Thermodynamic Functions in the Fermi Statistics

Let the value of 0 at the absolute zero of temperature be c0o. We know how to find it from Sec. 3, simply by counting up N levels from the lowest level. First we shall try to find how co depends on temperature. We shall assume that the number of energy levels between e and e + de is 

The term —1 in the first integral takes care of the summation at the absolute zero, where the Fermi function is unity for energies less than coo, 

In the first integral, through the very small range from e0o — o to e0 — 0o, we can replace the integrand by its value when u = 0, or . Thus the first term becomes (dN/de)0( 0 — e0o). To reduce the second integral 

Equation (4.5) represents e0 by the first two terms of a power series in the temperature, and the approximations we have made give the term in T2 correctly, though we should have to be more careful to get higher terms. We see, as we should expect from the last section, that if (d2N/de2)o = 0, so that the distribution of energy levels is uniform at o, 0 will be independent of temperature to the approximation we are using. 

let us find the internal energy in the same sort of way. Written as a summation, it is 

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