INFN School on Electron Accelerators. Beam Acceleration Cavity Field and Concepts

Transcription

1 INFN Shool on leton Aeleatos -4 Septembe 7, INFN Sezione di Pisa Letue b Beam Aeleation Cavity Field and Conepts Calo Pagani Univesity of Milano INFN Milano-LASA & GD

2 Linea Collide Coneptual Sheme Final Fous Demagnify and ollide beams Bunh Compesso Redue σ z to eliminate houglass effet at IP Damping Ring Redue tansvese phase spae (emittane) so smalle tansvese IP size ahievable leton Gun Delive stable beam uent Main Lina Aeleate beam to IP enegy without spoiling DR emittane Positon Taget Use eletons to paipodue positons Calo Pagani

3 RF to tansfe negy to the Beam To give enegy to a haged patile beam, apat fom details, you need to let him move aoss a egion in whih an eleti field exists and is dieted as the patile motion. Δ patile F Loentz ds v In the aeleato wold RF takes ae of all the vaiety of items that ae equied to aomplish this task of eating a egion filled of eletomagneti enegy that an be suked by the beam while ossing it. An RF powe soue is used to fill, via a ouple, the RF avity, o esonato that is the e.m. enegy ontaine fom whih the beam is taking its enegy. What we ask to a good avity? High Q fo losses: U stoed enegy P diss dissipated powe U Q ω P diss q dt Small R s fo high Q: R s sufae esistane G avity geometial fato Q G Rs Calo Pagani 3

4 Competing Lina Tehnologies Taveling wave V ph and Vg < NLC/GLC: CLIC f.4 GHz f 3 GHz Nomal-Conduting Standing wave V ph and Vg TSLA: f.3 GHz Supe-Conduting π mode Ratio between Nb and Cu Rs f [MHz] K 4. K Remembeing that the powe dissipated on the avity walls to sustain a field is: P diss R s S H ds standing wave ase a pulsed opeation is equied to edue the time in whih the maximum allowable field is podued to aeleate the patiles Calo Pagani 4

5 The Lina Aeleato Conept An RF soue is used to geneate an eleti field in a egion of a esonant metalli stutue The patiles of the beam need to be loalized in bunhes and popely phased with espet to the field so that the beam is aeleated d ( γm ) q ( s t) ds z, In ode to keep aeleation along the lina this synhonism ondition needs to be maintained. z bunhes leti field Calo Pagani 5

6 Calo Pagani 6 Maxwell quations and Waves t B t B B ε μ B t B letomagneti fields ae desibed by Maxwell quations that in empty spae ae: Fom Maxwell quation we obtain the Wave quations fo leti and Magneti Fields ε μ whee:

7 Calo Pagani 7 Plane Wave: quations ) ( kz t i e ω ) ( kz t i e H H ω ) ( kz t i e t ω ) ( kz t i e H t H ω Defining the z-axis paallel to the dietion of popagation, we an solve the wave equation as a supeposition of tavelling plane waves: Absene of boundaies (isotopi, homogeneous vauum) equies that vetos and H be onstant fo all time and spae Applying the wave equation to the eleti and magneti fields yields: Sine and H ae onstant, both the time deivative and Laplaian opeate only on the omplex exponential. Afte anellation of onstant fatos, we find: k k k ω ω ω i.e., a plane wave with a phase veloity and a goup veloity ε μ

8 Fee-Spae Solution Go bak and apply Maxwell s equations to this solution: z ik That is: k o,z must be zeo! z i( ωt kz), ze k: tivial solution, no wave!,z : eleti field aeleates beam tansvese to dietion of wave popagation! If beam is aeleated in x while wave moves in z, then wave will fist aeleate, then deeleate, the beam! No good fo aeleation! Calo Pagani 8

9 Plana Wave: Pitoial View letomagneti wave in empty spae The enegy tanspot pe unit aea is desibed by the Poynting veto S S μ B The Phase veloity v ph is the veloity of an obseve sitting at onstant phase Goup veloity v g is the veloity of the enegy popagation v ph v g Calo Pagani 9

10 Bounded Solution to the Wave quation Apply some kind of boundaies in x and y, so that non-zeo x and y deivatives of the eleti field an anel z deivative (i.e. pemits nonzeo,z while still obeying Maxwell). Ty a onduting pipe of adius b, oiented along z axis: y x b z i( ωt kz) i( ωt kz) e H He This time vetos and H ae funtions of tansvese oodinates x and y (o and θ) but not z o t. Thus we an simplify some deivatives: ik, k z z + iω, ω t t Calo Pagani

11 Pefet Conduto Solution - Using ylindial oodinates we have, at the bounday, i.e. at b, the nomal omponent of B and the tangential omponent of ae ontinuous. if the onduto is pefet, then within the onduto the eleti and magneti field ae identially zeo. Thus at b, H, z, and θ. sine θ, the θ omponent of the magneti ul equation must go to zeo. In total θ z H H z n B n With some algeba and aneling the ommon omplex omponent (time dependene) we get fom the wave equation the longitudinal eleti field: i( ωt kz) e t, z anj n ( k)os( nθ + θ n ) n Whee: J n ae the Bessel funtions k ( ω k n must be is an intege ) Calo Pagani

12 Pefet Conduto Solution - 3 Beause: b, We an set: k b z np, whee z np is the p th zeo of J n. As a esult: k >, z anp J n ( k, np)os( nθ + θ np p n y x b b z ) k z ω np, np k b k beause k must be eal fo popagation and fo k we have the utoff fequeny: t i( ωt kz) e z ω b np, np ω Cutoff fequeny But also: ω>ω ω < ω Taveling wave: popagation vanesent wave: an t popagate ω ω ω vg < k ω ω ω vph + > k k Calo Pagani

13 TM and T Modes A simila solution is available fo the magneti field veto In geneal a wave with a given phase and goup veloity annot have both a longitudinal eleti field and a longitudinal magneti field! Waves with H,z ae alled TM (tansvese magneti) modes; waves with,z ae alled T (tansvese eleti) modes. Usually the modes ae efeed to with thei index numbes, T uv o TM np TM mode has nonzeo z,, H θ omponents only Calo Pagani 3

14 Real Aeleating Stutues: Cavities Imposing bounday ondition in the longitudinal dietion, z, we have fo eah mode (fo example the TM ) two waves: ightwad (+z)-popagating wave and a leftwadpopagating wave The ombination an give a wave with phase veloity v ph Taveling wave stutue V ph and Vg < Standing wave stutue V ph and Vg π mode Calo Pagani 4

15 Realisti multiell avities standing wave: esonant stutue In ode to effiiently aeleate the beam, multiell esonatos ae used, by peiodially epeating the esonant stutue and poviding oupling between the diffeent ells. Any geomety an be omputed with existing numeial odes The simplest oupling is epesented by the field though the beam hole (apaitive oupling) The beam needs to keep the elative phase with the field λ RF β L z [MV/m] z [m] Cavità β.5 Cavità ideale Calo Pagani 5

16 letons and Potons eleton and poton masses,poton,eleton A poton vaies its veloity on a muh highe kineti enegy ange Synhonous ondition fo a multiell avity: βv/ T ½m v v leton β(t) leton β(t) T ½m v,poton Poton β(t).5 MeV,eleton 938 MeV L λ RF β The ell length depends on the patile veloity. Synhonism is exat only fo a given veloity value. Cavities opeated in a veloity ange.. Poton β(t) T [MeV] T [GeV] Fo eletons all RF avities ae idential Fo potons, avity geometies follow the patile veloity, that is the patile β. Below β.5, speial stutues ae equied Calo Pagani 6

17 Modes of the multiell avity (] ]>FP@ b) A oupled system of N esonant osillatos at ω an osillate in N nomal modes, fist band with eigenfequenies lose to ω (] >9P@ b) ]>FP@ b) ω ω n nπ + k os N Whee k is the oupling oeffiient (] >9P@ N N ]>FP@ b) ω/ω..8 ω 4π/5 ω π U (] >9P@ ]>FP@ z [MV/m] (m) [MV/m].6.4. ω π/5 ω 3π/5 (+k) / -~k b). ω π/5 - - ]>FP@ Δφnπ/N Calo Pagani 7

18 negy gain and dissipated powe To aeleate patiles effiiently, vey high eleti field is equied In any stutue (avity) holding an eletomagneti field, both dissipated powe and stoed enegy sale quadatially with the fields The effiieny of a avity depends fom: Its quality fato, Q diven by the sufae esistane, R s Its shunt impedane, Δ ΔT funtion of the avity geomety and of the sufae esistane, R s F Lo Q ωu P diss ds q ( ΔV ) v Fo effiient aeleation Q, and /Q must all be as high as possible P diss dt U is the enegy stoed in the avity P diss is the powe dissipated on its sufae ΔV is the voltage seen by the beam Q ( ΔV ) ωu Good mateial fo maximum Q and (that is minimum P diss ) Good design fo maximum /Q ove Q is puely a geometial fato Calo Pagani 8

19 Cavity lumped iuit model and R S A avity at the fundamental mode has an equivalent esonant lumped iuit L R C Q detemines the fequeny band Δf Δ f Q f,, 3dB ω Q ω RC LC fi Δf ΔI If ω πf V P diss R R popotional to Q detemines P diss R depends invesely fom the avity R s though a geometial fato R Q R R s In patie, fo a given geomety and a given aeleating field the sufae esistane R s plays the uial ole of detemining the dissipated powe, that is the powe equied to sustain the field Rs Calo Pagani 9

20 Supeondutivity wheneve possible Fo a good but not pefet onduto ( ρ ), the fields and uents penetate into the onduto in a small laye at the avity sufae (the skin depth, δ) With RF fields, a SC avity dissipate powe, not all eletons ae in Coope pais. R s ρ δ P diss R s S H ds Nb Cu R s [ n Ω ] 9 η C T T T 4 Ratio between Nb and Cu Rs η /7 fo T 3K,T 4.K /5 fo T 3K,T K tot T [ m Ω ] 7.8 f [ GHz ] R s In NC lina a huge amount of powe is deposited in the oppe stutue: MW to have MV Pulsed opeation and Low Duty Cyle Supeondutivity, dastially edues the dissipated powe. But some dawbaks Highe omplexity: efigeation and yomodules Highe tehnology: avity teatments Canot and efigeation plant effiienies Simple geometies: lowe shunt impedane And two big advantages: Lage boe adius: less beam losses CW o high duty ile pefeed f [ GHz ] [ ] [ ] exp K T K η η C th f [MHz] SC SupeConduting NC o RT NomalConduting η th 5W at 3K fo W at T 4.K 8W at 3K fo W at T K K 4. K 5 3% at T 4.K 5 % att K Calo Pagani

21 ILC Positon Captue Cavity Pototype Goal: Powe with 5 MW, mse pulses to podue 5 MV/m gadient Calo Pagani

22 SRF avities fo SNS,+ β.6 SNS 6-ell avity esult,+ β.8 SNS 6-ell avity esult Test # Test #,+,+ Q Design Goal Q Design goal,+9,+9, a [MV/m], a [MV/m] Calo Pagani

23 SRF Cavities fo ADS p / a 3.57 B p / a 5.88 mt/(mv/m) β.47 at INFN Milano Z5 TJNAF 3/3/4 Z5 Salay 4/6/4 Test # limited by stong field emission Z5 Z5 - befoe onditioning Z5 - afte onditioning Design Value Q multipating baies stat of eleton emission a [MV/m] Calo Pagani 3

24 SRF Cavities at low beta Today Spoke avity is the Sta But still othe ompetitos Vey pomising esults but it is vey diffiult to ompae them Results fom elliptial avities ae peditable Sepaate vauum is mandatoy fo low beta geometies Miophonis must be ued (piezo-tunes?) go to K? To ompae with elliptial same leanness level must be poven Calo Pagani 4

25 SRF avities fo ings CSR KKB LHC SOLIL Calo Pagani 5