The Practical Side of Radiation Safety – Quick Estimation
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The Practical Side of Radiation Safety – Quick Estimation

By Dr. Zoomie

Dear Dr. Zoomie – there’s a few places where it looks like I need to do some calculation or to record some measurements, but I’m not sure how accurate I need to be. How many decimal places do I need to have, for example when I’m trying to calculate the decayed activity of my sources, expected radiation levels when I add shielding, and that sort of thing?

I used to stare at the dial on my radiation instrument, trying to figure out the precise reading it was giving me. The problem was (and remains) that radiation happens randomly, so the count rate is always changing – the needle (or, today, the digital display) isn’t going to settle out on an unchanging number…which means that whatever number I write down is going to be a guess of some sort, and even the best guess I can make will have only a limited accuracy. And that’s OK, because something that was pointed out to me around the same time is that most radiation instruments are allowed to be off by up to 20% from the actual radiation dose rate or count rate. This suggests a good practical rule of thumb – that how you read your instrument doesn’t need to be much more accurate than the instrument’s accuracy.

Say, for example, that you’re trying to measure a radiation field of 10 mR/hr – as long as your instrument is reading between 8 and 12 mR/hr then you meter is OK (although we prefer to aim for somewhere between 9 and 11 mR/hr). So if you’re studying your instrument dial, which is fluctuating between 9.7 and 10.2 mR/hr, you don’t really need to figure out if the reading is 9.7, 9.8, 9.9, 10, 10.1, or 10.2 – they’re all close enough to 10 mR/hr that I’d be inclined to just call it 10, write that down, and move on to the next survey location.

The same holds true with calculations. I learned early on, for example, that there’s no need to remember more than the first digit of pi (you might remember pi – 3.14159 – from your high school math class, but it shows up repeatedly in science, including in radiation safety). I used to scoff at the occasional state government that attempted to legislate that pi be equal to exactly 3…until a colleague pointed out that, for our purposes, 3 is close enough. Similarly, if you’re trying to design radiation shielding (or to calculate the effects of radiation shielding), to calculate radiation dose from a contaminated floor or a wall, or whatever it is that you’re trying to figure out, you don’t necessarily need to crunch numbers forever – just long enough to be within the accuracy of your radiation instruments. In my case, I do a lot of my calculations in spreadsheets, which will churn out as many digits as you care for (I just cranked out the first 100 places to the right of the decimal point and could have kept going) – when colleagues review or check my work they sometimes have to remind me that I only really need two or three digits, and I can often only justify one or two. I usually default to using only two or three digits unless there’s a good reason to use more or less.

Another way to (sometimes) save some time on a calculation is to do a worst-case estimate and see how bad that is – if the worst-case estimate is OK then I know that the results from a more detailed calculation will be even further from a limit. For example, say I’m trying to determine whether or not I need to clean up a radioactive spill involving a short-lived isotope. One rule of thumb is that, after 10 half-lives, radiation dose rate will drop by a factor of about 1000 (and the total amount of radioactivity will decrease by the same amount). So when our nuclear pharmacy spilled some Tc-99m on a Friday afternoon I asked them what dose rate they were getting – when they told me it was about 1 mR/hr I told them to just lock up the room with the spill until Monday and everything should be OK. My thinking here starts with Tc-99m’s half-life of 6 hours; 10 half-lives is 60 hours, or 2 ½ days. Ten half-lives of decay will take dose rates from 1 mR/hr down to 1 microR/hr, which is far too low to cause any concerns due to exposure.

If contamination is a concern I can assume that Tc-99m has a gamma constant of about 0.1 mR/hr per mCi at a distance of 1 meter. So 1 microR/hr at waist height would come from about 10 microCi of nuclide. One of the odd facts rattling around in my brain is that 1 microCi has a decay rate of 2.22 million decays per minute so 10 microCi undergoes about 25 million decays per minute, or 25 million dpm. That’s not much dose, but it’s a lot of contamination since our goal is to get down to about 1000 dpm. On the other hand, after another 2 ½ days the activity will be down to 25,000 dpm, one more day (4 half-lives) will bring it down to about 2000 dpm, and another 6 hours will get us down to 1000.

Using these quick estimates, if I’m able to just wait for a little more than 6 days – maybe we can lock the room to keep people out – then I don’t need to do any decontamination. If I can’t take the room out of circulation for 6 days then I have two options – I can decontaminate the area with the spill, or I can cover it over (I’ve used plastic sheets and plywood in the past). I should say, too, that I left a bit of cushion in these calculations. For example, 10 half-lives actually reduces activity (and dose rate and count rate) by a factor of 1024, which is slightly more than the factor of 1000 I assumed, and the 1000 dpm goal I mentioned is actually the limit for 100 square cm (about 4” x 4”). But if the time frame I come up with (6 days – Thursday) is something I can live with then I don’t need to spend the extra time to find a precise answer. On the other hand, if I really need to get that room back into circulation on Monday then it might be worth spending a half hour doing some more precise calculations, pulling out the plastic, or taking a few hours to decontaminate the area.

I could keep going, but it gets sort of repetitive fairly quickly. The bottom line is that you can save yourself a lot of time by coming up with ways of making quick estimates of contamination or radiation levels – you only need to do more precise calculations if the quick estimate shows you might be close to a limit. And (going back to the first few paragraphs), knowing that your instrument reading is likely no more accurate than about plus or minus 10% of the actual reading, there’s no reason to spend time hammering out extra decimal places. Not to mention that there’s no need to pore over your meter, trying to figure the dose (or count) rate exactly – as long as you can determine the first two digits then you’ll usually be OK.