Miller index (crystallography)

Miller indices are a notational convention used in crystallography to specify points within a crystal or lattice structure. They define the manner in which a plane in the object intersects with its internal axes, using integers to represent each value. These integers define the intercepts of the specified internal location, enabling an observer to identify any exact point, direction, family of directions, plane, or family of planes contained within the structure.

The number of integers in a Miller index corresponds with the number of dimensions in the object under study. One-dimensional objects have a one-digit index, two-dimensional objects have a two-digit index, and three-dimensional objects have a three-digit index. Standard crystallographic Miller indices include three digits, but four-digit systems are sometimes used to define points within three-dimensional objects with hexagonal structures.

Miller indices were developed by the British mineralogist William Hallowes Miller. He published his indexing system in his 1839 Treatise on Crystallography.

Background

Crystallography is a scientific discipline that studies the manners and means by which atoms in crystalline structures form bonds and establish their positional locations. The United Kingdom-based Institute of Physics traces the formal history of the branch to 1912, when a research team led by German physicist Max von Laue (1879–1960) conducted a series of experiments that directed X-ray beams through different types of crystals. At the time, scientists had not yet determined whether X-rays were a type of electromagnetic wave. Von Laue’s studies not only proved that X-rays were waves but also demonstrated that X-rays diffracted in varied but specific directions when beamed through different types of crystals. The scientific community interpreted this phenomenon as convincing empirical evidence that crystals featured lattice-shaped internal atomic structures, in which atoms arrange themselves into sets of unique groupings that repeat in sequential order, with each grouping displaying identical structures each time it repeats. Von Laue was awarded the 1914 Nobel Prize in Physics for his discovery.

Von Laue’s findings gave Miller’s earlier work a new application. Miller was a mineralogist based at the University of Cambridge, and his research into minerals and crystals led him to develop an indexing system for identifying specific points within solid crystalline structures. His 1839 Treatise on Crystallography is considered a foundational work in the nascent scientific discipline, which at the time was in its earliest and most rudimentary stages.

During Miller’s lifetime, the science of crystals mainly revolved around expert debates regarding the exact nature of the order and symmetry observed within their structures. As early forms of modern atomic theory developed during the nineteenth century, Miller and other crystallographers applied its principles to crystal structures, correctly theorizing that crystals were defined by tiny particle arrangements organized in ordered forms. Von Laue’s X-ray discoveries in the early 1910s yielded convincing proof of the theory’s accuracy, enabling later generations of crystallographers to identify the precise forms and features of their atomic structures. Miller indices proved useful in defining specific individual points within crystalline lattice structures and are still used in this capacity in the twenty-first century.

Overview

Miller indices identify individual points as well as directions and planes, both of which have specific definitions within the context of crystallography. A crystallographic direction refers to the spatial orientation of formative growth within the crystal structure and can also describe the direction of a crystal’s original face, which is a flat surface that defines and encloses the crystal’s form. Crystallographic planes are features of lattice structures that run individually or in parallel families that intersect with one another at periodic points. The integers used in Miller indices identify the direction and planes along which specific locations within the crystalline structure are situated, thus allowing observers to define any particular point, direction, plane, directional family, or planar family within it.

Miller indices use five specific types of notation that identify individual points, individual directions, families of directions, individual planes, and families of planes. The notational form for describing an individual point is written as (a,b,c), in round parentheses with each given value separated by an unspaced comma. The integers a, b, and c each represent a particular value along the x-, y-, and z-axes that define three-dimensional space, respectively.

To define a direction, Miller indices use the form [abc], in square brackets with each integer presented separately with no comma separation or spaces. A family of directions is indicated by < abc > in angled brackets with no comma separation or spaces.

Individual planes are notated as (abc) in round parentheses without commas or spaces separating the integer values. A family of planes is written as {abc}, in curly brackets without comma separation or spaces between the integers.

Points within a three-dimensional crystalline structure are defined by their positions along the x-, y-, and z-axis, with each integer describing the number of unit-lengths along each axis that must be traveled to reach the point’s location. Directions are determined by vectors, which pass from their point of origin to a specific destination within the lattice. The [abc] values within an individual direction indicate the basic vector value that applies to each dimension within a three-dimensional object. To determine the (abc) values for an individual plane, the observer must first locate the position of the plane intercepts on each of the x-, y-, and z-axes. These intercepts are then converted into their fractional coordinates, and the reciprocals of each fractional intercept become the integer expressed in the index.

Miller indices can also define points, directions, and planes that define lattice-shaped objects of fewer than three dimensions. In these cases, the number of integers presented in each index corresponds to the number of dimensions in the object. For example, a point in a two-dimensional object is written as (a,b), while an individual direction is given as [ab], an individual plane is written as (ab), and so on.

Bibliography

“Crystallography.” Institute of Physics, 2021, www.iop.org/explore-physics/physics-stepping-stones/crystallography. Accessed 18 June 2021.

“Crystallography: Miller Indices.” University of Georgia,www.clay.uga.edu/courses/8550/millerindices.html. Accessed 18 June 2021.

Glazer, A.M. “A Brief History of Crystallography.” Oxford University Press, 27 May 2016, blog.oup.com/2016/05/a-brief-history-of-crystallography/. Accessed 18 June 2021.

Kelly, Anthony and Kevin M. Knowles. Crystallography and Crystal Defects. John Wiley & Sons, 2020, pp. 3–40.

Khan, Allaudin and Shumsun Naher Begum. Solid State Physics: Structure and Properties of Materials. GRIN Verlag, 2020.

Haynes, Adam. Fundamentals of Crystallography. Research Press, 2019.

Li, Shunde, et al. "Coherent Growth of High-Miller-Index Facets Enhances Perovskite Solar Cells." Nature, 14 Oct. 2024, doi.org/10.1038/s41586-024-08159-5. Accessed 8 Nov. 2024.

“Miller Indices.” Libre Texts,28 Mar. 2021, chem.libretexts.org/Bookshelves/Physical‗and‗Theoretical‗Chemistry‗Textbook‗Maps/Book%3A‗Surface‗Science‗(Nix)/01%3A‗Structure‗of‗Solid‗Surfaces/1.02%3A‗Miller‗Indices‗(hkl). Accessed 18 June 2021.

Nelson, Stephen A. “Axial Ratios, Parameters, Miller Indices.” Tulane University,7 Sept. 2016, www.tulane.edu/~sanelson/eens211/axial‗ratios‗paramaters‗miller‗indices.htm. Accessed 18 June 2021.