Finding Distances to Brillouin Zone Borders from Origo

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Discussion Overview

The discussion revolves around understanding the distances from the origin to the Brillouin zone (BZ) boundaries in a body-centered cubic (bcc) lattice. Participants explore the relationships between direct and reciprocal lattice constants and how to calculate these distances in three-dimensional space.

Discussion Character

  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant expresses confusion about the distances from the origin to the BZ plane borders, noting that their initial assumption about the distances being half the basis vectors applies only to 2-D square lattices.
  • Another participant suggests that in 3D, the distances should be calculated as |G-vector|/2, using the smallest possible G-vector.
  • A follow-up question asks if the previously mentioned distance applies to all 12 directions in the BZ.
  • A response clarifies that the calculation was specifically for four directions in the BZ, as illustrated in a reference figure, and indicates that the remaining eight directions can also be calculated.
  • Another participant specifies their interest in the distance along the 110-direction from the origin to a point in the Σ-direction.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the generality of the distance calculations across all directions in the BZ, as some clarify specific cases while others seek broader applicability.

Contextual Notes

There are limitations regarding the assumptions made about the lattice constants and the specific directions being discussed, which may affect the calculations presented.

rubertoda
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Hi, i have problems understanding the Brilluoin zone to the bcc lattice

First, i wonder what the distances are, from the origo to the brillouin zone plane borders.
I firts thought it would be half the distance of the basis vectors (mine are 2pi/a, where a is my lattice constant, BUT its only true for 2-D calculations for square lattices.

I know that the distances in 3D should be |G-vector|/2, and i take the smallest possible G.


But i still don't know

ANyone who knows how to find the distances from origo to the first BZ zone boundaries, knowig that the basis vector in the reciprocal space is 2pi/2?are they all the same??
 
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I think if the direct lattice constant is a, then the reciprocal lattice constant would be2\pi/a hence your favorite distance is (\pi/\sqrt2)a.
 
Thanks a lot. and this goes eveforryone of the 12 directions in the BZ?
 
No, it was for the four direction located in the middle of the figure 4.16 Ashcroft & Mermin. you can easily calculate it for remained 8 directions (see the figure).
 
ok thanks. I am jut interested in the 110-direvtion, from origo to N,in the Ʃ-direction
 

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