Hello Taslimur,

the momentum conservation is an esssential rule for transitions. But you must consider the whole momentum (orbital and spin) j = l +-s. Contrary to single atoms, you have energy bands with neighboured spin up and spin down states. This makes spin exchange very easy due to electron electron interaction within the band. Furthermore, not only dipol transitions but quadrupol transitions are possible, of couse with lower probability.

With Regard

R. Mitdank

Because of the breaking of the full translational symmetry in a crystal, the states you are referring to cannot be purely d states: there are admixtures of other atomic states, even though following your analysis these turn out not to be as significant as one might think at first glance. Therefore, there is first-order optical inter-band transition between the states you are referring to, however with a relatively weak oscillator strength. You should take the residual components of the relevant wave functions along other atomic states than d states seriously.

Thanks a lot Professor Rüdiger Mitdank and Professor Behnam Farid.

Md. Taslimur Rahman : My pleasure.