When is a Lattice Not a Lattice? The changing meaning of the term lattice in crystallography and physics

Abstract

The concept of the lattice is central to the understanding of crystalline solids. However, usage of this word can very often differ between crystallographers, for whom a lattice is a mathematical object that describes the symmetry of a crystal, and physicists and chemists, for whom lattice is commonly used as a word for a regular array of particles. The similarity between the two definitions means they can easily be confused by students or inexperienced practitioners, but the fundamental differences can make the consequences of such confusion significant, an issue rarely tackled directly in popular textbooks. Here we examine the historical roots of this problem of context, and the changing understanding of the word 'lattice' over time. While the origins of the term lattice lie with the 19 th century mathematical crystallographers, their usage, both in terms of the words used and their meanings, was fluid, and no strong distinction between the lattice and the physical components of a crystal was made. Leading crystallographers in the early 20 th century regularly used the word lattice in a way that is unacceptable to some of their modern counterparts. We identify the decade after 1910 as the start of divergence between the physical and crystallographic meaning, catalysed by the discovery of X-ray diffraction and the development of lattice dynamics, although the current definitions did not become entrenched until the 1940s. While history has shown us that this classificatory issue is not just a matter of scientific disagreement and perhaps at its root is a disagreement in our understanding of classification itself. Lastly, we discuss possible resolutions to the matter.

The importance of definitions in crystallography

“…In Definition 3, any periodic crystal has a purely geometric part, which is a periodic set of zero-sized points at all atomic centers, and the physical part of atomic attributes of these points, see the history in Palgrave & Tobin (2021). Any lattice � can be considered a periodic point set whose motif M consists of a single point p, for example, at the origin of R n .…”