The calculation models require three different sort of information : statistics, detector response and physics of the decay processes.

### STATISTICS

It is assumed that the photoelectron emission at the output of the photocathode follows a Poisson law. When an electron with energy E interacts with the liquid scintillator a mean of m photoelectrons is going to be emitted by the photocathode. The probability for the emission of exactly n photoelectrons is :

The photomultiplier produces a signal at the output of the anode when the number of photoelectrons emitted by the photocathode is equal or greater than 1. That means that the detection efficiency may be obtained from :

Detector efficiency

as

Detection efficiency can also be obtained more easily from :

Detection efficiency

When the liquid scintillation spectrometer has two photomultipliers working in coincidence, the counting efficiency of the system, for monoenergetic particles interacting with the liquid scintillator, is the product of the detection efficiency corresponding to each photomultiplier. If we assume that the photomultipliers have identical response, the counting efficiency is :

Counting efficiency

To compute m we define a free parameter l as the fictitious number of electrons emitted by the photocathode in order to obtain a computed counting efficiency equal to the experimental one. The relationship between the free parameter l, the energy E of the interacting electron and the mean number m of photoelectrons emitted by the photocathode (for a LS spectrometer of 2 photomultipliers) is given by the following equation :

where Q(E)is the ionization quench factor which will be analyzed in the next paragraph. Since the light emitted by the vial is distributed between two photomultipliers, we write 2 to take into account this light distribution.

### DETECTOR RESPONSE

A characteristic of the spectrometers is their response to the interacting radiation. Liquid scintillation spectrometers present poor linearity for low energies. The ratio between energy and number of electrons at the output of the cathode is non-linear, when the energy of the interacting electrons is less than 10 keV. The non-linearity is due to the ionization quenching process.

Ionization quench depends on the concentration of the excited molecules of the solvent produced when the ionizing particle interacts with the liquid scintillator. Both excited molecules in the first singlet state or excitons S1, and those in higher singlet states Sn can contribute to the ionization quench process.

When two excitons are very close, the following reactions are highly probable :

where one of the molecules is in the fundamental state S0, while the other one is in a superexcited state M**. Superexcited states are bonded states with energy higher than the first ionization level. Consequently, the ionization quench will compete with dissociation and internal conversion to less excited electronic levels.

We have the same situation with the reactions :

The final result is the same in both situations. Excited molecules are lost and, consequently, the intensity of the flash generated by ionizing particles reduced.

In general, the response (number of fluorescence photons L(E)) to an ionizing particle which moves in a liquid scintillator is a non-linear function of the particle energy E. This non-linearity increases with the ionization power of the particle. Consequently, a beta ray of low energy will undergo higher ionization quenching than a higher energy beta ray. Birks formulated a semi-empirical model for the process and proposed the following equation for the specific fluorescence, dL/dx (number of emitted photons per unit distance along the path) :

where h0 is the scintillation efficiency (number of fluorescence photons emitted per unit of energy), k is the rate constant for ionization quenching, dE/dx is the stopping power and B(dE/dx) is the linear ionization density.

The response is given by :

where

is the ionization quench function. The computation of the ionization quench function presents three problems : the selection of the optimal value for kB, the stopping power values for energies lower than 1 keV and the composition of the commercial liquid scintillator, which atomic composition is not supplied by the firms.

### PHYSICS OF THE DECAY PROCESSES

The decay process determines the equations and the nuclear constants to be used into the models. In pure beta-ray emission the statistical beta-spectrum, the relativistic Fermi factor, the correction for finite nuclear radius and the shape factor for forbidden transition are the same for all equations required in the model.

For EC, the situation is complicated by the huge number of atomic rearrangement pathways. The detection of the emitted X-rays and Auger electrons complicates the model.

The physics of the decay process is an extremely important part of the model and requires a detailed analysis of the involved processes. The books in the bibliography give an extensive description of the matter.