PhotoElectric effect

The photoelectric effect is the ejection of an electron from a material after a photon has been absorbed by that material. 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 and Mean Free Path

The parameterization of the photoabsorption cross section proposed by Biggs et al. [1] was used :

$$\sigma(Z,E_{\gamma}) = \frac{a(Z,E_{\gamma})}{E_{\gamma}} + ... ...gamma})}{E_{\gamma}^3} + \frac{d(Z,E_{\gamma})}{E_{\gamma}^4}$$ (6.1)

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 [2]. As a rule, the boundaries of these intervals were equal to the corresponding photoabsorption edges.

In a given material the mean free path, $\lambda$, for a photon to interact via the photoelectric effect is given by :

$$\lambda(E_{\gamma}) = \left( \sum_i n_{ati} \cdot \sigma (Z_i,E_{\gamma}) \right)^{-1}$$ (6.2)

where $n_{ati}$ is the number of atoms per volume of the $i^{th}$ element of the material. The cross section and mean free path are discontinuous and must be computed 'on the fly' from the formulas 6.1 and 6.2.

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 :

$$T_{photoelectron} = E_{\gamma}-B_{shell}(Z_i)$$ (6.3)

Theta Distribution of the Photoelectron

The polar angle of the photoelectron is sampled from the Sauter-Gavrila distribution (for K-shell) [3], which is correct only to zero order in $\alpha Z$ :

$$\frac{d\sigma}{d(\cos\theta)} \sim \frac{\sin^2\theta}{(1-\b... ...} \gamma (\gamma-1)(\gamma-2)(1-\beta\cos\theta) \right\rbrace$$ (6.4)

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

$\cos\theta$ is sampled from the probability density function :

$$f(\cos\theta) = \frac{1-\beta^2}{2\beta} \frac{1}{(1-\beta\c... ...w \hspace{5mm} \cos\theta = \frac{(1-2r)+\beta}{(1-2r)\beta+1}$$ (6.5)

The rejection function is :

$$g(\cos\theta) = \frac{1-\cos^2\theta}{(1-\beta\cos\theta)^2} \left\lbrack 1+b(1-\beta\cos\theta) \right\rbrack$$ (6.6)

with $b=\gamma(\gamma-1)(\gamma-2)/2$
It can be shown that $g(\cos\theta)$ is positive $\forall \cos\theta \in [-1,\ +1]$, and can be majored by :

$$gsup = \gamma^2 \ \lbrack 1+b(1-\beta) \rbrack \mbox{ if } \gamma \in \ ]1,2]$$ (6.7)

$$= \gamma^2 \ \lbrack 1+b(1+\beta) \rbrack \mbox{ if } \gamma > 2$$

The efficiency of this method is $\sim 50\%$ if $\gamma < 2$, $\sim 25\%$ if $\gamma \in [2,\ 3]$.

Relaxation

In the current implementation the relaxation of the atom is not simulated, but instead is counted as a local energy deposit.

Status of this document

09.10.98 created by M.Maire.
08.01.02 updated by mma
22.04.02 re-worded by D.H. Wright
02.05.02 modifs in total cross section and final state (mma)
15.11.02 introduction added by D.H. Wright

Bibliography

1. Biggs F., and Lighthill R., Preprint Sandia Laboratory, SAND 87-0070 (1990)
2. Grichine V.M., Kostin A.P., Kotelnikov S.K. et al., Bulletin of the Lebedev Institute no. 2-3, 34 (1994).
3. Gavrila M. Phys.Rev. 113, 514 (1959).