Fission Product Yield Data for the Transmutation of Minor Actinide ...

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