Linear Dependence of Peak Width in $\chi (q,\omega )$ vs $T_{\mathit{c}}$ for ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{O}_{6+\mathit{x}}$ Superconductors

It is shown that the momentum space width of the peak in the spin susceptibility, $\mathrm{Im}\chi (\mathbf{q},\omega )$, is linearly proportional to the superconducting $T_c$: $T_c=ħv^{*}\Delta q$ with $ħv^{*}\simeq 35\mathrm{meV}Å$. This relation is similar to the linear relation between incommensurate peak splitting and $T_c$ in LaSrCuO superconductors, as first proposed by Yamada et al. [Phys. Rev. B 57, 6165 (1998)]. The velocity $ħv^{*}$ is smaller than Fermi velocity or the spin-wave velocity of the parent compound and remains the same for a wide doping range. This result points towards strong similarities in magnetic state of ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6+x}$ and LaSrCuO.