Wigner-Seitz = Brillouin Zone?

[andrewm]

I am confused about the relation between the Wigner-Seitz cell and the first Brillouin zone.

My teacher explained that to find the Wigner-Seitz cell in real space, one draws lines between the lattice points and connects the perpendicular bisecting planes. This constructs the volume nearer to any point in real space than any other point.

My confusion arises from people saying "The Wigner-Seitz cell in the reciprocal lattice is the first Brillouin zone". Isn't that unclear?

Are the two algorithms equivalent for finding the first Brillouin zone?

Algorithm 1:

1. Transform spatial vectors to reciprocal vectors. Put a "dot" at the tip of each reciprocal vector.
2. Find the Wigner-Seitz cell in this new lattice by finding the volume closer to any dot than any other dot.
3. This cell is the first Brillouin zone.


Algorithm 2:

1. Find the Wigner-Seitz cell in real space.
2. Draw a vector to each point on the Wigner-Seitz surface.
3. Transform these vectors to reciprocal vectors using the usual 3 cross- and triple-product equations
4. The surface made up of the tips of these vectors is the first Brillouin zone.



So, are these the same thing? Which is correct?

 

[malawi_glenn]

The Wigner-Seitz cell in the reciprocal lattice is the first Brillouin zone

Is a correct statement.

 

[andrewm]

Yes, but does it mean that the Wigner-Seitz cell is constructed in the reciprocal lattice and that is the zone? Or that the Wigner-Seitz cell, constructed in real space, is then transformed to k-space, and that is the zone?

 

[malawi_glenn]

You construct the wigner-seitz cell in the reciprocal lattice = 1st Brilloun zone.

So first obtain the reciprocal lattice, then perform the same algoritm as you would have done to get the wigner-seitz cell in the "position lattice", but in the reciprocal space instead of the position lattice space.

 

[andrewm]

Excellent, thanks.