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A package to calculate CG-coefficient, Racha coefficient, and Wigner 3j, 6j, 9j symbols. We offer three version API for all these coeddicients.

  • normal version: CG, threeJ, SixJ, Racah, nineJ. Their parameters can be Integer or Rational (half integer), and their result is simplified by default.
  • double parameters version: dCG, d3j, d6j, dRacah d9j. Their parameters means double of the real angular momentum, so can only be Integer. The result is not simplified. We will explain what is simplify later.
  • float version: fCG, f3j, f6j, fRacah, f9j. Their parameters is same as double parameter version. They use Float64 for calculation, so they are not exact but are fast for numeric calculation. Because these function use stored binomial result for speed up calculation, you should reserve space before calculate large angular momentum coefficients. See wigner_init_float function for details.

Install

Just install with Julia REPL and enjoy it.

pkg> add CGcoefficient

Usage

using CGcoefficient
sixJ(1,2,3,4,5,6)

\[\frac{1}{3}\sqrt{\frac{2}{715}}\]

In a markdown enviroment, such as jupyter notebook, it will give you a latex output.

d6j(2,4,6,8,10,12)

\[\frac{572}{3}\sqrt{\frac{1}{116968280}}\]

The d version functions do not simplify the result for the seek of speed, because simplify needs prime factorization which is slow. You can simplify the result explicitly,

simplify(d6j(2,4,6,8,10,12))

\[\frac{1}{3}\sqrt{\frac{2}{715}}\]

You can also do some arithmetics with the result, thus do arithmetics using the SqrtRational type. The result is also not simplified

x = sixJ(1,2,3,4,5,6) * exact_sqrt(1//7) * exact_sqrt(1//13) * iphase(2+3+5+6)
simplify(x)

\[\frac{1}{39}\sqrt{\frac{2}{385}}\]

In a console enviroment it will give out a text output.

julia> nineJ(1,2,3,5,4,3,6,6,0)1//39√(2//385)

You can also use print function to force print a text output.

print(Racah(1,2,3,2,1,2))
1//5√(1//21)

About

This package is inspired by Ref [1]. See CENS-MBPT for details.

The idea is to simplify 3nj Symbols to sum combinations of binomial coefficients. We can calculate binomial coefficients by Pascal's Triangle, and store them first. Then we calculate 3nj Symbols using the stored binomial coefficients.

In this package, we just use the builtin binomial function for exact calculation. Only the float version uses stored binomials.

Index