  We report high-resolution resonant inelastic x-ray scattering (RIXS)
measurements of the collective spin fluctuations in three compositions of the
superconducting cuprate system La2-xSrxCuO4. We have mapped out the excitations
throughout much of the 2-D (h,k) Brillouin zone. The spin fluctuations in
La2-xSrxCuO4 are found to be fairly well-described by a damped harmonic
oscillator model, thus our data allows us to determine the full wavevector
dependence of the damping parameter. This parameter increases with doping and
is largest along the (h, h) line, where it is peaked near (0.2,0.2). We have
used a new procedure to determine the absolute wavevector-dependent
susceptibility for the doped compositions La2-xSrxCuO4 (x=0.12,0.16) by
normalising our data to La2CuO4 measurements made with inelastic neutron
scattering (INS). We find that the evolution with doping of the intensity of
high-energy excitations measured by RIXS and INS is consistent. For the doped
compositions, the wavevector-dependent susceptibility is much larger at
(1/4,1/4) than at (1/2,0). It increases rapidly along the (h,h) line towards
the antiferromagnetic wavevector of the parent compound (1/2,1/2). Thus, the
strongest magnetic excitations, and those predicted to favour superconductive
pairing, occur towards the (1/2,1/2) position as observed by INS.
