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- 3-1 Symmetry elements

(1) Rotation symmetry

- two fold (diad ) 2
- three fold (triad ) 3
- Four fold (tetrad ) 4
- Six fold (hexad ) 6

= Rotate by

，then invert.

(a) one fold rotation inversion (

(b) two fold rotation inversion (

= mirror symmetry (m)

x

= tetrahedral site in a

tetrahedron

(e) inversion hexad ()

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 ()

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

rotation symmetriesfor crystal with

translational symmetry.

graphically

The space cannot

be filled!

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!

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 cosn (= 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.

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.

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

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

(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?

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

(b) Center Rectangular lattice (; 90o)

Maintain mirror symmetry m

- (; 90o)

Rhombus cell

(Primitive unit cell)

; 90o

(c) Parallelogram lattice(; 90o)

Destroy mirror symmetry

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)

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

(2) Monoclinic system which the molecule is isotropic (

(; )

one diad axis

(only one axis perpendicular tothe drawing plane maintain 2-fold symmetry in a parallelogram lattice)

c which the molecule is isotropic (

b

a

c

b

(1) Primitive monoclinic lattice (P cell)

(2) Base centered monoclinic lattice

a

B-face centered monoclinic lattice

c which the molecule is isotropic (

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

(2) Body centered monoclinic lattice which the molecule is isotropic (

Body centered monoclinic = Base centered monoclinic

So monoclinic has two types which the molecule is isotropic (

1. Primitive monoclinic

2. Base centered monoclinic

c which the molecule is isotropic (

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

c which the molecule is isotropic (

b

a

(b) B- face centered orthorhombic

= A -face centered orthorhombic

(c) Body-centered orthorhombic (I- cell)

rectangular which the molecule is isotropic (

body-centered orthorhombic

based centered orthorhombic

c which the molecule is isotropic (

b

a

(2) Derived from centered rectangular lattice

- (; 90o)

(a) C-face centered Orthorhombic

C- face centered orthorhombic

= B- face centered orthorhombic

(b) Face-centered Orthorhombic (F-cell) which the molecule is isotropic (

Up & Down

Left & Right

Front & Back

Orthorhombic which the molecule is isotropic (has 4 types

1. Primitive orthorhombic

2. Base centered orthorhombic

3. Body centered orthorhombic

4. Face centered orthorhombic

(4) Tetragonal system which the molecule is isotropic (

(;

)

c

b

a

One tetrad axis

starting from square lattice

(; )

First layer which the molecule is isotropic (

Second layer

(; )

starting from square lattice

(1) maintain 4-fold symmetry

(a) Primitive tetragonal lattice

First layer which the molecule is isotropic (

Second layer

(b) Body-centered tetragonal lattice

Tetragonal has 2 types

1. Primitive tetragonal

2. Body centered tetragonal

(5) Hexagonal system which the molecule is isotropic (

a = b c;

= = 90o; = 120o

c

b

One hexad axis

a

starting from hexagonal lattice (2D)

a = b; = 120o

which the molecule is isotropic (

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

Hexagonal which the molecule is isotropic (has 1 types

1. Primitive hexagonal

Rhombohedral (trigonal)

2. Primitiverhombohedral (trigonal)

(6) Cubic system which the molecule is isotropic (

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) which the molecule is isotropic (Primitive cubic

= 90o

c

a = b = c;

= = =90o

b

a

- (b) Face centered cubic

= 60o

a = b = c;

= = =60o

cubic (isometric) which the molecule is isotropic (

Special case of orthorhombic with a = b = c

Primitive (P)

Body centered (I)

Face centered (F)

Base center (C)

Tetragonal (I)?

a = bc

Tetragonal (P)

Cubic has 3 types

1. Primitive cubic (simple cubic)

2. Body centered cubic (BCC)

3. Face centered cubic (FCC)

which the molecule is isotropic (

http://www.theory.nipne.ro/~dragos/Solid/Bravais_table.jpg

= P

= I

= T P

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