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

Category: Radiation Basics — Doses and Dose Calculations

The following question was answered by an expert in the appropriate field:


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.

Radionuclide Half-life Decay mode (fractional yield) Maximum beta energy, MeV
sup>233U 1.59 × 105y Alpha (1.0)  
229Th 7.3 × 103 y Alpha (1.0)  
225Ra 14.9 d Beta (0.72 and 0.28) 0.322 and 0.362
225Ac 10 d Alpha (1.0)  
221Fr 4.8 min Alpha (1.0)  
217At 32 ms Alpha (1.0)  
213Bi 45.6 min Beta (0.32, 0.64, 0.011, 0.007) 0.98, 1.42, 0.32, 1.13
213Po 4 µs Alpha (1.0)  
209Pb 3.25 hr Beta (1.0) 0.65
209Bi Stable    

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

Ask the Experts is posting answers using only SI (the International System of Units) in accordance with international practice. To convert these to traditional units we have prepared a conversion table. You can also view a diagram to help put the radiation information presented in this question and answer in perspective. Explanations of radiation terms can be found here.
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