BEAM INSTABILITIES IN LINEAR MACHINES 1.

BEAM INSTABILITIES IN LINEAR MACHINES 1 Massimo.Ferrario@LNF.INFN.IT CERN 4 November 2015

SELF FIELDS AND WAKE FIELDS The realm of collecdve effects Direct self fields Image self fields Space Charge Wake fields

OUTLINE The rms emijance concept rms envelope equadon Space charge forces Beam/Envelope emijance oscilladons Matching condidons in a linac and emijance compensadon

Trace space of an ideal laminar beam " x $ # x! = dx dz = p x $ % p z p x << p z X X

Trace space of a laminar beam X X

Trace space of non laminar beam X X

Geometric emittance: Ellipse equation: ε g γx 2 + 2αx x $ + β x $ 2 = ε g Twiss parameters: βγ α 2 = 1! Ellipse area: A = πε g β = 2α

Trace space evolution No space charge => cross over X With space charge => no cross over X X X

rms emittance σ x' x σ x ε rms + + f x,! x ( x )dx d x! =1 f! x, x! rms beam envelope: + + σ x 2 = x 2 = x 2 f ( x, x &)dxd x & Define rms emittance: ( ) = 0 γx 2 + 2αx $ x + β $ x 2 = ε rms such that: σ x = x 2 = βε rms σ x' = x! 2 = γε rms Since: α = β $ 2 β = x2 ε rms it follows: α = 1 2ε rms d dz x 2 = x x % ε rms = σ xx' ε rms

σ x = x 2 = βε rms σ x' = x '2 = γε rms σ xx' = x x! = αε rms It holds also the relation: γβ α 2 = 1 Substituting α,β,γ we get 2 2 σ x' σ x " σ xx' $ ε rms ε rms # ε rms % ' & 2 =1 We end up with the definition of rms emittance in terms of the second moments of the distribution: ε rms = ( x 2 ) σ x2 σ 2 x' σ 2 xx' = x 2 x" 2 x "

Which distribution has no correlations? σ xx' = x x! = αε rms = 0? x x

What does rms emittance tell us about phase space distributions under linear or non-linear forces acting on the beam? x 2 ε rms = x 2 x # 2 x x # 2 x a a a x a x Assuming a generic x, x " correlation of the type: x! = Cx n When n = 1 ==> ε rms = 0 2 ε rms = C 2 ( x 2 x 2n x n+1 2 ) When n = 1 ==> ε rms = 0

Constant under linear transformation only And without acceleration: " x = p x p z

Normalized rms emittance: ε n,rms Canonical transverse momentum: p x = p z x! = m o cβγ x! p z p ε n,rms = σ x2 σ 2 px σ 2 xpx = 1 m o c ( x 2 2 2 p x xp ) βγ ε x rms Liouville theorem: the density of particles n, or the volume V occupied by a given number of particles in phase space (x,p x,y,p y,z,p z ) remains invariant under conservative forces. It hold also in the projected phase spaces (x,p x ),(y,p y )(,z,p z ) provided that there are no couplings

OUTLINE The rms emijance concept rms envelope equadon Space charge forces Beam/Envelope emijance oscilladons Matching condidons in a linac and emijance compensadon

Envelope Equation without Acceleration Now take the derivatives: dσ x dz = d dz d 2 σ x dz 2 = d dz x 2 = 1 2σ x σ x x! = 1 dσ x x! σ x σ x dz d dz x2 = 1 2 x x! = σ x x! 2σ x σ x σ 2 x x! σ = 1 x! 2 + x x! 3 x σ x 2 x ( ) σ x! σ x 3 = σ! 2 x + x x!! 2 x σ x! 3 σ x σ x And simplify: σ!! x = σ 2 xσ 2 2 x' σ xx' + x x!! 3 σ x σ x = ε 2 rms σ + x x!! 3 x σ x We obtain the rms envelope equation in which the rms emittance enters as defocusing pressure like term. σ!! x x x!! σ x = ε 2 rms 3 σ x

σ!! x x x!! σ x = ε 2 rms 3 σ x Assuming that each particle is subject only to a linear focusing force, without acceleration: x!! + k 2 x x = 0 take the average over the entire particle ensemble 2 σ x " + k 2 x σ x = ε rms σ x 3 2 x x!! = k x x 2 We obtain the rms envelope equation with a linear focusing force in which the rms emittance enters as defocusing pressure like term. 2 ε rms σ T 3 x V P

S = kn log( πε)

OUTLINE The rms emijance concept rms envelope equadon Space charge forces Beam/Envelope emijance oscilladons Matching condidons in a linac and emijance compensadon

Space Charge: what does it mean? The net effect of the Coulomb interactions in a multi-particle system can be classified into two regimes: 1) Collisional Regime ==> dominated by binary collisions caused by close particle encounters ==> Single Particle Effects 2) Space Charge Regime ==> dominated by the self field produced by the particle distribution, which varies appreciably only over large distances compare to the average separation of the particles ==> Collective Effects

Continuous Uniform Cylindrical Beam Model I ρ = π R 2 v R J = I π R 2 Gauss s law ε o E ds = Ampere s law ρdv I E r = 2πε o R 2 v r for r R I 1 E r = 2πε o v r for r > R Ir B dl = µ o J ds B ϑ = µ o 2πR for r R 2 I B ϑ = µ o for r > R 2πr B ϑ = β c E r

Bunched Uniform Cylindrical Beam Model L(t) R(t) (0,0) s l v z = βc Longitudinal Space Charge field in the bunch moving frame: ρ = Q πr 2 L E z s,r = 0 ( ) = ρ 4πε o R 2 π L ( l s ) rdrdϕd l & ( l s ) 2 + r 2 3 / 2 ) 0 0 0 '( * + E z ( s,r = 0) = ρ % 2ε 0 &' ( ) R 2 + ( L s ) 2 R 2 + s 2 + 2 s L ( )*

Radial Space Charge field in the bunch moving frame by series representation of axisymmetric field: $ E r (r, s) ρ & % ε 0 s ' E z (0, s) ) r ( 2 + [ ] r3 16 + E r (r, s ) = ρ 2ε 0 % ' &' ( L s ) R 2 + ( L + s ) 2 s R 2 + s 2 ( * )* r 2

Lorentz Transformation to the Lab frame E z = E z E r = γ E r L = γl ρ = ρ γ s = γs E z (0,s) = ρ & R 2 + γ 2 (L s) 2 R 2 + γ 2 s 2 + γ 2s L γ 2ε 0 '( ( ) ) * + E r (r,s) = γρ & ( (L s) 2ε 0 ( + ' R 2 + γ 2 (L s) 2 s R 2 + γ 2 s 2 ) + r + 2 * It is still a linear field with r but with a longitudinal correlation s

Bunched Uniform Cylindrical Beam Model E z (0,s,γ ) = I 2πγε 0 R 2 βc h s,γ ( ) E r (r,s,γ ) = Ir 2πε 0 R 2 βc g s,γ ( ) γ= 1 γ = 5 γ = 10 R s (t) L(t) Δt

Lorentz Force F r = e( E r βcb ϑ ) = e( 1 β 2 )E r = ee r γ 2 is a linear function of the transverse coordinate dp r dt = F r = ee r γ 2 = eir 2πγ 2 ε 0 R 2 βc g s,γ ( ) The attractive magnetic force, which becomes significant at high velocities, tends to compensate for the repulsive electric force. Therefore space charge defocusing is primarily a non-relativistic effect. F x = eix 2πγ 2 ε 0 σ x2 βc g s,γ ( )

Envelope Equation with Space Charge Single particle transverse motion: dp x dt d dt x!! = = F x p x = p x! = βγm o c x! ( p x! ) = βc d dz p x! F x βcp ( ) = F x F x = eix 2πγ 2 ε 0 σ x2 βc g s,γ ( ) x!! = k sc s,γ ( ) σ x 2 x

Now we can calculate the term x x " that enters in the envelope equation σ!! x = ε 2 rms σ + x x!! 3 x σ x!! x = k sc σ x 2 x x x!! = k sc σ x 2 x 2 =k sc Including all the other terms the envelope equation reads: σ!! x + k 2 σ x = Space Charge De-focusing Force ε n 2 ( βγ) 2 σ + k sc 3 x σ x Emittance Pressure External Focusing Forces Laminarity Parameter: ( ρ = βγ ) 2 2 k sc σ x 2 ε n

The beam undergoes two regimes along the accelerator σ x " + k 2 σ x = ε n 2 ( βγ) 2 σ + k sc 3 x σ x ρ>>1 Laminar Beam σ x " + k 2 σ x = ε n 2 ( βγ) 2 σ + k sc 3 x σ x ρ<<1 Thermal Beam

OUTLINE The rms emijance concept rms envelope equadon Space charge forces Beam/Envelope emijance oscilladons Matching condidons in a linac and emijance compensadon

Surface charge density Surface electric field Restoring force Plasma frequency Plasma oscillations

Neutral Plasma Oscillations Instabilities EM Wave propagation Single Component Cold Relativistic Plasma Magnetic focusing Magnetic focusing

( ) σ " + k 2 s σ = k sc s,γ σ Equilibrium solution: ( ) = k sc( s,γ ) σ eq s,γ k s Single Component Relativistic Plasma k s = qb 2mcβγ Small perturbation: σ( ζ ) = σ eq ( s) +δσ ( s) δ σ # s ( ) + 2k 2 s δσ ( s) = 0 δσ ( s) = δσ o ( s)cos( 2k s z) Perturbed trajectories oscillate around the equilibrium with the same frequency but with different amplitudes: σ( s) = σ eq ( s) +δσ o ( s)cos( 2k s z)

σ eq ( s) Perturbed trajectories oscillate around the equilibrium with the same frequency but with different amplitudes: σ ( s) = σ eq ( s) +δσ o ( s)cos( 2k s z)

ε rms = σ x2 σ 2 x' σ 2 xx' sin( 2k s z)

Emittance Oscillations are driven by space charge differential defocusing in core and tails of the beam p x Projected Phase Space x Slice Phase Spaces

Perturbed trajectories oscillate around the equilibrium with the same frequency but with different amplitudes X X

Elliptical cross section bunch E y = I βcπε o y Y X + Y ( ) E r = I 2βcπε o R 2 r Y R* R * = X + Y 2 X E x = I βcπε o x X X + Y ( ) E x ( X, z) = E y ( Y,z) = E ( r R *,z)

OUTLINE The rms emijance concept rms envelope equadon Space charge forces Beam/Envelope emijance oscilladons Matching condidons in a linac and emijance compensadon

High Brightness Photo-Injector

Envelope Equation with Longitudinal Acceleration p o = γ o m o β o c p x << p o p = p o + p! z p! = ( βγ)! m o c dp x dt x!! +! = d ( p x! ) = βc d ( dt dz p x! ) = 0 p!! p x! = 0 ( x = βγ )! βγ x! x x!! ( = βγ )! βγ x x! ( = βγ )! βγ σ xx' σ!! x = ε 2 rms σ + x x!! 3 x σ x dσ x dz = σ! = σ xx' x σ x Space Charge De-focusing Force Adiabatic Damping ( )! σ!! x + βγ βγ σ! x + k 2 σ x = ε n 2 ( βγ) 2 σ + k sc 3 x σ x Emittance Pressure Other External Focusing Forces ε n = βγε rms

Beam subject to strong acceleration σ!! x + γ! σ! x + k 2 RF γ γ 2 σ = ε 2 n x γ 2 σ + k o sc 3 x γ 3 σ x We must include also the RF focusing force: 2 k RF =! γ 2 2 k o sc = 2I g( s,γ ) I A

σ x " + γ " σ " x + k RF γ γ 2 σ x = 2 ε 2 n γ 2 3 σ + x k o sc γ 3 σ x γ = 1 +αz ==> γ " = 0 Looking for an equilibrium solution σ inv = σ o γ n ==> all terms must have the same dependence on γ σ " inv = nσ o γ n 1 γ " σ inv " = n( n 1)σ o γ n 2 " γ 2 n( n 1)σ o γ n 2 γ % 2 + nσ o γ n 2 γ % 2 + k 2 RF σ o γ n 2 = k sc o γ 3 n σ x n 2 = 3 n n = 1 2

σ x " + γ " σ " x + k RF γ γ 2 σ x = 2 ε 2 n γ 2 3 σ + x k o sc γ 3 σ x γ = 1 +αz ==> γ " = 0 Looking for an equilibrium solution σ inv = σ o γ n ==> all terms must have the same dependence on γ Laminar beam Thermal beam ρ >> 1 n = 1 2 ρ << 1 n = 0 σ q = σ o γ σ ε = σ o

Space charge dominated beam (Laminar) σ q = 1 γ # 2I I A γ Emittance dominated beam (Thermal) σ ε = 2ε n γ $

σ q = 1 γ # 2I I A γ This solution represents a beam equilibrium mode that turns out to be the transport mode for achieving minimum emittance at the end of the emittance correction process

An important property of the laminar beam σ q = 1 γ # 2I I A γ σ q ' = 2I I A γ 3 Constant phase space angle: δ = γσ ' q σ q = γ & 2 p x p x δ X δ X

Laminarity parameter ρ = 2Iσ 2 γi A ε 2Iσ 2 q 2 n γi A ε = 4I 2 2 n γ ' 2 I 2 A ε 2 n γ 2 Transition Energy (ρ=1) γ tr = 2I # γ I A ε n ρ I=4 ka ε th = 0.6 µm E acc = 25 MV/m I=1 ka Potential space charge emittance growth I=100 A ρ = 1

Matching Conditions with a TW Linac γ tr = 2I # γ I A ε n 150 MeV σ ' = 0 σ q = 1 # γ 2I I A γ 25 MV/m

z=0.23891 0.05 z=1.5 z=10 0.04 0.04 0.02 0.02 Pr 0 Pr 0 pr_[rad] 0-0.02-0.02-0.04-0.05 0 0.001 0.002 0.003 0.004 0.005 0.006 3.5 0 0.0008 0.0016 0.0024 0.0032 0.004 R [m] R [m] -0.04 0 0.0008 0.0016 0.0024 0.0032 0.004 R_[m] 3 2.5 2 rms norm. emittance [um] rms beam size [mm] 1.5 1 0.5 0 0.004 0.0035 0.003 Gun Linac 0 2 4 6 8 10 z=0.23891 z=1.5 Z=10 0.004 0.004 0.0035 Z_[m] 0.0035 0.003 0.003 Rs [m] 0.0025 0.002 0.0015 0.001 0.0005 Rs [m] 0.0025 0.002 0.0015 0.001 0.0005 Rs [m] 0.0025 0.002 0.0015 0.001 0.0005 0-0.003-0.002-0.001 0 0.001 0.002 0.003 Zs-Zb [m] 0-0.003-0.002-0.001 0 0.001 0.002 0.003 Zs-Zb [m] 0-0.003-0.002-0.001 0 0.001 0.002 0.003 Zs-Zb [m]

Emittance Compensation for a SC dominated beam: Controlled Damping of Plasma Oscillations ε n oscillations are driven by Space Charge propagation close to the laminar solution allows control of ε n oscillation phase ε n sensitive to SC up to the transition energy

References:

Emittance and Entropy The entropy of the definition: distribution is by For large N, n, Stirling's formula gives: is the number of ways in which the points can be assigned to the cells to produce the given distribution If A is sufficiently small, the summation may be replaced by an integral to give: ρ dx d x! = N

Emittance and Entropy The entropy of the definition: distribution is by Consider a distribution in which the density is uniform and bounded by an ellipse of area πε and: ρ = n A = N πε ρ dx d x! = N S o = log N " ( ) log$ AN # % ' = log πε πε & ( ) log( A)

Envelope oscillations drive Emittance oscillations σ(z) ε(z) ε rms = ( x 2 ) sin 2k s z σ 2 x σ 2 x' σ 2 xx' = x 2 x % 2 x % ( )

energy spread induces decoherence σ(z) ε(z)