Quasi-Free Electronic Model of Solids

As we saw previously, see Sections 3.3.3, 3.4.3, and 3.5.3, in the solid state environment there seems to exists two equivalent types of wave-function 

In free eleetrons approximation we actually consider the effective cancellation of the bulk and valence potentials for whose their states assiuned as independent, i.e., orthogonal. Let s reformulate here the free solution so that to allow more realistic potential of the system. Firstly, let s separate the forth (+) and back (-) propagations by the normalized wave funetions  

the boundary conditions, at the extremes of the crystal domain provide the geometric quantification of the wave-vector, as  

However, due to the imit cell periodicity assumed (since observed) for a solid crystal, the whole domain (in ID direction here) may be considered portioned to become  

Nevertheless, this restriction has a fundamental consequence in establishing the so-ealled first Brillouin zone as the first (the so-ealled ground ) quantified interval for the wave veetor (of wave function 

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