Word Count : 498

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To show that the given wave function satisfies the Bloch theorem for an electron in a body-centered cubic (bcc) lattice, we need to prove that the wave function can be written in the form of a Bloch function: \[ \psi(x, y, z) = e^{i \mathbf{k} \cdot \mathbf{r}} u(\mathbf{r}), \] where \( \mathbf{k} \) is the wave vector and \( u(\mathbf{r}) \) is a periodic function with the periodicity of the lattice. Given wave function: \[ \psi(x, y, z) = 1 + \cos\left(\frac{2\pi}{b}(x + y)\right) + \cos\left(\frac{2\pi}{b}(y + z)\right) + \cos\left(\frac{2\pi}{b}(z + x)\right), \] where \( b \) is the _________ ______ ____ __________ _____ _________ ____ _________.
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