Ewald's sphere

The Ewald sphere is a geometric construction used in electron, neutron, and X-ray crystallography which demonstrates the relationship between:

• the wavevector of the incident and diffracted x-ray beams,
• the diffraction angle for a given reflection,
• the reciprocal lattice of the crystal

It was conceived by Paul Peter Ewald, a German physicist and crystallographer.[1] Ewald himself spoke of the sphere of reflection.[2]

Ewald's sphere can be used to find the maximum resolution available for a given x-ray wavelength and the unit cell dimensions. It is often simplified to the two-dimensional "Ewald's circle" model or may be referred to as the Ewald sphere.

Ewald construction

Ewald construction

A crystal can be described as a lattice of points of equal symmetry. The requirement for constructive interference in a diffraction experiment means that in momentum or reciprocal space the values of momentum transfer where constructive interference occurs also form a lattice (the reciprocal lattice). For example, the reciprocal lattice of a simple cubic real-space lattice is also a simple cubic structure. Another example, the reciprocal lattice of an FCC crystal real-space lattice is a BCC structure, and vice versa. The aim of the Ewald sphere is to determine which lattice planes (represented by the grid points on the reciprocal lattice) will result in a diffracted signal for a given wavelength, ${\displaystyle \lambda }$, of incident radiation.

The incident plane wave falling on the crystal has a wave vector ${\displaystyle K_{i}}$ whose length is ${\displaystyle 2\pi /\lambda }$. The diffracted plane wave has a wave vector ${\displaystyle K_{f}}$. If no energy is gained or lost in the diffraction process (it is elastic) then ${\displaystyle K_{f}}$ has the same length as ${\displaystyle K_{i}}$. The difference between the wave-vectors of diffracted and incident wave is defined as scattering vector ${\displaystyle \Delta {K}=K_{f}-K_{i}}$. Since ${\displaystyle K_{i}}$ and ${\displaystyle K_{f}}$ have the same length the scattering vector must lie on the surface of a sphere of radius ${\displaystyle 2\pi /\lambda }$. This sphere is called the Ewald sphere.

The reciprocal lattice points are the values of momentum transfer where the Bragg diffraction condition is satisfied and for diffraction to occur the scattering vector must be equal to a reciprocal lattice vector. Geometrically this means that if the origin of reciprocal space is placed at the tip of ${\displaystyle K_{i}}$ then diffraction will occur only for reciprocal lattice points that lie on the surface of the Ewald sphere.

Applications

Small scattering-angle limit

When the wavelength of the radiation to be scattered is much smaller than the spacing between atoms, the Ewald sphere radius becomes large compared to the spatial frequency of atomic planes. This is common, for example, in transmission electron microscopy. In this approximation, diffraction patterns in effect illuminate planar slices through the origin of a crystal's reciprocal lattice. However, it is important to note that while the Ewald sphere may be quite flat, a diffraction pattern taken perfectly aligned down a zone axis (high-symmetry direction) contains precisely zero spots that exactly satisfy the Bragg condition. As one tilts a single crystal with respect to the incident beam, diffraction spots wink on and off as the Ewald sphere cuts through one zero order Laue zone (ZOLZ) after another.

See also

• Bragg's law
• Laue equations
• Kikuchi line (solid state physics)

References

1. Ewald, P. P. (1921). "Die Berechnung optischer und elektrostatischer Gitterpotentiale". Annalen der Physik. 369 (3): 253–287. Bibcode:1921AnP...369..253E. doi:10.1002/andp.19213690304.
2. Ewald, P. P. (1969). "Introduction to the dynamical theory of X-ray diffraction". Acta Crystallographica Section A. 25 (1): 103–108. Bibcode:1969AcCrA..25..103E. doi:10.1107/S0567739469000155.

External links

• Origin of the Ewald Sphere in scattering (TEM)
• See also Chapter 5 in this web site