# A106: Twinning

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Tim Grne

Advanced Macromolecular Structure Determination

TwinningTim Grne

Dept. of Structural Chemistry, University of Gttingen

March 2011

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

tg@shelx.uni-ac.gwdg.de

A106: Advanced Macromolecular Crystallography 1/52

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

Tim Grne

Space Groups, Point Groups and Other Classifications

A106: Advanced Macromolecular Crystallography 2/52

Tim Grne

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.

A106: Advanced Macromolecular Crystallography 3/52

Tim Grne

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.

A106: Advanced Macromolecular Crystallography 4/52

Tim Grne

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

A106: Advanced Macromolecular Crystallography 5/52

Tim Grne

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.

A106: Advanced Macromolecular Crystallography 6/52

Tim Grne

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.

A106: Advanced Macromolecular Crystallography 7/52

Tim Grne

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

A106: Advanced Macromolecular Crystallography 8/52

Tim Grne

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)

A106: Advanced Macromolecular Crystallography 9/52

Tim Grne

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

A106: Advanced Macromolecular Crystallography 10/52

Tim Grne

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

A106: Advanced Macromolecular Crystallography 11/52

Tim Grne

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.

A106: Advanced Macromolecular Crystallography 12/52

Tim Grne

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.

A106: Advanced Macromolecular Crystallography 13/52

Tim Grne

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)

A106: Advanced Macromolecular Crystallography 14/52

Tim Grne

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.

A106: Advanced Macromolecular Crystallography 15/52

Tim Grne

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.

A106: Advanced Macromolecular Crystallography 16/52

Tim Grne

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 1: 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

fje2i(hxj+kyj+lzj)

=

N/2fje2i(hxj+kyj+lzj) +

N/2fje2i(h(xj)+k(yj)+l(zj))

=

N/2fje2i(hxj+kyj+lzj) +

N/2fje

2i(hxj+kyj+lzj)

= 2

N/2fj cos(2(hxj + kyj + lzj))

In a centrosymmetric space group, all structure factors are real numbers, and can only have

a phase = 0 or = 180!

A106: Advanced Macromolecular Crystallography 17/52

Tim Grne

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.

A106: Advanced Macromolecular Crystallography 18/52

Tim Grne

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

A106: Advanced Macromolecular Crystallography 19/52

Tim Grne

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.

A106: Advanced Macromolecular Crystallography 20/52

Tim Grne

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. Friedels Law makes reciprocal space centrosymmetric:

I(hkl) = I(hkl)

A106: Advanced Macromolecular Crystallography 21/52

Tim Grne

The Laue Groups (2/2)

Friedels Law reduces the 32 point groups to the following 11 Laue groups [4].

Crystal Class Possible Laue Groupstriclinic 1monoclinic 2/morthorhombic mmmtetragonal 4/m 4/mmmtrigonal 3 3m1, 31mhexagonal 6/m 6/mmmcubic m3 m3m

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.

A106: Advanced Macromolecular Crystallography 22/52

Tim Grne

Twinning

A106: Advanced Macromolecular Crystallography 23/52

Tim Grne

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.

A106: Advanced Macromolecular Crystallography 24/52

Tim Grne

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

Usua

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