Landau theory

Landau theory is a beautifully simple recipe for understanding a continuous structural transition without knowing the atomic details. Its one idea: near the transition the order parameter eta is small, so write the crystal's free energy as a power series in eta and see where it is lowest. Because eta is small you keep only the first few terms, and symmetry tells you which terms are even allowed. It is like describing a ball settling in a bowl by the shape of the bowl near the bottom, without tracking every atom.

For the commonest case symmetry forbids odd powers, so F(eta, T) = F0 + (1/2) a eta^2 + (1/4) b eta^4 + ..., with the key assumption that the quadratic coefficient changes sign at the transition: a = a0 (T minus Tc), while b > 0. Above Tc, a is positive and the minimum sits at eta = 0 (disordered phase). Below Tc, a turns negative, the eta = 0 point becomes a hilltop, and new minima appear at eta^2 = minus a/b = (a0/b)(Tc minus T). So eta proportional to (Tc minus T)^(1/2) — the order parameter grows continuously from zero, exactly the second-order behaviour, and the theory even predicts the jump in heat capacity at Tc.

The strength of Landau theory is that it is driven by SYMMETRY: the allowed terms in the expansion are fixed by the group-subgroup relation between parent and product, so it classifies which transitions can be continuous and how many domain variants appear. Its honest limitation is that it is a mean-field theory — it ignores fluctuations, so very close to Tc (the critical region) its exponents (like the 1/2) are not exactly right, and real critical behaviour needs the renormalisation group. Away from that narrow region, Landau theory is remarkably successful, and it is the backbone of how we think about ferroelectric, ferroelastic and ordering transitions.