Other meanings of Ewald sphere
Crystallography
The Ewald sphere is a geometric construction in crystallography that describes the conditions for diffraction in reciprocal space. It is a sphere of radius 1/λ (where λ is the X-ray wavelength) centered on the crystal, and a diffraction peak occurs when a reciprocal lattice point lies exactly on its surface. This construction, introduced by Paul Peter Ewald in 1913, is fundamental to interpreting diffraction patterns and designing crystallographic experiments.
The Ewald sphere is a sphere of radius 1/λ in reciprocal space, centered on the crystal's origin. The incident X-ray beam defines a direction from the crystal to the sphere's center, and the scattered beam direction is represented by a vector from the crystal to a point on the sphere. Diffraction occurs when a reciprocal lattice point (representing a set of crystal planes) lies exactly on the sphere's surface, satisfying the Laue condition and the Bragg equation 2d sinθ = λ.1
Because the sphere's radius is inversely proportional to wavelength, changing λ changes which reciprocal lattice points intersect the sphere, thus selecting which reflections are observed. In practice, the crystal is rotated to bring different reciprocal lattice points onto the sphere, which is why single-crystal diffraction experiments involve rotating the crystal.2
The Ewald sphere is central to interpreting X-ray, neutron, and electron diffraction patterns. In X-ray crystallography, the rotation method uses the Ewald sphere to predict which reflections will appear as the crystal rotates; the sphere's intersection with the reciprocal lattice defines the observable reflections.
In electron diffraction, the Ewald sphere is nearly flat due to the very short wavelength, which leads to the formation of Kikuchi lines and the need for careful alignment in transmission electron microscopy. In neutron diffraction, the sphere's radius is comparable to that of X-rays, but the scattering factors differ, affecting the intensity of reflections.
The Ewald sphere is also used in the analysis of diffuse scattering, where deviations from perfect periodicity cause intensity away from the sphere's surface. In protein crystallography, the sphere's curvature is exploited to collect complete data sets by rotating the crystal, and the 'incomplete' sphere (the region of reciprocal space not accessible) is a practical limitation.
Historically, Ewald's construction was developed in his 1912–1913 doctoral work, and it was later extended to dynamical diffraction theory, which accounts for multiple scattering. The concept also appears in the study of quasi-crystals, where the reciprocal lattice is not periodic, and in the design of X-ray free-electron laser experiments, where the sphere's radius changes with the extremely short pulses.3
The Ewald sphere is intimately connected to the reciprocal lattice and the Laue equations. It provides a visual way to understand the Bragg condition, and it is used in the construction of the 'limiting sphere'—the sphere of radius 2/λ that encloses all possible reflections for a given wavelength.4
In powder diffraction, the Ewald sphere intersects the Debye-Scherrer cones, producing rings on the detector. The concept is also used in the analysis of grazing-incidence diffraction and in the interpretation of precession photographs. Modern software for crystallographic data processing often uses the Ewald sphere to index reflections and refine unit-cell parameters.5
The Ewald sphere is a cornerstone of crystallographic theory, bridging the gap between the physical crystal and the observed diffraction pattern.
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