Deformation density static model

When observed structure factors are used, the thermally averaged deformation density, often labeled the dynamic deformation density, is obtained. An attractive alternative is to replace the observed structure factors in Eq. (5.8) by those calculated with the multipole model. The resulting dynamic model deformation map is model dependent, but any noise not fitted by the muitipole functions will be eliminated. It is also possible to plot the model density directly using the model functions and the experimental charge density parameters. In that case, thermal motion can be eliminated (subject to the approximations of the thermal motion formalism ), and an image of the static model deformation density is obtained, as discussed further in section 5.2.4. 

The static model deformation density corresponding to the multipole refinement results is given by... 

Figure 5.12 shows both the dynamic and the static model deformation densities in the plane of the oxalic acid molecule, based on the data set also used for Fig. 5.2. The increase in peak height, due to higher resolution, and reduction in background noise relative to the earlier maps is evident. The model acts as a noise filter because the noise is generally not fitted by the model functions during the minimalization procedure.

Static deformation density maps can be compared directly with theoretical deformation densities. For tetrafluoroterephthalonitrile (l,4-dicyano-2,3,5,6-tetra-fluorobenzene) (Fig. 5.13), a comparison has been made between the results of a density-functional calculation (see chapter 9 for a discussion of the density-functional method), and a model density based on 98 K data with a resolution of (sin 0//)max = 1.15 A -1 (Hirshfeld 1992). The only significant discrepancy is in the region of the lone pairs of the fluorine and nitrogen atoms, where the model functions are clearly inadequate to represent the very sharp features of the density distribution. 

FIG. 11.3 Comparison of the ab-initio local density functional deformation density for Si with the experimental static model deformation density. Contour interval is 0.025 e A-3. Negative contours are dashed lines. Source Lu and Zunger (1992), Lu et al. (1993). 

Once the multipole analysis of the X-ray data is done, it provides an analytical description of the electron density that can be used to calculate electrostatic properties (static model density, topology of the density, dipole moments, electrostatic potential, net charges, d orbital populations, etc.). It also allows the calculation of accurate structure factors phases which enables the calculation of experimental dynamic deformation density maps [16] ... 

Having secured a set of n values for phosphorus, the pseudoatom model was fitted to the four simulated data sets to test the effectiveness of the pseudoatoms model s formal deconvolution of multipolar valence density features from thermal vibrations smearing. Results are illustrated in Figure 4 as maps of the model static deformation densities ... 

Figure 4. Model static deformation densities of H3PO4 in the 0=P-0(H) plane from simulated structure factors with U = 0 (a), at 75K (b), at 150K (c), at 300K (d). Contours as in Figure 3.

Fig. 5.36. Electron-density distributions in borates — deformation density maps calculated for BlOHlj-. (a) static model map calculated from pseudoatom model (b) theoretical deformation density map from ab initio Hartree-Fock configuration-interaction calculation (after Gajhede et al., 1986 reproduced with the publisher s permission).

Experimental investigation of the electronic charge density in pyroelectric lithium sulphate monohydrate was reported in [65, 66]. In these experiments lithium sulphate monohydrate single crystal was studied by neutron [65] and X-ray diffraction [66] at different temperatures. Static deformation and charge density model refinements allowed an estimation of the differences in electron densities between 80 K and 298 K. Changes in the sulphate oxygen atom lone-pair deformation densities, caused by contraction in S04 - Li" contacts between 298 K and 80 K were also found. 

From a single crystal X-ray diffraction study with the data set refined by a multipole expansion model to R = 0.0054 and R = 0.0056, electron density maps have been obtained. The static deformation density distribution resulting from the multipole refinement is depicted in Fig. 4-25, p. 50 [10, 11]. 

External stress, locally applied, can have nonlocal static effects in ferroelastics (see Fig. 4 of Ref. [7]). Dynamical evolution of strains under local external stress can show striking time-dependent patterns such as elastic photocopying of the applied deformations, in an expanding texture (see Fig.5 of Ref. [8]). Since charges and spins can couple linearly to strain, they are like internal (unit-cell) local stresses, and one might expect extended strain response in all (compatibility-linked) strain-tensor components. Quadratic coupling is like a local transition temperature. The model we consider is a (scalar) free energy density term... 

It is demonstrated that the quasi-static stress-strain cycles of carbon black as well as silica filled rubbers can be well described in the scope of the theoretic model of stress softening and filler-induced hysteresis up to large strain. The obtained microscopic material parameter appear reasonable, providing information on the mean size and distribution width of filler clusters, the tensile strength of filler-filler bonds, and the polymer network chain density. In particular it is shown that the model fulfils a plausibility criterion important for FE applications. Accordingly, any deformation mode can be predicted based solely on uniaxial stress-strain measurements, which can be carried out relatively easily. 

The relation of thermal effects with strain processes has been tackled since the last 160 years, being, the studies of Kelvin, the first steps in the research of this topic[l]. The estimation of heat released during the processes of strain by observing temperature levels is a problem that has been dealt by many authors. Models developed by metallurgical engineers allow to evaluate the evolution of some microstructural parameters, such as dislocation density or the grain size. However, when strain processes are quasi-static or the deformation speed (Sd) is slow, the temperature variations are small enough so that they can be considered as isothermal processes. 

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