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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 343 records · Page 19
All sourcesNASA NTRSDOE OSTI
DOE OSTI2026

A New 1D Model for Thermal Mixing and Stratification in Advanced Reactor Transients

Thermal mixing and stratification in large pools and enclosures play a critical role in the safety and performance of pool-type nuclear reactors, particularly during transient scenarios involving significant temperature differences between incoming and bulk coolant. Accurate modeling of these phenomena is essential for predicting system behavior and supporting passive safety features such as natural circulation. Here, this paper presents a new 1D model for thermal mixing and stratification, developed and implemented in the SAM code. The model represents a large pool as 1D coolant jet channels and zero-dimensional bulk pool volumes, enabling the simulation of a wide range of flow configurations, including hot and cold jet interactions, stratified layers, and the influence of complex geometries such as ceilings, free surfaces, and internal obstacles. Heat exchange between jet and pool regions is governed by closure relations calibrated against 3D computational fluid dynamics (CFD) simulations. The model improves upon earlier approaches by incorporating time-dependent jet characteristics and capturing the associated delay effects more accurately. Code-to-code comparisons and validation against experimental data from the Thermal Stratification Test Facility demonstrate the model’s accuracy and flexibility. This work offers two key contributions: (1) an efficient and robust method for simulating thermal mixing and stratification at the system level, eliminating the need for external coupling between system analysis codes and CFD, and (2) a significant enhancement of SAM’s capabilities to analyze thermal stratification phenomena in advanced reactor systems.

SAM↗
DOE OSTI2020

A status review on the thermal stratification modeling methods for Sodium-cooled Fast Reactors

The thermal stratification phenomenon plays a crucial role in the safety of various nuclear systems, including the Gen-III + Light Water Reactors (LWR) and the Gen-IV reactors. The phenomenon is of particular importance for the pool-type Sodium-cooled Fast Reactors (SFRs) because it may cause neutronic and thermal-hydraulic in- stabilities in the reactor core, or lead to damages of both the reactor vessel and in-vessel components due to the growth of thermal fatigue cracking. More significantly, thermal stratification could impede the establishment of the natural circulation during accidental scenarios and introduce uncertainties to the core safety of SFRs. Efforts for modeling of the thermal stratification in SFRs have been made for decades to prevent or mitigate the damage caused by the phenomenon. This paper gives a review of the advances that have been made in recent 10 years on the computational modeling methods for thermal stratification phenomenon in SFRs. These methods can be generally drawn into two categories. The first one is the system-level methods which provide fast-running but approximate calculations, and the second one is the CFD methods which provide high-resolution calculations at high computational expense. Finally, after introducing the efforts that have been made to improve the one- dimensional (1-D) models, the paper envisioned the possible research directions that could be pursued to enhance the modeling of thermal stratification in the near future.

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗
DOE OSTI2023

Predeployment progress of the Canister Deposition Field Demonstration

This report updates the high-level test plan for evaluating surface deposition on three commercial 32PTH2 spent nuclear fuel (SNF) canisters inside NUTECH Horizontal Modular Storage (NUHOMS) Advanced Horizontal Storage Modules (AHSMs) from Orano (formerly Transnuclear Inc.) and provides a summary of the surface sampling activities that have been conducted to date. The details contained in this report represent the best designs and approaches explored for testing as of this publication. Given the rapidly developing nature of this test program, some of these plans may change to accommodate new objectives or requirements. One goal of this testing is to collect defensible and detailed dust deposition measurements from the surface of dry storage canisters in a marine coastal environment to guide chloride-induced stress corrosion cracking (CISCC) research. Another goal is to provide data for the validation of computational fluid dynamics (CFD) based deposition modeling. To facilitate surface sampling, the otherwise highly prototypic dry storage systems will not contain SNF but rather will be electrically heated to mimic the decay heat and thermal hydraulic environment. Test and heater design is supported by detailed CFD modeling. Instrumentation throughout the canister, storage module, and environment will provide extensive information about the thermal-hydraulic behavior of horizontal dry cask storage systems. Manual sampling over a comprehensive portion of the canister surface at regular time intervals will offer detailed quantification and composition of the deposited particulates from a realistic storage environment. Discussions of a potential host site for the Canister Deposition Field Demonstration (CDFD) are ongoing. Until a host site is chosen, testing of key CDFD hardware components including the heater assemblies, power skid, and remote data acquisition system will continue. Functional testing of the finalized heater assemblies and test apparatus started this fiscal year. These initial heater tests have shown the assemblies are performing within design specifications. Staged surface sampling of a mockup of a canister outside the AHSM on a transfer skid was also performed. Refinements to the sampling procedures and techniques were captured from observation of these activities and lessons-learned debriefs. These updated sampling procedures and techniques are planned to be tested again in the field using the mockup in order to assure personnel are using the most accurate and repeatable methods possible prior to deployment for actual CDFD testing.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗
DOE OSTI2023

Efficient Thermal Energy Storage with Radial Flow in Packed Beds

Thermal energy storage (TES) is a way to store excess heat in order to generate power later. These types of systems are valuable in the solar industry, where power production can still occur when sunlight is not available. This not only increases system flexibility, it drives down the cost of electricity. One option to store thermal energy is with a packed bed where the storage media resides in a cylindrical container. Flow from one end of the cylinder to the other deposits or recovers heat (axial flow). While a promising technology in terms of energy storage, it exhibits a high pressure drop that lowers overall system efficiency. This project seeks to flow the heat transfer fluid through the storage media in the radial direction. This system offers the potential to retain reasonable thermal performance while substantially lowering the pressure drop. The overall goal was to assess the performance of radial flow experimentally and numerically. Three main designs were considered in this work. The first system utilized a central pipe and four receiving pipes near the wall. The second system utilized a central pipe and an annulus with holes near the wall. The last approach considered segmenting the bed so only select zones would receive flow in the radial direction. In a high-aspect ratio system, the hole patterns in the piping are important for ensuring even flow into the bed, and a major design effort was testing holes patterns to promote even flow. The four pipe system was shown to be not feasible, as preferential flow paths occur that do not allow even and full heating of the packed bed. The annular system can lead to even flow, but the results show this is better accomplished through larger piping instead of considering variations in the hole pattern along the axial length of the pipe. Segmenting the bed can lead to similar exergetic performance when compared to axial or radial flow. However, the pressure drop in the segmented design is likely too high for practical implementation. A baseline commercial design showed exergetic efficiencies for axial, radial, and radial segments of 81.6%, 82.8%, and 80.2%, respectively. Pressure drop for the axial and radial results were 2.36 psi and 2.45 psi, respectively, with segments being nearly an order of magnitude higher. At these large scales, the aspect ratio of the system is important. From a baseline of 0.64, an aspect ratio of 0.32 for radial flow showed an exergetic efficiency of 86% but a pressure drop of 5.39 psi. All of these results provide new insights into packed bed thermal energy storage with radial flow. Competing effects must be considered when designing a radial system, and results show a radial design can show strong thermal results at the expense of system efficiencies from pressure drop. The COMSOL models used in this analysis are available to the public and other researchers. Public benefits include a potential increase in the efficiency of packed bed thermal energy storage. Higher efficiency in storage promotes green energy technologies while reducing costs.

14 SOLAR ENERGY↗
DOE OSTI2025

Fluoride-Cooled High-Temperature Pebble-Bed Reactor Reference Plant Model Updates

This work presents the latest improvements to, and investigations performed with, the Fluoride-Cooled High-Temperature Pebble-Bed Reactor reference plant models for the United States Nuclear Regulatory Commission. These models, developed with the Comprehensive Reactor Analysis Bundle, or BlueCRAB, serve as the foundation for the future development of detailed design evaluation models based on license applications. BlueCRAB is the code suite proposed for non-light-water reactor systems safety analyses, and it incorporates various simulation tools developed by the Nuclear Energy Advance Modeling and Simulation program, including the Griffin code for reactor physics, the Pronghorn and SAM codes for core thermal fluids, the BISON code for solid conduction and fuel performance, and the SAM code for system analysis. The primary objective of this work is to assess the level of readiness of BlueCRAB for modeling fluoride-cooled high-temperature pebble-bed reactors. We first developed numerical models in BlueCRAB that include the key physics for this technology, ensuring an adequate level of fidelity for modeling the core performance during accident scenarios. This was followed by simulation of transient scenarios, two loss-of-forced-cooling events (one protected and one unprotected), and two control rod withdrawal events (one delayed and one prompt supercritical reactivity insertion). The analysis includes comparisons between the 2-D thermal fluid porous media models in Pronghorn and SAM, comparisons between coupled Pronghorn-Griffin and coupled SAM-Griffin models for two loss-of-forced cooling events and one control rod withdrawal event, and comparisons between SAM single-solve and domain-overlapping approaches for multi-scale thermal fluid coupling. In addition, we performed comparisons between 3-D, 2-D, and 0-D neutronic models for the two control rod withdrawal scenarios with Pronghorn-Griffin. The results show that the BlueCRAB models led to physically intuitive solutions for the scenarios examined. The changes in the various scalar and vector fields such as the neutron flux, power, temperatures, densities, pressures, and velocities are all within the expected ranges, and their distributions can be explained from the system response of the transients and the geometric and material variations. Several comparisons suggest that the porous media models in Pronghorn and SAM can lead to similar solutions, even though they are based on different methodologies. The simulations demonstrate that there are differences between the various levels of fidelity, and it is advisable to have flexible tools that can cover the breadth and depth of needs that may arise in future technical evaluations. We believe that BlueCRAB’s capabilities represent a significant asset for confirmatory analyses aimed at resolving important safety questions.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2026

Scaling considerations for supercritical carbon dioxide cycles including turbomachinery loss models

A modeling framework for the supercritical carbon dioxide recompressed closed Brayton cycle was developed. Unlike typical models, this effort incorporated generalized empirical turbomachinery loss models. Aerodynamic, windage, and leakage losses were considered in order to address the limitations of conventional constant-efficiency turbomachinery assumptions without relying on machine-specific or computationally expensive simulations. The model enables system-level exploration of optimal cycle design across a range of power scales, including smaller scales that are relevant to microreactors and extraterrestrial power applications. Parametric studies and multi-objective optimizations are used to evaluate the trade-offs between thermal efficiency and system compactness based on an analytical heat exchanger scaling model, yielding Pareto-optimal fronts across a range of operating pressures. Results reveal that at small power scales, the Pareto-optimal compressor inlet pressure becomes subcritical due to the increasing influence of density-dependent turbomachinery losses. Here, the relative contributions of each loss mechanism are quantified, and design recommendations are provided for key parameters such as recompression split ratio and generator cavity pressure across varying power scales.

Multi-objective optimization↗
DOE OSTI2020

Analysis of the thermal recovery of HTS cables after an overcurrent fault

High temperature superconducting (HTS) power transmission cables are cooled to operating temperatures typically below 80 K. A valuable feature of some HTS cables is that they can limit short-circuit fault currents, which can be more than ten times the maximum operating current of a cable. When a fault occurs the HTS wire is no longer superconducting and the heating which occurs will raise the temperature above the allowable operating point. To recover from the fault, the conductor must be cooled down before returning to service. The recovery is affected by the refrigeration system's performance which varies with load and temperature. A transient numerical model of an HTS cable is applied to investigate the thermal recovery after a fault for some HTS cable system configurations and operating conditions.

Demko, Jonathan A.↗
DOE OSTI2026

Optimal Co-Design of Integrated Thermal-Electrical Networks and Control Systems for Grid-interactive Efficient District (GED) Energy Systems

This project advances a unified, open-source framework for the optimal co-design of thermal, electrical, and control systems in grid-interactive efficient districts (GEDs). As communities integrate growing levels of distributed energy resources, traditional approaches that model thermal and electrical networks independently lead to reduced efficiency, limited flexibility, and missed opportunities for coordinated operation. To address these challenges, the research team developed a comprehensive suite of physics-based models, control algorithms, and software tools that enable holistic simulation, optimization, and demonstration of district-scale energy systems.

14 SOLAR ENERGY↗
DOE OSTI2022

A multiphysics model of the versatile test reactor based on the MOOSE framework

The traditional modeling approach for sodium fast reactor cores relies on separate physics models, where the fuel performance, thermal–hydraulics, and neutronics calculations required to predict the core physics characteristics for nominal conditions are decoupled by relying on user-imposed boundary conditions. Here, this paper aims at evaluating the impact of multiphysics simulations for predicting the core characteristics of the Versatile Test Reactor, which is being designed as a 300-MWt sodium-cooled fast reactor. The purpose of the Versatile Test Reactor is to accelerate the testing of advanced nuclear materials in the United States. The proposed multiphysics model relies on the Griffin reactor physics code, the SAM thermal–hydraulic system code, the BISON fuel performance code, as well as generic Multiphysics Object-Oriented Simulation Environment capabilities implemented in the open-source tensor mechanics module. For k eff calculations, the introduction of a tight coupling between the neutronics, thermo-mechanical and thermal–hydraulics models induces a change of around 543 pcm in the eigenvalue, compared to the traditional standalone neutronics calculation where approximate temperature profiles are used. The multiphysics model is then employed for quantifying the impact of the thermal conductivity uncertainties on some of the key figures of merit, such as the fuel centerline temperature, assembly powers, and keff for nominal core conditions. As anticipated, uncertainties on fuel thermal conductivity mostly impact the fuel centerline temperature, and to a lesser extend the k eff .

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2020

Local spin polarizations in relativistic heavy-ion collisions

Here, based on a generalized side-jump formalism for massless chiral fermions, which naturally takes into account the spin-orbit coupling in the scattering of two chiral fermions and the chiral vortical effect in a rotating chiral fermion matter, we have developed a covariant and total-angular-momentum-conserved chiral transport model to study both the global and local polarizations of this matter. For a system of massless quarks of random spin orientations and finite vorticity in a box, we have demonstrated that the model can exactly conserve the total angular momentum of the system and dynamically generate the quark spin polarization expected from a thermally equilibrated quark matter. Using this model to study the spin polarization in relativistic heavy-ion collision, we have found that the local quark spin polarizations depend strongly on the reference frame where they are evaluated as a result of the nontrivial axial charge distribution caused by the chiral vortical effect. We have further shown that because of the anomalous orbital or side-jump contribution to the quark spin polarization, the local quark polarizations calculated in the medium rest frame are qualitatively consistent with the local polarizations of Lambda hyperons measured in experiments.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2024

Radiation and heat transport in divergent shock–bubble interactions

Shock–bubble interactions (SBIs) are important across a wide range of physical systems. In inertial confinement fusion, interactions between laser-driven shocks and micro-voids in both ablators and foam targets generate instabilities that are a major obstacle in achieving ignition. Experiments imaging the collapse of such voids at high energy densities (HED) are constrained by spatial and temporal resolution, making simulations a vital tool in understanding these systems. In this study, we benchmark several radiation and thermal transport models in the xRAGE hydrodynamic code against experimental images of a collapsing mesoscale void during the passage of a 300 GPa shock. We also quantitatively examine the role of transport physics in the evolution of the SBI. This allows us to understand the dynamics of the interaction at timescales shorter than experimental imaging framerates. We find that all radiation models examined reproduce empirical shock velocities within experimental error. Radiation transport is found to reduce shock pressures by providing an additional energy pathway in the ablation region, but this effect is small (∼1% of total shock pressure). Employing a flux-limited Spitzer model for heat conduction, we find that flux limiters between 0.03 and 0.10 produce agreement with experimental velocities, suggesting that the system is well-within the Spitzer regime. Higher heat conduction is found to lower temperatures in the ablated plasma and to prevent secondary shocks at the ablation front, resulting in weaker primary shocks. Finally, we confirm that the SBI-driven instabilities observed in the HED regime are baroclinically driven, as in the low energy case.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2025

A hybrid surrogate modeling framework for the Digital Twin of a Fluoride-salt-cooled High-temperature Reactor (FHR)

While nuclear energy is a non-greenhouse-gas emitting energy source, expensive operational costs due to the high-level of safety requirements decreases their competitiveness in the sustainable energy market. Advanced reactor concepts paired with Digital Twins aim to increase the commercialization gains of nuclear energy by reducing operational costs, increasing reactor reliability and enhancing power generation. To support Digital Twin tasks such as real-time autonomous control, proactive maintenance monitoring or optimizing power demand operations, a fast and accurate virtual representation of the Nuclear Power Plant (NPP) is required. The computational cost of high-fidelity, physics-based models are unsuitable for real-time analysis or scalability. Here, in this work, a hybrid surrogate modeling framework is developed fora Fluoride-salt-cooled High-temperature Reactor (FHR) that leverages physics-inspired models for key reactor components and uses data-driven methods for rapid system state space prediction. The Xenon reactivity feedback model is integrated to inform the surrogate model about the reactor core and the homologous pump theory model is the basis for representing pump degradation. Using a detailed, two dimensional thermal hydraulics model to generate data on the FHR, we train a network of Vectorized Autoregressive Moving-Average with eXogenous input (VARMAX) models to predict the remaining state values. The result is a surrogate model that provides a detailed reactor state representation of 41 system states and a pump degradation analysis. The framework is applied to Load Follows profiles, yielding high accuracy and a speedup that is more than 4000x faster compared to the higher- fidelity thermal hydraulics model, enabling real-time operational intelligence and applications in long horizon predictions. While the surrogate model framework is demonstrated for the particular case of FHR, the hybrid physical/data-driven modeling approach including the network of surrogates and the underlying modularity has the potential to be applied to other physical asset systems.

Digital Twins↗
DOE OSTI2024

Sensitivity Analyses of Natural Convection in the AP1000 Passive Containment Cooling System Following LBLOCA Using CFD

The AP1000 power plant has implemented multiple layers of passive safety systems to enhance reactor safety and ensure system integrity during postulated scenarios, such as loss of coolant accident (LOCA) and main steam line break (MSLB). Among these, the passive containment cooling system (PCCS) is a safety-related system designed to prevent the containment from exceeding the design limits of both pressure and temperature following a postulated design basis accident, by transferring heat from the steel containment vessel to the atmosphere. During LOCA, natural air convection plays a vital role in removing heat from the steel containment vessel to the atmosphere. This is integrated with the condensate film from the Passive Containment Cooling Water Storage Tank (PCCWST). Air natural convection is also the only heat removal mechanism after the dry-out of the PCCWST and during the loss of shutdown decay heat removal (LOSDHR) event. Therefore, it is essential to investigate the thermal hydraulics behavior of the containment under transient conditions to ensure its safety and integrity. This paper presents a simplified CFD/ANSYS FLUENT model developed to analyze the impact of different parameters, such as air relative humidity, air temperature, and steel containment temperature, on the natural convection capability to remove the decay heat following LOCA. The model aims to examine the thermal behavior of the containment and provides a comprehensive understanding of the system's natural convection capability during postulated accidents. The results obtained from the model can be used to improve the safety and integrity of the containment system and provide valuable insights for future research in the field.

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗
DOE OSTI2020

Development and Validation of a Simulation Testbed for the Intelligent Building Agents Laboratory (IBAL) Using TRNSYS

This paper documents the development and validation of a dynamic primary cooling and thermal storage system simulation testbed. The system simulation testbed, sIBAL, is based on the Intelligent Building Agents Laboratory (IBAL) at the National Institute of Standards and Technology (NIST), which is a research infrastructure and testbed for the development, evaluation, and demonstration of intelligent control algorithms. The sIBAL testbed developed in this project will serve as a virtual twin of the real facility for future control algorithm development. The details of the methodologies used to develop and validate the simulation testbed, which replicates the dynamic behaviors of the primary cooling and ice storage system in the IBAL facility, are presented. The simulation testbed was developed in TRNSYS using built-in component models and MATLAB functions to replicate the two water-cooled chillers, a thermal storage tank, pumps, valves and other components for four different operation modes. Experiments on IBAL components were designed and executed to generate experimental data for model development and verification of the simulation platform. The validation of the simulation results was carried out in two phases: 1) independent component simulations for the chillers and thermal storage tank, and 2) a combined testbed simulation of the entire hydronic system. Comparison of simulation results to the experimental data obtained from the IBAL facility showed errors within 1 °C for the temperatures outputs of both the chiller and the thermal storage model. The error is within an acceptable range for further intelligent control algorithms development. The findings from the study are summarized and presented along with areas where additional research is needed. In addition, data filtering procedures and model refinement measures utilized to improve the accuracy and accelerate the computation time of the simulation are presented.

cooling systems↗
DOE OSTI2025

Anomalous thermal transport in Eshelby twisted van der Waals nanowires

Dislocations in van der Waals (vdW) layered nanomaterials induce strain and structural changes that substantially impact thermal transport. Understanding these effects could enable the manipulation of dislocations for improved thermoelectric and optoelectronic applications, but experimental insights remain limited. Here, in this study, we use synthetic Eshelby twisted vdW GeS nanowires (NWs) with single screw dislocations as a model system to explore the interplay between dislocation-induced structural modifications and lattice thermal conductivity. Our measurements reveal a monoclinic structure stabilized by the dislocation, leading to a substantial drop in thermal conductivity for larger-diameter NWs (70% at room temperature), supported by first-principles calculations. Interestingly, we also find an anomalous enhancement of thermal conductivity with decreasing diameter in twisted NWs, contrary to typical trends in non-twisted GeS NWs. This is attributed to increased conductivity near the NW cores due to compressive strain around the central dislocations, and aligns with a density-functional-theory-informed core–shell model. Our results highlight the critical role of dislocations in thermal conduction, providing fundamental insights for defect and strain engineering in advanced thermal applications.

36 MATERIALS SCIENCE↗
DOE OSTI2021

A measure of active interfaces in supported catalysts for high-temperature reactions

Formulating knowledge of structure-function relationships in heterogeneous catalysis is central to the design of efficient catalysts; yet, the elucidation of dominant reaction sites has remained as a challenge. Here, in this paper, we present a methodology that can be used to visualize metal-gas and metal-oxide-gas interfaces in three dimensions and to quantify their catalytic activity levels. As a case study, CH 4 oxidation occurring in a Pt/CeO 2 system is chosen. By employing thermally robust Pt@CeO 2 model catalysts with size-tunable and monodisperse cores, and gas-permeable shells, we reconstruct a series of structures in 3D via electron tomography and match the information to activity data and theoretical calculations. This strategy reveals that two different interfaces catalyze the CH 4 oxidation and that their contribution to the overall rate changes with the Pt size, temperature, and gas atmosphere. Our results provide an analytic platform on which to explore reaction pathways and mechanisms applicable to multiple reactions and materials.

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

Comparison of carrier localization effects between InAs quantum dashes and quantum dots in a DWELL (dashes- or dots-in-a-well) configuration

The optical properties of InAs quantum dashes (QDashes) grown on InP and InAs quantum dots (QDots) grown on GaAs in a dashes- or dots-in-a-well (DWELL) configuration are comparatively investigated using temperature-dependent photoluminescence (PL) measurements. The trends in PL characteristics such as exciton energy, spectral bandwidth and integrated intensity with respect to temperature are found to be distinctly dissimilar between the two systems. A rate-equation model involving exciton recombination and thermal transfer in a localized-state ensemble is used to quantitively interpret the experimental data. The results of this study suggest that QDashes in this configuration exhibit PL properties more consistent with a lower degree of carrier localization compared to QDots. A preliminary structural analysis highlighting the shape/size differences between the two nanostructures is also presented.

carrier localization↗
DOE OSTI2024

Critical slowing of the spin and charge density wave order in thin film Cr following photoexcitation

We report on the evolution of the charge density wave (CDW) and spin density wave (SDW) orders of a chromium film following photoexcitation with an ultrafast optical laser pulse. The CDW is measured by ultrafast time-resolved x-ray diffraction of the CDW satellite that tracks the suppression and recovery of the CDW following photoexcitation. We find that as the temperature of the film approaches a discontinuous phase transition in the CDW and SDW orders, the time scales of recovery increase exponentially from the expected thermal time scales. We extend a Landau model for SDW systems to account for this critical slowing with the appropriate boundary conditions imposed by the geometry of the thin film system. This model allows us to assess the energy barrier between the available CDW/SDW states with different spatial periodicities.

36 MATERIALS SCIENCE↗