  We theoretically investigate the microscopic mechanism of conversion between
the electron spin and the microscopic local rotation of atoms in crystals. In
phonon modes with angular momenta, the atoms microscopically rotate around
their equilibrium positions in crystals. In a simple toy model with phonons, we
calculate the spin expectation value by using the adiabatic series expansion.
We show that the time-averaged spin magnetization is generated by the
microscopic local rotation of atoms via the spin-orbit interaction. On the
other hand, in the system with a simple vibration of atoms, time-averaged spin
magnetization becomes zero due to the time-reversal symmetry. Moreover, the
magnitude of the time-averaged spin magnetization depends on the inverse of the
difference of instantaneous eigenenergy, and we show that it becomes smaller in
band insulators with a larger gap.
