Lattice Design II: Insertions Bernhard Holzer, ERN β x, y D
.) Reminder: equation of motion x'' + K( s)* x= K = k+ ρ single particle trajectory considering both planes " x(s) % " x(s ) % $ ' $ ' $ x'(s) ' = M * $ x'(s ) ' $ y(s) ' $ y(s ) ' $ ' $ ' # y'(s) # y'(s ) e.g. matrix for a quadrupole lens: APS Light Source # cos( k s k sin( k s % ( % ( # x S x % k sin( k s cos( k s ( % ( M foc = % ( = % ' x S' x ( % cosh( k s % k sinh( k s ( % y S y ( ( % ( $ ' y S' y ' % ( $ k sinh( k s cosh( k s '
.) Dispersion momentum error: Δp p Δp xʹ ʹ + x( k) = ρ p ρ general solution: xs () = x() s + x() s h xi () s Ds () = Δp p i Δp xs () = xβ () s + Ds () p x S D x x' = ' S' D' * x' p p Δ Δ p p S
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 D l = l M D = = s~ )ds ~ D( s ) S( s )* ( ( s )* S( s~ )d ~ s ρ( s~ ) ρ( s~ ) #"! #"! = =...in similar way for quadrupole matrices,!!! in a quite different way for dipole matrix (see appendix)
Dispersion in a FoDo ell: QF QD l D L!! we have now introduced dipole magnets in the FoDo: à we still neglect the weak focusing contribution /ρ à but take into account /ρ for the dispersion effect assume: length of the dipole = l D alculate 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 l Q, l D = L!f = f Q
Matrix of the half cell M = M * M * M Halfell QD B QF! # M Halfell = # # "!f $! *# " % l! $ * # # % # "!f $ %! M Halfell = # " ' S S '! # $ = # # % # "!f!f +! f $ % calculate the dispersion terms D, D from the matrix elements = s~ )ds ~ D( s ) S( s )* ( ( s )* S( s~ )d ~ s ρ( s~ ) ρ( s~ )
l D(l) = l* ρ ( s )ds ( l ) f! f " ρ sds S(s) (s) (s) S(s) l D(l) = l l (l ρ f " ) ( l )* f " ρ * l = l ρ l3 " f ρ l ρ + l3 " f ρ D(l) = l ρ latest news: TLEP E=75 GeV = 8km l = 5m, ρ = 94m D = 4cm in full analogy on derives for D : D'(s) = l! # ρ + l " f! $ %
and we get the complete matrix including the dispersion terms D, D! # M halfell = # # " S D ' S ' D'! # $ # # = # # % # # # " l " f l l ρ l "f + l " f l ρ (+ l! f ) $ % boundary conditions for the transfer from the center of the foc. to the center of the defoc. quadrupole ˆ D D = M/ * β D
Dispersion in a FoDo ell D = ˆD( l " f )+ l ρ ˆ D = l ρ * (+ sinψ cell ) sin ψ cell D ( = l ρ * sinψ cell ) sin ψ cell = l " f * ˆD + l ρ (+ l " f ) where ψ cell denotes the phase advance of the full cell and l/f = sin(ψ/) D max ( µ ) D min ( µ ).5 8 6 4 D Dˆ 3 6 9 5 8 µ 8 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 < 8!!!! life is not easy
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 transfer matrix for a drift: S s M = = ' S ' β() s = β α s+ γ s α() s = α γ s γ() s = γ refers to the position of the last lattice element s refers to the position in the drift
location of the waist: β beam waist: α = l s given the initial conditions α, β, γ : where is the point of smallest beam dimension in the drift or at which location occurs the beam waist? beam waist: α() s = α = γ * s l α = γ beam size at that position: γ () l = γ + ( ) γ () l α l = = α() l = β() l β() l β () l = γ
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 + α α =, γ = = β β and we get for the β function in the neighborhood of the symmetry point s β() s = β + β!!! 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, 89-88
Alternative description: The Beam Matrix Once more unto the breach dear friends: Transformation of Twiss parameters just because it is mathematical more elegant... let s define a beam matrix: B = β α ) ( + ' α γ * " X = x % and a orbit vector: $ ', X T = x, x' # x' ( ) the product X T * B * X = (x, x' ) * γ α ) ( + * x ) ( + = γ ' α β * ' x' x + α x x' +β x ' = ε *... is constant transformation of the orbit vector: X = M * X and so we get: ε = X T * B * X = X T M T (M T ) B M M X = X T M T {(M T ) B M }M X
Transformation of Twiss parameters and using A T B T = (BA) T and A B = (BA) ε = X T M T {(M T ) (MB ) } M X = X T M T { MB M T } M X = (MX ) T { MB M T } MX = X T { MB M T } X but we know already that ε = const = X T * B * X = X T * B * X and in the end and after all we learn that... in full equivalence to... B = β % β( α ) ( + = M * B ' α γ * M T ' * α * ' * γ ) s % m m m m ( % β( ' * ' * = ' m m m m + m m m m ** α ' m m m m * ' * ) γ ) s
Transformation of Twiss parameters Example again the drift space... starting from α = " M = s % $ ' # B = β α ) ( + = β ) ( + ' α γ * ' /β * Beam parameters after the drift: " B = MB M T = s % " $ '* β % " $ '* % $ ' # # /β # s # % β + s β = % % $ s β s β β ( ( ( ' β = β + s β
A bit more in detail: β-function in a Drift If we cannot fight against Liouvuille theorem... at least we can optimise Optimisation of the beam dimension: β() l = β + l β Find the β at the center of the drift that leads to the lowest maximum β at the end: l l d ˆ β dβ l = = β β =l ˆ β = β * β If we choose β = l we get the smallest β at the end of the drift and the maximum β is just twice the distance l
But:... unfortunately... in general... clearly there is another problem!!! high energy detectors that are Example: installed Luminosity that optics drift at spaces LH: β * = 55 cm are for a little smallest bit bigger β max we than have a to few limit centimeters the overall... length and keep the distance s as small as possible.
Luminosity Minibeta Insertion R = L * Σ react L = 4πe f * nb I σ p x I σ p y Example: Luminosity run at LH β ε σ x, y x, y x, y =.55 m = 5 = 7 µ m rad m f n b =.45 khz = 88 I p = 584 ma L =. 34 cm s production rate of events is determined by the cross section Σ react and the luminosity that is given by the design of the accelerator
Mini-β Insertions: Betafunctions A mini-β insertion is always a kind of special symmetric drift space.!greetings from Liouville * α = * γ + α = = * β β x εγ α ε γ α ε β σ * ʹ = ε * β εβ x * β * σ = σ ʹ * at a symmetry point β is just the ratio of beam dimension and beam divergence.
size of β at the second quadrupole lens (in thin lens approx): after some transformations and a couple of beer β() l = β + l β l * ll β() s = + * β + l * + l+ f β f β * *
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 ( + ) β L Φ L ( s) = ds arctan β + s / β = β 9 φ(s) 7 5 3 3 5 7 9 5 4 3 3 4 5 L / β 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.
Are there any problems? sure there are... * large β values at the doublet quadrupoles! large contribution to chromaticity Q and no local correction Q " = 4π * aperture of mini β quadrupoles limit the luminosity K(s)β(s)ds 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: ΔQ = s + l Δ K( s) β ( s) ds π 4 s! keep distance s to the first mini β quadrupole as small as possible
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 ʹ 8 individually powered quad magnets are needed to match the insertion (... at least)
5.) Dispersion Suppressors There are two rules 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 Δ xs () = x() s Ds ()* p β + p beam size at the IP dispersion trajectory σ * =7µm D =.5m Δp p.* 4 x D =65µm
Dispersion Suppressors = s~ )ds ~ D( s ) S( s )* ( ( s )* S( s~ )d ~ s ρ( s~ ) ρ( s~ ) β x β y optical functions of a FoDo cell without dipoles: D = D = Remember: Dispersion in a FoDo cell including dipoles ˆ D = l ρ * $ + sinψ ' cell ) % ( sin ψ cell D = l ρ * % sinψ ( cell ' * ) sin ψ cell
FoDo cell including the effect of the bending magnets β D Dispersion Suppressor Schemes.) 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 ) 6 additional quadrupole β x( s ), α x( s ) lenses required β ( s ), ( s ) y α y
Dispersion Suppressor Quadrupole Scheme 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
.) The Missing Bend Dispersion Suppressor turn it the other way round: Start with D(s) =, D!(s) = and create dispersion using dipoles - in such a way, that it fits exactly the conditions at the centre of the first regular quadrupoles:! + D(s) = S(s)* ρ(!s) (!s)d!s (s)* ρ(!s) S(!s)d!s l ˆD = ρ * sin ψ cell # " at the end of the arc: add m cells without dipoles followed by n regular arc cells. sin ψ cell $ %, D' = empty cell at the HERA dispersion suppressor the He-pipe vacuumchamber are simply passed through.
.) The Missing Bend Dispersion Suppressor conditions for the (missing) dipole fields: nφ sin =, k =,... m+ n π Φ = (k + ) nφ sin =, k =,3... or m = number of cells without dipoles followed by n regular arc cells. β Example: phase advance in the arc Φ = 6 number of suppr. cells m = number of regular cells n = D
3.) The Half Bend Dispersion Suppressor condition for vanishing dispersion: nφ * δ supr *sin ( c ) = δ arc so if we require we get and equivalent for D = δ supr = * δ nφ sin ( c ) = sin( nφ c ) = arc sin(φ) sin ( Φ/) sin( φ) sin φ " " %% # # $ $ ''.8.6.4. nφ c = k * π, k =,3,.....4.6.8..4.6.8..4.6.8 Φ φ π 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 suppr = / δ arc Example: phase advance in the arc Φ = 6 number of suppr. cells n = 3
TLEP: km, 4 Arcs, dispersion free straight sections, half bend dispersion suppre
6.) Resume.) Dispersion in a FoDo cell: small dispersion large bending radius short cells strong focusing ˆ D = l ρ * $ + sinψ ' cell ) % ( sin ψ cell D = l ρ * % sinψ ( cell ' * ) sin ψ cell.) hromaticity of a cell: small Q weak focusing small β Q total " = 4π { K(s) md(s) }β(s)ds 3.) Position of a waist at the cell end: α, β = values at the end of the cell l α = γ β () l = γ 4.) β function in a drift β() s = β α s+ γ s 5.) Mini β insertion small β short drift space required phase advance 8 β() s s = β + β
Appendix I: Dispersion solution of the inhomogenious equation of motion the dispersion function is given by = s~ )ds ~ D( s ) S( s )* ( ( s )* S( s~ )d ~ s ρ( s~ ) ρ( s~ ) proof: ~ ~ ( ~ s ) ~ ~ S( ~ s ) D'( s) = S'( s)* ~ ( s ) ds + S( s)* ~ '( s)* ~ S( s ) ds ( s) ρ( s ) ρ( s ) ρ( s ) ρ( ~ s ) S D'( s) = S'( s)* ds ~ '( s)* d ~ s ρ ρ D ''( s) = S''( s)* ds ~ + S' ''( s)* ρ ρ S S ds ~ ' ρ ρ D' '( s) = S''( s)* ~ ds ''( s)* + )( S' S ρ ρ ~ S ~ D' '( s) = S''( s)* ds ''( s)* ρ ds + ρ ρ ') = det( M) = now the principal trajectories S and fulfill the homogeneous equation S ''( s) = K * S, ''( s) = K *
and so we get: ~ S ~ D' '( s) = K * S( s)* ds + K * ( s)* ρ ds + ρ ρ D' '( s) = K * D( s) + D' '( s) + K * D( s) = ρ ρ qed.
Appendix II: Dispersion Suppressors... the calculation of the half bend scheme 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 =
.) calculate the dispersion D in the periodic part of the lattice transfer matrix of a periodic cell: M S β S (cosφ+ αsin φ) βsβ sinφ β = ( α αs)cos φ ( + ααs)sinφ βs (cosφ αs sin φ) ββ β S for the transformation from one symmetriy point to the next (i.e. one cell) we have: Φ = phase advance of the cell, α = at a symmetry point. The index c refers to the periodic solution of one cell. cos Φ βsin Φ Dl ( ) S D Mell = ' 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: D(l) = S(l)* l ρ(!s) (!s)d!s (l)* l ρ(!s) S(!s)d!s D'(l) = S '(l)* l ρ(!s) (!s)d!s '(l)* l ρ(!s) S(!s)d!s
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. 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 ΔΦ= 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 : D(l) = S(l)* l ρ(!s) (!s)d!s (l)* l ρ(!s) S(!s)d!s D(l) = β sinφ * L ρ * β m β *cos( Φ ±ϕ m ) cosφ * L ρ β m β *sin(φ ±ϕ m )
Φ Φ Dl ( ) = 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 Dl ( ) = δ βmβ sin Φ* cos *cosϕm cos Φ* sin *cosϕm Dl ( ) δ βmβ *cosϕ m sin *cos Φ * cos *sin Φ = Φ Φ remember: sin x= sin x*cos x cos cos sin x= x x Dl ( ) δ βmβ *cosϕ sin Φ *cos Φ m (cos Φ sin Φ )*sin Φ =
Φ Φ Φ Φ Dl ( ) = δ βmβ *cos ϕm*sin cos cos + sin Φ Dl ( ) = δ β β *cos ϕ *sin m m in full analogy one derives the expression for D : D' ( l) = δ β / β *cosϕ m c m Φ *cos c 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 ' = D D Dʹ M* Dʹ = With these boundary conditions the Dispersion in the FoDo is determined: D Φ *cos Φ + δ β β *cos ϕ *sin = D m m
(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 = 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: D(l) = S(l)* l ρ(!s) (!s)d!s (l)* l ρ(!s) S(!s)d!s
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 n βm Dn = βsin nφ* δsupr* cos( iφ Φ ± ϕm)* i= β n cos nφ * δ * β β *sin( iφ Φ ± ϕ ) supr m m i= Φ Φ D *sin n * * cos((i ) ) * *cos n sin((i ) ) n n n = βmβ 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 δsupr* βmβ *sin n * cos(( ) )*cosϕm i= n Φ δsupr* βmβ *cos nφ sin((i ) )*cosϕm i=
n n Φ Φ Dn = δsupr* βmβ *cosϕm cos((i ) )*sinnφ sin((i ) )*cosnφ i= i= nφ nφ nφ nφ sin *cos sin *sin Dn = δsupr* βmβ *cosϕm sin nφ 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 { sup r m m D sin cos *cos sin (cos sin ) sin n = x x x x x x x Φ sin }
(A) and in similar calculations: D n D' δ * β β *cosϕ nφ = *sin Φ sin supr m m δ * β β *cosϕ supr m m n = *sinnφ Φ sin 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: à equating (A) and (A) gives the conditions for the matching of the periodic dispersion in the arc to the values D = D = afte the suppressor. D δ * β β *cosϕ nφ cosϕ = *sin = δ β β * sin sin supr m m m n arc m Φ Φ
nφ = sin( nφ ) = δsupr sin ( ) 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,...