Plastic Scintillator Calculation

Below is the calculations done to determine the probability of pair production depending on thickness of the scintillator.

Molecules per [math] cm^3 = \frac{grams CH_{2}}{cm^3} * \frac{mol}{gram} * {N[A]}[/math] (NOTE: [math] \frac{gram}{cm^3} [/math] is just the density of the scintillator material and N[A] is Avogadro's number)

Molecules per [math] cm^2 (K) = \frac{Molecules}{cm^3} * Thickness [/math]

Weighted cross-section [math] (\sigma_w) = (\sigma_{elec}C + \sigma_{nucleus}C) + 2(\sigma_{elec}H + \sigma_{nucleus}H)[/math]

Probability of interaction (%) [math]= \sigma_w * K * 100%[/math]

All cross sections listed here are pair production cross-sections

For carbon [math]\sigma_{nucleus} = 9.645*10^{-2} barns[/math] or [math]9.645*10^{-26}cm^2[/math]

For carbon [math]\sigma_{elec} = 1.030*10^{-2} barns[/math] or [math]1.030*10^{-26}cm^2[/math]

For hydrogen [math]\sigma_{nucleus} = 2.688*10^{-3} barns[/math] or [math]2.688*10^{-27}cm^2[/math]

For hydrogen [math]\sigma_{elec} = 1.716*10^{-3} barns[/math] or [math]1.716*10^{-27}cm^2[/math]

Avogadro's number [math] = \frac{6.022*10^{23}molecules}{mol}[/math]

Molecular formula for PVT [math] = C_{10}H_{11} [/math]

Density of polyvinyl toluene (a common scintillator material) [math] = \frac{1.02grams}{cm^3}[/math] (NOTE: this value is from Rexon RP 200 [1])

or is it [math]\rho_{BC408} = \frac{1.032grams}{cm^3}[/math]  H/C = 11/10  [2] (TF)

For the sample calculation the thickness will be set to 1 cm just to get probability per cm

So entering all the numbers into the 4 initial equations gives the following answers:

Molecules per [math] cm^3 = \frac{1.02g PVT}{cm^3} * \frac{1 mol}{131 g} * \frac{6.022*10^{23}molecules}{mol} = \frac{4.689*10^{21}molecules PVT}{cm^3} [/math]

Molecules per [math] cm^2 (K) = \frac{4.689*10^{21}molecules PVT}{cm^3} * 1cm = \frac{4.689*10^{21}molecules PVT}{cm^2} [/math]

Weighted cross-section [math] (\sigma_w) = (10*(1.030*10^{-26}cm^2 + 9.645*10^{-26}cm^2)) + (11*(1.716*10^{-27}cm^2 + 2.688*10^{-27}cm^2)) = 1.116*10^{-24}cm^2[/math]

Probability of interaction (%) [math]= 1.116*10^{-24}cm^2 * \frac{4.689*10^{21}molecules PVT}{cm^2} * 100% = 0.5233%[/math]

Doing the same calculations using the Bicron BC 408 PVT with anthracene [3] for the material yields a probability of [math]0.5294%[/math]

A different way to calculate probability of interaction

I checked out a few of the physics material supply sites and most of them list with their products the amounts of each individual atom per [math] cm^3[/math]. Therefore there is a quicker way to calculate the probability of interaction which is listed below.

Probability of interaction [math]= ((\frac{NumCarbonAtoms}{cm^3} *(\sigma_{elec}C + \sigma_{nucleus}C)) + (\frac{NumCarbonAtoms}{cm^3} *(\sigma_{elec}H + \sigma_{nucleus}H)))*100%[/math]

Using this method of calculation for Rexon RP 200 yields a probability [math] = 0.5234%[/math]

Using this method for Bicron BC 408

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