Periodic Boundary Conditions, Reciprocal Lattices and Brillouin Zones

3 Periodic Boundary Conditions, Reciprocal Lattices and Brillouin Zones 

We started our discussion in this chapter by assuming an infinitely extended crystal. Now we shall introduce the Born-von Karman or periodic boundary conditions as we did for the linear chain in Chap.2. For this purpose we subdivide the infinitely large crystal into macrocrystals . Each macrocrystal is a parallelepiped defined by the vectors N a, N a, N a, where 1 2 3 primitive translation vectors and N, N2, are large 

The periodic boundary conditions require that the atomic displacements for atoms separated by a translation N. a., or a sum of such translations, must be the same 

Equation (3.34) specifies the possible values of q. In order to express these conditions in a simple manner, we introduce the reciprocal lattice. The primitive translation vectors of the reciprocal lattice are the three vectors 2, definded by [3.1] 

From (3.35) it follows that an expression for q which satisfies (3.34) is given by 

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