Linear and Nonlinear Resonances

• Linear coupling resonances – skew quadrupoles; solenoids; vertical closed orbit in sextupoles

• Parametric Resonances: mν=ℓ, ℓ=integer.• Coupling resonances: Order of resonance = |m|+|n|

Sum resonances: mνx+nνz=ℓDifference resonances: mνx-nνz=ℓ

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octupole; skew : octupole, : sextupole; skew : sextupole, :

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For hs=1 or =, one obtains

Here B0 is the reference magnetic field, and r0 is the reference radius of the magnet aperture. The field quality is expressed in terms of the normal and skew harmonic coefficients, bn and an in unit of 10−4.

Note that the Hamiltonian Ĥ is s-dependent, and the Hamiltonian has different value at different locations. We can remove this flutter of Hamiltonian by making a Canonical transformation, and by employing the orbital angle θ=s/R as the independent coordinate.

),,()(21'

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The new Hamiltonian becomes JJJR

JRs

FHRH

1~ 2

s dss0

)(

0 ,

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ddJ

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dd

The conjugate phase space coordinates are

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zzz

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VI Linear CouplingThe vector potentials for skew quadrupoles and solenoids are

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/1)1(~ 22

Let (x, px/p; z, pz/p) be the conjugate phase space coordinates. The betatronmotion can be derived from the linearized Hamiltonian:

Floquet transformation to the linear coupling potential:

Here (Jx, x) and (Jz, z) are pairs of conjugate phase-space coordinates, and νx, νzare betatron tunes. Since Vlc(s) is a periodic function of s, it can be expanded in Fourier harmonics as

If the linear coupling kernel Alc∓ satisfies a periodic condition similar to that in a synchrotron with P superperiods, the resonance coupling coefficient G1,∓1,ℓ will be zero unless ℓ is an integer multiple of P. If ℓ is an integer multiple of P, each superperiod contributes additively to the linear coupling resonance strength. This is called the systematic linear coupling resonance. For example, since the superperiodicity of the LEP lattice is 8, the difference between the integer part of the horizontal and vertical betatron tunes should not be 0, 8, 16, · ·, to minimize the effect of the systematic linear coupling resonance. The strength of the linear coupling resonance due to random errors such as quadrupole roll and vertical closed orbit in sextupoles is smaller. It occurs at all integer ℓ. Near a difference linear coupling resonance, the horizontal and vertical betatron motions are coupled. The coupling resonance can cause beam size increase and decrease the beam lifetime. Thus the linear-coupling resonance-strength should be minimized, and the resonance strength is usually small.

A. Effective Hamiltonian for a Single Linear Coupling Resonance

The betatron tunes are separated by λ, and the minimum separation between the normal mode tunes is |G1,−1,ℓ|. The graph shows an example of measured betatron tunes vs the strength of one quadrupole at the IUCF cooler ring across a linear coupling resonance νx−νz+1≈0 (ℓ = −1 for the IUCF cooler).

Near an isolated coupling resonance νx − νz = ℓ, the linear coupling Hamiltonian in action-angle phase space coordinates, is approximately

The Hamiltonian Eq. (2.220) corresponds to two coupled linear oscillators, which can be expanded in terms of two normal modes with tunes

The measured coherent betatron oscillations excited by a horizontal kicker. The linear coupling gives rise to beating between the horizontal and vertical betatron oscillations. The tune of beating is equal to λ=((νx−νz)2+|G1,−1,ℓ|2)½. The beat period were measured to be about 120 revolutions, which corresponds to λ ≈ 0.0083.

B. Resonance precessing frame and Poincar´e surface of section

Transform the Hamiltonian into a “resonant precessing frame” by using the generating function

zz

xx

FJFJJF

JF

22

2

22

1

21 , , ,

where δ1 = νx−νz−ℓ is the resonance proximity parameter. The system is integrable with two invariants J2 and H1 = E1. Since J2 is invariant, we obtain

For a given J2, all tori can be described by a single parameter E1 that is determined from the initial condition. The particle motion in the resonant precessing frame is determined completely by the condition of a constant J2and a constant Hamiltonian value H1(J1, ϕ1, J2) = E1.

C. Initial horizontal orbitWe first consider a simple orbit with “energy” E1=δ1J2, which corresponds to an initial horizontal betatron action with J1,max=J2. The particle trajectory satisfies H1=δ1J2. Expressed in terms of the normalized coordinates: Q=√2J1cosφ1, P=−√2J1sinφ1, we can decompose the trajectories in two ellipses: a Courant Synder circle and a coupling ellipse:

J.Y. Liu et al., Phys. Rev. E49, 2347 (1994).

VII Nonlinear Resonances

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The Poincar´e maps for betatron motion perturbed by a single sextupole magnet at a tune below (left) and above (right) a third order resonance. The integrated sextupole strength is S=0.5m−2 with lattice parameters βx=20m, and αx=0. Arrows indicate directions of motion near a separatrix.

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B. The leading order resonances driven by sextupoles

The third order resonance at 3νx = ℓThe Hamiltonian near a third-order resonance at 3νx = ℓ is

where G3,0,ℓ is the resonance strength, Jx, ϕx are conjugate phase-space coordinates, θ is the orbiting angle serving time coordinate, νx is the horizontal betatron tune.

Transform the phase space coordinate to a resonance rotating frame with a generating function to obtain new phase-space coordinates:

The new Hamiltonian and Hamilton’s equations of motion are

where δ=νx−ℓ/3 is the resonance proximity parameter.

The fixed points (FPs) of the Hamiltonian are determined by dJ/dθ=0 and dϕ/dθ=0. Without nonlinear detuning, there is no stable fixed point for the third order resonance. The action and Hamiltonian value at the UFP, and small amplitude motion near the UFP are

The motion near the fixed point is hyperbolic. Because of nonlinear term in the Equation above, the amplitude will grow faster than an exponential. The direction of particle motion near a separatrix is marked with arrows in the Figure.

Without a nonlinear detuning term, the third-order resonance appears at all values of δ. The stable motion is bounded by the curve of J1/2

UFP. For a given aperture Jmax the width of the third-order betatron resonance is

SeparatrixThe separatrix is the Hamiltonian torus that passes through the UFP, i.e. H = EUFP. The separatrix orbit, for δ/G3,0,ℓ > 0,

Nonlinearity in accelerators has been employed to provide • Beam manipulations such as slow extraction, beam dilution• Landau damping for collective beam instabilities• Overcoming spin depolarization resonances

Nonlinear detuning parameters: Accelerator magnets may have many nonlinear magnetic multipoles. Some of them can introduce nonlinear perturbation to betatron motion, e.g.

With Floquet transformation, the Hamiltonian becomes

The coefficients α’s are called nonlinear detuning parameters

Betatron detuning:

+…..

chromaticity

octupole

sextupole

The bifurcation of third-order resonance islands occurs at 16αδ ≤ 9G3,0,ℓ

2. The Figure shows αJUFP

1/2/|G3,0,ℓ| vs αδ/G3,0,ℓ2

for the bifurcation of third-order resonance.

Effect of nonlinear detuningNonlinear magnetic multipoles also generate nonlinear betatron detuning, i.e. the betatron tunes depend on the betatron actions. Including the effect of nonlinear betatron detuning, the Hamiltonian near a third-order resonance is

With nonlinear detuning, stable fixed points appear. The fixed points of the Hamiltonian for á > 0 and G > 0 are

Sextupole 3rd resonance

Nonlinear beam dynamics on resonance crossing It appears that sextupoles will not produce resonances higher than the third order

ones listed esarlier. However, strong sextupoles are usually needed to correct chromatic aberration. Concatenation of strong sextupoles can generate high-order resonances such as 4νx, 2νx±2νz, 4νz, 5νx, . . . , etc. The Figure below shows the Poincar´e maps of the single sextupole model at νx=3.7496 and νx=3.795, i.e. a single sextupole can also drive the fourth and higher order resonances. One can use a canonical perturbation method to explain the tracking result. Since resonance islands only exist with νx<3.75 or νx<3.8, the effective nonlinear detuning must be positive. The largest phase space map marks the boundary of stable motion.

Near a weak fourth-order 1D resonance, the Hamiltonian can normally be approximated by

The solid lines are the Hamiltonian tori with parameters αxx=650(πm)−1, G4,0,15=80(πm)−1, and νx−3.75=−7.8×10−4.

The betatron phase space can be visualized as a space filled by invariant tori, even near a nonlinear resonance.For a difference resonance, the invariant is bounded!

νx–2 νz=ℓ

• The studies of sum resonances are not as successful. We have constructed a tune jump quadrupole to move betatron tunes onto a sum resonance x+2z and observed betatron amplitude growth obeying the invariant at the resonance. However, the data is not in excellent quality.

• Take 2x+2z resonance as an example, we expect to see particle loss through tori as shown in the graph below. This means that the betatron phase space is filled with resonance lines, where particles that locked onto a resonance will leak out to a large amplitude betatron motion through these resonance tori. The invariant tori are unbounded for sum resonances!

• Experiments has yet to be carried out!

Linear resonances

Resonances up to 4th order

Up to 8th order resonances

Space charge resonances in high power accelerators Almost all accelerators encounter one kind of nonlinear resonance or another! These resonances can limit dynamical aperture, affect beam life time, and cause other effects. At the same time, nonlinearity in beam has many useful applications such as Landau damping, beam manipulations, etc.

If the accelerator has a superperiod P and the sextupole field satisfies a similar periodic condition, the resonance strength G3,0,ℓ is zero unless ℓ is an integer multiple of P. For example, the systematic third-order resonance strength for the AGS will be zero except for ℓ = 12, 24, etc. Thus the nonlinear resonances are classified into systematic and random resonances. Systematic nonlinear resonances are located at ℓ = P×integer. At a systematic resonance, the contribution of each superperiod is coherently additive to the resonance strength. Since the chromatic sextupoles are usually arranged according to the superperiod of the machine, one should pay great attention to the systematic sextupolar nonlinear resonances. Random sextupole fields induce nonlinear resonances at all integer ℓ, and their resonance strengths are usually weak. Nevertheless, the betatron tunes shouldavoid low-order nonlinear resonances.

System and random resonances