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Superfluid Density, Penetration Depth, Condensate Density

Fascination with the concept of superconducting (SC) {\it superfluid density} $ρ_s$ has persisted since the beginning of superconductivity theory, with numerical values of an actual density rarely provided. Over time $ρ_s$, addressed mostly in cuprate and following high temperature superconductors, has become synonymous with the normalized (unitless) inverse square of the magnetic penetration depth $λ_L$ (the London expression, with superfluid density denoted $n_s$), with interest primarily on its temperature $T$ dependence that is expected to reflect the T-dependence of the SC gap amplitude and gap symmetry. In conventional superconductors, generalized expressions from the London penetration depth via Ginzburg-Landau theory, then to BCS theory provide updated pictures of the supercurrent density-vector potential relationship. The BCS value $λ_{band}$ is distinct from any particle density, instead involving particle availability at the Fermi surface and Fermi velocity as the determining factors, thus providing a basis for a more fundamental theory and understanding of what is being probed in penetration depth studies. The number density of superconducting electrons ${\cal N}_s(T$=0) -- the scalar SC {\it condensate density} -- is provided, first from a phenomenological estimate but then supported by BCS theory. A straightforward relation connecting ${\cal N}_s(0)$ to the density of dynamically transporting carriers in the normal state at $T_c$ is obtained. Numerical values of relevant material parameters including $λ_{band}$ and ${\cal N}_s$ are provided for a few conventional SCs.