Integral form of the beam envelope equation for electron beam propagation in vacuum and its relation to the K-V equation

The study of the beam envelope radius—a parameter characterizing the transverse size and evolution of a particle beam along its propagation path—is fundamental to the particle accelerators application and the execution of space-borne experiments employing artificial relativistic electron beams. In this paper, we investigate the propagation of electron beams in vacuum and derive an integral form of the beam envelope equation. This equation is equivalent to the simplified differential form of the Kapchinsky-Vladimirsky (K-V) equation excluding the effects of external forces and radial emittance. The integral equation is validated by the widely used ASTRA (A Space Charge Tracking Algorithm) simulation code. The effect of electron energy on beam envelope radius is uncertain and depends on whether internal force or external force dominates. When external force dominates, a decrease in electron energy results in a smaller beam envelope radius. Conversely, when internal force dominates, an increase in electron energy leads to a smaller beam envelope radius. This study is a bridge between integral form and differential form of the envelope equation, and will provide a better understanding for the K-V equation and a scientific basis for researching beam propagation technology.