Second Order Transitions

Thermodynamic properties of systems near a second order phase transition are of great interest because one observes great simplicity and universality in their behaviour. A ferromagnetic system is an example of one that displays a second-order phase transition and will be used below for detailed discussion. At low temperature, an appreciable fraction of the atomic spins (the "atomic magnets") in some metals (e.g. iron) become spontaneously polarized in the same direction, thus giving rise to a net measurable magnetic field. However as the temperature is raised, the spins become randomized due to thermal agitation and the magnetization is lost. The critical temperature at which the change occurs is called in this system the Curie point. (For iron this is $1044K$). The spatial symmetries of the system in the two phases are different. In the high-temperature phase the system is ‘disordered’, with no net magnetization but with complete rotational symmetry (isotropy). At low temperature, the system becomes ‘ordered’ and the net magnetization defines a preferred direction in space. The material thus becomes anisotropic at low temperature, breaking rotational symmetry. The low-temperature ordered phase is therefore less symmetrical and to describe it fully one needs to introduce an extra parameter called the order parameter, which in this case is just the magnetisation vector $\vec{M}$. For simplicity, we will work below with a single component $M$, called the scalar magnetization (you can think of this as the magnetization in the $z$-direction of a planar ferromagnetic system). When $M=0$ one is in the disordered high-temperature phase while for $M \neq 0$, one is in the low-temperature ordered phase with net magnetization. (A more familiar example is the ice to liquid water phase transition. The molecules in liquid water are in a disordered state and the system is homogeneous and isotropic. As water is cooled it freezes into ice whose properties are no longer isotropic as the molecules are now arranged in a regular lattice which defines fixed directions. That is, the solid state is more ordered but has less symmetry than the liquid state. Unfortunately this liquid-solid transition is an example of a first order phase transition, while our interest below is in continuous or second-order phase transitions. The liquid-vapour transition on the other hand does have a second-order phase transition at the critical point as we will discuss briefly later.) When one discusses ferromagnetic systems, often it is in the presence of some external magnetic field $\cal{H}$, which actually defines the direction of $M$ from one of the many otherwise equivalent possibilities. The susceptibility $\chi$ measures the change of the system to a change in the external field and is given by

$$\chi \equiv {1 \over V} {\partial M \over \partial \cal{H}}.$$ (4.28)

Note: Just as the pressure $P$ and volume $V$ were thermodynamic parameters for a gas, the relevant parameters for a ferromagnetic system are $\cal{H}$ and $M$. In fact $\cal{H}$ plays a role analogous to the intensive parameter $P$ and $-M$ (note the minus sign) plays a role similar to the extensive parameter $V$. For example the work done by the ferromagnetic system in an infinitesimal change is given by

$$dW = - {\cal{H}} dM .$$ (4.29)

Although one can do so more rigorously, simply using the above analogy allows us to obtain the relation

$$M = - \left({\partial G \over \partial \cal{H} }\right)_{T}$$ (4.30)

Where $G$ is now the Gibbs free energy of the ferromagnetic system with $\cal{H}$ and $M$ the relevant parameters. Another quantity of interest is the heat capacity given by

$$C_{\cal{H}} = \left({ \partial Q \over \partial T}\right)_{... ... \left( { \partial^2 G \over \partial T^2} \right)_{\cal{H}}$$ (4.31)