Reciprocal lattice

Reciprocal lattice

A reciprocal lattice is a conceptual lattice used in the field of solid state physics and crystallography. It is defined as the set of all wave vectors that yield plane waves with the periodicity of a given Bravais lattice. The reciprocal lattice plays a crucial role in the analysis of diffraction patterns and the electronic structure of crystals.

Definition

The reciprocal lattice is constructed from the Bravais lattice vectors. If the Bravais lattice is defined by the primitive vectors \(\mathbf{a}_1\), \(\mathbf{a}_2\), and \(\mathbf{a}_3\), the corresponding reciprocal lattice vectors \(\mathbf{b}_1\), \(\mathbf{b}_2\), and \(\mathbf{b}_3\) are given by: \[ \mathbf{b}_1 = 2\pi \frac{\mathbf{a}_2 \times \mathbf{a}_3}{\mathbf{a}_1 \cdot (\mathbf{a}_2 \times \mathbf{a}_3)} \] \[ \mathbf{b}_2 = 2\pi \frac{\mathbf{a}_3 \times \mathbf{a}_1}{\mathbf{a}_1 \cdot (\mathbf{a}_2 \times \mathbf{a}_3)} \] \[ \mathbf{b}_3 = 2\pi \frac{\mathbf{a}_1 \times \mathbf{a}_2}{\mathbf{a}_1 \cdot (\mathbf{a}_2 \times \mathbf{a}_3)} \]

Properties

The reciprocal lattice has several important properties:

• The reciprocal of the reciprocal lattice is the original Bravais lattice.
• The volume of the unit cell in the reciprocal lattice is inversely proportional to the volume of the unit cell in the direct lattice.
• The Miller indices of a plane in the direct lattice correspond to the coordinates of a point in the reciprocal lattice.

Applications

The concept of the reciprocal lattice is fundamental in the analysis of X-ray diffraction, electron diffraction, and neutron diffraction experiments. It is also essential in the study of the band structure of solids, where the Brillouin zone, a uniquely defined primitive cell in the reciprocal lattice, is used to describe the allowed energy levels of electrons in a crystal.

Brillouin Zone

The Brillouin zone is a uniquely defined primitive cell in the reciprocal lattice. It is the Wigner-Seitz cell of the reciprocal lattice and is used to describe the fundamental properties of waves in periodic structures. The first Brillouin zone is particularly important in the study of electronic band structure.

Ewald Construction

The Ewald construction is a graphical method used to visualize the conditions for diffraction in a crystal. It involves the reciprocal lattice and is used to determine the directions in which X-rays will be diffracted by a crystal.

See Also

• Bravais lattice
• Miller indices
• Brillouin zone
• Diffraction
• Solid state physics
• Crystallography