Introduction to Transverse. Beam Optics II. Reminder: the ideal world. x x x x. Bernhard Holzer, DESY-HERA. 1 cos( Ks) sin( Ks.

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1 Introduction to Transverse Beam Otics II Bernhard Holzer, DESY-HERA Reminder: the ideal world x x = M x x s 0 ẑ M foc 1 cos( Ks) sin( Ks = K sin( ) cos( ) K K s K s 0 θ ρ s x z

2 The Beta Function Beam arameters of a tyical high energy ring: I = 100 ma articles er bunch: N Examle: HERA Bunch attern... question: do we really have to calculate some single article trajectories? xs () = ε β()cos( s ψ() s + φ) ψ () s s = 0 ds β() s

3 Beam Emittance and Phase Sace Ellise x ε = γ()* s x() s+ 2 α()() sxsx () s+ β() sx () s 2 2 εγ α ε β εβ x x Usually we get in a quadruole α ( s) = 0 Inside foc quadruoles β reaches maximum largest aerture needed x

4 the not so ideal world 1.) Emittance... so sorry ε const. According to Hamiltonian mechanics: q = osition = x hase sace diagram relates the variables q and = momentum = mcγβ Liouvilles Theorem: dq = const for convenience (i.e. because we are lazy bones) we use in accelerator theory: x dx ds dx dt dt ds β β x = = = where β = v/c = = x = dq const mc γβ dx mcγβ x dx ε = xdx 1 βγ the beam emittance shrinks during acceleration ε ~ 1 / γ

5 2.) Disersion Momentum error: Δ 0 Question: do you remember yesterday on age 11 sure you do: Force acting on the articles 2 2 = d ( + ) mv F m x ρ = eb 2 zv dt x + ρ 1 x eb x (1 ) 0 ρ ρ = mv + exg mv Δ = (1 ) mv +Δ neglecting higher order terms 1 Δ 1 x + x( k) = 2 ρ ρ Momentum sread of the beam adds a term on the r.h.s. of the equation of motion. inhomogeneous differential equation.

6 1 Δ 1 x + x( k) = 2 ρ ρ general solution: x() s = x () s + x () s h i x () s + K() s x () s = 0 h 1 Δ xi () s + K() s xi() s = ρ h Normalise with resect to /: xi () s Ds () = Δ Disersion function D(s) * is that secial orbit, an ideal article would have for / = 1 * the orbit of any article is the sum of the well known x β and the disersion * as D(s) is just another orbit it will be subject to the focusing roerties of the lattice

7 Disersion Examle: uniform diole field x β Closed orbit for / > 0. ρ Δ xi () s = D() s Matrix formalism: Δ xs () = xβ () s + Ds () x C S x Δ D = + Δ x C S x D xs () = Cs () x0 + Ss () x0 + Ds () s 0

8 x C S D x x = C S D x Δ Δ s 0 Examle HERA x β = mm Ds ( ) m Δ Amlitude of Orbit oscillation contribution due to Disersion beam size Calculate D, D s1 s1 1 1 D() s = Ss () Csds () Cs () Ssds () ρ ρ % % % % s0 s0

9 Examle: Drift M M Drift Drift = 1 l l 0 = s1 s1 1 1 D() s = Ss () Csds () Cs () Ssds () ρ ρ % % % % s0 s0 = 0 = 0 M Examle: Diole Diole l l cos ρ sin ρ ρ = 1 l l sin cos ρ ρ ρ l Ds () = ρ (1 cos ) ρ l D () s = sin ρ

10 3.) Momentum Comaction Factor: The disersion function relates the momentum error of a article to the horizontal orbit coordinate. inhomogeneous differential equation x + K( s)* x = 1 Δ ρ x β ρ general solution x( s) = x ( s) + D( s) β Δ Butitdoesmuchmore: it changes the length of the off - energy - orbit!!

11 article with a dislacement x to the design orbit ath length dl... dl ds = ρ + x ρ x dl ds article trajectory dl = 1 + x ρ ( s) ds ρ design orbit circumference of an off-energy closed orbit xδe lδe = o dl = o 1 + ds ρ ( s) remember: x ( s) = D( s) ΔE Δ δ l ΔE Δ D( s) = o ds ρ ( s) * The lengthening of the orbit for off-momentum articles is given by the disersion function and the bending radius.

12 Definition: δ l L ε = α c Δ 1 D( s) α c = o ds L ρ ( s) For first estimates assume: 1 ρ = const dioles D( s) ds = ldioles D diole α c = ldioles D 2πρ D L ρ = L ρ 2π α c D L D R Assume: v c δ T δ lε Δ = = α c T L α c combines via the disersion function the momentum sread with the longitudinal motion of the article.

13 Tune and Quadruoles Question: what will haen, if you do not make too many mistakes and your article erforms one comlete turn? Transfer Matrix from oint 0 in the lattice to oint s : M β s ( cosψs + α0sinψs) βsβ0 sinψs β0 = ( α0 αs)cos ψs (1 + α0αs)sinψs β0 ( cosψ sin ) s αs ψs ββ s 0 β s

14 Matrix for one comlete turn the Twiss arameters are eriodic in L: β( s + L) = β( s) α( s + L) = α( s) γ( s + L) = γ( s) M turn C S cos ψturn + αsin ψturn βsin ψturn = = C S γsinψturn cos ψturn αsinψturn Definition: hase advance of the article oscillation er revolution in units of 2π is called tune x(s) Q Δψ turn μ = = 2π 2π 0 s

15 Quadruole Error in the Lattice otic erturbation described by thin lens quadruole 1 0 cos ψ + αsin ψ β sin ψ turn turn 0 turn dist = Δk 0 = Δkds 1 γsinψturn cos ψturn αsinψturn M M M quad error ideal storage ring M dist = cosψ+ αsin ψ βsin ψ Δkds (cos ψ + αsin ψ) γsinψ Δkds βsin ψ+ cos ψ αsin ψ rule for getting the tune Trace( M) = 2cosψ= 2cosψ +Δkds βsinψ 0 0 ψ= ψ +Δψ 0 Quadruole error Tune Shift

16 sin 0 cos( ψ0+δ ψ) = cos ψ Δkdsβ ψ 0+ 2 remember the old fashioned trigonometric stuff and assume that the error is small!!! sin 0 ψ0 Δψ ψ0 Δ ψ= ψ Δkdsβ ψ 0+ cos cos sin sin cos 1 Δψ 2 Δ ψ = Δkdsβ 2 and referring to Q instead of ψ: ψ = 2πQ s0 + l Δ Δ Q= s 0 k() s β() s ds 4π

17 a quadruol error leads to a shift of the tune: s0+ l Δkβ () s Δklquad β Δ Q= ds 4π 4π s0! the tune shift is roortional to the β-function at the quadruole!! field quality, ower suly tolerances etc are much tighter at laces where β is large!!! mini beta quads: β 1900 arc quads: β 80!!!! β is a measure for the sensitivity of the beam Examle: measurement of β in a storage ring: GI06 NR y = x Qx,Qy tune sectrum y = -3E-12x k*l tune shift as a function of a gadient change

18 Chromaticity: ξ Influence of external fields on the beam: ro. to magn. field & ro. zu 1/ diole magnet α = B* dl e x D = Δ D()* s focusing lens k = g e article having... to high energy to low energy ideal energy

19 Chromaticity: ξ k = g e = 0 +Δ in case of a momentum sread: e g e Δ k = (1 ) g = k0 Δk +Δ Δ Δ k = k 0 0 whichacts like a quadruole error in the machine and leads to a tune sread: Δ 1 dq= k0 β () s ds 4π 0 definition of chromaticity: Δ Q = ξ Δ 0

20 Problem: chromaticity is generated by the lattice itself!! ξ is a number indicating the size of the tune sot in the working diagram, ξ is always created if the beam is focussed it is determined by the focusing strength k of all quadruoles 1 ξ : = * o k( s) β( s) ds 4π k = quadruole strength β = betafunction indicates the beam size and even more the sensitivity of the beam to external fields Examle: HERA HERA-: ξ = / = 0.5 *10-3 Q = Some articles get very close to resonances and are lost

21 Correction of ξ: 1.) sort the articles acording to their momentum xd ( s) = D( s) Δ 2.) aly a magnetic field that rises quadratically with x (sextuole field) B x = gxz % B z = g( x z ) 2 % B z x B x z = = gx % linear rising gradient : Sextuole Magnets: normalised quadruole strength: N S S N gx k sext = = m sext. x % / e Δ ksext = msext. D corrected chromaticity: ξ 1 = { k ( s) md( s) } β ( s) ds 4π o

22 Chromaticity in the FoDo Lattice 1 ξ = β()* s ksds () 4π β β Ÿ β-function in a FoDo ˆ β = μ (1+ sin ) L 2 sin μ β = μ (1 sin ) L 2 sin μ ξ 1 N π * ˆ β β f 4 Q ξ = Q μ μ L(1+ sin ) L(1 sin ) 1 1 N * * 2 2 4π f sinμ

23 using some TLC transformations... ξ can be exressed in a very simle form: ξ = μ 2 L sin 1 1 N 2 4π f sin μ Q μ L sin 1 1 ξ = N 2 4π f μ μ Q sin cos 2 2 remember... x x sin x = 2sin cos 2 2 ξ Cell = μ L tan 1 * 2 4π f μ Q sin 2 utting... μ sin = 2 L 4 f Q ξ Cell 1 μ = * tan π 2 contribution of one FoDo Cell to the chromaticity of the ring:

24 Chromaticity 1 ξ = o k ( s) β( s) ds 4π question: main contribution to ξ in a lattice? interaction region

25 Resume : beam emittance ε 1 βγ disersion orbit x( s) = x ( s) + D( s) β Δ momentum comaction δ l L ε = α c Δ 2π α c D L D R quadruole error Δ Q= s0 + l Δ s 0 k() s β() s ds 4π chromaticity 1 ξ : = * o k ( s) β( s) ds 4π

26 Question: after all the very exciting question: why do our articles not obey gravity and just fall down in the storage ring???? Answer will be discussed in the evening, having a good glass of red wine at the bar.