Fission barrier heights and lifetimes for heavy and superheavy nuclei

Fission barrier heights and lifetimes for heavy
and superheavy nuclei
J. Bartel*, A. Dobrowolski''', B. Nerlo-Pomorska^, K. Pomorski''' and
F.A. Ivanyuk**
*IPHC-DRS, CNRS-IN2P3 and University of Strasbourg, Strasbourg, France
^TheoreticalPhysics Division, MCS University, Lublin, Poland
** Institute for Nuclear Research, Kiev, Ukraine
Abstract. Ground-state masses, fission barrier heights and a lifetimes for actinide and transactinide
nuclei are determined in the framework of the macroscopic-microscopic model with the
Lublin-Strasbourg Drop, the Strutinsky shell-correction method and the Modified Funny-Hills shape
parametrization accounting for elongation, neck formation, left-right asymmetry and non-axality
together with the Yukawa folding procedure. Fission barrier height are nicely reproduced in our
approach which contains no adjustable parameter
Keywords: Macroscopic-microscopic model. Modified Funny-Hills shape parametrization.
Optimal-shapes variational procedure, Lublin-Strasbourg Drop
PACS: 21.10.Tg, 21.60.-n, 24.10.-i, 24.75.+i, 25.85.-w
INTRODUCTION
One of the fundamental problems in nuclear structure is the adequate description of
nuclear shapes. In the study of heavy-ion reactions, in fusion, fission, nuclear rotations
and collective vibrations the nuclear deformation-energy surface plays a predominant
role. It is then obvious that the parametrisation of nuclear shapes needs to be both
simple (involving only a few relevant collective parameters) and flexible, i.e. allowing
for a reliable description of the large variety of the above phenomena. A very successful
parametrisation of the nuclear surface in cylindrical coordinates, in particular for the
description of fissioning nuclei, is known as ''Funny-Hills" (FH) parametrisation [1].
We will use a slightly modified version thereof [2] which allows for an extension to
non-axially deformed shapes and gives at the same time a better description of fission
barriers in the variational sense.
Nuclear deformation energies will be calculated in a macroscopic-microscopic model
with a dominant contribution of liquid-drop model (LDM) type and a quantum correction
accounting for shell and pairing effects, described in the BCS approximation with
a monopole pairing strength Gpair, determined in the so-called uniform-gap method [1]
as function of the uniform gap and the average level density at the Fermi surface.
Since the basic idea of Weizsacker and Bethe [3, 4] to describe the average nuclear
energy in terms of volume, surface and Coulomb energy of a charged liquid drop, various
new terms have been included in a LDM type expression, but the basic concept remains
the same. It is worth mentioning that, already in 1953, Hill and Wheeler [5] concluded
on the basis of the Fermi-gas model, that a curvature term proportional to A^l^ should
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231

e part of such a leptodermous expansion. This term was studied in Ref. [6], where
its magnitude was adjusted to the at that time experimentally known fission-barrier
heights. Since the traditional liquid-drop model performed already astonishingly well
without that term, one could have expected that its inclusion plays a rather limited role.
It turns out, however, that it leads to a significant improvement of the LDM formula, as
demonstrated in the so called Lublin-Strasbourg Drop (LSD) [7] with a r.m.s. deviation
with experimental masses of {5M) = 0.698 MeV, as compared to 0.732 MeV within
the traditional approach and, even more spectacular, with a r.m.s. deviation for fissionbarrier
heights for nuclei with Z > 70 of {5VB) = 0.88 MeV as compared to 5.58 MeV
[7] without the inclusion of a curvature term. The LSD mass estimates of Ref. [7] were
evaluated with the microscopic energy corrections taken from the mass table of Moller
at al. [8]. The above quoted discrepances in the estimates of the fission barrier heights
can be slightly reduced (see Ref. [9]) when one introduces a deformation dependent
congruence energy as proposed in Ref. [10]. It is in this approach that fission barriers
are going to be evaluated in what follows.
SHAPE PARAMETRIZATIONS FOR FISSIONING NUCLEI
A shape parametrization fulfilling the above requirement of being both accurate and
simple and which has proven extremely sucessfull in the context of nuclear fission are
the so-called "Funny-Hills" shapes [1]
2 j Rlc^ {l-u^){A + au + Bu^) , B>0
'^'^"'^\Rlc^{l-u^){A+au)exp{Bc^u^),
B

ensures volume conservation of the deformed nucleus. We refer hereafter to (3) as the
''ModifiedFunny-Hills" (MFH) parametrisation. In a similar way as for the original FH
parametrisation we can define a neck parameter h e.g. by
h = \[B-{c-\] (5)
such that /i = 0 corresponds for actinide nuclei roughly to the average LDM path to
fission in the {c, h} deformation space.
Apart from the deformation parameters c, B and a which have a similar (although not
identical) meaning as in the original FH parametrisation, a new parameter a has been
introduced in Eq. (3). A minimization of the LDM energy along the fission paths yields
a value of a in the vicinity of a = 1.
An alternative way to describe nuclear shapes, which is not relying on any shape
parametrization, was proposed by V. Strutinsky [11]. It consists in defining a profile
function, like Ps{z), and searching for the minimum of the LDM energy under the
constraint that the volume V and the elongation are fixed
5p, {E^OM-hV-^Rn)=^ (6)
where Aj and Aj are the corresponding Lagrange multipliers and the elongation parameter
_Kj2 can be chosen e.g. as the distance between the centers of mass of left and right
part of the nucleus, or any other measure of the elongation (as the mass quadrupole
moment). The minimization with respect to the profile function leads then to an integrodifferential
equation which can be solved iteratively [11] (see Ref. [12] for details).
. . . . 1 . . . . 1 . . . . 1 . . . .
/ ^ = ^ \D=O-75 :
LU
LU
0.010
0.005
0.000
.
/ \ "
/ \'- .
FH, B-minimization
MFH, B-minimization
"optimal" shapes
0.5 1.0 1.5 2.0
R.„/R„
V\
\ -
2.5
FIGURE 1. LDM deformation energy obtained with Strutinsky's optimal shapes as compared to Funny
Hills and Modified Funny Hills shapes for a Liquid-Drop fissility parameter of x = 0.75.
Applying this method enables us to test the quality of a given shape parametrization
as done in Ref. [12]. It turns out that (after minimization in B) the MFH shapes are
closest to the so-called optimal shapes obtained by the Strutinsky variational procedure
as shown in Fig. 1. Other shape parametrizations, like the expansion in spherical
harmonics or the Cassinian ovaloids [14], on the contrary, fail to reproduce the optimal
shapes of the fission barriers, unless a very large number of terms is included.
233

The parametrisations (1) and (3) which we introduced till now are restricted to axially
deformed shapes. It is well known, however, that in the actinide region, non-axial shapes
are favoured near the first saddle. Without attempting to describe such shapes in the fully
general case, we assume that the cross section perpendicular to the z axis is an ellipse
with half axes a^ and Uy which, of course, depend on z. In polar coordinates
2 2
^^'(^'

a diffuseness correction and an exchange contribution of the N-body wavefunction. A
modern version of the Liquid-Drop model is presented below.
The fluctuating part 5E^^ of the nuclear energy is usually evaluated in the form
^•E'mic = ^•E'shell + ^•E'pair ' (11)
where the first term, refered to as the shell-correction energy, contains that part of the
fluctuation which is due to the nuclear shell structure, and where the second term is due
to the pairing correlations between the nucleons.
For a system like the atomic nucleus which is composed of two sorts of fermions,
protons and neutrons, the calculation of shell-correction and pairing energy needs to be
carried out for both types of nucleons with 8E^^^^^ and 8E^^^ getting contributions from
neutrons and protons.
To determine single-particle spectra of a given nucleus at a given deformation, as is
needed both in the Strutinsky shell correction energy and in the BCS pairing treatment,
one has to define single-particle potentials for any shape. To do that we prefer not to
use the prescription of ref. [1] which defined single-particle densities and potentials as
Fermi type functions perpendicular to the equivalent sharp surface, given e.g. by Eq.
(9), since such a prescription does not allow for an unambiguously definition of these
quantities for necked-in shapes appearing in the vicinity of the scission configuration.
Neutron and proton densities as well as the corresponding single-particle potentials can,
however, be unambiguously defined by folding the deformed equivalent sharp surface
with an appropriate folding function [16] (see also Ref. [13] for details).
The Lublin-Strasbourg Drop Model
In our macroscopic-microscopic model the average nuclear binding energy is determined
using the Lublin-Strasbourg-Drop (LSD) [7] as a function of mass and charge
number (via the isospin parameter I) and the nuclear deformation
£^,(Z,7V;def) =flvoi(l - K-vo/)^ + asur(l -
K,^f)A^I^B,„,{def)
+ acur (1 - Kcur/2)^'''^5cur(def) + | ^ -^5coul(def) - C4 |- -£'cong , (12)
where the parameters, listed in Table 1, are adjusted to all 2766 experimental binding
energies taken from Ref. [17] using the microscopic corrections from [8] and the congruence
energy iicong estimated according to [18]. The functions isur(def), icur(def),
-5coui(def) defined as the variation, relative to the spherical shape, of the nuclear surface,
curvature and Coulomb energies, can be evaluated for any nuclear shape [19] and, in
particular, for the MFH shapes used in this work.
It has been demonstrated that Eq. (10) v^'ithE^^^ (12) and 8E^:^^ taken from [8] is not
only able to describe the experimental binding energies of all presently known nuclei
with a very good accuracy [7], but experimental fission barrier heights as well [9] as
pointed out above.
235

TABLE 1.
avoi =-15.4920 MeV
asur= 16.9707 MeV
acur = 3.8602 MeV
To = 1.21725 fm
Parameters entering the LSD model
K-voi= 1.8601
Ksui= 2.2938
Kcur = -2.3764
C4 =0.9181 MeV
FISSION BARRIER HEIGHTS AND a-DECAY LIFETIMES
Using a gradient method (see Ref. [13] for details), we have located all physically
relevant stationary points for a sample of 18 actinide nuclei. Fig. 2 displays the fissionbarrier
heights for these nuclei calculated as the difference between the total ground
state and the saddle-point energy. In the lower part of the figure the difference of the
experimental and theoretical barrier heights is given. We conclude that our macroscopic-
>
^
CD
>
2
m
X
CO
7.5
7
6.5
6
5.5
5
4.5
4
2
1
0
-1
-
-3
n 23ep^
n
232T-„
n 234y
+ 2S*U
+ 232Th
+ ^*PU
,
D
D
D
23Bp^
236^
+ 2S*Th
+ 23Bp„
B
D
+
^
238^
238..
240^ni
H 2lJgn
240y
+ 2*"U
140 142 144 146 148 150
,
N
+ »«cm
+ »*Pu
,
n 24ep^
B gSgm
* 2Sg["
+ Pu
,
n 260(,m
+ 26"cm
,
J
-
-
•:
-
-,
i
-
152 154 156
FIGURE 2. Theoretical barrier heights Sth of actinide nuclei from ^^^Th to ^'"Cf (upper) and difference
between experimental [20] and theoretical barrier heights (lower part) as function of neutron number
microscopic approach with the LSD macroscopic energy, the Strutinsky shell correction
and the pairing treatment in terms of the BCS approach with the uniform-gap method
for the determination of the pairing strength is able to reproduce the fission barrier
heights on the average within less than 1 MeV. Since our theoretical estimates for the
barrier heights are almost all higher than the experimental values, this discrepancy will
still be reduced e.g. when taking the effect of different proton-neutron deformations
into account, as has been demonstrated in Ref. [13] where it was shown that such
an additional degree of freedom in the macroscopic-microscopic approach can lead
to an energy gain up to about 1 MeV. Note that in our calculations no zero-point
vibration energy is included since the macroscopic-microscopic model or the Hartree-
Fock-Bogoliubov type approach are based on a variational method which by definition
yields an upper bound of the nuclear ground-state energy and no further corrections are
therefore needed. The obtained accuracy is practically of the same quality as the barrier
236

heights estimates of Ref [20]. Keep in mind, however, that the parameters of the finite
range LDM approach [21] were re-adjusted to reproduce experimental barriers in the
best possible way. The parameters of the LSD model used in our approch were, on the
contrary, adjusted only to experimentally known gound-state masses. In this respect it is
quite gratifying to notice the good capacity of our method to reproduce fission barriers,
which seems to indicate that the essential physics is correctly contained in our approach
and this, as pointed out before, with a minimal number of adjustable parameters.
801 6
801 8
802 0
802 2
802 4
802 6
802 8
803 0
803 2
803 4
803 6
803 8
804 0
804 2
804 4
804 6
804 8
805 0
805 2
805 4
805 6
805 8
806 0
FIGURE 3. Energy landscape of the nucleus Pu at a deformation corresponding to the second saddle
point as function of the left-right asymmetry parameter a and the non-axiality parameter 77.
It is also quite interesting to investigate the importance of taking left-right asymmetry
and non axiality into account when calculating fission barrier heights. For that purpose
we have calculated the energy gain obtained including, in addition to the elongation and
neck parameters c and /i, also the left-right asymmetry and non-axiality parameters a
and 77 • It is generally admitted that non-axiality plays a role in the vicinity of the first
barrier and that left-right asymmetry is of mayor importance at the second barrier. It
turns out, however, in our study that both these degrees of freedom are important as
well at the first as at the second barrier and that the commulative effect of both is of
the order of 1-2 MeV at the first, but reaches up to 7 MeV at the second barrier. We
would like to attract the reader's attention to the fact that even if at a given deformation
point, as at the second saddle, the minimal energy turns out to be axially symmetric,
the energy landscape in our 4-dimensional deformation space might show another local
minimum at almost the same energy which exhibits a substantial non-axiality, as shown
in Fig. 3 for the nucleus ^'^''Pu at a deformation corresponding to the second saddle. This
second local minimum found here at fixed values of c and h is believed to belong to
a different fission valley and that one would need to carry out a dynamical calculation
taking mass parameters into account to make any reasonable predictions about a fission
path or fission lifetime.
Motivated by our quite encouraging results on fission-barrier heights caused, as we
believe, by a correct description of nuclear deformation energies, we can make some
predictions on a-decay half lives. For such an estimate the Viola-Seaborg expression
[22] is often used. Recently an even simpler expression [23, 24] has been proposed
which for even-even nuclei reads
log,oTa{Z,N)
aZ
VQa{Z,N)
-hZ- (13)
237

where a = 1.5372,b = —0.1607,c = —44.0718. With this expression we obtain the
results displayed in Fig. 4 which show again a nice agreement with experimental data.
FIGURE 4.
Comparison between experimental and theoretical a-decay half lives.
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