Miller Indices

Given the importance of lattice planes to the description and understanding of crystalline systems, it is helpful to have a simple and concise means to refer to the various planes in a given lattice system. The absolute location of the plane is not so important in this context; only the orientation is needed. The reciprocal lattice provides a convenient means to this end. Each vector on the reciprocal lattice is normal to a plane on the direct lattice, so to specify a direct-lattice plane it is sufficient to specify its reciprocal-lattice vector. Any lattice vector is expressible as a sum of integer multiples of three primitive vectors, so three integers are sufficient to specify the reciprocal vector, and hence the direct-lattice plane. These three integers are known as the Miller indices of the plane.

The conventional notation for the Miller indices encloses them in parentheses: (hkl). For example, the (100) plane is that having normal vector n = (1)a* + (0)b* + (0)c* = a* (again, a*, b*, c* are the primitive vectors of the reciprocal lattice). Here one is reminded of two important points: (1) the Miller indices are multipliers of primitive vectors of the reciprocal lattice, not the direct lattice; (2) the reciprocal lattice vectors, and hence the unit normal, and hence the plane's orientation, depend on the structure of the direct lattice, so the same Miller indices applied to different lattices might refer to planes of different orientation.

It is conventional to choose Miller indices with elements having no common multiplier, because multiplication of all indices by the same integer does not change the orientation of the plane being described. Thus the (111) and (222) planes are equivalent. Also, it is conventional to use an overbar to describe a negative index. In this module it is easier instead to put the minus sign before the integer, so we do not use the overbar notation.

The reciprocal of the three integers in the Miller index are the intercepts of the plane with the a, b, and c axes, respectively. Sometimes the Miller index is defined in these terms, rather than as coordinates of the normal in the basis of the reciprocal-lattice primitives.

Hexagonal lattices employ a variant of the three-index scheme. For these systems the index comprises four integers: (hklm). The extra one is among the first three and is redundant, conveying no additional information. In fact the first three indices must sum to zero. Referring to the figure (which looks down the c vector), the (other two) primitive vectors a and b are shown in red, and the reciprocal-lattice primitives are in blue. An alternative choice for the primitive vector b is shown in yellow, and use of it changes the reciprocal vector a* to the one shown in green. The third miller index is the multiplier for a* if the yellow b primitive were used. Miller indices for three planes are indicated using the four-index scheme.

A direction in a lattice is distinct from the concept of a plane. Unlike plane specifications, directions are described in terms of multiples of the direct-lattice primitive vectors. The notation for directions uses square brackets: [hkl]. So [100] is in the direction of a, [120] is a + 2b, etc.