Two-dimensional derivative lattices

Billiet, Y.; Bertaut, E. F.

doi:10.1107/97809553602060000529

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

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International Tables for Crystallography (2006). Vol. A. ch. 13.2, p. 844

Section 13.2.3. Two-dimensional derivative lattices

Y. Billiet^a ^‡ and E. F. Bertaut^b ^§

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

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). 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

International Tables for Crystallography (2006). Vol. A. ch. 13.2, p. 844