Unit Cells in Space or Spaces for Unit Cells

TL;DR

This paper explores how to define a space for unit cells in crystallography to measure relationships and distances between them.

Contribution

The paper introduces new approaches to represent and measure distances between unit cells in a multidimensional space.

Findings

Unit cells can be represented in a 6-dimensional space using lengths and angles.

A direct metric in this space is not feasible due to incommensurate dimensions.

Alternative 6 or 7-dimensional spaces are proposed for better measurement.

Abstract

Crystallographers think about unit cells as 3 lengths and 3 angles. In other words, as 6 numbers,often in a row. But in order to talk about the relationships between unit cells, we need to know where they are in the space of unit cells, whatever that is. Ideally, we would like to be able to measure the "distances" between 2 or more unit cells, so we would like to have a metric space. The lengths and angles can be treated as a 6- dimensional space, but the they are not commensurate, so there is not a 'nice" measure. Other 6 or 7 dimensional spaces are needed.