Introduction to Hemihedral Twinning 
Twinning is a crystal growth disorder in which the specimen is composed
of distinct domains whose orientations differ but are related in a
particular, welldefined way. Sometimes nonconvex crystal morphology
implicates twinning, but microscopic examination is
often uninformative. Twinning usually prevents a successful structure
determination unless it is detected and either avoided or
corrected. In order to know when one is at risk, it is necessary to
understand the symptoms of the different types of twinning
and the symmetries where they may arise.
There are two fundamentally different
categories of twinning, epitaxial and merohedral. In epitaxial twinning, different
domains are typically oriented so that their molecular spacings
match at the crystal face where they meet, but their crystal lattices are
not superimposable. In other words, their crystal lattices superimpose in
fewer than threedimensions. This leads to distinct, interpenetrating
reciprocal lattices in the diffraction pattern. Since epitaxial twinning is
recognizable in the diffraction pattern, attempts can be made to integrate
reflections from a single lattice, or untwinned crystals can be sought. We
do not consider epitaxial twinning further.
Merohedral twinning refers to the special cases where the lattices of
the different domains overlap in three dimensions. Domains
whose orientations are crystallographically distinct can have superimposable
lattices in cases where the rotational symmetry of the lattice exceeds
the rotational symmetry of the crystal space group. This situation arises
in several point symmetries: 3, 32 (hexagonal but not rhombohedral),
4, 6, and 23.
We focus on the most common
type of merohedral twinning, hemihedral,
which involves just two different domain orientations.
As a result of hemihedral twinning, each observed diffraction intensity is
actually a weighted sum of two crystallographically distinct,
twinrelated, reflections. Two twinrelated reflections contribute
to an observed diffraction peak with weights determined by the
fractional volume of the specimen represented by each domain orientation.
This is the twin fraction, alpha. The observed intensity is equal to
alpha x I(hkl) + (1alpha) x I(h'k'l'), where hkl and h'k'l' are
twinrelated indices.
Depending on the hemihedral twinfraction, there are two different
scenarios that must be understood. If the twinfraction is nearly
equal to onehalf, the observed diffraction pattern acquires the additional
symmetry imposed by the twinning operation. The data obey an erroneously
high symmetry and are processed as such. This has been
called 'perfect twinning'. The true crystallographic
intensities cannot be recovered from the observed measurements, but
it is still possible to solve the structure by molecular replacement
(in practice) and by MIR (in theory).
Clues that diffraction data may be from a perfectly
twinned specimen include: (a) a unit cell that is too small to contain the known
molecule under the apparent space group symmetry, and (b)
an intensity distribution
that does not follow Wilson statistics. The latter test is generally applicable
and is performed by this Webserver. Several related expressions can be
computed. One whose expected values are rational (and therefore easy
to remember) is <I^{2}> / <I>^{2}. The expected value is 1.5 for
(acentric) twinned data and 2.0 for (acentric) untwinned data. The test
must be performed on normalized data or in thin shells. Anisotropic
diffraction is a complicating factor, as it tends to have the opposite effect
from twinning.
The situation in which the hemihedral twin fraction is not onehalf has been
called 'partial twinning'. The result is not higher apparent symmetry,
but observed intensities that contain contributions from distinct
reflections. This effect can be reversed to give the true crystallographic
intensities if: (1) the twin fraction is significantly different from onehalf
and (2) the twinfraction can be estimated accurately. The important
problem of
estimating alpha has been covered by numerous investigators. On this
Web site we estimate alpha by a statistical evaluation of the
similarity between twinrelated observations. Pseudo or noncrystallographic
symmetry can be a complicating factor, as it tends in special cases to
mimic twinning (at least at lower resolution) by leading to similarity
between crystallographically distinct reflections. Measurement errors
lead to a systematic underestimation of alpha, so the test should be
performed over a resolution range where the data are wellmeasured.
For more information on twinning and how to proceed if twinning is detected,
the reader is referred to the review by
Yeates
(T. O. Yeates (1997),
Detecting
and Overcoming Crystal Twinning. Methods in Enzymology 276, 344358),
which is a suitable citation for those using this Webbased server. Queries by
email are also
welcome. I am especially interested in hearing about cases where twinning
is detected.
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