  Motivated by recent inelastic neutron scattering (INS) experiments on
La-based cuprates and based on the fermiology theories, we study the spin
susceptibility for La-based (e.g., La$_{2-x}$Sr$_x$CuO$_4$) and Y-based (e.g.,
YBa$_2$Cu$_3$O$_y$) cuprates, respectively. The spin excitation in
YBa$_2$Cu$_3$O$_y$ is dominated by a sharp resonance peak at the frequency 40
meV in the superconducting state. Below and above the resonance frequency, the
incommensurate (IC) peaks develop and the intensity of the peaks decreases
dramatically. In the normal state, the resonant excitation does not occur and
the IC peaks are merged into commensurate ones. The spin excitation of
La$_{2-x}$Sr$_x$CuO$_4$ is significantly different from that of Y-based ones,
namely, the resonance peak does not exist due to the decreasing of the
superconducting gap and the presence of the possible spin-stripe order. The
spectra are only enhanced at the expected resonance frequency (about 18 meV)
while it is still incommensurate. On the other hand, another frequency scale at
the frequency 55 meV is also revealed, namely the spectra are commensurate and
local maximum at this frequency. We elaborate all the results based on the
Fermi surface topology and the d-wave superconductivity, and suggest that the
spin-stripe order be also important in determining the spin excitation of
La-based cuprates. A coherent picture for the spin excitations is presented for
Y-based and La-based cuprates.
