nf_angularMomentumCoupling.cc File Reference

#include <stdlib.h>
#include <cmath>
#include "nf_specialFunctions.h"

Go to the source code of this file.

Functions
double	nf_amc_log_factorial (int n)
double	nf_amc_factorial (int n)
double	nf_amc_wigner_3j (int j1, int j2, int j3, int j4, int j5, int j6)
double	nf_amc_wigner_6j (int j1, int j2, int j3, int j4, int j5, int j6)
double	nf_amc_wigner_9j (int j1, int j2, int j3, int j4, int j5, int j6, int j7, int j8, int j9)
double	nf_amc_racah (int j1, int j2, int l2, int l1, int j3, int l3)
double	nf_amc_clebsh_gordan (int j1, int j2, int m1, int m2, int j3)
double	nf_amc_z_coefficient (int l1, int j1, int l2, int j2, int s, int ll)
double	nf_amc_zbar_coefficient (int l1, int j1, int l2, int j2, int s, int ll)
double	nf_amc_reduced_matrix_element (int lt, int st, int jt, int l0, int j0, int l1, int j1)

Function Documentation

nf_amc_clebsh_gordan()

double nf_amc_clebsh_gordan	(	int	j1,
		int	j2,
		int	m1,
		int	m2,
		int	j3
	)

Definition at line 288 of file nf_angularMomentumCoupling.cc.

                                                                      {
/*
*       Clebsh-Gordan coefficient 
*           = <j1,j2,m1,m2|j3,m1+m2>
*           = (-)^(j1-j2+m1+m2) * std::sqrt(2*j3+1) * / j1 j2   j3   \
*                                                \ m1 m2 -m1-m2 /
*
*   Note: Last value m3 is preset to m1+m2.  Any other value will evaluate to 0.0.
*/
 
    int m3, x1, x2, x3, y1, y2, y3;
    double cg = 0.0;
 
    if ( j1 < 0 || j2 < 0 || j3 < 0) return( 0.0 );
    if ( j1 + j2 + j3 > 2 * MAX_FACTORIAL ) return( INFINITY );
 
    m3 = m1 + m2;
 
    if ( ( x1 = ( j1 + m1 ) / 2 + 1 ) <= 0 ) return( 0.0 );
    if ( ( x2 = ( j2 + m2 ) / 2 + 1 ) <= 0 ) return( 0.0 );
    if ( ( x3 = ( j3 - m3 ) / 2 + 1 ) <= 0 ) return( 0.0 );
 
    if ( ( y1 = x1 - m1 ) <= 0 ) return( 0.0 );
    if ( ( y2 = x2 - m2 ) <= 0 ) return( 0.0 );
    if ( ( y3 = x3 + m3 ) <= 0 ) return( 0.0 );
 
    if ( j3 == 0 ){
        if ( j1 == j2 ) cg = ( 1.0 / std::sqrt( (double)j1 + 1.0 ) * ( ( y1 % 2 == 0 ) ? -1:1 ) );
    }
    else if ( (j1 == 0 || j2 == 0 ) ){
        if ( ( j1 + j2 ) == j3 ) cg = 1.0;
    }
    else {
        if( m3 == 0 && std::abs( m1 ) <= 1 ){
            if( m1 == 0 ) cg = cg1( x1, x2, x3 );
            else          cg = cg2( x1 + y1 - y2, x3 - 1, x1 + x2 - 2, x1 - y2, j1, j2, j3, m2 );
        }
        else if ( m2 == 0 && std::abs( m1 ) <=1 ){
                          cg = cg2( x1 - y2 + y3, x2 - 1, x1 + x3 - 2, x3 - y1, j1, j3, j3, m1 );
        }
        else if ( m1 == 0 && std::abs( m3 ) <= 1 ){
                          cg = cg2( x1, x1 - 1, x2 + x3 - 2, x2 - y3, j2, j3, j3, -m3 );
        }
        else              cg = cg3( x1, x2, x3, y1, y2, y3 );
    }
 
    return( cg );
}

Referenced by nf_amc_reduced_matrix_element(), nf_amc_wigner_3j(), nf_amc_z_coefficient(), and nf_amc_zbar_coefficient().

nf_amc_factorial()

double nf_amc_factorial

(

int

n

)

Definition at line 96 of file nf_angularMomentumCoupling.cc.

                                  {
/*
*       returns n! for pre-computed table. INFINITY is return if n is negative or too large.
*/
    return G4Exp( nf_amc_log_factorial( n ) );
}

nf_amc_log_factorial()

double nf_amc_log_factorial

(

int

n

)

Definition at line 85 of file nf_angularMomentumCoupling.cc.

                                      {
/*
*       returns ln( n! ).
*/
    if( n > MAX_FACTORIAL ) return( INFINITY );
    if( n < 0 ) return( INFINITY );
    return nf_amc_log_fact[n];
}

Referenced by nf_amc_factorial().

nf_amc_racah()

double nf_amc_racah	(	int	j1,
		int	j2,
		int	l2,
		int	l1,
		int	j3,
		int	l3
	)

Definition at line 253 of file nf_angularMomentumCoupling.cc.

                                                                      {
/*
*       Racah coefficient definition in Edmonds (AR Edmonds, "Angular Momentum in Quantum Mechanics", Princeton (1980) is
*       W(j1, j2, l2, l1 ; j3, l3) = (-1)^(j1+j2+l1+l2) * { j1 j2 j3 }
*                                                         { l1 l2 l3 }
*       The call signature of W(...) appears jumbled, but hey, that's the convention.
*
*       This convention is exactly that used by Blatt-Biedenharn (Rev. Mod. Phys. 24, 258 (1952)) too
*/
 
    double sig;
 
    sig = ( ( ( j1 + j2 + l1 + l2 ) % 4 == 0 ) ?  1.0 : -1.0 );
    return sig * nf_amc_wigner_6j( j1, j2, j3, l1, l2, l3 );
}

Referenced by nf_amc_wigner_9j(), nf_amc_z_coefficient(), and nf_amc_zbar_coefficient().

nf_amc_reduced_matrix_element()

double nf_amc_reduced_matrix_element	(	int	lt,
		int	st,
		int	jt,
		int	l0,
		int	j0,
		int	l1,
		int	j1
	)

Definition at line 473 of file nf_angularMomentumCoupling.cc.

                                                                                               {
/*
*       Reduced Matrix Element for Tensor Operator
*           = < l1j1 || T(YL,sigma_S)J || l0j0 >
*
*   M.B.Johnson, L.W.Owen, G.R.Satchler
*   Phys. Rev. 142, 748 (1966)
*   Note: definition differs from JOS by the factor sqrt(2j1+1)
*/
    int    llt;
    double x1, x2, x3, reduced_mat, clebsh_gordan;
 
    if ( parity( lt ) != parity( l0 ) * parity( l1 ) ) return( 0.0 );
    if ( std::abs( l0 - l1 ) > lt || ( l0 + l1 ) < lt ) return( 0.0 );
    if ( std::abs( ( j0 - j1 ) / 2 ) > jt || ( ( j0 + j1 ) / 2 ) < jt ) return( 0.0 );
 
    llt = 2 * lt;
    jt *= 2;
    st *= 2;
 
    if( ( clebsh_gordan = nf_amc_clebsh_gordan( j1, j0, 1, -1, jt ) ) == INFINITY ) return( INFINITY );
 
    reduced_mat = 1.0 / std::sqrt( 4 * M_PI ) * clebsh_gordan / std::sqrt( jt + 1.0 )         /* BRB jt + 1 <= 0? */
                * std::sqrt( ( j0 + 1.0 ) * ( j1 + 1.0 ) * ( llt + 1.0 ) )
                * parity( ( j1 - j0 ) / 2 ) * parity( ( -l0 + l1 + lt ) / 2 ) * parity( ( j0 - 1 ) / 2 );
 
    if( st == 2 ){
        x1 = ( l0 - j0 / 2.0 ) * ( j0 + 1.0 );
        x2 = ( l1 - j1 / 2.0 ) * ( j1 + 1.0 );
        if ( jt == llt ){
            x3 = ( lt == 0 ) ? 0 :  ( x1 - x2 ) / std::sqrt( lt * ( lt + 1.0 ) );
        }
        else if ( jt == ( llt - st ) ){
            x3 = ( lt == 0 ) ? 0 : -( lt + x1 + x2 ) / std::sqrt( lt * ( 2.0 * lt + 1.0 ) );
        }
        else if ( jt == ( llt + st ) ){
            x3 = ( lt + 1 - x1 - x2 ) / std::sqrt( ( 2.0 * lt + 1.0 ) * ( lt + 1.0 ) );
        }
        else{
            x3 = 1.0;
        }
    }
    else x3 = 1.0;
    reduced_mat *= x3;
 
    return( reduced_mat );
}

nf_amc_wigner_3j()

double nf_amc_wigner_3j	(	int	j1,
		int	j2,
		int	j3,
		int	j4,
		int	j5,
		int	j6
	)

Definition at line 105 of file nf_angularMomentumCoupling.cc.

                                                                          {
/*
*       Wigner's 3J symbol (similar to Clebsh-Gordan)
*           = / j1 j2 j3 \
*             \ j4 j5 j6 /
*/
    double cg;
 
    if( ( j4 + j5 + j6 ) != 0 ) return( 0.0 );
    if( ( cg = nf_amc_clebsh_gordan( j1, j2, j4, j5, j3 ) ) == 0.0 ) return ( 0.0 );
    if( cg == INFINITY ) return( cg );
    return( ( ( ( j1 - j2 - j6 ) % 4 == 0 ) ?  1.0 : -1.0 ) * cg / std::sqrt( j3 + 1.0 ) );          /* BRB j3 + 1 <= 0? */
}

nf_amc_wigner_6j()

double nf_amc_wigner_6j	(	int	j1,
		int	j2,
		int	j3,
		int	j4,
		int	j5,
		int	j6
	)

Definition at line 121 of file nf_angularMomentumCoupling.cc.

                                                                          {
/*
*       Wigner's 6J symbol (similar to Racah)
*               = { j1 j2 j3 }
*                 { j4 j5 j6 }
*/
    int i, x[6];
 
    x[0] = j1; x[1] = j2; x[2] = j3; x[3] = j4; x[4] = j5; x[5] = j6;
    for( i = 0; i < 6; i++ ) if ( x[i] == 0 ) return( w6j0( i, x ) );
 
    return( w6j1( x ) );
}

Referenced by nf_amc_racah().

nf_amc_wigner_9j()

double nf_amc_wigner_9j	(	int	j1,
		int	j2,
		int	j3,
		int	j4,
		int	j5,
		int	j6,
		int	j7,
		int	j8,
		int	j9
	)

Definition at line 226 of file nf_angularMomentumCoupling.cc.

                                                                                                  {
/*
*       Wigner's 9J symbol
*             / j1 j2 j3 \
*           = | j4 j5 j6 |
*             \ j7 j8 j9 /
*
*/
    int i, i0, i1;
    double rac;
 
    i0 = max3( std::abs( j1 - j9 ), std::abs( j2 - j6 ), std::abs( j4 - j8 ) );
    i1 = min3(     ( j1 + j9 ),     ( j2 + j6 ),     ( j4 + j8 ) );
 
    rac = 0.0;
    for ( i = i0; i <= i1; i += 2 ){ 
        rac += nf_amc_racah( j1, j4, j9, j8, j7,  i )
            *  nf_amc_racah( j2, j5,  i, j4, j8, j6 )
            *  nf_amc_racah( j9,  i, j3, j2, j1, j6 ) * ( i + 1 );
        if( rac == INFINITY ) return( INFINITY );
    }
 
    return( ( ( (int)( ( j1 + j3 + j5 + j8 ) / 2 + j2 + j4 + j9 ) % 4 == 0 ) ?  1.0 : -1.0 ) * rac );
}

nf_amc_z_coefficient()

double nf_amc_z_coefficient	(	int	l1,
		int	j1,
		int	l2,
		int	j2,
		int	s,
		int	ll
	)

Definition at line 437 of file nf_angularMomentumCoupling.cc.

                                                                             {
/*
*       Biedenharn's Z-coefficient coefficient
*           =  Z(l1  j1  l2  j2 | S L )
*/
    double z, clebsh_gordan = nf_amc_clebsh_gordan( l1, l2, 0, 0, ll ), racah = nf_amc_racah( l1, j1, l2, j2, s, ll );
 
    if( ( clebsh_gordan == INFINITY ) || ( racah == INFINITY ) ) return( INFINITY );
    z = ( ( ( -l1 + l2 + ll ) % 8 == 0 ) ? 1.0 : -1.0 )
        * std::sqrt( l1 + 1.0 ) * std::sqrt( l2 + 1.0 ) * std::sqrt( j1 + 1.0 ) * std::sqrt( j2 + 1.0 ) * clebsh_gordan * racah;
 
   return( z );
}

nf_amc_zbar_coefficient()

double nf_amc_zbar_coefficient	(	int	l1,
		int	j1,
		int	l2,
		int	j2,
		int	s,
		int	ll
	)

Definition at line 453 of file nf_angularMomentumCoupling.cc.

                                                                                {
/*
*       Lane & Thomas's Zbar-coefficient coefficient
*           = Zbar(l1  j1  l2  j2 | S L )
*           = (-i)^( -l1 + l2 + ll ) * Z(l1  j1  l2  j2 | S L )
*
*       Lane & Thomas Rev. Mod. Phys. 30, 257-353 (1958).
*       Note, Lane & Thomas define this because they did not like the different phase convention in Blatt & Biedenharn's Z coefficient.  They changed it to get better time-reversal behavior.
*       Froehner uses Lane & Thomas convention as does T. Kawano.
*/
    double zbar, clebsh_gordan = nf_amc_clebsh_gordan( l1, l2, 0, 0, ll ), racah = nf_amc_racah( l1, j1, l2, j2, s, ll );
 
    if( ( clebsh_gordan == INFINITY ) || ( racah == INFINITY ) ) return( INFINITY );
    zbar = std::sqrt( l1 + 1.0 ) * std::sqrt( l2 + 1.0 ) * std::sqrt( j1 + 1.0 ) * std::sqrt( j2 + 1.0 ) * clebsh_gordan * racah;
 
   return( zbar );
}