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Modern Physics to Which He Did This Observation, Usually to Within About 0.1% Chapter 12 NUCLEAR STRUCTURE AND RADIOACTIVITY Radioactive isotopes have proven to be valuable tools for medical diagnosis. The photo shows gamma-ray emission from a man who has been treated with a radioactive element. The radioactivity concentrates in locations where there are active cancer tumors, which show as bright areas in the gamma-ray scan. This patient’s cancer has spread from his prostate gland to several other locations in his body. 370 Chapter 12 | Nuclear Structure and Radioactivity The nucleus lies at the center of the atom, occupying only 10−15 of its volume but providing the electrical force that holds the atom together. Within the nucleus there are Z positive charges. To keep these charges from flying apart, the nuclear force must supply an attraction that overcomes their electrical repulsion. This nuclear force is the strongest of the known forces; it provides nuclear binding energies that are millions of times stronger than atomic binding energies. There are many similarities between atomic structure and nuclear structure, which will make our study of the properties of the nucleus somewhat easier. Nuclei are subject to the laws of quantum physics. They have ground and excited states and emit photons in transitions between the excited states. Just like atomic states, nuclear states can be labeled by their angular momentum. There are, however, two major differences between the study of atomic and nuclear properties. In atomic physics, the electrons experience the force provided by an external agent, the nucleus; in nuclear physics, there is no such external agent. In contrast to atomic physics, in which we can often consider the interactions among the electrons as a perturbation to the primary interaction between electrons and nucleus, in nuclear physics the mutual interaction of the nuclear constituents is just what provides the nuclear force, so we cannot treat this complicated many- body problem as a correction to a single-body problem. We therefore cannot avoid the mathematical difficulties in the nuclear case, as we did in the atomic case. The second difference between atomic and nuclear physics is that we cannot write the nuclear force in a simple form like the Coulomb force. There is no closed-form analytical expression that can be written to describe the mutual forces of the nuclear constituents. In spite of these difficulties, we can learn a great deal about the properties of the nucleus by studying the interactions between different nuclei, the radioactive decay of nuclei, and the properties of some nuclear constituents. In this chapter and the next we describe these studies and how we learn about the nucleus from them. 12.1 NUCLEAR CONSTITUENTS The work of Rutherford, Bohr, and their contemporaries in the years between 1911 and 1920 showed that the positive charge of the atom is confined in a very small nuclear region at the center of the atom, that the nucleus in an atom of atomic number Z has a charge of +Ze, and that the nucleus provides most (99.9%) Enrico Fermi (1901–1954, Italy- of the atomic mass. It was also known that the masses of the atoms (measured in United States). There is hardly a field atomic mass units) were very nearly integers; a glance at Appendix D confirms of modern physics to which he did this observation, usually to within about 0.1%. We call this integer A the mass not make contributions in theory or number. It was therefore reasonable to suppose that nuclei are composed of a experiment. He developed the statis- number A of more fundamental units whose mass is very close to 1 u. Because tical laws for spin- 1/2 particles, and the only particle known at that time with a mass close to1uwastheproton in the 1930s he proposed a theory (the nucleus of hydrogen, with a mass of 1.0073 u and a charge of +e), it was of beta decay that is still used today. postulated (incorrectly, as we shall see) that the nucleus of an atom of mass He was the first to demonstrate the number A contained A protons. transmutation of elements by neutron Such a nucleus would have a nuclear charge of Ae rather than Ze; because bombardment (for which he received A > Z for all atoms heavier than hydrogen, this model gives too much positive the 1938 Nobel Prize), and he directed charge to the nucleus. This difficulty was removed by the proton-electron model, the construction of the first nuclear in which it was postulated (again incorrectly) that the nucleus also contained reactor. (A − Z) electrons. Under these assumptions, the nuclear mass would be about 12.1 | Nuclear Constituents 371 A times the mass of the proton (because the mass of the electrons is negligible) and the nuclear electric charge would be A(+e) + (A − Z)(−e) = Ze,in agreement with experiment. However, this model leads to several difficulties. First, as we discovered in Chapter 4 (see Example 4.7), the presence of electrons in the nucleus is not consistent with the uncertainty principle, which would require those electrons to have unreasonably large (∼ 19 MeV) kinetic energies. A more serious problem concerns the total intrinsic spin of the nucleus. From measurements of the very small effect of the nuclear magnetic moment on the atomic transitions (called the hyperfine splitting), we know that the proton has an intrinsic spin of 1/2, just like the electron. Consider an atom of deuterium, sometimes known as “heavy hydrogen.” It has a nuclear charge of +e, just like ordinary hydrogen, but a mass of two units, twice that of ordinary hydrogen. The proton-electron nuclear model would then require that the deuterium nucleus contain two protons and one electron, giving a net mass of two units and a net charge of one. Each of these three particles has a spin of 1/2, and the rules for adding angular momenta in quantum mechanics would lead to a spin of deuterium of either 1/2 or 3/2. However, the measured total spin of deuterium is 1. For these and other reasons, the hypothesis that electrons are a nuclear constituent must be discarded. The resolution of this dilemma came in 1932 with the discovery of the neutron, a particle of roughly the same mass as the proton (actually about 0.1% more massive) but having no electric charge. According to the proton-neutron model, a nucleus consists of Z protons and (A − Z) neutrons, giving a total charge of Ze and a total mass of roughly A times the mass of the proton, because the proton and neutron masses are roughly the same. The proton and neutron are, except for their electric charges, very similar to one another, and so they are classified together as nucleons. Some properties of the two nucleons are listed in Table 12.1. The chemical properties of any element depend on its atomic number Z, but not on its mass number A. It is possible to have two different nuclei with the same Z but with different A (that is, with the same number of protons but different numbers of neutrons). Atoms of these nuclei are identical in all their chemical properties, differing only in mass and in those properties that depend on mass. Nuclei with the same Z but different A are called isotopes. Hydrogen, for example, has three isotopes: ordinary hydrogen (Z = 1, A = 1), deuterium (Z = 1, A = 2), and tritium (Z = 1, A = 3). All of these are indicated by the chemical symbol H. When we discuss nuclear properties it is important to distinguish among the different isotopes. We do this by indicating, along with the chemical symbol, the atomic number Z, the mass number A, and the neutron number N = A − Z in the following format: A ZXN where X is any chemical symbol. The chemical symbol and the atomic number Z give the same information, so it is not necessary to include both of them in the isotope label. Also, if we specify Z then we don’t need to specify both N and TABLE 12.1 Properties of the Nucleons Name Symbol Charge Mass Rest Energy Spin + 1 Proton p e 1.007276 u 938.28 MeV /2 1 Neutron n 0 1.008665 u 939.57 MeV /2 372 Chapter 12 | Nuclear Structure and Radioactivity A. It is sufficient to give only the chemical symbol and A. The three isotopes of 1 2 3 1 hydrogen would be indicated as 1H0, 1H1, and 1H2, or more compactly as H, 2H, and 3H. In Appendix D you will find a list of isotopes and some of their properties. Example 12.1 Give the symbol for the following: (a) the isotope of helium (b) Again from the periodic table, we know that for tin with mass number 4; (b) the isotope of tin with 66 neu- (Sn), Z = 50. We are given N = 66, so A = Z + N = 116. 116 116 trons; (c) an isotope with mass number 235 that contains The symbol is 50Sn66 or Sn. 143 neutrons. (c) Given that A = 235 and N = 143, we know that = − = Solution Z A N 92. From the periodic table, we find that (a) From the periodic table, we find that helium has Z = 2. this element is uranium, and so the proper symbol for this 235 235 With A = 4, we have N = A − Z = 2. Thus the symbol isotope is 92U143 or U. 4 4 would be 2He2 or He. 12.2 NUCLEAR SIZES AND SHAPES Like atoms, nuclei lack a hard surface or an easily definable radius. In fact, different types of experiments can often reveal different values of the radius for the same nucleus.
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