Try here for all the answers
http://en.wikipedia.org/wiki/Electronic_band_structure
A crystal lattice with added 'doping' atoms is not anything like the isolated atoms that make up that lattice.
This explains the difference between energy of ionization and band gap of a crystal lattice made up of these atoms.
The computer you are using right now exists because some smart blokes at Bell labs were looking at the fundamentals of crystal lattice structure of Germanium and Silicon in the forties. The practical result was the transistor.
Just shows how fundamental research leads to totally unexpected results. Wheras goal oriented reseach just gets you to your forseen goal!
Bert