Symmetry Breaking, Landau Theory, and Domains

Symmetry breaking: the daughter keeps only part of what the parent had

Guide 1 of this rung ended with a promise: one idea unifies almost every structural transition, and that idea is symmetry. Here is the whole story in one sentence. Cooling from a high-temperature parent phase to a low-temperature daughter almost always LOWERS the symmetry — the daughter keeps only some of the parent's symmetry operations, so its space group is a subgroup of the parent's, a group-subgroup relation. Picture a perfectly round dinner table before anyone reaches for the bread: every seat is equivalent, the arrangement looks the same however you rotate it — maximum symmetry. The instant one person picks up the roll on their left, that rotational symmetry is broken; the table now has a definite handedness. Nothing about the physics preferred left over right, yet the system had to commit to one. That commitment is symmetry breaking, and the daughter crystal is exactly the table after everyone has chosen.

To measure how far the symmetry has broken we use an order parameter, usually written eta (the Greek letter eta). By construction it is exactly zero in the symmetric parent and grows away from zero in the daughter — a single number that says how committed the crystal is to its lower-symmetry choice. Its physical identity changes from transition to transition, and naming it is the first move in any analysis: for a displacive change it is the amplitude of the atomic shift; for an order-disorder transformation like Cu3Au it is the degree of long-range chemical order (the Bragg-Williams order you met in guide 4); for a ferroelastic change it is the spontaneous strain; for a ferroelectric change it is the spontaneous electric polarization. Different quantities, one job: eta measures the broken symmetry.

Name the parent and the daughter, and note which is the more symmetric — cooling almost always runs high-symmetry parent to lower-symmetry daughter.
Write the group-subgroup relation: the daughter's space group is a subgroup of the parent's, and the LOST operations are the whole point.
Identify the order parameter eta — the physical quantity (a displacement, a chemical order, a strain, a polarization) that is zero above and grows below.
Decide the order of the transition: does eta jump (first-order) or grow continuously from zero (second-order)? Landau theory reads this off the free energy.
Count the domains: each lost symmetry operation breeds an equivalent variant, and the number of variants is the index of the daughter's group in the parent's.

Landau theory: a free energy written as a polynomial in eta

Lev Landau's insight was disarmingly simple. Near a continuous transition the order parameter eta is small, so instead of solving the impossible many-atom problem, just expand the free energy as a power series in eta — and let symmetry decide which powers are allowed to appear. If the parent's symmetry makes eta and -eta physically equivalent (a leftward shift is as good as a rightward one), then only EVEN powers can survive, and to lowest order Landau theory writes F(eta, T) = F0 + A eta^2 + B eta^4. The masterstroke is where the temperature hides: put ALL of it in the quadratic coefficient, A(T) = a times (T - Tc), with a and B fixed positive constants and Tc the transition temperature. Everything now follows from the sign of that one coefficient.

LANDAU FREE ENERGY   F(eta) = F0 + A eta^2 + B eta^4   (B > 0,  A = a(T - Tc))

    T > Tc   (A > 0)         T = Tc   (A = 0)         T < Tc   (A < 0)
    one well at eta = 0       just flattening         a DOUBLE well

      \         /              \       /              \           /
       \       /                \     /                \         /
        \     /                  \   /                  \_     _/
         \   /                    \ /                     \   /
          \_/                      V                       \_/  <- two minima
    -------o-------          ------o------          ----o-------o----
           0    eta                0                   -e0  0  +e0

    parent stable            marginal / soft         daughter appears:
    (only eta = 0)           (spring gone)           eta = +/- sqrt(a(Tc - T)/2B)
                                                     TWO equal wells = TWO domains

The Landau landscape as temperature falls. Above Tc a single well pins eta at zero (parent). At Tc the well flattens. Below Tc it splits into two equal minima at eta = plus or minus sqrt(a(Tc - T)/2B): the daughter, growing continuously from zero — and the two equally deep wells are already the seed of two domains.

Read the landscape and the physics falls out. Above Tc the coefficient A is positive, so F has a single well with its bottom at eta = 0: the symmetric parent is the only stable state. Below Tc, A turns negative and the curve buckles into a double well; minimizing F (setting dF/deta = 2A eta + 4B eta^3 to zero) gives eta = plus or minus sqrt(a times (Tc - T) / 2B). So the order parameter grows continuously out of zero, as (Tc - T)^(1/2) — a smooth square-root onset with no jump and no latent heat at Tc. That is precisely a second-order (continuous) transition, and Landau theory delivers it from three symmetry-chosen terms. Notice too that the double well has TWO symmetric minima, +e0 and -e0: two equally valid daughters. Hold that thought — it is where domains are born.

Be honest about what this beautiful machine assumes. Landau theory is a mean-field theory: it treats eta as one uniform number for the whole crystal and ignores the way eta actually fluctuates from place to place. Very close to a genuine continuous transition those fluctuations take over, and the true onset exponent is nearer eta proportional to (Tc - T)^0.33 (the 3D Ising value) than the mean-field (Tc - T)^0.5. Yet Landau theory remains the workhorse of structural transitions, because the elastic strains and electric dipoles that couple across the crystal are LONG-ranged — they average the fluctuations away — so mean-field predictions often hold accurately over a wide temperature range. And the same polynomial handles first-order changes too: let B go negative (with a positive eta^6 term to catch the bottom), or let symmetry permit an eta^3 term, and the minima appear at a jump rather than growing from zero.

Soft modes: a vibration that freezes into the new structure

Landau's eta is abstract, but for a displacive transformation it wears a gorgeous physical face: a lattice vibration. Recall from guide 2 that displacive means the atoms only shift a small fraction of a bond length, all together and cooperatively, without breaking a single bond. Among all the crystal's vibrations there is one special pattern of atomic motion whose restoring force — the effective spring holding the atoms in the parent arrangement — grows weaker and weaker as you cool. This is the soft mode, and its frequency obeys omega^2 proportional to (T - Tc): the spring softens, the vibration slows, and at Tc its frequency reaches zero.

When the frequency hits zero the lattice has lost all resistance to that one pattern of displacement — so the atoms simply lean into it and it FREEZES in. And here is the unification: the frozen displacement pattern (the mode's eigenvector) is exactly the distortion that turns parent into daughter, and its amplitude IS Landau's order parameter eta. The soft mode and the order parameter are therefore the same thing seen from two sides — a living, slowing vibration above Tc, and a static, frozen-in distortion below it. A displacive transition, in this light, is simply a phonon that stopped oscillating and stayed put.

Barium titanate, BaTiO3, is the textbook example and it also introduces the next theme. Above about 120 degrees C it is a cubic perovskite — a Ti ion centred in an octahedron of oxygens, no net dipole, fully symmetric (this hot phase is called paraelectric). As it cools, a transverse optic soft mode softens; at the Curie point that mode freezes, the Ti and oxygen sublattices shift oppositely, the Ti settles OFF-centre, and every unit cell acquires a permanent electric dipole. The crystal is now tetragonal and carries a spontaneous polarization — it has become ferroelectric. One honest nuance: real BaTiO3 is not a picture-perfect soft-mode crystal. The Ti also hops among off-centre sites even above the Curie point, so the transition has an order-disorder streak mixed into its displacive character. Nature rarely reads only one page of the textbook.

Ferroic order parameters you can feel — and switch

When the order parameter is itself a macroscopic property that you can measure with an instrument AND flip with an applied field, the transition earns the name ferroic. Two headliners run this show. In a ferroelectric transition eta is the spontaneous electric polarization P, and an electric field can switch its direction. In a ferroelastic transition eta is the spontaneous strain — the crystal's built-in distortion — and a mechanical stress can switch which way it leans. Both are usually displacive, both are driven by a soft mode, and both are pure examples of the structure-property relationship: the symmetry-broken structure does not merely differ from the parent, it CARRIES a useful, switchable property that the parent could not have.

Why can you feel it only in the daughter? Because the high symmetry of the parent FORBIDS it. A cubic crystal has too much symmetry to point a net polarization in any one direction — every axis is equivalent, so the dipoles must cancel. Only by breaking symmetry to a polar daughter (say cubic to tetragonal) does the crystal single out one axis and allow a spontaneous polarization to survive along it. This is the same crystallographic accounting you met in the symmetry rung: a spontaneous polarization is possible only in a polar point group, so a ferroelectric transition must carry the crystal from a non-polar parent to a polar daughter. The property appears precisely because, and exactly when, the forbidding symmetry is removed.

The payoff is a shelf of real devices. Ferroelectrics — BaTiO3, and the workhorse PZT (lead zirconate titanate) — are the high-permittivity dielectrics inside ceramic capacitors, the piezoelectric hearts of ultrasound transducers and inkjet actuators, and the storage cells of ferroelectric memory (polarization up or down is a stored 1 or 0). Ferroelastics are just as concrete, and they close a loop with guide 3: the transformation twins of a martensitic transformation ARE ferroelastic domains. When a shape-memory alloy like Nitinol is bent, you are pushing those ferroelastic twin variants to reorient (detwinning); warming it drives the diffusionless transformation back and the wire snaps to its remembered shape. The abstract symmetry story of this rung is, quite literally, what makes a self-bending stent work.

Domains: broken symmetry cannot help but make patches

Now for the deepest consequence, the one the double well already hinted at. Because the parent possessed symmetry operations the daughter lacks, each LOST operation maps the daughter onto a different-but-equally-valid version of itself — a distinct variant. A real crystal does not transform in one tidy sweep; it nucleates the daughter in many places at once, and different regions independently pick different variants. Where two variants meet, they cannot match, and you get a domain wall. The crucial honesty here: domains are NOT a symptom of careless processing you could avoid by working more slowly. They are FORCED by symmetry. A field can bias the choice, but the multiplicity of equal options is written into the group-subgroup relation itself.

You can even count them. The number of distinct domain variants equals the INDEX of the daughter's symmetry group in the parent's — simply order(parent) divided by order(daughter). Lost point-group operations give orientation (twin) variants; lost lattice translations give antiphase (translation) variants. Work a real case: cooling a cubic crystal (point group m-3m, order 48) to tetragonal (4/mmm, order 16) gives 48/16 = 3 orientation variants — the unique tetragonal axis can lie along x, y, or z. Those three variants are ferroelastic domains, and the walls between them are exactly the transformation twins and twin boundaries you met in guide 3. (In ferroelectric BaTiO3 each of those three axes also splits into two polarization senses, giving six domain states in all.)

The translational flavour closes the loop with guide 4. When disordered FCC copper-gold orders into the Cu3Au superlattice, it loses the face-centring translations of its parent — so there are FOUR ways to place the ordered pattern, related by exactly those lost vectors. Order the crystal in separate places and neighbouring patches often start out of step; where two of these out-of-step ordered regions meet, you get an antiphase domain boundary, a seam of wrong-neighbour bonds that raises the energy just like a twin wall does. Same principle as the tetragonal twins, only now the broken symmetry is translational rather than orientational — and the count, four variants, again equals the index of the ordered group in the disordered one.

Domains are not a curiosity; they run technology, and applying a field lets you steer them. Because the variants are energetically equal, an external field grows the favoured ones at the expense of the rest — a stress detwins a ferroelastic (that is shape memory), an electric field poles a ferroelectric (that is how you write a memory bit and align a piezoelectric), and a magnetic field sweeps the magnetic domains that make transformer steel useful. And do not forget the last knob from guide 1: pressure. Squeeze a crystal hard and it restacks into a denser polymorph — silicon turns metallic under a diamond anvil, and the minerals of the deep Earth are high-pressure forms of familiar ones. Some of these are reconstructive and first-order; others are clean displacive symmetry-breakings. Temperature, chemical order, field, and pressure are four knobs turning the very same contest: which broken-symmetry structure wins.