12th Training Course on Symmetry and Group Theory (a student’s perspective)

I first heard about this course from a colleague, an experienced experimentalist in diffraction-based characterisation, at the end of a long day of working together on a structural refinement that was proving troublesome. Like many PhD students in solid-state chemistry, or condensed matter physics (the same thing in my opinion), I had studied X-ray diffraction, and a small amount of group theory through the lens of molecular symmetry, during my undergraduate degree. “Excellent!” I thought, “I am fully prepared to tackle complex structural characterisation of novel materials and their phase transitions, based on symmetry relations, group theory, and space groups…” Of course not.

When I started my PhD, I had a limited grasp of what a space group even was. Rietveld refinements were a wonderous black box that you fed data, then marvelled at the structure that popped out (which was, more often than not, an unphysical mess). Programs such as TOPAS, GSAS, and FullProf are incredibly powerful and valuable tools to the experimental structural scientist. However, they are infamously poor at dealing with symmetry, particularly in the hands of a student with limited understanding of the underlying concepts.

“Garbage in, garbage out.”

This is, of course, not for lack of trying. There are numerous resources (both online and in print) for the learning of crystallography. However, these can often seem impenetrable, riddled with confusing conventions, field specific jargon, and mathematics one may not have encountered in many years (if at all). Further, experimentally focused research groups don’t necessarily have theoreticians on hand, or experimentalists with the specific expertise (or free time) required to build up the knowledge and confidence of new students. This often adds up to make crystallography seem insurmountable to the budding crystallographer, and is one of the main reasons I was so excited when I heard about this course. The training course on symmetry and group theory has run for many years now, primarily in Japanese, but additionally in English since 2019, in response to the demand shown by students who are not native Japanese speakers. Primarily, the course was motivated by the many obstacles commonly faced by early career researchers, owed to the somewhat limited crystallographic training provided by (already busy) undergraduate curricula. The course is hosted at the High Energy Accelerator Organisation site in Tsukuba (KEK), and is provided free of charge.¹ It is sponsored by the Crystallographic Society of Japan, the Institute of Materials Structure Science, and supported by many other Japanese scientific societies and associations. From 2019, it has been certificated as a common lecture by SOKENDAI (The Graduate University for Advanced Studies). All lectures and exercises are delivered by Professor Massimo Nespolo (Université de Lorraine, France).

Unlike any other course I have previously studied, which required students to make unsatisfying concessions, accept axioms without explaining how they arise, and learn relations by rote, Professor Nespolo starts truly from first principles, with basic algebra. This lays necessary foundations, which are built up procedurally and developed throughout the course into a rich understanding, with many satisfying relations that become apparent as more advanced concepts are introduced. Further, the opportunity to participate in frequent guided exercises, where the taught concepts are explored in greater detail, allows participants to really absorb the content of the course and solidify their understanding. The condensed nature of the course facilitates a breadth of topics to be taught, starting from the required fundamental algebra, building to point/space groups and their constituent properties from a geometric/graphical perspective. This then facilitates understanding of the physical meanings of the following crystallographic calculations via matrices, and how all of this adds up to macroscopic properties and phase transitions in the materials, which is the subject of our research.

I study battery cathode (positive electrode) materials. As alkali ions are removed from between transition metal oxide layers, this class of materials undergo a range of complex phase transitions, including layer glides, vacancy orderings, and static/dynamic distortions (such as the well-known Jahn-Teller effect). Each of these factors has the potential to result in symmetry lowering effects, from the original high-symmetry parent cell. As an example from my own work, combined refinement of structural models against X-ray and neutron diffraction data has allowed us to find that removing 1/3 of the sodium ions from sodium nickel oxide (NaNiO₂ -> Na_2/3NiO₂) results in symmetry lowering from C2/m to P2₁/c. Electrochemically desodiating the material in this way results in multiple extra symmetrically distinct crystallographic sites (where only single Na, Ni, and O sites existed in the parent), and is associated with sodium/vacancy ordering and transition metal orbital ordering, both of which combine to affect the resultant structure of the new cell.

Participants receive certificates for completing the course.

I look forward to applying what I have learned regarding fundamental crystallographic symmetry, and group-subgroup relations, to solving the remaining novel structures throughout the Na_1-xNiO₂ phase-space, and would gladly participate in the advanced course (provided my Japanese improves over the next few years!). I strongly recommend this course to any other students who have struggled to understand the underlying crystallographic concepts that inform the necessary decisions required to devise physically meaningful (and symmetrically sensible) Rietveld refinements for novel structural elucidation.