513.001 Molecular and Solid State Physics

Average electron energy in a metal

For a metal at a temperature of T = 0 K, the conduction electrons at the bottom of the band have an energy E = 0 and the conduction electrons with the highest energy have an energy E = E_f. Assuming that the density of states is given by D(E) = 3n√E/(2E_F^3/2), what is the average energy of the conduction electrons?

An expression for the average energy should be given in terms of the Fermi energy Ef, the electron density n, pi = 3.14159... and numerical constants. The functions sin, cos, tan, asin, acos, exp, ln, and sqrt can be used. For instance, 2*sin(h*m)*exp(Ef) is a valid (but wrong) answer. Be sure to include a * to indicate multiplication; write 2*h*m not 2hm to multiply 2 times h times m.

Hint: Not all of the constants n, Ef, and pi appear in the answer.