The Average Transport Path Length in Scattering Media

Bardsley, J. N.; Dubi, A.

doi:10.1137/0140005

Cited by 28 publications

(29 citation statements)

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“…neutrons) into the volume of interest (e.g. fuel assembly parallelepiped) and no absorption or particle production in the medium [15]. It should be noted that the 4V/S result holds true for a wide range of situations for which scattering centers are present in the volume of interest [15].…”

Section: Appendix A: Analysis Of Analytical Self-shielding Functionmentioning

confidence: 90%

“…That is, the probability that a neutron is absorbed along an average, straight chord length of r, assuming that the neutron will not scatter, that the neutron enters from outside the volume (i.e. not produced within the volume), and that the material is homogenous within the volume is If one considers a right parallelepiped fuel volume with length H and edge L, such that the volume is given by and that the surface area, S, is given by , and that the assumptions under which the mean Dirac chord length are valid [15], such that , then (A.5) is equivalent to (6), considering that .…”

Section: Appendix A: Analysis Of Analytical Self-shielding Functionmentioning

confidence: 99%

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Lead Slowing-Down Spectrometry Time Spectral Analysis for Spent Fuel Assay: FY11 Status Report

Kulisek¹,

Anderson²,

Bowyer³

et al. 2011

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AcknowledgmentsThe authors would like to thank all the project collaborators including:Mark Shaver (PNNL) and Eric Smith (currently at the International Atomic Energy Agency).We would also like to acknowledge the Department of Energy Fuel Cycle Research and Development Program, Material Protection, Accounting, and Control Technology (MPACT) Campaign for their support in funding this work. PNNL-20769iii Executive SummaryDeveloping a method for the accurate, direct, and independent assay of the fissile isotopes in bulk materials (such as used fuel) from next-generation domestic nuclear fuel cycles is a goal of the Office of Nuclear Energy, Fuel Cycle R&D, Material Protection and Control Technology (MPACT) Campaign. To meet this goal, MPACT supports a multi-institutional collaboration, of which PNNL is a part, to study the feasibility of Lead Slowing Down Spectroscopy (LSDS). This technique is an active nondestructive assay method that has the potential to provide independent, direct measurement of Pu and U isotopic masses in used fuel with an uncertainty considerably lower than the approximately 10% typical of today's confirmatory assay methods. This document is a progress report for FY2011 PNNL analysis and algorithm development.Progress made by PNNL in FY2011 continues to indicate the promise of LSDS analysis and algorithms applied to used fuel. PNNL developed an empirical model based on calibration of the LSDS to simulated responses generated from well-characterized used fuel. The empirical model accounts for self-shielding effects using empirical basis vectors calculated from the singular value decomposition (SVD) of a matrix containing the true self-shielding functions of the used fuel assembly models. The potential for the direct and independent assay of the sum of the masses of 239 Pu and 241 Pu to within approximately 3% over a wide used fuel parameter space was demonstrated. PNNL also continued to develop an analytical model. Such efforts included the addition of six more non-fissile absorbers in the analytical shielding function and the nonuniformity of the neutron flux across the LSDS assay chamber. These improvements in the algorithm did not significantly improve the results of the analytical approach. A hybrid analytical-empirical approach was developed to determine the mass of total Pu (sum of the masses of 239 Pu, 240 Pu, and 241 Pu), which is an important quantity in safeguards. Results using this hybrid method were of approximately the same accuracy as the pure empirical approach. In addition, total Pu was determined with much better accuracy with the hybrid approach than the pure analytical approach.In FY2012, PNNL will continue efforts to optimize its empirical model and minimize its reliance on calibration data. In addition, PNNL will continue to develop an analytical model, considering effects such as neutron-scattering in the fuel and cladding, as well as neutrons streaming through gaps between fuel pins in the fuel assembly. PNNL-20769iv

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Section: Appendix A: Analysis Of Analytical Self-shielding Functionmentioning

confidence: 90%

Section: Appendix A: Analysis Of Analytical Self-shielding Functionmentioning

confidence: 99%

Lead Slowing-Down Spectrometry Time Spectral Analysis for Spent Fuel Assay: FY11 Status Report

Kulisek¹,

Anderson²,

Bowyer³

et al. 2011

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“…It was observed in the periodic Lorentz gas [Golse 2012] and is also the reason why the Milne problem in a binary mixture has two definitions/solutions [Pomraning 1989]. The relationship in Equation 1 corresponds to known relationships between chord lengths and lineal path functions in heterogeneous materials [Torquato and Lu 1993], and was first suggested in a bounded random flight [Mazzolo 2009] in order to preserve a remarkable invariance property of chord lengths in convex uniformly-illuminated random media, first proven for exponential media [Bardsley and Dubi 1981;Blanco and Fournier 2003] and then extended to nonexponential free paths [Mazzolo 2009;Mazzolo et al 2014], and more recently observed in the laboratory [Savo et al 2017].…”

Section: Related Workmentioning

confidence: 99%

A Reciprocal Formulation of Nonexponential Radiative Transfer. 2: Monte Carlo Estimation and Diffusion Approximation

d’Eon¹

2019

Journal of Computational and Theoretical Transport

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When lifting the assumption of spatially-independent scattering centers in classical linear transport theory, collision rate is no longer proportional to vector flux / radiance because the macroscopic cross-section Σ t (s) depends on the distance s to the previous collision or boundary. This creates a nonlocal relationship between collision rate and flux and requires revising a number of familiar deterministic and Monte Carlo methods. We generalize collision and track-length estimators to support unbiased estimation of either flux integrals or collision rates in generalized radiative transfer (GRT). To provide benchmark solutions for the Monte Carlo estimators, we derive the four Green's functions for the isotropic point source in infinite media with isotropic scattering. Additionally, new moment-preserving diffusion approximations for these Green's functions are derived, which reduce to algebraic expressions involving the first four moments of the free-path lengths between collisions.

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“…Then, a reasonable choice is to assume an isotropic distribution. For the particular case of exponentially distributed Pearson walks, the isotropy condition can be actually relaxed by demanding detailed balance to be satisfied at each collision [10].…”

Section: A Cauchy's Formula For the Average Number Of Collisionsmentioning

confidence: 99%

Cauchy's formulas for random walks in bounded domains

Mazzolo

Mulatier

Zoia

2014

Journal of Mathematical Physics

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Cauchy's formula was originally established for random straight paths crossing a body $B \subset \mathbb{R}^{n}$ and basically relates the average chord length through $B$ to the ratio between the volume and the surface of the body itself. The original statement was later extended in the context of transport theory so as to cover the stochastic paths of Pearson random walks with exponentially distributed flight lengths traversing a bounded domain. Some heuristic arguments suggest that Cauchy's formula may also hold true for Pearson random walks with arbitrarily distributed flight lengths. For such a broad class of stochastic processes, we rigorously derive a generalized Cauchy's formula for the average length travelled by the walkers in the body, and show that this quantity depends indeed only on the ratio between the volume and the surface, provided that some constraints are imposed on the entrance step of the walker in $B$. Similar results are obtained also for the average number of collisions performed by the walker in $B$, and an extension to absorbing media is discussed.Comment: 12 pages, 6 figure

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The Average Transport Path Length in Scattering Media

Cited by 28 publications

References 1 publication

Lead Slowing-Down Spectrometry Time Spectral Analysis for Spent Fuel Assay: FY11 Status Report

Lead Slowing-Down Spectrometry Time Spectral Analysis for Spent Fuel Assay: FY11 Status Report

A Reciprocal Formulation of Nonexponential Radiative Transfer. 2: Monte Carlo Estimation and Diffusion Approximation

Cauchy's formulas for random walks in bounded domains

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