Crystals are repeating patterns
In a crystal the atoms sit in a pattern that repeats in all three directions. The smallest box that builds the whole crystal when it is copied side by side is the unit cell. A cubic cell is described by one length, the lattice constant a, a few hundred picometers for most solids; a hexagonal cell needs two, a and c. Metals, salts, ice, minerals and most solid elements are crystals; glass and many plastics are not.
Counting atoms per cell
Atoms on the surface of a cell are shared with its neighbors. A corner atom belongs to 8 cells, an atom on an edge to 4, and an atom on a face to 2, so
Z = N_corner ÷ 8 + N_edge ÷ 4 + N_face ÷ 2 + N_inside
- Simple cubic (polonium, the only element with this structure): atoms on the 8 corners, Z = 1. Each atom touches 6 neighbors.
- Body-centered cubic (iron at room temperature, chromium, tungsten, sodium): corners plus one atom in the middle, Z = 2, 8 neighbors.
- Face-centered cubic (copper, aluminum, silver, gold, nickel): corners plus the middle of each face, Z = 8 × ⅛ + 6 × ½ = 4, 12 neighbors.
- Hexagonal close-packed (magnesium, zinc, titanium): layers stacked ABAB, Z = 2 in the small cell, 12 neighbors.
The number of nearest neighbors is the coordination number.
Packing fraction
Treating atoms as hard spheres that touch along the shortest distance, the share of the cell they fill is
packing fraction = Z × (4/3)πr³ ÷ V
In simple cubic the atoms touch along the edge, r = a ÷ 2, giving π ÷ 6 = 52.4%. In body-centered cubic they touch along the body diagonal, r = √3 a ÷ 4, giving 68.0%. In face-centered cubic they touch along the face diagonal, r = √2 a ÷ 4, giving 74.0%, the same as hexagonal close-packed. Kepler guessed in 1611 that 74% is the densest any packing of equal spheres can be; Thomas Hales proved it in 1998.
Ionic and covalent crystals
A salt is two interpenetrating lattices, one of each ion. Which one forms depends roughly on the radius ratio r₊ ÷ r₋: from 0.414 to 0.732 each ion sits among 6 others, above 0.732 among 8, and from 0.225 to 0.414 among 4.
- Rock salt, NaCl: two face-centered lattices shifted by half a cell edge; 6:6 coordination, 4 formula units per cell. Na⁺ ÷ Cl⁻ = 102 ÷ 181 = 0.56.
- Cesium chloride, CsCl: a Cs⁺ in the middle of a cube of Cl⁻; 8:8 coordination, 1 formula unit per cell. Cs⁺ ÷ Cl⁻ = 174 ÷ 181 = 0.96. It is not body-centered cubic, because the middle atom is a different kind.
- Zinc blende, ZnS: two face-centered lattices shifted by a quarter of the body diagonal; each atom has 4 neighbors at the corners of a tetrahedron.
- Diamond: the zinc blende structure with every atom carbon. It fills only 34% of space, because covalent bonds point in fixed directions. Silicon and germanium have the same structure.
Density from the unit cell
The mass in one cell divided by its volume gives the density:
ρ = Z M ÷ (N_A V)
with M the molar mass of one formula unit, N_A = 6.022 × 10²³ per mole and V = a³ for a cubic cell or (√3 ÷ 2) a² c for the hexagonal one. Run backward, with a measured by X-rays and ρ by weighing, this is one of the ways the Avogadro constant has been measured.
Miller indices
Planes through a crystal are named by three whole numbers (h k l). The planes of the family cut the cell edges at a ÷ h, a ÷ k and a ÷ l; a 0 means they run parallel to that edge. So (1 0 0) are the faces of the cube, (1 1 0) cut diagonally across it, and (1 1 1) cut off the corners. In a cubic crystal the spacing between neighboring planes is
d = a ÷ √(h² + k² + l²)
and in a hexagonal one 1 ÷ d² = 4(h² + hk + k²) ÷ 3a² + l² ÷ c². Hexagonal crystals are often given four indices (h k i l), with i = −(h + k), so that planes related by the sixfold symmetry look alike.
Bragg's law
X-rays have wavelengths about the size of the spacing between atoms. In 1912 Max von Laue showed that crystals diffract them; in 1913 William Henry Bragg and his son William Lawrence Bragg explained the angles. X-rays reflected from neighboring planes travel paths that differ by 2d sin θ, where θ is the angle between the beam and the planes. They add up only when that difference is a whole number of wavelengths:
n λ = 2 d sin θ
The detector sits at 2θ from the incoming beam. Since sin θ cannot exceed 1, planes closer together than λ ÷ 2 never reflect. A second-order reflection (n = 2) from (1 1 1) is the same as a first-order reflection from planes half as far apart, so it is usually written (2 2 2).
The structure factor
Bragg's law says where a peak could be; whether it appears depends on every atom in the cell:
F = Σ f_j e^(2πi (h x_j + k y_j + l z_j))
where x_j, y_j and z_j are the atom's position as fractions of the cell edges and f_j is how strongly it scatters, roughly its number of electrons at low angles and less at high angles. The intensity goes as |F|². This gives the systematic absences:
- Body-centered: F = 0 unless h + k + l is even. Iron has no (1 0 0) peak; its first is (1 1 0).
- Face-centered: h, k and l must be all odd or all even. Copper shows (1 1 1), (2 0 0), (2 2 0), (3 1 1), (2 2 2).
- Diamond: as face-centered, and also no peak when h + k + l is twice an odd number, so (2 0 0) and (2 2 2) vanish.
- Rock salt: peaks with all-odd indices have F = 4(f_Cl − f_Na), so (1 1 1) is weak. In KCl, K⁺ and Cl⁻ have the same 18 electrons, those peaks almost disappear, and the pattern looks like that of a simple cubic crystal with half the cell edge.
- Hexagonal close-packed: no peak when h + 2k is a multiple of 3 and l is odd, such as (0 0 1) and (1 1 1).
In a powder, tiny crystals face every way, so each family of planes gives a ring at its own 2θ. The list of angles and intensities is the powder pattern, a fingerprint used to identify materials.
Worked examples
- The density of copper: face-centered, Z = 4, a = 361.5 pm = 3.615 × 10⁻⁸ cm, so V = 4.724 × 10⁻²³ cm³. ρ = 4 × 63.55 ÷ (6.022 × 10²³ × 4.724 × 10⁻²³) = 8.94 g/cm³. The measured value is 8.96.
- Copper's first peak: for (1 1 1), d = 361.5 ÷ √3 = 208.7 pm. With copper Kα X-rays, λ = 154.06 pm, sin θ = 154.06 ÷ (2 × 208.7) = 0.369, θ = 21.66° and the peak is at 2θ = 43.32°.
- Iron's missing peak: body-centered with a = 286.65 pm. The (1 0 0) peak would be at 2θ = 31.2°, but h + k + l = 1 is odd, so F = 0. The first peak is (1 1 0): d = 286.65 ÷ √2 = 202.7 pm and 2θ = 44.67°.
- A lattice constant from a peak: salt shows a strong peak at 2θ = 31.70° with copper Kα. d = 154.06 ÷ (2 sin 15.85°) = 282.0 pm. If it is the (2 0 0) peak of a face-centered lattice, a = 2d = 564.0 pm.
- Packing in face-centered cubic: 4 spheres of r = √2 a ÷ 4 give 4 × (4/3)π(√2 a ÷ 4)³ ÷ a³ = π ÷ (3√2) = 0.740.
Using the simulation
In the Crystal Lattice 3D Viewer, things to try:
- Step through simple cubic, body-centered and face-centered cubic in space-filling view and compare the packing fractions; then cut the atoms at the faces to see the eighths and halves that make up Z.
- Choose body-centered iron and set (1 0 0): the reflection is absent. Set (1 1 0) and find its peak.
- Choose diamond and set (2 2 2): absent. Compare its powder pattern with copper's, which has the same face-centered lattice.
- Choose rock salt and compare the weak (1 1 1) peak with the strong (2 0 0).
- Choose simple cubic and set (3 3 3): with copper radiation λ is longer than 2d, so no angle works. Switch to molybdenum.
- Drag the lattice constant down until the atoms overlap, and watch the density rise as 1 ÷ a³ and the peaks move to higher angles.
- Turn on the 3 × 3 × 3 block to see how one cell builds the crystal and how the planes run through it.
What the model assumes
- A perfect, infinite crystal: every cell identical, no vacancies, impurities, dislocations or surfaces.
- Hard spheres: atoms are spheres of fixed radius: metallic radii for the metals, ionic radii for NaCl and CsCl, covalent radii for zinc blende and diamond. The packing fraction and the overlap warning depend on that choice.
- One lattice constant per structure, measured near room temperature; magnesium keeps its measured ratio c ÷ a = 1.624 when you change a.
- A simple scattering factor: f = Z e^(−2s²) ÷ (1 + 3s²), with s = sin θ ÷ λ in Å⁻¹. It starts at the number of electrons and falls with angle, a fit that roughly matches measured patterns of copper and salt; real values come from tables.
- Intensity is |F|² times the Lorentz-polarization factor (1 + cos² 2θ) ÷ (sin² θ cos θ), summed over all the planes with the same spacing. Absorption, preferred orientation and the thermal motion of atoms are not modeled separately.
- One sharp wavelength: Kα is treated as a single line, and each peak is drawn as a narrow curve of fixed width.
Edge cases
- All three Miller indices 0: no planes, so no d and no peak.
- Absent reflections: the planes are drawn and d is shown, but F = 0 and there is no peak. Body-centered (1 0 0), diamond (2 0 0) and (2 2 2), magnesium (1 1 1).
- λ > 2d: high indices in a small cell, such as (3 3 3) in polonium with copper radiation; sin θ would have to exceed 1. Molybdenum's shorter wavelength reaches more planes.
- Peaks beyond 150°: the pattern stops at 150°, near the limit of many diffractometers.
- Lattice constant far from the real one: the density and peaks follow, but the atoms keep their radii, so a squeezed cell shows packing above 74% or even above 100% and an overlap warning, and a stretched one has atoms that no longer touch. Neither is a real crystal.
- Changing the structure resets the lattice constant to that example's measured value.
- Shared atoms in the 3 × 3 × 3 block: the count Z is always for one cell.
- Hexagonal indices: the three-index (h k l) is also shown as the four-index (h k i l); planes such as (1 0 0) and (0 1 0) are equivalent there.
Where the model stops being right
- Atoms are not hard spheres. Radii change with the number of neighbors and the type of bonding, and ions are squeezed and polarized. Radius-ratio rules predict the structure of many salts wrongly.
- Real crystals have defects. Vacancies, impurities and grain boundaries make measured densities differ a little from the X-ray density, and are what make metals bend.
- Temperature. Crystals expand on heating (copper by about 17 parts per million per kelvin), and atoms vibrate, which weakens high-angle peaks. Some change structure: iron turns face-centered above 912 °C.
- Bonding electrons. The structure factor treats each atom as a round cloud. In diamond the electrons in the bonds make the forbidden (2 2 2) peak appear weakly, and the shapes of real electron clouds follow orbitals like those in the Atomic Orbitals 3D Viewer.
- Real X-ray sources and samples. Copper Kα is a doublet that splits peaks at high angles; small crystals broaden peaks; elements such as iron fluoresce under copper radiation; and crystals lying in a preferred direction distort the intensities. Measured patterns are fitted with full models (Rietveld refinement) that include all of these.
- Not everything is a simple crystal. Alloys with atoms mixed at random, quasicrystals with order but no repeat, and glasses with no long-range order at all, which give broad humps instead of sharp peaks.
Related tools
The Molar Mass Calculator gives M for the density equation, and How to Calculate Molar Mass explains it. The Scientific Notation Converter handles numbers like 10⁻²³ cm³, and the Length Converter turns picometers into ångströms and nanometers. Diffraction from a crystal is interference, as in the Double-Slit Experiment Simulator. For the electrons inside each atom, see Atomic Orbitals Explained: s, p, d and f; for single molecules rather than crystals, the 3D Molecule Viewer.