Lattice vibrations harmonic approximation

Some of lowest-order diagrams for the temperature GF (A3.6) are shown in Fig. A3.1. The dashed and the solid lines represent the GFs of high-frequency and low-frequency vibrations of a planar lattice in the harmonic approximation ... 

Fig. A3.1. Some lowest-order diagrams for the temperature GF (A3.6). The dashed and solid lines correspond to the GF for high-frequency and resonance low-frequency vibrations of a molecular planar lattice in the harmonic approximation (see Eq. (A3.9) and (A3.10)). Each vertex is associated with the factor -y/N, the integration and summation being performed over each vertex coordinates r, from 0 to / , and over all internal wave vectors K. At ptiClK 1, the main contribution is provided by a-type diagrams.184...

A first impression of collective lattice vibrations in a crystal is obtained by considering one-dimensional chains of atoms. Let us first consider a chain with only one type of atom. The interaction between the atoms is represented by a harmonic force with force constant K. A schematic representation is displayed in Figure 8.4. The average interatomic distance at equilibrium is a, and the equilibrium rest position of atom n is thus un =na. The motion of the chain of atoms is described by the time-dependent displacement of the atoms, un(t), relative to their rest positions. We assume that each atom only feels the force from its two neighbours. The resultant restoring force (F) acting on the nth atom of the one dimensional chain is now in the harmonic approximation... 

The heat capacity models described so far were all based on a harmonic oscillator approximation. This implies that the volume of the simple crystals considered does not vary with temperature and Cy m is derived as a function of temperature for a crystal having a fixed volume. Anharmonic lattice vibrations give rise to a finite isobaric thermal expansivity. These vibrations contribute both directly and indirectly to the total heat capacity directly since the anharmonic vibrations themselves contribute, and indirectly since the volume of a real crystal increases with increasing temperature, changing all frequencies. The constant volume heat capacity derived from experimental heat capacity data is different from that for a fixed volume. The difference in heat capacity at constant volume for a crystal that is allowed to relax at each temperature and the heat capacity at constant volume for a crystal where the volume is fixed to correspond to that at the Debye temperature represents a considerable part of Cp m - Cv m. This is shown for Mo and W [6] in Figure 8.15. 

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... 

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 ... 

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... 

In the simplest adiabatic case with an orbital singlet term, potential energy of the crystal lattice is parabolic with one minimum point. At low temperatures, vibrations of the lattice are localized at the bottom of this well, and as a rule, the so-called harmonic approximation applies. This corresponds to the so-called polaron effect and brings us to the concept of electrons coated with phonons. 

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). 

Theory of Lattice Dynamics in the Harmonic Approximation by A. A. Maradudin, E. W. Montroll, G. H. Weiss and I. P. Ipatova, Academic Press, New York New York, 1971. A formidable volume with not only a full accounting for the fundamentals of vibrations in solids, but also including issues such as the effects of defects on vibrations in solids. 

The Contribution of Harmonic Lattice Vibrations to the Free Energy. Recall the treatment of lattice vibrations in chap. 5. In that discussion, we noted that in the harmonic approximation, the Hamiltonian for harmonic vibrations of a crystalline... 

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 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). 

Phonons are normal modes of vibration of a low-temperature solid, where the atomic motions around the equilibrium lattice can be approximated by harmonic vibrations. The coupled atomic vibrations can be diagonalized into uncoupled normal modes (phonons) if a harmonic approximation is made. In the simplest analysis of the contribution of phonons to the average internal energy and heat capacity one makes two assumptions (i) the frequency of an elastic wave is independent of the strain amplitude and (ii) the velocities of all elastic waves are equal and independent of the frequency, direction of propagation and the direction of polarization. These two assumptions are used below for all the modes and leads to the famous Debye model. 

In the quantum mechanical treatment of this model, the equations of motion in the harmonic approximation become analogous to those for electromagnetic waves in space [2-4]. Thus, each wave is associated with a quantum of vibrational energy hu and a crystal momentum hq. By analogy to the photon for the electromagnetic quantum, the lattice vibrational quantum is called a phonon. The amplitude of the wave reflects the phonon population in the vibrational mode (i.e., the mode with frequency co and... 

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. 

Typical lattice vibrations involve small atomic excursions of the order of 10 pm or smaller, thus we may expand the expression for potential energy 0 into the Taylor series about the equilibrium position of the ions. We restrict ourselves by the powers of displacements not exceeding the second one. This is the so-called harmonic approximation. We obtain... 

The free energy due to harmonic lattice vibrations (or equivalently the Debye temperature) is approximately the same for bcc, fee, and hep structures but with a significant tendency for the bcc value to be a few percent lower. The more open bcc structure has a transverse phonon mode with a particularly low frequency which causes a more rapid decrease in the free energy with temperature. On cooling, sodium and lithium transform partially from bcc to hep at very low temperatures (0.1-0.2 Tm). Calcium, strontium, beryllium, and thallium transform to a bcc phase at high temperatures (0.66-0.98 Tm) when there is a considerable anharmonic contribution to the free energy. 

Computation of vibrational frequencies for crystalline phases can be carried out with various methods. Perhaps the most common is the to use the quasi-harmonic approximation in lattice dynamics calculations (see Parker, this volume). Some excellent examples of this type of study are Cohen et al. 1987, Hemley et al. (1989), Wolf and Bukowinski (1987), and Chaplin et al. (1998). In general, however, such calculations serve as a validation of the modeling technique rather than as a method to interpret frequencies. Vibrational modes in crystalline solids are readily assigned because the structure is known from X-ray diffraction studies. In fact, isochemical crystalline solids are used frequently to help interpret spectra of glasses (e.g., McMillan 1984). 

In Section II of this review we discuss the different forms of classical lattice dynamical treatments which have been applied to molecular solids. The applications to specific systems and comparison of results with experiment will then be taken up. In Section III we give a short treatment of quantum lattice dynamics, which has been developed to deal with quantum solids as helium and hydrogen. Classical approaches in the harmonic approximation fail for these systems. In Section IV, intensities of infrared and Raman spectra in the lattice vibration region are discussed. A group theoretical appendix has been added for the reader who is not familiar with this aspect. 

Another technique to obtain the effects of the anharmonic terms on the excitation frequencies and the properties of molecular crystals is the Self-Consistent Phonon (SCP) method [71]. This method is based on the thermodynamic variation principle, Eq. (14), for the exact Hamiltonian given in Eq. (10), with the internal coordinates not explicitly considered. As the approximate Hamiltonian one takes the harmonic Hamiltonian of Eq. (18). The force constants in Eq. (18) are not calculated at the equilibrium positions and orientations of the molecules as in Eq. (19), however. Instead, they are considered as variational parameters, to be optimized by minimization of the Helmholtz free energy according to Eq. (14). The optimized force constants are found to be the thermodynamic (and thus temperature dependent) averages of the second derivatives of the potential over the (harmonic) lattice vibrations ... 

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