Topics in D-1 Dimensional Statistical Mechanics: 1. Statistical Mechanics of Equilibrium Crystal Shapes: Interfacial Phase Diagrams and Phase Transitions. 2. Symmetry Classifications of Two-Dimensional Phase Transitions
Rottman, Craig Alan
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https://hdl.handle.net/2142/77365
Description
Title
Topics in D-1 Dimensional Statistical Mechanics: 1. Statistical Mechanics of Equilibrium Crystal Shapes: Interfacial Phase Diagrams and Phase Transitions. 2. Symmetry Classifications of Two-Dimensional Phase Transitions
Author(s)
Rottman, Craig Alan
Issue Date
1983
Department of Study
Physics
Discipline
Physics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Physics, Condensed Matter
Language
eng
Abstract
This thesis consists of two parts. The first part, contained in Chapter 1, concerns interfacial phase diagrams and phase transitions associated with equilibrium crystal shapes. The second portion, contained in Chapter 2, deals with symmetry classifications of phase transitions in two dimensions.
In Chapter 1 we review the present status of the statistical mechanical theory of equilibrium crystal shapes. Special emphasis is placed on the relation between singularities occurring in the shapes of three-dimensional (d = 3) crystals and the phase transitions of certain d = 2 models. We exploit the thermodynamic conjugacy of the Wulff plot and the equilibrium crystal shape to give interfacial phase diagrams in both density and field variables. From this perspective, sharp edges or points on the crystal shape correspond to first-order phase transitions, while smooth joining of curved and planar ("faceted") regions corresponds to second-order phase transitions. Equilibrium crystal shapes of a simple-cubic (Ising) lattice gas with nearest-neighbor (attractive) and next-nearest-neighbor (nnn) interactions are considered in detail. Typical equilibrium crystal shapes at nonzero temperature consist of facets and smoothly curved surfaces. When nnn interactions are attractive, only second-order transitions occur. Both Kosterlitz-Thouless ("roughening") and Pokrovsky-Talapov (Gruber-Mullins) universality classes are represented. When nnn interactions are repulsive, first-order transitions and tricritical behavior also occur. The present experimental situation is summarized.
Chapter 2 begins with statistical mechanical background. It then discusses properties of the phases which form the focus of attention in the remainder of the chapter. A statistical mechanical model of the transitions between these phases and the solution within the approximation known as Landau theory are discussed. The reasons why Landau theory, despite its shortcomings, may be useful in describing these phase transitions are presented. The group-theoretical calculations of this study predict which phase transitions may be second order and also identify the universality classes to which these belong. Other phase transitions are predicted to be first order. These calculations extend the work of others to include all symmetries within certain classes of two-dimensional space groups, namely, usual space groups and magnetic space groups.
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