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: