Deformation Parameters and Collective Temperature Changes in Photofission Mass Yields of Actinides Within the Systematic Statistical Scission Point Model

The photofission fragment mass yields of actinides are evaluated using a systematic statistical scission point model. In this model, all energies at the scission point are presented as a linear function of the mass numbers of fission fragments. The mass yields are calculated with a new approximated relative probability for each complementary fragment. The agreement with the experimental data is quite good, especially with a collective temperature T col of 2 MeV at intermediate excitation energy and T col = 1 MeV for spontaneous fission. This indicates that the collective temperature is greater than the value obtained by the initial excitation energy. The generalized superfluid model is applied for calculating the fragment temperature. The deformation parameters of fission fragments have been obtained by fitting the calculated results with the experimental values. This indicates that the deformation parameters decrease with increasing excitation energy. Also, these parameters decrease for fissioning systems with odd mass numbers.


INTRODUCTION
Since the fission discovery, the experimental and theoretical fission mass yields have been continuously developed. The most widely used theoretical model to study mass yields is the statistical model which was founded by Fong and Wilkins [1,2]. This model has been developed in many branches, such as the Gaussian model [3,4] and modified scission point models [6][7][8][9][10][11][12][13]. The time-dependent model has been significantly developed by Randrup [14][15][16] and others [17][18][19][20][21][22] to predict the shape of mass yields (symmetric or asymmetric modes). Because all of them have sophisticated computations, a systematic method is needed to evaluate the mass distribution of fission fragments in an easy way.
Although the statistical model can predict transitions between symmetric and asymmetric modes in the region of heavy actinides, the calculated results are inaccurate compared to the experimental data. This problem is found where the calculated results were smeared (refined) by the Gaussian model with the width 1.5 amu to obtain a smoother curve [8].
On the other hand, some researchers [6][7][8][9] added some terms to neutron kinetic energy or gamma endpoint energy, E (as the initial excitation energy), to obtain the excitation energy of the fissioning nucleus, E*, for example, Pasca [7,9] added the Q-factor and the difference between the potential energy of the fissioning nucleus and the potential energy of one of the fragments at the scission point to the initial excitation energy to obtain excitation energy (i.e., E* Q+E+U cn −U i ). Some others [10,11] took available energy as the difference between the potential energy of the fissioning system at the scission point and the energy of the excited compound nucleus, which is the sum of Q-factor and initial excitation energy (E* Q+E). This addition of energies to the initial excitation energy is more pronounced in time-dependent models [14] when nuclear excitation energy measured relative to potential energy (E* E-U). Also, fragment temperature, instead of collective temperature, has been used to calculate the mass yield of 238 U in my previous work [13]. Also, collective temperature is usually calculated by the excitation energy of fissioning systems. Here, the role of collective temperature is examined by selecting two values (1 and 2 MeV) independently.
In the next section, a systematic method is presented to calculate mass yields for actinides; the fewer parameters and refinements are included in this framework. For this reason, all energies at the scission point are formulated as a function of the mass number of fission fragments. Then, all mass yields for photofission of actinides are calculated and compared with the experimental data (Section 3).

THEORETICAL FRAMEWORK
The relative yield is usually calculated as the ratio between the probabilities of a given fragmentation and the sum of probabilities of all possible fragmentations as follows: where P(A i , Z i ) is the relative probability of formation of any fission fragment. In the statistical method, the relative probability of any fission fragment pair is given by Ref. 2.
where V(N, Z, s, β 1 , β 2 ) is the total potential energy of the fissioning system at the scission point; T coll is the collective temperature of the fissioning system; s is the spherical coaxial distance; and β i s (i L, H) are the deformation parameters for the light and heavy fragments. The total potential energies at the scission point are defined as follows: The first term is the interaction energy between complementary fragments, V Interac V Coul + V prox . The individual energy includes the macroscopic energy (V Mac ), shell correction energy, and pairing energy as microscopic energies (V Mic ), that is, V indiv V Mac,i + E shell,i + E pair,i . In the following sections, these energies are presented.

Interaction Energy
The nuclear proximity potential (V prox ) is presented [23,24] by the following equation: where the width (diffuseness) of the nuclear surface is b ≈ 0.88. C 1 and C 2 are the Süsmann central radii of fragments that is related to the sharp radius R i as follows: R 0,i is the net radius of each fission fragment obtained through a semi-empirical equation that is a function of the mass number of fission fragments [24] as follows: MeV fm 2 ; where N i , Z i , and A i are neutron, atomic, and mass numbers of any fission fragments, respectively. In Eq. 4, Φ is the universal proximity relation which is a function of distance between two interaction fragments and is defined [23] as follows: Coulomb energy is written as [4,12] follows: where r s + R 1 + R 2 , n 1 3 2 5π √ r 2 , n 2 3 7πr 2 , n 3 9 14πr 4 , and n 4 27R 2 1 R 2 2 10πr 4 . R i is the radii of deformed nuclei that can be evaluated using the deformation-dependent expansion of nuclear radii as follows: where θ is the angle made by the axis of symmetry with fission axis, β i is the quadrupole deformation parameter of fragments, and Y 20 is the spherical harmonic functions.

Individual Energies
The individual energies include the macroscopic energy, shell correction energy, and pairing energy. The macroscopic energy is calculated in the framework of the FRLDM model from Ref. 26, and the spheroidally deformed is applied as [27].
where the quantity B1 is the relation generalized surface or nuclear energy in a model that accounts for the effect of infinite range of nuclear force. The quantity f is the proton form factor correction to Coulomb energy. Relative Coulomb energy B3 is used in the first approximate. All parameters are chosen from Ref. 26. The pairing correction energy (E pair in MeV) is calculated by a simple relation as follows: 0 for Z i and N i even 12 Z i for Z i even and N i odd 12 N i for N i even and Z i odd 24 A i for Z i and N i odd.
The shell correction energy is calculated according to Ref. 13. The systematic formalism of this method is used to calculate the shell correction energy of fragments. Also, the results of this method agree with the results of the Strutinsky model [28].

Temperature
The temperature-dependent relation for pairing energy is given by Ref. 29.
Temperature dependence of shell energy is applied as [29] follows: where E shell (T) is the shell correction energy for each excitation energy and E shell (0) is the zero excitation energy. Also, S 0 2.5 MeV −1 , τ 2π 2 A 1/3 T/41, E 1 −18.54 MeV, and E 0 42.28 MeV, and E is energy corresponding to temperature.
Temperature (T) usually calculate with Fermi gas relation, but we used the generalized superfluid model as follows: where E is the excitation energy and E con is the condensation energy for the even-even nucleus. We have where E pair,0 12/ A √ and the level density parameters are given by

RESULTS AND DISCUSSION
Similar to Refs. 30 and 31, the atomic number of fission fragments are obtained with the unchanged charge density distribution as [32] follows: where Z cn is the atomic number of compound nucleus, A cn is the mass number of compound nucleus, and ν is post-scission neutrons and is defined by Refs. 33 and 34. Pasca [9] added the height of the fission barrier to the potential energy at the scission point to calculate the collective temperature. In this study, the height of the fission barrier is chosen as the excitation energy for spontaneous fission. Thus, for photofission, the height of the fission barrier is added to the initial excitation energy (the bremsstrahlung endpoint energy) (i.e., 6+E MeV for 238 U). Thus, E* in Eqs 15, 16 is the height of the fission barrier of the target nucleus plus the initial excitation energies.
On the other hand, the collective temperature is not dependent on the excitation energy in this study such as Ref. 2. Therefore, here, the excitation energy only affects the total potential energy of the fissioning system. These calculations indicate that the excitation energy has little change in the values of fission fragment mass distributions. For example, by increasing the excitation energy by 20 MeV, the mass distribution changes by less than 1 percent. Therefore, this small effect indicates that the major effect of the excitation energy in mass yield values is due to the change in collective temperature. Of course, the excitation energies are divided between the fragments proportional to their masses.
Equation 2 is an exponential function that is strongly upwardsloping, and the collective temperature is usually taken to be constant in Refs. 2 and 7 (the change in collective temperature is discussed later). Therefore, the values of deformation parameters provide the minimal value of the total potential energy. On the other hand, the minimal values of the total potential energy of the fissioning system at the scission point correspond to the minimal values of the deformation parameters. Therefore, only the maximum values of deformation parameters could be considered in Eq. 2 as follows:

Investigation on Fission of 238 U
The mass yield for spontaneous fission of 238 U is presented in Figure 1. In the left side of Figure It can be seen that β m,L values increase with increasing mass numbers of fission fragments. This indicates that the probability of the formation of symmetric fragments is reduced, which shows the dominance of the asymmetric fission mode. When T coll 2 MeV, the deformation parameters are obtained as β m,H 0.5 and also, for A L 96, β m,L equals to 0.43. However, β m,L values increase significantly for mass numbers between 102 and 96. This increase could be due to the magic neutron number (N L 50) of fission fragments with the mass number A L 96. Figure 1 shows that the calculated results for mass yield are in good agreement with the experimental data for two collective temperatures. But, as the collective temperature increases, the order of β m,L values is disturbed for spontaneous fission of 238 U. This indicates that T coll 1 is better for spontaneous fission of 238 U. Also, the obtained β m,L values, especially for fragments with a mass number around 98, are close to the results of Ref. 13, which used Eq. 2 and close to the results of Ref. 38 obtained from the study of the total kinetic energy. However, in the recent reference, the values of deformation parameters are the same for the two complementary fission fragments, which caused differences in the β values for some fragments.
The photofission mass yield of 238 U is presented in Figure 2 also, when A L 96 and 97, β m,L equals to 0.44. It can be seen that β m,L values increase with increasing mass number of fission fragments. Also, the light fission fragments around mass number 96 are zirconium, which has the semi-magic number in the proton number (Z L 40). Figure 2 shows that the mass yield has a good agreement to the experimental data for T coll 2 MeV. In Figure 3, the photofission mass yield of 238 U is presented at 67.8 MeV bremsstrahlung endpoint energy. In the left and right

Investigation on the Plutonium Isotopes
The photofission mass yield of 240 Pu is presented in Figure 4 Also, for A L 94, β m,L equals to 0.36; for A L 96 and A L 97, β m,L equals to 0.32; for A L 99, A L 101, and A L 103, β m,L equal to 0.42; and for A L 104, β m,L equals to 0.40. Also, the β m,L values in plutonium-239 fission are lower than the β m,L values in plutonium-240 fission. This decrease can be due to both an increase in excitation energy and the odd effect of the plutonium-239 nucleus. Of course, since the compound nucleus does not absorb neutrons in the photofission process, the number of neutrons remains odd; it shows that the fission of nucleus with the odd number of neutrons can be easier than the fission of an even nucleus. To examine the odd-even effect, we investigate the neptunium nucleus in the next section.

Investigation on Neptunium Isotopes
The photofission mass yield of 237 Np is presented in Figure 6 at 28 MeV bremsstrahlung endpoint energy. In the left side of Figure 6, the results of systematic calculation are presented for It can be seen that β m,L values increase with increasing mass numbers of fission fragments like in other studies [42]. As can be seen, the β m,L values for photofission of 237 Np are higher than the β m,L values for photofission of 239 Pu, while the excitation energy The calculated results for the neptunium even-odd nucleus are in good agreement with the experimental values, and the deformation parameters have the least variations. Also, the deformation parameters reduce oscillations when T coll 2 MeV. These conditions are true for all actinides, so it is best to use T coll 2 MeV for intermediate energy of photofission. Therefore, by using the fragment temperature, instead of collective temperature in Eq. 2, the values of mass distribution are better obtained like in Ref. 13. Unlike the Gaussian models to calculate mass yields [5], where the excitation energy (bremsstrahlung endpoint energy) is used to evaluate mass distribution, in the statistical scission point model, the Q-factor energy and the height of the fission barrier are better added to the initial excitation energy.
Naik and Pomme [39,44,45] showed that the fission fragment mass yields around mass numbers 94-95, 99-100, and 104-105 are higher than other fission fragment mass yields for photofission of 238 U and 240 Pu in the low-energy region. Therefore, our calculated results confirm the changes in the mass yields of experimental data in some fragments. But these changes are seen for all nuclei studied in this systematic study.

SUMMARY
The mass yield for spontaneous fission and photofission of actinides are calculated within a systematic scission point model. The calculated results are compared with the available experimental data. There is good agreement with the experimental data, especially for a collective temperature of 2 MeV.
For intermediate excitation energy, the calculated results with T coll 2 MeV have better agreement with the experimental data, so it is better to add initial excitation energy (E) with the height of the fission barrier (and other energies such as Q-factor) to evaluate the mass yields. But for spontaneous fission, it is better not to change the excitation energy because the calculated results with T coll 1 are in good agreement with FIGURE 7 | Calculated mass yield for photofission of 237 Np at 9.5 MeV bremsstrahlung endpoint energy, accompanied by experimental data [43]. In the left and right sides are presented the calculated results obtained for T coll 1 MeV and T coll 2 MeV, respectively.
Frontiers in Physics | www.frontiersin.org February 2021 | Volume 9 | Article 629978 the experimental data. Therefore, by using the fragment temperature, instead of collective temperature and adding other energies to initial excitation energy, the values of mass yield are closer to the experimental data. In this study, the collective temperature is constant and also the change in mass distribution values was small with the change in excitation energy; therefore, the major effect of excitation energy (in other studies) is due to change in collective temperature.
The deformation parameters of fission fragments are presented by fitting the calculated results to the experimental data. There are close to the values in other studies obtained by the total kinetic energy and the integral form. The deformation values increase with increasing mass numbers of fission fragments (symmetric fragments) for all fissioning systems, which is due to the dominance of the asymmetric fission mode for photofission of actinides. On the other hand, the deformation parameter values decrease with increasing excitation energy. This increase in excitation energy also causes the deformation parameter changes to be irregular. Also, these parameters decrease for odd mass number fissioning systems. Also, the fissioning systems with odd neutron numbers have less deformation parameter values than the fissioning systems with even neutron numbers. The mass yield values for photofission of other actinides can be predicted by this method.
It is seen that the higher values in mass numbers of fission fragments around 104-105 and 99-100 in experimental data for photofission are related to the potential energy of the fissioning system at the scission point, and it can be seen for all photofission of nuclei actinide. But for some nuclei, these peaks are so small that they are not seen in the measurements.

DATA AVAILABILITY STATEMENT
The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.