  The vibrational spectra of solids, both ordered and amorphous, in the
low-energy regime, control the thermal and transport properties of materials,
from heat capacity to heat conduction, electron-phonon couplings, conventional
superconductivity etc. The old Debye model of vibrational spectra at low energy
gives the vibrational density of states (VDOS) as proportional to the frequency
squared, but in many materials the spectrum departs from this law which results
in a peak upon normalizing the VDOS by frequency squared, which is known as the
"boson peak". A description of the VDOS of solids (both crystals and glasses)
is presented starting from first principles. Without using any assumptions
whatsoever about the existence and nature of "disorder" in the material, it is
shown that the boson peak in the VDOS of both ordered crystals and glasses
arises naturally from the competition between elastic mode propagation and
viscous damping. The theory explains the recent experimental observations of
boson peak in perfectly ordered crystals, which cannot be explained based on
previous theoretical frameworks. The theory also explains, for the first time,
how the vibrational spectrum changes with the atomic density of the solid, and
explains recent experimental observations of this effect.
