Page 238 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics
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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 ÿ ÿ (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.