Extended X-ray absorption fine structure (EXAFS)

Definition

Extended X-ray absorption fine structure (EXAFS) is the portion of XAFS spectra well above an absorption edge – typically starting ~ 50 eV above the absorption edge. EXAFS can be interpreted as due to scattering of the photoelectron ejected from the absorbing atom by the photo-electric effect. The photo-electron will scatter from surrounding atoms and a portion of it will return coherently to the absorbing atom still in its excited state (before the hole in the core electron level has been refilled). The amplitude of the scattered photo-electron at the absorbing atom will modify the probability of creating a photo-electron, and so the probability of X-ray absorption.

EXAFS can be modelled with the EXAFS equation:

[math]\chi(k) = \sum\limits_j {{N_jS_0^2}\over{kR_j^2}}F_j(k)e^{-2R_j/\lambda_j(k)}e^{-2k^2\sigma_j^2}\sin\Big[2kR_j + \Phi_j(k)\Big][/math]

[math]k = {2\pi\over\lambda} = \sqrt{{{2m_e(E-E_0)}\over{\hbar^2}}}[/math]

where [math]k[/math] is the photoelectron wavenumber, [math]E[/math] is the X-ray energy, [math]E_0[/math] is the energy of the absorption edge (q.v.), and [math]m_e[/math] is the electron mass. Note that [math]k[/math] is really the 'photoelectron momentum index’ and differs from the physical momentum (see Rehr, Albers cited in History below). The normalised value of [math]\chi(k)[/math] is dimensionless, because the (standard) expression above has [math]F[/math] in units of [math]1/k[/math] (i.e. length). Other definitions for [math]F[/math] being dimensionless (as a form factor or scattering amplitude) have [math](kR)^2[/math] in the denominator to maintain consistency of units.

The sum in the EXAFS equation is most simply over shells of atoms of a particular type [math]j[/math] and similar distances from the origin of the initial photoelectron. Then [math]N_j[/math] is the coordination number, [math]R_j[/math] the interatomic distance, and [math]\sigma_j^2[/math] represents the mean-square disorder in the distance for the [math]j[/math]th shell. [math]F_j[/math] is the photoelectron (back-)scattering amplitude and [math]\Phi_j(k)[/math] is the corresponding (back-)scattering phase for the [math]j[/math]th atomic shell. [math]S_0^2[/math] is an amplitude reduction factor accounting for relaxation of the absorbing atom due to the presence of the empty core level and multi-electron excitations. [math]\lambda_j(k)[/math] is the photoelectron inelastic mean free path, which has a strong dependence upon [math]k[/math], and has values in the range of 1 to 100 Å over the XAFS regime.

The crude approximation of [math]\Phi_j(k) \approx -2a_0k[/math] ([math]a_0[/math] is the Bohr radius) works for many systems and causes peaks for a particular shell in the Fourier transform of [math]\chi(k)[/math] to be shifted ~ 0.5 Å below the actual interatomic distance. Both [math]F_j(k)[/math] and [math]\Phi_j(k)[/math] depend upon the atomic number [math]Z[/math] of the scattering atom, and have non-linear dependence on [math]k[/math].

The [math]e^{-2k^2\sigma_j^2}[/math] term is often referred to as the EXAFS isotropic or effective Debye–Waller factor, including thermal vibration and static disorder. The sum over shells and use of [math]\sigma_j^2[/math] in the standard EXAFS equation can be generalized to an integral over the partial pair distribution function [math]g(R)[/math] in which one atom is always the absorbing atom.

The sum in the EXAFS equation can be generalized to be over photo-electron scattering paths instead of shells of atoms. This formalism allows the inclusion of multiple scattering paths for the photo-electron, which can give important contributions in many systems. The interpretation of many components of the EXAFS equation are then slightly modified, so that [math]R_j[/math] is then half the path length, and [math]F_j(k)[/math] and [math]\Phi_j(k)[/math] become the (multiple) scattering amplitude and phase-shift for the entire path.

The EXAFS equation allows the numerical determination of the local structural parameters [math]N_j[/math] and [math]R_j[/math], and [math]\sigma_j^2[/math] knowing the scattering amplitude [math]F_j(k)[/math] and [math]\Phi_j(k)[/math] for a small number (typically 1 to 10) of shells or paths. It breaks down at low [math]k[/math] (the XANES region) as the [math]1/k[/math] term increases, [math]\lambda_j(k)[/math] increases, the disorder terms do not strongly dampen the EXAFS, and the EXAFS picture of single particle scattering is no longer a good approximation.

History

The history is reviewed in Lytle, F. W. (1999), J. Synchrotron Rad. 6, 123 and Stumm von Bordwehr, R. (1989), Ann. Phys. (Paris) 14, 377. EXAFS scattering theory: Sayers, Stern and Lytle [(1971), Phys. Rev. Lett. 27, 1204] developed a quantitative parametrization of the central EXAFS region from the Kronig short-range order theory. Their EXAFS equation has become standard for much current work; an extensive review [Rehr and Albers (2000), Rev. Mod. Phys. 72, 621–654 covers significant developments of the scattering perspective, especially in terms of theoretical calculations for EXAFS used in quantitative analysis and the inclusion of spherical outgoing photoelectron waves.

See also

• Absorption edge
• Multi-electronic excitation
• X-ray absorption fine structure (XAFS)