Laplace transforms of the Hulthén Green's function and their application to potential scattering

U. Laha   ; S. Ray   ; S. Panda   ; J. Bhoi

Teoretičeskaâ i matematičeskaâ fizika, Tome 193 (2017) no. 1, pp. 104-114

Résumé

We derive closed-form representations for the single and double Laplace transforms of the Hulthén Green's function of the outgoing wave multiplied by the Yamaguchi potential and write them in the maximally reduced form. We use the expression for the double transform to compute the low-energy phase shifts for the elastic scattering in the systems $\alpha$–nucleon, $\alpha$–$\text{He}^3$, and $\alpha$–$\text{H}^3$. The calculation results agree well with the experimental data.

Détail

Keywords: Hulthén Green's function, integral transform of Hulthén Green's function, scattering phase shift computation, $\alpha$–nucleon system, $\alpha$–nucleus system.

Bibliographie

[1] L. Hulthén, “Über die Eigenlösungen der Schrödinger–Gleichung des Deuterons”, Ark. Mat. Astron. Fysik A, 28:5 (1942), 1–12 | Zbl

[2] Y. Yamaguchi, “Two-nucleon problem when the potential is nonlocal but separable. I”, Phys. Rev., 95:6 (1954), 1628–1634 | DOI | Zbl

[3] R. Nyuton, Teoriya rasseyaniya voln i chastits, Mir, M., 1969 | MR

[4] A. W. Babister, Transcendental Functions Satisfying Non-Homogeneous Linear Differential Equations\, MacMillan, New York, 1967 | MR

[5] L. J. Slater, Generalized Hypergeometric Functions, Cambridge Univ. Press, London, 1966 | MR

[6] G. Beitmen, A. Erdeii, Vysshie transtsendentnye funktsii, v. 1, Mir, Nauka, 1973 | MR

[7] H. van Haeringen, Charged Particle Interactions. Theory and Formulas, The Coulomb Press, Leyden, 1985

[8] U. Laha, C. Bhattacharyya, K. Roy, B. Talukdar, “Hamiltonian hierarchy and the Hulthén potential”, Phys. Rev. C, 38:1 (1988), 558–560 | DOI

[9] W. N. Balley, Generalised Hypergeometric Series, Cambridge Univ. Press, London, 1935

[10] C. S. Warke, R. K. Bhaduri, “On the Jost function of a non-local potential”, Nucl. Phys. A, 162:2 (1971), 289–294 | DOI | MR

[11] B. Talukdar, U. Laha, T. Sasakawa, “Green's function for motion in Coulomb-modified separable nonlocal potentials”, J. Math. Phys, 27:10 (1986), 2080–2086 | DOI | MR | Zbl

[12] U. Laha, B. J. Roy, B. Talukdar, “Transforms of the Coulomb Green function by the form factor of the Graz potential”, J. Phys. A: Math. Gen., 22:17 (1989), 3597–3604 | DOI

[13] B. Talukdar, U. Laha, S. R. Bhattaru, “Double Laplace transform of the Coulomb Green function”, J. Phys. A: Math. Gen., 18:7 (1985), L359–362 | DOI | MR

[14] U. Laha, J. Bhoi, “Off-shell Jost solutions for Coulomb and Coulomb-like interactions in all partial waves”, J. Math. Phys., 54:1 (2013), 013514, 22 pp. | DOI | MR | Zbl

[15] U. Laha, J. Bhoi, “Integral transform of the Coulomb Green's function by the Hankel function and off-shell scattering”, Phys. Rev. C, 88:6 (2013), 064001, 10 pp. | DOI | MR

[16] D. K. Ghosh, S. Saha, K. Niyogi, B. Talukdar, “Laplace transform method for off–shell scattering on nonlocal potentials”, Czech. J. Phys. B, 33:5 (1983), 528–539 | DOI

[17] J. Bhoi, U. Laha, “Integral transforms and their applications to scattering theory”, Int. J. Appl. Phys. Math., 4:6 (2014), 386–405 | DOI

[18] U. Laha, J. Bhoi, “Higher partial-wave potentials from supersymmetry-inspired factorization and nucleon–nucleus elastic scattering”, Phys. Rev. C, 91:3 (2015), 034614, 6 pp. | DOI

[19] J. Bhoi, U. Laha, “Elastic scattering of light nuclei through a simple potential model”, Phys. Atomic Nucl., 79:4 (2016), 370–374 | DOI

[20] P. Mohr, “Low-energy $^3\text{He}(\alpha,\alpha)$–$^3\text{He}$ elastic scattering and the $^3\text{He}(\alpha,\gamma)^7\text{Be}$ reaction”, Phys. Rev. C, 79:6 (2009), 065804, 10 pp. | DOI

[21] J. Dohet-Eraly, D. Baye, “Microscopic cluster model of $\alpha+n$, $\alpha+p$, $\alpha+{}^3\text{He}$, and $\alpha+\alpha$ elastic scattering from a realistic effective nuclear interaction”, Phys. Rev. C, 84:1 (2011), 014604, 13 pp. | DOI

[22] G. R. Satchler, L. W. Owen, A. J. Elwin, G. L. Morgan, R. L. Walter, “Microscopic study of nucleon–${}^4\text{He}$ scattering and effective nuclear potentials”, Nucl. Phys. A, 112:1 (1968), 1–31 | DOI

[23] R. J. Spiger, T. A. Tombrello, “Scattering of $\text{He}^3$ by $\text{He}^4$ and of $\text{He}^4$ by Tritium”, Phys. Rev., 163:4 (1967), 964–984 | DOI

[24] Y. Li Xu, H. Rui Guo, Y. Lu Han, Q. Biao Shen, “Applicability of the systematic helium-3 potential for triton–nucleus reactions”, Internat. J. Modern Phys. E, 24:1 (2015), 1550005, 14 pp. | Zbl

[25] G. L. Morgan, R. L. Walter, “Neutron–Helium Interaction. II. Angular Distributions and phase shifts from 0.2 to 7.0 MeV”, Phys. Rev., 168:4 (1968), 1114–1129 | DOI

[26] W. R. Boykin, S. D. Baker, D. M. Hardy, “Scattering of ${}^3\text{He}$ and ${}^4\text{He}$ from polarized ${}^3\text{He}$ between 4 and 10 MeV”, Nucl. Phys. A, 195:1 (1972), 241–249 | DOI

[27] D. M. Hardy, R. J. Spiger, S. D. Baker, Y. S. Chen, T. A. Tombrello, “Scattering of ${}^3\text{He}$ and ${}^4\text{He}$ from polarized ${}^3\text{He}$ between 7 and 18 MeV”, Nucl. Phys. A, 195:1 (1972), 250–256 | DOI

[28] H. van Haeringen, “Off-shell $T$ matrix for Coulomb plus simple separable potentials for all $l$ in closed form”, J. Math. Phys., 24:5 (1983), 1274–1281 | DOI | MR | Zbl

[29] B. Talukdar, D. K. Ghosh, T. Sasakawa, “Coulomb-modified nuclear scattering. III”, J. Math. Phys, 25:2 (1984), 323–326 | DOI | MR

[30] U. Laha, A. K. Jana, T. K. Nandi, “Phase-function method for Hulthén-modified separable potentials”, Pramana, 37:5 (1991), 387–393 | DOI

[31] C. V. Sukumar, “Supersymmetry, factorisation of the Schrodinger equation and a Hamiltonian hierarchy”, J. Phys. A: Math. Gen., 18:2 (1985), L57–L62 | DOI | MR

[32] J. Lindhard, A. Winther, “Transient fields acting on heavy ions during slowing-down in magnetized materials”, Nucl. Phys. A, 166:3 (1971), 413–435 | DOI

[33] W. D. Kraeft, M. Luft, A. A. Mihajlov, “Scattering properties and electrical conductivity for the Coulomb cut-off potential”, Physica A: Stat. Mech. Appl., 120:1–2 (1983), 263–278 | DOI