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

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SUMMARY

The discussion clarifies the origin of the factor of pi in the calculation of the Brillouin zone (BZ) for a body-centered cubic (BCC) structure. The correct expression for the closest face of the BZ is +/- (2*pi) / (sqrt(2)*a), where 'a' represents the cubic lattice parameter. The factor of pi arises from the transformation to the reciprocal lattice, specifically due to the definition of the reciprocal lattice vectors and the relationship between BCC and face-centered cubic (FCC) lattices.

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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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