Lattice vibrations harmonic theory

One of the most valuable features of Raman spectroscopy is the well-known effect of local strain on the optical phonons (at q k. 0). The most basic approach to the theory of lattice vibrations assumes that interatomic forces in the crystal are linear functions of the interatomic displacement so that they obey a form of Hooke s Law. Under this harmonic approximation, the frequency m for mode j is given by ... 

Depending on the character of the molecular motions, one can distinguish several physical situations. In most cases, the molecules are trapped in relatively deep potential wells. Then, they perform small translational and orientational oscillations around well-defined equilibrium positions and orientations. Such motions are reasonably well described by the harmonic approximation. The collective vibrational excitations of the crystal, which are considered as a set of harmonic oscillators, are called phonons. Those phonons that represent pure angular oscillations, or libra-tions, are called librons. For some properties it turns out to be necessary to look at the effects of anharmonicities. Anharmonic corrections to the harmonic model can be made by perturbation theory or by the self-consistent phonon method. These methods, which are summarized in Section III under the name quasi-harmonic theories, can be considered to be the standard tools in lattice dynamics calculations, in addition to the harmonic model. They are only applicable in the case of fairly small amplitude motions. Only the simple harmonic approximation is widely used the calculation of anharmonic corrections is often hard in practice. For detailed descriptions of these methods, we refer the reader to the books and reviews by Maradudin et al. (1968, 1971, 1974), Cochran and Cowley (1967), Barron and Klein (1974), Birman (1974), Wallace (1972), and Cali-fano et al. (1981). 

A scheme as described here is indispensable for a quantum dynamical treatment of strongly delocalized systems, such as solid hydrogen (van Kranendonk, 1983) or the plastic phases of other molecular crystals. We have shown, however (Jansen et al., 1984), that it is also very suitable to treat the anharmonic librations in ordered phases. Moreover, the RPA method yields the exact result in the limit of a harmonic crystal Hamiltonian, which makes it appropriate to describe the weakly anharmonic translational vibrations, too. We have extended the theory (Briels et al., 1984) in order to include these translational motions, as well as the coupled rotational-translational lattice vibrations. In this section, we outline the general theory and present the relevant formulas for the coupled... 

To proceed further, three major approximations to the theory are made [44] First, that the transition operator can be written as a pairwise summation of elements where the index I denotes surface cells and k counts units of the basis within each cell second, that the element is independent of the vibrational displacement and, third, that the vibrations can all be treated within the harmonic approximation. These assumptions yield a form for w(kf, k ) which is equivalent to the use of the Bom approximation with a pairwise potential between the probe and the atoms of the surface, as above. However, implicit in these three approximations, and therefore also contained within the Bom approximation, is the physical constraint that the lattice vibrations do not distort the cell, which is probably tme only for long-wavelength and low-energy phonons. 

A. A. Maradudin, E. W. Montroll, G. H. Weiss, and I. P. Ipatova, Theory of Lattice Vibrations in the Harmonic Approximation (Academic Press, New York, 1971), Solid State Physics, Supplement 3. 

The theory of lattice vibrations which we discussed in the preceeding chapters has been based on the harmonic approximation which neglects all terms in the expansion of the potential energy (3.6) higher than the second-order terms. The most important consequences of the harmonic approximations are ... 

Einstein9 was the first to propose a theory for describing the heat capacity curve. He assumed that the atoms in the crystal were three-dimensional harmonic oscillators. That is, the motion of the atom at the lattice site could be resolved into harmonic oscillations, with the atom vibrating with a frequency in each of the three perpendicular directions. If this is so, then the energy in each direction is given by the harmonic oscillator term in Table 10.4... 

Vibrations in a real crystal are described by the lattice dynamical theory, discussed in section 2.1, rather than by the atomic oscillator model. Each harmonic phonon mode with branch index k and wavevector q then has, analogous to Eq. 

Maradudin AA, Montroll EW, Weiss GH, Ipatava P (1971) Theory of lattice dynamics in the harmonic approximation. Academic Press, New York March N (1992) Electro Density Theory of atoms and molecules. Academic Press Marcott C, Havel HA, Hedlund B, Overend J, Moscowitz A (1979) A vibrational rotational strength of extraordinary intensity. Azidomethemoglobin A. In Mason SF (cd) Optical activity and chiral discrimination, Reidel, Dordrecht, p 289... 

At normal temperatures the lattice dynamics involves predominantly low amplitude atomic motions that are well described in a harmonic approximation. Therefore, potential models widely used in the theory of molecular vibration, such as a generalized valence force field (GVFF) model, may be of use for such studies. In a GVFF the potential energy of a system is described with a set... 

For phonons such a residual interaction is anharmonicity, which is commonly ignored in the calculation of the frequencies and amplitudes of the normal vibrations of the crystal lattice. In this harmonic approximation, advanced at the very beginning of the development of present-day solid-state theory (1)—(3), the excited states of the lattice are associated with sets of various numbers of phonons of one kind or another. The energy, for example, of the excited state of a lattice with two phonons equals... 

The lattice component was calculated using the harmonic approximation, in which all the acoustic and low-frequency optical vibrations are included with the help of a single Debye fimction, while high-frequency crystal vibrations are taken into accoimt by Einstein s equation. According to Kelley s derivations (Gurvich et al., 1978-1984) based on the Born-von Karman d)mamic crystal lattice theory, we therefore have... 

Next step in molecular spectroscopy was the detailed understanding of the vibrational spectra with a meticulous assignment of the vibrational (or vibrorotational or ro-vibronic) transitions in terms of group theory, molecular and/or lattice dynamics, molecular harmonic or an-harmonic force fields, etc. [4-7]. These works aimed at the detailed understanding of the molecular dynamics but were limited to relatively small and highly symmetrical molecules. 

The theory of lattice specific heat was basically solved by Einstein, who introduced the idea of quantized oscillation of the atoms. He pointed out that, because of the quantization of energy, the law of equipartition must break down at low temperatures. Improvements have since been made on this model, but all still include the quantization of energy. Einstein treated the solid as a system of simple harmonic oscillators of the same frequency. He assumed each oscillator to be independent. This is not really the case, but the results, even with this assumption, were remarkably good. All the atoms are assumed to vibrate, owing to their thermal motions, with a frequency v, and according to the quantum theory each of the three degrees of freedom has an associated energy of which replaces the kT as postulated by... 

The background of the theory of the observed IR-spectroscopic effect in crystals can be summarized in the following way. If in the harmonic approximation an isolated impurity molecule could be characterized by two adjacent levels (p and q), then in an anhannonic condition we get a new pair of more remote levels (/and g). As experiments show [80], in vibrational spectra the value of FR for isolated molecules is often comparable with the value of the DS for the molecular lattice (about several tens of cm ). Therefore, in zero" approximation one can formulate expressions for the wave function of an excited impurity crystal. Using these expressions, Lisitsa and... 

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