Minitest 3: Dimensionality error 1a and b

Physics Q&A
1

On the minitest i made the mistake of interpreting “basis: A = (0, 0, 0),(1/2, 0, 0),(0, 1/2, 0)” as a notation for x,y,z dimension. This howewer is a notation for a1, a2, a3 right? I was wondering if when no axis is mentioned you can just assume it’s the (a1,a2,a3) dimensionality?

Following this, my next question is about the shared answers of the minitest: in the answer of question 1b is stated “b3 = 2π/c(0, 0, 1)” without any concrete mention of the dimensionality we’re talking about here. Is it correct that 2π/c(0, 0, 1) is in x,y,z (instead of a1,a2,a3)? I would like to know if that’s true in order to verify I understood what I’ve done wrong.

2

There are two ways to specify the lattice basis: Cartesian coordinates and fractions of lattice vectors. The basis specified in the problem can only be correctly interpreted as the basis in fractions of lattice vectors because otherwise it would have a wrong dimensionality. The expression \mathbf{b}_3 = (2\pi/c)\times (0, 0, 1) has the dimensionality of inverse length like it should: the prefactor is 1/length, and the vector components itself are dimensionless.

Does that answer your questions?