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Radioactivity©98 EXPERIMENT 11 Objective: To observe the radioactive decay of certain isotopes, to measure the penetrating power of their radiations, and to calculate their half-lives and decay constants. DISCUSSION: The nuclei of certain isotopes are unstable, eventually breaking up spontaneously into stable particles. The most frequent products of this decay are alpha particles (helium-4 nuclei), beta particles (electrons), gamma rays (very high energy photons) and whatever is left of the nucleus after decay has occurred. It is not possible to predict when a given nucleus will decay. However, given a very large number of a given nucleus, it is possible to predict that a certain fraction of them will remain unchanged after a certain lapse of time. Specifically, if initially there are n0 nuclei, then after a lapse of time t, the number n remaining is given by n n0 e t , (1) whereis a constant that depends upon the isotope undergoing the nuclear transformation. This kind of relationship is known as exponential decay. It is also encountered, among other places, in the expression that gives the current delivered by a capacitor while discharging through a resistor. A characteristic of exponential decay is that the time in which any given fraction of the nuclei disintegrates is always the same. For example, if after 50 years, 15% of the nuclei have decayed, then in another 50 years, 15% of the remaining nuclei will decay. An indirect way of expressing the value of the decay constantis to give the timein which half of the sample decays. This time is called the half-life. In this case n = n0 /2, and so e = 2. The value offor which eis 2 is the natural logarithm of 2, 0.69315… That is, the half-life is given by 0.69315 . (2) The value ofcan be determined experimentally by observing the decay of the radioactive substance. Suppose, for example, that the sample is polonium-210, which decays by the emission of a helium nucleus to become lead-206. In a certain time interval, say one minute, just a small part of the polonium nuclei disintegrate, but this number is proportional to the total number of nuclei in the sample. As this total number gets smaller, so also does the number which decay. That is, the number of disintegrating nuclei is directly proportional to the total number in the sample. Thus Eq. (1) can also be written in terms of the rates of decay. Letting ro be the initial number of disintegrations per unit time, and r the number at time t, we have 11-1 r n n0 , (3) r r e t. 0 (4) 1 r0 . t r (5) r0 and so, Taking the inverse of Eq. (4),we have ln We next consider the penetrating power of the three kinds of radiation. Alpha particles consist of two protons and two neutrons, bound together in a very stable arrangement. The alpha particle, with its charge of 2 protons, usually does not possess much kinetic energy as it leaves the parent nucleus. Because of its charge and low speed, it generally interacts very quickly with matter and loses what little kinetic energy it had. It picks up two stray electrons and becomes the helium atom, which then dissipates into the atmosphere. Beta particles are electrons. They generally leave the decaying nucleus at high speeds. But because of their charge and small mass, they also are quickly brought to rest when interacting with matter. However, they are more penetrating than alpha particles. Gamma rays are very high-energy photons. Because these photons are uncharged and carry much energy, they do not react with matter as easily as alpha and beta particles. Nevertheless, their penetrating power is radically limited by lead shielding. If the half-life of a radioactive sample is very long, then during the course of a few hours, the number of disintegrations per unit time, r, will not change appreciably. In this case, the number of particles that are able to penetrate matter drops off exponentially with the thickness of material. Letting 0 be the number of particles impinging on the surface of an absorbing medium and the number of particles emerging from the medium after having traversed the distance x, we have that 0 e k , (6) where k is the absorption constant. Its value depends upon the nature of the radiation and the absorber. Eq. (6) is identical in form to that for the decrease in counting rate with time, Eq. (4). There is a particular thickness such that e-k is 0.5. This thickness is called the half-value thickness for the material for the particular radiation involved. The value of can be determined by first finding k and setting equal to 0.69315/k. 11-2 APPARATUS SETUP: 1. Setup the Signal Interface as shown in Fig. 1. a. Plug the power brick into the back of the Interface, then into the strip on your table. b. Connect the white cable into the back of the Interface then into your computer’s USB port. c. Plug the Geiger Tube into Digital Channel 1. d. Turn on the Interface (switch is on the back). Figure 1: PASCO Interface setup. 2. Start DataStudio. It should be located somewhere under your start menu. 3. When prompted, choose ‘Create Experiment’. 4. Set up the software to recognize the hardware: a. In the software, add a “Geiger Counter” Digital Channel 1. b. Notice that there is a list of Data options on the upper left (usually). Also notice that there is a list of Display options on the bottom left (usually). c. Click and drag the “Table” icon to “Geiger Counts, Ch1 (counts)”. d. In the Table window, click on the Statistics button ‘Standard Deviation’. and choose ‘Mean’ and 11-3 e. Move and resize the table window so you can see everything, making the table long. EXERCISES FOR HALF-VALUE THICKNESS: Shielding of Alpha, Beta, and Gamma Radiation: We will be setting the timer to count for five 30sec intervals. Several counts are needed so that a mean and standard deviation of the measured quantity can be found, thus more accurately representing the actual number of decays in a given time period. The individual results will be listed in a table and the average will be calculated there. 1. Set up the length of sample time and the automatic stop: a. Set the Sample Rate interval 30 sec. b. Under the “Sampling Options…” choose the “Automatic Stop” tab. c. Choose ‘Time’ and set the interval to 150 sec (5*30sec). 2. Background Radiation. Background radiation is just that, radiation ever present in the background. It can come from many sources including the sun and naturally occurring radioactive processes. This background count is to be subtracted from all other data taken during this experiment as it adds a systematic error to the results. a. Adjust the Geiger detector so that it is approximately 3/4 of an inch above the table top. b. Remove the cap from the Geiger detector. c. Click ‘Start’ in the DataStudio window and let the counter count for its 5, 30 sec intervals. The counter should stop recording automatically. Note: you will only see data show up in the table after each 30 sec interval. d. Record the mean and standard deviation for the background radiation. e. Save your background radiation data. 3. Alpha Radiation. a. Under “Experiment” menu, choose “Delete ALL Data Runs”. b. Place a source of alpha particles under the Geiger tube, hole side up. c. Click ‘Start’ in the main window and let the counter count for the 5, 30 sec intervals. d. Record the mean and standard deviation for the alpha radiation without shielding. 11-4 e. Place a piece of paper between the source and the tube. f. Click ‘Start’ in the Science Workshop window and let the counter count again for 5 intervals. g. Record the mean and standard deviation for the alpha radiation with shielding. h. Save your alpha radiation data. 4. Beta Radiation. Measure the transmission of beta radiation through successively thicker absorbers. a. REPLACE THE CAP ON THE GEIGER TUBE! b. Under “Experiment” menu, choose “Delete ALL Data Runs”. c. Place a source of beta particles under the Geiger tube. d. Click the ‘Start” button in the main window and let the counter count for its 5 intervals. e. Record the mean for beta radiation without any absorber. This value is v0. f. Place, in succession, each of the polyethylene absorbers between the source and the Geiger tube and record the mean v and absorber thickness for each absorber. v g. In Excel, Graph ln 0 verses and determine the absorption constant k from v the slope of the line. h. Save your beta radiation data. 5. Gamma Radiation. Repeat Exercise 4 for gamma rays using the lead absorbers. Save your gamma radiation data. 6. Discuss Exercises 3, 4 and 5 and what the results tell you about the nature of alpha, beta and gamma radiation. 11-5 EXERCISES FOR RADIOACTIVE DECAY: Lifetime of Barium Nucleus: Do not procure your radioactive sample until you are ready to take data – read the instructions below FIRST. 1. Get rid of old setup. a. Under “Experiment” menu, choose “Delete ALL Data Runs”. b. Close the table. c. Click and drag the “Graph” icon to “Geiger Counts, Ch 1 (counts/sample)”. 2. Change the sample rate to 15 sec. 3. Set up the counter for 48, 15sec intervals (48x15 sec = 720 sec). a. Find the “Set Sampling Options…” and choose the “Automatic Stop” tab. b. Change the “Time” interval to 720 sec. 4. Data Acquisition a. Place the tray with the radioactive liquid on the table with the Geiger counter 3/4 inch above the sample. The cap on the G-M tube should be removed. b. Click the ‘Start’ button to begin taking data. Datastudio will automatically update the Table display and the Graph as each new sample point is taken. It will automatically stop at the end of 48 samples. c. Go get a cup of coffee. d. Save your half life data. DATA ANALYSIS: Determine the half-life of the sample using Eq. (2) and the graph generated. 11-6