CCP4i: Graphical User Interface
	Reindexing

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.
??? Should there be something about real and reciprocal axes? Look at both - see if any are preserved????

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).

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 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



• 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 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



• 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 number	space group	point group	crystal 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 number	space group	point group	crystal 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 number	space group	point group	crystal 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 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



• 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 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



• 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 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



• 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
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