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

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56 3 Particle Accelerators and Particle Detection

primary beam

primary

beam

secondary beam

accelerator

primary

target

collimator

detector

separator

counter

target

focussing

lense

Fig. 3.5 (a) Layout of a ﬁxed target experiment at a protonsynchrotron: once accelerated, the

protons are extracted in order to form the primary beam. The primary beam is then sent onto a

target where many new hadrons are produced in the collision. The electrically charged hadrons of

interest are collimated, focused and separated in momentum. In this way, one or more secondary

beams are obtained: the one sketched here was used for a measurement of total hadron-hadron

cross-sections. (b) Optical layout of a secondary beam. C

are collimators; L

,and

are quadrupole lenses (magnetic quadrupoles); M, S are dipolar magnets

In the following paragraphs, the technique of a time-of-ﬂight system in a mag-

netic spectrometer and the method used to measure the mass composition of

a secondary beam will be introduced; the procedure for recognizing particles

produced in interactions in a bubble chamber will also be presented.

The layout shown in Fig. 3.5 is valid for any ﬁxed target experiment. In proton-

antiproton storage rings, the two beams travel in almost circular orbits in opposite

directions and are deﬂected in order to make them intersect in one or more

interaction points. Collisions occur between protons and antiprotons moving in

3.5 Particle Production in a Secondary Beam 57

opposite directions. In this case, it is not possible to directly measure the number

of particles before they interact. Instead, the number of collisions per unit of time

can be measured thanks to a system of detectors that surrounds the interaction zone.

These techniques will be discussed later. For example, in Chap. 9, electron-positron

interactions will be reviewed.

3.4.2 Baryonic Number Conservation

The proton and the particles that decay into a ﬁnal state containing a proton are

called baryons. In terms of subconstituents (Chap. 7), a baryon is formed by three

quarks. The baryonic number C1 is attributed to baryons. The antifermions that

contain a

p in the ﬁnal state of the decay chain are antibaryons; they have the

baryonic number equal to 1. All the other particles have a baryonic number equal

to zero. It has been veriﬁed that until now, the total baryonic number is conserved in

any reactions and decays. A good example comes from the 

decay: 

! p



The total baryonic number is C1 before the decay (since the 

is made of three

quarks) and C1 in the ﬁnal state (after the decay): the baryonic number conservation

is satisﬁed and the reaction is possible, at least as long as this conservation law is

concerned.

The proton is the lightest known baryon. Due to the baryonic number con-

servation, the proton should not decay and therefore should be rigorously stable.

However, Grand Uniﬁed theories (GUT) of the interactions (Chap. 13) foresee the

possibility of the proton decay, and therefore the violation of the baryon number

conservation. Until now, no reliable evidence for proton decay has been found; the

measured proton lifetime is much longer than the age of the Universe.

3.5 Particle Production in a Secondary Beam

3.5.1 Time-of-Flight Spectrometer

Charged particles in a secondary beam may be identiﬁed using a time-of-ﬂight

system in a magnetic spectrometer (see Fig. 3.6). The electrically charged particles

produced in proton-proton collisions and emitted at a given angle are counted by

the counter C

and separated in momentum by the magnet M. The magnet M

acts on charged particles as an optical prism acts on light, i.e., by deﬂecting the

particles of an angle ˇ. In the case of light, the angle ˇ depends on the wavelength

of the light (the prism separates the light in its various colors); in the case of

particles with electric charge q and momentum p, the angle ˇ depends on p=q.

The particles with a given momentum selected by the magnet reach the counter C

The distance C

D ` and the time t employed by the particles to go from

58 3 Particle Accelerators and Particle Detection

, K

, p

Primary

beam

Fig. 3.6 A primary proton beam interacts with the neutrons and protons of the target B; the

charged particles emitted at an angle ˛ are counted with the counter C

, separated in momentum

with the magnet M and again counted with the counter C

. The time used by every particle to travel

the distance ` D C

is measured. (In real experiments, it is necessary to add collimators,

quadrupole lenses and a ﬁrst stage momentum separation)

to C

are measured; the measurements of p; `; t allow one to determine the

mass m. Nonrelativistically, one has



p D mv

t D `=v

(3.14)

from which

m D p=v D pt=` ! t D m`=p: (3.15)

With a given momentum p and a given distance `, the time of ﬂight t between C

and C

only depends on the particle mass which can therefore be obtained from the

measurement of t. Practically, the particles have a very high velocity, comparable

to the light velocity; it is therefore necessary to modify the ﬁrst equation of (3.14)

according to the special relativity formulas. Therefore, the momentum becomes p D

mv, ˇ D v=c D `=tc and one has

m D

v

1 

! t D

C p

/: (3.16)

The time needed to cover the distance ` with respect to the time needed to cover the

same distance by a particle traveling at the light speed is

t D





 1



1 C 



 1

(3.17)

where  D p=mc () p=m if one sets c D 1). Indeed,

ˇ D

vc

c

C m

p=mc

1 Cp



1 C 

(3.18)

3.5 Particle Production in a Secondary Beam 59

Numerical example. Let us assume that the beam analyzed by the magnet

M contains positive electrons .m

D 0:911  10

27

g D 0:511 MeV/, 

mesons .m



D 139:6 MeV D 273 m

/, K

D 493:7 MeV D 966 m

)

and protons (m

D 938:3 MeV D 1; 836 m

/ in equal proportions. Suppose

also that the momentum of these particles is p D 1 GeV/c and that ` D

D 10 m. By calculating , t and t for every particle, one has

W  D 1:957  10

D `=c D 33:3 ns .t

' 0/



W  D 7:16 t



D t D

1C



D 33:6 ns .t



D 0:3 ns/

W  D 2:03 t

D 37:2 ns .

D 3:9 ns/

p W  D 1:066 t

D 45:7 ns .t

D 12:4 ns/

(3.19)

(1ns D1 nanosecond D10

9

s). The time zero is deﬁned as the time employed

by massless particles traveling at the light speed. Positrons cover the distance

` D 10 m almost at the light speed. The additional time with respect to the

time zero is given by (3.17).

Note that t only depends on  D p=m.The

mesons take an

additional time t



D 0:3 ns, the K

take an extra t

D 3:9 ns and the

protons take an extra t

D 12:4 ns. Figure 3.7 shows the time distribution

(and therefore the mass distribution) observed if the four types of particles are

produced in equal proportions. It should be noted that the relation between the

time of ﬂight and mass is not linear; it is therefore difﬁcult to separate peaks

due to small mass particles (electrons, pions, muons if they would be present).

With common electronic systems, it is easy to distinguish time differences of

the order of 0.1–0.5 ns. 

mesons can therefore be easily separated from

protons for momenta up to a few GeV.

Figure 3.8 shows the results of an analysis in mass of a positive beam of about

2 GeV/c at the CERN PS in the 1960s. The mass spectrometer using a time-of-ﬂight

system allows one to measure the charge and the mass of new particles; there is no

information on other quantum numbers. In Fig.3.8, some peaks corresponding to

already known particles are present; they are peaks due to electrons (e



), positrons

) and protons (p). Besides, peaks due to new particles directly produced in

the proton-nucleon interaction can be observed: they are peaks due to 

, 



mesons with a mass of 139.6 MeV; K

, K



mesons with mass of 493.7 MeV

and antiprotons, having the same mass as the proton, i.e., 938.3 MeV. Particles

not directly produced in the proton-nucleon interaction are also visible; these

particles come from the  and K meson decays and are, for example, the muons

( 

, 



) with a mass of 105:7 MeV D207 m

; these particles can also be

60 3 Particle Accelerators and Particle Detection

ΔN

Δt

0481216

Δt (ns)

Fig. 3.7 Time distribution foreseen for particles analyzed with the time-of-ﬂight system described

in Fig. 3.6, assuming that the beam is composed of e

;

and p in equal proportions. The

peak width is only due to the experimental resolution

–

0.5 106 139.6 494 938.2

Mass (MeV)

Log. N

Fig. 3.8 Mass distribution of particles produced in the forward direction in pN collisions at

26 GeV and analyzed with a time-of-ﬂight spectrometer. Muons and electrons are not directly

produced in the pN collisions. The y-coordinate is an arbitrary logarithmic scale

produced in secondary interactions (electrons and positrons). Moreover, the follow-

ing comments can be made:

1. The particles more abundantly produced are  mesons, followed by K mesons

(about 10–100 times less) and ﬁnally by the antiprotons (1,000 times less).

2. The positive particles are more abundantthan the negative. This is only partly due

to the electric charge conservation (the initial state can be pp with charge C2, or

pn with charge C1). The K

mesons are at least twice as abundant than the K



mesons: strange particles as the K mesons are produced in pair .K



/,butthe

can also be produced in association with a strange baryon

(see Sect. 8.14).

3. Particles and antiparticles rigorously have the same mass.

This is the reason why positively charged secondary cosmic rays (see Chap. 13) on the Earth’s

surface are more abundant that the negatively charged ones.

3.6 Bubble Chambers in Charged Particle Beams 61

4. The peak width in Fig. 3.8 is due to the experimental resolution and is not an

intrinsic width.

5. The peaks due to electrons and muons can vary in intensity (height) according to

the beam composition. This is due to the fact that electrons and muons are not

secondary particles produced in the proton-nucleon interaction, but are at least

tertiary particles.

6. Electrons and muons can be separated from pions only using a spectrometer with

an extraordinary time resolution.

3.6 Bubble Chambers in Charged Particle Beams

3.6.1 Conservation Laws

It is quite instructive to analyze bubble chamber photographs. Each photo contains a

wealth of information and it is sufﬁcient enough to qualitatively analyze a few care-

fully chosen photos in order to verify the existence of new particles and to establish

some of their properties. A quantitative analysis can also be done by performing

measurements on the photos. This obviously requires the appropriate measurement

tools. Before starting such analyses, we shall recall some formulas of relativistic

mechanics, concepts on conservation laws, practical formulas, measurement units

and orders of magnitude (see Appendix A.2).

Relativistic Mechanics Formulas

(for c D 1)

Momentum p D m

v p D m

v

Kinetic energy T D .  1/m

T D .  1/m

Mass energy E

D m

Total energy E D T C E

D m

E D m

C m

where m

is the particle rest mass, v its velocity, c is the light speed in vacuum,

ˇ D v=c,  D 1=

1  ˇ

Conservation laws. In any interaction, the following quantities must be conserved:

1. The momentum

2. The total energy (and the kinetic energy in an elastic collision)

3. Theelectriccharge

4. The baryonic number

5. Separately, the electron, muon and tau leptonic numbers. In the reactions due

to the strong interaction, parity, strangeness, charm quantum number and other

quantum numbers (see Chap. 7) must also be conserved

62 3 Particle Accelerators and Particle Detection

Lorentz force. A particle with an electric charge q and momentum p moving in

a magnetic ﬁeld B perpendicular to its velocity v, is subject to the Lorentz force

having an intensity F D qvB; the particle curves following a circumference arc

with radius R such that

p D qRB (3.20a)

(this is a classical formula which is also valid relativistically; in the cgs system, one

has F D qvB=c, p D qRB=c). This equation allows one to determine the ratio p=q

through the measurement of the curvature radius R, when the value of the magnetic

ﬁeld B is known. In practical units, one has (for q Djej =protoncharge)

p.GeV=c/ D 0:30R.m/B.T/: (3.20b)

The momentum expressed in (GeV/c) is equal to the product of a constant (0.30) by

the curvature radius R expressed in meters and by the magnetic ﬁeld B expressed

in Tesla. In the cgs system, one has p(MeV/c) ' 0:30R(cm) B(kG).

Energy unit. Expressing an energy in electronVolt multiples, one has

1J D 1=



1:6022  10

19



eV D 1=



1:6022  10

13



MeV

D 6:241  10

eV D 6:241  10

GeV: (3.21)

Mass unit. A mass can be expressed in energy units using the formula E D m

For the proton,

one has m

D 1:6726  10

27

.2:9979  10

=.1:6022  10

13

/ D

938:27 MeV/c

; with c D 1, one has m

= 938.27MeV. As an order of magnitude,

one can write m

' 1GeV.

Momentum unit. Expressing mass and momentum in energy units and using c D

1, one has

D p

C m

cD1

! p

C m

: (3.22)

Energy and momentum. In chemical reactions, the involved energies are of the

order of a few eV/atom. In nuclear reactions, the energies are of the order of the

binding energy of a nucleon inside a nucleus, that is, of the order of a few MeV.

In high energy physics, particles used as projectiles have an associated wavelength

given by the de Broglie equation. This wavelength can be smaller than the proton

(or neutron) size, that is,  D h=p < 1 fm. If  D 1 fm D 10

15

p D



.4:136  10

21

MeV s/.3  10

cm=s/

13

D 1:24 GeV=c: (3.23)

D 1:6726  10

27

kg, c D 2:9979 10

m/s.

3.6 Bubble Chambers in Charged Particle Beams 63

300

240

121

146

180

210

270

ELECTRON

330 MeV/c

PROTON

420

380

340

310

280

250

220

Fig. 3.9 Simulation of a particle trajectory (track) in a hydrogen bubble chamber inside a 2 T

magnetic ﬁeld due to (a) a 330 MeV/c electron; (b) a 470 MeV/c proton (dark track in the bottom

part). (c) Deﬁnition of sagitta s and curvature radius r

Practical considerations. From formula (3.20b), a particle with 1 GeV/c momen-

tum in a 2T magnetic ﬁeld describes a circumference having 1.67m radius. If the

trajectory is measured for a path AB D 50 cm (see Fig. 3.9c), it corresponds to a

sagitta

of length s ' AB

=8R D 50

=.8  167/ D 2 cm, which can be easily

measured. For a particle with 10 GeV/c momentum, the sagitta becomes 2 mm,

which is more difﬁcult to measure. For didactic purposes, it is therefore better

to use photos with particles having momenta of the order of 1GeV/c or smaller.

In addition, it is important to choose photos where the trajectories are in a plane

perpendicular to the optical axis of the ﬂash-camera system, with such an axis

parallel to the magnetic ﬁeld. Otherwise, we should make a tridimensional spatial

reconstruction using at least two photos taken with different cameras. In the general

case, a particle trajectory is a helix (having the direction of the magnetic ﬁeld

as an axis), not a circumference. One should also take into account that usually,

For an arc of circumference of radius R and a chord length y, the sagitta length s is given by

R D

64 3 Particle Accelerators and Particle Detection

the photos are not of natural size; there is an enlargement factor g (generally

smaller than one). The curvature radius R is determined through the measurement

of many points along the trajectory which is then reconstructed using computer

programs.

Energy loss. A fast charged particle constantly interacts through the Coulomb

interaction with the atoms of the crossed medium ionizing and exciting them.

Energy loss through ionization per path length unit depends on the atomic number

Z of the crossed medium and on the square of the electric charge of the fast particle

(2.10). At velocities much smaller than c, the dependence on the velocity in the

Bethe–Block formula is of the type 1=v

; at larger velocities, the curve reaches

a minimum followed by a slight relativistic rise. Such an energy loss is small in

comparison to the kinetic energy of the fast particle. As a result of the energy loss,

positive and negative ions are produced through ionization in the crossed medium

along the trajectory of the particle. Around these ions, little bubbles may form and

can be photographed. A relativistic charged particle produces 5–10 bubbles per

centimeter in a bubble chamber; a slower particle loses more energy, producing

a “black” track which may even stop in the bubble chamber. An electron always

has a velocity close to that of light and loses little energy through ionization (thus

producing a small number of bubbles per centimeter); however, it loses energy

through radiation in a discontinuous way. Consequently, the radius of curvature of

an electron track quickly decreases. The number of bubbles per cm of track are only

roughly measured. However, it is possible to obtain estimates on the mass of heavy

particles measuring the curvature radius and the number of bubbles per cm. In a

liquid hydrogen bubble chamber, the energy loss is dE=dx ' 0:27 MeV/cm, that

is, about 4 MeV cm

/g; in a material different from hydrogen, it is about 2 MeV

/g. Let us now analyze a few photos, beginning with the simplest cases.

3.6.2 The Electron “Spiral”

The photo of Fig. 3.10 shows a series of bubbles (the so-called “track”) produced

by an electron in a bubble chamber: it is a typical track with a spiral form. Tracks

produced by electrons are easily recognizable because no other particle can leave a

track with a small number of bubbles per cm describing a circumference with a small

radius of curvature. The small number of bubbles indicates that the electron velocity

is high, very close to the speed of light in vacuum. The small radius of curvature of

the trajectory tells us that the particle rest mass is very small. The electron constantly

loses energy; therefore, its energy and momentum, and consequently the radius of

curvature continuously decreases. Figure 3.9a shows theoretical tracks calculated

by only using the energy loss through ionization for a proton with a momentum of

470 MeV/c, and for an electron with a 330 MeV/c momentum in a bubble chamber

immersed in a 2 T magnetic ﬁeld. Notice that the proton (which produces a track

with a large number of bubbles per cm, a “black” track) stops in the chamber.

3.6 Bubble Chambers in Charged Particle Beams 65

Fig. 3.10 Spiral track produced by an electron in a bubble chamber. The chamber is immersed in

a magnetic ﬁeld B D 0:12 T whose direction is perpendicular to and going out of the sheet plane.

The moving electron is subject to a centripetal force, constraining it to describe a circumference

whose radius of curvature is related to the electron momentum. In this picture, the initial electron

momentum (p,inMeV/c)isp D 3:6 R with R expressed in cm. The energy loss decreases the

curvature radius of the trajectory which becomes a spiral (Photo: Harvard project, Elementary

Particles)

The trajectory of the electron in Fig. 3.10 is different from that of Fig. 3.9;the

electron loses energy in a discontinuousway also through bremsstrahlung; therefore,

it loses more energy than predicted in Fig. 3.9. In conclusion, we can state that the

track in Fig. 3.10 is surely due to an electron.

3.6.3 Electron-Positron Pair

In Fig. 3.11, besides the beam particle tracks, two spiral tracks starting from the

same point are visible. The top one on the right is due to an electron as in Fig. 3.10.

However, another spiral that rotates in opposite direction is also present. Accurate

measurements establish that this spiral is due to the motion of a particle with a

mass exactly equal to that of the electron and with the same electric charge but of

opposite sign: it is the track produced by a positron. The presence of the two spirals

with origin in a single vertex indicates that the following reaction took place:

 C nucleus ! e



C nucleus: (3.24)

An e



pair was created by a high energy photon in the Coulomb ﬁeld of a

nucleus. In a bubble chamber, neutral particles such as the photon are not visible and

the recoil nucleus travels too small of a distance to be observed. For high energy ,

the pair creation process dominates over the photoelectric and Compton effects.