An energy gap is, in principle, a dominant parameter in superconductivity. However, this view has been challenged for the case of high-Tc cuprates, because anisotropic evolution of a d-wave-like superconducting gap with underdoping has been difficult to formulate along with a critical temperature Tc. Here we show that a nodal-gap energy 2ΔN closely follows 8.5 kBTc with underdoping and is also proportional to the product of an antinodal gap energy Δ* and a square-root superfluid density √Ps for Bi2Sr2CaCu2O8+δ, using low-energy synchrotron-radiation angle-resolved photoemission. The quantitative relations imply that the distinction between the nodal and antinodal gaps stems from the separation of the condensation and formation of electron pairs, and that the nodal-gap suppression represents the substantial phase incoherence inherent in a strong-coupling superconducting state. These simple gap-based formulae reasonably describe a crucial part of the unconventional mechanism governing Tc.
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