# Inverse Square Law in Beta Radiation

Setup: A radioactive source is placed at alternating distances from a Geiger Counter. Counts per second for 10 different distances (each spaced 1 cm apart) were taken for preset times ranging from 200 to 2000 seconds (depending on how far the source is from the counter).

Objective and Question: We are plotting counts per second vs source-to-counter distance for our sample, which emits beta radiation. We expect it to be $\frac{1}{r^2}$ dependance by the inverse square law, however we are getting less than $\frac{1}{r^2}$ dependance for larger distances (i.e) we are getting fewer counts than predicted by the inverse square law. At first we thought it was the air particles interacting with the beta particles, but that shouldn't have anything to do with it (according to the instructor), and the lab question hints it should have something to do with the geometry of the experiment. Can someone please help me understand this problem?

• Betas will range out in air given enough room, but without knowing something about your setup we don't have a chance of advising you on what might be affecting your acceptance. Commented Oct 15, 2013 at 16:45
• @dmckee I edited my question--what else do you need to know about my setup? Commented Oct 15, 2013 at 16:55
• Why do you think the air propagation distance won't decrease the counts? What is the characteristic penetration depth into air for the beta-particles you are using? How are you integrating your counts? Are you dividing the result by the detection time at each point? Commented Oct 15, 2013 at 16:58
• @DumpsterDoofus My instructor said it's not the air particles. For each run, I am dividing by the preset time for that run. For example, if I have 10000 counts for 200 seconds, my counting rate is then 10000/200=100 cps (counts per second). Commented Oct 15, 2013 at 17:06
• What are the physical dimensions of the GM tube and how is it oriented relative the source? Commented Oct 15, 2013 at 17:10

## 2 Answers

The reason why the inverse square law doesn't hold for a beta emitter is due to the fact that with a beta emitter we are dealing with a whole energy spectrum of beta particles (ranging from 0 eV to the maximum beta energy; in this case 2.28 MeV for 90Y). The low-energy betas are absorbed in the intermediate layer of air and the further you get from the source the more betas are absorbed by the air until you reach the maximum range in air (for the maximum beta energy 2.28 MeV of 90Y this range is 8.8 meter, for the maximum beta energy 0.546 MeV of 90Sr this range is 2 meter) and all betas are absorbed. The inverse square law for a beta emitter is valid up to approx. 30 cm in air, at greater distances the inverse square law deviates further and further.

Note: because the half-life of the daughter 90Y (64 hours) is very much shorter than that of the mother 90Sr (29 years), there is absolute equilibrium between mother and daughter, i.e. the activity of the daughter 90Y is equal to that of the mother 90Sr.

I don't know if you have already solved the problem, but the same think happened to me when I ran the experiment. It turned out that whenever using a 90Sr source, you actually have a 90Sr/90Y radioactive source and they both undergo beta decay, but with different energies (2.28 MeV and 0.546 MeV). So for short distances you have the sum of the two spectra and for larger distances only one. If you calculate the range of the 0.546 Mev particle, then you can calculate at which point the spectrum belongs to the 2.28 MeV particle only.