LECTURE 17. Linear coupling (continued) Linear coupling (continued) Pretzel Orbits

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1 LETURE 7 Linear coupling (continued) oupling coefficient for ditribution of kew quadrupole and olenoid Pretzel Orbit Motivation and application Implication Linear coupling (continued) oupling coefficient for ditribution of kew quadrupole and olenoid The previou dicuion focued on a ingle kew quadrupole, for implicit. Actual machine tpicall have a ditribution of kew quadrupole, and alo ma include olenoid. The aial olenoid field couple to the lope of the trajector; the end field couple to the trajector itelf: (c.f., Lecture 3, p :) = Τ + = Τ Τ ; + Τ Τ =, Τ = ρ ρ /3/ USPAS Lecture 7 /3/ USPAS Lecture 7 Let ee how to calculate the coupling coefficient for an arbitrar ditribution of kew quadrupole and olenoid trength around the ring. We ll call the location at which we want to evaluate the trajectorie =. At ome other point in the ring,, let the kew quadrupole trength be k, and the olenoid trength Τ ( ). For the moment, we aume that thi i the onl point of coupling in the ring. At the end of the dicuion, we ll integrate over the whole ring to get the reult for a ditribution of trength. The incremental kick delivered to a trajector at thi point b thee field, which etend a ditance, i ξ α ξ β ξ Τξ = T k T = = + T k T = = In Floquet coordinate, we have β β Τ ξ αξ = k + = kξ β + Τ β Thi give /3/ USPAS Lecture 7 3 /3/ USPAS Lecture 7 4

2 ξ = κ ξ κ + ξ, ξ = κ3ξ κ κ α ββ T β α k T = + β T β β T = κ = β β 3 β β in which everthing i evaluated at the point. There are imilar equation for, in which and are interchanged, and T->-T. Now conider a trajector which tart at =, with phae pace coordinate ξ = r coφ ξ ( ) = r in φ ξ = r coφ ξ ( ) = r in φ at that point. It travel to, at which the betatron phae i Φ = Φ( ). The phae pace coordinate there are ξ ( ) = r co φ + Φ ξ ( ) r in φ Φ = ( + ) = ( + Φ ) ξ ( ) = r co φ + Φ ξ ( ) r in φ The change in the Floquet coordinate at thi point are then /3/ USPAS Lecture 7 5 /3/ USPAS Lecture 7 6 ξ = κ rco φ + Φ κ rin φ Φ, ξ = κ3rco φ + Φ ξ = κ r co φ + Φ κ rin φ Φ, ξ = κ r co φ + Φ 3 ( + ) ( + ) We then continue from thi point to =, where we tarted. The Floquet coordinate at = are given b ξ in π Φ co( π Φ ) = ξ in π Φ co π ( ) Φ with a imilar equation for. ξ( ) + ξ ξ ( ) + ξ /3/ USPAS Lecture 7 7 We then calculate the change in the phae-amplitude variable over the turn, uing dr ξ ξ ( ) = ξ ξ + dn dφ ξ ξ = tan dn tan ξ ( ) ξ ( ) In the following reult, the parameter κ are aumed to be mall, o onl the linear term are retained. The trigonometric function have alo been epanded, onl term driving the difference reonance have been retained, and the change of variable to the rotating coordinate tem ha been made. The reulting equation are /3/ USPAS Lecture 7 8

3 dr r = ( ϑco( φ φ ) + ϑin( φ φ )) dn dφ r = πδ ϑco( φ φ) ϑin φ φ dn r ϑ = κin Φ Φ ( κ κ3) co Φ Φ ) ( ) ( ϑ = κ co Φ Φ κ κ in Φ Φ 3 with δ=. There are imilar equation for, obtained b interchanging and. At thi point, we can generalize to a ditribution of kew quadrupole and etupole trength around the ring. The above equation give the contribution from an element of length at : /3/ USPAS Lecture 7 9 for a ditribution of kew quadrupole and olenoid trength around the ring, we make the following replacement: α T β α β T ϑ d ββk + β β T β β in( Φ( ) Φ( )) + β β co( Φ ( ) Φ ( )) α T β ϑ d ββk α β T + β β T β β co( Φ( ) Φ( ))+ + β β in( Φ ( ) Φ ( )) A before, we now introduce the comple quantitie /3/ USPAS Lecture 7 w = rep ( iφ ) w = rep( iφ ) The reult i the pair of comple equation dw i = i ϑ w δ π + ϑ w dn dw iϑ ϑ = iδπw w dn The equivalent matri equation i r dw r + iπδmw=, M= dn ϑ iϑ δπ ϑ iϑ δπ Subtituting from above, we can write the matri a Θ δπ M = * Θ δπ in which ββ k T + Θ= ( ( Φ Φ d i α β ep )) β α i T β β + β β The eigenvalue are λ = + ε λ = + ε β + β /3/ USPAS Lecture 7 /3/ USPAS Lecture 7

4 in which ε = Θ πδ The minimum tune plit, on the difference reonance, i Θ ( ) = min π orrection of coupling. For a difference reonance correponding to = m+δ, we can approimate = + Φ ( ) Φ ( ) θ θ m δ θ π in which θ = i the azimuthal angle. Then, for mall δ, the coupling coefficient become T + Θ ββk + α β α β π β β d ep im i T β β + β β The coefficient which drive the = m difference reonance are the mth Fourier component of the coupling trength. To correct a general et of coupling error, at leat two corrector are needed, to cancel the two Fourier harmonic (real and /3/ USPAS Lecture 7 3 /3/ USPAS Lecture 7 4 imaginar part of Θ). If the coupling error and the lattice function have uperperiodicit N, thi will uppre Fourier harmonic which do not atif m= jn, for integral j. Pretzel Orbit Motivation and application The term pretzel orbit refer to the deliberate introduction of cloed orbit ditortion, through the ue of electric field, in order to provide orbit eparation at undeired colliion point in multiple bunch particle-antiparticle collider. Pretzel orbit were invented and firt developed at ESR. The are now in ue here, and alo in LEP at ERN, and in the Tevatron at Fermilab, to allow multiple bunch operation and higher luminoit. Wh do more bunche give higher luminoit? Recall, Lecture, p 38, luminoit formula: L = f c N b 4πσ Here N b =number of particle per colliding bunch, and f c =colliion frequenc. If there are bunche per pecie, then fc = f, where f i the revolution frequenc, and o N b L = f 4πσ If there i ome limit on N b (e.g, the beam-beam limit, which i proportional to N b ), then more bunche will give more luminoit. /3/ USPAS Lecture 7 5 /3/ USPAS Lecture 7 6

5 If, however, I can make N b a big a I want, but have a fied total number of particle N = N b, then I can write Nb f N L = f = 4 πσ 4πσ and I want to make a mall a I can (i.e., ) to maimize luminoit. The tpical ituation in particle-antiparticle collider i operation at the beam-beam limit, and we want to have a man bunche a poible. However, bunche have colliion point, while tpicall there are onl one or two detector. At each colliion point, we uffer from the beam-beam interaction, o we want to minimize the number of colliion point. Thu, we want to eparate the bunche everwhere in the machine, o the do not collide, ecept at the colliion point where we have detector. Thi i the purpoe of pretzel orbit λ p unche λ λ λ olliion point /3/ USPAS Lecture 7 7 /3/ USPAS Lecture 7 8 The figure above illutrate a poible ideal realization of the baic idea, providing two colliion point with 8 bunche. Two cloed orbit ditortion are generated, of wavelength λ and amplitude p. The bunch pacing i equal to λ. The bunche are arranged a hown, o that while two are at the colliion point, the other are at the pretzel antinode. The orbit ditortion i generated uing electric field (tpicall electrotatic eparator), o that the oppoitel charged, counter-rotating bunche follow an orbit with the oppoite ign. The bunche paing at the pretzel antinode are eparated b a eparation p, while thoe at the colliion point collide. The cheme accommodate = λ bunche, where λ i the betatron wavelength. Since λ, the value of the tune et the maimum number of bunche. /3/ USPAS Lecture 7 9 Thi limitation ha been overcome at ESR and LEP b uing train of bunche, with a pacing much maller than λ. The train mut be hort enough to fit in the region of pretzel antinode. A mall croing angle i introduced in the traight ection to prevent undeired colliion for bunche in a ingle train. The pretzel hown above i mmetric about each colliion point. An antimmetric pretzel i alo poible, and in fact deireable: /3/ USPAS Lecture 7

6 The following figure illutrate the orbit eparation cheme. (Animation of thee figure are available in the animation folder). Left: colliion at two point, other bunche at pretzel antinode Right: after colliion, mot bunche at pretzel node. Left: All bunche near pretzel antinode /3/ USPAS Lecture 7 /3/ USPAS Lecture 7 Right: two colliion, other bunche at pretzel antinode. Implication: There are a number of iue aociated with pretzel orbit operation. Long-range beam-beam colliion. The long-range colliion caue cloed orbit error, tune hift, beta function ditortion, and reonance ecitation. The need to limit thee effect et the ize of the pretzel amplitude p, upon which all other effect depend. Aperture. The deformed orbit, plu betatron ocillation around them, mut fit into the good field region of the magnet aperture. /3/ USPAS Lecture 7 3 Pretzel cloure. If the orbit deformation leak into the colliion region, the colliding bunche ma fail to collide head-on, or even mi each other. Diperion. The deformed cloed orbit generate diperion; thi will be vertical diperion if the pretzel i vertical, and will contribute to quantum ecitation of the vertical emittance in an electron machine. Path length change. The path length on the deformed orbit will change. Thi can reult in an energ difference between the colliding beam. Setupole effect: If the pretzel i horizontal: The cloed orbit deformation in the etupole caue horizontal dipole error, which will modif the /3/ USPAS Lecture 7 4

7 cloed orbit. It alo caue quadrupole error in both plane, which in turn reult in tune hift, beta function ditortion, and econd order reonance enhancement. If the pretzel i vertical: The cloed orbit deformation in the etupole caue horizontal dipole error, and kew quadrupole error in both plane, which increae the coupling. Particle-antiparticle energ difference: If the pretzel i preent in the rf cavitie, and the rf field varie with poition, there ma be energ difference between the two beam. Nonlinear reonance from field error. The large amplitude ecurion of the beam ma allow them to enter nonlinear field region, increaing the enitivit to reonance. Injection. During the damped betatron ocillation which occur after injection, the eparation between the bunche ma be reduced, potentiall leading to beam lo. Electrotatic eparator. The requirement on thee device are challenging. In addition to having to provide high electric field (tpicall > kv/cm), for high current electron-poitron machine, the mut have low impedance. For proton-antiproton collider, the mut be ver reliable, a park often caue lo of the tored beam. Machine that operate with flat beam mut trictl limit the amount of vertical diperion and coupling, in order to minimize the vertical emittance. Vertical pretzel cloure error at the colliion point are alo ver damaging, becaue of the mall /3/ USPAS Lecture 7 5 /3/ USPAS Lecture 7 6 vertical beam ize. Hence, electron collider tpicall chooe the pretzel to be in the horizontal plane. Let eamine ome of thee effect quantitativel, for the cae of horizontall eparated orbit. Long-range beam-beam colliion. To etimate the effect of thee colliion, we need to know the field produced b a bunch. Imagine the bunch to have a length L along the direction of motion. We will be eeking the longrange field, at a ditance from the bunch large compared to it tranvere ize. So, we imagine the bunch to have a ver mall tranvere ize. v L The bunch i taken to be compoed of ultra-relativitic point charge, which have flattened field that are directed perpendicular to the direction of motion (ee figure above). To find the electric field at a point a ditance r from the bunch, we urround the bunch with a Gauian urface a hown: E /3/ USPAS Lecture 7 7 /3/ USPAS Lecture 7 8

8 v r Gauian urface Appling Gau Law to find the field give r r E da = E( πrl) = E = ε πrlε To find the magnetic field at r, ue Ampere Law r Amperian loop r r dl = ( π r) = µ I = µ d dt d dt t v v v = = = = µ L π rl /3/ USPAS Lecture 7 9 /3/ USPAS Lecture 7 3 Now conider a point charge -e, moving oppoite to the bunch, at the point r. The effect of the long-range field of the bunch on the trajector of thi particle i given b (ee Lect, p. 35): e ee e ee = = p vp p vp r θ E E E E E r r So the total change in lope of the trajector produced b the field of the bunch i e = p ee vp e = r p e = m πε γc L r ee vp e = m πε γc L r For mall θ, we have /3/ USPAS Lecture 7 3 /3/ USPAS Lecture 7 3

9 c L c =L/ From t= the figure to effective t=l/c length of For a bunch with N b particle of charge e, the angular kick are the right, we ee that =L/: the the field een b the particle i half the bunch length. Ne = b N = b r N r = b m c πγ ε r γ r γ r, in which e r = 4 πε mc =.8-5 m i the claical electron radiu. On pretzel orbit, the beam are eparated b a ditance p. Hence, we have r = ( p+ ) +, where and meaure the betatron ocillation about the pretzel orbit. Thu N = b r N = b r γ p+ + γ p+ + Tpicall(, )<<p, o we can epand p p p + and to lowet order in (,) we have /3/ USPAS Lecture 7 33 /3/ USPAS Lecture 7 34 Nr = b γ p 4 p Nr = b γ 4 p The firt term in parenthee in the -equation correpond to a dipole error. Since it i linear in p, the error will have different ign for particle and antiparticle, reulting in differential orbit change and pretzel cloure error. In principle, thi can be corrected b adjuting the eparator. The econd term in, and the onl term in, i a quadrupole error. The effective focal length i f N = = γ b r p defocuing for both tpe of particle, in, and focuing in. For bunche, producing - long-range croing, the tune hift due to the long-range croing i βi Nb βr, = 4π f 4π γ p LR i i in which β i a tpical lattice function at the croing. For a given tolerable tune hift, the required pretzel amplitude i p = Nb βr 4π γ LR, Eample: We want LR to be mall compared to a tpical maimum head-on tune hift, which might be HO =.5. Taking LR to be.5, for ESR parameter β=3 m, =3, N b =, γ= 4, we have p = 6 mm, which require a full aperture of 3 mm plu room for betatron ocillation. /3/ USPAS Lecture 7 35 /3/ USPAS Lecture 7 36

10 In practice, it i not thi tune hift itelf which caue problem, but rather maller, higher order nonlinear effect which are difficult to correct. Neverthele, thi imple etimate correctl et the cale of the required pretzel eparation. Setupole effect of horizontal pretzel orbit The vertical field of a etupole i = ( + ). Let the cloed orbit deformation produced b the pretzel be p(). Then, on the pretzel, the etupole field i = ( ) = ( + p() ) + p () p () in which (,) now refer to betatron ocillation about the pretzel orbit. We ee that the effect of the etupole i to produce a dipole p field error, which i the ame for both pecie. Thi error can be corrected with tandard correction dipole. There i alo a tune p hift due to the quadrupole error k = mp ( ρ) =, in which m i the etupole trength. The total tune hift, integrated around the ring, i = dm() β() p(). π 4 /3/ USPAS Lecture 7 37 /3/ USPAS Lecture 7 38 The tune hift per unit pretzel amplitude i called the tonalit. Thi tune hift will have oppoite ign for particle and antiparticle. If the ring ha uperperiodicit two, and the pretzel i antimmetric about the mmetr point, ( p + = p )then = dm() β() p() + dm() β() p() 4π dm p dm p = () β() () + ( ) β ( π ) ( ) = dm() β() p() dm() β() p() 4 = π The tonalit i zero to lowet order. The quadrupole error produce a lattice function ditortion (from Lect 8, p ) /3/ USPAS Lecture 7 39 /3/ USPAS Lecture 7 4

11 β() β () = d m( ) p( ) β ( )co Φ Φ ( ) π inπ [ ] For tune near a half-integer, thi perturbation i maimall antimmetric about /. The tonalit, calculated uing the perturbed lattice function, will thu be non-zero in net to lowet order in pretzel amplitude. /3/ USPAS Lecture 7 4 Path length change. In one of the homework problem, it wa hown that a dipole error θ at a location where the diperion i η produce a path length change = ηθ. If the eparator that produce the pretzel are located at diperive point, then the path length change on the pretzel will be η θ = i where the um i over all the pretzel kick. Thi change i oppoite for the two pecie of particle. Since the circumference i fied b the rf wavelength and harmonic number, the path length change on the pretzel reult in an energ change given b δ =. The two α /3/ USPAS Lecture 7 4 i i pecie will then have different energie, which can be a problem if there i reidual vertical diperion at the interaction point. To lowet order in the pretzel amplitude (i.e., neglecting the change in η due to the pretzel itelf) i zero for an antimmetric pretzel in a uperperiod lattice /3/ USPAS Lecture 7 43