# Calculating Impedances for a Particle Accelerator

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Two-dimensional resistive-wall impedance with finite thickness: Its mathematical structures and their physical meanings

(PTEP Editors' Choice)

Prog. Theor. Exp. Phys. **2022**, 053G01 (2022).

In a new study, scientists provide a more comprehensive picture of dealing with the impedances of a two-dimensional conductive cylindrical beam pipe with walls of finite thickness and space-charge forces simultaneously.

Particle accelerators use electromagnetic fields to accelerate successive charged beam pulses close to the speed of light. As one beam moves inside the vacuum chamber with limited conductivity, the vessel generates electromagnetic fields. The interaction, which is referred to as resistive-wall impedance, can make subsequent beams unstable in the longitudinal or transverse directions. Meanwhile, the respective pulses have their intrinsic repulsive forces, called space-charge impedance.

In a new study published in *Progress of Theoretical and Experimental Physics*, researchers from the Japan Proton Accelerator Research Complex provide a more comprehensive picture of dealing with the resistive-wall as well as space-charge impedances, simultaneously.

The impedances were calculated for a cylindrical resistive chamber that was enclosed by a vacuum and, further, a perfectly conductive chamber. This setup enabled researchers to find the new image of the resistive-wall impedance with space-charge impedance.

In the conventional picture, the longitudinal resistive-wall impedance for relativistic beams monotonically approaches zero for an extremely thin chamber, whereas it is unclear what principle makes the space-charge impedance modified accordingly.

In their new calculations, the researchers rewrite the resistive-wall impedance with space-charge force based on a parallel circuit theory and demonstrate how the impedances are determined to minimize the energy loss of the beam under given circumstances. This picture naturally explains how the resistive-wall and space-charge impedances change for the chamber with various thicknesses. Owing to this principle, the transverse impedance and the longitudinal impedance for non-relativistic beams have characteristics different from the longitudinal impedance for relativistic beams.

By introducing a more comprehensive concept of the impedance, the findings of this study can help improve the performance of accelerators that are used not only to study particle physics but also in medical applications and other industries.

Two-dimensional resistive-wall impedance with finite thickness: Its mathematical structures and their physical meanings

(PTEP Editors' Choice)

Prog. Theor. Exp. Phys. **2022**, 053G01 (2022).