# Lattice Design of an Intermediate-Energy Electron Storage Ring Dedicated to Materials Research
## 3 Nonlinear Optics Optimization  
The DDBA lattice employs three families of sextupole magnets, namely SF, SD1, and SD2, for chromaticity correction, and two families of harmonic sextupole magnets, S1 and S2. Octupole magnets are not utilized, resulting in a total of three knobs for nonlinear optimization (SF, SD1 with fixed strength used for chromaticity correction). During optimization, we utilized the _OPA_ code [27] to optimize the resonance driving terms (RDT) while fixing the corrected chromaticity to (1, 1). The optimized solutions were imported into the Elegant code [28] for tracking. Figure 4 shows the on- and off-momentum dynamics aperture (DA) in six dimensions (6D) at the injection point (at the center of LSS), including synchrotron radiation and RF cavity. The 6D DA is large and maintains no significant reduction even for off-momentum, making it highly suitable for off-axis injection. \begin{table}
\begin{tabular}{l l l l l} \hline \hline
**Magnet** & **Name** & **Length** & **Strength** & **Bore Diameters** \\ \hline Combined dipoles & B & 1.50 m & 0.87 T, \(-\)5.27 T/m & - \\ Quadrupoles & Q1 & 0.30 m & 22.37 T/m & 50 mm \\  & Q2 & 0.10 m & \(-\)7.37 T/m & 50 mm \\  & Q3 & 0.40 m & 24.33 T/m & 50 mm \\  & Q4 & 0.10 m & 7.37 T/m & 50 mm \\  & Q5 & 0.40 m & 22.12 T/m & 50 mm \\ Sextupoles & S1 & 0.15 m & 687.58 T/m\({}^{2}\) & 56 mm \\  & S2 & 0.15 m & \(-\)824.57 T/m\({}^{2}\) & 56 mm \\  & SD1 & 0.20 m & \(-\)368.87 T/m\({}^{2}\) & 56 mm \\  & SF & 0.40 m & 430.02 T/m\({}^{2}\) & 56 mm \\  & SD2 & 0.40 m & \(-\)483.10 T/m\({}^{2}\) & 56 mm \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Specifications of main magnets. Figure 4: 6D DA of on- and off-momentum at the injection point (at the center of LSS). Frequency map analysis (FMA) [29] was employed to analyze the DDBA lattice. FMA is a numerical method based on fine Fourier techniques, which provides a clear representation of the global dynamics of many multidimensional systems. FMA transforms the particle coordinate space (x, y) into the frequency space (v\({}_{x}\), v\({}_{y}\)), where regular motion appears as a fixed point and irregular motion diffuses, manifesting as changes in tunes with the number of turns. The diffusion rate is defined as the following formula:  
\[D_{r}=\log_{10}(\frac{\sqrt{\Delta v_{x}^{2}+\Delta v_{y}^{2}}}{N}), \tag{2}\]  
where \(\Delta v_{x}\) and \(\Delta v_{y}\) are, respectively, the differences in x and y tunes from the first and second half of the tracking, and \(N\) is the number of turns. And a smaller \(D_{r}\) indicates more stable particle motion. Figure 5 displays the results of FMA, where particles were tracked for 1024 turns, and \(D_{r}\) was calculated based on the tunes from the first 512 turns and the last 512 turns. In Figure 5b, resonance lines up to the fifth order are plotted, with the blue lines representing second-order resonances, red lines representing third-order resonances, black lines representing fourth-order resonances, and green lines representing fifth-order resonances. From Figure 5b, it can be found that the particles do not cross any potentially dangerous resonance lines, which indicates that the lattice has good nonlinear dynamics performance. Figure 6 demonstrates the tune shift with momentum deviation ranging from \(-8\%\) to \(8\%\). It can be seen that the vertical tune of the lattice crosses the half-integer resonance when the momentum deviation approaches \(-8\%\). Combined with the FMA results of off-momentum shown in Figure 7, it can be observed that no particle loss occurs when crossing the half-integer resonance [30], indicating that the lattice has good momentum acceptance and excellent off-momentum dynamics. We also performed 4D and 6D tracking for the local momentum aperture (LMA), and the results are presented in Figure 8. It can be seen that the 4D LMA is very large, with a momentum acceptance of approximately \(11\%\) at the injection LSS. However, the 6D tracking of the LMA is limited by the RF acceptance, which is approximately \(4\%\).