Birks' law[1][2] (named after British physicist John B. Birks)[3] is an empirical formula for the light yield per path length as a function of the energy loss per path length for a particle traversing a scintillator, and gives a relation that is not linear at high loss rates.
The relation is: where L is the light yield, S is the scintillation efficiency, dE/dx is the specific energy loss of the particle per path length, k is the probability of quenching,[1] and B is a constant of proportionality linking the local density of ionized molecules at a point along the particle's path to the specific energy loss;[1] "Since k and B appear only as a product, they act as one parameter, kB, called Birks' coefficient, which has units of distance per energy.
[6] Birks speculated that the loss of linearity is due to recombination and quenching effects between the excited molecules and the surrounding substrate.
[7] A compilation of Birks' constant for various materials can be found in Semi-empirical calculation of quenching factors for ions in scintillators.
[8][9] A more complete theory of scintillation saturation, that gives Birks' law when only unimolecular de-excitation is included, can be found in a paper by Blanc, Cambou, and De Laford.