source: trunk/documents/UserDoc/DocBookUsersGuides/PhysicsReferenceManual/latex/hadronic/theory_driven/FermiBreakup/FermiBreakSim.tex @ 1211

\section{Fermi break-up simulation for light nuclei.}
\hspace{1.0em} The GEANT4 Fermi break-up model is capable to predict 
final states as result of an excited nucleus with atomic number $A < 17$ 
statistical break-up.

For light nuclei  the values of excitation
energy per nucleon are often comparable with nucleon binding
energy. Thus a light excited nucleus breaks into two or more fragments
with branching given by available phase space.  To describe a process of
nuclear disassembling the so-called Fermi break-up model is used
\cite{Fermi50}, \cite{Kretz61}, \cite{EG67}.  This statistical approach
was first used by Fermi \cite{Fermi50} to describe the multiple
production in high energy nucleon collision.



\subsection{ Allowed channel.} 

\hspace{1.0em}The channel will be allowed for decay, if the total
kinetic energy $E_{kin}$ of all fragments of the given channel at the
moment of break-up is positive. This energy can be calculated according
to equation:
\begin{equation}
\label{FBS1}E_{kin} = U+M(A,Z)-E_{Coulomb} - \sum_{b=1}^{n}(m_b+\epsilon_{b}),
\end{equation} 
$m_{b}$ and $\epsilon_{b}$ are masses and excitation energies of fragments, 
respectively, $E_{Coulomb}$ is the Coulomb barrier for a given channel. It 
is approximated by
\begin{equation}
\label{FBS2}E_{Coulomb} = \frac{3}{5} \frac{e^2}{r_{0}}(1 + 
\frac{V}{V_{0}})^{-1/3}
(\frac{Z^2}{A^{1/3}}-\sum_{b=1}^{n}\frac{Z^2}{A_b^{1/3}}),
\end{equation}
where $V_0$ is the volume of the system corresponding to the normal
nuclear matter density and $\kappa = \frac{V}{V_0}$ is a parameter (
$\kappa = 1$ is used).

\subsection{Break-up probability.} 

\hspace{1.0em}The total  probability  for nucleus to
break-up into $n$ componets  (nucleons, deutrons, tritons, alphas etc) 
in the final state is given by
\begin{equation}
\label{FBS3}W(E,n) = (V/\Omega)^{n-1}\rho_{n}(E),
\end{equation}
where $\rho_{n}(E)$ is the density of a number of final states, $V$ is
the volume of decaying system and $\Omega = (2\pi \hbar)^{3}$ is the
normalization volume.  The density $\rho_{n}(E)$ can be defined as a
product of three factors:
\begin{equation}
\label{FBS4}\rho_{n}(E)=M_{n}(E)S_nG_n.
\end{equation}
The first one is the phase space factor defined as
\begin{equation}
\label{FBS5}M_{n} = \int_{-\infty}^{+\infty}...\int_{-\infty}^{+\infty}
\delta(\sum_{b=1}^{n} {\bf p_{b}}) \delta(E-\sum_{b=1}^{n}\sqrt{p^2+m^2_b})
\prod_{b=1}^{n} d^3p_b,
\end{equation}
where ${\bf p_b}$ is fragment $b$ momentum. The second one is the spin
factor
\begin{equation}
\label{FBS6} S_n = \prod_{b=1}^{n}(2s_b+1),
\end{equation}
which gives the number of states with different spin orientations.  The
last one is the permutation factor
\begin{equation}
\label{FBS7}G_n = \prod_{j=1}^{k}\frac{1}{n_j !},
\end{equation}
which takes into account identity of components in final state. $n_j$ is
a number of components of $j$- type particles and $k$ is defined by $n =
\sum_{j=1}^{k}n_{j}$).

In non-relativistic case (Eq. ($\ref{FBS10}$) the integration in
Eq. ($\ref{FBS5}$) can be evaluated analiticaly (see e. g. \cite{BBB58}).
The probability for a nucleus with energy $E$ disassembling into $n$
fragments with masses $m_b$, where $b = 1,2,3,...,n$ equals
\begin{equation}
\label{FBS8} W(E_{kin},n) = 
S_nG_n (\frac{V}{\Omega})^{n-1}(\frac{1}{\sum_{b=1}^{n}m_b}
\prod_{b=1}^{n}
m_{b})^{3/2}
 \frac{(2\pi)^{3(n-1)/2}}{\Gamma(3(n-1)/2)}E_{kin}^{3n/2-5/2}, 
\end{equation}
where $\Gamma(x)$ is  the gamma function.

\subsection{Fermi break-up model parameter.} 

\hspace{1.0em}Thus the Fermi break-up model has only one free parameter
$V$ is the volume of decaying system, which can be calculated as
follows:
\begin{equation}
\label{FBS9} V = 4\pi R^3/3 = 4\pi r_{0}^3 A/3,
\end{equation}
where $r_{0} = 1.4 $ fm is used.

\subsection{ Fragment characteristics.}

We take into account the formation of fragments in their ground and
low-lying excited states, which are stable for nucleon
emission. However, several unstable fragments with large lifetimes:
$^{5}He$, $^{5}Li$, $^{8}Be$, $^{9}B$ etc are also considered.  Fragment
characteristics $A_b$, $Z_b$, $s_b$ and $\epsilon_b$ are taken from
\cite{AS81}.


\subsection{ MC procedure.} 

\hspace{1.0em}The nucleus break-up is described by the Monte Carlo (MC)
procedure. We randomly (according to probability Eq. ($\ref{FBS8}$) and
condition Eq. ($\ref{FBS1}$)) select decay channel. Then for given
channel we calculate kinematical quantities of each fragment according
to $n$-body phase space distribution:
\begin{equation}
\label{FBS10}M_{n} = \int_{-\infty}^{+\infty}...\int_{-\infty}^{+\infty}
\delta(\sum_{b=1}^{n} {\bf p_{b}}) \delta(\sum_{b=1}^{n}
\frac{p^2_b}{2m_b}-E_{kin})
\prod_{b=1}^{n} d^3p_b.
\end{equation}
The Kopylov's sampling procedure \cite{Kopylov70} is applied.  The angular
distributions for emitted fragments are considered to be isotropical.