Basic Physics of Nuclear Medicine/Print version

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Mathematical Model

A simple mathematical model will be presented below which will help us get a better handle on the performance of a scintillation detector. We will do this by quantifying the performance of the scintillator, the photocathode and the dynodes.

Let's use the following symbols to characterize each stage of the detection process:

m: number of light photons produced in crystal
k: optical efficiency of the crystal, that is the efficiency with which the crystal transmits light
l: quantum efficiency of the photocathode, that is the efficiency with which the photocathode converts light photons to electrons
n: number of dynodes
R: dynode multiplication factor, that is the number of secondary electrons emitted by a dynode per primary electron absorbed.

Therefore the charge collected at the anode is given by the following equation:

$Q = m k l R^n e\,\!$

where e: the electronic charge.

For example supposing a 100 keV gamma-ray is absorbed in the crystal. The number of light photons produced, m, might be about 1,000 for a typical scintillation crystal. A typical crystal might have an optical efficiency, k, of 0.5 - in other words 50% of the light produced reaches the photocathode which might have a quantum efficiency of 0.15. A typical PMT has ten dynodes and let us assume that the dynode multiplication factor is 4.5.

Therefore

$Q = 1000(0.5)(0.15)(4.5^{10})(1.6 \cdot 10^{-19})\ \text{C}$

$\therefore Q = 41 \cdot 10^{-12}\ \text{C}$

$\therefore Q \approx 40\ \text{pC}$

This amount of charge is very small. Even though we have used a sophisticated photodetector like a PMT we still end up with quite a small electrical signal.

A very sensitive amplifier is therefore needed to amplify this signal. This type of amplifier is generally called a pre-amplifier and we will refer to it again later.