Pressure Sensitivity Characterization of Superconducting Spoke Cavities

The following proposal illustrates a method to characterize the pressure sensitivity behavior of superconducting spoke cavities. This methodology relies on evaluating the variation of resonant frequency of a cavity by observing only the displacements at designed regions of the cavity. The proposed method permits a reduced computational burden and a systematic approach to achieve a minimum value of pressure sensitivity in a complex system of dressed cavity. This method has been used to characterize the superconducting spoke cavities type-1 (SSR1), under development for Project X, and to design the helium containment vessel in such way to reduce the pressure sensitivity value to zero. INTRODUCTION The cavity sensitivity to Helium pressure is an important parameter which must be taken in consideration during the design of a dressed cavity system. The pressure fluctuations in the Helium bath cause cavity detuning by elastic deformations and micro-oscillations of the cavity walls. Any small shift from the resonant frequency of the cavity requires significant increase in power to maintain the electromagnetic field constant and, at the same time, it produces phase errors that affect the beam. For a cavity on resonance, the electric and magnetic stored energies are equal. If a small perturbation is made on the cavity wall, this will generally produce an unbalance of the electric and magnetic energies, and the resonant frequency will shift to restore the balance. The Slatter perturbation theorem [1] describes the shift of the resonant frequency, when a small volume ∆ is removed from the cavity of volume . From the theorem is understandable that the frequency increases if the magnetic field is large where the walls are pushed in, and it decreases if the electric is large there. This result is easier to remember if one identifies a decrease in the effective inductance where the magnetic field is large and an increase in the effective capacitance where the electric field large. The traditional evaluation of ⁄ involves a series of electromagnetic and structural analyses that can be performed in parallel with multiphysics software, such as Comsol [2] or Ansys multiphysics [3]. The goal is to evaluate the resonant frequency of the cavity under two arbitrary pressure loads. In this way it is possible to calculate the df/dp as: * Operated by Fermi Research Alliance, LLC under Contract No. De-AC02-07CH11359 with the U.S. Department of Energy. # email: donato@fnal.gov = − Where:  , are the resonant frequency, respectively, after and before the application of the pressure fluctuation that deform the volume of the cavity;  is the pressure load applied to simulate the fluctuation of the helium bath. It is of great importance to have a sense of how this sensitivity is affected by the different shapes of the helium vessel. During the phases of mechanical design of the vessel, its design is typically changed several times for engineering purposes and the iteration process is reduced considerably if there is a methodology to optimize and to estimate quickly the behavior of the system, in terms of sensitivity of resonant frequency to the pressure. METHODOLOGY To describe the methodology let’s consider a simple case at first. Figure 1: Sketch of a cavity connected to a helium vessel (a) by two interfaces (b) where the directional displacements are probed to study the characterization. An RF cavity enclosed in a helium vessel with only 2 areas of interface( = 2), Figure 1. The definition and identification of such interfaces (DOF) is of great importance for this methodology. The electromagnetic behavior of the RF cavity is probed in relation with the displacements at such interfaces. The goal is to be able to evaluate the electromagnetic behavior of the RF cavity by observing only the displacements at such interfaces. The latter can be done by simple structural analyses of the complete assembly (cavity and vessel) under the same pressure loads ( ) used to extract the characteristic equation. The structural (a) Helium Vessel Helium bath Cavity Interface_1