Debye-Waller attenuation factor

Distortions of the First Kind and Thermal Disorder. In crystallography the best-known example for a lattice distortion of the first kind is the reduction of peak intensity from random temperature movement of the atoms. In materials science a frozen-in thermal disorder of nanostructures25 is observed as well. The result of this kind of disorder is a multiplicative26 attenuation of the scattering intensity by the Debye-Waller factor... 

The variation with the electron wave vector k is associated with an intensity variation in the experimentally observed polar and azimuth angles. (In order to include vibrational attenuation of interference effects, each scattered wave has to be multiplied by the temperature dependant Debye-Waller factor.)... 

Structure factor, we assumed that the atoms had a certain position defined by the vector r. We now see that the positions of these atoms are constantly changing. As a result, the diffracted amplitude is modified. This modification depends in theory on what family of planes considered because each atom is moving inside a potential well with an anisotropic shape, since it depends on the ciystallographic direction. The complete description of this effect will not be given here. We will simply point out that the diffracted intensity is attenuated by a factor, called the Debye or Debye-Waller factor, smaller than 1, and whose value usually depends on which family of planes is being considered. This factor will be denoted by D. Finally, the intensity diffiacted by a polyciystalline sample is therefore written ... 

The Debve-Waller factor is an amplitude term in any scattering experiment that takes account of the movements of the scatterers about their average positions. This results in attenuation of the scattering which increases with scattering vector. For EXAFS analysis the appropriate Debye-Waller factor takes account of variations in the absorber-scatterer distance, and thus depends on how much the motion of this pair of atoms is correlated. 

In EXAFS experiments as well as in other EXAFS-like methods, variations in the sample temperature are well described by the Debye-Waller factor and lead to the exponential attenuation of line structure when the sample temperature increases. The temperature dependence of the SEFS spectrum is also described by the Debye-Waller factor— more exactly, by two Debye-Waller factors corresponding to the interference terms of the final and intermediate states [Eq. (38)). Since these interference terms are determined by different wave numbers, p and q, the change of the sample temperature results in a change of the relative intensity of the oscillating terms, which reveals itself in the unusual dependence behavior of SEFS. 

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