Phase function metod for elastic nucleon-nucleon scattering using Hellmann plus Coulomb potential


  • Bidhan Khirali National Institute of Technology Jamshedpur
  • B. Swain National Institute of Technology Jamshedpur
  • A. K. Behera National Institute of Technology Jamshedpur
  • U. Laha National Institute of Technology Jamshedpur



Hellmann potential; phase function method; scattering phase shifts; scattering cross section; (n-p) & (p-p) systems


The phase function method/variable phase approach to potential scattering is exploited to calculate the phase shifts for nucleon-nucleon systems in low and intermediate energy regions by representing nuclear part of the interaction by the Hellmann potential while the electromagnetic part by the Coulomb one.  In addition, the differential and total scattering cross sections are estimated with our phase parameters. Results reproduced by the concerned potential are in good agreement with the previous works in the literature.


B. H. Yazarloo, H. Mehraban, and H. Hassanabadi, Relativistic scattering states of the Hellmann potential, Acta Physica Polonica A, 127 (2015) 684.

A. Arda and R. Sever, PT-/non-PT-symmetric and nonHermitian Hellmann potential: approximate bound and scattering states with any values, Physica Scripta, 89 (2014) 105204.

A. Arda, Analytical solution of two-body spinless Salpeter equation for Hellman potential, Indian Journal of Physics, 91 (2017) 903.

H. Hellmann, A combined approximation procedure for calculation of energies in the problem of many electrons, Acta Physicochim, U.R.S.S. 1 (1935) 913

H. Hellmann, A New Approximation Method in the Problem of Many Electrons, J. Chem. Phys., 3 (1935) 61.

H. Hellmann and W. Kassatotchkin, Metallic Binding According to the Combined Approximation Procedure, J. Chem. Phys., 4 (1936) 324.

H. Tezuka, Positive energy bound state solutions in relativistic Coulomb potential, Jpn. J. Ind. Appl. Math., 14 (1997) 39.

S. H. Dong, On the Bound States of the Dirac Equation with a Coulomb Potential in 2+1 Dimensions, Phys. Scr., 67 (2003) 89.

G. Kocak, O. Bayrad and I. Boztosum, Arbitrary l-state solutions of the Hellmann potential, J. Theor. Comput. Chem., 06 (2007) 893,

R. Dutt, U. Mukherji, Y.P. Varshni, Shifted large-N expansion for the bound states of the Hellmann potential, Phys. Rev. A, 34 (1986) 777.

A. K. Roy, A. F. Jalbout, E. I. Proynov, Accurate calculation of the bound states of Hellmann Potential, J. Math. Chem., 44 (2008) 260.

J. Adamowski, Bound eigenstates for the superposition of the Coulomb and the Yukawa Potentials, Phys. Rev. A, 31 (1985) 43.

S. M. Ikhdair, R. Sever, A perturbative treatment for the bound states of the Hellmann Potential, J. Mol. Struct., 809 (2007) 103.

E.S. William, E.P. Inyang, and E.A. Thompson, Arbitrary -solutions of the Schrödinger equation interacting with Hulthen-Hellmann potential model. Rev. Mex. de Fis., 66 (2020) 730

C. O. Edet, K. O. Okorie, H. Louis, and N. A. NzeataIbe, Any l-state solutions of the Schrödinger equation interacting with Hellmann-Kratzer potential model, Indian J. Phys., 94 (2019) 243. s12648-019-01467-x

C. A. Onate, M. C. Onyeaju, A. N. Ikot and O. Ebomwonyi, Eigen solutions and entropic system for Hellmann potential in the presence of the Schrödinger equation, Eur. Phys. J. Plus, 132 (2017) 462.

R. L. Hall, and Q. D. Katatbeh, Spectral bounds for the Hellmann potential, Phys. Lett. A, 287 (2001) 183.

I. Nasser, M. S. Abdelmonem, and A. Abdel-Hady, Scaling behavior of the Hellmann potential with different strength parameters, Mol. Phys., 112 (2014) 2608.

A. Arda, and R. Sever, Pseudospin and Spin Symmetric Solutions of the Dirac Equation: Hellmann Potential, WeiHua Potential, Varshni Potentia, Zeitschrif fur Naturforschung A, 69 (2014) 163.

M. Hamzavi, K. E. Tylwe, and A. A. Rajabi, Approximate Bound States Solution of the Hellmann Potential, Commun. Theor. Phys., 60 (2013) 1.

C. A. Onate, J. O, A. Ojonubah, E. J. Eweh, and M. Ugboja, Approximate eigen solutions of DKP and Klein-Gordon equations with Hellmann potential, African Rev. Phys., 9 (2014) 497

B. H. Yazarloo, H. Mehraban, and H. Hassanabadi, Relativistic scattering states of the Hellmann potential, Acta Phys. Pol. A, 127 (2015) 684.

S. Hassanabadi, M. Ghominejad, B. H. Yazarloo, M. Solaimani, and H. Hassanabadi, Approximate solution of scattering states of the spinless Salpeter equation with the Yukawa potential, Chinese J. Phys., 52 (2014) 1194.

J. Callaway, P.S. Laghos, Application of the pseudopotential method to atomic scattering, Phys. Rev., 187 (1969) 192.

G. McGinn, Atomic and Molecular Calculations with the Pseudopotential Method. VII One ValenceElectron Photoionization Cross Sections, J. Chem. Phys., 53 (1970) 3635.

V. K. Gryaznov et al., Thermal Properties of Working Media of a Gas-Phase Nuclear Reactor, Eh. Eksp. Teor. Fiz., 78 (1980) 573

J.N. Das, and S. Chakravarty, Atomic inner-she11 ionization, J. Phys. Rev. A, 32 (1985) 176.

Y.P. Varshni and R.C. Shukla, Alkyle hydride molecules: Potential energy curves and the nature of their binding, Rev. Mod. Phys. 35 (1963) 130.

J. G. Philips and L. Kleinmann, New method for calculating wave functions in crystals and molecules, Phys. Rev. A, 116 (1959) 287.

A. J. Hughes and J. Callaway, Energy Bands in Body-Centered and Hexagonal Sodium, Phys. Rev. A, 136 (1964) 1390.

F. Calogero, Variable Phase Approach to Potential Scattering (New York: Academic, (1967)

U. Laha, A. K. Jana, and T. K. Nandi, Phase-function method for Hulth n-modified searable potentials, Pramana - J. Phys., 37 (1991) 387.

J. Bhoi, R. Upadhyay, and U. Laha, Parameterization of Nuclear Hulthen Potential for Nucleus-Nucleus Elastic Scattering, ´ Commun. Theor. Phys., 69 (2018) 203.

U. Laha, and J. Bhoi, Parameterization of the nuclear Hulthen potentials, Phys. At. Nucl., 79 (2016) 62.

P. Sahoo, A. K. Behera, B. Khirali, and U. Laha, Nuclear Hulthen potentials for F and G Partial waves Research & Reviews: J. Phys., 10 (2021) 31.

A. K. Behera, J. Bhoi, U. Laha, and B. Khirali, Commun. Behera, Study of nucleon - nucleon and alpha- nucleon elastic scattering by the Manning - Rosen potential, Theor. Phys. 72 (2020) 075301.

A. K. Jana, J. Pal, T. Nandi, and B. Talukdar, Phase-function method for complex potentials, Pramana- J. Phys. 39 (1992) 501.

A. K. Behera, U. Laha, M. Majumder, and J. Bhoi, EnergyMomentum Dependent Potentials and np Scattering, Research and Reviews: J. Phys. 8 (2019) 2265

A. K. Behera, U. Laha, M. Majumder, and J. Bhoi, Applicability of Phase-Equivalent Energy-Dependent Potential. Case Studies, Phys. At. Nucl. 85 (2022) 124.

J. M. Watson, A Treatise on the Theory of Bessel Functions (London: Cambridge University Press,1922)

R Navarro Perez, J E Amaro and E Ruiz Arriola, The low-energy structure of the nucleon- nucleon interaction: statistical versus systematic uncertainties, J. Phys. G: Nucl. Part. Phys. 43 (2016) 114001.

F. Gross, and A. Stadler, Covariant spectator theory of np scattering : Phase shifts obtained from precision fits to data below 350 MeV, Phys. Rev. C 78 (2008) 014005.

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

J. R. Taylor, Scattering Theory: The Quantum theory on nonrelativistic collisions (Mineola, New York, 1972)

R. G. Newton, Scattering Theory of Waves and Particles (New York: McGraw-Hill, 1982).

C. L. Bailey, W. E. Bennett, T. Bergstralth, R. G. Nuckolls, H. T. Richards, and J. H. Williams, The neutron-proton and neutron-carbon scattering cross sections for fast 28 neutrons, Phys. Rev. 70 (1946) 583.

F. F. Chen, C. P. Leavitt, and A. M., Total p-p and p-n cross sections at cosmotron energies, Phys. Rev. 103 (1956) 211.

R. A. Arndt, W. J. Briscoe, A. B. Laptev, I. I. Strakovskyt, and R. L. Workman, Absolute total np and pp cross-section determinations, Nucl. Sci. Eng. 162 (2009) 312.

B. H. Daub et al.,, Measurements of the neutron-proton and neutron-carbon total cross section from 150 to 800 keV, Phys. Rev. C 87 (2013) 014005.

J. D. Jackson and J. M. Blatt, The interpretation of low energy proton-proton scattering, Reviews of Modern Physics, 22 (1950) 77.

R. J. Slobodrian, H. E. Conzett, E. Shield and W. F. Tivol, Proton-proton elastic scattering between 6 and 10 MeV. Physical Review, 174 (1968) 1122.

A. N. Mitra, In Advances in Nuclear Physics- The Nuclear Three-Body Problem ed. ByM. Baranger and E. Vogt (NewYork: Plenum, 1969) Vol. 3. 1




How to Cite

B. Khirali, B. Swain, A. K. Behera, and U. Laha, “Phase function metod for elastic nucleon-nucleon scattering using Hellmann plus Coulomb potential”, Rev. Mex. Fís., vol. 69, no. 6 Nov-Dec, pp. 061201 1–, Nov. 2023.