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Difference between revisions of "Reflection conditions"

From Online Dictionary of Crystallography

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(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:
 
(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''.
 
  
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The resulting conditions apply to the whole three-dimensional set of reflections hkl. Accordingly, they are called ''[[integral reflection conditions]]''. They are given in [[integral reflection conditions| Table 1]].
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*''Centred cells''
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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''.
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<table border cellspacing=0 cellpadding=5 align=center>
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<caption align=top> '''Table 1. Integral reflection conditions for centred lattices.''' </caption>
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<tr align=left>
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<th>Reflection<br> condition </th>
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<th>Centring type of cell </th>
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<th>Centring symbol</th>
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</tr>
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<tr align=left>
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<td>None</td> <td> Primitive</td> <td> ''P''<br>
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''R'' (rhombohedral axes)</td>
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</tr>
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<tr align=left>
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<td>''h'' + ''k'' = 2''n''</td> <td>''C''-face centred</td> <td>''C''</td>
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</tr>
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<tr align=left>
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<td>''k'' + ''l'' = 2''n''</td> <td>''A''-face centred</td> <td>''A''</td>
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</tr>
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<tr align=left>
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<td>''l'' + ''h'' = 2''n''</td> <td>''B''-face centred</td> <td>''B''</td>
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</tr>
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<tr align=left>
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<td>''h'' + ''k'' + ''l'' = 2''n''</td> <td>body centred</td> <td>''I''</td>
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</tr>
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<tr align=left>
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<td>''h'' + ''k'', ''h'' + ''l'' and<br>
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''k'' + ''l'' = 2''n'' or:<br>
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''h'', ''k'', ''l'' all odd or all<br>
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even (‘unmixed’)</td> <td>all-face centred</td> <td> ''F''</td>
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</tr>
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<tr align=left>
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<td> &minus; ''h'' + ''k'' + ''l'' = 3''n''</td> <td> rhombohedrally<br>
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centred, obverse<br>
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setting (standard)</td>
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<td rowspan=2>''R'' (hexagonal axes)</td></tr>
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<tr align=left>
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<td> ''h'' &minus; ''k'' + ''l'' = 3''n''</td><td> rhombohedrally<br>
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centred, reverse<br>
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setting </td></tr>
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<tr align=left>
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<td> ''h'' &minus; ''k'' = 3''n''</td> <td>hexagonally centred</td> <td> ''H''</td></tr>
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</table>
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*''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,
 
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):
 
they are called ''[[zonal reflection conditions]]''. For instance, for a glide plane parallel to (001):

Revision as of 16:09, 16 May 2017

Conditions de réflexion (Fr). Condizioni di diffrazione (It). 消滅則 (Ja).

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 lattices.
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
  • 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 reflectionreflection condition glide vector glide plane
0kl k = 2 n b/2 b
l = 2 nc/2 c
k + l = 2 nb/2 + c/2 n
k + l = 4 n
k, l = 2n		b/4 ± c/4 d

The zonal reflection conditions are listed in Table 2.2.13.2 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 h00, 0k0, 00l). They are called serial reflection conditions. For instance, for a screw axis parallel to [001], the reflection conditions are:

type of reflectionreflection condition screw vector screw axis
00l l = 2 n c/2 21; 42
l = 4 nc/4 41; 43
000l l = 2 n c/2 63
l = 3 n c/3 41; 31; 32; 62; 64
l = 6 nc/661;65

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.

See also

Section 1.6.4 of International Tables of Crystallography, Volume A, 6th edition

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