Submitted by Christopher Stanley George Cousins to the University of Exeter as a thesis for the degree of Doctor of Philosophy in Physics (March 2001).

Following a brief and selective history of elasticity, the general theory of the role of relative sublattice displacements on the elasticity of single-crystalline material is elaborated in Chapter 1. This involves the definition of (a) rotationally-invariant inner displacements and (b) the internal strain tensors that relate those inner displacements to the external strain. The total elastic constants of such materials can then be decomposed into partial and internal parts, the former free of, and the latter involving, the inner displacement(s). Six families of inner elastic constants are needed to characterize the internal parts of the second- and third-order constants. The relation of the second-order inner elastic constants to the longwave coupling constants of lattice dynamics is shown, and a new form of secular equation for the frequencies and eigenvectors of the optic modes at the zone centre is given. In Chapter 2 the point-group symmetry implications for the inner elastic constants are explored in detail.

Chapter 3 is an interlude in which the measurement of the internal strain in cubic diamond is described.

In Chapters 4 and 5 the general formalism is applied to cubic and hexagonal diamond and to hexagonal and rhombohedral graphite. Space-group symmetry implications are described in detail and the formalism is extended to cover effective constants, pressure derivatives, elastic compliances and compressibilities. The allotropes are treated individually in terms of the Keating model in the following four Chapters. Cubic diamond is treated in Chapter 6 in terms of the original model. A shortcoming of the model non-transferability of its parameters to alternative descriptions of unit cell geometry is overcome by redefining both the Keating strain and the Keating parameters. The modified Keating model is then extended rigorously and successfully to a non-cubic material, hexagonal graphite, for the first time in Chapter 7. Chapter 8 presents a completely plausible account of the elasticity and zone-centre optic modes in hexagonal diamond by transferring the modified parameters from cubic diamond. The little that is known experimentally, the bulk modulus and three Raman frequencies, is predicted exactly. Chapter 9 extends Keating to the rhombohedral form of graphite using transferred parameters and provides a detailed picture of its transformation to cubic diamond. In Chapter 10 the relation of bond-order potentials to the Keating model is explored. An

Appendix contains a generalised method of homogeneous deformation, developed to relate the computationally-friendlyinfinitesimal strain approach to the thermodynamically-rigorous finite strain formalism, and the associated computational protocols needed to determine all elastic and inner elastic constants, and hence all derived quantities, of the allotropes discussed.