CERN Accelerator School. Beam Cooling. Håkan Danared Manne Siegbahn Laboratory. Daresbury, 27 September2007

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1 CERN Acclrator School Bam Cooling Håkan Danard Mann Sigbahn Laboratory Darsbury, 7 Sptmbr007

2 Bam Cooling, and Why? Th momntum sprad, transvrs and longitudinal, of particls in a bam can b sn as a thrmal motion, and cooling is to rduc th momntum sprad. Rducing th momntum sprad also mans incrasing th phas-spac dnsity. Bam cooling can b usd for many purposs: Accumulation of antiprotons or othr particls Incras of luminosity in collidrs Countraction of hating in intrnal targts Improvd prcision in masurmnts Emittanc control at dclration...

Mthods for Bam Cooling Diffrnt cooling mthods ar usd for diffrnt kinds of bams, but only thr kinds of activ cooling hav bn implmntd so far: Elctrons cool thmslvs

antiprotons and ions Lasr cooling for a fw kinds of atomic ions Ionization cooling, soon, for muons *) all coold particls will b rfrrd to as ions Grsh Budkr, 1966

3 Mthods for Bam Cooling Diffrnt cooling mthods ar usd for diffrnt kinds of bams, but only thr kinds of activ cooling hav bn implmntd so far: Elctrons cool thmslvs by mitting radiation, this is for anothr lctur Elctron cooling for low/mdium-nrgy protons, antiprotons and ions * Stochastic cooling for mdium/high-nrgy protons, antiprotons and ions Lasr cooling for a fw kinds of atomic ions Ionization cooling, soon, for muons *) all coold particls will b rfrrd to as ions Grsh Budkr, 1966 Эффективный метод демпфирования колебаний частиц в протонных и аитипротонных накопителях Simon van dr Mr, 197 Stochastic damping of btatron oscillations in th ISR

4 Liouvill s Thorm For a systm of N particls, Liouvill s thorm can b formulatd: in μ spac, which is th 6-dimnsional phas spac whr ach particl is rprsntd by a point (6 coordinats bcaus ach particl has x, y, z, p x, p y and p z ) in Γ spac, which is th 6N-dimnsional phas spac whr th whol systm (bam) is rprsntd by a singl point In ithr cas, th thorm says that th phas-spac dnsity, masurd along th trajctory of a givn point, dos not chang in tim. Fin print: Liouvill s thorm in μ spac rquirs non-intracting particls in a consrvativ dynamical systm, volving according to Hamilton s quations. In Γ spac th thorm is valid also for intracting particls. In practic it oftn works in μ spac vn if particls intract, as long as hard collisions can b nglctd.

5 Liouvill s Thorm, in Simpl Words No fancy schm of magnts or lctrostatic filds can shrink th bam mittanc

6 Elctron Cooling Elctron cooling is lik pouring cold lctrons on th warm ions. Hat is transfrrd from th circulating ions to frsh, cold lctrons through Coulomb intractions. Principl Practic

dipol * Vacuum chambr Support fram Rturn yok NEG pump *) Th lctrons

7 CRYRING Elctron Coolr Cryostat Suprconducting solnoid Elctron gun Ion bam Magnt winding (NC) Intraction rgion Cryopump Collctor Corrction dipol * Vacuum chambr Support fram Rturn yok NEG pump *) Th lctrons ar dflctd vrtically in th toroids, but th ions dflct horizontally. Why?

8 Elctron Cooling Forc In a microscopic pictur, th ions ar slowd down in collisions with th lctrons, xprincing a (non-consrvativ) friction forc. How larg is th friction? This can b stimatd by studying binary ncountrs btwn ions and lctrons. A starting point is th Ruthrford cross sction: 4 Z q σ ( θ) = 4 4 4(4πε ) μ u sin 0 1. ( θ / ) Hr, Z is th ion charg stat and μ is th rducd mass m i m /(m i +m ). From th figur it is sn that θ u Δu = u(1 cosθ) = z u sin ( θ / ), Δu z = Δu σ( θ) u dω = Ω z π θ θ max min Δu z σ( θ) u sin θ dθ.

9 Elctron Cooling Forc Th intgral is simpl to solv, and th rsult is LC Δuz 0 μ u Zq = 4π 4πε, (and Δu = Δu 0 ) x y = whr w hav introducd th Coulomb logarithm L C = ln sin( θ sin( θ max min / ) / ) sin ln sin θ θ max min ln b b max min. b W ar mor intrstd in th chang in ion vlocity v i than in th rlativ vlocity u, and w us th fact that ions ar much havir than lctrons: Zq LC vi v Δvi = 4π 3 4πε 0 mim vi v.

10 Elctron Cooling Forc Elctron Cooling Forc This corrsponds to a forc acting on th ions i i C 0 i v v v v m L πε Zq π F = Th forc has to b intgratd ovr th (flattnd Maxwllian) vlocity distribution of th lctrons, giving, d ) ( i i C 0 i v v v v v v f m L n πε Zq π F = whr n is th lctron dnsity, L C is tratd as a constant (around 10) and. xp ) ( 1/ = v kt m v kt m πkt m πkt m v f

Elctron Cooling Forc Th lctrons ar mittd from a cathod at about 900 C (kt 100 mv).

Th transvrs lctron tmpratur can b rducd down to kt 1 mv by xpanding th lctron bam btwn th gun and th intraction rgion.

Effcts of th magntic fild ntr into th Coulomb logarithm, and this is whr th thory bcoms difficult!

11 Elctron Cooling Forc Th lctrons ar mittd from a cathod at about 900 C (kt 100 mv). Whn thy ar acclratd, th longitudinal nrgy sprad in th bam fram of rfrnc shrinks (show this!) down to kt 0.1 mv. Th transvrs lctron tmpratur can b rducd down to kt 1 mv by xpanding th lctron bam btwn th gun and th intraction rgion. Coldr lctrons in principl cool bttr, but th magntic fild also can mak cooling strongr. Effcts of th magntic fild ntr into th Coulomb logarithm, and this is whr th thory bcoms difficult! Rlativity maks lctron cooling slowr: For xampl, th longitudinal drag rat is ~1/γ, and th transvrs cooling tim is ~γ 5. That is on rason why lctron cooling has bn usd for rlativistic bams only in on cas, so far.

Elctron cooling is thus fast and charactrizd by a tim constant or rat for small v, but it is slowr for larg v.

12 Elctron Cooling Forc Th drag forc is proportional to th rlativ vlocity v for small v and to 1/v for v larg compard to th thrmal lctron vlocitis. Elctron cooling is thus fast and charactrizd by a tim constant or rat for small v, but it is slowr for larg v. Transvrs and longitudinal drag forc as functions of th corrsponding rlativ vlocity componnt, calculatd with kt = 100 mv, kt = 0.1 mv and without ffcts of th magntic fild. For singly chargd ions and n = m -3.

Rcombination Rcombination btwn highly chargd ions and lctrons can rduc th bam liftim. An xtrm xampl is Pb 53+, whr a 53 ma lctron bam rducd th liftim in CRYRING from 1.8 to 1.

13 Rcombination Rcombination btwn highly chargd ions and lctrons can rduc th bam liftim. An xtrm xampl is Pb 53+, whr a 53 ma lctron bam rducd th liftim in CRYRING from 1.8 to 1. s du to a dilctronic rsonanc nar 0 V. Mannrvik t al., PRL 81, 313 (1998) Th pak shap of dilctronic rcombination rsonancs can b usd to dtrmin th lctron tmpratur vry accuratly. Hr, th good fit btwn xprimnt (blu) with C 3+ ions and thory (rd) shows that th lctron tmpratur is kt = 9.4 mv and kt = 0.08 mv.

14 Coold Bams Longitudinal cooling of dutrons in CRYRING. Transvrs cooling of H - in CRYRING, Δt = 300 ms. Not fastr cooling of cor.

Elctron Cooling, Summarizd In a macroscopic pictur, hat is transfrrd from a hot ion bam to a cold lctron bam Microscopic pictur includs Coulomb intractions and binary collisions Friction-lik forc on

15 Elctron Cooling, Summarizd In a macroscopic pictur, hat is transfrrd from a hot ion bam to a cold lctron bam Microscopic pictur includs Coulomb intractions and binary collisions Friction-lik forc on ions Tim scal from millisconds to hours Difficult for high nrgis and hot ion bams Hardwar is mainly magnts (high-quality fild), lctron optics with gun and collctor, and vacuum Now in opration at CERN, GSI, Hidlbrg, Jülich, Århus, Stockholm, Tokyo, Chiba, Kyoto, Frmilab and Lanzhou ERL-basd 54 MV coolr plannd for RHIC

16 Stochastic Cooling Liouvill s thorm of cours dosn t hold if an xtrnal dvic dtcts particl positions and movs th particls onto bttr orbits. Stochastic cooling uss bam pickups and kickrs to prform this. Principl Practic R. Pasquinlli, COOL03

17 Transvrs Stochastic Cooling Assum that w hav a pickup, amplifir, tc., with a bandwidth W. According to th Nyquist thorm, th bam is thn sampld with a tim rsolution T s =1/(W). Th numbr of particls passing through th dtctor during that tim is N s =N T s /T= N/(WT) if N is th total numbr of particls in th ring and T thir rvolution tim. In a simpl modl of transvrs (or btatron) cooling, a diffrnc pickup masurs th cntr of gravity of th sampl which has a displacmnt x = 1 N s N s i= 1 x i, and a kickr corrcts th position of th sampl, moving it by Δx= -<x>, such that its cntr of gravity nds up on th nominal orbit. (Th kickr actually changs th dirction of motion of th particls, of cours.)

18 Stochastic Cooling Rat Lt s sparat a tst particl from all othr particls and writ 1 xt 1 xt * x = xi = + xi + x. N N N N s i s s i t Th tst particl, lik all othr particls, is displacd by Δx= -<x>. W can thus writ Δx t xt = x = x N s *. Clarly, th displacmnt of th tst particl dpnds on its own position but also on all th othr particls in th sampl. If w tak th (drastic) stp of nglcting th influnc of all othr particls, i.., th last trm, w find a cooling rat Δx t /x t =1/N s pr turn. Aftr multiplication with th rvolution frquncy w nd up with th cooling rat pr scond: 1 1 = τ TN s W =. N s

19 Stochastic Cooling Rat Surprisingly, this simpl drivation ovrstimats th actual, optimal cooling rat by only a factor of. A mor rigorous drivation givs a rat W = [ g g ( M + U )]. N 1 τ This is th cooling rat for th amplitud x. Th cooling rat for th mittanc, proportional to x, is twic as big. W hav now also introducd th gain g, th mixing factor M and th nois-to-signal ratio U. Th gain is proportional to th lctronic gain, and it is dfind as th ratio -Δx/<x>. Th mixing factor tlls how many turns it taks for th sampls to gt randomizd aftr a corrction. U dpnds on both th bam and th pickup (signal strngth) and on th lctronics (nois). Th so-calld bad mixing is nglctd hr. Optimal cooling rat is obtaind whn g=1/(m+u) and is qual to 1 W 1 = τ N M + U.

Longitudinal Stochastic Cooling Thr ar two basic typs of momntum cooling:

Th othr tchniqu, filtr cooling, uss notch filtrs that hav a priod qual to

20 Longitudinal Stochastic Cooling Thr ar two basic typs of momntum cooling: On tchniqu, Palmr cooling, uss th corrlation btwn momntum and position in rgions of high disprsion. A transvrs diffrnc pickup connctd to a longitudinal kickr, or rf gap, thn can provid cooling. Th xprssion for cooling rat is th sam as for th transvrs cas. Th othr tchniqu, filtr cooling, uss notch filtrs that hav a priod qual to th dsird rvolution frquncy and a gain that changs sign at th cntr of ach Schottky band. A signal from a sum pickup abov th cntr frquncy givs a dclrating fild in th kickr and vic vrsa.

Stochastic Cooling, Summarizd Th bst would b a Maxwll s dmon that puts individual particls onto corrct orbits But it is nough to tak sampls of a largr numbr of particls and put thir avrag positions

21 Stochastic Cooling, Summarizd Th bst would b a Maxwll s dmon that puts individual particls onto corrct orbits But it is nough to tak sampls of a largr numbr of particls and put thir avrag positions right, if th procss is rpatd many tims. Hardwar is mainly pickups, amplifirs, filtrs and kickrs Important things ar bandwidth, powr, signal strngth and nois Extnsion to optical frquncis? Now in opration at CERN, Frmilab, GSI, FZ Jülich, BNL

On rason is that th particls mov rapidly past th dtctors and kickrs and thir charg distribution bcoms Lorntz contractd.

Th cost and complxity of lctron acclrators dpnds strongly on bam nrgy.

22 Stochastic Cooling vs. Elctron Cooling Stochastic cooling works bst with rlativistic particls bcaus on can thn rach highr bandwidths, up to 4-8 GHz. On rason is that th particls mov rapidly past th dtctors and kickrs and thir charg distribution bcoms Lorntz contractd. Compard to lctron cooling, th cost for stochastic cooling is not vry sntitiv to bam nrgy. Th cost and complxity of lctron acclrators dpnds strongly on bam nrgy. v c v c Othr diffrncs and similaritis Stochastic cooling is fastr for wak bams than for intns bams, whil lctron cooling is indpndnt of bam intnsity. Stochastic cooling is bttr for hot bams, bcaus of bttr signal, whil lctron cooling gts slowr for hot bams. With both mthods, highly chargd ions cool fastr. In stochastic cooling bcaus of bttr signal-to-nois ratio. Elctron cooling is asir in cas of variabl bam vlocitis.

Lasr cooling works with ions having closd, twostat transitions at optical wavlngths (or transitions that ar Dopplr shiftd to

23 Lasr Cooling In lasr cooling, a vlocity-dpndnt (non-consrvativ) forc is obtaind whn ions absorb photons, and thus momntum, dpnding on thir vlocity through th Dopplr shift. Momntum is absorbd from on dirction but rmittd isotropically. A nt momntum transfr rsults. Lasr cooling works with ions having closd, twostat transitions at optical wavlngths (or transitions that ar Dopplr shiftd to optical wavlngths). Lasr cooling rquirs spcific ions, is difficult transvrsally, but is strong and fastr than lctron or stochastic cooling.

This can b a achivd with,.g., a countrpropagating lasr, th rf or an induction acclrator.

24 Lasr Cooling Th rang of th lasr forc is short, so on nds to swp th lasr frquncy, or chang th ion vlocity, to covr th whol ion vlocity distribution. To rach a cold quilibrium on also nds a rstoring forc. This can b a achivd with,.g., a countrpropagating lasr, th rf or an induction acclrator. So far, lasr cooling has only bn applid to 7 Li +, 9 B +, 4 Mg + and, mor rcntly, C 3+ whr th transition is Dopplr-shiftd into th optical wavlngth rang.

Ionization (Muon) Cooling With ionization cooling, all

mattr, but longitudinal momntum only is rstord in rf cavitis.

Ionization cooling is particularly good for muons.

Also, mittanc incras du to scattring must not b fastr than

25 Ionization (Muon) Cooling With ionization cooling, all momntum componnts ar rducd whn th particls pass through mattr, but longitudinal momntum only is rstord in rf cavitis. Th nt rsult is a rduction of transvrs momntum. Ionization cooling is particularly good for muons. Hadrons mak nuclar ractions and lctrons produc brmsstrahlung. Also, mittanc incras du to scattring must not b fastr than cooling. Longitudinal cooling through disprsion and wdg absorbrs. MICE, Muon Ionization Cooling Exprimnt

26 Th End

High Energy Physics. Lecture 5 The Passage of Particles through Matter

High Energy Physics. Lecture 5 The Passage of Particles through Matter High Enrgy Physics Lctur 5 Th Passag of Particls through Mattr 1 Introduction In prvious lcturs w hav sn xampls of tracks lft by chargd particls in passing through mattr. Such tracks provid som of th most