Indications that dynamics of atoms in a crystal are important: failure of the static lattice approximation

Crystallography is generally concerned with the static properties of crystals, describing features such as the average positions of atoms and the symmetry of a crystal. Solid state physics takes a similar line as far as elementary electronic properties are concerned. We know, however, that atoms actually move around inside the crystal structure, since it is these motions that give the concept of temperature, and the structures revealed by X-ray diffraction or electron microscopy are really averaged over all the motions. The only signature of these motions in the traditional crystallographic sense is the temperature factor (otherwise known as the Debye-Waller factor (Debye 1914; Waller 1923, 1928) or displacement amplitude), although diffuse scattering seen between reciprocal lattice vectors is also a sign of motion (Willis and Pryor 1975). 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 X-ray, electron and neutron beams, elec­tronic structure and electrical properties, etc. There are, however, a number of properties that cannot be explained by a static model. These include:

• thermal properties, e.g. heat capacity ;

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• 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 phenomena at low frequencies;

• interaction of radiation (e.g. light and thermal neutrons) with matter.

Are the atomic motions that are revealed by these features random, or can we find a good description for the dynamics of the crystal lattice? The answer is that the motions are not random; rather they are determined by the forces that atoms exert on each other. The aim of this book is to show that in fact we have a very good idea of the way atoms move inside a crystal lattice. This is the essence of the subject of lattice dynamics.

The classical motions of any atom are simply determined by Newton’s law of mechanics: force = mass x acceleration. Formally, if rp) is the position of atom j at time t, then

(1.1)

where nij is the atomic mass, and (p/Xp 0 is the instantaneous potential energy of the atom. Equation (1.1) is our key equation. We therefore need some knowledge of the nature of the atomic forces found in a crystal. The potential energy in equation (1.1) arises from the instantaneous interaction of the atom with all the other atoms in the crystal. We will often assume that this can be written as a sum of separate atom-atom interactions that depend only on the distances between atoms:

(1.2)

where r_tj is the distance between atoms i and j, and (pyir^ is a specific atom-atom interaction. The sum over i in equation (1.2) gives the interactions with all other atoms in the crystal.

Of course, quantum mechanics rather than classical mechanics determines the motions of atoms. But we will see that the main features of lattice dynamics follow exactly from the classical equation (1.1), whilst quantum effects are pri­marily revealed in the subsequent thermodynamic properties.

Our aim in this chapter is to set the scene for the rest of the book. In the first part we will consider some elementary ideas associated with interatomic

potentials. These are essential if we are to use real examples to illustrate the basic ideas we will develop in the following chapters. In the second part we will present the basic formalism for describing the motions of waves in crys­tals, which we will build upon in the rest of the book.