Lattice parameters from direct-space images at two tilts

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Lattices in three dimensions are oft studied from the ``reciprocal space'' perspective of diffraction. Today, the full lattice of a crystal can often be inferred from direct-space information about three sets of non-parallel lattice planes. Such data can come from electron-phase or Z contrast images taken at two tilts, provided that one image shows two non-parallel lattice periodicities, and the other shows a periodicity not coplanar with the first two. We outline here protocols for measuring the 3D parameters of cubic lattice types in this way. For randomly-oriented nanocrystals with cell side greater than twice the continuous transfer limit, orthogonal +/-15 deg and +/-10 deg tilt ranges might allow one to measure 3D parameters of all such lattice types in a specimen from only two well-chosen images. The strategy is illustrated by measuring the lattice parameters of a 10 nm WC₁₋ₓ crystal in a plasma-enhanced chemical-vapor deposited thin film.