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nf_angularMomentumCoupling.cc
Go to the documentation of this file. Or view the newest version in sPHENIX GitHub for file nf_angularMomentumCoupling.cc
1 /*
2 * calculate coupling coefficients of angular momenta
3 *
4 * Author:
5 * Kawano, T <kawano@mailaps.org>
6 *
7 * Modified by David Brown <dbrown@bnl.gov>
8 * No longer must precompute the logarithm of the factorials.
9 * Also renamed things to make more Python friendly.
10 * Finally, fixed a bunch of bugs & confusing conventions
11 *
12 * Functions:
13 *
14 * Note that arguments of those functions must be doubled, namely 1/2 is 1, etc.
15 *
16 * wigner_3j(j1,j2,j3,j4,j5,j6)
17 * Wigner's 3J symbol (similar to Clebsh-Gordan)
18 * = / j1 j2 j3 \
19 * \ j4 j5 j6 /
20 *
21 * wigner_6j(j1,j2,j3,j4,j5,j6)
22 * Wigner's 6J symbol (similar to Racah)
23 * = { j1 j2 j3 }
24 * { j4 j5 j6 }
25 *
26 * wigner_9j(j1,j2,j3,j4,j5,j6,j7,j8,j9)
27 * Wigner's 9J symbol
28 * / j1 j2 j3 \
29 * = | j4 j5 j6 |
30 * \ j7 j8 j9 /
31 *
32 * racah(j1, j2, l2, l1, j3, l3)
33 * = W(j1, j2, l2, l1 ; j3, l3)
34 * = (-1)^(j1+j2+l1+l2) * { j1 j2 j3 }
35 * { l1 l2 l3 }
36 *
37 * clebsh_gordan(j1,j2,m1,m2,j3)
38 * Clebsh-Gordan coefficient
39 * = <j1,j2,m1,m2|j3,m1+m2>
40 * = (-)^(j1-j2+m1+m2) * std::sqrt(2*j3+1) * / j1 j2 j3 \
41 * \ m1 m2 -m1-m2 /
42 *
43 * z_coefficient(l1,j1,l2,j2,S,L)
44 * Biedenharn's Z-coefficient coefficient
45 * = Z(l1 j1 l2 j2 | S L )
46 *
47 * reduced_matrix_element(L,S,J,l0,j0,l1,j1)
48 * Reduced Matrix Element for Tensor Operator
49 * = < l1j1 || T(YL,sigma_S)J || l0j0 >
50 *
51 * References:
52 * A. R. Edmonds, Angular Momentum in Quantum Mechanics, Princeton University Press 1974.
53 * E. Condon, and G. Shortley, The Theory of Atomic Spectra, Cambridge, 1935.
54 */
55 
56 #include <stdlib.h>
57 #include <cmath>
58 
59 #include "nf_specialFunctions.h"
60 
61 #if defined __cplusplus
62 #include <cmath>
63 #include "G4Exp.hh"
64 namespace GIDI {
65 using namespace GIDI;
66 #endif
67 
68 static const int MAX_FACTORIAL = 200; // maximal factorial n! (2 x Lmax)
69 /*static const double ARRAY_OVER = 1.0e+300; // force overflow */
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};
71 
72 static int parity( int x );
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 );
81 /*static double triangle( int, int, int );*/
82 /*
83 ============================================================
84 */
85 double nf_amc_log_factorial( int n ) {
86 /*
87 * returns ln( n! ).
88 */
89  if( n > MAX_FACTORIAL ) return( INFINITY );
90  if( n < 0 ) return( INFINITY );
91  return nf_amc_log_fact[n];
92 }
93 /*
94 ============================================================
95 */
96 double nf_amc_factorial( int n ) {
97 /*
98 * returns n! for pre-computed table. INFINITY is return if n is negative or too large.
99 */
100  return G4Exp( nf_amc_log_factorial( n ) );
101 }
102 /*
103 ============================================================
104 */
105 double nf_amc_wigner_3j( int j1, int j2, int j3, int j4, int j5, int j6 ) {
106 /*
107 * Wigner's 3J symbol (similar to Clebsh-Gordan)
108 * = / j1 j2 j3 \
109 * \ j4 j5 j6 /
110 */
111  double cg;
112 
113  if( ( j4 + j5 + j6 ) != 0 ) return( 0.0 );
114  if( ( cg = nf_amc_clebsh_gordan( j1, j2, j4, j5, j3 ) ) == 0.0 ) 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 ) ); /* BRB j3 + 1 <= 0? */
117 }
118 /*
119 ============================================================
120 */
121 double nf_amc_wigner_6j( int j1, int j2, int j3, int j4, int j5, int j6 ) {
122 /*
123 * Wigner's 6J symbol (similar to Racah)
124 * = { j1 j2 j3 }
125 * { j4 j5 j6 }
126 */
127  int i, x[6];
128 
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 ) );
131 
132  return( w6j1( x ) );
133 }
134 /*
135 ============================================================
136 */
137 static double w6j0( int i, int *x ) {
138 
139  switch( i ){
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 );
143  x[5] = x[4]; break;
144  case 2: if ( ( x[0] != x[1] ) || ( x[3] != x[4] ) ) return( 0.0 );
145  break;
146  //TK fix bug and add comment on 17-05-23
147  //This is the case of 6.3.2 of A. R. Edmonds, Angular Momentum in Quantum Mechanics, Princeton University Press 1974.
148  case 3: if ( ( x[1] != x[5] ) || ( x[2] != x[4] ) ) return( 0.0 );
149  x[5] = x[0]; x[0] = x[4]; x[3] = x[1]; break;
150  case 4: if ( ( x[0] != x[5] ) || ( x[2] != x[3] ) ) return( 0.0 );
151  x[5] = x[1]; break;
152  case 5: if ( ( x[0] != x[4] ) || ( x[1] != x[3] ) ) return( 0.0 );
153  x[5] = x[2]; break;
154  }
155 
156  if( ( x[5] > ( x[0] + x[3] ) ) || ( x[5] < std::abs( x[0] - x[3] ) ) ) return( 0.0 );
157  if( x[0] > MAX_FACTORIAL || x[3] > MAX_FACTORIAL ) { /* BRB Why this test? Why not x[5]? */
158  return( INFINITY );
159  }
160 
161  return( 1.0 / std::sqrt( (double) ( ( x[0] + 1 ) * ( x[3] + 1 ) ) ) * ( ( ( x[0] + x[3] + x[5] ) / 2 ) % 2 != 0 ? -1 : 1 ) );
162 }
163 /*
164 ============================================================
165 */
166 static double w6j1( int *x ) {
167 
168  double w6j, w;
169  int i, k, k1, k2, n, l1, l2, l3, l4, n1, n2, n3, m1, m2, m3, x1, x2, x3, y[4];
170  static int a[3][4] = { { 0, 0, 3, 3},
171  { 1, 4, 1, 4},
172  { 2, 5, 5, 2} };
173 
174  w6j = 0.0;
175 
176  for ( k = 0; k < 4; k++ ){
177  x1 = x[ ( a[0][k] ) ];
178  x2 = x[ ( a[1][k] ) ];
179  x3 = x[ ( a[2][k] ) ];
180 
181  n = ( x1 + x2 + x3 ) / 2;
182  if( n > MAX_FACTORIAL ) {
183  return( INFINITY ); }
184  else if( n < 0 ) {
185  return( 0.0 );
186  }
187 
188  if ( ( n1 = n - x3 ) < 0 ) return( 0.0 );
189  if ( ( n2 = n - x2 ) < 0 ) return( 0.0 );
190  if ( ( n3 = n - x1 ) < 0 ) return( 0.0 );
191 
192  y[k] = n + 2;
193  w6j += nf_amc_log_fact[n1] + nf_amc_log_fact[n2] + nf_amc_log_fact[n3] - nf_amc_log_fact[n+1];
194  }
195 
196  n1 = ( x[0] + x[1] + x[3] + x[4] ) / 2;
197  n2 = ( x[0] + x[2] + x[3] + x[5] ) / 2;
198  n3 = ( x[1] + x[2] + x[4] + x[5] ) / 2;
199 
200  k1 = max4( y[0], y[1], y[2], y[3] ) - 1;
201  k2 = min3( n1, n2, n3 ) + 1;
202 
203  l1 = k1 - y[0] + 1; m1 = n1 - k1 + 1;
204  l2 = k1 - y[1] + 1; m2 = n2 - k1 + 1;
205  l3 = k1 - y[2] + 1; m3 = n3 - k1 + 1;
206  l4 = k1 - y[3] + 1;
207 
208  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]
209  - nf_amc_log_fact[m1] - nf_amc_log_fact[m2] - nf_amc_log_fact[m3] ) * ( ( k1 % 2 ) == 0 ? -1: 1 );
210  if( w6j == INFINITY ) return( INFINITY );
211 
212  if( k1 != k2 ){
213  k = k2 - k1;
214  m1 -= k-1; m2 -= k-1; m3 -= k-1;
215  l1 += k ; l2 += k ; l3 += k ; l4 += k;
216 
217  for ( i = 0; i < k; i++ )
218  w6j = w - w6j * ( ( k2 - i ) * ( m1 + i ) * ( m2 + i ) * ( m3 + i ) )
219  / ( ( l1 - i ) * ( l2 - i ) * ( l3 - i ) * ( l4 - i ) );
220  }
221  return( w6j );
222 }
223 /*
224 ============================================================
225 */
226 double nf_amc_wigner_9j( int j1, int j2, int j3, int j4, int j5, int j6, int j7, int j8, int j9 ) {
227 /*
228 * Wigner's 9J symbol
229 * / j1 j2 j3 \
230 * = | j4 j5 j6 |
231 * \ j7 j8 j9 /
232 *
233 */
234  int i, i0, i1;
235  double rac;
236 
237  i0 = max3( std::abs( j1 - j9 ), std::abs( j2 - j6 ), std::abs( j4 - j8 ) );
238  i1 = min3( ( j1 + j9 ), ( j2 + j6 ), ( j4 + j8 ) );
239 
240  rac = 0.0;
241  for ( i = i0; i <= i1; i += 2 ){
242  rac += nf_amc_racah( j1, j4, j9, j8, j7, i )
243  * nf_amc_racah( j2, j5, i, j4, j8, j6 )
244  * nf_amc_racah( j9, i, j3, j2, j1, j6 ) * ( i + 1 );
245  if( rac == INFINITY ) return( INFINITY );
246  }
247 
248  return( ( ( (int)( ( j1 + j3 + j5 + j8 ) / 2 + j2 + j4 + j9 ) % 4 == 0 ) ? 1.0 : -1.0 ) * rac );
249 }
250 /*
251 ============================================================
252 */
253 double nf_amc_racah( int j1, int j2, int l2, int l1, int j3, int l3 ) {
254 /*
255 * Racah coefficient definition in Edmonds (AR Edmonds, "Angular Momentum in Quantum Mechanics", Princeton (1980) is
256 * W(j1, j2, l2, l1 ; j3, l3) = (-1)^(j1+j2+l1+l2) * { j1 j2 j3 }
257 * { l1 l2 l3 }
258 * The call signature of W(...) appears jumbled, but hey, that's the convention.
259 *
260 * This convention is exactly that used by Blatt-Biedenharn (Rev. Mod. Phys. 24, 258 (1952)) too
261 */
262 
263  double sig;
264 
265  sig = ( ( ( j1 + j2 + l1 + l2 ) % 4 == 0 ) ? 1.0 : -1.0 );
266  return sig * nf_amc_wigner_6j( j1, j2, j3, l1, l2, l3 );
267 }
268 
269 /*
270 ============================================================
271 */
272 /*
273 static double triangle( int a, int b, int c ) {
274 
275  int j1, j2, j3, j4;
276 
277  if ( ( j1 = ( a + b - c ) / 2 ) < 0 ) return( 0.0 );
278  if ( ( j2 = ( a - b + c ) / 2 ) < 0 ) return( 0.0 );
279  if ( ( j3 = ( -a + b + c ) / 2 ) < 0 ) return( 0.0 );
280  j4 = ( a + b + c ) / 2 + 1;
281 
282  return( std::exp( 0.5 * ( nf_amc_log_fact[j1] + nf_amc_log_fact[j2] + nf_amc_log_fact[j3] - nf_amc_log_fact[j4] ) ) );
283 }
284 */
285 /*
286 ============================================================
287 */
288 double nf_amc_clebsh_gordan( int j1, int j2, int m1, int m2, int j3 ) {
289 /*
290 * Clebsh-Gordan coefficient
291 * = <j1,j2,m1,m2|j3,m1+m2>
292 * = (-)^(j1-j2+m1+m2) * std::sqrt(2*j3+1) * / j1 j2 j3 \
293 * \ m1 m2 -m1-m2 /
294 *
295 * Note: Last value m3 is preset to m1+m2. Any other value will evaluate to 0.0.
296 */
297 
298  int m3, x1, x2, x3, y1, y2, y3;
299  double cg = 0.0;
300 
301  if ( j1 < 0 || j2 < 0 || j3 < 0) return( 0.0 );
302  if ( j1 + j2 + j3 > 2 * MAX_FACTORIAL ) return( INFINITY );
303 
304  m3 = m1 + m2;
305 
306  if ( ( x1 = ( j1 + m1 ) / 2 + 1 ) <= 0 ) return( 0.0 );
307  if ( ( x2 = ( j2 + m2 ) / 2 + 1 ) <= 0 ) return( 0.0 );
308  if ( ( x3 = ( j3 - m3 ) / 2 + 1 ) <= 0 ) return( 0.0 );
309 
310  if ( ( y1 = x1 - m1 ) <= 0 ) return( 0.0 );
311  if ( ( y2 = x2 - m2 ) <= 0 ) return( 0.0 );
312  if ( ( y3 = x3 + m3 ) <= 0 ) return( 0.0 );
313 
314  if ( j3 == 0 ){
315  if ( j1 == j2 ) cg = ( 1.0 / std::sqrt( (double)j1 + 1.0 ) * ( ( y1 % 2 == 0 ) ? -1:1 ) );
316  }
317  else if ( (j1 == 0 || j2 == 0 ) ){
318  if ( ( j1 + j2 ) == j3 ) cg = 1.0;
319  }
320  else {
321  if( m3 == 0 && std::abs( m1 ) <= 1 ){
322  if( m1 == 0 ) cg = cg1( x1, x2, x3 );
323  else cg = cg2( x1 + y1 - y2, x3 - 1, x1 + x2 - 2, x1 - y2, j1, j2, j3, m2 );
324  }
325  else if ( m2 == 0 && std::abs( m1 ) <=1 ){
326  cg = cg2( x1 - y2 + y3, x2 - 1, x1 + x3 - 2, x3 - y1, j1, j3, j3, m1 );
327  }
328  else if ( m1 == 0 && std::abs( m3 ) <= 1 ){
329  cg = cg2( x1, x1 - 1, x2 + x3 - 2, x2 - y3, j2, j3, j3, -m3 );
330  }
331  else cg = cg3( x1, x2, x3, y1, y2, y3 );
332  }
333 
334  return( cg );
335 }
336 /*
337 ============================================================
338 */
339 static double cg1( int x1, int x2, int x3 ) {
340 
341  int p1, p2, p3, p4, q1, q2, q3, q4;
342  double a;
343 
344  p1 = x1 + x2 + x3 - 1; if ( ( p1 % 2 ) != 0 ) return( 0.0 );
345  p2 = x1 + x2 - x3;
346  p3 =-x1 + x2 + x3;
347  p4 = x1 - x2 + x3;
348  if ( p2 <= 0 || p3 <= 0 || p4 <= 0 ) return( 0.0 );
349  if ( p1 >= MAX_FACTORIAL ) return( INFINITY );
350 
351  q1 = ( p1 + 1 ) / 2 - 1; p1--;
352  q2 = ( p2 + 1 ) / 2 - 1; p2--;
353  q3 = ( p3 + 1 ) / 2 - 1; p3--;
354  q4 = ( p4 + 1 ) / 2 - 1; p4--;
355 
356  a = nf_amc_log_fact[q1]-( nf_amc_log_fact[q2] + nf_amc_log_fact[q3] + nf_amc_log_fact[q4] )
357  + 0.5 * ( nf_amc_log_fact[ 2 * x3 - 1 ] - nf_amc_log_fact[ 2 * x3 - 2 ]
358  + nf_amc_log_fact[p2] + nf_amc_log_fact[p3] + nf_amc_log_fact[p4] - nf_amc_log_fact[p1] );
359 
360  return( ( ( ( q1 + x1 - x2 ) % 2 == 0 ) ? 1.0 : -1.0 ) * G4Exp( a ) );
361 }
362 /*
363 ============================================================
364 */
365 static double cg2( int k, int q0, int z1, int z2, int w1, int w2, int w3, int mm ) {
366 
367  int q1, q2, q3, q4, p1, p2, p3, p4;
368  double a;
369 
370  p1 = z1 + q0 + 2;
371  p2 = z1 - q0 + 1;
372  p3 = z2 + q0 + 1;
373  p4 = -z2 + q0 + 1;
374  if ( p2 <= 0 || p3 <= 0 || p4 <= 0) return( 0.0 );
375  if ( p1 >= MAX_FACTORIAL ) return( INFINITY );
376 
377  q1 = ( p1 + 1 ) / 2 - 1; p1--;
378  q2 = ( p2 + 1 ) / 2 - 1; p2--;
379  q3 = ( p3 + 1 ) / 2 - 1; p3--;
380  q4 = ( p4 + 1 ) / 2 - 1; p4--;
381 
382  a = nf_amc_log_fact[q1] - ( nf_amc_log_fact[ q2 ] + nf_amc_log_fact[ q3 ] + nf_amc_log_fact[ q4 ] )
383  + 0.5 * ( nf_amc_log_fact[ w3 + 1 ] - nf_amc_log_fact[ w3 ]
384  + nf_amc_log_fact[ w1 ] - nf_amc_log_fact[ w1 + 1 ]
385  + nf_amc_log_fact[ w2 ] - nf_amc_log_fact[ w2 + 1 ]
386  + nf_amc_log_fact[ p2 ] + nf_amc_log_fact[ p3 ] + nf_amc_log_fact[ p4 ] - nf_amc_log_fact[ p1 ] );
387 
388  return( ( ( ( q4 + k + ( mm > 0 ) * ( p1 + 2 ) ) % 2 == 0 ) ? -1.0 : 1.0 ) * 2.0 * G4Exp( a ) );
389 }
390 /*
391 ============================================================
392 */
393 static double cg3( int x1, int x2, int x3, int y1, int y2, int y3 ) {
394 
395  int nx, i, k1, k2, q1, q2, q3, q4, p1, p2, p3, z1, z2, z3;
396  double a, cg;
397 
398  nx = x1 + x2 + x3 - 1;
399  if ( ( z1 = nx - x1 - y1 ) < 0 ) return( 0.0 );
400  if ( ( z2 = nx - x2 - y2 ) < 0 ) return( 0.0 );
401  if ( ( z3 = nx - x3 - y3 ) < 0 ) return( 0.0 );
402 
403  k1 = x2 - y3;
404  k2 = y1 - x3;
405 
406  q1 = max3( k1, k2, 0 );
407  q2 = min3( y1, x2, z3 + 1 ) - 1;
408  q3 = q1 - k1;
409  q4 = q1 - k2;
410 
411  p1 = y1 - q1 - 1;
412  p2 = x2 - q1 - 1;
413  p3 = z3 - q1;
414 
415  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 ]
416  + nf_amc_log_fact[ z1 ] + nf_amc_log_fact[ z2 ] + nf_amc_log_fact[ z3 ]
417  + nf_amc_log_fact[ x1 - 1 ] + nf_amc_log_fact[ x2 - 1 ] + nf_amc_log_fact[ x3 - 1 ]
418  + nf_amc_log_fact[ y1 - 1 ] + nf_amc_log_fact[ y2 - 1 ] + nf_amc_log_fact[ y3 - 1 ] )
419  - nf_amc_log_fact[ p1 ] - nf_amc_log_fact[ p2 ] - nf_amc_log_fact[ p3 ]
420  - nf_amc_log_fact[ q1 ] - nf_amc_log_fact[ q3 ] - nf_amc_log_fact[ q4 ] ) * ( ( ( q1 % 2 ) == 0 ) ? 1 : -1 );
421  if( cg == INFINITY ) return( INFINITY );
422 
423  if ( q1 != q2 ){
424  q3 = q2 - k1;
425  q4 = q2 - k2;
426  p1 = y1 - q2;
427  p2 = x2 - q2;
428  p3 = z3 - q2 + 1;
429  for( i = 0; i < ( q2 - q1 ); i++ )
430  cg = a - cg * ( ( p1 + i ) * ( p2 + i ) * ( p3 + i ) ) / ( ( q2 - i ) * ( q3 - i ) * ( q4 - i ) );
431  }
432  return( cg );
433 }
434 /*
435 ============================================================
436 */
437 double nf_amc_z_coefficient( int l1, int j1, int l2, int j2, int s, int ll ) {
438 /*
439 * Biedenharn's Z-coefficient coefficient
440 * = Z(l1 j1 l2 j2 | S L )
441 */
442  double z, clebsh_gordan = nf_amc_clebsh_gordan( l1, l2, 0, 0, ll ), racah = nf_amc_racah( l1, j1, l2, j2, s, ll );
443 
444  if( ( clebsh_gordan == INFINITY ) || ( racah == INFINITY ) ) return( INFINITY );
445  z = ( ( ( -l1 + l2 + ll ) % 8 == 0 ) ? 1.0 : -1.0 )
446  * std::sqrt( l1 + 1.0 ) * std::sqrt( l2 + 1.0 ) * std::sqrt( j1 + 1.0 ) * std::sqrt( j2 + 1.0 ) * clebsh_gordan * racah;
447 
448  return( z );
449 }
450 /*
451 ============================================================
452 */
453 double nf_amc_zbar_coefficient( int l1, int j1, int l2, int j2, int s, int ll ) {
454 /*
455 * Lane & Thomas's Zbar-coefficient coefficient
456 * = Zbar(l1 j1 l2 j2 | S L )
457 * = (-i)^( -l1 + l2 + ll ) * Z(l1 j1 l2 j2 | S L )
458 *
459 * Lane & Thomas Rev. Mod. Phys. 30, 257-353 (1958).
460 * 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.
461 * Froehner uses Lane & Thomas convention as does T. Kawano.
462 */
463  double zbar, clebsh_gordan = nf_amc_clebsh_gordan( l1, l2, 0, 0, ll ), racah = nf_amc_racah( l1, j1, l2, j2, s, ll );
464 
465  if( ( clebsh_gordan == INFINITY ) || ( racah == INFINITY ) ) return( INFINITY );
466  zbar = std::sqrt( l1 + 1.0 ) * std::sqrt( l2 + 1.0 ) * std::sqrt( j1 + 1.0 ) * std::sqrt( j2 + 1.0 ) * clebsh_gordan * racah;
467 
468  return( zbar );
469 }
470 /*
471 ============================================================
472 */
473 double nf_amc_reduced_matrix_element( int lt, int st, int jt, int l0, int j0, int l1, int j1 ) {
474 /*
475 * Reduced Matrix Element for Tensor Operator
476 * = < l1j1 || T(YL,sigma_S)J || l0j0 >
477 *
478 * M.B.Johnson, L.W.Owen, G.R.Satchler
479 * Phys. Rev. 142, 748 (1966)
480 * Note: definition differs from JOS by the factor sqrt(2j1+1)
481 */
482  int llt;
483  double x1, x2, x3, reduced_mat, clebsh_gordan;
484 
485  if ( parity( lt ) != parity( l0 ) * parity( l1 ) ) return( 0.0 );
486  if ( std::abs( l0 - l1 ) > lt || ( l0 + l1 ) < lt ) return( 0.0 );
487  if ( std::abs( ( j0 - j1 ) / 2 ) > jt || ( ( j0 + j1 ) / 2 ) < jt ) return( 0.0 );
488 
489  llt = 2 * lt;
490  jt *= 2;
491  st *= 2;
492 
493  if( ( clebsh_gordan = nf_amc_clebsh_gordan( j1, j0, 1, -1, jt ) ) == INFINITY ) return( INFINITY );
494 
495  reduced_mat = 1.0 / std::sqrt( 4 * M_PI ) * clebsh_gordan / std::sqrt( jt + 1.0 ) /* BRB jt + 1 <= 0? */
496  * std::sqrt( ( j0 + 1.0 ) * ( j1 + 1.0 ) * ( llt + 1.0 ) )
497  * parity( ( j1 - j0 ) / 2 ) * parity( ( -l0 + l1 + lt ) / 2 ) * parity( ( j0 - 1 ) / 2 );
498 
499  if( st == 2 ){
500  x1 = ( l0 - j0 / 2.0 ) * ( j0 + 1.0 );
501  x2 = ( l1 - j1 / 2.0 ) * ( j1 + 1.0 );
502  if ( jt == llt ){
503  x3 = ( lt == 0 ) ? 0 : ( x1 - x2 ) / std::sqrt( lt * ( lt + 1.0 ) );
504  }
505  else if ( jt == ( llt - st ) ){
506  x3 = ( lt == 0 ) ? 0 : -( lt + x1 + x2 ) / std::sqrt( lt * ( 2.0 * lt + 1.0 ) );
507  }
508  else if ( jt == ( llt + st ) ){
509  x3 = ( lt + 1 - x1 - x2 ) / std::sqrt( ( 2.0 * lt + 1.0 ) * ( lt + 1.0 ) );
510  }
511  else{
512  x3 = 1.0;
513  }
514  }
515  else x3 = 1.0;
516  reduced_mat *= x3;
517 
518  return( reduced_mat );
519 }
520 /*
521 ============================================================
522 */
523 static int parity( int x ) {
524 
525  return( ( ( x / 2 ) % 2 == 0 ) ? 1 : -1 );
526 }
527 /*
528 ============================================================
529 */
530 static int max3( int a, int b, int c ) {
531 
532  if( a < b ) a = b;
533  if( a < c ) a = c;
534  return( a );
535 }
536 /*
537 ============================================================
538 */
539 static int max4( int a, int b, int c, int d ) {
540 
541  if( a < b ) a = b;
542  if( a < c ) a = c;
543  if( a < d ) a = d;
544  return( a );
545 }
546 /*
547 ============================================================
548 */
549 static int min3( int a, int b, int c ) {
550 
551  if( a > b ) a = b;
552  if( a > c ) a = c;
553  return( a );
554 }
555 
556 #if defined __cplusplus
557 }
558 #endif