Editing Electron diffraction (section)

=== Waves, diffraction and quantum mechanics ===
{{See also|Introduction to quantum mechanics|matter wave}}
{{anchor|Figure 4}}[[File:Wave packet propagation (phase faster than group, nondispersive).gif|thumb|Figure 4: Propagation of a wave packet demonstrating the movement of a bundle of waves; see [[group velocity]] for more details.|alt=A video illustrating a wavepacket of electrons, a small bundle.]]
Independent of the developments for electrons in vacuum, at about the same time the components of quantum mechanics were being assembled. In 1924 [[Louis de Broglie]] in his PhD thesis ''Recherches sur la théorie des quanta''<ref name=Broglie>{{cite web |last1=de Broglie |first1=Louis Victor |title=On the Theory of Quanta |url=https://fondationlouisdebroglie.org/LDB-oeuvres/De_Broglie_Kracklauer.pdf |access-date=25 February 2023 |website=Foundation of Louis de Broglie |edition=English translation by A.F. Kracklauer, 2004.}}</ref> introduced his theory of [[electron]] waves. He suggested that an electron around a nucleus could be thought of as [[standing wave]]s,<ref name="Broglie" />{{Rp|pages=Chpt 3}} and that electrons and all matter could be considered as waves. He merged the idea of thinking about them as particles (or corpuscles), and of thinking of them as waves. He proposed that particles are bundles of waves ([[wave packet]]s) that move with a [[group velocity]]<ref name="Broglie" />{{Rp|location=Chpt 1-2}} and have an [[Effective mass (solid-state physics)|effective mass]], see for instance [[#Figure 4|Figure 4]]. Both of these depend upon the energy, which in turn connects to the [[Wave vector|wavevector]] and the relativistic formulation of [[Albert Einstein]] a few years before.<ref>{{Cite book |last=Einstein |first=Albert |url=https://en.wikisource.org/wiki/Relativity:_The_Special_and_General_Theory |title=Relativity: The Special and General Theory}}</ref>

This rapidly became part of what was called by [[Erwin Schrödinger]] ''undulatory mechanics'',<ref name="Schroedinger">{{Cite journal |last=Schrödinger |first=E. |date=1926 |title=An Undulatory Theory of the Mechanics of Atoms and Molecules |url=https://link.aps.org/doi/10.1103/PhysRev.28.1049 |journal=Physical Review |language=en |volume=28 |issue=6 |pages=1049–1070 |doi=10.1103/PhysRev.28.1049 |bibcode=1926PhRv...28.1049S |issn=0031-899X|url-access=subscription }}</ref> now called the [[Schrödinger equation]] or wave mechanics. As stated by [[Louis de Broglie]] on September 8, 1927, in the preface to the German translation of his theses (in turn translated into English):<ref name="Broglie" />{{Rp|page=v}}<blockquote>''M. Einstein from the beginning has supported my thesis, but it was M. E. [[Erwin Schrödinger|Schrödinger]] who developed the propagation equations of a new theory and who in searching for its solutions has established what has become known as “Wave Mechanics”.''</blockquote>
The Schrödinger equation combines the kinetic energy of waves and the potential energy due to, for electrons, the [[Coulomb potential]]. He was able to explain earlier work such as the quantization of the energy of electrons around atoms in the [[Bohr model]],<ref>{{Cite journal |last=Bohr |first=N. |date=1913 |title=On the constitution of atoms and molecules |url=https://www.tandfonline.com/doi/full/10.1080/14786441308634955 |journal=The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science |language=en |volume=26 |issue=151 |pages=1–25 |doi=10.1080/14786441308634955 |bibcode=1913PMag...26....1B |issn=1941-5982|url-access=subscription }}</ref> as well as many other phenomena.<ref name="Schroedinger" /> Electron waves as hypothesized<ref name="Broglie" />{{Rp|location=Chpt 1-2}} by de Broglie were automatically part of the solutions to his equation,<ref name="Schroedinger" /> see also [[introduction to quantum mechanics]] and [[matter waves]].

Both the wave nature and the undulatory mechanics approach were experimentally confirmed for electron beams by experiments from two groups performed independently, the first the [[Davisson–Germer experiment]],<ref name="DG0">{{Cite journal |last1=Davisson |first1=C. |last2=Germer |first2=L. H. |date=1927 |title=The Scattering of Electrons by a Single Crystal of Nickel |url=http://dx.doi.org/10.1038/119558a0 |journal=Nature |volume=119 |issue=2998 |pages=558–560 |doi=10.1038/119558a0 |bibcode=1927Natur.119..558D |s2cid=4104602 |issn=0028-0836|url-access=subscription }}</ref><ref name="DG1">{{Cite journal |last1=Davisson |first1=C. |last2=Germer |first2=L. H. |date=1927 |title=Diffraction of Electrons by a Crystal of Nickel |journal=Physical Review |volume=30 |issue=6 |pages=705–740 |doi=10.1103/physrev.30.705 |bibcode=1927PhRv...30..705D |issn=0031-899X|doi-access=free }}</ref><ref name="DG2">{{Cite journal |last1=Davisson |first1=C. J. |last2=Germer |first2=L. H. |date=1928 |title=Reflection of Electrons by a Crystal of Nickel |journal=Proceedings of the National Academy of Sciences |language=en |volume=14 |issue=4 |pages=317–322 |doi=10.1073/pnas.14.4.317 |issn=0027-8424 |pmc=1085484 |pmid=16587341|bibcode=1928PNAS...14..317D |doi-access=free }}</ref><ref name=":0">{{Cite journal |last1=Davisson |first1=C. J. |last2=Germer |first2=L. H. |date=1928 |title=Reflection and Refraction of Electrons by a Crystal of Nickel |journal=Proceedings of the National Academy of Sciences |language=en |volume=14 |issue=8 |pages=619–627 |doi=10.1073/pnas.14.8.619 |issn=0027-8424 |pmc=1085652 |pmid=16587378 |bibcode=1928PNAS...14..619D |doi-access=free }}</ref> the other by [[George Paget Thomson]] and Alexander Reid;<ref>{{Cite journal |last1=Thomson |first1=G. P. |last2=Reid |first2=A. |date=1927 |title=Diffraction of Cathode Rays by a Thin Film |journal=Nature |language=en |volume=119 |issue=3007 |pages=890 |doi=10.1038/119890a0 |bibcode=1927Natur.119Q.890T |s2cid=4122313 |issn=0028-0836|doi-access=free }}</ref> see note{{efn|name=Wlength}} for more discussion. Alexander Reid, who was Thomson's graduate student, performed the first experiments,<ref>{{Cite journal |last=Reid |first=Alexander |date=1928 |title=The diffraction of cathode rays by thin celluloid films |journal=Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character |language=en |volume=119 |issue=783 |pages=663–667 |doi=10.1098/rspa.1928.0121 |bibcode=1928RSPSA.119..663R |s2cid=98311959 |issn=0950-1207|doi-access=free }}</ref> but he died soon after in a motorcycle accident<ref>{{Cite journal |last=Navarro |first=Jaume |date=2010 |title=Electron diffraction chez Thomson: early responses to quantum physics in Britain |url=https://www.cambridge.org/core/product/identifier/S0007087410000026/type/journal_article |journal=The British Journal for the History of Science |language=en |volume=43 |issue=2 |pages=245–275 |doi=10.1017/S0007087410000026 |s2cid=171025814 |issn=0007-0874|url-access=subscription }}</ref> and is rarely mentioned. These experiments were rapidly followed by the first non-relativistic diffraction model for electrons by [[Hans Bethe]]<ref name="Bethe" /> based upon the Schrödinger equation,<ref name="Schroedinger" /> which is very close to how electron diffraction is now described. Significantly, [[Clinton Davisson]] and [[Lester Germer]] noticed<ref name="DG2" /><ref name=":0" /> that their results could not be interpreted using a [[Bragg's law]] approach as the positions were systematically different; the approach of [[Hans Bethe]]<ref name="Bethe" /> which includes the refraction due to the average potential yielded more accurate results. These advances in understanding of electron wave mechanics were important for many developments of electron-based analytical techniques such as [[Seishi Kikuchi]]'s observations of lines due to combined elastic and inelastic scattering,<ref name=":17">{{Cite journal |last=Kikuchi |first=Seishi |date=1928 |title=Diffraction of cathode rays by mica |url=https://scholar.google.com/scholar?output=instlink&q=info:sxVYQV4VcTcJ:scholar.google.com/&hl=en&as_sdt=0,14&as_ylo=1927&as_yhi=1929&scillfp=7509118820046091375&oi=lle |journal=Proceedings of the Imperial Academy |volume=4 |issue=6 |pages=271–274 |doi=10.2183/pjab1912.4.271 |s2cid=4121059 |via=Google Scholar|doi-access=free }}</ref><ref name=":18" /> [[gas electron diffraction]] developed by [[Herman Francis Mark|Herman Mark]] and Raymond Weil,<ref>{{Cite journal |last1=Mark |first1=Herman |last2=Wierl |first2=Raymond |date=1930 |title=Neuere Ergebnisse der Elektronenbeugung |url=http://dx.doi.org/10.1007/bf01497860 |journal=Die Naturwissenschaften |volume=18 |issue=36 |pages=778–786 |doi=10.1007/bf01497860 |bibcode=1930NW.....18..778M |s2cid=9815364 |issn=0028-1042|url-access=subscription }}</ref><ref>{{Cite journal |last1=Mark |first1=Herman |last2=Wiel |first2=Raymond |date=1930 |title=Die ermittlung von molekülstrukturen durch beugung von elektronen an einem dampfstrahl |journal=Zeitschrift für Elektrochemie und angewandte physikalische Chemie |volume=36 |issue=9 |pages=675–676|doi=10.1002/bbpc.19300360921 |s2cid=178706417 }}</ref> diffraction in liquids by Louis Maxwell,<ref name=":20">{{Cite journal |last=Maxwell |first=Louis R. |date=1933 |title=Electron Diffraction by Liquids |url=https://link.aps.org/doi/10.1103/PhysRev.44.73 |journal=Physical Review |language=en |volume=44 |issue=2 |pages=73–76 |doi=10.1103/PhysRev.44.73 |bibcode=1933PhRv...44...73M |issn=0031-899X|url-access=subscription }}</ref> and the first electron microscopes developed by [[Max Knoll]] and [[Ernst Ruska]].<ref name="Knoll1">{{Cite journal |last1=Knoll |first1=M. |last2=Ruska |first2=E. |date=1932 |title=Beitrag zur geometrischen Elektronenoptik. I |url=http://dx.doi.org/10.1002/andp.19324040506 |journal=Annalen der Physik |volume=404 |issue=5 |pages=607–640 |doi=10.1002/andp.19324040506 |bibcode=1932AnP...404..607K |issn=0003-3804|url-access=subscription }}</ref><ref name="Knoll2">{{Cite journal |last1=Knoll |first1=M. |last2=Ruska |first2=E. |date=1932 |title=Das Elektronenmikroskop |url=http://link.springer.com/10.1007/BF01342199 |journal=Zeitschrift für Physik |language=de |volume=78 |issue=5–6 |pages=318–339 |doi=10.1007/BF01342199 |bibcode=1932ZPhy...78..318K |s2cid=186239132 |issn=1434-6001|url-access=subscription }}</ref>