Space Group Notation

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The space group symbols used throughout this CD-ROM follow the Hermann-Mauguin notation. The equivalent Schoeflies symbols are not provided.

The initial letter of a space group symbol represents the lattice type which may primitive (P), single-face centred (A, B, or C), all-face centred (F), body-centred (I), or rhomohedrally centred (R). (For rhomohedral space groups, a primitive unit cell may also be chosen, but the symbol R is still used so as to distinguish these space groups from the primitive trigonal space groups based on hexagonal axes.)

Note that in the discussion given below, the bar above an axis (indicating an inversion axis) is written with a minus sign in front of the rotation axis symbol.

Triclinic: Symbol type P-1.

Following the lattice type, the remaining symbol is either 1 or -1 showing the absence or presence of an inversion centre, respectively.

Monoclinic: Symbol types P2, Pm, P2/m.

After the lattice type, the remaining part of the space group symbol indicates symmetry with respect to the unique axis direction, i.e. axes parallel to the unique axis or planes perpendicular to it. Note that the short form of the space group symbol omits the two "1"s for the symmetry with respect to the other two axes.

Orthorhombic: Symbol types P222, Pmm2 (or Pm2m or P2mm), Pmmm.

After the lattice type, there are three parts to the space group symbol indicating the symmetry with respect to the x, y, and z axis directions, respectively. Thus, for example, the space group symbol Pnma indicates a primitive lattice with an n-glide plane perpendicular to the x axis, a mirror plane perpendicular to the y axis, and an a-glide plane perpendicular to the z axis.

Tetragonal: Symbol types P4, P-4, P4/m, P422, P4mm, P-42m (or P-4m2), P4/mmm.

The tetragonal space groups may be subdivided into two groups: Those with and without additional symmetry with respect to the x and y axes. The fourfold symmetry is always chosen to lie parallel to the z axis and is specified second in the space group symbol after the lattice type. For those space groups with symmetry along other the other axes, e.g. as in P-42m, the next part of the symbol indicates the symmetry with respect to both the x and y axes. The remaining part of the symbol indicates the symmetry with respect to both of the diagonals between the x and y axes.

Note that for the enlarged tetragonal unit cells, i.e. C or F centred unit cells, the symbols chosen to describe the symmetry with respect to the x and y axes, and with respect to the face-diagonal directions is simply the same as that used for the equivalent P and I centred cells, but with the order of the symbols reversed e.g. the symbol C4/mmb is used for the enlarged unit cell of space group P4/mbm. A more logical symbol in this instance would be C4/mmn.

Trigonal & Rhombohedral: Symbol types P3, P-3, P321, P312, P3m1, P31m, P-3m1, P-31m.

The threefold symmetry is always chosen to lie parallel to the z axis and the symbol for it follows the lattice type in the space group symbol. Additional symmetry elements may be lie parallel to the x and y axes (e.g. the twofold rotation axes in P321) or perpendicular to them (e.g. the twofold rotation axes in P312). The order of the last two components of the space group symbol is used to distinguish the two possibilities.

Hexagonal: Symbol types P6, P-6, P6/m, P622, P6mm, P-62m (or P-6m2), P6/mmm.

The order of the symbols in the hexagonal space group symbols is similar to that of the trigonal space group symbols except that the symmetry with respect to the z axis is now of order six.

Cubic: Symbol types P23, Pm-3, P432, P-43m, Pm3m.

The symbols of the cubic space group symbols refer to the lattice type (P, F, or I) followed by symmetry with respect to the x, y, and z axes, then the threefold symmetry of the body diagonals, followed lastly by any symmetry with respect to the face diagonals if present.

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