Introduction to Accelerator Physics
Beam Dynamics for „Summer Students“
Bernhard Holzer,
CERN-LHC
The Ideal World
IP5
I.) Magnetic Fields and Particle Trajectories
IP8
IP2
IP1
*
Luminosity Run of a typical storage ring:
LHC Storage Ring: Protons accelerated and stored for 12 hours
distance of particles travelling at about v ≈ c
L = 1010-1011 km
... several times Sun - Pluto and back
intensity (1011)
3 -
2 1 -


guide the particles on a well defined orbit („design orbit“)
focus the particles to keep each single particle trajectory
within the vacuum chamber of the storage ring, i.e. close to the design orbit.
1.) Introduction and Basic Ideas
„ ... in the end and after all it should be a kind of circular machine“
 need transverse deflecting force
F  q *( E  v  B)
Lorentz force
typical velocity in high energy machines:
v  c  3*108 m s
Example:
B  1T
 F  q  3 108
F  q  300
equivalent el. field ...
m Vs
1 2
s
m
MV
m
E
technical limit for el. field:
E 1
MV
m
old greek dictum of wisdom:
if you are clever, you use magnetic fields in an accelerator wherever
it is possible.
y
The ideal circular orbit
θ
ρ
●
s
circular coordinate system
condition for circular orbit:
Lorentz force
centrifugal force
FL  e v B
Fcentr
 m0 v 2


 m0 v 2
 ev B

p
 B
e
B ρ = "beam rigidity"
2.) The Magnetic Guide Field
Dipole Magnets:
B 
define the ideal orbit
homogeneous field created
by two flat pole shoes
0 n I
h
Normalise magnetic field to momentum:
p
 B
e
1


convenient units:
eB
p
 Vs 
B  T    2 
m 
 GeV 
p
 c 
Example LHC:
B  8. 3 T
GeV
p  7000
c
1

1

e
8.3 Vs
2
m
7000 *109 eV
 0.333
8.3 1
7000 m

c
8.3 s * 3 *108 m
7000 *109 m 2
s
The Magnetic Guide Field
ρ
α
ds
field map of a storage ring dipole magnet
  2.53 km
rule of thumb:
2πρ = 17.6 km
≈ 66%
1

 0.3
B T 
p GeV / c 
B  1 ... 8 T
„normalised bending strength“
2.) Focusing Properties – Transverse Beam Optics
classical mechanics:
pendulum
there is a restoring force, proportional
to the elongation x:
d 2x
m * 2  c * x
dt
general solution: free harmonic oszillation
x(t )  A *cos(t   )
Storage Ring: we need a Lorentz force that rises as a function of
the distance to ........ ?
................... the design orbit
F ( x)  q * v * B ( x)
Quadrupole Magnets:
required:
focusing forces to keep trajectories in vicinity of the ideal orbit
linear increasing Lorentz force
linear increasing magnetic field
By  g x
Bx  g y
normalised quadrupole field:
k
simple rule:
g
p/e
k  0.3
g (T / m )
p(GeV / c )
LHC main quadrupole magnet
g  25 ... 220 T / m
what about the vertical plane:
... Maxwell
E
B = j 
0
t

By Bx

g
x
y
Focusing forces and particle trajectories:
normalise magnet fields to momentum
(remember: B*ρ = p / q )
Dipole Magnet
Quadrupole Magnet
B
B
1


p / q B 
g
k :
p/q
3.) The Equation of Motion:
B( x )
1
1
1

 k x 
m x 2  n x 3  ...
p/e

2!
3!
only terms linear in x, y taken into account dipole fields
quadrupole fields
Separate Function Machines:
Split the magnets and optimise
them according to their job:
bending, focusing etc
Example:
heavy ion storage ring TSR
*
man sieht nur
dipole und quads  linear
ŷ
The Equation of Motion:
●
*
Equation for the horizontal motion:
x  x (
*
1

2
 k)  0
θ
ρ
y
●
x
s
Equation for the vertical motion:
1
2
k
0
no dipoles … in general …
 k
quadrupole field changes sign
y  k y  0
y
x
4.) Solution of Trajectory Equations
Define … hor. plane:
… vert. Plane:
K 1 2  k
x  K x  0
K k
Differential Equation of harmonic oscillator … with spring constant K
Ansatz:
Hor. Focusing Quadrupole K > 0:
x( s)  x0  cos(
K s)  x0 
x( s)   x0 
K  sin(
1
sin(
K s)
K
K s)  x0  cos(
K s)
For convenience expressed in matrix formalism:
x
x
   M foc *  
 x  s1
 x  s 0
M foc



 cos K l


  K sin K l

1

sin
K
cos




Kl 


K l 

s=0
s = s1
hor. defocusing quadrupole:
x  K x  0
Ansatz: Remember from school
x(s)  a1  cosh( s)  a2  sinh(  s)
M defoc





cosh
Kl
1
sinh
K
K sinh
Kl
cosh

K l


K l 
drift space:
K=0
x(s)  x
0 *s

!
1 l 
M drift  

 0 1
with the assumptions made, the motion in the horizontal and vertical planes are
independent „ ... the particle motion in x & y is uncoupled“
Transformation through a system of lattice elements
combine the single element solutions by multiplication of the matrices
focusing lens
M total  M QF * M D * M QD * M Bend * M D*.....
x 
x 
   M(s2,s1 ) *  
x's2
x's1
dipole magnet
defocusing lens
court. K. Wille
in each accelerator element the particle trajectory corresponds to the movement of a
harmonic oscillator „

x(s)
0
typical values
in a strong
foc. machine:
x ≈ mm, x´ ≤ mrad
s
5.) Orbit & Tune:
Tune: number of oscillations per turn
64.31
59.32
Relevant for beam stability:
non integer part
LHC revolution frequency: 11.3 kHz
0.31*11.3  3.5kHz
LHC Operation: Beam Commissioning
First turn steering "by sector:"
One
beam at the time
Beam
through 1 sector (1/8 ring),
correct trajectory, open collimator and move on.
Question: what will happen, if the particle performs a second turn ?
... or a third one or ... 1010 turns
x
0
s
II.) The Ideal World:
Particle Trajectories, Beams & Bunches
Bunch in a Storage Ring
( Z  X  Y)
Astronomer Hill:
differential equation for motions with periodic focusing properties
„Hill‘s equation“
Example: particle motion with
periodic coefficient
equation of motion:
x( s )  k ( s ) x( s )  0
restoring force ≠ const,
k(s) = depending on the position s
k(s+L) = k(s), periodic function
we expect a kind of quasi harmonic
oscillation: amplitude & phase will depend
on the position s in the ring.
6.) The Beta Function
„it is convenient to see“
... after some beer ... general solution of Mr Hill
can be written in the form:
Ansatz:
x(s)   *  (s) *cos( (s)   )
ε, Φ = integration constants
determined by initial conditions
β(s) periodic function given by focusing properties of the lattice ↔ quadrupoles
 ( s  L)   ( s )
ε beam emittance = woozilycity of the particle ensemble, intrinsic beam parameter,
cannot be changed by the foc. properties.
scientifiquely spoken: area covered in transverse x, x´ phase space … and it
is
constant !!!
Ψ(s) = „phase advance“ of the oscillation between point „0“ and „s“ in the lattice.
For one complete revolution: number of oscillations per turn „Tune“
Qy 
1
ds

2  ( s)
7.) Beam Emittance and Phase Space Ellipse
   ( s)* x 2 ( s)  2 ( s) x( s) x( s)   ( s) x( s) 2
x´

 
●

Liouville: in reasonable storage rings
area in phase space is constant.
●
 

A = π*ε=const
●
●
●
●

x
x(s)
s
ε beam emittance = woozilycity of the particle ensemble, intrinsic beam parameter,
cannot be changed by the foc. properties.
Scientifiquely spoken: area covered in transverse x, x´ phase space … and it is constant !!!
Particle Tracking in a Storage Ring
Calculate x, x´ for each linear accelerator
element according to matrix formalism
plot x, x´as a function of „s“
●
… and now the ellipse:
note for each turn x, x´at a given position „s1“ and plot in the
phase space diagram
Emittance of the Particle Ensemble:
( Z  X  Y)
Emittance of the Particle Ensemble:
x(s)    (s)  cos((s)   )
xˆ (s)    (s)
Gauß
Particle Distribution:
 ( x) 
N e
e
2  x

particle at distance 1 σ from centre
↔ 68.3 % of all beam particles
single particle trajectories, N ≈ 10 11 per bunch
LHC:
  180 m
  5 *1010 m rad
   *   5 *10 10 m *180 m  0.3 mm

aperture requirements: r 0 = 12 * σ
1 x2
2  x2
III.) The „not so ideal“ World
Lattice Design in Particle Accelerators
 x, y
D
1952: Courant, Livingston, Snyder:
Theory of strong focusing in particle beams
Recapitulation: ...the story with the matrices !!!
Equation of Motion:
x  K x  0
Solution of Trajectory Equations
K  1  2  k … hor. plane:
K k
x
x
   M *  
 x   s1
 x  s 0
… vert. Plane:
1 l 

M drift  
0
1


M foc
M defoc

 cos( K l )


  K sin( K l )

 cosh( K l )


 K sinh( K l )

K l)
K


cos( K l ) 
1
sin(

K l)
K


cosh( K l ) 
1
sinh(
M total  M QF * M D * M B * M D * M QD * M D * ...
8.) Lattice Design:
„… how to build a storage ring“
Geometry of the ring: B *   p / e
p = momentum of the particle,
ρ = curvature radius
Bρ= beam rigidity
Circular Orbit: bending angle of one dipole

ds


dl


Bdl
B
The angle run out in one revolution
must be 2π, so for a full circle

 Bdl  2
B
p
 Bdl  2 q
… defines the integrated dipole field around the machine.
Example LHC:
7000 GeV Proton storage ring
dipole magnets N = 1232
l = 15 m
q = +1 e
 B dl  N l B  2
p/e
2 7000 109 eV
B
 8.3 Tesla
m
1232 15 m 3 108
e
s
FoDo-Lattice
A magnet structure consisting of focusing and defocusing quadrupole lenses in
alternating order with nothing in between.
(Nothing = elements that can be neglected on first sight: drift, bending magnets,
RF structures ... and especially experiments...)
Starting point for the calculation: in the middle of a focusing quadrupole
Phase advance per cell μ = 45°,
 calculate the twiss parameters for a periodic solution
9.) Insertions
 x, y
D
β-Function in a Drift:
 ( )  0 
2
l
l
0
*
β0
At the end of a long symmetric
drift space the beta function
reaches its maximum value in the
complete lattice.
-> here we get the largest beam
dimension.
-> keep l as small as possible
7 sima beam size iside a mini beta quadrupole
... unfortunately ... in general
... clearly there is another problem !!!
high energy detectors that are
installed
in that
driftatspaces
Example:
Luminosity
optics
LHC: β* = 55 cm
arefor
a little
bit bigger
centimeters
smallest
βmax wethan
haveatofew
limit
the overall...length
and keep the distance “s” as small as possible.
The Mini-β Insertion:
R  L * react
production rate of events
is determined by the
cross section Σreact
and a parameter L that is given
by the design of the accelerator:
… the luminosity
I1 * I 2
1
L
* *
2
4 e f 0 b  x * * y
10.) Luminosity
p2-Bunch
10
11
particles
p1-Bunch
IP
10
11
±σ
particles
Example: Luminosity run at LHC
 x , y  0.55 m
f 0  11.245 kHz
 x , y  5 10 10 rad m
nb  2808
 x , y  17 m
I p  584 mA
L  1.0 1034 1
cm 2 s
I p1 I p 2
1
L
*
2
4e f 0 nb  x y
Mini-β Insertions: Betafunctions
A mini-β insertion is always a kind of special symmetric drift space.
greetings from Liouville
 /
x´
the smaller the beam size
the larger the bam divergence
●
●
●
●
●
●

x
Mini-β Insertions: some guide lines
* calculate the periodic solution in the arc
* introduce the drift space needed for the insertion device (detector ...)
* put a quadrupole doublet (triplet ?) as close as possible
* introduce additional quadrupole lenses to match the beam parameters
to the values at the beginning of the arc structure
parameters to be optimised & matched to the periodic solution:
8 individually
powered quad
magnets are
needed to match
the insertion
( ... at least)
x , x
y, y
Dx , Dx
Qx , Qy
IV) … let´s talk about acceleration
Electrostatic Machines
(Tandem –) van de Graaff Accelerator
creating high voltages by mechanical
transport of charges
* Terminal Potential: U ≈ 12 ...28 MV
using high pressure gas to suppress discharge ( SF6 )
Problems: * Particle energy limited by high voltage discharges
* high voltage can only be applied once per particle ...
... or twice ?
* The „Tandem principle“: Apply the accelerating voltage twice ...
... by working with negative ions (e.g. H-) and
stripping the electrons in the centre of the structure
Example for such a „steam engine“: 12 MV-Tandem van de Graaff
Accelerator at MPI Heidelberg
12.) Linear Accelerator
Energy Gain per „Gap“:
1928, Wideroe
+ -̶
+
-̶
+
-̶
+
W  q U 0 sin  RF t
drift tube structure at a proton linac
(GSI Unilac)
500 MHz cavities in an electron storage ring
* RF Acceleration: multiple application of
the same acceleration voltage;
brillant idea to gain higher energies
13.) The Acceleration
Where is the acceleration?
Install an RF accelerating structure in the ring:

E

c
z
B. Salvant
N. Biancacci
14.) The Acceleration for Δp/p≠0
“Phase Focusing”
ideal particle


particle with Δp/p < 0 
particle with Δp/p > 0


below transition
faster
  
slower
Focussing effect in the
longitudinal direction
keeping the particles
close together
... forming a “bunch”

oscillation frequency: f s  f rev 
h s qU 0 cos s
*
2
Es
≈ some Hz
... so sorry, here we need help from Albert:
E
  total2 
mc
1
v
mc 2
 1 2
c
E
v2
1 2
c
v/c

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
... some when the particles
do not get faster anymore
0.2
0.1
.... but heavier !
0
0
500
1000
1500
2000
2500
3000
3500
kinetic energy of a proton
4000
4500
5000
15.) The Acceleration for Δp/p≠0
“Phase Focusing” above transition
ideal particle


particle with Δp/p < 0 
particle with Δp/p > 0
heavier
  
lighter



Focussing effect in the longitudinal direction
keeping the particles close together ... forming a “bunch”
... and how do we accelerate now ???
with the dipole magnets !
The RF system: IR4
S34
ACS ACS
S45
ADT
D3
D4 Q5
Q6
Q7
B2
420 mm
194 mm
B1
ACS
ACS
Bunch length (4)
ns
1.06
Energy spread (2)
10-3
0.22
Synchr. rad. loss/turn keV
Synchr. rad. power
kW
3.6
RF frequency
M
Hz
400
Harmonic number
4xFour-cavity cryo module 400 MHz, 16 MV/beam
Nb on Cu cavities @4.5 K (=LEP2)
Beam pipe diam.=300mm
7
35640
RF voltage/beam
MV
16
Energy gain/turn
keV
485
Synchrotron
frequency
Hz
23.0
LHC Commissioning: RF
( Z  X  Y)
a proton bunch: focused longitudinal by
the RF field
~ 200 turns
Bunch length ~ 1.5 ns  ~ 45 cm
RF off
RF on,
phase optimisation
RF on, phase adjusted,
beam captured
Problem: panta rhei !!!
(Heraklit: 540-480 v. Chr.)
( Z  X  Y)
Bunch length of Electrons ≈ 1cm
just a stupid (and nearly wrong) example)
U0
t
  500 MHz
c 
  60 cm
  60 cm
sin( 90o )  1
sin( 84o )  0.994
U
 6.0 10 3
U
typical momentum spread of an electron bunch:
p
 1.0 103
p
17.) Dispersion and Chromaticity:
Magnet Errors for Δp/p ≠ 0
Influence of external fields on the beam: prop. to magn. field & prop. zu 1/p
dipole magnet
B dl


focusing lens
k
p/e
x D ( s )  D( s )
p
p
g
p
e
particle having ...
to high energy
to low energy
ideal energy
Dispersion
Example: homogeneous dipole field
xβ
Closed orbit for Δp/p > 0
xβ
. ρ
xi ( s)  D( s) 
Matrix formalism:
x( s)  x ( s)  D( s) 
p
p
x( s)  C ( s)  x0  S ( s)  x0  D( s) 
p
p
 x   C S  x  p  D 
   
  
 




x
C
S
x
p
 s 
  0
 D  0
p
p
or expressed as 3x3 matrix
 x  C S D  x 

 
  




 x   C S D  x 
 p p   0 0 1   p p 
 

s 
0

D
Example
x  1...2 mm
D( s )  1...2 m
p  1 103
p
Amplitude of Orbit oscillation
contribution due to Dispersion ≈ beam size
 Dispersion must vanish at the collision point
Calculate D, D´: ... takes a couple of sunny Sunday evenings !
!
V.) Are there Any Problems ???
sure there are
Some Golden Rules to Avoid Trouble
I.) Golden Rule number one:
do not focus the beam !
 (s) *
Problem: Resonances
x co (s) 
Assume: Tune = integer

Qualitatively spoken:

1

 s1 * cos( s1  s  Q) ds
s1
Q 1 
2sin Q
0
Integer tunes lead to a resonant increase
of the closed orbit amplitude in presence of
the smallest dipole field error.
Tune and Resonances
m*Qx+n*Qy+l*Qs = integer
Tune diagram up to 3rd order
Qy =1.5
… and up to 7th order
Qy =1.3
Homework for the operateurs:
find a nice place for the tune
where against all probability
the beam will survive
Qy =1.0
Qx =1.0
Qx =1.3
Qx =1.5
II.) Golden Rule number two: Never accelerate charged particles !
Fdef
Transport line with quadrupoles
x   K(s)x  0
Transport line with quadrupoles and space charge
x   (K(s)  K SC (s))x  0

2r I 
x    K(s)  2 03 3 x  0
ea β γ c 

KSC
Golden Rule number two:
Never accelerate charged particles !
Q x, y  
Tune Shift due to Space Charge Effect
Problem at low energies
r0 N
2x, y   2
v/c

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
... at low speed the particles
repel each other
0.2
0.1
0
0
1000
2000
Linac 2 Ekin=60 MeV
Linac 4 Ekin=150 MeV
3000
4000
5000
Ekin of a proton
III.) Golden Rule number three:
Never Collide the Beams !
25 ns
the colliding bunches influence each other
 change the focusing properties of the ring !!
most simple case:
linear beam beam tune shift
 x* * rp * N p
Qx 
2  p ( x   y ) *  x
Qx
and again the resonances !!!

Courtesy W. Herr
Qx
IV.) Golden Rule Number four: Never use Magnets
“effective magnetic length”
lmag
B * leff

 Bds
0
Clearly there is another problem ...
... if it were easy everybody could do it
Again: the phase space ellipse
for each turn write down – at a given
position „s“ in the ring – the
x
x
single partilce amplitude x
   M turn *  
and the angle x´... and plot it.  x   s1
 x  s 0
●
A beam of 4 particles
– each having a slightly different emittance:
Installation of a weak ( !!! ) sextupole magnet
The good news: sextupole fields in accelerators
cannot be treated analytically anymore.
 no equatiuons; instead: Computer simulation
„ particle tracking “
●
Effect of a strong ( !!! ) Sextupole …
 Catastrophy !
●
„dynamic aperture“
Golden Rule XXL: COURAGE
and with a lot of effort from Bachelor / Master / Diploma / PhD
and Summer-Students the machine is running !!!
thank’x for your help and have a lot of fun