Deformation mapping

Most of the relevant features of the charge density distribution can be elegantly elucidated by means of the topological analysis of the total electron density [43] nevertheless, electron density deformation maps are still a very effective tool in charge density studies. This is especially true for all densities that are not specified via a multipole model and whose topological analysis has to be performed from numerical values on a grid. 

Conventional implementations of MaxEnt method for charge density studies do not allow easy access to deformation maps a possible approach involves running a MaxEnt calculation on a set of data computed from a superposition of spherical atoms, and subtracting this map from qME [44], Recourse to a two-channel formalism, that redistributes positive- and negative-density scatterers, fitting a set of difference Fourier coefficients, has also been made [18], but there is no consensus on what the definition of entropy should be in a two-channel situation [18, 36,41] moreover, the shapes and number of positive and negative scatterers may need to differ in a way which is difficult to specify. 

Thanks to the particular choice made for the NUP, taken equal to a superposition of spherical atoms, it is for the first time possible within the present approach to compute MaxEnt deformation maps in a straightforward manner. Once the Lagrange multipliers X have been obtained, the deformation density is simply... 

The MaxEnt deformation density in the COO- plane is shown in Figure 6(a). The deformation map shows correct qualitative features differences between the single C-C bond and the C-0 bonds are clearly visible, and so are the lone-pair maxima on the oxygen atoms. If compared to the conventional dynamic deformation density... 

Figure 6. Experimental deformation map for nitromalonamide. The contour interval is 0.1 e/A3. The dotted line is the zero contour. Solid lines are positive contours, broken lines are negative contours sinS/k < 0.7. The plane shown is the one spanned by the Ql)-O(l) and the C(l)-C(3) vectors. 

Figure 6.9 (a) Standard deformation density of tetrafluoroterephthalonitrile in the molecular plane. Contour interval is 0.1 e A-3, terminated at 1.5 e A 3. (b) Molecular diagram with a box around the fragment shown in the deformation map (a). (Reproduced with permission from F. L. Hirshfeld, Acta Crystallogr., B40, 613, 1984.)... 

Deformable bodies, flow past, 11 775-777 Deformation, defined, 21 702 Deformation maps, 13 479-480 Deformation processing, of metal-matrix composites, 16 169-171 Deformation strain, 13 473 Defrost controllers, in refrigeration systems, 21 540... 

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 X-N technique is sensitive to systematic errors in either data set. As discussed in chapter 4, thermal parameters from X-ray and neutron diffraction frequently differ by more than can be accounted for by inadequacies in the X-ray scattering model. In particular, in room-temperature studies of molecular crystals, differences in thermal diffuse scattering can lead to artificial discrepancies between the X-ray and neutron temperature parameters. Since the neutron parameters tend to be systematically lower, lack of correction for the effect leads to sharper atoms being subtracted, and therefore to larger holes at the atoms, but increases in peak height elsewhere in the X-N deformation maps (Scheringer et al. 1978). 

Hirshfeld (1984) found the electrostatic charge balance at the F nuclei, based on the experimental deformation density, to be several times more repulsive (i.e., anti-bonding) than that of the promolecule. Very sharp dipolar functions at the exocyclic C, N, and F atoms, oriented along the local bonds, were introduced in a new refinement in which the coefficients of the sharp functions were constrained to satisfy the electrostatic Hellmann-Feynman theorem (chapter 4). The electrostatic imbalance was corrected with negligible changes in the other parameters of the structure. The model deformation maps were virtually unaffected, except for the innermost contour around the nuclear sites. 

Fig. 16 Deformation map for iPP with Mw of 200 kg mol Different regimes of microdeformation behaviour are indicated on a plot of Kc vs. T for various strain rates [19, 26]...

Fig. 17 Deformation map for iPP. Different regimes of microdeformation behaviour are plotted as a function of thermal degradation times at 130 °C and the initial Mw [26, 104]...

For dynamic deformation maps, < >m differs from s when the crystal is acentric. Neglecting this phase difference can underestimate the deformation density of a covalent bond by 0.21 A-3 which represents something like one-third to one-half of the deformation density [76]. 

In conclusion, almost quantitative agreement is obtained between experimental static deformation maps and extended triple- -plus polarization maps. 

In the multipole refinement of TPPFe(THF)2, a D4h local symmetry was imposed on the iron atom which explains that only four dt population parameters were derived inspection of Table 4 leads to the same conclusion derived qualitatively from the examination of the deformation maps i.e., the 5 2g state is the main contributor to the ground state of the complex. This interesting calculation of d electron population calculations was also performed on other coordination compounds like metal carbonyls [38] and metal carbynes [39]. 

M. F. Ashby and H. Frost, Deformation Maps, Pergamon Press, Oxford, U.K., 1982. 

Figures 19 and 20 summarize the fracture behavior of the previously mentioned PP with MFR 12 dg min 1 in between -30 and 60 °C. Details concerning the experimental procedure, the F-d curves and the data reduction according to the principles of the LEFM are given in [77]. The aim of this section is to correlate the relative capacity of both systems to absorb input energy up to a deformation corresponding to Fmax (Gini) and up to fracture (Gtot Fig. 19) with their deformation maps deduced from their Fd-curves and careful observation of their fracture surfaces (Fig. 20). To take implicitly into account variations in specimen stiffness, arising from variations in temperature and rate dependence of the modulus the test rate was sometimes expressed in terms of the crack tip loading rate, dK/dt, given by ...

Fig. 20 Deformation maps of a a non-nucleated iPP and b its -nucleated homologue (MFR 12dgmin ) for different temperatures and crack tip loading rates as deduced from the fracture surfaces of compact tension specimens. A rough indication of the test speed is provided by the upper scale. Same grades as in Fig. 19...

Fig. 39 Deformation map as a function of the temperature and the pressure of a random copolymer and a /S-nudeated PP. Data adapted from Greenshields [216]...

Here we have introduced a tensor F known as the deformation gradient tensor whose components reflect the various gradients in the deformation mapping and are given by... 

Use of the fact that the deformation mapping may be written as x(t) = X + u(X, t) reveals that the deformation gradient may be written alternatively as... 

For simplicity, we begin by considering the case in which the shear is in the Xi-direction on a plane with normal n = (0, 0, 1), an example of which is shown in fig. 2.4. In this case, the deformation mapping is... 

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