Editing X-ray photoelectron spectroscopy (section)

====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>