We report a comprehensive Cu $K$-edge RIXS investigation of $rm La_{2-x}Sr_xCuO_4$ (LSCO) for 0$leq$x$leq$0.35, stripe-ordered $rm La_{1.875}Ba_{0.125}CuO_4$ (LBCO), and $rm La_{2}Cu_{0.96}Ni_{0.04}O_4$ (LCNO) crystals. The RIXS spectra measured at three high-symmetry momentum transfer (textbf{q}) positions are compared as a function of doping and for the different dopants. The spectra in the energy range 1-6 eV can be described with three broad peaks, which evolve systematically with increased doping. The most systematic trend was observed for textbf{q}=($pi$, 0) corresponding to the zone boundary. As hole doping increased, the spectral weight transfer from high energies to low energies is nearly linear with emph{x} at this textbf{q}. We interpret the peaks as interband transitions in the context of existing band models for this system, assigning them to Zhang-Rice band$rightarrow$upper Hubbard band, lower-lying band$rightarrow$upper Hubbard band, and lower-lying band$rightarrow$Zhang-Rice band transitions. The spectrum of stripe-ordered LBCO was also measured, and found to be identical to the correspondingly doped LSCO, except for a relative enhancement of the near-infrared peak intensity around 1.5-1.7 eV. The temperature dependence of this near-infrared peak in LBCO was more pronounced than for other parts of the spectrum, continuously decreasing in intensity as the temperature was raised from 25 K to 300 K. Finally, we find that 4% Ni substitution in the Cu site has a similar effect on the spectra as does Sr substitution in the La site.
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