I have seen this image multiple times already, but mainly without context about what it’s for. However I can guess that it’s used for calculating the amount of electron states in a material. Shouldn’t the factor 3 in front of the formula at N_states,3D then be a factor 2. Because it should signify the spin degeneracy? (I don’t see in which context this should be a 3)
This formula is for phonon modes in the Debye model, not for electron states.
The prefactor counts the number of phonon polarizations: 1 in 1D, 2 in 2D, and 3 in 3D. In 3D these are one longitudinal and two transverse modes.
The factor 2 in the 1D integral is not spin; it counts the two directions, +k and -k.
So your intuition is right for electrons: for electronic states the degeneracy factor would be 2_s from spin, not 3.
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