Transcript half-life
http://www.nearingzero.net Monday, May 2, go to slide 12 Nuclear Transformations Radioactive Decay Half Life Radioactive Series “Plutonium keeps better in small pieces.”—Lester Del Rey. Chapter 12 Nuclear Transformations Radioactivity occurs because some nuclei are unstable and spontaneously decay. Important aspects of radioactivity: • Elements transform into other, different elements. • The energy released in radioactive decay comes from mass which is converted to energy. • Radioactivity is a quantum phenomenon. Radioactive decay is a statistical process. 12.1 Radioactive Decay There are five kinds of radioactive decay. Figure 12.3 shows them and gives the reasons for their occurrence. Understand figure 12.3. Starting on the next slide are the five kinds of radioactive decay. We will go into more detail for each in later sections in this chapter. (1) Gamma decay. • Occurs when a nucleus has excess energy. • A gamma ray (packet of energy) is emitted from the nucleus. • The parent and daughter nuclides are the same. • Example: 87 38 Sr * 87 38 Sr + . The * in the reaction denotes an excited nuclear state. (2) Alpha decay. • Occurs when the nucleus is too large. • An alpha particle is emitted, reducing the size of the nucleus. • The daughter nucleus has an atomic number 2 less and an atomic mass 4 less than the parent nucleus. • Example: 238 92 4 U 234 Th + 90 2 He . (3) Beta decay. • Occurs because the nucleus has too many neutrons relative to protons. • A neutron changes into a proton and emits an electron. • The daughter nucleus has an atomic number 1 more and an atomic mass the same as the parent nucleus. • Example: 14 6 C 147 N + e- . Later we will find there is something missing from this reaction. (4) Electron capture. • Occurs because a nucleus has too many protons relative to neutrons. • A proton captures an electron and changes into a neutron. • The daughter nucleus has an atomic number 1 less and an atomic mass the same as the parent nucleus. • Example: 64 29 Cu + e- 64 28 Ni . Again, we will find something is missing from this reaction. (5) Positron emission. • As with electron capture, this occurs because a nucleus has too many protons relative to neutrons. • A proton emits a positron and changes into a neutron. • The daughter nucleus has an atomic number 1 less and an atomic mass the same as the parent nucleus. • Example: 64 29 Cu 64 28 Ni + e+ . Guess what? Something is missing from this reaction! Radioactive decay involves an unstable nucleus giving off a particle or ray, and in the process becoming a more stable nucleus. There are several ways to detect what the particle/ray is. Detect the radiation after it passes through a magnetic field. Positive and negative charged particles will be deflected in different directions. Neutral particles or rays go straight through. See what it penetrates. A piece of paper can stop alpha rays. Beta particles can be stopped by a sheet of aluminum. Even lead may not stop gamma rays. The activity of a radioactive sample is the rate at which atoms decay. If N(t) is the number of atoms present at a time t, then the activity R is R =- dN . dt dN/dt is negative, so the activity is a positive quantity. The SI unit of activity is the becquerel: 1 becquerel = 1 Bq = 1 event/second. Another unit of activity is the curie (Ci) defined by 1 curie = 1 Ci = 3.70x1010 events/s = 37 GBq. This section also contains interesting paragraphs about the relation between the geology of the earth and radioactivity, and about radiation hazards. I won't lecture on them, but they are testable on quizzes or the final exam, so be sure to read them. 12.2 Half-Life Experimental measurements show that the activities of radioactive samples fall off exponentially with time. *Empirically: R = - R 0 e-λt . is called the “decay constant” of the decaying nuclide. Each radioactive nuclide has a different decay constant. *Argh! The half-life, T½, is the time it takes for the activity to drop by ½. We can find a relationship between and T½: R0 -λΤ = - R 0 e 1/2 2 activity after T½ original activity 1 -λΤ = e 1/2 2 +λΤ 1/2 e =2 Τ1/2 = ln 2 = ln 2 Τ 1/2 = 0.693 Τ 1/2 Here's a plot of the activity of a radionuclide. The initial activity was chosen to be 1000 for this plot. The half-life is 10 (in whatever time units we are using). All decay curves look like this; only the numbers on the axes will differ, depending on the radionuclide (which determines the half-life) and the amount of radioactive material (which determines the initial activity). Hyperphysics is a good place to go for supplementary material. Here’s their plot of radioactive decay (they use A instead of R for activity). One more picture, from Physics2000. We’ll visit their decay simulator later. What a crummy graph— look how bumpy it is! What’s this? (Answers later!) Remember, empirically… R = - R 0 e-λt . Let’s fix this! The empirical activity law can be derived if we assume that is the probability per unit time for the decay of a nucleus. Then dt is the probability that the nucleus will undergo decay in a time dt. If a sample contains N undecayed nuclei, then the number dN that will decay in the time dt is just N times the probability of decay, dN = -N dt . This equation can be integrated to give N = - N0 e-λt . which you should recognize as looking like the activity law with N's instead of R's. The activity R of a sample of N radioactive nuclei is just R = N. What’s the difference between R = - R 0 e-λt and N = - N0e -λt Other than the fact that one talks about rates and the other about numbers? R = - R 0 e-λt is empirical, and you should always be suspicious of empirical equations, which may or may not have any physical meaning. N = - N0 e-λt was derived under the assumption that is the decay probability per unit time, and is part of a testable theory. Big difference! Important! The equation for activity R in terms the number of nuclei present R = N involves , which is a probability. Radioactive decay is inherently probabilistic in nature! That tells you there must be quantum mechanics lurking behind it. That also tells you why this curve should not be smooth, even if you could eliminate all experimental sources of error! Here’s a worthwhile radioactivity applet. Example: radon (nasty stuff*) has a half-life of 3.8 days. If you start with 1 mg of radon, after 3.8 days you will have 0.5 mg of radon. Days 0 3.8 7.4 11.4 Radon Left (mg) 1 0.5 0.25 0.125 The mean lifetime of a nucleus is different than its half-life. It turns out that T = 1.44 T1/2 . See Beiser for details but don’t worry about this for the final. *But perhaps not as dangerous as once believed. http://www.nearingzero.net radiometric dating Carbon-14 dating is the best-known example. Carbon-14 is formed in the atmosphere by the reaction 14 7 N + 01 n 146 C + 11 H . This reaction is continually taking place in the atmosphere, and the carbon-14 atoms are continually beta decaying to nitrogen14, with a half-life of 5760 years. Because carbon-14 is continually being created and decaying, we eventually reach a steady state condition, where there is a constant amount of carbon-14 in the atmosphere. Living things take up carbon-14 as long as they are alive, and have the same ratio of carbon-14 to carbon-12 as does the atmosphere. When living things die, they stop taking up carbon-14, and the radioactive carbon-14 decays. If we compare the carbon-14 to carbon-12 ratio in a dead organism with a living one, we can tell how long the carbon-14 has been decaying without replenishing, and therefore how long the organism has been dead. This assumes he carbon-14 to carbon-12 ratio in the atmosphere is the same now as it was when the organism died. It also assumes living organisms now are essentially the same in their carbon content as were similar organisms long ago. Carbon-14 dating takes us back a relatively short time, and both assumptions seem to be valid. The formula for radiocarbon dating, derived from R = R0 e-t, is R0 1 t = ln . R We need to know the activity R0 of the organism at death, which is the reason for the second assumption on the previous slide. Radiocarbon dating is good for a few half-lives of carbon-14, or 50,000 or so years. A similar approach can be taken with radioactive potassium, rubidium, or uranium, to go back much further in time. We have to find parent-daughter decay schemes that give us unique daughter nuclei; i.e., they could have only come from decay of the parent. If we assume the daughter nuclei came only from the original radioactive nuclei, we can calculate the original number, and then calculate the decay time. We measure the time back to some event caused the clock to start "ticking;" i.e., an event that froze into the sample the particular number of parent atoms which resulted in the observed number of daughter atoms. Radiocarbon example. A piece of wood has 13 disintegrations per minute per gram of carbon. The activity of living wood is 16 dpm per gram. How long ago did the tree die? t = R 1 ln 0 . R 5760 years 16 t = ln = 1726 years . 0.693 13 12.3 Radioactive Series FS2003—skip this section (good idea to read it anyway). Most radionuclides belong to one of four radioactive series. The masses of the nuclides in these four series are given by A = 4n A = 4n+1 A = 4n+2 A = 4n+3. The fact that there are only four decay series may seem surprising at first, but it's not, if you think about it. The kinds of radioactive decay we mentioned above involve either a change in mass number A of 4 (alpha decay) or of 0 (all of the others). Thus, if we start with a nuclide of mass number A0, it can only decay into nuclides of mass number A0-4, A0-8, etc. This table in Beiser summarizes radioactive series further: Mass Numbers Stable End Product Series Parent 4n thorium 90Th 4n+1 neptunium 93Np 4n+2 uranium 239 92U 82Pb 206 4n+3 actinium 235 92U 82Pb 207 I used old-fashioned nuclide notation whole lot easier in powerpoint! 232 237 82Pb 83Bi 208 209 232 because it’s a Th 90 The thorium series. note: decay decay branch at 216Po http://hyperphysics.phy-astr.gsu.edu/hbase/nuclear/radser.html Visit http://hyperphysics.phy-astr.gsu.edu/hbase/ nuclear/radser.html to see charts of the other three series. We can calculate the number of daughter nuclei present as a function of time in these decay series. For this semester I am skipping the calculation.