I am slightly confused about how we write the basis in fractional coordinates. For example in the FCC lattice when we consider the conventional unit cell with orthogonal lattice vectors, we see that we have 4 lattice points in the unit cell so it’s not conventional. But we don’t actually have 4 points fully contained within the lattice, we have fractions of corners that add up to 4. So how do we decide which of these points should be in the basis? For example in the FCC for the warm up exc we have 4 points in the basis, one in the origin and one halfway each planar direction in 3D. But how should I know its these 4 points I consider in the basis if all atoms at these points are of the same type. How do you know which points you include in your basis?
since there is in total one corner atom (8*1/8) in the unit cell, you have to put one corner atom in the basis. You can choose any of the 8 corner atoms. The most direct choice seems to be the one at the origin, but any choice is ok. Similarly, since there are 3 face atoms in the unit cell, you have to put 3 of them in the basis. Which 3 is again up to you. The ones closest to the origin seem to be a reasonable choice
I still don’t fully understand the concept of the basis. I often “know” the right basis but as of now that mostly feels like a guess. If for example, we want to determine the basis of the lattice in minitest 3 2022, I would assume you would need 1 atom of both layers. Or would any 2 atoms work? It’s not really clear to me.
If you have found a valid combination of lattice + basis, you should be able to construct the crystal structure using the lattice in combination with the basis. So do check that. Note that there are many different sets of basis vectors that allow you to do that, and there is no hard rule for choosing one set of basis vectors over the other. However, it is conventional to choose atoms that are closest to a lattice point and that reside at least partly within the unit cell. Therefore, to construct the fcc structure using a conventional unit cell, you typically put the atoms at 0,0 and those at (0.5,0.5,0), (0.5,0,0.5), (0,0.5,0.5) in the basis.