Revised Hamiltonian near third-integer resonance and implications for an electron storage ring

In electron storage rings, an accurate description of particle dynamics near third-integer resonance is crucial for various applications. The conventional approach is to extrapolate far-resonance dynamics to near resonance, but the difficulty arises because the nonlinear detuning parameter diverges at this critical point. Here we derive, via a suitable application of the canonical perturbation theory, a revised detuning parameter that is well behaved near resonance. The resultant theory accurately describes the morphology of resonance islands for a wide range of parameter space and facilitates its optimization.

Figure 1

The Frenet-Serret coordinate system used in this study. The particle moves along the trajectory line, with the position denoted by the vector $\overrightarrow{r}$. The origin of the Frenet-Serret coordinate system is denoted by the vector $\overrightarrow{r_0}$. The ideal orbit is represented by the dotted curved line. The bending radius is denoted by $\rho$, and the unit vectors of each axis are denoted by $\widehat{x}$, $\widehat{y}$, and $\widehat{s}$.

Figure 2

Example of phase space configuration with ${\mathcal{H}}_5$.

Figure 3

The horizontal (blue) and vertical (red) beta functions, and the horizontal dispersion function (black) in PLS-II lattice. PLS-II is a “double bend achromat (DBA)” lattice. The initial horizontal beta function is ${\beta }_{x,0}=7.006\mathrm{m}$ and the tune is ${\nu }_x=1.3325$. The rectangular inset illustrates the magnet distribution within PLS-II, where the red denotes the quadrupole magnets, the blue the bending magnets, and the green the sextupole magnets $S_{1-4}$.

Figure 4

Nonlinear detuning parameter vs tune. The original detuning parameter ${\alpha }_0$ is plotted by red line. The revised detuning parameter ${\alpha }_{-1}$ is plotted by blue line. The circles denote the values of detuning parameters when the tune is ${\nu }_x=1.3325$ for PLS-II lattice.

Figure 5

(a) Phase-space trajectories with contour plots of the Hamiltonian in Eq. (51) where ${\delta }_{\nu }=-8.33\times {10}^{-4}$, $g_{3,0,l_{3{\nu }_x}}=-0.923$ and $j=-1$ (gray line). Red dots represent phase-space trajectories from tracking results. Normalized coordinates are defined as $(x_n=\frac{x}{\sqrt{2{\beta }_{x,0}}},p_n=-\sqrt{\frac{{\beta }_{x,0}}{2}}{x}^{\prime })$ where ${\beta }_{x,0}$ is the beta function at the starting position of the lattice. (b) Same as (a), but for $j=0$.

Figure 6

The empirical nonlinear detuning parameter ${\alpha }_1$ (blue dots) when (a) $S_1$, (b) $S_2$, (c) $S_3$, and (d) $S_4$ sextupole strength are varied. Also shown are ${\alpha }_{-1}$ (red lines) and ${\alpha }_0$ (black dashed lines).

Figure 7

Fixed points (blue dots) for varying (a) $S_1$, (b) $S_2$, (c) $S_3$, and (d) $S_4$. The theoretical prediction of Eq. (60) with ${\alpha }_{-1}$ is plotted in red solid line and that with ${\alpha }_0$ is plotted in black dashed line.

Figure 8

Fixed points (blue dots) vs tune. The theoretical prediction of Eq. (60) with ${\alpha }_{-1}$ is plotted in red solid line and that with ${\alpha }_0$ is plotted in black dashed line.

Figure 9

(a) $S_4$ vs ${\alpha }_{-1}$, (b) $S_i$ ($i=1$, 2, 3) vs $S_4$.

Figure 10

Plot of coefficients and fittings. (a) Plot of ${\alpha }_{-1},{\alpha }_0$ and ${\alpha }_6$ vs ${\delta }_{\nu }$ (b)–(d) precise plot of each coefficients and fitting results. Fitting was performed up to second order.

Figure 11

${\alpha }_1$ calculating process (a) Tracking results of electron for 1500 cells and contour plot. The tracking was performed for every three cells to confine the results to a single island. The blue line represents the tracking results near the center of the islands. (b) Oscillation part of ${\mathcal{H}}_4$ for blue line of Fig. 11. It is defined by $({\mathcal{H}}_4-⟨{\mathcal{H}}_4⟩)$ (c) We calculated the standard deviation of the oscillation part of ${\mathcal{H}}_4$ for the blue line in Fig. 11 and defined the value of $\alpha$ that minimizes the function ${\sigma }_{{\mathcal{H}}_4-⟨{\mathcal{H}}_4⟩}$ as ${\alpha }_1$. ${\alpha }_{-1}$ in $x$ axis is 1058.1 and ${\alpha }_1$ is 981.9. (d) Graph of tracking results and Hamiltonian contours in (a) and plot of Hamiltonian defined by using ${\alpha }_1$.