Livermore Model

Three model classes are available G4LivermorePhotoElectricModel, G4LivermorePolarizedPhotoElectricModel, and G4LivermorePolarizedPhotelectricGDModel.

Cross sections

The total photoelectric and single shell cross-sections are based on either EPICS2014 or EPICS2017 data as described in Low Energy Livermore Model. They are tabulated from threshold to the low energy limit of the fit. Above it, they are parameterised in two different energy intervals, as following:

\[\sigma(E) = \frac{a_1}{E}+\frac{a_2}{E^2}+\frac{a_3}{E^3}+\frac{a_4}{E^4}+\frac{a_5}{E^5}+\frac{a_6}{E^6}.\]

The intervals ranges are set dynamically and they depend on the atomic number of the element and the corresponding k-shell binding energy. The accuracy of such parameterisation is better than 1%. To avoid tracking problems for very low-energy gamma the photoelectric cross section is not zero below first ionisation potential but stay constant, so all types of media are not transparent for gamma.

Sampling of the final state

The incident photon is absorbed and an electron is emitted.

The electron kinetic energy is the difference between the incident photon energy and the binding energy of the electron before the interaction. The sub-shell, from which the electron is emitted, is randomly selected according to the relative cross-sections of all subshells, determined at the given energy. The interaction leaves the atom in an excited state. The deexcitation of the atom is simulated as described in Atomic relaxation.

Angular distribution of the emitted photoelectron

For sampling of the direction of the emitted photoelectron by default the angular generator G4SauterGavrilaAngularDistribution is used. The algorithm is described in PhotoElectric Effect.

For polarized models alternative angular generators are applied.

G4LivermorePolarizedPhotoElectricModel uses the G4PhotoElectricAngularGeneratorPolarized angular generator.

This model models the double differential cross section (for angles \(\theta\) and \(\phi\)) and thus it is capable of account for polarization of the incident photon. The developed generator was based in the research of Sauter in 1931 [RHPA64, Sau31]. Sauter’s formula was recalculated by Gavrila in 1959 for the K-shell [Gav59] and in 1961 for the L-shells [Gav61]. These new double differential formulas have some limitations, \(\alpha\ Z \ll 1\) and have a range between \(0.1 <\beta< 0.99 c\).

The double differential photoeffect for K–shell can be written as [Gav59]:

\[\frac{d\sigma}{d \omega}(\theta,\phi) = \frac{4}{m^2}{\alpha^6}{Z^5}\frac{\beta^3(1-\beta^2)^3}{\left[1-(1-\beta^2)^{1/2}\right]} \left(F\left(1-\frac{\pi\alpha Z}{\beta}\right)+ \pi\alpha Z G\right)\]

where

\[\begin{split}F &= \frac{\sin^2 \theta \cos^2 \phi}{(1-\beta \cos \theta)^4} - \frac{1-(1-\beta^2)^{1/2}}{2(1-\beta^2)}\frac{\sin^2\theta\cos^2\phi}{(1-\beta\cos\theta)^3} \\ &+ \frac{\left[1-(1-\beta^2)^{1/2}\right]^2}{4(1-\beta^2)^{3/2}}\frac{\sin^2\theta}{(1-\beta\cos\theta)^3}\end{split}\]

\[\begin{split}G &= \frac{[1-(1-\beta^2)^{1/2}]^{1/2}}{2^{7/2} \beta^2 (1-\beta \cos \theta)^{5/2}}\left[\frac{4\beta^2}{(1-\beta^2)^{1/2}} \frac{\sin^2 \theta \cos^2 \phi}{1-\beta\cos\theta} + \frac{4\beta}{1-\beta^2}\cos \theta \cos^2 \phi - \right.{} \\ &- 4 \left.\frac{1-(1-\beta^2)^{1/2}}{1-\beta^2}(1-\cos^2\phi)-\beta^2\ \frac{1-(1-\beta^2)^{1/2}}{1-\beta^2} \frac{\sin^2 \theta}{1-\beta \cos \theta} - \right.{} \\ &+ \left.4\beta^2\frac{1-(1-\beta^2)^{1/2}}{(1-\beta^2)^{3/2}} - 4\beta \frac{\left[ 1-(1-\beta^2)^{1/2}\right]^2}{(1-\beta^2)^{3/2}}\right] \\ &+ \frac{1-(1-\beta^2)^{1/2}}{4\beta^2(1-\beta\cos\theta)^2}\left[\frac{\beta}{1-\beta^2}-\frac{2}{1-\beta^2}\cos\theta\cos^2\phi + \frac{1-(1-\beta^2)^{1/2}}{(1-\beta^2)^{3/2}}\cos\theta \right.{} \\ &- \left.\beta \frac{1-(1-\beta^2)^{1/2}}{(1-\beta^2)^{3/2}}\right]\end{split}\]

where \(\beta\) is the electron velocity, \(\alpha\) is the fine–structure constant, \(Z\) is the atomic number of the material and \(\theta\), \(\phi\) are the emission angles with respect to the electron initial direction.

The double differential photoeffect distribution for L1–shell is the same as for K–shell aside from a constant [Gav61]:

\[B = \xi \frac{1}{8}\]

where \(\xi\) is equal to 1 when working with unscreened Coulomb wave functions as it is done in this development.

Since the polarized Gavrila cross–section is a 2–dimensional non–factorized distribution an acceptance–rejection technique was the adopted [LP03]. For the Gavrila distribution, two functions were defined \(g_1(\phi)\) and \(g_2(\theta)\):

\[\begin{split}g_1(\phi) &= a \\ g_2(\theta) &= \frac{\theta}{1+c\theta^2}\end{split}\]

such that:

\[A g_1(\phi)g_2(\theta) \ge \frac{d^2 \sigma}{d\phi d\theta}\]

where A is a global constant. The method used to calculate the distribution is the same as the one used in Low Energy 2BN Bremsstrahlung Generator, being the difference \(g_1(\phi) = a\).

G4LivermorePolarizedPhotoElectricGDModel uses its own methods to produce the angular distribution of the photoelectron. The method to sample the azimuthal angle \(\phi\) is described in [DL06].

Bibliography

DL06

G.O. Depaola and F. Longo. Measuring polarization in the x-ray range: new simulation method for gaseous detectors. NIMA, 566():590, 2006.

Gav59(1,2)

M. Gavrila. Relativistic k-shell photoeffect. Phys. Rev., 113(2):, 1959.

Gav61(1,2)

M. Gavrila. Relativistic l-shell photoeffect. Phys. Rev., 124(4):, 1961.

LP03

A. Trindade L. Peralta, P. Rodrigues. Monte carlo generation of 2bnbremsstrahlung distribution. CERN EXT-2004-039, (039):, July 2003.

RHPA64

R L Pexton R H Pratt, R D Levee and W Aron. K-shell photoelectric cross sections from 200 kev to 2 mev. Phys. Rev., 134(4A):, 1964.

Sau31

Fritz Sauter. Über den atomaren photoeffekt bei großer härte der anregenden strahlung. Annalen der Physik, 401(2):217–248, 1931. URL: https://doi.org/10.1002/andp.19314010205, doi:10.1002/andp.19314010205.