Problems and Solutions on Thermodynamics and Statistical Mechanics

224 Problem3 d Solutiom on thermodynamic^ d Statistical Mechanics

Solution:
(a) The partition function is

J

= empHIkT = sinh

m=- J

(b) The average magnetization is

• M=-(g)T=NkT(&lni)
  T
  = 2 [(2J + 1) coth (2.7 + 1)- - coth
  2 [ 2kT 2kT

(c) When kT >> pH, using

cothxa?(l+%), 2 for x<1

we get

M - w -NJ(J^1 + 1)- P2 H

3 kT

2058

Two dipoles, with dipole moments MI and M2, are held apart at a

separation R, only the orientations of the moments being free. They are in

thermal equilibrium with the environment at temperature T. Compute the

mean force F between the dipoles for the high temperature limit __

kT R3

1. The system is to be treated classically.
   Remark: The potential energy between two dipoles is:

MlM2 <<

(3(M1 .R)(M2 .R) - (MI .M2)R2)

4= R5

(Princeton)

Solution:

Taking the z-axis along the line connecting'M1 and M2, we hatre

[2 cos 81 cos e2 - sin 81 sin 82 cos(pl - p2)].

4=- Ml M2

R3