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Detected-neutron multiplication factor measured by neutron source multiplication method Tomohiro Endo ⇑, Akio Yamamoto, Yoshihiro Yamane Department of Materials, Physics and Energy Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8603, Japan

a r t i c l e

i n f o

Article history: Received 3 September 2010 Received in revised form 10 December 2010 Accepted 5 January 2011 Available online 24 August 2011 Keywords: Neutron source multiplication method Subcriticality Effective neutron multiplication factor Detected-neutron multiplication factor Detector importance

a b s t r a c t An alternative deﬁnition of neutron multiplication factor measured by the neutron source multiplication (NSM) method is newly proposed. This newly deﬁned neutron multiplication factor, kdet, is derived on the basis of neutron detection process in a subcritical system with an external neutron source. The deﬁnition of kdet is expressed as a ratio of total number of detected ﬁssion-neutrons to total number of detected all neutrons. In this paper, a heuristic derivation of kdet is presented, and another interpretation of kdet is explained by using the detector importance function. Based on the idea of kdet, the measurement principle of NSM method is reinterpreted, and the correction factors in the NSM method are clariﬁed. In order to verify our proposed NSM method, numerical analysis of the NSM method is carried out. The numerical results suggest that target neutron multiplication factors of the NSM method can be well estimated even without any corrections by putting a neutron detector where the effective neutron multiplication factor keff is well approximated by kdet. Ó 2011 Published by Elsevier Ltd.

1. Introduction In this paper, we newly propose an alternative deﬁnition of a neutron multiplication factor measured by the neutron source multiplication method. In nuclear facilities where ﬁssile materials are treated, there are potential hazards of criticality accidents. Therefore, criticality safety design and operation are inevitable in order to prevent criticality accident under all possible conditions. In order to prevent criticality accidents, we need to design and control the mass, concentration, geometry, and arrangement of ﬁssile materials. Here, subcriticality is not directly measured in the present operation, e.g., the real-time monitoring of criticality is not carried out. The application of a subcriticality measurement technique in real operation can conﬁrm that a neutron multiplication system is kept in a subcritical state. If the subcritical state is conﬁrmed by the realtime measurement, we can quantify safety margins and guarantee the safety operation for general public to acquire public acceptance of peaceful use of nuclear energy. Based on the real-time measurement of subcriticality, we can deal with reasonable reductions of excessive safety margins in the criticality safety design due to the uncertainties of nuclear data, geometry, material, the calculation errors of design codes, and so on. Many subcritical measurement techniques have been proposed. The proposed subcritical measurement techniques are classiﬁed ⇑ Corresponding author. Tel.: +81 52 789 3606; fax: +81 52 789 3608. E-mail address: [email protected] (T. Endo). 0306-4549/$ - see front matter Ó 2011 Published by Elsevier Ltd. doi:10.1016/j.anucene.2011.01.007

into three methods: Steady-state analyses such as the neutron source multiplication method (Mukaiyama et al., 1975; Mizoo, 1977; Tsuji et al., 2003; Misawa and Unesaki, 2003; Cacuci, 2010) and the exponential experiment technique (Suzaki, 1991); kinetics analyses such as the pulse neutron source method (Simmons and King, 1958; Sjöstrand, 1956), the source jerk method (Hashimoto and Miki, 1995) and the inverse kinetic method (Hoogenboom and Van Der Sluijs, 1988); and reactor noise analyses such as the Feynman-a method (de Hoffmann, 1949; Feynman et al., 1956), the Rossi-a method (Orndoff, 1957), the 252Cfsource-driven noise analysis method (Mihalczo et al., 1978), and the third order neutron-correlation technique (Furuhashi and Izumi, 1968; Dragt, 1967). Each of method has advantages and disadvantages. For example, the pulsed neutron source method (Sjöstrand, 1956) and the 252Cf-source-driven noise analysis method (Mihalczo et al., 1978) can measure an absolute value of subcriticality. Here, absolute measurement of subcriticality means that a negative reactivity can be evaluated by only given information about the measuring multiplication system. In other words, absolute value of subcriticality can be measured without conﬁrmation of critical state by adding positive reactivity, or without comparison with subcritical state where the subcriticality is known beforehand. Although the pulsed neutron source method and the 252Cfsource-driven noise analysis method have such preferable merit to measure absolute value of subcriticality, these techniques inevitably require special experimental equipments such as the pulsed neutron source or the 252Cf installed ionization chamber, respectively. Moreover, the third order neutron-correlation technique

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(Furuhashi and Izumi, 1968; Dragt, 1967; Endo et al., 2006a,b) can also measure the absolute value of subcricicality. This technique, however, is based on higher order reactor noise analysis, so a long measurement time is required in order to reduce the statistical error of measured subcriticality. On the contrary, the neutron source multiplication (NSM) method is a simple measurement technique and has feasibility of realtime measurement. The NSM method does not require any special equipment other than a stationary external neutron source and an ordinary neutron detector. Additionally, the NSM method is based on steady-state analysis, so that this technique is very suitable for real-time measurement. Although the NSM method has such advantages, the absolute value of subcriticality cannot be measured by the NSM method, because the NSM method enables us to measure a relative reactivity difference from a reference state where the absolute value of subcriticality, or the effective neutron multiplication factor keff, is known beforehand by another measurement technique or by virtue of numerical analysis imitating the reference state as much as possible. Furthermore, in the NSM method, the correction factors play important roles in order to accurately estimate subcriticality from the measured neutron count rates (Mukaiyama et al., 1975; Mizoo, 1977; Tsuji et al., 2003; Cacuci, 2010). In order to evaluate these correction factors, we need the following numerical analyses for the reference and target subcritical states:

For simplicity, the generation-dependency of k is neglected, i.e., the ratio of neuron production to annihilation is constant for every generation. By using the deﬁnition of k, the S neutrons emitted by the external neutron source newly produce kS ﬁssion-neutrons by the sacriﬁce of their own lives, consequently kS neutrons are introduced into the neutron multiplication system as a next generation. In a similar way, the produced kS neutrons reproduce k2S neutrons, and continue to reproduce ﬁssion-neutrons in the same manner. Thus, the total number of neutrons in the neutron multiplication system is expressed by the inﬁnite summation of geometrical series: 2

In this paper, we will pay attention to the NSM method, and reconsider the deﬁnition and the physical meaning of neutron multiplication factor measured by the NSM method. The objective of this paper is to clarify how to correct the subcriticality measured by the NSM method, the physical meaning of the correction factors, and how to reduce the impact of correction factors. Firstly in Section 2, we review the concept of the NSM method and refer to the principle of measurement and the approximation in the simple NSM method. Secondly in Section 3, we reconsider the neutron multiplication factor measured by the NSM method, and newly introduce an alternative deﬁnition, namely detected-neutron multiplication factor, kdet. We also consider the physical meanings of the multiplication factor kdet by the aid of detector importance function (Bell and Glasstone, 1970). Additionally, the NSM method is reinterpreted based on the newly introduced kdet. Finally, in Section 4, the numerical analysis of the NSM method is carried out for simple subcritical cores by a three-dimensional 2-group transport calculation, in order to verify our proposed NSM method. 2. Concept of neutron source multiplication method In this section, the concept of NSM method is reviewed. For a simple derivation of the NSM method, the one-point reactor approximation without neutron energy dependency is assumed. Let us suppose that S neutrons are emitted per unit time by a decay of an external neutron source. The injected neutrons ﬂy in a subcritical neutron multiplication system, collide with nuclides, and then cause some kind of nuclear reactions, e.g., scattering, capture, ﬁssion, and so on. Here, let us introduce a neutron multiplication factor k:

Total number of fission-neutrons in next neutron generation k¼ : Total number of neutrons belonging to current neutron generation ð1Þ

1 X

i

Sk :

ð2Þ

i¼0

In cases of critical and supercritical states, i.e., k P 1, the total number of neutrons, N, diverges to inﬁnity. On the other hand, in a case of subcritical state, k < 1; therefore N converges to the following value at steady-state:

N¼

S : 1k

ð3Þ

Then, a count rate CR in the steady-state is expressed as follows:

CR ¼ (1) External source problems, which is forward neutron ﬂux calculation with actual external neutron source, to estimate spatial and energetic dependencies of actual neutron ﬂuxes for reference and target subcritical states. (2) Forward and adjoint keff-eigenvalue problems for reference and target states, to extract fundamental mode components from the measured neutron count rates.

3

N ¼ S þ Sk þ Sk þ Sk þ ¼

eS ; 1k

ð4Þ

where e means a detector efﬁciency, which is the detection probability for neutrons existing in this system. Here, components of count rate are classiﬁed into two parts:

CR ¼ eS þ

eSk ; 1k

ð5Þ

where the ﬁrst term means a non-ﬁssion component due to direct detection of neutrons originated from external neutron source decays; the second term means a ﬁssion component due to ﬁssionneutrons originated from ﬁssion reactions. By utilizing Eq. (4), which indicates the relationship between the count rate and the neutron multiplication factor, relative reactivity difference from a reference state can be measured as follows. Let us suppose that a neutron multiplication factor is known beforehand at the reference state. The reference multiplication factor and measured neutron count rate are denoted by kref and CRref, respectively. From Eq. (4), the product of detector efﬁciency and source strength, eS, which corresponds to the non-ﬁssion component of count rate, is expressed as follow:

eS ¼ ð1 kref ÞCRref :

ð6Þ

Eq. (6) is used for a kind of calibration in the NSM method. Supposing that a neutron count rate, CRtarget, is measured for an unknown neutron multiplication system with same neutron source and detector, the relationship of CRtarget and the multiplication factor ktarget is approximated by Eq. (4), i.e.,

ktarget ¼ 1

eS : CRtarget

ð7Þ

Here, let us approximate that the value eS of the unknown neutron multiplication system is equal to that of the reference state. By substituting the Eq. (6) into the right-hand side of Eq. (7) to eliminate eS, the multiplication factor for the unknown neutron multiplication system can be obtained as follows:

ktarget ¼ 1 ð1 kref Þ

CRref : CR target

ð8Þ

As derived above, it is noted that the simple NSM method is based on the following approximation:

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(1) One point reactor approximation: For the simplicity of derivation, the spatial and energetic dependency of neutron density is not taken into account. The one-point reactor approximation can be improved by advanced NSM methods (Mukaiyama et al., 1975; Mizoo, 1977; Tsuji et al., 2003; Misawa and Unesaki, 2003; Cacuci, 2010). In these advanced techniques, a fundamental mode component of k-eigenfunction is extracted from the measured neutron count rate, and the spatial and energetic dependency is corrected based on the fundamental mode of k-eigenfunction. The deﬁnition of k-eigenfunction will be described in Section 3.1. (2) Approximation of neutron multiplication factor: The effective neutron multiplication factor keff is usually, and implicitly, used for the multiplication factor k measured by the NSM method. It is noted that the advanced NSM methods attempt to extract the fundamental mode component of k-eigenfunction. Consequently, the multiplication factor k in Eq. (4) turns out to be the fundamental eigenvalue k0, i.e., the effective multiplication factor keff. (3) Approximation of eS: The product of eS, whose physical meaning corresponds to the non-ﬁssion component of the neutron count rate, is approximated as a constant value for both reference and target subcritical states. The advanced NSM methods explicitly correct this difference by more accurately estimating the detector efﬁciency e and the effective neutron source S, by virtue of the adjoint ﬂux of k-eigenfunction. 3. Deﬁnition of neutron multiplication factor for NSM method 3.1. Detected-neutron multiplication factor In this subsection, a new expression of neutron multiplication factor measured by the NSM method is introduced by a heuristic method. ~Þ is described by At a steady-state, angular neutron ﬂux wð~ r; E; X the following Boltzmann equation:

~Þ ¼ Fwð~ ~Þ þ Sð~ ~ Þ; Awð~ r; E; X r; E; X r; E; X

ð9Þ

~Þ means an external neutron source strength at where Sð~ r; E; X ~Þ; the operators A and F are neutron annihilation and producð~ r; E; X tion operators, respectively, denoted by

Z 1 Z 0 ~r þ Rt ð~ ~0 ! X ~Þ; AX r; EÞ dE dX0 Rs ð~ r; E0 ! E; X 0 4p Z Z vð~r; EÞ 1 0 dE dX0 mRf ð~ r; E0 Þ; F 4p 0 4p

ð10Þ ð11Þ

ð12Þ

In Eq. (12), the ﬁrst term means the non-ﬁssion component which is produced by primary neutrons emitted by the external neutron source without ﬁssion. In this paper, ws is named as source-ﬂux. Thus, the source-ﬂux is described by the following Boltzmann equation without ﬁssion-neutron production reaction:

~Þ ¼ Sð~ ~Þ: Aws ð~ r; E; X r; E; X

~Þ ¼ Fw ð~ ~ ~ ~ ~ ~ Awf ð~ r; E; X f r; E; XÞ þ Fws ðr; E; XÞ ¼ Fwðr; E; XÞ:

ð14Þ

It can be conﬁrmed that the sum of Eqs. (13) and (14) is equal to the original Boltzmann equation denoted by Eq. (9). In order to compare the concept of NSM method described in Section 2, let us divide the ﬁssion-ﬂux into the terms of generation-wise ﬂux Wf,i resulting from i-th generation ﬁssion-neutrons. A primary neutrons emitted by external source produce a next generation ﬁssion-neutrons, denoted by Fws. The ﬁrst produced ﬁssion-neutrons Fws reproduce FWf,1 ﬁssion-neutrons, where Wf,1 satisﬁes the following Boltzmann equation:

~Þ ¼ Fw ð~ ~ AWf;1 ð~ r; E; X s r; E; XÞ:

ð15Þ

If ws is expanded by the k-eigenfunctions (Henry, 1975), then

~Þ ¼ ws ð~ r; E; X

1 X n¼0

kn

fny S

fny Ffn

~Þ þ Rð~ ~Þ; r; E; X r; E; X fn ð~

ð16Þ

where kn is k-eigenvalue of n-th mode; fn and fny represent the forward and adjoint k-eigenfunctions corresponding to kn, described as follows:

~Þ ¼ 1 Ffn ð~ ~Þ; Afn ð~ r; E; X r; E; X kn ~Þ ¼ 1 Fy f y ð~ ~Þ; Ay fny ð~ r; E; X r; E; X kn n

ð17Þ ð18Þ

the operators A and F are adjoint operators of neutron annihilation and production:

Z 1 Z 0 ~r þ Rt ð~ ~!X ~0 Þ; Ay X r; EÞ dE dX0 Rs ð~ r; E ! E0 ; X 0 4p Z 1 Z vð~r; E0 Þ 0 r; EÞ dE dX0 : Fy mRf ð~ 4p 4p 0

ð19Þ ð20Þ

In (16), R means a residual term which cannot be expanded by keigenfunctions:

~Þ ¼ w ð~ ~ Rð~ r; E; X s r; E; XÞ

y fn S ~Þ: kn y fn ð~ r; E; X fn Ffn n¼0

1 X

ð21Þ

According to the reference, the residual term R can be expressed by a summation of eigenfunctions belonging to zero eigenvalue (Henry, 1975).

~Þ ¼ Rð~ r; E; X

1 X

ð0Þ ~ ~ að0Þ n fn ðr; E; XÞ;

ð22Þ

n¼0

where Rt ð~ r; EÞ is the macroscopic total cross section; ~0 ! X ~Þ the macroscopic scattering cross section from Rs ð~ r; E0 ! E; X ~0 to energy E and direction X ~; vð~ energy E0 and direction X r; EÞ the ﬁssion spectrum, where the spatial dependency of vð~ r; EÞ expresses that of ﬁssile nuclides. Here, let us classify the angular ﬂux into two terms:

~Þ ¼ w ð~ ~ ~ ~ wð~ r; E; X s r; E; XÞ þ wf ðr; E; XÞ:

sion-ﬂux. The ﬁssion-ﬂux is obtained by eliminating ws from total angular ﬂux w, hence is described by the following equation:

ð13Þ

On the other hand, the second term means the ﬁssion component whose root is ﬁssion reaction. In this paper, wf is named as ﬁs-

ð0Þ

where fn

is a solution of

Ffnð0Þ ð~ r; E; ~Þ

X ¼ 0;

ð23Þ

ð0Þ an

and is the expansion coefﬁcient corresponding to and (23) lead to

~Þ ¼ FRð~ r; E; X

1 X

ð0Þ ~ ~ að0Þ n Ffn ðr; E; XÞ ¼ 0:

ð0Þ fn .

Eqs. (22)

ð24Þ

n¼0

Namely, the ﬁrst produced ﬁssion-neutrons Fws do not include the residual term:

~Þ ¼ Fws ð~ r; E; X

1 X n¼0

kn

fny S

fny Ffn

~Þ: r; E; X Ffn ð~

ð25Þ

By substituting Eq. (25) into Eq. (15), Wf,1 can be expressed as follows:

~Þ ¼ Wf;1 ð~ r; E; X

y fn S ~Þ: k2n y fn ð~ r; E; X Ffn f n n¼0

1 X

ð26Þ

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T. Endo et al. / Annals of Nuclear Energy 38 (2011) 2417–2427

Subsequently, next generation ﬂux Wf,i reproduced by FWf,i1 is derived as follows:

1 X

kj ¼

kjþ1 n

n¼0

~Þ ¼ FWf;i1 ð~ ~Þ: AWf;i ð~ r; E; X r; E; X

) !,( ! y 1 X fny S hRd fn i j f n S h Rd f n i þ hRd Ridj1 ; kn y y fn Ffn fn Ffn n¼0

ð27Þ

ð36Þ

In a recurrent manner, Wf,i can be expanded by k-eigenfunctions as follows:

where dj1 is a Kronecker delta, i.e., kj includes the residual term hRdRi only if j = 1. From Eq. (36), we can see that the limit of kj as j approaches inﬁnity is the effective neutron multiplication factor keff, which is fundamental mode of k-eigenvalue k0, because of kn/ k0 < 1 for n P 1: !, ! y y 1 jþ1 1 j X X fn S hRd fn i fn S hRd fn i kn kn j lim kj ¼ lim kjþ1 k 0 0 j!1 j!1 k0 k0 fny Ffn fny Ffn n¼0 n¼0 !, ! y y f S hRd f0 i f0 S hRd f0 i ð37Þ ¼ k0 0 y ¼ k0 ¼ keff y f0 Ff0 f0 Ff0

~Þ ¼ Wf;i ð~ r; E; X

1 X

kniþ1

n¼0

fny S

fny Ffn

~Þ: r; E; X fn ð~

ð28Þ

Above mentioned ﬁssion chain reaction terminates depending on the subcriticality. Consequently, by using Eqs. (16) and (28), wf and w can be expressed by the summation until the neutron chain reaction dies away:

~Þ ¼ wf ð~ r; E; X

1 X

~Þ ¼ Wf;i ð~ r; E; X

1 X n¼0

i¼1

y fn S k2n ~Þ; r; E; X y fn ð~ 1 kn fn Ffn

~ ~ Þ ¼ w ð~ ~ ~ wð~ r; E; X s r; E; XÞ þ wf ðr; E; XÞ y 1 X fn S kn ~Þ þ Rð~ ~Þ; ¼ r; E; X r; E; X y fn ð~ 1 kn fn Ffn n¼0

ð29Þ

In the simple measurement principle of NSM method, the dependency of kj on the neutron generation is not explicitly taken into account. Here, we introduce a geometric mean kGM as a kind of generation-averaged value of kj:

ð30Þ

N Y

kGM ¼ lim

N!1

Then, the total neutron count rate can be expressed by the following formula:

hRd wi ¼ hRd ws i þ hRd wf i ¼ hRd ws i þ

1 X

hRd Wf;i i;

ð31Þ

dV

V

Z

1

dE

Z

dXRd ð~ r; EÞwð~ r; E; XÞ:

ð32Þ

4p

0

It is noted that Rd includes not only the macroscopic reaction cross section in a detector but also the detection probability for charged particles induced by the neutron detection reaction. For example, if we utilize a BF3 counter for the detection of neutrons, then Rd can be estimated by the product of the macroscopic (n, a) reaction cross section of 10B and the detection probability of the BF3 counter for alpha particles induced by (n, a) reaction. Thus, hRdwi turns out to represent a measured neutron count rate. Based on the concept of NSM method, let hRdwi and hRd wf i rewrite as follows:

hRd wi ¼ hRd ws i þ

1 X hRd Wf;i i hRd Wf;i1 i hRd Wf;2 i h hRd Wf;1 i R d Wf;i1 i hRd Wf;i2 i i¼1

hRd Wf;1 i hRd ws i hRd ws i

¼ hRd ws i þ

1 i X Y

1 X

hRd wi ¼

i¼1

hRd wf i ¼

1 i X Y i¼1

i kGM

¼1þ

i¼0

1 i X Y i¼1

! kj :

ð40Þ

j¼1

The left hand side of Eq. (40) is equal to the inﬁnite summation of geometrical series where the geometrical ratio is kGM: 1 X

i

kGM ¼

i¼0

1 : 1 kGM

ð41Þ

From Eqs. (40) and (41), another expression of kGM can be obtained as

P1 Qi kGM ¼

j¼1 kj

i¼1

1þ

P1 Qi

j¼1 kj

i¼1

kGM ¼ ð33Þ

!

:

ð42Þ

hRd ws i þ

ð34Þ

i Q

!

kj hRd ws i

j¼1

P1

i¼1

i Q

!

:

ð43Þ

kj hRd ws i

j¼1

Finally, by substituting Eq. (34) into Eq. (43), the expression of kGM can be rewritten as

j¼1

if j ¼ 1

> :

if j P 2

hRd Wf;j1 i

1 X

j¼1

8 hR W i d f;1 > < hRd w s i hRd Wf;j i

By comparing Eq. (39) with Eq. (33), the value of kGM should satisfy the following identical condition:

i¼1

!

kj hRd ws i;

ð39Þ

By multiplying the numerator and denominator of right-hand side of Eq. (42) by hRdwsi, then

where kj is a generation-dependent multiplication factor and deﬁned as follows:

kj

i

kGM hRd ws i:

P1 kj hRd ws i;

ð38Þ

i¼0

where Rd means a macroscopic detection cross section; the bracket means total integral for spatial, energetic, and angular variables, e.g.,

Z

kj

j¼1

Then, Eq. (33) can be transformed into

i¼1

hRd wi ¼

!1=N

kGM ¼

hRd wf i hRd wf i ¼ ; hRd ws i þ hRd wf i hRd wi

ð44Þ

or by using Eqs. (29) and (30), k-eigenfunction expansion formula of kGM is

;

ð35Þ

or, by using Eqs. (16) and (28), k-eigenfunction expansion formula of kj is

kGM ¼

1 X n¼0

) !,( ! y y 1 X fn S hRd fn i fn S hRd fn i k2n kn þ hRd Ri : y y 1 kn 1 kn fn Ffn fn Ffn n¼0

ð45Þ

Based on above mentioned derivation process, let us newly deﬁne detected-neutron multiplication factor kdet as follows:

T. Endo et al. / Annals of Nuclear Energy 38 (2011) 2417–2427

kdet

hRd wf i : hRd wi

ð46Þ

As shown in Eq. (46), the physical meaning of kdet is a ratio of total number of detected ﬁssion-neutrons to total number of detected all neutrons. As we can understand from Eq. (46), if we measure neutron counts for a non-ﬁssile system, the value of kdet is equal to zero because of hRdwfi = 0. On the other hand, if we measure for a neutron multiplication system with nearly critical state, is nearly equal to hRdwfi, therefore kdet 1. In general, the value of kdet differs from the effective neutron multiplication factor keff, this fact is more clariﬁed by another expression of kdet described in next Section 3.2. By utilizing the deﬁnition of kdet, the measured neutron count rate can be simply and exactly expressed as the follows:

hRd wi ¼

hRd ws i : 1 kdet

ð47Þ

By comparing Eq. (47) with Eq. (4), it is clear that the somewhat ambiguous values eS and kin Eq. (4) are rigorously deﬁned as hRdwsi and kdet, respectively. 3.2. Another interpretation of kdet by detector importance function The deﬁnition of detected-neutron multiplication factor kdet by Eq. (46) does not explicitly express a balance of neutron production and annihilation, although Eq. (46) implies the balance in the ratio of two neutron count rates, hRdwfi/hRdwi. In this subsection, let us further consider the physical meaning of kdet from another point of view, which is based on the neutron balance of production and annihilation. For this purpose, let us introduce detector importance function. The detector importance function is described by the following adjoint Boltzmann equation:

~Þ ¼ Fy Iy ð~ ~Þ þ Rd ð~ Ay Iy ð~ r; E; X r; E; X r; EÞ;

ð48Þ

~Þ denotes an adjoint function involved with an inﬂuwhere I ð~ r; E; X ence for detector reaction, namely, detector importance function (Bell and Glasstone, 1970). The physical meaning of I is an expected value of neutron counts that are detected by neutron detector until a ~Þ ﬁssion chain, or a neutron family, caused by one neutron at ð~ r; E; X dies out completely. Rd ð~ r; EÞ is the same as described in previous subsection. By multiplying both sides of forward ﬂux equation, Eq. (9), by adjoint importance function I , and integrating over all variables, we can obtain that y

Iy Aw ¼ Iy Fw þ Iy S :

ð49Þ

On the contrary, by multiplying both sides of adjoint importance equation, Eq. (48), by the forward total angular ﬂux w, and integrating over all variables, we can obtain that

hwAy Iy i ¼ hwFy Iy i þ hRd wi:

ð50Þ

Here, by utilizing the mathematical property of adjoint operator, the following equalities are satisﬁed: y

y y

hI Awi ¼ hwA I i;

ð51Þ

hIy Fwi ¼ hwFy Iy i:

ð52Þ

Therefore, we can conﬁrm the following well-known relationship between the count rates of detector and the inner product of detector importance function and external neutron source strength:

hRd wi ¼ hIy Si:

ð53Þ

Now, we apply the same manner of derivation of Eq. (49) to Eq. (13) which describes the behavior of source-ﬂux ws. Namely, by multiplying both sides of Eq. (13) by I , and integrating over all variables, we can obtain that

hIy Aws i ¼ hIy Si:

2421

ð54Þ

By comparing Eq. (53) with Eq. (54), we can derive the following importance relationship involved with the source-ﬂux ws:

hRd wi ¼ hIy Si ¼ hIy Aws i

ð55Þ

Next, we pay attention to Eq. (14) which describes the behavior of ﬁssion-ﬂux wf. In Eq. (14), we can regard the ﬁssion-source term due to ws, or Fws, as a kind of external neutron source for wf, i.e.,

~Þ ¼ Fw ð~ ~ ~ ~ Awf ð~ r; E; X f r; E; XÞ þ Sf ðr; E; XÞ;

ð56Þ

where

~Þ Fw ð~ ~ Sf ð~ r; E; X s r; E; XÞ:

ð57Þ

By comparing Eq. (56) with Eq. (9), we can suppose that Eq. (56) describes the Boltzmann equation relevant to the ﬁssion-ﬂux wf, and the external neutron source for wf is given by Sf. In other words, if we replace S in Eq. (9) with Sf, we can obtain the ﬁssion-ﬂux wf from this replaced Boltzmann equation. Accordingly, if we replace S in the right-hand side of Eq. (53) with Sf, the neutron ﬂux of left hand side of Eq. (53) shall become the ﬁssion-ﬂux wf. Hence, we can derive the following importance relationship involved with the ﬁssion-ﬂux wf:

hRd wf i ¼ hIy Sf i ¼ hIy Fws i:

ð58Þ

Finally, by substituting Eqs. (55) and (58) into the deﬁnition of kdet, Eq. (46), the following interesting expression for kdet can be also derived:

kdet

hIy Fws i hIy Aws i

ð59Þ

As we can understand from above expression, the multiplication factor kdet means the ratio of weighted integration of neutron production to that of annihilation, where weighting function is detector importance function. Furthermore, in the deﬁnition of kdet, the neutron production and annihilation rates are originated from the source-ﬂux. Eq. (59) clariﬁes that the value of kdet is a ratio of production to annihilation relevant to non-ﬁssion-neutron with focus on detected neutrons, or emphasized by the detector importance function. Obviously, the value of kdet differs from that of keff unless the spatial and energetic dependencies of ws and I are equal to the forward and adjoint fundamental mode of k-eigenfunctions, respectively. Interestingly, Eq. (59) is similar to the deﬁnition of subcritical multiplication factor, ksub (Kobayashi and Nishihara, 2000; Nishihara et al., 2003; Shahbunder et al., 2010a,b,c):

ksub

hFwi ; hAwi

ð60Þ

or, the deﬁnition of generalized multiplication factor introduced by Gandini (Gandini, 2002):

kgen

hw Fwi ; hw Awi

ð61Þ

where w⁄ is an arbitrary non-null positive weighting function. It is noted that the deﬁnition of kdet includes the detector importance function as weighting function, and the neutron ﬂux in the production and annihilation reaction rates is the source-ﬂux ws, not total angular ﬂux w. It is quite natural expression to include the detector importance function I in the deﬁnition of multiplication factor by the NSM method, because we must detect neutrons in order to measure the neutron multiplication factor.

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T. Endo et al. / Annals of Nuclear Energy 38 (2011) 2417–2427

3.3. Reinterpretation for measurement principle of NSM method Based on the Eq. (47), let us reinterpret the measurement principle of the NSM method. Firstly, the neutron count rates at a reference and target subcritical states are described as follows:

hRd wref i ¼

Rd ws;ref

ð62Þ

; 1 kdet;ref hRd ws;target i hRd wtarget i ¼ ; 1 kdet ;target

ð63Þ

where the subscript ref and target mean the values at reference and target subcritical states, respectively. By taking the ratio of Eqs. (62) and (63), we can obtain that

hRd wtarget i 1 k det;ref ; ¼ fs 1 kdet;target hRd wref i

ð64Þ

where fs is a ratio of hRdwsi at the reference and target subcritical states:

fs

hRd ws;target i : hRd ws;ref i

ð65Þ

By transforming Eq. (64) in contrast with Eq. (8), we can obtain that

kdet ;target ¼ 1 fs ð1 kdet;ref Þ

hRd wref i : hRd wtarget i

ð66Þ

Furthermore, by introducing a factor fc to convert from kdet to keff, the following formula for the NSM method can be ﬁnally derived:

fc;target keff;target ¼ 1 fs ð1 fc;ref keff;ref Þ

hRd wref i ; hRd wtarget i

ð67Þ

where fc,ref and fc,target are conversion factors at the reference and target states, respectively:

fc;ref fc;target

kdet;ref ; keff;ref kdet;target : k eff;target

conversion factor fc, i.e., kdet keff, we should carefully consider the arrangement of external neutron source and detector. Namely, it is important to ﬁnd out particular positions of neutron source and detector where kdet keff, then the value of fc is nearly equal to unity, thereby the conversion factor can be negligible.

ð68Þ ð69Þ

Based on Eq. (67), the correlation factors in the NSM method are essentially classiﬁed into the following two terms: (1) Source-ﬂux correlation factor fs: The correction factor fs derives from the difference between non-ﬁssion component of the neutron count hRdwsi at the reference and target subcritical states. If the difference is not negligible, fs should be taken into account for the correction factor in the NSM method. In order to reduce the impact of fs, i.e., fs 1, the neutron count rates due to primary neutrons emitted by the external source should be identical as much as possible for both states. For example, we should use same neutron source and detector for the reference and targets state with similar geometrical conﬁguration. Furthermore, it is preferable that a reactivity perturbation between both states does not affect hRdwsi. For example, in the reference state, if we put a neutron absorber in between neutron source and detector, the value of fs would be less than unity because hRdwsi is decreased by the neutron absorber. (2) Conversion factor fc: The multiplication factors in the Eq. (66) are detectedneutron multiplication factor kdet. Therefore, we should convert kdet into the effective factor keff, if we want to just estimate keff. Thus, the ratio of kdet/keff, namely fc, is the correction factor for conversion of two multiplication factors in Eq. (67). In order to reduce the impact of the

3.4. Calculation methodology of correction factors for proposed NSM method By the aid of numerical analysis, the source-ﬂux correlation factor fs and the conversion factor fc in our proposed NSM method, Eq. (67), can be evaluated as follows: (1) Source-ﬂux correlation factor fs: Firstly, the source-ﬂux of ws,ref and ws,target for the reference and target states can be estimated by external source problems for both states with actual external neutron source, but without ﬁssion neutron source. Namely, as described in Eq. (13), external source problem for ws is numerically solved under ﬁctitious non-production condition, where the ﬁssion-neutrons production operator F = 0, e.g., macroscopic production cross section mRf is set to zero although absorption cross section Ra keeps actual value. It is noted that the amplitude of source-ﬂux ws is uniquely determined for that of a given source strength S. Then, by using the obtained source-ﬂux ws and the detection cross section Rd corresponding to the neutron detector, count rates due to source-ﬂux, hRd ws;ref i and hRd ws;target i, can be estimated for the both states. Finally, the correlation factor fs can be evaluated by the ratio of these count rates, as deﬁned in Eq. (65). Since fs is deﬁned by the ratio, if the same external source and detector are used for both states, the absolute values of S and Rd are eliminated in the correction factor fs. Thus, the relative spatial and energetic distribution of S and Rd are important to evaluate fs. (2) Conversion factor fc: In order to evaluate the conversion factor fc, not only conventional effective neutron multiplication factor keff but also detected-neutron multiplication factor kdet are needed. The value of keff can be obtained from the keff-eigenvalue problem. On the other hand, the value of kdet can be obtained from the following two external source problems with actual external source: One is the conventional external source problem under actual production condition to get the total angular ﬂux w; the other under ﬁctitious non-production condition to get the source-ﬂux ws. It is noted that the spatial and energetic distributions of external source strength are identical for the two external source problems. After these two calculations, the ﬁssionﬂux wf can be also estimated from the difference between w and ws. Consequently, the value of kdet can be estimated from the count rate ratio of hRd wfi/hRdwi, as deﬁned in Eq. (46). As we can see from Eq. (46), the absolute values of S and Rd are eliminated in the evaluation of kdet, since external source and detector are identical for hRdwfi and hRdwi. Finally, the conversion factor fc can be evaluated by the ratio of kdet/keff. It is noted that above mentioned correction factors fs and fc can be evaluated by only forward ﬂux calculations. Namely, adjoint calculations, which are necessary in the advanced NSM methods, are not necessary to evaluate fs and fc. Thus, these correction factors could be also estimated by continuous energy Monte Carlo codes which can statistically solve the forward ﬂux with faithful reproduction of geometrical conﬁgurations.

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T. Endo et al. / Annals of Nuclear Energy 38 (2011) 2417–2427 Table 1 Two energy group constants for subcritical cores. Core C30 Macroscopic cross section (1/cm) Fast Rt,1 0.21100 Ra,1 0.00320 mRf,1 0 Rs,1?1 0.19000 Rs,1?2 0.01780 Thermal

Fission spectrum (–) Fast Thermal

Reﬂector C35

C45

0.21600 0.00286 0 0.19194 0.02120

0.22200 0.00237 0 0.19423 0.02540

0.23600 0 0 0.18840 0.04760

Rt,2 Ra,2 mRf,2 Rs,2?1 Rs,2?2

1.2300 0.0930 0.1680 0 1.1370

1.4100 0.0850 0.1490 0 1.3250

1.6400 0.0724 0.1210 0 1.5676

2.8500 0.0191 0 0 2.8309

v1 v2

1.0 0.0

1.0 0.0

1.0 0.0

– –

4. Numerical analysis 4.1. Calculation model and condition In order to promote an understanding of the detected-multiplication factor kdet, and to verify our proposed NSM method described in Section 3.3, we carried out the numerical analysis by using THREEDANT, which is a three-dimensional multi-group discrete ordinates transport code (Alcouffe et al., 1995). The calculation model and condition of numerical analysis are as follows:

(1) Homogeneous rectangular parallelepiped core. The dimension is 41 cm in x-direction, 33 cm in y-direction, and 49 cm in z-direction, respectively. The core is surrounded by a 20 cm thick reﬂector. (2) Two energy group constants are listed in Table 1. The original cross sections are quoted from reference (Misawa et al., 2010). Three subcritical cores are analyzed by changing the group constants of core region. (3) Spatial distributions of external neutrons source are following three cases: (a) Uniform source in the core region. (b) Point-wise source at (x, y, z) = (0.0, 0.0, 0.0), i.e., at the center of core. (c) Point-wise source at (x, y, z) = (31, 0.0, 0.0) in the reﬂector region. In addition, the energy spectrum of external source is the same as ﬁssion spectrum. (4) Point-wise detector for thermal neutron. (5) Total number of spatial meshes is 81 73 89 for x-, y-, and z-directions. (6) EO8 quadrature set is used for SN solid angle quadrature set (Endo et al., 2007). The total number of discrete directions of EO8 set is 80, same as that of conventional S8 level symmetric quadrature set. By utilizing EO8 set, we can exactly calculate a numerical integration of the Xix Xjy Xkz over an octant of the sphere in the range of 0 6 i + j + k 6 4, where Xx, Xy, and Xz mean direction cosines for x-, y-, and z-directions, respectively. (7) Convergence criteria is 1.0 106 for inner iteration.

Table 2 Numerical results of neutron multiplication factor by proposed NSM method. Uniform source detector at (14, 10, 0)

keff (reference) (–) kdet (–) Souce-ﬂux correction factor fs (–) Conversion factor fc (–) k1 (method 1) (–) k1/keff 1 (%dk/k) k2 (method 2) (–) k2/keff 1 (%dk/k) k3 (method 3) (–) k3/keff 1 (%dk/k) k4 (method 4) (–) k4/keff 1 (%dk/k) ⁄

Point-wise source at center detector at (12, 0, 0)

Point-wise source at reﬂector detector at (9, 0, 0)

C45

C35⁄

C30

C45

C35⁄

C30

C45

C35⁄

C30

0.99273 0.99263 1.28937 0.99990 0.99430 0.16 0.99265 0.01 0.99263 0.01 0.99273 0.00

0.96639 0.96630 1.00000 0.99990 – – – – – – – –

0.92311 0.92339 0.83162 1.00029 0.90813 1.62 0.92360 0.05 0.92339 0.03 0.92311 0.00

0.99273 0.99321 1.26798 1.00049 0.99459 0.19 0.99314 0.04 0.99321 0.05 0.99273 0.00

0.96639 0.96675 1.00000 1.00037 – – – – – – – –

0.92311 0.92054 0.84164 0.99722 0.90457 2.01 0.91968 0.37 0.92054 0.28 0.92311 0.00

0.99273 0.99121 1.27162 0.99847 0.99364 0.09 0.99191 0.08 0.99121 0.15 0.99273 0.00

0.96639 0.96347 1.00000 0.99698 – – – – – – – –

0.92311 0.92452 0.84264 1.00152 0.91758 0.60 0.93055 0.81 0.92452 0.15 0.92311 0.00

Reference subcritical state.

Fig. 1. Fundamental mode ﬂux of keff-eigenfunctuon at (y, z) = (0, 0).

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T. Endo et al. / Annals of Nuclear Energy 38 (2011) 2417–2427

4.2. Numerical results First of all, keff-eigenvalue problems are numerically solved for three subcritical cores (C30, C35, C45) to evaluate reference values of effective neutron multiplication factors. Fig. 1 and Table 2 show numerical results of fundamental mode ﬂux and keff which are obtained by forward keff-eigenvalue problems, respectively. As shown in Fig. 1, the spatial distributions of fundamental mode ﬂux are almost same for three subcritical states, since the reactivity change results from the homogeneous change of cross sections in core region. Secondary, we analyzed the external source problems under the actual production and the ﬁctitious non-production conditions (mRf = 0 in the THREEDANT input ﬁles), thereby the conventional total ﬂux / and source-ﬂux /s were calculated, respectively, where / and /s are solid angle integrals of w and ws, respectively. After

that, the ﬁssion-ﬂux /f were also estimated from the difference between / and /s. Fig. 2 shows spatial distributions of fast and thermal ﬂux / and /f on the x-axis at (y, z) = (0, 0). In the case of uniform source, the spatial distribution of / is similar to that of /f, and these relative distributions are similar to fundamental mode ﬂux as shown in Fig. 1. On the other hand, in the cases of point-wise source, the spatial distributions of / have sharp peaks at the source positions and the sharpness becomes larger for deeper subcritical state. It is noted that such ﬂux peaks do not appear in ﬁssion-ﬂux /f. The disappearance of spatial ﬂux peak results from the fact that /f is formed by neutrons having experiences with at least one ﬁssion reaction. In the case of the point-wise detector for thermal neutron, the value of kdet can be evaluated by thermal ﬂux /2 ð~ r d Þ and /f;2 ð~ rd Þ at the detector position ~ rd :

Fig. 2. Total neutron ﬂux at (y, z) = (0, 0).

T. Endo et al. / Annals of Nuclear Energy 38 (2011) 2417–2427

kdet ¼

rd Þ hRd wf i /f;2 ð~ hRd wi /2 ð~ rd Þ

ð70Þ

Fig. 3 shows the numerical results of kdet and the relative differences between kdet and keff on the x-axis at (y, z) = (0, 0). As shown in Fig. 3, the value of kdet tends to increase as keff approaches to unity, i.e., the subcritical systems approach to the critical state. However, it is noted that newly introduced kdet has spatial dependencies on external neutron source distribution and detector position, and the dependencies become much larger as the subcriticality becomes deeper. In addition, we can recognize some particular behaviors of kdet. For example, one of the behaviors appears in the neighboring region of the point-wise source, where kdet is considerably underestimated compared with keff. In the neighboring region of the point neutron source, the source-ﬂux is dominant, because neutrons

2425

generated from point-wise neutron source tend to be detected without ﬁssion. Hence, the underestimation of kdet is caused by the large contribution of source-ﬂux, in other words, the small contribution of ﬁssion-ﬂux. Another behavior appears when the point-wise source exists in the reﬂector region. If the neutron detector is placed in the reﬂector region between the core region and the point-wise source, kdet is extremely underestimated as compared with keff, and approaches zero by increasing the distance between the core region and the detector. The reason is the same as that for the neighboring region of point-wise source. On the contrary, if the neutron detector is placed in the opposite side of reﬂector region where external source does not exist, kdet is extremely overestimated and nearly equal to unity. In this case, the detected neutrons mainly originated from the ﬁssion-neutrons in core region, i.e., primary neutrons emitted by the point-wise source hardly reach to the

Fig. 3. Detected-neutron multiplication factor at (y, z) = (0, 0).

2426

T. Endo et al. / Annals of Nuclear Energy 38 (2011) 2417–2427

detector position across the core region without any ﬁssion reactions. Therefore, the neutron count rate hRdwi is almost equal to hRdwfi, consequently, kdet 1. The relative differences between kdet and keff are relevant to the conversion factor fc in our proposed NSM method. In the case of uniform source, the relative differences are small, less than 1%, and slightly larger at reﬂector region than core region. On the other hand, in the cases of point-wise source, it is very interesting to note that we can ﬁnd particular detector positions where the values of kdet are approximately equal to keff. Speciﬁcally in Fig. 3, the such detector positions are x ±12, 9 for the point-wise source at the center of core and at the reﬂector, respectively. Furthermore, the particular detector positions are quite stationary for subcritical states, although the positions depend on the spatial distribution of external source. It is because the fundamental mode ﬂux is almost same for three subcritical states, as shown in Fig. 1. The values of kdet and keff can be evaluate for each subcritical state, therefore we can search the detector positions of kdet keff for only one certain subcritical state, e.g., the reference state.

C35

C45 0.5

40

40

40

20

20

20

0

0

0

-20

-20

-20

-40

-40

-40 -40

-2 0

20

0

-40

40

0.0

-20

20

0

-40

40

-20

20

0

40

kdet/keff - 1 [%]

y [cm]

C30

In order to search the detector positions of kdet keff in two dimensions, Fig. 4 plots the relative difference between kdet and keff at z = 0 plane within the range of 0.5 % to 0.5%. In Fig. 4, green colored regions can be regard as the detector regions of kdet keff. As shown in Fig. 4, the detector regions of kdet keff can be ﬁnd out not only core regions but also reﬂector regions for the uniform source and the point-wise source at reﬂector. On the other hand, for the point-wise source at center of core, the detector positions of kdet keff are limited in core region and the points exist on the circumference of circle in core region of which radius is approximately 12 cm. Finally, we examined our proposed NSM method at the detector positions of kdet keff. In this analysis, we selected preferable point-wise detector positions at (x, y, z) = (14, 10, 0), (12, 0, 0), and (9, 0, 0) for the uniform source, the point-wise source at center of core, and the point-wise source at reﬂector, respectively, as shown in Fig. 4. In this veriﬁcation, C35 subcritical state was regarded as the reference state, target multiplication factors ktarget for C30 and C45 states were estimated by following four methods:

-0.5

x [cm]

(a) uniform source C35

C45 0.5

40

40

20

20

20

0

0

0

-20

-20

-20

-40

-40

-40 -40

-20

20

0

-40

40

0.0

-20

20

0

-40

40

-20

20

0

40

kdet/keff - 1 [%]

y [cm]

C30 40

-0.5

x [cm]

(b) point-wise source at center of core C30

C35 Reflector

0.5

Reflector

Reflector

Core

Core

Core

0

0

0

-20

-20

-20

-40

-40

-40 -40

-20

0

20

40

0.0

-40

-20

0

20

40

-40

-20

0

20

40

kdet/keff - 1 [%]

20

20

20

y [cm]

C45 40

40

40

-0.5

x [cm]

(c) point-wise source at reflector Fig. 4. Speciﬁc point-wise detector regions of kdet keff at z = 0 plane (circle and cross mean a point-wise source and a selected detector position).

T. Endo et al. / Annals of Nuclear Energy 38 (2011) 2417–2427

Method 1: Without correction.

k1;target 1 ð1 keff;ref Þ

hRd wref i hRd wtarget i

ð71Þ

Method 2: With source-ﬂux correction factor fs.

k2;target 1 fs ð1 keff;ref Þ

hRd wref i hRd wtarget i

ð72Þ

Method 3: With source-ﬂux correction factor fs and only conversion factor fc,ref.

k3;target 1 fs ð1 fc;ref k

eff;ref Þ

hRd wref i hRd wtarget i

ð73Þ

Method 4: With source-ﬂux correction factor fs, and conversion factors fc,ref and fc,target.

k4;target ¼

1 fc;target

(

hRd wref i 1 fs ð1 fc;ref keff;ref Þ hRd wtarget i

)

ð74Þ

Table 2 shows estimated ktarget by above mentioned four methods. By virtue of detector positions where kdet keff, the neutron multiplication factors are well estimated even without corrections (Method 1). By taking into account of the source-ﬂux correction factor fs, the estimated neutron multiplication factors can be improved, especially for C30 core which is deepest subcritical state (Method 2). Although point-wise detector positions are selected as kdet is equal to keff as much as possible, there are slight discrepancies between kdet and keff, because of spatial discretization. Therefore, in order to ﬁx the slight discrepancies, Method 3 explicitly takes into account of only the conversion factor fc,ref at the reference state. As shown in Table 2, the estimated values of k3,target by Method 3 are obviously equal to detected-neutron multiplication factors kdet,target. Furthermore, by virtue of quite stationary detector positions for three subcrtical states, the estimated k3,target by Method 3 can be slightly improved in C30 core, especially for point-wise source at reﬂector. In the case of Method 3, the relative difference between k3,target and keff,target is equivalent to that of kdet,target and keff,target. Finally, if the conversion factors fc,target at the target states can be taken into account, the estimated values of ktarget are perfectly equal to effective neutron multiplication factors keff,target at the target states (Method 4). In conclusion, it is veriﬁed that we can accurately estimate neutron multiplication factor of NSM method by setting a detector at a preferable position kdet keff depending on the spatial distribution of external neutron source. By utilizing the source-ﬂux correction factor fs and conversion factor fc, we can further improve the accuracy of NSM method. 5. Conclusion In this paper, we reconsidered the neutron multiplication factor measured by the NSM method. Detected-neutron multiplication factor kdet is newly introduced, this factor is deﬁned by the ratio of total number of detected ﬁssion-neutrons to total number of detected all neutrons. Based on the idea of kdet, it is clear that correlation factors of the NSM method consist of the source-ﬂux correction factor fs to ﬁx the difference of non-ﬁssion component of neutron count rate, and the conversion factor fc to convert from kdet to the effective multiplication factor keff. The physical meanings of these correction factors are straightforward, moreover, these factors can be evaluated by only forward neutron ﬂux calculations without adjoint calculations. Through the numerical analysis of three-dimensional two-group transport calculations for simple geometry, the validity of our proposed NSM method was veriﬁed. Numerical results suggest that we can ﬁnd out a prefera-

2427

ble detector position where kdet keff for only one certain subcritical state, e.g., the reference state, depending on the external neutron source distribution. By putting a neutron detector at such a particular position, the target neutron multiplication factors can be well estimated even without any corrections. Obviously, we can further improve the accuracy of NSM method by utilizing the factors fs and fc. References Alcouffe, R.E., et al., 1995. DANTSYS: A Diffusion Accelerated Neutral Particle Transport Code System. Los Alamos National Laboratory report LA-12969-M. Bell, G.I., Glasstone, S., 1970. Nuclear Reactor Theory. Van Nostrand Reinhold Company. pp. 256–258. Cacuci, D.G., 2010. Handbook of nuclear engineering. Reactor Analysis, vol. 3. Springer. pp. 2147–2152. de Hoffmann, F., 1949. Statistical aspects of pile theory. The Science and Engineering of Nuclear Power, vol. II. Addison Wesley Press, Cambridge, Mass. pp. 116–119. Dragt, J.B., 1967. Threefold correlations and third order moments in reactor noise. Nukleonik 10, 7–13. Endo, T., Yamane, Y., Yamamoto, A., 2006a. Space and energy dependent theoretical formula for the third order neutron correlation technique. Ann. Nucl. Energy 33, 521–537. Endo, T., Yamane, Y., Yamamoto, A., 2006b. Derivation of theoretical formula for the third order neutron correlation technique by using importance function. Ann. Nucl. Energy 33, 857–868. Endo, T., Yamane, Y., Yamamoto, A., 2007. Development of new solid angle quadrature sets to satisfy even- and odd-moment conditions. J. Nucl. Sci. Technol. 44 (10), 1249–1258. Feynman, R.P., de Hoffmann, F., Serber, R., 1956. Dispersion of the neutron emission in U-235 ﬁssion. J. Nucl. Energy 3, 64–69. Furuhashi, A., Izumi, A., 1968. Third moment of the number of neutrons detected in short time intervals. J. Nucl. Sci. Technol. 5 (2), 48–59. Gandini, A., 2002. On the multiplication factor and reactivity deﬁnitions for subcritical reactor systems. Ann. Nucl. Energy 29, 645–657. Hashimoto, K., Miki, R., 1995. Space-dependent effect observed in subcriticality measurements for loosely coupled-core system. J. Nucl. Sci. Technol. 32 (10), 1054–1060. Henry, A.F., 1975. Nuclear-Reactor Analysis. MIT Press. pp. 346–351. Hoogenboom, J.E., Van Der Sluijs, A.R., 1988. Neutron source strength determination for on-line reactivity measurements. Ann. Nucl. Energy 15, 553–559. Kobayashi, K., Nishihara, K., 2000. Deﬁnition of subcriticality using the importance function for the production of ﬁssion neutrons. Nucl. Sci. Eng. 136, 272–281. Mihalczo, J.T., Paré, V.K., Ragan, G.L., Mathis, M.V., Tillett, G.C., 1978. Determination of reactivity from power spectral density measurements with californium-252. Nucl. Sci. Eng. 66, 29–59. Misawa, T., Unesaki, H., 2003. Measurement of subcriticality by higher mode source multiplication method. In: Proc. Int. Conf. on Nuclear Criticality Safety (ICNC2003), Tokai-mura, Japan, October 20–24, 2003. Misawa, T., Unesaki, H., Pyeon, C., 2010. Nuclear Reactor Physics Experiments. Kyoto University Press. Mizoo, N., 1977. Reliability of a new modiﬁed neutron source multiplication method. JAERI-M 7135, Japan Atomic Energy Research Institute (JAERI). Mukaiyama, T., Nakano, M., Mizoo, N., Cho, M., 1975. Reactivity measurement in a far-subcritical fast system (II) neutron source multiplication method. JAERI-M 6067, Japan Atomic Energy Research Institute (JAERI). Nishihara, K. et al., 2003. A new static and dynamic one-point equation and analytic and numerical calculations for a subcritical system. J. Nucl. Sci. Technol. 40 (7), 481–492. Orndoff, J.D., 1957. Prompt neutron periods of metal critical assemblies. Nucl. Sci. Eng. 2, 450–460. Shahbunder, H., Pyeon, C.H., Misawa, T., Shiroya, S., 2010a. Experimental analysis for neutron multiplication by using reaction rate distribution in acceleratordriven system. Ann. Nucl. Energy 37, 592–597. Shahbunder, H., Pyeon, C.H., Misawa, T., Lim, J.Y., Shiroya, S., 2010b. Subcritical multiplication factor and source efﬁciency in accelerator-driven system. Ann. Nucl. Energy 37, 1214–1222. Shahbunder, H., Pyeon, C.H., Misawa, T., Lim, J.Y., Shiroya, S., 2010c. Effects of neutron spectrum and external neutron source on neutron multiplication parameters in accelerator-driven system. Ann. Nucl. Energy 37, 1785–1791. Simmons, B.E., King, J.S., 1958. A pulsed neutron technique for reactivity determination. Nucl. Sci. Eng. 3, 595–608. Sjöstrand, N.G., 1956. Measurement on a subcritical reactor using a pulsed neutron source. Arkiv för Fysic 11, 233–246. Suzaki, T., 1991. Subcriticality determination of low-enriched UO2 lattices in water by exponential experiment. J. Nucl. Sci. Technol. 28 (12), 1067–1077. Tsuji, M., Suzuki, N., Shimazu, Y., 2003. Subcriticality measurement by neutron source multiplication method with a fundamental mode extraction. J. Nucl. Sci. Technol. 40 (3), 158–169.

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