The Ewald sphere is a geometric construction used in electron, neutron, and x-ray diffraction which shows the relationship between:

• the wavevector of the incident and diffracted 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. Ewald himself spoke of the sphere of reflection. It is often simplified to the two-dimensional "Ewald's circle" model or may be referred to as the Ewald sphere.

Ewald construction

A crystal can be described as a lattice of atoms, which in turn leads to the reciprocal lattice with a wavevector as shown in the diagram after the wave has been diffracted by the atoms.

The energy of the waves (electron, neutron or x-ray) depends upon the magnitude of the wavevector, so if there is no change in energy (elastic scattering lies on the Ewald sphere, which is the condition for Bragg diffraction depends upon the Fourier transform of the shape of the sample,, the structure factor for the relevant reciprocal lattice vector, and also whether the scattering is weak or strong. For neutrons and x-rays the scattering is generally weak so there is mainly Bragg diffraction, but it is much stronger for electron diffraction.

See also

• Bragg's law

• Electron diffraction

• Laue equations

• Structure factor

• X-ray crystallography

References

Notes

External links

• Origin of the Ewald Sphere in scattering (TEM)

• See also Chapter 5 in this web site