Temperature factor anharmonic

As first shown by Dawson (1967), Eq. (11.3) can be generalized by inclusion of anharmonicity of the thermal motion, which becomes pronounced at higher temperatures. We express the anharmonic temperature factor of the diamond-type structure [Chapter 2, Eq. (2.45)] as 71(H) = TC(H) -f iX(H), in analogy with the description of the atomic scattering factors. Incorporation of the temperature... 

The third-rank tensors, as occur in the expression for the anharmonic temperature factor (chapter 2), the restrictions may be derived by use of the transformation law ... 

J From population parameters in Stevens et al (1980) total Fe population fixed at 6 electrons with anharmonic cumulant temperature factors. 

For vanadium, the ratios are smaller, and the dynamic density maps do not show a distinct maximum in the cube direction. The difference is attributed to anharmonicity of the thermal motion. Thermal displacement amplitudes are larger in V than in Cr, as indicated by the values of the isotropic temperature factors, which are 0.007 58 and 0.00407 A2 respectively. As in silicon, the anharmonic displacements are larger in the directions away from the nearest neighbors, and therefore tend to cancel the asphericity of the electron density due to bonding effects. 

Since the anharmonic term is small relative to the leading harmonic term, the corresponding temperature factor can be written as... 

The great potential of the X-ray data for obtaining motional information has recently led to a molecular dynamics test197 of the standard refinement techniques that assume isotropic and harmonic motion. Since simulations have shown that the atomic fluctuations are highly anisotropic and, in some cases, anharmonic (see Chapt. VI.A.1), it is important to determine the errors introduced in the refinement process by their neglect. A direct experimental estimate of the errors resulting from the assumption of isotropic, harmonic temperature factors is difficult because sufficient data are not yet available for protein crystals. Moreover, any data set includes other errors that would obscure the analysis, and the specific correlation of temperature factors and motion is complicated by the need to account for static disorder in the crystal. As an alternative to an experimental analysis of the errors in the refinement of proteins, a purely theoretical approach has been used.197 The basic idea is to generate X-ray data from a molecular dynamics simulation... 

The crystal structure of synthetic hibonite, CaAli20i9, was determined by Kato and Saalfeld (1968) and was refined by Utsunomiya et al. (1988). Since anharmonicity of the thermal vibration of the A12 atom was observed, refinement of the temperature factors including anharmonic terms up to the fourth order was done (Utsunomiya et al. 1988). The result supports a split atom model where statistically occupies one of the two equivalent A12 sites at room temperature. 

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. 

It is clear that nonconfigurational factors are of great importance in the formation of solid and liquid metal solutions. Leaving aside the problem of magnetic contributions, the vibrational contributions are not understood in such a way that they may be embodied in a statistical treatment of metallic solutions. It would be helpful to have measurements both of ACP and A a. (where a is the thermal expansion coefficient) for the solution process as a function of temperature in order to have an idea of the relative importance of changes in the harmonic and the anharmonic terms in the potential energy of the lattice. 

With data averaged in point group m, the first refinements were carried out to estimate the atomic coordinates and anisotropic thermal motion parameters IP s. We have started with the atomic coordinates and equivalent isotropic thermal parameters of Joswig et al. [14] determined by neutron diffraction at room temperature. The high order X-ray data (0.9 < s < 1.28A-1) were used in this case in order not to alter these parameters by the valence electron density contributing to low order structure factors. Hydrogen atoms of the water molecules were refined isotropically with all data and the distance O-H were kept fixed at 0.95 A until the end of the multipolar refinement. The inspection of the residual Fourier maps has revealed anharmonic thermal motion features around the Ca2+ cation. Therefore, the coefficients up to order 6 of the Gram-Charlier expansion [15] were refined for the calcium cation in the scolecite. 

Further Mossbauer effect studies (304 — 12 K) and magnetic susceptibility measurements (301 - 1K) on the neutral complex [Fe(papt)2 ] have been performed recently 1S9 The magnetic data are shown in Fig. 34. The values of - In f(s T2) and - In f(J Aj) have been found to follow the high-temperature approximation of the Debye model above 105 K and 140 K, respectively, if anharmonic corrections have been introduced. No simple model is available at present, which would be capable to account for the complete temperature dependence of the Debye-Waller factors in this crossover system. 

In myoglobin, we find that the anharmonic contribution significantly enhances thermal conduction over that in the harmonic limit, by more than a factor of 2 at 300 K. Moreover, the thermal conductivity rises with temperature for temperatures beyond 300 K as a result of anharmonicity, whereas it appears to saturate around 100 K if we neglect the contribution of anharmonic coupling of vibrational modes. The value for the thermal conductivity of myoglobin at 300 K is about half the value for water. The value for the thermal diffusivity that we calculate for myoglobin is the same as the value for water. Thermal transport coefficients for other proteins will be presented elsewhere. 

---