Outline
Physics of particle
accelerators
1.
2.
3.
4.
5.
6.
IDPASC School
Basic principles of particle accelerators
Several ways of accelerating particles
Accelerator zoology
Colliders
Some more complex topics
Conclusions
LPNHE, Paris, 8-21 February, 2015
Patrick Puzo
Laboratoire de l’Accélérateur Linéaire, Université Paris-Sud
puzo@lal.in2p3.fr
P. Puzo
IDPASC School, February 2015
2
References


I assume you have prerequisite on classical electrodynamics and
special relativity
So:
Total Energy = γ m c2
107
Protons

105
An introduction to the Physics of Particle Accelerators, World
Scientific Publishing, M. Konte & W. Mackay
Accelerator Physics, World Scientific Publishing, S. Lee
104
103

2
Momentum = γ m β c ≈ γ m c
Particle Accelerator Physics, Springer, H. Wiedemann
Electrons
106
102
Kinetic Energy = (γ −1) m c
€


We will always deal with
ultra-relativistic particles
Lorentz
Facteurfactor
de Lorentz

Cern Accelerator Schools (specialized) and Cern Summer
Student Lectures (introduction)
10
1
1
10
102
103
104
105
106
107
Energie totale (MeV)
Total energy (MeV)

€
Some pictures will be taken from a book on Accelerators I am
currently writing with André Tkatchenko (in French)
€
P. Puzo
IDPASC School, February 2015
3
A bit of vocabulary

P. Puzo
IDPASC School, February 2015
4
Outline
Accelerators/colliders
Past
•  CERN (Geneva): Sp pS (p, p ), LEP (e+, e-)
•  SLAC (Stanford): PEP (e+, e-), SLC (e+, e-), PEP II (e+, e-)
•  DESY (Hamburg): HERA (e, p)
€
€
•  Fermilab (Chicago): Tevatron (p, p)
•  KEK (Tsukuba): KEK-B (e+, e-)
  Present
€ Au)
•  BNL (New York): RHIC (Au,
•  GANIL (Caen): SPIRAL (ions)
•  CERN: LHC (p, p)
  (Far) future
•  ILC (e+, e-)
1.  Basic principles of particle accelerators

P. Puzo
IDPASC School, February 2015
2.
3.
4.
5.
6.
5
Several ways of accelerating particles
Accelerator zoology
Colliders
Some more complex topics
Conclusions
P. Puzo
IDPASC School, February 2015
6
1
Lorentz force : action of E field
Driving equations in accelerator physics

Particle accelerators use electromagnetism and special relativity
Maxwell equations

  ρ
∇.E =
ε0

 
∂B
∇×E = −
∂t
 
∇.B= 0

 
 1 ∂E
∇ × B = µ0 J + 2
c ∂t
€


∫ F .dt
The kinetic energy gain is:
 
ΔEkin = ∫ F €
. ds

F = q ( E + v × B)
Lorentz force
Electric force
Magnetic force
 
with ds = v dt

  
and F = q ( E + v × B)
 
 
dEkin
⇒ ΔEkin = q ∫ E . ds ⇒
= q E .v
dt
 
 
⇒ Ekin = Cste if E = 0 or E ⊥ v
€
€
To produce high energy photons or neutrons, one needs first to
accelerate charged particles and create neutral particles via
nuclear reaction. For instance: 9 Be(d, n)10B or e+ + e − → q + q + γ
  Neutral beam cannot be well collimated
€ School, February 2015
IDPASC
P. Puzo
€

7


 
dp d(γ m v )
=
= qv×B
dt
dt
Trajectory is curved in B field:
  Projecting: (ρ: local radius)
dv
=0
dt
γm
€
P. Puzo

 No speed
€ variation
v2
p
=qvB ⇒ Bρ=
ρ
q

Magnetic rigidity

Numerically (for a quantum charge):

Particles turn around B field at cyclotron frequency
€
Classical case
ωc =
P. Puzo
qB
m
Accelerating
a charged particle is always done by an electric
€
field
8
IDPASC School, February 2015
Relative effects of E and B
Lorentz force: action of B field

The momentum variation of a charge under the Lorentz force is:

Δp =
Lorentz: Action of E(t) and B(t) on a charge q (moving at
speed v):






pGeV/c = 0,3 BT ρ m
€
qB
or ωc =
γm

If B and v are perpendicular:

E
Fe
E
=   =
Fm v × B β c B
If E = 107 V/m, then the B field to have the same force is 3 T if
β = 0,01 and 0,03 T if β = 1
€
Relativistic case
9
IDPASC School, February 2015
Limits:
  Emin ≈ 0 and Bmin ≈ BEarth
  Emax in vacuum: ≈ 100 kV/cm or 107 V/m
  Bmax ≈ 2 T (classical magnet) or ≈ 10 T (superconducting
magnet)

At high energy, guiding of high energy particles will only be done
by B fields. E fields are only used at low energy
P. Puzo
10
IDPASC School, February 2015
€
Charged particle optics


Optics is guiding of charged particles. To 1st order (as in
geometrical optics), it is linear
A
x : position
x’ : angle
⎛1 L ⎞
C = ⎜
⎟
⎝0 1 ⎠
⎛ x ⎞ ⎛ e f ⎞ ⎛a b ⎞ ⎛ x0 ⎞
⎜ ⎟ = ⎜
⎟ ⎜
⎟ ⎜ ⎟
⎝ xʹ′ ⎠ ⎝ g h ⎠ ⎝ c d ⎠ ⎝ x0ʹ′ ⎠
B
⎛a b ⎞
A = ⎜
⎟
⎝ c d ⎠
P. Puzo
⎛ e f ⎞
B = ⎜
⎟
⎝ g h ⎠

⎛ x ⎞
⎛ x ⎞
⇒ ⎜ ⎟ = B A ⎜ 0 ⎟
⎝ xʹ′ ⎠
⎝ xʹ′0 ⎠
⎛ x ⎞
⎛ x ⎞
⇒ ⎜ ⎟ = B C A ⎜ 0 ⎟
⎝ xʹ′ ⎠
⎝ xʹ′0 ⎠
An accelerator merges drift spaces and elements where B field
€
€
guides
charged particles:
  Usually dipoles, quadrupoles and sextupoles
€
IDPASC School, February 2015
€
B
L
For two elements:
€
A
€
Between elements, we have drift spaces:
An element is a region where B ≠ 0. For one element:
⎛ x ⎞ ⎛a b ⎞ ⎛ x0 ⎞
⎜ ⎟ = ⎜
⎟ ⎜ ⎟
⎝ xʹ′ ⎠ ⎝ c d ⎠ ⎝ xʹ′0 ⎠


11
P. Puzo
IDPASC School, February 2015
12
€
2
Fringe
field
Dipoles




Orthogonal B field usually is uniform
if pole geometry is under control

1 qB qBc
=
≈
ρ
p γ m c2
A charged particle is bent on a circular
trajectory with radius ρ:
A particle with another momentum
will be more or less deviated:
  Particles with Δp will not be kept
on magnetic axis
p + Δp
€
p

p - Δp
For an angle θ and a radius of curvature ρ :
Matrix for a dipole
P. Puzo

In a ring, all particles come back to the same longitudinal
position s after one turn (but not necessary at the same
transverse position)
If all dipoles have the same B field value (or the same radius of
curvature ρ), the ring is isomagnetic
Be careful of the difference between the
dipole radius of curvature ρ and the ring
circumference R
⎛1 ρ θ ⎞
A = ⎜
⎟
⎝0 1 ⎠
13
IDPASC School, February 2015
€
P. Puzo
14
IDPASC School, February 2015
Yoke
Culasse

g

8000
µ = f(B)
Bobine
d’excitation
Each
coil
parcourues par le
circulates
courant NI/2
μr
max
6000
Be = f(μ)
4000
2000
NI/2
μr >> 1
Be
μr
Vacuumà chamber
Chambre
vide
Classical dipoles are usually
in iron to keep magnetic
field lines
  g is the gap
0
0.4
0.8
1.2
1.6 B (T)

For a C-shape magnet iron

Real example:
to 1st order
Yoke
Culasse
Several types:
  C-shape magnets
  H-shape magnets
  Window magnets
Culasse
Chambre à vide
From theory
to practice
Chambre à vide
Vacuum chamber
IDPASC School, February 2015
LEP
dipole
Bobine d’excitation
parcourues par le
courant NI/2
g
P. Puzo
NI (Ampères-tours)
μr >> 1
15
P. Puzo
16
IDPASC School, February 2015
Quadrupoles

B field proportional to the axis: the further away from the axis,
the higher deviation

A set of two orthogonal quadrupoles focusing in each plane is
globally focusing:
1 1
1
L
=
+
−
f fF f D fF f D
∂B ∂B
Bx = k y and By = k x with k = x = y
∂y
∂x
€



€
Geometry of the poles matters a lot for the field homogeneity

Each quadrupole is focusing in one plan and defocusing in the
other but …
P. Puzo
IDPASC School, February 2015
17
Matrices are:
⎛ 1
AFoc = ⎜
⎝−1/ fF
0 ⎞
⎟ et
1 ⎠
⎛ 1
ADéfoc = ⎜
⎝1/ fD
0 ⎞
⎟
1 ⎠
Convention: a focusing quadrupole is focusing in horizontal (and
defocusing in vertical) and a defocusing quadrupole is defocusing
€
in horizontal (and focusing in vertical)
P. Puzo
IDPASC School, February 2015
18
3
Sextupoles

Field is non linear:

Bx =
∂2 By
∂x 2
x y et By =

B field on€horizontal axis:
Bx =

B field on vertical axis:
€
Bx =

∂2 By
∂x 2
∂2 By
∂x 2
∂2 By
∂x 2
Classical dipole schematics:
z
(x
2
−y
2
)
Bobine
x0 Δy and By =
x
⎞
∂2 By ⎛ x02
⎜ − x0 Δx ⎟
∂x 2 ⎝ 2
⎠
y0 Δx and By = −
⎞
∂2 By ⎛ y02
⎜ − y0 Δy ⎟
∂x 2 ⎝ 2
⎠
From theory
to practice
N
S
S
N

Classical sextupole schematics:
z
A sextupole is equivalent to a dipole combined with a quadrupole.
It creates some coupling between horizontal and vertical planes
S
€
N
N
S
S
From theory
to practice
x
N
P. Puzo
19
IDPASC School, February 2015
Combined function magnets
20
IDPASC School, February 2015
Superconducting magnets
z
z
Z
P. Puzo
I(θ)
Chambre à vide
Vacuum chamber
N
X, x
x
N
d
Combine focusing and
deviation

S
S
12,5 kA
21
IDPASC School, February 2015

Quadrupole (Tm)
Classical magnet
≈2
≈ 20
Superconducting magnet
≈8
≅ 100

Limits: Joule effect or high B threshold for Sc phase
Differences:
  Classical: conductor position does not matter. Quality of the
B field is given by the geometry of the polar pieces
  Superconducting: conductor position gives quality of the field.
The yoke purpose is only to channel the B lines outside and
has no influence on the field in the center
P. Puzo
P. Puzo
IDPASC School, February 2015
IDPASC School, February 2015
22
Large Hadron Collider (LHC)
High field limits:
Dipole (T)

Sextupôle
I(θ) = I0 cos(3θ)
Squew quadrupole:
combines H and V
focusing
Superconducting magnets

Quadrupôle
I(θ) = I0 cos(2θ)
No longer used nowadays on purpose, but can be created by
misalignment ..
   To correct for the coupling, one uses sextupoles or squew
quadrupoles
P. Puzo

Dipôle
I(θ) = I0 cos(θ)
23
p-p collider (E = 7 TeV) built in
the LEP tunnel (2πR = 27 km)
Superconducting magnets
(dipoles: 8,4 T)
  90 tons of liquid helium II
Helium
phase
diagram
(no field)
p (bar)
(S)
30
(L)
He II
10
P. Puzo
He I
C
hg
(V)
0
0

(L)
hs
20
1
2
3
4
5
T (K)
« The largest refrigerator in the world »
IDPASC School, February 2015
24
4
LHC dipole magnets
P. Puzo
25
IDPASC School, February 2015
P. Puzo
26
IDPASC School, February 2015
Source

Simplest ion source
1st part of an accelerator is always a source delivering stable
particles (electrons, protons, ions)
Anode

Simplest electron source
Simplest proton source
H+
Best performances are obtained with more sophisticated
sources using:
  Plasmas
  Chemistry for surfaces
(breakdowns, ..)
  Vacuum technology
  Lasers (photocathodes)
More realistic example
Plasma (obtained
by heating)
P. Puzo

27
IDPASC School, February 2015
Vacuum chambers




NEG coating

Proton
Lead
shield
P. Puzo
Photons emitted by ultra-relativistic electrons in a cone
centered on the trajectory with opening angle θ = 1/γ
θ=
New technology since LEP: distributed pumping using NEG pumps
(Non Evaporable Getter)
Al
€
Energy loss on one turn by ultra-relativistic electrons:
(ΔE )[ keV ] = 88,5
LHC vacuum chamber
IDPASC School, February 2015
29
1
γ
First evidence in 1947 on an electron synchrotron from GE built
by Langmuir
Cooling
LEP vacuum chamber
28
IDPASC School, February 2015
Synchrotron radiation
Vacuum is needed to limit interactions with residual gas (elastic
or inelastic collisions) and allow perfect running of specific
systems (sources, accelerating cavities, ..)
-5 to 10-13 mbar (lower than on the Moon)
  In practice: from 10
Cooling
Ion and p beam properties are fixed after the source
P. Puzo
P. Puzo
4
E [GeV
]
ρ [m]
IDPASC School, February 2015
Radius of
curvature
30
€
5



For e± and protons of the same
Momentum on the same orbit, one has:
e
PRS
⎛ m ⎞4 ⎛ 1 ⎞4
−13
= ⎜⎜ e ⎟⎟ = ⎜
⎟ ≈ 10
⎝1836 ⎠
⎝ m p ⎠
Synchrotron radiation is only visible on electron machines, and
barely on proton machines
€
E (GeV)
e±
protons

p
PRS
50
1,1
LEP 2
108
15,6
7000
0,0073
Synchrotron power can be
very damaging
P. Puzo

PSR (MW)
LEP 1
LHC

31
IDPASC School, February 2015
Power stored in the beam is enormous
Energy (GeV)
P(MW) = E(GeV) × I(mA)
Total current (mA)
Power (GW)
60
60
1000
€
HERA p
920
HERA e
27,5
20
0,55
RHIC
250
0,013
0,003
LHC
7000
500
3500
SOLEIL
2,75
400
1,1
P. Puzo
IDPASC School, February 2015
32
60
1.  Basic principles of particle accelerators
2.  Several ways of accelerating particles
55
3.
4.
5.
6.
33
IDPASC School, February 2015
Static electric field
Accelerator zoology
Colliders
Some more complex topics
Conclusions
P. Puzo
Bertozzi experiment (1964)
ΔL = 8 m
  Evidence of relativistic effects
in the electron gun of a microscope
v2/c 2
Classical theory:
IDPASC School, February 2015
34
Static electric field



Obviously, the synchrotron radiation limitation vanishes (quite
completely) for a linear accelerator
Safety aspects are crucial
P. Puzo

Synchrotron radiation is looked forward in some cases
  Dedicated accelerators to produce synchrotron radiation will
be compact
Outline
Tevatron

Synchrotron radiation is a limit if the energy is the key point
  HEP accelerators will be large
Example of the lead
shielding of the LEP
polarimeter
Beam power



Practical limitation due to breakdowns (order of MV/m)
  Can push limit up to 10 MV/m using SF6, but not more
How to reach higher energies ? Would it be possible to chain
these accelerating cells ?
1.2
1
0.8

Relativistic theory:
γ = 1+
Ec
m c2
0.6
0.2
0
0
€
P. Puzo
Description
relativiste
Relativistic theory Classical theory classique
Description
Experimental
dataerimentales
Donn
ees exp
0.4
1
2
IDPASC School, February 2015
3
4
5
6
7
8
9
10
Ec/mc 2
35
P. Puzo
IDPASC School, February 2015
36
6
Time varying electric field


VRF is the accelerating voltage (time
dependant) across the gap

A particle crossing the gap will receive:
+g /2
ΔE = q ∫ −g
/2 E z (z, t) dz

Time varying magnetic field
Accelerating cavity

What€matters only in the field
when the particle goes through
the cavity (10 cm ≈ 300 ps)
Neglecting B variation on one turn, total force F and
accelerating force Facc are:
⇒ Facc =
Half accelerating cell
P. Puzo
Acceleration via the induced
electric field
37
IDPASC School, February 2015

e r dB
2 dt
Sparsely used
€
P. Puzo
38
IDPASC School, February 2015
Plasma based acceleration




1979: first proposal (Tajima and Dawson)
1983: first acceleration (UCLA and LULI laboratories)
2004: first bunches (RAL and LOA laboratories)

An intense laser beam creates a plasma wave: some electrons can
experience very high gradients
Electric field
v
Electron density


One can reach in principle 10 GV/m (compared to < 50 MV/m in a
classical structure)
P. Puzo

Laser wake field accelerator
  Plasma created by laser pulse
IDPASC School, February 2015
LWFA example:

39
PWFA example (FFTB)
42 GeV electrons in
plasma cell Nature 445 741 15-Feb-2007
v
moving charge
a =3
P. Puzo

a = 10
IDPASC School, February 2015
40
Plasma is probably (part of) the future of high energy
accelerators
  Timescale ?
Co
nv
en
t
ion
al
ac
ce
ler
at
or
s

Plasma wake field accelerator Particule au repos
a = 1.2
  Plasma created by ultra-relativistic
electron beam
v
Filed
line for fast

P. Puzo
Next step: 40 J / 40 fs 
10 GeV
Some electrons
double their
energy in 84 cm
IDPASC School, February 2015
41
P. Puzo
IDPASC School, February 2015
42
7
More exotic scheme

Outline
CLIC project: energy transfer from low energy high intensity
beam through high energy low intensity beam
1.  Basic principles of particle accelerators
2.  Several ways of accelerating particles
3.  Accelerator zoology
4.  Colliders
5.  Some more complex topics
6.  Conclusions

Concept tested since several years. 80 MV/m achieved, but with
damages in the transfer structure. Will it be operational one
day ?
P. Puzo
IDPASC School, February 2015
43
P. Puzo
IDPASC School, February 2015
44
Outline


Repetition rate can be much higher on a circular machines than
on a linear one (50 kHz versus 100 Hz typically)
  Consequences for a collider on beam size
1.  Basic principles of particle accelerators
2.  Several ways of accelerating particles
3.  Accelerator zoology
1.  Circular accelerators
2.  Linear accelerators
Is synchrotron radiation a concern ?
4.  Colliders
5.  Some more complex topics
6.  Conclusions
P. Puzo
IDPASC School, February 2015
45
P. Puzo
IDPASC School, February 2015
46
Cyclotron



Principle due to Ernest Lawrence
(1929)
  Beam is curved by B field in
« Ds » produced by classical
magnets
  Alternative E field in the gap
  Time to travel in one D is
independent of the radius
One could show that:
ω=

1st cyclotron (80 keV)

A modern one: the 590 MeV from PSI (Zurich)
Cavities
qB
γm
Magnets
Used for ions and protons, not
for electrons
€
P. Puzo
IDPASC School, February 2015
47
P. Puzo
IDPASC School, February 2015
48
8
Synchrotron


For years, cyclotrons were the most popular accelerators in the
world (simplicity, cost, compactness, ..)
Synchrotron
€
To go higher than the GeV, something else is needed

TRIUMPH cyclotron
(used for heavy ion
research)

P. Puzo
1 qB
=
ρ
p
Cyclotron
49
IDPASC School, February 2015
On the opposite to the cyclotron, B field is applied on the
circumference only
Principle:
  Particles are running on a circular path and gain energy by
going through accelerating cavities
  The higher the energy, the higher the cavity frequency
  B field has to increased simultaneously to keep the particle on
the circular orbit
P. Puzo
IDPASC School, February 2015
50
Example with an injector


Guiding and focusing are done by
different elements
Dynamical range of a classical
mgnet is limited (≈ Bmax/20, Bmax)
  Examples:
•  LEP: from 20 tp 110 GeV
•  LHC: from 0,45 to 7 TeV
  Synchrotron concept requires
the use of an injector

Acceleration chain at DESY
(920 GeV protons and 27,5
GeV e+/e-)
P. Puzo

51
IDPASC School, February 2015
Two types of synchrotrons:
  Proton synchrotrons
•  The 1st one discovered the antiproton (Bevatron, Berkeley,
1954)
•  SppS (CERN) discovered W± and Z0
•  LHC discovered Higgs boson
  Electron synchrotrons
•  LEP was a Z0 and W± factory. Determined the number of
neutrino families and 10s of precise standard model
measurements
Most of the HEP discoveries since 1955 were done on
synchrotrons
P. Puzo
IDPASC School, February 2015
52
Betatron


Principle is to used time varying B field to
accelerate particles vie induced E field
(formerly only with protons)
One has:
Fem =
e r dB
2 dt

F
Particle
trajectory

F
€
B field
€

Modern
use: X ray production (with electrons)
€
Radiography
P. Puzo
Tumor
treatment
IDPASC School, February 2015
53
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