Penelope Model

The G4PenelopeIonisation class calculates the continuous energy loss due to electron and positron ionisation and simulates the \(\delta\)-ray production by electrons and positrons. The electron production threshold \(T_{c}\) for a given material is used to separate the continuous and the discrete parts of the process. The simulation of inelastic collisions of electrons and positrons is performed on the basis of a Generalized Oscillation Strength (GOS) model (see Ref. [eal01] for a complete description). It is assumed that GOS splits into contributions from the different atomic electron shells.

Electrons

The total cross section \(\sigma^{-} (E)\) for the inelastic collision of electrons of energy \(E\) is calculated analytically. It can be split into contributions from distant longitudinal, distant transverse and close interactions,

\[\sigma^{-} (E) = \sigma_{dis,l} + \sigma_{dis,t} + \sigma^{-}_{clo}.\]

The contributions from distant longitudinal and transverse interactions are

(127)\[\sigma_{dis,l} = \frac{2 \pi e^{4}}{m_{e}v^{2}} \sum_{shells} f_{k} \frac{1}{W_{k}} \ln \Big( \frac{W_{k}}{Q^{min}_{k}} \ \frac{Q^{min}_{k}+2m_{e}c^{2}}{W_{k}+2m_{e}c^{2}} \Big) \Theta (E-W_{k})\]

and

(128)\[\sigma_{dis,t} = \frac{2 \pi e^{4}}{m_{e}v^{2}} \sum_{shells} f_{k} \frac{1}{W_{k}} \Big[ \ln \Big( \frac{1}{1-\beta^{2}} \Big) - \beta^{2}-\delta_{F} \Big] \Theta (E-W_{k})\]

respectively, where:

\[\begin{split}m_e &= \mbox{mass of the electron;} \\ v &= \mbox{velocity of the electron;} \\ \beta &= \mbox{velocity of the electron in units of } c;\\ f_k &= \mbox{number of electrons in the } k\mbox{-th atomic shell;}\\ \Theta &= \mbox{Heaviside step function;} \\ W_k &= \mbox{resonance energy of the } k\mbox{-th atomic shell oscillator;}\\ Q^{min}_{k} &= \mbox{minimum kinematically allowed recoil energy for energy transfer } W_k \\ &= \sqrt{\Big[ \sqrt{E(E+2m_{e}c^{2})}-\sqrt{(E-W_{k})(E-W_{k}+ 2m_{e}c^{2})} \Big]^{2}+m_{e}^{2}c^{4}}-m_{e}c^{2} ; \\ \delta_F &= \mbox{Fermi density effect correction}.\end{split}\]

\(\delta_F\) is computed as described in Ref. [Fan63].

The value of \(W_{k}\) is calculated from the ionisation energy \(U_{k}\) of the \(k\)-th shell as \(W_k=1.65 \ U_k\). This relation is derived from the hydrogenic model, which is valid for the innermost shells. In this model, the shell ionisation cross sections are only roughly approximated; nevertheless the ionisation of inner shells is a low probability process and the approximation has a weak effect on the global transport properties 1.

The integrated cross section for close collisions is the Møller cross section

(129)\[\sigma^{-}_{clo} = \frac{2 \pi e^{4}}{m_{e}v^{2}} \sum_{shells} f_{k} \int_{W_{k}}^{\frac{E}{2}} \frac{1}{W^{2}} F^{-}(E,W) dW,\]

where

\[F^{-}(E,W) = 1+ \Big( \frac{W}{E-W} \Big)^{2} - \frac{W}{E-W} + \Big( \frac{E}{E+m_{e}c^{2}} \Big)^{2} \Big( \frac{W}{E-W} + \frac{W^{2}}{E^{2}} \Big).\]

The integral of Eq.(129) can be evaluated analytically. In the final state there are two indistinguishable free electrons and the fastest one is considered as the “primary”; accordingly, the maximum allowed energy transfer in close collisions is \(E/2\). The GOS model also allows evaluation of the spectrum \(d \sigma^{-}/d W\) of the energy \(W\) lost by the primary electron as the sum of distant longitudinal, distant transverse and close interaction contributions,

(130)\[\frac{d\sigma^{-}}{dW} = \frac{d\sigma^{-}_{clo}}{dW} + \frac{d\sigma_{dis,l}}{dW} + \frac{d\sigma_{dis,t}}{dW}.\]

In particular,

(131)\[\frac{d\sigma_{dis,l}}{dW} = \frac{2 \pi e^{4}}{m_{e}v^{2}} \sum_{shells} f_{k} \frac{1}{W_{k}} \ln \Big( \frac{W_{k}}{Q_{-}} \ \frac{Q_{-}+2m_{e}c^{2}}{W_{k}+2m_{e}c^{2}} \Big) \delta(W-W_{k}) \Theta (E-W_{k}),\]

where

\[Q_{-} = \sqrt{\Big[ \sqrt{E(E+2m_{e}c^{2})}-\sqrt{(E-W)(E-W+ 2m_{e}c^{2})} \Big]^{2}+m_{e}^{2}c^{4}}-m_{e}c^{2},\]

(132)\[\frac{d\sigma_{dis,t}}{dW} = \frac{2 \pi e^{4}}{m_{e}v^{2}} \sum_{shells} f_{k} \frac{1}{W_{k}} \Big[ \ln \Big( \frac{1}{1-\beta^{2}} \Big) - \beta^{2}-\delta_{F} \Big] \Theta (E-W_{k}) \delta(W-W_{k})\]

and

(133)\[\frac{d \sigma^{-}_{clo}}{dW} = \frac{2 \pi e^{4}}{m_{e}v^{2}} \sum_{shells} f_{k} \frac{1}{W^{2}} F^{-}(E,W) \Theta (W-W_{k}).\]

Eqs.(127), (128) and (129) derive respectively from the integration in \(dW\) of Eqs.(131), (132) and (133) in the interval \([0,W_{max}]\), where \(W_{max}=E\) for distant interactions and \(W_{max}=E/2\) for close. The analytical GOS model provides an accurate average description of inelastic collisions. However, the continuous energy loss spectrum associated with single distant excitations of a given atomic shell is approximated as a single resonance (a \(\delta\) distribution). As a consequence, the simulated energy loss spectra show unphysical narrow peaks at energy losses that are multiples of the resonance energies. These spurious peaks are automatically smoothed out after multiple inelastic collisions. The explicit expression of \(d\sigma^{-}/dW\), Eq.(130), allows the analytic calculation of the partial cross sections for soft and hard ionisation events, i.e.

\[\sigma^{-}_{soft} \ = \ \int_{0}^{T_{c}} \frac{d\sigma^{-}}{dW} dW \quad \textrm{and} \quad \sigma^{-}_{hard} \ = \ \int_{T_{c}}^{W_{max}} \frac{d\sigma^{-}}{dW} dW.\]

The first stage of the simulation is the selection of the active oscillator \(k\) and the oscillator branch (distant or close). In distant interactions with the \(k\)-th oscillator, the energy loss \(W\) of the primary electron corresponds to the excitation energy \(W_{k}\), i.e. \(W\)=\(W_{k}\). If the interaction is transverse, the angular deflection of the projectile is neglected, i.e. \(\cos \theta =1\). For longitudinal collisions, the distribution of the recoil energy \(Q\) is given by

\[\begin{split}P_{k} (Q) = \left\{ \begin{array}{lll} \frac{1}{Q [1+Q/(2m_{e}c^{2})]} & \textrm{if} \ Q_{-} < Q < W_{max} \\ 0 & \textrm{otherwise} \end{array} \right.\end{split}\]

Once the energy loss \(W\) and the recoil energy \(Q\) have been sampled, the polar scattering angle is determined as

\[\cos \theta = \frac{E(E+2m_{e}c^{2})+(E-W)(E-W+2m_{e}c^{2})- Q(Q+2m_{e}c^{2})}{2\sqrt{E(E+2m_{e}c^{2})(E-W)(E-W+2m_{e}c^{2})}}.\]

The azimuthal scattering angle \(\phi\) is sampled uniformly in the interval \((0,2\pi)\). For close interactions, the distributions for the reduced energy loss \(\kappa \equiv W/E\) for electrons are

\[P^{-}_{k}(\kappa) = \Big[ \frac{1}{\kappa^{2}}+\frac{1}{(1-\kappa)^2} - \frac{1}{\kappa(1-\kappa)} + \Big( \frac{E}{E+m_{e}c^{2}} \Big)^{2} \Big( 1+\frac{1}{\kappa(1-\kappa)} \Big) \Big] \Theta(\kappa-\kappa_{c}) \Theta(\frac{1}{2}-\kappa)\]

with \(\kappa_{c} = \max(W_{k},T_{c})/E\). The maximum allowed value of \(\kappa\) is 1/2, consistent with the indistinguishability of the electrons in the final state. After the sampling of the energy loss \(W= \kappa E\), the polar scattering angle \(\theta\) is obtained as

\[\cos^{2} \theta = \frac{E-W}{E} \ \frac{E+2m_{e}c^{2}}{E-W+2m_{e}c^{2}}.\]

The azimuthal scattering angle \(\phi\) is sampled uniformly in the interval \((0,2\pi)\). According to the GOS model, each oscillator \(W_{k}\) corresponds to an atomic shell with \(f_{k}\) electrons and ionisation energy \(U_{k}\). In the case of ionisation of an inner shell \(i\) (K or L), a secondary electron (\(\delta\)-ray) is emitted with energy \(E_{s}=W-U_{i}\) and the residual ion is left with a vacancy in the shell (which is then filled with the emission of fluorescence x-rays and/or Auger electrons). In the case of ionisation of outer shells, the simulated \(\delta\)-ray is emitted with kinetic energy \(E_{s}=W\) and the target atom is assumed to remain in its ground state. The polar angle of emission of the secondary electron is calculated as

\[\cos^{2} \theta_{s} = \frac{W^{2}/\beta^{2}}{Q(Q+2m_{e}c^{2})} \Big[ 1+ \frac{Q(Q+2m_{e}c^{2})-W^{2}}{2W(E+m_{e}c^{2})} \Big]^{2}\]

(for close collisions \(Q=W\)), while the azimuthal angle is \(\phi_{s} = \phi + \pi\). In this model, the Doppler effects on the angular distribution of the \(\delta\) rays are neglected. The stopping power due to soft interactions of electrons, which is used for the computation of the continuous part of the process, is analytically calculated as

\[S^{-}_{in} = N \int_{0}^{T_{c}} W \frac{d\sigma^{-}}{dW} dW\]

from the expression (130), where \(N\) is the number of scattering centers (atoms or molecules) per unit volume.

Positrons

The total cross section \(\sigma^{+} (E)\) for the inelastic collision of positrons of energy \(E\) is calculated analytically. As in the case of electrons, it can be split into contributions from distant longitudinal, distant transverse and close interactions,

\[\sigma^{+} (E) = \sigma_{dis,l} + \sigma_{dis,t} + \sigma^{+}_{clo}.\]

The contributions from distant longitudinal and transverse interactions are the same as for electrons, Eq.(127) and (128), while the integrated cross section for close collisions is the Bhabha cross section

(134)\[\sigma^{+}_{clo} = \frac{2 \pi e^{4}}{m_{e}v^{2}} \sum_{shells} f_{k} \int_{W_{k}}^{E} \frac{1}{W^{2}} F^{+}(E,W) dW,\]

where

\[F^{+}(E,W) = 1- b_{1}\frac{W}{E} + b_{2} \frac{W^{2}}{E^{2}} - b_{3} \frac{W^{3}}{E^{3}} + b_{4} \frac{W^{4}}{E^{4}};\]

the Bhabha factors are

\[\begin{split}b_{1} &= \Big( \frac{\gamma-1}{\gamma} \Big)^{2} \ \frac{2(\gamma+1)^{2}-1} {\gamma^{2}-1}, \\ b_{2} &= \Big( \frac{\gamma-1}{\gamma} \Big)^{2} \ \frac{3(\gamma+1)^{2}+1} {(\gamma+1)^{2}}, \\ b_{3} &= \Big( \frac{\gamma-1}{\gamma} \Big)^{2} \ \frac{2(\gamma-1)\gamma} {(\gamma+1)^{2}}, \\ b_{4} &= \Big( \frac{\gamma-1}{\gamma} \Big)^{2} \ \frac{(\gamma-1)^{2}} {(\gamma+1)^{2}},\end{split}\]

and \(\gamma\) is the Lorentz factor of the positron. The integral of Eq.(134) can be evaluated analytically. The particles in the final state are not indistinguishable so the maximum energy transfer \(W_{max}\) in close collisions is \(E\). As for electrons, the GOS model allows the evaluation of the spectrum \(d \sigma^{+}/d W\) of the energy \(W\) lost by the primary positron as the sum of distant longitudinal, distant transverse and close interaction contributions,

(135)\[\frac{d\sigma^{+}}{dW} = \frac{d\sigma^{+}_{clo}}{dW} + \frac{d\sigma_{dis,l}}{dW} + \frac{d\sigma_{dis,t}}{dW},\]

where the distant terms \(\frac{d\sigma_{dis,l}}{dW}\) and \(\frac{d\sigma_{dis,t}}{dW}\) are those from Eqs.(131) and (132), while the close contribution is

\[\frac{d \sigma^{+}_{clo}}{dW} = \frac{2 \pi e^{4}}{m_{e}v^{2}} \sum_{shells} f_{k} \frac{1}{W^{2}} F^{+}(E,W) \Theta (W-W_{k}).\]

Also in this case, the explicit expression of \(d\sigma^{+}/dW\), Eq.(135), allows an analytic calculation of the partial cross sections for soft and hard ionisation events, i.e.

\[\sigma^{+}_{soft} = \int_{0}^{T_{c}} \frac{d\sigma^{+}}{dW} dW \quad \textrm{and} \quad \sigma^{+}_{hard} = \int_{T_{c}}^{E} \frac{d\sigma^{+}}{dW} dW.\]

The sampling of the final state in the case of distant interactions (transverse or longitudinal) is performed in the same way as for primary electrons, see Electrons. For close positron interactions with the \(k\)-th oscillator, the distribution for the reduced energy loss \(\kappa \equiv W/E\) is

\[P^{+}_{k}(\kappa) = \Big[\frac{1}{\kappa^{2}} - \frac{b_{1}}{\kappa}+b_{2} -b_{3}\kappa + b_{4} \kappa^{2} \Big] \Theta(\kappa-\kappa_{c}) \Theta(1-\kappa)\]

with \(\kappa_{c} = \max(W_{k},T_{c})/E\). In this case, the maximum allowed reduced energy loss \(\kappa\) is 1. After sampling the energy loss \(W= \kappa E\), the polar angle \(\theta\) and the azimuthal angle \(\phi\) are obtained using the equations introduced for electrons in Electrons. Similarly, the generation of \(\delta\) rays is performed in the same way as for electrons. Finally, the stopping power due to soft interactions of positrons, which is used for the computation of the continuous part of the process, is analytically calculated as

\[S^{+}_{in} = N \int_{0}^{T_{c}} W \frac{d\sigma^{+}}{dW} dW\]

from the expression (135), where \(N\) is the number of scattering centers per unit volume.

Bibliography

eal01

F. Salvat et al. Penelope - a code system for monte carlo simulation of electron and photon transport. Technical Report, Workshop Proceedings Issy-les-Moulineaux, France; AEN-NEA, 5-7 November 2001.

Fan63

U. Fano. Penetration of protons, alpha particles and mesons. Ann. Rev. Nucl. Sci., 13():1, 1963.

1

In cases where inner-shell ionisation is directly observed, a more accurate description of the process should be used.