Robust intrinsic electronic superconducting phases in underdoped La_(2-x)Sr_xCuO4 single crystals

We have measured the superconducting critical temperature (Tc) and the diamagnetic susceptibility of La2-xSrxCuO4 single crystals in various magnetic fields. We observed a field-induced evolution from an apparent Tc phase to an intrinsic Tc1 = 15 K or Tc2 = 30 K phase characterized by "magic" hole concentration which is commensurate with that of a two dimensional electronic lattice. The onset Tc of the intrinsic superconducting phases remains robust up to H = 5 Tesla. We suggest that the intrinsic superconducting phases at "magic" doping concentrations are the pristine electronic phases of high temperature superconductivity.