Using different lasers source (e.g. green or red ones) to probe the same phenomena, for example adsorption of some species on certain electrode. Should we expect the same raman shift? I mean, the same peak position in terms of cm-1?
Yes, the difference in-between the vibrations should be the same. You might also compare the quantum yield of the detector which influences the measured intensities.
If you are nonresonant, your shift should always be the same. In some samples , different wavelength lasers are resonant with different species in an (inhomogeneously broadened) absorption profiles and these species can have different Raman shifts. A classic example is carbon nanotube samples (see John's comment above), which can contain mixtures of semiconducting/metallic tubes of different chiralities and diameters, all of which have different UV/Vis NIR absorption profiles and Raman spectra.
Raman shift is the same for different wavelength of the laser. The raman shift corresponds to the energy difference of the initial state and the final state. However when you are going to high precision measurements (resolution in the range of the natural linewidth) then you can find a dependence between laserwavenlength/intensity and the absolute position. But thats not relevant for standard analytical applications. What is more relevant is, that the intensity of the transition is linear in the laser intensity but is depending on the laser wavenumber something like I = v^3 or I=v^4. The laserwavelength will also influence the background (e.g. fluorescence).
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