# Photoelectric effect — quantum mechanical treatment

I'm curious if anyone knows a paper or a book where authors computed the probability of exciting an electron by sending a photon on a solid.

For example, my photon starts in state $| \sigma_+ \rangle$ and scatters on the solid which is in the ground state $|0 \rangle$, and they end up with the photon destroyed, i.e. $|0_{ph}\rangle$ and the solid with one excited electron, like $| 1\rangle$. It would be nice to know the probability of that happening depending on the photon's initial state ($\alpha |\sigma_-\rangle +\beta |\sigma_+ \rangle$).

• Beyond Fermi's golden rule? – Mikael Kuisma Jul 26 '16 at 10:39
• The probability of the photoelectric effect occurring is measured by the cross section, σ. There is an approximation for this σ for photons above the energy of the highest atomic binding energy. Presumably that is not what you want. However, in case you did, you can read that here: ocw.mit.edu/courses/nuclear-engineering/… – Matt Jul 26 '16 at 10:43
• @Mikael Kuisma The Fermi golden rule computes the probability that your system scatters from state $| i \rangle$ to state $| f \rangle$, assuming a given scattering potential. So, the creation/destruction of particles is not there. The details of the scatterer are, if I remember correctly, encoded in the details of the potential. – Magicsowon Jul 26 '16 at 11:08
• @user3653831 The scattering potential in this case is A.p You can then second quantize the vector potential A if you wish, and calculate the matrix element between second quantized states, but I believe it adds nothing. – Mikael Kuisma Jul 26 '16 at 11:48
• Look for theory papers of numerical simulations of ARPES measurements. There is the initial effect, multiple scattering and some of the papers might use diagrams and second quantization as well. – Mikael Kuisma Jul 26 '16 at 11:50

## 1 Answer

Your starting place should be E.O. Kane "Theory of Photoelectric Emission from Semiconductors', Phys. Rev. 127(1) 131-141 (1962). This is done from a density of states perspective, considering both volume and surface states.

The following paper in Phys Rev, G.W. Gobeli and F. G. Allen, 'Direct and Indirect Excitation Processes in Photoelectric Emission from Silicon', Phys. Rev. 127(1) 141-149 (1962) is also worth the read, since it is linked directly with Kane's paper (coupled theory/experiment papers, both from Bell Labs folks).