Statistical Phyaica 221
Solution:
(a) The partition function is
z = exp(x) + exp(-x) ,
where x = pgH/kT.
(b) The total magnetic moment is
a
aH
M = po(N+ - N-) = NkT- Inz
= Npo tanh(x).
(c) The entropy of the system is
S = Nk(ln z - pa In ./a/?)
= Nk(ln 2 + In(cosh z)) - z tanh(z).
2055
A paramagnetic system consists of N magnetic dipoles. Each dipole
carries a magnetic moment p which can be treated classically. If the system
at a finite temperature T is in a uniform magnetic field H, find
(a) the induced magnetization in the system, and
(b) the heat capacity at constant H.
(UC, Berkeley)
Solution:
(a) The mean magnetic moment for a dipole is
Jpcos6exp(xcos6)dfl
s exp (x cos 6) dn
p s: cos 6 exp(z cos 6) sin 6d6
s: exp (z cos 0) sin 6d6
(’) =
=pcoth~-- ,
[ :I
where x = pH/kT. Then the induced magnetization in the system is
(M) = N(p) = Np