External vs. ``internal'' pressure effect on the anti-ferromagnetic superexchange energy, $J$, in LnBa$_2$Cu$_3$O$_6$ (Ln=La,Nd,...,Lu)

What causes the difference between the effect of ``internal'' pressure, as caused by ionic substitution, and external pressure on $T_c^{max}$ in the cuprates [1]? Is it the density of states, the pairing boson energy scale ($\omega_B$), condensation energy (which governs fluctuations), or ...? Many models of high temperature superconductivity put the energy scale of $\omega_B$ as the anti-ferromagnetic super-exchange energy, $J$, between adjacent Cu(2) ions in the CuO$_2$ plane. We therefore investigated Raman $B_{1g}$ two-magnon scattering in high quality LnBa$_2$Cu$_3$O$_6$ (Ln123) single crystals, Ln(=La, Nd, Sm, Eu, Gd, Dy, Yb, Lu), at ambient pressure to determine the effect of internal pressure on $J$. Comparing with measurements of $J$ under external pressure reveals that internal and external pressure have \emph{quantitatively} the same effect on $J$. However, and most surprisingly, we find an anticorrelation between $J$ and $T_c^{max}$ when ion size or internal pressure is the implicit variable. Given the opposite effects of internal and external pressure on $T_c^{max}$, this result suggests that some energy scale other than short range anti-ferromagnetic interactions has a more dominant effect on $T_c^{max}$.\\[4pt] [1] e.g. M. Marezio, Physica C, 341-348, 375 (2000)