59 #include "nf_specialFunctions.h"
61 #if defined __cplusplus
70 static const double nf_amc_log_fact[] = {0.0, 0.0, 0.69314718056, 1.79175946923, 3.17805383035, 4.78749174278, 6.57925121201, 8.52516136107, 10.6046029027, 12.8018274801, 15.1044125731, 17.5023078459, 19.9872144957, 22.5521638531, 25.1912211827, 27.8992713838, 30.6718601061, 33.5050734501, 36.395445208, 39.3398841872, 42.3356164608, 45.3801388985, 48.4711813518, 51.6066755678, 54.7847293981, 58.003605223, 61.261701761, 64.557538627, 67.8897431372, 71.2570389672, 74.6582363488, 78.0922235533, 81.5579594561, 85.0544670176, 88.5808275422, 92.1361756037, 95.7196945421, 99.3306124548, 102.968198615, 106.631760261, 110.320639715, 114.034211781, 117.7718814, 121.533081515, 125.317271149, 129.123933639, 132.952575036, 136.802722637, 140.673923648, 144.565743946, 148.477766952, 152.409592584, 156.360836303, 160.331128217, 164.320112263, 168.327445448, 172.352797139, 176.395848407, 180.456291418, 184.533828861, 188.628173424, 192.739047288, 196.866181673, 201.009316399, 205.168199483, 209.342586753, 213.532241495, 217.736934114, 221.956441819, 226.190548324, 230.439043566, 234.701723443, 238.978389562, 243.268849003, 247.572914096, 251.89040221, 256.22113555, 260.564940972, 264.921649799, 269.291097651, 273.673124286, 278.06757344, 282.474292688, 286.893133295, 291.323950094, 295.766601351, 300.220948647, 304.686856766, 309.16419358, 313.65282995, 318.15263962, 322.663499127, 327.185287704, 331.717887197, 336.261181979, 340.815058871, 345.379407062, 349.954118041, 354.539085519, 359.13420537, 363.739375556, 368.354496072, 372.979468886, 377.614197874, 382.258588773, 386.912549123, 391.575988217, 396.248817052, 400.930948279, 405.622296161, 410.322776527, 415.032306728, 419.7508056, 424.478193418, 429.214391867, 433.959323995, 438.712914186, 443.475088121, 448.245772745, 453.024896238, 457.812387981, 462.608178527, 467.412199572, 472.224383927, 477.044665493, 481.87297923, 486.709261137, 491.553448223, 496.405478487, 501.265290892, 506.132825342, 511.008022665, 515.890824588, 520.781173716, 525.679013516, 530.584288294, 535.49694318, 540.416924106, 545.344177791, 550.278651724, 555.220294147, 560.169054037, 565.124881095, 570.087725725, 575.057539025, 580.034272767, 585.017879389, 590.008311976, 595.005524249, 600.009470555, 605.020105849, 610.037385686, 615.061266207, 620.091704128, 625.128656731, 630.172081848, 635.221937855, 640.27818366, 645.340778693, 650.409682896, 655.484856711, 660.566261076, 665.653857411, 670.747607612, 675.84747404, 680.953419514, 686.065407302, 691.183401114, 696.307365094, 701.437263809, 706.573062246, 711.714725802, 716.862220279, 722.015511874, 727.174567173, 732.339353147, 737.509837142, 742.685986874, 747.867770425, 753.05515623, 758.248113081, 763.446610113, 768.6506168, 773.860102953, 779.07503871, 784.295394535, 789.521141209, 794.752249826, 799.988691789, 805.230438804, 810.477462876, 815.729736304, 820.987231676, 826.249921865, 831.517780024, 836.790779582, 842.068894242, 847.35209797, 852.640365001, 857.933669826, 863.231987192};
73 static int max3(
int a,
int b,
int c );
74 static int max4(
int a,
int b,
int c,
int d );
75 static int min3(
int a,
int b,
int c );
76 static double w6j0(
int,
int * );
77 static double w6j1(
int * );
78 static double cg1(
int,
int,
int );
79 static double cg2(
int,
int,
int,
int,
int,
int,
int,
int );
80 static double cg3(
int,
int,
int,
int,
int,
int );
89 if( n > MAX_FACTORIAL )
return( INFINITY );
90 if( n < 0 )
return( INFINITY );
91 return nf_amc_log_fact[
n];
113 if( ( j4 + j5 + j6 ) != 0 )
return( 0.0 );
115 if( cg == INFINITY )
return( cg );
116 return( ( ( ( j1 - j2 - j6 ) % 4 == 0 ) ? 1.0 : -1.0 ) * cg / std::sqrt( j3 + 1.0 ) );
129 x[0] = j1; x[1] = j2; x[2] = j3; x[3] = j4; x[4] = j5; x[5] = j6;
130 for( i = 0; i < 6; i++ )
if ( x[i] == 0 )
return(
w6j0( i, x ) );
137 static double w6j0(
int i,
int *
x ) {
140 case 0:
if ( ( x[1] != x[2] ) || ( x[4] != x[5] ) )
return( 0.0 );
141 x[5] = x[3]; x[0] = x[1]; x[3] = x[4];
break;
142 case 1:
if ( ( x[0] != x[2] ) || ( x[3] != x[5] ) )
return( 0.0 );
144 case 2:
if ( ( x[0] != x[1] ) || ( x[3] != x[4] ) )
return( 0.0 );
145 if ( x[3] != x[4] )
break;
146 case 3:
if ( ( x[1] != x[5] ) || ( x[2] != x[4] ) )
return( 0.0 );
147 x[5] = x[0]; x[0] = x[4]; x[3] = x[1];
break;
148 case 4:
if ( ( x[0] != x[5] ) || ( x[2] != x[3] ) )
return( 0.0 );
150 case 5:
if ( ( x[0] != x[4] ) || ( x[1] != x[3] ) )
return( 0.0 );
154 if( ( x[5] > ( x[0] + x[3] ) ) || ( x[5] < std::abs( x[0] - x[3] ) ) )
return( 0.0 );
155 if( x[0] > MAX_FACTORIAL || x[3] > MAX_FACTORIAL ) {
159 return( 1.0 / std::sqrt( (
double) ( ( x[0] + 1 ) * ( x[3] + 1 ) ) ) * ( ( ( x[0] + x[3] + x[5] ) / 2 ) % 2 != 0 ? -1 : 1 ) );
167 int i, k, k1, k2,
n, l1, l2, l3, l4, n1, n2, n3, m1,
m2,
m3, x1, x2, x3, y[4];
168 static int a[3][4] = { { 0, 0, 3, 3},
174 for ( k = 0; k < 4; k++ ){
175 x1 = x[ ( a[0][k] ) ];
176 x2 = x[ ( a[1][k] ) ];
177 x3 = x[ ( a[2][k] ) ];
179 n = ( x1 + x2 + x3 ) / 2;
180 if( n > MAX_FACTORIAL ) {
181 return( INFINITY ); }
186 if ( ( n1 = n - x3 ) < 0 )
return( 0.0 );
187 if ( ( n2 = n - x2 ) < 0 )
return( 0.0 );
188 if ( ( n3 = n - x1 ) < 0 )
return( 0.0 );
191 w6j += nf_amc_log_fact[n1] + nf_amc_log_fact[n2] + nf_amc_log_fact[n3] - nf_amc_log_fact[n+1];
194 n1 = ( x[0] + x[1] + x[3] + x[4] ) / 2;
195 n2 = ( x[0] + x[2] + x[3] + x[5] ) / 2;
196 n3 = ( x[1] + x[2] + x[4] + x[5] ) / 2;
198 k1 =
max4( y[0], y[1], y[2], y[3] ) - 1;
199 k2 =
min3( n1, n2, n3 ) + 1;
201 l1 = k1 - y[0] + 1; m1 = n1 - k1 + 1;
202 l2 = k1 - y[1] + 1; m2 = n2 - k1 + 1;
203 l3 = k1 - y[2] + 1; m3 = n3 - k1 + 1;
206 w6j = w =
G4Exp( 0.5 * w6j + nf_amc_log_fact[k1] - nf_amc_log_fact[l1] - nf_amc_log_fact[l2] - nf_amc_log_fact[l3] - nf_amc_log_fact[l4]
207 - nf_amc_log_fact[m1] - nf_amc_log_fact[m2] - nf_amc_log_fact[m3] ) * ( ( k1 % 2 ) == 0 ? -1: 1 );
208 if( w6j == INFINITY )
return( INFINITY );
212 m1 -= k-1; m2 -= k-1; m3 -= k-1;
213 l1 += k ; l2 += k ; l3 += k ; l4 += k;
215 for ( i = 0; i < k; i++ )
216 w6j = w - w6j * ( ( k2 - i ) * ( m1 + i ) * ( m2 + i ) * ( m3 + i ) )
217 / ( ( l1 - i ) * ( l2 - i ) * ( l3 - i ) * ( l4 - i ) );
224 double nf_amc_wigner_9j(
int j1,
int j2,
int j3,
int j4,
int j5,
int j6,
int j7,
int j8,
int j9 ) {
235 i0 =
max3( std::abs( j1 - j9 ), std::abs( j2 - j6 ), std::abs( j4 - j8 ) );
236 i1 =
min3( ( j1 + j9 ), ( j2 + j6 ), ( j4 + j8 ) );
239 for ( i = i0; i <= i1; i += 2 ){
243 if( rac == INFINITY )
return( INFINITY );
246 return( ( ( (
int)( ( j1 + j3 + j5 + j8 ) / 2 + j2 + j4 + j9 ) % 4 == 0 ) ? 1.0 : -1.0 ) * rac );
251 double nf_amc_racah(
int j1,
int j2,
int l2,
int l1,
int j3,
int l3 ) {
263 sig = ( ( ( j1 + j2 + l1 + l2 ) % 4 == 0 ) ? 1.0 : -1.0 );
296 int m3, x1, x2, x3, y1, y2, y3;
299 if ( j1 < 0 || j2 < 0 || j3 < 0)
return( 0.0 );
300 if ( j1 + j2 + j3 > 2 * MAX_FACTORIAL )
return( INFINITY );
304 if ( ( x1 = ( j1 + m1 ) / 2 + 1 ) <= 0 )
return( 0.0 );
305 if ( ( x2 = ( j2 + m2 ) / 2 + 1 ) <= 0 )
return( 0.0 );
306 if ( ( x3 = ( j3 - m3 ) / 2 + 1 ) <= 0 )
return( 0.0 );
308 if ( ( y1 = x1 - m1 ) <= 0 )
return( 0.0 );
309 if ( ( y2 = x2 - m2 ) <= 0 )
return( 0.0 );
310 if ( ( y3 = x3 + m3 ) <= 0 )
return( 0.0 );
313 if ( j1 == j2 ) cg = ( 1.0 / std::sqrt( (
double)j1 + 1.0 ) * ( ( y1 % 2 == 0 ) ? -1:1 ) );
315 else if ( (j1 == 0 || j2 == 0 ) ){
316 if ( ( j1 + j2 ) == j3 ) cg = 1.0;
319 if( m3 == 0 && std::abs( m1 ) <= 1 ){
320 if( m1 == 0 ) cg =
cg1( x1, x2, x3 );
321 else cg =
cg2( x1 + y1 - y2, x3 - 1, x1 + x2 - 2, x1 - y2, j1, j2, j3, m2 );
323 else if ( m2 == 0 && std::abs( m1 ) <=1 ){
324 cg =
cg2( x1 - y2 + y3, x2 - 1, x1 + x3 - 2, x3 - y1, j1, j3, j3, m1 );
326 else if ( m1 == 0 && std::abs( m3 ) <= 1 ){
327 cg =
cg2( x1, x1 - 1, x2 + x3 - 2, x2 - y3, j2, j3, j3, -m3 );
329 else cg =
cg3( x1, x2, x3, y1, y2, y3 );
337 static double cg1(
int x1,
int x2,
int x3 ) {
339 int p1, p2, p3, p4, q1, q2, q3, q4;
342 p1 = x1 + x2 + x3 - 1;
if ( ( p1 % 2 ) != 0 )
return( 0.0 );
346 if ( p2 <= 0 || p3 <= 0 || p4 <= 0 )
return( 0.0 );
347 if ( p1 >= MAX_FACTORIAL )
return( INFINITY );
349 q1 = ( p1 + 1 ) / 2 - 1; p1--;
350 q2 = ( p2 + 1 ) / 2 - 1; p2--;
351 q3 = ( p3 + 1 ) / 2 - 1; p3--;
352 q4 = ( p4 + 1 ) / 2 - 1; p4--;
354 a = nf_amc_log_fact[q1]-( nf_amc_log_fact[q2] + nf_amc_log_fact[q3] + nf_amc_log_fact[q4] )
355 + 0.5 * ( nf_amc_log_fact[ 2 * x3 - 1 ] - nf_amc_log_fact[ 2 * x3 - 2 ]
356 + nf_amc_log_fact[p2] + nf_amc_log_fact[p3] + nf_amc_log_fact[p4] - nf_amc_log_fact[p1] );
358 return( ( ( ( q1 + x1 - x2 ) % 2 == 0 ) ? 1.0 : -1.0 ) *
G4Exp( a ) );
363 static double cg2(
int k,
int q0,
int z1,
int z2,
int w1,
int w2,
int w3,
int mm ) {
365 int q1, q2, q3, q4, p1, p2, p3, p4;
372 if ( p2 <= 0 || p3 <= 0 || p4 <= 0)
return( 0.0 );
373 if ( p1 >= MAX_FACTORIAL )
return( INFINITY );
375 q1 = ( p1 + 1 ) / 2 - 1; p1--;
376 q2 = ( p2 + 1 ) / 2 - 1; p2--;
377 q3 = ( p3 + 1 ) / 2 - 1; p3--;
378 q4 = ( p4 + 1 ) / 2 - 1; p4--;
380 a = nf_amc_log_fact[q1] - ( nf_amc_log_fact[ q2 ] + nf_amc_log_fact[ q3 ] + nf_amc_log_fact[ q4 ] )
381 + 0.5 * ( nf_amc_log_fact[ w3 + 1 ] - nf_amc_log_fact[ w3 ]
382 + nf_amc_log_fact[ w1 ] - nf_amc_log_fact[ w1 + 1 ]
383 + nf_amc_log_fact[ w2 ] - nf_amc_log_fact[ w2 + 1 ]
384 + nf_amc_log_fact[ p2 ] + nf_amc_log_fact[ p3 ] + nf_amc_log_fact[ p4 ] - nf_amc_log_fact[ p1 ] );
386 return( ( ( ( q4 + k + ( mm > 0 ) * ( p1 + 2 ) ) % 2 == 0 ) ? -1.0 : 1.0 ) * 2.0 *
G4Exp( a ) );
391 static double cg3(
int x1,
int x2,
int x3,
int y1,
int y2,
int y3 ) {
393 int nx, i, k1, k2, q1, q2, q3, q4, p1, p2, p3, z1, z2, z3;
396 nx = x1 + x2 + x3 - 1;
397 if ( ( z1 = nx - x1 - y1 ) < 0 )
return( 0.0 );
398 if ( ( z2 = nx - x2 - y2 ) < 0 )
return( 0.0 );
399 if ( ( z3 = nx - x3 - y3 ) < 0 )
return( 0.0 );
404 q1 =
max3( k1, k2, 0 );
405 q2 =
min3( y1, x2, z3 + 1 ) - 1;
413 a = cg =
G4Exp( 0.5 * ( nf_amc_log_fact[ x3 + y3 - 1 ] - nf_amc_log_fact[ x3 + y3 - 2 ] - nf_amc_log_fact[ nx - 1 ]
414 + nf_amc_log_fact[ z1 ] + nf_amc_log_fact[ z2 ] + nf_amc_log_fact[ z3 ]
415 + nf_amc_log_fact[ x1 - 1 ] + nf_amc_log_fact[ x2 - 1 ] + nf_amc_log_fact[ x3 - 1 ]
416 + nf_amc_log_fact[ y1 - 1 ] + nf_amc_log_fact[ y2 - 1 ] + nf_amc_log_fact[ y3 - 1 ] )
417 - nf_amc_log_fact[ p1 ] - nf_amc_log_fact[ p2 ] - nf_amc_log_fact[ p3 ]
418 - nf_amc_log_fact[ q1 ] - nf_amc_log_fact[ q3 ] - nf_amc_log_fact[ q4 ] ) * ( ( ( q1 % 2 ) == 0 ) ? 1 : -1 );
419 if( cg == INFINITY )
return( INFINITY );
427 for( i = 0; i < ( q2 - q1 ); i++ )
428 cg = a - cg * ( ( p1 + i ) * ( p2 + i ) * ( p3 + i ) ) / ( ( q2 - i ) * ( q3 - i ) * ( q4 - i ) );
440 double z, clebsh_gordan =
nf_amc_clebsh_gordan( l1, l2, 0, 0, ll ), racah =
nf_amc_racah( l1, j1, l2, j2, s, ll );
442 if( ( clebsh_gordan == INFINITY ) || ( racah == INFINITY ) )
return( INFINITY );
443 z = ( ( ( -l1 + l2 + ll ) % 8 == 0 ) ? 1.0 : -1.0 )
444 * std::sqrt( l1 + 1.0 ) * std::sqrt( l2 + 1.0 ) * std::sqrt( j1 + 1.0 ) * std::sqrt( j2 + 1.0 ) * clebsh_gordan * racah;
461 double zbar, clebsh_gordan =
nf_amc_clebsh_gordan( l1, l2, 0, 0, ll ), racah =
nf_amc_racah( l1, j1, l2, j2, s, ll );
463 if( ( clebsh_gordan == INFINITY ) || ( racah == INFINITY ) )
return( INFINITY );
464 zbar = std::sqrt( l1 + 1.0 ) * std::sqrt( l2 + 1.0 ) * std::sqrt( j1 + 1.0 ) * std::sqrt( j2 + 1.0 ) * clebsh_gordan * racah;
481 double x1, x2, x3, reduced_mat, clebsh_gordan;
484 if ( std::abs( l0 - l1 ) > lt || ( l0 + l1 ) < lt )
return( 0.0 );
485 if ( std::abs( ( j0 - j1 ) / 2 ) > jt || ( ( j0 + j1 ) / 2 ) < jt )
return( 0.0 );
491 if( ( clebsh_gordan =
nf_amc_clebsh_gordan( j1, j0, 1, -1, jt ) ) == INFINITY )
return( INFINITY );
493 reduced_mat = 1.0 / std::sqrt( 4 * M_PI ) * clebsh_gordan / std::sqrt( jt + 1.0 )
494 * std::sqrt( ( j0 + 1.0 ) * ( j1 + 1.0 ) * ( llt + 1.0 ) )
495 *
parity( ( j1 - j0 ) / 2 ) *
parity( ( -l0 + l1 + lt ) / 2 ) *
parity( ( j0 - 1 ) / 2 );
498 x1 = ( l0 - j0 / 2.0 ) * ( j0 + 1.0 );
499 x2 = ( l1 - j1 / 2.0 ) * ( j1 + 1.0 );
501 x3 = ( lt == 0 ) ? 0 : ( x1 - x2 ) / std::sqrt( lt * ( lt + 1.0 ) );
503 else if ( jt == ( llt - st ) ){
504 x3 = ( lt == 0 ) ? 0 : -( lt + x1 + x2 ) / std::sqrt( lt * ( 2.0 * lt + 1.0 ) );
506 else if ( jt == ( llt + st ) ){
507 x3 = ( lt + 1 - x1 - x2 ) / std::sqrt( ( 2.0 * lt + 1.0 ) * ( lt + 1.0 ) );
516 return( reduced_mat );
523 return( ( ( x / 2 ) % 2 == 0 ) ? 1 : -1 );
528 static int max3(
int a,
int b,
int c ) {
537 static int max4(
int a,
int b,
int c,
int d ) {
547 static int min3(
int a,
int b,
int c ) {
554 #if defined __cplusplus
static double cg3(int, int, int, int, int, int)
static int max4(int a, int b, int c, int d)
double nf_amc_racah(int j1, int j2, int l2, int l1, int j3, int l3)
double nf_amc_z_coefficient(int l1, int j1, int l2, int j2, int s, int ll)
double nf_amc_wigner_9j(int j1, int j2, int j3, int j4, int j5, int j6, int j7, int j8, int j9)
const G4double w[NPOINTSGL]
double nf_amc_clebsh_gordan(int j1, int j2, int m1, int m2, int j3)
double nf_amc_wigner_3j(int j1, int j2, int j3, int j4, int j5, int j6)
static double cg1(int, int, int)
static const int MAX_FACTORIAL
static int min3(int a, int b, int c)
static const double nf_amc_log_fact[]
double nf_amc_reduced_matrix_element(int lt, int st, int jt, int l0, int j0, int l1, int j1)
static double cg2(int, int, int, int, int, int, int, int)
double nf_amc_wigner_6j(int j1, int j2, int j3, int j4, int j5, int j6)
G4double G4Exp(G4double initial_x)
Exponential Function double precision.
double nf_amc_zbar_coefficient(int l1, int j1, int l2, int j2, int s, int ll)
static double w6j0(int, int *)
const G4double x[NPOINTSGL]
double nf_amc_factorial(int n)
double nf_amc_log_factorial(int n)
static double w6j1(int *)
static int max3(int a, int b, int c)