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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) Reflection (mirror) symmetry m LH RH mirror (3) Inversion symmetry (center of symmetry) 1 RH LH (4) Rotation-Inversion axis 360o = Rotate by ，then invert. (a) one fold rotation inversion (1) (b) two fold rotation inversion (2) = mirror symmetry (m) (c) inversion triad (3) 3 = Octahedral site in an octahedron (d) inversion tetrad (4) x = tetrahedral site in a tetrahedron (e) inversion hexad (6) x Hexagonal close-packed (hcp) lattice 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 ( 1 ) Proof: There are only 1, 2, 3, 4, and 6 fold rotation symmetries for crystal with translational symmetry. graphically The space cannot be filled! T Start with the translation A lattice point Add a rotation lattice point A lattice point pT T: scalar T T A T T A translation vector connecting two lattice points! It must be some integer of T or we contradicted the basic Assumption of our construction. p: integer Therefore, is not arbitrary! The basic constrain has to be met! B’ B b T T T A T tcos A’ T tcos T To be consistent with the original translation t: b pT b T 2T cos pT p 1 2 cos 2 cos 1 p M p M 4 -3 3 -2 2 -1 1 0 0 1 -1 2 cos -1.5 -1 -0.5 0 0.5 1 -- 2/3 /2 /3 0 p must be integer M must be integer n (= 2/) b -2 3 4 6 (1) M >2 or M<-2: no solution -3T 2T T Allowable rotational 0 symmetries are 1, 2, 3, 4 and 6. -T Look at the case of p = 2 n = 3; 3-fold pT 2T = T1 T2 T2 T1 T2 120o T1 120o angle 3-fold lattice. Look at the case of p = 1 n = 4; 4-fold pT T = 90o T1 T2 T2 T1 4-fold lattice. T1 T2 90o Look at the case of p = 0 n = 6; 6-fold pT 0T T2 = T1 T2 T1 60o T1 T2 60o Exactly the same as 3-fold lattice. Look at the case of p = 3 n = 2; 2-fold pT 3T Look at the case of p = -1 pT 1T 1 2 n = 1; 1-fold 1-fold 2-fold 3-fold 4-fold 6-fold Parallelogram T1 T2 T1 T2 general Hexagonal Net T1 T2 T1 T2 120 Square Net T1 T2 o T1 T2 90o Can accommodate 1- and 2-fold rotational symmetries Can accommodate 3- and 6-fold rotational symmetries Can accommodate 4-fold rotational Symmetry! These are the lattices obtained by combining rotation and translation symmetries? How about combining mirror and translation Symmetries? Combine mirror line with translation: T2 T1 constrain m m Unless Or 0.5T T1 T2 T1 T2 90o Primitive cell centered rectangular Rectangular 5 lattices in 2D (1) Parallelogram (Oblique) T1 T2 T1 T2 general (2) Hexagonal T1 T2 (3) Square T1 T2 T1 T2 120o T1 T2 90o (4) Centered rectangular T1 T2 T1 T2 90o (5) Rectangular T1 T2 T1 T2 90o Double cell (2 lattice points) Primitive cell Symmetry elements in 2D lattice Rectangular = center rectangular? 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 1 (a) Rectangular lattice ( ≠ ; = 90o) Maintain mirror symmetry ( ≠ ; = 90o) m (b) Center Rectangular lattice ( ≠ ; = 90o) Maintain mirror symmetry m ( ≠ ; = 90o) Rhombus cell (Primitive unit cell) = ; ≠ 90o (c) Parallelogram lattice ( ≠ ; ≠ 90o) Destroy mirror symmetry (d) Square lattice ( = ; = 90o) b a (e) hexagonal lattice ( = ; = 120o) 3-2-3. 3-D lattice: 7 systems, 14 Bravais lattices Starting from parallelogram lattice ( ≠ ; ≠ 90o) (1)Triclinic system c a 1-fold rotation (1) b ( ≠ ≠ ; ≠ ≠ ≠ 90o) lattice center symmetry at lattice point as shown above which the molecule is isotropic (1) (2) Monoclinic system ( ≠ ≠ ; = = 90o ≠ ) one diad axis (only one axis perpendicular to the drawing plane maintain 2-fold symmetry in a parallelogram lattice) (1) Primitive monoclinic lattice (P cell) c b a (2) Base centered monoclinic lattice c a b B-face centered monoclinic lattice The second layer coincident to the middle of the first layer and maintain 2-fold symmetry Note: other ways to maintain 2-fold symmetry c a b A-face centered monoclinic lattice If relabeling lattice coordination a b b a A-face centered monoclinic = B-face centered (2) Body centered monoclinic lattice Body centered monoclinic = Base centered monoclinic So monoclinic has two types 1. Primitive monoclinic 2. Base centered monoclinic (3) Orthorhombic system c a b ( ≠ ≠ ; = = = 90o) 3 -diad axes (1) Derived from rectangular lattice ( ≠ ; = 90o) to maintain 2 fold symmetry The second layer superposes directly on the first layer (a) Primitive orthorhombic lattice c a b (b) B- face centered orthorhombic = A -face centered orthorhombic c a b (c) Body-centered orthorhombic (I- cell) rectangular body-centered orthorhombic based centered orthorhombic (2) Derived from centered rectangular lattice ( ≠ ; = 90o) (a) C-face centered Orthorhombic a c b C- face centered orthorhombic = B- face centered orthorhombic (b) Face-centered Orthorhombic (F-cell) Up & Down Left & Right Front & Back Orthorhombic has 4 types 1. Primitive orthorhombic 2. Base centered orthorhombic 3. Body centered orthorhombic 4. Face centered orthorhombic (4) Tetragonal system c b a ( = ≠ ; = = = 90o) One tetrad axis starting from square lattice ( = ; = 90o) starting from square lattice ( = ; = 90o) (1) maintain 4-fold symmetry (a) Primitive tetragonal lattice First layer Second layer (b) Body-centered tetragonal lattice First layer Second layer Tetragonal has 2 types 1. Primitive tetragonal 2. Body centered tetragonal (5) Hexagonal system c a b a = b c; = = 90o; = 120o One hexad axis starting from hexagonal lattice (2D) a = b; = 120o (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 Hexagonal has 1 types 1. Primitive hexagonal Rhombohedral (trigonal) 2. Primitive rhombohedral (trigonal) (6) Cubic system c a b a = b = c; = = = 90o 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 = 90o (a) Primitive cubic c a b a = b = c; = = = 90o (b) Face centered cubic = 60o a = b = c; = = = 60o (c) body centered cubic = 109o a = b = c; = = = 109o cubic (isometric) Special case of orthorhombic with a = b = c Primitive (P) Body centered (I) Face centered (F) Base center (C) Tetragonal (I)? Tetragonal (P) a = b c Cubic has 3 types 1. Primitive cubic (simple cubic) 2. Body centered cubic (BCC) 3. Face centered cubic (FCC) =P =TP http://www.theory. nipne.ro/~dragos/S olid/Bravais_table. jpg =I