A typical definition of a twinned crystal is the
following: "Twins are regular aggregates consisting of crystals of the
same species joined together in some definite mutual orientation" (Giacovazzo,1992).
So for the description of a twin two things are necessary: a description
of the orientation of the different species relative to each other (twin
law) and the fractional contribution of each component. The *twin law*
can be expressed as a matrix that transforms the *hkl* indices of
one species into the other.

where osf is the overall scale factor, k_{m} is the fractional
contribution of twin domain m and F_{c}mis
the calculated structure factor of twin domain m. The sum of the fractional
contributions k_{m} must be unity, so (n-1) of them can be refined
and k_{1} is calculated by:

In SHELXL two kinds of twins are distinguished:

**(a)** For twins in which the reciprocal lattices
exactly coincide (twinning by merohedry or pseudo-merohedry), the procedure
is relatively simple. The command TWIN r_{11} r_{12} r_{13}
r_{21} r_{22} r_{23} r_{31} r_{32}
r_{33} n defines the twin law. **R**_{ }as the matrix
that transforms the hkl indices of one component into the other and n is
the number of twin domains. **R** is applied (n-1) times; the default
value of n is 2.

**(b)** In cases where only some reflections have
contributions from more than one domain (non-merohedral twins or twinning
by reticular merohedry) the *.hkl* file must be edited and the index
transformations applied to individual contributors, which are also assigned
component numbers. The code HKLF 5 is used to read in this file; no TWIN
command should be used.

In both cases, starting values of the fractional
contributions are input with the instruction BASF k_{2} ... k_{n};the
k_{m} values will be refined. Note that (in the new version of
SHELXL) linear restraints may be applied to these k values by means of
SUMP instructions; this can be very useful to prevent instabilities in
the early stages of refinement. For this purpose k_{2}...k_{n}
are assigned parameter numbers immediately following the free variables.

In the course of the final structure factor calculation,
the program estimates the absolute structure parameter *x* (Flack,
1983) and its esd. *x* is the fractional contribution of the inverted
component of a 'racemic twin'; it should be zero if the absolute structure
is correct, unity if it has to be inverted, and somewhere between 0 and
1 if racemic twinning is really present. Thus the above formulas apply
with n=2 and **R** = (-1 0 0, 0 -1 0, 0 0 -1).

It is a bonus of the refinement against *F*^{2}
that this calculation is a 'hole in one' and doesn't require expensive
iteration. A comparison of *x* with its esd provides an indication
as to whether the refined absolute structure is correct or whether it has
to be 'inverted'; the program prints a suitable warning should this be
necessary. This attempt to refine *x* 'on the cheap' is reliable when
the true value of *x* is close to zero, but may produce a (possibly
severe) underestimate of *x* for structures which have to be inverted,
because *x* is correlated with positional and other parameters which
have not been allowed to vary. Effectively these parameters have adapted
themselves to compensate for the wrong (zero) value of *x* in the
course of the refinement, and need to be refined with *x* to eliminate
the effects of correlation. These effects will tend to be greater when
the correlation terms are greater, e.g. for polar space groups and for
poor data to parameter ratios (say less than 8:1). *x* can be refined
at the same time as all the other parameters using the TWIN instruction
with the default matrix **R** = (-1 0 0, 0 -1 0, 0 0 -1) and BASF with
one parameter (*x*); this implies racemic twinning and so is refined
exactly as for other simple cases of twinning. Refinement of racemic twinning
should normally only be attempted towards the end of the refinement after
all non-hydrogen atoms have been located. 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.

For most space groups 'inversion' of the structure
simply involves inserting an instruction 'MOVE 1 1 1 -1' before the first
atom. Where the space group is one of the 11 enantiomorphous pairs [e.g.
P3_{1} and P3_{2}] the translation parts of the symmetry
operators need to be inverted as well to generate the other member of the
pair. There are seven cases for which, if the standard setting of the International
Tables for Crystallography has been used, inversion in the origin does
**not**
lead to the inverted absolute structure. This problem was probably first
described in print by Parthe & Gelato (1984) and Bernardinelli &
Flack (1985), but had been investigated previously by D. Rogers (personal
communication to GMS, ca. 1980).

The offending space groups and corresponding correct MOVE instructions are:

**Fdd2 MOVE .25 .25 1 -1 I4 _{1}cd
MOVE 1 .5 1 -1**

**I4 _{1} MOVE 1 .5 1 -1 I2d
MOVE 1 .5 .25 -1**

**I4 _{1}22 MOVE 1 .5 .25 -1 F4_{1}32
MOVE .25 .25 .25 -1**

**I4 _{1}md MOVE 1 .5 1 -1**

**6.3 Refinement against powder
data**

Refinement of twinned crystals and refinement against ** F^{2}**-values
derived from powder data are similar in that several reflections with different
indices may contribute to a single

Although SHELXL 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 powder data the least-squares refinement fits the overall scale
factor (**osf ^{2} **where osf is given on the FVAR instruction)
times the multiplicity weighted sum of calculated intensities to

**( F_{c}^{2})* = osf^{2} [ m_{1}F_{c1}^{2}
+ m_{2} F_{c2}^{2} + m_{3} F_{c3}^{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.

**6.4 Frequently encountered twin laws**

The following cases are relatively common:

(a) Twinning by merohedry. 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 the c-axis unique except for cubic):

**TWIN 0 1 0 1 0 0 0 0 -1**

plus one BASF parameter if the twin components are
not equal in scattering power. If they are equal, i.e. the twinning is
perfect, as indicated by the R_{int} for the higher symmetry Laue
group, then the BASF instruction can be omitted and k_{1} and k_{2}
are fixed at 0.5.

**(b)** Orthorhombic with **a** and **b**
approximately equal in length may emulate tetragonal:

**TWIN 0 1 0 1 0 0 0 0 -1**

plus one BASF parameter for unequal components.

**(c)** Monoclinic with beta approximately 90
°
may emulate orthorhombic:

**TWIN 1 0 0 0 -1 0 0 0 -1**

plus one BASF parameter for unequal components.

**(d)** Monoclinic with **a** and **c**
approximately equal and beta approximately 120 degrees may emulate hexagonal
[P2_{1}/c would give absences and possibly also intensity statistics
corresponding to P6_{3}]. There are three components, so n must
be specified on the TWIN instruction 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 3**

plus 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)** Rhombohedral obverse/reverse twinning
on hexagonal axes.

**TWIN -1 0 0 0 -1 0 0 0 1**

**6.5 Combined general and racemic
twinning**

If general and racemic twinning are to be refined simultaneously, n (the last parameter on the TWIN instruction) should be doubled and given a negative sign, and there should be |n|-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. Sometimes it is necessary to use this approach to distinguish between possible twin laws for non-centrosymmetric structures, when they differ only in an inversion operator In a typical example (an organocesium compound), when the TWIN instruction was input as:

**TWIN 0 1 0 1 0 0 0 0 -1 -4**

The BASF parameters refined to:

**BASF 0.33607 0.00001 0.00455**

Which means that the last two components (the ones
involving inversion) can be ignored, and the final refinement performed
with the '-4' deleted from the end of the TWIN instruction, and a single
BASF parameter. The introduction of twinning reduced the *R*1-value
from 18% to 1.8% in this example. Note that the program does not allow
the BASF parameters to become negative, since this would be physically
meaningless (this explains the 0.00001 above).

**6.6 Processing of twinned and
powder data**

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 before refinement. 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.

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 *F*_{o}^{2} by the ratio of the *F*_{c}^{2}
for the prime reflection divided by the total *F*_{c}^{2},
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 the one with |m| = 1).

**6.7 The warning signs for twinning**

Experience shows that there are a number of characteristic warning signs for twinning. Of course not all of them can be present in any particular example, but if one finds several of them the possibility of twinning should be given serious consideration.

**(b)** The R_{int}-value for the higher
symmetry Laue group is only slightly higher than for the lower symmetry
Laue group.

**(c)** The mean value for |*E*^{2}-1|
is much lower than the expected value of 0.736 for the non-centrosymmetric
case. If we have two twin domains and every reflection has contributions
from both, it is unlikely that both contributions will have very high or
that both will have very low intensities, so the intensities will be distributed
so that there are fewer extreme values.

**(d)** The space group appears to be trigonal
or
hexagonal.

**(e)** There are impossible or unusual systematic
absences.

**(f) **Although the data appear to be in order,
the structure cannot be solved.

**(g)** The Patterson function is physically impossible.

**(i)** There are problems with the cell refinement.

**(j) **Some reflections are sharp, others split.

**(k) **K = mean(*F*_{o}^{2})
/ mean(*F*_{c}^{2}) is systematically high for the
reflections with low intensity.

**(l)** For all of the 'most disagreeable' reflections,
*F*_{o}
is much greater than *F*_{c}.

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 Hoenle & von Schnering (1988). 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'! Indeed, it has been suggested
by G.B. Jameson that all structures (including proteins) that are solved
in space groups (such as P3_{1}) that could be merohedrally twinned
without changing the systematic absences should be tested for such twinning
(possible only present to a minor extent) by:

**TWIN 0 1 0 1 0 0 0 0 -1**

**BASF 0.1**

Refinement of twinned crystals often requires the full arsenal of constraints and restraints, since the refinements tend to be less stable, and the effective data to parameter ratio may well be low. In the last analysis chemical and crystallographic intuition may be required to distinguish between the various twinned and disordered models, and it is not easy to be sure that all possible interpretations of the data have been considered.

I should like to thank Regine Herbst-Irmer who wrote most of this chapter.