Editing X-ray photoelectron spectroscopy (section)

===Quantum mechanical treatment===

When a photoemission event takes place, the following energy conservation rule holds:

:<math> h\nu =|E_{b}^{v}|+E_{kin} </math>

where <math>h\nu</math> is the photon energy, <math>|E_{b}^{v}|</math> is the electron binding energy (with respect to the vacuum level) prior to ionization, and <math>E_{kin}</math> is the kinetic energy of the photoelectron. If reference is taken with respect to the Fermi level (as it is typically done in [[photoelectron spectroscopy]]) <math>|E_{b}^{v}|</math> must be replaced by the sum of the binding energy relative to the Fermi level, <math>|E_{b}^{F}|</math>, and the sample work function, <math>\Phi_{0}</math> .

From the theoretical point of view, the photoemission process from a solid can be described with a semiclassical approach, where the electromagnetic field is still treated classically, while a quantum-mechanical description is used for matter.
The one—particle Hamiltonian for an electron subjected to an electromagnetic field is given by (in [[International System of Units|SI units]]):

:<math> i\hbar \frac{\partial \psi}{\partial t} = \left[\frac{1}{2m}\left(\hat{\mathbf{p}} - e\hat{\mathbf{A}}\right)^2 + \hat{V} \right]\psi = \hat{H}\psi 
</math>,

where <math>\psi</math> is the electron wave function,
<math>e, m </math> are the electron charge and mass, <math>\mathbf{A}</math> is the vector potential of the electromagnetic field, and <math>V</math> is the unperturbed potential of the solid.
In the Coulomb gauge (<math>\nabla \cdot \mathbf{A}=0</math>), the vector potential commutes with the momentum operator 
(<math>[\mathbf{\hat{p}}, \mathbf{\hat{A}}]=0 </math>), so that the expression in brackets in the Hamiltonian simplifies to:

:<math> \left(\hat{\mathbf{p}} - e \hat{\mathbf{A}}\right)^2 = \hat{p}^2 - 2e\hat{\mathbf{A}}\cdot\hat{\mathbf{p}} + e^2\hat{A}^2 </math>

Actually, neglecting the <math>\nabla\cdot\mathbf{A}</math> term in the Hamiltonian, we are disregarding possible photocurrent contributions.<ref>{{Cite book |last=Hüfner |first=Stefan |url=http://link.springer.com/10.1007/978-3-662-09280-4 |title=Photoelectron Spectroscopy |date=2003 |publisher=Springer Berlin Heidelberg |isbn=978-3-642-07520-9 |series=Advanced Texts in Physics |location=Berlin, Heidelberg |doi=10.1007/978-3-662-09280-4}}</ref> Such effects are generally negligible in the bulk, but may become important at the surface.
The quadratic term in <math>\mathbf{A}</math> can be instead safely neglected, since its contribution in a typical photoemission experiment is about one order of magnitude smaller than that of the first term .

In first-order perturbation approach, the one-electron Hamiltonian can be split into two terms, an unperturbed Hamiltonian <math>\hat{H}_{0}</math>, plus an interaction Hamiltonian <math>\hat{H}'</math>, which describes the effects of the electromagnetic field:

:<math> \hat{H}' = -\frac{e}{m} \hat{\mathbf{A}}\cdot \hat{\mathbf{p}} 
</math>

In time-dependent perturbation theory, for an harmonic or constant perturbation, the transition rate between the initial state <math>\psi_{i}</math> and the final state <math>\psi_{f}</math> is expressed by [[Fermi's Golden Rule]]:

:<math> \frac{dw_{if}}{dt} \propto \frac{2\pi}{\hbar}|\langle \psi_{f}|\hat{H}'|\psi_{i} \rangle |^2 \delta (E_{f}-E_{i}-h\nu) </math>,

where <math>E_{i}</math> and <math>E_{f}</math> are the eigenvalues of the unperturbed Hamiltonian in the initial and final state, respectively, and <math>h\nu</math> is the photon energy. [[Fermi's Golden Rule]] uses the approximation that the perturbation acts on the system for an infinite time. This approximation is valid when the time that the perturbation acts on the system is much larger than the time needed for the transition. It should be understood that this equation needs to be integrated with the density of states <math>\rho(E)</math> which gives:<ref name="Fermi's Golden Rule">{{cite book|last1=Sakurai|first1=J.|title=[[Modern Quantum Mechanics]]|date=1995|publisher=Addison-Wesley Publishing Company|isbn=0-201-53929-2|edition=Rev.|page=[https://archive.org/details/modernquantummec00saku/page/n344 332]}}</ref>

:<math> \frac{dw_{if}}{dt} \propto \frac{2\pi}{\hbar}|\langle \psi_{f}|\hat{H}'|\psi_{i} \rangle |^2 \rho(E_{f})=|M_{fi}|^2 \rho(E_{f}) </math>

In a real photoemission experiment  the ground state core electron binding energy cannot be directly probed, because the measured one incorporates both initial state and final state effects, and the spectral linewidth is broadened owing to the finite core-hole lifetime (<math>\tau</math>).

Assuming an exponential decay probability for the core hole in the time domain (<math> \propto \exp{-t/\tau} </math>), the spectral function will have a Lorentzian shape, with a FWHM (Full Width at Half Maximum) <math>\Gamma</math> given by:

:<math> I_{L}(E)=\frac{I_{0}}{\pi}\frac{\Gamma /2}{(E-E_{b})^2+(\Gamma /2)^2} </math>

From the theory of Fourier transforms, <math>\Gamma</math> and <math>\tau</math> are linked by the indeterminacy relation:

<math> \Gamma \tau \geq \hbar </math>

The photoemission event leaves the atom in a highly excited core ionized  state, from which it can decay radiatively (fluorescence) or non-radiatively (typically by ''Auger'' decay).
Besides  Lorentzian broadening, photoemission spectra are also affected by a Gaussian broadening, whose contribution can be expressed by

:<math> I_{G}(E)=\frac{I_{0}}{\sigma \sqrt{2}}\exp{\left( -\frac{(E-E_{b})^2}{2\sigma^2}\right)}</math>

The Gaussian linewidth <math>\sigma</math> of the spectra depends on the experimental energy resolution, vibrational and inhomogeneous broadening.
The first effect is caused  by the non perfect monochromaticity of the photon beam, resulting in a finite bandwidth, and by the limited resolving power of the analyzer. The vibrational component is produced by the excitation of low energy vibrational modes both in the initial and in the final state. Finally, inhomogeneous broadening can originate from the presence of unresolved core level components in the spectrum.