Unfolding the fission prompt gamma-ray energy and multiplicity distribution measured by DANCE Page: 8 of 15
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LLNL-TR-460216
The colored representations of the M-response matrix (Fig. 5 and 6) show that the
multiplicity range spans up to M = 23. The number of BaF2 crystals in DANCE is 160, but
the multiplicity cannot go that high for two reasons: a cluster may contain several crystals,
and clusters may overlap thus reducing Mc1 value. That's why the matrix is truncated to the
size of 23 for the practical purpose. The columns of the matrix become increasingly linearly
dependent on each other, i.e., as My increases, the dependence of Mc1 on My decreases. The
more linearly dependent matrix is, the more difficult is to solve it. That is, solving the
matrix becomes nearly impossible once its determinant closes to zero within the limits of
computer accuracy. Fig. 4 is a 3D view of the M-response matrix.
It is necessary to mention that the present matrix is built for the 1-MeV mono energetic
photons. The same matrix for the 2-MeV photons would look slightly different because the
DANCE detection efficiency weakly depends on the a-ray energy Ey (see Ref. [3]), where
the probabilities for detecting the M=1 events with the 1 MeV and 2 MeV energies are
87% and 85% correspondingly. For the higher multiplicity case such as My=5, the largest
Mc1 values varies less than 5%, but the total area of the Mc1 distribution, which is the total
efficiency, changes only less than 2%. The majority of a-rays produced in the neutron capture
and neutron induced fission reactions on many nuclei lay within 100 keV to several MeV.
Thus, for the first approximation, this M-response matrix can be used to obtain true My
distributions without knowing the exact a-ray energy if we assume the M-response matrix
for 1 MeV is the same for other energies.
Multiplicity unfolding
The M-response matrix can be used to derive the true a-ray multiplicity My from the
measured cluster multiplicity Mc1, the so-called unfolding procedure. Also the same matrix
can be used for folding the true My into the measured Mc1.
The folding procedure can be demonstrated in Fig 7. The Brunson distribution [4] for
the 252Cf fission prompt a-ray output (red) represents a true My distribution. The Mc1
distribution (black) is the simulated DANCE response to the Brunson distribution, i.e., the
Brunson distribution is fed into the GEANT4 model of DANCE as an input and the Mc1
distribution is the output. The folded Mfold distribution (blue) is the matrix product. The
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Chyzh, A.; Wu, C. Y.; Bredeweg, T.; Couture, A.; Jandel, M.; Ullmann, J. et al. Unfolding the fission prompt gamma-ray energy and multiplicity distribution measured by DANCE, report, October 16, 2010; Livermore, California. (https://digital.library.unt.edu/ark:/67531/metadc841722/m1/8/: accessed April 18, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.