Few-Body Systems

, Volume 52, Issue 1–2, pp 97–123 | Cite as

Four-Nucleon Scattering with a Correlated Gaussian Basis Method

  • S. Aoyama
  • K. Arai
  • Y. Suzuki
  • P. Descouvemont
  • D. Baye
Article

Abstract

Elastic-scattering phase shifts for four-nucleon systems are studied in an ab-initio type cluster model in order to clarify the role of the tensor force and to investigate cluster distortions in low energy d+d and t+p scattering. In the present method, the description of the cluster wave function is extended from (0s) harmonic-oscillator shell model to a few-body model with a realistic interaction, in which the wave functions of the subsystems are determined with the Stochastic Variational Method. In order to calculate the matrix elements of the four-body system, we have developed a Triple Global Vector Representation method for the correlated Gaussian basis functions. To compare effects of the cluster distortion with realistic and effective interactions, we employ the AV8′ potential + a three nucleon force as a realistic interaction and the Minnesota potential as an effective interaction. Especially for 1S0, the calculated phase shifts show that the t+p and h+n channels are strongly coupled to the d+d channel for the case of the realistic interaction. On the contrary, the coupling of these channels plays a relatively minor role for the case of the effective interaction. This difference between both potentials originates from the tensor term in the realistic interaction. Furthermore, the tensor interaction makes the energy splitting of the negative parity states of 4He consistent with experiments. No such splitting is however reproduced with the effective interaction.

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References

  1. 1.
    Wildermuth K., Tang Y.C.: A unified theory of the nucleus. Vieweg, Braunschweig (1977)CrossRefGoogle Scholar
  2. 2.
    Kamada H., Nogga A., Glöckle W., Hiyama E., Kamimura M. et al.: Benchmark test calculation of a four-nucleon bound state. Phys. Rev. C 64, 044001 (2001)ADSCrossRefGoogle Scholar
  3. 3.
    Varga K., Suzuki Y., Lovas R.G.: Microscopic multicluster description of neutron-halo nuclei with a stochastic variational method. Nucl. Phys. A 571, 447 (1997)ADSCrossRefGoogle Scholar
  4. 4.
    Varga K., Ohbayasi K., Suzuki Y.: Stochastic variational method with noncentral forces. Phys. Lett. B 396, 1 (1997)ADSCrossRefGoogle Scholar
  5. 5.
    Varga K., Usukura J., Suzuki Y.: Second bound state of the positronium molecule and biexcitons. Phys. Rev. Lett. 80, 1876 (1998)ADSCrossRefGoogle Scholar
  6. 6.
    Usukura J., Varga K., Suzuki Y.: Signature of the existence of the positronium molecule. Phys. Rev. A 58, 1918 (1998)ADSCrossRefGoogle Scholar
  7. 7.
    Suzuki Y., Varga K.: Stochastic variational approach to quantum-mechanical few-body problems. Lecture notes in physics. Springer, Berlin (1998)Google Scholar
  8. 8.
    Varga K., Suzuki Y.: Precise solution of few-body problems with the stochastic variational method on a correlated Gaussian basis. Phys. Rev. C 52, 2885 (1995)ADSCrossRefGoogle Scholar
  9. 9.
    Suzuki Y., Horiuchi W., Orabi M., Arai K.: Global-vector representation of the angular motion of few-particle systems II. Few-Body Syst. 42, 33 (2008)ADSCrossRefGoogle Scholar
  10. 10.
    Varga K., Suzuki Y., Usukura J.: Global-vector representation of the angular motion of few-particle systems. Few-Body Syst. 24, 81 (1998)ADSCrossRefGoogle Scholar
  11. 11.
    Carlson J., Schiavilla R.: Structure and dynamics of few-nucleon systems. Rev. Mod. Phys. 70, 743 (2008)ADSCrossRefGoogle Scholar
  12. 12.
    Pudliner B.S., Pandharipande V.R., Carlson J., Pieper S.C., Wiringa R.B.: Quantum Monte Carlo calculations of nuclei with A<7. Phys. Rev. C 56, 1720 (1997)ADSCrossRefGoogle Scholar
  13. 13.
    Navratil P., Kamuntavicius G.P., Barrett B.R.: Few-nucleon systems in a translationally invariant harmonic oscillator basis. Phys. Rev. C 61, 044001 (2000)ADSCrossRefGoogle Scholar
  14. 14.
    Viviani M.: Transformation coefficients of hyperspherical harmonic functions of an A-body system. Few-Body Syst. 25, 177 (1998)ADSCrossRefGoogle Scholar
  15. 15.
    Feldmeier H., Neff T., Roth R., Schnack J.: A unitary correlation operator method. Nucl. Phys. A 632, 61 (1998)ADSCrossRefGoogle Scholar
  16. 16.
    Neff T., Feldmeier H.: Tensor correlations in the unitary correlation operator method. Nucl. Phys. A 713, 311 (2003)ADSMATHCrossRefGoogle Scholar
  17. 17.
    Arai K., Aoyama S., Suzuki Y.: Microscopic cluster model study of 3He+p scattering. Phys. Rev. C 81, 037301 (2010)ADSCrossRefGoogle Scholar
  18. 18.
    Phitzinger B., Hofmann M., Hale G.M.: Elastic p-3He and n-3H scattering with two- and three-body forces. Phys. Rev. C 64, 044003 (2001)ADSCrossRefGoogle Scholar
  19. 19.
    Deltuva A., Fonseca A.C.: Four-nucleon scattering: Ab initio calculations in momentum space. Phys. Rev. C 75, 014005 (2007)ADSCrossRefGoogle Scholar
  20. 20.
    Deltuva A., Fonseca A.C.: Four-body calculation of proton-3He scattering. Phys. Rev. Lett 98, 162502 (2007)ADSCrossRefGoogle Scholar
  21. 21.
    Quaglioni S., Navratil P.: Ab initio many-body calculations of nucleon-nucleus scattering. Phys. Rev. C 79, 044606 (2009)ADSCrossRefGoogle Scholar
  22. 22.
    Quaglioni S., Navratil P.: Ab initio many-body calculations of n-3H, n-4He, p-3,4He, and n-10Be scattering. Phys. Rev. Lett. 08, 092501 (2008)ADSCrossRefGoogle Scholar
  23. 23.
    Viviani M., Rosati S., Kievsky A.: Neutron-3H and proton-3He zero energy scattering. Phys. Rev. Lett. 81, 1580 (1998)ADSCrossRefGoogle Scholar
  24. 24.
    Viviani M., Kievsky A., Rosati S., George E.A., Knulson L.D.: The Ay problem for p-3He elastic scattering. Phys. Rev. Lett. 86, 3739 (2001)ADSCrossRefGoogle Scholar
  25. 25.
    Viviani M., Kievsky A., Girlanda L., Marcucci L.E., Rosati S.: Neutron-triton elastic scattering. Few-Body Syst. 45, 119 (2009)ADSCrossRefGoogle Scholar
  26. 26.
    Lazauskas R., Carbonell J., Fonseca A.C., Viviani M., Kievsky A., Rosati S.: Low energy n-3H scattering: a novel testground for nuclear interactions. Phys. Rev. C 71, 034004 (2005)ADSCrossRefGoogle Scholar
  27. 27.
    Fisher B.M. et al.: Proton-3He elastic scattering at low energies. Phys. Rev. C 74, 034001 (2006)ADSCrossRefGoogle Scholar
  28. 28.
    Arriaga A., Pandharipande V.R., Schiavilla R.: Variational Monte Carlo calculations of the 2H(d, γ)4He reaction at low energies. Phys. Rev. C 43, 983 (1991)ADSCrossRefGoogle Scholar
  29. 29.
    Sabourov K. et al.: Experimental and theoretical study of the 2H(d, γ)4He reaction below Ec.m.=60 keV. Phys. Rev. C 70, 064601 (2004)ADSCrossRefGoogle Scholar
  30. 30.
    Hofmann H.M., Hale G.M.: 4He experiments can serve as a database for determining the three-nucleon force. Phys. Rev. C 77, 044002 (2008)ADSCrossRefGoogle Scholar
  31. 31.
    Hofmann H.M., Hale G.M.: Microscopic calculation of the 4He system. Nucl. Phys. A 613, 69 (1997)ADSCrossRefGoogle Scholar
  32. 32.
    Hofmann H.M., Hale G.M.: Microscopic calculation of the spin-dependent neutron scattering lengths on 3He. Phys. Rev. C 68, 021002 (2003)ADSCrossRefGoogle Scholar
  33. 33.
    Deltuva A., Fonseca A.C.: Ab initio four-body calculation of n-3He, p-3H, and d-d scattering. Phys. Rev. C 76, 021001 (2007)ADSCrossRefGoogle Scholar
  34. 34.
    Deltuva A., Fonseca A.C., Sauer P.U.: Four-nucleon system with Δ-isobar excitation. Phys. Lett. B 660, 471 (2008)ADSCrossRefGoogle Scholar
  35. 35.
    Lazauskas R., Carbonell J.: Ab-initio calculations of four-nucleon elastic scattering. Few-Body Syst. 34, 105 (2004)ADSGoogle Scholar
  36. 36.
    Ciesielski F., Carbonell J., Gignoux C.: Low energy n  +  t scattering and the NN forces. Phys. Lett. B 447, 199 (1999)ADSCrossRefGoogle Scholar
  37. 37.
    Assenbaum H.J., Langanke K.: Low-energy 2H(d,γ)4He reaction and the D-state admixture in the 4He ground state. Phys. Rev. C 36, 17 (1987)ADSCrossRefGoogle Scholar
  38. 38.
    Fowler W.A., Caughlan G.R., Zimmerman B.A.: Thermonuclear reaction rates. Annu. Rev. Astron. Astrophys. 5, 525 (1967)ADSCrossRefGoogle Scholar
  39. 39.
    Baye D., Heenen P.H., Libert-Heinemann M.: Microscopic R-matrix theory in a generator coordinate basis: (III). Multi-channel scattering. Nucl. Phys. A 291, 230 (1977)ADSCrossRefGoogle Scholar
  40. 40.
    Kanada H., Kaneko T., Saito S., Tang Y.C.: Microscopic study of the d+α scattering system with the multi-channel resonating-group method. Nucl. Phys. A 444, 209 (1985)ADSCrossRefGoogle Scholar
  41. 41.
    Arai K., Descouvemont P., Baye D.: Low-energy 6He+p reactions in a microscopic multicluster model. Phys. Rev. C 63, 044611 (2001)ADSCrossRefGoogle Scholar
  42. 42.
    Descouvemont P., Baye D.: The R-matrix theory. Rep. Prog. Phys. 73, 036301 (2010)MathSciNetADSCrossRefGoogle Scholar
  43. 43.
    Navratil P., Quaglioni S.: Ab initio many-body calculations of deuteron-4He scattering and 6Li states. Phys. Rev. C 83, 044609 (2011)ADSCrossRefGoogle Scholar
  44. 44.
    Pudliner B.S., Pandharipande V.R., Carlson J., Pieper S.C., Wiringa R.B.: Quantum Monte Carlo calculations of nuclei with A≤7. Phys. Rev. C 56, 1720 (1997)ADSCrossRefGoogle Scholar
  45. 45.
    Hiyama E., Gibson B.F., Kamimura M.: Four-body calculation of the first excited state of 4He using a realistic NN interaction: 4 He(e, e′)4He(0+ 2) and the monopole sum rule. Phys. Rev. C 70, 031001 (2003)ADSCrossRefGoogle Scholar
  46. 46.
    Thompson D.R., LeMere M., Tang Y.C.: Systematic investigation of scattering problems with the resonating-group method. Nucl. Phys. A 286, 53 (1977)ADSCrossRefGoogle Scholar
  47. 47.
    Boys S.F.: The integral formulae for the variational solution of the molecular many-electron wave equations in terms of Gaussian functions with direct electronic correlation. Proc. R. Soc. London. Ser. A 258, 402 (1960)MathSciNetADSMATHCrossRefGoogle Scholar
  48. 48.
    Singer K.: The use of Gaussian (exponential quadratic) wave functions in molecular problems. I. General formulae for the evaluation of integrals. ibid. 258, 412 (1960)ADSMATHGoogle Scholar
  49. 49.
    Suzuki Y., Usukura J.: Excited states of the positronium molecule. Nucl. Inst. Method B 171, 67 (2000)ADSCrossRefGoogle Scholar
  50. 50.
    Suzuki Y., Usukura J., Varga K.: New description of orbital motion with arbitrary angular momenta. J. Phys. B 31, 31 (1998)ADSCrossRefGoogle Scholar
  51. 51.
    Horiuchi W., Suzuki Y.: Inversion doublets of 3N+N cluster structure in excited states of 4He. Phys. Rev. C 78, 034305 (2008)ADSCrossRefGoogle Scholar
  52. 52.
    Tilley D.R., Weller H.R., Hale G.M.: Energy levels of light nuclei A=4. Nucl. Phys. A 541, 1 (1992)ADSCrossRefGoogle Scholar
  53. 53.
    Tamagaki R.: Potential models of nuclear forces at small distances. Prog. Theor. Phys. 39, 91 (1968)ADSCrossRefGoogle Scholar
  54. 54.
    Santos F.D., Arriaga A., Eiró A.M., Tostevin J.A.: 4He D-state effect in the d(d, γ)4He reaction. Phys. Rev. C 31, 707 (1985)ADSCrossRefGoogle Scholar
  55. 55.
    Wachter B., Mertelmeier T., Hofmann H.M.: The 2H(d, γ)4He reaction and the D-state of the alpha particle: a microscopic study. Phys. Lett. B 200, 246 (1988)ADSCrossRefGoogle Scholar
  56. 56.
    Angulo C., Arnould M., Rayet M., Descouvemont P., Baye D. et al.: A compilation of charged-particle induced thermonuclear reaction rates. Nucl. Phys. A 656, 3 (1999)ADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • S. Aoyama
    • 1
  • K. Arai
    • 2
  • Y. Suzuki
    • 3
    • 4
  • P. Descouvemont
    • 5
  • D. Baye
    • 5
    • 6
  1. 1.Center for Academic Information ServiceNiigata UniversityNiigataJapan
  2. 2.Division of General EducationNagaoka National College of TechnologyNagaoka, NiigataJapan
  3. 3.Department of PhysicsNiigata UniversityNiigataJapan
  4. 4.RIKEN Nishina CenterWakoJapan
  5. 5.Physique Nucléaire Théorique et Physique MathématiqueUniversité Libre de Bruxelles (ULB)BrusselsBelgium
  6. 6.Physique QuantiqueUniversité Libre de Bruxelles (ULB)BrusselsBelgium

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