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# More on symmetry - PowerPoint PPT Presentation

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

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Presentation Transcript

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

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

Below

Back to stereograms and point symmetry

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

Example: 2-fold rotation in plane (2)

Example: mirror in plane (m)

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

Example: 3 perpendicular 2-fold rotations (222)

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.

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

General positions

Special positions

Equivalent positions

b

x, y, z

a

In terms of axes…

Again, from workshop:

• Take a point at (x y z)

• Simple mirror in bc plane

(x’ y’ z’)

r’

r

(x y z)

b

a

General convention

• Right hand rule

• (x y z)  (x’ y’ z’)

or r’ = Rr

R represents the matrix of the point operation

b

x, y, z

a

Back to the mirror…

• Take a point at (x y z)

• Simple mirror in bc plane

roto-inversion around z

Left as an example to show with a diagram.

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

b

a

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

etc.

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

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

Centrosymmetric – have a centre of symmetry

Enantiomorphic – opposite, like a hand and its mirror

* - polar, or pyroelectric, point groups

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

Can be described by the Seitz operator:

e.g.

a

,

c

,

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-axisThis gives symmetry operator ½+x, y, -z.

n glide = Diagonal glide

Here the translation vector has components in two (or sometimes three) directions

a

+

+

,

-

b

+

+

So for example the translations would be (a  b)/2

Special circumstances for cubic & tetragonal

+

+

,

-

b

+

+

n glide

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

Symmetry operator ½+x, ½+y, -z

d glide = Diamond glide

Here the translation vector has components in two (or sometimes three) directions

,

,

-

-

a

+

+

,

-

+

,

-

,

-

b

+

+

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)

Symmetry operator ¼+x, ¼+y, -z

Studied (briefly) in the workshop

Combinations of point symmetry and glide planes

E. S. Fedorov (1881)

Build up from one point:

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

2 rotation axis

21 screw axis

Note e.g. 31 and 32 give different handedness

Looking down from above

• P42 (tetragonal) – any additional symmetry?

4 fold rotation and translation of ½ unit cell

Carry this on….

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!

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 glide

if l = 2n c clide

if k+1 = 2n n glide

• P2/c

Equivalent positions:

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

Equivalent 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

• 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

Handout of Structure and Space group

• Most atoms lie on special positions

• YBa2Cu3O7 is the orthorhombic phase

• Space group: Pmmm