Implementation

A photoabsorption ionization Monte Carlo code provided to us by users has been implemented in GEANT. Tables are prepared at initialization time and sampling from these tables is done each step at running time. An important thing to note is the dependence of the $(dN/dx)$_>E

on the photoelectric cross-sections which, when taken from different sources, may differ considerably, especially at low energies.

To fill the tables, the integration of equation has to be done. Before the integration over the energy, the integral over the oscillator strength function $f(k,\omega )\; \sim \sigma$_γ(ω) (last term in equation ) and the dielectric constant (equations and ) $\epsilon$ have to be computed. These integaration can be done analytically when the parameterized cross-sections are used.

The oscillator strength function should fulfill the Bethe sum rule []: $\int$₀^&inf;f(k,ω) dω= 1

To respect this condition, in the PAI model, one has to add some contribution from the excitation processes to low-energy region of the photoelectric cross-sections []. In GEANT, the photoelectric cross-section parameterization of the Sandia [] report is used. In order not to change the tabulated parameterization, the sum rule is taken into account by lowering the energy limit for the integration below the ionization limit. This will not satisy the equation in all cases. In the program, however, the Bethe sum rule is forced by dividing the integrated cross-section from E to

by the value given by the integration over the whole range.

Figure: Above, the integrated and normalized cross-section of the gas mixture 70% Xe - 10% $CO$₂ - 20% $CF$₄

for two different data sets. The low energy limit of the cross-sections from the Sandia data in chosen to be 11 eV. In the figure below, the low energy limit is the ionization limit for each component of the mixture

In figure , the integrated cross-section (oscillator strength function) $\int$_E^E_maxσ_γ/ ∫_E0^E_maxσ_γ

is given for two different sets of photoelectric cross-section data for a gas mixture 70% Xe - 10% $CO$₂ - 20% $CF$₄ . In the figure below, the continuous line is the result of the integration tabulated from [] [] [] and the dashed line is from the integration of GEANT cross-sections considering the lower limit of the cross-section the ionization energy (12.3 eV for Xe, 11.26 eV for C, 13.6 for O, and 17.42 for F). In the figure above, the continuous line is as before, but for the dashed line, the lower limit of the cross-sections is considered to be 11 eV for all elements of the mixture to compensate for the sum rule.

The imaginary part of the complex dielectric constant can then be computed from the integrated cross-section. To compute the real part of $\epsilon$ , one needs to take the Cauchy principal value which is basically a limiting process cancelling the two infinite contribution around the pole $(x$₀-δ,x₀+δ) . The derivation of the real part with the Sandia parametrization is shown in detail in the appendix.

Figure: Above, the integrated and normalized cross-section of the gas mixture 70% Xe - 10% $CO$₂ - 20% $CF$₄

for two different data sets. The low energy limit of the cross-sections from the Sandia data in chosen to be 11 eV. In the figure below, the low energy limit is the ionization limit for each component of the mixture

Knowing the values of complex dielectric constant and the integrated cross-section, the integration of equation can be done. This is done for several values of Lorentz factor during the initialization time in subroutine GSTINI. The value of energy loss is sampled from these tables in subroutine GSTREN.

One should remember the dependence of this method on the photoelectric cross section data. Figure shows the results for the two different data sets. The difference in the low-energy part is very significative as the sampling is done from these tables and, according the inverse transverse method, the number of collisions is sampled between the maximum and minimum of the curves in figure .

F.Carminati PHYS337