Download - A106: Twinning
Tim Grüne
Advanced Macromolecular Structure Determination
TwinningTim Grüne
Dept. of Structural Chemistry, University of Göttingen
March 2011
http://shelx.uni-ac.gwdg.de
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Space Groups, Point Groups and Other Classifications
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Molecules as Crystals
Molecules which form a crystal are held together by molecular forces such that the compound
is built up by the repetition of a unit cell.
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Crystals are Regular
“Crystalline” means: shifting the unit cell by an integer multiple of one of more of the unit cell
vectors brings the unit cell in superposition with another unit cell.
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The Crystal Lattice
By picking one point per unit cell and connecting corresponding points, one can see the
underlying crystal lattice, defined by the unit cell vectors ~a, ~b, and ~c
b
a
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Unit Cell(s)
The choice of the unit cell is not unique and the unit cell does not need to contain a chemically
sensible molecule (but e.g. two parts which constitute the whole molecule upon translation).
Ch. 9 in [2] lists certain conventions the unit cell should fulfil, like the presentation of the
symmetry of the crystal. They restrict the choices of unit cells in certain space groups.
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Space Groups
The aforementioned fact of “repetition by shifting” is common to all crystals: All crystals pos-
sess translational symmetry.
Some (actually, most) crystals possess more than translational symmetry.
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Basic Symmetry Elements
Possible Symmetries in Crystals include
• rotational Symmetry (2-, 3-,
4-, and 6-fold)
• mirror planes
• inversion centres
Glycerol and solvent structure around a 2-fold axis inPDB-ID 1OFC. Solvent molecules “like” to sit in specialpositions
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Basic Symmetry Elements
Possible Symmetries in Crystals include
• rotational Symmetry (2-, 3-,
4-, and 6-fold)
• mirror planes
• inversion centres
Asymmetric unit and full molecule with mirror
plane (space group P2/m)
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Basic Symmetry Elements
Possible Symmetries in Crystals include
• rotational Symmetry (2-, 3-,
4-, and 6-fold)
• mirror planes
• inversion centres
Inversion centre of Ciprofloxacin, P21/c. Cour-
tesy J. Holstein
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Symmetry Elements
The combination of the “basic” symmetry elements with each other, or with the translational
symmetry of the lattice results in further symmetry elements, like
• Screw axes (P43)
• Glide mirror planes
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The Space Group
A space group consists of a selection of symmetry operations which are compatible with each
other and the translation of the (infinite) crystal lattice.
Compatible: e.g. 3-fold
axis within P6
The unit cell with P6 symmetry (one 6-fold
axis along ~c) automatically contains 3-fold
axes within the unit cell and 2-fold axes.
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The Space Group
A space group consists of a selection of symmetries which are compatible with each other
and the translation of the crystal lattice.
Incompatible: 5-fold axis
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cover the whole space.
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The Space Group
A space group consists of a selection of symmetries which are compatible with each other
and the translation of the crystal lattice.
Another incompatibility is
a 3-fold axis along a 4-fold
axis (unless it is a 12-fold
axis which does not exist
for crystals)
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Compatible Symmetry Operators
N.B. The cube does have a 3-fold axis — it only depends on the point of view:
3-fold axis if the cube
along the space diagonal.
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Compatible Symmetry Operators
N.B. The cube does have a 3-fold axis — it only depends on the point of view:
3-fold axis if the cube
along the space diago-
nal - corresponding cor-
ners coloured accordingly.
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Space Group and Reciprocal Space
The symmetry of a crystal not only influences the reciprocal lattice points, but also the re-
flections (structure factors and notably intensities).
One simple example: P 1̄: for each atom in the crystal at (x, y, z) there is the same type ofatom at (−x,−y,−z). The structure factor calculates as
F (hkl) =Natoms∑
fje−2πi(hxj+kyj+lzj)
=
N/2∑fje−2πi(hxj+kyj+lzj) +
N/2∑fje−2πi(h(−xj)+k(−yj)+l(−zj))
=
N/2∑fje−2πi(hxj+kyj+lzj) +
N/2∑fje
2πi(hxj+kyj+lzj)
= 2
N/2∑fj cos(2π(hxj + kyj + lzj))
In a centrosymmetric space group, all structure factors are real numbers, and can only have
a phase φ = 0◦ or φ = 180◦!
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Systematic Absences
Similar calculations in other space groups show that some reflections are absent (I(hkl) =
0), notably in space groups with screw axis and glide planes.
The international tables [2] list the opposite, namely the reflections which are present, e.g. in
P21:
Only reflections along the b∗-axis of the type 02n0 are present, (010), (030), (050), . . . ,
should have zero intensity. i.e.
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The Point Groups (1/2)
Symmetries of a crystal allow conclusions about its macroscopic properties, especially on its
optical and electromagnetic properties [3, 4].
Often for a material to possess some property, a certain symmetry must be absent, e.g.
piezoelectric crystal⇒ no centrosymmetric space group
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The Point Groups (2/2)
These macroscopic properties, however, are generally independent of translational parts in
the symmetry, i.e. it does not matter whether or not e.g. a 4-fold axis is a screw axis 41 or
not.
From this point of view, only space groups without their translational components are of inter-
est.
Stripping a space group off its translational components means
• removing the Bravais Symbol (P, F, I, . . . )
• removing screw components (41→ 4, . . . )
• changing glide planes into mirror planes (a, b, c, n→ m)
This reduces the 230 space groups to the 32 point groups or crystal classes.
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The Laue Groups (1/2)
The reciprocal lattice is related to the “real” lattice.
How does the symmetry of the “real” lattice reflect to the reciprocal lattice?
Two things to consider:
1. The reciprocal lattice is translationally invariant (the origin in the Ewald sphere construc-
tion is arbitrary)
Therefore the diffraction pattern has (at most) the symmetry of the point group of the
crystal.
2. Friedel’s Law makes reciprocal space centrosymmetric:
I(hkl) = I(h̄k̄l̄)
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The Laue Groups (2/2)
Friedel’s Law reduces the 32 point groups to the following 11 Laue groups [4].
Crystal Class Possible Laue Groupstriclinic 1̄monoclinic 2/morthorhombic mmmtetragonal 4/m 4/mmmtrigonal 3̄ 3̄m1, 3̄1mhexagonal 6/m 6/mmmcubic m3̄ m3̄m
Determination of the Laue group is often the first step in data processing after determination
of the unit cell, and data are processed taking the Laue group into account, not the space
group.
We need point and Laue groups in order to understand merohedral twinning.
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Twinning
According to G. Friedel[??], 1904
“A twin is a complex crystalline edifice built up of two or more homogeneous portions
of the same crystal species in contact (juxtaposition) and oriented with respect to each
other according to well-defined laws.”
Courtesy M. Sevvana
Single insulin crystal and two
insulin crystal grown into each
other as an example of non-
merohedral twins.
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Twin Law
A two-dimensional twin:
idea from R. Herbst-Irmer
Twin law:(−1 0
0 1
)(rotation by 180◦)
Contribution of orientation 1 to diffraction: 4/9
Contribution of orientation 2 to diffraction: 5/9
Usually each orientation forms domains of a large
number of unit cells
⇒ not one larger unit cell
⇒ both orientations contribute independently to
diffraction pattern
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How to Deal with Twinned Crystals
Life is easier with a single, non-twinned crystal. When faced with a twin it is worth putting
some effort into improving the crystallisation conditions and growing a single crystal.
But sometimes one does not have a choice.
Do not throw away your twinned crystal — the data can often be integrated and refined.
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Types of Twinning
One can distinguish four different types of twinning in crystallography:
1. non-merohedral twinning
2. Twinning by merohedry
3. Twinning by pseudo-merohedry
4. Twinning by reticular merohedryMineralogists distinguish many more phenotypes and subgroups of twins
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Non-Merohedral Twinning
• Two or more crystals attached together
• Arbitrary orientation (not related to crystal’s symmetry)
• Often: “satellites”, small crystals attached to one main crystal
→ try to separate crystals before mounting, e.g. with a cats whisker. Sometimes
crystals break apart during transfer into cryo-solution.
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Example of Non-Merohedral Twinning
courtesy T. Graewert
• Main lattice
• Small spots interspersed in-
between main lattice
⇒ Small’ish satellite crystal
⇒ Most spots do not overlap
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Example of Non-Merohedral Twinning
courtesy C. Grosse
View of reciprocal lattice of
spots from the first few im-
ages.non-“suspicious” direction
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Example of Non-Merohedral Twinning
courtesy C. Grosse
View of reciprocal lattice of
spots from the first few im-
ages.“suspicious” direction:
systematic absences?
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Example of Non-Merohedral Twinning
courtesy C. Grosse
View of reciprocal lattice of
spots from the first few im-
ages.Correct indexing with two
domains in two orienta-
tions
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Non-Merohedral Twinning
Easiest solution: ignore smaller domain, only integrate reflections belonging to main/ one
crystal.
Works ok if only very few spots overlap. One could even integrate the two different lattices
separately.
Overlapping spots are diffi-
cult to integrate for software:
difficult assignment of pixel
to correct spot.
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Merohedral Twinning
The reciprocal lattice of merohedral twins overlap perfectly, because the
twin law is a symmetry operator of the crystal system, but not the point group.
The crystal system describes the symmetry of the pure lattice (triclinic, monoclinic, orthorhom-
bic, tetragonal, tri-/hexagonal, and cubic).
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Merohedral Twinning
Merohedral Twinning can only occur when the real space group belongs to the lower Laue
group, i.e.
Crystal System Real Laue “Twinned” Lauetetragonal 4/m 4/mmm
trigonal 3̄ 3̄m1hexagonal 6/m 6/mmm
cubic m3̄ m3̄m
Merohedral twinning cannot occur in monoclinic, triclinic, or orthorhombic space groups —
they only contain 1 Laue group.
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Merohedral Twinning: Example
Lattice with 4-fold symmetry perpendicular
to plane also has mirror planes.
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Merohedral Twinning: Example
Lattice with 4-fold symmetry perpendicular
to plane also has mirror planes.
Lattice: mirror planes (strong lines)
Unit cell (and intensities) with 4-fold sym-
metry, but not a mirror plane.
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Merohedral Twinning: Example
1st domain 2nd domain mirrored
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Merohedral Twinning: Example
Observed data: sum of two crystals with perfect overlap of the reciprocal lattices.
Each spot on detector =
sum of two reflections from
two “crystals”.
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Twinning by Reticular Merohedry
Obverse Reverse
from Massa [4]
• E.g. twin consisting of obverse and reverse set-
tings of rhombohedral crystals
• About 1/3 of reflections overlap perfectly, the
remaining do not overlap
• Once detected, structure solution less difficult
than for merohedral twins
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Pseudo-Merohedral Twins
The twinned crystal pretends a higher symmetry than it really has.
Overlaps are incomplete, i.e. not perfect as they are in the case of merohedral twinning
(hence the name).
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Data Processing with Twinning
Twinned data are best coped with as early as possible, i.e. at the data processing step.
Keep in mind: processing software predicts the position of the reflections on the detector from
the experimental parameters (unit cell, wavelength, crystal orientation, . . . ).
Three different situations occur.
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1) Non-overlapping reflections
Especially in the case of non-merohedral twinning, a large number of reflections do not over-
lap and can be treated as normal reflections:
1. Predict position from appropriate cell and orientation
2. Subtract background
3. Measure intensity
The integration software, however, has be to set-up for twinning and be capable of dealing
with more than one orientation matrix for the crystal.
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2) Partially Overlapping Reflections
Some of the reflections of a non-merohedral twin overlap partially and the main problem is to
assign the detector values at the border between the overlapping pixels.
The same phenomenon can occur with
crystals of large unit cell parameters, and
can be improved by increasing detector
distance to the crystal.
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2) Partially Overlapping Reflections
The pixels in the overlapping region contain photons from both spots. The contributions have
to be split up correctly.
30
40
50
60
70
80
90
0 5 10 15 20 25 30 35 40
Inte
nsity
Detector
databackground
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2) Partially Overlapping Reflections
The pixels in the overlapping region contain photons from both spots. The contributions have
to be split up correctly.
This may be achieved by profile fitting.
−10
0
10
20
30
40
50
60
0 5 10 15 20 25 30 35 40
Inte
nsity
Detector
data−backgroundfitted profiles
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3) Perfectly Overlapping reflections
0
1
2
3
4
5
6
7
8
9
10
Inte
nsity
1+212
Perfect overlaps happen e.g. in the case of merohedral
twins.
Deconvolution during data processing is not possible.
Data can be handled during refinement, refinement in-
cludes the twin fraction α:
I1+2(hkl) = (1− α)I1(hkl) + αI2(h′k′l′)
α: Volume of domain one / crystal volume
1− α: Volume of domain two / crystal volume
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Dealing with Twinned Data (1/2)
(Obviously) Twinned data are best dealt with starting at the data processing stage.
SAINT (from Bruker AXS) in combination with TWINABS for scaling and SHELXL for refine-
ment are probably the most sophisticated work flow.
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Dealing with Twinned Data (2/2)
Unfortunately, most data processing software do not support twinned data. In case of one
major domain, one can
1. Ignore twinning
• poor results
2. Omit all overlapping reflections
• May lead to (very) incomplete data
3. Omit partial overlaps, split exact overlaps (refinement of twin fraction)
• May still lead to incomplete data
• Special file format required (SHELXL HKLF5 format)
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Warning Signs for (Pseudo-) Merohedral Twinning
• Laue group appears higher than true Laue group
• The matching of symmetry equivalent reflections (Rmerge) in the higher (pre-
tended) Laue group is only slightly worse than in the lower (correct) Laue group
• Twinning disturbs the expected intensity distribution of the reflections: The mean
of |E2 − 1| is lower than expected (� 0.736)
• Trigonal and hexagonal space groups are suspicious
• systematic extinctions are not consistent with any space group
• Refinement: Structure cannot be solved or R-values very high
list compiled by G. Sheldrick
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Warning Signs for Non-Merohedral Twinning
• Some split reflections, others are sharp
• Difficult indexing/ determination of cell dimensions
• unusually long cell edge (“supercell” for near-180◦ domain rotation)
• No suitable space group
• Refinement: Many outliers |Fobs| � |Fcalc|, especially for reflections with low
|Fcalc| (integration of two spots as one)
• Strange left-over electron density, which cannot be assigned to disorder or sol-
vent molecules (especially for small-molecule structures)
list compiled by G. Sheldrick
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References
1. C. Giacovazzo (ed.), Fundamentals of Crystallography, Union of Crystallography, Oxford
University Press (2002)
2. T. Hahn (ed.), International Tables for Crystallography, Vol. A, Union of Crystallography
3. A. Authier (ed.), International Tables for Crystallography, Vol. D, Union of Crystallography
4. W. Massa, Crystal Structure Determination, Springer (2004)
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