Introduction to Transverse Beam Dynamics
I Linear Beam Optics
Bernhard Holzer, DESY-HERA
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
Æ
Æ
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“
Æ need transverse deflecting force
Lorentz force
r
r r r
F = q * (E + v × B)
typical velocity in high energy machines:
v ≈ c ≈ 3*108 m s
old greek dictum of wisdom:
if you are clever, you use magnetic fields in an accelerator wherever
it is possible.
But remember: magn. fields act allways perpendicular to the velocity of the particle
Æ only bending forces, Æ no „beam acceleration“
The ideal circular orbit
ẑ
θ
ρ
●
s
circular coordinate system
condition for circular orbit:
Lorentz force
centrifugal force
FL = e * v * B
FZentr
γ m0 v 2
=
ρ
γ m0v2
= e*v * B
ρ
p
= B* ρ
e
I.) The Magnetic Guide Field
B =
ρ
α
μ0 n I
h
ds
field map of a storage ring dipole magnet
p
= B* ρ
e
„radius of curvature,
normalised bending strength“
B [T ]
= 0.2998 *
ρ
p [GeV / c]
1
Example HERA:
B = 5.2 Tesla
p = 920 GeV/c
1/ρ = 1.7178*10 -3 /m
ρ ~ 600m
2πρ ~ 3.7 km
Quadrupole Magnets:
required: focusing forces to keep trajectories in vicinity of the ideal orbit
linear increasing Lorentz force
linear increasing magnetic field
Bz = g * x
normalised quadrupole field:
gradient of a
quadrupole magnet:
g=
2 μ0 nI
r2
k=
… what about the vertical plane:
Maxwell:
g
p/e
r
r r
r ∂E
∇×B = j +
=0
∂t
⇒
∂B x ∂B z
=
∂z
∂x
Bx = g * z
II.) The equation of motion:
Linear approximation:
* ideal particle
Æ design orbit
* any other particle Æ coordinates x, z small quantities
x,z << ρ
Æ 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) = Bz0
dB z
1 d 2 B z 2 1 eg ´´
+
x+
x +
+ ...
2
3
dx
2! dx
3! dx
B0
B (x)
g*x
1 eg ´
1 eg ´´
=
+
+
+
+ ...
p/e
B0 ρ
p/e
2! p / e 3! p / e
normalise to momentum
p/e = Bρ
The Equation of Motion:
B (x)
1
=
ρ
p/e
+ k*x +
1
mx
2!
2
+
1
nx
3!
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 Æ 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:
d 2ρ
⎛ dθ ⎞
ar = 2 − ρ ⎜
⎟
dt
⎝ dt ⎠
2
Ideal orbit:
ρ = const ,
2
Force:
⎛ dθ ⎞
2
ρω
F = mρ ⎜
=
m
⎟
⎝ dt ⎠
F = mv 2 / ρ
general trajectory: ρ Æ ρ + x
d2
mv 2
F = m 2 (x + ρ) −
= e Bz v
dt
x+ρ
dρ
=0
dt
mv 2
d2
= e Bz v
F = m 2 (x + ρ) −
x+ρ
dt
ẑ
2
1
ρ
z
x
●
s
2
2
1
d
d
(
x
+
ρ
)
=
x
2
2
dt
dt
2
remember: x ≈ mm , ρ ≈ m … Æ develop for small x
1
1
x
≈ (1 − )
ρ
x+ρ ρ
… as ρ = const
Taylor Expansion
( x − x0 )
( x − x0 ) 2
′
f ( x ) = f ( x0 ) +
* f ( x0 ) +
* f ′′( x 0 ) +
1!
2!
d 2 x mv 2
x
m 2 −
(1 − ) = eBz v
ρ
ρ
dt
...
guide field in linear approx.
Bz = B0 + x
∂Bz
∂x
∂B ⎫
d 2 x mv 2
x
⎧
(1 − ) = ev ⎨ B0 + x z ⎬
m 2 −
ρ
ρ
∂x ⎭
dt
⎩
: m
e v B0
d 2x v2
x
ev x g
−
−
=
+
(
1
)
dt 2
ρ
ρ
m
m
independent variable: t → s
dx
dx ds
=
*
dt
ds
dt
d 2x
d ⎛ dx ds ⎞
d ⎛ dx ds ⎞ ds
=
*
=
*
⎟
⎜
⎟
⎜
2
dt
dt ⎝ ds dt ⎠ ds ⎝ ds dt ⎠ dt
d 2x
dx dv
2
′
′
*
*
*v
x
v
=
+
dt 2
ds ds
x′
v
x ′′v −
2
v2
ρ
(1 −
x
ρ
)=
e v B0
ev x g
+
m
m
: v2
x ′′ −
x ′′ −
x ′′ −
1
ρ
x
(1 −
1
ρ
1
ρ
+
+
ρ
x
ρ
2
x
ρ
2
)=
=
e B0
e xg
+
mv
mv
B0
xg
+
p/e
p/e
= −
1
ρ
normalize to momentum of particle
B0
1
= −
ρ
p/e
+ k x
x′′ + x(
*
mv=p
1
ρ
2
g
= k
p/e
− k) = 0
Equation for the vertical motion:
1
ρ2
k
=0
no dipoles … in general …
↔ −k
quadrupole field changes sign
z ′′ + k ⋅ z = 0
Remarks:
*
x′′ + (
1
ρ
2
− k) ⋅ x = 0
… there seems to be a focusing even without
a quadrupole gradient
„weak focusing of dipole magnets“
k =0
⇒
x′′ = −
1
ρ
2
*x
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. (!)
Mass spectrometer: particles are separated
according to their energy
and focused due to the 1/ρ
effect of the dipole
III.) Solution of Trajectory Equations
Define … hor. plane:
… vert. Plane:
K =1 ρ2 −k
K =k
y ′′ + K * y = 0
Differential Equation of harmonic oscillator … with spring constant K
Ansatz:
x( s ) = a1 ⋅ cos(ω s ) + a2 ⋅ sin(ω s )
general solution: linear combination of two independent solutions
x′( s ) = −a1ω sin(ω s ) + a2ω cos(ω s )
x′′( s ) = −a1ω 2 cos(ω s ) − a2ω 2 sin(ω s ) = −ω 2 x( s )
general solution:
x( s ) = a1 cos( K s ) + a2 sin( K s )
ω= K
determine a1 , a2 by boundary conditions:
x (0) = x0
s=0
x ′ ( 0 ) = x 0′
a1 = x 0
,
,
a2 =
x 0′
K
Hor. Focusing Quadrupole K > 0:
x ( s ) = x 0 ⋅ co s(
K s ) + x 0′ ⋅
x ′( s ) = − x 0 ⋅
K ⋅ s in (
1
K
sin (
K s ) + x 0′ ⋅ c o s (
For convenience expressed in matrix formalism:
s = s0
⎛x⎞
⎛x⎞
⎜⎜ ⎟⎟ = M foc * ⎜⎜ ⎟⎟
⎝ x′ ⎠ s 0
⎝ x′ ⎠ s1
M
foc
⎛
⎜ cos( K s )
=⎜
⎜⎜
⎝ − K sin( K s )
1
K
⎞
K s⎟
⎟
⎟
K s ) ⎟⎠
0
sin(
cos(
K s)
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(ω s) + a2 ⋅ sinh(ω s )
Ansatz:
M defoc
drift space:
K=0
!
⎛
⎜
=⎜
⎜⎜
⎝
⎛1
M drift = ⎜
⎝0
cosh
K sinh
K l
K l
1
K
sinh
cosh
⎞
K l⎟
⎟
⎟
K l ⎟⎠
l⎞
⎟
1⎠
with the assumptions made, the motion in the horizontal and vertical planes are
independent „ ... the particle motion in x & z is uncoupled“
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) * ⎜ ′ ⎟
⎝ x ⎠s 2
⎝ x ⎠ s1
dipole magnet
defocusing lens
court. K. Wille
x(s)
0
typical values
in a strong
foc. machine:
x ≈ mm, x´ ≤ mrad
s
IV.) 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
The Particle Ensemble:
HERA Bunch Pattern:
N
b
N
b
N
b
=
100m A τ rev
*
180
e
100 * 10− 3 C b
21 * 10− 6
s
*
*
*
=
− 19
s
Cb
180
1.6 * 10
= 7.3 * 10 10
… müssen wir wirklich einen Sack voll 1012 Flöhen hüten ?
… do we really have to calculate 1012 single particle trajectories ????
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.
V.) The Beta Function
General solution of Hill´s equation:
(i)
x(s) = ε β (s) ⋅ cos(ψ (s) + φ )
ε, Φ = integration constants determined by initial conditions
β(s) periodic function given by focusing properties of the lattice ↔ quadrupoles
β (s + L) = β (s)
Inserting (i) into the equation of motion …
s
ds
β ( s)
0
ψ ( 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∫
2π β (s)
Beam Emittance and Phase Space Ellipse
general solution of
Hill equation
(1) x(s) = ε * β (s) *cos(ψ (s) + φ )
(2) x′(s) = −
ε
β ( s)
*{α (s)*cos(ψ (s) + φ ) + sin(ψ (s) + φ )}
from (1) we get
cos(ψ (s) + φ ) =
x( s )
ε * β ( s)
Insert into (2) and solve for ε
ε = γ (s)* x2 (s) + 2α (s) x(s) x′(s) + β (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
−1
β ′(s)
2
1 + α ( s)2
γ ( s) =
β ( s)
α ( s) =
Beam Emittance and Phase Space Ellipse
ε = γ (s)* x2 (s) + 2α (s) x(s) x′(s) + β (s) x′(s)2
−α ε
x´
εγ
●
γ
Liouville: in reasonable storage rings
area in phase space is constant.
●
−α ε
β
A = π*ε=const
●
●
●
●
εβ
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“
10
xtx
●
0
10
0
20
40
60
80
100
60
80
100
t
10
yx´
t
0
10
0
20
40
t
… and now the ellipse:
note for each turn x, x´at a given position „s1“ and plot in the
phase space diagram
10
10
5
xx´t
0
5
− 10
10
10
− 10
5
0
yxt
5
10
10