Fission process simulation.

Atomic number distribution of fission products.

As follows from experimental data [VR73] mass distribution of fission products consists of the symmetric and the asymmetric components:

\[F(A_f) = F_{sym}(A_f) + \omega F_{asym}(A_f),\]

where \(\omega(U,A,Z)\) defines relative contribution of each component and it depends from excitation energy \(U\) and \(A,Z\) of fissioning nucleus. It was found in [eal93] that experimental data can be approximated with a good accuracy, if one take

\[F_{sym}(A_f) = \exp \left[-\frac{(A_f - A_{sym})^2}{2\sigma_{sym}^2} \right]\]

and

\[\begin{split}F_{asym}(A_f) &= \exp \left[ -\frac{(A_f - A_2)^2}{2\sigma_2^2} \right] + \exp \left[-\frac{A_f - (A - A_2)^2}{2\sigma_2^2} \right] + \\ {} &+ C_{asym} \left\{ \exp \left[-\frac{(A_f - A_1)^2}{2\sigma_1^2} \right] + \exp \left[-\frac{A_f - (A - A_1)^2}{2\sigma_2^2} \right] \right\} ,\end{split}\]

where \(A_{sym} = A/2\), \(A_1\) and \(A_2\) are the mean values and \(\sigma^2_{sim}\), \(\sigma^2_1\) and \(\sigma^2_2\) are dispersion of the Gaussians respectively. From an analysis of experimental data [eal93] the parameter \(C_{asym} \approx 0.5\) was defined and the next values for dispersions:

\[\sigma^2_{sym} = \exp{(0.00553U + 2.1386)},\]

where \(U\) in MeV,

\[2\sigma_1 = \sigma_2 = 5.6 \mbox{ MeV}\]

for \(A \leq 235\) and

\[2\sigma_1 = \sigma_2 = 5.6 + 0.096 (A - 235) \mbox{ MeV}\]

for \(A > 235\) were found.

The weight \(\omega(U,A,Z)\) was approximated as follows

\[\omega = \frac{\omega_{a} - F_{asym}(A_{sym})} {1 - \omega_a F_{sym}((A_1 + A_2)/2)}.\]

The values of \(\omega_a\) for nuclei with \(96 \geq Z \geq 90\) were approximated by

\[\omega_a(U) = \exp{(0.538U - 9.9564)}\]

for \(U \leq 16.25\) MeV,

\[\omega_a(U) = \exp{(0.09197U - 2.7003)}\]

for \(U > 16.25\) MeV and

\[\omega_a(U) = \exp{(0.09197U - 1.08808)}\]

for \(z = 89\). For nuclei with \(Z \leq 88\) the authors of [eal93] constracted the following approximation:

\[\omega_a(U) = \exp{[0.3(227 - a)]} \exp{ \{0.09197[U - (B_{fis} - 7.5)] - 1.08808 \}},\]

where for \(A > 227\) and \(U < B_{fis} - 7.5\) the corresponding factors occuring in exponential functions vanish.

Charge distribution of fission products.

At given mass of fragment \(A_f\) the experimental data [VR73] on the charge \(Z_f\) distribution of fragments are well approximated by Gaussian with dispertion \(\sigma^2_{z} = 0.36\) and the average \(\langle Z_f \rangle \) is described by expression:

\[\langle Z_f \rangle = \frac{A_f}{A}Z + \Delta Z,\]

when parameter \(\Delta Z = -0.45\) for \(A_f \geq 134\), \(\Delta Z = - 0.45(A_f -A/2)/(134 - A/2)\) for \(A - 134 < A_f < 134\) and \(\Delta Z = 0.45\) for \(A \leq A - 134\).

After sampling of fragment atomic masses numbers and fragment charges, we have to check that fragment ground state masses do not exceed initial energy and calculate the maximal fragment kinetic energy

\[T^{max} < U + M(A,Z) - M_1(A_{f1}, Z_{f1}) - M_2(A_{f2}, Z_{f2}),\]

where \(U\) and \(M(A,Z)\) are the excitation energy and mass of initial nucleus, \(M_1(A_{f1}, Z_{f1})\), and \(M_2(A_{f2}, Z_{f2})\) are masses of the first and second fragment, respectively.

Kinetic energy distribution of fission products.

We use the empirically defined [EKM85] dependence of the average kinetic energy \(\) (in MeV) of fission fragments on the mass and the charge of a fissioning nucleus:

\[\langle T_{kin} \rangle = 0.1178 Z^2/A^{1/3} + 5.8.\]

This energy is distributed differently in cases of symmetric and asymmetric modes of fission. It follows from the analysis of data [eal93] that in the asymmetric mode, the average kinetic energy of fragments is higher than that in the symmetric one by approximately \(12.5\) MeV. To approximate the average numbers of kinetic energies \(\langle T_{kin}^{sym} \rangle \) and \(\langle T_{kin}^{asym} \rangle \) for the symmetric and asymmetric modes of fission the authors of [eal93] suggested empirical expressions:

\[\langle T_{kin}^{sym} \rangle = \langle T_{kin} \rangle - 12.5 W_{asim},\]

\[\langle T_{kin}^{asym} \rangle = \langle T_{kin} \rangle + 12.5 W_{sim},\]

where

\[W_{sim} = \omega \int F_{sim}(A)dA \bigg/ \int F(A)dA\]

and

\[W_{asim} = \int F_{asim}(A)dA \bigg/ \int F(A)dA,\]

respectively. In the symmetric fission the experimental data for the ratio of the average kinetic energy of fission fragments \(\langle T_{kin}(A_f)\rangle \) to this maximum energy \(\langle T^{max}_{kin} \rangle \) as a function of the mass of a larger fragment \(A_{max}\) can be approximated by expressions

\[\langle T_{kin}(A_f) \rangle/\langle T^{max}_{kin} \rangle = 1 - k [(A_f - A_{max})/A]^2\]

for \(A_{sim} \leq A_f \leq A_{max} + 10\) and

\[\langle T_{kin}(A_f) \rangle / \langle T^{max}_{kin} \rangle = 1 - k(10/A)^2 - 2 (10/A)k(A_f - A_{max} - 10)/A\]

for \(A_f > A_{max} + 10\), where \(A_{max} = A_{sim}\) and \(k = 5.32\) and \(A_{max} = 134\) and \(k = 23.5\) for symmetric and asymmetric fission respectively. For both modes of fission the distribution over the kinetic energy of fragments \(T_{kin}\) is choosen Gaussian with their own average values \(\langle T_{kin}(A_f) \rangle = \langle T_{kin}^{sym}(A_f) \rangle \) or \(\langle T_{kin}(A_f) \rangle = \langle T_{kin}^{asym}(A_f) \rangle\) and dispersions \(\sigma^2_{kin}\) equal \(8^2\) MeV² or \(10^2\) MeV² for symmetrical and asymmetrical modes, respectively.

Calculation of the excitation energy of fission products.

The total excitation energy of fragments \(U_{frag}\) can be defined according to equation:

\[U_{frag} = U + M(A,Z) - M_1(A_{f1}, Z_{f1}) - M_2(A_{f2}, Z_{f2}) - T_{kin},\]

where \(U\) and \(M(A,Z)\) are the excitation energy and mass of initial nucleus, \(T_{kin}\) is the fragments kinetic energy, \(M_1(A_{f1}, Z_{f1})\), and \(M_2(A_{f2}, Z_{f2})\) are masses of the first and second fragment, respectively.

The value of excitation energy of fragment \(U_f\) determines the fragment temperature (\(T = \sqrt{U_f/a_f}\), where \(a_f \sim A_f\) is the parameter of fragment level density). Assuming that after disintegration fragments have the same temperature as initial nucleus than the total excitation energy will be distributed between fragments in proportion to their mass numbers one obtains

\[U_f = U_{frag} \frac{A_f}{A}.\]

Excited fragment momenta.

Assuming that fragment kinetic energy \(T_f= P^2_f/(2(M(A_{f},Z_{f}+U_f)\) we are able to calculate the absolute value of fragment c.m. momentum

\[P_f=\frac{(M_1(A_{f1},Z_{f1}+U_{f1})(M_2(A_{f2},Z_{f2}+U_{f2})}{ M_1(A_{f1},Z_{f1})+U_{f1} + M_2(A_{f2},Z_{f2})+U_{f2}}T_{kin}.\]

and its components, assuming fragment isotropical distribution.

Bibliography

EKM85

Viola V. E., Kwiatkowski K., and Walker M. Phys. Rev. C, 31:1550, 1985.

eal93(1,2,3,4,5)

Adeev G. D. et al. Technical Report, INR, Moscow, Russia, 1993. Preprint INR 816/93.

VR73(1,2)

Huizenga J. R. Vandenbosch R. Nuclear Fission. Academic Press, New York, 1973.