Thermal Differential EXAFS describes those DiffEXAFS measurements taken with mod- ulation of a sample’s temperature. In this particular situation, the physics of DiffEXAFS is more complicated than that described by equation (2.12) since it is not just the mean scattering path lengthsjthat is dependent on sample temperature, but also the variance in scattering path length σ2
j.
It is also important, therefore, to consider the form of the atomic pair-correlation func- tion. In order for thermal expansion, or any other such thermally induced strain, to exist within a material, the pair-correlation function must be asymmetric. This in turn re- quires anharmonicity to be considered, and thus the (harmonic) Gaussian Debye-Waller
factor must be replaced. Commonly, this results in the fine-structure function being ma- nipulated in terms of a cumulant expansion [72]. However, for DiffEXAFS, temperature changes are very small, and so anharmonic contributions to the fine-structure from any source other than thermal expansion are negligible. In this case it is possible to adopt the quasi-harmonic approximation of Leibfried & Ludwig [42], whereby the Gaussian form of the pair-correlation function is retained, but the centroid of that Gaussian displaced to model thermal expansion.
Under these conditions, the Taylor expansion of (2.11) becomes
∆χ=X j Aj(k)e−2k 2 σ2 j kcos ksj+φj(k)∆sj −2k2sin ksj +φj(k)∆σ2j (2.13)
The Thermal Differential Fine-structure Function therefore contains two signals super- imposed upon one another. The first, as in (2.12) is characterised by ∆sj. In the absence of any non-linear phenomena such as phase-transitions, this just arises from thermal expansion in the sample. The second, new term is characterised by ∆σ2j, and so describes changes to thermal disorder.
The difference between this function and that of (2.12) can be seen in Figure 2.1, where a typical Joule magnetostriction DiffEXAFS signal is plotted for a 90◦ rotation in
sample magnetisation, and a typical Thermal DiffEXAFS signal plotted for a 1K change in sample temperature; both at the Fe-K edge.
Examining (2.13), it is clear first and foremost that the disorder term retains the sin phase dependency of the original fine-structure function (2.11), whereas the expansion term has changed to acosdependency. Contributions from thermal disorder are therefore in phase with the conventional EXAFS, whilst contributions from thermal expansion are in quadrature. This difference is key in providing the ability to resolve one term from the other in an experimental DiffEXAFS spectrum.
It can also be seen that both terms scale with photoelectron wavevector; expansion by k1 and disorder by k2. This indicates that both terms are amplified relative to the conventional EXAFS as x-ray energy increases, resulting in more high-k oscillations being present in the DiffEXAFS compared to the conventional fine-structure. This in
Figure 2.1: DiffEXAFS signals at the Fe-K edge for magnetisation modulation of FeCo (provided by R.F. Pettifer) and thermal modulation of Fe foil. EXAFS for the pure Fe sample is shown, which is virtually identical to the FeCo structure. As can be seen, the modulation of different sample properties results in very different signals. The magnetisation signal only contains one component through magnetostrictive strain, whereas the thermal signal contains components from expansion of the crystal lattice and changes to atomic vibrational amplitudes.
turn allows DiffEXAFS data to be acquired further from the edge, with the structure not being washed-out till kis typically around 15 to 20˚A−1.
Now, inserting the thermal expansion coefficient for each pathαj, and considering the possibility of non-unit temperature modulation, (2.13) becomes
∆χ ∆T = X j Aj(k) ksjcos ksj+φj(k) αj −2k2sin ksj+φj(k) ∆σ2 j ∆T (2.14)
true for large changes in temperature, but is acceptable when working with DiffEXAFS, since temperature modulation is only of the order of few Kelvin2
. This expression will also hold when other strains, not related to thermal expansion, must be considered, so long as the components ofαjinclude the contributions from all source of thermal strain. Each αj may be analysed in the context of the geometry of path j in order to obtain the second-rank thermal expansion tensorαmn. Depending on the type of crystal under study, αmn may contain up to nine independent parameters, describing atomic strains along different crystallographic directions. Each coefficient must be determined by the analysis of a scattering path with geometry sensitive to strains along the same direction described by the coefficient. Some paths, particularly multiple scattering paths, may be sensitive to strains described by two or more coefficients.
However, the point group crystal symmetry of a chosen sample material can be exploited through von Neumann’s Principle to reduce the number of independent coefficients [53]. For instance, with crystals of cubic symmetry, the tensor is isotropic; all off-diagonal elements are zero, and all diagonal elements equal. This reduces the number of inde- pendent coefficients to one andαmn toα.
Note also, that in inserting αj into (2.13) an additional coefficient, sj, is needed. This reveals the last key property of the differential fine-structure function: larger scattering paths are relatively amplified compared to shorter ones. High-order paths therefore hold relatively greater significance than they would do in conventional EXAFS. Critically, the thermal disorder term does not scale withsj. As a result, whensj is large, the thermal expansion component of the differential fine-structure becomes a greater fraction of the total observed signal than when it is small. This allows expansion to be more easily detected in high-order scattering paths.
2
Even so, it is still reasonable to expect a different values to be obtained when the absolute temper- ature, about which the DiffEXAFS measurements are made, is significantly altered.