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Zero-current longitudinal beam dynamics (in linacs) Jean-Michel Lagniel (CEA/GANIL)

Zero-current longitudinal beam dynamics (in linacs) Jean-Michel Lagniel (CEA/GANIL) Longitudinal focalization = nonlinear forces => separatrix, tune spread Acceleration => damping of the phase oscillations.

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Zero-current longitudinal beam dynamics (in linacs) Jean-Michel Lagniel (CEA/GANIL)

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  1. Zero-current longitudinal beam dynamics (in linacs) Jean-Michel Lagniel (CEA/GANIL) Longitudinal focalization = nonlinear forces => separatrix, tune spread Acceleration => damping of the phase oscillations. The longitudinal beam dynamics is complex,even when the nonlinear space-charge forces are ignored. The three different ways to study and understandthis zero-current longitudinal beam dynamics will be presented and compared. J-M Lagniel ESS-Lund Jan 29, 2014

  2. The 3 ways to study the longitudinal beam dynamics Synchronous particle and oscillations around the synchronous particle      J-M Lagniel ESS-Lund Jan 29, 2014

  3. Program I- EoM integration in field maps  Mappings (Transit-Time-Factor) II- Mappings  2nd order differential EoM (Smooth approximation) III- Longitudinal beam dynamics without damping (Phase-space portraits @ Poincaré surface of section) Smooth approximation EoM / Mapping / Integration in field map IV- Longitudinal beam dynamics with damping (Basin of attraction, bifurcation diagrams) V- Concluding remarks J-M Lagniel ESS-Lund Jan 29, 2014

  4. I- From EoM integration in field maps to mappings Numerical integration (dz) of the EoM in field maps (, W) phase-space J-M Lagniel ESS-Lund Jan 29, 2014

  5. I- From EoM integration in field maps to mappings Mappingi+1 = i +  Wi+1 = Wi + W => Integration over one accelerating cell - cavity  (z = 0) Ez mean value (Panofsky 1951) The Transit-Time-Factor contains all the information on the field map and speed + radial evolution over the accelerating cell / cavity Without approximation More complicated than the original  only useful with approximations J-M Lagniel ESS-Lund Jan 29, 2014

  6. I- From EoM integration in field maps to mappings Odd function of z  Ez(r,z) = even function + constant speedover the cell Constant speed and radius over the cell The TTF of each particle is a function of the particle mean radius and velocities (input values in practice) (not function of the particle radius and speed evolution over the cell) Under this form, the TTF is the an component of the Ez(z) Fourier transform with n = Lc / βλ => n = h ( loss of information on the shape of Ez(z) ) J-M Lagniel ESS-Lund Jan 29, 2014

  7. I- From EoM integration in field maps to mappings   Allows to find analytical expressions of the TTF for particular field distributions J-M Lagniel ESS-Lund Jan 29, 2014

  8. I- From EoM integration in field maps to mappings  Using the approximated formula to evaluate the particle TTF we have found a practical way to build a mapping J-M Lagniel ESS-Lund Jan 29, 2014

  9. I- From EoM integration in field maps to mappings  Using an approximated formula to evaluate the particle TTF  we have found a practical way to build a mapping NO !  This mapping is not (by far) symplectic (area preserving)  when the TTF is calculated taking into account the particle mean speed and radius A phase correction must be added to obtain a symplectic mapping (1st order) Pierre Lapostolle et al 1965 – 1975 (B.C. age). J-M Lagniel ESS-Lund Jan 29, 2014

  10. I- From EoM integration in field maps to mappings The only way to produce a simple symplectic mapping is to consider the synchronous particle TTF for every particle TTF analytical expression => neglect the evolution of the velocity in the cell Simple symplectic mapping => neglect the effect of the particle velocity spread on the TTF (Phase and energy evolution with respect to the synchronous particle) (Mapping used for the comparison with the other methods) J-M Lagniel ESS-Lund Jan 29, 2014

  11. II- From mappings to 2nd order differential EoM Smooth approximation considering the mapping without phase correction  Long term behavior Large amplitude oscillations Low amplitude oscillations J-M Lagniel ESS-Lund Jan 29, 2014

  12. II- From mappings to 2nd order differential EoM Error on the longitudinal phase advance per focusing period induced by the smooth approximation Smooth approximation Mapping Twiss matrix J-M Lagniel ESS-Lund Jan 29, 2014

  13. III- Longitudinal beam dynamics without damping Smooth approximation(2nd order Differential EoM obtained using the smooth approximation) Particle trajectories – separatrix vs synchronous phase s = -90° s = -30° J-M Lagniel ESS-Lund Jan 29, 2014

  14. III- Longitudinal beam dynamics without damping Smooth approximation Choice of s Beam size vs longitudinal aperture -20°  -15° Long. Acc. / 2 €€€ The temptation is high to increase the synchronous phase €€€ High-power LINAC designers (and managers) must bringas much attention to the longitudinal beam size / longitudinal aperture ratioas they bring to the radial beam size / radial aperture ratio J-M Lagniel ESS-Lund Jan 29, 2014

  15. III- Longitudinal beam dynamics without damping Smooth approximationvs Mapping s = -90° J-M Lagniel ESS-Lund Jan 29, 2014

  16. III- Longitudinal beam dynamics without damping Mapping s = -90° 0l* > 50° More and more resonances => resonance overlaps=> larger choatic area 82°, 86°, 90° / lattice => real phase advance value higher than the one given by the smooth approximation J-M Lagniel ESS-Lund Jan 29, 2014

  17. III- Longitudinal beam dynamics without damping As 0l* increases the phase-space portraits plotted using the mapping show more and more resonances  more and more resonance overlaps  larger and larger choatic areas  longitudinal acceptance reduction [P. Bertrand, EPAC04] Is it true or is it a spurious effect of the mapping ? ( periodic error = excitation of the resonances ? ) ... If yes, why ? Check making a direct integration of the EoM J-M Lagniel ESS-Lund Jan 29, 2014

  18. III- Longitudinal beam dynamics without damping Longitudinal toy Direct integration of the EoMs = -90° h  frf TTF Field map = First-harmonic-model 0l*  Epic J-M Lagniel ESS-Lund Jan 29, 2014

  19. III- Longitudinal beam dynamics without damping Phase-space portraits plotted using the Longitudinal Toy 0l* = 80° Lc = L h = 1 Ez(z) = pure sinusoid (first-harmonic) 1/4 resonance not excited !!!! Stable oscillations aroundthe “inverted pendulum position” 0l* = 90° 0l* = 95° J-M Lagniel ESS-Lund Jan 29, 2014

  20. III- Longitudinal beam dynamics without damping Phase-space portraits plotted using the Longitudinal Toy 0l* = 50° Lc = L/4 h = 4 Ez(z) with harmonics > 1 The resonances are excited  Mapping  0l* = 90° 0l* = 70° J-M Lagniel ESS-Lund Jan 29, 2014

  21. III- Longitudinal beam dynamics without damping summary EoM @ smooth approximation Ez(z) = Constant => Resonances not excited ... but essential to understand the longitudinal beam dynamics Physics Mapping Ez(z) = Dirac comb (period L) => FT[Ez(z)] = Dirac comb (1/L) All resonances excited EoM using field maps Ez(z) = Field map => FT[Ez(z)] = some harmonics (1/L) Some resonances excited (need more work !) The longitudinal acceptance is significantly reduced at high 0l* J-M Lagniel ESS-Lund Jan 29, 2014

  22. IV- Longitudinal beam dynamics with damping (, d/ds) plane Damped linear harmonic oscillator Attractor = 0 d/ds = 0 Linac = under-damped regime (RFQ ?) Longitudinal phase advance  J-M Lagniel ESS-Lund Jan 29, 2014

  23. IV- Longitudinal beam dynamics with damping Smooth approximationSmall amplitude oscillations (, W) phase-plane Phase-space area preservationif adiabatic evolution Attractor = W axis W   Damping = energy spread J-M Lagniel ESS-Lund Jan 29, 2014

  24. IV- Longitudinal beam dynamics with damping Smooth approximation (, d/ds) plane Attractor = (0, 0) s = -30° Basin of attraction Acceptances + separatrix K = 0 J-M Lagniel ESS-Lund Jan 29, 2014

  25. IV- Longitudinal beam dynamics with damping Mapping (, d/ds) plane Attractors = (0, 0) and the 1/4 resonance islands 0l* = 82° J-M Lagniel ESS-Lund Jan 29, 2014

  26. IV- Longitudinal beam dynamics with damping Mapping Basin of attraction (, d/ds) plane K = 0.02 K = 0.10 0l* = 60° K = 0.01 K = 0.10 0l* = 70° Attractors : (0, 0) (1/6 resonance) J-M Lagniel ESS-Lund Jan 29, 2014

  27. IV- Longitudinal beam dynamics with damping Mapping (fractal) Basin of attraction (, d/ds) plane K = 0.05 K = 0.01 0l* = 82° Attractors(0, 0)(1/4 resonance) K = 0.20 K = 0.10 J-M Lagniel ESS-Lund Jan 29, 2014

  28. IV- Longitudinal beam dynamics with damping ESS linac (2012) K = 0.36 … 0.10 … 0.04 DTL 0.015 … 0.005 highenergy J-M Lagniel ESS-Lund Jan 29, 2014

  29. IV- Longitudinal beam dynamics with damping SPIRAL 2 superconducting linac K = 0.05 … 0.08 … 0.12 … 0.19 … 0.16 … 0.08 … 0.05 J-M Lagniel ESS-Lund Jan 29, 2014

  30. IV- Longitudinal beam dynamics with damping summary Damping induced by the acceleration  Phase-width reduction and energy-spread growth  The stable fix points of the resonance islandsact as main attractors at low damping rates  The damping can annihilate the effect of the resonances  J-M Lagniel ESS-Lund Jan 29, 2014

  31. V- Five points to keep in mind When the Transit-Time-Factor is used (linac designs and optimizations, understanding of the basic physics…) (i.e. as soon as a direct numerical integration of the EoM is abandoned !) the longitudinal beam dynamics must be computed in such a way that the longitudinal motion in the (δφ, δW) phase plane remains symplectic (area preserving) • Option #1 = Use the synchronous-particle TTF for all the particles(but keep in mind the consequences of this approximation) • Option #2 = Follow the work done at the B.C. age(A.C. age  prefer numerical integrations !) J-M Lagniel ESS-Lund Jan 29, 2014

  32. V- Five points to keep in mind The nonlinear character of the accelerating field induces aphase-advance spread (tune shift) which must be considered when the phase width is important (or halo) • This nonlinear character makes thelongitudinal beam dynamics much more complicated than the radial one ! • Space-charge induced nonlinearities • will obviously complicate the situation ! • Think phase advanceevolutionwith amplitude J-M Lagniel ESS-Lund Jan 29, 2014

  33. V- Five points to keep in mind The results obtained using the “classical approximations” (TTF & smooth approximation) are very useful to understand the longitudinal beam dynamics(including the large-amplitude motions and damping) BUT • For longitudinal phase advances greater than 60° / period,these approximations induce errors on the values of the parameters calculated using them (e.g. 0l *) AND Hide the resonances excited by the nonlinear accelerating field localized in the cavities = acceptance reduction J-M Lagniel ESS-Lund Jan 29, 2014

  34. V- Five points to keep in mind To understand the longitudinal beam dynamics in linacsit is essential to take into account the damping induced by the accelerationwhen the damping rate is significant with respect to the period of the longitudinal oscillations. • k should be considered as an important parameter • to analyze a linac design and understand its longitudinal beam dynamics J-M Lagniel ESS-Lund Jan 29, 2014

  35. V- Five points to keep in mind s = ??? -20°  -15° Longitudinal Acceptance / 2 • A systematic and well defined ruleto choose the synchronous phaseshould be defined taking into accountboth risk and project economy J-M Lagniel ESS-Lund Jan 29, 2014

  36. Concluding remarks Hope you are now convinced thatthe zero-current longitudinal beam dynamics is complex ! At least more complex than what is taught inclassical Accelerator Books and Accelerator Schools … Several questions still open  Why no (nearly no) excitation of the resonances for Lc = L and h = 1 ?(numerical integration of the EoM) •  How different Fourier spectra of the longitudinal repartition of Ez(z)act on the beam dynamics ? (series of multi-cell cavities…) •  Which second order differential equation can be used as modelto study, understand and predict the effect (resonances) of this longitudinal repartition ? •  What happen when the space-charge forces are added ? J-M Lagniel ESS-Lund Jan 29, 2014

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