  Light propagating in a nondispersive medium is accompanied by a mass density
wave (MDW) of atoms set in motion by the optical force of the field itself
[Phys. Rev. A 95, 063850 (2017)]. This recent result is in strong contrast with
the approximation of fixed atoms, which assumes that atoms are fixed to their
equilibrium positions when light propagates in a medium and which is deeply
rooted in the conventional electrodynamics of continuous media. In many
photonic materials, the atoms carry the majority of the total momentum of light
and their motion also gives rise to net transfer of medium mass with a light
pulse. In this work, we use optoelastic continuum dynamics combining the
optical force field, elasticity theory, and Newtonian mechanics to analyze the
angular momentum carried by the MDW. We show that accounting for the MDW in the
analysis of the angular momentum gives for the field's share of the total
angular momentum of light a quantized value that is generally a fraction of
$\hbar$. In contrast, the total angular momentum of the mass-polariton (MP)
quasiparticle, which is a coupled state of the field and the MDW, and also the
elementary quantum of light in a medium, is an integer multiple of $\hbar$. The
same coupling is found for orbital and spin angular momentum components. The
physical picture of the angular momentum of light emerging from our theory is
fundamentally more general than earlier theoretical models, in which the total
angular momentum of light is assumed to be carried by the electromagnetic field
only or by an electronic polariton state, which also involves dipolar
electronic oscillations. These models cannot describe the MDW shift of atoms
associated with light. We simulate the MDW of Laguerre-Gaussian pulses in
silicon and present a schematic experimental setup for measuring the
contribution of the atomic MDW to the total angular momentum of light.
