# Reflection conditions

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Conditions de réflexion (*Fr*). Auslöschungsgesetze (*Ge*). Condizioni di diffrazione (*It*). 消滅則 (*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.

Reflection condition |
Centring type of cell | Centring symbol |
---|---|---|

None | Primitive | PR (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
| all-face centred | F |

− h + k + l = 3n | rhombohedrally centred, obverse |
R (hexagonal axes) |

h − k + l = 3n | rhombohedrally centred, reverse | |

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 *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 |
---|---|---|---|

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 nk, 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 *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 |
---|---|---|---|

00l | l = 2 n | c/2 | 2_{1}; 4_{2} |

l = 4 n | c/4 | 4_{1}; 4_{3} | |

000l | 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 2.1.3.7 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