Four-dimensional Langevin approach to ﬁssion with Cassini shape parameterization

. We apply the Cassini shape parameterization to the dynamical study of actinide ﬁssion using the multi-dimensional Langevin equation. The fragment mass distributions for ﬁssion of 236 U and 256 , 258 Fm are calculated with the 4D Langevin equation. The 4D collective space is formed by adding either α 3 or α 6 to the basic 3D { α, α 1 , α 4 } space. We investigate the role of α 3 and α 6 by comparing the results of 4D ( α 3 or α 6 ) with that of 3D. For 236 U and 256 Fm, where asymmetric ﬁssion is predominant, the fragment mass distributions calculated in the 4D ( α 3 ) space are in good agreement with experimental data. On the other hand, for 258 Fm, which is characterized by symmetric ﬁssion, 4D ( α 6 ) is e ﬀ ective for describing the shape of compact symmetric fragments. We demonstrate the advantages of the Cassini shape parameterization for the dynamical study of actinide ﬁssion.


Introduction
The dynamical approach using the multi-dimensional Langevin equation is widely accepted as a practical method in the study of nuclear fission [1,2]. In this approach, the nuclear shape is represented by a shape parameterization; a set of deformation parameters such as elongation, mass asymmetry and neck radius. The fission process is described by the multi-dimensional Langevin equation as the time evolution of the deformation parameters. Since this is not a self-consistent calculation, the choice of the shape parameterization determines the model space and the results of the Langevin calculation are largely dependent on it. Therefore, it is important to select the shape parameterization, where a small number of deformation parameters covers a variety of deformed nuclear shapes relevant to the fission process.
The original Cassini ovals are suitable for describing shapes of fissioning nuclei because they can draw spherical, elongated, and even two separated objects depending on the elongation parameter α. To allow more flexibility, Pashkevich proposed the generalized Cassini ovals, in which the surface is defined by a Legendre expansion [3]. Here, the deformation parameters are the coefficients α n of the n-th order Legendre polynomial. The potential energies of deformation for 208 Pb, 230 Th, 236 U, 240 Pu, 252 Cf were calculated by the macroscopic-microscopic approach in the (α, α 1 ) plane [3]. In particular, macroscopic energy and shell and pairing correction of 236 U around scission were analyzed in detail [4]. Later, by considering statis-tical equilibrium in the deformation space, Carjan et al. established a method to calculate the fragment mass and total kinetic energy distributions from the potential and computed them for Fm to Rf [5]. Recently, they reported the fragment mass and total kinetic energy distributions in the super-heavy nuclear region [6,7]. They have shown in these studies that not only α 1 but also α 3 , α 4 and α 6 are essential parameters for describing the fragment shapes. However, these are calculated with the elongation parameter α fixed at scission, and the dynamical effects are neglected. As a next step, it is definitely important to use the Cassini shape parameterization in the framework of the Langevin equation and to calculate the time evolution of {α, α n } from the ground state to scission.
In the present paper, we will demonstrate the effectiveness of the Cassini shape parameterization by solving the four-dimensional (4D) Langevin equation. We will use {α, α 1 , α 3 , α 4 } or {α, α 1 , α 4 , α 6 } as a collective space. We will further investigate the role of α 3 and α 6 in the description of fragment shapes by comparing the fragment mass distributions and the average scission shapes between 3D {α, α 1 , α 4 } and 4D (3D + α 3 or α 6 ). In Sect. 2, we will describe the Cassini shape parameterization and the multidimensional Langevin equation. In Sect. 3, we will show the calculation results for neutron-induced fission of U and spontaneous fission of Fm. In Sect. 4, we will give the summary.

Cassini shape parameterization
In the Cassini shape parameterization [3], the lemniscate coordinate (R, x) is used as an orthogonal system to de-scribe nuclear shapes. The R = const lines correspond to the surface of Cassini ovals which represents the division of a spherical shape into separated objects by varying the elongation parameter α. The original Cassini oval at α = 0 shows the spherical shape, and α = 1 shows a zero-neck configuration. Thus, α is considered as the main fission coordinate.
To include various deformations into the original Cassini ovals, the Legendre expansion of the surface is introduced by where R 0 is the radius of the spherical nucleus and P n (x) is the Legendre polynomials. α n are considered as the deformation parameters in the Cassini shape parameterization. In particular, α 1 and α 4 are important parameters that should be incorporated in calculations for any system because α 1 directly corresponds to the mass asymmetry of fragments at scission and α 4 is necessary for describing the symmetric deformation such as the ground state, the 1st barrier and the 2nd minimum. {α, α 1 , α 4 } is the standard 3D parameter set.
In this study, we focus on α 3 and α 6 as the fourth parameter. In the following, we show the role of α 3 and α 6 in describing important shapes in the fission process. Figure  1 shows typical shapes that can be described by the Cassini shape parameterization. The top left figure is drawn using only α 1 and shows that both fragments have similar deformations. On the other hand, the bottom left figure shows a nearly spherical fragment and a largely deformed fragment, depicted using only α 3 . Thus, α 3 can represent the shape-asymmetric deformation. In the 4D calculation the mass-asymmetry is determined by the combination of α 1 and α 3 . α 6 corresponds to the octupole deformation of fragments and is related to the distance between the centers of mass of the fragments. As shown in two figures on the right side of Fig. 1, α 6 can describe outwardly elongated or compact shapes depending on the sign. We expect that α 6 is necessary to describe the shapes corresponding to the super-long and super-short symmetric fission.

Multi-dimensional Langevin equation
The multi-dimensional Langevin equations for describing the time evolution of collective coordinate q i and conju-gate momentum p i of q i are given as where V is the potential energy of deformation, m i j is the macroscopic inertia tensor from [8], and γ i j is the macroscopic friction tensor from [9]. The potential energy of deformation is computed with the microscopic-macroscopic method [10] and includes the damping effect of the microscopic potential [11]: where a is the constant level density parameter taken from [12], the damping factor E D = 25 MeV is used in this study and T is the nuclear temperature computed with T = √ E int /a based on the Fermi gas model. The internal energy E int is calculated by the condition of the total energy conservation at each time step of the Langevin calculation.
g i j is determined by the Einstein relation T * γ i j = g ik g k j , where T * is effective nuclear temperature [13]. It is related to T by T * = (ℏω/2) coth(ℏω/2T ) with ℏω = 1.2 MeV. The random number R j is characterized by the white noise with ⟨R i (t) where ⟨⟩ denotes average over an ensemble.

Numerical results
We approach to fission by solving the 3D Langevin equation with a standard parameter set {α, α 1 , α 4 }. We also perform the 4D Langevin calculation with additional parameters α 3 and α 6 . In order to obtain the fragment mass distributions with a sufficient accuracy, we simulate the fission process for two million nuclei at the ground state until a half of them reach the scission condition α = 0.985.

Neutron-induced fission of 235 U
In this section, we focus on the neutron-induced fission of 235 U. It is well known that the fragment mass distribution of U has a characteristic of asymmetric fission. The combination of fragments consists of a spherical nucleus and the remaining nucleus with deformation, because the double magic nucleus 132 Sn shows strong stability when it is spherical. In the calculation using Cassini shape parameterization, α 3 is expected to play an important role. Therefore, we compare the results of the fragment mass distributions in two deformation spaces: 3D {α, α 1 , α 4 } and 4D {α, α 1 , α 3 , α 4 }. We consider that the two cases of incident neutron energy: thermal (almost 0 MeV) and 14 MeV. Figure 2 shows the fragment mass distributions of thermal and 14 MeV neutron-induced fission calculated with the 3D and 4D Langevin equations. The experimental values are also shown [14,15]. As can be seen, all distributions indicate that asymmetric fission is dominant; especially the distributions for the thermal neutron-induced fission shows almost no symmetric component. Compared with the experimental data, the 3D results show that the peak position is found to be too mass asymmetric and the distribution is narrower. On the other hand, it is found that the 4D calculation with the addition of α 3 gives distributions with the peak position shifted to mass symmetric division in better agreement with the experimental data than the 3D results.  To demonstrate the role of α 3 to the fragment shape, we pick out the trajectories of asymmetric fission from the events that reached scission and calculate the average values of the Cassini parameters at scission. We show in Fig.  3 the average shapes for thermal and 14 MeV neutroninduced fission of 235 U. The average shape is described using the average value of the Cassini parameters at scission, which indicates fragment masses corresponding to the peak position of the distribution in Fig. 2. The fragment masses and the average values of the Cassini parameters are listed below the average shapes. The 3D results show that the average shapes consist of two fragments with similar deformations. On the other hand, 4D results show that the heavy fragment is almost spherical and the light fragment is prolate. The difference between 3D and 4D can be understood to be due to the fact that α 3 describes the shape asymmetry of the fragments, which strongly reflects the shell effect of 132 Sn. Therefore, we can conclude that the shape asymmetry of fragments α 3 is an effective parameter when dealing with asymmetric fission.

Spontaneous fission of 256 Fm and 258 Fm
As another example of actinide nuclei, we examine spontaneous fission of Fm isotopes. In this element, it is known that the fragment mass distribution in spontaneous fission changes dramatically with neutron number, being asymmetric at N=156 and symmetric at N=158 [15,16]. To investigate the effectiveness of the Cassini shape parameterization, we calculate the fission process of 256 Fm and 258 Fm at low excitation energy. Figure 4 shows the fragment mass distributions for the fission of 256 Fm and 258 Fm. At the beginning, we focus on the distributions obtained using 3D and 4D (3D + α 3 ). In the 256 Fm results, as in 236 U, the addition of α 3 moves the peak position of the distribution inward and reproduces the experimental data well. On the other hand, in 258 Fm, the peak of the distribution at symmetric separation is eliminated by the addition of α 3 . As a result, it is completely different from the experimental data.
Since 258 Fm is characterized by symmetric fission, α 6 is expected to play an important role in improving the fragment mass distribution. We solve the Langevin equation in 4D {α, α 1 , α 4 , α 6 } space using α 6 instead of α 3 . The results are shown by the triangles in Fig. 4. The distribution of 258 Fm shows that the peak at asymmetric fission is lower than that of 3D and symmetric fission is dominant. It is found that the distribution of 258 Fm is in better agreement with experimental data. On the other hand, the 256 Fm result shows the same trend as the 258 Fm one when α 6 is added, which is not consistent with the experimental data. Therefore, α 6 is considered to be effective in treating systems characterized by symmetric fission.
To investigate the contribution of the additional parameters α 3 and α 6 to the fragment shapes, we focus on the average shapes of fragments. For 256 Fm, the average shape by 4D in Fig. 5 indicates the deformation of shape asymmetry. As in 236 U, α 3 is important for the description of asymmetric deformation. For the 4D averaged shape of 258 Fm, we use α 6 instead of α 3 and fix the mass number of fragments at 129, corresponding to symmetric fission. As can be seen, the shape is more compact in 4D. The distance between the centers of masses calculated from the Cassini parameters in Fig. 5 is 2.34R 0 in 3D and 2.15R 0 in 4D. This suggests that α 6 can describe the super-short symmetric fission. We assume that this effect is essential in discussing symmetric fission.
From the discussion with the results of 256 Fm and 258 Fm, it is found that α 3 and α 6 are important for describing the fragment shapes. In this study, we treat α 3 and α 6 separately, but it is desirable to use both α 3 and α 6 for the fission of 256 Fm and 258 Fm consistently. We will do this in the next study.

Summary
We apply the Cassini shape parameters to the Langevin equation, which can describe various shapes of fission fragments. We obtain the fragment mass distribution by solving the 3D Langevin equation with {α, α 1 , α 4 } as the standard parameter set and the 4D Langevin equation with α 3 and α 6 added to 3D. Results are given for thermal and 14 MeV neutron-induced fission of 235 U. In the distribution of 4D {α, α 1 , α 3 , α 4 }, the mass-asymmetry at the peak position is inside the 3D, which better reproduces the experimental data. From the analysis of the shapes of fission fragments, it is found that α 3 represents the shape asymmetry of fragments and is important to reflect the shell effect of 132 Sn. We also present results for the sponta-neous fission of Fm isotopes. For 256 Fm, which is characterized by asymmetric fission, α 3 is important for describing the fragment shapes. On the other hand, for 258 Fm, which is characterized by symmetric fission, α 6 is appropriate instead of α 3 . In conclusion, the five Cassini parameters α, α 1 , α 3 , α 4 and α 6 are important in the Langevin calculation of fission in the actinide region. As a next step, we will perform a 5D Langevin calculation that includes all five parameters.