# III Crystal Symmetry 3-1 Symmetry elements (1) Rotation symmetry

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III Crystal Symmetry 3-1 Symmetry elements (1) Rotation symmetry

(2) Reflection (mirror) symmetry m
RH LH (3) Inversion symmetry (center of symmetry) 1 LH RH

(4) Rotation-Inversion axis
= 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
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!

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!

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.

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.

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

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?

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

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

Symmetry elements in 2D lattice
Rectangular = center rectangular?

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

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

(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

(c) Parallelogram lattice (𝑎≠𝑏; 𝛾≠ 90o)
Destroy mirror symmetry

(d) Square lattice (𝑎=𝑏; 𝛾= 90o)
b (e) hexagonal lattice (𝑎=𝑏; 𝛾= 120o)

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)

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

(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)

(1) Primitive monoclinic lattice (P cell)
b c (2) Base centered monoclinic lattice c b a 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 a c b A-face centered monoclinic lattice If relabeling lattice coordination 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
a (𝑎≠𝑏≠𝑐; 𝛼=𝛽=𝛾 = 90o) 3 -diad axes

(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

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

body-centered orthorhombic  based centered orthorhombic
rectangular body-centered orthorhombic  based centered orthorhombic

(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

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

starting from square lattice (𝑎=𝑏; 𝛾 = 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 a = b  c;  =  = 90o;  = 120o c b One hexad axis a 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 a b c 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 = 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

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

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

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)

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