Synchrotron Motion with Space-Charge

Transcription

1 Synchrotron Motion with Space-Charge

2 Basic Principles without space-charge RF resonant cavity providing accelerating voltage V (t). Often V = V 0 sin(φ s + ω rf t), where ω rf is the angular frequency synchronised with the arrival time of beam particles. Particle with phase φ = φ s at revolution period T 0 and momentum p 0 is called the synchronous particle. Synchronous particle synchronizes with rf wave with a frequency ω rf = h, where = β 0 c/r 0 is revolution frequency and h is harmonic number. It encounters the rf voltage at phase angle φ s on every revolution. The acceleration rate for synchronous particle is de 0 dt = 2π qv sin φ s. E 0 is the synchronous energy

3 Non-synchronous particles have small deviations of rf parameters: ω = + ω, φ = φ s + φ, θ = θ s + θ p = p 0 + p, E = E 0 + E. θ is the azimuthal orbital angle, where φ = φ φ s = h θ so ω = d dt θ = 1 h d dt φ = 1 h dφ dt Energy gain per revolution for non-synchronous particle is qv sin φ Thus = de dt = ω qv sin φ 2π ( ) E d dt = 1 2π qv ( sin φ sin φ s ) Since p p 0 = 1 β 2 E E, an equivalent form is d dt p p 0 = 2πβ 2 E qv ( sin φ sin φ s )

4 ω = βc R = ω = β α p is momentum compaction, α p = 1 = β 0 to the transition energy m 0 γ t c 2 Write η = 1 γ 2 t R R 0 ( ) 1 p γ 2 α p = p 0 γ 2 t 1 γ 2 ( 1 γ 2 1 γ 2 t, where γ t corresponds (slip factor) ) p p 0 Then dφ dt = h ω = hη p p 0 = hω2 0η β 2 E ( E )

5 ( φ, E ) Hamiltonian Formulation are conjugate coordinates in longitudinal phase-space Hamiltonian H = 1 2 hηω 2 0 β 2 E Hamilton s equations are d dt ( E dφ dt ( ) E ) 2 + qv 2π = [ cos φ cos φs +(φ φ s ) sin φ s ] H ( E/ ) = hηω2 0 β 2 E ( E = H φ = 1 2π qv [ sin φ sin φ s ] d 2 ( ) Combined: φ φs dt 2 = hω2 0qV η [ sin φ sin φs 2πβ 2 ] E linearised hω2 0qV η cos φ s ( ) φ φs 2πβ 2 E = stable oscillations for η cos φ s < 0 (McMillan & Veksler) Below transition, γ < γ t, η < 0, so require 0 < φ s < 1 2 π Above transition, γ > γ t, η > 0, so shift synchronous phase to π φ s )

6 Synchrotron Mapping Equation Hamiltonian formalism = uniform distribution of rf. In reality, finite number of cavities, localised in short sections of synchrotrons. Then use symplectic mapping equations: After one revolution of ring, in time T 0 = 2πR 0 βc = 2π, (φ n, E n ) (φ n+1, E n+1 ), where E n+1 = E n + qv ( sin φ n sin φ s ) φ n+1 = φ n + 2πhη β 2 E E n+1 Here E = E 0,n+1 = E 0 + qv sin φ s, γ = E/m 0 c 2, β = 1 1/γ 2 and η = α p 1/γ 2 Symplectic = Jacobian ( E n+1, φ n+1 ) ( E n, φ n ) =1

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8 Fields created by an unbunched (coasting) beam Assume a round beam of mean radius a in a circular pipe of radius b Velocity of beam = βc Line density (number of particles per unit length) = λ(s) s Er βc b a Fields are: E r = qλ 2πɛ 0 1 r ; E r = qλ 2πɛ 0 r a 2 ; B θ = µ 0qλβc 2π B θ = µ 0qλβc 2π 1 r ; r a 2 ; r a r a

9 Chamber E r (s) E w E r (s + s) Beam E s s βc Chamber Beam induces charges on inner surface of chamber wall, which form a current I w equal and opposite to the AC component of beam current.

10 Chamber E r (s) E w E r (s + s) Beam E s s βc Chamber Faraday s Law: E dl = B ds t = (E s E w ) s + q (1 + 2 ln b ) (λ(s ) + s) λ(s) = 4πɛ 0 a ( s ln b ) µ0 qβ c λ a 4π t where β is speed of disturbance, maybe not the same as β but very close: β β.

11 Since λ t = β c λ s βc λ s, longitudinal field is E s = qg 0 4πɛ 0 ( 1 β 2 ) λ s + E w qg 0 λ 4πɛ 0 γ 2 s + E w where g 0 = ln b a 1. Perfectly conducting smooth wall, E w = 0, g is geometry factor = space charge field E s = qg 0 λ 4πɛ 0 γ 2 s 2. Inductive wall (common in accelerators), inductance L/2πR per unit length = E w = L 2πR [ = E s = q di w dt qβ 2 c 2 L 2πR g 0 4πɛ 0 γ 2 β2 c 2 L 2πR λ s ] λ s = qβc 2π [ g0 2β Z 0 = 1 ɛ 0 c 377 Ω ] Z 0 λ γ 2 L s 3. Smooth resistive wall = skin effect; gives an impedance µ = permeability Z skin = (1 i) µ 0R ω b 2µµ 0 σ σ = conductivity

12 Hamiltonian H = 1 2 hηω 2 0 β 2 E ( E ) 2 + qv 2π [ cos φ cos φs +(φ φ s ) sin φ s ] For general rf H = 1 2 hηω 2 0 β 2 E ( E ) 2 q 2π U(φ) where U(φ) = φ φ s [ V (φ) V (φs ) ] dφ = φ φ s V (φ)dφ V (φ s )(φ φ s ) Space-charge will alter the effective voltage, generally in a non-linear way. But if the line density λ U, the self forces ( λ s λ ) caused by φ space-charge and inductive wall are proportional to the external force, giving a stationary distribution.

13 Local elliptic energy distribution Write W = E, so that H = hω2 0η 2β 2 E W 2 q 2π U(φ) A necessary and sufficient condition for a stationary particle distribution is that the phase-space density f(w, φ) can be written as a function of the Hamiltonian f = f(h). The elliptic distribution is given by: f(w, φ) = d2 N dw dφ = f(h) =c 1 H0 H, where H 0 is the Hamiltonian of the extreme (boundary) particles. Bucket extremities given by W = 0 or φ1 φs φ2 U(φ 1 )=U(φ 2 )= 2π q H 0 Then λ(φ) = dn dφ = f(w, φ)dw H 0 + q 2π U(φ) U(φ) U(φ 2) φ2

14 The line density is λ(φ) =c 2 [ U(φ) U(φ2 ) ] and has the same shape as the potential. More precisely: λ(φ) =N b U(φ) U(φ 2 ) u(φ 1, φ 2 ) where N b is the number of particles in the bunch and u(φ 1, φ 2 )= φ2 φ 1 [ U(φ) U(φ2 ) ] dφ The bunch current is I(φ) =q dn dφ dφ dt =2πhI b U(φ) U(φ 2 ) u(φ 1, φ 2 ). where I b = qn b /2π is the mean current per bunch.

15 Recall that the field in the beam from space-charge and inductance is E s = qβc 2π [ g0 2β ] Z 0 λ γ 2 L s corresponding to a voltage per turn U s = qβcr [ g0 2β ] Z 0 λ γ 2 L s. At low frequencies (long bunches), this gives a reactive coupling impedance [ Z e n = i L g ] 0Z 0 2βγ 2 = i L e where n = ω/ and L e is the effective inductance. So space-charge is equivalent to a negative, energy-dependent wall inductance. BNL-AGS BNL-RHIC Fermilab Booster Fermilab MI KEK-PS γ t Z sc /n [Ω]

16 For the elliptic distribution, total voltage seen by the beam is V (φ) 2πh V t (φ) = 2 I b Im {Z e /n} V (φ) V (φ s), φ 1 < φ < φ 2 u(φ 1, φ 2 ) V (φ) elsewhere The induced voltage has the same shape as the applied voltage 1. Below transition η < 0 and u(φ 1, φ 2 ) < 0, so a space-charge dominated beam with Im { Z e n } < 0 leads to a reduced focusing force. 2. Above transition η > 0, u(φ 1, φ 2 ) > 0, so a dominating inductive wall impedance results in reduced focusing, while a space-charge dominated beam sees enhanced focusing.

17 Voltage seen by a particle changes from V (φ) V (φ s ) to V t (φ) V (φ s )= ( ) 1 2πh2 I b Im {Z e /n} u(φ 1, φ 2 ) [V (φ) V (φ s )] and bucket area is changed relative to its low intensity value A 0 to V t (φ) V (φ s ) A t = A 0 V (φ) V (φ s ) = A 0 1 2πh2 I b Im {Z e /n} u(φ 1, φ 2 ) Gives limiting intensity: Î b = u(φ 1, φ 2 ) 2πh 2 Im {Z e /n}. At this intensity, induced voltage cancels the applied voltage, bucket area goes to zero and phase-space density is infinite.

18 Negative Mass (microwave) Instability Assume a (wakefield) disturbance to the distribution of the form f n e i(ωt nθ) (θ = orbiting angle,n= mode number). Satisfying Vlasov s equation gives: Stability requires real Ω i Z e n η = ( ) 2 Ω = i qi 0Z e /n nω 2πβ 2 E η For space-charge, Z e /n is capacitive (negative inductive), so require η < 0 for stability. There is an effective frequency shift without producing damaging collective effects. Above transition, η > 0, there is a space-charge driven instability, known as the negative mass or microwave instability c.f. H 1 2 hηω 2 0 β 2 E ( E Z e /n capacitive inductive resistive Below transition η < 0 stable unstable unstable Above transition η > 0 unstable stable unstable ) 2 + q 2π U(φ) [ L g 0Z 0 2βγ 2 ] η > 0

19 Theory shows that the microwave instability can be avoided for reasonably long bunches (e.g. SNS) provided I b < 0.4Îb. The induced (space-charge) voltage should never exceed 40% of the applied voltage. The corresponding reduction in bucket area is 23%.

20 Evolution of an initially uniform coasting beam, with no rf but under the effects of space charge and noise, showing formation of bunches and bunch merging.

21 Sinusoidal Voltage Sinusoidal voltage, V (φ) =V 0 sin φ gives U(φ) =V 0 [ cos φs cos φ (φ φ s ) sin φ s ] = u(φ 1, φ 2 )= φ2 φ 1 [ U(φ) U(φ2 ) ] dφ = V 0 f(φ 1, φ 2 ) where f(φ 1, φ 2 ) = sin φ 2 sin φ (φ 2 φ 1 )(cos φ 1 + cos φ 2 ). Note: uses U(φ 1 )=U(φ 2 ) to eliminate φ s. Line density is λ(φ) = N b [ ] cos φ + φ sin φs cos φ 2 φ 2 sin φ s. f(φ 1, φ 2 ) For short bunches, φ l = φ 2 φ 1 and f(φ 1, φ 2 ) 1 12 φ3 l cos φ s. Then potential: U(φ) 1 2 (φ φ s) 2 V 0 cos φ s line density: λ(φ) 6N b φ 3 l (φ 2 φ)(φ φ 1 ) quadratic, resulting in linear fields.

22 Longitudinal Envelope Equation dφ dt = hω2 0η β 2 E ( E ), d dt U s (φ) = qrg 0 2ɛ 0 γ 2 λ s = qh2 g 0 2ɛ 0 γ 2 R ( E In terms of distance from the synchronous particle, ) = qv 0 2π λ φ qh2 g 0 2ɛ 0 γ 2 R 12N b φ 3 l ( sin φ sin φs ) + q 2π U s(φ) (φ φ s ) = d2 z dt 2 = 0η Rω2 β 2 E dz dt = R d θ dt ( E d dt = R h ) d φ dt = qrω2 0η 2πβ 2 E = Rω2 0η β 2 E E { V 0 cos φ s 6qN bh 2 g 0 ɛ 0 Rγ 2 φ 3 l } (φ φ s ) = d2 z ds 2 + k zz K l z 3 m z =0 Bunch has an approximately parabolic line density between [ z m,z m ] where k z = qhηv 0 2πR 2 β 2 E cos φ s and K l = 3 2 r 0 g 0 N b η β 2 γ 3

23 Corresponding envelope equation is z m + k z z m K l z 2 m ɛ2 zz z 3 m =0 ɛ zz is the total (unnormalised) emittance of the bunch in the moving frame. Note: (i) Longitudinal equations for a straight channel can be obtained by setting α p = 0, η = 1/γ 2 (ii) Below transition, η < 0, so for φ s < 1 2 π, k z > 0 and K l > 0; analogous to a normal linear accelerator. (iii) Above transition, η > 0, so longitudinal focusing requires a change φ s π φ s. Now the perveance K l < 0 and space-charge is focusing. Space-charge electric field increases the energy of a particle at the front of the bunch. This increases orbit radius, which slows down angular motion and hence reduces distance z from bunch centre.

24 (iv) Synchrotron tune is Q 2 z = Q 2 z0 R2 K l For small differences, this gives z 2 m = Q 2 z r 0 g 0 N b R 2 η β 2 γ 3 z 2 m. Q z = 3 4 r 0 g 0 N b R 2 η β 2 γ 3 z 2 mq z0 Longitudinal tune shift is negative below transition and positive above transition.

25 ISIS Synchrotron Magnetic field in magnets B(t) =B 0 B 1 cos(2πft) (f=rep.rate, 50 Hz) has to match acceleration from cavities for synchronous particle. p dp Rigidity = Bρ = = qρḃ q dt E 2 /c 2 = p 2 + m 2 0c 2 = de dt = pc2 E dp dt Energy gain per revolution V (φ s ) = 2πR βc 1 q de dt =2πRρḂ ISIS pre 2006 used V (φ,t)=v 0 (t) sin φ, with V 0 modulated in time. = sin φ s = 2πR 2πRρ ρḃ = V 0 (t) V 0 (t) 2πfB 1 sin(2πft) R = 26 m, ρ =7m, f = 50 Hz, B 1 = T = sin φ s 93.5 kv V 0 (t) sin(2πft)

26 sin φ s 93.5 kv V 0 (t) sin(2πft) 0 t 10 ms Peak voltage V 0 (t) has to be modulated so that RHS is less than 1.

27 Peak current steadily increases, though transverse space-charge effects (e.g. tune depression) are greatest at low energy, immediately after injection.

28 ISIS dual harmonic rf scheme: V (φ,t)=v 0 (t) [ sin φ δ sin(2φ + θ) ] where δ, the ratio of the h =2:h =4 voltages, and θ, the relative phase, are functions of time. sin φ s δ sin(2φ s +θ) = 93.5 kv V 0 (t) sin(2πft) Produces lengthening of the bunches, flattening of the line-density, and reduction in the peak current; and gives improved bunching factor B f = mean current peak current. reduced transverse space-charge forces

29 Simulation of the ISIS acceleration cycle, showing the formation of the double oscillation centre and merging to give the final beam on target.