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35 // 35 //
36 // %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 36 // %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
37 // 37 //
38 // MODULE: G4Bessel.cc 38 // MODULE: G4Bessel.cc
39 // 39 //
40 // Version: B.1 40 // Version: B.1
41 // Date: 15/04/04 41 // Date: 15/04/04
42 // Author: P R Truscott 42 // Author: P R Truscott
43 // Organisation: QinetiQ Ltd, UK 43 // Organisation: QinetiQ Ltd, UK
44 // Customer: ESA/ESTEC, NOORDWIJK 44 // Customer: ESA/ESTEC, NOORDWIJK
45 // Contract: 17191/03/NL/LvH 45 // Contract: 17191/03/NL/LvH
46 // 46 //
47 // %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 47 // %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
48 // 48 //
49 // CHANGE HISTORY 49 // CHANGE HISTORY
50 // -------------- 50 // --------------
51 // 51 //
52 // 18 Noevmber 2003, P R Truscott, QinetiQ Ltd 52 // 18 Noevmber 2003, P R Truscott, QinetiQ Ltd, UK
53 // Created. 53 // Created.
54 // 54 //
55 // 15 March 2004, P R Truscott, QinetiQ Ltd, U 55 // 15 March 2004, P R Truscott, QinetiQ Ltd, UK
56 // Beta release 56 // Beta release
57 // 57 //
58 // 06 August 2015, A. Ribon, CERN 58 // 06 August 2015, A. Ribon, CERN
59 // Migrated to G4Exp, G4Log and G4Pow. 59 // Migrated to G4Exp, G4Log and G4Pow.
60 // 60 //
61 // %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 61 // %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
62 ////////////////////////////////////////////// 62 ////////////////////////////////////////////////////////////////////////////////
63 // 63 //
64 #include "G4Bessel.hh" 64 #include "G4Bessel.hh"
65 #include "G4PhysicalConstants.hh" 65 #include "G4PhysicalConstants.hh"
66 66
67 #include "G4Exp.hh" 67 #include "G4Exp.hh"
68 #include "G4Log.hh" 68 #include "G4Log.hh"
69 #include "G4Pow.hh" 69 #include "G4Pow.hh"
70 ////////////////////////////////////////////// 70 ////////////////////////////////////////////////////////////////////////////////
71 // 71 //
72 G4Bessel::G4Bessel () 72 G4Bessel::G4Bessel ()
73 {;} 73 {;}
74 ////////////////////////////////////////////// 74 ////////////////////////////////////////////////////////////////////////////////
75 // 75 //
76 G4Bessel::~G4Bessel () 76 G4Bessel::~G4Bessel ()
77 {;} 77 {;}
78 ////////////////////////////////////////////// 78 ////////////////////////////////////////////////////////////////////////////////
79 // 79 //
80 G4double G4Bessel::I0 (G4double x) 80 G4double G4Bessel::I0 (G4double x)
81 { 81 {
82 const G4double P1 = 1.0, 82 const G4double P1 = 1.0,
83 P2 = 3.5156229, 83 P2 = 3.5156229,
84 P3 = 3.0899424, 84 P3 = 3.0899424,
85 P4 = 1.2067492, 85 P4 = 1.2067492,
86 P5 = 0.2659732, 86 P5 = 0.2659732,
87 P6 = 0.0360768, 87 P6 = 0.0360768,
88 P7 = 0.0045813; 88 P7 = 0.0045813;
89 const G4double Q1 = 0.39894228, 89 const G4double Q1 = 0.39894228,
90 Q2 = 0.01328592, 90 Q2 = 0.01328592,
91 Q3 = 0.00225319, 91 Q3 = 0.00225319,
92 Q4 =-0.00157565, 92 Q4 =-0.00157565,
93 Q5 = 0.00916281, 93 Q5 = 0.00916281,
94 Q6 =-0.02057706, 94 Q6 =-0.02057706,
95 Q7 = 0.02635537, 95 Q7 = 0.02635537,
96 Q8 =-0.01647633, 96 Q8 =-0.01647633,
97 Q9 = 0.00392377; 97 Q9 = 0.00392377;
98 98
99 G4double I = 0.0; 99 G4double I = 0.0;
100 if (std::abs(x) < 3.75) 100 if (std::abs(x) < 3.75)
101 { 101 {
102 G4double y = G4Pow::GetInstance()->powN(x/ 102 G4double y = G4Pow::GetInstance()->powN(x/3.75, 2);
103 I = P1+y*(P2+y*(P3+y*(P4+y*(P5+y*(P6+y*P7) 103 I = P1+y*(P2+y*(P3+y*(P4+y*(P5+y*(P6+y*P7)))));
104 } 104 }
105 else 105 else
106 { 106 {
107 G4double ax = std::abs(x); 107 G4double ax = std::abs(x);
108 G4double y = 3.75/ax; 108 G4double y = 3.75/ax;
109 I = G4Exp(ax) / std::sqrt(ax) * 109 I = G4Exp(ax) / std::sqrt(ax) *
110 (Q1+y*(Q2+y*(Q3+y*(Q4+y*(Q5+y*(Q6+y*(Q7+ 110 (Q1+y*(Q2+y*(Q3+y*(Q4+y*(Q5+y*(Q6+y*(Q7+y*(Q8+y*Q9))))))));
111 } 111 }
112 return I; 112 return I;
113 } 113 }
114 ////////////////////////////////////////////// 114 ////////////////////////////////////////////////////////////////////////////////
115 // 115 //
116 G4double G4Bessel::K0 (G4double x) 116 G4double G4Bessel::K0 (G4double x)
117 { 117 {
118 const G4double P1 =-0.57721566, 118 const G4double P1 =-0.57721566,
119 P2 = 0.42278420, 119 P2 = 0.42278420,
120 P3 = 0.23069756, 120 P3 = 0.23069756,
121 P4 = 0.03488590, 121 P4 = 0.03488590,
122 P5 = 0.00262698, 122 P5 = 0.00262698,
123 P6 = 0.00010750, 123 P6 = 0.00010750,
124 P7 = 0.00000740; 124 P7 = 0.00000740;
125 const G4double Q1 = 1.25331414, 125 const G4double Q1 = 1.25331414,
126 Q2 =-0.07832358, 126 Q2 =-0.07832358,
127 Q3 = 0.02189568, 127 Q3 = 0.02189568,
128 Q4 =-0.01062446, 128 Q4 =-0.01062446,
129 Q5 = 0.00587872, 129 Q5 = 0.00587872,
130 Q6 =-0.00251540, 130 Q6 =-0.00251540,
131 Q7 = 0.00053208; 131 Q7 = 0.00053208;
132 132
133 G4double K = 0.0; 133 G4double K = 0.0;
134 if (x <= 2.0) 134 if (x <= 2.0)
135 { 135 {
136 G4double y = x * x / 4.0; 136 G4double y = x * x / 4.0;
137 K = (-G4Log(x/2.0)) * I0(x) + 137 K = (-G4Log(x/2.0)) * I0(x) +
138 P1+y*(P2+y*(P3+y*(P4+y*(P5+y*(P6+y*P7))) 138 P1+y*(P2+y*(P3+y*(P4+y*(P5+y*(P6+y*P7)))));
139 } 139 }
140 else 140 else
141 { 141 {
142 G4double y = 2.0 / x; 142 G4double y = 2.0 / x;
143 K = G4Exp(-x) / std::sqrt(x) * 143 K = G4Exp(-x) / std::sqrt(x) *
144 (Q1+y*(Q2+y*(Q3+y*(Q4+y*(Q5+y*(Q6+y*Q7)) 144 (Q1+y*(Q2+y*(Q3+y*(Q4+y*(Q5+y*(Q6+y*Q7))))));
145 } 145 }
146 return K; 146 return K;
147 } 147 }
148 ////////////////////////////////////////////// 148 ////////////////////////////////////////////////////////////////////////////////
149 // 149 //
150 G4double G4Bessel::I1 (G4double x) 150 G4double G4Bessel::I1 (G4double x)
151 { 151 {
152 const G4double P1 = 0.5, 152 const G4double P1 = 0.5,
153 P2 = 0.87890594, 153 P2 = 0.87890594,
154 P3 = 0.51498869, 154 P3 = 0.51498869,
155 P4 = 0.15084934, 155 P4 = 0.15084934,
156 P5 = 0.02658733, 156 P5 = 0.02658733,
157 P6 = 0.00301532, 157 P6 = 0.00301532,
158 P7 = 0.00032411; 158 P7 = 0.00032411;
159 const G4double Q1 = 0.39894228, 159 const G4double Q1 = 0.39894228,
160 Q2 =-0.03988024, 160 Q2 =-0.03988024,
161 Q3 =-0.00362018, 161 Q3 =-0.00362018,
162 Q4 = 0.00163801, 162 Q4 = 0.00163801,
163 Q5 =-0.01031555, 163 Q5 =-0.01031555,
164 Q6 = 0.02282967, 164 Q6 = 0.02282967,
165 Q7 =-0.02895312, 165 Q7 =-0.02895312,
166 Q8 = 0.01787654, 166 Q8 = 0.01787654,
167 Q9 =-0.00420059; 167 Q9 =-0.00420059;
168 168
169 G4double I = 0.0; 169 G4double I = 0.0;
170 if (std::abs(x) < 3.75) 170 if (std::abs(x) < 3.75)
171 { 171 {
172 G4double ax = std::abs(x); 172 G4double ax = std::abs(x);
173 G4double y = G4Pow::GetInstance()->powN(x/ 173 G4double y = G4Pow::GetInstance()->powN(x/3.75, 2);
174 I = ax*(P1+y*(P2+y*(P3+y*(P4+y*(P5+y*(P6+y 174 I = ax*(P1+y*(P2+y*(P3+y*(P4+y*(P5+y*(P6+y*P7))))));
175 } 175 }
176 else 176 else
177 { 177 {
178 G4double ax = std::abs(x); 178 G4double ax = std::abs(x);
179 G4double y = 3.75/ax; 179 G4double y = 3.75/ax;
180 I = G4Exp(ax) / std::sqrt(ax) * 180 I = G4Exp(ax) / std::sqrt(ax) *
181 (Q1+y*(Q2+y*(Q3+y*(Q4+y*(Q5+y*(Q6+y*(Q7+ 181 (Q1+y*(Q2+y*(Q3+y*(Q4+y*(Q5+y*(Q6+y*(Q7+y*(Q8+y*Q9))))))));
182 } 182 }
183 if (x < 0.0) I = -I; 183 if (x < 0.0) I = -I;
184 return I; 184 return I;
185 } 185 }
186 ////////////////////////////////////////////// 186 ////////////////////////////////////////////////////////////////////////////////
187 // 187 //
188 G4double G4Bessel::K1 (G4double x) 188 G4double G4Bessel::K1 (G4double x)
189 { 189 {
190 const G4double P1 = 1.0, 190 const G4double P1 = 1.0,
191 P2 = 0.15443144, 191 P2 = 0.15443144,
192 P3 =-0.67278579, 192 P3 =-0.67278579,
193 P4 =-0.18156897, 193 P4 =-0.18156897,
194 P5 =-0.01919402, 194 P5 =-0.01919402,
195 P6 =-0.00110404, 195 P6 =-0.00110404,
196 P7 =-0.00004686; 196 P7 =-0.00004686;
197 const G4double Q1 = 1.25331414, 197 const G4double Q1 = 1.25331414,
198 Q2 = 0.23498619, 198 Q2 = 0.23498619,
199 Q3 =-0.03655620, 199 Q3 =-0.03655620,
200 Q4 = 0.01504268, 200 Q4 = 0.01504268,
201 Q5 =-0.00780353, 201 Q5 =-0.00780353,
202 Q6 = 0.00325614, 202 Q6 = 0.00325614,
203 Q7 =-0.00068245; 203 Q7 =-0.00068245;
204 204
205 G4double K = 0.0; 205 G4double K = 0.0;
206 if (x <= 2.0) 206 if (x <= 2.0)
207 { 207 {
208 G4double y = x * x / 4.0; 208 G4double y = x * x / 4.0;
209 K = G4Log(x/2.0)*I1(x) + 1.0/x * 209 K = G4Log(x/2.0)*I1(x) + 1.0/x *
210 (P1+y*(P2+y*(P3+y*(P4+y*(P5+y*(P6+y*P7)) 210 (P1+y*(P2+y*(P3+y*(P4+y*(P5+y*(P6+y*P7))))));
211 } 211 }
212 else 212 else
213 { 213 {
214 G4double y = 2.0 / x; 214 G4double y = 2.0 / x;
215 K = G4Exp(-x) / std::sqrt(x) * 215 K = G4Exp(-x) / std::sqrt(x) *
216 (Q1+y*(Q2+y*(Q3+y*(Q4+y*(Q5+y*(Q6+y*Q7)) 216 (Q1+y*(Q2+y*(Q3+y*(Q4+y*(Q5+y*(Q6+y*Q7))))));
217 } 217 }
218 return K; 218 return K;
219 } 219 }
220 ////////////////////////////////////////////// 220 ////////////////////////////////////////////////////////////////////////////////
221 // 221 //
222 G4double G4Bessel::pI0 (G4double x) 222 G4double G4Bessel::pI0 (G4double x)
223 { 223 {
224 const G4double A0 = 0.1250000000000E+00, 224 const G4double A0 = 0.1250000000000E+00,
225 A1 = 7.0312500000000E-02, 225 A1 = 7.0312500000000E-02,
226 A2 = 7.3242187500000E-02, 226 A2 = 7.3242187500000E-02,
227 A3 = 1.1215209960938E-01, 227 A3 = 1.1215209960938E-01,
228 A4 = 2.2710800170898E-01, 228 A4 = 2.2710800170898E-01,
229 A5 = 5.7250142097473E-01, 229 A5 = 5.7250142097473E-01,
230 A6 = 1.7277275025845E+00, 230 A6 = 1.7277275025845E+00,
231 A7 = 6.0740420012735E+00, 231 A7 = 6.0740420012735E+00,
232 A8 = 2.4380529699556E+01, 232 A8 = 2.4380529699556E+01,
233 A9 = 1.1001714026925E+02, 233 A9 = 1.1001714026925E+02,
234 A10 = 5.5133589612202E+02, 234 A10 = 5.5133589612202E+02,
235 A11 = 3.0380905109224E+03; 235 A11 = 3.0380905109224E+03;
236 236
237 G4double I = 0.0; 237 G4double I = 0.0;
238 if (x == 0.0) 238 if (x == 0.0)
239 { 239 {
240 I = 1.0; 240 I = 1.0;
241 } 241 }
242 else if (x < 18.0) 242 else if (x < 18.0)
243 { 243 {
244 I = 1.0; 244 I = 1.0;
245 G4double y = x * x; 245 G4double y = x * x;
246 G4double q = 1.0; 246 G4double q = 1.0;
247 for (G4int i=1; i<101; i++) 247 for (G4int i=1; i<101; i++)
248 { 248 {
249 q *= 0.25 * y / i / i; 249 q *= 0.25 * y / i / i;
250 I += q; 250 I += q;
251 if (std::abs(q/I) < 1.0E-15) break; 251 if (std::abs(q/I) < 1.0E-15) break;
252 } 252 }
253 } 253 }
254 else 254 else
255 { 255 {
256 G4double y = 1.0 / x; 256 G4double y = 1.0 / x;
257 I = G4Exp(x) / std::sqrt(twopi*x) * 257 I = G4Exp(x) / std::sqrt(twopi*x) *
258 (1.0 + y*(A0+y*(A1+y*(A2+y*(A3+y*(A4+y*( 258 (1.0 + y*(A0+y*(A1+y*(A2+y*(A3+y*(A4+y*(A5+y*(A6+y*(A7+y*(A8+y*(A9+y*(A10+y*A11))))))))))));
259 } 259 }
260 260
261 return I; 261 return I;
262 } 262 }
263 ////////////////////////////////////////////// 263 ////////////////////////////////////////////////////////////////////////////////
264 // 264 //
265 G4double G4Bessel::pI1 (G4double x) 265 G4double G4Bessel::pI1 (G4double x)
266 { 266 {
267 const G4double A0 = -0.3750000000000E+00, 267 const G4double A0 = -0.3750000000000E+00,
268 A1 = -1.1718750000000E-01, 268 A1 = -1.1718750000000E-01,
269 A2 = -1.0253906250000E-01, 269 A2 = -1.0253906250000E-01,
270 A3 = -1.4419555664063E-01, 270 A3 = -1.4419555664063E-01,
271 A4 = -2.775764465332E-01, 271 A4 = -2.775764465332E-01,
272 A5 = -6.7659258842468E-01, 272 A5 = -6.7659258842468E-01,
273 A6 = -1.9935317337513E+00, 273 A6 = -1.9935317337513E+00,
274 A7 = -6.8839142681099E+00, 274 A7 = -6.8839142681099E+00,
275 A8 = -2.7248827311269E+01, 275 A8 = -2.7248827311269E+01,
276 A9 = -1.2159789187654E+02, 276 A9 = -1.2159789187654E+02,
277 A10 = -6.0384407670507E+02, 277 A10 = -6.0384407670507E+02,
278 A11 = -3.3022722944809E+03; 278 A11 = -3.3022722944809E+03;
279 279
280 G4double I = 0.0; 280 G4double I = 0.0;
281 if (x == 0.0) 281 if (x == 0.0)
282 { 282 {
283 I = 0.0; 283 I = 0.0;
284 } 284 }
285 else if (x < 18.0) 285 else if (x < 18.0)
286 { 286 {
287 I = 1.0; 287 I = 1.0;
288 G4double y = x * x; 288 G4double y = x * x;
289 G4double q = 1.0; 289 G4double q = 1.0;
290 for (G4int i=1; i<101; i++) 290 for (G4int i=1; i<101; i++)
291 { 291 {
292 q *= 0.25 * y / i / (i+1.0); 292 q *= 0.25 * y / i / (i+1.0);
293 I += q; 293 I += q;
294 if (std::abs(q/I) < 1.0E-15) break; 294 if (std::abs(q/I) < 1.0E-15) break;
295 } 295 }
296 I *= 0.5 * x; 296 I *= 0.5 * x;
297 297
298 } 298 }
299 else 299 else
300 { 300 {
301 G4double y = 1.0 / x; 301 G4double y = 1.0 / x;
302 I = G4Exp(x) / std::sqrt(twopi*x) * 302 I = G4Exp(x) / std::sqrt(twopi*x) *
303 (1.0 + y*(A0+y*(A1+y*(A2+y*(A3+y*(A4+y*( 303 (1.0 + y*(A0+y*(A1+y*(A2+y*(A3+y*(A4+y*(A5+y*(A6+y*(A7+y*(A8+y*(A9+y*(A10+y*A11))))))))))));
304 } 304 }
305 305
306 return I; 306 return I;
307 } 307 }
308 ////////////////////////////////////////////// 308 ////////////////////////////////////////////////////////////////////////////////
309 // 309 //
310 G4double G4Bessel::pK0 (G4double x) 310 G4double G4Bessel::pK0 (G4double x)
311 { 311 {
312 const G4double A0 = 0.1250000000000E+00, 312 const G4double A0 = 0.1250000000000E+00,
313 A1 = 0.2109375000000E+00, 313 A1 = 0.2109375000000E+00,
314 A2 = 1.0986328125000E+00, 314 A2 = 1.0986328125000E+00,
315 A3 = 1.1775970458984E+01, 315 A3 = 1.1775970458984E+01,
316 A4 = 2.1461706161499E+02, 316 A4 = 2.1461706161499E+02,
317 A5 = 5.9511522710323E+03, 317 A5 = 5.9511522710323E+03,
318 A6 = 2.3347645606175E+05, 318 A6 = 2.3347645606175E+05,
319 A7 = 1.2312234987631E+07; 319 A7 = 1.2312234987631E+07;
320 320
321 G4double K = 0.0; 321 G4double K = 0.0;
322 if (x == 0.0) 322 if (x == 0.0)
323 { 323 {
324 K = 1.0E+307; 324 K = 1.0E+307;
325 } 325 }
326 else if (x < 9.0) 326 else if (x < 9.0)
327 { 327 {
328 G4double y = x * x; 328 G4double y = x * x;
329 G4double C = -G4Log(x/2.0) - 0.57721566490 329 G4double C = -G4Log(x/2.0) - 0.5772156649015329;
330 G4double q = 1.0; 330 G4double q = 1.0;
331 G4double t = 0.0; 331 G4double t = 0.0;
332 for (G4int i=1; i<51; i++) 332 for (G4int i=1; i<51; i++)
333 { 333 {
334 q *= 0.25 * y / i / i; 334 q *= 0.25 * y / i / i;
335 t += 1.0 / i ; 335 t += 1.0 / i ;
336 K += q * (t+C); 336 K += q * (t+C);
337 } 337 }
338 K += C; 338 K += C;
339 } 339 }
340 else 340 else
341 { 341 {
342 G4double y = 1.0 / x / x; 342 G4double y = 1.0 / x / x;
343 K = 0.5 / x / pI0(x) * 343 K = 0.5 / x / pI0(x) *
344 (1.0 + y*(A0+y*(A1+y*(A2+y*(A3+y*(A4+y*( 344 (1.0 + y*(A0+y*(A1+y*(A2+y*(A3+y*(A4+y*(A5+y*(A6+y*A7))))))));
345 } 345 }
346 346
347 return K; 347 return K;
348 } 348 }
349 ////////////////////////////////////////////// 349 ////////////////////////////////////////////////////////////////////////////////
350 // 350 //
351 G4double G4Bessel::pK1 (G4double x) 351 G4double G4Bessel::pK1 (G4double x)
352 { 352 {
353 G4double K = 0.0; 353 G4double K = 0.0;
354 if (x == 0.0) 354 if (x == 0.0)
355 K = 1.0E+307; 355 K = 1.0E+307;
356 else 356 else
357 K = (1.0/x - pI1(x)*pK0(x)) / pI0(x); 357 K = (1.0/x - pI1(x)*pK0(x)) / pI0(x);
358 return K; 358 return K;
359 } 359 }
360 ////////////////////////////////////////////// 360 ////////////////////////////////////////////////////////////////////////////////
361 // 361 //
362 362