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The Ewald sphere is a geometric construction used in
electron The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary charge, elementary electric charge. It is a fundamental particle that comprises the ordinary matter that makes up the universe, along with up qua ...
,
neutron The neutron is a subatomic particle, symbol or , that has no electric charge, and a mass slightly greater than that of a proton. The Discovery of the neutron, neutron was discovered by James Chadwick in 1932, leading to the discovery of nucle ...
, and
x-ray An X-ray (also known in many languages as Röntgen radiation) is a form of high-energy electromagnetic radiation with a wavelength shorter than those of ultraviolet rays and longer than those of gamma rays. Roughly, X-rays have a wavelength ran ...
diffraction which shows the relationship between: * the
wavevector In physics, a wave vector (or wavevector) is a vector used in describing a wave, with a typical unit being cycle per metre. It has a magnitude and direction. Its magnitude is the wavenumber of the wave (inversely proportional to the wavelength), ...
of the incident and diffracted beams, * the diffraction angle for a given reflection, * the
reciprocal lattice Reciprocal lattice is a concept associated with solids with translational symmetry which plays a major role in many areas such as X-ray and electron diffraction as well as the energies of electrons in a solid. It emerges from the Fourier tran ...
of the
crystal A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macros ...
. 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 A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macros ...
can be described as a lattice of atoms, which in turn leads to the
reciprocal lattice Reciprocal lattice is a concept associated with solids with translational symmetry which plays a major role in many areas such as X-ray and electron diffraction as well as the energies of electrons in a solid. It emerges from the Fourier tran ...
. With electrons, neutrons or x-rays there is diffraction by the atoms, and if there is an incident plane wave \exp(2 \pi i \mathbf\cdot \mathbf) with a
wavevector In physics, a wave vector (or wavevector) is a vector used in describing a wave, with a typical unit being cycle per metre. It has a magnitude and direction. Its magnitude is the wavenumber of the wave (inversely proportional to the wavelength), ...
\mathbf, there will be outgoing wavevectors \mathbf and \mathbf 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 Elastic scattering is a form of particle scattering in scattering theory, nuclear physics and particle physics. In this process, the internal states of the Elementary particle, particles involved stay the same. In the non-relativistic case, where ...
) these have the same magnitude, that is they must all lie on the Ewald sphere. In the Figure the red dot is the origin for the wavevectors, the black spots are reciprocal lattice points (vectors) and shown in blue are three wavevectors. For the wavevector \mathbf the corresponding reciprocal lattice point \mathbf lies on the Ewald sphere, which is the condition for Bragg diffraction. For \mathbf the corresponding reciprocal lattice point \mathbf is off the Ewald sphere, so \mathbf = \mathbf + \mathbf + \mathbf where \mathbf is called the excitation error. The amplitude and also intensity of diffraction into the wavevector \mathbf depends upon the
Fourier transform In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...
of the shape of the sample, the excitation error \mathbf, the
structure factor In condensed matter physics and crystallography, the static structure factor (or structure factor for short) is a mathematical description of how a material scatters incident radiation. The structure factor is a critical tool in the interpretation ...
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 Electron diffraction is a generic term for phenomena associated with changes in the direction of electron beams due to elastic interactions with atoms. It occurs due to elastic scattering, when there is no change in the energy of the electrons. ...
.


See also

*
Bragg's law In many areas of science, Bragg's law — also known as Wulff–Bragg's condition or Laue–Bragg interference — is a special case of Laue diffraction that gives the angles for coherent scattering of waves from a large crystal lattice. It descr ...
*
Electron diffraction Electron diffraction is a generic term for phenomena associated with changes in the direction of electron beams due to elastic interactions with atoms. It occurs due to elastic scattering, when there is no change in the energy of the electrons. ...
* Laue equations *
Structure factor In condensed matter physics and crystallography, the static structure factor (or structure factor for short) is a mathematical description of how a material scatters incident radiation. The structure factor is a critical tool in the interpretation ...
*
X-ray crystallography X-ray crystallography is the experimental science of determining the atomic and molecular structure of a crystal, in which the crystalline structure causes a beam of incident X-rays to Diffraction, diffract in specific directions. By measuring th ...


References


Notes


External links


Origin of the Ewald Sphere in scattering (TEM)


{{Crystallography Diffraction Crystallography