Now we need to know how the electrons are distributed among the various quantum states at any given temperature. The Maxwell-Boltzmann distribution states that the average number of particles in a state of energy E is proportional to e-EltT (page 1250). However, it wouldn’t be right to use the Maxwell-Boltzmann distribution, for two very important reasons. The first reason is the exclusion principle. At absolute zero the Maxwell-Boltzmann function predicts that all the electrons would go into the two ground states of the system, with n. = ny = n, = 1 and m, = ± t. But the exclusion principle allows only one electron in each state. At absolute zero the electrons can fill up the
lowest available states, but there’s not enough room for all of them to go into the lowest states. Thus a reasonable guess as to the shape of the distribution would be Fig. 44-21. At absolute zero temperature the states are filled up to some value Bro, and all states above this value are empty.
The second reason we can’t use the Maxwell-Boltzmann distribution is more subtle. That distribution assumes that we are dealing with distinguishable particles. It might seem that we could put a tag on each electron and know which is which. But overlapping electrons in a system such as a metal are indistinguishable. Suppose we have two electrons; a state in which the first is in an energy level EI and the second is in level E2 is not distinguishable from a state in which the two electrons are reversed, because we can’t tell which electron is which. The statistical distribution function that emerges from the exclusion principle and the indistinguishably requirement is called (after its inventors) the Ferml-Dirac distribution. Because of the exclusion principle, the probability that a particular state with energy E is occupied by an electron is the same as feE), the fraction of states with that eriergy that are occupied:
Example 44-9 shows that the average energies and Fermi speeds of free electrons in metals are generally determined almost entirely by the exclusion principle; even at ordinary temperature the behavior is very far from what the equipartition principle predicts. We could make a similar analysis to determine the electronic contributions to heat capacities in a solid. IT there is one conduction electron per atom, the equipartition principle would predict an additional molar heat capacity at constant volume of 3Rf2 from electron kinetic energies. But when kT is much smaller than ~ the usual situation in metals. then only the few electrons near the Fermi level can find empty states and change energies appreciably when the temperature changes. The number of such electrons is
proportional to kTIEf” so we should expect the electron heat capacity at constant volume to be proportional to the product (kTIEr»(3Rf2) = (3kT/2EF)R. A more detailed analysis shows that in fact the electron contribution to the molar heat capacity at constant volume ofa metal is not far from our prediction. We invite you to verify that if T = 290 K (kT = 1/40 eV) and if EF = 7.0 eV, then Cv = O.OlSR, which is only 1.2% of the 3R12 prediction of the equipartition principle. Fermi energies for metals typically fall in the range from 1.6 to 14 eV. The Fermi temperature TFis defined as TF= EF/k. Fermi temperatures for metals are typically in the range of I.S to 16 x 10· K, and typical Fermi speeds are O.S to 2.2 X 106 mls.