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Crystallographic symmetries in 2dx
Nikhil Biyani edited this page Aug 17, 2016
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A crystallographic space group is the mathematical group of symmetry operations which apply to both the given unit cell and the crystal array. Only a finite number of crystal packing arrangements are possible which counts to 230 possible crystallographic space groups in 3D and reduces to 65 for proteins and chiral molecules.
17 plane groups describe all the possible symmetry arrangements in projection images of 2D crystals. These plane groups are different (but correlate somewhat trivially) to the 17 2D space groups which describe all possible 2D crystal arrangements.
NOTE: These are still to be implemented in 2dx, probably in coming versions
We internally use symmetry codes inspired by MRC software package. The possible symmetries are described below:
| 2dx/MRC code | 1 |
| Summary | Identical |
| Symmetry Operators | (x,y,z) |
| Additional treatment | None |
| Systematic absences | None |
| CCP4 equivalent | Number 1; Space Group P1 |
| 2dx/MRC code | 2 |
| Summary | Two fold symmetry with dyad along Z |
| Symmetry Operators | (x,y,z), (-x,-y,z) |
| Additional treatment | All phases for l=0 should be 0 or 180 (whichever closer) |
| Systematic absences | None |
| CCP4 equivalent | Number 1003; Space Group P121 |
| 2dx/MRC code | 3a |
| Summary | Two fold symmetry with dyad along X |
| Symmetry Operators | (x,y,z), (x,-y,-z) |
| Additional treatment | None |
| Systematic absences | None |
| CCP4 equivalent | Number 1003; Space Group P121 |
| 2dx/MRC code | 3b |
| Summary | Two fold symmetry with dyad along Y |
| Symmetry Operators | (x,y,z), (-x,y,-z) |
| Additional treatment | All phases for k=0 & l=0 should be 0 or 180 (whichever closer) |
| Systematic absences | Amplitudes for (0, 2k+1, 0) = 0 |
| CCP4 equivalent | Number 3; Space Group P121 |
| 2dx/MRC code | 4a |
| Summary | Two fold screw along X |
| Symmetry Operators | (x,y,z), (1/2+x,-y,-z) |
| Additional treatment | None |
| Systematic absences | None |
| CCP4 equivalent | Number 1004; Space Group P1211 |
| 2dx/MRC code | 4b |
| Summary | Two fold screw along X |
| Symmetry Operators | (x,y,z), (-x,1/2+y,-z) |
| Additional treatment | All phases for k=0 & l=0 should be 0 or 180 (whichever closer) |
| Systematic absences | Amplitudes for (0, 2k+1, 0) = 0 |
| CCP4 equivalent | Number 4; Space Group P1211 |
| 2dx/MRC code | 5a |
| Summary | Two fold with dyad along X and a phase shift |
| Symmetry Operators | (x,y,z), (x,-y,-z) |
| Additional treatment | phase shift (+1/2, +1/2, 0) |
| Systematic absences | Amplitudes for spots with mod(h+k,2) equals 1 are 0 |
| CCP4 equivalent | Number 1005; Space Group C2 |
| 2dx/MRC code | 5b |
| Summary | Two fold with dyad along Y and a phase shift |
| Symmetry Operators | (x,y,z), (-x,y,-z) |
| Additional treatment | phase shift (+1/2, +1/2, 0) and all phases for k=0 & l=0 should be 0 or 180 (whichever closer) |
| Systematic absences | Amplitudes for (0, 2k+1, 0) and if mod(h+k,2) equals 1 are 0 |
| CCP4 equivalent | Number 5; Space Group C2 |
| 2dx/MRC code | 6 |
| Summary | Orthorhombic symmetry |
| Symmetry Operators | (x,y,z), (-x,-y,z), (x,-y,-z), (-x,y,-z) |
| Additional treatment | All phases for l=0 should be 0 or 180 (whichever closer) |
| Systematic absences | None |
| CCP4 equivalent | Number 16; Space Group P222 |
| 2dx/MRC code | 7a |
| Summary | Two fold screw along X and 2nd axis along Y |
| Symmetry Operators | (x,y,z), (1/2+x,-y,-z), (-x,y,-z), (1/2-x,-y,z) |
| Additional treatment | None |
| Systematic absences | Amplitude for (2h+1, 0, 0) = 0 |
| CCP4 equivalent | Number 2017; Space Group P2221 |
| 2dx/MRC code | 7b |
| Summary | Two fold screw along Y and 2nd axis along X |
| Symmetry Operators | (x,y,z), (x,-y,-z), (-x,1/2+y,-z), (-x,1/2-y,z) |
| Additional treatment | All phases for l=0 should be 0 or 180 (whichever closer) |
| Systematic absences | Amplitude for (0, 2k+1, 0) = 0 |
| CCP4 equivalent | Number 1017; Space Group P2221 |
| 2dx/MRC code | 8 |
| Summary | Twofold down Z |
| Symmetry Operators | (x,y,z),(1/2+x,1/2-y,-z),(1/2-x,1/2+y,-z),(-x,-y,z) |
| Additional treatment | All phases for l=0 should be 0 or 180 (whichever closer) |
| Systematic absences | Amplitude for (2h+1, 0, 0) and (0, 2k+1, 0) = 0 |
| CCP4 equivalent | Number 18; Space Group P21212 |
| 2dx/MRC code | 9 |
| Summary | Orthrhombic with additional phase shift |
| Symmetry Operators | (x,y,z),(1/2+x,1/2-y,-z),(1/2-x,1/2+y,-z),(-x,-y,z) |
| Additional treatment | phase shift (1/2,1/2,0) and all phases for l=0 should be 0 or 180 (whichever closer) |
| Systematic absences | Amplitudes for spots with mod(h+k,2) equals 1 are 0 |
| CCP4 equivalent | Number 21; Space Group C222 |
| 2dx/MRC code | 10 |
| Summary | Four fold |
| Symmetry Operators | (x,y,z),(-x,-y,z),(-y,x,z),(y,-x,z) |
| Additional treatment | All phases for l=0 should be 0 or 180 (whichever closer) |
| Systematic absences | None |
| CCP4 equivalent | Number 75; Space Group P4 |
| 2dx/MRC code | 11 |
| Summary | Four fold |
| Symmetry Operators | (x,y,z),(-x,-y,z),(-y,x,z),(y,-x,z),(x,-y,-z),(-x,y,-z),(-y,-x,-z),(y,x,-z) |
| Additional treatment | All phases for l=0 should be 0 or 180 (whichever closer) |
| Systematic absences | None |
| CCP4 equivalent | Number 89; Space Group P422 |
| 2dx/MRC code | 12 |
| Summary | Four fold up and down |
| Symmetry Operators | (x,y,z), (-x,-y,z), (1/2-y,1/2+x,z), (1/2+y,1/2-x,z), (1/2+x,1/2-y,-z), (1/2-x,1/2+y,-z), (-y,-x,-z), (y,x,-z) |
| Additional treatment | All phases for l=0 should be 0 or 180 (whichever closer) |
| Systematic absences | Amplitude for (2h+1, 0, 0) and (0, 2k+1, 0) = 0 |
| CCP4 equivalent | Number 90; Space Group P4212 |
| 2dx/MRC code | 13 |
| Summary | Three fold |
| Symmetry Operators | (x,y,z), (-y,x-y,z), (-x+y,-x,z) |
| Additional treatment | None |
| Systematic absences | None |
| CCP4 equivalent | Number 143; Space Group P3 |
| 2dx/MRC code | 14 |
| Summary | Three fold |
| Symmetry Operators | (x,y,z), (-y,x-y,z), (-x+y,-x,z), (-y,-x,-z), (-x+y,y,-z), (x,x-y,-z) |
| Additional treatment | All phases for h=k & l=0 should be 0 or 180 (whichever closer) |
| Systematic absences | None |
| CCP4 equivalent | Number 149; Space Group P312 |
| 2dx/MRC code | 15 |
| Summary | Three fold |
| Symmetry Operators | (x,y,z), (-y,x-y,z), (-x+y,-x,z), (y,x,-z), (x-y,-y,-z), (-x,-x+y,-z) |
| Additional treatment | All phases for (k=0 & l=0) + (h=0 & l=0) should be 0 or 180 (whichever closer) |
| Systematic absences | None |
| CCP4 equivalent | Number 150; Space Group P321 |
| 2dx/MRC code | 16 |
| Summary | Six fold |
| Symmetry Operators | (x,y,z), (-y,x-y,z), (-x+y,-x,z), (-x,-y,z), (x-y,x,z), (y,-x+y,z) |
| Additional treatment | All phases for l=0 should be 0 or 180 (whichever closer) |
| Systematic absences | None |
| CCP4 equivalent | Number 168; Space Group P6 |
| 2dx/MRC code | 17 |
| Summary | Six fold |
| Symmetry Operators | (x,y,z), (-y,x-y,z), (-x+y,-x,z), (-x,-y,z), (x-y,x,z), (y,-x+y,z), (y,x,-z), (x-y,-y,-z), (-x,-x+y,-z), (-y,-x,-z), (-x+y,y,-z), (x,x-y,-z) |
| Additional treatment | All phases for l=0 should be 0 or 180 (whichever closer) |
| Systematic absences | None |
| CCP4 equivalent | Number 177; Space Group P622 |
Per Bullough, University Sheffield, UK (Thanks a lot, Per!!!)