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As a stationary solution of the Vlasov-Fokker-Planck equation, the Haissinskiequation predicts the equilibrium line density of a bunch that circulates in astorage ring for a given wake function. This paper shows that some equationsregarding the centroid shift of the bunch, the peak position of the bunchprofile, bunch length, and extraction of impedance from the bunch profile canbe derived from the Haissinski equation in a self-consistent manner. Inparticular, a generalized quadratic equation for potential-well bunchlengthening is obtained to accommodate any absolute impedance model, expandingupon Zotter's cubic equation, which is primarily applicable to inductiveimpedance. The equations derived in this paper are tested using computedimpedance models for some electron storage rings, showing machine-dependentproperties of impedance effects. We conclude that these equations can beemployed in electron storage rings to effectively bridge the gap betweenimpedance computations and beam-based measurements.

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