Parallel lattice planes

This plane intercepts at -a on the b/2 on the y-axis and c/2 on the z-axis

It is a (122) plane.

Suppose we have a plane parallel to this, which intercepts the x-axis at -2a, the y-axis at b and the z-axis at c.

Taking reciprocals gives this plane an h index of -1/2, a k index of 1 and an l index of 1.

To remove the fraction we have to multiply throughout by 2.

This gives the Miller Index of (122).

This has the same index as the first plane.

We can draw in the rest of these (122) planes in these 2 unit cells.

This is a family of parallel planes with the same index.

Looking at the last of these planes, we can move the axes and index it again.

This plane intercepts the x-axis at a, the y-axis at -b/2 and the z-axis at -c/2.

This gives the Miller Index of (122), the same as before, but multiplied through by -1. Indices related in this way refer to identical planes.

These planes are separated by a distance, d_hkl, between each pair of planes.

In an orthogonal system, with lattice parameters a, b, and c, for a plane (hkl), this is given by:

\[{\left( {\frac{1}{{{d_{hkl}}}}} \right)^2} = {\left( {\frac{h}{a}} \right)^2} + {\left( {\frac{k}{b}} \right)^2} + {\left( {\frac{l}{c}} \right)^2}\]

For this example;
a = b = c = 5Å,
h = -1, k = 2, l = 2

\[{\left( {\frac{1}{{{d_{\bar{1}22}}}}} \right)^2} = {\left( {\frac{{ - 1}}{5}} \right)^2} + {\left( {\frac{2}{5}} \right)^2} + {\left( {\frac{2}{5}} \right)^2}\]

d₁₂₂ = 5/3 Å

We can draw in the (244) planes

\[{\left( {\frac{1}{{{d_{2244}}}}} \right)^2} = {\left( {\frac{{ - 2}}{5}} \right)^2} + {\left( {\frac{4}{5}} \right)^2} + {\left( {\frac{4}{5}} \right)^2}\]

d₂₄₄ = 5/6 Å

When you double the indices you get planes parallel to the original with half the d-spacing.