Quasi-harmonic lattice dynamics

For the description of the temperature and stress-related behavior of the crystal we used the method of consistent quasi-harmonic lattice dynamics (CLD), which permits the determination of the equilibrium crystal structure of minimum free energy. The techniques of lattice dynamics are well developed, and an explanation of CLD and its application to the calculation of the minimum free-energy crystal structure and properties of poly(ethylene) has already been presented. ... 

III. Harmonic and Quasi-harmonic Theories of Lattice Dynamics. 149... 

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). 

Since it became clear from various observations that the librational motions of the molecules, even in the ordered a and y phases of nitrogen at low temperature, have too large amplitudes to be described correctly by (quasi-) harmonic models, we have resorted to the alternative lattice dynamics theories that were described in Section IV. Most of these theories have been developed for large-amplitude rotational oscillations, hindered or even free rotations, and remain valid when the molecular orientations become more and more localized. 

Lattice dynamics calculations on the plastic /3-nitrogen phase are relatively scarce because, obviously, the standard (quasi-) harmonic theory cannot be applied to this phase. Classical Monte Carlo calculations have been made by Gibbons and Klein (1974) and Mandell (1974) on a face-centered cubic (a-nitrogen) lattice of 108 N2 molecules, while Mandell has also studied a 32-molecule system and a system of 96 N2 molecules on a hexagonal close-packed (/3-nitrogen) lattice. Gibbons and Klein used 12-6... 

This simplified equilibration process holds strictly for a cubic crystal in a general non-cubic case, the volume, static pressure and bulk modulus have to be replaced by the strain components (related to unit-cell parameters), stress components and elastic constants, respectively. The computer program PARAPOCS[17] performs the lattice-dynamical, thermodynamical and quasi-harmonic calculations in the general tensorial formalism, and has been used to obtain all results reported below for calcite and aragonite. 

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). 

At low temperatures, if most of the anharmonic effects are due to lattice expansion, the quasi-harmonic approximation can be successfully applied. However, if the average displacement of the atoms is so large that the potential energy cannot be approximated by quadratic terms anymore, the approximation fails. In such cases, we can use a classical simulation method such as molecular dynamics to sample the phase space and calculate observables using these samples. We should note that this is strictly valid only in case of high temperatures, where Tmd Tqm-... 

---