BMS Institute of Technology and Management, Bangalore

Department of Physics

Study Material for Case study and topic of Innovative Question of III internal test

Date: - 06.12.2019

Case study topic:- Quantum Free electron theory of Metals

Conduction in metals must follow Ohm's Law, which states that the current is directly proportional to the electric field applied to the metal. The law, named after German physicist Georg Ohm, appeared in 1827 in a published paper laying out how current and voltage are measured via electrical circuits. The key variable in applying Ohm's Law is a metal's resistivity. Resistivity is the opposite of electrical conductivity, evaluating how strongly a metal opposes the flow of electric current. This is commonly measured across the opposite faces of a one- meter cube of material and described as an ohm meter (Ω⋅m). Resistivity is often represented by the Greek letter rho (ρ). Electrical conductivity, on the other hand, is commonly measured by siemens per meter (S⋅m−1) and represented by the Greek letter sigma (σ). One siemens is equal to the reciprocal of one ohm.

Conductivity of Metals

Material Conductivity σ(S/m) at 20°C Silver 6.30x107 Copper 5.98x107 Annealed Copper 5.80x107 Gold 4.52x107 Aluminum 3.5x107 Magnesium 2.15x107 Zinc 1.682x107 Cobalt 1.60x107 Cadmium 1.467 Nickel (electrolytic) 1.46x107 Iron 1.04x107 Quantum free electron theory of metals:  Quantum free electron theory was proposed by Arnold Sommerfeld.  As per his investigation, the behavior of free electron in different possible energy states is explained, how the large number of electrons are disturbed in the energy states.

Assumptions  The energies of free electrons are quantized.  The distribution of electrons is as per the Pauli's exclusion principle.  Electrons travel under constant potential and confine to the boundaries of metal.  All the attractive and repulsive forces are neglected.  lectrons on this energy level are also called Fermions and obey Pauli's exclusion principle.  Each energy level can accommodate at most two electrons.

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 The highest occupied energy level by the electron at zero kelvin is called Fermi energy level.  At zero kelvin energy levels below the Fermi level are completely filled and above Fermi level all the energy levels are empty.  If the temperature increases above zero kelvin, the probability occupation of electrons at that level is half.  That particular energy of that level is called Fermi energy.  The value of Fermi energy in metal is 5eV.

Temperature dependence of electrical conductivity: We know that the electrical conductivity is given by

We know the relaxation time can be written as from the above equation we can write,

Substituting this equation in electrical conductivity expression we get,

In this equation there is no direct relation between the electrical conductivity and the temperature but we know and

So, from both the above equations we get,

Dependence of electrical conductivity on the concentration of electron: As per quantum theory the electrical conductivity is given by,

from this equation we have, 휆 and 휎 훼 푣퐹 It is clear that copper and aluminium the value of n for aluminium is 2.13 times higher than that of copper. But the value of λ/vf for copper is about 3.73 times higher than that of aluminium. Thus the conductivity of copper exceeds that of aluminium.

Innovative question topic:- Laser and its Military applications.

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