# Lattice Design of an Intermediate-Energy Electron Storage Ring Dedicated to Materials Research
## 4 Intra-Beam Scattering (IBS) and Touschek Scattering  
The IBS effect is caused by small-angle Coulomb scattering between charged particles in a bunch [32; 33]. It is generally more severe in MBA lattices due to ultra-low emittance, which leads to a high density of electron bunches. This results in severe IBS effects, subsequently increasing the emittance and energy spread of the beam in its final equilibrium state. To decrease the natural emittance of the storage ring, the MBA lattice usually designs the bending magnets with weak fields. This results in smaller synchrotron radiation losses and longer damping times. The IBS effect is closely related to the damping time, and the longer the damping time, the more severe the IBS effect. For a specific lattice structure, the natural emittance of the beam is proportional to the square of its energy, while the IBS growth rate is inversely proportional to the cube of its energy. Therefore, under the combined effects of IBS, synchrotron radiation damping, and quantum excitation, there exists an optimized energy value at which the equilibrium emittance of the storage ring is minimized. The relationship between beam energy and equilibrium emittance under different beam currents was simulated based on the storage ring parameters presented in Table 1. The IBS effect of this DDBA lattice was simulated in 80% bunch filling mode with 1% and 0.1% transverse coupling, respectively, using the Elegant code to calculate the IBS effect, and the results are shown in Figure 9. As can be seen in Figure 9, the minimum equilibrium emittance of the DDBA lattice occurs at a beam energy of approximately 1.5-2.0 GeV as the beam current increases at the transverse coupling of 1%. At the transverse coupling of 0.1%, the minimum equilibrium emittance of the DDBA lattice occurs at a beam energy of about 1.8-2.2 GeV as the beam current increases. At lower beam energies, the equilibrium emittance is primarily determined by the combined effects of synchrotron radiation damping and IBS effect due to the small horizontal emittance and high IBS growth rate. At higher energies, the equilibrium emittance is primarily determined by quantum excitation and synchrotron radiation damping, which is approximately equal to the natural emittance of the beam at zero beam current. In this paper, we selected a beam energy of 3 GeV because, at this energy, the IBS effect is negligible, and it is suitable for producing hard X-rays. Due to the low magnet strength of this DDBA lattice, the ability to boost the beam energy of this storage ring to 4 GeV is retained for subsequent upgrades. Touschek scattering is a large-angle elastic scattering process between electrons inside a bunch. The Touschek effect is related to IBS, but specifically refers to scattering events that result in a significant transfer of momentum from the transverse to the longitudinal planes. In contrast, IBS refers to multiple small-angle scattering events. If the change in  
Figure 9: Equilibrium emittance versus beam energy for different beam currents at (**a**) 1% transverse coupling; (**b**) 0.1% transverse coupling. longitudinal momentum is significant, the momentum deviation of one or both electrons may exceed the momentum acceptance of the storage ring, resulting in the loss of electrons from the bunch. In low-emittance storage rings, Touschek scattering is the primary factor that limits beam lifetime. We calculated and analyzed the Touschek lifetime of the beam for various beam currents, transverse couplings, and momentum acceptances, and the findings are presented in Figure 10. For Figure 10a, the momentum acceptance of the storage ring was set to a fixed value of 4.06% in the calculation, which is consistent with the nonlinear dynamics performance of the designed DDBA lattice, as shown in Figure 8. For Figure 10b, the beam current was fixed at 1000 mA in the calculation. The RF acceptance is about 5.26% when the RF voltage is 6.1 MV.