3.22: Reflection conditions
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:
- Centered 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.
- Glide planes .
The resulting conditions apply only to two-dimensional sets of reflections, i.e. to reciprocal-lattice nets containing the origin (such as hk 0, h 0 l , 0 kl , 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 |
|---|---|---|---|
| 0 kl | 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 = 2 n |
b /4 ± c /4 | d |
The zonal reflection conditions are listed in Table 2.2.13.2 of International Tables of Crystallography, Volume A .
- Screw axes .
The resulting conditions apply only to one-dimensional sets of reflections, i.e. reciprocal-lattice rows containing the origin (such as h 00, 0 k 0, 00 l ). 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 |
|---|---|---|---|
| 00 l | l = 2 n | c /2 | 2 1 ; 4 2 |
| l = 4 n | c /4 | 4 1 ; 4 3 | |
| 000 l | l = 2 n | c /2 | 6 3 |
| l = 3 n | c /3 | 4 1 ; 3 1 ; 3 2 ; 6 2 ; 6 4 | |
| l = 6 n | c /6 | 6 1 ;6 5 |
The serial reflection conditions are listed in Table 2.2.13.2 of International Tables of Crystallography, Volume A .
(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.