International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by Th. Hahn

International Tables for Crystallography (2006). Vol. A, ch. 13.2, pp. 843-844
https://doi.org/10.1107/97809553602060000529

Chapter 13.2. Derivative lattices

Y. Billieta and E. F. Bertautb§

aDépartement de Chimie, Faculté des Sciences et Techniques, Université de Bretagne Occidentale, Brest, France, and bLaboratoire de Cristallographie, CNRS, Grenoble, France

The lattices that correspond to the isomorphic subgroups of space group P1 and plane group p1 are termed derivative lattices. Formulae for the construction of three- and two-dimensional derivative lattices are given, and the three- and two-dimensional lattices of indices 2 to 7 are tabulated.

13.2.1. Introduction

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The three-dimensional subgroups of space group P1 and the two-dimensional subgroups of plane group p1 are all isomorphic subgroups; i.e. these subgroups are pure translation groups and correspond to lattices. In the past, these lattices have often been called `superlattices' (the term `sublattice' perhaps would be more precise). To avoid confusion, the lattices that correspond to the isomorphic subgroups of P1 and p1 are designated here as derivative lattices.

The number of derivative lattices (both maximal and nonmaximal) of a lattice is infinite and always several derivative lattices of index [[i] \geq 2] exist. Only for prime indices are maximal derivative lattices obtained; for any prime p, there are [(p^{2} + p + 1)] three-dimensional derivative lattices of P1, whereas there are [(p + 1)] two-dimensional derivative lattices of p1. The number of nonmaximal derivative lattices is given by more complicated formulae (cf. Billiet & Rolley Le Coz, 1980[link]).

13.2.2. Construction of three-dimensional derivative lattices

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It is possible to construct in a simple way all three-dimensional derivative lattices of a lattice (Table 13.2.2.1).[link] Starting from a primitive unit cell defined by a, b, c, each derivative lattice possesses exactly one primitive unit cell defined by [{\bf a}'], [{\bf b}'], [{\bf c}'] by means of the following relation[\eqalign{&{\bf a}' = p_{1} {\bf a},\quad {\bf b}' = p_{2}{\bf b} + q_{1}{\bf a},\quad {\bf c}' = p_{3}{\bf c} + r_{1}{\bf a} + q_{2}{\bf b}\cr &(p_{1}, p_{2}, p_{3}\ \hbox{positive integers, not necessarily prime};\cr &\hbox{index} = p_{1}p_{2}p_{3} \gt 1; q_{1}, q_{2}, r_{1}\ \hbox{integers};\cr &- p_{1}/2 \lt q_{1} \leq p_{1}/2; - p_{2}/2 \lt q_{2} \leq p_{2}/2;\cr &- p_{1}/2 \lt r_{1} \leq p_{1}/2).}]

Table 13.2.2.1| top | pdf |
Three-dimensional derivative lattices of indices 2 to 7

The entry for each derivative lattice starts with a running number which is followed, between parentheses, by the appropriate basis-vector relations. It should be noted that the seven derivative lattices of index 2 are also recorded in the space-group table of P1 (No. 1) under Maximal isomorphic subgroups of lowest index but in the slightly different sequence 1, 2, 3, 6, 5, 4, 7.

Index 2 [1(2{\bf a},{\bf b},{\bf c})]; [2({\bf a},2{\bf b},{\bf c})]; [3({\bf a},{\bf b},2{\bf c})]; [4(2{\bf a},{\bf b} + {\bf a},{\bf c})]; [5(2{\bf a},{\bf b},{\bf c} + {\bf a})]; [6({\bf a}, 2{\bf b}, {\bf c} + {\bf b})]; [7(2{\bf a}, {\bf b} + {\bf a},{\bf c} + {\bf a})]
Index 3 [1(3{\bf a}, {\bf b}, {\bf c})]; [2({\bf a}, 3{\bf b}, {\bf c})]; [3({\bf a}, {\bf b}, 3{\bf c})]; [4(3{\bf a}, {\bf b} + {\bf a}, {\bf c})]; [5(3{\bf a}, {\bf b}, {\bf c} + {\bf a})]; [6({\bf a}, 3{\bf b}, {\bf c} + {\bf b})]; [7(3{\bf a}, {\bf b} - {\bf a}, {\bf c})]; [8(3{\bf a}, {\bf b}, {\bf c} - {\bf a})]; [9({\bf a}, 3{\bf b}, {\bf c} - {\bf b})]; [10(3{\bf a}, {\bf b} + {\bf a},{\bf c} + {\bf a})]; [11(3{\bf a}, {\bf b} + {\bf a},{\bf c} - {\bf a})]; [12(3{\bf a}, {\bf b} - {\bf a},{\bf c} + {\bf a})]; [13(3{\bf a}, {\bf b} - {\bf a},{\bf c} - {\bf a})]
Index 4 [1(4{\bf a}, {\bf b}, {\bf c})]; [2({\bf a}, 4{\bf b}, {\bf c})]; [3({\bf a}, {\bf b}, 4{\bf c})]; [4(4{\bf a}, {\bf b} + {\bf a}, {\bf c})]; [5(4{\bf a}, {\bf b}, {\bf c} + {\bf a})]; [6({\bf a}, 4{\bf b}, {\bf c} + {\bf b})]; [7(4{\bf a}, {\bf b} - {\bf a}, {\bf c})]; [8(4{\bf a}, {\bf b}, {\bf c} - {\bf a})]; [9({\bf a}, 4{\bf b}, {\bf c} - {\bf b})]; [10(4{\bf a}, {\bf b} + 2{\bf a}, {\bf c})]; [11(4{\bf a}, {\bf b}, {\bf c} + 2{\bf a})]; [12(2{\bf a}, 2{\bf b} + {\bf a}, {\bf c})]; [13({\bf a}, 4{\bf b}, {\bf c} + 2{\bf b})]; [14(2{\bf a}, {\bf b}, 2{\bf c} + {\bf a})]; [15({\bf a}, 2{\bf b}, 2{\bf c} + {\bf b})]; [16(4{\bf a}, {\bf b} + {\bf a},{\bf c} + {\bf a})]; [17(4{\bf a}, {\bf b} + {\bf a},{\bf c} - {\bf a})]; [18(4{\bf a}, {\bf b} - {\bf a},{\bf c} + {\bf a})]; [19(4{\bf a}, {\bf b} - {\bf a},{\bf c} - {\bf a})]; [20(4{\bf a}, {\bf b} + {\bf a},{\bf c} + 2{\bf a})]; [21(4{\bf a}, {\bf b} + 2{\bf a},{\bf c} + {\bf a})]; [22(2{\bf a}, 2{\bf b} + {\bf a},{\bf c} + {\bf b})]; [23(4{\bf a}, {\bf b} - {\bf a},{\bf c} + 2{\bf a})]; [24(4{\bf a}, {\bf b} + 2{\bf a},{\bf c} - {\bf a})]; [25(2{\bf a}, 2{\bf b} + {\bf a},{\bf c} + {\bf a} + {\bf b})]; [26(4{\bf a}, {\bf b} + 2{\bf a},{\bf c} + 2{\bf a})]; [27(2{\bf a}, 2{\bf b} + {\bf a},{\bf c} + {\bf a})]; [28(2{\bf a}, {\bf b} + {\bf a}, 2{\bf c} + {\bf a})]; [29(2{\bf a}, 2{\bf b}, {\bf c})]; [30(2{\bf a}, {\bf b}, 2{\bf c})]; [31({\bf a}, 2{\bf b}, 2{\bf c})]; [32(2{\bf a}, 2{\bf b}, {\bf c} + {\bf a})]; [33(2{\bf a}, 2{\bf b}, {\bf c} + {\bf b})]; [34(2{\bf a}, {\bf b} + {\bf a}, 2{\bf c})]; [35(2{\bf a}, 2{\bf b}, {\bf c} + {\bf a} + {\bf b})]
Index 5 [1(5{\bf a}, {\bf b}, {\bf c})]; [2({\bf a}, 5{\bf b}, {\bf c})]; [3({\bf a}, {\bf b}, 5{\bf c})]; [4(5{\bf a}, {\bf b} + {\bf a},{\bf c})]; [5(5{\bf a}, {\bf b}, {\bf c} + {\bf a})]; [6({\bf a}, 5{\bf b}, {\bf c} + {\bf b})]; [7(5{\bf a}, {\bf b} - {\bf a}, {\bf c})]; [8(5{\bf a}, {\bf b}, {\bf c} - {\bf a})]; [9({\bf a}, 5{\bf b}, {\bf c} - {\bf b})]; [10(5{\bf a}, {\bf b} + 2{\bf a}, {\bf c})]; [11(5{\bf a}, {\bf b}, {\bf c} - 2{\bf a})]; [12({\bf a}, 5{\bf b}, {\bf c} + 2{\bf b})]; [13(5{\bf a}, {\bf b} - 2{\bf a}, {\bf c})]; [14(5{\bf a}, {\bf b}, {\bf c} + 2{\bf a})]; [15({\bf a}, 5{\bf b}, {\bf c} - 2{\bf b})]; [16(5{\bf a}, {\bf b} + {\bf a},{\bf c} + {\bf a})]; [17(5{\bf a}, {\bf b} + {\bf a},{\bf c} - {\bf a})]; [18(5{\bf a}, {\bf b} - {\bf a},{\bf c} + {\bf a})]; [19(5{\bf a}, {\bf b} - {\bf a},{\bf c} - {\bf a})]; [20(5{\bf a}, {\bf b} + {\bf a},{\bf c} + 2{\bf a})]; [21(5{\bf a}, {\bf b} + {\bf a},{\bf c} - 2{\bf a})]; [22(5{\bf a}, {\bf b} + 2{\bf a},{\bf c} + {\bf a})]; [23(5{\bf a}, {\bf b} - 2{\bf a},{\bf c} + {\bf a})]; [24(5{\bf a}, {\bf b} + 2{\bf a},{\bf c} - 2{\bf a})]; [25(5{\bf a}, {\bf b} - 2{\bf a},{\bf c} + 2{\bf a})]; [26(5{\bf a}, {\bf b} + 2{\bf a},{\bf c} + 2{\bf a})]; [27(5{\bf a}, {\bf b} - 2{\bf a},{\bf c} - {\bf a})]; [28(5{\bf a}, {\bf b} - {\bf a},{\bf c} - 2{\bf a})]; [29(5{\bf a}, {\bf b} - 2{\bf a},{\bf c} - 2{\bf a})]; [30(5{\bf a}, {\bf b} + 2{\bf a},{\bf c} - {\bf a})]; [31(5{\bf a}, {\bf b} - {\bf a},{\bf c} + 2{\bf a})]
Index 6 [1(6{\bf a}, {\bf b}, {\bf c})]; [2({\bf a}, 6{\bf b}, {\bf c})]; [3({\bf a}, {\bf b}, 6{\bf c})]; [4(6{\bf a}, {\bf b} + {\bf a}, {\bf c})]; [5(6{\bf a}, {\bf b}, {\bf c} + {\bf a})]; [6({\bf a}, 6{\bf b}, {\bf c} + {\bf b})]; [7(6{\bf a}, {\bf b} - {\bf a}, {\bf c})]; [8(6{\bf a}, {\bf b}, {\bf c} - {\bf a})]; [9({\bf a}, 6{\bf b}, {\bf c} - {\bf b})]; [10(6{\bf a}, {\bf b} + 2{\bf a}, {\bf c})]; [11(6{\bf a}, {\bf b}, {\bf c} + 2{\bf a})]; [12({\bf a}, 6{\bf b}, {\bf c} + 2{\bf b})]; [13(3{\bf a}, 2{\bf b} + {\bf a}, {\bf c})]; [14({\bf a}, 3{\bf b}, 2{\bf c} + {\bf b})]; [15(3{\bf a}, {\bf b}, 2{\bf c} + {\bf a})]; [16(6{\bf a}, {\bf b} - 2{\bf a}, {\bf c})]; [17(6{\bf a}, {\bf b}, {\bf c} - 2{\bf a})]; [18({\bf a}, 6{\bf b}, {\bf c} - 2{\bf b})]; [19(3{\bf a}, 2{\bf b} - {\bf a}, {\bf c})]; [20({\bf a}, 3{\bf b}, 2{\bf c} - {\bf b})]; [21(3{\bf a}, {\bf b}, 2{\bf c} - {\bf a})]; [22(6{\bf a}, {\bf b} + 3{\bf a}, {\bf c})]; [23(6{\bf a}, {\bf b}, {\bf c} + 3{\bf a})]; [24({\bf a}, 6{\bf b}, {\bf c} + 3{\bf b})]; [25(2{\bf a}, 3{\bf b} + {\bf a}, {\bf c})]; [26({\bf a}, 2{\bf b}, 3{\bf c} + {\bf b})]; [27(2{\bf a}, {\bf b}, 3{\bf c} + {\bf a})]; [28(6{\bf a}, {\bf b} + {\bf a},{\bf c} + {\bf a})]; [29(6{\bf a}, {\bf b} + {\bf a},{\bf c} - {\bf a})]; [30(6{\bf a}, {\bf b} - {\bf a},{\bf c} + {\bf a})]; [31(6{\bf a}, {\bf b} - {\bf a},{\bf c} - {\bf a})]; [32(6{\bf a}, {\bf b} + {\bf a},{\bf c} + 2{\bf a})]; [33(6{\bf a}, {\bf b} + {\bf a},{\bf c} - 2{\bf a})]; [34(3{\bf a}, 2{\bf b} + {\bf a},{\bf c} + {\bf b})]; [35(3{\bf a}, 2{\bf b} - {\bf a},{\bf c} + {\bf b})]; [36(6{\bf a}, {\bf b} + 2{\bf a},{\bf c} + {\bf a})]; [37(6{\bf a}, {\bf b} - 2{\bf a},{\bf c} + {\bf a})]; [38(6{\bf a}, {\bf b} - {\bf a},{\bf c} + 2{\bf a})]; [39(6{\bf a}, {\bf b} + 2{\bf a},{\bf c} - {\bf a})]; [40(3{\bf a}, 2{\bf b} + {\bf a},{\bf c} + {\bf a} + {\bf b})]; [41(6{\bf a}, {\bf b} - {\bf a},{\bf c} - 2{\bf a})]; [42(6{\bf a}, {\bf b} - 2{\bf a},{\bf c} - {\bf a})]; [43(3{\bf a}, 2{\bf b} - {\bf a},{\bf c} - {\bf a} + {\bf b})]; [44(6{\bf a}, {\bf b} + {\bf a},{\bf c} + 3{\bf a})]; [45(6{\bf a}, {\bf b} + 3{\bf a},{\bf c} + {\bf a})]; [46(2{\bf a}, 3{\bf b} + {\bf a},{\bf c} + {\bf b})]; [47(6{\bf a}, {\bf b} - {\bf a},{\bf c} + 3{\bf a})]; [48(6{\bf a}, {\bf b} + 3{\bf a},{\bf c} - {\bf a})]; [49(2{\bf a}, 3{\bf b} + {\bf a},{\bf c} - {\bf b})]; [50(6{\bf a}, {\bf b} + 2{\bf a},{\bf c} + 2{\bf a})]; [51(3{\bf a}, 2{\bf b} + {\bf a},{\bf c} - {\bf a})]; [52(3{\bf a}, {\bf b} - {\bf a}, 2{\bf c} + {\bf a})]; [53(6{\bf a}, {\bf b} - 2{\bf a},{\bf c} - 2{\bf a})]; [54(3{\bf a}, 2{\bf b} - {\bf a},{\bf c} - {\bf a})]; [55(3{\bf a}, {\bf b} - {\bf a}, 2{\bf c} - {\bf a})]; [56(6{\bf a}, {\bf b} + 2{\bf a},{\bf c} - 2{\bf a})]; [57(3{\bf a}, 2{\bf b} - {\bf a},{\bf c} + {\bf a})]; [58(3{\bf a}, {\bf b} + {\bf a}, 2{\bf c} + {\bf a})]; [59(6{\bf a}, {\bf b} - 2{\bf a},{\bf c} + 2{\bf a})]; [60(3{\bf a}, 2{\bf b} + {\bf a},{\bf c} + {\bf a})]; [61(3{\bf a}, {\bf b} + {\bf a}, 2{\bf c} - {\bf a})]; [62(6{\bf a}, {\bf b} + 2{\bf a},{\bf c} + 3{\bf a})]; [63(2{\bf a}, 3{\bf b} + {\bf a},{\bf c} + {\bf a} - {\bf b})]; [64(3{\bf a}, 2{\bf b}, {\bf c} - {\bf a} + {\bf b})]; [65(6{\bf a}, {\bf b} + 3{\bf a},{\bf c} + 2{\bf a})]; [66(3{\bf a}, 2{\bf b} + {\bf a},{\bf c} - {\bf a} + {\bf b})]; [67(2{\bf a}, 3{\bf b}, {\bf c} + {\bf a} - {\bf b})]; [68(6{\bf a}, {\bf b} - 2{\bf a},{\bf c} + 3{\bf a})]; [69(2{\bf a}, 3{\bf b} + {\bf a},{\bf c} + {\bf b} + {\bf a})]; [70(3{\bf a}, 2{\bf b}, {\bf c} + {\bf a} + {\bf b})]; [71(6{\bf a}, {\bf b} + 3{\bf a},{\bf c} - 2{\bf a})]; [72(3{\bf a}, 2{\bf b} - {\bf a},{\bf c} + {\bf a} + {\bf b})]; [73(2{\bf a}, 3{\bf b}, {\bf c} + {\bf a} + {\bf b})]; [74(6{\bf a}, {\bf b} + 3{\bf a},{\bf c} + 3{\bf a})]; [75(2{\bf a}, 3{\bf b} + {\bf a},{\bf c} + {\bf a})]; [76(2{\bf a}, {\bf b} + {\bf a}, 3{\bf c} + {\bf a})]; [77(3{\bf a}, 2{\bf b}, {\bf c})]; [78(3{\bf a}, {\bf b}, 2{\bf c})]; [79(2{\bf a}, 3{\bf b}, {\bf c})]; [80(2{\bf a}, {\bf b}, 3{\bf c})]; [81({\bf a}, 3{\bf b}, 2{\bf c})]; [82({\bf a}, 2{\bf b}, 3{\bf c})]; [83(3{\bf a}, 2{\bf b}, {\bf c} + {\bf a})]; [84(3{\bf a}, {\bf b} + {\bf a}, 2{\bf c})]; [85(2{\bf a}, 3{\bf b}, {\bf c} + {\bf b})]; [86(3{\bf a}, 2{\bf b}, {\bf c} - {\bf a})]; [87(3{\bf a}, {\bf b} - {\bf a}, 2{\bf c})]; [88(2{\bf a}, 3{\bf b}, {\bf c} - {\bf b})]; [89(3{\bf a}, 2{\bf b}, {\bf c} + {\bf b})]; [90(2{\bf a}, 3{\bf b}, {\bf c} + {\bf a})]; [91(2{\bf a}, {\bf a} + {\bf b}, 3{\bf c})]
Index 7 [1(7{\bf a}, {\bf b}, {\bf c})]; [2({\bf a}, 7{\bf b}, {\bf c})]; [3({\bf a}, {\bf b}, 7{\bf c})]; [4(7{\bf a}, {\bf b} + {\bf a}, {\bf c})]; [5(7{\bf a}, {\bf b}, {\bf c} + {\bf b})]; [6({\bf a}, 7{\bf b}, {\bf c} + {\bf b})]; [7(7{\bf a}, {\bf b} - {\bf a}, {\bf c})]; [8(7{\bf a}, {\bf b}, {\bf c} - {\bf a})]; [9({\bf a}, 7{\bf b}, {\bf c} - {\bf b})]; [10(7{\bf a}, {\bf b} + 2{\bf a}, {\bf c})]; [11(7{\bf a}, {\bf b}, {\bf c} - 3{\bf a})]; [12({\bf a}, 7{\bf b}, {\bf c} + 2{\bf b})]; [13(7{\bf a}, {\bf b} - 3{\bf a}, {\bf c})]; [14(7{\bf a}, {\bf b}, {\bf c} + 2{\bf a})]; [15({\bf a}, 7{\bf b}, {\bf c} - 3{\bf b})]; [16(7{\bf a}, {\bf b} - 2{\bf a}, {\bf c})]; [17(7{\bf a}, {\bf b}, {\bf c} + 3{\bf a})]; [18({\bf a}, 7{\bf b}, {\bf c} - 2{\bf b})]; [19(7{\bf a}, {\bf b} + 3{\bf a}, {\bf c})]; [20(7{\bf a}, {\bf b}, {\bf c} - 2{\bf a})]; [21({\bf a}, 7{\bf b}, {\bf c} + 3{\bf b})]; [22(7{\bf a}, {\bf b} + {\bf a},{\bf c} + {\bf a})]; [23(7{\bf a}, {\bf b} + {\bf a},{\bf c} - {\bf a})]; [24(7{\bf a}, {\bf b} - {\bf a},{\bf c} + {\bf a})]; [25(7{\bf a}, {\bf b} - {\bf a},{\bf c} - {\bf a})]; [26(7{\bf a}, {\bf b} + {\bf a},{\bf c} + 2{\bf a})]; [27(7{\bf a}, {\bf b} + {\bf a},{\bf c} - 2{\bf a})]; [28(7{\bf a}, {\bf b} + 2{\bf a},{\bf c} + {\bf a})]; [29(7{\bf a}, {\bf b} - 2{\bf a},{\bf c} + {\bf a})]; [30(7{\bf a}, {\bf b} + 3{\bf a},{\bf c} - 3{\bf a})]; [31(7{\bf a}, {\bf b} - 3{\bf a},{\bf c} + 3{\bf a})]; [32(7{\bf a}, {\bf b} + {\bf a},{\bf c} + 3{\bf a})]; [33(7{\bf a}, {\bf b} + {\bf a},{\bf c} - 3{\bf a})]; [34(7{\bf a}, {\bf b} + 3{\bf a},{\bf c} + {\bf a})]; [35(7{\bf a}, {\bf b} - 3{\bf a},{\bf c} + {\bf a})]; [36(7{\bf a}, {\bf b} + 2{\bf a},{\bf c} - 2{\bf a})]; [37(7{\bf a}, {\bf b} - 2{\bf a},{\bf c} + 2{\bf a})]; [38(7{\bf a}, {\bf b} + 2{\bf a},{\bf c} + 2{\bf a})]; [39(7{\bf a}, {\bf b} - 3{\bf a},{\bf c} - {\bf a})]; [40(7{\bf a}, {\bf b} - {\bf a},{\bf c} - 3{\bf a})]; [41(7{\bf a}, {\bf b} - 2{\bf a},{\bf c} - 2{\bf a})]; [42(7{\bf a}, {\bf b} + 3{\bf a},{\bf c} - {\bf a})]; [43(7{\bf a}, {\bf b} - {\bf a},{\bf c} + 3{\bf a})]; [44(7{\bf a}, {\bf b} + 2{\bf a},{\bf c} + 3{\bf a})]; [45(7{\bf a}, {\bf b} + 3{\bf a},{\bf c} + 2{\bf a})]; [46(7{\bf a}, {\bf b} - 2{\bf a},{\bf c} - 3{\bf a})]; [47(7{\bf a}, {\bf b} - 3{\bf a},{\bf c} + 2{\bf a})]; [48(7{\bf a}, {\bf b} + 2{\bf a},{\bf c} - 3{\bf a})]; [49(7{\bf a}, {\bf b} - 3{\bf a},{\bf c} - 2{\bf a})]; [50(7{\bf a}, {\bf b} - 2{\bf a},{\bf c} + 3{\bf a})]; [51(7{\bf a}, {\bf b} + 3{\bf a},{\bf c} - 2{\bf a})]; [52(7{\bf a}, {\bf b} + 3{\bf a},{\bf c} + 3{\bf a})]; [53(7{\bf a}, {\bf b} - 2{\bf a},{\bf c} - {\bf a})]; [54(7{\bf a}, {\bf b} - {\bf a},{\bf c} - 2{\bf a})]; [55(7{\bf a}, {\bf b} - 3{\bf a},{\bf c} - 3{\bf a})]; [56(7{\bf a}, {\bf b} + 2{\bf a},{\bf c} - {\bf a})]; [57(7{\bf a}, {\bf b} - {\bf a},{\bf c} + 2{\bf a})]

Note that the vector [{\bf a}'] has the same direction as the vector a and the plane [({\bf a}', {\bf b}')] is parallel to the plane (a, b), i.e. the matrix of the transformation is triangular. Equivalent formulae can be derived by permutations of the vectors a, b, c which keep the directions of [{\bf b}'] or [{\bf c}'] and which preserve the parallelism of the planes [({\bf b}', {\bf c}')] with (b, c) or [({\bf a}', {\bf c}')] with (a, c).

Example

One can derive easily the 7 derivative lattices of index 2.[\eqalign{&1(2{\bf a},{\bf b},{\bf c});\ 2({\bf a}, 2{\bf b},{\bf c});\ 3({\bf a},{\bf b}, 2{\bf c});\ 4(2{\bf a},{\bf b} + {\bf a},{\bf c});\cr &5(2{\bf a},{\bf b},{\bf c} + {\bf a});\ 6({\bf a}, 2{\bf b},{\bf c} + {\bf b});\ 7(2{\bf a},{\bf b} + {\bf a},{\bf c} + {\bf a}).}]

Another primitive cell of a given derivative lattice is obtained if one of the following three elementary transformations is performed on the vectors of a primitive cell of this derivative lattice:

  • (i) the sign of a vector is changed;

  • (ii) two vectors are interchanged;

  • (iii) to a vector is added q times another vector (q integer).

(i)[link] and (ii)[link] are left-handed transformations, (iii)[link] is right-handed.

Example

The primitive cell [{\bf a}''], [{\bf b}''], [{\bf c}''] ([{\bf a}'' = - 2{\bf b} - {\bf a}], [{\bf b}'' = 9{\bf a} - 2{\bf c} - 2{\bf b}], [{\bf c}'' = {\bf c} - {\bf a} + 3{\bf b}]) belongs to the derivative lattice of index 10 given by the primitive cell [{\bf a}',{\bf b}',{\bf c}'\ ({\bf a}' = 5{\bf a},\ {\bf b}' = 2{\bf b} + {\bf a},\ {\bf c}' = {\bf c} - 2{\bf a} + {\bf b})] because these two cells are related by the following sequence of elementary transformations:[\eqalignno{&(- 2{\bf b} - {\bf a},\ 9{\bf a} - 2{\bf c} - 2{\bf b},\ {\bf c} - {\bf a} + 3{\bf b}){\buildrel \rm{(iii)} \over \rightarrow}\cr &(- 2{\bf b} - {\bf a},\ 9{\bf a} - 2{\bf c} - 2{\bf b},\ {\bf c} - 2{\bf a} + {\bf b}){\buildrel \rm{(iii)} \over \rightarrow}\cr &(- 2{\bf b} - {\bf a},\ 5{\bf a},\ {\bf c} - 2{\bf a} + {\bf b}){\buildrel \rm{(i)} \over \rightarrow} (2{\bf b} + {\bf a},\ 5{\bf a},\ {\bf c} - 2{\bf a} + {\bf b})\cr &{\buildrel \rm{(ii)} \over \rightarrow} (5{\bf a},\ 2{\bf b} + {\bf a},\ {\bf c} - 2{\bf a} + {\bf b});}] ([{\bf a}''], [{\bf b}''], [{\bf c}'']) and ([{\bf a}'], [{\bf b}'], [{\bf c}']) have the same handedness.

13.2.3. Two-dimensional derivative lattices

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All previous considerations are valid also for two-dimensional lattices and their derivative lattices (Table 13.2.3.1).[link] The relevant formula for any index is[\eqalign{&{\bf a}' = p_{1}{\bf a},\quad {\bf b}' = p_{2}{\bf b} + q{\bf a}\cr &[{\bf a}\ \hbox{direction is kept}]\cr &(p_{1}, p_{2}\ \hbox{positive integers, not necessarily prime;}\cr &\hbox{index} = p_{1}p_{2} \gt 1;\ q\ \hbox{any integer;} - p_{1}/2 \lt q \leq p_{1}/2).}] A similar formula is obtained by interchange of a and b.

Table 13.2.3.1| top | pdf |
Two-dimensional derivative lattices of indices 2 to 7

The entry for each derivative lattice starts with a running number which is followed, between parentheses, by the appropriate basis-vector relations.

Index 2 [1(2{\bf a}, {\bf b});\ 2({\bf a}, 2{\bf b});\ 3(2{\bf a}, {\bf b} + {\bf a})]
Index 3 [1(3{\bf a}, {\bf b});\ 2({\bf a}, 3{\bf b});\ 3(3{\bf a}, {\bf b} + {\bf a});\ 4(3{\bf a}, {\bf b} - {\bf a})]
Index 4 [1(4{\bf a}, {\bf b});\ 2({\bf a}, 4{\bf b});\ 3(4{\bf a}, {\bf b} + {\bf a});\ 4(4{\bf a}, {\bf b} - {\bf a});\ 5(4{\bf a}, {\bf b} + 2{\bf a})];
  [6(2{\bf a}, 2{\bf b} + {\bf a});\ 7(2{\bf a}, 2{\bf b})]
Index 5 [1(5{\bf a}, {\bf b});\ 2({\bf a}, 5{\bf b});\ 3(5{\bf a}, {\bf b} + {\bf a});\ 4(5{\bf a}, {\bf b} - {\bf a});\ 5(5{\bf a}, {\bf b} + 2{\bf a});]
  [6(5{\bf a}, {\bf b} - 2{\bf a})]
Index 6 [1(6{\bf a}, {\bf b});\ 2({\bf a}, 6{\bf b});\ 3(6{\bf a}, {\bf b} + {\bf a});\ 4(6{\bf a}, {\bf b} - {\bf a});\ 5(6{\bf a}, {\bf b} + 2{\bf a});]
  [6(3{\bf a}, 2{\bf b} + {\bf a});\ 7(6{\bf a}, {\bf b} - 2{\bf a});\ 8(3{\bf a}, 2{\bf b} - {\bf a});\ 9(6{\bf a}, {\bf b} + 3{\bf a});]
  [10(2{\bf a}, 3{\bf b} + {\bf a});\ 11(3{\bf a}, 2{\bf b});\ 12(2{\bf a}, 3{\bf b})]
Index 7 [1(7{\bf a}, {\bf b});\ 2({\bf a}, 7{\bf b});\ 3(7{\bf a}, {\bf b} + {\bf a});\ 4(7{\bf a}, {\bf b} - {\bf a});\ 5(7{\bf a}, {\bf b} + 2{\bf a});]
  [6(7{\bf a}, {\bf b} - 3{\bf a});\ 7(7{\bf a}, {\bf b} - 2{\bf a});\ 8(7{\bf a}, {\bf b} + 3{\bf a})]

References

Billiet, Y. & Rolley Le Coz, M. (1980). Le groupe P1 et ses sous-groupes. II. Tables de sous-groupes. Acta Cryst. A36, 242–248.








































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