Editing Electron diffraction (section)

=== Dynamical diffraction ===

While kinematical diffraction is adequate to understand the geometry of the diffraction spots, it does not correctly give the intensities and has a number of other limitations. For a more complete approach one has to include multiple scattering of the electrons using methods that date back to the early work of Hans Bethe in 1928.<ref name="Bethe">{{Cite journal |last=Bethe |first=H. |date=1928 |title=Theorie der Beugung von Elektronen an Kristallen |url=https://onlinelibrary.wiley.com/doi/10.1002/andp.19283921704 |journal=Annalen der Physik |language=de |volume=392 |issue=17 |pages=55–129 |doi=10.1002/andp.19283921704|bibcode=1928AnP...392...55B |url-access=subscription }}</ref> These are based around solutions of the Schrödinger equation<ref name="Schroedinger" /> using the relativistic effective mass <math>m^*</math> described earlier.<ref name="Fujiwara" /> Even at very high energies dynamical diffraction is needed as the relativistic mass and wavelength partially cancel, so the role of the potential is larger than might be thought.<ref name="Fujiwara" /><ref name="AHDiss" />{{anchor|Figure 7}}[[File:CBED-EFiltered.png|thumb|Figure 7: CBED patterns using all the electrons, with just those which have not lost any energy and those which have excited one or two [[plasmons]]|left|alt=Diagram of convergent-beam diffraction patterns with different energy filters. The ones where energy losses have been removed are clearer.]]
The main components of current dynamical diffraction of electrons include:
* Taking into account the scattering back into the incident beam both from diffracted beams and between all others, not just single scattering from the incident beam to diffracted beams.<ref name="Bethe" /> This is important even for samples which are only a few atoms thick.<ref name="Bethe" /><ref name="CowleyII" />
* Modelling at least semi-empirically the role of inelastic scattering by an imaginary component of the potential,<ref name="Yoshioka">{{Cite journal |last=Yoshioka |first=Hide |date=1957 |title=Effect of Inelastic Waves on Electron Diffraction |url=https://journals.jps.jp/doi/10.1143/JPSJ.12.618 |journal=Journal of the Physical Society of Japan |language=en |volume=12 |issue=6 |pages=618–628 |doi=10.1143/JPSJ.12.618 |bibcode=1957JPSJ...12..618Y |issn=0031-9015|url-access=subscription }}</ref><ref name="HowieII">{{Cite journal | first1=Archibald| last1=Howie | first2=Michael | last2=Whelan |date=1961 |title=Diffraction contrast of electron microscope images of crystal lattice defects – II. The development of a dynamical theory |url=http://dx.doi.org/10.1098/rspa.1961.0157 |journal=Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences |volume=263 |issue=1313 |pages=217–237 |doi=10.1098/rspa.1961.0157 | bibcode=1961RSPSA.263..217H | s2cid=121465295 |issn=0080-4630|url-access=subscription }}</ref><ref name="PHInel">{{Cite journal |date=1963 |title=Inelastic scattering of electrons by crystals. I. The theory of small-angle in elastic scattering |url=http://dx.doi.org/10.1098/rspa.1963.0017 | last1=Hirsch | first1=Peter | last2=Whelan | first2=Michael | journal=Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences |volume=271 |issue=1345 |pages=268–287 |doi=10.1098/rspa.1963.0017 |bibcode=1963RSPSA.271..268H |s2cid=123122726 |issn=0080-4630|url-access=subscription }}</ref> also called an "optical potential".<ref name="Peng" />{{Rp|location=Chpt 13}} There is always inelastic scattering, and often it can have a major effect on both the background and sometimes the details, see [[#Figure 7|Figure 7]] and [[#Figure 18|18]].<ref name="Yoshioka" /><ref name="HowieII" /><ref name="PHInel" />
* Higher-order numerical approaches to calculate the intensities such as [[multislice]],<ref name=MS1/><ref>{{Cite journal |last=Ishizuka |first=Kazuo |date=2004 |title=FFT Multislice Method—The Silver Anniversary |url=https://www.cambridge.org/core/product/identifier/S1431927604040292/type/journal_article |journal=Microscopy and Microanalysis |language=en |volume=10 |issue=1 |pages=34–40 |doi=10.1017/S1431927604040292 |pmid=15306065 |bibcode=2004MiMic..10...34I |s2cid=8016041 |issn=1431-9276|url-access=subscription }}</ref> matrix methods<ref>{{Cite book |last=Metherell |first=A. J. |title=Electron Microscopy in Materials Science: Part II |publisher=Commission of the European Communities |year=1975 |pages=397–552 |url=https://op.europa.eu/en/publication-detail/-/publication/9da8f73f-c340-40ee-b3cf-d4bacfcc4fd7}}</ref><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>{{Rp|location=Sec 4.3}} which are called Bloch-wave approaches or [[Muffin-tin approximation|muffin-tin]] approaches.<ref>{{Cite journal |last=Berry |first=M V |date=1971|title=Diffraction in crystals at high energies |url=https://iopscience.iop.org/article/10.1088/0022-3719/4/6/006 |journal=Journal of Physics C: Solid State Physics |volume=4 |issue=6 |pages=697–722 |doi=10.1088/0022-3719/4/6/006 |bibcode=1971JPhC....4..697B |issn=0022-3719|url-access=subscription }}</ref> With these diffraction spots which are not present in kinematical theory can be present, e.g.<ref name="Gjønnes 65–67">{{Cite journal |last1=Gjønnes |first1=J. |last2=Moodie |first2=A. F. |date=1965 |title=Extinction conditions in the dynamic theory of electron diffraction |url=https://scripts.iucr.org/cgi-bin/paper?S0365110X65002773 |journal=Acta Crystallographica |volume=19 |issue=1 |pages=65–67 |doi=10.1107/S0365110X65002773 |bibcode=1965AcCry..19...65G |issn=0365-110X|url-access=subscription }}</ref>
* Contributions to the diffraction from [[Elasticity (physics)|elastic strain]] and  [[crystallographic defect]]s, and also what [[Jens Lindhard]] called the string potential.<ref>{{Cite journal |last=Lindhard |first=J. |date=1964 |title=Motion of swift charged particles, as influenced by strings of atoms in crystals |url=https://linkinghub.elsevier.com/retrieve/pii/0031916364911333 |journal=Physics Letters |language=en |volume=12 |issue=2 |pages=126–128 |doi=10.1016/0031-9163(64)91133-3|bibcode=1964PhL....12..126L |url-access=subscription }}</ref>
* For [[transmission electron microscopy|transmission electron microscopes]] effects due to variations in the thickness of the sample and the normal to the surface.<ref name="Cowley95" />{{Rp|location=Chpt 6}} 
* Both in the geometry of scattering and calculations, for both [[Electron diffraction#Low-energy electron diffraction (LEED)|LEED]]<ref name="McRae">{{Cite journal |last=McRae |first=E. G. |date=1966 |title=Multiple-Scattering Treatment of Low-Energy Electron-Diffraction Intensities |journal=The Journal of Chemical Physics |language=en |volume=45 |issue=9 |pages=3258–3276 |doi=10.1063/1.1728101 |bibcode=1966JChPh..45.3258M |issn=0021-9606|doi-access=free }}</ref> and [[Electron diffraction#Reflection high-energy electron diffraction (RHEED)|RHEED]],<ref name="Collela">
{{cite journal
 |last=Colella |first=R.
 |date=1972
 |title=n-Beam dynamical diffraction of high-energy electrons at glancing incidence. General theory and computational methods
 |url=https://scripts.iucr.org/cgi-bin/paper?S0567739472000026
 |journal=Acta Crystallographica Section A |volume=28 |issue=1 |pages=11–15
 |doi=10.1107/S0567739472000026 |bibcode=1972AcCrA..28...11C |issn=0567-7394
|url-access=subscription}}</ref><ref name="Maksym">{{Cite journal |last1=Maksym |first1=P.A. |last2=Beeby |first2=J.L. |date=1981 |title=A theory of RHEED |url=https://linkinghub.elsevier.com/retrieve/pii/003960288190649X |journal=Surface Science |language=en |volume=110 |issue=2 |pages=423–438 |bibcode=1981SurSc.110..423M |doi=10.1016/0039-6028(81)90649-X|url-access=subscription }}</ref> effects due to the presence of surface steps, [[surface reconstruction]]s and other atoms at the surface. Often these change the diffraction details significantly.<ref name="McRae" /><ref name="Collela" /><ref name="Maksym" /> 
* For [[Electron diffraction#Low-energy electron diffraction (LEED)|LEED]], use more careful analyses of the potential because contributions from [[Exchange interaction|exchange]] terms can be important.<ref name=Pendry71>{{Cite journal |last=Pendry | first=J B | date=1971 |title=Ion core scattering and low energy electron diffraction. I |url=http://dx.doi.org/10.1088/0022-3719/4/16/015 |journal=Journal of Physics C: Solid State Physics |volume=4 |issue=16 |pages=2501–2513 |doi=10.1088/0022-3719/4/16/015 | bibcode=1971JPhC....4.2501P |issn=0022-3719|url-access=subscription }}</ref> Without these the calculations may not be accurate enough.<ref name="Pendry71" />