Advertisement

The Generator Coordinate Method

  • Walid Younes
  • Daniel Marc Gogny
  • Jean-François Berger
Chapter
Part of the Lecture Notes in Physics book series (LNP, volume 950)

Abstract

This chapter describes in detail the formalism of the Generator Coordinate Method and its use in constructing both static and dynamic collective states of the nucleus, starting from constrained Hartree-Fock-Bogoliubov solutions. The full formalism is then reduced to a Schrödinger-like equation, and the calculation of its inertial tensor and zero-point energies is presented. Next, the calculations are extended beyond the adiabatic approximation and the Schrödinger Collective Intrinsic Model is derived. The multi-\(O\left (4\right )\) schematic model introduced in Chap.  2 is used throughout this chapter to illustrate the formalism.

Notes

Acknowledgements

This chapter was prepared by a contractor of the U.S. Government under contract number DE-AC52-06NA27344. Accordingly, the U.S. Government retains a nonexclusive, royalty-free license to publish or reproduce the published form of this contribution, or allow others to do so, for U.S. Government purposes.

References

  1. 1.
    Hill, D.L., Wheeler, J.A.: Phys. Rev. 89, 1102 (1953)ADSCrossRefGoogle Scholar
  2. 2.
    Griffin, J.J., Wheeler, J.A.: Phys. Rev. 108, 311 (1957)ADSCrossRefGoogle Scholar
  3. 3.
    Schunck, N., Robledo, L.M.: arXiv:1511.07517v2, submitted to Rep. Prog. Phys. (2016)Google Scholar
  4. 4.
    Haider, Q., Gogny, D.: J. Phys. G 18, 993 (1992)ADSCrossRefGoogle Scholar
  5. 5.
    Fredholm, E.I.: Acta Math. 27, 365 (1903)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Wyld, H.W.: Mathematical Methods for Physics. Addison-Wesley (1976)Google Scholar
  7. 7.
    Chattopadhyay, P., Dreizler, R.M., Trsic, M., Fink, M.: Z. Phys. A285, 7 (1978)ADSCrossRefGoogle Scholar
  8. 8.
    Bonche, P., Dobaczewski, J., Flocard, H., Heenen, P.-H., Meyer, J.: Nucl. Phys. A510, 466 (1990)ADSCrossRefGoogle Scholar
  9. 9.
    Ring, P., Schuck, P.: The Nuclear Many-Body Problem. Springer, Heidelberg (1980)CrossRefGoogle Scholar
  10. 10.
    Robledo, L.M., Bertsch, G.F.: Phys. Rev. C 84, 054302 (2011)ADSCrossRefGoogle Scholar
  11. 11.
    Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes in C: the Art of Scientific Computing, 2nd edn., Cambridge University Press (1992)Google Scholar
  12. 12.
    Duguet, T., Bonche, P.: Phys. Rev. C 67, 054308 (2003)ADSCrossRefGoogle Scholar
  13. 13.
    Robledo, L.M.: J. Phys. G 37, 064020 (2010)ADSCrossRefGoogle Scholar
  14. 14.
    Reinhard, P.-G.: Nucl. Phys. A252, 120 (1975)ADSCrossRefGoogle Scholar
  15. 15.
    Reinhard, P.-G., Grümmer, F., Goeke, K.: Z. Phys. A317, 339 (1984)ADSCrossRefGoogle Scholar
  16. 16.
    Reinhard, P.-G., Goeke, K., Rep. Prog. Phys. 50, 1 (1987)ADSCrossRefGoogle Scholar
  17. 17.
    Weinberg, S.: In: Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. Wiley, New York/London/Sydney/Toronto (1972), Part 2, Chaps. 3–4Google Scholar
  18. 18.
    Podolsky, B.: Phys. Rev. 32, 812 (1928)ADSCrossRefGoogle Scholar
  19. 19.
    Pauli, W.: Handbuch der Physik, vol. XXIV, p. 120. Springer, Berlin (1933)Google Scholar
  20. 20.
    Bernard, R., Goutte, H., Gogny, D., Younes, W.: Phys. Rev. C 84, 044308 (2011)ADSCrossRefGoogle Scholar
  21. 21.
    Didong, M., Müther, H., Goeke, K., Faessler, A.: Phys. Rev. C 14, 1189 (1973)ADSCrossRefGoogle Scholar
  22. 22.
    Müther, H., Goeke, K., Allaart, K., Faessler, A.: Phys. Rev. C 15, 1467 (1977)ADSCrossRefGoogle Scholar
  23. 23.
    Tajima, N., Flocard, H., Bonche, P., Dobaczewski, J., Heenen, P.-H.: Nucl. Phys. A551, 409 (1993)ADSCrossRefGoogle Scholar
  24. 24.
    Holzwarth, G.: Nucl. Phys. A185, 268 (1972)ADSCrossRefGoogle Scholar
  25. 25.
    Kerman, A.K., Koonin, S.: Phys. Scripta 10A, 118 (1974)ADSCrossRefGoogle Scholar
  26. 26.
    Libert, J., Girod, M., Delaroche, J.-P.: Phys. Rev. C 60, 054301 (1999)ADSCrossRefGoogle Scholar
  27. 27.
    Li, Z.P., Nikšic, T., Ring, P., Vretenar, D., Yao, J.M.: J. Meng Phys. Rev. C 86, 034334 (2012)Google Scholar
  28. 28.
    Peierls, R.E., Yoccoz, J.: Proc. Phys. Soc. Lond. A70, 381 (1957)ADSCrossRefGoogle Scholar
  29. 29.
    Peierls, R.E., Thouless, D.J.: Nucl. Phys. 38, 154 (1962)CrossRefGoogle Scholar
  30. 30.
    Rouhaninejad, H., Yoccoz, J.: Nucl. Phys. 78, 353 (1966)CrossRefGoogle Scholar
  31. 31.
    Girod, M., Grammaticos, B.: Nucl. Phys. A330, 40 (1979)ADSCrossRefGoogle Scholar
  32. 32.
    Robledo, L.M., Egido, J.L., Nerlo-Pomorska, B. Pomorski, K.: Phys. Lett. B201, 409 (1988)ADSCrossRefGoogle Scholar
  33. 33.
    Baranger, M., Vénéroni, M.: Ann. Phys. (N.Y.) 114, 123 (1978)Google Scholar
  34. 34.
    Villars, F.M.H.: In: Ripka, G., Porneuf, M. (eds.) Proceedings of the International Conference on Nuclear Selfconsistent Fields, Trieste, 1975. North Holland, Amsterdam (1975)Google Scholar
  35. 35.
    Villars, F.M.H.: Nucl. Phys. A285, 269 (1977)ADSMathSciNetCrossRefGoogle Scholar
  36. 36.
    Goeke, K., Reinhard, P.G., Ann. Phys. (N.Y.) 124, 249 (1980)Google Scholar
  37. 37.
    Baran, A., Sheikh, J.A., Dobaczewski, J., Nazarewicz, W., Staszczak, A.: Phys. Rev. C 84, 054321 (2011)Google Scholar
  38. 38.
    Baran, A., Kowal, M., Reinhard, P.-G., Robledo, L.M., Staszczak, A., Warda, M.: Nucl. Phys. A944, 442 (2015)ADSCrossRefGoogle Scholar
  39. 39.
    Reinhard, P.-G., Goeke, K.: Phys. Rev. C20, 1546 (1979)ADSGoogle Scholar
  40. 40.
    Holzwarth, G., Yukawa, T.: Nucl. Phys. A219, 125 (1974)ADSCrossRefGoogle Scholar
  41. 41.
    Krieger, S.J., Goeke, K.: Nucl. Phys. A 234, 269 (1974)Google Scholar
  42. 42.
    Brink, D.M., Giannoni, M.J., Veneroni, M.: Nucl. Phys. A 258, 237 (1976)Google Scholar
  43. 43.
    Goeke, K., Reinhard, P.G.: Ann. Phys. 112, 328 (1978)ADSCrossRefGoogle Scholar
  44. 44.
    Giannoni, M.J., Quentin, P.: Phys. Rev. C 21, 2060 (1980)Google Scholar
  45. 45.
    Giannoni, M.J., Quentin, P.: Phys. Rev. C 21, 2076 (1980)Google Scholar
  46. 46.
    Dobaczewski, J., Skalski, J.: Nucl. Phys. A 369, 123 (1981)Google Scholar
  47. 47.
    Grümmer, F., Goeke, K., Reinhardt, P.-G.: Quantized ATDHF: theory and realistic applications to heavy ion fusion. In: Goeke, K., Reinhard, P.-G. (eds.) Time-Dependent Hartree-Fock and Beyond: Proceedings of the International Symposium Held in Bad Honnef, Germany, 7–11 June 1982. Lecture Notes in Physics, vol. 171, pp. 323. Springer, Berlin/Heidelberg (1982)Google Scholar
  48. 48.
    Goeke, K., Grümmer, F., Reinhard, P.G.: Ann. Phys. 150, 504 (1983)ADSCrossRefGoogle Scholar
  49. 49.
    Nazarewicz, W.: Nucl. Phys. A 557, 489c (1993)Google Scholar
  50. 50.
    Goutte, H., Berger, J.F., Casoli, P., Gogny, D.: Phys. Rev. C 71, 024316 (2005)Google Scholar
  51. 51.
    Skalski, J.: Phys. Rev. C 77, 064610 (2008)Google Scholar
  52. 52.
    Warda, M., Egido, J., Robledo, L.M., Pomorski, K.: Phys. Rev. C 66, 014310 (2002)Google Scholar
  53. 53.
    Schindzielorz, N., Erler, J., Klüpfel, P., Reinhard, P.-G., Hager, G.: Int. J. Mod. Phys. E 18, 773 (2009)ADSCrossRefGoogle Scholar
  54. 54.
    Inglis, D.R.: Phys. Rev. 103, 1786 (1956)ADSCrossRefGoogle Scholar
  55. 55.
    Belyaev, S.T.: Mat. Fys. Medd. Dan. Vid. Selsk. 31, 11 (1959)Google Scholar
  56. 56.
    Bès, D.R., Szymański, Z.: Nucl. Phys. 28, 42 (1961)CrossRefGoogle Scholar
  57. 57.
    Nilsson, S.G., Tsang, C.-F., Sobiczewski, A., Szymański, Z., Wycech, S., Gustafson, C., Lamm, I.-L., Möller, P., Nilsson, B.: Nucl. Phys. A 131, 1 (1969)Google Scholar
  58. 58.
    Sobiczewski, A., Szymański, Z., Wycech, S., Nilsson, S.G., Nix, J.R., Tsang, C.F., Gustafson, C., Möller, P., Nilsson, B.: Nucl. Phys. A 131, 67 (1969)Google Scholar
  59. 59.
    Brack, M., Damgård, J., Jensen, A.S., Pauli, H.C., Strutinsky, V.M., Wong, C.Y.: Rev. Mod. Phys. 44, 320 (1972)ADSCrossRefGoogle Scholar
  60. 60.
    Inglis, D.R.: Phys. Rev. 96, 1059 (1954)ADSCrossRefGoogle Scholar
  61. 61.
    Yuldashbaeva, E.Kh., Libert, J., Quentin, P., Girod, M.: Phys. Lett. B461 1 (1999)ADSCrossRefGoogle Scholar
  62. 62.
    Goeke, K., Grümmer, F., Reinhard, P.G.: Z. Phys. A317, 339 (1984)ADSGoogle Scholar
  63. 63.
    Thouless, D.J., Valatin, J.G.: Nucl. Phys. 31, 211 (1962)CrossRefGoogle Scholar
  64. 64.
    Girod, M., Delaroche, J.-P.M., Berger, J.-F., Libert, J.: Phys. Lett. B 325, 1 (1994)Google Scholar
  65. 65.
    Terasaki, J., Heenen, P.-H., Bonche, P., Dobaczewski, J., Flocard, H.: Nucl. Phys A593, 1 (1995)ADSCrossRefGoogle Scholar
  66. 66.
    Egido, J.L., Robledo, L.M.: Phys. Rev. Lett. 85, 1198 (2000)ADSCrossRefGoogle Scholar
  67. 67.
    Duguet, T., Bonche, P., Heenen, P.-H.: Nucl. Phys. A679, 427 (2001)ADSCrossRefGoogle Scholar
  68. 68.
    Laftchiev, H., Samsoen, D., Quentin, P., Piperova, J.: Eur. Phys. J. A12, 155 (2001)ADSCrossRefGoogle Scholar
  69. 69.
    Afanasjev, A.V., Khoo, T.L., Frauendorf, S., Lalazissis, G.A., Ahmad, I.: Phys. Rev. C 67, 024309 (2003)Google Scholar
  70. 70.
    L. Próchniak, Quentin, P., Samsoen, D., Libert, J.: Nucl. Phys. A 730, 59 (2004)Google Scholar
  71. 71.
    Bianco, D., Knapp, F., Lo Iudice, N., Veselý, P., Andreozzi, F., De Gregorio, G., Porrino, A.: J. Phys. G 41, 025109 (2014)ADSCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Walid Younes
    • 1
  • Daniel Marc Gogny
    • 2
  • Jean-François Berger
    • 3
  1. 1.Department of PhysicsLawrence Livermore National LaboratoryLivermoreUSA
  2. 2.Lawrence Livermore National LaboratLivermoreUSA
  3. 3.Centre DAM-Ile de FranceRetired from Commissariat à l’Energie AtomiqueBruyères-le-Châtel, ArpajonFrance

Personalised recommendations