Heavy-Ion Fusion Mechanism And Predictions of Super-Heavy Elements Production

Abstract

Fusion process is shown to firstly form largely deformed mono‐nucleus and then to undergo diffusion in two‐dimensions with the radial and mass‐asymmetry degrees of freedom. Examples of prediction of residue cross sections are given for the elements with

Z
= 117

and 118.

Full text

Heavy-Ion Fusion Mechanism And Predictions
Of Super-Heavy Elements Production
Yasuhisa Abea , Caiwan Shenb , David Boilleyc , Bertrand G. Giraudd
and Grigory Kosenkoe
aRCNP, Osaka University, Ibaraki (Osaka), 567-0047, Japan.
bSchool of Science, Huzhou Teachers College, Huzhou (Zhejiang), 313000, China,
cGANIL,CEA/DSM-CNRS/IN2P3, BP 55027, F-14076, France,
and Univ. Caen, BP 5186, F-14032 Caen, France,
dIPT, CEA/DSM, CEA-Saclay, Gif-sur-Yvette, F-91191, France,
eDepartment of Physics, Omsk University, Omsk, RU-644077, Russia
Abstract. Fusion process is shown to firstly form largely deformed mono-nucleus and then to
undergo diffusion in two-dimensions with the radial and mass-asymmetry degrees of freedom.
Examples of prediction of residue cross sections are given for the elements with Z=117 and 118.
Keywords: Heavy-ion Fusion; Fusion hindrance; Super-heavy elements; Cross section.
PACS: 25.70, Jj, 25.70. Lm, 27.90. +b
INTRODUCTION
Theoretical prediction of optimum incident system, optimum incident energy, and
absolute value of maximum cross section for Super-Heavy Elements (SHE)
production is a long-standing challenging problem in nuclear physics. It becomes
more and more important when we go to heavier elements, because residue cross
section becomes smaller and smaller, down to the order of pb to fb. Why such
extremely small cross sections? One immediately thinks of fragility of SHE, that is,
the fact that there is no barrier against fission within the Liquid Drop Model (LDM),
and only so-called shell correction energy in the ground state sustains nucleus of SHE
against fission decay, as is understood by the fissility being close to 1. The feature is
correctly taken into account through Ignatyuk prescription of the level density
parameter.1,2 As expected, the survival probability for SHE is very small, even if
compound nucleus is formed. The small cross section, however, is not only due to
the survival probability, but also due to small fusion probability, which is expected
from so-called fusion hindrance.3,4 Existence of the hindrance has been known
experimentally since many years ago, but its mechanism was not understood yet.
Recently the mechanism has been clarified.5,6,7 The theory is based on the
observation that di-nucleus configuration formed by the incident projectile and target
is located outside the fission saddle or the ridge-line, because the configuration has a
very large deformation as a compound nucleus, while the saddle point configuration is
in2p3-00384520, version 1 - 2 Sep 2009
Author manuscript, published in "Nuclear Structure and Dynamics (NSD09), Dubrovnik : Croatia (2009)"
DOI : 10.1063/1.3232052

close to the spherical shape in heavy nuclei with fissility close to 1. One more
essential point in the theory is an assumption that dissipation is very strong, strong
enough for the relative kinetic energy to dissipate already at the contact distance of the
projectile and the target, which is confirmed with the Surface Friction Model (SFM).8,9
Thus, we employ Smoluchowski or over-damped Langevin equation for the multi-
dimensional fusion dynamics for overcoming of the saddle point or the ridge-line to
the spherical compound nucleus.
DYNAMICS FROM DI-NUCLEUS TO MONO-NUCLEUS10, 11
For the description of nuclear shapes of the composite system formed by the incident
channel, we employ Two-Center Parameterization (TCP), which encompasses di-
nucleus as well as mono-nucleus configurations.12,13,14
There are three essential parameters: distance between two centers of the oscillator
potentials R, mass-asymmetry
α
, and neck correction
ε
. The neck correction
ε
is
defined as a ratio between the smoothed peak height at the connection point of the
right and left oscillator potentials and the height of the potential without correction,
i.e., that of the potential spike at the connection point. So,
ε
can vary between 1 and 0.
The value 1 corresponds to the smoothed peak with the same height as the original
spike, while the value 0 does to no peak, i.e., to a single wide flat potential. In other
words, the former describes the touching configuration of the incident channel, while
the latter does mono-nucleus with very large deformation. In the TCP, the three
degrees of freedom are almost independent, especially in di-nucleus configurations,
though they are not normal modes. Thus, it is meaningful to analyze each degree of
freedom separately. Of course, there are couplings between them, and friction tensor
also induces couplings, for which we use so-called One-Body Dissipation model
(OBD).15 The coupling effects will be discussed elsewhere, starting with multi-
dimensional Smoluchowski equation.16
Firstly, we take up the case with mass-symmetric entrance channel, which makes
the problem simpler with only two degrees of freedom left. The radial motion is
already solved and analyzed in detail.17, 18 Fusion probability, i.e., a probability for
passing over the saddle point into the spherical shape is shown to be given by the
fluctuation of Langevin trajectories, or by a tail of diffusion, and to be well
approximated by an error function. In cases with strong dissipation such as OBD, the
function can be approximated by an Arrhenius function, exp(
−
V
s
ad /
T
), where Vsad
denotes the saddle point height measured from the energy of the touching
configuration and T the temperature of the composite system. This clearly explains the
feature of the fusion hindrance, i.e., an extremely small and slow increase of the fusion
probability known experimentally.3, 4 Here, it is worth to notice that Vsad =0.0 defines a
border between the normal and the hindered fusions. Interestingly, the border line
obtained is found to be consistent with the measured data on 100Mo+100Mo (normal)
and 110Pd+110Pd (hindered) systems.19 Next, the time evolution of the radial fusion was
analyzed, which shows that the fusion process is undertaken in time scale around
several in unit of h/MeV.18
During the diffusion process in the radial motion, how does the neck degree of
freedom evolve? In order to answer that question, we solve the neck motion with the
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starting point
ε
equal to 1.0. The radial variable is fixed at the contact distance of two
nuclei for the moment. The potential for
ε
is calculated with LDM, which is almost
linear, and the friction is calculated with OBD. Corresponding 1-dimensional
Smoluchowski equation was already solved by him.20
We apply it to our present case and find that the neck degree of freedom quickly
reaches to the equilibrium distribution in the space [0.0, 1.0], in one order of
magnitude shorter than the time scale of the radial diffusion.10 The average value of
the neck variable is about 0.1, near the bottom of the potential. This indicates that the
fusion process proceeds to firstly filling-out of the neck cleft or crevice of the di-
nucleus toward the formation of the mono-nucleus with the very large deformation,
and then to diffusion in the radial degree of freedom.
The mass-asymmetry degree of freedom is also analyzed in the same way for 48Ca
induced reactions, and turns out that its time scale is close to that of the radial one, not
to the neck one.11
PRELIMINARY RESULTS OF RESIDUE CROSS SECTIONS
In realistic calculations of fusion probability, we numerically solve two-dimensional
Langevin equation for radial and mass-asymmetry degrees of freedom21 with neck
parameter being fixed at the average value of 0.1. We also need to know the
probability for overcoming of the usual Coulomb barrier in the entrance channel.
There are two ways to employ: the empirical formula for capture cross sections22 and
Langevin calculations of trajectories with SFM.8,9 As for the survival probability, we
use the theory of statistical decay for particle emission and fission decay.23, 24
20 30 40 50
10
-2
10
-1
10
0
σ
res
(pb)
E* (MeV)
5
n
4
n
2
n
48Ca + 249Bk
3
n
FIGURE 1. Predictions of xn residue cross sections for Z=117 and 118 elements. No arbitrary
parameter is introduced except the reduction factor for the shell correction energy in the calculations of
the survival probabilities
Preliminary studies are made on 48Ca + Bk isotopes25 as well as 48Ca + 249Cf
systems, where mass table predicted by P. Möller et al. is used.26 In Fig. 1, the results
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for the systems for Z=117 and 118 are shown, where the empirical formula is used for
the capture probability with certain modification of the parameters suitable for 48Ca
induced reactions, and a part of the code HIVAP for the statistical decay.
Since there are still ambiguities in fusion probabilities, we premise that absolute
values of residues cannot be reproduced. We adjust it expediently by reducing the
shell correction energy, i.e., fission barrier height and thus the survival probability.
The calibration of the factor is made with the Dubna data on 48Ca + 248Cm system.27
The factor turns out to be 0.45, which we keep in use for the other systems.25 In
order to eliminate the factor, more detailed investigations are necessary on fusion
dynamics in very heavy systems, which are under way.16
ACKNOWLEDGMENTS
The present work is supported by JSPS grant No. 18540268 and National Science
Foundation of China grant No. 10675046.
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