Bernhard Holzer, CERN-LHC

Introduction to Transverse Beam Dynamics

*

I.) Magnetic Fields and Particle TrajectoriesThe Ideal World

Largest storage ring: The Solar System

astronomical unit: average distance earth-sun1AE ≈ 150 *106 kmDistance Pluto-Sun ≈ 40 AE

AE

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

Luminosity Run of a typical storage ring:

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.

Lorentz force *( )F q E v B

„ ... 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

Transverse Beam Dynamics:

0.) Introduction and Basic Ideas

Example:

2

8 11031m

Vs

s

mqFTB

m

MVqF 300

technical limit for el. field:

m

MVE 1

Eequivalent el. field …

circular coordinate system

* *LF e v B

condition for circular orbit:

20

Zentr

m vF

Lorentz force

centrifugal force

20 * *

m ve v B

The ideal circular orbit

ρ

s

θ

z

*Be

p

1.) The Magnetic Guide Field

Normalise magnetic field to momentum:

Dipole Magnets:

define the ideal orbit

homogeneous field created

by two flat pole shoes

convenient units:

2m

VsTB

c

GeVp

h

InB 0

29

8

9

2

10*7000

10*33.8

10*7000

3.81

m

sms

ceV

mVs

eTB 3.8

c

GeVp 7000

Example LHC:

m1

7000

3.8333.0

1

Be

p

p

Be1

field map of a storage ring dipole magnet

ρ

α

ds

„normalised bending strength“

2πρ = 17.6 km

≈ 66%

The Magnetic Guide Field

km53.2

cGeVp

TB

/3.0

1rule of thumb:

TB 8...1

Calculation of the Quadrupole Field:

gradient of a

quadrupole field:

court. K. Wille

coil N * I

integration path

0

2

2

1

1

0

0 * INHdsdsHdsHdsH Fe

INdsH *

now we know that 0

BH

rgrB *)(and we require

INdrrg

drB

dsH

aa

**

0 0

1

0 0 0

00

2

0 **2

r

INg

HFe = H0 / μFe H ┴ ds

μFe ≈ 1000

2.) Quadrupole Magnets:

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

E B = 0

tj

ygBxgB xy

LHC main quadrupole magnet

mTg /220...25simple rule:

)/(

)/(3.0

cGeVp

mTgk

ep

gk

/

2

02

r

nIg

y

B

x

Bxy

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:

normalise to momentum

p/e = Bρ

.../

´´

!3

1

/

´

!2

1

/

*

/

)(

0

0

ep

eg

ep

eg

ep

xg

B

B

ep

xB

...´´

!3

1

!2

1)(

3

2

2

2

0dx

egx

dx

Bdx

dx

dBBxB zz

zz

Example:

heavy ion storage ring TSR

Separate Function Machines:

Split the magnets and optimise

them according to their job:

bending, focusing etc

...!3

1

!2

1*

1

/

)( 32 nxmxxkep

xB

The Equation of Motion:

only terms linear in x, z taken into account dipole fields

quadrupole fields

* man sieht nur

dipole und quads linear

Equation of Motion:

z

x

ρ

s

θ

z

Consider local segment of a particle trajectory

... and remember the old days:

(Goldstein page 27)

radial acceleration:

22

2r

d da

dtdtIdeal orbit: , 0

dconst

dt

Force:

2

2dF m m

dt

2 /F mv

general trajectory: ρ ρ + x

vBex

mvx

dt

dmF z

2

2

2

)(

vBex

mvx

dt

dmF z

2

2

2

)(

xdt

dx

dt

d2

2

2

2

)(

zρ

s

z

x

)1(11 x

x

remember: x ≈ mm , ρ ≈ m … develop for small x

veBxmv

dt

xdm z)1(

2

2

2

1

1 … as ρ = const

2

2

Taylor Expansion

...)(*!2

)()(*

!1

)()()( 0

2

00

00 xf

xxxf

xxxfxf

guide field in linear approx.

0z

z

BB B x

x2 2

02(1 ) zBd x mv x

m ev B xxdt

: m

m

gxve

m

Bvexv

dt

xd 0

2

2

2

)1(

dt

ds

ds

dx

dt

dx*

dt

ds

dt

ds

ds

dx

ds

d

dt

ds

ds

dx

dt

d

dt

xd**

2

2

independent variable: t → s

vx

vds

dv

ds

dxvx

dt

xd*** 2

2

2

m

gxve

m

Bvexvvx 0

22 )1( : v 2

2

1( ) 0x x k

mv

gxe

mv

Bexx 0)1(

1

ep

gx

ep

Bxx

//

1 0

2

m v = p

1

/

0

ep

B

kep

g

/

xkx

x11

2

normalize to momentum of particle

Equation for the vertical motion:

0z k z

*

01

2

kk

no dipoles … in general …

quadrupole field changes sign

Remarks:

2

1( ) 0x k x

… there seems to be a focusing even without

a quadrupole gradient

„weak focusing of dipole magnets“*

xxk *1

02

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. (!)

Hard Edge Model:*

0)(*)()(

1)(

2sxsk

ssx

bending and focusing fields … are functions

of the independent variable „s“

Inside a magnet the focusing

properties are constant !

const1 constk

magl

eff BdslB0

*

„effective magnetic length“

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:

21K k* 0y K y

K k

)cos()sin()( 21 sasasx

)()sin()cos()( 22

2

2

1 sxsasasx K

)sin()cos()( 21 sKasKasx

general solution:

)sin()cos()( 21 sasasx

Hor. Focusing Quadrupole K > 0:

0 0

1( ) cos( ) sin( )x s x K s x K s

K

0 0( ) sin( ) cos( )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 = s0s = s1

01

*s

foc

sx

xM

x

x

0s

K

xaxx

xaxx

020

010

,)0(

,)0(determine a1 , a2 by boundary conditions:

1cosh sinh

sinh cosh

defoc

K l K lKM

K K l K l

hor. defocusing quadrupole:

drift space:

K = 0

1

0 1drift

lM

! with the assumptions made, the motion in the horizontal and vertical planes are

independent „ ... the particle motion in x & z is uncoupled“

s = s1s = 0

0*xKx

Ansatz: )sinh()cosh()( 21 sasasx

Remember from school:

,)cosh()( ssf )sinh()( ssf

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:

1q

q

f lkl

... focal length of the lens is much bigger than the length of the magnet

0l kl const=limes: while keeping

1 0

11xM

f

1 0

11zM

f

... useful for fast (and in large machines still quite accurate) „back on the envelope

calculations“ ... and for the guided studies !

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

court. K. Wille

2 1

s2,s1( )*

s s

x xM

x x

x(s)

s

0

typical values

in a strong

foc. machine:

x ≈ mm, x ≤ mrad

Tune: number of oscillations per turn

31.292

32.297

Relevant for beam stability:

non integer part

0.292*47.3 13.81kHz kHz

5.) Orbit & Tune:

HERA revolution frequency: 47.3 kHz

Question: what will happen, if the particle performs a second turn ?

x

... or a third one or ... 1010 turns

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: ( ) ( ) ( ) 0x 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 s

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

( )( )

sds

ss

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“

1

2 ( )y

dsQ

so

( ) ( )s L s

(i)

Beam Emittance and Phase Space Ellipse

(1) ( ) * ( ) *cos( ( ) )x s s s

(2) ( ) * ( )*cos( ( ) ) sin( ( ) )( )

x s s s ss

( )cos( ( ) )

* ( )

x ss

s

general solution of

Hill equation

from (1) we get

Insert into (2) and solve for ε

2 2( )* ( ) 2 ( ) ( ) ( ) ( ) ( )s x s s x s x s s x s

* 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( ) ( )

2

1 ( )( )

( )

s s

ss

s

2 2( )* ( ) 2 ( ) ( ) ( ) ( ) ( )s x s s x s x s s x s

Beam Emittance and Phase Space Ellipse

x´

x

x(s)

s

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 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

Bq

bending strength of a dipole:1 00.2998 ( )1

( / )

B Tm

p GeV c

focusing strength of a quadrupole:2 0.2998

( / )

gk m

p GeV c

2 0

2

20.2998

( / )r

nIk m

p GeV c a

focal length of a quadrupole:1

q

fk l

equation of motion:1 p

x Kxp

matrix of a foc. quadrupole: 2 1s sx M x

1cos sin

,

sin cos

K l K lKM

K K l K l

1 0

11

M

f

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

Bibliography