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Multipactor discharge in an accelerating superconducting elliptic cavity occurs usually near its equator. As simulations show, the dimensions of the trajectories of multipacting electrons are very small compared to the dimensions of a cavity. This feature gives a way for solving explicit equations of motion instead of cumbersome simulations. Electric and magnetic fields near the cavity equator are presented in a form of expansions up to the third power of coordinates. Comparisons with numerical calculations of fields made with the SLANS code for the TESLA cavity cells, as well as with the analytical solution for a spherical cavity, are done. These fields are used for solving the equations of motion of electrons in crossed rf fields near the equator. Based on the analysis of these equations, general features of multipacting in this area are obtained. Results are compared with simulations and experimental data. The experimental formulas for multipacting zones are explained and their dependence on the cavity geometries is shown. Because of small sizes of electron trajectories, the influence of the weld seams is taken into account. This suggests a possible explanation of multipacting in a cavity which was not found by simulations. The developed approach allows evaluation of multipacting in a cavity without its simulations but after an analysis of fields in the equatorial region. These fields can be computed by any code used for cavity calculation.

The accelerating gradient performance of superconducting niobium cavities is rapidly progressing as a result of reduced field emission due to improvement in surface preparation techniques. Beyond the field emission limitation, there exists a fundamental limit imposed by the critical magnetic field of niobium. One way to tackle this limit is to reduce the ratio of the peak magnetic field to the accelerating gradient so that a higher accelerating gradient is possible while the cavity is still superconducting. New cavity shapes of reentrant type have been proposed and optimized [1]. A single cell 1300MHz cavity of this new class of shapes has been fabricated. Because of the reentrant geometry, the fabrication and surface cleaning of the cavity becomes challenging. In this paper, we present some calculation prerequisites and the fabrication and preparation results for this new cavity.

An algorithm for optimization of the multicell cavity cells is proposed. Inner cells are optimized for minimal losses or minimal magnetic field, when the aperture diameter, E pk =E acc-the ratio of peak electric field to the accelerating field, and the wall slope angle are given. Optimization of the end cells is done for minimal losses or maximal acceleration in them. Two shapes of the end cells-with and without the end irises-are analyzed. This approach facilitates further optimization for higher order modes extraction because it permits keeping the achieved optimal values nearly the same while changing some dimensions of the cells. Comparison of the proposed cavity geometry with the TESLA cavity geometry illustrates the traits of the presented approach. It is also shown that lower values of the wall slope angle, which lead to the reentrant shape for the inner cells, are also beneficial for the end cells. For the Cornell Energy Recovery Linac most dangerous are dipole modes causing the beam breakup (BBU). Minimization of power of higher order modes (HOMs) in a multicell cavity was done using derivatives of the BBU parameter with respect to geometric parameters of the cavity cells. As a starting point of optimization, the shape with minimal losses at the fundamental mode was taken. Further changing the shape for better propagation of HOMs was done with degradation of the fundamental mode loss parameter G Á R sh =Q within 1% while decrease of the BBU parameter was nearly 3 orders of magnitude. The BBU threshold current tends to be inversely proportional to this parameter.