Debye-Waller thermal parameter

Subscript 1 refers to M-N parameters and subscript 2 refers to M-S or M-Cl parameters. The functions Fj(kj) and 0j(kj) used in this study are the theoretically calculated amplitude and phase functions for the j back-scatterer (4H. Four parameters are least-squares refined for each term the scale factor, B the Debye-Waller thermal parameter, a the interatomic distance, r and the energy threshold, AEq. 

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. 

If the difference in atomic number between the absorber element and the backscattering element is >10 and if only one kind of element backscatters, EXAFS spectra can be analyzed readily to provide local structural data on adsorbed species. However, because the electron mean free path, thermal and static disorder parameters (Debye-Waller factors), and coordination number for an absorber environment cannot be determined a priori with sufficient accuracy, EXAFS data for suitable reference compounds of known molecular structure must be used to help interpret the EXAFS spectrum for an interfaeial region. 

The inadequacy of the Debye approximation in describing the details of the frequency distribution function in a real solid is well known. This results in noticeable disparities between Debye temperatures derived from the results of different experimental techniques used to elucidate this parameter on the same solid, or over different temperature ranges. Substantial discrepancies may be expected in solids containing two (or more) different atoms in the unit cell. This has been demonstrated by the Debye-Waller factors recorded for the two different Mdssbauer nuclei in the case of Snl4,7 or when the Debye-Waller factor has been compared with the thermal shift results for the same Mdssbauer nucleus in the iron cyanides.8 The possible contribution due to an intrinsic thermal change of the isomer shift may be obscured by an improper assignment of an effective Debye temperature. 

Mardalen et al. [56,60] adopted a similar basic model, since they did not observe additional peaks requiring cell doubling, but the model could not account for the observed relative intensities, unless extremely large thermal damping parameters (Debye-Waller exponents) were associated with the side chains (for PHT and POT). This was physically interpreted in terms of a large degree of positional disorder of the side chains, and the model of side chain disorder found some support in the absence in their data of hkl-reflections with h and k or I finite. Finite h only in reflections of the type hOO was taken as evidence for disorder between layers of polymer chains stacked on top of each other in the b-c plane. 

In a crystal, displacements of atomic nuclei from equilibrium occur under the joint influence of the intramolecular and intermolecular force fields. X-ray structure analysis encodes this thermal motion information in the so-called anisotropic atomic displacement parameters (ADPs), a refinement of the simple isotropic Debye-Waller treatment (equation 5.33), whereby the isotropic parameter B is substituted by six parameters that describe a libration ellipsoid for each atom. When these ellipsoids are plotted [5], a nice representation of atomic and molecular motion is obtained at a glance (Fig. 11.3), and a collective examination sometimes suggests the characteristics of rigid-body molecular motion in the crystal, like rotation in the molecular plane for flat molecules. Lattice vibrations can be simulated by the static simulation methods of harmonic lattice dynamics described in Section 6.3, and, from them, ADPs can also be estimated [6]. 

The atomic scattering factors are the Fourier transforms of the spherical atomic electron distributions. They are considered as known from quantum-chemical calculations. The site occupation parameters may assume values different from unity if the structure is disordered. The Debye-Waller factors allow for the atomic thermal motions. They are functions of the atomic displacement parameters W. Omitting the atom index n and representing the Miller indices and lengths of the reciprocal lattice vectors by and a, respectively ... 

The vacancies in transition metal carbides and nitrides induce lattice distortions in their neighborhood and the thermal motion of metal atoms adjacent to the vacancies become asymmetric and anharmonic. Temperature-dependent X-ray diffraction experiments yield reliable information about the thermal vibrations under the assumption that the static part of the Debye-Waller (D-W) factor is temperature independent—i.e., the concentrations of vacancies and lattice distortions remain constant within the given temperature range—but it is very sensitive to the local atomic arrangement. The mean value of the Debye temperature averaged over the temperature range 623 to 1273 K is 9m = 498 9 K (37), so that the thermal vibrations in ZrCo.98 can be described by the quasi-harmonic one-point potential (OPP) mode in the temperature range 295 to 1273 K. The very weak variation with temperature of the Debye temperature indicates that the potential parameters are temperature independent. 

XAS is an interesting tool to obtain in situ results on the near range order of passive layers on metals even for highly disordered or amorphous films. Experimental requirements have been shortly described above. From the reflectivity data, one may calculate with the Fresnel equations the absorption spectra x(E) of the film (see Refs. [140,141] for details). The oscillation of p(E) above the absorption edge, the so-called extended absorption fine structure (EXAFS), leads to the EXAFS fimction %(E) = ( X - MoVMo with the background value Po- The Fourier transform of % E) to the real space yields the structural parameters, i.e., the radius of the coordination shells Rj, their coordination numbers Nj, i.e. the number of neighbors for a given coordination shell and the Debye-Waller factor Oj as a measure of the structural (and thermal) disorder. 

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