Nuclear Physics in a Nutshell: 4 - Hardcover

Bertulani, Carlos A.

Hardcover

ISBN 10: 0691125058 ISBN 13: 9780691125053

Verlag: Princeton University Press, 2007

Inhaltsangabe

Nuclear Physics in a Nutshell provides a clear, concise, and up-to-date overview of the atomic nucleus and the theories that seek to explain it. Bringing together a systematic explanation of hadrons, nuclei, and stars for the first time in one volume, Carlos A. Bertulani provides the core material needed by graduate and advanced undergraduate students of physics to acquire a solid understanding of nuclear and particle science. Nuclear Physics in a Nutshell is the definitive new resource for anyone considering a career in this dynamic field. The book opens by setting nuclear physics in the context of elementary particle physics and then shows how simple models can provide an understanding of the properties of nuclei, both in their ground states and excited states, and also of the nature of nuclear reactions. It then describes: nuclear constituents and their characteristics; nuclear interactions; nuclear structure, including the liquid-drop model approach, and the nuclear shell model; and recent developments such as the nuclear mean-field and the nuclear physics of very light nuclei, nuclear reactions with unstable nuclear beams, and the role of nuclear physics in energy production and nucleosynthesis in stars. Throughout, discussions of theory are reinforced with examples that provide applications, thus aiding students in their reading and analysis of current literature. Each chapter closes with problems, and appendixes address supporting technical topics.

Über diesen Titel

Über die Autorin bzw. den Autor

Carlos A. Bertulani is Research Professor of Physics at the University of Tennessee and the Oak Ridge National Laboratory. He is the author of "Physics of Radioactive Nuclear Beams, Introduction to Nuclear Physics", and "Introduction to Nuclear Reactions".

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"The particular attraction of this book is the detail with which it provides, in one place, all of the essential physics required for an understanding of the field. An excellent piece of scholarship, it will come to be regarded as an essential text for beginning graduate physics study. Indeed, I know of no other modern treatment other than this that goes to such lengths, and within this context it is indeed a tour de force."--David A. Bradley, University of Surrey

"This book does a fine job of developing three topics--hadrons, nuclei, and stars--that are often covered separately, and bringing them together in an appealing way in the context of real physical systems. Remarkably self-contained, with helpful, unobtrusive appendices, it develops most physical concepts from start to finish. It can be used for a course on stellar physics, nuclear physics, or advanced quantum mechanics."--Savas Dimopoulos, Stanford University

Aus dem Klappentext

"The particular attraction of this book is the detail with which it provides, in one place, all of the essential physics required for an understanding of the field. An excellent piece of scholarship, it will come to be regarded as an essential text for beginning graduate physics study. Indeed, I know of no other modern treatment other than this that goes to such lengths, and within this context it is indeed a tour de force."--David A. Bradley, University of Surrey

"This book does a fine job of developing three topics--hadrons, nuclei, and stars--that are often covered separately, and bringing them together in an appealing way in the context of real physical systems. Remarkably self-contained, with helpful, unobtrusive appendices, it develops most physical concepts from start to finish. It can be used for a course on stellar physics, nuclear physics, or advanced quantum mechanics."--Savas Dimopoulos, Stanford University

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Nuclear Physics in a Nutshell

By Carlos A. Bertulani

PRINCETON UNIVERSITY PRESS

Copyright © 2007 Princeton University Press
All right reserved.
ISBN: 978-0-691-12505-3

Contents

Introduction...........................................................11 Hadrons..............................................................42 The Two-Nucleon System...............................................313 The Nucleon-Nucleon Interaction......................................714 General Properties of Nuclei.........................................985 Nuclear Models.......................................................1196 Radioactivity........................................................1707 Alpha-Decay..........................................................1858 Beta-Decay...........................................................1959 Gamma-Decay..........................................................21810 Nuclear Reactions—I...........................................25811 Nuclear Reactions—II..........................................29812 Nuclear Astrophysics................................................33413 Rare Nuclear Isotopes...............................................385Appendix A Angular Momentum............................................401Appendix B Angular Momentum Coupling...................................419Appendix C Symmetries..................................................432Appendix D Relativistic Quantum Mechanics..............................440Appendix E Useful Constants and Conversion Factors.....................459References.............................................................461Index..................................................................469

Chapter One

Hadrons

1.1 Nucleons

The scattering experiments made by Rutherford in 1911 [Ru11] led him to propose an atomic model in which almost all the mass of the atom was contained in a small region around its center called the nucleus. The nucleus should contain all the positive charge of the atom, the rest of the atomic space being filled by the negative electron charges.

Rutherford could, in 1919 [Ru19], by means of the nuclear reaction

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)

detect the positive charge particles that compose the nucleus called protons. The proton, with symbol p, is the nucleus of the hydrogen atom; it has charge +e of the same absolute value as that of the electron, and mass

m_p = 938.271998(38) MeV/c², (1.2)

where the values in parentheses are the errors in the last two digits.

From study of the hydrogen molecule one can infer that the protons in the molecule can be aligned in two different ways. The spins of the two protons can be parallel, as in orthohydrogen, or antiparallel, as in parahydrogen. Each proton has two possible orientations relative to the spin of the other proton, and like the electron the proton has spin 1&frac;2.

In orthohydrogen the wavefunction is symmetric with respect to the interchange of the spins of the two protons, since they have the same direction, and experiments show that the wavefunction is antisymmetric with respect to the interchange of the spatial coordinates of the protons. This justifies the wavefunction being antisymmetric with respect to the complete interchange of the protons. In parahydrogen the wavefunction is also antisymmetric with respect to the complete interchange of the two protons, being antisymmetric with respect to the interchange of the spins of the protons and symmetric with respect to the interchange of their spatial coordinates. This shows that the protons obey Fermi-Dirac statistics; they are fermions and the Pauli exclusion principle is applicable to them. At most one proton can exist in a given quantum state.

The neutron, with symbol n, has charge zero, spin 1&frac;2, and mass

m_n = 939.565330(38) MeV/c². (1.3)

In 1930, Bothe and Becker [BB30] discovered that a very penetrating radiation was released when boron, beryllium, or lithium was bombarded with α-particles. At that time it was thought that this penetrating radiation was γ-rays (high-energy photons). In 1932, Curie and Joliot [CJ32] figured out that the radiation was able to pull out protons from a hydrogen-rich material. They suggested that this was due to Compton scattering, that is, the protons recoiled after scattering the γ-rays. This hypothesis, however, meant that the radiation consisted of extremely energetic ? -rays, and no explanation could be given for the origin of such high energies. Also in 1932, Chadwick [Ch32] showed, by means of an experiment conducted at the Cavendish laboratory in Cambridge, that the protons ejected from the hydrogen-rich material had collided with neutral particles with mass close to the mass of the proton. These were neutrons, the neutral particles that composed the penetrating radiation discovered by Bothe and Becker. The reaction that occurred when beryllium was bombarded with a-particles was

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.4)

The existence of the neutron was also necessary to explain some features of the molecular spectrum showing that the wavefunctions of nitrogen molecules were symmetric with respect to interchange of the two ¹⁴N nuclei. As a consequence, the ¹⁴N nuclei were bosons. This could not be explained if the 14N nucleus were composed only of protons and electrons, since 14 protons and 7 electrons are needed for that, which means an odd number of fermions. A system made up of an odd number of fermions is a fermion, since the interchange of two systems of this type can be made by the interchange of each of their fermions, and each change of two fermions changes the sign of the total wavefunction. In the same way, we can say that a system composed of an even number of fermions is a boson. This shows that if the ¹⁴N nucleus is formed by 7 protons and 7 neutrons it is a boson, assuming that the neutron is a fermion. In this way, the study of the N2 molecule led Heitler and Hertzberg [HH29] to conclude that atomic nuclei are composed of protons and neutrons and not of protons and electrons.

Several other studies established that neutrons obey the Pauli principle and thus are fermions, having spin 1&frac;2. We recall that particles with fractional spin (2n + 1)/2 are fermions, and that particles with integer spin are bosons. Protons and neutrons have similar properties in several aspects, and it is convenient to utilize the generic name nucleon for both.

1.2 Nuclear Forces

The origin of the Coulomb force between charged particles is the exchange of photons between them. This is represented by the Feynman diagram (a) of figure 1.1. In this diagram lines oriented up represent the direction in which time increases. At some instant of time the particles exchange a photon, which gives rise to attraction or repulsion between them. The photon has zero mass and the Coulomb force is a long-range force.

The force that keeps the nucleus bound is the nuclear force. It acts between two nucleons of any type and, in contrast to the Coulomb force, it is of short range. In 1935 Yukawa [Yu35] suggested that the nuclear force has its origin in the exchange of particles with finite rest mass between the nucleons. These particles are called mesons, and this situation is described by the Feynman diagram of figure 1.1(b). In the emission of a meson with rest mass M, the total energy of the nucleon-nucleon system is not conserved by the amount ΔE = Mc². From Heisenberg's uncertainty principle, ΔE Δt [??] [??], the exchanged meson can exist during a time _t (in which violation of energy conservation is allowed), such that

Δt [??] [??]/ΔE = [??]/Mc². (1.5)

During this time the exchanged meson can travel at most a distance

R = cΔt [??] [??]/Mc, (1.6)

since the velocity of light c, is the maximum velocity. Then, if the nuclear force can be described by meson exchange, the mesons would exist "virtually" during a time permitted by the uncertainty principle. The nuclear force range would be approximately [??]/Mc. Experimentally one finds that the nuclear force range is R [??] 10^-13 cm. Thus, an estimate for the meson mass is

M [??] [??]/Rc [??] 0.35 x 10^-24 g [??] 200 MeV, (1.7)

where 1 MeV/c² = 1.782 x 10^-27 g (for brevity, one normally omits c²).

In 1936, Anderson and Neddermeyer [AN36] observed cosmic rays in a bubble chamber and found a particle with mass approximately equal to that predicted by Yukawa. These particles were investigated during the next ten years but, because their interaction with nucleons was extremely weak, they could not be the Yukawa meson. This puzzle was solved by Lattes, Muirhead, Powell, and Ochialini [La47]. They discovered that there are two types of mesons: the μ-mesons and the π-mesons. The π-meson interacts strongly with nucleons, but has a very short lifetime and decays into a μ-meson, the particle identified previously by Anderson. The muon, as it is known today, has a longer lifetime and does not interact strongly with other particles. The muon does not enter into the description of the nuclear force and is classified among the leptons, the family of light particles to which the electron belongs.

The π-meson, known as the pion, is the particle predicted by Yukawa. Pions were produced in the laboratory for the first time by Gardner and Lattes in 1948 [GL48], using 340 MeV a-particles from the University of California synchrocyclotron.

1.3 Pions

The pion exists in three charge states, π⁺, π⁰, and π^-. The π⁺ and π^- have the same mass, 139.56995(35) MeV, and the same mean lifetime, τ = 2.6 x× 10^-8 s, and decay almost exclusively by the process

π⁺ [right arrow] μ⁺ + v_μ, π^- [right arrow] μ^- + [??]_μ, (1.8)

where μ⁺, μ^- are the positive and negative muons, v_μ is the muonic neutrino, and [bar.v]_μ is the corresponding antineutrino. Only a small fraction, 1.2 x 10^-4, of pions decay by

π⁺ [right arrow] e⁺ + v_e, π^- [right arrow] e^- + [bar.v]_e, (1.9)

yielding, respectively, a positron (electron) and an electron neutrino (antineutrino). (Neutrinos are particles with zero charge and very small mass. Electron neutrinos have a significant role in the ß-decay theory; see chapter 8.)

The decay fraction in a given mode is called the branching ratio. Charged pions can also decay as

π⁺ [right arrow] μ⁺ + v_μ + γ, π^- [right arrow] μ^- + [bar.v]_μ + γ, (1.10)

also with a branching ratio of 1.2 x 10^-4.

The mass of the neutral pion π⁰ is 134.9764(6) MeV, a value 4.6 MeV smaller than that of the charged pions. π⁰ decays as

π⁰ [right arrow] γ + γ, (1.11)

with a branching ratio of 98.8 %, by

π⁰ [right arrow] e⁺ + e^- + γ, (1.12)

with a branching ratio of 1.2 %, and by other much less probable processes. The π⁰ total lifetime is (8.4 ± 0.6) x 10^-17 s.

The simplest way to produce pions involves collisions between nucleons:

p + p [right arrow] p + p + π⁰, p + p [right arrow] p + n + π⁺,

p + n [right arrow] p + p + π^-, p + n [right arrow] p + n + π⁰. (1.13)

Pion properties can also be investigated by reactions induced by them, like elastic scattering,

π^- + p [right arrow] π^- + p, (1.14)

inelastic scattering,

π^- + p [right arrow] π⁰ + π^- + p, (1.15)

or charge exchange reactions,

π^- + p [right arrow] π⁰ + n. (1.16)

The analysis of pion-nucleon and pion-deuteron reactions led to the conclusion that the pion spin is zero. The pions are bosons and obey Bose statistics, required for the treatment of particles with integer spin.

1.4 Antiparticles

For each particle in nature there is a corresponding antiparticle, with the same mass, and with charge of the same magnitude and opposite sign. This concept was established around 1930 with the development of relativistic quantum mechanics by Dirac and had its first experimental confirmation with the discovery of the positron (antielectron) by Anderson [An32] in 1932. The proton antiparticle (antiproton) was detected by Chamberlain and collaborators in 1955 [Ch55], using the 6 GeV bevatron at the University of California.

The first studies of the reaction [bar.p] + p, where [bar.p] represents the antiproton, have shown that in the great majority of cases this reaction leads to the annihilation of the p[bar.p] pair with production of pions, but in 0.3% of cases it is able to form the n[bar.n] pair, where [bar.n] is the antiparticle of the neutron, or antineutron. It was in this way that, in 1956, Cork and collaborators [Co56] first detected the antineutron, using antiprotons emerging from a beryllium target bombarded with 6.2 GeV protons.

Antiprotons and antineutrons are antinucleons. The magnitude of every quantity associated to some particle is identical to that of the corresponding antiparticle, but, as we shall see soon, there are, besides charge, other quantities for which the values for particles and antiparticles have opposite signs.

The mesons π⁺ and π^- are antiparticles of each other. In this case it is not important to define which is the particle and which is the antiparticle, since mesons are not normal constituents of matter. In the case of π⁰, particle and antiparticle coincide, since charge and magnetic moment are zero.

1.5 Inversion and Parity

Apart from rotational invariance (discussed in Appendix A), a system can have another important spatial symmetry, namely, that with respect to the inversion of coordinates. Let the quantum state of a particle be described by the wavefunction Ψ(r). The parity of this state is connected to the properties of the wavefunction by an inversion of coordinates

r [right arrow] -r. (1.17)

If

Ψ(-r) = +Ψ(r) (1.18)

we say that the state has positive parity, and if

Ψ(-r) = -Ψ(r) (1.19)

we say that the state has negative parity. An inversion of coordinates about the origin is represented in quantum mechanics by the operator Π, where

ΠΨ(r) = Ψ(-r). (1.20)

Π is called the parity operator. The eigenvalues of Π are ±1 (since Π² = 1):

ΠΨ(r) = ±Ψ(r). (1.21)

If the potential that acts on the set of particles is an even function, that is, V(r) = V(-r), the parity operator commutes with the Hamiltonian and the parity remains constant in time, that is, it is conserved.

From the analysis of a system of two particles 1 and 2 that do not interact, described by the product wavefunction

Ψ(r₁) Ψ(r₂), (1.22)

the parity of the system is the product of the parities of each particle. That is, the parity is a multiplicative quantum number.

Besides the parity connected to its spatial state, a particle can also have an intrinsic parity. In this case the total parity is the product of the intrinsic and spatial parities. In processes where no particle is created or destroyed, the intrinsic parities of the particles are irrelevant. In reactions where particles are created and destroyed, the application of parity conservation must include the intrinsic parities of the particles.

Since Π = Π^-1, there is no distinction between the active and passive viewpoints (see Appendix A). In Cartesian coordinates the inversion transformation means (x, y, z) [right arrow] (-x,-y,-z), whereas in spherical polar coordinates

(r, θ, φ) [right arrow] (r, π - θ, φ + π). (1.23)

Therefore sin θ does not change, cos θ changes sign, and the function e^-imφ acquires the factor (-)^m.

Rotations commute with inversion, as can be easily understood from the geometrical picture and checked formally. This implies that if a state belonging to a rotational multiplet has a certain parity, this quantum number should be the same for all members of the multiplet. For the spherical function Y_ll given by equations (A.7) and (A.89) of Appendix A, we find parity (-)^l. Therefore we conclude that

ΠY_lm(n) = Y_lm(-n) = (-)^lY_lm(n). (1.24)

If the particle has a positive intrinsic parity, the total parity will be (-1)^l. If the particle has a negative intrinsic parity, the total parity will be (-1)^l+1. The same result is clear for P_l which are polynomials of order l in cos θ. In particular, for the backward direction

P_l (-1) = (-)^l. (1.25)

The operators, such as r or p, acting on a state with a certain parity, change this value to the opposite one. They can be called Π-odd operators.

For a particle in a spherically symmetric field, stationary wavefunctions have a certain value of the orbital momentum l,

ψ(r) = R_l(r)Y_lm(n), (1.26)

where R_l(r) is a radial function. We see that parity for single-particle motion is uniquely determined by the orbital momentum. This is not the case in many-body systems, where total momentum and parity are independent in general.

From an analysis of pion and nucleon reactions, one concludes that the intrinsic parity of the former is Π_π = -1 and for the nucleons Π_n = Π_p = +1.

1.6 Isospin and Baryonic Number

The elementary particles exist in groups of approximately the same mass, but with different charges. The mass of the neutron, for example, is about the same as that of the proton, and the mass of the neutral pion, π⁰, is approximately equal to that of the charged pions, π⁺ and π^-. In 1932, Heisenberg [He32] suggested that the proton and the neutron could be seen as two charge states of a single particle, using the name nucleon to identify this particle.

In the theory of atomic spectra, a state that has multiplicity 2s + 1 has spin s¹. It is common, however, to refer to the spin quantum number s simply as spin s (for more details on multiplets, see Appendix A). This is also true for the orbital and total angular momentum. One example is the Zeeman effect, which is the energy splitting among the 2s + 1 states of an atom in a magnetic field. The spin s is identified with the angular momentum of the system, and operators for the components of this angular momentum, s_x, s_y, s_z, can be defined. The quantum commutation rules are defined as

[s_x, s_y] = ihs_z, [s_y, s_z] = ihs_x, [s_z, s_x] = ihs_y. (1.27)

(Continues...)

Excerpted from Nuclear Physics in a Nutshellby Carlos A. Bertulani Copyright © 2007 by Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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Details zur Ausgabe

• Verlag: Princeton University Press
• Erscheinungsdatum: 2007
• ISBN 10: 0691125058
• ISBN 13: 9780691125053
• Einband: Tapa dura
• Sprache: Englisch
• Anzahl der Seiten: 488
• Kontakt zum Hersteller: Nicht verfügbar