Classical Free Electron Theory (Drude Model) and Quantum Comparison

Posted on Nov 30, 2025 in Physics

Classical Free Electron Theory and Assumptions

The Classical Free Electron Theory (or Drude-Lorentz model) treats a metal as a container of free electrons (an “electron gas”) moving randomly within a fixed lattice of positive ions. When an external electric field is applied, these electrons experience a force and “drift” in the opposite direction, creating a current.

Assumptions:

Classical Mechanics:

The free electrons are treated as classical particles and obey Maxwell-Boltzmann statistics.

Free Electrons:

The valence electrons are “free” to move throughout the metal, while the positive ions are fixed in the lattice.

Negligible Interactions:

The repulsive forces between electrons and the attractive forces between electrons and ions are ignored, except during collisions.

Elastic Collisions:

Electrons collide only with the positive ion cores (not each other). These collisions are assumed to be instantaneous and elastic, and they reset the electron’s velocity randomly.

External Field:

Between collisions, electrons accelerate uniformly under the influence of the external electric field.

Definitions

I)

Drift Velocity (vd):

The average, net velocity gained by free electrons in a conductor when an electric field is applied. This slow, directional velocity is superimposed on their high-speed random thermal motion.

ii)

Mobility (μ):

A measure of how easily a charge carrier (like an electron) moves through a material under an electric field. It is defined as the drift velocity acquired per unit electric field: μ = vd / E.

iii)

Resistivity (ρ):

An intrinsic property of a material that measures its opposition to the flow of electric current. It is the inverse of conductivity: ρ = 1/σ.

iv)

Current Density (J):

A vector quantity representing the electric current (I) flowing per unit cross-sectional area (A). J = I/A.

V)

Electric Field (E):

A region of space around a charge where a force is exerted on other charges. In a conductor, it is the force per unit charge (E = F/q) that drives the flow of electrons.

vi)

Electrical Conductivity (σ):

An intrinsic property of a material that measures its ability to conduct electric current. It is the reciprocal of resistivity: σ = 1/ρ.

More Definitions

I)

Relaxation Time (τ):

The average time interval between two successive collisions of a free electron with the lattice ions.

ii)

Mean Free Path (λ):

The average distance an electron travels between two successive collisions.

iii)

Mean Collision Time:

This is just another name for Relaxation Time (τ).

iv)

Relation between relaxation time and mean collision time:

They are identical concepts, representing the same physical quantity.

4. Failures of Classical Free Electron Theory

• Heat Capacity:

  
  Classical theory predicts a large electronic specific heat (Cv = 3/2 R), suggesting electrons contribute significantly to the metal’s heat capacity.
  Experimentally, the electronic contribution is tiny (~1% of this value).

• Temperature Dependence of Resistivity:

  
  Classical theory predicts resistivity ρ ∝ √T. Experimental results show that ρ ∝ T at high temperatures.

• Paramagnetism:

  
  It fails to explain the weak, temperature-independent paramagnetism (Pauli paramagnetism) observed in metals.

• Quantum Phenomena:

  
  It cannot explain quantum effects like the Photoelectric effect, Compton effect, or the black-body spectrum.

5. Comparison of Assumptions: Classical vs. Quantum

6. Success of Quantum Free Electron Theory

The Quantum Free Electron Theory (QFET) successfully resolved the major failures of the classical model:

• Heat Capacity:

  
  By applying Fermi-Dirac statistics, QFET shows that only electrons very close to the Fermi Energy (EF) can be thermally excited. This small fraction of electrons correctly predicts the very low, experimentally observed electronic specific heat, which is proportional to T.

• Conductivity/Resistivity:

  
  QFET explains that resistivity arises from electrons scattering off lattice vibrations (phonons) and impurities, not every ion. This correctly predicts ρ ∝ T at high temperatures and a residual resistivity at 0K.

• Paramagnetism:

  
  QFET successfully explains Pauli paramagnetism as a result of the spin-flipping of electrons near the Fermi level.