Lattice Design II: Insertions Bernhard Holzer, DESY

Transcription

1 Lattice Design II: Insertions Bernhard Holzer, DESY

2 .) Reminder: equation of motion ẑ x'' + K( s)* x= 0 K = k+ ρ θ ρ s x z single particle trajectory xs () x0 = M * x '( s ) x ' 0 e.g. matrix for a quadrupole lens: M foc cos( K s) sin( K s = K sin( ) cos( ) K K s K s 0 DORIS storage ring

3 .) Dispersion momentum error: p p 0 p x + x( k) = ρ p ρ general solution: x() s = x () s + x () s h xi () s Ds () = p p i p xs () = xβ () s + Ds () p x S D x x ' = ' S' D' * x' p 0 0 p p p S 0

4 Dispersion: the dispersion function D(s) is (...obviously) defined by the focusing properties of the lattice and is given by: = s~ )ds ~ D( s ) S( s )* ( ( s )* S( s~ )d ~ s ρ( s~ ) ρ( s~ )! weak dipoles large bending radius small dispersion Example: Drift M l D = 0 l 0 M D = = s~ )ds ~ D( s ) S( s )* ( ( s )* S( s~ )d ~ s ρ( s~ ) ρ( s~ ) = 0 = 0...in similar way for quadrupole matrices,!!! in a quite different way for dipole matrix (see appendix)

5 Dispersion in a FoDo ell: QF QD l D L!! we have now introduced dipole magnets in the FoDo: we still neglect the weak focusing contribution /ρ but take into account /ρ for the dispersion effect assume: length of the dipole = l D.) calculate the matrix of the FoDo half cell in thin lens approximation: in analogy to the derivations of ˆ, β β * thin lens approximation: * length of quad negligible * start at half quadrupole f = >>l kl Q lq 0, l D = L f = f Q

6 matrix of the half cell M = M * M * M Halfell QD B QF M Half ell M Half ell 0 0 l = * * 0 f f l S l f = = ' S' l l + f f calculate the dispersion terms D, D from the matrix elements D( s = s~ )ds ~ ) S( s )* ( ( s )* S( s~ )d ~ s ρ( s~ ) ρ( s~ )

7 l l s l D() l = l* * ds * * s ds ρ 0 f f ρ l l l l l l l l D() l = l * * = + ρ f f ρ ρ f ρ ρ f ρ D() l = l ρ in full analogy on derives for D : l l D'( s) = + ρ f

8 and we get the complete matrix including the dispersion terms D, D l l l f ρ S D l l l l Mhalfell = ' S' D' = + ( + ) f f ρ f boundary conditions for the transfer from the center of the foc. to the center of the defoc. quadrupole ˆ D D 0 = M / * 0

9 Dispersion in a FoDo ell ˆ l l D= D ( ) + ρ f 0 = l * Dˆ ( ) f + l ρ + l f Dˆ = l * ρ µ ( + sin ) µ sin D = l * ρ µ ( sin ) µ sin where µ denotes the phase advance of the full cell and l/f = sin(µ/) 0 D max ( µ ) D min ( µ ) D Dˆ µ 80 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 µ µ< 80!!!! life is not easy

10 3.) Lattice Design: Insertions... the most complicated one: the drift space Question to the auditorium: what will happen to the beam parameters α, β, γ if we stop focusing for a while? β S S β α = ' S' + S ' SS ' * α γ ' S' ' S' γ S 0 transfer matrix for a drift: S s M = = ' S ' 0 β() s = β α s+ γ s α() s = α γ s 0 0 α() s = α γ s refers to the position of the last lattice element s refers to the position in the drift

11 location of the waist: β beam waist: α = 0 l s given the initial conditions α 0, β 0, γ 0 : where is the point of smallest beam dimension in the drift or at which location occurs the beam waist? beam waist: α( s) = 0 α = γ * s 0 0 l α0 = γ 0 beam size at that position: γ () l = γ 0 + ( ) γ () l α l = = α() l = 0 β() l β() l β () l = γ 0

12 β-function in a Drift: let s assume we are at a symmetry point in the center of a drift. β() s = β α s + γ s as + α0 α0 = 0, γ0 = = β β 0 0 and we get for the β function in the neighborhood of the symmetry point s β() s = β0 + β 0!!! Nota bene:.) this is very bad!!!.) 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,

13 β-function in a Drift: If we cannot fight against Liouvuille theorem... at least we can optimise Optimisation of the beam dimension: β() l = β0 + l β Find the β at the center of the drift that leads to the lowest maximum β at the end: 0 d ˆ β = = 0 dβ β 0 0 β =l l! 0 ˆ β = β 0 l l * β 0 If we choose β 0 = l we get the smallest β at the end of the drift and the maximum β is just twice the distance l

14 ... clearly there is another problem!!! Example: Luminosity optics at HERA: β * = 8 cm for smallest β max we have to limit the overall length of the drift to L = *l L = 36 cm But:... unfortunately... in general high energy detectors that are installed in that drift spaces are a little bit bigger...

15 The Mini-β Insertion: R = L* Σ react. production rate of (scattering) events is determined by the cross section Σ react and a parameter L that is given by the design of the accelerator: the luminosity ZEUS detector: inelastic scattering event of e+/p

16 The Mini-β Insertion: Event rate of a collider ring: R= σ * Luminosity: given by the total stored beam currents and the beam size at the collision point (IP) R L L = I * I * 4πe f b σ * σ * * 0 x y How to create a mini β insertion: * symmetric drift space (length adequate for the experiment) * quadrupole doublet on each side (as close as possible) * additional quadrupole lenses to match twiss parameters to the periodic cell in the arc

17 Mini-β Insertions: Betafunctions A mini-β insertion is always a kind of special symmetric drift space. greetings from Liouville * α = 0 x α ε γ * γ + α * = = β β εγ α ε β σ * = ε * β εβ x * β * σ = σ * at a symmetry point β is just the ratio of beam dimension and beam divergence.

18 size of β at the second quadrupole lens (in thin lens approx): after some transformations and a couple of beer IP * l l l * ll β( s) = + * β + l * + l + f β f * *

19 Mini-β Insertions: Phase advance By definition the phase advance is given by: Φ () s = ds β () s Now in a mini β insertion: s β() s = β ( + ) 0 β0 L Φ L ( s) = ds arctan β + s / β = β Φ(s) fl ( ) s L 4 onsider 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.

20 Are there any problems?? sure there are... * large β values at the doublet quadrupoles large contribution to chromaticity ξ and no local correction ξ = o 4π { ( ) β( )} K s s ds * aperture of mini β quadrupoles limit the luminosity beam envelope at the first mini β quadrupole lens in the HERA proton storage ring * field quality and magnet stability most critical at the high β sections effect of a quad error: s0 + l k() s β() s ds Q= 4 s 0 keep distance s to the first mini β quadrupole as small as possible π

21 Mini-β Insertions: some guide lines * 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: α, β D, x x x x α, β Q, D Q y y x y 8 individually powered quad magnets are needed to match the insertion (... at least)

22 Dispersion Suppressors There are two comments of paramount importance about dispersion:! it is nasty!! it is not easy to get rid of it. remember: oscillation amplitude for a particle with momentum deviation xs () = x() s Ds ()* p β + p beam size at the IP σ = 8 µ m, σ = 3 µ m * * x y dispersion trajectory Ds () =.5 p p 5*0 m 4 x D 0.75 mm

23 Dispersion Suppressors s s D() s = Ss () sds () s () Ssds () ρ ρ s0 s0 β x β y optical functions of a FoDo cell without dipoles: D=0 D = 0 Remember: Dispersion in a FoDo cell including dipoles µ ( + sin ) Dˆ = l * ρ µ sin D µ ( sin ) = l * ρ µ sin

24 FoDo cell including the effect of the bending magnets Dispersion Suppressor Schemes I.) The straight forward one: use additional quadrupole lenses to match the optical parameters... including the D(s), D (s) terms * Dispersion suppressed by quadrupole lenses, * β and α restored to the values of the periodic solution by 4 additional quadrupoles D( s ), D' ( s ) β x( s ), α x( s ) β ( s ), ( s ) y α y 6 additional quadrupole lenses required

25 Dispersion Suppressors periodic FoDo structure matching section including 6 additional quadrupoles dispersion free section, regular FoDo without dipoles 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

26 II.)The Half Bend Dispersion Suppressor s s D() s = S() s () s ds () s S() s ds ρ ρ s0 s0 reate dispersion in such a way, that it fits exactly the conditions at the centre of the first regular quadrupoles: Ds ( ) = Dˆ, D ( s) = 0 nφ δsuprsin ( ) = δarc δsupr = sin( nφ ) = 0 δ arc in the n suppressor cells the phase advance has to accumulate to a odd multiple of π strength of suppressor dipoles is half as strong as that of arc dipoles, δ = l dipole /ρ nφ = k* π, k =,3,... Example: phase advance in the arc Φ = 60 number of suppr. cells n = 3 δ suppr = / δ arc

27 II.)The Missing Bend Dispersion Suppressor at the end of the arc: add m cells without dipoles followed by n regular arc cells. condition for dispersion suppression: nφ sin =, k = 0,... m+ n π Φ = (k + ) nφ sin =, k =,3... or Example: phase advance in the arc Φ = 60 number of suppr. cells m = number of regular cells n =

28 Resume.) Dispersion in a FoDo cell: small dispersion large bending radius short cells strong focusing µ (+ sin ) Dˆ = l * ρ µ sin.) hromaticity of a cell: small ξ weak focusing small β ξ = o{ () () () () ()} total 4 K s β s m s D s β π + s ds 3.) Position of a waist at the cell end: α 0 β 0 = values at the end of the cell l α0 = γ 0, β () l = γ 0 4.) β function in a drift β() s = β α s + γ s ) Mini β insertion small β short drift space required phase advance 80 β() l = β0 + l β 0

29

30 Dispersion Suppressors... the calculation in full detail (for purists only).) 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

31 .) calculate the dispersion D in the periodic part of the lattice transfer matrix of a periodic cell: M 0 S β S (cosφ + α 0 sin φ) βs β0 sinφ β 0 = ( α0 α S)cos φ ( + α 0α S)sinφ βs (cosφ α S sin φ ) β β S β 0 0 for the transformation from one symmetriy point to the next (i.e. one cell) we have: Φ = phase advance of the cell, α = 0 at a symmetry point. The index c refers to the periodic solution of one cell. cos Φ β sin Φ D ( l) S D M ell = ' S ' D' sin cos D'( l) = Φ Φ β The matrix elements D and D are given by the and S elements in the usual way: l D () l = S()* l ( s ) ds ()* l S( s) ds ρ( s) ρ( s ) 0 0 l D '( l) = S '( l) * ( s ) ds '( l) * S( s) ds ρ( s) ρ( s ) 0 0 l l

32 here the values (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 (s) and S(s) is approximated by the values in the middle of the dipole magnet. dipole magnet dipole magnet -φ m φ m Φ / Transformation of (s) from the symmetry point to the center of the dipole: m βm βm Φ Φ = cos Φ = cos( ± ϕm) Sm = βmβsin( ± ϕm) β β where β is the periodic β function at the beginning and end of the cell, β m its value at the middle of the dipole and φ m the phase advance from the quadrupole lens to the dipole center. Now we can solve the intergal for D and D : l D () l = S()* l ( s ) ds ()* l S( s) ds ρ( s) ρ( s ) 0 0 L β m Φ L Φ Dl ( ) = β sin Φ * * *cos( ± ϕm) cos Φ * βmβ *sin( ± ϕm) ρ β ρ l

33 Φ Φ D( l) = δ βmβ sin Φ cos( + ϕm) + cos( ϕm) I have put δ = L/ρ for the strength of the dipole x + y x y remember the relations cos x+ cos y = cos *cos x + y x y sin x+ sin y = sin *cos cos sin( Φ ϕm) sin( Φ Φ ϕm) + + Φ Φ D( l) = δ βmβ sin Φ *cos *cosϕm cos Φ *sin *cosϕm D( l) δ β mβ * cosϕ m sin * cos Φ * cos *sin Φ = Φ Φ remember: sin x = sin x*cos x cos cos sin x = x x D( l) δ βmβ *cosϕ sin Φ m *cos Φ (cos Φ sin Φ )*sin Φ =

34 Φ Φ Φ Φ D( l) = δ βmβ * cos ϕm *sin cos cos + sin Φ D() l = δ β β *cos ϕ *sin m m in full analogy one derives the expression for D : D( l) = δ β / β *cos ϕ *cos m m Φ As we refer the expression for D and D to a periodic struture, namly a FoDo cell we require periodicity conditons: and by symmetry: D ' = 0 D D D M * D = With these boundary conditions the Dispersion in the FoDo is determined: D Φ *cos Φ + δ β β *cos ϕ *sin = D m m

35 (A) D = δ β β *cos ϕ m m Φ /sin This is the value of the periodic dispersion in the cell evaluated at the position of the dipole magnets. 3.) alculate 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. dipole magnet dipole magnet -φ m φ m Φ / The relation for D, generated in a cell still holds in the same way: l D () l = S()* l ( s ) ds ()* l S( s) ds ρ( s) ρ( s ) 0 0 l

36 as the dispersion is generated in a number of n cells the matrix for these n cells is cos nφ βsin nφ D n n Mn = M = sin nφ cos nφ D ' n β 0 0 n β m Dn = β sin nφ * δsup r * cos( iφ Φ ± ϕm)* i= β n cos nφ * δ * β β *sin( iφ Φ ± ϕ ) sup r m m i= Φ Φ D *sin n * * cos((i ) ) * *cos n sin((i ) ) n n n = ββ m Φ δsupr ± ϕm βmβ δsupr Φ ± ϕm i= i= x + y x y x + y x y remember: sin x+ sin y = sin *cos cos x+ cos y = cos *cos Φ D = nφ i n δsup r * βmβ *sin n * cos(( ) )*cosϕm i= n Φ δ sup r * βmβ *cos nφ sin((i ) )* cosϕm i=

37 n n Φ Φ Dn = δsupr * βmβ *cosϕm cos((i ) )*sin nφ sin((i ) )*cos nφ i= i= nφ nφ nφ nφ sin *cos sin *sin Dn = δsup r * βmβ *cosϕm sinnφ cos nφ * Φ sin Φ sin δsupr * βmβ *cosϕm nφ nφ nφ Dn = sin nφ *sin *cos cos nφ *sin Φ sin set for more convenience x = nφ / δ * β β *cosϕ { sup r m m D sin *sin *cos cos *sin n = x x x x x Φ sin } δ * β β *cosϕ { sin cos *cos sin (cos sin )sin sup r m m D n = x x x x x x x Φ sin }

38 (A) and in similar calculations: D n D ' δ * β β *cosϕ nφ = *sin Φ sin sup r m m δ * β β *cosϕ sin *sin supr m m n = nφ Φ 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 (A) and (A) gives the conditions for the matching of the periodic dispersion in the arc to the values D = D = 0 afte the suppressor. D δ * β β *cosϕ nφ cosϕ = *sin = δ β β * sin sin sup r m m m n arc m Φ Φ

39 nφ = sin( nφ ) = 0 δsuprsin ( ) δarc δsupr = δ arc and at the same time the phase advance in the arc cell has to obey the relation: nφ = k* π, k =,3,...