Theories derived from Haissinski equation and their applications to electron storage rings

TL;DR

This paper derives and tests equations from the Haissinski equation to analyze bunch properties and impedance effects in electron storage rings, providing a bridge between theoretical models and measurements.

Contribution

It introduces a generalized quadratic equation for bunch lengthening applicable to any impedance model, extending previous cubic models.

Findings

Derived equations accurately predict bunch properties.

Tested models show machine-dependent impedance effects.

Equations facilitate impedance measurement from beam data.

Abstract

As a stationary solution of the Vlasov-Fokker-Planck equation, the Haissinski equation predicts the equilibrium line density of a bunch that circulates in a storage ring for a given wake function. This paper shows that some equations regarding the centroid shift of the bunch, the peak position of the bunch profile, bunch length, and extraction of impedance from the bunch profile can be derived from the Haissinski equation in a self-consistent manner. In particular, a generalized quadratic equation for potential-well bunch lengthening is obtained to accommodate any absolute impedance model, expanding upon Zotter's cubic equation, which is primarily applicable to inductive impedance. The equations derived in this paper are tested using computed impedance models for some electron storage rings, showing machine-dependent properties of impedance effects. We conclude that these equations can be employed in electron storage rings to effectively bridge the gap between impedance computations and beam-based measurements.