Lattice-vibrational frequencies

The Bragg scattering of X-rays by a periodic lattice in contrast to a Mossbauer transition is a collective event which is short in time as compared to the typical lattice vibration frequencies. Therefore, the mean-square displacement (x ) in the Debye-Waller factor is obtained from the average over the ensemble, whereas (r4) in the Lamb-Mossbauer factor describes a time average. The results are equivalent. 

All solid state ionic conduction proceeds by means of hops between well-defined lattice sites. Ions spend most of their time on specific sites where their only movement is that of small oscillations at lattice vibrational frequencies (10 Hz). Occasionally, ions can hop into adjacent... 

Notions of high - and low -frequency limiting behavior depend on one s point of view, and the notation reflects this what is low frequency to an ultraviolet spectroscopist may be high frequency to an infrared spectroscopist. For insulating solids the value of c in the near infrared is often denoted as c0 by ultraviolet spectroscopists it refers to frequencies low compared with certain oscillators—electrons in this example—which may, however, be high compared with lattice vibrational frequencies. Consequently, this same limiting value is denoted as by infrared workers. 

Implicit in (9.40) is the assumption that co is small compared with lattice vibrational frequencies. The susceptibility in the frequency region where Debye relaxation is the dominant mode of polarization is therefore... 

The frequency factor, v, should be on the order of the lattice vibrational frequency, or 1013 s-1. The extreme simplicity of the model makes it very convenient for many applications. This theoretical model is now generally referred to as the image-hump model, or the Schottky-hump model, of field desorption. The potential energy curve of this model is not defined at all distances because of the crude nature of the argument it is nevertheless shown schematically in Fig. 2.9(a).48... 

Similar methods have been used to integrate thermodynamic properties of harmonic lattice vibrations over the spectral density of lattice vibration frequencies.21,34 Very accurate error bounds are obtained for properties like the heat capacity,34 using just the moments of the lattice vibrational frequency spectrum.35 These moments are known35 in terms of the force constants and masses and lattice type, so that one need not actually solve the lattice equations of motion to obtain thermodynamic properties of the lattice. In this way, one can avoid the usual stochastic method36 in lattice dynamics, which solves a random sample of the (factored) secular determinants for the lattice vibration frequencies. Figure 3 gives a typical set of error bounds to the heat capacity of a lattice, derived from moments of the spectrum of lattice vibrations.34 Useful error bounds are obtained... 

If the free-atom recoil energy is much greater than the characteristic energy for phonon excitation ha>h where cor is the associated lattice vibration frequency, then phonon creation represents another mode of energy loss, which destroys resonance (29, 30, 32). For R° less than or of the order of tuo, a significant fraction of the nuclear events (emission and absorption)... 

Therefore, diffusivity is basically the product of the lattice vibration frequency, vacancy concentration, and activated-lattice concentration (equation 26). 

Figure 3-4 (a) Raman spectra of the symmetric (vj) and the anti-symmetric (V3) stretching modes in solid H2S at various pressures. The phase transition occurs at about 11 GPa. (b) Pressure dependence of the intramolecular and the lattice vibrational frequencies in solid H2S at 300 K. (Reproduced with permission from Ref. 12.)... 

Starr, T. L. and Williams, D. E. (1977 ). Coulombic nonbonded interatomic potential functions derived from crystal-lattice vibrational frequencies in hydrocarbons. Acta Crystallogr A, 33, 771-6. [153]... 

Another experimental result which might elucidate directly the mechanism of superconductivity is the shift of transition temperature Tc with isotopic substitution. This phenomenon had provided the first suggestion for the role of phonons in superconductivity well before the birth of the BCS theory. If the mass M of all of the atoms in a sample is increased, the lattice vibration frequencies will all decrease as w Looking back to Eq. (1),... 

Cambier. P. (1986) Infrared study of goethite of varying crystallinity and particle size. I Interpretation of OH and lattice vibration frequencies. Clay Min. 21 191 -200. 

T. L. Starr and D. E. Williams, Acta Crystallogr. Ser. A, A106, 771 (1977). Coulombic Non-bonded Interatomic Potential Functions Derived from Crystal-Lattice Vibrational Frequencies in Hydrocarbons. 

Similarly to electronic polarization, other polarization mechanisms can be invoked molecular polarization, which concerns the displacement of the nuclei of the molecule where the charge resides, and lattice polarization, which involves movements of the entire lattice. The energies and times corresponding to these processes are estimated from the intramolecular and lattice vibration frequencies. The energy and time of the various polarization processes are summarized in Table 2.2.1. 

The observables which are most sensitive to the choice of adjustable parameters are lattice vibration frequencies, but experimental values are sparse their calculation requires a comparatively large effort and involves approximations whose validity cannot always be taken for granted (e.g. the harmonic approximation). 

By resonance transfer, we mean two radiators which are radiating at the same frequency. This is quite different from the example given above for radiative exchange. Note that we are not speaking of lattice vibrational frequencies but of radiation frequencies. We thus speak of two "coupled" oscillators. To further illustrate this concept, suppose we take two dipoles, there wUl be an electric moment, E 6), which is a vector product whose strength is a function of the angle, 6, between the vector moments of the dipoles ... 

Other observables also depend on the total energy derivatives, in particular on second-order derivatives, such as the bulk modulus, the elastic constants, and lattice vibration frequencies. 

Since the internal molecular frequencies are so much higher than the lattice vibrational frequencies, the two classes of motion are mostly separable. It is then possible to discuss the lattice vibrations separately, assuming the molecule to be a rigid body. The motions of these molecular units can be divided into two types translatory displacements of the... 

More recently, interest in the lattice vibrations of molecular solids has centered around the elucidation of intermolecular potential functions. If a pair potential is assumed, it can be tested by calculating the observables by application of the appropriate lattice dynamics. Dows (1962) was the first to attempt a calculation of lattice vibrational frequencies from an assumed potential. He treated solid ethylene and used a model which represented the pair interaction by repulsions between hydrogen atoms on neighboring molecules. 

Lattice Energy, Lattice Vibrational Frequencies, and Intensity Ratios for a-Nj [Frequencies are in units of cnr with half-intensity widths in parentheses. Calculated values are listed for two potential models (I) Lennard-Jones potential plus quadrupole-quadrupole term (Ron and Schnepp, 1967 Walmsley and Pople, 1964). Nearest neighbor interactions only (11) Diatomic Potential (Kuan, Warshel, and Schnepp, 1970) with 6-12 terms Raman assignments as in Cahill and Leroi (1969) and Brith, Ron, and Schnepp (1969).]... 

We take into account only the static electron polarizability q (0) inside one elementary unit. This simplification is justified because the studied lattice vibration frequencies (i.e. phonon frequencies) are small with respect to the Eigenfirequencies of the election polarizatioa The total polarization is given by the sum... 

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