Ewald's sphere

Concept in crystallography

Template:Crystallography

Ewald's Sphere

Diagram of Ewald's sphere showing the relationship between the incident beam, the crystal lattice, and the diffracted beams.

Ewald's sphere is a geometric construct used in crystallography to visualize the relationship between the incident beam, the crystal lattice, and the diffracted beams. It is named after the German physicist Paul Peter Ewald, who developed this concept to aid in the understanding of X-ray diffraction and electron diffraction patterns.

Concept

Ewald's sphere is a sphere in reciprocal space with a radius equal to the reciprocal of the wavelength of the incident radiation. The center of the sphere is placed at the tip of the incident wave vector, which represents the direction and magnitude of the incoming beam.

The crystal lattice is represented in reciprocal space as a series of points, known as reciprocal lattice points. When a reciprocal lattice point lies on the surface of Ewald's sphere, the conditions for diffraction are satisfied, and a diffracted beam is produced. This is known as the Laue condition.

Construction

To construct Ewald's sphere, follow these steps:

1. Draw the incident wave vector, \( \mathbf{k}_0 \), with a length equal to \( 1/\lambda \), where \( \lambda \) is the wavelength of the incident radiation. 2. Place the center of the sphere at the tip of \( \mathbf{k}_0 \). 3. The sphere's surface will intersect reciprocal lattice points that satisfy the diffraction condition.

Applications

Ewald's sphere is a fundamental tool in the analysis of diffraction patterns. It helps in determining which lattice planes will produce diffracted beams for a given orientation of the crystal. This is crucial in techniques such as X-ray crystallography, neutron diffraction, and electron diffraction.

Related Concepts

• Bragg's law
• Reciprocal lattice
• Diffraction pattern
• Laue equations

Related pages

• X-ray crystallography
• Electron diffraction
• Neutron diffraction
• Reciprocal space