STUDY OF NUCLEUS-NUCLEUS POTENTIAL BY COMBINED MEASUREMENT OF
DEEP SUB-BARRIER FUSION AND CLUSTER DECAY
S.P. Tretyakova1, A.A. Ogloblin2, R.N. Sagaidak1, S.V. Khlebnikov3, W. Trzaska4
1.Flerov Laboratory of Nuclear Reactions, JINR, Dubna, Russia,
2.Kurchatov Atomic Energy Institute, Moscow, Russia,
3. Khlopin Radium Institute, St-Petersburg, Russia,
4. JYFL, University of Jyväskylä, Jyväskylä, Finland
Nucleus-nucleus potential is one of the
main goals of study of nuclear dynamics. The
wide-spread tools used for its determination are
elastic scattering and fusion reactions. However,
these processes are sensitive, as a rule, only to the
external repulsive part of the potential formed
almost inclusively by Coulomb field. Information
about the internal attractive part is limited by
strong absorption in the surface region and suffers
in most cases from uncertainties of different
origin. Sub-barrier fusion could, in principle,
provide the required information about the
potential inside the top of the barrier. However,
typical measurements of the cross-sections going
down by a couple of orders of magnitude from the
above-barrier plateau in the excitation functions
usually cannot distinguish between different
theoretical potentials. For instance, a lot of the
data can be reproduced by the potential in the
form of the inverted parabola (Wong formula).
Obviously, the latter is adequate only in the region
very close to the top of the barrier. In order to
probe the potential at smaller distances one has to
go down from the cross-section plateau by at least
4 - 5 orders of magnitude.
There exist a process which depends on the
barrier penetrability in the region far below the
top of the barrier and completely inaccessible for
any nuclear reactions. This is cluster radioactivity
(CR). Until now CR was observed for about
twenty parent nuclides (Ra – Cm, A = 221-242)
decaying by the emission of the fragments from
14
C to 34Si. The question is if CR can be used for
the determination of the nucleus-nucleus
potentials of the interacting daughter nuclei.
The problem is that in its turn the CR
mechanism is, as a matter of fact, unknown and
depends not only on the barrier penetrability.
Different models describing CR probabilities
quite well predict completely different shapes of
the potential barriers [2]. Independent information
about the barriers, especially on their internal
parts, is of great importance for understanding the
mechanism of cluster decays. As both decay
products are practically always formed in their
ground states, the study of their deep sub-barrier
fusion can contribute in solving this problem as
well. These arguments prompted us to begin the
combined study of the extremely deep sub-barrier
fusion and the corresponding cluster decay of the
formed compound nuclei. The aim is to get
complimentary information both on nucleusnucleus potentials and mechanism of CR. The
criterion of the validity of such approach should
be the possibility of the extracted potential to
reproduce both fusion excitation function and
cluster decay probability.
Strictly speaking, nowadays there are only
two available projectile-target combinations
which lead to the formation of the compound
nuclei with the measured cluster decay
probabilities. These are
222
Ra 14C + 208Pb and 230U 22Ne + 208Pb
However, one can expect that cluster decay
probabilities for some nuclides can be predicted
rather reliably, e.g., for 220Ra or 224Th. This allows
considering the projectile-target combinations
12
C + 208Pb 220Ra and 16O + 208Pb 224Th
for which high quality fusion data exist.
Especially important feature of these excitation
functions is that some of them (e.g.,[1]) were
measured by solid state track detectors down to
the cross-sections ~ 10-33 cm2. This method [2] has
unique sensitivity for studying small fusionfission cross-sections.
This paper is dedicated to the analysis of
the 16O + 208Pb  224Th fusion reaction [1]
together with the predicted 224Th  16O cluster
decay. The excitation functions for fission and
evaporation residuals are shown in Fig.1. Their
sum corresponds to fusion cross-sections. The
experimental points were taken from [3, 5].
Including the data [1] and [6] for fission and ERs,
respectively, allows to probe the potential in the
extremely sub-barrier region. Especially important
feature of the data [1] is that they were obtained
by the solid state track detectors down
To the cross sections ~ 10 -33 cm2 [2].
Analysis of the data was performed in
the framework of the HIVAP code [7]. For
the calculation of the fusion cross section
10
3
sfus (mb)
16
10
2
10
1
10
0
d2(Ecmsfus) / dE2
cm (mb / MeV)
10
O+
208
BBass
Pb
Morton et al.
r0=1.12, V0=80, d=0.8, s(r0)/r0=2.0%
V 0=85 MeV/fm, d=0.75 fm, s(r0)/r0=0
r0=1.14 fm, V0=75, s(r0)/r0=2.3%
V 0=80 MeV/fm, s(r0)/r0=2.0%
-1
1000
800
600
400
200
Gauss fit
0
-200
70
75
80
85
Ecm (MeV)
Fig.1.The excitation functions for
fission and evaporation residuals.
the potential barrier passing model (PBPM) was
used. In such approach the couple channel effects
are reproduced by the fluctuating fusion barrier
(expressed as the percentage of the radius
parameter, r0) [8]. In the HIVAP code, PBPM is
incorporated with the standard statistical model
allowing the calculation of fission and ER cross
sections [7]. The fits to the data with different
parameter values of the nuclear exponential
potential are shown in Fig.1.
As usually, the simple one barrier model
underestimates the fusion cross-section. In fusion
studies, as it was shown in a number of works,
excitations of different collective degrees of
freedom through coupling to the relative motion
of the colliding nuclei, cause a splitting in energy
of the single fusion barrier resulting in a
distribution of barriers. This barrier distribution
drastically alters the fusion probability from its
value calculated assuming tunneling through a
single barrier. As was shown earlier [8], the
barrier distribution could be obtained from the
fusion cross-sections σ by taking the second
derivative with respect to the center-of-mass
energy Ec.m. of the quantity (Ec.m.σ). When the
effects of quantal tunneling are considered,
d 2 (Ec.m.σ)/dE 2c.m. becomes continuous, and each
barrier is smoothed in energy.
The fits to the experimental values of
d 2 (Ec.m.σ)/dE 2c.m. are presented in Fig.1( in the
down position). Good agreement with the data
provides another argument in favor of the validity
of our empirical fenomenological potential. The
average value of the barrier obtained with this
approximation could be considered as an
experimental estimate of the fusion barrier.
The obtained potential is presented in Fig.2.
It is characterized by steep fall at the distances
smaller the top of the barrier. Such behavior is
typical for the potentials describing elastic
scattering (e.g. folding model potentials,
Christensen - Winter potential [9] or deep WoodsSaxon potentials with the diffuse parameter a =
0.6 – 0.7). It is evident that the “fission-like”
potentials being much thicker than the obtained
cannot reproduce the experimental data, especially
in the deep sub-barrier region.
The potentials
normally obtained from fusion measurements can
be sometimes approximated by Woods-Saxon
form-factor but often require much larger
diffuseness, say a > 1 [9]. The comparison with
the two Woods-Saxon potentials obtained from
the study of 16O + 208Pb fusion [3] is shown in
Fig.2. These both potentials were considered [3]
to give the best fits to the data but, as the matter of
fact, did not reproduce the excitation function
quite well. We see, that the potential with the
diffuse parameter a = 0.65 is more or less close to
our exponential potential, whereas the potential
with a = 1.005 differs quite strongly.
The further selection of the potential
could be done by applying the condition that
fusion potential should satisfy the cluster decay
probability of the corresponding compound
nucleus, 224Th in the case under discussion. The
latter was not yet measured, but more or less
reliable estimates based on the different cluster
radioactivity models can be done. From [10] it
follows that that the half-period of 224Th is
expected to be logT1/2 (s) = 14.7.
In order to get logT1/2 from the fusion
measurements in the frame of the “alpha-decaylike” model one has to calculate the penetrability
P of the empirical barrier and multiply it by the
frequency ν and the spectroscopic factor S
according to the relation:
λ = 0.693/T1/2 = P . ν . S.
We used the typical value of ν = 1 . 1021
sec-1 and microscopical form-factor S (224Th 16O
+ 208Pb) = 3.2 . 10-12 calculated in [11]. The results
are: logT1/2 = 14.7 from exponential potential of
this work, 15.9 and 19.1 from WS potential with a
= 0.65 and 1.005 [3], respectively. We notice the
excellent agreement of the predicted cluster
decay probability with the potential obtained
in this paper and satisfactory agreement with
the “scattering-like” WS potential of [3]. On
the other hand, the WS potential with
abnormally large diffuseness, though often
successfully used for describing the fusion
cross-section, is not compatible with the
cluster radioactivity data.
The following conclusions can be
done from this work:
 The use of solid state track detectors
for fusion-fission studies allows getting data
much more sensitive to the shape of the
internal part of barrier than it is possible in the
tradition measurements.
 The combined analysis of the
extremely deep sub-barrier fusion and cluster
decay of the corresponding compound
nucleus really allows making the selection
between different fusion potentials and
provides important information on cluster
radioactivity mechanism.
 The potential of the 16O + 208Pb fusion
occurred to be similar to the one predicted by
the models describing the elastic scattering.
No change to the abnormally large diffuseness
is required.
 The evidence of the “alpha-decaylike” mechanism of the 224Th 16O + 208Pb
decay was obtained. This result confirms
some previous observations (fine structure of
223
Ra  14C decay , 12C+208Pb elastic
scattering ) concerning the cluster decays
mechanism of the light actinides.
Fig.3. Fusion potentials.
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