1.1 The lattice

Lithium niobate, LiNbO₃, is a crystal that does not occur in nature P. Its oxygen atoms form flat sheets stacked almost like close-packed spheres. Between the sheets sit octahedral holes, each caged by six oxygens. One third of the holes hold lithium, one third hold niobium, and one third are empty, and along the crystal's unique axis c they repeat in the order niobium, empty, lithium P.

The crystal has a threefold rotation axis along c and three mirror planes that contain it, 60° apart P. Nothing else: no inversion centre, no fourfold axis. That short list, point group 3m, fixes which electrical, optical and mechanical responses can exist at all (1.7).

Hexagonal to rhombohedral setting

Asymmetric unit, hexagonal axes, R3c :H

Coordinates: P-db. Expanded with the 18 operators of R3c :H, duplicates removed (counts and distances above).

Octahedral levels

O₆ centroids every c/6 in the cell expanded from the CIF derived: every level holds one Nb, one Li and one empty site, and along each column the order is Nb, empty, Li.

The octahedral column

Along +c in the column at (0, 0), the cations of the expanded cell and the empty sites run empty 0.146 → Li 0.281 → Nb 0.499 → empty 0.646 → Li 0.781 → Nb 0.999 derived: Nb, empty, Li, Nb, … going up c P, drawn as Li–Nb–(empty) chains of face-sharing octahedra P-comp. The O₆ centroids sit every c/6 (z = 0.146, 0.313, 0.479, …). Both cations sit off-centre toward +c: Nb 0.275 Å above its O₆ centroid, Li 0.713 Å above its O₃ face derived, the displacement that carries the polarization (1.2).

The c length equals two Nb–Li plus two Li–Nb spacings along the column: 2 × 3.922 Å + 2 × 3.010 Å = 13.864 Å P.

Structure determination. The full structure was solved at Bell Labs in 1966 by single-crystal X-ray diffraction (part 3) and neutron diffraction (part 4) at 24 °C, with polycrystal X-ray data from 24 to 1200 °C (part 5) P. The coordinates used here come from a later synchrotron refinement: 20 keV, 0.7 Å, 293 K, R = 0.015 P-db. The cell file expands the asymmetric unit with all 18 operators of R3c :H, removes duplicates, and locates the empty octahedra as O₆ centroids with no cation within 1.0 Å derived.

Three settings. Hexagonal (Z = 6, 30 atoms), rhombohedral (Z = 2, 10 atoms), and the orthohexagonal Cartesian frame used for tensors P-comp. A value of c near 13.86 Å is hexagonal; a value near 5.49 Å is the rhombohedral edge.

Mirrors are glides. The mirror operations of the space group carry a translation c/2 (operators such as −y, −x, ½ + z in the CIF), so in R3c they are c-glide planes; the point group 3m keeps only their rotational part, a mirror derived. The 3.054 Å Nb–Li distance is not perpendicular to c, correcting the 1966 description P.

Cleavage. The empty octahedra lie in the {01̅12} planes, midway between Li and Nb planes; weak bonding across that vacancy plane is the qualitative explanation P. In hexagonal fractions those planes are y + 2z = const; every empty site of the cell has y + 2z = 0.2917 (mod 1), the Li sites 0.5612 and the Nb sites 0.998 derived. The angle follows from the lattice alone: tan φ = c / (√3 a) for the plane normal, plane-to-c angle = 90° − φ = 32.749° derived.

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Unit cellR3c, hexagonal axes, Z = 6P-db derived

Nb 6aLi 6aO 18bempty 6a (dashed O₆)

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NbO₆LiO₆Empty sitesc, P_s

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Rhombohedral cellCleavage planeColumn

Hexagonal, 293 K

a 5.148 Å · c 13.863 Å

Rhombohedral, Z = 2

a_R 5.4943 Å · α 55.872°

(01̅12) to c

32.749° derived

Column

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Plane

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Turn on the rhombohedral cell: the 10-atom primitive cell grows from the corner and shares its long diagonal with c. The cleavage plane passes through empty octahedra only. With the column on, sweep the pointer along the cell edge: Nb, empty, Li light in order up c.