Calculating Radiation Exposure

Gary Schafer, 4 January 2015

You just never know what I might be interested in at any particular time. I don't even know, and I'm me. I have a front-row seat, as it were. Lately, I've been interested in radiation exposure. As a child, I was interested in nuclear energy. If it were not for my abysmal ability at the math involved in physics, I would probably have gone into nuclear engineering.

But I still like to dabble. Lately, I've been researching how to calculate radiation exposure given a certain amount of a radioactive element. To ensure I was understanding things properly, I decided to run some calculations and see how close I came. To do that, I needed to known values, some answers when I wasn't certain I understand the question correctly. That's where Rad Pro Calculator came in handy. I'd tripped across several websites that talk about radiation and radiation exposure, and many of them had links to this site. So I went to it, and used some numbers I'd come across on cobalt-60, a very strong gamma ray emitter.

This is where the problem begins. I cannot seem to reconcile the number I get from the Rad Pro Calculator with the number I get doing the calculation myself, which is different still from the numbers I get from the radiation material safety data sheet (MSDS) for cobalt-60.

Here's the problem. Cobalt-60 emits (essentially) two gamma rays each time an atom decays. Also, with it's fairly short half-life (1925.28 days, according to the National Nuclear Data Center), there are trillions of decays each second in just a gram of the stuff. A "Radioactive Material Safety Data Sheet from the University of Alaska Anchorage says that cobalt-60 provides an exposure rate of 370 microroentgens per hour per gigabequerel (uR/h/GBq) at a distance of 1 meter. This number is supported by information from Oak Ridge Associated Universities, which lists an exposure rate of 369.7 uSv/h/GBq. This is also supported a fact sheet from the Environmental Protection Agency, which lists a value of 370.3 uSv/h/GBq. (NOTE: Yes, I realize that I'm mixing exposure rates and equivalent dose rates. According to most radiation websites, the weighting factor for gamma rays is 1, so they're the same. Plus I'm on a roll. So shut up.) I'm going to assume that these three entities, a university, a US national laboratory, and a US governmental agency, are correct in their measurements and/or calculations. Therefore, when I do my own calculations, the equivalent dose-rate for 1 GBq of activity of cobalt-60 at 1 meter should 370 uSv/h.

Using an explanation from the Health Physics Society website, I was able to calculate the dose rate as follows:

  • X=(5.263e-6)*A*y*E*(μen/ρ)/r2, where:
    • X = exposure rate in roentgens / hour (R/h)
    • A = activity in bequerels (Bq)
    • y = number of photons emitted per decay.
    • E = energy emitted with each photon.
    • μen/ρ = mass energy-absorption coefficient
    • r = distance
  • First, set A = 1e9 because the already-known values (roughly 370 uSv/h/GBq) are on a per GBq (gigabequerel) basis.
  • y is not so simple. Cobalt-60 decays and releases two photons of unequal energy. One photon is 1.173 MeV, and the other is at 1.332 MeV. These are the values for E. They're important because the next value, μen/ρ, is energy-dependent.
  • Using the NIST (National Institute of Standards and Technology) website for mass energy-absorption coefficients of tissue-equivalent plastic, I had values of 0.03070 for 1 MeV, 0.02934 for 1.25 MeV, and 0.02805 for 1.5 MeV. Using linear interpolation, I obtained a value of 0.02976 at 1.173 MeV, and 0.02847 at 1.332 MeV. Combining this all together, I obtained the value for y*E*(μen/ρ) = [(1.173)(0.02976)+(1.332)(0.02847)] = 0.07283.
  • r = 100 cm (1 meter).
  • Putting this altogether, I get X = X=(5.263e-6)*A*y*E*(μen/ρ)/r2 = (5.263e-6)(1e9)(0.07283)/(1002) = 38.33 mR/h/GBq. Since this is a photon (gamma ray), and the weighting factor for photons is 1, this calculates to an equivalent dose of 38.33 mrem/h/GBq, which can be converted to 383.3 uSv/h/GBq. Note that my value (383.3 uSv/h/GBq) is off by roughly 4% from the values stated by the University of Alaska Anchorage, Oak Ridge National Labs, and the EPA. They all stated a value of 370 uSv/h/GBq. Mind you, it's higher, but that's okay. Conservatism in radiation safety is not a bad thing.

Which leads us to the Rad Pro Calculator website. I used their Gamma Emitter Dose - Activity Calculator. I entered the following values:

  • "Select Calculation" = "Activity and Dose-Rate"
  • "Add Shielding" = LEAVE UNCHECKED!
  • "Select Activity Calculation" = "Activity to Dose-Rate"
  • "Enter or Select Isotope" = "Co-60"
  • "Select Dose-Rate Units" = "uSv/hr"
  • "Select Activity Units" = "MBq"
  • "Enter Activity" = "1000" (NOTE: This is one GBq.)
  • "Select Distance Units" = "Centimeters"
  • "Enter Distance" = "100" (NOTE: This is 1 meter.)

When I pressed the "Calculate" button, I obtained a value of 306.37 uSv/h, which is the same as 306.37 uSv/h/GBq since I entered 1 GBq as my amount of activity. This value (306.37 uSv/h/GBq) is only 82.8% of the 370 uSv/h/GBq that the various academic and government agencies state, and slightly less than that of my calculated answer.

This, my friends, is my quandary. I can kinda / sorta see how I might have screwed up. I used "tissue-equivalent plastic" for the mass-energy absorption coefficient. Maybe I should have used something else? If so, I don't know what. Also, the calculations that the government agencies did were a lot more complicated. But that doesn't help with the HUUUGE difference I'm getting from the Rad Pro Calculator. I know, I know. It's the Internet and anyone can put up a website. This one was done by someone named Ray McGinnis. Unfortunately, I cannot find any information on Mr. McGinnis, nor his background or qualifications. I did find a "Contact Me" page on his personal website, and I've sent an inquiry as to why the discrepancy between his calculator and the published values. If I hear back, I'll update this post.

UPDATE: I got a reply from Mr. McGinnis. Long story short, Rad Pro Calculator uses George Chabot's equations from the Health Physics website, with the difference that he uses air absorption AND he accounts for attenuation in air, whereas this basic calculation does not. Thanks, Ray, for the feedback, and for helping me to better understand nuclear radiation.

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