That n=3, l=0 “band” of the outermost electron could hold two electrons, one spin up, one spin down. Na has only one electron in that state, so only half the states are filled. We say that the band is half filled. Then, since the states in the band are essentially a continuum, the slightest increase in energy of the topmost electron in the band can move it into a new state with no objection by the Pauli Principle. It can be moved easily by an electric field in particular, so Na is a conductor. In contrast take NaCl in solid form. The outer electron goes in the outer shell of Cl for binding the crystal, so we have a Na+ ion and a Cl- ion. Both are filled shells, so there is no nearby state:
That n=3, l=0 “band” of the outermost electron could hold two electrons, one spin up, one spin down. Na has only one electron in that state, so only half the states are filled. We say that the band is half filled. Then, since the states in the band are essentially a continuum, the slightest increase in energy of the topmost electron in the band can move it into a new state with no objection by the Pauli Principle. It can be moved easily by an electric field in particular, so Na is a conductor. In contrast take NaCl in solid form. The outer electron goes in the outer shell of Cl for binding the crystal, so we have a Na+ ion and a Cl- ion. Both are filled shells, so there is no nearby state: