Local structure and the intermediate-energy fine structure in x-ray Raman scattering from ice Ih

The structure of the various different equilibrium and nonequilibrium phases of water ice is a topic of considerable interest, with strong relevance for geophysics, atmospheric sciences, and space sciences.~ Recent advances in non-resonant x-ray Raman scattering (XRS) provide a new method for studying local structure of water ices in extreme environments including especially in high-pressure cells.~ Here, we investigate two pragmatic issues: the optimum choice of momentum transfer $q$ for these measurements and the usefulness of the intermediate-energy fine structure as a strong fingerprint of local atomic structure out to several coordination shells.~ To this end, we present new XRS measurements of ice Ih with greatly improved statistics over earlier work, and also present extensive full-multiple calculations of the dependence of the intermediate-energy fine structure on local structure.~ We find that XRS measurements at high $q$, where the XRS cross-section is largest but where multipole transitions can be important, show little difference from dipole-limited soft x-ray absorption studies.~ In addition, our calculations predict significant sensitivity of the XRS intermediate-energy fine structure to different ice structures.