Say we've a cubic crystal of unity edge length.
A set of planes in such a crystal is specified by their miller indice as $(3 2 0)$. One of these planes then has intercepts on the axes as $(2 ,3 ,\infty)$. It's well known that the interplanar distance between two adjacent planes in a cubic lattice is given
$d_{hkl} = \frac{a}{\sqrt{h^2+k^2+l^2}}$,in our case this gives us
$d_{320} = 1/\sqrt13$.
Now if I look at the planes which form the Miller indices of $ 3 2 0$ I find that apart from a plane having intercepts as $(2 ,3,0)$ an adjacent parallel plane to this plane having the same indices will pass via origin $0,0,0$ and the distance between them is $d_{320} = 6/\sqrt13$
Which contradicts the above result! Can anyone please point out what's wrong in my reasoning?
