Reflection conditions

Conditions de réflexion (Fr). Auslöschungsgesetze (Ge). Condizioni di diffrazione (It). 回折条件 (Ja). Ausencias sistemáticas (Sp).

Definition

The reflection conditions describe the conditions of occurrence of a reflection (structure factor not systematically zero). There are two types of systematic reflection conditions for diffraction of crystals by radiation:

(1) General conditions. They apply to all Wyckoff positions of a space group, i.e. they are always obeyed, irrespective of which Wyckoff positions are occupied by atoms in a particular crystal structure. They are due to one of three effects:

• Centred cells

The resulting conditions apply to the whole three-dimensional set of reflections hkl. Accordingly, they are called integral reflection conditions. They are given in Table 1.

Table 1. Integral reflection conditions for centred cells.

Reflection condition	Centring type of cell	Centring symbol
None	Primitive	P R (rhombohedral axes)
h + k = 2n	C-face centred	C
k + l = 2n	A-face centred	A
l + h = 2n	B-face centred	B
h + k + l = 2n	body centred	I
h + k, h + l and k + l = 2n or: h, k, l all odd or all even (‘unmixed’)	all-face centred	F
− h + k + l = 3n	rhombohedrally centred, obverse setting (standard)	R (hexagonal axes)
h − k + l = 3n	rhombohedrally centred, reverse setting
h − k = 3n	hexagonally centred	H
h + k + l = 3n	D centred	D

• Glide planes

The resulting conditions apply only to two-dimensional sets of reflections, i.e. to reciprocal-lattice nets containing the origin (such as hk0, h0l, 0kl, hhl). For this reason, they are called zonal reflection conditions. For instance, for a glide plane parallel to (001):

Type of reflection	Reflection condition	Glide vector	Glide plane
0kl	k = 2 n	b/2	b
	l = 2 n	c/2	c
	k + l = 2 n	b/2 + c/2	n
	k + l = 4 n k, l = 2n	b/4 ± c/4	d

The zonal reflection conditions are listed in Table 2.1.3.7 of International Tables for Crystallography, Volume A, 6th edition.

• Screw axes

The resulting conditions apply only to one-dimensional sets of reflections, i.e. reciprocal-lattice rows containing the origin (such as h00, 0k0, 00l). They are called serial reflection conditions. For instance, for a screw axis parallel to [001], the reflection conditions are:

Type of reflection	Reflection condition	Screw vector	Screw axis
00l	l = 2 n	c/2	21; 42
	l = 4 n	c/4	41; 43
000l	l = 2 n	c/2	63
	l = 3 n	c/3	41; 31; 32; 62; 64
	l = 6 n	c/6	61;65

The serial reflection conditions are listed in Table 2.1.3.7 of International Tables for Crystallography, Volume A, 6th edition.

(2) Special conditions ('extra' conditions). They apply only to special Wyckoff positions and occur always in addition to the general conditions of the space group.

See also

• Chapter 1.6.4 of International Tables for Crystallography, Volume A, 6th edition