Band Models and the Reciprocal Lattice

When electrons enter a solid lattice, they occupy certain energy levels arranged in well-defined zones. What this means is that there is an energy zone within the crystal where the electron wave density is high, and a resonance condition has arisen. The electrons are free to move within this zone and electroneutrality is preserved because the electron wave 

To understand this, consider a free electron with energy, e, and wave vector, k. As shown on the next page in 5.3.1., e varies with k. This curve is a potential energy diagram of the free electron. Note that we have converted sin 9 (the linear sinusoidal function) into jt (the radial function). If we apply a mono-atomic lattice, arranged in a linear manner, and having a lattice constant of a, to our electron, we can show that Bragg reflection occurs at  

Observe also that there are three allowed zones in 5.3.1., labeled as shown in the following  

It is thus evident that even in a simple linear structure (i.e.- a row of M-atoms), the electrons associated with the structure are restricted to an allowed band. 

Joining the atoms into a structure causes a perturbation of the electron wave-functions which results in restricted energy zones for the electrons associated with the structure. The general case is shown in Part la b of 

---