Specific heat Fermi-Dirac statistics

A major achievement of the free-electron model was to show why the contributions of the free electrons to the heat capacity and magnetic susceptibility of a metal are so small. According to Boltzmann statistics, the contribution to the former should be nkB per unit volume. According to Fermi-Dirac statistics, on the other hand, only a fraction of order kBT/ F of the electrons acquire any extra energy at temperature T, and these have extra energy of order kBT. Thus the specific heat is of order nfcBT/ F, and an evaluation of the constant gives... 

T law the result is a straight line, but this does not, in contrast to what is expected from Debye s theory, go through the origin. Hence, in this case, a linear contribution in T is included in the specific heat. This is attributable to the translation of the electron gas in the box and can be calculated from the box energy (Eq. (2.25)) by means of Fermi-Dirac statistics. Since metals are not at the focus of our interests we will omit any further discussion here . ... 

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