Lattice Design in Particle Accelerators Bernhard Holzer, CERN. 1952: Courant, Livingston, Snyder: Theory of strong focusing in particle beams

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1 Lattice Deign in Particle Accelerator Bernhard Holzer, CERN β, y D 95: Courant, Livington, Snyder: Theory of trong focuing in particle beam

2 Lattice Deign: how to build a torage ring High energy accelerator à circular machine omewhere in the lattice we need a number of dipole magnet, that are bending the deign orbit to a cloed ring Geometry of the ring: centrifugal force = Lorentz force e*v * B = e* B = mv ρ mv ρ = p/ρ B*ρ = p / e p = momentum of the particle, ρ = curvature radiu Bρ= beam rigidity Eample: heavy ion torage ring TSR 8 dipole magnet of equal bending trength

3 .) Circular Orbit: defining the geometry α ρ d α = ρ dl ρ d α = B * dl B * ρ field map of a torage ring dipole magnet The angle wept out in one revolution mut be π, o Bdl α = = π Bdl = π* B* ρ p q for a full circle Nota bene: Δ 4 i uually required!! B B

4 Eample LHC: 7 GeV Proton torage ring dipole magnet N = 3 l = 5 m q = + e B dl N l B = π p / e 9 π 7 ev B 8 m 3 5 m 3 e = 8.3 Tela

5 Focuing Force: ingle particle trajectorie y + K* y = ρ K = k+ / ρ K = k hor. plane vert. plane y dipole magnet quadrupole magnet = ρ k = B p / q g p / q Eample: HERA Ring: Bending radiu: ρ = 58 m Quadrupol Gradient: g = T/m k = 33.64* -3 /m /ρ =.97 * -6 /m For etimate in large accelerator the weak focuing term /ρ can in general be neglected Solution for a focuing magnet y y y K K K ( ) = *co( * ) + *in( * ) y( ) = y * K *in( K * ) + y *co( K * )

6 Or written more convenient in matri form: y y = * y y Hor. focuing Quadrupole agnet Hor. defocuing Quadrupole agnet QF QD co( K * l) in( K * l) K = K in( K * l) co( K * l) coh( K * l) inh( K * l) K = K inh( K * l) coh( K * l) Drift pace Drift = = * * * *... lattice QF D QD D QF

7 8.) Tranfer atri ye we had the topic already general olution of Hill equation { } () = ε β()co ψ() + φ ε ʹ ( ) = α( )co { ψ( ) + φ} + in { ψ( ) + φ} β () remember the trigonometrical gymnatic: in(a + b) = etc ( ) () = ε β coψ coφ inψ inφ ε ʹ ( ) = [ α co ψ co φ α in ψ in φ + in ψ co φ + co ψ in φ ] β tarting at point () =, where we put Ψ() = co φ =, εβ α in φ = ( ʹ β + ) ε β inerting above

8 β ( ) = co + in + in ʹ β { ψ α ψ} { ββ ψ} β ʹ ( ) = {( α α) co ψ ( + α α)inψ} + { coψ αinψ} ʹ ββ β which can be epreed... for convenience... in matri form = ʹ ʹ β ( coψ + αinψ) ββ inψ β = ( α α)co ψ ( + αα)inψ β ( coψ in ) α ψ ββ β * we can calculate the ingle particle trajectorie between two location in the ring, if we know the α β γ at thee poition. * and nothing but the α β γ at thee poition. *! * Äquivalenz der atrizen

9 9.) Periodic Lattice β ( coψ + αinψ) ββ inψ β = ( α α)co ψ ( + αα)inψ β ( coψ in ) α ψ ββ β Thi rather formidable looking matri implifie coniderably if we conider one complete revolution DELTA Electron Storage Ring coψ turn + α inψ ( ) = γ inψ turn turn coψ β inψ turn α inψ turn turn ψ turn = + L d β ( ) ψ turn = phae advance per period Tune: Phae advance per turn in unit of π Q = π d β ( )

10 Stability Criterion: Quetion: what will happen, if we do not make too many mitake and your particle perform one complete turn? atri for turn: β inψ turn coψ turn + α inψ turn = coψ turn α inψ turn γ inψ turn α β = coψ + in ψ γ α atri for N turn: J N N = ( coψ + J inψ ) = co Nψ + J in Nψ The motion for N turn remain bounded, if the element of N remain bounded ψ = real co ψ Tr ( )

11 tability criterion. proof for the dibelieving collegue!! atri for turn: coψturn + α inψturn β inψturn = γinψturn coψturn αinψturn α β = coψ + inψ γ α atri for turn: ( I coψ + J inψ )( I coψ J inψ ) = + I J = I coψ coψ + IJ coψ inψ + JI inψ coψ + now I = I I J = α J I = γ α * γ β * α J β α = α γ inψ inψ α = γ β α β α I J = α β α β α γβ αβ βα J = * = = = I γ α γ α γα + αγ α γβ J I = I co( ψ + ψ ) + J in( ψ + ψ ) = I co(ψ ) + J in(ψ )

12 .) Tranformation of α, β, γ conider two poition in the torage ring:, ince ε = cont (Liouville): = m m m m " # % & Betafunction in a Storage Ring... remember W = CS -SC = inerting into ε ort via, and compare the coefficient to get... * = γ α β ε γ α β ε + ʹ + ʹ = + ʹ + ʹ = = * = m m m m # % & ( = m m # # = m + m # ε = β (m m ) + α (m m )(m m ) + γ (m m )

13 The Twi parameter α, β, γ can be tranformed through the lattice via the matri element defined above. in matri notation: β() = m β m m α + m γ α() = m m β + (m m + m m )α m m γ γ() = m β m m α + m γ % β( * α * & γ ) % m m m m ( % β( * * = m m m m + m m m m ** α & m m m m * * ) & γ )!.) thi epreion i important.) given the twi parameter α, β, γ at any point in the lattice we can tranform them and calculate their value at any other point in the ring. 3.) the tranfer matri i given by the focuing propertie of the lattice element, the element of are jut thoe that we ued to calculate ingle particle trajectorie.... and here tart the lattice deign!!!

14 ot imple eample: drift pace particle coordinate " drift = m m % " = l % # m m & # & l = * l l () = + l* ( l) = tranformation of twi parameter: β l l β α = l * α γ γ l β() = β * l α + l * γ Stability...? trace( ) = + = à A periodic olution doen t eit in a lattice built ecluively out of drift pace.

15 Lattice Deign: Arc: regular (periodic) magnet tructure: bending magnet! define the energy of the ring main focuing & tune control, chromaticity correction, multipole for higher order correction Straight ection: drift pace for injection, diperion uppreor, low beta inertion, RF cavitie, etc and the high energy eperiment if they cannot be avoided

16 3.) The FoDo-Lattice A magnet tructure coniting of focuing and defocuing quadrupole lene in alternating order with nothing in between. (Nothing = element that can be neglected on firt ight: drift, bending magnet, RF tructure... and epecially eperiment...) QF QD QF L Starting point for the calculation: in the middle of a focuing quadrupole Phae advance per cell µ = 45,! calculate the twi parameter for a periodic olution

17 Periodic Solution of a FoDo Cell β y β Output of the optic program: Nr Type Length Strength β α φ β z α z φ z m /m m /π m /π IP,,,6,, 5,95,, QFH,5 -,54,8,54,4 5,488 -,78,7 QD 3,5,54 5,488 -,78,7,8,54,66 3 QFH 6, -,54,6,,5 5,95,,5 4 IP 6,,,6,,5 5,95,,5 QX=,5 QZ=,5. 5 * π = 45.5 * π = 45

18 Can we undertand what the optic code i doing? matrice QF co( K * lq) in( K * lq) K = K in( K * lq) co( K * lq), Drift l = d trength and length of the FoDo element K = +/-.54 m - lq =.5 m ld =.5 m The matri for the complete cell i obtained by multiplication of the element matrice = * * * * FoDo qfh ld qd ld qfh Putting the number in and multiplying out... FoDo =.6.77

19 The tranfer matri for period give u all the information that we need!.) i the motion table? trace( FoDo ) =.45 <.) Phae advance per cell () = coψ + α inψ cell cell ) ( γ inψ cell β inψ cell *, coψ cell α inψ cell + coψ cell = trace() =.77 ψ cell = co & % trace() ) = 45 ( 3.) hor β-function 4.) hor α-function β = m inψ cell =.6 m α = m coψ cell inψ cell =

20 Can we do a bit eaier? We can... in thin len approimation! atri of a focuing quadrupole magnet: QF co( K * l) in( K * l) K = K in( K * l) co( K * l) If the focal length f i much larger than the length of the quadrupole magnet, f = >> l kl Q Q the tranfer matri can be aproimated uing kl q = cont, l q = f

21 4.) FoDo in thin len approimation l D L l D = L /!f = f Calculate the matri for a half cell, tarting in the middle of a foc. quadrupole: = * * halfcell QD / ld QF /! # halfcell = # "!f! & & * l D # % "! & * # % #!f " & & % ~ note: f denote the focuing trength ~ of half a quadrupole, o = f f " halfcell = # l D!f l D l D! f + l D!f % & for the econd half cell et f! -f

22 FoDo in thin len approimation atri for the complete FoDo cell " = # + l D!f l D l D! f l D!f % " * & # l D!f l D l D! f + l D!f % & " = # l D l!f D (+ l D )!f ( l D!f l D 3 f ) l D!!f % & Now we know, that the phae advance i related to the tranfer matri by coψ cell = trace() = *( 4l d f ) = l d f After ome beer and with a little bit of trigonometric gymnatic co( ) = co ( ) in ( / ) = in ( )

23 we can calculate the phae advance a a function of the FoDo parameter coψ cell = in (ψ cell /) = l d f in(ψ cell /) = l d / f = L cell f in(ψ cell /) = L cell 4 f Eample: 45-degree Cell L Cell = l QF + l D + l QD +l D =.5m+.5m+.5m+.5m = 6m /f = k*l Q =.5m*.54 m - =.7 m - in(ψ cell /) = L cell 4 f =.45 ψ cell = 47.8! β =.4 m Remember: Eact calculation yield: ψ cell = 45! β =.6 m

24 Stability in a FoDo tructure " FoDo = # l D l!f D (+ l D )!f ( l D!f l D 3 f ) l D!!f % & Stability require: SPS Lattice Trace( ) < 4ld Trace( ) = ~ f < f > L cell 4 For tability the focal length ha to be larger than a quarter of the cell length... don t focu to trong!

25 Tranformation atri in Term of the Twi Parameter Tranformation of the coordinate vector (, ) in a lattice ( ) ( ) =, ρ General olution of the equation of motion ( ) = ε * β ( ) *co( ψ ( ) + ϕ) () = ε β() *{ α()co(ψ() + ϕ) + in(ψ() + ϕ) } Tranformation of the coordinate vector (, ) epreed a a function of the twi parameter β (coψ + α inψ ) β = ( α )co ( + )in α ψ αα ψ ββ β β β β inψ (coψ α inψ )

26 Tranfer atri for half a FoDo cell: halfcell l = l D D ~ f ~ f l l + D D ~ f l D L Compare to the twi parameter form of β (coψ + α inψ ) β = ( α )co ( + )in α ψ αα ψ ββ β (coψ β β β inψ α inψ ) In the middle of a foc (defoc) quadrupole of the FoDo we allway have α =, and the half cell will lead u from β ma to β min " = m " m % = # m m & # β β coψ cell inψ cell β β β β in ψ cell β β coψ cell % &

27 Solving for β ma and β min and remembering that. in ψ cell = l d f = L 4 f ( ) ( ) m = ˆ β m β = + l d / f l d / f = + in ψ cell / in ψ cell / m = ˆ β β = f = m l d in ψ cell / ( ) } ˆ β = β = (+ in ψ cell )L inψ cell ( in ψ cell )L inψ cell!! The maimum and minimum value of the β-function are olely determined by the phae advance and the length of the cell. Longer cell lead to larger β typical hape of a proton bunch in a FoDo Cell ( Z, X, Y)

28 5.) Beam dimenion: Optimiation of the FoDo Phae advance In both plane a gauian particle ditribution i aumed, given by the beam emittance ε and the β-function σ = εβ HERA beam ize In general proton beam are round in the ene that ε ε y So for highet aperture we have to minimie the β-function in both plane: r = εβ + εβ y y typical beam envelope, vacuum chamber and pole hape in a foc. Quadrupole len in HERA

29 Optimiation of the FoDo Phae advance r = εβ + εβ y y earch for the phae advance µ that reult in a minimum of the um of the beta ˆ β + β = (+ in ψ cell )L inψ cell + ( in ψ cell )L inψ cell ˆ β + β = L inψ cell d dψ cell ( L inψ cell ) = βge ( µ ) L in ψ cell * coψ cell = ψ cell = 9! µ 8

30 Electron are different electron beam are uually flat, ε y - % ε! optimie only β hor ψ d ( ˆ d L(+ in cell β ) = ) = ψ cell = 76! dψ cell dψ cell inψ cell βma ( µ ) 8 βmin ( µ ) red curve: β ma blue curve: β min a a function of the phae advance ψ µ 8

31 Orbit ditortion in a periodic lattice field error of a dipole/ditorted quadrupole d Bd δ ( mrad) = = ρ p/ e the particle will follow a new cloed trajectory, the ditorted orbit: β ( ) ( ) = * β ( ~ ) co( ( ~ ) ( ) Q) d ~ in( Q) ( ~ ψ ψ π π ρ ) * the orbit amplitude will be large if the β function at the location of the kick i large β(~ ) indicate the enitivity of the beam! here orbit corrector hould be placed in the lattice * the orbit amplitude will be large at place where in the lattice β() i large! here beam poition monitor hould be intalled

32 Orbit Corrector and Beam Intrumentation in a Storage Ring Ela ring, Bonn *

33 Réumé.) Dipole trength p Bd = N * B * leff = π q l eff effective magnet length, N number of magnet.) Stability condition Trace( ) < for periodic tructure within the lattice / at leat for the tranfer matri of the complete circular machine 3.) Tranfer matri for periodic cell () = coψ cell + α inψ cell ) ( γ inψ cell β inψ cell *, coψ cell α inψ cell + α,β,γ depend on the poition in the ring, µ (phae advance) i independent of 4.) Thin len approimation QF = f Q f Q = kl QQ focal length of the quadrupole magnet f Q = /(k Q l Q) >> l Q

34 5.) Tune (rough etimate) ψ period = +L d β() Tune = phae advance in unit of π Q = N * ψ period π = π * d β() π * πr β = R β R, β average radiu and β-function Q R β 6.) Phae advance per FoDo cell (thin len appro in ψ cell = l d f = L cell 4 f Q L Cell length of the complete FoDo cell, f Q focal length of the quadrupole, µ phae advance per cell 7.) Stability in a FoDo cell L f Q > (thin len appro) 4 Cell 8.) Beta function in a FoDo cell (thin len appro) ˆ β = (+ in ψ cell )L cell inψ cell β = ( in ψ cell )L cell inψ cell L Cell length of the complete FoDo cell, µ phae advance per cell