Acoustic branch

The Debye model is more appropriate for the acoustic branches of tire elastic modes of a hanuonic solid. For molecular solids one has in addition optical branches in the elastic wave dispersion, and the Einstein model is more appropriate to describe the contribution to U and Cj from the optical branch. The above discussion for phonons is suitable for non-metallic solids. In metals, one has, in addition, the contribution from the electronic motion to Uand Cy. This is discussed later, in section (A2.2.5.6T... 

The composition factor for the acoustic branch of the NIS spectrum is derived from (9.10) by assuming (in the approximation of total decoupling of inter- and intramolecular vibrations) that the msd in acoustic modes are identical for all the atoms in the molecular crystal ... 

In the molecular approximation used in (14) only the L = 3W — 6 (W is the number of atoms) discrete intramolecular vibrations of the molecular complex in vacuo are considered. In general these vibrations correspond to the L highest optical branches of the phonon spectrum. The intermolecular vibrations, which correspond to the three acoustical branches and to the three lowest optical branches are disregarded, i.e., the center of mass and - in case of small amplitudes - the inertial tensor of the complex are assumed to be fixed in space... 

The solution for the acoustic branch approaches zero as q goes to zero, and for small q ... 

Kieffer has estimated the heat capacity of a large number of minerals from readily available data [8], The model, which may be used for many kinds of materials, consists of three parts. There are three acoustic branches whose maximum cut-off frequencies are determined from speed of sound data or from elastic constants. The corresponding heat capacity contributions are calculated using a modified Debye model where dispersion is taken into account. High-frequency optic modes are determined from specific localized internal vibrations (Si-O, C-0 and O-H stretches in different groups of atoms) as observed by IR and Raman spectroscopy. The heat capacity contributions are here calculated using the Einstein model. The remaining modes are ascribed to an optic continuum, where the density of states is constant in an interval from vl to vp and where the frequency limits Vy and Vp are estimated from Raman and IR spectra. 

The motion of atoms in the lattice can be depicted as a wave propagation (phonon). By dispersion we mean the variation in the wave frequency as reciprocal space is traversed. The propagation of sound waves is similar to the translation of all atoms of the unit cell in the same direction hence the set of translational modes is commonly defined as an acoustic branch. The remaining vibrational modes are defined as optical branches, because they are capable of interaction with light (see McMillan, 1985, and Tossell and Vaughan, 1992, for more exhaustive explanations). 

Figure 3J Schematic representation of a vibrational spectrum of a crystalline phase. Dotted curves acoustic branches and optical continuum. Solid line total spectrum, and 0) 2 Einstein oscillators. Reprinted with permission from Kieffer (1979c), Review of Geophysics and Space Physics, 17, 35-39, copyright 1979 by the American Geophysical Union.

The nondimensionalized frequencies are related to linear and angular frequencies by equation 3.36. The conversion factor from linear frequencies in cm to undimension-alized frequencies is chik = 1.4387864 cm (where c is the speed of hght in vacuum). Acoustic branches for the various phases of interest may be derived from acoustic velocities through the guidelines outlined by Kieffer (1980). Vibrational modes at higher frequency may be derived by infrared (IR) and Raman spectra. Note incidentally that the tabulated values of the dispersed sine function in Kieffer (1979c) are 3 times the real ones (i.e., the listed values must be divided by 3 to obtain the appropriate value for each acoustic branch see also Kieffer, 1985). 

The bias observed between experimental measurements and Kieffer s model predictions is due to the relative paucity of experimental data concerning cutoff frequencies of acoustic branches, and also to the assumption that the frequencies of the lower optical branches are constant with K and equivalent to those detected by Raman and IR spectra (corresponding only to vibrational modes at K = 0). Indeed, several of these vibrational modes, and often the most important ones, are inactive under Raman and IR radiation (Gramaccioli, personal communication). The limits of the Kieffer model and other hybrid models with respect to nonempirical computational procedures based on the equation of motion of the Born-Von Karman approach have been discussed by Ghose et al. (1992). 

The Debye model assumes that there is a single acoustic branch, the frequency of which increases with constant slope (proportional to the average velocity of sound in the crystal) as q increases, up to the boundary of the Brillouin zone. The boundary is assumed to be of spherical shape, with a radius qD determined by the total number of normal modes of the crystal. Thus,... 

Fig. 2. Dispersion curves of the optical and acoustical branch in BZ 1 for the linear AB chain...

According to Eq. (II.7), co = 0 for k = 0 in the center of BZ 1. With these values Eqs. (II. 1)—(II.3) lead to the relation UA = UB. This means that both sets of atoms vibrate with the same amplitude and in phase (because they have the same sign). A translation of the whole chain results which corresponds to an acoustical wave with X = °°. This is called a longitudinal acoustical branch (LA). 

The regularity of the array of the atoms in ideal crystals permits the subdivision of the crystal space into equal and equally oriented regions of space, the so-called elementary cells. Each cell contains the same complex of atoms with the same orientation in space. Such an elementary cell may contain s atoms. Each atom has three degrees of freedom, one in each of the directions of the three coordinate axes. Therefore a space lattice has 3 s eigenfrequencies or modes. Of these 3 s modes for k = 0 i.e. in the center of BZ 1) three correspond to the translations into the directions of the coordinate axes. These have the frequency to = 0, which corresponds to the frequencies of the acoustical branches according to Eq. (II.7) for k = 0. The 3 s eigenfrequencies of a crystal with s atoms in the elementary cell correspond to 3 s - 3 optical and 3 acoustical branches. 

Charged point defects on regular lattice positions can also contribute to additional losses the translation invariance, which forbids the interaction of electromagnetic waves with acoustic phonons, is perturbed due to charged defects at random positions. Such single-phonon processes are much more effective than the two- or three phonon processes discussed before, because the energy of the acoustic branches goes to zero at the T point of the Brillouin zone. Until now, only a classical approach to account for these losses exists, which has been... 

Figure 1-38 (a) Dispersion relation for the optical and acoustic branches in solids, (b) Wave motion in an infinite diatomic lattice. (Reproduced with permission from Ref. 49.)... 

Phonons At least two phonon branches are involved in the observed absorption the acoustic phonons and the optical 46-cm "1 branch. Our model includes a single acoustic branch [with cutoff frequency f2max, and isotropic Debye dispersion hfiac q) = hQmaxq/qmax] and an optical dispersionless branch (Einstein s model, with frequency /20p). 

Brillouin scattering provides information about the acoustic branches of the dispersion curves of the material under study. The measured frequency shift of the radiation is equal to that of the phonon under consideration (EQN (1)), and its wave vector is deduced from EQN (2), so the sound velocity may be calculated by ... 

Indium nitride has twelve phonon modes at the zone centre (symmetry group Cev), three acoustic and nine optical with the acoustic branches essentially zero at k = 0. The infrared active modes are Ei(LO), Ei(TO), Ai(LO) and Ai(TO). A transverse optical mode has been identified at 478 cm 1 (59.3 meV) by reflectance [6] and 460 cm 1 (57.1 meV) by transmission [24], In both reports the location of a longitudinal optical mode is inferred from the Brout sum rule, giving respective values of 694 cm 1 (86.1 meV) and 719 cm 1 (89.2 meV). Raman scattering of single crystalline wurtzite InN reveals Ai(LO) and E22 peaks at 596 cm 1 and at 495 cm 1 respectively [25],... 

One can show also [10] that the density of states for the plasmon band is peaked at the upper (qz = 0) and the lower (qz = n/dc) branches. As a result, the plasmon band can be modeled as a set of two branches (see Fig. 1). The upper branch is similar to that in the 3D case. The most important factor is the existence of low acoustic branches ( electronic sound). 

The acoustic branch can contribute to the pairing in a way similar to usual acoustic phonons. Theoretically the pure acoustic plasmon mechanism was studied in [11]. Such a mechanism in conjunction with phonon contribution (phonon-plasmon mechanism) was proposed in [9], The equation for the order... 

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