⚛️Solid State Physics Unit 1 Review

Miller indices give you a compact, three-number label for any plane or direction inside a crystal lattice. They're the standard language crystallographers use to talk about plane orientations, and they connect directly to measurable quantities like diffraction peak positions and interplanar spacings.

This guide covers the notation, how to calculate Miller indices from scratch, and where they show up in practice.

Definition of Miller indices

Miller indices are a notation system for describing the orientation of planes and directions in a crystal lattice. Instead of trying to describe a plane's tilt in words, you reduce it to three integers that capture the plane's relationship to the crystallographic axes. Every plane in a crystal can be uniquely labeled this way, which makes it possible to compare results across experiments and communicate precisely about crystal geometry.

Notation for Miller indices

(hkl) format

Miller indices are written as three integers in parentheses: (hkl). Each integer comes from taking the reciprocal of the plane's fractional intercept along one of the three principal crystal axes (x, y, z). By convention, you write them without commas or spaces between the integers.

The parentheses matter. (hkl) refers to a single plane or a set of parallel planes. Curly braces {hkl} refer to all planes that are equivalent by the crystal's symmetry. Square brackets [hkl] denote a direction, and angle brackets ⟨hkl⟩ denote a family of equivalent directions.

Meaning of h, k, and l

h corresponds to the x-axis, k to the y-axis, and l to the z-axis.
Each index tells you the reciprocal of where the plane cuts that axis (in units of the lattice parameter).
A zero means the plane never intersects that axis, i.e., it runs parallel to it.

For example, (100) describes a plane that intercepts the x-axis at one lattice parameter and is parallel to both the y- and z-axes. (111) intercepts all three axes equally.

Conventions for negative indices

A negative index is written with a bar over the number: $(\bar{1}00)$ means the plane intercepts the negative x-axis.
Planes like $(100)$ and $(\bar{1}00)$ are parallel but face opposite directions. They're crystallographically equivalent in terms of their atomic arrangement.

Relationship to crystal planes

Plane orientation

Miller indices fully specify a plane's orientation relative to the crystal axes. All planes sharing the same (hkl) label are parallel to each other and have identical atomic arrangements. Changing even one index gives you a differently oriented set of planes.

Interplanar spacing

The distance between adjacent parallel planes with indices (hkl) is called the interplanar spacing, $d_{hkl}$ . For a cubic crystal with lattice constant $a$ :

$d_{hkl} = \frac{a}{\sqrt{h^2 + k^2 + l^2}}$

This formula shows why higher-index planes are more closely spaced. The (111) planes in a cubic crystal are separated by $a/\sqrt{3}$ , while the (100) planes are separated by $a$ . This spacing is what determines where diffraction peaks appear via Bragg's law.

Parallel planes

Planes whose indices are integer multiples of each other are parallel. For instance, (100), (200), and (300) all have the same orientation. However, (200) planes are spaced at $a/2$ rather than $a$ , so they represent a finer slicing of the lattice. In practice, (200) picks out every other (100)-type plane.

Calculation of Miller indices

Determining Miller indices from a plane's geometry is a straightforward recipe. Here's the step-by-step process:

Step-by-step procedure

Find the intercepts. Identify where the plane crosses the x, y, and z axes. Express each intercept as a multiple of the corresponding lattice parameter ( $a$ , $b$ , $c$ ). If the plane is parallel to an axis, that intercept is $\infty$ .
Take reciprocals. Invert each intercept. An intercept of $\infty$ becomes 0.
Clear fractions. Multiply all three reciprocals by the smallest number that converts them all to integers.
Reduce if needed. Divide by any common factor so the indices are the smallest set of integers with the same ratio.
Write in (hkl) format. Enclose the three integers in parentheses.

Worked examples for cubic crystals

Example 1: A plane intercepts at $(a,\; a,\; a)$ .

Fractional intercepts: (1, 1, 1)
Reciprocals: (1, 1, 1)
Miller indices: (111)

Example 2: A plane intercepts at $(a,\; a,\; \infty)$ .

Fractional intercepts: (1, 1, $\infty$ )
Reciprocals: (1, 1, 0)
Miller indices: (110)

Example 3: A plane intercepts at $(a,\; 2a,\; \infty)$ .

Fractional intercepts: (1, 2, $\infty$ )
Reciprocals: (1, 1/2, 0)
Multiply by 2 to clear the fraction: (2, 1, 0)
Miller indices: (210)

A common mistake is forgetting to take reciprocals first. If a plane hits the y-axis at $2a$ , the index is not 2. The reciprocal of 2 is 1/2, which becomes 1 after clearing fractions (depending on the other intercepts).

Significance in crystallography

Identification of crystal planes

Miller indices give every plane in a crystal a unique, standardized label. This lets researchers across different labs refer to the same plane unambiguously, whether they're discussing atomic arrangements, surface properties, or diffraction data.

Determination of crystal structure

X-ray diffraction is the primary technique for determining crystal structures, and Miller indices are central to interpreting the results. Each diffraction peak corresponds to a specific (hkl) plane. The peak's angular position tells you the interplanar spacing $d_{hkl}$ (through Bragg's law, $n\lambda = 2d\sin\theta$ ), and its intensity depends on what atoms sit on that plane and where. By indexing all the peaks, you can work backward to determine the lattice type, lattice parameters, and atomic positions.

(hkl) format, Miller indices - Online Dictionary of Crystallography

Relation to diffraction patterns

In any diffraction experiment (X-ray, electron, or neutron), each spot or peak maps to a set of Miller indices. The pattern of which indices appear and which are absent reveals the crystal's symmetry. For example, in a body-centered cubic lattice, reflections where $h + k + l$ is odd are systematically absent. These selection rules are a direct consequence of the lattice type and are one of the first things you check when indexing a pattern.

Miller indices in different crystal systems

Cubic vs. non-cubic lattices

In cubic systems (simple cubic, FCC, BCC), all three lattice parameters are equal ( $a = b = c$ ) and all angles are 90°. This makes the relationship between Miller indices and geometry especially clean. The interplanar spacing formula $d_{hkl} = a/\sqrt{h^2 + k^2 + l^2}$ only works for cubic crystals.

For non-cubic systems (tetragonal, orthorhombic, monoclinic, etc.), the $d$ -spacing formulas become more complex because $a \neq b \neq c$ and/or the axes aren't all at right angles. The procedure for finding Miller indices is the same, but interpreting the geometry requires the full lattice parameters.

Special cases for hexagonal crystals

Hexagonal crystals use a four-index system called Miller-Bravais indices, written as (hkil):

The first three indices (h, k, i) relate to the three equivalent axes ( $a_1$ , $a_2$ , $a_3$ ) in the hexagonal basal plane, which are 120° apart.
The fourth index (l) corresponds to the c-axis, perpendicular to the basal plane.
The third index is always redundant: $i = -(h + k)$ . This constraint ( $h + k + i = 0$ ) is built into the system.

The reason for using four indices instead of three is to make crystallographically equivalent planes look equivalent in the notation. With only three indices, planes that are physically identical in a hexagonal crystal can end up with very different-looking labels.

Applications of Miller indices

X-ray diffraction analysis

XRD patterns consist of peaks at specific $2\theta$ angles. Each peak is labeled with the (hkl) of the planes responsible for that reflection. From the peak positions, you extract $d$ -spacings and then lattice parameters. From the peak intensities, you get information about atomic positions within the unit cell. Miller indices are the organizing framework for all of this analysis.

Electron microscopy

In transmission electron microscopy (TEM), electron diffraction patterns are indexed using Miller indices to identify crystal structure and orientation. Selected-area diffraction patterns show a grid of spots, each labeled with (hkl). High-resolution TEM images show lattice fringes whose spacings correspond to specific $d_{hkl}$ values, letting you directly visualize particular sets of crystal planes.

Surface and interface studies

Crystal surfaces are labeled by the Miller indices of the terminating plane. The (111), (110), and (100) surfaces of the same material can have very different atomic densities, bonding environments, and reactivities. For instance, the (111) surface of an FCC metal is the most densely packed and often the most stable, while the (110) surface is more open and tends to be more reactive. Grain boundaries and heterostructure interfaces are also described by the Miller indices of the planes on each side of the boundary.

⚛️Solid State Physics Unit 1 Review