R e ¼ R e0 1 þ R eb þ R es
ð
Þ
ð 17:35Þ
where R e0 is the true value. Also, calculated neutronics parameters R c are expressed
by
R c ¼ R c0 1 þ R cb þ R cs þ SΔσ
ð
Þ
ð 17:36Þ
where R c0 is the true value, R cb is systematic error, R cs is statistical error from
calculation methods, and SΔσ is the error from cross-section error. To eliminate the
systematic errors in measurements and calculations, we consider the ratio of
measurement to calculation, called bias factors:
f ¼
R e
R c
¼
1 þ R eb þ R es
1 þ R cb þ R cs þ SΔσ
ð17:37Þ
Because the average of statistical errors becomes zero, the variance of f becomes
V f
ð Þ ¼ V R es
ð ÞþV R cs
ð Þþ SWS
T
ð17:38Þ
where W is the variance of nuclear data used. In deriving Eq. (17.38) it was assumed
that all the systematic and statistical errors are smaller than unity and that there is no
correlation between statistical errors of measurements and calculations. From
Eq. (17.38) we can say that if there is no statistical error, the bias factor f is within
the range of
1 À cσ < f < 1 þ cσ σ ¼
ffiffiffiffiffiffiffiffiffi
V f
ð Þ
p
ð17:39Þ
with the confidence level of 65 %(c ¼ 1), 95 % (c ¼ 2), or 99 %(c ¼ 3). Therefore, if
f is outside the range, we can say in the foregoing confidence level, there is a
systematic error of
R eb À R cb ¼ 1 À f
j
jÀ cσ
ð17:40Þ
For sodium void calculations, calculated values are the sum of positive
nonleakage components and negative leakage components. The negative leakage
components are difficult to estimate because the transport effect has to be considered in calculating the neutron steaming. Therefore, there may be a nonnegligible
systematic error in the leakage term R
L
cb when the void pattern is leaky. By
considering such a void pattern, we can discard the leakage term in systematic
errors. Thus, we can determine the systematic errors. After the removal of the
systematic errors, we can apply the cross-section adjustment method or the bias
factor method to improve the calculation accuracy. In the cross-section adjustment
method [13], the adjusted cross section is determined so as to minimize the
functional J
17 Method Development for Calculating Minor Actinide Transmutation in a Fast. . .
193
ð
Þ
ð 17:35Þ
where R e0 is the true value. Also, calculated neutronics parameters R c are expressed
by
R c ¼ R c0 1 þ R cb þ R cs þ SΔσ
ð
Þ
ð 17:36Þ
where R c0 is the true value, R cb is systematic error, R cs is statistical error from
calculation methods, and SΔσ is the error from cross-section error. To eliminate the
systematic errors in measurements and calculations, we consider the ratio of
measurement to calculation, called bias factors:
f ¼
R e
R c
¼
1 þ R eb þ R es
1 þ R cb þ R cs þ SΔσ
ð17:37Þ
Because the average of statistical errors becomes zero, the variance of f becomes
V f
ð Þ ¼ V R es
ð ÞþV R cs
ð Þþ SWS
T
ð17:38Þ
where W is the variance of nuclear data used. In deriving Eq. (17.38) it was assumed
that all the systematic and statistical errors are smaller than unity and that there is no
correlation between statistical errors of measurements and calculations. From
Eq. (17.38) we can say that if there is no statistical error, the bias factor f is within
the range of
1 À cσ < f < 1 þ cσ σ ¼
ffiffiffiffiffiffiffiffiffi
V f
ð Þ
p
ð17:39Þ
with the confidence level of 65 %(c ¼ 1), 95 % (c ¼ 2), or 99 %(c ¼ 3). Therefore, if
f is outside the range, we can say in the foregoing confidence level, there is a
systematic error of
R eb À R cb ¼ 1 À f
j
jÀ cσ
ð17:40Þ
For sodium void calculations, calculated values are the sum of positive
nonleakage components and negative leakage components. The negative leakage
components are difficult to estimate because the transport effect has to be considered in calculating the neutron steaming. Therefore, there may be a nonnegligible
systematic error in the leakage term R
L
cb when the void pattern is leaky. By
considering such a void pattern, we can discard the leakage term in systematic
errors. Thus, we can determine the systematic errors. After the removal of the
systematic errors, we can apply the cross-section adjustment method or the bias
factor method to improve the calculation accuracy. In the cross-section adjustment
method [13], the adjusted cross section is determined so as to minimize the
functional J
17 Method Development for Calculating Minor Actinide Transmutation in a Fast. . .
193