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: Students:

Chiar.mo Prof. Giovanni Martinelli

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 6

1.1 About the nucleus and its very existence . . . . . . . . . . . . . . . . 6

1.2 A brief history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 The Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4 Structure of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Structure of the nucleus 11

2.1 Nuclei and their constituents . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.1 Protons, Neutrons and Electrons . . . . . . . . . . . . . . . . 11

2.1.2 Mass defect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1.3 Nuclear radius . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.1.4 Nuclear stability . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2 Binding Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.1 Semi-empirical mass formula . . . . . . . . . . . . . . . . . . . 19

2.3 Forces that show saturation . . . . . . . . . . . . . . . . . . . . . . . 23

3 Strong Nuclear Interaction 27

3.1 Nuclear Shell Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2 Nuclear force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2.1 Main charateristics of Nuclear force . . . . . . . . . . . . . . . 29

3.2.2 Yukawa Potential . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.3 Quantum Chromodynamics (QCD) . . . . . . . . . . . . . . . . . . . 36

3.3.1 Particles and antiparticles . . . . . . . . . . . . . . . . . . . . 36

3.3.2 Color Charge . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4 Conclusions 44

5 Bibliography 46

Books . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

Articles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4

Websites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5

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 char-

acteristics. 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 pos-

itive charge with electrons embedded, similar to raisins in a pudding.

A few years later Moseley observed that the nuclear charge is times the pro-

Z

tons, where is which is roughly half of the atomic mass number

atomic number,

Z

In 1920 Rutherford supposed the existence of a neutral particle, which he called

A. however, it remained difficult to understand how could protons, which are

neutron;

positively charged and repel each other electromagnetically, be bound together in a

6

very tiny space.

In 1932 James Chadwick indeed discovered the neutron[3], 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 then protons without

being unstable, and also does not explain the fact that neutrons decay into protons,

electrons and antineutrino via as already observed in those years.

−

β decay,

The first successful model was suggested by Yukawa in 1935[32]. In his model

Yukawa provided a theoretical explanation for the strong nuclear force, predicting

the existence of as force carriers; the existence of mesons was later exper-

mesons

imentally confirmed in 1947. Yukawa’s theory is constructed in analogy with the

quantum theory of electromagetic interactions, but with a critical difference in the

range with in which the force carrier is able to propagate. In fact, the nuclear force

has a very rapid decrease beyond 1f m.

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 Electrodynamics (QED). First Murray Gell-

Quantum

Mann[7] and George Zweig[34] independently proposed that protons and neutrons

are composed of more fundamental particles, which were later called by Gell-

quarks

Mann. They also suggested that quarks interact via the Then,

strong interactions.

in 1972, QCD was formalized by David Gross, Frank Wilczek and Hugh Politzer[8],

introducing the concept of as the source of the strong force.

color charge

These historical milestones have led to modern nuclear physics, fitting within the

framework of the of particle physics, which provides a theoretical

Standard Model

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 as described in Section 3.3.1. These fundamental forces

boson, 7

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 frame-

work 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 has not yet been

graviton,

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 con-

siderably stronger than the gravitational force and weaker than the strong nuclear

force, and has an infinite range. The force carrier for electromagnetic interactions

is which has rest mass = 0. QED is the theory that completely explains

photon, m

0

the electromagnetic interaction; in fact, QED is fully experimentally verified and

has mathematical consistency that permits accurate prediction. QCD has been de-

veloped 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 nu-

clear force is responsible for radioactive decay, such as the and is also

−

β decay

responsible for interactions. The force carriers are both and bosons,

neutrino W Z

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

5

nuclear force. However, the strong force also has short range; in fact, this force acts

at a distance of around 10 meters, which is 1 femtometer ( 1f and corresponds

−15 m),

roughly to the size of a proton or a neutron. The force carrier of this interaction is

8

the which, similar to a photon, is massless; conversely to the photon, which

gluon,

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 tranlsate

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 which is a distinct kind of charge

color 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 behav-

ior: while being electrically neutral, electron dynamics between neighbouring atoms

may give rise, through permanent or instantaneuos 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

phenomenogical models, such as Yakawa 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, differ-

ent phenomenological nuclear models, such as the liquid drop model of Weizsäcker

or the shell model, are used depending on the specific purpose.

9

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.

10

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 (except

electrons, neutrons and protons

which does not have a neutron in its structure). Collectively, neutrons

hydrogen,

and protons are called this is due to the fact that they are tightly packed

nucleons;

together in the center of the atom forming the atomic which occupies a

nucleus,

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

(Z), which summed to the number of neutrons gives the

number atomic mass

N

(or simply

number 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 kg, while the proton and neutron have a similar

-31

mass, approximately 1.67 x 10 kg), that electrons contribute very little to the

-27

overall atomic mass.

Typically, in nuclear physics, mass is expressed in rather than

atomic mass units

kilograms (amu or as well as in (eV ) divided by .

2

u) electronvolts c

The relation between SI mass units and is :

u M eV

1 = 1.66054 = 931.494

27

u x10 kg 2

c

11

Mass (u) Mass (kg) Rest energy (MeV/c )

2

Electron [18] 5.4858 x 10 9.109 383 7139 x 10 0.5110

-4 -31

Proton [20] 1.007276 1.672 621 925 95 x 10 938.272

-27

Neutron [19] 1.008665 1.674 927 500 56 x 10-27 939.565

Table 2.1: Comparison of electrons, protons, and neutrons across different units of

measurement

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

M A

and atomic number is

Z (A + (2.1)

≃ −

M Z) M Z M

n p

where is the neutron mass and the proton mass. If that were correct, then

M M

n p

the mass of a simple nucleus such as a hydrogen isotope with one proton

deuteron,

and one neutron, would be

+ = 1.007276 + 1.008665 = 2.015941 (2.2)

M M u

p n

However, this value does not correspond to the experimental measure of the mass

of the deuteron, which is 2.014101 [16].

u

The difference between these two values is far from negligible,

∆M = 2.015941 2.014101 = 1.840 = 1713.94 (2.3)

2

− u M eV /c

and represents the ”mass” released in the form of energy during the formation of the

atom. The mass difference ∆M between the sum of the masses of the constituents of

any nucleus and its actual mass is called and corresponds to the

mass defect binding

of the nucleons divided by . Indeed, to separate the nucleons and isolate

2

energy E c

b

each one of its constituents it is necessary to supply at least as much energy as the

mass defect times .

2

c

In other words, there is a direct relation between the mass of a nucleus and its

binding energy, (A,

E Z)

b (2.4)

= + (A − −

M Z M p Z) M n 2

c

12

and using the experimentally determined , Equation (2.4) gives the mass de-

M

fect/binding energy as (A,

E Z)

b

∆M = = + (A (2.5)

− −

ZM Z)M M .

p n

2

c

It is particularly useful to introduce the as

average binding energy per nucleon f

∆M (2.6)

=

f A

and to study its behavior for varying nuclear species, i.e., for varying (Fig. 2.1),

A

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 [28].

As we can see from the figure, has quite a complicated behavior with growing

f A,

rather quickly for small mass number, with local maxima in correspondence of 4 He

and reaching a peak around and slowly decaying afterwards, for large

16 56

O, F e, A.

Indeed, for 56, nuclear stability is generally increasing with while beyond

A < A,

= 56 nuclear stability begins to decrease. Therefore, the nuclear reaction of fusion

A

is energetically convenient at low while becomes energetically convenient

fission

A, 13

at high , as shown by the arrows in Fig. 2.1.

A

The Q-value is used to calculate the amount of energy release in those reactions.

In a generic nuclear reaction + + the Q-value is defined as the difference

→

a A b B,

between the sum of the mass of reactants and the mass of the products, times .

2

c

= (m + )c (m + )c (2.7)

2 2

−

Q m m

a A b B

The Q-value,as mentioned above, yields information about the amount of energy

necessary or supplied in a nuclear reaction; indeed, when 0 means that the

Q >

reaction is exoenergetic and therefore the reaction releases energy, while conversely,

0 means that the reaction is endoenergetic and thus energy must be supplied

Q <

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 vs shows, for a rather large interval of

B/A A