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

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Discussion Overview

The discussion revolves around the representation of the wave vector q in the context of Bloch theorem as presented in Ashcroft and Mermin. Participants explore the mathematical formulation and implications of using q = k - K versus q = k + K, focusing on the conventions and choices in defining these vectors within the first Brillouin zone (B.Z).

Discussion Character

  • Technical explanation, Conceptual clarification, Debate/contested

Main Points Raised

  • One participant questions why the wave vector q is expressed as k - K, suggesting that using q = k + K might be more conventional.
  • Another participant argues that there is no difference between the two representations, noting that both k and K can take positive and negative values, thus covering all possibilities.
  • A different participant speculates that the choice of k - K might be conventional or might have some special significance in this context.
  • One participant asserts that there is nothing inherently special about the k - K representation, explaining that defining a new vector G = -K leads to the same mathematical outcome, indicating that it is a matter of personal preference.

Areas of Agreement / Disagreement

Participants express differing views on the significance of the choice between k - K and k + K, with some suggesting there may be a conventional preference while others maintain that both forms are equivalent. The discussion remains unresolved regarding any special implications of the k - K choice.

Contextual Notes

Participants do not clarify any assumptions regarding the definitions of k and K, nor do they address any potential limitations in the mathematical derivation of Bloch theorem that might affect the choice of representation.

pallab
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TL;DR
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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