From the beam-envelope matrix to synchrotron-radiation integrals

The equilibrium state of an electron in a storage ring can be described most accurately by the envelope matrix, as long as the electron motion is linear. The equilibrium envelope can be calculated in the same way as the equilibrium barycenter (closed orbit). This is suited for accurate numerical calculations. The ‘‘emittances’’ can be extracted from the envelope as approximate quantities. The radiation integrals, which express the emittances in terms of Twiss parameters, dispersions, and other optical parameters, are extended to cover general 6×6 dynamics. Without any coupling between modes, these reduce to those of Sands.