Physics Experimental Physics Temple University, Spring C. J. Martoff, Instructor

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1 Physics Experimental Physics Temple University, Spring C. J. Martoff, Instructor Physics 4796 Lab Writeup Counting Statistics (or, Is it Radioactive?) 0.1 Purpose of This Lab Exercise: Demonstrate principles of counting statistics including trial-to-trial variablity, significance tests, detecting a signal in the presence of background. Counting statistics are among the simplest example of statistical principles discussed in the text. Statistical fluctuations of the kind explored here are of crucial importance in finance (loan reserves), communication systems design (network capacity) and public policy debates on topics like the health risks of cellphones and nuclear power. Approximate time needed: one lab session for measurement, one for calculation and discussion with instructor. References: An Introduction to Error Analysis: John R. Taylor, Ch. 3 (particularly Section 3.2 ff). 0.2 Apparatus Setup The experiment uses a very simple Geiger-Muller counter radiation detector borrowed from the lower division labs. This should help focus your attention on the counting statistics rather than the apparatus. More refined nuclear counting gear will be used in a later lab this semester. The instructor will demonstrate the use of the control unit for the detector. The control unit is used to set the high voltage applied to the detector, and the desired length of time for which to measure. It has a display that shows the number of counts detected (number of energetic subnuclear particles interacting). These counts are the data in this exercise. Radioactive sources for this lab will be available from the instructor. If you lose one of these sources you will be condemning the instructor to endless Nuclear Regulatory Commission paperwork, with obvious (negative) 1

2 consequences for the naturally high regard in which the instructor initially holds you. The nuclear physics of the particular sources used will be explored in a later lab. For now we just regard the sources as a way to get counts that are randomly distributed in time and consequently obey the laws of counting statistics. These sources are so low level as to be quite safe. At a distance of 10 cm with no shielding, these sources give you a radiation dose rate about the same as natural background (several times 10 7 Sv/hr). For comparison, flying to Europe at a time of low solar activity gives you an additional radiation dose of at least Sv due to less atmospheric shielding between you and the solar and cosmic radiation. At the time of high solar activity or a big flare, this rate can be larger by a factor of 100 or more. Finding the HV Plateau Before using the detector, it is necessary to set the high voltage (HV) to a good value. Such a value would a) make the detector count particles efficiently b) make the counting rate for pulses from the source be reasonably independent of the HV setting. The second requirement implies that the exact HV setting used is not too critical. However the optimum setting may be a bit different for each Geiger-Muller tube. The process of determining the appropriate HV is called plateauing the counter. Place the source in the lowest slot of the detector stand and slip the detector into the top of the stand. Set the detector control unit to count for one minute. Measure the number of counts in one minute vs. the high voltage setting. These counts contain both signal plus background. Start from 500 V and go to 800 V in steps of 40 V. Write the measured counts in a data table as you go. Repeat with the radioactive source removed (at least 4 meters away from the detector) to measure the background by itself. Subtracting this background count from the (signal + background) measured before, gives the signal counts alone. Plot the counts for (signal + background), the background alone, and signal alone in your notebook as a function of HV. The (signal + background) plot should show a steep rise followed by a flat region and then another steep rise. Notice that the second steep rise occurs similarly in the (signal + background) and the pure background counts. It is therefore not due to counting of particles from the source. For the rest of the lab, set the HV approximately in the middle of the flat region of the signal plot ( on the plateau ). 0.3 Finally, Measure Some Counting Statistics Take a fifteen-minute count with the source present and one with the source absent. Use this data to calculate the time needed to get 225 counts with the source and without the source. Round this time off to the nearest 10 sec and call it T

3 Now measure the number of counts in T 225 seconds with the source and then again without the source. Do this 10 times and record the results. Analysis Pool the T 225 counting data from everyone in the lab. Prepare a frequency distribution histogram for the data. (See Section 5.1 of Introduction to Error Analysis.) Mark the number of expected counts (225) on the histogram. Compute the mean, standard deviation, and Full Width at Half Maximum (FWHM) of the data (most calculators and spreadsheet programs have a button for this) and mark these on the histogram in a sensible way. 0.4 Questions: Part I 1. Explain why the HV plateauing procedure leads to an HV setting, that satisfies the two criteria stated for a good setting. 2. What is the standard deviation of the number of counts in the set of ten T 225 measurements? What standard deviation would you expect from the discussion in the textbook? 3. The mean value of the ten T 225 counts probably did not come out 225 as you intended. There are two reasons for this: the systematic error caused by rounding off T 225 to the nearest 10 seconds, and the random errors caused by counting statistics. Use ratio and proportion to correct for the systematic error. (In other words, use your fifteen minute count and your rounded value for T 225 to calculate a corrected value for the expected number of counts.) Report the corrected value for the expected counts. Did the correction for systematic error improve the agreement with the data mean? 4. There is a formula for the statistical uncertainty in the mean of N measurements ( standard error of the mean )in the textbook. This formula predicts how much the mean of a set of ten T 225 counts would vary if several more sets of ten counts were made. Alternatively it gives an estimate for how far the mean of a set of ten counts should be expected to deviate from the true counting rate. What is the standard error of the mean for your ten T 225 counts? 5. Make a table of the means of the whole class sets of ten counts. How would you test whether these behave according to the prediction mentioned in the previous question? 6. Do your test. Do the observed mean values behave according to the prediction? 3

4 7. Assume that the true counting rate is given by the mean of the whole class fifteen-minute counts. Does the distribution of the means of sets of ten corrected T 225 counts agree with the prediction for standard error of the mean? Explain. Part II 8. Consider the problem of determining whether a well water sample is legal drinking water or not, according to EPA regulations. The regulations allow a MCL (Maximum Contamination Level) for betaand gamma- emitters of 5 pico Curie/liter (5 pci/l; 1 pci is decays per second). Say the results have to be back to the water utility within 24 hours, so you have a total of say 20 hours of counting time available. To simplify this exercise and focus attention on the counting statistics rather than the nuclear physics and detector technology, we will make some perhaps unrealistic assumptions about the measurement. We ll assume that when a 1 liter water sample is placed in contact with the Geiger-Muller tube, 1 out of a hundred gamma rays emitted from the sample actually gets counted. This is mainly due to the facts that the gamma rays go off in random directions from the sample, so most of them miss the detector completely. Even those that go through the detector don t all get counted, because gamma rays can go right through things without interacting. 9. With this assumption, estimate the rate of counts due to gamma ray interactions in the detector for a 1 liter sample containing gamma emitters at the MCL. 10. How much of an increase over the background counting rate does this interaction counting rate cause? 11. It can be shown that to measure such a small signal rate, the best strategy is to count with and without source (signal+background and background alone) for equal amounts of time. Based on your background measurements in class, how many background counts would you expect in a ten-hour count? 12. With this number of background counts, what is the minimum number of additional signal counts you could detect at the 95% confidence level (2σ level)? 13. How many pci/l would the sample have to have to give this many counts? 14. Say the detector s background counting rate were reduced by a factor of 100 while leaving everything else the same. How many pci/l could then be detected? 4

5 15. Say the detector were magically changed so that it counted 100% of the emitted gamma rays rather than 1%, while leaving everything else the same. How many pci/l could then be detected? 5