1. 3D Symmetry_1 (2 weeks)

2. Next we would move a step further into 3D symmetry. Leonhard Euler : http://en.wikipedia.org/wiki/Leonhard_Euler Google search: Euler

3. For convenience, set R = 1 Great circle (GC), R=1 Small circle R<1  AB Spherical trigonometry Distance:  AOB =  (GC) o Pole 90 o to arc AB. OP  plane defined by OAB AB o P Well defined angle B’  POA =  /2;  POB =  /2;  POB’ =  /2

4. A B C ab c arc BC = a arc AC = b arc AB = c. GC Spherical Angles A BC  BAC =  B’OC’  B’OC’ o GC ? Trigonometry: points on a surface of a sphere (directions that intersect the sphere) are connected using arcs of great circles OA  OB OA  OC OB  OC angle Center of the sphere C’ B’ A is the pole for plane defined by B’OC’

5. ABC and ABC are mutually polar! Proof: B: pole of arc AC  B is 90 o away from point A. C: pole of arc AB  C is 90 o away from point A.  A:pole of arc BC. Similarly, B: AC, C: pole of arc AB. A BC A B C Polar triangle A, pole of arc BC B, pole of arc AC C, pole of arc AB GC

6. A B C B C P Q Proof:  BAC = , arc BC = a,   + a = .  a B : pole of arc AC  arc BQ =  /2 C : pole of arc AB  arc CP =  /2  arc BQ + arc CP =  = (arc BP+ arc PQ) + (arc PQ+ arc QC) = (arc BP+ arc PQ+ arc QC) + arc PQ = a + arc PQ A A C o B Q C Why! See pictures of spherical angle in page 4 (bottom)! =  POQ = 

7. Law of cosines Plane Trigonometry A B C a b c How about law of cosines in spherical trigonometry? length angle Triangle is defined as C is spherical angle at point u.  vOu = a  wOu = b  vOw = c a, b, c 

8. u v w o y z b a 90 o (1) (2) 1 a (1) (2) (1)= tana 1/(2)= cosa  (2) = seca 1 b (3) (4) (3)= tanb 1/(4)= cosb  (4) = secb Unit circle (3) (4) http://en.wikipedia.org/wiki/Spherical_law_of_cosines OO u u z y (From  uyz) (From  oyz)

9. Stop here about spherical trigonometry! We obtain all the relations needed for further discussion of the 3D point groups!

10. Combination of two rotation operations in 3 D: AA BB 1 R 2 R 3 R CC A  : (1)  (2) B  : (2)  (3) (1) and (3) relation? 3-D: translation, reflection, rotation, and inversion. must be crystallographic c

11. AA BB c AB cAB c C Locate the position of axis C baAA BB CC c ab Euler construction: A B M M’ N’ NC’ C  A  : AM  AM’. B  : BN  BN’. C (the point unmoved). OC: the axis A symmetry element is the locus of a point that is left unmoved by an operation.

12. (1) A  : leave A unmoved. (2) B  : move A to A’. A B M N’ C A’  /2   ABC =  A’BC =  /2 AB = A’B  ABC =  A’BC   ACB =  A’CB   /2 A  /2 B C  /2  /2 c a b The law of cosine (spherical trigonometry)

13. A  /2 B C  /2  /2 c a b 180 o -a 180 o -b 180 o -c 180 o -  /2 180 o -  /2 180 o -  /2 Law of cosine to the polar triangle Polar triangle

15. All the rotation combinations possible in 3D that need to be tested: B A 1 1 111 112 113 114 116 2 212 213 214 216 313 314 3 316 2 222 223 224 226 323 326 324 4 414 416 6 616 346 626636646666 424 426 434 436 444 446 333 336 334

16. Axis at A, B, or C , , or   /2  /2  /2 1-fold 2-fold 3-fold 4-fold 6-fold 360 o 180 o 120 o 90 o 60 o 180 o 90 o 60 o 45 o 30 o 0 1/21/2 3 1/2 /2 0 1 3 1/2 /21/2 1/2

17. Case: 11n A: 1,  = 360 o, cos(  /2) = -1; sin(  /2) = 0 A  /2 B C  /2  /2 c a b B: 1,  = 360 o, cos(  /2) = -1; sin(  /2) = 0 C: n,  = 360 o /n, cos(  /2); sin(  /2) A B C 180 o 180 o /n c a b 180 o None exist! Except,  = 360 o  111

18. Case: 22n A: 2,  = 180 o, cos(  /2) = 0; sin(  /2) = 1 A  /2 B C  /2  /2 c a b B: 2,  = 180 o, cos(  /2) = 0; sin(  /2) = 1 C: n,  = 360 o /n, cos(  /2); sin(  /2) A B C 90 o 180 o /n c a b 90 o

19. 222 223 224 226 Angle between A and B axis A B A B A B A B C a b What are a and b?

20. A: 2,  = 180 o, cos(  /2) = 0; sin(  /2) = 1 B: 2,  = 180 o, cos(  /2) = 0; sin(  /2) = 1 C: n,  = 360 o /n, cos(  /2); sin(  /2)  a = 90 o.  b = 90 o.

21. 30 o 222 90 o 223 60 o 224 45 o 226 A B C A B C A B C A B C

22. Case: 23n A: 2,  = 180 o, cos(  /2) = 0; sin(  /2) = 1 B: 3,  = 120 o, cos(  /2) = 0.5; sin(  /2) = 3 0.5 /2 C: n,  = 360 o /n, cos(  /2); sin(  /2) A B C 90 o 360 o /n c a b 60 o None exist The rest of combination does not exist! 233 234 236

23. Case: 233 A: 2,  = 180 o, cos(  /2) = 0; sin(  /2) = 1 B: 3,  = 120 o, cos(  /2) = 0.5; sin(  /2) = 3 0.5 /2 C: 3,  = 120 o, cos(  /2) = 0.5; sin(  /2) =3 0.5 /2  a = 70 o 32.  b = 54 o 44.

24. 70 o 32’ 54 o 44’ 233 x y z 000 A B C Angle between A and B is Angle between A and C is Angle between B and C is

25. Case: 234 A: 2,  = 180 o, cos(  /2) = 0; sin(  /2) = 1 B: 3,  = 120 o, cos(  /2) = 0.5; sin(  /2) = 3 0.5 /2 C: 4,  = 90 o, cos(  /2) = 1/2 0.5 ; sin(  /2) = 1/2 0.5  a = 54 o 44.  b = 45 o.

26. 54 o 44’ 234 45 o 35 o 16’ x y z 000 A B C Angle between A and B is Angle between A and C is Angle between B and C is

27.  Geometry of the permissible nontrivial combination of rotations: Combination 2A =  2B =  2C =  cab 222 223 224 226 233 234 180 o 90 o 60 o 45 o 30 o 54 o 44 35 o 16 180 o 120 o 180 o 120 o 90 o 60 o 120 o 90 o 54 o 44 45 o 90 o 70 o 32 54 o 44

28. 222 International symbol 322 32(2) Just like 3m(m) Only one independent 2 fold rotation axis 422 (1) (2) (3) (1)(2) (3) 622 22 operation is basically on the plane!

29. Schonllies notation: T n22 222 32(2) 422 622 D n D 2 D 3 D 4 D 6 dihedral 233 Tetrahedral Schonllies notation different dihedral angle 23 is enough to specify the symmetry! 23

30. International symbol http://en.wikipedia.org/wiki/Tetrahedron

31. 54 o 44’ 234 or 432 45 o 35 o 16’ Schonllies notation: O Octahedron International symbol http://en.wikipedia.org/wiki/Octahedron

32. 11 axial combinations 1 2 3 4 6 222 322 422 622 233 432 11 axial combinations + Extender vertical m n horizontal m n 422 horizontal  vertical  diagonal  for D n, T, O Ways to add m: Not for C n Extender:  v,  h,  d, ! (+ extender  create new rotation axis!)

33. 1 2 3 4 6 222 32 422 622 23 432 vhdvhd See reading crystal4.pdf C nv, D nv T v, O v C nh, D nh T h, O h C ni, D ni T i, O i D nd T d, O d http://ocw.mit.edu/courses/materials- science-and-engineering/3-60-symmetry- structure-and-tensor-properties-of- materials-fall-2005/readings/crystal4.pdf We will explain it later

34. m m - 2mm - 3m4mm 1 2 3 4 6 222 32 vhdvhd C nv, D nv T v, O v C nh, D nh T h, O h C ni, D ni T i, O i D nd T d, O d -- 6mm -

35. Let’s look at some cases Two fold rotation (2) + horizontal mirror (  h ) R R L ?: (1)  (3) at the point of intersection 1 1 1 1 1 1 (1) R (2) R (3) L updown R L R L L R: right-handedness L: left-handedness

36. R L Four fold rotation (4) + horizontal mirror (  h ) updown R L Four fold rotation (4) + vertical mirror (  v ) updown R R R R L L L L The mirror that you put in Mirror 45 o with respect to the first mirror set

37. Six fold rotation (6) + horizontal mirror (  h )  R R  R R  R R  R R  R R  R R  Down  Up  L L  L L  L L  L L  L L  L L Six fold rotation (6) + vertical mirror (  v ) Group symmetry elements: 12

38. Three fold rotation (3) + horizontal mirror (  h )  R R  R R  R R updown  L L  L L  L L (1)   (1)  (1)   (2)  (1)   (3)  (1)   (1)  (1)   (2)  (1)   (3)  New two step operations Roto-inversion (1) (2) (3)

39.  (1) R (2) R (3) L Roto-reflection

40. Roto-inversion  (1) R (2) R (3) L The one used

41. (1) (2) (3) (4) (5) (6) (1)  (1) (1)  (2) (1)  (3) (1)  (4) (1)  (5) (1)  (6) Three fold rotation (3) + vertical mirror (  v ) Three fold rotation (3) + inversion ( ) R R R L L L  Down  Up

42.  down  up R L You can except

43. (1) (2) (1) To (1) (3) (1) To (2) (4) (1) To (3) (1) To (4) Sphenoid (Greek word for axe) Not tetrahedron Equal length The rest four: equal length. R L R L

44. How about ?  Down  Up  R R  L L  R R  L  R R is a special one that you have to add to the 11 axial combination

45. Add  h  L L R  R R  R R  R R  R R  R R  R R  R R  R R  L R L R  R L R L  L R L R  R L R L  R L R L  L R L R  R L R L  L R L R +  h Add  L L R Homework! R R  R L R L  R L R L

46. Look at the ppt file that I send you regarding to 222 + Extender (  v,  h,  d, ) as an example!

47. T 23 up down all down. all up. T 23 Add a horizontal mirror plane ThTh Create an inversion center inversion R R  R L R L  L  R  R L R L  L R L R

48. Crystal System Symmetry Direction PrimarySecondaryTertiary TriclinicNone Monoclinic[010] Orthorhombic[100][010][001] Tetragonal[001][100]/[010][110] Hexagonal/ Trigonal [001][100]/[010][120]/[1 0] Cubic [100]/[010]/ [001] [111][110]

49. Buerger’s book 3D crystallographic point group 2D lattices: chapter 7 (pg. 69-83) Euler’s construction: pg. 35-43 Some combination theorems: chapter 6 Points group: pg: 59-68