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This is to certify that the

dissertation entitled

SUBBARRIER FUSION IN LIGHT AND MEDIUM HEAVY SYSTEMS

presented by

Jian-qun Wu

has been accepted towards fulfillment
of the requirements for

Doctoral degree in Physics

 

 

 

"i7
,/ ? S 7
A“) i112- [/ (1.71:1

/ Major professor

 

Date May 12, 1986

 

MS U is an Affirmative Action/Equal Opportunity Institution 0-12771

 

 

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RETURNING MATERIALS:
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SUBBARRIER FUSION IN LIGHT AND MEDIUM HEAVY SYSTEMS

BY

Jian-qun Wu

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY

Department of Physics and Astronomy

1986

ABSTRACT

\
SUBBARRIER FUSION IN LIGHT AND MEDIUM HEAVY SYSTEMS

BY

Jian-qun Wu

The effects of nuclear structure on subbarrier fusion in some light
and medium heavy systems are studied in a two-dimensional barrier
penetration model. The low-lying intrinsic excitations are represented
by surface harmonic vibrations whose parameters are determined to
reproduce low-lying collective excitations. The intrinsic degrees of
freedom are coupled to the relative motion of two nuclei through a
proximity nuclear-nuclear potential which is a function of the closest
distance between two surfaces of the nuclei. We find that this model is
able to explain the enhancements in subbarrier fusion observed in a
variety of systems. Fusion in “O+“O is enhanced relative to “O+“O,
and this behavior is well described by the theory. In medium heavy
systems, the higher overtones of the low-lying vibrations are important.
We reproduce the fusion cross section in “‘Ti+’°Zr, 93Nb, s°Ti+’°Zr,
93Nb with vibrational parameters close to those that describe transition
strengths. We also studied the subbarrier fusion in the heavy systems
°‘Br+’°Zr, ’“Zr with only partial success. Without treating the effects

due to "extra push" or "extra-extra push", the relative subbarrier

 

Jian-qun WLJ

fusion of “Br+’°Zr and "Br+’“Zr are reproduced in the two-dimensional
barrier penetration model. This two-dimensional model, however, fails
to reproduce the absolute fusion cross section in these systems. In
some systems, two-neutron transfer may play a significant role. We
calculated the fusion cross section for the ‘“Ni+"Ni system including
particle transfer channels. The form factors were obtained using a
realistic microscopic model. However, the magnitude of the enhancement
in this case can hardly be accounted for without a more drastic
treatment of the combined effects of deformation and transfer than is

conventionally assumed.

ACKNOWLEDGMENTS

First, and foremost, I wish to thank my thesis advisor, Prof.
George Bertsch, for his guidance and the freedom he gave me through out
the course of my graduate study at NSCL. Second, I wish to thank the
remaining members of my guidance committee: Profs. Walter Benenson,
Jerry Nolan, and Dan Stump. Third, I wish to thank Dr. Henning Esbensen
for his constant encouragement and help over past three years. In
addition, I would like to thank Drs. Don Cha, Raad Raadi, Rainer Kaps,
Joe Molitoris, and Erich Ormand, who were my officemates at different
times during past three years, for the helps I received. I would
especially like to thank Dr. Erich Ormand for his assistance in writing
this thesis. I would like to express my gratitude to Shari Conroy for
her invaluable assistance and friendship through out the course of this
work.

I would like to take this space to thank the Nuclear Theory Group
at Oak Ridge National Laborotory and the Department of Physics &
Astronomy at University of Tennesse for the hospitality and the
intellectual stimulation I received during my visit from Nov., 198“ to
Jun., 1985. I greatly appreciate the encouragement and advice I

received from Prof. Z. R. Yu when he visited M. S. U. in 1985.

ii

TABLE OF CONTENTS

LIST OF FIGURESOOOI.0.0...O....0.0...O...OOOOOOOOOOOOOOOOOOOOOOO0.0. v

LIST OF TABLES. ....... O ...... 00.... OOOOOOOOOOOOOOOO 0.0... ....... O... x

Chapter 1 Introduction............................................. 1
1.1 Quantum Tunneling ......................... . ...... . ......... 1
1.2 Subbarrier Fusion.......................................... u
1.3 Multidimensional Tunneling....... .......... . ........... .... 6

1.4 Methods of Studying Subbarrier Fusion...................... 6

Chapter 2 Models of Zero-Point Motions................ ........... .. 9

2.1 Importance of Collective Excitation........................ 9

2.2 Effects of Rotational Excitations. ............ . ...... ...... 13
2.3 Vibrational Excitations.... ...... .. .......... .............. 1”
Chapter 3 Coupled Channel Method..... ..................... ......... 20
3.1 Coupled Channel Treatment. ..... ............................ 20
3.2 Numerical Methods and Approximations ....................... 23
303 TeChnical POintSoooooe ..... oooooooooe00000000000000.0000... 30
Chapter A Validity of ZPM and Simple Approximations.. .............. 36

”.1validitYOfZPMOOOOOOOOOOOOOOOOOOIOOOOOOOOOOOOOOOOOOOOOOOOO 36
u.2 Two Independent Degenerate Harmonic Oscillators ............ 38

”.3 Factorization of The Enhancement Factors................... N1

Chapter 5 Subbarrier Fusion in Light Nuclei. ........ . ..... ......... nu

iii

Chapter 6 Subbarrier Fusion in Medium Heavy Nuclei.................

6.1 Details of The Calculation........

6.2 Results and Discussions...............

Chapter 7 Particle Transfers in Subbarrier Fusion........
7.1 Introduction......
7.2 Form Factors for Neutron Transfer........

7.3 Details of The Calculation........

7.N The Interplay of Intrinsic Excitations and the Particle

TranSfers on SUbbarrier Fugionoooo00000000000000.oooooooooo

7.5 Simultaneous Two Neutron Transfer ...................... ....

7.6 Discussion.......

Chapter 8 Conclusions......

REFERENCES..

iv

52
52
56

67
67
69

71

76

76

78

80

81

FIGURE

2.3

LIST OF FIGURES

PAGE

Schematic plot of internuclear potential.................... 3
A possible deformation configuration of a system of

two nuclei which are spherical in their ground states....... 10
System of a spherical nucleus and a deformed nucleus

separated by a distance of r................................ 12
Surface vibrations of a system consisting of a "stiff"

(nucleus 1) and a "soft" (nucleus 2) nuclei................. 15
Calculated and measured fusion cross sections for the

reactions “O+Sm. The abcissa is the c. m. energy Ec.m.
The samarium isotopes used are indicated, and experimental
results agree with the calculations at smaller energies.

There are no experimental data for 1“Sin. The standard
deviations that have been used for the quadrupole and

octupole ZPM or static deformations and the values of AR in
eq.(2.7) are given in table 1 and 2 in ref.(Esb.81),
respectively................................................ 17
Effects of both target and projectile zero-point motion and
average dynamical deformation (Coulomb excitation) on the

fusion cross section for soNi+”Ni+”‘Ba. The solid circles

give the experimental data of ref.(Bec.80), the dashed curve
gives the result for a one-dimensional barrier corresponding

to spherical nuclei, and the solid curve gives final result

for a two-dimensional potentialeenergy surface, and the

dotedashed curve gives the result of ZPM without dynamical

“.2

“.3

5.2

deformation................................................. 32
Fusion probabilities for the reaction “O+‘“°Sm are shown

as function of the c. m. energy. The results are obtained

from coupled-channel calculations, for various values of

the one-phonon excitation energy hm. The standard deviation

of zero-point motion amplitude is a - 0.27 fm.............. 37
S-wave fusion excitation function is shown as function of

c. m. energy. The dashed curve is the result obtained by
combining two vibrational modes into single one through
eq.(u.1). The solid line is the result obtained by

solving exactly a three-dimensional coupled equation.

Due to numerical restrictions, the exact result can not

be obtained for energies lower than 98 MeV. The parameters

are given in Table A.1................ ..... .......... ..... . HO
Fusion enhancement factors are shown as functions of

c. m. energy. The solid line is the result obtained

by combining two vibrational modes into one through

eq.(u.1). The dashed curve is the result obtained by

virtue of factorization rule. The parameters used

are listed in Table M.1.... ..... ........................... M2
The measured fusion cross section for 1"Demo and

160+”0, from ref.(Tho.85) is shown by the experimental

points. The potential model fit to the 16O“O is shown

as the solid line, and the coupled-channel model with

rescaled potential is shown by the dashed line.... ..... .... us
Enhancement factors extracted by log-linear interpolation

of the data from ref.(Tho.85). The dashed line is the

vi

603

prediction of the rescaled potential, eq.(3.2). The

dotted line is the prediction of the potential eq.(5.3).

The solid line is the prediction of the coupled-channel

model with the rescaled potential.......................... H6
Effective one-dimensional potential barriers for l3C+”C
l"N+”N, 13C+“°0, “°Ca+“°Ca, ‘“Ni+‘“Ni, and ‘“Ni+’“Ge.

The outer turning point is determined from the KNS

potential readjusted as described in ref.(Bal.83) to

fit the peak positions. The thickness is inverted

from fusion cross sections and the shaded region

indicates the error. The short dashed line denotes

the point Coulomb potential, the long dashed curves

in the upper figures denote the unmodified KNS

potential, and for ease of presentation the energy is

plotted relative to the barrier height..................... 53
Fusion cross sections for s°Ti+9°Zr. The bold continuous

curve is the result of the one-dimensional barrier

penetration calculation. The two-channel coupling result

is shown by dash-dotted curve. The thin continuous curve

is the result obtained by solving 6 channel coupled

equation. The 6 channels correspond to the first 6

excited vibrational states assumed in the s°Ti with a

energy spacing of 1.55 MeV. The dashed curve is the

prediction from ZPM. All calculations include the

quadrupole Coulomb excitation and ignore the angular

momentum transfer. ...... ............... ...... .............. 57

Same plot as in Fig.6.2, but for “‘Ti+’°Zr. The parameter

vii

6.“

6.5

6.6

V0, Ad, AR are kept same as those for s°Ti+’°Zr. 0.89 Mev

the energy of the first excited state in “‘T1, is used as

the energy quanta for the vibrational mode. The a is

some how 0.39 fm, different from that quoted from the B(E2)
measurement................................................ 59
Fusion cross sections for “‘Ti+”Nb. The hm and a for

93Nb is taken as the average of those for 92Zr and 9“M0.

The excitations in both “‘Ti and 93Nb are combined into

one mode with hm - hw(“‘Ti) - hu(”Nb) and the 0 equals

 

/Ioi(“‘Ti)+o’(’3Nb). The number of channels coupled is 5.. 60
Fusion cross sections for s°Ti+”Nb. In the coupled

channel calculation, the hm is the average of those for

 

s°Ti and 93ND. The a is /—67(56Ti)+ozI"Nb)............... 61
Plot of the ratio of the fusion cross sections for °‘Br+

9“Zr to that for °‘Br+’°Zr. The experimental data is the
cross section of evaporation residual formation............ 63
Calculated fusion cross sections for 58Ni+“‘Ni compared to
the data of ref.(Bec.80). This plot is the copy of
Fig.2(b) in ref.(Bro.83). The dotted curve shows the limit
of no coupling, i.e., one-dimensional barrier penetration.
The dash-dotted curve is the result obtained by including
intrinsic excitations. The dashed curve includes additional
channel coupling, which was attributed to transfer reaction
channels.. .............. . ............ ... ................... 68
Form factor F(r) (solid line) is shown as function of the

separation r. F(r-s(r)) (dot-dashed curve) is the form

viii

7.3

7.“

factor in the semiclassical model. Also shown (dashed

curve) is the largest single-particle transfer amplitude

between two f5/2 orbits in ‘“N1 and soNi, respectively..... 7“
Fusion excitation functions of the ‘“Ni+’°Ni system.

Dotted curve is the result of the one-dimensional barrier
penetration model. The dashed curve is the result obtained

by including only intrinsic excitations with parameters

given in Table 7.1. The dot-dashed curve is the result

with both transfer and intrinsic excitation included. The
Fi(r-s(r))'s are the form factors used in obtaining the

result that reproduces data (solid curve).................. 75
Contours of equal density |w(r,s)|2 are shown in the (r,s)

plan. The wave function is the coupled-channel result

without transfer coupling for the reaction "Ni+‘“Ni at a

c. m. energy of 90 Mev with parameters given in Table 7.1.

The dotted curve is the semiclassical trajectory in which

the incoming flux is maximized in the s-direction.......... 77

TABLE

6.2

LIST OF TABLES

PAGE
Calculated s°Ti+9°Zr S-wave fusion cross sections at a
c. m. energy of 100 MeV as a function of number of
channels in the model space in different representations
with parameters given in chapter 6. ..... ................... 29

Parameters used in obtaining Fig.“.2 and Fig.“.3........... 39
Parameters of surface vibrations deduced from the

measurement of low-lying states. For odd nuclei they

are averages of those of two neighboring nuclei.

The o's (standard deviations) are calculated according

to eq.(2.7)................................................ 55
Parameters of surface vibrations used in the fusion
cross-section calculations of Figs.6.2,3,“,5,6. The

parameters in parenthesis were obtained by fitting to the

data. The rest were taken or calculated from Table 6.1.... 55
Parameters of the two-dimensional potential and the

surface harmonic vibrational mode obtained by fitting

the s"NH”NI fusion data .................................. 72

Chapter 1 Introduction

1.1 Quantum Tunnelling

One of most striking features of quantum physics, the description
of nature at the microscopic level, is that classically forbidden
processes can occur, though generally with a small probability. Such
processes are usually referred to as "tunneling" or "penetration".

Since tunneling is a classically forbidden process, it must take
place with a small probability because otherwise classical physics would
be of no use at all. It is indeed true that the probabilities are
small. In the one-dimensional WKB approximation, the penetrability is

given by (Sch.68)

 

P -= 1/{1+ exp(—2f:=/§M/h’TV(r)—s)dr)l. (1.1)
l

where V(r) is the potential energy, M is the reduced mass, E is the
total energy and r1 and r2 are classical turning points. One can see
from eq.(1.1) that P is very small unless the "action" (the integral in
the exponent) is small.

An analytic formula can be derived from eq.(1.1) for a parabolic

potential, i.e V(r)=V - l Mm2(r-Rb)2, where V is the barrier height and

b 2 D

RD is the barrier location. The penetrability is then given by (Hil.53)

l
1+exp[2n(Vb4E7/hm]'

 

P v (1.2)

This procedure is also useful for the three-dimensional problem with
spherical symmetry (three dimensional kinematics, but one-dimensional
dynamics). Because of the relative large masses involved, it is a
reasonable approximation for reactions involving heavy ions to assume
that the effective centrifugal potential only changes the height of the
potential barrier, and does not alter the position and the shape of the
barrier. Thus, the penetrability for each partial wave can be

calculated as

1

P22 1+exp[2n(vb£-E)/hw] . (1.3)

 

2
where V . V +-£££:LB1-.

2
b1 0 ZMRb

The total cross section (Won.73) is then

o-—E—r2 (22H) P

i=0 a

. E3;— I; 21m. P - szhm/ZE in{1+exp[2n(E-vb)/nu]}. (1.“)

2

Eq.(1.2) and therefore eq.(1.“) is valid for both E>Vb and Esvb.

In the case E>Vb, eq.(1.“) reduces to the well-known formula:
s 2 " °
0 an (l Vb/E). (l 5)
While for E<<Vb, we have

0 a szhw/ZE exp[2n(E*Vb)/hm]. (1.6)

Potential

 

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l

 

>
Rs
Separation Between Two nuclei

Fig.1.l. Schematic plot of internuclear potential.

Eq.(1.“)-(1.6) are very useful in computing the compound nuclear
formation (fusion) cross section. As we will see, eq.(1.6), however,
underestimates the fusion cross section in many cases. But it is the

starting point.

1.2 Subbarrier Fusion

Potential energies between two nuclei are usually approximated as
functions dependent only on the separation between the centers of mass
for each nucleus. They consist of a long range Coulomb interaction and
a short range nuclear interaction. A typical potential is shown in
Fig.1.1. The exact potential curve depends on the system, but roughly
speaking, the dip inside the barrier, called a pocket, becomes smaller
as the system gets heavier. This is because the nuclear interaction is
proportional to A, the number of nucleons, while the Coulomb part is a
22 a A2. For a superheavy system (e. g. U + U ), nuclear attraction is
not strong enough to overcome Coulomb repulsion, and a potential pocket
is not formed. For light and medium heavy systems, which are the
subject of this thesis, however, the dip occurs and a barrier in between
the inner and outer regions is formed. Fusion is usually assumed to
happen when the system passes the Coulomb barrier. With this
definition, one can use eq.(1.“) to compute the fusion cross section.
For E >>Vb, experiments appear to agree with the law of linear inverse
energy dependence given by eq.(1.5). For E < Vb, eq.(1.6) does not work
well in general. Observed fusion excitation functions (probability or
cross section as a function of bombarding energy) are much greater than

those given by eq.(1.6) for energies far below coulomb barrier. This is

not because the derivation of eq.(1.6) from eq.(1.“) is wrong, but the
one-dimensional approximation is insufficient. In the one-dimensional
barrier penetration model (BPM), both projectile and target are assumed
to be point (structureless) particles. However, nuclei may be excited
during the collision. Therefore, they can not be treated as point
particles.

Above-barrier fusion is a classically allowed process (in fact, it
is the only final state allowed in the one-dimensional model), and
intrinsic excitations of the projectile and/or the target does not
change the order of magnitude of the above-barrier fusion cross section.
Thus, eq.(1.5) can be used. Intrinsic excitations, however, may alter
other observations, such as elastic and inelastic angular distributions.
As far as above-barrier fusion is concerned, the one-dimensional model
is a good approximation.

How intrinsic excitations effect subbarrier fusion is an entirely
different story. It will be seen in chapter 2 that intrinsic
excitations can change the order of magnitude and the shape of
subbarrier fusion excitation function. There are some systems for which
eq.(1.5) works for energies just below the Coulomb barrier. In general,
however, heavy-ion subbarrier fusion is a multidimensional tunnelling
process (Lec.8“).

One motivation for studying heavy-ion subbarrier fusion is to
search for superheavy elements. These superheavy elements decay very
rapidly into smaller components, even in their ground state. So they
have to be made with energies as low as possible in order to make them
live longer to make experimental study feasible. This leads to the

study of subbarrier fusion. At the moment, the heaviest elements are

all very shortly lived, and can not be made as targets in experiments in
seeking heavier elements. Thus, heavy-ions (usually referred to any ion
heavier than a-particle) as both targets and bombarding particles are

required.

1.3 Multidimensional Tunneling

An appreciation of the underlying physics is crucial in deriving
the results of WKD approximation (eq.(1.1)). From a practitioner's
point of view, however, the approximation is not that crucial in the
sense that with modern computers one can perform a straightforward one-
dimensional integral to obtain the penetrability. The numerical
calculation of multidimensional tunnelling, on the other hand, is very
difficult. Therefore, although one can not obtain an analytical
formula, as in the WKB in one-dimensional case, approximations based on
the underlying physics are absolutely necessary.

0n the theoretical side, in addition to reproducing experimental
data, the study of multidimensional tunnelling is a interesting subject
in itself and subbarrier fusion provides a test on our understanding of

the subject.

1.“ Methods of Studying Subbarrier Fusion

For light and medium heavy systems, meaSurements of subbarrier
fusion can be made directly by measuring the evaporation residue because
particle emmision dominates fission in the compound decay. In such

experiments, one measures the charge or the mass or both of the reaction

products, directly or indirectly (e. g. Gut.73, PUh.75, Nat.70, Bro.75,
Sto.81). For heavy systems, fission is the dominant channel for
compound system decay. In this case, there is no evaporation residue,
so the fusion yield is measured by measuring fragment coincidences. In
such experiments, one must know beforehand the properties of the
fragmentation of the compound systems. In general, fusion experiments
of heavy systems consist of more sophisticated measurements than direct
measurements in lighter systems.

Theoretically, as mentioned in the previous section, exact
calculations are very difficult. Almost all subbarrier fusion
excitation functions fall off exponentially as bombarding energy
decreases except in very few systems where resonances exist. The key
parameter is the energy in the center of mass (c.m.) frame. This
enables one to make very useful approximations. As an example, we will
see in Chapter 3, that one can make approximations about angular
momentum algebra based on energy considerations.

The principle reason why eq.(1.6) fails is that nuclei may be
excited during the collision. Therefore one has to go beyond the one-
dimensional barrier penetration model (BPM). The two-dimensional zero‘
point motion models (ZPM) which take into account the barrier
fluctuations due to intrinsic excitations have been very successful in
describing subbarrier fusion for some systems. In next chapter, ZPM
will be described. Following that in chapter 3, we will discuss the
more general two-dimensional model, namely, coupled-channel formulations
and approximations used to solve the coupled equations. In chapter “,
the validity of ZPM will be tested, and approximations for reducing

three- or more-dimensional problems to two-dimensional ones will be

studied. Chapter 5 and chapter 6 contain the results of calculations
and discussions of subbarrier fusion in some light and medium heavy
systems. Effects due to particle transfer on subbarrier fusion will be
studied for system of Ni+Ni in chapter 7. Finally, we conclude in

chapter 8.

 

Chapter 2 Models of Zero-Point Motions

The simplest two-dimensional models are zero-point motion models.
In these models, the intrinsic excitation energy is assumed to be zero.
The idea is that only low-lying collective excitations are important, so
the excitation energies are small, and can be neglected completely.
There are two kinds of models associated with two different modes of
collective excitations, namely vibrations and rotations. In sec.2.1, we
will first discuss why collective excitations play an important role in
subbarrier fusion. Sec.2.2 contains a description of the rotational
averaging model which takes into account the rotational excitation in a
well-deformed nucleus. In sec.2.3, the vibrational zero-point motion

model will be discussed.

2.1 Importance of Collective Excitation.

There are two reasons why collective excitations are an important
degree of freedom in subbarrier fusion. One is that low-lying
collective states have smaller excitation energies than other kinds of
excitations, say single-particle excitations. If the excitation energy
is large, then the energy of the relative motion of the two nuclei is
much smaller than that in the entrance channel. Therefore, the
excitation gives little contribution to the process of subbarrier
fusion. One can see this more clearly in following way: if the internal
excitation happens during the collision of two nuclei, the potential is
no longer a local one-dimensional one, rather a multidimensional matrix.

When it is diagonalized, a number of eigenvalues will be smaller than

lO

 

 

Nucleus 1 Nucleus 2

Fig.2.l. A possible deformation configuration of a system
of two nuclei which are spherical in their ground states.

11

the potential in the entrance channel. It is this effect which can
enhance subbarrier fusion reactions. On the other hand, if the
internal excitation energy is large (the difference between two neighbor
diagonal elements in the potential matrix is large), the lowest
eigenvalue of the potential matrix would be essentially same as that in
the entrance channel. Thus, it is not possible for excitations to
contribute significantly.

The second reason is that collective excitations are more strongly
coupled to the relative motion than other intrinsic excitations. Many
particles move coherently in a collective excitation, which changes the
internuclear potential substantially. If the off-diagonal elements
(couplings) in the potential matrix are large, the lowest eigenvalue of
the potential matrix is much lower than the potential in the entrance
channel, and subbarrier fusion can be enhanced substantially.

There is a physical picture which explains why low-lying collective
excitations are important in subbarrier fusion.

Suppose two nuclei are deformed as shown in Fig.2.1. Though the
deformation take some energy away from the relative motion, if r is
within the range of nuclear force, the attractive nuclear potential is
much stronger than if there is no deformation. This "extra" attraction
makes the barrier lower than if the two nuclei were in their ground
states, and leads to enhancement in subbarrier fusion. In order to make
this mechanism work, the excitation energy must be small and the "extra"
nuclear attraction must over-compensate the loss of the energy for the
relative motion. Only low-lying collective excitation can fulfill these
two requirements. Note, however, that the Coulomb force works against

this 0

12

Nucleus 1 Nucleus 2

Fig.2.2 System of a spherical nucleus and a deformed
nucleus separated by a distance of r.

13

2.2 Effects of Rotational Excitations.

Now we consider a subbarrier fusion reaction between two heavy
ions. Here we take one nucleus to have a well-deformed ground state,
while the other possesses spherical symmetry. The lowest possible
internal excitation for this system is a rotational excitation of the
deformed nucleus. In this case, the potential is no longer one-
dimensional, but at least a function of two dimensions. As shown in
Fig.2.2, the potential is a function of the relative coordinate r and
the angle 9 between the symmetric axis of the deformed nucleus and the
relative coordinate.

Wong (Won.73) has calculated the fusion cross-section for such
systems. Instead of following the details of his work, we give a rough
estimate to show what effect averaging over the angle 9 has on
subbarrier fusion.

The Coulomb barrier height V can be written as

b

Vb = Vb + f(6)o

where 7b is the averaged barrier height and f(e) is some function with

Jf(6) d6 = o.

The subbarrier fusion cross section ofus , in the WKB
approximation, is an exponential function of Vb and Ecm (the energy of

the relative motion in the c. m. frame), namely

1n

‘(Vb(e)-Ecm)c, (2.1)

Ofuswb’ 6) a: e

where c is some positive number. The averaged cross section is then

-r(e) c
e . (2.2)

- ‘(V-E)cjd6
afus = I de Ofus(vb’e) « e b cm

 

Tde

where the uniform distribution over 9 is assumed and the reaction is
incoherent in e, that is, 9 does not change during a collision, which,

we will see later, corresponds to the zero excitation energy limit. If

 

J de e-f(6)c
is greater than one (it turns out that it is the case if
fde
Ecm< Vb), then afus > afus(vb)’ that is, the subbarrier fusion cross

section is enhanced.
In general the rotational excitation energy is not zero, but small,

and therefore can be approximated as zero for some heavy systems.

2.3 Vibrational Excitations.

For spherical nuclei, the lowest excitations are surface
vibrations. Thus subbarrier fusion of two spherical nuclei is enhanced
mainly due to vibrational excitations in the nuclei. In general, the
vibrational excitations are not purely harmonic, however, we will assume
that this is the case for the purpose of simplicity.

The picture here is the same as that in sec.2.2 : The potential is
a two variable function. The first variable is the distance between the
centers of the two nuclei r, while the second is the vibrational

surface-collective variable 3.

15

 

 

Nucleus 1 Nucleus 2

Fig.2.3. Surface vibrations of a system consisting
of a "stiff" (nucleus 1) and a "soft" (nucleus 2)

Nuclei.

16

The nucleus deforms around the equilibrium spherical shape.
Because of the isotopy of the space, on the average the nuclear surface
vibrates as shown by dashed circle in Fig.2.3. For a positive 3 there
is more nuclear attraction than if s-O, and there is even less nuclear,
attraction for negative 3. If the intrinsic excitation energy is small,
and can be approximated as zero, the fusion cross section-is that
averaged over 3 with some weight determined by the asymptotic boundary
condition. Again due to the exponential fall-off of subbarrier fusion
cross section (eq.(2.1), the subbarrier fusion is usually enhanced.

Esbensen (Esb.81) used this model to calculate subbarrier fusion
cross sections of ‘°O+Sm and Ar+Sm. This model was very successful in
explaining the experimental data, which were much larger than the
results given by conventional penetration model. In these systems, the
targets (Sm isotopes) have low-lying vibrational excitations, and the
intrinsic excitations were assumed to be only in the targets.

The Schrddinger equation in ZPM can be written as

2

.- P
l2m

 

+ V(r.3)] w(r.8) = Ecm w(r.s). (2.3)

Since there is no differential operator in s, this equation can be
solved by treating s as a parameter and solving the one-dimensional
equation for r. There is no correlation in s, so nuclei bombarding with
a different s scatter incoherently, i. e., s does not change during the
collision. This is sometimes called the frozen approximation. In this

approximation the penetrability is

P(Ecm) 8 I ds F(Ecm’ 5) 8(3), (2.“)

17

 

 

 

 

 

 

 

 

 

 

103
i
I.
102:-
E
3 >-
E
:3 10 I—
D C
I
e— / ‘3
t . < Sm 1
t 1
‘0" f i i a
55 60 65

Ed, (MeV)

Fig.2.“. Calculated and measured fusion cross sections for the
reactions 160+Sm. The abcissa is the c.m. energy E . The
samarium isotopes used are indicated, and experimegtgl results

agree with the calculations at smaller energies. There are no
experimental results for H"Sm. The standard deviations that have
been used for the quadrupole and octupole ZPM or static deformations
and the values of AR in eq. (2.6) are given in table 1 and in

table 2 in ref. (Esb.81), respectively.

18

where P(Ecm,s) is the penetration factor for the potential V(r,s), which
can be calculated by eq.(1.“), B(s) is determined by the asymtotic
boundary condition. Usually the target is in its ground state,

therefore 8(a) is a gaussian, i. e.

82
202

B(S)‘72—11;;—e o (2.5)

 

where o is the r. m. s. amplitude of the surface vibrational ground
state of the target. For subbarrier fusion, P(Ecm) is usually much
greater than that if there is no coupling of the surface motion to the
relative motion.

In Fig.2.“, the calculated fusion cross sections of 16O+Sm in this
model are compared with the data (Sto.80). The nuclear potential used

was

s RLRz l
U(r,s) V° R,+R2 1+expEIr-R1-R2~AR-s)/a] ° (2'6)

 

The parameters V0 and a were extracted from elastic scattering

data (Chr.76), and AR was adjusted to fit the fusion data. In the
calculation, the monopole, quadrupole, and octopole Coulomb interaction
were included in the potential. The a's were extracted from the low-

lying state data through (Esb.81)

R

B(EA) 1/2
0A " m((2l+1)'B-Sm) (2.7)

19

where A denotes the multipolarity, B(EA)/B&EA) is the electric
transition strength in terms of Weisskopf unit and R and Z are the
radius and charge number of the Sm-isotOpe, respectively. In this case,
angular momentum transfer was ignored. As can be seen in the figure,

the agreement is very good in the subbarrier region.

Chapter 3 Coupled Channel Method

There is one thing in common in Wong (Won.73) and Esbensen
(Esb.83)'s work: the intrinsic excitation energy is assumed to be zero.
This is a good approximation for some heavy systems, but questionable in
general. In these models, the two dimensional equation is essentially
one-dimensional, so it is easy to do numerical calculations.
Intuitively, the ZPM can only be an approximation. The more general
formulation of the two-dimensional model with finite intrinsic
excitation energy, namely the coupled channel formulation will be given
in sec.3.1. In sec.3.2 various numerical methods and approximations for
solving coupled-channel equations are presented. The Coulomb excitation

and angular momentum transfer will be discussed in sec.3.3.

3.1 Coupled Channel Treatment

We know from the previous chapter that collective excitations
enhance the subbarrier fusion. The simplest realistic Hamiltonian is at
least two-dimensional. In this section, we present the coupled-channel
formulation for a system in which the surface vibrational degree of
freedom is important.

The two-dimensional Hamiltonian can be written as

“2

H(r,s) = (- 25

V2 + Hin(8) + V(r.s) ). (3.1)

20

21

where m is the reduced mass, H1n(s) is the surface vibrational

Hamiltonian in nucleus 1. V(r.s) is composed of Coulomb and nuclear

interactions, 1. e.

2 2
V(r,s) . 21:26 + 3z,z,e Rls Rle

5r: + V erf(P‘RL‘Rz‘AR‘S)

°R,+ R2 a ' (3'2)
where the second term is the quadrupole Coulomb excitation. The
complementary error functional form was used for convenience in
numerical computation. The parameters V0, a, R1 and AR are taken as

Vo - -31.61 MeV, a = 1.“2 fm,

i _ ~43 (3.3)

AR=0.29 fm, and R1: 1.233 A 0.98 A ,

and were extracted from elastic scattering data (Chr.76).

To solve the two-dimensional Schrbdinger equation
H(r,s) w(r,s) - (E+§hw) w(r,s), (3.“)

where E is the total energy of both the relative motion in the c. m.

frame and the intrinsic excitation, we expand the wave function on a set

of oscillator eigenfunctions ¢n(s) of Hin(s)

Mme») = r11 wn(r') ¢n(s) (3.5)

and obtain the coupled equations

H2 82
{‘-EH— 5;? + <n|V|n> ‘ E + Ohm} Wn(r)‘ "min<mlvln> vm(P)- (3'6)

22

The matrix elements <n|V|m> are given by

<n|V|m> . Ids ¢n(s) V(r.s) ¢m(s). (3.7)

The boundary conditions are incoming waves at a certain c. m.

distance rmin near the minimum of the total interaction or decaying

states, when E-nhm is less than the total interaction energy at rmin and
Coulomb wave functions at r=rmax outside the barrier, where the nuclear

fields vanishes, i. e.

e-ikn(m1n) r

 

Pn rs rm1
w (r) > I _ (3.8)
n 6 w( )(r) + R w(+)(r) r; r
n0 0 n n c n ma
. . th
Here hkn(min) is the radial momentum at rmin in the n channel and

Wé:;(r) represents outgoing and incoming Coulomb wave function in
different channels with an energy of E-nhm in the relative motion.

There is only an incoming Coulomb wave function in the elastic channel.
The fusion probability P is the sum over all channels of the relative

fus

flux at rmin in each channel, that is
B 2 . .
Pfus E |Pn| hkn(m1n)//2Em. (3 9)

In principle, the coupled-channel formulation is equivalent to
eq.(3.1) if a complete set of oscillator eigenfunctions ¢n(s) is used in
the expansion. In practice, however, a finite set is used in order that

the calculation is feasible.

23

In next section, we will discuss several methods and approximations

for solving the coupled equations in eq.(3.6).

3.2 Numerical Methods and Approximations

We first start with the method of iteration. This method has been
used by Rhoades-Brown et al. (Rho.8“) in subbarrier fusion calculations.
It works like this: first, the solution satisfying the outer boundary
condition (at r-rma) in eq.(3.8) is guessed. The solution is used for
the right-hand side (RHS) of eq.(3.6). By calculating the LHS of
eq.(3.6). a new solution is obtained. Then, the new solution is used
for the RHS to obtain a newer solution. One repeats this process until
a pre-set convergence test is achieved. In the calculation, an
additional short range imaginary potential is added to the Hamiltonian
(eq.(3.1)) to replace the inner boundary condition. Unfortunately, the
method of iteration does not guarantee a converged solution. Various
tricks have been invented to improve this procedure. The method
described above is the simplest, called sequential iteration, in which
the RHS is kept fixed while solutions in all channels are obtained. An
improved method is called block iteration, in which a solution in a
channel is immediately substituted into the RHS with some modifications
when they are necessary to satisfy the boundary condition. In addition,
Pade approximments have also been used by RhoadesfiBrown et al. (Rho.80).
This method works for some problems which can not be solved by either
the simple sequential or block iteration techniques. But, again, there

is no guarantee. For subbarrier fusions, calculations are often time

2“

consuming, and a converged solution can not be obtained when the fusion
cross section is down below the 0.1 mb level (Rho.8“).

The direct matrix inversion method has been used by Ring et
al. (Rin.78) in calculating subbarrier fission penetrability. It has
also been used by Esbensen et al. (Esb.83) in subbarrier fusion
calculation. Instead of using an additional imaginary potential, an
incoming wave boundary condition is imposed at a distance inside the
Coulomb barrier. This represents the same physics as that represented
by the short range imaginary potential used in iteration method: Once
the system passes the Coulomb barrier, fusion take place with a 100%
probability. By solving eq.(3.6), one obtains a matrix connecting the
inner boundary condition and outer boundary condition. By inverting the
matrix, the fusion probability is obtained. Mathematically, it works as
follows. Suppose the total number of channels is N. So there are N
independent solutions to eq.(3.6) satisfying the inner boundary
conditions given in eq.(3.8). But only one solution satisfies both
inner and outer boundary conditions given in eq.(3.8) which can be
obtained in the following way:

One solves eq.(3.6) with

wi,n(r) = 6 exp(-ik£min) r) at rsrmi i-1,2,-oo- N, (3.10)

in

where i denotes the channel number and n denotes the specific boundary

condition. At rsrma, one obtains

(r) 3:1, 2,----N (3.11)

25

where n denotes the specific boundary condition in eq.(3.10) and j
denotes the channel number. n assumes values 1,2,----N, giving N
independent solutions. The general solution satisfying the inner

boundary condition given in eq.(3.8) is a linear combination of these

solutions

N
WJ(P)‘Z P
n=1

nvjmw). (3.12)

where the Pn's are coefficients. To achieve a solution that satisfies

the outer boundary condition, one finds coefficients P“, such that

 

 

 

 

 

 

r— -‘ - -
81,1 81,200.00... B1’N 1 P11 1}
J 82,182,200000000 82,N P2 0
OOOOOOOOOOOOOOOOO... O - o ' (3.13)
_BN,1 BN, 2. O O O O I O O BN’N PN-J 1b0-
and, therefore, we have
_P1 '17
1
P2 _1 0
. - B 0 . (3.1“)
P 0
L N. I. l

 

 

 

 

The fusion probability is then determined with eq.(3.9)-

26

With this procedure there is no convergence problem. However,
numerical precision can quickly become a limiting factor when the number
of channels increases.

Below the barrier the matrix to be inverted has elements which
differ by several orders of magnitude. This is because wave functions
in various channels are exponential functions with different decay
lengths. Therefore, elements of B differ by orders of magnitude for
energy far below the barrier. The transmission factor (penetrability)

can be written as

P = p + c (3.15)

where little p is the true value of the transmission factor and e is the
numerical error that comes out of a numerical calculation. A simple
estimate of the error can be made for a simplified model:

Suppose the potential barrier is a square potential characterized
by its height Vb and its width a. The error in the transmission factor
obtained by the direct matrix-inversion method described above is of the
order of

5 ~ exp((d'-a)a)’ (3.16)

10m

 

where a' and a are wave numbers under the barrier in channels with the
least and the largest energies for relative motion, respectively. And m
is the number of effective digits that can be handled by the computer

used. Substituting eq.(3.16) into eq.(3.15). one obtains

27

exp((a'-d)a))

T . t (1+
t 10m

 

(3.17)

As the energy goes down, the relative error becomes larger. At some
point, one can not use the direct matrix inversion method any more.
The precision limitation may be overcome by a step-wise inversion
method. That is, one divides the barrier into several intervals and
constructs transmission matrices for each segment. The total

transmission factor is

eXP((a'-o)a/n)) , 3 p [1, exp((a1-a)a/n))
m , i p m
i=1 10 i=1 i 10

 

(n-1)/n exp((a'-a)a))

P 1om

exp(-(a'-a)fl:la)

. P (1 + n P
n

. (3.18)

where n is the number of segments and little pi is the transmission
factor for 1th segment. In obtaining eq.(3.18), the barrier has been
sliced into equal parts and the transmission factor has been assumed to
be the same for each segment and P (- p?) is the true total transmission

(n71)/nexp(-(a'-a)fl%la) is less than one and

factor. The factor n P
decreases as n increases. Compare eq.(3.17) with eq.(3.18), one finds
that the error can be reduced by the step-wise matrix inversion method.

The adiabatic representation is also very efficient, and has been
used in subbarrier fusion calculations (Tan.85). To illustrate, let us
first rewrite eq.(3.6) in a more compact form:

n2 32
I 2M 5;? + V(r)-E) 1(r)=o. (3.19)

 

28

where w(r) is an N-dimensional vector (N is the total number of channels
explicitly treated in the calculation). Each component wi(r) is the
wave function in channel 1 and V(r) is a NxN potential matrix with off-
diagonal elements given by eq.(3.7). The diagonal elements are given by
eq.(3.7) plus the intrinsic excitation energies. The coupled channel
basis is transformed to diagonalize V(r) at each point. Then eq.(3.19)
becomes
(lg—M: D(r) 3;, Wm + D(r)V(r)D“1(r)-E) D(r)‘i‘(r) - o, (3.20)
where D(r) is the transformation matrix and D(r)V(r)D(r)--1 is diagonal.
In this representation, only the first few channels are needed to
accurately calculate the subbarrier fusion cross section. Therefore,
the basis is usually truncated to a few states with the lowest energies.
If the space is truncated to a single state and the derivatives of D.-1
are dropped in the kinetic energy operator, we obtain the well-known
adiabatic approximation. This limit is employed in the subbarrier
fusion calculations of Tanimura et al. (Tan.85). The advantage of using
the adiabatic representation can be seen in a numerical example
demonstrated in Table 3.1, comparing the S-wave fusion cross sections-
for s°Ti + 9°Zr under various truncations. The first 5 states in the
entrance channel representation define the space. One would expect the
results to be independent of the representation if the full space is
used. From the last row of the Table, we see that this is not quite the
case; the definition of an incoming-wave boundary condition depends
somewhat on the representation. The adiabatic representation shows that

a 2-state truncation has converged, although a 1-state truncation is

Table 3.1. Calculated s°Ti+9°Zr S-wave fusion cross sections at a c. m.

energy of 100 MeV as a function of number of channels in the model space

in different representations with parameters given in chapter 6

 

 

Representation Adiabatic
No. of cnannels Entrance Adiabatic Approximation
1 0.03 ub 18.“ ub “6.7 ub
2 0.98 ub 6.65 ub
3 3.115 ub 6.“5 pl)
5 5.61 1.10 6.51 ub

 

3O

quite poor. The adiabatic approximation, which is a 1 state truncation
with the entrance channel kinetic operator, works poorly in this case.
In the entrance channel representation, several states are needed. If a
factor of two is tolerable in the accuracy, the space can be truncated
to three channels. This is consistent with the findings of Takigawa, et

a1. (Tak.85)

Sec.3.3 Technical Points

In this section, we discuss the effects of the Coulomb excitation
and angular momentum transfer between intrinsic degree of freedom and
the relative motion. The reason we discuss these effects is that the
amount of computation can be reduced a great deal if one can make some
approximations about these. We start first by studying Coulomb
excitation.

Let us begin with a qualitative discussion. First, the Coulomb
field reduces the enhancement due to nuclear multipole forces. The
reason is that the nuclear force is attractive, while the coulomb
interaction between positive charges (protons) is repulsive. In a
subbarrier reaction, the nuclear force tends to change the shapes of the
nuclei to have a stronger attraction. The Coulomb force, on the other
hand, changes the shapes to reduce the repulsion. In other words, two
fields prefer different shapes. Due to the saturation of nuclear
interaction, the Coulomb force becomes more important as system becomes
heavier. Roughly speaking, the strength of the nuclear interaction is

proportional to A, the number of nucleons involved. While the Coulomb

31

strength goes as «A’. As a matter of fact, the effects of the extra-
push to be discussed in chapter 6 is mainly due to relative importance
of the Coulomb force in heavy systems.

Multipole components of the Coulomb force introduce some
complications in numerical calculations. Unless one uses the adiabatic
approximation (Rho.80). one has to integrate the coupled equations to a
large separation because of the long-range of the Coulomb interaction.
And if there were no Coulomb multipole force, the subbarrier fusion
would be easily estimated in a multichannel WKB approximation.

Landowne (Lan.81) showed numerically that Coulomb multipole
interactions work against nuclear attractions in subbarrier fusion
reactions. In his model, in classically allowed region he assumed a
classical trajectory in the two-dimensional space (r,s) for the system.
The zero-point fluctuations of s were assumed to be the same as without
multipole Coulomb forces, but the center of s was determined by the
trajectory. Under the barrier, the frozen approximation was used to
calculate the penetrability. Essentially the fusion cross section is
given by eq.(2.“) with s in the integrand replaced by s-so, so being the
value of the variable of the intrinsic degree of freedom at the
classical turning point. so is nonzero due to the Coulomb excitation.

In Fig.3.1, the results in different models are compared. The ZPM
without the inclusion of the Coulomb excitation (dash-dotted curve)
enhances the subbarrier fusion substantially. The ZPM with the Coulomb
excitation included (solid line) reduces the enhancement. The net
enhancement is almost negligible. Therefore in heavy systems, Coulomb

excitation is very important. However, in lighter systems, e.g. 0+0

32

 

103

I 1 IIIIIEI
q

.1

d

—

q

q

.

-1

d

u

q

q

—

uni

q

q

.1

l I lllll

1

   

- —-.=

E ' ,/ a

Z . .47 A

t . ° .I/ i

1 _ / /IKB __

:23» :1() E_- , I I’ g

E I I. I 1

E " *1 ' 1
\—I ’ IFII

b 100 g— 1 ""

1 .. ’O I

10— 5" I I. 1..

E 1 , —diznensional E

I 1' ' T.

 

 

 

10—2 LIA—LIA LTJ ILLLll LLL

90 95 100 105 1 10
Bonnie")

Fig.3.l. Effect of both target and projetile zero-point motion

and average dynamical deformation (Coulomb excitation) on the
fusion cross section for $8Ni+5°Ni+llsBa. The solid circles give
the experimental data of Becherman et al.(Bec.80), the dashed
curve gives the result for a one-dimensional barrier corresponding
to spherical nuclei, and the solid curv gives final result for a
two-dimensional potential-energy surface, and the dot-dashed curve
gives the result of ZPM without dynamical deformation.

33

systems which will be studied in chapter 5 one can ignore Coulomb
excitation.

Now we discuss the effect due to angular momentum transfer. In
general, there is an angular momentum transfer between intrinsic degrees
of freedom and the relative motion when one of or both of the nuclei are
excited during the collision in a fusion reaction. For a total angular
momentum J, there are several different orbital angular momenta
possible. The radial wave function of the relative motion in a channel

therefore can be labeled as Rn(r)J 2 s , where n denotes the nTH excited
I U 0

fl

intrinsic state(s), t the orbital angular momentum, 3n the intrinsic

spin coupled with A to J. For a fixed n and J, R(r) are different for

2 +1
different 2's. The differences come from the term gfi£££;;- in the

Hamiltonian which governs the radial motion. However, in heavy ion

2
reactions, the %M is a small number. For example, for s°Ti+°°Zr, the

2
53:1 at the Coulomb barrier is only about 6 KeV. So the differences in

h2i(2+1)
2 M r1

Hamiltonians for a definite J and n are approximately same. In

for 2's coupled with s to J are negligible. Thus the

subbarrier fusion reactions, the different 2's coupled with s to J don't
have any significance as far as the final product (compound nucleus) is
concerned. One would like to reduce the number of channels by not
distinguishing different 2's and the orientations of intrinsic spin for
same n and J, i.e., label a channel by n and J only. If this is
possible, the number of channels will be greatly reduced. For instance,
for s-2 and J22, the original 25 channels is reduced to a single

channel.

3“

We follow Takigawa (Tak.85) to show that it is possible. We
consider a system of two even-even nuclei, where either the projectile
or the target has one- and two-phonon states of a vibrational mode of
excitation with multipolarity A. We denote the wave function of the
relative motion in the entrance channel as x3(r), J being the initial
angular momentum. We distinguish the one-phonon channels by the angular
momentum A of the relative motion, and the two-phonon channels by the
total intrinsic angular momentum A in addition. Accordingly, we denote
the wave function of the relative motion in a one-phonon channel and in
two-phonon channel by xi(r) and XEA(r), respectively. The coupled

channel equations become

 

 

1112a2 112.1 1
(‘2M EFT + U(r) + ~2fi£%;-l ‘ E) x3(r) + f(r) i N£x£(r) s 0 (3.21)
a: d2 112.1 J1
(~54— (Tr—2 + U(r) + "2'M‘(‘F'+=")‘ - 13-11...) Xi”) + f(r) N2x3(r)
- 9.
+ ./2 f(r) XMLAXCAW) :- o (3.22)
L,A
and
112 (12 112.1 .11 - 11.
(~37 3;, + um + Wig—’— - E — 2m.) XII“ + /2 f(1")31'MI.IIXII.(r)
‘0 9 (3023)
where N1: /(2A+1)/“n /(22+1)/(2J+1) (3.2“)
and
MEA = (-1)J“L‘*/722+1>(2A+1)(2i+1)/un <AlOO|LO> {ifii}. (3.25)

35

Here, ho is the phonon energy and f(r) is the coupling form factor. The
couplings are assumed to be linear in the coordinate of the vibrational

mode of excitation. Also, the change in the centrifugal potential has

been ignored .

Now let

.. 1 ‘

x‘(r) = R % Nixi(r) _ (3.26)
'2 (r) = 1— z N MI 2 (r) (3 27)
x N2 1 LA XLA '

and use the sum rule for three-j and ClebschsGorden coefficients. Then

we have
0 1 o E o 0 7°
{D(r) + F(r) 1 o «5 - o E‘Hw o } 11 - o. (3.28)
o «2 o o o E-2Mm i2

_ h’ d2 h’J(J+1)
where D(r) - 2M EF7 + U(r) + ZM—F7———

and F(r) s /(2A+1)/“n f(r)

We now see that the number of channels has been reduced to 3, from

13 for i=2 and J22, for example.

Chapter “ Validity of ZPM and Simple Approximations

In ZPM the intrinsic excitation energy is assumed to be zero. The
two-dimensional equation in ZPM can easily be solved numerically. But
in general, ZPM tends to overestimate the subbarrier fusion cross
section. Because the barrier penetrability is very sensitive to the
energy, one would expect ZPM to overestimates subbarrier fusion cross
section considerably. In sec.“.1, we will study the validity of ZPM
through a numerical example using the coupled-channel method described
in chapter 3. Approximations used to combine two independent intrinsic
excitation modes into a single one are discussed in sec.“.2.
Factorization of the subbarrier fusion enhancement factors due to each

individual intrinsic excitation mode is studied in sec.“.3.

“.1 Validity of ZPM

We take the reaction 160+“’°Sm for head on collisions (i=0) as an
example to see the effects of finite excitations.

The fusion probability is shown in Fig.“.1 as function of the c. m.
energy. The calculations are carried out using the direct matrix
inversion method described in the previous chapter. The hm-O and hm-w
limits are shown together with the results for Mw=1 MeV and 10 MeV. The
value of a was o.27fm, which is close to the value obtained both for the
low-lying quadruploe and octupole states in 1”Sin. Since the excitation
energy hm for these states are 0.55 MeV and 1.16 Mev, respectively, we
can conclude that the dynamical effect due to the finite value of hm

will not lead to a major reduction of the subbarrier fusion cross

36

37

 

10-I P-

 

hw= 10 MeV

 

 

 

1.
0.3
00
IO'2 1- ._.
“3'3 1 1, 1 L I, 1 1 1 1
55 60 65

EC..M. (MCV)

Fig.4.l. Fusion probabilities for the reaction150+“”Sm are
shown as function of the c.m. energy. The results are obtained
from coupled-channel calculations, for various values of the
one-phonon excitation energy hm. The standard deviation of the
zero-point motion amplitude is 0 = 0.27 fm.

38

sections shown in sec.2.3. We see that ZPM is a good approximation when

hm - 1 MeV in this example.

“.2 Two Independent Degenerate Harmonic Oscillators

In some fusion reactions, more than one intrinsic mode may be
coupled to the relative motion. In such cases, the collective
coordinates r and s will not be enough to describe the system. Thus the
Schrddinger equation has more than two variables, and is numerically
very difficult to solve. If one expands the wave function in all
intrinsic eigenfunctions, one would find a multidimensional matrix
differential equation.

In Ref. (Wu.86) two independent degenerate harmonic vibrations have
been combined into one to calculate the fusion cross section. For two

modes with same hm, the o for the combined mode is given by

0 = J of + 0:. (“-1)

where 01 and 02 are the r. m. s. amplitudes for each mode,
respectively. We discuss this approximation in this section.

The energy spectrum of the combined mode is identical to the
original two. But transitions between states are different. First, we
show that transition matrix elements from the ground state to excited
states are same. The probability of the transition from the ground-

state to the nth excited state is pr0portional to

39

n n

 

I |<0.01<s.+s.>”|1,n-1>|= .2 (c‘?)2 it (mg) 03.1 022(n-j)23 2...,-
1'0 J'O J 23 2h J
' 2% (201+ 20%)n ' 2% (202)“, (“ 2)

and is the same as that in a single mode with a given by eq.(“.1). If
one neglect the contributions from higher multipoles in (s,+ 8,), the
combined mode gives same transition rate for the ground state as that
given by the two independent modes. However, the transitions between
excited states are different. For example, l<1|s|2>|2 in the combined

mode is 202 . 2(o§+o§), while the original two independent modes gives

-;—{[<0,1|(S,+82)IO.2>)2 + (<0,1|(s,+sz)|1.1>]2 +

(<1,0|(s,+s2)|2,0>)2 + (<1,0|(s,+s2)|1,1>)2 - 731' 2(af + oi), (“.3)

where the factor of % represents an averaging over the initial states.
In Fig.“.2, we show the numerical results for s°Ti+’°Zr, the

coupling parameters given in Table “.1 are arbitrary, and for the

Table “.1. Parameters used in obtaining Fig.“.2 and Fig.“.3.

 

System Vo a AR a1 02 o Mm, 111112 Mu)

 

s°Ti+9°2r -31.7Mev 0.63fm 0.35fm 0.2fm 0.2fm o.28rm 1.5MeV 1.5Mev 1.5Mev

 

“0

 

10—8 _lllLlllLlllllLlllLlllllllllllL'
94 96 98 100 102 104 106

Ec.u.(Mev)

Fig.4.2. S-wave fusion excitation function is shown as function
of c.m. energy. The dashed curve is the result obtained by com-
bining two vibrational modes into single one through eq.(4.1).
The solid line is the result obtained by solving exactly a three-
dimensional coupled equation. Due to numerical restrictions, the
exact result can not be obtained for enrgies lower than 98 MeV.
The parameters used are given in Table 4.1.

“1

purpose of comparison only. The combined mode gives the same results as
that given by a three-dimensional calculation. The three-dimensional
calculation in which two independent vibrations are simultaneously
coupled to the relative motion is very difficult and time consuming, and

can not be calculated for even lower energies.

“.3 Factorization of The Enhancement Factors

When several modes of intrinsic excitations are coupled to the
relative motion in a fusion reaction, the subbarrier fusion enhancement
over the conventional one-dimensional barrier penetration can be quite
large. In general, the calculation is very difficult, and it would be
nice if the total enhancement factor could be given by the product of
the enhancement factors obtained by coupling only one mode at a time.
If this were true, instead of doing a multidimensional penetration
calculation, one would only have to solve a number of two-dimensional
problems. Certainly this is not true for energies just below the
barrier, because the transmission probability can not exceed 1, while
the product of the enhancement factors may give a transmission
probability greater than 1. The same argument does not have to be true
for energies far below the barrier.

In Fig.“.3, we show the enhancement factors as functions of energy
in the center of mass frame. The total enhancement factors (solid
curve) is obtained by using eq.(“.1) to combine two identical vibrations
into a single one. The parameters used in the calculations are also

given in Table “.1. The 0's and hm's are arbitrary, but close to the

L12

 

IrrII—rrrrrTFIIFTIIIFTIIIITII

I
p
I
p
p
n

l 1 11111

  
 
    
     
 

/

I I IIIIIII
1 1 111111]

1 I 11111]

U(mb)
3.3.

l IIIIIIII

/

l I l ITIIII

.1
.11
.1
cut
at
.1

 

 

 

1.0—1 lLlllllll1111|1llllLLLlLlllfl
94 96 98 100 102 104 106

Ec.u.(MeV)

Fig.4.3. Fusion enhancement factors are shown as functions of
c.m. energy.fThe solid line is the result obtained by combining
two vibrational modes into one through eq.(4.1). The dashed curve
is the result obtained by virtue of factorization rule. The para;
meters used are listed in Table “.1.

“3

actual values. They were assigned for the purpose of discussion only.
The product of the two enhancement factors (taken to be same here) is
persistently larger than the calculated total enhancement factor.
Therefore one can use the factorization rule only to estimate the upper

limit of the enhancement.

Chapter 5 Subbarrier Fusion in Light Nuclei

In this and next two chapters, we study subbarrier fusion in
various systems. The systems treated in this chapter are 160+”,"0.
Since, in general, a realistic coupled channel calculation is
numerically difficult, it is important to find simpler cases which are
more amenable to a complete theoretical treatment. The study comparing
fusion cross sections of 180 and 160 on 160 targets provides a nice case
that can be treated without much ambiguity in a model. Therefore, it
provides a nice test for the model.

For “O+“0 system, the internal degrees of freedom are relatively
unimportant because of the high excitation energies of the inelastic
channels and particle transfer channels. The nucleus 1"0 on the other
hand has a 2+ state at 1.98 Mev excitation with a transition strength
16.6 of Weisskopf units from the ground state.

The experiment findings (Tho.85) is that the 180 fusion cross
section is enhanced over 160 in the subbarrier region. We show this in
Fig.5.2 with the cross section ratio plotted as a function of the c. m.
energy. The enhancement is small compared to that found in heavier
systems, but there is certainly an effect present. In ref. (Tho.85), it
is suggested that the difference is mainly due to the changes in the
potential barrier going from one system to another. Therefore, as a
first step in the analysis, we will examine the expected effect on the
potential by adding two neutrons to 160. We also expect a significant
effect from the low-lying 2+ state, which may be roughly estimated with

the frozen approximation mentioned in chapter 2. According to eq.(2“)

4“

“5

 

10°

cr(1nb)

10"1

10“2

 

 

10-3 _
6 8 10 12

Ec.u.(MeV)

Fig.5.1 The measured fusion cross section for 180+160 and
160+‘50, from ref. (Tho.85) is shown by the experimental points.
The potential model fit to the l60+160 is shown as the solid
line, and the coupled-channel model with rescaled potential is
shown by the dashed line.

“6

 

 

 

 

 

 

 

I I I I I I
6.0 8.0 10.0 12.0
Earn. (MeV)

 

Fig. 5.2. Enhancement factors extracted by log-linear
interpolation of the data from ref. (Tho.85). The.dashéd.
line is the prediction of the rescaled potential, eq.(3-2)--
The dotted line is the prediction of the potential eq.(5.3)-
The solid line is the prediction of the coupled-channel
model with the rescaled potential.

“7

in ref. (Esb.83), the enhancement in the subbarrier fusion may be
estimated as
“l 2nZ Z e2 )2]

P’Iztji"L—LT" O

woRb (5.1)

where R is the position of the Coulomb barrier and hmo - The

2:!
b Mr2 r-Rb'
constants in the above expression may be estimated as follows for the
‘°0+“0 system. The deformation length associated with the 0+- 2+
transition is determined from the transition strength by eq.(2.7). This
yields a value a . 0.33 fm. From the potential model fit to the barrier
described below, we find a barrier height of 9.7 MeV at a radius Rb- 8.9
fm. The thickness of the barrier at an energy of 6.75 Mev is 6 fm,
which corresponds a parabolic barrier having an oscillator frequency of
hm - 1.8 MeV. The predicted enhancement with these parameters is 2.5.
indicating that the excited state in 180 may significantly effect
subbarrier fusion.

The detailed calculations employ the coupled channel technique
described in chapter 3 to calculate the fusion cross section. The
potential in each channel is purely real, and an incoming wave boundary
condition is imposed at a small radius, which we take to be “.1 fm. The
coupled channel equations are solved by the direct matrix-inversion
method, which is much more reliable than iterative techniques for
energies below the barrier. As in chapter 3 we use the internuclear
potential of eq.(3.2). dropping the quadrupole Coulomb field. The two

nuclear radii R1 and R2 are determined by eq.(3.3). We found that the

“8

potential fits the fusion data of 160+“0 quite well with the parameter
set

Vo - -39.8 Mev/fm, AR - 0.“5 fm, and a - 1.65 fm. Since the potential
is of the proximity form, we can extrapolate to an 1"0+“50 potential
changing one of the radii according to eq.(3.3). The rescaled potential
is wider and deeper than the original 160 potential, giving more
subbarrier fusion. However, the enhancement is rather small, as may
seen by the dashed line in Fig.5.2. Of course, the proximity scaling
may not be a reliable way to extrapolate the influence of the valence
neutrons on the potential, so we also examined their effect in a folding

model. Namely, we determined the ‘°0+“0 potential according to

V(‘°0+"0) = V(“O+“O) + f p(r') U(r-r') d3r'. (5.2)

Here U(r) is a Woods-Saxon potential with parameters given in
ref. (Boh.69), and p is the valence neutron density determined from the
d—wave bound state in that Woods-Saxon potential. This method also
gives a small enhancement, which is shown by the dotted line in Fig.5.2.
Thus, we are inclined to reject the potential barrier variation as the
main mechanism for producing the 180 enhancement.

The potential matrix for the coupled channel calculation is
constructed by a macroscopic model of the excitation. We assume the
2+state to be a harmonic vibration of the surface deformation coordinate

s, and evaluate the matrix elements according to eq.(3.8), which yields,

<0|Vn(r-s)|0> + vC <0|Vn(r-s)|1>

(5.3)
<0|Vn(r-s)|1> <1|Vn(r-s)|1> + vc + 1.98

“9

In this equation, we have neglected the angular momentum transfer, an
effect treating the combined system in a rotating frame approximation.
The integral over 3 is easily performed for the error function, giving

<0|Vn(r-s)|0> - vn(r)|a 1/2

!‘(32+202)
(5.“)

a
<1|Vn(r-s)|0> - a afi<°IVn(r‘3)I°>°

The wave function for each entrance channel angular momentum is found by
integrating the coupled equations out from the inner boundary. For the
energies we study here, the sum over angular momentum can be truncated
at 2:12, with an accuracy of 1%. The predicted enhancement including
the potential and coupled channel effects is shown as the solid curve in
Fig.5.2. As expected from the frozen approximation, the inclusion of
channel coupling substantially enhances the cross section in the
subbarrier region. However at energies above the barrier, the theory
predicts the enhancement to persist, while experimentally equal cross
section are found. We will return to this later.

The coupled-channel model of eq.(5.3) is oversimplified in several
aspects, and the assumptions need to be examined and justified. First
the higher excitations in ‘°0 might increase the cross section even
more. In fact, such couplings are implicit in the harmonic model used
to derive the sudden approximation, eq.(5.1). However, the B(E2) from
the first 2+ state to the triplet states near “ MeV is lower than that
from the ground-state to the first 2+ state, while the harmonic model
require a larger B(E2). We find that including these channels in a

five-dimensional matrix has a negligible effect on the fusion cross

50

section. Another oversimplification in the model is the neglect of the
Coulomb excitation of the 2+ state. It is shown in chapter 3 that for
medium heavy systems the Coulomb part of the Coupling interferes
destructively with the nuclear part and reduces the subbarrier
enhancement. Indeed, we find that by including Coulomb excitation
according to eq.(3.2). the cross section is reduced by as much as 7% at
the lowest energy, becoming less important as the energy goes up. As
mentioned above, the coupled-channel model treats the system in a
rotating-frame approximation. The physical justification for this is
that we expect axially symmetric deformations to provide the dominant
path for fusion. The angular momentum transfer assumption was tested
with the code PTOLEMY, used to compare the cross sections treating the
excited state as L=2 or L-O. The cross section at subbarrier energies
comes out larger with angular momentum transfer, by an amount comparable
to the Coulomb correction. Since both effects are small and opposite in
sign, we are justified in neglecting them in this context. Nucleon
transfer effects are also not included in the formalism. One might
expect that the two neutron transfer, which has a zero Q-value, plays a
dominant role. However, for such processes, contributions in the even
and odd partial waves have opposite signs and cancel in the Q - 0 limit.
Above the barrier, the cross section is reduced by the angular
momentum transfer, but not enough to equalize the cross sections of the
two reactions. The experimental observation of equal cross sections is
inexplicable. A channel coupling under the barrier will always enhance
the cross section. For the low-partial waves that go over the barrier,
it is hard to see how the extra couplings in the ”0+“0 system could

reduce the transmission factor. It appears to be a general phenomenon

51

found also for other systems that the fusion cross sections above the
barrier come out too large when the model is fit to subbarrier fusion

data. We will see some of examples of this later.

Chapter 6 Subbarrier Fusion in Medium Heavy Nuclei

For heavy systems, one has to go beyond the potential barrier model
to reproduce the data. This was convincingly demonstrated in work by
Balantekin et al. (Bal.83), in which an effective one-dimensional
potential was extracted directly from the fusion data. Fig.6.1 shows
the extracted potential for various systems, from light to medium heavy.
For the ’“Ge+‘“Ni system one finds a barrier whose inner edge has an
unphysical negative slope. ‘Thus, one cannot fit with the data by
manipulating the parameters or forms of one-dimensional potentials. In
this chapter, a quantitative treatment of subbarrier fusion extending to
heavy nuclei is presented. The systems studied are Ti, Br + Zr, Nb and
Zr+Zr. In the following section we discuss the parameters of the

calculations, and the results are presented in sec.6.2.

6.1 Details of The Calculations

The formulation of the surface-coupling model follows that given in
sec.3.1. The internal degrees of freedom are described by a harmonic-
oscillator Hamiltonian with an internal coordinate s. The coupling to
the radial coordinate is through the ion-ion potential, and is given by
eq.(3.2). The parameters V0, R1, and a are given by eq.(3.3). For
the systems Ti+Zr and Ti+Nb, AR was adjusted to reproduce the
experimental fusion cross sections in the barrier region for s°Ti+9°Zr
and “‘Ti+’3Nb using the one-dimensional model, respectively. For Br+Zr
systems, we fit the subbarrier fusion data of °‘Br+9°Zr to determine AR

and o in "Br for this isotopic system. The parameter AR is found to be

52

53

 

   

v-BIMeV)

 

 

V-EBIMeV)

 

 

 

1 11
I4

 

r (1m)

Fig.6.l. Effective one-dimensional potential barriers for 13C+13C,
1"Ni-”N, 1304-160, “°Ca+9°Ca, °“Ni+5“Ni, and 6“Ni+7“Ge. The outer
turning point is idetermined from the KNS potential readjusted as
described in ref. (Ba1.83) to fit the peak positions. The thickness

is inverted from fusion cross sections and the shaded region indicates
the error. The short dashed line denotes the point Coulomb potential,
the long dashed curves in the upper figures denote the unmodified KNS
potential, and for ease of presentation the energy is plotted relative
to the barrier height.

5H

0.35 fm for Ti+Zr systems and 0.140 fm for all other systems. It should
be mentioned that these parameters are far from unique as far as fusion
is concerned. At most two parameters are determined by the above-
barrier cross section, namely the barrier height and the barrier
position. Other sets of V., a and AR give the same fit and the same
coupled channel results below the barrier.

The intrinsic oscillator Hamiltonian H1n(s) is specified by the
frequency hm and by a, the r.m.s. amplitude of the coordinate in the
ground state. The coupled-channel equation for the system is therefore
given by eq.(3.6). The internal Hamiltonian H1n(s) should be chosen to
reproduce the energy and transition strength of the lowest excitations
of the nuclei. For a single nucleus, the relation of o to the B(EL) of

a transition is given by eq.(2.7):

R

‘ B(EA))%
A Z(A+3)

._.... , (2.7)
Bw(EA)

a [(2i+1)

The excitation of both nuclei may be treated with a single coordinate 3,
if the energies are degenerate; in that case the effective amplitude of

the deformation is related to the individual amplitudes by

o= Jc¥+og (”.1)

Eq.(u.1) will also be applied in cases where the excitation
energies are not degenerate and both excitations are important. The
lower of two energies will be used if the difference is small.

Otherwise we take the average of two energies as oscillator energy. As

55

Table 6.1. Parameters of surface vibrations deduced from the measurement
of low lying state. For odd nuclei they are averages of those of two
neighboring even-even nuclei. The a's (standard deviations) are
calculated according to eq.(2.7).

 

 

Nucleus Energy of the First Excited State (MeV) s(fm)
9°Zr 2.18 0.16
’“Zr 0.91 0.16
saTi 1.55 0.25
“‘Ti 0.889 0.u9
81Br 0.7 0.36
93Nb 0.9 0.22

 

Table 6.2. Parameters of surface vibrations used in the fusion cross-
section calculations of Figs.6.2,3,h,5,6. The parameters in parenthesis
were obtained by fitting to the data. The rest were taken or calculated
from Table 6.1

 

 

 

 

 

Reaction hm (Mev) 0(fm)
5°Ti+9°Zr 1.55 o.3-/o.25’+0.162
“‘Ti+’°Zr 0.918 (0.39)

“‘Ti+°3Nb 0.9 o.u5-/o.222+(o.39)2
s°T1+°3Nb 1.2: % (o.9+1.5) o.32-=/o.222+o.252
°‘Br+’°Zr . (0.9) (0.32)

 

°‘Br+’“Zr 0.9 0.36-/(0.32)2+0.16I

 

56

we shall see this procedure gives the parameters for H1n that describe
the reaction s°Ti+°°Zr very well below the barrier. For the softer
system “‘Ti+’°Zr, the method gives too much subbarrier fusion and we
treat a as an adjustable parameter.

For an odd nuclei the hm and a of the collective excitation are
hard to extract from the data on low-lying states. They are chosen to
fit the fusion data for one system and kept same for other systems. We
expect the values of these parameters lie between those of two
neighboring even-even nuclei. For 93Nb, the averaged parameters a and
hm describe subbarrier fusion excitations of s°T1+”Nb and “‘Ti+’3Nb
very well.

The coupled-channel calculations were carried out using the direct
matrix-inversion method. The inner boundary was chosen at the point
where the potential in the entrance channel is the minimum. The results
do not sensitively depend on where the boundary is. The internal
excitation parameters extracted from experimental data are listed in

Table 6.1. Table 6.2 shows the coupling parameters used in the

calculations.

6.2 Results and Discussions

The fusion excitation function for 5°Ti+°°Zr calculated from the
model is compared with experiment (Ste.85) in Fig.6.2. As one can see,
the fusion is underestimated by the conventional barrier penetration
model (bold continuous curve), a factor of 60 too low at Ec m-100.6 MeV,

although they were fit in the barrier region. The fusion is enhanced by

 

57

 

 

 

I I F I I I r I
5°Ti +9°zr .
C
102*- +
0,. .
/
Io' - "’O ~
A ,I/ _
:13 1’ /
E x .I' ! DATA
‘73 l/ — WKB +
to- ,' . 2 —
b I0o -,’ ,I - --- 2 CHANNEL 0+3. 4
I
.I -— 6 CHANNEL
, I! COUPLING
IO" _/ ZPM -
'2' l 1 L L 1

 

IO

l ' L
|08 HZ “6
Earn} MCV)

IOO I04

Fig.6.2. Fusion cross sections for 5°Ti+9°Zr. The bold continuous

curve is the results of the one—dimensional barrier penetration cal—
culations. The two-channel coupling result is shown by dash-dotted
curve. The thin continuous curve is the result obtained by solving

6 channel coupled equation. The 6 channels correspond to the first 6
excited vibrational states assumed in 5°Ti with a energy spacing of
1.55 MeV. The dashed curve is the prediction from ZPM. All calculations
include the quadrupole Coulomb excitation and ignore the angular mo-
mentum transfer.

58

coupling the excitation of the first excited state to the relative
motion (dash-dotted curve). But the fusion is still underestimated by a
factor of 3 at the lowest energy. One is naturally led to expect that
the contributions from higher intrinsic excitations would account for
the difference. But the parameters of couplings of the higher
excitations to the relative motion is very hard to extract from the
spectrum. However, if one approximate these excitations as pure
harmonic vibrations, one would get the full-channel coupling results
shown by the thin continuous curve in Fig.6.2. The agreement below the
barrier is very good in this approximation. The above-barrier fusion is
slightly overpredicted by our model, though the (Z/A)eff of the system
does not exceed the extra-push threshold (Nix.77). The prediction from
ZPM is also shown. Although the orders of magnitude are right, the ZPM
overpredicts the fusion of the s°Ti+9°Zr for energies far below the
Coulomb barrier. Alternatively if the ZPM is fit to the subbarrier
data, then the potential barrier is too low at higher energies (Ste.85).
Rhoades-Brown et al. (Rho.8fl) calculated subbarrier fusion cross
sections for various medium heavy systems, using the coupled channel
version of the PTOLEMY code, which treats angular momentum algebra
exactly. A few low-lying states were coupled to the relative motion
with parameters extracted from the low-lying excitation measurements.
It was found that the subbarrier fusion was greatly enhanced compared to
the one-dimensional penetration approximation, but was still
underestimated for energies far below the Coulomb barrier, by orders of
magnitude in some systems. From our results for s°Ti+9°Zr, we feel that

if higher intrinsic couplings are treated correctly, one can reproduce

59

 

 

 

 

 

P I I I I If I
I 4 . ‘
.. 6T'+902r j
102- -
E :
: 1
3 d
E .
3'0 F 1
H- . a
b : I
' , I Data ‘
— Coupled Channel .
0..
'0 : --- ZPM j
: i
.0" I L l I l .
98 I02 106 no 114 "8

Ec_m.(MeV)

Fig.6.3. Same plot as in Fig.6.2, but for “6Ti+9°Zr. The
parameters Vo, Ad, AR are kept same as those for 5°Ti+9°Zr.
0.89 MeV, the energy of the first excited state in “6T1,

is used as the energy quanta for the vibrational mode. The
a is somehow 0.39 fm, different from that quoted from the
B(E2) measurement.

 

60

 

3
'° I I I I

r1IlIr
A
0’
:!
+
to
01
Z
0'

l 1111

I
\
1

I

 
  

2

l0: --1
q

" d

"' q

l" -I

I- a:

l' I

l

l

 

 

 

3
E .
.2 '° ‘:
b I
1
I0° _ 1
E I
u I DOIO j
L -—. Coupled Channel :
- ' " ' 2 PM «4
'0-1 L L l l
I00 I04 108 "2 II6 l20

Ec.m.( MeV)

Fig.6.4. Fusion cross sections for “5Ti+93Nb.

The hm and o for 93Nb is taken as the average

of those for 92Zr and 9"no. The excitations in
both “6T1 and 93Nb are combined into one mode

with fiw = fiw(“6Ti) = fiw(93Nb) and the 0 equals
/FEYK“5T1)+OZijfib). The number of channels coupled
is 5.

 

 

    
      

 

 

 

61
'03 : F I I I q
I- u
: 5°71 + 93NI: , :
I- .q .
I- -l
Ioz :- 1
l- Cl
l- 1
- 4
"3' :- .1
3 C 2
g " cl
:3 b ‘
W- I- u
b
«3° '5
I
I Data .
.. — Coupled Channel
K) ‘1
"- ZPM 5
cl
'0’2 _ i L l L
I00 I04 108 II2 116 I20

EC.m.(Me\/)

Fig.6.5. Fusion cross sections for soTi+93Nb. In the cou led
channel calculation, the fim is the average of those for °Ti
and 93Nb. The a is «62(SVI1)+02(’7Nb).

62

the data in the coupled-channel model.

In Fig.6.3 we compare the calculation for “‘Ti+°°Zr with the
data (Ste.85). Here a was reduced to achieve agreement with the data.
In other words, the subbarrier fusion would be over predicted if we take
the experimental value for a. This suggests that the excitations in
“‘Ti can not be approximated as pure harmonic vibrations. The ZPM is
more successful (closer to the coupled-channel result) for “‘Ti+’°Zr
because the excitation energy is lower, only 0.88 MeV. In Fig.6.“ and
Fig.6.5 we show the results for “‘Ti+”Nb and s°Ti+°3Nb. The hm and 0
for 93Nb were taken as the average of those for 92Zr and ’“Kr. Again
the agreement is good.

For heavier systems, fusion is inhibited at the configuration just
inside the Coulomb barrier (Nix.77, Swi.81). The threshold projectile-

target parameters for inhibition is given by (Bir.83)

K - (Zz/A)

thr- eff Chi": 35 {1‘1.78[(N,"Z,+N2-Zz)/(A1+A2)J2}.35.

The Ti+Zr, Nb systems have K ~25, well below the threshold value.

critical
The next fusion reactions we consider have values of K around the
threshold value in s-wave channel. The systems we consider are
°‘Br+’°,’“Zr, and 9°Zr+9° 9"Zr, measured by Beckerman et al. (Bec.8u).
In Fig.6.6 we plot the fusion cross section ratio of the alBr+"'Zr to
the alBr+’°Zr. The effects of 9°Zr excitation is assumed small compared
to that of 9"Zr, and we ignore it completely in the calculation. The 0

and hm for °‘Br is obtained by a subbarrier-fusion fit for °‘Br+9°Zr.

Then we investigate the isotopic effects of Zr. As can be seen, the

63

 

.aowumEHOM m:v«mmu.:o«umuoam>m mo aofimomm mmouo mzm mu mama Housmawumaxm mce .uNom+umqo

now umnu ou uNsm+um~o wow mcofimomm mmouo aoamam mam mo owumu mam no uon .c.o.wam

gazfiEdm
cm. on. 8.

 

l.

l
I

ll
lllllll
omohmnq n

(1206 "' 18m) ”ID/(12’s... Jale) ”'0

llIlI

 

 

 

_

I-
(3
CU

 

6N

coupled-channel calculations which include the excitation in 9"Zr (solid
line, reproduce the data reasonably well. While the results obtained by
treating both 9°Zr, ’“Zr as point particles with the rescaled potential
for the °‘Br+’“Zr system according to eq.(3.2) and eq.(3.3) are far away
from being in agreement with the data in the subbarrier region. The
above-barrier fusion is overpredicted by our coupled-channel
calculation.

In these heavy systems the extra-push energies are zero for s- I
waves, but not for higher partial waves. The maximum i-value with zero
extra-push energy is about 26 (Bir.83), and one should take this into
account in coupled-channel calculations. This was not done in our
calculation, because only first few partial waves contribute for
subbarrier fusion reactions.

Inside the Coulomb barrier, instead of fusing together, the system
may transfer a few nucleons and then fall apart. In this case a so-
called an "extra-extra-push" energy is needed for compound nucleus
formation (Bir.83). In the systems °‘Br+’°,’“Zr, the extra-extra-push
energies are not zero. This should affect the above-barrier fusion as
well as subbarrier fusion. In the following, we explain why we believe
that the relative subbarrier fusion can be investigated in our simple-
two~dimensional model, without treating these extra-push or extra-extra-
push energies explicitly. In the calculation, we impose an incoming
wave boundary condition at internuclear distance ro where the potential
is the minimum in the entrance channel. For subbarrier fusion this is a
reasonable approximation because as the system reaches ro it is unlikely
to "bounce" back, and go through another penetration. The situation is

quite different when the energy in the relative motion is greater than

65

the barrier height. The system energetically prefers to go to other
channels rather than to complete fusion, if the energy is greater than
the barrier height but not enough to overcome the extra-push (or extra-
extra-push) barrier. However, the isotopic difference in extra-push I
energies in Zr should be small. When we fit subbarrier fusion for
°‘Br+’°Zr, the extra-push effects are accounted for by adjusting the 0
and hm for alBr in subbarrier region. With such a a and hm for uBr the
subbarrier fusion difference between °‘Br+’°Zr and alBr+"‘Zr should
mainly be due to the isotopic difference in Zr. Because of the
complexity, the extra-push effects explicitly show up in above-barrier
energy region. These are qualitative arguments. In order to get a
better agreement with the data, one has to perform much more complicated
calculations in which many channels has to be included explicitly. This
is beyond the scope of this thesis.

We also calculated the fusion cross sections for the systems
9°Zr+9°Zr and 9"Zr+’°Zr. The parameters V0, a, and AR are fixed by
fitting the subbarrier fusion of 9°Zr+9°Zr. With these potential
parameters, however, the coupled channel calculations for ’“Zr+’°Zr
underestimate the subbarrier fusion by about two orders of magnitude at
the lowest energy measured. We believe that the particle transfers in
°“Zr+’°Zr reactions may contribute to the subbarrier fusion
significantly, and account for the differences between the coupled
channel calculations and the data (Bro.83).

We see that subbarrier fusion for systems below the extra-push
threshold can be quantitatively understood by a two-dimensional model.
For heavier systems that require an extra-push for fusion, the simple

two-dimensional model can reproduce the relative fusion cross section

66

for different isotopes. The absolute fusion cross section, however,

cannot be predicted by the model.

Chapter 7 Particle Transfers in Subbarrier Fusion

In this chapter, we study the effects of particle transfer on
subbarrier fusion. The systems considered here are soNi+5°Ni and
"Ni+‘“Ni. In general, a complete calculation is numerically very
difficult. But we make a simplified estimate, aiming to obtain results

which give the right order of magnitude for subbarrier fusion.

7.1 Introduction

When two nuclei approach each other in a collision, a few nucleons
may be transferred from one nucleus to another. This in general does
not contribute very much to subbarrier fusion because of the small
coupling amplitude. This is especially true for medium heavy or light
systems where a small neck is enough to guarantee complete fusion.
Particle transfers, however, may be important for systems which have
positive Q-values (Bro.83). Although the coupling strength for transfer
is small, the subbarrier fusion cross section may be enhanced because
the effective barrier in the transferred channel is lower than that in
the entrance channel. For energies well below the Coulomb barrier, the
small portion of the incoming flux in the transferred channel has a much
larger penetration probability, leading to an enhancement in the fusion
cross section. For some systems, there may be a large number of
transfer channels coupled to the relative motion, and although the
contribution due to each channel coupling is small, the enhancement due

to all channel couplings may be significant (Rho.85).

67

68

 

103

EIIFTIIIrIITIITIIIIT:
C ’1’5

. ar 1".
102 2’1 ” '2
2’ ’1 I
/II 2

{/9

q

II I l1llllll 1 I'llllll
\-
'~.\<-o-
. ..
lb
1 l lllllll

10°

0' (mb)

Ch
3?”
4*

CD
,2:

~ .

l illllfll

0
§ .
o

10""1

.J

I 0
pl 5
:1()-:3 .i I I. IAII:I_ I I I I I I,iLl I I, I I

90 95 100 105 110

Ec.u.(MeV)

r I I‘ll
l l l llllll

 

 

 

Fig.7.1. Calculated fusion cross sections for 5°Ni+‘“Ni compared

to the data of ref.(Bec.80). This plot is the copy of Fig.2(b) in
ref.(Bro.83). The dotted curve shows the limit of no-coupling, i.e.
one-dimensional barrier penetration. The dash-dotted curve is the
result obtained by including intrinsic excitations. The dashed
curve includes additional (1 MeV) strength at Q = +5 Mev, which was
attributed to transfer reaction channels.

69

In comparing soNi+5°Ni fusion data with that of °“Ni+"Ni, Broglia
et al. (Bro.83) found that in addition to the known low-lying intrinsic
excitations, an extra channel coupling with a positive Q-value was
needed to reproduce the ‘“Ni+’°Ni data. They found that a
phenomenological constant coupling of 1 MeV strength and a Q-value of 5
MeV was required. With this additional channel coupling, the relative
enhancement in the ‘“Ni+’°Ni fusion cross section was explained, and is
shown in Fig.7. 1 .

From Fig.7.1, one see that with all known low-lying excitations
coupled (dashed curve), the fusion is still underestimated at energies
far below the Coulomb barrier. But good agreement is obtained with the
additional coupling (continuous curve). This additional channel
coupling was interpreted as being due to two-neutron transfer (from ‘“Ni
to 58N1) for which the Q-value is positive.

It would be nice if the additional channel coupling can be
understood microscopically. In the following sections, we will study a
microscopic model for the two-neutron transfer form factor in the

58Ni+“’Ni system and its effects on subbarrier fusion.
7.2 Form Factors for Neutron Transfer

In our microscopic treatment, the neutron-transfer form factor is

calculated as

. . Ari A'+i A‘i+1 A'+i‘l
I I '9 '3 " , , .1
f1(JlJZJl J2 r) < “’1‘ WJZIVI $11 $32) (7 )

70

where lAmj> is the many-particle wave function for a Ni-isotope with a
spin J, and V is the single-particle potential field in the nucleus to
which the neutron is transferred. Also, i-1 and 1-2 label first- and
second-neutron transfer, respectively. The ground states of "Ni, ‘°Ni,
““Ni and ‘2 Ni have Jfl-O+ and the trivial indices indicating these
nuclei will be suppressed. Since V is a one-body operator, the form

factor can be written in second quantization form as

. 63 6'0 59 1. so
f1(j,,J,;r) ‘2;k <k|v|2> < wjilail w) < wjz|ak| w>
O (702)
f ( ) Z I I 62 I 63 so I 59
' ,’ ;r - < V > < a . > < a . >
2 J: J. p,q q p I pl le wl ql v32
where lk), ISL), |q> and |p> represent single-particle states in "Ni,
““Ni, 59Ni and 63Ni, respectively.
For 59Ni+"3Ni, there are 3 low-lying levels in each nucleus, j-(%-.
3‘, g-) for a total of 12 states (including m-degeneracy), and,
therefore, there are 1AA different internal states for the 59Ni+‘3Ni
system. Conservation of the angular momentum along the axis connecting
the two nuclei reduces this quantity to 28. However, this is still too
large because of the numerical difficulties in carrying out the
calculation. We shall make the approximation that these internal states
are degenerate, and can be treated as one state. From secondrorder
perturbation theory, the form-factor for one-neutron transfer is

gi ven by

 

 

F(r)=/ z <f|VII><IIVIi> s “d z
I

1in: f1 (J10323r)f2(~j1hj2;r) (7'3)

71

for both first- and second-neutron transfer. This is a good
approximation only when the Q-value for the first-neutron transfer is
large and negative (that is, first-order perturbation gives little
contribution). This, however, is not the case. Therefore, the form
factor F,(r) for first‘neutron transfer is the square root of the sum of

the squares of each individual form factors, that is,

 

 

Fl(r‘)’v/ 2 (II—Vl—ijz ‘ /J. 2:] f1(jl’J2;r)f (70“)
I 1’ 2
as one would expect from the first-order perturbation theory. While the

form factor for second-neutron transfer is
F,(r)=F2(r)/F,(r). (7.5)

The Q-values for one- and two-neutron transfer are taken from
binding energy data (Bro.83) and are -0.6 MeV and 3.9 MeV, respectively.
We assume that the neutrons are transferred one by one, i.e., no
simultaneous two neutron transfer. We will see later that this is

indeed a reasonable approximation.
7.3 Details of the Calculation

Both 58Ni and ‘“Ni are open-shell nuclei, and intrinsic collective
motions are important in subbarrier fusion. In order to reproduce the
fusion data, these couplings are needed along with the neutron transfer

coupling. However, since we are primarily interested in the effects due

72

to neutron transfer, we parameterize the intrinsic collective
excitations by fitting the fusion excitation function data in saNi+5°Ni.
The potential is again that of eq.(3.2). The strength V,, the barrier
shift AR, and the diffuseness a are parameters to be determined.

The intrinsic excitations are characterized by a harmonic-
oscillator spectrum in the variable 3, with a r. m. s. amplitude o and
excitation energy hm. We first fit the soNi+5°Ni fusion data by
adjusting these parameters. This is done using the coupled-channel
technique, where the number of channels is also a parameter in the fit.

The resulting parameters are shown in Table 7.1. This two-dimensional

Table 7.1 Parameters of the two-dimensional potential (eq.(3.2)) and the
surface harmonic-vibrational mode obtained by fitting the 56Ni+5°Ni

fusion data.

 

 

V° a No. of surface 0 (The r. m. s. amplitude hm
vibrational states of s in the ground state)
-36 Mev 0.7 fm 3 0.“ fm 3 MeV

 

potential is then used for the 58Ni+“'Ni system, where corrections due
to the larger size of ‘“Ni are applied. In doing this, we actually

assume that the intrinsic couplings are the same for both the 58Ni+5°Ni

73

and soNi+"’Ni systems. We justify this assumption by comparing our
results with those obtained with the more detailed treatment of Broglia
et al. (Bro.83) (without neutron transfer). We find generally good
agreement between the two procedures for energies far below the Coulomb
barrier.

In our calculation, the potential fields which generate the single
particle orbits and cause the neutron transfers are assumed to have a
Woods-Saxon form. The parameters used are those given by Bohr and
Mottelson (Boh.69). The spectroscopic factors, used in eq.(7.2), were
calculated with the Oxford-Buenos Aires-MSU Shell-Model Code (Rae.85)
using the fp-shell interaction of van Hees (Van.81) and Koops (Koo.77).
The shell-model configuration space consisted of the 0f

7/2’ 193/2'

0f5/2 and the 1p”2 orbits. Due to computational restrictions, no

particle-hole excitations of the closed 0f,”2 orbit were allowed. In
addition, corrections to eq.(7.2) due to the non-orthogonality of single
particle states in different nuclei have been neglected since the
overlaps are small. The form factors are shown in Fig.7.2.

When particle transfer and internal excitation are coupled with the
relative motion, the subbarrier fusion cross-section is greatly
enhanced. This can be seen in Fig.7.3. The dot-dashed curve is the
result obtained by including both intrinsic excitation and the neutron-
transfer couplings (three-dimensional calculations), and is greater than
the result obtained with only the intrinsic-excitation couplings (dashed
line), which is, in turn, much greater than the result given by the one-
dimensional barrier penetration model (dotted line). There is, however,

still a gap between this calculation and the experimental data for

energies far below the Coulomb barrier.

74

 

B IIIIIIII[ITIIIITITIITIIIIIIW
I- -l

h‘ —— -

F(r)

     

- ' - ° - F(r—s(r))

144141

I

d

Form Factors(MeV)

 

 

~
0 llllllillllllllllLiEiLl[-111ng

B 9 10 11 12 13 14
r(fm)

Fig.7.2. Form factor F(r)(solid line) is shown as function
of the separation r. F(r-s(r))(dot-dashed curve) is the
form factor in the semiclassical model. Also shown (dashed
curve) is the largest single-particle transfer amplitude
between two fS/Z orbits in 6“Ni andsaNi, respectively.

- 75

 

 

 

 

EI I‘l I'[ I I I' I [II' r’I I7 r I I IS
102 .5— “71':
; :
A 101 5— —=
.0 5 E
E : :
b 100 3.7" “a
i 5
’ a
__ . I

10 1 .’I —§
I I

1()-:3 _II I II I I IEI I I_ I I l I II I I I,

 

90 95 100 105 1 10
Ec.u.(MeV)

Fig.2.3 Fusion excitation functions of the 6“Ni-+581“ system. Dotted
curve is the result of the one-dimensional barrier penetration model.
The dashed curve is the result obtained by including only intrinsic
excitations with parameters given in Table 7.1. The dot-dashed curve
is the result with both transfer and intrinsic excitation included.
The F.(r-s(r))'s are the form factors used in obtaining the reSult
that reproduces data (solid line).

 

76

7.“ The Interplay of Intrinsic Excitations and the Particle Transfers on

Subbarri er Fusi on

The particle-transfer form factor was evaluated using eq.(7.2) for
nuclei in their ground states. However, the nuclei may be deformed
during the collision, with a preferred deformation such that the long
axis coincides with the axis connecting the two nuclei. This effect
enhances the neutronrtransfer form factor because the nuclei are
effectively closer than they are in their ground state. We take this
into account by evaluating eq.(7.2) along a semiclassical trajectory.
First, turning off the neutron transfer, we perform a two-dimensional
penetration calculation using the potential defined in eq.(3.2) and the
parameters in Table 7.1. The semiclassical trajectory is defined as
that in which the incoming flux is maximized in the direction of the
surface-vibrational degree of freedom, parameterized by s, and is shown
in Fig.7.”. In a semiclassical approximation, we assume that the system
only goes through this trajectory, and therefore, the form factor F1(r)
is replaced by F1(r-s(r)), s(r) being evaluated on the trajectory. With
this enhanced form factor, we find general good agreement between
theory (solid curve in Fig.7.3) and experiment. We note, however, that

the experimental data are underestimated somewhat for low energies.
7.5 Simultaneous Two Neutron Transfer

So far, we have neglected the simultaneous two-neutron transfer

mediated by the residual interaction. We estimate this contribution by

77

 

 

 

.coumumuvam mcm
ca vumuaaxma ma asau mcwaouau may :u«:3 :« muOmomnmuu Hmowmmmaouamm any ma m>u=o
vmmmov may .~.m m~nmh.ca am>ww mummmammma sues >0: cm «a mwumam .a.o n ma «z:m+H2on
:owuommu mam wow wcaaasou umumamuu usozuaz masmmu HmaamzoIvmaaaoo mam ma :owuocau
m>m3 mes .amaa Am.uv mam a“ macaw mum ~_Am.uva_ zuwmamu Hmaom mo mnsomcoo .¢.~.wfim

 

 

 

 

 

 

 

 

 

 

 

78

assuming the residual interaction to have a 6-function dependence, whose
strength is determined from the pairing energy in soNi. This residual
interaction coupled with shell-model two-particle parentage coefficients
yields two-neutron transfer form factors which change the fusion cross
section by only a few percent.

The residual interaction in 58Ni may be much more complicated in
form than the 6-function pairing described above. However, we believe
that the above pairing is the leading part of the residual interaction.
The above procedure should, at least, give the correct order of

magnitude of the effects due to simultaneous two-neutron transfer.

7.6 Discussions

We find that the relative enhancement of the saNi+“‘Ni fusion cross
section over that of 5°Ni+5°Ni is explained in our simple model (see
Fig.7.3). However, the calculation should only be taken seriously to
within a factor of two or so, as there are a few uncertainties in the
model. First, the intrinsic excitations in ‘“Ni are not identical to.
those of 5°Ni, although at energies far below the Coulomb barrier, our
results (with intrinsic coupling only) agree with those obtained with
the more detailed analysis of Broglia et al. Second, the 28 one-neutron
transferred states were treated as one intermediate state. Third, the
replacement F(r) with F(r-s(r)) is a semiclassical approximation.
Finally, the neutron-transfer form factors were evaluated using eg.(7.2)
with shell-model spetroscopic factors. There is evidence (Bay.82),
however, that this procedure may in fact underestimate contributions due

to step-by-step two-neutron transfer. Since two-neutron transfer is

79

very important at low energy, this may explain the discrepancy between
theory and experiment at low energy (see Fig.7.3).

It should be remarked that the system “Ni+°"Ni is nearly
symmetric, with only a few transfer channels contributing towards
fusion. However, in order to get more comprehensive appreciation of the
effects due to particle transfer on subbarrier fusion, a systematic
study of asymmetric systems where many particle-transfer channels are
open is necessary.

After extensive experimental and theoretical studies over past a
few years, the effects due to intrinsic excitations on heavy-ion
subbarrier fusion are well understood qualitatively. Quantitative
agreements can now be achieved in many cases. For light and medium
systems, coupled channel methods give a satisfactory description of
subbarrier fusion. However, there are some ambiguities associated with
choice of the parameters. As is discussed by Rhoades~Brown et al.
(Rho.8u), these uncertainties maybe reduced by constraining the
parameters to reproduce other data, such as inelastic and elastic
scattering, etc. There are no free parameters for neutron-transfer form
factor in our calculation. The parameters that come from a shell model
give the neutron-transfer form factors which describe subbarrier fusion.
It is also important to verify that the transfer form factors in this
work are consistent with data on the particle transfer reactions. This
has been measured in the Ni+Ni system (Van.86) and the analysis is under

way (Lan.86).

Chapter 8 Conclusions

The structure effect on subbarrier fusion was first described by
ZPM. ZPM provides a simple picture of the interplay between intrinsic
degree of freedom and internuclear motion. Also, calculations in ZPM
are easy. We found that ZPM is a good approximation for describing low-
lying intrinsic excitation effects on subbarrier fusion in medium heavy
systems (Fig.u.l). In general, ZPM gives the right order of magnitude
of the effects due to low-lying collective excitations. To describe
subbarrier fusion more accurately, however, finite energy excitation
formulation (coupled channel method) is needed in some systems
(Fig.6.2).

In light systems, only the first few intrinsic excitations need to
be included in the two-dimensional model to describe subbarrier fusion
because of the relatively large excitation energies. Structure effects
on subbarrier fusion in light systems can therefore be described well
without ambiguities in the theory (Fig.5.1,2). In the case of ‘°O+“0,
only the transition from the ground state to the first excited state in
the 18O is treated in the theory. In medium heavy systems, more
intrinsic excitations are needed in the model space in order to describe
subbarrier fusion (Fig.6.2). Subbarrier fusion in medium systems can
also be well understood in the two-dimensional model with vibrational
parameters close to those that describe low-lying collective
excitations.

Particle transfers were found to be important in the subbarrier
fusion of ‘“Ni+‘°Ni in a macroscopic model (Bro.83), and is also well

explained using the microscopic model of this work.

80

Bal.83

Bay.82

Bec.81

Bec.8u

Bir.83

Blo.71

Boh.69

Bro.75

Bro.83

Chr.76
Esb.81

Esb.83

Gut.73

Hil.53

Koo.77

Lan.81
Lec.8fl
Lan.86
Nat.70

Nix.77

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SSSSSSSSSS

”IllIIIIll;(IIlllllflllllllllll

31