PHYS 3719 – Spring 2015

Geiger-Müller Detector and Counting Statistics

*REPORT MUST FOLLOW THE FORMAT DISCUSSED IN LECTURE, NOT THAT OF THIS HANDOUT.* Note: it is expected that each question below be addressed in your report; e.g. the questions about how a Geiger-Müller counter works should be addressed in your Methods section. Note that there is not a linear mapping between the topics listed below and where you address them in your report; it is your responsibility to organize them into a report.

Purpose To understand how a Geiger-Müller detector works and, by counting radiation events; to test the model that the Binomial, Poisson and Gaussian probability distributions describe, in different regions of parameter space, the statistics of counting experiments; to test the model that the standard deviation of Gaussian and Poisson distributions is equal to the square root of the mean.

Introduction In this experiment we could potentially use α, β, γ, and µ radiation (the last of these are more widely known as “muons”). Since α particles have a very short penetration depth and there is no easily available source of muons, we will restrict our interest to β and γ. The sources we have available are 55Fe, 60Co, 137Cs, 133Ba, 14C, 22Na and 204Tl. Research the type of emission, energy and half life associated with each source. Be sure to cite any references used in preparing these descriptions. Have any of these sources lost a significant fraction of their activity since purchase? (Dates of manufacture are on each source). In this section of your report, describe why this experiment is important and the significance of “ionizing radiation” as it applies to this experiment. In the theoretical section of the Introduction, review the properties of the Binomial, Poisson and Gaussian distributions and their areas of applicability as well as expectations of the circumstances in which each apply in this experiment. Give the theoretical expression for the mean and standard deviation for each distribution.

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Experimental Setup Available Equipment Radiation sources Geiger-Müller (G-M) tube Electronics board for pulse processing Junction box: High voltage to G-M tube and pulse output (gray chassis box) High voltage power supply Oscilloscope Computer with automated counting software Digital voltmeter Familiarize yourself with the setup, starting with a comparison between the physical detector (Fig. 1 and 2) and the schematic diagram (Fig 3.). In this experiment we use model 712 Geiger-Müller tubes from LND Inc. Their specifications are given in Figure 4. You may wish to start by mastering the operation of the oscilloscope. A good place to start is with the internally generated 5 V sine wave. Operating instructions are given in a separate handout. How does a Geiger-Müller detector work? What is its basic construction? How does it produce a “pulse” to signal the passage of radiation? If you incorporate information from the handout in your report you must cite it. Let us investigate some characteristics in the pulse generation system: - What does the initial pulse look like? - Measure the pulse duration (it will be some microseconds). - Follow the pulse with the oscilloscope as it travels through the circuit. How does the signal

change throughout the circuit? Include images in your report. For a given source-detector configuration, are all pulses alike? What pulse parameters change with supply voltage? Document the pulse appearance as it goes through the circuit.

- How does the appearance of pulses as seen on the oscilloscope screen depend on the trigger threshold setting?

- Determine the effect of varying the comparator threshold voltage between 0.10 and 5.0 V. Taking enough measurements of the count rate to reduce the uncertainty in your measurements so that it is significantly smaller than the effect you are observing. (See the section on generating a response curve, below, for hints on how to do this.) Does the number of counts measured under fixed conditions of the radiation source depend on your setting of the comparator threshold voltage? Discuss in your report. What comparator setting will you use for the remainder of your experiments?

- Are the samples symmetric? Do you get the same number of counts from top and bottom?

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Figure 1. Geiger-Müller counter experiment general layout.

Figure 2. Geiger-Müller counter electronics board.

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Figure 3. Geiger-Müller counter circuit diagram.

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Figure 4. Geiger tube specifications (712).

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Experimental Procedure For a description of the operation of the Labview data collection program, see Appendix 1. 1. Establish an operating point for the G-M detector

Figure 5 shows an artist’s conception of the Geiger-Müller counter output count rate as a function of applied supply potential, Vs. The “plateau” region lies in the range 𝑉1 < 𝑉𝑠 < 𝑉2, where the output count rate remains approximately constant independent of 𝑉𝑠. The operating point (OP) should be in the plateau region, far removed from the “knee” of the curve and from the region of spontaneous ionization.

Explain the response curve of the Geiger-Müller counter in terms of the behavior of the electrons traveling through your circuit and the ionizing particles.

Measure the counting rate (or number of counts) as a function of the power supply voltage Vs, for Vs < 600 V (see NOTE below), for a given active source: - Assume that for a given source-Geiger tube configuration you can select a counting time

that gives you 500 counts in average. If your data are described by a Gaussian distribution, a) What will their standard deviation be?

- How many times will you have to repeat the measurement to have 95% confidence that the mean of your measurements lies within 5 counts (1%) of the value you would get if you took an infinite number of readings at each voltage?

- At each voltage, measure the number of counts for the number of measurements calculated in the previous step; make a histogram (see Appendix 2 for a suggestion on how to establish bin widths) and fit a Gaussian function to analyze if your data are

Figure 5. Response curve of a Geiger-Müller counter.

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effectively described by a Gaussian distribution; obtain the mean, the standard deviation and the standard deviation of the mean by “standard calculation”.

- Plot the mean of the number of counts with error bars as a function of the power supply voltage. This graph is the response curve of your Geiger-Müller counter.

- Perform a linear fit of the data in the plateau region, obtaining the slope and the intercept with corresponding uncertainties. Obtain the correlation coefficient and make a residual plot of the data as a test of linearity. Determine the operating point (OP).

- The manufacturer’s specification on the slope of the plateau is “less than 6% per 100 V”. How does this specification compare to your data?

NOTE: DO NOT OPERATE THE DETECTOR OVER 600V, even if you have NOT reached the upward swing located after V2 on the graph, because the Geiger-Müller tube will be ruined.

2. Detection of radiation from an active source

- For the operating point (OP) obtained, measure the count rate numerous times for a given

radioactive source to create a good distribution of values. Be sure to record the length of time for which the detector counts. Adjust the counting time and source-detector distance so that you get several hundred counts in each measurement.

- Plot the results in a histogram (see Appendix 2 for a suggestion on how to establish bin widths). Which type of statistics do you think best describe this distribution? Try fitting it with each of the three distributions we have studied. Which gives the best fit?

- Determine the mean and the standard deviation by “standard calculation”. Compare these values with those obtained using the predictions of this distribution for counting experiments.

- What is the reduced 𝜒2 of the fit? What is the p-value? Which of the models is supported by the data and which not?

- Repeat the measurements for different amounts of data you take, e.g., for 1, 2, 5, 10, 20, 50, 100, 200, 500 and 1000 measurements. Discuss how your results vary as functions of how many data points you take. Plot the mean, standard deviation, and standard deviation of the mean of your measurements as a function of the number of measurements. Discuss.

- For your data set with the largest number of measurements, plot the results as a function of

bin width. Which bin width gives the smallest value of reduced 2? Are your results in agreement with the “model” put forth in Appendix 2? Note that Origin not only bins your data but also “automatically” fits your binned data with Gaussian, Poisson and five other totally useless functions. At the top of the columns of fit data are the fit parameters

appropriate for each function. From these you can determine the value of the reduced 2 for each fit using the definition in Taylor eq. 12.7. To do this you will probably want to import your binned data into Excel, calculate the function value for each bin and insert the results into eq. 12.7

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3. Detection of background radiation In the absence of a source, your Geiger-Müller counter will detect background radiation at a rate of a count every few seconds. - Measure the count rate numerous times to create a good distribution of values for the

background radiation rate. - Plot the results. - Determine the mean and standard deviation by “standard calculation”. Compare these

values with those obtained using the predictions of this distribution. - Attempt to fit with all three types of distributions. Which gives the best fit? Quantify

your justification. Again, some options can be eliminated by judicious choice of experimental parameters.

4. Binomial distribution (Optional for extra credit) It is generally accepted among particle physicists that in any series of measurements, the

results of which could be binned, the uncertainty or error in the value in any bin was equal to

the square root of the value associated with that bin. Let’s try to understand where the

expression came from and determine under what circumstances we get in trouble for using it.

Let us begin by defining the result of an experiment to be a success if the outcome appears in

the ith bin of the distribution of measurement results. If the result shows up in any other bin

the experiment will be deemed a failure. Hence our binning experiment meets the

qualifications for a binomial distribution. In this case the number of successes is simply the

value associated with the ith bin: Ni, and the probability of a given measurement giving the

value Ni is simply given by pi = Ni /N, where N is the total number of measurements. The mean

number of measurements falling into the ith bin is trivially given by the usual expression: <N> =

N pi = Ni. The standard deviation in Ni is then given by the usual expression for binomial

distributions: 𝜎𝑖 = √𝑁𝑝𝑖(1 − 𝑝𝑖)

But Npi = Ni, just the number of counts in the bin. Normally you will have enough bins in your

distribution so that the probability of being in any one of them is relatively small, i.e. small

compared to one, so the (1-pi) factor above can be approximated by unity. Then we have,

𝜎 ≈ √𝑁𝑝, = 𝑠𝑞𝑢𝑎𝑟𝑒 𝑟𝑜𝑜𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑢𝑐𝑐𝑒𝑠𝑠𝑒𝑠.

Obviously this approximation fails if a large fraction of the total counts show up in only one bin;

for such a bin pi is no longer negligible compared to one. In fact, if pi ≈ 1, then we can

approximate the first factor of pi inside the square root by 1, and the answer becomes,

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𝜎 ≈ √𝑁(1 − 𝑝), = 𝑠𝑞𝑢𝑎𝑟𝑒 𝑟𝑜𝑜𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑓𝑎𝑖𝑙𝑢𝑟𝑒𝑠.

To test this, choose a time interval such that the mean number of observed background counts

is about 0.95; i.e., one or more counts occurs 95% of the time.

Since the probability p of observing one or more events is near unity, the statistics governing the number of observed zeroes are claimed to be binomial, not Poisson as one might expect. To test this, perform the following procedure under the conditions listed above: (a) Take 100 measurements of the number of counts measured in your short window. Record the observed number of measurements with one count or more; i.e., the number of measurements which result in successes. (b) Repeat the above step 20 times. (c) Compute the mean and standard deviation of the number of successes of your 20 sets

of 100 measurements. (d) Compare your observed standard deviation with the predictions of Binomial statistics

and Poisson statistics. Discuss the circumstances under which the uncertainty in the value associated with one bin in a histogram is equal to the square root of the value and when this is a bad approximation.

Writing the Lab Report The results of this experiment are embodied in 3 graphs, which must be included in the lab report. The graphs are: (1) the plateau curve of your G-M tube, with operating point (OP) indicated; (2) a good-statistics histogram of data with mean of several hundred counts, with a Gaussian function fit to the data superimposed on it; (3) a good-statistics histogram of data with mean about 1-3 counts, with Poisson and Gaussian function fits to the data superimposed; If you have opted to include the binomial test, plot a graph of your data mean shown three times, once with Poisson error bars, once with binomial error bars, and once with the standard deviation of the data. Comment on the relationship of the top error bar and a line of probability one. This is a fourth graph, which is optional. Of course your text should describe these results. Remember, keep the lab report short!!!

Reading Material [1] J.R. Taylor, An Introduction to Error Analysis, 2nd Ed., University Science Books, CA (1997),

Sect. 3.2, 5, 10, 11 and 12. [2] D. Scott, “On optimal and data-based histograms”, Biometrika 66, 605 (1979). Posted on

Canvas.

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[3] C. Melissinos and J. Napolitano, Experiments in Modern Physics, Academic Press, CA (2003), Sect. 8.3.4.

[4] W. H. Tait, Radiation Detection, Butterworths, London (1980), Sect. 7.4.

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Appendix 1. Automated data collection from a Geiger-Müller detector One of the primary strengths of computers is to perform a series of actions repeatedly and accurately. In the context of the Geiger-Müller counter experiments, this means automated data collection of counts. The computer program is written in Labview and is called FT232GeigerCounterArrivaltimes.vi. This should be located on the PC in c:/labviewvi and on the desktop. The front panel looks like the image shown in Figure 6. The “total reads” is where to enter the desired number of periods to collect data. The “Gate time (s)” is where to enter the counting duration. The indicator boxes “current count” and “mean” display data as the program is run. To run the program, first enter desired values in the “total reads” and “Gate time (s)” boxes then click the arrow button below the “Edit” menu. The program will begin by asking you for a file name. Start by defining a sub-directory for you and your lab partner, preferably not on the desktop but rather in the “users” directory. The program will run and plot a histogram with bin width equal to one during data collection. The counter board will flash one of the LED's during data transfer to the PC. The button labeled “stop” stops the counting and proceeds to the data saving screen. Prior to beginning data acquisition, a pop-up window will ask where to save data. Filename.dat contains the number of occurrences of each number of counts; it is directly useful for making a histogram. Filenameraw.dat contains the actual sequence of counts measured; it is most useful for transporting into a spreadsheet to calculate a mean and standard deviation.

Figure 6. Front panel of the program used for data collection from a Geiger-Müller detector.

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Appendix 2. Selecting a bin width 1) As a guide to choosing your histogram bin size, Reading Material [2] suggests the prescription

𝑊 =3.49𝜎

𝑁1/3 (4)

where W is the bin width, 𝜎 the standard deviation of the distribution and N the total number of histogram entries (measurements). As a practical matter, it may be preferable to give an integer width to your bins, close to the W recommended by the equation. 2) The procedure probably most commonly used: play with the bin size until your fit is to your liking.