Miller Indices

 

The use of Miller indices for describing planes and directions within a crystal lattice is very common.  To obtain the Miller indices describing a plane, you use the following four-step procedure given by Pierret and is as follows:

 

1.                                          Set up coordinate axes along the edges of the unit cell and then note where the plane to be indexed intercepts the axes.  Then divide each intercept value by the unit cell length along the respective coordinate axis.  Record the resulting normalized intercept sent in the order x,y,z.

2.                                          Invert the intercept values (i.e. 1/intercepts)

3.                                          Using an appropriate multiplier, convert the 1/intercept set to the smallest possible set of whole numbers

4.                                          Enclose the whole-number set in curvilinear brackets

 

 

The Miller indices for a direction for cubic crystals turn out to be the same as the Miller indices for the plane normal to the direction.  For noncubic crystals, the procedure is more lengthy but just a straightforward calculation where you find the components of the direction vector in terms of the basis vectors followed by multiplying these components by a whole number to get the smallest set of whole numbers possible.

 

There exists the following notation:

 

(hkl)           Crystal Plane

{hkl}          Equivalent Planes

[hkl]           Crystal Direction

<hkl>         Equivalent Directions

 

The distance between adjacent crystal planes is given by: