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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 1,261 records · Page 70
All sourcesNASA NTRSDOE OSTI
DOE OSTI2023

Transition from wall modes to multimodality in liquid gallium magnetoconvection

Coupled laboratory-numerical experiments of Rayleigh-Bénard convection in liquid gallium subject to a vertical magnetic field are presented. The experiments are carried out in two cylindrical containers with diameter-to-height aspect ratio Γ = 1.0 and 2.0 at varying thermal forcing (Rayleigh numbers 10 5 ≲ Ra ≲ 10 8 ) and magnetic field strength (Chandrasekhar numbers 0 ≲ Ch ≲ 3 × 10 5 ). Laboratory measurements and numerical simulations confirm that magnetoconvection in our finite cylindrical tanks onsets via nondrifting wall-attached modes, in good agreement with asymptotic predictions for a semi-infinite domain. With increasing supercriticality, the experimental and numerical thermal measurements and the numerical velocity data reveal transitions between wall mode states with different azimuthal mode numbers and between wall-dominated convection to wall and interior multimodality. These transitions are also reflected in the heat transfer data, which combined with previous studies connect onset to supercritical turbulent behaviors in liquid metal magnetoconvection over a large parameter space. The gross heat transfer behaviors between magnetoconvection and rotating convection in liquid metals are compared and discussed.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2021

Hexagonal Geometries in MPACT

The MPACT code is a high-fidelity light-water reactor analysis code using whole-core pin-resolved neutron transport calculations on modern parallel-computing hardware. MPACT uses the 2D/1D method to solve 3D neutron transport problems by decomposing the problem into a stack of 2D slices, each of which is solved independently using the method of characteristics (MOC). The slices are then coupled axially using the P3 nodal expansion method (NEM-P3) for the 1D axial calculations. MPACT also employs the coarse mesh finite difference (CMFD) method to accelerate calculations. This manuscript details work supporting advanced reactor designs using hexagonal pins and hexagonal assemblies such as the VVER-1000. If performed correctly, MOC is geometry agnostic. However, MPACT previously had optimizations in place for Cartesian geometries, specifically in the modularization and current calculations. Sections 2 and 3 detail the changes made to MPACT to support MOC and CMFD calculations on hexagonal geometries. Section 4 reports results demonstrating solution consistency for problems run with and without CMFD acceleration, results demonstrating solution consistency when run in serial and parallel, and pincell results using the Monte Carlo code, McCard’s benchmark results.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
NASA NTRS1996

Theoretical Basis for Finite Difference Extrapolation of Sonic Boom Signatures

Calculation of sonic boom signatures for aircraft has traditionally followed the methods of Whitham' and Walkden. The wave disturbance generated by the vehicle is obtained by area rule linearized supersonic flow methods, which yield a locally axisymmetric asymptotic solution. This solution is acoustic in nature, i.e., first order in disturbance quantities, and corresponds to ray acoustics. Cumulative nonlinear distortion of the signature is incorporated by using this solution to adjust propagation speed to first order, thus yielding a solution second order in disturbance quantities. The effects of atmospheric gradients are treated by Blokhintzov's method of geometrical acoustics. Both nonlinear signature evolution and ray tracing are applied as if the pressure field very close to the vehicle were actually that given by the source term (the 'F-function') of the asymptotic linearized flow solution. The viewpoint is thus that the flow solution exists at a small radius near the vehicle, and may be treated as an input to an extrapolation procedure consisting of ray tracing and nonlinear aging. The F-function is often regarded as a representation of a near-field pressure signature, and it is common for computational implementations to treat it interchangeably with the pressure signature. There is a 'matching radius' between the source function and the subsequent propagation extrapolation. This viewpoint has been supported by wind tunnel tests of simple models, and very typically yields correct results for actual flight vehicles. The assumption that the F-function and near-field signature are interchangeable is generally not correct. The flowfield of a vehicle which is not axisymmetric contains crossflow components which are very significant at small radii and less so at larger distances. From an acoustical viewpoint, the crossflow is equivalent to source diffraction portions of the wave field. Use of the F-function as a near field signature effectively assumes that the diminution of the crossflow/diffraction component may be applied all at once at the matching radius noted above. This approximation, though not rigorously validated, is responsible for the usual correct far-field results. On the other hand, if an actual near-field signature (either from wind tunnel or CFD data) is used at a starting point rather than one based on th effective source distribution, the predicted far-field signature is generally wrong.

Plotkin, Kenneth J.↗
DOE OSTI2021

Non-equilibrium effective field theory and second sound

We investigate the phenomenon of second sound in various states of matter from the perspective of non-equilibrium effective field theory (EFT). In particular, for each state of matter considered, we find that at least two (though sometimes multiple) qualitatively different EFTs exist at finite temperature such that there is always at least one EFT with a propagating second-sound wave and at least one with no such second-sound wave. To aid in the construction of these EFTs, we use the method of cosets developed for non-equilibrium systems. It turns out that the difference between the EFTs with and without second-sound modes can be understood as arising from different choices of a new kind of inverse Higgs constraint. Finally, we demonstrate that it is possible to bypass the need for new inverse Higgs constraints by formulating EFTs on a new kind of manifold that is like the usual fluid worldvolume, but with reduced gauge symmetries.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2019

Investigation of a passively Q-switched Raman laser at 1176 nm with Nd{sup 3+}:YAG/Cr{sup 4+}:YAG/YAG composite crystal and a coupled cavity

To reduce the thermal lens effect and increase the efficiency, a composite crystal of Nd{sup 3+}:YAG/Cr{sup 4+}:YAG/YAG and a coupled cavity structure were employed in our passively Q-switched YVO{sub 4}-based Raman laser system. Both the simulation and the experiment results showed that the undoped YAG crystal bonded in the end-face of Cr{sup 4+}:YAG crystal can speed up the heat transfer of Cr{sup 4+}:YAG crystal and reduce the temperature difference between its side surfaces and the center significantly, and the coupled cavity structure can improve the laser performance, especially at high pump power. The maximum output power of Raman laser of 0.92 W at 1176 nm was obtained with an optical-to-optical efficiency of 11.51%. The corresponding repetition rate and the pulse width is 53.5 kHz and 3.22 ns, respectively.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2019

Mechanical Response of Additively Manufactured Stainless Steel 304L Across a Wide Range of Loading Conditions

The mechanical response of additively manufactured (AM) stainless steel 304L has been investigated across a broad range of loading conditions, covering 11 decades of strain rate, and compared with the behaviors of traditional ingot-derived (wrought) material. In general, the AM material exhibits a greater strength and reduced ductility compared with the baseline wrought form. These differences are consistently found from quasi-static and high strain rate tests. A detailed investigation of the microstructure, the defect structure, the phase, and the composition of both forms reveals differences that may contribute to the differing mechanical behaviors. Compared with the baseline wrought material, dense AM stainless steel 304L has a more complex grain structure with substantial sub-structure, a fine dispersion of ferrite, increased dislocation density, oxide dispersions and larger amounts of nitrogen. In-situ neutron diffraction studies conducted during quasi-static loading suggest that the increased strength of AM material is due to its initially greater dislocation density. The flow strength of both forms is correlated with dislocation density through a square root dependence akin to a Taylor-like relationship. Neutron diffraction measurements of lattice strains also correlate with a crystal plasticity finite element simulations of the tensile test. Other simulations predict a significant degree of elastic and plastic anisotropy due to crystallographic texture. Hopkinson tests at higher strain rates $\dot{ε}$ = 500 and 2500 s -1 ) also show a greater strength for AM stainless steel 304L; although, the differences compared with wrought are reduced at higher strain rates. Gas gun impact tests, including reverse ballistic, forward ballistic and spall tests, consistently reveal a larger dynamic strength in the AM material. The Hugoniot Elastic Limit (HEL) of AM SS 304L exceeds that of wrought material although considerable variability is observed with the AM material. Forward ballistic testing demonstrates spall strengths of AM material (3.27 -- 3.91 GPa) that exceed that of the wrought material (2.63 -- 2.88 GPa). The Hugoniot equation-of-state for AM samples matches archived data for this metal alloy.

36 MATERIALS SCIENCE↗
NASA NTRS1994

Local dynamic subgrid-scale models in channel flow

The dynamic subgrid-scale (SGS) model has given good results in the large-eddy simulation (LES) of homogeneous isotropic or shear flow, and in the LES of channel flow, using averaging in two or three homogeneous directions (the DA model). In order to simulate flows in general, complex geometries (with few or no homogeneous directions), the dynamic SGS model needs to be applied at a local level in a numerically stable way. Channel flow, which is inhomogeneous and wall-bounded flow in only one direction, provides a good initial test for local SGS models. Tests of the dynamic localization model were performed previously in channel flow using a pseudospectral code and good results were obtained. Numerical instability due to persistently negative eddy viscosity was avoided by either constraining the eddy viscosity to be positive or by limiting the time that eddy viscosities could remain negative by co-evolving the SGS kinetic energy (the DLk model). The DLk model, however, was too expensive to run in the pseudospectral code due to a large near-wall term in the auxiliary SGS kinetic energy (k) equation. One objective was then to implement the DLk model in a second-order central finite difference channel code, in which the auxiliary k equation could be integrated implicitly in time at great reduction in cost, and to assess its performance in comparison with the plane-averaged dynamic model or with no model at all, and with direct numerical simulation (DNS) and/or experimental data. Other local dynamic SGS models have been proposed recently, e.g., constrained dynamic models with random backscatter, and with eddy viscosity terms that are averaged in time over material path lines rather than in space. Another objective was to incorporate and test these models in channel flow.

Cabot, William H.↗
NASA NTRS1995

On the breakup of viscous liquid threads

A one-dimensional model evolution equation is used to describe the nonlinear dynamics that can lead to the breakup of a cylindrical thread of Newtonian fluid when capillary forces drive the motion. The model is derived from the Stokes equations by use of rational asymptotic expansions and under a slender jet approximation. The equations are solved numerically and the jet radius is found to vanish after a finite time yielding breakup. The slender jet approximation is valid throughout the evolution leading to pinching. The model admits self-similar pinching solutions which yield symmetric shapes at breakup. These solutions are shown to be the ones selected by the initial boundary value problem, for general initial conditions. Further more, the terminal state of the model equation is shown to be identical to that predicted by a theory which looks for singular pinching solutions directly from the Stokes equations without invoking the slender jet approximation throughout the evolution. It is shown quantitatively, therefore, that the one-dimensional model gives a consistent terminal state with the jet shape being locally symmetric at breakup. The asymptotic expansion scheme is also extended to include unsteady and inerticial forces in the momentum equations to derive an evolution system modelling the breakup of Navier-Stokes jets. The model is employed in extensive simulations to compute breakup times for different initial conditions; satellite drop formation is also supported by the model and the dependence of satellite drop volumes on initial conditions is studied.

Papageorgiou, Demetrios T.↗
DOE OSTI2020

PT-symmetry entails pseudo-Hermiticity regardless of diagonalizability

We prove that in finite dimensions, a Parity-Time (PT)-symmetric Hamiltonian is necessarily pseudo-Hermitian regard-less of whether it is diagonalizable or not. This result is different from Mostafazadeh’s, which requires the Hamiltonian to be diagonalizable. PT-symmetry breaking often occurs at exceptional points where the Hamiltonian is not diagonalizable. Our result implies that PT-symmetry breaking is equivalent to the onset of instabilities of pseudo-Hermitian systems, which was systematically studied by Krein et al. in 1950s. In particular, we show that the mechanism of PT-symmetry breaking is the resonance between two eigen modes with opposite signs of actions.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2025

Exotic phases in finite-density ℤ 3 theories

Lattice ℤ 3 theories with complex actions share many key features with finite- density QCD including a sign problem and $\mathcal{CK}$ symmetry. Complex ℤ 3 spin and gauge models exhibit a generalized Kramers-Wannier duality mapping them onto chiral ℤ 3 spin and gauge models, which are simulatable with standard lattice methods in large regions of parameter space. The Migdal-Kadanoff real-space renormalization group (RG) preserves this duality, and we use it to compute the approximate phase diagram of both spin and gauge ℤ 3 models in dimensions one through four. Chiral ℤ 3 spin models are known to exhibit a Devil’s Flower phase structure, with inhomogeneous phases that can be thought of as ℤ 3 analogues of chiral spirals. Out of the large class of models we study, we find that only chiral spin models and their duals have a Devil’s Flower structure with an infinite set of inhomogeneous phases, a result we attribute to Elitzur’s theorem. We also find that different forms of the Migdal-Kadanoff RG produce different numbers of phases, a violation of the expectation for universal behavior from a real-space RG. We discuss extensions of our work to ℤ N models, SU(N) models and nonzero temperature.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
NASA NTRS2012

A Comprehensive Structural Dynamic Analysis Approach for Multi Mission Earth Entry Vehicle (MMEEV) Development

The anticipated NASA Mars Sample Return Mission (MSR) requires a simple and reliable method in which to return collected Martian samples back to earth for scientific analysis. The Multi-Mission Earth Entry Vehicle (MMEEV) is NASA's proposed solution to this MSR requirement. Key aspects of the MMEEV are its reliable and passive operation, energy absorbing foam-composite structure, and modular impact sphere (IS) design. To aid in the development of an EEV design that can be modified for various missions requirements, two fully parametric finite element models were developed. The first model was developed in an explicit finite element code and was designed to evaluate the impact response of the vehicle and payload during the final stage of the vehicle's return to earth. The second model was developed in an explicit code and was designed to evaluate the static and dynamic structural response of the vehicle during launch and reentry. In contrast to most other FE models, built through a Graphical User Interface (GUI) pre-processor, the current model was developed using a coding technique that allows the analyst to quickly change nearly all aspects of the model including: geometric dimensions, material properties, load and boundary conditions, mesh properties, and analysis controls. Using the developed design tool, a full range of proposed designs can quickly be analyzed numerically and thus the design trade space for the EEV can be fully understood. An engineer can then quickly reach the best design for a specific mission and also adapt and optimize the general design for different missions.

Perino, Scott↗
DOE OSTI2015

Multiscale Universal Interface: A concurrent framework for coupling heterogeneous solvers

Graphical abstract: - Abstract: Concurrently coupled numerical simulations using heterogeneous solvers are powerful tools for modeling multiscale phenomena. However, major modifications to existing codes are often required to enable such simulations, posing significant difficulties in practice. In this paper we present a C++ library, i.e. the Multiscale Universal Interface (MUI), which is capable of facilitating the coupling effort for a wide range of multiscale simulations. The library adopts a header-only form with minimal external dependency and hence can be easily dropped into existing codes. A data sampler concept is introduced, combined with a hybrid dynamic/static typing mechanism, to create an easily customizable framework for solver-independent data interpretation. The library integrates MPI MPMD support and an asynchronous communication protocol to handle inter-solver information exchange irrespective of the solvers' own MPI awareness. Template metaprogramming is heavily employed to simultaneously improve runtime performance and code flexibility. We validated the library by solving three different multiscale problems, which also serve to demonstrate the flexibility of the framework in handling heterogeneous models and solvers. In the first example, a Couette flow was simulated using two concurrently coupled Smoothed Particle Hydrodynamics (SPH) simulations of different spatial resolutions. In the second example, we coupled the deterministic SPH method with the stochastic Dissipative Particle Dynamics (DPD) method to study the effect of surface grafting on the hydrodynamics properties on the surface. In the third example, we consider conjugate heat transfer between a solid domain and a fluid domain by coupling the particle-based energy-conserving DPD (eDPD) method with the Finite Element Method (FEM)

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2020

Dissolution of a cylindrical disk in Hele-Shaw flow: a conformal-mapping approach

Here we apply conformal mapping to find the evolving shapes of a dissolving cylinder in a potential flow. Similar equations can be used to describe melting in a flowing liquid phase. Results are compared with microfluidic experiments and numerical simulations. Shapes predicted by conformal mapping agree almost perfectly with experimental observations, after a modest (20 %) rescaling of the time. Finite-volume simulations show that the differences with experiment are connected to the underlying assumptions of the analytical model: potential flow and diffusion-limited dissolution. Approximate solutions of the equations describing the evolution of the shape of the undissolved solid can be derived from a Laurent expansion of the mapping function from the unit circle. Asymptotic expressions for the evolution of the area of the disk and the shift in its centre of mass have been derived at low and high Péclet number. Analytic approximations to the leading-order Laurent coefficients provide additional insight into the mechanisms underlying pore-scale dissolution.

58 GEOSCIENCES↗
DOE OSTI2025

VARI3D & PERSENT: Perturbation and Sensitivity Analysis

The nodal diffusion method is one of the most widely used approaches in modern reactor analysis. In the nodal diffusion method, a coarse multi-group set of “homogenized” parameters is constructed such that the complex geometry of a reactor core along with the energy dependence of neutron and gamma ray cross sections in a nuclear reactor are conserved in the simpler geometry. The homogenization is typically done on a fuel assembly level as is the case in the DIF3D code developed at Argonne National Laboratory. The nodal methodology is used primarily to predict fuel cycle behavior of nuclear systems of which there is a substantial amount of validation in the literature. Another use of the nodal method is to obtain reactivity coefficients and kinetics parameters for use in a safety analysis of a given nuclear reactor. While there are many ways to obtain reactivity worth and kinetics parameters, the work presented in this manuscript is unique as it provides the user with the ability to compute reactivity worths, kinetics parameters, and cross section sensitivities with a Cartesian and hexagonal geometry based transport code. This manuscript serves as a single manual for two separate codes: VARI3D and PERSENT. The VARI3D code (VARIational 3D) is based upon the classic finite difference diffusion theory solver available in DIF3D. The PERSENT code (PERturbation and SENitivity for Transport) is based upon the variational nodal method employed in DIF3D termed VARIANT. The VARIANT solver was added to DIF3D in 1995 and has seen continued development and use for the last 18 years. Because VARI3D primarily uses deprecated coding practices, rather than incorporating the perturbation and sensitivity treatments for transport within VARI3D, a new coding development was built using modern Fortran coding. The primary purpose of this manual is to describe the theory behind PERSENT (and by convenience, that of VARI3D) and discuss the input and output of PERSENT along with giving potential users an idea of how to use it. While this manuscript does describe the input and output of VARI3D, the PERSENT code is intended to be the replacement capability of VARI3D as PERSENT can generate nearly identical (if not superior) diffusion theory results. In this manuscript, the relevant aspects of generalized perturbation theory and exact perturbation theory that apply to both VARI3D and PERSENT are covered. The input and output of VARI3D is displayed by excerpting several of the example problems. Similarly, the input and output of PERSENT is displayed along with tips on how best to use the code. Note that the input and output of the inhomogeneous solver wrapped around DIF3D (DIF3D_IFS) is also discussed as it is needed to carry out some of the sensitivities in PERSENT such as reaction rate ratios. This manuscript describes several perturbation and sensitivity problems, and the results computed using PERSENT. From these sections, potential users should find that PERSENT provides not only the typical tables of numbers desired in perturbation and sensitivity analysis work, but also can visually plot the result for a more thorough understanding of the space and energy distribution (Section 5). Overall, PERSENT is observed to produce accurate reactivity worths and sensitivities for the displayed set of test problems and clearly demonstrates the need to have a transport-based sensitivity capability as evident from the thousands of percent errors observed in the 21-group hexagonal fast reactor problem (covered in Section 7). The uncertainty calculation capability is described in Section 3 and demonstrated in Section 7.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2015

Development of Hexagonal Nodal Diffusion Codes at National Tsing-Hua University

The development of nodal methods for hexagonal geometry reactors at National Tsing-Hua University is reviewed. In particular, the development of the Hybrid Nodal Green's Function Method in hexagonal geometry is re-cast into the Direct-Coarse Mesh Finite Difference Method for efficient implementation with residual solution techniques such as GMRES. The method is further modified to include a rigorous treatment of non-physical terms that arise from the transverse integration procedure. The treatment follows the conformal mapping approach first developed by Y-A Chao and his co-workers. The overall development is implemented into a code written in C++ and tested on a number of well known benchmarks. The new code is shown to be very highly accurate while extremely efficient computationally.

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗
DOE OSTI2019

Is the Trotterized UCCSD Ansatz Chemically Well-Defined?

The variational quantum eigensolver (VQE) has emerged as one of the most enticing near-term quantum algorithms that can be used to simulate many-body systems such as molecular electronic structures. Serving as an attractive ansatz in the VQE algorithm, unitary coupled cluster (UCC) theory has seen a renewed interest in recent literature. Yet, unlike the original classical UCC theory, implementation on a quantum computer requires a finite-order Suzuki-Trotter decomposition to separate the exponentials of the large sum of Pauli operators. While previous literature has recognized the non-uniqueness of different orderings of the operators in the Trotterized form of UCC methods, the question of whether or not different orderings matter at the chemical scale has not been addressed. In this letter, we explore the effect of operator ordering on the Trotterized UCCSD ansatz, as well as the much more compact k-UpCCGSD ansatz recently proposed by Lee et al. We observe a significant, system-dependent variation in the energies of Trotterizations with different operator orderings. The energy variations occur on a chemical scale, sometimes on the order of hundreds of kcal/mol. This letter establishes the need to define not only the operators present in the ansatz, but also the order in which they appear. This is necessary for adhering to the quantum chemical notion of a ``model chemistry'', in addition to the general importance of scientific reproducibility. As a final note, we suggest a useful strategy to select out of the combinatorial number of possibilities, a single well-defined and effective ordering of the operators.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2015

Creation of problem-dependent Doppler-broadened cross sections in the KENO Monte Carlo code

In this paper, we introduce a quick method for improving the accuracy of Monte Carlo simulations by generating one- and two-dimensional cross sections at a user-defined temperature before performing transport calculations. A finite difference method is used to Doppler-broaden cross sections to the desired temperature, and unit-base interpolation is done to generate the probability distributions for double differential two-dimensional thermal moderator cross sections at any arbitrarily user-defined temperature. The accuracy of these methods is tested using a variety of contrived problems. In addition, various benchmarks at elevated temperatures are modeled, and results are compared with benchmark results. Lastly, the problem-dependent cross sections are observed to produce eigenvalue estimates that are closer to the benchmark results than those without the problem-dependent cross sections.

42 ENGINEERING↗
DOE OSTI2018

An assessment of atypical mesh topologies for low-Mach large-eddy simulation

An implicit, low-dissipation, low-Mach, variable density control volume finite element formulation is used to explore foundational understanding of numerical accuracy for large-eddy simulation applications on hybrid meshes. Detailed simulation comparisons are made between low-order hexahedral, tetrahedral, pyramid, and wedge/prism topologies against a third-order, unstructured hexahedral topology. Using smooth analytical and manufactured low-Mach solutions, design-order convergence is established for the hexahedral, tetrahedral, pyramid, and wedge element topologies using a new open boundary condition based on energy-stable methodologies previously deployed within a finite-difference context. A wide range of simulations demonstrate that low-order hexahedral- and wedge-based element topologies behave nearly identically in both computed numerical errors and overall simulation timings. Moreover, low-order tetrahedral and pyramid element topologies also display nearly the same numerical characteristics. Although the superiority of the hexahedral-based topology is clearly demonstrated for trivial laminar, principally-aligned flows, e.g., a 1x2x10 channel flow with specified pressure drop, this advantage is reduced for non-aligned, turbulent flows including the Taylor–Green Vortex, turbulent plane channel flow ( Re τ 395), and buoyant flow past a heated cylinder. With the order of accuracy demonstrated for both homogenous and hybrid meshes, it is shown that solution verification for the selected complex flows can be established for all topology types. Although the number of elements in a mesh of like spacing comprised of tetrahedral, wedge, or pyramid elements increases as compared to the hexahedral counterpart, for wall-resolved large-eddy simulation, the increased assembly and residual evaluation computational time for non-hexahedral is offset by more efficient linear solver times. Lastly, most simulation results indicate that modest polynomial promotion provides a significant increase in solution accuracy.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗