PhD Thesis

Inner Elasticity and the Higher-Order Elasticity of Some Diamond and Graphite Allotropes

Inner Elasticity and the Higher-Order Elasticity of Some Diamond and Graphite Allotropes

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.

You can download the entire thesis in PDF, (approx. 3.5 Mbytes).

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Last modified: Mon Sep 10 16:32:22 BST 2001 by Jonathan Goss