Brian Sontag
Lab Partner: Robert Spivey
April 23, 2012
Section 3
Radiation Counting
Introduction:
The purpose of this experiment is to measure the natural radioactivity and 𝛾-radiation that
is emitted by Cesium 137 using a Geiger tube. Ionizing radiation occurs when particles liberate
electrons from atoms which create charged ions. In this instance, Cesium 137 (Cs-137) emits
𝛾-radiation which is high energy electromagnetic radiation. These rays are emitted from gamma
decay when the excited daughter nucleus from alpha or beta decay drops from a higher energy
state to a lower energy state. This drop emits a high energy electromagnetic wave. The source of
this decay is Cesium 137 which is a radioactive isotope of Cesium. It has a half-life of 30.17
years, and its decay energy is 1.176 MeV. In order to measure the measure the 𝛾-radiation of
Cs-137, the level of natural radioactivity must be measured too. Natural radioactivity or
background radioactivity is the measurement of radiation from cosmic, natural, and man-made
sources. Some examples are the sun, uranium deposits, and smoke detectors. The natural
radiation and 𝛾-radiation will be measured using a Geiger tube. A Geiger tube is filled with inert
gas that will be ionized by the radiation. Once the gas has been ionized, the electric charge will
interact with the electrodes lining the tube, creating a potential difference. This potential
difference is what is used to calculate the total number of counts.
The acquired data is compared using Poisson and Normal distributions. Poisson
distribution is a calculation of probability density equation the probability that any count 𝑥 will
be exactly 𝑘 if the expected count value is 𝜆. This is given by the equation:
(1) 𝑓(𝑘; 𝜆) = Pr(𝑋 = 𝑘) =
𝜆𝑘 𝑒 −𝜆
𝑘!
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The Poisson distribution will only fit if four assumptions are correct:
1. The probability of observing a single event over a small interval is proportional to the
size of that interval.
2. The probability of two events occurring in the same narrow interval is negligible.
3. The probability of an event within a certain interval does not change over different
intervals.
4. The probability of an event in one interval is independent of the probability of an event in
any other non-overlapping interval.
(What is Poisson, Children’s Mercy Hospital)
Another method for determining the probability is by using normal or Gaussian
distribution. This creates a bell curve that is centered on the area of highest probability. The
highest probability of a result thus is located in the center with a decreasing probability as it
deviates from the center. The equation for the normal distribution is given by:
1 𝑥−𝜇 2
1
(2) 𝑓(𝑥) = 𝜎√2𝜋 exp (− 2 (
𝜎
) )
In Equation 2, 𝜇 is the mean value, which in this case, is the average number of counts. 𝜎 is the
standard deviation which is given by the equation:
(3) 𝜎 = √
Σ(𝑥−𝜇)2
𝑛
In Equation 3, 𝜎 is the standard deviation, 𝑥 is a value from the population, 𝜇 is the average
number of counts, and 𝑛 is the total number of 𝑥 in the population.
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Additionally the attenuation of 𝛾-radiation by wood is also measured. Attenuation is a
physical constant of a material that measures how easily the material can be penetrated by the
electromagnetic radiation; which in this case is 𝛾-radiation because the wood stops the alpha and
beta particles from penetrating it. It is most easily found by slightly altering the Beer-Lambert
law:
(4) 𝜅 =
𝐼
𝐼0
ln( )
𝑥
In equation 4, 𝐼 is the emergent radiation intensity, 𝐼0 is the incident radiation intensity, 𝑥 is the
distance travelled, and 𝜅 is the attenuation coefficient.
In summary, the goal of this experiment is to find the Poisson distribution and normal
distribution of the 𝛾-radiation that is released without disregarding the natural radioactivity.
Additionally, this experiment will calculate the attenuation coefficient by using Equation 4.
Method:
In order to begin the experiment, the group must first verify that all of the equipment is
present; this should include: a PC, Lab Pro, Lab Pro power supply, a Geiger tube, some cables,
Logger Pro 3.8.5, a ruler, two wooden blocks, and a Cesium 137 sample. To begin, Logger Pro
must be installed on the computer that will be used for the experiment. If it is not, a CD with the
proper installation material is located at the lab bench. The proper setup for the lab should have
the power supply connect to the Lab Pro device and to an electrical outlet. The Geiger tube is
connected to the Dig 1 port of the Lab Pro. The Lab Pro is connected to the PC via a supplied
USB cable. Once the computer recognizes the new hardware, open Logger Pro. The
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experimenter should open the Lab Pro menu and assign the Radiation Monitor to the appropriate
slot, which for this experiment is Dig 1. The group experienced some trouble with the device
disconnecting itself once the data was gathered, so the students need to take verify its connection
after every test. The time is changed by selecting the clock icon and selecting “time-based.” This
will allow the experimenter to set the sample rate (seconds/sample).
The first section of data collection is the Poisson distribution for natural radioactivity. For
this section, the goal is to measure the number of counts for every second due to natural
radioactivity. This should be repeated a total of 300 times. The measurements are saved as an
xmbl-file, and the data in the table is saved in a text-file. This step can be repeated to allow a
larger sampling size.
The second section of the experiment is the normal distribution which calculates the
number of counts per second due to Cesium-137’s radioactive decay. Once the sample is
obtained from the instructor, it should be placed next to the radiation counter such that number of
counts per second is between 100 − 200
counts
. This was found to be a distance of roughly 4cm.
second
The yellow label needs to be facing the opening of the Geiger counter to obtain the most accurate
result. The group should now start the data collection to determine the total number of counts per
second, and the measurement is repeated 300 times. As before, the measurements are saved in an
xmbl-file and the data in the table is saved in a text-file.
The final section of the lab is to measure the attenuation of 𝛾-radiation by wood. Like the
section before, the Cs-137 sample should be placed in front of the Geiger tube, but this time it
should be as close as possible to the mica window. The group will then measure the total number
of counts for 100 seconds. Once this measurement is complete, the group needs to measure the
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attenuation of 𝛾-radiation by using the two wooden blocks to create four different attenuation
lengths. The wooden blocks are in contact with the mica window of the Geiger tube, and the Cs137 sample is placed at the other end of block so that it is still in line with the window. The
group then will start radiation counting for 100 seconds.
Results:
The Poisson distribution section was run a total of 300 times to calculate the number of
counts per second for background radiation. This yielded the chat shown below in Table 1. There
were no recordings of anything higher than 2 counts/second. 0 counts/second was recorded much
more frequently than any other result. This information was compiled into a histogram which is
also shown below in Figure 1.
0 Counts/Second
1 Count/Second
2 Counts/Second
231
57
12
Frequency of result
Table 1: This shows how frequent a given number of counts/second was recorded during a test
that took a total of 300 measurements.
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Histogram of Background Radiation
Frequency of Result
250
200
150
100
50
0
1
2
3
Number of Counts per Second (Counts/Second)
Figure 1: This histogram offers a graphical analysis of the information given in Table 1.
Using the information from Table 1, the experimental mean is easily calculated to be 𝜇 =
counts
0.27 second. The expected number of counts per second can more accurately be found using the
Poisson distribution which is shown in Equation 1. This yields the graph shown below in Figure
3. As you can see, the probability that there will be 0 counts/second is much higher than it is for
either 1 or 2 counts/second.
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Figure 2: This is a histogram showing the calculated frequency of distribution of the counts with
the experimental distribution.
Qualitatively, the theoretical and experimental values shown in Figure 2 seem to be in
agreement with each other. The values are within 1 or 2 percentage points of probability. To
quantitatively determine their level of agreement, a Chi-square test is performed which is given
by the equation:
(5) χ2 = ∑𝑛𝑖=1
(𝑂𝑖 −𝐸𝑖 )2
𝐸𝑖
In Equation 5, 𝑂𝑖 is the observed frequency of counts, 𝐸𝑖 is the theoretical frequency of counts,
and 𝑛 is the degrees of freedom. In this experiment, there is only one degree of freedom. The 𝜒 2
value is 2.744. This is below the cutoff value of 3.84 if it were above this threshold, then the
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null hypothesis would need to be rejected. The p-value for the 𝜒 2 level is 0.4327 which is the
chance that the difference between the experiment and theory could have happened by chance
assuming the distribution is Poisson at 5% significance. Since the p-value is about the cutoff
value of 0.05, the null hypothesis is not rejected and the distribution is Poisson.
The second section of the experiment is determining the normal distribution. Out of a
total of 300 experiments, the five highest counts per second are: 154, 148, 146, 144, and 143, and
the five fewest counts per second are: 91, 92, 93, 94, and 94. Based on these numbers, a normal
distribution histogram can be seen below in Figure 3.
Figure 3: The normal distribution showing the frequency of the number of counts/second.
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From here, the normal distribution, the mean number of counts per second (𝑥̅ ) is
117.206 and the standard deviation (𝜎𝑥 ) is 10.897. For a normal distribution, about 68% of the
measurements should be within 𝑥̅ ± 𝜎𝑥 . In this instance 67.7% is inside 117.206 ± 10.897
which is roughly equivalent to 68%. Additionally, about 96% should be within 𝑥̅ ± 2𝜎𝑥 . In this
experiment, 94.7% of the measurements are inside 117.206 ± 21.794. This is somewhat larger
than the difference should be, but it is still does not deviate significantly. Finally, less than 1% of
the results should be outside of 𝑥̅ ± 3𝜎𝑥 . In this experiment 0% of the measurements are outside
of 117.206 ± 32.691. In summary, the measurements do agree with the prediction according to
the normal distribution. Additionally, when performing a Shapiro-Wilke test for the goodness of
fit, 𝑊 = 0.9946 and the is 0.098 which shows that the distribution is normal, and the results
need not be rejected.
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The final section of the experiment is to determine the attenuation coefficient of wood. In
order to properly determine the number of counts due to the Cesium and not the natural
radioactivity, the number of counts needs due to natural radioactivity needs to be subtracted from
the counts. This is shown below in Table 2.
𝑪𝒐𝒖𝒏𝒕𝒔
𝑪𝒐𝒖𝒏𝒕𝒔
Length (𝒎𝒎)
Counts (𝟏𝟎𝟎 𝑺𝒆𝒄𝒐𝒏𝒅𝒔)
Adjusted Counts (𝟏𝟎𝟎 𝑺𝒆𝒄𝒐𝒏𝒅𝒔)
0
14367
14331
43.8
904
868
80
365
329
87.6
260
225
123.8
165
129
160
96
60
Table 2: This table shows the recorded number of counts for the various lengths created using
the wooden blocks. This is then adjusted to take into account the amount of natural radioactivity.
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Table 2 can then be transformed into a graph shown below in Figure 4. The amount of
error for each point is given by the scale on it. These are equal to the standard deviation 𝜎 =
±√𝑐𝑜𝑢𝑛𝑡𝑠. From the slope of this graph can be determined to be wood constant (𝜅) =
1
32.49455 ± 5.094441 𝑚. The error is found using Equation 4.
Figure 4: This shows the number of counts per 100 seconds and its alteration based of the
distance of wood. The slope of this graph determines the attenuation constant. The error bars on
each point is given by the standard deviation.
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The lab also had some sources of error. For example, the background radiation could
change based on the time. One example of how this could happen is if the experimenter was
using a cellphone close to the Geiger counter. This occurred at one point, but it is not known if it
affected the results. Additionally, the number of counts per second is significantly higher when
the Cesium-137 sample is placed directly outside of the mica window on the Geiger tube. One
reason is because the alpha and beta particle are completely blocked by the wood, so only the
gamma radiation can penetrate the block. Another reason is because as the sample is moved
away, the arc length of the wave that is measured decreases. This means that less is measured
overall.
Conclusion:
The purpose of this lab was threefold: to determine the natural radioactivity using a
Cesium-137, a Geiger tube, and Poisson distribution, to determine the normal distribution of
radiation, and to determine the attenuation coefficient (𝜅). The goals of the experiment were all
successfully met. The Poisson distribution for natural radioactivity is shown in Figure 2. When a
𝜒 2 -test is performed on it, it yields a value of 2.7445 which is less than the value of 3.84 so the
null hypothesis that the distribution is Poisson is not rejected.
The tests for the normal distribution also agreed with the theory. 67.7% is close to the
68% that should be inside 𝑥̅ ± 𝜎𝑥 , 94.7% is slightly more than the 96% that should be
inside 𝑥̅ ± 2𝜎𝑥 , and 0% is much less than the 1% that should be outside 𝑥̅ ± 3𝜎𝑥 . In addition,
the Shapiro-Wilke test yields a p-value of 0.098. All of these show that the normal distribution
does not need to be rejected.
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Using Equation 4 and the graph shown in Figure 4 show that the attenuation constant is
1
32.49455 ± 5.094441 𝑚. In conclusion, despite some inherent errors in the experimental
procedure, the experiment still yielded all of the expected results.
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Citation:
Attenuation coefficient. (n.d.). Retrieved from http://www.ndted.org/EducationResources/
CommunityCollege/Radiography/Physics/attenuationCoef.htm
Degrees of freedom. (2011). Retrieved from
http://www.statsdirect.com/help/basics/degrees_of_freedom.htm
Normal distribution. In Merriam-Webster Retrieved from http://www.merriamwebster.com/dictionary/normal distribution
What is normal distribution?. (n.d.). Retrieved from
http://www.childrensmercy.org/stats/definitions/norm_dist.htm
What is poisson distribution?. (n.d.). Retrieved from
http://www.childrensmercy.org/stats/definitions/poisson.htm