*Acta Cryst.* (1992). A**48**, 727-732

By P. M. DE WOLFF (Chairman), *Meermanstraat 126, 2614 AM Delft, The Netherlands,*

J. D. H. DONNAY, *Department of Geological Sciences, McGill University, 3450 University Street, Montreal, Canada H3A 2A7,*

W. FISCHER, *Institut für Mineralogie, Petrologie und Kristallographie der Phillips-Universität, Lahnberge, D-3550 Marburg (Lahn), Germany,*

R. B. GALIULIN, *Institute of Crystallography, Academy of Sciences of Russia, Leninsky Prospekt 59, Moscow 117333, Russia,*

A. M. GLAZER, *Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU England,*

TH. HAHN, *Institut für Kristallographie, RWTH, Templergraben 55, D-5100 Aachen, Germany,*

MARJORIE SENECHAL, *Department of Mathematics, Smith College, Northampton, MA 01063, USA,*

D. P. SHOEMAKER, *Chemistry Department, Oregon State University, OR 97331, USA,*

H. WONDRATSCHEK, *Institut für Kristallographie, Universität Karlsruhe, Kaiserstrasse 12, D-7500 Karlsruhe 1, Germany,*

A. J. C. WILSON (*ex officio*, IUCr Commission on International Tables), *Crystallographic Data Centre, University Chemistry Laboratory, Lensfield Road, Cambridge CB2 1EW, England*

AND S. C. ABRAHAMS (*ex officio*, IUCr Commission on Crystallographic Nomenclature), *Department of Physics, Southern Oregon State College, 1250 Siskiyou Boulevard, Ashland, OR 97520-5047, USA*

*(Received 7 November 1991; accepted 9 March 1992)*

* Appointed 14 November 1984, modified 10 August 1987 [see * Acta Cryst.* (1986), A**42**, 64 for original membership] under ground rules outlined in *Acta Cryst.* (1979), A**35**, 1072. Final Report accepted 23 December 1991 by the IUCr Commission on Crystallographic Nomenclature and 9 March 1992 by the Executive Committee.

New or redefined printed symbols are proposed in the light of the recently accepted redefinition of symmetry elements [de Wolff *et al.* (1989). *Acta Cryst.* A**45**, 494-499]. In particular, the letter *e* covers certain glide planes which hitherto had no unique symbol, such as those called 'either *a* or *b*'. The use of * e* in the Hermann-Mauguin symbol of five different space groups is recommended. For *e* planes projected in a direction parallel to the plane, a graphical symbol is proposed which removes the ambiguity of their present designation. The letter *k* is proposed for a newly defined class of glide planes which until now were without specific symbol. The symbols for symmetry operations introduced in the space-group descriptions of *International Tables for Crystallography* (1989), Vol. A (Dordrecht: Kluwer Academic Publishers) are recommended for general use, with modifications only for glide reflection operations.

The *Ad-hoc* Committee appointed in 1980 to consider 'nomenclature problems concerning symmetry operations and symmetry elements in space groups' has issued two Reports entitled *Nomenclature for Crystal Families, Bravais-Lattice Types and Arithmetic Classes* (de Wolff *et al.*, 1985) and *Definition of Symmetry Elements in Space Groups and Point Groups* (de Wolff *et al.*, 1989). As noted in the 1989 Report, the only outstanding problem concerning symmetry operations is that of choosing appropriate symbols, since the concept is clear. A provisional notation has been adopted in *International Tables for Crystallography* (1983, 1989), referred to hereafter as *ITA*83.

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 *E*2'.

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. ITA*83), 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 | En_{j} | Line b | Screw rotation about b, angle , u =j/n times shortest latticetranslation 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 |

That is, all rotations and screw rotations with the same axis

In

An important new aspect of symbols like *Eb* may, however, be pointed out. According to *ITA*83, 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 * ITA*83 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 * ITA*83: '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.

The symbol is determined by two criteria. One criterion is the Bravais type ( | ||

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 | - |

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.^{}

The existing graphical symbols as defined and used in * ITA*83 are considered to be adequate by the *Ad-hoc* Committee with one exception. The situation is very different from that of printed symbols because in drawings the projection direction becomes an added parameter. In the terms `perpendicular projection' and `parallel projection' used below, perpendicular and parallel refer to the orientation of the projection direction with respect to the glide plane.

For an *Ee* plane the symbol exists for its perpendicular projection. The existing graphical symbols for the parallel projection of an *Ee*-glide plane, however, suffer from the same ambiguity as the printed symbols. They express the glide vector either parallel or perpendicular to the projection direction, but the choice is arbitrary. Therefore, a new symbol is proposed for the parallel projection of an *Ee* glide plane, namely

All glide planes parallel to two axes in space groups * Fmm*2, *Fmmm*, , and (Nos. 42, 69, 202, 225 and 226), as well as certain planes in the five space groups listed in §3.

Diagonally oriented *Ee* planes occur in space groups * I*4*mm*, *I*4*cm*, , *I*4/*mmm*, *I*4/*mcm*, and (Nos. 107, 108, 121, 139, 140, 217 and 229).

In *ITA*83 all these *Ee* planes are drawn rather arbitrarily as either dotted or dashed lines in their parallel projections. Fig. 4 shows the diagram of space group * I*4*cm* (No. 108), both as shown in *ITA*83 and with use of the new symbol.

Although *Ek* is a new print symbol, in the diagrams of * ITA*83 all glide planes *Ek* have been drawn satisfactorily with the same conventions as used for *Ea*/*b*/*c* glide planes. As in these cases, there is a single special glide vector (perpendicular to the crystal axis in net *N*, * cf.* Table 3). When this vector is, for instance, parallel to the plane of projection, the *Ek* plane is always drawn as the dashed line prescribed by the relevant convention. Therefore, no new graphical symbol is proposed for *Ek* glide planes.

Such planes are readily found in the diagrams of *ITA*83 as planes parallel to just one axis and projected as dashed lines, * e.g.* *xxz* in *P*4*bm* and (Nos. 100 and 113). In particular, Table 3 is exactly in conformity with the distinction shown between *Ek* and * En* planes. See, for instance, the dashed and dot-dashed lines for *R*3*m* and *R*3*c* (Nos. 160 and 161).

The characters appearing after the lattice letter in the Hermann-Mauguin (HM) symbol of a space group were originally meant to represent generating operations of the group. For instance, * b* was a *b*-glide reflection in a plane oriented according to its position in the HM symbol.

In practice, the popular though ill-defined symmetry elements took over from the operations. Thus, *b* came to be seen as a glide plane, *Eb* in our present terminology. There is no harm in that re-interpretation except when the operation belongs to an *Ee*-glide plane. If this holds, for instance, for the above *b*-glide reflection, then there is an *Ee* but no * Eb*-glide plane in the corresponding orientation. In this case, *b* becomes a very misleading character. Apart from this, the bias (given to *b* over *a* or *c*) is just as disturbing as in the case of the symmetry-element symbols treated in §1.

Therefore, it is proposed to *replace such misleading letters* *a* or *b* by *e* in all five HM symbols in which they occur:

Space group No. | 39 | 41 | 64 | 67 | 68 |

Symbol in ITA83: | Abm2 | Aba2 | Cmca | Cmma | Ccca |

New symbol: | Aem2 | Aea2 | Cmce | Cmme | Ccce. |

A further advantage of the proposed new symbols is that *e* - unlike *a* or *b* - is neutral and is therefore not changed upon axis permutation.

A complete set of print symbols was designed by W. Fischer & E. Koch (*ITA*83, §11.2) and was extensively applied in the *Symmetry Operations* sections of the space-group descriptions.

In short, each symbol consists of up to three parts. The first part is a single character (sometimes with an index) which describes the kind of operation. The following part(s) give(s) the components of any relevant shift or translation vector - always in parentheses - and the coordinates of the operation's geometric element, in that order.

The *Ad-hoc* Committee, after considering this system, wishes to introduce two modifications for *glide reflections*:

(i) instead of the present first character (which may be * a*, *b*, *c*, *n*, *d* or *g*), * always write the letter* *g*;

(ii) *always write the glide-vector components* (in parentheses) *in full*, in particular for the simple glide reflections in *a*-, *b*- or *c*-glide planes where they were previously omitted.

Rule (i) suppresses information about the kind of glide plane to which the operation belongs. Very often that information is irrelevant or even confusing. For *a*/*b*/*c* planes the suppression can destroy essential information, but the loss is restored by rule (ii) as shown in the example below.

By adopting these changes, the uniformity of symbols - also with respect to those for rotations - is greatly improved. For instance, the symbol of the glide reflection in the plane , with the unusual glide vector , namely , now falls in line with that for a simple *b*-glide reflection. In *ITA*83 the latter was denoted by , but this is changed by rule (ii) into .

The above rules apply equally to glide reflections belonging to the element set of a mirror plane. Thus, if the shift component of such an operation is (0, 1, 2), then its symbol begins with * g*(0, 1, 2), not with *m*(0, 1,2).

*International Tables for Crystallography* (1983). Vol. A, edited by TH. HAHN, especially §§ 1.1, 1.3, 1.4, 7 and 11. Dordrecht: Kluwer Academic Publishers.

*International Tables for Crystallography* (1989). Vol. A, 2nd, revised, ed., edited by TH. HAHN. Dordrecht: Kluwer Academic Publishers.

- ...vectors.
- The glide-plane symbols
*c*,*n*and*k*for the three diagrams (2), (9) and (14) of Fig. 3 have of course been assigned according to Table 3. One member of the*Ad-hoc*Committee (Professor Wondratschek) thinks that the symbol*n*for diagram (9) conflicts with traditional notions about*n*planes. He proposes the assignment of a special symbol to diagram (9). This proposal has, however, not received adequate support within the*Ad-hoc*Committee.