A106: Twinning

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Transcript of A106: Twinning

  • 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