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
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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.