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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 1,099 records · Page 61
All sourcesNASA NTRSDOE OSTI
NASA NTRS2006

What does Reflection from Cloud Sides tell us about Vertical Distribution of Cloud Droplet Sizes?

Cloud development, the onset of precipitation and the effect of aerosol on clouds depend on the structure of the cloud profiles of droplet size and phase. Aircraft measurements of cloud profiles are limited in their temporal and spatial extent. Satellites were used to observe cloud tops not cloud profiles with vertical profiles of precipitation-sized droplets anticipated from CloudSat. The recently proposed CLAIM-3D satellite mission (cloud aerosol interaction mission in 3-D) suggests to measure profiles of cloud microphysical properties by retrieving them from the solar and infrared radiation reflected or emitted from cloud sides. Inversion of measurements from the cloud sides requires rigorous understanding of the 3-dimentional(3-D) properties of clouds. Here we discuss the reflected sunlight from the cloud sides and top at two wavelengths: one nonabsorbing to solar radiation (0.67 microns) and one with liquid water efficient absorption of solar radiation (2.1 microns). In contrast to the plane-parallel approximation, a conventional approach to all current operational retrievals, 3-D radiative transfer is used for interpreting the observed reflectances. General properties of the radiation reflected from the sides of an isolated cloud are discussed. As a proof of concept, the paper shows a few examples of radiation reflected from cloud fields generated by a simple stochastic cloud model with the prescribed vertically resolved microphysics. To retrieve the information about droplet sizes, we propose to use the probability density function of the droplet size distribution and its first two moments instead of the assumption about fixed values of the droplet effective radius. The retrieval algorithm is based on the Bayesian theorem that combines prior information about cloud structure and microphysics with radiative transfer calculations.

Marshak, A.↗
NASA NTRS2005

What Does Reflection from Cloud Sides Tell Us About Vertical Distribution of Cloud Droplet Sizes?

Cloud development, the onset of precipitation and the effect of aerosol on clouds depend on the structure of the cloud profiles of droplet size and phase. Aircraft measurements of cloud profiles are limited in their temporal and spatial extent. Satellites were used to observe cloud tops not cloud profiles with vertical profiles of precipitation-sized droplets anticipated from Cloudsat. The recently proposed CLAIM-3D satellite mission (cloud aerosol interaction mission in 3D) suggests to measure profiles of cloud microphysical properties by retrieving them from the solar and infrared radiation reflected or emitted from cloud sides. Inversion of measurements from the cloud sides requires rigorous understanding of the 3-dimensional (3D) properties of clouds. Here we discuss the reflected sunlight from the cloud sides and top at two wavelengths: one nonabsorbing to solar radiation (0.67 micrometers) and one with liquid water efficient absorption of solar radiation (2.1 micrometers). In contrast to the plane-parallel approximation, a conventional approach to all current operational retrievals, 3D radiative transfer is used for interpreting the observed reflectances. General properties of the radiation reflected from the sides of an isolated cloud are discussed. As a proof of concept, the paper shows a few examples of radiation reflected from cloud fields generated by a simple stochastic cloud model with the prescribed vertically resolved microphysics. To retrieve the information about droplet sizes, we propose to use the probability density function of the droplet size distribution and its first two moments instead of the assumption about fixed values of the droplet effective radius. The retrieval algorithm is based on the Bayesian theorem that combines prior information about cloud structure and microphysics with radiative transfer calculations.

Marshak, Alexander↗
DOE OSTI2013

A Novel Graphical User Interface for High-Efficacy Modeling of Human Perceptual Similarity Opinions

We present a novel graphical user interface (GUI) that facilitates high-efficacy collection of perceptual similarity opinions of a user in an effective and intuitive manner. The GUI is based on a hybrid mechanism that combines ranking and rating. Namely, it presents a base image for rating its similarity to seven peripheral images that are displayed simultaneously following a circular layout. The user is asked to report the base image s pairwise similarity to each peripheral image on a fixed scale while preserving the relative ranking among all peripheral images. The collected data are then used to predict the user s subjective opinions regarding the perceptual similarity of images. We tested this new approach against two methods commonly used in perceptual similarity studies: (1) a ranking method that presents triplets of images for selecting the image pair with the highest internal similarity and (2) a rating method that presents pairs of images for rating their relative similarity on a fixed scale. We aimed to determine which data collection method was the most time efficient and effective for predicting a user s perceptual opinions regarding the similarity of mammographic masses. Our study was conducted with eight individuals. By using the proposed GUI, we were able to derive individual user profiles that were 41.4% to 46.9% more accurate than those derived with the other two data collection GUIs. The accuracy improvement was statistically significant.

Kress, James M↗
DOE OSTI2013

High-Efficacy Modeling of Human Perceptual Similarity Opinions

The computer code relates to human nature studies and more specifically to a novel graphical user interface (GUI) that facilitates high-efficacy collection of perceptual similarity opinions of a user in an effective and intuitive manner. The GUI presents a base image for rating its similarity to seven peripheral images that are displayed simultaneously following a circular layout. The user is asked to report the base image's pairwise similarity to each peripheral image on a fixed scale while preserving the relative ranking among all peripheral images. The collected data are then used to predict the user's subjective opinions regarding the perceptual similarity of images.

Xu, Songhua↗
DOE OSTI2020

Advances in Resin Management Using 3R-Scan - 20154

The most important factor underlying optimal waste management is developing a clear picture of the radioactivity content of the waste and its impact on waste disposal cost. For over 35 years since the publishing of 10CFR61, waste characterization has relied on sampling the final waste product after formation. In the days following 10CFR61, the cost of final disposal was marginal with only a small impact on the overall costs. Constraints were added by provisions of the Low Level Waste Policy Act of 1985 leading to increasingly limited access to those disposal sites that remained available. In addition, Nuclear Regulatory Commission (NRC) pressure promoting waste volume reduction led to disposal costs inevitably rising. Despite this, characterization practices in monitoring of waste generation for activity content still center on the same dated processes. This results in a disposal classification on the basis of endpoint sampling without consideration of the homogeneity of the waste mixture. It can also disregard consideration of the representativeness of the single or small sample base. As a minimum effort, a formalized sampling program of a fixed grouping of waste streams can be implemented that could account for more than 95% of all of the activity carried in solid waste products. The sample results for each radionuclide could then be trended as time passes to develop reasonable scaling factors for difficult to measure radionuclides. This process, identified in NRC guidance, has been rigorously followed by a relatively small number of facilities. The trended scaling factors serve to improve accuracy by identifying anomalous results that could otherwise go undetected. Direct monitoring of the accumulation of activity in process streams generating solid radwaste, including demineralizers and filter streams, is a more precise approach. Nearly all of these streams are monitored by plant chemistry on a regular schedule to maintain water quality. This paper discusses viable options for developing the basis for characterization through process monitoring of the accumulation of activity at the point of generation. Special focus is on resin bed tracking and how process knowledge of these streams can be brought together to form a consistent and precise solid waste radioactivity inventory. Some of the specific points covered in this paper include the merger of the fission product release computer program, 3R-STAT, with the radwaste sample analysis computer program, SCAN4 to create 3R-SCAN, the importance of individual waste stream influences on the overall source term, and the use of historic sample data to develop scaling factors using an automated process. (authors)

12 MANAGEMENT OF RADIOACTIVE AND NON-RADIOACTIVE W↗
DOE OSTI2024

A Method for Producing Hierarchical and Statistically Calibrated Predictions of Nuclear Material Properties from Existing Models

Computer vision-based analysis of micrographs of nuclear materials is an emerging technique for property prediction, synthetic route identification, and other material analysis tasks. These analysis tasks play a pivotal role in many material characterization applications such as signature development for treaty verification, process optimization, etc. The backbone in many of the recent computer vision-based techniques is a deep learning model, which takes a fixed-size set of pixels and provides a class prediction for that set of pixels. For example, previous work developed a deep convolutional neural network (CNN) to predict the synthetic route from a 256 px x 256 px patch taken from a larger image of uranium ore concentrates. In this work, we present several methods for first calibrating these models in a manner that they can provide accurate probabilities of their predictions’ veracity, and several methods of combining these probabilities. Overall, the combination of these two steps into a pipeline allows for full-image and even full-sample (where a sample has many images) predictions with associated confidence values. Finally, we show that one can also use the patch predictions and confidence to produce a visualization to map predicted constituents through the image. Results and examples for predicting and mapping uranium ore concentrates’ synthetic process from imagery will be presented.

artificial intelligence↗
NASA NTRS2013

Lunar Sulfur Capture System

The Lunar Sulfur Capture System (LSCS) protects in situ resource utilization (ISRU) hardware from corrosion, and reduces contaminant levels in water condensed for electrolysis. The LSCS uses a lunar soil sorbent to trap over 98 percent of sulfur gases and about two-thirds of halide gases evolved during hydrogen reduction of lunar soils. LSCS soil sorbent is based on lunar minerals containing iron and calcium compounds that trap sulfur and halide gas contaminants in a fixed-bed reactor held at temperatures between 250 and 400 C, allowing moisture produced during reduction to pass through in vapor phase. Small amounts of Earth-based polishing sorbents consisting of zinc oxide and sodium aluminate are used to reduce contaminant concentrations to one ppm or less. The preferred LSCS configuration employs lunar soil beneficiation to boost concentrations of reactive sorbent minerals. Lunar soils contain sulfur in concentrations of about 0.1 percent, and halogen compounds including chlorine and fluorine in concentrations of about 0.01 percent. These contaminants are released as gases such as H2S, COS, CS2,HCl, and HF during thermal ISRU processing with hydrogen or other reducing gases. Removal of contaminant gases is required during ISRU processing to prevent hardware corrosion, electrolyzer damage, and catalyst poisoning. The use of Earth-supplied, single-use consumables to entirely remove contaminants at the levels existing in lunar soils would make many ISRU processes unattractive due to the large mass of consumables relative to the mass of oxygen produced. The LSCS concept of using a primary sorbent prepared from lunar soil was identified as a method by which the majority of contaminants could be removed from process gas streams, thereby substantially reducing the required mass of Earth-supplied consumables. The LSCS takes advantage of minerals containing iron and calcium compounds that are present in lunar soil to trap sulfur and halide gases in a fixedbed reactor downstream of an in-ISRU process such as hydrogen reduction. The lunar-soil-sorbent trap is held at a temperature significantly lower than the operating temperature of the hydrogen reduction or other ISRU process in order to maximize capture of contaminants, but is held at a high enough temperature to allow moisture to pass through without condensing. The lunar soil benefits from physical beneficiation to remove ultrafine particles (to reduce pressure drop through a fixed bed reactor) and to upgrade concentrations of iron and/or calcium compounds (to improve reactivity with gaseous contaminants).

Berggren, Mark↗
NASA NTRSDate not supplied

A Study on Effect of Tuning Capacitor in Dynamic Radioisotope Power Systems

This study presents the effect of the tuning capacitor on stability and electrical power output of a gas-bearing based free-piston Stirling convertor (FPSC) when operating on a fixed-frequency type controller. It has been widely accepted that the tuning capacitor is an important circuit component of the FPSC to stabilize piston amplitude and maximize electrical power output by correcting the power factor. Models suggest the stability of the FPSC is strongly coupled to the tuning capacitance, while the coupling to the electrical power output is weak. This applies to cases where the operating frequency of the FPSC is fixed by a fixed-frequency controller and other operating conditions, such as the hot-end and cold-end temperatures, piston amplitude, and pressure are constant. To verify the modeling analysis results, an Advanced Stirling Convertor (ASC) was controlled by an AC power supply and the piston amplitude, hot-end temperature, cold-end temperature, and pressure are actively controlled to 4.3mm, 760˚C, 40˚C and 485psig, respectively, while the tuning capacitor value was changed from 840μF to 1040μF and 1240μF. Furthermore, the vibration test results of the ASC with different tuning capacitor values are also presented to show the effect of the tuning capacitor on stability. The experimental test results verified that the effect of tuning capacitor on electrical power output is negligible while the effect on stability is rather noticeable. Therefore, the ideal tuning capacitor value should be selected based on required stability of the FPSC.

Free-Piston Stirling Convertor Controller↗
DOE OSTI2021

Harnessing stratospheric diffusion barriers for enhanced climate geoengineering

Stratospheric sulfate aerosol geoengineering is a proposed method to temporarily intervene in the climate system to increase the reflectance of shortwave radiation and reduce mean global temperature. In previous climate modeling studies, choosing injection locations for geoengineering aerosols has, thus far, only utilized the average dynamics of stratospheric wind fields instead of accounting for the essential role of time-varying material transport barriers in turbulent atmospheric flows. Here we conduct the first analysis of sulfate aerosol dispersion in the stratosphere, comparing what is now a standard fixed-injection scheme with time-varying injection locations that harness short-term stratospheric diffusion barriers. We show how diffusive transport barriers can quickly be identified, and we provide an automated injection location selection algorithm using short forecast and reanalysis data. Within the first 7 d days of transport, the dynamics-based approach is able to produce particle distributions with greater global coverage than fixed-site methods with fewer injections. Additionally, this enhanced dispersion slows aerosol microphysical growth and can reduce the effective radii of aerosols up to 200–300 d after injection. While the long-term dynamics of aerosol dispersion are accurately predicted with transport barriers calculated from short forecasts, the long-term influence on radiative forcing is more difficult to predict and warrants deeper investigation. Statistically significant changes in radiative forcing at timescales beyond the forecasting window showed mixed results, potentially increasing or decreasing forcing after 1 year when compared to fixed injections. We conclude that future feasibility studies of geoengineering should consider the cooling benefits possible by strategically injecting sulfate aerosols at optimized time-varying locations. Our method of utilizing time-varying attracting and repelling structures shows great promise for identifying optimal dispersion locations, and radiative forcing impacts can be improved by considering additional meteorological variables.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2018

Natronomonas pharaonis halorhodopsin Ser81 plays a role in maintaining chloride ions near the Schiff base

Highlights: • S81A mutant of NpHR lose light-induced photocurrent. • The direction of the Thr126 side chain was fixed by a hydroxyl group of Ser81. • S81A mutation decrease the interaction between Thr126 and a chloride ion. • Ser81 plays a key role in chloride migration. Optogenetic technologies have often been used as tools for neuronal activation or silencing by light. Natronomonas pharaonis halorhodopsin (NpHR) is a light-driven chloride ion pump. Upon light absorption, a chloride ion passes through the cell membrane, which is accompanied by the temporary binding of a chloride ion with Thr126 at binding site-1 (BS1) near the protonated Schiff base in NpHR. However, the mechanism of stabilization of the binding state between a chloride ion and BS1 has not been investigated. Therefore, to identify a key component of the chloride ion transport pathway as well as to acquire dynamic information about the chloride ion-BS1 binding state, we performed a rough analysis of the chloride ion pathway shape followed by molecular dynamics (MD) simulations for both wild-type and mutant NpHR structures. The MD simulations showed that the hydrogen bond between Thr126 and the chloride ion was retained in the wild-type protein, while the chloride ion could not be retained at and tended to leave BS1 in the S81A mutant. We found that the direction of the Thr126 side chain was fixed by a hydroxyl group of Ser81 through a hydrogen bond and that Thr126 bound to a chloride ion in the wild-type protein, while this interaction was lost in the S81A mutant, resulting in rotation of the Thr126 side chain and reduction in the interaction between Thr126 and a chloride ion. To confirm the role of S81, patch clamp recordings were performed using cells expressing NpHR S81A mutant protein. Considered together with the results that the NpHR S81A-expressing cells did not undergo hyperpolarization under light stimulation, our results indicate that Ser81 plays a key role in chloride migration. Our findings might be relevant to ongoing clinical trials using optogenetic gene therapy in blind patients.

60 APPLIED LIFE SCIENCES↗
NASA NTRS2010

Developments and Validations of Fully Coupled CFD and Practical Vortex Transport Method for High-Fidelity Wake Modeling in Fixed and Rotary Wing Applications

A novel Computational Fluid Dynamics (CFD) coupling framework using a conventional Reynolds-Averaged Navier-Stokes (BANS) solver to resolve the near-body flow field and a Particle-based Vorticity Transport Method (PVTM) to predict the evolution of the far field wake is developed, refined, and evaluated for fixed and rotary wing cases. For the rotary wing case, the RANS/PVTM modules are loosely coupled to a Computational Structural Dynamics (CSD) module that provides blade motion and vehicle trim information. The PVTM module is refined by the addition of vortex diffusion, stretching, and reorientation models as well as an efficient memory model. Results from the coupled framework are compared with several experimental data sets (a fixed-wing wind tunnel test and a rotary-wing hover test).

Anusonti-Inthra, Phuriwat↗
DOE OSTI2021

Comment on “Review of experimental studies of secondary ice production” by Korolev and Leisner (2020)

Abstract. This is a comment on the review by Korolev and Leisner (2020, hereafter KL2020). The only two laboratory/field studies ever to measure the breakup in ice–ice collisions for in-cloud conditions were negatively criticised by KL2020, as were our subsequent theoretical and modelling studies informed by both studies. First, hypothetically, even without any further laboratory experiments, such theoretical and modelling studies would continue to be possible, based on classical mechanics and statistical physics. They are not sensitive to the accuracy of lab data for typical situations, partly because the nonlinear explosive growth of ice concentrations continues until some maximum concentration is reached. To a degree, the same final concentration is expected regardless of the fragment number per collision. Second, there is no evidence that both lab/field observational studies characterising fragmentation in ice–ice collisions are either mutually conflicting or erroneous such that they cannot be used to represent this breakup in numerical models, contrary to the review. The fact that the ice spheres of one experiment were hail sized (2 cm) is not a problem if a universal theoretical formulation, such as ours, with fundamental dependencies, is informed by it. Although both lab/field studies involved head-on collisions, rotational kinetic energy for all collisions generally is only a small fraction of the initial collision kinetic energy (CKE) anyway. Although both lab/field experiments involved fixed targets, that is not a problem since the fixing of the target is represented via CKE in any energy-based formulation such as ours. Finally, scaling analysis suggests that the breakup of ice during sublimation can make a significant contribution to ice enhancement in clouds, again contrary to the impression given by the review.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2024

Multiphysics Degradation Modeling of Energy Storage Materials via RKPM with a Neural Network-Enhancement

In energy storage materials, strong electrochemical-mechanical coupling and highly anisotropic material properties contribute to the formation and propagation of micro-cracking during charge/discharge cycling, resulting in reduced performance and service life. A coupled electro-chemo-mechanical reproducing kernel particle method (RKPM) formulation is developed, and a patch-test is formulated to certify optimal convergence of the proposed RKPM method for the coupled physics system. With microstructural images supplied by the National Renewable Energy Laboratory (NREL), pixel-based model construction by RKPM is then used to represent the complex material microstructures for modeling the coupled physics of these systems. Further, a neural network-enhanced reproducing kernel particle method (NN-RKPM) [1, 2] is introduced to effectively model damage and crack propagation in the material microstructures; the location, orientation, and solution transition near a localization are automatically captured by superimposed block-level NN optimizations. This NN enrichment approach allows for effective modeling of localizations via a fixed background discretization, relieving tedious efforts for adaptive refinement in traditional mesh-based methods. Applications to the heterogeneous microstructures of Li-ion battery cathodes will be presented to demonstrate the effectiveness of the proposed methods. Reference: [1] Baek, J., Chen, J. S., Susuki, K., "Neural Network enhanced Reproducing Kernel Particle Method for Modeling Localizations," International Journal for Numerical Methods in Engineering, Vol. 123, pp 4422-4454, https://doi.org/10.1002/nme.7040, 2022. [2] Baek, J., Chen, J. S., "A Neural Network-Based Enrichment of Reproducing Kernel Approximation for Modeling Brittle Fracture", Computer Methods in Applied Mechanics and Engineering Vol. 410, 116590, 2024.

electro-chemo-mechanical coupling↗
DOE OSTI2024

Leveraging a Neural Network-Enhanced Reproducing Kernel Particle Method for Multiphysics Degradation Modeling of Energy Storage Materials

Energy storage materials exhibit strong electro-chemo-mechanical coupling and highly anisotropic material properties, contributing to the formation and propagation of micro-cracking during charge/discharge cycling and resulting in reduced performance and service life. A coupled electro-chemo-mechanical reproducing kernel particle method (RKPM) formulation has been developed to analyze this system. With microstructural images supplied by the National Renewable Energy Laboratory (NREL), pixel-based model construction by RKPM is used to represent the complex material microstructures that dictate the coupled physics of these systems. Traditional electro-chemo-mechanical models rely on mesh-based finite element methods, which can lead to difficulties in meshing such complex geometries and capturing crack propagation due to mesh dependency. Here, a neural network-enhanced reproducing kernel particle method (NN-RKPM) [1, 2] is introduced to effectively model damage and crack propagation in the material microstructures; the location, orientation, and solution transition near a localization are automatically captured by superimposed block-level NN optimizations. This NN enrichment approach allows for effective modeling of localizations via a fixed background discretization, relieving tedious efforts for adaptive refinement in traditional mesh-based methods. Applications to the heterogeneous microstructures of Li-ion battery cathodes will be presented to demonstrate the effectiveness of the proposed methods. NN-RKPM is additionally used to inform how crack opening and closure in turn affect the coupled chemical equations and material microstructure. Reference: [1] Baek, J., Chen, J. S., Susuki, K., "Neural Network enhanced Reproducing Kernel Particle Method for Modeling Localizations," International Journal for Numerical Methods in Engineering, Vol. 123, pp 4422-4454, https://doi.org/10.1002/nme.7040, 2022. [2] Baek, J., Chen, J. S., "A Neural Network-Based Enrichment of Reproducing Kernel Approximation for Modeling Brittle Fracture", Computer Methods in Applied Mechanics and Engineering Vol. 410, 116590, 2024.

degradation↗
DOE OSTI2020

Dynamic Performance Enhancement for Nonlinear Stochastic Systems Using RBF Driven Nonlinear Compensation with Extended Kalman Filter

In this paper, a novel hybrid control method is proposed to enhance the control performance of the PI based control system for a class of nonlinear and non-Gaussian stochastic dynamic processes with unmeasurable states. Firstly, to enhance the tracking performance of the PI controller where the PI parameters are fixed in many actual control systems, the compensative signal is formed using the extended Kalman filter(EKF) based state estimator and driven by the radial basis function (RBF) neural network based compensator. In addition, the weights of RBF is trained to minimize the entropy criterion of tracking error as the process is subjected to non-Gaussian disturbances. Meanwhile, since the precise statistical property of noises is hard to obtain for many industrial processes, the kernel density estimation (KDE) technique is employed in this paper to estimate the entropy. The convergence of RBF network is discussed and the stability of the resulted closed-loop hybrid control system is analyzed in mean square sense. Finally, a numerical example and a practical system testing are given to illustrate the effectiveness of the proposed control method.

stochastic systems, PI Controller, Entropy, Extend↗
DOE OSTI2024

Direct Power Control of Back to Back Modular Multilevel Converter with Advanced Grid Support Functions for Grid Forming Application

Possibility of using back-to-back modular multilevel converter system with a direct power control architecture has been investigated in this paper. The rectifier side is controlled to be in grid following mode whereas the inverter side is controlled in grid forming mode. Both the sides’ local controllers use Lyapunov energy function based architecture to accomplish the objective of active and reactive power control as well as maintaining a fixed user defined voltage over the dc bus on the rectifier side. Therefore, the proposed direct power control architecture has embedded dc bus control architecture to maintain fixed user defined value of the capacitors. Similar architecture is implemented on the inverter side to accomplish either voltage control during grid forming or power control during grid following. Advanced grid support functionalities based on IEEE-1547-2018 is implemented and utilized to generate the reference values for either the rectifier or the inverter sides. The overall system is modeled based on MATLAB/Simulink and PLECS domain and various important case studies to verify the efficacy of the overall system has been presented.

direct power control (DPC)↗
DOE OSTI2017

Fluctuations in quantum mechanics and field theories from a new version of semiclassical theory. II.

This is the second paper on the semiclassical approach based on the density matrix given by the Euclidean time path integral with fixed coinciding end points. The classical path, interpolating between this point and the classical vacuum (called a “flucton”), as well as systematic one- and two-loop corrections were calculated in the first paper [M. A. Escobar-Ruiz, E. Shuryak, and A. V. Turbiner, Phys. Rev. D 93, 105039 (2016)] for a double-well potential. Here, we extend them for a number of quantum-mechanical problems, such as an anharmonic oscillator and the sine-Gordon potential. The method is based on a systematic expansion in Feynman diagrams and thus can be extended to quantum field theories (QFTs). We show that the loop expansion in quantum mechanics resembles the leading-log approximations in QFT. In this sequel, we present a complete set of results obtained using this method in a unified way. Alternatively, starting from the Schrödinger equation we derive a generalized Bloch equation whose semiclassical-like, iterative solution generates the loop expansion. We rederive the two-loop expansions for all three of the above potentials and extend them to three loops, which has not yet been done via Feynman diagrams. All results for both methods are fully consistent with each other. An asymmetric (tilted) double-well potential (nondegenerate minima) is also studied using the second method.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗