Introduction to Transverse Beam Dynamicscas.web.cern.ch/sites/cas.web.cern.ch/files/... · 2.)...
Transcript of Introduction to Transverse Beam Dynamicscas.web.cern.ch/sites/cas.web.cern.ch/files/... · 2.)...
Bernhard Holzer, CERN
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x(m),!x
(m),!y(m
)
x ! x ! y
Introduction to Transverse Beam Dynamics
The Ideal World I.) Magnetic Fields and Particle Trajectories
Lorentz force
„ ... in the end and after all it should be a kind of circular machine“ ! need transverse deflecting force
typical velocity in high energy machines: 83*10≈ ≈ m
sv c
1.) Introduction and Basic Ideas
28 11031
mVs
smqFTB ∗∗∗=→=
mMVqF 300∗=
mMVE 1≤
E
Example:
equivalent el. field
technical limit for el. field
Bvevm=
ργ 2
0
circular coordinate system condition for circular orbit:
Lorentz force
centrifugal force
The ideal circular orbit ρ
s
θ ●
y
BveFL =
ργ 2
0 vmFcentr =ρB
ep=
B ρ = "beam rigidity"
A word of wisdom ... if you are clever, you use magnetic fields in an accelerator wherever it is possible.
1.) The Magnetic Guide Field
Define the Geometry of the Ring:
Dipole Magnets:
define the ideal orbit homogeneous field created by two flat pole shoes
convenient units:
[ ] ⎥⎦
⎤⎢⎣
⎡== 2mVsTB ⎥⎦
⎤⎢⎣
⎡=cGeVp
hInB 0µ=
TB 3.8=
cGeVp 7000=
Example LHC:
2πρ = 17.6 km ≈ 66 %
θ
θ =lρ≈BlBBρ
=Bdl∫p / q
= 2π
ρ = 2.8 km
7000 GeV Proton storage ring dipole magnets N = 1232 l = 15 m q = +1 e
Teslae
smm
eVB
epBlNdlB
3.8103151232
1070002
/2
8
9
=≈
=≈∫
π
π
Example LHC:
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=
2.) Focusing Properties – Transverse Beam Optics
general solution: free harmonic oszillation
Classical Mechanics: pendulum
there is a restoring force, proportional to the elongation x:
€
F = m * d2xdt 2
= −k * x
( ) *cos( )x t A tω ϕ= +Ansatz
€
˙ x = −Aω *sin(ωt +ϕ)˙ x = −Aω 2 *cos(ωt +ϕ)
Solution
€
ω = k /m, x(t) = x0 *cos( km t +ϕ)
required: focusing forces to keep trajectories in vicinity of the ideal orbit
linear increasing Lorentz force
linear increasing magnetic field
Quadrupole Magnets:
normalised quadrupole field:
gradient of a quadrupole magnet:
what about the vertical plane: ... Maxwell
ygBxgB xy ==
LHC main quadrupole magnet
mTg /220...25≈simple rule: )/()/(3.0cGeVp
mTgk =
epgk/
=
202rnIg µ
=
yB
xB xy
∂
∂=
∂
∂⇒
3.) The equation of motion:
Linear approximation:
* ideal particle ! design orbit
* any other particle ! coordinates x, y small quantities x,y << ρ
! magnetic guide field: only linear terms in x & y of B have to be taken into account
Taylor Expansion of the B field:
normalise to momentum p/e = Bρ
.../´´
!31
/´
!21
/*
/)(
0
0 ++++=ep
egep
egepxg
BB
epxB
ρ
...´´!31
!21)( 3
22
2
0 ++++=dxegx
dxBd
xdxdB
BxB yyyy
Example: heavy ion storage ring TSR
Separate Function Machines: Split the magnets and optimise them according to their job: bending, focusing etc
...!31
!211
/)( 32 ++++= xnxmxkepxB
ρ
The Equation of Motion:
only terms linear in x, y taken into account dipole fields quadrupole fields
* man sieht nur dipole und quads ! linear
Equation of Motion:
●
y
x
ρ
s
θ ● Consider local segment of a particle trajectory ... and remember the old days: (Goldstein page 27)
radial acceleration:
22
2⎛ ⎞= − ⎜ ⎟⎝ ⎠
rd da
dtdtρ θ
ρ Ideal orbit: , 0= =dconstdtρ
ρ
Force: 2
2⎛ ⎞= =⎜ ⎟⎝ ⎠
dF m mdtθ
ρ ρω
2 /=F mv ρ
general trajectory: ρ ! ρ + x
vBexmvx
dtdmF y=
+−+=
ρρ
2
2
2
)(
y
vBexmvx
dtdmF y=
+−+=
ρρ
2
2
2
)(
xdtdx
dtd
2
2
2
2
)( =+ ρ
yρ
s ● x
)1(11ρρρx
x−≈
+
remember: x ≈ mm , ρ ≈ m … ! develop for small x
veBxmvdtxdm y=−− )1(
2
2
2
ρρ
1
1 … as ρ = const
2
2
Taylor Expansion
...)(!2)()(
!1)()()( 0
20
00
0 +ʹ′ʹ′−
+ʹ′−
+= xfxxxfxxxfxf
y
guide field in linear approx.
: m
dtds
dsdx
dtdx
=
dtds
dtds
dsdx
dsd
dtds
dsdx
dtd
dtxd
⎟⎠
⎞⎜⎝
⎛=⎟⎠
⎞⎜⎝
⎛=2
2
independent variable: t → s
vxʹ′v
dsdv
dsdxvx
dtxd
+ʹ′ʹ′= 22
2
mgxve
mBvexvvx +=−−ʹ′ʹ′ 0
22 )1(
ρρ: v 2
xB
xBB yy ∂
∂+= 0 ⎭
⎬⎫
⎩⎨⎧
∂
∂+=−−
xB
xBevxmvdtxdm y
0
2
2
2
)1(ρρ
mgxve
mBvexv
dtxd
+=−− 02
2
2
)1(ρρ
mvgxe
mvBexx +=−−ʹ′ʹ′ 0)1(1
ρρ
epgx
epBxx
//1 0
2 +=+−ʹ′ʹ′ρρ
m v = p
ρ1
/0 −=epB
kepg
=/
xkxx +−=+−ʹ′ʹ′ρρρ11
2
normalize to momentum of particle
Equation for the vertical motion: * 01
2 =ρ
kk −↔
no dipoles … in general …
quadrupole field changes sign
0=+ʹ′ʹ′ yky
0)1( 2 =−+ʹ′ʹ′ kxxρ
y
x
Differential Equation of harmonic oscillator … with spring constant K
Ansatz:
general solution: linear combination of two independent solutions
4.) Solution of Trajectory Equations
Define … hor. plane:
… vert. Plane:
21= −K kρ
=K k
)cos()sin()( 21 sasasx ωωωω +−=ʹ′
)()sin()cos()( 222
21 sxsasasx ωωωωω −=−−=ʹ′ʹ′ K=ω
)sin()cos()( 21 sKasKasx +=
general solution:
)sin()cos()( 21 sasasx ωω ⋅+⋅=
0=+ʹ′ʹ′ xKx
Hor. Focusing Quadrupole K > 0:
0 01( ) cos( ) sin( )ʹ′= ⋅ + ⋅x s x K s x K sK
0 0( ) sin( ) cos( )ʹ′ ʹ′= − ⋅ ⋅ + ⋅x s x K K s x K s
For convenience expressed in matrix formalism:
0
1cos( ) sin(
sin( ) cos( )
⎛ ⎞⎜ ⎟
= ⎜ ⎟⎜ ⎟⎜ ⎟−⎝ ⎠
foc
K s K sKM
K K s K s
s = s0 s = s1
01
*s
focs x
xM
xx
⎟⎟⎠
⎞⎜⎜⎝
⎛ʹ′
=⎟⎟⎠
⎞⎜⎜⎝
⎛ʹ′
0=+ʹ′ʹ′ xKx
4.) Solution of Trajectory Equations
1cosh sinh
sinh cosh
⎛ ⎞⎜ ⎟
= ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
defoc
K l K lKM
K K l K l
hor. defocusing quadrupole:
drift space: K = 0
10 1⎛ ⎞
= ⎜ ⎟⎝ ⎠
driftl
M
! with the assumptions made, the motion in the horizontal and vertical planes are independent „ ... the particle motion in x & y is uncoupled“
s = s1 s = 0
0=−ʹ′ʹ′ xKx
Ansatz: )sinh()cosh()( 21 sasasx ωω ⋅+⋅=
Remember from school:
,)cosh()( ssf = )sinh()( ssf =ʹ′
Thin Lens Approximation:
1cos sin
sin cos
⎛ ⎞⎜ ⎟
= ⎜ ⎟⎜ ⎟⎜ ⎟−⎝ ⎠
k l k lkM
k k l k l
matrix of a quadrupole lens
in many practical cases we have the situation:
1= >> q
qf l
kl ... focal length of the lens is much bigger than the length of the magnet
limes: while keeping
1 01 1
⎛ ⎞⎜ ⎟= ⎜ ⎟⎜ ⎟⎝ ⎠
xMf
1 01 1
⎛ ⎞⎜ ⎟= −⎜ ⎟⎜ ⎟⎝ ⎠
zMf
... useful for fast (and in large machines still quite accurate) „back on the envelope calculations“ ... and for the guided studies !
0→ql constlk q =
Combining the two planes:
hor foc. quadrupole lens
Clear enough ( hopefully ... ? ) : a quadrupole magnet that is focussing o-in one plane acts as defocusing lens in the other plane ... et vice versa.
matrix of the same magnet in the vert. plane:
1cosh sinh
sinh cosh
⎛ ⎞⎜ ⎟
= ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
defoc
K l K lKM
K K l K l
0
1cos( ) sin(
sin( ) cos( )
⎛ ⎞⎜ ⎟
= ⎜ ⎟⎜ ⎟⎜ ⎟−⎝ ⎠
foc
K s K sKM
K K s K s
! with the assumptions made, the motion in the horizontal and vertical planes are independent „ ... the particle motion in x & y is uncoupled“
focusing lens
dipole magnet
defocusing lens
Transformation through a system of lattice elements
combine the single element solutions by multiplication of the matrices
*.....* * * *= etotal QF D QD B nd DM M M M M M
x(s)
s
court. K. Wille
0 typical values in a strong foc. machine: x ≈ mm, x´ ≤ mrad
€
xx '"
# $ %
& ' s2
= M(s2,s1) *xx '"
# $ %
& ' s1
in each accelerator element the particle trajectory corresponds to the movement of a harmonic oscillator „
Tune: number of oscillations per turn 64.31
59.32
Relevant for beam stability: non integer part
5.) Orbit & Tune:
LHC revolution frequency: 11.3 kHz kHz5.33.11*31.0 =
First turn steering "by sector:" q One beam at the time q Beam through 1 sector (1/8 ring), correct trajectory, open collimator and move on.
LHC Operation: Beam Commissioning
Question: what will happen, if the particle performs a second turn ?
x
... or a third one or ... 1010 turns
0
s
Astronomer Hill:
differential equation for motions with periodic focusing properties „Hill‘s equation“
Example: particle motion with periodic coefficient
equation of motion: ( ) ( ) ( ) 0ʹ′ʹ′ − =x s k s x s
restoring force ≠ const, we expect a kind of quasi harmonic k(s) = depending on the position s oscillation: amplitude & phase will depend k(s+L) = k(s), periodic function on the position s in the ring.
6.) The Beta Function
General solution of Hill´s equation:
( ) ( ) cos( ( ) )= ⋅ +x s s sε β ψ φ
β(s) periodic function given by focusing properties of the lattice ↔ quadrupoles
ε, Φ = integration constants determined by initial conditions
Inserting (i) into the equation of motion …
0
( )( )
= ∫s dss
sψ
β
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“
( ) ( )s L sβ β+ =
(i)
∫=)(2
1sdsQy βπ
The Beta Function
Amplitude of a particle trajectory:
Maximum size of a particle amplitude
)()(ˆ ssx βε=
β determines the beam size ( ... the envelope of all particle trajectories at a given position “s” in the storage ring. It reflects the periodicity of the magnet structure.
€
x(s) = ε * β(s) *cos(ψ(s) +ϕ)
7.) Beam Emittance and Phase Space Ellipse
general solution of Hill equation
from (1) we get
Insert into (2) and solve for ε
* 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
2
1( ) ( )21 ( )( )
( )
−ʹ′=
+=
s s
sss
α β
αγ
β
))(cos()()()1( φψβε += sssx
{ }))(sin())(cos()()(
)()2( φψφψαβε
+++−=ʹ′ ssss
sx
)()())(cos(s
sxsβε
φψ =+
)()()()()(2)()( 22 sxssxsxssxs ʹ′+ʹ′+= βαγε
Beam Emittance and Phase Space Ellipse
x´
x εβ
●
●
●
●
●
●
Liouville: in reasonable storage rings area in phase space is constant. A = π*ε=const
ε beam emittance = woozilycity of the particle ensemble, intrinsic beam parameter, cannot be changed by the foc. properties. Scientifiquely speaking: area covered in transverse x, x´ phase space … and it is constant !!!
)()()()()(2)()( 22 sxssxsxssxs ʹ′+ʹ′+= βαγε
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
Résumé:
beam rigidity: ⋅ = pB qρ
bending strength of a dipole: 1 00.2998 ( )1( / )
− ⋅⎡ ⎤ =⎣ ⎦B Tm
p GeV cρ
focusing strength of a quadrupole: 2 0.2998( / )
− ⋅⎡ ⎤ =⎣ ⎦gk m
p GeV c
focal length of a quadrupole: 1
=⋅ q
fk l
equation of motion: 1 Δ
ʹ′ʹ′+ =px Kxpρ
matrix of a foc. quadrupole: 2 1= ⋅s sx M x
1cos sin,
sin cos
⎛ ⎞⎜ ⎟
= ⎜ ⎟⎜ ⎟⎜ ⎟−⎝ ⎠
K l K lKM
K K l K l
1 01 1
⎛ ⎞⎜ ⎟= ⎜ ⎟⎜ ⎟⎝ ⎠
Mf
1.) P. Bryant, K. Johnsen: The Principles of Circular Accelerators and Storage Rings Cambridge Univ. Press 2.) Klaus Wille: Physics of Particle Accelerators and Synchrotron
Radiation Facilicties, Teubner, Stuttgart 1992 3.) Peter Schmüser: Basic Course on Accelerator Optics, CERN Acc.
School: 5th general acc. phys. course CERN 94-01 4.) Bernhard Holzer: Lattice Design, CERN Acc. School: Interm.Acc.phys course,
http://cas.web.cern.ch/cas/ZEUTHEN/lectures-zeuthen.htm cern report: CERN-2006-002
5.) A.Chao, M.Tigner: Handbook of Accelerator Physics and Engineering, Singapore : World Scientific, 1999. 6.) Martin Reiser: Theory and Design of Chargged Particle Beams Wiley-VCH, 2008 7.) Frank Hinterberger: Physik der Teilchenbeschleuniger, Springer Verlag 1997 8.) Mathew Sands: The Physics of e+ e- Storage Rings, SLAC report 121, 1970 9.) D. Edwards, M. Syphers : An Introduction to the Physics of Particle Accelerators, SSC Lab 1990
Bibliography: