Alternate origins for the 65 enantiomorphic space groups

In a previous version of this document, all origins that are non-equivalent to the first origin listed for a particular space group were highlighted.  Here only origins which are non-equivalent to all other origins for the space group (i.e. the non-redundant set) are highlighted; this is the minimal set of alternate origin shifts which must be tested in order to ensure that two isomorphous crystals in the correct relative orientation can be superposed using one member of the set.  However, in order to superpose the correctly oriented asymmetric units of the two crystals, the entire redundant set must be tested, since this additionally includes all the possible lattice and centring translations.  Note that the redundant set is actually infinite, since it also includes the infinite set of origins generated by all the possible unit-cell translations of the crystal lattice.

Also note that an enantiomorphic space group is not the same as a non-centrosymmetric one.  The latter by definition includes any space group lacking a centre of symmetry, e.g. any space group possessing mirror or glide plane symmetry elements (such as Pm).  Pure enantiomeric molecules cannot crystallize in any space group containing either a centre of symmetry, or mirror or glide plane elements.

In reciprocal space an equivalent origin shift leaves both the amplitudes and phases, and therefore also the electron density function, invariant (unchanged).  A non-equivalent origin shift leaves only the amplitudes invariant: the phases in general will be different (though certain subsets of the phases may be invariant).  The new electron density function will be translated to the new origin, but will not be directly superposable on the unshifted function.

In primitive space groups all the alternate origins are non-equivalent; equivalent origins occur only centred lattices.  In the latter case the origins that are equivalent to the origin at (0,0,0) are the same as the lattice translations for the particular lattice type as follows:

NOTES

h.(I-R)t = m  for all h and for all R

... (1)

where h is a vector of reflection indices, I is the identity matrix, R is the rotation matrix component of a space-group symmetry operator, t is the origin shift and m is any integer.  Also, h must obey the condition:

h.s = n  for all s

... (2)

where s is a lattice translation vector in a centred space group, and n is any integer.  This is the condition that the reflection is observable, i.e. that the amplitude is not required to be identically zero because it is a systematic absence.

Example: F222

The lattice vectors for the F lattice from the table above are: (0, ½, ½),  (½, 0, ½),  and (½, ½,0), from which condition (2) implies the following conditions on h:

For orthorhombic space groups the R matrices consist of either (1, -1, -1),  (-1, 1, -1) or  (-1, -1, 1) on the diagonal and zeroes elsewhere, and so the matrices (I-R) consist of either (0, 2, 2),  (2, 0, 2),  or (2, 2, 0) on the diagonal and zeroes elsewhere.  Hence, for the origin shift (¼, ¼, ¼) as shown in the table, the vector (I-R)t will be either (0, ½, ½),  (½, 0, ½),  or (½, ½,0).  In each case, whatever the value of h, subject to the condition (2) (e.g. h, k, l all odd), the result of (1) will always be an integer.  The same argument clearly applies to all the alternate origins listed for F222; furthermore it can be seen that no other values of t satisfy condition (1).

Similar arguments apply to the other space groups.

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