Answer to Question #12633 Submitted to "Ask the Experts"

Is there an established rule-of-thumb formula to calculate beta dose rates in air at various distances if the activity of the nuclide is known? I am especially interested in ²³³U.

In short, yes, there is a well-established and often-used expression for estimating dose rates in air from some beta-emitting radionuclides. The expression I am referring to applies only to point isotropic beta-emitting sources and is typically written as

D = 300 A d^-2,

in which D is the dose rate in rads h^-1, A is the source activity in curies, and d is the distance (in feet) in air from the source.

For SI units, the equation would be written as

D = 7.53 × 10^-8 A d^-2,

in which D is the dose rate in gray per hour, A is the source activity in becquerel, and d is the distance in cm from the source.

This equation is easily derived by writing the dose rate as the product of the beta particle fluence rate from a point isotropic source and the average value of the mass stopping power for the beta radiation in air. If the beta yield is less than one (or more than one in the case of serial decay), the expression would be multiplied by the fractional beta yield. The above expression provides a reasonable, though inexact, estimate for beta emitters of relatively high energies (maximum energy > 1 MeV) at distances up to about a meter and for sources of small physical dimensions and low mass (so as to minimize beta particle attenuation in the source and to simulate a point source). For lower energy beta emitters, the equation becomes less applicable but may still provide estimative values at relatively close distances from the source as long as the point source assumption still applies.

You note that your specific interest is in ²³³U. Since this radionuclide is an alpha emitter, I am assuming that you are concerned with aged ²³³U, which would have some beta-emitting radioactive progeny present. Below is a summary table of the radioactive progeny of ²³³U and their pertinent decay characteristics.

As you are likely aware, ²³³U is not a naturally occurring radionuclide. It can be made by neutron capture in ²³²Th, followed by two subsequent beta decays to yield ²³³U. As such, the ²³³U so produced and separated is initially free of beta-emitting radionuclides. The first radionuclide produced by the ²³³U decay is ²²⁹Th, which has a half-life in excess of 7,000 years. Thus, the ²²⁹Th will grow in quite slowly, although once it has been produced the subsequent radioactive progeny will grow in relatively rapidly, all having half-lives less than 15 days. The ²²⁵Ra, ²¹³Bi, and ²⁰⁹Pb may all contribute to beta particle emission. The reality is, though, that over any realistic time intervals since the ²³³U was originally produced, only a tiny fraction of the potential equilibrium of quantity of ²²⁹Th will have been produced. Production of ²³³U began in the United States in the 1940s, but most of the material was produced between the mid-1950s and early 1970s in support of the defense effort, the proposition being that ²³³U was a fissile material that had some advantages over ²³⁹Pu for nuclear weapons applications. Some ²³³U was also produced in conjunction with attempts to develop and use the ²³²Th-²³³U fuel cycle for ultimate commercial power generation.

To demonstrate how the simple dose rate equation might apply we shall assume a one-gram quantity of ²³³U produced in 1958, this would represent an activity of 3.57 ×10⁸ Bq. As of 2018, a 60-year period since production of the ²³³U, a tiny amount of this would have decayed, and the activity of ²²⁹Th that we would expect to be present would be (3.57 × 10⁸ Bq)(1 – exp((-ln2/(7.3 × 10³ y))(60 y)) = 2.03 × 10⁶ Bq. This would also represent the approximate activity of each of the three beta-emitters shown above in the list of decay products. Neglecting beta particle attenuation and assuming a point source geometry, we obtain, using the simple expression cited above, an expected dose rate at 30.5 cm (one foot) from the one gram of material of 

D = (7.53 × 10^-8) (2.03 × 10⁶)(3)/(30.5)² = 0.5 mGy h^-1

The factor of 3 in the above calculation is to account for the fact that there is one beta particle emitted per disintegration of each of the three beta-emitting progeny.

In reality, because many of the beta particles emitted would be rather low in energy, even the one-gram mass of material, as well as intervening air between the source and dose point, would produce some attenuation of the beta radiation. For the case of ²³³U, it is clear that you must be aware of the history of the material on hand—especially its age, in order to relate the measured dose rate to activity. Naturally, this simple dose equation is limited in application. If the source has extended dimensions and mass, the point source assumption may not apply, and attenuation may be very significant. In such instances, more sophisticated methods would be necessary to make meaningful dose rate predictions. We should also note that, depending on the steps taken in the production of the ²³³U, there are frequently radioactive contaminants present; the most likely such is ²³²U, which is a byproduct of the thorium irradiation process and has a 70-year half-life. This ultimately decays to stable ²⁰⁸Pb through several progeny that include three beta emitters, ²¹²Pb, ²¹²Bi, and ²⁰⁸Tl that could possibly add to detected beta radiation.

I hope this is helpful to you.

George Chabot, PhD