In this article we will discuss about:- 1. Introduction to Free Electron Theory 2. Electrical Conductivity as per Free Electron Theory 3. Limitation.
Introduction to Free Electron Theory:
According to Drude, the metals consists of positive ions cores with the valence electrons moving freely among these cores. The electrons are, therefore bound to move within the metal due to electrostatic attraction between the positive ion cores and the electrons.
The potential field of these ion cores, which is responsible for such an interaction, is assumed to be constant throughout the metal and the mutual repulsion among the electrons is neglected.
The behaviour of free electrons moving inside the metals is considered to be similar to that of atoms or molecules in perfect gas. These free electrons are, therefore, also referred to as free electron gas and the theory is accordingly named as free electron gas model. The free electron gas, however, differs from an ordinary gas in some respects.
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Firstly, the free electron gas is negatively charged whereas the molecules of an ordinary gas are mostly neutral. Secondly, the concentration of electrons in an electron gas is quite large as compared to the concentration of molecules in an ordinary gas. The valence electrons are also called the conduction electrons and obey the Pauli’s exclusion principle. These electrons are responsible for conduction of electricity through metals.
Since the conduction electrons move in a uniform electrostatic field of ion cores, their potential energy remains constant and is normally taken as zero. Thus the total energy of a conduction electron is equal to its kinetic energy. Also since the movement of conduction electrons is restricted to within the crystal only, the potential energy of a stationary electron inside a metal is less than the potential energy of an identical electron just outside it.
This energy difference, V0, serves as a potential barrier and stops the inner electrons from leaving the surface of the metal. Thus, in free electron gas model, the movement of a free electron gas inside a potential energy box which in one dimensional case is represented by a potential well as shown in Fig. 5.1.
In 1909, H.A. Lorentz postulated that the electrons constituting the electron gas obey Maxwell-Bultzmann statistics under equilibrium conditions. This idea along with Drude consideration leads to constitute the Drude-Lorentz theory. This theory is also known as classical theory.
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The free electrons in a metal move isotropically and move in a particular direction on an application of electric field. Because of elastic collisions, the magnitude of
steady state current is proportional to the voltage applied provided the temperature of metal remains constant. This lead to Ohms law.
Moreover, as the free electrons can move easily, the metals exhibit high electrical and thermal conductivities. The ratio of the electrical conductivity ‘σ’ to the thermal conductivity ‘K’ should be constant for all metals at a constant temperature i.e., σ/K = constant. This is called Weidemann-Franz law, which has been realised even in practice also.
This theory also explains the luster and opaque properties of metals. When an electromagnetic radiation falls on the surface of metal, the free electrons are set into forced oscillation. The electrons return to their normal states by emitting the same amount of energy in all directions, and hence produce metallic luster.
Electrical Conductivity as per Free Electron Theory:
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On the basis of free electron theory, the electrons move freely in solid, so it is interesting to see the effect of external electric field on these electrons. Let E be the applied electric field, m be the mass of the electron and e be the charge on the electron. The force F due to applied field will be-
F = eE
Also F = ma, where a is the acceleration.
∴ a = eE/m
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Because of collisions of electrons during motion, the electrons will not get accelerated indefinitely. In fact, their velocity will fall to zero. If τ be the relaxation time (collision time) then the average electronic velocity known as drift velocity is given by-
vd = aτ = (eE/m) τ … (i)
Let I be the current carried by a conductor on application of electric field E corresponding to drift velocity vd. In time dt, the electrons will travel a distance vd dt and the number of electrons crossing any cross-sectional area A in time dt will be contained in volume Avd dt. Thus, if there are n electrons per unit volume of the conductor, the total charge flowing through the section in time dt is-
dQ = enAvd dt
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or I = dQ/dt = en Avd
And current density J = I/A = en vd … (ii)
Using equation (i), we have,
J = en (eEτ/m) = ne2 τE/m … (iii)
For a particular material, the quantity ne2τ/m in (iii) is constant at particular temperature and is known as electrical conductivity ‘σ’ of the material.
∴ J = ne2τE/m = σE … (iv)
or σ = ne2τ/m … (v)
We know that the resistance R of a conductor is given by-
R = r l/A where r is resistivity of material and l is the length of conductor. Also r = 1/σ, therefore, from equations (i) and (iv), we have,
I = JA = σEA = EA/ᵨ = El/R
Also E = V/l
∴ I = Vl/Rl = V/R
which is nothing but Ohm’s law. That is why equation (iv) and (v) are also known as Ohm’s law.
Equation (v) can also be written as-
Where, µ (= eτ/m) is the mobility acquired by electrons in the presence of electric field. Using equation (i), the mobility of electrons can also be expressed as-
Therefore, the mobility of electron in the metal is defined as the steady state drift velocity per unit electric field.
Let λ be the mean free path and V’ be the root mean square velocity of electrons, then the relaxation time (commonly known as mean time between collisions) τ is given by-
Now the electrical conductivity ‘σ’ can be expressed as-
Since electrical resistivity r is the reciprocal of electrical conductivity ‘σ’.
or ρ ∝ √T
Above result shows that the resistivity varies as √T whereas actually it is found to vary linearly with temperature.
Limitations of Free Electron Theory:
Following limitations have been observed:
(a) It cannot explain why only some crystals are metallic in nature.
(b) It cannot explain why the metals prefer only certain structure e.g. Fe is cubic while Zn is hexagonal.
(c) It cannot explain why the observed specific heat of metals is only 1% of the calculated value (i.e., 3/2 N KB; N are number of free electrons per gram atom).
(d) It cannot explain the temperature variation of the electrical conductivity.
(e) It cannot explain the paramagnetic behaviour of metals.
(f) It also could not explain the occurrence of long mean free paths at low temperatures.