Dependence of loss rate of electrons due to elastic gas scattering on the shape of the vacuum chamber in an electron storage ring

The beam lifetime in an electron storage ring is also limited by the loss rate of the stored electrons due to the elastic coulomb scattering of electrons with the nuclei of residual gas atoms. The contribution to the beam lifetime due to this elastic scattering depends upon the shape factor which is governed by the shape of the vacuum chamber. In this paper, analytical expressions for the shape factor for a rectangular and an elliptical vacuum chamber as a function of longitudinal position along the circumference in a storage ring are derived using an approach in which the position of electrons at the focusing quadrupole is transformed to the location of defocusing quadrupole and vice versa to define the parts of the vacuum chamber, where the loss of electrons takes place at the location of quadrupoles. The expressions available in the literature are for the average shape factors. The expression of shape factor for a rectangular chamber derived in this paper are similar to the expression for average shape factor quoted in the literature, whereas a new expression for elliptical shape of vacuum chamber having no resemblance with the available expression for average shape factor is obtained. A comparative study of shape factors at each scattering location in the ring obtained from derived expressions and with the existing expressions are reported using Indus-2 lattice parameters for rectangular and elliptical shapes of the chamber. A comparison of average shape factors for these two shapes using derived as well as existing expressions is also reported. These studies indicate that the effect of the rectangular and elliptical shape of vacuum chamber on beam lifetime due to elastic coulomb scattering between electrons and nuclei of residual gas molecules is nearly same.