## 3/4) Decipher the twin law

As stated in the introduction, twinning by pseudo-merohedry may occur in
crystals where the cell dimensions mimic a higher-symmetry crystal system.
Here we have a monoclinic crystal masquerading as orthorhombic by virtue
of a β angle that is almost exactly 90°. The question is: what
is the twin law, and how do we describe it in a mathematically rigorous
way ? The answer in this case lies in the difference
between orthorhombic and monoclinic symmetry. For a monoclinic crystal we
could have 2-fold rotation, 2_{1} screw, mirror, and *c*-axial
or *n*-diagonal glide planes associated *only* with the *b*
axis (assuming standard conventions are respected). For orthorhombic,
these symmetry elements may be associated with *a*, *b*, and
*c*. In reciprocal space, translational parts of screws and glides
show up only as systematic absences, so in deciphering the twin law, we
need only consider rotation, reflection and inversion. Since our structure
is centrosymmetric, that limits us to 2-fold rotation and mirror operations
associated with the *a* and *c* axes.
A two-fold rotation changes the sign of two indices, while a mirror flips
just one sign. The allowable twin operations, expressed as (3x3) transformation
matrices, are as follows:

In the above, (i) and (iii) define reflection perpendicular to *a* and
*c* while (ii) and (iv) describe rotation about *a* and *c*,
respectively. Since the structure is centrosymmetric, (i) and (ii) are
equivalent, as are (iii) and (iv). Similarly, since monoclinic symmetry
has either *m*, *2* or *2/m* point symmetry associated with
the *b* axis, we can flip the sign of *b* (mirror perpendicular
to *b*) or flip the signs of *a* and *c* (2-fold about
*b*). Thus (i), (ii), (iii) and (iv) are equivalent: any of the four
matrices will accomplish the same result. To illustrate the superposition
of monoclinic twin components when β = 90°, the image below shows
two unit cells related by a mirror perpendicular to the *a* axis.

If you roll your mouse cursor over the image, the unit cell boxes should
converge, and exactly superimpose. The superposition is perfect only if
*bc* is perpendicular to *a*, *i.e.* if β = 90°.
This is the essence of twinning exhibited by the structure in question.
The twin components can co-exist side-by-side, with minimal interference
or distortion. Without looking at the structure and analysing molecular
contacts, however, we cannot tell if the twinning in real space is by a
mirror or by 2-fold rotation. We can't even tell which axis, *a* or
*c* is involved, but for the sake of refinement it doesn't matter:
in reciprocal space the four twin operations are equivalent for this crystal.
Nevertheless, unless there is some compelling reason to choose otherwise,
it makes most sense to use a symmetry operation of the first kind. In
other words, choose a proper rotation rather than a reflection
(or roto-inversion).

The twin operations given above can be translated into *SHELXL* commands as follows:

(i) TWIN -1 0 0 0 1 0 0 0 1

(ii) TWIN 1 0 0 0 -1 0 0 0 -1

(iii) TWIN 1 0 0 0 1 0 0 0 -1

(iv) TWIN -1 0 0 0 -1 0 0 0 1

For inclusion in the *SHELXL* structure model, you should also add
a *BASF* (batch scale factor) instruction with an educated guess at
the occupancy factor of the main component. In the final segment, we'll
add *TWIN* and *BASF* instructions to the model and refine the
structure to convergence.