Lattice dynamics quantum solids

The Raman spectra of solids have a more or less prominent collision-induced component. Rare-gas solids held together by van der Waals interactions have well-studied CILS spectra [656, 657]. The face-centered, cubic lattice can be grown as single crystals. Werthamer and associates [661-663] have computed the light scattering properties of rare-gas crystals on the basis of the DID model. Helium as a quantum solid has received special attention [654-658] but other rare-gas solids have also been investigated [640]. Molecular dynamics computations have been reported for rare-gas solids [625, 630, 634]. 

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... 

Organic Molecular Solids m. Quantum Lattice Dynamics... 

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. 

Guyer (1969) has reviewed the field of quantum lattice dynamics and applications to solid helium in great detail. He also compared the theoretical with experimental results quite exhaustively. We shall here review the applications to this system only briefly. 

First conclusion. The combination of quasiharmonic lattice dynamics in the quantum regime, for T < 0jjo3> together with molecular dynamics in the classical regime, T > 0j(o3> provides a simple and reasonably accurate representation of the vibrational thermodynamics of a nonquantum solid. 

The next section of this chapter describes the results of significant studies by Yushchenko et al. [71-74]. These studies provided insights into the molecular nature of the Rehbinder effect and constituted the first steps toward a numerical simulation of the elementary acts of deformation and bond rupture in the lattice of a solid. The studies also dealt with the influence of adsorption on this process (Sections 1.1 and 4.2). For detailed discussions of these molecular dynamic experiments, the reader is referred to the original works and reviews [71-83]. The quantum mechanical studies of Rehbinder effect were also done by Ab-initio calculations (c-c bond cleavage) [84]. 

The solution of the dynamical problem for the gas and surface atoms requires in principle solution of the quantum mechanical equations of motion for the system. Since this problem has been solved only for 3-4 atomic systems we need to incorporate some approximations. One obvious suggestion is to treat the dynamics of the heavy solid atoms by classical rather than quantum dynamical equations. As far as the lattice is concerned we may furthermore take advantage of the periodicity of the atom positions. At the surface this periodicity is, however, broken in one direction and special techniques for handling this situation are needed. Lattice dynamics deals with the solution of the equations of motion for the atoms in the crystal. As a simple example we consider first a one-dimensional crystal of atoms with identical masses. If we include only the nearest neighbor interaction, the hamiltonian is given by ... 

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. 

The state of polarization, and hence the electrical properties, responds to changes in temperature in several ways. Within the Bom-Oppenheimer approximation, the motion of electrons and atoms can be decoupled, and the atomic motions in the crystalline solid treated as thermally activated vibrations. These atomic vibrations give rise to the thermal expansion of the lattice itself, which can be measured independendy. The electronic motions are assumed to be rapidly equilibrated in the state defined by the temperature and electric field. At lower temperatures, the quantization of vibrational states can be significant, as manifested in such properties as thermal expansion and heat capacity. In polymer crystals quantum mechanical effects can be important even at room temperature. For example, the magnitude of the negative axial thermal expansion coefficient in polyethylene is a direct result of the quantum mechanical nature of the heat capacity at room temperature." At still higher temperatures, near a phase transition, e.g., the assumption of stricdy vibrational dynamics of atoms is no... 

For the partially deuterated benzoic acid (C6D5COOH), the solid state H NMR spectrum is dominated by the intra-dimer H- H dipole-dipole interaction. In a single crystal, both tautomers A and B are characterised by a well-defined interproton vector with respect to the direction of the magnetic field (Fig. 1). Proton motion modulates the H- H dipole-dipole interactions, which in turn affects the H NMR lineshape and the spin-lattice relaxation time. It has been shown that spin-lattice relaxation times are sensitive to the proton dynamics over the temperature range from 10 K to 300 K, and at low temperatures incoherent quantum tunnelling characterises the proton dynamics. A dipolar splitting of about 16 kHz is observed at 20 K. From the orientation dependence of the dipolar splitting, the... 

A scheme as described here is indispensable for a quantum dynamical treatment of strongly delocalized systems, such as solid hydrogen (van Kranendonk, 1983) or the plastic phases of other molecular crystals. We have shown, however (Jansen et al., 1984), that it is also very suitable to treat the anharmonic librations in ordered phases. Moreover, the RPA method yields the exact result in the limit of a harmonic crystal Hamiltonian, which makes it appropriate to describe the weakly anharmonic translational vibrations, too. We have extended the theory (Briels et al., 1984) in order to include these translational motions, as well as the coupled rotational-translational lattice vibrations. In this section, we outline the general theory and present the relevant formulas for the coupled... 

The present account of the computational techniques will be brief since comprehensive reviews are given elsewhere.In general, three classes of techniques have been employed in the study of solid state ionic materials atomistic (static lattice), molecular dynamics (MD) and ab initio quantum mechanical methods. 

A wide variety of different models of the pure water/solid interface have been investigated by Molecular Dynamics or Monte Carlo statistical mechanical simulations. The most realistic models are constructed on the basis of semiempirical or ab initio quantum chemical calculations and use an atomic representation of the substrate lattice. Nevertheless, the understanding of the structure of the liquid/metal surface is only at its beginning as (i) the underlying potential energy surfaces are not known very well and (ii) detailed experimental information of the interfacial structure of the solvent is not available at the moment (with the notable exception of the controversial study of the water density oscillations near the silver surface by Toney et al. [140, 176]). 

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