Determination of an Effective Detector Position for Pulsed-Neutron-Source Alpha Measurement by Time-Dependent Monte Carlo Neutron Transport Simulations

A simple method using the time-dependent Monte Carlo (TDMC) neutron transport calculation is presented to determine an effective detector position for the prompt neutron decay constant (α) measurement through the pulsed-neutron-source (PNS) experiment. In the proposed method, the optimum detector position is searched by comparing amplitudes of detector signals at different positions when their α estimates by the slope fitting are converged. The developed method is applied to the Pb-Bizoned ADS experimental benchmark at Kyoto University Critical Assembly. The α convergence time estimated by the TDMC PNS simulation agrees well with the experimental results. The α convergence time map and the corresponding signal amplitude map predicted by the developed method show that polyethylene moderator regions adjacent to fuel region are better positions than other candidates for the PNS αmeasurement.


Introduction
Since the early 1990s, accelerator-driven subcritical systems (ADS) for transmutation of radioactive wastes and energy production have been proposed and designed throughout the world with their advantages of high flexibility of fuel compositions and the enhanced safety concept [1][2][3].The neutronic characteristics of the subcritical reactor have been extensively studied theoretically [4,5] and experimentally [6][7][8].The prompt neutron decay constant (hereafter referred to as ) of a subcritical system is a fundamental kinetics parameter which represents its asymptotic behavior ignoring the delayed neutron effect.Moreover  can be directly measured [9,10] by injecting a short burst of neutrons in the system, called the pulsed-neutron-source (PNS) experiment.Since Simmons and King [9] applied an exponential regression to neutron detector signals from the PNS experiment, this  measurement method has been popularly employed because it can provide  results independent of the positioning and energy characteristics of the detector and neutron source [9,11,12] by reducing higher-mode contaminations on the exponential fitting [13,14].
In practice, however, the PNS  measurement may yield considerably different results at different detector positions and neutron sources, as reported in the experimental benchmarks on an ADS at Kyoto University Critical Assembly (KUCA) [15,16].This measurement dependency on the detector position and the neutron source can be attributed mostly to the signal contamination [11,16] by the highermode components of the prompt neutron flux, which is caused by taking detector signals before the higher-mode components fully decay out.It is difficult, however, to obtain confident detector signals after the prompt neutron flux converges to the fundamental mode in a deep subcritical system where the prompt neutron flux decreases rapidly.Therefore, it is necessary to determine effective detector positions where the prompt neutron flux converges fast with larger signal strength than other candidate positions.
The objective of this paper is to devise a simple but practical way to determine an optimum detector position for the  measurement through the PNS experiment using the time-dependent Monte Carlo (TDMC) neutron transport analyses [17][18][19].In the TDMC calculations, the combing algorithm [17,20] is applied to maintain the time-bin-wise neutron population because an exponential decrease of the neutron population in an analog TDMC calculation of a subcritical system causes large statistical uncertainties.In the proposed method, the optimum detector position is searched by comparing the strength of detector signal at each spatial position when the  estimate at the position is converged.The position-dependent  convergence is diagnosed by a slope fitting to the detector signals obtained from the TDMC calculations.The proposed methods are implemented in a Seoul National University continuous-energy Monte Carlo (MC) code, McCARD [21], and applied to the Pb-Bi-zoned ADS experimental benchmark at KUCA [22].

Determination of an Optimum Detector
Position through the TDMC Analysis where

𝑖,𝑗
means the neutron energy of the last flight  of history  at time bin .After the th time-bin TDMC simulations for all histories, the number of neutrons for the next time-bin simulations is increased to be the user-inputted number of histories by splitting according to the number of surviving neutrons at  +1 with conserving the total weight.

𝛼 Estimation by the Slope Fitting.
The time-dependent detector signals from prompt neutrons can be represented by MC responses of the reaction rate in the detector volume   at r during time interval (   − Δ/2,    + Δ/2),   (r,    ), defined as where   , , and  are the time-step, isotope, and reaction type index.  denotes the prompt neutron flux.
Then  corresponding to the detector position r can be estimated by an exponential fitting to the TDMC results of   (r,    ) as [13] where  1 and  2 are fitting constants and  and   are the time after the neutron burst and the beginning time of the fitting interval, respectively. est (r |   ) indicates an estimate of  from a neutron detector located at r using   .In this study,  est (r |   ) are calculated with increasing   from 0.0 ms to 3.9 ms by 0.1 ms and setting the fitting interval to 1.0 ms.
An onset time of the convergence of  est (r),  0 (r) is determined when the relative error of a mean value of  est (r |   ) comparing to its reference, denoted by  ref , becomes less than a prescribed value  as where  is the number of replicas with different random number sequences. est, (r |   ) is an  estimate of the th replica calculation. of 0.05 is used for this convergence diagnosis.
Here  ref is calculated by the MC -iteration method [23] which is developed to solve the -mode eigenvalue equation expressed as ]  Σ  (r  ,   ) Σ  (r  ,   )   (  , Ω  → , Ω) where the subscript  indicates prompt neutron.  is named the time source [23].V() is a neutron speed corresponding to its energy .]  and ]  denote the average numbers of neutrons emitted from reaction type  and prompt fission neutrons, respectively.  (  , Ω  → E, Ω)Ω is the probability that a collision of type  by a neutron of direction Ω  and energy   will produce a neutron in direction interval Ω about Ω with energy in  about .Other notations follow convention.By directly applying the power iteration method [24] for (8), it is demonstrated [23] to stably estimate  even for a deep subcritical system.

Determination of an Optimum Detector Position.
The amplitude of neutron signals used for the exponential regression when  est (r) is converged can be defined as where Δ denotes the fitting time interval.Then the optimum detector position for the PNS  measurement can be determined as a position r where   (r) becomes maximized because the statistical uncertainty of the detector signals during [ 0 (r),  0 (r) + Δ] is assumed to be inversely proportional to the signal amplitude at the position by following the Poisson distribution.

Pb-Bi-Zoned Experimental Benchmark.
The developed method to determine the optimum detector position for the PNS  measurement is applied for the Pb-Bi-zoned ADS experimental benchmark at KUCA [22].The benchmark provides 6 different subcritical cores comprised of Pb-Bi loaded enriched uranium fuel and polyethylene moderator and reflector.The spallation neutron source is generated in the center of the core by injecting 100 MeV protons to the Pb-Bi target.The PNS  measurement is conducted with three optical fiber detectors in different positions.Case 6 among the six cores is chosen for an application of the developed method and its core configuration is shown in Figure 1.

Spallation Source Treatment.
The spallation neutron source information is obtained from MCNPX2.6.0 [25] proton source simulations.The spallation neutron spectra from the Pb-Bi target are tallied with respect to angle between the outgoing direction of neutrons and proton beam.The angle bin is equally divided by 15 degrees.The MCNPX calculation is done with 10,000,000 histories and la150h proton library provided.The neutron spectra and relative angular flux distribution are given in Figure 2. One can see that the neutron spectra tend to be more hardened as its direction is more forwarded and the overall neutron yield is biased to the forward direction.The direction and energy of the spallation neutrons are inputted in the form of histograms and uniformly sampled in each bin at the beginning of the McCARD TDMC simulations.

Detector Modeling.
The neutron detector used in the experiment is a small-sized optical fiber detector [26] with 1 mm diameter which makes it available to be inserted into gaps between assemblies.The detector consists of a mixture of 6 LiF neutron converter and ZnS scintillator of which signals are induced by charged particles emitted from (, ) and (, ) reactions.Since the real size of the detector is too small to obtain confident tally results in the TDMC simulations, the detector size is enlarged to cover the active core region at each intersection of air gaps.The tally region of the detector is shown in Figure 3.In the detector regions, the detector signals are tallied as a sum of (, ) and (, ) reaction rates while the neutron simulation is conducted as if the detectors are filled with air to prevent them distorting the MC neutron tracking.

Searching the Optimum Detector Positions.
To verify the feasibility of the devised method,  est (r |   ) is estimated at the two detector positions which are marked as optical fibers #1 and #2 in Figure 1. est (r |   ) at each detector position is compared with  ref calculated by the -iteration method and  exp (r |   ) which is estimated by the exponential regression of experimental detector signals.Note that comparison results for optical fiber #3 are omitted because its detector signals might be contaminated with gamma-ray induced by high energy neutron sources. est (r |   ) is estimated with 100 replicas of TDMC simulation using 1,000,000 histories and 0.1 ms time bin up to 5 ms. ref is calculated by the -iteration method with 100,000 histories and 100 active iterations.ENDF/B-VII.1 cross section libraries are used for both calculations.
Figure 4 shows comparison results for the two detector positions.The solid lines and the dashed lines are the TDMC and experimental results at detector positions.The value of  ref is estimated to be 1950.0with its standard deviation of 2.0.From the figure, one can see that the  estimates converge to the reference value with different convergence rates depending on their positions.Also one can see a discrepancy of the initial convergence trends of detector #2 between the TDMC and experimental results, which can be attributed to a difference of detector signal yields sensitive to neutron energy range.The convergence times detected by (5) using the TDMC tally results are 0.8 ms for detector #1 and 0.4 ms for detector #2, whereas those from the experiments are 1.5 ms for detector #1 and 1.1 ms for detector #2.Although the convergence times estimated from the TDMC calculation differ from the experiments' by 0.7 ms for both detectors due to initial effects of the higher-mode components, it is noteworthy that their differences between the two detector positions are the same as 0.4 ms.This implies that the proposed method based on the TDMC calculation can predict quite well the sensitivity of the  convergence time depending on detector positions.
The TDMC  estimations are conducted for all possible detector positions in the air gaps between assemblies to search the optimum detector positions.Figure 5 shows the convergence time and amplitude of neutron signal at each candidate position.The white colored positions in the convergence time map are where the  estimates do not converge until 4.0 ms.Both the convergence time and the neutron signal map show that the polyethylene moderator regions adjacent to fuel region converge faster and give higher neutron signals than other regions.It is expected to obtain more reliable detector signals for the PNS  measurement at these optimum detector positions.

Conclusions
A simple method to determine an effective detector position for the PNS  measurement is proposed by comparing signal amplitudes at different detector positions estimated by the TDMC neutron transport calculations when their  estimates by the slope fitting are converged.The developed method is implemented in McCARD and applied to case 6 core in the KUCA Pb-Bi-zoned ADS experimental benchmarks.From the comparisons with experimental results, it is shown that the TDMC calculation predicts the  convergence time quite well.The proposed method provides the  convergence time map and the corresponding signal amplitude map for case 6 core, which can be used to determine effective detector positions and to validate experimental results in the PNS  measurement.

Figure 3 :Figure 4 :
Figure 3: Tally region of the optical fiber detector (red region).

Figure 5 :
Figure 5: Convergence time and amplitude of neutron signal maps.