R-matrix

The term R-matrix has several meanings, depending on the field of study. Its original use has been to mathematically describe nuclear reactions. ^[1] 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 Wigner ^{[2] [3]} , Eisenbud, Breit ^[4] , Blatt, Weisskopf, and others. ^[5] Related theories are U-matrix, S-matrix, by M-matrix, or T-matrix. ^[6]

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 ^[7] and C. N. Yang in 1967 ^[8] and in the group algebra ${\displaystyle \mathbb {C} [S_{n}]}$ of the symmetric group in the work of A. A. Jucys in 1966. ^[9] The classical R-matrix arises in the definition of the classical Yang–Baxter equation. ^[10]

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. ^[11]

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 atoms. ^[12] This approach was later adapted for electron, positron and photon scattering by molecules. ^{[13] [14] [15]}

Other applications

R-matrix method is used in UKRmol ^[16] and UKRmol+ ^[17] 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.

See also

• Compound nucleus
• Distorted Waves Born Approximation
• Optical model

References

1. Lane, A. M.; Thomas, R. G. (1958-04-01). "R-Matrix Theory of Nuclear Reactions". Reviews of Modern Physics. 30 (2): 257–353. doi:10.1103/RevModPhys.30.257. ISSN   0034-6861.
2. Wigner, Eugene P. (1946-11-01). "Resonance Reactions". Physical Review. 70 (9–10): 606–618. doi:10.1103/PhysRev.70.606.
3. Wigner, Eugene P. (1946-07-01). "Resonance Reactions and Anomalous Scattering". Physical Review. 70 (1–2): 15–33. doi:10.1103/PhysRev.70.15.
4. Breit, G.; Wigner, E. (1936-04-01). "Capture of Slow Neutrons". Physical Review. 49 (7): 519–531. doi:10.1103/PhysRev.49.519.
5. Wigner, E. P.; Eisenbud, L. (1947-07-01). "Higher Angular Momenta and Long Range Interaction in Resonance Reactions". Physical Review. 72 (1). American Physical Society (APS): 29–41. Bibcode:1947PhRv...72...29W. doi:10.1103/physrev.72.29. ISSN   0031-899X.
6. Paetz gen. Schieck, Hans (2014). Nuclear Reactions: An Introduction. Lecture Notes in Physics. Vol. 882. Berlin, Heidelberg: Springer Berlin Heidelberg. doi:10.1007/978-3-642-53986-2. ISBN   978-3-642-53985-5.
7. McGuire, J. B. (1964-05-01). "Study of Exactly Soluble One-Dimensional N-Body Problems". Journal of Mathematical Physics. 5 (5). The American Institute of Physics (AIP): 622–636. Bibcode:1964JMP.....5..622M. doi:10.1063/1.1704156. ISSN   0022-2488.
8. Yang, C. N. (1967-12-04). "Some Exact Results for the Many-Body Problem in one Dimension with Repulsive Delta-Function Interaction". Physical Review Letters. 19 (23). American Physical Society (APS): 1312–1315. Bibcode:1967PhRvL..19.1312Y. doi:10.1103/PhysRevLett.19.1312. ISSN   0031-9007.
9. Jucys, A. A. (1966). "On the Young operators of the symmetric group" (PDF). Lietuvos Fizikos Rinkinys. 11 (23). Gos. Izd-vo Polit. i Nauch. literatury.: 163–180.
10. Kupershmidt, Boris A. (1999). "What a Classical r-Matrix Really Is". Journal of Nonlinear Mathematical Physics. 6 (4). Informa UK Limited: 448–488. arXiv: math/9910188 . Bibcode:1999JNMP....6..448K. doi:10.2991/jnmp.1999.6.4.5. ISSN   1402-9251.
11. Li, Lifeng (1994-11-01). "Bremmer series, R-matrix propagation algorithm, and numerical modeling of diffraction gratings". Journal of the Optical Society of America A. 11 (11). The Optical Society: 2829–2836. Bibcode:1994JOSAA..11.2829L. doi:10.1364/josaa.11.002829. ISSN   1084-7529.
12. Burke, P G; Hibbert, A; Robb, W D (1971). "Electron scattering by complex atoms". Journal of Physics B: Atomic and Molecular Physics. 4 (2). IOP Publishing: 153–161. Bibcode:1971JPhB....4..153B. doi:10.1088/0022-3700/4/2/002. ISSN   0022-3700.
13. Schneider, Barry (1975). "R-matrix theory for electron-atom and electron-molecule collisions using analytic basis set expansions". Chemical Physics Letters. 31 (2). Elsevier BV: 237–241. Bibcode:1975CPL....31..237S. doi:10.1016/0009-2614(75)85010-x. ISSN   0009-2614.
14. Schneider, Barry I. (1975-06-01). "R-matrix theory for electron-molecule collisions using analytic basis set expansions. II. Electron-H₂ scattering in the static-exchange model". Physical Review A. 11 (6). American Physical Society (APS): 1957–1962. Bibcode:1975PhRvA..11.1957S. doi:10.1103/physreva.11.1957. ISSN   0556-2791.
15. 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
16. Carr, J.M.; Galiatsatos, P.G.; Gorfinkiel, J.D.; Harvey, A.G.; Lysaght, M.A.; Madden, D.; Mašín, Z.; Plummer, M.; Tennyson, J. (2012). "The UKRmol program suite". Eur. Phys. J. D (66): 58. doi:10.1140/epjd/e2011-20653-6.
17. Mašín, Zdeněk; Benda, Jakub; Gorfinkiel, Jimena D.; Harvey, Alex G.; Tennyson, Jonathan (2019-12-07). "UKRmol+: A suite for modelling electronic processes in molecules interacting with electrons, positrons and photons using the R-matrix method". Computer Physics Communications. 249: 107092. arXiv: 1908.03018 . doi:10.1016/j.cpc.2019.107092.