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

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6 1 Historical Notes and Fundamental Concepts

KT D

mhv

i (1.1)

where K is the Boltzmann constant and T is the absolute temperature. The

mean velocity of the nitrogen .N

/ molecules at the room temperature is

3K T

3  1:38  10

23

 293

28  1:66  10

27

' 510 ms

1

: (1.2)

The mean kinetic energy of the nitrogen molecules is

mhv

KT ' 3:8  10

2

electronVolt .eV/: (1.3)

Molecules traveling in the vacuum collide with the walls of the container

and rebound, producing a pressure on the container walls. The collisions

between two molecules are relatively rare and are completely elastic; the long

distance forces between molecules are negligible. Under these conditions, the

molecules can be considered as fundamental objects, namely, the “Democri-

tus atom.”

Suppose that we raise the temperature of the gas. At the submicroscopic

level, the effect is the increases of the molecules mean velocity and thus

of their kinetic mean energy. The disordered motion is the same, though

everything happens more quickly; in particular, the impact on the walls is

more frequent and consequently the pressure on the walls is higher.

We continue to increase the temperature: except for the increasing pres-

sure, nothing new happens for a little while. When the temperature becomes

very high (as the kinetic energy of molecules), molecules can be excited after

collision. This corresponds to an inelastic collision: part of the kinetic energy

is transformed into the excitation energy of a molecule. Shortly after being ex-

cited (a fraction of a millionth of a second), the molecule de-excites, emitting

infrared or visible electromagnetic radiation. Our gas is now more complex: it

is constituted of molecules, excited molecules and electromagnetic radiation.

If we continue to raise the temperature, one reaches a given critical value

(order of a thousand of Kelvin degrees) for the nitrogen molecules. At this tem-

perature, the mean kinetic energy of the molecules is about a tenth of eV; but a

small fraction of molecules have an energy notably greater than that average.

It can therefore happen that in a collision, one or both nitrogen molecules

fragment into two nitrogen atoms. We have therefore a situation with nitrogen

molecules and atoms, excited molecules and electromagnetic radiation.

For higher temperatures, a simpliﬁcation occurs: when every single

molecule has fragmented into atoms, one can say that the gas is only formed

1.3 The Concept of the Atom and Indivisibility 7

of nitrogen atoms plus visible electromagnetic radiation. In this situation, we

may believe that the “Democritus atom” is the nitrogen atom.

Let us again raise the temperature, and therefore the mean kinetic energy

of the nitrogen atoms, see Fig. 1.1. Again, nothing new happens for a little

while, then the process which happened for the molecules repeats itself for

the atoms. First, the collision energy is large enough to excite the atoms;

then, it becomes large enough to ionize them, i.e., to extract an electron from

an atom. Our gas is now made of electrons, positive ions of nitrogen (i.e.,

atoms with a missing electron) and electromagnetic radiation which is more

energetic than that present to lower temperatures. A gas of this type, with

electrons, positive ions and photons, is present in the external layers of the

Sun, particularly in the photosphere.

Proceeding with our ideal experiment, by continuing to increase the

temperature, we will reach other phases corresponding to nitrogen atoms

without two electrons, then without three, etc., until all the electrons will

be stripped from the nitrogen atom. The new type of gas is made of positive

nitrogen nuclei, electrons and high-energy photons (X-rays). This complex

gas is called a plasma, i.e., a state made of positive and negative electric

charges and electromagnetic radiation. This is the fourth state of matter

which is abundant in the Universe because it is present in the stars. At this

point, we could say that the “Democritus atoms” are the electrons and the

nuclei. The plasma cannot be contained because the container walls would

be destroyed by collision of the gas particles (if the number density of gas

particles is large). It must be “conﬁned” with the mean of magnetic ﬁelds

using, for example, a Tokamak apparatus, where the nuclear fusion and

magnetic conﬁnement is studied (Sect. 14.11).

A further increase of the collision energy induces the fragmentation of the

nitrogen nuclei into neutrons and protons. (From an experimental point of

view, it is however easier to reach the same observation using high energy

collisions between an electron and a nucleus, or even between two nuclei.)

While protons are stable, neutrons decay after few minutes in a proton, an

electron and an electron antineutrino, n ! pe





. Our gas is now made of

electrons, protons, electromagnetic radiation (-rays) plus



. Antineutrinos

do not have electric charge and have a tiny probability to interact with

other particles; therefore, they quickly leave the region where they have been

produced.

The proton, neutron, electron, photon and the antineutrino were for a

long time all considered “elementary particles.” It can be hypothesized that

in some part of the Universe, there exists a plasma of protons, neutrons

and electrons (perhaps at the center of the neutron stars). Protons and

neutrons are not really elementary. One believes that by further raising the

temperature, another transition should be observed, passing to a new type

of gas that should have as elementary objects the leptons, the quarks, the

8 1 Historical Notes and Fundamental Concepts

–10

1 TeV

1 GeV

1 MeV

10 eV

1 eV

(°K)

<10

–17

–15

–14

–10

–9

DIMENSIONS

(m)

ATOMS

TEMPERATURE

IONIZATION

ENERGY

QUARKS

ELECTRONS

PROTONS

NEUTRONS

NUCLEI

ATOMS

MOLECULE

OF PROTONS

OF NUCLEI

OF ATOMS

OF MOLECULES

COSMIC

TIME

(s)

Fig. 1.1 Scale of dimensions in meters for various types of “atoms” and scale of “ionization”

energies, i.e., the necessary energies to extract a constituent from the considered “atom.” The

temperatures corresponding to the considered energies and the cosmic time when the Universe

had that temperature starting from the Big Bang are also shown

photons and the gluons, see Fig. 1.1. This new state of the matter should be

reachable for temperatures larger than 10

K (i.e., for collision energies

higher than 100 GeV), under the condition of large density. The state with

quarks and gluons is called quark–gluon plasma. (Could it be the ﬁfth state

of the matter?) The discovery of this possible ﬁfth state of matter is one

of the objectives of the new LHC (Large Hadron Collider) accelerator at

CERN.

We have perhaps reached the end of our ideal experiment in the sense that in

the collisions between two particles, it is not possible to reach higher collision

energies, that is, higher temperatures. At this level of knowledge, “Democritus

atoms” have been identiﬁed; it is necessary to specify that some of them, the

leptons and the quarks, can be really thought of as “atoms,” while the photons and

gluons can respectively be thought of as the transmitters of the electromagnetic and

strong interactions (all can be considered as point-like). Here, a more elaborated

discussion shall pursue concerning the fundamental interactions (gravitational,

weak, electromagnetic and strong), their uniﬁcation and the possible symmetries

amongst the “matter particles” (the leptons and the quarks) and the “force particle”

(the photon, the gluons and the intermediate bosons W



and Z

Proceeding with our ideal experiment at even more elevated temperatures, can a

gas of new, smaller, and more “elementary” particles be hypothesized?Some believe

that is the case.

1.4 The Standard Model of Microcosm – Fundamental Fermions and Bosons 9

It is reasonable to think that a gas of particles constituted the primitive Universe

immediately following the Big Bang. Indeed, one can imagine that the primitive

Universe was a warm gas of tiny, but very massive particles. The gas has cooled

down with increasing time, leading successively to a gas composed of less heavy

particles, then to a “gas” of quarks, gluons and leptons, to a “gas” of protons and

electrons, to a gas of atomic nuclei and electrons and ﬁnally to a gas of atoms and

molecules. Therefore, the study of high temperature gas allows us to study situations

typical of the early Universe: the higher the temperature, the closer we are to the

initial conditions of the Big Bang. The higher temperatures reached in laboratory in

collisions between two leptons or quarks are obtained at energies in the center of

mass of hundreds of GeV, i.e., around 10

K, corresponding to the temperature that

existed slightly less than a billionth of a second (see Eq. 1.5) after the Big Bang.

Figure 1.1 illustrates “atoms” found in our ideal experience proceeding towards

smaller dimensions. A scale of temperature is also illustrated, together with the

corresponding energy, determined through the relationship

Kinetic energy .eV/ D 3K T =2 ' 1:3  10

4

T.K/; (1.4)

i.e., 1 eV = 7,740 K.

Finally, a temporal scale is also shown starting from the Big Bang using the

relationship [W08]

t.s/ ' 2:25  10

/: (1.5)

1.4 The Standard Model of Microcosm – Fundamental

Fermions and Bosons

According to the so-called Standard Model of the Microcosm (SM), the ultimate

constituents (fundamental) of matter are quarks and leptons, i.e., point-like fermions

with spin 1/2. We can consider them as the smaller known objects. The quarks and

the leptons can be classiﬁed into three “families” as shown in Table 1.1 and in

Table 1.2. The quarks u;c;thave electric charge equal to C2=3 times that of the

proton, while the quarks d; s; b have electric charge equal to 1=3. The neutrinos



, 



and 



have a null electric charge. The ﬁrst family includes the quarks u;d

and the leptons 

, e



. The ordinary matter is constituted of quarks u;dand of

electrons e



. The second and third families seem to be “replicas” of the ﬁrst one.

The quarks and the leptons of the second and third families can be produced in

collisions between high energy particles.

The particles that mediate the four fundamental forces are the photon for the

electromagnetic interaction, the intermediate vector bosons W



and Z

for the

weak interaction, and the eight gluons for the strong interaction. These particles,

unlike those listed before, are bosons, that is, they have integer spin equal to one.

10 1 Historical Notes and Fundamental Concepts

Table 1.2 (a) Leptons and

quarks (spin 1/2 fermions)

ﬁrst family. (b) Leptons and

quarks second and

third families

First family

Symbol Q L

Leptons 

01



1 1 

Quarks u C2=3 C1=3

d 1=3 C1=3

Q is the electric charge in unit of the proton charge, L

is the

electronic lepton number, B the baryonic number.















The Standard Model does not consider the gravitational interaction that may be

neglected in the microcosm at the attainable energies.

The four fundamental forces seem to be each very different from the other.

In collisions between high energy particles, it was possible to verify that the

electromagnetic and weak interaction are tightly entangled and that one therefore

can speak of electroweak interaction. The attainable energies with accelerators

(actual and future) do not allow one to directly verify the theories that foresee

a further uniﬁcation of the electroweak force with the strong one (Theories of

Grand Uniﬁcation (GUT)). One can only hope to observe indirect effects through

experiments without accelerators.

The Higgs boson, H

is also listed in Table 1.1. The Higgs boson has not yet

been experimentally observed; its existence is postulated to resolve inconsistencies

in theoretical physics and attempts are being made to ﬁnd the particle by experiment,

using the Large Hadron Collider (LHC) at CERN and the Tevatron at Fermilab.

All particles with electric charge are subject to the electromagnetic interaction.

Lepton and quarks are also subject to the weak interaction. In particular, the

neutrinos, being uncharged, are only subject to the weak interaction. The particles

composed of quarks (the hadrons) are subject to the strong interaction. Hadrons

are known in two topologies: those constituted by three quarks (the baryons,asthe

proton and neutron) and those constituted by a quark-antiquark pair (the mesons).

As for leptons, antiquarks also exist and particles composed of three antiquarks are

called antibaryons. As will be discussed later, the number of baryons and leptons is

conserved. This means that, as described by the relationship E D mc

, the energy

can be converted in mass in the form of particles; nevertheless, the total number of

baryons and leptons must be conserved. If an electron is produced, it must be created

in association with a positron (its antiparticle, with electric charge and leptonic

number of opposite sign) as expected from the Dirac theory.

Chapter 2

Particle Interactions with Matter and Detectors

2.1 Introduction

What does it mean to see a particle? From the epistemological point of view,

to observe an object means to detect the light reﬂected by its surface. The light

is nothing else than a component of the electromagnetic radiation that can be

detected by the human eye. The dimensions of the particles are such that the

electromagnetic wave is undisturbed: the visible light wavelength ranges from 400

to 700nm (corresponding to the color range from violet to red) and is quite larger

than the dimension of an atom (0.1 nm). The only possibility is therefore to detect

the emitted radiation when the particles interact with matter. A particle detector

is a translator that connects, through adequate ampliﬁcations, one of our sensory

organs (or a computer) with the effect produced by the interaction of the particle to

be detected with matter. Particle physics is based on experiments in which particle

interactions are studied thanks to the use of more or less sophisticated detectors.

In this chapter, the main processes occurring in the interaction of radiation with

matter and the experimental techniques to detect them are brieﬂy described. The

term radiation refers to both charged (electrons, protons, etc.) and neutral particles

(photons, neutrons, etc.) with energies greater than the keV. The study of these

processes of radiation-matter interaction is of fundamental importance since they

are at the base of the methods of particle detection; they determine the sensitivity

and the efﬁciency of any experimental apparatus, and are therefore necessary in

order to understand the operation of detectors and experiments.

The radiation “sees” the matter as an aggregate of constituents (electrons, atomic

nuclei, nucleons, quarks, etc.) separated by distances larger than their dimensions.

The radiation-matter interaction depends on the probability of interaction between

an incident particle and a target particle, i.e., on the type of fundamental interaction

involved (e.g., strong, electromagnetic, weak) and on the incident particle energy.

Sometimes, the target may be a whole atom; more often, it is an atomic electron.

For instance, for a beam of low energy neutrons (up to a few MeV), the most

probable collision arises between a neutron and an atomic nucleus. However, at

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

12 2 Particle Interactions with Matter and Detectors

higher energies, it may arise between a neutron and a nucleon of an atomic nucleus;

at even higher energies, the interaction can occur between a quark of the neutron

and a quark of a nucleon.

Particle detectors are based on the fact that when particles cross a given medium,

they excite and ionize it. The gas counters (e.g., the Geiger counter) detect the

electrons produced through ionization of the medium atoms; these electrons are

then accelerated in a strong electric ﬁeld, producing a small measurable electric

signal. The scintillation counters are instead based on the detection of the light

emitted through de-excitation of the medium atoms. For this reason, the principles

of particle – matter interactions through excitation and/or ionization are described in

the ﬁrst part of this chapter. The detection techniques are in continuous evolution and

in the second part we shall synthetically illustrate the basic principles of operation of

the main particle detectors, making references to specialized texts for more detailed

studies (e.g., [L87]).

2.2 Passage of Charged Particles Through Matter

2.2.1 Energy Loss Through Ionization and Excitation

A fast charged particle that moves in a given medium loses energy almost constantly

and is slightly deﬂected from its initial direction. These two effects are the result of

two types of collisions:

1. Inelastic collisions with the atomic electrons of the medium, particularly with

those more external; these collisions produce ionization and/or excitation of the

atoms of the medium; an excited atom de-excites, emitting one or more photons.

These collisions are the main source of energy loss of the incident charged

particle.

2. Elastic collisions with the nuclei. These collisions are less frequent; in practice,

they do not cause a loss of energy, but a variation in the direction of the incident

particle.

The collisions described in 1 and 2 statistically occur a large number of times per

path length unit of the incident particle. For particles heavier than the electron (or

positron), the cumulative effect of the collisions may be considered as a continuous

effect along the whole trajectory of the particle. The energy loss in every collision

(of the order of a few tens of eV) is a tiny fraction of the total kinetic energy of the

incident particle. However, since the number of collisions in a dense medium is very

large, it results in a measurable energy loss which for fast particles is of the order

of 2 MeV g

1

of crossed material (that is, 2 MeV for every cm of crossed

material with the water density). Moreover, such an average energy loss has small

ﬂuctuations and varies slowly with the incident particle energy.

2.2 Passage of Charged Particles Through Matter 13

–v

2πbdb

Fig. 2.1 Scheme of a Coulomb interaction between a particle and the electrons of a medium in a

cylindrical layer at a distance b from the trajectory of the particle

The lightest relativistic particles, particularly the electrons, besides the normal

energy loss through ionization and excitation, have another important source of

energy loss: the bremsstrahlung process, that is, the emission of a high-energy

photon. For this reason, they are discussed in a dedicated section.

2.2.2 “Classical” Calculation of Energy Loss Through Ionization

Let us consider a heavy particle with electric charge ze,massM , and speed v

that travels through some medium with atomic number Z and density .Wemust

consider the collisions with the atomic electrons. Figure 2.1 sketches the collision

of the projectile particle with a target electron located at a distance b (the so-called

impact parameter). If the incident particle has a mass M  m

where m

is the

electron mass, it is practically not deﬂected and may be considered in motion on

a rectilinear trajectory. The speed of the atomic electron is much smaller than the

speed of the particle. Therefore, the electron can be regarded as stationary during

the collision. This hypothesis is easily demonstrable from the atomic theory: the

speed of the atomic electrons is of the order of .c˛

=n/,where˛

D 1=137 is

the ﬁne structure constant and n is the principal quantum number. In the collision,

the impulse delivered to the target electron is

I D

Fdt D e

dt D e

dx D e

: (2.1)

We have only taken into account the component of the electromagnetic ﬁeld, E

perpendicular to the trajectory of the fast particle thanks to the symmetry of the

problem. Using the Gauss theorem on a cylinder of radius b and inﬁnite length, one

can write (in the Gauss-cgs system)

2bdx D 4ze )

dx D

2ze

(2.2)

14 2 Particle Interactions with Matter and Detectors

and therefore (if v D constant),

I D

2ze

: (2.3)

The energy ıE received by the electron located at a distance b is

ıE.b/ D

: (2.4)

When the projectile particle travels a short distance dx in a medium with electron

density N

, the energy imparted to the electrons in a layer of the medium at a

distance from b to b C db from the trajectory is

 dE.b/ D ıE.b/N

d v D

4z

dx: (2.5)

The elementary volume is dv D 2bdbdx. The total energy loss per path length

unit is obtained, integrating (2.5) from a value b

min

to b

max

. The choice of b

min

max

has to be made in order to prevent the value of the integral to become inﬁnite,

which is obviously unphysical: b

min

has to be slightly larger than zero because (2.5)

diverges for b

min

0; b

max

should not be inﬁnite because in this case, the collision

would last indeﬁnitely, contrarily to the hypothesis that the electron stays stationary.

The integration gives



4z

max

min

: (2.6)

The negative sign indicates that the incident particle loses energy. The problem is

now only connected to the determination of b

min

and b

max

To determine b

min

, we consider the maximum energy that the electron can

receive in a collision. Classically, in a central collision, the electron can gain the

energy

.2v/

. Keeping special relativity in mind, such a quantity becomes

2

,where D .1  ˇ

1=2

;ˇ D v=c, p D m

v. Placing this value in

(2.4), one has

min

D 2

) b

min

m

: (2.7)

For b

max

, let us recall that the electrons are bound in atoms with orbital frequen-

cies . If the whole process of collision happens in a time comparable or greater

than the revolution period  D 1=, the collision is adiabatic and the electron does

not receive any energy (it receives it in the ﬁrst phase and loses it in the second

one). It is therefore necessary to hypothesize that the collision occurs in a brief time

compared to the period . The typical time of interaction is classically t D b=v and

relativistically t= D b=.v/. From this last relation, one has

v

  D



) b

max

v



: (2.8)

2.2 Passage of Charged Particles Through Matter 15

In an atom, there are different bound states, each with a different frequency ;

we have thus used an average frequency

 in (2.8). The maximum value for b is

consequently b

max

D v=. Placing this value and that of (2.7)in(2.6), one has



4z





: (2.9)

This is the classical Bohr’s formula. It gives a reasonable description of the energy

loss for helium nuclei (˛ particles) and heavier nuclei. However, it does not properly

describe the energy loss of lighter particles, for example, the protons, despite the fact

that it contains the essential characteristics of the energy loss due to collisions with

atomic electrons.

A better approximation including relativistic effects is given by the Bethe-

Bloch formula. Equation (2.9) is modiﬁed to take into account the mean ionization

potential of the medium and the maximum energy transferred to the electron.

Moreover, one has to consider two additional terms: the ı correction known as the

density effect and the C shell correction. After these changes, (2.9) becomes



D 2N







max



 2ˇ

 ı  2



(2.10)

where

D e

D 2:818  10

13

cm classical electron radius

D N

 Z =A

2N

D 0:1535 MeV g

1

= electron mass D 0:55110 MeV =c

D 9:110  10

31

D Avogadro’s number D 6:022  10

mol

1

I D mean ionization (excitation) potential of the target

Z; A D atomic number and atomic weight of the absorber medium

 D material density

ze D charge of the incident particle

ˇ D v=c of incident particle

 D 1=

1  ˇ

ı D density effect correction (important at high energy)

C D shell correction(already important at low energy)

max

D maximum kinetic energy imparted to an e



in a single collision

' 2m

.ˇ /

,forM  m

Note that:

•Themean ionization (excitation) potential I has a value of 10 eV, and it is

actually difﬁcult to calculate. It is estimated on the basis of dE=dx experimental

measurements. In a medium with Z<13, one has a semi-empirical formula

I=Z ' 12 C7=Z (eV).