Why Use Homogeneous Dispersion for Group Velocity in Inhomogeneous Lattice?

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Discussion Overview

The discussion centers around the use of homogeneous dispersion to calculate the group velocity of an electron wavepacket in an inhomogeneous lattice. Participants explore the implications of applying a formula typically associated with homogeneous systems to a context that includes impurities, raising questions about the validity of this approach.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant presents the formula for group velocity, vgroup(k) = ∇kEk, and questions the appropriateness of using homogeneous dispersion in an inhomogeneous lattice context.
  • Another participant seeks clarification on the definitions of homogeneous dispersion and inhomogeneous lattice, indicating a need for further explanation.
  • A third participant defines homogeneous as a system without impurities and inhomogeneous as one with impurities, contributing to the understanding of the terms used.
  • One participant asserts that the formula for group velocity is a definition that applies to any dispersion relation, suggesting that the system's characteristics may not affect its validity, but raises the question of how results might differ between homogeneous and inhomogeneous systems.

Areas of Agreement / Disagreement

Participants express differing levels of understanding regarding the application of homogeneous dispersion in inhomogeneous lattices, and there is no consensus on the implications of this approach.

Contextual Notes

There are unresolved questions about the impact of impurities on the group velocity calculation and the assumptions underlying the use of homogeneous dispersion in this context.

Niles
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Hi

The group velocity of an electron wavepacket in a homogeneous lattice is

vgroup(k) = ∇kEk,

where Ek is the dispersion. I have just read an article, where they use this to find the group velocity of a wavepacket in an inhomogeneous lattice, but they use the homogeneous dispersion. I don't quite understand why this is allowed. Can you clarify this?


Niles.
 
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would u clarify what you mean by homogeneous dispersion? what is an inhomogeneous lattice?
 
I mean homogeneous when there is no impurity. Likewise I mean inhomogeneous when there is an impurity.
 
icic... I am not sure if I understand your question. But your given formula is actually the definition of the group velocity, and it applies to any dispersion relation. This group velocity means the velocity of the overall shape of the wave and it doesn't matter what your system is.

I guess what matters, probably in your problem, is how different it would be when you calculate it in homogeneous and inhomogeneous system.
 

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