--------------- Although FLUKA is able to perform fully analogue particle transport calculations (i.e. to reproduce faithfully actual particle histories), in many cases of very non-uniform radiation fields, such as those encountered in shielding design, only a very small fraction of all the histories contributes to the desired response (dose, fluence) in the regions of interest, for instance behind thick shielding. In these cases, the user's concern is not to simulate exactly what occurs in reality, but to estimate in the most efficient way the desired response. This can be obtained by replacing the actual physical problem with a mathematically equivalent one, i.e. having the same solution but faster statistical convergence. A rigorous mathematical treatment of such variance-reduction techniques can be found in several textbooks (see for instance those of Lux and Koblinger [Lux91] or Carter and Cashwell [Car75]). In the present practical introduction we will only issue a few important warnings: 1. In the limit of the number of histories tending to infinity, the value of all calculated quantities tend EXACTLY to the same average in the analogue and in the corresponding biased calculation. In other words, biasing is mathematically correct and implies no approximation. However, an acceleration of convergence in certain regions of phase space (space/energy/angle) will generally be paid for by a slower convergence in other regions. Because an actual calculation does not use an infinite number of particles, but is necessarily truncated after a finite number of histories, results must be considered with some judgment. For instance, if the number of histories is too small, it can happen that total energy is not conserved (check the energy budget summary at the very end of main output!) 2. A bad choice of biasing parameters may have consequences opposite to what is desired, namely a slower convergence. A long experience, and often some preliminary trial-and-error investigation, are needed in order to fully master these techniques (but some biasing algorithms are "safer" than others). 3. Because biasing implies replacing some distributions by others having the same expectation values but different variance (and different higher moments), biasing techniques in general do not conserve correlations and cannot describe correctly fluctuations. The simplest (and safest) biasing option offered by FLUKA is importance biasing, which can be requested by option BIASING. Each geometry region is assigned an "importance", namely a number between 1.0E-4 and 1.0E+4, proportional to the contribution that particles in that region are expected to give to the desired result. The ratio of importances in any two adjacent regions is used to drive a classical biasing algorithm ("Splitting" and "Russian Roulette"). In a simple, monodimensional attenuation problem, the importance is often set equal to the inverse of the expected fluence attenuation factor for each region. In electron accelerator shielding, two other biasing options are commonly employed: EMF-BIAS and LAM-BIAS. The first one is used to request leading particle biasing, a technique which reduces considerably the computer time required to handle high-energy electromagnetic showers. With this option, CPU time becomes proportional to primary energy rather than increasing exponentially with it. Option LAM-BIAS is necessary in order to sample with acceptable statistics photonuclear reactions which have a much lower probability than competing electromagnetic photon reactions, but are often more important from the radiological point of view. Other important options are those which set weight window biasing (WW-FACTOr WW-THRESh and WW-PROFIle) but their use requires more experience than assumed here for a beginner. Particle importances, weight windows and low-energy neutron biasing parameters are reported for each region in standard output. On user's request (expressed asSDUM= PRINT in a BIASING card), Russian Roulette and Splitting counters are printed for each region on standard output before the final summary. Such counters can be used for a better tuning of the biasing options.
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