It is quite common to find that the diffraction from subsequent crystals for a protein do not apparently merge well. There are many physical reasons for this, but before throwing the data away it is sensible to consider whether another indexing regime could be used. For illustrations and examples see HKLVIEW-examples below. For documentation on re-indexing itself, and some hints, see also REINDEX.

For **orthorhombic** crystal forms with different cell dimensions
along each axis you can usually recognise if the next crystal is the same as the last
and see how to transform it (remember to keep your axial system right-handed!).

In **P1** and **P21** there are many
ways of choosing axes,
but they should all generate the same crystal volume. Use
MATTHEWS_COEF or some other
method to check this - if the volumes *are not* the same, or at least related by
integral factors, you have a new form.
If they *are* the same it is recommended to plot some sections of the reciprocal
lattice; you can often see that the patterns will match if you rotate in some way
(see
HKLVIEW-examples below). A common change
in **P21**
or **C2** where the twofold axis will be constant, is that
a**new* = a**old*
+ c**old*, and c**new*
must be chosen carefully.
One very confusing case can arise if the length of (a*+nc*) is almost equal
to that of a* or nc*, but it should be possible to sort out from the diffraction pattern
plots.

Confusion arises mostly when two or more axes are the same length, as in the
**tetragonal**, **trigonal**, **hexagonal** or
**cubic** systems.
In these cases any of the following definitions of axes is equally valid and likely
to be chosen by an auto-indexing procedure. The classic description of this is that
these are crystals where the Laue symmetry is of a lower order than the apparent
crystal lattice symmetry.

real axes: | (a,b,c) | or | (-a,-b,c) | or | (b,a,-c) | or | (-b,-a,-c) |

reciprocal axes: | (a*,b*,c*) | or | (-a*,-b*,c*) | or | (b*,a*,-c*) | or | (-b*,-a*,-c*) |

*The crystal symmetry may mean that some of these systems are already equivalent:*

*For instance, if *(h,k,l)* is equivalent to *(-h,-k,l)*, the axial system
pairs *[(a,b,c) and (-a,-b,c)]* and *[(b,a,-c) and (-b,-a,-c)]

- run1 55.76 84.62 70.96 90 112.71 90
- run2 56.04 85.11 71.57 90 113.15 90

Some basic trigonometry shows the new cell dimensions to be:

- run1 (reindexed) 55.76 84.62 71.33 90 113.42 90

Note the transformation `-h, -k, h+l` preserves the hand, i.e. the
corresponding matrix has a determinant of +1.

Here are details for the possible systems:

- All
**P4i**and related**4i**space groups:

(h,k,l) equivalent to (-h,-k,l) so we only need to check:real axes: (a,b,c) and (b,a,-c) reciprocal axes: (a*,b*,c*) and (b*,a*,-c*) *i.e.*check if reindexing (h,k,l) to (k,h,-l) gives a better match to previous data sets. - For all
**P4i2i2**and related**4i2i2**space groups:

(h,k,l) is equivalent to (-h,-k,l)*and*(k,h,-l)*and*(-k,-h,-l) so any choice of axial system will give identical data. - All
**P3i**and**R3**:

(h,k,l)*not*equivalent to (-h,-k,l)*or*(k,h,-l) or (-k,-h,-l) so we need to check all 4 possibilities:real axes: (a,b,c) and (-a,-b,c) and (b,a,-c) and (-b,-a,-c) reciprocal axes: (a*,b*,c*) and (-a*,-b*,c*) and (b*,a*,-c*) and (-b*,-a*,-c*) *i.e.*reindex (h,k,l) to (-h,-k,l)*or*(h,k,l) to (k,h,-l)*or*(h,k,l) to (-k,-h,-l).

*N.B. For trigonal space groups, symmetry equivalent reflections can be conveniently described as*(h,k,l)*,*(k,i,l)*and*(i,h,l)*where*i=-(h+k). Replacing the 4 basic sets with a symmetry equivalent gives a bewildering range of possibilities!*.* - All
**P3i12**:

(h,k,l) already equivalent to (-k,-h,-l) so we only need to check:real axes: (a,b,c) and (b,a,-c) reciprocal axes: (a*,b*,c*) and (b*,a*,-c*) *i.e.*reindex (h,k,l) to (k,h,-l) which is equivalent here to reindexing (h,k,l) to (-h,-k,l). - All
**P3i21**and**R32**:

(h,k,l) already equivalent to (k,h,-l) so we only need to check:real axes: (a,b,c) and (-a,-b,c) reciprocal axes: (a*,b*,c*) and (-a*,-b*,c*) *i.e.*reindex (h,k,l) to (-h,-k,l). - All
**P6i**:

(h,k,l) already equivalent to (-h,-k,l) so we only need to check:real axes: (a,b,c) and (b,a,-c) reciprocal axes: (a*,b*,c*) and (b*,a*,-c*) *i.e.*reindex (h,k,l) to (k,h,-l). - All
**P6i2**:

(h,k,l) already equivalent to (-h,-k,l)*and*(k,h,-l)*and*(-k,-h,-l) so we do not need to check. - All
**P2i3**and related**2i3**space groups:

(h,k,l) already equivalent to (-h,-k,l) so we only need to check:real axes: (a,b,c) and (b,a,-c) reciprocal axes: (a*,b*,c*) and (b*,a*,-c*) *i.e.*reindex (h,k,l) to (k,h,-l). - All
**P4i32**and related**4i32**space groups:

(h,k,l) already equivalent to (-h,-k,l)*and*(k,h,-l)*and*(-k,-h,-l) so we do not need to check.

space group number | space group | point group | crystal system |
---|---|---|---|

75 | P4 | PG4 | TETRAGONAL |

76 | P41 | PG4 | TETRAGONAL |

77 | P42 | PG4 | TETRAGONAL |

78 | P43 | PG4 | TETRAGONAL |

79 | I4 | PG4 | TETRAGONAL |

80 | I41 | PG4 | TETRAGONAL |

space group number | space group | point group | crystal system |
---|---|---|---|

89 | P422 | PG422 | TETRAGONAL |

90 | P4212 | PG422 | TETRAGONAL |

91 | P4122 | PG422 | TETRAGONAL |

92 | P41212 | PG422 | TETRAGONAL |

93 | P4222 | PG422 | TETRAGONAL |

94 | P42212 | PG422 | TETRAGONAL |

95 | P4322 | PG422 | TETRAGONAL |

96 | P43212 | PG422 | TETRAGONAL |

97 | I422 | PG422 | TETRAGONAL |

98 | I4122 | PG422 | TETRAGONAL |

space group number | space group | point group | crystal system |
---|---|---|---|

143 | P3 | PG3 | TRIGONAL |

144 | P31 | PG3 | TRIGONAL |

145 | P32 | PG3 | TRIGONAL |

146 | R3 | PG3 | TRIGONAL |

space group number | space group | point group | crystal system |
---|---|---|---|

149 | P312 | PG312 | TRIGONAL |

151 | P3112 | PG312 | TRIGONAL |

153 | P3212 | PG312 | TRIGONAL |

space group number | space group | point group | crystal system |
---|---|---|---|

150 | P321 | PG321 | TRIGONAL |

152 | P3121 | PG321 | TRIGONAL |

154 | P3221 | PG321 | TRIGONAL |

155 | R32 | PG32 | TRIGONAL |

space group number | space group | point group | crystal system |
---|---|---|---|

168 | P6 | PG6 | HEXAGONAL |

169 | P61 | PG6 | HEXAGONAL |

170 | P65 | PG6 | HEXAGONAL |

171 | P62 | PG6 | HEXAGONAL |

172 | P64 | PG6 | HEXAGONAL |

173 | P63 | PG6 | HEXAGONAL |

space group number | space group | point group | crystal system |
---|---|---|---|

177 | P622 | PG622 | HEXAGONAL |

178 | P6122 | PG622 | HEXAGONAL |

179 | P6522 | PG622 | HEXAGONAL |

180 | P6222 | PG622 | HEXAGONAL |

181 | P6422 | PG622 | HEXAGONAL |

182 | P6322 | PG622 | HEXAGONAL |

space group number | space group | point group | crystal system |
---|---|---|---|

195 | P23 | PG23 | CUBIC |

196 | F23 | PG23 | CUBIC |

197 | I23 | PG23 | CUBIC |

198 | P213 | PG23 | CUBIC |

199 | I213 | PG23 | CUBIC |

space group number | space group | point group | crystal system |
---|---|---|---|

207 | P432 | PG432 | CUBIC |

208 | P4232 | PG432 | CUBIC |

209 | F432 | PG432 | CUBIC |

210 | F4132 | PG432 | CUBIC |

211 | I432 | PG432 | CUBIC |

212 | P4332 | PG432 | CUBIC |

213 | P4132 | PG432 | CUBIC |

214 | I4132 | PG432 | CUBIC |

Test to see if the other hand is the correct one:

Change x,y,z for (cx-x, cy-y, cz-z)

Usually (cx,cy,cz) = (0,0,0).

Remember you need to change the twist on the screw-axis stairs for P3i, P4i, or P6i!

P2_{1} - to P2_{1}; For the half step of 2_{1} axis,
the symmetry stays the same.

P3_{1} - to P3_{2}

P3_{2} - to P3_{1}

P4_{1} to P4_{3}

(P4_{2} - to P4_{2}: Half c axis step)

P4_{3} -to P4_{1}

P6_{1} to P6_{5}

P6_{2} - to P6_{4}

(P6_{3} - to P6_{3})

etc.

In a few non-primitive spacegroups, you can change the hand and not change the spacegroup by a cunning shift of origin:

- I4
_{1} - (x,y,z) to (-x,1/2-y,-z)
- I4
_{1}22 - (x,y,z) to (-x,1/2-y,1/4-z)
- F4
_{1}32 - (x,y,z) to (1/4-x,1/4-y,1/4-z)

Plus some centric ones:

- Fdd2
- (x,y,z) to (1/4-x,1/4-y,-z)
- I4
_{1}md - (x,y,z) to (1/4-x,1/4-y,-z)
- I4
_{1}cd - (x,y,z) to (1/4-x,1/4-y,-z)
- I4bar2d
- (x,y,z) to (1/4-x,1/4-y,-z)

Full size versions of the example pictures can be viewed by clicking on the iconised ones.

Prepared for CCP4 by Maria Turkenburg, University of York, England

Some additional material from Martyn Winn, Daresbury Lab.

- [Bernhardinelli(1985)]
- G. Bernardinelli and H. D. Flack (1985). Least-squares absolute-structure refinement. Practical experience and ancillary calculations. Acta Cryst. A41, 500-511.