Atomic displacements related

It i usually convenient to work with a set of internal displacement coordinates, 5, as they have chemical significance. In the limit of small amplitudes of atomic displacements, the two sets of coordinates are linearly related. Thus,... 

Most spectroscopic properties are related to second derivatives of the total energy. As a simple illustrative example, vibrational modes, which arise from the harmonic oscillations of atoms around their equilibrium positions, are characterized by the quadratic variation of the total energy as a function of the atomic displacements SRy... 

As a first approximation, we may represent the stress-displacement relation by a sine function (Figure 5.10b), since when atom A in Figure 5.10a is displaced to the point where it is directly over atom B in the plane below, an unstable equilibrium exists and... 

Macroscopic treatments of diffusion result in continuum equations for the fluxes of particles and the evolution of their concentration fields. The continuum models involve the diffusivity, D, which is a kinetic factor related to the diffusive motion of the particles. In this chapter, the microscopic physics of this motion is treated and atomistic models are developed. The displacement of a particular particle can be modeled as the result of a series of thermally activated discrete movements (or jumps) between neighboring positions of local minimum energy. The rate at which each jump occurs depends on the vibration rate of the particle in its minimum-energy position and the excitation energy required for the jump. The average of such displacements over many particles over a period of time is related to the macroscopic diffusivity. Analyses of random walks produce relationships between individual atomic displacements and macroscopic diffusivity. 

The atomic displacement is related to the normal coordinate Q through... 

In VFF the molecular vibrations are considered in terms of internal coordinates qs (s = 1..3N — 6, where N is the number of atoms), which describe the deformation of the molecule with respect to its equilibrium geometry. The advantage of using internal coordinates instead of Cartesian displacements is that the translational and rotational motions of the molecule are excluded explicitly from the very beginning of the vibrational analysis. The set of internal coordinates q = qs is related to the set of Cartesian atomic displacements x = Wi by means of the Wilson s B-matrix [1] q = Bx. In the harmonic approximation the B-matrix depends only on the equilibrium geometry of the molecule. 

In the description of the technique, the particular aspects that make it different of other schemes aimed at the computation of IFCs in solids or molecules [1-9] will be emphasized. These aspects are connected to the central use of a variational principle in order to find the changes in the wavefunctions due to atomic displacements, on one hand, and to find the change in electronic energy due to the changes in wavefunctions, on the other hand. Some technical details, related with the presence of relatively long-ranged interatomic force constants, caused by... 

In reaction (d) the transition state probably resembles the reactants, and we may assume that the relative reactivity of the two nucleophilic atoms is related to the ratio of their charge densities in the ambident ion (5o/qp). Thus for a displacement reaction in general,... 

A variety of defect formation mechanisms (lattice disorder) are known. Classical cases include the - Schottky and -> Frenkel mechanisms. For the Schottky defects, an anion vacancy and a cation vacancy are formed in an ionic crystal due to replacing two atoms at the surface. The Frenkel defect involves one atom displaced from its lattice site into an interstitial position, which is normally empty. The Schottky and Frenkel defects are both stoichiometric, i.e., can be formed without a change in the crystal composition. The structural disorder, characteristic of -> superionics (fast -> ion conductors), relates to crystals where the stoichiometric number of mobile ions is significantly lower than the number of positions available for these ions. Examples of structurally disordered solids are -> f-alumina, -> NASICON, and d-phase of - bismuth oxide. The antistructural disorder, typical for - intermetallic and essentially covalent phases, appears due to mixing of atoms between their regular sites. In many cases important for practice, the defects are formed to compensate charge of dopant ions due to the crystal electroneutrality rule (doping-induced disorder) (see also -> electroneutrality condition). 

Within the dimer, each monomeric bis(dithiolene) unit retains similar values for bond distances and angles when compared to related, nondimeric species. However, metal atoms are drawn out of the monomeric unit plane by 0.1-0.45 A and the coordination geometry about the central atom is best described as square pyramidal. A consequence of metal atom displacement is that all c — M—c angles (Scheme 3) in dimeric structures are smaller than those for monomeric... 

Full support for that hypothesis came with information concerning the position of atoms in enzymes, their complexes with substrates and inhibitors, and the values of so called B-factors that relate to the average amplitude of atoms displacement. Structural investigations with the use of such physical methods as NMR and Raman resonance spectroscopy, theoretical calculations, in particular, also produce evidence in favour of this concept. 

Another factor to be taken into account is the degree of over determination, or the ratio between the number of observations and the number of variable parameters in the least-squares problem. The number of observations depends on many factors, such as the X-ray wavelength, crystal quality and size, X-ray flux, temperature and experimental details like counting time, crystal alignment and detector characteristics. The number of parameters is likewise not fixed by the size of the asymmetric unit only and can be manipulated in many ways, like adding parameters to describe complicated modes of atomic displacements from their equilibrium positions. Estimated standard deviations on derived bond parameters are obtained from the least-squares covariance matrix as a measure of internal consistency. These quantities do not relate to the absolute values of bond lengths or angles since no physical factors feature in their derivation. 

The intensity variation along the rod (i.e. as a function of or /) is solely contained in the structure factor it is thus related to the z-co-ordinates of the atoms within the unit-cell of this quasi-two dimensional crystal. In general, the rod modulation period gives the thickness of the distorted layer and the modulation amplitude is related to the magnitude of the normal atomic displacements. This is the case of a reconstructed surface, for which rods are found for fractional order values of h and k, i.e. outside scattering from the bulk. 

Of central importance for understanding the fundamental properties of ferroelec-trics is dynamics of the crystal lattice, which is closely related to the phenomenon of ferroelectricity [1]. The soft-mode theory of displacive ferroelectrics [65] has established the relationship between the polar optical vibrational modes and the spontaneous polarization. The lowest-frequency transverse optical phonon, called the soft mode, involves the same atomic displacements as those responsible for the appearance of spontaneous polarization, and the soft mode instability at Curie temperature causes the ferroelectric phase transition. The soft-mode behavior is also related to such properties of ferroelectric materials as high dielectric constant, large piezoelectric coefficients, and dielectric nonlinearity, which are extremely important for technological applications. The Lyddane-Sachs-Teller (LST) relation connects the macroscopic dielectric constants of a material with its microscopic properties - optical phonon frequencies ... 

Before carrying out the energy diffusion calculations on the water cluster, it is useful to determine the speed of sound in glassy water, which we compute from the dispersion relation for the water cluster. To compute a dispersion relation, we need to assign a wave number, k, to a normal mode of frequency, go. We have obtained dispersion relations for proteins [111] via computation of the correlation function for the direction of atomic displacements as a function of distance for individual normal modes of the protein, a function that was studied in earlier work by Nishikawa and Go [112], Computation of the correlation function allows us to match a wave number of a plane wave to a normal mode... 

Inelastic neutron scattering spectroscopy is characterized by completely different intensities because the neutron scattering process is entirely attributable to nuclear interactions [110] Each atom features its nuclear cross section, which is independent of its chemical bonding. Then the intensity for any transition is simply related to the atomic displacements scaled by scattering cross sections. And because the cross section of the proton is about one order of magnitude greater than that for any other atom, the method is able to record details of quantum dynamics of proton transfer. 

We may have defined Equation (44) in a slightly different manner as is usual in the literature. Instead of writing R in the argument of the exponential function, one can write R/ = R/ + [similarly to f,- = x vectors in Equation (29) for atom displacements]. In such a case the Fourier coefficients, Tiy-, of the new expression are related to those of Equation (44) by a phase factor, Sk/ = Tiyexp(—27rikxy), that depends on the atom positions inside the unit cell. We shall see that the convention we have adopted is more convenient for a unified description of commensurate and incommensurate magnetic structures. 

A crystal structure is described by a collection of parameters that give the arrangement of the atoms, their motions and the probability that each atom occupies a given location. These parameters are the atomic fractional coordinates, atomic displacement or thermal parameters, and occupancy factors. A scale factor then relates the calculated structure factors to the observed values. This is the suite of parameters usually encountered in a single crystal structure refinement. In the case of a Rietveld refinement an additional set of parameters describes the powder diffraction profile via lattice parameters, profile parameters and background coefficients. Occasionally other parameters are used these describe preferred orientation or texture, absorption and other effects. These parameters may be directly related to other parameters via space group symmetry or by relations that are presumed to hold by the experimenter. These relations can be described in the refinement as constraints and as they relate the shifts, Ap,-, in the parameters, they can be represented by... 

While the structure and force field uniquely determine the vibrational frequencies of the molecule, the structure cannot in general be obtained directly from the spectrum. However, to a useful approximation, the atomic displacements in many of the vibrational modes of a large molecule are concentrated in the motions of atoms in small chemical groups, and these localized modes are to a good approximation transferable between molecules. Therefore, in the early studies of peptides and proteins (Sutherland, 1952), efforts were directed mainly to the identification of such characteristic frequencies and the determination of their relation to the structure of the molecule. This kind of analysis depended on empirical correlations of the spectra of chemically similar molecules. 

In the binding of the metal ion to the carbohydrate moiety, charge also appears to play a role, and this is related to a difference in solvation enthalpy in polar solvents. Oxygen donor atoms displace with more difficulty the solvation shell for a trivalent ion in a polar solvent than for a divalent ion. This difference is removed when the study is performed in a nonpolar solvent. Consequently, it would appear that, for studies of carbohydrates in aqueous solution, the charge on the molecule must be considered. 

This relation, established in 1963 by Krivoglaz [KRI 63, KRI 69], is the fundamental equation for describing the expression of the total intensity racted by a crystal containing a concentration c of dislocations. As you can see, this intensity corresponds to the one diffracted by a ciystal free of any dislocations multiplied by a factor (e ) smaller than 1 and that decreases when T increases, and hence when the dislocation density increases (see equation [5.29]). This generic form of the effect of dislocations on the diffracted intensity is similar to the one describing the effect of temperature, which actually corresponds to variations in atomic mobility and therefore to a certain form of atom displacements with respect to their reference position. 

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