As the title states, can a Miller index for a cubic structure have 4 digits? If I have a structure with intercepts (2,8,3) on the x-y-z axes respectively, the following Miller index would be (12,3,8), which is not 3 digits.
[Physics] 4 digit Miller Index for a cubic structure
crystalssolid-state-physics
Related Solutions
From the diagram it is obvious that the Miller-Bravais index has redundant information, as the indices point along three directions that are 120° apart. That means that you can do a simple geometrical derivation using this diagram as your guide:
The third equation follows immediately from the vector addition of $\vec u$ and $\vec v$ - you can see that $\vec t$ points in the opposite direction.
It is equally obvious that, in the way I drew the picture, $\vec u = \vec{v'} + \frac12 \vec{u'}$ and $\vec v = \frac12 \vec{u'} - \vec v'$
Simple manipulation of these equations gets you to the expressions you quote.
Calculating the packing factor for a monoclinic lattice does seem a strange thing to do, but whatever the lattice the packing factor is simply the volume of all the atoms in the unit cell divided by the volume of the unit cell:
$$ \text{APF} = \frac{N_\text{atoms}V_\text{atoms}}{V_\text{cell}} $$
And for a primitive cell obviously $N_\text{atoms}=1$.
Best Answer
Four digit Miller index is sometimes used for hexagonal lattices. The idea is to have useful property: permutation of indexes gives an equivalent direction.
It is mentioned in wikipedia, by the way. So you could start from there.
Four-digit Miller indexes are never used for cubic structure. If you could provide the citation where you saw them the answer could be better.