Photon absorption requirements If I blast a material with photons at a frequency strictly greater than the difference between any two energy levels of all electrons existing in that material, will any of these electrons jump to a higher energy state?  Some say the energy match must be exact.  Some say the balance moves on as a lower energy photon, which might be eternal if lower than any bottom level  Intuitively, the material should heat up. How is this heat manifested?  Thank you
 A: There is Compton scattering

the X-rays are scattered through an angle $θ$ and emerge at a different wavelength related to $θ$  . Although classical electromagnetism predicted that the wavelength of scattered rays should be equal to the initial wavelength, multiple experiments had found that the wavelength of the scattered rays was longer (corresponding to lower energy) than the initial wavelength.


These Feynman diagrams can be used for modeling the interaction of photons with particles and/or fields.
The models are complicated, example :

Direct determinations of electron momentum densities in solids by measuring Compton scattered photons 
  in coincidence with recoil electrons will allow more precise testing of electronic structure calculations.

In the case of solid matter, the electrons in the diagram may be  virtual, representing the interaction of the photon with the field and the loss of energy of the photon. How the energy is dissipated will depend on the solid, its lattice rotation and vibration levels as the excess energy turns into kinetic energy and heat.
A: Given a particular material, one would find that the material has a particular absorption spectrum. This represents a function with the probability density for a photon to be absorbed by that material as a function of the frequency (wavelength) of the photon. So, if your photon has a frequency that does not match the energy differences between the energy bands of the electrons in that material, then the probability for absorption would be low. Therefore, the material would be transparrent for the photon at that frequency.
However, one would find that the absorption peaks in the spectrum have a finite width. So, even if the frequency does not match the energy difference exactly, there is still a nonzero probability that the material can absorb the photon. What happens with the difference in energy? Well, that can go into (or be borrowed from) other forms of energy, such as thermal vibrations in the material.
