R-matrix

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{{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 }} and C. N. Yang in 1967{{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 }} and in the group algebra $\backslash mathbb\{C\}\; [S\_n]$ of the symmetric group in the work of A. A. Jucys in 1966.{{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}} The classical R-matrix arises in the definition of the classical Yang–Baxter equation.{{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 }}

In quasitriangular Hopf algebra, the R-matrix is a solution of the Yang–Baxter equation.

The numerical modeling of diffraction gratings in optical science can be performed using the R-matrix propagation algorithm.{{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 }}

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 atoms.{{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 }} This approach was later adapted for electron, positron and photon scattering by molecules.{{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 }}{{cite journal | last=Schneider | first=Barry I. | title=R-matrix theory for electron-molecule collisions using analytic basis set expansions. II. Electron-H₂ 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 }}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

R-matrix method is used in UKRmol{{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 }} and UKRmol+{{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 }} code suits. The user-friendly software Quantemol Electron Collisions (Quantemol-EC) and its predecessor Quantemol-N are based on UKRmol/UKRmol+ and employ MOLPRO package for electron configuration calculations.

• Compound nucleus
• Distorted Waves Born Approximation
• Optical model

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Category:Matrices (mathematics)

Category:Nuclear physics