Post on 27-Dec-2015
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) bystatic 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
S. Nagaitsev, Feb 7, 2013 4
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
S. Nagaitsev, Feb 7, 2013 5
Kepler problem:
In spherical coordinates:
Example of this system: the Solar system
kV
r
2
22 r
mkH
J 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
S. Nagaitsev, Feb 7, 2013 6
S. Nagaitsev, Feb 7, 2013 7
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.
S. Nagaitsev, Feb 7, 2013 8
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 3
U 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;
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S. Nagaitsev, Feb 7, 2013 10
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
S. Nagaitsev, Feb 7, 2013 11
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, 2013 12
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Strong Focusing – our standard method since 1952
)()(
0)(
0)(
,, sKCsK
ysKy
xsKx
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
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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:
S. Nagaitsev, Feb 7, 2013 15
Courant-Snyder invariant
yxz
zsKz
or
,0)("
Equation of motion for
betatron oscillations
3
1)( where
sK
221 ( )
( )2 ( ) 2z
sI z z s z
s
Invariant (integral) of motion,a conserved qty.
S. Nagaitsev, Feb 7, 2013 16
Simplest accelerator focusing elements
Drift space: L – length
Thin quadrupole lens:
0
'
'
L
x x x L
x x
y y y L
y y
0
'
'F
xx x
xx Fy y
y yy
F
S. Nagaitsev, Feb 7, 2013 17
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
LD
D
S. Nagaitsev, Feb 7, 2013 18
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
L
F
L
D
L
F
L
D
L
F
L
D
particle(x, x')
s
D
S. Nagaitsev, Feb 7, 2013 19
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
L
F
L
D
L
F
L
D
L
F
L
D
particle(x, x')
s
D
0 2L
F
1 1
1
1 1 0 1 1 0
' 0 1 1 0 1 1 'i i
x L L x
x F F x
2Trace M Stability:
Phase space trajectories: x’ vs. x
S. Nagaitsev, Feb 7, 2013 20
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 z
s
-- Courant-Snyder invariants describe phase-space ellipses
S. Nagaitsev, Feb 7, 2013 21
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, 2013 22
Most accelerators rely on both
LHC: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…
S. Nagaitsev, Feb 7, 2013 23
L
F
L
D
L
F
L
D
L
F
L
D
particle(x, x')
s
D
1 1 3
1
1 1 0 1 1 0
' 0 1 1 0 1 1 'i i
x L L x
x F F x x
add a cubicnonlinearity
S. Nagaitsev, Feb 7, 2013 24
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
S. Nagaitsev, Feb 7, 2013 25
Beam-beam effects
One of most important limitations of all past, present and future colliders
S. Nagaitsev, Feb 7, 2013 26
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, 2013 27
dynamic aperturelimitations lead to reduced beamlifetime
S. Nagaitsev, Feb 7, 2013 28
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
S. Nagaitsev, Feb 7, 2013 30
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, 2013 31
)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)
S. Nagaitsev, Feb 7, 2013 32
Frequency Map Analysis (FMA)
S. Nagaitsev, Feb 7, 2013 33
LHC
b*=15cm q=590mrad Dp/p=0, sz=7.5cmLHC head-on collisions (D Shatilov, A.Valishev)
FMA
Advanced Superconductive Test Accelerator at Fermilab
S. Nagaitsev, Feb 7, 2013 34
IOTA
ILC CryomodulesPhoto injector
Integrable Optics Test Accelerator
S. Nagaitsev, Feb 7, 201335
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, 2013 36
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, 2013 37
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, 2013 38
S. Nagaitsev, Feb 7, 2013 39
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
xfxp
px
CBxAx
DxBxxf
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!
S. Nagaitsev, Feb 7, 2013 40
CBxAx
DxBxxf
2
2
)(xC
Dxf )(
C
D1
10
C
D
2acos
2
1
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 = const
1. 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
S. Nagaitsev, Feb 7, 2013 41
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, 2013 42
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
S. Nagaitsev, Feb 7, 2013 43
Example: Tevatron electron lens
150 MeV beam
solenoid
Electron lens current density:
22( )
1
In r
ar
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, 2013 44
2( )
1
krr
ar
electronlens
0 0 0
10 0 0
0 0 0
10 0 0
cI sI
sI cI
cos( )
sin( )
1 0
0 1
c
s
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 resonance
S. 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
S. Nagaitsev, Feb 7, 2013 46
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
S. Nagaitsev, Feb 7, 2013 47
A particle collides with a bunch (charge distribution)
Integrablebunch distribution
Gaussiannon-integrable
FMA comparison
S. Nagaitsev, Feb 7, 2013 48
Integrable
Gaussiannon-integrable
Main ideas
S. Nagaitsev, Feb 7, 2013 49
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
S. Nagaitsev, Feb 7, 2013 50
),(22
2222
yxUyxpp
H yx
),(22 yxDCppBpApI yyxx
,
,2
,
2
22
axC
axyB
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 functions
Also, to make it a magnet we need satisfy theLaplace equation:
S. Nagaitsev, Feb 7, 2013 51
033222 yxxyyyxx xUyUUcxyUUxy
22
)()(),(
gf
yxU
c
ycxycxc
ycxycx
2
22222
2222
0 yyxx UU
Nonlinear integrable lens
S. Nagaitsev, Feb 7, 2013 52
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:
t
t
21
21
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 iy
c c c c
B
Transverse forces
S. Nagaitsev, Feb 7, 2013 53
-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
S. Nagaitsev, Feb 7, 2013 54
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.Kashikhin
8
14
S. Nagaitsev, Feb 7, 2013 55
2-m longmagnet
Examples of trajectories
S. Nagaitsev, Feb 7, 2013 56
Ideal nonlinear lens
A single 2-m long nonlinear lens creates a tune spread of ~0.25.
S. Nagaitsev, Feb 7, 2013 57
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
S. Nagaitsev, Feb 7, 2013 58
2 rH J J J
S. Nagaitsev, Feb 7, 2013 59
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, 2013 60
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, 2013 61
Extra slides
S. Nagaitsev, Feb 7, 2013 62
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