Interazioni nucleari forti e forza nucleare

Università degli Studi Guglielmo Marconi

Facoltà di Ingegneria

Corso di laurea in ingegneria industriale - L9

Strong interaction and nuclear forces

Supervisor: Chiar.mo Prof. Giovanni Martinelli

Students: Carlo Iazeolla Matricola 0020816

Academic Year 2023/2024

A mia mamma che ci ha sempre sperato...

"Io devo studiare sodo e preparare me stesso, perché prima o poi verrà il mio momento."

"I will study and prepare myself, and someday my chance will come." - Abraham Lincoln

Contents

• 1. Introduction
  • 1.1 About the nucleus and its very existence
  • 1.2 A brief history
  • 1.3 The Standard Model
  • 1.4 Structure of the thesis
• 2. Structure of the nucleus
  • 2.1 Nuclei and their constituents
    • 2.1.1 Protons, neutrons and electrons
    • 2.1.2 Mass defect
    • 2.1.3 Nuclear radius
    • 2.1.4 Nuclear stability
  • 2.2 Binding Energy
    • 2.2.1 Semi-empirical mass formula
  • 2.3 Forces that show saturation
• 3. Strong Nuclear Interaction
  • 3.1 Nuclear Shell Model
  • 3.2 Nuclear force
    • 3.2.1 Main characteristics of Nuclear force
    • 3.2.2 Yukawa Potential
  • 3.3 Quantum Chromodynamics (QCD)
    • 3.3.1 Particles and antiparticles
    • 3.3.2 Color Charge
• 4. Conclusions
• 5. Bibliography
  • Books
  • Articles
  • Websites

Chapter 1

Introduction

1.1 About the nucleus and its very existence

The atomic nucleus is made up of neutrons and protons, which are bound together, and they are also the main foundation of matter, determining most of atom’s characteristics. In fact, nuclei contain the largest percentage of the total mass of atoms. However, the force that holds neutrons and protons together remains one of the most complex topics in modern physics. Protons are positively charged hence they experience a repulsive force, and therefore, considering that nuclei not only exist but are remarkably stable, it means that must exist a much stronger force which overcomes the electromagnetic repulsion. This thesis tries to answer the question providing a brief explanation of "What is the force that binds nucleons together?", the principle governing nuclear interactions.

1.2 A brief history

The study of the atomic nucleus began in 1911, when Ernest Rutherford discovered the dense core of the nucleus through his gold foil experiment; the dense core was completely unexpected. In fact, before this, the prevailing atomic model was the "plum pudding" model, which suggested that atoms were composed of a diffuse positive charge with electrons embedded, similar to raisins in a pudding.

A few years later Moseley observed that the nuclear charge is Z times the protons, where Z is which is roughly half of the atomic mass number. In 1920 Rutherford supposed the existence of a neutral particle, which he called neutron; however, it remained difficult to understand how could protons, which are positively charged and repel each other electromagnetically, be bound together in a very tiny space.

In 1932 James Chadwick indeed discovered the neutron, and he also tried to explain the stability of nuclei by proposing the idea that neutrons act as a glue, capable of counteracting the electrostatic repulsion between protons. However, this theory could not explain why some nuclei have more neutrons than protons without being unstable, and also does not explain the fact that neutrons decay into protons, electrons, and antineutrino via β^- decay, as already observed in those years.

The first successful model was suggested by Yukawa in 1935. In his model Yukawa provided a theoretical explanation for the strong nuclear force, predicting the existence of mesons as force carriers; the existence of mesons was later experimentally confirmed in 1947. Yukawa's theory is constructed in analogy with the quantum theory of electromagnetic interactions, but with a critical difference in the range within which the force carrier is able to propagate. In fact, the nuclear force has a very rapid decrease beyond 1fm.

Later in the 1960s and 1970s, due to the work of many scientists, the theory was expanded, giving birth to the theory, which is Quantum Chromodynamics (QCD) the currently theory that explains the nuclear interaction as a residual force, using a framework similar to that of Quantum Electrodynamics (QED). First, Murray Gell-Mann and George Zweig independently proposed that protons and neutrons are composed of more fundamental particles, which were later called quarks by Gell-Mann. They also suggested that quarks interact via the strong interactions.

In 1972, QCD was formalized by David Gross, Frank Wilczek, and Hugh Politzer, introducing the concept of color charge as the source of the strong force. These historical milestones have led to modern nuclear physics, fitting within the framework of the Standard Model of particle physics, which provides a theoretical quantum framework that describes all the fundamental interactions governing the universe except gravity.

1.3 The Standard Model

The Standard Model identifies four different fundamental forces, each with its own features connected to those of the particle which carries the force. Each of these particles is called boson, as described in Section 3.3.1. These fundamental forces are gravity, electromagnetism, the weak nuclear force, and strong nuclear force. All other forces that are commonly seen can be attributed to these four fundamental interactions; for example, the friction between two components sliding against each other dissipates heat, showing one effect of the electromagnetic interaction, which acts at the microscopic level between interacting atoms. In fact, electromagnetic interaction causes resistance to motion, which is manifested as friction.

The gravitational force is the weakest force; nevertheless, its effects reach over infinite distances. At microscopic level its effects are negligible, thus, in the framework of the Standard Model it does not play any role for the particle interaction. However, the gravitational force governs the attraction between objects with mass, and thus it is dominant on macroscopic scales, such as the motion of planets and stars. The force carrier of gravitational forces, named graviton, has not yet been observed; however, it is assumed to be massless, as the photon, because of its infinite range.

The electromagnetic force is the most easily observable force which underlies most of the commonly observable phenomena, such as electricity or magnetism. It is considerably stronger than the gravitational force and weaker than the strong nuclear force, and has an infinite range. The force carrier for electromagnetic interactions is photon, which has rest mass = 0. QED is the theory that completely explains the electromagnetic interaction; in fact, QED is fully experimentally verified and has mathematical consistency that permits accurate prediction. QCD has been developed in analogy to QED.

The weak nuclear force is stronger than gravitational force; nevertheless, is weaker compared to the electromagnetic force and the strong nuclear force. The weak nuclear force is responsible for radioactive decay, such as the β^- decay and is also responsible for neutrino interactions. The force carriers are both W and Z bosons, which, being massive, determine only a short range interaction.

The last interaction is the strong nuclear force, which is responsible for holding the quarks together. It is the strongest of the four fundamental interactions, 100 times stronger than the electromagnetic force and 10 times stronger than the weak nuclear force. However, the strong force also has short range; in fact, this force acts at a distance of around 10^-15 meters, which is 1 femtometer (1fm) and corresponds roughly to the size of a proton or a neutron. The force carrier of this interaction is the gluon, which, similar to a photon, is massless; conversely to the photon, which only carries the force, the gluon not only carries the force but is indeed charged. This leads the gluons to interact with each other, making the strong force highly complex. The fact that the gluon has vanishing rest mass however does not translate into an infinite interaction range: in this case, the range is very tiny due to the fact the gluon is itself color charged, and as a consequence, gluon field lines attract each other, and in turn, the attraction between quarks grows as distance increases, as better explained in Section 3.3, generating a phenomenon which is called quark confinement.

Quarks and gluons interact through color charge, which is a distinct kind of charge that comes in three different types: red, green, and blue. When quarks are bound together to form protons or neutrons, the total color charge of the combination is neutral, which means that it collectively contains one of each colors. In fact, protons and neutrons are made up of three different quarks. Quantum chromodynamics, described in Section 3.3, is the theory that explains the interaction between quarks; also the protons and neutron interaction is explained in QCD in terms of residual interaction between nucleons and gluons.

In fact, gluons mediate the force between quarks; since protons and neutrons have an internal quark-gluon dynamics, they create a field of charge in which mesons and gluons are exchanged between neighboring nucleons, generating an attractive force which overcomes the electromagnetic repulsion. Atoms exhibit similar behavior: while being electrically neutral, electron dynamics between neighboring atoms may give rise, through permanent or instantaneous dipole interactions, to residual forces which are called Van der Waals forces.

Actually, a key challenge in nuclear physics is to bridge the gap between QCD and phenomenological models, such as Yukawa potential. In fact, while the complete QCD Lagrangian is known, it is nonetheless not yet possible to mathematically describe problems involving nucleons starting from the fundamental interactions between quarks, due to the highly complicated nature of the color interaction. Thus, different phenomenological nuclear models, such as the liquid drop model of Weizsäcker or the shell model, are used depending on the specific purpose.

1.4 Structure of the thesis

This thesis aims at exploring all the cited questions. In chapter 2 an overview of the foundation concepts of nuclear physics is presented, describing many properties of nuclei such as mass, mass defect, nuclear radius and stability. The focus will then move to the binding energy, presenting Weizsäcker’s formula and explaining the logic behind every term.

In chapter 3 strong nuclear interactions are qualitatively described. First, it is explained how the nuclear shell model works; then, the main characteristics of the nuclear force, and the Yukawa potential are presented, concluding with a qualitative description of Quantum Chromodynamics and color interactions.

In the end, in chapter 4 a summary of the main notions discussed in the thesis is presented, and together with a look at the future challenges of particle physics.

Chapter 2

Structure of the nucleus

2.1 Nuclei and their constituents

2.1.1 Protons, Neutrons and Electrons

As is well known, every atom is composed of electrons, neutrons and protons (except hydrogen, which does not have a neutron in its structure). Collectively, neutrons and protons are called nucleons; this is due to the fact that they are tightly packed together in the center of the atom forming the atomic nucleus, which occupies a very tiny volume compared to the total dimension of the atom, and makes up 99.9% of its total mass.

From the Moseley experiment, in 1913, we know that nuclei have a positive charge equal to the number of protons. The number of protons of an atom is called atomic number (Z), which summed to the number of neutrons gives the atomic mass number (or simply mass number A).

The number of electrons contained in an atom is also Z, because they need to balance the positive charge of the protons. However, the electron mass is so tiny compared to that of the nucleons (9.1 x 10^-31 kg, while the proton and neutron have a similar mass, approximately 1.67 x 10^-27 kg), that electrons contribute very little to the overall atomic mass.

Typically, in nuclear physics, mass is expressed in atomic mass units rather than kilograms (amu or u) as well as in electronvolts (eV/c²). The relation between SI mass units and u is:

1u = 1.66054 x 10^-27 kg = 931.49427 MeV/c²

2.1.2 Mass defect

As mentioned above, the nucleons are responsible for almost the entire mass of an atom. Thus, one may expect that the total mass of an atom of mass number A and atomic number Z is approximately:

M ≈ (A - Z)M_n + ZM_p

where M_n is the neutron mass and M_p the proton mass. If that were correct, then the mass of a simple nucleus such as a hydrogen isotope with one proton and one neutron, deuteron, would be:

M_p + M_n = 1.007276 + 1.008665 = 2.015941 u

However, this value does not correspond to the experimental measure of the mass of the deuteron, which is 2.014101 u.

The difference between these two values is far from negligible,

ΔM = 2.015941 - 2.014101 = 1.840 x 10^-3 u = 1713.94 MeV/c²

and represents the "mass" released in the form of energy during the formation of the atom. The mass difference ΔM between the sum of the masses of the constituents of any nucleus and its actual mass is called mass defect and corresponds to the binding energy of the nucleons divided by c². Indeed, to separate the nucleons and isolate each one of its constituents it is necessary to supply at least as much energy as the mass defect times c².

In other words, there is a direct relation between the mass of a nucleus and its binding energy, E_b(A, Z)

ΔM = (A - Z)M_p + ZM_n - M

It is particularly useful to introduce the average binding energy per nucleon f:

f = ΔM / A

and to study its behavior for varying nuclear species, i.e., for varying A (Fig. 2.1), as such behavior reveals the stability of the various nuclei.

Figure 2.1: Variation of binding energy per nucleon against A. The fact that in a region around 60 nucleons are included all the most stable nuclei is an explanation of their prevalence in the universe.

As we can see from the figure, f has quite a complicated behavior with growing A, rather quickly for small mass number, with local maxima in correspondence of ⁴He, ¹⁶O, ⁵⁶Fe, reaching a peak around A = 56, and slowly decaying afterwards, for large A. Indeed, for A < 56, nuclear stability is generally increasing with A while beyond A = 56 nuclear stability begins to decrease. Therefore, the nuclear reaction of fusion is energetically convenient at low A while fission becomes energetically convenient at high A.

The Q-value is used to calculate the amount of energy release in those reactions. In a generic nuclear reaction a + A → b + B, the Q-value is defined as the difference between the sum of the mass of reactants and the mass of the products, times c².

Q = (m_a + m_A)c² - (m_b + m_B)c²

The Q-value, as mentioned above, yields information about the amount of energy necessary or supplied in a nuclear reaction; indeed, when Q > 0, it means that the reaction is exoenergetic and therefore the reaction releases energy, while conversely, when Q < 0, it means that the reaction is endoenergetic and thus energy must be supplied to the reactants. Typically, fusion and fission have a positive Q-value.

Fusion consists of a nuclear reaction in which two nuclei combine to form a third, more stable nucleus; different from fusion, fission involves the splitting of a heavy nucleus into two lighter, more stable nuclei, resulting in the release of energy.

That nucleons are tightly bound together in stable nuclei despite the high mutual repulsion of the protons shows that there must be very intense attractive forces at play between them. As the plot of B/A vs A shows, for a rather large interval of A...