Landau theory is a phenomenological theory used by
mineralogists to describe the
order-disorder phase transitions of a system. It is based on writing the "free energy" F of the system as a power series of an
order parameter Q which measures how ordered the system is. A system acts so as to minimise its free energy, so by minimising the approximation for F we can estimate properties of the system.
In the simplest version of Landau theory, the equation used for F is:
F = A(T-Tc)Q2 + BQ4.
Here A and B are positive constants, T is the temperature, and Tc is the critical temperature at which the system starts to become ordered.
- Why does the equation only have even powers of Q? Because this model applies to the type of ordering where states with order parameter Q and -Q are obviously equivalent. E.g., if we're arranging atoms of types A and B on a square lattice, here are two perfectly ordered states. One has Q=1 and the other has Q=-1, but they are mirror images of each other.
A B A B B A B A
B A B A A B A B
A B A B B A B A
B A B A A B A B
Q = 1 Q = -1
There are other types of order parameter, some of which need to be expressed as a vector rather than just one number, but these need different equations. You can also have more than one order parameter, or include terms in Q6 and beyond.
- Why does T only appear where it does in the equation? Because free energy is defined as E - TS (E is internal energy, S is entropy). Internal energy and entropy are determined by things like the positions of the atoms (which we are rolling up into the single number Q) rather than temperature. We leave out the temperature-dependent part of the Q4 term because as long as we are only covering a narrow range of temperature, the coefficient of Q4 won't change very much.
To find out the order parameter at a given temperature, we minimise F with respect to Q. Above Tc, F is minimised at Q=0, but below Tc, it is minimised at Q = sqrt(A(T-Tc)/2B). Other properties like the heat capacity can also be estimated from this model.