# Space group determination

### Space groups as combinations of symmetry elements

Space group determination entails the following steps:

1. determine the Laue group: this is the symmetry of the intensity-weighted point lattice (diffraction pattern). 1,2,3,4,6=n-fold rotation axis; -n means inversion centre; m means mirror.
2. find out about Bravais type: the second letter signifies centering (P=primitive; C=centered on C-face; F=centered on all faces; I=body-centered; R=rhombohedral). The first letter (a=triclinic; m=monoclinic; o=orthorhombic; h=trigonal or hexagonal; c=cubic) is redundant since it can be inferred from the Laue group.
3. possible spacegroups are now given in columns 3 and 4; CORRECT always suggests the one given in column 3 but this is no more likely than those in column 4.
4. choose according to screw axis; this may result in two possibilities (enantiomorphs).
5. determine correct enantiomorph - this usually means that one tries to solve the structure in both spacegroups, and only one gives a sensible result (like helices that are right-handed, amino acids of the L type).

Laue group Bravais type spacegroup
number
suggested by
CORRECT
other possibilities alternative indexing
possible?
choosing among all spacegroup possibilities
-1 aP 1 -
2m mP 3 4 screw axis extinctions let you decide
2m mC 5 - The pointless article discusses why I2 (mI) should not be used.
mmm oP 16 17, 18, 19 screw axis extinctions let you decide. The pointless article discusses why CCP4's 1017 (P2122), 2017 (P2212), 2018 (P21221) , 3018 (P22121) are not needed and should not be used.
mmm oC 21 20 screw axis extinctions let you decide
mmm oF 22 -
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 yes screw axis extinctions let you decide, except between 76/78 enantiomorphs
4/m tI 79 80 yes 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 tI 97 98 screw axis extinctions let you decide
-3 hP 143 144, 145 4 possibilities screw axis extinctions let you decide, except between 144/145 enantiomorphs
-3 hR 146 - yes, and obverse/reverse
-3/m hP 149 151, 153 yes 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 yes 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
6/m hP 168 169, 170, 171, 172, 173 yes 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
m-3 cP 195 198 yes screw axis extinctions let you decide
m-3 cF 196 - yes
m-3 cI 197 199 yes 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 cF 209 210 screw axis extinctions let you decide
m-3m cI 211 214 screw axis extinctions let you decide

If you find an error in the table please send an email to kay dot diederichs at uni-konstanz dot de !

## The 65 space groups in which proteins composed of L-amino acids can crystallize

The mapping of numbers and names is:

```****** LATTICE SYMMETRY IMPLICATED BY SPACE GROUP SYMMETRY ******

BRAVAIS-           POSSIBLE SPACE-GROUPS FOR PROTEIN CRYSTALS
TYPE                     [SPACE GROUP NUMBER,SYMBOL]
aP      [1,P1]
mP      [3,P2] [4,P2(1)]
mC,mI    [5,C2]
oP      [16,P222] [17,P222(1)] [18,P2(1)2(1)2] [19,P2(1)2(1)2(1)]
oC      [21,C222] [20,C222(1)]
oF      [22,F222]
oI      [23,I222] [24,I2(1)2(1)2(1)]
tP      [75,P4] [76,P4(1)] [77,P4(2)] [78,P4(3)] [89,P422] [90,P42(1)2]
[91,P4(1)22] [92,P4(1)2(1)2] [93,P4(2)22] [94,P4(2)2(1)2]
[95,P4(3)22] [96,P4(3)2(1)2]
tI      [79,I4] [80,I4(1)] [97,I422] [98,I4(1)22]
hP      [143,P3] [144,P3(1)] [145,P3(2)] [149,P312] [150,P321] [151,P3(1)12]
[152,P3(1)21] [153,P3(2)12] [154,P3(2)21] [168,P6] [169,P6(1)]
[170,P6(5)] [171,P6(2)] [172,P6(4)] [173,P6(3)] [177,P622]
[178,P6(1)22] [179,P6(5)22] [180,P6(2)22] [181,P6(4)22] [182,P6(3)22]
hR      [146,R3] [155,R32]
cP      [195,P23] [198,P2(1)3] [207,P432] [208,P4(2)32] [212,P4(3)32]
[213,P4(1)32]
cF      [196,F23] [209,F432] [210,F4(1)32]
cI      [197,I23] [199,I2(1)3] [211,I432] [214,I4(1)32]
```

## Space group selected by XDS: ambiguous with respect to enantiomorph and screw axes

In case of a crystal with an unknown space group (SPACE_GROUP_NUMBER=0 in XDS.INP), XDS (since version June 2008) helps the user in determination of the correct space group, by suggesting possible space groups compatible with the Laue symmetry and Bravais type of the data, and by calculating the Rmeas for these space groups.

XDS (or rather, the CORRECT step) makes an attempt to pick the correct space group automatically: it chooses that space group which has the highest symmetry (thus yielding the lowest number of unique reflections) and still a tolerable Rmeas compared to the Rmeas the data have in any space group (which is most likely a low-symmetry space group - often P1).

In some cases the automatic choice is the correct one, and re-running the CORRECT step is then not necessary. However, neither the correct enantiomorph nor screw axes (see below) are determined automatically by XDS.

## Space group selected by user

In case the space group selected by XDS should be incorrect, the resulting list (in CORRECT.LP) should give the user enough information to pick the correct space group herself (alternatives are in the big table above!). The user may then put suitable lines with SPACE_GROUP_NUMBER=, UNIT_CELL_CONSTANTS= into XDS.INP and re-run the CORRECT step to obtain the desired result. (The REIDX= line is no longer required; XDS figures the matrix out.)

## Influencing the selection by XDS

### keywords and parameters

The automatic choice is influenced by a number of decision constants that may be put into XDS.INP but which have defaults as indicated below:

• MAX_CELL_AXIS_ERROR= 0.03 ! relative deviation of unconstrained cell axes from those constrained by lattice symmetry
• MAX_CELL_ANGLE_ERROR= 3.0 ! degrees deviation of unconstrained cell angles from those constrained by lattice symmetry
• TEST_RESOLUTION_RANGE= 10.0 5.0 ! resolution range for calculation of Rmeas
• MIN_RFL_Rmeas= 50 ! at least this number of reflections are required
• MAX_FAC_Rmeas= 2.0 ! factor to multiply the lowest Rmeas with to still be acceptable

The user may experiment with adjusting these values to make the automatic mode of space group determination more successful. For example, if the crystal diffracts weakly, all Rmeas values will be high and no valid decision can be made. In this case, I suggest to use e.g.

```TEST_RESOLUTION_RANGE= 50.0 10.0
```

### dealing with alternative indexing

There are two ways to have XDS choose an indexing consistent with some other dataset:

• using UNIT_CELL_A-AXIS=, UNIT_CELL_B-AXIS=, UNIT_CELL_C-AXIS= from a previous data collection run with the same crystal

One can also manually force a specific indexing, using the REIDX= keyword, but this is error-prone.

## Screw axes

The current version makes no attempt to find out about screw axes. It is assumed that the user checks the table in CORRECT.LP entitled REFLECTIONS OF TYPE H,0,0 0,K,0 0,0,L OR EXPECTED TO BE ABSENT (*), and identifies whether the intensities follow the rules

1. a two-fold screw axis along an axis in reciprocal space (theoretically) results in zero intensity for the odd-numbered (e.g. 0,K,0 with K = 2*n + 1) reflections, leaving the reflections of type 2*n as candidates for medium to strong reflections (they don't have to be strong, but they may be strong!).
2. similarly, a three-fold screw axis (theoretically) results in zero intensity for the reflections of type 3*n+1 and 3*n+2, and allows possibly strong 3*n reflections . 31 and 32 cannot be distinguished - they are enantiomorphs.
3. analogously for four-fold and six-fold axes.

Once screw axes have been deduced from the patterns of intensities along H,0,0 0,K,0 0,0,L , the resulting space group should be identified in the lists of space group numbers printed in IDXREF.LP and CORRECT.LP, and the CORRECT step can be re-run. Those reflections that should theoretically have zero intensity are then marked with a "*" in the table. In practice, they should be (hopefully quite) weak, or even negative.

## Notes

• There is still the old Wiki page Old way of Space group determination
• To prevent XDS from trying to find a better space group than the one with the lowest Rmeas (often P1), you could just use MAX_FAC_Rmeas= 1.0
• Space group determination is not a trivial task. There is a number of difficulties that arise -
• ambiguity of hand of screw axis (e.g. 31 versus 32, or 62 versus 64) - see the table above!
• twinning may make a low-symmetry space group look like a high-symmetry one
• pointless helps with identification of screw axes, and its identification of Laue group is more sensitive than that implemented in CORRECT. However it is not fail-safe, and cannot tell the correct enantiomorph.
• only when the structure is satisfactorily refined can the chosen space group be considered established and correct. Until then, it is just a hypothesis and its alternatives (see the table) should be kept in mind.