The Physics of Accelerators C.R. Prior Rutherford Appleton Laboratory and Trinity College, Oxford...

The Physics of Accelerators

C.R. Prior

Rutherford Appleton Laboratory

and Trinity College, Oxford

CERN Accelerator School Loutraki, Greece, Oct 2000

C.R. Prior, Rutherford Appleton Laboratory

Contents• Basic concepts in the study of Particle

Accelerators (including relativistic effects)

• Methods of Acceleration– linacs and rings

• Controlling the beam– confinement, acceleration, focusing– animations– synchrotron radiation



Introduction• Basic knowledge for the study of particle

beams:– applications of relativistic particle dynamics– classical theory of electromagnetism (Maxwell’s

equations)

• More advanced studies require– Hamiltonian mechanics– optical concepts– quantum scattering theory, radiation by charged

particlesCERN Accelerator School Loutraki, Greece, Oct 2000




Applications of Accelerators• Based on

– possibility of directing beams to hit specific targets

– production of thin beams of synchrotron light

• Bombardment of targets used to obtain new materials with different chemical, physical and mechanical properties

• Synchrotron radiation covers spectroscopy, X-ray diffraction, x-

ray microscopy, crystallography of proteins. Techniques used to manufacture products for aeronautics, medicine, pharmacology, steel production, chemical, car, oil and space industries.

• In medicine, beams are used for Positron Emission Tomography (PET), therapy of tumours, and for surgery.

Basic ConceptsEnergy of a relativistic particle

where c=speed of light = 3x108

m/s

is the “relativistic -factor”

For v<<c,

20 cmE

2

21

1

cv

202

120 vmcmE

Rest energy Classical kinetic energy



Relativistic kinetic energy 120 cmT

MomentumConnection between E and p:

so for ultra-relativistic particlesUnits: 1eV = 1.6021 x 10-19 joule.

E[eV] = m0c2/e where e=electron charge

Rest energies: Electron E0 = 511 keV

Proton E0 = 938 MeV

vmp 0

220

22

2cmp

cE


C.R. Prior, Rutherford Appleton Laboratory Basic Concepts

pcE

Velocity v. Kinetic Energy

Low energies:

E E0+½m0v2

High energies:

2

201

EEcv



At low energies, Newtonian mechanics may be used; relativistic formulae necessary at high energies

Equation of motion and Lorentz force

Acceleration in direction of constant E-field

Constant energy and spiralling about constant B-field.

BvEqdt

pd

EpE

cq

dt

dE

2

qB

p



v

E

qBc

2

Radius FrequencyqvBvm

2

0

Methods of AccelerationLinacsLinacs

Vacuum chamber with one or more DC

accelerating structures with E-field aligned in

direction of motion.– Avoids expensive magnets– no loss of energy from synchrotron radiation– but many structures, limited energy gain/metre– large energy increase requires long accelerator



Fermilab linac

CyclotronCyclotron

Use magnetic fields to force particles to pass

through accelerating cells periodically

R.F. electric field

•Constant B, constant accelerating frequency f.

•Spiral trajectories

• For synchronism

=> approximately constant energy, so only possible at low velocities ~1.

Use for heavy particles (protons, deuterons, -particles)


C.R. Prior, Rutherford Appleton Laboratory Methods of Acceleration

~

B

nf

qB

p

v

E

qBc

2

Isochronous CyclotronIsochronous CyclotronHigher energies => relativistic effects

=> no longer constant.

Particles get out of phase with accelerating

fields; eventually no overall acceleration.

Solution: vary B to compensate and keep f

constant.

Thomas (1938): need both radial (because varies) and azimuthal B-field variation for stable orbits.

Construction difficulties.



v

E

qBc

2

nf

qB

p

Synchro-cyclotronSynchro-cyclotronModulate frequency f of accelerating structure instead.

McMillan & Veksler (1945): oscillations are stable.

Betatron (Kerst 1941)Betatron (Kerst 1941)Particles accelerated by rotational electric field generated by time varying B:

Theory of “betatron” oscillations.Overtaken by development of synchrotron.

t

BE



qB

p

v

E

qBc

2

nf

SynchrotronSynchrotron

Principle of frequency modulation but in addition variation in time of B-field to match increase in energy and keep revolution radius constant.

Magnetic field produced by several dipoles, increases linearly with momentum. At high energies:

B= p/q E/qc

E[GeV]=0.29979 B[T] [m]

per unit charge. Limitations of magnetic fields => high energies only at large radius

e.g. LHC E = 8 TeV, B = 10 T, = 2.7 km



qB

p

v

E

qBc

2

nf

(i) storage rings: accumulate particles and keep circulating for long periods; used for high intensity beams to inject into more powerful machines or synchrotron radiation factories.

(ii) colliders: two beams circulating in opposite directions, made to intersect; maximises energy in centre of mass frame.



Example of variation of parameters with time in a synchrotron:

Important types of Synchrotron:

B

E



Confinement, Acceleration and Focusing of Particles

Confinement

By increasing E (hence p) and B together, possible to maintain cyclotron at constant radius and accelerate a beam of particles.

In a synchrotron, confining magnetic field comes from a system of several magnetic dipoles forming a closed arc. Dipoles are mounted apart, separated by straight sections/vacuum chambers including equipment for focusing, acceleration, injection, extraction, experimental areas, vacuum pumps.

qB

p


C.R. Prior, Rutherford Appleton Laboratory Confinement

ISIS dipole (RAL)


C.R. Prior, Rutherford Appleton Laboratory Confinement

Hence mean radius of ring R>

e.g. CERN SPS

R=1100m, =225m

Can also have large machines with a large number of dipoles each of small bending angle.

e.g. CERN SPS

744 magnets, 6.26m long, angle =0.48o

Injection

Extraction

Collimation

R.F.

Dipoles

Focusing elements

Dipoles


C.R. Prior, Rutherford Appleton Laboratory Acceleration

Acceleration

A positive charge crossing uniform E-fields is

accelerated according to Eqdt

pd

Simple model of an RF cavity: uniform field between parallel plates of a condenser containing small holes to allow for the passage of the beam

Revolution period when vc.

Revolution frequency Hz.

If several bunches in machine, introduce RF cavities in

straight sections with oscillating fields

h is the harmonic number.

Energy increase E when particles pass RF cavities can

increase energy only so far as can increase B-field in dipoles

to keep constant .

Magnetic rigidity



c

L

v

R

2

L

c

1

L

hchrf

q

pB

Bunch passing cavity: centre of bunch called the synchronous particle.

Particles see voltage

For synchronous particle s = 0 (no acceleration)

Particles arriving early see < 0

Particles arriving late see > 0

energy of those in advance is decreased and vice versa.

To accelerate, make 0 < s< so that synchronous particle gains energy CERN Accelerator School Loutraki, Greece, Oct 2000


tVtV rf sin2sin 00

sqVE sin0

Bunching effect



Not all particles are stable. There is a limit to the stable region (the separatrix or “bucket”) and, at high intensity, it is important to design the machine so that all particles are confined within this region and are “trapped”.

Weak FocusingParticles injected horizontally into a uniform magnetic field follow a circular orbit.

Misalignment errors, difficulty in perfect injection cause particles to drift vertically and radially and to hit walls.

severe limitations to a machine.

Require some kind of stability mechanism:

Vertical stability requires negative field gradient.



rifqBvr

vm 20 i.e. horizontal restoring force

is towards the design orbit.

10

d

dB

Bn Weak focusing

Overall stability:

Focusing

Strong FocusingAlternating gradient (AG) principle (1950’s)A sequence of focusing-defocusing fields provides a

stronger net focusing force.Quadrupoles focus horizontally, defocus vertically or vice

versa. Forces are proportional to displacement from axis.

Focusing



A succession of opposed elements enable particles to follow stable trajectories, making small oscillations about the design orbit.

Technological limits on magnets are high.


C.R. Prior, Rutherford Appleton Laboratory Focusing

Fermilab quadrupole

Sextupoles are used to correct longitudinal momentum errors.



Thin lens analogy of AG focusing:

x

finout

x

x

fx

x

1

101

F-drift-D system:

f

l

f

ll

f

l

f

fl

1

1

11

01

10

11

101

2

Thin lens of focal length f 2/l, focusing overall, if l f. Same for D-drift-F (f -f ), so system of AG lenses can focus in both planes simultaneously.

lB

B

f

1

f

xx

Drift: unchanged, xlxxx inout

x

xl

x

x

10

1

Focusing



Matched beam oscillations in simple FODO cell

Matched beam oscillations in a proton driver for a

neutrino factory



Typical example of ring design:

• basic lattice

• beam envelopes

• phase advances

• phase space non-linearities

• Poincaré maps

Electrons and Synchrotron Radiation

• Particles radiate when they are accelerated, so charged particles crossing the magnetic dipoles of a lattice in a ring (centrifugal acceleration) emit radiation in a direction tangential to their trajectory.

• After one turn of a circular accelerator, total energy loss by synchrotron radiation is

• Proton mass : electron mass = 1836. For the same energy and radius,



4

20

18

/GeVm

10034.6GeV

cm

GeVEE

1310: pe EE

Synchrotron Radiation



• In electron machines, strong dependence of radiated energy on E.

• Losses must be compensated by cavities

• Technological limit on maximum energy a cavity can deliver upper band for electron energy in an accelerator:

• Better to have larger accelerator for same power from RF cavities at high energies.

• To reach twice LEP energy with same cavities would require a machine 16 times as large.

Synchrotron Radiation

4/1maxmax m10GeV EE

e.g. LEP 50 GeV electrons, = 3.1 km, circumference = 27 km.

Energy loss per turn = 0.18 GeV per particle

• Radiation within a light cone of angle

• For electrons in the range 90 MeV to 1 GeV, is in the range 10-4 - 10-5 degs.

• Such collimated beams can be directed with high precision to a target - many applications.


C.R. Prior, Rutherford Appleton Laboratory Synchrotron Radiation

cE

toclosespeedsforkeV

5111

Luminosity



Area, A

• Measures interaction rate per unit cross section - an important concept for colliders.

• Simple model: Two cylindrical bunches of area A. Any particle in one bunch sees a fraction N/A of the other bunch. (=interaction cross section). Number of interactions between the two bunches is N2 /A.

Interaction rate is R=f N2 /A, and• Luminosity

• CERN and Fermilab colliders

have L ~ 1030 cm-2s-1. SSC was

aiming for L ~ 1033 cm-2s-1

A

Nf

2

L

pp

Reading• E.J.N. Wilson: Introduction to Accelerators• S.Y. Lee: Accelerator Physics• M. Reiser: Theory and Design of Charged Particle

Beams• D. Edwards & M. Syphers: An Introduction to the

Physics of High Energy Accelerators• M. Conte & W. MacKay: An Introduction to the

Physics of Particle Accelerators• R. Dilao & R. Alves-Pires: Nonlinear Dynamics in

Particle Accelerators• M. Livingston & J. Blewett: Particle Accelerators



The Physics of Accelerators C.R. Prior Rutherford Appleton Laboratory and Trinity College, Oxford...

Documents

Transcript of The Physics of Accelerators C.R. Prior Rutherford Appleton Laboratory and Trinity College, Oxford...