Introduction

Abstract

In considering the structure of solid matter, crystallographers begin with the strict mathematical formulation of an atomic pattern repeating endlessly with exact regularity. The truly stable form for a chemical compound at a particular pressure and temperature has the lowest value for the Gibbs energy function G = U — TS + PV (U internal energy, T temperature, S entropy, P pressure, V volume). At zero Kelvin only one atomic pattern at a given pressure could have minimum internal energy. Because attractive forces between atoms lead to a condensed state at lowered temperature, the stable form at the limit consists of a single edifice composed of regularly repeating units. At elevated temperatures, the entropy increases due to thermal motion and to substitutional and positional disorder. A structural variant with disorder is favored over an ordered form if the entropy term outweighs the increase of internal energy caused by the disordered atoms. As the heat motion increases, the difference between the internal energies of ordered and disordered forms becomes less significant in relation to the entropy term, and certain types of disorder become favored, culminating in breakdown to the liquid or vapor state. Thus the idealized concept of a strictly regular crystal structure becomes less applicable as the temperature rises, and it is the resultant disorder which provides a petrogenetic value to crystallographic studies of minerals. However mathematical analysis of the idealized regular structure is fundamental to all mineralogical studies.