Zerocurrent longitudinal beam dynamics in linacs JeanMichel Lagniel

Zero-current longitudinal beam dynamics (in linacs) Jean-Michel Lagniel (GANIL) The longitudinal beam dynamics is complex even when the nonlinear space-charge forces are ignored The three different ways to study and understand the zero-current longitudinal beam dynamics will be presented and compared. J-M Lagniel CERN Eu. CARD 2/MAX March 20 -21, 2014 Page 1

The 3 ways to study the longitudinal beam dynamics Synchronous particle and oscillations around the synchronous particle J-M Lagniel CERN Eu. CARD 2/MAX March 20 -21, 2014 Page 2

Program I- Eo. M integration in field maps II- Mappings 2 nd order differential Eo. M III- Longitudinal beam dynamics without damping Smooth approximation vs Mapping vs Integration in field map IV- Longitudinal beam dynamics with damping Smooth approximation vs Mapping vs Integration in field map V- Concluding remark J-M Lagniel CERN Eu. CARD 2/MAX March 20 -21, 2014 Page 3

I- From Eo. M integration in field maps to mappings Numerical integration (dz) of the Eo. M in field maps ( , W) phase-space J-M Lagniel CERN Eu. CARD 2/MAX March 20 -21, 2014 Page 4

I- From Eo. M 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 ! J-M Lagniel only useful with approximations CERN Eu. CARD 2/MAX March 20 -21, 2014 Page 5

I- From Eo. M integration in field maps to mappings Constant speed and radius over the cell Odd function of z Ez(r, z) = even function + 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) but not function of the particle radius and speed evolution over the cell J-M Lagniel CERN Eu. CARD 2/MAX March 20 -21, 2014 Page 6

I- From Eo. M integration in field maps to mappings Allows to find analytical expressions of the TTF for particular field distributions J-M Lagniel CERN Eu. CARD 2/MAX March 20 -21, 2014 Page 7

I- From Eo. M 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 (1 st order) Pierre Lapostolle et al 1965 – 1975 (B. C. age). J-M Lagniel CERN Eu. CARD 2/MAX March 20 -21, 2014 Page 8

I- From Eo. M 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) J-M Lagniel CERN Eu. CARD 2/MAX March 20 -21, 2014 Page 9

II- From mappings to 2 nd order differential Eo. M Smooth approximation considering the mapping without phase correction Large amplitude oscillations Long term behavior Low amplitude oscillations Error J-M Lagniel CERN Eu. CARD 2/MAX March 20 -21, 2014 Page 10

IIISmooth approximation Longitudinal beam dynamics without damping (2 nd order Differential Eo. M obtained using the smooth approximation) Particle trajectories – separatrix vs synchronous phase s = -90° s = -30° J-M Lagniel CERN Eu. CARD 2/MAX March 20 -21, 2014 Page 11

III- Longitudinal beam dynamics without damping Smooth approximation Choice of s -20° -15° Long. Acceptance / 2 €€€ The temptation is high to increase the synchronous phase €€€ High-power LINAC designers (and managers) must bring as much attention to the longitudinal beam size / longitudinal aperture ratio as they bring to the radial beam size / radial aperture ratio J-M Lagniel CERN Eu. CARD 2/MAX March 20 -21, 2014 Page 12

III- Longitudinal beam dynamics without damping Smooth approximation J-M Lagniel vs CERN Eu. CARD 2/MAX Mapping March 20 -21, 2014 s = -90° Page 13

III- Longitudinal beam dynamics without damping Mapping s = -90° 0 l* > 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 CERN Eu. CARD 2/MAX March 20 -21, 2014 Page 14

III- Longitudinal beam dynamics without damping As 0 l* 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 [ Fateev & Ostroumov NIM 1984 ] [ Bertrand, EPAC 04 ] 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 Eo. M J-M Lagniel CERN Eu. CARD 2/MAX March 20 -21, 2014 Page 15

III- Longitudinal beam dynamics without damping Longitudinal toy Direct integration of the Eo. M h s = -90° frf TTF Field map = First-harmonic-model 0 l* Epic J-M Lagniel CERN Eu. CARD 2/MAX March 20 -21, 2014 Page 16

III- Longitudinal beam dynamics without damping Phase-space portraits plotted using the Longitudinal Toy Lc = L 0 l* = 80° h=1 Ez(z) = pure sinusoid (first-harmonic) 1/4 resonance not excited !!!! 0 l* = 90° J-M Lagniel 0 l* = 95° CERN Eu. CARD 2/MAX March 20 -21, 2014 Page 17

III- Longitudinal beam dynamics without damping Phase-space portraits plotted using the Longitudinal Toy Lc = L/4 0 l* = 50° h=4 Ez(z) with harmonics > 1 The resonances are excited Mapping 0 l* = 90° 0 l* = 70° J-M Lagniel CERN Eu. CARD 2/MAX March 20 -21, 2014 Page 18

III- Longitudinal beam dynamics without damping summary Eo. M @ 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 Eo. M using field maps Ez(z) = Field map => FT[Ez(z)] = some harmonics (1/L) Some resonances excited (need more work !) The longitudinal acceptance can be significantly reduced at high 0 l* J-M Lagniel CERN Eu. CARD 2/MAX March 20 -21, 2014 Page 19

IV- Longitudinal beam dynamics with damping Smooth approximation ( , ’) plane Attractor = (0, 0) s = -30° Basin of attraction Acceptances + separatrix K = 0 J-M Lagniel CERN Eu. CARD 2/MAX March 20 -21, 2014 Page 20

IV- Longitudinal beam dynamics with damping Mapping ( , ’) plane Attractors = (0, 0) and the 1/4 resonance islands 0 l* = 82° J-M Lagniel CERN Eu. CARD 2/MAX March 20 -21, 2014 Page 21

IV- Mapping Longitudinal beam dynamics with damping Basin of attraction ( , ’) plane K = 0. 02 K = 0. 10 0 l* = 60° K = 0. 01 K = 0. 10 0 l* = 70° Attractors : (0, 0) (1/6 resonance) J-M Lagniel CERN Eu. CARD 2/MAX March 20 -21, 2014 Page 22

IV- Mapping Longitudinal beam dynamics with damping (fractal) Basin of attraction K = 0. 01 ( , d /ds) plane K = 0. 05 0 l* = 82° Attractors (0, 0) (1/4 resonance) K = 0. 20 K = 0. 10 J-M Lagniel CERN Eu. CARD 2/MAX March 20 -21, 2014 Page 23

IV- Longitudinal beam dynamics with damping ESS linac (2012) K= 0. 36 … 0. 10 … 0. 04 DTL J-M Lagniel CERN Eu. CARD 2/MAX 0. 015 … 0. 005 high energy March 20 -21, 2014 Page 24

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 CERN Eu. CARD 2/MAX March 20 -21, 2014 Page 25

IV- Longitudinal beam dynamics with damping Summary The stable fix points of the resonance islands act as main attractors at low damping rates The damping can annihilate the effect of the resonances k should be considered as an important parameter to analyze a linac design and understand its longitudinal beam dynamics J-M Lagniel CERN Eu. CARD 2/MAX March 20 -21, 2014 Page 26

Concluding remark Hope you are now convinced that the zero-current longitudinal beam dynamics is complex ! !!! At least more complex than what is taught in classical Accelerator Books and Accelerator Schools !!! J-M Lagniel CERN Eu. CARD 2/MAX March 20 -21, 2014 Page 27