Editing Electron diffraction (section)

== 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. |url=http://worldcat.org/oclc/1001251352 |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=J. |first=Heinrich, K. F. |url=http://worldcat.org/oclc/801808484 |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=F. |first=Egerton, R. |url=http://worldcat.org/oclc/706920411 |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 |url=https://dx.doi.org/10.1016/0304-3991%2892%2990213-4 |journal=Ultramicroscopy |language=en |volume=41 |issue=4 |pages=335–348 |doi=10.1016/0304-3991(92)90213-4 |issn=0304-3991|url-access=subscription }}</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. |url=https://www.jstor.org/stable/j.ctvqc6g7s |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 lneses 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}}</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 |url=http://dx.doi.org/10.1143/jpsj.9.184 |journal=Journal of the Physical Society of Japan |volume=9 |issue=2 |pages=184–198 |doi=10.1143/jpsj.9.184 |issn=0031-9015|url-access=subscription }}</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 |url=http://dx.doi.org/10.1016/j.ultramic.2006.10.007 |journal=Ultramicroscopy |volume=107 |issue=6–7 |pages=507–513 |doi=10.1016/j.ultramic.2006.10.007 |pmid=17234347 |issn=0304-3991|url-access=subscription }}</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 |url=http://dx.doi.org/10.1016/j.ultramic.2009.01.011 |journal=Ultramicroscopy |volume=109 |issue=6 |pages=758–765 |doi=10.1016/j.ultramic.2009.01.011 |pmid=19269095 |issn=0304-3991|url-access=subscription }}</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 |url=https://dx.doi.org/10.1016/0022-3093%2878%2990098-4 |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|url-access=subscription }}</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 |url=http://dx.doi.org/10.1088/0034-4885/45/6/001 |journal=Reports on Progress in Physics |volume=45 |issue=6 |pages=587–629 |doi=10.1088/0034-4885/45/6/001 |issn=0034-4885|url-access=subscription }}</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 |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}}</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 |url=http://dx.doi.org/10.1116/1.573160 |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|url-access=subscription }}</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 |pages=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 |url=http://dx.doi.org/10.1103/physrevb.15.643 |journal=Physical Review B |volume=15 |issue=2 |pages=643–658 |doi=10.1103/physrevb.15.643 |bibcode=1977PhRvB..15..643J |issn=0556-2805|url-access=subscription }}</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. |url=https://www.worldcat.org/oclc/45485010 |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 |pages=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=K. Oura |author2=V. G. Lifshifts |author3=A. A. Saranin |author4=A. V. Zotov |author5=M. Katayama |title=Surface Science |url=https://archive.org/details/surfacesciencein00oura_931 |url-access=limited |publisher=Springer-Verlag, Berlin Heidelberg New York |date=2003 |pages=[https://archive.org/details/surfacesciencein00oura_931/page/n10 1]–45|isbn=9783540005452 }}</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 |url=https://www.worldcat.org/oclc/1293917727 |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}}</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 |url=https://www.worldcat.org/oclc/54529276 |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 |pages=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 |url=https://www.worldcat.org/oclc/40857022 |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,-. |url=http://dx.doi.org/10.1002/bbpc.19890931027 |journal=Berichte der Bunsengesellschaft für physikalische Chemie |volume=93 |issue=10 |pages=1151–1152 |doi=10.1002/bbpc.19890931027 |issn=0005-9021|url-access=subscription }}</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 |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 |url=http://worldcat.org/oclc/902763902 |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}}