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Introduction

The Ad-hoc Committee was set up in 1980 to `consider nomenclature problems concerning symmetry operations and symmetry elements in space groups'. With regard to symmetry operations, the only problem is that of the appropriate symbols; however, a complete notation has been adopted in International Tables for Crystallography (1983) (referred to hereafter as IT A83).

The term `symmetry element ' is widely used only by crystallographers, mineralogists and spectroscopists. It is the collective designation for a number of concepts which - judging from their names - appear to be of a geometric nature: a rotation axis, a mirror plane etc. An essential feature of symmetry elements is the connection between such geometric items and one or more symmetry operations of a given space group or point group. This connection has consisted in the past of a traditional set of ad-hoc assignments of geometric items to symmetry operations. It has been the foremost goal of our Ad-hoc Committee to put that set on the footing of a precise and unified general definition.

In the successive editions of the International Tables for Crystallography, only IT A83 and later editions contain an explicit description of symmetry elements; see its Table 1.3. The last column of that table is headed: `Generating symmetry operation with glide or screw vector'. The term `generating' is not explained; it is taken to mean that if a structure has a symmetry operation listed here, then the corresponding `symmetry element' listed in the second column, with its symbol as in the first, is present. However, that interpretation leads to several problems:

(1) The table reflects the usual ad-hoc assignment; again there is no clue to a unified definition.

(2) The geometric item is obvious for a (glide-) reflection, a (screw-) rotation and an inversion. For a rotoinversion, however, the term `rotoinversion axis' in column 2 is unclear. It suggests a mere line, but that is not sufficient to determine the operation.

(3) According to this table, a plane such as xy0 in space group Cmma (67), for instance, is both an a- and a b-glide plane. The current symbol `a' is awkwardly biased. More serious is the resulting apparent ambiguity of the concept `glide plane'.

(4) In Cmmm (65), two possible `generating' symmetry operations share a plane. One is the reflection (x, y, -z), the other is an n-glide reflection ($\frac{1}{2} + x, \frac{1}{2} + y, -z$). The table identifies the plane xy0 as a mirror plane as well as an n-glide plane, and the current symbol `m' may hence seem inappropriate. Again the question arises: what is `a glide plane'? And what if it can also be a mirror plane?

(5) There are also glide planes for which there is no conventional symbol in IT A83. Corresponding glide reflections are labelled `g' in IT A83. They occur, for example, in space groups P4bm (100), P4/nbm (125), P4₂/nnm (134), R3m (160) and $Fm\bar{3}m$ (225).

Problem 1, in particular the uncertain meaning of `a glide plane', formed a major difficulty for the Ad-hoc Committee when it attempted to design an improved nomenclature for symmetry elements. We therefore undertook the formation of an unambiguous definition of `symmetry element'. Since this term is sometimes confused with `group element', adoption of a new name without the word `element' was contemplated. The original term was nevertheless finally retained since it is so firmly rooted in the crystallographic literature.

The Ad-hoc Committee, however, has introduced the auxiliary new concepts `geometric element' and `element set', in the expectation that they will allow use of the term `symmetry element' in the sense precisely specified below, which does not seriously conflict with the previous use of the term. In $\S$5 we reexamine the above problems 2 to 5 in the light of the new definition.

Copyright © 1989, 1998 International Union of Crystallography