Recent Progress on Nonlinear Integrability Theoretical work, simulations and experimental results

Transcription

1 Recent Progress on Nonlinear Integrabilit Theoretical work, simulations and experimental results Alexander Valishev ICFA Mini-Workshop on Nonlinear Dnamics and Collective Effects in Particle Beam Dnamics NOCE017. Arcidosso, Ital 1 September 017

2 Acknowledgments D.Broemmelsiek, A.Burov, K.Carlson, A.Didenko, N.Edd, J.Eldred, V.Kashikhin, V.Lebedev, J.Leibfritz, M.McGee, S.Nagaitsev, L.Nobrega, H.Piekarz, E.Prebs, A.Romanov, G.Romanov, V.Shiltsev, G.Stancari, J.Thangaraj, R.Thurman-Keup, S.Wesseln, D.Wolff (FNAL) S.Chattopadha, B.Freemire (NIU) D.Shatilov (BINP) S.Antipov (CERN) D.Bruhwiler, N.Cook, S.Webb (RadiaSoft) F.O Shea, A.Murokh (RadiaBeam) R.Kishek, K.Ruisard (UMD) C.Mitchell (LBNL) A. Valishev NOCE017 - Nonlinear Integrabilit 09/1/017

3 Modern Accelerator Lattices The backbone of all present accelerators linear focusing lattice: particles have nearl identical betatron tunes b design. Such machines are built using dipoles and quadrupoles. All nonlinearities (both natural and speciall introduced) are perturbations 3 A. Valishev NOCE017 - Nonlinear Integrabilit 09/1/017

4 A. Valishev NOCE017 - Nonlinear Integrabilit 4 Strong Focusing Standard Approach Since 195 ) ( ) ( 0 ) ( 0 ) (,, s K C s K s K x s K x x x x = + î í ì = + = + -- piecewise constant alternating-sign functions Christofilos (1949); Courant, Livingston and Snder (195) 09/1/017

5 Stabilit of Linear Lattices in Light Sources Low beam emittance requires Strong Focusing Strong Focusing leads to strong chromatic aberrations Chromaticit correction demands strong sextupole magnets Strong sextupoles introduce significant nonlinear perturbation integrabilit is ruined Dnamical aperture limitation leads to reduced lifetime, issues with injection, etc. There are was to (partiall) mitigate these issues High periodicit Proper arrangement of sextupole phase advances 5 A. Valishev NOCE017 - Nonlinear Integrabilit 09/1/017

6 Stabilit of Linear Lattices in High Intensit Machines 1965 Priceton-Stanford CBX: First mention of an 8-pole magnet Observed vertical resistive wall instabilit With octupoles, increased beam current from ~5 to 500 ma CERN PS: In 1959 had 10 octupoles; not used until 1968 At 10 1 protons/pulse observed (1 st time) head-tail instabilit. Octupoles helped. Once understood, chromaticit jump at transition was developed using sextupoles. More instabilities were discovered; helped b octupoles, fb LHC has 336 octupoles that are used at close to full current 6 A. Valishev NOCE017 - Nonlinear Integrabilit 09/1/017

7 Focusing: Linear vs. Nonlinear Accelerators are linear sstems b design (frequenc is independent of amplitude). In accelerators, nonlinearities are either intentionall introduced (chromaticit correction sextupoles, Landau damping octupoles) or unavoidable (space charge, beambeam, magnet imperfections). All nonlinearities (in present rings) lead to resonances and dnamic aperture limits. Are there magic nonlinearities with zero resonance strength? The answer is es (we call them integrable ) 7 A. Valishev NOCE017 - Nonlinear Integrabilit 09/1/017

8 Do Accelerators Need to be Linear? Search for solutions that are strongl nonlinear et stable Orlov (1963) McMillan (1967) 1D solution ü Perevedentsev, Danilov (1990) generalization of McMillan case to D, round colliding beams. Require non-laplacian potentials to realize Round colliding beams possess 1 invariant VEPP-000 at BINP (Novosibirsk, Russia) commissioned in 006. Record-high beambeam tune shift ~0.5 attained in 013 Danilov, Shiltsev (1998) Non-linear low energ electron lenses suggested, FNAL-FN-0671 Chow, Car (1994) ü Nonlinear Integrable Optics: Danilov and Nagaitsev solution for nonlinear lattice with invariants of motion that can be implemented with Laplacian potential, i.e. with special magnets Phs. Rev. ST Accel. Beams 13, (010) 8 A. Valishev NOCE017 - Nonlinear Integrabilit 09/1/017

9 What is Common to These Solutions? One begins from a conventional linear lattice accelerator (albeit speciall designed and carefull controlled) and then adds the special nonlinear element (Laplacian or E-Lens) and to make the lattice -D integrable. Does not require much new technolog a significant portion of accelerator circumference is made with common quadrupoles and dipoles 9 A. Valishev NOCE017 - Nonlinear Integrabilit 09/1/017

10 D Generalization of McMillan Mapping Thin lens 1D thin lens kick p i x i = = -x i-1 p i-1 + f ( x ) D a thin lens solution can be carried over to D case in axiall smmetric sstem i f ( x) Ax p Bx + Dx - Ax + Bx + C = ( x p + xp ) + C( x + p ) + = const + B Dxp 1. The ring with transfer matrix æ 0 b 0 0ö ç ç æ ci si öç b ç ç è-si ci øç b ç 1 ç è b ø c = cos( f) s = sin( f) æ1 0ö I = ç è0 1ø. Axiall-smmetric thin kick kr q () r = ar + 1 can be created with electron lens 10 A. Valishev NOCE017 - Nonlinear Integrabilit 09/1/017

11 D Generalization of McMillan Mapping The sstem possesses two integrals of the motion (transverse): Angular momentum: xp - p = const McMillan-tpe integral, quadratic in momentum x 4 4 px i 0 p i x i i For large amplitudes, the fractional tune is 0.5 For small amplitude, the electron (defocusing) lens can give a tune shift of 0.5 per cell! 11 A. Valishev NOCE017 - Nonlinear Integrabilit 09/1/017

12 D Generalization of McMillan Mapping Q A Q x A x Numerical tracking + Frequenc Map Analsis All excited resonances have the form k (Q x + Q ) = m. The do not cross each other, so there are no stochastic laers and diffusion 1 A. Valishev NOCE017 - Nonlinear Integrabilit 09/1/017

13 Implementation with Electron Lens Low-energ electron beam with nonlinear densit, confined in strong longitudinal magnetic field interacts with circulating beam Used at Tevatron and RHIC 13 A. Valishev NOCE017 - Nonlinear Integrabilit 09/1/017

14 Another Class of Solutions 1. Remove time dependence from Hamiltonian thus making it an integral of the motion One integral alread gives a better degree of regularit in the motion (round colliding beams, 1/- integer working point in colliders, crab-crossing at DAFNE). Shape the nonlinear potential to find a second integral 14 A. Valishev NOCE017 - Nonlinear Integrabilit 09/1/017

15 Time-Independent Hamiltonian 09/1/017 A. Valishev NOCE017 - Nonlinear Integrabilit 15 Start with a Hamiltonian Choose s-dependence of the nonlinear potential such that H is time-independent in normalized variables This results in H being the integral of motion Note there was no requirement on V can be made with an conventional magnets, i.e. octupoles ( ) ),, ( s x V x s K p p H x + ø ö ç ç è æ = H N = p xn + p N + x N + N +U x N, N,ψ ( ), ) ( ) ( ) (, ) ( s z s s p p s z z N N b b b b - = = ( ) ) ( ), ( ), ( ) ( b b b s x V x p p H N N N N N xn N =

16 Recipe for Time-Independent Hamiltonian 1 Start with a round axiall-smmetric linear lattice (FOFO) with the element of periodicit consisting of a. Drift L b. Axiall-smmetric focusing block T-insert with phase advance n p Add special nonlinear potential V(x,,s) in the drift Tue Feb 14:40: OptiM - MAIN: - C:\Users\nsergei\Documents\Papers\Invariants\Round lens\quad line1.opt BETA_X&Y[m] 0 DISP_X&Y[m] BETA_X BETA_Y DISP_X DISP_Y A. Valishev NOCE017 - Nonlinear Integrabilit 09/1/017

17 Axiall Smmetric Time-Independent Hamiltonian Linear optical sstem consisting of a. Section with a constant beta function. For a 150-MeV electron beam, a solenoid 0.5T gives -m beta functions (constant over the solenoid length, L) b. Matched focusing block T-insert with a phase advance of n π (+ arbitrar rotation) The fractional tunes are: Q 0 = L πβ or L πβ Add axiall-smmetric nonlinear potential in the drift (electron lens) V(x,, s) = V(x, ) ß(s) æ 1 b(s) ç - k ç ç 0 ç è k L 0ö 0 0 1ø T insert æ ö ç ç ç ç è ø s 17 A. Valishev NOCE017 - Nonlinear Integrabilit 09/1/017

18 Axiall Smmetric Time-Independent Hamiltonian The sstem is integrable. Two integrals of motion (transverse): Angular momentum: xp - p = const The total transverse energ (Hamiltonian is time-independent). A ver interesting case is near (just above) the integer resonance x The sstem can lose linear (small amplitude) stabilit but retain the large amplitude stabilit 18 A. Valishev NOCE017 - Nonlinear Integrabilit 09/1/017

19 Axiall Smmetric Time-Independent Hamiltonian Q A Q x A x The footprint with one defocusing lens crosses integer resonance 19 A. Valishev NOCE017 - Nonlinear Integrabilit 09/1/017

20 Henon-Heiles Tpe Sstems For example, build V with Octupoles Onl one integral of motion H Tune spread limited to ~1% of Q 0 A. Valishev NOCE017 - Nonlinear Integrabilit 0 ( ) ( ) ø ö ç ç è æ - + = 3 4 4,, x x s s x V b k ( ) ) ( 1 ) ( 1 x x k x p p H x = ø ö ç ç è æ - + = N N N N x x U k 09/1/017

21 Henon-Heiles Tpe Sstem with Octupoles Q A Q x A x While dnamic aperture is limited, the attainable tune spread is large ~0.03 compare to created b LHC octupoles 1 A. Valishev NOCE017 - Nonlinear Integrabilit 09/1/017

22 Special Potential Second Integral of Motion Find potentials that result in the Hamiltonian having a second integral of motion quadratic in momentum All such potentials are separable in some variables (cartesian, polar, elliptic, parabolic) First comprehensive stud b Gaston Darboux (1901) Darboux equation ( ) + x + c x U xx U ( )U x + 3U x 3xU = 0 General solution was found, which satisfies the Laplace equation (Phs. Rev. ST Accel. Beams 13, 08400, 010) A. Valishev NOCE017 - Nonlinear Integrabilit 09/1/017

23 Stabilit of Nonlinear Lattices Nonlinear sstems can be more stable! 1D sstems: non-linear (unharmonic) oscillations can remain stable under the influence of periodic external force perturbation. Example: D: The resonant conditions z!!+w 0 sin( z) = asin( w0t) are valid onl for certain amplitudes. kw ( J, J ) + lw ( J, J ) = m Nekhoroshev s condition guaranties detuning from resonance and, thus, stabilit. An Exponential Estimate of the Time of Stabilit of Nearl- Integrable Hamiltonian Sstems. Russian Math. Surves 3:6 (1977) from Uspekhi Mat. Nauk 3:6 (1977) 3 A. Valishev NOCE017 - Nonlinear Integrabilit 09/1/017

24 Stabilit of Nonlinear Lattices Proposed solutions have ver tight tolerances on the implementation Linear maps (1% in b, in phase) Nonlinear potential (3D implementation) We perform rigorous studies of stabilit to Chromatic aberrations Nonlinearities and other imperfections in the arcs Effect of 3 rd degree of freedom Space charge Some implementations are more stable Elliptic potential does not satisf Nekhoroshev s criterion Hamiltonian for Electron lens is steep 4 A. Valishev NOCE017 - Nonlinear Integrabilit 09/1/017

25 Investigation of kinematic nonlinearities and nd /3 rd -order dipole effects in the IOTA ring (nonlinear insert ON) IOTA Lattice (.5 MeV p) nonlinear inserts ON L=1.8 m, t=0.45, c = m 1/, µ 0 =0.3 Matched nonlinear KV beam ε 0 = 3.9 mm-mrad, σ δ = 0 Rapid mixing over the first -3 turns, followed b slow diffusion. H/hHi [%] H/hHi [%] Diffusion of the H invariant (IMPACT-Z) phase mixing first 50 turns 3 rd order smplectic dipoles w/ exact drifts linear smplectic tracking w/ exact drifts linear smplectic tracking Slow relative growth in <H> ~ 0.15%. - constant if H invariant is preserved - zero for a nonlinear KV distribution Kinematic effects plateau at 0.%. The corresponding value of σ I /<I> (for the second invariant) grows b 1% over 8000 turns. Expected to be less problematic for a large ring such as the proposed Rapid Ccling Snchrotron. 5 A. Valishev NOCE017 - Nonlinear Integrabilit 09/1/017

26 Proof of Principle Experiments IOTA We are building the Integrable Optics Test Accelerator To test the Nonlinear Integrable Optics To become a machine for proof-of-principle R&D 6 A. Valishev NOCE017 - Nonlinear Integrabilit 09/1/017

27 IOTA Phsics Drivers 1. Nonlinear Integrable Optics Experimental demonstration of NIO lattice in a practical accelerator. Space Charge Compensation Suppression of SC-related effects in high intensit circular accelerators Nonlinear Integrable Optics Electron lenses Electron columns Circular betatron modes 3. Optical Stochastic Cooling Proof-of-principle demonstration 4. Beam collimation Technolog development for hollow electron beam collimation 5. Electron Cooling Advanced techniques Laser-Plasma Accelerator Demonstration of injection into snchrotron Quantum Phsics Localization of single electron wave function 7 A. Valishev NOCE017 - Nonlinear Integrabilit 09/1/017

28 Fermilab Accelerator Science and Technolog Facilit FAST Schematic not to scale! 8 A. Valishev NOCE017 - Nonlinear Integrabilit 09/1/017

29 IOTA Laout 9 A. Valishev NOCE017 - Nonlinear Integrabilit 09/1/017

30 IOTA Laout 30 A. Valishev IOTA Status 6/6/017

31 IOTA Laout 31 A. Valishev IOTA Status 6/6/017

32 Nonlinear Magnet Joint effort with RadiaBeam Technologies (Phase I and II SBIR) FNAL Concept: -m long nonlinear magnet RadiaBeam full 1.8-m magnet designed, fabricated and delivered to IOTA in Phase II 3 A. Valishev NOCE017 - Nonlinear Integrabilit 09/1/017

33 IOTA Goals for Integrable Optics The IOTA experiment has the goal to demonstrate the possibilit to implement nonlinear integrable optics with a large betatron frequenc spread DQ>1 and stable particle motion in a realistic accelerator design 33 A. Valishev NOCE017 - Nonlinear Integrabilit 09/1/017

34 IOTA Staging Phase I Phase I will concentrate on the academic aspect of single-particle motion stabilit using e- beams Achieve large nonlinear tune shift/spread without degradation of dnamic aperture b painting the accelerator aperture with a pencil beam Suppress strong lattice resonances = cross the integer resonance b part of the beam without intensit loss Investigate stabilit of nonlinear sstems to perturbations, develop practical designs of nonlinear magnets The measure of success will be the achievement of high nonlinear tune shift = A. Valishev NOCE017 - Nonlinear Integrabilit 09/1/017

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36 Collaboration Toda 5 Partners: ANL, Berkele, BNL, BINP, CERN, Chicago, Colorado State, IAP Frankfurt, JINR, Kansas, LANL, LBNL, ORNL, Marland, Universidad de Guantajuato Mexico, Michigan State, NIU, Oxford, RadiaBeam Technologies, RadiaSoft LLC, Tech-X, Tennessee, Vanderbilt 013 NIU-FNAL: Joint R&D Cluster Vladimir Shiltsev IOTA/FAST Meeting, 06/06/17 015

37 Training and Universit Collaboration Excellent connection to the universit communit through the Joint Fermilab/Universit PhD program Alread 9 graduate students doing thesis research at FAST/IOTA Opportunities for summer internships 37 A. Valishev NOCE017 - Nonlinear Integrabilit 09/1/017

38 Summar IOTA/FAST offers a vast and unique program in accelerator science IOTA/FAST experiments are a great opportunit to explore something trul novel with circular accelerators IOTA/FAST will be a strong driver of national and international collaboration and training 38 A. Valishev NOCE017 - Nonlinear Integrabilit 09/1/017

39 IOTA Parameters Nominal kinetic energ e - : 150 MeV, p+:.5 MeV Nominal intensit e - : , p+: Circumference 40 m Bending dipole field 0.7 T Beam pipe aperture 50 mm dia. Maximum b-function (x,) 1, 5 m Momentum compaction Betatron tune (integer) 3 5 Natural chromaticit Transverse emittance r.m.s. e - : 0.04 µm, p+: µm SR damping time 0.6s ( turns) RF V,f,q e - : 1 kv, 30 MHz, 4 Snchrotron tune e - : Bunch length, momentum spread e - : 1 cm, A. Valishev NOCE017 - Nonlinear Integrabilit 09/1/017

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