Bright, Matthew
(2024)
Continuous Spaces of Low Dimensional Lattices.
PhD thesis, University of Liverpool.
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Abstract
Crystal structures are typically classified discretely by their symmetry groups - however, given the huge growth in deposited crystal structures, and the ability of crystal structure prediction methods (CSP) to generate thousands of potential structures, many of which may be nearly identical in a geometric sense, for a given set of input molecules classification of this nature is potentially too coarse to allow careful exploration of the space of existing or potential structures. Our research aim is therefore to develop a continuous classification of periodic structures In this work we focus on the simplest of these structures- the two dimensional lattice, defined as the infinite set of integer combinations of a basis of the Euclidean plane. Our approach in this work is to consider any two lattices lattices identical if they are related by isometry - or, if we wish to distinguish such structures which are related by reflection, by a rigid motion. We seek to map any given lattice to some mathematical object which is the same for any pair of lattices related by these transformations, and different otherwise - a complete invariant. We want this invariant to vary continuously, in the mathematical sense, under perturbations of the lattice, such that we may map lattices to a metric space, in which their geometric similarity may be rigorously compared. The key contribution of this thesis is a full solution of this problem for two dimensional lattices and the development as a result of a visualisation of all possible lattice configurations as a compact space. We have implemented this solution and applied it to large datasets of actually existing lattices from the Cambridge Structural Database and elsewhere. To our knowledge the various maps shown in this thesis give the first visualisations of the set of real world periodic structures in a continuous space.
| Item Type: | Thesis (PhD) |
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| Uncontrolled Keywords: | Crystallography, Lattice, Metric Spaces, Periodic Geometry |
| Divisions: | Faculty of Science and Engineering > School of Electrical Engineering, Electronics and Computer Science |
| Depositing User: | Symplectic Admin |
| Date Deposited: | 18 Mar 2024 08:45 |
| Last Modified: | 18 Mar 2024 08:45 |
| DOI: | 10.17638/03178964 |
| Supervisors: |
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| URI: | https://livrepository.liverpool.ac.uk/id/eprint/3178964 |
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