International
Tables for Crystallography Volume C Mathematical, physical and chemical tables Edited by E. Prince © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. C. ch. 4.2, pp. 216-217
Section 4.2.3.4.1.1. Theory
D. C. Creaghb
|
The theory that will be outlined here has evolved through the efforts of many workers over the past decade. The oscillatory part of the X-ray attenuation relative to the `background' absorption may be written as where
is the measured value of the linear attenuation coefficient at a photon energy E and
is the `background' linear attenuation coefficient. This is sometimes the extrapolation of the normal attenuation curve to the edge energy, although it is usually found necessary to modify this extrapolation somewhat to improve the matching of the higher-energy data with the XAFS data (Dreier, Rabe, Malzfeldt & Niemann, 1984
). In most computer programs, the normal attenuation curve is fitted to the data using cubic spline fitting routines.
The origin of XAFS lies in the interaction of the ejected photoelectron with electrons in its immediate vicinity. The wavelength of a photoelectron ejected when a photon is absorbed is given by λ = 2π/k, where
This outgoing spherical wave can be back-scattered by the electron clouds of neighbouring atoms. This back-scattered wave interferes with the outgoing wave, resulting in the oscillation of the absorption rate that is observed experimentally and called XAFS. Equation (4.2.3.8) was written with the assumption that the absorption rate was directly proportional to the linear absorption coefficient.
It is conventional to express in terms of the momentum of the ejected electron, and the usual form of the theoretical expression for χ(k) is
Here the summation extends over the shells of atoms that surround the absorbing atom,
representing the number of atoms in the ith shell, which is situated a distance
from the absorbing atom. The back-scattering amplitude from this shell is
for which the associated phase is
. Deviations due to thermal motions of the electrons are incorporated through a Debye–Waller factor,
, and ρ is the mean free path of the electron.
The amplitude function depends only on the type of back-scattering atom. The phase, however, contains contributions from both the absorber and the back-scatterer:
where l = 1 for K and
edges, and l = 2 or 0 for
and
edges. The phase is sensitive to variations in the energy threshold, the magnitude of the effect being larger for small electron energies than for electrons with considerable kinetic energy, i.e. the effect is more marked in the neighbourhood of the absorption edge. Since the position of the edge varies somewhat for different compounds (Azaroff & Pease, 1974
), some impediment to the analysis of experimental data might occur, since the determination of the interatomic distance
depends upon the precise knowledge of the value of
.
In fitting the experimental data based on an empirical value of threshold energy using theoretically determined phase shifts, the difference between the theoretical and the experimental threshold energies cannot produce a good fit at an arbitrarily chosen distance
, since the effect will be seen primarily at low k values
, whereas changing
affects
at high k values
. This was first demonstrated by Lee & Beni (1977
).
The significance of the Debye–Waller factor should not be underestimated in this type of investigation. In XAFS studies, one is seeking to determine information regarding such properties of the system as nearest- and next-nearest-neighbour distances and the number of nearest and next-nearest neighbours. The theory is a short-range-order theory, hence deviations of atoms from their expected positions will influence the analysis significantly. Thus, it is often of value, experimentally, to work at liquid-nitrogen temperatures to reduce the effect of atomic vibrations.
Two distinct types of disorder are observed: vibrational, where the atom vibrates about a mean position in the structure, and static, where the atom occupies a position not expected theoretically. These terms can be separated from one another if the variation of XAFS spectra with temperature is studied, because the two have different temperature dependences. A discussion of the effect of a thermally activated disorder that is large compared with the static order has been given by Sevillano, Meuth & Rehr (1978). For systems with large static disorders, e.g. liquids and amorphous solids, equation (4.2.3.10)
has to be modified somewhat. The XAFS equation has to be averaged over the pair distribution function g(r) for the system:
Other factors that must be taken into account in XAFS analyses include: inelastic scattering (due to multiple scattering in the absorbing atom and excitations of the atoms surrounding the atom from which the photoelectron was ejected) and multiple scattering of the photoelectron. Should multiple scattering be significant, the simple model given in equation (4.2.3.10) is inappropriate, and more complex models such as those proposed by Pendry (1983
), Durham (1983
), Gurman (1988
, 1995
), Natoli (1990
), and Rehr & Albers (1990
) should be used. Several computer programs are now available commercially for use in personal computers (EXCURVE, FEFF5, MSCALC). Readers are referred to scientific journals to find how best to contact the suppliers of these programs.
References









