Phys 243 Lab 7: Radioactive Half-life Dr. Robert MacDonald The King s University College Winter 2013 Abstract In today s lab you ll be measuring the half-life of barium-137, a radioactive isotope of barium. The lab will involve data analysis and a bit of reference researching. Contents 1 Objective 1 2 Background 2 2.1 Nuclear decay........................................ 2 2.2 Decay rate.......................................... 2 2.3 Half-life........................................... 3 2.4 Barium-137 generation and decay............................. 4 2.5 Statistical comparisons................................... 4 3 Procedure 5 3.1 Pre-questions........................................ 5 3.2 Data collection....................................... 5 3.3 Analysis........................................... 5 4 Conclusion 6 5 Checklist 7 1 Objective To determine the half-life of barium-137, accounting for background radiation levels. 1
2 Background 2.1 Nuclear decay Nuclear decay, and the structure of the nucleus itself, were described successfully using the shell model of the nucleus; this model is still used today. (It was first proposed by Maria Goeppert- Mayer in 1949; it earned her a Nobel Prize in 1963.) Have a look at Section 43.4 in your textbook for more details. (Everything you need for today s lab is below, but this stuff is really cool, so go read the textbook later too. :) A nucleus is similar to an atom in many ways. In particular, a nucleus has stationary states configurations of nucleons (protons and neutrons) with different energy levels for each state. The nucleus occupies one of these states, much like an atom. An unstable nucleus is essentially in a sort of excited state. Sooner or later, it will spontaneously transition to a lower-energy state, and in the process emit some sort of radiation. Unlike an atom, though, which can only emit photons, there s so much energy bound up in a nucleus that there are several different things it can emit, depending on which nucleus we re talking about. There are three types of nuclear radiation (table from your textbook): Radiation Identification Charge Stopped by Alpha, α 4 He nucleus +2e Sheet of paper Beta, β Electron ±e Few mm of aluminum Gamma, γ High-energy photon 0 Many cm of lead Some nuclei emit alpha rays, some beta rays, some gamma rays, and some nuclei can emit more than one type of radiation. (Some nuclei can also emit neutrons, but those are uncommon.) 2.2 Decay rate Nuclear decay is of course a quantum mechanical process, so it happens spontaneously and at random. You can never tell exactly when a specific nucleus is going to decay. Decays happen faster for some nuclei than others, which simply means that a given nucleus is more likely to decay sooner rather than later. A Geiger counter (see textbook Section 43.5) produces an electric pulse whenever it is hit with ionizing radiation, which includes α, β, and γ rays ; it s often hooked up to a speaker, so you hear a click for each detection. If you hold a Geiger counter near some radioactive material, there are a few things you can notice: You hear a more or less steady stream of clicks, which slows down over time as the source decays away. (More on that shortly.) The clicks are not completely steady, but come in bursts and at random intervals. You hear more (faster) clicks as you hold the detector closer to the material. 2
In fact, nuclear radiation follows an inverse square law, just like light or sound, since the radiation is being emitted equally in all directions. This is the most important thing to know about safely handling radioactive sources: Keep your hands off as much as reasonably possible. The decay rate of a source (a sample of radioactive material) is the average number of decays per second. Each decay is the de-excitation of a single nucleus, so if you have N nuclei in your sample then the decay rate of the sample is dn dt. All the nuclei are the same, so each nucleus has the same chance of decaying in the next, say, 1 second. This means that the number of nuclei that do decay is proportional to the number you start with; that is, dn dt = rn (1) where r is what we call the decay rate for this nucleus. It has units of decays per second or s 1. Solve this differential equation for N to get the number of nuclei remaining at time t: N(t) = N 0 e rt = N 0 e t/τ, (2) i.e. N follows an exponential decay. The number τ = 1/r (Greek letter tau ) is called the lifetime of the nucleus, and is measured in seconds. N 0 is the number of nuclei you had at t = 0 (whenever you chose that to be). After one lifetime (t = τ), you have N 0 /e 0.37N 0 roughly a third of what you started with. Note that this is true no matter how many nuclei you started with! The (average) rate of clicks from the Geiger counter is not the same as dn dt, but it s proportional to this. That means you can measure τ (or r) from a plot of counts detected vs time. (You ll actually be using a histogram; ask me for details.) However, your sample material is not the only source of radiation in the room! Cosmic rays are nuclear radiation coming from space, and they pass through all the time. In addition, trace amounts of uranium, thorium, radon, potassium, and other radioactive materials are present everywhere around us (especially in concrete); these are in low enough amounts to be harmless, 1 but the Geiger counter detects them. All this background radiation will increase your counts detected by a constant amount. 2.3 Half-life The half-life of a nucleus, t 1/2, is the time it takes for half of the atoms in a sample to decay. (Again, note that it s the same no matter how many atoms you have! To think about: How many atoms have decayed after two half-lives?) In other words, it s defined as: N(t 1/2 ) = N 0 2 The half-life can be determined using eq.(2). N(t) can be written in terms of the half-life if we want to: N(t) = N 0 ( 1 2) t/t1/2. (4) (3) 1 For fun some time, look up banana equivalent dose. 3
There is a practical reason to slightly prefer eq.(2) when writing the equation into a computer: The function exp(x) calculates e x in an efficient way. There s no simple equivalent to raise anything else to a power. 2.4 Barium-137 generation and decay The radioisotope we ll be working with today is barium-137. The 137 identifies its atomic mass A, and the chemical name can tell you the number of protons Z (its atomic number ). In standard chemical notation you would write 137 Ba. This material can t simply be stored, since its half-life is so short, so it has to come from a generator. Our generator is a stock of cesium-137 (Cs-137), embedded in a lattice in a container. Cs-137 has a half-life of about 30 years, and decays into Ba-137, so as the radioactive Ba-137 decays away it is continually replenished. A solvent is washed through the container, dissolving barium but not cesium (or the lattice), so what comes out is a liquid loaded with Ba-137. It will be up to you to determine (through research in various references) what is decaying, what the final daughter nucleus is, and what type of radiation is being emitted. 2.5 Statistical comparisons Say you have a measurement, w (just to pick a letter), with an uncertainty σ w (Greek letter sigma ), and you want to compare it to a known value w 0. (For example, comparing a measured half-life against the accepted value from some reference material.) A measurement is consistent with a number if it s less than two errors away from that number; that is, it s consistent if w w 0 < 2σ w (5) Now say you have two measurements, x 1 and x 2, with uncertainties σ 1 and σ 2, and you want to check if they re consistent with each other. (Say, two measurements of a background count rate.) The proper way to do this is to ask whether the difference is consistent with zero. So first you find the difference x 1 x 2. The uncertainty in the difference is given by σ diff = σ1 2 + σ2 2. (6) As before, the difference is consistent with zero if x 1 x 2 < 2σ diff. (7) 4
3 Procedure 3.1 Pre-questions First, answer these questions based on Sec. 2: Question 1. Which of the three types of nuclear radiation result in a different element after it is emitted? Explain. Question 2. Explain why the rate detected by the Geiger counter is not the same as the decay rate of the sample (specifically dn dt, not r), even if there were no background. How does this affect your experimental method? Question 3. Show that N(t) in eq.(2) satisfies the differential equation in eq.(1). Question 4. Determine a formula that relates the half life t 1/2 to the lifetime τ, starting from eq.(2). Show your work. 3.2 Data collection Open the lifetime experiment file in LoggerPro. It will record the data from the radiation monitor. (There may already be a computer or two for taking data; in that case you ll share the data.) First, record five minutes of background data (counting with no source present). This will be used as a cross-check later. When you re ready, I ll prepare a sample of Ba-137 and place it about 1 cm below the radiation monitor. Take data for 20-30 minutes, until the count rate flattens out visibly. The counts shown in this data are proportional to N(t). 3.3 Analysis Fit a constant to your background data; this is the simplest way to measure the background count rate with an uncertainty. State the result in your report; include the uncertainty, and use appropriate significant figures. Please also include the graph in your report, showing the fit. (Yes, I also want you to state the result separately from the graph.) Be sure you have the proper units on both axes, as well as on the background count rate you measure! Now bring up your N(t) data; you ll use this to measure τ and t 1/2. I want you to simultaneously (i.e. as part of the same fit) determine the background rate B. In principle you could take the background rate you find using the no-source data and subtract it from the source data. However, finding it as part of the N(t) fit is important because of the way curve fitting works: To the fitter, adding a constant background to an exponential decay looks a lot like what you d get if you increased N 0 and τ instead. (This is one of the reasons for taking source data until the rate levels off.) 5
Since the background from the no-source data has an uncertainty, there s basically zero chance that you d subtract exactly the right background rate from the source data, so this would bias your other fit results in an unknown way. On the other hand, if the fitter is allowed to adjust the background as it goes, it can take that relationship into account and give more accurate results (and more accurate uncertainties). Question 5. What equation will you fit to your N(t) data in order to determine τ (and t 1/2 ) and simultaneously measure the background B? Use LoggerPro s curve fit tool to fit the equation you just came up with. (The equation might already be in the list, actually.) Include the graph with the fit results in your report, and state what the generic fit parameters in the graph actually correspond to (which one is N 0, which one is τ, etc). As a check of the fit quality, compare this background to the background rate you measured from the no-source data, using the consistency calculation from sec. 2.5. (If they re inconsistent, that usually indicates that the fitter had trouble sorting things out, and your results may be off.) Using your result for the lifetime τ, calculate the half-life t 1/2. They re proportional, so the percent uncertainty is the same; convert this to an uncertainty in seconds for t 1/2. Now, time for a bit of research. Question 6. Figure out what is decaying, what the final daughter nucleus is, and what type of radiation is being emitted. (There s slightly more to answering what is decaying than just saying Ba-137.) Also find out the accepted value of the half-life. 4 Conclusion For your conclusion, answer these questions: Question 7. Is your result consistent with the accepted value? Show your (very brief) work. Question 8. What were some possible sources of error in this experiment? 6
5 Checklist Answers to all questions (including pre-questions ). Graph of background data, with constant fit. Statement of measured background rate B. Equation to fit N(t) (source) plus background. Graph of source data, with fit, and explanation of parameters. Statement of decay lifetime τ. Comparison between your two background measurements. Statement of measured half-life (with units and uncertainty). Comparison between your measured half-life and accepted value. Sources of error. 7