Phonon density of states (or vibrational density of states) is defined in exactly the same way as the electronic densities of state, see the DOS equation. The integral over the Brillouin zone goes over all 3N phonon bands, where N is the number of atoms in the cell. There are two popular conventions regarding normalization of the phonon DOS. Either it can be normalized to unity or to the total number of vibrational modes, 3N.
The partial (or projected) phonon density of states is determined as a contribution from the given atom to the total phonon DOS. This is a useful concept that allows one to understand the nature of various branches in the phonon spectrum. The contribution to the partial density of states on atom i, from each phonon band, is evaluated using:
where ej is the eigenvector (normalized to unit length) associated with the mode of energy Ej. The projected density of states is then obtained by summation of these contributions over all phonon bands. By construction all the projected phonon DOS sum up to the true phonon DOS.
Theory in CASTEP
Density of states and partial density of states