Encyclopedia of Nuclear Physics and its Applications E-Book
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- Herausgeber: Wiley-VCH
- Kategorie: Wissenschaft und neue Technologien
- Serie: Encyclopedia of Applied Physics
- Sprache: Englisch
This book fills the need for a coherent work combining carefully reviewed articles into a comprehensive overview accessible to research groups and lecturers. Next to fundamental physics, contributions on topical medical and material science issues are included.
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Seitenzahl: 1526
Veröffentlichungsjahr: 2013
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Table of Contents
Related Titles
Title Page
Copyright
Preface
List of Contributors
Part A: Fundamental Nuclear Research
Chapter 1: Nuclear Structure
1.1 Introduction
1.2 General Nuclear Properties
1.3 Nuclear Binding Energies and the Semiempirical Mass Formula
1.4 Nuclear Charge and Mass Distributions
1.5 Electromagnetic Transitions and Static Moments
1.6 Excited States and Level Structures
1.7 Nuclear Models
Glossary
References
Further Readings
Chapter 2: Nuclear Reactions
2.1 Introduction
2.2 Basic Principles
2.3 Statistical Reactions
2.4 Direct Reactions
2.5 Heavy Ion Reactions
2.6 Electromagnetic Probes
2.7 Relativistic Nuclear Collisions
2.8 Nuclear Reactions in Stars
2.9 Reactions with Radioactive Nuclear Beams
Acknowledgments
References
Further Readings
Chapter 3: Electrostatic Accelerators
3.1 Introduction
3.2 Types of Electrostatic Accelerators
3.3 Applications of Low-Energy Electrostatic Accelerators
3.4 High-Energy Electrostatic Accelerators
3.5 Summary
Acknowledgments
Glossary
References
Further Readings
Chapter 4: Linear Accelerators
4.1 Introduction
4.2 General Principles and Brief History
4.3 Electron Linear Accelerators
4.4 Proton Linear Accelerators
4.5 Heavy-Ion Linear Accelerators
4.6 Other Types of Linear Accelerators and New Methods of Acceleration
4.7 Engineering Aspects
4.8 Concluding Remarks
Glossary
References
Chapter 5: Exotic Nuclear Beam Facilities
5.1 Introduction
5.2 Production of Exotic Nuclei
5.3 Basic Separation Principles
5.4 Facilities
5.5 Milestones and Outlook
Glossary
References
Chapter 6: Superheavy Nuclei
6.1 Introduction
6.2 Stability of Superheavy Nuclei
6.3 Experimental Techniques
6.4 Experiments on SHN
6.5 Summary and Perspectives
References
Further Readings
Chapter 7: Nuclear -Spectroscopy and the -Spheres
7.1 Introduction
7.2 An Example: the Spectroscopy of Er through the Decades
7.3 Modern High-Resolution -Ray Spectroscopy
7.4 Large High-Resolution -Ray Detector Arrays
7.5 Future Outlook
References
Chapter 8: γ Optics and Nuclear Photonics
8.1 Introduction
8.2 The Grenoble Measurement of the Index of Refraction of γ-Rays
8.3 Nonperturbative High-Field Quantum Electro Dynamics (QED)
8.4 The New γ Optics
8.5 New Nuclear Spectroscopy
8.6 Nuclear Photonics
8.7 Conclusions and Outlook
8.8 Key Topics of γ Optics
References
Chapter 9: The Proton
9.1 Introduction
9.2 The Proton as Building Block of Matter
9.3 Models and Theories
9.4 Modern Description of the Proton
9.5 Wee Partons and Other Mysteries
References
Further Readings
Chapter 10: Physics of the Neutron
10.1 Introduction
10.2 Basic Properties of the Neutron
10.3 Neutron Sources and Neutron Beams
10.4 Neutron Interactions with the Atomic Nucleus
10.5 Neutrons in Particle Physics
10.6 Some Applications and Trends
Acknowledgment
References
Further Readings
Chapter 11: Neutrino Astrophysics
11.1 Introduction
11.2 Solar Neutrinos
11.3 Atmospheric Neutrinos
11.4 Supernova Neutrinos and Nucleosynthesis
11.5 Neutrinos and Nucleosynthesis
11.6 Neutrino Cooling and Red Giants
11.7 High-Energy Astrophysical Neutrinos
Acknowledgments
References
Chapter 12: Nuclear Astrophysics
12.1 Introduction
12.2 Connecting Nuclear Physics and Astrophysics
12.3 Nuclear Physics for Astrophysics
12.4 Outlook
References
Chapter 13: Relativistic Nucleus–Nucleus Collisions
13.1 Introduction
13.2 Phases of QCD Matter
13.3 Soft-Particle Production
13.4 Spectra and Flow
13.5 Hard Probes
13.6 Outlook
References
Part B: Applied Nuclear Physics
Chapter 14: Neutron Stars
14.1 Introduction: Supernovae and Neutron Stars
14.2 The Outer Layers of a Neutron Star
14.3 The Inner Structure of a Neutron Star
14.4 Summary and Outlook
References
Chapter 15: Supernovae and Their Nucleosynthesis
15.1 Introduction
15.2 Nuclear Burning in Stars
15.3 End Stages of Stellar Evolution
15.4 Core-Collapse Supernovae
15.5 Type Ia Supernovae
15.6 Summary and Outlook
References
Chapter 16: Accelerator Mass Spectrometry and its Applications
16.1 Introduction
16.2 General Principles and Instrumentation
16.3 The Trend Toward Smaller Accelerators
16.4 Applications
16.5 Conclusions and Future Perspectives
Acknowledgment
References
Further Readings
Chapter 17: Nuclear Medicine
17.1 Introduction
17.2 Current Medical Use of Radionuclides and Radiopharmaceuticals
17.3 Modes of Radioisotope Production
17.4 Preclinical Radiolabeling and Biodistribution Validation
17.5 Pre-clinical and Clinical Dosimetry
17.6 Clinical Validation of Radiopharmaceuticals
References
Chapter 18: Cancer Therapy with Ion Beams
18.1 Introduction
18.2 Physical Advantages of Ion Beams for Therapy
18.3 Passive and Active Beam Delivery Systems
18.4 Quality Control and Patient Flow
18.5 PET Detection of Instable Isotopes is Used for Quality Control
18.6 Radiobiological Advantages of Carbon Ions
18.7 Treatment Planning
18.8 Clinical Results and the International Situation
18.9 Conclusions
References
Further Readings
Part C: Nuclear Power
Chapter 19: The Physics of Nuclear Power from Fission and Fusion
19.1 Introduction
19.2 Binding Energy [1(1)],[2(1)]
19.3 Physics of Neutron Chain Fission Reactors
19.4 Physics of Fusion Plasmas
19.5 Fusion–Fission Hybrids
References
Further Readings
Chapter 20: Fundamentals of Controlled Nuclear Fission and Essential Characteristics of Pressurized-Water Reactors
20.1 Introduction
20.2 Reactor Developments
20.3 Principle of Controlled Nuclear Fission, Chain Reaction
20.4 Basic Principle of a Nuclear Reactor
20.5 Reactor Core Design
20.6 Thermohydraulics of the Reactor Core
20.7 Design, Function, and Safety Concepts as Demonstrated by a Pressurized-Water Reactor (PWR)
20.8 Radioactive Waste Management
20.9 Decommissioning and Dismantling of Nuclear Power Plants
20.10 Outlook on New Reactor Technologies of the Third and Fourth Generations
References
Further Readings
Chapter 21: Generation IV Nuclear Reactors
21.1 Introduction
21.2 Outline of the Generation IV Reactor Concepts
21.3 Outlook
References
Chapter 22: Transmutation of High-Level Nuclear Waste by Means of Accelerator Driven System (ADS)
22.1 Introduction
22.2 Partitioning and Transmutation
22.3 ADS Technology
22.4 Conclusions
References
Further Readings
Chapter 23: Fusion Energy by Magnetic Confinement
23.1 Introduction
23.2 The Role of Fusion in the Universe
23.3 Processes for Technical Fusion
23.4 The Conditions for Controlled Nuclear Fusion
23.5 The Tasks of Fusion Research and the Basic Features of Fusion Energy
23.6 The Basics of Magnetic Confinement
23.7 Characteristics of High-Temperature Fusion Plasmas
23.8 Status of Fusion Energy Development
23.9 The Next Development Steps
23.10 Technical Issues
23.11 Safety Issues
23.12 Outlook
References
Chapter 24: Heavy Ion Inertial Fusion
24.1 Introduction
24.2 Overview: Basic Issues and Key Parameters for an IFE Plant Driven by Heavy Ion Beams
24.3 Target Physics and Target Design
24.4 The Heavy Ion Inertial Fusion Driver
24.5 The Inertial Fusion Reactor
24.6 Conclusions
References
Index
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The Editor
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Preface
With this volume, WILEY-VCH continues a series of reeditions of the “Wiley Encyclopedia of Applied Physics” that was edited by Professor G. L. Trigg in the early 1980s. It united more than 600 monographic articles that embraced physics in the widest conceivable way, from “Accelerators” via “Oceanography” to “Xerography.” The individual articles were addressed at a broad scientific/technical readership, written in a didactical style combined with a focus on new and exciting developments.
About 25 years have elapsed since this monumental undertaking, and the fields of physics have undergone substantial evolution. In this volume, we revisit Nuclear Physics, an extremely broad field of multifaceted fundamental research, and a wealth of application.
Nuclear Physics phenomena present a universal testing ground for processes and structures that are related to the three fundamental forces/interactions: strong, electromagnetic, and weak. In a certain sense, it could play the role of a paradigmatic realization of the entire Standard Model of fundamental interaction, if it was not for the fact that the strong force manifestations in nuclei and nuclear interactions represent the remotest realizations of the elementary strong force that is described by the quantum chromodynamics (QCD) theory – in the sense that nuclear forces are a strictly nonperturbative QCD effect of higher order that is essentially inaccessible to analytic or even numerical methods. This is why we do not have, even today, a “nuclear structure QCD approach.” This QCD approach exists, however, in the consideration of nucleon structure. The many-nucleon patterns of nuclear properties and reactions can be confronted with effective nucleon–nucleon forces that have been familiar for many decades but can now be, at least qualitatively, understood as a QCD analogy to, for example, van der Waals-like forces. At the limit of ultrarelativistic accelerator energies, nuclear collisions come under explicit governance of QCD, creating the primordial state of matter consisting of quarks and gluons: the so-called quark–gluon plasma. This is one of the present day fundamental research themes of the field, which, furthermore, features some of the most intricate many-body problems known to science.
Applications of Nuclear Physics could fill many such books as this. The selection of topics presented here is guided loosely by their importance in the intuitive view of the scientifically and technically interested public. We have selected topics ranging from genuinely fundamental interdisciplinary science (such as nuclear and neutrino astrophysics, neutron stars and supernovae) via nuclear medicine and matter analysis finally to the techniques of nuclear power generation and its future. We have omitted nuclear weapons, not for the sake of political correctness (we build them, so we should in principle also understand them) but for the reason that classification forbids reasonable monographic public presentation. In any case, the presentation chosen here cannot be all-encompassing. However, to accomplish a certain level of comprehensiveness, we have included a number of updated articles from the previous volume edited in the same style (Encyclopedia of Applied High Energy and Particle Physics, Wiley-VCH 2009), with a closely related dedication.
I wish to thank all the authors for their outstanding devotion. Special thanks go to Dr. A. Mueller of IN2P3 for his very essential help with the entire section on nuclear power, which resulted in a unique spectrum of presentations.
Special thanks go to my editors at WILEY, Mrs. Vera Palmer and Anja Tschörtner, for their unbending encouragement and patience.
Frankfurt
December 2012
Reinhard Stock
List of Contributors
A: Fundamental Nuclear Research
1
Nuclear Structure
Jan Jolie
1.1 Introduction
1.2 General Nuclear Properties
1.2.1 Properties of Stable Nuclei
1.2.2 Properties of Radioactive Nuclei
1.3 Nuclear Binding Energies and the Semiempirical Mass Formula
1.3.1 Nuclear Binding Energies
1.3.2 The Semiempirical Mass Formula
1.4 Nuclear Charge and Mass Distributions
1.4.1 General Comments
1.4.2 Nuclear Charge Distributions from Electron Scattering
1.4.3 Nuclear Charge Distributions from Atomic Transitions
1.4.4 Nuclear Mass Distributions
1.5 Electromagnetic Transitions and Static Moments
1.5.1 General Comments
1.5.2 Electromagnetic Transitions and Selection Rules
1.5.3 Static Moments
1.5.3.1 Magnetic Dipole Moments
1.5.3.2 Electric Quadrupole Moments
1.6 Excited States and Level Structures
1.6.1 The First Excited State in Even–Even Nuclei
1.6.2 Regions of Different Level Structures
1.6.3 Shell Structures
1.6.4 Collective Structures
1.6.4.1 Vibrational Levels
1.6.4.2 Rotational Levels
1.6.5 Odd-A Nuclei
1.6.5.1 Single-Particle Levels
1.6.5.2 Vibrational Levels
1.6.5.3 Rotational Levels
1.6.6 Odd–Odd Nuclei
1.7 Nuclear Models
1.7.1 Introduction
1.7.2 The Spherical-Shell Model
1.7.3 The Deformed Shell Model
1.7.4 Collective Models of Even–Even Nuclei
1.7.5 Boson Models
Glossary
References
Further Readings
1.1 Introduction
The study of nuclear structure today encompasses a vast territory from the study of simple, few-particle systems to systems with close to 300 particles, from stable nuclei to the short-lived exotic nuclei, from ground-state properties to excitations of such energy that the nucleus disintegrates into substructures and individual constituents, from the strong force that hold the atomic nucleus together to the effective interactions that describe the collective behavior observed in many heavy nuclei.
After the discovery of different kinds of radioactive decays, the discovery of the structure of the atomic nucleus begins with the fundamental paper by Ernest Rutherford [1], in which he explained the large-angle alpha (α)-particle scattering
All of these structure suggestions occurred before James Chadwick [9] discovered the neutron, which not only explained certain difficulties of previous models (e.g., the problems of the confinement of the electron or the spins of light nuclei), but opened the way to a very rapid expansion of our knowledge of the structure of the nucleus. Shortly after the discovery of the neutron, Heisenberg [10] proposed that the proton and neutron are two states of the nucleon classified by a new spin quantum number, the isospin. It may be difficult to believe today, 60 years after Chadwick's discovery, just how rapidly our knowledge of the nucleus increased in the mid-1930s. Hans A. Bethe's review articles [11, 12], one of the earliest and certainly the best known, discuss many of the areas that not only form the basis of our current knowledge but that are still being investigated, albeit with much more sophisticated methods.
The organization here will begin with general nuclear properties, such as size, charge, and mass for the stable nuclei, as well as half-lives and decay modes (α, β, γ, and fission) for unstable systems. Binding energies and the mass defect lead to a discussion of the stability of systems and the possibility of nuclear fusion and fission. Then follow details of the charge and current distributions, which, in turn, lead to an understanding of static electromagnetic moments (magnetic dipole and octupole, electric quadrupole, etc.) and transitions. Next follows the discussion of single-particle and collective levels for the three classes of nuclei: even–even, odd-A, and odd–odd (i.e., odd Z and odd N). With these mainly experimental details in hand, a discussion of various major nuclear models follows. These discussions attempt, in their own way, to categorize and explain the mass of experimental data.
1.2 General Nuclear Properties
1.2.1 Properties of Stable Nuclei
The discovery of the neutron allowed each nucleus to be assigned a number, A, the mass number, which is the sum of the number of protons (Z) and neutrons (N) in the particular nucleus. The atomic number of chemistry is identical to the proton number Z. The mass number A is the integer closest to the ratio between the mass of a nucleus and the fundamental mass unit. This mass unit, the unified atomic mass unit, has the value 1 u = 1.660538921(73) ×10−27 kg = 931.494061(21) MeV c−2. It has been picked so that the atomic mass of a 12C6 atom is exactly equal to 12 u. The notation here is AXN, where X is the chemical symbol for the given element, which fixes the number of electrons and hence the number of protons Z. This commonly used notation contains some redundancy because A = Z + N but avoids the need for one to look up the Z-value for each chemical element. From this last expression, one can see that there may be several combinations of Z and N to yield the same A. These nuclides are called isobars. An example might be the pair 196Pt118 and 196Au117. Furthermore, an examination of a table of nuclides shows many examples of nuclei with the same Z-value but different A- and N-values. Such nuclei are said to be isotopes of the element. For example, oxygen (O) has three stable isotopes: 16O8, 17O9, and 18O10. A group of nuclei that have the same number of neutrons, N, but different numbers of protons, Z (and, of course, A), are called isotones. An example might be 38Ar20, 39K20, and 40Ca20. Some elements have but one stable isotope (e.g., 9Be5, 19F10, and 197Au118), others, two, three, or more. Tin (Z = 50) has the most at 10. Finally, the element technetium has no stable isotope at all. A final definition of use for light nuclei is a mirror pair, which is a pair of nuclei with N and Z interchanged. An example of such a pair would be 23Na12 and 23Mg11.
The nuclear masses of stable isotopes are determined with a mass spectrometer, and we shall return to this fundamental property when we discuss the nuclear binding energy and the mass defect in Section 1.3. After mass, the next property of interest is the size of a nucleus. The simplest assumption here is that the mass and charge form a uniform sphere whose size is determined by the radius. While not all nuclei are spherical or of uniform density, the assumption of a uniform mass/charge density and spherical shape is an adequate starting assumption (more complicated charge distributions are discussed in Section 1.4 and beyond). The nuclear radius and, therefore, the nuclear volume or size is usually determined by electron-scattering experiments; the radius is given by the relation
1.1
which, with r0 = 1.25 fm, gives an adequate fit over the entire range of nuclei near stability. An expression such as Eq. (1.1) implies that nuclei have a density independent of A, that is, they are incompressible. A somewhat better fit to the nuclear sizes can be obtained from the Coulomb energy difference of mirror nuclei, which covers but a fifth of the total range of A. This yields r0 = 1.22 fm. Even if the charge and/or mass distribution is neither spherical nor uniform, one can still define an equivalent radius as a size parameter.
Two important properties of a nuclide are the spin J and the parity π, often expressed jointly as Jπ, of its ground state. These are usually listed in a table of isotopes and give important information about the structure of the nuclide of interest. An examination of such a table will show that the ground state and parity of all even–even nuclei is 0+. The spin and parity assignments of the odd-A and odd–odd nuclei tell a great deal about the nature of the principal parts of their ground-state wave functions. A final property of a given element is the relative abundance of its stable isotopes. These are determined again with a mass spectrograph and listed in various tables of the nuclides.
1.2.2 Properties of Radioactive Nuclei
A nucleus that is unstable, that is, it can decay to a different or daughter nucleus, is characterized not only by its mass, size, spin, and parity but also by its lifetime τ and decay mode or modes. (In fact, each level of a nucleus is characterized by its spin, parity, lifetime, and decay modes.) The law of radioactive decay is simply
1.2
where N(0) is the number of nuclei initially present, λ is the decay constant, and its reciprocal τ is the lifetime. Instead of the lifetime, often the half-life T1/2 is used. It is the time in which half of the nuclei decay. By setting N(T1/2) = N(0)/2 in Eq. (1.2), one obtains the relation
1.3
The decay mode of ground states can be α, β, or spontaneous fission. Excited states mostly decay by γ-emission. More exotic decays are observed in unstable nuclei far from stability where nuclei decay takes place by emission of a proton or neutron.
In α-decay, the parent nucleus emits an α-particle (a nucleus of 4He2), leaving the daughter with two fewer neutrons and protons:
1.4
The α-particle has zero spin, but it can carry off angular momentum. In β-decay the weak interaction converts neutrons into protons (β−-decay) or protons into neutrons (β+-decay). Which of the two decays takes place depends strongly on the masses of the initial and final nuclei. Because a neutron is heavier than a proton, the free neutron is unstable against β−-decay and has a lifetime of 878.5(10) s. The mass excess in β−-decay is released as kinetic energy of the final particles. In the case of the free neutron, the final particles are a proton, an electron, and an antineutrino, denoted by . All of these particles have spin 1/2 and can also carry off angular momentum. In the case of β+-decay, the final particles are a bound proton, an antielectron or positron, and a neutrino. Finally, as an alternative to β+-decay the initial nucleus can capture an inner electron. In this so-called electron capture decay, only a neutrino, ν, is emitted by the final nucleus. In general, the decays can be written as
1.5a
1.5b
1.5c
One very rare mode of decay is double β-decay, in which a nucleus is unable to β-decay to a Z + 1 daughter for energy reasons but can emit two electrons and make a transition to a Z + 2 daughter. An example is 82Se48 → 82Kr46 with a half-life of (1.7 ± 0.3) × 1020 years. Double β-decay is observed under the emission of two neutrinos. Neutrinoless double β-decay is intensively searched for in 76Ge because it is forbidden for massless neutrinos with definite helicities. Enriched Ge is hence used as it allows the use of a large single crystal as source and detector (for a review see [13]).
In spontaneous fission, a very heavy nucleus simply breaks into two heavy pieces. For a given nuclide, the decay mode is not necessarily unique. If more than one mode occurs, then the branching ratio is also a characteristic of the radioactive nucleus in question.
An interesting example of a multimode radioactive nucleus is 242Am147. Its ground state (Jπ = 1−, T1/2 = 16.01 h) can decay either by electron capture (17.3% of the time) to 242Pu148 or by β− decay (82.7% of the time) to 242Cm146. On the other hand, a low-lying excited state at 0.04863 MeV (Jπ = 5−, T1/2 = 152 years) can decay either by emitting a γ-ray (99.52% of the time) and going to the ground state or by emitting an α-particle (0.48% of the time) and going to 238Np145. There is an excited state at 2.3 MeV with a half-life of 14.0 ms that undergoes spontaneous fission [14]. The overall measured half-life of 242Am147 is then determined by that of the 0.04863 MeV state. Such long-lived excited states are known as isomeric states. From this information on branching ratios, one easily finds the several partial decay constants for 242Am147. For the ground state, λec = 2.080 × 10−6 s−1 and λβ− = 9.944 × 10−6 s−1, while for the excited state at 0.04863 MeV, λγ = 1.439 ×10−10 s−1 and λα = 6.639 × 10−13 s−1 and for the excited state at 2.3 MeV, λSF =49.5 s−1.
1.3 Nuclear Binding Energies and the Semiempirical Mass Formula
1.3.1 Nuclear Binding Energies
One of the more important properties of any compound system, whether molecular, atomic, or nuclear, is the amount of energy needed to pull it apart, or, alternatively, the energy released in assembling it from it constituent parts. In the case of nuclei, these are protons and neutrons. The binding energy of a nucleus AXN can be defined as
1.6
where MH is the mass of a hydrogen atom, Mn the mass of a neutron, and MX(Z,A) the mass of a neutral atom of isotope A. Because the binding energy of atomic electrons is very much less than nuclear binding energies, they have been neglected in Eq. (1.6). The usual units are atomic mass units, u. Another quantity that contains essentially the same information as the binding energy is the mass excess or the mass defect, Δ = M(A) − A. (Another useful quantity is the packing fraction P = [M(A) − A]/A = Δ/A.) The most interesting experimental quantity B(A,Z)/A is the binding energy per nucleon, which varies from somewhat more than 1 MeV nucleon−1 (1.112 MeV nucleon−1) for deuterium (2H1) to a peak near 56Fe30 of 8.790 MeV nucleon−1 and then falls slowly until, at 235U143, it is 7.591 MeV nucleon−1. Except for the very light nuclei, this quantity is roughly (within about 10%) 8 MeV nucleon−1. A strongly bound light nucleus is the α-particle, as for 4He2 the binding energy is 7.074 MeV nucleon−1. It is instructive to plot, for a given mass number, the packing fraction as a function of Z. These plots are quite accurately parabolas with the most β-stable nuclide at the bottom. The β− emitters will occur on one side of the parabola (the left or lower two side) and the β+ emitters on the other side. For odd-A nuclei, there is but one parabola, the β-unstable nuclei proceeding down each side of the parabola until the bottom or most stable nucleus is reached. For the even-A nuclei, there are two parabolas, with the odd–odd one lying above the even–even parabola. The fact that the odd–odd parabola is above the even–even one indicates that a pairing force exists that tends to increase the binding energy of the even–even nuclei. See Figure 1.1 for the A = 100 mass chain. Other indications of the importance of this pairing force are the before-mentioned 0+ ground states of all even–even nuclei and the fact that only four stable odd–odd nuclei exist: 2H1, 6Li3, 10B5, and 14N7. For even-A nuclei, the β-unstable nuclei zig-zag between the odd–odd parabola and the even–even parabola until arriving at the most β-stable nuclide, usually an even–even one. If the masses for each A are assembled into a three-dimensional plot (with N running along one long axis, Z along a perpendicular axis, and M(A,Z) mutually perpendicular to these two), one finds a “landscape” with a deep valley running from one end to the other. This valley is known as the valley of stability.
Figure 1.1 The packing fraction Δ/A plotted against the nuclear charge Z for nuclei with mass number A = 100. Note that the odd–odd nuclei lie above the even–even ones. The β− transitions are indicated by - ⋅ -, the β+ transitions by ---, while the double β-decay is denoted by ⋅ ⋅ ⋅.
Data from [14]. The double β-decay from [15].
The immediate consequence of the behavior of B(A,Z)/A is that a very large amount of energy per nucleon is to be gained from combining two neutrons and two protons to form a helium nucleus. This process is called fusion. The release of energy in the fission process follows from the fact that B(A,Z)/A for uranium is less than for nuclei with more or less half the number of protons. Finally, the fact that the binding energy per nucleon peaks near iron is important to the understanding of those stellar explosions known as supernovae. In Figure 1.2, the packing fraction, P = Δ/A, is plotted against A for the most stable nuclei for a given mass number. Note that P has a broad minimum near iron (A = 56) and rises slowly until lawrencium (A = 260). This shows most clearly the energy gain from the fission of very heavy elements.
Figure 1.2 The packing fraction Δ/A plotted against the mass number A for all nuclei from to .
Data from [14].
1.3.2 The Semiempirical Mass Formula
The semiempirical mass formula may be looked upon as simply the expansion of B(A,Z) in terms of the mass number. Because B(A,Z)/A is nearly constant, the most important term in this expansion must be the term in A. From Eq. (1.1) relating the nuclear radius to A1/3, we see that a term proportional to A is a volume term. However, this term overbinds the system because it assumes that each nucleon is surrounded by the same number of neighbors. Clearly, this is not true for surface nucleons, and so a surface term proportional to A2/3 must be subtracted from the volume term. (One might identify this with the surface tension found in a liquid drop.) Next, the repulsive Coulomb forces between protons must be included. As this force is between pairs of protons, this term will be of the form Z{Z − l)/2, the number of pairs of Z protons, divided by a characteristic nuclear length or A1/3. Two other terms are necessary in this simple model. One term takes into account that, in general, Z ∼ A/2, clearly true for stable light nuclei, and less so for heavier stable nuclei where more neutrons are needed to overcome the mutual repulsion of the protons. This term is generally taken to be of the form asym (N − Z)2/A. The other term takes into account the fact, noted in Section 1.3.1, that even–even nuclei are more tightly bound than odd–odd nuclei because all of the nucleons of the former are paired off. This is done by adding a term δ/2 that is positive for even–even systems and negative for odd–odd systems and zero for odd-A nuclei. Thus, the two parabolas for even A are separated by δ. From Eq. (1.6), the semiempirical Bethe–Weisäcker mass formula then becomes M(A,Z) = ZMH + NMn − B(A,Z) with
1.7
Originally, the constants were fixed by the measured binding energies and adjusted to give appropriate behavior with the mass number [16]. Myers and Swiatecki [17] (see also [18]) have included other terms to account for regions of nuclear deformation, as well as an exponential term of the form −aaAexp(−γA1/3), for which they provide no physical explanation beyond the fact that it reduces the deviation from experiment. Their model evolved into the macroscopic-microscopic global mass formula, called the finite-range droplet model (see [19]) and the DZ-model proposed by Duflo and Zuker [20], and more microscopic models, called HFB [21]. The many adjustable parameters of the available mass formulas are then fitted to masses of 1760 atomic nuclei [22]. The formulas fit binding energies quite well with errors below 1%, but still have problems to predict masses far from stability. As those are important for nuclear astrophysics, the measurement of masses of exotic nuclei is an important field today.
A number of consequences flow from even a superficial examination of Eq. (1.7). The fact that the binding energy per nucleon, B/A, is essentially constant with A implies that the nuclear density is constant and, thus, the nuclear force saturates. That is, nucleons interact only with a small number of their neighbors. This is a consequence of the very short range of the strong force. If this were not so, then each nucleon would interact with all others in the given nucleus (just as the protons interact with all other protons), and the leading term in B(A,Z) would be proportional to the number of pairs of nucleons, which is A(A − l)/2 or roughly A2. This would imply that B/A would go as A. Thus, not only does the nuclear force saturate (the Coulomb force does not) but it is also of very short range (that of the Coulomb force is infinite) as the sizes of nuclei are of the order of 3.0 fm (recall Eq. (1.1)).
1.4 Nuclear Charge and Mass Distributions
1.4.1 General Comments
In his 1911 paper, Rutherford was able to conclude that the positive charge of the atom was concentrated within a sphere of radius <10−14 m (10 fm). This result came from α-particle scattering. However, for energetic enough α-particles, the scattering result will contain a component due to nuclear interactions of the α-particle, as well as the Coulomb interaction. For probing the structure of nuclei, electrons have the advantage that their scattering is purely Coulombic; however, to determine details of the internal nuclear structure, electron energies must be well over 100 MeV for their de Broglie wavelengths to be less than nuclear dimensions. Well before the existence of such high-energy electron beams, nuclear structure effects were extracted from information provided by optical hyperfine spectra. In particular, nuclear charge distributions (electric quadrupole moments) and current distributions (magnetic dipole moments) were deduced from very accurate optical measurements (see the following section). A result involving the innermost electrons of heavy atoms is the isotope shift, which can be observed in atomic X-rays. This arises because the nuclear radii for two different isotopes of the same atom will produce slightly different binding energies of their K-shell electrons. Thus, the K X-rays of these isotopes will be very slightly different in energy. As an example, the isotopic pair 203T1122 and 205T1124 have an isotope shift of about 0.05 eV. Another early method to determine the charge radius is to take the difference between the binding energies of two mirror nuclei (cf. Section 1.2.1). This leads to an expression that only involves ac and, thus, the nuclear radius. This is useful for light nuclei for which mirror pairs occur.
With the advent of copious beams of negative muons, much more accurate optical-type hyperfine spectrum studies could be made. The process is quite simple, and the advantages obvious. By stopping negative muons in a target, an exotic atom is formed in which the muon replaces an orbital electron and transitions to the muonic K-shell follow. These transitions of the muon to the ls1/2 state emit photons of the appropriate (but high) energies. (As the muon is more than 200 times as massive as the electron, the radii of the muon orbits are reduced by that amount, so that electron-shielding problems are much reduced.) The energies of the photons are such that the 2pl/2–23/2 splitting is easily measured (in 116Sn66, it is 45.666 keV). Thus, both nuclear charge radii and isotope shifts are quite accurately determined. In some recent experiments, root mean square (RMS) charge radii have been measured with a precision of 2 × 10−18 m. As electron scattering and muonic atoms are the two methods of measuring characteristics of the nuclear charge radius most susceptible of the greatest accuracy, they will be discussed in turn.
1.4.2 Nuclear Charge Distributions from Electron Scattering
In any scattering experiment, what is measured is the differential cross section (dσ/dΩ). Rutherford developed an expression for α-particle scattering that can be used for low-energy, spinless particles incident on a spinless target. Both incident and target particles are assumed to be point particles. The differential cross section for scattering relativistic electrons off point-charged particles leads to the expression for Mott scattering, while, if the target particle has nonzero spin (there is then a magnetic contribution), one obtains the Dirac scattering formula for (dσ/dΩ). However, real nuclei are not point particles, so one needs to make use of the charge form factor with the transferred momentum. The form factor is then the Fourier transform of the charge density :
1.8
If one restricts the problem to spherically symmetric distributions, the angular integration of the Fourier integral follows at once, so that
1.9
If the target nucleus has zero spin (applicable to all even–even nuclei), then the differential cross sections for a point target and a finite-sized target are related by
1.10
With the charge form factor determined experimentally, the inverse transform yields the radial charge density
1.11
If the target nucleus is not of spin zero, then an additional term containing the so-called transverse form factor, FT(q), is needed. (The form factor defined in Eq. (1.9) is sometimes called the longitudinal form factor.) In any event, the charge distribution must be normalized to the number of protons (Z) in the target nucleus.
At this point, there are two ways to proceed. The first is a model-independent analysis of the form factor, or, second, one can assume a model with several parameters and fit these to the data. Limiting oneself to small momentum transfers, one can obtain the form factor as a power series in q2 by expanding sin(qr) in Eq. (1.9) in a power series of its argument. Keeping only the lowest term of order q2, one obtains
1.12
with
1.13
the RMS radius of the charge distribution. It should be noted that this is not the nuclear radius R, which is usually taken as the radius of the constant-density sphere. This yields
1.14
Data compilations [23] show that for most stable nuclei,
1.15
As was stated in Section 1.2.1, a better way to describe the charge distribution is to use a Fermi distribution which takes account of the constant charge density, ρ0, at the center of the nucleus and the gradual decrease near the surface. This is achieved by
1.16
with R1/2 the radius at half density and a the diffuseness parameter indicating the distance at which the density falls from 90 to 10% of the constant density ρ0. For heavy nuclei, the following parameterization holds:
1.17
There are enough experimental electron-scattering data available throughout all regions of the stable nuclei that quite accurate charge parameters exist for almost all of the systems. The compendium by de Vries et al. [23] lists these parameters fitted to the data for several distribution functions in addition to the two-parameter Fermi functions.
1.4.3 Nuclear Charge Distributions from Atomic Transitions
During the last decades, tremendous progress was obtained in the study of atomic transitions using high-precision laser spectroscopy. This allows the measurement of nuclear charge radii and also of nuclear moments for stable and even unstable isotopes. This is because the difference between a point nucleus and a finite-size nucleus causes a very small change in the Coulomb potential the atomic electrons feel. A small energy difference on the atomic levels results when we assume that the nucleus is a sphere with constant charge density. For 1s electrons one obtains
1.18
with a0 the Bohr radius. Because no point nucleus exists and the theoretical calculations are not accurate enough to calculate the small shift exactly, one generally measures isotope shifts as the frequency difference of atomic transitions measured in two isotopes of a given element. This then yield the differences in the nuclear radius. Starting from known radii of stable isotopes, it is then possible to determine the radii of unstable nuclei on which one cannot perform electron scattering. It is also possible to measure isotope shifts using optical transitions. Because these are caused by the outermost electrons, the shifts are very small in the order of parts per million. As indicated above, they are still within reach of modern laser techniques.
The small shift for 1s binding energies is related to the large difference between the Bohr radius and the nuclear radius. On replacing one electron by a muon, the muonic orbits shrink by a factor of 207, the mass difference between the heavy muon and the light electron. At the same time, the muon binding energy is increased by a similar factor, making the transition energies in the mega electron volt region. The energies are so high (in 238U146, the measured 2p–1s transition is about 6.1 MeV) that one must generate Dirac solutions for the muon moving in a Coulomb potential generated by a non-point-charge distribution. To these initial Dirac solutions, one must add corrections, which, in order of size, are vacuum polarization, nuclear polarization, the Lamb shift, and relativistic recoil. Electronscreening corrections are often included, but they are very tiny (for the ls muonic state in 238U146, this correction has the value of 11 eV).
As many nuclei are not spherical, several studies have used as the appropriate charge distribution a slightly modified form of Eq. (1.16), which includes the deformations
1.19
Here the βn are deformation parameters that determine the nuclear shape. As an example, the nuclear mean square radius may be expressed as
1.20
Experiments to fit a, c, and, in deformed regions, βn have been made throughout the periodic table with results consistent with the electron data. However, to combine the results of electron-scattering experiments with those from muonic atoms, it is necessary to use the so-called Barrett moment
1.21
where k and α are fitted to the experimental data. The muonic data are equivalent to data from electron-scattering experiments at low momentum transfer. The inclusion of the muonic Barrett moment improves the overall fit by reducing normalization errors. This then reduces the uncertainties over what would be obtained by fitting either the electron-scattering or the muonic atom data alone. Extensive tables of data fitted by various charge distribution models as well as model independent analyzes can be found in de Vries et al. [23].
1.4.4 Nuclear Mass Distributions
While the measurement of the charge distribution can be made using electromagnetic probes, this is not possible for the mass distribution because of the uncharged neutron. Instead, the nuclear strong force has to be used. This is more complicated as mostly both Coulomb force and strong force are present. Nevertheless, from α-scattering experiments, information of the mass distribution is obtained. There are also indirect ways in which one can get information on nuclear mass radii. One example is the dependence of α-decay rate on the nuclear radius that defines the Coulomb barrier. In deformed nuclei, this causes an anisotropy because the Coulomb barrier is lower in the direction of the longest axis, making the tunnel probability enhanced. A second way is to use pions instead of muons. These interact with the nucleus through both the Coulomb force and strong force, which, in comparison to muonic atoms, causes an extra shift that allows the determination of the mass radius. The result of these experiments on stable nuclei finds that the charge and mass radii are equal to within about 0.1 fm. This somewhat surprising result can be understood as a balance between the proton Coulomb repulsion that tends to push the protons to the outside and a strongly attractive neutron–proton strong force that tends to pull the extra neutrons to the inward.
Recently, the common opinion that the radii scale with A1/3 was found to be heavily violated in more exotic nuclei.
Especially in light nuclei with a large neutron number, so-called halo-nuclei, strong deviations were observed (an early review is given in [24]). Using the radioactive beam techniques, very neutron-rich He, Li, and Be isotopes can be created and studied in the laboratory. It turned out that these loosely bound nuclei show very extended neutron radii whereby two neutrons are moving at radii similar to the radii of Pb isotopes. Moreover, as is the case of 11Li8, the bound system consists of three entities: two neutrons and a 9Li6 that cannot exist two by two, as the dineutron and 10Li7 are unbound. The research on exotic nuclei is still in its infancy and more exotic features such as proton halos or neutron skins are expected. They are of importance as they may influence the creation of the elements under astrophysical conditions.
1.5 Electromagnetic Transitions and Static Moments
1.5.1 General Comments
Static electromagnetic nuclear moments played an important role in the unscrambling of the detailed measurements of atomic optical hyperfine structure well before the gross components of atomic nuclei were in hand. Almost a decade before the discovery of the neutron, Pauli [7] suggested that the optical hyperfine splitting might be due, in part, to the interaction with a nuclear magnetic moment (μ). This suggestion lay fallow until 1930, when Goudsmit and Young, using the spectroscopic data of Schiller and of Granath, deduced the nuclear magnetic moment of 7Li to be μ = 3.29μN, where the nuclear magneton equals
1.22
This value is quite close to the currently accepted value (μ = 2.327μN). Because of the existence, by then, of extensive hyperfine optical spectroscopic data, Goudsmit, in 1933, was able to publish a table of some 20 nuclear magnetic moments ranging from 7Li to 209Bi. In 1937, Schmidt published a simple, single-particle model of nuclear magnetic moments and supported it with the experimental moments of 32 odd-proton nuclei and 15 odd-neutron nuclei. This simple model yields what is now known as the Schmidt limits, within which almost all nuclear magnetic moments lie (see the following).
The suggestion that the nuclear electric quadrupole moments (Q) might also play an important role in optical hyperfine structure was again made before the discovery of the neutron. Racah [8] was the first to work out the theory associated with “nuclear charge asymmetry” and the interaction with the atomic electrons. Casimir [25], sometime later, developed the theory of nuclear electric quadrupole hyperfine interaction and applied it to 151Eu and 153Eu. In this paper, Casimir mentions work by Schiller and Schmidt, who determined Q for 175Lu. A short time later, Gollnow [26] obtained Q = 5.9 b for this nucleus, quite close to the currently accepted value of 5.68 b. This very large quadrupole moment (very much larger than can be accounted for by the single-particle shell model) was to provide, 20 years later, strong impetus for the development of the collective model of the nucleus. In 1954, Schwartz [27] extended the theory of nuclear hyperfine structure to examine the magnetic octupole hyperfine interaction and calculated the first four nuclear magnetic octupole moments (O) from data of the hyperfine structure of the nuclear ground states. The next nuclear moment is the hexadecapole (H); however, no direct measurements of such static moments exist. What is known about these moments comes mainly from electromagnetic transitions of electrons and negative muons. For an in-depth theoretical study of all of these moments and how they can be used to test various nuclear models see, in particular, the text by Castel and Towner [28].
Nowadays, the measurement of moments is still very important to assess the single-particle structure of exotic nuclei and several powerful techniques have been developed in this domain [29]. Most information is, however, gathered via the determination of electromagnetic transitions by γ-ray spectroscopy. This is to a large extent due to the availability of large-volume semiconductor detectors for γ-ray detection and the high computing power that allows one to analyze more and more complex measured spectra using coincidence conditions. The recent development of γ-ray tracking detectors out of segmented Ge-detectors offers very high perspectives in the field of exotic nuclei [30].
1.5.2 Electromagnetic Transitions and Selection Rules
Without going into a detailed discussion on how matrix elements are calculated, we review here the calculation of electromagnetic transitions. The interested reader can find more details in Heyde [31]. The calculation of transitions and also moments involves the wavefunctions of nuclear states and forms a very sensitive probe for nuclear structure research. On the other hand these transitions and moments are electromagnetic in nature making the interaction very well understood. Using the long-wavelength approximation λR and a multipole expansion of the electromagnetic operators, the transition rates per unit of time can be expressed as
1.23
with L the multipolarity, ω the angular frequency of the radiation such that up to a small nuclear recoil correction, and B(L) the reduced transition probability
1.24
in units of e2bL and μNbL−2 for electric and magnetic B(LM) values. The labels α identify the initial and final states and O(LM) is the electric or magnetic multipole operator of rank LM. As all states have good angular momentum, one can now use for the transition rates the Wigner–Eckart theorem to remove all reference to the M projections. This yields
1.25
The electric multipole operator for a number of point charges becomes
1.26
with eeff(i) the effective charge of the ith nucleon. Here it is anticipated that owing to core polarization effects and truncations of the model space, other values than the free charges +e(0) for proton (neutron) need to be used. Instead of the operator (Eq. (1.25)), one can also use a similar operator but using the nuclear electric charge density ρch and an integration over the nuclear volume. These are used as several models describe the nucleus as a droplet (Section 1.7). For the magnetic multipole operator, we have
1.27
with the effective gyromagnetic ratio gs which also may differ from the free ones.
The multipole expansion and the fact that states in atomic nuclei have good angular momentum and parity leads to several selection rules. The first one is related to the vector coupling of the angular momentum and states |Ji − Jf| ≤ L ≤ Ji + Jf. The second is due to the parity of the operators, which clearly is (−1)L for the electric and (−1)L−1 for the magnetic operator. Owing to this, electric and magnetic transitions of order L cannot take place at the same time between states, and moments such as the electric dipole moment are forbidden. While the selection rules allow the determination of the spin and parities of nuclear excited states, they are also (at a higher level) invaluable to test nuclear models (Section 1.7).
Weisskopf has estimated the so-called single-particle values or Weisskopf units (W.u.) by assuming that a single-particle makes a transition with multipole L from a state with spin L + 1/2 toward a state with spin 1/2, that the radial part of the wavefunction can be approximated by a constant value up to the radius R, and that certain values for the effective charges hold. This leads to the following estimates for the half-lives corresponding to the single-particle values:
1.28
One notices that transition rates of the lower multipolarities are faster than the higher by orders of magnitude and that for a given multipolarity, the electric ones are about 100–1000 faster as the magnetic ones. Owing to the selection rules and enhanced quadrupole collectivity, only E2 and M1 transitions happen on similar timescales. In this case, one has a transition of mixed multipolarity.
The Weisskopf estimates are very crucial to determine whether a transition is caused by a single nucleon changing orbits or by several nucleons acting in a collective way. The measurement of transition rates of excited states delivers very important information on nuclear structure, but is also quite involved. One needs to measure the lifetime of a state, the (mixed) multipolarity, and the energy and intensities of the transitions deexiting a given state. To this end, γ-ray arrays consisting of several Ge-detectors are appropriate. Besides this, electromagnetic decay can also take place with the emission of conversion electrons. These electrons allow one to determine the multipolarity as well as to observe the by γ-emission forbidden E0 transitions (due to the fact that the photon has spin 1 with projection +1 and −1).
1.5.3 Static Moments
In contrast to the transition rates, the multipole moments are generally defined as the matrix element of the M = 0 component of the moment operator for a single state with magnetic projection M = +J. Of the moments, the two lowest are the most important.
1.5.3.1 Magnetic Dipole Moments
If one assumes that the magnetic properties are associated with the individual nucleons, then the magnetic moment is defined as
1.29
where the sum extends over all of the A nucleons. Generally, this will not be needed; for instance, the magnetic moment of an odd-A nucleus will be generated by the last neutron and proton as the adjacent even–even ground state has no magnetic moment. In Eq. (1.29), gl and gs are the orbital and spin gyromagnetic ratios. They are often chosen as 0.7gfree where the free-particle values are gl = l, gs = 5.587 for protons and gl = 0, gs = −3.826 for neutrons (all in μN). It is instructive to calculate the magnetic moment for a single nucleon (which, as explained earlier, is a good approximation for the ground state of odd-A nuclei). Using the total angular momentum and the Wigner–Eckart theorem, one deduces the single-particle magnetic moments for aligned and antialigned orbital momentum and spin:
1.30a
1.30b
The magnetic moments plotted as a function of spin form the so-called Schmidt lines. It is interesting that, when plotted, almost all of these moments lie between the Schmidt lines. The moments that lie outside these limits occur mainly for some very light nuclei. One may conclude that the single-particle model does possess some validity. Another set of limits, the Margenau–Wigner (M-W) limits, is obtained by replacing the free-particle values for the orbital gyromagnetic ratios, gl, by the uniform value Z/A. The justification for this is that one is in effect averaging over all states that lead to the correct nuclear spin. This calculation represents an early attempt to account for core contributions to the dipole-moment operator. Figure 1.3 and Figure 1.4 show plots of a number of the ground-state magnetic moments for odd-Z (Figure 1.3) and odd-N (Figure 1.4) nuclei.
Figure 1.3 Nuclear magnetic moments in units of the nuclear magneton (μ/μN) plotted against the nuclear spin (I) for a number of odd-Z nuclei. The Schmidt limits, as well as the Margenau–Wigner (M-W) limits, are shown as solid lines.
Data from [14].
Figure 1.4 Nuclear magnetic moments in units of the nuclear magneton (μ/μN) plotted against the nuclear spin (I) for a number of odd-N nuclei. The Schmidt limits, as well as the Margenau–Wigner (M-W) limits, are shown as solid lines.
Data from [14].
Besides the ground state, excited states can have magnetic moment and their measurement is often used to extract information on the underlying single-particle structure. Common in nuclear structure physics is the use of g-factors that are analogous to the single-particle gyromagnetic ratios, except that they are dimensionless. They are defined as
1.31
One way to determine the g-factor is to measure the Lamor frequency when excited nuclei are placed in an external magnetic field B. Then,
1.32
The main problem hereby is to align the spins of an ensemble of atomic nuclei. This has to be done by the nuclear reaction used, cooling in an external magnetic field, or via the observation of changes in angular correlations.
1.5.3.2 Electric Quadrupole Moments
As a general definition of the quadrupole moment, we have the expectation value of (3z2 − r2). Using the proportionality of the quantity with Y20(θ,φ) one gets
1.33
or, using the Wigner–Eckart theorem,
1.34
The quadrupole moment can also be calculated for a single nucleon in an orbit j = J. This yields
1.35
One thus obtains a negative quadrupole moment for a single nucleon. If the orbit is filled up to a single hole, a quadrupole moment as in Eq. (1.35) but with a positive sign is expected. We would like to illustrate the application of Eq. (1.35) with examples near the doubly magic nucleus 16O8 (see also Section 1.7). The orbit that the ninth nucleon can occupy has j = 5/2 and we can use Eqs. (1.13) and (1.14) to estimate <r2>. This yields for 17F8 using the free-proton charge Q = −5.9 fm2, which is in excellent agreement with the experimental absolute value given in Firestone et al. (1996) of |Q| = 5.8(4) fm2. For the odd neutron nucleus 17O9, one finds experimentally Q = −2.578 fm2, which shows the need for effective charges and can be reproduced using 0.44e as neutron effective charge. Finally, if we place five neutrons in the j = 5/2 orbit, we have the N = 13 isotones and expect moments of Q = +2.6 fm2. Experimentally, one finds Q = 20.1(3) fm2 in 25Mg13 and Q = 10.1(2) fm2 in 23Ne13. This observation of much larger quadrupole moments occurs for most atomic nuclei having several nucleons in an orbit or in several orbits, indicating that the model is too simple and that all of the electric charges must be considered. This holds especially if the core is not spherically symmetric and the motion becomes collective.
1.6 Excited States and Level Structures
1.6.1 The First Excited State in Even–Even Nuclei
The most obvious characteristic of the various nuclei is that all of the even–even nuclei have ground-state spins and parities of 0+. This not only categorizes one large group of nuclei but also indicates that the nuclear force is such that it couples, preferentially, pairs of like nucleons to angular momentum zero. The second observation is that the first excited state in even–even nuclei is almost always a 2+ excitation. This can be understood by the combination of good total angular momentum and the Pauli principle. If one couples two nucleons in orbit j and implies the antisymmetrization, one obtains
1.36
with the Clebsch–Gordan coupling coefficient for the angular momentum coupling. This can, however, be rewritten as the m-values are in the summation. Using the symmetry properties, one arrives at
1.37
The phase factor, and the fact that 2j is an odd number, imply that states with odd values of J do not exist. One might wonder whether this is an essential property of fermions, but surprisingly it is not. Consider bosons with integer angular momentum . Then, owing to symmetrization, Eq. (1.36) becomes
1.38
