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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

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**(2) Reflection (mirror) symmetry m**

RH LH (3) Inversion symmetry (center of symmetry) 1 LH RH

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**(4) Rotation-Inversion axis**

= Rotate by ，then invert. (a) one fold rotation inversion ( 1 ) (b) two fold rotation inversion ( 2 ) = mirror symmetry (m)

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

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**(d) inversion tetrad ( 4 )**

x = tetrahedral site in a tetrahedron (e) inversion hexad ( 6 ) x Hexagonal close-packed (hcp) lattice

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**3-2. Fourteen Bravais lattice structures**

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 )

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**Proof: There are only 1, 2, 3, 4, and 6 fold**

rotation symmetries for crystal with translational symmetry. graphically The space cannot be filled!

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**Start with the translation**

Add a rotation lattice point lattice point lattice point A translation vector connecting two lattice 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!

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**To be consistent with the original translation t:**

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

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**Look at the case of p = 2 n = 3; 3-fold 3-fold lattice.**

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

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**Exactly the same as 3-fold lattice. Look at the case of p = 3 **

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

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**These are the lattices obtained by combining **

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?

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**Combine mirror line with translation:**

constrain m m Unless Or 0.5T Primitive cell centered rectangular Rectangular

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**5 lattices in 2D (1) Parallelogram (Oblique) (2) Hexagonal (3) Square**

(4) Centered rectangular Double cell (2 lattice points) (5) Rectangular Primitive cell

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**Symmetry elements in 2D lattice**

Rectangular = center rectangular?

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

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**(a) Rectangular lattice (𝑎≠𝑏; 𝛾= 90o)**

Maintain mirror symmetry m (𝑎≠𝑏; 𝛾= 90o)

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**(b) Center Rectangular lattice (𝑎≠𝑏; 𝛾= 90o)**

Maintain mirror symmetry m (𝑎≠𝑏; 𝛾= 90o) Rhombus cell (Primitive unit cell) 𝑎=𝑏; 𝛾≠ 90o

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**(c) Parallelogram lattice (𝑎≠𝑏; 𝛾≠ 90o)**

Destroy mirror symmetry

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**(d) Square lattice (𝑎=𝑏; 𝛾= 90o)**

b (e) hexagonal lattice (𝑎=𝑏; 𝛾= 120o)

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**3-2-3. 3-D lattice: 7 systems, 14 Bravais lattices**

Starting from parallelogram lattice (𝑎≠𝑏; 𝛾≠ 90o) (1)Triclinic system fold rotation (1) c b a (𝑎≠𝑏≠𝑐; 𝛼≠𝛽≠𝛾≠ 90o)

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**lattice center symmetry at lattice point as shown above which the molecule is isotropic ( 1 )**

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(2) Monoclinic system (𝑎≠𝑏≠𝑐; 𝛼=𝛽=90o≠𝛾) one diad axis (only one axis perpendicular to the drawing plane maintain 2-fold symmetry in a parallelogram lattice)

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**(1) Primitive monoclinic lattice (P cell)**

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 (𝑎≠𝑏≠𝑐; 𝛼=𝛽=𝛾 = 90o) 3 -diad axes

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**(1) Derived from rectangular lattice (𝑎≠𝑏; 𝛾= 90o)**

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) Body-centered orthorhombic (I- cell)

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**body-centered orthorhombic based centered orthorhombic**

rectangular body-centered orthorhombic based centered orthorhombic

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**(2) Derived from centered rectangular lattice (𝑎≠𝑏; 𝛾= 90o)**

(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; 𝛼=𝛽=𝛾 = 90o) c 𝛽 𝛼 b a 𝛾 One tetrad axis starting from square lattice (𝑎=𝑏; 𝛾 = 90o)

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**starting from square lattice (𝑎=𝑏; 𝛾 = 90o)**

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

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**(1) maintain 6-fold symmetry**

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

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

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(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

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

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**(c) body centered cubic**

a = b = c; = = = 109o

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**1. Primitive cubic (simple cubic) 2. Body centered cubic (BCC) **

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)

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

= T P

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