View source for Electron diffraction

{{Short description|Bending of electron beams due to electrostatic interactions with matter}}
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{{anchor|Figure 1}}[[Image:Austenite ZADP.jpg|thumb|Figure 1: Selected area diffraction pattern of a twinned [[austenite]] crystal in a piece of [[steel]]|alt=Electron diffraction pattern showing white spots on a dark background, as a general example.]]

'''Electron diffraction''' is a generic term for phenomena associated with changes in the direction of [[electron beams]] due to [[Elastic collision|elastic]] interactions with [[atoms]].{{efn|name=Diff}} It occurs due to [[elastic scattering]], when there is no change in the energy of the electrons.<ref name="Cowley95" />{{Rp|location=Chpt 4}}<ref name="Reimer">{{Cite book |last=Reimer |first=Ludwig |title=Transmission Electron Microscopy: Physics of Image Formation and Microanalysis. |date=2013 |publisher=Springer Berlin / Heidelberg |isbn=978-3-662-13553-2 |oclc=1066178493}}</ref>{{Rp|location=Chpt 5}}<ref name="Form" /><ref name=":11">{{Cite journal |last=Humphreys |first=C J |date=1979 |title=The scattering of fast electrons by crystals |url=https://iopscience.iop.org/article/10.1088/0034-4885/42/11/002 |journal=Reports on Progress in Physics |volume=42 |issue=11 |pages=1825–1887 |doi=10.1088/0034-4885/42/11/002 |s2cid=250876999 |issn=0034-4885|url-access=subscription }}</ref> The negatively charged electrons are scattered due to [[Coulomb's law|Coulomb forces]] when they interact with both the positively charged atomic core and the negatively charged electrons around the atoms. The resulting map of the directions of the electrons far from the sample is called a diffraction pattern, see for instance [[#Figure 1|Figure 1]]. Beyond patterns showing the directions of electrons, electron diffraction also plays a major role in the contrast of images in [[electron microscope]]s.

This article provides an overview of electron diffraction and electron diffraction patterns, collectively referred to by the generic name electron diffraction. This includes aspects of how in a [[#A primer on electron diffraction|general way]] electrons can act as waves, and diffract and interact with matter. It also involves the extensive [[#History|history]] behind modern electron diffraction, how the combination of developments in the 19th century in understanding and controlling [[#Electrons in vacuum|electrons in vacuum]] and the early 20th century developments with [[#Waves, diffraction and quantum mechanics|electron waves]] were combined with early [[#Electron microscopes and early electron diffraction|instruments]], giving birth to electron microscopy and diffraction in 1920–1935. While this was the birth, there have been a large number of [[#Subsequent developments in methods and modelling|further developments]] since then.

There are many [[#Types and techniques|types and techniques]] of electron diffraction. The most common approach is where the electrons [[#In a transmission electron microscope|transmit]] through a thin sample, from 1&nbsp;nm to 100&nbsp;nm (10 to 1000 atoms thick), where the results depend upon how the atoms are arranged in the material, for instance a [[#Selected area electron diffraction|single crystal]], [[#Polycrystalline pattern|many crystals]] or [[#Multiple materials and double diffraction|different types]] of solids. Other cases such as [[#Bulk and surface superstructures|larger repeats]], [[#Aperiodic materials|no periodicity]] or [[#Diffuse scattering|disorder]] have their own characteristic patterns. There are many different ways of collecting diffraction information, from parallel illumination to a [[#Convergent beam electron diffraction|converging beam]] of electrons or where the beam is [[#Precession electron diffraction|rotated]] or [[#4D STEM|scanned]] across the sample which produce information that is often easier to interpret. There are also many other types of instruments. For instance, in [[#In a scanning electron microscope|a scanning electron microscope]] (SEM), [[electron backscatter diffraction]] can be used to determine crystal orientation across the sample. Electron diffraction patterns can also be used to characterize molecules using [[#Gas electron diffraction|gas electron diffraction]], liquids, surfaces using lower energy electrons, a technique called [[#Low-energy electron diffraction (LEED)|LEED]], and by reflecting electrons off surfaces, a technique called [[#Reflection high-energy electron diffraction (RHEED)|RHEED]].

There are also many levels of analysis of electron diffraction, including:
# The simplest approximation using the de Broglie wavelength<ref name="Broglie" />{{Rp|location=Chpt 1-2}} for electrons, where only the [[#Plane waves, wavevectors and reciprocal lattice|geometry]] is considered and often [[Bragg's law]]<ref name=":7" />{{Rp|pages=96–97}} is invoked. This approach only considers the electrons far from the sample, a far-field or [[Fraunhofer diffraction|Fraunhofer]]<ref name="Cowley95" />{{Rp|pages=21–24}} approach.
# The first level of more accuracy where it is approximated that the electrons are only scattered once, which is called [[#Kinematical diffraction|kinematical diffraction]]<ref name="Cowley95" />{{Rp|location=Sec 2}}<ref name="HirschEtAl" />{{Rp|location=Chpt 4-7}} and is also a far-field or Fraunhofer<ref name="Cowley95" />{{Rp|pages=21–24}} approach. 
# More complete and accurate explanations where multiple scattering is included, what is called [[#Dynamical diffraction|dynamical diffraction]] (e.g. refs<ref name="Cowley95" />{{Rp|location=Sec 3}}<ref name="HirschEtAl" />{{Rp|location=Chpt 8-12}}<ref name="Peng" />{{Rp|location=Chpt 3-10}}<ref name="Pendry71" /><ref name="Maksym" />). These involve more general analyses using relativistically corrected [[Schrödinger equation]]<ref name="Schroedinger" /> methods, and track the electrons through the sample, being accurate both near and far from the sample (both [[Fresnel diffraction|Fresnel]] and [[Fraunhofer diffraction|Fraunhofer]] diffraction).

Electron diffraction is similar to [[X-ray crystallography|x-ray]] and [[neutron diffraction]]. However, unlike x-ray and neutron diffraction where the simplest approximations are quite accurate, with electron diffraction this is not the case.<ref name="Cowley95" />{{Rp|location=Sec 3}}<ref name="Reimer" />{{Rp|location=Chpt 5}} Simple models give the geometry of the intensities in a diffraction pattern, but dynamical diffraction approaches are needed for accurate intensities and the positions of diffraction spots.

== A primer on electron diffraction ==
All matter can be thought of as [[matter wave]]s,<ref name="Broglie" />{{Rp|location=Chpt 1-3}} from small particles such as electrons up to macroscopic objects – although it is impossible to measure any of the "wave-like" behavior of macroscopic objects. Waves can move around objects and create interference patterns,<ref name="Born & Wolf">
{{cite book
 |last1=Born |first1=M. |author1-link=Max Born
 |last2=Wolf |first2=E. |author2-link=Emil Wolf
 |year=1999
 |title=[[Principles of Optics]]
 |publisher=[[Cambridge University Press]]
 |isbn=978-0-521-64222-4
}}</ref>{{Rp|location=Chpt 7-8}} and a classic example is the [[Young's two-slit experiment]] shown in [[#Figure 2|Figure 2]], where a wave impinges upon two slits in the first of the two images (blue waves). After going through the slits there are directions where the wave is stronger, ones where it is weaker – the wave has been [[Diffraction|diffracted]].<ref name="Born & Wolf"/>{{Rp|location=Chpt 1,7,8}} If instead of two slits there are a number of small points then similar phenomena can occur as shown in the second image where the wave (red and blue) is coming in from the bottom right corner. This is comparable to diffraction of an [[#Waves, diffraction and quantum mechanics|electron wave]] where the small dots would be atoms in a small crystal, see also note.{{efn|name=Diff}} Note the strong dependence on the relative orientation of the crystal and the incoming wave.{{anchor|Figure 2}}

{{multiple image
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| align             = right
| height            = 150
| image1            = Doubleslit.gif
| image2            = Bragg Diffraction.gif
| footer            = Figure 2: Young's double slit experiment, showing the wave in blue and the two slits in yellow; the other Figure with red and blue waves is similar from a small array of white atoms.
| alt1              = An image showing the result of a double-slit diffraction and interference experiment
| alt2              = An image that illustrates electron diffraction from a very small, ordered array of atoms.
| total_width       = 
}}
Close to an aperture or atoms, often called the "sample", the electron wave would be described in terms of near field or [[Fresnel diffraction]].<ref name="Born & Wolf"/>{{Rp|location=Chpt 7-8}} This has relevance for imaging within [[electron microscope]]s,<ref name="Cowley95"/>{{Rp|location=Chpt 3}}<ref name="Reimer"/>{{Rp|location=Chpt 3-4}} whereas electron diffraction patterns are measured far from the sample, which is described as far-field or Fraunhofer diffraction.<ref name="Born & Wolf"/>{{Rp|location=Chpt 7-8}} A map of the directions of the [[#Plane waves, wavevectors and reciprocal lattice|electron waves]] leaving the sample will show high intensity (white) for favored directions, such as the three prominent ones in the Young's two-slit experiment of [[#Figure 2|Figure 2]], while the other directions will be low intensity (dark). Often there will be an array of spots (preferred directions) as in [[#Figure 1|Figure 1]] and the other figures shown later.

== History ==
The historical background is divided into several subsections. The first is the general background to electrons in vacuum and the technological developments that led to [[cathode-ray tube]]s as well as [[vacuum tube]]s that dominated early television and electronics; the second is how these led to the development of electron microscopes; the last is work on the nature of electron beams and the fundamentals of how electrons behave, a key component of [[quantum mechanics]] and the explanation of electron diffraction.

=== Electrons in vacuum ===
{{See also|Cathode ray|Electron#History|label 2=History of the electron}}
{{anchor|Figure 3}}{{multiple image
| direction         = vertical
| align             = right
| width             = 200
| image1            = Katódsugarak mágneses mezőben(1).jpg
| image2            = Katódsugarak mágneses mezőben(2).jpg
| footer            = Figure 3: A Crookes tube – without emission (top, grey background) and with emission and a shadow due to the [[cross pattée]] blocking part of the electron beam (bottom, black background); see also [[Cathode ray | cathode ray tube]]
| alt1              = Image of a Crookes tube when it is not actively being used.
| alt2              = Image of a Crookes tube when it is operating, showing luminescence when the electrons hit the glass walls.
}}

Experiments involving electron beams occurred long before the discovery of the electron; [[wiktionary:ἤλεκτρον|ēlektron]] (ἤλεκτρον) is the Greek word for [[amber]],<ref name="DictOrigins">
{{cite book
 | last = Shipley | first = J.T.
 | title = Dictionary of Word Origins
 | page = 133
 | publisher = [[The Philosophical Library]] | year = 1945
 | isbn = 978-0-88029-751-6
 | url = https://archive.org/details/dictionaryofword00ship/page/133
 | url-access = registration
}}</ref> which is connected to the recording of electrostatic charging<ref name="Lacks">{{Cite journal |last1=Iversen |first1=Paul |last2=Lacks |first2=Daniel J. |date=2012 |title=A life of its own: The tenuous connection between Thales of Miletus and the study of electrostatic charging |url=https://www.sciencedirect.com/science/article/pii/S0304388612000216 |journal=[[Journal of Electrostatics]] |language=en |volume=70 |issue=3 |pages=309–311 |doi=10.1016/j.elstat.2012.03.002 |issn=0304-3886|url-access=subscription }}</ref> by [[Thales of Miletus]] around 585 BCE, and possibly others even earlier.<ref name="Lacks"/>

In 1650, [[Otto von Guericke]] invented the [[vacuum pump]]<ref name="Harsch 2007">
{{cite journal
| last=Harsch
| first=Viktor
| date=2007
| title=Otto von Gericke (1602–1686) and his pioneering vacuum experiments
| journal=Aviation, Space, and Environmental Medicine
| volume=78| issue=11
| pages=1075–1077
| doi=10.3357/asem.2159.2007
| issn=0095-6562| pmid=18018443
}}</ref> allowing for the study of the effects of high voltage electricity passing through [[rarefied air]]. In 1838, [[Michael Faraday]] applied a high voltage between two metal [[electrode]]s at either end of a glass tube that had been partially evacuated of air, and noticed a strange light arc with its beginning at the [[cathode]] (negative electrode) and its end at the [[anode]] (positive electrode).<ref name=":1">Faraday, Michael (1838) [https://books.google.com/books?id=ypNDAAAAcAAJ&pg=PA125 "VIII.  Experimental researches in electricity. — Thirteenth series.,"] ''Philosophical Transactions of the Royal Society of London'', '''128''' :  125–168.</ref> Building on this, in the 1850s, [[Heinrich Geissler]] was able to achieve a pressure of around 10<sup>−3</sup> [[Atmosphere (unit)|atmospheres]], inventing what became known as [[Geissler tube]]s. Using these tubes, while studying electrical conductivity in [[rarefied]] gases in 1859, [[Julius Plücker]] observed that the radiation emitted from the negatively charged cathode caused phosphorescent light to appear on the tube wall near it, and the region of the phosphorescent light could be moved by application of a magnetic field.<ref name=":3">{{Cite journal|last=Plücker|first=M.|date=1858|title=XLVI. Observations on the electrical discharge through rarefied gases|journal=The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science|volume=16|issue=109|pages=408–418|doi=10.1080/14786445808642591|issn=1941-5982}}</ref>

In 1869, Plücker's student [[Johann Wilhelm Hittorf]] found that a solid body placed between the cathode and the phosphorescence would cast a shadow on the tube wall, e.g. [[#Figure 3|Figure 3]].<ref name="Martin 1986">{{Citation |last=Martin |first=Andre |title=Advances in Electronics and Electron Physics, Volume 67 |pages=183–186 |year=1986 |editor-last=Hawkes |editor-first=Peter |contribution=Cathode Ray Tubes for Industrial and Military Applications |publisher=Academic Press |isbn=978-0-08-057733-3}}
</ref> Hittorf inferred that there are straight rays emitted from the cathode and that the phosphorescence was caused by the rays striking the tube walls. In 1876 [[Eugen Goldstein]] showed that the rays were emitted perpendicular to the cathode surface, which differentiated them from the incandescent light. [[Eugen Goldstein]] dubbed them [[cathode ray]]s.<ref>{{Cite book |last=Goldstein |first=Eugen |url=https://books.google.com/books?id=7-caAAAAYAAJ&pg=PA279 |title=Monatsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin |date=1876 |publisher=The Academy |pages=279–295, pp 286 |language=de}}</ref><ref name="Whittaker">
{{cite book
 |last=Whittaker
 |first=E.T.
 |author-link=E. T. Whittaker
 |title=[[A History of the Theories of Aether and Electricity]]
 |volume=1
 |publisher=Nelson |place=London
 |year=1951
}}</ref> By the 1870s [[William Crookes]]<ref name=":2">{{Cite journal |last=Crookes |first=William |date=1878 |title=I. On the illumination of lines of molecular pressure, and the trajectory of molecules |url=https://royalsocietypublishing.org/doi/10.1098/rspl.1878.0098 |journal=Proceedings of the Royal Society of London |language=en |volume=28 |issue=190–195 |pages=103–111 |doi=10.1098/rspl.1878.0098 |s2cid=122006529 |issn=0370-1662|url-access=subscription }}</ref> and others were able to evacuate glass tubes below 10<sup>−6</sup> atmospheres, and observed that the glow in the tube disappeared when the pressure was reduced but the glass behind the anode began to glow. Crookes was also able to show that the particles in the cathode rays were negatively charged and could be deflected by an electromagnetic field.<ref name=":2" /><ref name="Martin 1986" />

In 1897, [[J. J. Thomson|Joseph Thomson]] measured the mass of these cathode rays,<ref>{{Cite journal |last=Thomson |first=J. J. |date=1897 |title=XL. Cathode Rays |journal=The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science |volume=44 |issue=269 |pages=293–316 |doi=10.1080/14786449708621070 |issn=1941-5982}}</ref> proving they were made of particles. These particles, however, were 1800 times lighter than the lightest particle known at that time – a [[hydrogen]] atom. These were originally called ''corpuscles'' and later named electrons by [[George Johnstone Stoney]].<ref>{{Cite journal |last=Stoney | first=George Johnstone |url=https://www.biodiversitylibrary.org/item/51466 |title=Cause of Double Lines in Spectra| journal=The Scientific Transactions of the Royal Dublin Society |year=1891 |volume=4 |location=Dublin |pages=563, pp 583}}</ref>

The control of electron beams that this work led to resulted in significant technology advances in electronic amplifiers and television displays.<ref name="Martin 1986" />

=== 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 |journal=Nature |volume=119 |issue=2998 |pages=558–560 |doi=10.1038/119558a0 |bibcode=1927Natur.119..558D |s2cid=4104602 |issn=0028-0836}}</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 |page=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 }}{{Dead link|date=September 2025 |bot=InternetArchiveBot }}</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 |journal=Die Naturwissenschaften |volume=18 |issue=36 |pages=778–786 |doi=10.1007/bf01497860 |bibcode=1930NW.....18..778M |s2cid=9815364 |issn=0028-1042}}</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 |journal=Annalen der Physik |volume=404 |issue=5 |pages=607–640 |doi=10.1002/andp.19324040506 |bibcode=1932AnP...404..607K |issn=0003-3804}}</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>

=== Electron microscopes and early electron diffraction ===
{{See also|Transmission Electron Microscopy#History|label 1=History of transmission electron microscopy}}

In order to have a practical microscope or diffractometer, just having an electron beam was not enough, it needed to be controlled. Many developments laid the groundwork of [[electron optics]]; see the paper by Chester J. Calbick for an overview of the early work.<ref>{{Cite journal |last=Calbick |first=C. J. |date=1944 |title=Historical Background of Electron Optics |url=http://aip.scitation.org/doi/10.1063/1.1707371 |journal=Journal of Applied Physics |language=en |volume=15 |issue=10 |pages=685–690 |doi=10.1063/1.1707371 |bibcode=1944JAP....15..685C |issn=0021-8979|url-access=subscription }}</ref> One significant step was the work of [[Heinrich Hertz]] in 1883<ref>{{Citation |last=Hertz |first=Heinrich |title=Introduction to Heinrich Hertz's Miscellaneous Papers (1895) by Philipp Lenard |date=2019 |work=Heinrich Rudolf Hertz (1857–1894) |pages=87–88 |publisher=Routledge |doi=10.4324/9780429198960-4 |isbn=978-0-429-19896-0 |s2cid=195494352 }}</ref> who made a cathode-ray tube with electrostatic and magnetic deflection, demonstrating manipulation of the direction of an electron beam. Others were focusing of electrons by an axial magnetic field by [[Emil Wiechert]] in 1899,<ref>{{Cite journal |last=Wiechert |first=E. |date=1899 |title=Experimentelle Untersuchungen über die Geschwindigkeit und die magnetische Ablenkbarkeit der Kathodenstrahlen |url=https://onlinelibrary.wiley.com/doi/10.1002/andp.18993051203 |journal=Annalen der Physik und Chemie |language=de |volume=305 |issue=12 |pages=739–766 |doi=10.1002/andp.18993051203|bibcode=1899AnP...305..739W }}</ref> improved oxide-coated cathodes which produced more electrons by [[Arthur Wehnelt]] in 1905<ref>{{Cite journal |last=Wehnelt |first=A. |date=1905 |title=X. On the discharge of negative ions by glowing metallic oxides, and allied phenomena |url=https://www.tandfonline.com/doi/full/10.1080/14786440509463347 |journal=The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science |language=en |volume=10 |issue=55 |pages=80–90 |doi=10.1080/14786440509463347 |issn=1941-5982}}</ref> and the development of the electromagnetic lens in 1926 by [[Hans Busch]].<ref>{{Cite journal |last=Busch |first=H. |date=1926 |title=Berechnung der Bahn von Kathodenstrahlen im axialsymmetrischen elektromagnetischen Felde |url=https://onlinelibrary.wiley.com/doi/10.1002/andp.19263862507 |journal=Annalen der Physik |language=de |volume=386 |issue=25 |pages=974–993 |doi=10.1002/andp.19263862507|bibcode=1926AnP...386..974B |url-access=subscription }}</ref>

{{anchor|Figure 5}}[[File:Ernst Ruska Electron Microscope - Deutsches Museum - Munich-edit.jpg|Figure 5: Replica built in 1980 by Ernst Ruska of the original electron microscope, in the Deutsches Museum in Munich|thumb|alt=An images of a replica of one of the original electron microscopes which is now in a museum in Germany]]

Building an electron microscope involves combining these elements, similar to an [[optical microscope]] but with magnetic or electrostatic lenses instead of glass ones. To this day the issue of who invented the transmission electron microscope is controversial, as discussed by Thomas Mulvey<ref name=Mulvey/> and more recently by Yaping Tao.<ref>{{Cite book |last=Tao |first=Yaping |title=Proceedings of the 3rd International Conference on Contemporary Education, Social Sciences and Humanities (ICCESSH 2018) |date=2018 |publisher=Atlantis Press |isbn=978-94-6252-528-3 |series=Advances in Social Science, Education and Humanities Research |pages=1438–1441 |language=en |chapter=A Historical Investigation of the Debates on the Invention and Invention Rights of Electron Microscope |doi=10.2991/iccessh-18.2018.313 |chapter-url=https://www.atlantis-press.com/proceedings/iccessh-18/25898208 |doi-access=free}}</ref> Extensive additional information can be found in the articles by Martin Freundlich,<ref>{{Cite journal |last=Freundlich |first=Martin M. |date=1963 |title=Origin of the Electron Microscope: The history of a great invention, and of a misconception concerning the inventors, is reviewed. |url=https://www.science.org/doi/10.1126/science.142.3589.185 |journal=Science |language=en |volume=142 |issue=3589 |pages=185–188 |doi=10.1126/science.142.3589.185 |pmid=14057363 |issn=0036-8075|url-access=subscription }}</ref> [[Reinhold Rudenberg|Reinhold Rüdenberg]]<ref name="Rüdenberg">{{cite book |last=Rüdenberg |first=Reinhold |chapter=Origin and Background of the Invention of the Electron Microscope |date=2010 |title=Advances in Imaging and Electron Physics |volume=160 |pages=171–205 |publisher=Elsevier |doi=10.1016/s1076-5670(10)60005-5 |isbn=978-0-12-381017-5 }}.</ref> and Mulvey.<ref name=Mulvey>{{Cite journal |last=Mulvey |first=T |date=1962 |title=Origins and historical development of the electron microscope |url=https://iopscience.iop.org/article/10.1088/0508-3443/13/5/303 |journal=British Journal of Applied Physics |volume=13 |issue=5 |pages=197–207 |doi=10.1088/0508-3443/13/5/303 |issn=0508-3443|url-access=subscription }}</ref>

One effort was university based. In 1928, at the [[Technische Hochschule]] in Charlottenburg (now [[Technische Universität Berlin]]), {{ill|Adolf Matthias|de|Adolf Matthias (Elektrotechniker)}} (Professor of High Voltage Technology and Electrical Installations) appointed [[Max Knoll]] to lead a team of researchers to advance research on electron beams and cathode-ray oscilloscopes. The team consisted of several PhD students including [[Ernst Ruska]]. In 1931, Max Knoll and Ernst Ruska<ref name="Knoll1" /><ref name="Knoll2" /> successfully generated magnified images of mesh grids placed over an anode aperture. The device, a replicate of which is shown in [[#Figure 5|Figure 5]], used two  [[magnetic lens]]es to achieve higher magnifications, the first electron microscope. (Max Knoll died in 1969,<ref>{{Cite web |title=Max Knoll |url=https://www.ancientfaces.com/person/max-knoll-birth-1897-death-1969-europe/18955684 |access-date=2023-09-26 |website=AncientFaces |language=en}}</ref> so did not receive a share of the [[Nobel Prize in Physics]] in 1986.)

Apparently independent of this effort was work at [[Siemens-Schuckertwerke|Siemens-Schuckert]] by [[Reinhold Rudenberg]]. According to patent law (U.S. Patent No. 2058914<ref>{{Cite web |last=Rüdenberg |first=Reinhold |title=Apparatus for producing images of objects |url=https://image-ppubs.uspto.gov/dirsearch-public/print/downloadPdf/2058914 |access-date=24 February 2023 |website=Patent Public Search Basic}}</ref> and 2070318,<ref>{{Cite web |last=Rüdenberg |first=Reinhold |title=Apparatus for producing images of objects |url=https://image-ppubs.uspto.gov/dirsearch-public/print/downloadPdf/2070318 |access-date=24 February 2023 |website=Patent Public Search Basic}}</ref> both filed in 1932), he is the inventor of the electron microscope, but it is not clear when he had a working instrument. He stated in a very brief article in 1932<ref>{{Cite journal |last=Rodenberg |first=R. |date=1932 |title=Elektronenmikroskop |url=http://link.springer.com/10.1007/BF01505383 |journal=Die Naturwissenschaften |language=de |volume=20 |issue=28 |page=522 |doi=10.1007/BF01505383 |bibcode=1932NW.....20..522R |s2cid=263996652 |issn=0028-1042|url-access=subscription }}</ref> that Siemens had been working on this for some years before the patents were filed in 1932, so his effort was parallel to the university effort. He died in 1961,<ref>{{Cite journal |date=April 1962 |title=Orbituary of Reinhold Rudenberg |url=https://pubs.aip.org/physicstoday/article/15/4/106/422766/Reinhold-Rudenberg |access-date=2023-09-26 |website=pubs.aip.org |doi=10.1063/1.3058109|doi-access=free |url-access=subscription }}</ref> so similar to Max Knoll, was not eligible for a share of the Nobel Prize.

These instruments could produce magnified images, but were not particularly useful for electron diffraction; indeed, the wave nature of electrons was not exploited during the development. Key for electron diffraction in microscopes was the advance in 1936 where {{ill|Hans Boersch|de}} showed that they could be used as micro-diffraction cameras with an aperture<ref>{{Cite journal |last=Boersch |first=H. |date=1936 |title=Über das primäre und sekundäre Bild im Elektronenmikroskop. II. Strukturuntersuchung mittels Elektronenbeugung |url=https://onlinelibrary.wiley.com/doi/10.1002/andp.19364190107 |journal=Annalen der Physik |language=de |volume=419 |issue=1 |pages=75–80 |doi=10.1002/andp.19364190107|bibcode=1936AnP...419...75B |url-access=subscription }}</ref>—the birth of [[#Selected area electron diffraction|selected area electron diffraction]].<ref name="HirschEtAl" />{{Rp|location=Chpt 5-6}}

Less controversial was the development of [[#Low-energy electron diffraction|LEED]]—the early experiments of Davisson and Germer used this approach.<ref name=DG1/><ref name=DG2/> As early as 1929 Germer investigated gas adsorption,<ref>{{Cite journal |last=Germer |first=L. H. |date=1929 |title=Eine Anwendung der Elektronenbeugung auf die Untersuchung der Gasadsorption |url=http://link.springer.com/10.1007/BF01375462 |journal=Zeitschrift für Physik |language=de |volume=54 |issue=5–6 |pages=408–421 |doi=10.1007/BF01375462 |bibcode=1929ZPhy...54..408G |s2cid=121097655 |issn=1434-6001|url-access=subscription }}</ref> and in 1932 Harrison E. Farnsworth probed single crystals of copper and silver.<ref>{{Cite journal |last=Farnsworth |first=H. E. |date=1932 |title=Diffraction of Low-Speed Electrons by Single Crystals of Copper and Silver |url=https://link.aps.org/doi/10.1103/PhysRev.40.684 |journal=Physical Review |language=en |volume=40 |issue=5 |pages=684–712 |doi=10.1103/PhysRev.40.684 |bibcode=1932PhRv...40..684F |issn=0031-899X|url-access=subscription }}</ref> However, the vacuum systems available at that time were not good enough to properly control the surfaces, and it took almost forty years before these became available.<ref name="VanHove">{{cite book |last1=Van Hove |first1=Michel A. |url=https://www.springer.com/gp/book/9783642827235 |title=Low-Energy Electron Diffraction |last2=Weinberg |first2=William H. |last3=Chan |first3=Chi-Ming |date=1986 |publisher=Springer-Verlag, Berlin Heidelberg New York |isbn=978-3-540-16262-9 |pages=13–426}}</ref><ref>{{Cite book |title=Fifty years of electron diffraction: in recognition of fifty years of achievement by the crystallographers and gas diffractionists in the field of electron diffraction |date=1981 |publisher=Published for the International Union of Crystallography by D. Reidel |editor=Goodman, P. (Peter) |isbn=90-277-1246-8 |location=Dordrecht, Holland |oclc=7276396}}</ref> Similarly, it was not until about 1965 that Peter B. Sewell and M. Cohen demonstrated the power of [[Electron diffraction#Reflection high-energy electron diffraction (RHEED)|RHEED]] in a system with a very well controlled vacuum.<ref>{{Cite journal |last1=Sewell |first1=P. B. |last2=Cohen |first2=M. |date=1965 |title=The Observation Of Gas Adsorption Phenomena By Reflection High-Energy Electron Diffraction |url=http://aip.scitation.org/doi/10.1063/1.1754284 |journal=Applied Physics Letters |language=en |volume=7 |issue=2 |pages=32–34 |doi=10.1063/1.1754284 |bibcode=1965ApPhL...7...32S |issn=0003-6951|url-access=subscription }}</ref>

=== Subsequent developments in methods and modelling ===
Despite early successes such as the determination of the positions of hydrogen atoms in NH<sub>4</sub>Cl crystals by W. E. Laschkarew and I. D. Usykin in 1933,<ref>{{Cite journal |last1=Laschkarew |first1=W. E. |last2=Usyskin |first2=I. D. |date=1933 |title=Die Bestimmung der Lage der Wasserstoffionen im NH4Cl-Kristallgitter durch Elektronenbeugung |url=http://link.springer.com/10.1007/BF01331003 |journal=Zeitschrift für Physik |language=de |volume=85 |issue=9–10 |pages=618–630 |doi=10.1007/BF01331003 |bibcode=1933ZPhy...85..618L |s2cid=123199621 |issn=1434-6001|url-access=subscription }}</ref> boric acid by [[John M. Cowley]] in 1953<ref name="CowleyII">{{Cite journal |last=Cowley |first=J. M. |date=1953 |title=Structure analysis of single crystals by electron diffraction. II. Disordered boric acid structure |url=https://scripts.iucr.org/cgi-bin/paper?S0365110X53001423 |journal=Acta Crystallographica |volume=6 |issue=6 |pages=522–529 |doi=10.1107/S0365110X53001423 |bibcode=1953AcCry...6..522C |s2cid=94391285 |issn=0365-110X|doi-access=free |url-access=subscription }}</ref> and orthoboric acid by [[William Houlder Zachariasen]] in 1954,<ref>{{Cite journal |last=Zachariasen |first=W. H. |date=1954 |title=The precise structure of orthoboric acid |url=https://scripts.iucr.org/cgi-bin/paper?S0365110X54000886 |journal=Acta Crystallographica |volume=7 |issue=4 |pages=305–310 |doi=10.1107/S0365110X54000886 |bibcode=1954AcCry...7..305Z |issn=0365-110X|doi-access=free }}</ref> electron diffraction for many years was a qualitative technique used to check samples within electron microscopes. [[John M. Cowley|John M Cowley]] explains in a 1968 paper:<ref>{{Cite journal |last=Cowley |first=J.M. |date=1968 |title=Crystal structure determination by electron diffraction |url=https://linkinghub.elsevier.com/retrieve/pii/0079642568900236 |journal=Progress in Materials Science |language=en |volume=13 |pages=267–321 |doi=10.1016/0079-6425(68)90023-6|url-access=subscription }}</ref> <blockquote>''Thus was founded the belief, amounting in some cases almost to an article of faith, and persisting even to the present day, that it is impossible to interpret the intensities of electron diffraction patterns to gain structural information.''</blockquote>This has changed, in transmission, reflection and for low energies. Some of the key developments (some of which are also described later) from the early days to 2023 have been:
* Fast numerical methods based upon the Cowley–Moodie [[multislice]] algorithm,<ref name=MS1>{{Cite journal |last1=Cowley |first1=J. M. |last2=Moodie |first2=A. F. |date=1957 |title=The scattering of electrons by atoms and crystals. I. A new theoretical approach |url=https://scripts.iucr.org/cgi-bin/paper?S0365110X57002194 |journal=Acta Crystallographica |volume=10 |issue=10 |pages=609–619 |doi=10.1107/S0365110X57002194 |bibcode=1957AcCry..10..609C |issn=0365-110X|url-access=subscription }}</ref><ref>{{Cite journal |last=Ishizuka |first=Kazuo |date=2004 |title=FFT Multislice Method—The Silver Anniversary |url=https://academic.oup.com/mam/article/10/1/34/6912350 |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> which only became possible<ref>{{Cite journal |last1=Goodman |first1=P. |last2=Moodie |first2=A. F. |date=1974 |title=Numerical evaluations of N -beam wave functions in electron scattering by the multi-slice method |url=https://scripts.iucr.org/cgi-bin/paper?S056773947400057X |journal=Acta Crystallographica Section A |volume=30 |issue=2 |pages=280–290 |doi=10.1107/S056773947400057X |bibcode=1974AcCrA..30..280G |issn=0567-7394|url-access=subscription }}</ref> once the fast Fourier transform ([[FFT]]) method was developed.<ref>{{Cite journal |last1=Cooley |first1=James W. |last2=Tukey |first2=John W. |date=1965 |title=An algorithm for the machine calculation of complex Fourier series |url=https://www.ams.org/mcom/1965-19-090/S0025-5718-1965-0178586-1/ |journal=Mathematics of Computation |language=en |volume=19 |issue=90 |pages=297–301 |doi=10.1090/S0025-5718-1965-0178586-1 |issn=0025-5718|doi-access=free |url-access=subscription }}</ref> With these and other numerical methods Fourier transforms are fast,<ref>{{Cite journal |title=The fast Fourier transform |journal=IEEE Spectrum |date=1967 |language=en-US |doi=10.1109/mspec.1967.5217220 |last1=Brigham |first1=E. O. |last2=Morrow |first2=R. E. |volume=4 |issue=12 |pages=63–70 |bibcode=1967IEEES...4l..63B |s2cid=20294294 }}</ref> and it became possible to calculate accurate, [[#Dynamical diffraction|dynamical]] diffraction in seconds to minutes with laptops using widely available [[Multislice#Available software|multislice programs]].
* Developments in the [[convergent-beam electron diffraction]] approach.  Building on the original work of [[Walther Kossel]] and [[Gottfried Möllenstedt]] in 1939,<ref name=KM/> it was extended by Peter Goodman and Gunter Lehmpfuhl,<ref name=":4">{{cite journal |last1=Goodman |first1=P. |last2=Lehmpfuhl |first2=G. |title=Observation of the breakdown of Friedel's law in electron diffraction and symmetry determination from zero-layer interactions |journal=Acta Crystallographica Section A |date=1968 |volume=24 |issue=3 |pages=339–347 |doi=10.1107/S0567739468000677|bibcode=1968AcCrA..24..339G }}</ref> then mainly by the groups of [[John Steeds (scientist)|John Steeds]]<ref name="Buxton1">{{cite journal |last1=Buxton |first1=B. F. |last2=Eades |first2=J. A. |last3=Steeds |first3=John Wickham |last4=Rackham |first4=G. M. |last5=Frank |first5=Frederick Charles |title=The symmetry of electron diffraction zone axis patterns |journal=Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences |date=1976 |volume=281 |issue=1301 |pages=171–194 |doi=10.1098/rsta.1976.0024 |bibcode=1976RSPTA.281..171B |s2cid=122890943 }}</ref><ref name=":5">{{Cite journal |last1=Steeds |first1=J. W. |last2=Vincent |first2=R. |date=1983 |title=Use of high-symmetry zone axes in electron diffraction in determining crystal point and space groups |url=https://scripts.iucr.org/cgi-bin/paper?S002188988301050X |journal=Journal of Applied Crystallography |volume=16 |issue=3 |pages=317–324 |doi=10.1107/S002188988301050X |bibcode=1983JApCr..16..317S |issn=0021-8898|url-access=subscription }}</ref><ref>{{Cite journal |last=Bird |first=D. M. |date=1989 |title=Theory of zone axis electron diffraction |url=https://onlinelibrary.wiley.com/doi/10.1002/jemt.1060130202 |journal=Journal of Electron Microscopy Technique |language=en |volume=13 |issue=2 |pages=77–97 |doi=10.1002/jemt.1060130202 |pmid=2681572 |issn=0741-0581|url-access=subscription }}</ref> and Michiyoshi Tanaka<ref name=":6">{{Cite journal |last1=Tanaka |first1=M. |last2=Saito |first2=R. |last3=Sekii |first3=H. |date=1983 |title=Point-group determination by convergent-beam electron diffraction |url=https://scripts.iucr.org/cgi-bin/paper?S010876738300080X |journal=Acta Crystallographica Section A |volume=39 |issue=3 |pages=357–368 |doi=10.1107/S010876738300080X |bibcode=1983AcCrA..39..357T |issn=0108-7673|url-access=subscription }}</ref><ref>{{Cite journal |last1=Tanaka |first1=M. |last2=Saito |first2=R. |last3=Watanabe |first3=D. |date=1980 |title=Symmetry determination of the room-temperature form of LnNbO 4 (Ln = La,Nd) by convergent-beam electron diffraction |url=https://scripts.iucr.org/cgi-bin/paper?S0567739480000800 |journal=Acta Crystallographica Section A |volume=36 |issue=3 |pages=350–352 |doi=10.1107/S0567739480000800 |bibcode=1980AcCrA..36..350T |s2cid=98184340 |issn=0567-7394|url-access=subscription }}</ref> who showed how to determine [[point group]]s and [[space group]]s. It can also be used for higher-level refinements of the electron density;<ref>{{Cite book |last1=Spence |first1=J. C. H. |url=http://link.springer.com/10.1007/978-1-4899-2353-0 |title=Electron Microdiffraction |last2=Zuo |first2=J. M. |date=1992 |publisher=Springer US |isbn=978-1-4899-2355-4 |location=Boston, MA |language=en |doi=10.1007/978-1-4899-2353-0|s2cid=45473741 }}</ref>{{Rp|location=Chpt 4}} for a brief history see [[Convergent-beam electron diffraction#History|CBED history]]. In many cases this is the best method to determine symmetry.<ref name="Buxton1" /><ref name="Atlas" />
* The development of new approaches to reduce dynamical effects such as [[precession electron diffraction]] and three-dimensional diffraction methods. Averaging over different directions has, empirically, been found to significantly reduce dynamical diffraction effects, e.g.,<ref name="LDMPD">{{Cite book |last=Marks |first=Laurence |url=https://link.springer.com/10.1007/978-94-007-5580-2 |title=Uniting Electron Crystallography and Powder Diffraction |date=2012 |publisher=Springer Netherlands |isbn=978-94-007-5579-6 |editor-last=Kolb |editor-first=Ute |series=NATO Science for Peace and Security Series B: Physics and Biophysics |location=Dordrecht |pages=281–291 |language=en |doi=10.1007/978-94-007-5580-2 |bibcode=2012uecp.book.....K |editor-last2=Shankland |editor-first2=Kenneth |editor-last3=Meshi |editor-first3=Louisa |editor-last4=Avilov |editor-first4=Anatoly |editor-last5=David |editor-first5=William I.F}}</ref> see [[Precession electron diffraction#Historical development|PED history]] for further details. Not only is it easier to identify known structures with this approach, it can also be used to solve unknown structures in some cases<ref name="White" /><ref name="LDMPD" /><ref name="Lukas1" /> – see [[precession electron diffraction]] for further information.
* The development of experimental methods exploiting [[ultra-high vacuum]] technologies (e.g. the approach described by {{ill|Daniel J. Alpert|de|Daniel Alpert}} in 1953<ref name="Alpert">{{Cite journal |last=Alpert |first=D. |date=1953 |title=New Developments in the Production and Measurement of Ultra High Vacuum |url=http://aip.scitation.org/doi/10.1063/1.1721395 |journal=Journal of Applied Physics |language=en |volume=24 |issue=7 |pages=860–876 |doi=10.1063/1.1721395 |bibcode=1953JAP....24..860A |issn=0021-8979|url-access=subscription }}</ref>) to better control surfaces, making [[Electron diffraction#Low-energy electron diffraction (LEED)|LEED]] and [[Electron diffraction#Reflection high-energy electron diffraction (RHEED)|RHEED]] more reliable and reproducible techniques. In the early days the surfaces were not well controlled; with these technologies they can both be cleaned and remain clean for hours to days, a key component of [[surface science]].<ref name="Alpert" /><ref name="Oura" /> 
* Fast and accurate methods to calculate intensities for [[Electron diffraction#Low-energy electron diffraction (LEED)|LEED]] so it could be used to determine atomic positions, for instance references.<ref>{{Cite journal |last=Kambe |first=Kyozaburo |date=1967 |title=Theory of Low-Energy Electron Diffraction |journal=Zeitschrift für Naturforschung A |volume=22 |issue=3 |pages=322–330 |doi=10.1515/zna-1967-0305 |s2cid=96851585 |issn=1865-7109|doi-access=free }}</ref><ref>{{Cite journal |last=McRae |first=E.G. |date=1968 |title=Electron diffraction at crystal surfaces |url=https://linkinghub.elsevier.com/retrieve/pii/0039602868900587 |journal=Surface Science |language=en |volume=11 |issue=3 |pages=479–491 |doi=10.1016/0039-6028(68)90058-7|url-access=subscription }}</ref><ref name="Pendry71" /> These have been extensively exploited to determine the structure of many surfaces, and the arrangement of foreign atoms on surfaces.<ref name="LEEDB" />
* Methods to simulate the intensities in [[Electron diffraction#Reflection high-energy electron diffraction (RHEED)|RHEED]], so it can be used semi-quantitatively to understand surfaces during growth and thereby to control the resulting materials.<ref name="Ichimiya" /> 
* The development of advanced [[detectors for transmission electron microscopy]] such as [[charge-coupled device]]<ref name="SpenceZuo">{{Cite journal |last1=Spence |first1=J. C. H. |last2=Zuo |first2=J. M. |date=1988 |title=Large dynamic range, parallel detection system for electron diffraction and imaging |url=http://aip.scitation.org/doi/10.1063/1.1140039 |journal=Review of Scientific Instruments |language=en |volume=59 |issue=9 |pages=2102–2105 |doi=10.1063/1.1140039 |bibcode=1988RScI...59.2102S |issn=0034-6748|url-access=subscription }}</ref> and direct electron detectors,<ref name="PDetect">{{Cite journal |last1=Faruqi |first1=A. R. |last2=Cattermole |first2=D. M. |last3=Henderson |first3=R. |last4=Mikulec |first4=B. |last5=Raeburn |first5=C. |date=2003 |title=Evaluation of a hybrid pixel detector for electron microscopy |url=https://www.sciencedirect.com/science/article/pii/S0304399102003364 |journal=Ultramicroscopy |language=en |volume=94 |issue=3 |pages=263–276 |doi=10.1016/S0304-3991(02)00336-4 |pmid=12524196 |issn=0304-3991|url-access=subscription }}</ref> which improve the accuracy and reliability of intensity measurements. These have efficiencies and accuracies that can be a thousand or more times that of the photographic film used in the earliest experiments,<ref name="SpenceZuo" /><ref name="PDetect" /> with the information available in real time rather than requiring [[photographic processing]] after the experiment.<ref name="SpenceZuo" /><ref name="PDetect" />

== Core elements of electron diffraction ==
=== Plane waves, wavevectors and reciprocal lattice ===
What is seen in an electron diffraction pattern depends upon the sample and also the energy of the electrons. The electrons need to be considered as waves, which involves describing the electron via a wavefunction, written in crystallographic notation (see notes{{efn|name=Pi}} and{{efn|name=RecP}}) as:<ref name="Form" /><math display="block">\psi (\mathbf r) = \exp(2\pi i \mathbf k \cdot \mathbf r)</math>for a position <math>\mathbf r</math>. This is a [[quantum mechanics]] description; one cannot use a classical approach. The vector <math>\mathbf k</math> is called the wavevector, has units of inverse nanometers, and the form above is called a [[plane wave]] as the term inside the exponential is constant on the surface of a plane. The vector <math>\mathbf k</math> is what is used when drawing ray diagrams,<ref name="Cowley95" />{{Rp|location=Chpt 3}} and in vacuum is parallel to the direction or, better, group velocity<ref name="Broglie" />{{Rp|location=Chpt 1-2}}<ref name=":21">{{Cite book |last=Schiff |first=Leonard I. |title=Quantum mechanics |date=1987 |publisher=McGraw-Hill |isbn=978-0-07-085643-1 |edition=3. ed., 24. print |series=International series in pure and applied physics |location=New York}}</ref>{{Rp|page=16}} or [[probability current]]<ref name=":21" />{{Rp|pages=27, 130}} of the plane wave. For most cases the electrons are travelling at a respectable fraction of the speed of light, so rigorously need to be considered using relativistic quantum mechanics via the [[Dirac equation]],<ref>{{Cite journal |last1=Watanabe |first1=K. |last2=Hara |first2=S. |last3=Hashimoto |first3=I. |date=1996 |title=A Relativistic n -Beam Dynamical Theory for Fast Electron Diffraction |url=https://scripts.iucr.org/cgi-bin/paper?S0108767395015893 |journal=Acta Crystallographica Section A |volume=52 |issue=3 |pages=379–384 |doi=10.1107/S0108767395015893 |bibcode=1996AcCrA..52..379W |issn=0108-7673|url-access=subscription }}</ref> which as spin does not normally matter can be reduced to the [[Klein–Gordon equation]]. Fortunately one can side-step many complications and use a non-relativistic approach based around the Schrödinger equation.<ref name="Schroedinger" /> Following Kunio Fujiwara<ref name="Fujiwara">{{Cite journal |last=Fujiwara |first=Kunio |date=1961 |title=Relativistic Dynamical Theory of Electron Diffraction |url=https://journals.jps.jp/doi/10.1143/JPSJ.16.2226 |journal=Journal of the Physical Society of Japan |language=en |volume=16 |issue=11 |pages=2226–2238 |doi=10.1143/JPSJ.16.2226 |bibcode=1961JPSJ...16.2226F |issn=0031-9015|url-access=subscription }}</ref> and [[Archibald Howie]],<ref name="AHDiss">{{Cite journal |last=Howie |first=A |date=1962 |title=Discussion of K. Fujiwara's paper by M. J. Whelan |journal=Journal of the Physical Society of Japan |volume=17(Supplement BII) |page=118}}</ref> the relationship between the total energy of the electrons and the wavevector is written as:<math display="block">E = \frac{h^2 k^2}{2m^*}</math>with<math display="block">m^* = m_0 + \frac{E}{2c^2}</math>where <math>h</math> is the [[Planck constant]], <math>m^*</math> is a relativistic [[Effective mass (solid-state physics)|effective mass]] used to cancel out the relativistic terms for electrons of energy <math>E</math> with <math>c</math> the speed of light and <math>m_0</math> the rest mass of the electron. The concept of effective mass occurs throughout physics (see for instance [[Ashcroft and Mermin]]),<ref name=":7">{{Cite book |last1=Ashcroft |first1=Neil W. |title=Solid state physics |last2=Mermin |first2=N. David |date=2012 |publisher=Brooks/Cole Thomson Learning |isbn=978-0-03-083993-1 |edition=Repr |location=South Melbourne}}</ref>{{Rp|location=Chpt 12}} and comes up in the behavior of [[quasiparticles]]. A common one is the [[electron hole]], which acts as if it is a particle with a positive charge and a mass similar to that of an electron, although it can be several times lighter or heavier. For electron diffraction the electrons behave as if they are non-relativistic particles of mass <math>m^*</math> in terms of how they interact with the atoms.<ref name="Fujiwara" />

The wavelength of the electrons <math>\lambda</math> in vacuum is from the above equations<math display="block"> \lambda = \frac 1 k = \frac{h}{\sqrt{2m^* E}} = \frac{h c}{\sqrt{E(2 m_0 c^2 + E)}},</math>and can range from about {{val|0.1|ul=nm}}, roughly the size of an atom, down to a thousandth of that. Typically the energy of the electrons is written in [[electronvolt]]s (eV), the voltage used to accelerate the electrons; the actual energy of each electron is this voltage times the [[electron charge]]. For context, the typical energy of a [[chemical bond]] is a few eV;<ref>{{Cite web |date=2013-10-02 |title=Bond Energies |url=https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)/Chemical_Bonding/Fundamentals_of_Chemical_Bonding/Bond_Energies |access-date=2023-09-26 |website=Chemistry LibreTexts |language=en}}</ref> electron diffraction involves electrons up to {{val|5,000,000|u=eV}}.

The magnitude of the interaction of the electrons with a material scales as<ref name="Cowley95">{{Cite book |last=Cowley |first=John M. |title=Diffraction physics |date=1995 |publisher=Elsevier |isbn=0-444-82218-6 |oclc=247191522}}</ref>{{Rp|location=Chpt 4}}<math display="block">2 \pi \frac{m^*}{h^2 k} = 2\pi\frac{ m^* \lambda} {h^2} = \frac \pi {hc} \sqrt{\frac{2m_0 c^2}{E} + 1}.</math>While the wavevector increases as the energy increases, the change in the effective mass compensates this so even at the very high energies used in electron diffraction there are still significant interactions.<ref name="Fujiwara" />

The high-energy electrons interact with the Coulomb potential,<ref name="Bethe" /> which for a crystal can be considered in terms of a [[Fourier series]] (see for instance [[Ashcroft and Mermin]]),<ref name=":7" />{{Rp|location=Chpt 8}} that is<math display="block">V(\mathbf r) = \sum V_g \exp(2 \pi i \mathbf g \cdot \mathbf r)</math>with <math>\mathbf g</math> a [[reciprocal lattice]] vector and <math>V_g</math> the corresponding Fourier coefficient of the potential. The reciprocal lattice vector is often referred to in terms of [[Miller indices]] <math>(h k l)</math>, a sum of the individual reciprocal lattice vectors <math>\mathbf A,\mathbf B,\mathbf C</math> with integers <math>h, k, l</math> in the form:<ref name="Form" /><math display="block">\mathbf g = h \mathbf A + k \mathbf B + l \mathbf C</math>(Sometimes reciprocal lattice vectors are written as <math>\mathbf a^*</math>,  <math>\mathbf b^*</math>, <math>\mathbf c^*</math> and see note.{{efn|name=RecP}}) The contribution from the <math>V_g</math> needs to be combined with what is called the shape function (e.g.<ref>{{cite book |last=Vainstein |first=B.K. |chapter=Experimental Electron Diffraction Structure Investigations |date=1964 |title=Structure Analysis by Electron Diffraction |pages=295–390 |publisher=Elsevier |doi=10.1016/b978-0-08-010241-2.50010-9 |isbn=978-0-08-010241-2 }}</ref><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 |bibcode=1950AcCry...3..316R |issn=0365-110X|doi-access=free }}</ref><ref name="Cowley95" />{{Rp|location=Chpt 2}}), which is the [[Fourier transform]] of the shape of the object. If, for instance, the object is small in one dimension then the shape function extends far in that direction in the Fourier transform—a reciprocal relationship.<ref>{{Cite web |title=Kevin Cowtan's Book of Fourier, University of York, UK |url=http://www.ysbl.york.ac.uk/~cowtan/fourier/crys1.html |access-date=2023-09-26 |website=www.ysbl.york.ac.uk}}</ref>

{{anchor|Figure 6}}[[File:EwaldS2.png|thumb|Figure 6: Ewald sphere construction for transmission electron diffraction, showing two of the Laue zones and the excitation error|alt=Illustration of how the wavevectors and diffraction from reciprocal lattice vectors is connected, called an Ewald sphere construction. This example is for transmission electron diffraction.]]

Around each reciprocal lattice point one has this shape function.<ref name="Cowley95" />{{Rp|location=Chpt 5-7}}<ref name="HirschEtAl">{{Cite book |last1=Hirsch |first1=P. B. | last2=Howie | first2=A. | last3=Nicholson| first3=R. B.| last4=Pashley | first4=D. W. | last5=Whelan | first5=M. J.|title=Electron microscopy of thin crystals |date=1965 |publisher=Butterworths |isbn=0-408-18550-3 |location=London |oclc=2365578}}</ref>{{Rp|location=Chpt 2}} How much intensity there will be in the diffraction pattern depends upon the intersection of the [[Ewald sphere]], that is energy conservation, and the shape function around each reciprocal lattice point—see [[#Figure 6|Figure 6]], [[#Figure 20|20]] and [[#Figure 22|22]]. The vector from a reciprocal lattice point to the Ewald sphere is called the excitation error <math>\mathbf s_g</math>.

For transmission electron diffraction the samples used are thin, so most of the shape function is along the direction of the electron beam. For both [[Electron diffraction#Low-energy electron diffraction (LEED)|LEED]]<ref name="LEEDB" /> and [[Electron diffraction#Reflection high-energy electron diffraction (RHEED)|RHEED]]<ref name="Ichimiya" /> the shape function is mainly normal to the surface of the sample. In [[#Low-energy electron diffraction|LEED]] this results in (a simplification) back-reflection of the electrons leading to spots, see [[#Figure 20|Figure 20]] and [[#Figure 21|21]] later, whereas in [[#Reflection high-energy electron diffraction|RHEED]] the electrons reflect off the surface at a small angle and typically yield diffraction patterns with streaks, see [[#Figure 22|Figure 22]] and [[#Figure 23|23]] later. By comparison, with both x-ray and neutron diffraction the scattering is significantly weaker,<ref name="Cowley95" />{{Rp|location=Chpt 4}} so typically requires much larger crystals, in which case the shape function shrinks to just around the reciprocal lattice points, leading to simpler Bragg's law diffraction.<ref name="Bragg">{{Cite journal |last1=Bragg |first1=W.H. |last2=Bragg |first2=W.L. |date=1913 |title=The reflection of X-rays by crystals |journal=Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character |language=en |volume=88 |issue=605 |pages=428–438 |doi=10.1098/rspa.1913.0040 |bibcode=1913RSPSA..88..428B |s2cid=13112732 |issn=0950-1207|doi-access=free }}</ref>

For all cases, when the reciprocal lattice points are close to the Ewald sphere (the excitation error is small) the intensity tends to be higher; when they are far away it tends to be smaller. The set of diffraction spots at right angles to the direction of the incident beam are called the zero-order Laue zone (ZOLZ) spots, as shown in [[#Figure 6|Figure 6]]. One can also have intensities further out from reciprocal lattice points which are in a higher layer. The first of these is called the first order Laue zone (FOLZ); the series is called by the generic name higher order Laue zone (HOLZ).<ref name="Reimer" />{{Rp|location=Chpt 7}}<ref>{{Cite web |title=higher-order Laue zone (HOLZ) reflection {{!}} Glossary {{!}} JEOL Ltd. |url=https://www.jeol.com/ |access-date=2023-10-02 |website=higher-order Laue zone (HOLZ) reflection {{!}} Glossary {{!}} JEOL Ltd. |language=en}}</ref>

The result is that the electron wave after it has been diffracted can be written as an integral over different plane waves:<ref name="Peng" />{{Rp|location=Chpt 1}}<math display="block"> \psi (\mathbf r) = \int \phi (\mathbf k) \exp(2 \pi i \mathbf k \cdot \mathbf r) d^3\mathbf k ,</math>that is a sum of plane waves going in different directions, each with a complex amplitude <math>\phi (\mathbf k)</math>. (This is a three dimensional integral, which is often written as <math>d\mathbf k</math> rather than <math>d^3\mathbf k</math>.) For a crystalline sample these wavevectors have to be of the same magnitude for elastic scattering (no change in energy), and are related to the incident direction <math>\mathbf k_0</math> by (see [[#Figure 6|Figure 6]])
<math display="block">\mathbf k = \mathbf k_0 + \mathbf g + \mathbf s_g.</math>
A diffraction pattern detects the intensities<math display="block"> I(\mathbf k) = \left| \phi(\mathbf k) \right| ^2 .</math>For a crystal these will be near the reciprocal lattice points typically forming a two dimensional grid. Different samples and modes of diffraction give different results, as do different approximations for the amplitudes <math>\phi (\mathbf k)</math>.<ref name="Cowley95" /><ref name="Reimer" /><ref name=":11" />

A typical electron diffraction pattern in TEM and [[Electron diffraction#Low-energy electron diffraction (LEED)|LEED]] is a grid of high intensity spots (white) on a dark background, approximating a projection of the reciprocal lattice vectors, see [[#Figure 1|Figure 1]], [[#Figure 9|9]], [[#Figure 10|10]], [[#Figure 11|11]], [[#Figure 14|14]] and [[#Figure 21|21]] later. There are also cases which will be mentioned later where diffraction patterns are [[#Aperiodic materials|not periodic]], see [[#Figure 15|Figure 15]], have additional [[#Diffuse scattering|diffuse]] structure as in [[#Figure 16|Figure 16]], or have rings as in [[#Figure 12|Figure 12]], [[#Figure 13|13]] and [[#Figure 24|24]]. With conical illumination as in [[#Convergent beam electron diffraction|CBED]] they can also be a grid of discs, see [[#Figure 7|Figure 7]], [[#Figure 9|9]] and [[#Figure 18|18]]. [[RHEED]] is slightly different,<ref name="Ichimiya" /> see [[#Figure 22|Figure 22]], [[#Figure 23|23]]. If the excitation errors <math>s_g</math> were zero for every reciprocal lattice vector, this grid would be at exactly the spacings of the reciprocal lattice vectors. This would be equivalent to a Bragg's law condition for all of them. In TEM the wavelength is small and this is close to correct, but not exact. In practice the deviation of the positions from a simple Bragg's law<ref name="Bragg" /> interpretation is often neglected, particularly if a column approximation is made (see below).<ref name="Peng" />{{Rp|page=64}}<ref name="HirschEtAl" />{{Rp|location=Chpt 11}}<ref name="Tanaka" />

=== Kinematical diffraction ===
In Kinematical theory an approximation is made that the electrons are only scattered once.<ref name="Cowley95" />{{Rp|location=Sec 2}}  For transmission electron diffraction it is common to assume a constant thickness <math>t</math>, and also what is called the Column Approximation (e.g. references<ref name="HirschEtAl" />{{Rp|location=Chpt 11}}<ref name="Tanaka">{{Citation |last=Tanaka |first=Nobuo |title=Column Approximation and Howie-Whelan's Method for Dynamical Electron Diffraction |date=2017 |work=Electron Nano-Imaging |pages=293–296 |place=Tokyo |publisher=Springer Japan |doi=10.1007/978-4-431-56502-4_27 |isbn=978-4-431-56500-0 }}</ref> and further reading). For a perfect crystal the intensity for each diffraction spot <math>\mathbf g</math> is then:<math display="block">I_{g} = \left|\phi(\mathbf k)\right|^2 \propto \left|F_{g}\frac{\sin(\pi t s_z)}{\pi s_z}\right|^2
</math>where <math>s_z</math> is the magnitude of the excitation error <math>|\mathbf s_z|</math> along z, the distance along the beam direction (z-axis by convention) from the diffraction spot to the [[Ewald's sphere|Ewald sphere]], and <math>F_{g}</math> is the [[structure factor]]:<ref name="Form" /><math display="block">F_{g} = \sum_{j=1}^N f_j \exp{(2 \pi i \mathbf g \cdot \mathbf r_j -T_j g^2)} </math>the sum being over all the atoms in the unit cell with <math>f_j</math> the form factors,<ref name="Form">{{Citation |last1=Colliex |first1=C. |title=Electron diffraction |date=2006 |url=https://xrpp.iucr.org/cgi-bin/itr?url_ver=Z39.88-2003&rft_dat=what%3Dchapter%26volid%3DCb%26chnumo%3D4o3%26chvers%3Dv0001 |work=International Tables for Crystallography |volume=C |pages=259–429 |editor-last=Prince |editor-first=E. |edition=1 |place=Chester, England |publisher=International Union of Crystallography |doi=10.1107/97809553602060000593 |isbn=978-1-4020-1900-5 |last2=Cowley |first2=J. M. |last3=Dudarev |first3=S. L. |last4=Fink |first4=M. |last5=Gjønnes |first5=J. |last6=Hilderbrandt |first6=R. |last7=Howie |first7=A. |last8=Lynch |first8=D. F. |last9=Peng |first9=L. M.|url-access=subscription }}</ref> <math>\mathbf g</math> the [[reciprocal lattice]] vector, <math>T_j</math> is a simplified form of the [[Debye–Waller factor]],<ref name="Form" /> and <math>\mathbf k</math> is the wavevector for the diffraction beam which is:<math display="block">\mathbf k = \mathbf k_0 + \mathbf g + \mathbf s_z</math>for an incident wavevector of <math>\mathbf k_0</math>, as in [[#Figure 6|Figure 6]] and [[Electron diffraction#Plane waves, wavevectors and reciprocal lattice|above]]. The excitation error comes in as the outgoing wavevector <math>\mathbf k</math> has to have the same modulus (i.e. energy) as the incoming wavevector <math>\mathbf k_0</math>. The intensity in transmission electron diffraction oscillates as a function of thickness, which can be confusing; there can similarly be intensity changes due to variations in orientation and also structural defects such as [[dislocations]].<ref>{{Cite journal | last1=Hirsch | first1=Peter | last2=Whelan | first2=Michael | date=1960 |title=A kinematical theory of diffraction contrast of electron transmission microscope images of dislocations and other defects |url=https://royalsocietypublishing.org/doi/10.1098/rsta.1960.0013 |journal=Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences |language=en |volume=252 |issue=1017 |pages=499–529 |doi=10.1098/rsta.1960.0013 | bibcode=1960RSPTA.252..499H | s2cid=123349515 |issn=0080-4614|url-access=subscription }}</ref> If a diffraction spot is strong it could be because it has a larger structure factor, or it could be because the combination of thickness and excitation error is "right". Similarly the observed intensity can be small, even though the structure factor is large. This can complicate interpretation of the intensities. By comparison, these effects are much smaller in [[x-ray diffraction]] or [[neutron diffraction]] because they interact with matter far less and often Bragg's law<ref name="Bragg" /> is adequate.

This form is a reasonable first approximation which is qualitatively correct in many cases, but more accurate forms including multiple scattering (dynamical diffraction) of the electrons are needed to properly understand the intensities.<ref name="Cowley95" />{{Rp|location=Sec 3}}<ref name="Peng" />{{Rp|location=Chpt 3-5}}

=== 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 |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}}</ref><ref name="PHInel">{{Cite journal |date=1963 |title=Inelastic scattering of electrons by crystals. I. The theory of small-angle in elastic scattering | 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}}</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. |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 |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}}</ref> Without these the calculations may not be accurate enough.<ref name="Pendry71" />

=== Kikuchi lines ===
{{main|Kikuchi lines}}

Kikuchi lines,<ref>{{Cite journal |last=Kainuma |first=Y. |date=1955|title=The Theory of Kikuchi patterns |url=https://scripts.iucr.org/cgi-bin/paper?S0365110X55000832 |journal=Acta Crystallographica |volume=8 |issue=5 |pages=247–257 |doi=10.1107/S0365110X55000832|bibcode=1955AcCry...8..247K |doi-access=free }}</ref><ref name="Reimer" />{{Rp|pages=311–313}} first observed by [[Seishi Kikuchi]] in 1928,<ref name=":17" /><ref name=":18">{{Cite journal |last=Kikuchi |first=Seishi |date=1928 |title=Electron diffraction in single crystals |journal=Japanese Journal of Physics |volume=5 |issue=3061 |pages=83–96}}</ref> are linear features created by electrons scattered both inelastically and elastically. As the electron beam interacts with matter, the electrons are diffracted via [[elastic scattering]], and also scattered [[inelastic scattering|inelastically]] losing part of their energy. These occur simultaneously, and cannot be separated – according to  the [[Copenhagen interpretation]] of quantum mechanics, only the probabilities of electrons at detectors can be measured.<ref name=":12">{{Citation |last=Faye |first=Jan |title=Copenhagen Interpretation of Quantum Mechanics |date=2019 |url=https://plato.stanford.edu/archives/win2019/entries/qm-copenhagen/ |encyclopedia=The Stanford Encyclopedia of Philosophy |editor-last=Zalta |editor-first=Edward N. |access-date=2023-09-26 |edition=Winter 2019 |publisher=Metaphysics Research Lab, Stanford University}}</ref><ref name=":13" /> These electrons form Kikuchi lines which provide information on the orientation.<ref name="Morniroli 2004"/>{{anchor|Figure 8}}[[File:KMapFCC.png|thumb|Figure 8: Kikuchi map for a [[face centered cubic]] material, within the stereographic triangle|alt=A Kukuchi map, which is a collage of diffraction patterns used to both determine crystal orientation and also to tilt to different orientations.]]

Kikuchi lines come in pairs forming Kikuchi bands, and are indexed in terms of the crystallographic planes they are connected to, with the angular width of the band equal to the magnitude of the corresponding diffraction vector <math>|\mathbf g|</math>. The position of Kikuchi bands is fixed with respect to each other and the orientation of the sample, but not against the diffraction spots or the direction of the incident electron beam. As the crystal is tilted, the bands move on the diffraction pattern.<ref name="Morniroli 2004"/> Since the position of Kikuchi bands is quite sensitive to crystal [[Orientation (geometry)|orientation]], they can be used to fine-tune a zone-axis orientation or determine crystal orientation. They can also be used for navigation when changing the orientation between zone axes connected by some band, an example of such a map produced by combining many local sets of experimental Kikuchi patterns is in [[#Figure 8|Figure 8]]; Kikuchi maps are available for many materials.

== Types and techniques ==
=== In a transmission electron microscope ===
{{anchor|Figure 9}}[[File:Difrakce.png|thumb|300px|Figure 9: Diffraction patterns (below, black background) with different crystallinity (above, diagrams) and beam convergence. From left: spot diffraction (parallel illumination), [[CBED]] (converging), and ring diffraction (parallel with many grains).|alt=Electron diffraction patterns from different types of crystals and different incident beam convergence.]]

Electron diffraction in a [[Transmission electron microscopy|TEM]] exploits controlled electron beams using electron optics.<ref name=":8">{{Cite book |last1=Hawkes |first1=Peter |url=https://www.sciencedirect.com/book/9780081022566/principles-of-electron-optics |title=Principles of Electron Optics Volume Two: Applied Geometric Optics |last2=Kasper |first2=Erwin |publisher=Elsevier |year=2018 |isbn=978-0-12-813369-9 |edition=2nd |pages=Chpts 36, 40, 41, 43, 49, 50}}</ref> Different types of diffraction experiments, for instance [[#Figure 9|Figure 9]], provide information such as [[lattice constants]], symmetries, and sometimes to solve an unknown [[crystal structure]].

It is common to combine it with other methods, for instance images using selected diffraction beams, [[High-resolution transmission electron microscopy|high-resolution images]]<ref>{{Cite book |last=Spence |year=2017 |first=John C. H. |title=High-resolution electron microscopy |publisher=Oxford University Press |isbn=978-0-19-879583-4 |oclc=1001251352}}</ref> showing the atomic structure, chemical analysis through [[energy-dispersive X-ray spectroscopy|energy-dispersive x-ray spectroscopy]],<ref>{{Cite book |last=Heinrich |first=J. K. F. |title=Energy dispersive x-ray spectrometry. |date=1981 |publisher=National Technical Information Service |oclc=801808484}}</ref> investigations of electronic structure and bonding through [[electron energy loss spectroscopy]],<ref>{{Cite book |last=Egerton |first=F. R. |title=Electron energy-loss spectroscopy in the electron microscope |date=2011 |publisher=Springer |isbn=978-1-4419-9582-7 |oclc=706920411}}</ref> and studies of the electrostatic potential through [[electron holography]];<ref>{{Cite journal |last=Cowley |first=J. M. |date=1992 |title=Twenty forms of electron holography |journal=Ultramicroscopy |language=en |volume=41 |issue=4 |pages=335–348 |doi=10.1016/0304-3991(92)90213-4 |issn=0304-3991}}</ref> this list is not exhaustive. Compared to [[x-ray crystallography]], TEM analysis is significantly more localized and can be used to obtain information from tens of thousands of atoms to just a few or even single atoms.

==== Formation of a diffraction pattern ====

{{anchor|Figure 10}}[[File:ElmagLensScheme.png|left|thumb|300px|Figure 10: Imaging scheme of magnetic lens (center, colored ray diagram) with image (left) and diffraction pattern (right, black background)|alt=Simple comparison of imaging, ray diagram and diffraction in an electron microscope.]]

In TEM, the electron beam passes through a thin film of the material as illustrated in [[#Figure 10|Figure 10]]. Before and after the sample the beam is manipulated by the [[electron optics]]<ref name=":8" /> including [[magnetic lens]]es, deflectors and [[apertures]];<ref name="Pella">{{Cite web |title=Apertures, Electron Microscope Apertures |url=https://www.tedpella.com/apertures-and-filaments_html/apertures-overview.aspx |access-date=2023-02-11|website=www.tedpella.com}}</ref> these act on the electrons similar to how glass lenses focus and control light. Optical elements above the sample are used to control the incident beam which can range from a wide and parallel beam to one which is a converging cone and can be smaller than an atom, 0.1&nbsp;nm. As it interacts with the sample, part of the beam is diffracted and part is transmitted without changing its direction. This occurs simultaneously as electrons are everywhere until they are detected ([[Wave function collapse|wavefunction collapse]]) according to the [[Copenhagen interpretation]].<ref name=":12" /><ref name=":13">{{Cite book |last=Gbur |first=Gregory J. |title=Falling Felines and Fundamental Physics |date=2019 |publisher=Yale University Press |isbn=978-0-300-23129-8 |pages=243–263 |doi=10.2307/j.ctvqc6g7s.17|jstor=j.ctvqc6g7s |s2cid=243353224 }}</ref>

Below the sample, the beam is controlled by another set of magnetic lenses and apertures.<ref name=":8" /> Each set of initially parallel rays (a [[#Geometrical considerations|plane wave]]) is focused by the first lens ([[Objective (optics)|objective]]) to a point in the [[back focal plane]] of this lens, forming a spot on a [[Detectors for transmission electron microscopy|detector]]; a map of these directions, often an array of spots, is the diffraction pattern. Alternatively the lenses can form a magnified image of the sample.<ref name=":8" /> Herein the focus is on collecting a diffraction pattern; for other information see the pages on [[transmission electron microscopy|TEM]] and [[scanning transmission electron microscopy]].

==== Selected area electron diffraction ====

The simplest diffraction technique in TEM is selected area electron diffraction (SAED) where the incident beam is wide and close to parallel.<ref name="HirschEtAl" />{{Rp|location=Chpt 5-6}} An aperture is used to select a particular region of interest from which the diffraction is collected. These apertures are part of a thin foil of a heavy metal such as [[tungsten]]<ref name="Pella" /> which has a number of small holes in it. This way diffraction information can be limited to, for instance, individual crystallites. Unfortunately the method is limited by the spherical aberration of the objective lens,<ref name="HirschEtAl" />{{Rp|location=Chpt 5-6}} so is only accurate for large grains with tens of thousands of atoms or more; for smaller regions a focused probe is needed.<ref name="HirschEtAl" />{{Rp|location=Chpt 5-6}}

If a parallel beam is used to acquire a diffraction pattern from a [[single-crystal]], the result is similar to a two-dimensional projection of the crystal reciprocal lattice. From this one can determine interplanar distances and angles and in some cases crystal symmetry, particularly when the electron beam is down a major zone axis, see for instance the database by Jean-Paul Morniroli.<ref name="Atlas">{{Cite book |last=Morniroli |first=Jean-Paul |url=https://www.electron-diffraction.fr/software_059.htm |title=The atlas of electron diffraction zone axis patterns |year=2015 |location=Webpage and hardcopy |archive-date=2022-10-07 |access-date=2023-02-14 |archive-url=https://web.archive.org/web/20221007145433/https://www.electron-diffraction.fr/software_059.htm }}</ref> However, projector lens aberrations such as [[Barrel Distortion|barrel distortion]] as well as dynamical diffraction effects (e.g.<ref>{{Cite journal |last1=Honjo |first1=Goro |last2=Mihama |first2=Kazuhiro |date=1954 |title=Fine Structure due to Refraction Effect in Electron Diffraction Pattern of Powder Sample Part II. Multiple Structures due to Double Refraction given by Randomly Oriented Smoke Particles of Magnesium and Cadmium Oxide |journal=Journal of the Physical Society of Japan |volume=9 |issue=2 |pages=184–198 |doi=10.1143/jpsj.9.184 |issn=0031-9015}}</ref>) cannot be ignored. For instance, certain diffraction spots which are not present in x-ray diffraction can appear,<ref name="Atlas" /> for instance those due to [[Jon Gjønnes|Gjønnes]]-Moodie extinction conditions.<ref name="Gjønnes 65–67"/> {{anchor|Figure 11}}[[File:Crystal orientation and diffraction.gif|thumb|300px|Figure 11: Diffraction pattern of [[magnesium]] simulated using CrysTBox for various crystal orientations. Note how the diffraction pattern (white/black) changes with the crystal orientation (yellow).|alt=A pair of image showing how diffraction patterns change with the orientation of the crystal.]]

If the sample is tilted relative to the electron beam, different sets of crystallographic planes contribute to the pattern yielding different types of diffraction patterns, approximately different projections of the reciprocal lattice, see [[#Figure 11|Figure 11]].<ref name="Atlas" /> This can be used to determine the crystal orientation, which in turn can be used to set the orientation needed for a particular experiment. Furthermore, a series of diffraction patterns varying in tilt can be acquired and processed using a [[diffraction tomography]] approach. There are ways to combine this with [[direct methods (crystallography)|direct methods]] algorithms using electrons<ref name="Sufficient" /><ref name="White" /> and other methods such as charge flipping,<ref name="Lukas1">{{Cite journal |last=Palatinus |first=Lukáš |date=2013 |title=The charge-flipping algorithm in crystallography |url=https://scripts.iucr.org/cgi-bin/paper?S2052519212051366 |journal=Acta Crystallographica Section B: Structural Science, Crystal Engineering and Materials |volume=69 |issue=1 |pages=1–16 |doi=10.1107/S2052519212051366 |pmid=23364455 |bibcode=2013AcCrB..69....1P |issn=2052-5192|doi-access=free }}</ref> or automated diffraction tomography<ref>{{Cite journal |last1=Kolb |first1=U. |last2=Gorelik |first2=T. |last3=Kübel |first3=C. |last4=Otten |first4=M.T. |last5=Hubert |first5=D. |date=2007 |title=Towards automated diffraction tomography: Part I—Data acquisition |journal=Ultramicroscopy |volume=107 |issue=6–7 |pages=507–513 |doi=10.1016/j.ultramic.2006.10.007 |pmid=17234347 |issn=0304-3991}}</ref><ref>{{Cite journal |last1=Mugnaioli |first1=E. |last2=Gorelik |first2=T. |last3=Kolb |first3=U. |date=2009 |title="Ab initio" structure solution from electron diffraction data obtained by a combination of automated diffraction tomography and precession technique |journal=Ultramicroscopy |volume=109 |issue=6 |pages=758–765 |doi=10.1016/j.ultramic.2009.01.011 |pmid=19269095 |issn=0304-3991}}</ref> to solve crystal structures.

==== Polycrystalline pattern ====
{{anchor|Figure 12}}[[File:SpotToRingDiffraction.gif|thumb|Figure 12: Relation between spot and ring diffraction illustrated on 1 to 1000 grains of [[MgO]] using simulation engine of [[CrysTBox]]. Corresponding experimental patterns can be seen in '''Figure 13.''' |alt=A pattern showing how diffraction patterns from different grain build up to yield a ring pattern.]]

Diffraction patterns depend on whether the beam is diffracted by one [[single crystal]] or by a number of differently oriented crystallites, for instance in a polycrystalline material. If there are many contributing crystallites, the diffraction image is a superposition of individual crystal patterns, see [[#Figure 12|Figure 12]]. With a large number of grains this superposition yields diffraction spots of all possible reciprocal lattice vectors. This results in a pattern of [[concentric]] rings as shown in [[#Figure 12|Figure 12]] and [[#Figure 13|13]].<ref name="HirschEtAl" />{{Rp|location=Chpt 5-6}}

{{anchor|Figure 13}}{{multiple image
| align             = right
| width             = 150
| image1            = ringGUI input.png
| image2            = ringGUI quadrant.png
| footer            = Figure 13: Ring diffraction image of [[MgO]] as recorded (left) and processed with CrysTBox ringGUI (right, with indexing). Corresponding simulated pattern can be seen in '''Figure 12'''.
| alt1              = Experimental ring pattern from magnesium oxide.
| alt2              = A computer model of a ring diffraction pattern to go with the other image.
}}

Textured materials yield a non-uniform distribution of intensity around the ring, which can be used to discriminate between nanocrystalline and amorphous phases. However, diffraction often cannot differentiate between very small grain polycrystalline materials and truly random order amorphous.<ref>{{Cite journal |last1=Howie |first1=A. |last2=Krivanek |first2=O. L. |last3=Rudee |first3=M. L. |date=1973 |title=Interpretation of electron micrographs and diffraction patterns of amorphous materials |url=http://www.tandfonline.com/doi/abs/10.1080/14786437308228927 |journal=Philosophical Magazine |language=en |volume=27 |issue=1 |pages=235–255 |doi=10.1080/14786437308228927 |bibcode=1973PMag...27..235H |issn=0031-8086|url-access=subscription }}</ref> Here [[high-resolution transmission electron microscopy]]<ref>{{Cite journal |last=Howie |first=A. |date=1978 |title=High resolution electron microscopy of amorphous thin films |journal=Journal of Non-Crystalline Solids |series=Proceedings of the Topical Conference on Atomic Scale Structure of Amorphous Solids |volume=31 |issue=1 |pages=41–55 |doi=10.1016/0022-3093(78)90098-4 |bibcode=1978JNCS...31...41H |issn=0022-3093}}</ref> and [[fluctuation electron microscopy]]<ref>{{Cite journal |last1=Gibson |first1=J. M. |last2=Treacy |first2=M. M. J. |date=1997 |title=Diminished Medium-Range Order Observed in Annealed Amorphous Germanium |url=https://link.aps.org/doi/10.1103/PhysRevLett.78.1074 |journal=Physical Review Letters |language=en |volume=78 |issue=6 |pages=1074–1077 |doi=10.1103/PhysRevLett.78.1074 |bibcode=1997PhRvL..78.1074G |issn=0031-9007|url-access=subscription }}</ref><ref>{{Cite journal |last1=Treacy |first1=M M J |last2=Gibson |first2=J M |last3=Fan |first3=L |last4=Paterson |first4=D J |last5=McNulty |first5=I |date=2005 |title=Fluctuation microscopy: a probe of medium range order |url=https://iopscience.iop.org/article/10.1088/0034-4885/68/12/R06 |journal=Reports on Progress in Physics |volume=68 |issue=12 |pages=2899–2944 |doi=10.1088/0034-4885/68/12/R06 |bibcode=2005RPPh...68.2899T |s2cid=16316238 |issn=0034-4885|url-access=subscription }}</ref> can be more powerful, although this is still a topic of continuing development.

==== Multiple materials and double diffraction ====
In simple cases there is only one grain or one type of material in the area used for collecting a diffraction pattern. However, often there is more than one. If they are in different areas then the diffraction pattern will be a combination.<ref name="HirschEtAl" />{{Rp|location=Chpt 5-6}} In addition there can be one grain on top of another, in which case the electrons that go through the first are diffracted by the second.<ref name="HirschEtAl" />{{Rp|location=Chpt 5-6}} Electrons have no memory (like many of us), so after they have gone through the first grain and been diffracted, they traverse the second as if their current direction was that of the incident beam. This leads to diffraction spots which are the vector sum of those of the two (or even more) reciprocal lattices of the crystals, and can lead to complicated results. It can be difficult to know if this is real and due to some novel material, or just a case where multiple crystals and diffraction is leading to odd results.<ref name="HirschEtAl" />{{Rp|location=Chpt 5-6}}

==== Bulk and surface superstructures ====
Many materials have relatively simple structures based upon small unit cell vectors <math>\mathbf a,\mathbf b,\mathbf c</math> (see also note{{efn|name=RecP}}). There are many others where the repeat is some larger multiple of the smaller unit cell (subcell) along one or more direction, for instance <math>N\mathbf a, M\mathbf b, \mathbf c</math>. which has larger dimensions in two directions. These [[Superstructure (condensed matter)|superstructures]]<ref name=Janner77 /><ref name="Bak">{{Cite journal |last=Bak |first=P |date=1982 |title=Commensurate phases, incommensurate phases and the devil's staircase |journal=Reports on Progress in Physics |volume=45 |issue=6 |pages=587–629 |doi=10.1088/0034-4885/45/6/001 |issn=0034-4885}}</ref><ref name=Jannsen2006/> can arise from many reasons:
# Larger unit cells due to electronic ordering which leads to small displacements of the atoms in the subcell. One example is [[antiferroelectricity]] ordering.<ref>{{Cite journal |last1=Randall |first1=Clive A. |last2=Fan |first2=Zhongming |last3=Reaney |first3=Ian |author4-link=Long-Qing Chen |last4=Chen |first4=Long-Qing |last5=Trolier-McKinstry |first5=Susan |date=2021 |title=Antiferroelectrics: History, fundamentals, crystal chemistry, crystal structures, size effects, and applications |url=https://ceramics.onlinelibrary.wiley.com/doi/10.1111/jace.17834 |journal=Journal of the American Ceramic Society |language=en |volume=104 |issue=8 |pages=3775–3810 |doi=10.1111/jace.17834 |s2cid=233534909 |issn=0002-7820|url-access=subscription }}</ref>
# Chemical ordering, that is different atom types at different locations of the subcell.<ref>{{Cite journal |last1=Heine |first1=V |last2=Samson |first2=J H |date=1983 |title=Magnetic, chemical and structural ordering in transition metals |url=https://iopscience.iop.org/article/10.1088/0305-4608/13/10/025 |journal=Journal of Physics F: Metal Physics |volume=13 |issue=10 |pages=2155–2168 |doi=10.1088/0305-4608/13/10/025 |bibcode=1983JPhF...13.2155H |issn=0305-4608|url-access=subscription }}</ref>
# Magnetic order of the spins. These may be in opposite directions on some atoms, leading to what is called [[antiferromagnetism]].<ref>{{Cite web |date=2019-09-13 |title=6.8: Ferro-, Ferri- and Antiferromagnetism |url=https://chem.libretexts.org/Bookshelves/Inorganic_Chemistry/Book%3A_Introduction_to_Inorganic_Chemistry_(Wikibook)/06%3A_Metals_and_Alloys-_Structure_Bonding_Electronic_and_Magnetic_Properties/6.08%3A_Ferro-_Ferri-_and_Antiferromagnetism |access-date=2023-09-26 |website=Chemistry LibreTexts |language=en}}</ref>

{{anchor|Figure 14}}[[File:Transmission electron diffraction pattern of Si (111) 7x7.png|thumb|Figure 14: Electron diffraction from a thin silicon (111) sample with a 7x7 reconstructed surface|left|alt=An electron diffraction pattern from a silicon surface with a reconstructed surface]]
In addition to those which occur in the bulk, superstructures can also occur at surfaces. When half the material is (nominally) removed to create a surface, some of the atoms will be under coordinated. To reduce their energy they can rearrange. Sometimes these rearrangements are relatively small; sometimes they are quite large.<ref>{{Cite journal |last1=Andersen |first1=Tassie K. |last2=Fong |first2=Dillon D. |last3=Marks |first3=Laurence D. |date=2018 |title=Pauling's rules for oxide surfaces |journal=Surface Science Reports |language=en |volume=73 |issue=5 |pages=213–232 |doi=10.1016/j.surfrep.2018.08.001|bibcode=2018SurSR..73..213A |s2cid=53137808 |doi-access=free }}</ref><ref>{{Cite book |title=Surface science: an introduction; with 16 tables |date=2003 |publisher=Springer |isbn=978-3-540-00545-2 |editor-last=Oura |editor-first=Kenjiro |edition= |series=Advanced texts in physics |location=Berlin Heidelberg |editor-last2=Lifšic |editor-first2=Viktor G. |editor-last3=Saranin |editor-first3=A. A. |editor-last4=Zotov |editor-first4=A. V. |editor-last5=Katayama |editor-first5=Masao}}</ref> Similar to a bulk superstructure there will be additional, weaker diffraction spots. One example is for the silicon (111) surface, where there is a supercell which is seven times larger than the simple bulk cell in two directions.<ref name=":15">{{Cite journal |last1=Takayanagi |first1=K. |last2=Tanishiro |first2=Y. |last3=Takahashi |first3=M. |last4=Takahashi |first4=S. |date=1985 |title=Structural analysis of Si(111)-7×7 by UHV-transmission electron diffraction and microscopy |journal=Journal of Vacuum Science & Technology A: Vacuum, Surfaces, and Films |volume=3 |issue=3 |pages=1502–1506 |doi=10.1116/1.573160 |bibcode=1985JVSTA...3.1502T |issn=0734-2101}}</ref> This leads to diffraction patterns with additional spots some of which are marked in [[#Figure 14|Figure 14]].<ref>{{Cite journal |last1=Ciston |first1=J. |last2=Subramanian |first2=A. |last3=Robinson |first3=I. K. |last4=Marks |first4=L. D. |date=2009 |title=Diffraction refinement of localized antibonding at the Si(111) 7 × 7 surface |url=https://link.aps.org/doi/10.1103/PhysRevB.79.193302 |journal=Physical Review B |language=en |volume=79 |issue=19 |article-number=193302 |doi=10.1103/PhysRevB.79.193302 |arxiv=0901.3135 |bibcode=2009PhRvB..79s3302C |issn=1098-0121}}</ref> Here the (220) are stronger bulk diffraction spots, and the weaker ones due to the surface reconstruction are marked 7 × 7—see note{{efn|name=RecP}} for convention comments.

==== Aperiodic materials ====
{{anchor|Figure 15}}[[File:Al-Cu-Fe-Cr_decagonal_quasicrystal_diffraction_pattern.tif|thumb|Figure 15: Electron diffraction pattern of a decagonal quasicrystal|alt=An electron diffraction pattern from a quasicrystal showing features not seen in patterns from regular crystals.]]
In an [[aperiodic crystal]] the structure can no longer be simply described by three different vectors in real or reciprocal space. In general there is a substructure describable by three (e.g. <math>\mathbf a, \mathbf b, \mathbf c</math>), similar to supercells above, but in addition there is some additional periodicity (one to three) which cannot be described as a multiple of the three; it is a genuine additional periodicity which is an [[irrational number]] relative to the subcell lattice.<ref name=Janner77>{{Cite journal |last1=Janner |first1=A. |last2=Janssen |first2=T. |date=1977 |title=Symmetry of periodically distorted crystals |journal=Physical Review B |volume=15 |issue=2 |pages=643–658 |doi=10.1103/physrevb.15.643 |bibcode=1977PhRvB..15..643J |issn=0556-2805}}</ref><ref name="Bak" /><ref name=Jannsen2006>{{Citation |last1=Janssen |first1=T. |title=Incommensurate and commensurate modulated structures |date=2006 |url=https://xrpp.iucr.org/cgi-bin/itr?url_ver=Z39.88-2003&rft_dat=what%3Dchapter%26volid%3DCb%26chnumo%3D9o8%26chvers%3Dv0001 |work=International Tables for Crystallography |volume=C |pages=907–955 |editor-last=Prince |editor-first=E. |access-date=2023-03-24 |edition=1 |place=Chester, England |publisher=International Union of Crystallography |doi=10.1107/97809553602060000624 |isbn=978-1-4020-1900-5 |last2=Janner |first2=A. |last3=Looijenga-Vos |first3=A. |last4=de Wolff |first4=P. M.|url-access=subscription }}</ref> The diffraction pattern can then only be described by more than three indices.

An extreme example of this is for [[quasicrystals]],<ref>{{Cite journal |last1=Shechtman |first1=D. |last2=Blech |first2=I. |last3=Gratias |first3=D. |last4=Cahn |first4=J. W. |date=1984 |title=Metallic Phase with Long-Range Orientational Order and No Translational Symmetry |journal=Physical Review Letters |language=en |volume=53 |issue=20 |pages=1951–1953 |doi=10.1103/PhysRevLett.53.1951 |bibcode=1984PhRvL..53.1951S |issn=0031-9007|doi-access=free }}</ref> which can be described similarly by a higher number of Miller indices in reciprocal space—but not by any translational symmetry in real space. An example of this is shown in [[#Figure 15|Figure 15]] for an Al–Cu–Fe–Cr decagonal quasicrystal grown by magnetron sputtering on a sodium chloride substrate and then lifted off by dissolving the substrate with water.<ref>{{Cite journal |last1=Widjaja |first1=E.J. |last2=Marks |first2=L.D. |date=2003 |title=Microstructural evolution in Al–Cu–Fe quasicrystalline thin films |url=https://linkinghub.elsevier.com/retrieve/pii/S0040609003009039 |journal=Thin Solid Films |language=en |volume=441 |issue=1–2 |pages=63–71 |doi=10.1016/S0040-6090(03)00903-9|bibcode=2003TSF...441...63W |url-access=subscription }}</ref> In the pattern there are pentagons which are a characteristic of the aperiodic nature of these materials.

==== Diffuse scattering ====
{{anchor|Figure 16}}[[File:NbCoSb showing diffuse scattering.png|thumb|Figure 16: Single frame extracted from a video of a Nb<sub>0.83</sub>CoSb sample showing diffuse intensity (snake-like) due to vacancies at the Nb sites|alt=Diffraction pattern showing extra features (wavy lines here) due to disorder.]]
A further step beyond superstructures and aperiodic materials is what is called ''diffuse scattering'' in electron diffraction patterns due to disorder,<ref name="Cowley95" />{{Rp|location=Chpt 17}} which is also known for x-ray<ref>{{Cite journal |last=Welberry |first=T. R. |date=2014 |title=One Hundred Years of Diffuse X-ray Scattering |url=http://link.springer.com/10.1007/s11661-013-1889-2 |journal=Metallurgical and Materials Transactions A |language=en |volume=45 |issue=1 |pages=75–84 |doi=10.1007/s11661-013-1889-2 |bibcode=2014MMTA...45...75W |s2cid=137476417 |issn=1073-5623|url-access=subscription }}</ref> or neutron<ref>{{Cite book |last=Nield |first=Victoria M. |title=Diffuse neutron scattering from crystalline materials |date=2001 |publisher=Clarendon Press |others=David A. Keen |isbn=0-19-851790-4 |location=Oxford |oclc=45485010}}</ref> scattering. This can occur from inelastic processes, for instance, in bulk silicon the atomic vibrations ([[phonon]]s) are more prevalent along specific directions, which leads to streaks in diffraction patterns.<ref name="Cowley95" />{{Rp|location=Chpt 12}} Sometimes it is due to arrangements of [[point defect]]s. Completely disordered substitutional point defects lead to a general background which is called ''Laue monotonic scattering.''<ref name="Cowley95" />{{Rp|location=Chpt 12}} Often there is a [[probability distribution]] for the distances between point defects or what type of substitutional atom there is, which leads to distinct three-dimensional intensity features in diffraction patterns. An example of this is for a Nb<sub>0.83</sub>CoSb sample, with the diffraction pattern shown in [[#Figure 16|Figure 16]]. Because of the vacancies at the niobium sites, there is diffuse intensity with snake-like structure due to correlations of the distances between vacancies and also the relaxation of Co and Sb atoms around these vacancies.<ref>{{Cite journal |last1=Roth |first1=N. |last2=Beyer |first2=J. |last3=Fischer |first3=K. F. F. |last4=Xia |first4=K. |last5=Zhu |first5=T. |last6=Iversen |first6=B. B. |date=2021 |title=Tuneable local order in thermoelectric crystals |url=https://journals.iucr.org/m/issues/2021/04/00/fc5055/ |journal=IUCrJ |language=en |volume=8 |issue=4 |pages=695–702 |doi=10.1107/S2052252521005479 |issn=2052-2525 |pmc=8256708 |pmid=34258017|arxiv=2103.08543 |bibcode=2021IUCrJ...8..695R }}</ref>

==== Convergent beam electron diffraction ====
{{main|Convergent-beam electron diffraction}}
{{anchor|Figure 17}}[[File:CBED sketch.png|thumb|Figure 17: Schematic of CBED technique. Adapted from W. Kossel and G. Möllenstedt.<ref name=KM>{{Cite journal |last1=Kossel |first1=W. |last2=Möllenstedt |first2=G. |date=1939 |title=Elektroneninterferenzen im konvergenten Bündel |url=https://onlinelibrary.wiley.com/doi/10.1002/andp.19394280204 |journal=Annalen der Physik |language=de |volume=428 |issue=2 |pages=113–140 |doi=10.1002/andp.19394280204|bibcode=1939AnP...428..113K |url-access=subscription }}</ref>|alt=Experimental setup for convergent beam electron diffraction.]]
In convergent beam electron diffraction (CBED),<ref name=":4" /><ref name=":5" /><ref name=":6" /> the incident electrons are normally focused in a converging cone-shaped beam with a crossover located at the sample, e.g. [[#Figure 17|Figure 17]], although other methods exist. Unlike the parallel beam, the convergent beam is able to carry information from the sample volume, not just a two-dimensional projection available in SAED. With convergent beam there is also no need for the selected area aperture, as it is inherently site-selective since the beam crossover is positioned at the object plane where the sample is located.<ref name="Morniroli 2004"/>

{{anchor|Figure 18}}[[File:CBEDThickness.png|thumb|Figure 18: Variations in CBED due to dynamical diffraction, with thickness increasing from a)-d) for Si [110]|left|alt=Changes in CBED patterns for different thicknesses of the sample, showing that they get more complicated with thicker samples.]]

A CBED pattern consists of disks arranged similar to the spots in SAED. Intensity within the disks represents dynamical diffraction effects and symmetries of the sample structure, see [[#Figure 7|Figure 7]] and [[#Figure 18|18]]. Even though the zone axis and lattice parameter analysis based on disk positions does not significantly differ from SAED, the analysis of disks content is more complex and simulations based on dynamical diffraction theory is often required.<ref>{{Cite journal |last1=Chuvilin |first1=A. |last2=Kaiser |first2=U. |date=2005 |title=On the peculiarities of CBED pattern formation revealed by multislice simulation |url=https://linkinghub.elsevier.com/retrieve/pii/S0304399105000483 |journal=Ultramicroscopy |language=en |volume=104 |issue=1 |pages=73–82 |doi=10.1016/j.ultramic.2005.03.003|pmid=15935917 |url-access=subscription }}</ref> As illustrated in [[#Figure 18|Figure 18]], the details within the disk change with sample thickness, as does the inelastic background. With appropriate analysis CBED patterns can be used for indexation of the crystal point group, space group identification, measurement of lattice parameters, thickness or strain.<ref name="Morniroli 2004"/>

The disk diameter can be controlled using the microscope optics and apertures.<ref name=":8" /> The larger is the angle, the broader the disks are with more features. If the angle is increased to significantly, the disks begin to overlap.<ref name="KM" /> This is avoided in large angle convergent electron beam diffraction (LACBED) where the sample is moved upwards or downwards. There are applications, however, where the overlapping disks are beneficial, for instance with a [[ronchigram]]. It is a CBED pattern, often but not always of an amorphous material, with many intentionally overlapping disks providing information about the [[optical aberrations]] of the electron optical system.<ref>{{Cite journal |last1=Schnitzer |first1=Noah |last2=Sung |first2=Suk Hyun |last3=Hovden |first3=Robert |date=2019 |title=Introduction to the Ronchigram and its Calculation with Ronchigram.com |journal=Microscopy Today |volume=27 |issue=3 |pages=12–15 |doi=10.1017/s1551929519000427 |s2cid=155224415 |issn=1551-9295|doi-access=free }}</ref>

==== Precession electron diffraction ====
{{main|Precession electron diffraction}}
{{anchor|Figure 19}}[[File:Precession Electron Diffraction (White).gif|Figure 19: Geometry of electron beam in precession electron diffraction. Original diffraction patterns collected by C.S. Own at Northwestern University<ref name="thesis">Own, C. S.: PhD thesis, System Design and Verification of the Precession Electron Diffraction Technique, Northwestern University, 2005, http://www.numis.northwestern.edu/Research/Current/precession.shtml</ref>|thumb|300x300px|alt=An animation showing how rotating the incident beam direction can build up in a precession experiment.]]
Precession electron diffraction (PED), invented by Roger Vincent and [[Paul Midgley]] in 1994,<ref>{{Cite journal |last1=Vincent |first1=R. |last2=Midgley |first2=P.A. |date=1994 |title=Double conical beam-rocking system for measurement of integrated electron diffraction intensities |url=https://linkinghub.elsevier.com/retrieve/pii/0304399194900396 |journal=Ultramicroscopy |language=en |volume=53 |issue=3 |pages=271–282 |doi=10.1016/0304-3991(94)90039-6|url-access=subscription }}</ref> is a method to collect electron diffraction patterns in a [[transmission electron microscope]] (TEM). The technique involves rotating (precessing) a tilted incident electron beam around the central axis of the microscope, compensating for the tilt after the sample so a spot diffraction pattern is formed, similar to a SAED pattern. However, a PED pattern is an integration over a collection of diffraction conditions, see [[#Figure 19|Figure 19]]. This integration produces a quasi-kinematical [[diffraction pattern]] that is more suitable<ref>{{Cite journal |last1=Gjønnes |first1=J. |last2=Hansen |first2=V. |last3=Berg |first3=B. S. |last4=Runde |first4=P. |last5=Cheng |first5=Y. F. |last6=Gjønnes |first6=K. |last7=Dorset |first7=D. L. |last8=Gilmore |first8=C. J. |date=1998|title=Structure Model for the Phase AlmFe Derived from Three-Dimensional Electron Diffraction Intensity Data Collected by a Precession Technique. Comparison with Convergent-Beam Diffraction |url=https://scripts.iucr.org/cgi-bin/paper?S0108767397017030 |journal=Acta Crystallographica Section A |volume=54 |issue=3 |pages=306–319 |doi=10.1107/S0108767397017030|bibcode=1998AcCrA..54..306G |url-access=subscription }}</ref> as input into [[direct methods (crystallography)|direct methods]] algorithms using electrons<ref name="Sufficient">{{Cite journal |last1=Marks |first1=L.D. |last2=Sinkler |first2=W. |date=2003 |title=Sufficient Conditions for Direct Methods with Swift Electrons |url=https://www.cambridge.org/core/product/identifier/S1431927603030332/type/journal_article |journal=Microscopy and Microanalysis |language=en |volume=9 |issue=5 |pages=399–410 |doi=10.1017/S1431927603030332 |pmid=19771696 |bibcode=2003MiMic...9..399M |s2cid=20112743 |issn=1431-9276|url-access=subscription }}</ref><ref name="White">{{Cite journal |last1=White |first1=T.A. |last2=Eggeman |first2=A.S. |last3=Midgley |first3=P.A. |date=2010 |title=Is precession electron diffraction kinematical? Part I |url=https://linkinghub.elsevier.com/retrieve/pii/S030439910900240X |journal=Ultramicroscopy |language=en |volume=110 |issue=7 |pages=763–770 |doi=10.1016/j.ultramic.2009.10.013|pmid=19910121 |url-access=subscription }}</ref> to determine the [[crystal structure]] of the sample. Because it avoids many dynamical effects it can also be used to better identify crystallographic phases.<ref>{{Cite journal |last1=Moeck |first1=Peter |last2=Rouvimov |first2=Sergei |date=2010 |title=Precession electron diffraction and its advantages for structural fingerprinting in the transmission electron microscope |journal=Zeitschrift für Kristallographie |language=en |volume=225 |issue=2–3 |pages=110–124 |doi=10.1524/zkri.2010.1162 |bibcode=2010ZK....225..110M |s2cid=52059939 |issn=0044-2968|doi-access=free }}</ref>

==== 4D STEM ====
{{main|4D scanning transmission electron microscopy}}
4D scanning transmission electron microscopy (4D STEM)<ref name=":9">{{Cite journal |last=Ophus |first=Colin |date=2019 |title=Four-Dimensional Scanning Transmission Electron Microscopy (4D-STEM): From Scanning Nanodiffraction to Ptychography and Beyond |journal=Microscopy and Microanalysis |language=en |volume=25 |issue=3 |pages=563–582 |doi=10.1017/S1431927619000497 |pmid=31084643 |bibcode=2019MiMic..25..563O |s2cid=263414171 |issn=1431-9276|doi-access=free }}</ref> is a subset of [[scanning transmission electron microscopy]] (STEM) methods which uses a pixelated electron detector to capture a [[convergent beam electron diffraction]] (CBED) pattern at each scan location; see the main page for further information. This technique captures a 2 dimensional reciprocal space image associated with each scan point as the beam rasters across a 2 dimensional region in real space, hence the name 4D STEM. Its development was enabled by better STEM detectors and improvements in computational power. The technique has applications in diffraction contrast imaging, phase orientation and identification, strain mapping, and atomic resolution imaging among others; it has become very popular and rapidly evolving from about 2020 onwards.<ref name=":9" />

The name 4D STEM is common in literature, however it is known by other names: 4D STEM [[EELS]], ND STEM (N- since the number of dimensions could be higher than 4), position resolved diffraction (PRD), spatial resolved diffractometry, momentum-resolved STEM, "nanobeam precision electron diffraction", scanning electron nano diffraction, nanobeam electron diffraction, or pixelated STEM.<ref>{{cite web |title=4D STEM {{!}} Gatan, Inc. |url=https://www.gatan.com/techniques/4d-stem |access-date=2022-03-13 |website=www.gatan.com |language=en}}</ref> Most of these are the same, although there are instances such as momentum-resolved STEM<ref>{{Cite journal |last1=Hage |first1=Fredrik S. |last2=Nicholls |first2=Rebecca J. |last3=Yates |first3=Jonathan R. |last4=McCulloch |first4=Dougal G. |last5=Lovejoy |first5=Tracy C. |last6=Dellby |first6=Niklas |last7=Krivanek |first7=Ondrej L. |last8=Refson |first8=Keith |last9=Ramasse |first9=Quentin M. |date=2018 |title=Nanoscale momentum-resolved vibrational spectroscopy |journal=Science Advances |language=en |volume=4 |issue=6 |article-number=eaar7495 |doi=10.1126/sciadv.aar7495 |issn=2375-2548 |pmc=6018998 |pmid=29951584|bibcode=2018SciA....4.7495H }}</ref> where the emphasis can be very different.

=== Low-energy electron diffraction (LEED) ===
{{main|Low-energy electron diffraction}}{{anchor|Low-energy electron diffraction}}
{{anchor|Figure 20}}{{anchor|Figure 21}}{{Multiple image
| total_width       = 250
| align             = right
| direction         = vertical
| image1            = Ewald construction for electron diffraction on a two-dimensional lattice, side view.svg
| caption1          = Figure 20:  Ewald sphere construction for LEED, with the shape function streaks indicated, <math>k_i</math> the incident beam and <math>k_f</math> one of the diffracted beams.
| image2            = Si100Reconstructed.png
| caption2          = Figure 21: LEED pattern of a Si(100) reconstructed surface. The underlying lattice is a square lattice, while the surface reconstruction has a 2x1 periodicity. Also seen is the electron gun that generates the primary electron beam; it covers up parts of the screen.
| alt1              = Connection between the wavevectors for low energy electrons and reciprocal space.
| alt2              = Experimental LEED pattern from a reconstructed silicon surface.
}}

Low-energy electron diffraction (LEED) is a technique for the determination of the surface structure of [[single crystal|single-crystalline]] materials by bombardment with a [[collimated beam]] of low-energy electrons (30–200 eV).<ref name="Oura">{{cite book |author1=Oura |first=K. |url=https://archive.org/details/surfacesciencein00oura_931 |title=Surface Science |author2=Lifshifts |first2=V. G. |author3=Saranin |first3=A. A. |author4=Zotov |first4=A. V. |author5=Katayama |first5=M. |date=2003 |publisher=Springer-Verlag, Berlin Heidelberg New York |isbn=978-3-540-00545-2 |pages=[https://archive.org/details/surfacesciencein00oura_931/page/n10 1]–45 |url-access=limited}}</ref> In this case the Ewald sphere leads to approximately back-reflection, as illustrated in [[#Figure 20|Figure 20]], and diffracted electrons as spots on a fluorescent screen as shown in [[#Figure 21|Figure 21]]; see the main page for more information and references.<ref name="VanHove" /><ref name="LEEDB">{{Cite book |last1=Moritz |first1=Wolfgang |title=Surface structure determination by LEED and X-rays |last2=Van Hove |first2=Michel |date=2022 |isbn=978-1-108-28457-8 |location=Cambridge, United Kingdom |pages=Chpt 3–5 |oclc=1293917727}}</ref> It has been used to solve a very large number of relatively simple surface structures of metals and semiconductors, plus cases with simple chemisorbants. For more complex cases transmission electron diffraction<ref name=":15" /><ref>{{Cite journal |last1=Gilmore |first1=C.J. |last2=Marks |first2=L.D. |last3=Grozea |first3=D. |last4=Collazo |first4=C. |last5=Landree |first5=E. |last6=Twesten |first6=R.D. |date=1997 |title=Direct solutions of the Si(111) 7 × 7 structure |url=https://linkinghub.elsevier.com/retrieve/pii/S0039602897000629 |journal=Surface Science |language=en |volume=381 |issue=2–3 |pages=77–91 |doi=10.1016/S0039-6028(97)00062-9|bibcode=1997SurSc.381...77G |url-access=subscription }}</ref> or surface x-ray diffraction<ref>{{Cite journal |last=Robinson |first=I. K. |date=1983 |title=Direct Determination of the Au(110) Reconstructed Surface by X-Ray Diffraction |url=https://link.aps.org/doi/10.1103/PhysRevLett.50.1145 |journal=Physical Review Letters |language=en |volume=50 |issue=15 |pages=1145–1148 |doi=10.1103/PhysRevLett.50.1145 |bibcode=1983PhRvL..50.1145R |issn=0031-9007|url-access=subscription }}</ref> have been used, often combined with [[scanning tunnelling microscopy|scanning tunneling microscopy]] and [[density functional theory]] calculations.<ref>{{Cite journal |last1=Enterkin |first1=James A. |last2=Subramanian |first2=Arun K. |last3=Russell |first3=Bruce C. |last4=Castell |first4=Martin R. |last5=Poeppelmeier |first5=Kenneth R. |last6=Marks |first6=Laurence D. |date=2010 |title=A homologous series of structures on the surface of SrTiO3(110) |url=https://www.nature.com/articles/nmat2636 |journal=Nature Materials |language=en |volume=9 |issue=3 |pages=245–248 |doi=10.1038/nmat2636 |pmid=20154691 |bibcode=2010NatMa...9..245E |issn=1476-4660|url-access=subscription }}</ref>

LEED may be used in one of two ways:<ref name="VanHove" /><ref name="LEEDB" />
# Qualitatively, where the diffraction pattern is recorded and analysis of the spot positions gives information on the symmetry of the surface structure. In the presence of an [[adsorbate]] the qualitative analysis may reveal information about the size and rotational alignment of the adsorbate unit cell with respect to the substrate unit cell.<ref name="VanHove" />
# Quantitatively, where the intensities of diffracted beams are recorded as a function of incident electron beam energy to generate the so-called I–V curves. By comparison with theoretical curves, these may provide accurate information on atomic positions on the surface.<ref name="LEEDB" />

=== Reflection high-energy electron diffraction (RHEED) ===
{{main|RHEED}}{{anchor|Reflection high-energy electron diffraction}}
{{anchor|Figure 22}}{{anchor|Figure 23}}{{Multiple image
| total_width       = 250
| align             = left
| direction         = vertical
| image1            = Ewald sphere construction in Reflection high-energy electron diffraction (RHEED).svg
| caption1          = Figure 22: Ewald sphere in_RHEED, where higher-order Laue zones matter.
| image2            = Si111 7x7 ReconstructionB.png
| caption2          = Figure 23: RHEED pattern of a silicon (111) surface with a 7x7 reconstruction.
| alt1              = Connection between the electron wavevectors and reciprocal lattice vectors for reflection.
| alt2              = Experimental reflection electron diffraction pattern from a silicon surface
}}

Reflection high energy electron diffraction (RHEED),<ref name="Ichimiya">{{Cite book |last1=Ichimiya |first1=Ayahiko |title=Reflection high-energy electron diffraction |last2=Cohen |first2=Philip |date=2004 |publisher=Cambridge University Press |isbn=0-521-45373-9 |location=Cambridge, U.K. |pages=Chpt 4–19 |oclc=54529276}}</ref> is a [[analytical technique|technique]] used to characterize the surface of [[crystalline]] materials by reflecting electrons off a surface. As illustrated for the Ewald sphere construction in [[#Figure 22|Figure 22]], it uses mainly the higher-order Laue zones which have a reflection component. An experimental diffraction pattern is shown in [[#Figure 23|Figure 23]] and shows both rings from the higher-order Laue zones and streaky spots.<ref name="Peng" />{{Rp|location=Chpt 5}} RHEED systems gather information only from the surface layers of the sample, which distinguishes RHEED from other [[material characterization|materials characterization]] methods that also rely on diffraction of [[electrons]]. Transmission electron microscopy samples mainly the bulk of the sample, although in special cases it can provide surface information.<ref>{{Cite journal |last1=Kienzle |first1=Danielle M. |last2=Marks |first2=Laurence D. |date=2012 |title=Surface transmission electron diffraction for SrTiO3 surfaces |url=http://xlink.rsc.org/?DOI=c2ce25204j |journal=CrystEngComm |language=en |volume=14 |issue=23 |page=7833 |doi=10.1039/c2ce25204j |bibcode=2012CEG....14.7833K |issn=1466-8033|url-access=subscription }}</ref> [[Low-energy electron diffraction]] (LEED) is also surface sensitive, and achieves surface sensitivity through the use of low energy electrons. The main uses of RHEED to date have been during thin film growth,<ref name=":16">{{Cite book |last=Braun |first=Wolfgang |title=Applied RHEED: reflection high-energy electron diffraction during crystal growth |date=1999 |publisher=Springer |isbn=3-540-65199-3 |location=Berlin |pages=Chpts 2–4, 7 |oclc=40857022}}</ref> as the geometry is amenable to simultaneous collection of the diffraction data and deposition. It can, for instance, be used to monitor surface roughness during growth by looking at both the shapes of the streaks in the diffraction pattern as well as variations in the intensities.<ref name="Ichimiya" /><ref name=":16" />

=== Gas electron diffraction ===
{{main|Gas electron diffraction}}
{{anchor|Figure 24}}[[File:GED C6H6 diff pattern.jpg|thumb|Figure 24: Gas electron diffraction pattern of [[benzene]].|alt=Experimental gas electron diffraction pattern, showing diffuse rings.]]

[[Gas electron diffraction]] (GED) can be used to determine the [[molecular geometry|geometry]] of [[molecule]]s in gases.<ref name=":14">{{Cite journal |last=Oberhammer |first=H. |date=1989 |title=I. Hargittai, M. Hargittai (Eds.): The Electron Diffraction Technique, Part A von: Stereochemical Applications of Gas-Phase Electron Diffraction, VCH Verlagsgesellschaft, Weinheim, Basel. Cambridge, New York 1988. 206 Seiten, Preis: DM 210,-. |journal=Berichte der Bunsengesellschaft für physikalische Chemie |volume=93 |issue=10 |pages=1151–1152 |doi=10.1002/bbpc.19890931027 |issn=0005-9021}}</ref> A gas carrying the molecules is exposed to the electron beam, which is diffracted by the molecules. Since the molecules are randomly oriented, the resulting diffraction pattern consists of broad concentric rings, see [[#Figure 24|Figure 24]]. The diffraction intensity is a sum of several components such as background, atomic intensity or molecular intensity.<ref name=":14" />

In GED the diffraction intensities at a particular diffraction angle <math>\theta</math> is described via a scattering variable defined as<ref name=":10" /><math display="block"> |s| = \frac{4\pi}{\lambda} \sin \left(\frac\theta 2\right).</math>The total intensity is then given as a sum of partial contributions:<ref name="Seip">{{Cite journal |last1=Seip |first1=H.M. |last2=Strand |first2=T.G. |last3=Stølevik |first3=R. |date=1969 |title=Least-squares refinements and error analysis based on correlated electron diffraction intensities of gaseous molecules |url=https://linkinghub.elsevier.com/retrieve/pii/0009261469851250 |journal=Chemical Physics Letters |language=en |volume=3 |issue=8 |pages=617–623 |doi=10.1016/0009-2614(69)85125-0 |bibcode=1969CPL.....3..617S |url-access=subscription }}</ref><ref name="Andersen">{{Cite journal |last1=Andersen |first1=B. |last2=Seip |first2=H. M. |last3=Strand |first3=T. G. |last4=Stølevik |first4=R. |last5=Borch |first5=Gunner |last6=Craig |first6=J. Cymerman |date=1969 |title=Procedure and Computer Programs for the Structure Determination of Gaseous Molecules from Electron Diffraction Data. |journal=Acta Chemica Scandinavica |language=en |volume=23 |pages=3224–3234 |doi=10.3891/acta.chem.scand.23-3224 |issn=0904-213X|doi-access=free }}</ref><math display="block"> I_\text{tot}(s) = I_a(s) + I_m(s) + I_t(s) + I_b(s) ,</math>where <math>I_a(s)</math> results from scattering by individual atoms, <math>I_m(s)</math> by pairs of atoms and <math>I_t(s)</math> by atom triplets. Intensity <math>I_b(s)</math> corresponds to the background which, unlike the previous contributions, must be determined experimentally. The intensity of atomic scattering <math>I_a(s)</math> is defined as<ref name=":14" /><math display="block"> I_a(s) = \frac{K^2}{R^2} I_0 \sum_{i=1}^N |f_i(s)|^2 ,</math>where <math>K = (8 \pi ^2 me^2)/h^2</math>, <math>R</math> is the distance between the scattering object detector, <math>I_0</math> is the intensity of the primary electron beam and <math>f_i(s)</math> is the scattering amplitude of the atom <math>i</math> of the molecular structure in the experiment. <math>I_a(s)</math> is the main contribution and easily obtained for known gas composition. Note that the vector <math>s</math> used here is not the same as the excitation error used in other areas of diffraction, see [[#Geometrical considerations|earlier]].

The most valuable information is carried by the intensity of molecular scattering <math>I_a(s)</math>, as it contains information about the distance between all pairs of atoms in the molecule. It is given by<ref name=":10">{{Cite journal |last=Schåfer |first=Lothar |date=1976 |title=Electron Diffraction as a Tool of Structural Chemistry |url=http://journals.sagepub.com/doi/10.1366/000370276774456381 |journal=Applied Spectroscopy |language=en |volume=30 |issue=2 |pages=123–149 |doi=10.1366/000370276774456381 |bibcode=1976ApSpe..30..123S |s2cid=208256341 |issn=0003-7028|url-access=subscription }}</ref><math display="block"> I_m(s) = \frac{K^2}{R^2} I_0 \sum_{i=1}^N \sum_{\stackrel{j=1}{i\neq j}}^N \left| f_i(s) \right| \left| f_j(s)\right| \frac{\sin [s(r_{ij}-\kappa s^2)]}{sr_{ij}} e^{-(1/2 l_{ij} s^2)} \cos [\eta _i (s) - \eta _i (s)] ,</math>where <math>r_{ij}</math> is the distance between two atoms, <math>l_{ij}</math> is the mean square amplitude of vibration between the two atoms, similar to a [[Debye–Waller factor]], <math>\kappa</math> is the anharmonicity constant and <math>\eta</math> a phase factor which is important for atomic pairs with very different nuclear charges. The summation is performed over all atom pairs. Atomic triplet intensity <math>I_t(s)</math> is negligible in most cases. If the molecular intensity is extracted from an experimental pattern by subtracting other contributions, it can be used to match and refine a structural model against the experimental data.<ref name=":10"/><ref name="Seip" /><ref name="Andersen" />

Similar methods of analysis have also been applied to analyze electron diffraction data from liquids.<ref>{{Cite journal |last1=Lengyel |first1=SáNdor |last2=KáLmáN |first2=Erika |date=1974 |title=Electron diffraction on liquid water |url=https://www.nature.com/articles/248405a0 |journal=Nature |language=en |volume=248 |issue=5447 |pages=405–406 |doi=10.1038/248405a0 |bibcode=1974Natur.248..405L |s2cid=4201332 |issn=0028-0836|url-access=subscription }}</ref><ref>{{Cite journal |last1=Kálmán |first1=E. |last2=Pálinkás |first2=G. |last3=Kovács |first3=P. |date=1977 |title=Liquid water: I. Electron scattering |url=https://www.tandfonline.com/doi/full/10.1080/00268977700101871 |journal=Molecular Physics |language=en |volume=34 |issue=2 |pages=505–524 |doi=10.1080/00268977700101871 |issn=0026-8976|url-access=subscription }}</ref><ref>{{Cite journal |last1=de Kock |first1=M. B. |last2=Azim |first2=S. |last3=Kassier |first3=G. H. |last4=Miller |first4=R. J. D. |date=2020-11-21 |title=Determining the radial distribution function of water using electron scattering: A key to solution phase chemistry |journal=The Journal of Chemical Physics |language=en |volume=153 |issue=19 |article-number=194504 |doi=10.1063/5.0024127 |pmid=33218233 |bibcode=2020JChPh.153s4504D |s2cid=227100401 |issn=0021-9606|doi-access=free |hdl=21.11116/0000-0007-6FBC-A |hdl-access=free }}</ref>

=== In a scanning electron microscope ===
{{Main|Electron backscatter diffraction}}
{{anchor|Figure 25}}[[File:EBSD (001) Si.png|thumb|Figure 25: Kikuchi lines in an EBSD pattern of [[silicon]].|alt=Kikuchi pattern, a set of line-like features from a scanning electron microscope.]]

In a [[scanning electron microscope]] the region near the surface can be mapped using an electron beam that is scanned in a grid across the sample. A diffraction pattern can be recorded using [[electron backscatter diffraction]] (EBSD), as illustrated in [[#Figure 25|Figure 25]], captured with a camera inside the microscope.<ref>{{Cite journal |last1=Dingley |first1=D. J. |last2=Randle |first2=V. |date=1992 |title=Microtexture determination by electron back-scatter diffraction |url=http://link.springer.com/10.1007/BF01165988 |journal=Journal of Materials Science |language=en |volume=27 |issue=17 |pages=4545–4566 |doi=10.1007/BF01165988 |bibcode=1992JMatS..27.4545D |s2cid=137281137 |issn=0022-2461|url-access=subscription }}</ref> A depth from a few nanometers to a few microns, depending upon the electron energy used, is penetrated by the electrons, some of which are diffracted backwards and out of the sample. As result of combined inelastic and elastic scattering, typical features in an EBSD image are [[Kikuchi lines]]. Since the position of Kikuchi bands is highly sensitive to the crystal orientation, EBSD data can be used to determine the crystal orientation at particular locations of the sample. The data are processed by software yielding two-dimensional orientation maps.<ref>{{Cite journal |last1=Adams |first1=Brent L. |last2=Wright |first2=Stuart I. |last3=Kunze |first3=Karsten |date=1993 |title=Orientation imaging: The emergence of a new microscopy |url=http://link.springer.com/10.1007/BF02656503 |journal=Metallurgical Transactions A |language=en |volume=24 |issue=4 |pages=819–831 |doi=10.1007/BF02656503 |bibcode=1993MTA....24..819A |s2cid=137379846 |issn=0360-2133|url-access=subscription }}</ref><ref>{{Cite journal |last=Dingley |first=D. |date=2004 |title=Progressive steps in the development of electron backscatter diffraction and orientation imaging microscopy: EBSD AND OIM |url=https://onlinelibrary.wiley.com/doi/10.1111/j.0022-2720.2004.01321.x |journal=Journal of Microscopy |language=en |volume=213 |issue=3 |pages=214–224 |doi=10.1111/j.0022-2720.2004.01321.x|pmid=15009688 |s2cid=41385346 |url-access=subscription }}</ref> As the Kikuchi lines carry information about the interplanar angles and distances and, therefore, about the crystal structure, they can also be used for [[Phase (matter)|phase]] identification<ref name=":19" />{{Rp|location=Chpts 6–7}} or [[Electron backscatter diffraction#Strain measurement|strain analysis]].<ref name=":19">{{Cite book |last1=Schwartz |first1=Adam J |title=Electron backscatter diffraction in materials science |last2=Kumar |first2=Mukul |last3=Adams |first3=Brent L |last4=Field |first4=David P |publisher=Springer New York |year=2009 |isbn=978-1-4899-9334-2 |oclc=902763902}}</ref>{{Rp|location=Chpt 17}}

== Notes ==
{{notelist|refs=
{{efn|name=Diff|Sometimes electron diffraction is defined similar to light or water wave diffraction, that is interference or bending of (electron) waves around the corners of an obstacle or through an aperture. With this definition the electrons are behaving as waves in a general sense, corresponding to a type of Fresnel diffraction. However, in every case where electron diffraction is used in practice the obstacles of relevance are atoms, so the general definition is not used herein.}}
{{efn|name=Wlength|In their first, shorter paper in Nature Davisson and Germer stated that their results were consistent with the de Broglie wavelength. Similarly Thomson and Reid used the de Broglie wavelength to explain their results. However, in their subsequently, more detailed papers Davisson and Germer specifically stated that their work was consistent with ''undulatory mechanics'', and not consistent with the de Broglie wavelength. More importantly, the (non-relativistic) wavelength comes automatically from the Schrödinger equation, as do the equations for the amplitudes of electron diffraction; these cannot be derived from the de Broglie wavelength. As cited in the main text, Davisson and Germer were able to demonstrate that the diffraction angles were different from those of [[Bragg's Law]], needing a proper treatment which includes the average potential inside the material. Since all theoretical models start from the Schrödinger equation (with relativistic terms included) this is really the key to electron diffraction, not the ''de Broglie wavelength''. See [[matter waves]] for more discussion.}}
{{efn|name=Pi|Herein crystallographic conventions are used. Often in physics a plane wave is defined as <math>\exp(i \mathbf k \cdot \mathbf r)</math>. This changes some of the equations by a factor of <math>2 \pi</math>, for instance <math>\hbar</math> appears instead of <math>h</math>, but nothing significant.}}
{{efn|name=RecP|Notations differ depending upon whether the source is crystallography, physics or other. In addition to <math>\mathbf A, \mathbf B, \mathbf C</math> for the reciprocal lattice vectors as used herein, sometimes <math>\mathbf a^*, \mathbf b^*, \mathbf c^*</math> are used. Less common, but still sometimes used, are <math>\mathbf a_1 ,\mathbf a_2,\mathbf a_3</math> for real space, and <math>\mathbf b_1, \mathbf b_2,\mathbf b_3</math> for reciprocal space. Also, sometimes reciprocal lattice vectors are written with capitals as <math>G</math> not <math>g</math>, and the length can differ by a factor of <math>2 \pi</math> as mentioned above if <math>\exp(i\mathbf k \cdot \mathbf r)</math> is used for plane waves. (Different notations also exist for the wavevectors <math>\mathbf k</math>, <math>\mathbf \chi</math> or <math>\mathbf q</math>.) Similar notation differences can occur with aperiodic materials and superstructures. Furthermore, when dealing with surfaces as in [[#Low-energy electron diffraction|LEED]], normally two-dimensional real and reciprocal lattice vectors in the surface are used, defined in terms of a matrix multiplier of the simple surface unit cell when there are reconstructions.  To make things slightly more complicated, frequently four [[Miller indices]] are used for hexagonal systems even though only three are needed.}}
}}

== References ==
<references>
<ref name="Morniroli 2004">{{cite book
| last=Morniroli
| first=Jean Paul
| title=Large-Angle Convergent-Beam Electron Diffraction Applications to Crystal Defects
| publisher=Taylor & Francis
| year=2004
| isbn=978-2-901483-05-2
| doi=10.1201/9781420034073
| page=}}
</ref>
</references>

== Further reading ==
* {{cite book |last=Cowley |first=John M. |title=Diffraction physics |date=1995 |publisher=Elsevier |isbn=0-444-82218-6 |oclc=247191522}}. Contains extensive coverage of kinematical and other diffraction.
* {{cite book |last=Reimer |first=Ludwig |title=Transmission Electron Microscopy: Physics of Image Formation and Microanalysis. |date=2013 |publisher=Springer Berlin / Heidelberg |isbn=978-3-662-13553-2 |oclc=1066178493}} Large coverage of many different areas of electron microscopy with large numbers of references.
* {{cite book |last1=Hirsch |first1=P. B. | last2=Howie | first2=A. | last3=Nicholson| first3=R. B.| last4=Pashley | first4=D. W. | last5=Whelan | first5=M. J.|title=Electron microscopy of thin crystals |date=1965 |publisher=Butterworths |isbn=0-408-18550-3 |location=London |oclc=2365578}}, often called the bible of electron microscopy.
* {{cite book |last1=Spence |first1=J. C. H. |url=http://link.springer.com/10.1007/978-1-4899-2353-0 |title=Electron Microdiffraction |last2=Zuo |first2=J. M. |date=1992 |publisher=Springer US |isbn=978-1-4899-2355-4 |location=Boston, MA |language=en |doi=10.1007/978-1-4899-2353-0|s2cid=45473741 }}, a large coverage of topic related to dynamical diffraction and [[CBED]]
* {{cite book |last1=Peng |first1=L.-M. |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}}. Very extensive coverage of modern dynamical diffraction.
* {{cite book |title=Transmission electron microscopy: diffraction, imaging, and spectrometry |date=2016 |publisher=Springer |isbn=978-3-319-26649-7 |editor-last=Carter |editor-first=C. Barry |location=Cham, Switzerland |editor-last2=Williams |editor-first2=David B. |editor-last3=Thomas |editor-first3=John M.}}, a recent textbook with many images, stronger on experimental aspects.
* {{cite book |last=Edington | first=Jeffrey William |title=Practical electron microscopy in materials science |date=1977 |publisher=Techbooks |oclc=27997701}}, an older source for experimental details, albeit hard to find.

{{Crystallography|state=collapsed}}
{{Electron microscopy|state=collapsed}}
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[[Category:Applied and interdisciplinary physics]]
[[Category:Crystallography]]
[[Category:Diffraction]]
[[Category:Electron]]
[[Category:Electron microscopy]]
[[Category:Materials science]]
[[Category:Quantum mechanics]]
[[Category:Scattering]]