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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'.
Symmetry operation | Geometric element | Additional parameters |
---|---|---|
Identity | Not required | None |
Translation | Not required | Vector |
Reflection in plane A | Plane A | None |
Glide reflection =reflection in plane A and translation parallel to A | Plane A | Glide vector |
Rotation about line b | Line b | Angle and sense of rotation |
Screw rotation = rotation about line b and translation parallel to b | Line b | Angle and sense of rotation screw vector |
Rotoinversion = rotation about line b and inversion through point P on b | Line b and point P on b | Angle (not equal to ) 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.
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 ,n = 2, 3, 4 or 6 | 1st,,(n - 1)th powers of d.o. and their coaxial equivalents |
Screw axis | Enj | Line b | Screw rotation about b, angle , u =j/n times shortest lattice translation along b, right-hand screw; n = 2, 3, 4 or 6, |
1st,,(n - 1)th powers of d.o. and their coaxial equivalents |
Rotoinversion axis | Line b and point P on b | Rotoinversion: rotation about b, angle , and inversion through P; n = 3, 4 or 6 | D.o. and its inverse | |
Center | Point P | Inversion through P | D.o. only |
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 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.
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.
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.
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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