Braibant S., Giacomelli G., Spurio M. Particles and Fundamental Interactions: An Introduction to Particle Physics

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412 13 Microcosm and Macrocosm

life.” The WMAP (see Sect. 13.5) measurement of the temperature differences

in the Cosmic Microwave Background radiation refers to this moment.

17. t

' 1 billion of years. The average particle energy is E

 0:1 eV. The

galaxy formation starts. Galaxies and galaxy clusters form, then the ﬁrst stars.

Clouds of matter were formed by ﬂuctuations in the matter spatial distribution;

gravitation starts to form protostars. Over time, as the gas cloud compresses, the

temperature at the center increases until it becomes so high that thermonuclear

reactions can be initiated. Hydrogen is burned, and as a consequence, helium is

produced as “ash.” The starlight starts to illuminate the Universe. The average

wavelength of the cosmic microwave background became large, corresponding

to infrared radiation.

18. t

' few billion years. The ﬁrst supernovae explode, launching a large

quantity of material containing heavy elements (synthesized within the stars)

in the interstellar space.

19. t

' 9 billion years. The Sun and its planets, including the Earth, start to

form by gravitational contraction. The material collected by our cloud mostly

contains hydrogen and helium, the material produced at the beginning of

the Universe; there are also signiﬁcant quantities of materials such as iron,

synthesized in massive stars which have already exploded.

20. t

' 13:7 billion years. Approximately 1 million years ago (or less), the

Homo sapiens began to wonder how our Universe was made.

Current size of the Universe. In conclusion, it can be assumed that the Universe

has a radius of about 13.7 billion light years and consists of about 100 billion

galaxies; each galaxy is made of about 100 billion stars. Taking into account the

mass of each star, the observable Universe is made of about 10

protons. However,

this should account for less than 5% of matter and energy: most of the matter/energy

is still invisible to us. The number of cosmic background radiation photons is about

a billion times the number of protons.

One might ask: what existed before the Big Bang? There can be no answer that

physically makes sense because you cannot get any information for the time before

the Big Bang. However, the inﬂation mechanism might suggest that there are many

“parallel Universes.”

Expansion accelerated or decelerated. In 1998, two astrophysicist teams began

to study the motion of distant galaxies by observing the ﬁnal stage of some of their

bright stars (type I supernovae). They found that these galaxies are moving away

from us more slowly than expected from the Hubble law [13S04]. The observed light

has left the galaxies a few billion years ago: looking further away in the Universe,

we observe objects at increasingly younger ages. In the spirit of the standard model

of the Big Bang, gravity slows the motion of bodies that are moving away from

each other. Therefore young “objects” should move faster away from us than older

objects. The results from type I supernovae indicate the opposite, namely, that the

Universe is now expanding faster than in the past. This is an outstanding result

whose consequences are not completely understood in the standard models of the

Microcosm and Macrocosm.

13.6 The Big Bang and the Primordial Universe 413

To obtain this situation, a ﬁeld that tends to accelerate objects in the Universe

should probably be present. This (unknown) energy ﬁeld would contribute to

approximately 70% of the Universe energy density. The name of quintessence or

dark energy is given to this energy ﬁeld. In the future, if the type I supernovae

results are deﬁnitively conﬁrmed by other independent methods, several concepts

about the Universe, particularly those relating to its future evolution, will probably

need to be revised.

The future of the Universe. What can be said concerning the future evolution of

the Universe? If the actual density of matter is smaller (or equal) than the critical

value, then the Universe will continue to forever expand. If it is larger, the Universe

would reach a maximum expansion and then have a contraction, leading to a ﬁnal

implosion. The recent results from WMAP and others indicate that the actual density

of matter is exactly equal to the critical density, and that the Universe is ﬂat. In the

far future, the stars, having ﬁnished their nuclear fuel, will go off one after the other,

and the Universe would become dark, without visible light. In the very far future,

one could have the proton decays and even later blacks hole “evaporation.”

Chapter 14

Fundamental Aspects of Nucleon Interactions

14.1 Introduction

The periodic table of elements (the Mendeleev table, Appendix A.1) is the most

extraordinary demonstration of the interconnection between the microcosm and

macrocosm, or between particle physics, astrophysics and cosmology. Following

the Standard Model of the macrocosm (Sect. 13.6), 3 min after the Big Bang, matter

in the Universe was primarily composed of hydrogen (92%), helium (8%) nuclei

(primordial nuclei) and electrons. Today, on Earth, every element of the periodic

table is present, from hydrogen .Z D 1/ to uranium .Z D 92/.Asweshallshow

in this chapter, all nonprimordial nuclei up to iron are formed inside stars, in the

processes of stellar nucleosynthesis. The release of these elements in the Universe

occurs through the gravitational collapse of massive stars, followed by the envelope

expulsion (supernova). Nuclei heavier than iron are formed through neutron capture

processes, followed by ˇ-decay. Most of the produced nuclei are radioactive. Only

those with long or very long lifetimes have survived. All the others were transformed

into stable nuclei.

In this chapter, we shall review the basic properties of nuclei, their structure

(distributions of electric charge and mass, volume) and the interactions between

nucleons (i.e., between neutrons and protons) to form the nuclei. The interaction

between nucleons cannot be described by a simple potential energy function. As

the molecules are formed by neutral objects (the atoms) and bound by residual

electromagnetic interactions, nuclei are formed in a similar way by objects which

are neutral for the strong interactions. Protons and neutrons have no color charge.

Nuclei are bound by a kind of residual strong interaction. Their interaction cannot

be formalized in a simple way from the mathematical point of view.

However, we shall see that a formula that parameterizes a fundamental quantity,

the nuclear binding energy, can be written. The higher the binding energy, the more

stable is the nucleus. The properties of the nuclear binding energy not only inﬂuence

the physics of nuclei, but also the structure and the stellar evolution.

S. Braibant et al., Particles and Fundamental Interactions: An Introduction to Particle

Physics, Undergraduate Lecture Notes in Physics, DOI 10.1007/978-94-007-2464-8

14,

415

416 14 Fundamental Aspects of Nucleon Interactions

Table 14.1 Distribution of

stable nuclei as a function of

the atomic number Z

(number of protons), the

number of neutrons N and

the mass number

A D Z C N

Number of

AZND A  Z stable nuclei

Even Even Even 157

Odd Even Odd 53

Odd Odd Even 50

Even Odd Odd 4

Total 264

It is not necessary to consider nuclear physics in terms of quarks and gluons,

even if protons and neutrons are made of quarks. In classical nuclear physics, the

existence of quarks can be ignored as well as the existence of meson and hadron

resonances. A nucleus consists of nucleons that somehow behave as almost free

particles, although they are in a high density medium (about 10

nucleons/cm

The average kinetic energies of nucleons in the nucleus are of the order of 20 MeV

(Sect. 14.3.1), which is considerably smaller than the energy scales of elementary

particles. The hadron composition in terms of quarks and gluons should only be

taken into account in a deeper description of nuclear physics. Moreover, under

particular conditions of energy density, such as those that can be reached during

the collision between two very high energy heavy nuclei, quarks could become free

and nuclear matter should behave as a plasma of quarks and gluons.

The chemical elements existing in nature are a ﬁnite number: they are those

that appear in the periodic table of elements. Each element has a nucleus with a

deﬁnite electric charge (i.e., number of protons). In the laboratory, it was possible

to artiﬁcially create some nuclei, called transuranic. In the Mendeleev table, their

position is higher than that occupied by uranium. However, these artiﬁcial nuclei

have relatively small lifetimes. The stable nuclei observed in Nature are 264; the

number of those unstable is larger than 1,500. Their number is increasing every year,

as more reﬁned experimental techniques allow one to observe unstable nuclei with

shorter and shorter lifetimes. The nuclei can be classiﬁed in terms of the number

of protons Z (the atomic number), the number of neutrons N and the number of

nucleons A (A D Z C N D Z protons plus N neutrons) (the mass number). By

sorting the nuclei on the basis of Z and N , the stable nuclei are distributed as shown

in Table 14.1. The largest number of stable nuclei occur when both Z and N are

even. The number of nuclei with Z even and N odd is approximately equal to that

with Z odd and N even. The content of Table 14.1 is evidence that the nuclear force

is independent of whether the nucleons are protons or neutrons.

The fundamental properties of nuclear physics are more complicated than those

of atomic physics. Our knowledge of nuclear interactions is, in any case, advanced

enough to conclude that the chemical elements everywhere in the Universe are the

same as those found on Earth. It does not exist, therefore, some elusive stable

element with physics properties unknown on Earth (e.g., the unobtanium on the

Pandora planet of the Avatar movie).

The physics laws are universal; from nuclear physics, we know that the nuclei

(and thus the atoms) on Earth are the same as those that found elsewhere in the

14.2 General Properties of Nuclei 417

Universe. From nuclear, atomic physics and astrophysics we also know that the

chemical composition of planetary systems may not be very different from that of

our Solar System. We do not know indeed if there is intelligent life beyond our Solar

System, anywhere else in the Universe. We are also unaware of the biology of this

hypothetical intelligent species, nor of their reproduction modality. Reproduction

can occur similar to life on Earth, with the union of two individuals of different

sexes: having seen the success and repeatability of natural laws, this assumption

might be plausible. In this case, the hypothetical aliens will probably exchange a

golden gift and not an iron object as a token of love. Also, in this planetary system,

the ratio between the number of gold nuclei with respect to iron nuclei will be

6

–10

5

, and gold will be a precious good. If in addition, there is on the gift a

brilliant transparent stone made of carbon atoms in a particular lattice arrangement,

we think that the hypothetical individuals of this planet will have a good chance of

successfully transmitting their genes .

14.2 General Properties of Nuclei

In 1911, Rutherford, studying the transmission of alpha particles (He nuclei) in

a thin layer of gold (Au), realized that they were also scattered at large angles

(up to 180

, i.e., backward). At that time, the Thomson model of the atom was

generally accepted: it assumed that the electrons are drowned in a positive charge

distributed throughout the volume of the atom. Such a charged sphere is not able to

signiﬁcantly deviate a particle with a mass equal to 7,300 electron masses. To obtain

deviations such as those observed, it must be hypothesized that the positive charge

is concentrated in a much smaller volume.

The interpretation of emission spectra of atoms and the Rutherford experiment

are at the base of the Bohr–Sommerfeld atomic model:

• The atom consists of a nucleus of charge C Ze.

• Z electrons, each with a charge e, are bound to the nucleus by the Coulomb

potential.

• The nucleus mass is much greater than the electron mass.

• The electric charge of the nucleus is concentrated in a region of space much

smaller than the size of the atom.

After the discovery of the neutron (1932), it was realized that the nuclei are bound

states of subconstituents nearly equal in mass, with N neutrons and Z protons.

The interaction between the nuclear constituents has characteristics very different

from the electromagnetic interaction. In particular, the forces holding the nucleus

together are called nuclear; nuclear forces do not depend on the electric charge and

they have an interaction range of about 10

15

m. The order of magnitude of the

nuclear binding energy is the MeV; it is straightforward to estimate that to maintain

two protons at a distance of r  1 fm against the Coulomb repulsion, an energy of

U>e

=r  1 MeV is needed. Quantities that characterize the atomic nuclei and

418 14 Fundamental Aspects of Nucleon Interactions

ΔV

Detector

Fig. 14.1 Operation principle of a mass spectrometer. The ionized nuclei are emitted from the

source S; they enter, through a collimator, in a region with electric ﬁeld E and magnetic ﬁeld B.

Because of E, the ion is subject to an upward force, whose module is ZeE. Meanwhile, due to the

magnetic ﬁeld, it is deﬂected downward with a force ZevB (c.g.s. units). The second collimator

selects only those particles for which the two forces cancel each other (those for with v D E=B).

After the second collimator, only the magnetic ﬁeld B is present. The Lorentz force deﬂects the

ions with a curvature radius R which can be experimentally measured

give information on their structure are: mass, radius, spin; the electric charge, the

magnetic dipole moment, the electric quadrupole moment.

Nuclei are indicated with the symbolic name of the element X (e.g., H for

hydrogen, Fe for iron). The atomic number Z (and therefore the electric charge

of the nucleus) is written as a subscript on the left. The mass number A is placed on

the top left:

X: (14.1)

Electric Charge of Nuclei. The electric charge of nuclei is measured by studying

the X-ray emission spectra of electrons in inner orbitals (K orbital); these electrons

are not affected by the electromagnetic shielding effect by the electrons arranged

in the outer orbitals. In 1913, Moseley established a relationship between the

frequencies of X-rays and the atomic number, that is,

h D

.Z 1/

(14.2)

where R

D m

=2 D 13:6 eV is the Rydberg constant. Moseley’s law ordered

all known elements in the Mendeleev table, showing that the nuclear electric charge

is a multiple of the elementary electric charge e.

Mass of the nuclei. The mass of the nuclei is determined by measuring their

trajectory in an electric and a magnetic ﬁeld. The mass spectrometer, ﬁrstly

developed by Aston in 1920 (Nobel laureate in 1922), was gradually improved to

achieve a precision measurement of M=M  10

6

. The operating principle is

shown in Fig. 14.1. Ions emitted from the source S are accelerated by an electric

ﬁeld and introduced in the area between two collimators where there is an electric

ﬁeld E orthogonal to a magnetic ﬁeld B.BothE and B are also orthogonal to the

14.2 General Properties of Nuclei 419

ion trajectory in order to only select nuclei with electric charge q D Ze and speed

v D E=B; after the second collimator, there is only the magnetic ﬁeld. The mass M

of the nucleus is determined by measuring the curvature radius R of the trajectory

curved by the Lorentz force (qvB D M v

=R)

M D

R D

R: (14.3)

To reduce systematic uncertainties, the measurements are usually done by compar-

ison amongst nuclei that have very small mass difference. An instructive online

applet of a mass spectrometer can be found in [14w1].

The mass spectrometer is also used to separate isotopes of the same element and

to measure their relative abundance. Carbon, for example, exists in Nature as two

isotopes with relative abundance of 98.89%





and 1.11%





. The atomic

weight of natural carbon corresponds to the mean value

A D12.01. For historical

reasons, the atomic mass unit (AMU), denoted as u, is often used in nuclear physics.

It is deﬁned by the relation 12u D mass of the isotope 12 of carbon. In these units,

the mass of the hydrogen atom is





D 1:007825 u D 938:783 MeV/c

The conversion factor between the units is

1 u D 931:494 MeV/c

: (14.4)

The number at the bottom of each element of the Mendeleev table in Appendix A.1

shows the value of the nuclear mass in AMU. For practical reasons, the proton rest

mass (m

D 938:271MeV/c

)isalsobeusedasmassunit.

14.2.1 The Chart of Nuclides

Nuclei with the same value of Z and different values of A have the same atomic

properties: chemical reactions depend almost exclusively on Z. For this reason,

nuclei with the same number of protons are called isotopes (because they occupy

the same position in the Mendeleev table of elements). Nuclei with the same value

of A and different values of Z are called isobars (because they have approximately

equal mass). In terms of variables .N; Z/, the stable nuclei are concentrated in a

narrow band, called stability valley (Fig.14.2), indicating a correlation between the

electric charge and the number of constituents.

The bond due to the strong interaction between .n–n/ and .p–n/ pairs is

identical. It is different for .p–p/ pairs because the protons repel electromagnet-

ically. For low values of A, nuclei have N D Z; starting from A ' 40, the fraction

of neutrons increases.

420 14 Fundamental Aspects of Nucleon Interactions

number of neutrons, N

number of protons, Z

Z=N

N=20

Z=20

N=8

Z=8

N=28

Z=28

N=82

N=50

Z=50

z=82

N = 126

Fig. 14.2 Chart of nuclides. The nuclide lifetime increases as the darkness of the square. The

stable nuclei are shown in black in the middle. The bars show the nuclei with magic numbers,

Sect. 14.3.3 (Adapted from the interactive and color chart, which is available at the Brookhaven

National Laboratory website [14w2])

14.2.2 Nuclear Binding Energy

The equivalence between mass and energy (E Dmc

) is not appreciable in atomic

physics. The reason is that the binding energies (of the order of eV, i.e., 2 10

5

)

are much smaller than the masses of the particles. The relation E D mc

becomes important in nuclear physics, where the energies involved may represent

a signiﬁcant fraction of the rest masses of the nucleons. For this reason, it is

sometimes convenient to express both energies and masses in natural units „D

c D 1 (see Appendix A.2), that is in MeV.

In the formation of a bond (atomic or nuclear), energy is gained because a

more stable system is obtained. The energy released should be compensated by

the decrease of the ﬁnal mass, with respect to the sum of the initial masses of

the elements. For example, the bound state with the smaller nuclear mass is the

deuterium nucleus (deuteron); the deuteron is a hydrogen isotope made by one

proton and one neutron (Z D 1; A D 2). In this case, the mass is decreased by

2.224MeV; it is a small amount compared to m

C m

, but not quite negligible

(0.2% m

14.2 General Properties of Nuclei 421

235

238

0 30 60 90 120 150 180 210 240 270

Mass number A

Average binding energy per nucleon (MeV)

Fig. 14.3 The measured binding energy (BE) per nucleon of stable nuclei measured as a function

of A. The peaks correspond to particularly stable nuclei. The curve has a maximum at A  60

The binding energy (BE) is deﬁned as the difference between the mass of the

nucleus and the sum of the masses of the constituent nucleons:

nucleus

kD1

 BE D .Zm

C Nm

/  BE: (14.5)

The helium nucleus

He (also called ˛ particle) is a particularly stable conﬁguration

whose binding energy is equal to 28.298 MeV. The binding energy of nuclei with

small A is not a regular function. For A>12, the binding energy is approximately

proportional to the number of nucleons (Fig. 14.3), with

 8 MeV/nucleon: (14.6)

This relation has a justiﬁcation in the framework of the nuclear drop model (see

Sect. 14.3.2).

14.2.3 Size of the Nuclei

The term nuclear radius needs to be correctly deﬁned through the operational

speciﬁcation of what is measured and of the measurement method. Information

422 14 Fundamental Aspects of Nucleon Interactions

on spatial distribution of nuclear matter andonthenuclear interaction range are

obtained with different methods: diffusion experiments (with ˛ particles, neutrons,

protons, electrons); spectroscopy of nuclear levels; analysis of nuclear binding

energies; study of nuclear decays. In this section, we shall mainly focus on the

scattering of particles on nuclei.

In Rutherford scattering (Sect.4.7.1), the closest approach of a particle depends

on the scattering angle  . Rutherford and Chadwick already noted a large deviation

from the expected Coulomb cross-section for a point-like charge when the trans-

ferred momentum is high, i.e., when the distance of closest approach is comparable

with the range of nuclear forces (see Sect. 7.1.1).

Since the ﬁrst measurements, it was found that the nuclear radius R is propor-

tional to the cubic root of the atomic weight A:

R D R

1=3

' 1:2  10

13

cm: (14.7)

Using particle accelerators, it was possible to reach higher transferred momenta,

and to study the structure of nuclei and nucleons in more detail (see Chap. 10).

The information obtained depends on the type of particle used as a probe. ˛

particles and protons are subject both to the Coulomb interaction and to the nuclear

interaction. The neutrons are subject to the nuclear interaction only (the magnetic

dipole interaction is negligible at low energy).

Distribution of the electric charge density. The electrons do not have nuclear

interactions and electron scattering experiments give detailed information on the

electric charge distribution, on the nucleus electromagnetic radius and on their mag-

netization. Using high energy electrons, E

D 100–1,000MeV, the electromagnetic

form factors of nuclei (Sect. 10.4) were measured. From the form factors, the electric

charge density .r/ and the nucleus magnetization M.r/ can be deduced. The

elastic cross-section of electrons on nuclei decreases as the transferred momentum

increases. This is roughly compatible with the expected behavior from a uniform

charge distribution in a sphere of radius R. A more accurate parameterization of the

electric charge density inside the nucleus is obtained with a spherically symmetric

distribution, that is,

.r/ D



1 C e

.rR/=t

: (14.8)

This is known as the Woods–Saxon distribution. It depends on two parameters:

R represents the value of the radius where the charge density is .r/  

=2; t

measures the thickness of the region where the charge density decreases rapidly.

For nuclei with large A, one approximately has

R D .1:18A

1=3

 0:48/ fm;tD 0:55 fm:

Distribution of nuclear matter. Neutron–nucleus scattering experiments are used

to study the distribution of nuclear matter and of the nucleus mean square radius.