We report on the determination of high-pressure elasticity from thermal diffuse x-ray scattering of magnesium oxide at pressures up to 28 GPa. We find that the full elasticity tensor as a function of pressure can be extracted from x-ray scattering of single crystals. This enables the simultaneous measurement of elasticity and crystal structure from a single experiment and thus marks a decisive step towards establishing reliable calibration for extreme pressures. Elastic constants can be extracted with this method from small crystals of arbitrary symmetry, shape and optical properties and will allow for significant progress in our understanding of the elastic behaviour of planets interior at geophysically relevant conditions, structural phase transitions and fundamental interactions of phonons with other quasi-particles.
We present a method for the precise determination of the full elasticity tensor from a single crystal diffraction experiment using monochromatic X-rays. For the two benchmark systems calcite and magnesium oxide we show that the measurement of thermal diffuse scattering in the proximity of Bragg reflections provides accurate values of the complete set of elastic constants. This approach allows for a reliable and model free determination of the elastic properties and can be performed together with crystal structure investigation in the same experiment.
We combine density functional theory within the local density approximation, the quasiharmonic approximation, and vibrational density of states to calculate single crystal elastic constants, and bulk and shear moduli of diamond at simultaneous high pressures and temperatures in the ranges of 0-500 GPa and 0-4800 K. Comparison with experimental values at ambient pressure and high temperature shows an excellent agreement for the first time with our first-principles results validating our method. We show that the anisotropy factor of diamond increases to 40% at high pressures and becomes temperat…
Comment on "On the importance of the free energy for elasticity under pressure"
Marcus等人（Marcus P，Ma H和Qiu S L 2002 J. Phys.：康登斯Matter 14 L525）声称，在压力下材料的热力学性质必须使用吉布斯自由能$G$计算，而不是内能$E$。Marcus等人指出，"$G$而不是$E$的最小值给出了平衡结构; $G$而不是$E$相对于平衡结构处的应变的二阶导数给出了平衡弹性常数。“这两种说法都不正确。
We report measurements of the evolution of the diffuse scattering in a single crystal PbMg1/3Nb2/3O3 as a function of hydrostatic pressure. Upon applying pressure the diffuse scattering intensity decreases and is suppressed at about 3 GPa, while no change in the line shape is observed. Correlations between Pb displacements, diffuse scattering and relaxor properties are discussed.
The pressure-induced phase transition, elasticity behavior, thermodynamic properties, and $P\mathtt{-}T$ phase diagram of $α$, $ω$, and $β$ equiatomic HfTi alloy are investigated using first-principles density-functional theory (DFT). The simulated pressure-induced phase transition of the alloy follows the sequence of $α\mathtt{\rightarrow}ω\mathtt{\rightarrow}β$, in agreement with the experimental results of Hf and Ti metals. Our calculated elastic constants show that the $α$ and $ω$ phases are mechanically stable at ambient pressure, while the $β$ phase is unstable, where a critical pressure…
Experimental phonon imaging in diamond anvils cell is demonstrated to be an adequate tool to extract the complete set of elastic constants of single-crystalline silicon up to the ZB$\rightarrow β-$Sn transition (10 GPa). Contrary to what was commonly admitted, we demonstrate that the development of the strain-energy density in terms of strains cannot be stopped, for silicon, after the terms containing the third-order elastic constants. Nonlinear elasticity, degree of anisotropy and pressure-induced mechanical stability of the cubic silicon structure are thus revisited and investigated in more …
tructural transformation, pressure dependent elasticity behaviors, phonon, and thermodynamic properties of the equiatomic TiZr alloy are investigated by using first-principles density-functional theory. Our calculated lattice parameters and equation of state for $α$ and $ω$ phases as well as the phase transition sequence of $α$$\mathtt{\rightarrow}$$ω$$\mathtt{\rightarrow}$$β$ are consistent well with experiments. Elastic constants of $α$ and $ω$ phases indicate that they are mechanically stable. For cubic $β$ phase, however, it is mechanically unstable at zero pressure and the critical pressu…
A general nonlinear theory for the elasticity of pre-stressed single crystals is presented. Various types of elastic moduli are defined, their importance is determined, and relationships between them are presented. In particular, B moduli are present in the relationship between the Jaumann objective time derivative of the Cauchy stress and deformation rate and are broadly used in computational algorithms in various finite-element codes. Possible applications to simplified linear solutions for complex nonlinear elasticity problems are outlined and illustrated for a superdislocation. The effect …
Silica is an abundant component of the Earth whose crystalline polymorphs play key roles in its structure and dynamics. As the simplest silicates, understanding pure silica is a prerequisite to understanding the rocky part of the Earth, its majority. First principle density functional theory (DFT) methods have often been used to accurately predict properties of silicates. Here, we study silica with quantum Monte Carlo (QMC), which until now was not computationally possible for such complex materials, and find that QMC overcomes the failures of DFT. QMC is a benchmark method that does not rely …
To study temperature dependent elastic constants, a new computational method is proposed by combining continuum elasticity theory and first principles calculations. A Gibbs free energy function with one variable with respect to strain at given temperature and pressure was derived, hence the full minimization of the Gibbs free energy with respect to temperature and lattice parameters can be put into effective operation by using first principles. Therefore, with this new theory, anisotropic thermal expansion and temperature dependent elastic constants can be obtained for crystals with arbitrary …
Using a generalization of vector calculus for space with non-integer dimension, we consider elastic properties of fractal materials. Fractal materials are described by continuum models with non-integer dimensional space. A generalization of elasticity equations for non-integer dimensional space, and its solutions for equilibrium case of fractal materials are suggested. Elasticity problems for fractal hollow ball and cylindrical fractal elastic pipe with inside and outside pressures, for rotating cylindrical fractal pipe, for gradient elasticity and thermoelasticity of fractal materials are sol…
We consider the mechanism of elastic strains and stresses as the main controlling factor of structure change under the influence of temperature, magnetic field, hydrostatic pressure. We should take into account that the energy of elastic deformation is commensurate to the energy of electric interactions and that is much higher than the rest of the bonds of lower energy value. Besides, the energy elastic stresses are of long range, so it forms the linearity in magnetization and bulk change. These regularities requires a fundamental understanding of the laws of interaction with respect to accept…
High-pressure and variable temperature single crystal synchrotron X-ray measurements combined with first-principles based molecular dynamics simulations study diffuse scattering in the relaxor ferroelectric system PSN (PbSc$_{1/2}$Nb$_{1/2}$O$_3$). Constant temperature experiments show pressure induced transition to the relaxor phase at different temperatures characterized by butterfly and rod shaped diffuse scattering around the $\{$h00$\}$ and $\{$hh0$\}$ Bragg spots, respectively. The simulations reproduce the observed diffuse scattering features as well as their pressure-temperature behavi…
Fe and Al are two of the most important rock-forming elements other than Mg, Si, and O. Their presence in the lower mantle's most abundant minerals, MgSiO_3 bridgmanite, MgSiO_3 post-perovskite and MgO periclase, alters their elastic properties. However, knowledge on the thermoelasticity of Fe- and Al-bearing MgSiO_3 bridgmanite, and post-perovskite is scarce. In this study, we perform ab initio molecular dynamics to calculate the elastic and seismic properties of pure, Fe^{3+}- and Fe^{2+}-, and Al^{3+}-bearing MgSiO_3 perovskite and post-perovskite, over a wide range of pressures, temperatur…
The effect of pressure on the zone-center optical phonon modes and the elastic properties of hafnium has been studied by both experimental (Raman scattering) and theoretical (DFT) approaches. It was found an anomaly dependence of the phonon frequency of E$_{2g}$ mode in the pressure range from 0 to 67 GPa. The calculated electronic structure of hafnium shows significant changes under pressure, which have a pronounced spatial anisotropy similar to the anisotropy of the observed phonon properties. The dependencies of the elastic properties on pressure including the components of the elasticity t…
In this paper, we did a systematic comparative study on the accuracy of two computational methods of elastic constants combined with the density functional theory (DFT), the stress-strain method and the energy-strain method. We took metal Ru as a prototype to compare its high-pressure elastic constants calculated by our present stress-strain method with the previous energy-strain results by others. Although the two methods yielded almost the same accuracy of high-pressure elastic constants for Ru, our stress-strain method directly based on the Hooke's law of elasticity theory is much straightf…
The third-order elastic moduli and pressure derivatives of the second-order elastic constants of novel B2-type AlRE (RE=Y, Pr, Nd, Tb, Dy, Ce) intermetallics are presented from first-principles calculations. The elastic moduli are obtained from the coefficients of the polynomials from the nonlinear least-squares fitting of the energy-strain functions. The calculated second-order elastic constants of AlRE intermetallics are consistent with the previous calculations. To judge that our computational accuracy is reasonable, the calculated third-order constants of Al are compared with the available…
Anomalous pressure dependence of the atomic displacements in the relaxor ferroelectric PbMg$_{1/3}$Ta$_{2/3}$O$_3$
本文用粉末中子衍射方法研究了PbMg $_{1/3}$ Ta $_{2/3}$ O $_3$（PMT）弛豫铁电体在高达$\sim 7$ GPa的静水压力下的晶体结构。我们发现一个急剧的压力引起的铅位移从反转中心，这与增加了$\sim$ 50%的各向异性的氧温度因子相关的减少。相比之下，Mg/Ta的振动对压力相当不敏感。我们属性这些变化负责先前报道的压力引起的抑制的反常介电常数和漫散射弛豫铁电体。
Third-order elasticity (TOE) theory is predictive of strain-induced changes in second-order elastic coefficients (SOECs) and can model elastic wave propagation in stressed media. Although third-order elastic tensors have been determined based on first principles in previous studies, their current definition is based on an expansion of thermodynamic energy in terms of the Lagrangian strain near the natural, or zero pressure, reference state. This definition is inconvenient for predictions of SOECs under significant initial stresses. Therefore, when TOE theory is necessary to study the strain de…
The present paper deals with the analysis of experimental results taking into account mechanisms brought by the bulk elastic energy transformed by the thermodynamic parameters, temperature, magnetic fields, high hydrostatic pressure (T-H-P). An effect of the external parameters is considered through the separation of critical lines and points of the cooling-heating effect in the course of analysis of linear elastic evolution of properties in sign alternating and crossing effects of elastic stress energy as well as their value for structural reorganizations at reversible volume changes (structu…
Floppy Networks (FNs) provide valuable insight into the origin of anomalous mechanical and thermal properties in soft matter systems, from polymers, rubber, and biomolecules to glasses and granular materials. Here, we use the very same FN concept to construct a quantitative microscopic theory of empty perovskites, a family of crystals with ReO$_3$ structure, which exhibit a number of unusual properties. One remarkable example is ScF$_3$, which shows a near-zero-temperature structural instability and large negative thermal expansion (NTE). We trace these effects to an FN-like crystalline archit…
