Reciprocal Lattice

A reciprocal lattice is a discrete lattice of vectors that describes the translational periodicity of a crystal in reciprocal space, rather than ordinary position space. Its vectors identify plane waves whose phases remain unchanged under every translation of the crystal’s Bravais lattice. It is fundamental to crystallography and condensed-matter physics, connecting crystal geometry with diffraction patterns and the behavior of waves in periodic materials. Reciprocal-lattice vectors have dimensions of inverse length; their points do not represent physical atomic positions. (iucr.org)

Mathematical definition

Let three primitive translation vectors $\mathbf a_1,\mathbf a_2,\mathbf a_3$ generate a three-dimensional direct lattice:

These vectors define a primitive unit cell. Using the convention common in physics, the reciprocal basis vectors satisfy

where the dot denotes the Euclidean inner product and $\delta_{ij}$ is the Kronecker delta. Every reciprocal-lattice vector is an integer linear combination

Equivalently, the reciprocal lattice consists precisely of the vectors satisfying

for every direct-lattice translation $\mathbf R$. This condition expresses the invariance of a plane wave under lattice translations. (ocw.mit.edu)

Writing $\Omega=\mathbf a_1\cdot(\mathbf a_2\times\mathbf a_3)$, the reciprocal basis is

Crystallographic texts frequently omit $2\pi$, instead defining $\mathbf a_i\cdot\mathbf b_j=\delta_{ij}$. The two conventions describe the same geometry with different numerical scales. The reciprocal basis is a scaled representation of the dual basis, identified with vectors through the Euclidean inner product. Applying the reciprocal construction twice, consistently using either convention, recovers the direct lattice. (dictionary.iucr.org)

Geometry and lattice planes

A nonzero reciprocal vector is perpendicular to a corresponding family of parallel planes. Their orientation is specified by Miller indices $(hkl)$. In the $2\pi$ convention, the indexed plane spacing is

For relatively prime indices, these are successive lattice planes. Multiplying all indices by an integer gives a reciprocal vector farther along the same direction, representing a higher-order spatial harmonic. Larger real-space spacings therefore correspond to smaller reciprocal-vector magnitudes. (dictionary.iucr.org)

For a simple cubic lattice with lattice parameter $a$, the reciprocal lattice is simple cubic with parameter $2\pi/a$. Face-centered and body-centered cubic lattices are reciprocal to one another. The construction depends on the complete translation lattice, not merely the shape of a chosen conventional cell; centering translations must be included when determining which indexed points belong to the reciprocal lattice. (dictionary.iucr.org)

Fourier representation

The reciprocal lattice supplies the allowed spatial frequencies of a lattice-periodic function. For example, a periodic electron density can be expanded as a Fourier series:

Thus reciprocal space is the natural domain for a crystal’s Fourier transform. For an ideal infinite periodic crystal, the transform contains discrete contributions at reciprocal-lattice nodes. Their weights encode the arrangement of matter within the repeating cell, rather than only its translation geometry. (iucr.org)

In diffraction, these weights are described by the structure factor, which specifies the amplitude and phase of each reflection. Within the kinematic approximation, reflection intensity is proportional to the squared magnitude of that factor. A geometrically possible reflection can consequently have zero intensity because contributions from different atoms cancel. Systematic absences arise from translational symmetry elements, including centering, glide planes, and screw axes. (dictionary.iucr.org)

Diffraction and the Ewald construction

For elastic scattering, let the incident and outgoing wave vectors be $\mathbf k_{\mathrm{in}}$ and $\mathbf k_{\mathrm{out}}$. Constructive interference from the translation lattice requires

Elasticity also requires equal wave-vector magnitudes, $2\pi/\lambda$. Together, these conditions are equivalent to Bragg’s law, conventionally written $2d\sin\theta=n\lambda$. They explain why crystal diffraction occurs at selected directions rather than continuously. (ocw.mit.edu)

The Ewald sphere expresses these conditions geometrically. A sphere of radius $2\pi/\lambda$, positioned so that the reciprocal origin lies on its surface, identifies accessible reflections wherever another reciprocal-lattice point intersects the surface. Rotating the crystal changes these intersections; changing wavelength changes the sphere’s radius. This construction underlies the interpretation of X-ray crystallography measurements. (iucr.org)

Brillouin zones and periodic waves

The first Brillouin zone is the Wigner–Seitz cell of the reciprocal lattice: the region closer to the origin than to any other reciprocal-lattice point. Its boundaries are perpendicular bisectors between reciprocal nodes, and translated copies tile reciprocal space. It is a primitive reciprocal cell, not the entire reciprocal lattice. (wikis.mit.edu)

Bloch’s theorem describes waves in periodic media as a plane-wave factor multiplied by a lattice-periodic function. Wave-vector labels differing by $\mathbf G$ have the same translation phases, allowing distinct modes to be organized within one Brillouin zone. This framework applies to electronic states and to electromagnetic waves in periodic structures. The zone is continuous: the wave vectors labeling modes inside it need not themselves be reciprocal-lattice nodes. (ocw.mit.edu)