  GNU-Darwin Web

# Reindexing (CCP4: General)

## NAME

reindexing - information about changing indexing regime

### General Remarks

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*)
N.B. There are alternatives where other pairs of symmetry operators are used, but this is the simplest and most general set of operators. For example: in P4i (-a,-b,c) is equivalent to (-b,a,c), or in P3i (-a,-b,c) is equivalent to (-b,a+b,c).

.N.B. In general you should not change the hand of your axial system; i.e. the determinant of the transformation matrix should be positive, and only such transformations are discussed here.

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)] are indistinguishable. This is the case for all tetragonal, hexagonal and cubic spacegroups.
If (h,k,l) is equivalent to (k,h,-l), the axial system pairs [(a,b,c) and (b,a,-c)] and
[(-a,-b,c) and (-b,-a,-c)] are indistinguishable. This is true for P4i2i2, P3i2, P6i22 and some cubic spacegroups.
If (h,k,l) is equivalent to (-k,-h,-l), the axial system pairs [(a,b,c) and (-b,-a,-c)] and
[(-a,-b,c) and (b,a,-c)] are indistinguishable. This is only true for P3i12 spacegroups.
See detailed descriptions below.

### Lookup tables

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.
• space group numberspace grouppoint groupcrystal 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

• 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.
• space group numberspace grouppoint groupcrystal 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

• 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!.
• space group numberspace grouppoint groupcrystal system
143 P3 PG3 TRIGONAL
144 P31 PG3 TRIGONAL
145 P32 PG3 TRIGONAL
146 R3 PG3 TRIGONAL

• 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).
• space group numberspace grouppoint groupcrystal system
149 P312 PG312 TRIGONAL
151 P3112 PG312 TRIGONAL
153 P3212 PG312 TRIGONAL

• 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).
• space group numberspace grouppoint groupcrystal system
150 P321 PG321 TRIGONAL
152 P3121 PG321 TRIGONAL
154 P3221 PG321 TRIGONAL
155 R32 PG32 TRIGONAL

• 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).
• space group numberspace grouppoint groupcrystal 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

• 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.
• space group numberspace grouppoint groupcrystal 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

• 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).
• space group numberspace grouppoint groupcrystal system
195 P23 PG23 CUBIC
196 F23 PG23 CUBIC
197 I23 PG23 CUBIC
198 P213 PG23 CUBIC
199 I213 PG23 CUBIC

• 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 numberspace grouppoint 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

### Changing hand

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!

P21 - to P21; For the half step of 21 axis, the symmetry stays the same.

P31 - to P32
P32 - to P31

P41 to P43
(P42 - to P42: Half c axis step)
P43 -to P41

P61 to P65
P62 - to P64
(P63 - to P63)
etc.

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

I41
(x,y,z) to (-x,1/2-y,-z)
I4122
(x,y,z) to (-x,1/2-y,1/4-z)
F4132
(x,y,z) to (3/4-x,1/4-y,3/4-z)

Plus some centric ones:

Fdd2
(x,y,z) to (1/4-x,1/4-y,-z)
I41md
(x,y,z) to (1/4-x,1/4-y,-z)
I41cd
(x,y,z) to (1/4-x,1/4-y,-z)
I4bar2d
(x,y,z) to (1/4-x,1/4-y,-z)

## PICTURES

Full size versions of the example pictures can be viewed by clicking on the iconised ones. A P32 data set indexed h,k,l The same P32 data set, reindexed -h,-k,l The same P32 data set, reindexed -k,-h,l The same P32 data set, reindexed k,h,-l Monoclinic data set, HKLVIEW h,2,l The same monoclinic data set, reindexed -h,-k,h-l

## AUTHORS

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