Doping Dependence of Bilayer Resonant Spin Excitations in $bf (Y,Ca)Ba_2Cu_3O_{6+x}$

Resonant magnetic modes with odd and even symmetries were studied by inelastic neutron scattering experiments in the bilayer high-$T_c$ superconductor $rm Y_{1-x}Ca_{x}Ba_2Cu_3O_{6+y}$ over a wide doping range. The threshold of the spin excitation continuum in the superconducting state, deduced from the energies and spectral weights of both modes, is compared with the superconducting d-wave gap, measured on the same samples by electronic Raman scattering in the $B_{1g}$ symmetry. Above a critical doping level of $delta simeq 0.19$, both mode energies and the continuum threshold coincide. We find a simple scaling relationship between the characteristic energies and spectral weights of both modes, which indicates that the resonant modes are bound states in the superconducting energy gap, as predicted by the spin-exciton model of the resonant mode.