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Method to extracting the penetration field in superconductors from DC magnetization data

The lower critical field, Bc1, is one of the fundamental quantities of a superconductor which directly manifests the Cooper pair bulk density in the material. Despite this field can be measured by several techniques, the most conventional way is to calculate this field from experimentally measured DC penetration field, Bp, which is defined as the starting point of the deviation of the DC magnetization curve, M(Bappl), from a linear dependence. Surprisingly enough we found that there is no mathematical routine which describes how this starting point of deviation can be found. Here we proposed to extract Bp data from the fit of M(Bappl) dataset to the power law, where the threshold criterion Mc can be established by a convention. The advantage of this approach is that the procedure extracts one additional characteristic parameter, which is the power-law exponent. We demonstrated the applicability of the approach for polycrystalline ThIr3, WB4.2, BaTi2Bi2O, Th4H15, for Pb thin films and for Nb single crystal. Due to in majority of reports Bc1(T) analysis is limited by the extracting of the London penetration depth, here we advanced the analysis to extract primary thermodynamic superconducting parameters (i.e., the ground state superconducting energy gap, $Δ(0)$, the relative jump in electronic specific heat at transition temperature, $ΔC/C$, and the gap-to-transition temperature ratio, $2Δ(0)/(k_B T_c )$ from Bc1(T) data. This extraction was shown for Nb, ThIr3, TaRh2B2 and NbRh2B2.