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Landau damping

Landau damping. effect occurs in systems consisting of a high number of oscillators having different oscillation frequencies and performing a collective motion. L.D. Landau, 1946, J. Phys. USSR 10 (1946) plasma physics N.G. van Kampen, 1955, Physica 21 (1955) mathematics

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Landau damping

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  1. Landau damping effect occurs in systems consisting of a high number of oscillators having different oscillation frequencies and performing a collective motion L.D. Landau, 1946, J. Phys. USSR 10 (1946) plasma physics N.G. van Kampen, 1955, Physica 21 (1955) mathematics C.E. Nielsen, A.N. Sessler, K.R. Symon, HEACC Geneva (1959) accelerator physics R.D. Kohaupt, “What is Landau Damping? Plausibilities, Fundamental Thoughts, Theory” DESY M-86-02 accelerator physics

  2. Landau damping from bunch-to-bunch frequency spread, rf modulation, gap Y.H. Chin, K. Yokoya, “Landau Damping of a Multi-Bunch Instability due to Bunch-to-Bunch Tune Spread,” DESY 86-097 (1986) multi-bunch Landau damping

  3. if a system of many oscillators (protons) with different oscillation frequencies is excited (kicked) their centroid motion decays in time as a result of the frequency spread, which extinguishes the coherent motion on the other hand, a beam can be driven unstable by self-excited electromagnetic fields (impedance) which act back on the beam; instability rise time tg the beam is Landau damped if the decay time due to the frequency spread is ‘shorter’ than the instability rise time

  4. we can replace discrete sum by integral, for times which are not extremely long

  5. frequency distribution centroid linear “wake force” coherent frequency shift in the absence of frequency spread

  6. special case: single frequency ansatz harmonic oscillation solution exists for arbitrary small W

  7. general case Fourier-Laplace-transform Bromwich integral (all singularities above the path C) z C a

  8. initial conditions cut –w(hmax)<Rez<w(hmax) singularities at zk analytic otherwise zk cut C

  9. singularities zk in the upper z-plane describe coherent stability and those in the lower z-plane coherent instability; the zk correspond to solutions of the integral has a finite value for any reasonable frequency distribution for all z values there is no solution to (1) for small W! interpretation: system cannot organize a collective motion if interaction not strong enough

  10. example: Lorentz spectrum Imh wx+iDw -W+ia X X Reh W-ia X X wx-iDw

  11. dispersion relation a>0: instability growth rate, a->0: border of stability above relation may not apply for a<0 consider a=0 absolute value of dispersion relation simplifies to |Dwcoh|=Dw if coherent tune shift less than Dw, the beam is stable

  12. including incoherent frequency shift dispersion relation where now instability threshold does not change

  13. with coherent and incoherent wake field: Fourier-Laplace-transform of impulse excitation: Yokoya factors for flat chamber

  14. single-turn multi-bunch instability, general impedance where

  15. dispersion relation including Yokoya factors

  16. general case Fourier-Laplace transform

  17. integral over frequencies is solved as (see A. Chao’s book): example: Lorentz spectrum can we find a solution to this equation?

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