International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by M. I. Aroyo

International Tables for Crystallography (2016). Vol. A, ch. 3.1, p. 713

Table 3.1.3.2 

P. M. de Wolffc

Table 3.1.3.2| top | pdf |
Lattice characters described by relations between conventional cell parameters

Under each of the roman numerals below `Lattice characters in', numbers of characters (cf. Table 3.1.3.1[link], first column) are listed for which the key parameter p lies in the interval defined by the same roman numeral below `Intervals of p'. For instance, a lattice with character No. 15 under IV has [p = c/a]; so it falls in the interval IV with [2^{1/2} \,\lt \,c/a\ (\lt\, \infty)]; No. 33 under II has [p = b]; therefore the interval [a - c] for II yields the relation [a \,\lt \,b\, \lt\, c].

Lattice symmetryBravais type of latticep = key parameterLattice characters inIntervals of pConventions
IIIIIIIVIIIIIIIV 
Tetragonal tP [c/a] 21 11 0   1        
Tetragonal tI [c/a] 18 6 7 15 0   [(2/3)^{1/2}]   1   [2^{1/2}]    
Hexagonal hP [c/a] 22 12 0   1        
Rhombohedral hR [c/a] 24 4 2 9 0   [(3/8)^{1/2}]   [(3/2)^{1/2}]   [6^{1/2}]   Hexagonal axes
Orthorhombic oP 32 no relations         [a \,\lt\, b\, \lt \,c]
Orthorhombic oS                              
[\quad b \,\lt\, a\sqrt 3]   c 23 13 0   d§       [a\, \lt\, b]
[\quad b\, \gt\, a\sqrt 3]   c 40 36 38 0   a   d§      
Orthorhombic oI r 8 19 42 0   a   b     [a \,\lt\, b\, \lt\, c]
Orthorhombic oF [b/a] 16 26 1   [3^{1/2}]       [a \,\lt\, b \,\lt\, c]
Monoclinic mP b 35 33 34 0   a   c     [a \,\lt\, c]††
Monoclinic mS                              
[\matrix{{\rm Centred}\hfill\cr{\rm net}\hfill}\left\{\phantom{\let\normalbaselines\relax\openup10pt\matrix{=1\hfill\cr=2\hfill\cr=2,3\hfill\cr=1,2,3\hfill\cr=3\hfill}}\right.] = 1‡‡   [b/a] [\cases{28\cr 29\cr}] [\left.{\phantom{\let\normalbaselines\relax\openup10pt\matrix{\cr\cr\cr\cr}}}\right\}] 0   [(1/3)^{1/2}]   1   [3^{1/2}]   C centred††
= 2   [b/a] 30
= 2, 3   [b/a] [\cases{37\cr 41\cr}] 20 25
= 1, 2, 3   [b/a] [\cases{27\cr 39\cr}] [\cases{10\cr 17\cr}] 14
= 3   [b/a] 43 1         I centred††
Triclinic aP [\alpha, \beta, \gamma] 31 44 60°   90°   120°     [a\, \lt\, b \,\lt\, c]
Cubic cP [\left.\matrix{3\cr 5\cr 1\cr}\right\}\hbox{ no relations}]          
cI          
cF          
The symbols for Bravais types of lattices were adopted by the International Union of Crystallography in 1985; cf. de Wolff et al. (1985[link]).
These conventions refer to the cells obtained by the transformations of Table 3.1.3.1[link]. They have been chosen for convenience in this table.
§[d = {1 \over 2} (a^{2} + b^{2})^{1/2}].
[r = {1 \over 2} (a^{2} + b^{2} + c^{2})^{1/2}].
††Setting with unique axis b; [\beta\, \gt\, 90^{\circ}]; [a\, \lt \,c] for both P and I cells, [a\, \lt\, c] or [a\, \gt\, c] for C cells.
‡‡This number specifies the centred net among the three orthogonal nets parallel to the twofold axis and passing through (1) the shortest, (2) the second shortest, and (3) the third shortest lattice vector perpendicular to the axis. For example, `2, 3' means that either net (2) or net (3) is the centred one.