Landau Damping of Beam Instabilities by Electron Lenses

Modern and future particle accelerators employ increasingly higher intensity and brighter beams of charged particles and become operationally limited by coherent beam instabilities. Usual methods to control the instabilities, such as octupole magnets, beam feedback dampers, and use of chromatic effects, become less effective and insufficient. We show that, in contrast, Lorentz forces of a low-energy, magnetically stabilized electron beam, or “electron lens,” easily introduce transverse nonlinear focusing sufficient for Landau damping of transverse beam instabilities in accelerators. It is also important to note that, unlike other nonlinear elements, the electron lens provides the frequency spread mainly at the beam core, thus allowing much higher frequency spread without lifetime degradation. For the parameters of the Future Circular Collider, a single conventional electron lens a few meters long would provide stabilization superior to tens of thousands of superconducting octupole magnets.

Figure 1

The incoherent tune shift by the round electron lens, $\delta {\nu }_x/\delta {\nu }_{\max }$, versus the particle transverse amplitudes, Eq. (2).

Figure 2

Stability diagram for the 7 TeV proton beams in LHC at the maximal strength of the Landau octupoles.

Figure 3

Electron lens stability diagrams are presented for various electron beam sizes (noted in units of the proton beam rms size), assuming the same current density at the center.

Figure 4

Frequency map analysis (FMA) and dynamic aperture modeling of LHC proton dynamics with comparable strength Landau damping provided by octupole magnets (a) and by the electron lens (b). Horizontal and vertical axes—initial particle amplitudes $A_x$, $A_y$ in units of the rms beam size varying from $0{\sigma }_p$ (core) to $8{\sigma }_p$ (halo). Brighter colors indicate exponentially stronger tune modulation indicating resonances (see color palette). 100 000 turns DA is shown in cyan lines.