2.1 Reciprocal lattice

The reciprocal lattice is a mathematical construction that re-expresses a crystal's real-space periodicity in Fourier (wavevector) space. It turns questions about plane spacings, diffraction geometry, and electron states into simple point-lattice problems, making it one of the most-used tools in solid state physics.

Definition of reciprocal lattice

The reciprocal lattice is the set of all wavevectors $\vec{G}$ for which a plane wave $e^{i\vec{G}\cdot\vec{r}}$ has the same periodicity as the real-space crystal. Equivalently, it's the Fourier transform of the direct lattice. Every point in the reciprocal lattice maps to a family of parallel lattice planes in real space, and the distance from the origin to that reciprocal lattice point is inversely proportional to the spacing of those planes.

Relationship to real space lattice

Three key connections link the two lattices:

• Each reciprocal lattice point $\vec{G}_{hkl}$ corresponds to the set of $(hkl)$ planes in real space.
• The magnitude $|\vec{G}_{hkl}|$ equals $2\pi / d_{hkl}$, where $d_{hkl}$ is the interplanar spacing. Closely spaced planes produce reciprocal points far from the origin, and vice versa.
• The direction of $\vec{G}_{hkl}$ is perpendicular to the $(hkl)$ planes.

Reciprocal lattice vectors

The reciprocal lattice basis vectors $\vec{b}_1, \vec{b}_2, \vec{b}_3$ are built from the real-space primitive vectors $\vec{a}_1, \vec{a}_2, \vec{a}_3$:

$\vec{b}_1 = 2\pi \frac{\vec{a}_2 \times \vec{a}_3}{\vec{a}_1 \cdot (\vec{a}_2 \times \vec{a}_3)}$

$\vec{b}_2 = 2\pi \frac{\vec{a}_3 \times \vec{a}_1}{\vec{a}_1 \cdot (\vec{a}_2 \times \vec{a}_3)}$

$\vec{b}_3 = 2\pi \frac{\vec{a}_1 \times \vec{a}_2}{\vec{a}_1 \cdot (\vec{a}_2 \times \vec{a}_3)}$

Note that the denominator is the same in all three expressions: it's the unit cell volume $V_{\text{cell}} = \vec{a}_1 \cdot (\vec{a}_2 \times \vec{a}_3)$.

These vectors satisfy the orthogonality condition:

$\vec{a}_i \cdot \vec{b}_j = 2\pi\,\delta_{ij}$

where $\delta_{ij}$ is the Kronecker delta (1 when $i=j$, 0 otherwise). This condition is what guarantees that the plane wave $e^{i\vec{G}\cdot\vec{R}}=1$ for every direct lattice vector $\vec{R}$.

Construction of reciprocal lattice

To build the reciprocal lattice from a known real-space lattice:

1. Identify the primitive vectors $\vec{a}_1, \vec{a}_2, \vec{a}_3$ of the direct lattice.
2. Compute the cell volume $V_{\text{cell}} = \vec{a}_1 \cdot (\vec{a}_2 \times \vec{a}_3)$.
3. Calculate $\vec{b}_1, \vec{b}_2, \vec{b}_3$ using the cross-product formulas above.
4. Generate reciprocal lattice points as all integer combinations $\vec{G} = h\vec{b}_1 + k\vec{b}_2 + l\vec{b}_3$.

Primitive vectors in reciprocal space

The vectors $\vec{b}_1, \vec{b}_2, \vec{b}_3$ are the primitive vectors of the reciprocal lattice. They span reciprocal space the same way $\vec{a}_1, \vec{a}_2, \vec{a}_3$ span real space. The reciprocal unit cell they define has volume $(2\pi)^3 / V_{\text{cell}}$.

Reciprocal lattice for cubic lattices

The cubic systems produce a clean and important pattern:

• Simple cubic (SC) with lattice constant $a$ → reciprocal lattice is SC with constant $2\pi/a$.
• Body-centered cubic (BCC) → reciprocal lattice is FCC. This is a result you'll use constantly; it follows from plugging the BCC primitive vectors into the formulas.
• Face-centered cubic (FCC) → reciprocal lattice is BCC.

The BCC ↔ FCC duality is worth memorizing. It shows up in diffraction selection rules, Brillouin zone shapes, and band structure calculations.

Reciprocal lattice for hexagonal lattices

The reciprocal lattice of a 2D hexagonal net is also hexagonal but rotated 30° relative to the direct lattice. For a hexagonal lattice with constants $a$ (in-plane) and $c$ (out-of-plane), one standard choice of reciprocal basis is:

• $\vec{b}_1 = \frac{2\pi}{a}\left(\frac{2}{\sqrt{3}},\; 0,\; 0\right)$
• $\vec{b}_2 = \frac{2\pi}{a}\left(\frac{1}{\sqrt{3}},\; 1,\; 0\right)$
• $\vec{b}_3 = \frac{2\pi}{c}\left(0,\; 0,\; 1\right)$

The exact components depend on how you orient the real-space vectors, so always derive them from the cross-product definitions rather than memorizing a single set of coordinates. The key physical fact is the 30° rotation and the $2\pi/a$, $2\pi/c$ scaling.

Properties of reciprocal lattice

Periodicity in reciprocal space

Because the direct lattice is periodic, any function with the crystal's translational symmetry (electron density, potential, etc.) can be expanded in a Fourier series over reciprocal lattice vectors $\vec{G}$. The reciprocal lattice itself is infinite and periodic, with its periodicity set by $\vec{b}_1, \vec{b}_2, \vec{b}_3$.

Reciprocal lattice and Brillouin zones

The first Brillouin zone (1st BZ) is the Wigner-Seitz cell constructed around the origin of the reciprocal lattice. You build it by drawing perpendicular bisector planes between the origin and every neighboring reciprocal lattice point; the smallest enclosed volume is the 1st BZ.

• It contains every wavevector $\vec{k}$ that is closer to the origin than to any other reciprocal lattice point.
• Higher-order Brillouin zones are formed by the regions bounded by the next set of bisector planes.
• The volume of each Brillouin zone equals $(2\pi)^3 / V_{\text{cell}}$, the same as the reciprocal unit cell.

Brillouin zones are central to electronic band structure because Bloch's theorem restricts unique electron states to a single zone.

Reciprocal lattice and diffraction patterns

Diffraction experiments (X-ray, electron, neutron) produce patterns that are direct images of the reciprocal lattice. The Laue condition for constructive interference is:

$\vec{k}' - \vec{k} = \vec{G}$

where $\vec{k}$ and $\vec{k}'$ are the incident and scattered wavevectors. This is equivalent to Bragg's law $2d\sin\theta = n\lambda$, but expressed in reciprocal-space language. Each diffraction spot corresponds to a reciprocal lattice point, and the spot's intensity depends on the structure factor $S(\vec{G})$, which encodes the atomic arrangement within the unit cell.

Applications of reciprocal lattice

Reciprocal lattice in X-ray diffraction

An X-ray diffraction pattern is essentially a map of the reciprocal lattice weighted by the structure factor. From the positions of diffraction peaks you extract lattice parameters and symmetry; from the intensities you determine atomic positions within the basis. The Ewald sphere construction provides a geometric way to predict which reciprocal lattice points satisfy the diffraction condition for a given wavelength and crystal orientation.

Reciprocal lattice in electron diffraction

Electron diffraction works on the same reciprocal-lattice principles but with some practical differences. Electrons have much shorter wavelengths (at typical TEM energies), so the Ewald sphere is nearly flat, and many reciprocal lattice points are excited simultaneously. This makes electron diffraction patterns look like 2D slices through the reciprocal lattice, which is useful for quick symmetry identification.

Reciprocal lattice and band structure

Electronic band structure $E(\vec{k})$ is plotted as a function of wavevector $\vec{k}$ within the first Brillouin zone. The shape of the BZ (and therefore the reciprocal lattice) determines:

• The high-symmetry points and paths along which bands are conventionally plotted (e.g., $\Gamma$, $X$, $L$, $K$ for FCC).
• Where band gaps open due to Bragg reflection at zone boundaries.
• The topology of the Fermi surface.

Without the reciprocal lattice framework, computing or interpreting band structures would be far more cumbersome.

Reciprocal lattice vs real space lattice

Fourier transform relationship

Any function with the periodicity of the lattice, such as the electron density $\rho(\vec{r})$, can be expanded as:

$\rho(\vec{r}) = \sum_{\vec{G}} \rho_{\vec{G}}\, e^{i\vec{G} \cdot \vec{r}}$

The Fourier coefficients are:

$\rho_{\vec{G}} = \frac{1}{V_{\text{cell}}} \int_{V_{\text{cell}}} \rho(\vec{r})\, e^{-i\vec{G} \cdot \vec{r}}\, d\vec{r}$

This is the mathematical backbone of the reciprocal lattice concept. Real-space periodicity becomes a discrete sum over $\vec{G}$ vectors, turning differential equations into algebraic ones.

Duality of real and reciprocal space

The two lattices are dual: the reciprocal lattice of the reciprocal lattice gives back the original direct lattice (up to a factor of $(2\pi)^2$ depending on convention). The orthogonality relation $\vec{a}_i \cdot \vec{b}_j = 2\pi\,\delta_{ij}$ is the formal statement of this duality. Large real-space cells produce finely spaced reciprocal lattices, and small real-space cells produce coarsely spaced ones.

Advantages of reciprocal lattice representation

• Diffraction: Scattering conditions reduce to a single vector equation ($\Delta\vec{k} = \vec{G}$) instead of geometric arguments about plane families.
• Band structure: Bloch's theorem naturally lives in $\vec{k}$-space, so the reciprocal lattice and BZ are the native coordinate system for electronic states.
• Phonons: Dispersion relations $\omega(\vec{k})$ are likewise defined in reciprocal space.
• Simplicity: Many problems that involve sums over lattice planes or periodic potentials become straightforward when recast as sums over $\vec{G}$.