Accelerator Physics Statistical and Collective Effects

Transcription

1 Acceleato Phsics Statistical and Collective Effects A. Bogacz, G. A. Kafft, and T. Zolkin Jeffeson Lab Lectue 4 USPAS Acceleato Phsics June 3

2 Statistical Teatments of Beams Distibution Functions Defined Statistical Aveaging Eamples Kinetic Equations Liouville Theoem Vlasov Theo Collision Coections Self-consistent Fields Collective Effects KV Equation Landau Damping Beam-Beam Effect USPAS Acceleato Phsics June 3

3 Beam ms Emittance Teat the beam as a statistical ensemble as in Statistical Mechanics. Define the distibution of paticles within the beam statisticall. Define single paticle distibution function ( ) ψ,, whee ψ(,')dd' is the numbe of paticles in [,+d] and [','+d'], and statistical aveaging as in Statistical Mechanics, e. g. ψ q q,, dd / N ψ q q,, dd/ N M USPAS Acceleato Phsics June 3

4 Closest ms Fit Ellipses Fo zeo-centeed distibutions, i.e., distibutions that have zeo aveage value fo and ' ε ms β = = ε ε σ ms ms α = ε ms γ = = ε ε σ ms ms USPAS Acceleato Phsics June 3

5 Case: Unifoml Filled Ellipse ψ γ + α + β, = Θ πε ε Θ hee is the Heavside step function, fo positive values of its agument and zeo fo negative values of its agument εβ σ = = 4 εα = 4 ε σ = = + 4β ε ε ms = 4 ( α ) Gaussian models (HW) ae good, especiall fo lepton machines USPAS Acceleato Phsics June 3

6 Dnamics? Stat with Liouville Thm. Genealization of the Aea Theoem of Linea Optics. Simple Statement: Fo a dnamical sstem that ma be descibed b a conseved eneg function (Hamiltonian), the elevant phase space volume is conseved b the flow, even if the foces ae non-linea. Stat with some simple geomet! ( q, p ) ( q, p) ( q, p ) 3 3 Aea ( q q )( p p ) ( q q )( p p ) Δ= 3 3 ( acute angle has line clockwise wt line 3) Δ Δp Δq Δ In phase space Aea Befoe=Aea Afte USPAS Acceleato Phsics June 3

7 Liouville Theoem H H q p q + q p Δ t+ p q p Δ t+ L p q (, ) (, ) L, (, ) H H q +Δq p q +Δ q+ q +Δq p Δ t+ p q +Δq p Δ t+ L p q (, ) (, ) L, (, ) H H q p +Δp q + q p +Δp Δ t+ p +Δp q p +Δp Δ t+ L p q H q +Δ q+ ( q +Δ q, p +Δp) Δ t+ L, p +Δ +Δ H p +Δp ( q +Δ q, p +Δp) Δ t+ L q (, ) (, ) L, (, ) ( q q, p p) Aea Δ = Δ t det H H H H Δ q+ ( q +Δq, p) ( q, p) Δt ( q +Δq, p) ( q, p) Δt p p q q H H H H ( q, p +Δp) ( q, p) t p ( q, p p) ( q, p) p p Δ Δ +Δ t q q Δ ( q p) Δ Δ H H ( ΔqΔ p) + ( q, p) ( q, p) Δ t = pq pq USPAS Acceleato Phsics June 3

8 Likewise H H H H Δ q+ ( q +Δq, p) ( q +Δ q, p +Δp) t ( q q, p) ( q q, p p) t p p Δ +Δ +Δ +Δ q q Δ Aea Δ = det H H H H ( q, p +Δp) ( q +Δ q, p +Δp) t p ( q, p p) ( q q, p p) t p p Δ Δ +Δ +Δ +Δ q q Δ H H ( ΔqΔp) ( ΔqΔ p) + ( q +Δ p, p +Δp) ( q +Δ p, p +Δp) Δ t = Δ t pq pq Because the stating point is abita, phase space aea is conseved at each location in phase space. In thee dimensions, the full 6-D phase volume is conseved b essentiall the same agument, as is the sum of the pojected aeas in each individual pojected phase spaces (the so-called thid Poincae and fist Poincae invaiants, espectivel). Defeat it b adding non- Hamiltonian (dissipative!) tems late. USPAS Acceleato Phsics June 3

9 Phase Space Plot of dnamical sstem state with coodinate along abscissa and momentum along the odinate Linea Oscillato p H p = + mω m USPAS Acceleato Phsics June 3

10 Liouville Theoem Aea in phase space is peseved when the dnamics is Hamiltonian p t = t t = t Aea = Aea USPAS Acceleato Phsics June 3

11 ( +Δ ) D Poof dv V t t V t = lim dt Δ t Δt d H ( s, t +Δ t) ( s) + Δ t + L ( s) + ( ( s), p ( s) ) Δ t + L dt p (, ) L, c s, t c s, t ( +Δt) t dp H p s t+δ t p s + Δ t+ p s s p s Δ t + L dt t L d V() t = pd= p( st, ) ( st, ) ds ds L d V ( t+δ t) = p d = p( s, t +Δ t) ( s, t +Δt) ds ds L H p( s) + ( ( s), p( s) ) Δt d ds H ( s) ( ( s), p ( s) ) Δt ds p USPAS Acceleato Phsics June 3

12 L dv d H H d s = p( s) ( ( s), p( s) ) ( ( s), p( s) ) ds dt + ds p ds (, ) L dp s H H d s = ( ( s), p( s) ) + ( ( s), p( s) ) ds ds p ds (wh is bounda tem of integation b pats zeo?) H = dp + p c s t H d B Geen's Thm. = when the diffeential is an eact diffeential H H i.e., =, in othe wods alwas p p note the integand above is eall dh, so H is a "potential" fo phase space!!! USPAS Acceleato Phsics June 3

13 3D Poincae Invaiants In a thee dimensional Hamiltonian motion, the 6D phase space volume is conseved (also called Liouville s Thm.) V6 D = dpdpdpzdddz V6 Additionall, the sum of the pojected volumes (Poincae invaiants) ae conseved dp d + dp d + dp dz z poj poj poj ( V ) ( V ) ( V ) dp dp ddz + dp dp dzd + dp dp dd z z poj poj poj ( V ) ( V ) ( V ) Emittance (phase space aea) echange based on this idea Moe complicated to pove, but ae tue because, as in D H H = q p p q i i i i USPAS Acceleato Phsics June 3

14 γ is a loop in 6D phase space γ ( t) = p( s, t), q( s, t) 3 L 3 di pidi = pi( s) ( s) p( s) + s p s d d H H (, ) (, ) dt () t i i ds pi i ds γ = = L 3 dpi s H H di s = ( ( s), p( s) ) + ( ( s), p( s) ) = dh = i= ds pi i ds γ () t fo an suface in 6D phase space V i i i i i i V i= V i= i= poj V 3 i i = z + z + i= 3 3 dpidi = z i=, with pd = dpd = dpd γ = V dp d dp dp ddz dp dp dzd dp dp dd dp dp dp dddz ( s) USPAS Acceleato Phsics June 3

15 Vlasov Equation B intepetation of ψ as the single paticle distibution function, and because the individual paticles in the distibution ae assumed to not coss the boundaies of the phase space volumes (collisions neglected!), ψ must evolve so that dψ = as the distibution evolves dt dψ ψ ( t+ δt, q( t+ δt), p( t+ δt) ) ψ ( t, q( t), p( t) ) = lim = dt δt δt whee the equation fo ANY (this is what makes it had to solve in geneal!) individual obits though phase space is given b q() t, p() t ψ dq ψ dp ψ + + = t dt q dt p USPAS Acceleato Phsics June 3

16 N t Consevation of Pobabilit = ψ t q p d d p 3 3 ;, is a conseved quantit continuit equation fo ψ is ( q& ψ) p( p& ψ) ψ + q + = t ψ dq ψ dp ψ H H ψ q p t dt q dt p = p q fo the Hamiltonian sstem ψ dq ψ dp ψ + + t dt q dt p = ψ q& L (, q, L) (, p, L) ψ p& L Δq Δp (, +Δ, L) (, +Δ, L) ψ q& L q q ψ p& L p p USPAS Acceleato Phsics June 3

17 Jean s Theoem The independent vaiable in the Vlasov equation is often changed to the vaiable s. In this case the Vlasov equation is ψ dq ψ dp ψ + + = s ds q ds p The equilibium Vlasov poblem, ψ / t =, is solved b an function of the constants of the motion. This esult is called Jean s theoem, and is the stating point fo instabilit analsis as the unpetubed poblem. ( L) If ψ = f A, B, C,, whee A, B, C, L ae constants of the motion d ψ dp ψ f d A dp A f d B dp B + = dt dt p A dt dt p B dt dt p f d C dp C f da f db f dc L = + + C dt dt p A dt B dt C dt + L = USPAS Acceleato Phsics June 3

18 Eamples -D Hamonic oscillato Hamiltonian. Bi-Mawellian distibution is a stationa distibution mω ψ = ep ( H / kt ) ep ( p / mkt ) ep ( m ω / kt ), π = πkt As is an othe function of the Hamiltonian. Contous of constant ψ line up with contous of constant H D tansvese Gaussians, including focusing stuctue in ing ( ) ( ( s) + ( s) + ( s) ) + + ψ s ;, ;, ep γ s α s β s / ε ( γ ) α β ε ep / Contous of constant ψ line up with contous of constant Couant- Snde invaiant. Stationa as paticles move on ellipses! USPAS Acceleato Phsics June 3

19 Solution b Chaacteistics Moe subtle: a solution to the full Vlasov equation ma be obtained fom the distibution function at some the initial condition, povided the paticle obits ma be found unambiguousl fom the initial conditions thoughout phase space. Eample: -D hamonic oscillato Hamiltonian. () cos ( ) sin ( ) / ( ) cos ( ) sin ( ) / = = () ωsinω( ) cosω( ) ( ) ωsinω( ) cosω( ) (, ; t= t) = f(, ) ψ ( t) = f( ω( t t) ω( t t) ω ω ω( t t) + ω ( t t ) ) f d( t;, ) f d ( t;, ) t ω t t ω t t ω t ω t t ω t t ω t t t t t t t t t t ψ Let, ; cos sin /, sin cos ψ = + t dt dt f f = ωsinω( t t) cosω( t t) + ω cosω( t t) ωsinω( t t) d ψ f f = cosω( t t) + ωsinω( t t) dt d ψ f f dψ = ω sin ω( t t) / ω+ cosω( t t) dt = dt USPAS Acceleato Phsics June 3

20 Beathing Mode ' ' Quate Oscillation ' ' ' The paticle envelope beaths at twice the evolution fequenc! USPAS Acceleato Phsics June 3

21 Sachee Theo Assume beam is acted on b a linea focusing foce plus additional linea o non-linea foces + k F = + k F = Fo space chage eample we'll see F Now qe ( β ) qe = = 3 γmc β γ mc β + k F = + k F = Assume distibutions zeo-centeed and let % = % = % = % = USPAS Acceleato Phsics June 3

22 ( ) Also = = % = %% = % = %% = %% + % = + = k + F = = %% + % = k + F / % = % %% + + k % F = % k 3 % % F % + + % = ε F ms % + k% "Envelope" equation 3 = % % USPAS Acceleato Phsics June 3

23 ms Emittance Conseved ( ) = + = + + k F ( ) = + + k F k F = F F Fo linea foces deivative vanishes and ms emittance conseved. Emittance gowth implies non-linea foces. USPAS Acceleato Phsics June 3

24 Space Chage and Collective Effects Collective Effects Billouin Flow Self-consistent Field KV Equation Bennet Pinch Landau Damping USPAS Acceleato Phsics June 3

25 Simple Poblem How to account fo inteactions between paticles Appoach : Coulomb sums Use Coulomb s Law to calculate the inteaction between each paticle in beam Unfavoable N in calculation but pehaps most ealistic moe and moe ealistic as computes get bette Appoach : Calculate EM field using ME Need pocedue to define chage and cuent densities Tack paticles in esulting field USPAS Acceleato Phsics June 3

26 Unifom Beam Eample Assume beam densit is unifom and ai-smmetic going into magnetic field n( ) = a Electic Field qn qn E = E = ε ε Self-Magnetic Field b Ampee's Law qnβc πbθ = μqnβcπ Bθ = μ USPAS Acceleato Phsics June 3

27 Billouin Flow Total Collective Foce on beam paticle ( ) qn F = q E+ v B ( β ) ε effective (de)focussing stength ω qn k = = β γ ε m p whee the non-elativistic "plasma fequenc" is ω 3 p c B pevious wok with solenoids in the otating fame, can have equilibium (foce balance) when ω p Ωc =Ω c non-elativistic plasma and ccloton fequencies ωl = γ γ This state is known as Billouin Flow and neglects beam tempeatue (fluid flow) USPAS Acceleato Phsics June 3

28 Comments Some authos, Reise in paticula, define a elativistic plasma fequenc ω p = qn 3 ε γ m Lawson s book has a nice discussion about wh it is impossible to establish a elativistic Billouin flow in a device whee beam is etacted fom a single cathode at an equipotential suface. In this case one needs to have eithe sheeing of the otation o non-unifom densit in the selfconsistent solution. USPAS Acceleato Phsics June 3

29 K-V Distibution Single value fo the tansvese Hamiltonian = ε β ε β ψ ρ K E ( α β ) + ( α+ β ) (,,, ) δ ( C ) ( z) I = πβcxy I = I 3 3 = I βγ I = πε βcy X ( + Y) 4πε mc q 3 C USPAS Acceleato Phsics June 3

30 K-V Envelope Equation E E I = πε βcx X Y I = πε βcy X ( + ) ( + Y) K + k = X X Y ( + ) K + k = Y X ( + Y) Envelope Equation K ε X + k X = 3 X Y X ( + ) K ε Y + ky = 3 X Y Y ( + ) USPAS Acceleato Phsics June 3

31 Luminosit and Beam-Beam Effect Luminosit Defined Beam-Beam Tune Shift Luminosit Tune-shift Relationship (Kafft-Ziemann Thm.) Beam-Beam Effect USPAS Acceleato Phsics June 3

32 Events pe Beam Cossing In a nuclea phsics epeiment with a beam cossing though a thin fied taget Taget Numbe densit n Beam Pobabilit of single event, pe beam paticle passage is σ is the coss section fo the pocess (aea units) l P = nσ l USPAS Acceleato Phsics June 3

33 Collision Geomet Beam Beam Pobabilit an event is geneated b a single paticle of Beam cossing Beam bunch with Gaussian densit* ( ) ( σ ) σ N ep / ep / ( P = σ ep z / ) 3/ σ z dz π σ σ σ ( ) = z ( ) ( σ ) σ N ep / ep / πσ σ σ * This epession still coect when elativit done popel USPAS Acceleato Phsics June 3

34 Collide Luminosit Pobabilit an event is geneated b a Beam bunch with Gaussian densit cossing a Beam bunch with Gaussian densit Event ate with equal tansvese beam sizes Luminosit P L = = N N π σ + σ σ + σ dn dt fnn 4πσ σ f N N fn N = σ = L 4πσ σ σ 33 ~ sec cm, fo = MHz, = =, σ = σ = micons σ USPAS Acceleato Phsics June 3

35 Beam-Beam Tune Shift As we ve seen peviousl, in a ing acceleato the numbe of tansvese oscillations a paticle makes in one cicuit is called the betaton tune Q. An deviation fom the design values of the tune (in eithe the hoizontal o vetical diections), is called a tune shift. Fo long tem stabilit of the beam in a ing acceleato, the tune must be highl contolled. * cos sin * μ β μ M tot = / f sin μ / β cos μ * cos μ β sin μ = * ( * cos μ / f sin μ / β cos μ β / f ) sin μ USPAS Acceleato Phsics June 3

36 * β T M cos( μ +Δ μ) = tot = cos μ sin μ f ξ * * =Δ = = β << Q Δμ β π 4π f f USPAS Acceleato Phsics June 3

37 Bessetti-Eskine Solution -D potential of Bi-Gaussian tansvese distibution Q ρ (, ) = ep ep πσ σ σ σ Potential Theo gives solution to Poisson Equation φ (, ) = = (, ) Bassetti and Eskine manipulate this to φ ρ ε ep ep Q σ q σ q + + dq 4πε σ + q σ + q USPAS Acceleato Phsics June 3

38 σ σ + i Q + i σ σ E = Im w ep w ( ) ( ε ) ( π σ σ σ σ σ σ σ σ ) σ σ + i Q i E Re w + σ σ = ep w ε π ( σ ) σ σ σ σ σ ( σ σ ) w z Comple eo function We need -D linea field fo small displacements φ Q E (,) = dq πε σ + q σ + q 3/ USPAS Acceleato Phsics June 3

39 Can do the integal analticall dq = ( σ ) + q σ + q σ + σ ( σ σ + q ) ( σ σ + q ) σ σ + q ( σ σ ) q = dq = 3/ / / σ + σ ( q ( σ σ ) ) ( σ σ ) ( q ( σ σ ) ) ( q ( σ σ ) ) σ + σ = + σ σ σ σ σ Similal fo the -diection E σ σ + q dq σ σ + σ + σ + σ σ + = = σ σ σ σ σ σ ( σ σ ) σ σ ( + ) (, ) φ Q = πε σ σ + σ σ + σ USPAS Acceleato Phsics June 3

40 Linea Beam-Beam Kick Linea kick eceived afte inteaction with bunch Δ mc = q E + v B t, t dt () ( γ β ) () z z b elativit, fo oppositel moving beams Δ γβ mc = q + β β E t, t dt Following linea Bassetti-Eskine model q E (,, z, t) = ep q moves with () t = (,, β zct) ( z β ct) z πε σ σ + σ σ πσ z USPAS Acceleato Phsics June 3

41 Δ γβ mc = / f Linea Beam-Beam Tune Shift ( + β β ) ( + β β ) ( + ) z z z z = = γ βz + β z σ 4 σ πε + σ mc / Both beams elativistic f q z z q β + β πε c σ σ + σ N e N γσ σ σ Fom linea Bassetti-Eskine model, and eplacing the beam size ξ N N = ξ = i πγ ε σ σ ε σ σ σ σ ( + / ) πγ ( + / )( / ) Agument entiel smmetic wt choice of bunch and ξ i N i i i N i i = ξ = i πγ ε σ σ ε σ σ σ σ i i + / πγ ( / )( / ) i + USPAS Acceleato Phsics June 3

42 L Luminosit Beam-Beam tune-shift elationship Epess Luminosit in tems of the (lage!) vetical tune shift (i eithe o ) = fn ξ γ + = ξ γ e + i i i i I i * * iβi iβi i σ / σ ( σ / σ ) Necessa, but not sufficient, fo self-consistent design Epessed in this wa, and given a known limit to the beam-beam tune shift, the onl vaiables to manipulate to incease luminosit ae the stoed cuent, the aspect atio, and the β* (beta function value at the inteaction point) Applies to ERL-ing collides, stoed beam (ions) onl USPAS Acceleato Phsics June 3

43 Luminosit-Deflection Theoem Luminosit-tune shift fomula is lineaized vesion of a much moe geneal fomula discoveed b Kafft and genealized b V. Ziemann. Relates eas calculation (luminosit) to a had calculation (beam-beam foce), and contains all the standad esults in beam-beam inteaction theo. Based on the fact that the elativistic beam-beam foce is almost entiel tansvese, i. e., -D electostatics applies. USPAS Acceleato Phsics June 3

44 -D Electostatics Theoem Q E = 4πε F = F = ρ d d ρ πε on n ρ / Q = n = ρ + b / Q zeo centeed Q i = ρi d b = ρ d / Q Q Q + b F = F = n n d d πε + b USPAS Acceleato Phsics June 3

45 + b = πδ b + b = + b ( b ) δ ( b ) ρ F ρ b d ε πε c ρ Genealizes E = take + b ε ( ) ρ δ Tansvese inteaction in the beam-beam poblem qq Δ p = USPAS Acceleato Phsics June 3

46 D( b) =Δ γ β = Δmγ β / m qq b = n ( ) n ( ) mc d d b b D b = N n b n d = 4 π e = = D( b) = ( γ ) Δ β e 4πε mc L b N N n b n d L b L b N 4π e N 4π e b b e USPAS Acceleato Phsics June 3

47 D γ / σ σ b = D f b N γξ L / as befoe = + * e β ( σ σ ) Maimum when D D =, = b b b b USPAS Acceleato Phsics June 3

48 Luminosit-Deflection Pais Round Beam Fast Model D b L( b) Gaussian Macopaticles D b D b Smith-Laslett Model = N e b NN σ σ + b = π σ + = Bassetti _ Eskine ; σ + σ ; σ + σ L b D b L b ( b ) b NN b = ep ep π σ + σ σ + σ σ + σ σ + σ 4 ( bˆ bˆ ) ( 4 ) ( 4 ˆ ˆ ) ( bˆ ) bˆ bˆ ( bˆ ) 5/ 4 4 ( 4 ˆ ˆ ) ( 4 ˆ ˆ ) 4 ˆ ˆ3 ˆ ˆ N eb + 4b b 3b b = sinh sinh ˆ 4 3/ b AB bˆ bˆ b + b N N ˆ 3 ˆ ˆ b b b sinh sinh π AB b + b b + b 3 = + + bˆ b b = + A B USPAS Acceleato Phsics June 3