Symmetry

1. Symmetry Translations (Lattices) A property at the atomic level, not of crystal shapes Symmetric translations involve repeat distances The origin is arbitrary 1-D translations = a row

2. Symmetry Translations (Lattices) A property at the atomic level, not of crystal shapes Symmetric translations involve repeat distances The origin is arbitrary 1-D translations = a row a  a is the repeat vector

3. Symmetry Translations (Lattices) 2-D translations = a net

4. Symmetry Translations (Lattices) 2-D translations = a net Unit cell Unit Cell: the basic repeat unit that, by translation only, generates the entire pattern How differ from motif ??

5. Symmetry b a Translations (Lattices) 2-D translations = a net Pick any point Every point that is exactly n repeats from that point is an equipoint to the original

6. Translations Exercise: Escher print 1. What is the motif ? 2. Pick any point and label it with a big dark dot 3. Label all equipoints the same 4. Outline the unit cell based on your equipoints 5. What is the unit cell content (Z) ?? Z = the number of motifs per unit cell Is Z always an integer ?

7. Which unit cell is correct ?? • Conventions: • 1. Cell edges should, whenever possible, coincide with symmetry axes or reflection planes • 2. If possible, edges should relate to each other by lattice’s symmetry. • 3. The smallest possible cell (the reduced cell) which fulfills 1 and 2 should be chosen Translations

8. Translations The lattice and point group symmetry interrelate, because both are properties of the overall symmetry pattern

9. Translations The lattice and point group symmetry interrelate, because both are properties of the overall symmetry pattern Good unit cell choice. Why? What is Z? Are there other symmetry elements ?

10. Translations The lattice and point group symmetry interrelate, because both are properties of the overall symmetry pattern This is why 5-fold and > 6-fold rotational symmetry won’t work in crystals

11. Translations There is a new 2-D symmetry operation when we consider translations The Glide Plane: A combined reflection and translation repeat Step 2: translate Step 1: reflect (a temporary position)

12. Translations There are 5 unique 2-D plane lattices.

13. There are 5 unique 2-D plane lattices. There are also 17 2-D Plane Groups that combine translations with compatible symmetry operations. The bottom row are examples of plane Groups that correspond to each lattice type

14. Plane Group Symmetry Combining translations and point groups

15. Plane Group Symmetry p211

16. Tridymite: Orthorhombic C cell

17. 3-D Translations and Lattices • Different ways to combine 3 non-parallel, non-coplanar axes • Really deals with translations compatible with 32 3-D point groups (or crystal classes) • 32 Point Groups fall into 6 categories

18. +c +a b g a +b Axial convention: “right-hand rule” 3-D Translations and Lattices • Different ways to combine 3 non-parallel, non-coplanar axes • Really deals with translations compatible with 32 3-D point groups (or crystal classes) • 32 Point Groups fall into 6 categories

19. c c c b b b a P P I = C a a Triclinic Monoclinic a ¹ b ¹ g a = g = 90 ¹ b o ¹ ¹ a b c ¹ ¹ a b c c b a P C F I Orthorhombic a = b = g = 90 ¹ ¹ a b c o

20. c a 2 a P I 1 Tetragonal a = b = g = 90 ¹ a = a c o 1 2 a 3 a 2 a P 1 F I Isometric a = b = g = 90 a = a = a o 1 2 3

21. 3-D Translations and Lattices Triclinic: • No symmetry constraints. • No reason to choose C when can choose simpler P • Do so by convention, so that all mineralogists do the same • Orthorhombic: • Why C and not A or B? • If have A or B, simply rename the axes until  C

22. 3-D Symmetry +c +a b g a +b Axial convention: “right-hand rule” Crystal Axes

23. 3-D Symmetry

24. 3-D Symmetry

25. 3-D Symmetry

26. 3-D Symmetry

27. 3-D Space Groups • As in the 17 2-D Plane Groups, the 3-D point group symmetries can be combined with translations to create the 230 3-D Space Groups • Also as in 2-D there are some new symmetry elements that combine translation with other operations • Glides: Reflection + translation • 4 types. Fig. 6.52 in Klein • Screw Axes: Rotation + translation • Fig. 5.67 in Klein