Experiment 21 Radioactive Decay

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Feb 20, 2024

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Report for Experiment #21 Radioactive Decay TA: June 23, 2023

Introduction Nuclear physics is the study that explains the atomic nucleus. There are stable nuclei, but unstable nuclei exist as well. As a result of being unstable, these nuclei decay and form other nuclei. Decay continues until a stable nucleus is formed. Radiation is emitted with each decay, which is why the unstable nuclei are considered radioactive. Radioactive materials are used in everyday household technology and we are regularly exposed to radiation. The atomic nucleus is the densely positively charged center of any atom which contains protons and neutrons. While protons and nuclei have the same atomic mass, protons are positively charged and neutrons are neutral in charge. Both particles are called nucleons. The sum of the nucleons defines the isotope, and the number of protons defines the atom. Electrons are smaller, negatively charged particles that orbit the nucleus with a large radius. During radioactive decay, there are several possible types of emissions. The emissions in today’s lab are alpha particle emissions, beta particle emissions, and gamma radiation. Alpha particles are helium nuclei (42He), containing two protons and two neutrons. When an alpha particle emission occurs the resulting nucleus has two less protons and neutrons than the parent nucleus. The alpha particle does not travel very far as it is stopped by multiple collisions with electrons. The electrons are knocked out of the atomic orbit as part of the ionization trail left in the alpha particle's wake. A beta particle is the electron emitted when a neutron spontaneously turns into a proton. The resulting nucleus has one more proton and one less neutron. Beta decay is the most common form of radioactive decay. Once a beta particle is emitted it travels farther than an alpha particle and collides with other particles and loses energy as it ionizes them. The last type of emission is gamma radiation. The number of protons and neutrons does not change during gamma emission. It does however usually occur alongside beta emission. Gamma rays also collide

mostly with atomic electrons, and when this occurs the electron acquires all of the energy. As a result, the electron is ejected with great kinetic energy. All nuclear radiation and emissions leave a trail of ionization. A Geiger tube, a metal tube with a wire through the center, detects radiation by detecting that trail. Electrons freed by ionization in the tube accelerate towards the wire in the middle of the Geiger tube. The electrons gather enough energy to cause further ionization and produce more free electrons. The cycle continues and a majority of the gas in the tube ionizes rather quickly. The gas is then conducting and the current pulse causes a break in the high voltage. This is registered by the computer as a ‘count’. The number counted by the computer is proportional to the amount of particles that go through the machine. However, for gamma emissions the number registered by the Geiger tube is less than the actual amount of gamma particles. This occurs because a gamma ray can pass easily through the window without interacting with the detector. The efficiency of the detector, E, can be defined by: ε = N counts N decay This equation represents the ratio of the counts to the number of decays of the radioactive source that reach the counter. Gamma rays typically go undetected because they do not interact with the gas. Investigation 1 The first investigation involved the use of a SPECTECH ST-350 box. The voltage was set to 850 V, and the time was made 30 seconds. The count was taken 20 times and all entries were recorded. From the data, the average number of background counts was calculated and found to be 18, using the following equation: average = f 1 + f 2 + ⋯ + f n n The standard deviation was found to be 4.26738 and the error of the average was also calculated and discerned as 0.930054. The following was used where σ is standard deviation and N is the number of samples: δaverage = σ √ N This is interesting given that 4 of the 20 values, 20%, had at least 5 count/30s difference from the calculated average. None of the values were 10 or more counts away from the average. This is indicative of how much the values may or may not fluctuate. Table 1: Derived Variables from the Measured Data in Investigation 1 Standard Deviation Average Count Error of the Average 4.267379326 18 0.930053762

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Investigation 2 For the second investigation, the ST-350 was adjusted to count for 10 seconds, as opposed to 30. Additionally, cesium ( 137 Cs) was put into the Geiger so that its emissions and radiation could be studied. The rest of the procedure remained unchanged, and 60 measurements were recorded for the investigation. Using the data collected, a histogram was created to visually show concentrations and patterns. Figure 1: Histogram displaying the concentrations of counts measured in Investigation 2 The data was also used to calculate the average counting rate of the cesium, in Counts/10s, the square root of the average, and the standard deviation. Table 2: Derived Data from the Measured Counts in Investigation 2 Avg. Counting Rate ( n s ) Square Root of Avg ( √ n s ) Standard Deviation Error of Avg. ( δ n s ) 819.4 28.62516375 27.43201149 3.541457455

Clearly the data seems to be supported by the fact that there is only a 1.193152268 difference between the square root of the average and the derived standard deviation. Additionally, 42 of the data points, which equates to 70%, fall within n s ± √ n s which is approximately equal to plus or minus one standard deviation. More importantly, 100% of the data points fall within ±2 standard deviations of the mean. This not only meets the 95% standard but is higher. This indicates little error within the experiment and strongly supports any findings from the data. It is also important to correct for the average baseline radiation to find the accurate emissions from the cesium. This would be the average found in investigation 1 subtracted from the average found in this investigation. The result is 813.9 counts/10 seconds. Investigation 3 It is given that 50% of gamma rays interact with electrons inside an absorber of x thickness, which is known as the half-value layer (λ). The intensity of the gamma ray decreases exponentially with the thickness of the absorber. This can be demonstrated using: I = I 0 e − μx where I is the intensity after passing through a thickness (x) of absorber, I 0 is the intensity (count rate) incident on the absorber, and µ is the absorption coefficient of the material. This coefficient is connected to the half-value layer, λ, by: μ = ln2 λ ≈ 0.693 λ The investigation was conducted exactly the same way as before, only this time lead slides acted as absorbers and time was set to 60 seconds. There were 4 different lead slides given, all of different thicknesses: 0.25, 0.125, 0.064, 0.032 in inches. Using the 4 different slides, there were 8 combinations to produce different thicknesses that were tested. During the first investigation, the average baseline radiation was established to be 18 counts/30 seconds or 36 counts/60 seconds. The corrected values were put into the table below. Table 3: The thicknesses in inches of lead plates and the resulting counts from the Geiger Thickness (in) I corr Counts 0.032 318 0.064 207 0.125 164 0.25 128 0.314 124

0.157 191 0.282 130 Given the data, a plot was made of the natural log of I corr against the thickness of the slides. 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 6 f(x) = − 2.86 x + 5.64 Natural Log of I corr against Thickness (in) of the Absober Thickness (in) ln (I corr) Figure 2: The Correlation between the natural log of I and the absorbance coefficient The trendline is linear and the slope is equal to µ because: ln I =− μx + K So, µ must be equal to -2.8586 and K (represented as the y-intercept) must be 5.6384 according to the data. Given this information and a previously stated equation, it is clear that -2.8586 ≈ 0.693/λ. Therefore, the half-layer value, λ = -0.2421. This is inches, so when converted to centimeters, λ = 0.6149 cm which is fairly close to the given and known value of λ for lead, 0.635 cm. It is still within the experimental uncertainty. Conclusion In conclusion, the aim of this lab was to research the statistical character of radioactive decay and to study the propagation and absorption of Gamma-rays from a radioactive source. Our aim was to precisely count the number of each lead absorber thickness and determine the average and standard deviation for the resulting data. The lead absorbers not being directly above the cesium source was the only probable fault. The lab was successful overall since we were able to collect data for each count precisely and determine the average and standard deviation.

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Questions 1. a. Navg = 750 counts/ 5 mins = 150 counts/min, Stdev = √ 150 / min = 12.247/min b. Relative error is 12.247/150 = 0.0816 = 8.16% c. Relative error is 1/ ( √ 150 0.01 min ¿ = 8.16 min 2. P(x<n) = sum(i = 0 to n-1)(P(x=i)) λ (10 s) = 486/6 = 81 counts, n = 72 counts P(x<72) = 0.144951185332563 λ (60 s) = 486 counts, n = 432 counts P(x<432) = 0.00598602193551374’ <432 in 60 s about 4% as likely as <72 in 10 s 3. ln I = ln I 0 - µx plot of I vs x = linear with a slope of -µ e µx = 3 µx = ln 3 2.6457 cm 4. 1.5625% 5. 10.34 m Acknowledgments I would like to thank my TA for taking time out of her schedule to help me and assisting me throughout the lab.

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