critical exponent
Suppose you measure how fast a magnet's magnetism grows as you cool it just past the point where it first appears. You will find it does not grow at any old rate, but follows a precise mathematical curve — and a single number controls exactly how steep that curve is. That governing number is a critical exponent.
Near a critical point, quantities like the order parameter, the correlation length, and the material's responsiveness do not vary smoothly; they shoot toward zero or toward infinity following power laws — expressions where one quantity is another raised to some fixed power. The critical exponent is precisely that power. Each exponent describes a different behavior: how quickly order builds up, how violently fluctuations grow, how sharply a quantity diverges. Measure them and you have a fingerprint of the transition.
Critical exponents matter because of a stunning fact: their values are almost entirely independent of the messy microscopic details and depend only on a few broad features like the dimensionality of the system. This is the heart of universality. The honest caveat is that these clean power laws hold only very close to the critical point; step away and ordinary, material-specific behavior takes over again.
As a magnet cools just below its Curie temperature, its magnetization grows roughly as the temperature distance below the transition raised to a power near 0.33. That number — the exponent — is found to be the same for many quite different magnetic materials, even though their atoms, spacings and bonds differ wildly.
A single critical exponent fixes how fast magnetization rises below the Curie point — and it is shared across many materials.
Different exponents are traditionally labeled with Greek letters — for the order parameter, the correlation length, the response to a field, and so on. They are not independent: deep scaling relations tie them together, so knowing a couple often lets you deduce the rest.