band filling
/ band FIL-ing /
Think of a theatre with rows of seats filling from the front. With just a few patrons, only the front rows are taken and there is plenty of room to shuffle around. Pack the place so a block of rows is completely full, and those rows are frozen — nobody can move without an empty seat to slide into. How full the seats are, not how many there are, decides whether the crowd can move. Band filling is exactly this idea applied to a crystal's electrons.
Band filling describes how completely electrons occupy the available states in a crystal's energy bands. Electrons fill bands from the lowest energy upward, two at a time per state. If the topmost occupied band is left only partly full, there are empty seats right beside the filled ones and the material conducts — a metal. If electrons exactly fill a band to its brim and then stop at a gap, every seat is taken and conduction is blocked — an insulator or semiconductor.
Band filling matters because it is the deciding factor that sorts materials into metals versus insulators, far more directly than the band shapes alone. A common misconception is that having more electrons makes a better conductor; what actually matters is whether the filling lands inside a band or right at a gap. This is also why counting electrons per atom — odd versus even — often predicts at a glance whether a simple solid will conduct.
Sodium has one outer electron per atom, which half-fills its top band, so it is a fine metal. Magnesium has two, which might be expected to fill a band exactly and make it an insulator — yet its bands overlap, the filling spills into a second band, and magnesium conducts after all. Band filling, read together with band overlap, tells the real story.
Sodium's half-filled band makes it a metal; band overlap rescues magnesium too.
The simple even-versus-odd electron count is only a rule of thumb; band overlap and the splitting of bands can override it, which is why some elements with an even count, like magnesium, still turn out to be metals.