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Symmetry, pointgroups

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A crystal has translational symmetry by definition. - If r (r) is the electron ... Macromolecules can occupy only 65 space groups due to their chirality. ... – PowerPoint PPT presentation

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Title: Symmetry, pointgroups


1
  • Symmetry, point-groups space groups
  • Crystallography is based upon the idea of
    translational, rotational inversion or mirror,
    symmetry.

2
Crystal definition A crystal is an object with
translational symmetry r(r) r(r) u a v
b w c
Has crystal symmetry
Doesnt have crystal symmetry
3
  • Crystal Lattice
  • A crystal has translational symmetry by
    definition.
  • - If r (r) is the electron density within a
    crystal at r then there exists vectors a, b c
    such that
  • r (r) r (r u a v b w c)
  • where u, v w are integers.
  • Each identical copy is called a unit cell.
  • a, b c are called unit cell vectors.
  • Unit cell vector lengths are a a, b b, c
    c.
  • a, b g describe the angles between unit cell
    vectors.
  • Use a right handed coordinate system.

4
  • Fractional coordinates.
  • Any position within the crystal can be described
    by
  • r (u x) a (v y) b (w z) c
  • where u, v w are integers 0 lt x, y, z lt 1.
  • x, y z are called fractional coordinates
    describe a position within the unit cell.

5
  • Lattice Planes
  • Lattice planes are planes which pass through the
    lattice points ?
  • Labled after the fractional position where they
    first cross the a, b c axes.
  • If a lattice plane crosses the axes at the
    fractional coordinates (x, y, z) then the lattice
    plane is given the indices (h, k, l) equal to
    (1/x, 1/y, 1/z).

6
  • Other crystallographic symmetry
  • In addition to translational symmetry there is
    frequently other symmetry within the crystal
    lattice
  • - Rotational symmetry.
  • - Mirror plans.
  • - Inversion.
  • - Combinations of rotation or mirror planes plus
    translation.

7
  • Rotational symmetry.
  • Twofold rotation (360o/2) written 2 or
  • Threefold rotation (360o/3) written 3 or
  • Fourfold rotation (360o/4) written 4 or
  • Six-fold rotation (360o/6) written 6 or
  • To have 4, two axes must have equal length one
    angle 90o.
  • Also holds for 6 (or 3), but one angle must
    be120o.
  • Not so restricted to have 2.

8
  • Which rotations are consistent with a lattice?
  • Suppose f 360o/n then (from figure)
  • 2a cos f u a
  • since it must be possible to get from one point
    to the next by the lattice translation symmetry.
  • Only possible solutions are u -2, -1, 0, 1 or
    2 only.

2 a cos f
f
Rotates to here
a
f
a
9
  • Only 1, 2, 3, 4 6 rotations consistant with
    translational symmetry.
  • Eg. 5 or 10 are never seen as crystal rotational
    symmetries!

10
  • Rotation matrix.
  • r2 Rn r1.
  • Eg. For rotational symmetry 2 about the y axis
  • Hence (x, y, z) ? (-x, y, -z)

11
  • Mirror planes
  • The mirror plane normal is parallel with an axis
    of the lattice.
  • If the mirror plane is parallel to z in a right
    angle system
  • r2 Mz r1.
  • where
  • Hence (x, y, z) ? (x, y, -z).

c
b
a
12
  • Inversion
  • An inversion obeys the relation
  • r2 - I r1.
  • where
  • Note I R2 M

13
  • Screw Axes
  • A screw axis is a translation by a fraction of
    the unit cell followed by a rotation.
  • mn means translate n/m of the unit cell then
    rotate 360o/m.
  • 21 means translate 1/2 of the unit cell rotate
    180o
  • 31 means translate 1/3 of the unit cell rotate
    120o.
  • 42 means translate 2/4 of the unit cell rotate
    90o.
  • 41 means translate 1/4 of the unit cell rotate
    90o.

c
eg. 42 about b
b
a
14
  • Screw axes
  • 21 drawn
  • 31 drawn
  • 42 drawn
  • 63 drawn
  • Red indicates translated ½ unit cell into the
    page.

15
  • Glide planes
  • A glide plane means a translation followed by a
    reflection in a plane (not certain how plane
    defined).
  • a is a glide translation of ½ along a.
  • b is a glide translation of ½ along b.
  • c is a glide translation of ½ along c.
  • n is a glide translation ½ along a face
    diagional.
  • d is a glide translation ¼ along a face
    diagional.

eg. b
Note need to check reflection correct!
16
  • Primitive
    crystal systems
  • Have 2, 3, 4 6 fold rotational symetry
    possible.
  • At least two axes must be equal for 3, 4 or 6
    fold rotational symmetry.
  • - These ideas underly the division into seven
    primitive unit-cells.

Triclinic contains no extra symmetry. a ? b ?
c a ? b ? g ? 90o
Monoclinic contains one 2-fold rotation. a ? b
? c two of a or b or g 90o
17
Orthorhombic a ? b ? c a b g 90o Has
three 2-fold axes.
Tetragonal a b ? c a b g 90o Has a
4-fold axes.
Cubic a b c a b g 90o Has three
4-fold axes.
18
Trigonal Rhombohedral axes a b c a b
g ? 90o Has a 3-fold axis.
Hexagonal a b ? c a b 90o g 120o
has a 6-fold axis.
Trigonal Hexagonal axes a b ? c a b 90o
g 120o but has only 3-fold axis.
19
  • Non-primative unit cell.
  • Four clases
  • - Primitive unit cell (P).
  • - Plane centred unit cell (A, B or C).
  • - Body centred unit cell (I).
  • - Face centred unit cell (F).
  • In combination with previous 7 primitive cells
    leads to 14 Bravais lattices.

P
A
I
F
20
Why bother with non-primitive cells?
  • The blue unit-cell is a primitive unit cells
    with ½ the volume of the A unit cell.
  • - However some symmetry of the total unit cell is
    lost in making the choice of the blue cell.
  • - a ? b ? g in this choice but a b g 90o in
    the other.

c
b
a
21
  • Number of ways of packing a crystal.
  • 14 Bravais lattices.
  • Additional crystallographic symmetries
  • - Rotation, Inversion, mirror planes.
  • Form the set of 32 point-groups.
  • - Addition of glide screw axes.
  • Leads to 230 space-groups!

22
  • Equivalent positions
  • Suppose (x, y, z) is a normal fractional
    coordinate.
  • For each symmetry operation within the
    space-group there exists another position (x,
    y, z) within the unit cell which is rellated to
    (x, y, z) by symmetry.
  • (x, y, z) is called an equivalent position.
  • eg. 2-fold rotation about a takes (x, y, z) out
    of the unit cell.
  • A translation by b c returns it into the unit
    cell.
  • This is (x, y, z) is an equivalent position
    to (x, y, z)

23
  • The assymeteric unit.
  • The smallest non-repeating unit within a crystal
    is called the assymeteric unit.
  • A unit-cell may have several copies of an
    asymmeteric unit, all of which are related by
    (non-translational) crystallographic symmetry.
  • Each atom within the assymeteric unit is related
    by a symmetry operation (ie. equivalent position)
    to an identical atom of another assymeteric unit.
  • The number of assymeteric units is determined by
    the number of equivalent positions within the
    space group!

24
  • Special positions.
  • A special position is a place where a point is
    taken to itself under a symmetry operation.
  • ie. (x, y, z) ? (x,y,z) (x, y, z).
  • - Electron density at a special position can be
    problematic to interpret.
  • - Direct methods for phasing dont work when the
    atoms lie on special positions.
  • eg. 2-fold rotation about a leaves (0, y, z) on
    the a-axis.
  • Thus (0, y, z) is a speical position.

25
  • Non-crystallographic symmetry.
  • In addition to the crystallographic symmetry
    molecules may repeat with non-crystallographic
    symmetry.
  • eg. A virus capsid has 5-fold symmetry, which is
    not a crystallographic symmetry.
  • Two molecules may pack against each other in a
    slightly skewed manner fail to give a perfect
    crystallographic symmetry.

26
  • Notation.
  • Space groups are first labelled according to
    lattice ie. P, A (B or C), F or I.
  • Next labelled relative to the highest symmetry
    axes.
  • - P6 means a primitive lattice with one 6-fold
    axis.
  • - P222 means a primitive lattice with three
    two-fold axes.
  • - F means a face-centred lattice with no
    additional symmetry.
  • A subscript then is used to denote screw axes.
  • - P63 means a six-fold screw axis.
  • - P212121 means three two-fold screw axes.

27
  • Packing of protein crystals.
  • The assymeteric unit may contain
  • One molecule.
  • Several molecules with different conformations.
  • Identical domains of one molecule.
  • Amino acids are chiral.
  • Inversion mirror symmetry operations are not
    allowed.
  • Only 65 of the possible 230 space groups are
    available.
  • Protein crystals are loosely packed.
  • Approximately 50 to 80 of a protein crystal
    is solvent.
  • Crystal conformations well approximate
    physiological conformations.

28
  • Summary
  • To fully describe a crystal packing you need
  • Length of unit cell axes, a, b c.
  • Unit cell angles, a, b g.
  • Crystal packing Primitive, plane centred, body
    centred or face centred?
  • Additonal symmetry rotation, screw axes,
    mirror, inversion glide symmetries.
  • Must identify the assymeteric unit.
  • Must identify any non-crystallographic symmetry.
  • There are 14 possible crystal lattices.
  • 230 possible space groups.
  • Macromolecules can occupy only 65 space groups
    due to their chirality.
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