More on symmetry

1
More on symmetry

• Learning Outcomes
• By the end of this section you should
• have consolidated your knowledge of point groups
  and be able to draw stereograms
• be able to derive equivalent positions for
  mirrors, and certain rotations, roto-inversions,
  glides and screw axes
• understand and be able to use matrices for
  different symmetry elements
• be familiar with the basics of space groups and
  know the difference between symmorphic
  non-symmorphic

2
The story so far

• In the lectures we have discussed point symmetry
• Rotations
• Mirrors

• In the workshops we have looked at plane symmetry
  which involves translation ? ua vb wc
• Glides
• Screw axes

3
Back to stereograms and point symmetry

• Example 2-fold rotation perpendicular to plane
  (2)

4
More examples

• Example 2-fold rotation in plane (2)

Example mirror in plane (m)
5
Combinations

• Example 2-fold rotation perpendicular to mirror
  (2/m)

Example 3 perpendicular 2-fold rotations (222)
6
Roto-Inversions

• A rotation followed by an inversion through the
  origin (in this case the centre of the stereogram)

Example bar 4 inversion tetrad
More examples in sheet.
7
Special positions

• When the object under study lies on a symmetry
  element ? mm2 example

General positions
Special positions
Equivalent positions
8
In terms of axes

• Again, from workshop
• Take a point at (x y z)
• Simple mirror in bc plane

9
General convention

• Right hand rule
• (x y z) ? (x y z)

or r Rr R represents the matrix of the point
operation
10
Back to the mirror

• Take a point at (x y z)
• Simple mirror in bc plane

11
Other examples
roto-inversion around z
Left as an example to show with a diagram.
12
More complex cases

• For non-orthogonal, high symmetry axes, it
  becomes more complex, in terms of deriving from a
  figure. 3-fold example

b
a
13
3-fold and 6-fold
etc.

• It is obvious that 62 and 64 are equivalent to
  3 and 32, respectively.

14
32 crystallographic point groups

• display all possibilities for the symmetry of
  space-filling shapes
• form the basis (with Bravais lattices) of space
  groups

Enantiomorphic Enantiomorphic Centrosymmetric Centrosymmetric
Triclinic 1
Monoclinic 2 2/m m
Orthorhombic 222 mmm mm2
Tetragonal 4 422 4/m 4/mmm 4mm 2m
Trigonal 3 32 3m
Hexagonal 6 622 6/m 6/mmm 6mm 2m
Cubic 23 432 m m m 3m
15
32 crystallographic point groups
Enantiomorphic Enantiomorphic Centrosymmetric Centrosymmetric
Triclinic 1
Monoclinic 2 2/m m
Orthorhombic 222 mmm mm2
Tetragonal 4 422 4/m 4/mmm 4mm 2m
Trigonal 3 32 3m
Hexagonal 6 622 6/m 6/mmm 6mm 2m
Cubic 23 432 m m m 3m

• Centrosymmetric have a centre of symmetry
• Enantiomorphic opposite, like a hand and its
  mirror
• - polar, or pyroelectric, point groups

16
Space operations

• These involve a point operation R (rotation,
  mirror, roto-inversion) followed by a translation
  ?
• Can be described by the Seitz operator

e.g.
17
Glide planes

• The simplest glide planes are those that act
  along an axis, a b or c
• Thus the translation is ½ way along the cell
  followed by a reflection (which changes the
  handedness )

Here the a glide plane is perpendicular to the
c-axis This gives symmetry operator ½x, y, -z.
18
n glide

• n glide Diagonal glide
• Here the translation vector has components in two
  (or sometimes three) directions

So for example the translations would be (a ?
b)/2 Special circumstances for cubic tetragonal
19
n glide

• Here the glide plane is in the plane xy
  (perpendicular to c)

Symmetry operator ½x, ½y, -z
20
d glide

• d glide Diamond glide
• Here the translation vector has components in two
  (or sometimes three) directions

So for example the translations would be (a ?
b)/4 Special circumstances for cubic tetragonal
21
d glide

• Here the glide plane is in the plane xy
  (perpendicular to c)

Symmetry operator ¼x, ¼y, -z
22
17 Plane groups

• Studied (briefly) in the workshop
• Combinations of point symmetry and glide planes

E. S. Fedorov (1881)
23
Another example

• Build up from one point

24
Screw axes

• Rotation followed by a translation
• Notation is nx where n is the simple rotation, as
  before
• x indicates translation as a fraction x/n along
  the axis

?
? /2
21 screw axis
2 rotation axis
25
Screw axes - examples
Looking down from above

• Note e.g. 31 and 32 give different handedness

26
Example

• P42 (tetragonal) any additional symmetry?

27
Matrix

• 4 fold rotation and translation of ½ unit cell

Carry this on.
28
Symmorphic Space Groups

• If we build up into 3d we go from point to plane
  to space groups

From the 32 point groups and the different
Bravais lattices, we can get 73 space groups
which involve ONLY rotations, reflection and
rotoinversions.
Non-symmorphic space groups involve translational
elements (screw axes and glide planes). There are
157 non-symmorphic space groups 230 space groups
in total!
29
Example of Symmorphic Space group
30
Example of Symmorphic Space group
31
Systematic Absences 2

• Systematic absences in (hkl) reflections ?
  Bravais lattices
• e.g. Reflection conditions hkl 2n ? Body
  centred

• Similarly glide screw axes associated with
  other absences
• 0kl, h0l, hk0 absences glide planes
• h00, 0k0, 00l absences screw axes

Example 0kl glide plane is perpendicular to
a if k2n b glide if l 2n c clide if
k1 2n n glide
32
Space Group example

• P2/c

Equivalent positions
33
Space Group example

• P21/c note glide plane shifted to y¼ because
  convention likes inversions at origin

Equivalent positions
34
Special positions

• Taken from last example
• If the general equivalent positions are

• Special positions are at
• ½,0,½ ½,½,0
• 0,0,½ 0,½,0
• ½,0,0 ½,½, ½
• 0,0,0 0,½,½

35
Space groups

• Allow us to fully describe a crystal structure
  with the minimum number of atomic positions
• Describe the full symmetry of a crystal structure
• Restrict macroscopic properties (see symmetry
  workshop) e.g. BaTiO3
• Allow us to understand relationships between
  similar crystal structures and understand
  polymorphic transitions

36
Example YBCO

• Handout of Structure and Space group
• Most atoms lie on special positions
• YBa2Cu3O7 is the orthorhombic phase
• Space group Pmmm