Chapter 5 - WebRing

CHAPTER 5. MAGNETIC SYSTEMS 263
Because the fluctuations become more important as the system approaches the critical point,
we expect that mean-field theory breaks down for T close to Tc. A criterion for the range of
temperatures for which mean-field theory theory is applicable is discussed in Section 9.1, page 435,
where it is shown that the fluctuations can be ignored if
ξ −d
0 |ǫ|(d/2)−2 ≪ 1, (5.128)
where ξ0 is the correlation length at T = 0 and is proportional to the effective range of interaction.
The inequality in (5.128) is always satisfied for d > 4 near the critical point where ǫ ≪ 1. That is,
mean-fieldtheoryyieldsthecorrectresultsforthecriticalexponentsinhigherthanfourdimensions.
(In four dimensions the power law behavior is modified by logarithmic factors.) In a conventional
superconductor such as tin, ξ0 ≈ 2300˚A, and the mean-field theory of a superconductor (known
as BCS theory) is applicable near the superconducting transition for ǫ as small as 10 −14 .
5.7.1 *Phase diagram of the Ising model
Nature exhibits two qualitatively different kinds of phase transitions. The more familiar kind,
which we observe when ice freezes or water boils, involves a discontinuous change in various thermodynamic
quantities such as the energy and the entropy. For example, the density as well as the
energy and the entropy change discontinuously when water boils and when ice freezes. This type
of phase transition is called a discontinuous or first-order transition. We will discuss first-order
transitions in the context of gases and liquids in Chapter 7.
The other type of phase transition is more subtle. In this case thermodynamic quantities such
as the energy and the entropy are continuous, but various derivatives such as the specific heat and
the compressibility of a fluid and the susceptibility of a magnetic system show divergent behavior
at the phase transition. Such transitions are called continuous phase transitions.
We have seen that the Ising model in two dimensions has a continuous phase transition in
zero magnetic field such that below the critical temperature Tc there is a nonzero spontaneous
magnetization, and above Tc the mean magnetization vanishes as shown by the solid curve in
the phase diagram in Figure 5.14. The three-dimensional Ising model has the same qualitative
behavior, but the values of the critical temperature and the critical exponents are different.
The behavior of the Ising model is qualitatively different if we apply an external magnetic
field H. If H = 0, the magnetization m is nonzero at all temperatures and has the same sign as H
(see Figure 5.15). The same information is shown in a different way in Figure 5.14. Each point in
the unshaded region corresponds to an equilibrium value of m for a particular value of T and H. 10
At a phase transition at least one thermodynamic quantity diverges or has a discontinuity.
For example, both the specific heat and the susceptibility diverge at Tc for a ferromagnetic phase
10 In a ferromagnetic material such as iron, nickel, and cobalt, the net magnetization frequently vanishes even
below Tc. In these materials there are several magnetic domains within which the magnetization is nonzero. These
domains are usually oriented at random, leading to zero net magnetization for the entire sample. Such a state is
located in the shaded region of Figure 5.14. The randomization of the orientation of the domains occurs when the
metal is formed and cooled below Tc and is facilitated by crystal defects. When a piece of iron or similar material
is subject to an external magnetic field, the domains align, and the iron becomes “magnetized.” When the field is
removed the iron remains magnetized. If the iron is subject to external forces such as being banged with a hammer,
the domains can be randomized again, and the iron loses its net magnetization. The Ising model is an example of
a single-domain ferromagnet.