polymer compositeband theory-1

Band Theory • This is a quantum-mechanical treatment of bonding in solids, especially metals.. • The spacing between energy levels is so minute in metals that the levels essentially mer

Band Theory

• This is a quantum-mechanical treatment of bonding in

solids, especially metals

• The spacing between energy levels is so minute in metals

that the levels essentially merge into a band.

• When the band is occupied by valence electrons, it is

called a valence band.

• A partially filled or low lying empty band of energy levels,

which is required for electrical conductivity, is a

conduction band.

• Band theory provides a good explanation of metallic luster

and metallic colors

<Ref> 1 “The Electronic Structure and Chemistry of Solids” by P.A Cox

2 “Chemical Bonding in Solids” by J.K Burdett

Magnesium metal

Constructive Interference for bonding orbital

The electron density is given by

ρ(x) = Ψ*(x) Ψ(x) =|Ψ(x)| 2

k k

k

E

ψ ψ

8

9

Lewis StructureHetero-nuclear Diatomic Molecule

Band Overlap in Magnesium

Valence bandConduction band

16

The π-Molecular Orbitals of Benzene

π-M.O of benzene

21

Linear Conjugated Double Bonds

E

π-M.O

Bonding

bonding

Anti-One-dimensional chain with n π -orbitals, jth level

Ej = α + 2 β cosj π /(n+1) , j =1, 2, 3 …

24

Elementary Band Theory

If Ψ(x) is the wave function along the chain

Periodic boundary condition:

The wavefunction repeats after N lattice spacings

This can be achieved only if Ψ(x+ a) = µ Ψ(x) (4)

µ is a complex number µ* µ = 1 (5)

Through n number of lattice space Ψ(x+ na) = µn Ψ(x) (6)

Through N number of lattice space Ψ(x+ Na) = µN Ψ(x) (7)

Since Ψ(x+ Na) = Ψ(x), µN = 1 (8)

⇒ µ = exp(2πip/ N) = cos(2πp/ N) + i sin(2πp/ N) (9)

Where, i = √-1, and p is an integer or quantum number

Define another quantum number k (Wave number or Wave vector)

Ψ(x+ a) = µ Ψ(x) = exp(ika) Ψ(x) (12)

Free electron wave like Ψ(x)= exp(ikx) = cos(kx) + i sin(kx) (13)

can satisfy above requirement

A more general form of wave function

Bloch function Ψ(x) = exp(ikx) µ(x) (14)

and, µ(x+a) = µ(x) a periodic function, unaltered by

moving from one atom to anothere.g atomic orbitals

⇒The periodic arrangement of atoms forces the wave functions of

e- to satisfy the Bloch function equation

E

? = 8

? = 2a

Wave vector (Wave number) k = 2π/λ

1 Determining the wavelength of a crystal orbital

2 In a free electron theory, k α momentum of e- ? conductivity

3 -π/a = k = +π/ a often called the First Brillouin Zone

Anti-bonding between all nearby atoms

nodenode

χn (x) : atomic orbital of atom n

C n : coefficient C n = exp(ikx) = exp(ikna)

x ) exp( ) ( )

( χ Bloch sums of atomic orbitals

From eq (10), k = 2πp/(N a) for quantum number p of repeating

unit N

Consider a value k’, corresponding to a number of p + N

k’ = 2π(p + N)/(N a) = k + 2π/a

C n ’ = exp{i(k + 2π/a )na}= exp(ikna)? exp(i2πn) = C n

A range of 2π/a contains N allowed values of k

However, Since k can be negative, usually let -π/a = k = +π/ a

(15)

(16)

(17)

Bloch function Ψk = Σn e-ikna Xn

where X n atomic wavefunction

k value

Index of translation between 0 – π/a

or, 0 – 0.5 a* (a* = 2π/a)

1-D Periodic

X 0 X 1 X 2 X 3 X 4 X 5 X 6

a

k k

k

E

ψ ψ

*

)(

exp

ψ ψ

σ-bond

1st Brillouin zone

DOS(E)dE

= # of levels between E and E + dE

39

k = 0 → 0.5a*

a

π/aπ/2a

π/3a

λ2a

∞4a

6a

∞

∞2a2a

center

r of orbital

C

C

jr r

n

r

r jr j

π π

ψ

π- bond

The evolution of the π energy levels of an infinite

one-dimensional chain (-CH-)n.

n k

b k ikna a A b B

1

) ( )

( )

n k

a k ikna b A a B

1

) ( )

( )

χ(A) = s- orbital, χ(B) = σ p- orbital

n k

n k

Ψ(0) = Σn e0 X n = X 0 + X 1 + X 2 + X 3 + X 4 + X 5 + X 6 + …

No effective overlap between orbitals ⇒ non-bonding

Effective overlap between orbitals ⇒ bonding

Ψ (π/a) = Σn e-inπ X n = X 0 - X 1 + X 2 - X 3 + X 4 - X 5 + X 6 - …

B bandE

χ(A) = s- orbital, χ(B) = σ p- orbital

n k

n k

Ψ(0) = Σn e0 X n = X 0 + X 1 + X 2 + X 3 + X 4 + X 5 + X 6 + …

Ψ (π/a) = Σn e-inπ X n = X 0 - X 1 + X 2 - X 3 + X 4 - X 5 + X 6 - …

Antibonding between neighbor orbitals

No effective overlap between orbitals ⇒ non-bonding

49

Nearly-free electron model

1st Brillouin zone

Energy gap is produced due to

periodic potential

52

Schematic showing the method of generating the band structure of the solid.

A comparison of the change in the energy levels and energy bands

associated with (a) the Jahn-Teller distortion of cyclobutadiene

and (b) the Peierls distortion of polyacetylene

Effect of Distortion

chain

56

σ bond