A semi-empirical formula for the dependence of upper critical field with component and strain in Nb₃Sn

We include the effects of Sn content b in the scaling law to generalize the upper critical field B_c2(T, e) function of temperature and strain in Nb₃Sn. The basis for this generalization comes from a theory developed by the same author [Scientific Reports 7, 1133 (2017)]. The important assumption in the current work is that, in determining the upper critical field B_c2, the contributions of strain, temperature and Sn content are independent of each other. It is followed that we write down a semi-empirical expression B_c2(T, e, b) = B_c2 (T₀, e₀, b)s(e)(1 - t^1.52) where the strain function s(e) and t ºT/T_c (e) are affected only by strain and temperature, respectively, and the variation of tin content b enters in the function B_c2 through B_c2(T₀, e₀, b). This separable form of B_c2(T, e, b) allows one to incorporate Sn content b in the scaling law in a natural way, while this variable is not considered by traditional scaling laws. In addition to this advance, one can see a more flexible function with triple variables which could be an effective tool to fit or predict experiments. As the last but most important outcome, this semi-empirical function instructs one how to understand the role of component content played in scaling behaviors of Nb₃Sn, so as to reduce the off-stoichiometric composition inducing discrepancy between the results by scaling laws and experiments.