Einstein Model Research Articles

Emergent universe from an unstable de Sitter phase

In the Emergent scenario, the Universe should evolve from a nonsingular state replacing the typical singularity of General Relativity, for any initial condition. For the scalar field model by Ellis and Maartens [Class. Quantum Grav. 21 (2003) 223] we show that only a set of measure zero of trajectories leads to emergence, either from a static state (an Einstein model), or from a de Sitter state. Assuming a scenario based on CDM interacting with a Dark Energy fluid, we show that in general flat and open models expand from a nonsingular unstable de Sitter state at high energies; for some closed models this state is a transition phase with a bounce, other closed models are cyclic. A subset of these models are qualitatively in agreement with the observable Universe, accelerating at high energies, going through a matter-dominated decelerated era, then accelerating toward a de Sitter phase.

Unveiling the boson peaks in amorphous phase-change materials

The Boson peak is a universal phenomenon in amorphous solids. It can be observed as an anomalous contribution to the low-temperature heat capacity over the Debye model. Amorphous phase-change materials (PCMs) such as Ge-Sb-Te are a family of poor glass formers with fast crystallization kinetics, being of interest for phase-change memory applications. So far, whether Boson peaks exist in PCMs is unknown and, if they do, their relevance to PCM properties is unclear. Here, we investigate the thermodynamic properties of the pseudo-binary compositions on the tie-line between Ge15Te85and Ge15Sb85from a few Kelvins to the liquidus temperatures. Our results demonstrate the evidence of the pronounced Boson peaks in heat capacity below 10 K in the amorphous phase of all compositions. By fitting the data using the Debye model combined with a modification of the Einstein model, we can extract the characteristic parameters of the Boson peaks and attribute their origin to the excess vibrational modes of dynamic defects in the amorphous solids. We find that these parameters correlate almost linearly with the Sb-content of the alloys, despite the nonmonotonic behaviors in glass forming abilities and thermal stabilities. In a broader context, we show that the correlations of the characteristic parameters of the Boson peaks with Tg and kinetic fragility, vary according to the type of bonding. Specifically, metallic glasses and conventional covalent glasses exhibit distinct patterns of dependence, whereas PCMs manifest characteristics that lie in between. A deeper understanding of the Boson peaks in PCMs holds the promise to enable predictions of material properties at higher temperatures based on features observed in low-temperature heat capacity.

Machine-learning assistant DFT study of half-metallic full-Heusler alloy N2CaNa: Structural, electronic, mechanical, and thermodynamics properties

We studied the structural, electronic, mechanical, and thermodynamic properties of N2CaNa full Heusler alloys using density functional theory (DFT). Results for the structural analysis establish structural stability with a minimum formation energy of 29.9 eV. The compound is brittle and mechanically stable, having checked out with the Pugh criteria. The B/G ratio of bulk modulus B to shear modulus G for N2CaNa is 4.766, hence the material is ductile. N2CaNa alloy is ductile in nature. The Debye model correctly predicts the low-temperature dependence of heat capacity, which is proportional to Debye’s T3 law. Just like the Einstein model, it also recovers the Dulong–Petit law at high temperatures, suggesting the thermodynamic stability of the compounds at moderate temperatures. The results demonstrate potential N2CaNa for applications in spintronics, structural engineering, and other fields requiring materials with tailored properties.

A Third Generation Calphad Description of Pure Lithium.

This study, based on the analysis of existing experimental data, presents a third generation Calphad description of lithium, covering all temperature ranges, using nonlinear least squares in Matlab. We have expanded the SGTE database's description of lithium phases (face-centered cube, body-centered cube, liquid) down to 0 K with reasonable accuracy, taking into account the significant effort required to reconstruct the database for each element. During the evaluation process, it was determined that the low-temperature phase of lithium is fcc. The heat capacity of crystalline Li was accurately described using the extended Debye model. The third generation Calphad description of lithium utilized the two-state model and the extended Einstein model, leading to improved agreement with experimental data compared to previous assessments.

Particularities of optical behavior of Ag8SiS6 single crystal

Ag8SiS6 single crystal was grown by directional crystallization from melt. A crystal sample was investigated by optical ellipsometry and spectroscopy. This sample had nonlinear spectral dependences of the refractive index n and the extinction coefficient k. The presence of a sharp maximum in the spectral dependence of the refractive index and a rather sharp decrease in the values of the extinction coefficient k at the wavelength of 780 nm were found. The behavior of the optical absorption edge of the Ag8SiS6 single crystal in the temperature range of 77…300 K was studied. An exponential dependence of the absorption coefficient α obeying the Urbach rule was observed at all the investigated temperatures. The optical pseudo-gap Eg* and the Urbach energy EU were calculated from the obtained experimental data. An increase in temperature of the Ag8SiS6 crystal was found to lead to a monotonic, almost linear decrease in Eg* (1.853…1.615 eV) and a monotonic nonlinear increase in EU (44.32…55.01 meV). The contributions of the temperature-independent (EU)X,C and temperature-dependent (EU)T components to the total Urbach energy EU for Ag8SiS6 were evaluated within the Einstein model.

On ablation characteristics based on two-temperature model involving a multivariate lattice heat capacity by ultrashort laser-irradiated in zirconia

To explore the mechanism of material removal, this paper conducts a systematic study on picosecond laser processing of zirconia by using both theory and experiment. Comparing the multivariate lattice heat capacity of the Einstein and Debye models, two-temperature model (TTM) is developed to improve the accuracy of the temperature field. Then, to verify the effectiveness of TTM proposed, ablation experiments are performed by single-pulse picosecond laser on zirconia at different laser energy density. The results show that the measured craters profiles are well agree with the simulated melting/vaporization temperature distribution. Micro-morphology is significantly affected by phase transition induced by temperature rise. Moreover, the results confirmed that increased temperature can lead to transition in zirconia crystal structure and oxygen vacancies. Finally, the effect of coupling temperature variations on the elemental and physical phase variations are focused to investigate, which can help to optimize the processing quality due to crystalline phase transitions. This study provides guidance for optimizing picosecond laser processing of zirconia.

Use of third generation data for the pure elements to model the thermodynamics of binary alloy systems: Part 3 – The theoretical prediction of the Al–Si–Zn system

In a previous paper a method was developed to define Einstein temperatures for metastable phases of the elements and their relation to the so-called lattice stabilities used in the past, and also the variation of the Einstein temperature with composition to account for the composition dependence of the excess entropy. This approach was demonstrated successfully for the Al–Zn system. In this paper this approach is extended to cover the Al–Si and Si–Zn binary systems. The phase diagram for the Al–Si–Zn ternary system was then predicted from the thermodynamic description of the binary subsystems only without any ternary interaction parameters. Agreement with the experimental data is shown to be very good.

Intermediate statistics: Addressing the thermoelectric properties of solids.

We study the thermodynamics of a crystalline solid by applying intermediate statistics obtained by deforming known solid state models using the mathematics of qanalogs. We apply the resulting qdeformation to both the Einstein and Debye models and study the deformed thermal and electrical conductivities and the deformed Debye specific heat. We find that the qdeformation acts in two different ways-but not necessarily as independent mechanisms. First, it acts as an effective factor of disorder or impurity, modifying the characteristics of a crystalline structure, which are phenomena described by qbosons. Second, it also manifests intermediate statistics, namely, the Banyons (or B-type systems). For the latter case we have identified the Schottky effect, normally associated with high-T_{c} superconductors in the presence of rare-earth-ion impurities. We also find that it increases the specific heat of the solids beyond the Dulong-Petit limit at high temperature. Such an effect is usually related to anharmonicity of interatomic interactions. Alternatively, since in the q-boson's case the statistics are in principle maintained, the effect of the deformation acts more slowly due to a small change in the crystal lattice. On the other hand, Banyons that belong to modified statistics are more sensitive to the deformation. The results reported here may be verified experimentally, for instance, in experimental samples by inserting impurities, or changes in pressure or temperature if one assumes these tuning quantities are related with the q-deformation parameter.

Solid-liquid phase boundary of oxide solid solutions using neural network potentials

We propose a cost-effective computational approach for predicting the phase boundary of oxide solid solutions, i.e., melting point, by identifying the point where the free energies of the solid and liquid phases are equivalent. To evaluate the melting point, we computed the temperature-dependent free energies of both phases using the thermodynamic integration (TI) method. This was combined with the reverse-scaling technique, employing the Einstein model for the solid and the scaled Uhlenbeck–Ford model for the liquid as the reference systems. The change in free energy between the real and reference states in the present TI method was calculated as the ensemble average of the configurations sampled from molecular dynamics (MD) simulations using neural network potentials (NNPs) trained on first-principles density functional theory (DFT) data. Initially, we benchmarked copper (Cu) as a typical metal. Tungsten (W) and magnesium oxide (MgO) were then tested as typical high melting-point metals and oxides, respectively, achieving good agreement with both ab initio MD (AIMD) results and experimental data. We then applied our established approach to a BaO–CaO oxide solid solution, observing that the computed phase boundary aligns well with Calphad predictions from previous studies. The integration of NNPs into phase boundary prediction is essential for reducing computational costs while ensuring accuracy, compared with AIMD-based approaches.

Calculation of the Second EXAFS Cumulant of Si Using the Anharmonic Correlated Einstein Model

The second expanded X-ray absorption fine structure (EXAFS) cumulant of the crystalline silicon (Si) has been studied in the temperature-dependent. This is calculated in explicit forms using the anharmonic correlated Einstein (ACE) model developed from the correlated Einstein model based on the anharmonic effective potential and the quantum statistical theory. The numerical results of Si in the temperature range from 0 to 1200 K are in good agreement with those obtained by the other theoretical models and experiments at several temperatures. The analytical results show that the ACE model is suitable for analyzing the experimental EXAFS data of diamond cubics.

Investigation of Debye temperature and temperature-dependent EXAFS cumulants of Zn, Zr, α-Ti, Ru and Hf metals

In this work, the anharmonic Einstein model is developed to determine the Debye temperature and investigate the temperature effects on the extended x-ray absorption fine structure (EXAFS) cumulants of hexagonal close-packed (hcp) metals. We have derived the analytical expressions of the anharmonic effective potential, the effective force constant, the Debye temperature and the first four EXAFS cumulants as a function of axial ratio e = c/a. Numerical calculations have been conducted for hcp Zn, Zr, α-Ti, Ru and Hf metals up to temperature 800 K. Our findings indicate that the anharmonicity of thermal lattice vibrations significantly influences the EXAFS cumulants, particularly at high temperatures. Ru atoms have the strongest coupling force causing a phenomenon that Ru lattice shows a smaller thermal disorder, and Zn has a greater thermal disorder. Additionally, we highlight the significant contributions of thermal disorder to the mean-square relative displacement at high temperatures due to thermal lattice vibrations. Moreover, our Debye temperatures derived from the developed model align reasonably well with those reported in previous studies.

Investigating Anharmonic XAFS Debye-Waller Factor of Crystalline Tungsten Based on Einstein Model

The anharmonic X-ray absorption fine structure (XAFS) Debye-Waller (DW) factor of the crystalline tungsten (W) has been investigated in the temperature-dependent. This DW factor is calculated in explicit forms using the quantum anharmonic correlated Einstein model developed from the correlated Einstein model based on the anharmonic effective potential and the quantum statistical theory. The numerical results of W in the temperature range from 0 to 900 K are in good agreement with those obtained by the other theoretical models and experiments at several temperatures. The analytical results show that the anharmonic correlated Einstein model is suitable for analyzing the experimental XAFS data of metals.

Anharmonic Thermal Expansion Coefficient of Crystalline Iron

The anharmonic thermal expansion (TE) coefficient of crystalline iron (Fe) has been calculated and analysed in the temperature-dependent. The thermodynamic parameters of the crystal lattice are derived from the influence of the thermal vibrations of all atoms. The calculation model is developed from the correlated Einstein model and quantum-statistical perturbation theory using the anharmonic effective potential. The obtained expression of the temperature-dependent TE coefficient of Fe is an explicit form. The numerical results of Fe agree well with those obtained from the experiments at various temperatures in the range from 0 K to 900 K. The obtained results show that the present model is efficient in investigating the TE coefficient of Fe.

Revisiting the thermodynamic properties of the ZrCr2 Laves phases by combined approach using experimental and simulation methods

A comprehensive study of ZrCr2 Laves phases is important for both fundamental research and technological applications. The ZrCr2 compound is a typical precipitate observed in Zr-based alloys including the newly developed Cr-coated Zr cladding known as Accident Tolerant Fuel cladding. The brittle behavior of the Laves phases and the low melting point at the interface zone between the precipitates and the Zr matrix are considered as governing the thermomechanical behaviors of these newly developed nuclear fuel claddings for normal operating conditions but more dramatically in case of Loss Of Coolant Accident (LOCA) conditions. The aim of the present study is to investigate the various ZrCr2 Laves polymorphs and their thermodynamic features particularly where conflicts exist in the literature. Based on diffraction experiments of annealed samples, it has been established that the transformation from the C14 high-temperature form to the C15 low-temperature form is of displacive nature without the C36 polymorph forming as an intermediate phase. The stable and metastable domains of C14 and C15 and their thermodynamic properties have been determined. The specific heat of both C14 and C15 types was measured over a wide range of temperatures from 2 to 1063 K by coupling relaxation calorimetry and DSC. The experimental data were fitted using a modified Einstein model. The room temperature entropies of the C14 and C15 phases were evaluated. The enthalpy of formation was measured for the first time using drop solution calorimetry in liquid Al at 1173 K in a Tian–Calvet calorimeter. The experimental value is in good agreement with Density Functional Theory (DFT) calculations. Finally, all these new results are discussed regarding the abundant literature on the ZrCr2 Laves phases.

Vehicle Engine Cooling System: Review Research

This study reveals that nano-refrigerants can improve the overall performance of these systems, particularly when used as nanoparticles in conjunction with a base refrigerant. The results show that nano refrigerants outperform base liquids in warm conductivity, and the size of the nanoparticles affects this conductivity. The thickness of nano-refrigerants shows a vertical pattern as the volume of particles increases, while it decreases with temperature increases. Traditional models, such as the Hamilton-Crosser and Einstein models, fail to accurately predict the warm conductivity and consistency of nanoliquids when temperature is considered. Even a small amount of nanoparticles can significantly improve the base liquid’s conductivity. The use of nano-fluids results in an improved convective intensity transfer coefficient for all volume concentrations of nanoparticles compared to water under different working conditions. This study delves into the characterization of nanofluids for vehicle engine cooling systems, focusing on their thermal properties and heat transfer capabilities. Through an analysis of thermal conductivity, heat transfer coefficients, viscosity, and nanoparticle size, the research aims to optimize the design and implementation of nanofluid-based cooling systems to enhance engine performance and fuel efficiency. By investigating the impact of different nanoparticles and concentrations on these properties, the study provides insights into the potential of nanofluids to improve cooling efficiency in automotive applications. The findings highlight the importance of understanding the relationship between nanoparticle characteristics and thermal properties for the effective utilization of nanofluids in vehicle cooling systems. By synthesizing findings from previous studies, this review aims to provide insights into the potential benefits of utilizing nanofluids in enhancing cooling efficiency and overall engine performance in automotive systems. The analysis underscores the importance of considering nanoparticle characteristics in optimizing nanofluid formulations for effective heat transfer in vehicle cooling systems.

Rigid covalent bond of α-sulfur investigated via temperature-dependent EXAFS

This study performs extended x-ray absorption fine structure (EXAFS) measurements for the S-K edge in the temperature range of 10 and 300 K in the transmission mode using a photodiode to detect the transmitted x-rays. It provides the first report of temperature variations in the structural parameters of α-S. As the temperature increases from 10 to 300 K in the Fourier transform of kχ(k) the first peak corresponding to the covalent bond of the eight-membered ring becomes slightly low anomalously despite thermal disturbances. However, as in normal materials, the second peak at 300 K decreases to approximately half of that at 10 K, which contains several intra- and inter-ring correlations. All structural parameters of the covalent bond obtained by nonlinear least squares fitting exhibit missing temperature variations. A value of zero for the asymmetric parameter in the EXAFS (C 3) implies that the potential of the covalent bond is symmetric, and the constant value of the mean square relative displacement (MSRD) with temperature implies that the potential is extremely high. The Einstein model fitting for the temperature variation in the MSRD yields an Einstein temperature of 942 K and force constant (K) of 405 N m−1. The value of K is the largest among those of chalcogen elements.

Analytic solutions of Eliashberg gap equations at superconducting critical temperature

We perform simple mathematical analysis of the Eliashberg gap equations, and derive the analytic expression at the superconducting critical temperature. In order to perform exact summation of LTc=∑λm12m−1, we use the Einstein and Debye models for the Eliashberg spectral function. This quantity is in good agreement with data derived from experimental and DFT data. To the leading-term approximation, the analytic expression of the superconducting critical temperature is of the form Tc=2eγ−1πωxexp−1+μ*ln4eγmcλ, where ωx=ω2 or ωln. The frequencies ω2 and ωln come out naturally from the models. These equations contain mc as an adjustable parameter. There are no other free parameters and no empirical formula involved. We compare the results of these equations with the celebrated Allen-Dynes modified McMillan equation together with the data from experiments and DFT calculations. The data show strong correlation between our analytic expression and the celebrated Allen-Dynes equation.

Development of size and shape dependent model for various thermodynamic properties of nanomaterials

The models developed in this work are used to study various thermodynamic properties of nanomaterials. A better agreement between theory and experiment indicates the validity of the proposed model. Since the model is simple and easy to use, we extend the theory to understand, how the thermodynamic properties of different nanomaterials depend on size and shape. The results were compared with previous theoretical work and experimental data. We have concluded that the model predicts a better agreement with previous theoretical work and experimental studies. Current research produces thermodynamic models for the Debye frequency, melting entropy, melting enthalpy, and Einstein temperature. To the best of our knowledge, such models are not yet available in the literature that works well.

Thermal equation of state of rhodium to 191 GPa and 2700 K using double-sided flash laser heating in a diamond anvil cell

The phase behavior of rhodium (Rh) metal has been studied to 191 GPa and 2700 K using a combination of room-temperature isothermal compression and double-sided flash laser heating experiments. The isothermal compression data have been fitted with a second-order adapted polynomial of order L equation of state (EoS) with best-fitting parameters of V0=13.764(2)Å3/atom, K0=258(3)GPa, and K′=5.36(9). Two-dimensional maps of the uniaxial stress component t are presented for Rh at different pressures showing the spatial distribution of the local stress state of a relatively high-yield strength material encased in a Bi pressure medium. In addition, a simple, thermal pressure equation-of-state model, based on a single Einstein temperature, has been fitted to the high-pressure-temperature data up to 2700 K at 148 GPa and ambient-pressure thermal expansion data up to 1982 K. Also determined are the best-fitting parameters to reproduce the thermal EoS within the two-dimensional integration software. The optimized parameters are V0=13.764Å3/atom, K0=260.54GPa, K′=5.114, αT=2.99×10−5 K−1, ∂αT/∂T=1.27×10−9 K−2, ∂K0/∂T=−6.43×10−5GPa/K, and ∂K′/∂T=−9.3×10−10K−1. Published by the American Physical Society 2024

High-pressure high-temperature EXAFS and Debye–Waller factor of platinum

The temperature and pressure effects on the mean-square relative displacement (MSRD) corresponding to the extended X-ray absorption fine structure (EXAFS) Debye–Waller factor of platinum have been studied based on the statistical moment method in quantum statistical mechanics. We implement numerical calculations for platinum up to pressure of 400 GPa. Examining the temperature effects on the EXAFS Debye–Waller factor at ambient pressure we show that theoretical MSRD curve is in good agreement with the previous measurement and calculations up to nearly 800 K. When pressure increases, our pressure-dependent MSRD curve follows very well with the prediction based on the numerical integration along the Hugoniot. Using the theoretical EXAFS Debye–Waller factor, we obtain the Hugoniot temperatures which are highly consistent with recent experimental results. This research advances the correlated Einstein model on the calculation of MSRDs of materials at high temperature and high pressure, specifically, it includes the zero-point vibrations at low temperature and contains the anharmonicity contributions caused by thermal lattice vibrations at high temperature.