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The term '''R-matrix''' has several meanings, depending on the field of study. Its original use has been to mathematically describe [[Nuclear reaction|nuclear reactions]].<ref>{{Cite journal |last=Lane |first=A. M. |last2=Thomas |first2=R. G. |date=1958-04-01 |title=R-Matrix Theory of Nuclear Reactions |url=https://link.aps.org/doi/10.1103/RevModPhys.30.257 |journal=Reviews of Modern Physics |language=en |volume=30 |issue=2 |pages=257–353 |doi=10.1103/RevModPhys.30.257 |issn=0034-6861}}</ref> In particular the general problem of nuclear reactions is to relate the values of the [[scattering]] or collision matrix elements (which in principle can be obtained from measurements) to the (slow) dynamics of [[nuclear structure]]. The R-matrix formalism describes the effects of the interaction of the nucleus with the outside world. Its interior is not specified, i.e. it is considered a "black box". The original formulations of the theory came from nuclear scientists [[Eugene Wigner|Wigner]]<ref>{{Cite journal |last=Wigner |first=Eugene P. |date=1946-11-01 |title=Resonance Reactions |url=https://journals.aps.org/pr/abstract/10.1103/PhysRev.70.606 |journal=Physical Review |volume=70 |issue=9-10 |pages=606–618 |doi=10.1103/PhysRev.70.606}}</ref><ref>{{Cite journal |last=Wigner |first=Eugene P. |date=1946-07-01 |title=Resonance Reactions and Anomalous Scattering |url=https://journals.aps.org/pr/abstract/10.1103/PhysRev.70.15 |journal=Physical Review |volume=70 |issue=1-2 |pages=15–33 |doi=10.1103/PhysRev.70.15}}</ref>, [[Leonard Eisenbud|Eisenbud]], [[Gregory Breit|Breit]]<ref>{{Cite journal |last=Breit |first=G. |last2=Wigner |first2=E. |date=1936-04-01 |title=Capture of Slow Neutrons |url=https://journals.aps.org/pr/abstract/10.1103/PhysRev.49.519 |journal=Physical Review |volume=49 |issue=7 |pages=519–531 |doi=10.1103/PhysRev.49.519}}</ref>, [[John M. Blatt|Blatt]], [[Victor Weisskopf|Weisskopf]], and others.<ref>{{cite journal | last1=Wigner | first1=E. P. | last2=Eisenbud | first2=L. | title=Higher Angular Momenta and Long Range Interaction in Resonance Reactions | journal=Physical Review | publisher=American Physical Society (APS) | volume=72 | issue=1 | date=1947-07-01 | issn=0031-899X | doi=10.1103/physrev.72.29 | pages=29–41| bibcode=1947PhRv...72...29W }}</ref> Related theories are U-matrix, [[S-matrix]], by [[M-matrix]], or [[T-matrix method|T-matrix]].<ref>{{Cite book |last=Paetz gen. Schieck |first=Hans |url=https://link.springer.com/10.1007/978-3-642-53986-2 |title=Nuclear Reactions: An Introduction |date=2014 |publisher=Springer Berlin Heidelberg |isbn=978-3-642-53985-5 |series=Lecture Notes in Physics |volume=882 |location=Berlin, Heidelberg |language=en |doi=10.1007/978-3-642-53986-2}}</ref>

== Other meanings and uses ==
The term ''R-matrix'' is used in connection with the [[Yang–Baxter equation]], first introduced in the field of [[statistical mechanics]] in the works of J. B. McGuire in 1964<ref>{{cite journal | last=McGuire | first=J. B. | title=Study of Exactly Soluble One-Dimensional N-Body Problems | journal=Journal of Mathematical Physics | publisher=The American Institute of Physics (AIP) | volume=5 | issue=5 | date=1964-05-01 | issn=0022-2488 | doi=10.1063/1.1704156 | pages=622–636 | bibcode=1964JMP.....5..622M }}</ref> and [[C. N. Yang]] in 1967<ref>{{cite journal | last=Yang | first=C. N. | title=Some Exact Results for the Many-Body Problem in one Dimension with Repulsive Delta-Function Interaction  | journal=Physical Review Letters | publisher=American Physical Society (APS) | volume=19 | issue=23 | date=1967-12-04 | issn=0031-9007 | doi=10.1103/PhysRevLett.19.1312 | pages=1312–1315 | bibcode=1967PhRvL..19.1312Y }}</ref> and in the [[group ring|group algebra]] <math>\mathbb{C} [S_n] </math> of the [[symmetric group]] in the work of [[Algimantas Adolfas Jucys| A. A. Jucys]] in 1966.<ref>{{cite journal | last=Jucys | first=A. A. | title=On the Young operators of the symmetric group | journal=Lietuvos Fizikos Rinkinys | publisher=Gos. Izd-vo Polit. i Nauch. literatury. | volume=11 | issue=23 | date=1966 | pages=163–180 | url=https://www.lietuvos-fizikai.lt/chessidr/straipsniai/LietFizRink/LFR-1971-v11-p5-AlgJucys-Factorization_of_Young_projection_operators_for_the_symmetric_group.pdf}}</ref> The classical R-matrix arises in the definition of the classical Yang&ndash;Baxter equation.<ref>{{cite journal | last=Kupershmidt | first=Boris A. | title=What a Classical r-Matrix Really Is | journal=Journal of Nonlinear Mathematical Physics | publisher=Informa UK Limited | volume=6 | issue=4 | year=1999 | issn=1402-9251 | doi=10.2991/jnmp.1999.6.4.5 | pages=448–488| bibcode=1999JNMP....6..448K | arxiv=math/9910188 }}</ref>

In [[quasitriangular Hopf algebra]], the R-matrix is a solution of the [[Yang&ndash;Baxter equation]].

The numerical modeling of diffraction gratings in [[optical science]] can be performed using the R-matrix propagation algorithm.<ref>{{cite journal | last=Li | first=Lifeng | title=Bremmer series, R-matrix propagation algorithm, and numerical modeling of diffraction gratings | journal=Journal of the Optical Society of America A | publisher=The Optical Society | volume=11 | issue=11 | date=1994-11-01 | issn=1084-7529 | doi=10.1364/josaa.11.002829 | pages=2829–2836| bibcode=1994JOSAA..11.2829L }}</ref>

==R-matrix method in quantum mechanics==

There is a method in computational [[quantum mechanics]] for studying scattering known as the R-matrix. Using the original R-matrix theory as a basis, a method was developed for [[electron]], [[positron]] and [[photon]] scattering by [[atom]]s.<ref>{{cite journal | last1=Burke | first1=P G | last2=Hibbert | first2=A | last3=Robb | first3=W D | title=Electron scattering by complex atoms | journal=Journal of Physics B: Atomic and Molecular Physics | publisher=IOP Publishing | volume=4 | issue=2 | year=1971 | issn=0022-3700 | doi=10.1088/0022-3700/4/2/002 | pages=153–161| bibcode=1971JPhB....4..153B }}</ref> This approach was later adapted for electron, positron and photon scattering by [[molecule]]s.<ref>{{cite journal | last=Schneider | first=Barry | title=R-matrix theory for electron-atom and electron-molecule collisions using analytic basis set expansions | journal=Chemical Physics Letters | publisher=Elsevier BV | volume=31 | issue=2 | year=1975 | issn=0009-2614 | doi=10.1016/0009-2614(75)85010-x | pages=237–241| bibcode=1975CPL....31..237S }}</ref><ref>{{cite journal | last=Schneider | first=Barry I. | title=R-matrix theory for electron-molecule collisions using analytic basis set expansions. II. Electron-H<sub>2</sub> scattering in the static-exchange model | journal=Physical Review A | publisher=American Physical Society (APS) | volume=11 | issue=6 | date=1975-06-01 | issn=0556-2791 | doi=10.1103/physreva.11.1957 | pages=1957–1962| bibcode=1975PhRvA..11.1957S }}</ref><ref>C J Gillan, J Tennyson, and P G Burke, in Computational Methods for Electron-Molecule Collisions, eds. W M Huo and F A Gianturco, (Plenum, New York, 1995), p. 239</ref>

== Other applications ==
R-matrix method is used in [[UK Molecular R-matrix Codes|UKRmol]]<ref>{{cite journal | last1=Carr | first1=J.M. |last2=Galiatsatos | first2=P.G. | last3=Gorfinkiel |first3=J.D.| last4=Harvey |first4=A.G. | last5=Lysaght |first5=M.A. |last6=Madden |first6=D. | last7=Mašín |first7=Z. | last8=Plummer |first8=M.| last9=Tennyson |first9=J. | title=The UKRmol program suite. | journal=Eur. Phys. J. D |year=2012 | issue=66 | pages=58| doi=10.1140/epjd/e2011-20653-6 }}</ref> and UKRmol+<ref>{{cite journal | journal=Computer Physics Communications | volume=249 | pages=107092 | last1=Mašín | first1=Zdeněk | last2=Benda | first2=Jakub | last3=Gorfinkiel | first3=Jimena D. | last4=Harvey |first4=Alex G.| last5=Tennyson |first5=Jonathan | title=UKRmol+: A suite for modelling electronic processes in molecules interacting with electrons, positrons and photons using the R-matrix method | date=2019-12-07 | doi=10.1016/j.cpc.2019.107092| arxiv=1908.03018 }}</ref> code suits. The user-friendly software [[Quantemol | Quantemol Electron Collisions (Quantemol-EC)]] and its predecessor[[Quantemol | Quantemol-N]] are based on UKRmol/UKRmol+ and employ [[MOLPRO]] package for electron configuration calculations.

== See also ==

* [[Nuclear reaction|Compound nucleus]]
* [[Born approximation|Distorted Waves Born Approximation]]
* [[Nuclear force|Optical model]]

==References==
{{Reflist|3}}

[[Category:Matrices]]
[[Category:Nuclear physics]]