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Substitution of (9) into (8), and use of equation (10) yields:
∑
(2)
( Σ Φ )
D2
D m +
n (1, w)
= −
Φ
m
(1)
m ( iw + α
m
) N
m
From the fundamental mode approximation, we obtain:
Considering (13) the CPSD 12
(w) will be:
∑
(2)
( Σ Φ )
D
D 0 +
n
0 2 ( 1, w)
= −
Φ
0
(1)
m ( iw + α
0
) N
0
N
∑ − 1
D1
( CPSD12
( w))
0
= Sog
n
( exp( −iw(
tf1
− nT )) Wn
1
⎢
p
n= 0 +
Φ Φ ⎣(
(
0
0
,
0
)
v
1
(12)
(13)
⎡ C(
D ⎤
2
)
⎥
α + iw)
(14)
⎦
D1
Where W
n
, is a weight factor that depends on the transfer function of the detector D 1
(source
detector).
∫
D 1 D1
D1
W = dw2
exp( −iw2(
tf1
− nT )) h ( w2
) = h ( tf1
− nT
n
)
(15)
We note that h D1 ( tf nT 1
− ) becomes zero by the causality condition when nT > tf1.
And C(D 2
) depends of the system detector D 2
:
3
∫ d r∫ S
( r´)
∫ dv´
∫
C ρ (16)
(2)
+
( D2 ) = dΩ´
f
sp
( v´,
Ω´)
Σ
D
Φ
0
(1) Φ
0
(1)
Being detector D 2
one of the system detectors and C(D 2
) is a function of the location of this
detector, the source neutron distribution ρ , inside the target, and the direct and adjoint fluxes Φ 0
(1)
S
+
and Φ
0
(1)
. From the phase and amplitude diagram of CPSD 12
(w), we can get the α
0
value and from
this value and the previous calculated Λ value the k eff
. From this methodology we can know the pole
( α
0
) location independently of the source, it means that can be used for all sub-critical systems,
always with k eff
values ranging from 0.94 to 1.
The graphical representation of the Equation (14) is showed on the Figure 1 and with data from
the simulation as we will show later.
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