Internal displacement parameter

This expression has been evaluated by using the internal displacement parameter of Table 8-4 and, in Table 9-4, the resulting e, are compared with values derived from experimental piezoelectric constants compiled by Martin (1972a). The... 

Experimental vibrational frequencies, in units of 10 radians per second, force constants deduced from them, in eV, and C, the Kleinman internal displacement parameter. 

Kinetic energy of electrons, 3 free-electron gas, 348f local approximation, 351, 377f, 541 Kleinman s internal displacement parameter, 198f tables, 196, 208, 220 Kohn anomalies, 395f Kohn-Sham exchange, 540 Koster-Slaler tables, 481 Kramers-Kronig relations, 99 Krypton, properties of. See Inert gas solids... 

Table 2. v, g and are, respectively, the ftequency of the zone-center optic phonon, the coupling coefficent vdth the transverse acoustic phonon and the internal displacement parameter of Kleinman. The experimental data are given between parenthesis (see Ref. 16). 

An extraordinarily easy thermal racemization was observed for aryl arenethiosulfinates (256). It occurs at a convenient rate at about 50°C. The following activation parameters were estimated for the racemization of p-tolyl p-toluenethiosulfinate 218 A// = 23 kcal/mol A5 = -4 e.u. Thus, the rate of racemization is about 10" times greater than that of diaryl sulfoxides. An internal displacement of sulfenyl sulfur rather than pyramidal inversion was proposed as the mechanism. Recent studies on the chemistry and stereochemistry... 

Figure 4.25 Comparison of the different order parameters for the 151 K second-order transition of PrAlOj. (After Sturge et al., 1975.) Unbroken line is the smooth curve through the internal displacement order parameter, cos 2(, from ESR measurements. Black circles represent the electronic order parameter from optical absorption studies. Squares represent the reduced macroscopic strain from elastic neutron scattering.

The straight solid line - the consequence of equations (3.52) and (3.53) at — 0 -depicts the analytical result for the Rouse dynamics. The solid curves represents the displacement for a macromolecule of length M = 25Me (% = 0.04) with corresponding (according to relations (3.25) and (3.29)) values of parameters B = 429 and ijj = 8.27 and the values of parameters of local anisotropy ae = 0 and ae = 0.3. For the isotropic situation (ae = 0), the curve can be calculated analytically according to equation (5.5), but for the parameter of local anisotropy ae = 0.3, the displacement ought to be calculated numerically. Internal resistance (parameters E and cq) does not affect mobility of macromolecular coil. Adapted from Pokrovskii (2006). 

We saw in Section 8-C that the strain 64 introduced four complications in the problem that were not present in the strain ei- The valence force field bypasses three of these, but leaves us with internal displacements. These are of interest in their own right and must be included if we wish to predict C44 in terms of the valence force field and parameters obtained from c, and c,2. That will be an interesting prediction since it gives some measure of the validity of the valence force field model, so let us proceed with it. 

Brock, C. P., Schweizer, W. B., and Dunitz, J. D. Internal molecular motion of triphenylphosphine oxide analysis of atomic displacement parameters for orthorhombic and monoclinic crystal modifications at 100 and 150 K. J. Amer. Chem. Soc. 107, 6964-6970 (1985). 

Anisotropic displacement parameters are a measure of the uncertainty of the atomic position and this arises fi om two sources the vibrational motion of the atoms (both internal and external vibrations, also called... 

Eventually we shall include all the vibrational contributions, both internal and external, and the resulting totai should then equal the anisotropic displacement parameters discussed above ( 2.4).)... 

The external displacements of benzene are much greater in the zeolite than in the crystal, hence the more intense external mode region. We may compare this value with that extracted from diffraction data taken on the same system [19]. The diffraction data were analysed under the constraints of the rigid molecule approximation, the reported anisotropic displacement parameter tensor must, therefore, include both internal and external modes... 

Once least-squares methods came into general use it became standard practice to refine not only atomic positional parameters but also the anisotropic thermal parameters or displacement parameters (ADPs), as they are now called [22]. These quantities are calculated routinely for thousands of crystal structures each year, but they do not always get the attention they merit. It is true that much of the ADP information is of poor quality, but it is also true that ADPs from reasonably careful routine analyses based on modem point-by-point or area diffractometer measurements can yield physically significant information about atomic motions in solids. We may tend to think of crystal structures as static, but in reality the molecules undergo translational and rotational vibrations about their equilibrium positions and orientations, as well as internal motions. Cruickshank taught us in 1956 how analysis of ADPs can yield information about the molecular rigid-body motion [23], and many improvements and modifications have been introduced since then. In particular, various computer programs are available to estimate the amplitudes of simple postulated types of internal molecular motion e.g., torsion-... 

From only two calculations of force and stress one can determine the optic mode frequency (the restoring force fo the internal displacements), the internal strain parameter f, and the elastic constant. This may be compared with a very arduous task of fitting multiple parameter curves if one has only the energy, as was done by Harmon, et al. ... 

The stress theorem determines the stress from the electronic ground state of any quantum system with arbitrary strains and atomic displacements. We derive this theorem in reciprocal space, within the local-density-functional approximation. The evaluation of stress, force and total energy permits, among other things, the determination of complete stress-strain relations including all microscopic internal strains. We describe results of ab-initio calculations for Si, Ge, and GaAs, giving the equilibrium lattice constant, all linear elastic constants Cy and the internal strain parameter t,. 

This program is for a thermal motion analysis of the anisotropic displacement parameters (ADPs), is carried out in terms of the LST rigid-body approximation, and considers the correlations in the internal and overall motions. THMA is rewritten from the 1992 version of THMA14. ... 

Table 6 Analysis of the Anisotropic Displacement Parameters (ADPs) in Terms of LST Rigid-body and Internal Motions...

ISO 14243-1. Implants for surgery—wear oftotal knee-joint prostheses —Part 1 loading and displacement parameters for wear-testing machines with load control and corresponding environmental conditions for test. International Organization for Standardization 2002. 

Build geometry, reference coordinate system atom 4 in origo, atom 7 on x-axis, atom 1 in xy--plane, do the transformation after minimisation, charge zero, symmetry number six, fifty steepest descents, no davidons, up to twenty newtons, do this, print normal coordinates in sorted internal displacements, with IR intensity (only when you have charge parameters), do thermodynamics for gas at BOOK, put in five torsions to start with,..., statistical weight one (trivial here, inportant when more than one conformer or isomer are present.). ... 

In order to simplify the expression for G, one has to employ a sufficiently simple model for the vibrational modes of the system. In the present case, the solvent contribution to the rate constant is expressed by a single parameter E, the solvent reorganization energy. In addition, frequency changes between the initial and final states are neglected and it is assumed that only a single internal mode with frequency co and with the displacement Ar is contributing to G. Thus the expression for G reduces to [124] ... 

A more complex example may be represented by TaSe2 its modification called 2H-TaSe2 is hexagonal (space group P6 mmc, with two formula units in the unit cell). This layered compound shows a displacive 2D modulation (defined by two vectors) its symmetry may be therefore described in terms of a supergroup in a 5D superspace (Janner and Janssen 1980). A general point is therefore denoted by the 5 parameters x,y,z, t, u, and a position vector by the five components xa + yb + zc + td + ue of the superspace, with a, b, c basis in the position space and d and e in the internal subspace . 

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