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As suggested by Uhrig [6], a method to obtain the k eff
, values was from the plot of the crosspower
spectral density versus the frequency. The simpler way is to find out the value of the eigenvalue
α
0
, from the break frequency value. The break frequency value for a single pole, can be obtained
from the plot of the phase ( Φ
12
( w)
) versus frequency, looking for the frequency which has a phase
equal to ( − π ). This method can be applied to the simulations performed with LAHET/MCNP-DSP
4
by Y. Rugama et al. [7] see Figure 2.
The simulations performed with LAHET + MCNP-DSP confirm that for the case of one
accelerator proton pulse per data block, one obtain the correct value of the eigenvalue from the
CPSD 12
(w) module and phase vs. frequency plots and then from the break frequency f
b
in Hz. The
eigenvalue is given by:
α0 = 2πf b
(17)
The multiplication constant of the system can be obtained from the expression:
α 0
Keff −1
keff
=
Λ
(18)
Where Λ is obtained from some previous calculation or determination. The mean generation
time for the 233 U/ 232 Th FEA cooled by liquid metal, from a previous calculation, was equal
to: Λ = 8.25846 × 10 -7 s.
We have checked that this method gives the correct value of the multiplication constant k eff
with
an acceptable error. For instance from the phase diagram obtained with LAHET + MCNP-DSP we get
a value of the Keff = 0. 985983 while with MCNP-4A, the K eff
from the sub-critical systems was
equal to 0 .96627 ± 0. 00067 .
The α
0
eigenvalue of the system for the fundamental mode, so it will be independent of the
numbers of proton pulses per block. In this study the data will be given using one pulse per data block
because as we can observe in Figure 3. The pole location using 1 pulse per data block or 5 pulses is at
the same point, but only for 1 pulse, the frequency at the phase equal to − π will give us the exactly
4
value for the break frequency.
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