Ewald's sphere

Ewald's sphere is a concept used in X-ray crystallography, a technique for determining the atomic and molecular structure of a crystal. The sphere is named after Paul Peter Ewald, a German physicist who made significant contributions to the understanding of crystal lattice structures through X-ray diffraction. Ewald's sphere plays a crucial role in the analysis of diffraction patterns, which are essential for elucidating the three-dimensional arrangement of atoms within a crystal.

Overview

In X-ray crystallography, a crystal is bombarded with X-rays, which diffract off the crystal lattice planes. The pattern of these diffracted rays can be captured on a detector. The arrangement and intensity of the spots formed (known as the diffraction pattern) provide information about the structure of the crystal. Ewald's sphere is a geometric construct that helps in interpreting these diffraction patterns.

Construction of Ewald's Sphere

Ewald's sphere is constructed in reciprocal space, which is a mathematical representation used to describe the diffraction patterns of crystals. The sphere itself represents all possible directions and wavelengths of diffracted X-rays that can be observed from a crystal subjected to X-ray bombardment. The center of Ewald's sphere is located at the point where the incident X-ray beam intersects the reciprocal lattice, and its radius is equal to 1/λ, where λ is the wavelength of the incident X-rays.

Significance in X-ray Crystallography

The intersection of Ewald's sphere with the reciprocal lattice points (which represent the inverse of the crystal's lattice spacings) determines the condition for X-ray diffraction. Only when a reciprocal lattice point lies on the surface of Ewald's sphere does diffraction occur, and the corresponding lattice planes are said to be in the diffracting condition. This concept is central to the derivation of the Bragg's law, which relates the wavelength of electromagnetic radiation to the diffraction angle and the lattice spacing in a crystalline sample.

Applications

Ewald's sphere is not only fundamental in understanding the principles behind X-ray crystallography but also in the practical analysis of crystal structures. It aids in the interpretation of complex diffraction patterns and the determination of crystal orientations. Furthermore, the concept of Ewald's sphere has been extended to other types of diffraction studies, such as electron and neutron diffraction, highlighting its broad applicability in the field of crystallography and materials science.

See Also

• X-ray diffraction
• Crystal structure
• Reciprocal space
• Bragg's law
• Paul Peter Ewald