Correlated Sampling and Source Iteration

Most Monte Carlo codes determine the eigenvalue of a system by the source iteration method. Application of the source iteration method encounters difficulties when used to calculate eigenvalue perturbations because perturbed weights are propagated from one generation to another. Experience has shown [Rie89, Rie84, Gal95] that the statistical fluctuations in the propagated weights grow considerably, and for most cases any useful information regarding the perturbation is lost over many generations. We have solved the same problems given in table 3.1 by applying the correlated sampling technique to the source iteration method of eigenvalue calculation. The actual Monte Carlo simulation is done in the reference system, with cross sections determined as the average of the unperturbed and perturbed systems. As discussed earlier, a $\delta$-scatter cross section is added to the total cross section for each of the reference, unperturbed, and perturbed systems to avoid fluctuations in the adjusting weight factors. For illustration, the value of $\delta^{ref}$ is given along with the perturbation results in table 3.2. The standard deviation of a single generation $\Delta$K is given by,

$$\sigma_s = \sqrt {{{\sum\limits _{n=1}^{I_a}}{\Delta}{K_n^2}... ..._{n=1}^{I_a}}{\Delta}{K_n})}^2 \over {I_a(I_a-1)}}} \;\;\;,$$ (71)

where I_a is the active number of fission generations up to and including the current generation. The standard deviation of the mean is,  

$$\sigma = {\sigma_s \over {I_a}^{1 \over 2}}\;\;\;,$$ (72)

and is provided with the numerical Monte Carlo results in table 3.2.

 

1|cUnperturbed 1ccross sections: 1c 1c|

$$\Sigma_t \nu\Sigma_f \Sigma_s$$ 1c|

$$\Delta \Delta$$ 1|cProblem # 1|c|Perturbed cross sections

$$\Delta\Sigma_t \Delta\Sigma_a \mp$$ 1 -.008807

$$\delta^{ref}$$

$$\Delta\Sigma_t \Delta\Sigma_a \mp$$ 2 -.000888

$$\delta^{ref}$$

$$\Delta\Sigma_f {\pm}$$ 3 .000903

$$\delta^{ref}$$

Similar to the table 3.1 results, the Monte Carlo runs for table 3.2 are for 30 inactive batches, 70 active batches, and 2000 neutrons per batch. We observe from table 3.2 that for the problems analyzed, the source iteration method has difficulty estimating the differential effect in eigenvalue. As mentioned before in section 3.2.1, the problems of table 3.2 will be solved (in section 3.3.4) using the CSFM method and will provide significantly improved results.