Abstract
We present a systematic study of the interplay between lattice parameters and the energy of the optical phonons as well as the antiferromagnetic coupling strength, $J$, in the high-$T_{\text{c}}$ superconducting cuprate $R$Ba$_2$Cu$_3$O$_{6+\delta}$ (\mbox{R-123}, $R =$ Y, Dy, Gd, Sm, Nd) with hole-doping $p$ ($0.00<p \lesssim 0.04$). The energy of the $B_{1g}$-mode at $\nu_{B1g}\approx~335$ cm$^{-1}$ has been found to relate systematically to the inverse of the lattice parameter $a$. Our results confirm the temperature dependent phonon splitting for Nd-123 at low doping, which has been reported for optimally-doped Nd-123. Surprisingly, $J$ is independent of $a$ for the first four $R$ families, and a general consistency between $T_{\text{c}}^{\text{max}}$ and $J$, as suggested in a previous investigation, could not be confirmed.
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