Magnetic critical field H_c and spin polarization energy limit field H_p in a superconductor

By the BCS framework [Bardeen, Cooper and Schrieffer, Phys. Rev., 108, 1175 (1957); Nam, Phys. Rev., 156, 470 (1967)], we present novel results, at zero temperature, H_c(x) = [4πN(0)]^1/22Δ/(1+x), H_c(1) = H_c(BCS) and H_p(x) = 2^1/22Δ/(1+x)(gμ_B), H_p(1) = H_p(CC) [Chandrasekhar, Appl. Phys. Lett., 1, 7 (1962); Clogston, Phys. Rev. Lett., 9, 266 (1962)], with the superfluid density x = ρ_s/ρ = tanh(G/2) [Nam, APSMAR22/A61.4 (2022), Phys. Rev. B3, 2946 (1971)], G= [1/N(0)V] [Nam, Phys. Lett. A193, 111 (1994); (E) ibid A197, 458 (1995)] = Σ_j (2/w_j)tan^-1(z/w_j), w_j=(π/2)j (odd integer), z=T_d/2T_c, Δ/2T_c = A(z) = z/sinhG, where N(0), V, and T_d are the density of states/spin at a reference energy, the BCS interaction energy, and the pairing interaction energy range (the cut-off energy), respectively. A(z>>1) (BCS) =0.88 and A(z<<1) = 1. The upper value [H_p(x)/T_c] =[2/0.88] [H_p(CC)/T_c]. For UTe2, T_c = 1.6K, H_p(0.53) is found to be 4.5T contrast to 3T = H_p(CC) by Ran et al, Science, 365, 684 (2019). Superconducting state [x<1] is a miscible liquid state with both ρ_s and ρ_n. Superconductors are multiconnected [Nam, Phys. Lett. A198, 447 (1995)].