a second-order transition
Phase transitions come in two temperaments. A first-order transition is abrupt — like ice melting: the two phases coexist, there is a latent heat, and a property jumps discontinuously. A second-order (or continuous) transition is gentle: there is no latent heat, no two phases coexisting at the transition, and the change grows in smoothly from zero rather than jumping. The order parameter eases away from zero continuously as you cross the critical temperature, so the crystal is never part-old-phase, part-new-phase — it is one phase that continuously distorts into another.
The name comes from Ehrenfest's classification: at a first-order transition the first derivatives of the free energy (entropy, volume) are discontinuous; at a second-order transition those first derivatives are continuous, but SECOND derivatives (heat capacity, thermal expansion, compressibility) jump or diverge. So there is no sudden absorption of heat, but the heat capacity spikes at Tc. The order parameter typically behaves like eta proportional to (Tc minus T)^(1/2) just below Tc — rising with a vertical initial slope, but from zero.
Structurally, continuous transitions require a group-subgroup relation between the two phases (the low-symmetry structure must be a symmetry subgroup of the high-symmetry one) — that is Landau's insight, and it is what allows the order parameter to grow smoothly from zero. Real examples include the beta-brass ordering transition, many displacive ferroelectric and ferroelastic transitions (often driven by a soft mode), and magnetic ordering. An honest note: many transitions that look continuous are actually weakly first-order on close inspection; truly continuous transitions are special.
The ordering of beta-brass is close to a second-order transition: as temperature falls below 454 degrees C the order parameter rises continuously from zero, with no latent heat and no two-phase coexistence. Its fingerprint is a lambda-shaped spike in the heat capacity at Tc — the first derivative of the free energy (entropy) is continuous, while the second derivative (heat capacity) jumps sharply there.
A second-order (continuous) transition: no latent heat, no coexisting phases, order parameter rising continuously from zero, with a heat-capacity spike at Tc.
A second-order transition has no latent heat and no two-phase coexistence, with the order parameter rising continuously from zero; the free energy's first derivatives (entropy, volume) stay continuous while second derivatives (heat capacity, expansion) jump or diverge — and a continuous transition requires a group-subgroup relation.