Skip to main content
SpringerLink
Log in

Variational Bounds in N-Particle Scattering Using the Faddeev–Yakubovskii Equations: Deuteron-Deuteron S=2 Scattering

  • Published:
Few-Body Systems Aims and scope Submit manuscript

Abstract

A variational-bound formulation of the N-particle scattering problem has been developed based on the Yakubovskii–Faddeev chain-of-partition equations. It is shown that the scattering amplitude for the elastic, rearrangement or break-up processes satisfies a Lippmann–Schwinger type integral equation in which the kernel integration is over the open channels and the closed channels enter through the effective potential. In the case where only two (three)-cluster open channels are allowed, the integral equation for the transition amplitude involves integration over only one (two) momentum vector(s). A variational estimate for the effective potential input to the integral equation is obtained when the closed channel partition Green’s functions are estimated variationally. It is shown that the variational estimates for the closed-channel Green’s function also provide upper and lower bounds that can be used as a subsidiary extremum principle to determine optimum parameters in the trial function. Several methods are provided for simplifying the determination of the effective potential. The inclusion of Coulomb potentials in the Yakubovskii–Faddeev (YF) chain-of-partition formalism is also described. In this approach the inter-particle potential is not assumed to be separable as in typical quasi-particle schemes. The many-body dependence of the effective potential is included via expectation values involving an inter-particle potential and spatially decaying trial functions. The N-body scattering problem is therefore reduced to: (a) solving a two-body scattering problem (three-body in the case of break-up) and (b) a bound-state type calculation to determine the effective potential. As an initial application, the method is applied to the case of low-energy elastic deuteron-deuteron scattering (including the Coulomb force) and compared to a recent cluster-reduction calculation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adhikari S. K.: Variational Principles and the Numerical Solution of Scattering Problems. Wiley, New York (1998)

    Book  Google Scholar 

  2. Faddeev L.: Mathematical Problems of the Quantum Theory of Scattering for a Three-Particle System. Daniel Davey, New York (1965)

    Google Scholar 

  3. Carew J., Rosenberg L.: Lower bounds on phase shifts for three-body systems: n-d quartet scattering. Phys. Rev. 177, 2599–2603 (1969)

    Article  ADS  Google Scholar 

  4. Rosenberg L.: Three-cluster states in reaction theory. Phys. Rev. C 13, 1406–1419 (1976)

    Article  ADS  Google Scholar 

  5. Carew J., Rosenberg L.: Upper and lower bounds on phase shifts for three-body systems: n-d scattering. Phys. Rev. C 5, 658–664 (1972)

    Article  ADS  MathSciNet  Google Scholar 

  6. Yakubovskii O.A.: On the integral equations in the theory of N-particle scattering. Sov. J. Nucl. Phys. 5, 937–942 (1967)

    Google Scholar 

  7. Ciesielski F., Carbonell J.: Solutions of the Faddeev-Yakubovsky equations for the four nucleon scattering states. Phys. Rev. C 58, 58–74 (1998)

    Article  ADS  Google Scholar 

  8. Lazauskas R., Carbonell J.: Testing nonlocal nucleon-nucleon interactions in four-nucleon systems. Phys. Rev. C 70, 044002–044002-12 (2004)

    Article  ADS  Google Scholar 

  9. Filikhin I.N., Yakovlev S.L.: Investigation of low-energy scattering in the nnpp system on the basis of differential equations for Yakubovsky components in configuration space. Phys. At. Nucl. 63(5), 5–68 (2000)

    Article  Google Scholar 

  10. Benoist-Gueutal P., L’Huillier M.: Properties of solutions for N-body Yakubovskii-Faddeev equations. J. Math. Phys. (N.Y.) 23, 1823–1834 (1982)

    Article  ADS  MathSciNet  Google Scholar 

  11. Cattapan G., Vanzani V.: New developments in N-body scattering theory, III-exact effective few-cluster reductions of N-body Faddeev-Yakubovskii equations. Nuovo Cim A89, 29–54 (1985)

    Article  ADS  Google Scholar 

  12. Cattapan G., Canton L.: πNNN-NNN problem: connectedness, transition amplitudes, and quasi-particle approximations. Phys. Rev. C 56, 689–701 (1997)

    Article  ADS  Google Scholar 

  13. Adhikari S.K., Kowalski K.L.: Dynamical Collision Theory and its Applications. Academic, San Diego (1991)

    Google Scholar 

  14. Rosenberg L.: Generalized Faddeev integral equations for multiparticle scattering amplitudes. Phys. Rev. B 140, 217–225 (1965)

    Article  ADS  Google Scholar 

  15. Weinberg S.: Systematic solution of multiparticle scattering problems. Phys. Rev. B 133, 232–251 (1964)

    Article  ADS  MathSciNet  Google Scholar 

  16. Kouri D.J., Levin F.S.: Channel T-operators and K-operators and Heitler damping equation for identical-particle scattering. Phys. Rev. A 10, 1616–1622 (1974)

    Article  ADS  Google Scholar 

  17. Mitra A.N., Gillespie J., Sugar R., Panchapakesan N.: Faddeev formalism for four-particle systems. Phys. Rev. B 140, 1336–1345 (1965)

    Article  ADS  MathSciNet  Google Scholar 

  18. Alt E.O., Grassberger P., Sandhas W.: Reduction of the three-particle collision problem to multi-channel two-particle Lippmann-Schwinger equations. Nucl. Phys. B 2, 167–180 (1967)

    Article  ADS  Google Scholar 

  19. Viviani M., Kievsky A., Rosati S., George E.A., Knutson L.D.: The Ay problem for p-3He elastic scattering. Phys. Rev. Lett. 86, 3739–3742 (2001)

    Article  ADS  Google Scholar 

  20. 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, 034004–034004-8 (2005)

    Article  ADS  Google Scholar 

  21. Viviani L. M., Girlanda L., Kievsky A., Marcucci L.E.: Recent progress in Ab-initio four-body scattering calculations. Few-Body Syst. 54, 647–656 (2013)

    Article  ADS  Google Scholar 

  22. Blackford L.S., Choi J., Cleary A., D’Azevedo E., Demmel J., Dhillon I., Dongarra J., Hammarling S., Henry G., Petitet A., Stanley K., Walker D., Whaley R.C.: ScaLAPACK Users’ Guide. Society for Industrial and Applied Mathematics (SIAM) Press, Philadelphia (1997)

    Book  MATH  Google Scholar 

  23. Dongarra J.J., Duff I.S., Sorensen D.C., van der Vorst H.A.: Numerical Linear Algebra for High-Performance Computers. Society for Industrial and Applied Mathematics (SIAM) Press, Philadelphia (1998)

    Book  MATH  Google Scholar 

  24. Duff I.S.: MA57 A code for the solution of sparse symmetric indefinite systems. ACM Trans. Math. Softw. 30(2), 118–144 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  25. Aaron R., Amado Y.D., Yam Y.Y.: Calculation of neutron deuteron scattering. Phys. Rev. 140, B1291–1303 (1965)

    Article  ADS  Google Scholar 

  26. Uzu E., Oryu S., Tanifuji M.: Calculation of low energy 2H(d, p)3H reaction by the four-body Faddeev-Yakubovsky equation. Few-Body Syst., Suppl. 12, 491–703 (2000)

    Article  Google Scholar 

  27. Crowe B.J. III, Brune C.R. , Geist W.H., Karwowski H.J., Ludwig E.J., Veal K.D., Fonseca A.C., Hale G.M., Fletcher K.A.: Analyzing powers for H-2(d, d)H-2 at deuteron energies of 3.0, 4.75 and 6.0 MeV. Phys. Rev. C 61, 034006–034006-12 (2000)

    Article  ADS  Google Scholar 

  28. Fonseca A.C.: Contribution of nucleon-nucleon P waves to nt-nt, dd-pt and dd-dd scattering observables. Phys. Rev. Lett. 83, 4021–4024 (1999)

    Article  ADS  Google Scholar 

  29. Fonseca A.C.: Microscopic calculation of four-nucleon scattering observables in dd-dd and dd-p(3H). Nucl. Phys. A 631, 675–679 (1998)

    Article  ADS  Google Scholar 

  30. Friar J.L., Payne G.L., Glöckle W., Hüber D., Witala H.: Benchmark solutions for n-d breakup amplitudes. Phys. Rev. C 51, 2356–2359 (1995)

    Article  ADS  Google Scholar 

  31. Felsher P.D., Howell C.R., Tornow W., Roberts M.L., Hanly J.M., Weisel G.J., Al Ohali M., Walter R.L., Slaus I., Lambert J.M., Treado P.A., Mertens G., Fonseca A.C., Soldi A., Vlahovic B.: Analyzing power measurements for the d+d-d+p+n breakup reaction at 12 MeV. Phys. Rev. C 56, 38–49 (1997)

    Article  ADS  Google Scholar 

  32. Fonseca A.C.: Four-body calculation of dd-dd and p-3H tensor analyzing powers. Phys. Rev. Lett. 63, 2036–2039 (1989)

    Article  ADS  Google Scholar 

  33. Kamada H., Koike Y., Glöckle W.: Complex energy method for scattering processes. Prog. Theor. Phys. 109, 869–874 (2003)

    Article  ADS  MATH  Google Scholar 

  34. Uzu E., Kamada H., Koike Y.: Complex energy method in four-body Faddeev-Yakubovsky equations. Phys. Rev. C 68, 061001(R)–061001-3 (2003)

    Article  ADS  Google Scholar 

  35. Alt E.O., Sandhas W., Ziegelmann H.: Coulomb effects in three-body reactions with two charged particles. Phys. Rev. C 17, 1981–2005 (1978)

    Article  ADS  MathSciNet  Google Scholar 

  36. Alt E.O.: Coulomb effects on few-body scattering states. Few-Body Syst. Suppl. 1, 79–87 (1986)

    Article  Google Scholar 

  37. Deltuva A., Fonseca A.C.: Ab initio four-body calculation of n-3He, p-H, and d-d scattering. Phys. Rev. C 76, 021001–021001-4 (2007)

    Article  ADS  Google Scholar 

  38. Noble J.V.: Three-body problem with charged particles. Phys. Rev. 161, 945–953 (1967)

    Article  ADS  Google Scholar 

  39. Bencze Gy.: An approximate treatment of Coulomb effects in the nuclear three-body problem. Nucl. Phys. A 196, 135–144 (1972)

    Article  ADS  Google Scholar 

  40. Merkuriev P.: On the three-body Coulomb scattering problem. Ann. Phys. (N.Y.) 130, 395–426 (1980)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  41. Rosenberg L.: Variational principles for breakup amplitudes: three charged clusters. Phys. Rev. A 75, 032708–032708-6 (2007)

    Article  ADS  Google Scholar 

  42. Rotenberg M.: Application of Sturmian functions to the Schroedinger three body problem: e+-H scattering. Ann. Phys. (N.Y.) 19, 262–278 (1962)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  43. Rotenberg M.: Theory and application of Sturmian functions. Adv. At. Mol. Phys. 6, 233–268 (1970)

    Article  ADS  Google Scholar 

  44. Papp Z.: Three-potential formalism for the three-body Coulomb scattering problem. Phys. Rev. C 55, 1080–1082 (1997)

    Article  ADS  Google Scholar 

  45. Christian R.S., Gammel J.L.: Elastic scattering of protons and neutrons by deuterons. Phys. Rev. 91, 100–121 (1953)

    Article  ADS  MATH  Google Scholar 

  46. Bencze Gy., Chandler C., Friar J.L., Gibson A.G., Payne G.L.: Low energy scattering theory for Coulomb plus long range potentials. Phys. Rev. C 35, 1188–1200 (1987)

    Article  ADS  Google Scholar 

  47. Timm, W., Stingl, M.: Coulomb effects in proton-deuteron scattering near threshold. J. Phys. G: Nucl. Phys. 2(8) (1976)

  48. Kharchenko V.F., Navrotsky M.A., Katerinchhuk P.A.: Coulomb effects in the proton-deuteron scattering and radiative capture processes at zero energy. Nucl. Phys. A 552, 378–400 (1993)

    Article  ADS  Google Scholar 

  49. Adhikari S.K., Das T.K.: Effect of polarization potential in proton deuteron scattering. Phys. Rev. C 37, 1376–1378 (1988)

    Article  ADS  Google Scholar 

  50. Wiringa R.B., Stoks V.G.J., Schiavilla R.: Accurate nucleon-nucleon potential with charge independence breaking. Phys. Rev. C 51, 38–51 (1995)

    Article  ADS  Google Scholar 

  51. Filikhin I.N., Yakovlev S.L.: Microscopic calculation of low-energy deuteron-deuteron scattering on the basis of the cluster-reduction method. Phys. At. Nucl. 63(2), 216–222 (2000)

    Article  Google Scholar 

  52. Friar J.L., Gibson B.F., Payne G.L.: Configuration space Faddeev continuum calculations: p-d s-wave scattering length. Phys. Rev. C 28, 983–994 (1983)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Nuclear Science and Technology, Brookhaven National Laboratory, Upton, NY, 11973-5000, USA

    John F. Carew

Authors
  1. John F. Carew
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to John F. Carew.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Carew, J.F. Variational Bounds in N-Particle Scattering Using the Faddeev–Yakubovskii Equations: Deuteron-Deuteron S=2 Scattering. Few-Body Syst 55, 171–190 (2014). https://doi.org/10.1007/s00601-014-0844-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00601-014-0844-0

Keywords

Search

Navigation