Bloch Theorem and Periodic Boundary Conditions

A real crystal is a finite macroscopic object made of a finite, although extremely large, number of atoms. However, the ratio of the number of atoms at the surface to the total number of atoms in the crystal, N, is very small, and proportional to. When N is large and the surface is neutral, the perturbation caused by the presence of the boundary is limited to only a few surface layers and, therefore, has no influence on the bulk properties. For this reason, a macroscopic crystal mostly exhibits properties and features of the bulk material, and unless attention is deliberately focused onto the crystal boundary, surface effects can be thoroughly neglected. If this is the case, the crystallographic model of an infinite and translation-invariant crystal fits in the aim of studying bulk properties. 

The potential energy of such a crystal must be a periodic function with the same periodicity as the lattice, so that for a translation by any direct lattice vector g, the potential energy does not change 

However, by using the Bloch theorem and considering that H(r — g) is equivalent to H r) as an obvious consequence of the form of the crystalline potential (Eq. [5]), we obtain Eq. [6] again 

What is the form of Bloch functions Equation [8] implies that a Bloch function can be written as the product of a plane wave and a periodic function u(r k) with the same periodicity of the lattice  

Bloch functions span an infinite crystal and do not decay to zero at infinity. To circumvent the problem of normalizing a wave function with infinite extent, it is easier to consider a finite crystal consisting ofN = Ni x N2 x N3 cells and then let N grow to infinity. To preserve periodicity, periodic boundary conditions are imposed, which can be stated in the following form If Ny cells exist along the -th direction (/ = 1,2,3) in the macroscopic crystal, it must happen that for any integer m and every j 

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