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In a real accelerator, there is another important source of e.m. fields to be considered, the beam itself, which circulating inside the pipe, produces additional e.m. fields called "self-fields“: Direct self fields Image self fields Wake fields SELF FIELDS AND WAKE FIELDS
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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
vaIrrEoo
r
Gauss’s law
Ampere’s law
Bdl o J dS
arfor 22 2
aIrJrB o
o
rEc
B
lrErl ro222
JlrlB o 2
J Ia2
Ia2v
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
)(
ar
crE
cr
arrE
arfor
o
or
o
or
Transverse Incoherent EffectsTransverse Incoherent Effects
We take the linear term of the transverse force in the betatron equation:
xx
Fmm
Fxdt
xd
xx
FzxF
x
csx
csx
x
x
csxcs
x
0
..
00
..20
22
20
....
1
),(
xF
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:
222
2221
302
bghRI
ecr scp
inc
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
2
2
Estimates for EDM MachinemcmbN 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 22222200
324
Impedance of a Step (S. Kheifets and S. Heifets)
2
0
20
2
2/
FWHML
npp
Iecm
qAF
nZ
FWHMTT RIccm
qAFZ
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