Until a few years ago, it was difficult to measure the number of gamma rays of a particular energy that were being emitted by a mixture of radioactive isotopes unless there were only a few such gamma rays with very different energies. Today instruments are available that can really pick them out of a complex mixture. Thus it is usually possible to “separate” with electronic instruments the radioactive element we are interested in measuring. Some of the examples below will show how this might be accomplished. Each radioactive nuclide[4] also has a characteristic half-life,[5] which is a measure of how fast the radioactive atoms change (transmute) to atoms of another element. In a reactor, even while they are being produced in the target, atoms of the radioactive nuclide are decaying with the particular half-life of the nuclide. The mathematical laws that govern this process tell us that the number of atoms determines the amount of decay; i.e., the more atoms there are, the greater the amount of decay in a given period of time. (The fraction that decays in that time is constant.) As a result, the target eventually becomes “saturated”, that is, the rate of production equals the rate of decay. When the irradiation is first begun, the number of radioactive atoms increases steadily. But eventually, this rate of increase slows down until, at saturation, further irradiation no longer increases the number of radioactive atoms present in the target. [Illustration: The energy of a gamma ray is equal to the energy difference between the two levels involved in the gamma-ray emission.] An _energy level diagram_. The slanted arrows indicate radioactive decay by beta-particle emission. In each case, manganese-56 decays to a certain energy level of iron-56. On the right the energy of each level is indicated. Following a beta emission to a high-energy (excited) state in iron-56, one or more gamma rays are emitted until the nucleus is de-excited to the level marked zero. The vertical arrows indicate gamma rays emitted during the de-excitation process. The energy of each gamma ray is the difference between the levels involved in the change. The numbers above the vertical arrows indicate the relative proportions of gamma rays of different energies emitted from that level. The mathematical relationship that describes the irradiation process exactly is: A₀ = Nφσ (1 - e^{-λt})