Reciprocal Lattice

The reciprocal lattice vectors are defined by:

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

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

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

Every reciprocal lattice point is at:

$\mathbf{G} = m_1\mathbf{b}_1 + m_2\mathbf{b}_2 + m_3\mathbf{b}_3, \quad m_i \in \mathbb{Z}$

Key property: $\mathbf{G} \cdot \mathbf{R} = 2\pi \times \mathrm{integer}$So $e^{i\mathbf{G}\cdot\mathbf{R}} = 1$.

Proof of key property.

$\mathbf{G} \cdot \mathbf{R} = (m_1\mathbf{b}_1 + m_2\mathbf{b}_2 + m_3\mathbf{b}_3) \cdot (n_1\mathbf{a}_1 + n_2\mathbf{a}_2 + n_3\mathbf{a}_3)$

By the orthogonality relation $\mathbf{b}_i \cdot \mathbf{a}_j = 2\pi\delta_{ij}$:

$\mathbf{G} \cdot \mathbf{R} = 2\pi(m_1 n_1 + m_2 n_2 + m_3 n_3) = 2\pi \times \mathrm{integer}$

$\blacksquare$

The first Brillouin zone is the Wigner-Seitz cell of the reciprocal lattice. It is the set of Points closer to the origin than to any other reciprocal lattice point.

For FCC (real space), the reciprocal lattice is BCC. The first Brillouin zone is a truncated octahedron. For BCC (real space), the reciprocal lattice is FCC.

Important volumes:

$V_{\mathrm{BZ} = (2\pi)^3 / V_{\mathrm{cell}}}$.

Theorem 2.1. The reciprocal lattice vector $\mathbf{G}_{hkl} = h\mathbf{b}_1 + k\mathbf{b}_2 + l\mathbf{b}_3$ Is perpendicular to the real-space planes $(hkl)$And $\lvert\mathbf{G}_{hkl}\rvert = 2\pi/d_{hkl}$ where $d_{hkl}$ is the interplanar spacing.

Proof. The plane $(hkl)$ has intercepts $\mathbf{a}_1/h$, $\mathbf{a}_2/k$, $\mathbf{a}_3/l$. Two vectors in this plane are $\mathbf{a}_2/k - \mathbf{a}_1/h$ and $\mathbf{a}_3/l - \mathbf{a}_1/h$.

$\mathbf{G}_{hkl} \cdot (\mathbf{a}_2/k - \mathbf{a}_1/h) = (2\pi/k)k - (2\pi/h)h = 0$.

Similarly $\mathbf{G}_{hkl} \cdot (\mathbf{a}_3/l - \mathbf{a}_1/h) = 0$. Hence $\mathbf{G}_{hkl}$ is Perpendicular to the plane.

For the spacing: the plane through the origin has equation $\mathbf{G}_{hkl} \cdot \mathbf{r} = 0$. The next parallel plane is $\mathbf{G}_{hkl} \cdot \mathbf{r} = 2\pi$ (since $e^{i\mathbf{G}\cdot\mathbf{r}} = 1$ On all lattice planes). The distance from the origin to this plane is $d_{hkl} = 2\pi/\lvert\mathbf{G}_{hkl}\rvert$. $\blacksquare$

The Ewald sphere provides a geometric criterion for when diffraction occurs. Given an incident Wave vector $\mathbf{k}$ (with $\lvert\mathbf{k}\rvert = 2\pi/\lambda$) and the reciprocal lattice:

1. Draw $\mathbf{k}$ terminating at the origin of reciprocal space.
2. Construct a sphere of radius $k = 2\pi/\lambda$ centred at the start of $\mathbf{k}$.
3. Diffraction occurs for every reciprocal lattice point $\mathbf{G}$ that lies on the sphere, since then $\mathbf{k}" = \mathbf{k} + \mathbf{G}$ also has $\lvert\mathbf{k}'\rvert = k$ (elastic scattering condition).

Implications:

• For a fixed wavelength and a single crystal, very few reciprocal lattice points lie on the Ewald sphere. The crystal must be rotated to bring different points onto the sphere.
• As $\lambda$ decreases (shorter wavelength), the Ewald sphere radius increases and more points satisfy the condition.
• For $\lambda \to 0$ (e.g., electron diffraction), the Ewald sphere is effectively flat, and the Laue condition reduces to a planar section through reciprocal space.