Ellwanger U. From the Universe to the Elementary Particles: A First Introduction to Cosmology and the Fundamental Interactions

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Chapter 1

Overview

The phenomena and the properties of the related physical objects that will be

discussed in more detail in later chapters are introduced here: the Big Bang and

the expansion of the Universe, and the constituents of matter. Consideration of the

composition of matter leads from atoms to nuclei, protons, neutrons, and quarks. The

various forms of radioactivity and the forces between elementary particles indicate

the presence of new fundamental interactions, the strong and weak interactions. The

presently known fundamental forces, and the elementary particles we are made of,

are summarized.

1.1 The Universe

The largest possible physical object is the Universe. Since its dimensions and its

dynamics far exceed our everyday experiences, and hence our imagination, it can be

conceived only with the help of numbers and formulas. High powers of ten—and the

ability to compute correctly with such numbers—are unavoidable in cosmology.

The visible part of the Universe consists of several hundred billion galaxies.

Usually, a galaxy consists of 10

–10

stars similar to our Sun. Our Milky Way, one

of the larger galaxies, contains about 3 ×10

stars.

Galaxies are distributed approximately homogeneously throughout the Universe.

However, there exist sparsely populated regions called voids, which are separated by

more densely populated ﬁlaments.

Distances within and between galaxies are speciﬁed in light years (ly): the speed

of light is given by

c = 299 792 458 m s

−1

 3 ×10

−1

. (1.1)

Since a year has about 3.1536 ×10

s, a light year corresponds to about

1ly 0.9461 ×10

m  10

km. (1.2)

U. Ellwanger, From the Universe to the Elementary Particles,1

Undergraduate Lecture Notes in Physics, DOI: 10.1007/978-3-642-24375-2_1,

2 1Overview

Fig.1.1 Schematic picture

of the expanding Universe

(An alternative unit is the parsec: 1 parsec=1pc  3.262 ly, 1kpc  3.262 ×10

and 1Mpc  3.262 ×10

ly.)

A typical size of a galaxy is 5–50 kpc or (1.5–15) ×10

ly; the diameter of our

Milky Way is about 10

ly. The distances between galaxies are on the order of 10

ly;

the distance to the next galaxy, Andromeda, is about 2.9×10

ly. The largest distance

observed up to now (to a supernova, a very luminous explosion of a star) is about

ly.

It is striking that the galaxies are veering away from us with a velocity v that

increases in proportion to their distance d:

v  H

d, (1.3)

where H

is the Hubble constant. This behavior is represented schematically in

Fig.1.1, where our position is in the center, and the arrows denote the velocity

vectors of the observed galaxies.

Traditionally, velocities v of galaxies are given in kilometers per second, and

distances d—for historical reasons—in megaparsecs. For the Hubble constant one

ﬁnds

 70

Mpc

. (1.4)

However, formula (1.3) can be applied only to sufﬁciently far galaxies, whose

velocities are large enough that “local” variations of up to 600 km s

−1

can be

neglected.

In practice one often presumes the validity of the formula (1.3), and uses it to

deduce the distances of galaxies from their radial velocities with respect to the Earth.

These radial velocities are determined with the help of the Doppler effect: the wave-

length of light emitted by an object moving away from us is larger (“redshifted”) than

the wavelength of light emitted by an object at rest. The increase of the wavelength

1.2 The Structure of Matter 3

is measurable, and allows the radial velocity of the object with respect to the Earth

to be determined.

Practically all other methods for the determination of distances (essentially with

the help of the luminosity, measured on Earth, of galaxies of known intrinsic bright-

ness) are in agreement with (1.3), and allow the Hubble constant to be determined.

In any case, the most important result is that the Universe is expanding.

What is the history of the Universe? Looking back in time, about 10

years ago

the Universe was compressed and hot. At that time, neither the galaxies nor the

stars had formed, and the Universe was an exploding hot, dense gas. This process is

called the Big Bang. The stars and the galaxies formed only during the subsequent

expansion, dilution, and cooling.

The detailed study of the expansion rate of the Universe as a function of time, the

different forms of matter, the temperature, the curvature of space and space-time,

etc. is the subject of research in cosmology. The employed formalism is the theory

of general relativity, from which the gravitational forces can be derived as well (see

Chap.3).

1.2 The Structure of Matter

Let us jump from the cosmological to the atomic scale. The orders of magnitude of

intermediate objects—which we will not discuss here—are as follows:

Systems of planets: the distance Earth–Sun is ∼1.5 ×10

Stars: the radius of the Sun is ∼7 ×10

Planets: the radius of the Earth is ∼6.4 ×10

Rocks, humans, ...: ∼1m

Grains of sand: ∼10

−3

Viruses: ∼10

−7

Simple molecules: ∼10

−9

Atoms: ∼10

−10

Observable forces in everyday life are

(a) the force of gravity,

(b) forces between bodies, the force of wind, water, combustion engines, friction

forces ....

All forces listed under (b) can be traced back to forces between atoms and mole-

cules, and are ultimately of electric origin.

1.2.1 The Structure of Atoms

An atom consists of a cloud of electrons, which are negatively charged particles.

The diameter of this cloud is ∼10

−10

m, which corresponds to the diameter of the

atom. In its center there is a positively charged nucleus with a diameter of some

4 1Overview

Fig.1.2 Schematic picture

of an atom (not to scale)

Nucleus

Electron cloud

−15

m(seeFig.1.2). The angstrom, named after the Swedish physicist Anders

Jonas Ångström, is deﬁned as 1 Å = 10

−10

m (approximately the size of an atom),

and the fermi (after the nuclear physicist Enrico Fermi) or femtometer as 1 fm =

−15

m (approximately the size of a nucleus).

The (negative) charge q

of an electron is denoted as q

=−e; the value of the

elementary charge e is e  1.60 × 10

−19

C (C stands for coulomb). The electric

charge of a nucleus is always positive and a multiple of e:

nucleus

=+Ze, Z = integer (1.5)

The number of electrons of an atom is equal to Z (except for ionized atoms, where

one or more electrons have been torn off). Hence, an intact atom is neutral:

atom

= q

nucleus

+ Zq

= Ze + Z(−e) = 0 (1.6)

The electrons are bound to the nucleus by the electric force, since objects of

opposite charge attract each other. The number of electrons—given by the charge of

the nucleus—determines the chemical properties of the element. Accordingly, the

charge of the nucleus deﬁnes the element:

hydrogen: q

nucleus

= 1e

helium: q

nucleus

= 2e

lithium: q

nucleus

= 3e

...

uranium: q

nucleus

= 92e

plutonium: q

nucleus

= 94e

...

A detailed understanding of the structure of the electron cloud and the

consequential chemical properties of atoms is possible only within the framework

of quantum mechanics.

1.2 The Structure of Matter 5

Electrons(2)

Nucleus(1) Nucleus(2)

Electrons(1)

Fig.1.3 Two atoms approaching each other

When two atoms approach each other (see Fig.1.3), repulsive forces act between

the electrons of atom 1 and the electrons of atom 2 as well as between the nucleus

of atom 1 and the nucleus of atom 2, but attractive forces act between the electrons

of atom 1 and the nucleus of atom 2 as well as between the electrons of atom 2 and

the nucleus of atom 1.

These forces compensate each other at large distances (compared to the size of

the atoms). At smaller distances of some 10

−10

m, the balance between the forces is

no longer exact, and—depending on the form of the electron clouds—the following

cases are possible:

(a) The repulsion between the electrons dominates, resulting in a smallest possible

distance between the atoms. For this reason one body cannot penetrate another,

and our hand feels, for example, a wall.

(b) The attraction between the electrons of atom 1 and the nucleus of atom 2 and

between the electrons of atom 2 and the nucleus of atom 1 dominates. In this

case the atoms remain bonded, share their electron clouds, and form a molecule

(see Fig.1.4).

The forces between atoms (or molecules) are called van der Waals forces. They

are complicated, but can be deduced from the (electric) forces between electrons and

nuclei.

Hence, the physical phenomena in our everyday life follow from just two funda-

mental forces: the electric (or electromagnetic) force and gravity.

6 1Overview

Electrons

Nucleus(1)

Nucleus(2)

Fig.1.4 Two atoms forming a molecule

1.3 The Structure of Nuclei

Nuclei consist of protons (with electric charge q

=+e) and neutrons (q

= 0).

(Neutrons were discovered by J. Chadwick in 1932, for which he was awarded the

Nobel prize in 1935.) Protons and neutrons are called baryons. The composition of

a nucleus is written in the form

X, where X is the chemical symbol of the element,

Z is the number of protons, and A the atomic number, which is equal to the number

of baryons (the sum of protons and neutrons). For instance:

hydrogen:

H (1 proton),

deuterium (chemically identical!):

H (1 proton, 1 neutron),

helium:

He (2 protons, 2 neutrons),

iron:

Fe (26 protons, 30 neutrons),

uranium:

238

U (92 protons, 146 neutrons).

Nuclei that differ only in the number of neutrons are called isotopes of an element.

We know that positively charged protons repel each other under the action of the

electric force. Therefore, what keeps the protons (and neutrons) inside a nucleus

together? Here a new force (or “interaction”) enters the game, called the strong

interaction. The strong interaction between baryons is always attractive. At small

distances of a few fermi it is stronger than the electric repulsion, but it decreases

rapidly at larger distances. The strong interaction is approximately independent of

the nature of the baryons, that is, it is approximately identical between two protons,

a proton and a neutron, and two neutrons.

The mass of a proton is m

∼1.67 ×10

−24

g. The mass of a neutron m

is nearly

the same; a neutron is just about 0.17% heavier. An electron is about 2000 times

lighter: m

∼9.1 ×10

−28

g. Hence, the mass of an atom is essentially given by the

mass of its nucleus.

1.3 The Structure of Nuclei 7

The mass of a nucleus differs from the sum of the masses of the protons and

neutrons by the binding energy, which contributes to the total mass, according to

Einstein (E = mc

nucleus

= Zm

+ (A − Z)m

−

binding

. (1.7)

The contribution of the binding energy is always negative, and of the order of some

−2

. The calculation of binding energies of various nuclei, which explain the

masses of the nuclei, is one of the tasks of nuclear physics (see the exercise at the

end of the chapter).

1.3.1 Radioactivity

Depending on its decomposition, a nucleus is more or less stable. An unstable nucleus

can emit particles and mutate into another nucleus, which is always lighter than the

original nucleus. This follows from the law of conservation of energy, which here has

to be applied including the contributions of the masses to the total energy. The decay

of a nucleus and the corresponding emission of particles is denoted as radioactivity,

for the discovery and the study of which A.H. Becquerel and Marie and Pierre Curie

were awarded the Nobel prize in 1903.

Let us consider the following situation: a nucleus with mass M, originally at rest,

decays into n decay products with masses M

, i = 1 ...n. After the decay, the decay

products ﬂy apart with different velocities v

, and therefore possess kinetic energies

i kin

v

. (Here we neglect relativistic corrections, which become important

only for |v|∼c, see the end of Sect. 3.1.) We denote the sum of all kinetic energies

by E

kin

. Then the law of conservation of energy reads

M =



i=1

kin

. (1.8)

Since E

kin

is always positive, (1.8) implies an inequality stating that the sum of the

massesofthen decay products must be smaller than M.BothM and M

have to be

determined from (1.7) above, including the corresponding binding energies.

For historical reasons, the nomenclature of the radioactivities is as follows.

1.3.2 α Radiation

α denotes a helium nucleus, α = 2p2n. The binding energy of this nucleus is partic-

ularly large, hence it is relatively light and can frequently appear as decay product.

In particular heavy nuclei decay under the emission of α particles (see Fig. 1.5).

8 1Overview

p n

n p

Z Z−2

A−4

Fig.1.5 α decay of a nucleus

Generally the α decay of a nucleus X consisting of A baryons (including Z protons)

into a nucleus Y can be written as

X →

A−4

Z−2

Y +

. (1.9)

Once (1.7) is inserted for the masses of the nuclei X, Y, and α in the law of

conservation of energy (1.8), one ﬁnds that the masses of the protons and neutrons

cancel. From the positivity of E

kin

one can derive a necessary condition on the

binding energies of the nuclei X, Y, and α for the decay to be possible:

binding

(X)<E

binding

(Y) + E

binding

(

). (1.10)

Whenever this condition is satisﬁed for some nucleus Y, the original nucleus X is

unstable. However, due to the laws of quantum mechanics it is impossible to predict

precisely the moment when the nucleus will decay; one can only measure (and

attempt to compute) a half-life τ

1/2

after which, on average, half of the nuclei have

decayed.

1.3.3

Radiation

stands for an electron. The emission of an electron occurs mainly in the case of

nuclei with more neutrons than protons (see Fig.1.6). The emission of an electron

is always accompanied by the emission of an (anti) neutrino ¯v. Neutrinos are very

light neutral particles that are very difﬁcult to detect. However, an emitted neutrino

carries energy and momentum, which provides an indication that a neutrino has been

emitted. W. Pauli ﬁrst inferred the existence of neutrinos from the conservation of

total energy and total momentum in

decays in 1930. However, their existence was

proven only in 1956, for which F. Reines was awarded the Nobel prize in 1995.

After the insertion of (1.7)into(1.8), the law of conservation of energy implies

+(A − Z)m

−

binding

(X)

= (Z + 1)m

+ (A − Z − 1)m

−

binding

(Y) + m

kin

. (1.11)

1.3 The Structure of Nuclei 9

p n

ν+ e +

Z+1

pn*

p n

pp*

Fig.1.6

decay of a nucleus

We see that the proton and neutron masses no longer cancel. The condition on the

binding energies for the feasibility of

decay now reads

binding

(X) − E

binding

(Y)<



− m



. (1.12)

Inside a nucleus,

radiation corresponds to the conversion of a neutron into a

proton, an electron, and a(n) (anti)neutrino:

n → p +e

−

+¯ν. (1.13)

In Fig.1.6, the neutron in the original nucleus X and the proton in the newly generated

nucleus Y are indicated by a star.

This process has nothing to do with the strong interaction, which—just like the

electric force (or the electric interaction)—cannot change the nature of neutrons

(or protons). Therefore we are dealing with a new phenomenon, the weak interaction.

(“Weak” means “relatively rare”.) A free neutron decays this way with a half-life

of about 10min (614s). However, a neutron inside a nucleus can decay only if the

binding energies of the nuclei X and Y satisfy the inequality (1.12)!

1.3.4

Radiation

stands for a photon. Photons are the constituents of electromagnetic radiation, i.e.,

X-rays, light, thermal radiation, microwaves, radio waves, etc. The equivalence of

electromagnetic radiation and particles in the form of photons manifests itself in the

photoelectric effect, in particular for energetic radiation such as X-rays. Amongst

other things, this ledto thedevelopmentof quantummechanics:according to quantum

mechanics, waves of a ﬁeld—such as the electromagnetic ﬁeld—are equivalent to a

beam of the corresponding particles, e.g. photons.

It was for the corresponding interpretation of the photoelectric effect—and not

for the development of the theory of relativity—that Albert Einstein was awarded

the Nobel prize in 1921.

The emission of photons occurs in the case of “excited” nuclei. Nuclei are termed

excited if their binding energy is less than its maximally possible value. In that case

the binding energy can increase abruptly, and the released energy is emitted in the

10 1Overview

Fig.1.7 Quark content of a

proton and of a neutron

form of a photon. In this process the decomposition of the nucleus is left unchanged

in terms of protons and neutrons.

There exist other types of radioactivity, such as

radiation, corresponding to the

emission of a positron; a positron is the antiparticle of an electron and has positive

electric charge. The existence of positrons was postulated in 1928 by P. Dirac (Nobel

prize in 1933), and proven in 1932 by C.D. Anderson (Nobel prize in 1936).

The constituents of nuclei, the protons and neutrons, are still not indivisible

“elementary particles”:

1.4 The Structure of Baryons

Today we know that baryons are bound states of three quarks. There exist several

types of quarks, including the u quark with electric charge q

e, and the d

quark with charge q

=−

e. A proton consists of two u quarks and one d quark,

and a neutron of two d quarks and one u quark, see Fig.1.7.

This agrees with the corresponding electric charges:

= 2 ×q

= 2 ×

e −

e = e, (1.14)

and

= q

+ 2 ×q

e +2 ×



−



e = 0. (1.15)

Which force is responsible for the binding of quarks? This is—again—the strong

interaction. The attractive force between baryons can be derived from the “funda-

mental” force between quarks, similar to the van der Waals forces between atoms

and molecules, which follow from the electric force between nuclei and electrons.

Later we will see that there exist additional (unstable) quarks, as well as additional

baryons.

The inner structure of baryons makes it necessary to reconsider the nature of the

weak interaction: if we compare the constituents (quarks) of a neutron to those of a

proton, we ﬁnd that the decay of a neutron is induced by the decay of a d quark into

a u quark: