*Acta Cryst.* (1985). A**41**, 278-280

# Nomenclature for crystal families, Bravais-lattice types and arithmetic classes

## Report of the International Union of Crystallography *Ad-Hoc* Committee on the nomenclature of symmetry*

BY P. M. DE WOLFF (Chairman), *Department of Applied Physics, Delft University of Technology, Lorentzweg
1, 2628 CJ Delft, The Netherlands*

N. V. BELOV,^{+} * Institute of Crystallography, Academy of Sciences of the
USSR, Leninsky pr. 59, Moscow 117333, USSR*

E. F. BERTAUT, *Laboratoire de Cristallographie, CNRS
Grenoble, 25 Avenue des Martyrs, BP 166X Centre de Tri, 38042 Grenoble CEDEX, France*

M. J. BUERGER,
*PO Box 361, Lincoln Center, MA 01773, USA*

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 Philipps-Universität, Lahnberge, D-3350 Marburg (Lahn), Federal Republic of Germany*

TH. HAHN, *Institut für Kristallographie, RWTH, Templergraben 55, D-5l00 Aachen, Federal Republic
of Germany*

V. A. KOPTSIK, *Moscow State University, Department of Physics, Leninskiye Gory, Moscow
117234, USSR*

A. L. MACKAY, *Department of Crystallography, Birkbeck College, London Wi E 7HX, England*

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

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

AND S. C. ABRAHAMS (*ex officio*, IUCr Commission on Crystallographic Nomenclature), *AT&T Bell Laboratories,
Murray Hill, NJ 07974, USA*

*(Received 13 August 1984; accepted 13 November 1984)*

### Abstract

Standard symbols representing crystal families, two- and three-dimensional Bravais-lattice types and arithmetic classes are recommended for use by the IUCr. The six crystal families are designated by lower-case letters. The family letter in the symbol of each of the 14 lattice types is followed by an upper-case letter to distinguish different lattice types within the family. Arithmetic classes are indicated by modified symbols of the corresponding symmorphic space groups.### Crystal families

Six lower-case letters, which originate from words denoting the shape of commonly used unit cells, are recommended as the standard designation of*crystal families*(Table 1). A 'crystal family' is a well defined and unambiguous concept in crystallography [

*International Tables for Crystallography*(1983), sections 2.1 and 8.2.6]. Comparing the classification into crystal families with existing classifications into 'crystal systems', it is noted that they coincide except for the hexagonal family in three dimensions, which contains all 'hexagonal', 'trigonal' and 'rhombohedral' space groups. Although this set is often regarded as a single 'crystal system' in the American and Russian literature, other widespread usage divides it into two different systems. Therefore, association of the lower-case letters with crystal systems (instead of families) is apt to cause confusion and is not recommended by the

*Ad-hoc*Committee.

Adoption of these symbols yields a clear interpretation of the lattice symbols, see below. Direct applications are not envisaged but future use is likely, since the ambiguity of the term 'crystal system' (see above) has so far prevented the general acceptance of a suitable nomenclature for this scale of classification. No such difficulty exists with the term 'crystal families'.

Table 1. *Standard symbols for the crystal families*

Two-dimensional Three-dimensional Symbol crystal family crystal familya- Triclinic (anorthic)mOblique MonoclinicoRectangular OrthorhombictSquare TetragonalhHexagonal Hexagonalc- Cubic

### Bravais-lattice types

Symbols for designating Bravais-lattice types independently of any description by a particular unit cell with labelled axes are presented in Table 2. The initial lower-case letter characterizes the crystal family (see above) to which the Bravais-lattice type belongs. The second letter designates the type of centring. These symbols are identical with those listed in*International Tables for Crystallography*(1983) and used in the 60-year index to

*Structure Reports*(1976),

^{++}with the exception of

*mS*and

*oS*, which replace

*mC*and

*oC*.

The letter *C* in the monoclinic and orthorhombic
Bravais-lattice-type symbols is ambiguous since *C* is
closely associated with one of the common setting
and/or unit-cell choices for the lattice, namely *A*, *C*
or *I* centring for the monoclinic family and *A*, *B* or
*C* centring for the orthorhombic family. The *Ad-hoc*
Committee has chosen the latter *S*, derived from
'side-face centred' (*i.e. seitenflächenzentriert*), to
replace *C*. This letter, which is also used to designate
similarly centred four-dimensional lattices (Wondratschek,
Bülow & Neubüser, 1971), is clearly appropriate
for *oS*-type lattices. In the case of *mS*-type lattices,
the letter *S* evokes neither an *A*- nor a *C*-centred in
preference to an *I*- centred unit cell: the symbol for
the monoclinic centred lattice remains *mS* regardless
of the basis actually used.

A similar comment applies to the symbol *hR*, which
designates the rhombohedral lattice regardless of
whether that lattice is described (using rhombohedral
coordinates) by a primitive rhombohedral unit cell
or (using hexagonal coordinates) by a hexagonal unit
cell.

Table 2. *Standard symbols for the Bravais lattice types*

Two-dimensional Three-dimensional Symbol lattice type Symbol lattice typempObliqueaPTriclinicopRectangular primitivemPMonoclinic primitiveocRectangular centredmSMonoclinic centredtpSquareoPOrthorhombic primitivehpHexagonaloSOrthorhombic single-face centredoIOrthorhombic body- centredoFOrthorhombic all faces centredtPTetragonal primitivetITetragonal body-centredhPHexagonal primitivehRRhombohedralcPCubic primitivecICubic body-centredcFCubic face-centred

### Arithmetic classes

The*Ad-hoc*Committee recommends that an

*arithmetic class*be designated by the modified symbol of the symmorphic space group contained in that class. The modification consists of interchanging the lattice letter and the point-group symbol. For example, instead of , write for the arithmetic class.

An arithmetic class is a class of space groups. Two space groups belong to the same arithmetic class if the following procedure leads to identical results for both:

(*a*) In the Hermann-Mauguin symbol replace
screw axes by rotation axes and glide planes by mirror
planes.

(*b*) Write the ensuing symbol in standard form,
*e.g. Cmm*2 for *A*2*mm*.

The result is the (standard) symbol of a symmorphic space group. It follows that the arithmetic class can be represented by the type of symmorphic space groups that it contains.

The arithmetic class is a concept widely used in
symmetry classification and in mathematical crystallography.
The proposed code not only characterizes
the class adequately, but is also recognizable as a
code for just this purpose.^{+++}

For further discussion of arithmetic classes, see
*International Tables for Crystallography* (1983).

### References

*International Tables for Crystallography* (1983). Vol. A, especially
sections 2.1 and 8.2. Dordrecht: Reidel.

Laves, F. (1966). Report to Seventh General Assembly of IUCr, p. 27,
Appendix *D(a)*, Moscow.

Pearson, W. B. (1967). *A Handbook of Lattice Spacings and Structures
of Alloys*, Vol. 2. New York:Pergamon Press.

*Structure Reports* (1976). *60-Year Structure Index 1913-1973. A.
Metals and Inorganic Compounds.* Utrecht: Bohn, Scheltema & Holkema.

Wondratschek, H., Bülow, R. & Neubüser, J. (1971). *Acta
Cryst.* A**27**, 523-535.

* Appointed 18 March 1980 under ground rules outlined in *Acta
Cryst.* (1979), A**35**, 1072. Final Report accepted 17 August 1984
by the IUCr Commission on Crystallographic Nomenclature and
13 November 1984 by the Executive Committee.
^{+} Deceased 6 March 1982.
^{++} The first use of this nomenclature was in an example cited by
Laves (1966) as chairman of the IUCr Subcommittee on Structure
Type Designation. All 14 symbols were published and extensively
used by Pearson (1967).
^{+++} Note added in proof by the Chairman: The new symbol for
classes with symmorphic group symbol *Pm* and *Cm* is identical
(*mP*) or very similar (*mC*) to one of the new lattice type symbols
given in the preceding section. In view of the context in which the
symbols will be used, confusion is not expected to occur.

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