Experiment #4: Radiation Counting Statistics

Transcription

1 Experiment #4: Radiation Counting Statistics NUC E Radiation Detection and Measurement Spring 2014 Report Prepared By: Christine Yeager Lab Preformed By: Christine Yeager Martin Gudewicz Connor Dickey Lab Preformed On: February 27, 2014 Report Due Date: March 20, 2014 Lab Submitted On: March 20, 2014

2 TABLE OF CONTENTS Summary Introduction Theory 3 Equipment.. 3 Procedure... 4 Data Analysis of Data Conclusion Suggestions for Future Work References 10 Appendices

3 Summary The experiment Radiation Counting Statistics is to learn more about statistics and radioactivity is random but predictable. The normal distribution statistics, ratio test, Chauvenet s Criterion, Chi- Square, and the Poisson distribution have various ways of looking at the data collected. The radioactive material is defiantly random, but from statistics it can be predicted as to when the radioactive will decay. Introduction This experiment is to determine errors and statistics of radiation counting in the experiment. The two types of errors that could happen in an experiment is systematic and random error. Systematic error is in the measurement of data. Random error is from the randomness of radioactive decay. Plotting and analyzing the data collected allows seeing the probability of the data collected. Theory The different tests that can be performed on data creates a way to understand the material better. The normal distribution statistics, ratio test, Chauvenet s Criterion, Chi-Square, and the Poisson distribution all allow a different look at the data. The equations used in this experiment are found in the appendices of the lab manual experiment #4. Equipment The experiment was performed in room 112 of the Academic Projects Building. All the equipment and computers used to complete the lab are found there. In this experiment, the Geiger Mueller detector system that included equipment found in Table 1 was setup. In Figure 1 the GM detector system is shown schematically how it was setup. During the experiment a gamma source counts were measured for various times and multiple trials. Table 1: Equipment Equipment Model Model Number Serial Number Oscilloscope Tektronix TDS 1002 C GM Detector Ludlum 1A0-9 PR Pulse Inverter Ortec - - Single Channel Analyzer Ortec Amplifier Canberra Timer and Counter Ortec NIM Bin and Power Supply Ortec 4001C Detector High Voltage Supply Canberra 3002D Radioactive Gamma Source - - Low Activity Beta Source 40 KCl 3

4 Figure 1 Pulser Inverter Oscilloscope Timer & Counter GM Tube Amplifier SCA Shelf Box High Voltage Procedure The GM counting system with the oscilloscope was setup as shown in the Lab Manual in Figure 1. Then the background was counted. A beta source was then used as a reference to see if the equipment was working properly. A gamma source was then counted for a 20 second count for 20 trials and recorded. Then a low activity beta source was counted for 5 seconds 200 times. The data was then entered in EXCEL and showed to the professor or TA to make sure the data was in the general area of being correct. The background is then counted again to be used in the analysis. Data Table 2: 20 Trials Data for 20 Seconds Each Trial Counts Trial Counts Table 3: 200 Trials Data for 5 Seconds Each Found in Appendices 4

5 Analysis of Data Table 3: 20 Trials Standard Deviation, Theoretical Standard Deviation, and Statistical Tests from supplied spreadsheet 20 Trials Data: Theoretical Stdev Ratio Test Chauvenet's Crit Sample Mean: Standard Deviation: Chi-Square Table 4: 200Trials Sample Mean and Standard Deviation from supplied spreadsheet First Second 100 Sample Mean Sample Standard Deviation Table 5: Theoretical Standard Deviation for each trial given in Table 3 Found in Appendices 5

6 Occurences Radiation Counting Statistics Yeager March 20, 2014 Table 6: Histogram Development for the 200 Trial Sample Bin Actual Poisson Figure 2: 200 Trial Histogram Developments Trial Hisogram Development Actual Poisson Counts per 5 Seconds A. 20-Sample 1. Compute Xe and S for this sample, compare these results with those obtained using the computer. Also convert these values into units of counts per minute (cpm). From the experimental mean compute the best estimate of the true standard deviation, both in counts and 6

7 cpm. Compare the experimental standard deviation with the expected standard deviation computed from Xe. The results calculated by the computer and by hand are about the same, the rounding makes the results slightly different. 2. Compute σi for the first five of the 20 trials, assuming a Poisson distribution. Use Equation (7) in Appendix A. Compare these results with those obtained using the computer. Should they differ significantly from S or σ values computed in A-1 above? Explain any differences. σ 2 =30.32 σ 3 =31.70 σ4=31.30 σ 5 =30.55 They should not differ significantly from S or σ. 3. Apply the Ratio Test to the first two data points in the sample to test for statistically improbable behavior. How does this value compare with the corresponding computer value? The value calculated by hand is the same as the value calculated by the computer. 4. Apply the Chauvenet's Criterion to the first five trial results and to any trials that were identified by the Excel spreadsheet as not meeting the criteria. How well do your manually calculated values compare to the results obtained using the computer spreadsheet? What do these results tell you? τ 2 =1.35 τ 3 =1.42 τ 4 =0.613 τ 5 =0.903 The results calculated by hand and by the computer are about the same. These results say the equations in EXCEL are correct, and they all meet the Chauvenet s Criterion. 5. Compute the Chi-square for the 20 trial sample used by the Excel spreadsheet for this calculation. How well does your manually calculated value compare with that obtained by the Excel spreadsheet? What does your value say about the quality of your sample? cp20sec The calculated values from the Excel spreadsheet and by hand are about the same. The value of the quality of your sample is 10% probability the calculated value of Chi-square will be equal to or greater than. 7

8 Counts per Minute Radiation Counting Statistics Yeager March 20, If you had to reject a trial point from your sample as a result of applying Chauvenet's Criterion, recalculate a new experimental mean and experimental standard deviation and reapply the Chi-square test for the resulting set of the now reduced number of trials. Do these results agree any better with your estimate of the value of the true standard deviation? Has your Chisquare value improved over that obtained from the complete (20 point) data set? Explain any changes observed. If I had to reject a trial point from the sample, because it was the furthest from the average any value. The results are not that much different from one another, but it does agree slightly better with the estimated value of the true standard deviation, and the Chi-square value improved. This happens because since there are fewer values further away from the average the Chi-square value improves. 7. Sum the three backgrounds and sum all of the 20 sample counts, obtaining in this way an equivalent 10-minute background count and an equivalent 400 second source + background count. Assuming Poisson statistics, computers ± (σsr) for the net count rate, and express it in units of cpm. How do these results compare with those obtained in A-1 of this section? Explain any differences. These results are about the same as in A-1 section. The slight change could be from the counts that only had background and no source. The results from this a lower than A-1 section because of the background/ no source counts. 8. Use your 20 data points set to create a control chart. Evaluate the data set using the 4 different criteria given to you in the lectures and determine whether or not your counting system is operating correctly. Figure 3: Control Chart for 20 Trial Data Control Chart for 20 Trial Data Trials 8

9 The counting system is operating correctly because the LCL is 4300 cpm, the LWL is 4400cpm, UWL at 5100cpm, and UCL is 5200cpm. 9. As a result of these tests, state your conclusion as to whether or not your sample belongs to the same random distribution. Provide the basis upon which your conclusion is made. The sample used in this experiment belongs in the random distribution, because the number of cpm varies by 500cpm in some areas. Five hundred cpm is a large number in the statistics of this lab report. B. 200-SAMPLE 1. Compute the experimental mean for this data set (note: you can simplify this calculation by determining and making use of the frequency distribution of all recorded trial values). How well does your frequency distribution and experimental mean compare with that generated by the Excel spreadsheet? The experimental mean for this data set is 3.49cp5sec. The frequency distribution and experimental mean very close. The values are almost the same in the experimental mean and the Excel spreadsheet. 2. Compare the frequency distribution of your data to the Poisson distribution calculated from the experimental mean. Plot both distributions together on the same plot and comment on similarities and differences. The plot is in the Data section in Figure 2. The plots are not equal but they are close to each other. Often the Poisson is higher than the actual, and this could be from random errors. 3 Compute the theoretical standard deviation (σ) for your sample. How does this value compare with that obtained experimentally? The computed theoretical standard deviation and the experimentally computed value are similar. The standard deviations are similar enough to be considered the same. 4. Based on your background data, what is the lower limit of detection of your system? The lower limit of the detection system is 0.37 Problems 1. A rule of thumb used many times in counting is that the standard deviation should 9

10 not exceed 1% of X. Show that X = 10 4 counts satisfies this rule of thumb. Stdv = 0.864% for X= If 30 minutes of total counting time are available, calculate the t+ and tb which will minimize (σsr) for rb = 30 cpm and r+ = 100 cpm. 3. Based on your background data, what is the lower limit of detection of your detector system? The lower limit of the detection system is 0.37 Conclusion Radioactive materials will not decompose a definite way, but they can be predicted. Statistics show how probable it is for something to happen. The many ways to do statistics and to preform tests on collected data creates a way to understand the material. The radioactive material tested is random because of its radioactivity, but it is similar to previous tests and experiments. Since it is similar the results can be compared and a more accurate prediction can be calculated. Suggestions for Future Work For this experiment there are no suggestions for future work. This experiment accomplishes the goal of expanding our knowledge of statistics. References Nuclear Engineering 450 Radiation Detection and Measurement Laboratory Manual, by Dr. J. S Brenzier, Dr. I. Jovanovic, Dr. R. M. Edwards, Dr. W. A. Jester, Dr. M. H. Vonth, and Dr. K. Unlu. Radiation Detection and Measurement 4 th edition, by Glenn F. Knoll Appendices Table 3: 200 Trials Data for 5 Seconds Each Run Value Run Value Run Value Run Value

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12 Table 5: Theoretical Standard Deviation for each trial given in Table 3 Run Value Run Value Run Value Run Value

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