U501 Beta-Term in Wigner's D-Function

Function subprograms RDJMNB and DDJMNB calculate the $\beta$ -term $d$_mn^j(β) in the matrix element of the finite rotation operator (Wigner's D-function)

$D$_mn^j(α,β,γ) = e^-imα d_mn^j(β) e^inγ

by using the formula (Ref. 1, No. 4.3.1(3))
$d$_mn^j(β) = (-1)^j+m(j+m)!(j-m)!(j+n)!(j-n)! x
$\sum$_k (-1)^k{cos^2k-m-n({12}β) sin^2j+m+n-2k({12}β)k!(j+m-k)!(j+n-k)!(k-m-n)!}
for arbitrary (either all integer or all half-integer) values of j,m,n such that $j\; \ge 0,\; |m|\; \le j$ and $|n|\; \le j$ . The summation over k runs from $max(0,m+n)$ to $min(j+m,j+n)$ .

On computers other than CDC or Cray, only the double-precision version DDJMNB is available. On CDC and Cray computers, only the single-precision version RDJMNB is available.

Structure:

FUNCTION subprograms
User Entry Names: RDJMNB, DDJMNB
Obsolete User Entry Names: DJMNB $\equiv$ RDJMNB
Files Referenced: Unit 6
External References: MTLMTR (N002), ABEND (Z035)

Usage:

In any arithmetic expression, RDJMNB(AJ,AM,AN,BETA) or DDJMNB(AJ,AM,AN,BETA) has the value $d$_mn^j(β) ,

where $AJ=\; j$ , $AM=\; m$ , $AN=\; n$ and $BETA=\; \beta$ . RDJMNB is of type REAL, DDJMNB is of type DOUBLE PRECISION, and AJ, AM, AN, BETA have the same type as the function name. BETA has to be given in degrees.

Restrictions:

$0\; \le AJ\; \le 25$ , $|AM|\; \le AJ$ , $|AN|\; \le AJ$ , $0\; \le BETA\; \le 360$ .

Accuracy:

Approximately full single- or double-precision machine accuracy, at least for small values of the indices.

Error handling:

Error U501.1: If any of the restrictions is not satisfied, the function value is set equal to zero, and a message is written on Unit 6, unless subroutine MTLSET (N002) has been called.

References:

1. D.A. Varshalovich, A.N. Moskalev and V.K. Khersonskii, Quantum theory of angular momentum, (World Scientific, Singapore 1988) 76

Random Numbers and General Purpose Utilities

V100