Basics The absorption of an X-ray photon by the atom excites a core level electron, thus generating a core hole. This generates a spherical electron wave with the excited atom as the center. The wave propagates outwards and get scattered off from the neighbouring atoms and is turned back towards the central ionized atom. The oscillatory component of the photoabsorption originates from the coupling of this reflected wave to the initial state via the dipole operator
Mfs as in (1). The Fourier transform of the oscillations gives the information about the spacing of the neighboring atoms and their chemical environment. This phase information is carried over to the oscillations in the Auger signal because the transition time in Auger emission is of the same order of magnitude as the average time for a photoelectron in the energy range of interest. Thus, with a proper choice of the absorption edge and characteristic Auger transition, measurement of the variation of the intensity in a particular Auger line as a function of incident photon energy would be a measure of the photoabsorption cross section. This excitation also triggers various decay mechanisms. These can be of radiative (fluorescence) or nonradiative (Auger and
Coster–Kronig) nature. The intensity ratio between the Auger electron and X-ray emissions depends on the atomic number
Z. The yield of the Auger electrons decreases with increasing
Z.
Theory of EXAFS The cross section of photoabsorption is given by
Fermi's golden rule, which, in the dipole approximation, is given as :P=\frac{2\pi}{\hbar}\sum_f |M_{fs}|^2 \delta (E_i + \hbar \omega - E_f), :M_{fs}= \langle f|e\mathbf\epsilon\cdot\mathbf r|i\rangle, where the initial state,
i with energy
Ei, consists of the atomic core and the Fermi sea, and the incident radiation field, the final state, ƒ with energy
Eƒ (larger than the Fermi level), consists of a core hole and an excited electron.
ε is the polarization vector of the electric field,
e the electron charge, and
ħω the x-ray photon energy. The photoabsorption signal contains a peak when the core level excitation is neared. It is followed by an oscillatory component which originates from the coupling of that part of the electron wave which upon scattering by the medium is turned back towards the central ionized atom, where it couples to the initial state via the dipole operator,
Mi. Assuming single-scattering and small-atom approximation for
kRj >> 1, where
Rj is the distance from the central excited atom to the
jth shell of neighbors and
k is the photoelectrons wave vector, : k = \frac{1}{\hbar} \sqrt{[2m (\hbar ( \omega - \omega_T )+ V_o)]}, where
ħωT is the absorption edge energy and
Vo is the inner potential of the solid associated with exchange and correlation, the following expression for the oscillatory component of the photoabsorption cross section (for K-shell excitation) is obtained: :\chi(k)=k^{-1}|f(k,\pi)|\sum_j\ W_j\sin[2kR_j+\alpha(k)]\exp(-\gamma R_j-2\sigma_j^2k^2), where the atomic scattering factor in a partial wave expansion with partial wave phase-shifts
δl is given by :f(k,\theta)= (1/k)\sum_{l=0}^\infty\ (2l+1)[\exp(2i\delta_{l}(k))-1]P_{l}(\cos\theta).
Pl(
x) is the
lth Legendre polynomial, γ is an attenuation coefficient, exp(−2
σi2
k2) is a
Debye–Waller factor and weight
Wj is given in terms of the number of atoms in the
jth shell and their distance as :W_{j} = \frac{N_j}{R_j^2}. The above equation for the
χ(
k) forms the basis of a direct, Fourier transform, method of analysis which has been successfully applied to the analysis of the EXAFS data.
Incorporation of EXAFS-Auger The number of electrons arriving at the detector with an energy of the characteristic
WαXY Auger line (where
Wα is the absorption edge core-level of element
α, to which the incident x-ray line has been tuned) can be written as :N_{T} = N_{W_{\alpha}XY}(\hbar \omega) + N_{B}(\hbar \omega), where
NB(
ħω) is the background signal and N_{W_{\alpha}XY}(\hbar \omega) is the Auger signal we are interested in, where N_{W_{\alpha}XY}(\hbar \omega) = (4 \pi)^{-1}\psi_{W_{\alpha} XY}[1-\kappa]\int_\Omega \int_0^\infty \ \rho_{\alpha}(z)\,\,P_{W_{\alpha}}(\hbar \omega; z)\exp\left[\frac {-z}{\lambda(W_{\alpha} XY)}\cos\theta\right]\ dzd\Omega, where \psi_{W_{ \alpha} XY} is the probability that an excited atom will decay via
WαXY Auger transition,
ρα(
z) is the atomic concentration of the element
α at depth
z,
λ(
WαXY) is the mean free path for an
WαXY Auger electron,
θ is the angle that the escaping Auger electron makes with the surface normal and
κ is the photon emission probability which is dictated the atomic number. As the photoabsorption probability, P_{W_{\alpha}}(\hbar \omega; z) is the only term that is dependent on the photon energy, the oscillations in it as a function of energy would give rise to similar oscillations in the N_{W_{\alpha}XY}(\hbar \omega). ==Notes==