MASS ATTENUATION COEFFICIENT OF LEAD

Transcription

1 OBJECTIVE MASS ATTENUATION COEFFICIENT OF LEAD The objective of this experiment is to measure the mass attenuation coefficient of lead by manipulating Beer-Lambert s law of attenuation. INTRODUCTION Background & Theory Gamma rays can penetrate through material much further than alpha or beta particles, due to the high energy it carries and its lack of electric charge. Though this is the case, gamma photons can get attenuated in matter, and they do so in one of three possible processes: 1) Photoelectric effect: Gamma photons can interact with electrons initially bound to an atom to eject an electron from the atom. This effect can only take place if the gamma s energy is greater than the binding potential of the electron. All of the gamma s energy is transferred to this electron (hence, called a photoelectron); the kinetic energy of the ejected photoelectron is the difference of energies of the incident gamma and the bound electron potential. 2) Compton effect: Gamma photons can collide with the free electrons in a material and scatter, imparting some of its energy to the electron. The result is a deflected photon of longer wavelength (ie. less energetic) and an electron with additional kinetic energy due to the collision with the photon. The energies imparted to an electron are related to the photon s angle of incidence, and can be calculated using the Compton Scattering formula. 3) Pair Production: Gamma photons, when in the vicinity of the Coulomb field of an atomic nucleus, can materialize into an electron-positron pair. This phenomenon can only occur if the energy of the gamma photon is at least twice the electron rest mass energy (1.022 MeV). On average, the kinetic energies of the electron and positron are (each) half the excess energy of the incident gamma photon. The attenuation coefficient of any element is a summation of the attenuation-contributions of each of these processes; hence, the larger the coefficient, the more probability of attenuating the gamma radiation. The relationship between attenuation and probability of penetration without interaction is expressed via the Beer-Lambert law: N(t) = N ' (t)e )*+ (Eq. 1) Where N is the number of un-attenuated photons (per unit time), N ' is the initial number of photons (per unit time), when traversing a distance x in a material of linear absorption coefficient μ. 1

2 The mass-absorption coefficient is obtained by dividing μ by the density of that material. The following mathematical manipulation can be done on Eq. (1) in order to include the massabsorption coefficient: N(t) = N ' (t)e ). / 0+ (Eq. 2) where the product ρx is the surface density of the material. Surface densities are practically useful, especially when dealing with non-homogeneous materials that may vary in both density and thicknesses; unlike Eq. (1), Eq. (2) can have the same mass attenuation coefficient for varying surface densities of a particular material. Additional Information 1) Cs-137 is an unstable nucleus with a half-life of years. Through spontaneous beta decay, it decays to a metastable form state of barium Ba-137. As shown in Figure 1 below, there are two possible beta decays to the stable state of Ba-137. The first, with probability 94.6%, has maximum beta energy of MeV; the second, with probability 5.4%, has energy of MeV. The metastable state of Ba-137 has a very short half-life, 2.55 minutes, and decays into the ground state via gamma emission. The energy of the gamma photons emitted by * Ba-137 is MeV. Cs yrs MeV 94.6% (meta) MeV 5.4% 2.55 min MeV Stable FIGURE 1: DECAY SCHEME FOR 137 Cs 2

3 2) For accurate (and digitized) values of μ ρ, please access the NIST website for mass attenuation coefficients: 3) The detector is located at the top of the opening located in the lower right hand side of the apparatus. The mica window at the entrance of this opening is very thin and fragile. Do not touch it under any circumstances! 4) One of the materials you will be using in this experiment is lead. Lead is a toxic substance, especially if ingested, so be sure to wash your hands with soap immediately after the lab. EQUIPMENT The ST150 Nuclear Lab Station (Figure 2 below) provides a self-contained unit that includes a versatile timer/counter, GM tube and source stand; High voltage is fully variable from 0 to +800 volts. Associated software that allows for operation of ST150 and data collection. 137 Cs radioactive source that must be signed out and returned. Precut lead pieces to place into gratings of Geiger counter for shielding. FIGURE 2: THE ST- 150 NUCLEAR LAB STATION: Arrow indicates location of shielding blocks 3

4 PROCEDURE On the desktop, open Launchpad and open STX; this is the software program used to operate the ST-150 lab station. Open MATLAB to prepare for data collection; refer to the Intro to STX & Data Extraction video for guidance. NOTE 1: Adjust the high voltage to the operating voltage value obtained in the Experiment 1: The Geiger Counter. (Make sure you are using the same model number!) Part 1: Source Counts as a Function of Lead Absorbers 1. Sign out a 137 Cs source from your TA and place it on a plastic tray. 2. Slide the source tray into the 4 th closest grating to the window (from top) of the detector. 3. Run a 1-minute trial five times at the operating voltage. 4. With the source tray still on the 4 th grating, take out the #11 shielding piece of lead from the GM counter and place it onto the 3 rd grating. NOTE 2: See Figure 2 for location of lead absorbers. 5. Run a 1-minute trial five times at the operating voltage. 6. With the source tray still on the 4 th grating and the #11 lead shield on the 3 rd grating, take the #10 shielding piece of lead from the GM counter and place it onto the 2 nd grating. 7. Run a 1-minute trial five times at the operating voltage. 8. With the source tray still on the 4 th grating, the #11 lead shield on the 3 rd grating and the #10 lead shield on the 2 nd grating, take the #9 shielding piece of lead from the GM counter and place it onto the 1 st grating. 9. Run a 1-minute trial five times at the operating voltage. Part 2: Importing Surface Densities of Lead Absorbers 1. Look at the inner side of the door enclosing the lead absorbers in the Geiger Counter (Figure 2). The thickness [inches] and surface density [mg/cm 2 ] of each shielding number is displayed. 2. On MATLAB, clear your workspace. 3. Create variables for the surface density values for each lead absorber. As a suggestion, name these variables as surf_den_11, surf_den_10, surf_den_09. 4

5 Part 3: Importing Counts and Saving Workspace 1. Import the 3 source-only counts from Part 1A (#3) as an array; refer to the Intro to STX & Data Extraction video for guidance. As a suggestion, name this array source. 2. Import the 3 source + #11 counts, Part 1A (#5). As a suggestion, name this array source_ Import the 3 source + #11 + #10 counts, Part 1A (#7). As a suggestion, name this array source_11_ Import the 3 source + #11 + #10 + #09 counts, Part 1A (#9). As a suggestion, name this array source_11_10_ From the workspace, save these arrays, as well as the surface density variables as a.mat file. As a suggestion, this workspace variable can be named lab4. Refer to the Saving arrays and workspace variables video for guidance. ** For your report: ** Prelab All answers provided in full sentences (mathematical q s should be typed). Data Analysis: Q1: Provide the answer in full sentences. Q2: Paste the associated m-file code. Q3: Paste the associated m-file code. Q4: Paste the associated m-file code. Q5: Paste the associated m-file code. Q6: Paste the associated m-file code. Q7: Paste the associated m-file code. Q8: Paste the associated m-file code. Q9: Paste the associated m-file code. Q10: Paste the associated m-file code. Q11: Paste the associated m-file code. Q12: Provide the graph, with linear fit and equation displayed. Q13: Provide the answer in full sentences. Q14: Provide the answer in full sentences. Q15: Provide the answer in full sentences. Q16: Paste the associated m-file code. Q17: Provide the graph, labelled with error bars displayed. Q18: Provide the answer in full sentences. Q19: Provide the answer in full sentences. Q20: Provide the answer in full sentences. 5

6 DATA ANALYSIS 1. In your Report, provide: a. The operating voltage used for the experiment b. The serial number of your GM counter c. The name and ID of your source Create & Save an m-file as Lab4 _ firstname1_firstname2_firstname3.m ; this script must: 2. Load the lab4 workspace variable. Refer to the Creating m.files and loading workspace variables video for guidance. 3. Create 3 arrays that take the mean of the counts. As a suggestion, name these arrays source_none_mean, source_11_mean, source_11_10_mean & source_11_10_9_mean. 4. Create 3 arrays that take the standard deviation of the counts. As a suggestion, name these arrays source_none_std, source_11_std, source_11_10_std & source_11_10_9_std. 5. Create an array that stores the 4 mean values in order of [source_none_mean,, source_11_10_9_mean]. As a suggestion, name this array mean_source. 6. Create an array that stores the 4 standard deviation values in order of [source_none_std,, source_11_10_9_std]. As a suggestion, name this array std_source. 7. Create an array of increasing surface density, in the order of [0, surf_den_11, surf_den_11+ surf_den_10, surf_den_11+ surf_den_10+ surf_den_09]. As a suggestion, name this array surf_den_mg. 8. Create an array that converts surf_den_mg to grams; as a suggestion, name this array surf_den_g. NOTE 3: The instructions outlined in Q 9-11 below require the omission of the first (counts_only) data point in each array; on MATLAB, this can be done by selecting the 2 nd entry until the end. For example: without_first_entry = all_entries(2:end). 9. Create an array, mean_lead that contains the mean values of the counts of lead absorbers only. To achieve this, use the mean_source array and exclude the 1 st value. 10. Create an array, surf_den_lead that contains the mean values of surface densities for lead absorbers only. To achieve this, use the surf_den_g array and exclude the 1 st value. NOTE 4: Now that variables are assigned for the lead absorber data only, you can plot and solve for the mass attenuation coefficient of lead. 11. Create a figure that plots surf_den_lead (Q10) versus the natural logarithm of mean_lead (Q9). 6

7 NOTE 5: On MATLAB, the log command is defaulted as the natural logarithm 12. Run your code. On the figure, Click on Tools and Basic fitting. Click on linear and show equations. This will overlay a linear fit to your data and display its equation. 13. Calculate the percentage error of your experimental (equation-predicted) value of the mass attenuation coefficient w.r.t. the tabulated value. NOTE 6: For tabulated mass attenuation coefficients, refer to the NIST website (Prelab 4, Q4). 14. Determine the equation-predicted value of number of counts for no absorbers. 15. Calculate the percentage error of your equation-predicted value of the counts with no absorbers w.r.t. the experimental value, extracted from mean_source (Q5). 16. In your m-file, create a variable x that ranges from 0 to the maximum value of surf_den_g (Q8) in intervals of NOTE 7: Type help errorbar on the command window to learn how to plot with error bars 17. Create a figure that: a. Plots surf_den_g (Q8) with the natural logarithm of mean_source (Q5). b. Contains error bars for the mean counts, using std_source (Q6); since the natural logarithm of the counts are plotted, the values of std_source must reflect error propagation for the natural logarithm: y = ln x y = x x c. In a different color, plot x (Q16) with the fitted equation acquired from Q12. d. Contains an appropriate legend, title and axes labels. 18. Observing the error bars and the fitted equation, how well would you say the experimental data overlaps the predicted values? From this, would you say that the percentage errors found in Qs 13 & 15 are significant or negligible? 19. Provide two factors that could have contributed error in the lab. 20. Describe how you could improve the impact of the two factors (Q19) to reduce error in the lab. ** Before you leave the lab, confirm with your TA that you have sent the following ** 1) Report (All Prelab and Analysis Qs) 2) 1 m-file: Lab4 _ firstname1_firstname2_firstname3.m 3) 1.mat file: lab4 7