Is the question asking for the primitive translation vector of simple cubic or reciprocal lattice?

Question

Can anyone please give me a clue on what the question wants? Based on the question, I am clueless if it asks for primitive translation vector of simple cubic or reciprocal lattice? Because the form of the given $\mathbf k_1$, $\mathbf k_2$, $\mathbf k_3$ is very different from simple cubic one.

Answer 1

From what you wrote I understand the question as follows.

You first need to determine what the Brillouin zone is for a simple cubic lattice. I recommend you try to do this yourself, and you should find that for a simple cubic lattice of lattice parameter $a$, then the Brillouin zone is also simple cubic with reciprocal lattice parameter $2\pi/a$.

Once you have determined the boundaries of your Brillouin zone cube, then the question is asking: are the given $\mathbf{k}_1$, $\mathbf{k}_2$, and $\mathbf{k}_3$ inside the Brillouin zone? If they already are inside the Brillouin zone, then that's it. If they are not, then you are asked to construct an equivalent $\mathbf{k}$ vector that is inside the Brillouin zone.

So the final question you need to ask is: how do I build an equivalent $\mathbf{k}$ vector inside the Brillouin zone? To do this, you need to use the fact that, due to periodicity of the reciprocal lattice, you can add any linear combination of reciprocal lattice vectors to a given $\mathbf{k}$ vector and you will obtain an equivalent $\mathbf{k}$ vector. What you need to do is figure out which linear combination of reciprocal lattice vectors to add to make sure that the resulting vector is inside the Brillouin zone.

Answer 2

We assume that the cubic lattice has side length $a$. Then the first Brillouin zone will look like BZ $=(-\frac{\pi}{a},\frac{\pi}{a}]^3$ (why?) with periodic boundary conditions along all three directions. Because of the periodic boundary conditions, we know that k-vectors are only physical modulo any reciprocal lattice vector, since adding such a vector will take you back to the same pooint. So we want to find the reciprocal lattice vectors that add to $k_1$, $k_2$, and $k_3$ such that all of them are in the Brillouin zone. Now it's just a matter of trial and error to find the correct vector. \begin{align} k_i \equiv k_i + \frac{2\pi}{a}(l,m,n) \in BZ \end{align} where $l,m,n \in \mathbb{Z}$. These are the equivalent $k$-vectors inside the Brillouin zone.

Brillouin Zone https://eng.libretexts.org/Bookshelves/Materials_Science/Supplemental_Modules_(Materials_Science)/Electronic_Properties/Brillouin_Zones#:~:text=A%20Brillouin%20zone%20is%20defined,vectors%20drawn%20from%20the%20origin.