SPACE CHARGE EFFECTSSPACE CHARGE EFFECTSMassimo FerrarioMassimo Ferrario

INFNINFN--LNFLNF

Darmstadt, 27 Sept. Darmstadt, 27 Sept. -- 9 Oct., 20099 Oct., 2009

EQUATION OF MOTIONEQUATION OF MOTION

The motion of charged particles is governed by the Lorentz force :

d m v

dtFe.m.

ext e E v B

Where m is the rest mass,

the relativistic factor and v the particle velocity

Charged particles are accelerated, guided and confined by external electromagnetic fields.

Acceleration is provided by the electric field of the RF cavity

Magnetic fields are produced in the bending magnets for guiding the charges on the reference trajectory (orbit), in the quadrupoles for the transverse confinement, in the sextupoles for the chromaticity correction.

There is another important source of e.m. fields :

the beam itself

Direct self fields

Image self fields

Wake fields

SELF FIELDS AND WAKE FIELDSSELF FIELDS AND WAKE FIELDS

Space Charge

• energy loss

• energy spread and emittance degradation

• shift of the synchronous phase and frequency (tune)

• shift of the betatron frequencies (tunes)

• instabilities.

These fields depend on the current and on the charges These fields depend on the current and on the charges velocity.velocity.

They are responsible of many phenomena of beam dynamics:They are responsible of many phenomena of beam dynamics:

(wake-fields)

Space Charge: What does it mean?Space Charge: What does it mean?The net effect of the CoulombCoulomb interactions in a multi-particle system can be

classified into two regimes:

1)1) Collisional Collisional RegimeRegime ==> dominated by binary collisionsbinary collisions caused by close particle encounters ==> Single Particle EffectsSingle Particle Effects

2) 2) Space Charge RegimeSpace Charge Regime ==> 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

Neutral Plasma

Magnetic focusing

Magnetic focusing

Single Component Relativistic Plasma

•Oscillations

•Instabilities

•EM Wave propagation

Electron beam

A measure for the relative importance of collisional versus collective effects is the

Debye Debye Length Length DD

Let consider a nonnon--neutralizedneutralized 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

r C

r

C e

4o

sr ?

Drp e

rCr /

D okBTe2nkB

= Boltzman constantT = TemperaturekB

T = average kinetic energy of the particlesn = particle density (N/V)

The effective potential of a test charge can be defined as the sum of the potential of the uniform distribution and a “perturbed”

term.

2s

r e

o

r e

o

nr Poisson Equation

n r nee s

r / kBT n

e sr

kBT

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

The charges sourrounding the test particles have a screening effect

s

r C

rer / D

sr

r for r D

sr

r for r D

Smooth functions for the charge and field distributions can be used as long as the Debye length remains small compared to the particle

bunch size

DD

x,y, z D

x,y, z D

Fields of a point charge with uniform motion

o

z

x,x’

y

o’

z’

y’

v

q

E q

4o

r r 3

B 0

• In the moving frame O’

the charge is at rest• The electric field is radial with spherical symmetry• The magnetic field is zero

34 rxqE

ox

34 ryqE

oy

34 rzqE

oz

vt is the position of the point charge in the lab. frame O.

Ex E xEy ( E y v B z )

Ez ( E z v B y )

Bx B xBy ( B y v E z / c 2 )

Bz ( B z v E y / c 2 )

r x 2 y 2 z 2 1 / 2

r 2 ( x vt )2 y 2 z 2 1/2

Relativistic transforms of the fields from O’

to O

1

1 2

x ( x vt )y yz z

c t ct vc

x

Ex E x q

4o

x r 3

q4o

( x vt )

2 ( x vt )2 y 2 z 2 3 / 2

Ey E y q

4o

y r 3

q4o

y

2 ( x vt )2 y2 z 2 3 / 2

Ez E z q

4o

z r 3

q4o

z

2 ( x vt)2 y2 z 2 3 / 2

The fields have lost the spherical symmetry

E q

4o

r

2 x2 y2 z 2 3 / 2

The field pattern is moving with the charge and it can be observed at t=0:

t=0

x

y

z

r

2222 sin

cos

rzy

rx

E q

4o

1 2 r2 1 2 sin2 3 / 2

r r

E q

4o

r

2 x 2 y 2 z2 3 / 2

2 x2 y2 z2 r2 2 1 2 sin2

>1

>>1

0 E q

4o

1r2

r r

0 E// q

4o

1 2r2

r r

0

2 E

q4o

r2

r r

E q

4o

1 2 r2 1 2 sin2 3 / 2

r r

11 2

2

2

/

/

0

cvEB

cvEB

B

yz

zy

x

B is transverse to the direction of motion

2cEvB

→∞

B 0

Continuous Uniform Cylindrical Beam ModelContinuous Uniform Cylindrical Beam Model

oE dS dVGauss’s law

Ampere’s law

B dl o J dS B oIr

2a2 for r a

B oI

2r for r a

B c

Er

J Ia2

Ia2v

a

Er I

2oa2v

r for r a

Er I

2ov1r

for r a

Lorentz Lorentz ForceForce

Fr e Er cB e 1 2 Er eEr

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

D o

2kBTe2n

In a charged particle beam moving at a longitudinal relativistic velocity, assuming that the random transverse motion in the beam

is non-relativistic, the Debye length has the following form:

vx kBTm

c

vz c

z

x

R=1mm, L=3mm

Q=1nC, T=103 K

DD[mm][mm]

Electron bunch

Surface charge density Surface

electric field

Restoring force

Plasma frequency

Plasma oscillations

Bunched Uniform Cylindrical Beam ModelBunched Uniform Cylindrical Beam Model

Longitudinal Space Charge field in the bunch moving frame:

˜ E z ˜ s , r 0 ˜ 4o

˜ l ˜ s ˜ l ˜ s 2 r 2

3 / 20

˜ L

0

2

0

R

rdrdd˜ l

L(t)R(t)

s

vz c

˜ E z (˜ s , r 0) ˜

20R2 ( ˜ L ˜ s )2 R2 ˜ s 2 2˜ s ˜ L

(0,0) l

˜ QR2 ˜ L

˜ E r (r,˜ s ) ˜ 0

s

˜ E z (˜ s ,0)

r2 r 3

16

˜ E r (r,˜ s ) ˜

20

( ˜ L ˜ s )

R2 ( ˜ L ˜ s )2

˜ s

R2 ˜ s 2

r2

Radial Space Charge field in the bunch moving frame

by series representation of axisymmetric

field:

Ez(0,s) 20

R2 2 (L s)2 R2 2s2 2s L

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

Er(r,s) 20

(L s)

R2 2 (L s)2

s

R2 2s2

r2

Lorentz Lorentz Transformation to the Lab frameTransformation to the Lab frame

Ez ˜ E zEr ˜ E r

˜ L L ˜

˜ s s

= 1 = 5

= 10

L(t)Rs

(t) t

Er(r,s, ) Ir20R2c

g s, Ez(0,s, ) I20R2c

h s,

Fr eEr

2 eIr

2 20R2cg s,

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

ks2

ksc s,

ksc s, 2IIA 3 g s,

IA 4omoc

3

e 17kA

-M. Reiser, Theory and design of charged particle beams, Wiley, 1994

Single Component Relativistic Plasma ks

2 ksc s,

ks qB

2mc

s 2ks2 s 0

eq s, ksc s,

ks

Equilibrium solution:

eq s s Small perturbation:

s eq s s eq s cos 2ksz

Perturbed trajectories oscillate around the equilibrium with the

same frequency but with different amplitudes:

Space Charge differential defocusing in core and tails of the Space Charge differential defocusing in core and tails of the beam drive Reversible beam drive Reversible Emittance Emittance Oscillations Oscillations

x

px

Projected Phase Space Slice Phase Slice Phase SpacesSpaces

rms x2 x'

2 xx'2 x 2 x 2 x x 2 sin 2ksz

There is another important source of e.m. fields :

the beam itself

Direct self fields

Image self fields

Wake fields

SELF FIELDS AND WAKE FIELDSSELF FIELDS AND WAKE FIELDS

Space Charge

Effects of conducting or magnetic screens Effects of conducting or magnetic screens

Let us consider a point charge close to a conducting screen.

The electrostatic field can be derived through the "image method". Since the metallic screen is an equi-potential plane, it can be removed provided that a "virtual" charge is introduced such that the potential is constant at the position of the screen

q q - q

A constant current

in the free space produces circular magnetic field.

If r

1, the material, even in the case of a good conductor, does not affect the field lines.

I

However, if the material is of ferromagneticferromagnetic typetype, with

rr >>1>>1, due to its

magnetisation, the magnetic field lines are strongly affected, inside and outside the material. In particular a very high magnetic permeability makes the tangential magnetic field zero at the boundary

so that the magnetic field is perpendicular to the surface, just like the electric field lines close to a conductor.

In analogy with the image method for charges close to conducting

screens, we get the magnetic field, in the region outside the material, as superposition of the fields due to two symmetric equal currents flowing in the same direction.

I I I

It is necessary to compare the wall thicknesswall thickness and the skin depth (region of penetration of the e.m. fields) in the conductor.

If the fields penetrate

and pass through the material, we are practically in the static boundary conditions case. Conversely, if the skin depth is very small, fields do not penetrate, the electric filed lines are perpendicular to the wall, as in the static case, while the magnetic field line are tangent to the surface.

I -II -I

w

w

Time-varying fields

w 2

Circular Perfectly Conducting Pipe (Beam at Center)

In the case of charge distribution, and , the electric field lines are perpendicular to the direction of motion. The transverse fields intensity can be computed like in the static case, applying the Gauss and Ampere laws.

o a2

2)/()( arr o

J c

I Ja2 c0( A) 2

2

2

)()(B

2

)(

ar

crE

cr

arrE

arfor

o

or

o

or

2222

21

areeEEecBEeF

o

orrrr

there is a cancellation of the electric and magnetic forces

In some cases, the beam pipe cross section is such that we can consider only the surfaces closer to the beam, which behave like two parallel plates. In this case, we use the image method to a charge distribution of radius a between two conducting

plates 2h apart. By applying the superposition principle we get the total

image field at a position y inside the beam.

Eyim(z ,y) (z )

2 o

(1)n

n1

12nh y

1

2nh y

Eyim(z ,y) (z )

2 o

(1)n

n1

2y2nh 2 y2

(z)

4 oh2

2

12y

Where we have assumed h>>a>y.

For d.c. or slowly varying currents, the boundary condition imposed by the conducting plates does not affect the magnetic field. As a consequence there is no

cancellation effect for the fields produced by the "image" charges.

2h q

Parallel Plates (beam at center)

-q

-q 2h

q

q 3h

y

From the divergence equation we derive also the other transverse

component, notice the opposite sign:

x

Exim

y

Eyim Ex

im(z ,x) (z )4 oh

2

2

12x

Including also the direct space charge force, we get:

Fx(z ,x) e(z )x o

12a2 2

2

48h2

Fy(z ,x) e(z )y o

12a2 2

2

48h2

Therefore, for >>1, and for d.c. or slowly varying currents the cancellation effect applies only for the direct space charge forces. There is no cancellation of the electric and magnetic forces due to the "image" charges.

Usually, the frequency beam spectrum is quite rich of harmonics, especially for bunched beams.

It is convenient to decompose the current into a d.c. component,

I, for which w

>>w

, and an a.c. component, Î, for which w

<< w

.

While the d.c. component of the magnetic field does not perceives the presence of the material, its a.c. component is obliged to be tangent at the wall.

For a charge density

we have I=v.

We can see that this current produces a magnetic field able to cancel the effect of the electrostatic force.

Parallel Plates (Beam at Center) a.c. currents

w

w

˜ E y (z, x) ˜ (z)y o

2

48h2 ; ˜ B x(z,x) c

˜ E y(z,x)

˜ F y(z,x) e ˜ (z)y o

22

48h2

˜ F x (z, x)e ˜ (z)x

2 o2

1a2

2

24h2

˜ F y(z,x) e ˜ (z)y2 o

21a2

2

24h2

There is cancellation of the electric and magnetic forces !!

I

-I

-I

I

I

Parallel Plates Parallel Plates -- General expression of the force General expression of the force

Taking into account all the boundary conditions for d.c. and a.c. currents, we can write the expression of the force as:

Fu e

2 o

1 2

1a2

2

24h2

2 2

24h2 2

12g2

u

where

is the total current, and

its d.c. part. We take the sign (+) if u=y, and the sign (–) if u=x.

-L. J. Laslett, LBL Document PUB-6161, 1987, vol III

Space charge effects in storage rings

When the beam is located at the centre of symmetry of the pipe, the e.m. forces due to space charge and images cannot affect the motion of the centre of mass (coherent), but change the trajectory of individual charges in the beam (incoherent).(incoherent).

These force may have a complicate dependence on the charge position. A simple analysis is done considering only the linear expansion of the self-fields forces

around the equilibrium trajectory.

Incoherent and Coherent Transverse Effects

Consider a perfectly circular accelerator with radius x

. The beam circulates inside the beam pipe. The transverse single particle motion in the linear regime, is derived from the equation of motion. Including the self field forces in the motion equation, we have

d m v

dt F ext r F self r

O

x

y

x

z

dvdt

F ext r F self r

m

Self Fields and betatron motion

In the analysis of the motion of the particles in presence of the self field, we will adopt a simplified model where particles execute

simple harmonic oscillations around the reference orbit. This is the case where the focussing term is constant. Although this condition in never fulfilled in a real accelerator, it provides a reliable model for the description of the beam instabilities

x (s) Kxx(s) 1 2Eo

Fxself (x)

x ( s) Q x

x

2

x( s) 1 2Eo

Fxself ( x , s)

Kx Qx

x

2

Qx Betatron tune: n. of betatron oscillations per turn

Transverse Incoherent Effects

We take the linear term of the transverse force in the betatron equation:

Fxs.c. ( x,z) Fx

s.c.

x

x0

x

x Qx

x

2

x 1 2Eo

Fxs.c.

x

x0

x

Qx Qx 2 Qx2 2QxQx Qx

x2

2 2EoQx

Fxs.c.

x

The betatron shift is negative since the space charge forces are defocusing on both planes. Notice that the tune shift is in general function of “z”, therefore there is a tune spread inside the beam.

x Qx

x

2

1

2Eo

Fxs.c.

x

x0

x 0

oxooo

xx lQEa

NeQ 222

22

4

Example: Incoherent betatron tune shift for an uniform electron beam of radius a, length lo , inside circular perfectly conducting Pipe

2222

..

2

2

ae

axe

xxF

o

o

o

ocs

x

)01 53.1 : ,01 82.2:( 4

18152

2

, mprotonsmelectronscm

er

oope

oxo

pexx lQa

NrQ 322

,2

For a real bunched beams the space charge forces, and the tune shift depend on the longitudinal and radial position of the charge.

Transverse Coherent Effects

The whole beam can execute betatron oscillations.

The environment affects this motion, causing a shift of the coherent tune which is generally different from the incoherent one.

For instance, in the case of circular pipe with an off-centre beam, there is a transverse force which tends to further deflect the beam, producing a defocusing coherent effect.

Circular Perfectly Conducting Pipe (Beam Off- Center)

Only in the case of pure cylindrical symmetry, the external pipe

does not affect the transverse space charge fields. When the symmetry is broken, the image charges give a significant contribution.

If the beam is displaced by r1 from the axis, by applying the image theorem we find that the image charge is located at the position r2 such that r1 r2 =b2. Assuming that r1 <<b

and that the beam radius

a<< b, we find the electric field:

Er (r ,z ) (z)

2 o r1

b2

r1 r2

br

Qxc.

x2

2Qu2Eo

Fus.c.

u

c.

ex

2o

4o2EoQxb

2

Parallel Plates (Beam Off- Center)

If the whole beam is displaced from the axis by yc

, the image charges produce a transverse field which leads to coherent effects:

Eyimc (z, yc ) (z)

4 oh2 2

4yc

Qyc.

x2

2Qu2Eo

Fus.c.

u

c.

x

2

2Qy2Eo

0

oh2 2

16

ych

Consequences of the space charge tune shifts

In circular accelerators the values of the betatron tunes should not be close to rational numbers in order to avoid the crossing of linear and non-linear resonances where the beam becomes unstable.

The tune spread induced by the space charge force can make hard to satisfy this basic requirement. Typically, in order to avoid major resonances the stability requires

Qu 0.3

END

PS Booster, accelerate proton bunchesFrom 50 to 800 MeV

in about 0.6 s. The tunes occupied by the particle are indicated in the diagram by the shaded area. As time goes on, the energy increase and the space charge tune spread gets smaller covering at t=100 ms the tune area shown by the darker area. The point of highest tune correspond to the particles which are least affected by the space charge. This point moves in the Q diagram since the external focusing is adjusted such that the reduced tune spread lies in a region free of harmful resonances.

Finally, the small dark area shows the situation at t=600 ms when the beam hasReached the energy of 800 MeV. The tune spread reduction is lower than expected with the energy increase (1/) dependence since the bunch dimensions also decrease during the acceleration.

r xx ˆ e x yˆ e yv Ý x e x Ý y e y o xx ˆ e za Ý Ý x o

2 xx e x Ý Ý y e y Ý o xx 2oÝ x ˆ e z

Following the same steps already seen in the "transverse dynamics" lectures, we write:

For the motion along x: Ý Ý x o2 xx 1

mFx

ext Fxself

which is rewritten with respect to the azimuthal

position s=vz

t:

Ý Ý x vz2 x o

2 xx 2 x

x 1xx

1

mvz2

Fxext Fx

self

We assume small transverse displacements x with respect to the closed orbit, and only dipoles for bending and quadrupole

to keep

the beam around the closed orbit:

xxFx

ext Foext

Fxext

x

x0

x

Around the closed orbit, putting vz

= c, we get

)(1112

022 xF

Ex

xF

Ex self

xox

extx

ox

where Eo

is the particle energy. This equation expressed as function of “s”

reads:

x (s) 1x

2(s) Kx(s)

x(s) 1

2Eo

Fxself (x,s)