Problem: Inversion Symmetry

        Silicon has acoustic and optical modes in its phonon spectrum. However, the optical modes are not ``IR-active" in the sense that IR radiation does not excite these phonons.   

(a) Show that the diamond lattice has inversion symmetry. Describe the position of the inversion symmetry points in the crystal. 

(b) A vibrational mode is infrared active, if the atomic displacements create an electric dipole moment. Why is the optical mode of silicon not IR active?