Simulated Radioactive Decay Using Dice Nuclei Purpose: In a radioactive source containing a very large number of radioactive nuclei, it is not possible to predict when any one of the nuclei will decay. Although the decay time for any one particular nucleus cannot be predicted, the average rate of decay of a large sample of radioactive nuclei is highly predictable. This laboratory uses 20-sided dice to simulate the decay of radioactive nuclei. When a 1 or a 20 is facing up after a throw of the dice, it represents a decay of that nucleus. Measurements on a collection of these dice will be used to accomplish the following objectives: 1. Demonstration of the analogy between the decay of radioactive nuclei and the decay of dice nuclei 2. Demonstration that both the number of nuclei not yet decayed (N) and the rate of decay (dn/dt) both decrease exponentially 3. Determination of experimental and theoretical values of the decay probability constant λ for the dice nuclei 4. Determination of the experimental and theoretical values for the half-life of the dice nuclei Equipment: ƒ 20-sided dice Theory: One of the most noticeable differences between classical physics known prior to 1900 and modern physics since that time is the increased role that probability plays in modern physical theories. The exact behavior of many physical systems cannot be predicted in advance. On the other hand, there are some systems that involve a very large number of
possible event, each of which is not predictable; and yet, the behavior of the system as a whole is quite predictable. One example of such a system is a collection of radioactive nuclei that emit α-, β-, or γ-radiation. It is not possible to predict when any one radioactive nucleus will decay and emit a particle. However, since any reasonable sample of radioactive material contains a large number of nuclei (say at least 1012 nuclei), it is possible to predict the average rate of decay with high probability. A basic concept of radioactive decay is that the probability of decay for each type of radioactive nuclide is constant. In other words, there are a predictable number of decays per second even though it is not possible to predict which nuclei among the sample will decay. A quantity called the decay constant, or λ, characterizes this concept. It is the probability of decay per unit time for one radioactive nucleus. The fundamental concept is that because λ is constant, it is possible to predict the rate of decay for a radioactive sample. The value of the constant λ is, of course, different for each radioactive nuclide. Consider a sample of N radioactive nuclei with a decay constant of λ. The rate of decay of these nuclei dn dt is related to λ and N by the equation dt dn = λ Eq. 1 The symbol dn dt stands for the rate of change of N with time t. The minus sign in the equation means that dn dt must be negative because the number of radioactive nuclei is decreasing. The number of radioactive nuclei at time t = 0 is designated as N0. The question of interest is how many radioactive nuclei N are left at some later time t. The answer to that question is found by rearranging Eq. 1 and integrating it subject to the condition that N = N0 at t = 0. The result of that procedure is t N N e λ = 0 Eq. 2
Equation 2 states that the number of nuclei N at some later time t decreases exponentially from the original number N0 that are present. A second question of interest is the value of the rate of decay dn dt of the radioactive sample. That can be found by substituting the expression for N from Eq. 2 back into Eq. 1. The result is t N e dt dn λ λ = 0 Eq. 3 Furthermore, the expression for dn dt in Eq. 1 can then be substituted into Eq. 3, leading to t N N e λ λ λ = 0 Eq. 4 The quantity λn is the activity of the radioactive sample. Since λ is the probability of decay for one nucleus, the quantity λn is the number of decays per unit time for N nuclei. Typically, λ is expressed as the probability of decay per second; so in that case, λn is the number of decays per second from a sample of N nuclei. The symbol A is used for activity ( A = λn ); thus, Eq. 4 becomes: t A A e λ = 0 Eq. 5 Equations 2 and 5 thus state that both the number of nuclei N and the activity A decay exponentially according to the same exponential factor. For measurements made on real radioactive nuclei, the activity A is the quantity that is usually measured. An important concept associated with radioactive decay processes is the concept of half-life. The time for the sample to go from the initial number of nuclei N0 to half that value N0 2 is defined as the half-life t½. If Eq. 2 is solved for the time t when N = N0 2 the result is λ λ ln(2) 0.693
t1 2 = = Eq. 6 This same result could also be obtained by considering the time for the activity to go from A0 to A0 2. Figure 1 shows graphs of activity of a radioactive sample versus time. Figure 1(a) shows a semi-log graph with the activity scale logarithmic and the time scale linear. The time scale is simply marked in units of the half-life. Note that the graph is linear on this 2 of 7 semi-log plot. The half-life is the time to go from any given value of activity to half that activity. Figure 1(b) shows the shape of the activity versus time graph if linear scales are chosen for both quantities. The laboratory exercise to be performed does not involve the decay of real radioactive nuclei. Instead, it is designed to illustrate the concepts described above by a simulated decay of dice nuclei. In the exercise, radioactive nuclei are simulated by a collection of 20-sided dice. The dice are shaken and thrown, and a dice nuclei has decayed if either a 1 or a 20 is face-up after the throw. In this simulation, the decay constant λ is equal to the probability of 2 out of 20 of a particular face coming up. Thus, the theoretical decay constant λ is 0.100. A unique aspect of this simulation experiment is that measurements can be taken on both the remaining number N and the number that decay. The number that decay is analogous to the activity A. For real radioactive nuclei, N cannot be measured directly but is inferred from measurements of the activity. 0.1 1 0 0.1
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Activity (counts/s) 0 2 t½ t½ 0 t½ 2 t½ 0.9 0.8 0.7 0.6 0.5 0.4 0.3 Activity (counts/s) 0.2 Time Figure 1 Graph of activity versus time on semilog and on linear scales. Time Experiment: 1. Depending on how many dice there are, each die may represent 5 or more nuclei.
Each time interval may have several rounds of rolling the dice to create a large enough sample of nuclei. You will need to keep careful track of how many dice decay in each round, and be sure to exclude that number from the next time interval. Your instructor will tell you how many nuclei to start with. You will determine how many rounds you need to reach that number. 3 of 7 Example: You instructor tells you that your radioactive sample contains 500 nuclei. You count your dice and find there are 50. This means you will roll all 50 dice in 10 rounds to simulate 500 nuclei. 2. Place all of the dice in a cardboard box or other container. Shake the dice and gently pour them out onto the table (or into a larger box). Count and record how many dice show a 1 or a 20 on their top-most face (this record keeping may be done on scratch paper and does not need to be included with your final report, unless your instructor directs you otherwise). When all the rounds are thrown, count and record the total number of decayed nuclei, and how many nuclei are left. Determine how many rounds will be thrown in the next time interval, and whether any rounds will contain fewer than the total number of dice. Example: You throw ten rounds of 50 dice each, and find the following number decay in each round: Round Number of Decays 1 5 2 7 3 3 4 0 5 10
6 6 7 3 8 3 9 8 10 7 A total of 52 dice decayed in that time interval. Thus, in the next time interval you start out with 500 52 = 448 dice. You will throw 8 rounds of 50 dice, and one round of 48 dice. 3. Repeat this process until you have fewer than 5 dice remaining for the time interval. If you wish, you may go until you have zero dice remaining, but this may take a while! Analysis: 1. Enter your data into an Excel spreadsheet. You should have two columns: Activity (number of decays), and N (number at the beginning of each throw) 2. Plot N on both a Cartesian and a semi-log graph. 3. Plot Activity on both a Cartesian and a semi-log graph. 4. Calculate the ratio of the number of dice removed (activity) after a given throw to the number shaken for that throw. These ratios will give an experimental value for the decay rate, λ. Note that number shaken is not the same as the number left after the throw it is the number left after the previous throw. 4 of 7 5. Calculate the average of these values, and record it as λexp. 6. The theoretical value of λ is 0.100. Calculate the percent error in the value of λexp as compared to λtheo. 7. If you have not done so already, activate one of the plots of N vs. throw, and click on
Chart > Add Trendline. Choose an exponential trendline, and under Options, choose Display equation on chart. Compare the value of λ returned by Excel to your calculated value of λexp. 8. Calculate the theoretical half-life from Eq. 6, using the value of λ = 0.100. Record that value as (t1/2)theo. For the purposes of this calculation, assume that a fractional throw is possible. 9. From the exponential curve of N vs. throw, determine the number of throws needed to go from N0 to ½ N0. Record that number as (t1/2)exp. For purposes of this determination, consider a fractional throw as possible. 10. Calculate the percentage error in the value (t1/2)exp compared to the value of (t1/2)theo. Record that percentage error. 11. Optional (possible Extra Credit? Ask your instructor!): Write an Excel program that simulates this experiment. Start with the same N0, and the same decay constant, λ. But notice for some of the throws A was greater than λn, and for some throws A was less than λn. You will need to figure out how to make your program do this. Graph the results, and insert a trendline. Notice how the curve fit changes with each iteration of the simulation. Change N0 what happens to the returned value of λ as N0 gets larger or smaller? Your instructor may elaborate on these instructions. Results: Write at least one paragraph describing the following: what you expected to learn about the lab (i.e. what was the reason for conducting the experiment?) your results, and what you learned from them Think of at least one other experiment might you perform to verify these results
Think of at least one new question or problem that could be answered with the physics you have learned in this laboratory, or be extrapolated from the ideas in this laboratory. 5 of 7 Clean-Up: Before you can leave the classroom, you must clean up your equipment, and have your instructor sign below. How you divide clean-up duties between lab members is up to you. Clean-up involves: Completely dismantling the experimental setup Removing tape from anything you put tape on Drying-off any wet equipment Putting away equipment in proper boxes (if applicable) Returning equipment to proper cabinets, or to the cart at the front of the room Throwing away pieces of string, paper, and other detritus (i.e. your water bottles) Shutting down the computer Anything else that needs to be done to return the room to its pristine, pre lab form. I certify that the equipment used by has been cleaned up. (student s name),. (instructor s name) (date) 6 of 7 Pre-Lab Assignment Read the experiment and answer the following questions before coming to
class on lab day. 1. A typical sample of radioactive material would contain as a lower limit approximately how many nuclei? (a) 1000, (b) 106, (c) 1012, or (d) 1023 2. The theory of radioactive decay can predict when each of the radioactive nuclei in a sample will decay. (a) true (b) false 3. State the definition of the decay constant l. What are its units? 4. A radioactive decay process has a decay constant. There are radioactive nuclei in the sample at t = 0. How many radioactive nuclei are present in the sample 1 hour later? Show your work. 4 1 1.50 10 λ = s 12 5.00 10 5. For the radioactive sample described in question 4, what is the activity A (in decays per second) at t = 0? What is the activity 1 hour later? Show your work. 7 of 7 VIDEO: http://youtu.be/qlfn0q35yow