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Theory of Hyperspherical Sturmians for Three-Body Reactions†

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Departamento de Física, Universidad Nacional del Sur and Consejo Nacional de Investigaciones Científicas y Técnicas, 8000 Bahía Blanca, Buenos Aires, Argentina
Instituto de Astronomía y Física del Espacio, Consejo Nacional de Investigaciones Científicas y Técnicas and Departamento de Física, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires. C.C. 67, Suc. 28, (C1428EGA) Buenos Aires, Argentina
División Colisiones Atómicas, Centro Atómico Bariloche and Consejo Nacional de Investigaciones Científicas y Técnicas, 8400 S. C. de Bariloche, Río Negro, Argentina
†Part of the “Vincenzo Aquilanti Festschrift”.
* Corresponding author, [email protected]
Cite this: J. Phys. Chem. A 2009, 113, 52, 14573–14582
Publication Date (Web):September 29, 2009
https://doi.org/10.1021/jp9040869
Copyright © 2009 American Chemical Society
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    Abstract

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    In this paper we present a theory to describe three-body reactions. Fragmentation processes are studied by means of the Schrödinger equation in hyperspherical coordinates. The three-body wave function is written as a sum of two terms. The first one defines the initial channel of the collision while the second one describes the scattered wave, which contains all the information about the collision process. The dynamics is ruled by an nonhomogeneous equation with a driven term related to the initial channel and to the three-body interactions. A basis set of functions with outgoing behavior at large values of hyperradius is introduced as products of angular and radial hyperspherical Sturmian functions. The scattered wave is expanded on this basis and the nonhomogeneous equation is transformed into an algebraic problem that can be solved by standard matrix methods. To be able to deal with general systems, discretization schemes are proposed to solve the angular and radial Sturmian equations. This procedure allows these discrete functions to be connected with the hyperquatization algorithm. Finally, the fragmentation transition amplitude is derived from the asymptotic limit of the scattered wave function.

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    Cited By

    This article is cited by 18 publications.

    1. George Rawitscher, Victo dos Santos Filho, Thiago Carvalho Peixoto. Sturmian Functions. 2018, 153-176. https://doi.org/10.1007/978-3-319-42703-4_11
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    3. M. Ambrosio, D. M. Mitnik, G. Gasaneo, J. M. Randazzo, A. S. Kadyrov, D. V. Fursa, I. Bray. Convergent close coupling versus the generalized Sturmian function approach: Wave-function analysis. Physical Review A 2015, 92 (5) https://doi.org/10.1103/PhysRevA.92.052518
    4. M. J. Ambrosio, L. U. Ancarani, D. M. Mitnik, F. D. Colavecchia, G. Gasaneo. A Generalized Sturmian Treatment of (e, 3e) Processes Described as a Three-Body Coulomb Problem. Few-Body Systems 2014, 55 (8-10) , 825-829. https://doi.org/10.1007/s00601-014-0831-5
    5. D M Mitnik, G Gasaneo, L U Ancarani, M J Ambrosio. Collision problems treated with the Generalized Hyperspherical Sturmian method. Journal of Physics: Conference Series 2014, 488 (1) , 012049. https://doi.org/10.1088/1742-6596/488/1/012049
    6. M. J. Ambrosio, G. Gasaneo, F. D. Colavecchia. Insights from the zero-angular-momentum wave in single and double ionization of He by fast electrons. Physical Review A 2014, 89 (1) https://doi.org/10.1103/PhysRevA.89.012713
    7. G. Gasaneo, D. M. Mitnik, J. M. Randazzo, L. U. Ancarani, F. D. Colavecchia. S -model calculations for high-energy-electron-impact double ionization of helium. Physical Review A 2013, 87 (4) https://doi.org/10.1103/PhysRevA.87.042707
    8. D M Mitnik, G Gasaneo, L U Ancarani. Use of generalized hyperspherical Sturmian functions for a three-body break-up model problem. Journal of Physics B: Atomic, Molecular and Optical Physics 2013, 46 (1) , 015202. https://doi.org/10.1088/0953-4075/46/1/015202
    9. G. Gasaneo, L.U. Ancarani, D.M. Mitnik, J.M. Randazzo, A.L. Frapiccini, F.D. Colavecchia. Three-Body Coulomb Problems with Generalized Sturmian Functions. 2013, 153-216. https://doi.org/10.1016/B978-0-12-411544-6.00007-8
    10. J M Randazzo, D M Mitnik, L U Ancarani, G Gasaneo, F D Colavecchia, A L Frapiccini. Hyperspherical versus spherical treatment of asymptotic conditions for three-body scattering problems. Journal of Physics: Conference Series 2012, 388 (2) , 022090. https://doi.org/10.1088/1742-6596/388/2/022090
    11. G Gasaneo, D M Mitnik, L U Ancarani, J M Randazzo, F D Colavecchia, A L Frapiccini. An analytically solvable model to test the hyperspherical Sturmian approach for break up processes. Journal of Physics: Conference Series 2012, 388 (4) , 042028. https://doi.org/10.1088/1742-6596/388/4/042028
    12. L. U. Ancarani, G. Gasaneo, D. M. Mitnik. An analytically solvable three-body break-up model problem in hyperspherical coordinates. The European Physical Journal D 2012, 66 (10) https://doi.org/10.1140/epjd/e2012-30353-4
    13. G Gasaneo, L U Ancarani. A spectral approach based on generalized Sturmian functions for two- and three-body scattering problems. Journal of Physics A: Mathematical and Theoretical 2012, 45 (4) , 045304. https://doi.org/10.1088/1751-8113/45/4/045304
    14. George Rawitscher. Iterative solution of integral equations on a basis of positive energy Sturmian functions. Physical Review E 2012, 85 (2) https://doi.org/10.1103/PhysRevE.85.026701
    15. M J Ambrosio, J A Del Punta, K V Rodriguez, G Gasaneo, L U Ancarani. Mathematical properties of generalized Sturmian functions. Journal of Physics A: Mathematical and Theoretical 2012, 45 (1) , 015201. https://doi.org/10.1088/1751-8113/45/1/015201
    16. J. M. Randazzo, F. Buezas, A. L. Frapiccini, F. D. Colavecchia, G. Gasaneo. Solving three-body-breakup problems with outgoing-flux asymptotic conditions. Physical Review A 2011, 84 (5) https://doi.org/10.1103/PhysRevA.84.052715
    17. John Scales Avery, James Emil Avery. Sturmians and Generalized Sturmians in Quantum Theory. 2011, 53-99. https://doi.org/10.1007/430_2011_53
    18. L U Ancarani, G Gasaneo. Derivatives of any order of the hypergeometric function p F q ( a 1 , …, a p ; b 1 , …, b q ; z ) with respect to the parameters a i and b i . Journal of Physics A: Mathematical and Theoretical 2010, 43 (8) , 085210. https://doi.org/10.1088/1751-8113/43/8/085210
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