Mass and activity

Where the number comes from

Specific activity follows from two facts and no measurements. A gram of a pure nuclide contains N_A/M atoms, where M is the molar mass in grams per mole. Each atom decays with probability λ = ln 2 / T½ per second. Multiplying gives a = ln 2 · N_A / (T½ · M) becquerel per gram. The only inputs are the half-life and the molar mass, and the calculator approximates the molar mass by the mass number, which is within 0.03% for every nuclide here.

The inverse relationship with half-life is the whole story. Tritium, with a half-life of 12.3 years, has a specific activity of 3.6 × 10¹⁴ Bq/g. Uranium-238, at 4.5 billion years, has 1.2 × 10⁴ Bq/g — ten orders of magnitude lower. A microgram of tritium is a serious source; a kilogram of depleted uranium is a paperweight that happens to be radioactive.

Why plutonium and uranium need care

The calculator returns the mass of the pure isotope. Real nuclear material never is one. Weapons-grade plutonium is mostly Pu-239 with several percent Pu-240 and smaller quantities of Pu-238, Pu-241 and Am-241 grown in from Pu-241 decay. Reactor-grade plutonium has far more of the heavier isotopes. Because Pu-238 and Pu-241 have specific activities thousands of times higher than Pu-239, a few percent of them dominates the total activity of the mixture even though they contribute almost nothing to its mass.

The consequence for accountancy is that grams and becquerels are not interchangeable through a single number unless the isotopic vector is known. To convert a real sample, apply this calculation to each isotope with its own mass fraction and add the activities. The reverse direction — inferring a total mass from a measured total activity — requires the vector too, and is much more sensitive to it.

Americium ingrowth

Pu-241 beta-decays with a 14-year half-life to Am-241, which is a strong 59.5 keV photon emitter. Plutonium that was almost purely an alpha and low-energy problem when it was separated becomes a measurable gamma source over a couple of decades. Any mass-to-activity figure for aged plutonium that omits the americium is wrong in a way that grows with time.

Practical uses

Three come up constantly. Converting a waste inventory between the mass basis used by a chemical process and the activity basis used by a disposal limit. Checking whether a quoted source strength is physically possible for the stated mass — a claim that fails this check is usually a units error. And estimating the carrier mass in a tracer preparation, where the radioactive atoms may be a vanishing fraction of the material actually present.

The last case is where intuition fails most often. A 37 MBq Tc-99m generator elution contains about two nanograms of Tc-99m. Everything else in the vial is carrier, saline and the long-lived Tc-99 ground state that the metastable state decays into.