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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

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

Back to stereograms and point symmetry

- Example 2-fold rotation perpendicular to plane

(2)

More examples

- Example 2-fold rotation in plane (2)

Example mirror in plane (m)

Combinations

- Example 2-fold rotation perpendicular to mirror

(2/m)

Example 3 perpendicular 2-fold rotations (222)

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.

Special positions

- When the object under study lies on a symmetry

element ? mm2 example

General positions

Special positions

Equivalent positions

In terms of axes

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

General convention

- Right hand rule
- (x y z) ? (x y z)

or r Rr R represents the matrix of the point

operation

Back to the mirror

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

Other examples

roto-inversion around z

Left as an example to show with a diagram.

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

3-fold and 6-fold

etc.

- It is obvious that 62 and 64 are equivalent to

3 and 32, respectively.

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

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

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.

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.

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

n glide

- Here the glide plane is in the plane xy

(perpendicular to c)

Symmetry operator ½x, ½y, -z

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

d glide

- Here the glide plane is in the plane xy

(perpendicular to c)

Symmetry operator ¼x, ¼y, -z

17 Plane groups

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

E. S. Fedorov (1881)

Another example

- Build up from one point

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

Screw axes - examples

Looking down from above

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

Example

- P42 (tetragonal) any additional symmetry?

Matrix

- 4 fold rotation and translation of ½ unit cell

Carry this on.

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!

Example of Symmorphic Space group

Example of Symmorphic Space group

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

Space Group example

- P2/c

Equivalent positions

Space Group example

- P21/c note glide plane shifted to y¼ because

convention likes inversions at origin

Equivalent positions

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

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

Example YBCO

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

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