Lattice Parameter Mismatch Research Articles

The influence of hydrostatic pressure on the synthesis of colloidal core-shell quantum dots with the mismatch of lattice parameters

The influence of hydrostatic pressure on the synthesis of colloidal core-shell quantum dots with the mismatch of lattice parameters

Electrodeposition of Non-Noble High-Entropy Alloys for Effective Hydrogen Evolution Electrocatalysts

As society works towards reducing its carbon footprint, various sectors such as energy, steelmaking and fertilizer production have begun to look increasingly towards decarbonization. Hydrogen production through electrolysis, when linked to renewables (solar, wind), provides a promising low-emission alternative as an energy carrier and chemical feedstock. With the decreasing cost of said renewables, greater focus is now on the capital cost of the electrolyser, and particularly the membrane/electrode structures. Conventional Proton Exchange Membrane (PEM) electrolysers rely on the use of electrocatalysts made with expensive noble metals such as platinum, ruthenium, and palladium [1]. Anion Exchange Membrane (AEM) electrolysers present an effective means of alkaline water splitting with low carbon emissions and can effectively utilize non-noble metal electrocatalysts for the Hydrogen Evolution Reaction (HER). A novel class of materials, High-Entropy Alloys (HEAs), made of non-noble metals or with reduced noble metal content have shown great promise as active electrocatalytic materials [2]. HEAs are typically defined as metal alloys made of five or more constituent elements each comprising at least 5% of the total material [3]. Having such a large number of principal alloying elements can lead to enhanced phase stability due to an increased configurational entropy, have unique chemical clustering, and can help prevent degradation of the material in harsh environments [2]. Additionally, the large lattice parameter mismatch between constituent elements can result in a high degree of strain which can help shift electronic states in a way that is favourable to hydrogen electrocatalysis [4]. These properties along with the so called “cocktail effect,” whereby combining multiple elements can result in unexpected synergistic interactions, makes HEAs uniquely suited to electrocatalytic applications [5].In this work, we investigate the electrocatalytic performance of electrodeposited FeNiCoMoW HEAs. The goal is to combine three hyper-d transition metals with two hypo-d transition metals to alter the electronic structure and improve electrocatalytic performance. Using chronoamperometric measurements, the Tafel slopes for FeNiCoMoW HEAs electrodeposited at pH 5, 6 and 7 were determined, with the pH 5 HEA demonstrating the smallest average Tafel slope of approximately 83 mV/dec. Compared to some electrodeposited binary alloys reported on in the literature, this marks an improvement and may be the result of unexpected interactions within the compositionally complex alloy. In the FeNiCoMoW HEA, the magnitude of the Tafel slope increased in tandem with the electrochemically active surface area (ECSA) as determined by Cyclic Voltammetry (CV) which suggests that the composition may be more important for improving performance. As the pH of the electrodeposition solution increased, the amount of cobalt in the as-deposited HEA decreased while the amounts of molybdenum and tungsten remained constant. The composition of nickel and iron appeared to vary inversely, reaching a maximum and minimum respectively at pH 6. From these results it seems that maximizing cobalt content and minimizing nickel content could potentially help to improve HER performance in the FeNiCoMoW system.

Twisted MoSe2 Homobilayer Behaving as a Heterobilayer.

Heterostructures (HSs) formed by the transition-metal dichalcogenide materials have shown great promise in next-generation (opto)electronic applications. An artificially twisted HS allows us to manipulate the optical and electronic properties. In this work, we introduce the understanding of the energy transfer (ET) process governed by the dipolar interaction in a twisted molybdenum diselenide (MoSe2) homobilayer without any charge-blocking interlayer. We fabricated an unconventional homobilayer (i.e., HS) with a large twist angle (∼57°) by combining the chemical vapor deposition (CVD) and mechanical exfoliation (Exf.) techniques to fully exploit the lattice parameter mismatch and indirect/direct (CVD/Exf.) bandgap nature. These effectively weaken the interlayer charge transfer and allow the ET to control the carrier recombination channels. Our experimental and theoretical results explain a massive HS photoluminescence enhancement due to an efficient ET process. This work shows that the electronically decoupled MoSe2 homobilayer is coupled by the ET process, mimicking a "true" heterobilayer nature.

Amorphous B coated Mg nanopowder induces low angle grain boundaries and enhances J c of MgB2 wire

In this work, amorphous B coated Mg nanopowder (BCMN) is synthesized and transport property of MgB2 superconducting wire is significantly enhanced with different contents of BCMN. BCMN has high reactivity since it contains nanoscale Mg and amorphous B. It allows to obtain MgB2 nanocrystals at only 400 °C with the compression of a lattice parameter and expansion of c lattice parameter compared to MgB2 formed by micron-sized Mg mixed with amorphous B (Mg + B) powders. These MgB2 nanocrystals serve as crystal nuclei and promote the crystallization and growth of MgB2. The mismatch of different lattice parameters prepared using BCMN and M + B powders induces low angle grain boundaries (LAGBs) embedded in MgB2 grains. LAGBs act as plane defects, leading to a dominant surface pinning mechanism and an enhancement in the critical current density on the magnetic field (J c(H)). At 4.2 K in 6 T, transport critical current density (J ct) of wire with 20 wt.% BCMN is 6.7 × 104 A·cm−2, approximately 1.8 times wire with 0 wt.% BCMN.

Assessment of the potential structure and nucleation efficacy of multi-component heterogeneous substrate

In recent years, nucleation of multi-component heterogeneous interface structures has attracted increasing attention due to their excellent grain refinement and anti-poisoning capabilities in Al-Si alloy. In this research, a general methodology was proposed to predict the surface structure and meanwhile evaluate the grain refinement efficacy of the multi-component boride nucleants based on a comprehensive experimental and theoretical study of a novel Al-Si alloy refiner, i.e., Al-3.5 Nb-1 V-1B. By using high-resolution electron microscopy, the (Nb, V)B2 particle with a characteristic “core-shell” surface structure was observed, attributed to V adsorption driven by interfacial energy minimization, validated by ab initio calculations. A general interfacial energy-based model was then proposed which predicts the multi-component boride substrate structure, well consistent with the experimental findings. The nucleation potency of different substrates was analyzed in atomic scale and only if the interfacial atomic vibration was considered together with the nucleation-influencing factors (i.e., the lattice mismatch and the anti Si-poisoning effect), the predicted nucleation potency order can agree well with the experimental results. A thermodynamics-kinetics combined analytic model was summarized by coupling the lattice mismatch parameter, the interfacial Si adsorption energy and the interfacial atomic vibration amplitude, which accurately evaluates the multi-component boride grain refiners for the Al-Si alloys.

Revealing the lattice engineering of Delafossite in core–shell CuRhO2@CuGaO2 heterostructure for efficient visible light photocatalytic activity

The establishment of heterojunctions at phase interfaces represents a pivotal strategy for enhancing photocatalytic performance. However, the inferior interfacial contact caused by the mismatch of crystal structure or lattice parameters seriously hampers the rapid and smooth migration of charge carriers in traditional heterojunctions. In this work, a tightly integrated Delafossite-based core–shell CuRhO2@CuGaO2 heterojunction was judiciously designed and constructed via a hydrothermal approach. Experimental and density-functional theory calculation results co-unravel that the built-in electric field is established in the lattice-matched heterogeneous interfaces, thereby providing smooth interface charge separation and transfer via the Z-scheme pathways. Benefitting from the heterojunction effect and unique core–shell coupling architecture, the strong visible light absorption and fast spatial charge transfer are realized in the CuRhO2@CuGaO2 heterostructure. The photocurrent density of the CuRhO2@CuGaO2 heterojunction is nearly 1.25 and 3.75 times those of pristine CuRhO2 and CuGaO2, respectively. As a result, the CuRhO2@CuGaO2 heterostructure reveals an excellent photocatalytic degradation rate of tetracycline hydrochloride, achieving a 3.02 and 2.38-fold improvement than those of the CuRhO2 and CuGaO2, respectively. This work is expected to pave the way for novel insights into the design of efficient heterojunction photocatalysts to steer the rapid photocarrier flows across the heterointerface.

Self-Competitive Growth of CsPbBr3 Planar Nanowire Array.

Semiconductor planar nanowire arrays (PNAs) are essential for achieving large-scale device integration. Direct heteroepitaxy of PNAs on a flat substrate is constrained by the mismatch in crystalline symmetry and lattice parameters between the substrate and epitaxial nanowires. This study presents a novel approach termed "self-competitive growth" for heteroepitaxy of CsPbBr3 PNAs on mica. The key to inducing the self-competitive growth of CsPbBr3 PNAs on mica involves restricting the nucleation of CsPbBr3 nanowires in a high-adsorption region, which is accomplished by overlaying graphite sheets on the mica surface. Theoretical calculations and experimental results demonstrate that CsPbBr3 nanowires oriented perpendicular to the boundary of the high-adsorption area exhibit greater competitiveness in intercepting the growth of nanowires in the other two directions, resulting in PNAs with a consistent orientation. Moreover, these PNAs exhibit low-threshold and stable amplified spontaneous emission under one-, two-, and three-photon excitation, indicating their potential for an integrated laser array.

Enhanced room-temperature magnetic field effect on spin-polarized excitons of cesium lead bromine-magnetite nanocomposites

The generation and manipulation of spin-polarized electrons are the basis of spintronic applications. Spin injection from a ferromagnetic surface or heterojunction to a semiconductor adjacent layer is often used to create spin-polarized carriers; however, for halide perovskites, due to the mismatch of lattice parameters with most ferromagnets at room temperature, this spin injection is challenging. In this study, we synthesize all-inorganic perovskite CsPbBr3 surrounded by Fe3O4 magnetic nanoparticles. It is found that this nanocomposite shows both magnetization and magnetic field-induced circularly polarized photoluminescence at room temperature. Specifically, with the attachment of Fe3O4 nanoparticles closely on the surface of CsPbBr3 nanocrystals, both the degree of circular polarization and g-factor enhanced by 3.5 times compared with that of pure CsPbBr3. The phenomenon should be due to the formation of exciton magnetic polaron through the coupling of the magnetic aligned nanoparticles with the excitonic state of the host semiconductor in the external magnetic field. The effective spin injection provides a method of controlling the excitonic spin polarization of all-inorganic halide perovskites for spintronic applications.

Effect of High Al Content AlGaN Buffer Layer on the Properties of GaN‐on‐Silicon Materials

The lattice parameter and thermal expansion coefficient mismatch between the silicon substrate and GaN lead to high tensile stress, which makes the GaN epitaxial layer prone to cracking. Effective compensation of tensile stress on GaN to prevent cracking is an important issue in GaN epitaxial growth. In this work, GaN‐on‐silicon materials with different AlGaN buffer layer structures are prepared by metal‐organic chemical vapor deposition (MOCVD). The GaN epitaxial material with a smooth and crack‐free surface is fabricated by inserting 7 AlGaN buffer layers. The crystal quality of GaN is characterized using high resolution X‐ray diffraction (HRXRD). The full‐width half maximum (FWHM) value of GaN(002) and GaN(102) crystal plane is 398 and 780 arcsec, respectively. The surface root mean square (RMS) roughness of GaN material is 0.31 nm, and the vertical breakdown voltage (BV) of the epitaxial wafer reaches 918 V. The results show that high Al content of the AlGaN buffer layer can effectively reduce the tensile stress and dislocation density of the GaN layer when the entire AlGaN layer thickness remains constant. Suppressing the generation of surface cracks and improving the crystal quality can improve the vertical breakdown voltage and two‐dimensional electron gas (2‐DEG) characteristics of GaN epitaxial wafers.

Elastic Field Evolution in Al–X (X = Sc, Zr, Er) Alloy During Heat Treatment, Insights From 3D-Multiphase Field Study

Abstract Microstructural evolution and resulting stress, strain, and concentration field distribution during Al3X (X = Sc, Zr, Er) precipitation in Al matrix are investigated in this work using the 3D-multiphase field method. Depending on the heat treatment, modulus mismatch, lattice parameter mismatch, and interfacial free energy, precipitate developed to rhombicuboctahedron, and near cuboidal morphologies. The composition distribution and Al–Al3X transformation driving force map identified a difference in precipitation kinetics for each alloy. The precipitation mechanism in the three systems is analyzed in detail with temporal evolution plots of energy components during phase transformation. Al3Er precipitate exhibits the highest growth rate due to Er's high diffusivity and significant lattice parameter mismatch in the Al–Er system. The system has a high chemical and elastic driving force for particle growth, thus attaining quasi-static equilibrium at a relatively lower temperature and time. Therefore, this system observes high magnitude stress, strain, and strain energy field around the Al matrix. The theoretical simulation results obtained from the present study will benefit aluminum multicomponent alloy design for high-strength applications.

Microstructural and mechanical properties of precipitation-strengthened Al-Mg-Zr-Sc-Er-Y-Si alloys

We study the precipitation behavior of two L12-strengthened alloys with Mg and Y additions: ultralow-Sc, Si-free, Al-1Mg-0.09Zr-0.007Sc-0.006Er-0.02Y-0.01Si, and low-Sc, Si-added, Al-1Mg-0.09Zr-0.013Sc-0.006Er-0.02Y-0.08Si (at.%), and their ambient temperature strength and high-temperature creep resistance. Scanning transmission electron microscopy analyses reveal that β-Mg2Si precipitates or their precursors (β’, β’’) form in the Si-added alloy at ∼ 200°C, which act as preferential nucleation sites for the L12-nanoprecipitates and cause partial depletion of Si solute atoms, which decelerates L12 precipitation-kinetics, resulting in a microhardness peak at the same isochronal temperature, 475°C, in both alloys. Atom-probe tomography analyses reveal that the L12-nanoprecipitates in both alloys exhibit Sc/Er/Y-rich cores and Zr-rich shells, as well as Mg segregation at the interfacial regions. The L12-nanoprecipitates in the Si-added alloy has a smaller Sc/Er/Y and higher Zr and Si concentrations. The Si-free alloy exhibits superior creep properties at 300°C, due to a larger lattice parameter mismatch of the L12-nanoprecipitates with the Al(f.c.c.) matrix provided by higher Sc/Er/Y concentrations. Both alloys exhibit slower L12 nanoprecipitation-kinetics, extremely high coarsening resistance, and similar high-temperature creep resistance compared to a recently developed Mg/Y-free low-Sc, Al-0.08Zr-0.014Sc-0.008Er-0.09Si (at.%) alloy. The extremely high coarsening resistance of the Si-added alloy is attributed to the smaller Si concentration in the Al(f.c.c.) matrix. Silicon is scavenged by the β-Mg2Si precipitates, thermodynamically stable at < 475°C, which is then not available in the matrix to accelerate the coarsening of the L12-nanoprecipitates.

Elastic, anisotropic, lattice dynamics and electronic properties of XNiM and XNi2M (X = Ti, Zr, Hf; M = Sn, Ge, Si): DFT comparison study

The ab-initio calculations were performed using density functional theory (DFT) as implemented in the Quantum ESPRESSO (QE) code. Lattice parameter mismatch was minimal (2.5 % - 3 %) in TiNiSn/TiNi2Sn, ZrNiSn/ZrNi2Sn, and HfNiSn/HfNi2Sn compounds. Half-Heusler compounds had shorter bond lengths and smaller bulk moduli than their full-Heusler counterparts. Directional dependencies of the Young’s and shear moduli were investigated using elastic constants. TiNiGe, HfNiSn, and ZrNiSn had a greater universal anisotropy index among the half-Heusler compounds. Half-Heusler, TiNi2Sn, TiNi2Si, ZrNi2Sn, and HfNi2Sn compounds are reported to be dynamically stable. The electronic structure computations show that TiNiSn, ZrNiSn, and HfNiSn compounds have the least indirect energy gaps of 0.4508 eV, 0.5019 eV, and 0.3847 eV respectively. TiNiSn, ZrNiSn, and HfNiSn compounds are good candidates for minimizing the thermal conductivity of thermoelectric devices. TiNi2Sn, ZrNi2Sn, and HfNi2Sn are good candidates as possible contact electrodes for TiNiSn, ZrNiSn, and HfNiSn.

Effects of lattice and mass mismatch on primary radiation damage in W-Ta and W-Mo binary alloys

The fundamental mechanisms of radiation damage in metal alloys of equiatomic composition are of great research interest due to the possibility to use them as new materials for radiation-hard environments. With W-Ta and W-Mo alloys as examples, this study investigates the effect of two factors, namely the difference in lattice parameters and the difference in masses of the constituent elements, on primary radiation damage by means of atomistic simulations. We compare the trends obtained with the available embedded atom method (EAM) interatomic potential and the tabulated Gaussian approximation potential (tabGAP). We observe a similar trend in the number of surviving defects as a function of alloy composition in W-Mo for both potentials, while the absolute numbers are fairly different. The tabGAP predicts higher resistance to irradiation, i.e. fewer surviving defects after the cascades. On the contrary, the trends of the dependence of surviving defects on alloy composition in W-Ta alloys are very different in the two potentials. We explain this difference by analyzing the threshold displacement energies (TDE) and melting temperatures in these alloys. The mass difference in W-Mo alloys results in Mo atoms with higher TDE in W-Mo alloys. The effect of the lattice parameter mismatch in W-Ta alloys is visible in the change of the number of surviving defects with increase of the W content in W-Ta alloys.

Development of TaC-based transition metal carbide superlattices via compound target magnetron sputtering

Transition metal carbides belong to ultra-high temperature ceramics and are particularly valued for their high thermal and mechanical stability as well as melting points of even above 4000 °C. However, a considerable limitation of these materials is their high inherent brittleness. Inspired by the success of nanolayered superlattice architecture—shown to enhance both hardness and toughness of transition metal (TM) nitrides—we developed superlattice films with TM carbides. Among these, density functional theory calculations suggest TaC/HfC and TiC/TaC to have a similar shear modulus mismatch of 23 and 19 GPa, but a different lattice parameter mismatch of 4.2 and 2.4%, respectively. Detailed transmission electron microscopy and X-ray diffraction show a pronounced superlattice structure for TiC/TaC with nominal bilayer periods Λnom of 2, 6, and 10 nm. Contrary, the TaC/HfC showed a more solid solution like characteristic for Λnom = 2 nm, and a clear superlattice structure for Λnom = 6 and 10 nmjap. While the hardness of the TaC/HfC coatings is between those of their constituents TaC (33.3 ± 1.9 GPa) and HfC (37.4 ± 3.2 GPa), the TiC/TaC superlattices outperform their constituents and clearly show a superlattice-effect with a peak of 44.1 ± 3.4 GPa at Λnom = 2 nm (TiC has 37.6 ± 3.1 GPa). Also the qualitative fracture behavior investigation with 450-mN-loaded cube corner indentation yield the TiC/TaC superlattices to be superior to the TaC/HfC as well as the monolithically prepared TiC, TaC, and HfC coatings.

The Modeling of Self-Consistent Electron–Deformation–Diffusion Effects in Thin Films with Lattice Parameter Mismatch

In our work, the model of self-consistent electron–deformation–diffusion effects in thin films grown on substrate with the mismatch of lattice parameters of the contacting materials is constructed. The proposed theory self-consistently takes into account the interaction of the elastic field (created by the mismatch of lattice parameters of the film and the substrate, and point defects) with the diffusion processes of point defects and the electron subsystem of semiconductor film. Within the framework of the developed model, the spatial distribution of deformation, concentration of defects, conduction electrons and electric field intensity is investigated, depending on the value of the mismatch, the type of defects, the average concentrations of point defects and conduction electrons. It is established that the coordinate dependence of deformation and the concentration profile of defects of the type of stretching (compression) centers, along the axis of growth of the strained film, have a non-monotonic character with minima (maxima), the positions of which are determined by the average concentration of point defects. It is shown that due to the electron–deformation interaction in film with a lattice parameter mismatch, the spatial redistribution of conduction electrons is observed and n-n+ transitions can occur. Information about the self-consistent spatial redistribution of point defects, electrons and deformation of the crystal lattice in semiconductor materials is necessary for understanding the problems of their stability and degradation of nano-optoelectronic devices operating under conditions of intense irradiation.

A unified bond energy model for size-dependent melting temperature of freestanding and embedded nanomaterials

The bond energy (BE) model is a simple and practical model that has been developed to predict the melting temperature of nanomaterials. The model contains a material parameter, whose value is assumed to be 0.5 or 0.75. However, the exact value of the material parameter is not known. Available experimental data reveal that 0.5 or 0.75 are not suitable for all materials. In this research, a unified BE model has been developed to calculate the melting temperature of freestanding and embedded nanoparticles. In addition, using the available experimental data, a semi-empirical relationship has been proposed to estimate the value of the material parameter as a function of atomic distances. The results show that for freestanding nanoparticles the value of the material parameter can vary between 0.1 and 1.6. It is also found that the material parameter for embedded nanoparticles correlated with the lattice parameter mismatch between the nanoparticle and the matrix. The results show that if lattice parameter mismatch is less than 11.5%, the material parameter of freestanding and embedded nanoparticles are equal, but increasing lattice parameter mismatch, increases the material parameter of embedded nanoparticles. In addition to the melting temperature of nanoparticles, the proposed model was successfully used to calculate the melting temperature of nanowires.

Electrostatically Driven Polarization Flop and Strain-Induced Curvature in Free-Standing Ferroelectric Superlattices.

The combination of strain and electrostatic engineering in epitaxial heterostructures of ferroelectric oxides offers many possibilities for inducing new phases, complex polar topologies, and enhanced electrical properties. However, the dominant effect of substrate clamping can also limit the electromechanical response and often leaves electrostatics to play a secondary role. Releasing the mechanical constraint imposed by the substrate can not only dramatically alter the balance between elastic and electrostatic forces, enabling them to compete on par with each other, but also activates new mechanical degrees of freedom, such as the macroscopic curvature of the heterostructure. In this work, an electrostatically driven transition from a predominantly out-of-plane polarized to an in-plane polarized state is observed when a PbTiO3 /SrTiO3 superlattice with a SrRuO3 bottom electrode is released from its substrate. In turn, this polarization rotation modifies the lattice parameter mismatch between the superlattice and the thin SrRuO3 layer, causing the heterostructure to curl up into microtubes. Through a combination of synchrotron-based scanning X-ray diffraction imaging, Raman scattering, piezoresponse force microscopy, and scanning transmission electron microscopy, the crystalline structure and domain patterns of the curved superlattices are investigated, revealing a strong anisotropy in the domain structure and a complex mechanism for strain accommodation.

The baric coefficient of CdSe quantum dot with a three-layer ZnS/CdS/ZnS shell

The model of CdSe quantum dot with a multilayer shell that undergoes mechanical deformation is constructed. This theory takes into account the influence of external hydrostatic pressure, the Laplace surface pressure and the mismatch of lattice parameters of contacting materials of the nanoheterosystem. Within the framework of the developed model, the baric coefficient of CdSe/ZnS/CdS/ZnS quantum dot was calculated for different values of the CdSe core radius and at different thicknesses of the ZnS/CdS/ZnS shell layers. It is established that for small cores (with a diameter of less than 4.8 nm) of quantum dots with a three-layer shell, two values of the baric coefficient are possible depending on the magnitude of external pressure. The analysis of use of the offered model for research of interaction of quantum dots with elements of biological systems through an elastic field of deformation is carried out.

Ultra-High Temperature Creep of Ni-Based SX Superalloys at 1250 °C

Very high temperature creep properties of twelve different Ni-based single crystal superalloys have been investigated at 1250 °C and under different initial applied stresses. The creep strength at this temperature is mainly controlled by the remaining γ′ volume fraction. Other parameters such as the γ′ precipitate after microstructure evolution and the γ/γ′ lattice parameter mismatch seem to affect the creep strength to a lesser degree in these conditions. The Norton Law creep exponent lies in the range 6–9 for most of the alloys studied, suggesting that dislocation glide and climb are the rate limiting deformation mechanisms. Damage mechanisms in these extreme conditions comprise creep strain accumulation leading to pronounced necking and to recrystallization in the most severely deformed sections of the specimens.

Microelasticity model of random alloys. Part I: mean square displacements and stresses

In concentrated solid solutions, the random distribution of elements of different sizes induces characteristic displacement and stress fields at the root of solid solution strengthening. The aim of this two-part article is to derive the statistical properties of these elastic fields. The present Part I focuses on the variance of the elastic fields, while Part II addresses their spatial correlations. In this first part, we develop two elastic models of random solid solutions, based respectively on real-space and spectral methods, to derive the mean square displacement and shear stress. Both approaches hold advantages and drawbacks that are discussed here. We show in particular that both the mean square displacement and shear stress are directly proportional to the variance of the atomic eigenstrains, which embodies the atomic size differences and simplifies to the classical lattice mismatch parameter if the alloy satisfies Vegard’s law. This allows to clarify the scaling relations between various quantities (mismatch parameter, mean square displacement and yield stress) and to bridge energy-based and stress-based models of concentrated solid solution strengthening proposed in the literature. The elastic predictions for the mean-square displacement and stresses are also successfully compared with atomistic results obtained for a model Lennard-Jones system and with a more complex Al–Mg interatomic potential.