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Fourier transforms of single-particle wave functions in cylindrical coordinates

  • M. RizeaEmail author
  • N. Carjan
Regular Article - Theoretical Physics

Abstract.

A formalism and the corresponding numerical procedures that calculate the Fourier transform of a single-particle wave function defined on a grid of cylindrical (\(\rho\), z) coordinates is presented. Single-particle states in spherical and deformed nuclei have been chosen in view of future applications in the field of nuclear reactions. Bidimensional plots of the probability that the nucleon's momentum has a given value \(K=\sqrt{k_{\rho}^{2}+k_{z}^{2}}\) are produced and from them the K -distributions are deduced. Three potentials have been investigated: a) a sharp surface spherical well (i.e., of constant depth), b) a spherical Woods-Saxon potential i.e., diffuse surface) and c) a deformed potential of Woods-Saxon type. In the first case the momenta are as well defined as allowed by the uncertainty principle. Depending on the state, their distributions have up to three separated peaks as a consequence of the up to three circular ridges of the bidimensional probabilities plots. In the second case the diffuseness allows very low momenta to be always populated thus creating tails towards the origin (K = 0). The peaks are still present but not well separated. In the third case the deformation transforms the above mentioned circular ridges into ellipses thus spreading the K-values along them. As a consequence the K-distributions have only one broad peak.

References

  1. 1.
    M.K. Banerjee, in Nuclear Spectroscopy, edited by Fay Ajzenberg-Selove (Academic Press, New York, London, 1960) Chapt. 2, p. 695Google Scholar
  2. 2.
    P.G. Hansen, J.A. Tostevin, Annu. Rev. Nucl. Part. Sci. 53, 219 (2003)ADSCrossRefGoogle Scholar
  3. 3.
    C.A. Bertulani, P.G. Hansen, Phys. Rev. C 70, 034609 (2004)ADSCrossRefGoogle Scholar
  4. 4.
    N. Carjan, M. Rizea, Phys. Lett. B 747, 178 (2015)ADSCrossRefGoogle Scholar
  5. 5.
    M.G. Mayer, J.H.D. Jensen, Elementary Theory of Nuclear Shell Structure (Wiley, New York, 1955)Google Scholar
  6. 6.
    S.G. Nilsson, Mat. Fys. Medd. Dan. Vidensk. Selsk. 29, 1 (1955) issue No. 1MathSciNetGoogle Scholar
  7. 7.
    L.D. Landau, E.M. Lifshitz, Quantum Mechanics (Pergamon Press, Oxford, 1977)Google Scholar
  8. 8.
    W. Rosenheinrich, Tables of Some Indefinite Integrals of Bessel Functions (Jena, Germany, 2016) www.eah-jena.de/~rsh/Forschung/Stoer/besint.pdf
  9. 9.
    J.P. Coleman, Comput. Phys. Commun. 21, 109 (1980)ADSCrossRefGoogle Scholar
  10. 10.
    A.J. MacLeod, ACM Trans. Math. Softw. 22, 288 (1996)MathSciNetCrossRefGoogle Scholar
  11. 11.
    R. Piessens, Comput. Phys. Commun. 25, 289 (1982)ADSCrossRefGoogle Scholar
  12. 12.
    V. Magni, G. Cerullo, S. DeSilvestri, J. Opt. Soc. Am. A 9, 2031 (1992)ADSCrossRefGoogle Scholar
  13. 13.
    R. Barakat, B.H. Sandler, Comput. Math. Appl. 40, 1037 (2000)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Rajesh K. Pandey, Vineet K. Singh, Om P. Singh, Commun. Comput. Phys. 8, 351 (2010)MathSciNetGoogle Scholar
  15. 15.
    B. Briggs, E. Henson, The FFT: An Owner's Manual for the Discrete Fourier Transform (SIAM, Philadelphia, 1995)Google Scholar
  16. 16.
    W.H. Press, B.P. Flannery, S.A. Teukolsky, W.T. Vetterling, Numerical Recipes (Cambridge University Press, 1986) pp. 451--453Google Scholar
  17. 17.
    M. Rizea, V. Ledoux, M. Van Daele, G. Vanden Berghe, N. Carjan, Comput. Phys. Commun. 179, 466 (2008)ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    M. Rizea, N. Carjan, Commun. Comput. Phys. 9, 917 (2011)CrossRefGoogle Scholar
  19. 19.
    D. Sorensen, R. Lehoucq, Chao Yang, K. Maschhoff, www.caam.rice.edu/software/ARPACK (1996)
  20. 20.
    M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions, 8th ed. (Dover, New York, 1972) eq. 25.4.14, p. 886Google Scholar
  21. 21.
    N. Carjan, M. Rizea, Phys. Rev. C 82, 014617 (2010)ADSCrossRefGoogle Scholar
  22. 22.
    R. Capote, N. Carjan, S. Chiba, Phys. Rev. C 93, 024609 (2016)ADSCrossRefGoogle Scholar
  23. 23.
    V. Pashkevich, Nucl. Phys. A 169, 275 (1971)ADSCrossRefGoogle Scholar

Copyright information

© SIF, Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.National Institute of Physics and Nuclear Engineering, “Horia Hulubei”BucharestRomania
  2. 2.Joint Institute for Nuclear Research, FLNRDubnaRussia
  3. 3.CENBGUniversity of BordeauxGradignan CedexFrance

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