In minitest 3 they ask to give the primitive lattice for the structure of boron and nitrogen.
In the answer they take the boron atoms as lattice vectors, however I solved it using a middle point lattice like shown below (and as recommended in the lecture). This however gives very different answers compared to the exercise. Is this approach also valid?:
Absolutely, this approach is valid. The lattice points do not have to sit on atoms; they only need to form a Bravais lattice whose points all have the same environment. The centers of the hexagons do that, so choosing the “middle-point” lattice is allowed. We of course accept any valid answer as correct.
The reason your answer looks very different from the official one is mostly that you chose a different origin. The official solution puts lattice points on one type of atom, while you put them in the center of the hexagon. Both describe the same crystal, but the basis coordinates change.
There is one important thing to watch out for: the basis must describe the full 3D crystal, not only one layer. Since the next layer has B above N and N above B, a translation by h in the z direction is not a lattice vector for boron nitride. It swaps the two atom types. Therefore the primitive vector in the vertical direction should have length 2h, and the basis must include atoms in both layers.
So if your lower-layer basis is, for example,
B: (1/3,1/3,0), \qquad N: (-1/3,-1/3,0),
then you should also include the atoms in the next layer:
N: (1/3,1/3,1/2), \qquad B: (-1/3,-1/3,1/2),
where the last coordinate is fractional with respect to the lattice vector of length 2h.
So the short answer is: the middle-point lattice is fine, but make sure the basis contains all atoms needed to reconstruct the stacked boron nitride crystal. Different choices of lattice points can give different-looking lattice vectors and basis coordinates, but they are equivalent if they reconstruct the same crystal.
Thanks for the swift answer. I see now that I made a mistake with the basis, however it’s good to know that these lattice vectors are also fine to use.
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