Livermore Model

The class G4LivermoreIonisationModel calculates the continuous energy loss due to electron ionisation and simulates \(\delta\)-ray production by electrons. The \(\delta\)-electron production threshold for a given material, \(T_c\), is used to separate the continuous and the discrete parts of the process. The energy loss of an electron with the incident energy, \(T\), is expressed via the sum over all atomic shells, \(s\), and the integral over the energy, \(t\), of \(\delta\)-electrons:

\[{dE\over dx}=\sum_{s}\left(\sigma_s(T){{\int^{T_c}_{0.1eV}t{d\sigma\over dt}dt} \over{\int^{T_{max}}_{0.1eV}{d\sigma\over dt}dt}}\right),\]

where \(T_{max} = 0.5T\) is the maximum energy transferred to a \(\delta\)-electron, \(\sigma_s(T)\) is the total cross-section for the shell, \(s\), at a given incident kinetic energy, \(T\), and 0.1 eV is the low energy limit of the EEDL data. The \(\delta\)-electron production cross-section is a complementary function:

\[\sigma(T)=\sum_{s}\left(\sigma_s(T){{\int^{T_{max}}_{T_c}{d\sigma\over dt}dt}\over {\int^{T_{max}}_{0.1eV}{d\sigma\over dt}dt}}\right).\]

The partial sub-shell cross-sections, \(\sigma_s\), are obtained from an interpolation of the evaluated cross-section data in the EEDL library [STPerkins89], according to the formula (3) in Generic Calculation of Total Cross Sections.

The probability of emission of a \(\delta\)-electron with kinetic energy, \(t\), from a sub-shell, \(s\), of binding energy, \(B_s\), as the result of the interaction of an incoming electron with kinetic energy, \(T\), is described by:

\[{d\sigma \over dt} = {P(x) \over x^2}, \;\; \mbox{with} \;\; x={t + B_s \over T + B_s},\]

where the parameter \(x\) is varied from \(x_{min} = (0.1eV + B_s)/(T + B_s)\) to 0.5. The function, \(P(x)\), is parametrised differently in 3 regions of \(x\): from \(x_{min}\) to \(x_1\) the linear interpolation with linear scale of 4 points is used; from \(x_1\) to \(x_2\) the linear interpolation with logarithmic scale of 16 points is used; from \(x_2\) to \(0.5\) the following interpolation is applied:

(138)\[P(x) = 1 - gx +(1 - g)x^2 + {x^2 \over 1-x} \left({1 \over 1-x} - g \right) + A*(0.5 - x)/x,\]

where \(A\) is a fit coefficient, \(g\) is expressed via the gamma factor of the incoming electron:

(139)\[g = (2\gamma - 1) / \gamma^2.\]

For the high energy case (\(x \gg 1\)) the formula ((138)) is transformed to the Möller electron-electron scattering formula [MC70, eal93].

The value of the coefficient, \(A\), for each element is obtained as a result of the fit on the spectrum from the EEDL data for those energies which are available in the database. The values of \(x_1\) and \(x_2\) are chosen for each atomic shell according to the spectrum of \(\delta\)-electrons in this shell. Note that \(x_1\) corresponds to the maximum of the spectrum, if the maximum does not coincide with \(x_{min}\). The dependence of all 24 parameters on the incident energy, \(T\), is evaluated from a logarithmic interpolation (3).

The sampling of the final state proceeds in three steps. First a shell is randomly selected, then the energy of the \(\delta\)-electron is sampled, finally the angle of emission of the scattered electron and of the \(\delta\)-ray is determined by energy-momentum conservation taken into account electron motion on the atomic orbit.

The interaction leaves the atom in an excited state. The deexcitation of the atom is simulated as described in Atomic relaxation. Sampling of the excitations is carried out for both the continuous and the discrete parts of the process.

Bibliography

eal93

René Brun et al. GEANT: Detector Description and Simulation Tool; Oct 1994. CERN Program Library. CERN, Geneva, 1993. Long Writeup W5013. URL: https://cds.cern.ch/record/1082634.

MC70

H. Messel and D. Crawford. Electron-Photon shower distribution. Pergamon Press, 1970.

STPerkins89

S.M.Seltzer S.T.Perkins, D.E.Cullen. Tables and graphs of electron-interaction cross-sections from 10 ev to 100 gev derived from the llnl evaluated electron data library (eedl), z=1-100. Technical Report UCRL-50400 Vol.31, Lawrence Livermore National Laboratory, 1989.