Simplified recursive algorithm for Wigner<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mn>3</mml:mn><mml:mi>j</mml:mi></mml:math>and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mn>6</mml:mn><mml:mi>j</mml:mi></mml:math>symbols

Luscombe, James H.; Luban, Marshall

doi:10.1103/physreve.57.7274

Cited by 32 publications

(27 citation statements)

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“…The 3-j symbols and Gaunt coefficients admit a number of recurrence relation and their computation is a routine task in physics (e.g. Luscombe & Luban 1998;Xu 1996;Sébilleau 1998). A reliable software for the 3-j coefficients can be found in the open source SHTOOLS 1 package, although major computer algebra systems implement the 3-j symbols as one of standard functions.…”

Section: Discussionmentioning

confidence: 99%

“…They can be evaluated by means of the simple recurrence The Gaunt coefficients present in the final expressions are defined in terms of the Wigner 3‐j symbols in , so and The 3‐j symbols and Gaunt coefficients admit a number of recurrence relation and their computation is a routine task in physics (e.g. Luscombe & Luban 1998; Xu 1996; Sébilleau 1998). A reliable software for the 3‐j coefficients can be found in the open source shtools 1 package, although major computer algebra systems implement the 3‐j symbols as one of standard functions.…”

Section: Mean Values Of the Yorp Torque Projectionsmentioning

confidence: 99%

“…From the three-term recurrence identities related with the Wigner 3-j symbols (Luscombe & Luban 1998), we can deduce useful relation (l 3 (l 3 + 1) − l 2 (l 2 + 1) − l 1 (l 1 + 1)) G m,0,−m l 1 ,l 2 ,l 3 = l 2 (l 2 + 1) (l 1 − m)(l 1 + m + 1)G m+1,−1,−m l 1 ,l 2 ,l 3 + (l 1 + m)(l 1 − m + 1)G m−1,+1,−m l 1 ,l 2 ,l 3 .…”

Section: A P P E N D I X a : S P E C I A L F U N C T I O N Smentioning

confidence: 99%

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YORP torque as the function of shape harmonics

Breiter

Michalska

2008

Monthly Notices of the Royal Astronomical Society

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The second order analytical approximation of the mean YORP torque components is given as an explicit function of the shape spherical harmonics coefficients for a sufficiently regular minor body. The results are based upon a new expression for the insolation function, significantly simpler than in previous works. Linearized plane parallel model of the temperature distribution derived from the insolation function allows to take into account a nonzero conductivity. Final expressions for the three average components of the YORP torque related with rotation period, obliquity, and precession are given in a form of Legendre series of the cosine of obliquity. The series have good numerical properties and can be easily truncated according to the degree of Legendre polynomials or associated functions, with first two terms playing the principal role. The present version fixes the errors discovered in the text that appeared in Monthly Notices RAS (388, pp. 297-944).Comment: 19 pages, 3 figures, published Mon. Not. R.A.S with minor errors that are corrected in the present versio

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Section: Discussionmentioning

confidence: 99%

Section: Mean Values Of the Yorp Torque Projectionsmentioning

confidence: 99%

Section: A P P E N D I X a : S P E C I A L F U N C T I O N Smentioning

confidence: 99%

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YORP torque as the function of shape harmonics

Breiter

Michalska

2008

Monthly Notices of the Royal Astronomical Society

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“…where the Wigner 3j symbol is calculated through recursion [43]. As the only term dependent on the distance between the spherical centers, h, is the spherical Bessel function j w (h), the derivatives of the interaction coefficients are not much more complicated.…”

Section: Appendixmentioning

confidence: 99%

Mie scattering by soft core-shell particles and its applications to ellipsometric light scattering

Ross

Sigel

2012

Phys. Rev. E

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Through the use of Mie theory generalized to multiple spheres, the derivatives of the scattering matrix elements and ellipsometric scattering variables are found as a function of shell thickness and nonconcentricity for core-shell particles. In particular, for the case of a core-shell sphere system where the centers are not concentric, the derivatives are taken with respect to the line segment describing the distance between spherical centers. The derivatives of the scattering matrix elements can be used to calculate the changes in ellipsometric light scattering, allowing for sensitivity and precision in quantitative models of fluctuations in core-shell systems. Computed results giving model contrast for a variety of sizes and fluctuation modes are used to design and guide novel light-scattering experiments currently underway.

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“…It can be better, in the case of large arguments in the 9j symbols, to work with the ratio of consecutive elements [24] …”

Section: Defining For Instancementioning

confidence: 99%

A note on recursive calculations of particular 9jcoefficients

Pain¹

2011

Lith. J. Phys.

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The calculation of angular-momentum coupling transformation matrices can be very time consuming and alternative methods, even if they apply only in special cases, are helpful. We present a recursion relation for the calculation of particular 9j symbols used in the quantum theory of angular momentum.Keywords: angular momentum, 9j coefficient, recursion relation 03.65.Fd, The coupling of N angular momenta is related to the definition of a 3(N − 1)j symbol. The 9j symbols, characterizing the coupling of four angular momenta, are involved for instance in the computation of the matrix elements of the products of tensor operators, and in the transformation from LS to jj coupling [1,2]. The triple sum series of Jucys and Bandzaitis [3] is the simplest known algebraic form for the 9j coefficient. Fourty years ago, Ališauskas and Jucys [4][5][6][7][8] derived an algebraic expression, in which 9j symbols are written as the threefold summation of multiplications and divisions of factorials. More recently, Wei [9,10] proposed to express 9j symbols as a summation of the products of binomial coefficients and devised an algorithm to calculate the binomial coefficients recursively. Although some relationships between particular 9j symbols have been obtained in special cases [11], it is quite difficult to find recursion formulas for the 9j symbol itself, due to the fact that it contains nine arguments. PACSThe general recursion relations for arbitrary 9j symbols were introduced in [12], [3] (with incorrect phase factors), and in [13] (in corrected form). However, as a rule in these relations, several angular momentum parameters are changing. It seems that relations with changing single momentum parameter are possible only if some other parameters of the 9j accept extreme or fixed small values. For instance, the following 9j symbol:

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Simplified recursive algorithm for Wigner3jand6jsymbols

Cited by 32 publications

References 5 publications

YORP torque as the function of shape harmonics

YORP torque as the function of shape harmonics

Mie scattering by soft core-shell particles and its applications to ellipsometric light scattering

A note on recursive calculations of particular 9jcoefficients

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