In the solutions of lecture 9: “crystal structure” question 1.2 a primitive unit cell is given for the BCC lattice. However, in lecture 10 we need to calculate the reciprocal lattice for a BCC lattice (exercise 1.3). There a different unit cell is used. Why is that? Because if I use the lattice calculated in lecture 9 I get a different answer.
There are many possible choices of primitive lattice vectors. They all span the same lattice. Same is true for the reciprocal lattice.
Sometimes it can be difficult to see that different sets of primitive lattice vectors span the same lattice. Drawing lattice points in the first octant of the axes system based on your found lattice vectors may help to visualize this.
yes, I know that, but is there a specific reason why they use another primitive unit cell in lecture 10 than in lecture 9? Because the one used in lecture 10 is not as trivial as the one used in lecture 9. So did they use another one just to show that you can also use other primitive unit cells? Or is there a more specific reason to use another primitive unit cell because of the situation or the question?
There is no specific reason for the different choices. The only thing we need to be careful about is that the direct and reciprocal lattice vectors have to correspond to each other, so if we choose the lattice vectors of FCC like in the exercise 1.2, the reciprocal lattice vectors won’t be like in 1.3 because those wouldn’t satisfy the condition a_i b_j = 2\pi \delta_{ij}.