Editing Quasicrystal (section)

==History==
[[File:Darb-i Imam shrine spandrel.JPG|thumb|Girih-tile subdivision found in the decagonal girih pattern on a spandrel from the Darb-i Imam shrine, Isfahan, Iran (1453 C.E.). A subdivision rule to construct perfect quasi-crystalline tilings has been identified<ref>{{Cite journal |last1=Lu |first1=Peter J. |last2=Steinhardt |first2=Paul J. |date=2007-02-23 |title=Decagonal and Quasi-Crystalline Tilings in Medieval Islamic Architecture |url=https://www.science.org/doi/10.1126/science.1135491 |journal=Science |language=en |volume=315 |issue=5815 |pages=1106–1110 |doi=10.1126/science.1135491 |pmid=17322056 |bibcode=2007Sci...315.1106L |s2cid=10374218 |issn=0036-8075 |access-date=2023-05-30 |archive-date=2024-09-18 |archive-url=https://web.archive.org/web/20240918002534/https://www.science.org/doi/10.1126/science.1135491 |url-status=live }}</ref>]]
The first representations of perfect quasicrystalline patterns can be found in several early [[Islamic art|Islamic works of art and architecture]] such as the Gunbad-i-Kabud tomb tower, the [[Darb-e Imam]] shrine and the [[Al-Attarine Madrasa]].<ref>{{Cite book |last=Al Ajlouni |first=Rima |title=Aperiodic Crystals |chapter=Octagon-Based Quasicrystalline Formations in Islamic Architecture |date=2013 |editor-last=Schmid |editor-first=Siegbert |editor2-last=Withers |editor2-first=Ray L. |editor3-last=Lifshitz |editor3-first=Ron |chapter-url=https://link.springer.com/chapter/10.1007/978-94-007-6431-6_7 |language=en |location=Dordrecht |publisher=Springer Netherlands |pages=49–57 |doi=10.1007/978-94-007-6431-6_7 |isbn=978-94-007-6431-6 |access-date=2023-05-30 |archive-date=2023-05-30 |archive-url=https://web.archive.org/web/20230530081621/https://link.springer.com/chapter/10.1007/978-94-007-6431-6_7 |url-status=live }}</ref><ref>{{Cite web |title=Islamic Quasicrystal Tilings {{!}} Paul J. Steinhardt |url=https://paulsteinhardt.org/islamic-tilings/ |access-date=2023-05-29 |website=paulsteinhardt.org |language=en-US |archive-date=2023-05-29 |archive-url=https://web.archive.org/web/20230529082126/https://paulsteinhardt.org/islamic-tilings/ |url-status=live }}</ref> On July 16, 1945, in Alamogordo, New Mexico, the [[Trinity (nuclear test)|Trinity]] nuclear bomb test produced icosahedral quasicrystals. They went unnoticed at the time of the test but were later identified in samples of red [[trinitite]], a glass-like substance formed from fused sand and copper transmission lines. Identified in 2021, they are the oldest known anthropogenic quasicrystals.<ref name=":3">{{cite journal |last1=Bindi |first1=Luca |title=Accidental synthesis of a previously unknown quasicrystal in the first atomic bomb test |journal=Proceedings of the National Academy of Sciences |date=2021-06-01 |volume=118 |issue=22 |pages=e2101350118 |doi=10.1073/pnas.2101350118 |pmid=34001665 |pmc=8179242 |bibcode=2021PNAS..11801350B |doi-access=free }}</ref><ref name="redtrinitite">{{cite news |last=Mullane |first=Laura |date=May 18, 2021 |title=Newly discovered quasicrystal was created by the first nuclear explosion at Trinity Site |url=https://phys.org/news/2021-05-newly-quasicrystal-nuclear-explosion-trinity.html |work=Phys.org |location= |access-date=May 21, 2021 |archive-date=June 21, 2021 |archive-url=https://web.archive.org/web/20210621144831/https://phys.org/news/2021-05-newly-quasicrystal-nuclear-explosion-trinity.html |url-status=live }}<!--Note that the author is from the same university as the lead researcher and co-author of the paper, so this is effectively a primary source.--></ref>

[[File:Penrose Tiling (Rhombi).svg|thumb|A [[Penrose tiling]]]]
In 1961, [[Hao Wang (academic)|Hao Wang]] asked whether determining if a set of tiles admits a tiling of the plane is an [[algorithmically unsolvable problem]] or not. He conjectured that it is solvable, relying on the hypothesis that every set of tiles that can tile the plane can do it ''periodically'' (hence, it would suffice to try to tile bigger and bigger patterns until obtaining one that tiles periodically). Nevertheless, two years later, his student [[Robert Berger (mathematician)|Robert Berger]] constructed a set of some 20,000 square tiles (now called [[Wang tiles]]) that can tile the plane but not in a periodic fashion. As further aperiodic sets of tiles were discovered, sets with fewer and fewer shapes were found. In 1974 [[Roger Penrose]] discovered a set of just two tiles, now referred to as [[Penrose tiling|Penrose tiles]], that produced only non-periodic tilings of the plane. These tilings displayed instances of fivefold symmetry. One year later [[Alan Lindsay Mackay|Alan Mackay]] showed theoretically that the diffraction pattern from the Penrose tiling had a two-dimensional [[Fourier transform]] consisting of sharp '[[Dirac delta function|delta]]' peaks arranged in a fivefold symmetric pattern.<ref name="r2" /> Around the same time, [[Robert Ammann]] created a set of aperiodic tiles that produced eightfold symmetry.

In 1972, R. M. de Wolf and W. van Aalst<ref name="r6" /> reported that the diffraction pattern produced by a crystal of [[sodium carbonate]] cannot be labeled with three indices but needed one more, which implied that the underlying structure had four dimensions in [[Reciprocal lattice|reciprocal space]]. Other puzzling cases have been reported,<ref name="Kleinert" /> but until the concept of quasicrystal came to be established, they were explained away or denied.<ref name="r7" /><ref name="r8" />

Dan Shechtman first observed ten-fold [[electron diffraction]] patterns in 1982, while conducting a routine study of an [[aluminium]]–[[manganese]] alloy, Al<sub>6</sub>Mn, at the US [[National Bureau of Standards]] (later NIST).<ref name=":4">{{Cite web | url=http://www.quasi.iastate.edu/discovery.html | archive-url=https://web.archive.org/web/20111007225727/http://www.quasi.iastate.edu/discovery.html | url-status=dead | archive-date=2011-10-07 | title=QC Hot News}}</ref> Shechtman related his observation to Ilan Blech, who responded that such diffractions had been seen before.<ref>{{Cite journal |last1=Ino |first1=Shozo |last2=Ogawa |first2=Shiro |date=1967 |title=Multiply Twinned Particles at Earlier Stages of Gold Film Formation on Alkalihalide Crystals |url=https://journals.jps.jp/doi/10.1143/JPSJ.22.1365 |journal=Journal of the Physical Society of Japan |language=en |volume=22 |issue=6 |pages=1365–1374 |doi=10.1143/JPSJ.22.1365 |bibcode=1967JPSJ...22.1365I |issn=0031-9015 |access-date=2023-03-24 |archive-date=2023-03-24 |archive-url=https://web.archive.org/web/20230324203315/https://journals.jps.jp/doi/10.1143/JPSJ.22.1365 |url-status=live }}</ref><ref>{{Cite journal |last1=Allpress |first1=J.G. |last2=Sanders |first2=J.V. |date=1967 |title=The structure and orientation of crystals in deposits of metals on mica |url=https://linkinghub.elsevier.com/retrieve/pii/0039602867900623 |journal=Surface Science |language=en |volume=7 |issue=1 |pages=1–25 |doi=10.1016/0039-6028(67)90062-3 |bibcode=1967SurSc...7....1A |access-date=2023-03-24 |archive-date=2023-12-03 |archive-url=https://web.archive.org/web/20231203003246/https://linkinghub.elsevier.com/retrieve/pii/0039602867900623 |url-status=live }}</ref><ref>{{Cite journal |last=Gillet |first=M |date=1977 |title=Structure of small metallic particles |url=https://linkinghub.elsevier.com/retrieve/pii/0039602877903752 |journal=Surface Science |language=en |volume=67 |issue=1 |pages=139–157 |doi=10.1016/0039-6028(77)90375-2 |bibcode=1977SurSc..67..139G |access-date=2023-03-24 |archive-date=2024-09-18 |archive-url=https://web.archive.org/web/20240918002514/https://www.sciencedirect.com/unsupported_browser |url-status=live }}</ref> Around that time, Shechtman also related his finding to [[John W. Cahn]] of the NIST, who did not offer any explanation and challenged him to solve the observation. Shechtman quoted Cahn as saying: "Danny, this material is telling us something, and I challenge you to find out what it is".<ref name= cahn>{{cite journal| url = https://www.nist.gov/nist-and-nobel/dan-shechtman/nobel-moment-dan-shechtman| title = NIST and the Nobel (September 30, 2016, Updated November 17, 2019) The Nobel Moment: Dan Shechtman| journal = NIST| date = 30 September 2016}}</ref>

The observation of the ten-fold diffraction pattern lay unexplained for two years until the spring of 1984, when Blech asked Shechtman to show him his results again. A quick study of Shechtman's results showed that the common explanation for a ten-fold symmetrical diffraction pattern, a type of [[crystal twinning]], was ruled out by his experiments. Therefore, Blech looked for a new structure containing cells connected to each other by defined angles and distances but without translational periodicity. He decided to use a computer simulation to calculate the diffraction intensity from a cluster of such a material, which he termed as "multiple [[polyhedral symmetry|polyhedral]]", and found a ten-fold structure similar to what was observed. The multiple polyhedral structure was termed later by many researchers as icosahedral glass.<ref>{{cite book | chapter-url=https://www.sciencedirect.com/science/article/pii/B9780120406036500076 | doi=10.1016/B978-0-12-040603-6.50007-6 | chapter=The Icosahedral Glass Model | title=Extended Icosahedral Structures | series=Aperiodicity and Order | year=1989 | last1=Stephens | first1=Peter W. | volume=3 | pages=37–104 | isbn=9780120406036 | access-date=2021-05-07 | archive-date=2021-05-07 | archive-url=https://web.archive.org/web/20210507114808/https://www.sciencedirect.com/science/article/pii/B9780120406036500076 | url-status=live }}</ref>

Shechtman accepted Blech's discovery of a new type of material and chose to publish his observation in a paper entitled "The Microstructure of Rapidly Solidified Al<sub>6</sub>Mn", which was written around June 1984 and published in a 1985 edition of ''[[Metallurgical Transactions A]]''.<ref>{{cite journal|last=Shechtman|first=Dan|author2=I. A. Blech |title=The Microstructure of Rapidly Solidified Al<sub>6</sub>Mn|journal=[[Metall Mater Trans A]]|year=1985|volume=16A|pages=1005–1012|doi=10.1007/BF02811670|issue=6|bibcode = 1985MTA....16.1005S |s2cid=136733193}}</ref> Meanwhile, on seeing the draft of the paper, John Cahn suggested that Shechtman's experimental results merit a fast publication in a more appropriate scientific journal. Shechtman agreed and, in hindsight, called this fast publication "a winning move". This paper, published in the ''[[Physical Review Letters]]'',<ref name="s" /> repeated Shechtman's observation and used the same illustrations as the original paper.

Originally, the new form of matter was dubbed "Shechtmanite".<ref>{{cite news|title=Impossible' Form of Matter Takes Spotlight In Study of Solids|newspaper=New York Times|url=https://www.nytimes.com/1989/09/05/science/impossible-form-of-matter-takes-spotlight-in-study-of-solids.html?pagewanted=all&src=pm|first=Malcolm W.|last=Browne|date=1989-09-05|access-date=2017-02-12|archive-date=2017-12-01|archive-url=https://web.archive.org/web/20171201135113/http://www.nytimes.com/1989/09/05/science/impossible-form-of-matter-takes-spotlight-in-study-of-solids.html?pagewanted=all&src=pm|url-status=live}}</ref> The term "quasicrystal" was first used in print by [[Paul Steinhardt]] and [[Dov Levine]]<ref name="r9" /> shortly after Shechtman's paper was published.

Also in 1985, T. Ishimasa ''et al.'' reported twelvefold symmetry in Ni-Cr particles.<ref name="r3" /> Soon, eightfold diffraction patterns were recorded in V-Ni-Si and Cr-Ni-Si alloys.<ref name="r4" /> Over the years, hundreds of quasicrystals with various compositions and different symmetries have been discovered. The first quasicrystalline materials were thermodynamically unstable: when heated, they formed regular crystals. However, in 1987, the first of many stable quasicrystals were discovered, making it possible to produce large samples for study and applications.<ref>{{Cite journal|last=Day|first=Charles|date=2001-02-01|title=Binary Quasicrystals Discovered That Are Stable and Icosahedral|url=https://physicstoday.scitation.org/doi/10.1063/1.1359699|journal=Physics Today|volume=54|issue=2|pages=17–18|doi=10.1063/1.1359699|bibcode=2001PhT....54b..17D|issn=0031-9228|access-date=2021-01-10|archive-date=2021-11-03|archive-url=https://web.archive.org/web/20211103024025/https://physicstoday.scitation.org/doi/10.1063/1.1359699|url-status=live}}</ref>

In 1992, the [[International Union of Crystallography]] altered its definition of a crystal, reducing it to the ability to produce a clear-cut diffraction pattern and acknowledging the possibility of the ordering to be either periodic or aperiodic.<ref name="bloomberg" /><ref>{{Cite web |title=Quasicrystal – Online Dictionary of Crystallography |url=https://dictionary.iucr.org/Quasicrystal |access-date=2024-04-04 |website=dictionary.iucr.org |archive-date=2024-04-04 |archive-url=https://web.archive.org/web/20240404024622/https://dictionary.iucr.org/Quasicrystal |url-status=live }}</ref>

[[File:Al71Ni24Fe5 TEM.jpg|thumb|Atomic image of a micron-sized grain of the natural Al<sub>71</sub>Ni<sub>24</sub>Fe<sub>5</sub> quasicrystal (shown in the inset) from a [[Khatyrka meteorite]] fragment. The corresponding [[:File:Al71Ni24Fe5 diffraction.jpg|diffraction patterns]] reveal a ten-fold symmetry.<ref name="natq" />]]
[[File:Zn-Mg-HoDiffraction.JPG|thumb|[[Electron crystallography|Electron diffraction pattern]] of an icosahedral [[holmium–magnesium–zinc quasicrystal|Ho–Mg–Zn quasicrystal]]]]

In 2001, Steinhardt hypothesized that quasicrystals could exist in nature and developed a method of recognition, inviting all the mineralogical collections of the world to identify any badly cataloged crystals. In 2007 Steinhardt received a reply by [[Luca Bindi]], who found a quasicrystalline specimen from [[Khatyrka (river)|Khatyrka]] in the [[University of Florence]] Mineralogical Collection. The crystal samples were sent to Princeton University for other tests, and in late 2009, Steinhardt confirmed its quasicrystalline character. This quasicrystal, with a composition of Al<sub>63</sub>Cu<sub>24</sub>Fe<sub>13</sub>, was named [[icosahedrite]] and it was approved by the [[International Mineralogical Association]] in 2010. Analysis indicates it may be meteoritic in origin, possibly delivered from a carbonaceous chondrite asteroid. In 2011, Bindi, Steinhardt, and a team of specialists found more icosahedrite samples from Khatyrka.<ref>{{Cite journal | doi = 10.1073/pnas.1111115109 | pmid = 22215583 | last = Bindi | first = Luca |author2=John M. Eiler |author3=Yunbin Guan |author4=Lincoln S. Hollister |author5=Glenn MacPherson |author6=Paul J. Steinhardt |author7=Nan Yao | title = Evidence for the extraterrestrial origin of a natural quasicrystal | journal = Proceedings of the National Academy of Sciences | date = 2012-01-03 |bibcode = 2012PNAS..109.1396B | volume = 109 | issue = 5 | pages = 1396–1401 |pmc=3277151| doi-access = free }}</ref> A further study of Khatyrka meteorites revealed micron-sized grains of another natural quasicrystal, which has a ten-fold symmetry and a chemical formula of Al<sub>71</sub>Ni<sub>24</sub>Fe<sub>5</sub>. This quasicrystal is stable in a narrow temperature range, from 1120 to 1200 K at ambient pressure, which suggests that natural quasicrystals are formed by rapid quenching of a meteorite heated during an impact-induced shock.<ref name="natq" />

Shechtman was awarded the [[Nobel Prize in Chemistry]] in 2011 for his work on quasicrystals. "His discovery of quasicrystals revealed a new principle for packing of atoms and molecules," stated the Nobel Committee and pointed that "this led to a paradigm shift within chemistry."<ref name="bloomberg" /><ref>{{cite news|url=https://www.bbc.co.uk/news/science-environment-15181187|title=Nobel win for crystal discovery|newspaper=[[BBC News]]|access-date=2011-10-05|date=2011-10-05|archive-date=2011-10-05|archive-url=https://web.archive.org/web/20111005104716/http://www.bbc.co.uk/news/science-environment-15181187|url-status=live}}</ref> In 2014, Post of Israel issued a stamp dedicated to quasicrystals and the 2011 Nobel Prize.<ref>[https://www.iycr2014.org/countries/events-by-country/israel/iycr-philately-day Crystallography matters ... more!] {{Webarchive|url=https://web.archive.org/web/20181221134716/https://www.iycr2014.org/countries/events-by-country/israel/iycr-philately-day |date=2018-12-21 }} iycr2014.org</ref>

While the first quasicrystals discovered were made out of [[intermetallic]] components, later on quasicrystals were also discovered in [[Soft matter|soft-matter]] and [[Molecule|molecular]] systems. Soft quasicrystal structures have been found in supramolecular dendrimer liquids<ref>{{Cite journal |last1=Zeng |first1=Xiangbing |last2=Ungar |first2=Goran |last3=Liu |first3=Yongsong |last4=Percec |first4=Virgil |last5=Dulcey |first5=Andrés E. |last6=Hobbs |first6=Jamie K. |date=March 2004 |title=Supramolecular dendritic liquid quasicrystals |url=https://www.nature.com/articles/nature02368 |journal=Nature |language=en |volume=428 |issue=6979 |pages=157–160 |doi=10.1038/nature02368 |pmid=15014524 |s2cid=4429689 |issn=1476-4687 |access-date=2022-11-06 |archive-date=2022-11-06 |archive-url=https://web.archive.org/web/20221106132759/https://www.nature.com/articles/nature02368 |url-status=live }}</ref> and ABC Star Polymers<ref>{{Cite journal |last1=Hayashida |first1=Kenichi |last2=Dotera |first2=Tomonari |last3=Takano |first3=Atsushi |last4=Matsushita |first4=Yushu |date=2007-05-08 |title=Polymeric Quasicrystal: Mesoscopic Quasicrystalline Tiling in $ABC$ Star Polymers |url=https://link.aps.org/doi/10.1103/PhysRevLett.98.195502 |journal=Physical Review Letters |volume=98 |issue=19 |pages=195502 |doi=10.1103/PhysRevLett.98.195502 |pmid=17677627 |bibcode=2007PhRvL..98s5502H |access-date=2022-11-06 |archive-date=2024-09-18 |archive-url=https://web.archive.org/web/20240918002513/https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.98.195502 |url-status=live }}</ref> in 2004 and 2007. In 2009, it was found that thin-film quasicrystals can be formed by [[self-assembly]] of uniformly shaped, nano-sized molecular units at an air-liquid interface.<ref>{{cite journal|doi=10.1038/nature08439|pmid=19829378|title=Quasicrystalline order in self-assembled binary nanoparticle superlattices|journal=Nature|volume=461|issue=7266|pages=964–967|year=2009|last1=Talapin|first1=Dmitri V.|last2=Shevchenko|first2=Elena V.|last3=Bodnarchuk|first3=Maryna I.|last4=Ye|first4=Xingchen|last5=Chen|first5=Jun|last6=Murray|first6=Christopher B.|bibcode=2009Natur.461..964T|s2cid=4344953}}</ref> It was demonstrated that these units can be both inorganic and organic.<ref>{{cite journal|doi=10.1126/science.aav0790 |pmid=30573624 |title=Single-component quasicrystalline nanocrystal superlattices through flexible polygon tiling rule |journal=Science |volume=362 |issue=6421 |pages=1396–1400 |year=2018 |last1=Nagaoka |first1=Yasutaka |last2=Zhu |first2=Hua |last3=Eggert |first3=Dennis |last4=Chen |first4=Ou |bibcode=2018Sci...362.1396N |doi-access=free|hdl=21.11116/0000-0002-B8DF-4 |hdl-access=free }}</ref> Additionally in the 2010s, two-dimensional molecular quasicrystals were discovered, driven by [[Intermolecular force|intermolecular interactions]]<ref>{{Cite journal |last1=Wasio |first1=Natalie A. |last2=Quardokus |first2=Rebecca C. |last3=Forrest |first3=Ryan P. |last4=Lent |first4=Craig S. |last5=Corcelli |first5=Steven A. |last6=Christie |first6=John A. |last7=Henderson |first7=Kenneth W. |last8=Kandel |first8=S. Alex |date=March 2014 |title=Self-assembly of hydrogen-bonded two-dimensional quasicrystals |url=https://www.nature.com/articles/nature12993 |journal=Nature |language=en |volume=507 |issue=7490 |pages=86–89 |doi=10.1038/nature12993 |pmid=24598637 |bibcode=2014Natur.507...86W |s2cid=4401013 |issn=1476-4687 |access-date=2022-11-06 |archive-date=2024-09-18 |archive-url=https://web.archive.org/web/20240918002622/https://www.nature.com/articles/nature12993 |url-status=live }}</ref> and interface-interactions.<ref>{{Cite journal |last1=Paßens |first1=M. |last2=Caciuc |first2=V. |last3=Atodiresei |first3=N. |last4=Feuerbacher |first4=M. |last5=Moors |first5=M. |last6=Dunin-Borkowski |first6=R. E. |last7=Blügel |first7=S. |last8=Waser |first8=R. |last9=Karthäuser |first9=S. |date=2017-05-22 |title=Interface-driven formation of a two-dimensional dodecagonal fullerene quasicrystal |journal=Nature Communications |language=en |volume=8 |issue=1 |article-number=15367 |doi=10.1038/ncomms15367 |pmid=28530242 |pmc=5458153 |bibcode=2017NatCo...815367P |s2cid=22736155 |issn=2041-1723}}</ref>

In 2018, chemists from Brown University announced the successful creation of a self-constructing lattice structure based on a strangely shaped quantum dot. While single-component quasicrystal lattices have been previously predicted mathematically and in computer simulations,<ref>{{Cite journal|last1=Engel|first1=Michael|last2=Damasceno|first2=Pablo F.|last3=Phillips|first3=Carolyn L.|last4=Glotzer|first4=Sharon C.|date=2014-12-08|title=Computational self-assembly of a one-component icosahedral quasicrystal|journal=Nature Materials|language=en|volume=14|issue=1|pages=109–116|doi=10.1038/nmat4152|issn=1476-4660|pmid=25485986}}</ref> they had not been demonstrated prior to this.<ref>{{Cite journal|last1=Chen|first1=Ou|last2=Eggert|first2=Dennis|last3=Zhu|first3=Hua|last4=Nagaoka|first4=Yasutaka|date=2018-12-21|title=Single-component quasicrystalline nanocrystal superlattices through flexible polygon tiling rule|journal=Science|language=en|volume=362|issue=6421|pages=1396–1400|doi=10.1126/science.aav0790|issn=0036-8075|pmid=30573624|bibcode=2018Sci...362.1396N|doi-access=free|hdl=21.11116/0000-0002-B8DF-4|hdl-access=free}}</ref>