Space Charge Mi-ga-on

Transcription

1 Space Charge Mi-ga-on Hamburg June nd 016

2 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

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

4 Trace space of a laminar beam X X

5 Trace space of non laminar beam X X

6 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

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

9 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

10 = 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 "

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

12 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

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

14 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

15 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 & = m quantum limit '

16 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

17 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

18 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

19 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 )

20 k B T beam,x = m o c ( βγ) ( γ x ε x,rms ) S = kn log πε ( )

21 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

22 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

23 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

24 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 / ) '( * + E z ( s,r = 0) = ρ % ε 0 &' ( ) R + ( L s ) R + s + s L ( )*

25 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 ( + [ ] r E r (r, s ) = ρ ε 0 % ' & ' ( L s ) R + ( L + s ) s R + s ( * ) * r

26 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

27 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

28 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 ( )

29 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

30 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

31 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

32 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

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

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

35 ( ) 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)

36 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)

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

38 Energy spread induces decoherence σ(z) ε(z)

39 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

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

41 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

42 High Brightness Photo-Injector

43 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

44 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

45 " + 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

46 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

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

48 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

49 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

50 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

51 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

52 z= z=1.5 z= Pr 0 Pr 0 pr_[rad] R [m] R [m] R_[m] 3.5 rms norm. emittance [um] rms beam size [mm] Gun Linac z= z=1.5 Z= Z_[m] Rs [m] Rs [m] Rs [m] Zs-Zb [m] Zs-Zb [m] Zs-Zb [m]

53 Energy spread induces decoherence σ(z) ε(z)

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55 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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