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Light is a transverse wave. Therefore, light in the optical range (i.e. visible light) couples to transverse collective excitations of a material when measuring the optical conductivity for instance. Likewise, X-rays are also transverse waves, but somehow couple to longitudinal collective excitations of a material.

I understand that it is because X-ray photons couple to the electron density as detailed in this paper, but what I am looking for is a plausible intuitive explanation with some physical insight as to why this might be so.

(Sorry if you can't view the paper from the publisher without incurring a fee, but I couldn't find it anywhere else. I think the question is clear enough without the link, though.)

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The waves are transverse, but depending on their polarization when they are incident on the material (S vs P polarization), can have a component normal to the interface of the medium. That component can excite longitudinal waves. –  user3814483 Jul 29 at 15:18
    
Even at normal incidence, where the polarization is parallel to the interface, one still excites longitudinal waves. –  Xcheckr Jul 29 at 16:34

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