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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 exchange Brennan Goddard (presented by Malika Meddahi) CERN Injection, extraction and transfer • An accelerator has limited dynamic range • Chain of stages needed to reach high energy • Periodic re-filling of storage rings, like LHC • External experiments, like CNGS Transfer lines transport the beam between accelerators, and onto targets, dumps, instruments etc. LHC: SPS: AD: ISOLDE: PSB: PS: LINAC: LEIR: CNGS: Large Hadron Collider Super Proton Synchrotron Antiproton Decelerator Isotope Separator Online Device Proton Synchrotron Booster Proton Synchrotron LINear Accelerator Low Energy Ring CERN Neutrino to Gran Sasso CERN Complex Normalised phase space • Transform real transverse coordinates x, x’ by X x 1 N S x' X' X X' 1 S 1 S 1 S x S x S x' 0 x S x' Normalised phase space Real phase space x’ x'max 1 1 Normalised phase space X' X' max x X Area = p Area = p xmax x 2 2 x x' x'2 X max X 2 X' 2 General transport Beam transport: moving from s1 to s2 through n elements, each with transfer matrix Mi y s y s 2 x 1 x x2 x1 C S x1 x ' M12 x' C ' S ' x' 1 2 1 Twiss parameterisation M12 n M12 Mn 2 cos sin 1 1 1 1 2 1 2 cos 1 1 2 sin i 1 1 cos sin 2 2 1 2 sin Circular Machine Circumference = L One turn M12 M 0 L sin 2pQ cos 2pQ sin 2pQ 1 2 1 sin 2 p Q cos 2 p Q sin 2 p Q • The solution is periodic • Periodicity condition for one turn (closed ring) imposes 1=2, 1=2, D1=D2 • This condition uniquely determines (s), (s), (s), D(s) around the whole ring Circular Machine • Periodicity of the structure leads to regular motion – Map single 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) x’ x'max a a x 2 2 x x' x'2 1 2 x Area = pa xmax a 1 2 Circular Machine • For a location with matched ellipse (, , an injected beam of emittance , characterised by a different ellipse (*, *) generates (via filamentation) a large ellipse with the original , , but larger x’ , Turn 1 Turn 2 x , , After filamentation Turn 3 , , Turn n>>1 Matched ellipse determines beam shape Transfer line One pass: x2 x1 x ' M12 x ' 2 1 x2 x' 2 x1 x' 1 M12 2 cos sin 1 1 1 1 2 1 2 cos 1 1 2 sin 1 cos sin 2 2 1 2 sin • No periodic condition exists • The Twiss parameters are simply propagated from beginning to end of line • At any point in line, (s) (s) are functions of 1 1 Transfer line • On a single pass there is no regular motion – Map single particle coordinates at entrance and exit. – Infinite number of equally valid possible starting ellipses for single particle ……transported to infinite number of final ellipses… x’ 1, 1 x1 x' 1 1, 1 x Entry M0L x’ x2 x' 2 2, 2 x Transfer Line Exit 2, 2 Transfer Line • Initial , defined for transfer line by beam shape at entrance , x’ x’ , x x Gaussian beam Non-Gaussian beam (e.g. slow extracted) • Propagation of this beam ellipse depends on line elements • A transfer line optics is different for different input beams Transfer Line • The optics functions in the line depend on the initial values Horizontal optics 350 300 - Design x functions in a transfer line x functions with different initial conditions BetaX [m] 250 200 150 100 50 0 1500 2000 2500 S [m] • Same 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 3000 Transfer Line • Another difference….unlike a circular ring, a change of an element in a line affects only the downstream Twiss values (including dispersion) Horizontal optics 350 300 - Unperturbed x functions in a transfer line x functions with modification of one quadrupole strength BetaX [m] 250 10% change in this QF strength 200 150 100 50 0 1500 2000 2500 S [m] 3000 Linking Machines • Beams have to be transported from extraction of one machine to injection of next machine – Trajectories must be matched, ideally in all 6 geometric degrees of freedom (x,y,z,q,f,y) • Other important constraints can include – Minimum bend radius, maximum quadrupole gradient, magnet aperture, cost, geology Linking Machines Extraction Transfer 1x, 1x , 1y, 1y s x(s), x(s) , y(s), y(s) 2x, 2x , 2y, 2y Injection The Twiss parameters can be propagated when the transfer matrix M is known x2 x1 C S x1 x ' M12 x ' C ' S ' x ' 1 2 1 2 2CS S 2 1 2 C CC ' CS ' SC ' SS ' 1 2 2 C '2 2C ' S ' S '2 1 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 (x x Dx D’x and y y Dy D’y) – Maximum and D values are imposed by magnet apertures – Other constraints can exist • 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 mixture of theory, experience, intuition, trial and error, … 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 [m] – Initial and final matching sections – independently powered quadrupoles, with sometimes irregular spacing. 300 BETX BETY 250 200 150 100 50 0 0 250 SPS 500 Initial matching section 750 1000 1250 1500 1750 2000 2250 2500 Regular lattice (FODO) (elements all powered in series with same strengths) 2750 3000 Final matching section 3250 LHC SPS to LHC Transfer Line (3 km) Extraction point Injection point 3500 3750 4000 S [m ] 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 using monitors (pick-ups) downstream of the corrector (p/2, 3p/2, …) Pickup Corrector dipole QF Trajectory QF QD QD p/2 • Horizontal and vertical elements are separated • H-correctors and pick-ups located at F-quadrupoles (large x ) • V-correctors and pick-ups located at D-quadrupoles (large y) Trajectory correction • Global correction can be used which attempts to minimise the RMS offsets at the BPMs, using all or some of the available corrector magnets. • Steering in matching sections, extraction 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 Trajectory correction Uncorrected trajectory. y growing as a result of random errors in the line. The RMS at the BPMs is 3.4 mm, and ymax is 12.0mm Corrected trajectory. The RMS at the BPMs is 0.3mm and ymax is 1mm Trajectory correction • Sensitivity 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 Correction with some monitors disabled With poor BPM phase sampling the correction algorithm produces a trajectory with 185mm ymax Note the change of vertical scale Steering (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 • Convenient to express injection error in s (includes x and x’ errors) a [s] = ((X2+X’2)/) x2 2xx’+ x’2/ a X' Septum X Bumper magnets kicker Mis-steered injected beam Steering (dipole) errors • Static 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 Blow-up from steering error • Consider a collection of particles with amplitudes A • The beam can be injected with a error in angle and position. • For an injection error ay (in units of sigma = ) the mis-injected beam is offset in normalised phase space by L = ay Matched particles X' Misinjected beam A X L Blow-up from steering error • The new particle coordinates in normalised phase space are X new X 0 Lcosq ' X new X '0 Lsin q Matched particles • For a general particle distribution, where A denotes amplitude in normalised phase space A 2 X 2 X '2 X' Misinjected beam A q L A2 / 2 X Blow-up from steering error • So if we plug in the new coordinates…. 2 2 '2 A new X new X new X 0 Lcosq X '0 Lsin q 2 X 02 X '02 2L X 0cosq X '0sin q L2 2 A new X 02 X '02 2L X 0cosq X '0sin q L2 0 0 2 0 2L cosq X 0 sin q X '0 L2 2 0 L2 • Giving for the emittance increase new A new 2 / 2 0 L2 / 2 0 1 a 2 / 2 2 Blow-up from steering error A numerical example…. Consider an offset a of 0.5 sigma for injected beam X' Misinjected beam new 0 1 a 2 / 2 1.125 0 0.5 X Matched Beam 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. X' Mismatched beam X Matched beam Blow-up from betatron mismatch General betatron motion x 2 2 2 sin(f f o ), x' 2 2 2 cos(f f o ) 2 sin(f f o ) applying the normalising transformation for the matched beam X2 X ' 2 1 1 1 1 0 x2 1 x' 2 an ellipse is obtained in normalised phase space A 2 X 22 1 2 2 1 2 1 1 2 X' 22 2 2X 2 X' 2 2 1 2 1 1 2 1 2 characterised by new, new and new, where new 2 1 , 1 2 1 2 new 2, 1 new 1 2 1 1 2 2 1 2 2 Blow-up from betatron mismatch From the general ellipse properties a A 2 H 1 H 1 , b A 2 H 1 H 1 Mismatched beam X' where 1 H new new 2 2 1 1 2 1 2 1 2 2 2 1 2 1 a X A giving 1 2 H 1 H 1 , X new A sin( f f1 ), 1 1 2 X' new b H 1 H 1 1 A cos(f f1 ) generally aA b A Matched Beam Blow-up from betatron mismatch We can evaluate the square of the distance of a particle from the origin as 2 2 A 2new X new X' new 2 A 02 sin2 (f f1 ) 1 2 A 02 cos 2 (f f1 ) The new emittance is the average over all phases new 1 1 1 2 A new 2 A 02 sin 2 (f f1 ) 2 A 02 cos 2 (f f1 ) 2 2 0.5 0.5 1 2 sin 2 (f f1 ) 2 cos 2 (f f1 ) 1 A 02 2 1 2 1 0 2 2 If we’re feeling diligent, we can substitute back for to give 2 1 2 1 1 1 2 1 2 new 0 2 H 0 0 1 2 2 2 2 1 2 1 where subscript 1 refers to matched ellipse, 2 to mismatched ellipse. Blow-up from betatron mismatch A numerical example….consider b = 3a for the mismatched ellipse b/a 3 X' Mismatched beam Then new 0 2 1 2 1 2 1.67 0 a b=3a o X Matched Beam Emittance and mismatch measurement • A profile monitor is needed to measure the beam size – e.g. beam screen (luminescent) provides 2D density profile of the beam • Profile fit gives transverse beam sizes s. • In a ring, is ‘known’ so can be calculated from a single screen Emittance and mismatch measurement • Emittance measurement in a line needs 3 profile measurements in a dispersion-free region • Measurements of s0,s1,s2, plus the two transfer matrices M01 and M12 allows determination of , and s0 s1 M12 s2 M23 s 02 s 12 s 22 0 1 2 s0 s1 s2 Emittance and mismatch measurement 2C1 S 1 S 12 0 1 C12 C C ' C S ' S C ' S S ' 0 1 1 1 1 1 1 1 1 1 1 C1 ' 2 2C1 ' S 1 ' S 1 ' 2 0 We have where so that Using we find where C1 C ' 1 1 cos sin 0 S1 0 S1' 1 11 0 1 cos 1 01 sin 1 , 1 C 0 2C1S1 0 S12 s 02 0 , 0, 2 1 s 1 1 s0 0 2 2 0 0 cos sin 1 1 0 1 sin 2 C 0 2C2 S 2 0 s 2 2 s0 2 2 2 0 1 2 0 0W s 2 / s 0 2 / S 22 s 1 / s 0 2 / S12 C2 / S 2 2 C1 / S1 2 W C1 / S1 C2 / S 2 S 22 0 1 2 0 Emittance and mismatch measurement Some algebra with above equations gives 0 1 s 2 / s 0 2 / S22 C2 / S2 2 WC2 / S2 2 W 2 / 4 And finally we are in a position to evaluate and 0 s 02 0 1 2 0 0W Comparing measured o, 0 with expected values gives measurement of mismatch Blow-up from thin scatterer • Scattering elements are sometimes required in the beam – Thin beam screens (Al2O3,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 Coulomb scattering. qs rms angle increase: 14.1 L L q [mrad ] Z inc 1 0.11 log 10 c p[ MeV / c] Lrad Lrad 2 s c = v/c, p = momentum, Zinc = particle charge /e, L = target length, Lrad = radiation length Blow-up from thin scatterer Each particles gets a random angle change qs but there is no effect on the positions at the scatterer X' Ellipse after scattering X new X 0 ' X new X '0 q s X After filamentation the particles have different amplitudes and the beam has a larger emittance 2 A new /2 Matched ellipse Blow-up from thin scatterer 2 2 '2 A new X new X new X 02 X '0 q s X' 2 X 02 X '02 2 X '0q s q s2 Ellipse after filamentation uncorrelated 2 A new X 02 X '02 2 X '0q s q s2 X 0 2 0 2 X '0 q s q s2 2 0 q s2 new 0 2 Matched ellipse q s2 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.) Blow-up from charge stripping foil • For LHC heavy ions, Pb53+ is stripped to Pb82+ at 4.25GeV/u using a 0.8mm thick Al foil, in the PS to SPS line is minimised with low- insertion (xy ~5 m) in the transfer line • • Emittance increase expected is about 8% TT10 optics 120 beta X Stripping foil beta Y 100 Beta [m] 80 60 40 20 0 0 50 100 150 S [m] 200 250 300 Emittance exchange insertion • Acceptances of circular accelerators tend to be larger in horizontal plane (bending dipole gap height small as possible) • Several multiturn extraction process produce beams which have emittances which are larger in the vertical plane larger losses • We can overcome this by exchanging the H and V phase planes (emittance exchange) Low energy machine After multi-turn extraction After emittance exchange High energy machine y x In the following, remember that the matrix is our friend… Emittance exchange Phase-plane exchange requires a transformation of the form: x1 0 x'1 0 y m 1 31 y' m 1 41 0 m13 0 m 23 m 32 0 m42 0 m14 x 0 m 24 x' 0 0 y 0 0 y' 0 A skew quadrupole is a normal quadrupole rotated by an angle q. The transfer matrix S obtained by a rotation of the normal transfer matrix Mq: S = R-1MqR where R is the rotation matrix cos q 0 sinq 0 0 cos q sinq 0 0 cos q sinq 0 0 sinq 0 cos q (you can convince yourself of what R does by checking that x0 is transformed to x1 = x0cosq y0sinq, y0 into -x0sinq y0cosq, etc.) Emittance exchange 1 M q 0 0 For a thin-lens approximation So that cos q 0 S R 1M q R sinq 0 0 sinq cos q 0 0 cos q sinq 0 0 1 0 0 0 1 0 0 0 1 0 1 sinq 0 0 cos q 0 1 cos 2q 0 sin 2q 0 (where = kl = 1/f is the quadrupole strength) 0 0 1 0 0 1 0 0 0 1 sin 2q 0 1 0 cos 2q 0 cos q 0 0 0 sinq 1 0 0 sinq cos q 0 0 cos q sinq 0 0 0 0 1 45º skew quad For the case of q = 45º, this reduces to 1 0 S 0 0 0 0 1 0 0 1 0 0 0 1 0 sinq 0 cos q Normal quad Emittance exchange The transformation required can be achieved with 3 such skew quads in a lattice, of strengths 1, 2, 3, with transfer matrices S1, S2, S3 A B Skew quad Skew quad 1 Skew quad 2 3 The transfer matrix without the skew quads is C = B A . x C 0 0 0 0 C Cx 0 0 0 0 y C x2 cos f x x1 sin f x x1 cos f 1 sin f x1 x x1 x 2 x x2 x1 x 2 cos f x x 2 sin f x x2 x1 x 2 sin f x x1 and similar for Cy Emittance exchange With the skew quads the overall matrix is M = S3B S2A S1 c11 b12 a 34 1 2 c12 c 21 b22 a 34 1 2 c 22 b34 a12 2 3 3 c 34 1 b34 a11 2 M c 34 1 b34 a11 2 a12 b34 2 c b a 3 11 12 34 1 2 c b a 12 3 44 12 2 c b a 22 34 1 2 44 1 c12 1 b12 a 33 2 b12 a 34 2 c 22 1 b22 a 33 2 b22 a 34 2 c 34 3 c b a 34 12 1 2 3 33 c 33 b34 a12 1 2 c 34 3 c12 1 b12 a 33 2 c 44 b12 a 34 2 3 c 43 b44 a12 1 2 Equating the terms with our target matrix form 0 0 m 31 m 41 0 m13 0 m 23 m 32 0 m 42 0 m14 m 24 0 0 a list of conditions result which must be met for phase-plane exchange. Emittance exchange 0 c12 0 c 34 0 c11 b12 a 34 1 2 0 c 22 b34 a12 2 3 0 c 33 b34 a12 1 2 0 c 44 b12 a 34 2 3 0 c 21 b22 a 34 1 2 3 c 34 1 b34 a11 2 0 c 43 b44 a12 1 2 3 c12 1 b12 a 33 2 The simplest conditions are c12 = c34 = 0. Looking back at the matrix C, this means that fx and fy need to be integer multiples of p (i.e. the phase advance from first to last skew quad should be 180º, 360º, …) We also have for the strength of the skew quads 1 2 c c11 33 b12 a 34 b34 a12 2 3 c 22 c 44 b34 a12 b12 a 34 Emittance exchange Several solutions exist 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 90º FODO lattice A B (=A) From symmetry A = B, and the values of and at all skew quads are identical. Therefore A x B x cos f x x sin f x 1 x2 sin f x x x sin f x cos f x x sin f x The matrix C is similar, but with phase advances of 2f with the same form for y Emittance exchange Since we have chose a 90º FODO phase advance, fx = fy = p/2, and 2fx = 2fy = p which means we can now write down A,B and C: x 1 x2 x A B 0 0 x 0 x 0 0 y 0 1 2 y y 0 0 y y 0 0 1 0 0 0 1 0 C 0 0 1 0 0 0 0 1 we can then write down the skew lens strength as For the 90º FODO with half-cell length L, F D i.e. 180º across the insertion in both planes 1 2 3 s 2 , L s 1 L 2 1 x y Summary • 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 extremes 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 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 – SVD trajectory analysis – Kick-response optics measurement techniques in transfer lines – Optics measurements including dispersion and p/p with >3 screens – Different phase-plane exchange insertion solutions