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Parallel lattice planes

lattice parallel

 

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.

lattice parallel Layer bar122

 

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.

lattice parallel Layer bar122

 

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.

lattice parallel Layer bar122

 

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.

lattice parallel Layer bar122

 

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.

lattice parallel Layer bar122

 

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

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

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

lattice parallel Layer bar122 d hkl

 

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

lattice parallel Layer bar122

 

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.

lattice parallel Layer bar244 Layer 9

 

lattice parallel Layer bar244 Layer 9 -->