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1. Printed symbols for symmetry elements

The definition of symmetry elements as given in the 1989 Report (de Wolff et al.) will be used throughout the present Report. Here we repeat the essence:

For any given symmetry operation its geometric element (plane, point and/or line) is defined. A symmetry element is the combination of the geometric element of one of the symmetry operations in a given space group with the set (called `element set') of all symmetry operations in that space group which share this geometric element.

Explicit definitions of geometric elements and descriptions of the ensuing symmetry elements as well as their symbols are given in Tables 1 and 2. (These are identical to Tables 1 and 2 in the 1989 Report except for glide planes and are repeated here for completeness, see below). Each symmetry element is represented by a symbol consisting of two characters. The first character is an upper-case E for all symmetry elements. It serves to show that the symbol refers to a symmetry element and not, for instance, to a symmetry operation. If this is clear already from the context, then the E may be omitted, e.g. `an axis 2' instead of `an axis E2'.

 

Table 1. Geometric elements of symmetry operations in point groups and space groups

Symmetry operation	Geometric element	Additional parameters
Identity	Not required	None
Translation	Not required	Vector $\mathbf{t}$
Reflection in plane A	Plane A	None
Glide reflection =reflection in plane A and translation $\mathbf{v}$ parallel to A	Plane A	Glide vector $\mathbf{v}$
Rotation about line b	Line b	Angle and sense of rotation
Screw rotation = rotation about line b and translation $\mathbf{u}$ parallel to b	Line b	Angle and sense of rotation screw vector $\mathbf{u}$
Rotoinversion = rotation about line b and inversion through point P on b	Line b and point P on b	Angle (not equal to $\pi$) and sense of rotation
Inversion through point P	Point P	None

The symbol Eg listed in the 1989 Report can be used for glide planes if one merely wants to show that the symmetry element is a glide plane. On the other hand, if it belongs to one of the special kinds which have long been denoted by an appropriate letter (a, b, c, n or d; cf. ITA83), then that letter replaces g in Eg.

 

Table 2. Symmetry elements in point groups and space groups

Name of symmetry element	Symbol	Geometric element	Defining operation (d.o.)	Operations in element set
Mirror plane	Em	Plane A	Reflection in A	D.o. and its coplanar equivalents*
Glide plane	Eg	Plane A	Glide reflection in A, 2v (not v) a lattice translation	D.o. and its coplanar equivalents*
Rotation axis	En	Line b	Rotation about b, angle $2\pi/n$,n = 2, 3, 4 or 6	1st,$\dots$,(n - 1)th powers of d.o. and their coaxial equivalents
Screw axis	Enj	Line b	Screw rotation about b, angle $2\pi/n$, u =j/n times shortest lattice translation along b, right-hand screw; n = 2, 3, 4 or 6, $j=1,\dots,(n-1)$	1st,$\dots$,(n - 1)th powers of d.o. and their coaxial equivalents
Rotoinversion axis	$E\bar n$	Line b and point P on b	Rotoinversion: rotation about b, angle $2\pi/n$, and inversion through P; n = 3, 4 or 6	D.o. and its inverse
Center	$E\bar 1$	Point P	Inversion through P	D.o. only

* That is, all glide reflections with the same reflection plane, with glide vectors differing from that of the d.o. (taken to be zero for a reflection) by a lattice translation vector.
That is, all rotations and screw rotations with the same axis b, the same angle and sense of rotation and the same screw vector $\mathbf{u}$ (zero for a rotation) up to a lattice translation vector.
In Eg, g is replaced by a, b, c, n, d, e or k for specific kinds of glide planes, cf. §2.

An important new aspect of symbols like Eb may, however, be pointed out. According to ITA83, denoting a plane by b merely meant that a glide reflection in the plane with a glide component b/2 along the b axis is a symmetry operation. This definition certainly applies to the situation depicted in Fig. 1.

 

Fig. 1. (After W. Fischer.) The element set of an Eb-glide plane, shown as a set of points above (+) and below (-) the plane produced by glide reflections in the plane, starting for instance from the + sign at upper left. The net N of translations parallel to the plane (+$\cdots$ + vectors) is indicated by a mesh, which in this case happens to be rectangular. Both pairs of edges are parallel to crystal axes. There is a glide reflection with its glide vector (+ $\cdots$ -) along the b axis.

Fig. 1 is adapted, as are Figs. 2 and 3, from a set of similar figures designed by Ad-hoc Committee member W. Fischer as an inventory of all types of glide plane. Although the set was presented to the Committee in 1980, long before the 1989 Report came out, each of its figures shows precisely the `element set' of the glide plane as defined in that Report (cf. the above summary). For a glide plane, the element set consists of all glide reflections having the plane as their common geometric element. Their action is shown in projection upon this plane. From the starting position of any + sign, each - sign results from one of the glide reflections of the set. All of these are shown within an elementary mesh of the resulting two-dimensional periodic pattern of + and - signs.

We shall often refer to the net N formed by all translations parallel to the plane; this net is easily visualized by looking at + signs only. These vectors are to be distinguished sharply from the vectors connecting a + sign with any - sign, each of which is the glide vector of a glide reflection belonging to the element set.

The new aspect arises because, in some cases, by the ITA83 definition, the b-glide plane is also an a-glide plane; see Fig. 2. Clearly this happens only if the net N is orthogonal centred, because then the a glide can be changed into a b glide (and vice versa) by adding a centring translation. The practice so far has been to call such a glide plane arbitrarily either a or b, thus causing an unjustified bias and a lack of uniqueness in these symbols. Therefore, we propose that the case of Fig. 2 be covered by a separate symbol.

 

Fig. 2. (After W. Fischer.) The element set of an Ee-glide plane. Cf. caption of Fig. 1. Note that the net N here is orthogonal centred.

The scope of this symbol should then be extended to glide planes in a diagonal orientation, that is, parallel to just one crystal axis, provided that the glide plane has a glide vector along that axis and that the net N is orthogonal centred. For such planes there is not the ambiguity of the above a-b random choice, but the extended scope of the new symbol is in line with that of all existing symbols (namely a, b, c, n and d). Each of these is used for a glide plane with both one and two crystal axes in the net N, cf. Fig. 3.

The letter e is proposed for the new symbol. Thus, Ee will apply to glide planes with orthogonal centred nets N and at least one glide vector along a crystal axis. A new criterion is hence necessary: namely the orientation of glide vectors with respect to the conventional axes of the crystal. Since the latter are along symmetry directions, whereas every glide plane is parallel to a mirror plane of the lattice, it is not surprising that there is always at least one conventional crystal axis in N. If there is only one such axis, then perpendicular to it there is always another translation in N.

The new symbol e as well as old symbols a, b, c, d, n will now be redefined in terms of this new criterion and of the Bravais type of net N. This net is monoclinic or orthogonal or tetragonal primitive (mp or op or tp) or orthogonal centred (oc). [The Bravais-net-type symbols are those introduced in the Ad-hoc Committee's first Report (de Wolff et al., 1985).] Only oc-type nets N allow an Ee-glide plane. The symbol En is applicable to nets N of the Bravais type mp or op, whereas Ed is for oc-type nets N. (As stated in a footnote to Table 1.3 in ITA83: `Glide planes d occur only in orthorhombic F space groups, in tetragonal I space groups, and in cubic I and F space groups. They always occur in pairs with alternating glide vectors'.) In contrast to Ea, Eb, Ec and Ee planes, however, for En and Ed planes there is no glide vector either parallel or perpendicular to a conventional axis in N.

The ensuing definitions of the glide planes of the above kinds are summarized in lines (i) and (ii) of Table 3, and more explicitly in Fig. 3.

 

Table 3. Printed symbols for special kinds of glide planes

Number of glide vectors parallel or perpendicular to crystal axes in net N	Bravais type of net N
	m, op, tp	oc
(i) One or two parallel	Ec(Ea, Eb)	Ee
(ii) None parallel, none perpendicular	En	Ed
(iii) None parallel, one perpendicular	Ek	-

* As defined in ITA83, § 9.1; however, for rhombohedral space groups, hexagonal axes only are used here.

All remaining glide planes were previously without specific symbol. They each have a diagonal orientation (just one conventional crystal axis in the net N). Among the glide reflections in their element set, there is none with a glide vector along that axis. However, one glide vector is (by symmetry) perpendicular to it. A symbol seems desirable, so again a new letter is proposed: k. The new symbol Ek is briefly defined in line (iii) of Table 3 and is fully illustrated in the lower block of Fig. 3. Some examples are given in §2.

 

Fig. 3. (Adapted from W. Fisher's drawings.) All possible aspects of the element sets of glide planes shown as in Fig. 1, but independent of axis labels. The diagrams are grouped in columns headed by the Bravais-net-type symbol (top line) of their nets N, cf. Table 3. The other criteria of that table are verified by looking first at the double lines showing the directions of crystal axes in the plane. One edge (vertical) of the mesh of N shown is always chosen along such an axis. The other edge is horizontal except in (1) and (8). For diagrams (1),$\dots$,(5), the glide-plane symbol is the label a, b or c of the vertical axis; for the others it is the encircled letter in the outlined block containing the diagram. Note the vertical glide vectors in diagrams (1),$\dots$,(7), the horizontal ones in (6), (7), (14), (15) and the absence of either in (8),$\dots$,(13). An example of occurrence is given below each diagram by the space-group symbol and the coordinate triplet of the plane.

In Fig. 3, Fischer's inventory of all types of glide plane is shown in an abbreviated - though still complete - fashion in which more graphical prominence has been given to the crystal axes. For each diagram, one example of its occurrence in a space group is listed.

In some rhombohedral space groups, diagonally oriented Ec, En and Ek planes occur with mp-type nets N which can be described by threefold centring of an orthogonal net. A rectangular triple mesh of the net N is shown for these types of glide planes in Fig. 3, diagrams (2), (9) and (14). In diagram (9), the similarity to other n diagrams such as (8) or (10) is recognized if in (9) a monoclinic primitive mesh of net N is considered with diagonal glide vectors.

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