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)

Above Below

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

44

4

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

x, y, z

-x, y, z

a

b

General convention

• Right hand rule• (x y z) (x’ y’ z’)

a

b(x y z)

(x’ y’ z’)

rr’

c

z

y

x

ccc

bbb

aaa

z

y

x

333231

232221

131211

'

'

'

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

z

y

x

z

y

x

z

y

x

100

010

001

'

'

'

100

010

001

100mx, y, z

-x, y, z

a

b

Other examples

z

x

y

z

y

x

z

y

x

100

001

010

'

'

'

4

Left as an example to show with a diagram.

roto-inversion around z

z

y

x

z

x

y

z

y

x

100

001

010

'

'

'

z

x

y

z

y

x

z

y

x

100

001

010

'

'

'

100

001

010

0014

More complex cases

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

z

yx

y

z

y

x

z

y

x

100

011

010

'

'

'

z

x

xy

z

yx

y

z

y

x

100

011

010

'

'

'

a

b

3-fold and 6-fold

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

z

x

yx

z

y

x

z

y

x

100

001

011

'

'

'

etc.

32 crystallographic point groups

• display all possibilities for the symmetry of space-filling shapes

• form the basis (with Bravais lattices) of space groups

Enantiomorphic 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

1

3

33

4

4

4

6 6

Enantiomorphic 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

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

1

3

33

4

4

4

6 6

Space operations

These involve a point operation R (rotation, mirror, roto-inversion) followed by a translation

Can be described by the Seitz operator:

RrrR |

e.g.

r0,0,|0012 21

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 ½+x, y, -z.

z

y

x

z

y

x

r,,|m2

12

1

21

0

0

100

010

001

00001

a

c ,

,

,

n glide

n glide = Diagonal glideHere 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

a

b

-

+

+ +,

+

n glide

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

Symmetry operator ½+x, ½+y, -z

z

y

x

z

y

x

r,,|m 21

21

21

21

21

21

0100

010

001

0001

a

b-

+

+ +

,+

d glide

d glide = Diamond glideHere 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

a

b

-

+,

+

++

-,+

-,-,-

,

d glide

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

a

b-

+

,

+

++

-,

+

-

,

-,

-

,

z

y

x

z

y

x

rm 41

41

41

41

41

41

0100

010

001

0,,|001

Symmetry operator ¼+x, ¼+y, -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 axis2 rotation axis

Screw axes - examples

Note e.g. 31 and 32 give different handedness

Looking down from above

Example

• P42 (tetragonal) – any additional symmetry?

Matrix

4 fold rotation and translation of ½ unit cell

z

x

y

z

y

x

r

21

21

21 0

0

100

001

010

,0,0|0014

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 h+k+l = 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 k=2n b glideif l = 2n c clideif k+1 = 2n n glide

Space Group example

• P2/c

zyxzyxzyxzyx ,,,,,, ,2

1,

2

1

Equivalent positions:

Space Group example

P21/c : note glide plane shifted to y=¼ because convention “likes” inversions at origin

2

1

2

1,

2

1

2

1, ,,,,,, zyxzyxzyxzyx

Equivalent positions:

Special positions

Taken from last example

If the general equivalent positions are:

2

1

2

1,

2

1

2

1, ,,,,,, zyxzyxzyxzyx

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