source: trunk/documents/UserDoc/UsersGuides/PhysicsReferenceManual/latex/hadronic/theory_driven/Fission/FissionCrossSection.tex @ 1208

\section{Nuclear fission cross section.}

\hspace{1.0em}The probability $P_{n}^{fis}$ that fission occurs at any
step of evaporation chain with $n$ evaporated fragments can be defined
as 
\begin{equation}
\label{FCS1}P_{n}^{fis} = 1-P_{n}, 
\end{equation} 
where $P_{n}$ is the probability of a transition from an excited state
to the ground state for the nucleus only by evaporation of $n$
fragments. The probability $P_{n}$ can be calculated using equation:
\begin{equation}
\label{FCS2}P_{n}=\prod_{i=1}^{n}[1-W_{fis}(E^{*}_i,A_i,Z_i)/W_{tot}(E^{*}_i,A_i,Z_i)],
\end{equation}
where $W_{fis}$ fission probability (per unit time) in the Bohr and
Wheeler theory of fission \cite{BW39}. It is assumed to be proportional
to the level density $\rho_{fis}(T)$ at the saddle point:
\begin{equation}
\label{FCS3}W_{fis}=\frac{1}{2\pi \hbar \rho_c(U_c)}
\int_{0}^{U_f-B_{fis}}
\rho_{fis}(U_f-B_{fis}-T)dT,
\end{equation}
where $U_f= E^{*} - \Delta_f$ and pairing energy
\begin{equation}
\label{FCS3a} \Delta_{f} = \kappa \frac{14}{\sqrt{A}} \ [MeV]
\end{equation} 
In Eq. ($\ref{FCS3}$)  $B_{fis}$ is the fission barrier height.
  $W_{tot}$
is total decay probability (per unit time) of a nucleus:
\begin{equation}
\label{FCS4} W_{tot}=W_{fis}+\sum_{b=1}^{6}W_{b}
\end{equation}
and $W_{b}$ is the probability to evaporate fragment of type $b$.  In
the Weisskopf and Ewing theory of particle evaporation \cite{WE40}:
\begin{equation}
\label{FCS5}W_{b}(T_b) = \sigma_{b}(T_b)\frac{(2s_b+1)m_b}{\pi^2 \hbar^3}
\frac{\rho_b(U_b - Q_b-T_b)}{\rho_c(U_c)}T_b,
\end{equation}
where $\sigma_{b}(T_b)$ is the inverse (absorption of particle $b$)
reaction cross section, $s_b$ and $m_b$ are particle spin and mass,
  $\rho_c$ and $\rho_b$ are level densities of compound nucleus  and
nucleus after particle evaporation, respectively.
The energies $U_b$ and $U_c$ are defined as $U_b = E^{*} - \Delta_b$ 
and $U_c = E^{*} - \Delta_c$, where $\Delta_{b,c}$ are pairing energies 
$\Delta_{Pair}$
of the compound and residual nuclei, respectively.
The pairing energy $\Delta_{Pair}$ is calculated according to
\begin{equation}
\label{FCS5a} \Delta_{Pair} = \kappa \frac{12}{\sqrt{A}} \ [MeV]
\end{equation}
with $\kappa = 0$, $1$, or $2$ for odd-odd, odd-even or even-even 
nuclei, respectively.

The Eq. ($\ref{FCS1}$) gives us a possibility to calculate numericaly
 the so-called fissility of nucleus $P_{fis} =
\sigma_{fis}/\sigma_{in}$ (see e.g. \cite{ICC80}), where $\sigma_{in}$
is the inelastic nuclear reaction cross section and hence the fission
cross section $\sigma_{fis}$. E.g.
\begin{equation}
\label{FCS2} \sigma_{fis}=\sigma_{in}P_{fis}=\sigma_{in}\frac{1}{N_{ch}}
\sum_{n=1}^{N_{ch}}P^{fis}_{n},
\end{equation}
where $N_{ch}$ is the number of fragment evaporation chains, which is
used for averaging.
 
As one can see from Eq. ($\ref{FCS3}$) the fission barrier height
$B_{fis}$ and the parameter of the level density of a nucleus $a_{fis}$
at saddle point are the basic ingredients of model, which are necessary
for the calculation of fission cross section.