1. Objectives By the end of this section you should: be able to recognise rotational symmetry and mirror planes know about centres of symmetry be able to identify the basic symmetry elements in cubic, tetragonal and orthorhombic shapes understand centring and recognise face- centred, body-centred and primitive unit cells. Know some simple structures (Fe, Cu, NaCl, CsCl)

3. Note for Symmetry experts! Crystallography uses a different notation from spectroscopy! In spectroscopy, this has C 4 symmetry In crystallography, it has 4 symmetry

4. Symmetry everywhere Pictures from Dr. John Reid

5. Symmetry everywhere Pictures from Dr. John Reid

6. Mirror Plane Symmetry Arises when one half of an object is the mirror image of the other half Symbol m

7. Left: Symmetrical face using the left half of the original face. Middle: Original face. Right: Symmetrical face using the right half of the original face. http://www.uni-regensburg.de/Fakultaeten/phil_Fak_II/Psychologie/Psy_II/beautycheck/english/symmetrie/symmetrie.htm Mirror Plane Symmetry How symmetrical is a face?

8. Mirror Plane Symmetry This molecule has two mirror planes One is horizontal, in the plane of the paper - bisects the Cl-C-Cl bonds Other is vertical, perpendicular to the plane of the paper and bisects the H-C-H bonds

9. Symmetry Something possesses symmetry if it looks the same from >1 orientation Rotational symmetry Can rotate by 120° about the C-Cl bond and the molecule looks identical - the H atoms are indistinguishable This is called a rotation axis - in particular, a three fold rotation axis, as rotate by 120° (= 360/3) to reach an identical configuration

10. All M.C. Escher works (c) Cordon Art-Baarn-the Netherlands. All rights reserved.

11. In general: n-fold rotation axis = rotation by (360/n)° We talk about the symmetry operation (rotation) about a symmetry element (rotation axis) ? Think of examples for n=2,3,4,5,6…

12. Rotational symmetry n = 2 n = 5 n = 6 360/2360/5360/6 180 o 72 o 60 o

13. Centre of Symmetry present if you can draw a straight line from any point, through the centre, to an equal distance the other side, and arrive at an identical point (phew!) Centre of symmetry at SNo centre of symmetry

14. Combinations - the plane point groups Carefully look at what symmetry is present in the whole pattern The blue pattern has rotational symmetry, but the yellow dots break this - therefore there are two mirror planes perpendicular to one another = mm Now try the examples on the sheet...

15. Combinations - the plane point groups

16. Symmetry in 3-d In handout 1 we said that a crystal system is defined in terms of symmetry and not by crystal shape. Thus we need to look at all the symmetry arising from different shapes of unit cell. From this we can deduce essential symmetry.

17. Unit cell symmetries - cubic 4 fold rotation axes (passing through pairs of opposite face centres, parallel to cell axes) TOTAL = 3

18. Unit cell symmetries - cubic 4 fold rotation axes TOTAL = 3 3-fold rotation axes (passing through cube body diagonals) TOTAL = 4

19. Unit cell symmetries - cubic 4 fold rotation axes TOTAL = 3 3-fold rotation axes TOTAL = 4 2-fold rotation axes (passing through diagonal edge centres) TOTAL = 6

20. Mirror planes - cubic 3 equivalent planes in a cube 6 equivalent planes in a cube

21. Tetragonal Unit Cell a = b c ; = = = 90 c a, b elongated / squashed cube

22. Reduction in symmetry CubicTetragonal Three 4-axesOne 4-axis Two 2-axes Four 3-axesNo 3-axes Six 2-axesTwo 2-axes Nine mirrorsFive mirrors See Q3 in handout 2.

23. Essential Symmetry Essential symmetry is that which defines the crystal system (i.e. is unique to that shape).

24. Cubic Unit Cell a=b=c, = = =90 a c b Many examples of cubic unit cells: e.g. NaCl, CsCl, ZnS, CaF 2, BaTiO 3 All have different arrangements of atoms within the cell. So to describe a crystal structure we need to know: the unit cell shape and dimensions the atomic coordinates inside the cell (see later)

25. Primitive and Centred Lattices Copper metal is face-centred cubic Identical atoms at corners and at face centres Lattice type F also Ag, Au, Al, Ni...

26. Primitive and Centred Lattices -Iron is body-centred cubic Identical atoms at corners and body centre (nothing at face centres) Lattice type I from German, innenzentriert Also Nb, Ta, Ba, Mo...

27. Primitive and Centred Lattices Caesium Chloride (CsCl) is primitive cubic Different atoms at corners and body centre. NOT body centred, therefore. Lattice type P Also CuZn, CsBr, LiAg

28. Primitive and Centred Lattices Sodium Chloride (NaCl) - Na is much smaller than Cs Face Centred Cubic Rocksalt structure Lattice type F Also NaF, KBr, MgO….

29. Another type of centring Side centred unit cell Notation: A-centred if atom in bc plane B-centred if atom in ac plane C-centred if atom in ab plane

30. Unit cell contents Counting the number of atoms within the unit cell Many atoms are shared between unit cells

31. AtomsShared Between:Each atom counts: corner 8 cells 1/8 face centre 2 cells 1/2 body centre 1 cell 1 edge centre 4 cells 1/4 Unit cell contents Counting the number of atoms within the unit cell Thinking now in 3 dimensions, we can consider the different positions of atoms as follows

32. Question 4, handout lattice typecell contents P1 [=8 x 1/8] I F C e.g. NaCl Na at corners: (8 1/8) = 1 Na at face centres (6 1/2) = 3 Cl at edge centres (12 1/4) = 3 Cl at body centre = 1 Unit cell contents are 4(Na + Cl - ) 2 [=(8 x 1/8) + (1 x 1)] 4 [=(8 x 1/8) + (6 x 1/2)] 2 [= 8 x 1/8) + (2 x 1/2)]

33. Summary Crystals have symmetry Each unit cell shape has its own essential symmetry In addition to the basic primitive lattice, centred lattices also exist. Examples are body centred (I) and face centred (F)