Problems and Solutions on Thermodynamics and Statistical Mechanics

Statistical Physic8 223

(b) The average magnetization of the gas at temperature T is

At high temperatures, PH - << 1. Let PH - = x, and expand
kT kT

xx

sox w -H. w2
3kT
At low temperatures, x >> 1, then

1

coth x - - w 1

X

and a w np.

(c) The magnetic susceptibility of the system is

• np2/3kT, at high temperature
  at low temperature.

There is spontaneous magnetization in the limit of low temperatures.

2057
A material consists of n independent particles and is in a weak external
magnetic field H. Each particle can have a magnetic moment rnp along
the magnetic field, where rn = J, J - 1,... , -J + 1, - J, J being an integer,
and p is a constant. The system is at temperature T.
(a) Find the partition function for this system.
(b) Calculate the average magnetization, a, of the material.
(c) For large values of T find an asymptotic expression for M.
(MITI