the phase fraction
If a microstructure is a mixture — say hard particles scattered through a soft matrix — one of the first questions is: how much of each? The phase fraction answers it: the proportion of the material that is a given phase, usually expressed as a volume fraction between 0 and 1 (or a percentage). Half hard phase and half soft is a phase fraction of 0.5 each.
The everyday way to measure it is beautifully indirect and rests on a stereological fact: on a random polished section, the area fraction a phase occupies equals its volume fraction, and (Delesse's principle) both equal the fraction of random test points that land on it. So you overlay a grid of points on the micrograph, count what fraction fall on the phase of interest, and that point fraction estimates the volume fraction directly — no 3D reconstruction needed. For example, of 500 grid points, 90 land on carbide, giving a carbide volume fraction of 90/500 = 0.18, or 18 percent. Phase fractions can also be predicted from a phase diagram by the lever rule.
Phase fraction matters because properties often scale with it: the strength of a two-phase alloy, the amount of hard wear-resistant carbide, the fraction of ductile phase for toughness, all track how much of each phase is present and are tuned by composition and heat treatment. Honest caveat: the area-equals-volume rule requires a truly random, representative section and enough counts — a biased cut or too few points gives a poor estimate — and it gives volume fraction, which must be converted with densities if you want a weight (mass) fraction.
Point-counting a duplex stainless steel with a 400-point grid gives 210 points on austenite and 190 on ferrite, so the austenite volume fraction is 210/400 = 0.53 and the ferrite 0.47 — close to the intended 50:50 balance that gives this steel its combination of strength and toughness.
Fraction of random points landing on a phase estimates its volume fraction directly — Delesse's area-equals-volume principle at work.
Point- or area-counting gives the volume fraction, not the weight fraction; convert with the phases' densities if you need mass. And the estimate is only as good as the section is random and the count is large.