Special or `extra' reflection conditions

Hahn, Th.; Looijenga-Vos, A.

doi:10.1107/97809553602060000505

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. 2.2, p. 32

Section 2.2.13.2. Special or `extra' reflection conditions

Th. Hahn^a ^* and A. Looijenga-Vos^b ^‡

^a Institut für Kristallographie, Rheinisch-Westfälische Technische Hochschule, Aachen, Germany, and ^bLaboratorium voor Chemische Fysica, Rijksuniversiteit Groningen, The Netherlands
Correspondence e-mail: hahn@xtl.rwth-aachen.de

2.2.13.2. Special or `extra' reflection conditions

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These apply either to the integral reflections hkl or to particular sets of zonal or serial reflections. In the space-group tables, the minimal special conditions are listed that, on combination with the general conditions, are sufficient to generate the complete set of conditions. This will be apparent from the examples below.

Examples

(1) $[P4_{2}22\ (93)]$

General position $[8p\!: 00l\!\!:\;l = 2n]$ , due to $[4_{2}]$ ; the projection on [001] of any crystal structure with this space group has periodicity $[{1 \over 2}c]$ .

Special position $[4i\!\!: hkl\!\!:\;h + k + l = 2n]$ ; any set of symmetrically equivalent atoms in this position displays additional I centring.

Special position $[4n\!\!: 0kl\!\!:\;l = 2n]$ ; any set of equivalent atoms in this position displays a glide plane $[c \perp [100]]$ . Projection of this set along [100] results in a halving of the original c axis, whence the special condition. Analogously for : .
(2) (15, unique axis b, cell choice 1)

General position $[8f\!\!: hkl\!\!:\;h + k = 2n]$ , due to the C-centred cell.

Special position $[4d\!\!: hkl\!\!:\;k + l = 2n]$ , due to additional A and B centring for atoms in this position. Combination with the general condition results in $[hkl\!\!:\;h + k, h + l, k + l = 2n]$ or hkl all odd or all even; this corresponds to an F-centred arrangement of atoms in this position.

Special position $[4b\!\!:\;hkl\!\!: l = 2n]$ , due to additional halving of the c axis for atoms in this position. Combination with the general condition results in $[hkl\!\!:\;h + k,l = 2n]$ ; this corresponds to a C-centred arrangement in a cell with half the original c axis. No further condition results from the combination.
(3) (15, unique axis b, cell choice 3)

For the description of space group No. 15 with cell choice 3 (see Section 2.2.16 and space-group tables), the reflection conditions appear as follows:

General position $[8f\!\!: hkl\!\!:\; h + k + l = 2n]$ , due to the I-centred cell.

Special position $[4b\!\!:\;hkl\!\!:\;h = 2n]$ , due to additional halving of the a axis. Combination gives $[hkl\!\!: h,k + l = 2n]$ , i.e. an A-centred arrangement of atoms in a cell with half the original a axis.

An analogous result is obtained for position 4d.
(4) Fmm2 (42)

General position $[16e\!\!: hkl\!\!: h + k,h + l,k + l = 2n]$ , due to the F-centred cell.

Special position $[8b\!\!: hkl\!\!:\;h = 2n]$ , due to additional halving of the a axis. Combination results in $[hkl\!\!: h,k,l = 2n]$ , i.e. all indices even; the atoms in this position are arranged in a primitive lattice with axes $[{1 \over 2}a,\; {1 \over 2}b]$ and $[{1 \over 2}c]$ .

For the cases where the special reflection conditions are described by means of combinations of `OR' and `AND' instructions, the `AND' condition always has to be evaluated with priority, as shown by the following example.

Example: $[P\bar{4}3n]$ (218)

Special position $[6d\!\!: hkl\!\!: h+k+l = 2n]$ or [h = 2n+1] , [k = 4n] and [l = 4n+2] .

This expression contains the following two conditions:

(a) $[hkl\!\!: h+k+l=2n]$ ;

(b) [h = 2n+1] and [k = 4n] and [l = 4n+2] .

A reflection is `present' (occurring) if either condition (a) is satisfied or if a permutation of the three conditions in (b) are simultaneously fulfilled.

References