Debye-Waller function

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. 

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. 

By Fourier transforming the EXAFS oscillations, a radial structure function is obtained (2U). The peaks in the Fourier transform correspond to the different coordination shells and the position of these peaks gives the absorber-scatterer distances, but shifted to lower values due to the effect of the phase shift. The height of the peaks is related to the coordination number and to thermal (Debye-Waller smearing), as well as static disorder, and for systems, which contain only one kind of atoms at a given distance, the Fourier transform method may give reliable information on the local environment. However, for more accurate determinations of the coordination number N and the bond distance R, a more sophisticated curve-fitting analysis is required. 

According to the model, a perturbation at one site is transmitted to all the other sites, but the key point is that the propagation occurs via all the other molecules as a collective process as if all the molecules were connected by a network of springs. It can be seen that the model stresses the concept, already discussed above, that chemical processes at high pressure cannot be simply considered mono- or bimolecular processes. The response function X representing the collective excitations of molecules in the lattice may be viewed as an effective mechanical susceptibility of a reaction cavity subjected to the mechanical perturbation produced by a chemical reaction. It can be related to measurable properties such as elastic constants, phonon frequencies, and Debye-Waller factors and therefore can in principle be obtained from the knowledge of the crystal structure of the system of interest. A perturbation of chemical nature introduced at one site in the crystal (product molecules of a reactive process, ionized or excited host molecules, etc.) acts on all the surrounding molecules with a distribution of forces in the reaction cavity that can be described as a chemical pressure. 

Fig. 3.16 Scaling presentation of the dynamic structure factor from a M =36,000 PE melt at 509 K as a function of the Rouse scaling variable. The solid lines are a fit with the reptation model (Eq. 3.39). The Q-values are from above Q=0.05,0.077,0.115,0.145 A The horizontal dashed lines display the prediction of the Debye-Waller factor estimate for the confinement size (see text)...

The errors in the fitting parameters may be obtained from the covariance matrix of the fit if it is available, but they are more commonly estimated by varying one parameter away from its optimal value while optimizing all other parameters until a defined increase in the statistical function is obtained. However, the statistical error values obtained do not represent the true accuracies of the parameters. In fact, it is difficult to determine coordination numbers to much better than 5%, and 20% is more realistic when the data are collected at room temperature taking into account the strong coupling between the coordination number and Debye Waller terms, the error in the latter may be 30%. 

Informations on the vibrational and electron mean free path properties. Such analysis is possible only if the interface phase is very well defined, and if temperature dependent measurements are done and compared. Debye Waller effects can be tangled with ordering transformation of the interface phase as a function of temperature and so on. If a single phase interface with order at least to the second nearest neighbour is recognised, then a temperature dependent Debye Waller, and mean free path analysis can be attempted. 

These maxima in the Fourier transform data, which correspond to the different chromium coordination shells, were isolated using a filter window function. The inverse transform of each peak was generated and fitted using a non-linear least squares program. The amplitude and phase functions were obtained from the theoretical curves reported by Teo and Lee (2 ). The parameters which were refined included a scale factor, the Debye-Waller factor, the interatomic distance, and the threshold energy difference. This process led to refined distances of 1.97(2) and 2.73(2) A which were attributed to Cr-0 and Cr-Cr distances, respectively. Our inability to resolve second nearest neighbor Cr-Cr distances may be a consequence of the limited domain size of the pillars. 

Two of the more direct techniques used in the study of lattice dynamics of crystals have been the scattering of neutrons and of x-rays from crystals. In addition, the phonon vibrational spectrum can be inferred from careful analysis of measurements of specific heat and elastic constants. In studies of Bragg reflection of x-rays (which involves no loss of energy to the lattice), it was found that temperature has a strong influence on the intensity of the reflected lines. The intensity of the scattered x-rays as a function of temperature can be expressed by I (T) = IQ e"2Tr(r) where 2W(T) is called the Debye-Waller factor. Similarly in the Mossbauer effect, gamma rays are emitted or absorbed without loss of energy and without change in the quantum state of the lattice by... 

From equation 5, it is apparent that each shell of scatterers will contribute a different frequency of oscillation to the overall EXAFS spectrum. A common method used to visualize these contributions is to calculate the Fourier transform (FT) of the EXAFS spectrum. The FT is a pseudoradial-distribution function of electron density around the absorber. Because of the phase shift [< ( )], all of the peaks in the FT are shifted, typically by ca. —0.4 A, from their true distances. The back-scattering amplitude, Debye-Waller factor, and mean free-path terms make it impossible to correlate the FT amplitude directly with coordination number. Finally, the limited k range of the data gives rise to so-called truncation ripples, which are spurious peaks appearing on the wings of the true peaks. For these reasons, FTs are never used for quantitative analysis of EXAFS spectra. They are useful, however, for visualizing the major components of an EXAFS spectrum. 

Flere (1 - f) and (1 - S) are the losses brought about by the corresponding process and gfastj/3 (t -> oo) = 0 (see Fig. 6). The factor / can be regarded as a generalized non-ergodicity parameter and, hence, it is expected to show a similar anomaly as the Debye-Waller factor /g (see Fig. 5). Such decomposition of the correlation function is useful in spin-lattice relaxation studies, as will be discussed in Section 3.2.4. 

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