Editing Electron diffraction (section)

=== 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>