For a lattice plane of any orientation there will be a set of parallel planes which can be drawn such that they pass through all points other lattice points. For obvious reasons all parallel lattice planes are termed a set of lattice planes:
Miller indices of lattice planes:
A set of lattice planes intercept each of the crystallographic axes a finite number of times per unit length of axis. By convention, the number of intercepts of planes per unit length along the x-axis is termed h, per unit length along the y-axis is termed k and per unit length along the z-axis is termed l. The unit length along each axis is taken to be the length of the unit cell parallel to that axis, that is a for the x-axis, b for the y-axis and c for the z-axis. The Miller Indices of the plane are then (hkl). Thus the set of lattice planes (hkl) divides a into h parts, b into k parts and c into l parts. An alternative way of conceptualising Miller indices is to consider the first plane out from the origin in any set of planes. This plane will make the intercepts a/h, b/k, c/l (where h, k and l are integers), on the x-axis, y-axis and z-axis respectively.
In some sets of lattice planes the intercepts the
first plane out from the origin intercepts one or more of the axes in the
negative direction along those axes. The Miller indices with respect to
those axes are then written with a bar over the top of the index
hkl) (which is read bar "h", bar "k" bar "l").
Lattice planes parallel to one of the axes will have no intercept with that axis and therefore have the Miller index of 0 with repect to that axis. Thus, in terms of Miller indices the unit-cell can be described as the parallelepiped bounded by adjacent lattice planes of the set (100), (010) and (001).
The intersection of two non-parallel lattice planes defines a lattice row. Such a row is indexed with reference to the parallel row passing through the origin. The indices of the line [UVW] are taken to be the coordinates of the first lattice point out from the origin through which that line passes. Note that lattice row reference indices are always enclosed in square brackets "" while plane reference indices are enclosed in round brackets"()".