Change in Laslett tune shift due to a dielectric vacuum chamber

In high intensity synchrotrons, the beam loss and thus the tune shift is of major concern. The one of the sources of this tune shift is the environment presented by the vacuum chamber and magnets, which form the electric and magnetic images of the beam. The shift in the tune due to these images is computed in the terms of Laslett coefficients. In rapid cycling synchrotrons for reducing the eddy currents, the chamber is made of dielectric (ceramic) with thin metallic coating inside. In an insulated case, due to this dielectric, the images formed by the chamber are modified and thus the formulation of Laslett tune shift also changes accordingly. In the present paper, we formulate theoretically the Laslett tune shift in the presence of this dielectric. We calculate the Laslett coefficient for both incoherent and coherent cases. The present analytical formulation is for a very simple parallel plane geometry and for a pencil like, continuous uniform beam. The additional change in tune shift due to the inclusion of the effect of a dielectric slab is up to almost 4%.