Introduction to Transverse Beam Dynamics

Transcription

1 Introduction to Transverse Beam Dynamics Bernhard Holzer, CERN BE-ABP The not so ideal world III.) Acceleration and and Momentum * Momentum Sread

2 The not so ideal world 5.) The / Problem ideal accelerator: all articles will see the same accelerating voltage. / nearly ideal accelerator: Cockroft Walton or van de Graaf / -5 Vivitron, Straßbourg, inner structure of the acc. section MP Tandem van de Graaf Accelerator at MPI for Nucl. Phys. Heidelberg

3 Linear Accelerator 98, Wideroe schematic Layout: Energy Gain er Ga : W q U sinω RF t drift tube structure at a roton linac 5 MHz cavities in an electron storage ring * RF Acceleration: multile alication of the same acceleration voltage; brillant idea to gain higher energies

4 Problem: anta rhei!!! (Heraklit: v. Chr.) ( Z, X, Y ) Examle: HERA RF: Bunch length of Electrons cm U t ν c 5 λ ν MHz λ 6 cm λ 6 cm sin(9 sin(84 o o ) ).994 U U 6. 3 tyical momentum sread of an electron bunch:. 3

5 6.) Disersion: trajectories for / Question: do you remember last session, age? sure you do ŷ Force acting on the article d mv F m ( x + ) e By v y dt x + x s remember: x mm, m develo for small x d x m dt mv x ( ) eb y v consider only linear fields, and change indeendent variable: t s B y B + B y x x x e B x ( ) + mv e x mv g + but now take a small momentum error into account!!!

6 Disersion: Disersion: develo for small momentum error + << x k xeg xeg eb B e x x + + kx x x + ) ( k x x + Momentum sread of the beam adds a term on the r.h.s. of the equation of motion. inhomogeneous differential equation. x k x k eb x x * * * ) ( * + + +

7 Disersion: x + x( k) general solution: x( s) x ( s) + x ( s) h i x ( s) + K( s) x ( s) h xi ( s) + K ( s) xi ( s) h Normalise with resect to /: xi ( s) D( s) Disersion function D(s) * is that secial orbit, an ideal article would have for / * 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

8 Disersion Examle: homogeneous diole field x β x β Closed orbit for / >. xi ( s) D( s) Matrix formalism: x( s) xβ ( s) + D( s) x C S x D + x C S x D x( s) C( s) x + S( s) x + D( s) s

9 or exressed as 3x3 matrix x C S D x x C S D x s β D Examle HERA x β... mm D( s)... m 3 Amlitude of Orbit oscillation contribution due to Disersion beam size Disersion must vanish at the collision oint! Calculate D, D s s D( s) S( s) C( s% ) ds% C( s) S( s) ds % % s s (roof: see aendix)

10 Examle: Drift M Drift l s s D( s) S( s) C( s% ) ds% C( s) S( s) ds % % s s Examle: Diole M foc cos( K s) sin( K s K sin( ) cos( ) K K s K s K s l B k M Diole l l cos sin l l sin cos l D( s) ( cos ) D ( s) sin l

11 Examle: Disersion, calculated by an otics code for a real machine x D D( s) * D(s) is created by the diole magnets and afterwards focused by the quadruole fields Mini Beta Section, no dioles!!! D(s) m s

12 Disersion is visible HERA Standard Orbit dedicated energy change of the stored beam closed orbit is moved to a disersions trajectory HERA Disersion Orbit x D D( s )* Attention: at the Interaction Points we require DD

13 7.) Momentum Comaction Factor: α article with a dislacement x to the design orbit ath length dl... dl ds + x x dl ds article trajectory design orbit dl + x ( s) ds circumference of an off-energy closed orbit x E l E o dl o + ds ( s) remember: x ( s) D ( s) E δ l E D ( s) o ds ( s) * The lengthening of the orbit for off-momentum articles is given by the disersion function and the bending radius.

14 Definition: l L δ ε α D( s) α ds L ( s) For first estimates assume: const. D( s) ds l Σ ( dioles) dioles D diole α L l Σ ( dioles) D π D L α π L D D R Assume: v c δt T δ l L ε α α combines via the disersion function the momentum sread with the longitudinal motion of the article.

15 Introduction to Transverse Beam Otics Bernhard Holzer IV.) Errors in Field and Gradient The überhaut nicht ideal world

16 8.) Quadruole Errors β D x

17 Quadruole Errors go back to Lecture I, age single article trajectory x x MQF * x x Solution of equation of motion x x cos( k lq ) + x sin( q k k l ) M QF cos( k k sin( l q k ) l q ) sin( k k cos( k l q lq) ), M thinlens f M turn M QF * M D * MQD * M D * M QF... x(s) Definition: hase advance of the article oscillation er revolution in units of π is called tune ψ turn Q π s

18 Matrix in Twiss Form Transfer Matrix from oint in the lattice to oint s : βs (cosψ s + α sinψ s ) β M (s) ( ) α α s cos(ψ s ( + αα s ) sinψ s β s β β (cos(ψ s α sinψ s ) βs β s β sinψ s For one comlete turn the Twiss arameters have to obey eriodic bundary conditions: β s sinψ turn cosψ turn + α s sinψ turn M (s) γ s sinψ s cosψ turn α s sinψ turn β ( s + L) β ( s ) α ( s + L) α ( s ) γ ( s + L) γ ( s )

19 Quadruole Error in the Lattice otic erturbation described by thin lens quadruole M dist cosψ turn + αsinψ turn β sinψ turn M k M kds γ sinψ turn cosψ turn αsinψ turn quad error ideal storage ring ẑ M dist cosψ + αsinψ kds(cosψ + αsinψ ) γ sinψ β sinψ kdsβ sinψ + cosψ αsinψ x s z rule for getting the tune Trace( M ) cosψ cosψ + kdsβ sinψ

20 Quadruole error Tune Shift ψ ψ + ψ kds β sinψ cos( ψ + ψ ) cosψ + remember the old fashioned trigonometric stuff and assume that the error is small!!! ψ kds β kdsβ sinψ cosψ cos ψ sinψ sin ψ cosψ + ψ and referring to Q instead of ψ: ψ πq Q s + l k( s) β ( s) ds π 4 s! 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 9 m arc quads: β 8 m!!!! β is a measure for the sensitivity of the beam

21 a quadruol error leads to a shift of the tune: s+ l kβ ( s) klquad β Q ds 4π 4π s Examle: measurement of β in a storage ring: tune sectrum.35 GI6 NR y x Qx,Qy y -3E-x k*l

22 Quadruole error: Beta Beat s+ l β β ( s ) β ( s) K cos( ψ s ψ s πq )ds sin πq s ( roof: see aendix ) β

23 9.) Chromaticity: A Quadruole Error for / Influence of external fields on the beam: ro. to magn. field & ro. zu / diole magnet α B dl / e x D ( s) D( s) focusing lens k g e article having... to high energy to low energy ideal energy

24 Chromaticity: Q' k g e + in case of a momentum sread: k eg e ( ) g k + + k k k which acts like a quadruole error in the machine and leads to a tune sread: Q kβ ( s) ds 4π definition of chromaticity: Q Q' ; Q ' k( s) β ( s) ds 4π

25 Where is the Problem?

26 Tunes and Resonances avoid resonance conditions: m Q x +n Q y +l Q s integer for examle: Q x

27 and now again about Chromaticity: Problem: chromaticity is generated by the lattice itself!! Q' is a number indicating the size of the tune sot in the working diagram, Q' is always created if the beam is focussed it is determined by the focusing strength k of all quadruoles Q ' 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-: Q' -7-8 /.5 * -3 Q Some articles get very close to resonances and are lost in other words: the tune is not a oint it is a ancake

28 Ideal situation: cromaticity well corrected, ( Q' ) Tune signal for a nearly uncomensated cromaticity ( Q' )

29 Tune and Resonances m*q x +n*q y +l*q s integer Qy.5 HERA e Tune diagram u to 3rd order and u to 7th order Qy.3 Qy. Qx. Qx.3 Qx.5 Homework for the oerateurs: find a nice lace for the tune where against all robability the beam will survive

30 Correction of Q': Need: additional quadruole strength for each momentum deviation /.) sort the articles according to their momentum x ( s) D ( s) D using the disersion function

31 Correction of Q':.) aly a magnetic field that rises quadratically with x (sextuole field) B x gxy ~ B y g ~ ( x y ) B y B x x y ~ gx linear rising gradient : Sextuole Magnet: normalised quadruole strength: gx % k sext m sext. x / e k sext m sext. D corrected chromaticity: Q ' { k( s) md ( s) } β ( s) ds 4π

32 sextuole magnet in a storage ring laced close to the quadruole lens quadruole magnet sextuole magnet N S S N

33 .) Insertions β x, y D

34 Insertions... the most comlicated one: the drift sace Question to the audience: what will haen to the beam arameters α, β, γ if we sto focusing for a while? β α γ S C SC S β CC ' SC ' + S ' C SS ' * α C' S ' C' S ' γ transfer matrix for a drift: C S s M C ' S ' β ( s) β α s + γ s α( s) α γ s γ ( s) γ

35 β-function in a Drift: let s assume we are at a symmetry oint in the center of a drift. β ( s) β α s + γ s as + α α, γ β β and we get for theβfunction in the neighborhood of the symmetry oint β ( s) β + s β!!! Nota bene:.) this is very bad!!!.) this is a direct consequence of the conservation of hase sace density (... in our words: ε const) and there is no way out. 3.) Thank you, Mr. Liouville!!! s * β s

36 But:... unfortunately... in general... clearly there is another roblem!!! high energy detectors that are Examle: installed Luminosity that otics drift at saces LHC: β * 55 cm arefor a little smallestβ bit bigger max we than have a to few limit centimeters the overall... length and kee the distance s as small as ossible.

37 Examle of a long Drift: The Mini-β Insertion: Luminosity: given by the total stored beam currents and the beam size at the collision oint (IP) L 4πe f rev n b I σ * x I σ * y β How to create a mini β insertion: * symmetric drift sace (length adequate for the exeriment) * make the beat values as small as ossible σ εβ *... where is the limit???

38 Mini-β Insertions: some guide lines * calculate the eriodic solution in the arc * introduce the drift sace needed for the insertion device (detector...) * ut a quadruole doublet (trilet?) as close as ossible * introduce additional quadruole lenses to match the beam arameters to the values at the beginning of the arc structure arameters to be otimised & matched to the eriodic solution: α, β D, x x x x α, β Q, D Q y y x y 8 individually owered quad magnets are needed to match the insertion (... at least)

39 and now back to the Chromaticity Q ' k( s) β ( s) ds 4π question: main contribution to Q' in a lattice? mini beta insertions

40 Resume Resume : : quadruole error: tune shift beta beat chromaticity π β π β 4 ) ( 4 ) ( ) ( quad l s s l s k ds s s k Q + ( )ds Q k s Q s s s l s s π ψ ψ β π β β ) ( cos ) ( sin ) ( + momentum comaction ds s s k Q ) ( ) ( 4 ' β π Q Q ' L l α δ ε R D D L π α

41 Aendix I: Disersion: Solution of the inhomogenious equation of motion Ansatz: s s D( s) S( s) C( s% ) ds% C( s) S( s) ds % % s s D ( s) S * C dt + S C C * S dt C S C S D ( s) S * dt C * dt C C S S D ( s) S * ds ~ + S C * ds ~ C C S S * ds ~ C * ds ~ + ( CS S C ) det M remember: for Cs) and S(s) to be indeendent solutions the Wronski determinant has to meet the condition W C C S S

42 and as it is indeendent of the variable s dw ds d ( CS SC ) CS SC K( CS SC) ds we get for the initital C, C conditions that we had chosen S, S W C C S S C ~ S ~ D S * ds C * + ds remember: S & C are solutions of the homog. equation of motion: S + K * S C + K * C C ~ S D K * S * + * * ~ ds K C + ds C ~ S * ~ D K S ds + C + ds D(s) D K * D + or D + K * D qed

43 Aendix II: Quadruole Error and Beta Function a change of quadruole strength in a synchrotron leads to tune sift: Q s + l s k ( s) β ( s) 4π ds k ( s)* l 4π quad * β GI6 NR.35 y x Qx,Qy tune sectrum....8 y -3E-x k*l tune shift as a function of a gradient change But we should exect an error in theβ-function as well shouldn t we???

44 Quadruole Errors and Beta Function a quadruole error will not only influence the oscillation frequency tune but also the amlitude beta function slit the ring into arts, described by two matrices A and B M turn B * A a A a b B b b b a a B ẑ s A s matrix of a quad error between A and B M dist m m * * m m * * B kds A M dist a B kdsa + a a kdsa + a M dist b a + b ( kdsa + ~ ~ ~ a )

45 the beta function is usually obtained via the matrix element m, which is in Twiss form for the undistorted case m β sin πq and including the error: m * ba + b a b a kds m β sin πq * ( ) m β sin πq a b kds As M* is still a matrix for one comlete turn we still can exress the element m in twiss form: * ( ) m ( β + dβ ) * sin π ( Q + dq ) Equalising () and () and assuming a small error β sin πq ab kds ( β + dβ ) * sin π ( Q + dq ) β sin πq ab kds ( β + dβ ) * sin πq cos πdq + cos πq sin πdq πdq

46 β sin πq ab kds β sin πq + β πdq cos πq + dβ sin πq + dβ πdq cos πq ignoring second order terms a b kds β πdq cos πq + dβ sin πq remember: tune shift dq due to quadruole error: (index refers to location of the error) β dq k ds 4π β kβds a b kds cos πq + dβ sin πq solve for dβ dβ π sin πq { a b + β β cos Q} kds exress the matrix elements a, b in Twiss form M β s ( cosψ s + α sinψ s ) βsβ sinψ s β ( α αs )cos ψ s ( + αα s )sinψ s β ( cosψ sin ) s αs ψ s β s sβ β

47 dβ π sin πq { a b + β β cos Q} kds a b β β sin ψ β β sin( πq ψ ) β β dβ π sin πq { sin ψ sin( πq ψ ) + cos Q} kds ψ after some TLC transformations cos( Q) π s+ l β β ( s ) β ( s) k cos( ( ψ s ψ s ) πq)ds sin πq s Nota bene:! the beta beat is roortional to the strength of the error k!! and to the β function at the lace of the error,!!! and to the β function at the observation oint, ( remember orbit distortion!!!)!!!! there is a resonance denominator