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it is not too dif cult to calculate C for any given (E , ), calculating C for all (E , ) required to evaluate the denominator in (1017) is not practical For this reason, it is problematical to sample C; in practice Monte Carlo transport codes use other approaches The TRIPOLI code [13] sometimes samples an approximation to C by sampling C n times and evaluating Ik = I (Ek , k ) The j th sample is picked with probability and weight multiplier pj = Ij n k=1 Ik =
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1 n Ij , n k=1 Ik
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(1018)
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Note that the denominator in the rightmost equality for pj is a Monte Carlo estimate of the integral in (1017) This is computationally expensive and therefore rarely used in TRIPOLI-4 runs [17] The MCBEND sampling [15] ignores C altogether and samples the biased collision output from p(E, E, )= I (E , ) I (E , )dE d , (1019)
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and the scattering physics associated with C is incorporated with the weight multiplication w= C(E, p(E, E, E, ) ) (1020)
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(The I function is a piecewise constant function in MCBEND and the importance function used is actually energy-independent as well) MCNP s dxtran method [31] accomplishes angle biasing in tandem with an expected value penetration technique The basic dxtran method consists of de ning a spherical region of interest The collision sampling proceeds as in Section 106, except that (after the collision nuclide, interaction type, and energy have been sampled) the dxtran method splits a particle into a dxtran particle that crosses the dxtran sphere (before its next collision) and a non-dxtran particle that does not The non-dxtran particle is sampled the same way, with the same weight, as it would have been without dxtran, except that the non-dxtran particle is killed if it reaches the sphere surface Thus, for events not including crossing the sphere, the weight of particles executing any next event is identical The dxtran particle s angle is sampled from an arbitrary density p( ) (usually a constant and always non-zero only in the cone of directions toward the sphere) with a weight multiplication such that the expected weight is preserved, wm p( ) = p( ), (1021)
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where p( ) is the unbiased density Of the particles scattered at that arrive at the surface of the sphere is e
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S( ) ( )d 0
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, the fraction
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where S( ) is the distance to the sphere in the sampled direction dxtran particle s weight at the sphere is wdxtran = p( ) e p( )
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Note that the dxtran particle has weight zero from the collision point until it reaches the sphere; it makes no tallies as these are already accounted for by the non-dxtran particle (MCBEND has a similar method called forced ight [15])
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For the sample problem here, a 100-cm radius dxtran sphere is placed at (0, 2000, 0); that is, at the top boundary of the void cylinder in Figure 101 All the runs in Table 103 were 60-minute runs with the same generated window and a transform parameter p = 06 Without dxtran, the fom is 12 and with dxtran the fom is 96 The history variance is decreasing with dxtran, but the fractional error in the mean reported in Table 103 has increased The problem is that dxtran takes too much time, 193 113 compared to 811 398 histories per hour Transport codes contain a wealth of sampling information beyond just the estimates themselves, and a look at this information indicates that the zones near the source put little weight on the dxtran sphere and the zones at the top of the concrete put weights of about 10 10 on the sphere It thus makes little sense to follow particles whose weights get lower than 10 10 The exponential factor in (1023) is accumulated zone by zone by moving the dxtran particle through each zone When the dxtran particle has been exponentially attenuated through a distance S ( ) < S( ) its nal weight at the dxtran sphere is known to be Table 103 Applying dxtran to improve tally F4 (60-minute runs per window and p = 06 ) Histories 811398 193113 341834 562564 Run type no dxtran dxtran dxtran/RR1 dxtran/RR 1,2 Mean 10 15 70365 74593 73325 74355 Fractional error 00372 00417 00315 00253 Variance of Tail Figure the variance slope of merit 00102 00060 00036 00043 33 100 85 53 12 96 17 26
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