PCS352

RYERSON PHYSICS

LAB #2

2: RADIOACTIVE DECAY & BEER’S LAW OBJECTIVE The objective of Part 1 is to determine the relative efficiency of a Geiger tube for the detection of gamma radiation using the Cs-137 source and shielding apparatus. The objective of Part 2 is to determine how the Cs-137 radiation behaves as the source is distanced from the detector. INTRODUCTION Background & Theory Electrically charged particles like alpha or beta, and gamma radiation that enter a GM detector can be detected if they have sufficient energy to ionize the gas. Because the detection efficiency depends on the energy of the charged particles or of the gamma radiation, a GM detector has a much higher efficiency for detecting beta particles than for detecting gamma rays. Beta particles do not travel far, but if they are able to enter the active portion of the detector, they have a higher probability of ionizing the gas inside it and being detected. Gamma radiation, on the other hand, has enough energy to enter the active portion, but not always enough to produce ionization. Cs-137 is an unstable nucleus with a half-life of 30.17 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 0.514 MeV; the second, with probability 5.4%, has energy of 1.176 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 0.662 MeV.

Cs 30.07 yrs

0.514 MeV 94.6% (meta) 2.55 min 0.662 MeV 1.176 MeV 5.4%

Stable FIGURE 1: DECAY SCHEME FOR 137Cs

1

PCS352

RYERSON PHYSICS

LAB #2

Radioactive Decay Law & Beer’s Law The activity of a radioactive source naturally decays as a function of time; and can be calculated at any time using the following equation: (Eq. 1) Where is the initial activity of the radioactive source, is the initial activity (at ). is the decay constant of the radioactive source, and is related to the half-life, by the relation: (Eq. 2) As with Cs-137, if the products of a radioactive source are unstable, a second decay is established; the production rate of the “daughter” nuclei is dependent on the production rate of the “parent” nuclei. Multiple equations can be derived depending on the relative ratios of the half-lives of the parent and daughter nuclei. One scenario, known as Secular Equilibrium, occurs when the half-life of the parent is much greater than that of the daughter: . Letting denote the initial activity of Cs-137, and denote the activity of the metastable product *Ba-137, the following secular-equilibrium equation may be used to determine the activity of *Ba-137 at a particular time : (

)

(Eq. 3)

Here is the decay constant *Ba-137, and denotes the initial activity of *Ba-137; for simplicity, the second expression on the RHS can be ignored by assuming , since initially it is assumed that only Cs-137 is present. Just like the relationship of a radioactive material’s activity under natural radioactive decay, the number of photons or particles traversing a medium follows an exponential decay law that is attributed to the attenuation in the medium. Beer’s law is not time dependent but rather spatially dependent; the number of particles/photons at location can be found through the following equation: (Eq. 4) Where are the initial number of particles/photons emanating from the source, and is the attenuation coefficient of the medium. The law is more acquainted with the fluence of a radiation source, however can be simplified to the display in equation 4. For values of ⁄ , please refer to the Turner book, Pages 189/190. For more accurate values, please access the NIST website for x-ray mass attenuation coefficients: http://www.nist.gov/pml/data/xraycoef/

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PCS352

RYERSON PHYSICS

LAB #2

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.

FIGURE 2: THE ST-150 NUCLEAR LAB STATION

PROCEDURE Part 1: Determination of GM Counter Efficiency 1. Obtain the same model number of the Geiger counter that was used in Lab #1. Connect the Geiger counter to a power source and to the lab computer. 2. On the desktop, open the Applications folder and open STX; this is the software program used to operate the ST-150 lab station. Open MATLAB to prepare for data collection. 3. Sign out a Cs-137 source from your TA (please follow the signing rules) - Make sure you note the following for later data analysis: i. Activity of the Source ii. Date that the activity was recorded 4. Place the source on a source tray and slide it into the 3rd closest grating to the window (ie. from top) of the GM detector. 5. Set the High Voltage of the GM detector to the operating voltage obtained in Lab #1. 6. Run a five-minute trial and repeat five times. 7. Save the five data values on an array in MATLAB; this data set corresponds to the number of gamma photons and beta particles measured by the GM detector 8. Take a shielding piece of Aluminum from your TA and carefully place it into the topmost grating. 9. Repeat steps 6 & 7; this second array corresponds to the number of gamma photons only, since the aluminum attenuates Cs-137 beta particles. 10. Return the Aluminum shielding piece back to your TA. 3

PCS352

RYERSON PHYSICS

LAB #2

Part 2: Investigation of Cs-137 Radiation as a function of Distance 1. 2. 3. 4.

Set the High Voltage of the GM detector to the operating voltage obtained in Lab #1. Place the source tray carrying the source on the topmost grating. Run a one-minute trial and repeat five times. Save the five data values on an array in MATLAB; this data set corresponds to the gamma photons and beta particles measured by the GM detector in the topmost grating. 5. Repeat these steps to capture five data values for all the other gratings, and record your data on MATLAB. 6. Once data collection is complete, set the voltage to 0V and switch off the GM tube. **When you have completed Lab 2, return all sources to your TA**

DATA ANALYSIS (**Refer to Video 2A & Video 2B**) 1. In a MATLAB m-file: a. Create an variable , that stores values increasing by 10 minutes, starting at 0 and ending at 200. b. Create a variable , that stores the initial activity of your Cs-137 source in Curies. c. Create a variable , that stores the half-life value of your Cs-137 source in minutes. d. Create a variable , that calculates the decay constant using T1. e. Create a variable that calculates the current activity, for all values of . f. Save and run the m-file, and confirm that you see values for all your variables. Note: When multiplying/diving an array, such as variable , you need to add a “.” next to the operation: “.*” or “./”; this way, the multiplication occurs to all elements of the array. 2. In the same m-file: a. Create a variable , that stores the half-life value of *Ba-137 in minutes. b. Create a variable , that calculates the decay constant using T1. c. Create a variable that calculates the current activity, for all values of . d. Save and run the m-file, and confirm that you see values for all your variables. 3. In the same m-file: a. Create one figure that plots v/s in one color, and vs in another color. b. Set the limits of the x and y axes to 0-100 min and 0-10e-6 Ci respectively. c. Title, label the graph and include a legend. d. Save and run this m-file: using the data cursor, click on the point where the two graphs just start to merge. e. Record the time displayed in the previous task; calculate how many half-lives of * Ba-137 is this value? 4. In your report, include the m-file code, the graph and the values determined in 3e. 4

PCS352

RYERSON PHYSICS

LAB #2

5. Using your “gamma+beta” values (in counts/min), calculate the mean and standard deviation of this data set, and include all three pieces of data in your report. 6. Repeat task 5 for the “gamma-only” data. 7. Calculate the “beta-only” counts/minute from the mean values obtained in tasks 1 & 2. Find the standard deviation using the appropriate error propagation formula; provide the values of all parameters used, and the result. ** See Appendix** 8. Calculate the relative efficiency of the tube for detecting gamma radiation, by taking the ratio of the gamma counts to the theoretical number of photons emitted and entering the tube. The theoretical number of photons entering the tube should be calculated with the following in mind: i. What is the current activity of the sample? ii. What percentage of this activity is attributed to gamma emission? iii. In what media did the gamma radiation traverse prior to entering the GM tube, and how much attenuation occurred in the media? ** Provide a detailed calculation for task 8 – hard or soft copy will be accepted ** 9. Find the mean and standard deviation for the data corresponding to each grating. Provide these values and the associated grating in your report. 10. Measure the distance of each grating to the mica window; please, DO NOT touch the mica window! Include the error with your value. 11. Save the mean values in an array called counts_per_min, and the distance values in an array called distance. 12. Open cftool ; a. Use the arrays formed in task 11 and fit an exponential curve to the graph. b. Include graph and parameter values in your report (print-screen is acceptable) c. What is the value of the attenuation coefficient from the curve fit parameter? Can this value be compared with the theoretical value? Please explain. d. A pervasive concept in physics is the inverse square phenomenon; the gravity, electric field strength, intensity etc. experienced at a particular location follow an ⁄ dependence to the distance from the source. Investigate if the radiation emanating from your source is also governed by an inverse square dependence; to do this, fit a power curve to your data. e. Do the parameters obtained in task 12d suggest a dependence? Include graph and parameter values in your report (print-screen is acceptable) f. How well does the power curve fit your data versus the exponential curve? What conclusions can you make? 13. For each part of the experiment, state two issues that produced inaccurate or imprecise results and state how these issues can be resolved

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PCS352

RYERSON PHYSICS

LAB #2

APPENDIX

Propagation of standard deviations (errors) The standard deviation associated with a variable z which is a function of other 2

 f  f f variables, eg. z= f(x,y,z) if given by  z    x     y     z   x   y   z  2

2

Particular cases: 2

1. z  ax

  f  f   f   z    x     y     z    z  a x  x   y   z 

x 2. z  a

  f   f   f   z    x     y     z    z  x a  x   y   z 

2

2

2

2

2

  f  f   f   z    x     y     z    z   x   y   z  2

3. z  x  y

2

2

 x 2   y 2

4. z  xy

  f     f   f   z    x     y     z    z  xy  x    y  x   y   z   x   y

x 5. z  y

  f 1  x   y   f   f   z    x     y     z    z      xy  x   y   x   y   z 

2

2

2

2

6

2

2

2

2

  

2

2