Lattice Dynamics and Specific Heats

Introduction

The static lattice model which is only concerned with the average positions of atoms and neglects their motions can explain a large number of material features such as:

· Chemical properties;

· Material hardness;

· Shapes of crystals;

· Optical properties;

· Bragg scattering of R-ray, electron and neutron beams;

· Electronics structure as well as electrical properties.

There are, however, a number of properties that cannot be explained by a static model. These include:

· Thermal properties such as heat capacity;

· Effects of temperature on the lattice, e.g. thermal expansion;

· The existence of phase transitions, including melting;

· Transport properties, e.g. thermal conductivity, sound propagation;

· The existence of fluctuations, e.g. the temperature factor;

· Certain electrical properties, e.g. superconductivity;

· Dielectric phenomenon at low frequencies;

· Interaction of radiation with matter, e.g. light and thermal neutrons.

Are the atomic motions that are revealed by these factors random, or can we find a good description for the dynamics of the crystal lattice? The answer is that the motions are not random but are constrained and determined by the forces that atoms exert on each other.

Vibrations in solids.

We will examine the vibrational behaviour of atoms in solids. The vibrations are thermally activated with a characteristic activation energy kBT.
Vibrational excitations are collective modes: all atoms in the material take part in the vibrational mode. The influence of translational periodicity characteristics of the structure of crystals has a dramatic effect on the vibrational behaviour when the wavelength of the vibrations becomes comparable to the size of the unit cell. When the vibration wavelength is much larger than the structural variation of the material, the solid may be considered as an elastic continuum (continuum approximation).

Representation

How can we visualise a vibration wave travelling through a crystal, where the space that vibrates is not continuous (like a string on a musical instrument) but is composed of discrete atoms? The answer is to think of the wave as representing displacements, u(x,t), of the atoms from their equilibrium position.

Continuous Media and Sound Waves

A sound wave is simply an elastic wave travelling in a medium. For a material regarded as an elastic continuum, the sound velocity is then directly related to the elastic modulus of the material. The sound produces a spatially varying stress s which in turn causes an instantaneous displacement u. If the sound is propagating in the x direction within a cube of material of mass density r, the net force acting on the volume element is:

is a measure of the velocity of a wave packet, composed of a group of plane waves, and having a narrow spread of frequencies about some mean value, w. For acoustic waves with long wavelengths (k » 0), i.e. in the elastic continuum limit, the phase and group velocities are equal. In a liquid, only longitudinal vibrations (modes) are supported (shear modulus is zero). The situation is more complicated in solids , where more than one elastic modulus is non-zero. As a consequence, both longitudinal and transverse acoustic modes exist even in isotropic solids, having in general different sound velocities. The situation is even more complicated for anisotropic crystals.

Counting vibrational states

The wavevector k characterises the vibrational wave. In the general solution to the wave equation, Eqn. 5, all k values are allowed. Restrictions on the allowed values of k appear through the imposition of boundary conditions. Two types of boundary conditions can be envisaged, depending on whether standing waves or propagating waves are involved.
For standing waves, and considering a cube of material of side L, the appropriate boundary condition for vibrational waves reflected from mechanically free surfaces is that an antinode of the vibration amplitude should exist at each surface. This corresponds to there being an integral number of half-wavelengths of the standing wave along the length of the cube. The allowed values of the standing wave vectors are given by

Schematic 1D illustration of a standing wave set up between the free surfaces of a cube of an elastic continuum with antinodes at the free surfaces.
Each allowed standing-wave solution of the wave equation consistent with the boundary conditions is represented by a point in the reciprocal space containing the k-vectors. The spacing between allowed k-values is Dks=p/L, and so the volume of k-space corresponding to the one k-value (standing-wave state).

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BIBLIOGRAFIA:

www.plato.ul.ie/academic/Vincent.Casey/PH4607SS1/LatticeDynamics.doc

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