Structure and Diffraction Properties of Disordered Systems
Thesis DisciplineElectrical Engineering
Degree GrantorUniversity of Canterbury
Degree NameDoctor of Philosophy
In many systems of interest, both physical and biological, disorder inhibits the organization and cooperative properties of the system. Disorder can originate from a variety of system defects and the degree of disorder also varies. Geometric frustration introduces disorder into a system in which all the preferred interactions between the elements of the system cannot be satisfied due to the topology of an underlying lattice that describes the position of these elements. Recently, geometric frustration has been recognized as an important organizing principle in a diverse range of systems from superconducting networks to neural computation. The correlation behavior of such systems is often complicated and poorly understood. The myosin lattice of higher vertebrate muscle is a geometrically frustrated system, and the presence of this kind of disorder has prevented a rigorous interpretation of X-ray diffraction patterns from muscle fibres for the purposes of studying muscle molecular structure.
This thesis investigates the correlation behavior of two geometrically frustrated systems, the triangular Ising antiferromagnet (TIA) and the fully frustrated square Ising model (FFS), and its use to interpret X-ray fibre diffraction patterns. A combination of numerical evaluation of exact expressions and Monte Carlo simulation is used to study a number of aspects of the two-point correlation function of the TIA and FFS. In the case of the TIA, a simple functional expression is developed that allows accurate calculation of the correlation function. Theory is developed for calculating diffraction by polycrystalline fibres of helical molecules, in which the constituent crystallites contain correlated substitution disorder. The theory was used to study the characteristics of diffraction by fibres with TIA-type substitution disorder statistics. A quantitative model of the disorder in the myosin filament array is developed and the above theory is used to calculate X-ray fibre diffraction from low resolution models of the myosin filament array in higher vertebrate muscle. The calculated diffraction is compared to measured diffraction data, showing good agreement.