THE GEIGER-MULLER TUBE AND THE STATISTICS OF RADIOACTIVITY

Transcription

1 GMstats. THE GEIGER-MULLER TUBE AN THE STATISTICS OF RAIOACTIVITY This experiment examines the Geiger-Muller counter, a device commonly used for detecting and counting ionizing radiation. Various properties of the counter are measured, and the analysis of counting experiments in general is studied. Theory: The Geiger-Muller Counter The counter consists of a cylindrical chamber (tube) with a wire stretched along its longitudinal axis and insulated from its walls. The chamber walls act as the cathode, and a positive voltage is applied to the wire (usually tungsten), making it the anode. The cylinder is filled with a low pressure gas mixture of argon and ethyl alcohol. When an ionizing particle passes through the gas in the counter it liberates electrons by collision, and these are attracted toward the centre wire (anode). As the electrons approach the high electric field near the centre wire (within a few diameters) they begin to pick up energy between collisions with gas atoms. Provided the applied voltage between cathode and anode is above the threshold value, the electrons pick up enough energy to ionize the atoms. The electrons produced in these ionizing collisions can also participate in the ionization process, causing an 'avalanche' of ion formation to occur. Atoms are excited by collisions with high-speed electrons at the avalanche site. The atoms generate photons as they decay to a lower energy state. These photons can ionize other alcohol molecules, and the resulting photoelectrons can cause further avalanches. Thus the discharge spreads along the tube, resulting in a positive ion sheath around the wire anode. Eventually the ion sheath reduces the electric field at the wire to such an extent that no further electron 'multiplication' can occur. When this happens, production of new avalanches ceases. The electrons striking the anode and the change in tube voltage due to the ion sheath produce a pulse at the anode which is amplified and counted. The ion sheath moves toward the cathode. uring this time, the counter is inoperative because of the reduced tube voltage due to the ion sheath. The time after one discharge has occurred until another can occur is called the dead time of the counter and is typically of the order of hundreds of microseconds. As the ion sheath moves toward the cylinder, collisions between argon ions and alcohol molecules result in the formation of neutral argon atoms and alcohol ions. The reverse process is energetically impossible. Thus when the sheath reaches the cylinder wall it consists only of alcohol ions. On striking the cylinder, the alcohol ion neutralizes and dissociates into two uncharged atomic groups. If, on the other hand, argon ions had been able to reach the cylinder, secondary electrons would have been produced. These electrons could then trigger another discharge, and the counter would recycle indefinitely. Counters containing polyatomic gases such as alcohol are called 'self-quenching', since the action of the gas prevents recycling, or continuous discharge, both by interaction with the argon ions and by absorption of photoelectrons. If, however, the operating voltage of the counter is set too high, the counter will go into continuous discharge because of the greater number of ions formed, not all of which can be neutralized by the alcohol before they reach the cathode.

2 GMstats. Once the ion sheath has dissipated to an extent that the counter voltage rises above the threshold value, passage of another radioactive particle through the tube will cause another discharge. If the ion sheath from a previous discharge has not completely dissipated when another discharge forms, the resulting voltage pulse will be smaller than if the tube had been in a neutral state. The time during which output pulses are smaller than normal is called the recovery time of the counter. Figure shows a schematic diagram of the Geiger counter; and the output of the counter, over a period of time, as displayed on an oscilloscope. Figure The operating characteristics of a G-M tube are commonly displayed by plotting counts per unit time (count rate) vs. anode voltage for a given source. The voltage at which the tube begins to count is called the starting voltage. The flat portion of the curve is called the plateau and corresponds to the Geiger region, the range of voltages over which "avalanching" occurs without continuous discharge (recycling) of the counter. A good counter has a plateau at least 00 V long with a small slope, thus fluctuations in the supply voltage will not affect the count rate. As the tube ages, the plateau shortens and its slope increases due to the increased probability of recycling as the alcohol gas is used up. A number of factors must be taken into account when analyzing counting experiments. Three of these factors are background, counter losses, and statistics. Background: the term applied to counts that are registered even when no radiation source is present. This is due to contamination in the lab from other experiments, building materials, the soil, and cosmic rays. The background counting rate must be subtracted from the total rate to obtain that due to the radiation source alone.

3 GMstats.3 Counter losses: As already mentioned, during the development of a discharge the voltage at the anode is lowered to such an extent that a second particle passing through at that time will not be counted. This dead time is denoted. If is the observed counting rate, the actual rate is given approximately by m m () m Since is small, of the order of 00 microseconds, this correction usually need only be applied for counting rates higher than 00/sec (i.e. when is significant compared to, say > 0.0). It is possible to determine the dead time of a counter by means of an operational technique. Suppose two sources S and S, when placed one at a time a certain distance from the counter, give true counting rates m and m. The counter, however, registers and due to dead time losses. When both sources are present the counter registers < + because of the higher counting rate and hence higher count loss to dead time. However, the actual rates should obey = +. Using equation () = + () m m m m m m (3) which can be solved for : m ( m m m m ) / mm (4) Since + is small compared to,, and, this may be expanded to yield m m m mm (5) Statistics of Radioactivity The count rate obtained in any time interval will fluctuate from the average counting rate over a long period of time according to the laws of probability. Counting experiments involving radioactive decay, or gamma radiation absorption obey a Poisson statistical distribution. For a single measurement of n counts the standard deviation is equal to n, and the result of a measurement is quoted as n ± n. (In some cases, the result of a measurement is quoted as n ± n since an uncertainty of corresponds to 95% confidence in the result). To obtain a relative uncertainty of %, for example, one must have n /n = 0.0, or n = 0,000 counts. One must always count long enough to obtain sufficient counts in order to have the desired accuracy. If N

4 GMstats.4 amount of runs are made under the same conditions, the results are given by n n / N where n is the mean. That is, the standard deviation of the mean of N runs, n, is reduced by a factor of N. The probable error for a single measurement is This means that if a series of measurements was made under the same conditions and the deviation from the mean was calculated, half of the deviations would be greater than 0.67 and half would be less. The Gaussian approximation to the Poisson distribution states that when counting radioactive decays (and other similar random events), the results of a series of count measurements repeated under identical conditions should be distributed according to the function: ( nn) / n ( n n) f( n) Ae Aexp n where n = mean (average) number of counts measured A = height of the distribution (Ideally, this should correspond to the 'number of occasions' that the average number of counts was observed, i.e. the largest # of occasions should occur for a count of n.) f(n) = predicted 'number of occasions' for a count of n to be observed Let N = total number of count measurements taken (i.e. number of count intervals) pi = number of occasions that a count ni was obtained then where n pn N i i of i i denotes count measurements with non-zero 'number of occasions'. Applications: The scientific applications of the GM counter are limited since it does not have any energy resolution. However, in situations where the radioactive species are already known, the GM counter can be used to monitor ambient radiation levels for safety purposes. A solid understanding of radiation statistics is important for anyone doing research in nuclear or particle physics. A reported value is only as good as its uncertainty, and the statistics of radioactive processes are an integral component of error calculations. Apparatus: The apparatus consists of the Geiger-Muller counter tube mounted above adjustable-height shelves on which the source may be placed, a SPECTECH ST360 counter/power supply/interface, and a personal computer (PC).

5 GMstats.5 The ST360 supplies the high voltage for the G-M tube and counts the pulses from the anode. The high voltage is adjustable from within the STX software. In addition to simply being counted, output pulses from the Geiger-Muller tube are stored in the PC for further analysis. In this experiment, the PC will be used to record the activity (count rate) of a radioactive source during a series of measurements made under identical conditions in order to investigate the statistical distribution of the count rates. A 36 Cl radiation source will be used in this experiment. It will be used to determine the starting and operating voltages and plateau curve for the G-M tube and to measure the dead time of the G- M tube. The PC will be used to study the statistical distribution of count rates from the 36 Cl source. The 36 Cl disk source should be handled with the tweezers provided. Procedure and Experiment: As mentioned in the theory, G-M counter tubes have finite lifetimes due to consumption of the alcohol gas, and excessive voltage will damage the counter. O NOT EXCEE 900 VOLTS. O NOT TOUCH THE EN WINOW OF THE COUNTER TUBE. etermination of Starting and Operating Voltages. Log-in to the computer (there is no password).. Turn on the ST360 controller/counter. 3. Open the STX program by double-clicking on the desktop icon.

6 GMstats.6 4. Place the 36 Cl source in the second shelf below the counter tube. 5. The starting and operating voltages will be determined by using the Plateau function of the STX software: a. In the Experiments menu, click Plateau b. Enter the following parameters: o Start: 750 o End: 900 o Step Voltage: 5 o Time per Step: 30 o Select the checkbox for Graph Results c. Click Run 6. Save the data as a tab-separated values file by clicking on the Save button. Give the data file a descriptive name. 7. Choose the operating voltage of the G-M counter to correspond to the middle of the plateau region, which should be between 830 and 840 volts. If you are not sure where to set the voltage, consult your instructor. 8. Set the operating voltage by going to the HV Setting item in the Setup menu. Also be sure to set the Step Voltage to 0 and the Step Voltage Enable to OFF. 9. Remove the source and determine the background radiation by making a 5 minute count measurement. (Go to the Preset menu and set the Time to 300 s and the Number of Runs to.) Measurement of ead Time. Use the following two source method (see equation () in Theory) to measure the dead time. Note that your 36 Cl source consists of two pieces. Measure m (the counts in 60 s for one half), then measure m (the counts in 60 s for both halves), then measure m (the counts in 60 s for the other half). The two halves must not be moved relative to the counter tube (other than to place them in the holder or remove them) in order to prevent errors due to altering the source geometry with respect to the counter tube. Taking n as the absolute error in a count n, determine the error in the dead time. Study of Statistics of Radioactivity. Place the 36 Cl source in the third shelf below the G-M tube. 3. ata will now be acquired for 3600 trials of s count measurements. Go to the Preset menu and set the Time to s and the Number of Runs to ata acquisition will stop automatically following completion of the 3600th trial. Note that due to lag time between the ST360 and the PC it may take longer than 60 minutes to acquire the 3600 trials.

7 GMstats.7 5. Save the data as a tab-separated values file by clicking the Save button (File menu). Give the data file a descriptive name. 6. Close the STX program and turn off the ST Open the data file in Excel. 8. Sort the data from lowest to highest according to the Counts column. In a blank column of the spreadsheet, enter the numbers starting with the lowest number of counts and ending with the highest number of counts, in increments of. 9. From the Tools menu, select ata Analysis, Histogram. For the input range select your Counts data, for the bin range select your lowest to highest number of counts, and click OK. 0. The new worksheet that is created contains the number of occurrences (Frequency) of each s count (Bin) that you measured.. This histogram data is now analysed using the radstats000.xls spreadsheet. Analysis: As mentioned in the theory, it is expected that the distribution of a series of radioactivity measurements made under identical conditions will follow a Poisson distribution. This will be tested by fitting a Poisson curve to the data, and by calculating various parameters from the data and seeing how well they compare with the corresponding Poisson parameters. An Excel spreadsheet that performs a complete analysis of the data (i.e. data consisting of a series of radioactivity count measurements made under identical conditions) is available for your use. In order to do the analysis, the spreadsheet requires that you input the lowest count with nonzero occasions, the highest count with non-zero occasions, and the number of occasions for each count from lowest to highest. The spreadsheet is obtained from the lab manual web page: :. Load the spreadsheet radstats000.xls. Be sure to save the file once it has loaded. If you are using a university computer with a network connection, save the file to your home directory (likely on the h: drive) by selecting Save As... from the File menu. The columns of the spreadsheet contain the following values: column A: a counting column (numbers increasing in sequence from ) column B: counts obtained in the selected time interval ( sec in this case) from lowest counts with non-zero occasions to highest counts with non-zero occasions. column C: number of occasions that the count listed in column B occurred. (Obtained from your data histogram and entered manually or by cut and paste.)

8 GMstats.8 column : theoretical number of occasions predicted by the Gaussian-approximated Poisson distribution, normalized to fit the total number of occasions and the average counts obtained experimentally. Columns F through J contain the results of intermediate calculations and need not be printed. column F: product of counts and number of occasions (column B entry times column C entry). column G: difference of actual counts and average counts column H: negative of number of occasions, if column G entry is greater than 0.67 times theoretical standard deviation. column I: number of occasions, if column G entry is less than 0.67 times theoretical standard deviation. column J: product of column C entry and square of column G entry (used to calculate the experimental standard deviation).. The data required by the spreadsheet is entered as follows: cell C4: lowest counts with non-zero occasions (as soon as the value is entered in cell C4, column B is updated to the proper range of count values). cell C7: highest counts with non-zero occasions column C: number of occasions for all counts values between these two values, inclusive. Once the data has been entered, the theoretical distribution is normalized to the experimental data to obtain the best fit. This normalization is done as follows: Using only integer values for # of occasions for peak of theoretical fit (cell 7), which corresponds to A, the height of the distribution, in the equation ( nn) / n ( n n) f( n) Ae Aexp n the value for cell 7 is chosen by trial and error such that the Total # of Occasions for Theoretical fit (cell 5) matches the Total # of Expt. Occasions (# of s trials) (cell 4) as closely as possible. The spreadsheet automatically generates a graph of the experimental and theoretical (fitted, Gaussian-approximated, Poisson) count distributions. (Click on the Chart tab to view the graph.) The numerical results of the spreadsheet analysis are presented in rows 4 to. Remember to save your file. The standard deviation,, of a Poisson distribution is = n. Compare this theoretical value (cell ) with the experimental standard deviation, exp, (cell ) as calculated by the spreadsheet using exp pi( ni ni) N i,

9 GMstats.9 For a Poisson distribution, the most probable error is 0.67 n (= 0.67). If the data obeys a Poisson distribution, half of the deviations from the mean should be greater than 0.67 n and half should be less. Compare cell 9, the total number of occasions for which n n < 0.67 n, with cell 8, the total number of occasions for which n n > 0.67 n. Are these numbers approximately the same? iscuss the result in terms of the previous paragraph. View the graph of the experimental and theoretical (fitted, Gaussian-approximated, Poisson) count distributions as produced by the spreadsheet. Qualitatively, how well does the theoretical curve fit your experimental data? o you feel justified in assuming that radioactive decay count measurements obey a Poisson distribution? NOTE: No error calculations are required for this part of the experiment (the statistics of counting experiments). The purpose is to determine whether or not the theoretical statistical uncertainty in a Poisson distribution ( n ) can be applied to experimental results obtained in counting experiments. If it is decided that counting experiments do obey Poisson statistics, then if a counting measurement is made once, and the value n is obtained, this value n is the best estimate for the expected mean count (if the experiment were repeated) and the best estimate for the standard deviation is n. That is, if one measurement is made of the number of events in some time interval, and the value obtained is n, then the value for the expected mean count for that time interval is n ± n. References: Bleuler & Goldsmith, Experimental Nucleonics, QC 784 Fretter, Introduction to Experimental Physics, QC 4 Halliday, Introductory Nuclear Physics, QC 73 Melissinos, Experiments in Modern Physics, QC 33 Taylor, An Introduction to Error Analysis, QA 75