Two-Beam Instability in Electron Cooling

The drift motion of cooling electrons makes them able to respond to transverse perturbations of a cooled ion beam. This response may lead to dipole or quadrupole transverse instabilities at specific longitudinal wave numbers. While the dipole instabilities can be suppressed by a combination of the Landau damping, machine impedance, and the active damper, the quadrupole and higher order modes can lead to either emittance growth, or a lifetime degradation, or both. The growth rates of these instabilities are strongly determined by the machine x− y coupling. Thus, tuning out of the coupling resonance and / or reduction of the machine coupling can be an efficient remedy for these instabilities.


Ion-electron interactions
• Electron cooling is a method to increase a phase space density of a hot (ion / pbar) beam by merging it with a cold electron beam.
• Cooling is caused by interactions of individual ions with individual electrons.
• Ion beam with electron beam two-beam instability (coherent beam-beam, strong-strong) Any of the 3 channels may be damaging for ions. 3 Channel 1: More Cooled Less Stable • The more beam is cooled the less stable it is.
-Landau damping → 0 -Beam stability parameter: • Instabilities can be driven by: -Machine impedance; -Stochastic cooling system; -Damper; -Structure resonance of the tune-depressed envelope modes; -Interaction with residual ions and/or e-clouds; -Collective interaction with e-beam.
• For RR w/o damper, the stability limit D=0.7-0.8. Channel 3: two-beam instability • Actually, the talk is beginning from here…
• A general idea was introduced by V. Parkhomchuk in ~1992. He posed a problem about this phenomenon.
• Transverse impedance of e-beam was calculated by myself in 1997 for : the impedance seemed too small for typical cases.

∞ → Ω Le
• In e-drift approximation, the problem was simultaneously treated by myself (1999) and P. Zenkevich & A. Bolshakov (1999) -AB: The growth rate is proportional to a coupling parameter : -Z&B: "For realistic parameters, the instability is impossible in our model".
Their model was essentially the same as mine.
• In 2000, V. Parkhomchuk & V. Reva applied the same ideas for an uncoupled revolution matrix with identical X and Y blocks and got • All the 3 papers were in contradiction with each other, and every of them seemed to be arithmetically correct… Actually, the theory of IEI starts from this slide…  • The dipole ion-electron motion does not depend on the ions emittance as soon as the ions are mainly inside the e-beam; thus, • The quadrupole ion-electron growth rate relates to the dipole rate in the same way as for the conventional impedance: From here, a matrix of the inner cooler is to be found; In practice, all the 3 phases (ie, iL, ed) are small, 1 << ≡ kl ψ So, a perturbation approach is proper.
The interaction parameter: Entire solenoid matrix: diagonal part: coupling part: Note: own / alien ~ small phases .
The perturbation is coupling-dominated.
Step Aside: Perturbation Theory for 4D Optics To get eigen-values for the perturbed revolution matrix, the perturbation formalism has to be developed for 4D optics.
These basis vectors are orthogonal through the symplectic unit matrix U : The perturbation theory is constructed very similar to the Quantum Mechanics.
The complex phase shifts: The growth rates: Result : as in QM, the tune shift is given by the diagonal mat Result : as in QM, the tune shift is given by the diagonal matrix element. rix element.
Useful relation:

Ion-Electron Growth Rates
The entire revolution: planar (uncoupled) modes, the force, acting back on the ions, is orthogonal to the ions velocity. The resulted work is zero, and thus the rate is zero too (at the lowest order over the small phases).
• From formal point of view, for the planar modes, the diagonal matrix elements of the drift perturbation are zero (at lowest order).
-The unperturbed matrix of PR2000 is DEGENERATED .
-For degenerated states, the perturbation formulae have to be used in a specific way (remember Quantum Mechanics!).
-Any combination of the orthogonal degenerated states is an eigen-state as well. There are infinite possibilities for the eigenvectors, and every one of them gives its own result for the growth rate.
• The recipe (from the QM, again): -Correct eigen-vectors are those which make the perturbation diagonal.
-Those are the circular modes (100% coupled ! ) Substituted in my perturbation formula, they immediately give the PR result!
• So, even without the machine coupling, there is an area around the coupling resonance, where even a small ion-electron interaction makes the optics 100% coupled.
• However, for practical cases, the width of this area is extremely small, E-4 -E-5, so practically the width is zero. Without machine coupling, the growth rate is zero everywhere, apart of that separate punctured point.
• PR2000 result follows from mathematically correct calculations for that punctured point. It was not realized, however, that even a tiny step out of that special point would suppress the rate by orders of magnitude if there is no machine coupling.

Z01
• A problem about rate dependence on the tune separation was formulated by P. Zenkevich in 2001 (Z01).
• He treated numerically the eigen-value problem for the following form of the revolution matrix: where is an inverse half-drift matrix, C is the perturbed solenoid matrix, and

Growth Rate as a Resonance Curve
For ACR (RIKEN) numbers, he calculated the following dependence of the growth rate on the tune separation: Growth rate @ 1 KturnD , by Z01 rad , y x µ µ − The maximal peak value here is in exact agreement with PR2000.
What factor does determine the width? -this problem were not raised in Z01.

Width of the Resonance
Careful look at Z01 revolution matrix reveals that it DOES include a machine coupling, since it contains an uncompensated solenoid C : So, it has to be that the width is determined by the ion Larmor phase advance. Taking the numbers, it looks pretty consistent with the figure above.
Working a bit more, the following formula can be checked to be an exact description of the Fig. Z01: If the solenoid is compensated, the revolution matrix is written as: Numerical eigen-value calculation for the same parameters as Z01, but with this compensated coupling, reduces the width ~ 300 times! See Fig. below Quadrupole IEI • The quadrupole ion-electron growth rate relates to the dipole rate in the same way as for the conventional impedance: • Similar to the dipole mode, the Landau damping does vanishes at • The resistive damping for these quadrupole fast modes is reduced much more, than the IEI rate. Thus, it might easily be insufficient to damp the quadrupole IEI motion.
• There are no such thing as a quadrupole damper (yet, at least).
• With , the fast-modes quadrupole IEI has all the reasons to develop, if the coupling is significant.
• If the coupling is significant, the quadrupole IEI easily develops.
• At some level, it is stopped by its own non-linearity, and stays forever.
• Due to vanished Landau-damping, transfer of this coherent motion into incoherent motion strongly suppressed.
• This suppressed energy transfer still goes, and gives slow emittance growth and lifetime degradation.
• To fix this problem, step a bit away from the coupling resonance, or / and reduce the coupling area.

RR experience
• Recycler (RR) used to stay at the coupling resonance, ~0.42 for the both tunes, having ~100% coupling.
• There was emittance growth, strongly correlated with the electron current and the pbar linear density.
• At "mining" state (max bunching), and 100 mA of e-beam, the typical rate was ~ 30 pi mm mrad/hr, or ~0.001 of the calculated quadrupole IEI rate for these parameters.
• The described theory pushed me to insist on more separation of the tunes.
• For the new WP, the coupling parameter reduced ~10-20 times .
• My thanks to A. Valishev for his help with the RR optical file.
• My special thanks are to Valeri Lebedev for his multiple remarks related to various aspects of this paper.