Editing X-ray photoelectron spectroscopy (section)

===Theory of core level photoemission of electrons===

====Inelastic mean free path====

In a solid, inelastic scattering events also contribute to the photoemission process, generating electron-hole pairs which show up as a tail on the high energy side of the main photoemission peak. Due to scattering, the electron intensity can be written in the [[Beer–Lambert law|Beer–Lambert form]]
:<math>
I(z) = I_0 e^{-z/\lambda}
</math>
where <math>\lambda</math> is the electronic '''[[inelastic mean free path]]''' ('''IMFP'''). Here, <math>z</math> is the distance to the sample surface. The IMFP generally depends rather weakly on material, but strongly on the photoelectron kinetic energy <math>E_\text{kin}</math>. Quantitatively we can fit the IMFP by<ref>{{cite book |last1=Attard |first1=Gary |last2=Barnes |first2=Colin |date=1998 |title=Surfaces |publisher=Oxford Chemistry Primers |page=27 |isbn=978-0198556862 }}</ref><ref>{{cite journal|last1=Seah|first1=M. P.|last2=Dench|first2=W. A.|date=1979|title=Quantitative Electron Spectroscopy of Surfaces: A Standard Data Base for Electron Inelastic Mean Free Paths in Solids|journal=Surf. Interf. Anal.|volume=1|number=1|page=2{{hyphen}}10|doi=10.1002/sia.740010103}}</ref><ref>{{cite web|url = http://www.lasurface.com/xps/imfp.php |title = XPS: The Mean Free Path|website = lasurface.com}}</ref>
:<math>
\lambda = 538\,\text{nm}\,\left( E_\text{kin}/\text{eV}\right)^{-2} + 0.41\,\text{nm}\,\left(a / \text{nm}\right)^{1/2} \left(E_\text{kin} / \text{eV}\right)^{1/2}
</math>
where <math>a</math> is the thickness of one monolayer, as given by the number density <math>\rho</math> as <math>a = \rho^{-1/3}</math>. The above formula is a fit to a compilation of experimental data for pure elements. For anorganic and organic compounds, its numerical factors are different, see the paper by Seah and Dench (1979).

====Plasmonic effects====

In some cases, energy loss features due to [[plasmon]] excitations are also observed. This can either be a final state effect caused by core hole decay, which generates quantized electron wave excitations in the solid ('''intrinsic plasmons'''), or it can be due to excitations induced by photoelectrons travelling from the emitter to the surface ('''extrinsic plasmons''').
Due to the reduced [[coordination number]] of first-layer atoms, the plasma frequency of bulk and surface atoms are related by the following equation:  
:<math> \omega_\text{surface} = \frac{\omega_\text{bulk}}{\sqrt{2}}</math>,
so that surface and bulk plasmons can be easily distinguished from each other.
Plasmon states in a solid are typically localized at the surface, and can strongly affect IMFP.

====Vibrational effects====

Temperature-dependent atomic lattice vibrations, or [[phonon]]s, can broaden the core level components and attenuate the interference patterns in an '''X-ray photoelectron diffraction''' ('''XPD''') experiment. The simplest way to account for vibrational effects is by multiplying the scattered single-photoelectron wave function <math>\phi_{j}</math> by the [[Debye–Waller factor]]:
:<math>W_{j}= \exp{(-\Delta k_{j}^2 \bar{U_{j}^2})}</math>,
where <math>\Delta k_{j}^2</math> is the squared magnitude of the wave vector variation caused by scattering,
and <math>\bar{U_{j}^2}</math> is the temperature-dependent one-dimensional vibrational [[mean squared displacement]] of the <math>j^{th}</math> emitter. In the Debye model, the mean squared displacement is calculated in terms of the Debye temperature, <math>\Theta_{D}</math>, as:
:<math> \bar{U_{j}^2}(T) = 9 \hbar ^2 T^2 / m k_{B} \Theta_{D} </math>