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To save time, a roulette game was played when w < 10 10 because this was consistent with the typical dxtran weights from collisions near the top of the concrete cylinder That is, the particle either survived with probability w /10 10 with weight 10 10 or was killed This saves time because far fewer zones have to be tracked through When this roulette game (RR1) is played, the number of histories increases to 341 834 and the fom is now more than without dxtran In the zones near the source the particles will be subject to a large exponential attenuation, so it makes little sense to start a dxtran particle, track it through a number of zones until its weight becomes too low, and then roulette the particle If one knows this is happening (this is known from summary information from the transport code), then one can play a roulette game even before producing a dxtran particle For each cell the user supplies a roulette probability q If the dxtran particle survives the roulette game, it s weight is multiplied by q 1 , otherwise it is killed Near the top of the concrete cylinder there will be little attenuation so there we take q = 1 At the bottom we expect the attenuation factor to be at least 0001, so we take q = 01 there In between, we geometrically space them resulting in the q probabilities in the concrete cylinder: 001, 0013, 0017, 0023, 0030, 0039, 0051, 0067, 0087, 011, 015, 020, 026, 034, 044, 058, 076, 10 Note from Table [103] that this second roulette game increases the particles per hour to 562 564 and the fom improves to 26
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To estimate the ux at a point (the F5 tally), at each collision a tally of
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is made where R and are the distance and direction from the collision site to the detector point To make any F5 estimate, a collision must occur in the tiny low density concrete cell at the top of Figure 101 To increase the number of collisions in this cell, the particle is split into an uncollided weight fraction e L (that goes the full distance L through the cell) and a collided weight fraction 1 e L The collision point is sampled from the conditional density (given that a collision is known to occur in 0 < < L) pc (s)ds = e d 1 e L (1026)
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This ensures that every particle track that enters makes a contribution to the F5 tally The result is given in Table 104 The reference F5 tally without forced
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Table 104 Final results applying forced collisions to improve tally F5 (60-minute run ) Histories Tally type 562564 526600 ref F5 tally F1 tally F4 tally F5 Mean 43727 43196 75105 44836 10 18 10 8 10 15 10 18 Fractional Variance of Tail Figure error the variance slope of merit 00330 00263 00259 00259 00087 00028 00027 00027 43 100 100 100 15 24 25 25
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collision ( rst line) is from the run reported on the last line of Table 103 The fom improved from 15 to 25 This concludes the illustration of rare event simulation on a sample problem
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MCNP does not use the comb technique, so the comb has not been demonstrated on the sample problem just concluded Nonetheless, the comb is an old transport method and has some interesting aspects The comb [16] inputs K tracks (of the same history) of varying weights wi > 0 with sum W = w1 + + wK and outputs M tracks of constant weight L = W/M Suppose that one has K particles and one decides that one would rather follow M particles The K particles are combed into M particles using an M-toothed comb Figure 102 shows a comb with K = 6 and M = 4 The length of the comb is the sum of the particle weights
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