Lattice dynamical method

The density of states (DOS) of lattice phonons has been calculated by lattice dynamical methods [111]. The vibrational DOS of orthorhombic Ss up to about 500 cm has been determined by neutron scattering [121] and calculated by MD simulations of a flexible molecule model [118,122]. 

The dispersion curves of surface phonons of short wavelength are calculated by lattice dynamical methods. First, the equations of motion of the lattice atoms are set up in terms of the potential energy of the lattice. We assume that thejxitential energy (p can be expressed as a function of the atomic positions 5( I y in the semi-infinite crystal. The location of the nth atom can be... 

Allan et describe how Monte Carlo and lattice dynamics methods can be... 

In most cases, the crystal potential is not known a priori. The usual procedure is to introduce some model potential containing several parameters, which are subsequently found by fitting the calculated crystal properties to the observed data available. This procedure has the drawback that the empirical potential thus obtained includes the effects of the approximations made in the lattice dynamics model, which is mostly the harmonic model. It is very useful to have independent and detailed information about the potential from quantum-chemical ab initio calculations. Such information is available for nitrogen (Berns and van der Avoird, 1980) and oxygen (Wormer and van der Avoird, 1984), and we have chosen the results calculated for solid nitrogen and solid oxygen to illustrate in Sections V and VI, respectively, the lattice dynamics methods described in Sections III and IV. Nitrogen is the simplest typical molecular crystal as such it has received much attention from theorists and... 

The second part deals with the actual calculation of the spectra and the bulk properties of molecular solids. The (lattice dynamics) methods to perform such calculations are outlined and it is demonstrated that a complete ah initio treatment of simple molecular crystals yields new and interesting insight in the behavior of these solids. Finally, it is illustrated how lattice dynamics calculations based on empirical atom-atom potentials are useful for the interpretation of the measurements on more complex molecular crystals. 

An overview of the current lattice dynamics methods is given in Sect. 2.2 (see also Ref [27]). In practically all cases, up to now, these methods have used semi-empirical intermolecular potentials, mostly of the atom-atom type. The parameters occurring in these potentials are usually fitted to the properties of interest, such as the lattice structure, the cohesion energy and the phonon frequencies. This procedure hides the flaws which are present in the intermolecular potentials as well as in the lattice dynamics method. In studies of solid [49-53], solid [41, 54-56] and solid [57] ab initio potentials have been used, however, which contain detailed information on the anisotropy of the potential and, in the case of O, also on its spin-dependence. Illustrative results of these studies are described in Sect. 2.3. The final Sect. 2.4, shows some typical phenomena occurring in more complex molecular crystals, such as phonon-vibron mixing, the dispersion and shifts of vibron bands and the effects of isotopic substitution, i.e. changes of nuclear masses, on the lattice- and internal vibrations. These phenomena are illustrated by results obtained on solid tetra-cyano-ethene [58] and on several chlorinated-benzene crystals [59, 60]. 

Any quantum mechanical lattice dynamics method has to start by giving the crystal Hamiltonian. For a molecular crystal the dynamical coordinates are the center of mass displacements of the molecules A from their equilibrium positions, the... 

The intermolecular potential as it is given in Eq. (3) for example, does not depend explicitly on the (external) molecular displacements or on the (internal) normal coordinates as required by Eq. (10). The atom-atom potential in Eq. (5) does not even depend explicitly on the molecular orientations Q. All these dependencies have to be brought out, by expansion and transformation of the potentials in Eq. (3) and Eq. (5), before these can actually be used in lattice dynamics calculations. The way this is performed depends on the lattice dynamics method chosen (see below). If one is not interested in the internal molecular vibrations, the free-molecule Hamiltonians may be omitted from Eq. (10) and the potential may be averaged over the molecular vibrational states. The effective potential thus obtained no longer depends on the coordinates and Q. ... 

The lattice dynamical method, which was originally proposed by Born and Huang [59], to calculate the velocity of sound waves in elastic solids and hence their elastic constants. The equation of motion of an atom is given by... 

Equation [27] is sketched in Eigure 13, and the dispersion relation is in qualitative agreement with dispersion curves calculated by lattice dynamics method for xenon hydrate (cf. Eigure 11a). 

A fiirther theme is the development of teclmiques to bridge the length and time scales between truly molecular-scale simulations and more coarse-grained descriptions. Typical examples are dissipative particle dynamics [226] and the lattice-Boltzmaim method [227]. Part of the motivation for this is the recognition that... 

Stassis C 19. Lattice Dynamics. In Skald and D L Price (Editors) Methods of Experimental Physics Volume 23 Neutron Scattering Part A. Orlando, Academic Press, pp. 369-440. 

In principle, we could find the minimum-energy crystal lattice from electronic structure calculations, determine the appropriate A-body interaction potential in the presence of lattice defects, and use molecular dynamics methods to calculate ab initio dynamic macroscale material properties. Some of the problems associated with this approach are considered by Wallace [1]. Because of these problems it is useful to establish a bridge between the micro-... 

These apparent restrictions in size and length of simulation time of the fully quantum-mechanical methods or molecular-dynamics methods with continuous degrees of freedom in real space are the basic reason why the direct simulation of lattice models of the Ising type or of solid-on-solid type is still the most popular technique to simulate crystal growth processes. Consequently, a substantial part of this article will deal with scientific problems on those time and length scales which are simultaneously accessible by the experimental STM methods on one hand and by Monte Carlo lattice simulations on the other hand. Even these methods, however, are too microscopic to incorporate the boundary conditions from the laboratory set-up into the models in a reahstic way. Therefore one uses phenomenological models of the phase-field or sharp-interface type, and finally even finite-element methods, to treat the diffusion transport and hydrodynamic convections which control a reahstic crystal growth process from the melt on an industrial scale. 

At the same time, many lattice dynamics models have been constructed from force-constant models or ab-initio methods. Recently, the technique of molecular dynamics (MD) simulation has been widely used" " to study vibrations, surface melting, roughening and disordering. In particular, it has been demonstrated " " " that the presence of adatoms modifies drastically the vibrational properties of surfaces. Lately, the dynamical properties of Cu adatoms on Cu(lOO) " and Cu(lll) faces have been calculated using MD simulations and a many-body potential based on the tight-binding (TB) second-moment aproximation (SMA). " ... 

One of the consequences of the suppression of the phase transition is the presence of a special critical point, Tc = 0 K. This point, called the quantum displacive limit, is characterized by special critical exponents. Its presence gives rise to classical quantum crossover phenomena. Quantum suppression and the response at and near this limit, Tc = 0 K, have been extensively studied on the basis of lattice dynamic models solved within the framework of both classical and quantum statistical mechanics. Figure 8 is a log-log plot of the 6 T) results for ST018 [15]. The expectation from theory is that in the quantum regime, y = 2 at 0.7 kbar, after which y should decrease. The results in Fig. 8 quantitatively show the expected behavior however, y is < 2 at 0.70 kbar. Despite the difference in the methods to suppress Tc in ST018, the results in Fig. 4a and Fig. 8 are quite similar. As shown in the results in Fig. 3b, uniaxial pressure also can be a critical parameter S for the evolution of ferroelectricity in STO. 

For the description of die temperature and stress-related behavior of the crystal we used die method of consistent quasi-hannonic lattice dynamics (CLD), which permits die 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. ... 

Similar methods have been used to integrate thermodynamic properties of harmonic lattice vibrations over the spectral density of lattice vibration frequencies.21,34 Very accurate error bounds are obtained for properties like the heat capacity,34 using just the moments of the lattice vibrational frequency spectrum.35 These moments are known35 in terms of the force constants and masses and lattice type, so that one need not actually solve the lattice equations of motion to obtain thermodynamic properties of the lattice. In this way, one can avoid the usual stochastic method36 in lattice dynamics, which solves a random sample of the (factored) secular determinants for the lattice vibration frequencies. Figure 3 gives a typical set of error bounds to the heat capacity of a lattice, derived from moments of the spectrum of lattice vibrations.34 Useful error bounds are obtained... 

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