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5. Comparison with IT A83

The essential feature of the definition of symmetry elements in Table 1.3 of IT A83 is that they are defined by a `generating symmetry operation'. In mathematics, an operation `generates' a certain group which is clearly not meant here. Hence, `generating' is taken as having the sense of `characteristic for'. Thus, Table 1.3 leads to the ambiguous or uncertain problems 2 to 5 of the Introduction.

By contrast, the concept `symmetry element' in the present proposal depends primarily not on a single operation but on a geometric element (GE). Given that GE, it is the element set (consisting of the operations sharing that GE) that determines the nature of (and, eventually, the symbol for) the symmetry element. There is always just one such set, so that no ambiguity can exist and it is only the symbol which may be open to discussion.

Since the difficulties with the IT A83 description occur mainly with glide planes, these are now re-examined. In Fig. 1(a) we may imagine that, for a given crystal structure, $P \rightarrow Q$ represents a symmetry reflection through the plane A.

It may be recalled that there also exists an infinite number of glide reflections (all of them symmetry operations) with respect to the same plane: $P \rightarrow R, S$ etc. (These are the coplanar equivalents of the operation $P \rightarrow Q$ mentioned in Table 2.) All these operations have the plane A as their common GE; together with the reflection they form the element set. In the present definition, the symmetry element in this case is always a mirror plane. In particular, problem 4 of the Introduction falls in this category. The new definition does not allow the xy0 plane in space group Cmmm to be called `a mirror and also an n-glide plane'. The fact that one of the coplanar equivalents happens to have the shift (a + b)/2, which is a lattice translation, does not change the nature (mirror plane) of the symmetry element.

The other alternative occurs if a glide reflection $P \rightarrow R$ is given through plane A, and none of its coplanar equivalents [discussed in note (iv) to Table 1.3 in IT A83] is a reflection (Fig. 1b). Then, plane A is always a glide plane. The present Report recognizes only geometrically distinct symmetry elements. Hence, in the case of problem 3 of the Introduction, an interpretation such as `both an a- and a b-glide plane' should no longer be possible: here the plane xy0 is just one single glide plane, whatever symbol is assigned to it. It is recommended that the symbols used to express this primary differentiation between mirror and glide planes. This can be done simply by reserving `m' exclusively for mirror planes, as was done by calling them Em. Any differentiation between various kinds of glide planes falls beyond the scope of the present Report.

The `special' glide planes of problem 5 in the Introduction need no further discussion beyond the statement that they are true glide planes, without special distinction in the sense of this Report.

A very similar situation exists for twofold axes. If, for a given twofold axis, a 180$^{\circ}$ screw rotation $P \rightarrow R$ is a symmetry operation, then combination with all symmetry translations parallel to the axis generates the element set consisting of infinitely many operations. If, among these, there is a rotation of 180$^{\circ}$ such as $P \rightarrow Q$ in Fig. 1(a) (where A now represents the axis), then the symmetry element is a twofold rotation axis E2. If there is no such operation among them (Fig. 1b), then the symmetry element is a two-fold screw axis E2₁. An axis cannot be both simultaneously, as a strict application of Table 1.3 in IT A83 would require for twofold axes parallel to (110) in structures with cF lattices; such axes can only be rotation axes.

Problem 2 of the Introduction has been solved in this Report by taking the conventional `line-plus-point' GE for a rotoinversion.

 

Figure 1: (a) Array related by the mirror plane A. (b) Array related by the glide plane A.

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