Signal and Background in the Beta-Gamma Lab

The two major measurements you are asked to make in this lab are the range of the beta's and the absorption of the gamma rays. The basic technique is the same in both cases: Measure the counting rate as a function of the amount of material (Al for the beta's, Pb for the gamma's) between the source and the detector. But the method of analysis is somewhat different. To understand this, it's helpful to think about what it means to measure something in a 'counting' experiment when there is a background source of counts:

Signal and Background: Suppose that the background rate is 1 count per second (about what I measured), and that you have enough material in place so that only the (few)  most energetic electrons (recall Figure 2 in the lab manual) make it to the Geiger tube.  For definiteness, suppose that 'a few' means 1 beta every 10 seconds. Then in a 100 second measurement, we would expect 100 counts from the background, and 10 counts from the beta's, for a total of 110. (In any given measurement, we would expect fluctuations about the 100 and the 10 consistent with the Poisson distribution.)The 'signal' is the estimated number of beta counts, which is the Actual number counted A minus the Background B:



The error on S is given by our standard result for error propagation:


So in the particular case considered here, the error would


Thus, our error on the signal is  BIG, because the background is big, and we are trying to subtract two large (fluctuating) numbers and believe the small difference between them.
In this little model, the 10 signal counts were assumed to be measured in a 100 second interval, so we could equivalently report the counting rate R for the signal as



You should be able to show that if your counted for 1000 seconds, you could reduce the expected error to



and



for 10,000 seconds (i.e., about 3 hours!).

MORAL: Measuring small signals in the presence of large backgrounds takes lots of time. (Time that might be better spent designing an experiment with a smaller background!)

Now let's see how this enters into our analysis of the beta and gamma data:

Beta's: In this case you measure the counting rate as a function of the Al thickness, and try to estimate where it "levels off", that is, where the counting rate becomes compatible with the background. This is not quite as trivial as it sounds, since at values near the end of the range, you will be trying to understand if 110 counts is 100 backgrond + 10 signal counts, or 110 background + 0 signal counts. Don't be too surprised if your measurement of the range is somewhat low. Do try to discuss this in your lab write-up.

Gamma's: Here the method of analysis is somewhat different. For each thickness of Pb absorber, you need to caculated the true signal rate by subtracting the estimated background from the number of observed counts and propagating the error (as above when we calculated S). A plot of the S's and their errors as a function of the amount of absorber should then show a nice exponential (i.e, a straight line on semi-log paper).  
If you don't subtract the background, it won't be a (roughly) straight line. If you do subtract the background, but don't propagate the errors correctly, the center of the data points will be (roughly) on the straight line, but without the 'appropriate' amount of scatter. The expected amount of scatter can be specified rigorously (via the chi-squared distribution); a rough rule of thumb is that about 1/3 of the error bars should miss the best-fit line (this assumes that the error-bars are '1-sigma' errors and each error bar is statistically independent of all others).