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8.6 Bose±Einstein condensation                 229

                        where k B is Boltzmann's constant, and typically ranges from 18 000 K to
                        90 000 K for metals. At temperatures up to the melting temperature, we have
                        the relationship
                                                       k B T   E F
                        Thus, even at temperatures well above absolute zero, the electrons are
                        essentially all in the lowest possible energy states. As a result, the electronic
                        heat capacity at constant volume, which equals dE tot =dT, is small at ordinary
                        temperatures and approaches zero at low temperatures.
                          The free-electron gas exerts a pressure on the walls of the in®nite potential
                        well in which it is contained. If the volume v of the gas is increased slightly by
                                                                     in equation (8.56) decrease
                        an amount dv, then the energy levels E n x ,n y ,n z
                        slightly and consequently the Fermi energy E F in equation (8.60) and the total
                        energy E tot in (8.62) also decrease. The change in total energy of the gas is
                        equal to the work ÿP dv done on the gas by the surroundings, where P is the
                        pressure of the gas. Thus, we have

                                               dE tot   3N dE F   2NE F   2E tot
                                        P ˆÿ        ˆÿ          ˆ       ˆ                 (8:68)
                                                dv       5 dv       5v     3v
                        where equations (8.60) and (8.62) have been used. For a typical metal, the
                                                    6
                        pressure P is of the order of 10 atm.



                                            8.6 Bose±Einstein condensation
                        The behavior of a system of identical bosons is in sharp contrast to that for
                        fermions. At low temperatures, non-interacting fermions of spin s ®ll the
                        single-particle states with the lowest energies, 2s ‡ 1 particles in each state.
                        Non-interacting bosons, on the other hand, have no restrictions on the number
                        of particles that can occupy any given single-particle state. Therefore, at
                        extremely low temperatures, all of the bosons drop into the ground single-
                        particle state. This phenomenon is known as Bose±Einstein condensation.
                          Although A. Einstein predicted this type of behavior in 1924, only recently
                        has Bose±Einstein condensation for weakly interacting bosons been observed
                                                  2
                        experimentally. In one study, a cloud of rubidium-87 atoms was cooled to a
                        temperature of 170 3 10 ÿ9  K (170 nK), at which some of the atoms began to
                        condense into the single-particle ground state. The condensation continued as
                        the temperature was lowered to 20 nK, ®nally giving about 2000 atoms in the
                                                                                        3
                        ground state. In other studies, small gaseous samples of sodium atoms and of
                        2  M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell (1995) Science 269, 198.
                        3  K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle
                         (1995) Phys. Rev. Lett. 75, 3969.
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