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Chapter 43
Molecules and Solids (Cont.)
Dr. Jie Zou
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Outline
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Free electron theory of metals
Band theory of solids
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Free-electron theory of metals
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Classical free-electron theory of electrical
conduction in metals:
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Model: Treats a metal as an electron gas and uses
the kinetic theory of gases.
Predicts Ohm’s law
Difficulties: Does not predict the correct values of
electrical and thermal conductivities.
Quantum-based free-electron theory of
metals:

Model: The outer-shell electrons are free to move
through the metal but are trapped within a threedimensional box formed by the metal surfaces.
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Fermi-Dirac distribution
function
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Quantum statistics: Required by the
Pauli exclusion principle that each
state of the system can be occupied by
only two electrons (one with spin-up
and the other with spin down).
The probability that a particular state
having energy E is occupied by one of
the electrons in a solid is given by
1
f ( E )  ( E  E F ) / k BT
e
1
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Function f (E) is called the Fermi-Dirac
distribution function. EF is called the
Fermi energy.
Quick quiz: Physical meaning of the two
plots on the left.
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Electron in a three-dimensional
box

For a particle in a one-dimensional box of length L,
the allowed value of energy is
 2 2 2
En 

2
2mL
n
n  1,2,3,...
For one electron in a solid cube of sides L and
volume L3, the energy for such an electron is (see
Problem 30)
 2 2 2
2
2
En 

2
2me L
(nx  n y  nz ) (nx , n y , nz ) are integers  1
For example, the ground state, nx = ny = nz =1 and E =
3ħ22/2meL2, can be occupied by two electrons
corresponding to spin-up and spin-down (ms = +1/2
and ms = -1/2).
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Density-of-states function

The number of allowed
states per unit volume that
have energies between E
and E + dE is
8 2me
g ( E )dE 
h3
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3/ 2
E1/ 2 dE
Function g(E) is called the
density-of-states function.
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Quick quiz

The Fermi energy for silver is 5.48 eV.
Near which of these energies are the
energy levels closer together?
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(a) 2 eV. (b) 6 eV. (c) The spacing of
energy levels is the same near both
energies.
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Electron distribution function
versus energy
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Set N(E)dE = The number of
electrons per unit volume that
have energy between E and E +
dE, where
N ( E )dE  f ( E ) g ( E )dE
8 2me
E 1/ 2 dE

h3
e ( E  E F ) / k BT  1
N(E) is called the electron
distribution function.
3/ 2

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Problem 35
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(a) Consider a system of electrons confined
to a three-dimensional box. Find the ratio of
the number of allowed energy levels at 8.50
eV to the number at 7.00 eV.
(b) Copper has a Fermi energy of 7.0 eV at
300 K. Calculate the ratio of the number of
occupied levels at an energy of 8.50 eV to
the number at the Fermi energy.
(c) Compare answers for (a) and (b).
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Electron concentration

Set ne = The total number of electrons per
unit volume. ne is called the electron
concentration.

8 2me
E 1/ 2 dE
ne   N ( E )dE 
0 e( E  EF ) / kBT  1
0
h3
Find Fermi energy at T = 0 K from
3/ 2

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8 2me
ne 
h3
2 8 2me
1/ 2
E
dE

3
0
3
h
2/3
2
3
n
h  e
E
(
0
)



Solve for F
2me  8 
3/ 2
EF
3/ 2
EF
3/ 2
See Table 43.4 for the values of ne for different
metals.
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Band theory of solids
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Energy bands of a sodium
crystal
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Blue represents energy bands
occupied by the sodium electrons.
Gold represents energy bands that
are empty.
Energy gaps or forbidden
energies (white regions)
between the allowed bands;
electrons cannot occupy states
that lie in these gaps.
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Homework

Chapter 43, P. 1436, Problems: #33,
35.
Dr. Jie Zou
PHY 1371
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