PC1144 Physics IV Radioactivity 1 Purpose Investigate the analogy between the decay of dice nuclei and radioactive nuclei. Determine experimental and theoretical values of the decay constant λ and the half-life for the dice nuclei. Measure the count rate versus voltage for a Geiger counter to determine its appropriate operating voltage. Investigate how well the observed distributions of counts compares to that predicted by the normal distribution. 2 Equipment 200 dice to simulate radioactive nuclei Geiger counter 210 Po α radiation source 90 Sr β radiation source 3 Theory A basic concept of radioactive decay is that the probability of decay for each radioactive nucleus 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 λ characterizes this concept. It is the probability of decay per unit time for one radioactive nucleus. Because λ is constant, it is possible to predict the rate of decay for a radioactive sample. The value of the constant λ is different for each radioactive nucleus. Consider a sample of N radioactive nuclei with a decay constant λ. The rate of decay of these nuclei dn/dt is related to λ and N by the equation dn dt = λn (1) Page 1 of 8
Radioactivity Page 2 of 8 The negative sign in the equation means that dn/dt is negative because the number of radioactive nuclei is decreasing. The number of radioactive nuclei at t = 0 is designated as N 0. At some later time t, the number of radioactive nuclei that are still there N is given by N = N 0 e λt (2) Equation (2) states that the number of radioactive nuclei N at some later time t decreases exponentially from the original number N 0 that originally present. The quantity λn is the number of decays per second and it is called the activity A of the sample. It can be shown that it obeys the equation A = A 0 e λt (3) Equations (2) and (3) state that both N and A decay exponentially with the same exponential factor. For measurements made on real radioactive nuclei, the activity A is usually all that can be measured. The time it takes for N 0 to be reduced to N 0 /2 and the time it takes for A 0 to be reduced to A 0 /2 are the same. It is called the half-life T 1/2 of the decay. It is related to λ by T 1/2 = ln(2) λ = 0.693 λ (4) Figure 1: Diagram of the essential elements of a Geiger tube. To study these radioactive processes, we must detect the presence of these particles that are the product of the decay. We can build devices in many form to accomplish the detection, but there all would have one feature in common. Every practical device that detects radiation allows the particles to interact with matter and then uses that interaction as basis for detection. The particular device we will use is called a Geiger counter. It consists of a tube in which the incident particle interacts and a scaling circuit to count the pulses of electricity produced. A diagram of a Geiger tube is shown in Figure1. The Geiger tube is a small metal cylinder with a thin self-supporting wire along the axis of the cylinder. The wire is insulated from the cylinder. The cylindrical wall of the tube serves as the negative electrode (cathode) and the wire along the axis is the positive electrode (anode). At the entrance end of the tube, there is a thin window formed by a very thin piece
Radioactivity Page 3 of 8 of fragile mica. Inside the counter is a special gas mixture that is ionized by any radiation that penetrates the window. In operation, a voltage is applied across the electrodes. The particular voltage for each tube must be determined experimentally. The applied voltage creates a large electric field in the tube and the field is especially large in the region near the central wire. When radiation passes through the window and ionizes the gas, the large electric field causes an acceleration of the free electrons. These accelerated electrons cause additional ionizations that creates an avalanche effect. The total number of ion-electron pairs created by a single incident particle is of the order of one million. The electrons are move mobile and drift toward the positive central wire. When they arrive at the wire, their negative charge causes the voltage of the wire to be lowered and this sudden drop in voltage creates a pulse that is counted by the electronic circuitry. Each pulse counted signifies the passage of a particle through the counter. The ions then recombine with electrons, leaving the gas neutral again and ready for the passage of another particle. The whole process takes a time of the order of 300 µ mathrms and during that time period if another particle goes through the counter, it may not be counted. Thus, one disadvantage of Geiger counter is this dead time during which counts may be missed. This is a negligible effect unless the count rate is very high. The count rate of a Geiger counter is a function of the voltage applied across the electrodes. Therefore, the counter should be operated in a region where the rate at which the count rate changes with voltage is a minimum. This is accomplished experimentally by measuring the count rate of some fixed source of radiation as a function of the voltage across applied to the tube. A graph of the count rate versus the voltage will be made from the data and the operating point will be chosen to be some voltage where the count rate versus voltage curve is nearly flat as possible. This is referred to as the plateau region. Figure 2: Typical count rate versus voltage for a Geiger tube. The counter needs a minimum voltage to produce pulses at all. Both this minimum voltage and the operating voltage are quite variable and depend upon the dimensions of the Geiger tube and on the particular gas used in the tube. Therefore, the exact nature of the count rate versus voltage curve depends on the particular tube used. A typical count rate versus voltage
Radioactivity Page 4 of 8 curve for a Geiger tube is shown in Figure 2. If all other sources of error are removed from a nuclear counting experiment, there remains an uncertainty due to the random nature of the nuclear decay process. It is assumed that there exists some true mean value of the count which shall be designated as m. However, we, emphasize, do not assume that there is a true value for any individual count C i. Although m is assumed to exist, it can never be known exactly. Instead, one can approach knowledge of the true mean m by a large number of observations. It can be shown that the best approximation to the true mean m is the mean C which is given by C = 1 n n C i (5) i=1 where C i stands for the ith value of the count obtained in n trials. The standard deviation from the mean σ and the standard error σ C are defined in the usual manner as σ = 1 n 1 n ( ) 2 C Ci and σ C = σ (6) n i=1 The way in which the measurements C i are distributed around the mean C depends upon the statistical distribution. The binomial distribution is the fundamental law for statistics of all random events including radiative decay. Calculations are difficult with this distribution and it is often approximated by another integral distribution called the Poisson distribution. For cases of m greater than 20, both the binomial and the Poisson distributions can be approximated by the Gaussian distribution. It has the advantage that it deals with continuous variables and thus calculations are much easier with the Gaussian distribution. For most nuclear counting problems of interest, the Gaussian distribution predicts the same results for nuclear counting that have been assumed for measurements in general. Approximately 68.3% of the measured values of C i should fall within C ±σ and approximately 95.5% of the measured values of C i should fall within C ± 2σ. There is one statistical idea valid for nuclear counting experiments that is not true for measurements in general. For any given single measurement of the count C in a nuclear counting experiment, an approximation to the standard deviation from the mean σ is given by σ C (7) For a series of repeated trials of a given count, the most accurate determination is given by C ± σ C. If only a single measurement of the count is made, the most accurate statement that can be made is given by C ± C.
Radioactivity Page 5 of 8 4 Experimental Procedure Part I: Simulated Radioactive Decay P1. Place all 200 of the dice in the box provided. Place the cover over the dice and shake them vigorously. Remove all dice that come to rest with ONE point face up. Remove only those that have ONE point that points directly upward. Record in Data Table 1 the number of dice that decay (are removed) on the first throw of the dice. Also record the number of dice that remain after the ones that decay are removed. P2. Place the cover on the dice and shake the remaining dice vigorously. Remove the dice that come to rest with ONE point face up on the second throw. Record in Data Table 1 the number of dice removed on the second throw and also record the number of dice that remain after the ones that decay are removed. P3. Continue this process of shaking the dice, removing the ones that have ONE point face up, and recording the number of dice removed and the number of dice left for each throw. Continue this procedure for a total of 20 throws of the dice; or until all of the dice have been removed. P4. Each group should record its data on the whiteboard so that the session results can be plotted as a set of data with better statistics. Sum the total number of dice thrown originally for the entire session and sum the number removed at each throw of the dice. Record this session data in Data Table 1. Part II: Characteristics Curve of the Geiger Counter P1. All Geiger tubes have a maximum permissible operating voltage above which they are subject to breakdown. This may permanently damage the tube. Check with your demonstrator for the specific maximum voltage of the particular Geiger tube used. DO NOT ever exceed this maximum voltage. Before plugging in the power cord of your instrument, make sure that the high voltage control is turned to the minimum setting. P2. Place the 210 Po α source as close as possible to the window of the Geiger tube. Once the source is positioned, take all the measurements for this procedure without changing the position of the source relative to the detector. P3. Turn on the power to the instrument, reset the counter to zero and start the counter in a continuous count mode with the high voltage still set to a minimum. Slowly increase the high voltage setting until the counts begin to register on the counter. Leave the voltage at this setting for which the Geiger tube just begins to count. This is called the threshold voltage.
Radioactivity Page 6 of 8 P4. Reset the counter to zero, start the counter and let it count for 1 minute. Repeat this procedure two more times for a total of three 1-minute counts at this voltage setting. Record the voltage and the three values of the count in Data Table 2. P5. Raise the high voltage setting by 25 V, repeat step P4 at this new high voltage setting and record the high voltage and the counts for these three trials. P6. Continue this process up to the maximum permissible voltage of the Geiger tube. Be sure to obtain the proper maximum voltage for your tube from your laboratory demonstrator. Part III: Counting Statistics P1. Set the Geiger counter to the proper operating voltage. Place the 90 Sr β source about 2 or 3 cm from the window of the Geiger tube. Once the source is positioned, take all the measurements for this procedure without changing the position of the source relative to the detector. P2. Reset the counter to zero, start the counter and let it count for 30 seconds. Record the value of the count in Data Table 3. P3. Repeat the count for a total of 50 trials. Make no changes whatsoever in the experimental arrangement for these 50 trials. Record each count in the Data Table 3.
Radioactivity Page 7 of 8 5 Data Processing Part I: Simulated Radioactive Decay D1. For each of the 20 shakes of the dice (group data), calculate the ratio of the number of dice removed after a given thrown to the number shaken for that thrown. Hint: Note carefully that this ratio must be calculated with data from two different rows in Data Table 1. For example, the number of dice thrown on the fourth throw is listed as the number of dice remaining after the third throw. Thus the ratio is calculated with the number removed on each row to the number remaining in the preceding row. D2. Determine your best experimental value for decay constant λ with the corresponding uncertainty. D3. Use percentage discrepancy to compare your experimental value for decay constant λ with the theoretical value. Hint: The percentage discrepancy is defined as Percentage discrepancy = Experimental value Theoretical value Theoretical value 100% D4. Perform a suitable linear least squares fit to your data (N, #throw) in Data Table 1 (session data). Determine the slope and intercept with the corresponding uncertainties of the least squares fit to the data. D5. Plot suitable graph as well as the best fitted line obtained above. D6. Based on your results from the least squares fit above, determine the experimental value of half-life T 1/2 (number of throws) with the corresponding uncertainty. Use percentage discrepancy to compare your experimental value of T 1/2 with the theoretical value. Hint: The experimental half-life is the number of throws needed to go from any point on the line to one-half that value. Part II: Characteristics Curve of the Geiger Counter D1. Calculate the mean for the three trials of the count at each voltage. D2. Plot the mean count rate (counts/min) versus voltage to produce graph like the one shown in Figure 2.
Radioactivity Page 8 of 8 Part III: Counting Statistics D1. Calculate the mean count C, the standard deviation from the mean σ and the standard error σ C for the 50 trials of the count. D2. For each count C i, calculate C i C /σ. D3. Determine what percentage of the counts C i are further from C than σ by counting the number of times a value of C i C /σ > 1 occurs. Express this number divided by 50 at a percentage. D4. Count the number of times that C i C /σ > 2 occurs. Express this number divided by 50 as a percentage. D5. Calculate C. 6 Questions Q1. For the simulation of a radioactive decay using dice, what are the analogous quantities to the real quantities listed below undecayed nucleus, decayed nucleus, time, decay constant? Q2. What quantity can be measured for the simulation laboratory that cannot normally be directly measured in a true radioactive decay laboratory? Q3. Compare the percentage of trials that have C i C /σ > 1 with that predicted by the normal distribution. Compare the percentage of trials that have C i C /σ > 2 with that predicted by the normal distribution. Q4. Calculate the percentage difference between C and σ C. Do the results confirm the expectations of equation (7)? Hint: The percentage difference is defined as Percentage difference = E 1 E 2 ( E 1 + E 2 )/2 100%