Transactions of the Korean Nuclear Society Spring Meeting
Taebaek, Korea, May 25-27, 2011

Search Strategy of Detector Position For Neutron Source Multiplication Methodby Using Detected-Neutron Multiplication Factor
aTomohiro ENDO a∗
Department of Materials, Physics and Energy Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya,Japan, 464-8603*Corresponding author: t-endo@nucl.nagoya-u.ac.jp
1. Introduction

By the aid of the definition of kdet,

In this paper, an alternative definition of a neutron
multiplication factor, detected-neutron multiplication
factor kdet, is produced for the neutron source
multiplication method (NSM). By using kdet, a searchstrategy of appropriate detector position for NSM isalso proposed.
The NSM is one of the practical subcriticalmeasurement techniques, i.e., the NSM 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 quasi
real-time measurement. It is noted that the correction
factors play important roles in order to accurately
estimate subcriticality from the measured neutron countrates [1-3]. The present paper aims to clarify how tocorrect the subcriticality measured by the NSM method,
the physical meaning of the correction factors, and how
to reduce the impact of correction factors by setting aneutron detector at an appropriate detector position.
2. Methods and Results
2.1 Detected-Neutron Multiplication Factor
The steady state of subcritical system with external
neutron source is assumed. In this subcritical system,
neutron count rate Σ d ψ is measured by detector,
where ψ is neutron flux, and Σ d is macroscopic
detection cross-section. Let us classify the neutron flux
into two terms:
rrrrrr(1)
ψ ( r , E , Ω ) = ψ s ( r , E , Ω ) + ψ f ( r , E , Ω) ,where ψ s is source-flux due to the external source S:
r
r
rr
(2)
Aψ s ( r , E , Ω ) = S ( r , E , Ω ) ;ψ f is fission-flux due to the fission source:
rr
rr
(3)
Aψ f ( r , E , Ω ) = F ψ ( r , E , Ω ) .
In Eqs. (2) and (3), A and F are neutron annihilationand production operators, respectively. Now, thedetected neutron multiplication factor kdet is defined asfollows:Σ ψ(4)
kdet ≡ d f ,Σd ψ
The physical meaning of kdet is a ratio of total numberof detected fission-neutrons to total number of detected
all neutrons.

Σd ψisexpressed as follows:Σ ψ(5)Σd ψ = d s .
1 − kdet
Let us suppose that neutron count rates are measured atboth reference and target subcritical states. If theeffective multiplication factor keff at the reference stateis known beforehand, the effective multiplication factor
keff at the target is estimated as follows:Σ ψ
f c, target keff, target = 1 − f s (1 − f c, ref keff, ref ) d ref , (6)Σ d ψ target
where the subscripts ref and target mean the values atreference and target subcritical states, respectively; fs isa source-flux correction factor,(7)
f s ≡ Σ d ψ s,target Σ d ψ s,ref ;fc is conversion factor from kdet to keff,
f c = k det k eff .

(8)

It is noted that these factors fs and fc can be evaluated byonly forward neutron flux calculations without adjointcalculations.
2.2 Search Strategy of Detector Position for NSM
If the neutron detector is set at an appropriate
position where kdet ≈ keff , the conversion factor fc isnearly equal to unity, thus it is expected that fc can benegligible. Based on this idea, the appropriate detector
position is predicted from the numerical analysis at thereference state.
(1) keff-eigenvalue calculation is carried out to estimate
the value of keff,ref at the reference state.
(2) External source problem is carried out under theactual fission-neutron production condition toestimate spatial distribution of ψ(3) External source problem is carried out under thefictitious non-production conditions condition(νΣ f = 0 ) to estimate spatial distribution of sourceflux ψ s(4) Fission-flux ψ f is evaluated by subtracting ψ sfrom ψ(5) By supposing a point-wise neutron detector, countrates Σ d ψ and Σ d ψ f are evaluated at thedetector position.
(6) The spatial distribution of k det,ref is evaluated by Eq.

- 131 -(4)Transactions of the Korean Nuclear Society Spring Meeting
Taebaek, Korea, May 25-27, 2011

(7) Appropriate detector position where k det , ref ≈ k eff,ref

C30

C3540y [cm]

2.3 Calculation Model and Condition
The numerical analysis was carried out by using
THREEDANT, which is a three-dimensional multigroup discrete ordinates transport code [4]. 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 ydirection, and 49 cm in z-direction, respectively.
The core is surrounded by a 20 cm thick reflector.
(2) Two energy group constants are quoted fromreference [5]. Three subcritical cores (C30, C35,C40) are analyzed by changing the group constantsof core region.
(3) Point-wise source at ( x, y, z ) = (−31, 0, 0) in thereflector region. In addition, the energy spectrum of
external source is the same as fission spectrum.
(4) Point-wise detector for only thermal neutron.
(5) Total number of spatial meshes is 81 × 73 × 89 forx-, y-, and z-directions, respectively.
(6) EO8 quadrature set is used for SN solid angle
quadrature set [6].
(7) Convergence criteria is 1.0×10-6 for inner iteration.
2.4 Numerical Results
In order to search appropriate detector positions in
two dimensions, Fig. 1 plots the relative difference
between kdet and keff at z = 0 plane within the range of−0.5 % to 0.5%. In Fig.1, green colored regions can beregard as the detector positions of kdet ≈ keff . As shownin Fig.1, the detector positions of kdet ≈ keff can be findout not only core regions but also reflector regions.
Furthermore, these detector positions are quite
stationary for subcritical states. For example, a pointwise detector is set at ( x, y, z ) = (−9, 0, 0) (Fig. 1). As anumerical simulation of NSM, C35 subcritical state wasregarded as the reference state, target multiplication
factors keff, target for C30 and C45 states were estimatedby NSM without correction factor fs and fc (Table I). By
virtue of appropriate detector positions of kdet ≈ keff , theneutron multiplication factors are well estimated even
without corrections. In other word, by setting detector
at the position where kdet ≈ keff , the impact ofcorrection factor can be reduced. Of course, if theconversion factors fs and fc at both reference and target
states can be taken into account, the estimatedmultiplication factors are nearly equal to effective
neutron multiplication factors keff, target at the targetstates.

Reflector

Reflector

2020CoreCore

Core000-20

-20

-20

-40
-40-4040 -40-20020-20020-4040 -400.0-2020040-0.5x [cm]

Fig. 1: Point-wise detector positions of kdet ≈ keff at z = 0
plane (circle and cross mean a point-wise source and aselected detector position)
Table I: Estimated multiplication factor by NSM
C45C35 (ref.)C30

keff [-]
kdet [-]

0.99273
0.991210.96639
0.96347

0.923110.92452fs [-]

1.271621.00000

0.84264fc [-]
estimated k by NSM
without correction[-]rel. dif. [%dk/k]

0.99847

0.99698

1.00152

0.993640.966390.91758

0.09

-0.60

3. Conclusions
In this paper, based on the idea of kdet, it is clear thatcorrelation factors of the NSM method consist of thesourc-flux correction factor fs to fix the difference of
non-fission component of neutron count rate, and theconversion factor fc to convert from kdet to the effective
multiplication factor keff. Through the numericalanalysis of three-dimensional two-group transport
calculations for simple geometry, appropriate detector
positions where kdet ≈ keff could be find out for only one
certain subcritical state, e.g., the reference state. By
putting a neutron detector at such a particular position,
the target neutron multiplication factors can be wellestimated even without any corrections.
REFERENCES[1] T. Mukaiyama, et al., Reactivity Measurement in a FarSubcritical Sast System [II] Neutron source multiplication
method, JAERI-M 6067, Japan Atomic Energy ResearchInstitute (JAERI), 1975.
[2] N. Mizoo, Reliability of a New Modified Neutron Source
Multiplication Method, JAERI-M 7135, Japan Atomic Energy
Research Institute (JAERI), 1977.
[3] D. G. Cacuci, Handbook of Nuclear Engineering: Vol.3
Reactor Analysis, Springer, pp. 2147-2152, 2010.
[4] R. E. Alcouffe, et al., DANTSYS: A Diffusion
Accelerated Neutral Particle Transport Code Systemm, Los
Alamos National Laboratory report LA-12969-M, 1995.
[5] T. Misawa, H. Unesaki, C. Pyeon, Nuclear Reactor
Physics Experiments, Kyoto University Press, 2010.
[6] T. Endo, A. Yamamoto, Development of New Solid Angle
Quadrature Sets to Satisfy Even- and Odd-moment Conditions,J. Nucl. Sci. Technol., 44, pp. 1249-1258, 2007.

- 132 -kdet/keff - 1 [%]

200.5

40

Reflector

can be searched.

C4540