Introduction to Transverse Beam Dynamics
Bernhard Holzer,
CERN-LHC
The Ideal World
I.) Magnetic Fields and Particle Trajectories
*
Largest storage ring: The Solar System
astronomical unit: average distance earth-sun
1AE ≈ 150 *106 km
Distance Pluto-Sun ≈ 40 AE
AE
Luminosity Run of a typical storage ring:
HERA 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
 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.
Transverse Beam Dynamics:
0.) Introduction and Basic Ideas
„ ... in the end and after all it should be a kind of circular machine“
 need transverse deflecting force
F
Lorentz force
q *( E v B)
typical velocity in high energy machines:
v
c 3*108 m s
Example:
B 1T
F
F
q 3 108
q 300
m Vs
1 2
s m
MV
m
equivalent el. field … E
technical limit for el. field:
E 1
MV
m
The ideal circular orbit
ẑ
θ
ρ
s
circular coordinate system
condition for circular orbit:
Lorentz force
centrifugal force
FL
e*v * B
FZentr
m0v 2
m0 v 2
e*v * B
p
e
B*
1.) The Magnetic Guide Field
Dipole Magnets:
define the ideal orbit
homogeneous field created
by two flat pole shoes
0
B
nI
h
Normalise magnetic field to momentum:
p
e
B
1
convenient units:
eB
p
B
Vs
m2
T
p
GeV
c
Example LHC:
B 8.3T
GeV
p 7000
c
1
1
8.3 Vs
e
m
7000*109 eV
0.333
8.3 s 3 *108 m
2
8.3 1
7000 m
c
s
7000*109 m 2
The Magnetic Guide Field
ρ
α
ds
field map of a storage ring dipole magnet
2πρ = 17.6 km
≈ 66%
2.53 km
rule of thumb:
1
0.3
BT
p GeV / c
B 1 ... 8 T
„normalised bending strength“
2.) Quadrupole Magnets:
Calculation of the Quadrupole Field:
integration path
coil N * I
H ds
N *I
1
Hds
2
H 0ds
0
H Feds
0
Hds
1
2
HFe = H0 / μFe
μFe ≈ 1000
now we know that
N *I
H
court. K. Wille
H ┴ ds
B
1
0
and we require
gradient of a
quadrupole field:
B(r )
g
H 0ds
g *r
2
0
a
*N *I
r2
0
0
B0
0
a
dr
0
g *r
0
dr
N *I
Quadrupole Magnets:
required:
focusing forces to keep trajectories in vicinity of the ideal orbit
linear increasing Lorentz force
By
linear increasing magnetic field
gx
Bx
gy
normalised quadrupole field:
gradient of a
quadrupole magnet:
simple rule:
g
2 0 nI
r2
k
g
p/e
k
g (T / m)
0.3
p(GeV / c)
what about the vertical plane:
... Maxwell
B= j
LHC main quadrupole magnet
g 25 ... 220 T / m
E
t
0
By
x
Bx
y
3.) The equation of motion:
Linear approximation:
* ideal particle
 design orbit
* any other particle  coordinates x, z small quantities
x,z << ρ
 magnetic guide field: only linear terms in x & z of B
have to be taken into account
Taylor Expansion of the B field:
Bz ( x)
Bz 0
dBz
x
dx
B( x)
p/e
B0
B0
g*x
p/e
1 d 2 Bz 2
x
2
2! dx
1 eg´
2! p / e
1 eg´´
...
3
3! dx
1 eg´´
3! p / e
...
normalise to momentum
p/e = Bρ
The Equation of Motion:
B( x )
p/e
1
k*x
1
mx 2
2!
1
nx3
3!
...
only terms linear in x, z 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
ẑ
Equation of Motion:
θ
Consider local segment of a particle trajectory
... and remember the old days:
ρ
z
x
s
(Goldstein page 27)
radial acceleration:
ar
d2
dt 2
d
dt
2
const ,
Ideal orbit:
Force:
F
F
general trajectory: ρ  ρ + x
F
d2
m 2 (x
dt
)
mv 2
x
e Bz v
m
d
dt
mv 2 /
d
dt
0
2
m
2
F
d2
m 2 (x
dt
mv 2
x
)
ẑ
e Bz v
2
1
z
ρ
x
s
2
2
1
d
(x
2
dt
2
remember: x ≈ mm , ρ ≈ m …  develop for small x
1
x
d
x
2
dt
)
1
(1
x
… as ρ = const
Taylor Expansion
)
f ( x)
d 2x
m 2
dt
mv 2
(1
x
) eBz v
f ( x0 )
(x
x0 )
* f ( x0 )
1!
(x
x0 ) 2
* f ( x0 )
2!
...
guide field in linear approx.
Bz
B0
x
Bz
x
d 2x
m 2
dt
d 2x
dt 2
mv 2
v2
(1
(1
x
x
) ev B0
x
Bz
x
)
e v B0
m
ev x g
m
)
e v B0
m
ev x g
m
: m
independent variable: t → s
dx
dt
d 2x
dt 2
d 2x
dt 2
dx ds
*
ds dt
d dx ds
*
dt ds dt
x * v2
d dx ds ds
*
ds ds dt dt
dx dv
*
*v
ds ds
x
x v
v
2
v2
(1
x
: v2
x
x
x
1
x
(1
1
)
B0
p/e
x
2
1
e B0
mv
x
e xg
mv
mv=p
xg
p/e
1
normalize to momentum of particle
B0
p/e
kx
2
g
p/e
x
*
x(
1
2
k) 0
Equation for the vertical motion:
1
2
k
no dipoles … in general …
0
k
quadrupole field changes sign
z
k z
0
1
k
Remarks:
x
*
k
(
1
2
k) x
0
… there seems to be a focusing even without
a quadrupole gradient
0
x
„weak focusing of dipole magnets“
1
*x
2
even without quadrupoles there is a retriving force
(i.e. focusing) in the bending plane of the dipole magnets
… in large machines it is weak.
*
bending and focusing fields … are functions
of the independent variable „s“
Hard Edge Model:
x ( s)
1
k ( s ) * x( s ) 0
2
( s)
Inside a magnet the focusing
properties are constant !
1
const
(!)
k
const
„effective magnetic length“
l mag
B * leff
Bds
0
4.) Solution of Trajectory Equations
Define … hor. plane:
K
1
… vert. Plane:
K
k
2
k
y
K*y
0
Differential Equation of harmonic oscillator … with spring constant K
Ansatz:
x(s) a1 cos( s) a2 sin( s)
general solution: linear combination of two independent solutions
x ( s)
a1 sin( s) a2 cos( s)
x ( s)
a1
2
cos( s) a2
2
sin( s )
2
x( s )
general solution:
x(s) a1 cos( K s) a2 sin( K s)
K
determine a1 , a2 by boundary conditions:
s 0
x(0)
x0
,
a1
x (0)
x0
,
a2
x0
x0
K
Hor. Focusing Quadrupole K > 0:
x( s )
x0 cos(
x (s)
x0
K s)
K
1
x0
s1
x
M foc *
x
cos(
sin(
K s)
x0 cos(
s = s0
s0
K s)
1
sin(
Ks
K
M foc
K sin(
K s)
cos(
K s)
K
For convenience expressed in matrix formalism:
x
x
sin(
K s)
0
K s)
s = s1
s=0
s = s1
hor. defocusing quadrupole:
x
K *x 0
Remember from school:
f (s) cosh(s) ,
f (s) sinh(s)
x(s) a1 cosh( s) a2 sinh( s)
Ansatz:
cosh
Kl
K sinh
K=0
!
M drift
sinh
Kl
K
M defoc
drift space:
1
Kl
cosh
Kl
1 l
0 1
with the assumptions made, the motion in the horizontal and vertical planes are
independent „ ... the particle motion in x & z is uncoupled“
Thin Lens Approximation:
cos
matrix of a quadrupole lens
1
kl
sin
kl
k
M
k sin
kl
cos
kl
in many practical cases we have the situation:
f
limes:
1
klq
l
lq
0
Mx
... focal length of the lens is much bigger than the length of the magnet
while keeping kl = const
1
0
1
f
1
Mz
1
0
1
f
1
... useful for fast (and in large machines still quite accurate) „back on the envelope
calculations“ ... and for the guided studies !
Transformation through a system of lattice elements
combine the single element solutions by multiplication of the matrices
M total
M QF * M D * M QD * M Bend * M D*.....
focusing lens
x
x
M (s2,s1) *
s2
x
x
dipole magnet
s1
defocusing lens
court. K. Wille
x(s)
0
typical values
in a strong
foc. machine:
x ≈ mm, x ≤ mrad
s
5.) Orbit & Tune:
Tune: number of oscillations per turn
31.292
32.297
Relevant for beam stability:
non integer part
HERA revolution frequency: 47.3 kHz
0.292*47.3 kHz
13.81 kHz
Question: what will happen, if the particle performs a second turn ?
... or a third one or ... 1010 turns
x
0
s
19th century:
Ludwig van Beethoven: „Mondschein Sonate“
Sonate Nr. 14 in cis-Moll (op. 27/II, 1801)
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
General solution of Hill´s equation:
(i)
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)
Inserting (i) into the equation of motion …
s
(s)
0
ds
(s)
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
o
2
( s)
Beam Emittance and Phase Space Ellipse
general solution of
Hill equation
(1) x( s)
(2) x ( s)
*
( s) *cos( ( s)
( s)
*
)
( s)*cos( ( s)
) sin( ( s)
)
from (1) we get
cos( ( s)
)
1
( s)
2
1 ( s)2
( s)
( s)
( s)
x( s)
* ( s)
Insert into (2) and solve for ε
(s)* x2 (s) 2 ( s) x( s) x ( s)
( s) x ( s) 2
* ε is a constant of the motion … it is independent of „s“
* parametric representation of an ellipse in the x x‘ space
* shape and orientation of ellipse are given by α, β, γ
Beam Emittance and Phase Space Ellipse
(s)* x2 (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“
●
x
x'
… and now the ellipse:
note for each turn x, x´at a given position „s1“ and plot in the
phase space diagram
7.) Résumé:
beam rigidity:
p
B
1
bending strength of a dipole:
focusing strength of a quadrupole:
q
1
m
k m
2
k m
2
focal length of a quadrupole:
f
1
k lq
equation of motion:
x
Kx
xs 2
matrix of a foc. quadrupole:
1
cos K l
sin
K sin
Kl
0.2998 g
p(GeV / c)
0.2998 2 0 nI
p(GeV / c) ar2
1 p
p
M xs1
Kl
K
M
0.2998 B0 (T )
p(GeV / c)
cos K l
,
M
1
0
1
f
1
Bibliography
1.) Klaus Wille,
Physics of Particle Accelerators and Synchrotron
Radiation Facilicties, Teubner, Stuttgart 1992
2.) M.S. Livingston, J.P. Blewett: Particle Accelerators,
Mc Graw-Hill, New York,1962
3.) H. Wiedemann, Particle Accelerator Physics
(Springer-Verlag, Berlin, 1993)
4.) A. Chao, M. Tigner, Handbook of Accelerator Physics and Engineering
(World Scientific 1998)
5.) Peter Schmüser: Basic Course on Accelerator Optics, CERN Acc.
School: 5th general acc. phys. course CERN 94-01
6.) Bernhard Holzer: Lattice Design, CERN Acc. School: Interm.Acc.phys course,
http://cas.web.cern.ch/cas/ZEUTHEN/lectures-zeuthen.htm
7.) Frank Hinterberger: Physik der Teilchenbeschleuniger, Springer Verlag 1997
9.) Mathew Sands: The Physics of e+ e- Storage Rings, SLAC report 121, 1970
10.) D. Edwards, M. Syphers : An Introduction to the Physics of Particle
Accelerators, SSC Lab 1990