C.R. Prior, Rutherford Appleton Laboratory
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
CERN Accelerator School
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C.R. Prior, Rutherford Appleton Laboratory
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
particles
CERN Accelerator School
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C.R. Prior, Rutherford Appleton Laboratory
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, xray 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.
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C.R. Prior, Rutherford Appleton Laboratory
Basic Concepts
Energy of a relativistic particle
E  m0 g c
1
g 
1 v
2
where c=speed of light = 3x108 m/s
is the “relativistic g-factor”
2
c2
E  m0 c  m0 v
2
For v<<c,
Rest energy
Relativistic kinetic energy
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1
2
2
Classical kinetic energy
T  m0 c g  1
2
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Basic Concepts
C.R. Prior, Rutherford Appleton Laboratory
p  m0g v
Momentum
Connection between E and p:
E
so E  pc
2
 p m c
2
c
2
2 2
0
for ultra-relativistic particles
Units: 1eV = 1.6021 x 10-19 joule.
E[eV] = m0gc2/e
where e=electron charge
Rest energies:
Electron E0 = 511 keV
Proton
E0 = 938 MeV
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Basic Concepts
Velocity v. Kinetic Energy
At low energies,
Newtonian
mechanics may be
used; relativistic
formulae necessary
at high energies
High energies:
v  c 1
E02
E2
Low energies:
E E0+½m0v2
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Basic Concepts
C.R. Prior, Rutherford Appleton Laboratory
Equation of motion and Lorentz force

  
dp
 q EvB
dt


Acceleration in direction of constant E-field
dE
c  
q
pE
dt
E
2
Constant energy and spiralling about
constant B-field.
2
2
qBc
v
p
m0gv

Frequency  
 qvB  Radius  
qB
E


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Methods of Acceleration
Linacs
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
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Fermilab linac
C.R. Prior, Rutherford Appleton Laboratory
Methods of Acceleration
Cyclotron
Use magnetic fields to force particles to pass
through accelerating cells periodically
•Constant B, constant accelerating
frequency f.
B
•Spiral trajectories
• For synchronism
~
R.F. electric field
p

qB
qBc 2 v


E

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f  n
=> approximately constant energy, so
only possible at low velocities g~1.
Use for heavy particles (protons,
deuterons, -particles)
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C.R. Prior, Rutherford Appleton Laboratory
Methods of Acceleration
Isochronous Cyclotron
Higher energies => relativistic effects
p
=>  no longer constant.

qB
Particles get out of phase with accelerating
qBc 2 v fields; eventually no overall acceleration.


E
 Solution: vary B to compensate and keep f
constant.
Thomas (1938): need both radial (because 
f  n
varies) and azimuthal B-field variation for
stable orbits.
Construction difficulties.
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C.R. Prior, Rutherford Appleton Laboratory
Methods of Acceleration
Synchro-cyclotron
p

qB
Modulate frequency f of accelerating structure
instead.
McMillan & Veksler (1945): oscillations are
stable.
qBc 2 v Betatron (Kerst 1941)


E
 Particles accelerated by rotational electric
field generated by time varying
 B:

f  n
B
E 
t
Theory of “betatron” oscillations.
Overtaken by development of synchrotron.
CERN Accelerator School
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C.R. Prior, Rutherford Appleton Laboratory
Methods of Acceleration
Synchrotron
Principle of frequency modulation but in addition
variation in time of B-field to match increase in energy
and keep revolution radius constant.
qBc 2
v
p
 


f  n
qB
E

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
CERN Accelerator School
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C.R. Prior, Rutherford Appleton Laboratory
Methods of Acceleration
Example of variation of
parameters with time in a
synchrotron:
B
E

Important types of Synchrotron:
(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.
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C.R. Prior, Rutherford Appleton Laboratory
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
p
beam of particles.
 
qB
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.
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C.R. Prior, Rutherford Appleton Laboratory
Confinement
ISIS dipole (RAL)
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Confinement
C.R. Prior, Rutherford Appleton Laboratory
Hence mean radius of
ring
R>
Dipoles
Injection
e.g. CERN SPS
R.F.
R=1100m, =225m
Collimation
Can also have large
machines with a large
number of dipoles each
of small bending angle.
Focusing elements
e.g. CERN SPS
Extraction
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Dipoles
744 magnets, 6.26m
long, angle =0.48o
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Acceleration
C.R. Prior, Rutherford Appleton Laboratory
Acceleration
A positive charge crossing uniform E-fields is
accelerated according to
dp
 qE
dt
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
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Acceleration
C.R. Prior, Rutherford Appleton Laboratory
2 R L

v
c
1
c
  

L

Revolution period
Revolution frequency
when vc.
Hz.
If several bunches in machine, introduce RF cavities in
straight sections with oscillating fields
h is the harmonic number.
hc
 rf  h 
L
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
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p
B 
q
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C.R. Prior, Rutherford Appleton Laboratory
Acceleration
Bunch passing cavity: centre of bunch called the
synchronous particle.
Particles see voltage V0 sin 2rf t  V0 sin  t 
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
Bunching effect
vice versa.
To accelerate, make 0 < s<  so that synchronous
E  qV0 sin  s
particle gains energy
CERN Accelerator School
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C.R. Prior, Rutherford Appleton Laboratory
Acceleration

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”.
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C.R. Prior, Rutherford Appleton Laboratory
Focusing
Weak Focusing
Particles 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:
m0 g v 2
i.e. horizontal restoring force
 qBv if r  
is towards the design orbit.
r
Vertical stability requires negative field gradient.
Overall
stability:
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 dB
0n
1
B d
Weak focusing
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Focusing
C.R. Prior, Rutherford Appleton Laboratory
Strong Focusing
Alternating 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.
A succession of opposed
elements enable particles to
follow stable trajectories,
making small oscillations
about the design orbit.
Technological limits on
magnets are high.
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C.R. Prior, Rutherford Appleton Laboratory
Focusing
Fermilab quadrupole
Sextupoles are used to
correct longitudinal
momentum errors.
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Focusing
C.R. Prior, Rutherford Appleton Laboratory
Thin lens analogy of AG focusing:
x 
x
 1
 x
     1
 x   out  f
x
f
f
Drift: x  x  x l , x  unchanged
F-drift-D system:
1
1

f
0  x 
  1   B   l
1  x   f  B 


  in
 x
 1 l  x 
   
 
 x   out  0 1 x   in
0  1 l  1

 1

1  0 1 
 f

l
l
0   1  f

1   l 1 
  f 2


l
f 
 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.
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C.R. Prior, Rutherford Appleton Laboratory
Matched beam oscillations
in simple FODO cell
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Focusing
Matched beam oscillations
in a proton driver for a
neutrino factory
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Focusing
C.R. Prior, Rutherford Appleton Laboratory
Typical example of ring
design:
• basic lattice
• beam envelopes
• phase advances
• phase space nonlinearities
• Poincaré maps
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Synchrotron Radiation
C.R. Prior, Rutherford Appleton Laboratory
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
E GeV  
6.034  10
 m
18
 EGeV 

 m GeV / c 2
 0






4
• Proton mass : electron mass = 1836. For the same energy
and radius,
13
Ee : E p  10
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C.R. Prior, Rutherford Appleton Laboratory
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:
Emax GeV   10  m Emax 
1/ 4
• Better to have larger accelerator for same power from RF cavities
at high energies.
e.g. LEP 50 GeV electrons,  = 3.1 km,
circumference = 27 km.
Energy loss per turn = 0.18 GeV per particle
• To reach twice LEP energy with same cavities would require a
machine 16 times as large.
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C.R. Prior, Rutherford Appleton Laboratory
Synchrotron Radiation
• Radiation within a light cone of angle
1
511
 
for speeds close to c
g EkeV
• 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.
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C.R. Prior, Rutherford Appleton Laboratory
Luminosity
• 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
N2
• Luminosity L  f
A
Area, A
• CERN and Fermilab p  p colliders
have L ~ 1030 cm-2s-1. SSC was
aiming for L ~ 1033 cm-2s-1
CERN Accelerator School
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C.R. Prior, Rutherford Appleton Laboratory
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
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