Integrable Systems for Accelerators

Integrable Systems for Accelerators

Sergei NagaitsevFeb 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 = 4x108

revolutions

Kinetic energy of a typical semi truck at 60 mph: ~7 MJ S. Nagaitsev, Feb 7, 2013 2

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 inmomentum.

The particle motion is confined by these integrals.

S. Nagaitsev, Feb 7, 2013 3

1 1 2 2 3 3H J J J 1

2J pdq

-- particle’s action

Ideal Penning trap


An ideal Penning trap is a LINEAR and integrable system It is a harmonic 3-d oscillator

1 1 2 2 3 3H J J J

Kepler problem – a nonlinear integrable system


Kepler problem:

In spherical coordinates:

Example of this system: the Solar system

kVr

2

22 r

mkHJ J J

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



Non-integrable systems At the end of 19th 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 (1st 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.


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 > 0.125 trajectories become chaotic

Same nature as Poincare’s “homoclinic tangle”

Michel Henon (1988): 2 2 2 31 1( , )

2 3U r z r z r z z

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;



The race for highest beam energy Cosmotron (BNL, 1953-66): 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


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)



Strong Focusing – our standard method since 1952

)()(

0)(0)(

,, sKCsK

ysKyxsKx

yxyx

y

x

s is “time”

13

-- piecewise constant alternating-sign functions

Particle undergoes betatron oscillations

Christofilos (1949); Courant, Livingston and Snyder (1952)

Strong focusing


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

( , , ) 0x y z

2 2( , )x y x y

Specifics of accelerator focusing:


Courant-Snyder invariant

yxzzsKz

or ,0)("

Equation of motion for

betatron oscillations

3

1)( where

sK

221 ( ) ( )

2 ( ) 2zsI z z s z

s

Invariant (integral) of motion,a conserved qty.


Simplest accelerator focusing elements

Drift space: L – length

Thin quadrupole lens:

0

'

'

L

x x x Lx xy y y Ly y

0

'

'F

xx xxx Fy yy yy

F


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)?

LF

LD

LF

LD

LF

LD

L

F

…Equivalent to:

particle(x, x')

s

LD

D


Particle stability in a simple channel Possible answers:

A. Always stable B. Stable only for some initial conditionsC. Stable only for certain L and F

LF

LD

LF

LD

LF

LD

particle(x, x')

s

D


Particle stability in a simple channel Correct answer:

A. Always stable B. Stable only for some initial conditionsC. Stable only for certain L and F

LF

LD

LF

LD

LF

LD

particle(x, x')

s

D

0 2LF

1 11

1 1 0 1 1 0' 0 1 1 0 1 1 'i i

x L L xx F F x

2Trace M Stability:

Phase space trajectories: x’ vs. x


F = 0.49, L = 1

7 periods,unstable traject.

1

23

4

5

6

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 = 11000 periodsstable traject.

10 5 0 5 10

50

0

50

pxj k

xj kx

x’

2 1 0 1 210

5

0

5

10

pxj k

xj kx

x’

20 10 0 10 2010

5

0

5

10

pxj k

xj k

…And, ALL particles are isochronous: they oscillate with the same frequency (betatron tune)! 2

21 ( ) ( )2 ( ) 2z

sI z z s zs

-- Courant-Snyder invariants describe phase-space ellipses


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.


Most accelerators rely on bothLHC:

Has a transverse feedback systemHas 336 Landau Damping Octupoles

Octupoles (an 8-pole magnet):

Potential: cubic nonlinearity (in force)

4 4 2 2( , ) 6x y x y x y

Let’s add a weak octupole element…


LF

LD

LF

LD

LF

LD

particle(x, x')

s

D

1 1 31

1 1 0 1 1 0' 0 1 1 0 1 1 'i i

x L L xx F F x x

add a cubicnonlinearity


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 99.999% (or more) of particles are distorted


Beam-beam effects One of most important limitations of all past,

present and future colliders


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


dynamic aperturelimitations lead to reduced beamlifetime


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, 2013 29

mlk yx

1 2 3( , , )H F J J J

x

y

Our research goal


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.


)sin()sin( 020 tazz

1 1 2 2 1 2( , ) ( , )k J J l J J m

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)


Frequency Map Analysis (FMA)


LHC

*=15cm =590mrad p/p=0, sz=7.5cmLHC head-on collisions (D Shatilov, A.Valishev)

FMA

Advanced Superconductive Test Accelerator at Fermilab


IOTAILC Cryomodules

Photo injector

Integrable Optics Test Accelerator


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


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.


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.



On the way to integrability: McMillan mapping

In 1967 E. McMillan published a paper

Final report in 1971. This is what later became known as the “McMillan mapping”:

Generalizations (Danilov-Perevedentsev, 1992-1995)

)(1

1

iii

ii

xfxppx

CBxAxDxBxxf

2

2

)(

const 222222 DxppxCxppxBpAx

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 50-100% ispossible!


CBxAxDxBxxf

2

2

)(xCDxf )(

CD1

10

CD

2acos

21

0)( xf

01

10

4 2 0 2 4

4

2

0

2

4

pn k

xn k

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: xpy - ypx = const1. Radial McMillan kick: r/(1 + r2) -- Can be realized

with an “Electron lens” or in beam-beam head-on collisions

2. Radial McMillan kick: r/(1 - r2) -- 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


Accelerator integrable systems Two types of integrable systems with nonlinear

lenses Based on the electron (charge column or colliding

beam) lens

Based on electromagnets


2 0U

2 0U Major limitingfactor!

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


Example: Tevatron electron lens

150 MeV beam

solenoid

Electron lens current density:

22( )

1

In rar

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


2( )1

krrar

electronlens

0 0 01 0 0 0

0 0 010 0 0

cI sIsI cI

cos( )sin( )1 00 1

cs

I

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 resonanceS. Nagaitsev, Feb 7, 2013 45

y xxp yp const

4 2 0 2 44

2

0

2

4

pxi

xi

4 2 0 2 44

2

0

2

4

py i

y i

Practical McMillan round lens


All excited resonances have the form k ∙ (x + y) = mThey 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


A particle collides with a bunch (charge distribution)

Integrablebunch distribution

Gaussiannon-integrable

FMA comparison


Integrable

Gaussiannon-integrable

Main ideas


1. 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, 084002 (2010)

),(

22

2222

NNNNyNxN

N yxUyxpp

H

)(,)(,)()(22

2222

syxVyxpp

H NNNNyNxN

N

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


),(22

2222

yxUyxppH yx

),(22 yxDCppBpApI yyxx

,,2,

2

22

axCaxyB

cayA

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 functionsAlso, to make it a magnet we need satisfy theLaplace equation:


033222 yxxyyyxx xUyUUcxyUUxy

22

)()(),(

gfyxU

c

ycxycxc

ycxycx

2

22222

2222

0 yyxx UU

Nonlinear integrable lens


2 2 2 2

( , )2 2

x yp p x yH U x y

Multipole expansion :

For c = 1|t| < 0.5 to provide linear stability for smallamplitudes

For t > 0 adds focusing in x

Small-amplitude tune s:

tt

2121

2

1

This potential has two adjustable parameters:t – strength and c – location of singularities

For |z| < c 2 4 6 82 2 4 6

2 8 16, Im ( ) ( ) ( ) ( ) ...3 15 35

tU x y x iy x iy x iy x iyc c c c

B

Transverse forces


-300

-200

-100

0

100

200

300

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

-50

-40

-30

-20

-10

0

10

20

30

40

50

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

Fx Fy

Focusing in x Defocusing in y

x y


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 c2×c

V.Kashikhin8

14


2-m longmagnet

Examples of trajectories


Ideal nonlinear lens A single 2-m long nonlinear lens creates a tune

spread of ~0.25.


FMA, fractional tunes

Small amplitudes(0.91, 0.59)

Large amplitudes

0.5 1.00.5

1.0

νx

νy

Collaboration with T. Zolkin (grad. student, UChicago)

A focusing system, separable in polar coordinates


2 rH J J J


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.


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)


Extra slides


System: linear FOFO; 100 A; linear KV w/ mismatchResult: quickly drives test-particles into the halo

500 passes; beam core (red contours) is mismatched; halo (blue dots) has 100x lower density

arXiv:1205.7083

System: integrable; 100 A; generalized KV w/ mismatchResult: nonlinear decoherence suppresses halo

500 passes; beam core (red contours) is mismatched; halo (blue dots) has 100x lower density

arXiv:1205.7083

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Integrable Systems for Accelerators