| The Geiger-Muller counter is used to study the properties of alpha, beta and gamma radiation. The statistical nature of counts from random sources is investigated. The Gaussian and Poisson distributions are compared with experimental distributions of counts, using least-squares fitting. |
The source used for most of the measurements in this experiment is a cobalt-60 gamma-ray source, encapsulated in a disk of orange plastic. Radiation levels of up to 10 or 15 mR/hr may be observed near the source. We will also use a beta source, Tl-204 (green plastic tabs), and an alpha-source, Am-241. Because the range of alpha particles is so small, the alpha source is less well encapsulated than the others, so be especially careful not to scratch the active surface.
Use the green survey meter (Technical Associates model PUG 1AB, with P-11 probe) to monitor radiation levels in the lab, and wear one of the XETEX model 415A personal dosimeters while in the lab. Record the dosimeter reading at the start of the lab period and at the end. Usual laboratory safety practices should be observed (e.g., no eating in the lab, wash your hands afterwards.)
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A. Setting up.
METHOD I: SCIENCE WORKSHOP
This method uses a computer running the Science Workshop software,
a Science Workshop interface, and a PASCO Geiger counter. The
electronics is pretty much a black box. Open Science Workshop, select
the Geiger method; connect the digital plug to A, and drag a data table;
select "counts in time interval;" figure out how to set the time interval
to 10 seconds. You are ready to go.
METHOD II: NUCLEAR ENTERPRISES SCALAR-TIMER
Here you should just try to get the equipment working, so
that you can measure some counting rates. Later on we will
investigate the principles of
operation of the Geiger-Muller tube.
You need a pulse counter and a source of high voltage for the Geiger tube. There are two suitable scaler-timers available - the Nuclear Enterprises model ST-5, and the Baird-Atomic model 955-151. For the purposes of this experiment they are equivalent. Get one of them and turn it on. Put it in test mode, and figure out how it works. You should be able to make it count for a preset time, and then stop, so that you can read the number of counts. Don't worry about all the other bells and whistles.
In using the Nuclear Enterprises counter-timers in digital-rate mode, note that, to get the actual number of counts, (a) the decimal point should be ignored, and (b) if a scale with a 3 is used, the result should be multiplied by three.
Now find a geiger tube, complete with sample holder and high-voltage cable. Put the Cobalt-60 source in the sample holder, as close to the window of the G-M tube as possible. Be careful of the mylar thin window - it is easily damaged. With the high voltage turned off, connect the cable to the "G.M." connector on the back of the scaler-timer. Set a very large preset time on the scaler-timer and start it running. Set the scaling threshold as low as it will go. Now carefully turn up the high voltage. Do not go over 1200 volts, or the G-M tube will be destroyed. At somewhere around 900 volts the scaler should suddenly start counting. You are counting single nuclear decays!
B. Alpha, beta, and gamma radiation.Here is a good time to play around. Take the source out. Do the counts stop? Can you detect
cosmic rays? (The flux of cosmic rays at sea level is about one per minute per square centimeter.)
(Remember, you should be recording what you do in your lab book.)
Get a 60Co
source (gamma, E = 1.3 MeV), a 204Tl
source (beta, no gamma), and a 241Am source (alpha). Try stopping each of these sources with
aluminum and lead absorbers. What do you see? How can you characterize
difference between
a,
b, and g
rays.
C. Counting statistics. Using the cobalt-60 source, do ten counts of ten seconds each. (Be sure that you are recording the total number of counts, not the number per second.) Using the "S" feature of Science Workshop (Method I) or Excel (Method II), calculate the average of these numbers and their standard deviation. This standard deviation, called sset, is the error on a single 10-second count. It should be about equal to the square root of the number of counts. Is it? This is the famous "square root of N" law of statistics, namely:
D. The Gaussian Distribution. You can use the random decays from a radioactive source to generate a Gaussian distribution, as follows. Make a histogram of a lot of values (at least 100) for one-second counts. (Method I: transfer the count values to Excel; Method II: make a histogram on the fly, without writing the values down, and enter the counts in each bin into Excel.) Use a bin width of one for your histogram. (You can always combine bins later.) Does the distribution look Gaussian? [Or does it look Poissonian? If it does, you'd better change something and repeat it. Ask the instructor if you are not sure.]
Draw in a "eyeball best-fit" Gaussian curve by hand, and estimate the central value x and the FWHM. Calculate sset. (Recall: HWHM = 1.18 s for a Gaussian.) Is this value consistent with the value expected from the square-root-of-N law?
There are some other checks you can do that the distribution is really Gaussian. (Method I: Set up to do a really high-statistics distribution while you are thinking.) Using your estimates for x and s, determine the 1 s, 2 s, and 3 s limits. Count the number outside, then use Table 1 (below) to predict the number outside. Does it agree, within statistical errors?
E. A Least-Squares Fit. It would be nice to test the hypothesis that the counts that you just recorded are Gaussianly distributed. You can do that by carrying out a least-squares fit to your histogram.
The general form for a Gaussian is
Here the central value x and the rms deviation about that value, s, are not assumed to have any particular relation. Use the least-squares technique to find the best estimate for x and F, and their errors. In our lab we can perform least-squares fits with the program MINSQ.
The least-squares fit yields values and errors for its parameters. Summarize the results. Then use these results to test a wide range of hypotheses concerning statistical nature of the distribution and the properties of radioactive decay.
Remember: testing a hypothes involves comparing discrepancies with errors and inferring the quality of the agreement, using Table I.
Get a printout of the best-fit curve and the data. You may want to use this as a figure in your first report.
F. The Poisson Distribution. While you are thinking about the implications of your Gaussian curve, set up for a high-statistics Poisson distribution (easiest for Method I). Reduce the average counting rate to between one and two counts per second by putting the source further away. Do another histogram of one-second counts. Can you see how its shape differs from the Gaussian distribution? Calculate the average count x for this distribution. This permits calculating the theoretical Poisson distribution for this histogram, using equation (2) and again multiplying by the total number of events. Try this. Does the theoretical prediction fit the data?
G. Pulse Shape (Method II only.). Do you still not have any idea what these pulses are that you are counting? There is a way to "see" them. Connect the Geiger tube to the scaler-timer through a splitter box, as shown in figure 1. This box provides a capacitativily coupled output for the
Roughly how long are the pulses? Are the all about the same size? Try varying the high voltage and see what happens.
A. The Geiger-Muller Tube. The Geiger-Muller tube is shown schematically in figure 2. It consists of a grounded outer cylinder with a wire along its axis. The wire is maintained at positive high voltage (about 1000 volts). The tube is filled with gas at a fraction of atmospheric pressure.
When charged particles pass through the tube they ionize the gas, and the electrons are attracted to the central wire. In the mode in which we use this tube, the electrons are accelerated to speeds such that they can ionize other atoms, and a spark in the gas results. The drop in voltage of the central wire is easily detected, and the tube is said to have counted.
The tube is dead for a couple of hundred microseconds after the discharge, until the electrons and ions recombine. This is a rather long dead time, making this type of counter suitable only for fairly low counting rates, up to a few thousand counts per second. In fact, a GM tube stops counting at all if the flux of charged particles is too large. When the Geiger counter stops clicking, look out!
B. Counting Statistics. Here we will sketch the derivation of the binomial, Gaussian and Poisson distributions.
The basic problem may be phrased as follows: Suppose we have n radioactive nuclei, with known half-life and probability of counting in a detector when they decay. What is the probability of recording k counts during a set counting interval? The general answer is provided by the binomial distribution. The Poisson distribution describes the large-n limit (k large or small), and the Gaussian distribution applies in the large-n, large-k limit.
Let p be the probability of a nucleus decaying and being counted during a set interval of time. Let . Then, the probability of nuclei one through k counting, and nuclei k+1 through n not counting, is pkqn-k. Some other set of k nuclei could also give the same number of counts, with the same probability. The other ways of getting k counts correspond to the distinct permutations of the labels of the n nuclei. There are n!/(k!(n-k)!) such permutations. (Can you show this?) So, the probability of counting k nuclei out of n is
(1)The large-n limit gives the Poisson distribution. Let us call the average number of counts x; it is fairly clear that, since p is the average number of counts from a single nucleus, then x = pn. Now, if n is large compared to k,
.We can also, in the limit k << n, approximate
.So,
.or
, k = 0,1,2, . . . .
(Poisson) (2)This is the Poisson distribution. It is applied, generally speaking, to situations where only a few events are counted. It is equally valid when large numbers of events are counted, but the Gaussian distribution can be used then, and it is easier to use.
For n, k, and n - k all large, we obtain the Gaussian distribution, given here without derivation:
, k = 0,1,2, . . . .
(Gaussian) (3)Examples of the Poisson and Gaussian distributions are shown in figure 3. The Poisson
C. Properties of the Gaussian distribution. The Gaussian is the famous Bell Curve of probability theory. It tells you how random events are distributed about their mean value.
For instance, it is easy to shown that the rms deviation of the distribution of k about x, its average value, is s = x1/2, the famous "square root of N" law of counting statistics. One can also show that the full width of the curve at half maximum (FWHM) is related to the standard deviation F by the formula
. (4)Often you want to estimate the standard deviation of a histogram - say the midterm grades for a class of 150 people. To do this, you plot the histogram, approximate it by a smooth curve like that of figure 3, measure the FWHM, and use the formula above to calculate s.
Once you know the standard deviation for a normally distributed variable, you can predict how many events (grades) should fall outside of certain limits. Table I tells the percentage of events outside of certain chosen limits. It tells you, for example, that 32 % of events are more than one standard deviation away from the average of the distribution.
| Table I. % of events falling outside of certain limits, for a normal distribution. (From the Particle Properties Data Booklet.) | |||||
|---|---|---|---|---|---|
| limit | % outside (low side or high side) | Quality of agreement | | | % outside (low side or high side) | limit |
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Method II