Description
The beauty of crystals has fascinated humankind throughout the ages. In its simplest form, a crystal is a structure of molecules repeating periodically in three directions and thus forming a crystal lattice with its unit cell. A challenging problem in crystallography is to resolve the internal structure of a crystal, that is, to identify the atoms and their positions in its unit cell. Since 1900, 32 Nobel prizes in Chemistry, Physics and Physiology or Medicine have been awarded for discoveries related to Crystallography, ranging from Konrad Roentgen (1901) to Crick/Watson/Wilkins (1962) for the helix structure of DNA to Shechtman (2011) for the discovery of quasicrystals. The theory of crystal structure determination via Xray diffraction is at the interface of many different mathematical areas:
- 3-dimensional Euclidean Geometry and Symmetry Groups: The constraint that unit cells of crystals are stacked together without gaps leads to the fact that there is only a finite number of possible symmetry groups based on translations, (screw) rotations, (roto)inversions and glide reflections. In fact there are precisely 230 so called crystallographic space groups.
- Fourier Analysis: The interference pattern of Xrays reflected by a crystal allows to determine the dimension of the unit cell. Its intensities (Bragg Peaks) are the absolute values of the Fourier coefficients of the electron density of the crystal. The fundamental problem in crystal structure determination, called the Phase Problem, is that these Fourier coefficients are complex numbers of which only their absolute values can be experimentally measured.
- Structure refinement via Optimisation: Once the initial structure of a crystal with its approximate atomic positions in the unit cell is determined, it requires a refinement process to improve the agreement between the observations and the theoretically computed quantities. This is done via a Least Squares Minimisation process.
The aim of this project is to learn the fundamental principles of crystallography and then to branch out into specific topics. Possible topics are:
- Diffraction patterns: Indexing the peaks and determining the unit cell dimensions and the symmetry group of a crystal
- Initial structure solution: Various methods like the Patterson function or charge-flipping to derive the initial crystal structure
- Detecting twins laws: Twins are intergrown crystals, exhibiting a superposition of their diffraction patterns; a challenge is to find the underlying rotation matrices relating the twin components.
- Dealing with atomic interactions and bonds: Study of quantum mechanical methods for the derivation of molecular electron densities for the structure refinement or other theoretical uses to improve the refinement process
- Quasicrystals: Investigating their experimental background (ten-fold diffraction patterns which cannot appear in periodic structures) and the corresponding mathematical theory (aperiodic tilings and Penrose tilings, Delone and Meyer sets)
Students in this topic do not need any background in chemistry or physics, only some curiosity about the nature of the microscopic world around us. All relevant non-mathematical aspects will be gently introduced and discussed during the course of the project.
If you are wondering about the origins/motivation of this project, there is a Durham University spin-off company, OlexSys, whose flagship product is the crystallographic software Olex2. A large proportion of all small molecule structure determinations is currently performed using this software. The company is always looking for young talented researchers who may be interested to work on further projects in this topic.
If you are interested in software development, there is also the opportunity to contribute to this open source software and see your code actually being used in the crystallographic community.
Resources
Some useful references on this topic are the following sources:
- Peter Engel: Geometric Crystallography, An axiomatic introduction to Crystallograhy, D. Reidel Publishing Company,1986
- Mois I. Aroyo (editor): International Tables for Crystallography, Teaching Edition: Crystallographic Symmetry, Wiley, 2021
- Werner Massa: Crystal Structure Determination, BoD – Books on Demand, Norderstedt, Germany, 2016
- Jenny Pickworth Glusker, Kenneth N. Trueblood: Crystal Structure Analysis, a Primer, Oxford Science Publications, 2010
- Luc J. Bourhis, Oleg V. Dolomanov, Richard J. Gildea, Judith A. K. Howard, Horst Puschmann (2015): The anatomy of a comprehensive constrained, restrained refinement program for the modern computing environment – Olex2 dissected. Acta Cryst. A71(1), 59-75
- Oleg V. Dolomanov, Luc J. Bourhis, Richard J. Gildea, Judith A. K. Howard, Horst Puschmann (2009): OLEX2: a complete structure solution, refinement and analysis program. J. Appl. Cryst. 42(2), 339-341
email: N
Peyerimhoff
|