The wave vector in 1st B.Z in Bloch theory

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SUMMARY

The discussion centers on the representation of the wave vector q in the proof of Bloch's theorem as q = k - K, where k is within the first Brillouin zone (B.Z.). Participants clarify that this choice is not due to any special mathematical property but rather a matter of convention. The alternative representation q = k + K is equally valid, as both forms cover all possible values of k and K. Ultimately, the choice between k - K and k + K simplifies the mathematical derivation without affecting the proof's integrity.

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  • Understanding of Bloch's theorem
  • Familiarity with the concept of the first Brillouin zone (B.Z.)
  • Knowledge of wave vectors in solid state physics
  • Basic grasp of vector mathematics
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  • Study the derivation of Bloch's theorem in Ashcroft and Mermin's "Solid State Physics"
  • Explore the implications of different wave vector representations in solid state physics
  • Learn about the mathematical properties of Brillouin zones
  • Investigate the role of reciprocal lattice vectors in crystallography
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Physicists, materials scientists, and students studying solid state physics who seek a deeper understanding of wave vector representations in Bloch's theorem.

pallab
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why q=k-K
why the general wave vector q (in the proof of Bloch theorem in Ashcroft Mermin) is represented by k-K, where k is in the 1st BZ ? why not q=k+K ( usual vector form) what is special about k-K?
 
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What's the difference ? k and K can take both positive and negative values so all the possibilities are covered. You choose the definition which simplifies the math
 
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no difference but I was thinking maybe k-K is the conventional one rather than k+K to use in this case or maybe there is "something special" with the k-K choice.
 
No, there is nothing special. When you derive Bloch theorem just start by defining a new vector G = -K and then you get to define q as k+G but the whole proof remains the same. It's a matter of taste.
 
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