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 Above Below 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 x, y, z -x, y, z a b

General convention Right hand rule (x y z)  (x’ y’ z’) or r’ = Rr b (x y z) (x’ y’ z’) r r’ c 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 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 * Cubic 23 432 m m m 3m

32 crystallographic point 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 * 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: ) , a c , Here the a glide plane is perpendicular to the c-axis This gives symmetry operator ½+x, y, -z.

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 glide Here the glide plane is in the plane xy (perpendicular to c) a b - + , Symmetry operator ½+x, ½+y, -z

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 glide Here the glide plane is in the plane xy (perpendicular to c) a b - + , 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 2 rotation axis 21 screw 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 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 P2/c Equivalent positions:

Space Group example P21/c : note glide plane shifted to y=¼ 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