Nuclear Physics

Nuclear Physics

Ali A. Abdulla

Professor of Physics

Physics Department

College of Science, University of Baghdad, Iraq

Copyright © 2015 by Ali A. Abdulla.

Library of Congress Control Number: 2015912134

ISBN: Hardcover 978-1-5035-9007-6

Softcover 978-1-5035-9006-9

eBook 978-1-5035-9005-2

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Contents

Preface

Chapter One

Introduction (Brief Necessary Review)

I-1—A General Look at Some Features of the Nucleus

I-2—Nuclear Size

I-3—Nuclear Mass

I-3-1—Semiempirical Mass Formula (Weisẵker Formula)

I-3-2—Nuclear Forces: Entity and Features

1-4 —Nuclear Separation Energy of Neutron ≡ Sn

I-5—Coulomb Forces and Mirror Nuclei

I-6—Charge Symmetry and Charge Independence of Nuclear Forces

I-6—Magnetic Moment and Schmidt Limits

I-6-1—Magnetic Moment of the Deuteron (2D1)

I-6-2—Schmidt Limits and More Complicated Nuclei

Chapter Two

Representation of Three-Dimensional Rotation Group

II-1—Introduction

II-2—Properties of D-Matrix

II-3—Addition of Angular Momentum

II-3-1—Coupling of Two Angular Momenta

II-3-2—Strong Coupling

II-3-3—Properties of C-G Coefficients

II-3-4—Wigner j-Symbols

II-4—Some Important Properties of 6j-Symbols

II-4-1—Symmetry Relations

II-4-2—Fundamental Relations with CGC

II-5—Symmetry Relations

II-6—Irreducible Tensor: Definition and Properties

II-6-1—Commutation Relations with J (TAM)

II-6-2—Hermitian Conjugate of Tq(k)

II-6-4—Special Tensor Operators: Identity Operator

II-6-4—Tensor of Rank (Λ = 2)

II-6-5—Reduction of Tensor

II-6-6—Rotational Problem

II-6-7—Spherical Bases Vectors

II-8—Theory Tackling in a General Way

II-7—Scalar Product of Two Tensors

II-7-1—Wigner-Eckart Theorem

II-7-2—Some General Remarks on D-Matrices (Summary)

II-7-3—Explicit Evaluation of Dmm’(j) (α, β, γ)

II-7-4—Symmetry Properties of 171792.jpg and 171787.jpg

II-7-5—D-Matrix and Spherical Harmonics Functions Ylm 171797.jpg

II-7-6—Type of Matrix Elements

II-7-7—Reflection in Space, Parity Operator

Chapter Three

Electromagnetic Properties of Nuclei

III-1—Introduction

III-2—Shape of the Nucleus

III-3—Magnetic Dipole Interaction (MDI)

III-4—Electrostatic Interactions

III-4-1—Electric Multipole Moments

III-4-2—Matrix Elements of Scalar Product of Two Commuting Tensor Operators

III-4-3—Application to Hyperfine Structure of Free Atoms

Chapter Four

Emission of Electromagnetic Radiations

IV-1—Introduction

IV-2—Hamiltonian of the Interaction H

IV-2-1—Vector Field Properties

IV-2-2—Parity of the Vector Spherical Harmonics (VSH)

IV-3—The Multipole Fields (MF)

IV-3-1—Dual Fields (DF)

Chapter Five

Emission of Multipole Radiations (EMR)

V-1—Selection Rules (SR)

V-1-1—Some Remarks to Be Noted

V-2—The Sources of Multipole Radiation, the Dynamic Multipole Moments

V-3—Angular Distribution of the Multipole Radiations

V-3-1—Emission (Transition) Probability of Multipole Radiation:

V-3-2—Internal Conversion

V-3-2-1—The General Idea of Computing Icc

V-3-3—Oriented Nuclei

V-3-3-1—Angular Correlation of Multipole Radiations

Chapter Six

Nuclear Beta Decay

VI-1—Introduction

VI-2—β-decay Processes

VI-2-1—Simple Theory of β-decay

VI-2-2—Fermi Theory of β-Decay

VI-2-3—Superallowed Transitions (SAT)

VI-2-4—Antineutrino Absorption and Characters

VI-2-5—Violation of Parity Conservation (in Weak Interaction)

VI-2-6—Two Components Theory of the Neutrino (c = c`)

VI-2-7—Experimental Evidences for Parity Nonconservation in Weak Interaction

VI-2-8—Mu-Meson Decay, the Universal Weak Interaction

Chapter Seven

Nuclear Reactions

VII-1—Introduction

VII-2—Wigner Channels of Interaction

VII-2-1—Types of Nuclear Reactions

VII-3—Cross Section, Definition, and Formalism

VII-3-1—Introduction

VII-3-2—Differential Cross Section

VII-3-3—Formation of Cross Section ( 171814.jpg )

VII-3-3-1—Properties of ( 171818.jpg )

VII-3-4—Nuclear Fission

VII-3-4-1—Spontaneous and Induced Fission

VII-3-4-2—Energy Released of the Fission

VII-3-4-3—Chain Reaction and Fission Reactor

VII-3-4-4—Nuclear Reactors

VII-3-5—Fusion

VII-3-5-1—Fusion Process

VII-3-5-2—Energy Production in the Stars

VII-3-5-3—The Fusion Reactor

Chapter Eight

High-Energy Physics: Nuclear Particles

VIII-1—Introduction

VIII-2—Force and Conservation Laws

VIII-2-1—Classification of Interactions

VIII-2-2—Conservation Laws

VIII-2-3—Bootstrap Dynamics in Creating Particles

VIII-3—The Concept of Strangeness Theory

VIII-4—Parity and Particles

VIII-5—Observed Nuclear Particles, Entity, and Properties

VIII-5-1—Introduction

VIII-5-2—Some Historical Remarks on These Particles

1—The Electron and the Positron

2—The Positron

3—Photon

4—Proton

5—Neutron

6—Neutrino

7— 171831.jpg -Meson (Muon)

8—Pions (π+ π0 π−)

9—K-Meson (Theton 171836.jpg and Taun τ)

10—Hyperons

11—Resonance Particles

VIII-5-3—Classification of These Physical Particles

VIII-5-4—Regge Theory for Particles

VIII-6—The Quark Model for Hadron Structures

Chapter Nine

Weak Interaction

IX-1—Introduction

IX-2—The Weak Interaction Process

IX-3—Classes of Weak Interaction

1— leptonic processes such as the decay of 171842.jpg ;

2— semi-leptonic processes with ∆Y = ∆S = 0 such as 171860.jpg + n 171852.jpg p + 171847.jpg ;

3— semi-leptonic processes with l∆Yl = 1;

4— hadronic processes such as 171864.jpg p + π−;

5— CP violating processes

IX-4—Construction of Weak Interaction Theory

IX-5—The Selection Rule l∆Yl 171871.jpg 1

IX-6— 171882.jpg - 171876.jpg Puzzle

IX-7—Non-Relativistic Analysis 171891.jpg P + 171886.jpg

Appendix (A)

Some Solved Problems

The References

Curriculum Vitae of the Author

Preface

Nuclear physics is an important branch of physics. It was developed after the discovery of the nucleus of the atom in 1911, as a result of a scattering of alpha particle (α) by a gold foil target, performed by the famous physicist Rutherford. This experiment defined the status of an atom as a nucleus of a positive charge followed by the discovery of the proton and the neutron, as constituents of this nucleus. The mass of the nucleus was found to be almost more than 99% of the mass of the atom. Alsoits size in terms of its radius was determined by Rutherford scattering about 1 fm (fermion), later scattering experiments and other different methods confirmed the radius of the nucleus is related to its mass number (A) as 37708.jpg , where ro is the basic radius of the nucleus; the lowest radius is the radius of the lowest electron orbit in the hydrogen ground state. It is called Bohr radius; it is about 0.53AO. The average of ro was measured or calculated by these different methods to be 1.2 fm. The force holding the nucleons bound inside the nucleus size is called a nuclear force which is not quite well-known as the electromagnetic force or the gravitational force, which are very well-known. But the main features of the nuclear force are almost known. These features were observed from the nuclear reactions and the nucleon- nucleon interactions and observed properties of nuclei such as the electromagnetic properties of a nucleus and the configuration of nucleons inside the nucleus volume. These nucleons are distributed under the effect of the exclusion principle. The experiment data plays the main role in developing the theory of nuclear physics. From this information, the nuclear force, which is related to the potential as (F = − 122732.jpg V), takes many possible forms. These forms are affected by the spins and orbital angular momenta of different nucleons and also by their isotopic spins, where for the motion of nucleons system (nucleus) there are a spacial coordinates, spin-space coordinates, independent of ( 122753.jpg ), and charge or isospace, which depends only on the isotopic spin (τ). Therefore, there are different forms of the Hamiltonian of the nuclear system, which might be of scalar form, vector form, tensor form, or a mixture of two or more of these forms. That depends on the required explanation to the results obtained experimentally or predicted theoretically. However, the nuclear force is quite stronger than the gravitational, the electromagnetic, and the nuclear weak interaction (β-decay, μ-decay) forces. It is claimed that the nuclear force is about 10³⁹ times the gravitational force, 10¹³ times the nuclear weak interaction force, and 10² times the electromagnetic force. Therefore, gravitational force can be neglected inside the nucleus compared to nuclear force. The nucleus is a microscopic system cannot be described by the classical mechanics. This fact activated scientists to develop a new mechanics to able to deal or to tackle the new physical phenomenon of atomic, nuclear, and molecular system in view of the proposed wave property of a moving system developed by de- Broglie in1924. So during the period 1926–1927, a new mechanics was introduced considering the wave and quantal properties. The wave mechanics was developed by Schrödinger, and the quantum mechanics was developed by Heisenberg, who based his quantum mechanics on the matrix element theory in mathematics, considering the quanta as numbers. But both developments lead to the result its self . Nuclear theory is greatly advanced experimentally and theoretically after the discovery of the nuclear energy in 1939, which was directed to the unhumanitarian uses. The detailed structures of the nucleus were studied for nuclei; the light and the heavy ones during the period 1939–2013 brought out huge information and helped a lot in developing other branches of science. Finally, this book is based on lectures in nuclear physics taught by the author for the last twenty five years at the graduate level. A brief and clear demonstration to the subjects was considered, because the detailed story of the development of nuclear theory is tackled by tens of books. The references considered for these lectures are so many therefore, some of the important onewill be listed at the end of the book. The book contains the following,

1-Chapter One

Introduction

2- Chapter Two

Representation of Three-Dimensional Rotation Group

Theory and Applications to Nuclear Physics

3- Chapter Three

Electromagnetic Properties of Nucleus

4- Chapter Four

Emission of Electromagnetic Radiation

(Electromagnetic Transitions)

5- Chapter Five

Emission of Multipole Radiation (EMR)

(Electromagnetic Radiations)

6- Chapter Six

Nuclear β-Decay

7-Chapter Seven

Nuclear Reactions

8- Chapter Eight

High-Energy Physics: Nuclear Particles

9- Chapter Nine

Weak Interactions

10- Appendix: Solved Problems

11- References

12- Author CV

Chapter One

Introduction (Brief Necessary Review)

I-1—A General Look at Some Features of the Nucleus

To start with, one might give a comparison between the different classes of matter according to their characteristic forces, where these forces determine in general the class of the matter (atoms, molecules, solids, nuclei, or hadrons). These classes of matter differ actually in the force of interactions, range of this force, the relativity importance in their dynamicalal characters, energy excitation, and the number of particles constituting it. These features are shown in table 1 below

Table 1: Classification of Matter

(de-Shalit and Fastback Vol. I)

From table 1, it is clear that nuclear physics is concerned with a matter of moderate force compared with that of hadrons (high-energy physics). But compared with atoms, molecules, and solids, it is of strong force. As it is well-known to physicists and students of physics, the nuclear forces responsible for binding protons with neutrons, and neutrons with neutrons, and protons with protons are quite strong compared with electromagnetic force, gravitational force, and the weak interaction force of β-decay. Nuclear forces equated to these forces are represented respectively as 10², 10³⁹ –10⁴⁰, and 10¹³ since the energy is related physically to the force, whether this force is related to the structure of the nucleus, to the nuclear reaction, or to the creation of a new nuclear particle. Therefore, the field of nuclear physics can be classified (according to the energy) as follows

(1)  0 ~ 1 MeV—the study will be concerned with nuclear structure such as the spin I, magnetic dipole moment μ, quadrupole moment Q, and parity (π).

(2)  1 ~ 10 MeV—this deals with the nuclear reactions of all types under 10 MeV.

(3)  > 1000 MeV—this is known as high-energy physics, which deals with nuclear particles, creation, or specifications. This book is specially designed for the first two (1, 2). Although the nuclear force is not well defined, but due to the fact that nuclear physics is an open field, one can deduce in brief the following features of the nuclear force and nuclear system:

1.  The force range is about 10−13 cm 37838.jpg 10−15 m.

2.  Force law is not quite well-known.

3.  The system is of many body problems.

4.  It is a microscopic system.

5.  Quantum mechanics is the good tool to describe the nuclear system.

6.  Hamiltonian (H = KE + PE) is not quite well defined for the nucleus, due to the fact that the force is not very well-known, which implies that potential 37845.jpg is not well defined as shape and quantity.

7.  There are two important distinguished models in describing the nuclear system; they are as follows

a.  Model-independent

b.  Model-dependent

As an example, take an excited state for a nucleus such as

It is known (as it will be shown) that the transition probability−1 from state ψi to state ψf is given by the well-known golden rule of transition 37933.jpg dρ (E) or in general 37947.jpg B (model) f (θ, I1, I2) where B is model dependent factor and f (θ, I1, I2) is shape dependent.

For a brief clarification, consider the nucleus; here, τ is a model-dependent (lifetime) that can be found by using Wigner-Eckart theorem (to be shown in the next chapters).

After this brief demonstration to the general description of the nucleus, the question is , what is the real nucleus after ninety-four years since its discovery by Rutherford’s α-particle experiment? After the formulations of quantum mechanics in its nonrelativistic and relativistic pictures and then quantum the field theory, the physics of the twentieth century has been concerned with the quantum structure of matter. Each system, such as an atom, a nucleus, or a hadron, studied by high-energy physics, has a ground state and a spectrum of excited states, depending on the energy of the excitation, that are specified by a set of internal quantum numbers, such as spins, in addition to their energies. This claim can be represented by the following diagram:

This diagram shows a nucleus of ground state ( 37997.jpg ), a first excited state, ( 38004.jpg ), and a second excited state ( 38011.jpg ). It is a simple system of excited nucleus, because some cases may be of so many excited states that might be of hundred excited states with transitions between these excited states and also between them and the ground state of the excited nucleus. These transitions may be a type of electromagnetic radiations (γ-transitions) or a type of particles transitions (α, β transitions) or a mixture of these types transitions such as α→β→γ or α→β, α→γ or β→γ. All these will be clarified in the chapter of α, β, γ decays related to the properties of nuclei in different status.

So at the present status of a nucleus, one claims with confidence that the nucleus entity is composed of two types of particles: a proton of known mass and charge (mp 38018.jpg 1.67 × 10−24 gm, charge = e+) and a neutron of mn > mp and a neutral charge. From table 1, the nucleus is under a force effect of a moderate strength and a short range. The effect of relativity is of some importance and the energies of excitation are in the range (0.5–10) MeV. The number of the particles as constituents of the nucleus starts from two (nucleus of deuteron) to many. Therefore, a nucleus is a many-body system, where any dealing with it is dealing with a many-body problem, which is not so easy to tackle without the approach of approximations. Hence, the well known approximation methods play an important role here. These approximation methods such as WKB, perturbation, vibrational, are well treated by quantum mechanics books (I. Schiff, Merzbacher, Messiah, Saxon, Alonso and Valk and the author. The constituents of the nucleus, the protons and neutrons are treated as one particle of different states, proton state and neutron state, within a defined space called charge space, or isotopic space, isobaric space, or isospace. This will be clarified later. The mass difference between mn and mp is about 0.78 MeV, but this does not affect the physical conception of a nucleon proposal for both proton and neutron, because the physical description of the nucleon as proton or neutron is obeying the intrinsic isobaric or isotopic spin defined as 38028.jpg or in some books 38040.jpg , where N is the number of the neutrons in the nucleus and Z is the number of the protons in the nucleus. (It is equivalent to the electrons’ number, and it represents the atomic number in the periodic table of elements. So here, there is a nucleus of a defined size (R = 10−15 m) and particles’ constituents situated in definite states (levels), according to the Pauli exclusion principle. Then the nucleon, in the state of the proton, is called a proton, and when it is in the neutron state, it is called a neutron. So this nucleon has two degrees of freedom in the so-called isospace, i.e. charge (or isospin space) is of two dimensions such as

So an element X is written as 38049.jpg . These particles are nuclear main particles that can be studied by nuclear physics and quantum mechanics. The hadrons are strongly interacting particles,it includes the baryons (nucleons and strange baryons), such as Λ, Σ, Ξ, and Ω, and the mesons, such as π, ρ, κ etc., so the concept of nuclei can be generalized by including in its domain all systems with two baryons (deuteron) or more. When these baryons are nucleons, the systems are the ordinary nuclei (such as H, He, B, Cu, etc.). If some baryons are strange baryons as well as nucleons, the systems are called hypernuclei. It is important to notice from table 1 that the nuclei are the only system that consists of a limited number of particles. The most massive known nucleus consists of 259 nucleons, where the least massive nucleus is the deuteron that consists of two nucleons, with moderately strong forces acting between the particles.

I-2—Nuclear Size

It is well established that the atom is no longer an undividable entity since 1897, where J.J.Thompson discovered the electron as a particle with specified characteristics. Then in 1911, Rutherford had discovered that the atom has a positively charged center with a mass of about 99.9% of the total mass of the atom. This central point of the atom is called a nucleus of the atom. Rutherford has discovered this by performing a scattering experiment of α-particle (2p + 2n) as an incident particle on a target of a gold (Au) foil. The impact parameter (b) was calculated to be about b ≈ 10−12 cm, which, physically, is the least distance between the points, where α-particle deviates from its straight path and the position where the nucleus is located. The scattering experiments were developed greatly to go more deeply inside the nucleus to study the charge and the current distributions within the nucleus. For this, high-energy (≈ 100 MeV) electrons were used as probes, by the scattering of these electrons, as projectiles, and the nucleus under study, as a target.

This nucleus is surrounded by a cloud of electrons. At this time, the atom shape is well established as the nucleus constitutes of protons and neutrons and a cloud of electrons moving in their orbits around the nucleus. Also, the size of the atom is 10−8 cm (in terms of radius). The radius (R) of a nucleus is given by 38056.jpg , where 38063.jpg cm = 1.2 fermi. So R depends on the mass number (A), which implies that the radius of each nucleus differs according to its mass number. The density of the electrons in the atoms was found with a little change over the10−8 cm dimension. Therefore, in many cases, the atomic spectroscopic data are accurate enough to trace even the effects of the shape of the nuclear change distribution on the dynamic of its surrounding electrons. Due to this fact, it is quite possible to determine the electric quadruple moment (Q) of the nucleus, where Q measures the extent to which the charge distributions in the nucleus deviate from the spherically symmetrical shape and acquire an ellipsoidal shape. Also, the magnetic dipole moments (μ), which reflect the current and spin distributions in nuclei, have generally dramatic effects on atomic spectra and, with further refined measurement, yielded information on nuclear magnetic octupole moments too. The coupling of atomic spins with nuclear spins gives the hyperfine structures for the nuclear spectra as well. The importance of measuring Q is represented by the following

All the features mentioned before were confirmed by the experimental measurements or studies for the last one hundred years. As mentioned previously, the scattering experiments were of important results concerning the charge, current, and spin distributions in the nucleus. In addition to that, the spectra, magnetic moments, and quadruple moments were measured for many nuclei. Using the electrons as powerful probes with high energy (≈ 100 MeV), where the wavelengths of the electrons’ waves are so close to the dimensions of the investigated nuclei, helps a lot to study the charge and current distributions in the nuclei. This is due to the fact that the forces acting on the electrons penetrating the nucleus depend on the details of the charge distributions in the nucleus; in addition to this, the electromagnetic interactions are quite well understood. Therefore, the data obtained from electron scattering experiments provide us with fairly accurate information about electromagnetic properties of the nucleus from all kinds of measurements, such as electron-scattering, atomic spectroscopy data, mu-mesic X-rays (muonic atoms, where ū-meson is captured in an atomic orbit that is even much better than ē, for probing the nucleus for charge distribution study). The mass of ū is about 105.66 MeV. This indicates that mū 38254.jpg 200 me, so the Bohr radius for this muonic atom is two hundred times less than that of ē, whence ū-meson probes the nuclear charge distribution from a much closer distance.

Mu-mesic X-rays have greatly helped in clarifying the charge distribution and the current distribution in nuclei. All these experiments and related others conforming the mass of the nucleus (A = mass number) are independent of nuclear density, which imply that the radius of the nucleus in general is given by the following

38261.jpg , where ro ≈ 1.12–1.2 fm. And A is the mass number of the nucleus (Z + N). Figure 1 gives the general charge distribution inside the nucleus.

I-3—Nuclear Mass

The mass of a nucleus can be determined from so many experiments such as nuclear reactions. But it was found that this measured mass is quite different from either the carbon mass C¹² (A = 12) or the oxygen mass O¹⁶ (A = 16). Here, C¹² is chosen as a reference of scale, where the atomic mass is 12, and one atomic unit is (1.66043 ± 0.00002) 38381.jpg 10−24 gm, which is equivalent to (931.478 ± 0.005) MeV/C². On this base, the following important data found

MH = 1.00782522 atomic mass units, hydrogen atom mass

MO = 15.99491494 ± 0.00000028 atomic mass units, oxygen atom mass

Therefore, the bare nuclear mass is given by the following

Mnuc = Mat. – (zme + Be (z)) (1)

where me is the electron mass and Be is the binding energy of the Z-electrons in a neutral atom, where it is estimated to be about 15.73 z⁷/³ eV (from Fermi-Thomas model). The mass of an electron is measured to be (5.48597 ± 0.00003) 38390.jpg 10−4 atomic mass units, which is equivalent to 0.511 MeV. Using equation (1) and the estimated value of Be (Z), one can get the following:

MP (proton mass) = (1.00727663 ± 0.00000008) amu ≡ (938.256 ± 0.005) MeV

MN (neutron mass) = (1.0086654 ± 0.0000004) amu ≡ (939.550 ± 0.005) MeV

Now the binding energy of the nucleus B (Z, N) is defined as the following:

38397.jpg (2)

In (2), a small correction, due to the electronic binding energy, is neglected.

M (Z, N) is the atomic mass of the nucleus, with Z = protons number, and N = A − Z is the neutrons number. So the binding energy B (Z, N) ≡ B (Z, A − Z) is the energy required to break up the nucleus into its nucleons: B > 0.

The binding energy B (Z, N) is an increasing function, generally speaking, of Z and N, i.e., 38404.jpg constant for the range A = 12