International
Tables for
Crystallography
Volume E
Subperiodic groups
Edited by V. Kopský and D. B. Litvin

International Tables for Crystallography (2006). Vol. E, ch. 1.2, p. 16   | 1 | 2 |

Section 1.2.13. Reflection conditions

V. Kopskýa and D. B. Litvinb*

aDepartment of Physics, University of the South Pacific, Suva, Fiji, and Institute of Physics, The Academy of Sciences of the Czech Republic, Na Slovance 2, PO Box 24, 180 40 Prague 8, Czech Republic, and bDepartment of Physics, Penn State Berks Campus, The Pennsylvania State University, PO Box 7009, Reading, PA 19610-6009, USA
Correspondence e-mail:  u3c@psu.edu

1.2.13. Reflection conditions

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The Reflection conditions are listed in the right-hand column of each Wyckoff position. There are two types of reflection conditions:

  • (i) General conditions. These conditions apply to all Wyckoff positions of the subperiodic group.

  • (ii) Special conditions (`extra' conditions). These conditions apply only to special Wyckoff positions and must always be added to the general conditions of the subperiodic group.

The general reflection conditions are the result of three effects: centred lattices, glide planes and screw axes. For the nine layer groups with centred lattices, the corresponding general reflection condition is [h + k = 2n]. The general reflection conditions due to glide planes and screw axes for the subperiodic groups are given in Table 1.2.13.1[link].

Table 1.2.13.1| top | pdf |
General reflection conditions due to glide planes and screw axes

(a) Layer groups.

(1) Glide planes.

Reflection conditionOrientation of planeGlide vectorSymbol
hk: h = 2n (001) a/2 a
hk: k = 2n (001) b/2 b
hk: [h+k = 2n] (001) a/2 + b/2 n
0k: k = 2n (100) b/2 b
h0: h = 2n (010) a/2 a

(2) Screw axes.

Reflection conditionDirection of axisScrew vectorSymbol
h0: h = 2n [100] a/2 21
0k: k = 2n [010] b/2 21

(b) Rod groups.

(1) Glide planes.

Reflection conditionOrientation of planeGlide vectorSymbol
l: l = 2n Any orientation parallel to the c axis c/2 c

(2) Screw axes.

Reflection conditionDirection of axisScrew vectorSymbol
l: l = 2n [001] c/2 21, 42, 63
l: l = 3n [001] c/3 31, 32, 62, 64
l: l = 4n [001] c/4 41, 43
l: l = 6n [001] c/6 61, 65

(c) Frieze groups, glide plane.

Reflection conditionOrientation of planeGlide vectorSymbol
h: h = 2n (10) a/2 g

Example: The layer group p4bm (L56)

General position 8d: [0k\!: k = 2n] and [h0\!: h = 2n] due respectively to the glide planes b and a. The projections along [100] and [010] of any crystal structure with this layer-group symmetry have, respectively, periodicity b/2 and a/2.

Special positions 2a and 2b: [hk\!: h + k = 2n]. Any set of equivalent atoms in either of these positions displays additional c-centring.








































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