CIENCIAS NUCLEARES
Alpha decay half-lives of superheavy nuclei in the WKB approximation
Períodos de semidesintegración alfa de núcleos superpesados en el marco de la aproximación WKB
Frank Bello Garrote, Javier Aguilera Fernández, Oscar Rodríguez Hoyos
Instituto Superior de Tecnologías y Ciencias Aplicadas (InSTEC)
Ave. Salvador Allende, esq. Luaces, Plaza. La Habana, Cuba
frankl@instec.cu
ABSTRACT
Alpha decay half-lives of superheavy nuclei are obtained in the context of barrier penetration theory built with the use of Coulomb and proximity potentials, taking into account the quadrupole deformations of nuclei. It is estimated from a classical viewpoint, a possible maximum value of the angular momentum of alpha particles emitted from odd and odd-odd nuclei. Masses and deformations of nuclei are obtained from the macro-microscopic method, with the use of the twocenter shell model. Alpha-decay half-lives are compared with recent experimental results
Key words: alpha decay, transactinide elements, half life, WKB approximation, potentials, ODD-ODD nuclei
RESUMEN
Se obtienen períodos de semidesintegración alfa en el marco de la teoría de penetración de barrera, esta última construida con el uso de los potenciales de proximidad y de Coulomb, teniendo en cuenta la deformación cuadrupolar de los núcleos. Se estima, desde el punto de vista clásico, el máximo valor posible del momento angular de las partículas alfa emitidas por núcleos impares e impar-impar. Las masas y las deformaciones de los núcleos se obtienen según el método macromicroscópico, con el uso del modelo de capas de dos centros. Los períodos de semidesintegración alfa se comparan con resultados experimentales.
Palabras claves: desintegración alfa, elementos transactinidos , período de desintegración, aproximación WKB, potenciales, núcleos impar-impar
Introduction
One of the main problems of modern nuclear physics
is the extension of the periodic system into the
islands of stability of superheavy elements (SHE).
For the synthesis of these nuclei fusion-evaporation
reactions are used and two approaches have been
successfully employed: cold and hot fusion. The former
have been used to produce new elements and
isotopes up to Z = 113 [1,2]; the latter have been
used to produce more neutron rich isotopes of elements
up to Z = 118 [3]. The identification of SHEs in
cold fusion reactions is based on the identifi cation of
the decay products via alpha correlations with known
alpha emitters at the end of the decay sequences,
but in hot fusion reactions the nuclei at the end of the
decay sequences are neutron rich isotopes that have
not been obtained yet in other kind of experiments;
thus, in this type of reactions, alpha-decay half-lives
systematics based on theoretical calculations provide
a useful tool for an ulterior identification of the
reaction products. Most of alpha decay half-lives
calculations are performed with the aid of semi-empirical
relationships [4-8]; alternatively, calculations
in the framework of quantum mechanical tunneling
have been done using the density-dependent M3Y
interaction model [9,10], the proximity potential model
[11] or using the relativistic mean-fi eld model to
calculate the interaction potential [12-14]. However,
in most works the infl uence of deformed shapes of
nuclei in the results of the half-lives calculations has
been neglected. In this work is presented a method
for obtaining alpha decay half-lives in the framework
of WKB approximation using the proximity potential
model, which takes into account quadrupole deformations
of nuclei. Besides the fact that alpha-decay
half-lives calculations can be used to identify new
nuclei in experiments, they can be used as a way to
test other theoretical results by comparison with experiment.
In this work, new theoretical values of masses and deformations, calculated from the macro-microscopic
method using the two center shell model,
are used in order to obtain the alpha-decay half-lives,
and the comparison of this half-lives values with experimental
ones, can be useful to test the veracity of
the calculation of masses and deformations.
Methods
Half-live Calculation
In the quantum tunneling theory of alpha decay,
the decay constant can be expressed as the product
of the alpha particle pre-formation probability
, by the number of assaults on the barrier per second
n, by the barrier penetration probability P.
The half-live , the main result of this paper, is
related to the decay constant as
The barrier penetration probability is calculated using one-dimensional WKB approximation
where μ is the reduced mass. The potential energy V is the sum of the Coulomb , nuclear
and centrifugal
energy.
In the above expressions, z and r are, respectively,
the distances between the surfaces and between
the centers of the alpha particle and the residual
nucleus, both measured along an axis parallel
to the vector which describes the relative motion; in
the present work is considered that the alpha particle
is emitted from the farthest point of the nuclear
surface (see Fig. 1), because in this way the alpha
particle face a lesser Coulomb barrier. The turning
points and
are determined from the equation
The barrier penetration probability P, is the most
important factor in the half-live calculation, however,
rather rough values of and
can reduce significantly the accuracy of the results. For
and v, values in [8] have been taken; they have been obtained
recently by a fit with a selected set of experimental data. For even-even, even-odd, odd-even
and odd-odd nuclei we have
Proximity Potential
The nucleus is a leptodermous distribution, i. e., a distribution essentially homogeneous except for its surface. The strong attraction between two nuclei occurs when their surfaces approach to a distance comparable to the surface width b; the energy of this interaction can be described by the proximity potential [15].
Here is the nuclear surface tension coefficient,
is the reciprocal of the square root of the Gaussian
curvature of the function that determines the distance between two points of the surfaces, evaluated at the point of closest approach, and
is an adimensional function called universal proximity potential.
The Lysekil formula is used for the surface tension coefficient [15].
Here ≈0.95 MeV/
and I = (N – Z)/A, where
N, Z and A refer to the set of both nuclei. Calculation
of the Gaussian curvature of a function which depends
on the shape of the surfaces of two deformed
nuclei can be difficult;
in (8) can be replaced by
a simple expression that depends on the principal
curvatures
,
y of the surfaces of both nuclei.
For the expansion of the nuclear surface in
spherical harmonics , generally are only taken
into account quadrupole deformations; therefore,
for a nucleus with axial symmetry, the radius
can
be expressed as follows, depending on the parameter
:
Here C is the radius of a spherical nucleus with
the same volume as the deformed nucleus. To define the radius of a leptodermous distribution there
are several parameters, the best known is the sharp
radius, usually taken as . However, when
the proximity potential is used, it is preferable to take
the radius of the nucleus as the central radius [16],
which is determined mostly by the characteristics of
the surface of the nucleus and not by the value of
the density distribution function inside the nucleus.
The central radius is related with the sharp radius by the expression:
The next formula can be used for the sharp radius
as it takes into account an R0 dependence with A
(see ref. [15]).
From (11), the principal curvatures of the nucleus in the emission point of the alpha particle are calculated; for prolate nuclei (Fig. 1 (a))
and for oblate nuclei (Fig. 1 (b))
The curvature of the alpha particle is equal to
the inverse of its radius, in this case, the central radius;
we take its sharp radius as = 1.671 fm. The
universal proximity potential [17] was obtained from
the Thomas-Fermi model with the inclusion of a momentum
dependent nucleon-nucleon interaction potential;
it reads:
Coulomb Energy
The Coulomb interaction for two axially symmetric nuclei with quadrupole deformations can be expressed analytically [18]; in the case of the nucleus and the alpha particle we have
where the F(r) are form factors and
is the angle
between the nucleus symmetry axis and the direction
of relative motion (see Fig. 1). The form factors are:
Centrifugal Potential
The orbital quantum number of the emitted
alpha particle is the fundamental factor in determining
the centrifugal barrier
In this paper is considered that both the parent
nucleus and the residual nucleus are in the ground
state, therefore, for even-even nuclei = 0. For odd
and odd-odd nuclei, from the classical definition of
angular momentum, we can make an argument that
leads to estimate a maximum value for
(
), whose
fundamental idea is that the impact parameter of
the alpha particle can not be greater than the radius
of the emitter nucleus; from here we obtain
Masses and Deformations
in the Ground State
The masses of nuclei in the ground state were
calculated by the macro-microscopic method, i.e.
the shell correction method of Strutinsky [19]. The
macroscopic part of the calculation was performed
by means of a version of the liquid drop model which
takes into account the fi nite range of the nuclear forces
(FRLDM) [20, 21]; the two center shell model
(TCSM) [22] was used for the microscopic part of
the calculation. The energy of the system depends
on fi ve parameters (see Fig. 2); fixing three of them
(= 1,
= 0 and
1 =
=
) a three-dimensional potential
surface whose minimum point corresponds
to the ground state can be constructed [23]. The
deformation parameter
was obtained from (11) once the minimum energy state was found and the shape of the nucleus was known.
Results and Discussion
Even-even nuclei
Figure 3 compares calculated half-lives with experimental
values obtained at JIRN, of the chains of
nuclei and
[3]. It shows the results obtained
from the Viola-Seaborg formula (VSS) [4] using
masses and deformations calculated by means of the
TCSM; it shows too the results obtained from the formalism
of the barrier penetration theory (BPT), using
masses and deformations calculated by means of the
TCSM and also using the masses and deformations
reported by Möller [21]. Parameters of the VSS formula were taken from [7]. The best result is reached for the nuclei
and
, in which both, BPT and
VSS calculations using TCSM masses and deformations, are in very good agreement with experiment. In the case of
, the result from Möller is better, as in the case of
, but for the last nucleus, BPT calculation
differs from the result of VSS formula for TCSM,
what indicates that there is something wrong with the
TCSM deformation. In the method to fi nd the masses
and deformations of nuclei, there is a probability for a
local minimum to be found in the search for a global minimum, and it could have similar energy but different deformation, and so distorts the results.
Odd and Odd-odd Nuclei
Figure 4 compares calculated half-lives with experimental
values [3] of the chains of nuclei ,
,
,
,
,
and
. BPT calculations using TCSM masses and deformations were performed with l = 0 and with l =
; the results
from the VSS formula and from BPT calculations
using Möller masses and deformations with
l = 0 also appear in this fi gure. In general, BPT calculation
for l = 0 differs from VSS formula more widely
than in the case of even-even nuclei, because
in general, l has nonzero value. As can be seen, in general, the value given by VSS is included in the range determined by the variation of l. The previous
results are in good enough agreement with experiment,
with the exception of a few, for example,
some isotopes of meitnerium, taking into account the margin of error that causes the variation of angular momentum.
Taking into account all of the nuclei of the chains mentioned so far (including even-even nuclei), the standard deviation s of the calculated half-lives with respect to the experimental ones can be taken as a way of comparison
In Möller case = 2.51 (
= 0.82 for even-even
nuclei) and in TCSM case
= 1.61 (
= 0.70 for eveneven
nuclei). In all cases we take values for l = 0.
Conclusions
A method for obtaining alpha-decay half-lives
which is based in the WKB approximation was developed
in the present work. This method takes
into account the cuadrupole deformation parameter,
which has a significant roll in the half-life value,
as was seen in section 3.1. If good enough
theoretical values of masses and deformations
are used for calculations, the present method can
be used as an additional way to predict or confirm experimental results in the region of superheavy nuclei.
Acknowledgements
Authors want to thank Yaser Martinez and Luis Felipe from Prof. W Greiner work group at FIAS for the ground states masses and deformations calculated using the two center shell model.
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Recibido: 14 de marzo de 2011
Aceptado: 28 de abril de 2011