Chapter 5 - WebRing

CHAPTER 5. MAGNETIC SYSTEMS 251
E
NJ
–0.4
–0.8
–1.2
–1.6
–2.0
0.0 1.0 2.0 3.0 4.0
kT/J
5.0
(a)
C
Nk
4.0
3.0
2.0
1.0
0.0
0.0 1.0 2.0 3.0 4.0
kT/J
5.0
Figure 5.9: (a) Temperature dependence of the energy of the Ising model on the square lattice
accordingto(5.88). NotethatE(T)isacontinuousfunctionofkT/J. (b) Temperaturedependence
of the specific heat of the Ising model on the squarelattice accordingto (5.90). Note the divergence
of the specific heat at the critical temperature.
The heat capacity can be obtained by differentiating E(T) with respect to temperature. It
can be shown after some tedious algebra that
C(T) = Nk 4
(βJ coth2βJ)2 K1(κ)−E1(κ)
π
−(1−tanh 2
π
2βJ)
2 +(2tanh2
2βJ −1)K1(κ)
, (5.90)
where
E1(κ) =
π/2
0
(b)
dφ
1−κ 2sin 2 φ. (5.91)
E1 iscalledthecompleteellipticintegralofthesecondkind. AplotofC(T)isgiveninFigure5.9(b).
The behavior of C near Tc is given by
C ≈ −Nk 2
2J
2
ln
T
π kTc
1−
Tc
+constant (T near Tc). (5.92)
An important property of the Onsager solution is that the heat capacity diverges logarithmically
at T = Tc:
C(T) ∼ −ln|ǫ|, (5.93)
where the reduced temperature difference is given by
ǫ = (Tc −T)/Tc. (5.94)