- The Fermi-Dirac distribution function for the free electrons
in a metal cannot be approximated by the Maxwell-Boltzmann
function at STP (see Exercise 49) for energies in the
neighborhood of kT. Verify this by using the method of
Exercise 49 to show that A 1 in copper if f() AekT.
As calculated in Sec. 9.9 NV8.48 1028 electrons/m^3 for
copper. Note that Eq. (9.55) must be used unchanged here.
9.11 Dying Stars- The sun has a mass of 2.0 1030 kg and a radius of 7.0
108 m. Assume it consists of completely ionized hydrogen at a
temperature of 10^7 K. (a) Find the Fermi energies of the proton
gas and of the electron gas in the sun. (b) Compare these ener-
gies with kTto see whether each gas is degenerate (kTF,
so that few particles have energies over F) or nondegenerate
(kT F, so that few particles have energies below Fand the
gas behaves classically).
53. Consider a white dwarf star whose mass is half that of the sun
and whose radius is 0.01 that of the sun. Assume it consists of
completely ionized carbon atoms (mass 12 u), so that there are
six electrons per nucleus, and its interior temperature is 10^7 K.
(a) Find the Fermi energies of the carbon nucleus gas and of the
electron gas. (b) Compare these energies with kTto see whether
each gas is degenerate or nondegenerate, as in Exercise 52.
54. The gravitational potential energy of a uniform-density
sphere of mass Mand radius Ris Eg^35 GM^2 /R. Consider
a white dwarf star that contains Nelectrons whose Fermi
energy is F. Since kTF, the average electron energy is,
from Eq.(9.51), about ^35 Fand the total electron energy is
Ee^35 NF. The energies of the nuclei can be neglected
compared with Ee. Hence the total energy of the star is E
EgEe.(a) Find the equilibrium radius of the star by letting
dEdR0 and solving for R. (b) Evaluate Rfor a star whose
mass is half that of the sun and consists of completely
ionized carbon atoms, as in Exercise 53.
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