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

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mirror RHLH RH (2) Reflection (mirror) symmetry m

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= Rotate by ， then invert. = mirror symmetry (m) (4) Rotation-Inversion axis

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3 = Octahedral site in an octahedron

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= tetrahedral site in a tetrahedron x Hexagonal close-packed (hcp) lattice x

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D lattice 3-2. Fourteen Bravais lattice structures

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

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

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T T T tcos b To be consistent with the original translation t: p must be integer M must be integer p M cos n (= 2 / ) b 2 3T /3 3 2T 0 0 /2 4 T / (1) -T M >2 or M<-2: no solution T TT AA’ B’B Allowable rotational symmetries are 1, 2, 3, 4 and 6.

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

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Look at the case of p = 0 = 60 o n = 6; 6-fold Look at the case of p = 3n = 2; 2-fold Look at the case of p = -1n = 1; 1-fold 12 Exactly the same as 3-fold lattice.

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1-fold 2-fold 3-fold 4-fold 6-fold Parallelogram Hexagonal Net Can accommodate 1- and 2-fold rotational symmetries 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?

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

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

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Symmetry elements in 2D lattice Rectangular = center rectangular?

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Two ways to repeat 1-D 2D (1) maintain 1-D symmetry (2) destroy 1-D symmetry D lattice

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Maintain mirror symmetry m

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Destroy mirror symmetry

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a b

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D lattice: 7 systems, 14 Bravais lattices (1)Triclinic system 1-fold rotation (1) b a c

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one diad axis (only one axis perpendicular to the drawing plane maintain 2-fold symmetry in a parallelogram lattice) (2) Monoclinic system

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

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

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

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

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(3) Orthorhombic system a b c 3 -diad axes

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(1) Derived from rectangular lattice to maintain 2 fold symmetry The second layer superposes directly on the first layer (a) Primitive orthorhombic lattice a b c

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

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rectangular body-centered orthorhombic based centered orthorhombic

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(2) Derived from centered rectangular lattice (a) C-face centered Orthorhombic a b c C- face centered orthorhombic = B- face centered orthorhombic

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

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

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

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(1) maintain 4-fold symmetry (a) Primitive tetragonal lattice First layer Second layer

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

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(5) Hexagonal system a = b c; = = 90 o ; = 120 o b a c One hexad axis starting from hexagonal lattice (2D) a = b; = 120 o

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(1) maintain 6-fold symmetry Primitive hexagonal lattice b a c (2) maintain 3-fold symmetry a b c 1/3 2/3 1/3 2/3 a = b = c; = = 90 o

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

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(6) Cubic system 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 b a c a = b = c; = = = 90 o

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(a) Primitive cubic = 90 o b a c a = b = c; = = = 90 o (b) Face centered cubic = 60 o a = b = c; = = = 60 o

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(c) body centered cubic = 109 o a = b = c; = = = 109 o

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

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nipne.ro/~dragos/S olid/Bravais_table. jpg = P= I = T P

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