Linear dependence of peak width in $χ(\bq, ω)$ vs T_c for YBCO superconductors

It is shown that the momentum space width of the peak in the spin susceptibility, Im$\chi(q,\omega)$, is linearly proportional to the superconducting $T_c$: $T_c = \hbar v^*\Delta q$ with $\hbar v^* \simeq 35 meV$\AA. 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 $\hbar v^*$ 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 YBCO and LaSrCuO.