Collective electron specific heat and spin paramagnetism in metals

In two previous papers expressions have been obtained for the susceptibility and specific heat of free electrons in forms appropriate for low and high temperatures. With slight interpolation these expressions are adequate to give the temperature variation over the whole temperature range. The results are given in such a form that they are also applicable to electrons not strictly free, as long as the number of states per unit energy range in the unfilled energy band involved is proportional to the square root of the energy, as for free electrons. In this paper more general expressions are obtained which show how the specific heat and the spin paramagnetism, and their temperature variation, depend on the band form. It is only the low temperature range which will be considered, as usually for metals ordinary temperatures fall well within the “low” range. Although the equivalents of some of the results obtained have been given before, and it has been noticed that there is a relation between the electronic specific heat and the spin paramagnetism, the general question has not been treated in any detail. Expressions are first obtained for collective electrons neglecting interchange interaction effects, and given in a convenient form for application. The difficult question of the susceptibility arising from the “orbital” motion of collective electrons will not here be discussed. The conditions necessary for a very large orbital effect do not usually hold; and for the more strongly paramagnetic metals, which are discussed here, it will be sufficient to consider the spin effect alone.