Problems and Solutions on Thermodynamics and Statistical Mechanics

222 Problems d Solutiona on Thermodynamics d Statistical Mechanice

a(u) - -hT- - - Nk( 1 - z2csch2z2).

aT

(b) c = - -

dT

2056
Consider a gas of spin 1/2 atoms with density n atoms per unit volume.
Each atom has intrinsic magnetic moment p and the interaction between
atoms is negligible.
Assume that the system obeys classical statistics.
(a) What is the probability of finding an atom with p parallel to the
applied magnetic field H at absolute temperature T? With p anti-parallel
to H?

(b) Find the mean magnetization of the gas in both the high and low

(c) Determine the magnetic susceptibility x in terms of p.

temperature limits?

(SUNY, Bufulo)

Solution:
(a) The interaction energy between an atom and the external magnetic
field is E = -p. H. By classical Boltzmann distribution, the number of
atoms per unit volume in the solid angle element dfl in the direction (0, p),
is
gexp(-PE)dfl= gexp(pHcosO/kT)dfl ,

where 0 is the angle between p and H and g is the normalization factor
given by

i.e.,

2?rg ln e-@'sin 6dO = n ,

nCLH
PH *
9=
47rkT sinh -
kT
Hence the probability density for the magnetic moment of an atom to be
parallel to H is

and that for the magnetic moment to be antiparallel to H is