Why do we use a non-primitive unit cell for the brillouin zone of these metals?

Question

I'm reading through a textbook and they defined a primitive unit cell as a unit cell with only one lattice point in it. The Brillouin zone is defined as a primitive unit cell of the reciprocal lattice (or equivalently the reciprocal unit cell to the real space primitive unit cell).

Then later on there are diagrams and discussions about the fermi surfaces of some metals such as Potassium and Calcium. Potassium is said to be in a BCC structure, and Calcium in an FCC structure. However, neither of these are their primitive unit cells (Potassium's BCC unit cells has 2 lattice points per unit cell).

I can't spot what I've misunderstood and can't find much online as it's not a very specific question.

Here's an example of the diagram. The wire framing is the Brillouin zone boundary.

Answer 1

When we are discussing about metals in real space, we use conventional unit cell to talk with. For example if we want to talk about Potassium without using conventional unit cell BCC, we would have to talk BCC with Truncated Octahedron(BCC Wigner-Seitz unit cell) which is hard to grasp the image. Therefore when we are discussing with other people we use conventional unit cell(BCC, FCC).

Whereas the BZ is defined as any primitive unit cell of the reciprocal lattice, having that ugly shape to discuss with. because this is defined using primitive unit cell, we have to think of the reciprocal lattice having the smallest cell, which in turn looks like a Truncated Dodecahedron(FCC wigner-Seitz unit cell).

So in result when we are talking about real space metals, we use the term in conventional unit cell, whereas when we are discussing about BZ in reciprocal space, we have to use primitive unit cell.