Local-limit disorder characteristics of niobium-based superconducting radio-frequency cavities

Nowadays $\mathrm{Nb}$-based superconducting radio-frequency cavities represent fundamental tools used for (Standard Model) particle acceleration, (beyond Standard Model) particle probing, and long-lifetime photon preservation. We study the superconducting radio-frequency shift in the vicinity of the critical temperature ${T}_{c}$ and the quality factor mainly at low temperatures within the Dynes superconductor model. We scrutinize and use the local limit response to the external electromagnetic field. Our approach allows for a finer analysis of the peculiar behavior of the resonant frequency shift immensely close to ${T}_{c}$, observed in recent experiments. In several regimes, we analytically elaborate on the width and depth of the resulting dip. Studying the sign of the slope of the resonant frequency shift at ${T}_{c}$ in the moderately clean regime clarifies the role of the pair-breaking and pair-conserving disorder. Next, to find the relevance of our description, we compare and also fit our results with the recent experimental data from the $\mathrm{N}$-doped $\mathrm{Nb}$ sample presented by Zarea et al. [Front. Supercond. Mater. 3, 1 (2023)]. Our analysis complies with the experimental findings, especially concerning the dip width. We offer a straightforward, homogeneous-disorder-based interpretation within the moderately clean regime. Comparative analysis for three other cavities with different resonant frequencies reported by Ueki et al. [Prog. Theor. Exp. Phys. 5, 053I02 (2025)] points toward a similar regime. Assuming the same regime at low temperatures, we address details of the high-quality plateaus. Summing all up, this work presents (and studies the limits of) the simple, effective description of the complex problem corresponding to the electromagnetic response in the superconductors, combining homogeneous conventional pairing and two different kinds of disorder scattering.