Editing Ewald's sphere

{{Short description|Energy conservation during diffraction by atoms}}
The '''Ewald sphere''' is a geometric construction used in [[electron diffraction|electron]], [[neutron diffraction|neutron]], and [[X-ray crystallography|x-ray]] diffraction which shows the relationship between: 
* the [[wavevector]] of the incident and diffracted beams, 
* the [[Bragg angle|diffraction angle]] for a given reflection, 
* the [[reciprocal lattice]] of the [[crystal]].

It was conceived by [[Paul Peter Ewald]], a German physicist and crystallographer.<ref>{{cite journal |title=Die Berechnung optischer und elektrostatischer Gitterpotentiale |lang=de |year=1921 |author=Ewald, P. P. |journal=Annalen der Physik |volume=369 |issue=3 |pages=253–287 |bibcode=1921AnP...369..253E |doi=10.1002/andp.19213690304 |url=https://zenodo.org/record/1424363}}</ref> Ewald himself spoke of the '''sphere of reflection'''.<ref>{{cite journal |doi=10.1107/S0567739469000155 |title=Introduction to the dynamical theory of X-ray diffraction |year=1969 |author=Ewald, P. P. |journal=Acta Crystallographica Section A |volume=25 |issue=1 |pages=103–108 |bibcode = 1969AcCrA..25..103E |doi-access=free}}</ref> It is often simplified to the two-dimensional "Ewald's circle" model or may be referred to as the Ewald sphere.

== Ewald construction ==
[[File:Ewald3.png|thumb|right|Ewald sphere construction]]
A [[crystal]] can be described as a [[lattice (group)|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 <math>\exp(2 \pi i \mathbf{k_0}\cdot \mathbf{r})</math>{{efn|In some physics texts the <math>2 \pi</math> is omitted}} with a [[Wave vector|wavevector]] <math>\mathbf{k_0}</math>, there will be outgoing wavevectors <math>\mathbf{k_1}</math> and <math>\mathbf{k_2}</math> as shown in the diagram<ref name="Cowley95">{{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}}</ref> 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 <math>\mathbf{k_1}</math> the corresponding reciprocal lattice point <math>\mathbf{g_1}</math> lies on the Ewald sphere, which is the condition for [[Bragg diffraction]]. For <math>\mathbf{k_2}</math> the corresponding reciprocal lattice point <math>\mathbf{g_2}</math> is off the Ewald sphere, so <math>\mathbf{k_2} = \mathbf{k_0} + \mathbf{g_2} + \mathbf{s}</math> where <math>\mathbf{s}</math> is called the excitation error. The amplitude and also intensity of diffraction into the wavevector <math>\mathbf{k_2}</math> depends upon the [[Fourier transform]] of the shape of the sample,<ref name="Cowley95"/><ref>{{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 }}</ref> the excitation error <math>\mathbf{s}</math>, 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]].<ref name="Cowley95"/><ref name="Peng">{{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}}</ref>

==See also==
* [[Bragg's law]]
* [[Electron diffraction]]
* [[Laue equations]]
* [[Structure factor]]
* [[X-ray crystallography]]

== References ==
{{Reflist}}

==Notes==
{{notelist}}

== External links ==
* [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]]