Harmonic crystal, vibrational energy

The statistical treatment of the vibrational degrees of freedom of crystals is far more difficult compared to gases. Let us initially consider a monoatomic crystal. An atom in a crystal vibrates about its equilibrium lattice position. In the simplest approach, three non-interacting superimposed linear harmonic oscillators represent the vibrations of each atom. The total energy, given by the sum of the kinetic and potential energies for the harmonic oscillators, is... 

In the perfectly harmonic crystal the total vibrational energy equals the sum of the energies of the simple harmonic oscillators or normal modes which have infinitesimal amplitude and ignore each other. The introduction of anharmonic coupling between oscillators leads initially to small shifts in... 

Kelvin (the zero point motion). This latter effect is explained by quantum mechanics, and it can in turn explain absorption features of impurities in crystalline matrices. The presentation of the fundamental vibrational modes of crystals is based on the harmonic approximation, where one only considers the interactions between an atom or an ion and its nearest neighbours. Within this approximation, an harmonic crystal made of N ions can be considered as a set of 3N independent oscillators, and their contribution to the total energy of a particular normal mode with pulsation ivs (q) is ... 

Microscopically, to understand the concept of thermal entropy, or heat capacity for that matter, one needs to appreciate that the vibrational energy levels of atoms in a crystal are quantized. If the atoms are assumed to behave as simple harmonic oscillators, i.e., miniature springs, it can be shown that their energy will be quantized with a spacing between energy levels given by... 

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... 

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 ... 

This model assumes that all of the vibrational normal modes act like harmonic oscillators with the same frequency. It is used to represent a crystal of a monatomic substance such as a solidified inert gas, a metal, or a network covalent crystal such as diamond. If a monatomic crystal has N atoms its vibrational energy is given by... 

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... 

It has already been stated that, theoretically, A atoms in a crystal have 3 A possible vibrational modes. Obviously, if we knew the energy associated with each vibrational mode at all T and could sum the energy terms in the manner discussed in section 3.1, we could define the internal energy of the crystal as a function of T, Cy could then be obtained by application of equation 3.27, and (harmonic) entropy could also be derived by integration of Cy in dTiT. 

In the harmonic approximation the potential energy of a crystal in which the atoms are vibrating about their equilibrium positions differs from 0, the potential energy with each atom on its equilibrium site, by... 

Considering nonlinear effects in the vibrations of a crystal lattice (see, for example, [8]) it is necessary to take into account anharmonicity only in the terms connected to the largest interatomic forces, while the potential energy of weak forces of interlayer (or interchain) interactions, as well as noncentral forces should be considered in the harmonic approach. Therefore in (1) it is possible to neglect the summands, containing correlators of the atom displacements from various layers or chains, i.e. the correlators of... 

When the harmonic approximation is dropped, potential-energy terms in r3, r4,... couple the phonon modes. These terms are responsible for processes such as (at low temperatures) phonon fission or vibration fission into phonons. To take account of the variation of the frequencies and of the equilibrium positions with temperature, the phenomenological quasi-harmonic approach is often used, in which the eigenfrequencies Qks are functions of the crystal volume.42... 

Such a potential energy function gives rise to the famihar parabolic curve (Figure 22) where the curvature of the function is related to the force constant. The success of this simple harmonic model in treating surface atom vibrations lies in the relatively small displacement of surface atoms during a period of vibration. For some crystal properties, such as thermal expansion at elevated temperature, anharmoitic contributions to the potential must be included for an accurate description. 

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. 

A lot of theoretical work on displacive phase transitions has focussed on a simple model in which atoms are connected by harmonic forces to their nearest neighbors, and each neighbor also sees the effect of the rest of the crystal by vibrating independently in a local potential energy well (Bruce and Cowley 1980). For a phase transition to occur, this double well must have two minima, and can be described by the following function ... 

In describing the normal modes of a protein, it is instructive to compare them conceptually with those of a simple model of a polymer, such as a chain of atoms, both periodic and aperiodic. In a harmonic periodic chain, the normal modes carry energy without resistance from one end of the ID crystal to the other. On the other hand, the vast majority of normal modes of an aperiodic chain are spatially localized [138]. Protein molecules, which are of course not periodic, can be better characterized as an aperiodic chain of atoms, and most normal modes of proteins are likewise localized in space [111,112,126-128]. If a normal mode a is exponentially localized, then the vibrational amplitude of atoms in mode a decays from the center of excitation, Ro, as... 

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