This review emphasizes the fundamental physical principles which describe the growth and decay of the members of a series of radioactive substances. The mathematical procedures are held to the minimum essentials. Graphs and tables are provided for the illustrative case of the “daughter products” of radon (RaA, RaB, RaC and RaC) whose interrelationships dictate many engineering requirements in underground uranium mining.
The mean life τ of a radioactive substance, or “radionuclide”, is often a more meaningful measure of its behavior than is its half-period T or its radioactive decay constant λ. For each radionuclide τ = 1/λ = 1.44T. From a long-lived parent the initial growth of the first radioactive decay product is linear with time, and is equal to the parent activity divided by the mean life τ of the first decay product. The initial growth of the second decay product is proportional to the square of time, and that of the third product is proportional to the cube of time. After these initial growths, and at time intervals which are comparable with the mean lives of the decay products, the growth of activity of each member of the series becomes more complicated. The nth activity always can be described by the algebraic sum of a series of n exponential terms, each term being characterized by the mean life of one of the n radionuclides in the series.
The “working level” (WL) unit of radon daughter product concentration and α-radiation exposure rate is defined and is shown to depend strongly on the airborne concentration of RaB, even though RaB emits no α-radiation. In air which initially contains radon but none of its decay products, the daughter product WL increases during the first 40 min approximately as the 0.85 power of time.
When a single filterable air-borne radionuclide of mean life τ is passed through an appropriate filter bed the activity I of this nuclide which accumulates in the filter in a time t is I = QCτ(1 – e-t/τ) where C is its filterable concentration in the air stream, and Q is the volumetric flow rate. The maximum activity QCτ in the filter just equals the activity flowing into the filter during one mean life. If the air contains 2 or more radionuclides, say RaA and RaB, at equal concentrations then the equilibrium activities of these 2 radionuclides in the filter will not be equal, but instead will be in the ratio of their mean lives, which for RaB/RaA is about 9. The growth in a filter of any mixture of air-borne RaA, RaB and RaC can be represented by a systematic array of six elementary growth curves, 1 for RaA, 2 for RaB and 3 for RaC.
The radioactive decay of any mixture of RaA, RaB and RaC can be determined without explicit mathematical formulation, using only the classical growth curves. This simplification of an otherwise complicated situation is accomplished by using the “summation rules” which connect growth and decay as complimentary processes, and which view any radioactive source as a summation of independent components, for example as a mixture of “red” atoms and “white” atoms. The summation rules are applicable to nonequilibrium mixtures, and to activities, number of atoms, disintegration energíes, components of working levels, interrupted collections, starting and stopping transients of activity in filters such as the human lung and dilution ventilation disequilibria.
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