Order-Disorder and Superlattices

From a random alloy to a patterned one

The last three guides watched a crystal change its structure by changing its lattice — iron's BCC-to-FCC flip, martensite's diffusionless shear. Here the lattice barely moves at all; what changes is who sits where on it. Start with a substitutional solid solution of three parts copper to one part gold. At high temperature it is a face-centered cubic crystal in which every site is a coin-toss: about one chance in four of holding a gold atom, three in four a copper atom, scattered at random like salt and pepper. The lattice is perfectly periodic, but the decoration on it is a mess.

Now cool slowly. At a sharp temperature — about 390 degrees C for this alloy, its ordering temperature Tc — something remarkable happens: the atoms stop tossing coins and take assigned seats. Every gold atom moves to a cube corner; every copper atom takes a face centre. The random solid solution has become an ordered structure, Cu3Au, with a repeating gold-corner, copper-face motif. Because that motif repeats with the full period of the cube — a longer, richer pattern than the bare FCC points alone — we call it a superlattice, and the change from random to patterned is an order-disorder transformation. Think of general-admission seating (sit anywhere) suddenly becoming reserved seating (everyone in their assigned chair).

What kind of transition is this? Unlike the diffusionless martensitic shear of guide 3, ordering demands that atoms actually swap places — a gold atom on a face centre must diffuse to a corner — so it is a slow, diffusion-controlled re-sorting, frozen out if you quench too fast. Yet, like a displacive change, it keeps the parent lattice skeleton intact: no bonds are torn, no atoms travel far. That places order-disorder in its own interesting corner of the transformation map — diffusional in mechanism, but gentle and often continuous in the way symmetry is lost, which is exactly why it makes the perfect bridge to the Landau picture of the final guide.

One number for how ordered it is

Ordering is rarely all-or-nothing, so we need a dial that reads from 0 (fully random) to 1 (perfectly ordered). That dial is the long-range order parameter S. Pick the sublattice that gold is supposed to occupy — the cube corners, which are one quarter of all sites. Let r be the fraction of those corner sites actually holding a gold atom, and let x = 1/4 be gold's overall fraction. Then S = (r - x) / (1 - x). Perfect order puts gold on every corner: r = 1, so S = (1 - 0.25)/(1 - 0.25) = 1. Total randomness makes a corner just as likely as anywhere to hold gold: r = 1/4, so S = 0. Half-ordered sits in between.

As you warm an ordered crystal back up, S does not hold at 1 and then snap: thermal agitation knocks a few atoms onto wrong sites, so S slips gradually from 1, sags faster as Tc nears, and finally reaches 0 at the ordering temperature, where the last trace of long-range order dissolves. The shape of that S-versus-temperature curve is the fingerprint of the transition — whether it glides smoothly to zero or drops off a cliff is exactly the first-order-versus-second-order distinction we return to at the end. For now, hold the picture: S measures how faithfully the atoms keep to their assigned seats, averaged across the whole crystal.

How diffraction catches the pattern: superlattice reflections

Here is where the structure of materials shows its power, because ordering leaves an unmistakable diffraction signature. In the disordered alloy, every site scatters as an average atom, with an effective scattering factor f_avg = 0.75 f_Cu + 0.25 f_Au — a smeared blend of copper and gold. To a diffraction camera the crystal is then a plain FCC lattice of identical average scatterers, and the FCC structure factor rule bites: a reflection survives only when h, k, l are all even or all odd. Mixed-index reflections like (100) and (110) vanish by systematic absence — the four atoms in the cell interfere exactly to zero. So the disordered crystal shows only the fundamental FCC peaks: (111), (200), (220), (311).

CU3AU:  DISORDERED  vs  ORDERED   (X-ray powder)

DISORDERED (A1, random FCC)
  every site = one AVERAGE atom,  f_avg = 0.75 f_Cu + 0.25 f_Au
  FCC rule: a peak survives only if h,k,l are all-even OR all-odd
  mixed hkl  (100)(110)(210)...  ->  SYSTEMATICALLY ABSENT

      |111      |200        |220      |311        (fundamentals only)
  ----+---------+-----------+---------+---------->  2-theta

ORDERED (L1_2)  Au at (0,0,0);  Cu at (1/2 1/2 0)(1/2 0 1/2)(0 1/2 1/2)
  F(unmixed) = f_Au + 3 f_Cu   ->  FUNDAMENTAL   (unchanged, strong)
  F(mixed)   = f_Au -  f_Cu    ->  SUPERLATTICE  (new, weak)

  |100 |110 |111 |200 |210|211|220      |311
  --+----+----+----+----+---+---+--------+-------->  2-theta
    ^^^^^^^^^^  weak superlattice lines SWITCH ON

  intensity(superlattice)  proportional to  S^2  x  (f_Au - f_Cu)^2

Ordering leaves the fundamental FCC reflections untouched (F = f_Au + 3 f_Cu either way) but switches ON new, weak superlattice reflections at the once-forbidden mixed indices, with structure factor F = f_Au - f_Cu.

Now order the crystal, and gold and copper occupy distinct sites, so their scattering no longer averages away. Work out the structure factor for a mixed-index reflection and you get F = f_Au - f_Cu — not zero, but the difference of the two scattering powers. The forbidden reflections switch on as faint superlattice reflections at (100), (110), (210), and so on. Two honest consequences follow. First, the fundamental peaks are completely unchanged — their structure factor is f_Au + 3 f_Cu whether the crystal is ordered or not — so ordering adds new lines without touching the old ones. Second, the superlattice intensity is proportional to S^2 times (f_Au - f_Cu)^2, so the ratio of a superlattice peak to a fundamental one gives a direct, quantitative read of the order parameter S. Diffraction does not just detect order; it measures it.

In practice, a superlattice reflection is only as strong as the contrast (f_Au - f_Cu) allows. For copper and gold the atomic numbers are 29 and 79, a big gap, so Cu3Au superlattice lines are weak but easily seen. But try to spot ordering in an alloy of two periodic-table neighbours — iron and cobalt (Z = 26 and 27), say — and to X-rays f_Fe and f_Co are nearly equal, so (f_Fe - f_Co) is tiny and the superlattice reflections almost vanish. The fix is to change your probe: neutron diffraction scatters off nuclei, and neighbouring elements (or different isotopes) can have wildly different neutron scattering lengths, restoring the contrast. One more bonus — a superlattice peak is broadened by the finite size of the ordered regions, so its width, read through the Scherrer idea, measures how big those regions have grown.

Where the pattern gets out of step: antiphase domains

When a crystal orders, the pattern does not switch on everywhere at once from a single starting point. Ordering nucleates independently at many separate places, and each nucleus faces a choice with no right answer: in Cu3Au the gold could take any one of the four equivalent FCC positions as 'its' corner sublattice. Different nuclei make that choice at random. Picture a crowd of tilers laying the same black-and-white checkerboard across a floor, but each starting from a different square: every patch is a perfect checkerboard, yet neighbouring patches need not agree on which squares are black.

Cool below Tc, and ordering nucleates independently at many separate points scattered through the disordered crystal.
Each nucleus picks, at random, one of the four equivalent ways to lay the Cu3Au pattern down — which of the four FCC sites gold will call its corner sublattice.
The ordered patches — the antiphase domains — grow outward, each extending its own pattern, until they collide with their neighbours.
Where two domains that made out-of-step choices meet, the ordering jumps by an antiphase vector, so like atoms face each other across a thin mismatch surface: an antiphase boundary.
Keep annealing, and those boundaries, which cost energy, migrate to shrink their total area — small domains are swallowed, the pattern coarsens, and the superlattice peaks sharpen, just like grain growth but for order.

These antiphase domains are not a curiosity — you can photograph them. In a transmission electron microscope, form the image using a single superlattice reflection (dark-field imaging), and each antiphase boundary shows up as a sharp dark fringe, mapping the ordered mosaic directly. The domains and their boundaries carry real consequences too: because a boundary forces wrong neighbours together, it costs energy and it obstructs dislocations, which is one route by which ordered alloys strengthen. And notice the deep pattern here — a lower-symmetry ordered phase, born from a higher-symmetry parent, is forced to appear in several equivalent variants that meet at walls. That is the universal fingerprint of a symmetry-breaking transition, and it is exactly the thread the final guide picks up.

First order, second order, and lost symmetry

Not all ordering transitions have the same character. Cu3Au orders with a jolt: at Tc the order parameter S drops discontinuously from a sizeable value straight to zero, the transition releases latent heat, and ordered and disordered regions briefly coexist — the hallmarks of a first-order transition. Beta-brass, the roughly equal copper-zinc alloy, does the opposite: cooling its BCC solid solution below about 460 degrees C, copper gathers on the cube corners and zinc on the body centres in a smooth, continuous slide, with S growing gently from zero. That is a textbook second-order transition — no latent heat, no coexistence, just symmetry quietly lowering as the crystal cools.

Underneath both cases is a symmetry story, and it explains the domains cleanly. The disordered Cu3Au is space group Fm-3m: its four FCC sites are all equivalent, tied together by the face-centering translations. On ordering, gold and copper are no longer interchangeable, so those centering translations stop being symmetries — the space group drops to Pm-3m, a subgroup of the parent. Exactly three centering translations are lost, and that lost count is why there are four antiphase-domain variants: the number of domains equals the number of symmetry operations the crystal gave up. Ordering can even change the crystal system — CuAu orders into a tetragonal structure with alternating copper and gold layers, its cube squashed slightly along the stacking axis, so a purely chemical re-sorting has produced a genuine shape change.

Why is one transition abrupt and another smooth? The answer is not thermodynamic bookkeeping but symmetry itself: whether a continuous, second-order path is even allowed depends on how the parent and daughter symmetries are related, a rule made precise by Landau theory — the subject of the last guide in this rung, which builds the order parameter S into a free-energy landscape and reads the transition off its shape. Carry two ideas forward. First, order-disorder is the cleanest illustration of the rung's refrain that the lattice is not the crystal: the FCC lattice points never moved, yet by changing only the motif — who sits where — we created a new phase, new diffraction peaks, and a new symmetry. Second, whenever a crystal drops to lower symmetry, it pays in domains. Hold both, and the final guide's Landau picture will feel like a homecoming.