Download THERMAL FLUCTUATIONS IN HEAVY

Nuclear Physics A491 (1989) 301-313
North-Holland,
Amsterdam
THERMAL
FLUCTUATIONS
IN HEAVY-ION
FUSION
(I). One-dimensional models
C.E. AGUIAR,
V.C. BARBOSA,
Insiituto de Fisica, Universidade
R. DONANGELO
and
REACTIONS
S.R. SOUZA
Federal do Rio de Janeiro, C. P. 68528, 21945 Rio de Janeiro R.J., Brazil
Received 9 May 1988
(Revised 12 September 1988)
Abstract: We consider the effect of thermal fluctuations
on the heavy-ion fusion probabilities
calculated
within different models of the nuclear potential
and dissipation
function.
We show that the
introduction
of fluctuations
has effects that depend considerably
on the model utilized, and which
may therefore help decide among the different theoretical
descriptions
now available.
1. Introduction
Our present understanding
of heavy-ion fusion reactions has taken much benefit
from classical trajectory models ‘,‘). The main ingredients
of these models are:
(a) a small number of collective degrees of freedom which evolve in time according
to classical equations of motion,
(b) conservative
forces, which give rise to a potential barrier between projectile
and target that must be overcome to bring these nuclei into contact, and
(c) dissipative forces, that exchange energy between macroscopic and microscopic
degrees of freedom.
The collective coordinates
are usually associated
to the distance between the
centers-of-mass
of the two nuclei, their orbital and intrinsic rotations, and, sometimes, their deformations
or other shape parameters.
The conservative
forces are
the Coulomb
repulsion
from the coupling
of the system.
and the nuclear
between
the collective
attraction,
and the frictional
and the single-particle
degrees
forces arise
of freedom
It is well known from nonequilibrium
statistical mechanics that dissipative processes are always accompanied
by thermal fluctuations.
It has been pointed out that
fluctuations,
other than those associated with mass and charge asymmetry, have not
received much attention
in model calculations
3), although their influence in the
fusion of very heavy nuclei is quite noticeable 435).The purpose of the present work
is to investigate the effect of fluctuations
in the radial motion on the fusion behavior
of a heavy-ion
system. We have preferred to introduce
them through use of the
Langevin approach, following Barbosa er al. 6), Abe et al. ‘) and Frobrich and Xu 8),
and not the equivalent
Fokker-Planck
treatment proposed earlier by Nijrenberg “)
because of the difficulties involved in extending
the latter method to cases with
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302
C.E. Aguiar et al. / Thermalfluctuations
(I)
more degrees of freedom which will be treated in the second part of the present
work. The inclusion of the fluctuating forces into the equations of motion is presented
in sect. 2. The time evolution of the trajectory ensemble obtained when ~uctuations
are considered is studied in sect. 3 for the case of two widely used models of nuclear
interaction and dissipation. The results obtained indicate that the introduction of
fluctuations leads to markedly differentiated effects according to the model
employed. This could be used to help lifting, at least in part, the present ambiguities
in the nuclear potential and dissipation. The question is explored in sects. 4 and 5
where we look at the effect of fluctuations on the fusion probabilities predicted by
these models in the case of intermediate and very heavy systems, respectively. In
the last section we present a summary and further discussion,
2. Equations of motion including Langevin forces
The classical equations that describe the radial motion of two colliding nuclei are:
dr P
-=-dt
/.L’
where r indicates the separation between the centers-of-mass of the two nuclei, p
the relative radial momentum, p their reduced mass, V(r) the sum of the long-range
Coulomb repulsion V,(r) and the short-range nuclear attraction V,(r) between the
nuclei, and r(r) the friction coefficient. As it was mentioned in the introduction,
there is considerable ambiguity in the choice of V,(r) and r(r). This comes about
because most heavy-ion reaction data is sensitive to the details of these two quantities
on a narrow region around the nuclear surface. It is then possible to provide quite
different prescriptions for V,(r) and T(r) which balance in such a way as to give
similar results. This is the case for two well-known descriptions. One is the proximity
model (PM) of Blocki et al. lo) and Randrup ‘l)
VN(r)=4&b@[(r-C,-Cd/b],
I’(r) = 4moCbly[(r
- C, - C,)/b]
,
(24
where C, and C2 are the half-density radii of the two nuclei, c = C,C,/( C, + C,),
b the surface thickness, y the surface tension, n, the bulk nucleon current, and @
and Y are universal functions described in refs. ‘Ozl’).We have taken the values of
Ci, b and no from these references, but in the case of y we took the value utilized
in the work of Back et al. 12). This choice of y yields proximity model Coulomb
barriers in agreement with the experimental results, while that originally provided
by ref. lo) overestimates the barrier height by approximately 4% [ref. ‘“>I.
C.E. Aguiur
The
et al. / 7’hermal_iluctuations
303
(I)
other model considered here is that of Gross and Kalinowski 14) (GKM)
where I? = r~(A~/3+A~‘3), and the values of r,, a, a, and r, are given in ref. 14).
In fig. 1 we exhibit the total potential energies for the 58Ni-t 58Ni frontal collision
case calculated within the two models given above. Also illustrated in this figure
are the classical trajectories which overcome the potential barriers with minimum
incident energy in each case. We observe that although the PM potential barrier is
considerably higherthan the one in the GKM (101 MeVversus 95 MeV, respectively),
the friction is markedly stronger in the latter case. Thus the fusion energy thresholds
are 101 MeV in the PM calculation and 100 MeV in the GKM, which means they
are in quite reasonable agreement with each other.
r (fm)
r (fm)
Fig. 1. Potential barriers in the proximity and Gross-Kalinowski
models. The critical trajectories
overcome these barriers with minimum incident energy are shown in each case.
which
The trajectory descriptions of eqs. (l), (2) disregard fluctuation effects. It is,
however, a quite familiar non equilibrium statistical physics result that the interaction
arising from the coupling of a system with a heat bath has two components: an
average friction force and a fluctuating random interaction known as the Langevin
force. If we assume that the microscopic degrees of freedom of the nuclear system
act as a heat bath for the collective variables, we must add this random force to the
304
C.E. Aguiar
et at. / ~eTmai~uctuat~ons
(I)
conservative and dissipative ones. The equations of motion (1) then read
dr P
-=dt p’
dp
-=
dt
-g-r;+L(t),
where t(t) is the Langevin force.
The random force t(t) has a null ensemble average value, and if we further
assume it has a markovian character its time correlation function is a &function:
(a)
= 0,
{L( t)L( t’)} = Zf)s( t - t’) .
(4)
The diffusion coefficient D appearing in eq. (4) determines the intensity of the
Langevin force. It is related to the friction coefficient r by the dissipation-fluctuation
theorem result,
D=Tl-
(5)
known as Einstein’s relation ‘*), in which T stands for the temperature of the heat
bath. This temperature is calculated here assuming that the kinetic energy that was
dissipated until time t, Q(t), is entirely transformed into internal nuclear excitation
energy, E* = Q(t), and accepting the Fermi-gas relation between this energy and
the temperature T,
E*=aT’.
(6)
In eq. (6) the value of the level density parameter a was taken to be A/10 MeV,
where A = Al +A, is the total number of nucleons in the system. Quantum
mechanical effects may be included in eq. (5) by use of an effective temperature
T* = $Aw coth (liw/27’), where w is some characteristic frequency of the system.
We have preferred not to include neither this nor other quantum effects, such as
tunneling, in order not to transform our classical trajectory calculations into a less
transparent hybrid. In any case one should note that since T” > T our use of the
classical Einstein relation does not overestimate the role of fluctuations.
The algorithm we employed to solve the equations of motion (3) is as follows.
We integrate both sides of eq. (3) between times t and f + At to obtain the iteration
relations
(7)
The impulse associated to the Langevin force acting during the time interval At,
r+At
1(t)
L(
=
If
t’)
dt’,
(8)
C.E. Aguiar et al. / Thermalfluctuations
has a normal
distribution
with zero mean value, (I(t))
(I)
305
= 0, and a variance
2DAt [ref. “)I. For the sake of simplicity in eq. (7) we dropped
in At, in spite of the possibly longer computational
time involved.
though,
that
Abe
et al. ‘) have
derived
higher
order
employed in this problem.
Since the equations
(7) for the time evolution
element
I(t)
we have to study the time development
higher order terms
We should remark,
procedures
of the system
which
contain
of an ensemble
associated to identical initial conditions
of the system. We
tions in such a way that the distance between the nuclei is
the dissipative
forces are effective and therefore where
present. In the next sections we show the result of such
(I(t)*) =
may be
a random
of trajectories
choose the initial condilarger than that at which
fluctuations
are not yet
calculations
for several
systems.
3. Time evolution of a trajectory ensemble
The study of sub-barrier
fusion reactions has stimulated
an intense interest in
systems within the intermediate
mass region. One of the cases more intensely studied
was the 58Ni+ 58Ni used in the example shown in the previous section. We then
start by considering the effect of fluctuations
on this system, which is approximately
representative
of the general behavior in this mass region. In fig. 2 we show the
result of running fifty trajectories
under the effect of the random Langevin forces,
at an energy above the fusion threshold
for both models. In the PM case the
dissipation
begins to be effective only after the trajectories
have gone beyond the
Coulomb
barrier radius. The resulting
fluctuations
are not able to change the
outcome, and all trajectories
of the ensemble
fall inside the potential
well thus
Gross- Kollnowskl
Model
,
i
IO
r
(fm)
50
15
5
IO
r
15
(fm)
Fig. 2. Effects of thermal fluctuations
on the trajectories.. The incident energy is well above the fusion
threshold in the absence of fluctuations.
For each calculation
we show an ensemble of 50 trajectories.
306
C.E. Aguiar
et al. / ‘Tkrmal$uctuations
58Ni + ‘*Ni
at
Proximity Model
(I)
E,, =I10 MeV
Gross- Kolinowski
Model
DISTANCE ffm)
Fig. 3. Time evolution in phase space of an ensemble of 2000 trajectories in the proximity and GrossKalinowski descriptions, respectively. The upper frame in each row shows the trajectory calculated using
these models without inclusion of flu~uations. The numbers besides the dots indicate the time in units
of 10ez3 s. The frames below it show the time evolution of the trajectory ensemble when fluctuations are
included.
C.E. Aguiar
et al. / Thermal fluctuations
307
(I)
corresponding to fusion. The GKM results display a very different behavior. In this
case the dissipation starts acting before the separation between the nuclear centers
reaches the Coulomb barrier radius. The strong fluctuations in this region cause
some trajectories to fall into the outwardly leading part of the Coulomb barrier.
These trajectories correspond to a deep inelastic collision process. The quite dissimilar effect of fluctuations on the trajectory ensemble is further illustrated by the
58.
NI +
Proximity
58Ni
Ecm = 110 MeV
at
Gross -
Model
Kolinowski
Model
t=a
~
Lt
t =6C
=60
VELOCITY
Fig. 4. Time evolution
of the velocity
distribution,
(fm I 10-23st
obtained
of fig. 3 on the velocity
by projection
axis.
of the trajectory
ensembles
308
C.E. Aguiar et al. j Thermal fluctuations
(I)
next two figures. In fig. 3 we show the time evolution in phase space of an ensemble
of 2000 trajectories treated within the two approaches discussed here. At the top of
the figure we show the trajectories obtained in the absence of fluctuations. Although
both of them correspond to a fusion process, we already notice here that the strong
dissipation in the GKM very quickly absorbs the C.M. translational kinetic energy,
while in the PM the process takes longer and presents oscillations. These oscillations
appear because the trajectory bounces several times against the potential walls, as
already seen in fig. 1. When the fluctuations are included the difference in the
dissipation models leads to a quite diverse behavior of the trajectory ensembles.
During the initial stages of the collision the GKM packet spreads out faster than
the PM one. Later it breaks up into two very different parts: one that retains a more
or less compact shape while slowly moving inwards, and which may be associated
to the fusion process, and another that scatters away under the influence of the
Coulomb potential, and which should therefore be considered to represent a deep
inelastic process. Fig. 4 further emphasises these contrasting conducts. There we
show the time dependence of the velocity probability distribution function for each
model. The growth of a separated packet in the GKM case is clearly seen.
The calculations presented in this section demonstrate that when fluctuations are
incorporated into classical trajectory models the results depend very strongly on
the dissipation employed. This should provide us with additional information which
could be helpful in removing at least in part the present ambiguity in the existing
nuclear potential and dissipation models. In the following two sections we exemplify
this approach by comparing the models discussed above with experimental results
in the medium and very heavy nuclear mass regions.
4. Medium heavy systems
From the calculations presented in the previous section it is obvious that the
GKM fusion probability for head-on collisions has a smoother energy dependence
when fluctuations are included, while the PM results are practically unaltered. This
effect is illustrated in fig. 5, where both calculations are presented. The inclusion
of fluctuations in the GKM then results in a considerable reduction of the fusion
cross sections for energies starting at a few MeV above the threshold, as already
noticed by Friibrich and Xu 8)_ On the other hand the PM is relatively insensitive
with regard to the inclusion of ~uctuations. Since in their absence both models
reproduce about equally well the experimental data in this mass region, we expect
that when fluctuations are included the present version of the GKM becomes
inadequate to represent the behavior of such fusing systems. It appears that a
reduction in the dissipation strength would help to improve the GKM predictions.
This coincides with the trend of the PM description, according to which dissipation
effects are not very important in the region before the Coulomb barrier. It is unlikely
that these conclusions will change significantly if rotational degrees of freedom are
C.E. Aguiar et al. / Thermal fluctuations
IO-
g
d
2
&
n
08-
I
I
309
(I)
PM
I
I
I
I
t
GKM
0.6-
OZ-
110
130
E,,
Fig. 5. Fusion
probabilities
calculated
IF
(MeV)
with thermal flucutations
Kalinowski
(GKM) models.
for the proximity
(PM)
and Gross-
considered and calculations
are performed for partial waves other than I = 0, as the
main reason for the differences
between the two models, i.e. strong dissipation
before or after reaching the barrier, will remain valid. A similar comment applies
to the inclusion of deformation
degrees of freedom or quantum mechanical
effects,
which are not expected to modify the results at the higher energies where the
differences are quite significant (see fig. 5).
5. Very heavy systems
In the last few years the experimental
and theoretical study of the fusion process
in the case of very heavy systems has been actively pursued. It has been recently
suggested that fluctuation effects could play an important role in these reactions 435).
We therefore proceed to investigate the effect of thermal fluctuations
on the fusion
process for heavier systems. We consider the data of ref. “) which deals with the
zero angular momentum fusion probability
of the systems 86Kr on 1213123Sband lz4Sn
on 9oV92,94396Zr.
The fusion probabilities
for these systems increase very slowly with
energy in the region of the Coulomb barrier. This small rate of increase cannot be
accounted for by the kind of processes frequently used to describe sub-barrier fusion
cross sections such as the coupling to other reaction channels.
Fig. 6 shows a comparison
of the experimental
data with the fusion probabilities
calculated with the GKM including thermal fluctuations.
We see that the calculations
follow the main trends of the data, but significant discrepancies
can also be noticed.
In particular we observe that the reduction of the dissipation
strength discussed in
C.E. Aguiar
IO’
4
I
86
IOO-
t
et al. / Thermalfluctuations
IO’
I
(I)
I
loo_
+
.
.
*
10-I -
I
e6Krt ‘23Sb
Kr + 12’Sb
*
I
1
c
t
‘.
.
lo-‘-
$
lo-2_
IO-3_
t
c
:
&
a
lo-2_
$
10-3
_:
:
lo-4E
“b
lO-5
190
I
I
210
I
230
250
lo-5_
I90
I
I
I
210
I
230
(MeV)
Ecm (MeV)
E,_,, (MeV)
Ecm (MeV)
E,,
Fig. 6. Comparison
E th
“b
th
of experimental
fusion probability
data for very heavy systems
of the Gross-Kalinowski
model including thermal fluctuations.
I
250
with the predictions
C. E. Aguiar
et al. / Thermal
flucruations
5 lo-‘_
5 lo-‘_
I5
3
g lo-2_
23
g lo-2_
$
g
2
10-3_
i?
10-3_
10-4.-
230
210
E,,
50
(I)
.t*
I;
210
230
EC,
(MeV)
2:
(MeV)
Fig. 6-continued
IO’-
I
I
*6Kr+ ‘23Sb
IOO..
10-l
IO-*_
IO"_
-.-
10-4
-
_ r,=2
r,=o
10-5_
19
E,_,, (MeV)
Fig. 7. Effect of reducing the dissipation
strength parameter
r, on the fusion probability.
The full curve
shows the results obtained with use of the standard value of I-,, and the others those found with smaller
values. r,, is given in units of 10dz3 s/MeV.
312
C.E. Aguiar et al. / Thermaljluctuations
the previous
section
does not help to improve
cases. The situation
changing
that as we reduce
region
is more clearly
the dissipation
strength
the agreement
explained
we improve
the quality
be expected
since we are reducing
approaching
the step-like
behavior
with the data in many
in fig. 7, where the consequence
in one of the systems
this parameter
at the cost of damaging
(I)
is illustrated.
the agreement
in the higher
of the fit at lower energies.
the importance
of the frictionless
of fluctuations
of
We observe
energy
This could
and therefore
case. However,
these results
clearly suggest that, in contrast with the case of not-so-heavy
systems discussed in
the previous section, dissipation,
and consequently
thermal fluctuations,
play an
important
role in the fusion behavior of very heavy systems.
The GKM with fluctuations describes, although only qualitatively,
the main trends
of the data. On the other hand, the PM with fluctuations
fails completely
in
reproducing
this data. This is because fluctuation effects are very small in this model,
while the data clearly requires an important
contribution
from them.
As we have seen this one-dimensional
proximity model cannot account for such
fluctuations.
But it has been shown by Swiatecki i6) and co-workers that for these
very heavy systems it is mandatory
to include additional
degrees of freedom in
order to describe two heavy nuclei in contact. Particularly
important
are variables
that describe the size of the neck connecting
the two nuclei and the asymmetry of
the system, besides the radial separation
considered
here. When these additional
degrees of freedom are included,
one obtains that for sufficiently heavy systems
fusion is no longer decided at the Coulomb
barrier, as it was the case in the
one-dimensional
models, but at the unconditional
saddle point in the multidimensional
configuration
space defined by all macroscopic
degrees of freedom.
The classical trajectory leading to the saddle point region feels a very strong one-body
dissipation
according to this model 16). We accordingly expect that fluctuations play
an important role in the fusion process for such heavy systems once the additional
degrees of freedom are incorporated
into the classical description.
In the case of a
not-so-heavy
system,
fusion
is decided
at the Coulomb
barrier
and fluctuations,
although still present, are not able to influence the outcome of the collision since
they develop after the tendency towards fusion is irresistible. In this way the results
of the one-dimensional
PM for this system are compatible
from what one expects
from a multidimensional
description
of the fusion process.
6. Conclusions
The introduction
of thermal fluctuations
into the classical description
of a heavyion fusion reaction was observed to have qualitatively
different effects according to
the dissipation model employed. The proximity description leads to small fluctuation
effects, which appears to be adequate for not too heavy systems but not at all in
the case of very heavy systems. The model of Gross and Kalinowski,
having a much
stronger dissipation,
introduces
larger fluctuation
effects than the proximity model.
C. E. Aguiar et al.
j ~er~ai~~efuaijo~s
(I)
313
The results are reasonable
for very heavy but not for lighter systems. We believe
that a global description, valid over the whole systems size region, may be obtained
by including additional shape degrees of freedom, e.g. in the manner of Swiatecki’s
model 16). We have already included these fluctuations into the schematic version
of this model 17).The results will be presented in the second part of the present work.
We would like to thank R.S. Comes and J. Lopes Neto for their help and advice.
This work has been partially supported by FINEP and CNPq.
References
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
11)
12)
13)
14)
15)
16)
17)
J.R. Birkelund, L.E. Tubbs, J.R. Huizenga, J.N. De and D. Sperber, Phys. Reports 56 (1979) 107
P. Friibrich, Phys. Reports 116 (1984) 337
L.M. Schmieder, W.U. Schroder and M.A. Butler, Phys. Rev. C37 (1988) 139
C.-C. Sahm, H.-G. Clerc, K.-H. Schmidt, W. Reisdorf, P. Armbruster,
F.P. Hessberger,
J.G. Keller,
G. Miinzenberg
and D. Vermeulen, Nucl. Phys. A441 (1985) 316
W.Q. Shen, J. Albinski, A. Gobbi, S. Gralla, K.D. Hildenbrand,
N. Herrmann, J. Kuzminski, W.F.J.
Miiller, H. Stelzer, J. Take, B.B. Back, S. Bjornholm and S.P. Sorensen, Phys. Rev. C36 (1987) 115
V.C. Barbosa, P.C. Soares, E.C. de Oliveira and L.C. Gomes, Revista Brasileira de Fisica 15 (1985)
211
Y. Abe, C. Gregoire and H. Delagrange,
J. de Phys. C4 (1986) 329
P. Frijbrich and S.Y. Xu, Nucl. Phys. A477 (1988) 143
W. Norenberg,
Phys. Lett. B52 (1974) 289
J. Biocki, J. Randrup, W.J. Swiatecki and CF. Tsang, Ann. of Phys. 105 (1977) 427
J. Randrup, Ann. of Phys. 112 (1978) 356
B.B. Back, R.R. Betts, J.E. Gindler, B.D. Wilkins, S, Saini, M.B. Tsang, C.K. Gelbke, W.G. Lynch,
M.A. McMahan
and P.A, Baisden, Phys. Rev. 62 (1985) 195
L.C. Vaz, J.M. Alexander and G.R. Satchler, Phys. Reports 69 (1981) 373
D.H.E. Gross and H. Kalinowski,
Phys. Reports 45 (1978) 175
N.G. van Kampen, Stochastic processes in physics and chemistry (North-Holland,
Amsterdam,
1981)
W.J. Swiatecki, Phys. Scripta 42 (1981) 113
R. Donangelo
and L.F. Canto, Nucl. Phys. A451 (1986) 349