TWIN Instruction

TWIN 3x3 matrix [-1 0 0 0 -1 0 0 0 -1] ncomp [2]

ncomp is the number of twin components (2 or greater) and the matrix is applied (iteratively if |ncomp| > 2) to generate the indices of the twin components from the input reflection indices, which apply to the first (prime) component. If a transformation matrix is also given on the HKLF instruction, it is applied first before the (iterative) application of the TWIN matrix. This method of defining twinning allows the standard HKLF 4 format to be used for the '.hkl' file, but can only be used when the reciprocal lattices for all twinned components are metrically superimposable. In other cases HKLF 5 format must be used. The Fo^2 values are fitted to the sum of k[m]*Fc[m]^2 multiplied by the overall scale factor, where k[1] is one minus the sum of k[2], k[3], .. and the starting values for the remaining twin fractions k[2], k[3], .. are specified on a BASF instruction. Only ONE TWIN instruction is allowed. If BASF is omitted the TWIN factors are all assumed to be equal (i.e. 'perfect' twinning).

For example, if a structure in the space group P2(1)/c with a and c almost equal and beta close to 120 degrees is pseudohexagonally twinned so that the space group appears to be P6(3) (with the pseudo-6(3) axis along b), refinement could be performed with the instructions:

 TWIN  0 0 1  0 1 0  -1 0 -1  3
 BASF .35 .25

The CELL, LATT and SYMM instructions would give the true monoclinic cell (in the conventional setting with the 2(1) axis along b). A full set of monoclinic data would be prerequisite for a satisfactory refinement. If the twinning is 'perfect', the BASF instruction would be left out, and a unique hexagonal set of data should suffice. If the data had been collected on a hexagonal cell in this example, an HKLF conversion matrix would be needed as well to make b the 2(1) axis first, e.g.:

 HKLF 4  1  1 0 0  0 0 1  0 -1 0

Refinement of racemic twinning may be performed with:

 TWIN  -1 0 0  0 -1 0  0 0 -1  2 (or just TWIN -defaults)
 BASF 0.4

so that 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). In this case the program does not calculate a separate Flack parameter in the final structure factor calculation, but uses the BASF parameter and its esd for the Flack parameter in the '.cif' output.

If the racemic twinning is present at the same time as normal twinning, ncomp should be doubled (because there are twice as many components as before) and given a negative sign (to indicate to the program that the inversion operator is to be applied multiplicatively with the specified TWIN matrix). The number of BASF parameters, if any, should be increased from m-1 to 2m-1 where m is the original number of components (equal to the new |ncomp| divided by 2). The TWIN matrix is applied m-1 times to generate components 2 ... m from the prime reflection (component 1); components m+1 ... 2m are then generated as the Friedel opposites of components 1 ... m. In such a case the program will estimate a Flack parameter in the final structure factor cycle, but it should be zero because it has already been taken into account. This is done because the esd of this number is still of interest even when there is no longer a single racemic twinning parameter. It should be noted that because of the way the twin component factors are defined, there will inevitably be very large (e.g. 0.99) correlation coefficients between BASF parameters j and j+m in this treatment of combined normal and racemic twinning.

For both the TWIN and HKLF 5 treatments, the data are reduced to the prime component by multiplying Fo^2 and Fc^2 by the ratio Fc^2(prime) / Fc^2(total) before performing the analysis of variance and 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 the component given with positive m (i.e. last, but not necessarily with |m| = 1).