Miller indices

Introduction

Named after the British crystallographer and mineralogist William Hallowes Miller in 1839^[1], the Miller indices are a convenient tool to characterize sets of lattice planes in crystalline materials. They were derived from the Weiss indices which are determined by how the lattice planes intercept the crystallographic axis. The indices can also be used to describe directions in the crystal. Therefore, the notation is important to distinguish between the different meanings (planes, directions) of the indices. For the special case of hexagonal systems, the Miller indices evolved to the Bravais-Miller indices to describe the system even better.

Especially in diffraction techniques the Miller indices are essential to understand and interpret the results.^[2]

Determination

Weiss indices

Weiss indices are a method to define a specific lattice plane in crystals by how it intercepts the crystallographic axis of the unit cell. Figure 1 shows a 2D scheme of how to determine the Weiss indices with of a square lattice.

Figure 1. Scheme of a square lattice with lattice lines (blue) and the corresponding Weiss indices (Figure: Elisabeth Albrecht).

The first number in the brackets is the interception of the lattice line with the a-axis. The second number states the interception with the b-axis. The interceptions are given in terms of the lattice parameters a and b. In 3D there would be a third number for the c-axis.^[3]

Negative numbers for negative interceptions are also possible, they are marked with a bar over the number, like \( (\bar{3} 2 2) \) . Often it is more convenient to use the negative sign instead:  \( (-3 2 2) \) .

Miller indices

In practical use, Weiss indices are often not very convenient. The numbers can get very big, depending on the origin of the axis and since the lattice is repetitive, planes with Weiss indices that are multiple of each other are equivalent. Due to where the origin is set, for instance the (111) and the (222) planes are equal. Another point is, that planes exist, that have no interception with one or two axis at all and therefore the axis is marked with an infinity symbol, ∞ (see Figure 1, (2 ∞)), which often is quite difficult in use.

To make the indices more convenient, Miller used the reciprocal values of the plane-axis interceptions and took the smallest multiple for which all numbers are integers as indices. Table 1 shows a few examples of Weiss indices and the corresponding Miller indices.

Table 1. Weiss indices and the corresponding Miller indices.

With the Miller indices, not only one lattice plane but a whole set of parallel, equivalent lattice planes is indexed. Since the smallest integer values are used, the indices for parallel planes are the same. Additionally there are no fractions anymore and the ∞ turns into 0, which makes the use much more convenient.^[2][4] Figure 2 shows some Miller indices for a cubic crystal.

Figure 2. Some of the most commonly used Miller indices for lattice planes in a cubic system. (Figure: Nea Möttönen, inspired by reference^[5]

Bravais-Miller indices

Hexagonal systems in general can be described by Miller indices like all other crystal systems, however they are often described by Bravais-Miller indices. Those will be further explained in the “Notation” section, in “Hexagonal systems”.^[3]

Directions

Not only lattice planes but also crystallographic directions can be indexed by the Miller indices. Miller indices define the direction that is perpendicular to the corresponding lattice plane. For instance, in a cubic crystal, the [100] direction would be along the a-axis of the lattice and perpendicular to the plane (100) that only intersects the a-axis. Figure 3 shows the [11] direction, which is the diagonal in a 2D lattice.^[4]

Figure 3. Diagonal direction in a 2D lattice with the corresponding Miller indices (Figure: Elisabeth Albrecht).

Notation

The generalized Miller indices are called h, k and l, for the a-, b- and c-axis interception, respectively.

For sets of parallel planes conventionally round brackets (hkl) are used whereas for directions square brackets [hkl] are used.^[2]

Cubic systems

In the special case of cubic systems, where all lattice parameters a, b and c are the same, not only planes which indices are multiples of each other are equivalent but also planes, where the order of the indices is shifted. The simplest example are the (100), (010) and (001) planes. Those are no multiples of each other but are equivalent planes due to the same repetition in all three directions. To clarify the content, different notations are used, that are shown in Table 2.

Table 2. Notation of Miller indices for crystallographic planes and directions.^[3]

In other words, the three sets of lattice planes (100), (010) and (001) can be described as a set of equivalent lattice planes {100}. Also for directions, [120], [012] and [201] as an example can be summed up to the equivalent directions <120>.

Hexagonal systems

It is common to use an additional index i for hexagonal systems. This set of indices (hkil) is called Bravais-Miller-indices. The index i is not an independent one, it corresponds to the -a -b -axis of a hexagonal lattice (Figure 4).

Figure 4. Hexagonal 2D lattice mesh with a-, b- and -a -b -axis (Figure: Elisabeth Albrecht).

It depends in the following way on the Miller indices:

-h – k = i

Other systems

For other crystal systems, only the (hkl) for planes and [hkl] for directions perpendicular to the corresponding planes are used, since they not necessarily repeat in all directions the same way.^[2][3][4]

Plane spacing

Once one has determined the Miller indices of a certain set of planes, those can be used to calculate the distance d between two planes (Fig. 5).

Figure 5. Shortest distance d between two planes of the set of planes (11). (Figure: Elisabeth Albrecht).

For each crystal system there is a special formula, including the Miller indices, the lattice parameters and the angles between the unit cell axis (in case they are not 90°). in Table 2 the crystal systems with the referring equations are listed. Here a, b, c are the lattice parameters, α, β and γ are the angles between the axis and d is the distance between two planes (see Figure 5).

Table 2. Plane spacing equations for the different crystal systems.^[6]

Use

The Miller indices are not only used to define lattice planes and directions theoretically. In practical, they can be seen in diffraction patterns, especially in X-ray diffraction methods (Figure 6). A diffraction pattern does not show the real lattice but the reciprocal lattice and each dot corresponds to a whole set of lattice planes, which is why Miller indices are a useful tool to describe them. The Miller indices are defined in reciprocal space, each dot on the pattern is defined by a specific combination of (hkl).

Figure 6. Diffraction pattern of a cubic lattice along the [100] direction. (License: CC BY 3.0)^[7]

References