1952: Courant, Livingston, Snyder:Theory of strong focusing in particle beams

D

yx ,β

Lattice Design in Particle AcceleratorsBernhard Holzer, DESY

Geometry of the ring:centrifugal force = Lorentz force

2

* * m ve v Bρ

=

p = momentum of the particle,ρ = curvature radius

Bρ= beam rigidity

* /mve B p ρρ

→ = =

* /B p eρ→ =

Example: heavy ion storage ring TSR8 dipole magnets of equal bending strength

LatticeLattice Design:Design: „… „… howhow to to buildbuild a a storagestorage ring“ring“

High energy accelerators circular machinessomewhere in the lattice we need a number of dipole magnets, that are bending the design orbit to a closed ring

CircularCircular Orbit:Orbit:„… „… definingdefining thethe geometrygeometry““

**

ds dl

B dlB

αρ ρ

αρ

= ≈

=

2 2*

Bdl* pBdl

B qα π π

ρ= = → =∫ ∫

The angle swept out in one revolutionmust be 2π, so

field map of a storage ring dipole magnet

ρ

α

ds

… for a full circle

Nota bene: 410BB

−Δ≈ is usually required !!

920 GeV Proton storage ringdipole magnets N = 416

l = 8.8mq = +1 e Tesla.

e*m.*sm**

eV**B

q/pB*l*NBdl

15588103416

109202

2

8

9

≈≈

∫ =≈

π

π

Example HERA:

00

0 0

'( ) *cos( * ) *sin( * )

'( ) * *sin( * ) ' *cos( * )

yy s y K s K sK

y s y K K s y K s

= +

= − +

„„ FocusingFocusing forcesforces … … singlesingle particleparticle trajectoriestrajectories““

'' * 0y K y+ =●

ρ

Solution for a focusing magnet

y21 /K k ρ= − +

K k=

hor. plane

vert. plane

qpB/

1=

ρdipole magnet

qpgk/

=quadrupole magnet

Example: HERA Ring:Bending radius: ρ = 580 mQuadrupol Gradient: g = 110 T/m

k = 33.64*10-3 /m2

1/ρ2 = 2.97 *10-6 /m2

For estimates in large accelerators the weak focusing term 1/ρ2 canin general be neglected

1cos( * ) sin( * )

sin( * ) cos( * )QF

K l K lKM

K K l K l

⎛ ⎞⎜ ⎟= ⎜ ⎟⎜ ⎟−⎝ ⎠

1cosh( * ) sinh( * )

sinh( * ) cosh( * )QD

K l K lKM

K K l K l

⎛ ⎞⎜ ⎟= ⎜ ⎟⎜ ⎟⎝ ⎠

10 1Drift

sM ⎛ ⎞

= ⎜ ⎟⎝ ⎠

Hor. focusing Quadrupole Magnet

Hor. defocusing Quadrupole Magnet

Drift space

1 1 1 2* * * * ...lattice QF D QD D QFM M M M M M=

Or written more convenient in matrix form: 0

⎟⎠

⎞⎜⎝

⎛=⎟

⎠

⎞⎜⎝

⎛'y

y*M

'yy

s

which can be expressed ... for convenience ... in matrix form0

⎛ ⎞ ⎛ ⎞=⎜ ⎟ ⎜ ⎟′ ′⎝ ⎠ ⎝ ⎠s

x xM

x x

( )

( )

0 00

0 0 0

0

cos sin sin

( ) cos (1 )sin cos sin

⎛ ⎞+⎜ ⎟

⎜ ⎟= ⎜ ⎟− − +⎜ ⎟−⎜ ⎟⎝ ⎠

ss s s s

s s s ss s s

s

M

s

β ψ α ψ β β ψβ

α α ψ α α ψ β ψ α ψββ β

* we can calculate the single particle trajectories between two locations in the ring, if we know the α β γ at these positions.

* and nothing but the α β γ at these positions.

* … !

VII.) Transfer Matrix MVII.) Transfer Matrix M

In the case of periodic lattices the transfer matrix can be expressedas a function of a set of periodic parameters α, β, γ

cos sin sin( )

sin cos( ) sinS S

S S

M sμ α μ β μγ μ μ α μ+⎛ ⎞

= ⎜ ⎟− −⎝ ⎠μ = phase advance per period:

( )

s L

s

dtt

μβ

+

= ∫

In terms of these new periodic parameters the solution of the equationof motion is

{ }

( ) * ( ) *cos( ( ) )

'( ) * sin( ( ) ) cos( ( ) )

y s s s

y s s s

ε β δ

ε δ α δβ

= Φ −

−= Φ − + Φ −

For stability of the motion in periodic latticestructures it is required that

( ) 2trace M <

PeriodicPeriodic LatticesLattices

VIII.) Transformation of VIII.) Transformation of αα, , ββ, , γγ

consider two positions in the storage ring: s0 , s0

⎛ ⎞ ⎛ ⎞= ⋅⎜ ⎟ ⎜ ⎟′ ′⎝ ⎠ ⎝ ⎠s s

x xM

x xsince ε = const:

2 2

2 20 0 0 0 0 0 0

2

2

x xx x

x x x x

ε β α γ

ε β α γ

′ ′= + +

′ ′= + +

express x0 , x´0 as a function of x, x´.

1

0

−⎛ ⎞ ⎛ ⎞= ⋅⎜ ⎟ ⎜ ⎟′ ′⎝ ⎠ ⎝ ⎠s

x xM

x x

... remember W = CS´-SC´ = 1

0

0

′ ′= −′ ′ ′= − +

x S x Sxx C x Cx

inserting into ε 2 22′ ′= + +x xx xε β α γ

2 20 0 0( ) 2 ( )( ) ( )′ ′ ′ ′ ′ ′ ′ ′= − + − − + −Cx C x S x Sx Cx C x S x Sxε β α γ

sort via x, x´and compare the coefficients to get ....

●ρ

sθ

z

●

s0

1− ′ −⎛ ⎞= ⎜ ⎟′−⎝ ⎠

S SM

C C

2 20 0 0( ) 2s C SC Sβ β α γ= − +

0 0 0( ) ( )′ ′ ′ ′= − + + −s CC SC S C SSα β α γ2 2

0 0 0( ) 2′ ′ ′ ′= − +s C S C Sγ β α γ

in matrix notation:

2 20

02 2

0

2

2

⎛ ⎞− ⎛ ⎞⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟ ′ ′ ′ ′= − + − ⋅⎜ ⎟ ⎜ ⎟⎜ ⎟

⎜ ⎟ ⎜ ⎟⎜ ⎟′ ′ ′ ′−⎝ ⎠ ⎝ ⎠⎝ ⎠s

C SC SCC SC CS SS

C S C S

ββα αγ γ

!

1.) this expression is important

2.) given the twiss parameters α, β, γ at any point in the lattice we can transform them and calculate their values at any other point in the ring.

3.) the transfer matrix is given by the focusing properties of the lattice elements, the elements of M are just those that we used to calculate single particle trajectories.

4.) go back to point 1.)

The new parameters α, β, γ can be transformed through the lattice via the matrix elements defined above.

2 2

2 20

2' ' ' ' *

' 2 ' ' 'S

C SC SCC SC S C SSC S C S

β βα αγ γ

⎛ ⎞−⎛ ⎞ ⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟= − + −⎜ ⎟⎜ ⎟ ⎜ ⎟

⎜ ⎟ ⎜ ⎟⎜ ⎟−⎝ ⎠ ⎝ ⎠⎝ ⎠

Question: „ What does that mean ???? “

... and here starts the lattice design !!!

Most simple example: drift space

⎟⎠

⎞⎜⎝

⎛=⎟

⎠

⎞⎜⎝

⎛=

101 l

'S'CSC

M

0

1*

' 0 1 'l

x l xx x

⎛ ⎞ ⎛ ⎞ ⎛ ⎞=⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠0 0

0

( ) * ''( ) '

x l x l xx l x

= +

=

particle coordinates

transformation of twiss parameters:

2

0

1 20 1 *0 0 1

l

l ll

β βα αγ γ

⎛ ⎞−⎛ ⎞ ⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟= −⎜ ⎟⎜ ⎟ ⎜ ⎟

⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠

20 0 0( ) 2 * *s l lβ β α γ= − +

Stability ...?

( ) 1 1 2trace M = + =A periodic solution doesn‘texist in a lattice built exclusivelyout of drift spaces.

→

LatticeLattice Design:Design:

Arc: regular (periodic) magnet structure: bending magnets define the energy of the ringmain focusing & tune control, chromaticity correction,multipoles for higher order corrections

Straight sections: drift spaces for injection, dispersion suppressors, low beta insertions, RF cavities, etc....

... and the high energy experiments if they cannot be avoided

TheThe FoDoFoDo--LatticeLattice

A magnet structure consisting of focusing and defocusing quadrupole lenses in alternating order with nothing in between.(Nothing = elements that can be neglected on first sight: drift, bending magnets, RF structures ... and especially experiments...)

Starting point for the calculation: in the middle of a focusing quadrupolePhase advance per cell μ = 45°,

calculate the twiss parameters for a periodic solution

L

QF QFQD

L

QF QFQD

Periodic solution of a FoDo Cell

Output of the optics program:

°= 4521250 π*.

xβ

yβ

Nr Type Length Strength βx αx φx βz αz φz

m 1/m2 m 1/2π m 1/2π

0 IP 0,000 0,000 11,611 0,000 0,000 5,295 0,000 0,000

1 QFH 0,250 -0,541 11,228 1,514 0,004 5,488 -0,781 0,007

2 QD 3,251 0,541 5,488 -0,781 0,070 11,228 1,514 0,066

3 QFH 6,002 -0,541 11,611 0,000 0,125 5,295 0,000 0,125

4 IP 6,002 0,000 11,611 0,000 0,125 5,295 0,000 0,125

QX= 0,125 QZ= 0,125

Can we understand, what the optics code is doing?

1cos( * ) sin( * ),

sin( * ) cos( * )

q q

QF

q q

K l K lKM

K K l K l

⎛ ⎞⎜ ⎟= ⎜ ⎟⎜ ⎟−⎝ ⎠

10 1Drift

d

lM

⎛ ⎞= ⎜ ⎟⎝ ⎠

strength and length of the FoDo elements K = +/- 0.54102 m-2

lq = 0.5 mld = 2.5 m

* * * *FoDo qfh ld qd ld qfhM M M M M M=

0.707 8.2060.061 0.707FoDoM

⎛ ⎞= ⎜ ⎟−⎝ ⎠

Putting the numbers in and multiplying out ...

The matrix for the complete cell is obtained by multiplication of the element matrices

matrices

The transfer matrix for 1 period gives us all the information that we need !

1.) is the motion stable? ( ) 1.415FoDotrace M = → < 2

2.) Phase advance per cell

cos sin sin( )

sin cos( ) sinM s

μ α μ β μγ μ μ α μ+⎛ ⎞

= →⎜ ⎟− −⎝ ⎠

1cos( ) * ( ) 0.7072

trace Mμ = =

1cos( * ( )) 452

arc trace Mμ = = °

3.) hor β-function

(1,2) 11.611sin( )M mβ

μ= =

4.) hor α-function

(1,1) cos( ) 0sin( )

M μαμ−

= =

Can we do it a little bit easier ?We can: … the „thin lens approximation“

Matrix of a focusing quadrupole magnet:1cos( * ) sin( * )

sin( * ) cos( * )QF

K l K lKM

K K l K l

⎛ ⎞⎜ ⎟= ⎜ ⎟⎜ ⎟−⎝ ⎠

If the focal length f is much larger than the length of the quadrupole magnet,

1Q

Qf lkl= >>

the transfer matrix can be aproximated using

1 01 1M

f

⎛ ⎞⎜ ⎟=⎜ ⎟⎝ ⎠

, 0q qkl const l= →

FoDoFoDo in in thinthin lenslens approximationapproximation

Calculate the matrix for a half cell, starting in the middle of a foc. quadrupole:

/ 2 / 2* *halfCell QD lD QFM M M M=

2

1

1

DD

halfCellD D

l lf

Ml l

f f

⎛ ⎞−⎜ ⎟= ⎜ ⎟

−⎜ ⎟+⎜ ⎟⎝ ⎠

%

% %

/ 2,

2Dl L

f f

=

=%

1 0 1 01* *1 11 10 1

DhalfCell

lM

f f

⎛ ⎞ ⎛ ⎞⎛ ⎞⎜ ⎟ ⎜ ⎟= ⎜ ⎟ −⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠% %

for the second half cell set f -f

lD

LL

note: denotes the focusing strengthof half a quadrupole, so f~

ff 2~=

FoDoFoDo in in thinthin lenslens approximationapproximation

2 2

1 1*

1 1

D DD D

D D D D

l ll lf f

Ml l l l

f f f f

⎛ ⎞ ⎛ ⎞+ −⎜ ⎟ ⎜ ⎟= ⎜ ⎟ ⎜ ⎟

− −⎜ ⎟ ⎜ ⎟− +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

% %

% % % %

Matrix for the complete FoDo cell:

2

2

2 2

3 2 2

21 2 (1 )

2( ) 1 2

D DD

D D D

l llf f

Ml l lf f f

⎛ ⎞− +⎜ ⎟

⎜ ⎟=⎜ ⎟

− −⎜ ⎟⎜ ⎟⎝ ⎠

% %

% % %

Now we know, that the phase advance is related to the transfer matrix by

2 2

2 2

4 21 1cos ( ) *(2 ) 122D Dl ltrace M

f fμ = = − = −

% %

After some beer and with a little bit of trigonometric gymnastics

2 2 2cos( ) cos ( ) sin ( / 2) 1 2sin ( )2 2x xx x= − = −

we can calculate the phase advance as a function of the FoDo parameter …

22

2

2cos( ) 1 2sin ( / 2) 1

sin( / 2) /2

D

CellD

lf

Ll ff

μ μ

μ

= − = −

= =

%

%%

Example: 45-degree Cell

LCell = lQF + lD + lQD +lD = 0.5m+2.5m+0.5m+2.5m = 6m

1/f = k*lQ = 0.5m*0.541 m-2 = 0.27 m-1

sin( / 2) 0.4054

CellLf

μ ≈ =

47.811.4m

μβ

→ ≈ °→ ≈

4511.6m

μβ= °=

Remember:Exact calculation yields:

sin( / 2)4

CellLf

μ =

StabilityStability in a in a FoDoFoDo structurestructure

2

2

2 2

3 2 2

21 2 (1 )

2( ) 1 2

D DD

FoDoD D D

l llf f

Ml l lf f f

⎛ ⎞− +⎜ ⎟

⎜ ⎟=⎜ ⎟

− −⎜ ⎟⎜ ⎟⎝ ⎠

% %

% % %

Stability requires:

4cellLf→ >

For stability the focal lengthhas to be larger than a quarter of the cell length !!

SPS Lattice

2~42)( 2

2

<−=flMTrace d

2)( <MTrace

Transformation Matrix in Terms of the Twiss parameters

Transformation of the coordinate vector (x,x´) in a lattice

General solution of the equation of motion

Transformation of the coordinate vector (x,x´) expressedas a function of the twiss parameters

●ρ

x

s1s2

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛

0

02,1 ')('

)(xx

Msxsx

ss

))(cos(*)(*)( ϕψβε += sssx

{ }))(sin())(cos()(*)()(' ϕψϕψαβε +++= sssssx

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−+−−

+=→

)sin(cossin)1(cos)(

sin)sin(cos

122122

1

21

12211221

1221121121

2

21

ψαψββ

ββψααψαα

ψββψαψββ

M

Transfer matrix for half a FoDo cell:

In the middle of a foc (defoc) quadrupole of the FoDo we allways have α = 0, and the half cell will lead us from βmax to βmin

Compare to the twissparameter form of M

lD

LL

1 2

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

+−

−=

fl

fl

lf

l

MDD

DD

halfcell~1~

~1

2

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−+−−

+=→

)sin(cossin)1(cos)(

sin)sin(cos

122122

1

21

12211221

1221121121

2

21

ψαψββ

ββψααψαα

ψββψαψββ

M

⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

−=⎟⎟

⎠

⎞⎜⎜⎝

⎛=

∨

∧

∨∧

∧∨

∧

∨

2cos

2sin1

2sin

2cos

'' μ

β

βμ

ββ

μββμ

β

β

SCSC

M

Solving for βmax and βmin and remembering that ….

1 sinˆ 1 /' 21 / 1 sin

2

D

D

l fSC l f

μβ

μβ∨

++= = =

− −

%

%

22

2

ˆ' sin

2

DlS fC

β β μ∨

= = =%

sin2 4

Dl Lff

μ= =%

(1 sin )2ˆ

sin

Lμ

βμ

+=

(1 sin )2

sin

Lμ

βμ

∨ −=

The maximum and minimum values of the β-function are solely determined by the phase advance and the length of the cell.

Longer cells lead to larger β

→

Z X, Y,( )

typical shape of a protonbunch in the HERA FoDo Cell

!

!

BeamBeam dimensiondimension::OptimisationOptimisation of of thethe FoDoFoDo Phase Phase advanceadvance::

In both planes a gaussian particle distribution is assumed, given by the beamemittance ε and the β-function

2x x y yr ε β ε β= +

σ εβ=

In general proton beams are „round“ in the sense that

x yε ε≈So for highest aperture we have to minimise the β-functionin both planes:

typical beam envelope, vacuum chamber and pole shape in a foc. Quadrupole lens in HERA

HERA beam size

OptimisingOptimising thethe FoDoFoDo phasephase advanceadvance

2x x y yr ε β ε β= +

search for the phase advance μ that results in a minimum of the sum of the beta’s

(1 sin )* (1 sin )*2 2ˆ

sin sin

L Lμ μ

β βμ μ

∨ + −+ = +

2( ) 0sind L

d μμ=

2 *cos 0 90sin

L μ μμ

= → = °

2ˆsin

Lβ βμ

∨

+ =

0 30 60 90 120 150 18010

12

14

16

18

2020

10

βges μ( )

1800 μ

electron beams are usually flat, εy ≈ 2 - 10 % εxoptimise only βhor

(1 sin )2ˆ( ) 0 76

sin

Ld dd d

μ

β μμ μ μ

+= = → ≈ °

red curve: βmax blue curve: βmin

as a function of the phase advance μ

ElectronsElectrons areare differentdifferent

1 36.8 72.6 108.4 144.2 1800

6

12

18

24

3030

0

βmax μ( )

βmin μ( )

1801 μ

field error of a dipole/distorted quadrupole

( )/

Bdsdsmradp e

δρ

→ = = ∫

the particle will follow a new closed trajectory, the distorted orbit:

* the orbit amplitude will be large if the β function at the location of the kick is largeindicates the sensitivity of the beam here orbit correctors should beplaced in the lattice

* the orbit amplitude will be large at places where in the lattice β(s)is large here beam position monitors should be installed

)~(sβ

● ●

o

●

Orbit Orbit distortionsdistortions in a in a periodicperiodic latticelattice

sdQsss

sQ

ssx ~))()~(cos(

)~(1)~(*

)sin(2)(

)( πψψρ

βπ

β∫ −−=

Elsa ring, Bonn

Orbit Orbit CorrectionCorrection and and BeamBeam InstrumentationInstrumentation in a in a storagestorage ringring

*

ResuméResumé: :

1.) Dipole strength:0* * 2eff

pBds N B lq

π= =∫leff effective magnet length, N number of magnets

2.) Stability condition: ( ) 2Trace M <

for periodic structures within the lattice / at least for the transfermatrix of the complete circular machine

3.) Transfer matrix for periodic cell

α,β,γ depend on the position s in the ring, μ (phase advance) is independent of s

cos ( )sin ( )sin( )

( )sin cos( ) ( )sins s

M ss s

μ α μ β μγ μ μ α μ+⎛ ⎞

= ⎜ ⎟− −⎝ ⎠

4.) Thin lens approximation:

1 01 1QF

Q

Mf

⎛ ⎞⎜ ⎟= ⎜ ⎟⎜ ⎟⎝ ⎠

1Q

Q Q

fk l

=

focal length of the quadrupole magnet fQ = 1/(kQlQ) >> lQ

5.) Tune (rough estimate):

RQβ

≈,R β

6.) Phase advance per FoDo cell(thin lens approx)

sin2 4

Cell

Q

Lf

μ=

LCell length of the complete FoDo cell, fQ focal length of thequadrupole, μ phase advance per cell

7.) Stability in a FoDo cell(thin lens approx) 4

CellQ

Lf >

8.) Beta functions in a FoDo cell(thin lens approx)

LCell length of the complete FoDo cell, μ phase advance per cell

(1 sin )2ˆ

sin

CellLμ

βμ

+=

(1 sin )2

sin

CellLμ

βμ

∨ −=

( )

s L

s

dtt

μβ

+

= ∫1 1 2: * * *

2 2 ( ) 2ds R RQ N

sμ π

βπ π β π β= = ≈ =∫o

Tune = phase advancein units of 2π

average radiusand β-function

ConclusionConclusion::* „the arc“ of a storage ring is usually built out of a periodic sequence

of single magnet elements eg. FoDo sections

* a first guess of the main parameters of the beam in the arc isobtained by the settings of the quadrupole lenses in this section

* we can get an estimate of the beam parameters using a selectionof „rules of thumb“

Usually the real beam properties will not differ too much from these estimatesand we will have a nice storage ring and a beautifull beam and everybody ishappy around.

And then someone comes and spoils it all by sayingsomething stupid like installing a tiny little piece of detectorin our machine …

INSERTIO

NS

INSERTIO

NS