Editing Electron diffraction (section)

==== 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.