Science and Technology of Nuclear Installations

Volume 2018, Article ID 2350458, 7 pages

https://doi.org/10.1155/2018/2350458

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

Seoul National University, 1 Gwanak-ro, Gwanak-gu, Seoul 08826, Republic of Korea

Correspondence should be addressed to Hyung Jin Shim; rk.ca.uns@jhmihs

Received 31 January 2018; Accepted 15 March 2018; Published 2 May 2018

Academic Editor: Eugenijus Ušpuras

Copyright © 2018 Sang Hoon Jang and Hyung Jin Shim. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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 () 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 estimates by the slope fitting are converged. The developed method is applied to the Pb-Bi-zoned ADS experimental benchmark at Kyoto University Critical Assembly. The convergence time estimated by the TDMC PNS simulation agrees well with the experimental results. The 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 measurement.

#### 1. 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 [1–3]. The neutronic characteristics of the subcritical reactor have been extensively studied theoretically [4, 5] and experimentally [6–8]. 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 higher-mode 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 [17–19]. 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].

#### 2. Determination of an Optimum Detector Position through the TDMC Analysis

##### 2.1. TDMC PNS Simulation

The population of prompt neutrons induced from a fast neutron burst in a subcritical system decreases exponentially. Thus a special population control technique is necessary for an efficient TDMC calculation. Here we adopt an analog MC simulation of the branching process in which extra neutrons from fission are sampled and tracked accompanied with the combing technique [17]. In the TDMC simulations with the combing technique, the time domain is split into time bins and each neutron is simulated time-bin-by-time-bin with updating its time variable whenever its track is sampled by [19]where ( or ), , and are the time after the th flight, the length, and the neutron energy of the th track of history at time bin . is the neutron mass. If the sampled time is greater than the upper time bound of the th time bin, that is, , then the track length of and time after the last flight of history , denoted by and , respectively, becomewhere 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 with conserving the total weight.

##### 2.2. 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 during time interval , , defined aswhere , , and are the time-step, isotope, and reaction type index. denotes the prompt neutron flux.

Then corresponding to the detector position can be estimated by an exponential fitting to the TDMC results of as [13]where and are fitting constants and and are the time after the neutron burst and the beginning time of the fitting interval, respectively. indicates an estimate of from a neutron detector located at using . In this study, 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 , is determined when the relative error of a mean value of comparing to its reference, denoted by , becomes less than a prescribed value aswhere is the number of replicas with different random number sequences. is an estimate of the th replica calculation. of 0.05 is used for this convergence diagnosis.

Here is calculated by the MC -iteration method [23] which is developed to solve the -mode eigenvalue equation expressed aswhere the subscript indicates prompt neutron. is named the time source [23]. is a neutron speed corresponding to its energy . and denote the average numbers of neutrons emitted from reaction type and prompt fission neutrons, respectively. is the probability that a collision of type by a neutron of direction and energy will produce a neutron in direction interval about 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.

##### 2.3. Determination of an Optimum Detector Position

The amplitude of neutron signals used for the exponential regression when is converged can be defined aswhere denotes the fitting time interval.

Then the optimum detector position for the PNS measurement can be determined as a position where becomes maximized because the statistical uncertainty of the detector signals during is assumed to be inversely proportional to the signal amplitude at the position by following the Poisson distribution.

#### 3. Application Results

##### 3.1. 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.