Multiple Quasi-invariant surfaces for dynamic aperture optimization in low-emittance synchrotrons

Synchrotron light research is a rapidly evolving field, primarily due to the unique properties of the radiation. These properties have been further enhanced by the implementation of multi-bend achromat cells. A key challenge in the design of these systems is the optimization of the dynamic aperture, for which various techniques have been developed. These techniques, commonly based on particle tracking, require substantial computational resources. Recent studies have shown that a tracking-free approach, based on a quasi-invariant, can improve nonlinear dynamics in the horizontal phase space of such systems. The present work extends this approach by demonstrating that it also provides meaningful information about vertical dynamics. Through detailed analysis, it is shown that the surface of section of a dynamical system can be mimicked by the quasi-invariant framework, reproducing the main features of the nonlinear dynamics. This information is used to propose an objective function for optimizing the dynamic aperture in synchrotron models. The effectiveness of the objective function is validated using a model with 79 pm rad emittance, achieving a dynamic aperture of approximately 5 mm for momentum deviations within the (¿3,3)% interval. The results demonstrate that the proposed quasi-invariant method enables dynamic aperture optimization for a fourth-generation synchrotron in a tracking-free manner and with low computational demands. © 2025 The Authors.