# Difference between energy levels and bands of energy

As per the notes of my Solid state physics, band gap arises when the two atoms come close to each other so that their discrete energy levels split and become continuous which gives rise to bands of energy. So then, what is the difference between energy levels and energy bands. Also,what about the allowed bands and forbidden bands. While solving Kronig penney model, I have known a litle bit(in surface level) about the formation of band gaps, but the only knowledge I have about it is that- the allowed bands are the regions of energy E for which the solution of Schrodinger equation exists and the forbidden bands are the ones for which the solution does not exist. But, how ???

• Two atoms won't form a band, you need a whole (in the model infinite) crystal lattice for that. So the band is basically nothing else than a sufficiently dense collection of discrete levels. If you were to add a finite temperature into the mix, of course, the coupling to the excited phonons will smear out the electronic energy levels sufficiently to make individual energy levels indistinguishable. – CuriousOne Dec 16 '14 at 15:28
• And if you don't take temperature into account, even then the levels will be widened by natural widening, i.e. due to finite lifetime on the level (though this widening is much smaller). – Ruslan Dec 17 '14 at 15:06

The band gaps are always present even in lonely atoms. But the allowed bands are something which forms when atoms meet and form a periodic lattice. What happens is that the energy levels of each atom split because of overlapping of electrons' wavefunctions. Thus adding a new atom to your lattice effectively adds yet another level to each band. When there're few atoms, the bands are actually sets of discrete levels. As the number of atoms grows, these levels make more and more dense system until some form of spectral line widening (simplest would be natural widening, but there're more) finally makes them form a continuous band. In any case, taking the number of atoms to infinity will give you continuous bands even without any widening.

Here's an animation which may help you see how this actually goes. It starts from a single atom, then additional atoms are added until there're 6 of them (here's up to 12). I hope you can mentally extrapolate what will happen when there're even more atoms. Here black lines are potential energy, and blue are probability density (they are a bit not symmetric, this is a result of discretization error). If you consider a single energy level in an atom, e.g. the $1s$ state of hydrogen, then if you bring two atoms together the level will split into two states (often called the bonding and anti-bonding orbitals). If you bring in a third atom the two states will split again giving four states, and so on. Grouping $n$ atoms will split the original distinct level into $2^{n-1}$ levels.

The number of atoms doesn't have to be very big before the the spacing between all these levels is far less than $kT$, and at that point we have a band i.e. effectively we have a continuous range of energies available to electrons.

So the bands, and therefore the band gaps, arise due to bringing many atoms together, not just two.

• No, there'll be exactly $n$ levels, not that many as you say! Each new atom adds one additional level. That can be easily seen if you take Born-von Karmann boundary conditions on a finite crystal with $n$ cells. There'll be $n$ points in the Brillouin zone, which correspond to splitting of each originally single level to $n$ levels. – Ruslan Dec 17 '14 at 15:02