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

mirror

RHLH

LH

RH

(2) Reflection (mirror) symmetry m

(3) Inversion symmetry (center of symmetry)

= Rotate by ， then invert. 360o

𝑛(a) one fold rotation inversion (

(b) two fold rotation inversion ( = mirror symmetry (m)

(4) Rotation-Inversion axis

(c) inversion triad ()

3

= Octahedral site in an octahedron

(d) inversion tetrad ()

= tetrahedral site in a tetrahedron

x

(e) inversion hexad ()

Hexagonal close-packed (hcp) lattice

x

3-2-1. 1-D lattice3 types of symmetry can be arranged in a 1-D lattice

(1) mirror symmetry (m)(2) 2-fold rotation (2)(3) center of symmetry ()

3-2. Fourteen Bravais lattice structures

Proof: There are only 1, 2, 3, 4, and 6 foldrotation symmetries for crystal withtranslational symmetry.

The space cannotbe filled!

graphically

Start with the translation T

Add a rotation A

A

lattice point

lattice point

latticepoint

A T

TT

T: scalar

Tp

T

A translation vector connecting twolattice points! It must be some integerof or we contradicted the basicAssumption of our construction.

T

p: integer

Therefore, is not arbitrary! The basic constrain has to be met!

T

T T

tcos tcos

b

To be consistent with theoriginal translation t:

pTb p must be integer cos21cos2 ppTTTb

Mp 1cos2 M must be integer

p M cos n (= 2/) b-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

M >2 orM<-2:no solution

T

T T

A A’

B’B

43210-1

Allowable rotationalsymmetries are 1, 2,3, 4 and 6.

Look at the case of p = 2

= 120o

TTp

2

1T

2T

21 TT

o21 120 TT

angle

Look at the case of p = 1

n = 3; 3-fold

n = 4; 4-fold

= 90o

TTp

21 TT

o21 90 TT

1T

2T

3-fold lattice.

4-fold lattice.

Look at the case of p = 0

= 60o

TTp

0

1T

2T 21 TT

o21 60 TT

n = 6; 6-fold

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

TTp

3

Look at the case of p = -1 n = 1; 1-fold

1 2

TTp

1

Exactly the same as 3-fold lattice.

1-fold2-fold3-fold4-fold6-fold

Parallelogram21 TT

general21 TT

Hexagonal Net21 TT

o21 201 TT

Can accommodate1- and 2-foldrotational symmetries

Can accommodate3- and 6-foldrotational symmetriesSquare Net

21 TT

o21 90 TT

Can accommodate4-fold rotationalSymmetry!

These are the lattices obtained by combiningrotation and translation symmetries?

How about combining mirror and translationSymmetries?

Combine mirror line with translation:

m

Unless

0.5T

centered rectangular

constrain

Or21 TT

o21 09 TT

Primitive cell

Rectangular

1T

2T

m

Parallelogram (Oblique)21 TT

general21 TT

Hexagonal21 TT

o21 201 TT

Square21 TT

o21 90 TT

(1)

(2)

(3)

21 TT

o21 09 TT

Double cell (2 lattice points)

Centered rectangular(4)

21 TT

o21 09 TT

Primitive cell

Rectangular(5)

5 lattices in 2D

Symmetry elements in 2D lattice

Rectangular = center rectangular?

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

3-2-2. 2-D lattice

m 2

(a) Rectangular lattice (; 90o)

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

(d) Square lattice (; 90o)

ab

(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)

  b

a

c

(; 90o)

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

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

(2) Monoclinic system

(; )

(1) Primitive monoclinic lattice (P cell)

ab

c

(2) Base centered monoclinic lattice

c

ba

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

bA-face centeredmonoclinic lattice

If relabeling lattice coordination

A-face centered monoclinic = B-face centered

b

a b

a

(2) Body centered monoclinic lattice

Body centered monoclinic = Base centered monoclinic

So monoclinic has two types1. Primitive monoclinic 2. Base centered monoclinic

(3) Orthorhombic system

a b

c

(; )

3 -diad axes

(1) Derived from rectangular lattice(; 90o)

to maintain 2 fold symmetryThe second layer superposes directly on the first layer

(a) Primitive orthorhombic lattice

ab

c

(b) B- face centered orthorhombic = A -face centered orthorhombic

c

a b

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

rectangular

body-centered orthorhombic based centered orthorhombic

(2) Derived from centered rectangular lattice

(; 90o)

(a) C-face centered Orthorhombic

abc

C- face centered orthorhombic= B- face centered orthorhombic

(b) Face-centered Orthorhombic (F-cell)

Up & Down

Left & RightFront & Back

Orthorhombic has 4 types1. Primitive orthorhombic 2. Base centered orthorhombic3. Body centered orthorhombic4. Face centered orthorhombic

(4) Tetragonal system

ab

c𝛼𝛾𝛽

(; )

One tetrad axis

starting from square lattice

(; )

starting from square lattice (; )

(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 types1. Primitive tetragonal 2. Body centered tetragonal

(5) Hexagonal system

a = b c; = = 90o; = 120o

b

a

c

One hexad axis

starting from hexagonal lattice (2D)

a = b; = 120o

(1) maintain 6-fold symmetry

Primitive hexagonal lattice

b

a

c

(2) maintain 3-fold symmetry

ab

c

1/32/3

1/3

2/3

a = b = c; = = 90o

Hexagonal has 1 types1. Primitive hexagonal

Rhombohedral (trigonal)2. Primitive rhombohedral (trigonal)

(6) Cubic system

4 triad axes ( triad axis = cube diagonal )

Cubic is a special form of Rhombohedral latticeCubic system has 4 triad axes mutually inclined along cube diagonal

ba

c

a = b = c;

= = = 90o

(a) Primitive cubic = 90o

ba

c

a = b = c;

= = = 90o

(b) Face centered cubic = 60o

a = b = c; = = = 60o

(c) body centered cubic = 109o

a = b = c; = = = 109o

cubic (isometric)

Primitive (P)Body centered (I)Face centered (F)Base center (C)

Special case of orthorhombic with a = b = c

a = b cTetragonal (P)

Tetragonal (I)?

Cubic has 3 types1. 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