Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

More documents

Recommendations

Info

306 Monte Carlo Particle Transport Methods: Neutron and Photon CalculationsConsequently, their difference is expected to have a smaller uncertainty than the correspondingdifference in the uncorrected simulation. This is the main idea behind the correlatedMonte Carlo method discussed in this Chapter.Let the perturbed system be characterized by the transition kernel T(P,P') and collisionkernel C(P',P"), and let the corresponding kernels in the perturbed system be T(P,P') andC(P',P"). For the sake of simplicity, we assume that both systems are finite and nonmultiplying.Extension to multiplying cases goes along the lines followed in Chapter 5.III.Unless otherwise stated, we shall assume that the correlated simulation is performed accordingto the unperturbed kernels, and the parallel game in the perturbed system is playedby introducing appropriate statistical weights, for convenience, we shall always talk abouttwo particles migrating in parallel (along identical trajectories). One is analog, which scoresin the unperturbed system and has a weight of unity; the other is a nonanalog particle,which should migrate in the perturbed system and therefore carries a statistical weight thataccounts for the difference between the unperturbed and perturbed systems.Thus, the game is analog as far as the estimate in the unperturbed system is concernedand nonanalog with respect to the estimation in the perturbed system. Such a correlatedgame will be called an unperturbed analog game. In the opposite case, when the game isplayed according to the perturbed kernels and statistical weights are introduced in the estimationof the unperturbed reaction rate, we talk about a perturbed analog game.Let I|J(P) and t\i(P) be the collision densities in the unperturbed and perturbed systems,respectively, and assume that the quantity to be determined is the reaction-rate difference8R = JdPiKP)f(P) -JdPiKP)f(P)If the source densities in the unperturbed and perturbed systems are denoted by Q(P) andQ(P), respectively, then according to the results in Section 5.1.A8R = JdPQ(P)M 1(P) IdPQ(P)M 1(P)dPQ(P)Q(P)Q(P)M 1(P) -M 1(P)where M 1(P) and M 1(P) are the expected scores in the unperturbed and perturbed systems,respectively, due to a unit weight starter. Now, starting the nonanalog particle with a weightW-- Q(P)/Q(P)from the analog source density and recording the difference of the nonanalog and analogscores, the variation of the reaction rate due to the perturbation is estimated. The varianceof this estimate, however, does not follow as easily as the first moment from the variancesof the uncorrelated ("single-particle") simulation. A slight generalization of the singleparticlemoment equations, nevertheless, will solve this problem. Equations describing momentsof functions of correlated scores are derived in the next section. The conditions ofthe feasibility of a correlated game are established in Section P>. Estimators of a correlatedscore difference are briefly discussed in Section C. The variance of the difference of thecorrelated scores is investigated in Section 1), and conditions are established under whichthe unperturbed analog game yields a lower variance than the perturbed analog game. Finally,alternative methods of calculating perturbations are reviewed in Sections F and G.
3©7A. CORRELATED MOMENT EQUATIONSIn the considerations below, we assume that the contribution function in both the analogand nonanalog (unperturbed and perturbed) games is f(P,P'), i.e., that a free flight from apoint P = (r,
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: 256 Monte Carlo Particle Transport
Page 317: 305Chapter 6SPECIAL GAMESIn the maj
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:
Page 456 and 457:
Page 458 and 459:
Page 460 and 461:
Page 462 and 463:
Page 464 and 465:
Page 466 and 467:
Page 468 and 469:
Page 470 and 471:
Page 472 and 473:
Page 474 and 475:
Page 476 and 477:
Page 478 and 479:
Page 480 and 481:
Page 482 and 483:
Q = iy.M 2 2 (1/T, - 1/1,..,)/1, £
Page 484 and 485:
Page 486 and 487:
Page 488 and 489:
Page 490 and 491:
Page 492 and 493:
48© Monte Carlo Particle Transport
Page 494 and 495:
Page 496 and 497:
Page 498 and 499:
Page 500 and 501:
Page 502 and 503:
Page 504 and 505:
Page 506 and 507:
Page 508 and 509:
Page 510 and 511:
Page 512 and 513:
Page 514 and 515:
Page 516 and 517:
Page 518 and 519:
a ( X506 Monte Carlo Particle Trans
Page 520 and 521:
Page 522 and 523:
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?