Fission Product Yield Data for the Transmutation of Minor Actinide ...
Fission Product Yield Data for the Transmutation of Minor Actinide ...
Fission Product Yield Data for the Transmutation of Minor Actinide ...
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
Frequently, <strong>the</strong> yields Y i (M) are approximated by<br />
Gaussians.<br />
The adoption <strong>of</strong> ei<strong>the</strong>r this function or any<br />
o<strong>the</strong>r to approximate Y i (M) is not necessary. The<br />
multi-component analysis <strong>of</strong> <strong>the</strong> mass and energy<br />
distributions can be fulfilled without any<br />
assumptions about <strong>the</strong> shapes <strong>of</strong> Y i (M), which could<br />
serve as an important source <strong>of</strong> in<strong>for</strong>mation about<br />
<strong>the</strong> <strong>for</strong>mation <strong>of</strong> distinct fission modes. As outlined<br />
below, we have proposed two versions <strong>of</strong> such an<br />
approach to analyse <strong>the</strong> mass and energy distributions<br />
on <strong>the</strong> basis <strong>of</strong> Eq. (4.5.4) from <strong>the</strong> following<br />
considerations:<br />
The following equations:<br />
Â<br />
Â<br />
Y (M) = Y ( M,E )<br />
exp exp k<br />
Ek<br />
= Y(M) Y ( E ) = Y(M),<br />
i<br />
 Â<br />
i i,M k<br />
Ek<br />
n<br />
n<br />
k exp<br />
Ek<br />
k exp k<br />
Ek<br />
exp k<br />
E ( M) = E Y ( M,E ) / Y ( M,E ) =<br />
 ÂEn k<br />
Y ( M)( Y<br />
i<br />
i Ek<br />
Â<br />
i<br />
n<br />
k i<br />
 Â<br />
E Y (M ) /Y ( M )<br />
i<br />
i, M k exp k<br />
Ek<br />
exp<br />
Â<br />
depend linearly on Y i(M). There<strong>for</strong>e, <strong>for</strong> every mass<br />
M and any total number L <strong>of</strong> independent modes<br />
one can build a system <strong>of</strong> L linear equations where<br />
<strong>the</strong> power n in <strong>the</strong> second relation has values from 1<br />
to L-1.<br />
In <strong>the</strong> three-modal case (S1, S2 and S),<br />
Eqs (4.5.5) reduce to <strong>the</strong> equation system outlined<br />
in Refs [4.5.6, 4.5.25]:<br />
=Â<br />
Y (M) Y (M),<br />
exp i<br />
i<br />
Y(M) i<br />
E k,exp(M)<br />
= Â E k,i(M)<br />
,<br />
Y (M)<br />
s<br />
2<br />
E,exp<br />
i<br />
exp<br />
i<br />
(E )) / Y ( M,E ) =<br />
Ï Y(M) i 2 ¸<br />
Ô s E,i(M)<br />
Y exp(M)<br />
Ô<br />
Ô<br />
Ô<br />
Ô Y i(M)Y j(M)<br />
Ô<br />
Ô<br />
Ô<br />
(M) = Â Ì+<br />
Â<br />
¥<br />
2 ˝<br />
Y j exp(M)<br />
i Ô<br />
Ô<br />
Ô<br />
2 Ô<br />
ÔÈ<br />
Î<br />
E k,i(<br />
M) - E k,j(M)<br />
˘ Ô<br />
˚<br />
ÓÔ<br />
˛Ô<br />
i<br />
(4.5.5)<br />
(..) 456<br />
E – k,i (M) and s 2 E,i (M) are <strong>the</strong> average total kinetic<br />
energy and variance as a function <strong>of</strong> fragment mass<br />
M, where <strong>the</strong> indices i and j define <strong>the</strong> modes S1, S2<br />
or S.<br />
Solutions to Eqs (4.5.5) or (4.5.6) require<br />
definitions <strong>of</strong> <strong>the</strong> expressions <strong>for</strong> Yi,M (Ek ) that are<br />
proposed in our work [4.5.6] and described in detail<br />
below. However, it should be noted that any<br />
ma<strong>the</strong>matical approximation <strong>of</strong> Yi,M (Ek ) does not<br />
take into account <strong>the</strong> statistical scatter <strong>of</strong> <strong>the</strong><br />
experimental MEDs, and if an approximate value is<br />
used <strong>for</strong> Yi,M (Ek ), Eq. (4.5.4) also becomes an<br />
approximation:<br />
Y (M,E ) ª Â Y (M)Y (E )<br />
exp k i i,M k<br />
i<br />
that can lead to undesired dependences <strong>of</strong> <strong>the</strong><br />
analysis results on <strong>the</strong> statistical uncertainties <strong>of</strong> <strong>the</strong><br />
experimental MEDs.<br />
We have suggested a more general method to<br />
avoid this effect [4.5.25] that allows <strong>the</strong> yields Y i (M)<br />
to be determined with <strong>the</strong> experimental uncertainties<br />
<strong>of</strong> Y exp,M (E k ) taken into account. This<br />
approach is based on <strong>the</strong> least squares method,<br />
where<br />
ÏÔ<br />
¸Ô<br />
ÌY<br />
exp(M,E k ) -Â<br />
Y i(M)Y i,M(E k ) ˝<br />
ÓÔ<br />
i<br />
˛Ô<br />
is minimized. The functional <strong>of</strong> c 2 (M) can be written<br />
as:<br />
c<br />
2<br />
(M)=<br />
 Ek<br />
Ï 1<br />
¸<br />
Ô ¥<br />
2 d (M,E k )<br />
Ô<br />
Ô<br />
Ô<br />
Ì<br />
2 ˝<br />
ÔÈ<br />
˘ Ô<br />
ÔÍÂ Y i(M)Y i,M(E k ) - Y exp(M,E k ) ˙ Ô<br />
ÓÎÍ<br />
i<br />
˚˙<br />
˛<br />
(4.5.7)<br />
in which d(M,E k ) is <strong>the</strong> experimental uncertainty <strong>of</strong><br />
Y exp,M(E k).<br />
c 2 (M) is minimized if <strong>the</strong> conditions ∂c 2 (M)/<br />
∂Y i (M) = 0 are fulfilled simultaneously <strong>for</strong> all i.<br />
Since <strong>the</strong> derivatives depend linearly on Y i(M), <strong>the</strong><br />
determination <strong>of</strong> optimum values <strong>for</strong> Y i (M) is<br />
reduced to <strong>the</strong> solution <strong>of</strong> <strong>the</strong> following system <strong>of</strong><br />
linear equations:<br />
2<br />
191