So I'm a bit confused about this question
This question asks for the miller indices for the "families of planes". Is there a single set of Miller indices for each cubic unit cell which I can use to present all of the planes for that unit cell?
For a) I have: $(1, 0, 0)$, $(-1, 0, 0)$
For b) I have: $(0, 1, 0)$, $(0, -1, 0)$, $(0, 3, 0)$, $(0, -3, 0)$
For c) I have: $(3, 2, 0)$, $(-3, -2, 0)$ and I have no idea how to find the others for this one.
Also I noticed that the planes for each cubic unit cell has the same direction. I know that enclosing miller indices in square brackets represents a direction but isn't this just a vector, not a representation of a family of planes? Thank you.
Best Answer
Assume a 3D lattice and denote its reciprocal lattice basis vectors as $\vec{b}_{1,2,3}$. The symbol $\left(h,k,l\right)$ stands for all the planes orthogonal to the vector $h\vec{b}_{1}+k\vec{b}_{2}+l\vec{b}_{3}$ (also written $\left[h,k,l\right]$ as you stated), so in fact there is no difference between $\left(1,0,0\right)$ and $\left(-1,0,0\right)$ for instance.