Debye—Waller factor

In the procedure of X-ray refinement, the positions of the atoms and their fluctuations appear as parameters in the structure factor. These parameters are varied to match the experimentally determined strucmre factor. The term pertaining to the fluctuations is the Debye-Waller factor in which the atomic fluctuations are represented by the atomic distribution tensor ... 

Here Pyj is the structure factor for the (hkl) diffiaction peak and is related to the atomic arrangements in the material. Specifically, Fjjj is the Fourier transform of the positions of the atoms in one unit cell. Each atom is weighted by its form factor, which is equal to its atomic number Z for small 26, but which decreases as 2d increases. Thus, XRD is more sensitive to high-Z materials, and for low-Z materials, neutron or electron diffraction may be more suitable. The faaor e (called the Debye-Waller factor) accounts for the reduction in intensity due to the disorder in the crystal, and the diffracting volume V depends on p and on the film thickness. For epitaxial thin films and films with preferred orientations, the integrated intensity depends on the orientation of the specimen. 

Thermal properties of overlayer atoms. Measurement of the intensity of any diffracted beam with temperature and its angular profile can be interpreted in terms of a surface-atom Debye-Waller factor and phonon scattering. Mean-square vibrational amplitudes of surfece atoms can be extracted. The measurement must be made away from the parameter space at which phase transitions occur. 

Where, /(k) is the sum over N back-scattering atoms i, where fi is the scattering amplitude term characteristic of the atom, cT is the Debye-Waller factor associated with the vibration of the atoms, r is the distance from the absorbing atom, X is the mean free path of the photoelectron, and is the phase shift of the spherical wave as it scatters from the back-scattering atoms. By talcing the Fourier transform of the amplitude of the fine structure (that is, X( )> real-space radial distribution function of the back-scattering atoms around the absorbing atom is produced. 

The used S5mbols are K, scale factor n, number of Bragg peaks A, correction factor for absorption P, polarization factor Jk, multiplicity factor Lk, Lorentz factor Ok, preferred orientation correction Fk squared structure factor for the kth reflection, including the Debye-Waller factor profile function describing the profile of the k h reflection. 

The Bragg scattering of X-rays by a periodic lattice in contrast to a Mossbauer transition is a collective event which is short in time as compared to the typical lattice vibration frequencies. Therefore, the mean-square displacement (x ) in the Debye-Waller factor is obtained from the average over the ensemble, whereas (r4) in the Lamb-Mossbauer factor describes a time average. The results are equivalent. 

An important accessory in many applications of Mossbauer spectroscopy is a cryostat for low temperature and temperature-dependent measurements. This may be necessary to keep samples frozen or to overcome small Debye-Waller factors of the absorbers at room temperature in the case of an isotope with high y-energy. Paramagnetic samples are measured at liquid-helium temperatures to slow down... 

Mossbauer nuclei per square centimetre, the Debye-Waller factor/a of the absorber material, and the resonance cross-section Gq of the Mossbauer isotope. For a multiline spectrum, the result must be split into separate values for each line, which are obtained by weighting t with the relative transition probability of each line. 

Chemical Information from Debye-Waller Factor Measurements... 

Conroy and Perlow [235] have measured the Debye-Waller factor for W in the sodium tungsten bronze Nao.gWOs. They derived a value of/= 0.18 0.01 which corresponds to a zero-point vibrational amplitude of R = 0.044 A. This amplitude is small as compared to that of beryllium atoms in metallic beryllium (0.098 A) or to that of carbon atoms in diamond (0.064 A). The authors conclude that atoms substituting tungsten in bronze may well be expected to have a high recoilless fraction. 

Table 7.8 Summary of results obtained for the four Os Mossbauer transitions studied. The absorber thickness d refers to the amount of the resonant isotope per unit area. The estimates of the effective absorber thickness t are based on Debye-Waller factors / for an assumed Debye temperature of 0 = 400 K. For comparison with the full experimental line widths at half maximum, Texp, we give the minimum observable width = 2 S/t as calculated from lifetime data.

Mossbauer spectroscopy with started only in 1965, when Harris et al. [322] measured the Mossbauer absorption spectra of the 99 keV transition of Pt in platinum metal as a function of temperature (between 20 and 100 K) and of absorber thickness and derived the temperature dependence of the Debye-Waller factor. 

Neutron-irradiated platinum Debye-Waller factor of Au in Pt after low-temperature neutron irradiation, lattice defect... 

Photoionization and therefore EXAFS takes place on a time scale that is much shorter than that of atomic motions so the experiment samples an average configuration of the neighbors around the absorber. Therefore, we need to consider the effects of thermal vibration and static disorder, both of which will have the effect of reducing the EXAFS amplitude. These effects are considered in the so-called Debye-Waller factor which is included as... 

Melroy and co-workers88 recently reported on the EXAFS spectrum of Pb underpotentially deposited on silver (111). In this case, no Pb/Ag scattering was observed and this was ascribed to the large Debye-Waller factor for the lead as well as to the presence of an incommensurate layer. However, data analysis as well as comparison of the edge region of spectra for the underpotentially deposited lead, lead foil, lead acetate, and lead oxide indicated the presence of oxygen from either water or acetate (from electrolyte) as a backscatterer. 

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... 

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