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Both INS and IXS can be used to study phonon dispersion relations. While INS requires large sample size due to low inelastic scattering cross section, IXS using synchrotron x-ray sources do not require a small sample as extremely intense synchrotron x-rays can be incident on small sample area. Therefore, is INS any good now to study phonon dispersion relations, or is it obsolete since synchrotron sources have come?

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While inelastic xray scattering is a powerful technique, it shouldn't come as a surprise that it has some limitations, and cases where neutrons are advantageous. So it's certainly too simplistic to call INS obsolete for these purposes.

There's a useful discussion in sec. 1.4.1 of Alfred Q. R. Baron's Introduction to High-Resolution Inelastic X-Ray Scattering notes on arXiv, which I think is an updated version of the chapter Baron A.Q.R. (2016) High-Resolution Inelastic X-Ray Scattering II: Scattering Theory, Harmonic Phonons, and Calculations. In: Jaeschke E., Khan S., Schneider J., Hastings J. (eds) Synchrotron Light Sources and Free-Electron Lasers. Springer.

Here follows two quotes from the mentioned section, where I've highlighted advantages of neutrons. These may or may not outweigh the advantages of xrays, depending on the material you're interested in:

However, the large absorption cross section also means that, with x-rays, radiation damage can be a serious issue, even with the relatively (as compared to other synchrotron work) weak meV-bandwidth beams used for IXS: protein crystals and polymers show visible damage on the hour time scales of typical scans, and the author has observed changes in elastic intensities in IXS spectra on these time scales. (Note, the trick of freezing a sample as used in structural studies is of arguable efficacy for IXS, as, in principle, the dynamics can change as soon as particles are ionized). As the relationship of the isotopic species and neutron scattering cross-section is complex, as opposed to the $\sim Z^2$ scaling for x-ray scattering, there also can be advantages for each (x-rays or neutrons) depending on the atoms/isotopes in the sample. However, the x-ray analogue of isotope replacement in INS, which is tuning to atomic resonances to change cross-sections to help identify features of the scattering related to particular atomic species, is not yet possible with meV resolution (see the section on RIXS, below)

Neutrons are advantageous when very high-energy resolution is needed, as backscattering spectrometers provide ~ 10 µeV level resolution, at least for smaller energy transfers, and spin echo spectrometers can reach neV levels, sometimes even ~10 µeV for larger energy transfers (Aynajian et al.2008). In addition, whereas the energy resolution for most xray spectrometers is approximately Lorentzian and so has long tails, the resolution with neutrons often has shorter tails. This can make neutrons advantageous for observing weak modes near to stronger ones, or for measuring modes in the presence of strong elastic backgrounds. However, a new spectrometer design (see sections 4.7, 4.8) may also improve the situation for x-rays. Neutrons are extremely interesting when large single crystals of heavier materials are available, whereas x-rays are limited by the short penetration length into the sample due to the high photoelectric absorption. Also, modern time-of-flight neutron spectrometers (Kajimoto et al.2011)(Abernathy et al.2012) allow collection of a huge swath of momentum space at one time, and assuming modest (~0.1 cc) size samples are available, may offer advantages for survey measurements of all phonons in a material.

There's also a broader perspective in Belushkin, A.V., Kozlenko, D.P. & Rogachev, A.V. Synchrotron and neutron-scattering methods for studies of properties of condensed matter: Competition or complementarity?. J. Synch. Investig. 5, 828 (2011).

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