the mass-action law
Defects reach a balance the same way a chemical reaction does — some form, some recombine, until the two rates match and the populations settle. The mass-action law is the equation that pins down that balance: for any defect reaction at equilibrium, the product of the concentrations of the products, divided by the product of the concentrations of the reactants (each raised to its coefficient in the equation), equals a constant K that depends only on temperature. It is ordinary chemical equilibrium, applied to vacancies, interstitials, electrons, and holes.
Take the intrinsic electronic equilibrium, nil ⇌ e' + h•. Mass action gives n times p = K_i, where n and p are the electron and hole concentrations — the semiconductor's np product is fixed at a given temperature. Take reduction of an oxide, O_O^x ⇌ 1/2 O2(g) + V_O•• + 2 e'; mass action gives K = [V_O••] times n^2 times pO2^(1/2). Each constant carries an Arrhenius temperature dependence, K = K0 exp(-dH/kT), where dH is the reaction's formation enthalpy — so, like defect populations themselves, equilibrium constants rise steeply with temperature. To solve a real problem you write the mass-action equation for every reaction, add the electroneutrality condition, and solve the set simultaneously.
This is the quantitative engine of defect chemistry. Combine the reduction mass-action law with the charge-balance approximation 2[V_O••] = n, and a little algebra gives the famous result that the electron concentration (and hence n-type conductivity) scales as pO2^(-1/6) — a prediction you can test by measuring conductivity versus furnace atmosphere. Plotting these mass-action solutions on log-log axes is precisely what a Brouwer diagram does; the law is honest only insofar as the defects are dilute and non-interacting, so it bends at high defect concentrations where clustering sets in.
For reduced ceria or titania, combining the reduction constant K = [V_O••] n^2 pO2^(1/2) with electroneutrality n = 2[V_O••] gives n proportional to pO2^(-1/6). Measure the conductivity of the oxide at several oxygen pressures, plot log-conductivity against log-pO2, and a slope of -1/6 confirms this defect model is correct.
The mass-action law turns a balanced defect equation into a testable power-law prediction linking defect count to oxygen pressure.
Mass action assumes dilute, independent defects (an ideal-solution approximation). At high concentrations defects interact and associate, the 'constant' stops being constant, and the neat power-law slopes break down.