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Radiation Experiment 1 Random Emission of Radiation Prelab questions for this exercise can be found on Page R-11 Safety Make sure that you have read the safety notes in the introductory section of this manual before beginning any practical work. Do not, under any circumstances, attempt to repair any of the equipment. Rather, turn off the apparatus at the power point and consult your demonstrator. While it is a general rule that you may not eat or drink in the laboratories, it is especially important in the Radiation laboratory. Radioactive materials taken internally, particularly emitters of -radiation can be very dangerous. If you need a drink go to the fountain at the eastern end of the laboratories. Whenever handling radioactive sources you should minimise your exposure to the radiation. This general precaution applies regardless of the strength of the sources. To do this you should try to: maximise your separation from the radioactive source minimise the time that you are exposed to the radioactive source shield the sources wherever possible (use the lead blocks provided to do this) wash hands after the laboratory session. Paradoxically, the lead that is used to shield the radioactive sources is also toxic. For this reason it is doubly important that you wash your hands after the laboratory session (especially before lunch). The radiation detectors used in this laboratory are powered by a high voltage power supply. As a result of this you should not re-wire any of the circuit modules without consulting a demonstrator. If any of the components or leads are damaged these should be brought to the attention of the demonstrator. Outline In this experiment you will learn about the random nature of radioactive decay and measure a fundamental property of all radioactive nuclei, the half-life. You will establish whether your data follows a “normal distribution” by comparing it with theoretical expectations of this distribution. First you will use a computer simulation of a normal distribution and examine its various properties to familiarise yourself with its behaviour. The concept of percentage statistics is introduced. The techniques you explore here will be used in the next experiment to estimate uncertainties in the data you collect from radioactive samples. Secondly you will measure a half-life, which is one of the fundamental properties of a radioactive substance. You will plot your data, estimate uncertainties and finally fit a straight line in order to quote a value for the half-life. By plotting lines of maximum and minimum gradient you should be able to quote uncertainties for your result. Physics 121/2 and 141/2 Laboratory Manual R-5 References 121/2: Sections 45.3, 45.4, 42.3. 141/2: Sections 43.3 and 43.5 Section A Normal Distribution Introduction Radiation is emitted in a random way from radioactive materials. You cannot know when a particular nuclide will decay only its probability of decay. In spite of this, a collection of radioactive nuclei does emit radiation in a regular way that can be reasonably well described by a normal distribution. It is quite likely that you are already familiar with normal distributions, even if your knowledge developed in an informal way. In its simplest form two quantities, the mean and the standard deviation can describe a normal distribution. For example I might want to count the number of cars which enter the university between 9:00 AM and 10:00 PM on a Monday. Presumably this number would change a bit from week to week, but is probably roughly constant. If I were to collect this data over a series of weeks I would be able to tell you two things. The mean number of cars would simply be the average count. The standard deviation would be a number, which I could calculate, that would tell you about the spread of results. It would give an idea of how close you could expect an individual count to be to the mean of all counts. This sort of treatment is very useful when measuring something which is not perfectly regular, but remains “roughly” regular. We could represent the occurrence of such data as a histogram (as shown in figure 1 below). The theoretical distribution might look as shown in figure 2. Here the area under the curve between x = a and x = b represents the number of observations that have values between x = a and x = b. Frequency of X Frequency of X X a b Figure 1 X Figure 2 If we take N measurements of some quantity (let’s say x), then the mean, , is defined by N = x i i=1 N , and the standard deviation, , is defined by, N = R-6 (x i -) i=1 N -1 2 . Physics 121/2 and 141/2 Laboratory Manual Properties of the normal distribution — exploration using StatPlay A1: Double click on the StatPlay icon in the labtools group on your computer. You will explore how the quantities mean and standard deviation describe the distribution. Choose the continuous distribution explore window. You should now see a normal distribution with a mean of 0, and a standard deviation of 1. Exercise 1 – Exploring and Describe the effects on the distribution of moving the mean. This can be achieved by dragging the box labelled . Small labelled sketches will be the most effective way of communicating your findings here. Now, in a similar way, describe the effects of changing the standard deviation (achieved by dragging the box labelled ). From this distribution of values about the mean, it is possible to tell the likelihood (probability) of a single measurement occurring within a particular range of values. Exercise 2 – Exploring the distribution down to the tail Change the values for mean and standard deviation to 10 and 1 respectively. Do this by typing in the values on the right of the screen. Click on the Z-scores button (at the top - it is the button which shows a graph with vertical lines on it). This will put lines on the graph at intervals of one standard deviation. Now look at the probability of taking a measurement that is lower than one standard deviation below the mean. Do this by clicking on the Left Tail button, which shows a graph with the left edge coloured differently. By moving this p line about you can record the probability of a value being below the line. Use this technique to answer the following questions: Question (a) What is the probability of obtaining a value below ? Question (b) What is the probability of obtaining a value below -? Question (c) What is the probability of obtaining a value below -2? Question (d) What is the probability of obtaining a value below -3? Now use these measurements to calculate the percentage probability of a single measurement being within one standard deviation on either side of the mean. Repeat for the probability of being within 2 and 3 standard deviations of the mean. These answers should agree with the much used values which your demonstrator will be able to give you. Exercise 3 – The median Another quantity which statisticians will often talk about is the median, which is defined as the point on the probability distribution below which the probability is one half. Check your normal distribution to find the point that corresponds to the median. For a normal distribution the median and the mean should be the same. Compare the mean and median for the simulated normal distribution and comment. Physics 121/2 and 141/2 Laboratory Manual R-7 Properties of the Normal Distribution — the experiment A2: Imagine that you have set up a counter to detect the radiation from a long-lived source, that is one whose activity, the number of particles emitted per second, may be regarded as virtually constant. You record the number of particles detected in a certain time interval, the count x, and then repeat a series of measurements for the same time interval. You will find that the counts are not all equal but are distributed over a small range. This distribution follows a form known as a Poisson distribution, which can be described by a single quantity (the mean), and has a standard deviation, which is equal to the square root of the mean. That is Provided is large enough, the Poisson distribution becomes a normal distribution with the linking between mean and standard deviation shown by the equation above. This equation is not true for all normal distributions, only those that are also well described by Poisson statistics. Your basic aim in this experiment is to firstly observe the distribution obtained from a radioactive decay and, secondly, to analyse these data to determine whether they do indeed follow a normal, and Poisson, distribution. Experiment (i) Hints on Uncertainties Whenever we take an experimental measurement we must also consider any errors or uncertainties associated with it. In this experiment you will need to take into account: Statistical uncertainties – What is the uncertainty associated with the count rates you measure? Refer to page R-3 of your manual for help with this. Background – How do I incorporate background contributions into my error analysis? Refer to page R-3 and R-4 of your manual for an example of this. Graphical Errors – How do I apply errors on a graph to a value I read from it? Refer to page 22 of your manual for a discussion of this. Settings for the Electronics Before you can begin acquiring data you need to set up the electronics for the experiment. Set up the apparatus with the settings shown below. H.V. power supply Single channel analyser R-8 900 V Selector switch to "normal" Lower level at 4.0 V Upper level at 10.0 V Counter Discriminator at 2-3 V Amplifier -ve input and unipolar output Physics 121/2 and 141/2 Laboratory Manual Put a 137Cs source in front of the detector and switch on all the electronics for taking data. Ask your demonstrator to feed the amplifier output into the oscilloscope, and set the gain so that the largest pulses produced by your source are about 9 V in amplitude. Do not alter any settings after your demonstrator has set your gain levels. Record the settings in your logbook. Collection of Results Start up the counter program. Enter a name for the file into which you plan to save data – this is important so that your data is not simply added to the default data file. Set your source at a distance from the detector that will give about 1000 counts every few seconds. Take 100 such measurements and save them to your file. Analysis of Results Before analysing the data you have just collected you will need to make the data readable by the StatPlay program. To do this, read the output file from the counter program into Excel. Delete all the information apart from the actual counts. Save the file as tab delimited text. Restart the StatPlay program and choose the data distributions explore window. Go to the File option at the top left of screen and choose open. Remember your file is in the data directory, so move to there and open up your data. As soon as the data is read you will see three things; the data listed exactly as it was read in, next to this the data sorted from maximum to minimum, and then a histogram of the data. Make sure the numbers are as you would expect - if not, ask your demonstrator to have a look. The program has not only sorted your data. It has also divided the full range of your data into about 10 bins and grouped the measurements into those bins. Each data point in the range of the bin contributes one to the frequency of that bin. These bin groupings are shown on the histogram. If you click on the first column of the histogram you will see red dots appear next to the data that is in this “bin”. Question (e) Compare the shape of the histogram to the normal distributions observed earlier. What are the similarities and differences? Try changing the bin size by clicking on the buttons above the histogram (pictures). Comment on the change of shape as you change the bin size. Return to what you consider is the most useful bin size - discuss this with your demonstrator before proceeding. Now analyse the distribution and discuss how well it agrees with the ideal normal distribution. Make use of some of the previously used features of StatPlay (for showing tail probabilities and the mean and median). Be sure to address the following questions: Question (f) Does the median equal the mean? Question (g) How do the tail probabilities compare to the ones calculated earlier from the simulated normal distribution? Question (h) How closely does ? Physics 121/2 and 141/2 Laboratory Manual R-9 Summary of Section A: As a summary of the main points learned so far today, describe how you could estimate the uncertainty in the following situation. Suppose you are conducting an experiment with a radioactive source and your detector reading is 1241 counts in 30 seconds. What is the uncertainty in this measurement Consider an experiment where you are detecting a mean of 50 counts per minute. How long a counting time would you use to try to obtain an uncertainty of 3% in your result (ie. an estimated standard deviation that is 3% of the estimated mean)? Section B Half Life of Indium Introduction If we have a sample of many nuclei of a particular radioactive species we cannot predict when an individual nucleus will decay. However, all the nuclei have the same probability of decay in a given period of time. An equivalent way of stating this is to say that each radioactive species has its own characteristic half-life, ; that is, a time in which half the nuclei in a sample decay (roughly – remember the randomness). No matter what size sample we start with, after one halflife, half of the nuclei will have decayed and half will not have decayed. A knowledge of the halflife of a radioactive substance allows us to calculate what number of nuclei will be remain undecayed after time t, provided we know how many there were at the start. If we start with No nuclei at time zero, then after: and 1 half-life, No/2 remain; 2 half-lives, half of these, ie. No/22, remain; 3 half-lives, half of these, ie. No/23, remain; p half-lives, No/2p remain. Hence, to find how many remain after time t, we have only to express t as a certain number halflives, such that: t = p and the number remaining after time t, Nt, will be: N0 -t N 0 2 1/2 ... (1) -t.ln2 N t N 0 exp 1/2 ... (2) Nt 2p and the number which will have decayed, N t , will be: Nt (t) N 0 N t R-10 -t 1/ 2 N 0 1 2 ... (3) Physics 121/2 and 141/2 Laboratory Manual The decay rate, ie. number of decays per unit time is proportional to the number of nuclei present. So, if you have two samples of the same radioactive species, with the first having with twice as many radioactive nuclei as the second sample, then the rate of decay of the first sample will also be twice the value of the rate of decay of the second sample. It then follows from equations (1) and (2) that the decay rate, at any time t, is related to the rate at time zero by the equations: t dN dN 1/ 2 dt = dt 2 t 0 ... (4) dN -(t.l n2)/ 1/ 2 = dt e 0 ... (5) Equation (4) may be used to work back from measured values of the decay rate, dN/dt, at various times to find 1/2, and that is precisely what is done in this experiment. Prelab Questions: After reading through the Introduction to Section B above as well as the introduction to the radiation experiments, complete the following questions. Determining the half-life of 128I The table below shows some measurements of the decay rate of a sample of often used medically as a tracer. Background has already been subtracted. 128I, a radionuclide What would be the statistical uncertainty associated with each of the R values (complete the table)? How does background affect the uncertainty associated with R? (Refer to page R-3 and R-4 of the Radiation introduction for help with this). Suppose the background was 1.6 ± 0.1 counts/second. Calculate the new uncertainties in each R value. Sketch a graph of the data (paste this into your manual) and use it to find a value of the half-life for 128I. = Time (min) R counts/sec 0 392.2 32 161.4 64 65.5 96 26.8 128 10.9 Physics 121/2 and 141/2 Laboratory Manual R R (including background) R-11 Experiment (ii) Reduce the lower level on the single channel analyser to 0.5 V to improve the efficiency of detection. Make a measurement of background by counting pulses for intervals of 1 second over a period of 1 minute. After the 30-second neutron bombardment, quickly fasten your indium disc to the face of your scintillation counter with sticky tape. Beginning immediately, measure the counts from your source for intervals of 1 second, again for a period of 1 minute. Analysis of Results Plot a graph of counts versus time. Question (i) What criterion was used to decide when it was no longer worthwhile to collect data? From your graph, find the half-life of 116Inm, and express it as: = Question (j) ± seconds. In plotting your data did you take the time corresponding to each point to be the start, middle or some other point of the counting period? Is this important to your determination of ? Explain your answer fully. Summing up Section B Discuss your results fully in your report. R-12 Physics 121/2 and 141/2 Laboratory Manual A postscript The Indium Radioactive Source Natural indium is 4% 113 115 115 49 In and 96% 49 In . When it is irradiated with neutrons, 49 In may absorb a neutron, producing 116 49 In in one of its excited states. These excited states decay by -ray emission, leading to either the first excited state or the ground state of 116 49 In , which is a beta emitter. The first excited state is so different in structure from the ground state that decay from it to the ground state virtually never occurs. Instead, it decays by beta emission to life of order 1 hour. The ground state also decays by beta emission to much shorter—a matter of seconds. It is the short half-life of 116 49 116 50Sn , 116 50Sn with a half- but its half-life is very In in its ground state that is measured in this experiment. To make we need a neutron source. The californium isotope 116 m 49 In 252 98 Cf undergoes spontaneous fission, the products of which emit neutrons. This means that it breaks into two parts of comparable mass, either or both of which may release free neutrons. It has a half-life of 2.65 years and is the neutron source we use. It is kept in a tube immersed in a tank of water. The neutrons from the decay of are slowed down in the water as a result of collisions with 252 98 Cf the nuclei of the hydrogen atoms of the water (protons). This is done because slow neutrons are more readily captured by 115 49 In than are fast neutrons. The Indium radioactive sources you will be using are placed in the tank for about 30 seconds by your demonstrator. This is enough time to produce the short lived form of 115 In n 116 In* 116In. The process may be represented by the equation: Ings 116 Physics 121/2 and 141/2 Laboratory Manual Sn - 116 R-13 R-14 Physics 121/2 and 141/2 Laboratory Manual