Synchrotron radiation sources

Normally, the SEXAFS work is done using synchrotron radiation as it has highly collimated, plane-polarized and precisely pulsed X-ray sources, with fluxes of 10¹² to 10¹⁴ photons/sec/mrad/mA and greatly improves the signal-to-noise ratio over that obtainable from conventional sources. A bright source X-ray source is illuminating the sample and the transmission is being measured as the absorption coefficient as

${\displaystyle {\begin{aligned}\mu ={\frac {\ln(I)}{\ln(I_{o})}},\end{aligned}}}$

where I is the transmitted and I_o is the incident intensity of the X-rays. Then it is plotted against the energy of the incoming X-ray photon energy.

Electron detectors

In SEXAFS, an electron detector and a high-vacuum chamber is required to calculate the Auger yields instead of the intensity of the transmitted X-ray waves. The detector can be either an energy analyzer, as in the case of Auger measurements, or an electron multiplier, as in the case of total or partial secondary electron yield. The energy analyzer gives rise to better resolution while the electron multiplier has larger solid angle acceptance.

Signal-to-noise ratio

The equation governing the signal-to-noise ratio is

${\displaystyle {\frac {S}{N}}={\sqrt {\frac {({\frac {\Omega }{4\pi }}\epsilon _{n}\mu _{A})}{(1+{\frac {I_{b}}{I_{n}}}(\mu _{T}+n))}}}\left({\frac {\delta \mu _{A}}{\mu _{A}}}I_{o}^{1/2}\right),}$

where

• μ_A is the absorption coefficient;
• I_n is the nonradiative contribution in electron counts/sec;
• I_b is the background contribution in electron counts/sec;
• μ_A is the absorption by the SEXAFS-producing element;
• μ_T is the total absorption by all the elements;
• I_o is the incident intensity;
• n is the attenuation length;
• Ω/(4π) is the solid angle acceptance for the detector;
• ε_n is the nonradiative yield which is the probability that the electron will not decay radiatively and will actually get emitted as an Auger electron.

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 M_fs 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

${\displaystyle P={\frac {2\pi }{\hbar }}\sum _{f}|M_{fs}|^{2}\delta (E_{i}+\hbar \omega -E_{f}),}$

${\displaystyle M_{fs}=\langle f|e\mathbf {\epsilon } \cdot \mathbf {r} |i\rangle ,}$

where the initial state, i with energy E_i, consists of the atomic core and the Fermi sea, and the incident radiation field, the final state, ƒ with energy E_ƒ (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, M_i.

Assuming single-scattering and small-atom approximation for kR_j >> 1, where R_j is the distance from the central excited atom to the jth shell of neighbors and k is the photoelectrons wave vector,

${\displaystyle k={\frac {1}{\hbar }}{\sqrt {[2m(\hbar (\omega -\omega _{T})+V_{o})]}},}$

where ħω_T is the absorption edge energy and V_o 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:

${\displaystyle \chi (k)=k^{-1}|f(k,\pi )|\sum _{j}\ W_{j}\sin[2kR_{j}+\alpha (k)]\exp(-\gamma R_{j}-2\sigma _{j}^{2}k^{2}),}$

where the atomic scattering factor in a partial wave expansion with partial wave phase-shifts δ_l is given by

${\displaystyle f(k,\theta )=(1/k)\sum _{l=0}^{\infty }\ (2l+1)[\exp(2i\delta _{l}(k))-1]P_{l}(\cos \theta ).}$

P_l(x) is the lth Legendre polynomial, γ is an attenuation coefficient, exp(−2σ_i²k²) is a Debye–Waller factor and weight W_j is given in terms of the number of atoms in the jth shell and their distance as

${\displaystyle W_{j}={\frac {N_{j}}{R_{j}^{2}}}.}$

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 W_αXY Auger line (where W_α is the absorption edge core-level of element α, to which the incident x-ray line has been tuned) can be written as

${\displaystyle N_{T}=N_{W_{\alpha }XY}(\hbar \omega )+N_{B}(\hbar \omega ),}$

where N_B(ħω) is the background signal and ${\displaystyle N_{W_{\alpha }XY}(\hbar \omega )}$ is the Auger signal we are interested in, where

${\displaystyle N_{W_{\alpha }XY}(\hbar \omega )=(4\pi )^{-1}\psi _{W_{\alpha }XY}[1-\kappa ]\int _{\Omega }\int _{0}^{\infty }\ \rho _{\alpha }(z)\,\,P_{W_{\alpha }}(\hbar \omega ;z)\exp \left[{\frac {-z}{\lambda (W_{\alpha }XY)}}\cos \theta \right]\ dzd\Omega ,}$

where ${\displaystyle \psi _{W_{\alpha }XY}}$ is the probability that an excited atom will decay via W_αXY Auger transition, ρ_α(z) is the atomic concentration of the element α at depth z, λ(W_αXY) is the mean free path for an W_αXY 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, ${\displaystyle P_{W_{\alpha }}(\hbar \omega ;z)}$ 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 ${\displaystyle N_{W_{\alpha }XY}(\hbar \omega )}$.