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I am studying Solid state physics written by Kittel. I am stuck on chapter 2, section "Fourier analysis of the basis". In the book, $S_{\mathbf{G}}$ is an integral throughout the region of a single unit cell. However, the book defines the atomic form factor $f_j$, but this integral is conducted over all space. Why is this inconsistent?

Also another question arises in the next page. For bcc lattice, the book says that there are two atoms in the $x_1=y_1=z_1=0$ and $x_1=y_1=z_1=\frac12$. How about the other atoms, one of these is $(1,1,0)$? I wonder why we consider only two atoms. I guess this is related to the fact that at each corners one cell occupies 1/8 portion of the atom, and the 8 atoms at the corners are symmetric, so we should only consider one full atom at the corner and one atom in the center.

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I don't have the book with me, so you're going to have to be more specific about your first question. The second one I can answer:

When you look at the conventional unit cell of bcc, you see that the basis vectors are $\vec{R}_1=(a,0,0)$, $\vec{R}_2=(0,a,0)$, and $\vec{R}_3=(0,0,a)$, $a$ being the side length of the unit cell. If you start with an atom at the origin, and repeat it at every $\vec{R} = n \vec{R}_1 + m \vec{R}_2 + p \vec{R}_3$, with $n, m, p \in \mathbb{Z}$, you will get atoms at every corner of every unit cell. If you, however, put one atom at the origin and one at $(a/2, a/2, a/2)$ and repeat this pattern as before, you will end up with a bcc lattice.

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