Is there a repository where one can find unpolarized nuclear Compton scattering data $\gamma (Z,N)\rightarrow \gamma (Z,N)$ for specific nuclei $(Z,N)$? or even some parametrization of structure functions that loosely follows the data?
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Try exploring the National Nuclear Data Center. If I search the Evaluated Nuclear Structure Data File for "incident particle: g" and "outgoing particle: g" I get datasets for most, but not all, nuclei. There are other databases hosted by the NNDC as well, and all are well-referenced to the experimental literature.