Using Gamma-ray Counts to Calculate Element Concentration

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Written and developed by Prof. Tor Bjørnstad (IFE/UiO) 

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In principle, it is possible to perform quantitative NAA by using the general formula for activation:

[math]w=\frac{R_{x}\cdot e^{\lambda_{x}}\cdot t_{d}}{\varepsilon \cdot \sigma \cdot \phi \cdot N_{A} \cdot I_{N} \cdot (1-e^{-\lambda_{x}} \cdot t_{i})}[/math]                                          

w = mass of unknown element in sample
Rx = measured count rate of unknown element (cps)
M = atomic weight of unknown element
λx = decay constant of the measured radionuclide of the unknown element = ln2/T1/2
Where T1/2 is the half-life of this radionuclide (s-1)
td = decay time = time between irradiation end and midpoint of counting period (s)
[math]\varepsilon_{x}[/math] = counting efficiency of the detected gamma energy
 σ= thermal neutron reaction cross section of the unknown element (barn = 10-24
φ= neutron flux in irradiation position (ns-1cm-2).
NA = Avogadro’s number (6.023[math]\cdot[/math]1023)
IA = natural occurrence of the target isotope in the actual element (as fraction in the
range 0-1)
ti = irradiation time (s)

However, in practice, the parameters σ and φ are not easily determined exactly. Therefore, a simpler method is to carry out so-called comparative analysis where the activity in the sample is compared to the activity in a simultaneously irradiated standard of the same element. If the unknown sample and the comparator standard are both measured on the same detector, then one needs to correct the difference in decay between the two. One usually decay corrects the measured counts (or activity) for both samples back to the end of irradiation using the half-life of the measured isotope. The following relation may be put up:

[math]\frac{w_{p}}{w_{s}}= \frac{R_{xp}\cdot M \cdot e^{\lambda_{x} \cdot t_{dx}}}{\frac{\varepsilon_{xp} \cdot \sigma \cdot \phi \cdot N_{A} \cdot I_{A} \cdot (1-e^{-\lambda_{x} \cdot t_{i}})}{\frac{R_{xs} \cdot M \cdot e^{\lambda_{x} \cdot t_{ds}}}{\varepsilon_{xs} \cdot \sigma \cdot \phi \cdot N_{A} \cdot I_{A} \cdot (1-e^{-\lambda_{x} \cdot t_{i}})}}}[/math]

If the unknown sample and the comparator standard are both measured on the same detector, [math]\varepsilon_{xp} = \varepsilon_{xs}[/math] and the following simple expression results:

[math]\frac{w_{p}}{w_{s}}=\frac{R_{xp} \cdot e^{\lambda_{x} \cdot t_{dx}}}{R_{xs} \cdot e^{\lambda_{x} \cdot t_{ds}}}[/math]