Problems and Solutions on Thermodynamics and Statistical Mechanics

Statistid Physics 319

neighbors generate a ‘mean field’ at the site of each atom, where

Q is a parameter which characterizes the strength of the interaction.

(e) Use the mean field approximation together with the results of part

(d) to calculate the susceptibility x in the weak field (i.e., the high temper-

ature) limit. At what temperature, T,, does x become infinite?

(MITI

Solution

where q = gp~H/2kT.

(a) z = (,-gMBH/zkT + egl’BH/2kT)N = (2cosh v)N ,

(b) F = -IcTlnz, S = - (g) = Nk[ln(Zcoshv) -vtanhq].

H

When v >> 1,

When q << 1,

S M Nk(1 + 3~) exp(-Zv);

S M Nk In 2.

(c) During adiabatic demagnetization, the entropy of the system re-

mains constant, i.e., S1 = So. Thus vl = v~, i.e., TI = H,To/Ho. Hence

Ti < To.

aF = kT (Alnz)

T

X

NgPB tanh - SPB H

2 2kT

= M/H = - NgPB tanh

2H

9PB H

2kT *

In the weak field limit

so that

(e) From the definition given for the mean field, we have