Monte Carlo methods in radiative transfer and electron-beam processing

Monte Carlo methods (MCMs) are the most versatile approaches in solving the integro-differential equations. They are statistical in nature and can be easily adapted for simulation of the propagation of ensembles of quantum particles within absorbing, emitting, and scattering media. In this paper, we use MCM for the solution of the Boltzmann transport equation, which is the governing equation for both radiative transfer and electron-beam processing. We briefly outline the methodology for the solution of MCMs, and discuss the similarities and differences between the two different application areas. The focus of this paper is primarily on the treatment of different scattering phase functions.     

Image reconstruction in diffuse optical tomography using the coupled radiative transport-diffusion model

The coupled radiative transport-diffusion model can be used as light transport model in situations in which the diffusion equation is not a valid approximation everywhere in the domain. In the coupled model, light propagation is modelled with the radiative transport equation in sub-domains in which the approximations of the diffusion equation are not valid, such as within low-scattering regions, and the diffusion approximation is used elsewhere in the domain. In this paper, an image reconstruction method for diffuse optical tomography based on using the coupled radiative transport-diffusion model is developed. In the approach, absorption and scattering distributions are estimated by minimising a regularised least-squares error between the measured data and solution of the coupled model. The approach is tested with simulations. Reconstructions from different cases including domains with low-scattering regions are shown. The results show that the coupled radiative transport-diffusion model can be utilised in image reconstruction problem of diffuse optical tomography and that it produces as good quality reconstructions as the full radiative transport equation also in the presence of low-scattering regions.     

Ballistic–diffusive phonon transport and size induced anisotropy of thermal conductivity of silicon nanofilms

The effective thermal conductivity of nanofilms is size dependent due to the diffusive–ballistic transport of phonons. In this paper, we investigate the cross-plane phonon transport from the viewpoint of the phonon Boltzmann equation. A predictive model for the size dependent thermal conductivity is proposed and agrees well with the results of molecular dynamics simulation for silicon nanofilms. The ballistic transport has different effects on the heat conduction in the in-plane or cross-plane directions, which causes the anisotropy of thermal conductivity of nanofilms. Such anisotropy is also size dependent and vanishes with the increase of film thickness.     

Hybrid discrete ordinates—spherical harmonics solution to the Boltzmann Transport Equation for phonons for non-equilibrium heat conduction

The Boltzmann Transport Equation (BTE) for phonons has found prolific use for the prediction of non-equilibrium heat conduction phenomena in semiconductor materials. This article presents a new hybrid formulation and associated numerical procedures for solution of the BTE for phonons. In this formulation, the phonon intensity is first split into two components: ballistic and diffusive. The governing equation for the ballistic component is solved using two different established methods that are appropriate for use in complex geometries, namely the discrete ordinates method (DOM), and the control angle discrete ordinates method (CADOM). The diffusive component, on the other hand, is determined by invoking the first-order spherical harmonics (or P₁) approximation, which results in a Helmholtz equation with Robin boundary conditions. Both governing equations, referred to commonly as the ballistic-diffusive equations (BDE), are solved using the unstructured finite-volume procedure. Results of the hybrid method are compared against benchmark Monte Carlo results, as well as solutions of the BTE using standalone DOM and CADOM for two two-dimensional transient heat conduction problems at various Knudsen numbers. Subsequently, the method is explored for a large-scale three-dimensional geometry in order to assess convergence and computational cost. It is found that the proposed hybrid method is accurate at all Knudsen numbers. From an efficiency standpoint, the hybrid method is found to be superior to direct solution of the BTE both for steady state as well as for unsteady non-equilibrium heat conduction calculations with the computational gains increasing with increase in problem size.     

The effect of surface roughness on lattice thermal conductivity of silicon nanowires

A theoretic model is presented to take into account the roughness effects on phonon transport in Si nanowires (NWs). Based on the roughness model, an indirect Monte Carlo (MC) simulation is carried out to predict the lattice thermal conductivities of the NWs with different surface qualities. Through fitting the experimental data with the MC predictions, the scattering strength on phonons from the boundary, umklapp phonon-phonon processes and impurities can be estimated. It is found that the scattering on phonons by the roughness cell boundaries in a rough nanowire can reduce the phonon mean free path to be smaller than the nanowire diameter, the Casimir limit of the phonon mean free path in a flat nanowire for phonons engaged in completely diffused boundary scattering processes.     

二维Mo2C的晶格热导率的第一性原理研究

Mo2C是构建Mxene基器件的重要材料之一,对Mo2C二维材料声子输运的理解非常必要。文章结合第一性原理方法和声子玻尔兹曼输运方程,研究了二维Mo2C材料的晶格热导率。研究表明,室温下二维Mo2C导热系数非常低,其锯齿方向和扶手椅方向的晶格热导率分别为7.20 和 5.04 W/mK。计算了声学振动和光学振动模式对晶格热导率的贡献,揭示总热导率主要由面内声学横波的振动模式所贡献。还进一步计算了声子群速度、声子弛豫时间、三声子散射空间和模式格林艾森参数,发现二维Mo2C中的声子群速度和声子弛豫时间对晶格传输有重要的影响。     

Generalized form of the direct and adjoint radiative transfer equations

The main goal of this paper is to give a rigorous derivation of the generalized form of the direct (also referenced as forward) and adjoint radiative transfer equations. The obtained expressions coincide with expressions derived by Ustinov [Adjoint sensitivity analysis of radiative transfer equation: temperature and gas mixing ratio weighting functions for remote sensing of scattering atmospheres in thermal IR. JQSRT 2001;68:195-211]. However, in contrast to [Ustinov EA. Adjoint sensitivity analysis of radiative transfer equation: temperature and gas mixing ratio weighting functions for remote sensing of scattering atmospheres in thermal IR. JQSRT 2001;68:195-211] we formulate the generalized form of the direct radiative transfer operator fully independent from its adjoint. To illustrate the application of the derived adjoint radiative transfer operator we consider the angular interpolation problem in the framework of the discrete ordinate method widely used to solve the radiative transfer equation. It is shown that under certain conditions the usage of the solution of the adjoint radiative transfer equation for the angular interpolation of the intensity can be computationally more efficient than the commonly used source function integration technique.     

Two-dimensional phonon transport in graphene

Properties of phonons-quanta of the crystal lattice vibrations-in graphene have recently attracted significant attention from the physics and engineering communities. Acoustic phonons are the main heat carriers in graphene near room temperature, while optical phonons are used for counting the number of atomic planes in Raman experiments with few-layer graphene. It was shown both theoretically and experimentally that transport properties of phonons, i.e. energy dispersion and scattering rates, are substantially different in a quasi-two-dimensional system such as graphene compared to the basal planes in graphite or three-dimensional bulk crystals. The unique nature of two-dimensional phonon transport translates into unusual heat conduction in graphene and related materials. In this review, we outline different theoretical approaches developed for phonon transport in graphene, discuss contributions of the in-plane and cross-plane phonon modes, and provide comparison with available experimental thermal conductivity data. Particular attention is given to analysis of recent results for the phonon thermal conductivity of single-layer graphene and few-layer graphene, and the effects of the strain, defects, and isotopes on phonon transport in these systems.     

Numerical solution of the Fokker-Planck equation with variable coefficients

We study the numerical solution of the Fokker-Planck equation. This equation gives a good approximation to the radiative transport equation when scattering is peaked sharply in the forward direction which is the case for light propagation in tissues, for example. We derive first the numerical solution for the problem with constant coefficients. This numerical solution is constructed as an expansion in plane wave solutions. Then we extend that result to take into account coefficients that vary spatially. This extension leads to a coupled system of initial and final value problems. We solve this system iteratively. Numerical results show the utility of this method.     

Comparison of the radiative transfer equations for constant and spatially varying refraction

The radiative transfer equation for scattering media with constant refraction index (RTE) and the radiative transfer equation for scattering media with spatially varying refraction index (RTEvri) are compared by using the principle of conservation of energy. It is shown that the RTEvri, not only accounts for the spatial variations of refraction index, but also contains a term that accounts for the divergence of the rays. The latter term is missing in the RTE. A corrected RTE is proposed.     

Forward Monte Carlo computations of fully polarized microwave radiation in non-isotropic media

A completely forward Monte Carlo radiative transfer code has been developed with biasing techniques to efficiently solve the polarized radiative transfer equation for the full Stokes vector. The code has been adapted to accommodate plane parallel/3-D vertically/horizontally inhomogeneous scattering atmospheres in Cartesian geometries. Particular attention has been paid in stochastically treating the propagation, the emission and the scattering through anisotropic media particularly suited for clouds containing perfectly or partially oriented particles. Our modelling is very appealing because all its biasing techniques do not introduce unphysical Stokes vector. Numerical results and comparisons with benchmark tests are presented for verification.     

热辐射输运问题的隐式蒙特卡罗方法求解

热辐射与物质相互作用及辐射光子在物质中的传输是惯性约束聚变研究中的重要课题. 介绍了基于隐式蒙特卡罗方法的辐射输运方程,在该方程的积分-微分形式基础上,推导了利于蒙特卡罗方法模拟的等价的积分输运方程;基于积分方程设计数值模拟流程,编写三维蒙特卡罗数值模拟程序;针对热辐射输运典型问题及benchmark问题开展了数值实验,计算结果验证了方法的适应性及程序的正确性. 关键词： 热辐射 惯性约束聚变 输运方程 隐式蒙特卡罗     

An accurate solver for forward and inverse transport

This paper presents a robust and accurate way to solve steady-state linear transport (radiative transfer) equations numerically. Our main objective is to address the inverse transport problem, in which the optical parameters of a domain of interest are reconstructed from measurements performed at the domain’s boundary. This inverse problem has important applications in medical and geophysical imaging, and more generally in any field involving high frequency waves or particles propagating in scattering environments. Stable solutions of the inverse transport problem require that the singularities of the measurement operator, which maps the optical parameters to the available measurements, be captured with sufficient accuracy. This in turn requires that the free propagation of particles be calculated with care, which is a difficult problem on a Cartesian grid.     

多约束纳米结构的声子热导率模型研究

纳米技术的快速发展使得对微纳尺度导热机理的深入研究变得至关重要. 理论和实验都表明, 在纳米尺度下声子热导率将表现出尺寸效应. 基于声子玻尔兹曼方程和修正声子平均自由程的方法得到了多约束纳米结构的声子热导率模型, 可以描述多个几何约束共同作用下热导率的尺寸效应. 不同几何约束对声子输运的限制作用可以分开计算, 总体影响则通过马西森定则进行耦合. 对于热流方向的约束, 采用扩散近似的方法求解声子玻尔兹曼方程; 对于侧面边界约束, 采用修正平均自由程的方法计算边界散射对热导率的影响. 得到的模型能够预测纳米薄膜(法向和面向)及有限长度方形纳米线的热导率随相应特征尺寸的变化. 与蒙特卡罗模拟及硅纳米结构热导率实验值的对比验证了模型的正确性.     

On the combined heat transfer in the multilayer non-gray porous fibrous insulation

A detailed numerical modeling is performed to investigate coupled heat transfer of natural convection, radiation and conduction in high-temperature multilayer thermal insulation (MTI), which consists of high-porous, non-gray semitransparent fibrous materials and reflective foils. Radiation within fibers, radiation between fibers and the reflective foils, conduction within fibers and convection from the fibers to the surrounding fluid are considered. Macroscopic (porous media) modeling is used to determine velocity, pressure and temperatures fields for fibrous insulation with a random packing geometry under natural convection, whereas the radiative transfer equation (RTE) is used to solve the radiative heat flux for non-gray materials. Key features of the macroscopic model include two-dimensional effects, non-gray radiative exchange, and the relaxation of the local thermodynamic non-equilibrium (LTNE). This model was validated by comparison with experimental data and it was used to investigate natural convection of coupled heat transfer in multilayer insulation, numerical results showed that natural convection is more likely to occur when the heated/cooled rate is low, while natural convection can be ignored in simulating steady-state coupled heat transfer in MTI.     

Effect of reflecting modes on combined heat transfer within an anisotropic scattering slab

Under various interface reflecting modes, different transient thermal responses will occur in the media. Combined radiative-conductive heat transfer is investigated within a participating, anisotropic scattering gray planar slab. The two interfaces of the slab are considered to be diffuse and semitransparent. Using the ray tracing method, an anisotropic scattering radiative transfer model for diffuse reflection at boundaries is set up, and with the help of direct radiative transfer coefficients, corresponding radiative transfer coefficients (RTCs) are deduced. RTCs are used to calculate the radiative source term in energy equation. Transient energy equation is solved by the full implicit control-volume method under the external radiative-convective boundary conditions. The influences of two reflecting modes including both specular reflection and diffuse reflection on transient temperature fields and steady heat flux are examined. According to numerical results obtained in this paper, it is found that there exits great difference in thermal behavior between slabs with diffuse interfaces and that with specular interfaces for slabs with big refractive index.     

Band structure and scattering mechanisms in manganese monosilicide

The electrical and thermal conductivities, the thermal emf, and the Hall coefficient have been measured in manganese monosilicide at temperatures between 70 and 700 °K. Band parameters are calculated for MnSi on the basis of wide-band and narrow-band models. Scattering mechanisms for the current carriers and phonons are discussed; in particular, the anomalous temperature dependence of the thermal conductivity is attributed to the high efficiency for phonon scattering by conduction electrons.Translated from Izvestiya VUZ. Fizika, No. 12, pp. 108–112, December, 1969.     

ARTS, the atmospheric radiative transfer simulator

ARTS is a modular program that simulates atmospheric radiative transfer. The paper describes ARTS version 1.0, which is applicable in the absence of scattering. An overview over all major parts of the model is given: calculation of absorption coefficients, the radiative transfer itself, and the calculation of Jacobians. ARTS can be freely used under a GNU general public license.Unique features of the program are its scalability and modularity, the ability to work with different sources of spectroscopic parameters, the availability of several self-consistent water continuum and line absorption models, and the analytical calculation of Jacobians.     

Scaling algorithms for the calculation of solar radiative fluxes

We derived new scaling formulae based on the method of successive orders of scattering to calculate solar radiative flux. In this report, we demonstrate a multiple scaling method, in which we introduce scaling factors for each scattering order independently. The formula of radiative transfer by the method of successive orders of scattering cannot be solved rapidly except in the case of optically thin atmospheres. Then we further derived a double scaling method, which scales the ordinary radiative transfer equation by two scaling factors. We applied the double scaling method to two-stream and four-stream approximations of the discrete ordinates method. Comparing the results of the double scaling method with those of the delta-M method, we found that the double scaling method improved the accuracy of radiative fluxes at large solar zenith angles, especially in the optically thin region, and that in the region where multiple scattering dominates, its accuracy was comparable to that of the delta-M method. Once we determined the scaling factors appropriately, the double scaling method calculated radiative fluxes as rapidly as the delta-M method in the two-stream and four-stream approximations. This method, therefore, is useful for accurate computation of solar radiative fluxes in general circulation models.     

Maxwell's equations, radiative transfer, and coherent backscattering: A general perspective

This tutorial paper provides a general overview of the hierarchy of problems involving electromagnetic scattering by particles and clarifies the place of the radiative transfer theory and the theory of coherent backscattering in the context of classical electromagnetics. The self-consistent microphysical approach to radiative transfer is compared with the traditional phenomenological treatment.