BAND theory
due to the failure of free electron models to predict the semiconductor and insulator properties, new model was needed.
The new models are Kronig-Penney, Ziemann and Feynmann models.
failure of free electron model is due to neglect of potential created by
positive ion core
Using the energy band concept, the properties of solids can be described more satisfactorily
In isolated free atom, electron has discrete energy
levels , the available energy states in isolated atoms form bands when atoms are brough together to form solid
The top most bands are called the conduction and valence bands separated by forbidden gap (Energy gap).The width of the gap can be used to differentiate between insulators, semiconductors or metals.
in shorts, energy band is the overlap and boaden of energy levels of electron !
For 2s band there are N energy levels containing 2N electrons, (with 2 electrons at each energy level)
For 2p bands =3N energy levels and 6N electrons.
Kronig‐Penney Model , consider the model is in 1D and the potential energy of electron is simplified.
Thist ε‐k relation show that electron are no longer free. 
Equation of wave travel with energy of 
hence, electrron are allowed to have energy in allowed band and not allowed to have energy in forbidden band.
if potentia energy of electron increses, the allowd band become smaller while forbidden band become narrower and vice versa. 
when potential energy approach infinity, the allowed band become energy level, the allowed energy are 
when potential energy appoarch 0, the solution is smilar to free electron 
At the boundary of allowed band, there are discontinuities at 
, n=1,2,3
