I have a question about picking primitive lattice vectors and corresponding bases. I thought that you are free to pick whatever primitive lattice vectors you want, and that you would have to write your basis in those prim. lattice vectors.
When I look at exam question 2 of 2019 however, it seems that that is not the case.
If we were to pick our primitive lattice vectors as d(1/2,1/2,0) d(1/2,0,1/2) and d(0,1/2,1/2), wouldn’t our corresponding base be A(0,0,0) , B(1/2,1/2,1/2)? The current answer I thought would correspond to the prim vectors (1,0,0), (0,1,0), (0, 0, 1).
Vectors (1,0,0), (0,1,0), (0, 0, 1) would not be primitive. One should always be careful with non-orthogonal vectors: using the primitive lattice vectors d(1/2,1/2,0) d(1/2,0,1/2), d(0,1/2,1/2) would not give a basis atom B(1/2, 1/2, 1/2). Compare this problem with the diamond lattice exercise from the lecture notes.
But I’m still a bit confused. The primitive vectors are written in cartesian coordinates. Is the basis also written in cartesian coordinates? And if it is, isn’t there a term d missing? I don’t understand how to write down the basis.
By convention, the basis is written as coordinates of the atoms expressed as fractions of the primitive lattice vectors. Therefore, B: (1/4, 1/4, 1/4) means that there’s a B-atom at coordinates (\mathbf{a}_1 + \mathbf{a}_2 + \mathbf{a}_3)/4.
This is about the filling factor. I understand that the distance between the nearest neighbor is is 2r = (sqrt(3)*d)/4, and from there you can deduce r = (sqrt(3)*d)/8. For the filled volume, I’m a bit confused why the formula is 2 atoms * 4pi/3 * r^3. and then for the total volume, the formula is (d^3)/4. Is it true that we are looking at the following unit cell?
This is about the filling factor. I understand that the distance between the nearest neighbor is is 2r = (sqrt(3)*d)/4, and from there you can deduce r = (sqrt(3)*d)/8. For the filled volume, I’m a bit confused why the formula is 2 atoms * 4pi/3 * r^3. and then for the total volume, the formula is (d^3)/4. Is it true that we are looking at the following unit cell?
The volume you colored isn’t a unit cell at all: to check this observe that moving this unit cell by d/2 in x or y direction changes the crystal. To solve the problem you need to answer the following questions:
What is the volume of the primitive unit cell?
How many atoms are there in the primitive unit cell?
