Lattice wave propagation

The atoms of a crystal vibrate around their equilibrium position at finite temperatures. There are lattice waves propagating with certain wavelengths and frequencies through the crystal [7], The characteristic wave vector q can be reduced to the first Brillouin zone of the reciprocal lattice, 0 < q <7t/a, when a is the lattice constant. 

The fluid mechanics origins of shock-compression science are reflected in the early literature, which builds upon fluid mechanics concepts and is more concerned with basic issues of wave propagation than solid state materials properties. Indeed, mechanical wave measurements, upon which much of shock-compression science is built, give no direct information on defects. This fluids bias has led to a situation in which there appears to be no published terse description of shock-compressed solids comparable to Kormer s for the perfect lattice. Davison and Graham described the situation as an elastic fluid approximation. A description of shock-compressed solids in terms of the benign shock paradigm might perhaps be stated as ... 

Fig, 9.9 Wave propagation in the HPP lattice geis, starting from a localized disturbance. Notice the appearance of a circular wave front on the macro-scale despite the discrete anisotropy at the microscopic (i.e. dynamical) level. 

The motion of atoms in the lattice can be depicted as a wave propagation (phonon). By dispersion we mean the variation in the wave frequency as reciprocal space is traversed. The propagation of sound waves is similar to the translation of all atoms of the unit cell in the same direction hence the set of translational modes is commonly defined as an acoustic branch. The remaining vibrational modes are defined as optical branches, because they are capable of interaction with light (see McMillan, 1985, and Tossell and Vaughan, 1992, for more exhaustive explanations). 

Wave Propagation in a One-dimensional Crystal Lattice.—Let us consider N atoms, each of mass m, equally spaced along a line, with distance d between neighbors. Let the x axis be along the line of atoms. We may conveniently take the positions of the atoms to be at x = d, 2d, 3d,. . . Nd, with y = 0, z = 0 for all atoms. These are the equilibrium positions of the atoms. To study vibrations, we must assume that each atom is displaced from its position of equilibrium. Consider the jth atom, which normally has coordinates x = jdy y = z 0, and assume that it is displaced to the position x = jd + /, y = Vi, z = f so that Vi, f / are the three components of the displacement of the atom. If the neighboring atoms, the (j — l)st and the j -f- l)st, are undisplaced, we assume that the force acting on the jth atom has the components... 

The dispersion relationships of lattice waves may be simply described within the first Brillouin zone of the crystal. When all unit cells are in phase, the wavelength of the lattice vibration tends to infinity and k approaches zero. Such zero-phonon modes are present at the center of the Brillouin zone. The variation in phonon frequency as reciprocal k) space is traversed is what is meant by dispersion, and each set of vibrational modes related by dispersion is a branch. For each unit cell, three modes correspond to translation of all the atoms in the same direction. A lattice wave resulting from such displacements is similar to propagation of a sound wave hence these are acoustic branches (Fig. 2.28). The remaining 3N-3 branches involve relative displacements of atoms within each cell and are known as optical branches, since only vibrations of this type may interact with light. 

Draine, B.T., and Goodman, J.J. (1993) Beyond clausius-mossotti wave propagation on a polarizable point lattice and the discrete dipole approximation, Astrophysical J., 405 685-697. 

Such electrons will be reflected backward and forward between two (adjacent) core ions (Bragg reflection). Due to the periodic lattice, electron propagation is prohibited in the x direction when the wave vector is equal (or close) to kx =... 

Compared to uniform compression, uniaxial strain along one of the lattice vectors is more likely to lead to detonation. Dick has proposed [43] that detonation initiation in nitromethane is favored by shock-wave propagation in specific directions related to the orientation-dependent sterie hindrance to the shear flow. This proposal is based on a model according to which the sterically hindered shear process causes preferential excitation of optical phonons strongly coupled with vibrons. 

Oxygen adsorption effect on the Van der Waals interaction contribution to the elastic moduli of ordered carbon nanotube arrays was estimated. Mechanical instability of square lattice of nanotubes with respect to the transition to triangular lattice was demonstrated. Variation of the elastic moduli due to the adsorption was shown to be of the same order of magnitude as the moduli themselves. This leads to variation of phase velocities of acoustic waves propagating across the array which proved to be dozens of times greater than mass loading by adsorbate for the acoustic wave frequency range of 0.1-1.0 GHz. 

To check these analytical results, one can perform numerical simulations that mimic the dispersal-reaction processes microscopically, for every biased form of w x) considered above, and find the critical value r at which wave propagation stalls. One can consider a one-dimensional lattice where dispersal and reaction processes take place starting from step-like initial conditions as in [67]. Instead of studying the evolution of a large number of individual particles, as is usually done, one can consider a continuous density of particles /O (x, t) for every cell in the lattice. For the case w x) = qb x - a) + (1 - q) (x + a) dispersal works in this way a fraction q of the density p x, t) in a certain cell goes to the cell at x -I- a and a fraction 1 - q goes to the cell at x - a. (For nondiscrete forms of w(x) one must discretize the corresponding function.) After that, the reaction function is applied to each cell in the lattice. The time between successive steps in this scheme is equal to T. 

The crystal lattice vibration and the force coefficients are the subject of Chapter 12. We describe the experimental dispersion curves and conclusions that follow from their examination. The interplanar force constants are introduced. Group velocity of lattice waves is computed and discussed. It allows one to make conclusions about the interatomic bonding strength. Energy of atomic displacements during lattice vibration (that is propagation of phonons) is related to electron structure of metals. 

In a three-dimensional crystal lattice, 3 N waves propagate in three independent directions. The total kinetic energy of the crystal lattice consisting of N atoms of the same mass m can be found by the obvious formula. 

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