nuclear medicine. Two methods (SPECT and PET) are discussed that use
nuclear radiation to do imaging of the body, known as radiation tomography.
Nuclear Physics and Medical Applications
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In this concluding chapter, we first summarize our knowledge of the atomic nucleus and discuss the types of nuclear radiation emitted by nuclei and how they can be detected. The rest of the chapter focuses on a variety of applications of nuclear radiation in science and medicine. We start our discussion of applications by introducing the half-life and its use in radioactive dating. Then we introduce some important ideas on dosimetry and the biological effects of radiation, as well as some ideas on nuclear medicine. Two methods (SPECT and PET) are discussed that use nuclear radiation to do imaging of the body, known as radiation tomography. The chapter concludes with the processes of nuclear fission and fusion, two topics that should be understood at a basic level by everyone in this nuclear age.
1. NUCLEAR SIZE, STRUCTURE, AND FORCES The nucleus is an extremely small dense object in an otherwise nearly empty atom. As we’ve seen, atomic sizes are about 0.1 nm. The nucleus is typically several fm (1015 m), or about 100,000 times smaller than the atom. To appreciate these relative sizes, imagine that we scale the size of an atom up to the size of a football field (100 yd ~ 100 m). On this scale the nucleus would have a relative size of 100 m/ (100,000) 1 mm, so that it would be like the head of a pin, not in a haystack, but on a football field. This is truly astounding because almost all of the mass of an atom is located inside the nucleus. Matter consists of dense cores in mostly empty space; the head of a pin located within an empty three-dimensional football field of space. Remember that the nucleus contains the protons and neutrons (together known as nucleons) of the atom, representing nearly its entire mass, because protons and neutrons each have more than 1800 times the mass of an electron. Unlike the electron, which appears to be pointlike, having no measurable size, nucleons have a finite size of about 1 fm. Neutral atoms have equal numbers of protons and electrons, with this number known as the atomic number and represented by Z; the number of neutrons in a nucleus is known as the neutron number and represented by N. The total number of protons and neutrons in a nucleus added together is known as the mass number A, where A Z N.
(26.1)
The integer mass number is approximately equal to the atomic mass. Remember that atomic masses are measured in atomic mass units (u), defined as 1/12 the mass of the carbon-12 atom, or 1 u 1.66 1027 kg.
J. Newman, Physics of the Life Sciences, DOI: 10.1007/978-0-387-77259-2_26, © Springer Science+Business Media, LLC 2008
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FIGURE 26.1 Plot of the stable nuclides. The line drawn is for Z N and the vertical and horizontal lines indicate the most stable nuclides (see the discussion of magic numbers below).
A particular nuclear species is called a nuclide, and is represented by the chemical symbol of its neutral atom together with its value of A written as a presuperscript. For example, 13C represents the nuclide with 6 protons (because all carbon atoms have six protons), and N ( A Z) 13 6 7 neutrons. Sometimes the Z value will be written explicitly as 136C although this is unnecessary because Z is evident from the chemical symbol. Nuclides with the same number of protons but different numbers of neutrons are known as isotopes; for example, the two stable isotopes of carbon are 12C and 13C with 6 or 7 neutrons, respectively; other isotopes of carbon are radioactive, meaning that they are unstable and “decay” into other nuclides (see below). Figure 26.1 shows a plot of the known stable nuclides. Scientists have learned about the size and shape of nuclei from high-energy scattering experiments. Electrons are accelerated to energies large enough (200 MeV) so that their wavelengths become comparable to nuclear dimensions, and are then directed on targets of various nuclei. Recall that l h /p where p ≈ E /c for relativistic electrons (in this case m0 can be neglected in the expression E2 p2c2 m02c4), so that l hc / E
1.2 10 12 m . E (in MeV)
For the energies just mentioned, the electron wavelength is below 6 fm, small enough to probe nuclear dimensions. From such experiments it is known that almost all
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nuclei are nearly spherical (although many of the rare-earth element nuclei, those with Z 57 71, are ellipsoidal) with somewhat fuzzy boundaries and effective radii R that depend on the mass number A according to R R0 A1/3,
(26.2)
with R0 ⬵ 1.2 fm. Because the density of the nucleus is given by the ratio of its mass (proportional to A) to its volume (proportional to R3, and thus, according to Equation (26.2), also to A), perhaps unexpectedly we see that the density of all nuclei is the same. We can therefore calculate the nuclear density using A 1, to find that 1.67 1027kg/ [(4/3)(1.2 1015 m)3] ⬵ 2 1017 kg/m3. This is an extremely high density; note that the density of common materials, and thus of atoms, is only on the order of 103 kg/m3, so that nuclei are 1014 times denser than atoms! Both the greater mass of a nucleon compared to the electron and, even more, the tiny size of the nucleus compared to atoms are responsible for this. Our picture of the nucleus as a dense ball of nucleons that are essentially in contact with one another leads to the striking question of why the nucleus is ever stable. After all, the protons, all with the same positive charge, are extremely close together in the nucleus and their electrical repulsive force is huge. Two protons that are 2 fm apart would experience an electrical repulsive force given by F
1 e2 , 4pe0 r2
(26.3)
where e is the proton charge and r is the 2 fm separation distance. This force is almost equal to 60 N (about 13 lb), a huge force that would instantly rip the nucleus apart if it were the only force acting. In fact, the nucleus is held together by the strong nuclear force, one of two very short-range nuclear forces (the other, known as the weak nuclear force, is involved in radioactive decay). The strong force between two neighboring protons in a nucleus provides an attractive force roughly 100 times stronger than the electrical repulsion between the two. This attractive force is the same for all protons and neutrons, independent of their electric charge, so that two neighboring neutrons, protons, or a neutron and a proton all feel the same attractive force. However, the strong force rapidly vanishes at distances of even a few fm within the nucleus, and certainly outside the nucleus. A useful simple picture of the nucleus is the liquid drop model in which the nucleus is pictured as a tiny drop of liquid. This analogy is appropriate because both the nucleus and a liquid drop have a uniform density, are incompressible, and are held together by large forces: surface tension forces in the case of a liquid, strong forces in the nucleus. This model provides a useful way to look at the process of nuclear fission later in this chapter as analogous to a drop of liquid breaking into two smaller drops.
2. BINDING ENERGY AND NUCLEAR STABILITY The total energy of the nucleus is the sum of its kinetic and potential energy. Because the potential energy is negative and larger, in magnitude, than the kinetic energy, the total energy of the nucleus is negative, just as we have seen it is for a neutral atom. If the nucleus were disassembled into its constituent protons and neutrons, their total energy would be more than that of the nucleus. This is just the same as the case for atoms where energy is needed to ionize an atom, for example, in hydrogen to separate the electron and proton, so that the energy of the final separated electron and proton have greater energy than that of the ground state atom. This difference is due to the binding energy of the atom or nucleus and, in the case
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of the nucleus is a considerable amount of energy. For any nucleus of atomic and mass numbers Z and A, the (positive amount of) binding energy is given by Nuclear Binding Energy Zmpc2 Nmnc2 mc2,
(26.4)
where mp, mn, and m are the masses of the proton, neutron, and nucleus, respectively. Because the energy equivalent of 1 atomic mass unit is (1 u)c2 931.5 MeV (found from E mc2 (1.6605 1027 kg)(2.9979 108 m/s)2(1 eV/1.6022 1019 J) 9.315 108 eV 931.5 MeV (with energy conversion to eV)), we see that the nucleons each have an energy equivalent of about 930 MeV, whereas a nucleus of mass number A has an energy equivalent of about A 930 MeV. A comparable calculation for an atom shows that the atomic binding energy is only on the order of at most tens of eV.
Example 26.1 Calculate the binding energy of 2H, 4He, 197Au, and 238U. Their nuclear masses are, respectively, m 2.013552 u, 4.001503 u, 196.923090 u, and 238.000180 u. Also calculate the binding energy per nucleon for each of these. Solution: Using Equation (26.4), the Z and N values of each isotope, and the values of mp 1.00727 u and mn 1.00867 u, we find for 2H, for example, that the binding energy B is B (1 # 1.00727 1 # 1.00867 2.01355) # 931.5 2.226 MeV. Similarly we find B values for 4He of 28.30 MeV, for 197Au of 1560 MeV, and for 238U of 1802 MeV. On a per nucleon basis, these values are 1.113, 7.075, 7.919, and 7.571 MeV/nucleon.
The nuclear binding energy is about 8 MeV per nucleon for nearly all but the smallest nuclides. This implies that the nuclear binding energy represents about (8 MeV)/(930 MeV) ⬵ 1% of the total nuclear energy, quite a substantial amount. If each nucleon interacted with all the others in a nucleus we should expect the binding energy per nucleon to grow in proportion to A, since each nucleon would interact with (A1) others. The binding energy per nucleon remains fairly constant, thus this implies that each nucleon only interacts with its nearest neighbors agreeing with our discussion above of the very short range of the strong nuclear force. Figure 26.2 shows the binding energy per nucleon of some nuclides as a function of mass number. Note that the larger the binding energy, the more stable the nucleus is. We show that this figure explains the phenomena of both nuclear fission and fusion. Many large nuclei are unstable and will spontaneously fission into two smaller nuclei, each of which has a larger binding energy per nucleon and is more stable. Similarly, under the proper conditions, two protons or other very small nuclei can combine, or fuse, together to form a larger nucleus that is more stable. Both of these reactions liberate substantial amounts of kinetic energy. Fission and fusion are further discussed in the last section of this chapter. There have been more than 2500 nuclides identified, with only a small number of these (about 280) stable. What determines whether a particular nucleus is stable or unstable? This is a complex issue. Figure 26.1 shows that at small values of N and Z stable nuclides have equal numbers of protons and neutrons, but that as these numbers increase, stable nuclides tend to have significantly more neutrons than protons. We can understand this fact as a consequence of the Pauli exclusion principle and the proton–proton electric repulsion. Recall that the exclusion principle states that interacting identical fermions, those elementary particles with half-integral spin, must have distinct quantum numbers. Protons and neutrons both have spin 1⁄2 and therefore must separately satisfy this principle.
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FIGURE 26.2 The binding energy per nucleon as a function of A. Note the absolute maximum at 56Fe and the minor peak at 4He, as well as the average value of about 8.5 MeV/u.
For a nucleus, just as for an atom, there are discrete energy levels at which nucleons can reside (discussed further below). If we consider a sequence of increasing Z ground state nuclides, as more protons are found in the nucleus they must occupy higher energy levels because only a spin-up and a spin-down proton can occupy the same otherwise labeled quantum state. An identical situation occurs for neutrons in a sequence of increasing N ground state nuclides. However, because protons and neutrons are different particles, they can occupy the same energy level. This implies that the energy of a nucleus with Z protons and no neutrons will be greater than the energy of a nucleus with Z/2 protons and Z/2 neutrons, because these can occupy the same set of the lower half of the energy levels (see Figure 26.3). Thus, for a given A, those nuclides with roughly equal Z and N numbers will have the lowest energies based solely on the exclusion principle, and will therefore be more stable. As more protons are packed into the nucleus, there is a second consideration. The repulsive electrical force between the protons begins to destabilize the nucleus and so there is a tendency for more neutrons than protons to be found in larger stable nuclei. These additional neutrons do not contribute to the electrical potential energy, but they do tend to separate the protons, thus stabilizing the nucleus by reducing the Coulomb interaction energy. This effect explains why the data in Figure 26.1 fall from the Z N line at larger values. For Z 82 (lead) additional neutrons do not eliminate the destabilization of the nucleus from large numbers of protons and these nuclei are all unstable. Although nuclear energy levels are very complex and no complete theory of the nucleus yet allows their precise calculation, there are some similarities between atomic and nuclear energy levels. A third factor that affects whether a nucleus is stable has to do with its energy level structure. When we discussed atoms and the Periodic Table of the Elements, we saw that the noble gas elements in the right-hand column all have completed electron shells and are extremely inert and stable. A similar energy level shell structure exists in the nucleus and those nuclides with closed shells are particularly stable. The numbers of nucleons in such closed shells are dubbed the magic numbers 2, 8, 20, 28, 50, 82, 126, . . . and apply to both the numbers N and Z (see Figure 26.1). For example, the 4He nuclide (also known as the alpha particle) is extremely stable because it has the magic number 2 for both N and Z. This can be
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FIGURE 26.3 Schematic energy level diagram for eight protons (left) or four protons and four neutrons (right). The arrows represent protons (blue) or neutrons (green) with spin up or down.
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seen in Figure 26.2 where 4He has an unusually high binding energy in its region of the curve. Nuclei that are not completely stable are called radioactive and come apart at some point in time. In the next section we discuss properties of radioactivity, and later sections focus on a variety of applications of nuclear radiation.
3. TYPES OF RADIATION AND THEIR MEASUREMENT The science of nuclear physics was born in 1896 when Becquerel, while working with photographic plates, accidentally discovered that a mineral (containing uranium) was able to expose the plate while in the dark. Shortly after this, the Curies (Marie and Pierre) isolated two new elements, named polonium and radium, and characterized the radiation they emit. Rutherford and others found that there are three distinct classes of nuclear radiation, based on their penetrating power: one type (named alpha, , rays) can be stopped by several sheets of paper; a second more penetrating type (beta, , rays) can be stopped by several mm thickness of aluminum; and a third most penetrating type (gamma, , rays) can pass through several cm of lead or through thick concrete walls. It was subsequently discovered that rays are helium nuclei (4He), that rays are high-speed electrons, and that rays are high-energy photons. These radioactive particles are all emitted from radioactive nuclei. We might begin by asking why these three types of particles and no others are the products of natural radioactivity. The primary requirement for a nucleus to be radioactive is that it must have more total energy than its products. This requirement can be written as Q (mP 兺mi)c2 0,
(26.5)
where mP is the mass of the original nucleus, the so-called parent, and the summation is over the masses of all the products, mi. In , , or decay, where in addition to the nuclear radiation, one of the products has most of the mass, it is called the daughter. The excess energy Q is known as the decay energy (or simply the Q of the reaction). Single nucleons are not emitted in nuclear decays because either Q 0 for that reaction (the usual case), or, if the reaction is energetically possible, it occurs so rapidly that the parent nucleus has all decayed away and is not naturally found. An isolated unstable nucleus will decay by the emission of either , , or radiation. We also study another type of nuclear reaction in which a collision between a nucleus and another nucleon can result in the splitting of the nucleus into two fission products of roughly equal size. Alpha decay is the spontaneous emission of an alpha particle from a nucleus. Because the alpha particle 42He consists of two neutrons and two protons, if the parent nucleus has Z protons and N neutrons, the daughter nucleus will have Z 2 protons, N 2 neutrons, and A 4 nucleons in total. In alpha decay the original 4 atom undergoes transmutation, becoming another element: AzE n A4 Z2 E¿ 2He, where E is the parent and E the daughter nucleus. The Q for this decay would be given by Q (ME ME MHe)c2 and, assuming the parent is at rest, Q essentially represents the kinetic energy of the emitted alpha particle (the daughter will necessarily recoil—in order to conserve linear momentum—but will carry off only a small fraction of Q; see Problem 7). Often the daughter nucleus of an alpha decay itself also undergoes alpha decay. This process defines a radioactive series, whereby each subsequent alpha decay results in a daughter with 4 fewer nucleons, eventually culminating in a stable nucleus. There are four possible series for alpha decay, based on whether the starting nucleus has an A equal to 4n, 4n 1, 4n 2, or 4n 3. For example, one such
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FIGURE 26.4 The thorium alpha decay series. Note that the only naturally occurring nuclides in this series are 232Th and 208Pb because all of the others have relatively short half-lives. Alpha decays ( ) result in a decrease of the mass number A by 4 (and Z by 2 determining the nucleus name) whereas beta decays ( ) produce no change in mass number A (but a change in Z of 1, changing the nucleus name; see just below).
Thorium Decay Chain Thorium-232 14 Billion yrs
α
Lead-208 Stable
α
Polonium-212 0.3 μsec
β Thallium-208 3 min
Radium-228 5.8 yrs β
α α
Actinium-228 6 hrs β Thorium-228 1.9 yrs
β Bismuth 212 1 hr β Lead-212 11 hrs
α
Radium-224 3.7 days
β
Radon-224 56 sec
α
α Polonium-216 0.15 sec
series (4n) begins with thorium 232Th and ends with the stable lead nuclide 208Pb (see Figure 26.4). Of these four possibilities, only three appear on the Earth, because the longest living element of the 4n 1 series has completely decayed away to stable products. The alpha particle has both charge (2e) and is relatively massive, therefore once it is ejected from a nucleus it will interact with matter rather strongly compared to beta and gamma radiation and has relatively little penetrating power, being stopped by paper. On the other hand, because of its charge and mass, it tends to be the most ionizing type of radiation as it passes through matter. This is discussed further in Section 5. Beta decay is the spontaneous emission of a high-energy electron (or positron, the antiparticle to the electron with the same mass and charge magnitude, but positive) from a nucleus. The electron is not an orbital electron of the atom, but comes directly from inside the nucleus where it is created just before being ejected. Examples of beta decay are the transmutation of 14C according to
14 C 6
n 14 7 N e n and of
19Ne
19 according to 19 10 Ne n 9 F e n, where is a neutrino, a neutral, nearly massless particle that is very difficult to detect (there are several types of neutrinos; we make no distinctions here). Note carefully that this decay conserves charge as can be seen by adding up the charges or Z numbers on the right, where the electron e (positron e) is sometimes written as 10 e (10 e) to indicate its charge. Because there are no electrons e (or positrons, e) to be found in the nucleus, how does this occur? The answer lies in the conversion of one type of nucleon to another within the nucleus. Either a neutron within the nucleus can spontaneously convert to a proton according to
n n p e n,
(26.6)
or a proton can convert to a neutron according to p n n e n
(26.7)
Note that both Equations (26.6) and (26.7) conserve electric charge (in Equation (26.6), both the left and right sides have 0 net charge; in Equation (26.7), both sides have 1 charge) and conserve the number of nucleons. The new nucleon will remain in the daughter nucleus; thus, when a nucleus undergoes beta decay, the mass number A does not change, but the Z will increase (e emission) or decrease (e emission) by 1 with N correspondingly decreasing or increasing. The
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ejected beta particle (e or e) and neutrino together acquire essentially the total kinetic energy Q released in the decay (Q 艑 (MParent MDaughter)c2, because the beta particle has negligible mass), so that the electron can have any energy between essentially 0 and Q, whereas the neutrino gains the balance of Q in kinetic energy. The beta particle is identical to any electron, but is so named simply to indicate it originates in a nucleus. When beta decay was first characterized, the variable energy of the emitted beta particle was not understood because the neutrino had not been detected. In addition to an apparent violation of conservation of energy, the laws of conservation of momentum and angular momentum appeared to be violated as well. In 1934 Enrico Fermi worked out a detailed theory of beta decay, proposing not only the existence of the neutrino, but a fourth type of fundamental force in nature known as the weak nuclear force. It was not until 1953 that direct laboratory evidence for the neutrino was obtained, but it had been accepted long before based on scientists’ belief in the fundamental conservation laws. It is currently thought that neutrinos are the most ubiquitous of all particles in the universe. In 1998 the first experimental evidence was obtained for a very small, but nonzero, neutrino mass by an international team of 120 scientists working in Japan. These experiments are very difficult and still a bit controversial. If nonzero, even if extremely small, the vast numbers of neutrinos in the universe would contribute substantially toward the total mass of the universe.
Example 26.2 Calculate the Q for the following reactions: (i) the alpha decay of 238U to 234Th; and (ii) the decay of 234Th to 234Pa. Use the following data: the nuclear masses of m(238U) 238.00018 u, m(4He) 4.00150 u, m(234Th) 233.99409 u, and m(234Pa) 233.99325 u, and m( ) (9.11 1031/1.66 1027) 0.00055 u. Solution: (i) The alpha decay products of 238U are 4He 234Th. We calculate the Q for this reaction, to be Q [m(238U) m(234Th) m(4He)]c2 (0.00459)(931.5) 4.28 MeV. (ii) In this case the reaction is 234Th → 234Pa , where the protactinium (Pa) nucleus has one more proton formed in the beta decay. Here we can calculate the Q of the reaction ignoring the neutrino produced. Doing this, we find that Q (233.99409 233.99325 0.00055)(931.5) 0.270 MeV. This is the maximum kinetic energy the electron can have because otherwise the neutrino may carry off some energy as well.
Gamma decay, the third type of radioactivity, is the emission of a high-energy photon from a nucleus. The gamma ray is emitted when a nucleus makes a downward transition between two nuclear energy levels, just as a photon is emitted from an atom when it makes a downward transition between atomic energy levels. A major difference is that, because of the much larger energy spacing between nuclear energy levels, a gamma ray has a much higher energy, about a million times more than a photon from an atomic transition. This much larger energy corresponds to a much shorter wavelength for gamma rays, on the order of 1012 1015 m. Typically, gamma rays are emitted by daughter nuclei that are left in excited states after or decays as they relax back to their ground state. Because gamma rays have no charge, they are the most penetrating of the three types of radiation. Medical imaging techniques that use radioactive isotopes require the emitted radiation to escape from the body in order to be detected. These
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Wire electrode at positive high voltage
Metal chamber at negative high voltage
FIGURE 26.5 Schematic of a Geiger counter used to measure the presence of radiation.
window Ionized gas tracks
Current pulse to electronics Gas-filled tube
radiation
techniques use gamma emitters because or rays have such short penetrating distances that they will not escape from the body. This is discussed further in Section 7 below. We conclude this section with a discussion of the detection of nuclear radiation. There are several general methods to detect individual radiation particles as well as several methods to visualize the trajectory of these particles. One basic class of detectors is the ion collection detector, consisting of a high Z gas (typically xenon) filled chamber with a thin window through which radiation enters (Figure 26.5). Inside are two electrodes (a negative cathode and positive anode) across which a high voltage is applied. Ionizing radiation that enters the tube interacts with the gas to create ion pairs that travel to the electrodes and make up a current. If the applied voltage is high enough, the current generated is proportional to the amount of ionizing radiation. Such detectors are called proportional counters. At even higher applied voltages, a single ionization event will trigger an avalanche of subsequent ionizations of the gas and under these conditions the detector is called a Geiger–Muller counter (sometimes a Geiger tube or counter). Geiger counters are excellent for detecting small amounts of radiation because of the large degree of amplification. In general, ionization detectors have limited application in nuclear medicine because they have poor efficiency for gamma rays which are the primary information-containing decay product, as mentioned above. A second type of radiation monitor is a scintillation detector, consisting of a scintillator coupled to a photomultiplier tube (Figure 26.6). The scintillator, or phosphor, is a material (typically NaI crystals, plastics, or a liquid) that emits visible light when excited by radiation. These are dense materials that are very efficiently excited by radiation, including gamma rays, and have relatively fast response times. The number of photons produced is proportional to the energy of the incident radiation and the light produced is then detected by the photomultiplier tube (see the photoelectric effect discussion in Chapter 24) whose output photocurrent can be analyzed to determine the energy of the incident radiation. Semiconductor detectors that use p–n junctions (see Chapter 25) to detect ionization due to radioactive particles are a third type of detector. Electron–hole pairs created in the p–n junction by radioactive particles constitute an electric current proportional to the radiation energy. A number of devices allow one to visualize the path of a single charged particle. The simplest is a photographic emulsion in which a chemical change along the particle’s trajectory can be developed to visualize the path. Two other devices, the cloud chamber and bubble chamber, make use of either a supercooled gas (that is ready to condense on any ionized particle) or a superheated liquid (that is ready to boil along the path of an ion), respectively, to visualize the trajectory of a highenergy ion. Usually a magnetic field within the chamber causes the charged particles to travel in helical paths. (Do you remember why?) Photographs of the
scintillator Current pulse to electronics photomultiplier
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FIGURE 26.6 A scintillation detector, converting radiation to light in the scintillator, the light then being detected by a photomultiplier and converted to an electric current signal.
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charge track (Figure 26.7) can then be used to measure the radii of curvature to deduce the momentum and sign of the charge of the particle.
4. HALF-LIFE AND RADIOACTIVE DATING In a macroscopic collection of radioactive nuclei, each nucleus decays independently of all the others. In fact because each nucleus is shielded by its atomic electrons, even environmental conditions of pressure, temperature, and the like do not affect radioactivity. It is impossible to predict when any particular nucleus will undergo radioactive decay. The radioactive decay process is a purely random one. We can, however, make statistical predictions about the fraction of nuclei that will decay in a given time interval based on an assumption that the probability for a decay is the same in every equal time interval up until the nucleus actually does decay. Once the parent transmutes to the daughter nuclide, that particular nucleus cannot repeat the process. Only if the daughter is itself radioactive can it decay further, but that process is described by a different probability. This statistical notion allows us to write that the decrease N in the total number of N nuclei in a sample (N equal to the number of radioactive decays) in a short time interval t is proportional to the time interval and to the total number of nuclei in the sample. In symbols we have that
FIGURE 26.7 Bubble chamber photo showing several interaction sites (vertices where tracks meet) and spirals indicating long-lived charged particles undergoing energy loss.
¢N lN¢t,
(26.8)
where the proportionality constant is called the decay constant whose value depends on the particular radioactive nuclide. Equation (26.8) can be solved for the number of nuclei N at any time t using calculus (see box) to find N(t) N0 e lt,
(26.9)
where N0 is the number of nuclei at time t 0. Equation (26.9), plotted in Figure 26.8 normalized to the fraction remaining, is known as the law of radioactive decay. The time 1/ is known as the lifetime of the decay and represents the time for the number of parent nuclei to decay to N0/e N0/2.718, as shown in the figure. The
1.2 0 log fraction remaining
1 fraction remaining
FIGURE 26.8 Radioactive decay law, normalized to the fraction remaining after some time. The half-life and one and two lifetimes are indicated on the figure. The insert shows that a semilog plot of the natural logarithm of the fraction remaining is plotted versus time, the data decrease linearly.
1/e
0.8 0.6
–1 –2 –3 –4 –5 –6 0
1
2
1/e
0.4
3
4
5
6
time
2
0.2 0 0
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τ1/2 1
2
4
6
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number of parent nuclei decreases exponentially with time. In subsequent equal time intervals , the number of parent nuclei will continue to decrease by the same ratio of 1/e as indicated in the figure, so that after two lifetimes there will be N0/e 2 nuclei left, after three lifetimes N0/e3, and so on. More commonly the rate of decay is specified by the half-life, defined as the time for the number of parent nuclei to decrease by a factor of two, rather than a factor of e (see Figure 26.8). Using Equation (26.9), we can substitute N(t) N0/2,
Writing the s in Equation (26.8) as differentials, we have dN lNdt. Dividing by N and integrating both sides from time 0 with N0 nuclei to some arbitrary time t with N(t) nuclei we have N(t)
L N0
N0 /2 N0e lt1/2 and then solve for t1/2 by taking the logarithm of both sides to find that t1/2
loge 2 l
0.693 . l
(26.10)
After one half-life there are N0/2 nuclei remaining, after two half-lives there are (N0/2)/2 N0/22 N0/4 remaining, after three half lives (N0/4)/2 N0/23 N0/8 remaining, and so on. The half-lives of various radioactive isotopes are listed in Table 26.1. Half-lives in nature vary from vanishingly short (1022 s) to nearly everlasting (1021 years).
Table 26.1 Half-Lives of Some Radioactive Nuclides Isotope
Symbol
Uranium-238 Carbon-14 Radium-226 Strontium-90 Cobolt-60 Iodine-131 Fluorine-18 Barium-141 Krypton-92 Polonium-214
238U 14C 226Ra 90Sr 60Co 131I 18F 141Ba 92Kr 214Po
Half-Life 4.5 109 years 5730 years 1600 years 29 years 5.3 years 8 days 1.8 h 18.3 min 1.8 s 164 s
Radioactivity
,
,
,
,
,
,
, ,
t
dN l dt. N L0
Remembering that the integral on the left is the natural logarithm of N, we have loge(N(t)) loge(N0) loge a lt.
N(t) b N0
Then, using the definition of the logarithm, we can rewrite this as Equation (26.9). Also note that by differentiating Equation (26.9) we can obtain Equation (26.11) for the activity, dN dN N0le lt c d e lt, dt dt 0 where we have used the first equation in this box in the second step.
The rate at which radioactive nuclei decay, N/t, is called the activity and is measured in disintegrations/s, or bequerel (Bq), where 1 Bq 1 disintegration/s. A more common unit of activity is the curie (Ci), with 1 Ci 3.7 1010 Bq. The curie is a rather large unit of activity in nuclear medicine and the mCi and Ci are often used. Activity can be directly measured by detection of the decay products. Because the number of decays in a short time interval is proportional to the number N of parent nuclei (see Equation (26.8)), the activity also decays exponentially with time according to ¢N ¢N a b e lt, ¢t ¢t 0
(26.11)
where the subscript again indicates the zero-time value. This should make intuitive sense; if after 10 half-lives there are 1/210 fewer radioactive nuclei, then the rate at which decays occur would also be expected to be smaller by the same factor.
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One application of radioactivity is the dating of ancient materials. A commonly used method is 14C dating (carbon-14 dating) of the age of once living organisms. All living plants and animals are carbon-based. There are two stable isotopes of carbon with 12C representing close to 99% and 13C about 1%. Carbon-14, a beta emitter with a half-life of 5730 years, is formed in the upper atmosphere by the interaction of cosmic rays with nitrogen in the air. The amount of 14C is very small, roughly 1.3 1012 times as much as 12C, but its net amount has remained stable over many thousands of years due to the balance in its production in the atmosphere and its radioactive decay. All living material incorporates 14C, ultimately by the absorption of CO2 in the air during photosynthesis in plants. Animals incorporate 14C on eating plants or other animals that have eaten plants earlier in the food chain. However, when an organism dies, no new 14C is further incorporated so that the ratio of 14C to 12C steadily declines with age after death, by a factor of two for every 5730 years. Measurement of 14C activity can thus be used to date the age of the remains of such organisms. For objects older than about 60,000 years, carbon dating does not work because there is too little 14C activity left to measure accurately. By using other isotopes with much longer half-lives, such as 238U, the geological age of rock formations can be determined in much the same way. A measurement of the parent to daughter ratio can be used to date materials back billions of years. Dating the oldest rocks found, the age of the Earth has been measured to be about 4 billion years. The oldest fossils found date from about 3 billion years ago. Radioactive dating has been critical in a host of geological and evolutionary studies.
Example 26.3 Suppose that you wish to authenticate animal skin remains from one of the earliest known collections of animals, that of Shulgi, a Sumerian ruler of a territory now in Iraq, dating back to 2094 BC. You take a small sample of the skin and chemically analyze it for carbon. From a 10 g sample of carbon, what activity would you expect to measure if the sample is indeed authentic? Solution: First we need to find the number of carbon nuclei present in the 10 g sample. We do this by assuming that essentially all the carbon is 12C so that there are (10 g)(1 mol/12 g)(6.02 1023 nuclei/mol) 5.0 1023 nuclei present. Thus, when alive, the animal would have had about (1.3 1012)(5 1023) 6.5 1011 14C nuclei present. In that case, from Equations (26.8) and (26.10) and the fact that 5730 y 1.81 1011 s, we know the initial activity was N0 (ln 2/1/2)N0 (0.693/1.81 1011)(6.5 1011) 2.49 Bq. If the animal skin is indeed authentic, it would be nearly 4100 years old. According to Equation (26.11) then, the expected count rate would be ¢N 12.49 Bq21e 4100/57302 1.22 Bq. ¢t Note that this rate would need to be measured very precisely by averaging over long times to ensure a reliable value because the count rate is so low.
Our discussion of radioactive decay in this section thus far has been limited to a single radioactive species decaying away according to Equation (26.9). In a more typical situation, there are several radioactive nuclides that decay successively from one to another in a radioactive series such as the thorium decay chain discussed in the previous section. In this case the parent nuclide will decay according to Equation (26.9), but each of the other nuclides in the series will be produced by the preceding decay and so the populations of these nuclides need to be found from their production rates, shown schematically in Figure 26.9.
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Parent: No(t)
λ0
First Daughter: N1(t)
λ1
Second Daughter: N2(t)
λ2
FIGURE 26.9 Schematic for series of radioactive decays, where the N’s are the populations and the ’s are the decay constants.
Using a similar analysis to that above for Equation (26.8), the change in first daughter population will be given by ¢N1 (t) [N0 (t)l0 N1 (t)l1 ]¢t,
(26.12)
where the first term on the right-hand side is the rate at which the population of first daughters increases from decays of parent nuclides, and the second term on the right is the rate at which first daughters decrease from its own decay at rate 1. A similar equation will hold for each subsequent daughter population. These equations can be solved for a variety of interesting cases, but the most common situation is one in which the parent decay rate is the slowest. Then over very long times, the parent population will decrease exponentially, according to Equation (26.9). But over much shorter times, the parent population N0 will essentially remain constant and will thus supply the first daughter population at a constant rate. Now because the first daughter decay rate is much faster, its population N1 will remain constant at a value controlled by the parent supply of first daughters. Under this condition, after a sufficient equilibrium time, all the N(t) will be constant in time, so that the left-hand side of Equation (26.12) becomes equal to zero. Then we find that N0l0 N1l1
(26.13)
and we can find the first daughter population to be a constant N1 (0/1)N0 in terms of the constant parent population N0. The same story will follow for the second and all other daughter populations in terms of that of the first, or previous, daughter population. This analysis explains why it is possible to have naturally occurring very short lifetime alpha emitting nuclei, such as are found in the radioactive series discussed in the previous section. If you look back at Figure 26.4 you will see, for example, that polonium-212 decays to lead-208 by alpha emission with a half-life of 0.3 s. Why should there be any Po212 left in naturally occurring ores mined on the Earth? The answer is that Po212 is a daughter in the radioactive series that has thorium-232, with a 14 billion year lifetime, as parent. The series of nuclei produced from Th232 continually produce new Po212 at essentially a constant rate.
5. DOSIMETRY AND BIOLOGICAL EFFECTS OF RADIATION The interaction of nuclear radiation with matter leads to ionization; in fact, nuclear radiation (as well as uv and x-ray photons) is sometimes also referred to as ionizing radiation. Because energies of only tens of electron volts are sufficient to ionize atoms, , , and particles, with energies of MeV, are each able to ionize many thousands of atoms before losing their energy. It is this ionization that makes nuclear radiation dangerous to living organisms. Here we introduce various units to measure exposure, and discuss those doses and the relative biological effects of radiation. A unit of exposure, the roentgen (R), was first introduced to define the extent of ionization produced by x-rays, but is also used for gamma radiation. Defined as the total number of ion pairs produced in a volume of 1 cm3 of dry air under standard conditions (0°C and 1 atmosphere of pressure), one roentgen is given by 1 R 2.58 104 C/kg air. This is a unit of exposure, giving the ionization level in air, but it does not give any information about absorption of radiation by living tissue or its effects on that tissue. A measure of the absorbed dose of radiation, the absorbed energy per unit mass, is the gray (Gy), where 1 Gy 1 J/kg. An older unit, still commonly used today, is the rad, where 1 rad 0.01 Gy. For a given exposure, the absorbed dose will vary greatly depending on the absorption characteristics of the material and the type of radiation.
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Furthermore, the amount of damage produced by a constant absorbed dose will also vary depending on the type of radiation. To account for these, another type of quantity is introduced, the biological dose equivalent, measured in sieverts (Sv), and given by biological dose equivalent (in Sv) absorbed dose (in Gy) RBE,
(26.14)
where RBE is a dimensionless weighting factor, named for relative biological effectiveness, that depends on the type of radiation. Values for RBE are given in Table 26.2. These values are obtained by considering the dose of radiation needed to produce the same effect as a dose of 200 KeV x-rays. Their exact values are somewhat fuzzy, because the relative effects of radiation on biological tissue depend on the particular choice of assay. From the table, we see that and rays produce similar effects to these x-rays and nucleons or particles produce considerably more damage to biological tissue. Another unit commonly used for biological dose equivalent is the rem, where 1 rem 0.01 Sv (note that the rem is commonly used when the absorbed dose is measured in rads, so that the biological dose equivalent (in rem) absorbed dose (in rad) RBE). Table 26.2 Relative Biological Effectiveness (RBE) of Different Types of Radiation Type of Radiation
RBE
200 KeV x-rays
Neutrons (fast) Protons
1 1 1 20 10 10
Our environment has many sources of natural radioactivity. We are all exposed to radioactivity from a continual shower of cosmic rays on the Earth (varying with altitude and latitude), from certain minerals found in building materials, from naturally found radon gas in the Earth that can enter and accumulate in basements, and even from radioactive elements (notably 14C and 40K) within our bodies. In addition, radiation is produced by many of the manmade devices in our environment, including television and cathode ray tube (CRT) computer monitors (but not liquid crystal display (LCD) monitors), luminous dial watches, as well as common dental and medical x-rays. To evaluate health risks posed by exposure to radiation, scientists have measured typical human biological dose equivalents and the U.S. government has established guidelines for maximum permissible occupational exposure. Table 26.3 shows some typical radiation doses from a variety of sources. Table 26.3 Typical Human Radiation Doses Source
Annual Dose (Sv)
Cosmic rays Cosmic rays (in high altitude airplane) Radioactive ores (external exposure) Ingested materials (mainly potassium) Inhalation of radon Diagnostic x-rays
4 104 7 106 Sv/h 6 104 2 104 2 104 7 104
These doses should be compared to average annual doses that hospital radiologists receive of about 5 103 Sv or to the maximum natural exposure to cosmic rays in mountainous areas of Brazil of about 102 Sv/year. Studies of these populations show no effects of these higher doses on mortality statistics. For comparison purposes single whole-body
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radiation doses at higher levels do have significant effects at levels over 0.50 Sv. At levels up to about 2 Sv there is a significant reduction in blood platelet and white cell counts. Above this level there is severe blood damage, nausea, hair loss, hemorrhage, and shortterm death in many cases. Whole-body doses between 4 and 5 Sv result in death to about 50% of such a population, and doses over 6 Sv result in nearly universal death. Long-term effects of radiation can be due to short-term high exposure or to accumulated chronic lowlevel exposure. Federal standards indicate an individual maximum annual exposure of 5 103 Sv, excluding medical sources. This is increased a factor of 10 for people who work with radiation sources, such as radiation technologists. It is thought that radiation kills cells by damaging their DNA so that the cells cannot reproduce or by causing sufficient other damage to prevent the cell’s normal repair mechanisms from working effectively. In medicine, radiation is often used to destroy cancer cells in a limited area of the body. Of course radiation will also kill healthy cells, particularly those that turn over rapidly, such as blood platelets and white cells or the cells lining the intestinal wall. That’s why the typical symptoms of radiation sickness are GI problems due to effects on the intestinal wall, immunological suppression due to white cell kill-off, and general weakness due to red cell and platelet kill-off. By giving radiation over a period of time in repeated smaller doses, it is often possible to minimize damage to normal cells while still killing tumor cells. The chemical changes induced by radiation are caused by the formation of free radicals, enhanced by the presence of oxygen. Therefore the oxygen content of a particular tissue or cancer type will affect the success of the radiation treatment. In the following two sections we discuss nuclear medicine further, focusing on the use of radioisotopes for both therapy and diagnostics.
6. RADIOISOTOPES AND NUCLEAR MEDICINE The key to understanding the use of radioactive isotopes (radioisotopes) in biological studies and in medicine is the fact that chemistry and radioactivity are completely independent processes. Chemistry is based on valence electron interactions and does not depend at all on nuclear properties. As an example of this, hydrogen and its isotope deuterium (an atom made from a single electron and a nucleus with one neutron in addition to a single proton) have exactly the same chemistry. The only difference in these two is their mass difference of nearly a factor of two. Because of this deuterium is often used in science experiments (in various types of spectroscopy) as an indicator of the location of hydrogen atoms because they bind in the same way chemically. Incorporation of radioactive isotopes in cells or in the body at very low doses does not directly change the normal sequence of chemical events that occurs. This fact allows radiolabeling (also known as tagging or tracer studies) to follow a particular type of molecule in its pathway through an organism. In this section we discuss several aspects of nuclear medicine, including the production and types of radioisotopes in use, tracer studies and detection methods in biological research, and various diagnostic tests in medicine using radioisotopes. In order to safely use radioisotopes in medicine, not only must the dose be well controlled, but the half-life of the isotope must be relatively short so that the radioactivity is quickly reduced, causing no long-term problems. The typical dose used in diagnostic tests is so low (~108 Sv/h) that there is no danger from radiation. Some commonly used radioisotopes are listed in Table 26.4. Technetium(Tc)-99m is the most common of these and can be combined with many different molecules to act as a radiopharmaceutical. It has a half-life of only 6 h so that in order to have sufficient amounts available for hospital studies it must be freshly extracted from molybdenum-99, itself having a 67 h half-life—a useful life span of about a week—and itself usually prepared in-house in a major hospital as discussed just below. The 99Mo is bound to a solid matrix in a chromatography column and as the technetium-99m forms by beta decay it is washed from the column and then can be used directly or as a radiopharmaceutical when labeling another molecule. Technetium-99m does not emit beta particles and its gamma emission is at an energy of 140 keV, a relatively low energy so that many escape the body to be detected. Furthermore, it has a very versatile chemistry and can be
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incorporated into a wide range of biomolecules that can be used to target different organs or tissues in the body. These are introduced into the body by injection, ingestion, or inhalation and then imaged, as discussed in the next section. Table 26.4 Some Commonly Used Radioisotopes in Medicine Radioisotope
Half-Life
Technetium-99m Iodine-123 Carbon-11 Iodine-131 Phosphorus-32
*6
h 13 h 20 min *8.1 days *14 days
e
,
Thallium-201 Gallium-67 Chromium-51
74 h 78 h *28 days
* Produced
Radiation
Applications Most widely used SPECT brain imaging PET Thyroid disorders Large variety of uses in biology and medicine Heart imaging Tumor imaging Red blood cell survival
in nuclear reactors; otherwise produced in an accelerator.
When radiopharmaceuticals are used in human diagnostic studies, there are two important characteristic times to consider. First, there is the physical half-life of the parent radioisotope, 1/2, as discussed above, that is due solely to nuclear decay. A second time constant is also important in these studies, the biological half-life b equal to the time for the body to wash out half of the pharmaceutical. This latter time constant is not of the same well-defined character as the radioactive half-life, but has considerable variability. These two processes occur simultaneously so that the effective decay rate in the body is given by the sum of the two different rate constants. This should make sense since both paths, physical radioactivity and elimination from the body, act to decrease radioactivity within the body and hence the effective rate constant should be their sum. The rate constant is the reciprocal of the corresponding time constants, therefore the overall effective half-life e is given by a “parallel” combination of time constants (similar to the effective resistance of parallel combinations of resistors), 1 1 1 . te t1/2 tb
(26.15)
Thus, the effective half-life is shorter than either the physical or biological half-life, just as the effective net resistance is less than either resistance in parallel. The most dangerous of environmental sources of radiation are those that are ingested and have long effective half-lives. An example is strontium-90 that can replace calcium in bones. It has a long biological half-life (45 years) as well as a long physical half-life (29 years), with a corresponding effective half-life of over 17 years. The fact that radioisotopes used in medicine need to have short half-lives means that they must be constantly replenished for use in hospitals and other medical facilities (they really have a built-in shelf life!). Major hospitals have special supply arrangements or even in-house facilities for their production. Two methods are used to produce radioisotopes: nuclear reactors or accelerators. In nuclear reactors, either neutron beams are used to produce radioisotopes with excessive numbers of neutrons that primarily decay by beta, followed by gamma, emission, or the reactor fission products are isolated and purified. This latter method is the primary source for 99Mo, the parent nucleus for technetium-99m, the most often used radioisotope. Cyclotrons (see Problem 21 in Chapter 17) and linear accelerators with proton beams are used to produce proton-rich radioisotopes. The production sources of the radioisotopes listed in Table 26.4 are indicated. Medical research often uses radioactive tracers as an in vitro tool. When used in test tube studies, radioisotopes provide a variety of methods in cellular and subcellular work. Some of the earliest uses of tracers were to map out biochemical pathways. Radioactive tracers can be used to determine rates of metabolic processes,
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spin +
antibody
Antigen Red = labeled
Antibody-Antigen Complex forms
FIGURE 26.10 Radioimmunological assay to determine the amount of antigen present. Known amounts of radiolabeled antigen and unlabeled antibody are combined and spun to separate the antibody–antigen complex from the antigen. From the ratio of counts in the pellet to that in the supernatant, the amount of antigen originally present can be found.
predominant pathways for biosynthesis and metabolism reactions, as well as spatial localization information. These are done by various chemical testing methods combined with measuring radioactivity levels at various stages in separations. Tracers can also be used in amounts too small for chemical testing. For example, a radioimmunological assay can determine the amount of an antigen present even in tiny amounts (~nanograms). In this technique a minute measured amount of radiolabeled antigen is added to the sample along with a measured small amount of antibody, small enough that it is all fully bound with antigen (see Figure 26.10). The antigen will bind to the antibody independent of whether it is labeled. When centrifuged, the antibody–antigen complex can be physically separated from the unbound antigen and the activity of each fraction can be determined. Therefore the ratio of labeled-to-unlabeled antigen bound to the antibody will reflect the same ratio as found in solution. Because the amount of labeled antigen added is known, the amount of antigen in the original sample can simply be computed from that ratio. There are radioimmunological tests for literally hundreds of drugs or proteins found in the blood, urine, and other bodily fluids. These are available in kits that are commonly used in clinical laboratories. In radioassays, it is important to record as much of the radioactivity as possible. The best detector used in biological research is one in which the sample is directly immersed in the detector itself, in the technique of liquid scintillation counting (Figure 26.11). In this method, the sample is dissolved or suspended in a mixture of a special solvent and a fluorescent liquid, together known as a scintillation cocktail. A radioactive particle emitted from the sample will produce a brief flash of light that is then detected by a sensitive photomultiplier tube, whose output electrical current is then a measure of the radioactivity. But more than this, if two different radioisotopes are present in the cocktail they will result in different amplitude current pulses making up the output electric current. A so-called pulse-height analysis of the output current of the photomultiplier tube allows the relative amounts of the two isotopes to be determined.
7. SPECT AND PET: RADIATION TOMOGRAPHY In this section we discuss two different imaging methods that are based on radioisotopes: single photon emission computer tomography (SPECT) and positron emission tomography (PET). Both of these methods give time-dependent three-dimensional images of the location of radioisotopes. Earlier imaging methods, using gamma ray cameras, give two-dimensional projections of the locations of radioactive sources within the body. The gamma ray cameras are plane arrays of scintillator/photomultiplier detectors, each with a lead collimating channel to only allow radiation directed toward it to be detected. Lead shielding stops all other radiation so that the detected intensity at each photomultiplier is a measure of the net amount of radioisotope along its axis (see Figure 26.12), giving a projected image of the “object” or location of radioisotopes within the body. These images are relatively poor compared to
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FIGURE 26.11 Liquid scintillation counting. Radioactive decay particles produce light in a scintillation cocktail; the light is collected and detected by a photomultiplier.
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CT or MRI pictures, with resolution limited by multiple scattering of gamma rays as they leave the body and by limited detector resolution to about 1 cm at best. On the other hand, by monitoring the time dependence of the images, information on the metabolism of the radiopharmaceutical can be obtained. Examples of such uses include images of the heart, kidneys, lungs, urinary tract, and so on to determine fluid flow volumes. For imaging, the best radioisotopes are gamma emitters since these will effectively escape the body to be FIGURE 26.12 A gamma ray detected. camera for obtaining projected SPECT uses an imaging system similar to that of CT scans. Either multidetector or images of the location of radioisorotating gamma ray camera systems are used to capture a series of two-dimensional topes through the body. The images, although each image uses a focused collection arrangement to improve resoluchannels at each detector are formed by lead shielding. tion and contrast (or ratio of the signal-to-noise of the background radiation). Data are back-projected to reconstruct the three-dimensional image, allowing sequential slices to be imaged with a spatial resolution of about 5 mm at best, compared to the 1 mm resolution of CT scans. Although the resolution is better in CT images, they measure only x-ray absorption through the body, which then must be interpreted in terms of structure of internal organs. SPECT examines images of the distribution of radiopharmaceuticals and the time dependence of the radioactivity signal as well. Because this spatial distribution is determined by the specific binding of the drug to which the radioisotope is attached, clearly these images are directly related to function and not simply to structure. Most major hospitals have facilities to do SPECT and it is increasingly used since the advent of better detectors and radioisotopes. Some of the organs imaged most often using SPECT include the brain, heart, circulatory system, bones, and tumors, in general. In combination with MRI and CT, this technique offers doctors an excellent tool in making diagnoses. Positron emission tomography (or PET) is an important variation on SPECT that is becoming more common as the assoFIGURE 26.13 Patient about to have a PET scan, ciated costs decrease. The radiation source in this case is a surrounded by a ring of detectors within the housing to positron emitter radioisotope (e.g., fluorine-18 or gallium-68) look for coincident 180° detections of gamma rays. that is attached to a pharmaceutical and ingested. These positron emitters have short half lives and usually require a hospital to have an accelerator facility to prepare the radioisotopes. An emitted positron is very rapidly annihilated by an electron to form a pair of gamma rays. The energy and momentum of these gamma rays must satisfy the laws of conservation of energy and momentum. If both the electron and positron were at rest, then the total momentum must remain zero (hence the need for two identical gamma rays traveling in exactly opposite directions) and the total energy must equal the total rest energy of the electron and positron. This energy is equivalent to 511 KeV for each gamma ray. Thus the net result of each decay event is the production of a pair of 511 KeV gammas that leave the body in opposite directions. PET detectors 180° apart around the source to be imaged are set to look for the coincident arrival of 511 KeV gamma rays (Figure 26.13). These characteristic events are very clearly due to the positron emission and by projecting the accumulated data from a large number of scans at different angles, and using similar image reconstruction methods to SPECT, high-quality image slices of typically 5 mm resolution can be obtained. Spatial resolution is inherently limited by two facts: the initial kinetic energy and momentum of both the positron and electron is typically small but nonzero so that there is some variability in the 180° angle, and also the positron may travel a short (~1 mm) distance before annihilation. Both of these effects, as well as limits on detector resolution, tend to smear out the images decreasing resolution a bit (see the example image in Figure 26.14).
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FIGURE 26.14 (left) CT image of a patient with lung cancer that has spread to the lymph glands as clearly seen in the PET scan (right) of the same patient.
PET scans of the brain, in particular, have revealed physiological correlates to a variety of disorders. Some of the most spectacular images recorded with PET have been brain scans that show brain activity in real-time. By imaging blood flow or glucose or oxygen metabolism and monitoring changes in time as the person is stimulated in various ways (e.g., visually), biochemical events can be directly correlated with brain activity (Figure 26.15). Studies comparing “normal” brains with those of people known to have various psychological disorders have begun to reveal a physiological basis for some of these problems. FIGURE 26.15 PET scan of brain showing the effects of Ritalin (methylphenidate; a drug prescribed for millions of young people with attention deficit hyperactivity disorder) on the number of dopamine transporters available (red more, blue less). These recycle extracellular dopamine, a molecule that has been noted to give pleasure, allowing it to re-enter cells. Thus Ritalin causes an increase in extracellular dopamine levels that apparently correlates well with increased levels of attention and ability to concentrate without distraction.
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8. FISSION AND FUSION In Section 2 we saw in Figure 26.2 that the binding energy of nuclides with A numbers near iron (56) have more binding energy, and are therefore more stable, than either very low A or very high A nuclides. In most larger nuclei, such as uranium, the long-range Coulomb repulsion of the protons is in a precarious balance with the short-range strong nuclear attractive force between adjacent nucleons. If such a nucleus is perturbed, for example, through a collision with an external nucleon, a new short-lived “excited” nucleus forms. The added energy causes the “liquid drop” nucleus to begin to elongate and once the nucleus becomes sufficiently asymmetric, the Coulomb repulsion of the two portions causes the nucleus to be unstable and decay by dividing into two roughly equal fission products. The difference in net binding energy between the higher-energy original nucleus and the total lower energy of the products is given off as kinetic energy of the fission products. This energy is substantial; for example, uranium has a binding energy per nucleon of about 7.6 MeV/nucleon (remember that these binding energies are actually negative, so that a smaller binding energy means a higher energy state), whereas the fission products have values of close to 8.5 MeV/nucleon. The difference of 0.9 MeV/nucleon amounts to about 100 MeV of kinetic energy for each of the two fission products. Fission was first discovered in 1938 by Hahn and Strassmann, who bombarded uranium with a beam of neutrons and found two fission products, barium and krypton. For each starting nucleus, there are many different pairs of possible fission products, most of them radioactive. One example of a fission reaction for uranium-235 is the reaction 1 0
141 92 1 n235 92 U n 56 Ba36 Kr30 n.
(26.16)
The fact that there are often additional neutrons emitted, with an average of 2–3 per fission, caused scientists early on to propose that a chain reaction of neutron-activated fission could occur. Each fission would lead to two or three neutrons released, some of which would produce further fissions so that there would be a positive feedback and rapid growth in the energy released in fission products. By 1942 Fermi had demonstrated such a chain reaction in the first nuclear reactor. The first use of nuclear fission was in the form of two atomic bombs dropped over Hiroshima and Nagasaki to end World War II with Japan in 1945. War had united many of Europe’s finest scientists with those of the United States in a secret effort to develop the atomic bomb at Los Alamos, New Mexico. Although it is generally agreed that the use of these bombs shortened the war and reduced the total number of deaths, some of the leading scientists who worked on the development of the atomic bomb believed, in retrospect, that it was a mistake and spent much of their subsequent efforts in attempts to bring about nuclear disarmament. Enrico Fermi’s first nuclear reactor had as its main initial function the production of plutonium to be used in two atomic bombs. Today there are about 450 nuclear power reactors used to generate electricity in about 30 countries around the world. Although there are several different designs of these reactors, they all basically use nuclear energy to generate heat, producing steam then used to drive turbines, thereby generating electricity. There are several key problems to producing controlled nuclear fission in a nuclear reactor. The predominant uranium-238 isotope (representing over 99% of naturally occurring U) is relatively stable against fission, whereas uranium-235 (only about 0.7% abundance) undergoes fission very efficiently when slow neutrons are absorbed. Sometimes uranium ore is processed to enrich the 235U component to a few percent to provide a “richer” fuel. A minimum amount of fuel, the critical mass, typically on the order of kg, is needed to have a self-sustaining nuclear reaction. A second problem is that of the two or three neutrons produced in a single fission, only one is needed to sustain a controlled reaction. If more than one neutron from each fission leads to additional fissions, the reaction will “run away,” as in a nuclear bomb, whereas if this number is less than 1.0, the reaction will eventually die out. Only by maintaining this number very near to 1.0, by the escape or absorption of excess neutrons in a special device known as a control rod, can the reaction be kept at a steady rate. Control rods are made from materials that very effectively absorb
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neutrons without undergoing fission. Neutron absorption in 235U leading to fission is most effective for slow thermal neutrons, those that have lost energy often making numerous collisions in a special purpose material known as a moderator. Moderators are designed to effectively slow neutrons. Water is commonly used as a moderator in nuclear reactors, with heavy (deuterated) water sometimes used because it absorbs fewer neutrons eliminating the need to enrich the uranium. Despite huge investments in safety features, there have been two significant accidents at nuclear power plants: one at Three Mile Island, in Pennsylvania in 1979 which was contained, and one at Chernobyl in Ukraine in 1986 where 31 people were initially killed, most from radiation. The Chernobyl accident released about 3–4% of its radioactive material resulting in about 130,000 people receiving significant radiation doses leading to a sharp increase in thyroid cancer among children in that region, with other long-term health effects still unclear. Apart from safety issues of nuclear power plants, there are also literally tons of highly radioactive waste products produced in these plants that need to be safely and securely isolated from our environment for thousands of years. Because of these safety and environmental concerns, alternative sources of electricity other than nuclear fission power are needed. Along with solar, wind, hydroelectric, and other “green” sources of power, a possible longterm solution involves a second type of nuclear reaction. According to Figure 26.2, two very low mass number nuclides with a small binding energy per nucleon can fuse together to produce a larger nuclide with a much greater binding energy per nucleon, thus releasing a large amount of energy. This process, known as nuclear fusion, releases much more energy per nucleon than fission, as can be seen from the steep initial slope in the binding energy per nucleon curve in Figure 26.2. In other words, the magnitude of the energy of the larger fused nucleus is much less than the sum of the energy of the lighter starting nuclei and the difference is liberated in the fusion reaction. For example, in the fusion of deuterium and tritium, two isotopes of hydrogen, an alpha particle and a neutron, form according to 2 1H
31 H : 42 He 10 n.
(26.17)
Calculating the net difference between the initial and final energies (using the masses of each and the equivalence of mass and energy; see the example just below) gives a net energy release of about 17 MeV for each fusion. Because there are only 5 nucleons involved in this reaction, the energy per nucleon is 3.4 MeV/nucleon, much larger than the 0.9 MeV/nucleon released in fission. On an energy per unit mass basis, fusion is a much more productive process than fission. Nuclear fusion occurs naturally in stars, including our sun, at extremely high temperatures. These thermonuclear reactions in stars are believed to have been responsible for generating all of the larger mass nuclei in the universe starting from hydrogen. We believe that very early in the history of the universe the temperature was too hot for atoms to be stable. As the universe expanded and cooled, hydrogen atoms formed and then condensed locally under gravity to form stars. As stars became more compact due to the force of gravity, the interior temperatures and pressures increased, providing an environment in which nuclear fusion could occur. Stellar fusion first uses hydrogen as a fuel, but as hydrogen is depleted fusion of other light nuclei also occurs. Thus, all the other elements found on Earth and throughout the universe originated in such stellar fusion reactions; we ourselves are therefore made of stellar material. One fusion reaction is the so-called proton–proton cycle: 1 1H
AND
1 1H
21 H n 32 H photon
3 2H
32 H n 42 He 11 H 11 H.
(26.18)
411H n 42 He 2e 2n 2g
Net reaction:
FISSION
11H n 21 H e neutrino
FUSION
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FIGURE 26.16 Inside the Princeton Tokamak (note the man on the left to judge the scale).
The overall result of these reactions is that four protons have fused to produce one alpha particle plus two each of positrons, neutrinos, and photons, with a net release of 24.7 MeV. The positrons quickly annihilate with electrons to form four additional photons, each with 0.51 MeV, so that the total energy released in the proton–proton cycle is (24.7 4 0.51) 26.7 MeV per helium nucleus formed. In order for this reaction sequence to occur, protons must be brought very close together at very high temperature to overcome their mutual electrostatic repulsion and fuse together. Central cores of stars, including our sun, have temperatures and pressures high enough for fusion to occur. To produce fusion on the Earth, where the pressure is much lower than in the core of a star, even hotter temperatures are required. The first fusion reactions produced were those of hydrogen bombs in which an atomic (fission) bomb was detonated to produce the sufficiently hot temperature necessary to initiate fusion in a deuterium and tritium pellet. Different schemes to produce controlled conditions for nuclear fusion have been tried, each attempting to heat a deuterium–tritium fuel pellet to temperatures of 108 K, by either extreme electric currents or particle or laser beams, forming a plasma (ionized gas) confined in space for long enough so that fusion can take place. In one scheme, magnetic confinement, the plasma is trapped by the presence of a very strong magnetic field that exerts magnetic forces on the moving ions traveling around within a toroidal (doughnut) shaped solenoid. Figure 26.16 shows the Princeton Tokamak Fusion Test Reactor for magnetic confinement. A second alternative scheme, inertial confinement, uses many highpowered laser pulses that simultaneously strike a deuterium–tritium fuel pellet from different directions. The beams produce high temperature and pressure so rapidly that the inertia of the fuel does not allow it to escape and fusion occurs. Figure 26.17 shows the target chamber of the NOVA Laser Facility at Lawrence Livermore Laboratory, a facility currently being replaced by an even larger one at the National Ignition Facility (NIF). Short controlled pulses of energy from fusion have been produced by both of these schemes, but much work needs to be done before these become viable commercial sources of energy. Fusion offers a number of advantages over the current fission nuclear power plants. Fuel for fusion is much more abundant, cheaper, and yields more energy on a per mass basis. The oceans are a vast supply of deuterium fuel. Furthermore, unlike fission, there are no radioactive byproducts, so that there are no long-term storage problems with radioactive waste. There is also the fact that, unlike fission reactions
FIGURE 26.17 (left) The NOVA laser showing some of the arms through which the laser power is focused on the fuel pellet at the center. (right) View of the artificial ministar created by inertial confinement fusion in the NOVA.
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in which chain reactions can become uncontrolled if there is a malfunction of control rods producing a melt-down as has happened at Chernobyl and Three Mile Island, failures in fusion reactors would lead to a shut-down of the fusion reactions themselves and no possibility of an out-of-control chain reaction. For these reasons if commercially produced in a reactor, energy from fusion might be the ultimate cure to the world’s energy problem.
CHAPTER SUMMARY The atomic nucleus contains Z protons and N neutrons with a total number of nucleons A Z N. Nuclei are very small in size, having radii given by R R0A1/3,
(26.4)
and is typically about 8 MeV per nucleon in all but the smallest nuclei. Three types of nuclear radiation exist known as alpha, beta, and gamma radiation. Alpha radiation is the emission of helium-4 nuclei (2 protons 2 neutrons) from nuclei through a process of tunneling. Beta emission comes from the production of electrons or positrons within the nucleus because of neutron or proton decay, respectively, and these “beta particles” are emitted from the nucleus along with neutrinos at high energy. Gamma emission comes from transitions from excited nuclear states giving rise to high energy photons. Each of these types of radioactivity are characterized by their Q, or decay energy, Q (mP 兺mi)c2 0,
(26.5)
where P stands for parent nucleus and the sum is over all the products. Radioactive decay is governed by an exponential decay of the numbers of radioactive nuclei N, N(t) N0 e lt,
(26.9)
where N0 is the number of such nuclei at time zero and is the decay rate for the process. The half-life of the
C H A P T E R S U M M A RY
t1/2
loge 2 l
0.693 . l
(26.10)
(26.2)
with R0 1.2 fm. A nucleus of mass m has a nuclear binding energy given by Nuclear Binding Energy Zmpc2 Nmnc2 mc2,
reaction is the time for 1/2 of the nuclei to decay and is related to the decay rate by
Measures of exposure to radioactivity include: the Roentgen (R) which is simply a measure of the number of decays per unit volume; the Gray (Gy; 1 Gy 1 J/kg) or rad (1 rad 0.01 Gy) which are measures of the absorbed dose of radiation; or the biological dose equivalent, measured in sieverts (Sv), or in rem (1 rem 0.01 Sv), biological dose equivalent (in Sv) absorbed dose (in Gy) RBE, (26.14) where RBE (relative biological effectiveness) is a dimensionless weighting factor describing the effectiveness of different radiation to be absorbed by the body. Nuclear medicine involves the use of short-lived radioactive tracers (radiolabeling) to follow the path of a particular molecule through the body either by in vivo or in vitro studies. Two imaging methods that use radiotracers are SPECT (single photon emission computer tomography) and PET (positron emission tomography). Because the binding energy per nucleon for large nuclei is less in magnitude (~0.9 MeV) than for intermediate-sized nuclei, large nuclei can undergo fission releasing the excess energy of ~100 MeV for each of the products, along with several neutrons. Under controlled conditions this energy can be harnessed in nuclear power plants, whereas if left uncontrolled, this serves as the basis for a fission bomb. Fusion is the naturally occurring process in stars whereby hydrogen nuclei, or other small nuclei, are compressed and heated until they fuse to form larger nuclei, releasing large amounts of binding energy (~3.4 MeV per nucleon) in the process. Current research is attempting to produce controlled fusion in the laboratory as a means of generating energy that would be much cleaner than fission nuclear power plants. Very high power lasers are being used to attempt to achieve the very high temperatures and pressures needed to cause fusion.
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QUESTIONS 1. Carefully distinguish between Z, A, and N. Which of these change and which remain constant for , , and decay? 2. Why does Equation (26.2) lead to a picture of the nucleus as a dense-packed ball of nucleons in contact with each other? (Hint: How does the equation predict the nuclear mass will vary with radius?) 3. Compare the notion of nuclear binding energy to the ionization energy of an atom. How is it similar and how is it different? 4. What three factors determine the stability of nuclei? 5. Which of the following nuclei have net spin (indicate whether due to protons or neutrons) in their ground states: 1H, 12C, 13C, 15N, 31P? 6. What are magic numbers and how are they determined? 7. Compare the , , and decay particles in terms of penetrating power and radiation damage produced. 8. Complete the following nuclear processes by stating what the nucleon X represents: (a) 60Co
n 60Ni X; (b) X n 234Pab ; (c) X n 230Th 4He
9. Write the complete decay scheme for the decay of 126Sn and of 60Co. 10. Does it matter at what time a measurement starts in order to measure the half-life of a radioactive sample? That is, suppose two experimenters take the same sample of a radioactive material with a 3 min half-life and each independently tries to measure the half-life. Does it matter whether they start their measurements at the same time? 11. If a radioactive sample starts out with 1020 nuclei, how many will be left after 10 half-lives? After 10 lifetimes? 12. What are the fortuitous circumstances that allow 14C dating of once-living organisms? 13. What is the difference between exposure, absorbed dose, and biological dose equivalent? Which is most important in determining health risks? 14. Which has the longer effective half-life when used as a radiopharmaceutical, an isotope with a 45 h halflife that takes twice as long to wash out of the body as a second isotope with a 75 h half-life and a 7 day biological half-life? 15. In the radioimmunological assay, why is the ratio of the labeled-to-unlabeled antigen bound to the antibody the same as that ratio found in the supernatant? 16. In PET, how is it known that a particular detected gamma came from pair annihilation within the body? 17. In a nuclear fission power plant, what is the purpose of a control rod? A moderator? 18. Which would liberate more energy: assembling 14 protons and 14 neutrons to make one 28Si nucleus or two 14N nuclei?
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MULTIPLE CHOICE QUESTIONS 1. The nucleus is about this many times as large as the atom: (a) 102, (b) 103, (c) 104, (d) 105. 56 2. The nuclide of iron 26 Fe, has (a) 30 neutrons, (b) 56 neutrons, (c) 26 electrons, (d) 56 protons. 3. The Nobelium nucleus, 255 102 No, a very short-lived (3 min half-life) manmade nuclide, has an effective radius of about (a) 1.2 fm, (b) 7.6 fm, (c) 5.6 fm, (d) 0.19 fm. 4. Without the strong nuclear force, carbon-based life could not exist. This is primarily because the strong nuclear force (a) permits nuclei to consist of more than just a single proton, (b) keeps gravity from collapsing all matter into a single point, (c) keeps the electric force from attracting all electrons into the nucleus, (d) is responsible for covalent bonds between carbon atoms. 5. Very large nuclei are radioactive because of all but the following (a) the electrical repulsive force destabilizes them, (b) the total binding energy of two fission fragments is smaller than that of the original nucleus, (c) the excess neutrons do not sufficiently shield the repulsive force between protons, (d) it is relatively easy for a proton to escape (tunnel) out of the nucleus. 6. A nucleus with 20 neutrons is unusually stable because (a) it has an unusually low binding energy, (b) it has a closed nuclear shell, (c) it has all spin paired neutrons, (d) it has an unusually high propensity for decay. 7. The neutrino was first predicted in beta decay because of missing (a) spin, (b) angular momentum, (c) energy, (d) charge. 8. In a scintillation detector, incoming X’s are eventually converted into outgoing Y’s where X and Y could be (a) gammas and visible photons, (b) electrons and electrons, (c) electrons and visible photons, (d) visible photons and electrons. 9. Starting with 1012 radioactive nuclei, after 4 half-lives about (a) 2.5 1011, (b) 1.8 1010, (c) 108, (d) 6.3 1010 nuclei will remain. 10. In radioactive decay, compared to the activity, the total number of radioactive nuclei decays (a) exponentially with the same half-life, (b) logarithmically with the same half-life, (c) exponentially with a greater half-life, (d) exponentially with a smaller half-life. 11. Carbon-14 dating relies on all of the following assumptions except (a) the amount of 14C in the air has remained constant, (b) no new 14C is taken in after death, (c) 14C is the only radioactive form of carbon with a long half-life, (d) 14C is an alpha emitter with a 5730 year half-life. 12. One hundred hours of flying time in a high-altitude jet gives an equivalent radiation dose to a diagnostic x-ray. This large dose is primarily due to the fact that (a) there are fewer people at those altitudes to absorb
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13.
14.
15.
16.
the cosmic rays so each person gets a higher dose, (b) the Earth’s atmosphere shields us on the ground from most cosmic rays (c) the plane is that much closer to the sun, so the dose is higher, (d) cosmic rays interact with the plane’s metal and produce high doses of secondary radiation. Which of the following is not a desired feature of a radiopharmaceutical? (a) A relatively short (hours) half-life, (b) relatively low energy radiation, (c) a long biological half-life, (d) ability to bind to specific target tissue. In liquid scintillation counting which of the following is not true? (a) For each radioactive decay a single electron is detected, (b) a radioactive decay results in a photon of visible light, (c) different radioactive emissions result in different amplitudes of detected current pulses, (d) the solvent that the sample is dissolved or suspended in has a fluorescent component. The detected particles in PET are all but the following. (a) They are an electron and positron, (b) they are measured 180° apart along a line, (c) their energy is equal to the rest mass of the electron, (d) they are produced after ingesting a positron emitting source. Essential features of a nuclear fission power plant include all but the following. (a) Moderators, (b) control rods, (c) deuterium–tritium fuel pellets, (d) steam turbines.
PROBLEMS 1. The largest stable nucleus has a mass number of 209. Find the ratio of the radii, surface areas, and volumes of this largest nucleus to that of a hydrogen nucleus. 2. A neutron star has a diameter of about 20 km and has a density roughly that of the nucleus. What is its mass? How many solar masses is this (solar mass 2 1030 kg)? What is the mass number for the neutron star (i.e., how many nucleons does it contain)? 3. If the sun (mass 2 1030 kg, radius 7 108 m) collapsed until it had a density equal to that of nuclei, what would be its radius? (Actually, a star cannot collapse to nuclear densities unless its mass exceeds a critical mass, known as the Chandrasekhar mass, of about 1.4 solar masses to overcome the Pauli exclusion repulsion of the electrons.) 4. Using the numbers in Example 26.1 calculate the binding energy of radium 226 (m 225.97709 u), radium 228 (m 227.98275 u), and thorium 232 (m 231.98864 u). Also find their binding energy per nucleon. 232 5. Calculate Q for the -decay of 90 Th using data in the previous problem and in Example 26.1. 6. Calculate Q for the decay of 24Na given the following data: m(24Na) 23.98492 u, m(24Mg) 23.97845 u, m(24Ne) 23.98812 u, m( ) 5.49 104 u. What is the range of possible energies for the emitted beta particle?
QU E S T I O N S / P RO B L E M S
7. Show that in alpha decay from a stationary parent nuclide that conservation of energy and momentum lead to a relation between the Q for the nuclear reaction and the kinetic energy gained by the alpha particle, KE, given by Q KE £1
8.
9.
10. 11.
12.
m( 42 He) m(daughter)
≥.
Then look back at Example 26.2 and calculate the kinetic energy of the alpha emitted in the decay of 238U. The first successful experiment to detect the neutrino was done in 1953 by Reines who won the 1995 Nobel prize for this work. Neutrinos from the Hanford nuclear reactor were incident on a tank of 200 L of water in which they very infrequently interacted with the water protons to produce a neutron and a positron in the reaction: p → n . The positrons subsequently annihilated producing two signature gammas traveling 180° apart and the neutrons were captured by cadmium in the form of 40 kg of CdCl2 salt added to the water to produce several additional gammas. These gammas were detected by three scintillator layers and 110 photomultiplier tubes surrounding the water and Reines and co-workers carried out numerous checks to ensure that these gammas did originate from the above reaction and not from any other source. Although the neutrino flux was as much as 1013 /cm2-s, they detected on average only 0.027 events per hour per phototube in the entire detector. Assuming the cross-sectional dimensions of the water tank to be 2 m on a side and the neutrino flux to be uniform over this area, what fraction of the neutrinos interacted with the water in their detector, assuming 100% collection efficiency of the gammas? Suppose that the phototube of a scintillation detector has a gain of 5 105, representing the average number of electrons produced at the anode for each electron emitted at the photocathode. If a 10 Ci gamma emitting radioactive source is detected with 5% efficiency, find the average output current from the phototube. How long does it take for 90% of a 60Co sample originally present to decay? What mass of 90Sr is needed to have an activity of 1 mCi? How long will it take for the activity to decrease to 0.25 mCi? Iodine is selectively accumulated in the thyroid gland where it can build to dangerous levels. When radioactive materials have been released into the atmosphere from either nuclear power plant accidents, such as the major one at Chernobyl, or from nuclear testing, these materials tend to concentrate (through eating of plants by animals) and show up in food products, such as milk, a food which is preferentially eaten by children. Thus, even though the half-life of 131I is relatively
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13.
14.
15.
16.
17.
short, this isotope has caused thyroid cancer in many children in affected areas. Some children received up to 1000 rem from 131I release at Chernobyl. By what factor is this above the maximum annual exposure recommended? How long would it have taken for the radiation level to have decreased so that for the same consumption of milk the exposure would have been at 0.1 times the maximum annual recommendation? After the sudden release of radioactivity from the Chernobyl nuclear reactor accident in 1986, the radioactivity of milk in Poland rose to 2000 Bq/L due to iodine-131 present in the grass eaten by dairy cattle. Radioactive iodine, with a half-life 8.0 days, is particularly hazardous because the thyroid gland concentrates iodine. (a) What is the decay constant that characterizes the decay of 131I if it has a half-life of 8 days? (b) What is the storage time needed to decrease the 131I content of cheese produced from these cows’ milk to 15% of the original level? A bone fragment is found in the desert. If it has a mass of carbon (due to only 14C and 12C) of 200 g, how old is it if it has an activity of 15 decays per second? The ratio of 14C to 12C is 1.3 1012. To destroy a cancerous tumor, a dose of gamma radiation totaling an energy of 2.12 J is to be delivered in 30.0 days from implanted sealed capsules containing palladium-103. Assuming that this isotope has a halflife of 17.0 days and emits gamma rays of energy 21.0 keV, which are entirely absorbed within the tumor, what is the initial activity of the set of capsules, and what total mass of radioactive palladium should these “seeds” contain? 238U decays by the emission of an alpha particle. (a) What is the decay sequence? (b) What is the daughter nucleus? (c) What is the energy of the alpha particle (its mass is 4.0026 u)? (d) What is the velocity of the particle? (e) Is the alpha particle relativistic? Strontium is chemically similar to calcium and can replace calcium in bones. The radiation from 90Sr can damage the bone marrow where blood cells are
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18.
19.
20.
21. 22.
23.
24.
25.
produced, and lead to serious health problems. How long would it take for all but 0.01% of a sample of 90Sr to decay? Calculate the activity (in Bq) of one gram of radium226. (Hint: See Example 26.3; this is the definition of one curie.) An amateur archeologist finds a bone that he believes to be from a dinosaur. He sends a chip of it off to a laboratory for 14C dating. The lab finds that the chip contains 5 g of carbon and has an activity of 0.5 Bq. How old is the bone? Could it be from a dinosaur? An 85 kg person was exposed to a gamma source and received a whole body dose of 0.5 Sv. How much energy was deposited in the person’s body? Repeat this calculation if the radiation was from an alpha source. What dose (in Gy) of gammas produces the same biological effects as a 50 rad dose of alpha particles? What fraction of a 1 g sample of 90Sr sitting on a table will remain in 17 years? If the strontium had been ingested and all initially been absorbed into a person’s bones, what fraction would remain after 17 years? (The biological half-life of 90Sr is 45 years.) A small amount of phosphorus-32 was accidentally ingested and its activity carefully monitored over time. After 8 days, the activity had halved. The physical half-life of phosphorus-32 is given in Table 26.4. Find its biological half-life. In a radioimmunological assay 10 nM of a 125I labeled antigen and 1 nM of antibody were added to an unlabeled sample of the antigen. The solution was centrifuged and the activity of the supernatant and pellet were measured and found to be in the ratio of 5.4:1. How much antigen was originally present in the solution? Calculate the net energy released in each step of the proton–proton cycle shown in Equation (26.18). Then add up the net release for one step in the cycle being sure to use a balanced net reaction; recall that the mass of the neutrino and photon are zero. Check your result by a direct calculation of the energy released in the net overall reaction. Use the following masses: m(1H) 1.00728 u; m(2H) 2.01355 u; m(e) 5.49 104 u; m(3H) 3.01550 u; m(4He) 4.00151 u.
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