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3-1 Symmetry elements

III Crystal Symmetry. 3-1 Symmetry elements. (1) Rotation symmetry. two fold ( diad ) 2 three fold (triad ) 3 Four fold (tetrad ) 4 Six fold ( hexad ) 6. LH. RH. mirror. RH. LH. (2) Reflection (mirror) symmetry m. (3) Inversion symmetry (center of symmetry).

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3-1 Symmetry elements

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  1. III Crystal Symmetry • 3-1 Symmetry elements (1) Rotation symmetry • two fold (diad ) 2 • three fold (triad ) 3 • Four fold (tetrad ) 4 • Six fold (hexad ) 6

  2. LH RH mirror RH LH (2) Reflection (mirror) symmetry m (3) Inversion symmetry (center of symmetry)

  3. (4) Rotation-Inversion axis = Rotate by ，then invert. (a) one fold rotation inversion ( (b) two fold rotation inversion ( = mirror symmetry (m)

  4. (c) inversion triad () 3 = Octahedral site in an octahedron

  5. (d) inversion tetrad () x = tetrahedral site in a tetrahedron (e) inversion hexad () x Hexagonal close-packed (hcp) lattice

  6. 3-2. Fourteen Bravais lattice structures 3-2-1. 1-D lattice 3 types of symmetry can be arranged in a 1-D lattice (1) mirror symmetry (m) (2) 2-fold rotation (2) (3) center of symmetry ()

  7. Proof: There are only 1, 2, 3, 4, and 6 fold rotation symmetriesfor crystal with translational symmetry. graphically The space cannot be filled!

  8. Start with the translation Add a rotation lattice point lattice point   lattice point A translation vector connecting twolattice points! It must be some integer of or we contradicted the basic Assumption of our construction. T: scalar  p: integer Therefore,  is not arbitrary! The basic constrain has to be met!

  9. B B’ b T T T T   tcos tcos A A’ T T To be consistent with the original translation t: pmust be integer M must be integer M >2 or M<-2: no solution pM cosn (= 2/) b 4 3 2 1 0 -1 -3 -1.5 -- -- -- -2 -1  2 3T -1 -0.5 2/3 3 2T 0 0 /2 4 T 1 0.5 /3 6 0 2 1 0  (1) -T Allowable rotational symmetries are 1, 2, 3, 4 and 6.

  10. Look at the case of p = 2 n = 3; 3-fold  angle = 120o 3-fold lattice. Look at the case of p = 1 n = 4; 4-fold  = 90o 4-fold lattice.

  11. Look at the case of p = 0 n = 6; 6-fold  = 60o Exactly the same as 3-fold lattice. Look at the case of p = 3 n = 2; 2-fold Look at the case of p = -1 n = 1; 1-fold 1 2

  12. Parallelogram Can accommodate 1- and 2-fold rotational symmetries 1-fold 2-fold 3-fold 4-fold 6-fold Hexagonal Net Can accommodate 3- and 6-fold rotational symmetries Square Net Can accommodate 4-fold rotational Symmetry! These are the lattices obtained by combining rotation and translation symmetries? How about combining mirror and translation Symmetries?

  13. Combine mirror line with translation: constrain m m Unless Or 0.5T Primitive cell centered rectangular Rectangular

  14. 5 lattices in 2D (1) Parallelogram(Oblique) (2) Hexagonal (3) Square (4) Centered rectangular Double cell (2 lattice points) (5) Rectangular Primitive cell

  15. Symmetry elements in 2D lattice Rectangular = center rectangular?

  16. 3-2-2. 2-D lattice Two ways to repeat 1-D  2D (1) maintain 1-D symmetry (2) destroy 1-D symmetry m 2

  17. (a) Rectangular lattice (; 90o) Maintain mirror symmetry m • (; 90o)

  18. (b) Center Rectangular lattice (; 90o) Maintain mirror symmetry m • (; 90o) Rhombus cell (Primitive unit cell) ; 90o

  19. (c) Parallelogram lattice(; 90o) Destroy mirror symmetry

  20. b a (d) Square lattice (; 90o) (e) hexagonal lattice(; 120o)

  21. 3-2-3. 3-D lattice:7 systems, 14 Bravais lattices Starting from parallelogram lattice (; 90o) (1)Triclinic system 1-fold rotation (1) c    b a (; 90o)

  22. lattice center symmetry at lattice point as shown above which the molecule is isotropic ()

  23. (2) Monoclinic system (; ) one diad axis (only one axis perpendicular tothe drawing plane maintain 2-fold symmetry in a parallelogram lattice)

  24. c b a c b (1) Primitive monoclinic lattice (P cell) (2) Base centered monoclinic lattice a B-face centered monoclinic lattice

  25. c b a The second layer coincident to the middle of the first layerand maintain 2-fold symmetry Note: other ways to maintain 2-fold symmetry A-face centered monoclinic lattice If relabeling lattice coordination A-face centered monoclinic = B-face centered

  26. (2) Body centered monoclinic lattice Body centered monoclinic = Base centered monoclinic

  27. So monoclinic has two types 1. Primitive monoclinic 2. Base centered monoclinic

  28. (3) Orthorhombic system c b a (; ) 3 -diad axes

  29. c b a (1) Derived from rectangular lattice • (; 90o)  to maintain 2 fold symmetry The second layer superposes directly on the first layer (a) Primitive orthorhombic lattice

  30. c b a (b) B- face centered orthorhombic = A -face centered orthorhombic (c) Body-centered orthorhombic (I- cell)

  31. rectangular body-centered orthorhombic  based centered orthorhombic

  32. c b a (2) Derived from centered rectangular lattice • (; 90o) (a) C-face centered Orthorhombic C- face centered orthorhombic = B- face centered orthorhombic

  33. (b) Face-centered Orthorhombic (F-cell) Up & Down Left & Right Front & Back

  34. Orthorhombic has 4 types 1. Primitive orthorhombic 2. Base centered orthorhombic 3. Body centered orthorhombic 4. Face centered orthorhombic

  35. (4) Tetragonal system (; ) c b a One tetrad axis starting from square lattice (; )

  36. First layer Second layer (; ) starting from square lattice (1) maintain 4-fold symmetry (a) Primitive tetragonal lattice

  37. First layer Second layer (b) Body-centered tetragonal lattice Tetragonal has 2 types 1. Primitive tetragonal 2. Body centered tetragonal

  38. (5) Hexagonal system a = b  c;  =  = 90o;  = 120o c  b  One hexad axis  a starting from hexagonal lattice (2D) a = b;  = 120o

  39. c  b a (1) maintain 6-fold symmetry Primitive hexagonal lattice c  b   a (2) maintain 3-fold symmetry 2/3 2/3 1/3  1/3 a = b = c;  =  =   90o

  40. Hexagonal has 1 types 1. Primitive hexagonal Rhombohedral (trigonal) 2. Primitiverhombohedral (trigonal)

  41. (6) Cubic system c a = b = c;  =  =  =90o   b  a 4 triad axes ( triad axis = cube diagonal ) Cubic is a special form of Rhombohedral lattice Cubic system has 4 triad axes mutually inclined along cube diagonal

  42. (a) Primitive cubic = 90o c a = b = c;  =  =  =90o   b  a • (b) Face centered cubic = 60o a = b = c;  =  =  =60o

  43. (c) body centered cubic = 109o a = b = c;  =  =  =109o

  44. cubic (isometric) Special case of orthorhombic with a = b = c Primitive (P) Body centered (I) Face centered (F) Base center (C) Tetragonal (I)? a = bc Tetragonal (P) Cubic has 3 types 1. Primitive cubic (simple cubic) 2. Body centered cubic (BCC) 3. Face centered cubic (FCC)

  45. http://www.theory.nipne.ro/~dragos/Solid/Bravais_table.jpg = P = I = T P

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