Anharmonic thermal motion

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. 

The (/-orbital population analysis was performed with both the harmonic and a more complete anharmonic thermal motion treatment, as discussed in section 10.7.3. The harmonic map is shown here. 

Anharmonic Thermal Motion and the Bonding Anisotropy of Transition Metal Complexes... 

In low-spin transition metal complexes, the preferential occupancy of the d orbitals in the crystal field tends to create excess density in the voids between the bonds, which means that anharmonicity tends to reinforce the electron density asphericity. We will discuss, in the following sections, to what extent the two effects can be separated by combined use of aspherical atom and anharmonic thermal motion formalisms. 

More quantitatively, the effect of the thermal motion follows from the anharmonic thermal motion formalisms discussed in chapter 2. In the bcc structure, the relevant nonzero anharmonic term in the one-particle potential is the anisotropic, cubic site-symmetry allowed, part of uJuku um in expression (2.39). The modified potential for the cubic sites is given by (Willis 1969, Willis and Pryor 1975)... 

While the idea of LF explained the H2 data quite well [28], we were surprised by the magnitude of the oscillations in our I2 data [16], as, unlike H2,12 is not vibrationally cold at room temperature - the conditions for our experiment. Generally, thermal motion is detrimental to observing coherent motion. Thus, we took a long time scale run to get a more accurate measurement of the frequency of the vibrations, shown in Fig. 1.5. These data also exhibit a vibrational revival, from which the anharmonicity of the potential well can be determined. Indeed, the vibrational frequency accurately matched that of the ground state. 

Though the anharmonic components of the thermal motion decrease rapidly with temperature, as described in chapter 2, they will be present to some extent even if the motion is reduced to zero-point vibrations. 

Collection of diffraction data at liquid-helium temperatures is important to reduce thermal motion and its anharmonicity. Similarly, the use of shorter wavelengths at such low temperatures makes data at higher values of sin 6/X accessible, which facilitates deconvolution of thermal motion and bonding effects. Both very low temperatures and hard radiation are becoming more readily available, and are expected to play a crucial role in future studies. 

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 appearance of reflections in the diffraction pattern due to anharmonicity of thermal motion is not limited to the diamond-type structures, and is observed, for example, for the A 15-type structure of the low-temperature superconductor V3Si (Borie 1981), and for zinc (Merisalo et al. 1978). It has been described as thermal excitation of reflections, though no excitation in the spectroscopic sense of the word is involved. 

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. 

Care has to be exercised in ascribing significance to small differences in X-H bond lengths because of the effects of thermal motion and anharmonicity discussed in Part I, Chapter 3. 

Zucker, U. H., and Schulz, H. Statistical approaches for the treatment of anhar-monic motion in crystals. 1. A comparison of the most frequently used formalisms of anharmonic thermal vibrations. Acta Cryst. A38, 563-568 (1982). 

Various models are reported in the literature for correcting the bond lengths and angles of H2O molecules in solid hydrates for thermal motion and anharmonicity. However, as recently shown , both positive and negative terms exist, which partially compensate for each other. Therefore, data obtained by neutron diffraction do not show systematic errors larger than 3 pm and 2°, respectively, and, hence, the differences between the average water molecule in crystalline hydrates and that in the gas phase discussed above should be real. ... 

It is well known from small molecule crystallography that the effects of thermal motion must be included in the interpretation of the X-ray data to obtain accurate structural results. Detailed models have been introduced to take account of anisotropic and anharmonic motions of the atoms and these models have been applied to high-resolution measurements for small molecules.413 In protein crystallography, the limited data available relative to the large number of parameters that have to be determined have made it necessary in most cases to assume that the atomic motions are isotropic and harmonic. With this assumption the structure factor F(Q), which is related to the measured intensity by 7(Q) = F(Q) 2, is given by... 

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. 

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