Space Charge Mi-ga-on Massimo.Ferrario@LNF.INFN.IT Hamburg June nd 016
OUTLINE The rms emicance concept rms envelope equa-on Space charge forces Space charge induced emicance oscilla-ons Matching condi-ons and emicance compensa-on
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 + αx x $ + β x $ = ε g Twiss parameters: βγ α = 1! Ellipse area: A = πε g β = α
Trace space evolution No space charge => cross over X With space charge => no cross over X X X
rms emittance ' x ε rms + + f x,! x ( x )dx d x! =1 f! x, x! rms beam envelope: + + = x = x f ( x, x &)dxd x & Define rms emittance: ( ) = 0 γx + αx $ x + β $ x = ε rms such that: = x = βε rms ' = x! = γε rms Since: α = β $ β = x ε rms it follows: α = 1 ε rms d dz x = x x % ε rms = x' ε rms
= x = βε rms ' = x ' = γε rms x' = x x! = αε rms It holds also the relation: γβ α = 1 Substituting α,β,γ we get ' " x' $ ε rms ε rms # ε rms % ' & =1 We end up with the definition of rms emittance in terms of the second moments of the distribution: ε rms = ( x ) σ x' σ xx' = x x" x "
Which distribution has no correlations? x' = 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 ε rms = x x # x x # 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 ε rms = C ( x x n x n+1 ) 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 = 1 m o c σ xσ px σ xpx = 1 m o c ( x 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
Limits of single particle emittance Limits are set by Quantum Mechanics on the knowledge of the two conjugate variables (x,p x ). According to Heisenberg: σ px! This limitation can be expressed by saying that the state of a particle is not exactly represented by a point, but by a small uncertainty volume of the order of! 3 in the 6D phase space. In D it holds: ε n,rms = 1 m o c σ xσ px σ xpx = $ 0 classical limit & % 1! m o c = " c & =1.9 10 13 m quantum limit '
OUTLINE The rms emicance concept rms envelope equa-on Space charge forces Space charge induced emicance oscilla-ons Matching condi-ons and emicance compensa-on
Envelope Equation without Acceleration Now take the derivatives: d dz = d dz d dz = d dz x = 1 x! = 1 d x! dz d dz x = 1 x x! = x! σ x x! σ = 1 x! + x x! 3 x x ( )! 3 = σ! x + x x!! x! 3 And simplify: σ!! x = σ xσ x' x' + x x!! 3 = ε rms σ + x x!! 3 x We obtain the rms envelope equation in which the rms emittance enters as defocusing pressure like term. σ!! x x x!! = ε rms 3
σ!! x x x!! = ε rms 3 Assuming that each particle is subject only to a linear focusing force, without acceleration: x!! + k x x = 0 take the average over the entire particle ensemble " + k x = ε rms 3 x x!! = k x x We obtain the rms envelope equation with a linear focusing force in which the rms emittance enters as defocusing pressure like term. ε rms σ T 3 x V P
Beam Thermodynamics Kinetic theory of gases defines temperatures in each directions and global as: k B T x = m v x T = 1 ( 3 T x +T y +T z ) E k = 1 m v = 3 k BT Definition of beam temperature in analogy: k B T beam,x = m o v x v x = ( βγc) x! = ( βγc) ( γ x ε x,rms ) We get: k B T beam,x = m o c ( βγ) ( γ x ε x,rms )
k B T beam,x = m o c ( βγ) ( γ x ε x,rms ) S = kn log πε ( )
OUTLINE The rms emicance concept rms envelope equa-on Space charge forces Space charge induced emicance oscilla-ons Matching condi-ons and emicance compensa-on
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 ) 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 ρ = I π R v R J = I π R Gauss s law ε o E ds = Ampere s law ρdv E r = E r = I πε o R v r for r R I πε o v 1 r for r > R Ir B dl = µ o J ds B ϑ = µ o πr for r R I B ϑ = µ o for r > R πr 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 L E z s,r = 0 ( ) = ρ 4πε o R π L ( l s ) rdrdϕd l & ( l s ) + r 3 / ) 0 0 0 '( * + E z ( s,r = 0) = ρ % ε 0 &' ( ) R + ( L s ) R + s + 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 ( + [ ] r3 16 + E r (r, s ) = ρ ε 0 % ' & ' ( L s ) R + ( L + s ) s R + s ( * ) * r
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 + γ (L s) R + γ s + γ s L γ ε 0 '( ( ) ) * + E r (r,s) = γρ & ( (L s) ε + 0 ' ( R + γ (L s) s R + γ s ) + * + r 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 πγε 0 R βc h s,γ ( ) E r (r,s,γ ) = Ir πε 0 R 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 )E r = ee r γ is a linear function of the transverse coordinate dp r dt = F r = ee r γ = eir πγ ε 0 R 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 πγ ε 0 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 πγ ε 0 βc g s,γ ( ) x!! = k sc s,γ ( ) x k sc = I I A g s,γ ( ) I A = 4πε o m o c3 e
Now we can calculate the term x x " that enters in the envelope equation σ!! x = ε rms σ + x x!! 3 x!! x = k sc x x x!! = k sc x =k sc Including all the other terms the envelope equation reads: Space Charge De-focusing Force σ!! x + k = ε n ( βγ) σ + k sc 3 x Emittance Pressure External Focusing Forces Laminarity Parameter: ( ρ = βγ ) k sc ε n
The beam undergoes two regimes along the accelerator " + k = ε n ( βγ) σ + k sc 3 x ρ>>1 Laminar Beam " + k = ε n ( βγ) σ + k sc 3 x ρ<<1 Thermal Beam
OUTLINE The rms emicance concept rms envelope equa-on Space charge forces Space charge induced emicance oscilla-ons Matching condi-ons and emicance compensa-on
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 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 mcβγ Small perturbation: σ( ζ ) = σ eq ( s) +δσ ( s) δ σ # s ( ) + k s δσ ( s) = 0 δσ ( s) = δσ o ( s)cos( k 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( k 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( k s z)
Envelope oscillations drive Emittance oscillations σ(z) ε(z) ε rms = ( x ) sin k sz σ x σ x' σ xx' = x x % x % ( )
Energy spread induces decoherence σ(z) ε(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
OUTLINE The rms emicance concept rms envelope equa-on Space charge forces Space charge induced emicance oscilla-ons Matching condi-ons and emicance compensa-on
High Brightness Photo-Injector
Envelope Equation with Longitudinal Acceleration dp x dt x!! +! = d ( p x! ) = βc d ( dt dz p x! ) = 0 p p x! = 0 x!! = βγ βγ ( )! x! p = βγm o c σ!! x = ε rms σ + x x!! 3 x x x!! ( = βγ )! βγ x x! ( = βγ )! ( )! βγ x' = βγ βγ σ! x Space Charge De-focusing Force Adiabatic Damping ( )! σ!! x + βγ βγ σ! x + k = ε n ( βγ) σ + k sc 3 x Emittance Pressure Other External Focusing Forces ε n = βγε rms
Beam subject to strong acceleration σ!! x + γ! σ! x + k RF γ γ σ = ε n x γ σ + k o sc 3 x γ 3 We must include also the RF focusing force: k RF =! γ k o sc = I g( s,γ ) I A
" + γ " σ " x + k RF γ γ = ε n γ 3 σ + x k o sc γ 3 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 " γ n( n 1)σ o γ n γ % + nσ o γ n γ % + k RF σ o γ n = k o sc γ 3 n n = 3 n n = 1
γ = 1 +αz ==> γ " = 0 " + γ " σ " x + k RF γ γ = ε n γ 3 σ + x γ 3 Looking for an equilibrium solution σ inv = σ o γ n ==> all terms must have the same dependence on γ k o sc Laminar beam Thermal beam ρ >> 1 n = 1 ρ << 1 n = 0 σ q = σ o γ σ ε = σ o
Space charge dominated beam (Laminar) σ q = 1 γ # I I A γ Emittance dominated beam (Thermal) σ ε = ε n γ $
σ q = 1 γ # I 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 # I I A γ σ q ' = I I A γ 3 Constant phase space angle: δ = γσ ' q σ q = γ & p x p x δ X δ X
Laminarity parameter ρ = Iσ γi A ε Iσ q n γi A ε = 4I n γ ' I A ε n γ Transition Energy (ρ=1) γ tr = I # γ I A ε n ρ I=4 ka ε th = 0.6 µm E acc = 5 MV/m I=1 ka Potential space charge emittance growth I=100 A ρ = 1
Matching Conditions with a TW Linac γ tr = I # γ I A ε n 150 MeV σ ' = 0 σ q = 1 # γ I I A γ 5 MV/m
z=0.3891 0.05 z=1.5 z=10 0.04 0.04 0.0 0.0 Pr 0 Pr 0 pr_[rad] 0-0.0-0.0-0.04 3.5-0.05 0 0.001 0.00 0.003 0.004 0.005 0.006 0 0.0008 0.0016 0.004 0.003 0.004 R [m] R [m] -0.04 0 0.0008 0.0016 0.004 0.003 0.004 R_[m] 3.5 rms norm. emittance [um] rms beam size [mm] 1.5 1 0.5 0 0.004 0.0035 0.003 Gun Linac 0 4 6 8 10 z=0.3891 z=1.5 Z=10 0.004 0.004 0.0035 Z_[m] 0.0035 0.003 0.003 Rs [m] 0.005 0.00 0.0015 0.001 Rs [m] 0.005 0.00 0.0015 0.001 Rs [m] 0.005 0.00 0.0015 0.001 0.0005 0.0005 0 0-0.003-0.00-0.001 0 0.001 0.00 0.003-0.003-0.00-0.001 0 0.001 0.00 0.003 Zs-Zb [m] Zs-Zb [m] 0.0005 0-0.003-0.00-0.001 0 0.001 0.00 0.003 Zs-Zb [m]
Energy spread induces decoherence σ(z) ε(z)
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
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