Atomic displacement parameters anharmonic

Yet another level of complexity of vibrational motion is taken into account by using the so-called anharmonic approximation of atomic displacement parameters. One of the commonly used approaches is the cumulant expansion formalism suggested by Johnson, in which the structure factor is given by the following general expression ... 

A brief description of the anharmonic approximation is included here for completeness since rarely, if ever, it is possible to obtain reasonable atomic displacement parameters of this complexity from powder diffraction data the total number of atomic displacement parameters of an atom in the fourth order anharmonic approximation may reach 31 (6 anisotropic + 10 third order +15 fourth order). The major culprits preventing their determination in powder diffraction are uncertainty of the description of Bragg peak shapes, non-ideal models to account for the presence of preferred orientation, and the inadequacy of accounting for porosity. 

Figure 4 Temperature dependence of selected atomic displacement parameters for the Mg, Si, Al and O atoms in pyrope garnet. Analysis of the displacement parameters obtained from accurate single-crystal diffraction data allow evaluation of the zero-point energy contribution, separation of static and dynamic effects, and detection of anharmonic vibrational contributions to the atomic motion.

Just as in the case of the conventional anisotropic approximation, the maximum number of displacement parameters is only realized for atoms located in the general site position (site symmetry 1). In special positions some or all of the displacement parameters will be constrained by symmetry. For example, 7333, Yi 13, Y223 and Y123 for an atom located in the mirror plane perpendicular to Z-axis are constrained to 0. Furthermore, if an atom is located in the center of inversion, all parameters of the odd order anharmonic tensors (3, 5, etc.) are reduced to 0. 

As mentioned above, Garrett et al. [1] first demonstrated the equivalence of the bond valence sum mismatch pathways in a-Agl with pathways determined experimentally from an anharmonic atomic displacement refinement of neutron diffraction data as a justification for predicting transport pathways for their then new fast Ag" ion conductor Agig I12P 2O7 using the same approach. In our earliest work on pathway models for similar Ag" ion conducting oxyhalide systems [3, 18], we used a parameter set by Radaev et al. [19],... 

Here Hint is the anharmonic interaction in the collinear-configurational approximation (see Refs. [5,7]), Vmk = v/n,7i/2displacement operators of the host atoms with respect to the atom(s)of the mode (Y.i eimeim = M- We take into account that the strongly excited mode can be considered classically, and replace its coordinate operator by Q(t) = A cos( )/f), where A is the initial amplitude of the mode. Then... 

In the first step the positions of all atoms in the cell are optimized. Cell parameters are usually borrowed from experiment. In some cases they are optimized [84] and in some cases not [85]. Harmonic frequency calculations verify that the computed structure corresponds to the global PES minimum. In the second step the anharmonic OH stretching [83, 84] frequency is estimated using ID potential curves calculated as a function of the displacement for the hydrogen atom. In the third step classical molecular dynamics (MD) simulations are performed. The IR [85] or vibrational spectrum [82, 83] of the crystal is computed from the Fourier transform of the corresponding time correlation function (see Section 9.3.1). 

Figure 3. The energy of displacement of one atom in hep iron at a density of 13 Mg m (Steinle-Neumann et al. 2000) (symbols). The atom is displaced along the a-axis directly towards one of the nearest neighbors magnitude of displacement is measured in units of the a lattice parameter. The solid line is a quaitic fit. A quadratic fit (dashed line) illustrates the magnitude of anharmonicity.

The density of lattice vacancies can be determined, usually accurately, by the site occupation parameter. Thus if the local atomic potential is bifurcated and atoms are locally displaced, it may appear that partial occupation of the displaced site describes the system quite well. However, in such cases there is usually strong coupling between the occupation parameter and the DW factor during the process of refinement. Also, local displacement of an atom is usually accompanied by a lattice relaxation around the displaced atom and a much increased DW factor of the displaced site. Thus, in practice, it is very difficult to describe anharmonic local displacements through partial occupation of two closely spaced sites. 

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