What are the assumptions of "quantum free electron theory" !!!!

In 1929, Somerfield stated to apply quantum mechanics to explain conductivity phenomenon in metal. He has improved the Drude - Lorentz theory by quantizing the free electron energy and retained the classical concept of free motion of electron at a random.

ASSUMPTIONS:-

· The electrons are free to move within the metal like gaseous molecules. They are confined to the metal due to surface potential.

· The velocity distribution of the free electrons is described by Fermi-Dirac Statistics because electrons are spin half particles.

· The free electrons would go into the different energy levels by following Pauli’s exclusion Principle which states that no two electrons have same set of Quantum numbers.

· The motion of electrons is associated with a complex wave called matter wave, according to De-Broglie hypothesis.

· The electrons cannot have all energies but will have discrete energies according to the equation, E = n² h² / 8ma².

Drawbacks:

Conductivity: According to Quantum free electron theory, the conductivity of a metal is

σ = μne, here ‘μ’ is the mobility of electrons, ‘n’ is the free electron concentration and ‘e’ is the electron charge.

According to the above equation, polyvalent metals like Aluminum (Al) should be more conductive than monovalent metals like copper (Cu). But experimentally it is not so.

Hall coefficient: According to the free electron theory, the hall coefficients for all metals is negative where as there are certain metals like Be, Cd, Zn for which the Hall coefficient is + ve. Free electron theory could not explain why certain substances behave as insulators and some other substances as semiconductors; in spite of they have free electrons in them.

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