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

Подождите немного. Документ загружается.

46 3 Particle Accelerators and Particle Detection

Table 3.1 Orders of

magnitude of the minimum

energy needed and tools used

to explore distances of the

order of x

x (cm) E Tool

5

2 eV Microscopes

8

2keV X rays

11

2MeV' 4m

 rays

14

2GeV' 2m

Accelerators

16

200 GeV' 2m

W;Z

Accelerators

17

2 TeV Accelerators

Lab.c.m.s.

Fig. 3.1 Illustration of a collision in the center-of-mass (c.m.) system and in the laboratory

system (Lab.)

The uncertainty principle states that position x (with uncertainty x)and

momentum p

(with uncertainty p

) cannot simultaneously be known to a

precision better than xp

'„=2,wherex, p

are root mean squared

errors (standard deviations). A relation for the energy is obtained by multiplying c,

xE '„c=2, which gives numerically, E(MeV) ' 1:973  10

11

(MeV cm)

=2x.cm/. To explore dimensions of the order of x,anenergyofE.MeV/ '

2E.MeV/ '„c=x ' 1:973  10

11

.MeV cm/=x.cm/ is needed. The orders

of magnitude of the minimum energy needed to explore dimensions of the order

of x are summarized in Table 3.1.

The second reason why particle accelerators are built is related to the “creation

of particles.” This is a process of converting energy into mass based on the Einstein

equation E D mc

: the energy available in a collision between two particles can be

transformed into the mass of the created particles. This creation occurs according

to certain rules, obeying precise conservation laws. Typical particle masses are also

listed in Table 3.1. In particular, the mass of 100 GeV approximately corresponds to

the heaviest particle observed thus far (W



and Z

). In March of 2010, LHC

successfully started at a center-of-mass (c.m.) energy of 7 TeV (the world maximum

available c.m. energy); the design luminosity and the maximum energy of 14 TeV

should be reached within 2014.

The creation process allows one to study the characteristics of new and unstable

particles; many particles are indeed unstable, with a lifetime in the range from 10

23

(for the so called “strong resonances”) to 10

s (for the neutrons). Besides, some

stable particles, for example, the neutrinos, are produced through the decays of

unstable particles.

Let us consider a simple reaction in which a new particle is produced:

pp ! pp

: (3.2)

In this reaction, the electric charge and the baryonic number are conserved. We now

consider the same process in the lab system (lab) and in the center-of-mass system

(see Fig. 3.1).

3.1 Why Do We Need Accelerators? 47

3.1.1 The Center-of-Mass (c.m.) System

In the c.m. system, the four-momenta

of the two particles (1 and 2) are denoted as



D .E



; p/, p



D .E



; p/. It should be recalled that a four-vector squared is

a relativistic invariant; in the four-momentum case, the invariant quantity is the rest

mass of the particle (e.g., p



D E



 p

D m

). The squared sum of the two

particles four-momenta is also a relativistic invariant representing the total energy

squared denoted as s; in the c.m. system, and with „Dc D 1, one has

s D .p



C p



D .E



C E



; p p/

; (3.3)

from which one can write

s D E

D .E



C E



/: (3.4)

For a collider (Sect. 3.3) with identical colliding particles, as for the process (3.2),

one has E



D E



and the energy of each particle is equal to the rest mass plus

the c.m. kinetic energy: E



D m

C T

c:m:

. In this case, Eq. 3.4 becomes (m

D m

)

s D E

D 2T

c:m:

C 2m

: (3.5)

The reaction (3.2) is only possible if the energy available in the ﬁnal state is at least

the rest mass of the three present particles, that is,

thr

D .2m

C m



0 /

: (3.6)

It is evident from (3.5) that the minimum kinetic energy (the so-called threshold

energy) needed to form the 

in the c.m. is equal to

thr

D .2m

C m



0 / D 2T

c:m:

thr

C 2m

; (3.7a)

from which

c:m:

thr

D m



0 =2 D 67:5 MeV: (3.7b)

3.1.2 The Laboratory System

We now denote p

D .E

; p

/ and p

D .m

;0/the four-momenta of the particles

in the laboratory system in which particle one is moving and particle two is at rest.

Bold characters are used to indicate vectors.

48 3 Particle Accelerators and Particle Detection

Since a four-vector squared is a relativistic invariant, the total energy calculated

in (3.3) must be equal to that computed in the laboratory system, that is,

s D Œ.E

C m

 .p

C 0/

 D m

C m

C 2E

: (3.8)

Also, in the laboratory system, the energy can be divided into a mass term and a

kinetic energy term, that is, E

D T

lab

. We can write (3.8) as (assuming again

that particles 1 and 2 are protons, m

D m

)

s D .m

C m

C 2T

lab

D 4m

C 2T

lab

: (3.9)

The kinetic energy necessary to form the state (3.2) must be larger than the process

threshold (3.6)

lab

thr



thr

 4m



=2m

: (3.10a)

Using (3.6), we can calculate, in the laboratory system, the kinetic energy of particle

one in order to satisfy the condition on the threshold energy for the production

process of (3.2). The threshold kinetic energy (m

D 938 MeV, m



D 135 MeV) is

lab

thr



C m





4m

i

=2m

D 280 MeV; (3.10b)

which can be compared with the kinetic energy of 67.5 MeV needed in the c.m.

system (see Problem 3.5).

3.1.3 Fixed Target Accelerator and Collider

The above example is not only didactic, but it corresponds to the two experimental

methods used to produce new particles: those with ﬁxed target accelerators in which

a particle is accelerated and the target is at rest in the laboratory system; and those

with colliders, namely, accelerating devices in which both particles (in general,

electrons and positrons, or protons and protons, or protons and antiprotons) collide

with equal momenta of opposite sign.

The most important parameter of an accelerator is its energy: the larger the

energy, the better the possibility is to investigate phenomena at smaller dimensions;

besides, a larger energy allows the production of more massive particles. Another

important parameter is the beam intensity: the larger the number of accelerated par-

ticles, the larger the number of collisions that can be observed, and consequently, the

measurements that can be performed are more precise. Moreover, rarer phenomena

can be searched for.

The construction and operation costs of these large accelerators has become so

high that it can no longer be sustained by a single university or state. Therefore,

large laboratories of international dimensions have been created. A well known

example is the CERN, the European Centre for Elementary Particles Physics near

3.2 Linear and Circular Accelerators 49

Geneva (Switzerland) which has actually become a world laboratory. The number

of accelerators operating at the “frontiers” of high energy and high intensity is

inevitably always growing smaller.

The increase in energy has been obtained not only by increasing the accelerator

dimensions, but also by inventing new acceleration methods. The proton or electron

energies in accelerators have increased by about a factor of ten every 5 years

since 1930. Every “jump” in energy was realized thanks to the application of

some new idea. The electrostatic accelerators allowed one to obtain protons with

energies in the lab system from 1 to 10 MeV, synchrocyclotrons up to 700 MeV,

protonsynchrotrons up to 1,000GeV, and the storage rings allowed one to reach

energies up to 2,000GeV (D2 TeV) in the c.m. system. At LHC, 14 TeV will be

reached. It should be recalled that the energies involved in nuclear physics are of the

order of the MeV, while high energy physics deals with energies of the order of at

least the GeV.

The incident particles are characterized by their energy rather than their velocity

because in high energy physics, the relativistic effects are dominant (remember

E D m

3.2 Linear and Circular Accelerators

Electrically charged particles (usually protons or electrons) are accelerated through

electric and magnetic ﬁelds which may be constant or varying in time and space.

An accelerator is schematically constituted by an ion source,anaccelerating ﬁeld

and a guide ﬁeld that forces the particles to move on well deﬁned orbits. The

acceleration takes place in high vacuum in order to reduce the collisions with

molecules of the residual gas and therefore to minimize energy loss.

The ion source can be schematized as a small cavity at low pressure where a

discharge ionizes in continuation gaseous hydrogen. Electric ﬁelds then carry the

electrons or the protons towards the next accelerator.

Accelerators can be classiﬁed as linear or circular. In linear accelerators,

particles are accelerated in a straight line by electric ﬁelds. In circular accelerators,

a magnetic ﬁeld forces the particles to move on circular orbits; the acceleration is

performed through radio-frequency electric ﬁelds (or through increasing magnetic

ﬁelds). Let us brieﬂy illustrate the principle of accelerator operation.

3.2.1 Linear Accelerators

Low energy proton linear accelerators consist of a set of cylindrical electrodes

connected to either side of an oscillating voltage with constant frequency. As shown

in Fig. 3.2, the protons move from left to right inside a structure made of successive

tubes. Inside the tubes, the protons are in a ﬁeld-free region and therefore move

with a constant velocity while they are accelerated by an electric ﬁeld between two

50 3 Particle Accelerators and Particle Detection

(3)

(2)

(1)

(4)

Fig. 3.2 Layout of a linear proton accelerator. The various components indicated are (1) ion

source, (2) accelerating cylindrical electrodes, (3) radio-frequency and (4) vacuum pipe

neighboring tubes. If the tube length is chosen in such a way that the time needed

to cross it is equal to half the oscillator period, the protons are accelerated a second

time when passing from one tube to the next one, and so forth. The length of the

tubes must therefore increase in order to take into account the increase in proton

velocity.

In the case of an electron accelerator, the electrodes are all of the same

length because the electrons, even of low energy, travel at the speed of light. The

acceleration is performed by the electromagnetic wave traveling along the axis of

the pipe with a phase velocity equal to the electron velocity. Only electrons in phase

with the accelerator ﬁeld (a bunch with a transverse dimension of a few centimeters

and a length of some decimeters) are accelerated. We can imagine that the electron

bunch rides the crest of the electromagnetic wave produced by the radio-frequency

system.

The electron or proton beam produced by an electrostatic accelerator is constant

in time. In linear accelerators (and in others that will be described in the following),

accelerated particles instead arrive in bunches.

3.2.2 Circular Accelerators

Synchrotrons (schematized in Fig. 3.3) have supplanted other types of circular

accelerators such as the cyclotrons and synchrocyclotrons. In synchrotrons, the

use of big and expensive magnets is avoided and particles are accelerated through

electric ﬁelds on an orbit of constant radius. The magnetic ﬁeld necessary to deﬂect

the particles is produced by a set of magnets placed along the circumference. Before

being injected in the orbit, the protons are preaccelerated with an electrostatic

accelerator followed by a linear accelerator. The particles are accelerated through

radio-frequency electric ﬁelds (RF) in resonant cavities placed along the accelerator

circumference; the particles are forced to stay on the same orbit thanks to the

contemporaneous increasing magnetic ﬁeld B. This allows the “bunches” of

particles to move inside a large torus (the synchrotron vacuum chamber).

The electron-synchrotrons nearly have a constant RF since the e



travel at a

constant speed, approaching the light speed in vacuum.

3.2 Linear and Circular Accelerators 51

Fig. 3.3 Layout of a

protonsynchrotron. The

preaccelerator is an

electrostatic accelerator (1)

followed by a linear

accelerator (2). In the main

ring one ﬁnds (3) the

magnets, (4) an accelerating

cavity and (5) the “straight

sections” in which (6) the

targets are installed, from

which (7) the secondary beam

lines are produced for the

experiments

LEP/LHC

SPS

EPA

LIL

–

linacs

p (proton)

ion

(positron)

–

(electron)

Proton ion

linacs

BOOSTER

100 GeV

3.5

0.6

0.2 GeV

Fig. 3.4 Layout of the CERN accelerator complex in use with the e



LEP and pp .E

D 7

TeV) LHC colliders; the maximum energy obtained per beam is indicated for the various

accelerators. (When the accelerator complex is used for the pp LHC, every accelerator energy

per beam is higher). The SPS has also been used as a p

p collider

All accelerators currently in operation with energies larger that the GeV are

synchrotrons (for protons or electrons) or linear accelerators for electrons. These

accelerators can be deﬁned as conventional. More “advanced” accelerators use

superconducting magnets, superconducting RF cavities or new methods of

acceleration. A modern high energy accelerator is made of a succession of accelera-

tors. Every accelerator increases the particle energy by about an order of magnitude

(Fig. 3.4).

52 3 Particle Accelerators and Particle Detection

3.3 Colliders and Luminosity

In storage rings (colliders, accelerators with intersecting beams) beams of particles

circulating in opposite directions are accumulated and then deﬂected in order to

make them collide in one or more well deﬁned interaction points. To obtain high

energies, it is essential to use a circular accelerator (collider). In collisions produced

with storage rings the laboratory system is almost coincident with the center-of-

mass system; therefore, the energy in the c.m. system is equal to the sum of the

energies of the two beams (for beams of particles with identical mass). For example,

in the CERN proton-antiproton collider, collisions between antiprotons of 315 GeV

with protons of 315 GeV were produced. It was equivalent, in terms of collision

energy, to an accelerator of antiprotons of 201,000GeV sent against protons at rest

(veriﬁed using (3.10)). Such an accelerator would be currently impossible to build.

Due to relativistic effects, an even better advantage is obtained when using electrons.

Particle accelerators represent one of the more direct veriﬁcations of the validity of

the special relativity theory: if nonrelativistic formulas were used to design them,

they would not work.

The colliders have the disadvantage, with respect to the ﬁxed target accelerators,

of being less versatile and capable of producing a smaller number of interactions

per unit of time. A collider luminosity L is deﬁned as the number that, multiplied by

the total cross-section of a given process , gives the total number N of collisions

per unit of time, that is,

N D L: (3.11a)

The luminosity can be expressed in terms of the collider parameters. Let us consider

an accelerator with radius R; number of particles per bunch circulating in one

direction (i.e., protons or electrons) n

and number of circulating bunches N

;

number of particles per bunch circulating in the opposite direction (in most cases,

antiprotons or positrons) n

and the number of circulating bunches N

;average

transverse radius of each bunch r. Then,

L D

4r

; (3.11b)

f is the beam revolution frequency f D1=,  is the period  D2R=c. The factor

G (usually 1) takes into account the ﬁnite length of a bunch.

The ﬁrst proton storage ring (up to 31 GeV per beam) was the ISR at CERN

(1970s) where luminosities of 2  10

2

1

have been reached. Since the total

proton-proton cross-section at the ISR energy is 510

26

, about 10

interactions

per second were produced in every interaction region. Most of the events produced

in the interactions were not very interesting (the so-called minimum bias events).

It was therefore needed to select potentially interesting events (as for the experi-

ments at successive, more advanced and more complicated accelerators). This is

mainly done with an electronic logic of the data acquisition system (the trigger).

3.3 Colliders and Luminosity 53

At the CERN and Fermilab proton-antiproton colliders, luminosities of the order

of .6  10/  10

2

1

have been obtained. The Fermilab Tevatron, after some

technical improvements, has reached luminosities about ten times larger. The PEP2

and KEKB e



colliders, in operation since 2000, and the pp LHC collider have

a design luminosity of 10

–10

2

1

3.3.1 Example: the CERN Accelerator Complex

A high energy accelerator is made of a succession of accelerators, each increasing

the accelerated particle energy by roughly an order of magnitude. The SPS complex

at CERN is composed of ﬁve consecutive accelerators. Actually, the CERN system

is a complex of versatile accelerators that can be used for various purposes.

At LEP (Large Electron Positron Collider, Fig. 3.4), e



collisions were

produced at an energy in the c.m. of about 91 GeV, corresponding to the mass of

the Z

boson, one of the weak force mediators; in a second phase (LEP2), the c.m.

energy was raised to 209GeV.

To operate such an accelerator, it was necessary to produce adequate bunches of

electrons and positrons, and to accelerate them at the various requested energies.

In the ﬁrst phase, a linear accelerator (LIL), designed to accelerate electrons and

positrons (up to 200 MeV in its ﬁrst part to produce e

; up to 600 MeV in the

second part) and the “Electron Positron Accumulator,” (EPA) (600 MeV) were built.

In a second phase, the existing accelerators were modiﬁed, for example, the PS and

SPS, in order to make them able to accelerate electrons and positrons in opposite

directions (up to 3.5 GeV for the PS, up to 20 GeV for the SPS). In the third phase,

the new large ring measuring 27 km in circumference was put into operation; at

LEP, usually four“trains,” each consisting of four bunches of electrons and four

positrons were accelerated. After acceleration, the particle bunches were circulating

in the collider for a few hours. The particle bunches were deﬂected in eight regions;

in four regions, they were kept vertically separated; in the other four regions, they

were brought into collisions. Before making them collide, the beams were focused

in order to reduce the transverse dimensions as much as possible in order to obtain

the highest possible luminosity. Large detectors were installed around each of the

four interaction points, allowing for the detailed studies of the positron-electron

collisions (Chap. 9).

To reduce the energy loss through bremsstrahlung, the LEP magnetic ﬁeld was

very low, 0:2–0:4 Tesla. Such a ﬁeld was obtained with low cost special magnets

made of iron and concrete. Normal accelerating radio-frequencies were used in the

ﬁrst phase, while superconducting RF cavities were used in the LEP2 phase which

reached c.m. energies of 130–209GeV.

Starting in 2000, the LEP was dismantled in order to use the tunnel for the LHC

(Large Hadron Collider) installation. The LHC is designed to study pp and nucleus-

nucleus collisions. The LHC uses superconducting magnets with a magnetic ﬁeld

of 8.3 T. For the time being, protons are accelerated up to 3.5 TeV with a total c.m.

54 3 Particle Accelerators and Particle Detection

energy of 7 TeV. In the future, when a completely safe operation will be insured, the

LHC will accelerate protons up to the design value of 7 TeV, reaching an energy in

the c.m. of 14 TeV. The luminosity should reach values up to 10

2

1

.The

main reason for such a collider is to “understand the origin of the particle masses”

(Chap.11), that is, the search for the Higgs boson and possibly other new particles.

Besides, at the LHC, heavy ions can be accelerated (Pb C Pb collisions) with the

main purpose of searching for and studying quark-gluon plasma.

High energy physics has contributed by bringing together scientists from around

the world. Even during the Cold War, scientists from both sides collaborated in

European, Russian and American research centers. As already mentioned, this

type of research is today extremely expensive, and no country could afford the

management of completely national laboratories or experiments. Besides the CERN

European laboratory, other large accelerator complexes are:

– The Fermilab accelerators in Batavia, Illinois, USA. In particular, the Tevatron

is a protonsynchrotron with superconducting magnets which accelerates protons

and antiprotons up to 1 TeV (hence the name)

– The SLAC accelerator complex in Stanford, California. The main accelerator is

a 2 miles (about 3.2 km) long linear accelerator that can accelerate e

and e



beams up to 50 GeV. The laboratory is now converting the original linear

accelerator to a light source facility: the Linac Coherent Light Source (LCLS) is

a free electron laser facility which can deliver extremely intense X-ray radiation

for research in a number of areas. It achieved the ﬁrst lasing in April 2009

– The Brookhaven National Laboratory on Long Island near New York

– The DESY laboratory in Hamburg where a new light source facility, the European

X-ray free electron laser (European XFEL), is being built

– The Institute for High Energy Physics (IHEP) in Protvino, Serpukhov, Moscow

region

– The Japanese laboratory KEK

– The INFN National Frascati Laboratory, close to Rome, Italy

3.4 Conversion of Energy into Mass

In a collision of a high energy proton with a proton at rest, a fraction of the available

energy may be transformed into mass (Sect. 3.1); new particles can therefore be

produced, most of them being unstable. In every reaction, besides the electric

charge, other quantum numbers (which will be described later), for example, the

baryonic number, the strong isospin, the leptonic numbers, etc., must be conserved.

See, for instance, Problems 3.6 and 3.7, for the discovery of the antiproton.

Examples of reactions due to the strong interaction are:

pp ! pp





pn ! pp



C neutral particles

pp ! pp

pp C neutral particles; etc:

(3.12)

3.4 Conversion of Energy into Mass 55

Similar reactions due to the electromagnetic interaction, followed by the strong

one, are:



p ! e



p









p ! e



n

n ! 



! 







;ecc:

(3.13)

For the weak interaction, one has, for example, 



p ! 



p

3.4.1 Use of Fixed Target Accelerators

We shall now describe how a proton beam may be used in a ﬁxed target experiment.

At the end of the acceleration cycle, the accelerated protons are extracted from

the accelerator: they constitute the primary beam which is sent onto a target, for

example, a beryllium cylinder (see Fig. 3.5). In the collision of a proton with a

beryllium nucleus, various types of particles are produced, some of them with a

short or very short lifetime. Secondary beams can be produced with long lifetime

particles emitted at a certain angle, as illustrated in Fig. 3.5b. The target acts as a

“source;” magnetic quadrupoles act as focusing magnetic lenses. Magnetic dipoles

have the same function as optical prisms, separating by “color” (in momentum),

thus allowing one to select monoenergetic particle beams.

With 450 GeV incident protons, about ten charged and ﬁve neutral particles are

produced on average in one collision. Because of relativistic effects, most of these

particles are preferably emitted in a small cone in the forward direction and each

transports a fraction of the incident proton momentum. With protons of 25–30GeV

(as those coming from the PS of CERN and from the AGS of Brookhaven), the

mean number of charged particles produced (the mean charged multiplicity)ismuch

lower. Such particles have an average momentum of the order of a few GeV and are

produced in a larger angular cone.

A set of “collimators” is used to deﬁne the solid angle and the beam monochro-

maticity. Such a beam is monoenergetic, though it contains particles of different

mass (e.g., electrons, mesons, protons, etc.). It is possible to obtain beams containing

a single type of particle performing a separation in mass on the monoenergetic beam

using an electrostatic separator, that is, an apparatus that produces an electric ﬁeld

besides the magnetic one.

It is important that secondary particle beams have high intensity and relatively

long temporal duration (a few seconds for every acceleration cycle) to be proﬁtably

used in experiments with counters and electronic devices. When these secondary

beams were used with bubble chambers, they had to instead have low intensity

(10–20 particles) and brief temporal duration (maximum of a few milliseconds).

The lifetime of the particles of the secondary beams must be sufﬁciently long in

order to travel tens or hundreds of meters.