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Emittance exchange by anisotropy instabilities in the SPL

Emittance exchange by anisotropy instabilities in the SPL. Short introduction to Hofmann’s stability charts, 4 examples of realistic SPL layouts ( =0.52 & =0.7 sections, 22 periods), simulations results with IMPACT, results for mismatched beams, conclusion and outlook.

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Emittance exchange by anisotropy instabilities in the SPL

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  1. Emittance exchange by anisotropy instabilities in the SPL • Short introduction to Hofmann’s stability charts, • 4 examples of realistic SPL layouts (=0.52 & =0.7 sections, 22 periods), • simulations results with IMPACT, • results for mismatched beams, • conclusion and outlook. Frank Gerigk CERN RF/PS

  2. Derivation of 2D (x,y) stability charts I Vlasov’s equation KV - type solution add perturbation: • Linearize the perturbed Vlasov equation and solve for the eigenfrequencies w : Frank Gerigk CERN RF/PS

  3. Derivation of 2D (x,y) stability charts II 2’nd order modes: even modes  envelope modes Oscillating and non-oscillating 3’d and 4’th order modes Numerical analysis with 2D simulations for KV- and waterbag-distributions Non-oscillating modes Stability Charts Frank Gerigk CERN RF/PS

  4. Hofmann Chart for the SPL (emittance ratio of 2) growth rate  equipartitioned beam emittance ratio=2 Frank Gerigk CERN RF/PS

  5. Hofmann Chart for the SPL with four test cases growth rate SPL IIb case 3 case 2 case 1  equipartitioned beam emittance ratio=2 Frank Gerigk CERN RF/PS

  6. tune ratios sz/sx,y full current full current zero current zero current full current full current zero current zero current Frank Gerigk CERN RF/PS

  7. zero current phase advance sx,y sz sz sx,y sx,y sx,y sz sz Frank Gerigk CERN RF/PS

  8. tune depression longitudinal longitudinal transverse transverse longitudinal longitudinal transverse transverse Frank Gerigk CERN RF/PS

  9. Anisotropy (T=ezsz/etst) Frank Gerigk CERN RF/PS

  10. emittance evolution ez ez -25% ex,y ex,y +25% ez ez -20% ex,y ex,y +20% Frank Gerigk CERN RF/PS

  11. Stability charts for different emittance ratios different scale! Frank Gerigk CERN RF/PS

  12. emittance for 1.3 rx, 0.7 ry ez ez ex,y ex,y ez ez ex,y ex,y Frank Gerigk CERN RF/PS

  13. emittance for 1.3 rx,y, 0.7 rz ez ez ex,y ex,y ez ez ex,y ex,y Frank Gerigk CERN RF/PS

  14. Conclusions and outlook • Good agreement between simulations of 3D periodic accelerating structure and results of 2D Vlasov analysis, • anisotropic (non-equipartitioned) beams are possible, • stability charts are especially useful for the design of SC linacs (high longitudinal phase advance), • current SNS bd-design shall be refined in collaboration with Hofmann and his charts, • equipartitioned beams seem to offer the largest stable area in parameter space, • no halo particles seem to be produced by emittance exchange. • How is the evolution of the 99%, 100% emittance? • Is the observed r.m.s. emittance exchange harmful? • What happens during a “quick” passage of unstable regions? • Should we make an experiment? Frank Gerigk CERN RF/PS

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