# Fission fragment mass yields of Th to Rf even-even nuclei 111Supported by
the Polish National Science Center (Grant No. 2018/30/Q/ST2/00185) and by the
National Natural Science Foundation of China (Grant No. 11961131010 and
11790325).

Krzysztof Pomorski1222Email<EMAIL_ADDRESS>José M. Blanco1, Pavel
V. Kostryukov1, Artur Dobrowolski1,
Bożena Nerlo-Pomorska1, Michał Warda1, Zhigang Xiao2333Email:
<EMAIL_ADDRESS>Yongjing Chen3, Lile Liu3,
Jun-Long Tian4, Xinyue Diao2, Qianghua Wu2
1 Institute of Physics, Maria Curie Skłodowska University, 20-031 Lublin,
Poland
2 Department of Physics, Tsinghua University, Beijing 100084, China
3 China Institute of Atomic Energy, Beijing 102413, China
4 School of Physics and Electrical Engineering, Anyang Normal University,
Anyang 455000, China

Abstract:

Fission properties of the actinide nuclei are deduced from theoretical
analysis. We investigate potential energy surfaces and fission barriers and
predict the fission fragment mass-yields of actinide isotopes. The results are
compared with experimental data where available. The calculations were
performed in the macroscopic-microscopic approximation with the Lublin-
Strasbourg Drop (LSD) for the macroscopic part and the microscopic energy
corrections were evaluated in the Yukawa-folded potential. The Fourier nuclear
shape parametrization is used to describe the nuclear shape, including the
non-axial degree of freedom. The fission fragment mass-yields of considered
nuclei are evaluated within a 3D collective model using the Born-Oppenheimer
approximation.

Keywords: nuclear fission, mac-mic model, fission barrier heights, fragment
mass-yields

PACS: 21.10.Dr,25.70.Ji,25.85.-w,25.85.Ec

##  1 Introduction

Good reproduction of fission barrier heights and fission fragments mass-yields
is a test of the theoretical models describing the nuclear fission process. An
interesting review of the existing fission models can be found in Refs. [1, 2,
3]. Extended calculations of the fission barrier heights can be found in Refs.
[4, 5]. Readers who are interested in the theory of nuclear fission can find
more details in the textbook [6].

In the present paper, the fission fragment mass yields (FMY) are obtained by
an approximate solution of the eigenproblem of a three-dimensional collective
Hamiltonian, of which the coordinates correspond to the fission, neck, and
mass-asymmetry modes. Here presented model is described in details in Refs.
[7, 8, 9]. The potential energy surfaces (PES) of fissioning nuclei obtained
by the macroscopic-microscopic (mac-mic) method in which the Lublin-Strasbourg
Drop (LSD) model [10] has been used for the macroscopic part of the energy,
while the microscopic shell and pairing corrections are evaluated using
single-particle levels of the Yukawa-folded (YF) mean-field potential [11,
12]. The Fourier parametrization is used to describe shapes of fissioning
nuclei [13, 14]. It is shown in Ref. [15] that this parametrization describes
very well the shapes of the nuclei even close to the scission configuration.

The paper is organized in the following way. In Section 2, we present first
the details of the shape parametrization and the theoretical model. Then we
show the collective potential energy surface evaluated within the mac-mic
model for the selected isotopes and our estimates of the fission barrier
heights. The calculated FMY are compared with the existing experimental data
in Section 3. The estimates of FMY for Th isotopes and their dependence on two
adjustable parameters are further discussed in details in Section 4.
Conclusions and perspectives of further investigations are presented in
Section 5.

##  2 Model of the fission dynamics

The evolution of a nucleus from the equilibrium state towards fission is
described here by a simple dynamical approach based on the PES. We assume that
at large deformations, the shape of the nucleus should depend on three
collective degrees of freedom describing its elongation, left-right asymmetry,
and the neck-size. At smaller deformations, up to the second saddle, the non-
axial shapes are also considered. In the following subsection, we present
shortly a Fourier type parametrization of the nuclear shape which is used in
the paper.

Fig. 1: Shape of a very elongated fissioning nucleus.

###  2.1 Fourier nuclear shape parametrization

A typical shape of the nucleus on the way from the saddle to the scission
configuration is shown in Fig. 1, where $\rho(z)$, the distance from the
z-axis to the surface of the nucleus as a function of $z$, is plotted. Here by
the nuclear surface is treated the surface of the nuclear liquid drop, or the
half-density surface when the microscopic density distribution is considered.

The function $\rho(z)$ corresponding to the nuclear surface can be expanded in
the Fourier series in the following way [13]:

$\begin{array}[]{ll}\rho_{s}^{2}(z)\\!=\\!R_{0}^{2}\\!\sum\limits_{n=1}^{\infty}&\left[a_{2n}\cos\left(\frac{(2n-1)\pi}{2}\,\frac{z-z_{sh}}{z_{0}}\right)\right.\\\
&+\left.a_{2n+1}\sin\left(\frac{2n\pi}{2}\,\frac{z-z_{sh}}{z_{0}}\right)\right]~{}.\end{array}$
(1)

Here $z_{0}$ is the half-length of the total elongation of the nucleus and
$z_{\rm sh}$ locates the center of mass of the nucleus at the origin of the
coordinate system. The expansion parameters $a_{i}$ can serve as parameters
describing the shape of nucleus. The length parameter $c=z_{0}/R_{0}$ is fixed
by the volume conservation condition, where $R_{0}$ is the radius of spherical
nucleus having the same volume as the deformed one.

Contrary to frequently used spherical harmonics expansion (conf. e.g., Refs.
[4, 5]), the Fourier series converges much earlier for the realistic shape of
nuclei [13, 14] and only a few first terms are sufficient in practical use.
Although one can work directly with these Fourier expansion coefficients
treating them as free deformation parameters, it is more suitable to use their
combinations $\\{q_{n}\\}$, called optimal coordinates [14], as following

$\left\\{\begin{array}[]{l}q_{2}=a_{2}^{(0)}/a_{2}-a_{2}/a_{2}^{(0)}\\\\[4.30554pt]
q_{3}=a_{3}\\\\[4.30554pt]
q_{4}=a_{4}+\sqrt{(q_{2}/9)^{2}+(a_{4}^{(0)})^{2}}\\\\[4.30554pt]
q_{5}=a_{5}-(q_{2}-2)a_{3}/10\\\\[4.30554pt]
q_{6}=a_{6}-\sqrt{(q_{2}/100)^{2}+(a_{6}^{(0)})^{2}}\;\;.\end{array}\right.$
(2)

The functions $q_{n}(\\{a_{i}\\})$ were chosen in such a way that the liquid-
drop energy as a function of the elongation $q_{2}$ becomes minimal along a
trajectory that defines the liquid-drop path to fission. The $a^{(0)}_{2n}$ in
Eq. (2) are the expansion coefficients of a spherical shape given by
$a^{(0)}_{2n}=(-1)^{n-1}\frac{32}{\pi^{3}\,(2n-1)^{3}}$. The above relations
proposed in Ref. [14] transform the original deformation parameters $a_{i}$ to
the more natural parameters $q_{i}$, which ensure that only minor variations
of the liquid-drop fission paths occur around $q_{4}=0$. In addition, more and
more elongated prolate shapes correspond to decreasing values of $a_{2}$,
while oblate ones are described by $a_{2}>1$, which contradicts the
traditional definition of the elongation parameter. The parametrization (2) is
rapidly convergent. It was shown in Ref. [15] that the effect of $q_{5}$ and
$q_{6}$ on the macroscopic potential energy of nuclei is negligible for small
elongations of nuclei up to the saddle points and contributes within 0.5 MeV
around the scission configurations.

Non-axial shapes can easily be obtained assuming that, for a given value of
the $z$-coordinate, the surface cross-section (blue dashed oval in Fig. 1) has
the form of an ellipse with half-axes $a(z)$ and $b(z)$ [14]:

$\varrho_{s}^{2}(z,\varphi)=\rho^{2}_{s}(z)\frac{1-\eta^{2}}{1+\eta^{2}+2\eta\cos(2\varphi)}\hskip
8.5359pt\mbox{with}\hskip 8.5359pt\eta=\frac{b-a}{a+b}~{},$ (3)

where the parameter $\eta$ describes the non-axial deformation of the nuclear
shapes. The volume conservation condition requires that
$\rho_{s}^{2}(z)=a(z)b(z)$.

###  2.2 Potential energy surfaces

The nuclear potential energies of actinide nuclei are evaluated in the
following equidistant grid-points in the 4D collective space built on the
$q_{2},~{}q_{3},~{}q_{4}$, and $\eta$ deformation parameters:

$\begin{array}[]{lr}\displaystyle q_{2}=&-0.60~{}(0.05)~{}2.35~{},\\\
q_{3}=&0.00~{}(0.03)~{}0.21~{},\\\ q_{4}=&-0.21~{}(0.03)~{}0.21~{},\\\
\eta~{}=&0.00~{}(0.03)~{}0.21~{}.\end{array}$ (4)

Here, the numbers in the parentheses are the step size, while the numbers on
the left (right) side are the lower (upper) boundaries od the grid,
respectively. The energy of a nucleus is obtained in the mac-mic model, where
the smooth energy part is given by the LSD model [10], and the microscopic
effects have been evaluated using the YF single-particle potential [11, 12].
The Strutinsky shell-correction method [16, 17, 18] with a $6^{\mathrm{th}}$
order correctional polynomial and a smoothing width
$\gamma_{S}=1.2\hbar\omega_{0}$ is used to determine the shell energy
correction, where $\hbar\omega_{0}=41/A^{1/3}$ MeV is the distance between the
spherical harmonic-oscillator major shells. The BCS theory [19] with the
approximate GCM+GOA particle number projection method [20] is used for the
pairing correlations. An universal pairing strengths written as
$G{\mathcal{N}^{2/3}}=0.28\hbar\omega_{0}$, with ${\mathcal{N}}=Z,N$ for
protons or neutrons, was adjusted in Ref. [21] to the experimentally measured
mass differences of nuclei from different mass regions. It was assumed in Ref.
[21] that the “pairing window” contains $2\sqrt{15{\mathcal{N}}}$ single-
particle energy levels closest to the Fermi level. All the above parameters
were fixed in the past, and none of them was specially fitted to the
properties of actinide nuclei.

A typical PES for actinides is shown in Fig. 2, where two cross-sections
$(q_{2},\eta)$ and $(q_{2},q_{3})$ of the 4D potential energy surface of 240Pu
are shown.

Fig. 2: Potential energy surface of 240Pu minimized with respect to $q_{4}$ at
the $(q_{2},\eta)$ (top map) and $(q_{2},q_{3})$ (bottom) planes.

As one can see, the inclusion of the non-axial deformation is important up to
elongations corresponding to the second saddle ($q_{2}\leq 1.2$). Apart of
some neutron deficient actinide nuclei which have $q_{3}\neq 0$ in the ground-
state, the left-right asymmetry begins to play an important role at large
elongations of nuclei, from the second saddle ($q_{2}\approx 1$) up to the
scission configuration ($q_{2}\gtrsim 2$).

Fig. 3: Fission barrier heights of even-even actinide nuclei in our 4D mac-mic
model.

The first ($E_{A}$) and the second ($E_{B}$) fission barrier heights obtained
in our model for nuclei from Th to Rf are compared in Fig. 3 with the
experimental data taken from Ref. [22, 3]. The agreement of our estimates with
the data is rather satisfactory, and is comparable within an accuracy obtained
in other theoretical models. The largest deviation between our estimates and
the experimental values are observed in thorium isotopes, where they are
underestimated. The main origin of these discrepencies is mostly from the
inaccuracy of determining the ground-state masses in our model. To prove it,
we have estimated the fission barrier heights using the so called
topographical theorem of Myers and Świa̧tecki [23], where the barrier height
(the largest one) is defined as

$E_{\rm barr}=M_{\rm mac}^{\rm sadd}-M_{\rm g.s.}^{\rm exp}~{},$ (5)

where $M_{\rm mac}^{\rm sadd}$ is the first barrier saddle point mass
evaluated in the macroscopic model (i.e., without microscopic energy
correction) and $M_{\rm g.s.}^{\rm exp}$ is the experimental ground-state mass
of the nucleus.

Using the LSD model [10] to evaluate the macroscopic (read LD) mass one
obtains the ’Świa̧tecki’ estimates of barrier heights, which deviate from the
experimental data only by 310 keV on the average, as shown in Fig. 4.

Fig. 4: Fission barrier heights of even-even actinide nuclei evaluated using
the topographical theorem and the LSD model compared with the experimental
barrier heights as a function of mass number $A$ [24].

It means that additional work to improve the estimates of the ground-state
masses has to be done. In particular, to make a better fit of the pairing
strength. Our ”universal” pairing force [21], used in the present work,
reproduces on average the pairing gaps of nuclei from different mass regions,
but it might be that it does not reproduce perfectly the pairing properties in
actinides.

###  2.3 Simple collective model for fission

The present research is a continuation and extension of our previous works [7,
8, 9], where more detailed description of the collective fission model was
given. The fundamental idea of this approach is the use of the Born-
Oppenheimer approximation (BOA) to separate the relatively slow motion towards
fission, mainly in $q_{2}$ direction, from the fast vibrations in the
“perpendicular” $q_{3}$ and $q_{4}$ collective coordinates. The BOA allows us
to treat these both types of motion as decoupled, what leads, in consequence,
to the wave function in form of the following product:

$\Psi_{nE}(q_{2},q_{3},q_{4})=u_{nE}(q_{2})\,\phi_{n}(q_{3},q_{4};q_{2})~{}.$
(6)

The function $u_{nE}(q_{2})$ is the eigenfunction corresponding to the motion
towards fission, while the $\phi_{n}(q_{3},q_{4};q_{2})$ simulates the
$n-$phonon “fast” collective vibrations in the “perpendicular” to the fission
mode $\\{q_{3},q_{4}\\}$ plane.

To determine the $u_{nE}(q_{2})$ function for a single $q_{2}$ mode one can
use the WKB approximation as it has been done in Ref. [7]. To obtain the
function $\phi_{n}(q_{3},q_{4};q_{2})$, one has to solve numerically for each
value of $q_{2}$ the eigenproblem of the underlying Hamiltonian in the
perpendicular $\\{q_{3},q_{4}\\}$ space. However, for the low energy fission,
it is sufficient to take only the lowest wave function in the perpendicular
mode and evaluate the density of probability $W(q_{3},q_{4};q_{2})$ of finding
the system for a given elongation $q_{2}$ within the area of $(q_{3}\pm
dq_{3},q_{4}\pm dq_{4})$ as

$W(q_{3},q_{4};q_{2})=|\Psi(q_{2},q_{3},q_{4})|^{2}=|\phi_{0}(q_{3},q_{4};q_{2})|^{2}~{}.$
(7)

Further simplification we have made is to approximate the modulus square of
the total wave function in Eq. (7) by the Wigner function in the following
form

$W(q_{3},q_{4};q_{2})\propto\exp{\frac{V(q_{3},q_{4};q_{2})-V_{\rm
min}(q_{2})}{T^{*}}}~{},$ (8)

where $V_{\rm min}(q_{2})$ is the minimum of the potential for a given
elongation $q_{2}$ and $T^{*}$ is a generalized temperature [25] which takes
into account both thermal excitation of fissioning nucleus and the collective
zero-point energy $E_{0}$

$T^{*}=E_{0}/{\rm tanh}(E_{0}/T)~{}.$ (9)

The temperature ($T$) of nucleus with mass-number $A$ is evaluated from its
thermal excitation energy ($E^{*}$) using the phenomenological relation
$E^{*}=aT^{2}$, with $a=A$/(10 MeV). The generalized temperature $T^{*}$ is
approximately equal to the zero-point energy when $T$ is small while for
sufficiently high temperatures ($T\gg E_{0}$) it approaches to $T$. In the
following $E_{0}$ is treated as one of two adjustable parameters of our model.
Of course, one expects $E_{0}$ of the order of 1 to 2 MeV as implied by the
energy level positions of typical collective vibrational states.

To obtain the FMY for a given elongation $q_{2}$ one has to integrate the
probabilities (8) over the full range of the neck parameter $q_{4}$

$w(q_{3};q_{2})=\int W(q_{3},q_{4};q_{2})dq_{4}~{}.$ (10)

It is rather obvious that the fission probability may strongly depend on the
neck radius $R_{\rm neck}$. Following Ref. [7] one assumes the neck rupture
probability $P$ to be equal to

$P(q_{2},q_{3},q_{4})=\frac{k_{0}}{k}\,P_{\rm neck}(R_{\rm neck})~{},$ (11)

where $P_{\rm neck}$ is a geometrical factor indicating the neck breaking
probability proportional to the neck thickness, while $k_{0}/k$ describes the
fact that the larger collective velocity towards fission,
$v(q_{2})=\dot{q}_{2}$, implies that the neck rupture between two neighboring
$q_{2}$ configurations is getting less probable. The constant parameter
$k_{0}$ plays the role of scaling parameter which is finally eliminated in the
calculation of the resulting FMY. The expression for the geometrical
probability factor $P_{\rm neck}(R_{\rm neck})$ is chosen here in a form of
Gauss function [8]:

$P_{\rm neck}(R_{\rm neck})=\exp{[-\log(2)(R_{\rm neck}/d)^{2}]}~{},$ (12)

where $d$, our second adjustable parameter, is the “half-width” of the neck-
breaking probability. The momentum $k$ in Eq. (11) simulates the dynamics of
the fission process, which, as usual, depends both on the local collective
kinetic energy ($E_{\rm kin}$) and the inertia ($M$) towards the fission mode

$\frac{\hbar^{2}k^{2}}{2\bar{M}(q_{2})}=E_{\rm kin}=E-E^{*}-V(q_{2})~{},$ (13)

with $\bar{M}(q_{2})$ standing for the (averaged over $q_{3}$ and $q_{4}$
degrees of freedom) inertia parameter at a given elongation $q_{2}$, and
$V(q_{2})$ is the potential corresponding to the bottom of the fission valley.
In the further calculations we assume that the part of the total energy
converted into heat $E^{*}$ is negligibly small due to rather small friction
forces in the low energy fission. A good approximation of the inertia
$\bar{M}(q_{2})$, proposed in Ref. [26], is to use the irrotational flow mass
parameter $B_{\rm irr}$, which is derived initially as a function of the
distance between fragments $R_{12}$ and the reduced mass $\mu$ of both
fragments

$\bar{M}(q_{2})=\mu[1+11.5\,(B_{\rm irr}/\mu-1)]\bigg{(}\frac{\partial
R_{12}}{\partial q_{2}}\bigg{)}^{2}~{}.$ (14)

In order to make use of the neck rupture probability $P(q_{3},q_{4};q_{2})$ of
Eq. (11), one has to rewrite the integral over $q_{4}$ in probability
distribution (10) in the following form:

$w(q_{3};q_{2})=\int W(q_{3},q_{4};q_{2})P(q_{2},q_{3},q_{4})dq_{4}~{},$ (15)

in which the neck rupture probability is now taken into account. The above
approximation describes a very important fact that, for a fixed $q_{3}$ value,
the fission may occur within a certain range of $q_{2}$ deformations with
different probabilities. Therefore, to obtain the true fission probability
distribution $w^{\prime}(q_{3};q_{2})$ at a strictly given $q_{2}$, one has to
exclude the fission events occurred in the “previous” $q^{\prime}_{2}<q_{2}$
configurations, i.e.,

$w^{\prime}(q_{3};q_{2})=w(q_{3};q_{2})\frac{1-\int\limits_{q^{\prime}_{2}<q_{2}}w(q_{3};q^{\prime}_{2})dq^{\prime}_{2}}{\int
w(q_{3};q^{\prime}_{2})dq^{\prime}_{2}}~{}.$ (16)

The normalized mass yield is then obtained as the integral of partial yields
over $q_{2}$:

$Y(q_{3})=\frac{\int w^{\prime}(q_{3};q_{2})dq_{2}}{\int
w^{\prime}(q_{3};q_{2})dq_{3}\,dq_{2}}~{}.$ (17)

Since there is a one-to-one correspondence between $q_{3}$ deformation and the
masses of the left ($A_{L}$) and right ($A_{R}=A-A_{L}$) fission fragments,
the yield given by Eq. (17) can be directly compared with the experimental
FMY’s. Note that due to the normalization procedure (17), the scaling
parameter $k_{0}$ introduced in Eq. (11) does no longer appear in the
definition of mass yield.

So, there are only two free parameters in the above model, namely the zero-
point energy $E_{0}$ in Eq. (9) and the half-width parameter $d$ appearing in
the probability of neck rupture in Eq. (12).

##  3 Fission fragment mass-yields

Relatively good estimates of the FMY’s obtained in our previous works [8, 9]
for Pu and Pt to Ra isotopes encourage us to apply our model to describe and
predict the mass yield for the low-energy fission of actinide nuclei from Th
to Rf. Our main goal is to show that the innovatory Fourier shape
parametrization [13] and the mac-mic model based on the LSD macroscopic energy
[10] with the microscopic energy correction evaluated using the Yukawa-folded
potential [11, 12] describes well the existing fission valleys in the broad
region of nuclei. It is worth recalling that none of the model parameters,
apart from $E_{0}$ and $d$ described in the previous section, was modified
here to get a better description.

In general, the theoretical estimates of FMY depend weakly on the choice of
the free parameters $E_{0}$ and $d$. To obtain the best fit to the existing
experimental FMY’s, the following overlap of theoretical and experimental
yields:

$I(E_{0},d)=\sum\int\limits_{0}^{A}|Y_{exp}(A_{f})-Y_{th}(A_{f};E_{0},d)|dA_{f}~{}.$
(18)

is minimized with respect to $E_{0}$ and $d$. The sum in Eq. (18) runs over
the nuclei, where the experimental data exist. Contrary to the $\xi^{2}$ fit,
this fitting procedure does not overestimate the role of large deviations. The
optimised values $E_{0}=2.2$ MeV and $d=$1.6 fm are finally used to obtain the
FMY for all considered actinide nuclei.

It is well known that the FMY of a given nucleus is mainly determined by its
PES properties at large deformations. Typical examples of PES for 240Pu are
shown in Fig. 2, where the mac-mic energy minimized with respect to $q_{4}$ is
plotted on the $(q_{2},\eta)$ (top) and $(q_{2},q_{3})$ (bottom) planes. The
labels at the layers correspond to the energy of the deformed nucleus (in MeV)
measured with respect to the LSD macroscopic energy of the spherical nucleus.
The first saddle is visible around $q_{2}=0.55$ and $q_{3}=0$, while the
second one is at $q_{2}=1.10$ and $q_{3}=0.08$. As one can see in the upper
panel, the non-axial deformation $\eta$ does not influence the PES at lager
$q_{2}$ deformation. So, we do not take this degree of freedom into account in
our analysis of the FMY’s. Let us notice that each of the 2D energy maps shown
in Fig. 2 is only a projection of the full 4D PES, and one has to consider
other cross-sections in order to analyze the fission process in details.

Fig. 5: Fission fragment mass-yields of Th (top part) and U (bottom part)
isotopes. Experimental data (red stars) for Th isotopes are extracted from the
charge-yields of Refs. [28, 29] while the mass-yields for U isotopes (botton
part) are taken from Ref. [30, 31] for the thermal neutron induced fission
(th). Just to guide the eye we have used for 238U the post-neutron data (blue
crosses) taken from Ref. [3].

The fission fragment mass yields obtained in our model are presented in Figs.
5 to 8. Some experimental data for the FMY were obtained for the fission of
excited nuclei. In such a case, we take this excitation into account and
reduce the microscopic energy correction according to the prescription found
in Ref. [27]. Our estimates of FMY correspond to the so-called pre-neutron
yields, i.e., the mass yields before neutron emission from fragments and with
such data (red stars in Figs. 5 to 8) they have to be compared. In the case of
Th isotopes, we have used the fragment charge yields from Refs. [28, 29] and
to obtain the mass-yields it is assumed that the $Z/N$ ratio in the fragment
is the same as in the mother nucleus. In cases when the pre-neutron data were
not available, we have plotted the post-neutron data (blue crosses) just to
get piece of information about the experimental situation. It is shown that
for the Th isotopes, although the agreement of the estimates with the
experimental data is not very satisfactory, the general trend is reproduced,
i.e., a transition from symmetric to asymmetric fission is evidently
reproduced with a growing mass number of isotope. The best agreement was
achieved for 218Th and 228-230Th nuclei. The agreement with experimental data
in the Uranium chain, presented in the bottom part of Fig. 5, is much better.
Here the maxima and the widths of the fragment mass distribution are well
reproduced, and a similar transition between symmetric and asymmetric fission
as in Th isotopes is evident.

Fig. 6: Fission fragment mass yields of Pu (top part) and Cm (bottom part)
isotopes. Experimental data (red stars) are taken from Ref. [34] for Pu chain
and Refs. [3, 32] for Cm nuclei.

The prediction of the FMY’s for Pu and Cm isotopes are compared in Fig. 6 with
the experimental data. The pre-neutron experimental yields for Pu [34] and
246Cm [32] isotopes are obtained for the spontaneous fission case, while those
for 244Cm and 248Cm are post-neutron yields taken from Ref. [3]. A nice
agreement with the data obtained for the two lightest Pu isotopes 236Pu and
238Pu is slightly spoiled when the number of neutrons increases i.e. for 242Pu
and 244Pu. It is mainly because we have used here the globally optimized
values of $E_{0}$ and $d$, which are not fitted to Pu data only as done in
Ref. [8]. In all investigated Pu and Cm isotopes here, the asymmetric fission
is predicted with the mass of the heavy fragment $A\approx 140$.

Fig. 7: Fission fragment mass yields of Cf (top) and Fm (bottom) isotopes.
Experimental data for pre-neutron yields (red stars) are taken from Refs. [31,
33] while the post-neutron yields (blue crosses) origin from Ref. [3, 36].

The estimates of the FMY for Cf and Fm chains of isotopes are shown in Fig. 7.
All experimental data correspond to the spontaneous fission, but only for
252Cf [31] and 256Cf [33] have to do with pre-neutron yields. The rest of the
experimental yields presented in Fig. 7 corresponds to the post-neutron data
(blue crosses). One can see that for lighter Cf and Fm isotopes, the
asymmetric yields are predicted, while in the case of the heaviest Cf and Fm
nuclei, the symmetric fission is foreseen. As one can deduce from the above
results, our estimates are rather consistent with the experimental yields.

Fig. 8: Fission fragment mass yields of No (top) and Rf (bottom) isotopes.
Experimental data (crosses) are taken from Refs. [35, 36, 37].
Fig. 9: Potential corresponding to the bottom of fission valley as a function
of the elongation parameter $q_{2}$

A similar tendency as seen in the Cf and Fm chains can be observed in Fig. 8
for the No and Rf isotopes, where the asymmetric fission is predicted in the
lighter nuclei while the symmetric fission mode dominates for the isotopes
with $N>156$. The agreement with the experimental data of 256No and 262Rf [35,
37] is evident.

The overall good quality of our predictions in a broad mass region of the
actinide elements is probably due to the fact that in very heavy nuclei, the
fission barrier is very short, and the fission valley forms very early, i.e.,
at a relatively small elongation of the nucleus. An oposite situation occurs
in the thorium nuclei, where the fission barriers are very broad. Fig. 9
presents the fission valley potential as a function of the elongation
parameter $q_{2}$. It is shown that the average slope of the curve from the
last saddle to scission in the thorium nuclei is almost three times smaller
than in nobelium. Obviously, such a large difference in the slope towards
fission influences the fission dynamics in these both types of nuclei. This is
a main reason why one has to study in details the PES in Th nuclei to explain
here observed change in the FMY systematics.

Fig. 10: Potential energy surface cross-sections of 218-230Th isotopes
minimized with respect the neck parameter $q_{4}$ on the plane
$(q_{2},q_{3})$.

##  4 Specific discussions on Th nuclei

The agreement of our estimates of the FMY’s with the experimental data in the
Th chain of isotopes depicted in Fig. 5 is not quantitatively satisfactory.
So, in the present section, we would like to look for the origin of these
discrepancies. First, these yields for the Th nuclei are evaluated using
$E_{0}$ and $d$ obtained by the fit to the data for all nuclei. The PES’s for
Th nuclei are very much different from those for heavier nuclei. This can be
seen by comparing the PES of 240Pu shown in Fig. 2 (bottom) with the
corresponding maps for 218-230Th isotopes presented in Fig. 10.

Fig. 11: Potential energy surface cross-sections of 218Th on the plane
$(q_{3},q_{4})$. The panels from top to bottom correspond to elongations
$q_{2}=$ 1.8 to 2.3, respectively. The solid red lines drawn in the bottom
panels correspond to the neck radius equaling to the nuclear radius constant.

In 240Pu, the fission path goes directly from the saddle point to the
asymmetric fission valley in 240Pu while it is not the case in 218Th where the
system from the 3rd minimum at $q_{2}\approx 1.2$ has a much smaller barrier
towards symmetric fission ($<0.5$ MeV) than in asymmetric one, where the
barrier is slightly higher ($\approx 1$ MeV) and thicker. It means that 218Th
nucleus prefers the symmetric fission, which is confirmed by the experimental
yield. In 222Th the situation is similar, while beyond 226Th the path leading
to the asymmetric fission begins to be preferred. To better understand this
process, one has to study the PES’s in the full 3D deformation space. In Fig.
11 the ($q_{2},q_{3}$) cross-sections of the PES for 218Th corresponding to
different elongations ($q_{2}=$1.8, 2.0, 2.2, and 2.3) are shown.

Fig. 12: Fission fragment mass yields of Th sotopes reproduced by using
constants $E_{0}$ and $d$ fitted to the experimental data (red stars) taken
from Refs. [28, 29].

Two minima, one corresponding to the symmetric ($q_{3}=0$) and the other to
the asymmetric ($q_{3}\approx 0.12$) configuration, are visible in each cross-
section. At $q_{2}=1.8$ they are separated by a 4 MeV high barrier which
becomes smaller with growing elongation $q_{2}$, reaching finally 0.5 MeV
height at $q_{2}=2.1$. At such elongations, the transition between the
symmetric and asymmetric fission is possible. Both fission valleys are well
separated again at the largest deformations close to the scission line (red
line in the figure). So, the Th nuclei make the ”decision” where to go pretty
early, i.e., at an early stage far before the scission configuration. It means
that one has to modify the adjustable parameters $E_{0}$ and $d$ in order to
better describe the transition between the symmetric and asymmetric fission
modes observed in Th nuclei when the neutron number grows. The new fit
performed to the data for Th isotopes only gives $E_{0}$=1.5 MeV and $d=$2.5
fm. The resulting mass yields are compared in Fig. 12 with the experimental
data. This time the agreement is much more satisfactory. The new value of the
neck parameter $d$ is larger than that adjusted to all nuclei. It suggests
that in Th nuclei, the choice of the preferable fission mode is made at a
thicker neck, i.e., in a pretty early stage. The smaller value of $E_{0}$ used
for Th isotopes is probably related to the competition between the symmetric
and asymmetric minima.

##  5 Summary and conclusions

In order to briefly summarize our investigations, we can write:

  * •

The overall accordance of the theoretical FMY estimates with the experimental
data indicates that the mac-mic model with the LSD energy for the macroscopic
smooth part and the shell and pairing corrections evaluated on the basis of
the Yukawa-folded single-particle potential describes well the potential
energy surfaces of actinide nuclei,

  * •

Three-dimensional set of the Fourier deformation parameters used to describe
the shape of fissioning nuclei are fully capable to produce a wide variety of
the shapes of nuclei on their way to fission,

  * •

The collective 3D model based on the Born-Oppenheimer approximation and
comprising elongation, mass asymmetry, and neck modes reproduces well the
mains features of the fission fragment mass yields data,

  * •

The Wigner function used to approximate the probability distribution related
to the neck and mass asymmetry degrees of freedom simmulates in a proper way
this distribution for low-energy fission,

  * •

A neck-breaking probability depending on the size of the neck has to be
introduced to improve the accordence of our FMY estimates with the
experimentally measured values.

Our mac-mic model and the collective 3D approach, which couples fission mode,
neck, and mass asymmetry collective vibrations, can describe the main features
of the fission process in actinide nuclei. The estimated fission barrier
heights deviate not much from their experimental values. The measured fission
fragment mass yields are also reproduced in a satisfactory way. On the other
hand, one has to treat the presented collective model as a kind of rough tool
which allows to obtain the FMY by a relatively quick calculation. To get more
precise results, one has to use more advanced models in which the whole
fission dynamics and the energy dissipation will be taken into account. Such
calculations may use the Langevin dynamics (conf. Ref. [6]) or the improved
quantum molecular dynamics model (ImQMD). The latter method has been
successfully applied to describe the fission process in the heavy ion induced
fission reactions, where the excitation energy increases, leading possibly to
a shorter fission time scale and even to the occurrence of a ternary fission
[38, 39].

The Langevin type calculations, profiting of the PES generated in a mac-mic
approach together with the 3D Fourier shape parametrization as well with the
use of the self-consistent method, are carried out in parallel by our group.

Acknowledgments

The authors would like to thank Christelle Schmitt and Karl-Heinz Schmidt for
supplying us with a part of the experimental data.

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