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At least 343 records · Page 19
All sourcesNASA NTRSDOE OSTI
DOE OSTI2025

Design and Analysis of a Mutual Inductance Level Sensor for Liquid Metals

Here, this article describes the design and analysis of an electromagnetic level sensor for use in high-temperature liquid metal systems. The mutual inductance level sensor (MILS) described in this work was fabricated using two single-conductor mineral insulated cables wrapped in a bifilar fashion around a stainless steel tube core and was housed in an isolating thimble that preserved the pressure boundary of the test vessel. Two sensor variations were fabricated that differ only in active length, 1016 and 1778 mm. Experimental data were collected using the 1016-mm sensor (MILS-MKII-040) that demonstrated a sensitivity of 9.2 μ V/mm in a room temperature testing stand that used solid aluminum as a surrogate for liquid metal. Experimental data were collected using the 1778-mm sensor (MILS-MKII-070) that demonstrated a sensitivity of 6.9 μ V/mm in the high-temperature (300 ° C) liquid sodium environment at the mechanisms engineering test loop (METL) of Argonne National Laboratory. The sensor performance was found to be repeatable over the course of several months, with roughly ±1% deviation from nominal output. Finite element models were developed in COMSOL Multiphysics that fully describe each test setup, and the models were validated using experimental data. The validated COMSOL models were used to perform an array of analyses that examined the performance of the sensor in differing environments. Maximizing the coil diameter inside the isolating thimble was found to maximize the signal and sensitivity of the sensor. An optimal operating frequency was found near 1000 Hz using both experimental data and COMSOL. The influence of a metallic thimble surrounding the sensor and a metallic sensor core was quantified and found to be negligible at the optimal operating frequency. The sensitivity of the sensor was quantified when monitoring the level of additional liquid metals. These include lead, lead-bismuth eutectic (LBE), sodium-potassium alloy (NaK), and lithium (in addition to sodium). The sensitivities were quantified using liquid metal properties at 350 ° C and 650 ° C. The geometry of the test stand model, all material properties used in the model, and the results are presented in a manner that allows the reader can replicate the model and perform additional analyses.

COMSOL↗
DOE OSTI2021

Superconducting Magnet Qualification Methodology for the Material Plasma Exposure Experiment

The Material Plasma Exposure eXperiment (MPEX), currently under design, is a new linear plasma device to advance the understanding of plasma-material interactions through the generation and delivery of plasmas as they are expected in future fusion reactor divertors. MPEX will be a steady-state device to study high-fluence exposures of plasma-facing materials and components. The requirements for the magnetic field at the target and the heating stages make the application of superconducting coils necessary. Conceptual designs for the superconducting magnets have been developed, and multiple cryostats with warm bore diameters of either 65 cm or 156 cm are envisioned to facilitate their integrated and timely assembly with other systems such as vacuum, water cooling, and RF power. Although design, fabrication, and testing for the magnets as stand-alone units are straightforward, challenges will arise during the integration of the system. Two different field profiles will be used during operation. The magnetic field where the electron cyclotron heating occurs needs to operate at both 1.25 and 2.5 T. It is critical that the magnets all share the same magnetic axis and alignment. The mutual inductance between cryostats will affect the quench behavior of the system. Also, cryostat-to-cryostat forces can be as large as 700 kN, and the magnitude and direction will change depending on which coils are energized. The design of the system must take those characteristics into account along with the quench scenarios. This paper describes the qualification approach that will be used to determine whether stand-alone tests can be used to ensure the success of the integrated system. Fiducials will be used to define the location of the magnetic axis for each cryostat to ensure proper alignment. Quench tests of a single magnet will be performed at a current above the normal operating current to account for additional stored energy from the mutual inductance to adjacent cryostats. Also, a 1018 steel plate will be mounted on either end of a cryostat to simulate the cryostat-to-cryostat forces. Requirements for the size and location of the steel plates are described.

Burkhardt, Earle↗
DOE OSTI2023

Assessment and Performance of Flexible Quench Antenna Array Diagnostics for Superconducting Magnets

FNAL has been developing multiple versions of flexible quench antennas (flex-QA), including some specially optimized for high sensitivity and/or high resolution, to characterize quench events and transients during current ramping in superconducting magnets. A fundamental feature in our use of these is the creation of grid-like structures of sensitive elements to cover coil surfaces, with the aim of getting precise localization of magnetic flux-change events. The flex-QA are coupled with fast data-acquisition, allowing comprehensive analysis of signals at the relevant fine time scales. In addition to arrays of various flex-QA types being used during cryogenic testing of superconducting magnets, we also are utilizing a newly developed room temperature test stand to better understand QA response characteristics. The data from actual superconducting magnet tests, warm test stand measurements, and simulation data on the same QA designs allows us to draw conclusions on operational feasibility and plan better for improvements of our sensors. In this paper we present data from the multiple tests performed and analysis results. Flex-QA designs are compared, and their features, options, and optimization discussed.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2020

Plasma Facing Component Characterization and Correlation With Plasma Conditions in Lithium Tokamak Experiment-β

Lithium coatings in the Lithium Tokamak eXperiment (LTX) led to flat temperature profiles. The flat temperature profiles were observed along with a hot, low density edge, implying a broad, collisionless scrape-off layer (SOL). Additionally, in vacuo X-ray photoelectron spectroscopy (XPS) measurements established that lithium coatings evaporatively deposited onto high-Z plasma facing components (PFCs) became oxidized while retaining the ability to achieve good plasma performance long after lithium was applied to the PFCs. Longstanding theory predicted flat temperature profiles with low recycling walls, which was presumed to be due to hydrogen binding with elemental lithium to form lithium hydride. The presence of oxidized lithium, however, raised questions regarding the exact mechanism of hydrogen retention in LTX. To investigate these questions, the upgraded facility LTX- β includes a new sample exposure probe (SEP) for more detailed in vacuo analysis of PFC samples. The SEP is equipped with a vacuum suitcase capable of transporting samples representative of the LTX- β outer midplane PFCs to a stand-alone XPS system while maintaining pressures lower than the LTX- β base vacuum to limit the contamination between sample exposure and analysis. The low-energy resolution XPS system used in past experiments could only enable the determination of elemental percentages on the PFC sample surfaces. Because the new XPS system has higher energy resolution, it is more direct to assign chemical compounds to the measured binding energies. This capability has been confirmed by comparing XPS data from PFC test samples with measurements using a commercial high-resolution XPS system. Quartz crystal microbalances (QCMs) were used to quantify the thickness of the deposited lithium on the LTX- β PFCs. This article describes the application of the SEP to characterize the PFC surfaces using XPS and their relationship to plasma conditions.

Maan, Anurag↗
DOE OSTI2020

Shaped Beams from Diamond Field-Emitter Array Cathodes

Diamond field-emitter arrays (DFEAs) are arrays of diamond pyramids with exquisitely sharp tips and micrometer-scale bases that produce high current densities. These arrays can be fabricated in arbitrary shapes, ranging from single tips to many millions of tips, so that they produce an inherently shaped electron beam. Each tip emits a modest current, but the large dense array can produce many Amps. We are investigating these cathodes for use in dielectric wakefield accelerators; however, they may also be applicable to vacuum microwave tubes. Recently, shaped beam production and transport have been demonstrated in the 1.3-GHz RF gun at the Argonne Cathode Test Stand at the Argonne National Laboratory. The charge was measured on a Faraday cup and the beam imaged on a YAG screen with peak electric field gradients on the cathode ranging from 12 to 35 MV/m. Three cathode geometries were tested: one 1-mm equilateral triangle with 7-mu m base pyramids and 10-mu m pitch, one 1-mm equilateral triangle with 10-mu m base and 25-mu m pitch, and one sparse 5 x 5 square array with 20-mu m base and 400-mu m pitch. The two triangular arrays emitted 35 nC in an RF macropulse at 35 MV/m and 13-nC charge at 27 MV/m, respectively, while the sparse array emitted 0.060-nC charge at 15 MV/m. This article presents the results of the triangular array experiments, including damage due to breakdown in the RF gun and initial models of tip-to-tip shielding.

Cathodes↗
DOE OSTI2022

Fire, insect and disease‐caused tree mortalities increased in forests of greater structural diversity during drought

Abstract Structural diversity is an emerging dimension of biodiversity that accounts for size variations in organs among individuals in a community. Previous studies show significant effects of structural diversity on forest growth, but its effects on forest mortality are not known, particularly at a large scale. To address this knowledge gap, we quantified structural diversity using stem structural diversity (SSD) based on both tree diameter and height. We obtained U.S. Forest Service Forest Inventory and Analysis (FIA) data from over 2400 plots across southcentral U.S. forests that have suffered a recent drought. Using data from multiple sampling times, we calculated SSD and compared the relative importance of SSD, species diversity, functional diversity and other stand attributes in determining tree mortalities caused by fire, insects and diseases. We also used FIRETEC, a physics‐based fire model, to test the effect of SSD on canopy consumption by fire. Our results showed that (1) SSD was positively associated with tree mortalities caused by all three disturbances; (2) species richness was negatively associated with insect‐ and disease‐caused mortalities; (3) functional diversity was negatively associated with fire‐ and disease‐caused mortalities and (4) more phylogenetically related species had more similar mortality rates by insect and disease but not fire. Moreover, the FIRETEC model showed increasing canopy consumption by fire in stands with greater SSD. Together, the different tree mortalities during drought associated with SSD more consistently than the other biodiversity metrics were evaluated. Synthesis . Our results suggest that SSD could be considered in modelling forest dynamics and planning management to sustain forest health under disturbances.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2024

Wildfire‐induced increases in photosynthesis in boreal forest ecosystems of North America

Abstract Observations of the annual cycle of atmospheric CO 2 in high northern latitudes provide evidence for an increase in terrestrial metabolism in Arctic tundra and boreal forest ecosystems. However, the mechanisms driving these changes are not yet fully understood. One proposed hypothesis is that ecological change from disturbance, such as wildfire, could increase the magnitude and change the phase of net ecosystem exchange through shifts in plant community composition. Yet, little quantitative work has evaluated this potential mechanism at a regional scale. Here we investigate how fire disturbance influences landscape‐level patterns of photosynthesis across western boreal North America. We use Alaska and Canadian large fire databases to identify the perimeters of wildfires, a Landsat‐derived land cover time series to characterize plant functional types (PFTs), and solar‐induced fluorescence (SIF) from the Orbiting Carbon Observatory‐2 (OCO‐2) as a proxy for photosynthesis. We analyze these datasets to characterize post‐fire changes in plant succession and photosynthetic activity using a space‐for‐time approach. We find that increases in herbaceous and sparse vegetation, shrub, and deciduous broadleaf forest PFTs during mid‐succession yield enhancements in SIF by 8–40% during June and July for 2‐ to 59‐year stands relative to pre‐fire controls. From the analysis of post‐fire land cover changes within individual ecoregions and modeling, we identify two mechanisms by which fires contribute to long‐term trends in SIF. First, increases in annual burning are shifting the stand age distribution, leading to increases in the abundance of shrubs and deciduous broadleaf forests that have considerably higher SIF during early‐ and mid‐summer. Second, fire appears to facilitate a long‐term shift from evergreen conifer to broadleaf deciduous forest in the Boreal Plain ecoregion. These findings suggest that increasing fire can contribute substantially to positive trends in seasonal CO 2 exchange without a close coupling to long‐term increases in carbon storage.

Biodiversity & Conservation↗
DOE OSTI2020

Cut–and–chip harvester material capacity and fuel performance on commercial-scale willow fields for varying ground and crop conditions

Shrub willow (Salix spp.) is capable of producing commercially attractive amounts of biomass in short rotations, but harvesting costs and logistics remain a concern. There is a particular need for information about harvesting operations on larger, commercial short–rotation woody crop systems. Another recent issue on commercial fields in northern New York is commercial growers conducting harvests during the growing season rather than the recommended dormant season when fields may be too wet to harvest. This study evaluated and modeled the in–field performance of a cut–and–chip harvester for almost 700 wagonloads of chips operating in commercial willow fields in a wider array of crop and field conditions than have been previously reported. Analysis indicated that the time of harvest (leaf–on or leaf–off) and whether site conditions were wet or dry affected the harvester's material capacity. Mean material capacity was greatest for leaf–off, dry conditions (71.8 Mg/hr) and lowest for leaf–on harvests, which were similar for wet (30.4 Mg/hr) and dry conditions (29.7 Mg/hr). Mean crop specific fuel consumption ranged between 1.3 and 3.3 L/Mg, but can get considerably higher for standing biomasses below 40 Mg/ha. Wet ground conditions and leaf–on harvests tend to decrease material capacity and increase fuel consumption as the harvester has to divert power to forward movement and material processing. Relationships for material capacity and fuel consumption based on standing biomass, time of harvest and ground conditions will be essential for evaluating and modeling the economic and environmental impacts of commercial–scale willow operations.

09 BIOMASS FUELS↗
DOE OSTI2024

Switchgrass ( Panicum virgatum L.) cultivars have similar impacts on soil carbon and nitrogen stocks and microbial function

Abstract Switchgrass ( Panicum virgatum L.) production for biofuel has the potential to produce reasonable yields on lands not suited for conventional agriculture. We assessed nine switchgrass cultivars representing lowland and upland ecotypes grown for 11 years at a site in the upper Midwest USA for belowground differences in soil carbon and nitrogen stocks, soil organic matter fractions, and standing root biomass to 1 m depth. We also compared potential nitrogen mineralization and carbon substrate use through community‐level physiological profiling in surface soils (0–10 cm depth). Average yields and standing root biomass differed among cultivars and between ecotypes, but we found no significant cultivar‐related impacts on soil carbon and nitrogen stocks, on the distribution of particulate and mineral‐associated soil organic matter fractions, nor on potential nitrogen mineralization or microbial community‐level physiological profiles. That these traits did not differ among cultivars suggests that soil carbon and nitrogen gains under switchgrass are likely to be robust with respect to cultivar differences, and to this point not much affected by breeding efforts.

Agriculture↗
DOE OSTI2025

Switchgrass ( Panicum Virgatum ) and Miscanthus ( Miscanthus × Giganteus ) Long-Term Yield Patterns Reveal Consistent Productivity Declines

Perennial grasses like switchgrass ( Panicum virgatum ) and miscanthus ( Miscanthus × giganteus ) are expected to supply a substantial amount of the United States bioeconomy's feedstock demand. However, uncertainties around their long-term yields challenge the viability of their potential and limit their wider adoption. To resolve their long-term yield patterns, we analyzed over 200 plantings of switchgrass and miscanthus across Michigan and Wisconsin, USA, measured over 5–15 years. We found a consistent two-phase long-term yield dynamic; during a yield-building phase , peak yields occurred within 4–5 years after planting, followed by a yield-decline phase in which switchgrass and miscanthus lost 30%–47% and 14%–40% of peak yields, respectively. Among the potential drivers of this dynamic and the yield decline, we found that weather conditions had little impact, as the variation across years was not large enough to drive the observed yield differences. Added nitrogen increased peak yields by 10%–20% and attenuated the yield decline by 20%–50%. However, since fertilized stands still showed a yield decline, other factors became limiting as stands aged. This conserved long-term yield dynamic has direct implications on management. A farm-to-gate economic analysis suggests replanting switchgrass and miscanthus 5 and 9 years following their peak yields maximizes profit over a 30-year time horizon. Results call for further management and breeding strategies to mitigate the yield-decline phase, and for reparameterization of global bioenergy models with carbon capture and storage, which may overestimate yields and the economic and environmental benefits of crops grown for bioenergy feedstocks.

bioenergy↗
DOE OSTI2026

Impacts of Pasture Conversion to Sugarcane on Water Fluxes and Water Use Efficiency in the Southeastern US

The expansion of sugarcane (cane), a high-yielding perennial crop, will likely reshape the bioenergy landscape in the Southeastern US. However, its ecohydrological implications, particularly following conversion from grazed pastures, a dominant land use in the region, remain highly uncertain. We investigated the impact of cane expansion on evapotranspiration (ET) and its partitioning, and the mechanisms influencing both ET components and water use efficiency (WUE) across multiple scales and growth cycles in subtropical Florida. We combined eddy covariance, biometric measurements, and process-based stomatal conductance (g s ) models. ET was 1.7% lower in cane than in improved pasture (IMP) but exceeded that in semi-native pasture (SN) by 21%. Transpiration (T) followed a similar pattern, consistent with lower g s in cane relative to IMP. Cane had more conservative water use and greater sensitivity of g s to vapor pressure deficit (VPD) compared to IMP pasture, suggesting cane may be more tolerant of increasing atmospheric water demand. In contrast, SN showed lower g s and weaker stomatal sensitivity to VPD compared to cane, resulting in lower T. In cane, stomatal regulation and T varied across growth cycles, with stomata becoming less water conservative as stands matured, highlighting the importance of incorporating stand age-dependent stomatal regulation into hydrological models. Evaporation (E) was higher in cane than pastures (19%–26%), partially offsetting WUE gains. Cane exhibited higher intrinsic WUE (GPP/g s ; Gross Primary Productivity), ecosystem WUE (GPP/ET), and harvest WUE (harvest/ET) than both pasture types. Large-scale pasture-to-cane conversion could produce widely contrasting hydrological outcomes. The net regional impact will depend on the proportion of each pasture type converted and on cane's high g s sensitivity to VPD, which triggers tight stomatal regulation and conservative water use, both of which will become increasingly consequential under intensifying atmospheric water demand.

bioenergy↗
DOE OSTI2021

Individual tree damage dominates mortality risk factors across six tropical forests

Summary The relative importance of tree mortality risk factors remains unknown, especially in diverse tropical forests where species may vary widely in their responses to particular conditions. We present a new framework for quantifying the importance of mortality risk factors and apply it to compare 19 risks on 31 203 trees (1977 species) in 14 one‐year periods in six tropical forests. We defined a condition as a risk factor for a species if it was associated with at least a doubling of mortality rate in univariate analyses. For each risk, we estimated prevalence (frequency), lethality (difference in mortality between trees with and without the risk) and impact (‘excess mortality’ associated with the risk, relative to stand‐level mortality). The most impactful risk factors were light limitation and crown/trunk loss; the most prevalent were light limitation and small size; the most lethal were leaf damage and wounds. Modes of death (standing, broken and uprooted) had limited links with previous conditions and mortality risk factors. We provide the first ranking of importance of tree‐level mortality risk factors in tropical forests. Future research should focus on the links between these risks, their climatic drivers and the physiological processes to enable mechanistic predictions of future tree mortality.

Zuleta, Daniel↗
DOE OSTI2022

The influence of increasing atmospheric CO 2 , temperature, and vapor pressure deficit on seawater-induced tree mortality

We report increasing seawater exposure is killing coastal trees globally, with expectations of accelerating mortality with rising sea levels. However, the impact of concomitant changes in atmospheric CO 2 concentration, temperature, and vapor pressure deficit (VPD) on seawater-induced tree mortality is uncertain. We examined the mechanisms of seawater-induced mortality under varying climate scenarios using a photosynthetic gain and hydraulic cost optimization model validated against observations in a mature stand of Sitka-spruce (Picea sitchensis) trees in the Pacific Northwest, USA, that were dying from recent seawater exposure. The simulations matched well with observations of photosynthesis, transpiration, nonstructural carbohydrates concentrations, leaf water potential, the percentage loss of xylem conductivity, and stand-level mortality rates. The simulations suggest that seawater-induced mortality could decrease by ~16.7% with increasing atmospheric CO 2 levels due to reduced risk of carbon starvation. Conversely, rising VPD could increase mortality by ~5.6% because of increasing risk of hydraulic failure. Across all scenarios, seawater-induced mortality was driven by hydraulic failure in the first two years after seawater exposure began, with carbon starvation becoming more important in subsequent years. Changing CO 2 and climate appear unlikely to have a significant impact on coastal tree mortality under rising sea levels.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2022

Scalability Testing Approach for Internet of Things for Manufacturing SQL and NoSQL Database Latency and Throughput

The proliferation of low-cost sensors and industrial data solutions has continued to push the frontier of manufacturing technology. Machine learning and other advanced statistical techniques stand to provide tremendous advantages in production capabilities, optimization, monitoring, and efficiency. The tremendous volume of data gathered continues to grow, and the methods for storing the data are critical underpinnings for advancing manufacturing technology. This work aims to investigate the ramifications and design tradeoffs within a decoupled architecture of two prominent database management systems (DBMS): sql and NoSQL. A representative comparison is carried out with Amazon Web Services (AWS) DynamoDB and AWS Aurora MySQL. The technologies and accompanying design constraints are investigated, and a side-by-side comparison is carried out through high-fidelity industrial data simulated load tests using metrics from a major US manufacturer. The results support the use of simulated client load testing for comparing the latency of database management systems as a system scales up from the prototype stage into production. As a result of complex query support, MySQL is favored for higher-order insights, while NoSQL can reduce system latency for known access patterns at the expense of integrated query flexibility. Here, by reviewing this work, a manufacturer can observe that the use of high-fidelity load testing can reveal tradeoffs in IoTfM write/ingestion performance in terms of latency that are not observable through prototype-scale testing of commercially available cloud DB solutions.

AWS↗
DOE OSTI2023

Investigation of Theoretical Solutions to a Bottom-Raised Oscillating Surge Wave Energy Converter (OSWEC) Through Experimental and Parametric Studies

Experiments were conducted on a wave tank model of a bottom raised oscillating surge wave energy converter (OSWEC) model in regular waves. The OSWEC model shape was a thin rectangular flap, which was allowed to pitch in response to incident waves about a hinge located at the intersection of the flap and the top of the supporting foundation. Torsion springs were added to the hinge in order to position the pitch natural frequency at the center of the wave frequency range of the wave maker. The flap motion as well as the loads at the base of the foundation were measured. The OSWEC was modeled analytically using elliptic functions in order to obtain closed form expressions for added mass and radiation damping coefficients, along with the excitation force and torque. These formulations were derived and reported in a previous publication by the authors. While analytical predictions of the foundation loads agree very well with experiments, large discrepancies are seen in the pitch response close to resonance. These differences are analyzed by conducting a sensitivity study, in which system parameters, including damping and added mass values, are varied. The likely contributors to the differences between predictions and experiments are attributed to tank reflections, standing waves that can occur in long, narrow wave tanks, as well as the thin plate assumption employed in the analytical approach.

analytical↗
DOE OSTI2026

Characterizing degraded visual environments with scanning lenses to improve contrast recovery

Atmospheric aerosols, such as fog and smoke, degrade optical signals via the scattering and absorption of light, making it difficult to recover information from the surrounding environment. Here, we present a technique that measures the angular distribution of light in scattering environments to help recover these degraded optical signals. This is done by passing a collimated beam through the aerosol environment and collecting the scattered and unscattered light with an f-theta scan lens. The lens transforms the angular distribution of the scattered light into the linear domain, mapping the collected light onto a detector. The measured angular scattering distribution is then used to estimate the point spread function of the environment at an arbitrary stand-off distance. This point spread function can be used to deconvolve images blurred by the aerosol environment in which the angular scattering distribution was initially measured. With this approach, we demonstrate improvements in the effective resolution of deblurred images up to 26% over a 120-cm stand-off distance. Experimentation was performed in the Sandia National Laboratories tabletop fog chamber, a platform that generates aerosol environments analogous to real-world fogs. We intend to leverage this technique in future work to develop a low-SWaP system for the in situ characterization of aerosol environments.

Aerosols↗
DOE OSTI2020

Noninteracting electrons in a prototypical one-dimensional sinusoidal potential

A prototypical model of a one-dimensional metallic monatomic solid containing noninteracting electrons is studied, where the argument of the cosine potential energy, periodic with the lattice, contains the first reciprocal lattice vector G 1 =2π/a, where a is the lattice constant. The time-independent Schrödinger equation can be written in reduced variables as a Mathieu equation for which numerically exact solutions for the band structure and wave functions are obtained. The band structure has band gaps that increase with increasing amplitude q of the cosine potential. In the extended-zone scheme, the energy gaps decrease with increasing index n of the Brillouin-zone boundary ka=nπ, where k is the crystal momentum of the electron. The wave functions of the band electron are derived for various combinations of k and q as complex combinations of the real Mathieu functions with even and odd parity, and the normalization factor is discussed. The wave functions at the bottoms and tops of the bands are found to be real or imaginary, respectively, corresponding to standing waves at these energies. Irrespective of the wave vector k within the first Brillouin zone, the electron probability density is found to be periodic with the lattice. The Fourier components of the wave functions are derived versus q, which reveal multiple reciprocal-lattice-vector components with variable amplitudes in the wave functions unless q = 0. The magnitudes of the Fourier components are found to decrease exponentially as a power of n for n~3 to 45 for ka = π/2 and q = 2, and a precise fit is obtained to the data. The probability densities and probability currents obtained from the wave functions are also discussed. The probability currents are found to be zero for crystal momenta at the tops and bottoms of the energy bands, because the wave functions for these crystal momenta are standing waves. To conlcude, the band structure is calculated from the central equation and compared to the numerically exact band structure.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗