Difference between revisions of "Reflection conditions"
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(better Japanese translation: 消滅則(systematic absences) -> 回折条件(reflection conditions)) |
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| − | < | + | <font color="blue">Conditions de réflexion</font> (''Fr''). <font color="red">Auslöschungsgesetze</font> (''Ge''). <font color="black">Condizioni di diffrazione</font> (''It''). <font color="purple">回折条件</font> (''Ja''). <font color="green">Ausencias sistemáticas</font> (''Sp''). |
| + | |||
== Definition == | == Definition == | ||
| − | The reflection conditions describe the conditions of | + | 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: | (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: | ||
| − | |||
| − | The resulting conditions apply to the whole three-dimensional set of reflections hkl. Accordingly, they are called ''[[integral reflection conditions]]''. They are given in | + | *''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. | ||
| − | *''Glide planes'' | + | <table border cellspacing=0 cellpadding=5 align=center> |
| − | The resulting conditions apply only to | + | <caption align=top> '''Table 1. Integral reflection conditions for centred cells.''' </caption> |
| + | <tr align=left> | ||
| + | <th>Reflection<br> condition </th> | ||
| + | <th>Centring type of cell </th> | ||
| + | <th>Centring symbol</th> | ||
| + | </tr> | ||
| + | <tr align=left> | ||
| + | <td>None</td> <td> Primitive</td> <td> ''P''<br> | ||
| + | ''R'' (rhombohedral axes)</td> | ||
| + | </tr> | ||
| + | <tr align=left> | ||
| + | <td>''h'' + ''k'' = 2''n''</td> <td>''C''-face centred</td> <td>''C''</td> | ||
| + | </tr> | ||
| + | <tr align=left> | ||
| + | <td>''k'' + ''l'' = 2''n''</td> <td>''A''-face centred</td> <td>''A''</td> | ||
| + | </tr> | ||
| + | <tr align=left> | ||
| + | <td>''l'' + ''h'' = 2''n''</td> <td>''B''-face centred</td> <td>''B''</td> | ||
| + | </tr> | ||
| + | <tr align=left> | ||
| + | <td>''h'' + ''k'' + ''l'' = 2''n''</td> <td>body centred</td> <td>''I''</td> | ||
| + | </tr> | ||
| + | <tr align=left> | ||
| + | <td>''h'' + ''k'', ''h'' + ''l'' and<br> | ||
| + | ''k'' + ''l'' = 2''n'' or:<br> | ||
| + | ''h'', ''k'', ''l'' all odd or all<br> | ||
| + | even (‘unmixed’)</td> <td>all-face centred</td> <td> ''F''</td> | ||
| + | </tr> | ||
| + | <tr align=left> | ||
| + | <td> − ''h'' + ''k'' + ''l'' = 3''n''</td> <td> rhombohedrally<br> | ||
| + | centred, obverse<br> | ||
| + | setting (standard)</td> | ||
| + | <td rowspan=2>''R'' (hexagonal axes)</td></tr> | ||
| + | <tr align=left> | ||
| + | <td> ''h'' − ''k'' + ''l'' = 3''n''</td><td> rhombohedrally<br> | ||
| + | centred, reverse<br> | ||
| + | setting </td></tr> | ||
| + | <tr align=left> | ||
| + | <td> ''h'' − ''k'' = 3''n''</td> <td>hexagonally centred</td> <td> ''H''</td></tr> | ||
| + | <tr align=left> | ||
| + | <td> ''h'' + ''k'' + ''l'' = 3''n''</td> <td> D centred</td><td>D</td></tr> | ||
| + | </table> | ||
| + | |||
| + | *''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): | they are called ''[[zonal reflection conditions]]''. For instance, for a glide plane parallel to (001): | ||
<table border cellspacing=0 cellpadding=5 align=center> | <table border cellspacing=0 cellpadding=5 align=center> | ||
<tr> | <tr> | ||
| − | <th> | + | <th>Type of reflection</th><th>Reflection condition</th> <th>Glide vector</th><th>Glide plane</th> </tr> |
<tr> <td rowspan=4>0''kl''</td> <td>''k'' = 2 ''n''</td> <td>'''b'''/2</td><td> ''b''</td></tr> | <tr> <td rowspan=4>0''kl''</td> <td>''k'' = 2 ''n''</td> <td>'''b'''/2</td><td> ''b''</td></tr> | ||
<tr> | <tr> | ||
| Line 26: | Line 71: | ||
</table> | </table> | ||
| − | The zonal reflection conditions are listed in Table 2. | + | 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 | + | *''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: | ||
<table border cellspacing=0 cellpadding=5 align=center> | <table border cellspacing=0 cellpadding=5 align=center> | ||
<tr> | <tr> | ||
| − | <th> | + | <th>Type of reflection</th><th>Reflection condition</th> <th>Screw vector</th><th>Screw axis</th> </tr> |
<tr> <td rowspan=2>00''l''</td> <td>''l'' = 2 ''n''</td> <td>'''c'''/2</td><td> 2<sub>1</sub>; 4<sub>2</sub></td></tr> | <tr> <td rowspan=2>00''l''</td> <td>''l'' = 2 ''n''</td> <td>'''c'''/2</td><td> 2<sub>1</sub>; 4<sub>2</sub></td></tr> | ||
<tr> | <tr> | ||
| Line 43: | Line 89: | ||
</table> | </table> | ||
| − | The serial reflection conditions are listed in Table 2. | + | The serial reflection conditions are listed in Table 2.1.3.7 of ''International Tables for Crystallography, Volume A'', 6th edition. |
| − | (2) ''Special conditions'' ( | + | (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 == | ==See also == | ||
| − | + | *Chapter 1.6.4 of ''International Tables for Crystallography, Volume A'', 6th edition | |
| − | + | [[Category:X-rays]] | |
| − | [[Category:X-rays]] | ||
Latest revision as of 15:59, 24 February 2019
Conditions de réflexion (Fr). Auslöschungsgesetze (Ge). Condizioni di diffrazione (It). 回折条件 (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 | 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: | 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 hk0, h0l, 0kl, 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 n k, 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 h00, 0k0, 00l). 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 | 21; 42 |
| l = 4 n | c/4 | 41; 43 | |
| 000l | l = 2 n | c/2 | 63 |
| l = 3 n | c/3 | 41; 31; 32; 62; 64 | |
| l = 6 n | c/6 | 61;65 |
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