Problems and Solutions on Thermodynamics and Statistical Mechanics

246 Problems tY Solutions on Thermodpamics €4 Statistical Mechanics

90%. This gives an upper limit to the weak interaction cross section of
o < 2.6 x 10-42E, cm2/electron. With E, - kT we obtain T < 10' K.

(b) The energy density of the neutrino gas is p, M aT:, and that of
the photon gas is p., = aT4. As p, 5 1O-'p, we have T, 5 T/101.5. For
T N 3 K, we get T, 5 0.1 K.

(c) At the early age of the universe (when kTkrn,c2) neutrinos
and other substances such as photons are in thermal equilibrium with
T, = T7,p,, xa p., and both have energy distributions similar to that of
black body radiation. Afterwards, the neutrino gas expands freely with the
universe and its energy density has functional dependence p,(v/T), where

the frequency v a -, the temperature T a -, R being the "radius" of the

universe. Hence the neutrino energies always follw the black body spec-

trum, just like the photons. However, because of the formation of photons

by the annihilation of electron-position pairs, p., > p,, and the temperature

of the photon gas is slightly higher than that of the neutrino gas. As the

photon temperature at present is^3 K, we expect T, <^3 K.

1 1

R R

2078

Imagine the universe to be a spherical cavity, with a radius of lo2' cm

and impenetrable walls.

(a) If the temperature inside the cavity is 3K, estimate the total num-

ber of photons in the universe, and the energy content in these photons.

(b) If the temperature were 0 K, and the universe contained 10'' elec-

trons in a Fermi distribution, calculate the Fermi momentum of the elec-

trons.

( Columbia)

Solution:

(a) The number of photons in the angular frequency range from w to

w + dw is