# Spontaneous fission half-lives of actinides and super-heavy elements

J. Marin Blanco<EMAIL_ADDRESS>Institute of Physics, Maria
Curie–Skłodowska University, 20-031 Lublin, Poland  A. Dobrowolski
<EMAIL_ADDRESS>Institute of Physics, Maria Curie–Skłodowska
University, 20-031 Lublin, Poland  A. Zdeb<EMAIL_ADDRESS>Institute
of Physics, Maria Curie–Skłodowska University, 20-031 Lublin, Poland  J.
Bartel<EMAIL_ADDRESS>IPHC, UMR7178, University of Strasbourg,
Strasbourg, France

###### Abstract

Spontaneous fission half-lives of actinide and super-heavy nuclei are
calculated, using the least-action integral, of the WKB tunneling probability
through the barrier that appears in the deformation landscape obtained in the
macroscopic-microscopic potential-energy surface. This deformation-energy
landscape is obtained using a Fourier shape parametrization with 4 deformation
parameters, taking into account the nuclear elongation, left-right asymmetry,
neck formation and non-axiality degrees of freedom. The collective inertia
tensor entering the WKB half-life expression is taken from the so-called
irrotational flow model, whose components are scalled by an overall
multiplicative factor. For a comparisons, we have also used the so-called
phenomenological mass parameter depending only on the center-of-mass
difference of the nascent fission fragments. Our approach is shown to be able
to reproduce empirical fission half-lives of all here considered nuclei to
within 3 orders of magnitude.

spontaneous fission, potential energy surface, macroscopic-microscopic model,
actinides, SHE, half-lives, pairing correlations, least-action path

##  I Introduction

Nuclear fission, as a decay mode competitive with the emission of light
particles such as neutrons or protons, light clusters like $\alpha$ particles
or $\gamma$ quanta, plays an essential role in determining the stability of
heavy and super-heavy nuclei. The nuclear fission process, induced by the
absorption of neutrons has been observed for the first time in 1938 by Hahn
and Strassmann HS39 . The theoretical explanation of this new phenomenon was
given within a few weeks by Meitner and Frisch MF39 . The authors established
the basic features of the low-energy fission process, such as the energy
released in this process being equal to almost 200 MeV, as well as the fact,
that it results from the Coulomb repulsion of the fission fragments. In
addition, it has been estimated that the number of neutrons emitted in each
such fission event is larger than one, and that a chain reaction is thus
possible. The spontaneous fission of uranium was discovered one and a half
year later by Flerov and Petrzak FP40 . Since these early days, a continuous
interest in the theoretical description of the fission process has been
observed. Based on the first theoretical model of a nucleus as a charged drop
of liquid, fission was described as a collective motion of nucleons in which
the nuclear deformation evolves from a form close to a sphere to an elongated
shape BW39 . Such a shape evolution is associated with the change of the
nuclear deformation energy which grows with increasing deformation. When the
elongation exceeds a certain critical value, the energy decreases again up to
the point where the nuclear system splits into two separated fragments. In a
quantum mechanical description the fission process can be understood as a
tunneling through the potential energy barrier. The tunneling probability and,
as a consequence, the spontaneous fission half-life strongly depends on the
shape of the fission barrier, in particular its height and width. Over the
last decades, there have been numerous attempts to present a reliable model of
the fission process which allows to reproduce in particular the measured
spontaneous fission half-lives. What the quality of this reproduction is
concerned, one has to keep in mind, however, that already a very small change
of the barrier, in particular its height, will lead to a substantial change in
the fission half-life. Among the best known of these early attempts, within
which the global systematics of spontaneous fission half-lives has been
reproduced, the semi-empirical formula proposed in 1955 by W.J. Świątecki
would come immediately to the mind S55 . The main idea of this approach comes
from the observation of the strong correlation between the logarithm of the
spontaneous fission half-lives and the ground state microscopic corrections
due to shell effects and pairing correlations. Later on, this concept has been
applied to up-to-date experimental data KP15 ; KP22  within a modern version
of the liquid-drop model, which is now known as the Lublin-Strasbourg Drop
(LSD) PD03 . There are also various theoretical approaches often based on
mathematically quite advanced methods BL80 ; SP07 ; VW66 ; CD96 ; RB97 ;
baran81 ; smol97 ; staszczakPhysRevC.87.024320 ; wardaPhysRevC.86.014322 .
There have also been several attempts to apply fully microscopic, self-
consistent methods in order to reproduce spontaneous fission observables bsn11
; smn , yet the accuracy in the reproduction of the experimental data can
still not be considered as being fully satisfactory. To obtain a better
agreement with the experiment, one may consider pairing as a dynamical degree
of freedom (see Refs. robpair ; robpair2 ) and/or use improved approaches for
the collective inertia robmass . Such approaches turn out, however, to be
numerically very costly. In general, spontaneous fission half-life
calculations require not only an assessment of the collective potential energy
surface (evaluated in a purely microscopical approach or, as will be done is
what follows, within the macroscopic-microscopic model), but also of the
collective inertia tensor. Commonly, the latter is obtained within the
Adiabatic Time-Dependent Hartree-Fock-Bogoliubov (ATDHFB) atdhfb1 ; atdhfb2 ,
the Generator Coordinate Method (GCM) with the generalized Gaussian Overlap
Approximation (GOA) GPB85 ; apair  or evaluated within the so-called
"cranking" approximation NTS69 ; BDJPSW . In the present work the
irrotational-flow approach of Ref. DSN76  (see also BN19 ) will be used to
evaluate the inertia tensor.

The present manuscript will be entirely devoted to present spontaneous fission
half-lives, obtained within that approach, and their comparison with
experimental data.

Using the tunneling model of the WKB approximation for a multidimensional
potential-energy barrier WKB ; WKB1 ; WKB2 , we will analyze the half-lives
for this process for selected even-even actinide and super-heavy nuclei from
$Z$ = 90 to 110. The calculations will in particular concern the isotopes of
the following actinide isotopic chains: Th, U, Pu, Cm, Cf, Fm, No, as well as
for the superheavy elements Rf, Sg, Hs, and Ds. The obtained results are
compared with the available experimental data. Since this comparison turns out
to be quite satisfactory, we will also make predictions for half-lives of
nuclei where the measurements have not yet been performed.

In section II the theoretical framework of our approach will be presented with
the main ingredients which are the parametrization of the nuclear shape on one
side and, on the other side, the model used to describe the energy of the
nuclear system as function of the chosen deformation, which in our present
study is the macroscopic-microscopic approach together with the Lublin-
Strasbourg Drop model with the Strutinsky shell correction and a seniority
force BCS pairing treatment. Section III will explain how the spontaneous-
fission half-life can be evaluated in a WKB-type model, based on the least-
action path, before we present in section IV our results for such half-lives
for some actinide and super-heavy nuclei. Section V finally draws some
conclusions and gives an outlook on further studies which can be carried out
in our approach.

##  II Theoretical framework

To be able to describe very heavy nuclei and their de-excitation through
fission or particle emission, a study of the evolution of their energy with
deformation is mandatory. We will therefore investigate in what follows the
two main ingredients required for such an investigation of what is commonly
called the deformation energy of the nucleus, namely the parametrization of
the nuclear shape up to very large deformations as they may occur in the
fission process, and a model capable to give a reliable description of the
nuclear energy at a given deformation.

###  II.1 Nuclear shape parametrization

The description of the huge variety of shapes encountered all across the
nuclear chart when going from oblate deformations as they appear in the
transition region, generated by the progressive filling of the $pf$ shell, to
prolate shapes as found in the rare-earth region and in actinide nuclei
necessitates a sufficiently rich and flexible nuclear shape parametrization.
This demand is even tightened if one requires to describe the typically very
elongated and often necked-in shapes as they are encountered in the fission
process. To model the physical reality (as far as that could be identified) as
faithfully as possible, it is obviously required to involve a large number of
deformation parameters, depicting the involved degrees of freedom,
characterized e.g. by the multipole moments of the nuclear shape. For a
numerical treatment, on the other hand, a very large number of deformation
parameter would be prohibitive. It is thus the demanding task for the nuclear
physicist to identify the essential degrees of freedom of a nuclear shape and
to bring these into an analytical form. A very large number of shape
parametrizations have been proposed (see Ref. HM88  for an extensive review)
and are currently used to investigate all kind of nuclear properties. One of
the most widely used (see e.g. Ko1 ; Ko2 ; Ko3 ; Ko4 ) such parametrization is
the one Lord Rayleigh proposed already towards the end of the 19th century
Ra79 . Among other more recent shape parametrizations which have been used to
describe the fission process, one should mention the quadratic surfaces of
revolution (QSR) Ni69  of Ray Nix, the Cassini ovals Pa71 ; PR08  of
Pashkevich, the famous Funny-Hills parametrization FH72  of the Copenhagen
group and its improved version PB06 , as well as the expansion of the nuclear
surface in Legendre polynomials TKS80  of Trentalange, Koonin and Sierk. While
the Rayleigh shapes were defined through the radius vector of any surface
point in spherical coordinates $r_{s}(\theta,\varphi)$, an approach which is
certainly well adapted to the description of nuclear shapes reasonably close
to a sphere, it became rapidly clear that for the description of rather
elongated shapes, as they are encountered in the fission process, a
parametrization that defines a surface point in cylindrical coordinates in the
form $\rho_{s}(z,\varphi)$, as this has been done in the Funny-Hills
parametrization FH72 , is much better suited. This is e.g. demonstrated by the
fact that if one is interested in the description of the fission process and
in particular in fission barrier heights, the Rayleigh parametrization fails,
or rather converges very slowly as has been demonstrated in Ref. DPB07 .

As it has already been mentioned above, the description of nuclear shapes as
they appear along the way from the nuclear ground state to the pre-scission
configurations is obviously not a trivial task. For practical reasons, that
description should contain as few deformation parameters of the nucleus as
possible and, at the same time, reproduce at least, major classes of its shape
occurring on its path to fission. Such shapes should comprise, among others,
axially-symmetric and asymmetric deformations, elongated forms, characterised
in addition by a left-right symmetry or asymmetry, and the posible presence of
a neck forming between the two nascent fission fragments.

To make an expansion, in cylindrical coordinates $(\rho,z,\varphi)$ of the
distance $\rho_{s}(z)$ of any surface point to the symmetry z-axis goes back
to the seminal work of Ref. FH72  and has been developped further in TKS80
and PB06 , but to make an expansion of $\rho^{2}_{s}$ as function of the
cylindrical $z$ coordinate in a Fourier series as

${\displaystyle\frac{\rho^{2}_{s}(u)}{R_{0}^{2}}\\!=\\!\sum_{n=1}^{\infty}\\!\left[a_{2n}\cos\left(\\!\frac{2n-1}{2}\pi
u\\!\right)+a_{2n+1}\sin\left(\\!\frac{2n}{2}\pi u\\!\right)\right]},$ (1)

has been presented for the first time in Ref. Pom15 . Here $R_{0}$ is the
radius of the spherical nucleus having the same volume, while $2\,z_{0}$ is
the length of the nuclear shape along the symmetry $z-$axis. The dimensionless
variable $u=(z-z_{sh})/z_{0}$ that appears in (1) contains a parameter
$z_{sh}$ given by:

${\displaystyle{z_{sh}}=\frac{3z_{0}^{2}}{2\pi
R_{0}}\sum_{n}(-1)^{n}\,\frac{a_{2n+1}}{n}},$ (2)

which ensures that the center-of-mass of the shape is always located at the
origin of the coordinate system.

There are of course very many parametrizations to describe the shape of a
deformed nucleus. When one is interested in rather elongated shapes as they
occur in the fission process it is certainly beneficial to use a
parametrization defined in cylindrical rather than in spherical coordinates,
as explained above. That is also why the so-called Funny-Hills shapes of Ref.
FH72  had such a enormous succes when dealing with the fission process. The
main advantage of the Fourier shapes consists in the fact that the expansion
of Eq. (1) converges very rapidly and that its convergence can be tested
easily by carrying the expansion to higher orders, thus including higher
multipoles, which was not possible for the Funny-Hills parametrization FH72
or its extension PB06 .

Since our Fourier parametrization constitutes, when carried to infinite order,
a complete orthogonal series, it is clear that for a given underlying model
(like our macro-micro approach) the quality of the parametrization will
crutially depend on the convergence of the expansion, i.e. on the number of
required expansion coefficients. It turns out, however, that with a very
limited number (of the order of 2-3) of Fourier coefficients, which are going
to be our deformation parameters, one is able to describe the nuclear energy
along the fission path with a quite good accuracy of the order of half an MeV
as compared to the case when higher order deformation parameters are taken
into account as has been shown in Ref. kostr_cpc21 . It also turns out that
these higher-order terms will play any noticible role only for very large
elongations and mass-asymmetry deformations ($q_{2}\approx 2.0$ and
$q_{3}>0.15$), while they do not exceed a very small fraction of an MeV in the
vicinity of the ground state.

The above parametrization (1) is obviously limited to axially symmetric
shapes. Shapes breaking axial symmetry can, however, be easily taken into
account by assuming that the cross section perpendicular to the symmetry
$z-$axis is always of the form of an ellipse with half axes $a$ und $b$ (see
Fig. 1), such that $a\,b=\rho_{s}^{2}(z)$ which ensures volume conservation.
One then defines a non-axiality parameter:

$\eta=\frac{b-a}{a+b}.$ (3)

which is the relative difference of the half axes $a$ and $b$. Assuming that
this parameter stays the same all across the nuclear shape, the profile
function of the nucleus can then be written in the general case of an axially-
asymmetric shape as BPN17 ; SPN17 :

$\varrho_{s}^{2}(z,\varphi)=\rho_{s}^{2}(z)\,f_{\eta}(\varphi),$ (4)

where

$f_{\eta}(\varphi)=\frac{1-\eta^{2}}{1+\eta^{2}+2\,\eta\cos(2\varphi)}\,.$ (5)

Figure 1: Schematic visualization of the parameters entering the definition of
the profile function defined through Eqs. (1) $-$ (5) .

In order to relate the Fourier coefficients $a_{\nu}$, which are our original
deformation coordinates, to some more physical deformation parameters and make
them vanish, at the same time, for a spherical shape, we introduce new
collective coordinates $q_{\nu}$ Pom15  through

$\begin{split}q_{2}&=a_{2}^{(0)}/a_{2}-a_{2}/a_{2}^{(0)},\\\ q_{3}&=a_{3},\\\
q_{4}&=a_{4}+\sqrt{(q_{2}/9)^{2}+(a_{4}^{(0)})^{2}},\\\
q_{5}&=a_{5}-(q_{2}-2)\,\frac{a_{3}}{10},\\\
q_{6}&=a_{6}-\sqrt{(\frac{a_{2}}{100})^{2}+(a_{6}^{(0)})^{2}},\end{split}$ (6)

where the $a_{2n}^{(0)}$ defined by

$a_{2n}^{(0)}=(-1)^{n-1}32/[\pi(2n-1)]^{3}$ (7)

correspond to the values of the $a_{2n}$ for a sphere. In what follows, we
will limit ourselves to only four deformation parameters
$(q_{2},q_{3},q_{4},\eta)$, where the parameter $q_{2}$ determines the
elongation of the shape and therefore stands for the quadrupole degree of
freedom, $q_{3}$ for the octupole deformation and thus for the left-right
asymmetry and $q_{4}$ for the hexadecapole deformation and would be
responsible for the possible formation of a neck region. Higher order terms
would then define higher-order multipole moments.

###  II.2 The macroscopic-microscopic approach

Having defined above an analytical form of the parametrization of the nuclear
shape that is rapidly convergent, as already mentioned and as this has been
shown e.g. in Ref. SPN17 , we shall now present the macroscopic-microscopic
approach which will allow us to evaluate the nuclear energy for any
deformation that can possibly be defined through the above shape
parametrization. This macroscopic-microscopic approach relies on a
parametrization of the average, liquid-drop type energy in the spirit of the
Bethe-Weizsäcker mass formula We35 ; Be36 . The liquid-drop type approach that
we are using in what follows is what is known as the Lublin-Strasbourg Drop
(LSD) PD03 , which has the particularity to contain in the leptodermous
expansion a curvature $A^{1/3}$ term and a congruence energy term Cong ; MNM95
. The total nuclear energy is then given by:

$\begin{split}E_{LSD}=&b_{vol}(1-k_{vol}I^{2})A-\\\
&b_{surf}(1-k_{surf}I^{2})A^{2/3}B_{surf}(\\{q_{i}\\})\\\
&-b_{cur}(1-k_{cur}I^{2})A^{1/3}B_{cur}(\\{q_{i}\\})\\\
&-\frac{3}{5}e^{2}\frac{Z^{2}}{r_{0}^{ch}A^{1/3}}B_{Coul}(\\{q_{i}\\})+C_{4}\frac{Z^{2}}{A}\\\
&-10\,exp(-4.2|I|).\end{split}$ (8)

Taking into account, in addition, microscopic energy corrections, taken from
Ref. MNM95 , the thus obtained total nuclear energy, which had been fitted to
reproduce in the best possible way the ground-state masses of the 2766
isotopes with $N\geq 8$ and $Z\geq 8$ known at that time (2003), has been
quite successful. It has, indeed, been shown that it does not only yield an
excellent description of nuclear masses (with an r.m.s. deviation of 0.70 MeV
from the experimental data), but that it is also able to reproduce
experimentally determined fission-barrier heights with a very good accuracy
Pom13  . The coefficients of this leptodermous expansion are given in the
Table 1.

Table 1: Values of the parameters of the Lublin-Strasbourg Drop Model. $b_{vol}$ | 15.4920 MeV
---|---
$b_{surf}$ | 16.9707 MeV
$b_{cur}$ | 3.8602 MeV
$k_{vol}$ | 1.8601
$k_{surf}$ | 2.2038
$k_{cur}$ | -2.3764
$C_{4}$ | 0.9181 MeV
$r_{0}$ | 1.21725 fm

###  II.3 Shell and pairing corrections

Observing that the average nuclear energy can be approximated to some
reasonable extent by a macroscopic mass formula, like the one of Weizsäcker
and Bethe We35 ; Be36 , but that there exist quantum effects in such a
microscopic system, which are often responsible for the essential physical
phenomena, like the structure of the nuclear ground state, a description of
the influence of these quantum effects, associated with the existence of the
shell structure in nuclei, was introduced by Myers and Swiatecki in 1966 in
terms of energy corrections MS66  to the smooth liquid-drop energy given in
our case by Eq. (8).

An important contribution was then made by Strutinsky in 1968 Str66 ; Str67 ;
Str68  who proposed an efficient and fast method for evaluating the total
energy of a nucleus by a smoothing procedure of the single-particle spectrum
which at the same time takes into account in some approximate way the
influence of the energy levels lying in the continuum. The average nuclear
energy obtained in such a way can then be subtracted from the sum of the
single-particle levels, to yield the so-called Strutinsky shell-correction
energy

$\delta
E_{shell}=\sum_{\nu}\left[n_{\nu}-\tilde{n}_{\nu}\frac{}{}\\!\right]e_{\nu}$
(9)

where $n_{\nu}$ is a Heavyside step function, with values 0 or 1 depending on
whether $e_{\nu}$ is located above or below the Fermi energy and
$\tilde{n}_{\nu}$ is obtained by a Strutinsky smoothing procedure BP73 . The
main advantage of the Strutinsky method, is that it can be applied to an
arbitrary spectrum of single-particle states.

Another microscopic correction to the total energy of the nucleus has its
origin in the pairing correlations which exist in a BCS-type approach and for
the heavy nuclei in our study only between nucleons of the same type (protons
or neutrons). These pairing correlations cause nuclei having an even number of
protons or neutrons to be more bound. This pairing interaction is described
here by means of the superconducting approach proposed initially for the
correlations between electrons by Bardeen, Cooper and Schrieffer BCSpa  in the
framework of solid state physics. In order to obtain a many-body solution
which is an eigenstate of the particle number operator, an approximate
projection of the BCS wave functions onto good particle number is carried out
in our approach using the Generator Coordinate Method (GCM) with the Gaussian
Overlap Approximation (GOA), as presented e.g. in Ref. GCM+GOA . Let us recall
that, in general, the $n^{th}$ GCM many-body state $|\Psi_{n}(X)\rangle$ is
constructed as a function of the single-particle variables $X$ as

$|\Psi_{n}(X)\rangle=\int dq\,f_{n}(q)|X;q\rangle,$ (10)

where $f_{n}(q)$ is called a weight function and $|X;q\rangle$ is a generator
function (of HF or HFB eigensolutions or BCS many-body solutions) which
depends on the single-particle variables $X$ and parametrically on a certain
set of collective variables $\\{q\\}$ which can be taken simply as the nuclear
deformation parameters or other degrees of freedom describing nuclear
collective motions. To determine the weights $f_{n}(q)$, one assumes the
existence of stationary solutions $\varepsilon_{n}$ of a many-body Hamiltonian
$\hat{H}_{mb}$, with respect to variations $\delta f_{n}(q)$

$\langle\Psi_{n}(X)|\hat{H}_{mb}|\Psi_{n}(X)\rangle\approx\langle\Phi_{n}(q)|\mathcal{\hat{H}}_{coll}|\Phi_{n}(q)\rangle=\varepsilon_{n}.$
(11)

Such a prescription represents an approximate way of mapping the single-
particle fermionic space onto a collective one, spanned by collective wave
functions $|\Phi_{n}(q)\rangle$. For this purpose, one assumes that the
generator coordinates are continuous and the overlap of generating functions
$|X;q\rangle$ has the form of a multidimensional Gaussian function or may be
transformed into a Gaussian shape. Let us now choose a generator function
$|X;\phi,q\rangle$ of the form

$|X;\phi,q\rangle=e^{i\phi\mathcal{\hat{N}}}|X;q\rangle_{{\rm BCS}},$ (12)

where $\phi$ is the so called gauge angle and $\\{q\\}$ is the set of our
collective deformation parameters $(\eta,q_{2},q_{3},q_{4})$. The hermitian
operator $\mathcal{\hat{N}}$ describes the fluctuations of the particle number

$\mathcal{\hat{N}}=-i\frac{\partial}{\partial\phi}\equiv\hat{N}-\,{}_{{\rm
BCS}}\langle X;q|\hat{N}|X;q\rangle_{{\rm BCS}}.$ (13)

With the above assumptions, the generator function entering Eq. (10) may be
rewritten to the form

$|X;q\rangle_{m}=\int\limits_{0}^{2\pi}\\!d\phi\;e^{i(N+m)\phi}[e^{-i{\hat{N}}\phi}|X;q\rangle_{{\rm
BCS}}]$ (14)

with $m=N-\langle\hat{N}\rangle=0,\pm 2,\pm 4,...$ corresponding to a quantum
number of rotation in the gauge space. For $m=0$ we get the prescription for
the typical particle-number projected generator function of the ground state,
where no quasi-particle pair is excited. The only effect of the particle
number projection is then given by the appearance of a zero-point energy
correction $\epsilon_{0}$ given as

$\epsilon_{0}=\frac{\sum\limits_{\nu>0}\left[(e_{\nu}-\lambda)(u_{\nu}^{2}-v_{\nu}^{2})+2\Delta
u_{\nu}v_{\nu}+G\,v_{\nu}^{4}\right]/E_{\nu}^{2}}{\sum\limits_{\nu>0}E_{\nu}^{-2}},$
(15)

which subtracted from the BCS ground-state energy without the projection
effects, leads to a deeper ground-state energy at a slightly higher value of
the pairing gap $\Delta$ as compared to the corresponding value in the
original BCS approach without projection. In Eq. (15),
$E_{\nu}=\sqrt{(e_{\nu}-\lambda)^{2}+\Delta^{2}}$ is the quasi-particle energy
while $\lambda$ and $G$ are respectively the BCS Fermi energy and the constant
pairing strength. Let us recall that these equations need to be defined
independently for protons and neutrons. The summations in the above equations
runs over the single-particle states inside what is called a pairing window of
energy width $2\Omega$ around the Fermi energy
($\lambda-\Omega<e_{\nu}<\lambda+\Omega$). Since the pairing interaction takes
place between a pair of particles in time-reversed states and since these have
precisely the same energy, this summation runs only over states with one fixed
orientation of the total angular momentum (let us call these $k\\!>\\!0$),
excluding their time-reversed ($k\\!<\\!0$) partners. In the above equations
$v_{\nu}^{2}$ is the occupation probability of the single-particle state of
energy $e_{\nu}$ while $u_{\nu}^{2}$ denotes the probability that this state
is unoccupied. Obviously, $v_{\nu}^{2}+u_{\nu}^{2}=1$. The single-particle
energies $e_{\nu}$ are the eigenvalues of a mean-field Hamiltonian with a
mean-field potential chosen in a well adapted way to describe the nucleus
under study at the chosen deformation. In this work this is generally done by
folding the deformed shape, generated in our case by the Fourier expansion
described in section II.A, with a Yukawa-folded single-particle potential as
explained e.g. in Refs. Yuk1 ; Yuk2 .

The energy correction generated by the pairing correlations is in general
defined as the difference between the nuclear energy, obtained in the above
projected $BCS$ approach and the sum of the single particle energies up to the
last occupied level

$\delta E_{pair}=E_{{\rm BCS}}-\sum_{\nu}e_{\nu}-{\tilde{E}_{pair}},$ (16)

where $\tilde{E}_{pair}$ is the so-called average pairing energy which is not
included in the liquid drop formula. The ground state energy of the nucleus in
such an approximation can then be written as

$E_{{\rm
BCS}}=2\sum\limits_{\nu>0}e_{\nu}v_{\nu}^{2}-G(\sum\limits_{\nu>0}u_{\nu}v_{\nu})^{2}-G\sum\limits_{\nu>0}v_{\nu}^{4}-\epsilon_{0}\,.$
(17)

The average pairing energy, projected onto good particle number, then writes
as

$\begin{split}\tilde{E}_{pair}=&-\frac{1}{2}\tilde{g}\,\tilde{\Delta}^{2}+\frac{1}{2}\tilde{g}\,\tilde{\Delta}\,\arctan\left({\frac{\Omega}{\tilde{\Delta}}}\right)-\log\left({\frac{\Omega}{\tilde{\Delta}}}\right)\,\tilde{\Delta}\\\
&+\frac{3}{4}G\,\frac{\Omega/\tilde{\Delta}}{1+\left(\frac{\Omega}{\tilde{\Delta}}\right)^{2}}/\arctan\left({\frac{\Omega}{\tilde{\Delta}}}\right)-\frac{1}{4}G,\end{split}$
(18)

where $\tilde{g}$ is the average density of single-particle levels in the
$2\Omega$ energy window whereas $\tilde{\Delta}$ denotes the average pairing
gap corresponding to a given strength $G$ of pairing interaction pom06

$\tilde{\Delta}=2\Omega\,e^{-1/(G\tilde{g})}.$ (19)

In all above considerations one admits a pairing energy window of width
$2\Omega$, containing $2\sqrt{15N_{q}}\,\,\,(N_{q}\\!=\\!{\rm\,Z\,or\,N})$
single-particle levels around the Fermi level NTS69 .

###  II.4 Fitting the pairing strength

To be able to carry out the calculations in the above described model with
pairing correlations acting inside a pairing window of width $2\Omega$ around
the Fermi energy, one has to adjust the pairing strengths $G$ and through
that, the pairing gaps $\Delta(G)$ for protons and neutrons. The latter have
to reproduce as accurately as possible the experimental proton and neutron
pairing gaps $\Delta_{q}^{(exp)}$ calculated out of measured mass excesses of
neighbouring odd-even heavy and super-heavy nuclei.

Figure 2: Comparison between calculated (black triangles) and empirical (red
dots) neutron (a) and proton (b) pairing gaps for the discussed isotopic
chains from $Z=90$ up to $Z=100$. For a better visibility we present for a
given value of the mass number $A$ only the one isotope, for which the
discrepancy between the theoretical and experimental values is the largest.
Panel (c) displays the largest differences between experimental and calculated
ground-state masses, evaluated with the pairing strength of Eq. (24) (red
dots) and the one obtained from Eq. (23) (black triangles).

The energy gap $\Delta_{q}$, ($q\\!=\\!n$ or $p$) for neutrons or protons
produced by the pairing interaction can be expressed as
$\Delta_{q}\\!=\\!E^{(q)}_{int}/2$, with $E^{(q)}_{int}$ the interaction
energy between two nucleons of type $q$. For a given nucleus with particle
numbers $N_{q}=N$ or $Z$, and the corresponding separation energies $S(N_{q})$

$E^{(q)}_{int}=S(N_{q})-\frac{1}{2}[S(N_{q}+1)+S(N_{q}-1)],$ (20)

the pairing gaps are easily expressed in terms of the empirical mass excesses
$B(N_{q})$ taken from Ref. mass_tables  in the following way:

$\Delta_{q}^{(exp)}=\frac{1}{4}\left[2B(N_{q})-B(N_{q}+1)-B(N_{q}-1)\frac{}{}\\!\right].$
(21)

The pairing strengths $G$ can therefore be deduced, for the here considered 39
heavy nuclei with $Z=90-100$ for which the ground-state masses are known, by
requiring that the pairing gaps $\Delta_{q}(G)$ obtained in the BCS approach
(including the particle-number projection) are found as close as possible to
their empirical values calculated through Eq. (21). One then requires the
following expression

$\sum_{\rm set}|\Delta_{q}^{(exp)}-\Delta_{q}(G)|$ (22)

to be minimal, where the sum runs over the set of the considered nuclei.

In order to facilitate the above discussed calculation, one usually tries to
find, in practice, a simple analytical expression, depending on $N,Z$, which
is able to reproduce the pairing strength $G$ for both protons and neutrons in
the best possible way. Among many such phenomenological expressions, one which
has proven quite successful may be written in the following form:

$G\,A=g_{0}+g_{1}\,(N-Z)\,.$ (23)

This expression depends on only two free parameters $g_{0}$ and $g_{1}$ which
are fitted to the value of $G$ that renders the expression of Eq. (22)
minimal.

The optimal values for these two parameters have been found for our sample of
39 actinide nuclei to be $g_{0}=18.35$ MeV and $g_{1}=0.103$ MeV for protons
and $g_{0}=24.1$ MeV and $g_{1}=-0.135$ MeV for neutrons.

The quality of this fit is visualized in Fig. 2 (a) and (b) where the values
of pairing gaps calculated using Eq. (23) are compared with the empirical ones
obtained from Eq. (21). To make this comparison more transparent, we chose to
present for each isobaric chain only the one isotope, for which the
discrepancy between the theoretical and experimental values is the largest.
One finds that the largest deviation for neutrons does not exceed 0.35 MeV
(236Th and 250Cf) and for protons is always lower than 0.2 MeV. It is worth to
point out that on average, the largest deviations from the empirical pairing
gaps for both types of nucleons reach around $0.12$ MeV. Panel (c) of the Fig.
2 presents, for the nuclei of panels (a) and (b), the macroscopic-microscopic
ground state energy, relative to the experimental data, with the pairing
corrections obtained using the prescription of Eq. (23) (black triangles) and
a previous fit of the pairing strength (red dots) within the same projected
BCS-like formalism as presented above (see GfitS  and references therein),
where the nucleon number dependence of $G$ is given by:

$G_{q}\cdot N_{q}^{2/3}=g^{(0)}_{q},\quad q=\\{n,p\\}.$ (24)

The only parameter $g^{(0)}_{q}$ in this parametrisation of the pairing
strength is chosen as $g^{(0)}_{q}\\!=\\!0.28\hbar\omega_{0}$ with a value of
$\hbar\omega_{0}=41/A^{1/3}\,$MeV of e.g. Ref. Gqfit41 , widely used in
macroscopic-microscopic calculations, and common for both protons and
neutrons.

As demonstrated in Fig. 2, our new pairing-strength fit, Eq. (23), gives, for
most of the nuclei in question, an overall better reproduction of experimental
ground-state masses as compared to its older version (24), with the exception
of the uranium isotopes 222,226,228U and of 256Fm isotopes. Only in 228U does
the discrepancy between the two models reach half an MeV to the disadvantage
of the new fit.

Taking into account the above considerations, one can now write down the total
energy of the nuclear system in the macroscopic-microscopic approach simply as

$E(N,Z,def)=E_{LSD}+\sum_{q}\left[\delta E^{(q)}_{shell}+\delta
E^{(q)}_{pair}\right]$ (25)

with the shell and pairing corrections $\delta E^{(q)}_{shell}$ and $\delta
E^{(q)}_{pair}$ $(q=n,p)$ being given by Eqs. (9) and (16), respectively.

Using the above prescription, we determine the nuclear energy as function of
the deformation parameters $\eta$, $q_{2}$, $q_{3}$, $q_{4}$ introduced in
Section II.A which stand respectively for the non-axiality, elongation, left-
right asymmetry and neck formation of the nuclear shape. The collection of all
these energy points constitutes what we call the deformation energy or
potential energy surface (PES) of a given nucleus on a discrete four-
dimensional mesh. We have chosen a step length of $\Delta q_{2}=0.05$ for the
elongation parameter $q_{2}$ and a step length of $\Delta q_{j}=0.03$ for the
other 3 deformation parameters with a total mesh size of $n_{2}\times
n_{3}\times n_{4}\times n_{\eta}=60\times 8\times 15\times 8=57600$ nodes. We
have verified that within such a mesh size we are able to describe with a good
enough accuracy all physically relevant effects, like local minima, saddle
points, the formation of valleys and ridges.

##  III Multidimensional WKB method

The WKB method is a semi-classical approximation which is widely used in
quantum mechanical problems to find an approximate solution of the Schrödinger
equation implying a potential barrier that a particle has to overcome. The
main assumption is that, under the influence of the potential, the particle
wave function can still be expressed in terms of a plane wave, but with a
momentum $k(x)$ which is position-dependent and slowly varying with $x$.

###  III.1 Lifetimes for spontaneous fission

In our approach we have used a multidimensional version of the above
characterized WKB approximation to calculate the lifetime of a nucleus
undergoing spontaneous fission. This approach has been widely used in nuclear
physics for fission and particle or cluster emission to determine the
penetrability of a potential-energy barrier defined in a multidimensional
deformation space. In the following, the standard one-dimensional WKB method
will be generalised to the case of a four-dimensional deformation space, where
the deformation variables are the Fourier parameters $q_{i}$ introduced in Eq.
(6).

The first step to obtain a good-quality estimate of the lifetime of a system
undergoing spontaneous fission is to search for the so-called “least-action
path” (LAP) leading to fission in our 4-dimensional PES that a nucleus would
have to follow on its way to a splitting into fission fragments. Such an
approach treats a fission event as a dynamical process, characterized by the
collective motion of a large number of nucleons tending to elongate the
nuclear shape starting from some initial state, like the nuclear ground state
until the scission configuration is reached. Please note that the collective
space in which the fission process is simulated can generally be
multidimensional, curvilinear and non-Euclidean. In the framework of our
present approach the dynamical path to fission actually proceeds in the four-
dimensional space defined by the ($q_{1}\equiv\eta,q_{2},q_{3},q_{4}$) Fourier
deformation parameters defined through Eqs. (3)-(7). We have investigated that
when triaxiality $q_{1}$ is taken into account, one can observe along the
least-energy path a slight (up to 1 MeV) lowering of, in particular, the inner
fission barriers height.

One should keep in mind that in the here presented approach the energy
$E(q_{1},q_{2},q_{3},q_{4})$ is obtained in the macroscopic-microscopic model,
where the shell corrections in (25) are determined in the Strutinsky method,
and the correction for the residual pairing interaction in the BCS
approximation with projection onto good particle number obtained in the GCM
approach (see Ref. GCM+GOA ). In both these methods, single-particle states of
a folded-Yukawa mean-field potential ldens  are used.

###  III.2 Least-action fission path

The action in the above introduced 4-dimensional deformation space
$\\{q_{1},q_{2},q_{3},q_{4}\\}$ can be represented through the following
integral:

$\begin{split}S=\\!\\!\\!\\!\\!\\!\int\limits_{q_{2}^{(g.s.)}}^{q_{2}^{(exit)}}\\!\\!\\!\\!\\!dq_{2}\sqrt{\frac{2}{\hbar^{2}}\sum_{ij=1}^{4}\beta
B_{ij}(\\{q_{k}\\})\big{[}{\mathcal{E}}\\!-\\!E(g.s.)\big{]}\frac{\partial
q_{i}}{\partial q_{2}}\,\frac{\partial q_{j}}{\partial q_{2}}},\end{split}$
(26)

where ${\mathcal{E}}\\!=\\!E(\\{q_{k}\\})$ and $E(g.s.)$ stand respectively
for the potential energy of any configuration along the fission path and at
the ground state deformation. The integration extends from the nuclear ground
state deformation up to a so-called “exit point ” which has the same energy as
the ground state ($E_{exit}=E_{g.s.}$). In the 4-dimensional deformation space
one can obviously find many such turning points which fulfill this condition.
The problem now consists in the identification of the one particular exit
point and the corresponding path leading to it, which renders the action
integral minimal. At first sight, one may have the impression that the action
integral in Eq. (26) is computed only along a one-dimensional fission path
L($q_{2}$) instead of being defined in the multi-dimensional collective space.
That is, however, not the case, since that search for the minimal action path
is carried out in the full 4-dimensional deformation space. A similar kind of
approach has also been used e.g. in Refs. LS99 ; LBP03  to calculate the LA
integral in a multi-dimensional deformation space.

The deformation-dependent quantity $B_{ij}(\\{q_{k}\\})$, with indices $i,j$
($i=1,2,3,4$) referring to the pair of shape parameters $(q_{i},q_{j})$, is
the irrotational flow inertia tensor in the Werner-Wheeler approximation, of
Refs. DSN76 ; BN19  and presented in Fig. 3 for the diagonal components
$B_{11},\,B_{22}$ and $B_{33}$ as well as for the off-diagonal component
$B_{23}$ projected onto the $(q_{2},q_{1})$ and $(q_{2},q_{3})$ plane
respectively. The values of the remaining two deformation parameters
$q_{3},\,q_{4}$, respectively $q_{1},\,q_{4}$ are adjusted in such a way that
they minimize the action integral (26) along the least-action path.

Please notice that the deformation dependent parts of this purely macroscopic
hydrodynamical mass tensor is essentially identical for all nuclei and has
simply to be multiplied by a scaling factor proportional to $A^{5/3}$, (see,
e.g. Ref. BN19 ) to obtain the proper value for a given particular nucleus.

Figure 3: Hydrodynamical mass tensor components in the $(q_{2},q_{1})$ (a) and
$(q_{2},q_{3})$ (b-d) deformation planes. The LAP is indicated by the solid
red line.

As can be seen from Fig. 3, the components $B_{22}$ gradually increases with
the elongation coordinate $q_{2}$, but is only weakly dependent on the mass-
asymmetry parameter $q_{3}$ in the region where it would have the largest
influence on the action integral (26), namely the barrier region around
$q_{2}\leq 1$. The other crucial inertia component, $B_{33}$, changes
relatively slowly with $q_{2}$ in this region (below $q_{2}\\!\lesssim\\!1$),
but increases dramatically in the vicinity of the scission configuration
($q_{2}\gtrsim 1.5$). In contrast, the component $B_{11}$ decreases with
elongation $q_{2}$, but stays almost constant when $q_{1}$ increases. Since
the WKB action integral is determined by an interplay between potential energy
and inertia tensor, we may thus conclude that it is the potential energy
gradient in $q_{1}$ direction which will mostly contribute to the final value
of the action integral and the impact of $B_{11}$ is small in the presented
case. Please notice also that the absolute values of $B_{33}$ are much larger
than the ones of $B_{22}$, and thus may contribute substantially to the total
action (26) whenever a derivative $\frac{dq_{3}}{dq_{2}}\neq 0$, comes into
play. This essentially happens if the fission path leads to a mass-asymmetric
division, even if $q_{3}$ stays about constant in its final stage. If, in
turn, the exit point is mass symmetric ($q_{3}=0$), the observed local
variations of $q_{3}$ along the LAP are too weak to significantly contribute
to the action integral. Please notice also that for the here studied nuclei
the fission path usually starts heading towards mass asymmetric deformations
around the second minimum ($q_{2}\approx 0.7-0.8$) as this is shown for the
cases of $\,{}^{234}$U or $\,{}^{252}$No in Fig. 4. It is interesting to note
that changes of the off-diagonal mass component $B_{23}$ shown in Fig. 3(d)
favour asymmetric fragmentation.

The parameter $\beta$ in front of the $B_{ij}$ mass tensor in (26) gives the
possibility of re-scaling its six independent components all together in order
to reproduce within a couple of orders of magnitude the measured actinide
half-lives. Obviously, such an operation does not touch the relative values
between the tensor components what, in addition to the reliability of the PES,
is relevant for a realistic determination of the course of the LAP, and thus
of the resulting action value. Let us also recall that the differences in the
inertia components evaluated within several available macroscopic or
microscopic approaches may even differ by as much as one order of magnitude
robmass .

One of the distinguishing features of the macroscopic hydrodynamical mass
tensor used here as compared to its microscopic (e.g. cranking model)
counterpart lies in the fact that the latter is often a rapidly fluctuating
function of deformation, caused mainly by the microscopic shell effects. These
local variations are, to some extent, smoothed out by the method of the least-
action trajectory itself, where the corresponding fission path tends to omit
states associated with a sudden change of the potential energy or the inertia.
This has a clear impact on the stability of the numerical search for the
minimum of the action integral (26) in a multi-dimensional space of
variational parameters. For a comparison, we have applied, in addition,
another efficient prescription of the collective inertia effects simulated by
the so-called phenomenological mass parameter $B(R_{12})$, expressed in units
of the reduced mass $A_{L}A_{R}/(A_{L}+A_{R})$, with $A_{L}$ and $A_{R}$ being
respectively the mass number of the left and right nascent fission fragments
(see, e.g. Ref. DNS76 ).

$B(R_{12})=1+k\frac{17}{15}\exp\bigg{[}\lambda\,(R_{12}^{({\rm
sph})}-R_{12})\bigg{]}.$ (27)

The above phenomenological mass depends on a single parameter
$R_{12}\\!=\\!R_{12}(q_{2},q_{3},q_{4})$ (in units of the radius $R_{0}$ of
the spherical shape) describing the evolution towards fission and which is
given by the centers-of-mass distance (for a spherical shape one has
$R_{12}^{({\rm sph})}\\!=\\!0.75\,R_{0}$) of the nascent fission fragments.
The parameter $\lambda\\!=\\!0.408/R_{0}$ describes the descent rate of the
exponential function. For this purpose, in contrast to the calculations with
full hydrodynamical mass tensor, the 3D total potential energy function
$E(q_{2},q_{3},q_{4})=E(q_{1}^{0},q_{2},q_{3},q_{4})$, Eq. (25) is used, where
$q_{1}^{0}$ is the nonaxiality deformation parameter which minimizes the full
4D potential energy $E(q_{1},q_{2},q_{3},q_{4})$ at at given point in the 3D
$(q_{2},q_{3},q_{4})$ space.

Figure 4: Potential energy surfaces for 230U (a-c), 234U (d-f) and 252No (g-i)
isotopes projected onto the $(q_{2},q_{1})$, $(q_{2},q_{3})$ and
$(q_{2},q_{4})$ deformation subspaces. The projection is performed in such a
way that the other two variables $q_{k}(q_{2})$ and $q_{k^{\prime}}(q_{2})$
take values that minimize the action (26) between the equilibrium and the exit
points. The dashed red and solid black curves correspond respectively to the
LAP obtained with the hydrodynamical mass tensor and the least-energy path
(LEP).

Please notice that the magnitude of the center-of-mass distance $R_{12}$
depends essentially on the elongation $q_{2}$ and only weakly on the left-
right asymmetry and the neck formation parameters, $q_{3}$ and $q_{4}$,
respectively. For that reason the least-action fission path obtained with the
phenomenological mass of Eq. (27) cannot be called fully dynamical.

According to the main concept, the parameter $k$ in (27) is chosen so as to
ensure that the value of $B(R_{12})$ along the fission barrier (in the
vicinity of $q_{2}\approx 1$) is close to the value of the hydrodynamical mass
tensor in that area.

At the same time, it should reproduce the asymptotic behaviour of the rigid-
body inertia when a nucleus splits into two fragments, in which case the
inertia of the strongly elongated nucleus, close to the scission
configuration, should smoothly merge into the reduced mass of the two
fragments. It turned out that the optimal value of this parameter is $k=11.5$.

Let us now explain an efficient method to determine the least-action path in
our 4D deformation space with the full hydrodynamical mass tensor, which will
then be used to calculate the tunneling probability through the fission
barrier to determine the spontaneous fission half-lives, a method based on the
concept first introduced by Ritz ritz  and successfully used in particular to
study the spontaneous fission process (see e.g. Refs. baranritz ; baran81 ;
LS99 .

In order to define any path in this deformation space, one first of all
notices that any continuous and bounded function over a given finite interval
of its arguments can always be approximated by a Fourier type expansion,
involving only $sin$ functions on top of an average path whenever the
endpoints of that path are fixed. In our case these endpoints are the ground
state and the exit point which is characterized by the same energy as the
ground state. Defining that average path under the barrier by a straight line
in the 4D deformation space connecting the ground state and the chosen exit
point and considering the elongation parameter $q_{2}$ as the essential
variable responsable for the fission process, one can always approximate the
deformation parameters $q_{1}$, $q_{3}$ and $q_{4}$ along the least-action
path as functions of $q_{2}$ in the following way:

$\displaystyle q_{\nu}^{({\rm
LAP})}(q_{2})=\left[q_{{\nu}_{g.s.}}+\frac{(q_{{\nu}_{exit}}-q_{{\nu}_{g.s.}})(q_{2}-q_{{2}_{g.s.}})}{q_{{2}_{exit}}-q_{{2}_{g.s.}}}\right.\;\;\;\;$
$\displaystyle+\left.\sum_{\ell=1}^{N_{F}}a_{\ell}\,sin\left(\ell\pi\frac{q_{2}-q_{{2}_{g.s.}}}{q_{{2}_{exit}}-q_{{2}_{g.s.}}}\right)\right],\;\;\nu=1,3,4\;\;\;\;$
(28)

where the amplitudes $a_{\ell}$ of the series expansion are treated as
variational parameters relative to which the minimum of the action integral
(26) is being searched. The upper limit $N_{F}$ of the Fourier series
expansion in each direction of the least-action path (28) has to be chosen
such that the final result for the tunneling probability becomes essentially
independent of $N_{F}$. We have found that a value of $N_{F}=8$ turns out to
be sufficient to obtain a very good convergence of the Fourier series and thus
a well converged tunneling probability.

Having found the least-action integral value (with respect to the $a_{\ell}$
amplitudes), one thus obtains the evolution of this path for a given nucleus
in the considered 4D deformation space.

##  IV Results

In Figs. 4 we present, for 230U, 234U and 252No, the projections of the full
4D PES onto the 2D sub-spaces $(q_{2},q_{1})$, $(q_{2},q_{3})$ and
$(q_{2},q_{4})$, where the other two deformation parameters are the functions
of the elongation $q_{2}$ that minimize the action integral (26) between the
ground state and the true exit point. The evolution of the LAP obtained when
using the above discussed hydrodynamical mass tensor in these landscapes is
indicated by the dashed red line. These isotopes have been chosen to cover the
region from light to heavy actinides. As can be seen, the PES and the
associated LAP in these extreme cases have different characteristics.

In the lighter actinides, due to the importance of the shell effects, the PES
is showing a stronger deformation dependence than in the heavy No isotope.
Consequently, the fission barriers in uranium isotopes, unlike in nobelium,
are generally higher and longer before reaching the scission configuration.
Already from this quick qualitative analysis, one can expect a shorter half-
life for nobelium as compared to uranium, an analysis which turns out to be
consistent with the experiment. As can already be concluded from Eq. (26), the
final course of the LAP in the multi-dimensional deformation space is dictated
by the interplay between the deformation dependent PES and the inertia tensor.
This is one of the reasons why the LAP is usually shorter than the least-
energy path (LEP) and passes, in general, through higher energy configurations
(sometimes by as much as 2 MeV) as compared to the corresponding LEP. The
actions along both these trajectories can therefore differ significantly, thus
causing sometimes a difference of several order of magnitude in the estimates
of the fission half-lives.

As one can see from part (e) of Fig. 4 the LAP for 234U starts from the mass-
symmetric ground state, stays left-right symmetric ($q_{3}=0$) up to the
second minimum and then evolves towards asymmetric shapes around $q_{2}\approx
0.7$, leading finally to an asymmetric fission valley at a value of
$q_{3}\approx 0.08$. One thus concludes that beyond the second minimum it is
absolutely crucial to take into account the mass asymmetry degree of freedom.
One notices that for the 234U nucleus the LEP and the LAP stay fairly close to
each other in the $(q_{2},q_{3})$ plane. When looking at the $(q_{2},q_{4})$
subspace for the same nucleus (see panel (f) of Fig. 4), one finds that the
LAP shows only a small deviations from a linear behaviour between a compact
ground-state shape at $q_{4}\approx 0.08$ and a medium-elongated necked-in
shape at $q_{4}\approx-0.12$. Regarding the non-axiality in panel (d), the LAP
between the ground state and the second minimum goes through rather moderate
non-axial shapes and returns to the axial path at $q_{2}=0.65$. A similar
trend is observed in the super-heavy 252No nucleus, panels (g)-(i). However in
this case the LAP ends around $q_{2}=1.1$, and is thus much shorter than in
234U. For 230U (panels (a)-(c)), on the contrary, the $q_{3}$ deformation is
almost negligible in the initial and final stages of the LAP whereas at
intermediate elongations it reaches values beyond $q_{3}\approx 0.1$ to bypass
the energy maximum (barrier) peaked at $q_{2}\approx 1$. In both the discussed
uranium isotopes their LAPs $q_{4}(q_{2})$ presented in panels (c) and (f)
have comparable shapes. Interestingly, the LAP in this nucleus prefers fully
axial shapes throughout its course, even though the LEP passes through
energetically slightly lower, non-axial configurations at very small
deformations just beyond the equilibrium point. In order to keep the
computation time within reasonable limits, without making any compromise on
the precision of our results, we are able to consider up to
$N_{F}^{tot}=3\times N_{F}=24$ harmonic components of the Fourier series in
Eq. (28). In such a large number of dimensions, one may clearly encounter a
problem of distinguishing between some local and the global minimum of the
action integral. To avoid this behaviour, we start the calculations for each
nucleus with a low value of $N_{F}$, e.g. $N_{F}=3$ for each of the three
functions $q_{1}(q_{2})$, $q_{3}(q_{2})$ and $q_{4}(q_{2})$, on top of the
average path. The value of $N_{F}$ is then gradually increased, checking after
each step whether convergence is obtained. It turns out that restricting
ourselves to the first few components of these series, like e.g. $N_{F}=6-8$
leads to LAPs in Fig. 4 that visually cannot be distinguished from the ones
obtained with larger values of $N_{F}$.

Having calculated the values $S$ of the action, one can finally determine the
spontaneous fission lifetime using the standard WKB relations SSS95 .

$T_{1/2}^{({\rm sf})}=\frac{2\pi\ln(2)}{\omega_{0}}\left(1+e^{2S}\right),$
(29)

where $E_{\rm{ZPE}}\\!\approx\\!\frac{1}{2}\hbar\omega_{0}$ stands for the
zero-point vibration energy which is usually taken to be in the range of
$0.5-1\;$MeV. In the present work we have taken a value of
$E_{\rm{ZPE}}=0.5\,$MeV.

Spontaneous fission half-lives are determined for selected isotopes of some
actinide nuclei, namely thorium (Th), uranium (U), plutonium (Pu), curium
(Cm), califormium (Cf), and fermium (Fm) and for super heavy isotopes of
nobelium (Nb), rutherfordium (Rf), seaborgium (Sg), hassium (Hs) and
darmstadtium (Ds) for which experimental data are available NUDAT . The
results of the calculations obtained with the rescaled (adjusting the value of
$\beta$) hydrodynamical mass tensor and the phenomenological mass formula (27)
are presented in Fig. 5 together with the measured values. The data for the
different isotopes calculated within the above presented approach are given as
open blue circles and black triangles, while the experimental data are shown
as full red circles.

Figure 5: (Top panel) Comparison of spontaneous fission half-lives for
actinide nuclei obtained in our 4D WKB approach with the irrotational flow
hydrodynamical mass tensor (open circles) and the phenomenological inertia
with 3D PES, Eq. (27) (open triangles) with the experimental data (full red
circles). (Bottom panel) Same but for super-heavy elements from nobelium to
darmstadtium.

In order to obtain some systematics for the spontaneous fission half-lives
displayed in Fig. 5 for all isotopic chains of actinides and super-heavy
elements up to $Z\\!=\\!110$, we have adjusted the parameter $\beta$ in Eq.
(26) through a $\chi^{2}$ minimization to all the 39 measured half-lives of
actinide nuclei from thorium to fermium presented in this figure. It is found
that the half-lives in all presented actinides are well reproduced with a
value of $\beta\\!=\\!5$. By choosing a smaller sample of nuclei we have made
sure that the value of $\beta\\!=\\!5$ is, indeed, practically independent of
the chosen sample. Such a $\beta$ value ensures that the logarithm of the
evaluated half-lives in super-heavy nuclei stays within reasonable limits of
approximately $2-3$ (in $1/s$ units) which is comparable with other recent
evaluations wardaPhysRevC.86.014322 ; pomshe . Let us notice at this point
that, as shown e.g. in Ref. PomGozdz , the hydrodynamical inertia used in our
approach may differ from the commonly used microscopic mass tensors obtained
within the cranking or the GCM+GOA model by almost a factor of $3-5$ on
average, particularly for compact nuclear shapes, ($q_{2}\approx 1$) where the
fission barrier is located. Once the $\beta$ value has been fixed, the
spontaneous fission half-lives calculations are performed for super-heavy
elements and compared with the experimental data.

Let us mention in this connection that the hydrodynamical inertia tensor has
been successfully used in calculations of fission properties determined by
shapes lying close to the scission configuration, such as fragment mass or
charge distributions, whereas the barrier penetration occurs at significantly
lower elongations around the fission barrier (see e.g. Ref. chipom ). As seen
in Fig. 3, the inertia components $B_{22}$ and $B_{33}$, most crucial for the
barrier penetration, are much smaller in the vicinity of the barrier region
than the ones close to the scission point. One has to keep in mind, however,
that the pure hydrodynamical approach seems, in its original form, not really
well suited for a reliable description of the effective inertia near the
barrier. The phenomenological mass parameter (27), on the other hand, contains
the rigid-body inertia as the essential contribution together with a term
(controlled by the parameter $k$) determined by the difference between the
rigid body and the irrotational flow inertia, thus making it more reliable in
the fission-barrier region. Investigating through the results presented in
Fig. 5 the capacity of our approach using the hydrodynamical inertia tensor to
reproduce the experimental fission half-lives for actinides, one could be
quite satisfied. There are, however, a few cases, that stick out from their
isotopic systematics by several orders of magnitude, which we would like to
understand a little better. These are e.g. some isotopes of Cm and the
heaviest Cf nuclei. To explain these discrepancies one may refer e.g. to a
recent work KP22 , where it is shown, within a simple analytical 1D WKB
approach, similar to the so called Światecki-like systematics swiatecki55  of
spontaneous fission half-lives, that the main quantity determining the fission
half-life is the fission barrier height $E_{B}$. Its dependence on other
properties of the fission barrier, including the barrier width, is already
somehow absorbed in the adjustable function $f(E_{B})$ (given by equations
(24) and (25) of Ref. KP22 ), common for all heavy and super-heavy elements.
At this point it may be worth to recall that even a small change in the
fission barrier height, leads to a substantial decrease or increase of the
tuneling probability and, as a consequence, produces longer or shorter fission
half-lives. Another interesting case is the one of the 232-234U isotopes,
where our macroscopic-microscopic model underestimates the fission barrier
heights by about $1-2$ MeV (see also Ref. Josepaper ) causing an
underestimation of the resulting fission half-lives by as much as some 2-4
orders of magnitude. A similar effect can also be observed for the 242-246Cm
isotopes, where the discrepancies between the experimental and theoretical
first and second barriers are the largest throughout the whole isotopic chain.

The reason for the overestimation of the lifetimes for the superheavy
258-262No and 256-260Rf isotopes is presumably similar to that for the
aforementioned actinides except that we do not know yet the experimental
barriers to be able to make quantitative comparisons.

It is interesting to also present in Fig. 5 the results of half-lives
calculations for actinide nuclei obtained with the phenomenological inertia of
Eq. (27) at the three-dimensional PES, where the total 4D energy function is
minimized, for a given $(q_{2},q_{3},q_{4})$ point, with respect to the
nonaxiality parameter $q_{1}$. Comparing the results obtained with both here
presented approximations, one observes that the half-lives with the 4D WKB
dynamics and full hydrodynamical inertia tensor with a scalling factor
$\beta\\!=\\!5$ (common for all tensor components) are in a majority of cases
much closer to the experimental data than those obtained with the
phenomenological scalar mass given by Eq. (27). This shows, we believe, that
considering the full description of the inertial properties of a system at
each deformation point, as well as including as many degrees of freedom
relevant for the fission dynamics as ever possible leads to a substantial
improvement of the fission-lifetime estimates.

##  V Conclusions

Spontaneous fission half-lives for nuclei in the range
$90\\!\leq\\!Z\\!\leq\\!104$ have been determined in the macroscopic-
microscopic approach together with the Lublin-Strasbourg Drop model, a mean-
field generated by a Yukawa-folding procedure and a constant $G$ seniority BCS
pairing treatment with a GCM+GOA particle-number projection. The dynamics of
the fission process have been simulated by the semiclassical WKB method with
the least-action integral describing the evolution of the nucleus in a four-
dimensional deformation space given by the expansion coefficients of a Fourier
shape parametrization which stand for elongation, mass asymmetry, non-axiality
and neck degrees of freedom.

In order to take into account the variation of the collective inertia along
the fission path, we have used in the action-integral expression the
irrotational flow mass tensor which has been scaled by a factor of 5 in order
to reproduce fission half-lives in the actinide region. Since the resulting
least-action path to fission tends, to some extent, to omit states where the
inertia changes dramatically, due to the presence of shell effects, the use of
this effectively macroscopic model of collective inertia should reduce the
numerical instabilities in the search of the minimum of the action integral.
For a comparison, we have also performed similar calculations of fission
lifetimes with a collective mass parameter, Eq. (27), which has also led to a
quite reasonable reproduction of fission half lives over a wide range of
actinide and super-heavy nuclei.

One notices that both these inertia approaches generally yield quite close
values of the spontaneous fission half-lives $T_{1/2}$, particularly in super-
heavy nuclei and in the actinide isotopes of thorium, uranium, plutonium and
curium, while in californium and fermium, the use of the collective mass
parameter, Eq. (27) and the 3D PES, leads to a mean deviation reaching several
orders of magnitude relative to the experimental results. One should also be
aware of the fact that the phenomenological mass formula (27) is a kind of
hybrid approach which, in order to reproduce the spontaneous fission half-
lives, combines two different evaluations of the nuclear collective inertia,
namely the hydrodynamical and the rigid-body approaches, and requires, to do
that, 3 adjustable parameters, whereas our approach using the irrotational
flow mass tensor yields better results with a single adjustable parameter,
namely the parameter $\beta$ in the expression (26) for the action integral.
Let us mention at this point that no zero-point energy correction is needed in
our approach and would, in fact, be in disaccord with the philosophy of our
macroscopic-microscopic model which, similar to a mean-field Hartree-Fock type
framework, describes the nuclear energy by some kind of variational approach,
where there is no place for an artificial increase or lowering of the ground-
state energy.

One should also keep in mind that spontaneous-fission is only one of several
possible nuclear decay channels, competing with the emission of light
particles (like n or p), $\gamma$ quanta, or the emission of light clusters
(like e.g. $\alpha$ particles). The competition between fission and these
other processes is something we are presently working on, and will be the
subject of a forthcoming publication.

###### Acknowledgements.

This work is supported by the COPIN-IN2P3 agreement (Project No. 08-131)
between the Polish and French nuclear laboratories and the Polish National
Science Center (Project No. 2018/30/Q/ST2/00185). The research work of A.Z. is
part of the project No. 2021/43/P/ST2/03036 co-funded by the National Science
Centre and the European Union Framework Programme for Research and Innovation
Horizon 2020 under the Marie Skłodowska-Curie grant agreement no. 945339. For
the purpose of Open Access, the author has applied a CC-BY public copyright
licence to any Author Accepted Manuscript (AAM) version arising from this
submission.

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