Chapter 5 - WebRing

CHAPTER 5. MAGNETIC SYSTEMS 259
Equation (5.110) has two solutions:
and
m(T > Tc) = 0 (5.111a)
3
m(T < Tc) = ±
1/2
(Jq/kT) 3/2((Jq/kT)−1)1/2 . (5.111b)
The solution m = 0 in (5.111a) corresponds to the high temperature paramagnetic state. The
solution in (5.111b) corresponds to the low temperature ferromagnetic state (m = 0). How do we
know which solution to choose? The answer can be found by calculating the free energy for both
solutions and choosing the solution that gives the smaller free energy (see Problems 5.15 and 5.17).
If we let Jq = kTc in (5.111b), we can write the spontaneous magnetization (the nonzero
magnetization for T < Tc) as
m(T < Tc) = 3 1/2 T
Tc −T
Tc
Tc
1/2
. (5.112)
We see from (5.112) that m approaches zero as a power law as T approaches Tc from below. It is
convenient to express the temperature dependence of m near the critical temperature in terms of
the reduced temperature ǫ = (Tc −T)/Tc [see (5.94)] and write (5.112) as
m(T) ∼ ǫ β . (5.113)
From (5.112) we see that mean-field theory predicts that β = 1/2. Compare this prediction to the
value of β for the two-dimensional Ising model (see Table 5.1).
We now find the behavior of other important physical properties near Tc. The zero-field
isothermal susceptibility (per spin) is given by
∂m
χ = lim
H→0 ∂H =
1−tanh 2 Jqm/kT
kT −Jq(1−tanh 2 Jqm/kT) =
For T Tc we have m = 0 and χ in (5.114) reduces to
χ =
1
k(T −Tc)
1−m 2
kT −Jq(1−m 2 . (5.114)
)
(T > Tc, H = 0), (5.115)
where we have used the relation (5.108) with H = 0. The result (5.115) for χ is known as the
Curie-Weiss law.
For T Tc we have from (5.112) that m 2 ≈ 3(Tc −T)/Tc, 1−m 2 = (3T −2Tc)/Tc, and
1
χ ≈
k[T −Tc(1−m 2 )] =
1
k[T −3T +2Tc]
(5.116a)
1
=
2k(Tc −T)
(T Tc, H = 0). (5.116b)
We see that we can characterizethe divergenceof the zero-field susceptibility as the critical point is
approached from either the low or high temperature side by χ ∼ |ǫ| −γ [see (5.98)]. The mean-field
prediction for the critical exponent γ is γ = 1.