Charge polarization calculated with a microscopic model for the ﬁssion fragments of U-236

. We investigated theoretical methods to estimate the charge polarization of the ﬁssion fragments. The method is based on the microscopic mean-ﬁeld model with the nucleon degree of freedom, and is represented in the three-dimensional coordinate space for dealing with any nuclear deformation. We employed the Skyrme e ﬀ ective interaction for the calculations and uranium-236 as a target nucleus. We reported static and dynamic methods in which the potential energy surface of the ﬁssioning nucleus concerning the quadrupole and octupole deformations are used. Although the static method deduced the charge polarization which indicates the nuclei with the magic number and the octupole deformation, the quantities are not consistent with the existing data library (Wahl’s systematics). The dynamic method with the iso-energy initial condition on the potential energy surface showed the ﬁnite charge polarization of light and heavy ﬁssion fragments which is consistent with the data library. Furthermore, it indicates that the charge polarization has energy dependence of the initial states of the dynamic method.


Introduction
Charge distribution of fission fragments is one of the most important quantities to understand the fission mechanisms and also to design nuclear power systems since the distribution of fission fragments (FFs) affects the final distribution of radioactive isotopes, therefore it affects the reprocessing procedure and final geological repository. The charge distribution is required in not only the nuclear engineering field but also the nucleosynthesis in nuclear astrophysics. To prepare correctly the charge distributions for the nucleosynthesis is very difficult because the source of the FF is unknown and include unmeasured fissile nuclei which will appear at the end of the neutron-capture nucleosynthesis.
The charge distribution has been evaluated as the deviation from the unchanged charge distribution (UCD) assumption in which the fragments keep the neutron-proton ratio of the fissile parent nucleus. The deviation is named the charge polarization (CP). For the major fission reactions in the nuclear reactor, their CP have been compiled in a data library called Wahl's systematics [1,2]. Although the library is designed for the nuclear engineering field, this is not suitable to predict the fission fragments from unmeasured fissile nuclei. In order to provide the CP of FFs generated from unknown fissile nuclei without empirical ways, we propose methods to describe FF based on a microscopic nuclear theory.
In this work, we introduce two model calculations which are performed by a static and dynamic models based * e-mail: ebata@mail.saitama-u.ac.jp on the mean-field model. And we discuss the calculated CP comparing with the data of Wahl's systematics. We will show that the finite CP in light and heavy FFs mass region has an important role to reproduce neutron multiplicity on the 235 U(n, f ) reaction. Furthermore, the finite CP can be deduced by the dynamic method and has the energy dependence at the initial states, which is concluded in this work.

Method
To estimate the charge polarization of FF, we employ theoretical methods based on a static mean-field model and a dynamical model, namely the time-dependent meanfield model. The static mean-field model is the Hartree-Fock plus BCS model (HF+BCS), and the dynamical model is the canonical-basis time-dependent Hartree-Fock-Bogoliubov theory (Cb-TDHFB) [3]. These models can deal with nuclear pairing correlation fully selfconsistently, although the used pairing is BCS level. And then, these model calculations are carried out with the three-dimensional Cartesian coordinate space representation, in which any nuclear deformation can be described.

Static mean-field model
The HF+BCS equations can be obtained by applying the BCS many-body wave function to the quantum manybody system. The BCS state represented in the canonical basis is written as, where |−⟩ is the vacuum state and theâ † k is the creation operator of particle state with the quantum number k. We choose the canonical-basis for the orbitals in |Φ BCS ⟩, as usual, which diagonalizes the density matrix [4]. Thek means the pair and the time-reversed state with k. The coefficients u k and v k are the BCS factors which are obtained under the normalization condition: u 2 k + |v k | 2 = 1. Eventually, the HF+BCS equations can be obtained according to the variation δ⟨Φ BCS |Ĥ|Φ BCS ⟩ = 0, where the Hamiltonian includes the particle number constraints. The solution of HF+BCS equations can be obtained by that the HF and the gap equations are solved simultaneously and self-consistently.
We calculate the potential energy surface (PES) of the target nucleus concerning quadrupole and octupole deformations, with the constrained Skyrme HF+BCS (CSHF+BCS). The degrees of freedom for the PES are the m = 0 components of quadrupole (Q 2m ) and octupole (Q 3m ) moments, although we add the constraints to the PES calculations in which the expectation values of the other momentum are zero. The constraints termsQ lm are added as the quadratic forms to the Hamiltonian: where C lm ,Q lm , and q lm are Lagrange multiplier, constraint operators and required expectation values, respectively. We also use the constraints to fix the center of mass and principal axis; therefore, l ≥ 1. The data of the obtained PES is shown in Sec.2.3. The Skyrme effective interaction of SkM * parameter set is employed in the present calculations. For the pairing interaction, we choose the form of constant G model, however, the pairing strength G is self-consistently determined through the level density consistent with the meanfield [3,5].

Dynamical mean-field model
Here, we introduce briefly the Cb-TDHFB. The Cb-TDHFB is derived from the time-dependent Hartree-Fock+Bogolbov theory with the canonical-basis representation and the approximation for the pairing channel [3]. The Cb-TDHFB consists with three dynamical equations for the canonical basis, occupation probabilities |v k (t)| 2 , and pair densities κ k (t) ≡ u k (t)v k (t). In the limit of nopairing correlation, the Cb-TDHFB equations are equal to the TDHF, and in the limit of the static case, their solution is those of HF+BCS. And the particle number, the orthogonalization among single-particle states, and the total energy of the system are conserved in the dynamical calculation with the Cb-TDHFB. Therefore, the configuration obtained by the static models in Sec.2.1 can be used as the initial state of the Cb-TDHFB calculation. In this work, we choose the initial states on the PES with the isoenergies. The choice of the initial state might have important roles to reproduce the fission fragment configurations.

Fission fragments and Charge polarization
In Sec.3, we show the results calculated by static and dynamic models, although the condition to calculate nucleon numbers of the FF is the same in both models. Here, we explain calculating the charge polarization (CP) from the fission configuration in the static and dynamic model. The nucleon numbers are obtained from the density distributions on the fission configurations. In the calculation for the nuclear elongation with the static or dynamical model, we check a position of the thinnest density less than 2% of the maximum density (≃ 0.16 fm −3 ) between the FF candidates. We decide on the left and right fragments by the density threshold position and regard that the target nucleus has reached to fission. The particle numbers of the FF are estimated by each integration of nuclear density over the left and right sides. After getting their particle numbers, the CP dZ(A f ) is written as, where Z f and A f are proton and mass numbers of FF which can be deduced from the calculations. For the static model calculations, the fission configurations can be obtained on the PES concerning Q 20 and Q 30 . The PES is calculated with the constraint ranges from 435 to 20,880 (fm 2 ) for Q 20 and from 0 to 80,000 (fm 3 ) for Q 30 . The configurations are obtained at every [q 20 , q 30 ] = [435, 5,000] (fm 2 , fm 3 ), although the estimated particle number should be regarded as the average in the bin. In this work, we got the 16 fission configurations, therefore, can evaluate the CP of 32 points due to getting light and heavy fission configurations.
For the dynamic model calculations, the fission configuration can be obtained after the real-time evolution of nuclear density. The initial states of real-time calculation are calculated using the configurations on the PES which are elongated over the second fission barrier. We choose the states with the iso-energy (−1796 MeV) which is the same total binding energy of the local minimum on the PES. Furthermore, to check the energy dependence of the CP, the states with −1800 MeV are also employed in the initial conditions in this work. The nuclear densities elongated over the second fission barrier have the neck and the Coulomb repulsion force between FF candidates is induced. Therefore, the FF candidates split into two parts without extra force. The times to reach fission are different for each initial state. Figure 1 shows the CPs estimated by the static model calculations and compiled in the Wahl's systematics [2]. The symmetric FF has A f =118.00 and Z f =46.00, which is palladium-118 ( 118 Pd) and is just on the UCD line (dZ=0). The behavior of the CP concerning FF mass A f is the reflection symmetry at the center of the 118 Pd. The CP amplitude slightly increases around the symmetric FF, and it forms a peak with 0.67 at (A f , Z f ) = (104.57, 41.43) (or (131.43, 50.57)). And there appears a shoulder with an amplitude (∼0.29) at (A f , Z f ) = (98.75, 38.78) (or (137.25, 53.22)). The nucleon numbers of FF at the peak and shoulder indicate the shell effects for the spherical magic numbers (Z=50, N=82) and the octupole deformation (N=84), which appear on the fission path. And then, the amplitude goes down slowly to the UCD line from its peak. The finite CP (|dZ| ∼ 0.6) in A f < 90 and A f > 140 mass regions has not appeared in the static model calculations. We can see the large difference between the CPs of ours and Wahl's systematics in Fig. 1. There are a deep valley (at A f ∼ 110) and a big hump (at A f ∼ 126). They do not appear in ours. To check the role of the finite CPs in A f < 90 and A f > 140 mass regions, we compare the neutron multiplicities computed by Hauser-Feshbach code (BeoH) [6] with each condition (UCD, Wahl's systematics, ours, and ours + the finite CP), which are shown in Table.1. The "ours + the finite CP" composes of ours and the finite CP expressed by a hyperbolic tangent function fitted to the CP of Wahl's systematic. The difference among BeoH calculations is due to the difference in CP. Table.1 indicates the importance on the neutron multiplicity of the finite CPs in A f < 90 and A f > 140 mass regions, and also shows that the CP around symmetric FF does not so contribute to the neutron multiplicity. Therefore, we should know the mechanism to rise the finite CP in the light and heavy mass regions in order to understand the fission phenomena.

Results
Our static model calculations indicated that the effects of nuclear shell structure appear in the FF configurations. However, these calculations show only the results for the lowest energy at each deformed configuration, namely, the adiabatic results. We perform dynamic model calculations to consider the diabatic effects on the CP. Figure 2 shows the CP obtained by the Cb-TDHFB with the initial states of iso-energies; E = −1793 (triangle) and −1800 (circle) MeV. The dashed lines mean the two-parameter Fermi functional f (A f ) fitted to the estimated CPs: where A sym = 118 and C 1 + C 2 means the estimated maximum CP dZ max . The dZ max s for E = −1793 and −1800

Conclusion
We proposed theoretical methods to estimate the charge polarization (CP) of fission fragments (FFs).
The constraint Skyrme Hartree-Fock+BCS model and the canonical-basis time-dependent Hartree-Fock-Bogoliubov model were utilized as the static and dynamic methods for the estimation which were performed in the threedimensional Cartesian coordinate representation. The Skyrme effective interaction (SkM * parameter set) and the smoothed constant G pairing correlation were employed to the model calculations. We calculated the potential energy surface (PES) of 236 U concerning the quadrupole Q 20 and octupole Q 30 nuclear deformations.
In the static model calculations, we obtained 16 fission configurations on the PES and the particle numbers at each configuration which showed the CP indicating nuclear shell structures. However, the calculated CP was widely different from those of the data library (Wahl's systematics). We compared the effects of each CP (UCD, ours, Wahl's) on the neutron multiplicity using Hauser-Feshbach model calculation code (BeoH) which indicated that the finite CP (∼ ±0.6) appeared in light and heavy mass regions had a crucial role to decide the neutron multiplicity from the fission of 236 U. We performed dynamic model calculations to consider the diabatic effects on the CP. The real-time evolution of nuclear density distributions was computed using the initial states with iso-energies on the PES obtained in the static model calculation. The initial states were before the fission although they showed well-deformed density distributions. We chose E = −1793 and −1800 MeV for the iso-energies. The E = −1793 MeV corresponded to the total binding energy of the second minimum on the PES in this work. The E = −1800 MeV was chosen to check the energy dependence of the CP. We showed that the finite CP in the light and heavy mass fragments appeared in the dynamic model calculations. The maximum finite CPs were estimated as 0.79 and 0.44 for the calculations with E = −1793 and −1800 MeV, which indicate the energy dependence of the CP.
In future work, we should calculate several properties of FFs using the calculated CP, which will be necessary to evaluate the availability of the theoretical method. The static model calculations will be needed to estimate the excitation energies of FFs, although the CP in the calculations was not suitable to reproduce the neutron multiplicity from the FFs of 236 U. We will need to check both static and dynamic methods to estimate the FF configurations through the applications for several fission reactions because the CP of FF has a system energy dependence.