III Crystal Symmetry
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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. LH. RH. mirror. RH. LH. (2) Reflection (mirror) symmetry m. (3) Inversion symmetry (center of symmetry).

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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


LH

RH

mirror

RH

LH

(2) Reflection (mirror) symmetry m

(3) Inversion symmetry (center of symmetry)


(4) Rotation-Inversion axis

= Rotate by

,then invert.

(a) one fold rotation inversion (

(b) two fold rotation inversion (

= mirror symmetry (m)


(c) inversion triad ()

3

= Octahedral site in

an octahedron


(d) inversion tetrad ()

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!


Start with the translation

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’

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 cosn (= 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


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?


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


Maintain mirror symmetry m

  • (; 90o)


(b) Center Rectangular lattice (; 90o)

Maintain mirror symmetry m

  • (; 90o)

Rhombus cell

(Primitive unit cell)

; 90o


(c) Parallelogram lattice(; 90o)

Destroy mirror symmetry


b

a

(d) Square lattice (; 90o)

(e) hexagonal lattice(; 120o)


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)



(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


(3) Orthorhombic system which the molecule is isotropic (

c

b

a

(; )

3 -diad axes


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


  • (c) which the molecule is isotropic (body centered cubic

= 109o

a = b = c;

 =  =  =109o


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 = bc

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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