Beam Transfer Lines. Brennan Goddard CERN

Transcription

1 Beam Transfer Lines Distinctions between transfer lines and circular machines Linking machines together Trajectory correction Emittance and mismatch measurement Blow-up from steering errors, optics mismatch and thin screens Phase-plane echange Brennan Goddard ERN

2 Injection, etraction and transfer An accelerator has limited dynamic range. hain of stages needed to reach high energy Periodic re-filling of storage rings, like LH Eternal eperiments, like NG ERN omple Transfer lines transport the beam between accelerators, and onto targets, dumps, instruments etc. LH: P: AD: IOLDE: PB: P: LINA: LEIR: NG: Large Hadron ollider uper Proton ynchrotron Antiproton Decelerator Isotope eparator Online Device Proton ynchrotron Booster Proton ynchrotron LINear Accelerator Low Energy Ring ERN Neutrino to Gran asso

3 Normalised phase space Transform real transverse coordinates, by N

4 Normalised phase space Real phase space Normalised phase space ε γ ma ε γ ε ε γ ma ε Area πε Area πε ma ε ma ε ε γ ε

5 General transport M y y s s n i n M M Beam transport: moving from s to s through n elements, each with transfer matri M i ( ) ( ) ( ) [ ] ( ) µ µ µ µ µ µ µ M Twiss parameterisation

6 ircular Machine ircumference L One turn L M M πq πq πq ( ) πq πq πq The solution is periodic Periodicity condition for one turn (closed ring) imposes,, D D This condition uniquely determines (s), (s), µ(s), D(s) around the whole ring

7 ircular Machine Periodicity of the structure leads to regular motion Map gle particle coordinates on each turn at any location Describes an ellipse in phase space, defined by one set of and values Matched Ellipse (for this location) ma a a γ Area πa γ ma a

8 ircular Machine For a location with matched ellipse (, ), an injected beam of emittance ε, characterised by a different ellipse ( *, * ) generates (via filamentation) a large ellipse with the original,, but larger ε, Turn Turn ε ο,, After filamentation Turn 3 Turn n>> ε > ε ο,, Matched ellipse determines beam shape

9 Transfer line ( ) ( ) ( ) [ ] ( ) µ µ µ µ µ µ µ M M No periodic condition eists The Twiss parameters are simply propagated from beginning to end of line At any point in line, (s) (s) are functions of One pass:

10 Transfer line On a gle pass there is no regular motion Map gle particle coordinates at entrance and eit. Infinite number of equally valid possible starting ellipses for gle particle transported to infinite number of final ellipses,, M L, Transfer Line Entry Eit,

11 Transfer Line Initial, defined for transfer line by beam shape at entrance,, Gaussian beam Non-Gaussian beam (e.g. slow etracted) Propagation of this beam ellipse depends on line elements A transfer line optics is different for different input beams

12 Transfer Line The optics functions in the line depend on the initial values Design functions in a transfer line functions with different initial conditions 5 Beta [m] [m] ame considerations are true for Dispersion function: Dispersion in ring defined by periodic solution ring elements Dispersion in line defined by initial D and D and line elements

13 Transfer Line 35 3 Another difference.unlike a circular ring, a change of an element in a line affects only the downstream Twiss values (including dispersion) - Unperturbed functions in a transfer line functions with modification of one quadrupole strength Beta [m] 5 5 % change in this QF strength [m]

14 Linking Machines Beams have to be transported from etraction of one machine to injection of net machine Trajectories must be matched, ideally in all 6 geometric degrees of freedom (,y,z,,φ,ψ) Other important constraints can include Minimum bend radius, maimum quadrupole gradient, magnet aperture, t, geology

15 Linking Machines γ γ Etraction Transfer Injection,, y, y (s), (s), y (s), y (s) s The Twiss parameters can be propagated when the transfer matri M is known M,, y, y

16 Linking Machines Linking the optics is a complicated process Parameters at start of line have to be propagated to matched parameters at the end of the line Need to match 8 variables ( D D and y y D y D y ) Maimum and D values are imposed by magnet apertures Other constraints can eist phase conditions for collimators, insertions for special equipment like stripping foils Need to use a number of independently powered ( matching ) quadrupoles Matching with computer codes and relying on miture of theory, eperience, intuition, trial and error,

17 Linking Machines For long transfer lines we can simplify the problem by designing the line in separate sections Regular central section e.g. FODO or doublet, with quads at regular spacing, ( bending dipoles), with magnets powered in series Initial and final matching sections independently powered quadrupoles, with sometimes irregular spacing. [m] 3 5 BET BETY [m ] P Initial matching section Regular lattice (FODO) (elements all powered in series with same strengths) Final matching section LH P to LH Transfer Line (3 km) Etraction point Injection point

18 Trajectory correction Magnet misalignments, field and powering errors cause the trajectory to deviate from the design Use small independently powered dipole magnets (correctors) to steer the beam Measure the response ug monitors (pick-ups) downstream of the corrector (π/, 3π/, ) orrector dipole Pickup Trajectory QF QD QF QD π/ Horizontal and vertical elements are separated H-correctors and pick-ups located at F-quadrupoles (large ) V-correctors and pick-ups located at D-quadrupoles (large y )

19 Trajectory correction Global correction can be used which attempts to minimise the RM offsets at the BPMs, ug all or some of the available corrector magnets. teering in matching sections, etraction and injection region requires particular care D and functions can be large bigger beam size Often very limited in aperture Injection offsets can be detrimental for performance

20 Trajectory correction Uncorrected trajectory. y growing as a result of random errors in the line. The RM at the BPMs is 3.4 mm, and y ma is.mm orrected trajectory. The RM at the BPMs is.3mm and y ma is mm

21 Trajectory correction ensitivity to BPM errors is an important issue If the BPM phase sampling is poor, the loss of a few key BPMs can allow a very bad trajectory, while all the monitor readings are ~zero orrection with some monitors disabled With poor BPM phase sampling the correction algorithm produces a trajectory with 85mm y ma Note the change of vertical scale

22 teering (dipole) errors Precise delivery of the beam is important. To avoid injection oscillations and emittance growth in rings For stability on secondary particle production targets onvenient to epress injection error in σ (includes and errors) a [σ] (( )/ε) ((γ )/ε) eptum a Bumper magnets kicker Mis-steered injected beam

23 teering (dipole) errors tatic effects (e.g. from errors in alignment, field, calibration, ) are dealt with by trajectory correction (steering). But there are also dynamic effects, from: Power supply ripples Temperature variations Non-trapezoidal kicker waveforms These dynamic effects produce a variable injection offset which can vary from batch to batch, or even within a batch. An injection damper system is used to minimise effect on emittance

24 Blow-up from steering error onsider a collection of particles with amplitudes A The beam can be injected with a error in angle and position. For an injection error a y (in units of sigma ε) the mis-injected beam is offset in normalised phase space by L a y ε Matched particles Mijected beam A L

25 Blow-up from steering error The particle coordinates in normalised phase space are L L For a general particle distribution, where A denotes amplitude in normalised phase space Matched particles A Mijected beam A L ε A /

26 Blow-up from steering error o if we plug in the coordinates. ( ) ( ) A L L ( ) L L ( ) A L L ( ) ε L L ε L Giving for the emittance increase ε A / ε L / ε ( a / )

27 Blow-up from steering error A numerical eample. onsider an offset a of.5 sigma for injected beam Mijected beam ε ε ( a / ).5ε ε.5 ε Matched Beam

28 Blow-up from betatron mismatch Optical errors occur in transfer line and ring, such that the beam can be injected with a mismatch. Filamentation will produce an emittance increase. In normalised phase space, consider the matched beam as a circle, and the mismatched beam as an ellipse. Mismatched beam Matched beam

29 Blow-up from betatron mismatch [ ] ) ( ) ( ), ( o o o φ φ φ φ ε φ φ ε A General betatron motion applying the normalig transformation for the matched beam an ellipse is obtained in normalised phase space γ,, characterised by γ, and, where

30 Blow-up from betatron mismatch λ λ b a A A Mismatched beam Matched Beam a b ( ) ( ) H H A b H H A a, ( ) γ H generally From the general ellipse properties where giving ( ) ( ) H H H H λ λ, A ) ( ), ( φ φ λ φ φ λ A A

31 Blow-up from betatron mismatch ) ( ) ( φ φ λ φ φ λ A A A ε ε λ λ ε ε H We can evaluate the square of the distance of a particle from the origin as The emittance is the average over all phases If we re feeling diligent, we can substitute back for λ to give ) ( ) ( ) ( ) ( λ λ ε φ φ λ φ φ λ φ φ λ φ φ λ ε A A A A where subscript refers to matched ellipse, to mismatched ellipse..5.5

32 Blow-up from betatron mismatch A numerical eample.consider b 3a for the mismatched ellipse λ b / a 3 Then Mismatched beam ε ε.67ε ( λ λ ) ε o a b3a Matched Beam

33 Emittance and mismatch measurement A profile monitor is need to measure the beam size E.g. beam screen (luminescent) provides D density profile of the beam Profile fit gives transverse beam sizes σ. In a ring, is known so ε can be calculated from a gle screen

34 Emittance and mismatch measurement Emittance measurement in a line needs 3 profile measurements in a dispersion-free region Measurements of σ,σ,σ, plus the two transfer matrices M and M allows determination of ε, and s s s M M 3 σ σ σ ε σ σ σ

35 Emittance and mismatch measurement γ γ ( ) ( ), W We have so that Ug we find σ σ σ σ ε σ,, ( ) ( ) ( ) ( ) ( ) ( ) / / / / / / / / σ σ σ σ W where ( ) ( ) ( ) [ ] ( ) µ µ µ µ µ µ µ where

36 Emittance and mismatch measurement ome algebra with above equations gives ( σ / σ ) / ( / ) W( / ) W / 4 And finally we are in a position to evaluate ε and ε σ W omparing measured o, with epected values gives measurement of mismatch

37 Blow-up from thin scatterer cattering elements are sometimes required in the beam Thin beam screens (Al O 3,Ti) used to generate profiles. Metal windows also used to separate vacuum of transfer lines from vacuum in circular machines. Foils are used to strip electrons to change charge state The emittance of the beam increases when it passes through, due to multiple oulomb scattering. s rms angle increase: s mrad 4. L [ ] Z inc. log c p[ MeV / c] Lrad L L rad c v/c, p momentum, Z inc particle charge /e, L target length, L rad radiation length

38 Blow-up from thin scatterer Each particles gets a random angle change s but there is no effect on the positions at the scatterer Ellipse after scattering s After filamentation the particles have different amplitudes and the beam has a larger emittance ε A / Matched ellipse

39 Ellipse after filamentation Matched ellipse s ε ε uncorrelated Blow-up from thin scatterer ( ) ( ) s s s s s s s s ε ε A A Need to keep small to minimise blow-up (small means large spread in angles in beam distribution, so additional angle has small effect on distn.)

40 Blow-up from charge stripping foil For LH heavy ions, Pb 53 is stripped to Pb 8 at 4.5GeV/u ug a.8mm thick Al foil, in the P to P line ε is minimised with low- insertion ( y ~5 m) in the transfer line Emittance increase epected is about 8% tripping foil TT optics beta beta Y 8 Beta [m] [m]

41 Emittance echange insertion Acceptances of circular accelerators tend to be larger in horizontal plane (bending dipole gap height small as possible) everal multiturn etraction process produce beams which have emittances which are larger in the vertical plane larger losses We can overcome this by echanging the H and V phase planes (emittance echange) Low energy machine After multi-turn etraction After emittance echange High energy machine y In the following, remember that the matri is our friend

42 Emittance echange Phase-plane echange requires a transformation of the form: y y m m m m m m m m y y A skew quadrupole is a normal quadrupole rotated by an angle. The transfer matri obtained by a rotation of the normal transfer matri M q : R - M q R where R is the rotation matri (you can convince yourself of what R does by checking that is transformed to y, y into - y, etc.)

43 Emittance echange For a thin-lens approimation M q o that M R R q For the case of 45º, this reduces to (where kl /f is the quadrupole strength) Normal quad 45º skew quad

44 Emittance echange The transformation required can be achieved with 3 such skew quads in a lattice, of strengths,, 3, with transfer matrices,, 3 y 3 A B The transfer matri without the skew quads is B A. [ ] ( ) ( ) [ ] φ φ φ φ φ φ φ and similar for y kew quad kew quad kew quad

45 Emittance echange With the skew quads the overall matri is M 3 B A M c ba34 c ba c34 b34a 3 34 c44 ba34 ( c b a ) ( c b a ) ( c b a ) ( c b a ) c c b a 3 c 34 b a 34 b 44 a 3 c 3 c 3 c b c b 33 b b a a a a b c 44 ba34 a 34 c34 3 c34 b a34 3 Equating the terms with our target matri form m m 3 4 m m 3 4 m m 3 3 m m 4 4 a list of conditions result which must be met for phase-plane echange.

46 Emittance echange c c c c c c c c b b b b b b a a a a a a ( c34 b34a ) ( c b a ) The simplest conditions are c c Looking back at the matri, this means that φ and φ y need to be integer multiples of π (i.e. the phase advance from first to last skew quad should be 8º, 36º, ) We also have for the strength of the skew quads 3 c b a c b a c33 b a 34 c44 b a 34

47 Emittance echange everal solutions eist which give M the target form. One of the simplest is obtained by setting all the skew quadrupole strengths the same, and putting the skew quads at symmetric locations in a 9º FODO lattice A B (A) From symmetry A B, and the values of and at all skew quads are identical. Therefore A B ( φ φ ) ( ) φ φ ( φ φ ) with the same form for y The matri is similar, but with phase advances of φ

48 Emittance echange ( ) ( ) y y y y y B A y s 3 ince we have chose a 9º FODO phase advance, φ φ y π/, and φ φ y π which means we can now write down A,B and : we can then write down the skew lens strength as i.e. 8º across the insertion in both planes For the 9º FODO with half-cell length L,, L L s D F

49 ummary Transfer lines present interesting challenges and differences from circular machines No periodic condition mean optics is defined by transfer line element strengths and by initial beam ellipse Matching at the etremes is subject to many constraints Trajectory correction is rather simple compared to circular machine Emittance blow-up is an important consideration, and arises from several sources Phase-plane rotation is sometimes required - skew quads

50 Keywords for related topics Transfer lines Achromat bends Algorithms for optics matching The effect of alignment and gradient errors on the trajectory and optics Trajectory correction algorithms VD trajectory analysis Kick-response optics measurement techniques in transfer lines Optics measurements including dispersion and p/p with >3 screens Different phase-plane echange insertion solutions