# Ab initio Debye-Waller factors

### From FEFF

XAFS analysis can provide structural information, including average near-neighbor distances R,their mean square fluctuations σR2 ,and coordination numbers NR. The quantities σR2 which appear in the XAFS Debye Waller (DW) factor are crucial to the success of the modern theory of XAFS and its applications. The DW factor accounts for thermal and structural disorder and generally governs the ”melting” of the XAFS oscillations with respect to increasing temperature and their decay with respect to increasing photoelectron energy. In practice, the DW factors of the many multiple-scattering terms in the XAFS signal can significantly complicate the analysis. To overcome these difficulties, feff offers several ways to calculate the Debye-Waller factors and account for the effects of thermal disorder in the ab initio XAFS calculations. These are described below.

Fig. 3.17 illustrates the way DW factors enter the XAFS problem.

If one uses the multiple scattering path expansion (PE), the DW factors are added to each path individually in module ff2x. If one uses Full Multiple Scattering (FMS), the effect of finite temperature is approximated by multiplying each free propagator by e−σ2k2 . This is only exact for single scattering paths, but since the effect of thermal disorder is reduced in the near-edge region anyway, it’s probably adequate.

There are three ways to modify the Debye–Waller factor. The DEBYE card calculates a Debye–Waller factor for each path in PE or FMS. The SIG2 card adds a constant Debye–Waller factor to all paths in PE. Finally, you can edit ‘list.dat’ to add a Debye–Waller factor to a particular path in PE. In PE, these three Debye–Waller factors are summed, so if the DEBYE and SIG2 cards are present, and if you have added a Debye–Waller factor to a particular path, the Debye–Waller factor used will be the sum of all three.

After changing the DW factors, feff must be rerun starting with module ff2x for PE and starting with module fms for FMS.

The DEBYE card offers a choice between 5 different models for the DW factors:

• 0 Correlated-Debye method (default) (CD) • 1 Equations of Motion method (EM)

• 2 Recursion method (RM)

• 3 Classical Correlated-Debye method (CCD) • 4 Read from ”sig2.dat” file

• 5 Dynamical-Matrix method (DM) • ¡0 Do not calculate DW factors

• <0 Do not calculate DW factors

Only method CD and CCD can run without additional input. These Correlated Debye models are isotropic and can be very inaccurate for anisotropic materials. Methods 1, 2 and 5 require that the force constants or the dynamical matrix be provided. We will now describe these requirements from a computational point of view. For more information on the physics behind these calculations, we refer to Anna Poiarkova’s thesis and our paper on the DM method.