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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 1,477 records · Page 82
All sourcesNASA NTRSDOE OSTI
NASA NTRS1995

Spray combustion at normal and reduced gravity in counterflow and co-flow configurations

Liquid fuel dispersion in practical systems is typically achieved by spraying the fuel into a polydisperse distribution of droplets evaporating and burning in a turbulent gaseous environment In view of the nearly insurmountable difficulties of this two-phase flow, a systematic study of spray evaporation and burning in configurations of gradually increasing levels of complexity, starting from laminar sprays to fully turbulent ones, would be useful. A few years ago we proposed to use an electrostatic spray of charged droplets for this type of combustion experiments under well-defined conditions. In the simplest configuration, a liquid is fed into a small metal tube maintained at several kilovolts relative to a ground electrode few centimeters away. Under the action of the electric field, the liquid meniscus at the outlet of the capillary takes a conical shape, with a thin jet emerging from the cone tip (cone-jet mode). This jet breaks up farther downstream into a spray of charged droplets - the so-called ElectroSpray (ES). Several advantages distinguish the electrospray from alternative atomization techniques: (1) it can produce quasi-monodisperse droplets over a phenomenal size range; (2) the atomization, that is strictly electrostatic, is decoupled from gas flow processes, which provides some flexibility in the selection and control of the experimental conditions; (3) the Coulombic repulsion of homopolarly charged droplets induces spray self-dispersion and prevents droplet coalescence; (4) the ES provides the opportunity of studying regimes of slip between droplets and host gas without compromising the control of the spray properties; and (5) the compactness and potential controllability of this spray generation system makes it appealing for studies in reduced-gravity environments aimed at isolating the spray behavior from natural convection complications. With these premises, in March 1991 we initiated a series of experiments under NASA sponsorship (NAG3-1259 and 1688) in which the ES was used as a research tool to examine spray combustion in counter-flow and co-flow spray diffusion flames, as summarized below. The ultimate objective of this investigation is to examine the formation and burning of sprays of liquid fuels, at both normal and reduced gravity, first in laminar regimes and then in turbulent ones.

Gomez, Alessandro↗
NASA NTRSDate not supplied

Impact of Space Radiation with Other Combined Space Environmental Factors on Plant Seeds and Plant Development: from Arabidopsis thaliana to Crops

One of the major concerns for long-term exploration missions beyond Earth’s magnetosphere is radiation risk, primarily from solar particle events (SPE) and galactic cosmic rays (GCR). With the goal of manned Mars exploration, the production of fresh food during long-duration space missions provides critical nutritional supplementation and may also benefit astronauts’ behavioral health. However, the effects of space radiation on plants and plant propagules have not been sufficiently investigated and characterized. To evaluate the effects of space radiation on plant seeds and development, we explored several platforms, including ground-based radiation facilities, the MISSE platform outside the ISS, and other Low Earth Obit (LEO) opportunities. Radiation particle types ranged from neutrons, single charged particles, simulated galactic cosmic rays, simulated solar particle events, the radiation field outside the ISS (total 10 months on the ISS), and the field inside a LEO vehicle (over 3 years). Radiation doses ranged from around 10 cGy to 80 cGy with acute or chronic exposures. In these studies up to 15 types of model plant and crop seeds were evaluated. In addition, tests were conducted to determine whether the combination of radiation exposure and simulated microgravity could have synergetic effects. Exposure to space radiation at the levels tested had no significant impact on the germination rate in Arabidopsis and the seeds from multiple crop plants; however, radiation exposure alone or with other combined factors reduced the viability of some seed types. Overall, the changes within plants grown from irradiated seeds were dose- and ion quality- dependent, with heavier ions causing more severe damage. These changes ranged from cotyledon deformation, shortened root length, smaller seedling size, and other signs of stress, depending on the seed types. Radiation exposure also significantly altered transcriptomic profiles in seedlings grown from irradiated seeds. The impact of space radiation on seeds and plants potentially affects the ability of plants to adapt to other environmental stresses (e.g. microgravity, water stress, and hardware constraints) as well as susceptibility to plant diseases, which need to be furtherly investigated.

Y Zhang↗
DOE OSTI2020

A Synthetic Hydrogel Composite with the Mechanical Behavior and Durability of Cartilage

Here, we report the first hydrogel with the strength and modulus of cartilage in both tension and compression, and the first to exhibit cartilage-equivalent tensile fatigue strength at 100 000 cycles. These properties are achieved by infiltrating a bacterial cellulose (BC) nanofiber network with a poly(vinyl alcohol) (PVA)–poly(2-acrylamido-2-methyl-1-propanesulfonic acid sodium salt) (PAMPS) double network hydrogel. The BC provides tensile strength in a manner analogous to collagen in cartilage, while the PAMPS provides a fixed negative charge and osmotic restoring force similar to the role of aggrecan in cartilage. The hydrogel has the same aggregate modulus and permeability as cartilage, resulting in the same time-dependent deformation under confined compression. The hydrogel is not cytotoxic, has a coefficient of friction 45% lower than cartilage, and is 4.4 times more wear-resistant than a PVA hydrogel. The properties of this hydrogel make it an excellent candidate material for replacement of damaged cartilage.

36 MATERIALS SCIENCE↗
DOE OSTI2026

Probing late-stage hadronic interactions at high baryon density via $K^{⁎0}$ production in the RHIC beam energy scan program

A precision measurement of the $K^{⁎0}$ meson yield is reported in Au+Au collisions at $\sqrt{s_{NN}}$ = 7.7, 11.5, 14.6, 19.6, and 27 GeV using the high-statistics data sample collected by the STAR experiment during the Beam Energy Scan II (BES-II) program at RHIC. The transeverse momentum (p T )-integrated yield ratios $\large{(}K^{⁎0} + \overline{K^{⁎0}}\large{)}/(K^+ + K^{-})$ in central collisions show a suppression relative to peripheral collisions at the (1.7–3.6) σ level, while a thermal model without final-stage rescattering overpredicts this ratio with a deviation of (6.9–8.2) σ. These results indicate a loss of the measured $K^{⁎0}$ signal in central collisions due to re-scattering of its hadronic decay products in the hadronic phase. The p T -integrated yield of charged kaons exhibits an approximate scaling with charged-particle multiplicity, independent of collision energy and system size. A similar trend is observed for the short-lived $K^{⁎0}$ resonance, although significant deviations emerge at lower energies. At BES energies, the $K^{⁎0}/K$ ratio shows stronger suppression than at the highest RHIC and LHC energies within a given multiplicity bin, particularly in central and mid-central collisions. This behavior is consistent with changes in the effective hadronic interaction cross section and is supported by transport model calculations, which indicate dominant meson–baryon interactions at lower energies and meson–meson interactions at higher energies.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2026

Systematic Construction of Time-Dependent Hamiltonians for Microwave-Driven Josephson Circuits

Time-dependent electromagnetic drives are fundamental for controlling complex quantum systems, including superconducting Josephson circuits. In these devices, accurate time-dependent Hamiltonian models are imperative for predicting their dynamics and designing high-fidelity quantum operations. Existing numerical methods, such as black-box quantization (BBQ) and energy-participation ratio (EPR), excel at modeling the static Hamiltonians of Josephson circuits. However, these techniques do not fully capture the behavior of driven circuits stimulated by external microwave drives, nor do they include a generalized approach to account for the inevitable noise and dissipation that enter through microwave ports. Here, we introduce numerical techniques that leverage classical microwave simulations, efficiently executable in finite-element solvers, to obtain the time-dependent Hamiltonian of microwave-driven superconducting circuits with arbitrary geometries under charge, flux, or mixed electromagnetic modulation. Importantly, our techniques do not rely on a lumped-element description of the superconducting circuit, in contrast to previous approaches to tackling this problem. We demonstrate the versatility of our approach by characterizing the driven properties of realistic circuit devices in complex electromagnetic environments, including coherent dynamics due to charge and flux modulation, as well as drive-induced relaxation and dephasing. Our techniques offer a powerful toolbox for optimizing circuit designs and advancing practical applications in superconducting quantum computing.

Lu, Yao [Yale U.; Yale U. (main); Fermilab] (ORCID↗
DOE OSTI2020

Superuniversality from disorder at two-dimensional topological phase transitions

We investigate the effects of quenched randomness on topological quantum phase transitions in strongly interacting two-dimensional systems. We focus first on transitions driven by the condensation of a subset of fractionalized quasiparticles (“anyons”) identified with “electric charge” excitations of a phase with intrinsic topological order. All other anyons have nontrivial mutual statistics with the condensed subset and hence become confined at the anyon condensation transition. Furthermore, using a combination of microscopically exact duality transformations and asymptotically exact real-space renormalization group techniques applied to these two-dimensional disordered gauge theories, we argue that the resulting critical scaling behavior is “superuniversal” across a wide range of such condensation transitions and is controlled by the same infinite-randomness fixed point as that of the 2D random transverse-field Ising model. We validate this claim using large-scale quantum Monte Carlo simulations that allow us to extract zero-temperature critical exponents and correlation functions in (2+1)D disordered interacting systems. We discuss generalizations of these results to a large class of ground-state and excited-state topological transitions in systems with intrinsic topological order as well as those where topological order is either protected or enriched by global symmetries. When the underlying topological order and the symmetry group are Abelian, our results provide prototypes for topological phase transitions between distinct many-body localized phases.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2026

Multi-physics Preconditioning for Thermally Activated Batteries

Thermal batteries, also known as molten-salt batteries, are single-use reserve power systems activated by pyrotechnic heat generation, which transitions the solid electrolyte into a molten state. The simulation of these batteries relies on multiphysics modeling to evaluate performance and behavior under various conditions. This paper presents advancements in scalable preconditioning strategies for the Thermally Activated Battery Simulator (TABS) tool, enabling efficient solutions to the coupled electrochemical systems that dominate computational costs in thermal battery simulations. We propose a hierarchical block Gauss-Seidel preconditioner implemented through the Teko package in Trilinos, which effectively addresses the challenges posed by tightly coupled physics, including charge transport, porous flow, and species diffusion. The preconditioner leverages scalable subblock solvers, including smoothed aggregation algebraic multigrid (SA-AMG) methods and domain-decomposition techniques, to achieve robust convergence and parallel scalability. Strong and weak scaling studies demonstrate the solver’s ability to handle problem sizes up to 51.3 million degrees of freedom on 2048 processors, achieving near sub-second setup and solve times for the end-to-end electrochemical solve. These advancements significantly improve the computational efficiency and turnaround time of thermal battery simulations, paving the way for higher-resolution models and enabling the transition from 2D axisymmetric to full 3D simulations.

25 ENERGY STORAGE↗
DOE OSTI2026

Detecting and Characterizing Fracture Zones Using a Convolutional Neural Network

This project directly supports the Geothermal Technologies Office (GTO) objectives outlined in the Multi-Year Program Plan (MYPP) by advancing two key research areas: “Exploration and Characterization” and “Data, Modeling, and Analysis.” This project has successfully demonstrated a pre-drilling ability to image and characterize the distribution and connectivity of subsurface faults and fractures, key parameters for identifying permeable pathways that enable geothermal fluids to circulate and produce energy. Specifically, we developed and implemented innovative machine learning methodologies to enhance geothermal exploration. Large-scale faults were detected using a Convolutional Neural Network (CNN), while small-scale fractures were characterized using a novel Double-Beam Neural Network (DBNN). These tools have proven both technically effective and cost-efficient by reducing reliance on expensive exploratory drilling. Through collaboration with our geothermal industry partner, this research has significantly advanced techniques for identifying hidden geothermal systems and extending the productive lifespan of existing geothermal fields. We applied our methods to two geothermal fields—Soda Lake (Nevada) and Lightning Dock (New Mexico)—to identify shallow steam-charged fracture zones and characterize deep faults at depths of 1.5-2 km. The steam zone identified at the Soda Lake geothermal field showed excellent agreement with prior drilling data, validating the effectiveness of our approaches. In addition, the analysis revealed three new prospective drilling targets for further development and verification. The outcomes of this project improve our scientific understanding of geothermal reservoir behavior, enhance exploration efficiency, extend the economic life of existing geothermal plants. Ultimately, these advancements contribute to GTO’s goal of achieving more sustainable, affordable, and data-driven geothermal energy development across the United States.

15 GEOTHERMAL ENERGY↗
DOE OSTI2021

Charge doping effects on magnetic properties of single-crystal La 1-x Sr x (Cr 0.2 Mn 0.2 Fe 0.2 Co 0.2 Ni 0.2 )O 3 (0 ≤ x ≤ 0.5) high-entropy perovskite oxides

The observed influence of hole doping on magnetic properties is mapped for the compositionally complex high-entropy oxide ABO 3 perovskite La 1-x Sr x (Cr 0.2 Mn 0.2 Fe 0.2 Co 0.2 Ni 0.2 )O 3 (0 ≤ x ≤ 0.5). It is found that aliovalent A -site substitution is a viable means to manipulate the magnetically active B -site sublattice. A series of single-crystal films are synthesized and show a general trend toward stronger ferromagnetic response and a shift in magnetic anisotropy as the Sr concentration increases. Magnetometry demonstrates complex and nonuniform responses similar to rigid and uncoupled composites at intermediate dopings. This behavior points to the presence of locally inhomogeneous magnetic phase competition, where ferromagnetic and antiferromagnetic magnetic contributions create a frustrated matrix containing uncompensated spins at the boundaries between these regions. The observations are discussed in the context of known responses to hole doping in the less complex ternary LaT M O 3 (T M =Cr,Mn,Fe,Co,Ni) oxides, and they are found to be different from a simple sum of the doped parents. The results are summarized in a preliminary magnetic phase diagram.

36 MATERIALS SCIENCE↗
DOE OSTI2023

Observation of a massive phason in a charge-density-wave insulator

The lowest-lying fundamental excitation of an incommensurate charge density wave (CDW) material is widely believed to be a massless phason -- a collective modulation of the phase of the CDW order parameter. However, as first pointed out by Lee and Fukuyama, long-range Coulomb interactions should push the phason energy up to the plasma energy of the CDW condensate, resulting in a massive phason and a fully gapped spectrum. Whether such behavior occurs in a CDW system has been unresolved for more than four decades. Using time-domain THz emission spectroscopy, we investigate this issue in the material (TaSe 4 ) 2 I, a classical example of a quasi-one-dimensional CDW insulator. Upon transient photoexcitation at low temperatures, we find the material strikingly emits coherent, narrow-band THz radiation. The frequency, polarization and temperature-dependence of the emitted radiation imply the existence of a phason that acquires mass by coupling to long-range Coulomb interaction. Our observations constitute the first direct evidence of the massive "Lee-Fukuyama" phason and highlight the potential applicability of fundamental collective modes of correlated materials as compact and robust sources of THz radiation.

36 MATERIALS SCIENCE↗
DOE OSTI2024

Effects of Na deficiency on spin dynamics in the Mott insulating Na 4−x Ir 3 O 8

Here, we report on Raman scattering and 23 Na nuclear magnetic resonance (NMR) investigations of j eff = 1/2 hyperkagome antiferromagnet Na 4−x Ir 3 O 8 , which lies in the Mott insulating state. Our Raman scattering experiments unveil remarkable parallels between the magnetic excitations of the pristine Na 4 Ir 3 O 8 and the Kitaev honeycomb material Na 2 IrO 3 , which are characterized by dominant fractional spinon excitations. In the case of moderate Na-deficient Na 4−x Ir 3 O 8 , however, a substantial suppression of the magnetic excitations is observed, alluding to the notable influence of charge fluctuations on spin dynamics. In addition, our site-specific 23 Na NMR measurements offer further insights into the spin dynamics when a minor concentration of holes is introduced into a spin-orbit coupled Mott insulator. Specifically, the spin-lattice relaxation rate 1/ T 1 reveals the emergence of pseudogaplike correlations at the Na(2) site, alongside a critical slowing down behavior at the Na(1) and Na(3) sites. These findings showcase the intricate interplay between itinerant holes and magnetic correlations in a spin-liquid-like background.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
NASA NTRS1990

Hypervelocity atmospheric flight: Real gas flow fields

Flight in the atmosphere is examined from the viewpoint of including real gas phenomena in the flow field about a vehicle flying at hypervelocity. That is to say, the flow field is subject not only to compressible phenomena, but is dominated by energetic phenomena. There are several significant features of such a flow field. Spatially, its composition can vary by both chemical and elemental species. The equations which describe the flow field include equations of state and mass, species, elemental, and electric charge continuity; momentum; and energy equations. These are nonlinear, coupled, partial differential equations that were reduced to a relatively compact set of equations of a self-consistent manner (which allows mass addition at the surface at a rate comparable to the free-stream mass flux). The equations and their inputs allow for transport of these quantities relative to the mass-averaged behavior of the flow field. Thus transport of mass by chemical, thermal, pressure, and forced diffusion; transport of momentum by viscosity; and transport of energy by conduction, chemical considerations, viscosity, and radiative transfer are included. The last of these complicate the set of equations by making the energy equation a partial integrodifferential equation. Each phenomenon is considered and represented mathematically by one or more developments. The coefficients which pertain are both thermodynamically and chemically dependent. Solutions of the equations are presented and discussed in considerable detail, with emphasis on severe energetic flow fields. For hypervelocity flight in low-density environments where gaseous reactions proceed at finite rates, chemical nonequilibrium is considered and some illustrations are presented. Finally, flight where the flow field may be out of equilibrium, both chemically and thermodynamically, is presented briefly.

Howe, John T.↗
DOE OSTI2022

Correlated Insulating Behavior in Infinite-Layer Nickelates

Unlike their cuprate counterparts, the undoped nickelates are weak insulators without long-range antiferromagnetic order. Identifying the origin of this insulating behavior, found on both sides of the superconducting dome, is potentially a crucial step in the development of a coherent understanding of nickelate superconductivity. In this work, we study the normal-state resistivity of infinite-layer nickelates using high magnetic fields to suppress the superconductivity and examine the impact of disorder and doping on its overall temperature ( T ) dependence. In superconducting samples, the resistivity of Nd- and La-based nickelates continues to exhibit weakly insulating behavior with a magnitude and functional form similar to that found in underdoped electron-doped cuprates. We find a systematic evolution of the insulating behavior as a function of nominal hole doping across different rare-earth families, suggesting a pivotal role for strong electron interactions, and uncover a correlation between the suppression of the resistivity upturn and the robustness of the superconductivity. By contrast, we find very little correlation between the level of disorder and the magnitude and onset temperature of the resistivity upturn. Combining these experimental observations with previous Hall effect measurements on these two nickelate families, we consider various possible origins for this correlated insulator behavior and its evolution across their respective phase diagrams.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2022

Partitioning behavior and mechanisms of rare earth elements during precipitation in acid mine drainage

Rare earth elements (REEs) are frequently found concentrated in acid mine drainage (AMD). The recovery of REEs from AMD has been successfully achieved using selective chemical precipitation. However, a portion of the REEs is often lost to the precipitates of the dominant metal contaminant ions. To better understand the REE partitioning behavior and mechanisms during the precipitation process, a systematic study was performed on both natural and synthetic AMD solutions. Precipitation test results show that REE removal was noticeably elevated at pH 4.0 after adding H 2 O 2 to convert ferrous to ferric ions, causing nearly complete precipitation of iron. Solution equilibrium calculations suggested that the REE removal increase was realized through adsorption onto the surfaces of the ferric precipitates. The presence of aluminum species in the solutions reduced the adsorption of REEs on the ferric precipitates. Based on electro-kinetic test results, it was concluded that aluminum species neutralize the negative surface charge of the ferric precipitates and compete with REEs for the adsorption active sites. The presence of ferrous ions in the solutions reduced REE adsorption on the aluminum precipitates at lower pH values (e.g., 5.0) due to competitive adsorption. However, at higher pH values (e.g., 6.0), REE removal to the precipitate product increased due to the precipitation of ferrous ions. Additionally, to the electro-kinetic tests and solution equilibrium calculations, mineralogy characterization, specific surface area measurement, particle size analysis, and morphology analysis were also conducted to investigate and identify the partitioning mechanisms.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2022

Intrinsic and doping-enhanced superconductivity in monolayer 1 H − TaS 2 : Critical role of charge ordering and spin-orbit coupling

The interplay of superconductivity with charge density wave (CDW) in metallic transition-metal dichalcogenides has been widely debated, and viable strategies manipulating these quantum states in the two-dimensional (2D) limit remain unclear. Using the ab initio anisotropic Migdal-Eliashberg theory, we successfully explain the superconductivity observed in monolayer 1H-TaS 2 by simultaneously determining its precise CDW structure and treating the marked modification of electron-phonon interaction and critical temperature T c by spin-orbit coupling effects. With this paradigm, we further show that electron doping weakens the CDW order leading to increased T c up to 11 K, along with a single-gap to two-gap superconductivity transition due to the suppression of the CDW gap. By contrast, a low hole doping barely affects the CDW but still yields a significantly enhanced superconducting order, implying their good coexistence. Combined with the synergistic behavior of CDW and superconductivity, which cooperate upon TaS 2 thickness reduction causing an unusual rise of T c , our results unravel diversified interactions between the two collective orders in ultrathin TaS 2 , being competition, coexistence or cooperation depending on external stimuli, which provide key clues for controlling correlated states in devices based on 2D CDW superconductors.

36 MATERIALS SCIENCE↗
DOE OSTI2021

Role of Decomposition Product Ions in Hysteretic Behavior of Metal Halide Perovskite

Ion migration is one of the most debated mechanisms and credited with multiple observed phenomena and performance in metal halide perovskites (MHPs) semiconductor devices. However, to date, the migration of ions and their effects on MHPs are not still fully understood, largely due to a lack of direct observations of temporal ion migration. In this work, using direct observation of ion migration in-operando, we observe the hysteretic migration behavior of intrinsic ions (i.e., CH 3 NH 3 + and I – ) as well as reveal the migration behavior of CH 3 NH 3 + decomposition ions. We find that CH 3 NH 3 + decomposition products can be affected by light and accumulate at the interfaces under bias. These MHP decomposition products are tightly related to the device performance and stability. Complementary results of time-resolved Kelvin probe force microscopy (tr-KPFM) demonstrate a correlation between dynamics of these interfacial ions and charge carriers. Overall, we find that there are a number of mobile ions including CH 3 NH 3 + decomposition products in MHPs that need to be taken into account when measuring MHP device responses (e.g., charge dynamics) and should be considered in future optimization studies of MHP semiconductor devices.

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

Ultrafast spin-nematic and ferroelectric phase transitions induced by femtosecond light pulses

We report, optically induced phase transitions of the manganite Pr 1/3 Ca 2/3 MnO 3 have been simulated by using a model Hamiltonian that captures the dynamics of strongly correlated charge, orbital, lattice, and spin degrees of freedom. Its parameters have been extracted from first-principles calculations. Beyond a critical intensity of a femtosecond light pulse, the material undergoes an ultrafast and nonthermal magnetic phase transition from a noncollinear to collinear antiferromagnetic phase. The light-pulse excites selectively either a spin-nematic or a ferroelectric phase, depending on the light polarization. The behavior can be traced to an optically induced ferromagnetic coupling between Mn trimers, i.e., polarons which are delocalized over three Mn sites. The polarization guides the polymerization of the polaronic crystal into distinct patterns of ferromagnetic chains determining the target phase.

36 MATERIALS SCIENCE↗
DOE OSTI2021

Magnetic properties of alternating Hubbard ladders

We explore the Hubbard Hamiltonian on ladders where the number of sites per rung alternates between two and three. These geometries are bipartite with nonequal or equal number of sites on the two sublattices. Thus they share a key feature of the Hubbard model in a class of lattices which Lieb has shown analytically to exhibit long-range ferrimagnetic order while being amenable to powerful numeric approaches developed for quasi-one-dimensional geometries. The density matrix renormalization group (DMRG) method is used to obtain the groundstate properties, e.g., excitation gaps, charge and spin densities as well as their correlation functions at half filling. We show the existence of long-range ferrimagnetic order in the one-dimensional ladder geometries. Our work provides detailed quantitative results which complement the general theorem of Lieb for generalized bipartite lattices. It also addresses the issue of how the alternation between quasi-long-range order and spin liquid behavior for uniform ladders with odd and even numbers of legs might be affected by a regular alternation pattern.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗