3.5) are functions of the reaction angle O, = Oo-Oo,
that is the angle of a particle leaving the target relative to the beam direction, which
is, of course, the interesting nuclear reaction angle. This angle can be calculated by

the formula:

9?? - [00(95) + 01(0b)5 + C;(Oo)62]

a. = 00 - 01 = (9(9) + (aways + (00605“

 

(3.11)

where Opp is the angle (in mr) of a particle at the front position detector relative to
the reference beam and Co(O(,), 01(0),), 02(05) are the coefficients of a quadratic fit

of the relation Opp = f (6) for the 0° elastically scattered beam particles (that is, for

51

Oo = 95). So, the scattering angle of a given particle can be calculated according to

eq. (3.11) from its final angle Opp and its momentum deviation 6.

The 6-dependence of the reaction angle O, and of the curvature radius p(O,.) for
a particle indicates that a successive approximation calculation has to be performed
in order to obtain O, and p(O,.) from the measured quantities X pp and Opp. For this
calculation, an initial value of 6 for a particle was obtained from its position X pp
using eq. (3.2) and the relation 6 = 100?. The convergence of this iterration is

fast, i.e. in few steps.

Using these expressions and procedure, the curvature radius (thus the momen-
tum) and the reaction angle of the particles were calculated on an event-by-event
basis. The resolutions (FWHM) of the curvature radius and the reaction angle were

approximately 2.0% and 15 mr, respectively.

As is standard in spectrometer measurements, an A/ q parameter was constructed
using the expression for the curvature radius and the measured TOF. Substituting

the momentum p = 7Am,v in eq. (3.1), we have:

7Amnv

 

Bp = 3.3356 (3.12)

where 7 is the relativistic factor 7 = (1 — fin-)4”

, m, is the nucleon mass (in MeV/c2)
and v is the velocity of the particle (in units of c, the speed of light). For the velocity
of a particle we can write: :2 oc TIIZ)?’ where L is the flight path length between
the target and the timing detector, TOF is the measured time-of-flight (relative to
the RF cycle) and T is a constant (representing the number of RF periods elapsed

between a given reaction at the target and the next beam pulse after the start signal

for this event from the detector). Substituting this expression of v in eq. (3.12) and

52

solving for A/q we get:

(3.13)

from which the A/ q parameter can be calculated also on an event-by-event basis.

Isotope separation and identification were performed using a two-dimensional his-
togram of AE versus A/q, generated for each run. In this histogram, the groups
corresponding to the elastically scattered beam particles, when present, were used to
calibrate the A/q axis. For each isotope group on the AE vs A/q histogram a soft-
ware gate was drawn from which a O, vs p/ q histogram was generated. Unfortunately,
due to limited TOF resolution, the isotope groups were not completely separated and

thus, each isotope gate was contamimated by neighboring isotopes.

Up to this point, the offline analysis was performed with the N SCL data analysis
program SARA [Winf 91]. For the subsequent analysis, a program was developed for
manipulation of the O, vs p/ q spectra in order to obtain the momentum distributions
of the different isotopes. The analysis steps performed by this program are outlined
below. First, a “control” matrix was created for the program. This matrix had
elements of the form: gate(isotope,run); that is, the elements were the gate numbers
of the different isotopes observed in the different runs of the experiment. The matrix
was read by the program enabling it to collect, for every isotope, the O, vs p/q spectra
from all the runs in which the isotope was present. Then, for each O, vs p/ q spectrum,
the program performed a cut of O, in three regions with centers at 1° , 2° and 3° and
widths of 1°. For each of these reaction-angle regions the program generated a p/ q

spectrum.

For normalization purposes, for each isotope, the program created a matrix of the
form: counts(p/q,run), containing the number of counts of every p/ q channel for every

run. The ratio of counts for runs with overlapping rigidity settings provided a way to

53

obtain a relative normalization of the runs. Beam charge collection was satisfactory
for only few of the runs (in which the beam was bent enough to hit the aluminum-bar
Faraday-cup in the dipole) and thus, we had to use this relative normalization method

for the rest of the runs.

For every isotope at each angle (1°, 2° and 3°), the program normalized the p/q
spectra, converted them to p/A spectra and properly combined the latter to give the
final p/A distributions. Finally, it integrated the p/A distributions at each angle to

create the angular distributions of the isotopes.

As representative results of the present study, the momentum distributions of
13C, 15B and 11Be produced by the “N beam at 75 MeV/u on Al target and their
angular distributions from Al and Ta targets are shown in Fig. 3.6. The error-bars
are statistical and are omitted when they are smaller than the plotting symbols. We
conservatively estimate that a factor of approximately 2 accounts for the systematic
errors primarily due to the uncertainty in the normalization of different runs with
overlapping momenta (approximately :I:20%) and to uncertainties in the beam current

integration.

The present results exhibit the main characteristics of momentum and angular
distributions of pro jectile-like isotopes produced at intermediate energies at and near
0 degrees. Near-projectile fragments (one or two nucleons removed), like 13C, have
narrow momentum distributions peaked near the beam velocity. Their angular dis-
tributions show the effect of Coulomb deflection, since for the Al target they are
steeper than those for the Ta target [Bibb 79]. If more nucleons are removed from
the projectile, the fragments, like llBe, have wider momentum distributions peaked
again near the beam velocity. Nucleon-transfer (pick-up) products have also been
observed in the experiment. The momentum distributions of these products (e.g. 15B

in Fig. 3.6) are peaked at considerably lower velocities than that of the beam. More

54

”N (75 MeV/u) + Al, Ta

 

 

 

 

 

 

 

 

 

Al target

*3 E 13 3 300 -—;
’11 15 :— C \ ‘2 zoo -3
O : f, E] E
\ 10 r- 1 1 100 I
% r- I \ .. 70 a
: : a

2 5.— f ] ' '1 5° 3
d - O, \ : A 30 _
\ EIIIIIIIII°ILI|IIIIIISIIIIIIT! a; irmIiiniIiLiriiii T
v 0’4:— B “i “D 50 —‘
o 3 :— éi .5 \E/ \ j

F—1 . .. .. —I
>\ 0.2E / 9. i "U 10 _§
< 0.1 .— 96 I -—. E 5 1
\ =| : I I 32' I I E, 'o l I I ‘
g 0.0,. .1 02 +44 1 1 LI 1 1 L L l I 1 I l
V :_ _: 7 o _
1—1 3: 11Be 3 500 _-_
N\ I . \. .. 300 _
b 2 :— . . \ _-. 200 __E
'° : ,I e : E
1 E— 0 —j 100 ‘

/ I \R‘ : 7o TTT\T\$ 3

O llllll-II(Illlllllllllllllfill;l 50 I l l I I I I 1 l I 1 1 I l 1 l 1 T1—

320 340 360 380 400 420 0 1 2
p/A (MeV/c) 9 (degrees)

Figure 3.6: Momentum and angular distributions of several projectile-like isotopes
produced by a 1‘N beam at 75 MeV/u on 2"Al and ’8’Ta targets. The momen-
tum/ nucleon distributions are at 0° from the 2"Al target. The arrows correspond to
the momentum/ nucleon of the beam. The dashed lines are Gaussian functions fitted
to the data. The angular distributions are from both the Al (squares) and Ta (closed
points) targets. The lines are just to guide the eye.

55

detailed measurements of such transfer products are presented in chapter 4 and their

characteristics are analysed in chapter 5.

Summarizing, the quality of the results of this experiment was limited, due to
the following reasons: first, the poor TOF resolution which hindered a clear isotope
separation and second, the limited momentum and reaction-angle resolution of the
spectrometer. The latter was partially due to time constraints during the experiment,
which prevented us from fully investigating the optics of the spectrometer and per-
forming more detailed calibrations. Nevertheless, the analysis of this data set was very
instructive, leading us to develop procedures to manipulate and combine effectively
the large number of spectra involved in the construction of distributions of projectile
fragmentation products. The preparation and experience obtained from the present
study was crucial for our subsequent measurements with the A1200 beam analysis
device, which are described in the following chapter. Finally, from a practical stand-
point, the experiment was very informative, giving us insight into how to operate a
O-degree spectrometer for fragmentation measurements and indicating the problem
of the alignment of the beamline (relative to the direction of the beam as emerging
from the accelerator) and the critical nature of the stability of the RF timing for the

future experiments with the A1200.

Chapter 4

Projectile Fragmentation
Measurements with the A1200
Mass Separator

I Introduction—Description of the A1200

In this chapter, studies of the projectile fragmentation process with the A1200 Mass
Separator are described. This work represents a continuation of our preliminary study
with the interim K1200 beamline (described in chapter 3) in which more detailed mea-
surements of the momentum distributions of near-projectile fragments and especially

of few nucleon pick-up products from peripheral reactions at intermediate energy were

made.

The A1200 beam analysis device [Sher 90], built in the place of the interim K1200
beamline, connects the K1200 cyclotron with the experimental vaults. The A1200 can
be used either as a beam transport device or as a fragment separator. It can separate
projectile fragments from K1200 beams and deliver them to the various experimental
devices or it can be operated in a “stand-alone” mode to study these fragments. The
A1200 (fig. 4.1) consists of a large number of superconducting magnets [Zell 85],
which are: four 22.5" bending dipole magnets, four quadrupole doublets and two

quadrupole triplets. There are also four room-temperature sextupole magnets for

56

57

aberration corrections.

The design and the symmetry of the A1200 allow two basic modes of operation:
a dispersive mode and an achromatic mode. In the dispersive mode, intended for
accurate cyclotron beam analysis purposes, the dispersions of the two dipole pairs are
added to give a maximal resolving power but a small acceptance. In the achromatic
mode, primarily used for radioactive beam separation, the dispersions are canceled
at the end of the device to provide a final achromatic image, while creating two
intermediate dispersive images (labeled Image #1 and Image #2 in Fig. 4.1) between

the two pairs of dipoles.

Furthermore, in the achromatic mode, two different acceptance modes are possible:
an intermediate (or medium) acceptance mode, if the target (labeled OBJECT in Fig.
4.1) is placed at the beginning of the A1200, and a high acceptance mode, if the target
is placed near the entrance to the first quadrupole triplet. The characteristics of the

A1200 operation modes are summarized in Table 4.1.

The magnets of the A1200 can be conveniently set to a desired Bp value with
a computer code. For a given rigidity value the code calculates the currents for
the magnets and sends the values to the power supplies. This program requires a
minimum amount of time to set all 22 magnets of the spectrometer for a measurement
at a given rigidity value. This feature was crucial for our distribution studies, because
in order to cover a substantial momentum range for all the interesting isotopes, a large
number of data runs (e.g. ~20-30) of neighboring rigidity settings had to be performed
(since the A1200 has a momentum acceptance of only 3% in the medium acceptance
mode). For accurate rigidity measurements, the fields of the dipole magnets are
measured by N MR magnetometers. For beam charge collection, aluminum bars are
located in the first pair of dipoles, along with movable blockers at the exits of these

dipoles. The bars and the blockers can be connected in series to a current integrator.

58

Schematic Layout

W//////7/I

/////////////n .
’ ’¢ Inn/ago O 1

FINA
22.5. DlpOIOS I ‘ % SCXIUPOIOI IMAG‘E-
, ,/ oaaecr Superconducfln o I

Quodru ole Don Ms ‘ MO' 4 k
' Q // //

K” -- g
/ A-IZOO BEAM ANALYSIS %
/

DEVICE

 

 

 

 

 

 

 

 

 

 

 

Figure 4.1: Schematic layout of the A1200 beam-analysis device.

59

Table 4.1: Characteristics of the operation modes of the A1200 [Sher 90].

 

 

 

 

 

 

 

 

 

 

Parameter I] Operation Mode
H High Res Medium Acc. High Acc.

fl[msr] 0.1 0.8 4.3

A0[mr] 10 20 54

Aflmrj ll 10 40 so

AP/P[%] |[ 0.4 3.0 3.0
n Bp[T-m] H 5.4 7.2 5.4
[[ Resolution? 100001 3300 1300
" (X/X)

Int. Image [I 3.0 0.47 1.24

Focal Plane 1.0 0.99 1.0

(X/5)[cm/%l

Int. Image [1 15.0 1.67 1.80

Focal Plane 10.0 0.0 0.0

 

 

 

 

Jtlmlrn beam spot on the production target.
tat the final image, with half the resolution at the center image.

 

60

II Experiment and results

In the fragmentation measurements with the A1200, we operated the device as a 0-
degree spectrometer in the achromatic medium acceptance mode. The full angular
acceptance (A0 = 20 mr, A05 = 40 mr) around Oodegrees (i.e., beam direction)
was employed. Fragmentation targets were placed at the beginning of the device
(“OBJECT” in Fig. 4.1). The reaction products were focused at the end of the
device on to a detection array consisting of two X-Y position sensitive parallel—plate
avalance counters (PPACs) and a AE—E telescope. The AE detector was a totally
depleted ion-implanted silicon detector (briefly called “PIN -diode”) 500 microns thick
with an active area of 25 cm”. The E detector was a 5 cm thick plastic scintillator
viewed by two photomultiplier tubes, one on each end. The two PPACs before the
AE—E telescope, separated by ~60 cm, were used to monitor the focusing of the
particles: the first PPAC determined the position of the particles at the final image
and, combined with the second one, the final angle with respect to the optical axis of
the device. A third X-Y sensitive PPAC was positioned at the second intermediate
image (Image #2 in Fig. 4.1) of the A1200. At this dispersive image, the horizontal
position of the particles is linearly related to their rigidity, and thus the momentum

of the particles can be calculated.

The position signals from the PPACs were created by charge division of an electron
avalanche collected on a series of anode strips (which were connected in series with
resistors). Each detector has two sets of anode strips, one horizontal and one verti-
cal, thus providing X- and Y-position information. The principle of resistive charge
division is illustrated in Fig. 4.2. The PPACs were operated with isobutane gas at
a pressure of 5 torr and at an anode voltage of ~600 Volt. The position resolution

(FWHM) obtained with these detectors was approximately 1 mm.

61

RESISTIVE WIRE

 

 

¢ D
EVENT POSITION

QR
QL+ QR

 

X:

Figure 4.2: The principle of resistive charge division for position measurements. The
charge of an electron avalanche, created by a fast charged particle entering the gas
volume of the detector, is collected by a series of anode strips which are connected in
series with resistors. The charge is split into two fractions, which propagate towards
the two ends of the resistor series. The ratio of the signals at the ends of the resistor
series is directly related to position.

62

A calibration of rigidity versus Image #2 horizontal position (simply called posi-
tion in the following) was obtained by measuring the positions (channel numbers) of
the 0-degree elastically scattered beam particles with various charge states (see Fig.
4.7). An absolute measurement of the beam momentum was achieved by positioning
a known charge state of the beam at the center of Image #2 PPAC and recording the

magnetic fields of the first pair of dipoles.

Time-of-flight was measured against the RF cycle of the cyclotron, with a reso-
lution (F WHM) better than 2 us for the particles of interest. The AE measurement
with the PIN-diode provided very good Z separation and a combination of AE with

the TOF measurement provided unambiguous particle identification (see Fig. 4.8).

A schematic diagram of the electronic modules used in the present measurements
is shown in Fig. 4.3. The basic characteristics are similar to those described in the
previous chapter and are briefly described here. The four position signals (Left, Right,
Top, Bottom) obtained from each of the three PPACs were read by ADC’s. TOF
against the RF cycle was measured with a TAC. The AB detector signal was sent to an
ADC; the two E—scintillator PMT signals were sent to QDC’s and their timing signals
were AN D’edl. The event trigger (Master Gate) was defined by the coincidence of the
following signals: the AN D’ed Image #2 PPAC left and right signals, the AE signal
and the AN D’ed left and right scintillator signals. Again for diagnostic purposes,
TDC signals were recored for all the PPACs, the AE detector, the E scintillator and
the RF timing. A total of 31 signals were recorded by the front-end computer. Again,
the standard N SCL data acquisition system was used to read the electronics modules

and write the data to tape [Fox 89].

In our first experiment with the A1200, upon commissioning of the device, we
measured the fragmentation products from the reactions of a 2"N e beam at 85 MeV/u

with 181Ta (10 mg/cm’) and 9Be (3 mg/cmz) targets. The target thicknesses were

63

IMAGE #2 PPAC

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

SHAPING AMP
ADC
TOP TFA um
F TDC
0
ADC
son .-.. 5"
TF SCALER
F TDC
0
MAN
RIGHT 5‘
TF CVSCALER
E_TDC

 

 

 

 

 

 

 

 

 

 

 

 

 

IWV\_ADC an 2
LE” TFA C SCALER
E_mc

(FINAL IMAGE PPAC'S IDENTICAL)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 
  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

pRg ADC li—SCALER
.. A. 1...... . ~—-—"°::::::
PIN DIODE C —T0c CPU NOT BUSY—N ‘———[g'___oc]—
F D O.
0 u r—_l_roc START (common)
09° T ' L—L—mc srop (couuou)
._ AC START , .l T
LEFT i {5.10. —-—" 5 Ross
E 50'“. _ d]— B" 3
MASTER cm:
__ooc
r SEALER
mom . A c
E 5cm. F “3°
0
'5 scam
CYCLOTRON _ r we
RF 0 TACI srov
an I

Figure 4.3: Schematic diagram of the electronics for the fragmentation experiments
with the A1200 (for definitions of the acronyms see Fig. 3.3).

64

such that, for the particles of interest, the energy loss and the angular spreading due
to multiple scattering were negligible. A series of successive data acquisition runs were
performed at overlapping rigidity settings that covered the A/ q region of 2.1 to 3.0.
The rigidity steps were 2% and thus, an 1% overlap between runs with neighboring
rigidity settings was obtained (since the rigidity acceptance of the A1200 is 3%). In
addition to the 20Ne experiment, we performed the same series of measurements with

beams of 15N and 18O at 70 MeV/ u with the 181Ta target.

The analysis of these data sets was similar to the data analysis described in chapter
3. However, it should be pointed out that the reconstruction of important quantities
from these data was simpler and more accurate, due to the superior optical charac-

teristics of the A1200 compared to those of the simple interim beamline.

Isotope separation and identification were performed, as already mentioned, with
a two-dimensional histogram of AE vs A/q. The isotope groups were very well sep-
arated and unambiguously identified. For each isotope, a position spectrum was
generated. The position spectra were converted to p/ A spectra (using the rigidity-
position calibtation) with a modified version of the program for spectrum manip-
ulation discussed in chapter 3. Subsequently, with this program, the p/ A spectra
were normalized and properly combined to give the final p/ A distributions. Normal-
ization of runs with the beam current integrator was not successful, since in many
runs the unreacted beam was not completely collected by the Faraday cups inside the
first pair of dipoles. Again, we had to rely on a relative normalization of the runs,
which turned out to be partially successful (mainly due to the small overlap between
runs with neighboring rigidities). For this reason, only a few of the spectra were of
good quality. The best PLF momentum distributions from this analysis are shown in

Figures 4.4, 4.5 and 4.6.

65

In addition to the previous experiments, a detailed measurement of the fragmen-
tation products from an 18O beam at 80 MeV/u with an 27A1 (5 mg/cm’) target
was performed at a later time. As additional experience operating the A1200 had
been acquired, this measurement was more successful and complete. In this exper-
iment, beam charge normalization was obtained using a monitor detector placed a
short distance after the target at a small angle relative to the beam direction. For
a given run, the total number of counts recorded by this detector was pr0portional
to the beam charge of this run, thus providing a satisfactory normalization of the
different runs. The data analysis of this experiment was similar to the analysis of
the previous experiments. In addition, the momentum distributions of the isotopes
were also normalized by the corresponding efficiency of Image #2 PPAC relative to
the AE detector (whose efficiency was taken to be 1). The relative yields of the frag-
ments were obtained by integrating the normalized momentum distributions and are
analysed in chapter 5. Representative results of momentum distributions from this

measurement, along with Gaussian functions fitted to the data, are shown in Figures

4.9 and 4.10.

The momentum distributions obtained from the measurements with the A1200
were of better quality compared to the distributions obtained from the K1200 interim
beamline measurements. As also previously observed, along with nucleon-removal
products (whose distributions were centered very close to the beam velocity), nu-
cleon pick-up products have been observed in these measurements (with momentum
distributions peaking at velocities lower than that of the beam). An analysis and

discussion of these distributions is presented in chapter 5.

66

20Ne (85 MeV/u) + Ta, Be

 

 

 

 

 

Ta target Be target
25000 :—- 20F 1 200000 E— 20F . —1
'0‘ 20000 :- : -g 150000 I- ,\ —2
C: 15000 :_' , l —: : ’\ j
C: E E 100000 :- -:
3 10000 :— 7 ‘\ —: E f .\ S
o 5000 Z— ! a; —: 50°00 :— d j ‘1
L0 0. .1 O:llllllLlJIlllll 1141ik11Ll11‘
0) 000 '— '._‘ r; 2
Q. 5 5 190 . 30000 :- 190 . . _q
3 4000 .— P ‘1 : ,\ 2
g 3000 E— , ‘4 —§ 20000 :— N" _i
O E_ 1 __‘ : d 2
° 2000 E I q, 10000 _— I 1
P 1000 F' I ‘E : I j
.g o :Illll I I llllllLllllLI l hillli o :I ll 1 l [111! I ll 1 I [JAIL :
'Ei moi—200 —: 2005—200 1
E 50 E“ f j 150 Z— 0 -—j
° 60 5— h -e E if 1
Z 5 5 100 ;- Q '1‘
4° 5' ' i, "z 5 I k a
: I = 50 r' I ‘ -:
20 :- i -: _ .
W, lu14l1111l1u‘1111d111il1:

 

 

 

 

 

0 0
320 340 360 380 400 420 320 340 360 380 400 420
P/A (MeV/C)

Figure 4.4: Momentum/nucleon distributions of PLFs produced by a 20Ne (85
MeV/u) beam on 181Ta and 9Be targets. The arrows correspond to the momen-

tum/nucleon of the beam. The dashed lines are Gaussian functions fitted to the
measured distributions.

67

15N (70 MeV/u) + Ta

 

 

250000 :— 14C ‘ -=

... 200000 :— 0 \ —:
a.) 5 I 1
5 150000 :— , c. "'2’
2 100000 E- I’ if -—:
0 50000 g— l l -:
16 0 MW
e. 5000 :- 15 —:
In : C I :
75 4000 ’— f\ —‘
a E , ' 2
0 2000 :— l I, —5
'U r :
g OMILLfiQIIIIIIIIIIULI:
I: ','_ _:
g 1000 :— —:-
O :_ ,_:
z 750 + a
5005- I i l '1

250 I— I —

Pll [JAA I] l I I‘ll] l l l l l l I ll .

 

 

 

0320 340 360 380 400 420
p/A (MeV/c)

Figure 4.5: Momentum/ nucleon distributions of PLFs produced by a 15N (70 MeV/u)
beam on a 181Ta target. The arrows correspond to the momentum/ nucleon of the
beam. The dashed lines are Gaussian functions fitted to the measured distributions.

68

1E30 (70 MeV/u) + Ta

 

 

 

 

 

 

 

 

 

'2 40000 E“ 17N R 5 40000 E— 16C a 1
2 30000 3’ '. '3 30000:— , , —2
: a j : 5
f: 20000 :— 0: ‘, 3 20000 E— ,' f —:
° : : 2 a
2' 10000 :- I ‘0‘ -E 10000 .— II. $\ —:«
é OW 0ElmLJIIA/IIIIII‘llliIIIIIIJIE
8 12502—1810 0 ‘g 10003-17C 0 -—
'3 100°?" . “i 300:— ‘ —:
% 750} I ', ‘i 6003— {'3' _§
5.. _: 5__ \ __§
5 , \1 :33E I *1
z :— P‘ 5 IE— —.:.
'- llllLL 11in IJIIILIIIILLIII‘

o 1111 llL.L J o
320 340 360 350 400 420 320 340 360 380 400 420

p/ A (MeV/C)

Figure 4.6: Momentum/ nucleon distributions of PLFs produced by a 18O (70 MeV/ u)
beam on a 181Ta target. The arrows correspond to the momentum/ nucleon of the
beam. The dashed lines are Gaussian functions fitted to the measured distributions.

69

 

 

 

 

.- 1 I T 1 I I T fit T l I I r T I I I T I 1—1
3.150:- T
E g . 1007+ 1
(D 3.125:- \ T
s... ‘ i
- a... .
g : *\ I 1808+ ‘
:3 - 1":
o - \\
'8 3.075— ' *Q —
. +
m - \\
.3 " \I-h<
“g 3.050:- \ ‘
m _ i
3.025~1L_11,,,,1L...1...1‘
400 500 600 700 800

Position (Channel No)

Figure 4.7: Rigidity vs Image #2 position calibration for the 18O (80 MeV/u) + 27A1
experiment. The dashed straight line is a least-squares fit to the data points.

70

AE

 

 

 

 

 

Figure 4.8: An example of a AE vs A/q histogram from the 18O (80 MeV/u) + 2"M

experiment.

71

180 (80 MeV/u) + A1

 

 

 
 

 

 

 

 

8000 :1 19 5000 E— 15 :

_. 6000 .— 0 ,‘1 1 400° 2“ C "1 "E
<19 : 1 : = 6 a =
. l 1 3000 .— —:

E; 4000 _— . 1 ——, E I 9 a
d 7. I § 2 2000 5" f \ ‘1
4'4 2000 :— -§ :__ 1 _5
2 : ' 1i 1 1°00 5 ’ l ‘0 5
o WWI—Lu: 0 MW

0) 1- ..‘1 L'... _‘
g, 00000 E- ”N p; E 2500 E 16C A 5
1’3 50000 E- 1 1 1 200° ~3— s 9 '—
5 E I 1 E 1500 :1.- I 1 --f
0 40000 :- 9 1 1 ; I 1 :
o 5 , 9 3 1°00 :- 9 o 1
'0 30000 5" s I \ ‘3 500 E- I \ -E
0) pl I l :1, I h I a I C 1
E 1500 — 18N A 1 60 '— 17C ’1. —
I... - I Q j I I:
0 1000 E- ? 1 -_ 40 L. \ .4
z : I 1 : E 0 § 3
500 :— 0 I‘\ 1 20 E- f‘ 1

P 9 : :II I Q: I is I 3

0 WM 0 ,

320 340 360 380 400 420 320 340 360 380 400 420

p/A (MeV/C)

Figure 4.9: Momentum/ nucleon distributions of PLFs produced by a 18O (80 MeV/ u)
beam on an 2"Al target. The arrows correspond to the momentum/ nucleon of the
beam. The dashed lines are Gaussian functions fitted to the measured distributions.

2500
2000
1500
1000

500

1000

500

0
50
40
30
20
10

Normalized counts per channel

Figure 4.10: Momentum/nucleon distributions of PLFs produced by a 18O (80
MeV/u) beam on an 2"Al target. The arrows correspond to the momentum/ nucleon
of the beam. The dashed lines are Gaussian functions fitted to the measured distri-

butions.

1500'

72

18o (80 MeV/u) + A1

 

 

_13 .2
: B I“. :
I \ I
_. P Q __
E ‘ 1 1
. ' ‘ ..
.— 0/ "1
1| 123:."1i1isi
.. / "
E—-14 —i
5 B ‘x E
:" J 1 ‘2
E I ‘5 i
'2‘ 09 ‘:
:_ .I I \ —§
5 / \ :

 

 

0
320 340 360 380 400 420

2500
2000
1500
1000

500

150
100

50

508
400
300
200
100

 

 

 

 

 

5'10 0 'i
;. I363 0 .4
E "\ E
:— . \ '1
: 6 ‘0 :
.— 11» 5 ° ‘2
: / :
plilmiLfiaiiiiliiiiliLjil1

I- -1
TllBe I 1
7.. / \ .2
I \\ I
; 4
r' 0‘ ‘5
- 0.1-1133.!ij .1....1 1..9..1

:‘8 1
E 3
FL I ,6“. ‘2
E“ Q. n '1
5 +1}- Q _s
F’ +. 1
:— , (1'?
hllwl+lfllllllll Ill llLl l

 

0
320 340 360 380 400 420
P/A (MeV/C)

Chapter 5

Results and Discussion

I Introduction

In this chapter, the characteristics of the distributions of projectile-like fragments
obtained from the experiments described in chapters 3 and 4 are presented in a sys-
tematic way and are interpreted with simple models. The momentum distributions
of the fragments were fitted with Gaussian functions using a simple histogram ma-
nipulation code. Care was taken to exclude the tail on the low-momentum side of the
distributions, which is attributed to dissipative mechanisms [Blum 86, Borr 86]. The
centroids and the widths of the fitted Gaussian distributions were then summarized
and analysed. A detailed study of the centroids, the widths and the relative yields
of the fragments produced in the reaction of 18O (80 MeV/u) projectiles with 2"Al
target is presented in the following sections, as this data was the most complete of the
various systems studied in this work. The characteristics of the distributions from the
other measurements were consistent with the 18O (80 MeV/ u) results. A compilation
of the centroids of the momentum distributions of nucleon pick-up products from all

the reactions studied in this work is given at the end of section II.

73

74

II Study of the centroids of the momentum dis-
tributions

A Nucleon removal products

The centroids of the parallel momentum/ nucleon distributions of nucleon removal
products from the reaction of 18O (80 MeV/u) with 27A1 is presented in Fig. 5.1.
For comparison purposes, the centroids are divided by the momentum/ nucleon of the

projectile.

The centroids of these distributions are close to the beam momentum/ nucleon and
do not follow any monotonic dependence on the mass loss AA (the number of nucleons
lost by the projectile). This feature is in contrast to that observed in high energy
fragmentation reactions in which the centroids decrease with AA (approximately

linearly for near-projectile fragments) [Morr 89, Stép 91].

The dependence of the mean momentum/ nucleon on mass loss can be derived in
a simple way from energy conservation. We assume that the removal of AA nucleons
from the projectile requires some “unbinding energy” E, coming from the projectile
initial kinetic energy Tp, thus the projectile kinetic energy at the moment of frag-
mentation is reduced to: T}, = Tp - E, Furthermore, we assume that the fragment’s
mean energy per nucleon, (Tp) /Ap, is equal to the projectile energy per nucleon at

the moment of fragmentation:

ifil = 115% (5.1)

The fragment’s mean (parallel) momentum/ nucleon is, of course, related to its

mean kinetic energy/nucleon by the relativistic equation:

(1%) = 1 \[fTFl ((TF) +2mncz) (5.2)

.72 AF .4;-

 

 

75

 

1.04 I 1

 

 

 

I F 4

,

o .

1.02 l- 0

.0 ' .
2?

. .

E L 0 0 1

V 1'00 0".“ '~L . E ..... g a .......... .-

t " ‘-‘o‘--' . . 0 ...... ,. ‘

\ 1 , ~. . ,,,,, if ,
' - .. ..... ~ ‘ o ...... o. . . . . .........

FE T 5 s "0' 0 ‘
2 i‘ '-°‘ ‘ ‘ ‘ ‘ 4
\ 0.98 _— ~ - g ‘ ‘ T
3; ~ ~ 2 ‘ .
I... 0.96 — —1

* 0, : 10.0 20.0 25.0 40.7 27.0 20.2 30.7 10.2 33.7 j

.

- -q : 4.0 14.7 7.7 22.0 0.0 4.7 13.7 1.0 0.0 1
0.94 l l l l I I l l l l L

 

O
H
N
(.0
uh
0|
Q
‘1
CD
CD

10

Mass Loss (AA)

Figure 5.1: Variation of the centroids of the parallel momentum/ nucleon distributions
of nucleon-removal products from the reaction ‘80 (80 MeV/u) + 2"Al. The closed
circles are the experimental points. [The two closed squares are from the reaction 180
(70 MeV/ u) + 181Ta.] The solid line corresponds to the beam momentum/nucleon.
The dashed line and the diamonds correspond to calculations according to eq. (5.3)
(see text). The open circles represent calculations using eq. (5.5). For a given
fragment, the cluster separation energy E, and the Q-value are given (in MeV) below
the isotope symbol (see text).

76

where m" is the rest mass of a bound nucleon (m,,c2 = 931.5 MeV). Inserting eq.

(5.1) into (5.2), we get:

El—lJTp-E, (Tp-E. )
AF — C AP AP +2m,.c2 (5.3)

 

 

We can calculate the separation energy E, in two limits (as already mentioned in
chapter 2): first, an independent-particle picture of the projectile nucleus and second,
a cluster picture. Assuming that the nucleons are independent particles, they can be

removed one after the other and the total energy required is given by:
E, = EMA/1 . (5.4)

where EM is the average separation energy per nucleon (~8 MeV for medium and
heavy nuclei). In this case, the mean momentum/ nucleon of a fragment is directly
related to AA. The values calculated according to eq. (5.3), using eq. (5.4) for E,,

are given by the straight dashed line in Fig. 5.1.

On the other hand, in the cluster-stripping model, E, is the energy required to
separate the projectile into two bound clusters: the observed fragment F and the re-
moved part B. The cluster separation energies E, for the fragments can be calculated
from binding energies (see next section for details) and are listed in Fig. 5.1 below
the fragment symbols. The values of the mean momentum/ nucleon calculated with
this model (shown in Fig. 5.1 by the diamonds) seem to follow the trend of the data,

but they are also considerably lower than the experimental points.

The failure of the previous simple approaches to accurately describe the centroids
of the momentum distributions led us to take into account the effects of the target
in the nucleon removal process (which is traditionally ignored in descriptions of high-
energy reactions). In an attempt to describe the variation of the centroids of the

nucleon removal products that takes into consideration the presence of the target,

77

we substituted —Q in eq. (5.3) for E,, where Q is the ground-state Q-value of the
two-body pro jectile—target rearrangement reaction leading to a given fragment. Thus,

similar to eq. (5.3), we have:

(PF)__1_ TP+Q TP+Q m
WF-d/ Ap ( Ap +2 .c’) (55)

 

 

The calculated values of the centroids for the reaction of 18O (80 MeV/u) on 2"A1 are
given in Fig. 5.1 by the open circles. Except for 1"N and 16C, very good agreement
between the calculated values and the experimental points is obtained, particularly
for the variation from nucleus to nucleus. From this comparison we infer that the
mean momentum/ nucleon of the removal products approximately follows the vari-
ation of the Q-value of the corresponding pro jectile—target rearrangement reaction
(rather than the separation energy E" which is the case in high-energy fragmenta-
tion). Interestingly, the mean momentum/nucleon values of 1"N and 16C are higher
than that of the projectile (0.7% and 0.4%, respectively). This feature is attributed
to the importance of direct transfer (stripping) reaction channels in the observed
momentum distributions of these fragments. Such direct transfer contributions are

expected to decrease progressively for products further in mass from the projectile.

In order to calculate the mean momentum/ nucleon of a pro jectile-like fragment
produced by a two-body transfer reaction, we first apply energy conservation to the

pro jectile—target system and we have:

T p + Q = (Tr) + (T11) (5-6)

where (Ty) and (Tr) are the mean kinetic energies of a given fragment and the
corresponding target-like residue (which has accepted the nucleons removed from the
projectile), respectively. Because we measure the PLF distributions very close to 0

degrees in the lab system, we should expect (T11) < (Tp). Kinematics calculations

78

with the code “S320” [Winf 86] for the system 180 (80 MeV/u) + 2"Al showed that,
near 0 degrees, (T77) is about 0.5-1.0 MeV (whereas (Tp) is on the order of 1400
MeV). Thus, the PLF has essentially all the initial projectile energy (plus the small

negative Q-value due to the two-body rearrangement):
(T1?) = Tp + Q (5.7)

Substituting this expression in eq. (5.2), we get:

(PF)__1_ TP+Q TP+Q m
CRT—C‘f/ir ( AF +2 "62) (5.8)

 

 

Evaluating eq. (5.8) for 1"N, we get: f E ((pp)/Ap)/(pp/Ap) = 1.028 (see Fig.
5.1) and for 16C we get: f = 1.057. We see that these values of the mean momen-
tum/ nucleon of 1"N and 18C are higher than the experimental values, whereas the
values calculated using eq. (5.5) are lower. Consequently, the observed momen-
tum/ nucleon distributions of these products contain a significant contribution from
the direct stripping channels, which have mean momentum/ nucleon (velocity) higher
than that of the beam. Direct transfer (stripping) channels may contribute, to a lesser
extent, to the distributions of the other nucleon removal products, since the centroids

of their momentum distributions tend to be higher than those calculated according

to eq. (5.5).

In contrast to the 18O (80 MeV/u) + 2l’Al system, the centroids of the momen-
tum/ nucleon distributions of 17N and 16C from the reaction ‘80 (70 MeV/u) + 181Ta
are lower than the beam momentum/ nucleon, approximately following the predictions
of the high-energy model [eq. (5.3)] previously discussed. Consequently, the direct
transfer channels are suppressed for this system. Such suppression with the heavier
181Ta target may be due to the higher Coulomb barrier of the system (72.7 MeV in
the lab frame) compared to that of the 18O + 2"Al system (28.7 MeV).

79

B Nucleon pick-up products

The centroids of the momentum / nucleon distributions of the nucleon pick-up products
observed in the 18O (80 MeV/u) + 2I'Al experiment are shown in Fig. 5.2 (closed
circles). These centroids are considerably below the momentum/ nucleon of the beam,

in contrast to the case of nucleon-removal products previously discussed.

In order to calculate the centroids of the distributions of the pick-up products, eq.
(5.8) can be applied [as previously discussed, the final product is assumed to have
essentially all the projectile initial kinetic energy (actually (Tp) = Tp + Q)]. The

calculations according to eq. (5.8) are shown in Fig. 5.2 by the open circles.

For 190, the calculated value (f = 0.969) agrees very well with the experimental
value (f = 0.971). Thus, we conclude that 19O is mostly produced by the direct
reaction: 2A’Al(180,190)23A1 (Q = -9.1 MeV).

For 18N , the calculated value (f = 0.993) is ~l% higher than the experimental value
(f = 0.984). In this case, the exchange reaction 2"Al(180,‘8N)""'Si (Q = -18.7 MeV)
may only partially contribute to the observed momentum distribution. In addition, we
can suppose that 18N is also produced by a two-step mechanism consisting of a break-
up of 18O to 17N and a neutron pick-up by 17N. According to this mechanism, the
mean kinetic energy of 1"N produced in the first step can be calculated as discussed
previously for the nucleon removal products. For the second step, represented by
the pick-up reaction 2"Al("’N,“’N)"3Al (Q = —10.2 MeV), the mean kinetic energy
of ‘8N can be calculated from eq. (5.7) where, instead of Tp we insert the mean
kinetic energy of 17N calculated in the break-up step. Then, from eq. (5.2), the
mean momentum/nucleon of this procuct can be calculated and is found to be (f =
0.965) ~2% lower than the experimental value. Consequently, we conclude that (at

least) both mechanisms (the nucleon exchange mechanism and the break-up+pick-up

80

 

180 (80 MeV/u) + A1

 

 

 

 

1.00 (L -—
,2 1 ° 0
<1 . 0... .
E 0.90 - "'9 -
. N .
v .. (b)
\ 8 I 1
m . o o .
2 0.96 '" - -
\
a. 1
V . \ .
Ill 0.94 F (a) -
H i '1
_ 190 ION 170
0.92 - -Q : 9.1 10.7 31.9 :1
r 1
I 1 i 1 1 4
0 1 3

Mass Loss (AA)

Figure 5.2: Variation of the centroids of the parallel momentum / nucleon distributions
of nucleon pick-up products from the reaction 180 (80 MeV/u) + 2"Al. The closed
circles are the experimental points. [The two closed squares are from the reaction
180 (70 MeV/u) + 181Ta.] The horizontal straight line corresponds to the beam
momentum/ nucleon. The open circles are calculations assuming a single-step transfer
reaction [the Q-values are given (in MeV) below the isotope symbols]. The diamonds
are calculations assuming a two-step mechanism: break-up + pick-up (see text).
The inclined straight lines are calculations according to a simple model based on
momentum conservation (see text).

81

mechanism) contribute to the production of 18N nuclei.

A similar analysis can be followed for the 17C case and similar conclusions can
be drawn. Assuming the two-body transfer reaction 27'Al(130,”C)28I’ (Q = —31.9
MeV), the calculated mean momentum/nucleon (f = 1.018, not shown in Fig. 5.2)
is ~4% higher than the experimental value (f = 0.980). Assuming a two-step mech-
anism involving, first, a break-up of 18O to 16C and then a neutron pick-up by 16C:
27'Al("3C,”C)26Al (Q = -l2.3 MeV), the calculated value (f = 0.959) is ~2% lower
than the experimental one. Alternatively, assuming a different two-step reaction in-
volving, first, a break-up of 18O to 17N and then, a nucleon exchange reaction to 17N:
2"Al("N,”C)2"Si (Q = —18.0 MeV), the calculated value (f = 0.991) is now ~1%

higher than the experimental value.

It should be noted that the experimental values of the mean momentum/ nucleon
of 18N and 1"C from the 18O (70 MeV/u) + 181Ta reaction are well described by
a mechanism involving, first, the fragmenation (break-up) of 18O to 17N and 16C,
respectively, and second, a neutron pick-up by these intermediate fragments. The
mechanisms involving the exchange reactions previouly discussed are, once again,
suppressed in the case of the 181Ta target. From this we conclude that, apart from
the case of 190, more than one channel is involved in the production of the nucleon

pick-up products.

Considering the complicated nature of the production mechanism of the nucleon
pick-up products, a simple model able to predict the general behaviour of the mean
momentum/ nucleon of these products is desirable. For this purpose, we developed
a simple mechanical model of the pick-up process based on momentum conserva-

tion. According to this model, the mean (parallel) momentum (pp) of the observed

82

fragment F is written as:

(pr) = (PPF)+(P1) (5.9)

where (ppp) is the mean momentum of the projectile part (with mass Apr) of the
observed fragment which is assumed to have, on average, the momentum/ nucleon

(velocity) of the projectile. Thus:

(pPF) = APF fif- (5.10)
P

and (1),) is the mean momentum of the nucleon (or nucleons) picked-up from the
target. If we assume that the transfered nucleon is picked-up randomly from the target

(thus, (pg) = 0), then the mean momentum] nucleon of the fragment is, according to

eq. (5.9) and (5.10):

(pr) _ APP PL

AF — AF AP (5.11)

The calculated values according to this equation are represented by the solid line
labeled (a) in Fig. 5.2 and are much lower than the experimental values. However, if
the nucleon is assumed to be picked-up from the target with momentum, on average,
equal to the typical nuclear Fermi momentum (230 MeV/ c) oriented along the direc-

tion of the projectile motion (thus, (pt) = ppemg), then the mean momentum/ nucleon

of the fragment is:
L10 _ L ( P_r )
AF - AF APF AP + pFer'rm (512)

The values calculated according to eq. (5.12) are given by the solid line labeled
(b) in Fig. 5.2 and are in good agreement (within :1:0.5%) with the experimental
points. Consequently, the downshift of the mean momentum/nucleon of the pick-

up products can be described in a simple way by momentum conservation and the

83

additional assumption that the nucleon is picked-up from the target preferentially
with momentum equal to the nuclear Fermi momentum oriented along the direction

of the projectile motion.

This model also applies to pick-up products where more than one nucleon is picked-
up from the target. If the number of picked-up nucleons is AAg, then similar to eq.

(5.12), we have:

1
(71;? = X}; [(14]? — A143) if: '1' AA¢ pFermi] (5.13)

where we have used the relation: App = A; — AA. for the mass of the projectile part

of the final fragment. Rearranging this equation we get:
AA
M — '11P; - —t (£1: -' chrmi) (514)

which, for a given projectile energy, has a linear dependence of the mean momen-
tum/ nucleon on AAglAF. A compilation of the centroids of the momentum/ nucleon
distributions of few-nucleon (up to 3) pick-up products obtained from the experiments
described in chapters 3 and 4 is presented in Fig. 5.3. The centroids are divided by
the momentum/ nucleon of the corresponding beam and are given as function of the
ratio AAt/Ap. The experimental points are shown by the closed symbols and the
calculated values [according to eq. (5.14)] are shown by the corresponding open sym-
bols. The overall agreement of the calculation with the experimental values is very

good.

III Study of the widths of the momentum distri-
butions

The variation of the widths of the (parallel) momentum distributions of pro jectile-like

fragments from the reaction of 18O (80 MeV/u) with 2"Al is presented in Fig. 5.4 as

84

 

 

 

 

r T . . r r I . . . . I . r e s I . . . . I
1.025 “ '-
. J'N i
' '..-1°C 1
1.000 - G~ \ '1
D “a. x "N 10N «
’2 L 13C\ \ s. 3‘
\ 1 \ \ 1"C
2'} i 190: . I. 19 17
0.975 " ° ‘0 N N “
\ r a"I""'-"“'--1;o‘ \
11.. t ‘ \ :' 1
2? ' 9 fii ’
1- D \
3. 0.950 »- “’c ”0‘ I ‘ x \ —
l" AA.—0 AAt=1 AA‘=2 \ \ \ “‘33 l
a... 1 \ \ \
0.925 b \ w“ ‘
a";
103 ,
0.900 r4; 1 1 1 1 1 l 1 1 1 1 I 1 1 1 1 L 1 1 1 1 l l
0.00 0.05 0.10 0.15 0.20

Figure 5.3: Variation of the centroids of the momentum/nucleon distributions of
few-nucleon pick-up products from the reactions studied in the present work. The
experimental points are shown by the closed symbols. The open symbols are calcu-
lated according to eq. (5.14). Specifically, the symbols correspond to the different
reactions as follows: circles: 18O (80 MeV/u) + 27Al, squares: 20Ne (85 MeV/u) «1-
181Ta, diamonds: 15N (70 MeV/u) + 181Ta, and crosses: 1"N (75 MeV/u) + 27Al
(chapter 3). The straight line represents the general trend of the data.

85

a function of the mass loss AA (closed points). Both nucleon removal and nucleon
pick-up products are included in the figure. Calculations of the widths performed
with the simple models of Goldhaber and Friedman are presented in subsections A

and B, respectively.

A Calculations of the widths with Goldhaber’s model

According to the independent-particle model of Goldhaber [Gold 74] (discussed in
chapter 2), the width of the (parallel) momentum distribution of nucleon removal

products is given by:

AF(AP - AF)
2 ____ 2 .
a ”0 AP _ 1

 

(5.15)

‘ For the present system, we evaluated this equation using a reduced width 00 = 80
MeV/c. The calculated values of the widths are indicated in Fig. 5.4 by the lower
dashed line and are in very good agreement with the data. It should be pointed out
that the reduced width obtained from the present measurement agrees very well with
the values previously obtained from the fragmentation data of 16O and 12C projectiles

at relativistic energies [Grei 75] (see Table 2.1 and discussion in chapter 2).

Interestingly, the width of the momentum distribution of 1"N is ~20% higher than
predicted by eq. (5.15). This discrepancy is not present in the case of 17N fragments
produced by the reaction 180 (70 MeV/ u) + 181Ta (indicated in fig. 5.4 by the closed
square). The broadening of the momentum distribution of 17N fragments with 2"Al
target is once again attributed to a considerable contribution of a direct stripping
channel, along with the fragmentation (break-up) channel, to the momentum distri-
bution of 1"N. The component of the momentum distribution due to the stripping
channel [Merm 86] peaks at momentum higher than that of the fragmentation chan-

nel (see also previous section), thus the overall momentum distribution is broader

86

180(80 MeV/u) + A1

 

 

 

 

200 __ l I I I 1
190 18N 17c ,0”j xgm
150 - I .. . -
O 9 x3: 0
E 37 / I.
0 .37 2 / 8
z / l' . .
v ; 0/
* .1 s)" I
b 100 '— j . .'/ _‘
I "l
.I 0,
I
I
17N ICC 15C 14B 13B 123 11Be IOBe 8L1 i
r I. : 15.9 29.0 25.0 40.7 27.0 23.2 35.7 18.2 33.7 ‘
50 1 1 1 1 I 1 1 1 1 1 1

 

0 1 2 3 4 5 6 7 6 9 10

Mass Loss (AA)

Figure 5.4: Variation of the widths of the parallel momentum distributions of
projectile-like fragments from the reaction 180 (80 MeV/ u) + 2"A1. The closed circles
are the experimental points. [The closed square is from the reaction 180 (70 MeV/u)
+ 181Ta.] The lower dashed line is calculated with Goldhaber’s model [Gold 74]. The
upper dashed line is from an extended version of Goldhaber’s model for nucleon pick-
up products. The open circles are calculations with Friedman’s model [Frie 83]. For
the nucleon removal products, the cluster separation energies E. are given (in MeV)
below the isotope symbols.

87

than that of the fragmentation channel alone. The stripping channel is suppressed
when 181Ta target is used and the width is in agreement with the value calculated

with the fragmentation model.

The widths of the momentum distributions of the neutron pick-up products are
larger than their corresponding fragmentation products (e.g. in Fig. 5.4 compare the
widths of 18N and 1"C with those of ll'N and 16C, respectively). In order to interpret
the increase in the width of the momentum distribution of a product that has picked-
up a nucleon from the target, we propose an extension of Goldhaber’s model in which
we take into account the nucleon momentum distribution in the target nucleus, along
with that in the projectile nucleus. The momentum distribution of a pick-up prod-
uct is assumed to result from the folding of two distributions: first, the momentum
distribution of the projectile part of the final fragment and second, the momentum
distribution of the nucleon(s) picked-up from the target. These distributions are as-
sumed to be Gaussian distributions with widths expressed as in eq. (5.15). The
momentum distribution of the final product is also Gaussian that has a width given
by the quadratic addition of the values from the previous two distributions. Thus:

2 2 APF(AP - Apr)

0' = 0’ 2 AA¢(AT - AAg)
0 AP - l

+"° AT-l

 

 

(5.16)

where App = AF — AA. is the mass of the projectile part of the final product and
AA. is the number of nucleons picked-up from the target (AT is the mass of the
target). In writing eq. (5.16) we have implicitly assumed that the nucleon momentum
distribution in the target is the same as that in the projectile nucleus and therefore,
we have used the same reduced width 00 for both the projectile and the target. The
widths of the momentum distributions of the neutron pick-up products calculated
according to eq. (5.16) with do = 80 MeV/c are indicated in Fig. 5.4 by the upper

dashed line and are in good agreement with the experimental values.

88

From this discussion, we conclude that the independent-particle model can well
account for the widths of the parallel momentum distributions of both nucleon removal
and nucleon pick-up products. For the pick-up products, we must take into account

the nucleon momentum distributions in both the projectile and the target.

The behaviour of the widths of the parallel momentum distributions of nucleon
removal and nucleon pick-up products shows that the width observable is rather
insensitive to the detailed course of the production mechanism. This statement is
also true for the widths of fragmentation products obtained at high energies (see

chapter 2).

B Calculations of the widths with Friedman’s model

Apart from the calculations with the independent-particle model, we performed calcu-
lations of the widths using the cluster-stripping model of Friedman [Frie 83] (discussed
in chapter 2) which are shown in Fig. 5.4 by the open circles. For the nucleon re-
moval products, eq. (2.15) was applied. For a given fragment, the separation energy
E, of the system F—R was calculated from the change in binding energy between the
projectile and the system F—R: E, = Bp — (Bp + 83), where “B” stands for binding
energy. The cluster separation energies E. are given in Fig. 5.4 below the isotope

symbols. Examples of break-up patterns for several products are given in Table 5.1.

Table 5.1: Examples of break-up patterns of 18O projectiles. E, are the corresponding
separation energies. The last line is the one-neutron stripping pattern from 2"Al
target.

180 ——» 17N + p E, = 15.9 MeV
18O --1 ‘3B + ‘He + p E. = 27.0 MeV
18O --> 10Be + 2‘He E. = 18.2 MeV
18O —-+ 8Li + 10B E, = 33.7 MeV
27A1 -—-7 26Al + n E, = 13.0 MeV

89

In calculating 13., we chose the break-up pattern (if more than one for a given
fragment) which corresponded to the lowest E.; that is, we assumed that the clusters
stripped from the projectile are the most bound. The widths calculated according to
Friedman’s model are also in good agreement with the experimental data, especially
for products close to the projectile (up to 1‘B, see Fig. 5.4). For the other nucleon
removal products, the calculated widths are somewhat lower than the experimen-
tal values. For example, a low width value is calculated for 1"Be and is due to the
small separation energy required to break 180 into 1"Be and two alpha particles (E.
= 18.2 MeV). Thus, we conclude that the cluster stripping patterns assumed in the
calculation (having the lowest possible E.) are not the only fragmentation pathways
leading to the production of the fragments. Apart from the cluster stripping chan-
nels, other fragmentation channels involving first, the production of excited primary
fragments and second, their subsequent decay by particle evaporation contribute to

the production mechanism (see also chapter 2).

In order to calculate the widths of the momentum distributions of the nucleon
pick-up products using Friedman’s model, we proposed an extension of the model
similar to that of Goldhaber’s model discussed previously. We again assumed that
the momentum distribution of a pick-up product results from the combination of the
momentum distribution of the projectile part of the product [with width app given
by eq. (2.15)] and the momentum distribution of the nucleon (or cluster of nucleons)
picked-up from the target [with width 0‘. given again by eq. (2.15) applied to the
stripping of the nucleon (or cluster) from the target]. The width of the combined

momentum distribution is given by the quadratic addition of the two widths:
a'2 = of); + of. (5.17)

For the neutron pick-up products of the present data, app is taken from the

90

corresponding nucleon removal products and a. is calculated from eq. (2.15) applied
to the neutron stripping from the target (see Table 5.1). The widths calculated
according to eq. (5.17) are shown in Fig. 5.4 and are in good agreement with the

experimental values.

The agreement of the measured widths of the momentum distributions of nucleon
pick-up products with Goldhaber’s model and Friedman’s model calculations suggests
that the widths of these products can be very well predicted by taking into account
the momentum distributions of both the projectile part and the picked-up part of the

observed fragment.

IV Study of Relative Yields

The relative yields of projectile-like fragments produced by the reaction 180 (80
MeV/u) + 27A1 are presented in Fig. 5.5 by the closed circles. The yields were
obtained by integrating the measured momentum distributions of the fragments (and

are expressed in total normalized counts).

Calculations of the yields with the abrasion-ablation model are shown in Fig.
5.5 by the diamonds. For the abrasion stage of the calculation we assumed proton—
neutron correlations according the the giant-dipole-resonance (GDR) model [Morr 79]
(see also chapter 2). As natural for an abrasion-ablation calculation, no nucleon pick-
up products were predicted by the model. For the nucleon removal products, the

predictions of the code are not in agreement with the present data.

Calculations of the yields with the INC model are shown in Fig. 5.5 by the
crosses. The INC code “ISABEL” of Yariv and Fraenkel [Yari 79] (see also chapter
2) was used for the nucleon-removal step of the reaction. For the deexcitation step, a

statistical evaporation code was used to calculate the cross sections of the fragments.

 

91

 

 

 

 

 

 

   

 

 

 

_ I l I I I 1 1 I r 1 1 I l r 1
1
106 g- 0 N C 1
105 g- '1
104 .— 1
103 g- 1
102 r 1.
10’ r 1
1’. s 5
.2 i
>" 100 L 1 1 1 1 1 1 L L 1 4 1 1 1
g 18 19 20 21 22 16 17 18 19 20 14 15 18 17 18
23 .
g l r l I 1 r l l 1 l 1 r 1 I r
m 4
04 . .
106 g- B Be Li 1:
105 g' _,
C e.,_ .
104 1‘ 61 '° ‘1
103 gr' ,..\.9 "g
- l
102 g- 1,
101 r 1
100 1 1 1 1 1 1 1 1 1 1 1 1 1 L 1
11 12 13 14 15 9 10 11 12 13 7 5 9 10 11

Fragment mass (A...)

Figure 5.5: Relative yields of projectile-like fragments from the reaction 1”0 (80
MeV/u) + 2"Al. The closed circles are the experimental points. The diamonds
and the crosses are calculations with the abrasion-ablation model [Morr 79] and
the INC model [Yari 79], respectively. The squares are predictions of a high-energy
parametrization [Siimm 90].

92

Interestingly, nucleon pick-up products are predicted by the model for the present
intermediate-energy reaction (as a result of nucleon exchange due to nucleon—nucleon
collisions in the overlap stage of the process). However, the agreement of the calcu-
lated yields with the trend of the data is rather poor. It should also be mentioned
that the calculation predicted the centroids and the widths of the parallel momentum
distributions of the fragments; the centroids were much lower than the experimental
points and the widths were larger. From these results, we infer that the INC cal-
culation for the present system seems to overestimate the excitation energy of the
primary fragments [in contrast to the case of PLFs from the heavy beam 8“Kr (200

MeV/ u) [Stép 91] for which the code gave predictions in agreement with the data].

In addition, we performed calculations of the yields with the high-energy empirical
parametrization of Siimmerer and Morrissey [Siimm 90], shown in Fig. 5.5 by the .
squares. This parametrization has been obtained from target fragments produced at
high energies (>1 GeV/ u). The present calculations were performed with the code
“INTENSITY” [Wing 91] which, at present, uses this high-energy parametrization
(for lack of an appropriate intermediate-energy parametrization) to predict rates of
radioactive beams of PLFs separated by the A1200. The predictions of this calculation
agree with the trend of the data only for fragments far in mass from the projectile

(i.e. B, Be, Li).

Finally, we performed calculations of the relative yields of the nucleon removal
products with Friedman’s model [Frie 83] (see also chapter 2) using eq. (2.16). The
parameter b was obtained from the slope of the function W = ln(7m:1:8/ 5' (23, AR))
versus 201:0 [eq. (2.17)]. The plot of this function for the present data set is given
in Fig. 5.6 and shows a good linear behaviour for most of the fragments. A value
b = 0.57 for the cut-off parameter is obtained from the slope of the least-squares-fit

straight line. It should be noted that this value is higher than the value proposed

93

by Friedman (b = 0.4) for the fragmentation products of relativistic 12C and 16O
[Frie 83]. Also, the present data show a better linear correlation for W than the data

of relativistic 160 (see Fig. 4 in Frie 83).

Using the extracted value of b in eq. (2.16) and normalizing the calculated yields
with respect to the experimental yield of 17N, the relative yields of the nucleon removal
products were obtained (shown in Fig. 5.7 by the open circles) and are in good

agreement with the data.

In order to calculate the relative yields for the nucleon pick-up products, we as-

sumed that the yield 7p of a. given fragment can be expressed as:

7F = “m? 7: (5.18)

where 7p; is the yield of the projectile part of the fragment, and 7, is the yield for
the removal of a nucleon (or cluster) from the target. Both factors are calculated
according to eq. (2.16) applied to the corresponding system. Thus, the factor 7p;-
can be taken from the corresponding nucleon removal products and 7. is calculated for
the stripping of the neutron(s) from the target. Because each of the factors appearing
in eq. (5.18) requires [according to eq. (2.16)] a different normalization, we need one
more normalization factor for the nucleon pick-up products. In addition, for the case
of 1"O and 200, where no nucleon removal occurs, a different normalization should
be necessary. In this case we normalized the relative yields according to 19O. For
the relative yield calculation of 18N, 19N, 1"C and 18C, normalization of the yields
relative to 18N was performed. The calculated values, shown in Fig. 5.7 by the open
diamonds, are in good agreement with the data. Thus, an extension of Friedman’s
model in the form of eq. (5.18) gives a good description of the ratio of yields of pick-up
products with different numbers of nucleons transfered (here for 1 and 2 neutrons).

As already has been mentioned in chapter 2, absolute cross section predictions are

94

 

 

 

 

18. . 1 . . . . I . . . . I . . . . I . . . AI .
L .1
\ J
I- 7\&\ d
1
16_L N . \\ _
14 x
61‘ C 10.\\\ — i
n\ 1- N \. . 15C 13B12B
1? 14" ‘k 7
5' : 15C \\ . .
5 - 10Be \\
E . \ .
I 12— ’x\ “Be -
. \ .
3 ”Li ‘\
C .\
1 9m 1
-— ' \M
10_ 14B .
L 123e .
8[-1--..1m. .1...11....1-
5.0 7.5 10.0 12.5 15.0
2axo

Figure 5.6: Plot of the function W = ln(7e,,zg/S) versus 202:0 for the yields of
projectile-like fragments from the reaction 100 (80 MeV/u) + 2"Al. The dashed line

is a. least-squares fit to the data points.

95

 

   

 

 

 

 

 

 

   

 

f I I l j I I I I l 1: T I I l l 1
. 1: :
ii :
105 :‘ ‘ar ‘=
10" r ‘:‘ 1.
> if 4
103 F ‘1‘ 1.
: if 3
1°“ 5 ‘r 1.
; 1: :
v 10’ 1‘ 1? 1.
711‘ E SE i
“-0 . 4
>" 100 1 1 1 1 1 1 1 1 1 1 .i 1 1 1
2 10 19 20 21 22 10 17 10 19 20 14 15 10 17 10
33‘ .
.2 f I I I l l I I I I l I l l I
0 I 1
n: ; 1
105 r B Be Li 1
105 E” 1
1 :
104 r 1
‘ - 3
F N .
103 E- '3
102 .- 1,
1 l
101 Ir' ‘1
100 P l | l l l 1 1 l I L L l L J L

 

 

1112131415 910111213 7891011

Fragment mass (A?)

Figure 5.7: Comparison of the relative yields of PLFs from the reaction ’90 (80
MeV/ u) + "A1 with calculations using Friedman’s model [Frie 83]. The closed circles
are the experimental points. The open circles are Friedman’s model calculations
for the nucleon-removal products and the diamonds are calculations for the pick-up
products (see text).

96

not possible with Friedman’s model, due to the neglect of the impact parameter of

the collision.

In conclusion, the relative yields of projectile-like fragments cannot be well de-
scribed by either the abrasion-ablation model or the intranuclear cascade model. The
INC model predicts the production of nucleon pick-up products at the present energy
(80 MeV/ u), but the calculations are higher than the experimental values. Interest-
ingly, the cluster-stripping model of Friedman gives a good description of the ream;
yields in most of the cases and especially for fragments close in mass to the projec-
tile. From these observations we can also infer that the projectile-like fragments of
the present light beam are produced with a break-up (and/ or pick-up) mechanism
possibly involving low excitation energies. This conclusion is similar to that reached

for data from other light projectiles at lower bombarding energies (see chapter 2).

Chapter 6

Summary and Conclusions

In this dissertation, a study of the projectile fragmenation process in the intermediate
energy regime was presented. A review of the experimental and theoretical progress
concerning the fragmentation reactions was presented in chapter 2. The characteris-
tics of pro jectile-like fragments, the basic models developed to interpret the data and

our present understanding of the fragmentation mechanism were discussed.

The experimental measurements of the present work (presented in chapters 3
and 4) were motivated by the need of systematic data of projectile fragments near
0 degrees produced by intermediate-energy beams. Because of the presence of the
primary beam, accurate measurements at and near 0 degrees are especially difficult
and require the use of O-degree spectrometers. In the present work, two different
devices were operated as O-degree spectrometers: the K1200 interim beamline and

the A1200 mass separator.

The operation of the K1200 interim beamline as a spectrometer was presented
in chapter 3. Beam optics calculations showed that a moderate acceptance mode
was possible for this simple device. The fragments were detected at the end of the
beamline, inside the 92” scattering chamber with a detection system consisting of two

position-sensitive detectors and a AE-E plastic scintillator telescope. The reactions

97

98

studied were “N (75 MeV/u) + 2"Al and 181Ta. Momentum and reaction-angle of the
fragments were reconstructed. The momentum distributions of the fragments at 0°, 1°
and 2° and their angular distributions were obtained from the data. N ucleon-removal
and nucleon pick-up products were observed in the experiment. Unfortunately, the
quality of the results of these measurements was limited first, due to poor TOF mea-
surement which hindered a clear isotope identification and second, due to a limited
resolution of the reconstructed momentum and reaction angle of the particles. How-
ever, the experiment and the analysis of the data were very instructive and valuable,

providing preparation and experience for the subsequent experiments with the A1200.

The measurements of projectile fragmentation distributions with the A1200 beam
analysis device were presented in chapter 4. The A1200 was operated in the achro-
matic medium acceptance mode, which has two intermediate dispersive images and
a final achromatic image. Fragmentation targets were placed at the beginning of the
device and the products were detected at the final-image detector array consisting
of two X -Y position detectors and a AE—E telescope. Momentum information was
obtained from position measurement at the second intermediate image with a third
position detector. In a first series of measurements, PLF distributions from the follow-
ing reactions were measured: 20Ne (85 MeV/u) + 9Be and 181Ta, 15N (70 MeV/u) +
181Ta and 18O (70 MeV/u) + 181Ta. The results of these measurements were limited,
mainly due to problems in the relative normalization of the data runs. Nevertheless,
several good momentum distributions were obtained from these data sets. At a later
time, a detailed measurement of PLFs from the reaction 180 (80 MeV/u) + 27Al
was performed which focused on measuring the O-degree momentum distributions of
neutron-rich projectile-like fragments. As additional experience operating the A1200
had been obtained, this measurement turned out to be the most complete and suc-

cessful of the present work and a detailed analysis and interpretation of its results

99

were presented in chapter 5. The measured O-degree momentum distributions of nu-
cleon removal and nucleon picloup products were fitted with Gaussian functions and

the extracted centroids and widths were subsequently analysed.

The centroids of the momentum/nucleon distributions of the nucleon removal
products from the reaction 180 (80 MeV/u) + 2"Al were very close to the beam
momentum/ nucleon. They did not decrease monotonically with respect to the mass
loss (as has been observed at high energies), but they followed the variation of the
ground-state Q-value of the two-body rearrangement reactions leading to the different
fragments. For PLFs very close to the projectile, the centroids were higher than the
beam momentum/ nucleon. This observation was attributed to the contribution of
a direct stripping channel, along with the fragmentation (break-up) channel to the
production of these fragments. On the other hand, the centroids of the nucleon
pick-up products were remarkedly lower than the beam momentum/ nucleon. This
feature was interpreted by the kinematics of direct pick-up and/or break-up-l-pick-
up channels which, depending on the situation, were assumed to contribute to the
observed momentum distributions. In addition, the direct channels were found to be

dramatically suppressed when the heavy 181Ta target was used.

The general behaviour of the centroids of the nucleon pick-up products was found
to follow the predictions of a simple mechanical model based on momentum conserva-
tion and the additional assumption that the nucleon(s) picked-up from the target have
preferentially momenta (on average) equal to the nuclear Fermi momentum oriented
in the direction of the projectile motion. This behaviour was found to be common

for all the nucleon-pickup products observed in the experiments of the present work.

The widths of the momentum distributions of the nucleon removal products were
in good agreement with the predictions of the independent-particle model of Gold-

haber [Gold 74] with a reduced width 00 = 80 MeV/c. However, the widths of the

100

nucleon pick-up products were larger than the corresponding removal products and
found to be well described if the momentum distributions of both the projectile part
and the picked-up nucleon(s) were taken into account. Similar to Goldhaber’s model,
the cluster stripping model of Friedman [Frie 83] gave good predictions of the widths,

especially for fragments very close to the projectile.

Finally, the relative yields of pro jectile-like fragments from the reaction u’0 (80
MeV/u) + 2"Al were obtained. Calculations with the abrasion-ablation model and
the intranuclear cascade model gave a rather poor agreement with the trend of the
data. However, calculations of the relative yields with Friedman’s model were, in
most of the cases, in good agreement with the experimental values. These facts
imply that a break-up (and/or pick-up) mechanism possibly involving low excitation
energy is responsible for the production of projectile-like fragments from the present

light projectile.

In conclusion, the results of the present study demonstrated a number of important
features of projectile-like fragments produced from light projectiles at intermediate
energies. Along with the fragmentation (break-up) channel, direct transfer channels
(stripping and/or pick-up) have a considerable contribution to the production of the
fragments. Due to the inclusive character of the present measurements, a separation
of the different reaction channels was not possible. However, assuming their contribu-

tions and using simple models, we were able to adequately describe the observations.

A study of the relative contributions of the break-up and transfer channels to the
production of pro jectile-like fragments requires exclusive measurements. For example,
measurements of PLFs in coincidence with charged particles can yield quantitative
information about the various mechanisms. At present, very few measurements of

this type exist in the literature.

101

It should be pointed out that our present systematics and understanding of the
mechanisms of intermediate energy fragmentation are far from complete. Further
extended measurements of PLF distributions should be performed at and near 0
degrees in order to obtain detailed systematics of the observables, test the applicability
of existing models and motivate additional theoretical work on the reaction dynamics

of peripheral collisions at the intermediate energy regime.

Apart from a reaction dynamics standpoint, a detailed systematics and under-
standing of the production mechanism of PLFs at intermediate energies are extremely
valuable for the development of radioactive nuclear beams, allowing trustworthy pre-

dictions of production rates and other important properties to be made.

Appendix A

Derivation of Equations of
Fragmentation Models

In this appendix, a derivation of the equations of two simple fragmentation models,
namely, Goldhaber’s model and Friedman’s model are presented in detail. These

models have been discussed in chapter 2.

I Goldhaber’s Model

Assuming a sudden fragmentation picture and applying momentum conservation to
the fragmenting projectile nucleus (or, equivalently, target nucleus) in its own rest

frame, we have

AP
2152 = 0 . (A.1)

6:]

where if,- denotes the momentum of the i-th nucleon of the projectile. Taking the

square of this expression, we get

AP AP
£p?+25.~i2}=0. (A.2)
i=1 s$j

Taking the average of the terms of this equation and performing the summations

we get

Asp?) + (’2’) <1»:- 2) = o. (11.3)

102

103

For the momentum pp of a projectile fragment with mass AF, we have

Ar
= z ,1... (AA)

i=1
Squaring this equation and evaluating the average values of the resulting terms

we get
(1.3.) = AF (pa + (A2") <17.- -r2;->. (.15)

Solving eq. (A.3) for (p; - 133-), substituting into eq. (A.5), (implicitly assuming
that the internal nucleon motion of the nucleons does not change upon fragmentation),
and performing the algebraic manipulations with the combinatorials, we arrive at the

following expression for (p37) :

1» >= 1:» :>A““A”_ 1A.” (As)

 

This equation relates the average-square momentum (p3?) of the observed fragment
with the average-square momentum (p?) of the nucleon motion inside the projectile
(also denoted as (p3). Assuming isotropy, the projection of the fragment momentum
distribution on any axis will be a Gaussian function with width 0 related with the
average-square fragment momentun via: (pf?) = 302. Inserting this expression in eq.

(A.6), we get the following equation for the width:

AF(AP-AF)
2_ 2
a —00 Ap-l

where the reduced width do is defined by
= 9:39 (A.8)

 

(A.7)

If, instead of a sudden fragmentation picture, an excitation-break-up picture is
adopted, the total kinetic energy of the (excited) projectile nucleus at the moment of

its break-up is

Ep = E12 + E}; (A.9)

104

where BF and En are the kinetic energies of the fragment and the removed part,
respectively. Refering to the projection of the motion on one axis, say the z—axis

(along the beam direction), we have

10’ 12’
E3, = W + 2771710412 - AF) (A.10)

 

where mu is the nucleon mass and the fact that the momentum vectors of the fragment
and the removed part are antiparallel (because of momentum conservation) has been
used. This equation shows that the motion of the system along the z-axis corresponds
to one degree of freedom, namely p, Thus, according to the theorem of energy
equipartition, the average energy (Em) will be §kT, where k is Boltzmann constant
and T is the temperature of the projectile. Taking the average of eq. (A.10) and

using the above argument, we have

(P3) (P3) _ l
2m,AF + 2m,.(Ap — AF) ‘ 2 AT (AAA)

which, since (pi) = or“, finally gives the following expression for the width:

 

 

02 = mnkT AFM” " AF). (A.12)
AP
Comparing this equation with eq. (A.7), the reduced width 00 is expressed as:
0'3 = mnkT AP - 1. (A.13)
AP

II Friedman’s Model

The radial wave function 1/2p_R(r) which describes the relative motion of the removed
cluster R and the observed fragment F is approximated, near the nuclear surface, by

the expression

e—ar

 

Ibis—RU) O< (A.14)

r

105

where a is the wave-number of the relative motion:
211E, (A.15)

where p is the reduced mass of the system F-R, p = Ap(Ap — AF)/Ap, and E, is its

separation energy.

The form eq. (A.14) of the wave function corresponds to the quantum mechanical

problem of a particle moving in a potential of the form

_ —I/o If 0 S I" S 30
V0.) - { 0 if r > 2:0 (A'IG)

with energy E = —E, < 0. Specifically, eq. (A.14) is the radial solution of the

Schrodinger equation of the particle in the (classically forbidden) region r > 2:0.

We express the radial distance r in terms of the longitudinal z and the transverse

r 1 coordinates:

r: fizz-Fri. (A-17)

Because of the peripheral character of the fragmentation process, we are mostly
interested in the behaviour of the wave function ¢p_3(r) near the region where the
cluster to be removed (upon interaction with the target nucleus) just “touches” the
remaining fragment, that is near the point (z = 0, r L = :50), where to is the radius of
the fragment, taken to be 2:0 = 1.211113 (in fm). With this point in mind, we expand
eq. (A.17) in a Taylor series about that point and keeping the lowest (nonzero) order
terms in z and r 1., we get

2’
r: —+r_._. (A.18)

220

Inserting r from this expression into eq. (A.14) and separating the variables, we

get

 

¢F—R(Z,7‘.L) 0t (A.19)

106

Thus, the wave function has been separated into two factors, each one depending on
only one of the coordinates z and r 1.- Using this wave function, we can derive the
longitudinal momentum distribution of the fragment. For this purpose, we evaluate
the partial Fourier transform fp-a(p,,rj) of the wave function t/JF-R(Z,1‘ L) with

respect to z (k, = pz/h):

Inmate) = m]: ¢F—R(Z rue-“v.1.

 

J- as”

6.0, w -530 .

at 71+] _e_ 23 e""" dz
1 + a;

 

72L -oo

 

e {mud-5)], (11.20)

The above Fourier transform is a mixed coordinate-momentum representation of the
wave function. The last step has been obtained using the approximate expression
1/ l + f-E' z e-fio’ and evaluating the resulting Fourier integral, using Tables of
Integrals. The Fourier transform fp-n(p,,r 1.), eq. (A.20), is then integrated over

r 1 (simply integrating the rL-dependent term). Taking its square modulus, the

longitudinal momentum distribution f (12,) of the fragment is obtained:
11p.) «e13 (A21)
with a given by

0'2 = h” i (1 + i) . (A.22)

22:0 (no
In order to include the contribution of the Coulomb interaction between the fragment
and the removed part to the width of the parallel-momentum distribution, another

approximate form of the relative wave function, containing the Colomb interaction,

107

will be derived. This is accomplished using a WKB approximation to the wave func-
tion in the region of interest. For r 2 :0, the wave function can be written as (see

e.g. Merz 70, page 117):

 

exp (— / kc(r) dr) (A23)

1 1
¢F—R,WKB(T) 0< ; he“)

where

 

kc(r) = %\/2p(E. + 1:3) (11.24)

where b,3 is the Coulomb parameter of the system F -R: be = e2ZpZR. Defining as y
the ratio of the Coulomb energy at r = 2:0 to the separation energy E., y = (be/xo)/ E.,

and using eq. (A.15), the function kc(r) is expressed as

kc(r) = a,/1+ 3:3- (11.25)

Using this expression, the integral in eq. (A23) is evaluated to give approximately

/kc(r) dr :2: a “1 + 31-?- r (A26)

thus the wave function takes the form:

c’a'M'

$17—12, WKB(7') 0< r . (A27)

 

The form of this wave function is similar to that of eq. (A.14) (the r—dependence of
the square root near 2:0 is slow and can be neglected). Consequently, a treatment of
this wave function similar to that of eq. (A.14) gives again a Gaussian p,-momentum

distribution, but in this case the width is given by:

2_ 21 L)
or —h 230 ((/1+y + as: . (A28)

0

Note that this expression can be obtained from eq. (A22) with the substitution

dang/1+”? withr=zro.

108

According to Friedman’s model, the yield 7;: of a fragment is expressed as

7F 0< 5(ZR, AR) Id’F—RU‘ = 1'0)|2- (A29)

The factor S(ZR, AR) expresses the relative probability for the removed part R to
contain ZR protons and is given by a hypergeometric distribution. The square mod-
ulus of the wave function at r = :0 gives the probability for the system F—R to be

found at a “touching” configuration. Squaring eq. (A.14) we have

e-2ar

 

I¢p-3(1‘)|2 2 IV2 (A30)

r2
where N is a normalization constant. In order to evaluate N, we further assume
that the wave function is constant up to a radius R, smaller than :50, and varies

exponentially according to eq. (A.14) for larger distances. Under this assumption,

the normalization constant is found to be approximately

(A.31)

 

Inserting this expression of N into eq. (A.30) and furthermore parametrizing R" as
RC = 20(1— b), with 0 < b <1, we get:

e-2a(::o-R¢) e-Zazob

__ = __ (A.32)
$312.: 2:3(1 — b)°

libs-120‘ = 10W 1’

Substituting this equation in eq. (A29), the following expression for the yield is
obtained

~2azob
7F 0‘ 5(Z31AR) EEK—1:55. (A’33)

Appendix B

Supplement on Calculations for
the Interim K1200 Beamline
Experiment

In this appendix, a discussion of beam optics concepts relevant to spectrometer oper-
ations is presented and derivations of the equations developed for the analysis of the
data obtained from the experiment with the Interim K1200 Beamline (chapter 3) are

given.

I Beam Optics concepts

In order to describe a particle as it passes through an ion-optical system, transfer
coefficients can be introduced which express the change of the coordinates of the
particle from one position to another [Care 87]. A reference trajectory is assumed
for the system where the coordinate systems 203/on and zyz are defined at the be-
ginning and at the end of the system, respectively, with the z-axes pointing along
the direction of the particle motion (fig. 8.1). For simplicity, we assume that the
system is mechanically symmetric about the zz—plane and the magnetic field on that
plane is everywhere perpendicular to it (or zero). In the present system, this plane is

the horizontal plane, which is also the dispersive plane of the dipole. The reference

109

110

trajectory passes through the quadrupole axes and the central orbit of the dipole

magnet.

A particle is assumed to be emitted from the point (20, ya) on the plane 20 = 0 at
the beginning of the system with direction specified by the angles 00 and 4’0, the angles
with respect to the z-axis made by the projections of the velocity of the particle on the
$2 and yz planes, respectively. After traveling through the system, it will emerge at
the end of the system at the point (z,y) on the plane 2 = 0 with direction specified
by the angles 0 and 45. A reference (or central) momentum p0 is defined for the
system as the momentum necessary for a particle to follow the reference trajectory.
The momentum of any other particle is then described by its deviation 6 (in percent)

from the reference momentum.

In order to take into account the differences in the length of different trajecto-
ries, a variable I is introduced and defined as the path length difference between the

trajectory of a given particle and the central trajectory of the system.

The beam-optical system maps the initial coordinates 2:0, 00, yo, (330, 10, and 6 of
a particle that traverses it to the final ones 3:, 0, y, d), I, 6 (6 doesn’t change, unless
accelerating elements are present in the system). Thus, each coordinate of a particle
at the exit of the system is a function of all the initial coordinates. For the final

position, for example, we can write:

I = f(301001y01¢011016) (B°1)

The values of the coordinates x, 0, y, (b and I are in principle very small compared
to typical lengths or angles of the spectrometer (for instance, the curvature radius
and the bending angle of a dipole, respectively). Also, the momentum deviations 6
of the particles from the reference momentum, are usually chosen to be small. As is

usual in spectrometer operation, a large momentum range can be covered by changing

111

Yo‘i y I)

 

 

 

 

 

(a) (b)

Figure B]: Coordinate systems used in general beam optics calculations.

112

the fields of the magnets in sussessive steps, thus changing the value of the reference

momentum p0 (whereas keeping 6 a small fraction of pa).

Equation (B.l) can be expressed as a Taylor expansion in its variables, usually

written as:
a: = (2mm. + (now. + (zl6)6 + (25):):0
+ (elm-2):: + (zlx0)xooo + (2|0’)03
+ (zlz6):ro6 + memos + (xl62)62
+ (zly’)y3 + (x|y¢)yo¢o + ($|¢’)¢3

+ (#9559062 + . -. (13.2)

Each coefficient in this expression is a partial derivative of the function a: with
respect to the indicated variable(s); so, for instance, we have: (zlx) = %, (xIO) = 5%,
(zl96) = 515%, and so on. These coefficients are called transfer coefficients or matrix
elements (of the corresponding order). Speciffically, the coefficient (xlx) is the first
order (horizontal) magnification and (zl6) is the dispersion of the system. Also (zld’)
is called the geometric aberration and (zI06) is called the chromatic aberration. Due

to the symmetry of the system about the :2 plane, terms of the form (:rly“¢") with

p + u odd must be zero and are not included in expression (B2).

The transfer coefficients are in general determined analytically. They can, for
instance, be calculated by the matrix multiplication code TRANSPORT up to third
order. Expressions similar to (B2) can also be written for the coordinates, other than

1:, of the system.

If at the end of the system (zI0) = 0, we have a (first order) horizontal focus. In
this case, the first order momentum resolution of the system is defined as the percent
change in momentum needed to displace the first order image by a distance equal

to its width An: = (:rlz)A:ro, where A230 is the horizontal size of the beam-spot at

113

the beginning of the system. Thus, the (first order) momentum resolution R1 of the
system is expressed as:

_ (:rIz)Azo

R1 — (xl6) (B.3)

When (xl0) is not zero and highter order terms [i.e. (2'26) and (2406)] are present,
their effects have to be taken in to account in the calculation of the momentum
resolution. In this case, the aberration-limited momentum resolution of the system

can be written as:

_ 9:2
Rab "' (23”) (B4)

where Arab is the overall image size resulting from the contributions of all the dif-
ferent aberrations. An approximate expression for Arab for the interim beamline

spectrometer is given at the end of the next section.

II Derivation of the expressions for the momen-
tum and the reaction angle

The final position a: of a particle at the front counter is related to the initial position

2:0, initial angle 90 and 6 with the expression [Care 87] (see Fig. 3.5a and 3.5b):
a: = (zlz):1:o + (xl0)00 + (zl6)6 + (:rlz6)zo6 + (zl06)006 + (:1:|62)62 + (zl062)0062(B.5)

where only the most important for the present system matrix elements are included
(for example all y-dependent terms are dropped). After rearrangement, we can get
the following equation for 6:

6 __ a: — (zlx):ro - (3|0)00
- (zl6) + (xlz6):ro + (xl06)00 + (x|62)6 + (zl062)006

 

(B.6)

114

Substituting 6 = 100 (£- - I) in the left side of the above equation, omittimg the
(zl62) term (since it has a negligible contribution, see values in Table 32), and using

the abbreviated expressions:

(:cl06)’ = (1|06) + (:rl062)6 (B.7)
(zl6)’ = (:rl6) + (xlx6)a:o (B.8)
we get:

pg :1: — (zlx):ro — (zl0)00
, :r as '00
10° W) (l + ”—1..” Y

 

P = P0 + (B9)
Expanding the binomial in the denominator to first order in its argument (since it is

small), we get:

,, = p. + W (1 - $3,599) (as - (21m. - (3|9)9o) (8.10)

Now, referring to Fig. 3.5b), the position coordinate :c, (with respect to the coordinate
system 2:02 in the horizontal plane), is related to the measured position X pp (in
channels) as:

_XFP - X";
CF?

 

+a (8.11)

in which the negative sign is due to the fact that the direction of increasing position as
measured by the detector was taken to be opposite to that indicated by the coordinate
system 1:02. After substitution of this expression for 1: in the above equation and
some algebraic manipulation, keeping terms up to first order in 00, we arrive at the

relation:

[2(00) = Ao(00) - 141(00)pr (8.12)

115

 

 

where:
_ Po P0 (9519”
A°(°°) " (°° + 100(x|6)'°°”) wow)! (”I”) + (.45)! AW 9" (3'13)
and
_ po ($l05)’
AAA) ‘ 100mm» (A (21619“ (BAA)
with the definition:
Xre
3:,” = CF}: + a —(:1:|:r)xo (B.15)

By writing the equations of Ao(90) and A1(00) for 00 = 05 and subtracting the latter

from the former, we finally get:

110(1),) -_- 140(1),.) — TWP—:37 ((zIO) + $2,”) (190 — a.) (B.l6)
and
A,(oo) = A.(9.,) - °° (3'95" (00 — a.) (13.17)

 

1000" (11(6)2

In order to derive the equation for the scattering angle 0,, we start from the expres-
sion that relates the final angle 0 of a particle at the front counter with the initial

coordinates :20, 00 and 6:
0 = (0|:r)zo + (0|0)00 + (0|6)6 + (9|z6):co6 + (0|06)006 + (0|62)62 + (0|962)0062(B.18)

Referring again to Fig. 3.5b we can write: 9 = Opp + ,6, where Opp is the angle of a
given particle with respect to the angle of the reference 0° elastically scattered beam

particles and )3 is the angle of these arbitrarily chosen reference beam particles with

116

respect to the optical axis of the spectrometer. Substituting this into eq. (B.18) and

rearranging, we get:

on. = cow.) + 01(6),).5 + 02(Oo)6’ (3.19)
with:

00(00) = —B + (opp. + ((919)190 (3.20)

0.09.) = (0|6) + (9)133, +(9|195)190 (3.21)

02(1),) = (9|6’) + (0)0522), ‘ (3.22)

By rewriting each of these equations for O0 = Ob and subtracting from the correspond-

ing equation for O0, we find:

00(00) = 00(95) + (0|0)(00 - 05) (8.23)
01(90) = 01(05) '1' (9I06)(90 - 0b) (B24)
02(1).) = (32(11):,((1)19.s’)((51o — a.) (3.25)

Substituting these expressions into the equation for Opp, and solving for O, = O0 — Ob,

we get:

0 _ on» - (0103+ 01(2); + 0.09362)
' " (0(0) + (0|06)6 + (OIO6’)6’

 

(3.26)

As mentioned in the previous section, the aberration-limited momentum resolution

of the system can be expressed as

Azab

“1311)

(3.27)

117

where Azab is the image size, which can be estimated to a first approximation, by an
addition in quadrature of the contributions from the different aberrations. For the

present system, for a given value of 6, Arab can be estimated as:
A2,), = ([(zIzmzol’ + hummus)” + ((1)9)4901’ + [(zIO6)dOo6]2)l/2 (3.28)

The scattering angle uncertainty dOo can be estimated, to a first approximation, as:

1/2

do, = L (110%,. + [(olz)Azo]’) (3.29)

(M)

where dOpp is the uncertainty in the measured final angle Opp. Expressions (B28)
and (B29) follow from eqs. (3.5) and (B18) respectively, where the most important

terms are kept and a quadratic addition of the corresponding errors is performed.

118

References

[Alon 84] J. Alonso, Proc. of the Workshop on Research with Radioactive Beams,
Washington DC (1984), Lawrence Berkeley Laboratory Report 18187, unpub-
lished and J. Alonso, A. Chatterjee and C. A. Tobias, IEEE Trans. on Nucl.
Sci., NS—26 (1978) 3003.

[Anne 87] R. Anne, D. Bazin, A. C. Mueller, J. C. Jacmart and M. Langevin, Nucl.
Instrum. and Methods A257 , 215 (1987).

[Bazi 90] D. Bazin, D. Guerreau, R. Anne, D. Guillemaud-Mueller, A. C. Mueller,
and M. G. Saint-Laurent, Nucl. Phys. A515 (1990) 349.

[Bert 81] G. Bertsch, Phys. Rev. Lett. 46 (1981) 472.

[Bibb 79] K. Van Bibber, D. L. Hendrie, D. K. Scott, H. H. Weiman, L. S. Schroeder,
J. V. Geaga, S. A. Cessin, R. Treuhaft, Y. J. Grossiord, J. O. Rasmussen and
C. Y. Wong, Phys. Rev. Lett. 43 (1979) 840.

[Blan 76] M. Blann, OVERLAID ALICE, U.S. ERDA Report No. COO-349429,
Rochester, N .Y., 1976 (unpublished); also see F. Plasil and M. Blann, Phys.
Rev. C 11 (1975) 508.

[Blat 52] J. M. Blatt and V. F. Weisskopf, “Theoretical Nuclear Physics”, J. Wiley
8!. Sons, 1952, p. 120.

[Blum 86] Y. Blumenfeld, Ph. Chomaz, N. Frascaria, J. P. Garron, J. C. Jacmart
and J. C. Rounette, Nucl. Phys. A455 (1986) 357.

[Bond 71] J. P. Bondorf, F. Dickmann, D. H. E. Gross, and P.J. Siemens, J. Phys.
(Paris) Coll. C6 (1971) 145.

119

[Borr 86] V. Borrel, B. Gatty, D. Guerreau, J. Galin, and D. Jacquet, Z. Phys. A -—
Atomic Nuclei 324 (1986) 205.

[Bowm 73] J. D. Bowman, W. J. Swiatecki and C. F. Tsang, LBL report 2908 (1973),

unpublished.

[Brad 88] F. P. Brady, W. B. Christie, J. L. Romero, C. E. Tull, B. McEachern, M.
L. Webb, J. C. Young, H. J. Crawford, D. E. Greiner, P. J. Lindstrom, and
H. Sann, Phys. Rev. Lett. 60 (1988) 1699.

[Bres 82] A. Breskin, Nucl. Instrum. and Methods 196( 1982) 11.

[Care 87] D. C. Carey, The Optics of Charged Particle Beams, Harwood Academic
Publishers (1987).

[Care 88] D. C. Carey, Fermi National Accelerator Laboratory Technical Manual TM-
1546, 1988, unpublished.

[Chen 68] K. Chen, Z. Fraenkel, G. Friedlander, J. R. Grover, J. M. Miller, and Y.
Shimamoto, Phys. Rev. 166 (1968) 949.

[Egel 81] Ch. Egelhaaf, G. Bohlen, H. Fuchs, A. Camp, H. Homeyer, and H. Kluge,
Phys. Rev. Lett. 46 (1981) 813.

[Egel 83] Ch. Egelhaaf, M. Bt'irgel, H. Fuchs, A. Camp, H. Homeyer, D. Kovar, and
W. Rauch, Nucl. Phys. A405 (1983) 397.

[Fox 89] R. Fox, R. Au, and A. VanderMolen, IEEE Trans. N ucl. Sci. (1989) 1562.
[Frie 83] W. A. Friedman, Phys. Rev. C 27 (1983) 569.

[Frie 87] E. M. Friedlander and H. H. Heckman, in Treatise on Heavy-Ion Reactions,
edited by D. A. Bromley (Plenum, New York, 1987).

120

[Gall 88] J. Galin, Nucl. Phys. A488 (1988) 297.

[Gelb 78] C. K. Gelbke, C. Olmer, M. Buenerd, D. L. Hendrie, J. Mahoney, M. C.
Mermaz and D. K. Scott, Phys. Rep. 42 (1978) 311.

[Gold 48] M. Goldhaber and E. Teller, Phys. Rev. 74 (1948) 1046.
[Gold 74] A. S. Goldhaber, Phys. Lett. 53B (1974) 306.

[Gold 78] A. S. Goldhaber and H. H. Heckman, Ann. Rev. Nucl. Part. Sci. 28 (1978)
161.

[Grei 75] D. E. Greiner, P. J. Lindstrom, H. H. Heckman, Bruce Cork, and F. S.
Bieser, Phys. Rev. Lett. 35 (1975) 152.

[Guer 83] D. Guerreau, V. Borrel, D. Jacquet, J. Galin, B. Gatty and X. Tarrago,
Phys. Lett. 1313 (1983) 293.

[Harp 71] J. D. Harp and J. M. Miller, Phys. Rev. C 3 (1971) 1847.
[Harv 81] B. G. Harvey, Phys. Rev. Lett. 47 (1981) 454.
[Harv 85] B. G. Harvey, Nucl. Phys. A444 (1985) 498.

[Heck 72] H. H. Heckman, D. E. Greiner, P. J. Lindstrom and F. S. Bieser, Phys.
Rev. Lett. 28 (1972) 926.

[Home 84] H. Homeyer, M. Biirgel, Ch. Egelhaaf, H. Fuchs, and G. Thoma, Z. Phys.
A — Atoms and Nuclei 319 (1984) 143.

[Hufn 85] J. Hiifner, Phys. Rep. 125 (1985) 129.

[Kran 87] K. S. Krane, “Introductory Nuclear Physics”, J. Wiley & Sons, 1987, p.
194.

121

[Kubo 90] T. Kubo, M. Ishihara, N. Inabe, T. Nakamura, H. Okuno, K. Yoshida, S.
Simoura, K. Asahi, H. Kumagai and I. Tanihata, in Proceedings of the 1“
International Conference on Radioactive Nuclear Beams, Oct. 1989, Berkeley,
California, edited by W. D. Myers, J. M. N itschke, and E. B. Norman, (World
Scientific, 1990), p.563.

[McVo 80] K. W. McVoy and M. C. Nemes, Z. Phys. A - Atoms and Nuclei 295
(1980) 177.

[Merm 86] M. C. Mermaz, V. Borrel, D. Guerreau, J. Galin, B. Gatty, and D. Jacquet,
Z. Phys. A - Atomic Nuclei 324 (1986) 217.

[Merz 70] E. Merzbacher, “Quantum Mechanics”, 2”“ edition, J. Wiley & Sons, 1970.

[Maui 71] E. J. Moniz, 1. Sick, R. R. Whitney, J. R. Ficenec, R. D. Kephart, and W.
P. Trower, Phys. Rev. Lett. 26 (1971) 445.

[Morj 88] M. Morjean, J. Frehaut, D. Guerreau, J. L. Charvet, G. Duchéne, H.
Doubre, J. Galin, G. Ingold, D. Jacquet, U. Jahnke, D. X. Jiang, B. Lott,
C. Magnago, Y. Patin, J. Pouthas, Y. Pranal, and J. L. Uzureau, Phys. Lett.
B203 (1988) 215.

[Morr 78] D. J. Morrissey, W. R. Marsh, R. J. Otto, W. Loveland, and G. T. Seaborg,
Phys. Rev. C 18 (1978) 1267.

[Mort 79] D. J. Morrissey, L. F. Oliveira, J. O. Rasmussen, G. T. Seaborg, Y. Yariv,
and Z. Fraenkel, Phys. Rev. Lett. 43 (1979) 1139.

[Morr 80] D. J. Morrissey, W. Loveland, M. de Saint Simon, and G. T. Seaborg, Phys.
Rev. C 21 (1980) 1783.

[Morr 89] D. J. Morrissey, Phys. Rev. C 39 ( 1989) 460.

122

[Murp 83] M. J. Murphy, and R. G. Stokstad, Phys. Rev. C 28 (1983) 428.
[Murp 83a] M. J. Murphy et al., Phys. Lett. 1203 (1983) 75.

[Miinz 90] G. Miinzenberg, H. Geissel, P. Armbruster, K. H. Behr, B. Blank, K. H.
Burkard, H. G. Clerc, M. Chen, J. P. Dufour, B. Franczak, J. J. Gaimard, E.
Hanelt, R. Kirchner, O. Klepper, B. Langenbeck, F. Nickel, K. Poppensieker,
M. S. Pravikoff, E. Roeckl, D. Schardt, K. H. Schmidt, H. J. Schiitt, Th.
Schwab, B. M. Sherrill, K. Siimmerer and H. Wollnik, in Proceedings of the 1"
International Conference on Radioactive Nuclear Beams, Oct. 1989, Berkeley,
California, edited by W. D. Myers, J. M. Nitschke, and E. B. Norman, (World
Scientific, 1990), p. 91.

[Myer 77] W. D. Myers, W. J. Swiatecki, T. Kodama, L. J. El-Jaick, and E. R. Hilf,
Phys. Rev. C 15 (1977) 2032.

[Oliv 79] L. F. Oliveira, R. Donangelo, and J. O. Rasmussen, Phys. Rev. C 19 (1979)
826.

[0130 81] D. L. Olson, B. L. Berman, D. E. Greiner, H. H. Heckman, P. J. Lindstrom,
G. D. Westfall, and H. J. Crawford, Phys. Rev. C 24 (1981) 1529.

[0130 83] D. L. Olson, B. L. Berman, D. E. Greiner, H. H. Heckman, P. J. Lindstrom,
and H. J. Crawford, Phys. Rev. C 28 (1983) 1602.

[Path 72] R. K. Pathria, “Statistical Mechanics”, Pergamon Press, 1972, p. 248.

[Rami 84] F. Rami, J. P. Coffin, G. Guillaume, B. Heusch, P. Wagner, A. Fahli, and
P. Fintz, Z. Phys. A - Atomic Nuclei 318 (1984) 239.

[Rami 85] F. Rami, J. P. Coffin, G. Guillaume, B. Heusch, P. Wagner, A. Fahli, and
P. Fintz, Nucl. Phys. A444 (1985) 325.

123

[Sand 87] D. P. Sanderson, J. Nolen, R. Swanson, N. Mooney and R. Blue, National
Superconducting Cyclotron Laboratory Annual Report 1987, p. 227, unpub-
lished.

[Sher 87] B. M. Sherrill, N. Anantaraman, H. Blosser, R. Blue, S. Bricker, J. C.
Dekamp, J. Easley, J. D. Johnson, H. Laumer, D. Lawton, F. Marti, L. Morris,
J. A. Nolen, R. M. Ronninger, D. P. Sanderson, J. S. Winfield and A. F. Zeller,

National Superconducting Cyclotron Laboratory Annual Report 1987, p. 183,
unpublished.

[Sher 90] B. M. Sherrill, W. Benenson, D. Mikolas, D. J. Morrisey, J. A. Nolen and J.
A. Winger, in Proceedings of the 1" International Conference on Radioactive
Nuclear Beams, Oct. 1989, Berkeley, California, edited by W. D. Myers, J. M.
Nitschke, and E. B. Norman, (World Scientific, 1990), p. 72.

[Silk 88] J. D. Silk, H. D. Holmgren, D. L. Hendrie, T. J. M. Symons, G. D. Westfall,
P. H. Stelson, S. Raman, R. L. Auble, J. R. Wu and K. Van Bibber, Phys.
Rev. C 37 (1988) 158.

[Soul 91] G. A. Souliotis, D. J. Morrissey, and B. M. Sherrill, Rev. Sci. Instrum. 62
(1991) 342.

[Stép 91] C. Stéphan, L. Tassan-Got, D. Bachelier, C. O. Bacri, R. Rimbot, B. Bor-
derie, J. L. Boyard, F. Clapier, C. Donzaud, T. Hennino, M. F. Rivet, P.
Roussel, D. Bazin, C. Grunberg, D. Disdier and B. Lott, Phys. Lett. B 262
(1991) 6.

[Stok 84] R. G. Stokstad et al., LBL report 18240, presented at Int. Conf. on Heavy
Ion Physics, Mt. Fuji, Japan, Aug. 1984.

[Stok 85] R. G. Stokstad, Comments on Nucl. Part. Phys. 13 (1984) 231.

124

[Siimm 90] K. Siimmerer and D. J. Morrissey, in Proceedings of the 1" International
Conference on Radioactive Nuclear Beams, Oct. 1989, Berkeley, California,
edited by W. D. Myers, J. M. Nitschke, and E. B. Norman, (World Scientific,
1990), p. 122.

[Siimm 90a] K. Siimmerer, W. Briichle, D. J. Morrissey, M. Schidel, B. Szweryn, and
Y. Weifan, Phys. Rev. C 42 (1990) 2546.

[Tass 91] L. Tassan-Got and C. Stéphan, Nucl. Phys. A524 (1991) 121.

[Viyo 79] Y. P. Viyogi, T. J. M. Simons, P. Doll, D. E. Greiner, H. H. Heckman, D.
L. Hendrie, P. J. Lindstrom, J. Mahoney, D. K. Scott, K. Van Bibber, G. D.
Westfall, H. Wieman, H. J. Crawford, C. McParland, and C. K. Gelbke, Phys.
Rev. Lett. 42 (1979) 33.

[West 76] G. D. Westfall, J. Gosset, P. J. Johansen, A. M. Poskanzer, W. G. Meyer,
H. H. Gutbrod, A. Sandoval, and R. Stock, Phys. Rev. Lett. 37 (1976) 1202.

[West 79] G. D. Westfall, L. W. Wilson, P. J. Lindstrom, H. J. Crawford, D. E.
Greiner, and H. H. Heckman, Phys. Rev. C19 (1979) 1309.

[Winf 86] J. Winfield, code “S320” (NSCL).
[Winf 91] J. Winfield, SARA Manual (NSCL).

[Wing 91] J. A. Winger, B. M. Sherrill, and D. J. Morrissey, Proc. EMIS Conf. ( 1991),

in press.
[Yari 79] Y. Yariv and Z. Fraenkel, Phys. Rev. C 20 (1979) 2227.

[Zell 85] A. F. Zeller, R. T. Swanson, J. A. Nolen, D. Landry, M. J. Dubois and J.
C. Dekamp, in Proceedings of the 9‘“ International Conference on Magnet

125

Technology (1985), edited by C. Mazinucci and P. Weymuth (Swiss Institute
for Nuclear Research, Zurich, 1985), p. 160 and J. C. Dekamp, C. T. Magsig,
J. A. Nolen and A. F. Zeller, IEEE Trans. in Mag., MAG—23 (1987) 524.

 

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