Problems and Solutions on Thermodynamics and Statistical Mechanics

330 Problems E/ Sdutiom on Thermodynamic8 d Stati~ticd Mechanic8

Hence H = -~JNAA - 2(-PJ + p0Ho)N~ + (-PJ/2 + poHo)N.

(a) The partition function is

z= c exp(g)

all states

The free energy is F = -kTlnz.

(b) Using the mean-field approximation, the ratio of the number of
the nearest-neighbor pairs with spins upward to the total number of pairs
equals the probability that the spins are all upward in the nearest-neighbor
sites, i.e.,
-=(+) ~NAA ,

2

PN

Thus,

The partition function is then

Defining the magnetization by

we have

1 H1 l+m

• In z CJ -- - -(I + m) In -
  N NkT 2 2
  1 1-m


• -(I-m)In-.
  2 2

For a In z/am = 0, we obtain

p&~ PJ^1 l+m

kT kT 2 1-m

• -m - - In - = 0 ,