2. Quantum free electron theory is a
microscopic theory developed by
Sommerfeld in 1928. According to
this theory, the free electrons move
with a constant potential obeys
quantum laws (Fermi-Dirac statistics).
3. Zone theory was developed by
Bloch in 1928.
3. Sommerfield‘s Quantum Theory
In 1928, Sommerfield suggested that electron gas
should be treated quantum mechanically in the
light of Pauli's exclusion principle which states
that no two particles can have all the quantum
numbers identical. He used Fermi-Dirac statistics
rather than Maxwell-Boltzmann statistics and
succeeded in explaining many experimental
facts. We first find the possible energy states in
one-dimensional and then extend the idea to
three-dimensional case.
4. QUANTUM FREE ELECTRON THEORY OF
METALS
The free electrons move in a constant potential inside
the metal
and are confined within defined boundaries.
• The eigen values of the conduction electron are
quantized.
• The electrons are considered to posses wave nature.
• In the various allowed energy levels, distribution of
electrons takes
place according to Pauli’s exclusion principle.
• Mutual attraction between electrons and lattice ions
and the
repulsion between individual electrons may be
ignored.
5. FERMI–DIRAC DISTRIBUTION FUNCTION
The Fermi-Dirac distribution applies to
fermions, particles with half-integer spin
which must obey the Pauli exclusion
principle. Each type of distribution
function has a normalization term
multiplying the exponential in the
denominator which may be temperature
dependent.
6. Fermi energy level and Fermi energy:
Electrons on this energy level are also called
Fermions and obey Pauli's exclusion principle.
Each energy level can accommodate at most
two electrons.
The highest occupied energy level by the
electron at zero kelvin is called Fermi energy
level.
At zero kelvin energy levels below the Fermi
level are completely filled and above Fermi
level all the energy levels are empty.
If the temperature increases above zero
kelvin, the probability occupation of
electrons at that level is half.
That particular energy of that level is called
Fermi energy.
The value of Fermi energy in metal is 5eV.
7. Specific heat of solid:
At zero kelvin energy levels under Fermi energy
level are filled by electrons. When the
temperature is increased to room temperature
few number lo electrons conduct heat or
electricity.
If we consider one kilo mole of metal
When the temperature of the solid is
increased, the atoms are set into simple
harmonic vibrations about their mean
positions with a frequency v, which is
characteristic of a solid and is called
Einstein's frequency
8. Dependence of electrical conductivity on the
concentration of electron
This theory also fails to explain the direct relation
between the electrical conductivity and the electron
density.
. The concepts of electrical conductivity and thermal
conductivity are explained by this theory.
. The phenomenon of Ferro magnetism, Para
magnetism and susceptibility are explained by this
theory.
. The concept of photoelectric effect, Compton
effect and black body radiation are explained by this
theory.
9. Scattering of electron and electrical resistivity
The main factors effecting electrical
resistivity of solid are
1. Temperature.
2. Impurities.
In general the resistivity change is
represented by
This scattering results in the loss of the
additional energy and momentum that the
electron may have gained from an applied
electric field and hence impedes the
electron response to the applied field, i.e.,
provides for metallic resistivity.
10. Assumptions
The energies of free electrons are quantized.
The distribution of electrons is as per the Pauli's
exclusion principle.
Electrons travel under constant potential and
confine to the boundaries of metal.
All the attractive and repulsive forces are
neglected.
Valence electrons move freely in a
constant potential within the boundaries
of metal and is prevented from escaping
the metal at the boundaries (high
potential). Hence the electron is trapped in
a potential well.
11. conclusion
According to quantum free electron theory, the
free electrons always collides with the positive
ions or electrons present in the metal. The
scattering of conduction electrons are due to 1.
Effect of temperature 2. Defect, e.g. impurities,
imperfections, etc.
12. Anyone who has never made a mistake has never
tried anything new.”
― Albert Einstein
