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Magnetic Dependence of the Energy Gap: a Good Model to Fit Q Slope of Low Beta Cavities

The reasons why the intrinsic quality factor (noted Qo) of a superconducting cavity drops with the accelerating field (noted Eacc) are still not well understood. In an effort to explain this phenomenon, mainly for high beta cavities, many models have been developed in the community but few of them could fit experimental data whatever the material treatment or surface conditioning. In the specific case of low beta cavities made of bulk Niobium (i.e Spiral 2 Quarter Wave Resonator), a model based on a magnetic field dependence of the energy gap has been developed to fit experimental data. The evolutions of the model input parameters depending on the cavity treatment or test conditions are consistent with the changes described in the literature. The model will be described step by step and specific examples will be given. INTRODUCTION The aim of this paper is not to explain what is at the origin of the medium or high field Q-slope (MFQS and HFQS), but is to give some feedback on some observations and studies made during the tests of some low beta (low frequency) cavities like Spiral2 β=0.12 Quarter-wave resonator (QWR) [1]. The behaviour of low-frequency cavities (in opposition with high-beta elliptical cavities) is relatively different regarding MFQS and HFQS and thus deserves to be pointed out as, above all, these are very less studied. Tackling the problem from 2 different sides (high frequency side and low frequency side) could bring some more light and help to better understand the origin(s) of the Q-slope. This paper will first introduce the model used to fit experimental data of QWR and will highlight a point that is often forgotten when we start considering a magnetic field dependence of the surface resistance. The paper will then present different example of fits done. THE MODEL After describing the correction done on the relation between the quality factor and the surface resistance, the model will be developed. Formula Corrections As already mentioned in [2], applying a model with a magnetic field dependence on the well known formula (1) expressing the quality factor (Qo) versus the surface resistance (Rs) leads to significant errors as a strong assumption is made to factorize Rs term out of the integral (See Figure 2). s R G Q  0