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

[email protected]

A106: Advanced Macromolecular Crystallography 1/52

http://shelx.uni-ac.gwdg.de

Tim Grüne

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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4-, and 6-fold)

• mirror planes


Asymmetric unit and full molecule with mirror

plane (space group P2/m)


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4-, and 6-fold)

• mirror planes


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

with lattice translation ��

��

��

��

��

��

��

�� A 5-fold axis leaves gaps and cannot

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


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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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courtesy C. Grosse

View of reciprocal lattice of

spots from the first few im-

ages.non-“suspicious” direction


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courtesy C. Grosse



ages.“suspicious” direction:

systematic absences?


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courtesy C. Grosse



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).


Tim Grüne

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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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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1st domain 2nd domain mirrored


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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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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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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)


Tim Grüne

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


Tim Grüne

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)
