In question 1c they ask for a set of primitive lattice vectors in the case that delta is not 0. The given solution is (a, 0, 0), (0, a, 0), (0, 0, a) (see images). However I can’t see how that is a primitive set of lattice vectors. My own answer was: (a/2, a/2, delta), (a/2, 0, a/2), (0, a/2, a/2).
Could anyone clarify this?
I have another question about the follow up question 1d. According to my calculations the 3 in the bottom line in the solution shoud be a -1. Is this correct?
Actually if you take your basis and try to see what lattice shows when you apply it, it is not the one shown in the picture (talking into account the displacement). What happens now is that the periodicity doesn’t happen with the nearest neighbours which form the fcc but only the cubic lattice maintains that periodicity. To correctly describe this new solid you require a basis, that will account for this displaced atom and the others in the other faces. As they don’t ask you for this basis the answer is just that the primitive basis is the one of a cubic lattice with parameter a.
It’s a bit tricky and difficult to explain in text. Let me know if you’d like more clarification.
I believe you are correct yes.
I still don’t understand how a (a, 0, 0), (0, a, 0), (0, 0, a) is a primitive set of lattice vectors, because the unit cell that is spanned by these vectors contains more than 1 atom. I thought that a unit cell corresponding with a primitive set of lattice vectors should always contain precisely 1 atom.
That’s actually not fully correct. A primitive unit cell is that which contains only one lattice point. Which is not the same as an atom. For instance in multi atomic structures you never have only one atom in the primitive unit cell.
In this case in order to span the lattice you require to go beyond the first neighbours because one of the atoms is displaced. So you’d have only one lattice point per primitive unit cell but several atoms.
You can also remember that:
A lattice is a set of points, where the environment of each point is the same.
So you need to find those points which have the same environment. And in this case those would be the ones given in the answer.
Hope this clarifies a bit more☺️
