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! Bravais type | ! Bravais type | ||
! spacegroup <br> number <br> suggested by <br> CORRECT | ! spacegroup <br> number <br> suggested by <br> CORRECT | ||
! other possibilities | ! other possibilities (with screw axes) | ||
! alternative indexing <br> possible? | ! alternative indexing <br> possible? | ||
! choosing among all spacegroup possibilities | ! choosing among all spacegroup possibilities | ||
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| mmm || oI || 23 || 24 |||| screw axis extinctions do '''not''' let you decide because the I centering results in h+k+l=2n and the screw axis extinction 00l=2n is just a special case of that. 23/24 do '''not''' form an enantiomorphic, but a ''special'' pair (ITC A §3.5, p. 46 in the 1995 edition). | | mmm || oI || 23 || 24 |||| screw axis extinctions do '''not''' let you decide because the I centering results in h+k+l=2n and the screw axis extinction 00l=2n is just a special case of that. 23/24 do '''not''' form an enantiomorphic, but a ''special'' pair (ITC A §3.5, p. 46 in the 1995 edition). | ||
|- | |- | ||
| 4/m || tP || 75 || 76, 77, 78 || | | 4/m || tP || 75 || 76, 77, 78 ||k,h,-l|| screw axis extinctions let you decide, except between 76/78 enantiomorphs | ||
|- | |- | ||
| 4/m || tI || 79 || 80 || | | 4/m || tI || 79 || 80 ||k,h,-l|| screw axis extinctions let you decide | ||
|- | |- | ||
| 4/mmm || tP || 89 || 90, 91, 92, 93, 94, 95, 96 |||| screw axis extinctions let you decide, except between 91/95 and 92/96 enantiomorphs | | 4/mmm || tP || 89 || 90, 91, 92, 93, 94, 95, 96 |||| screw axis extinctions let you decide, except between 91/95 and 92/96 enantiomorphs | ||
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| 4/mmm || tI || 97 || 98 |||| screw axis extinctions let you decide | | 4/mmm || tI || 97 || 98 |||| screw axis extinctions let you decide | ||
|- | |- | ||
| -3 || hP || 143 || 144, 145 || | | -3 || hP || 143 || 144, 145 ||-h,-k,l; k,h,-l; -k,-h,-l|| screw axis extinctions let you decide, except between 144/145 enantiomorphs | ||
|- | |- | ||
| -3 || hR || 146 || - || | | -3 || hR || 146 || - ||k,h,-l, and obverse (-h+k+l=3n) / reverse (h-k+l=3n)|| | ||
|- | |- | ||
| -3/m || hP || 149 || 151, 153 || | | -3/m || hP || 149 || 151, 153 ||k,h,-l|| screw axis extinctions let you decide, except between 151/153 enantiomorphs. Note: the twofold goes along the diagonal between a and b. | ||
|- | |- | ||
| -3/m || hP || 150 || 152, 154 || | | -3/m || hP || 150 || 152, 154 ||-h,-k,l|| screw axis extinctions let you decide, except between 152/154 enantiomorphs. Note: compared to previous line, the twofold goes along a. | ||
|- | |- | ||
| -3/m || hR || 155 || - ||obverse/reverse|| | | -3/m || hR || 155 || - ||obverse/reverse|| | ||
|- | |- | ||
| 6/m || hP || 168 || 169, 170, 171, 172, 173 || | | 6/m || hP || 168 || 169, 170, 171, 172, 173 ||k,h,-l|| screw axis extinctions let you decide, except between 169/170 and 171/172 enantiomorphs | ||
|- | |- | ||
| 6/mmm || hP || 177 || 178, 179, 180, 181, 182 |||| screw axis extinctions let you decide, except between 178/179 and 180/181 enantiomorphs | | 6/mmm || hP || 177 || 178, 179, 180, 181, 182 |||| screw axis extinctions let you decide, except between 178/179 and 180/181 enantiomorphs | ||
|- | |- | ||
| m-3 || cP || 195 || 198 || | | m-3 || cP || 195 || 198 ||k,h,-l|| screw axis extinctions let you decide | ||
|- | |- | ||
| m-3 || cF || 196 ||-|| | | m-3 || cF || 196 ||-||k,h,-l|| | ||
|- | |- | ||
| m-3 || cI || 197 || 199 || | | m-3 || cI || 197 || 199 ||k,h,-l|| screw axis extinctions do '''not''' let you decide because the I centering results in h+k+l=2n and the screw axis extinction 00l=2n is just a special case of that. 197/199 do '''not''' form an enantiomorphic, but a ''special'' pair (ITC A §3.5, p. 46 in the 1995 edition). | ||
|- | |- | ||
| m-3m || cP || 207 || 208, 212, 213 |||| screw axis extinctions let you decide, except between 212/213 enantiomorphs | | m-3m || cP || 207 || 208, 212, 213 |||| screw axis extinctions let you decide, except between 212/213 enantiomorphs | ||
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|} | |} | ||
Alternative indexing possibilities taken from http://www.ccp4.ac.uk/html/reindexing.html (for R3 and R32, obverse/reverse are specified). | |||
If you find an error in the table please send an email to kay dot diederichs at uni-konstanz dot de ! | If you find an error in the table please send an email to kay dot diederichs at uni-konstanz dot de ! | ||
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== Subgroup and supergroup relations of these space groups == | === Subgroup and supergroup relations of these space groups === | ||
Compiled from [https://onlinelibrary.wiley.com/doi/book/10.1107/97809553602060000001 International Tables for Crystallography (2006) Vol. A1 (Wiley)]. Simply put, for each space group, a maximum ''translationengleiche'' subgroup has lost a single type of symmetry, and a minimum ''translationengleiche'' supergroup has gained a single symmetry type. Example: P222 is a supergroup of P2, and a subgroup of P422 (and P4222 and P23). Of course the | Compiled from [https://onlinelibrary.wiley.com/doi/book/10.1107/97809553602060000001 International Tables for Crystallography (2006) Vol. A1 (Wiley)]. Simply put, for each space group, a maximum ''translationengleiche'' subgroup has lost a single type of symmetry, and a minimum ''translationengleiche'' supergroup has gained a single symmetry type. Example: P222 is a supergroup of P2, and a subgroup of P422 (and P4222 and P23). Of course the sub-/supergroup relation is recursive, which is why P1 is also a (sub-)subgroup of P222 (but not a maximum ''translationengleiche'' subgroup). The table below does not show other types of relations, e.g. non-isomorphic ''klassengleiche'' supergroups which may result e.g. from centring translations, because I find them less relevant in space group determination. | ||
The table | |||
{| cellpadding=" | The table is relevant because in particular (perfect) twinning adds a symmetry type, and leads to an apparent space group which is the supergroup of the true space group. | ||
! spacegroup | {| cellpadding="0" cellspacing="0" border="1" | ||
! spacegroup number | |||
! maximum ''translationengleiche'' subgroup | ! maximum ''translationengleiche'' subgroup | ||
! minimum ''translationengleiche'' supergroup | ! minimum ''translationengleiche'' supergroup | ||
! name | ! spacegroup name | ||
|- | |- | ||
| 1 ||-|| 3, 4, 5, 143, 144, 145, 146 || P 1 | | 1 ||-|| 3, 4, 5, 143, 144, 145, 146 || P 1 | ||
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<pre> | <pre> | ||
SPACE_GROUP_NUMBER=5 | SPACE_GROUP_NUMBER=5 | ||
UNIT_CELL_CONSTANTS= 233.70 78.42 73.22 90 | UNIT_CELL_CONSTANTS= 233.70 78.42 73.22 90 105.34 90 | ||
</pre> | </pre> | ||
because this enforces just the correct cell constraints. | because this enforces just the correct cell constraints. | ||
== See also == | == See also == | ||
[http://pd.chem.ucl.ac.uk/pdnn/symm3/allsgp.htm The 230 3-Dimensional Space Groups] | [http://pd.chem.ucl.ac.uk/pdnn/symm3/allsgp.htm The 230 3-Dimensional Space Groups] |