Advanced Solid State Physics

14.5.2 First Order Phase Transitions

First order phase transitions are very complicated to deal with, because the order parameter isn’t
necessarily small in the closer neighbourhood ofTCand even shows a discontinuity atTC. There are
many different ways to look at first order phase transitions and we want to take a closer look at the
free energyfas before.

f=f 0 +α 0 (T−TC)m^2 +

1

2

βm^4 +

1

3

γm^6 +... (261)

Figure 145: The free energyfas a function of the order parametermfor first order transitions

Here the first constantα 0 is positive,β is negative andγkeeps the free energy from diverging at
extreme values forT. Fig. 145 shows a plot of the free energy as a function of the order paramter.
AtT > TC, there is only one minimum atm= 0but atT=TC, there are three minima at different
values form. By cooling the material even further, the central minimum slowly vanishes while the
outer minima become deeper. Because of that, processes like supercooling (lowering the temperature
of a liquid or gas below its freezing point without becoming a solid) or superheating (heating a liquid
above its boiling point without boiling) are possible, because the material is trapped in a metastable
state, e.g. the local minimum atm= 0of the free energy for some time.

Fig. 146 shows the polarization ofBaTiO 3 as a function of the temperature, which is a typical example
for a first order phase transitions.

We now minimize the free energy as in the last section and get the following solutions form:

m= 0,±

√

−β±

√

β^2 − 4 α 0 (T−TC)γ

2 γ

so far small values of the order parameter, the free energy is approximately:

f≈f 0 +α 0 (T−TC)

−β±

√

β^2 − 4 α 0 (T−TC)γ

2 γ

(262)