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Part a Fundamental Nuclear Research 1 Part A Fundamental Nuclear Research Encyclopedia of Nuclear Physics and its Applications, First Edition. Edited by Reinhard Stock. © 2013 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2013 by Wiley-VCH Verlag GmbH & Co. KGaA. 3 1 Nuclear Structure Jan Jolie 1.1 Introduction 5 1.2 General Nuclear Properties 6 1.2.1 Properties of Stable Nuclei 6 1.2.2 Properties of Radioactive Nuclei 7 1.3 Nuclear Binding Energies and the Semiempirical Mass Formula 8 1.3.1 Nuclear Binding Energies 8 1.3.2 The Semiempirical Mass Formula 10 1.4 Nuclear Charge and Mass Distributions 12 1.4.1 General Comments 12 1.4.2 Nuclear Charge Distributions from Electron Scattering 13 1.4.3 Nuclear Charge Distributions from Atomic Transitions 14 1.4.4 Nuclear Mass Distributions 15 1.5 Electromagnetic Transitions and Static Moments 16 1.5.1 General Comments 16 1.5.2 Electromagnetic Transitions and Selection Rules 17 1.5.3 Static Moments 19 1.5.3.1 Magnetic Dipole Moments 19 1.5.3.2 Electric Quadrupole Moments 21 1.6 Excited States and Level Structures 22 1.6.1 The First Excited State in Even–Even Nuclei 22 1.6.2 Regions of Different Level Structures 23 1.6.3 Shell Structures 23 1.6.4 Collective Structures 25 1.6.4.1 Vibrational Levels 25 1.6.4.2 Rotational Levels 26 1.6.5 Odd-A Nuclei 28 1.6.5.1 Single-Particle Levels 28 1.6.5.2 Vibrational Levels 28 Encyclopedia of Nuclear Physics and its Applications, First Edition. Edited by Reinhard Stock. © 2013 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2013 by Wiley-VCH Verlag GmbH & Co. KGaA. 4 1 Nuclear Structure 1.6.5.3 Rotational Levels 28 1.6.6 Odd–Odd Nuclei 28 1.7 Nuclear Models 29 1.7.1 Introduction 29 1.7.2 The Spherical-Shell Model 30 1.7.3 The Deformed Shell Model 32 1.7.4 Collective Models of Even–Even Nuclei 33 1.7.5 Boson Models 35 Glossary 40 References 41 Further Readings 42 5 1.1 different isotopes. With the availability of Introduction α-sources, due to the works of the Curies in Paris, Rutherford [5] was able to perform The study of nuclear structure today the first nuclear reactions on nitrogen. encompasses a vast territory from the study The first attempt at understanding the of simple, few-particle systems to systems relative stability of nuclear systems was with close to 300 particles, from stable made by Harkins and Majorsky [6]. This nuclei to the short-lived exotic nuclei, from model, like many others of the time, ground-state properties to excitations of consisted of protons and electrons. In such energy that the nucleus disintegrates 1924, Wolfgang Pauli [7] suggested that into substructures and individual con- the optical hyperfine structure might be stituents, from the strong force that hold explained if the nucleus had a magnetic the atomic nucleus together to the effec- dipole moment, while later Giulio Racah tive interactions that describe the collective [8] investigated the effect on the hyperfine behavior observed in many heavy nuclei. structure if the nuclear charge were not After the discovery of different kinds of spherically symmetric – that is, if it had an radioactive decays, the discovery of the electric quadrupole moment. structure of the atomic nucleus begins All of these structure suggestions with the fundamental paper by Ernest occurred before James Chadwick [9] discov- Rutherford [1], in which he explained ered the neutron, which not only explained the large-angle alpha (α)-particle scattering certain difficulties of previous models (e.g., from gold that had been discovered earlier the problems of the confinement of the by Hans W. Geiger and Ernest Marsden. electron or the spins of light nuclei), but Indeed, Rutherford shows that the atom opened the way to a very rapid expan- holds in its center a very tiny, positively sion of our knowledge of the structure of charged nucleus that contains 99.98% the nucleus. Shortly after the discovery of of the atomic mass. In 1914, Henry the neutron, Heisenberg [10] proposed that Moseley [2, 3] showed that the nuclear the proton and neutron are two states of the charge number Z equaled the atomic nucleon classified by a new spin quantum number. Using the first mass separators, number, the isospin. It may be difficult Soddy[4]wasabletoshowthatone to believe today, 60 years after Chadwick’s chemical element could contain atomic discovery, just how rapidly our knowledge nuclei with different masses, forming of the nucleus increased in the mid-1930s. Encyclopedia of Nuclear Physics and its Applications, First Edition. Edited by Reinhard Stock. © 2013 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2013 by Wiley-VCH Verlag GmbH & Co. KGaA. 6 1 Nuclear Structure Hans A. Bethe’s review articles [11, 12], one mass of a nucleus and the fundamental of the earliest and certainly the best known, mass unit. This mass unit, the unified discuss many of the areas that not only atomic mass unit, has the value 1 u = − form the basis of our current knowledge 1.660538921(73) ×10 27 kg = 931.494061 − but that are still being investigated, albeit (21) MeV c 2. It has been picked so that 12 with much more sophisticated methods. the atomic mass of a C6 atom is exactly A The organization here will begin with equal to 12 u. The notation here is XN, general nuclear properties, such as size, where X is the chemical symbol for the charge, and mass for the stable nuclei, given element, which fixes the number of as well as half-lives and decay modes electrons and hence the number of protons (α, β, γ, and fission) for unstable systems. Z. This commonly used notation contains Binding energies and the mass defect some redundancy because A = Z + N lead to a discussion of the stability of but avoids the need for one to look up systems and the possibility of nuclear the Z-value for each chemical element. fusion and fission. Then follow details From this last expression, one can see of the charge and current distributions, that there may be several combinations which, in turn, lead to an understanding of of Z and N to yield the same A.These static electromagnetic moments (magnetic nuclides are called isobars.Anexample 196 196 dipole and octupole, electric quadrupole, might be the pair Pt118 and Au117. etc.) and transitions. Next follows the Furthermore, an examination of a table of discussion of single-particle and collective nuclides shows many examples of nuclei levels for the three classes of nuclei: with the same Z-value but different A- even–even, odd-A, and odd–odd (i.e., and N-values. Such nuclei are said to odd Z and odd N). With these mainly be isotopes of the element. For example, 16 experimental details in hand, a discussion oxygen (O) has three stable isotopes: O8, 17 18 of various major nuclear models follows. O9,and O10. A group of nuclei that These discussions attempt, in their own have the same number of neutrons, N,but way, to categorize and explain the mass of different numbers of protons, Z (and, of experimental data. course, A), are called isotones.Anexample 38 39 40 might be Ar20, K20,and Ca20.Some elements have but one stable isotope (e.g., 9 19 197 1.2 Be5, F10,and Au118), others, two, General Nuclear Properties three, or more. Tin (Z = 50) has the most at 10. Finally, the element technetium has 1.2.1 no stable isotope at all. A final definition Properties of Stable Nuclei of use for light nuclei is a mirror pair, which is a pair of nuclei with N and Z The discovery of the neutron allowed each interchanged. An example of such a pair 23 23 nucleus to be assigned a number, A,the would be Na12 and Mg11. mass number, which is the sum of the The nuclear masses of stable isotopes number of protons (Z) and neutrons (N) are determined with a mass spectrometer, in the particular nucleus. The atomic and we shall return to this fundamental number of chemistry is identical to the property when we discuss the nuclear proton number Z. The mass number A is binding energy and the mass defect in the integer closest to the ratio between the Section 1.3. After mass, the next property 1.2 General Nuclear Properties 7 of interest is the size of a nucleus. The spectrograph and listed in various tables of simplest assumption here is that the mass the nuclides. and charge form a uniform sphere whose size is determined by the radius. While 1.2.2 not all nuclei are spherical or of uniform Properties of Radioactive Nuclei density, the assumption of a uniform mass/charge density and spherical shape A nucleus that is unstable, that is, it can is an adequate starting assumption (more decay to a different or daughter nucleus, complicated charge distributions are is characterized not only by its mass, size, discussed in Section 1.4 and beyond). The spin, and parity but also by its lifetime τ nuclear radius and, therefore, the nuclear and decay mode or modes. (In fact, each volume or size is usually determined by level of a nucleus is characterized by its electron-scattering experiments; the radius spin, parity, lifetime, and decay modes.) is given by the relation The law of radioactive decay is simply − − 1/3 = λt = t/τ R = r0A (1.1) N(t) N(0)e N(0)e (1.2) = which, with r0 1.25 fm, gives an adequate where N(0) is the number of nuclei initially fit over the entire range of nuclei near present, λ is the decay constant, and its stability.
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