Fermi break-up simulation for light nuclei

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 formulated [Fer50], [Kre61], [EGKR69, EpherreG67], [BBI+95]. This statistical approach was first used by Fermi [Fer50] to describe the multiple production in high energy nucleon collision. The Geant4 Fermi break-up model is capable to predict final states as result of an excited nucleus with \(Z < 9\) and \(A < 17\) statistical break-up.

Allowed channels

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:

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

\(U\) is primary fragment excitation, \(m_{b}\) 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

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

where \(V_0\) is the volume of the system corresponding to the normal nuclear matter density

\[V_{0} = 4\pi R^3/3 = 4\pi r_{0}^3 A/3,\]

where \(r_{0} = 1.3\) fm is used. Free parameter of the model is the ratio of the effective volume \(V\) to the normal volume, currently

\[\kappa = \frac{V}{V_0} = 6.\]

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

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

where \(\rho_{n}(E)\) is the density of a number of final states, \(\Omega = (2\pi \hbar)^{3}\) is the normalization volume. The density \(\rho_{n}(E)\) can be defined as a product of three factors:

\[\rho_{n}(E)=M_{n}(E)S_nG_n.\]

The first one is the phase space factor defined as

(259)\[M_{n} = \int_{-\infty}^{+\infty}...\int_{-\infty}^{+\infty} \delta \left(\sum_{b=1}^{n} {\bf p_{b}} \right) \delta \left(E-\sum_{b=1}^{n}\sqrt{p^2+m^2_b} \right) \prod_{b=1}^{n} d^3p_b,\]

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

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

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

\[G_n = \prod_{j=1}^{k}\frac{1}{n_j !},\]

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. (261) the integration in Eq. (259) can be evaluated analiticaly (see e. g. [BaravsenkovBarbavsevB58]). The probability for a nucleus with energy \(E\) disassembling into \(n\) fragments with masses \(m_b\), where \(b = 1,2,3,...,n\) equals

(260)\[W(E_{kin},n) = S_nG_n \left(\frac{V}{\Omega}\right)^{n-1} \left(\frac{1}{\sum_{b=1}^{n}m_b} \prod_{b=1}^{n} m_{b} \right)^{3/2} \frac{(2\pi)^{3(n-1)/2}}{\Gamma(3(n-1)/2)}E_{kin}^{3n/2-5/2},\]

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

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: ⁵He, ⁵Li, ⁸Be, ⁹B etc. are also considered. Fragment characteristics \(A_b\), \(Z_b\), \(s_b\) and \(\epsilon_b\) are taken from [AS81, AS82, AS83, AS84, AS85, Err83, Err84]. Recently nuclear level energies were changed to be identical with nuclear levels in the gamma evaporation database (see Section Photon evaporation).

Sampling procedure

The nucleus break-up is described by the Monte Carlo (MC) procedure. We randomly (according to probability Eq. (260) and condition Eq. (258) select decay channel. Then for given channel we calculate kinematical quantities of each fragment according to \(n\)-body phase space distribution:

(261)\[M_{n} = \int_{-\infty}^{+\infty}...\int_{-\infty}^{+\infty} \delta \left( \sum_{b=1}^{n} {\bf p_{b}} \right) \delta \left(\sum_{b=1}^{n} \frac{p^2_b}{2m_b}-E_{kin} \right) \prod_{b=1}^{n} d^3p_b.\]

The Kopylov’s sampling procedure [I70, Kop85, Kop73] is applied. The angular distributions for emitted fragments are considered to be isotropical.

Bibliography

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