I am studying Solid State Physics by Ashcroft and Mermin. In Chapter 9, it is explained that the Fermi surface in weak periodic potential can be constructed from free electron Fermi sphere. The leading order modifications take place where the original Fermi sphere of free electron intersects with Bragg planes. There, the degeneration splits up, and the spherical surface deforms as shown in Fig.9.6. The resulting Fermi surface can be presented in reduced zone scheme as shown in Figure 9.9-9.10.

Following the textbook, I did the Problem 1 which calculates $\rho_1,\rho_2$ in Figure 9.6. Also, I follow the arguments in the textbook why nth Brillouin zone defined at the bottom of P.163 is indeed a primitive cell. In particular, it can be shown that one can move each piece of the nth Brillouin zone into the 1st Brillouin zone by translation of integer times of reciprocal vectors, the resulting "zone" just fits perfectly into the 1st Brillouin zone without any overlapping nor empty space.

Figure 9.10 of the textbook

Now, my question concerns the comments shown in the caption of Figure 9.10, as shown above. It reads: "... However, the cell is actually the first zone (i.e., is centered on $\mathbf K=0$) only in the figures illustrating the second zone surfaces. In the first and third zone figures $\mathbf K =0$ lies at the center of one of the horizontal faces, while for the fourth zone figures it lies at the center of the hexagonal face on the upper right (or the parallel face opposite it (hidden)). The six tiny pockets of electrons constituting the fourth zone surface for valence 3 lie at the corners of the regular hexagon given by displacing that hexagonal face in the [111] direction by half the distance to the face opposite it..."

Why the point $\mathbf K=0$ does not lies at the center? It seems to me, if the original center $\mathbf K=0$ is concerned, then after the translation where the nth zone is translated to the 1st zone, the original center will be moved to the outside of the 1st Brillouin zone by definition, instead of on its surface. Unfortunately I cannot see what I have missed. Many thanks for the explanation!


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