Dynamic lattice

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  

Newton s second law, we get the equations of motion for the atom-displacements as 

The dispersion curve (Fig. 2.8) corresponds to the so-called low frequency acoustical mode. In a three-dimensional lattice this mode survives and appears as a surface wave. 

If the masses in the one-dimensional lattice are different with alternating values mi and m2, one obtains by solving the lattice dynamical equations the following dispersion relation  

FIGURE 2.8 The dispersion curves for a simple one-dimensional lattice with an optical and an acoustical branch. 

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Califano S, Sohettino V and Neto N 1981 Lattice Dynamics of Molecular Crystals (Berlin Springer)... 

Solid-state Systems Lattice Statics and Lattice Dynamics... 

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. 

Allan N L, G D Barrera, J A Purton, C E Sims and M B Taylor 2000. Ionic Solids at High Temperatures and Pressures Ah initio, Lattice Dynamics and Monte Carlo Studies. Physical Chemistry Chemical Physics 2 1099-1111. 

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). " ... 

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]. 

For a more detailed account of the recoil-free fraction and lattice dynamics, the reader is referred to relevant textbooks ([12-15] in Chap. 1). 

Much of the Pt Mossbauer work performed so far has been devoted to studies of platinum metal and alloys in regard to nuclear properties (magnetic moments and lifetimes) of the excited Mossbauer states of Pt, lattice dynamics, electron density, and internal magnetic field at the nuclei of Pt atoms placed in various magnetic hosts. The observed changes in the latter two quantities, li/ (o)P and within a series of platinum alloys are particularly informative about the conduction electron delocalization and polarization. 

The work of Wortmann et al. [65-67], Gavriliuk et al. [68, 69] and Sturhahn et al. [70] convincingly demonstrates the power of nuclear resonant scattering experiments with synchrotron radiation for high-pressure smdies of magnetism and lattice dynamics. An illustrative example was presented at the Fifth Seeheim Workshop by Wortmann [65] Fig. 9.28a shows NFS spectra of LuFe2 at 295 and... 

Elastomers are solids, even if they are soft. Their atoms have distinct mean positions, which enables one to use the well-established theory of solids to make some statements about their properties in the linear portion of the stress-strain relation. For example, in the theory of solids the Debye or macroscopic theory is made compatible with lattice dynamics by equating the spectral density of states calculated from either theory in the long wavelength limit. The relation between the two macroscopic parameters, Young s modulus and Poisson s ratio, and the microscopic parameters, atomic mass and force constant, is established by this procedure. The only differences between this theory and the one which may be applied to elastomers is that (i) the elastomer does not have crystallographic symmetry, and (ii) dissipation terms must be included in the equations of motion. 

H. BOttger, Principles of the Theory of Lattice Dynamics (Academie-Verlag, Berlin, 1983). 149, 176 ... 

With development of ultrashort pulsed lasers, coherently generated lattice dynamics was found, first as the periodic modulation in the transient grating signal from perylene in 1985 by De Silvestri and coworkers [1], Shortly later, similar modulation was observed in the reflectivity of Bi and Sb [2] and of GaAs [3], as well as in the transmissivity of YBCO [4] by different groups. Since then, the coherent optical phonon spectroscopy has been a simple and powerful tool to probe femtosecond lattice dynamics in a wide range of solid... 

Material response in THz frequency region, which corresponds to far- and mid-infrared electromagnetic spectrum, carries important information for the understanding of both electronic and phononic properties of condensed matter. Time-resolved THz spectroscopy has been applied extensively to investigate the sub-picosecond electron-hole dynamics and the coherent lattice dynamics simultaneously. In a typical experimental setup shown in Fig. 3.5, an... 

The absolute value of the entropy of a compound is obtained directly by integration of the heat capacity from 0 K. The main contributions to the heat capacity and thus to the entropy are discussed in this chapter. Microscopic descriptions of the heat capacity of solids, liquids and gases range from simple classical approaches to complex lattice dynamical treatments. The relatively simple models that have been around for some time will be described in some detail. These models are, because of their simplicity, very useful for estimating heat capacities and for relating the heat capacity to the physical and chemical... 

The Kieffer approach uses a harmonic description of the lattice dynamics in which the phonon frequencies are independent of temperature and pressure. A further improvement of the accuracy of the model is achieved by taking the effect of temperature and pressure on the vibrational frequencies explicitly into account. This gives better agreement with experimental heat capacity data that usually are collected at constant pressure [9],... 

M. T. Dove, Introduction to Lattice Dynamics. Cambridge Cambridge University Press, 1993. 

Elevated temperatures and thermal expansion Helmholtz, Gibbs energies and lattice dynamics... 

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