Plastic Scintillator Calculation
Below is the calculations done to determine the probability of pair production depending on thickness of the scintillator.
Molecules per [math]\displaystyle{ cm^3 = \frac{grams CH_{2}}{cm^3} * \frac{mol}{gram} * {N[A]} }[/math] (NOTE: [math]\displaystyle{ \frac{gram}{cm^3} }[/math] is just the density of the scintillator material and N[A] is Avogadro's number)
Molecules per [math]\displaystyle{ cm^2 (K) = \frac{Molecules}{cm^3} * Thickness }[/math]
Weighted cross-section [math]\displaystyle{ (\sigma_w) = (\sigma_{elec}C + \sigma_{nucleus}C) + 2(\sigma_{elec}H + \sigma_{nucleus}H) }[/math]
Probability of interaction (%) [math]\displaystyle{ = \sigma_w * K * 100% }[/math]
All cross sections listed here are pair production cross-sections
For carbon [math]\displaystyle{ \sigma_{nucleus} = 9.645*10^{-2} barns }[/math] or [math]\displaystyle{ 9.645*10^{-26}cm^2 }[/math]
For carbon [math]\displaystyle{ \sigma_{elec} = 1.030*10^{-2} barns }[/math] or [math]\displaystyle{ 1.030*10^{-26}cm^2 }[/math]
For hydrogen [math]\displaystyle{ \sigma_{nucleus} = 2.688*10^{-3} barns }[/math] or [math]\displaystyle{ 2.688*10^{-27}cm^2 }[/math]
For hydrogen [math]\displaystyle{ \sigma_{elec} = 1.716*10^{-3} barns }[/math] or [math]\displaystyle{ 1.716*10^{-27}cm^2 }[/math]
Avogadro's number [math]\displaystyle{ = \frac{6.022*10^{23}molecules}{mol} }[/math]
Density of polyvinyl toluene (a common scintillator material) [math]\displaystyle{ = \frac{1.02grams}{cm^3} }[/math]
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]\displaystyle{ cm^3 = \frac{1.02g PVT}{cm^3} * \frac{1 mol}{14 g} * \frac{6.022*10^{23}molecules}{mol} = \frac{4.387*10^{22}molecules PVT}{cm^3} }[/math]
Molecules per [math]\displaystyle{ cm^2 (K) = \frac{4.387*10^{22}molecules PVT}{cm^3} * 1cm = \frac{4.387*10^{22}molecules PVT}{cm^2} }[/math]
Weighted cross-section [math]\displaystyle{ (\sigma_w) = (1.030*10^{-26}cm^2 + 9.645*10^{-26}cm^2) + 2(1.716*10^{-27}cm^2 + 2.688*10^{-27}cm^2) = 1.1556*10^{-25}cm^2 }[/math]
Probability of interaction (%) [math]\displaystyle{ = 1.1556*10^{-25}cm^2 * \frac{4.387*10^{22}molecules PVT}{cm^2} * 100% = 0.5070% }[/math]