# Free Electron Theory

Classical free electron theory of metals

This theory was developed by Drude and Lorentz and hence is also known as Drude-Lorentz theory. According to this theory, a metal consists of electrons which are free to move about in the crystal like molecules of a gas in a container. Mutual repulsion between electrons is ignored and hence potential energy is taken as zero. Therefore the total energy of the electron is equal to its kinetic energy.

Drift velocity

If no electric field is applied on a conductor, the free electrons move in random directions. They collide with each other and also with the positive ions. Since the motion is completely random, average velocity in any direction is zero. If a constant electric field is established inside a conductor, the electrons experience a force F = -eE due to which they move in the direction opposite to direction of the field. These electrons undergo frequent collisions with positive ions. In each such collision, direction of motion of electrons undergoes random changes. As a result, in addition to the random motion, the electrons are subjected to a very slow directional motion. This motion is called drift and the average velocity of this motion is called drift velocity vd. Consider a conductor subjected to an electric field E in the x-direction. The force on the electron due to the electric field = -eE. By Newton’s law, -eE = mdvd/dt

dvd = -eEdt/m

Integrating,

Vd = -eEt/m + Constant

When t = 0, vd = 0 Therefore Constant = 0

Vd = -eEt/m --------------- (1)

Electrical conductivity

Consider a wire of length ‘dl’ and area of cross section ‘A’ subjected to an electric field E. If ‘n’ is the concentration of the electrons, the number of electrons flowing through the wire in dt seconds = nAvddt. The quantity of charge flowing in time dt = nAvddt.e

Therefore I = dq/dt = neAvd

Current density J = I/A = nevd

Subsittuting the value of vd from (1),

J = nee Et/m = ne2Et/m --------------- (2)

By Ohm’s law, J = s E

Therefore s = J/E = ne2t/m -------------- (3)

Mobility of a charge carrier is the ratio of the drift mobility to the electric field. µ = vd/E m2/Volt-Sec

Substituting vd from (1),

µ = et/m -------------- (4)

Substituting this in equation (3),

s = neµ ------------- (5)

Relaxation time and mean free path

When the field E is switched off, due to the collision of the electrons with lattice ions and lattice defects, their velocity will start to decrease. This process is called relaxation. The relaxation time(t) is the time required for the drift velocity to reduce to 1/e of its initial value. The average distance traveled by an electron between two consecutive collisions is called mean free path (l) of the electron. l = vdt -------------- (6)

Temperature dependence

The free electron theory is based on Maxwell-Boltzmann statistics. Therefore Kinetic energy of electron = ½ mvd2 = 3/2 KBT

Vd = Ö 3KBT/m

Substituting this in equation (6),

t = lÖ m/3KBT -------------- (7)

Since s = ne2t/m, s is proportional to Ö1/T

Or r is proportional to ÖT.

Wiedmann-Franz law

The ratio of thermal conductivity to electrical conductivity of a metal is directly proportional to absolute temperature.

K/s is proportional to T

Or, K/sT = L, a constant called Lorentz number.

L = 3KB2/2e2

Drawbacks of Classical free electron theory

1) According to this theory, r is proportional to ÖT. But experimentally it was found that r is proportional to T. 2) According to this theory, K/sT = L, a constant (Wiedmann-Franz law) for all temperatures. But this is not true at low temperatures. 3) The theoretically predicted value of specific heat of a metal does not agree with the experimentally obtained value. 4) This theory fails to explain ferromagnetism, superconductivity, photoelectric effect, Compton effect and blackbody radiation.

Quantum free electron theory

Classical free electron theory could not explain many physical properties. In 1928, Sommerfeld...

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