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

[philip041]

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.

 

[lanedance]

are you working the reciprocal lattice? i think the pi comes about in the transformation to reciprocal lattice

 

[nasu]

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))

 

[philip041]

cheers!