INTENSITY LIMITATIONS(Space Charge and Impedance)

M. Zobov

Is it important?

J.Bosser et al., NIM 441 (2000) 1-8

In a real accelerator, there is another important source of e.m. fields to be considered, the beam itself, which circulating inside the pipe, producesadditional e.m. fields called "self-fields“:

Direct self fields

Image self fields

Wake fields

SELF FIELDS AND WAKE FIELDS

• energy loss

• shift of the synchronous phase and frequency (tune) • shift of the betatron frequencies (tunes)

• energy spread and emittance degradation • instabilities.

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

They are responsible of many phenomena of beam dynamics:

What do we mean with space charge?What do we mean with space charge?

It is net effect of the CoulombCoulomb interactions in a multi-particle system

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

Example 1. Relativistic Continuous Uniform Cylindrical

oE dS dV

arfor 22 2

va

IrrE

oor

Gauss’s law

Ampere’s law

Bdl o J dS

arfor 22 2

a

IrJrB o

o

rEc

B

lrErl ro222

JlrlB o 2

J I

a2

I

a2v

a

Linear with r

L. Palumbo, JUAS

Lorentz ForceLorentz Force

Fr e E r cB e 1 2 E r eE r

2

The attractive magnetic force, which becomes significant at high velocities, tends to compensate for the repulsive electric force.

• has only radialradial component

• is a linearlinear function of the transverse coordinate

)/(2 mCao 2)/()( arr o

)(

)/(

02

2

AcaJI

mAcJ

2

2

2

)()(B

2

)(

a

r

crE

cr

a

rrE

arfor

o

or

o

or

Transverse Incoherent EffectsTransverse Incoherent Effects

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

xx

F

mm

Fx

dt

xd

xx

FzxF

x

csx

csx

x

x

csxcs

x

0

..

00

..20

22

2

0

....

1

),(

x

F

mvvvvvvv

csx

xxxxxxx

..

020

22

2

12

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.

Incoherent Tune ShiftThe tune shift for unbunched beams in a perfectly conducting vacuum chamber of half-height h, between perfect magnetic pole pieces at a distance g from the axis:

22222

21

302

bgh

RI

ec

r scpinc

For bunched beams we have to take into account a bunching factor B defined as the ratio of average to peak current

R

NB zbunches

22

Estimates for EDM Machine

mcmbN 1011011

8/11020 BNN bunchesbuckets

P = 0.7 GeV/c = 1.0674

P = 1.5 GeV/c = 1.2804

136.08017.0 inc

0138.0800172.0 inc

! To be compared with 18.03865.0 y

22ay

K

2. Optics functions change by changing the tune. This leads to the size changes, i.e. collective effects

1.

F q Ezˆ z Ex vBy ̂ x Ey vBx ̂ y F//F

there can be two effects on the test chargetest charge :

1) a longitudinal force which changes its energy,

2) a transverse force which deflects its trajectory.

Wake Potentials

n n nnnnoin kkgppkkpgZkZ 22222

20

21 / llmm lallmlm pJgJT

nlnmn

bnlmmlm

pppJpJpT

22222200324

Impedance of a Step (S. Kheifets and S. Heifets)

2

0

20

2

2/

FWHML

npp

I

ecm

q

AF

nZ

FWHMTT RIccm

q

AFZ

0

20

2/

4

Limits on Impedances

Longitudinal

Transverse

Similar to DANE ?

The wake fields can act as a positive feedback leading to instabilities. Nonlinearities damp them (Landau damping)

REFERENCES

1. L. Palumbo, “Space Charge Effects and Instabilities”, JUAS, 2003.

2. M. Zobov and A. Gallo, “Instabilities”, http://www.lnf.infn.it/acceleratori/dafne/seminary/dafne_zobov.pdf

3. L. Palumbo, V. Vaccaro and M. Zobov, “Wake Fields and Impedance”, CAS CERN 95-06, 1995