Answer to Question #8270 Submitted to "Ask the Experts"

How did scientists determine the half-life of ²³⁸U, a primordial radionuclide, to be about 4.5 billion years and what is the measurement error for this half-life?

As you have apparently inferred, when a radionuclide has a half-life that is long compared to the time interval over which radioactive decay observations are possible, the overall decay rate remains substantially the same and experimental measurements of the change in the activity of a given sample with time are not sufficiently precise to allow determination of the half-life. In such instances, one must employ alternative techniques to evaluate the half-life.

In the case of ²³⁸U and some other long-lived radionuclides, one approach that has been used is to separate a pure sample of the radionuclide in a known chemical form, weigh the sample, and then measure the activity, A (disintegration rate). The half-life is then determined from the fundamental definition of activity as the product of the radionuclide decay constant, λ, and the number of radioactive atoms present, N.

One solves for λ and gets the half-life from the relationship λ = ln2/T_1/2. The number of atoms is determined from the measured mass of the sample, its fractional mass content of the radionuclide, and Avogadro’s number, N_o = 6.022 x 10²³ atoms g^-1 atomic weight.

For example, suppose we isolate 5.00 mg of pure ²³⁸UO₂, which contains 4.41 mg of ²³⁸U. The uranium decays 100 percent of the time by alpha emission. If the 5 mg were deposited in a thin uniform layer and counted for its alpha activity, and we obtained a count rate of 16.9 cps with an alpha detection efficiency of 0.315 counts per disintegration (Bq-s), we would then calculate

A = 1014 cpm/0.315 c d^-1 = 3219 dpm = λ N = (ln2/T_1/2)(4.41 x 10^-3 g/238.03 g/g-atomic weight)(6.022 x 10²³ atoms/g-atomic weight).

If we solve for T_1/2 we obtain T_1/2 = 2.40 x 10¹⁵ minutes = 4.57 x 10⁹ years.

This would compare to the presently accepted value of 4.468 x 10⁹ years. See the Chart of the Nuclides on the Brookhaven National Laboratory site. The estimated uncertainty in this value is approximately 3 x 10⁶ years. Naturally, the numbers used in the example were contrived, and the uncertainty in the result would have to consider all the uncertainties involved in the measurement. We have also not considered the complication associated with possible interference from ²³⁴U, which also occurs in natural uranium and also decays by alpha emission. Counting using alpha particle energy spectrometry is effective in separating the alpha particles from the two uranium isotopes.

There are other methods for half-life evaluation as well. Some include allowing the short-lived progeny, ²³⁴Th and ²³⁴Pa, to grow into the separated ²³⁸U and to count some of the progeny radiations.

It is also possible to make theoretical estimations of some radionuclide half-lives using a quantum-mechanical approach described as the Geiger-Nuttal Law, which provides a means for estimating the decay constant associated with alpha decay. If this is of interest to you, you might want to review a paper by Perepelitsa and Pepper that does some comparisons between the theoretical evaluation and experimental measurements for some alpha emitters. Hope this is helpful to you.

George Chabot, PhD, CHP