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Different energy bands can be drawn in different zones in $k$-space is called Extended-Zone-Scheme, whereas if the different bands are drawn in first Brillouin zone, then its called Reduced Zone Scheme. If every band is drawn in every zone in $k$-space is known as Periodic Zone Scheme.

Is there any physical significance of reduced or periodic zone schemes? or is it just a geometrical convenience.

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  • $\begingroup$ Can you give some more background to the question? $\endgroup$ Nov 19, 2021 at 15:24
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    $\begingroup$ @BrendanDarrer No, The question says it all. $\endgroup$
    – 147875
    Nov 19, 2021 at 16:01
  • $\begingroup$ fine, no problem! $\endgroup$ Nov 19, 2021 at 16:27

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The lattice momenta that differ by a reciprocal lattice vector are equivalent - this is the basis for drawing reduced zone scheme in place of the extended zone scheme. The two are mathematically equivalent, but one typically discusses processes in a solid state in terms of the reduced scheme, where several energy states (bands) correspond to the same momentum - understanding that these momenta are limited to the first Brillouin zone.

Periodic zone scheme is redundant, as it shows the same states multiple times.

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    $\begingroup$ Does that mean it is a geometrical convenience? $\endgroup$
    – 147875
    Nov 19, 2021 at 16:00
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    $\begingroup$ @147875 One could say so... but it seems like complicating things beyond necessary. Why do you think that the extended scheme is more natural? $\endgroup$
    – Roger V.
    Nov 19, 2021 at 16:12
  • $\begingroup$ Extended zone scheme seems more natural because it is the effect of all the Bragg planes at each $\pm k$ on free electron parabola. Whereas in reduced zone scheme, you are re-drawing all bands in a first Brillouin zone. $\endgroup$
    – 147875
    Nov 19, 2021 at 17:20
  • $\begingroup$ The reduced zone scheme makes it easier to understand and follow interband transitions. In particular optical excitations, which have a selection rule of $\Delta k = 0$ $\endgroup$
    – rpsml
    Nov 29, 2021 at 22:57

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