Abstract
Sum of the squares of the Bessel function and the Neumann function of the same order of half-odd integer has been found to be very useful in addressing a puzzle in nuclear physics. One approximate formula available in the literature is valid for the complex argument whose real part is greater than zero, and the absolute value of error term is undefined for half-odd integers. Another approximate formula which is valid for all complex arguments has been obtained using sophisticated mathematical method called Barnes' method. However, the error in the formula is very difficult to calculate. We have obtained exact formula for the sum of the squares of Bessel and Neumann functions of the same order of half-odd integers which is valid for all complex arguments, and its proof is also given.Keywords: Bessel functions; Neumann functions; Anomalous absorption; Partial waves DOI: 10.4038/josuk.v4i0.2694J Sci.Univ.Kelaniya 4 (2008): 15-20
Highlights
INTRODUCTIONIn case of elastic scattering of neutrons on composite nuclei, it has been found (Kawai & Iseri, 1985) that the S-matrix element becomes zero for a special combination of energy (E), orbital angular momentum ( l ), total angular momentum (j) and composite target nuclei (A)
In case of elastic scattering of neutrons on composite nuclei, it has been found (Kawai & Iseri, 1985) that the S-matrix element becomes zero for a special combination of energy (E), orbital angular momentum ( l ), total angular momentum (j) and composite target nuclei (A). This phenomenon is called anomalous absorption of neutron partial waves by the nuclear optical potential. This phenomenon occurs is case of elastic proton scattering on composite target nuclei (Iseri & Kawai, 1986) and it has been found (Piyadasa, 1985) that this phenomenon is universal for light ion elastic scattering on composite nuclei
In finding the origin of systematic of the anomalous absorption of neutron partial waves by the nuclear optical potential, we have found that exact formula for the sum jl2
Summary
In case of elastic scattering of neutrons on composite nuclei, it has been found (Kawai & Iseri, 1985) that the S-matrix element becomes zero for a special combination of energy (E), orbital angular momentum ( l ), total angular momentum (j) and composite target nuclei (A). This phenomenon is called anomalous absorption of neutron partial waves by the nuclear optical potential. We have obtain exact formula for the sum of the square of the Bessel function and the Neumann function of the same order of half odd integer which is valid for all complex Z, and its proof is given
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