Source-quest 2026!
Doc – I’ve got a question for you. I’ve got a 1”x1” sodium iodide detector and work for a company with a soil density gauge. I’m trying to figure out how close I need to be to detect a lost, unshielded source with my detector, or if I need to get one with a bigger crystal. The thing is, I’m not even sure where to start. Got any thoughts?
Cool – calculations! But, having said that, understanding the problem is the first thing to do; the math comes later, and the button-pushing is the last thing to do. And I’ll tell you – this sounds like a complicated problem, but you can break it down into a few fairly simple steps. Let’s see how it works out.
Step 1: how many gamma ray photons is the source emitting?
Step 2: how many of those photons will pass through the detector?
Step 3: how many of the photons passing through the detector will result in a count?
Step 4: what’s the lowest count rate that is a detection, rather than a statistical fluctuation in normal background levels?
So let’s go through them one at a time!
Step 1: how many gamma ray photons is the source emitting?
For this, we need to know how much radioactivity we’re trying to find. So let’s assume (to make the calculations easy) the source is 1 GBq (about 27 mCi), undergoing 1 billion decays every second. We’ll also assume that it’s a Cs-137 source that gives off 1 gamma for every decay, so a 1 GBq source will give off 1 billion (109 ) gammas every second.
For this step, you need to keep in mind that not every radionuclide gives off one gamma for each decay. Co-60, for example, emits two gammas with each decay while K-40 only gives off a gamma one time in ten. You need to include these factors (if they apply) to calculate the number of gammas that the source is emitting, based on the activity and the emission probability for the gamma radiation.
Step 2: how many of those photons will pass through your detector?
To make this one a little easier, let’s assume that you’re 1 m from the source; then we can come up with a number that can be corrected depending on the actual distance.
The gammas given off by a source are given off randomly in all directions, so we can start with calculating the number of gammas passing through each square cm of space at a distance of 1 m. The formula for calculating the surface area of a sphere is SA=4 x πr2 (or you can use an online calculator). So at a distance of 1 m, we have 1 billion gammas emitted every second that are passing through a surface area of 12.6 x 104 square cm; dividing 1 billion gammas per second by 1,260,000 square cm gives us a gamma flux of about 794 (let’s make it easy and call it 800) gammas passing through every square cm of space every second.
So that’s the gamma photon flux in space a meter from the source, but we’re trying to figure out the flux through your 1”x1” detector. Roughly speaking, the cross-sectional area of your detector is 1” x 1” = 1 square inch, or about 6.5 square cm. So if there are 800 gammas passing through each square cm every second, there will be about…let’s call it 5000 gammas passing through your detector every second.
Step 3: how many of the photons passing through your detector will result in a count?
For this one you need to know the odds that any single gamma will be detected – the detector’s counting efficiency. For Cs-137 passing through sodium iodide a reasonable first guess that it’s going to be about 10%; with 5000 gammas passing through your detector every second, a 10% counting efficiency means that about 500 gammas will deposit enough energy to be registered as a count, giving a count rate of about 500 cps. Since there are 60 seconds in a minute, your count rate (as measured by most instruments in the US) will be about 500×60 = 30,000 cpm.
Step 4: what’s the lowest count rate that is a detection, rather than a random fluctuation in normal background levels?
There’s a statistical calculation we can do, but let’s start with a somewhat easier look at the matter. As a rough rule of thumb, background count rate can spike up by a factor of two, sometimes three, but just for a second or two and rarely any higher than that. So if your expected reading is more than two or three times normal background count rate with your detector then you ought to be able to “see” the source. With a 1”x1” sodium iodide detector, background count rate is usually around 1000 cpm or so, depending on altitude and local geology. So with an expected count rate of 30,000 cpm a meter away, you could easily detect your sources if you were a meter away from them. And, in fact, you can get up to about three meters away before the inverse square law drops the count rate down to the point where you’ll need a more statistical approach.
I’m not going to get into those details here, but if you want to get into the calculations there’s a nice NRC document you can refer to: Lower Limit of Detection. You can also find an online calculator that makes things easy, as long as you fill in the blanks properly. The thing to remember with any calculations is that they assume you’re summing up all of the counts for a few minutes or longer – if you’re using an instrument that reads out in CPM, just list the counting time as 1 minute and enter the meter reading.
So…if you go through this and the expected count rate from the source you’re trying to detect using your detector is more than two or three times as much as background radiation levels then your instrument and detector – and your survey plan – are up to the task. If the calculated count rate is lower than this, you can use an online calculator to see how close to an actual detection you might be – if you’re close with your detector, source, and survey plan then simply counting for a minute longer at each survey location or spacing your survey lines a little closer together ought to do the trick.