Crystal lattices and the unit cell

Result

Crystal lattices and the unit cell

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Statement

Describing periodic solids from a repeating motif.

Why it matters

A crystalline solid can contain on the order of \(10^{23}\) atoms, yet its entire structure is described completely by specifying just one small repeating unit and the rule for translating it through space. The unit cell is that repeating unit, and establishing how to define and count atoms within one is the essential first step this whole unit builds on: bragg-law's diffraction analysis measures the spacing between planes defined by the lattice established here, close-packing-efficiency computes how densely spheres fill specific lattice arrangements, band-theory's orbital overlap depends on the periodic arrangement fixed here, and born-lande-lattice-energy sums electrostatic interactions over exactly this repeating geometric arrangement of ions.

Before any of those quantitative results can be applied, a solid's structure has to be reduced to a lattice (an abstract, infinite array of points) decorated with a motif (the actual atom, ion, or group of atoms placed at each point) and packaged into the smallest repeating box, the unit cell, that generates the whole crystal by simple translation.

Hypotheses

The crystal is treated as perfectly periodic and infinite, ignoring surface effects, edge effects, and any defects in the real, finite sample.Real crystals always terminate somewhere and generally contain some concentration of defects (vacancies, dislocations, impurities); the unit-cell description is an idealisation of the bulk, periodic interior, valid as a model for the overwhelming majority of atoms in any macroscopic sample, where surface-to-volume ratio is negligible. Every lattice point is occupied by an identical motif in an identical orientation.This is what makes the structure genuinely periodic and describable by translation alone; without it, no single small repeating unit could regenerate the whole structure by simple translation, and a far larger, non-periodic description would be required instead. The seven crystal systems and fourteen Bravais lattices constitute an exhaustive classification: it is a mathematical fact, not an empirical observation, that no arrangement of translationally periodic points in three dimensions falls outside these fourteen lattice types, since they are derived from the complete set of point-symmetry-compatible ways of tiling three-dimensional space.

Proof

\text{Unit cell defined by lattice parameters } a, b, c, \alpha, \beta, \gamma

Three edge lengths and three inter-axial angles fully specify a parallelepiped-shaped repeating box; restricting the relationships allowed among these six parameters (e.g. requiring \(a=b=c\) and \(\alpha=\beta=\gamma=90^\circ\) for a cubic cell) generates exactly the seven distinct crystal systems, each a different symmetry-constrained subset of the general parallelepiped. A

\text{Bravais lattices: primitive (P), body-centred (I), face-centred (F), base-centred (C)}

Within a given crystal system, additional lattice points can be placed at the body centre, on all six faces, or on one pair of opposite faces, without breaking that system's required symmetry; enumerating every symmetry-consistent centring option across all seven crystal systems gives exactly fourteen distinct Bravais lattices, no more and no fewer. B

\text{Fractional atom count per cell: corner} \times\tfrac18,\ \text{edge}\times\tfrac14,\ \text{face}\times\tfrac12,\ \text{body (interior)}\times1

An atom sitting exactly on a corner is shared among the eight unit cells that meet at that corner, so only \(1/8\) of it belongs to any one cell; by identical reasoning an edge atom is shared among four cells (\(1/4\) each), a face atom between two cells (\(1/2\) each), and a body-interior atom belongs entirely (\(1\)) to the single cell containing it. Summing these fractional contributions over every atom position in the cell gives the true number of atoms per unit cell. A

\text{Simple cubic: } 8\times\tfrac18=1 \qquad \text{BCC: } 8\times\tfrac18+1=2 \qquad \text{FCC: } 8\times\tfrac18+6\times\tfrac12=4

Applying Step 3's counting rule to the three common cubic Bravais lattices: a simple cubic cell has atoms only at its eight corners (net one atom); a body-centred cubic (BCC) cell adds one full interior atom (net two); a face-centred cubic (FCC) cell adds one atom on each of its six faces (net four) — these three net counts (1, 2, 4 atoms per cell) are exactly the standard results used throughout the rest of this unit. A

Result

N_{\text{cell}} = \sum_{\text{positions}} n_i \times f_i, \qquad f_{\text{corner}}=\tfrac18,\ f_{\text{edge}}=\tfrac14,\ f_{\text{face}}=\tfrac12,\ f_{\text{body}}=1

Reading. The unit cell is the smallest repeating box, defined by six lattice parameters, that regenerates an entire crystal by pure translation; the true number of atoms it contains is found by summing fractional contributions according to exactly where in the cell each atom sits.

Scope. Applies to any crystal describable by one of the fourteen Bravais lattices with a single-atom motif; molecular or multi-atom motifs (most real ionic and molecular crystals) require multiplying the lattice-point count by the number of atoms in the motif, a direct extension covered in Problems.

Corollaries & converses

close-packing-efficiency computes packing fraction directly from the atom count and cell volume established here, showing FCC and the closely related hexagonal close-packed structure both achieve the maximum possible packing density for identical spheres.
bragg-law's diffraction planes are defined by Miller indices, themselves expressed relative to the same unit-cell axes \(a, b, c\) fixed by this result — diffraction measurement and unit-cell geometry are two sides of the same description.
Converse: measuring a crystal's density experimentally and comparing it against the mass-per-unit-cell computed from Step 3's atom count (together with the cell's measured dimensions) is a standard method for confirming which Bravais lattice a given crystalline solid actually adopts.

Fails without

Drop the infinite-periodicity assumption and apply unit-cell counting directly at a crystal's surface or a grain boundary: atoms at these locations do not sit within the idealised, fully repeating bulk environment the fractional-counting rule (Step 3) assumes, and a separate, explicit defect or surface treatment is required instead.
Assign different chemical species to what should be equivalent lattice points (as with CsCl, mistaken for a BCC lattice): since Cs+ and Cl− are not an identical motif, the corner and body-centre positions are not truly equivalent by translational symmetry, and CsCl is correctly described only as simple cubic with a two-ion motif, not as a genuine BCC Bravais lattice.

Common errors

Counting every atom drawn in a unit-cell diagram as a full, whole atom belonging entirely to that cell, ignoring the fractional-sharing rule of Step 3 for corner, edge, and face positions.
Confusing the seven crystal systems (symmetry classes defined by relationships among \(a,b,c,\alpha,\beta,\gamma\)) with the fourteen Bravais lattices (which further specify centring within a system) — the two classifications are related but not identical in count or meaning.
Assuming a base-centred (C) lattice is possible in every crystal system; centring types compatible with a given system's symmetry are restricted, which is exactly why the enumeration yields fourteen lattices rather than the naively expected larger number.
Forgetting to account for a multi-atom motif when a real ionic or molecular crystal is involved, applying the atom-counting rule as though a single atom, rather than a full formula unit or molecule, sat at each lattice point.

Discussion

Auguste Bravais proved in 1848 that exactly fourteen distinct lattice types are geometrically possible in three-dimensional space, a landmark result of pure geometry that predates any direct experimental means of observing atomic arrangements — X-ray diffraction, the technique that would eventually confirm crystal structures directly (bragg-law), was not developed until the 1910s, more than sixty years later.

The seven crystal systems are themselves grouped, on symmetry grounds, into six crystal families (trigonal and hexagonal share a family owing to their closely related symmetry), a further layer of classification generally introduced only once the basic seven-system, fourteen-lattice framework given here is secure.

Common misconception: that the unit cell shown in a textbook diagram is somehow the "true," physically bounded repeating chunk of the crystal, rather than simply the most convenient mathematical choice. Any of infinitely many differently shaped cells could in principle tile the same lattice; the conventional unit cell is chosen specifically to display the lattice's full symmetry as clearly and compactly as possible, not because it is the uniquely correct physical unit.

Worked examples

\text{Rock salt (NaCl): interpenetrating FCC lattices of Na}^+\text{ and Cl}^-

Treating Cl− ions as occupying an FCC lattice (Step 4: net 4 per cell) and Na+ ions as occupying the octahedral holes of that same lattice (equivalent to a second, offset FCC lattice, also net 4 per cell), the conventional NaCl unit cell contains \(4\) Na+ and \(4\) Cl− ions, i.e. \(4\) formula units of NaCl per unit cell. A

Z = 4 \text{ formula units of NaCl per conventional unit cell}

Reading. A multi-atom (here, two-ion) motif is handled by applying the fractional-counting rule (Step 3) separately to each ion's own sublattice, then combining the results into a single formula-unit count for the cell as a whole.

Scope. The identical approach, applied to whatever motif occupies each lattice point, gives the correct formula-unit count \(Z\) for any ionic or molecular crystal, not only the elemental, single-atom-motif cases of Step 4.

Problems

A polonium crystal adopts a simple cubic lattice with a monatomic motif. State the number of atoms per unit cell, and identify which of Step 4's three cases applies.
Solution
Simple cubic has lattice points only at the eight corners, each shared among eight cells: \(8\times\tfrac18=1\) atom per unit cell, matching the simple cubic case of Step 4 directly. Polonium is, in fact, the standard textbook example of an element crystallising in this comparatively rare, low-packing-efficiency structure.
Confirm that a body-centred cubic (BCC) unit cell contains exactly two atoms, by explicitly summing the fractional contributions from every corner and the body-centre position.
Solution
Eight corner atoms, each shared among eight cells: \(8\times\tfrac18=1\). One body-centre atom, entirely interior to this one cell: \(1\times1=1\). Total: \(1+1=2\) atoms per unit cell, matching the BCC case of Step 4.
Caesium chloride (CsCl) is sometimes mistaken for a body-centred cubic structure, but it is not, strictly, a BCC Bravais lattice. Explain why, referencing the Hypotheses' requirement that every lattice point carry an identical motif.
Solution
In CsCl, Cs+ sits at the cell corners while Cl− sits at the body centre (or vice versa) — two chemically different ions occupy geometrically distinct positions. A true BCC Bravais lattice requires every lattice point, corner and body-centre alike, to carry an identical motif (Hypotheses); since the corner and body-centre positions in CsCl are occupied by different ions, they are not equivalent by the lattice's own translational symmetry, and CsCl is correctly described as a simple cubic lattice with a two-ion (Cs+, Cl−) motif, not as a BCC lattice of a single ion type.