It was conceived by Paul Peter Ewald, a German physicist and crystallographer.[1] Ewald himself spoke of the sphere of reflection.[2] It is often simplified to the two-dimensional "Ewald's circle" model or may be referred to as the Ewald sphere.
A crystal can be described as a lattice of atoms, which in turn this 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 [a] with a wavevector, there will be outgoing wavevectors and as shown in the diagram[3] 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 and shown in blue are three wavevectors. For the wavevector the corresponding reciprocal lattice point lies on the Ewald sphere, which is the condition for Bragg diffraction. For the corresponding reciprocal lattice point is off the Ewald sphere, so where is called the excitation error. The amplitude and also intensity of diffraction into the wavevector depends upon the Fourier transform of the shape of the sample,[3][4] the excitation error , 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.[3][5]
^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.
^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.
^Rees, A. L. G.; Spink, J. A. (1950). "The shape transform in electron diffraction by small crystals". Acta Crystallographica. 3 (4): 316–317. doi:10.1107/s0365110x50000823. ISSN 0365-110X.
^Peng, L.-M.; Dudarev, S. L.; Whelan, M. J. (2011). High energy electron diffraction and microscopy. Oxford: Oxford University Press. ISBN978-0-19-960224-7. OCLC 656767858.
ewald, sphere, ewald, sphere, geometric, construction, used, electron, neutron, diffraction, which, shows, relationship, between, wavevector, incident, diffracted, beams, diffraction, angle, given, reflection, reciprocal, lattice, crystalit, conceived, paul, p. 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 crystalIt was conceived by Paul Peter Ewald a German physicist and crystallographer 1 Ewald himself spoke of the sphere of reflection 2 It is often simplified to the two dimensional Ewald s circle model or may be referred to as the Ewald sphere Contents 1 Ewald construction 2 See also 3 References 4 Notes 5 External linksEwald construction Edit Ewald sphere construction A crystal can be described as a lattice of atoms which in turn this 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 e x p 2 p i k 0 r displaystyle exp 2 pi ik 0 r a with a wavevector k 0 displaystyle k 0 there will be outgoing wavevectors k 1 displaystyle k 1 and k 2 displaystyle k 2 as shown in the diagram 3 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 and shown in blue are three wavevectors For the wavevector k 1 displaystyle k 1 the corresponding reciprocal lattice point g 1 displaystyle g 1 lies on the Ewald sphere which is the condition for Bragg diffraction For k 2 displaystyle k 2 the corresponding reciprocal lattice point g 2 displaystyle g 2 is off the Ewald sphere so k 2 k 0 g 2 s displaystyle k 2 k 0 g 2 s where s displaystyle s is called the excitation error The amplitude and also intensity of diffraction into the wavevector k 2 displaystyle k 2 depends upon the Fourier transform of the shape of the sample 3 4 the excitation error s displaystyle 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 3 5 See also EditBragg s law Electron diffraction Laue equations Structure factor X ray crystallographyReferences Edit 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 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 a b c John M Cowley 1995 Diffraction physics Elsevier ISBN 0 444 82218 6 OCLC 247191522 Rees A L G Spink J A 1950 The shape transform in electron diffraction by small crystals Acta Crystallographica 3 4 316 317 doi 10 1107 s0365110x50000823 ISSN 0365 110X Peng L M Dudarev S L Whelan M J 2011 High energy electron diffraction and microscopy Oxford Oxford University Press ISBN 978 0 19 960224 7 OCLC 656767858 Notes Edit In some physics texts the 2 p displaystyle 2 pi is omittedExternal links EditOrigin of the Ewald Sphere in scattering TEM See also Chapter 5 in this web site Retrieved from https en wikipedia org w index php title Ewald 27s sphere amp oldid 1146700312, wikipedia, wiki, book, books, library,
