Answer to Question #13419 Submitted to "Ask the Experts"

Category: Radiation Basics

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


A simple equation is provided to determine the dose-rate from beta-emitters with energies between 0.5 MeV and 3 MeV. How can the dose-rate from a low-energy beta-emitter, sulfur-35 (35S) in this case, be determined?


I shall attempt to provide information to you, but there is a bit of uncertainty on my part as to your specific concerns. You mention a simple equation to determine dose rates from moderate to high energy beta emitters, but you do not state whether the equation applies to dose from an external source or a source internal to the body. I shall assume that the simple equation you mention is for an external source; one of the most common of such equations is given by:

D = k A d-2,

where D is absorbed dose rate, k is a constant, A is source activity, and d is distance from an assumed point isotropic source to the dose point. In the original form of this equation, D was in pre-SI units of rads h-1, k had a value of 300 rads h-1 Ci-1 ft2, A was in curies, and d was in feet. You can find further discussion of this equation and its use in our response to an earlier question 12633 on the Health Physics Society/Ask the Experts website. It is true that this equation is not valid at lower beta energies because it does not take account of beta attenuation between the source and the dose point, and the value of the mass stopping power that was assumed to generate the equation would not apply at the lower energies for which the stopping power values would be greater than those for higher energy beta radiation.

The maximum beta energy for 35S is 0.167 MeV, and the average beta energy is about 0.049 MeV. The average energy beta particle would have a range of only about 4 cm in air at typical room temperature; the maximum energy particle would have a range in the same air of about 30 cm. Thus, if the individual using the 35S has no exposed skin within about 30 cm of the source, we would expect no dose. In fact, because skin dose is typically calculated below a dead skin thickness of 7 mg cm-2, the distance at which any dose would be received would be reduced to perhaps 20 to 25 cm. Finally, the beta spectrum includes very few particles close to the maximum beta energy so that the dose is dominated by lower energy beta particles whose range is notably less than that of the maximum energy particle. The fact that very little dose is expected from the low yield high energy particles, the distance at which any significant dose would accrue would be even less. Of course, any clothing covering an individual’s skin would also attenuate the beta radiation. Additionally, depending on the physical nature of the source—e.g., a liquid solution, a covered source, or some other configuration—attenuation in the source and surrounding material would likely reduce the distance in air at which any dose would accrue to an even lower value.

Unfortunately, even at close distances from a low-energy beta source, the estimation of external dose to exposed skin is not straightforward because beta radiation is attenuated. Complicating things beyond this is the fact that the beta radiation may also scatter appreciably as it traverses material, and this must often be accounted for in any meaningful dose estimations. There have been some equations developed to approximate dose rates from point isotropic beta-emitting sources. These analytical expressions may also be integrated to determine dose rates associated with non-point source geometries. One of the earliest of such equations was developed by Loevinger.  Another is an equation developed by Chabot et al. (1988), which allows for considerations of other materials between the skin and source. A presentation and discussion of these is beyond the scope of this response; however, if you decide you want to review them, please contact me, and I will send you additional information.

I will note here that there is a popular computer code, called VARSKIN, available for doing a variety of dose calculations involving beta-emitting sources. It has gone through several versions; the current one is VARSKIN 6. The US Nuclear Regulatory Commission (NRC) website has a brief VARSKIN 6 summary. The page also has a link to the complete manual for the code. The Varskin 6 code is not presently available through the NRC web site, but may become available in the future. If your facility has a budget for software, you can obtain Version 4 through the Radiation Safety Information Computational Center (RSICC). You will need to register at the site.

To demonstrate the results you might obtain, I used an earlier version of the Varskin code to generate some dose rates for a point isotropic 35S source of 3.7 x 107 Bq.  I assumed air gaps ranging from 0 to 10 cm of air between the source and skin surface and a thickness of 7 mg cm-2 for the dead skin layer. As per recommendations of the NRC Occupational Dose Limits, I assumed a disc-shaped area of 10 cm2 as the area over which the live skin dose (shallow dose) should be averaged. Here are the results:

Distance in air (cm) Dose rate (mGy h-1)
0 1210
1 778
2 359
3 161
4 76.1
5 38.4
6 19.8
7 10.3
8 5.3
9 2.7
10 0

From a plot of these data one could easily interpolate values for intermediate distances and adjust for different source activities. Of course, these results are likely not fully realistic in that I do not know what the actual source geometry is and what additional attenuation might apply.I hope this is helpful.

George Chabot, PhD


Chabot GE, Skrable KW, French CS. When hot particles are not on the skin. Radiation Protection Management 5: 31–42; 1988.

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