Problems and Solutions on Thermodynamics and Statistical Mechanics

Statistical Physics 32 1

The degeneracy of the quantized energy levels is given by

where the integral limits A1 represents 2p~B1 < (p2 +p”,)/Zm < 2/1~gB(Z+
l), and A2 represents 2/10 Bl < p2/2m < 2/10 B(1-t 1) with p~g = eA/2mc.
(b) In== xln(l+eBcL.e-@‘)

where X = exp(Pp).

In the high temperature limit, X << 1, hence

where XT = h/JM and x = pnB/kT.

where F is the free energy of the system.

Hence

M=- XV^1 x cosh x

XV x

BY

we have M = -xp~L(x), where L(x) = cothx - 1/x.

At high temperatures, kT >> /~BB or x << 1. Therefore,

11

3 45

L(z) = -x - -x3 +... ,

M M -xpi B/3kT ,

M

VB

xa, = ~ = -E/1;/3kT ,