Few-Body Systems

, 60:29 | Cite as

Coulomb Force Effects in Few-Nucleon Systems

  • A. DeltuvaEmail author
  • A. C. Fonseca
  • P. U. Sauer
Part of the following topical collections:
  1. Ludwig Faddeev Memorial Issue


Theoretical predictions for sample observables of three-nucleon and four-nucleon reactions are reviewed. The focus is on Coulomb force effects. The calculations are based on the Alt–Grassberger–Sandhas version of the Faddeev equations. The calculations are done in momentum space. The calculational technique used to include the Coulomb repulsion between protons screens the infinite Coulomb tail, renormalizes the results and thereby corrects them for screening. The competition between three-nucleon force and Coulomb force effects as well as the Coulomb domination in special kinematic situations of reactions are discussed. Reactions connected by charge symmetry are reviewed. Special reaction observables are studied, in search for the hadronic violation of charge symmetry in the nuclear interaction and for its competition with the charge-asymmetric Coulomb force.



A.D. acknowledges the support by the Alexander von Humboldt Foundation under Grant No. LTU-1185721-HFST-E.


  1. 1.
    A. Kievsky, M. Viviani, S. Rosati, Polarization observables in p–d scattering below 30 MeV. Phys. Rev. C 64, 024002 (2001)ADSCrossRefGoogle Scholar
  2. 2.
    L.D. Faddeev, Mathematical Aspects of the Three-Body Problem in the Quantum Scattering Theory. Academy of Sciences of the USSR Works of the Steklov Mathematical Institute Vol.69, Israel Program for Scientific Translations, Jerusalem (1965)Google Scholar
  3. 3.
    A. Deltuva, A.C. Fonseca, P.U. Sauer, Nuclear many-body scattering calculations with the Coulomb interaction. Annu. Rev. Nucl. Part. Sci. 58, 27 (2008)ADSCrossRefGoogle Scholar
  4. 4.
    S.P. Merkuriev, C. Gignoux, A. Laverne, Three-body scattering in configuration space. Ann. Phys. 99, 30 (1976)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    R. Lazauskas, Elastic proton scattering on tritium below the \(n\)-\({}^3\rm He\) threshold. Phys. Rev. C 79, 054007 (2009)ADSCrossRefGoogle Scholar
  6. 6.
    E.O. Alt, P. Grassberger, W. Sandhas, Reduction of the three-particle collision problem to multi-channel two-particle Lippmann–Schwinger equations. Nucl. Phys. B 2, 167 (1967)ADSCrossRefGoogle Scholar
  7. 7.
    E.O. Alt, W. Sandhas, H. Ziegelmann, Coulomb effects in three-body reactions with two charged particles. Phys. Rev. C 17, 1981 (1978)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    E.O. Alt, W. Sandhas, Coulomb effects in three-body reactions with two charged particles. Phys. Rev. C 21, 1733 (1980)ADSCrossRefGoogle Scholar
  9. 9.
    R. Machleidt, High-precision, charge-dependent Bonn nucleon–nucleon potential. Phys. Rev. C 63, 024001 (2001)ADSCrossRefGoogle Scholar
  10. 10.
    A. Deltuva, R. Machleidt, P.U. Sauer, Realistic two-baryon potential coupling two-nucleon and nucleon-\(\Delta \)-isobar states: fit and applications to three-nucleon system. Phys. Rev. C 68, 024005 (2003)ADSCrossRefGoogle Scholar
  11. 11.
    K. Hatanaka, Y. Shimizu, D. Hirooka, J. Kamiya, Y. Kitamura, Y. Maeda, T. Noro, E. Obayashi, K. Sagara, T. Saito, H. Sakai, Y. Sakemi, K. Sekiguchi, A. Tamii, T. Wakasa, T. Yagita, K. Yako, H.P. Yoshida, V.P. Ladygin, H. Kamada, W. Glöckle, J. Golak, A. Nogga, H. Witała, Cross section and complete set of proton spin observables in \(\vec{p}d\) elastic scattering at 250 MeV. Phys. Rev. C 66, 044002 (2002)ADSCrossRefGoogle Scholar
  12. 12.
    D.G. McDonald, W. Haeberli, L.W. Morrow, Polarization and cross section of protons scattered by \(He^{3}\) from 4 to 13 MeV. Phys. Rev. 133, B1178 (1964)ADSCrossRefGoogle Scholar
  13. 13.
    M .T. Alley, L .D. Knutson, Spin correlation measurements for p\({}^{3}\)He elastic scattering between 4.0 and 10.0 MeV. Phys. Rev. C 48, 1890 (1993)ADSCrossRefGoogle Scholar
  14. 14.
    S. Kistryn, E. Stephan, B. Klos, A. Biegun, K. Bodek, I. Ciepal, A. Deltuva, A. Fonseca, N. Kalantar-Nayestanaki, M. Kis, A. Kozela, M. Mahjour-Shafiei, A. Micherdzinska, P. Sauer, R. Sworst, J. Zejma, W. Zipper, Evidence of the Coulomb-force effects in the cross-sections of the deuteron proton breakup at 130 MeV. Phys. Lett. B 641, 23 (2006)ADSCrossRefGoogle Scholar
  15. 15.
    P.U. Sauer, Can the charge symmetry of nuclear forces be confirmed by nucleon–nucleon scattering experiments? Phys. Rev. Lett. 32, 626 (1974)ADSCrossRefGoogle Scholar
  16. 16.
    R.A. Brandenburg, S.A. Coon, P.U. Sauer, Nuclear charge asymmetry in the A = 3 nuclei. Nucl. Phys. A 294, 305 (1978)ADSCrossRefGoogle Scholar
  17. 17.
    K. Sagara, H. Oguri, S. Shimizu, K. Maeda, H. Nakamura, T. Nakashima, S. Morinobu, Energy dependence of analyzing power \(A_y\) and cross section for \(p+d\) scattering below 18 MeV. Phys. Rev. C 50, 576 (1994)ADSCrossRefGoogle Scholar
  18. 18.
    C.R. Howell, W. Tornow, K. Murphy, H.G. Pfützner, M.L. Roberts, A. Li, P.D. Felsher, R.L. Walter, I. Šlaus, P.A. Treado, Y. Koike, Comparison of vector analyzing-power data and calculations for neutron–deuteron elastic scattering from 10 to 14 MeV. Few Body Syst. 2, 19 (1987)ADSCrossRefGoogle Scholar
  19. 19.
    W. Glöckle, H. Witała, D. Hüber, H. Kamada, J. Golak, The three-nucleon continuum: achievements, challenges and applications. Phys. Rep. 274, 107 (1996)ADSCrossRefGoogle Scholar
  20. 20.
    J. Strate, K. Geissdörfer, R. Lin, W. Bielmeier, J. Cub, A. Ebneth, E. Finckh, H. Friess, G. Fuchs, K. Gebhardt, S. Schindler, Differential cross section of the 2H(n, nnp)-reaction at \(E_n = 13\) MeV. Nucl. Phys. A 501, 51 (1989)ADSCrossRefGoogle Scholar
  21. 21.
    H .R. Setze, C .R. Howell, W. Tornow, R .T. Braun, D .E. González Trotter, A .H. Hussein, R .S. Pedroni, C .D. Roper, F. Salinas, I. Šlaus, B. Vlahovic, R .L. Walter, G. Mertens, J .M. Lambert, H. Witała, W. Glöckle, Cross-section measurements of neutron–deuteron breakup at 13.0 MeV. Phys. Rev. C 71, 034006 (2005)ADSCrossRefGoogle Scholar
  22. 22.
    G. Rauprich, S. Lemaitre, P. Niessen, K.R. Nyga, R. Reckenfelderbäumer, L. Sydow, H. Paetz gen. Schieck, H. Witała, W. Glöckle, Study of the kinematically complete breakup reaction 2H(p, pp)n at \(E_p = 13\) MeV with polarized protons. Nucl. Phys. A 535, 313 (1991)Google Scholar
  23. 23.
    A.C. Fonseca, A. Deltuva, Numerical exact ab initio four-nucleon scattering calculations: from dream to reality. Few-Body Syst. 58, 46 (2017)ADSCrossRefGoogle Scholar
  24. 24.
    P. Doleschall, Influence of the short range nonlocal nucleon–nucleon interaction on the elastic n–d scattering: below 30 MeV. Phys. Rev. C 69, 054001 (2004)ADSCrossRefGoogle Scholar
  25. 25.
    W. Grüebler, V. König, P.A. Schmelzbach, B. Jenny, J. Vybiral, New highly excited 4He levels found by the 2H(d, p)3H reaction. Nucl. Phys. A 369, 381 (1981)ADSCrossRefGoogle Scholar
  26. 26.
    J .M. Blair, G. Freier, E. Lampi, W. Sleator, J .H. Williams, The angular distributions of the products of the D–D reaction: 1 to 3.5 Mev. Phys. Rev. 74, 1599 (1948)ADSCrossRefGoogle Scholar
  27. 27.
    W. Grüebler, V. König, P.A. Schmelzbach, R. Risler, R.E. White, P. Marmier, Investigation of excited states of 4He via the 2H(d, p)3H and 2H(d, n)3He reactions using a polarized deuteron beam. Nucl. Phys. A 193, 129 (1972)ADSCrossRefGoogle Scholar
  28. 28.
    V. König, W. Grüebler, R.A. Hardekopf, B. Jenny, R. Risler, H. Bürgi, P. Schmelzbach, R. White, Investigation of charge symmetry violation in the mirror reactions 2H(d, p)3H and 2H(d, n)3He. Nucl. Phys. A 331, 1 (1979)ADSCrossRefGoogle Scholar
  29. 29.
    O.A. Yakubovsky, On the integral equations in the theory of N particle scattering. Sov. J. Nucl. Phys. 5, 937 (1967)Google Scholar
  30. 30.
    L.S. Fereira, A.C. Fonseca, L. Streit, (eds). Models and Methods in Few-Body Physics. Lecture Notes in Physics, vol. 273 (1987)Google Scholar
  31. 31.
    M. Viviani, L. Girlanda, A. Kievsky, L.E. Marcucci, Effect of three-nucleon interactions in p-He\(_3\) elastic scattering. Phys. Rev. Lett. 111, 172302 (2013)ADSCrossRefGoogle Scholar
  32. 32.
    H. Pöpping, P.U. Sauer, Z. Xi-Zhen, The two-nucleon system above pion-threshold: a force model with \(\Delta \)-isobar and pion degrees of freedom. Nucl. Phys. A 474, 557 (1987) (Erratum Nucl. Phys. A 550, 563 (1992))Google Scholar
  33. 33.
    R. Machleidt, D.R. Entem, Chiral effective field theory and nuclear forces. Phys. Rep. 503, 1 (2011)ADSCrossRefGoogle Scholar
  34. 34.
    E. Epelbaum, H. Krebs, U.-G. Meißner, Improved chiral nucleon–nucleon potential up to next-to-next-to-next-to-leading order. Eur. Phys. J. A 51, 53 (2015)ADSCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Institute of Theoretical Physics and AstronomyVilnius UniversityVilniusLithuania
  2. 2.Institut für Theoretische Physik IIRuhr-Universität BochumBochumGermany
  3. 3.Centro de Física Nuclear da Universidade de LisboaLisbonPortugal
  4. 4.Leibniz Universität HannoverHannoverGermany

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