Solid Angle Contour Integrals, Series, and Tables

Masket, A. V.

doi:10.1063/1.1746479

Cited by 57 publications

(12 citation statements)

References 3 publications

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“…An analytical expression for Ω circ in terms of elliptic integrals due to Philip A. Macklin (Macklin, 1957) appears included as a footnote in Masket (1957). In the present work we show that also Ω cyl can be reduced to elliptic integrals and give, without derivation, expressions for Ω circ which can be deduced in a akin way and are different from that due to P.A.…”

Section: Introductionmentioning

confidence: 58%

“…1 Partially supported by Fundação para a Ciência e Tecnologia (Programa Praxis XXI -BD/15808/98) being exhaustive we give some examples of such works. Masket (1957) outlined a general procedure based on Stokes theorem to reduce the double integral Ω = sin θdθdϕ to a contour integral in a single variable (θ or ϕ). The method was used to express Ω circ and Ω cyl as single integrals which were numerically integrated.…”

Section: Introductionmentioning

confidence: 99%

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

Prata

2003

Radiation Physics and Chemistry

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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 (Ω cyl ) and the other by either of the end circles (Ω circ ). We derive an expression for Ω cyl in terms of elliptic integrals of the first and third kinds and give similar expressions for Ω 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.

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Section: Introductionmentioning

confidence: 58%

Section: Introductionmentioning

confidence: 99%

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

Prata

2003

Radiation Physics and Chemistry

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“…This problem, principally associated with a great variety of measuring methods developed in fields of nuclear and optical physics, was studied intensively during the middle decades of the current century [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15] and is still being considered with the use of some new analytical approaches applied to the different geometries of a point source and surface shapes. This problem, principally associated with a great variety of measuring methods developed in fields of nuclear and optical physics, was studied intensively during the middle decades of the current century [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15] and is still being considered with the use of some new analytical approaches applied to the different geometries of a point source and surface shapes.…”

Section: Introductionmentioning

confidence: 99%

A method for calculating the average solid angle subtended by a circular disk from uniformly distributed points within a coaxial circular plane

Tryka¹

1999

Review of Scientific Instruments

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Methods for calculating the solid angles subtended at a point by closed contours of some given objects are required in many areas of optical and nuclear physics to estimate the fluxes or particle beams of radiation. It is therefore of interest to derive solid angle equations for various closed contours and the location of such contours with respect to a single point or some points. In this article two methods are described for calculating the solid angles for objects having the shape of a circular disk. In the first method the formula for calculating the solid angle at a point subtended by a circular disk was presented in a more general form than the expressions reported by S. Tryka ͓Opt. Commun. 137, 317 ͑1997͔͒. In the second method this formula was used to derive another formula for calculating the average solid angle subtended by a circular disk from uniformly distributed multiple points lying on the planar circular surface coaxial to the disk. Both of these formulas were represented by superpositions of simple elementary functions with the complete Legendre-Jacobi elliptic integrals. These superpositions were given as dependencies on the radii of the disk and of the circular surface and on the distance between the disk and the surface. The influences of these radii and distance on the solid angle value were illustrated graphically by some three-dimensional surface plots. Some representative results were calculated to eight decimal places and given in tabular form. In addition, a similar table of results computed for the solid angle at a point subtended by a circular disk was included. It was shown that the formulas obtained are simple and directly applicable in some high-language programs. As an example of such an application all plots and calculations in this article were performed using MATHEMATICA 2.2.3 software.

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“…I would like to thank Professor John H. Hubbell for providing a copy of the works by A.V. Masket (Masket, 1957), A.H. Jaffey (Jaffey, 1954) and Hubbell et al (Hubbell et al , 1961). This work was partially supported by Fundação para a Ciência e Tecnologia (Grant BD/15808/98 -Programa Praxis XXI).…”

Section: Acknowledgementsmentioning

confidence: 99%

“…The case of an isotropic point source has been treated to great extent (Jaffey (1954), Macklin (1957), Masket et al (1956), Masket (1957), Gillespie (1970), Gardner and Verghese (1971), Green et al (1974), Prata (2003b)). …”

Section: Introductionmentioning

confidence: 99%

Analytical calculation of the solid angle defined by a cylindrical detector and a point cosine source with parallel axes

Prata¹

2004

Radiation Physics and Chemistry

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We derive analytical expressions for the solid angle subtended by a right finite circular cylinder at a point source with cosine angular distribution in the case where the source direction is parallel to the cylinder axis. As a subsidiary result, an expression for the solid angle subtended by a disc detector at a spread disc source is also provided, in the case where the two discs have a common symmetry axis which is also coincident with the source direction.

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Solid Angle Contour Integrals, Series, and Tables

Cited by 57 publications

References 3 publications

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

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

A method for calculating the average solid angle subtended by a circular disk from uniformly distributed points within a coaxial circular plane

Analytical calculation of the solid angle defined by a cylindrical detector and a point cosine source with parallel axes

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