x_opt and x_c in Cuprate Superconductors

The characteristic hole concentrations x_opt = 0.16 and x_c = 0.19 in the under- to overdoped transition regime of cuprate superconductors are shown to be intimately interrelated by mesoscale organization of bond-like RVB arrays extending over 3 x 4 copper sites and comprising 2 holes spaced by 3a_0. Columnar organization maximizes T_c but reduces the superfluid stiffness, while zigzag (tweedy) organization maximizes the superfluid stiffness at reduced T_c. The latter goes hand in hand with the closure of the pseudogap. Columnar and zigzag mesoscale organizations of the bond-like RVB arrays are displacively transformed in each other between x_opt and x_c by weak overdoping.