Lattice dynamics anharmonicity effects

A.V. Belushkin, M.A. Adams, A.I. Kolesnikov L.A. Shuvalov (1994). J. Phys. Condens. Matter, 6, 5823-5832. Lattice dynamics and effects of anharmonicity in different phases of cesium hydrogen sulphate. 

It is easily seen by inspection that the biorthogonal basis set definition (3.55) cmnddes with the definifion (3.18) ven in the discussion of the Lanczos method. We recall that the dynamics of operators (liouville equations) or probabilities (Fokker-Planck equations) have a mathematical structure similar to Eq. (3.29) and can thus be treated with the same techniques (see, e.g., Chapter 1) once an appropriate generalization of a scalar product is performed. For instance, this same formalism has been successfully adopted to model phonon thermal baths and to include, in principle, anharmonicity effects in the interesting aspects of lattice dynamics and atom-solid collisions. ... 

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). 

Introduction to Lattice Dynamics by M. Dove, Cambridge University Press, Cambridge England, 1993. I really like Dove s book. Unlike the present work, it has the virtue of being short and to the point. The discussion of anharmonic effects is instructive. 

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-... 

We are encountering a feature that is inherent in all "direct" approaches to lattice dynamics as the total energy can only be calculated with finite displacements u, the harmonic terms appear intertwined with the enharmonic contributions, and we have to treat all the expansion terms simultaneously from the very beginning. In addition to the frozen phonon frequencies, we then obtain detailed Information on the anharmonicity of the mode in question, data which are difficult to find by other means, both theoretical and experimental. In some cases, it can be verified that the displacement u is small enough and does not give rise to any noticeable enharmonic effects. With most displacement patterns, however, the total energy has to be evaluated for several magnitudes of the displacement (typically 5 to 25 values of ) ... 

The anisotropy of polymer erystals is eharacteristie of the orientation of intra- and intermoleeular interaetions, which is also reflected in a broad separation on the dynamical time scale. It is especially important in polymer crystals to be cognizant of the limitations imposed by either the assumptions on which a method is based (e.g., the quasiharmonic approximation for lattice dynamics) or the robustness of the simulation method (e.g., ergodicity of the simulation in Monte Carlo or molecular dynamics). In some instances, valuable checks on these assumptions and limitations (e.g., quantum mechanical vs. classical dynamics, finite size effects and anharmonicity) can and should be made by repeating the study using more than one method. 

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