Integrable Systems for Accelerators Sergei Nagaitsev Feb 7, 2013

Challenges of modern accelerators (the LHC case)  LHC: 27 km, 7 TeV per beam  The total energy stored in the magnets is HUGE: 10 GJ (2,400 kilograms of TNT)  The total energy carried by the two beams reaches 700 MJ (173 kilograms of TNT)  Loss of only one ten-millionth part (10 −7 ) of the beam is sufficient to quench a superconducting magnet  LHC vacuum chamber diameter : ~40 mm  LHC average rms beam size (at 7 TeV): 0.14 mm  LHC average rms beam angle spread: 2 µrad  Very large ratio of forward to transverse momentum  LHC typical cycle duration: 10 hrs = 4x10 8 revolutions  Kinetic energy of a typical semi truck at 60 mph: ~7 MJ S. Nagaitsev, Feb 7,

What keeps particles stable in an accelerator?  Particles are confined (focused) by static magnetic fields in vacuum.  Magnetic fields conserve the total energy  An ideal focusing system in all modern accelerators is nearly integrable  There exist 3 conserved quantities (integrals of motion); the integrals are “simple” – polynomial in momentum.  The particle motion is confined by these integrals. S. Nagaitsev, Feb 7, particle’s action

Ideal Penning trap S. Nagaitsev, Feb 7,  An ideal Penning trap is a LINEAR and integrable system  It is a harmonic 3-d oscillator

Kepler problem – a nonlinear integrable system S. Nagaitsev, Feb 7,  Kepler problem:  In spherical coordinates:  Example of this system: the Solar system

Other famous examples of integrable systems  Examples below can be realized with electro-magnetic fields in vacuum  Two fixed Coulomb centers – Euler (~1760)  separable in prolate ellipsoidal coordinates  Starting point for Poincare’s three-body problem  Vinti potential (Phys. Rev. Lett., Vol. 3, No. 1, p. 8, 1959)  separable in oblate ellipsoidal coordinates  describes well the Earth geoid gravitational potential S. Nagaitsev, Feb 7,

7 Non-integrable systems  At the end of 19 th century all dynamical systems were thought to be integrable.  1885 math. prize for finding the solution of an n-body problem (n>2)  However, nonintegrable systems constitute the majority of all real-world systems (1 st example, H. Poincare, 1895)  The phase space of a simple 3-body system is far from simple. This plot of velocity versus position is called a homoclinic tangle.

S. Nagaitsev, Feb 7, Henon-Heiles paper (1964)  First general paper on appearance of chaos in a Hamiltonian system.  There exists two conserved quantities  Need 3 for integrability  For energies E > trajectories become chaotic  Same nature as Poincare’s “homoclinic tangle” Michel Henon (1988):

Particle motion in static magnetic fields  For accelerators, there are NO useful exactly integrable examples for axially symmetric magnetic fields in vacuum:  Example 1: Uniform magnetic field  Example 2: Field of a magnetic monopole  Until 1959, all circular accelerators relied on approximate (adiabatic) integrability.  These are the so-called weakly-focusing accelerators  Required large magnets and vacuum chambers to confine particles; S. Nagaitsev, Feb 7,

10 The race for highest beam energy  Cosmotron (BNL, ): 3.3 GeV Produced “cosmic rays” in the lab Diam: 22.5 m, 2,000 ton  Bevatron (Berkeley, 1954): 6.3 GeV Discovery of antiprotons and antineutrons: 1955 Magnet: 10,000 ton  Synchrophasatron (Dubna,1957): 10 GeV Diam: 60 m, 36,000 ton Highest beam energy until 1959

Strong Focusing S. Nagaitsev, Feb 7,

CERN PS  In Nov 1959 a 28-GeV Proton Synchrotron started to operate at CERN  3 times longer than the Synchrophasatron but its magnets (together) are 10 times smaller (by weight) S. Nagaitsev, Feb 7,

S. Nagaitsev, Feb 7, Strong Focusing – our standard method since 1952 s is “time” piecewise constant alternating-sign functions Particle undergoes betatron oscillations Christofilos (1949); Courant, Livingston and Snyder (1952)

Strong focusing S. Nagaitsev, Feb 7,  Focusing fields must satisfy Maxwell equations in vacuum  For stationary fields: focusing in one plane while defocusing in another  quadrupole:  However, alternating quadrupoles results in effective focusing in both planes Specifics of accelerator focusing:

S. Nagaitsev, Feb 7, Courant-Snyder invariant Equation of motion for betatron oscillations Invariant (integral) of motion, a conserved qty.

S. Nagaitsev, Feb 7, Simplest accelerator focusing elements  Drift space: L – length  Thin quadrupole lens:

S. Nagaitsev, Feb 7, Simple periodic focusing channel (FODO)  Thin alternating quadrupole lenses and drift spaces  Let’s launch a particle with initial conditions x and x’  Question:  Is the particle motion stable (finite)? L F L D L F L D L F L D L F …Equivalent to: particle (x, x') s L D D

S. Nagaitsev, Feb 7, Particle stability in a simple channel  Possible answers: A.Always stable B.Stable only for some initial conditions C.Stable only for certain L and F L F L D L F L D L F L D particle (x, x') s D

S. Nagaitsev, Feb 7, Particle stability in a simple channel  Correct answer: A.Always stable B.Stable only for some initial conditions C.Stable only for certain L and F L F L D L F L D L F L D particle (x, x') s D Stability:

Phase space trajectories: x’ vs. x S. Nagaitsev, Feb 7, F = 0.49, L = 1 7 periods, unstable traject F = 0.51, L = 1 50 periods, stable traject. When this simple focusing channel is stable, it is stable for ALL initial conditions ! F = 1.2, L = periods stable traject. x x’ x …And, ALL particles are isochronous: they oscillate with the same frequency (betatron tune)! -- Courant-Snyder invariants describe phase-space ellipses

S. Nagaitsev, Feb 7, Report at HEAC 1971 Landau damping  Landau damping – the beam’s “immune system”. It is related to the spread of betatron oscillation frequencies. The larger the spread, the more stable the beam is against collective instabilities.  The spread is achieved by adding special magnets -- octupoles  External damping (feed-back) system – presently the most commonly used mechanism to keep the beam stable.

S. Nagaitsev, Feb 7, Most accelerators rely on both LHC: Has a transverse feedback system Has 336 Landau Damping Octupoles  Octupoles (an 8-pole magnet):  Potential:  cubic nonlinearity (in force)

Let’s add a weak octupole element… S. Nagaitsev, Feb 7, L F L D L F L D L F L D particle (x, x') s D add a cubic nonlinearity

S. Nagaitsev, Feb 7, The result of a nonlinear perturbation:  Betatron oscillations are no longer isochronous:  The frequency depends on particle amplitude  Stability depends on initial conditions  Regular trajectories for small amplitudes  Resonant islands (for larger amplitudes)  Chaos and loss of stability (for large amplitudes)

Example 2: beam-beam effects  Beams are made of relativistic charged particles and represent an electromagnetic potential for other charges  Typically:  0.001% (or less) of particles collide  % (or more) of particles are distorted S. Nagaitsev, Feb 7,

Beam-beam effects  One of most important limitations of all past, present and future colliders S. Nagaitsev, Feb 7, Beam-beam Force Luminosity beam-beam

Example 3: electron storage ring light sources  Low beam emittance (size) is vital to light sources  Requires Strong Focusing  Strong Focusing leads to strong chromatic aberrations  To correct Chromatic Aberrations special nonlinear magnets (sextupoles) are added S. Nagaitsev, Feb 7, dynamic aperture limitations lead to reduced beam lifetime

S. Nagaitsev, Feb 7, Summary so far…  The “Strong Focusing” principle, invented in 1952, allowed for a new class of accelerators to be built and for many discoveries to be made, e.g.:  Synchrotron light sources: structure of proteins  Proton synchrotrons: structure of nuclei  Colliders: structure of elementary particles  However, chaotic and unstable particle motion appears even in simplest examples of strong focusing systems with perturbations  The nonlinearity shifts the particle betatron frequency to a resonance (nω = k)  The same nonlinearity introduces a time-dependent resonant kick to a resonant particle, making it unstable. The driving term and the source of resonances simulteneosly

Linear vs. nonlinear  Accelerators are linear systems by design (freq. is independent of amplitude).  In accelerators, nonlinearities are unavoidable (SC, beam-beam) and some are useful (Landau damping).  All nonlinearities (in present rings) lead to resonances and dynamic aperture limits.  Are there “magic” nonlinearities with zero resonance strength?  The answer is – yes (we call them “integrable”) 3D: S. Nagaitsev, Feb 7,

Our research goal S. Nagaitsev, Feb 7,  Our goal is to create practical nonlinear accelerator focusing systems with a large frequency spread and stable particle motion.  Benefits:  Increased Landau damping  Improved stability to perturbations  Resonance detuning

Nonlinear systems can be more stable!  1D systems: non-linear (unharmonic) oscillations can remain stable under the influence of periodic external force perturbation. Example:  2D: The resonant conditions are valid only for certain amplitudes. Nekhoroshev’s condition guaranties detuning from resonance and, thus, stability. S. Nagaitsev, Feb 7,

Tools we use  Analytical  Many examples of integrable systems exist; the problem is to find examples with constraints (specific to accelerators)  Help from UChicago welcome  Numerical  Brute-force particle tracking while varying focusing elements;  Evolution-based (genetic) optimization algorithms;  Spectral analysis, e.g. Frequency Map Analysis;  Lyapunov exponent  Experimental  Existing accelerators and colliders;  A proposed ring at Fermilab: IOTA (Integrable Optics Test Accelerator) S. Nagaitsev, Feb 7,

Frequency Map Analysis (FMA) S. Nagaitsev, Feb 7, LHC  *=15cm  =590  rad  p/p=0,  z =7.5cm LHC head-on collisions (D Shatilov, A.Valishev) FMA

Advanced Superconductive Test Accelerator at Fermilab S. Nagaitsev, Feb 7, IOTA ILC Cryomodules Photo injector

Integrable Optics Test Accelerator S. Nagaitsev, Feb 7, Beam from linac

IOTA schematic  pc = 150 MeV, electrons (single bunch, 10^9)  ~36 m circumference  50 quadrupoles, 8 dipoles, 50-mm diam vac chamber  hor and vert kickers, 16 BPMs S. Nagaitsev, Feb 7, Nonlinear inserts Injection, rf cavity

Why electrons?  Small size (~50 um), pencil beam  Reasonable damping time (~1 sec)  No space charge  In all experiments the electron bunch is kicked transversely to “sample” nonlinearities. We intend to measure the turn-by-turn beam positions as well as synch light to obtain information about phase space trajectories. S. Nagaitsev, Feb 7,

Experimental goals with nonlinear lenses  Overall goal is to demonstrate the possibility of implementing nonlinear integrable optics in a realistic accelerator design  Demonstrate a large tune shift of ~1 without degradation of dynamic aperture  Quantify effects of a non-ideal lens  Develop a practical lens design. S. Nagaitsev, Feb 7,

S. Nagaitsev, Feb 7, On the way to integrability: McMillan mapping  In 1967 E. McMillan published a paper  Final report in This is what later became known as the “McMillan mapping”:  Generalizations (Danilov-Perevedentsev, ) If A = B = 0 one obtains the Courant-Snyder invariant

McMillan 1D mapping  At small x: Linear matrix: Bare tune:  At large x: Linear matrix: Tune: 0.25  Thus, a tune spread of % is possible! S. Nagaitsev, Feb 7, A=1, B = 0, C = 1, D = 2

What about 2D?  How to extend McMillan mapping into 2-D?  Two 2-D examples exist:  Both are for Round Beam optics: xp y - yp x = const 1.Radial McMillan kick: r/(1 + r 2 ) -- Can be realized with an “Electron lens” or in beam-beam head-on collisions 2.Radial McMillan kick: r/(1 - r 2 ) -- Can be realized with solenoids  In general, the problem is that the Laplace equation couples x and y fields of the non-linear lens  The magnetic fields of a dipole and a quadrupole are the only uncoupled example S. Nagaitsev, Feb 7,

Accelerator integrable systems  Two types of integrable systems with nonlinear lenses  Based on the electron (charge column or colliding beam) lens  Based on electromagnets S. Nagaitsev, Feb 7, Major limiting factor! The only known exact integrable accelerator systems with Laplacian fields: Danilov and Nagaitsev, Phys. Rev. RSTAB 2010

Experiments with an electron lens  5-kG, ~1-m long solenoid  Electron beam: ~0.5 A, ~5 keV, ~1 mm radius S. Nagaitsev, Feb 7, Example: Tevatron electron lens 150 MeV beam solenoid Electron lens current density:

Experiment with a thin electron lens  The system consists of a thin (L < β) nonlinear lens (electron beam) and a linear focusing ring  Axially-symmetric thin McMillan lens:  Electron lens with a special density profile  The ring has the following transfer matrix S. Nagaitsev, Feb 7, electron lens

Electron lens (McMillan – type)  The system is integrable. Two integrals of motion (transverse):  Angular momentum:  McMillan-type integral, quadratic in momentum  For large amplitudes, the fractional tune is 0.25  For small amplitude, the electron (defocusing) lens can give a tune shift of ~-0.3  Potentially, can cross an integer resonance S. Nagaitsev, Feb 7,

Practical McMillan round lens S. Nagaitsev, Feb 7, All excited resonances have the form k ∙ ( x + y ) = m They do not cross each other, so there are no stochastic layers and diffusion e-lens (1 m long) is represented by 50 thin slises. Electron beam radius is 1 mm. The total lens strength (tune shift) is 0.3 FMA analysis

Recent example: integrable beam-beam S. Nagaitsev, Feb 7, A particle collides with a bunch (charge distribution) Integrable bunch distribution Gaussian non-integrable

FMA comparison S. Nagaitsev, Feb 7, Integrable Gaussian non-integrable

Main ideas S. Nagaitsev, Feb 7, Start with a time-dependent Hamiltonian: 2.Chose the potential to be time-independent in new variables 3.Find potentials U(x, y) with the second integral of motion and such that ΔU(x, y) = 0 See: Phys. Rev. ST Accel. Beams 13, (2010) Integrable systems with nonlinear magnets

Integrable 2-D Hamiltonians  Look for second integrals quadratic in momentum  All such potentials are separable in some variables (cartesian, polar, elliptic, parabolic)  First comprehensive study by Gaston Darboux (1901)  So, we are looking for integrable potentials such that S. Nagaitsev, Feb 7, Second integral:

Darboux equation (1901)  Let a ≠ 0 and c ≠ 0, then we will take a = 1  General solution ξ : [1, ∞], η : [-1, 1], f and g arbitrary functions Also, to make it a magnet we need satisfy the Laplace equation: S. Nagaitsev, Feb 7,

Nonlinear integrable lens S. Nagaitsev, Feb 7, Multipole expansion : For c = 1 |t| < 0.5 to provide linear stability for small amplitudes For t > 0 adds focusing in x Small-amplitude tune s: This potential has two adjustable parameters: t – strength and c – location of singularities For |z| < c

Transverse forces S. Nagaitsev, Feb 7, FxFy Focusing in xDefocusing in y x y

S. Nagaitsev, Feb 7, Nonlinear Magnet  Practical design – approximate continuously-varying potential with constant cross-section short magnets Quadrupole component of nonlinear field Magnet cross section Distance to pole c 2×c V.Kashikhin 8 14

S. Nagaitsev, Feb 7, m long magnet

Examples of trajectories S. Nagaitsev, Feb 7,

Ideal nonlinear lens  A single 2-m long nonlinear lens creates a tune spread of ~0.25. S. Nagaitsev, Feb 7, FMA, fractional tunes Small amplitudes (0.91, 0.59) Large amplitudes νxνx νyνy

Collaboration with T. Zolkin (grad. student, UChicago)  A focusing system, separable in polar coordinates S. Nagaitsev, Feb 7,

S. Nagaitsev, Feb 7,

Summary  Some first steps toward resonance elimination are already successfully implemented in accelerators (round beams, crab- waist, dynamic aperture increase in light sources)  Next step (and a game-changer) should be integrable accelerator optics.  Examples of fully integrable focusing system exist for first ever implementations (IOTA ring), encouraging simulation results obtained by Tech-X;  More solutions definitely exist – unfortunately, the mathematics is not well-developed for accelerators – it includes solving functional or high order partial differential equations;  Virtually any next generation machine with nonlinearities can profit from resonance eliminations. S. Nagaitsev, Feb 7,

Acknowledgements  Many thanks to my colleagues:  V. Danilov (SNS)  A. Valishev (FNAL)  D. Shatilov (Budker INP)  D. Bruhwiler, J. Cary (U. of Colorado)  S. Webb (Tech-X)  T. Zolkin (U. of Chicago) S. Nagaitsev, Feb 7,

Extra slides S. Nagaitsev, Feb 7,

System: linear FOFO; 100 A; linear KV w/ mismatch Result: quickly drives test-particles into the halo 500 passes; beam core (red contours) is mismatched; halo (blue dots) has 100x lower density arXiv:

System: integrable; 100 A; generalized KV w/ mismatch Result: nonlinear decoherence suppresses halo 500 passes; beam core (red contours) is mismatched; halo (blue dots) has 100x lower density arXiv: