3.38 Sspd_chapter 1_part 15_quantum fluids_superconductivity &  (Page 17/6)

SSPD_Chapter 1_Part 15_ SUPER-FLUIDITY AND SUPERCONDUCTIVITY-A SUBCLASS OF QUANTUM FLUIDS.

[ Electrical Engineering Materials by A. J. dekkar, Publisher PHI ]

[The New Physics, edited by Pau; Davies, ‘Low Temperature Physics, superconductivity and superfluidity’ by Anthony Leggett, pp 268, Cambridge University Press, 1992 .]

We saw in Chapter 1_Part 3_Section 1.3.1 that below degeneracy temperature quantum mechanical effects are markedly evident. For solids and liquids, the atoms wave nature is exhibited below the temperature

T _degeneracy = h ² /(3kM×((m _p +m _n )/2)×(a×10 ^-10 m) ² )

= 633.3/(M×a ² ) Kelvin

where M is the mass number of the atom and ‘a’ is the distance between the scatterer in Angstrom. In order to observe super fluid and super conductors we must go to low enough temperatures to obtain quantum liquid. The following two conditions have to be satisfied to obtain quantum liquids:

1. cold enough to be treated as degenerate systems

that is the liquid should be below degeneracy temperature;

(ii) it should be a bosonic system.

An assembly of odd number of fermions is fermionic and an assembly of even number of fermions is bosonic.

In classical mechanics particles follow Maxwell-Boltzman statistics . In a large number of gas particles system of volume V and gas molecules number N, each particle has a velocity v and energy E =Kinetic Energy + Potential Energy = [(1/2)mv ² + 0] .Here we assume that Potential Energy is zero. Maxwell- Boltzman distribution tells that number of particles at energy E is n(E) where ;

n(E) = K× Exp[-E/kT] = K× Exp[-mv ² /(2kT)] ………………………………1.158

In a quantum mechanical system kinetic energy will be quantized just as the energy of electron in a potential well is quantized corresponding to the standing waves of matter wave of different quantum numbers. But here the well corresponds to the large volume hence quantum difference between the discrete energy states is miniscule therefore discrete energy states E _i can be treated as a continuous variable while plotting the graph. In this graph plot, n _i is the number per quantum state whereas in classical systems n is the number per unit energy .

Therefore n _i (E _i ) = K× Exp[-E _i /kT] …………………………………………………………… 1.159

In quantum mechanical systems, we have density of permissible quantum states N(E) per unit energy.

Hence n _i (E _i ) = N(E _i )P(E _i )dE = number of particles lying between E _i and (E _i + dE)

Where N(E _i ).d(E) = number of permissible quantum states between E _i and (E _i + dE)

And P(E _i ) = probability of occupancy at E _i = Exp[-E _i /kT] according to Maxwell- Boltzman Statistics ;

Therefore Equation(1.159) can be written as :

n _i (E _i ) = N(E _i )dE × Exp[-E _i /kT] ………………………………………………… 1.160

Plot of Equation (1.159) is given in Figure (1.82). The plot of Equation(1.160) will require N(E _i )dE to be accounted for . N(E _i )dE is not a constant but a function of E _i

Figure 1.82. The average number of particles per state at a given temperature for particles obeying Classical Statistics.