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Geiger-Müller Detector and Counting Statistics 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. We will also use an investigation of the electronic workings of the detector as a vehicle for learning the operation of an oscilloscope, an important tool used in many areas of physics. References. Read before starting the experiment. Melissinos(1966) Chapter 5 on Radiation and particle Detectors Tait Chapter 7 on Gas Counters; especially “Geiger-Mueller Counters” on pp. 191-193. The following sections of Review of Particle Properties Section 32.4 of http://pdg.lbl.gov/2014/reviews/rpp2014-rev-passage-particles-matter.pdf Section 36.6 of http://pdg.lbl.gov/2014/reviews/rpp2014-rev-particle-detectors-accel.pdf Section 37.4 of http://pdg.lbl.gov/2014/reviews/rpp2014-rev-probability.pdf Section 38 http://pdg.lbl.gov/2014/reviews/rpp2014-rev-statistics.pdf 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 the Introduction of your report, be sure to describe “ionizing radiation” as it applies to this experiment. In the Theory section, 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. Theory In addition to analysis of the results, there is a lot of good physics to be seen in the construction of the Geiger- Müller tube itself. Assume the G-M tube is a cylindrical capacitor whose dimensions are given in the spec sheet and/or measured by you in the lab. Calculate the capacitance and compare to the value in the manufacturer’s specification sheet (Fig.1). From Gauss’s Law, calculate the electric field inside the G-M tube. According to Tait, what determines the rise time for the pulse? Calculate the rise time for your tube. (This does not require physics more advanced than sophomore, but does include a numerical integration.) Does the rise time depend on the supply voltage or another parameter you control? Compare to your measurements. Melissinos claims (p. 177) that “…the whole gas becomes ionized, and a discharge takes place.” Estimate the number of electrons and ions in the tube if this occurs. (Use the Ideal Gas Law; assume the neon pressure is 75 Torr.) Compare to the number of electrons you measure in a

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typical pulse. (How will you estimate the number of electrons in a typical pulse?) Will this be a topic for discussion in your report? (This task is repeated later in conjunction with analysis of how a pulse propagates through the circuit. You need do it only once.) Available Equipment

1. Radiation Sources Geiger-Müller Tube 2. Electronics board for pulse processing 3. Junction box: High voltage to G-M tube and pulse output (gray chassis box) 4. High voltage power supply

5. Oscilloscope 6. Computer with automated Counting Software Digital voltmeter

Figure 1 Geiger Experiment General Layout

Experimental Setup Familiarize yourself with the setup, starting with a comparison between the physical detector (Fig. 1 and 2) and the schematic diagram (Fig 3.). How does a Geiger- Müller detector work? What is its basic construction? How does it produce a “pulse” to signal the passage of radiation? See Appendices 1 and 2 as well as the Wikipedia article “Geiger-Muller Tube” for details. If you incorporate information from any of these sources in your report you must cite them! (To paraphrase Barack Obama: “You didn’t think up these words and concepts yourself.”)

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The operation of an oscilloscope is explained in many locations on the internet. Think of the oscilloscope as software: you learn how it works by playing with it. Your report should contain several paragraphs describing the operation of the oscilloscope: enough information so that the next time that you have to use one you can use your report as a reference. This portion (like all others) should also contain sufficient detail to convince the grader that you understand how an oscilloscope works.

Figure 3 Geiger Counter Circuit Diagram

Figure 2 Geiger Experiment Electronics Board

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What does the initial pulse look like? How do the pulse shape and height depend on power supply (anode – cathode) voltage? In sophomore physics you learned about the RC time constant of a resistor-capacitor combination. (Series or parallel? Does it matter?) Exactly how does the charge collected from the ionized neon atoms in the G-M detector work its way through the initial parts of the circuit (up to the op amp)? Are the pulse rise time and duration related to circuit components? How? Do your calculations based on your “circuit analysis” agree with measurements from your ‘scope trace? Investigate some characteristic times in the pulse generation system. Measure the pulse duration. (It will be some microseconds.) Calculate (estimate) the amount of charge in a typical pulse. (The oscilloscope measures the voltage across what? How do you get a current, and subsequently charge, from this?) It is claimed that all of the gas in the G-M tube is ionized by the passage of a “particle” of ionizing radiation. Do your data support or contradict this “model”? At the percent level or within an order of magnitude? 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. Note that the oscilloscope has [at least] six menu buttons that change the options available on the right of the screen. Items that appear on the right of the screen are changed by pressing the buttons to their right. An important option for investigating the long-term behavior of the Geiger counter is found under the “Display” menu: Set the “Persist” option to “Infinite” with a sweep time of several hundred µsec and watch the signal develop over several minutes. In this configuration the ‘scope will give you information about the dead time of the system. In principle you can investigate the source of the dead time by changing Rs; in practice, since the supply voltage appears across Rs, this must be done with extreme care or your personal “dead time” will begin very soon and last very long!) How does the appearance of pulses as seen on the oscilloscope screen depend on the trigger threshold setting? (Does “threshold voltage” refer to an adjustment on the Geiger circuit, the oscilloscope or both?) Record for inclusion in your report the pulse shape coming from (a) the G-M tube, and (b) the counter input. How does the number of counts measured under fixed conditions of the radiation source and G-M tube juxtaposition, depend on your setting of the comparator threshold voltage? How should the number of counts depend on this setting? Find the average of 100 measurements of the count rate when the threshold voltage is 1.0, 5.0 and 10V. Discuss in your report. Your report should include at least a brief description of what an oscilloscope does and why/how it gives information and why we should trust your operation of the device. How did you decide on values of important operating parameters for both the oscilloscope and the Geiger counter circuit? (“That’s the way it was set up when we got here” is

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not an acceptable explanation!). In this experiment we use model 712 G-M tubes from LND Inc. Their specifications are given below.

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 GM 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, where the output count rate remains approximately constant independent of . The operating point

Figure 5: Response curve of Geiger-Mueller counter

OP should be in the plateau region, far removed from the “knee” of the curve and from the region of spontaneous ionization. How will you determine the plateau region? This is actually a very important exercise in both particle physics and general experimental design. Assume that for a given source – Geiger tube configuration you can select a counting time that gives you about 500 counts. Assume that your plateau region is 150 V wide. (This is a number you will determine, but let’s start with this guess; it is certainly within a factor of two.) The manufacturer’s specification on the slope of the plateau is “less than 6% per 100 V, so you would expect to see a maximum increase of 45 counts across the 150 volt plateau. Obviously if the uncertainty in your measurement of the number of counts at a given voltage is ± 50 counts, you are going to have a credibility issue trying to claim that you know the slope is less than 6% per 100 volts. How about ± 5 counts? If your data are described by a Gaussian distribution, what will be their standard deviation…or is it the standard deviation of the mean that is relevant? How many times will you have to repeat the measurement to have 68% confidence that the mean of your measurements lies within 5 counts of the value you would get if you took an infinite number of readings at each voltage? This is not a trivial concept, but the ability to do the calculation is very important.

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Being able to do calculations like this has a huge significance for experimental design. These considerations are not unique to particle physics: the cutting edge of nearly all experimental physics is determined by noise in your measuring apparatus; that noise contributes the same uncertainty as random variations in particle counts. Optical measurements in nano- physics are often done with single photon counting systems.

First consideration before starting serious data-taking: which source to use and which side should be facing the detector? We have about a dozen sources available. One day all were investigated with the same detector at the same supply voltage. The number of counts measured one day in a certain window varied between 3.8 and 1300. Is the inherent activity of the source relevant to any consideration in your experiment? If so, you may want to investigate several options for the source you choose and the side of the source you place nearest the detector.

DO NOT OPERATE THE DETECTOR OVER 600V, even if you have NOT reached the upward swing located after V2 on the graph. We have routinely ruined Geiger-Mueller tubes before the onset of spontaneous ionization. Is there a difference in count rate depending on which side of the sample faces up or down? To revisit a consideration mentioned above, the manufacturer specifies a maximum slope for the plateau region. How does this specification compare to your data? Explain what is meant by “6%/100V”. If you simply plot your data and fit it, is the slope of your fit equal to this? Explain.

Next issue. It is generally considered to be a scientific faux pas to publish results which are characteristic of the measuring apparatus rather than the physical system being studied. One opportunity for checking such a possibility is to measure the count rate of your source as a function of the time window for which you measure the number of counts. Obviously the emission rate of the source could care less for how long you take data. Calculate the emission rate from a sample in a specific configuration for different measuring times. Do you always get the same answer? How certain are you that your measurements are different or the same?

2. Detection of radiation from an active source

For a given radioactive source, measure the count rate numerous times (at least 100) to create a good distribution of values. Appendix 1 explains how to use the computer to take data for this part of the experiment. 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. Which type

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of statistics do you think best describe this distribution? Try fitting it with the different statistical distributions. Remember that the statistical distributions describe probability densities. Refer to Appendix 2 for a suggestion on how to establish bin widths. Was your prediction correct? What is the reduced χ2 of the fit? Do the data support the model or not? (Obviously if you have one set of data and three different models, not all will be applicable!) You should be able to determine νσ,v and νσ for each of the fits. How do they compare to the values of νσ,v and νσ that you obtain by straight calculation? Also observe νσ,v and νσ as a function of the amount of data you take. Do this, e.g., for 2, 5, 10, 20, 50, 100, 200, 500 and 1000 measurements. You are also encouraged to take one dataset overnight or even over a weekend with a very large number of measurements (like 20,000). Discuss how your results vary as functions of how many data points you take.

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 in a histogram. Which type of statistics do you think can be used to describe this distribution? Try fitting it with the different possible statistical distributions. Remember that the statistical distributions describe distributions of probability densities. Was your prediction correct? What is the χ2 of the fit? You should be able to determine νσ,v and νσ from the fits. How do they compare to the νσ,v and νσ you obtain by straight calculation?

4. [Optional for 10 points extra credit] Binomial Distribution

It is widely known and accepted that the error in a binning experiment is given by the square root of the number of measurement values in the bin. This follows from the fact that the standard deviation of a Poisson distribution is given by ( )pNp −= 1σ and if the probability of the event occurring is small, ( ) pp >>−1 , so the square root reduces to

Np=σ . But Np is simply the average value of N, the distribution average, μ. To investigate a distribution where approximating the error by the square root of the average is a bad idea because the underlying assumption is incorrect, choose a time interval, T , for measuring the arrival of background radiation such that the mean number of observed background counts, μ, is significantly less than one, e.g. 0.2; your time interval will probably be less than 1 sec. For a given counting configuration, the average number of counts in a short time will be the same at the average for a long time, which can be determined either with the manual start/stop and a stopwatch or with the computer. (There is actually a function in Excel for counting the number of occurrences of some value.) According to the

Poisson probability distribution, you should observe zero counts about 82% of the time under such circumstances. Since the probability, p, of observing zero is not small, the statistics governing the number of observed zeroes are claimed to be binomial, not Poissonian as one might expect. To test this, perform the following procedure under the conditions listed above: (a) Take 100 measurements of the rate. Record the observed number of zeroes. (b) Repeat the above step 10 times. (c) Compute the mean and standard deviation of the number of zeroes in your 10 sets of measurements. (d) Compare your observed standard deviation with the predictions of Binomial statistics and Poisson statistics. (e) Discuss why, despite the fact that this “experiment” is somewhat contrived, it illustrates one common circumstances where one can err in accepting the “common knowledge” that the square root of number of counts in a bin gives a good estimate of its error.

References

[1] J.R. Taylor, An Introduction to Error Analysis, 2nd Ed., University Science Books, Sausalito CA (1997). Copy in the Undergrad Lab.

[2] D. Scott, “On optimal and data-based histograms”, Biometrika 66 605 (1979). [3] C. Melissinos and J. Napolitano, Experiments in Modern Physics, Academic Press, San Diego, CA (2003). Copy in the Undergrad Lab.

[4] W. H. Tait, Radiation Detection, Butterworths, London (1980). Copy in the Undergrad Lab.

Appendix 1: Automated data collection from Geiger Detector One of the primary strengths of computers is to perform a series of actions repeatedly and accurately. In the context of the Geiger counter experiments, this means automated data collection of counts.

There are several computer programs to readout the Geiger Counter written in Labview. These programs should be located on the PC in c:/LabViewVi/FTDI-Geiger. The front panel looks like the image below. The 'total reads' is where to enter the desired number of periods to collect data. The 'period (sec)' 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 period 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. When finished, a pop-up window will ask where to save data. The resulting text file has 3 columns of data. There are the histogram bin values, the counts in the bins, and then the raw numbers. These raw numbers can then be used when plotting a histogram of the data.

Appendix 2: Selecting a bin width

As a guide to choosing your histogram bin size, Reference [2] suggests the prescription

3/1

49.3N

W σ=

(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 question. (In counting experiments this is the only option that makes sense.)

An alternative procedure is provided below. This is clearly more work but should produce a superior product.

II. Calculate the mean and variance of the number of events as

III. Compute a formula*, and *

Finally the procedure probably most commonly used: play with the bin size until your fit is to your liking. The above option is a mathematical formalism for accomplishing the same task!

Enjoy!

IV. Repeat i-iii while changing . Find that minimizes .

*VERY IMPORTANT: Do NOT use a variance that uses N-1 to divide the sum of squared errors. Use the biased variance in the method. *To obtain a smooth cost function, it is recommended to use an average of cost functions calculated from multiple initial partitioning positions.

Shimazaki and Shinomoto, Neural Comput 19 1503-1527, 2007

I. Divide the data range into enter the i'th bin.

bins of width . Count the number of events that