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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

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More on symmetry
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
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

Above

Below

Back to stereograms and point symmetry

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


More examples
More examples

Example: 2-fold rotation in plane (2)

Example: mirror in plane (m)


Combinations
Combinations

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

Example: 3 perpendicular 2-fold rotations (222)


Roto inversions
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
Special positions

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

General positions

Special positions

Equivalent positions


In terms of axes

-x, y, z

b

x, y, z

a

In terms of axes…

Again, from workshop:

  • Take a point at (x y z)

  • Simple mirror in bc plane


General convention

c

(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


Back to the mirror

-x, y, z

b

x, y, z

a

Back to the mirror…

  • Take a point at (x y z)

  • Simple mirror in bc plane


Other examples
Other examples

roto-inversion around z

Left as an example to show with a diagram.


More complex cases
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
3-fold and 6-fold

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

etc.


32 crystallographic point groups
32 crystallographic point groups

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

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


32 crystallographic point groups1
32 crystallographic point groups

Centrosymmetric – have a centre of symmetry

Enantiomorphic – opposite, like a hand and its mirror

* - polar, or pyroelectric, point groups


Space operations
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

,

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
n glide

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


N glide1

a

+

+

,

-

b

+

+

n glide

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

Symmetry operator ½+x, ½+y, -z


D glide
d glide

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


D glide1

,

,

-

-

a

+

+

,

-

+

,

-

,

-

b

+

+

d glide

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

Symmetry operator ¼+x, ¼+y, -z


17 plane groups
17 Plane groups

Studied (briefly) in the workshop

Combinations of point symmetry and glide planes

E. S. Fedorov (1881)


Another example
Another example

Build up from one point:


Screw axes
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

2 rotation axis

21 screw axis


Screw axes examples
Screw axes - examples

Note e.g. 31 and 32 give different handedness

Looking down from above


Example
Example

  • P42 (tetragonal) – any additional symmetry?


Matrix
Matrix

4 fold rotation and translation of ½ unit cell

Carry this on….


Symmorphic space groups
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!




Systematic absences 2
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 glide

if l = 2n c clide

if k+1 = 2n n glide


Space group example
Space Group example

  • P2/c

Equivalent positions:


Space group example1
Space Group example

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

Equivalent positions:


Special positions1
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
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
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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