Fermi break-up simulation for light nuclei.

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 [1], [2], [3]. This statistical approach was first used by Fermi [1] to describe the multiple production in high energy nucleon collision.

Allowed channel.

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:

$$E_{kin} = U+M(A,Z)-E_{Coulomb} - \sum_{b=1}^{n}(m_b+\epsilon_{b}),$$ (31.1)

$m_{b}$ 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

$$E_{Coulomb} = \frac{3}{5} \frac{e^2}{r_{0}}(1 + \frac{V}{V_... .../3} (\frac{Z^2}{A^{1/3}}-\sum_{b=1}^{n}\frac{Z^2}{A_b^{1/3}}),$$ (31.2)

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).

Break-up probability.

The total probability for nucleus to break-up into $n$ componets (nucleons, deutrons, tritons, alphas etc) in the final state is given by

$$W(E,n) = (V/\Omega)^{n-1}\rho_{n}(E),$$ (31.3)

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:

$$\rho_{n}(E)=M_{n}(E)S_nG_n.$$ (31.4)

The first one is the phase space factor defined as

$$M_{n} = \int_{-\infty}^{+\infty}...\int_{-\infty}^{+\infty} ... ...elta(E-\sum_{b=1}^{n}\sqrt{p^2+m^2_b}) \prod_{b=1}^{n} d^3p_b,$$ (31.5)

where ${\bf p_b}$ is fragment $b$ momentum. The second one is the spin factor

$$S_n = \prod_{b=1}^{n}(2s_b+1),$$ (31.6)

which gives the number of states with different spin orientations. The last one is the permutation factor

$$G_n = \prod_{j=1}^{k}\frac{1}{n_j !},$$ (31.7)

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. [5]). The probability for a nucleus with energy $E$ disassembling into $n$ fragments with masses $m_b$, where $b = 1,2,3,...,n$ equals

$$W(E_{kin},n) = S_nG_n (\frac{V}{\Omega})^{n-1}(\frac{1}{\su... ...\frac{(2\pi)^{3(n-1)/2}}{\Gamma(3(n-1)/2)}E_{kin}^{3n/2-5/2},$$ (31.8)

where $\Gamma(x)$ is the gamma function.

Fermi break-up model parameter.

Thus the Fermi break-up model has only one free parameter $V$ is the volume of decaying system, which can be calculated as follows:

$$V = 4\pi R^3/3 = 4\pi r_{0}^3 A/3,$$ (31.9)

where $r_{0} = 1.4$ fm is used.

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 [6].

MC procedure.

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:

$$M_{n} = \int_{-\infty}^{+\infty}...\int_{-\infty}^{+\infty} ... ..._{b=1}^{n} \frac{p^2_b}{2m_b}-E_{kin}) \prod_{b=1}^{n} d^3p_b.$$ (31.10)

The Kopylov's sampling procedure [7] is applied. The angular distributions for emitted fragments are considered to be isotropical.