# Description of the multidimensional potential energy surface in fission of
252Cf and 258No

A. Zdeb<EMAIL_ADDRESS>CEA, DAM, DIF, F-91297 Arpajon, France  Université
Paris-Saclay, CEA, LMCE, F-91680 Bruyères-le-Châtel, France  M. Warda
<EMAIL_ADDRESS>Instytut Fizyki, Uniwersytet Marii Curie–Skłodowskiej,
PL-20031 Lublin, Poland  L. M. Robledo<EMAIL_ADDRESS>Departamento de
Física Teórica and CIAFF, Universidad Autónoma de Madrid, Madrid, Spain
Center for Computational Simulation, Universidad Politécnica de Madrid, Campus
de Montegancedo, Bohadilla del Monte, E-28660 Madrid, Spain

###### Abstract

The microscopic studies on nuclear fission require the evaluation of the
potential energy surface as a function of the collective coordinates. A
reasonable choice of constraints on multipole moments should be made to
describe the topography of the surface completely within a reasonable amount
of computing time. We present a detailed analysis of fission barriers in the
self-consistent Hartree-Fock-Bogoliubov approach with the D1S parametrization
of the Gogny nucleon-nucleon interaction. Two heavy isotopes representing
different spontaneous fission modes - 252Cf (asymmetric) and 258No (bimodal) -
have been chosen for the analysis. We have shown the existence of complicated
structures on the energy surface that can not be fully described in two-
dimensional calculations. We analyze apparent problems that can be encountered
in this type of calculations: multiple solutions for given constraints and
transitions between various potential energy surfaces. We present possible
solutions on how to deal with these issues.

spontaneous fission, potential energy surface, microscopic methods, Cf-252,
No-258

##  I Introduction

The accurate description of collective nuclear motion from the ground state up
to the scission point represents a crucial step to understand the fission of
the atomic nucleus. An essential ingredient is the behavior of the binding
energy of the nucleus as a function of the parameters characterizing the
collective degrees of freedom. On its way from the initial ground state
configuration to scission, the nucleus has to tunnel through a potential
energy barrier determining the time scale of fission half-lives. Not only the
height but also the width and the shape of the fission barrier are important
for spontaneous fission half-lives estimations, while the topography of the
potential energy surface (PES) is essential to determine fission dynamics. A
detailed description of the theory of fission and challenges in this field can
be found in the review papers: Refs. Krappe and Pomorski (2012); Schunck and
Robledo (2016); Bender et al. (2020).

The first historical attempt to explain and describe the fission process was
based on the semi-empirical liquid drop model assumptions Bohr (1936) where
the interplay between Coulomb repulsion and surface energy of a charged and
deformed drop of nuclear matter is responsible for the splitting Meitner and
Frisch (1939). The success of this simple approach suggested that collective
variables (in an intrinsic frame of the nucleus), associated with the shape
evolution of a fissioning system, are a crucial ingredient in any microscopic
analysis of fission process. The adiabatic approximation - based on the
assumption that the time-scale of nucleons motion is much shorter than the
time-scale of shape changes - allows obtaining a consistent description by
considering just a small set of collective degrees of freedom well adapted to
the shape evolution involved in fission.

Many theoretical papers describing the PES in heavy actinides have been
published recently improving our knowledge of fission of heavy and super-heavy
nuclei Pashkevich et al. (2009); Möller et al. (2009); Mirea et al. (2010);
Abusara et al. (2010); Jachimowicz et al. (2012); Lu et al. (2014); Rodríguez-
Guzmán and Robledo (2014); Pomorski et al. (2015); Rodríguez-Guzmán and
Robledo (2016); Schmitt et al. (2017); Jachimowicz et al. (2017); Pomorski et
al. (2018); Chai et al. (2018); Jachimowicz et al. (2020a, b); Rodríguez-
Guzmán et al. (2020). In those articles, one can distinguish two different
approaches to describe the shape of the nucleus in the fission process. Both
are based on the minimization of the energy along the different paths
connecting the starting configuration (usually the ground state) with the
scission configurations. In the first approach, which is mostly used with
macroscopic-microscopic theories, a pre-defined class of nuclear shapes
defined uniquely in terms of a given set of deformation parameters is used to
define the set of accessible configurations to be used in the minimization of
the energy. The number of parameters defines the dimensionality of the
problem. An increasing number of dimensions increases the variety of the
shapes used and, therefore, usually improves the calculation’s quality by
providing lower energy solutions. This flexibility comes at the expense of
calculating the energy for a huge number of points in the deformation space.
On the other hand, the approach benefits from full control over the shapes
characterizing the evolution of a nucleus from the ground state to scission.
The alternative approach is mostly used by the microscopic self-consistent
methods. In the self-consistent procedure, the wave functions along the
fission path are determined by the minimization of the energy of the nucleus
within a given set of constraints and assumed symmetries. The constraints only
provide general guidelines for the nuclear density distribution and do not fix
its shape uniquely. For a given set of constraints, the nuclear shape is free
to take any form among the allowed ones as to minimize the energy of the
nucleus. The advantage of this procedure is that we do not have to worry if
our space of deformation parameters is large enough to describe the shape
adopted by the nucleus. Nevertheless, the results provided with this method
are not always unique Dubray and Regnier (2012) as one can easily land for a
given set of constraints in a local minimum instead of the absolute one. As a
consequence, the self-consistent minimization procedure can produce several
solutions with the same value of the constraints but different energies
depending, for instance, on the starting wave function used in the solution of
the variational equations. Those “multivalued” solutions are connected in a
higher dimensional deformation space by a barrier separating them. A typical
example, often seen in axially symmetric calculation constraining in the
quadrupole moment, is the existence of two solutions with the same quadrupole
moment but different values of the octupole moment in the region of the second
barrier. The two coexisting minima are linked by a path going through the
octupole moment in the parameter space. One of the unwanted consequences of
this kind of situation is the possibility to jump between configurations when
considering the behavior of quantities as a function of the current shape
parameter. The leap is unphysical because one is skipping the path connecting
the coexisting minima. When describing this situation, one talks about jumping
from one fission valley to another. Whether the dynamics of fission justify
this transition or it is just an artifact due to the limited number of
constraints used to describe the fission path is a delicate issue that will be
the subject of the present paper.

If the constraints are partially released, several different solutions may
exist as local minima of the potential energy. In this case, one has to decide
which of them should be considered in the analysis of fission. We may even
encounter a situation where on the edge between two valleys, a fake or missing
saddle can be found Dubray and Regnier (2012). The first case occurs when two
distinct nuclear shape configurations have got the same energy at some
coordinate, and the fission paths are incorrectly linked together. The missing
one means that minimizing the energy with a single constraint may pass over a
saddle leading to another valley. The consequence of a wrong interpretation of
the PES may have a non-negligible impact on the description of fission
dynamics. The barrier heights may be improperly evaluated, and the fission
valleys may be omitted or redundant.

It has been previously shown that non-geometrical constraints like the one
associated with pairing correlations may strongly influence the fission
dynamics scenario because of the strong dependence of the collective inertias
on the inverse of the pairing gap Urin and Zaretsky (1966); Moretto and
Babinet (1974); Sadhukhan et al. (2014); Giuliani et al. (2014); Rodríguez-
Guzmán and Robledo (2018); Schunck et al. (2014). As a consequence, the
inclusion of the pairing degree of freedom can substantially modify the
dynamical evolution of a collective wave packet. Moreover, this kind of
constraint can play a central role if the least action principle is used
instead of the least energy one to determine the fission path. The treatment
of spontaneous fission using the least action principle and including the
pairing degree of freedom leads to a substantial reduction of the spontaneous
fission life time Giuliani et al. (2014); Rodríguez-Guzmán and Robledo (2018);
Schunck et al. (2014); Sadhukhan et al. (2014). However, and despite its
relevance, we are not considering the pairing degree of freedom in the present
study.

This paper is devoted to the discussion of the choice of the multipole moments
to be used as constraints for the calculation of the PES leading to fission in
the self-consistent methods. The primary purpose is to determine which
subspace of collective coordinates in a deformation space is sufficient for an
accurate description of static properties of fissioning nuclei. We will
analyze the mechanism of creating multiple solutions on the PES in the limited
dimensionality of the space. The consequences of this behavior will be
discussed. We will indicate regions where the discontinuity of the PES may be
found.

To carry out our investigation, we have chosen two isotopes representing two
different fission modes - asymmetric (252Cf) cfe (2011); Göök et al. (2014);
Weber et al. (1981) and bimodal (258No) Hulet et al. (1986, 1989). With this
choice, we cover a broad and representative range of fission modes in heavy
nuclei. The total half-life of the radioactive isotope 252Cf is
$t_{1/2}=2.645(8)$ y with alpha decay as the dominant branch. Only 3.092(8)%
of the fissioning nuclides undergo spontaneous fission. On the other hand,
258No has a far shorter half live $t_{1/2}=1.2(2)$ ms and fission is the
principal decay channel Audi et al. (2017).

##  II Theoretical framework

The present study has been carried out using the self-consistent constrained
Hartree-Fock-Bogoliubov (HFB) method with the finite-range density-dependent
nucleon-nucleon Gogny interaction. We use the well-reputed parametrization D1S
Berger et al. (1990) that has been extensively used in the literature to
describe many different nuclear structure phenomena Péru and Martini (2014);
Robledo et al. (2018) including the microscopic description of fission Berger
et al. (1991); Warda et al. (2002); Delaroche et al. (2006); Baran et al.
(2015); Warda and Egido (2012).

In the HFB method Schunck and Robledo (2016); Robledo et al. (2018), the
nuclear states are obtained as the solutions of the HFB equation, which is
derived by requiring that the mean value of the routhian is a minimum:

$\delta(\left\langle{\Phi}\right|\widehat{H}-\lambda_{Z}\widehat{Z}-\lambda_{N}\widehat{N}-\sum\limits_{ij}\lambda_{ij}\widehat{Q}_{ij}\left|{\Phi}\right>)=0.$
(1)

Here $\widehat{H}$ is the microscopic Hamiltonian, $\lambda_{N}$ and
$\lambda_{Z}$ are the Lagrange multipliers used to fix the number of neutrons
$N$ and protons $Z$, while $\lambda_{ij}$ are the Lagrange multipliers
associated with the average value of the multipole moments $\hat{Q}_{ij}$ with
multipolarity $i$ and projection $j$. In this work the quadrupole $(Q_{20})$,
octupole $(Q_{30})$, hexadecapole $(Q_{40})$ and triaxial quadrupole
($Q_{22}$) deformation parameters are considered. The equation is solved by
expanding the creation and annihilation quasiparticle operators of the
Bogoliubov transformation in a harmonic oscillator basis with oscillator
length parameters optimized for each set of collective deformation parameters
as to minimize the binding energy. Most of the results are obtained in the
axial regime with an axially symmetric deformed oscillator basis with
$N_{\perp}=15$ and $N_{z}=22$. This basis is well suited to describe elongated
shapes along the $z$ axis as those typical of fission. Beyond mean-field, two-
body kinetic energy correction and rotational energy correction are included
in the calculation of the binding energy. In the calculations with non-zero
triaxial multipole moment $Q_{22}$ reflection symmetry is preserved, and
therefore odd multipole moments are zero by construction and not considered in
the discussion. For the “triaxial” calculations an oscillator basis containing
$N=18$ shells is used.

The multipole moment operators are defined as

$\hat{Q}_{lm}=\frac{1}{\sqrt{2(1+\delta_{m0})}}(\hat{M}_{lm}+r_{m}(-1)^{m}\hat{M}_{l-m})$
(2)

with $r_{m}=1$ if $m\geq 0$ and -1 if $m<0$. The raw multipole operators
$\hat{M}_{lm}$ are given by

$\displaystyle\hat{M}_{lm}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{4\pi}{2l+1}}r^{l}Y_{lm}(\theta,\varphi)$
$\displaystyle=$
$\displaystyle\sqrt{\frac{(l-m)!}{(l+m)!}}r^{l}P_{lm}[cos(\theta)]e^{im\phi}\;,$

where $Y_{lm}$ are spherical harmonics and $P_{lm}$ are associated Legendre
polynomials. Using the standard definition of the spherical harmonics we
obtain $\hat{Q}_{20}=z^{2}-\frac{1}{2}(x^{2}+y^{2})$,
$\hat{Q}_{30}=z^{3}-\frac{3}{2}(x^{2}+y^{2})z$,
$\hat{Q}_{40}=z^{4}-3(x^{2}+y^{2})z^{2}+\frac{3}{8}(x^{2}+y^{2})^{2}$ and
$\hat{Q}_{22}=\sqrt{3}/2(x^{2}-y^{2})$.

The computations were performed using the self-consistent HFB solver HFBaxial
that uses the approximate second order gradient method Robledo and Bertsch
(2011) to solve the HFB equations and the formulas of Egido et al. (1997) for
an accurate evaluation of the matrix elements of the Gaussian central
potential for the large basis required in fission. The program starts from an
initial wave function and iteratively minimizes the energy subject to a given
set of constraints. In this procedure, the shape of the nucleus is adjusted to
fulfill the conditions imposed by the constraint parameters. Usually, the
initial configuration is taken from a neighbor, previously computed, wave
function. As the initial configuration is expected to be “close” to the sought
solution this choice represents a way to decrease the number of iterations
required in the minimization process and therefore, represents substantial
computer time savings. The final solution may depend on the starting point in
specific cases. This feature will be thoroughly discussed below.

As it has been mentioned in the Introduction, the use of a limited set of
constraints always introduces some uncertainties in which wave functions
belong to the neighborhood of which local minima. To help in the
characterization of the different solutions of the HFB equation it is
convenient to use a technique based on searching for discontinuities in the
matter density distribution all over the PES. The so-called density distance
$D_{\rho\rho^{\prime}}$ Dubray and Regnier (2012) has been used to detect such
discontinuities in calculations considering the pair of shape parameters
$Q_{20}-Q_{30}$ as well as $Q_{30}-Q_{40}$. For given configurations with
matter density distributions $\rho$ and $\rho^{\prime}$, the density distance
is defined as:

$D_{\rho\rho^{\prime}}=\int|\rho({\bf r})-\rho^{\prime}({\bf r})|d\bf r\;.$
(4)

Density distance remains small when nuclear shapes are similar, and it
increases wherever there is a substantial change of nuclear shape for the two
densities considered.

##  III Results

###  III.1 The PES in many coordinates

Figure 1: Various fission paths for 252Cf (top) and 258No (bottom) are plotted
as a function of $Q_{20}$. See text for details.

Figure 2: The PES of 252Cf (top) and258No (bottom) as a function of $Q_{20}$
and $Q_{30}$ are shown as contour and color plots. Constant energy contours
are plotted every 1 MeV. Asymmetric fission path is represented by a blue
dashed line.

Figure 3:  The PES in $Q_{20}-Q_{22}$ plane for 252Cf (top) and 258No (bottom)
are shown as contour and color plots. The iso-energy contour lines are plotted
every 1 MeV. The dashed line shows the fragment of the fission path around the
first barrier with non-zero triaxial deformation.
Figure 4: The PES of 252Cf in a $Q_{30}-Q_{40}$ plane is shown as a contour
and color plot. Constant energy contours are plotted every 2 MeV. The energy
color scale is the same as in Fig. 2. See text for details. Figure 5: The same
as in Fig. 4 but for 258No.

The theoretical description of fission is based on the analysis of the
topography of the PES when represented as a function of the deformation
parameters related to the elongation of the nucleus. More elaborated studies
also include parameters related to reflection asymmetric shapes as they are
required for the description of the asymmetry in fission fragment mass
distribution. The inclusion of several shape parameters is also helpful in
better characterizing not only the height but also the shape of the fission
barrier. The shape of the fission PES of 252Cf and 258No are shown in Fig. 1
as a function of the quadrupole deformation. In Fig. 2, the PES maps in the
quadrupole-octupole plane are displayed for the same isotopes. The mesh points
used to obtain the maps are calculated every $\Delta Q_{20}$= 5 b and $\Delta
Q_{30}$=4 b3/2.

The results presented in Fig. 2 have been calculated in the axial regime.
However, non-axial shapes are crucial to describe the region of the first
fission barrier correctly, and therefore we present the influence of triaxial
deformations on the PES in Fig. 3. To simplify the calculation, reflection
symmetry is preserved in these triaxial calculations, and therefore all the
shapes have zero octupole moment. Usually, it is assumed that the information
obtained from the aforementioned one- and two-dimensional plots together with
the collective inertia is sufficient to describe the details of the fission
process from the ground state till scission. Of course, they contain the key
elements of the fission process like fission barrier heights and fragment mass
asymmetry. Nevertheless, there are several limitations to this approach. First
- the collective motion not only explores the minimum energy path (or the
minimum action path) but also its neighborhood. There is a finite probability
that the collective wave packet representing the evolution of the system
towards fission explores regions close to the classical trajectory Goutte et
al. (2005); Zdeb et al. (2017a); Bulgac et al. (2019). In this respect, the
width of the valley, or - in other words - the stiffness of the potential
energy, affects the spread of the collective wave packet and therefore the
fragment mass distribution. Departure from the classical least energy fission
path implies considering a broader set of nuclear shapes like those obtained
by modifying the neck width and length or the shape of the pre-fragments.
These variations can lead to a change of the obtained mas asymmetry at the
scission point. Special care should be placed on a possible reduction of the
neck thickness as it determines the scission configuration.

Another problem is that points on the $Q_{20}-Q_{30}$ map are defined as local
minima of the energy with given constraints on quadrupole and octupole
moments. There is no guarantee that the minimum is unique. In fact, it was
shown that sometimes there might exist multiple local minima in the energy
Tsekhanovich et al. (2019). These minima create different surfaces in the same
place of the $Q_{20}-Q_{30}$ map In such a case, one has to decide which of
them should be taken under consideration for fission dynamics and,
unfortunately, the answer is not straightforward.

Thus, we observe that other degrees of freedom may affect some aspects of the
theoretical description of fission. It is even the case when a two-dimensional
map in the self-consistent calculations is created from nuclear shapes
optimized in the procedure of energy minimization, not just created by a two-
parameter formula. To better describe and understand fission, one has to
increase the size of the considered space of constraints on which the PES is
spanned and look at it from a broader perspective. In this way, one should be
able to compare all the available valleys and deduce which path should be
preferred by the evolving system.

In order to extend the space of deformation, the most natural and most used
coordinate is the next term in the multipole expansion: the hexadecapole
moment $Q_{40}$, responsible for necking Tsekhanovich et al. (2019). By
decreasing the value of the hexadecapole moment, shapes with a thinner neck
are obtained. The alternative option is applying the constraint on the neck
parameter $Q_{N}$ Warda et al. (2002, 2012). It produces the same effect as
the hexadecapole moment, but it is more sensitive at the scission region and
less precise elsewhere.

To visualize a three-dimensional PES is a difficult task that can somehow be
facilitated if one of the variables is kept fixed and the PES for the
remaining two variables is plotted as a contour plot. Such a procedure has
been followed in Fig. 4 for 252Cf and Fig 5 for 258No where we show sections
of the PES for fixed values of $Q_{20}$ as maps in the $Q_{30}-Q_{40}$ space.
The black, blue, and red dots correspond to the least-energy fission paths
found on the PES spanned on the $Q_{20}-Q_{30}$ space. The black dots
correspond to the lowest-energy minimum, the blue ones stand for the next-in-
energy local minimum, and the red dots indicate the post-scission minimum
corresponding to two fragment solution. The blue, dashed lines show the
results of 2-dimensional calculations, where the hexadecapole moment is self-
consistently given by the minimization of the total HFB energy with double
constraints on $Q_{20}$ and $Q_{30}$. The thick black line represents scission
configurations and will be discussed below.

The analysis of the fission barrier in the next subsections will be based on
all the above mentioned plots of the PES.

###  III.2 Detailed description of the PES

The fission barriers and the PESs obtained for the two nuclei considered agree
with the expectations for nuclei in the heavy-actinides region. Both isotopes
are prolate in their ground states with $Q_{20}=16$ b. On the quadrupole-
octupole map of the PES, one can see a fission valley heading towards large
octupole deformation starting at the ground state. It describes super-
asymmetric fission strongly related to cluster radioactivity Warda and Robledo
(2011); Warda et al. (2018). The minimum of the energy corresponding to this
valley can be seen in Figs. 4 and 5 at $Q_{20}=30$, 40 and 50 b and large
octupole moments. The saddle point in the super-asymmetric valley reaches over
20 MeV and in heavy actinides, leads to an exotic decay mode Warda et al.
(2018), not observed experimentally. We will not discuss this type of fission
here.

The first reflection-symmetric barrier is located at around $Q_{20}=30-35$ b.
It is well known that triaxial deformation reduces its height Delaroche et al.
(2006); Abusara et al. (2010); Chai et al. (2018). In Figs. 1 and 3 we can see
that the barrier width stays unchanged by including triaxiality but the sharp-
peaked summit of the axial barrier is cut off by 4.7 MeV in 252Cf and 3.7 MeV
in 258No. The fission barriers including triaxiallity are 7.0 MeV and 7.3 MeV
high, respectively. The experimental value for 252Cf is 5.3 MeV Britt (1980);
Smirenkin (1993). Triaxial deformation of the fission path is relatively small
$\gamma\leq 12^{\circ}$. At around $Q_{20}=50$ b the nucleus goes back to
fission through axial shapes, and triaxiality is negligible for larger
elongations.

The modification of the landscape by including triaxial deformations is non-
negligible, but its influence on the spontaneous fission half-life depends on
the value of the collective inertia along the triaxial path as compared to the
axial one. There are indications that triaxiality should not affect fission
dynamics since the energy decrement at the saddle is compensated in the
calculation of the collective action by the increase of the collective inertia
and therefore, the tunneling probability is larger along the axial path Zhao
et al. (2015); Delaroche et al. (2006). Please note that despite the
similarity in barrier heights, the experimental fission half-lives of the two
isotopes differ by 12 orders of magnitude Wang et al. (2012). The key to
understanding this result, implying quite different barrier penetrabilities is
the existence of a second barrier in 252Cf that disappears in the 258No case,
reducing the effective width of the total barrier dramatically. In
consequence, theoretical fission half-lives are
$\log_{10}(t_{1/2}/\mathrm{s})=8.74$ for 252Cf
$\log_{10}(t_{1/2}/\mathrm{s})=-1.94$ for 258No (calculated in the axial
regime). These values are less than one order of magnitude away from
experimental data.

The second minimum can be found at an energy as low as 1.3 MeV above the
ground state at $Q_{20}=50$ b in 252Cf and 0.2 MeV below the ground state at
$Q_{20}=55$ b in 258No. In both isotopes, the shapes of the nucleus are
axially and reflection symmetric in the second minimum. A well pronounced
fission valley with non-zero octupole deformation opens up at larger
elongation in both isotopes. The topography of the PES beyond this point is
crucial for determining the fission fragments mass asymmetry and fission half-
lives. The key factor is whether the nucleus would prefer to stay in the
symmetric fission path or rather turn into the octupole valley. The
substantial discrepancy between 252Cf and 258No can be found in the shape of
the symmetric fission barrier and the shape of the energy surface around
$Q_{20}=100$ b for small octupole deformation. The difference is relatively
small in absolute values but provides important consequences for the fission
properties.

One can notice in the 252Cf PES that around deformation $(Q_{20},Q_{30})$ =
(100 b, 0 b3/2) a 4.5 MeV high second barrier arises, which blocks the
symmetric fission channel in this isotope. In fact, from $Q_{20}=70$ b, there
is no local symmetric minimum on the PES (there is a peak not saddle in the
two-dimensional plot), and hence the barrier is plotted with a dash-dotted
line in Fig 1. In this region, remaining at zero octupole moment is
energetically unfavorable as the potential energy grows 3.2 MeV above the
second minimum and the asymmetric valley is easily reachable with small energy
costs. We can see in Figs. 1 and 4 that the asymmetric valley starts already
at $Q_{20}=70$ b and the second saddle is at 3.1 MeV above the ground state
(1.8 MeV above second minimum). Moreover, the energy in the asymmetric fission
path rapidly decreases, reducing the barrier width.

In contrast, the second barrier in 258No, located at $Q_{20}=65$ b, is flat
with a height only 0.7 MeV above the second minimum (0.3 MeV above the ground
state). It is almost completely hidden below the ground state and therefore
does not contribute in a substantial way to the half-life, as it was mentioned
above. At $Q_{20}=90$ b the asymmetric valley opens up without any additional
barrier (see Fig. 5). The fission process may proceed in the asymmetric mode,
but the nucleus may as well stay at the reflection symmetric path without
energetic costs. This fact explains the experimentally observed bimodal
fission mass distribution with a small (5% abundance) component with high
kinetic energy and mass symmetric distribution Hulet et al. (1986, 1989).

The topography of the $Q_{30}$ \- $Q_{40}$ planes given for fixed quadrupole
moment and shown in Figs. 4 and 5 has quite a simple structure up to the
region of the second barrier. We can see only one – or at most two – local
minima in parabolic-shaped valleys. In the final phase of the fission process,
the PES is far much complex. Already at $Q_{20}=120$ b in both isotopes, we
can see several branches of the fission valley with two or three local minima
for the same quadrupole moment. The first one is octupole deformed. Its shape
is depicted in the top panel of Fig. 6. It leads to asymmetric fission-
fragment mass distribution, and therefore we refer to it as an asymmetric
mode. The second one with $Q_{30}=0$ b3/2 and small values of $Q_{40}$ (around
50 - 70 b2) is a natural continuation of the symmetric fission path. The
nuclear shape consists here of two almost spherical pre-fragments, see the
middle panel of Fig. 6. This one is often called a compact mode of fission. By
stretching this configuration, we increase the distance between pre-fragments
and make the neck thinner up to its disappearance. The compact path is
continued after scission with two separated fragments. For a two-fragment
solution, further increasing of the quadrupole moment leads to an increase of
the distance between daughter nuclei instead of a shape change. The minimum of
the energy is obtained for a configuration with a small mass asymmetry between
fragments and, in consequence, a small octupole deformation.

The third minimum also corresponds to reflection symmetric shapes, but the
hexadecapole moment takes much larger values, over 90 b2. In this
configuration, pre-fragments are not well disentangled. The shape is almost
cylindrical, cucumber-like with a small reduction of thickness in the region
of the neck (bottom panel of Fig. 6). This solution is called symmetric
elongated mode Staszczak et al. (2009, 2013). Small reflection asymmetry can
also be found here, especially for large quadrupole moments. The corresponding
fission path is the highest in energy, and it survives up to a very large
elongation of the nucleus as the neck is not formed and a rupture of the
system is not possible without a substantial energy cost. Calculations with
the Skyrme energy density functional suggest that this fission path is
energetically comparable with an asymmetric path around the second barrier in
the nuclei from the region of heavy actinides and may play an important role
in fission Staszczak et al. (2009, 2013).

We should stress here that the crossing of fission paths in Fig. 1 cannot be
interpreted as a possible place of bifurcation or configuration change. This
is a typical example of a fake saddle point Dubray and Regnier (2012). The
fact that two or even three lines have the same energy for the same quadrupole
moment does not mean that they represent the same or similar shapes of the
nucleus. As one can see in Fig. 6 the differences in the nuclear density
distribution between fission paths are usually huge.

Figure 6: The density distribution of 258No at $Q_{20}=140$ b (left) and
$Q_{20}=200$ b (right) in various fission paths: asymmetric (top), compact
before and after scission (middle) and elongated symmetric (bottom).

The existence of two local minima and surrounding local valleys (e.g. compact
and symmetric elongated) at the same quadrupole and octupole deformation leads
straight to the multiplication of the surfaces on the traditional
elongation–mass asymmetry maps, like in Fig. 2. The blue dashed lines in Figs.
4 and 5 indicate solutions corresponding to the local minimum of the energy
for fixed $Q_{20}$ and $Q_{30}$, i.e. these data could be used to create the
PES map in the ($Q_{20}$, $Q_{30}$) space. Mixing values coming from different
valleys may lead to an ambiguous and often erroneous interpretation of the
calculated results which might eventually suggest contradictory conclusions.
As the surfaces are often similar in energy, the choice of the surface would
likely be a matter of a random selection by a numerical procedure or an
arbitrary decision of the researcher. There is also the risk of an accidental
change of the surface. This effect could easily be omitted in the analysis of
fission as the graphical plotting programs are likely to smooth out sharp
ridges. The application of the density distance parameter described below in
Subsection III.3 can be useful to prevent these non-physical interpretations.

In the analysis of Figs. 4 and 5 various scenarios for reaching pre-scission
configuration can be envisaged. The scission line is visible in these plots
for $Q_{20}\geq 120$ b as a black line (consequence of plotting many closed
energy contour lines) separating single shapes (above) from two-fragments
configuration (below). The rapid change on the energy is a consequence of the
strong dependence of the neck with multipole moment parameters: slight changes
lead to a strong reduction of the neck and eventually to the splitting of the
nucleus (see discussion below).

Beyond the region of the second fission barrier ($Q_{20}=100$ b), the PES goes
down towards scission. The pre-scission line, i.e. the line of the most
elongated shapes before the rupture of the neck, is clearly seen at Fig. 2 as
a few MeV high cliff that separates the fission valley from the two-fragments
configuration. There are several ways of reaching a scission configuration:
The first is through a symmetric compact mode. Increasing elongation leads to
a relatively smooth shape evolution from pre- to post-scission configuration.
Nevertheless, even if a energy fall is not noticed, an abrupt change in the
nuclear density distribution around the neck region is visible (e.g. see Warda
and Zdeb (2015) for the 258Fm case). The two dimensional $Q_{30}-Q_{40}$
sections of the PES between $Q_{20}=120$ b and 130 b in 252Cf and between 150
b to 160 b in 258No do not differ substantially among them despite a few MeV
energy drop in the minimum of the valley. It indicates that the pre- and post-
scission density distributions of the whole system are relatively similar
despite the neck rupture. Another, most intuitive, way of reaching a scission
point from the asymmetric fission valley is by going along the fission path
(marked with the blue dashed line in Figs. 1 and 2 and black dots in Figs. 4
and 5) to the largest possible quadrupole moment. As the energy decreases
simultaneously with increasing elongation, the largest gradient indicates this
is the most probable fission scenario. The large mass asymmetry
$A_{H}/A_{L}\sim 140-142/112-116$ obtained before the neck rupture in this
point corresponds to the experimentally observed most probable fission
fragment asymmetry.

The two scenarios presented above for the evolution of the nuclear shape
assume that the system remains on the fission paths presented in Fig. 1.
Therefore, only one or two particular shapes of the nucleus are taken at
scission. Such analysis can explain the fission modes observed in the
experiment, but it is insufficient to reproduce the details of the fission
fragment mass distribution. The reason is that on its way from the saddle to
scission, the collective wave packet may explore different configurations away
from the lowest energy fission path, still fulfilling the condition of the
descending energy but not with the largest gradient. In this way, every
configuration of the scission line in the neighborhood of the asymmetric
fission valleys which are visible in Fig. 2 is accessible, of course with a
reduced probability Zdeb et al. (2017a). These configurations can be observed
in Figs. 4 and 5. For $Q_{20}\geq 140$ b the post-scission compact minimum in
the south-west corner of each panel has got lower energy than on the
asymmetric path. The latter valley seems to be soft in the direction towards
the first one, and the ridge separating them does not exceed 2 MeV. Those
figures indicate that the exit points from the asymmetric fission valley are
available already at relatively small octupole moments and with a smaller
asymmetry of nuclear shapes than at the end of the asymmetric fission path.

Two additional aspects of this scenario must be pointed out. First, a scission
point is accessible at much less elongated nuclei than at the end of the
asymmetric fission path. In some lighter nuclei, where a scission point is
above the ground state energy, this would reduce the width of the barrier. In
consequence, tunneling probability and the half-lives can be substantially
shortened Tsekhanovich et al. (2019). Second, at smaller quadrupole moment, an
exit point from the asymmetric valley at low octupole moment may be ended not
in fusion valley but in the compact fission valley before scission. This is
another mechanism of feeding the symmetric mode of fission. Thus, the
dynamical calculations of the fission mass yields for 252Cf, where only
quadrupole and octupole deformations were taken under consideration, showed a
small contribution for the symmetric mass division Zdeb et al. (2017b).

###  III.3 Surface discontinuity and density distance

Figure 7: The density distance function $D_{\rho\rho^{\prime}}$ (in
logarithmic scale) given by Eq. (4) between two neighboring configurations in
a deformation space $Q_{20}-Q_{30}$ of 252Cf (top) and 258No (bottom). Contour
lines for the energy are plotted every 5 MeV for better identification of the
different regions of the PES.  Figure 8: Fragments of the first barrier and
asymmetric valley at fixed $Q_{20}=140$ b of 252Cf. Figure 9: Fragments of the
PES of 252Cf at fixed $Q_{20}=50$ b, 70 b, 90 b and 110 b. Figure 10: The same
as in Fig. 9, but for 258No.

Figure 11: The same as in Fig. 7, but for the PES at fixed $Q_{20}$ in a
deformation space $Q_{30}-Q_{40}$ of 252Cf at $Q_{20}=70$ b (top), at
$Q_{20}=130$ b (middle) and 258No at $Q_{20}=160$ b (bottom). Contour lines of
equal energy are plotted every 5 MeV for better identification of the
different regions of the PES.

As mentioned in Section II, the density distance is an appropriate quantity to
check the consistency of the PES and pinpoint possible discontinuities of the
surface that may lead to false conclusions concerning fission dynamics. In
Fig. 7 we have plotted, for each point in the $(Q_{20},Q_{30})$ plane, the
largest of the two density distances computed for the two configurations
$(Q_{20}+\Delta Q_{20},Q_{30})$ and $(Q_{20},Q_{30}+\Delta Q_{30})$. The
figures look pretty similar for both isotopes and a few regions of large
$D_{\rho\rho^{\prime}}$ can be identified:

  1. A -

first barrier;

  2. B -

around the second minimum;

  3. C -

between symmetric and asymmetric valley at $Q_{30}\sim 15$ b3/2;

  4. D -

bordering asymmetric valley at large octupole deformation;

  5. E -

scission line.

We will denote these regions as “regions of discontinuities” due to the large
difference in densities corresponding to neighboring points indicating on
possible abrupt change of configuration.

The first region of discontinuity (A) is encountered already at the first
barrier. The nucleus has a double cone (diamond) shape on the up-going part of
the barrier, whereas a two-center structure is created beyond the maximum of
the barrier, as it can be seen in the left panels of Fig. 8. Both density
distributions are clearly different with a significant jump in hexadecapole
moment. A sharp peak of the first fission barrier also indicates a sudden
change of configuration, and the fake or missing saddle mentioned above is
usually present (Dubray and Regnier, 2012) because two solutions can be found
for the same quadrupole moment around the peak. The density distance
calculated in this region does not take into account the influence of
triaxiality discussed in Subsection III.1 that eliminates discontinuities.

In the right panels of Fig. 8, the region (D) of enhanced density distance is
observed. As we can see, there is no discontinuity here but only a change of
the hexadecapole moment can be observed in this region. The source of this
kink can be explained by looking at Figs. 4 and 5. The valleys plotted in the
$Q_{30}-Q_{40}$ planes around $Q_{20}=140$ b lie along a more or less straight
line with a slope of the order of 1. We determine the localization of the
bottom of these valleys by minimization with a constraint on the octupole
moment (marked by blue dashed lines). This procedure sometimes does not give
the correct value that is obtained by following the direction of the gradient
along $Q_{30}$ and $Q_{40}$ that determines the bottom of the valley. As a
consequence of the improper determination of the bottom of the valley using a
constraint on the octupole moment, large increases in the value of $Q_{40}$
can be noticed between $Q_{30}=50$ b 3/2 and 55 b 3/2 in Fig 8. No real
discontinuity of the surface is found here, but rather a problem with the
precise numerical evaluation of the bottom of the valley and therefore no
important physics is missed here.

The regions (C) and (D) should be discussed together as they separate the
compact symmetric and the asymmetric fission valleys. A first look at the PES
in Fig. 1 reveals a seemingly smooth surface: no sharp ridges or sudden energy
changes. Therefore, it comes as a surprise to find a discontinuity here. To
solve this puzzle, one has to use a magnifying glass: in Figs. 9 and 10 we
have plotted a blown-up view of the sections of both valleys in 252Cf and
258No, respectively. The step size in these calculations was reduced to
$\Delta Q_{30}=1$ b 3/2 with starting point from the nearest mesh point with
smaller or larger octupole moment.

In the left columns of Figs. 9 and 10, the PES in the second minimum is
plotted. In this case, the energy grows up smoothly with increasing mass
asymmetry. Only a small bending in octupole moment can be noticed. The PES
picture at $Q_{20}=70$ b, in the region (B) of discontinuity, is completely
different, see panels in the second column. We find a kink in the energy curve
at a relatively small octupole deformation. At this point, a clear change of
configuration takes place as the neck, clearly visible for near reflection
symmetry shapes, is not pronounced anymore in more asymmetric shapes. Also,
the hexadecapole moment jumps from $Q_{40}=25$ b2 to 35 b2. It is easy to
distinguish here the compact valley from the elongated asymmetric one.
Moreover, in 252Cf, the asymmetric valley can be extended towards lower
octupole moments up to zero. This is the germ of the symmetric elongated
fission valley. Discontinuity in the region (B) is not a significant problem
in the description of fission as a transfer from the second minimum to the
asymmetric valley requires a few additional MeV of energy, which practically
blocks such evolution.

Increasing the quadrupole moment, we find a space between regions (B) and (C)
in Fig. 7. Its impact is also visible in Figs. 9 and 10 at $Q_{20}=90$ b.
Here, the PES as a function of $Q_{30}$ is continuous again. Hexadecapole
moment gradually increases indicating a smooth connection between the compact
and asymmetric valleys. The transfer between configurations is possible as a
consequence of decreasing the energy of the asymmetric minimum. The tiny part
of the symmetric elongated surface can be noticed only very close to
reflection symmetric shapes.

The rightmost panels of Figs. 9 and 10 describe the region (C) of
discontinuity. Despite the same values of the quadrupole and octupole moments
and the similarity of energies, the hexadecapole moments and the shapes of the
nucleus are considerably distinct in both configurations. In the compact
symmetric mode, two pre-fragments are separated by a thin neck. In the
asymmetric mode, the density distribution is more uniform along the symmetry
axis. In both isotopes, the compact mode is limited to hexadecapole moments in
the range from $Q_{40}=50$ b2 to 60 b2 whereas the asymmetric mode is
described by much higher values, over 65 b2. In 252Cf, the compact valley
created by increasing the octupole moment ends with a sudden drop into the
asymmetric valley. In this nucleus, by decreasing octupole moment in the
calculations of consecutive mesh points in the asymmetric valley, one may
reach zero octupole moment in a configuration characteristic for the elongated
symmetric valley. The transition from the compact to the asymmetric valley is
an analogue of the scission line discontinuity described below. In 258No both
surfaces meet at the same energy, and the elongated symmetric part can not be
determined at $Q_{20}=110$ b. In region (C), in both isotopes, transfer to the
asymmetric valley is energetically favorable.

The transition between configurations in this region is usually overlooked. It
is easy to link both valleys when the mesh points are scattered by 4 or 5
b3/2, which is usually a reasonable choice while producing the PES maps. The
similarity of the energy slopes as well as the fact that increasing octupole
moment on the compact valley beyond its end, leads to a solution in the
asymmetric one (by applying the self-consistent energy minimization) enhances
the chances of making mistakes. The discontinuity in the region (C) signals
the frontier between the compact mode surface in the symmetric second minimum
region and the symmetric second barrier.

Looking at the energy section maps in Figs. 4 and 5 we find that the
discontinuity between symmetric and asymmetric surfaces discussed above is a
good example of missing saddles as discussed by Dubray and Regnier (Dubray and
Regnier, 2012). In three-dimensional PES, the problem of a rapid change of
configurations disappears.

Finally, a commonly known discontinuity is localized in the region (E) of the
scission line. There is a huge difference in pre-scission and post-scission
configurations. It is very hard or even impossible to find a continuous link
by using constraints on multipole moments to control the nuclear shape.
Applying constraints on the neck parameter $Q_{N}$ may help to provide the
continuous surface on the scission line Warda and Robledo (2011); Warda et al.
(2012).

Let us discuss now the source of the scission line cliff obtained in the self-
consistent calculations. The asymmetric valley in two-dimensional space is
build of configurations, which are local minima along all the directions
orthogonal to the ones of the constraints. In consequence, decreasing the neck
thickness (or hexadecapole moment) leads to increasing the energy of the
system even though the ruptured nucleus is energetically favorable. The height
of the barrier separating the asymmetric path from the no-neck solution
reduces to zero with increasing quadrupole moment. Beyond the pre-scission
line, any shape of the nucleus is unstable against neck rupture, i.e. the
energy monotonically decreases with decreasing neck thickness. The energy
minimization procedure cannot find a stable solution with a neck. The gradient
of the energy directs the system towards post-scission configuration with much
lower energy in the self-consistent process. Of course, the cliff on the
scission line does not mean that in Nature the neck disappears instantly, but
only that a further thinning of the neck should occur without increasing the
elongation of the system.

The concept of density distance can also be applied to the PES given as a
function of the octupole and hexadecapole moments for fixed $Q_{20}$ values.
An example of the results is presented in Fig. 11. Here again, the
discontinuity at the scission line is clearly visible. The density distance is
also enhanced in some other regions of the PES quite randomly scattered on the
surface. Its values are relatively small in comparison to the scission line
ones. The only characteristic region of larger density distance separates
elongated symmetric valley from the compact one.

###  III.4 Multiple solutions

We have already shown that the description of fission in terms of a one
dimensional or even a two-dimensional PES may lead to many misunderstandings.
Two or more different configurations can be obtained in the same location of
the PES in those cases. This problem has been observed while discussing the
region between the compact symmetric and the asymmetric valleys. Similar
issues can arise if one analyses the so-called fusion channel with two
separate fragments. Decreasing the quadrupole moment of the system in this
region leads to approaching fragments which are much closer than what can be
achieved in the scission configuration. In consequence, the fusion valley
covers a much larger area than presented in Fig. 2. This surface, mostly
”hidden” under the asymmetric fission valley from Fig 2, is shown in Fig. 12.
Only in the lowest quadrupole deformation region, the fragments are close to
each other, and Coulomb energy is so high that the fusion valley climbs up
above the fission one. Since the global energy minimum for a given quadrupole
moment may be in the post-scission configuration, an important physical
problem appears. Should we take this solution as part of the PES leading to
fission or rather keep staying in the fission valley as long as it is
possible.

Two alternative approaches can be found in Fig. 1 of Ref. Regnier et al.
(2019) by Regnier et al. and Fig. 1 of Ref. Warda and Zdeb (2015) by Warda et
al. Both Figures show the PES of 258Fm calculated with the same model and
interaction (HFB theory and Gogny D1S force). Nevertheless, the shape of the
asymmetric fission valley is different. In the first plot, it is narrow and
equipotential (the lines are mainly horizontal), whereas in the second case,
the asymmetric valley is wide and the lines of constant energy are rather
vertical. The source of the difference comes from the distinct strategy of
selecting the local minima for the surface. Regnier et al. selected the lowest
of the local minima for the given constraints. Warda et al. put more attention
to the continuity of the changes in the shape and preservation of the valley
in which a nucleus was in the previous phase of evolution. In the first
approach, the scission line is localized at a much lower elongation, and the
PES includes a larger part of the fusion valley.

We would like to stress that, unexpectedly, both figures give the correct
surface in two dimensions showing the importance of considering
multidimensional PESs. Restricting the deformation space to just one, two or
three dimensions simplifies the interpretation of the results as well as its
graphical representation, but it always hides information and can veil our
understanding of the nature of the fission process.

We would like to point out that one can observe an internal structure of the
PES in the fusion channel. The surface is usually smooth with mass and
deformations of the fragments gradually evolving with octupole moment. A rapid
drop in the energy indicates an abrupt change of the fragment configuration.

Figure 12: The same as in Fig. 2 but for the fusion valleys are plotted.

The extended fusion valley can also be seen in Fig. 13 where, for fixed
quadrupole moments, the PES as a function of the octupole and hexadecapole
moments are plotted. Again we can see the fusion valley even at low quadrupole
moments. It extends towards much larger hexadecapole moments than it was shown
in Figs. 4 and 5. This analysis provides one more, not very optimistic,
conclusion. Applying a triple constraint (on quadrupole, octupole and
hexadecapole moments) and preparing a three-dimensional surface in the self-
consistent calculation does not guarantee the uniqueness of the solution. We
still can obtain completely distinct shapes of the density distribution
depending on the initial configuration.

Figure 13: The same as in Fig. 12 but for cross-sections in $Q_{30}-Q_{40}$
plane.

##  IV Conclusions

We have investigated the fission barriers of two heavy actinide nuclei: 252Cf
and 258No using the self-consistent microscopic approach. The calculations
were made using multiple constraints on the quadrupole, octupole, and
hexadecapole moments. A detailed analysis has shown a complicated structure of
the potential energy surface at large quadrupole deformations of the nucleus.
The competition between local minima at a given quadrupole moment: compact,
elongated and symmetric elongated determines the fission mode and the
experimentally observed fragment mass distribution. We have shown that the
scission may occur at a quadrupole deformation smaller than the one at the end
of the fission path in the minimum of the valley on the potential energy
surface.

The calculations using three constraints give a much more complete description
of the potential energy surface. Nevertheless, it is possible to find distinct
solutions corresponding to the same values of the quadrupole, octupole, and
hexadecapole moments. These configurations create various layers of the
potential energy surface in the same place of two- or three-dimensional maps.
It makes the description of the fission even more complicated.

Reducing the full space of deformation to two-dimensions creates a potential
energy surface that is not continuous. Rapid changes of the configuration of
density distribution may be found even at seemingly smooth surfaces. We
therefore conclude that the jumps between surfaces must be discussed in the
analysis of the fission process.

The present analysis is based on the HFB approximation, but it also applies to
any constrained, self-consistent type of calculations. Problems with multiple
minima and surface discontinuities also affect macroscopic-microscopic models,
although it is much easier to control the whole spectrum of nuclear shapes
there.

The discussion of the determination of the potential energy surface is not the
only relevant issue in the analysis of fission. Many other aspects also have a
significant impact on the dynamics of this process. We should mention here the
influence of the inertia parameter and pairing degrees of freedom that shall
also be investigated in the future.

###### Acknowledgements.

M.W. acknowledges support by the Polish National Science Center under Contract
No. 2018/30/Q/ST2/00185. The work of LMR is supported by Spanish Ministry of
Economy and Competitiveness (MINECO) Grant No. PGC2018-094583-B-I00.

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