Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

More documents

Recommendations

Info

466 Monte Carlo Particle Transport Methods: Neutron and Photon Calculationsand
46?moment of the score isM 2= Mf[I + 2('/Ti ~ 1/T,. ,)/1,] (7..70)j = iThe variance of the score readsD 2 = M 2- Mf = Mf 2 (1/T, - 1 /T:. ,)/1, (7.71)In order to optimize the efficiency of the scheme, we have to find an estimate of theexpected number of collisions to be played in a history. Let C 1be the average number ofcollisions suffered by a particle in region i. Neglecting the collisions suffered by particlesleaving a region in a negative direction (cf. assumption 3), the total expected number ofcollisions isN " 2 C,(k!)or, from Equation (7.69)N - 2 C 1T 1.. ,1, (1 T>\Thus, from Equations (7.71) and (7.72), the quality factor of the game readsQ = D 2 N = MfI (1/T 1- 1/T 1_,)/!, (V C 1T,. JEquating the derivative of Q with respect to 1, to zero, we obtain the quasi-optimum importances:I 1= OtI(T 1_, ~ T 1V(C 1T 2 ..,T,)] 1/2 (7.73)where a is constant and is determined by the conditionI 1= 1The quality factor with the optimized importances becomesf m 1 2Q„P, = Mf J2 [C 1(T,.., ~ T,)/T,]" 2 J (7.74)The quasi-optimum importances in Equation (7.73) contain the bulk parameters T 1(transmission from x„ to x,) and C 1(number of collisions in region i). These bulk parameters canbe determined in a preliminary small-sized run and/or can be modified in a production run.Thus, for using the importances in Equation (7.73), one has to estimate 2m bulk parameterswhere m is the number of regions. Introducing some further approximations, the number ofbulk parameters can be reduced to just one. Let us first note that to a reasonable accuracy.
Page 2 and 3:
Library of Congress Cataloging-in-P
Page 7 and 8:
III. Statistical Considerations •
Page 9 and 10:
IV. Further Generalizations — 183
Page 11:
Appendix 6A:Unbiased Estimation of
Page 14 and 15:
2 Monte Carlo Particle Transport Me
Page 17 and 18:
Chapter 2INTRODUCTIONWhen we starte
Page 19 and 20:
thus,As it is proved" — and is so
Page 21 and 22:
9B. THE PROBABILITY MIXING METHODTh
Page 23 and 24:
11It is easy to prove 15by summing
Page 25 and 26:
13but the most probable values:x, =
Page 27 and 28:
ISand give a random sign (by the us
Page 30 and 31:
18 Monte Carlo Particle Transport M
Page 32 and 33:
Page 34 and 35:
Page 36 and 37:
Page 38 and 39:
Page 40 and 41:
Page 42 and 43:
Page 44 and 45:
">3
Page 46 and 47:
Page 48 and 49:
Page 50 and 51:
Page 52 and 53:
Page 54 and 55:
Page 56 and 57:
Page 58 and 59:
Page 60 and 61:
Page 62 and 63:
Page 64 and 65:
Page 66 and 67:
Page 68 and 69:
Page 70 and 71:
Page 72 and 73:
Page 74 and 75:
Page 76 and 77:
Page 78 and 79:
Page 80 and 81:
Page 82 and 83:
Page 84 and 85:
Page 86 and 87:
Page 88 and 89:
Page 90 and 91:
Page 92 and 93:
Page 94 and 95:
Page 96 and 97:
Page 98 and 99:
Page 100 and 101:
Page 102 and 103:
Page 104 and 105:
Page 106 and 107:
Page 108 and 109:
Page 110 and 111:
Page 112 and 113:
100 Monte Carlo Panicle Transport M
Page 114 and 115:
102 Monte Carlo Particle Transport
Page 116 and 117:
Page 118 and 119:
Page 120 and 121:
Page 122 and 123:
Page 124 and 125:
Page 126 and 127:
Page 128 and 129:
Page 130 and 131:
Page 132 and 133:
Page 134 and 135:
Page 136 and 137:
Page 138 and 139:
Page 140 and 141:
Page 142 and 143:
Page 144 and 145:
Page 146 and 147:
Page 148 and 149:
Page 150 and 151:
Page 152 and 153:
Page 154 and 155:
Page 156 and 157:
Page 158 and 159:
Page 160 and 161:
Page 162 and 163:
Page 164 and 165:
Page 166 and 167:
Page 168 and 169:
Page 170 and 171:
Page 172 and 173:
Page 174 and 175:
Page 176 and 177:
Page 178 and 179:
Page 180 and 181:
Page 182 and 183:
Page 184 and 185:
Page 186 and 187:
Page 188 and 189:
Page 190 and 191:
Page 192 and 193:
Page 194 and 195:
Page 196 and 197:
Page 198 and 199:
Page 200 and 201:
Page 202 and 203:
Page 204 and 205:
Page 206 and 207:
Page 208 and 209:
Page 210 and 211:
Page 212 and 213:
Page 214 and 215:
Page 216 and 217:
Page 218 and 219:
Page 220 and 221:
Page 222 and 223:
Page 224 and 225:
Page 226 and 227:
Page 228 and 229:
Page 230 and 231:
Page 232 and 233:
Page 234 and 235:
Page 236 and 237:
Page 238 and 239:
Page 240 and 241:
Page 242 and 243:
Page 244 and 245:
Page 246 and 247:
Page 248 and 249:
Page 250 and 251:
Page 252 and 253:
Page 254 and 255:
Page 256 and 257:
Page 258 and 259:
Page 260 and 261:
Page 262 and 263:
25!) Monte Carlo Particle Transport
Page 264 and 265:
Page 266 and 267:
Page 268 and 269:
Page 270 and 271:
Page 272 and 273:
Page 274 and 275:
Page 276 and 277:
Page 278 and 279:
Page 280 and 281:
Page 282 and 283:
Page 284 and 285:
Page 286 and 287:
Page 288 and 289:
Page 290 and 291:
Page 292 and 293:
Page 294 and 295:
Page 296 and 297:
Page 298 and 299:
Page 300 and 301:
Page 302 and 303:
Page 304 and 305:
Page 306 and 307:
Page 308 and 309:
Page 310 and 311:
Page 312 and 313:
Page 314 and 315:
Page 317 and 318:
305Chapter 6SPECIAL GAMESIn the maj
Page 319 and 320:
3©7A. CORRELATED MOMENT EQUATIONSI
Page 321 and 322:
309Equation (6.1) will be referred
Page 323 and 324:
311The second moment of the score d
Page 325 and 326:
313andC S(P',P")/C S(P',P") = y(P',
Page 327 and 328:
315introduced in Equation (6.1) for
Page 329 and 330:
317andL„(P 0,P;,...,Pi + 1) = L n
Page 331 and 332:
319whereB„(P„,P; P:, ,) = A*(P
Page 333 and 334:
E. EXAMPLES AND SPECIAL TECHNIQUESL
Page 335 and 336:
323The weight factors associated wi
Page 337 and 338:
325It is seen from Equation (6.37)
Page 339 and 340:
327G. PARAMETRIC PERTURBATIONS: INT
Page 341 and 342:
129Hall 31 gave a constructive deri
Page 343 and 344:
331kernels, which remain unchanged
Page 345 and 346:
333and from Equation (6.51)w';,,(i)
Page 347 and 348:
335According to Equations (6,55) th
Page 349 and 350:
337Minimization is performed by set
Page 351 and 352:
339and letdsdaLet us consider a cor
Page 353 and 354:
341whereis the length of the flight
Page 355 and 356:
and from Equation (6.79)I / ds \ 2
Page 357 and 358:
345Let us assume that we are intere
Page 359 and 360:
347Introduction of this factor also
Page 361 and 362:
3492. Let i|/ n (P) denote the solu
Page 363 and 364:
3SJNow, if ( "> > 0, then k = i an
Page 365 and 366:
353andk„ = S l "VS'" •" (6. if;
Page 367 and 368:
3SSLet k„ denote the normalizatio
Page 369 and 370:
3572. It is then proved that with t
Page 371 and 372:
and start new histories until N new
Page 373 and 374:
361The application of source iterat
Page 375 and 376:
363we haveD 2 [k] = < 2 (k, - k,) )
Page 377 and 378:
365neutron density, S 0 for every
Page 379 and 380:
367if the same relation holds for t
Page 381 and 382:
369Then the multiplication factors
Page 383 and 384:
371in the unperturbed system, the w
Page 385 and 386:
373The perturbation of the effectiv
Page 387 and 388:
375where P" = (r',E'). Multiplying
Page 389 and 390:
377iterate of the derivative of the
Page 391 and 392:
579where P* = (r*,w*,E) = (r*,E*).
Page 393 and 394:
381This estimator, unlike f, in Equ
Page 395 and 396:
383tends to some stable attracting
Page 397 and 398:
3§SIo the case of the next-event e
Page 399 and 400:
387LDFIGURE 6.1.Geometry in scoring
Page 401 and 402:
389Let us denoteThen Equation (6.18
Page 403 and 404:
391is to be increased in order to o
Page 405 and 406:
393Now, since bl(P) = g(P) is bound
Page 407 and 408:
395is scored at every collision wit
Page 409 and 410:
397and its expectation isR — R 1n
Page 411 and 412:
3994. If O > 0 m, then the particle
Page 413 and 414:
401plays a distinguished role), and
Page 415 and 416:
403Theorem 6.7 — If x and s denot
Page 417 and 418:
this a priori information, the best
Page 419 and 420:
40?Thus, the bias introduced into t
Page 421 and 422:
409In view of Equation (6.226) and
Page 423 and 424:
41.1This means that if we use the k
Page 425 and 426:
413Then the whole-sample and rare-s
Page 427 and 428: 41.5According to the results of Sec
Page 429 and 430: 41?(Note that vD 2 [|i|v] is indepe
Page 431 and 432: 419withkOn the other hand, § is an
Page 433 and 434: 421Thus, the expected ratio reads
Page 435 and 436: 423St follows from Equations (6.265
Page 437 and 438: 425Wc start from the observation th
Page 439 and 440: 427Nowi'6.77K)kj„,and therefore(V
Page 441 and 442: 42*3Similarly, for the fourth momen
Page 443 and 444: 43 sOn the other hand, multiplying
Page 445 and 446: 43.¾This will be proven in the lem
Page 447 and 448: 435ThenT.m = EJ.Iy.y.Rjeyiy,,,and s
Page 449 and 450: 6. Brown. F. B. and Martin, W. R.,
Page 451: 67. Rief, H. and Fioretti, A., Mont
Page 454 and 455: 442 Monte Carlo Particle Transport
Page 482 and 483: Q = iy.M 2 2 (1/T, - 1/1,..,)/1, £
Page 492 and 493: 48© Monte Carlo Particle Transport
Page 496 and 497: 484 Monte Carlo Panicle Transport M
Page 498 and 499: 486 Monte Carlo Panicle Transport M
Page 518 and 519: a ( X506 Monte Carlo Particle Trans
Page 525 and 526: 513INDEXAAbsorptiondefined, 25photo
Page 527 and 528: 515score probability in general tim
Page 529:
517path stretching, 447—455splitt
show all

Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

Create successful ePaper yourself

Delete template?

Save as template?