Ewald's sphere

File:Ewald3.png

A crystal can be described as a lattice of atoms, which in turn leads to the reciprocal lattice. With electrons, neutrons or x-rays there is diffraction by the atoms, and if there is an incident plane wave $\backslash exp(2\; \backslash pi\; i\; \backslash mathbf\{k\_0\}\backslash cdot\; \backslash mathbf\{r\})${{efn|In some physics texts the $2\; \backslash pi$ is omitted}} with a wavevector $\backslash mathbf\{k\_0\}$, there will be outgoing wavevectors $\backslash mathbf\{k\_1\}$ and $\backslash mathbf\{k\_2\}$ as shown in the diagram{{Cite book |last=John M. |first=Cowley |url=http://worldcat.org/oclc/247191522 |title=Diffraction physics |date=1995 |publisher=Elsevier |isbn=0-444-82218-6 |oclc=247191522}} 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) 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 $\backslash mathbf\{k\_1\}$ the corresponding reciprocal lattice point $\backslash mathbf\{g\_1\}$ lies on the Ewald sphere, which is the condition for Bragg diffraction. For $\backslash mathbf\{k\_2\}$ the corresponding reciprocal lattice point $\backslash mathbf\{g\_2\}$ is off the Ewald sphere, so $\backslash mathbf\{k\_2\}\; =\; \backslash mathbf\{k\_0\}\; +\; \backslash mathbf\{g\_2\}\; +\; \backslash mathbf\{s\}$ where $\backslash mathbf\{s\}$ is called the excitation error. The amplitude and also intensity of diffraction into the wavevector $\backslash mathbf\{k\_2\}$ depends upon the Fourier transform of the shape of the sample,{{Cite journal |last1=Rees |first1=A. L. G. |last2=Spink |first2=J. A. |date=1950 |title=The shape transform in electron diffraction by small crystals |journal=Acta Crystallographica |volume=3 |issue=4 |pages=316–317 |doi=10.1107/s0365110x50000823 |issn=0365-110X|doi-access=free |bibcode=1950AcCry...3..316R }} the excitation error $\backslash mathbf\{s\}$, 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.{{Cite book |last1=Peng |first1=L.-M. |url=https://www.worldcat.org/oclc/656767858 |title=High energy electron diffraction and microscopy |date=2011 |publisher=Oxford University Press |first2=S. L.| last2=Dudarev | first3=M. J. |last3=Whelan |isbn=978-0-19-960224-7 |location=Oxford |oclc=656767858}}

• Bragg's law
• Electron diffraction
• Laue equations
• Structure factor
• X-ray crystallography

{{Reflist}}

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• [http://www.rodenburg.org/theory/Ewaldsphere21.html Origin of the Ewald Sphere in scattering (TEM)]
• [http://www.xtal.iqfr.csic.es/Cristalografia/index-en.html See also Chapter 5 in this web site]

{{Crystallography}}

Category:Diffraction

Category:Crystallography