CERN 5 th TLEP Mini-Workshop 2013

1. CERN 5th TLEP Mini-Workshop 2013 On simple invariants conservation in TLEP V. Danilov SNS AP group S. Nagaitsev FNAL

2. Talk outline • Integrals of motion • Advantages to have simple invariants • TLEP specifics • Necessary conditions • Possible approaches to create approximately integrable motion to prevent particle loss, beam blow-up, etc. for TLEP • On mitigation of Telnov’s effect Presentation_name

3. Nonlinear systems in general 1) All solutions of classical mechanics, known by end of XIX century – integrable. Kepler;s problem – 2 integrals of motion Energy and angular momentum. 2) Nonintegrable systems constitute majority of all real systems (1st examples, H. Poincare, 1895) 3) In accelerators, any arbitrary nonlinearity (sextupoles, octupoles, beam-beam kick, etc.) is nonintegrable 4) They are characterized by infinite number of resonances, chaotic motion around unstable points (homoclinic and heteroclinic structures), diffusion, particle loss, and beam-blow-up (on the right, horizontal phase space of 2D motion in linear lattice with 1 octupole) 5) They all have integrals of motion (initial conditions), But they are extremely complex Presentation_name

4. On the way to integrability (resonance suppression) Important steps could be made toward elimination of resonances (integrability) 1) Colliding beams: a) Round – angular momentum conservation- 1D motion for r (Novosibirsk, V. Danilov et al, EPAC96, realized at VEPP2000, tune shift around 0.15 achieved); b) Crab waist - decoupling x and y motion (P. Raimondi (2006), tune shift 0.1 achieved at DAFNE), factor 2 increase is probable. c) Talman’sMoebius ring. • Angular momentum conservation makes the motion one – dimensional. Arnold diffusion is suppressed – any invariant line is impenetrable by particles from space it encompasses Presentation_name

5. TLEP specifics • Very flat beams • Strong synchrotron radiation • Short radiation times • High beam-beam tune shifts. ξx,y~0.1 • Asumption – relatively weak nonlinear fields can modify beam distribution due to strong synchrotron radiation Presentation_name

6. Necessary conditions (my talk at 4th TLEP workshop) • Since Closed Orbit Instability threshold is close, working point should be far from integer; • VLHC problem – detuning wake cause large spread of betatron tunes. It could be fatal for the beam; • Shape of the vacuum chamber determines it and its important to optimize it – probably it has to be round • No other tune spreads (due to chromaticity, etc.) Presentation_name

7. Example of distribution with integrable beam-beam motion This trick is used to get integrable lattice for IOTA that is under construction at Fermilab. Latest developments for beam-beam see A. Valishev (FNAL) NAPAC13 talk “Beam-beam limits for integrable system” How to find something similar for TLEP? Presentation_name

8. Integrable round beam-beam A particle collides with a bunch (charge distribution) weak-strong approximation Longitudinal bunch distribution (1/β) for integrable beam-beam Gaussian non-integrable Presentation_name

9. FMA comparison Integrable Bare tune: 0.3 Gaussian non-integrable Presentation_name

10. Possible TLEP approach • As for round beams integrability requires modification of longitudinal beam distribution function • For very flat beams it makes sense to decouple x and y motions and make them close to integrable separately • In this case each plane will have its own simple invariant of motion (next slide) • Unfortunately, synchrotron motion with high tune requires additional measures – for now its beyond the scope of our talk Presentation_name

11. Example of nearly integrable TLEP distribution ~1mm at IP I – unity matrix Modified transverse distribution Modified longitudinal distribution Horizontal beta ~100 time larger than vertical If the beta-function is close to constant along the beam distribution, the vertical motion is equivalent to 1D time-independent motion with Y Hamiltonian being an invariant. X tune ~ 0.5 + e, e<<1 In normalized variables horizontal field will look like very thin lens. The dynamic map consists of short kicks and a linear map in between. These dynamics is very close to one with time independent X Hamiltonian Presentation_name

12. Modified distributions in TLEP (nonlinear magnet) IOTA nonlinear quadrupole with changed polarity it will be nonlinear dipole we can create relatively good motion and large diffusion in the middle of the beam Longitudinal distribution reforming + large dispersion and flat beam to prevent large perturbation for vertical motion For the transverse distribution modification, the betatron horizontal size has to be larger than the dispersive contribution in size Presentation_name

13. Other possible methods • Noise can make the beam more flat if initial size is smaller than the design one • Lasers (not head on – 4gamma^2 quanta destroy particles) • Phase modulation or periodic horizontal dipole kick • ? Tan, Burov, FNAL PRST-AB 2013 Phase modulation flattens the longitudinal distribution – in this case makes it more parabolic Results are from Tevatron Presentation_name

14. Naive thoughts on Telnov’s effect mitigation • Metal or nanomaterial walls with high • conductivity for fields reflection; • 2) Have to be thin to prevent large scattering; • 3) Have to withstand a few picosecond pulses • with E~1010 V/m; • 4) Electrons from field emission occupy small • space, therefore the pinch effect has to be small 1 mm reflection E-field e- stream e- e+ reflection Presentation_name

15. Conclusion • There exist special distributions of beams producing nonlinear close-to-integrable motion • Their implementations may advance beam-beam tune shifts above 0.1 (0.08->0.12) • Telnov’s effect mitigation could be considered in terms of beam field cutting/reflection due to interaction with thin films (?) • This won’t change motion integrability for suggested distributions Presentation_name