Three phases, zero freedom
Guide 1 handed you a small piece of bookkeeping that turns out to run this whole guide: the Gibbs phase rule. For a solid-liquid system at fixed pressure it reads F = C - P + 1, where C is the number of components, P the number of phases sitting in equilibrium together, and F the degrees of freedom — the count of knobs (temperature, composition) you may freely turn while still keeping exactly that set of phases. In a binary alloy C = 2. Inside a single-phase field, P = 1 and F = 2, so you can roam in both temperature and composition. In a two-phase field, P = 2 and F = 1: pick a temperature and the two phase compositions are forced, which is exactly why the tie line and lever rule worked in guide 2.
Now push it one step further. What if three phases coexist at once? Then P = 3 and F = 2 - 3 + 1 = 0. Zero degrees of freedom means zero wiggle room: the temperature is locked to a single value, and all three phases are pinned to fixed, unchangeable compositions. A three-phase equilibrium in a binary cannot happen over a range of anything — it can only happen at one exact point. That is what the word invariant means, and a reaction that occurs there is an invariant reaction (also called isothermal, since it runs at one fixed temperature). On the diagram it is forced to draw as a horizontal line — the isotherm — stretched between the three locked compositions. Every eutectic, peritectic, and eutectoid you will ever meet is just this same zero-freedom event wearing a different arrangement of phases.
The eutectic: the family's first member
You already met the archetype in guide 3. In a binary eutectic system, cool a liquid of just the right composition and at one special temperature it does not freeze into a single solid — instead the whole liquid transforms at once into two different solids side by side. Written as a reaction: L -> alpha + beta on cooling. This is the eutectic reaction, and its three phases (one liquid, two solids) meeting at F = 0 are exactly the zero-freedom event from the last section. On the diagram it sits at the bottom of a V, the point where the two falling liquidus lines meet on the eutectic isotherm — the lowest melting point in the whole system, which is what the Greek eutektos, 'easily melted', is telling you.
Two things carry straight over from that guide and are worth re-pinning, because they hold for every reaction below. First, the reaction only runs to completion for an alloy sitting exactly at the eutectic composition; an off-composition alloy freezes some primary alpha (or primary beta) first, and only the leftover liquid — which drifts toward the eutectic point as it cools — undergoes the reaction. Second, the amounts of the two product solids are still set by the ordinary lever rule applied on the tie line just below the isotherm: the invariant reaction changes which phases exist, but it does not suspend the see-saw that weighs them. The fingerprint the eutectic leaves behind is a fine, often lamellar two-phase microstructure — alternating ribbons of alpha and beta grown together as the single liquid split.
The peritectic: a reaction that swallows a solid
The peritectic reaction rearranges the same three phases into a different sentence. On cooling, a liquid reacts with an already-solid phase to produce a third, new solid: L + alpha -> beta. Notice this reads backwards from the eutectic — instead of one phase splitting into two, two phases (a liquid and a solid) combine into one. The name is the giveaway: 'peri' is Greek for 'around', and it describes exactly how the reaction proceeds. The new beta forms first at the surface where liquid touches the alpha crystals, growing as a shell or rim that wraps around each grain of alpha.
And that shell is the peritectic's honest headache. Once beta has walled the alpha off from the liquid, the reaction can only continue if atoms diffuse all the way through the growing beta layer to keep feeding it — and solid-state diffusion is slow. Think of a chocolate bonbon whose shell sets before the frozen centre has thawed: the shell seals the two apart and the inside is left stranded. In real castings cooled at ordinary rates, peritectic reactions therefore rarely go to completion; you are left with cored grains — an unreacted alpha core wrapped in a beta rim — a textbook example of a microstructure that departs from the equilibrium diagram simply because kinetics ran out of time. Peritectics are common in commercial alloys: several appear in the copper-zinc system behind ordinary brass, and the iron-carbon diagram itself opens with one at its high-temperature top corner.
The eutectoid: all solid, and the heart of steel
Now take the eutectic reaction and simply demote the parent from liquid to solid. A single solid phase, on cooling, decomposes at one fixed point into two different solids: gamma -> alpha + beta. This is the eutectoid reaction, and the tell is entirely in its suffix. '-oid' means 'resembling' — a humanoid resembles a human without being one — so eutectoid means 'eutectic-like': the same one-splits-into-two pattern, the same V-bottom point, the same F = 0, but with a solid parent standing where the liquid stood. It is genuinely the most important reaction in this guide, because one particular eutectoid is the pivot on which almost all of steel turns.
That reaction is the one waiting in the iron-carbon diagram you will study in full next. Cool a steel of about 0.76 weight-percent carbon and at 727 degrees C the solid solution called austenite (gamma iron, holding all its carbon dissolved) decomposes in one stroke into two solids at once: soft, nearly carbon-free ferrite and hard, carbon-rich cementite (Fe3C). Because the two grow together in the eutectic manner, they interleave as fine alternating layers — the beautiful lamellar structure called pearlite. A steel sitting right at 0.76 percent carbon is called eutectoid steel, and on slow cooling it becomes essentially 100 percent pearlite. This single line is a large part of why steel is the most tunable structural material we have.
Put a real number on it with the lever rule, exactly as in guide 2. Just below 727 degrees C the two product solids have fixed compositions: ferrite holds only about 0.022 percent carbon, cementite a fixed 6.70 percent. The parent pearlite sits at 0.76 percent, so the fraction of cementite is (0.76 - 0.022) / (6.70 - 0.022) = 0.738 / 6.678, about 0.11 — roughly 11 percent cementite and 89 percent ferrite by mass. That is why pearlite, under the microscope, is mostly wide ferrite bands with thin cementite plates threaded through them, in almost exactly that eight-to-one ratio. Reasoning, not memorising: the same see-saw you learned on an isomorphous diagram tells you the make-up of the toughest, most storied microstructure in engineering.
The whole family on one card
INVARIANT REACTIONS -- every one: 3 phases meet, F = C - P + 1 = 2 - 3 + 1 = 0
reaction on cooling parents products the tell
------------ -------------------- ------------ ------------ ----------------------
eutectic L -> alpha + beta 1 liquid 2 solids liquid splits in two
eutectoid gamma -> alpha + beta 1 solid 2 solids '-oid' = solid parent
peritectic L + alpha -> beta liquid+solid 1 solid liquid eats a solid;
new solid wraps around
peritectoid alpha + beta -> gamma 2 solids 1 solid all-solid peritectic
monotectic L1 -> alpha + L2 1 liquid solid+liquid liquid splits, one stays
liquid
Read any horizontal line the same way: name the 3 phases, sort parents (above the
line on cooling) from products (below), and the pattern above tells you which it is.- Spot the horizontal line. Any isotherm (flat line) crossing a binary diagram is an invariant reaction — that flatness is F = 0 drawing itself.
- Read off the three phases touching the line — one field sits above it and two below, or two above and one below.
- Going down in temperature, call the phase(s) above the line the parents and those below the products, then write the reaction as parents -> products.
- Match the pattern: one liquid to two solids is eutectic; one solid to two solids is eutectoid; a liquid plus a solid to one solid is peritectic.
- Finally, drop a tie line just below the isotherm and apply the lever rule to get how much of each product forms.
One last piece completes the picture of why a real diagram can carry a whole string of these reactions. Many alloy systems form intermetallic compounds — ordered phases sitting at or near a fixed stoichiometry, like Mg2Pb or the several ordered phases inside brass — and each such compound behaves like a new component planted mid-diagram, effectively slicing one wide diagram into several narrow sub-diagrams stacked side by side. Every internal boundary between them can host its own eutectic, peritectic, or eutectoid, which is how a single real system racks up several invariant reactions at once. With the whole family named and the naming trick in hand, you are ready for the one diagram that stitches a peritectic, a eutectic, and the all-important eutectoid into a single chart — the iron-carbon diagram, and the story of steel, in guide 5.