The Frequency Shift and Q of Disordered Superconducting RF Cavities

Niobium superconducting radio-frequency (SRF) cavities for high energy accelerator applications have been greatly improved in terms of the quality factor $Q$ by techniques such as Nitrogen doping. However, the mechanisms leading improvement in $Q$ are still not fully understood. Quite recently the SRF group at Fermilab measured anomalies in the frequency shift of N-doped SRF Niobium cavities near the transition temperature. Here we report our theoretical analysis of these results based on the microscopic theory of superconductivity that incorporates anisotropy of the superconducting gap and inhomogeneous disorder in the screening region of the SRF cavities. We are able to account for frequency shift anomalies very close to $T_c$ of the order of fractions of a kHz. Our results for the frequency shift and Q are in good agreement with the experimental data reported for all four N-doped Nb SRF cavities by Bafia et al. We also compare our theory with an earlier report of on a Nb sample measured at 60 GHz. We also show that the quality factor calculated theoretically has a peak of upper convexity with the largest $Q$ at intermediate levels of disorder. For strong disorder, i.e. the dirty limit, pair breaking in the presence of disorder and screening currents limits the $Q$.