Chapter 37, Problem 63P

(a) Count the number of electron states N(E) with energy equal to or less than E in Equation 35.8 by finding the volume available to such states in the space with Cartesian coordinate axes n_x, n_y, n_z. (Hint: Consider each set of positive integers, at the corner of a unit cube, and that lies inside a radius $\sqrt{n_x^2+n_y^2+n_z^2}$, and remember that there are two spin values per state.) (b) Differentiate N(E) with respect to E to obtain Equation 37.5.