How Atoms Stack in Crystals: Unit Cells, Closest Packing and X-Ray Diffraction

Over 90% of natural and man-made solids are crystalline. Their particles settle into regular, repeating arrangements because packing them efficiently maximizes the attractions between them and minimizes their energy, and that atomic order often shows at a scale we can see.

The unit cell

A crystal is best described by its smallest repeating piece, the unit cell: a set of lattice points marking where atoms or ions sit, repeated in all three directions to build the whole solid.

A unit cell, repeated in every direction, makes the crystal. Image: OpenStax, CC BY 4.0.

How metals pack

Metals are the simplest case, since each is made of one kind of atom. Imagine stacking identical balls. There are three common cubic arrangements:

Atoms on the corners of a cube are shared with the eight cubes that meet there, so each counts as one-eighth; atoms on a face are shared by two, so each counts as half. So a simple cubic cell holds 8 × 1/8 = 1 atom, a BCC cell 1 + 1 = 2, and an FCC cell 1 + 6 × 1/2 = 4.

The three cubic unit cells: lattice points above, atoms below. Image: OpenStax, CC BY 4.0.

Closest packing. Because packing tightly lowers their energy, most metals pack as closely as possible, in layers of hexagonally arranged atoms. Stack the layers in a repeating ABCABC pattern and you get cubic closest packing, which is the same thing as FCC; stack them ABAB and you get hexagonal closest packing. Either way, each atom touches 12 neighbours: six in its own layer, three above and three below.

Hexagonal closest packing (ABAB) and cubic closest packing (ABCABC). Image: OpenStax, CC BY 4.0.

Sizes and densities from the unit cell

Knowing the structure and the length of a cell's edge gives the size of the atoms and the density of the metal.

• Polonium, simple cubic: neighbouring atoms touch along the edge, so the 336 pm edge equals two radii, and a polonium atom's radius is 168 pm. One atom's mass (its molar mass divided by Avogadro's number) inside a cube 336 pm on a side gives a density of 9.16 g/cm³.
• Calcium, FCC: here atoms touch across the diagonal of each face, which is four radii long. With an edge of 558.8 pm, the Pythagorean theorem gives a radius of 197.6 pm, and four atoms per cell give a density of 1.53 g/cm³.

The same arithmetic can rule a structure out: if nickel were simple cubic, its measured cell would give a density of 2.23 g/cm³, nowhere near its real 8.90.

Beyond cubes, a unit cell is defined by three edge lengths and the three angles between them; there are seven lattice systems and 14 kinds of unit cell in all.

Ionic crystals

Salts are harder to pack, because they combine ions of two sizes. Each ion attracts opposite charges equally in all directions, so stable structures surround every ion with as many oppositely charged ions as possible, all in contact. Usually the larger anions form a closest-packed array and the smaller cations tuck into the holes between them: small cations into tetrahedral holes (two for every anion), larger ones into octahedral holes (one per anion), and still larger ones into cubic holes in a looser, simple cubic array.

The bigger the cation, the bigger the hole it needs. Image: OpenStax, CC BY 4.0.

Which holes are filled, and how many, sets the formula. Zinc ions fill half the tetrahedral holes among sulfide ions: ½ × 2 = 1 zinc per sulfur, so ZnS. Aluminium ions fill two-thirds of the octahedral holes among oxide ions, a ratio of 2:3, so Al₂O₃, sapphire.

Three familiar structures:

• Cesium chloride (CsCl): ions of similar size (174 and 181 pm) in a simple cubic arrangement, with one ion at the centre of a cube of the other.
• Sodium chloride (NaCl): very different sizes (102 and 181 pm): chloride ions in an FCC array with sodium ions in the octahedral holes.
• Zinc blende (ZnS) and fluorite (CaF₂): FCC arrays with the smaller ions in tetrahedral holes, half of them in zinc blende, all of them in fluorite.

The sodium chloride structure. Image: OpenStax, CC BY 4.0.

Ionic radii can be worked out the same way as atomic ones (the chloride radius in lithium chloride comes out at 1.82 Å, or 0.182 nm), though such figures assume perfectly round ions and are only approximate.

Seeing the lattice: X-ray crystallography

X-rays have wavelengths about as long as the spacing between atoms in a crystal, so a crystal diffracts them. Waves scattered by neighbouring planes of atoms reinforce each other only at particular angles: when the extra distance one wave travels is a whole number of wavelengths. That condition is the Bragg equation, named after the English physicist W. H. Bragg:

n\lambda = 2d\sin\theta

where λ is the wavelength, d the spacing between the planes, θ the angle of the diffracted beam and n a whole number. A diffractometer measures those angles, and the equation turns them into distances between atoms: X-rays of 0.1315 nm diffracted by copper at 25.25° reveal planes 0.154 nm apart.

Constructive and destructive interference between X-rays scattered by two planes of atoms. Image: OpenStax, CC BY 4.0.

Rosalind Franklin

X-ray diffraction revealed one of biology's great secrets. The British chemist Rosalind Franklin, with doctoral student Raymond Gosling, found that DNA takes two forms, a long thin fibre when wet and a short wide one when dry, and her diffraction images gave Francis Crick and James Watson the crucial evidence that DNA is a double helix, which they described in 1953. Crick, Watson and Maurice Wilkins received the 1962 Nobel Prize. Franklin, who also did pioneering work on viruses and their RNA, died of ovarian cancer in 1958, aged 37.

An X-ray diffraction pattern like the ones Franklin recorded. Image: National Institutes of Health.

Sources

• Paul Flowers, William R. Robinson, Richard Langley and Klaus Theopold, Chemistry, "Lattice Structures in Crystalline Solids", OpenStax (Rice University), licensed under CC BY 4.0. Changed: rewritten in hubnx's own words and shortened, the worked examples summarized and the exercises left out; the figures are the book's, and the diffraction image is the National Institutes of Health's. This page is shared under the same licence.