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

In summary, the answer for the closest face of the BZ in a BCC structure is given as +/- (2*pi) / (sqrt(2)*a), where a is the cubic lattice parameter. The pi comes from the transformation to reciprocal lattice, where b1 = 2pi/Volume (b2xb3) for the reciprocal lattice of BCC. In FCC, which is the reciprocal lattice of BCC, the side is 4Pi/a and the nearest neighbors of a corner are the centers of adjacent faces at a distance of sqrt(2)/2 * (size of cube). The Brillouin zone is at half this distance, so it will be 1/2*(sqrt(2)/2)*(4
  • #1
philip041
107
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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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  • #2
are you working the reciprocal lattice? i think the pi comes about in the transformation to reciprocal lattice
 
  • #3
The reciprocal lattice of BCC is FCC. The 2pi comes from teh 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))
 
  • #4
cheers!
 

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

What is the Brillouin Zone of a BCC lattice?

The Brillouin Zone of a BCC (Body-Centered Cubic) lattice is a region in reciprocal space that represents all possible momentum vectors for electrons in the lattice. It is the first Brillouin Zone for a BCC lattice and is bounded by the faces of a rhombic dodecahedron.

How is the Brillouin Zone of a BCC lattice different from other lattices?

The Brillouin Zone of a BCC lattice is unique in that it is the only lattice with a first Brillouin Zone that is not a simple shape, such as a cube or a sphere. It is also the only lattice with a first Brillouin Zone that is not centered at the origin.

What is the significance of the Brillouin Zone of a BCC lattice?

The Brillouin Zone of a BCC lattice is important in understanding the electronic properties of materials with a BCC crystal structure. It helps to determine the allowed energy levels and electronic band structure of a material, which has implications for its physical properties and potential applications.

How is the Brillouin Zone of a BCC lattice calculated?

The Brillouin Zone of a BCC lattice can be calculated using the reciprocal lattice vectors and the symmetry of the BCC crystal structure. This involves finding the perpendicular bisectors of the reciprocal lattice vectors and using them to construct the faces of the rhombic dodecahedron that forms the Brillouin Zone.

Can the Brillouin Zone of a BCC lattice change under different conditions?

Yes, the Brillouin Zone of a BCC lattice can change under different conditions, such as changes in temperature or external pressure. These changes can alter the electronic band structure of a material, which can affect its electrical and magnetic properties.

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