The photoelectric effect is the ejection of an electron from a material after a photon has been absorbed by that material. In the standard model G4PEEffectFluoModel it is simulated by using a parameterized photon absorption cross section to determine the mean free path, atomic shell data to determine the energy of the ejected electron, and the K-shell angular distribution to sample the direction of the electron.

Cross Section

The parameterization of the photoabsorption cross section proposed by Biggs et al.[BL88] was used:

(6)\[\sigma(Z,E_{\gamma}) = \frac{a(Z,E_{\gamma})}{E_{\gamma}} + \frac{b(Z,E_{\gamma})}{E_{\gamma}^2} + \frac{c(Z,E_{\gamma})}{E_{\gamma}^3} + \frac{d(Z,E_{\gamma})}{E_{\gamma}^4}\]

Using the least-squares method, a separate fit of each of the coefficients \(a,b,c,d\) to the experimental data was performed in several energy intervals [VMAPeal94, AGU+00]. As a rule, the boundaries of these intervals were equal to the corresponding photoabsorption edges. The cross section (and correspondingly mean free path) are discontinuous and must be computed ‘on the fly’ from the formula (6). Coefficients are defined for each Sandia table energy interval.

If photon energy is below the lowest Sandia energy for the material the cross section is computed for this lowest energy, so gamma is absorbed by photoabsorption at any energy. This approach is implemented coherently for models of photoelectric effect of Geant4. As a result, any media become not transparent for low-energy gammas.

The class G4StaticSandiaData.hh contains the corrected data table for the cross-section applied according to the Sandia table with extra data taken from the Lebedev report. The coefficients are from Ref.[BL88].

The first energy intervals and coefficients for Xe are corrected to correspond perfectly to the data of J.B. West et al.[WM78]. The coefficients are checked to correspond perfectly to the data from B.L. Henke et al. [eal82]. The coeficients for Carbon are checked to correspond perfectly to the data of B.L. Henke et al. (as Xe). The first three energy intervals and coefficients for C are corrected to correspond perfectly to the data of Gallagher et al. [eal88]. The coefficients for Oxygen are checked to correspond perfectly to the data of B.L. Henke et al. (as Xe). The first two energy intervals and coefficients for O are corrected to correspond perfectly to the data of Gallagher et al. (as C). The coeficients for Hydrogen are checked to correspond perfectly to the data of B.L. Henke et al. (as Xe). The first three energy intervals and coefficients for H are corrected to correspond perfectly to the data of L.C. Lee et al.[eal77]. The first energy intervals and coefficients for He, Ne, Ar, and Kr are corrected to correspond perfectly to the data of G.V. Marr et al.[MW76].

The most of ionisation energies are taken from S. Ruben[Rub85]. Twenty-eight of the ionisation energies have been changed slightly to bring them up to date (changes from W.C. Martin and B.N. Taylor of the National Institute of Standards and Technology, January 1990). Here the ionisation energy is the least energy necessary to remove to infinity one electron from an atom of the element.

Final State

Choosing an Element

The binding energies of the shells depend on the atomic number \(Z\) of the material. In compound materials the \(i^{th}\) element is chosen randomly according to the probability:

\[Prob(Z_i,E_{\gamma}) = \frac{n_{ati} \sigma(Z_i,E_{\gamma})} {\sum_i [ n_{ati} \cdot \sigma_i (E_{\gamma})]} .\]

Shell

A quantum can be absorbed if \(E_{\gamma} > B_{shell}\) where the shell energies are taken from G4AtomicShells data: the closest available atomic shell is chosen. The photoelectron is emitted with kinetic energy:

\[T_{photoelectron} = E_{\gamma}-B_{shell}(Z_i)\]

Theta Distribution of the Photoelectron

The polar angle of the photoelectron is sampled from the Sauter-Gavrila distribution (for K-shell) [Gav59], which is correct only to zero order in \(\alpha Z\):

\[\frac{d\sigma}{d(\cos\theta)} \sim \frac{\sin^2\theta}{(1-\beta\cos\theta)^4} \left\lbrace 1 + \frac{1}{2} \gamma (\gamma-1)(\gamma-2)(1-\beta\cos\theta) \right\rbrace\]

where \(\beta\) and \(\gamma\) are the Lorentz factors of the photoelectron.

Introducing the variable transformation \(\nu = 1 - \cos\theta_{e}\), as done in Penelope, the angular distribution can be expressed as

\[p(\nu) = (2-\nu) \Big[ \frac{1}{A+\nu} + \frac{1}{2} \beta \gamma (\gamma - 1)(\gamma -2) \Big] \frac{\nu}{(A+\nu)^{3}},\]

where

\[\gamma = 1 + \frac{E_{e}}{m_{e}c^{2}}, \quad A = \frac{1}{\beta} - 1,\]

\(E_{e}\) is the electron energy, \(m_{e}\) its rest mass and \(\beta\) its velocity in units of the speed of light \(c\).

Though the Sauter distribution, strictly speaking, is adequate only for ionisation of the K-shell by high-energy photons, in many practical simulations it does not introduce appreciable errors in the description of any photoionisation event, irrespective of the atomic shell or of the photon energy.

Relaxation

Atomic relaxations can be sampled using the de-excitation module of the low-energy sub-package Atomic relaxation. For that atomic de-excitation option should be activated. In the physics_list sub-library this activation is done automatically for G4EmLivermorePhysics, G4EmPenelopePhysics, G4EmStandardPhysics_option3 and G4EmStandardPhysics_option4. For other standard physics constructors the de-excitation module is already added but is disabled. The simulation of fluorescence and Auger electron emission may be enabled for all geometry via UI commands:

/process/em/fluo true
/process/em/auger true
/process/em/pixe true

Please see further detailed information on atomic deexcitation here.

There is a possibility to enable atomic deexcitation only for G4Region by its name:

/process/em/deexcitation myregion true true false

where three Boolean arguments enable/disable fluorescence, Auger electron production and PIXE (deexcitation induced by ionisation).

Bibliography

AGU+00: J Apostolakis, S Giani, L Urban, M Maire, A.V Bagulya, and V.M Grichine. An implementation of ionisation energy loss in very thin absorbers for the geant4 simulation package. Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 453(3):597 – 605, 2000. URL: http://www.sciencedirect.com/science/article/pii/S0168900200004575, doi:https://doi.org/10.1016/S0168-9002(00)00457-5.
BL88(1,2): F Biggs and R Lighthill. Analytical approximations for x-ray cross sections III. Technical Report, Sandia Lab, aug 1988. Preprint Sandia Laboratory, SAND 87-0070. URL: https://doi.org/10.2172/7124946, doi:10.2172/7124946.
eal82: B.L. Henke et al. Low-energy x-ray interaction coefficients:photoabsorption, scattering, and reflection. Atom. Data Nucl. Data Tabl., 27:1–144, 1982.
eal88: Gallagher et al. J.Phys.Chem.Ref.Data, 1988.
eal77: Lee L.C. et al. Journ. of Chem. Phys., 67:1237, 1977.
Gav59: Mihai Gavrila. RelativisticK-shell photoeffect. Physical Review, 113(2):514–526, jan 1959. URL: https://doi.org/10.1103/PhysRev.113.514, doi:10.1103/physrev.113.514.
MW76: G.V. Marr and J.B. West. Absolute photoionization cross-section tables for helium, neon, argon, and krypton in the vuv spectral regions. Atom. Data Nucl. Data Tabl., 18:497–508, 1976.
Rub85: S. Ruben. Handbook of the Elements. Open Court, La Salle, IL, 3rd ed. edition, 1985.
VMAPeal94: Grichine V.M., Kostin A.P., and Kotelnikov S.K. et al. Bulletin of the Lebedev Institute, 1994.
WM78: J.B. West and J. Morton. Absolute photoionization cross-sections tables for xenon in the vuv and the soft x-ray regions. Atom. Data Nucl. Data Tabl., 22:103–107, 1978.