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NASA NTRS2018

Overview of Methods for Deriving the Radiative Transfer Theory from the Maxwell Equations. I: Approach Based on the Far-Field Foldy Equations

In this paper, we revisit, with further enhancements and clarifications, the self-consistent first-principles approach developed previously for deriving the vector radiative transfer theory for a discrete random medium with a sparse concentration of particles. We specifically consider the case of a plane-parallel particulate layer embedded in an otherwise homogeneous unbounded medium. The solution method is based on the far-field Foldy equations, an order-of-scattering expansion for the total field derived under the Twersky approximation, the computation of the coherent field by assuming that the positions of the particles are uncorrelated, and the ladder approximation for the coherency dyadic. The latter yields an integral equation for the diffuse specific coherency dyadic, defined through an angular spectrum repre- sentation for the coherency dyadic, which in turn, gives the vector radiative transfer equation for the diffuse specific intensity column vector. We analyze specifically the computation of the coherent field for inhomogeneous particulate media and multiple species of particles, the continuous extension of the far-field representation to the near field, the Foldy approximation, and the Foldy integral equation for the coherent field. Finally, we discuss the transition from the vector to the scalar radiative transfer equation.

Doicu, Adrian↗

NASA NTRS2012

Desert Research and Technology Studies (DRATS) 2010 Science Operations: Operational Approaches and Lessons Learned for Managing Science during Human Planetary Surface Missions

Desert Research and Technology Studies (Desert RATS) is a multi-year series of hardware and operations tests carried out annually in the high desert of Arizona on the San Francisco Volcanic Field. These activities are designed to exercise planetary surface hardware and operations in conditions where long-distance, multi-day roving is achievable, and they allow NASA to evaluate different mission concepts and approaches in an environment less costly and more forgiving than space.The results from the RATS tests allows election of potential operational approaches to planetary surface exploration prior to making commitments to specific flight and mission hardware development. In previous RATS operations, the Science Support Room has operated largely in an advisory role, an approach that was driven by the need to provide a loose science mission framework that would underpin the engineering tests. However, the extensive nature of the traverse operations for 2010 expanded the role of the science operations and tested specific operational approaches. Science mission operations approaches from the Apollo and Mars-Phoenix missions were merged to become the baseline for this test. Six days of traverse operations were conducted during each week of the 2-week test, with three traverse days each week conducted with voice and data communications continuously available, and three traverse days conducted with only two 1-hour communications periods per day. Within this framework, the team evaluated integrated science operations management using real-time, tactical science operations to oversee daily crew activities, and strategic level evaluations of science data and daily traverse results during a post-traverse planning shift. During continuous communications, both tactical and strategic teams were employed. On days when communications were reduced to only two communications periods per day, only a strategic team was employed. The Science Operations Team found that, if communications are good and down-linking of science data is ensured, high quality science returns is possible regardless of communications. What is absent from reduced communications is the scientific interaction between the crew on the planet and the scientists on the ground. These scientific interactions were a critical part of the science process and significantly improved mission science return over reduced communications conditions. The test also showed that the quality of science return is not measurable by simple numerical quantities but is, in fact, based on strongly non-quantifiable factors, such as the interactions between the crew and the Science Operations Teams. Although the metric evaluation data suggested some trends, there was not sufficient granularity in the data or specificity in the metrics to allow those trends to be understood on numerical data alone.

Eppler, Dean↗

NASA NTRS2017

Advanced Stirling Convertor Control Unit Testing at NASA Glenn Research Center in the Radioisotope Power Systems System Integration Laboratory

Future NASA missions could include establishing Lunar or Martian base camps, exploring Jupiters moons and travelling beyond where generating power from sunlight may be limited. Radioisotope Power Systems (RPS) provide a dependable power source for missions where inadequate sunlight or operational requirements make other power systems impractical. Over the past decade, NASA Glenn Research Center (GRC) has been supporting the development of RPSs. The Advanced Stirling Radioisotope Generator (ASRG) utilized a pair of Advanced Stirling Convertors (ASC). While flight development of the ASRG has been cancelled, much of the technology and hardware continued development and testing to guide future activities. Specifically, a controller for the convertor(s) is an integral part of a Stirling-based RPS. For the ASRG design, the controller maintains stable operation of the convertors, regulates the alternating current produced by the linear alternator of the convertor, provides a specified direct current output voltage for the spacecraft, synchronizes the piston motion of the two convertors in order to minimize vibration as well as manage and maintain operation with a stable piston amplitude and hot end temperature. It not only provides power to the spacecraft but also must regulate convertor operation to avoid damage to internal components and maintain safe thermal conditions after fueling. Lockheed Martin Coherent Technologies has designed, developed and tested an Engineering Development Unit (EDU) Advanced Stirling Convertor Control Unit (ACU) to support this effort. GRC used the ACU EDU as part of its non-nuclear representation of a RPS which also consists of a pair of Dual Advanced Stirling Convertor Simulator (DASCS), and associated support equipment to perform a test in the Radioisotope Power Systems System Integration Laboratory (RSIL). The RSIL was designed and built to evaluate hardware utilizing RPS technology. The RSIL provides insight into the electrical interactions between as many as 3 radioisotope power generators, associated control strategies, and typical electric system loads. The first phase of testing included a DASCS which was developed by Johns Hopkins UniversityApplied Physics Laboratory and simulates the operation and electrical behavior of a pair of ASCs in real time via a combination of hardware and software. Testing included the following spacecraft electrical energy storage configurations: capacitive, battery, and supercapacitor. Testing of the DASCS and ACU in each energy storage configuration included simulation of a typical mission profile, and transient voltage and current data during load turn-on/turn-off. Testing for these devices also included the initiation of several system faults such as short circuits, electrical bus over-voltage, under-voltage and a dead bus recovery to restore normal power operations. The goal of this testing was to verify operation of the ACU(s) when connected to a spacecraft electrical bus.

Dugala, Gina M.↗

NASA NTRS2004

AIN-Based Packaging for SiC High-Temperature Electronics

Packaging made primarily of aluminum nitride has been developed to enclose silicon carbide-based integrated circuits (ICs), including circuits containing SiC-based power diodes, that are capable of operation under conditions more severe than can be withstood by silicon-based integrated circuits. A major objective of this development was to enable packaged SiC electronic circuits to operate continuously at temperatures up to 500 C. AlN-packaged SiC electronic circuits have commercial potential for incorporation into high-power electronic equipment and into sensors that must withstand high temperatures and/or high pressures in diverse applications that include exploration in outer space, well logging, and monitoring of nuclear power systems. This packaging embodies concepts drawn from flip-chip packaging of silicon-based integrated circuits. One or more SiC-based circuit chips are mounted on an aluminum nitride package substrate or sandwiched between two such substrates. Intimate electrical connections between metal conductors on the chip(s) and the metal conductors on external circuits are made by direct bonding to interconnections on the package substrate(s) and/or by use of holes through the package substrate(s). This approach eliminates the need for wire bonds, which have been the most vulnerable links in conventional electronic circuitry in hostile environments. Moreover, the elimination of wire bonds makes it possible to pack chips more densely than was previously possible.

Savrun, Ender↗

NASA NTRS1990

Numerical modeling of the atmosphere with an isentropic vertical coordinate

A theta-coordinate model simulating the nonlinear evolution of a baroclinic wave is presented. In the model, vertical discretization maintains important integral constraints such as conservation of the angular momentum and total energy. A massless-layer approach is used in the treatment of the intersections of coordinate surfaces with the lower boundary. This formally eliminates the intersection problem, but raises other computational problems. Horizontal discretization of the continuity and momentum equations in the model are designed to overcome these problems. Selected results from a 10-day integration with the 25-layer, beta-plane version of the model are presented. It is concluded that the model can simulate the nonlinear evolution of a baroclinic wave and associated dynamical processes without major computational difficulties.

Hsu, Yueh-Jiuan G.↗

DOE OSTI2023

Continuous-flow reactor with superior production rate and stability for CO 2 reduction using semiconductor photocatalysts

Semiconductor photocatalyst approaches for solar CO 2 reduction are attractive due to their simplicity but have lagged in efficiency compared to less-integrated photoelectrochemical (PEC) approaches and to electrolysis reactors. We identify poor mass transport and catalyst deactivation as key constraints. To address them, we have developed a continuous-flow photocatalytic reactor system allowing us to control the triple-phase interface on the photocatalyst surface using the liquid and reactant gas flow rates. With the goal of selectively producing CO, the reactor is optimized by controlling the pressure and flow rates of the reactant gas and electrolyte in contact with both sides with the intermediately placed catalyst. In comparison to batch reactors with an immobile photocatalyst bed and gas phase CO 2 or CO 2 purged water, 10–24 times higher production rates are achieved for photocatalysts such as TiO 2 , ZnO, C 3 N 4 , and CdS by simply changing to the designed flow-type photoreactor without any catalyst modification. In addition, CO selectivity (93.2%) and long-term stability (>780 min) using the designed reactor are significantly enhanced compared to using the batch reactors (71.7%, <180 min for reduced 50% activity). Here, we propose that the enhanced mass transport on the photocatalyst surface accelerates the desorption of the initial photolysis product, CO, and prevents the poisoning effect from deactivating photocatalyst activity. This study has the potential to facilitate the utilization of semiconductor-based photocatalytic reactions for achieving superior performance wih gaseous reactants.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2023

Office of Nuclear Verification FY 2023 Quarterly Report TSVT Q1

This project encompasses the continued development and training of a U.S. operational team, the Test Site Verification Team (TSVT), supporting verification of nuclear testing activities. TSVT builds on decades of U.S. nuclear testing history and nuclear explosion monitoring experience. The Team maintains readiness to deploy internationally on short notice to provide field-based support of verification of declared or undeclared nuclear testing and associated activities, as well as follow-on activities including monitoring and capability disablement and dismantlement, as established by negotiated agreement or treaty. The roles and structure of the TSVT are integrated with other NA-243 deployable verification teams and the interagency. FY23 TSVT activities focus on continued capability buildup within the team, including Team trainings and exercises with a focus on missions in confined spaces (e.g., tunnels, mines, other underground facilities), increased familiarity with foreign nuclear weapons testing programs, demonstration of Team capacity to deploy, train, and practice sustained OPSEC in non-western locations, establishment of sustainable storage and maintenance of equipment, specification and procurement of additional equipment to support field observations, further evolution of concepts of operation documents (CONOPs), and mission coordination with Headquarters and associated Stakeholders. Activities will culminate with a full-scale domestic team exercise at the end of FY23 focusing on underground activities signatures/observations and safety including advanced outdoor safety and familiarity in working around explosive test environments. In addition, we will be further articulating approaches and capacity relevant to the identification of nuclear tests, as well as monitoring of nuclear testing activities and/or dismantlement of nuclear test sites and anticipate developing additional equipment requests in support of this evolution. The TSVT Team Leads will also coordinate with its Senior Advisor, the TSVT Logistics and Readiness (L&R) Training Lead, and NNSA Headquarters to draft a five-year TSVT training and exercise plan, that strategically and incrementally builds capacity and expertise in key areas of significance for the continuum of nuclear and nuclear-related testing activities that fall within the team’s mission space.

42 ENGINEERING↗

DOE OSTI2026

State of Innovation 2025: Progress in Accelerating Next-Generation Cement and Concrete Technologies

The cement and concrete sectors are entering a decisive period as next-generation technologies advance from laboratory research to demonstration, early deployment, and first-of-a-kind commercial plants. Building on the 2024 State of Innovation report, the 2025 outlook highlights both the rapid acceleration of innovation and the urgent need for coordinated action across the value chain. Venture capital activity into the cement and concrete space stabilized following the record surge of 2022-2023, yet landmark financings, such as Sublime Systems' $200 million round and Terra CO2's $124 million Series B, signal continued investor confidence in companies approaching commercialization. Corporate procurement has become a powerful new catalyst, with Microsoft, Amazon, and CRH (Cement Roadstone Holdings) Ventures providing long-term commitments that underpin the first wave of next-generation cementitious products. The sector is shifting from early-stage experimentation toward the scaling of well-capitalized leaders capable of bridging the critical "capitalization gap." Early innovators continue to expand the toolkit through novel binders, electrochemical cements, biogenic limestone, and carbonate mineralization pathways. Going into 2026, cost competitiveness, durability validation, and scalability enabled by resilient supply chains remain the decisive factors for market adoption. At the 2025 Next Generation Cement and Concrete Critical Technologies Meeting, attendees emphasized dual-track funding strategies that integrate federal grants with private capital as key to enabling market breakthrough. State programs and corporate demand are sustaining momentum, while successful companies increasingly demonstrate both economic value and reduced dependence on imported materials. The National Concrete Pavement Technology Center and others underscored that broad integration of next-generation materials will hinge on standards compatibility, verified field performance, and workforce readiness. Colorado continues to serve as a proving ground through pilot programs that combine supplier training, phased implementation, and real-world data to de-risk innovation and provide replicable models for other regions. The 2025 Cement and Concrete Critical Technologies Workshop reinforced that scaling next-generation materials will require alignment among technology innovation, performance validation, and market demand. Stakeholders must move beyond siloed efforts toward collaborative frameworks that coordinate standards, funding, and infrastructure deployment. As a neutral convener and technical validator, the National Laboratory of the Rockies (NLR) plays a pivotal role in bridging innovation and market adoption through collaborative research, technology validation, and entrepreneurship programs. By uniting innovators, incumbents, policymakers, contractors, and investors, NLR and its partners are helping chart a credible pathway toward widespread commercialization in the decade ahead.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗

DOE OSTI2024

Integrating crystallographic and computational approaches to carbon-capture materials for the mitigation of climate change

Here, this article presents an overview of the current state of the art in the structure determination of microporous carbon-capture materials, as discussed at the recent NIST workshop “Integrating Crystallographic and Computational Approaches to Carbon-Capture Materials for the Mitigation of Climate Change”. The continual rise in anthropogenic CO 2 concentration and its effect on climate change call for the implementation of carbon capture technologies to reduce the CO 2 concentration in the atmosphere. Porous solids, including metal–organic frameworks (MOFs), are feasible candidates for gas capture and storage applications. However, determining the structure of these materials represents a significant obstacle in their development into advanced sorbents. The existing difficulties can be overcome by integrating crystallographic methods and theoretical modeling. The workshop gathered experimentalists and theorists from academia, government, and industry to review this field and identify approaches, including collaborative opportunities, required to develop tools for rapid determination of the structures of porous solid sorbents and the effect of structure on the carbon capture performance. We highlight the findings of that workshop, especially in the need for reference materials, standardized procedures and reporting of sorbent activation and adsorption measurements, standardized reporting of theoretical calculations, and round-robin structure determination.

36 MATERIALS SCIENCE↗

NASA NTRS1992

A boundary integral method for an inverse problem in thermal imaging

An inverse problem in thermal imaging involving the recovery of a void in a material from its surface temperature response to external heating is examined. Uniqueness and continuous dependence results for the inverse problem are demonstrated, and a numerical method for its solution is developed. This method is based on an optimization approach, coupled with a boundary integral equation formulation of the forward heat conduction problem. Some convergence results for the method are proved, and several examples are presented using computationally generated data.

Bryan, Kurt↗

NASA NTRS1993

Advanced composite fuselage technology

Boeing's ATCAS program has completed its third year and continues to progress towards a goal to demonstrate composite fuselage technology with cost and weight advantages over aluminum. Work on this program is performed by an integrated team that includes several groups within The Boeing Company, industrial and university subcontractors, and technical support from NASA. During the course of the program, the ATCAS team has continued to perform a critical review of composite developments by recognizing advances in metal fuselage technology. Despite recent material, structural design, and manufacturing advancements for metals, polymeric matrix composite designs studied in ATCAS still project significant cost and weight advantages for future applications. A critical path to demonstrating technology readiness for composite transport fuselage structures was created to summarize ATCAS tasks for Phases A, B, and C. This includes a global schedule and list of technical issues which will be addressed throughout the course of studies. Work performed in ATCAS since the last ACT conference is also summarized. Most activities relate to crown quadrant manufacturing scaleup and performance verification. The former was highlighted by fabricating a curved, 7 ft. by 10 ft. panel, with cocured hat-stiffeners and cobonded J-frames. In building to this scale, process developments were achieved for tow-placed skins, drape formed stiffeners, braided/RTM frames, and panel cure tooling. Over 700 tests and supporting analyses have been performed for crown material and design evaluation, including structural tests that demonstrated limit load requirements for severed stiffener/skin failsafe damage conditions. Analysis of tests for tow-placed hybrid laminates with large damage indicates a tensile fracture toughness that is higher than that observed for advanced aluminum alloys. Additional recent ATCAS achievements include crown supporting technology, keel quadrant design evaluation, and sandwich process development.

Ilcewicz, Larry B.↗

NASA NTRS1995

Increasing cellular coverage within integrated terrestrial/satellite mobile networks

When applying the hierarchical cellular concept, the satellite acts as giant umbrella cell covering a region with some terrestrial cells. If a mobile terminal traversing the region arrives to the border-line or limits of a regular cellular ground service, network transition occurs and the satellite system continues the mobile coverage. To adequately assess the boundaries of service of a mobile satellite system an a cellular network within an integrated environment, this paper provides an optimized scheme to predict when a network transition may be necessary. Under the assumption of a classified propagation phenomenon and Lognormal shadowing, the study applies an analytical approach to estimate the location of a mobile terminal based on a reception of the signal strength emitted by a base station.

Castro, Jonathan P.↗

NASA NTRS1990

ECLSS Integration Analysis: Advanced ECLSS Subsystem and Instrumentation Technology Study for the Space Exploration Initiative

In his July 1989 space policy speech, President Bush proposed a long range continuing commitment to space exploration and development. Included in his goals were the establishment of permanent lunar and Mars habitats and the development of extended duration space transportation. In both cases, a major issue is the availability of qualified sensor technologies for use in real-time monitoring and control of integrated physical/chemical/biological (p/c/b) Environmental Control and Life Support Systems (ECLSS). The purpose of this study is to determine the most promising instrumentation technologies for future ECLSS applications. The study approach is as follows: 1. Precursor ECLSS Subsystem Technology Trade Study - A database of existing and advanced Atmosphere Revitalization (AR) and Water Recovery and Management (WRM) ECLSS subsystem technologies was created. A trade study was performed to recommend AR and WRM subsystem technologies for future lunar and Mars mission scenarios. The purpose of this trade study was to begin defining future ECLSS instrumentation requirements as a precursor to determining the instrumentation technologies that will be applicable to future ECLS systems. 2. Instrumentation Survey - An instrumentation database of Chemical, Microbial, Conductivity, Humidity, Flowrate, Pressure, and Temperature sensors was created. Each page of the sensor database report contains information for one type of sensor, including a description of the operating principles, specifications, and the reference(s) from which the information was obtained. This section includes a cursory look at the history of instrumentation on U.S. spacecraft. 3. Results and Recommendations - Instrumentation technologies were recommended for further research and optimization based on a consideration of both of the above sections. A sensor or monitor technology was recommended based on its applicability to future ECLS systems, as defined by the ECLSS Trade Study (1), and on whether its characteristics were considered favorable relative to similar instrumentation technologies (competitors), as determined from the Instrumentation Survey (2). The instrumentation technologies recommended by this study show considerable potential for development and promise significant returns if research efforts are invested.

Source record↗

DOE OSTI2024

Considerations for a Co-Located Aquaculture and Wave Energy Deployment: Quantifying Demand and Assessing Integration

The co-location of marine energy and aquaculture is a concept of increasing value and interest in the United States, as the desire for sustainably produced seafood and renewable energy continues to grow. With both industries being fairly nascent in their national development and facing challenges, the deployment of a wave-powered aquaculture farm requires numerous considerations. This report analyzes the energy demands of an integrated multi-trophic aquaculture system at various farm scales and discusses the preliminary technical, regulatory, and logistical considerations required for a co-located deployment. A pilot-scale deployment is outlined with potentially ideal characteristics, and an optimized planning and permitting approach is introduced.

16 TIDAL AND WAVE POWER↗

NASA NTRS1992

A new approach to the digital redesign of continuous-time controllers

This paper presents a new method for digital redesign of both a continuous-time state-feedback control law and a cascaded analog controller. The concept of the Law of Mean from the integral calculus is utilized for the development of the equivalent digital controllers from the available analog controllers. A tuning parameter has been introduced into the digitally redesigned control gains so that the digitally controlled states will closely match the original continuous-time states. The proposed digital controllers give excellent performance.

Shieh, Leang-San↗

DOE OSTI2021

Ab Initio Direct Dynamics

The reactivity and dynamics of molecular systems can be explored computationally by classical trajectory calculations. The traditional approach involves fitting a functional form of a potential energy surface (PES) to the energies from a large number of electronic structure calculations and then integrating numerous trajectories on this fitted PES to model the molecular dynamics. The ever-decreasing cost of computing and continuing advances in computational chemistry software have made it possible to use electronic structure calculations directly in molecular dynamics simulations without first having to construct a fitted PES. In this “on-the-fly” approach, every time the energy and its derivatives are needed for the integration of the equations of motion, they are obtained directly from quantum chemical calculations. This approach started to become practical in the mid-1990s as a result of increased availability of inexpensive computer resources and improved computational chemistry software. The application of direct dynamics calculations has grown rapidly over the last 25 years and would require a lengthy review article. The present Account is limited to some of our contributions to methods development and various applications. To improve the efficiency of direct dynamics calculations, we developed a Hessian-based predictor-corrector algorithm for integrating classical trajectories. Hessian updating made this even more efficient. Furthermore, this approach was also used to improve algorithms for following the steepest descent reaction paths. For larger molecular systems, we developed an extended Lagrangian approach in which the electronic structure is propagated along with the molecular structure. Strong field chemistry is a rapidly growing area, and to improve the accuracy of molecular dynamics in intense laser fields, we included the time-varying electric field in a novel predictor-corrector trajectory integration algorithm. Since intense laser fields can excite and ionize molecules, we extended our studies to include electron dynamics. Specifically, we developed code for time-dependent configuration interaction electron dynamics to simulate strong field ionization by intense laser pulses. Our initial application of ab initio direct dynamics in 1994 was to CH 2 O → H 2 + CO; the calculated vibrational distributions in the products were in very good agreement with experiment. In the intervening years, we have used direct dynamics to explore energy partitioning in various dissociation reactions, unimolecular dissociations yielding three fragments, reactions with branching after the transition state, nonstatistical dynamics of chemically activated molecules, dynamics of molecular fragmentation by intense infrared laser pulses, selective activation of specific dissociation channels by aligned intense infrared laser fields, angular dependence of strong field ionization, and simulation of sequential double ionization.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2020

Effective entropy of quantum fields coupled with gravity

Entanglement entropy, or von Neumann entropy, quantifies the amount of uncertainty of a quantum state. For quantum fields in curved space, entanglement entropy of the quantum field theory degrees of freedom is well-defined for a fixed background geometry. In this paper, we propose a generalization of the quantum field theory entanglement entropy by including dynamical gravity. The generalized quantity named effective entropy, and its Renyi entropy generalizations, are defined by analytic continuation of a replica calculation. The replicated theory is defined as a gravitational path integral with multiple copies of the original boundary conditions, with a co-dimension-2 brane at the boundary of region we are studying. We discuss different approaches to define the region in a gauge invariant way, and show that the effective entropy satisfies the quantum extremal surface formula. When the quantum fields carry a significant amount of entanglement, the quantum extremal surface can have a topology transition, after which an entanglement island region appears. Our result generalizes the Hubeny-Rangamani-Takayanagi formula of holographic entropy (with quantum corrections) to general geometries without asymptotic AdS boundary, and provides a more solid framework for addressing problems such as the Page curve of evaporating black holes in asymptotic flat spacetime. We apply the formula to two example systems, a closed two-dimensional universe and a four-dimensional maximally extended Schwarzchild black hole. We discuss the analog of the effective entropy in random tensor network models, which provides more concrete understanding of quantum information properties in general dynamical geometries. We show that, in absence of a large boundary like in AdS space case, it is essential to introduce ancilla that couples to the original system, in order for correctly characterizing quantum states and correlation functions in the random tensor network. Using the superdensity operator formalism, we study the system with ancilla and show how quantum information in the entanglement island can be reconstructed in a state-dependent and observer-dependent map. We study the closed universe (without spatial boundary) case and discuss how it is related to open universe.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2023

Calculation of Dynamical Response Functions Using a Bound-State Method

Abstract We investigate a method to extract response functions (dynamical polarisabilities) directly from a bound-state approach applied to calculations of perturbation-induced reactions. The use of a square-integrable basis leads to a response in the form of a sum of $$\delta $$ δ functions. We integrate this over energy and fit a smooth function to the resulting stepwise-continuous one. Its derivative gives the final approximation to the physical response function. We show that the method reproduces analytical results where known, and analyse the details for a variety of models. We apply it to some simple models, using the stochastic variational method as the numerical method. Albeit we find that this approach, and other numerical techniques, have some difficulties with the threshold behavior in coupled-channel problems with multiple thresholds, its stochastic nature allows us to extract robust results even for such cases.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗