Auto-correlation and variance-to-mean measurements in a subcritical core obeying multiple alpha-modes

Neutron noise measurements based on the Rossi-α and Feynman-α methodologies have been performed in a heterogeneous subcritical system. It is shown that the traditional single alpha-mode formulations of the Rossi-α and Feynman-α methods are not applicable due to the presence of higher alpha-modes. Formalisms taking into account multiple alpha-modes are applied resulting in satisfactory results. Three alpha-modes could be identified using the Rossi-α method, whereas only two could be obtained using the Feynman-α method. In the Feynman-α case, the possibility to obtain the fastest decaying alpha-mode was diminished due to detector dead time effects. It was found that the slowest decaying alpha-mode does not exactly correspond to the prompt decay found in pulsed neutron source measurements, which confirms the results of previous studies. Strengths and weaknesses of the multiple alpha-mode Rossi-α and Feynman-α methods observed in this study are pointed out.     

252Cf源驱动噪声分析法测量α初步研究

在一球形浓缩铀临界装置上,采用²⁵²Cf源驱动噪声分析方法对其次缓发临界状态下的α进行测量,在频域内分析数据得到α。对比频域内的两种数据处理方式（功率谱实虚部相除方法和功率谱取模直接拟合方法）,用互功率谱密度函数实虚部相除方法得到在-0.1$情况下,脉冲堆的α为0.58μs^-1,与Rossi-α方法的结果相吻合。     

Feynman-α and Rossi-α analyses for a subcritical reactor system driven by a pulsed spallation neutron source in Kyoto University Critical Assembly

ABSTRACT

For a subcritical reactor system driven by a periodically pulsed spallation neutron source in Kyoto University Critical Assembly (KUCA), the Feynman-α and the Rossi-α neutron correlation analyses were carried out to determine the prompt-neutron decay constant and quantitatively to confirm a non-Poisson characteristics of the neutron source. In these correlation analyses, a non-negligible contribution of delayed neutrons and a non-Poisson character of the source were considered, and each pulse was assumed to be a delta function. When a neutron counter was placed closely to the reactor core, the prompt-neutron decay constant determined from the present Feynman-α analysis well agreed with that done from a previous analysis for the same subcritical system driven by an inherent neutron source. However, the decay constant determined from the present Rossi-α analysis was in poor agreement with that done from the above previous analysis. This disagreement originated from an inevitable excitation of a higher mode. In the Rossi-α counting probability distribution, the excitation deformed a sharp cusp arising from the delta function to a smooth convex shape. When the data around the convex top were masked for least-squares fitting of the present Rossi-α formula, the disagreement could be successfully resolved. Compared with the previous Feynman-α and Rossi-α analyses under the Poisson inherent source, the non-Poisson spallation source definitely enhanced the respective prompt-neutron correlation amplitudes. The enhancement rate increased with an increase in subcriticality. Moreover, the Degweker’s factor (m ₂-m ₁ ²)/m ₁ ² of 0.067 ± 0.011, which indicated a non-Poisson character of the present spallation source, could be determined from the present correlation analysis and the non-zero value of the factor convinced us that the present source had a different statistical distribution from the Poisson.     

Monte Carlo simulation of Feynman-α and Rossi-α techniques for calculation of kinetic parameters of Tehran Research Reactor

Noise analysis techniques including Feynman-α (variance-to-mean) and Rossi-α (correlation) have been simulated by MCNP computer code to calculate the prompt neutron decay constant (α₀), effective delayed neutron fraction (β_eff) and neutron generation time (Λ) in a subcritical condition for the first operating core configuration of Tehran Research Reactor (TRR). The reactor core is considered to be in zero power (reactor power is less than 1 W) in the entire simulation process. The effect of some key parameters such as detector efficiency, detector position and its dead time on the results of simulation has been discussed as well. The results of proposed method in the current study are validated against both the experimental data and the results of MTR_PC computer code.     

Prompt alpha and reactivity measurements on fast metal assemblies

This paper summarizes and reviews the methods of reactivity determination and measurement of the prompt-neutron decay, briefly describes the equipment requirements for such measurements for unmoderated metal assemblies, and presents experimental results to illustrate the methods. These assemblies include fast metal critical assemblies and fast pulsed reactors. The primary reactivity determination methods used have been: (1) stable reactor period measurements which are usually used near delayed criticality to obtain the reactivity or to calibrate the reactivity prior to burst initiation; (2) prompt reactor period measurements which are useful to determine the reactivity early in superprompt critical excursions; (3) inverse kinetics rod drop measurements which obtain the reactivity as a function of time after a rod or reactor component is removed from a delayed critical assembly, and (4) prompt neutron decay constant measurements from which the reactivity can be obtained if corrections are made for changes in the neutron lifetime. Inverse kinetics and decay constant measurements are usually used below delayed criticality, although decay constant measurements have been performed above delayed critical.The decay constant is usually obtained by the traditional pulsed-neutron method, using a pulsed neutron source such as a Cockcroft-Walton accelerator, or by the Rossi-α method. The recent use of ²⁵²Cf has resulted in some new techniques for determining the decay constant and reactivity; this method has some unique advantages over the traditional methods. The theory of these new measurements is reviewed, and some recent results are presented.     

柱形金属铀次临界系统瞬发中子衰减常数测量

次临界核系统的瞬发中子衰减常数α与反应性有着重要联系。采用²⁵²Cf随机脉冲源法测量了一柱形金属次临界系统的瞬发中子衰减常数。为对源中子的影响进行分析,借助蒙特卡罗模拟方法建立模型进行了模拟,对源直穿中子和核系统瞬发中子时间分布特性进行了比较,分析了源中子对瞬发中子衰减曲线的影响。模拟结果表明,对该柱形金属铀系统,源中子注入100 ns后源直穿中子对核系统瞬发中子的影响可忽略。根据分析结果选取了合理起始道,对实验数据进行单指数最小二乘拟合,得到该次临界系统的α为15.5μs^-1。     

Application of Polarity Correlation Method to Graphite Moderated Reactor

The polarity correlation method has been applied to an experiment for measuring the prompt mode neutron decay constant of a reactor with a long neutron lifetime. Measurements have been performed in the SHE VIII-l at critical and several subcritical steady states (0~ — 1$).

The analog signal of detector output is converted to a polarity signal having only one-bit of information on the amplitude of the detected signal. The polarity correlation function is obtained as the correlation function between two polarity signals. Attention has been paid in the design of the correlator to obtain ample stability in operation. Using a digital computer, the prompt mode neutron decay constant is determined with 2% experimental accuracy from the decay curve of the polarity correlation function. A series of the prompt mode neutron decay constants is fitted to an approximated expression of inhour equation to obtain βl. The value of each prompt mode neutron decay constant was found to be in good agreement with that measured by the pulsed neutron technique. The polarity correlation method is particularly applicable to critical or near critical state where a pulsed neutron source cannot be effectively introduced.

In Appendix, an estimation is made for the error in the measured polarity correlation function due to unbalanced mean setting level in the experiment.     

252Cf随机脉冲源方法测量球形钚系统瞬发中子衰减常数

²⁵²Cf随机脉冲源方法由早期的重复脉冲源方法演变而来,是测量核系统瞬发中子衰减常数α的有效方法。采用该方法测量了钚球装配31 mm、29 mm厚钢反射层核系统的α,在有效信号和噪声的比例为1∶1的情况下,得到的瞬发中子衰减谱信噪比为7∶1,最小二乘拟合结果依次为2.25 μs^-1和3.00 μs^-1,拟合误差为±0.02 μs^-1。与Rossiα方法的测量结果进行了比较,两种测量方法的结果差异小于1.3%。     

Comparison of reactivity estimations obtained from rod-drop and pulsed neutron source experiments

Experimental estimates of a subcritical level using rod-drop and pulsed neutron source (PNS) techniques are presented. First, three methods were applied to the same rod-drop data set: the prompt drop, area and linear fit methods. Second, two methods were applied to the same PNS data set: the prompt decay and area methods. Except for the prompt decay PNS method, all the other methods are in good agreement. It is shown that the area PNS method is the least space-dependent and the most accurate.     

Power spectral analysis for a thermal subcritical reactor system driven by a pulsed 14 MeV neutron source

A series of power spectral analyses for a thermal subcritical reactor system driven by a pulsed 14 MeV neutron source was carried out at Kyoto University Critical Assembly (KUCA), to determine the prompt-neutron decay constant of the accelerator-driven system (ADS). The cross-power spectral density between time-sequence signal data of two neutron detectors was composed of a familiar continuous reactor noise component and many delta-function-like peaks at the integral multiple of pulse repetition frequency. The prompt-neutron decay constant inferred from the reactor noise component of the cross-power spectral density was consistent with that obtained by a pulsed neutron experiment. However, the reactor noise component of the auto-power spectral density of each detector was hidden by a white chamber noise in the higher-frequency range and this feature resulted in a considerable underestimation of the decay constant. For several runs with a low pulse-repetition frequency, furthermore, we attempted to infer the decay constant from point data of the delta-function-like peaks. The analysis for a run under a slightly subcritical state resulted in the consistent decay constant; however, those for other runs under significantly subcritical states underestimated the decay constant. Considering the contribution of a spatially higher mode to the point data, the above underestimation was solved to obtain the consistent decay constant. While the Feynman-α formula for a pulsed neutron source is too complicated to be fitted directly to variance-to-mean ratio data, the present analysis on frequency domain is much simpler and the conventional formula based on the first-order reactor transfer function is available for fitting to power spectral density data.     

多重性探测器中子衰减时间的标定

采用Rossi-α方法直接测量了多重性探测器中子衰减时间,并采用几个不同强度的锎源和不同质量的钚部件进行了Rossi-α验证测量,准确获得了某个多重性探测器的中子衰减时间。Rossi-α直接测量相比改变符合时间宽度间接标定的传统方法更直接可靠,且可在钚部件无损测量中作为辅助监测手段。     

Determination of prompt-neutron decay constant from phase shift between beam current and neutron detection signals for an accelerator-driven system in the Kyoto University Critical Assembly

A unique power spectral analysis for a subcritical reactor system driven by a pulsed 14 MeV neutron source was carried out at the Kyoto University Critical Assembly (KUCA). In this analysis, a complex cross-power spectral density between time-sequence signal data from an accelerator beam ammeter and a neutron detector was measured to determine the prompt-neutron decay constant of an accelerator-driven system (ADS) from the phase data of the spectral density. Assuming the one-point kinetics model, in theory, the decay constant can be arithmetically derived from the phase at the integral multiples of the pulse repetition frequency. However, the actual derivation from the phase at a pulse repetition frequency of 20 Hz considerably underestimated the prompt-neutron decay constant, compared with that obtained by a previous pulsed neutron experiment, and the derived decay constant apparently decreased with an increase in the multiple of the pulsed repetition frequency. Considering a lag time in detector response, the above underestimation and the above apparent decrease were solved to obtain the consistent decay constant. While both previous power spectral analysis and Feynman-α analysis for pulsed neutron source require non-linear least-squares fits of the respective complicated formulae, the present analysis makes the fitting unnecessary except at regular calibration of the lag time. This feature is advantageous for a robust online monitoring of subcritical reactivity of an actual ADS.     

Modal influence of the detector location for the noise calculation of the ADS

The purpose of this paper is to investigate the space dependence of neutron noise in a source-driven subcritical system. The noise produced by the fluctuations of the source are measured from the cross spectrum between the source and a detector located inside the system using the methodology of Muñoz-Cobo et al. (Muñoz-Cobo, J.L., Rugama, Y., Valentine, T., Mihalczo, J., Perez, R., 2002. Annals of Nuclear Energy in press). The prompt neutron decay constant obtained from the source-detector cross spectrum is dependent on the detector location because of the influence of higher modes of the neutron flux. One group diffusion theory is used to determine the eigenvalues and eigenfunctions of the fundamental and higher modes. This analytical approximation will be used to explain the detector location effect through quantitative evaluation of a specific model. The analytical system subcriticality will be obtained from the eigenvalue equation in the static case at two different subcritical situations.     

Measurement of large negative reactivity of an accelerator-driven system in the Kyoto University Critical Assembly

Large negative reactivity of a subcritical system driven by a pulsed 14 MeV neutron source has been measured in the Kyoto University Critical Assembly. The subcriticality of the accelerator-driven system (ADS) ranged in effective multiplication factor roughly from 0.98 to 0.92, which corresponded to an operational range of an actual ADS proposed by Japan Atomic Energy Agency. As the measurement technique, pulsed neutron method, power spectral analysis for pulsed neutron source, accelerator-beam trip method were employed. From neutron count decay data obtained by the pulsed neutron experiment, not only the prompt-neutron decay constant of fundamental mode but also a higher spatial mode could be derived. The subcriticality was also determined from the fundamental decay constant. The measured cross-power spectral density consisted of a familiar correlated reactor-noise component and many uncorrelated delta-function-like peaks at the integral multiple of pulse repetition frequency. The fundamental prompt-neutron decay constant, i.e., the subcriticality determined from the latter uncorrelated peaks was consistent with that obtained by the above pulsed neutron experiment. However, the magnitude of the former correlated component was reduced with an increase in the subcriticality and eventually this component became almost white at deeply subcritical state ranging in the multiplication factor under 0.95. Consequently, the determination of the decay constant from the correlated component was impossible under such a subcritical state. As data analysis method for the beam trip experiment, both the conventional integral count method and the least-squares inverse kinetics method (LSIKM) were employed. The LSIKM analysis led to the consistent subcriticality with that obtained by the pulsed neutron experiment, while the integral count method significantly underestimated the subcriticality. This underestimation originated from a residual background count, which was maintained after the beam trip. The LSIKM was mostly not influenced by such a slight count rate.     

A Generalized Analysis of Rossi-α Experiment

A generalized theory of the Rossi-α method is shown and is applied to the analysis of an experiment made with a remotely subcritical system.

Starting from the neutron transport equation, a general expression for the probability of detecting correlated neutrons is given, which is a function of the positions and sensitivities of the neutron detectors used in the experiment.

Some numerical examples for this expression are also given under the one-group diffusion approximation for comparison with the experiment. The agreement between them is satisfactory.     

Preliminary measurements of the prompt neutron decay constant in MASURCA

Pulse counting techniques have been used to measure the prompt decay constant = (β - ) / Λ in the MASURCA reactor of CEA at critical state. The data has been analyzed in time domain using Rossi- and Feynman- techniques, and in frequency domain using the cross power spectral density.

The Rossi- technique has been studied using one and two detectors. Due to the strong inherent spontaneous fission source, the one-detector variant gives a very strong white-noise signal, which is absent in the two-detector method. Because each neutron detected recorded not only a pulse, but also an echo after 120 ns, corrections had to be made to the theory applied.

The Feynman- technique is even more sensitive to the echo in the signals, and quite large corrections had to be made. Nevertheless the results obtained are in reasonable agreement with those of the correlation methods. For both measurement techniques, experiments of long duration are needed to get accurate results. The results obtained agree within 10% with calculations.

The prompt decay constant has also been measured with a continuous current technique. From the cross power spectral density thus obtained, the -value is in agreement with that of the pulse counting techniques.     

Effective delayed neutron fraction by Rossi-α method in accelerator-driven system experiments with 100 MeV protons at kyoto university critical assembly

In the accelerator-driven system (ADS), the effective delayed neutron fraction (β_eff) is a requisite for converting the subcriticality from dollar units to pcm units. To evaluate the accurate β_eff value in ADS, measurement of β_eff complements its calculation methodology and the nuclear data on delayed neutrons. Subcriticality measurements are carried out by the pulsed neutron source method in the Kyoto University Critical Assembly, and the neutron noise analyses are conducted by the Rossi-α method with the pulsed shape of the spallation neutron source. The value of β_eff is deduced with the combined use of measured subcriticality in dollar units and correction parameters by MCNP6.1 together with JENDL-4.0 and JENDL/HE-2007. A comparison between the calculated and the measured β_eff represents the acceptable accuracy within the subcriticality range of around k_eff = 0.93 in the ADS operations. Here, the applicability of the measurement methodology based on the Rossi-α method is demonstrated by varying the subcriticality.     

Calculation of the pulsed Feynman-alpha formulae and their experimental verification

An effective method of calculating the pulsed Feynman-alpha formula for finite width pulses is introduced and applied in this paper. The method is suitable for calculating both the deterministic and the stochastic Feynman-alpha formulae, while also being capable of treating various pulse shapes through very similar steps and partly identical formulae. In the paper both the deterministic and the stochastic cases are treated for square and Gaussian pulses. The solutions show a very good agreement with the results of currently performed experiments by some of the authors at the Kyoto University Critical Assembly (KUCA).The formulae obtained are also used for a quantitative evaluation of the prompt neutron decay constant from a large number of experiments made at the KUCA for a wide range of parameters such as subcritical reactivity, pulse repetition frequency and pulse width. The suitability of the formulae to determine the prompt neutron decay constant α by curve fitting to the measured data was investigated. It was found that, despite the larger deviation from the traditional Feynman Y(T)-curves from the traditional ones with a constant source (i.e., larger ripples superimposed on a smooth curve), the stochastic pulsing method is superior to the deterministic one in that it yields the correct α value for all subcriticalities. The deterministic method also works fine for most cases, but its application is not so straightforward.     

基于OpenMC的瞬发中子衰减常数计算模块开发与验证

瞬发中子基波衰减常数α可定量描述反应堆内中子随时间的变化,是计算绝对反应性所需的中子动力学参数之一,对次临界（特别是较深次临界）绝对反应性的精确测量具有重要意义。本文在开源程序OpenMC基础上,基于k α迭代方法,以中子径迹长度上的平均时间吸收权重修正作为k α迭代参数因子,在输运过程中对瞬发、缓发中子分别考虑,开发了具有瞬发α本征值问题计算功能的OpenMC PA模块。以Godiva衍生基准题和MUSE 4次临界实验装置为计算对象,对程序计算瞬发α本征值问题能力进行验证。结果表明,该计算模块有优于MCNP4C的计算速度与计算范围,计算值与参考值的相对误差小于05%。OpenMC PA能满足次临界系统瞬发α本征值和中子动力学参数计算需求。     

Measurement of kinetic parameters in the fast subcritical core MASURCA

In the MUSE shared cost action of the European Fifth Framework Program measurements have been performed to investigate the neutronic behavior of the fast subcritical core MASURCA coupled with the GENEPI accelerator. The aim is to examine the applicability of different measurement techniques for the determination of the main kinetic parameters. The measurement of Rossi-alpha distributions, recorded with the accelerator turned off, showed that the analysis of the obtained distributions is feasible for deep subcritical levels, but with strongly deteriorated statistics. From Rossi-alpha distributions, recorded with the pulsed neutron source in operation, the alpha decay constant was easily derived due to good statistics on the correlated signal resulting from the strong intensity of the neutron pulse. When applying the pulsed neutron source analysis, the reactivity (in dollars) together with the ratio of the mean neutron lifetime l and the effective delayed neutron fraction β_eff is immediately derived. Although these first results are very promising, further measurements are needed to qualify the method at larger subcritical levels which are representative for future ADS.