Solid angle subtended by a cylindrical detector at a point source in terms of elliptic inte.pdf

Solid angle subtended by a cylindrical detector at a point source in terms of elliptic inte.pdf

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Solid angle subtended by a cylindrical detector at a point source in terms of elliptic inte

a r X i v : m a t h - p h / 0 2 1 1 0 6 1 v 2 2 F e b 2 0 0 3 Solid angle subtended by a cylindrical detector at a point source in terms of elliptic integrals M. J. Prata 1 Instituto Tecnolo?gico e Nuclear (ITN), Sacave?m, Portugal Abstract The solid angle subtended by a right circular cylinder at a point source located at an arbitrary position generally consists of a sum of two terms: that defined by the cylindrical surface (?cyl) and the other by either of the end circles (?circ). We derive an expression for ?cyl in terms of elliptic integrals of the first and third kinds and give similar expressions for ?circ using integrals of the first and second kinds. These latter can be used alternatively to an expression also in terms of elliptic integrals, due to Philip A. Macklin and included as a footnote in Masket (Rev. Sci. Instr., 28 (3), 191-197, 1957). The solid angle subtended by the whole cylinder when the source is located at an arbitrary location can then be calculated using elliptic integrals. Key words: solid angle, point source, cylindrical detector, cylinder, elliptic integrals 1 Introduction The knowledge of the solid angle (?) subtended by a right, finite, circular cylinder at a point isotropic source is required in numerous problems in nu- clear and radiation physics. Generally, ? can be expressed as sum of two components: that subtended by the cylindrical surface (?cyl) and the other by either of the end circles (?circ). Through the years this calculation has been addressed by various authors using different methods. Without the worry of Email address: mjprata@sapo.pt (M. J. Prata). 1 Partially supported by Fundac?a?o para a Cie?ncia e Tecnologia (Programa Praxis XXI - BD/15808/98) Preprint submitted to Elsevier Science 5 February 2008 being exhaustive we give some examples of such works. Masket (1957) out- lined a general procedure based on Stokes theorem to reduce the double in- tegral ? = ∫∫ sin θdθd? to a contour integral in a single variable (θ

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