Classical Free Electron Theory of Metals - Civil Engineering (CE) PDF Download

Classical free electron theory of metals
In order to explain electrical conductivity in metals, Lorentz and Drude put forward a theory called free electron theory of metals. It is based on the following assumptions.

1. Free electrons in a metal resemble molecules of a gas. Therefore, Laws of kinetic theory of gasses are applicable to free electrons also. Thus free electrons can be assigned with “average velocity   “Mean free path λ” and “mean collision time τ ”.
2. The motion of an electron is completely random. In the absence of electric field, number of electrons crossing any cross section of a conductor in one direction is equal to number of electrons crossing the same cross section in opposite direction. Therefore net electric current is Zero.
3. The random motion of the electron is due to thermal energy. Average kinetic energy of the electron is given by

Average Velocity, k=Boltzman Constant, T=Absolute Temperature, m=mass of the electron.
4. Electric current in the conductor is due to the drift velocity acquired by the electrons in the presence of the applied electric field.
5. Electric field produced by lattice ions is assumed to be uniform throughout the solid.
6. The force of repulsion between the electrons and force of attraction between electrons and lattice ions is neglected.

The Drift Velocity
In the absence of the applied electric field, motion of free electron is completely random. During their motion electrons undergo collisions with the residual ions and during each collision direction and magnitude of their velocity changes in general. When electric field is applied, electrons experiences force in the direction opposite to the applied field. Therefore in addition to their random velocity, electron acquires velocity in the direction of the force. Since electrons continue to move in their random direction, with only a drift motion due to applied field, velocity acquired by the electrons in the direction opposite to the applied field is called Drift velocity and is denoted by v_d. Note the v_d is very small compared to   the average thermal velocity. Electric current in a conductor is primarily due to the drift velocity of the electrons.

Relaxation Time, Mean collision time and Mean free path
 Mean free path (λ):
The average distance traveled by electrons between two successive collisions during their random motion is called mean free path, it denoted by λ.

Mean collision Time τ :
The average time taken by an electrons between two successive collisions during their random motion is called mean collision time, it is denoted by τ . The relationship between λ and τ is given by λ =  τ .

Relaxation Time τr:
In the presence of an applied electric field, electrons acquire drift velocity v_d in addition to the thermal velocity  . if electric field is switched off, v_d reduces and becomes zero after some time. Let electric field is switched off at the instant t = 0, when drift velocity v_d=v₀. The drift velocity of the electron after the lapse of t seconds is given by

where τr is called relaxation time. Suppose t=τr, then v_d=v₀ e ⁻¹ = 1/e v₀

Thus the relaxation time is defined as the time during which drift velocity reduces to 1/e times its maximum value after the electric field is switched off. 
Relation between Relaxation time and Mean collision time:
The relationship between relaxation time and mean collision time is given by

where θ is scattering angle,   average value of cos θ taken over very large number of collision made by electrons.
It can be shown that, τ = τ_r for metals.

Ohms Law :
Consider a conductor of length l and area of cross section A. Let ρ be the electrical conductivity of the conductor, then resistance of the conductor is given by

Let a potential difference of V volts be applied between two ends of the conductor, which causes current I to flow through the conductor. Then, according to Ohms Law.

V = IR

where J = I/A is called current density

V/l = J/σ
If “E” is the electric field established inside the conductor, then

The above equation also represents Ohm’s law.

Expression for electrical conductivity
Consider a conductor of length l, area of cross section A, having n number of free electron per unit volume. Let a potential difference of V volt be applied between two ends of the conductor and E be the electric field established inside the conductor. Then according to Ohms law

J = σE

where σ is the electrical conductivity of the conductor and J is a current density given by

J = I/A

σ = J/E

Consider a cross section ‘X ‘of the conductor. Let I be the current flowing through the conductor. Then according to the definition, current I is given by the quantity of charge flowing across the cross section X per second.

I = Ne

where N is the number of electrons crossing the cross section X in one second. We know that, if v_d is drift velocity of the electrons. Electrons travel a distance equal to v_d in one second. Therefore number of electrons crossing the cross section X in one second is equal to number of electrons occupying the space between two cross sections X and Y separated by the distance v_d.

where n=number of electrons per unit volume between the cross sections X and Y
I = nAv_de

But J = I/A = nev_d

We know that free electrons undergo collisions with positive ions during their random motion. Let the drift velocity acquired by the electron becomes zero during each collision and let v_d be the drift velocity acquired by the electron just before next collision then
v_d = 0 + aτ

Where a is the acceleration acquired by the electron in the presence of electric field E and τ is the average time taken by the electron between two successive collisions. Force acting on the electron in the presence of electric field is given by

from free electron theory

Equation (3.4) gives expression for electric conductivity.
Note that, electrical conductivity of a conductor σ is proportional to
1. Number of electrons per unit volume and
2. Mean free path for free electrons
3. Inversely proportional to square root of the absolute temperature.

Mobility of Electrons:
The easiness with which electrons get drifted in the presence of applied electric field is called “Mobility”. It is defined as the drift velocity acquired by the electron per unit electric field. It is denoted by µ