Proving Greatest Density of Points in {111} & {110} Planes

[Chillguy]

Homework Statement

Prove that the lattice planes with the greatest densities of points are the {111} planes in a fcc bravis lattice and the {110} planes in a bcc bravis lattice.

Homework Equations

d/v=points per unit area where d is the spacing of planes and v is the unit volume.

The Attempt at a Solution

In the fcc case
$$d=\frac{2\pi}{hb_1+kb_2+lb_3}\\ b_1=\frac{2\pi}{a}(y-x+z)$$
Which is in terms of the reciprocal space. So we simply need to maximize this value. Does this immediately tell us it is in the {111} miller index because the reciprocal is given in terms of 3 coordinates?

 

[Asim Riaz]

In the bcc case, we have d=\frac{2\pi}{hb_1+kb_2+lb_3}\\b_1=\frac{2\pi}{a}\big(\frac{x+y+z}{2}\big)And we see that the same process applies in finding the maximum.