D

yx ,

Lattice Design in Particle AcceleratorsBernhard Holzer, CERN

Lattice Design: „… how to build a storage ring“

centrifugal force Lorentz force

p = momentum of the particle,ρ = curvature radiusBρ= beam rigidity

* /B p e

Example: heavy ion storage ring, 8 dipole magnets of equal bending strength

0.) Geometry of the 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

410B

B

2 2 **

Bdl pBdl

B q

The angle swept out in one revolution must be 2π, so

… for a full circle

is usually required !!Nota bene:

7000 GeV Proton storage ring dipole magnets N = 1232 l = 15 m q = +1 e

Teslae

sm

m

eVB

epBlNdlB

3.8103151232

1070002

/2

8

9

Example LHC:

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

1

0 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

Single particle trajectoryinside a lattice element is always (?) a part of a harmonic oscillation

1.) Focusing Forces: Single Element Matrices

€

x

x '

⎛

⎝ ⎜

⎞

⎠ ⎟f

= M *x

x'

⎛

⎝ ⎜

⎞

⎠ ⎟i

0 00

0 0 0

0

cos sin sin

( )cos (1 )sincos sin

ss s s s

s s s ss s s

s

M

s

* 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. * … !

2.) Transfer Matrix M ... as a function of the optics parameters

ψ turn = phase advance per period

turnsturnturns

turnsturnsturnsM

sincossin

sinsincos)(

Tune: Phase advance per turn in units of 2π

)(2

1

s

dsQ

Ls

s

turn s

ds

)(

3.) Periodic Lattices

x

0

s

4.) Transformation of α, β, γ

€

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟s2

=

m112 −2m11m12 m12

2

−m11m21 m12m21 + m22m11 −m12m22

m122 −2m22m21 m22

2

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟*

β

α

γ

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟s1

Relation between the two desriptions single particle trajectory

(x, x’), (y,y’) particle ensemble, called the beam

α, β, γ

… just as Big Ben

… and just as any harmonic pendulum

Most simple example: drift space

0

1*

' 0 1 'l

x l x

x x

0 0

0

( ) * '

'( ) '

x l x l x

x l x

particle coordinates

transformation of twiss parameters:

2

0

1 2

0 1 *

0 0 1l

l l

l

20 0 0( ) 2 * *s l l

Stability ...?

( ) 1 1 2trace M

A periodic solution doesn‘t exist in a lattice built exclusively out of drift spaces.

€

Mdrift =m11 m12

m21 m22

⎛

⎝ ⎜

⎞

⎠ ⎟=

1 l

0 1

⎛

⎝ ⎜

⎞

⎠ ⎟

HEP storage ring lattice

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

4521250 *.

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

Periodic Solution of a FoDo Cell

0.125 * 2π = 450

5.) The FoDo-Lattice

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...)

L

QF QFQD

L

QF QFQD

1cos( * ) sin( * )

,

sin( * ) cos( * )

q q

QF

q q

K l K lKM

K K l K l

1

0 1Driftd

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 8.206

0.061 0.707FoDoM

Putting the numbers in and multiplying out ...

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

matrices

Can we understand what the optics code is doing ?

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

1.) is the motion stable? ( ) 1.415FoDotrace M

2.) Phase advance per cell

3.) hor β-function

< 2

4.) hor α-function

€

M(s) =cosψ cell + α s sinψ cell β s sinψ cell

−γ s sinψ cell cosψ cell −α s sinψ cell

⎛

⎝ ⎜

⎞

⎠ ⎟

€

cosψ cell =1

2trace(M) = 0.707

€

cell = cos−1 1

2trace(M)

⎛

⎝ ⎜

⎞

⎠ ⎟= 45

€

=m12

sinψ cell

=11.611 m

€

=m11 − cosψ cell

sinψ cell

= 0

Matrix of a focusing quadrupole magnet:1

cos( * ) sin( * )

sin( * ) cos( * )QF

K l K lKM

K K l K l

Can we do a bit easier ?

6.) FoDo in thin lens approximation

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

1Q

Qf lkl

1 0

1 1M

f

, 0q qkl const l

the transfer matrix can be approximated by

but keeping its foc. properties

lD

LL

2

2

2 2

3 2 2

21 2 (1 )

2( ) 1 2

D DD

D D D

l ll

f fM

l l l

f f f

FoDo

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

€

sin(ψ cell /2) =Lcell

4 f

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

Remember:Exact calculation yields:

€

sin(ψ cell /2) =Lcell

4 f= 0.405

€

→ cell = 47.8o

→ β =11.4 m

€

→ cell = 45o

→ β =11.6 m

Stability in a FoDo structure

2

2

2 2

3 2 2

21 2 (1 )

2( ) 1 2

D DD

FoDo

D D D

l ll

f fM

l l l

f f f

Stability requires:

SPS Lattice

2)( MTrace

4cellL

f

2~4

2)(2

2

f

lMTrace d

For stability the focal length has to be larger than a quarter of the cell length ... don’t focus to strong !

Transformation Matrix in Terms of the Twiss Parameters

fl

fl

lf

l

MDD

DD

halfcell

~1~

~1

2

Transfer Matrix for half a FoDo cell (magnet parameters):

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

€

M =

β∨

β∧ cos

ψ cell

2β∨

β∧

sinψ cell

2

−1

β∧

β∨

sinψ cell

2

β∧

β∨ cos

ψ cell

2

⎛

⎝

⎜ ⎜ ⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟ ⎟

L

QF QFQD

L

QF QFQD

)sin(cossin)1(cos)(

sin)sin(cos

122122

1

21

12211221

1221121121

2

21

M

Transfer Matrix (Twiss parameters):

Solving for βmax and βmin and remembering that ….

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 proton bunch in a FoDo Cell

€

m22

m11

=ˆ β ( β

=1+ ld / ˜ f

1− ld / ˜ f =

1+ sin ψ cell /2( )1− sin ψ cell /2( )

€

m12

m21

= ˆ β ( β = ˜ f 2 =

ld2

sin2 ψ cell /2( )

!

!€

ˆ β =(1+ sin

ψ cell

2)L

sinψ cell

€

(1− sinψ cell

2)L

sinψ cell

€

sinψ cell

2=

ld

˜ f =

L

4 f

7.) scaling of Twiss parameters

8.) Beam dimension: Optimisation of the FoDo Phase advance:

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

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

x y

So for highest aperture we have to minimise the β-functionin both planes:

2x x y yr search for the phase advance μ that results in a minimum of the sum of

the beta’s

€

ˆ β +( β =

(1+ sinψ cell

2)L

sinψ cell

+(1− sin

ψ cell

2)L

sinψ cell

€

ˆ β +( β =

2L

sinψ cell

€

d

dψ cell

(2Lsinψ cell

) = 0

0 30 60 90 120 150 18010

12

14

16

18

2020

10

ges ( )

1800

€

L

sin2ψ cell

*cosψ cell = 0 → ψ cell = 90o

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

red curve: βmax

blue curve: βmin

as a function of the phase advance ψ

Electrons are different

1 36.8 72.6 108.4 144.2 1800

6

12

18

24

3030

0

max( )

min ( )

1801

€

d

dψ cell

( ˆ β ) =d

dψ cell

L(1+ sinψ cell

2)

sinψ cell

= 0 → ψ cell = 76o

problem of momentum „error“ in dipole magnets:

in case of non-vanishing momentum error we get an inhomogeneous differentail equation

0pp

2

1 1( )

px x k

p

9.) Dispersion:

general solution:

( ) ( ) ( )h ix s x s x s ( ) ( ) ( ) 0h hx s K s x s

1( ) ( ) ( )i i

px s K s x s

p

where the two parts xh and xi describe the solution of the hom. and inhom. equation

( )( ) i

pp

x sD s ( ) ( ) ( )

px s x s D s

p

normalising with respect to Δp/p we get the so-called dispersion function

€

x

x'

Δp / p

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟s

=

C S D

C' S' D'

0 0 1

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟*

x

x'

Δp / p

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟0

D and D‘ describe the disp[ersive properties of the lattice element (i.e. the magnet) and depend on it‘s bending and focusing properties.

Dispersion:

. ρ

xβ

Closed orbit for Δp/p > 0

( ) ( )ip

x s D sp

... and so what ... ?

Dispersion function D(s) * is that special orbit, an ideal particle would have for Δp/p = 1 * the orbit of any particle is the sum of the well known xβ and the dispersion* as D(s) is just another orbit it will be subject to the focusing properties of the lattice

e.g. matrix for a quadrupole lens:

€

M foc =cos( K s

1

Ksin( K s

− K sin( K s cos( K s

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟=

C S

C' S'

⎛

⎝ ⎜

⎞

⎠ ⎟

Calculate D, D´

1 1

0 0

1 1( ) ( ) ( ) ( ) ( )

s s

s s

D s S s C s ds C s S s ds

(proof: see appendix)

2

2

12

' ' ' 1 (1 )2

0 0 10 0 1

halfCell

fC S D

M C S Df f f

So we get the complete matrix including the dispersion terms D, D´

0 30 60 90 120 150 1800

2

4

6

8

1010

0.5

D max ( )

D min ( )

1801

Nota bene:

! small dispersion needs strong focusing → large phase advance !! ↔ there is an optimum phase for small β!!! ...do you remember the stability criterion? ½ trace = cos ψ ↔ ψ < 180°!!!! … life is not easy

boundary conditions for the transfer in a FoDo from the center of the foc. to the center of the defoc. quadrupole

1/ 2

ˆ

0 * 0

1 1

D D

M

€

ˆ D =l 2

ρ*

(1+1

2sin

ψ cell

2)

sin2 ψ cell

2

€

D∨

=l 2

ρ*

(1−1

2sin

ψ cell

2)

sin2 ψ cell

2

10.) Dispersion Suppressor SchemesBernhard Holzer: Lattice Design, CERN Acc. School:

CERN-2006-02

FoDo cell including the dispersive effect of dipoles

D

Example LHC

3

1...2

( ) 1...2

1 10

x mm

D s m

pp

Amplitude of Orbit oscillation contribution due to Dispersion ≈ beam size Dispersion must vanish at the collision point

1.) The straight forward one: Dispersion Suppressor Quadrupole Scheme use additional quadrupole lenses to match the optical parameters ... including the D(s), D´(s) terms

* Dispersion suppressed by 2 quadrupole lenses,

)s(),s(

)s(),s(

)s('D),s(D

yy

xx

6 additional quadrupole

lenses required

* β and α restored to the values of the periodic solution by 4 additional quadrupoles

Advantage:

! easy, ! flexible: it works for any phase advance per cell ! does not change the geometry of the storage ring, ! can be used to match between different lattice structures (i.e. phase advances)

Disadvantage:

! additional power supplies needed (→ expensive) ! requires stronger quadrupoles ! due to higher β values: more aperture required

Dispersion SuppressorQuadrupole Scheme

2(2 1)

2 2C

m nk

1sin , 0,2 ...

2 21

sin , 1,3 ...2 2

C

C

nk or

nk

The Missing Bend Dispersion Suppressor

conditions for the (missing) dipole fields:

Example:

phase advance in the arc ΦC = 60°number of suppr. cells m = 1 number of regular cells n = 1

m = number of cells without dipoles followed by n regular arc cells.

D

… turn it the other way round: Start at the IP with

and create dispersion – using dipoles - in such a way, that it fits exactly the conditions at the centre of the first regular quadrupoles:

ˆ( ) , ( ) 0D s D D s

1 1

0 0

1 1( ) ( ) ( ) ( ) ( )

s s

s s

D s S s C s ds C s S s ds

at the end of the arc: add m cells without dipoles followed by n regular arc cells.

and

The Half Bend Dispersion Suppressor

arc2

supr )2

(sin**2 cn

1)2

(sin 2 cn

0)sin( cn

condition for vanishing dispersion:

so if we require arcsupr *2

1

and equivalent for D‘=0

we get

...,3,1,* kkn c

strength of suppressor dipoles is half as strong as that of arc dipoles, δsuppr = 1/2 δarc

in the n suppressor cells the phase advancehas to accumulate to a odd multiple of π

Example: phase advance in the arc ΦC = 90°

number of suppr. cells n = 2

m

mrad

m

yx

yx

yx

17

105

55.0

,

10,

,

mAI p 584

2808

245.110

bn

kHzf

scmL 2

34 1100.1

Example: Luminosity run at LHC

yx

pp

b

II

nfeL

21

02

*4

1

€

R = L * Σreact

production rate of events is determined by the cross section Σreact and the luminosity that is

given by the design of the accelerator

11.) Lattice Design: Luminosity & Mini-Beta-Insertions

Lattice Design: Mini-Beta-Insertions

Twiss parameters in a drift:

2 2

2 2

0

2

' ' ' ' *

' 2 ' ' 'S

C SC S

CC SC S C SS

C S C S

1

' ' 0 1

C S sM

C S

with

„0“ refers to the position of the last lattice element

„s“ refers to the position in the drift

20 0 0

0 0

0

( ) 2

( )

( )

s s s

s s

s

starting in the middle of a symmetric drift where α = 0 we get

2

00

( )s

s

Nota bene: 1.) this is very bad !!! 2.) this is a direct consequence of the conservation of phase space density (... in our words: ε = const) … and there is no way out. 3.) Thank you, Mr. Liouville !!! Joseph Liouville

1809-1882

... clearly there is another problem !!!

Example: Luminosity optics at LHC: β* = 55 cm

for smallest βmax we have to limit the overall length and keep the distance “s” as small as possible.

But: ... unfortunately ... in general high energy detectors that are installed in that drift spaces

are a little bit bigger than a few centimeters ...

* calculate the periodic solution in the arc

* introduce the drift space needed for the insertion device (detector ...)

* put a quadrupole doublet (triplet ?) as close as possible

* introduce additional quadrupole lenses to match the beam parameters to the values at the beginning of the arc structure

parameters to be optimised & matched to the periodic solution: , ,

, ,x x x x

y y x y

D D

Q Q

-> 8 individually powered quad magnets are needed to match the insertion ( ... at least)

Mini-β Insertions: some guide lines

dublet mini-beta-structure (HERA-p) triplet mini-beta-structure (LHC-IP1)

Now in a mini β insertion:

Mini-β Insertions: Phase advance

By definition the phase advance is given by:1

( )( )

s dss

2

0 20

( ) (1 )s

s

Consider the drift spaces on both sides of the IP: the phase advance

of a mini β insertion is approximately π,

in other words: the tune will increase by half an integer.

2 20 0 00

1 1( ) arctan

1 /

L Ls ds

s

50 40 30 20 10 0 10 20 30 40 5090705030101030507090

lengt h (m)

90

90

L( )

5050 k L( )

)(s

/L

Mini-β Insertions: Betafunctions

A mini-β insertion is always a kind of special symmetric drift space.greetings from Liouville

at a symmetry point β is just the ratio of beam dimension and beam divergence.

* 0

x´

x

●

●

●

●

●

●

2*

*

1 1

**

*

**

The LHC Mini-Beta-Insertions

IP1 TA

S* Q1 Q2 Q3 D1

(1.38 T) TA

N*

D2 Q4(3.8 T)

Q5 Q6 Q7

4.5

K 1.9 KWarm

Separation/ Recombination

Matching Quadrupoles

Inner Triplet

1.9 K

ATLASR1

4.5

K

4.5

K

188 mm

Tertiary collimator

s

mini β opticsQ1

Q2

Q3

D1

High Light of the HEP-Year

ATLAS event display: Higgs => two electrons & two muons

€

R = L * Σreact ≈10−12b ⋅251

10−15b= some1000 H

production rate of events is determined by the cross section Σreact

and a parameter L that is given by the design of the accelerator:… the luminosity

€

Σreact ≈1 pb

€

L∫ dt ≈25 fb−1

The High light of the year

1 22 * *

0

*1*

4 b *x y

I IL

e f

The luminosity is a storage ring quality parameter and depends on beam size ( β !! ) and stored current

remember: 1b=10-24 cm2

sure there are...

* large β values at the doublet quadrupoles large contribution to chromaticity Q’ … and no local correction

* aperture of mini β quadrupoles limit the luminosity

* field quality and magnet stability most critical at the high β sections effect of a quad error:

beam envelope at the first mini β quadrupole lens in the HERA proton storage ring

keep distance „s“ to the first mini β quadrupole as small as possible

ls

s

dsssKQ

0

0 4

)()(

Are there any problems ?

€

€

Q =−1

4πK(s)β (s)ds∫

€

Qx =β x

* * rp * N p

2π γ p (σ x + σ y ) *σ x

the colliding bunches influence each other => change the focusing properties of the ring !! for LHC a strong non-linear defoc. effect

most simple case: linear beam beam tune shift

=> puts a limit to Np

Beam-Beam-Effect

observed particle losses when beams are brought into collision

12.) Luminosity Limits

-10 -5 0 5 10

amplitude (σ)

€

L =1

4πfrev N p nb( )

γ N p

εnβ *

⎛

⎝ ⎜

⎞

⎠ ⎟⋅F ⋅W

beam

-bea

m-f

orce

Luminosity Limits

2tan21

1

2

22

21

2

xx

s

F

<=>FLHC = 0.836

Φ

ρ1(x,y,s,-s0) ρ2(x,y,s,-s0)

x

... cannot be avoided... ϕ/2 has to increase with decreasing β*

)(2

)(2

22

1

212

xx

dd

eW

W factor due to beam offset

... can be avoided by careful tuningused for luminosity leveling (IP2,8)

€

L =1

4πfrev N p nb( )

γ N p

εnβ *

⎛

⎝ ⎜

⎞

⎠ ⎟⋅F ⋅W

Geometric Loss Factor F

bunches have to be separated at any parasitic encounter

Remember: 25ns Δs = 3.75m

crossing angle unavoidable: ϕ/2 = 142.5 μrad

13.) The LHC Luminosity Upgrade

Establish β* =10-15 cm at IP1 & 5 to reach a “virtual luminosity” of L = 2*1035

limits to overcome: matching quadrupoles -> ATSaperture in mini β quadrupoles ->Nb3Snlumi-loss due to crossing angle -> crab crossing

€

^

= 20 km !!!

€

5.5m injection optics

40cm pre-squeeze optics

15cm ATS optics

Standard low-beta-Squeeze

ATS-Squeeze

€

(s) = β *+s2

β *

Optics Transition Injection – Pre-Squeeze needs TLC optimisation

The LHC Luminosity Upgrade

gradient change for the squeeze without creating hysteresis problems

find a smooth and adiabatic transition without (too many) hysteresis problems,increase the crossing angle simultaneously to avoid beam beam encountersincrease the sextupoles to keep chromaticity compensated at any time

crossing angle bump for the case:β=15 cm, ε=3.0μm, +/- 10σwith location of parasitic 25ns encounters

Crossing Angles & Apertures

The LHC Luminosity Upgrade

Luminosity & Loss Factor

crossing angle ϕ = 590 μrad

★★★★★★

transv. deflecting cavity“crab-cavity”

The LHC Luminosity UpgradeCrab Crossing

leveling via closed Orbit Bumpsnon-linear beam beam effect !!

leveling via β* -> proof of principle, tricky

procedurefeed down -> orbit effect

2 vertices 20 vertices

A luminosity limit of its own:“Pile-up problem”

CAS

Trond-

heim

1.) Klaus Wille, Physics of Particle Accelerators and Synchrotron

Radiation Facilicties, Teubner, Stuttgart 1992 (Oxford Univ. Press)

2.) P. Bryant, The Principles of Circular Accelerators and Storage Rings, (Cambridge University Press)

3.) H. Wiedemann, Particle Accelerator Physics (Springer-Verlag, 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, CERN 2006-002 and CERN 2014- ???

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

18.) Bibliography

Appendix I: Dispersion: Solution of the Inhomogenious Equation of Motion

the dispersion function is given by

s~d)s~(S)s~(

*)s(Cs~d)s~(C)s~(

*)s(S)s(D

11

proof:

sdS

sCsdC

sSsD ~*)('~*)(')('

)~(

)~()(~)~(

)~(

1*)('

)~(

)~(*)(~)~(

)~(

1*)(')('

s

sSsCsdsS

ssC

s

sCsSsdsC

ssSsD

S

CsdS

sCC

SsdC

sSsD '~*)('''~*)('')(''

)'')(1

*)(''~*)('')('' CSCSsCsdC

sSsD

1)det( M

1~*)(''~*)('')('' sd

SsCsd

CsSsD

now the principal trajectories S and C fulfill the homogeneous equation

CKsCSKsS *)('',*)(''

and so we get:1~*)(*~*)(*)('' sd

SsCKsd

CsSKsD

1

)(*)('' sDKsD

1

)(*)('' sDKsD

qed.

Appendix II: Dispersion Suppressors... the calculation of the half bend scheme in full detail (for purists only)

1.) the lattice is split into 3 parts: (Gallia divisa est in partes tres)

* periodic solution of the arc periodic β, periodic dispersion D* section of the dispersion suppressor periodic β, dispersion vanishes

* FoDo cells without dispersion periodic β, D = D´ = 0

2.) calculate the dispersion D in the periodic part of the lattice

transfer matrix of a periodic cell:

0 00

0

0 0

00

(cos sin ) sin

( ) cos (1 )sin(cos sin )

SS

S

S S SS

S

M

for the transformation from one symmetriy point to the next (i.e. one cell) we have: ΦC = phase advance of the cell, α = 0 at a symmetry point. The index “c” refers to the periodic

solution of one cell.

cos sin ( )

1' ' ' sin cos '( )

0 0 10 0 1

C C C

Cell C CC

D lC S D

M C S D D l

0 0

1 1( ) ( )* ( ) ( )* ( )

( ) ( )

l l

D l S l C s ds C l S s dss s

The matrix elements D and D‘ are given by the C and S elements in the usual way:

0 0

1 1'( ) '( )* ( ) '( )* ( )

( ) ( )

l l

D l S l C s ds C l S s dss s

here the values C(l) and S(l) refer to the symmetry point of the cell (middle of the quadrupole) and the integral is to be taken over the dipole magnet where ρ ≠ 0. For ρ = const the integral over C(s) and S(s) is

approximated by the values in the middle of the dipole magnet.

Transformation of C(s) from the symmetry point to the center of the dipole:

cos cos( )2

m m Cm m

C C

C

sin( )

2C

m m C mS

where βC is the periodic β function at the beginning and end of the cell, βm its value at the middle of the dipole and φm the phase advance from the quadrupole lens to the dipole center.

Now we can solve the intergal for D and D’:

0 0

1 1( ) ( )* ( ) ( )* ( )

( ) ( )

l l

D l S l C s ds C l S s dss s

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

m C CC C m C m C m

C

L LD l

φm

ΦC /2

-φm

dipole magnet dipole magnet

( ) sin cos( ) cos( )2 2

C Cm C C m mD l

cos sin( ) sin( )2 2

C CC m m

remember the relations cos cos 2cos *cos2 2

x y x yx y

sin sin 2sin *cos2 2

x y x yx y

( ) sin *2cos *cos cos *2sin *cos2 2

C Cm C C m C mD l

( ) 2 *cos sin *cos * cos *sin2 2

C Cm C m C CD l

remember: sin 2 2sin *cosx x x2 2cos 2 cos sinx x x

2 2 2( ) 2 *cos 2sin *cos (cos sin )*sin2 2 2 2 2

C C C C Cm C mD l

I have put δ = L/ρ for the strength of the dipole

2 2 2( ) 2 *cos *sin 2cos cos sin2 2 2 2

C C C Cm C mD l

( ) 2 *cos *sin2

Cm C mD l

in full analogy one derives the expression for D‘:

As we refer the expression for D and D‘ to a periodic struture, namly a FoDo cell we require periodicity conditons:

*

1 1

C C

C C C

D D

D M D

and by symmetry: ' 0CD

With these boundary conditions the Dispersion in the FoDo is determined:

*cos *cos *2sin2

CC C m C m CD D

2cos*cos*/2)(' c

mcmlD

*cos / sin2

CC m C mD

This is the value of the periodic dispersion in the cell evaluated at the position of the dipole magnets.

3.) Calculate the dispersion in the suppressor part:

We will now move to the second part of the dispersion suppressor: The section where ... starting from D=D‘=0 the dispesion is generated ... or turning it around where the Dispersion of the arc is

reduced to zero.The goal will be to generate the dispersion in this section in a way that the values of the periodic cell

that have been calculated above are obtained.

0 0

1 1( ) ( )* ( ) ( )* ( )

( ) ( )

l l

D l S l C s ds C l S s dss s

The relation for D, generated in a cell still holds in the same way:

(A1)

φm

ΦC /2

-φm

dipole magnet dipole magnet

sup1

1sin * * cos( )*

2

nm

n C C r C C mi C

D n i

as the dispersion is generated in a number of n cells the matrix for these n cells is

cos sin

1sin cos '

0 0 1

C C C n

nn C C C n

C

n n D

M M n n D

sup1

1cos * * *sin( )

2

n

C r m C C C mi

n i

sup sup1 1

*sin * * cos((2 1) ) * *cos sin((2 1) )2 2

n nC C

n m C C r m m C r C mi i

D n i n i

remember: sin sin 2sin *cos2 2

x y x yx y

cos cos 2cos *cos

2 2

x y x yx y

sup1

* *sin * cos((2 1) )*2cos2

nC

n r m C C mi

D n i

sup1

* *cos sin((2 1) )*2cos2

nC

r m C C mi

n i

sup1 1

2 * *cos cos((2 1) )*sin sin((2 1) )*cos2 2

n nC C

n r m C m C Ci i

D i n i n

sup

sin *cos sin *sin2 2 2 22 * *cos sin cos *sin sin

2 2

C C C C

n r m C m C CC C

n n n n

D n n

sup 22 * *cossin *sin *cos cos *sin

2 2 2sin2

r m C m C C Cn C C

C

n n nD n n

set for more convenience x = nΦC/2

sup 22 * *cossin 2 *sin *cos cos 2 *sin

sin2

r m C mn

C

D x x x x x

sup 2 2 22 * *cos2sin cos *cos sin (cos sin )sin

sin2

r m C mn

C

D x x x x x x x

sup 22 * *cos*sin

2sin2

r m C m Cn

C

nD

and in similar calculations:

sup2 * *cos' *sin

sin2

r m C mn C

C

D n

This expression gives the dispersion generated in a certain number of n cells as a function of the dipole kick δ in these cells.

At the end of the dispersion generating section the value obtained for D(s) and D‘(s) has to be equal to the value of the periodic solution:

equating (A1) and (A2) gives the conditions for the matching of the periodic dispersion in the arc to the values D = D‘= 0 afte the suppressor.

(A2)

sup 22 * *cos cos*sin *

2sin sin2 2

r m C m C mn arc m C

C C

nD

2sup

sup

2 sin ( ) 12

2sin( ) 0

Cr arc

r arc

C

n

n

and at the same time the phase advance in the arc cell has to obey the relation:

* , 1,3, ...Cn k k