a soft mode
Every crystal hums with atomic vibrations — phonons, standing waves of atoms rocking back and forth, each with its own frequency (stiffness). A soft mode is one particular vibration whose frequency falls toward zero as you approach a structural transition. As that mode softens, the restoring force fighting its displacement pattern weakens; at the transition the frequency reaches zero, the atoms no longer spring back, and the vibration FREEZES IN — the momentary distortion becomes the new, permanent structure.
Here is the beautiful part: the frozen-in displacement pattern of the soft mode IS the atomic shift that takes the high-symmetry parent into the low-symmetry daughter. So a displacive transition can be read as a phonon going soft. In many perovskite ferroelectrics — barium titanate is the classic — a particular optical phonon (the Ti moving against the oxygen cage) softens as the crystal cools toward the Curie point near 120 degrees C; when it freezes, titanium sits permanently off-centre, giving the polar, ferroelectric tetragonal phase. In SrTiO3 a zone-boundary mode softens near 105 K and freezes into rotated oxygen octahedra. The mode frequency squared typically drops linearly toward the transition: omega^2 proportional to (T minus Tc).
The soft mode is the dynamical face of the order parameter: the order parameter is essentially the amplitude of the frozen soft-mode displacement, and its softening is why Landau's quadratic coefficient a passes through zero (a proportional to omega^2). Soft modes are measured directly by inelastic neutron or light (Raman) scattering, and they are the microscopic engine of displacive ferroelectric and ferroelastic transitions. Caveat: not every displacive transition is cleanly soft-mode driven — some are order-disorder in character or overdamped — but the soft-mode picture is the cleanest and most powerful.
As barium titanate (BaTiO3) cools toward its Curie point near 120 degrees C, a transverse optical phonon — titanium vibrating against the oxygen octahedron — softens toward zero frequency. When it freezes, the titanium atom stops permanently off-centre, and the cubic paraelectric phase becomes the polar tetragonal ferroelectric phase: one phonon going soft creates ferroelectricity.
A soft mode: a vibration whose frequency drops to zero at the transition; its frozen displacement pattern is exactly what turns parent into product (as in BaTiO3).
The frozen displacement pattern of a soft mode is exactly the shift that turns the high-symmetry parent into the low-symmetry daughter; the order parameter is essentially the frozen mode's amplitude, and its softening drives Landau's coefficient a through zero (a proportional to omega^2). Not every displacive transition is cleanly soft-mode driven.