Where Does the Pi Come From in the BCC Brillouin Zone Calculation?

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Homework Help Overview

The discussion revolves around the calculation of the Brillouin zone (BZ) for a body-centered cubic (BCC) structure, specifically addressing the origin of the factor of pi in the expression for the closest face of the BZ.

Discussion Character

  • Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants explore the relationship between the BCC structure and its reciprocal lattice, questioning how the factor of pi arises in the calculations. There is a discussion about the definitions and transformations involved in moving to the reciprocal lattice.

Discussion Status

Some participants have provided insights into the definitions of the reciprocal lattice and its implications for the BZ calculation. Multiple interpretations of the problem are being explored, particularly regarding the relationship between the BCC and FCC lattices.

Contextual Notes

There is an ongoing examination of the definitions and assumptions related to the reciprocal lattice and the specific parameters involved in the BZ calculation. The original poster expresses confusion over the expected outcome versus the provided answer.

philip041
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I'm answering a question where it becomes necessary to know the closest face of the BZ in a bcc structure. The answer is given as +/- (2*pi) / (sqrt(2)*a) where a is the cubic lattice parameter.

I would have thought the Answer would have been sqrt(3)*a / 4. Where does the pi come from?

Cheers in advance.
 
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are you working the reciprocal lattice? i think the pi comes about in the transformation to reciprocal lattice
 
The reciprocal lattice of BCC is FCC. The 2pi comes from the definition of the reciprocal lattice:
b1=2pi/Volume (b2xb3) etc.

The reciprocal FCC has a side of 4Pi/a. In FCC the nearest neighbors of a corner are the centers of adjacent faces, at distance sqrt(2)/2 * (size of cube).
The Brillouin zone is at half this distance so it will be
1/2*(sqrt(2)/2)*(4pi/a) = pi*sqrt(2)/a = 2pi/(a*sqrt(2))
 
cheers!
 

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