where A₀ is the radioactivity produced (disintegrations per cubic centimeter per second); N is the number of target atoms per cubic centimeter in the sample; φ is the neutron flux (neutrons per square centimeter per second); σ is the cross section for the reaction (square centimeters); λ is the disintegration constant[6] for the radioactive atoms produced (number per second); the number “e” is the base of natural logarithms; and t is the irradiation time in seconds. Note that for short irradiation times (t very small), 1-e^{-λt} approximates λt, while for long irradiations (t very large), 1-e^{-λt} approximates 1. [Illustration: Graph: Decay scheme for manganese-56] Maximum Energy % Betas 2.84 MeV 53 1.03 MeV 30 0.720 MeV 16 0.30 MeV 1 % Gammas 0.847 MeV 68 1.811 MeV 20 2.110 MeV 10 Other Energies 2 2.543 MeV 2.658 MeV 2.957 MeV 3.39 MeV (avge~ 1.4 gammas per beta) This summarizes what the decay scheme or energy level diagram shows in terms of the relative amounts of betas and gammas emitted in the decay of manganese-56. Thus, you could observe more than three times as many gamma rays having an energy of 0.847 MeV than of 1.811 MeV, etc. Note that while one, and only one, beta is emitted in the decay of one atom of manganese-56, two gammas can sometimes be emitted in one decay. Of course, when the target is removed from the reactor, the number of radioactive atoms begins to decrease according to the characteristic half-life of the nuclide. The mathematical expression that describes the process of radioactive decay of a single nuclide is: A_{t} = A₀e^{-λt} where A_{t} is the radioactivity of an isotope at some time, t, after the end of the irradiation, and A₀ is the radioactivity at the end of the irradiation. [Illustration: Fraction of saturation sodium-24 activity _vs_ Time of irradiation (hours)] The activation of sodium-23 to sodium-24, which has a half-life of 15 hours. The horizontal line marked 1.0 represents the “saturation” activity level for a sample of sodium of a certain size in a constant neutron flux. Note that after about 120 hours, the activity of the sample is within 1% of the value at saturation, which is the most active that sample will ever become at a given φ. Note also that after the first 15 hours (1 half-life) the sample is exactly half way to its value at saturation. Thus long irradiations are useful to increase the sensitivity of the analysis, but only up to a certain point. The result of all this is that the sensitivity of an analysis depends in practice on a number of practical as well as theoretical factors: