Miller Indices and Crystal Planes in Cubic Crystals

Crystal Planes and Miller Indices Quick Take

This concise overview explains how unit cells and atomic radii connect to packing, density, and plane orientation in cubic crystals, and introduces Miller indices as a compact labelling system for directions and planes. Real-world examples such as rock-salt and cubic-diamond illustrate how a lattice plus a basis yields actual materials.

• lattice versus basis concept
• Miller indices and direction brackets
• plane spacing formula and diffraction relevance
• anisotropy and packing in different planes

Introduction: Crystallography Foundations

The lecture revisits the three Bravais lattices and shows how knowing the unit cell, together with atomic radii, lets you estimate packing fractions, atomic packing fractions, and density. The density is expressed in terms of the cell parameters, the number of atoms per cell, grams per mole, and Avogadro's number, illustrating how microscopic geometry links to macroscopic material properties. The copper example demonstrates how to combine periodic-table data with lattice type to compute density and related properties.

"the lattice is a stamp without putting things there yet" - Lecturer

Lattice and Basis: Stamp and Repetition

The lecturer explains the distinction between the lattice (the stamp) and the basis (what you place at each stamp). This framework accounts for how crystals like rock-salt are formed when an FCC lattice is paired with a two-atom basis (sodium and chlorine). Similarly, cubic diamond uses an FCC lattice with a carbon pair that repeats, yielding a diamond structure. These examples emphasize that the same Bravais lattice can produce different crystals depending on the basis.

"The basis is what to repeat" - Lecturer

Miller Indices: Describing Directions in Crystals

The talk then introduces Miller indices as the crystallographer's language for directions. A vector from the origin is described by projections onto X, Y, Z (in units of the lattice constants A, B, C) and reduced to the smallest integer values with square brackets. Negative components are indicated by a bar, and equivalent directions form families such as the 100 family. This notation keeps crystallographers happy by staying within a standardized convention while navigating the cubic lattice symmetry.

"A Miller plane is not just one plane. A Miller plane is an infinite set of planes" - Lecturer

Miller Planes: Labeling Planes and Spacing

A Miller plane is represented by the triplet (h, k, l) and stands for an infinite, equally spaced set of planes. The reciprocal intercepts are reduced to integers by taking reciprocals and clearing fractions. The spacing between adjacent planes in a cubic cell follows d = a / sqrt(h^2 + k^2 + l^2). For example, the 100 plane has spacing a, while the 200 plane has spacing a/2. This distance rule underpins diffraction and anisotropic properties of crystals.

"the distance between two of them is the length of the edge of the unit cell A divided by the square root of the squares added together"

Plane Density and Anisotropy in Simple Cubic and Beyond

The density of atoms in a plane depends on how the plane cuts the lattice. In a simple cubic lattice, the 100 plane contains roughly one atom per unit area, giving a packing fraction of 1/A^2. The 110 plane has a different cross-sectional area and packing, illustrating how plane orientation and packing affect material properties such as fracture behavior and conductive pathways. The discussion extends to real materials like nickel and copper, where lattice type and plane orientation strongly influence mechanical response and transport properties.

This framework ties back to the broader goal of crystallography: understanding how atomic-scale arrangement governs macroscopic properties through geometry, symmetry, and packing in the crystal lattice.