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All sourcesNASA NTRSDOE OSTI
NASA NTRS2020

Variability of Lidar-Derived Particle Properties Over West Africa Due to Changes in Absorption: Towards an Understanding

Measurements performed in western Africa (Senegal) during the SHADOW field campaign are analyzed to show that spectral dependence of the imaginary part of the complex refractive index (CRI) of dust can be revealed by lidar-measured particle parameters. Observations in April 2015 provide good opportunity for such study, because, due to high optical depth of the dust, exceeding 0.5, the extinction coefficient could be derived from lidar measurements with high accuracy and the contribution of other aerosol types, such as biomass burning, was negligible. For instance, in the second half of April 2015, AERONET observations demonstrated a temporal decrease in the imaginary part of the CRI at 440 nm from approximately 0.0045 to 0.0025. This decrease is in line with a change in the relationship between the lidar ratios (the extinction-to-backscattering ratio) at 355 and 532 nm (S(355) and S(532)). For instance in the first half of April, S(355)∕S(532) is as high as 1.5 and the backscatter Angstrom exponent, A(β), is as low as −0.75, while after 15 April S(355)/S(532)=1.0 and A(β) is close to zero. The aerosol depolarization ratio δ(532) for the whole of April exceeded 30 % in the height range considered, implying that no other aerosol, except dust, occurred. The performed modeling confirmed that the observed S(355)∕S(532) and Aβ values match the spectrally dependent imaginary part of the refractive index as can be expected for mineral dust containing iron oxides. The second phase of the SHADOW campaign was focused on evaluation of the lidar ratio of smoke and estimates of its dependence on relative humidity (RH). For five studied smoke episodes the lidar ratio increases from 44±5 to 66±7 sr at 532 nm and from 62±6 to 80±8 sr at 355 nm, when RH varied from 25 % to 85 %. Performed numerical simulations demonstrate that observed ratio S(355)∕S(532), exceeding 1.0 in the smoke plumes, can indicate an increase in the imaginary part of the smoke particles in the ultraviolet (UV) range.

Lidar-Derived Particle Properties↗
DOE OSTI2023

Spectral induced polarization signatures of smoldering remediation enhanced with colloidal activated carbon: An experimental study

Monitoring the remediation of soil and groundwater contaminated by organic compounds remains highly challenging. The spectral induced polarization (SIP) method exhibits significant promise for monitoring changes to the electrical properties of soils that are undergoing remediation. Thermal treatments, such as smoldering combustion, have become established remediation techniques for destroying contaminants. Smoldering combustion is now being supported by colloidal activated carbon (CAC), with CAC able to adsorb contaminants and supplement the fuel source for destroying contaminants. The objective of this study is to investigate the potential of SIP for tracking the smoldering remediation of imitated field soils supplemented with CAC. SIP column experiments were first conducted to assess the response of SIP to varying concentrations of CAC in field soils that contain, or do not contain, organic material (OM). Here these results demonstrate that increasing OM and CAC contents increase both the real and imaginary components of the complex conductivity, with the imaginary conductivity also showing frequency dependence. Next, a suite of smoldering and SIP column experiments was conducted to investigate if SIP can detect changes in imitated field soils of varying OM and CAC contents that have been remediated by smoldering combustion. The SIP results on the examined soils both before and after smoldering show that SIP can detect changes in the real conductivity, and particularly the imaginary conductivity, between different soil compositions and different stages of the remediation process. High resolution scanning electron microscopy (SEM) imaging was performed on all samples to validate the SIP and smoldering experiments, confirming significant reductions in carbon after smoldering. Overall, this study suggests that SIP has potential to detect changes in the electrical properties of field soils due to the addition of remedial fluids like CAC and contaminant destruction by smoldering remediation.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2023

Nonlinear Loss Engineering in Near‐Zero‐Index Bulk Materials

Transparent conducting oxides (TCOs) show unprecedented optical nonlinearities in the near infrared wavelength range, where the real part of their linear refractive index approaches zero. More specifically, the Kerr nonlinearities of these materials have sparked widespread attention due to their magnitude and speed. However, due to the absorptive nature of these nonlinear processes, it is of fundamental interest to further investigate the imaginary component of the nonlinear index. The present work studies the nonlinear optical absorption properties of aluminium‐doped zinc oxide (AZO) thin films in their near‐zero‐index (NZI) spectral window. It is found that the imaginary part of the refractive index is reduced under optical excitation such that the field penetration depth more than doubles. An optically induced shift of the NZI bandwidth of ≈120 nm for a pump intensity of 1.3 TW cm −2 is also demonstrated. Looking into the optically induced spectral redistribution of the probe signal, local net gain is recorded, which is ascribed to a nonlinear adiabatic energy transfer. The present study adds key information about the fundamental interplay between real and imaginary nonlinear indices in NZI media, while advancing parametric amplification as viable direction for loss compensation.

36 MATERIALS SCIENCE↗
DOE OSTI2026

Capturing soil moisture and salinity changes in flooded coastal forests using electrical resistivity and induced polarization

Geophysical methods provide high-resolution spatial measurements of physical quantities sensitive to changes in soil moisture and salinity and can be used to monitor soil hydrological responses to flooding. However, extracting quantitative hydrological information from geophysical data remains challenging. In this study, we extended existing petrophysical models to estimate soil moisture and salinity from electrical measurements to address this challenge. We manipulated two hydrologically isolated 2000 m 2 experimental plots by simultaneously inundating them with 265 m 3 of either freshwater or estuarine water. Repeated electrical resistivity and induced polarization measurements were used to image the water and solute infiltration along two transects that are 100 and 42 m in length. Petrophysical models derived from laboratory multi-salinity electrical measurements were used to estimate changes in soil moisture and fluid salinity from field measurements of real and imaginary conductivity during the ecosystem-scale flooding experiment. The real conductivity increased by ∼100% in the freshwater plot and ∼570% in the saltwater plot. The change in imaginary conductivity in the freshwater plot was <1 mS/m, whereas that of the estuarine water plot was ∼5 mS/m. Real conductivity shows dependence on soil moisture content with a coefficient of determination (R 2 ) > 0.7, while the imaginary conductivity shows a dependence on soil salinity with R 2 > 0.6. The results validate the use of electrical resistivity for estimating changes in soil moisture content in response to flooding. Combining electrical resistivity imaging with induced polarization measurements provides the possibility to account for changes in pore fluid conductivity.

Adebayo, Moses B. [Univ. of Toledo, OH (United Sta↗
DOE OSTI2016

Non-associativity in non-geometric string and M-theory backgrounds, the algebra of octonions, and missing momentum modes

We propose a non-associative phase space algebra for M-theory backgrounds with locally non-geometric fluxes based on the non-associative algebra of octonions. Our proposal is based on the observation that the non-associative algebra of the non-geometric R-flux background in string theory can be obtained by a proper contraction of the simple Malcev algebra generated by imaginary octonions. Furthermore, by studying a toy model of a four-dimensional locally non-geometric M-theory background which is dual to a twisted torus, we show that the non-geometric background is “missing” a momentum mode. The resulting seven-dimensional phase space can thus be naturally identified with the imaginary octonions. This allows us to interpret the full uncontracted algebra of imaginary octonions as the uplift of the string theory R-flux algebra to M-theory, with the contraction parameter playing the role of the string coupling constant g s .

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2018

Variational study of fermionic and bosonic systems with non-Gaussian states: Theory and applications

We present a new variational method for investigating the ground state and out of equilibrium dynamics of quantum many-body bosonic and fermionic systems. Our approach is based on constructing variational wavefunctions which extend Gaussian states by including generalized canonical transformations between the fields. The key advantage of such states compared to simple Gaussian states is presence of non-factorizable correlations and the possibility of describing states with strong entanglement between particles. In contrast to the commonly used canonical transformations, such as the polaron or Lang–Firsov transformations, we allow parameters of the transformations to be time dependent, which extends their regions of applicability. We derive equations of motion for the parameters characterizing the states both in real and imaginary time using the differential structure of the variational manifold. The ground state can be found by following the imaginary time evolution until it converges to a steady state. Collective excitations in the system can be obtained by linearizing the real-time equations of motion in the vicinity of the imaginary time steady-state solution. Our formalism allows us not only to determine the energy spectrum of quasiparticles and their lifetime, but to obtain the complete spectral functions and to explore far out of equilibrium dynamics such as coherent evolution following a quantum quench. We illustrate and benchmark this framework with several examples: a single polaron in the Holstein and Su–Schrieffer–Heeger models, non-equilibrium dynamics in the spin-boson and Kondo models, the superconducting to charge density wave phase transitions in the Holstein model.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2022

The Role of Phonons and Oxygen Vacancies in Non-Cubic SrVO 3

Combining neutron diffraction with pair distribution function analysis, we have uncovered hidden reduced symmetry in the correlated metallic d 1 perovskite, SrVO 3 . Specifically, we show that both the local and global structures are better described using a GdFeO 3 distorted (orthorhombic) model as opposed to the ideal cubic ABO 3 perovskite type. Recent reports of imaginary phonon frequencies in the density functional theory (DFT)-calculated phonon dispersion for cubic SrVO 3 suggest a possible origin of this observed non-cubicity. Namely, the imaginary frequencies computed could indicate that the cubic crystal structure is unstable at T = 0 K. However, our DFT calculations provide compelling evidence that point defects in the form of oxygen vacancies, and not an observable symmetry breaking associated with calculated imaginary frequencies, primarily result in the observed non-cubicity of SrVO 3 . These experimental and computational results are broadly impactful because they reach into the thin-film and theoretical communities who have shown that SrVO 3 is a technologically viable transparent conducting oxide material and have used SrVO 3 to develop theoretical methods, respectively.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2016

Efficient Temperature-Dependent Green’s Functions Methods for Realistic Systems: Compact Grids for Orthogonal Polynomial Transforms

The Matsubara Green’s function that is used to describe temperature-dependent behavior is expressed on a numerical grid. While such a grid usually has a couple of hundred points for low-energy model systems, for realistic systems with large basis sets the size of an accurate grid can be tens of thousands of points, constituting a severe computational and memory bottleneck. In this paper, we determine efficient imaginary time grids for the temperature-dependent Matsubara Green’s function formalism that can be used for calculations on realistic systems. Here we show that, because of the use of an orthogonal polynomial transform, we can restrict the imaginary time grid to a few hundred points and reach micro-Hartree accuracy in the electronic energy evaluation. Moreover, we show that only a limited number of orthogonal polynomial expansion coefficients are necessary to preserve accuracy when working with a dual representation of the Green’s function or self-energy and transforming between the imaginary time and frequency domain.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2020

Characterizing Physical Properties of Streambed Interface Sediments Using In Situ Complex Electrical Conductivity Measurements

Streambed sediment physical properties such as surface area, are difficult to quantify in situ but exert a high-level control on a wide range of biogeochemical processes and sorption of contaminants. We introduce the use of complex electrical conductivity (CC) methods (also known as spectral-induced polarization (SIP)) that measure both real and imaginary conductivity to noninvasively and efficiently characterize shallow streambed sediments. We explore the method through synthetic modeling, laboratory, and field measurements to demonstrate the sensitivity of imaginary conductivity to sediment surface area, controlled in part by fine-grained iron oxides produced by anoxic groundwater discharge. Laboratory measurements verify expected relationships between CC parameters and sediment properties. Synthetic modeling using a 1D analytical model illustrates the influence of water layer depth and conductivity on the field CC measurements made at the streambed-stream water interface. Specifically, the inverted sediment imaginary conductivity is less impacted by uncertainty in the water layer depth and conductivity relative to the real conductivity and phase shift. Field CC measurements along a landfill-impacted river reveal discrete streambed zones with enhanced bulk surface area generally corresponding to anoxic groundwater discharge zones with high concentrations of fine-grained iron oxide precipitates.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2014

Complex refractive indices in the near-ultraviolet spectral region of biogenic secondary organic aerosol aged with ammonia

Atmospheric absorption by brown carbon aerosol may play an important role in global radiative forcing. Brown carbon arises from both primary and secondary sources, but the mechanisms and reactions for the latter are highly uncertain. One proposed mechanism is the reaction of ammonia or amino acids with carbonyl products in secondary organic aerosol (SOA). We generated SOA in situ by reacting biogenic alkenes (α-pinene, limonene, and α-humulene) with excess ozone, humidifying the resulting aerosol, and reacting the humidified aerosol with gaseous ammonia. We determined the complex refractive indices (RI) in the 360 – 420 nm range for these aerosols using broadband cavity enhanced spectroscopy (BBCES). The average real part (n) of the measured spectral range of the NH3-aged α-pinene SOA increased from n = 1.50 (±0.01) for the unreacted SOA to n = 1.57 (± 0.01) after a 1.5h exposure to 1.9 ppm NH3; whereas,the imaginary component (k) remained below k < 0.001 (± 0.002). For the limonene and α-humulene SOA the real part did not change significantly, and we observed a small change in the imaginary component of the RI. The imaginary component increased from k = 0.0 to an average k= 0.029 (± 0.021) for α-humulene SOA, and from k < 0.001 (± 0.002) to an average k = 0.032 (±0.019) for limonene SOA after a 1.5 h exposure to 1.3 and 1.9 ppm of NH3, respectively. Collected filter samples of the aged and unreacted α-pinene SOA and limonene SOA were analyzed off-line with nanospray desorption electrospray ionization high resolution mass spectrometry (nano-DESI/HR-MS), and in-situ with a Time-of-Fligh Aerosol Mass Spectrometer, confirming that the SOA reacted and that various nitrogen-containing reaction products formed. If we assume that NH3 aging reactions scale linearly with time and concentration, then a 1.5 h reaction with 1 ppm NH3 in the laboratory is equivalent to 24 h reaction with 63 ppbv NH3, indicating that the observed aerosol absorption will be limited to atmospheric regions with high NH3 concentrations.

20 refractive index↗
DOE OSTI2014

Ion-temperature-gradient sensitivity of the hydrodynamic instability caused by shear in the magnetic-field-aligned plasma flow

The cross-magnetic-field (i.e., perpendicular) profile of ion temperature and the perpendicular profile of the magnetic-field-aligned (parallel) plasma flow are sometimes inhomogeneous for space and laboratory plasma. Instability caused either by a gradient in the ion-temperature profile or by shear in the parallel flow has been discussed extensively in the literature. In this paper, (1) hydrodynamic plasma stability is investigated, (2) real and imaginary frequency are quantified over a range of the shear parameter, the normalized wavenumber, and the ratio of density-gradient and ion-temperature-gradient scale lengths, and (3) the role of inverse Landau damping is illustrated for the case of combined ion-temperature gradient and parallel-flow shear. We find that increasing the ion-temperature gradient reduces the instability threshold for the hydrodynamic parallel-flow shear instability, also known as the parallel Kelvin-Helmholtz instability or the D'Angelo instability. We also find that a kinetic instability arises from the coupled, reinforcing action of both free-energy sources. For the case of comparable electron and ion temperature, we illustrate analytically the transition of the D'Angelo instability to the kinetic instability as (a) the shear parameter, (b) the normalized wavenumber, and (c) the ratio of density-gradient and ion-temperature-gradient scale lengths are varied and we attribute the changes in stability to changes in the amount of inverse ion Landau damping. We show that near a normalized wavenumber k{sub ⊥}ρ{sub i} of order unity (i) the real and imaginary values of frequency become comparable and (ii) the imaginary frequency, i.e., the growth rate, peaks.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2016

Homogeneous and inhomogeneous broadenings and the Voigt line shapes in the phase-resolved and intensity sum-frequency generation vibrational spectroscopy

In this report we show that the ability to measure the sub-1 cm -1 resolution phase-resolved and intensity high-resolution broadband sum frequency generation vibrational spectra (HR-BB-SFG-VS) of the –CN stretch vibration of the Langmuir-Blodgett (LB) monolayer of the 4-n-octyl-4’-cyanobiphenyl (8CB) on the z-cut α-quartz surface allows for the first time the direct comparison and understanding of the homogeneous and inhomogeneous broadenings in the imaginary and intensity SFG vibrational spectral lineshapes in detail. The difference of the full width at half maxima (FWHM) of the imaginary and intensity SFG-VS spectra of the same vibrational mode is the signature of the Voigt lineshape and it measures the relative contribution to the overall lineshape from the homogeneous and inhomogeneous broadenings in SFG vibrational spectra. From the phase-resolved and intensity spectra, we found that the FWHM of the 2238.00 ±0.02 cm -1 peak in the phase-resolved imaginary and intensity spectra is 19.2 ± 0.2 cm -1 and 21.6 ± 0.4 cm -1 , respectively, for the –CN group of the 8CB LB monolayer on the z-cut α-quartz crystal surface. The FWHM width difference of 2.4 cm -1 agrees quantitatively with a Voigt lineshape with a homogeneous broadening half width of Γ = 5.29 ± 0.08 cm -1 and a inhomogeneous standard derivation width Δω = 5.42 ± 0.07 cm -1 . These results shed new lights on the understanding and interpretation of the lineshapes of both the phase-resolved and the intensity SFG vibrational spectra, as well as other incoherent and coherent spectroscopic techniques in general.

Environmental Molecular Sciences Laboratory↗
DOE OSTI2017

Non-uniqueness of quantum transition state theory and general dividing surfaces in the path integral space

Despite the fact that quantum mechanical principles do not allow the establishment of an exact quantum analogue of the classical transition state theory (TST), the development of a quantum TST (QTST) with a proper dynamical justification, while recovering the TST in the classical limit, has been a long standing theoretical challenge in chemical physics. One of the most recent efforts of this kind was put forth by Hele and Althorpe (HA) [J. Chem. Phys. 138, 084108 (2013)], which can be specified for any cyclically invariant dividing surface defined in the space of the imaginary time path integral. The present paper revisits the issue of the non-uniqueness of QTST and provides a detailed theoretical analysis of HA-QTST for a general class of such path integral dividing surfaces. While we confirm that HA-QTST reproduces the result based on the ring polymer molecular dynamics (RPMD) rate theory for dividing surfaces containing only a quadratic form of low frequency Fourier modes, we find that it produces different results for those containing higher frequency imaginary time paths which accommodate greater quantum fluctuations. This result confirms the assessment made in our previous work [Jang and Voth, J. Chem. Phys. 144, 084110 (2016)] that HA-QTST does not provide a derivation of RPMD-TST in general and points to a new ambiguity of HA-QTST with respect to its justification for general cyclically invariant dividing surfaces defined in the space of imaginary time path integrals. Finally, our analysis also offers new insights into similar path integral based QTST approaches.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2025

Validation of 3D cryoEM single-particle reconstruction correctness and handedness with Ewald's sphere correction

The correct description of quantum scattering places the observed scattering contributions on the Ewald's sphere and its Friedel mate copy. In electron microscopy, due to the large radius of the Ewald's sphere, these scattering contributions are typically merged during data analysis. We present an approach that separates and factorizes those contributions into real and imaginary components of the image. When an inverted solution is calculated, the map derived from the real component of the image generates an inverted solution, while the map derived from the imaginary component of the image generates an inverted and sign-flipped solution. Therefore, the sign of correlation between reconstructions derived from the real and imaginary components provides the automatic determination of handedness and additional validation for the quality of 3D reconstructions. The factorization and its implementation are robust enough to be routinely used in single-particle reconstructions, even at resolutions below the limit where the curvature of the Ewald's sphere affects the overall signal-to-noise ratio.

Chemistry↗
DOE OSTI2017

3-D decoupled inversion of complex conductivity data in the real number domain

Complex conductivity inversion, aslo called spectral induced polarization inversion, involves estimating the frequency dependent complex electrical conductivity distribution of the subsurface. The superior diagnostic capabilities provided by complex conductivity spectra have driven advancements in mechanistic understanding of complex conductivity as well as modelling and inversion appraoches over the past several decades. In this work, we demonstrate the theory and application for an approach to three-dimensional modeling and inversion of complex conductivity data in the real number domain. Beginning from first principles, we demonstrate how the equations for the real and imaginary components of the complex potential may be decoupled. This leads to a description of the real and imaginary source current terms, and a corresponding assessment of error arising from an assumption necessary complete the decoupled modeling. We show that for most earth materials, which exhibit relatively small phases (e.g. less than 0.2 radians) in complex conductivity, these errors become insignificant. For higher phase materials, the errors may be quantified and corrected through an iterative procedure. We demonstrate the accuracy of numerical forward solutions by direct comparison to corresponding analytic solutions. We demonstrate the inversion using both synthetic and field examples with data collected over a waste infiltration trench, at frequencies ranging from 0.5 to 7.5 Hz. In addition to insight provided by the decoupled equations, we see three primary advantages of the decoupled inversion in comparison to fully coupled inversion in complex number domain. First, memory requirements are reduced by a factor of two because it is unnecessary to simultaneously store both the real and complex components and associated Jacobian matrices. Second, data noise is specified separately for the real and imaginary components of potential (or magnitude and phase), which removes the ambiguity in data misfit residuals that exists in the complex number domain, thereby enabling the inversion to appropriately fit the data. Final, except for the source current terms, the modeling and inversion algorithms are equivalent to the direct current electrical resistivity tomography counterpart. In conclusion, this enables direct application of advancements and approaches developed originally for electrical resistivity tomography.

58 GEOSCIENCES↗
DOE OSTI2020

Scattering using real-time path integrals

Background: Path integrals are a powerful tool for solving problems in quantum theory that are not amenable to a treatment by perturbation theory. Most path integral computations require an analytic continuation to imaginary time. While imaginary time treatments of scattering are possible, imaginary time is not a natural framework for treating scattering problems. More importantly, quantum algorithms for calculating path integrals require real-time evolution. Purpose: Here, we test a recently introduced method for performing direct calculations of scattering observables using real-time path integrals in order to understand the challenges facing real-time path integral calculations of scattering observables. Method: The computations are based on a new interpretation of the path integral as the expectation value of a potential functional on cylinder sets of continuous paths with respect to a complex probability distribution. The method can in principle be applied to arbitrary short-range potentials. Results: The method is applied to compute matrix elements of Møller wave operators applied to narrow wave packets. These are used to calculate half-shell sharp-momentum transition matrix elements for one-dimensional potential scattering. The calculations for half-shell transition operator matrix elements converge to the numerical solution of the Lippmann-Schwinger equation. Conclusions: This work presents a proof in principle that scattering observables can be computed using real-time Feynman path integrals. While the computational method is not efficient, it can be improved. It provides a laboratory for studying quantum computational algorithms that are applicable to scattering problems.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2022

Frequency-dependent complex viscosity obtained for a liquid two-dimensional dusty plasma experiment

Strongly coupled plasmas in a liquid phase can be characterized by a complex viscosity η(ω), which is a function of frequency. Data from a single experiment with dusty plasma were analyzed to compare η(ω) obtained by two fundamentally distinct methods. In a nonequilibrium method, a pair of counterpropagating laser beams, separated by a gap, applied a sinusoidal shear to a two-dimensional liquid, and η(ω) was determined using the constitutive relation. In an equilibrium method, there was no externally applied shear, so η(ω) could be calculated with a generalized Green-Kubo relation. The results for these two methods are compared for the real and imaginary parts of η(ω). For both parts, it is confirmed that the two methods yield results that agree qualitatively in their trends with frequency, with the real part diminishing with ω and the imaginary part increasing with ω, as expected for viscoelastic liquids. Quantitatively, the values of η(ω) obtained by the two methods differ slightly. For the experiment that we analyze, values for the real and imaginary parts of η(ω) are substantially greater than those reported in an earlier experiment, which we attribute to shear thinning effects in the earlier experiment. As a result, the experiment we analyze was designed to minimize shear thinning, unlike the earlier experiment.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2016

Analytical Solution for Three-Dimensional, Unsteady Heat Conduction in a Multilayer Sphere

An analytical solution has been obtained for the transient problem of three-dimensional multilayer heat conduction in a sphere with layers in the radial direction. The solution procedure can be applied to a hollow sphere or a solid sphere composed of several layers of various materials. In general, the separation of variables applied to 3D spherical coordinates has unique characteristics due to the presence of associated Legendre functions as the eigenfunctions. Moreover, an eigenvalue problem in the azimuthal direction also requires solution; again, its properties are unique owing to periodicity in the azimuthal direction. Therefore, extending existing solutions in 2D spherical coordinates to 3D spherical coordinates is not straightforward. In a spherical coordinate system, one can solve a 3D transient multilayer heat conduction problem without the presence of imaginary eigenvalues. A 2D cylindrical polar coordinate system is the only other case in which such multidimensional problems can be solved without the use of imaginary eigenvalues. The absence of imaginary eigenvalues renders the solution methodology significantly more useful for practical applications. The methodology described can be used for all three types of boundary conditions in the outer and inner surface of the sphere. Lastly, the solution procedure is demonstrated on an illustrative problem for which results are obtained.

36 MATERIALS SCIENCE↗