source: trunk/documents/UserDoc/UsersGuides/PhysicsReferenceManual/latex/hadronic/theory_driven/Fission/ParamFission.tex @ 1292

\section{Parameterized Fission Model}

\subsection{Fission Process Simulation}

\subsubsection{Atomic number distribution of fission products}

\hspace{1.0em}As seen in experimental data \cite{VH73} the mass
distribution of fission products consists of symmetric and asymmetric 
components:
\begin{equation}
\label{FPS1} F(A_f) = F_{sym}(A_f) + \omega F_{asym}(A_f),
\end{equation}
where $\omega(U,A,Z)$ defines the relative contribution of each component
and depends on the excitation energy $U$ and $A,Z$ of the fissioning
nucleus.  It was found \cite{ABIM93} that experimental data can be
approximated with good accuracy, if one takes
\begin{equation}
\label{FPS2} F_{sym}(A_f) = \exp{[-\frac{(A_f - A_{sym})^2}{2\sigma_{sym}^2}]}
\end{equation}
and
\begin{equation}
\begin{array}{c}
\label{FPS3} F_{asym}(A_f) = \exp{[-\frac{(A_f - A_{2})^2}{2\sigma_{2}^2}]} + 
\exp{[-\frac{{A_f - (A - A_{2})}^2}{2\sigma_{2}^2}]} + \\
+ C_{asym}\{\exp{[-\frac{(A_f - A_{1})^2}{2\sigma_{1}^2}]} + 
\exp{[-\frac{{A_f - (A - A_{1})}^2}{2\sigma_{2}^2}]}\},
\end{array}
\end{equation}
where $A_{sym} = A/2$, $A_1$ and $A_2$ are the mean values and
$\sigma^2_{sym}$, $\sigma^2_1$ and $\sigma^2_2$ are the dispersions of the
Gaussians, respectively.  From an analysis of experimental data
\cite{ABIM93} the parameter $C_{asym} \approx 0.5$ was defined, followed by
the dispersion values:
\begin{equation}
\label{FPS4} \sigma^2_{sym} = \exp{(0.00553U + 2.1386)},
\end{equation} 
where $U$ is in MeV,
\begin{equation}
\label{FPS5} 2\sigma_1 = \sigma_2 = 5.6 \ MeV
\end{equation}
for $A \leq 235$ and 
\begin{equation}
\label{FPS6} 2\sigma_1 = \sigma_2 = 5.6 + 0.096 (A - 235) \ MeV
\end{equation}
for $A > 235$ were found.

The weight $\omega(U,A,Z)$ was approximated as follows
\begin{equation}
\label{FPS7} \omega = \frac{\omega_{a} - F_{asym}(A_{sym})}
{1 - \omega_a F_{sym}((A_1 + A_2)/2)}.
\end{equation}
The values of $\omega_a$ for nuclei with $96 \geq Z \geq 90$ were 
approximated by
\begin{equation}
\label{FPS8} \omega_a(U) = \exp{(0.538U - 9.9564)}
\end{equation}
for $U \leq 16.25$ MeV,
\begin{equation}
\label{FPS9} \omega_a(U) = \exp{(0.09197U - 2.7003)}
\end{equation}
for $U > 16.25$ MeV and 
\begin{equation}
\label{FPS10} \omega_a(U) = \exp{(0.09197U - 1.08808)}
\end{equation}
for $z = 89$. 
An approximation for nuclei with $Z \leq 88$ \cite{ABIM93} is given by:
\begin{equation}
\label{FPS11}\omega_a(U) = 
\exp{[0.3(227 - a)]} \exp{ \{0.09197[U - (B_{fis} - 7.5)] 
- 1.08808 \}},
\end{equation}
where for $A > 227$ and $U < B_{fis} - 7.5$ the corresponding factors occurring
in the exponential functions vanish.

\subsubsection{Charge distribution of fission products}

\hspace{1.0em} For a given mass of the fragment $A_f$ the experimental 
data \cite{VH73} on the charge $Z_f$ distribution of fragments are well 
approximated by a Gaussian with dispersion $\sigma^2_{z} = 0.36$ and the 
average $<Z_f>$ is described by the expression:
\begin{equation}
\label{FPS12} <Z_f> = \frac{A_f}{A}Z + \Delta Z, 
\end{equation}
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 the sampling of fragment atomic mass numbers and fragment charges, 
we must check that the fragment ground state masses do not exceed the 
initial energy.  The maximal fragment kinetic energy is calculated as 
\begin{equation}
\label{FPS13a}T^{max} < U + M(A,Z) - M_1(A_{f1}, Z_{f1}) - M_2(A_{f2}, Z_{f2}),
\end{equation}
where $U$ and $M(A,Z)$ are the excitation energy and mass of the initial
nucleus.  $M_1(A_{f1}, Z_{f1})$ and $M_2(A_{f2}, Z_{f2})$ are the masses
of the first and second fragment, respectively.

\subsubsection{Kinetic energy distribution of fission products}

\hspace{1.0em} The average kinetic energy $<T_{kin}>$ (in MeV) of fission 
fragments has an empiricaly defined \cite{VKW85} dependence on the mass and 
charge of a fissioning nucleus:
\begin{equation}
\label{FPS13}<T_{kin}> = 0.1178 Z^2/A^{1/3} + 5.8.
\end{equation}
This energy is distributed differently in cases of symmetric and
asymmetric modes of fission.  It follows from the analysis of data
\cite{ABIM93} that in the asymmetric mode, the average kinetic energy of
fragments is higher than that in the symmetric one by approximately
$12.5$ MeV.  Empirical expressions have been suggested \cite{ABIM93}
to approximate the average values of the kinetic energies $<T_{kin}^{sym}$ 
and $<T_{kin}^{asym}>$ for the symmetric and asymmetric modes of fission, 
\begin{equation}
\label{FPS14} <T_{kin}^{sym}> = <T_{kin}> - 12.5 W_{asim}, 
\end{equation}
\begin{equation}
\label{FPS15} <T_{kin}^{asym}> = <T_{kin}> + 12.5 W_{sim},
\end{equation} 
where 
\begin{equation}
\label{FPS16} W_{sim} = \omega \int F_{sim}(A)dA/\int F(A)dA
\end{equation}
and
\begin{equation}
\label{FPS17} W_{asim} = \int F_{asim}(A)dA/\int F(A)dA,
\end{equation} 
respectively.  For symmetric fission the experimental data for the
ratio of the average kinetic energy of fission fragments
$<T_{kin}(A_f)>$ to this maximum energy $<T^{max}_{kin}>$ as a function
of the mass of a larger fragment $A_{max}$, can be approximated by the
expressions
\begin{equation}
\label{FPS18} <T_{kin}(A_f)>/<T^{max}_{kin}> = 
1 - k [(A_f - A_{max})/A]^2
\end{equation}
for $A_{sim} \leq A_f \leq A_{max} + 10$ and 
\begin{equation}
\label{FPS19} <T_{kin}(A_f)>/<T^{max}_{kin}> = 
1 - k(10/A)^2 - 2 (10/A)k(A_f - A_{max} - 10)/A .
\end{equation}
These are valid for $A_f > A_{max} + 10$, where $A_{max} = A_{sim}$ and 
$k = 5.32$ for symmetric fission, and $A_{max} = 134$ and $k = 23.5$ for 
asymmetric fission.  For both modes of fission the distribution over the
kinetic energy of fragments $T_{kin}$ is chosen to be Gaussian with their 
own average values $<T_{kin}(A_f)>= <T_{kin}^{sym}(A_f)>$ or
$<T_{kin}(A_f)>=<T_{kin}^{asym}(A_f)>$, and dispersions $\sigma^2_{kin}$
equal $8^2$ MeV and $10^2$ MeV$^2$ for symmetrical and asymmetrical
modes, respectively. 

\subsubsection{Calculation of the excitation energy of fission products}

\hspace{1.0em} The total excitation energy of fragments $U_{frag}$ 
can be defined according to the equation
\begin{equation}
\label{FPS21} U_{frag} = U + M(A,Z) - M_1(A_{f1}, Z_{f1}) - M_2(A_{f2}, Z_{f2}) - 
T_{kin},
\end{equation}
where $U$ and $M(A,Z)$ are the excitation energy and mass of the initial
nucleus, $T_{kin}$ is the fragment kinetic energy, $M_1(A_{f1},
Z_{f1})$,  and $M_2(A_{f2}, Z_{f2})$ are the masses of the first and second 
fragments, respectively.

The value of the 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 the initial nucleus, then the total
excitation energy will be distributed between fragments in proportion to
their mass numbers.  One then obtains
\begin{equation}
\label{FPS22} U_f = U_{frag} \frac{A_f}{A}.
\end{equation}

\subsubsection{Excited fragment momenta}

\hspace{1.0em} Assuming that the 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 
\begin{equation}
\label{FPS23}
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}, 
\end{equation}
and its components, assuming the isotropic distribution of fragments.