Effects of systematic octupole coupling resonances

ACCELERATOR, RAY TECHNOLOGY AND APPLICATIONS

Effects of systematic octupole coupling resonances

Hong-Jin Zeng，

Shyh-Yuan Lee，

Shu-Xin Zheng，

Hong-Juan Yao，

Xue-Wu Wang，

Hui Ning，

Xiao-Jun Meng

Nuclear Science and Techniques

Vol.30, No.7

Article number 104

Published in print 01 Jul 2019

Available online 03 Jun 2019

DOI：10.1007/s41365-019-0629-8

34900

The Xi’an Proton Accelerator Facility synchrotron lattice has a systematic fourth-order resonance. The systematic octupole component in dipole magnets is found to have no adverse effect on the dynamic aperture in multiparticle tracking. The frequency map shows particles locked onto the 2νx-2νz=0 resonance. However, we will show that the instantaneous betatron tunes can vary widely around the resonance line for particles locked onto the resonance.

OctupoleCoupling resonanceDynamic apertureXiPAF

1 Introduction

The Xi’an Proton Accelerator Facility (XiPAF) is a 230 MeV high-intensity proton accelerator for medical and radiation-science applications[1]. The design is based on missing dipole FODO cells with six superperiods. The accelerator employs third-order resonant slow extraction for high duty cycle beam operation at the 3νx=5 resonance [2, 3]. The initial betatron tunes are νx=1.76 and νz=1.81 at injection[4], tunable up to a range of ±0.5 for the third-order resonance slow extraction[5]. Figure 1 shows an example of the XiPAF lattice at νx=1.757717 and νz=1.812286. The XiPAF synchrotron has systematic octupole resonances at 4νx=6, 4νz=6, 2νx+2νz=6, and 2νx-2νz=0. The betatron tunes are sufficiently far away from the octupole parametric and sum resonances, except for the systematic octupole difference resonance. The effect of the systematic Montague space-charge resonance [6-8] at 2νx-2νz=0 has been carefully studied at injection and during the beam acceleration [9].

Fig. 1.

The optical functions of the XiPAF lattice in one superperiod at νx=1.757717 and νz=1.812286.

In the optimization of the dipole-magnet design, there is a substantial systematic octupole component in all dipole magnets. This study focuses on the effect of the systematic octupole magnetic field on the dynamic aperture and its effects on the performance of the synchrotron.

The frequency-map analysis [10-13] has become a standard tool for particle-beam tracking and dynamic-aperture analysis. When beam particles encounter a nonlinear resonance, the betatron tunes of these particles seem to clump and stay on the resonance line[14], as emphasized by Schmidt in a recent International Committee for Future Accelerators (ICFA) mini-workshop on dynamic apertures of circular accelerators in 2017. This phenomenon is sometimes known as the devil’s staircase in nonlinear dynamics. In fact, when particles are trapped in a resonance island, they have a resonance tune. However, each particle in the resonance island may be individually modulated by their motion in the resonance island [15, 16]. Although the tune of particles trapped in resonance islands is the resonance tune, their actual turn-by-turn tune is modulated by the "island tune," exhibited as sidebands around the resonant tune.

This study is intended to examine the effect of the systematic octupole resonance at 2νx-2νz=0 on the dynamic aperture of the XiPAF synchrotron and compare the resonant trapping of the coupling resonance line and the actual particle tunes of these particles. We organize this paper as follows. Section 2 studies the effect of the systematic octupole magnetic field on the dynamic aperture. Section 3 details the multiparticle simulations to examine the effect of particles locked onto the coupling resonance. The conclusion is discussed in Section 4.

2 Effect of systematic octupoles on dynamic apertures

In the Frenet–Serret coordinate system, the Hamiltonian for particle motion in the presence of octupoles is

\begin{array}{l} H = \frac{1}{2} {x^{'}}^{2} + \frac{1}{2} K_{x} (s) x^{2} + \frac{1}{2} {z^{'}}^{2} + \frac{1}{2} K_{z} (s) z^{2} + V_{3} (s), \\ V_{3} (s) = \frac{K_{3} (s)}{24} (x^{4} - 6 x^{2} z^{2} + z^{4}), \end{array}

(1)

where K₃(s)=(³ Bz/x³)|_closed-orbit/Bρ, Bz is the vertical component of the magnetic field, and Bρ is the magnetic rigidity of the beam. Because the XiPAF lattice has six-fold symmetry, systematic octupole resonances occur at 4νx=6, 4νz=6, 2νx+2νz=6 and 2νx-2νz=0. The first three resonances are outside the operation range. However, the beam sits near the 2νx-2νz=0 resonance. The K₃(s) parameter in XiPAF can be as high as 40 m^-4 at injection. We study the effect of this coupling resonance at the XiPAF.

The Dynamic Aperture (DA) is normally explored by particle tracking via the magnitude of the "tune diffusion index": $D = \log_{10} [\sqrt{{(Δ ν_{x})}^{2} + {(Δ ν_{z})}^{2}}],$ where Δνx and Δνz are the differences of the betatron tunes evaluated for each particle between the first half and the second half of the tracking turns [10-13]. A stable motion is associated with a small tune diffusion index. The frequency map plots the tune diffusion index vs. the initial particle coordinates (x,z) or the betatron tune space (νx,νz). The resulting frequency map exhibits resonance lines associated with particle loss or high diffusion index in phase-space locations of the beam. In particular, it appears that particles are locked onto these resonance lines [14]. We use the ELEGANT code [17] to perform DA tracking for the XiPAF. The frequency-map analysis is integrated in the ELEGANT code[18-21]. Figure 2 shows the diffusion index vs. the betatron tunes at turn 100 and 1000 at the top, and the diffusion index vs. the horizontal and vertical coordinates at the bottom. Additional supporting material in the form of a.gif movie file is included to show the evolution of the frequency map of particles. The betatron tunes of a zero-betatron-amplitude particle are νx=1.82 and νz=1.85. Although the vacuum chamber is elliptically shaped, we calculate the frequency map for the rectangular area x∈(-50 mm, 50 mm) and y∈(0, 25 mm). We use 15,000 particles uniformly distributed in the area. Systematic dipole errors with K₁=-0.003037 m^-2, K₂=0.0575958 m^-3, and K₃=20 m^-4 are used in the simulation. Because the quadrupole and sextupole errors do not produce systematic resonances, their effects are not important. Only the systematic octupole field is important to the dynamics. There is no particle loss in this simulation. Here, we note that the betatron tunes are trapped on the resonance line with a substantial decrease in the diffusion index. The decrease of diffusion index is not a surprise owing to the trapping of these particles on the resonance line. A supplementary.gif file shows the evolution of the frequency map and diffusion index for every 100 revolutions. It shows the converging of the betatron tune toward the coupling line owing to the resonance and the decrease of diffusion index in the aperture. It would be interesting to know what happens to those particles trapped in the coupling resonance.

Fig. 2.

(Color online) Top plots: Diffusion index vs. the particle tunes at turn number 100 (left), and 1000 (right). Note that the tunes of the particles are trapped on the 2νx-2νz=0 resonance. The betatron tunes of the zero-amplitude particles are νx=1.82 and νz=1.85. Bottom plots: The corresponding diffusion index vs. the horizontal aperture ±50 mm and vertical aperture 0 - 25 mm.

3 Effects of the octupole coupling resonance on beam dynamics

To understand the effect of the octupole resonances, we perform a Floquet transform on the Hamiltonian in Eq. (1) [22]:

\begin{array}{l} x = \sqrt{2 β_{x} J_{x}} \cos (ψ_{x} + χ_{x} (s) - ν_{x} θ), \\ z = \sqrt{2 β_{z} J_{z}} \cos (ψ_{z} + χ_{z} (s) - ν_{z} θ), \end{array}

where (Jx,ψx) and (Jz,ψz) are conjugate action-angle coordinates, νx and νz are betatron tunes, and $χ_{x, z} = \int_{0}^{s} (1 / β_{x, z} (s)) d s$ are betatron phase advances. The resulting Hamiltonian is

\begin{array}{l} H = ν_{x} J_{x} + ν_{z} J_{z} + \frac{1}{2} α_{x x} J_{x}^{2} + α_{x z} J_{x} J_{z} + \frac{1}{2} α_{z z} J_{z}^{2} + \\ G_{2, - 2, l} J_{x} J_{z} \cos (2 ϕ_{x} - 2 ϕ_{z} - l θ + ξ_{2, - 2, l}) + \dots, \end{array}

(2)

where the orbiting angle, θ=s/R, serves as the "time coordinate," R is the mean radius, ≤ll is an integer, αxx, αxz, and αzz are nonlinear detuning parameters, and G_2,-2,≤ll and ξ_2,-2,≤ll are the resonance strength and its phase, given by

\begin{array}{l} α_{x x} = \frac{1}{16 π} \oint β_{x}^{2} K_{3} d s, α_{x z} = \frac{- 1}{8 π} \oint β_{x} β_{z} K_{3} d s, α_{z z} = \frac{1}{16 π} \oint β_{z}^{2} K_{3} d s, \\ G_{2, - 2, l} e^{j ξ_{2, - 2, l}} = \frac{1}{16 π} \oint β_{x} β_{z} K_{3} (s) \times e^{j [2 χ_{x} (s) - 2 χ_{z} (s) - (2 ν_{x} - 2 ν_{z} - l) θ]} d s, \end{array}

where βx and χx are the horizontal betatron function and betatron phase, and βz(s) and χz(s) are those of the vertical betatron motion. Figure 3 shows the detuning parameters and the octupole coupling resonance vs. the integrated octupole field in a dipole of the XiPAF, and the resonance phase is ξ_2,-2,0=60°. To understand the effect of the coupling resonance on beam dynamics, we perform a canonical transformation to this coupling Hamiltonian using the generating function:

Fig. 3.

The detuning parameters and the strength of the octupole coupling resonance for the XiPAF is drawn as a function of the octupole integrated field in the dipole.

F_{2} (ϕ_{x}, ϕ_{z}, J_{1}, J_{2}) = (ϕ_{x} - ϕ_{z} - \frac{1}{2} l θ + \frac{1}{2} ξ_{2, - 2, l}) J_{1} + ϕ_{z} J_{2} .

The coordinate transformation from the old phase-space coordinates to the new phase-space coordinates is

\begin{array}{l} ϕ_{1} = ϕ_{x} - ϕ_{z} - \frac{1}{2} l θ + \frac{1}{2} ξ_{1, - 2, l}, J_{x} = J_{1}, \\ ϕ_{2} = ϕ_{z}, J_{z} = - J_{1} + J_{2}, \end{array}

and the new Hamiltonian becomes $\tilde{H} = H_{1} (J_{1}, ϕ_{1}, J_{2}) + H_{2} (J_{2})$ , where $H_{2} (J_{2}) = ν_{z} J_{2} + \frac{1}{2} α_{22} J_{2}^{2}$ and

\begin{array}{l} H_{1} (J_{1}, ϕ_{1}, J_{2}) = δ J_{1} + \frac{1}{2} α_{11} J_{1}^{2} + α_{12} J_{1} J_{2} + \\ G_{2, - 2, l} J_{1} (J_{2} - J_{1}) \cos (2 ϕ_{1}) . \end{array}

(3)

Here, $δ = ν_{x} - ν_{z} - \frac{1}{2} l$ is the resonance proximity parameter, and the detuning parameters are α₁₁=αxx-2αxz+αzz, α₁₂=αxz-αzz, and α₂₂=αzz. Hamilton’s equations of motion are $\frac{d J_{2}}{d θ} = - \frac{\partial \tilde{H}}{_{2}} = 0,$ $\frac{d ϕ_{2}}{d θ} = \frac{\partial \tilde{H}}{\partial J_{2}},$ and

\frac{d J_{1}}{d θ} = - \frac{\partial \tilde{H}}{\partial ϕ_{1}} = + 2 G_{2, - 2, l} J_{1} (J_{2} - J_{1}) \sin (2 ϕ_{1}), 20 p t

(4)

\begin{array}{l} \frac{d ϕ_{1}}{d θ} = \frac{\partial \tilde{H}}{\partial J_{1}} = {δ + α_{12} J_{2}} + α_{11} J_{1} + \\ G_{2, - 2, l} (J_{2} - 2 J_{1}) \cos (2 ϕ_{1}) . \end{array}

(5)

Particle dynamics obey Eqs. (4) and (5) at constant J₂ and H₁, which are invariant if the bare betatron tunes νx and νz do not vary with time. Particle motion in the horizontal and the vertical planes are coupled as shown in the dynamic aperture study in Sec. 2, where particle tunes are shown to "damp" to the resonance line.

3.1 Fixed points and separatrices

Because the Hamiltonian H₁ is also invariant, each invariant torus in the phase-space coordinates,

X_{1} = \sqrt{2 β_{x} J_{1}} \cos (ϕ_{1}), P_{1} = - \sqrt{2 β_{x} J_{1}} \sin (ϕ_{1}),

(6)

of the resonance rotating frame will have constant H₁ and J₂ values. The tori can be analyzed by the fixed points and separatrix of the Hamiltonian.

The fixed points are determined by solving dJ₁/dθ=0 and dϕ₁/dθ=0:

2 G_{2, - 2, l} J_{1} (J_{2} - J_{1}) \sin (2 ϕ_{1}) = 0,

(7)

\begin{array}{l} {δ + α_{12} J_{2}} + α_{11} J_{1} + \\ G_{2, - 2, l} (J_{2} - 2 J_{1}) \cos (2 ϕ_{1}) = 0. \end{array}

(8)

The Unstable Fixed Points (UFP) are located at

J_{1, U F P} = J_{2}; \cos 2 ϕ_{1, U F P} = \frac{δ + (α_{12} + α_{11}) J_{2}}{G_{2, - 2, l} J_{2}},

(9)

provided that |δ+(α₁₂+α₁₁)J₂| le;|G_2,-2,J₂|. The separatrix is the Hamiltonian torus that passes through the UFP, i.e., J₁=J₂ and

δ + \frac{1}{2} α_{11} (J_{1} + J_{2}) + α_{12} J_{2} - G_{2, - 2, l} J_{1} \cos 2 ϕ_{1} = 0.

(10)

This is an ellipse in the normalized phase-space coordinates (X₁,P₁). The UFP of Eq. (9) is the intersection of the separatrix of Eq. (10) and the Courant-Snyder circle J₁ = J₂.

The Stable Fixed Points (SFP) are located at

δ + α_{12} J_{2} + α_{11} J_{1, S F P} \pm G_{2, - 2, l} (J_{2} - 2 J_{1, S F P}) = 0,

(11)

where the ± corresponds to ϕ_{1, SFP}=0 or π, and π/2 or 3π/4, respectively. Depending on the value of J₁ and J₂ of the beam particles, particles locked onto the resonance will have their own separatrices.

For particles not trapped on the resonance, their ellipses are simply regular Courant-Snyder invariants. Alternatively, for particles trapped on the resonance line, their ellipses are divided into two halves by a coupling ellipse of Eq. (10), where the curvature is determined by α₁₁ and G_2,-2,≤ll. Because particle actions Jx and Jz are coupled, the horizontal and vertical beam emittances can exchange unless the initial emittances are equal: єxi=єzi [23].

3.2 Numerical simulations

To demonstrate the effect of the coupling resonance on beam particles, we track a beam with 50 particles at an accelerator with parameters αxx=727.24 m^-1, αxz=-1256.78 m^-1, αzz = 727.24 m^-1, and |G_2,-2,0|=628.39 m^-1. The bare betatron tunes are δ=νx-νz=-0.02.

Figure 4 shows the "instantaneous tunes" and the normalized phase-space Poincaré maps in the resonance rotating frame of 50 particles every 100 turns. Here, the instantaneous fractional tune is defined as

Fig. 4.

Left: The vertical tune vs. the horizontal tune of 50 particles is plotted for every 100 revolutions for 6100 revolutions. Right: The evolution of the phase-space Poincaré map (X₁,P₁) of these 50 particles.

q_{x} (n) = \frac{ψ_{x} (n) - ψ_{x} (n - 1)}{2 π},

(12)

q_{z} (n) = \frac{ψ_{z} (n) - ψ_{z} (n - 1)}{2 π} .

(13)

Note that there are 10 particles locked onto the coupling resonance and their Poincaré maps have a characteristic resonance structure. Their instantaneous betatron tunes have the specific feature of locking onto the resonance line with tune modulation depending on their amplitude. Although the frequency-map analysis [10-12] put the tunes of these particles on the coupling line, as shown in the top plots of Fig. 2, the instantaneous betatron tunes oscillate around the resonance line, but not on the resonance line.

The right plot of Fig. 4 shows the normalized Poincaré map of these 50 particles at every 100 revolutions for 6100 revolutions. It is clear that particles locked onto resonances having invariant tori separated by their separatrices, as shown in Eq. (10).

4 Conclusion

We performed a dynamic-aperture analysis of the XiPAF in the presence of systematic octupoles in dipole magnets. Although the fourth-order coupling resonance is relatively strong, the coupling resonance will not affect the DA of this compact accelerator. Using the frequency-map analysis, we showed that the tune "damping" to the resonance line is associated with the principle frequency analysis of the software. The betatron tunes of these trapped particles are actually still varying largely turn-by-turn. The apparent tune "damping" is essentially the artifact of tune computation (see Fig. 2).

References

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S.X. Zheng, Q.Z. Xing, X.L. Guan et al.,

Overall design and progress of XiPAF project

, in Proceedings of SAP2017, Jishou, China, WECH2 (2017). DOI: 10.18429/JACoW-SAP2017-WECH2