Geant4  10.02.p02
nf_angularMomentumCoupling.cc
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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  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 );
149  x[5] = x[1]; break;
150  case 5: if ( ( x[0] != x[4] ) || ( x[1] != x[3] ) ) return( 0.0 );
151  x[5] = x[2]; break;
152  }
153 
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 ) { /* BRB Why this test? Why not x[5]? */
156  return( INFINITY );
157  }
158 
159  return( 1.0 / std::sqrt( (double) ( ( x[0] + 1 ) * ( x[3] + 1 ) ) ) * ( ( ( x[0] + x[3] + x[5] ) / 2 ) % 2 != 0 ? -1 : 1 ) );
160 }
161 /*
162 ============================================================
163 */
164 static double w6j1( int *x ) {
165 
166  double w6j, w;
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},
169  { 1, 4, 1, 4},
170  { 2, 5, 5, 2} };
171 
172  w6j = 0.0;
173 
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] ) ];
178 
179  n = ( x1 + x2 + x3 ) / 2;
180  if( n > MAX_FACTORIAL ) {
181  return( INFINITY ); }
182  else if( n < 0 ) {
183  return( 0.0 );
184  }
185 
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 );
189 
190  y[k] = n + 2;
191  w6j += nf_amc_log_fact[n1] + nf_amc_log_fact[n2] + nf_amc_log_fact[n3] - nf_amc_log_fact[n+1];
192  }
193 
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;
197 
198  k1 = max4( y[0], y[1], y[2], y[3] ) - 1;
199  k2 = min3( n1, n2, n3 ) + 1;
200 
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;
204  l4 = k1 - y[3] + 1;
205 
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 );
209 
210  if( k1 != k2 ){
211  k = k2 - k1;
212  m1 -= k-1; m2 -= k-1; m3 -= k-1;
213  l1 += k ; l2 += k ; l3 += k ; l4 += k;
214 
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 ) );
218  }
219  return( w6j );
220 }
221 /*
222 ============================================================
223 */
224 double nf_amc_wigner_9j( int j1, int j2, int j3, int j4, int j5, int j6, int j7, int j8, int j9 ) {
225 /*
226 * Wigner's 9J symbol
227 * / j1 j2 j3 \
228 * = | j4 j5 j6 |
229 * \ j7 j8 j9 /
230 *
231 */
232  int i, i0, i1;
233  double rac;
234 
235  i0 = max3( std::abs( j1 - j9 ), std::abs( j2 - j6 ), std::abs( j4 - j8 ) );
236  i1 = min3( ( j1 + j9 ), ( j2 + j6 ), ( j4 + j8 ) );
237 
238  rac = 0.0;
239  for ( i = i0; i <= i1; i += 2 ){
240  rac += nf_amc_racah( j1, j4, j9, j8, j7, i )
241  * nf_amc_racah( j2, j5, i, j4, j8, j6 )
242  * nf_amc_racah( j9, i, j3, j2, j1, j6 ) * ( i + 1 );
243  if( rac == INFINITY ) return( INFINITY );
244  }
245 
246  return( ( ( (int)( ( j1 + j3 + j5 + j8 ) / 2 + j2 + j4 + j9 ) % 4 == 0 ) ? 1.0 : -1.0 ) * rac );
247 }
248 /*
249 ============================================================
250 */
251 double nf_amc_racah( int j1, int j2, int l2, int l1, int j3, int l3 ) {
252 /*
253 * Racah coefficient definition in Edmonds (AR Edmonds, "Angular Momentum in Quantum Mechanics", Princeton (1980) is
254 * W(j1, j2, l2, l1 ; j3, l3) = (-1)^(j1+j2+l1+l2) * { j1 j2 j3 }
255 * { l1 l2 l3 }
256 * The call signature of W(...) appears jumbled, but hey, that's the convention.
257 *
258 * This convention is exactly that used by Blatt-Biedenharn (Rev. Mod. Phys. 24, 258 (1952)) too
259 */
260 
261  double sig;
262 
263  sig = ( ( ( j1 + j2 + l1 + l2 ) % 4 == 0 ) ? 1.0 : -1.0 );
264  return sig * nf_amc_wigner_6j( j1, j2, j3, l1, l2, l3 );
265 }
266 
267 /*
268 ============================================================
269 */
270 /*
271 static double triangle( int a, int b, int c ) {
272 
273  int j1, j2, j3, j4;
274 
275  if ( ( j1 = ( a + b - c ) / 2 ) < 0 ) return( 0.0 );
276  if ( ( j2 = ( a - b + c ) / 2 ) < 0 ) return( 0.0 );
277  if ( ( j3 = ( -a + b + c ) / 2 ) < 0 ) return( 0.0 );
278  j4 = ( a + b + c ) / 2 + 1;
279 
280  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] ) ) );
281 }
282 */
283 /*
284 ============================================================
285 */
286 double nf_amc_clebsh_gordan( int j1, int j2, int m1, int m2, int j3 ) {
287 /*
288 * Clebsh-Gordan coefficient
289 * = <j1,j2,m1,m2|j3,m1+m2>
290 * = (-)^(j1-j2+m1+m2) * std::sqrt(2*j3+1) * / j1 j2 j3 \
291 * \ m1 m2 -m1-m2 /
292 *
293 * Note: Last value m3 is preset to m1+m2. Any other value will evaluate to 0.0.
294 */
295 
296  int m3, x1, x2, x3, y1, y2, y3;
297  double cg = 0.0;
298 
299  if ( j1 < 0 || j2 < 0 || j3 < 0) return( 0.0 );
300  if ( j1 + j2 + j3 > 2 * MAX_FACTORIAL ) return( INFINITY );
301 
302  m3 = m1 + m2;
303 
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 );
307 
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 );
311 
312  if ( j3 == 0 ){
313  if ( j1 == j2 ) cg = ( 1.0 / std::sqrt( (double)j1 + 1.0 ) * ( ( y1 % 2 == 0 ) ? -1:1 ) );
314  }
315  else if ( (j1 == 0 || j2 == 0 ) ){
316  if ( ( j1 + j2 ) == j3 ) cg = 1.0;
317  }
318  else {
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 );
322  }
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 );
325  }
326  else if ( m1 == 0 && std::abs( m3 ) <= 1 ){
327  cg = cg2( x1, x1 - 1, x2 + x3 - 2, x2 - y3, j2, j3, j3, -m3 );
328  }
329  else cg = cg3( x1, x2, x3, y1, y2, y3 );
330  }
331 
332  return( cg );
333 }
334 /*
335 ============================================================
336 */
337 static double cg1( int x1, int x2, int x3 ) {
338 
339  int p1, p2, p3, p4, q1, q2, q3, q4;
340  double a;
341 
342  p1 = x1 + x2 + x3 - 1; if ( ( p1 % 2 ) != 0 ) return( 0.0 );
343  p2 = x1 + x2 - x3;
344  p3 =-x1 + x2 + x3;
345  p4 = x1 - x2 + x3;
346  if ( p2 <= 0 || p3 <= 0 || p4 <= 0 ) return( 0.0 );
347  if ( p1 >= MAX_FACTORIAL ) return( INFINITY );
348 
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--;
353 
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] );
357 
358  return( ( ( ( q1 + x1 - x2 ) % 2 == 0 ) ? 1.0 : -1.0 ) * G4Exp( a ) );
359 }
360 /*
361 ============================================================
362 */
363 static double cg2( int k, int q0, int z1, int z2, int w1, int w2, int w3, int mm ) {
364 
365  int q1, q2, q3, q4, p1, p2, p3, p4;
366  double a;
367 
368  p1 = z1 + q0 + 2;
369  p2 = z1 - q0 + 1;
370  p3 = z2 + q0 + 1;
371  p4 = -z2 + q0 + 1;
372  if ( p2 <= 0 || p3 <= 0 || p4 <= 0) return( 0.0 );
373  if ( p1 >= MAX_FACTORIAL ) return( INFINITY );
374 
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--;
379 
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 ] );
385 
386  return( ( ( ( q4 + k + ( mm > 0 ) * ( p1 + 2 ) ) % 2 == 0 ) ? -1.0 : 1.0 ) * 2.0 * G4Exp( a ) );
387 }
388 /*
389 ============================================================
390 */
391 static double cg3( int x1, int x2, int x3, int y1, int y2, int y3 ) {
392 
393  int nx, i, k1, k2, q1, q2, q3, q4, p1, p2, p3, z1, z2, z3;
394  double a, cg;
395 
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 );
400 
401  k1 = x2 - y3;
402  k2 = y1 - x3;
403 
404  q1 = max3( k1, k2, 0 );
405  q2 = min3( y1, x2, z3 + 1 ) - 1;
406  q3 = q1 - k1;
407  q4 = q1 - k2;
408 
409  p1 = y1 - q1 - 1;
410  p2 = x2 - q1 - 1;
411  p3 = z3 - q1;
412 
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 );
420 
421  if ( q1 != q2 ){
422  q3 = q2 - k1;
423  q4 = q2 - k2;
424  p1 = y1 - q2;
425  p2 = x2 - q2;
426  p3 = z3 - q2 + 1;
427  for( i = 0; i < ( q2 - q1 ); i++ )
428  cg = a - cg * ( ( p1 + i ) * ( p2 + i ) * ( p3 + i ) ) / ( ( q2 - i ) * ( q3 - i ) * ( q4 - i ) );
429  }
430  return( cg );
431 }
432 /*
433 ============================================================
434 */
435 double nf_amc_z_coefficient( int l1, int j1, int l2, int j2, int s, int ll ) {
436 /*
437 * Biedenharn's Z-coefficient coefficient
438 * = Z(l1 j1 l2 j2 | S L )
439 */
440  double z, clebsh_gordan = nf_amc_clebsh_gordan( l1, l2, 0, 0, ll ), racah = nf_amc_racah( l1, j1, l2, j2, s, ll );
441 
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;
445 
446  return( z );
447 }
448 /*
449 ============================================================
450 */
451 double nf_amc_zbar_coefficient( int l1, int j1, int l2, int j2, int s, int ll ) {
452 /*
453 * Lane & Thomas's Zbar-coefficient coefficient
454 * = Zbar(l1 j1 l2 j2 | S L )
455 * = (-i)^( -l1 + l2 + ll ) * Z(l1 j1 l2 j2 | S L )
456 *
457 * Lane & Thomas Rev. Mod. Phys. 30, 257-353 (1958).
458 * 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.
459 * Froehner uses Lane & Thomas convention as does T. Kawano.
460 */
461  double zbar, clebsh_gordan = nf_amc_clebsh_gordan( l1, l2, 0, 0, ll ), racah = nf_amc_racah( l1, j1, l2, j2, s, ll );
462 
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;
465 
466  return( zbar );
467 }
468 /*
469 ============================================================
470 */
471 double nf_amc_reduced_matrix_element( int lt, int st, int jt, int l0, int j0, int l1, int j1 ) {
472 /*
473 * Reduced Matrix Element for Tensor Operator
474 * = < l1j1 || T(YL,sigma_S)J || l0j0 >
475 *
476 * M.B.Johnson, L.W.Owen, G.R.Satchler
477 * Phys. Rev. 142, 748 (1966)
478 * Note: definition differs from JOS by the factor sqrt(2j1+1)
479 */
480  int llt;
481  double x1, x2, x3, reduced_mat, clebsh_gordan;
482 
483  if ( parity( lt ) != parity( l0 ) * parity( l1 ) ) return( 0.0 );
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 );
486 
487  llt = 2 * lt;
488  jt *= 2;
489  st *= 2;
490 
491  if( ( clebsh_gordan = nf_amc_clebsh_gordan( j1, j0, 1, -1, jt ) ) == INFINITY ) return( INFINITY );
492 
493  reduced_mat = 1.0 / std::sqrt( 4 * M_PI ) * clebsh_gordan / std::sqrt( jt + 1.0 ) /* BRB 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 );
496 
497  if( st == 2 ){
498  x1 = ( l0 - j0 / 2.0 ) * ( j0 + 1.0 );
499  x2 = ( l1 - j1 / 2.0 ) * ( j1 + 1.0 );
500  if ( jt == llt ){
501  x3 = ( lt == 0 ) ? 0 : ( x1 - x2 ) / std::sqrt( lt * ( lt + 1.0 ) );
502  }
503  else if ( jt == ( llt - st ) ){
504  x3 = ( lt == 0 ) ? 0 : -( lt + x1 + x2 ) / std::sqrt( lt * ( 2.0 * lt + 1.0 ) );
505  }
506  else if ( jt == ( llt + st ) ){
507  x3 = ( lt + 1 - x1 - x2 ) / std::sqrt( ( 2.0 * lt + 1.0 ) * ( lt + 1.0 ) );
508  }
509  else{
510  x3 = 1.0;
511  }
512  }
513  else x3 = 1.0;
514  reduced_mat *= x3;
515 
516  return( reduced_mat );
517 }
518 /*
519 ============================================================
520 */
521 static int parity( int x ) {
522 
523  return( ( ( x / 2 ) % 2 == 0 ) ? 1 : -1 );
524 }
525 /*
526 ============================================================
527 */
528 static int max3( int a, int b, int c ) {
529 
530  if( a < b ) a = b;
531  if( a < c ) a = c;
532  return( a );
533 }
534 /*
535 ============================================================
536 */
537 static int max4( int a, int b, int c, int d ) {
538 
539  if( a < b ) a = b;
540  if( a < c ) a = c;
541  if( a < d ) a = d;
542  return( a );
543 }
544 /*
545 ============================================================
546 */
547 static int min3( int a, int b, int c ) {
548 
549  if( a > b ) a = b;
550  if( a > c ) a = c;
551  return( a );
552 }
553 
554 #if defined __cplusplus
555 }
556 #endif
G4Exp.hh
m3
static const double m3
Definition: G4SIunits.hh:130
z
G4double z
Definition: TRTMaterials.hh:39
cg3
static double cg3(int, int, int, int, int, int)
Definition: nf_angularMomentumCoupling.cc:391
max4
static int max4(int a, int b, int c, int d)
Definition: nf_angularMomentumCoupling.cc:537
nf_amc_racah
double nf_amc_racah(int j1, int j2, int l2, int l1, int j3, int l3)
Definition: nf_angularMomentumCoupling.cc:251
nf_amc_z_coefficient
double nf_amc_z_coefficient(int l1, int j1, int l2, int j2, int s, int ll)
Definition: nf_angularMomentumCoupling.cc:435
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: nf_angularMomentumCoupling.cc:224
w
const G4double w[NPOINTSGL]
Definition: G4PreCompoundFragment.cc:44
nf_amc_clebsh_gordan
double nf_amc_clebsh_gordan(int j1, int j2, int m1, int m2, int j3)
Definition: nf_angularMomentumCoupling.cc:286
a
G4double a
Definition: TRTMaterials.hh:39
nf_amc_wigner_3j
double nf_amc_wigner_3j(int j1, int j2, int j3, int j4, int j5, int j6)
Definition: nf_angularMomentumCoupling.cc:105
parity
static int parity(int x)
Definition: nf_angularMomentumCoupling.cc:521
cg1
static double cg1(int, int, int)
Definition: nf_angularMomentumCoupling.cc:337
s
static const double s
Definition: G4SIunits.hh:168
MAX_FACTORIAL
static const int MAX_FACTORIAL
Definition: nf_angularMomentumCoupling.cc:68
min3
static int min3(int a, int b, int c)
Definition: nf_angularMomentumCoupling.cc:547
nf_amc_log_fact
static const double nf_amc_log_fact[]
Definition: nf_angularMomentumCoupling.cc:70
m2
static const double m2
Definition: G4SIunits.hh:129
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: nf_angularMomentumCoupling.cc:471
cg2
static double cg2(int, int, int, int, int, int, int, int)
Definition: nf_angularMomentumCoupling.cc:363
n
const G4int n
Definition: G4UrQMD1_3Interface.hh:144
nf_amc_wigner_6j
double nf_amc_wigner_6j(int j1, int j2, int j3, int j4, int j5, int j6)
Definition: nf_angularMomentumCoupling.cc:121
G4Exp
G4double G4Exp(G4double initial_x)
Exponential Function double precision.
Definition: G4Exp.hh:183
nf_amc_zbar_coefficient
double nf_amc_zbar_coefficient(int l1, int j1, int l2, int j2, int s, int ll)
Definition: nf_angularMomentumCoupling.cc:451
w6j0
static double w6j0(int, int *)
Definition: nf_angularMomentumCoupling.cc:137
x
const G4double x[NPOINTSGL]
Definition: G4PreCompoundFragment.cc:56
nf_amc_factorial
double nf_amc_factorial(int n)
Definition: nf_angularMomentumCoupling.cc:96
nf_amc_log_factorial
double nf_amc_log_factorial(int n)
Definition: nf_angularMomentumCoupling.cc:85
w6j1
static double w6j1(int *)
Definition: nf_angularMomentumCoupling.cc:164
mm
static const double mm
Definition: G4SIunits.hh:114
max3
static int max3(int a, int b, int c)
Definition: nf_angularMomentumCoupling.cc:528