SPACE CHARGE EFFECTS SPACE CHARGE EFFECTS IN PHOTO-INJECTORSIN PHOTO-INJECTORS

Massimo FerrarioMassimo Ferrario

INFN-LNFINFN-LNF

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Madison, June 28 - July 2

• Space Charge Dominated BeamsSpace Charge Dominated Beams

• Cylindrical Beams Cylindrical Beams

• Plasma Oscillations

Space Charge: What does it mean?Space Charge: What does it mean?

Space Charge RegimeSpace Charge Regime ==> The interaction beween particles is dominated by the self fieldself field produced by the particle distribution, which varies appreciably only over large distances compare to the average separation of the particles ==> Collective EffectsCollective Effects

A measure for the relative importance of collective effects is the

Debye Length Debye Length DD

Let consider a non-neutralizednon-neutralized system of identical charged particlesidentical charged particles

We wish to calculate the We wish to calculate the effective potentialeffective potential of a fixed of a fixed test charged particletest charged particle surrounded by other particlessurrounded by other particles that are statistically distributed. that are statistically distributed.

NN total number of particles

nn average particle density

€

Φ r

r ( ) =C

r

€

C =e

4πεo

€

Φs

v r ( ) = ?

The particle distribution around the test particle will deviate from a uniform distribution so that the effective potential of the test particle is now defined as

the sum of the original and the induced potential:

€

∇2Φs

r r ( ) =

e

εo

δr r ( ) +

e

εo

Δnr r ( )

€

Δnr r ( ) = nm

r r ( ) − n

€

nm

r r ( ) =

1

Neδ

r r −

r r i( )

i= 1

N

∑

Local deviation from n

Local microscopic distribution

Poisson Equation

Presupposing thermodynamic equilibrium,nm will obey the following distribution:

€

nm

r r ( ) = ne−eΦ s

r r ( ) / kBT

€

Δnr r ( ) = nm

r r ( ) − n = n e−eΦ s

r r ( ) / kBT − 1( ) ≈ −

eΦs

r r ( )

kBT

Where the potential energy of the particles is assumed to be much smaller than their kinetic energy

€

∇2Φs

r r ( ) +

Φs

r r ( )

λ D2

=e

εo

δr r ( )

€

D =εokBT

e2n

The solution with the boundary condition that Φs vanishes at infinity is:

€

Φs

r r ( ) =

C

re−r / λD

DD

Conclusion:Conclusion: the effective interaction range of the test particle is limited to the Debye lengthDebye length

The charges sourrounding the test particles have a screening effect

€

Φs

r r ( ) =

C

re−r / λD

€

Φs

r r ( ) ≈Φ

r r ( ) for r << λ D

€

Φs

r r ( ) <<Φ

r r ( ) for r ≥ λ D

Smooth functions for the charge and field distributions can be used

DD

If collisions can be neglected the Liouville’s theorem holds in the 6-D phase space (r,p). This is possible because the smoothed space-charge forces acting on a particle can be treated like an applied force. Thus the

6-D phase space volume occupied by the particles remains constant during acceleration.

In addition if all forces are linear functions of the particle displacement from the beam center and there is no coupling between degrees of

freedom, the normalized emittance associated with each plane (2-D phase space) remains a constant of the motion

Important consequencesImportant consequences

Continuous Uniform Cylindrical Beam ModelContinuous Uniform Cylindrical Beam Model

€

εoE ⋅dS = ρdV∫∫

€

E r =ρr

2εo

=Ir

2πεoa2v

for r ≤ a

Gauss’s law

Ampere’s law

€

B ⋅dl = μo J ⋅dS∫∫

€

Bϑ =μoJr

2= μo

Ir

2πa2 for r ≤ a

€

Bϑ =β

cE r

€

2πrlεoE r = ρ 2πr2 l

€

J =I

πa2

€

ρ =I

πa2v

€

2lBϑ = μoJlr

€

a

Linear with r

Lorentz ForceLorentz Force

€

Fr = e E r − βcBϑ( ) = e 1− β 2( )E r =

eE r

γ 2

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

has only radialradial component and

is a linearlinear function of the transverse coordinate

€

γm d2r

dt 2=

eE r

γ 2=

eI

2πγ 2εoa2v

r

Equation of motion:Equation of motion:

€

d2r

dz2=

eI

2πmγ 3εoa2v 3

r =K

a2r

€

d2r

dt 2= β 2c 2 d2r

dz2

€

K =eI

2πmγ 3εov3

=2I

Ioβ3γ 3

€

Io =4πεomc 3

eAlfven current

Generalized perveance

If the initial particle velocities depend linearly on the initial coordinates

then the linear dependence will be conserved during the motion, because of the linearity of the equation of motion.

This means that particle trajectories do not cross ==> Laminar BeamLaminar Beam€

dr 0( )dt

= Ar 0( )

Laminar BeamLaminar Beam

€

γx 2 + 2αx ′ x + β ′ x 2 = εrms

€

γβ −α 2 = 1€

x 2 = βεrms and ′ x 2 = γεrms

€

α =−′ β

2= −

1

2εrms

d

dzx 2 = −

x ′ x

εrms

€

εrms = x 2 ′ x 2 − x ′ x 2

x

x’

xrms

x’rms

What about RMS Emittance (Lawson)?What about RMS Emittance (Lawson)?

In the phase space (x,x’) all particles lie in the interval bounded by the points (a,a’).

x

x’

a

a’

What about the rms emittance?

€

€

εrms2 = x 2 ′ x 2 − x ′ x

2

€

′ x = Cx n

€

εrms2 = C2 x 2 x 2 n − x n +1 2

( )

When n = 1 ==> r = 0

When n = 1 ==> r = 0

x

x’

a

a’

The presence of nonlinear space charge forces can distort the phase space contours and causes emittance growth

€

ρ r( ) =I

πa2v1−

r2

a2

⎛

⎝ ⎜

⎞

⎠ ⎟

€

E r =ρr

2εo

=I

2πεoa2v

r −r3

a2

⎛

⎝ ⎜

⎞

⎠ ⎟

€

εrms2 = C2 x 2 x 2 n − x n +1 2

( ) ≠ 0

Space Charge induced emittance Space Charge induced emittance oscillationsoscillations

Bunched Uniform Cylindrical Beam ModelBunched Uniform Cylindrical Beam Model

Longitudinal Space Charge field in the bunch moving frame:

€

E z ˜ z ,r = 0( ) =˜ ρ

4πεo

˜ l − ˜ z ( )

˜ l − ˜ z ( )2

+ r2 ⎡ ⎣ ⎢

⎤ ⎦ ⎥3 / 2

0

˜ L

∫0

2π

∫0

R

∫ rdrdϕd˜ l

L(t)R(t)

z

€

vz = βc

€

E z(˜ z ,r = 0) =˜ ρ

2ε0

R2 + ( ˜ L − ˜ z )2 − R2 + ˜ z 2 + 2˜ z − ˜ L ( )[ ]

(0,0) l

€

˜ ρ =Q

πR2 ˜ L

€

E r(r,˜ z ) ≅˜ ρ

ε0

−∂

∂˜ z E z (˜ z ,0)

⎡

⎣ ⎢

⎤

⎦ ⎥r

2+ ⋅⋅⋅[ ]

r3

16+

€

E r(r,˜ z ) =˜ ρ

2ε0

( ˜ L − ˜ z )

R2 + ( ˜ L − ˜ z )2+

˜ z

R2 + ˜ z 2

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

r

2

Radial Space Charge field in the bunch moving frame

by series representation of axisymmetric field:

Lorentz Transformation to the Lab frameLorentz Transformation to the Lab frame

€

L =˜ L

γ ⇒ ρ = γ˜ ρ

€

E r = γ ˜ E r

€

Bϑ = γβ

c˜ E r =

β

cE r

€

Fr = eE r

γ 2

€

E z(z) =ρ

γ 2ε0

R2 + γ 2 (L − z)2 − R2 + γ 2z2 + γ 2z − L( )[ ]

=ρ

γ 2ε0

γL( )R2

γ 2L2+ (1−

z

L)2 −

R2

γ 2L2+

z2

L2+ 2

z

L− 1

⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

=ρL

2ε0

A + (1−ζ )2 − A + ζ 2 + 2ζ − 1( )[ ]

€

A =R

γL€

ζ =z

L

€

E r(r,z) =ρ

2ε0

(1−ζ )

A2 + (1−ζ )2+

ζ

A2 + ζ 2

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

r

2

=Ir

4πε0R2vg ζ( )

Beam Aspect RatioBeam Aspect Ratio

It is still a linear field with r but with a longitudinal correlation ζ

€

ρ =Q

πR2L=

I

πR2v

γ = 1 γ = 5 γ = 10

Ar,s ≡Rs γsL( )

L(t)Rs(t) Δt

Simple Case: Transport in a Long SolenoidSimple Case: Transport in a Long Solenoid

ks =qB

2mcβγ

€

K ζ( ) =2Ig ζ( )

Io βγ( )3

€

′ ′ R + ks2R =

K ζ( )R

==> Equilibrium solution ? ==>

€

′ ′ R = 0

€

Req ζ( ) =K ζ( )

ks

0 0.0005 0.001 0.0015 0.002 0.0025metri

0.5

0.6

0.7

0.8

0.9

gg(ζ

ζ

€

′ ′ R + ks2R =

K ζ( )R

€

R ζ( ) = Req ζ( ) + δr ζ( )

€

δ ′ ′ r + ks2 Req + δr( ) =

K ζ( )

Req + δr( )

€

δ ′ ′ r + ks2 Req + δr( ) =

K ζ( )

Req 1+δr

Req

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

=K ζ( )Req

1−δr

Req

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

€

δ ′ ′ r ζ( ) + 2ks2δr ζ( ) = 0

Small perturbations around the equilibrium solutionSmall perturbations around the equilibrium solution

Plasma frequency

€

δ ′ ′ r ζ( ) + 2ks2δr ζ( ) = 0

€

R ζ( ) = Req ζ( ) + δr ζ( )cos 2ksz( )

′ R ζ( ) = −δr ζ( ) sin 2ksz( )

€

kp = 2ks

x

px

Projected Phase Space Slice Phase Slice Phase SpacesSpaces

Emittance Oscillations are driven by space charge differential Emittance Oscillations are driven by space charge differential defocusing in core and tails of the beamdefocusing in core and tails of the beam

0 1 2 3 4 5metri

-0.5

0

0.5

1

1.5

2

envelopes

0 1 2 3 4 5metri

0

20

40

60

80

emi

R(ζ)

ε(z)

Envelope oscillations drive Emittance oscillationsEnvelope oscillations drive Emittance oscillations

δγγ

=0

€

ε z( ) = R2 ′ R 2 − R ′ R 2

÷ sin 2ksz( )

ks =qB

2mcβγ

X

X’

Perturbed trajectories oscillate around the equilibrium Perturbed trajectories oscillate around the equilibrium with the with the

same frequency but with different amplitudessame frequency but with different amplitudes

0

1

2

3

4

5

6

0 5 10 15

HBUNCH.OUT

sigma_x_[mm]enx_[um]

sigma_x_[mm]

z_[m]

Q =1 nCQ =1 nCR =1.5 mmR =1.5 mmL =20 psL =20 ps thth = 0.45 mm-mrad = 0.45 mm-mrad

EEpeakpeak = 60 MV/m (Gun) = 60 MV/m (Gun)

EEacc acc = 13 MV/m (Cryo1)= 13 MV/m (Cryo1)

B = 1.9 kG (Solenoid)B = 1.9 kG (Solenoid)

I = 50 AI = 50 AE = 120 MeVE = 120 MeV nn = 0.6 mm-mrad = 0.6 mm-mrad

nn

[mm-mrad][mm-mrad]

Z [m]

HOMDYN Simulation of a L-band photo-injector

6 MeV6 MeV

3.5 m