Problems and Solutions on Thermodynamics and Statistical Mechanics

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