Refinement of twinned crystals and refinement against F2-values derived from powder data are similar in that several reflections with different indices may contribute to a single F2 observation. For powder data this requires some small adjustments to the format of the '.hkl' file; the batch number becomes the multiplicity m, and where several reflections contribute to the same observation the multiplicity is made positive for the last reflection in the group and negative for the rest. A similar approach is possible for twinned crystals, except that the batch number is replaced by the twin component number, and the batch scale factors (BASF) may be refined to determine the fractional contributions of the components 2, 3, ... k1, the fraction of component 1, is refined as ( 1 - k2 - k3 - ... ). In simple cases, i.e. when the lattices of all components are always coincident, the normal format can be retained in the '.hkl' file, and the index transformation specified with a TWIN instruction. Although SHELXL-93 may be useful for some high symmetry and hence reasonably well resolved powder and fibre diffraction patterns - the various restraints and constraints should be exploited in full to make up for the poor data/parameter ratio - for normal powder data a Rietveld refinement program would be much more appropriate.
For both powder (HKLF 6) and twinned data (HKLF 5 or TWIN with HKLF 4), the reflection data are reduced to the 'prime' component, by multiplying Fo2 and Fc2 by the ratio Fc2(prime) / Fc2(total), before performing the analysis of variance and the Fourier calculations. Similarly OMIT h k l refers to the indices of the prime component. The prime component is the one for which the indices have not been transformed by the TWIN instruction (i.e. m = 1 ), or in the case of HKLF 5 or HKLF 6 the component given with positive m (i.e. the last contributor to a given intensity measurement, not necessarily with |m| = 1).
For powder data the least-squares refinement fits the overall scale factor (osf2 where osf is given on the FVAR instruction) times the multiplicity weighted sum of calculated intensities to Fo2:
(Fc2)* = osf2 [ m(1) * Fc(1)2 + m(2) * Fc(2)2 + m(3) * Fc(3)2 + ... ]
where the multiplicities of the contributors are given in the place of the batch numbers in the '.hkl' file. Since it is then not possible to define batch numbers as well, BASF cannot be used with powder data.
For twinned data (TWIN or HKLF 5) the expression becomes:
(Fc2)* = osf2 [ k(1) * Fc(1)2 + k(2) * Fc(2)2 + k(3) * Fc(3)2 + ... ]
where the starting values for the k(2), k(3), ... are given on the BASF instruction, and k(1) is defined such that Sigma[k(m)] = 1. If no BASF instruction is used, all the k(m) are made equal. m is the component number given in the place of the batch number in the '.hkl' file; if TWIN is used to generate the components, m is 1 for the initial indices, 2 after applying the TWIN matrix once, 3 after applying it twice, etc. The parameter ncomp must be given on the TWIN instruction if the matrix is to be applied more than once.
The following cases are relatively common:
(a) The lower symmetry trigonal, tetragonal, hexagonal or cubic Laue groups may be twinned so that they look (more) like the corresponding higher symmetry Laue groups (assuming c unique except for cubic):
TWIN 0 1 0 1 0 0 0 0 -1plus one BASF parameter if the twin components are not equal in scattering power.
(b) Orthorhombic with a and b approximately equal may emulate tetragonal:
TWIN 0 1 0 1 0 0 0 0 -1plus one BASF parameter for unequal components.
(c) Monoclinic with beta approximately 90 degrees may emulate orthorhombic:
TWIN 1 0 0 0 -1 0 0 0 -1plus one BASF parameter for unequal components.
(d) Monoclinic with a and c approximately equal and beta approximately 120 degrees may emulate hexagonal [P21/c would give absences and possibly also intensity statistics corresponding to P63]. There are three components, so ncomp must be specified and the matrix is applied once to generate the indices of the second component and twice for the third component. In German this is called a 'Drilling' as opposed to a 'Zwilling' (with two components):
TWIN 0 0 1 0 1 0 -1 0 -1 3plus TWO BASF parameters for unequal components. If the data were collected using an hexagonal cell, then an HKLF matrix would also be required to transform them to a setting with b unique:
HKLF 4 1 1 0 0 0 0 1 0 -1 0(e) Refinement of racemic twinning may be performed by adding the following two instructions to the '.ins' file (and retaining HKLF 4):
TWIN BASF 0.5since the default TWIN matrix inverts the indices. In this example, the BASF coefficient is the Flack absolute structure parameter x (H.D. Flack, Acta Cryst., (1983) A39, 876-881; G. Bernardinelli and H.D. Flack, Acta Cryst., A41 (1985) 500-511). Refinement of racemic twinning should normally only be attempted after all non-hydrogen atoms have been located AND the program suggests that it would be advisable. If racemic twinning is refined in this way, the automatic calculation of the Flack x parameter in the final structure factor cycle is suppressed, since the BASF parameter is x.
If general and racemic twinning are to be refined simultaneously, ncomp should be doubled and given a negative sign, and there should be |ncomp|-1 BASF twin component factors (or none, in the unlikely event that all are to be fixed as equal). The inverted components follow those generated using the TWIN matrix, in the same order. In such a case a single Flack x parameter is no longer appropriate; the program will still estimate a value, which should be zero since the effect has already been taken into account, but its esd gives a guide to the reliability of the racemic refinement.
The HKLF 5 and 6 instructions force MERG 0, i.e. neither a transformation of reflection indices into a standard form nor a sort-merge is performed. If twinning is specified using the TWIN instruction, any MERG instruction may be used and the default remains MERG 2. Although this is always safe for racemic twinning, there may be other forms of twinning for which it is not permissible to sort-merge first. Whether or not MERG is used, the program ignores all systematically absent contributions, with the result that a reflection is excluded from the data if it is systematically absent for all components.
Twinning usually arises for good structural reasons. When the heavy atom positions correspond to a higher symmetry space group it may be difficult or impossible to distinguish between twinning and disorder (of the light atoms); see W. Hoenle and H.G. von Schnering, Z. Krist., 184 (1988) 301-305. Since refinement as a twin usually requires only two extra instructions and one extra parameter, in such cases it should be attempted first, before investing many hours in a detailed interpretation of the 'disorder'!
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