Exponential transform with weight window

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The weight window generator was run for 5 minutes with a transform parameter of p = 07 and produced the weight window lower bounds in the 18 zones (from the bottom of the concrete to the top of the concrete) of 5000E-01, 1007E-01, 4147E-02, 1492E-02, 5008E-03, 1645E-03, 5485E-04, 1913E-04, 7117E-05, 2844E-05, 1199E-05, 5253E-06, 2356E-06, 1078E-06, 4891E-07, 2217E-07, 1007E-07, and 4610E-08 The survival weight and upper window bounds were 3 and 5 times the lower bounds respectively Table 102 shows results for 500 000 history runs with different transform parameters Comparing Tables 101 and 102, note that the particle histories per minute are roughly a factor of 10 less with the weight window On the other Table 102 Exponential transform with weight window Transform parameter 00 (analog) 01 02 03 04 05 06 07 08 09 099 (poor) Mean 10 8 42811 40905 41190 42112 41524 41000 40929 42260 40316 40399 40306 Fractional error 00335 00323 00308 00281 00272 00277 00264 00290 00334 00387 00670 Variance of Tail Figure Time the variance slope of merit (minutes) 00034 00041 00033 00023 00031 00042 00028 00058 00076 00101 00446 100 80 100 100 100 49 100 47 66 59 32 28 26 27 34 35 34 36 30 23 17 61 3200 3670 3826 3749 3880 3854 3993 3997 3953 3867 3659

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hand, the ef ciency (fom) is dramatically improved and every calculation with p 09 passes MCNP s ten statistical checks ([23], see also Section 1019) In particular, note that the tail slope estimate (k > 4) is indicating at least three nite moments for all calculations, so that use of the central limit theorem is possible, unlike the calculations in Table 101 The last line shows that even with a very poor choice of a very high transform parameter, the tail slope indicates that the variance is nite Elaborating a bit more, using the central limit theorem requires at least two nite moments Eventually, the high score tail of the density f (x) must decrease faster than 1/x 3 (slope 3), or else the second moment, x 2 f (x) dx, will not be nite Note that it is even better if the high score tail of the density f (x) decreases faster than 1/x 5 (slope 5), so that four nite moments exist When four moments exist, not only is the central limit theorem applicable, but also the sample variance is usually a good estimate of the true variance used in the central limit theorem Valid con dence intervals are thus much more likely when four moments exist A complete print (event log) of the largest scoring history for p = 06 indicates that one of the (relatively rare) high-energy 14 MeV source neutrons is responsible for the largest score At this point, one would attempt to bias the sampling of the source so that more 14 MeV neutrons were sampled, with correspondingly smaller weights One could proceed by using the weight window generator to estimate an energy-dependent weight window, then adjust the source energy bias so that the source particles are within their space-energy weight windows Space does not permit description here, but see [3] for an illustration of source energy bias and space-energy weight windows

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Collision biasing

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This chapter has shown how to get particles through a bulk concrete shield, but almost none of the penetrating particles will have just the right angle to stream up the void in Figure 101 Let the analog collision kernel be C(E, E , ) for a particle entering collision at E, and exiting at E , (The collision occurs at one spatial point in an instant of time, so space and time variables do not change upon collision) Using the expected score function I (E , ) one would like to sample from the biased collision density (given that exactly one particle is known to exit the collision) C(E, E, )= C(E, E , )I (E , ) C(E, E , )I (E , )dE d (1017)

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The trouble is that sampling C ef ciently is not straightforward Recall from the analog sampling section that the collision process is a complicated procedure that is composed of a number of steps Note that there are numerous ways to go from (E, E , ) One has to sum, over all possible reactions, on all possible collision nuclides, the probability densities for scattering to (E , ) Although

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