Effects of systematic octupole coupling resonances

Effects of systematic octupole coupling resonances 1 2 3 4 first-author 5 corresp shylee@indiana.edu shylee@indiana.edu shylee@indiana.edu 1 2 3 corresp zhengsx@tsinghua.edu.cn zhengsx@tsinghua.edu.cn zhengsx@tsinghua.edu.cn 1 2 3 1 2 3 4 4 Key Laboratory of Particle and Radiation Imaging (Tsinghua University), Ministry of Education, Beijing 100084, China Key Laboratory of Particle and Radiation Imaging (Tsinghua University), Ministry of Education, Beijing 100084, China Laboratory For Advanced Radiation Sources and Application, Tsinghua University, Beijing 100084, China Laboratory For Advanced Radiation Sources and Application, Tsinghua University, Beijing 100084, China Department of Engineering Physics, Tsinghua University, Beijing 100084, China Department of Engineering Physics, Tsinghua University, Beijing 100084, China Northwest Institute of Nuclear Technology, Xi’an, China Northwest Institute of Nuclear Technology, Xi’an, China Department of Physics, Indiana University, Bloomington, Indiana 47405, USA Department of Physics, Indiana University, Bloomington, Indiana 47405, USA 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 2 ν x -2 ν z =0 resonance. However, we will show that the instantaneous betatron tunes can vary widely around the resonance line for particles locked onto the resonance. Octupole Coupling resonance Dynamic aperture XiPAF 1 Introduction 1 The Xi’an Proton Accelerator Facility (XiPAF) is a 230 MeV high-intensity proton accelerator for medical and radiation-science applications[ 1 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 3 ν x =5 resonance [ 2 2 , 3 3 ]. The initial betatron tunes are ν x =1.76 and ν z =1.81 at injection[ 4 4 ], tunable up to a range of ±0.5 for the third-order resonance sl ow extraction[ 5 5 ]. Figure 1 Figure 1 shows an example of the XiPAF lattice at ν x =1.757717 and ν z =1.812286. The XiPAF synchrotron has systematic octupole resonances at 4 ν x =6, 4 ν z =6, 2 ν x +2 ν z =6, and 2 ν x -2 ν z =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 6 - 8 8 6-8 ] at 2 ν x -2 ν z =0 has been carefully studied at injection and during the beam acceleration [ 9 9 ]. 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 10 - 13 13 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 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 15 , 16 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 2 ν x -2 ν z =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 1 In the Frenet–Serret coordinate system, the Hamiltonian for particle motion in the presence of octupoles is <math display="block" id="M1"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>H</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><msup><mi>x</mi><mo>′</mo></msup><mn>2</mn></msup><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>K</mi><mi>x</mi></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><msup><mi>z</mi><mo>′</mo></msup><mn>2</mn></msup><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>K</mi><mi>z</mi></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><msup><mi>z</mi><mn>2</mn></msup><mo>+</mo><msub><mi>V</mi><mn>3</mn></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mn>,</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mi>V</mi><mn>3</mn></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msub><mi>K</mi><mn>3</mn></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow><mrow><mn>24</mn></mrow></mfrac><mo stretchy="false">(</mo><msup><mi>x</mi><mn>4</mn></msup><mo>−</mo><mn>6</mn><msup><mi>x</mi><mn>2</mn></msup><msup><mi>z</mi><mn>2</mn></msup><mo>+</mo><msup><mi>z</mi><mn>4</mn></msup><mo stretchy="false">)</mo><mn>,</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd></mtr></mtable></mrow></math> where K 3 ( s )=( 3 B z / x 3 )| closed-orbit / Bρ , B z 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 4 ν x =6, 4 ν z =6, 2 ν x +2 ν z =6 and 2 ν x -2 ν z =0. The first three resonances are outside the operation range. However, the beam sits near the 2 ν x -2 ν z =0 resonance. The K 3 ( 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": <math id="M2"><mrow><mi>D</mi><mo>=</mo><msub><mstyle mathsize="140%" displaystyle="true"><mrow><mi>log</mi></mrow></mstyle><mrow><mn>10</mn></mrow></msub><mrow><mo>[</mo><mrow><msqrt><mrow><msup><mrow><mo stretchy="false">(</mo><mi>Δ</mi><msub><mi>ν</mi><mi>x</mi></msub><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><mi>Δ</mi><msub><mi>ν</mi><mi>z</mi></msub><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></msqrt></mrow><mo>]</mo></mrow><mn>,</mn></mrow></math> where Δ ν x and Δ ν z are the differences of the betatron tunes evaluated for each particle between the first half and the second half of the tracking turns [ 10 10 - 13 13 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 ( ν x , ν z ). 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 14 ]. We use the ELEGANT code [ 17 17 ] to perform DA tracking for the XiPAF. The frequency-map analysis is integrated in the ELEGANT code[ 18 18 - 21 21 18-21 ]. Figure 2 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 ν x =1.82 and ν z =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 K 1 =-0.003037 m -2 , K 2 =0.0575958 m -3 , and K 3 =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. 3 Effects of the octupole coupling resonance on beam dynamics 1 To understand the effect of the octupole resonances, we perform a Floquet transform on the Hamiltonian in Eq. (1) [ 22 22 ]: <math display="block" id="M3"><mtable columnalign="left"><mtr><mtd><mi>x</mi><mo>=</mo><msqrt><mrow><mn>2</mn><msub><mi>β</mi><mi>x</mi></msub><msub><mi>J</mi><mi>x</mi></msub></mrow></msqrt><mi>cos</mi><mtext> </mtext><mo stretchy="false">(</mo><msub><mi>ψ</mi><mi>x</mi></msub><mo>+</mo><msub><mi>χ</mi><mi>x</mi></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>−</mo><msub><mi>ν</mi><mi>x</mi></msub><mi>θ</mi><mo stretchy="false">)</mo><mn>,</mn></mtd></mtr><mtr><mtd><mi>z</mi><mo>=</mo><msqrt><mrow><mn>2</mn><msub><mi>β</mi><mi>z</mi></msub><msub><mi>J</mi><mi>z</mi></msub></mrow></msqrt><mi>cos</mi><mtext> </mtext><mo stretchy="false">(</mo><msub><mi>ψ</mi><mi>z</mi></msub><mo>+</mo><msub><mi>χ</mi><mi>z</mi></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>−</mo><msub><mi>ν</mi><mi>z</mi></msub><mi>θ</mi><mo stretchy="false">)</mo><mn>,</mn></mtd></mtr></mtable></math> where ( J x , ψ x ) and ( J z , ψ z ) are conjugate action-angle coordinates, ν x and ν z are betatron tunes, and <math id="M4"><mrow><msub><mi>χ</mi><mrow><mi>x</mi><mn>,</mn><mi>z</mi></mrow></msub><mo>=</mo><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mn>0</mn><mi>s</mi></msubsup></mrow></mstyle><mo stretchy="false">(</mo><mn>1</mn><mo>/</mo><msub><mi>β</mi><mrow><mi>x</mi><mn>,</mn><mi>z</mi></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mtext>d</mtext><mi>s</mi></mrow></math> are betatron phase advances. The resulting Hamiltonian is <math display="block" id="M5"><mtable columnalign="left"><mtr><mtd><mi>H</mi><mo>=</mo><msub><mi>ν</mi><mi>x</mi></msub><msub><mi>J</mi><mi>x</mi></msub><mo>+</mo><msub><mi>ν</mi><mi>z</mi></msub><msub><mi>J</mi><mi>z</mi></msub><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>α</mi><mrow><mi>x</mi><mi>x</mi></mrow></msub><msubsup><mi>J</mi><mi>x</mi><mn>2</mn></msubsup><mo>+</mo><msub><mi>α</mi><mrow><mi>x</mi><mi>z</mi></mrow></msub><msub><mi>J</mi><mi>x</mi></msub><msub><mi>J</mi><mi>z</mi></msub><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>α</mi><mrow><mi>z</mi><mi>z</mi></mrow></msub><msubsup><mi>J</mi><mi>z</mi><mn>2</mn></msubsup><mo>+</mo></mtd></mtr><mtr><mtd><msub><mi>G</mi><mrow><mn>2,</mn><mo>−</mo><mn>2,</mn><mi mathvariant="script">l</mi></mrow></msub><msub><mi>J</mi><mi>x</mi></msub><msub><mi>J</mi><mi>z</mi></msub><mi>cos</mi><mo stretchy="false">(</mo><mn>2</mn><msub><mi>ϕ</mi><mi>x</mi></msub><mo>−</mo><mn>2</mn><msub><mi>ϕ</mi><mi>z</mi></msub><mo>−</mo><mi mathvariant="script">l</mi><mi>θ</mi><mo>+</mo><msub><mi>ξ</mi><mrow><mn>2,</mn><mo>−</mo><mn>2,</mn><mi mathvariant="script">l</mi></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><mo>⋯</mo><mn>,</mn></mtd></mtr></mtable></math> where the orbiting angle, θ = s/R , serves as the "time coordinate," R is the mean radius, ≤ll is an integer, α xx , α xz , and α zz are nonlinear detuning parameters, and G 2,-2, ≤ll and ξ 2,-2, ≤ll are the resonance strength and its phase, given by <math display="block" id="M6"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><msub><mi>α</mi><mrow><mi>x</mi><mi>x</mi></mrow></msub><mo>=</mo><mfrac><mn>1</mn><mrow><mn>16</mn><mi>π</mi></mrow></mfrac><mstyle displaystyle="true"><mrow><mo>∮</mo></mrow></mstyle><msubsup><mi>β</mi><mi>x</mi><mn>2</mn></msubsup><msub><mi>K</mi><mn>3</mn></msub><mtext>d</mtext><mi>s</mi><mn>,</mn><mtext> </mtext><msub><mi>α</mi><mrow><mi>x</mi><mi>z</mi></mrow></msub><mo>=</mo><mfrac><mrow><mo>−</mo><mn>1</mn></mrow><mrow><mn>8</mn><mi>π</mi></mrow></mfrac><mstyle displaystyle="true"><mrow><mo>∮</mo></mrow></mstyle><msub><mi>β</mi><mi>x</mi></msub><msub><mi>β</mi><mi>z</mi></msub><msub><mi>K</mi><mn>3</mn></msub><mtext>d</mtext><mi>s</mi><mn>,</mn><mtext> </mtext><msub><mi>α</mi><mrow><mi>z</mi><mi>z</mi></mrow></msub><mo>=</mo><mfrac><mn>1</mn><mrow><mn>16</mn><mi>π</mi></mrow></mfrac><mstyle displaystyle="true"><mrow><mo>∮</mo></mrow></mstyle><msubsup><mi>β</mi><mi>z</mi><mn>2</mn></msubsup><msub><mi>K</mi><mn>3</mn></msub><mtext>d</mtext><mi>s</mi><mn>,</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><msub><mi>G</mi><mrow><mn>2,</mn><mo>−</mo><mn>2,</mn><mi mathvariant="script">l</mi></mrow></msub><msup><mi>e</mi><mrow><mi>j</mi><msub><mi>ξ</mi><mrow><mn>2,</mn><mo>−</mo><mn>2,</mn><mi mathvariant="script">l</mi></mrow></msub></mrow></msup><mo>=</mo><mfrac><mn>1</mn><mrow><mn>16</mn><mi>π</mi></mrow></mfrac><mstyle displaystyle="true"><mrow><mo>∮</mo></mrow></mstyle><msub><mi>β</mi><mi>x</mi></msub><msub><mi>β</mi><mi>z</mi></msub><msub><mi>K</mi><mn>3</mn></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>×</mo><mtext> </mtext><msup><mi>e</mi><mrow><mi>j</mi><mo stretchy="false">[</mo><mn>2</mn><msub><mi>χ</mi><mi>x</mi></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>−</mo><mn>2</mn><msub><mi>χ</mi><mi>z</mi></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>−</mo><mo stretchy="false">(</mo><mn>2</mn><msub><mi>ν</mi><mi>x</mi></msub><mo>−</mo><mn>2</mn><msub><mi>ν</mi><mi>z</mi></msub><mo>−</mo><mi mathvariant="script">l</mi><mo stretchy="false">)</mo><mi>θ</mi><mo stretchy="false">]</mo></mrow></msup><mtext>d</mtext><mi>s</mi><mn>,</mn></mrow></mtd></mtr></mtable></mrow></math> where β x and χ x are the horizontal betatron function and betatron phase, and β z ( s ) and χ z ( s ) are those of the vertical betatron motion. Figure 3 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: <math display="block" id="M7"><mrow><msub><mi>F</mi><mn>2</mn></msub><mo stretchy="false">(</mo><msub><mi>ϕ</mi><mi>x</mi></msub><mn>,</mn><msub><mi>ϕ</mi><mi>z</mi></msub><mn>,</mn><msub><mi>J</mi><mn>1</mn></msub><mn>,</mn><msub><mi>J</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msub><mi>ϕ</mi><mi>x</mi></msub><mo>−</mo><msub><mi>ϕ</mi><mi>z</mi></msub><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi mathvariant="script">l</mi><mi>θ</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>ξ</mi><mrow><mn>2,</mn><mo>−</mo><mn>2,</mn><mi mathvariant="script">l</mi></mrow></msub><mo stretchy="false">)</mo><msub><mi>J</mi><mn>1</mn></msub><mo>+</mo><msub><mi>ϕ</mi><mi>z</mi></msub><msub><mi>J</mi><mn>2</mn></msub><mn>.</mn></mrow></math> The coordinate transformation from the old phase-space coordinates to the new phase-space coordinates is <math display="block" id="M8"><mtable columnalign="left"><mtr><mtd><msub><mi>ϕ</mi><mn>1</mn></msub><mo>=</mo><msub><mi>ϕ</mi><mi>x</mi></msub><mo>−</mo><msub><mi>ϕ</mi><mi>z</mi></msub><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi mathvariant="script">l</mi><mi>θ</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>ξ</mi><mrow><mn>1,</mn><mo>−</mo><mn>2,</mn><mi mathvariant="script">l</mi></mrow></msub><mn>,</mn><mtext> </mtext><mtext> </mtext><msub><mi>J</mi><mi>x</mi></msub><mo>=</mo><msub><mi>J</mi><mn>1</mn></msub><mn>,</mn></mtd></mtr><mtr><mtd><msub><mi>ϕ</mi><mn>2</mn></msub><mo>=</mo><msub><mi>ϕ</mi><mi>z</mi></msub><mn>,</mn><mtext> </mtext><mtext> </mtext><msub><mi>J</mi><mi>z</mi></msub><mo>=</mo><mo>−</mo><msub><mi>J</mi><mn>1</mn></msub><mo>+</mo><msub><mi>J</mi><mn>2</mn></msub><mn>,</mn></mtd></mtr></mtable></math> and the new Hamiltonian becomes <math id="M9"><mrow><mover accent="true"><mi>H</mi><mo>˜</mo></mover><mo>=</mo><msub><mi>H</mi><mn>1</mn></msub><mo stretchy="false">(</mo><msub><mi>J</mi><mn>1</mn></msub><mn>,</mn><msub><mi>ϕ</mi><mn>1</mn></msub><mn>,</mn><msub><mi>J</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mi>H</mi><mn>2</mn></msub><mo stretchy="false">(</mo><msub><mi>J</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow></math> , where <math id="M10"><mrow><msub><mi>H</mi><mn>2</mn></msub><mo stretchy="false">(</mo><msub><mi>J</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>ν</mi><mi>z</mi></msub><msub><mi>J</mi><mn>2</mn></msub><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>α</mi><mrow><mn>22</mn></mrow></msub><msubsup><mi>J</mi><mn>2</mn><mn>2</mn></msubsup></mrow></math> and <math display="block" id="M11"><mtable columnalign="left"><mtr><mtd><msub><mi>H</mi><mn>1</mn></msub><mo stretchy="false">(</mo><msub><mi>J</mi><mn>1</mn></msub><mn>,</mn><msub><mi>ϕ</mi><mn>1</mn></msub><mn>,</mn><msub><mi>J</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mi>δ</mi><msub><mi>J</mi><mn>1</mn></msub><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>α</mi><mrow><mn>11</mn></mrow></msub><msubsup><mi>J</mi><mn>1</mn><mn>2</mn></msubsup><mo>+</mo><msub><mi>α</mi><mrow><mn>12</mn></mrow></msub><msub><mi>J</mi><mn>1</mn></msub><msub><mi>J</mi><mn>2</mn></msub><mo>+</mo></mtd></mtr><mtr><mtd><msub><mi>G</mi><mrow><mn>2,</mn><mo>−</mo><mn>2,</mn><mi mathvariant="script">l</mi></mrow></msub><msub><mi>J</mi><mn>1</mn></msub><mo stretchy="false">(</mo><msub><mi>J</mi><mn>2</mn></msub><mo>−</mo><msub><mi>J</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mi>cos</mi><mo stretchy="false">(</mo><mn>2</mn><msub><mi>ϕ</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mn>.</mn></mtd></mtr></mtable></math> Here, <math id="M12"><mrow><mi>δ</mi><mo>=</mo><msub><mi>ν</mi><mi>x</mi></msub><mo>−</mo><msub><mi>ν</mi><mi>z</mi></msub><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi mathvariant="script">l</mi></mrow></math> is the resonance proximity parameter, and the detuning parameters are α 11 = α xx -2 α xz + α zz , α 12 = α xz - α zz , and α 22 = α zz . Hamilton’s equations of motion are <math id="M13"><mrow><mfrac><mrow><mtext>d</mtext><msub><mi>J</mi><mn>2</mn></msub></mrow><mrow><mtext>d</mtext><mi>θ</mi></mrow></mfrac><mo>=</mo><mo>−</mo><mfrac><mrow><mo>∂</mo><mover accent="true"><mi>H</mi><mo>˜</mo></mover></mrow><mrow><msub><mrow/><mn>2</mn></msub></mrow></mfrac><mo>=</mo><mn>0,</mn></mrow></math> <math id="M14"><mrow><mfrac><mrow><mtext>d</mtext><msub><mi>ϕ</mi><mn>2</mn></msub></mrow><mrow><mtext>d</mtext><mi>θ</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mo>∂</mo><mover accent="true"><mi>H</mi><mo>˜</mo></mover></mrow><mrow><mo>∂</mo><msub><mi>J</mi><mn>2</mn></msub></mrow></mfrac><mn>,</mn></mrow></math> and <math display="block" id="M15"><mrow><mfrac><mrow><mtext>d</mtext><msub><mi>J</mi><mn>1</mn></msub></mrow><mrow><mtext>d</mtext><mi>θ</mi></mrow></mfrac><mo>=</mo><mo>−</mo><mfrac><mrow><mo>∂</mo><mover accent="true"><mi>H</mi><mo>˜</mo></mover></mrow><mrow><mo>∂</mo><msub><mi>ϕ</mi><mn>1</mn></msub></mrow></mfrac><mo>=</mo><mo>+</mo><mn>2</mn><msub><mi>G</mi><mrow><mn>2,</mn><mo>−</mo><mn>2,</mn><mi mathvariant="script">l</mi></mrow></msub><msub><mi>J</mi><mn>1</mn></msub><mo stretchy="false">(</mo><msub><mi>J</mi><mn>2</mn></msub><mo>−</mo><msub><mi>J</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mi>sin</mi><mo stretchy="false">(</mo><mn>2</mn><msub><mi>ϕ</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mn>,</mn><mtext> </mtext><mn>20</mn><mi>p</mi><mi>t</mi></mrow></math> <math display="block" id="M16"><mtable columnalign="left"><mtr><mtd><mfrac><mrow><mtext>d</mtext><msub><mi>ϕ</mi><mn>1</mn></msub></mrow><mrow><mtext>d</mtext><mi>θ</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mo>∂</mo><mover accent="true"><mi>H</mi><mo>˜</mo></mover></mrow><mrow><mo>∂</mo><msub><mi>J</mi><mn>1</mn></msub></mrow></mfrac><mo>=</mo><mo>{</mo><mi>δ</mi><mo>+</mo><msub><mi>α</mi><mrow><mn>12</mn></mrow></msub><msub><mi>J</mi><mn>2</mn></msub><mo>}</mo><mo>+</mo><msub><mi>α</mi><mrow><mn>11</mn></mrow></msub><msub><mi>J</mi><mn>1</mn></msub><mo>+</mo></mtd></mtr><mtr><mtd><msub><mi>G</mi><mrow><mn>2,</mn><mo>−</mo><mn>2,</mn><mi mathvariant="script">l</mi></mrow></msub><mo stretchy="false">(</mo><msub><mi>J</mi><mn>2</mn></msub><mo>−</mo><mn>2</mn><msub><mi>J</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mi>cos</mi><mo stretchy="false">(</mo><mn>2</mn><msub><mi>ϕ</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mn>.</mn></mtd></mtr></mtable></math> Particle dynamics obey Eqs. (4) and (5) at constant J 2 and H 1 , which are invariant if the bare betatron tunes ν x and ν z 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 2 Because the Hamiltonian H 1 is also invariant, each invariant torus in the phase-space coordinates, <math display="block" id="M17"><mrow><msub><mi>X</mi><mn>1</mn></msub><mo>=</mo><msqrt><mrow><mn>2</mn><msub><mi>β</mi><mi>x</mi></msub><msub><mi>J</mi><mn>1</mn></msub></mrow></msqrt><mi>cos</mi><mo stretchy="false">(</mo><msub><mi>ϕ</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mn>,</mn><msub><mi>P</mi><mn>1</mn></msub><mo>=</mo><mo>−</mo><msqrt><mrow><mn>2</mn><msub><mi>β</mi><mi>x</mi></msub><msub><mi>J</mi><mn>1</mn></msub></mrow></msqrt><mi>sin</mi><mo stretchy="false">(</mo><msub><mi>ϕ</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mn>,</mn></mrow></math> of the resonance rotating frame will have constant H 1 and J 2 values. The tori can be analyzed by the fixed points and separatrix of the Hamiltonian. The fixed points are determined by solving d J 1 /d θ =0 and d ϕ 1 /d θ =0: <math display="block" id="M18"><mrow><mn>2</mn><msub><mi>G</mi><mrow><mn>2,</mn><mo>−</mo><mn>2,</mn><mi mathvariant="script">l</mi></mrow></msub><msub><mi>J</mi><mn>1</mn></msub><mo stretchy="false">(</mo><msub><mi>J</mi><mn>2</mn></msub><mo>−</mo><msub><mi>J</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mi>sin</mi><mo stretchy="false">(</mo><mn>2</mn><msub><mi>ϕ</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mn>0,</mn></mrow></math> <math display="block" id="M19"><mtable columnalign="left"><mtr><mtd><mo>{</mo><mi>δ</mi><mo>+</mo><msub><mi>α</mi><mrow><mn>12</mn></mrow></msub><msub><mi>J</mi><mn>2</mn></msub><mo>}</mo><mo>+</mo><msub><mi>α</mi><mrow><mn>11</mn></mrow></msub><msub><mi>J</mi><mn>1</mn></msub><mo>+</mo></mtd></mtr><mtr><mtd><msub><mi>G</mi><mrow><mn>2,</mn><mo>−</mo><mn>2,</mn><mi mathvariant="script">l</mi></mrow></msub><mo stretchy="false">(</mo><msub><mi>J</mi><mn>2</mn></msub><mo>−</mo><mn>2</mn><msub><mi>J</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mi>cos</mi><mo stretchy="false">(</mo><mn>2</mn><msub><mi>ϕ</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mn>0.</mn></mtd></mtr></mtable></math> The Unstable Fixed Points (UFP) are located at <math display="block" id="M20"><mrow><msub><mi>J</mi><mrow><mn>1,</mn><mtext>U</mtext><mi>F</mi><mi>P</mi></mrow></msub><mo>=</mo><msub><mi>J</mi><mn>2</mn></msub><mo>;</mo><mtext> </mtext><mi>cos</mi><mn>2</mn><msub><mi>ϕ</mi><mrow><mn>1,</mn><mtext>U</mtext><mi>F</mi><mi>P</mi></mrow></msub><mo>=</mo><mfrac><mrow><mi>δ</mi><mo>+</mo><mo stretchy="false">(</mo><msub><mi>α</mi><mrow><mn>12</mn></mrow></msub><mo>+</mo><msub><mi>α</mi><mrow><mn>11</mn></mrow></msub><mo stretchy="false">)</mo><msub><mi>J</mi><mn>2</mn></msub></mrow><mrow><msub><mi>G</mi><mrow><mn>2,</mn><mo>−</mo><mn>2,</mn><mi mathvariant="script">l</mi></mrow></msub><msub><mi>J</mi><mn>2</mn></msub></mrow></mfrac><mn>,</mn></mrow></math> provided that | δ +( α 12 + α 11 ) J 2 | le;| G 2,-2, J 2 |. The separatrix is the Hamiltonian torus that passes through the UFP, i.e., J 1 = J 2 and <math display="block" id="M21"><mrow><mi>δ</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>α</mi><mrow><mn>11</mn></mrow></msub><mo stretchy="false">(</mo><msub><mi>J</mi><mn>1</mn></msub><mo>+</mo><msub><mi>J</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mi>α</mi><mrow><mn>12</mn></mrow></msub><msub><mi>J</mi><mn>2</mn></msub><mo>−</mo><msub><mi>G</mi><mrow><mn>2,</mn><mo>−</mo><mn>2,</mn><mi mathvariant="script">l</mi></mrow></msub><msub><mi>J</mi><mn>1</mn></msub><mi>cos</mi><mn>2</mn><msub><mi>ϕ</mi><mn>1</mn></msub><mo>=</mo><mn>0.</mn></mrow></math> This is an ellipse in the normalized phase-space coordinates ( X 1 , P 1 ). The UFP of Eq. (9) is the intersection of the separatrix of Eq. (10) and the Courant-Snyder circle J 1 = J 2 . The Stable Fixed Points (SFP) are located at <math display="block" id="M22"><mrow><mi>δ</mi><mo>+</mo><msub><mi>α</mi><mrow><mn>12</mn></mrow></msub><msub><mi>J</mi><mn>2</mn></msub><mo>+</mo><msub><mi>α</mi><mrow><mn>11</mn></mrow></msub><msub><mi>J</mi><mrow><mn>1,</mn><mtext>S</mtext><mi>F</mi><mi>P</mi></mrow></msub><mo>±</mo><msub><mi>G</mi><mrow><mn>2,</mn><mo>−</mo><mn>2,</mn><mi mathvariant="script">l</mi></mrow></msub><mo stretchy="false">(</mo><msub><mi>J</mi><mn>2</mn></msub><mo>−</mo><mn>2</mn><msub><mi>J</mi><mrow><mn>1,</mn><mtext>S</mtext><mi>F</mi><mi>P</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mn>0,</mn></mrow></math> where the ± corresponds to ϕ 1, SFP =0 or π , and π /2 or 3 π /4, respectively. Depending on the value of J 1 and J 2 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 α 11 and G 2,-2, ≤ll . Because particle actions J x and J z are coupled, the horizontal and vertical beam emittances can exchange unless the initial emittances are equal: є xi = є zi [ 23 23 ]. 3.2 Numerical simulations 2 To demonstrate the effect of the coupling resonance on beam particles, we track a beam with 50 particles at an accelerator with parameters α xx =727.24 m -1 , α xz =-1256.78 m -1 , α zz = 727.24 m -1 , and | G 2,-2,0 |=628.39 m -1 . The bare betatron tunes are δ = ν x - ν z =-0.02. Figure 4 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 <math display="block" id="M23"><mrow><msub><mi>q</mi><mi>x</mi></msub><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msub><mi>ψ</mi><mi>x</mi></msub><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>−</mo><msub><mi>ψ</mi><mi>x</mi></msub><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><mn>2</mn><mi>π</mi></mrow></mfrac><mn>,</mn></mrow></math> <math display="block" id="M24"><mrow><msub><mi>q</mi><mi>z</mi></msub><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msub><mi>ψ</mi><mi>z</mi></msub><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>−</mo><msub><mi>ψ</mi><mi>z</mi></msub><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><mn>2</mn><mi>π</mi></mrow></mfrac><mn>.</mn></mrow></math> 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 10 - 12 12 10-12 ] put the tunes of these particles on the coupling line, as shown in the top plots of Fig. 2 Fig. 2 , the instantaneous betatron tunes oscillate around the resonance line, but not on the resonance line. The right plot of Fig. 4 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 1 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. 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Lett ., 110 , 094801 ( 2013 ). doi: 10.1103/PhysRevLett.110.094801 http://doi.org/10.1103/PhysRevLett.110.094801 Evolution of beam distribution in crossing a walkinshaw resonance 10.1007/s41365-019-0629-8 This work was supported by U.S. National Science Foundation (No. PHY-1504778). 2018-11-07 2019-02-22 2019-03-20 2019-06-03 2019-07-01 2021-04-11 © China Science Publishing & Media Ltd. (Science Press), Shanghai Institute of Applied Physics, the Chinese Academy of Sciences, ChineseNuclear Society and Springer Nature Singapore Pte Ltd. 2019 This is a post-peer-review, pre-copyedit version of an article published in Nuclear Science and Techniques. The final authenticated version is available online at: http://dx.doi.org/10.1007/s41365-019-0629-8 http://dx.doi.org/10.1007/s41365-019-0629-8 . 2019 Nuclear Science and Techniques 07 30