experiment #4: radiation counting...
TRANSCRIPT
Experiment #4: Radiation Counting Statistics
NUC E 450 - 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
Radiation Counting Statistics Yeager March 20, 2014
2
TABLE OF CONTENTS
Summary ……..………………………………………………………………………………….. 3
Introduction ……….……………………………………………………………………………... 3
Theory …………………………………………………………………………………………… 3
Equipment …………………..…………………………………………………………………… 3
Procedure ……………………….……………………………………………………………….. 4
Data ………….………………………..…………………………………………………………. 4
Analysis of Data ……………………………...………………………………………………….. 5
Conclusion ……………………………………...……………………………………………… 10
Suggestions for Future Work …………………...……………………………………………… 10
References ……………………………………………………………………………………… 10
Appendices ……………………………………...……………………………………………… 10
Radiation Counting Statistics Yeager March 20, 2014
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 C020574
GM Detector Ludlum 1A0-9 PR217743
Pulse Inverter Ortec - -
Single Channel Analyzer Ortec 550 1047
Amplifier Canberra 2022 07033170
Timer and Counter Ortec 974 866
NIM Bin and Power Supply Ortec 4001C 00225581
Detector High Voltage Supply Canberra 3002D 07033297
Radioactive Gamma Source - -
Low Activity Beta Source 40
KCl
Radiation Counting Statistics Yeager March 20, 2014
4
Figure 1
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
1 995 11 920
2 919 12 981
3 1005 13 992
4 980 14 985
5 933 15 992
6 900 16 934
7 979 17 933
8 933 18 978
9 989 19 916
10 964 20 990
Table 3: 200 Trials Data for 5 Seconds Each
Found in Appendices
Shelf
Box
GM
Tube
Pulser
Inverter
High
Voltage
Amplifier
Oscilloscope
SCA
Timer &
Counter
Radiation Counting Statistics Yeager March 20, 2014
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.
995 31.54 1.23 1.1
919 30.32 1.35
1005 31.7 1.42
980 31.3 0.62
933 30.55 0.9
900 30 1.96
979 31.29 0.58
933 30.55 0.9
989 31.45 0.91
964 31.05 0.1
920 30.33 1.32
981 31.32 0.65
992 31.5 1
985 31.38 0.78
992 31.5 1
934 30.56 0.87
933 30.55 0.9
978 31.27 0.55
916 30.27 1.45
990 31.46 0.94
Sample Mean:
Standard
Deviation: Chi-Square
960.9 33.14 21.72
Table 4: 200Trials Sample Mean and Standard Deviation from supplied spreadsheet
First 100 200 Second 100
Sample Mean 3.47 3.485 3.5
Sample Standard Deviation 1.909334 1.878141 1.85592145
Table 5: Theoretical Standard Deviation for each trial given in Table 3
Found in Appendices
Radiation Counting Statistics Yeager March 20, 2014
6
Table 6: Histogram Development for the 200 Trial Sample
Bin Actual Poisson
0 5 6.13
1 22 21.37
2 36 37.23
3 49 43.25
4 37 37.68
5 23 26.26
6 12 15.25
7 10 7.59
8 3 3.31
9 3 1.28
10 0 0.45
11 0 0.14
12 0 0.04
13 0 0.01
14 0 0
15 0 0
Figure 2: 200 Trial Histogram Developments
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
0
10
20
30
40
50
60
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Occ
ure
nce
s
Counts per 5 Seconds
200 Trial Hisogram Development
Actual
Poisson
Radiation Counting Statistics Yeager March 20, 2014
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.
√ √ σ2=30.32 σ3=31.70 σ4=31.30 σ5=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?
| |
√
| |
√ τ2=1.35 τ3=1.42 τ4 =0.613 τ5=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.
Radiation Counting Statistics Yeager March 20, 2014
8
6. 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 Chi-
square 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
4400
4500
4600
4700
4800
4900
5000
5100
0 5 10 15 20 25
Co
un
ts p
er
Min
ute
Trials
Control Chart for 20 Trial Data
Radiation Counting Statistics Yeager March 20, 2014
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
Radiation Counting Statistics Yeager March 20, 2014
10
not exceed 1% of X. Show that X = 104 counts satisfies this rule of thumb.
Stdv = 0.864% for X=104
2. 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 4th
edition, by Glenn F. Knoll
Appendices
Table 3: 200 Trials Data for 5 Seconds Each
Run Value Run Value Run Value Run Value
1 4 51 5 101 3 151 2
2 5 52 1 102 2 152 4
3 1 53 4 103 7 153 2
4 3 54 5 104 1 154 3
5 6 55 3 105 2 155 3
Radiation Counting Statistics Yeager March 20, 2014
11
6 6 56 3 106 3 156 4
7 1 57 4 107 4 157 2
8 5 58 2 108 0 158 4
9 1 59 3 109 4 159 1
10 1 60 3 110 4 160 2
11 3 61 5 111 5 161 4
12 3 62 8 112 2 162 2
13 4 63 1 113 4 163 4
14 1 64 1 114 4 164 4
15 6 65 5 115 3 165 5
16 2 66 2 116 1 166 3
17 2 67 2 117 4 167 4
18 3 68 3 118 1 168 5
19 1 69 5 119 8 169 7
20 2 70 9 120 6 170 3
21 2 71 3 121 3 171 5
22 2 72 2 122 3 172 2
23 6 73 2 123 6 173 3
24 3 74 5 124 3 174 5
25 3 75 2 125 2 175 2
26 4 76 2 126 0 176 3
27 4 77 4 127 3 177 5
28 4 78 1 128 6 178 4
29 5 79 4 129 5 179 9
30 9 80 5 130 4 180 3
31 2 81 2 131 3 181 3
32 3 82 2 132 4 182 2
33 1 83 3 133 3 183 3
34 3 84 7 134 2 184 4
35 5 85 3 135 3 185 2
36 5 86 3 136 4 186 1
37 7 87 2 137 4 187 7
38 7 88 3 138 4 188 5
39 5 89 5 139 5 189 4
40 1 90 4 140 4 190 3
41 7 91 1 141 3 191 3
42 0 92 3 142 7 192 1
43 3 93 3 143 2 193 7
44 3 94 4 144 4 194 0
Radiation Counting Statistics Yeager March 20, 2014
12
45 3 95 3 145 2 195 0
46 6 96 3 146 4 196 1
47 6 97 2 147 6 197 2
48 2 98 3 148 8 198 7
49 6 99 1 149 4 199 4
50 3 100 6 150 2 200 1
Table 5: Theoretical Standard Deviation for each trial given in Table 3
Run Value Run Value Run Value Run Value
1 2 51 2.24 101 1.73 151 1.41
2 2.24 52 1 102 1.41 152 2
3 1 53 2 103 2.65 153 1.41
4 1.73 54 2.24 104 1 154 1.73
5 2.45 55 1.73 105 1.41 155 1.73
6 2.45 56 1.73 106 1.73 156 2
7 1 57 2 107 2 157 1.41
8 2.24 58 1.41 108 0 158 2
9 1 59 1.73 109 2 159 1
10 1 60 1.73 110 2 160 1.41
11 1.73 61 2.24 111 2.24 161 2
12 1.73 62 2.83 112 1.41 162 1.41
13 2 63 1 113 2 163 2
14 1 64 1 114 2 164 2
15 2.45 65 2.24 115 1.73 165 2.24
16 1.41 66 1.41 116 1 166 1.73
17 1.41 67 1.41 117 2 167 2
18 1.73 68 1.73 118 1 168 2.24
19 1 69 2.24 119 2.83 169 2.65
20 1.41 70 3 120 2.45 170 1.73
21 1.41 71 1.73 121 1.73 171 2.24
22 1.41 72 1.41 122 1.73 172 1.41
23 2.45 73 1.41 123 2.45 173 1.73
24 1.73 74 2.24 124 1.73 174 2.24
25 1.73 75 1.41 125 1.41 175 1.41
26 2 76 1.41 126 0 176 1.73
27 2 77 2 127 1.73 177 2.24
28 2 78 1 128 2.45 178 2
29 2.24 79 2 129 2.24 179 3
Radiation Counting Statistics Yeager March 20, 2014
13
30 3 80 2.24 130 2 180 1.73
31 1.41 81 1.41 131 1.73 181 1.73
32 1.73 82 1.41 132 2 182 1.41
33 1 83 1.73 133 1.73 183 1.73
34 1.73 84 2.65 134 1.41 184 2
35 2.24 85 1.73 135 1.73 185 1.41
36 2.24 86 1.73 136 2 186 1
37 2.65 87 1.41 137 2 187 2.65
38 2.65 88 1.73 138 2 188 2.24
39 2.24 89 2.24 139 2.24 189 2
40 1 90 2 140 2 190 1.73
41 2.65 91 1 141 1.73 191 1.73
42 0 92 1.73 142 2.65 192 1
43 1.73 93 1.73 143 1.41 193 2.65
44 1.73 94 2 144 2 194 0
45 1.73 95 1.73 145 1.41 195 0
46 2.45 96 1.73 146 2 196 1
47 2.45 97 1.41 147 2.45 197 1.41
48 1.41 98 1.73 148 2.83 198 2.65
49 2.45 99 1 149 2 199 2
50 1.73 100 2.45 150 1.41 200 1