# How to find the Miller indices for a family of planes

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.

• do your parents know that sometimes you like to do crystal math?
– Jim
Nov 30, 2017 at 14:22
• I hope they don't Nov 30, 2017 at 14:30

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.
• You are correct for a) and c). For b) one usually writes $\left(0,3,0\right)$ because you represent planes that cross the unit cell $x$-axis at $\frac{1}{3}$, $\frac{2}{3}$ and etc.. Nov 30, 2017 at 14:34