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INFLUENCE OF DIFFERENT COLLECTIVE
COORDINATES
ON SPONTANEOUS FISSION PROCESS
IN Fm ISOTOPESfl
A. STASZCZAK and Z. LOJEWSKI
Department of Theoretical Physics, M. Curie-Sklodowska University
pi. M. Curie-Sklodowskiej 1, 20-031 Lublin, Poland
Abstract
CN , We study the role of different collective degrees of freedom in spon-
taneous fission process of even-even Fm isotopes. To find a proper
■ collective space we examine a reach collection of nuclear shape param-
CN . eters {f3\}, with A=2,3,4,5,6 and 8; as well as the paring degrees of
freedom i.e. proton A p and neutron A n pairing gaps. On the basis
£f) ■ of the multidimensional dynamic-programming method (MDPM) the
optimal collective space {/?2, /?4, Pe, A p , A„} for Fm isotopes was found.
1 Introduction
It is commonly known that experimental values of the spontaneous fission
half-lives (T 8 f) of nine even-even Fm isotopes (N = 142, 144, 158) form
approximately two sides of an acute-angled triangle with a vertex in N = 152.
The rapid changes in T s f on both sides of 252 Fm are particularly dramatic for
^ ■ the heavier Fm isotopes, where T s f descends by about ten orders of magnitude
when one passes from 254 Fm to 258 Fm. This strong nonlinear behaviour of
T s f vs. neutron number N produces good opportunity for testing theoretical
models.
The aim of this paper is to present T s f of even-even Fm isotopes, obtained
on the basis of a wholly dynamical analysis in different multidimensional col-
lective spaces.
The used method is described in sect. 2, the results and discussion are given
in sect. 3 and the conclusions are presented in sect. 4.
1 Contribution to the International Workshop XXIV on Gross Properties of Nuclei and
Nuclear Excitations, Hirschegg, Austria, Jan. 15-20, 1996.
1
2 Formulation of the Method
2.1 Spontaneous Fission Half— Life
The spontaneous fission half-lives T s f for the even-even Fm isotopes were
evaluated within the one-dimensional WKB semiclassical approximation
T sf [yr] = W- 28A [l + expS(L)], (1)
where S(L) is the action-integral along a fission path L(s) in the multi-
dimensional deformation space {X\}
S(L) = £ {^B cS (s)[V(s) - £]} V2 ds . (2)
An effective inertia associated with the fission motion along the path L(s) is
«-W = E^^. (3)
A.fl
In above equations ds defines the element of the path length in the {X\}
space. The integration limits s\ and S2 correspond to the entrance and exit
points of the barrier V(s), determined by a condition V(s) = E, where E =
V(Xy) + 0.5 MeV is defined as a sum of a ground-state energy V(X®) and
a zero-point energy in the fission direction at the equilibrium deformation X°
and denotes the energy of the fissioning nucleus.
2.2 Potential Energy and Inertia Tensor
The potential energy V is calculated by the macroscopic-microscopic model
very similar to that used in |]J . For the macroscopic part we used the Yukawa-
plus-exponential finite-range model and for microscopic part the Strutinsky
shell correction, based on the Woods-Saxon single-particle potential with "uni-
versal" variant of the parameters . The single-particle potential is extended
to involve residual pairing interaction, which is treated in the BCS approxima-
tion. The inertia tensor Bx x x x , which describes the inertia of the nucleus with
respect to change of its shape, is calculated in the cranking approximation (cf.
e.g. 1).
2.3 Multidimensional Dynamic— Programming Method (MDPM)
Dynamic calculations of the spontaneous-fission half-lives T s f are understood
as a quest for least-action trajectory L min which fulfills a principle of the
2
least-action 5[5(L)] = 0. To minimize the action integral (2) we used the
dynamic-programming method [|J. Originally this method was used only for
two-dimensional deformation space. We extended the method up to four di-
mensions.
In opposite to approximation used in ]I], where only two of coordinates
(P2 an d Pa) have been handled dynamically and the remaining degrees of free-
dom have been found only by minimization of the potential energy V, in our
multidimensional dynamic-programming method (MDPM) all coordinates are
treated dynamically as independent variables.
3 Results and Discussion
3.1 Effect of Higher Even— Multipolarity /3 6 and (3 8 Parameters on
the Spontaneous Fission Half— Lives
Fig. 1. Logarithms of the calculated in the different collective spaces spon-
taneous fission half-lives of the even-even Fm isotopes vs. neutron number.
The experimental values are shown for comparison.
To find the proper collective space for description of the fission process we
examine three kinds of deformation spaces: {p 2 , Pa, Pq, Ps}, {P2, Pa, P35, Pe}
and {p 2 , Pa, A p , A n }, where parameter p 35 defines an average trajectory in a
{Pzi P5) plane; A p and A n denotes proton and neutron pairing gap, respectively.
Fig. 1 shows the logarithm of the spontaneous fission half-lives T s f, given
in years, for even-even Fm isotopes; obtained when only deformations of the
3
even-multipolarities {/3\}, A=2,4,6,8 are considered. One can see that effect
of deformation /3 6 on T s f is stronger then deformation /3 g . This is in agreement
with earlier results (see e.g.
3.2 Role of the Reflection— Asymmetry Shape Parameter (3 35
To examine the role of the deformations with odd multipolarities on T s f we
collect the (3% and parameters in one parameter /3 35 =(/3 3 , 0^=0.^13^) and
perform the dynamical calculation of T s f in four-dimensional collective space
{02, 0a, P35, 0o\- Results of this study are presented in Fig. 2. It is easy to
see that reflection-asymmetry shape parameter 0^ does not change T s f in Fm
region.
The reason of this lies in the dynamical treatment of fission process. The
parameters /3 3 and 0$ shorten the static fission barriers (i.e. barriers along
static paths), however the effective inertia B e ff, eq. (3), along these static
paths are larger than along the path with 035=0.
Fig. 2. The same as in Fig.l, but obtained when reflection-asymmetry ^35
shape parameter (see text) is included in the collective space.
3.3 Influence of the Pairing Degrees of Freedom
The residual pairing interactions are usually treated in the stationary way
(i.e.in BCS approximation). The idea of dynamical calculations of T s f in mul-
tidimensional collective space spanned by the shape parameters as well as the
4
pairing-field parameters (A p , A„) was proposed in ref. |J and practically ap-
plied to the macroscopic-microscopic calculations with Nilsson single-particle
potential in JF], ||
Fig. 3. Comparison between T s f of even-even Fm isotopes, obtained in col-
lective spaces with and without pairing degrees of freedom i.e. proton (A p )
and neutron (A„) pairing gaps. The difference between two kinds of T s f is
shadowed.
In Fig. 3 we present T s f of even-even Fm isotopes obtained in four-dimensional
collective space {fo, @4, A p , A n } and in two-dimensional space {/^A}- The
difference between T s f got with and without pairing degrees of freedom is
shadowed in the figure.
This difference represents the dynamical effect of the pairing degrees of
freedom on the spontaneous fission half-lives T s f.
As it is seen, proton (A p ) and neutron (A„) pairing gaps reduce T s f for
Fm isotopes with N > 152 for about 3 orders of magnitude.
3.4 Optimal Multidimensional Deformation Space
Fig. 4 presents T s f of even-even Fm isotopes calculated in {j3 2 , /3a, (3§\ collective
space and corrected by the effect of the pairing degrees of freedom from Fig. 3.
This correction is strongly isotopic dependent and considerably improves
theoretical prediction of T s f. The dynamical study in different deformation
5
spaces of various dimensions has shown that optimal collective space for de-
scription of spontaneous fission half-lives in even-even Fm isotopes is five-
dimensional space {/?2, 04, 0e, A p , A„}.
Fig. 4. Logarithms of T s f of even-even Fm isotopes from Fig. 1 calculated in
{02, 04, /%} collective space and corrected by the effect of the pairing degrees
of freedom from Fig. 3.
4 Conclusions
The following conclusions may be drawn from the present study.
1. The contribution of parameter 8 to T s f is negligible.
2. In dynamical calculations the deformations with odd multipolarities 03
and 05 do not change T s f.
3. The proton A p and neutron A n pairing gaps reduce T s f for heavy even-
even Fm isotopes.
4. The optimal collective space for description of spontaneous fission half-
lives in even-even Fm isotopes is {0 2 , 04, 06, A p , A n }.
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