The symmetry and antisymmetry of distortions

Open Access
Vanleeuwen, Brian Kevin
Graduate Program:
Materials Science and Engineering
Doctor of Philosophy
Document Type:
Date of Defense:
March 31, 2015
Committee Members:
  • Venkatraman Gopalan, Dissertation Advisor
  • Long Qing Chen, Committee Member
  • Zhiwen Liu, Committee Member
  • Vincent Henry Crespi, Committee Member
  • Daniel Bernard Litvin, Special Member
  • symmetry
  • antisymmetry
  • crystallography
  • space groups
This work is about symmetry. In particular, it addresses the symmetry of distortions. Any parameterized path through configurational space is a distortion (distortion, distortion path, and pathway are used synonymously in this work). This goes beyond just the typical use of the term distortion that describes a path connecting a high symmetry prototype structure to a lower symmetry distorted structure; for instance, the tetragonal PbTiO3 structure is commonly described as a distorted cubic PbTiO3 structure. In this work, a distortion is any parameterized path through configurational space, including the paths that atoms takes when hopping between sites in a crystal lattice, the pathways taking reactants to product in chemical reactions and conformational changes, the paths taken in domain wall, dislocation, and grain boundary motion, and many other things. Clearly, understanding these paths is very important for many physical problems, e.g. an example is given in Chapter 1 where ignoring the consequences of distortion symmetry could result in overestimating the activation barrier for diffusion by a factor of ~5X (see Sections 1.1.2, 1.2.3, and 1.3.2). Fundamentally, distortion symmetry works because it recognizes the separation of the configurational space from the parameter of the pathway (herein referred to as the distortion parameter, λ). This separation is something like the separation between time and space in classical mechanics and so λ is seen as a “time-like” parameter. Because of the distortion parameter’s time-like nature, the time reversal operation has an analog that reverses the distortion parameter. This operation is called distortion reversal. Distortion symmetry comes from considering the conventional symmetry of the configurations along a pathway in conjunction with distortion reversal. The content of Chapter 1 is mostly independent of the other chapters and is written to be of interest to a more general audience than the others. Distortion reversal and time reversal are independent antisymmetry operations and so motivate the listing of the types of double antisymmetry space groups described in Chapter 3. The mathematical tools necessary to determine the types of double antisymmetry space groups are discussed in Chapter 2. Chapter 4 describes how the idea of rotation-reversal symmetry led to the development of distortion symmetry.