An isothermal equation of state for solids

An isothermal equation of state (EOS) for solids, recently suggested by the authors in the realistic form, V/V₀=f(P), with relative volume as the dependent and the pressure as the independent variable, was shown to have an advantage for some close-packed materials in that it allows B′_∞=(∂B_s/∂P)_s(P→∞) to be fitted, and this is where the usual standard equations fail. In the present study, our EOS is applied to a number of inorganic as well as organic solids, including alloys, glasses, rubbers and plastics; varying widely in their bonding and structural characteristics, as well as in their bulk modulus values. A very good agreement is observed between the data and fits. The results obtained are compared with those from two well-known equations, expressible in the realistic form, proposed by Murnaghan and Luban. Further, the results are also compared with those from the widely used two- and three-parameter EOSs, expressible in the unrealistic form only, P=f(V/V₀), proposed by Birch—and also with those from the EOS model of Keane in which B′_∞ is explicitly expressed as an equation of state parameter. The results obtained from our model compare well to these EOSs. Our EOS, in general, yields the smallest mean-squared deviations between data and fits. The values of B′_∞calculated from our EOS are compared with those from Keane's model. Further, we have studied the variation of B′_∞with temperature using the experimental isotherms of Mo and W at 10 different temperatures ranging from 100 to 1000 K, and observed that the values of B′_∞ yielded by our model and that of Keane vary, as expected, within a narrow range. Furthermore, our EOS is applied to study the stability of the fit parameters with variation in the pressure ranges with reference to the isothermal compression data on Mo and W—and also to study the variation of isothermal bulk modulus with pressure, with reference to the ultrasonic data on NaCl and noted a very good agreement with experiment. In addition, our model is applied, with B₀ and B′₀ constrained to the theoretical values, to the five theoretical isotherms of MgO at 300, 500, 1000, 1500 and 2000 K obtained on the basis of a first principles approach—a good agreement is observed with the predictions, and the values of B′_∞ inferred at different temperatures tend to converge to a constant value.     

On the limitations of the equation of state based on the Lennard-Jones potential

It is emphasized that any equation of state (EOS) based on the generalized Lennard-Jones potential or the Mie potential, suffers from two main shortcomings as pointed out by Stacey and Davis [2]. One of the shortcomings viz. the problem related to imaginary numbers for the exponents in the potential function, has been removed recently by Jiuxun [11] by using a relationship between the exponents. However, the modified EOS obtained by Jiuxun suffers from the second shortcoming viz. it gives lower values for −B ₀ B″₀, an important equation of state parameter related to the second pressure derivative of the bulk modulus. Values of B ₀ B″₀ obtained by Jiuxun are not consistent with those reported by Stacey and Davis.      

Shanker formulation needs modification at high pressures

It is found that the Shanker formulation widely used in the literature to study the thermal expansion of solids (at constant pressure) works under the effect of pressure (at constant temperature) up to a limited range (≈30 kbar). Large deviations occur, when the pressure range is increased, demonstrating the failure of the relation under high pressure (at constant temperature). We, therefore, propose the modification in the formulation on an empirical basis. The modified relation is used to study the compression behavior of ionic solids viz. NaF, NaCl, NaBr and NaI crystals. The results obtained with the modified relation are compared with the experimental data in the light of the results obtained from Shanker formulation and Birch-Murnaghan equation of state. A good agreement between theory and experiment demonstrates the validity of the modification presented in the present note.     

Experimental equations of state for the rare gas solids

Piston-displacement equations of state (EOS) for the rare gas solids neon, argon, krypton and xenon have been determined to 20 kbar at temperatures from 4.2 K to the triple point in each case, with V(P,T) relations which are believed to be accurate to roughly 0.1 per cent. The present paper describes the results for the three heavier solids, argon, krypton and xenon, and indicates consistency between these results and other low pressure experiments at all temperatures. In particular, the individual isotherms can be represented by P(V) relations which are suggested by the form of the Lennard-Jones potential, the bulk moduli are only slightly temperature dependent at constant volume, and the data for the three solids appear to obey a reduced EOS both at T = 0 and near the triple point.     

A perturbed hard-sphere equation of state extended to imidazolium-based ionic liquids

A perturbed hard-sphere equation of state (EOS) has been previously employed to predict pressure–volume–temperature properties of some ionic liquids (ILs) with phosphonium-, pyridinium-, and pyrrolidinium cations. In this work, we have extended the considered EOS to another class of ILs in compressed states. This class consists of 14 imidazolium-based ILs. The predicted densities were compared with those obtained from the experiment, over a broad pressure range from 0.1 to 200 MPa. From 1,122 data points examined for the aforementioned ILs, the total average absolute deviation was found to be 1.05%.     

Pressure dependence of thermodynamic properties of NaBr,NaI and AgCl

Various state-equations, derived by expanding energy as a function of volume in Taylor series and using different order Pade’s approximants, have been combined with quasi-harmonic approximation for free energy to reproduce the pressure dependence of thermodynamic properties of NaBr, NaI and AgCl crystals. We have used these state-equations to compute the reduced volume, the isothermal bulk modulus and the pressure derivative of isothermal bulk modulus for the three crystals at various pressures (up to 80 kbar) and at room temperature (T=298 K). The results obtained are reasonably good lending support to the state-equations and the technique used to extend their applicability. The significant results obtained in the present study include the unified reduced equation of state for the three crystals which generates almost a single curve for theP – V behaviour at room temperature.     

A new temperature-dependent equation of state of solids

In the present paper, a temperature-dependent equation of state (EOS) of solids is discussed which is found to be applicable in high-pressure and high-temperature range. Present equation of state has been applied in 18 solids. The calculated data are found in very good agreement with the data available from other sources.     

High Pressure Study of HfNi Crystallographic and Electronic Structure

The crystallographic structure and electronic properties of HfNi were studied as a function of pressure by combining X-ray diffraction results with the full potential linearized augmented plane wave (LAPW) calculations. No phase transition was observed up to a pressure of 35.3 GPa, with a total volume contraction of V/V ₀ = 0.85, a bulk modulus value of B ₀ = 52 ± 3 GPa and B ₀ ^′ = 1.29 ± 0.26. The calculated linear increase in the V _zz value as a function of the pressure induced volume reduction at the hafnium site was attributed mainly to the p–p contribution, while in the nickel site, a non negligible d–d contribution to V _zz is also observed, and attributed to the high 3d-partial DOS near the nickel nucleus. Based on the total electronic DOS at E _Fermi calculated for 0 K (N(E ⁰ _Fermi)), a value of 6.85 and 5.03 (mJ/mol/k²) was calculated for the band contribution (γ _band) to the electronic specific heat coefficient (γ) at a pressure of 0 and 35.3 GPa, respectively.     

高压下碱卤晶体的三参量等温状态方程

 由等温体积弹性模量的定义及其与压强关系的假设出发,导出了一个新的适用于高压下碱卤晶体的三参量等温状态方程。计算了室温下NaCl晶体在0～30 GPa压强范围内、CsCl晶体在0～40 GPa压强范围内的相对体积以及NaCl晶体在0～30 GPa压强范围内的等温体积弹性模量,计算结果与实验值一致。对等温体积弹性模量及其对压强的一阶、二阶导数与压强的关系进行了讨论,指出压强趋于无穷大时的等温体积弹性模量是常数。     

Equation of state for inert gas solids

The equation of state is a fundamental relation to analyse the thermophysical properties of different class of solids and it plays a key role in basic and applied condensed matter physics research. A lot of work has been done in the field of ionic solids, minerals and metals but a very little work is done in the field of inert gas solids. Most of the equations of state failed to explain the properties of inert gas solid because of their abnormal behavior in the low temperature range. In the present paper, Singh-Gupta equation of state has been used to study the properties of these solids. The results obtained using these equations have shown a good agreement with available experimental results. Thus it is shown that these equations of states successfully explain the behavior of inert gas solids.      

Thorium under strong compression—a test case for the evaluation of EOS data by different forms and procedures

Abstract

X-ray diffraction on Thorium under pressures to 300 GPa at ambient temperature provides new EOS data for both the (low pressure) fcc phase as well as for the (high pressure) bct phase. A detailed evaluation of these data and a comparison with previous results shows systematic differences in the fitted parameters resulting from the use of different EOS forms and from the correlations in the parameters used in all these fitting procedures. The absolute uncertainties in these parameters are elucidated and special attention is given to the compatibility of different EOS forms for phases which are related by second order transitions.     

Improved equation of state for ionic liquids using surface tension

The major objective of this work was the development of a reliable model to describe volumetric properties of ionic liquids (ILs). In this regard, we have applied the Ihm–Song–Mason equation of state (EOS) to some phosphonium- and imidazolium-based ILs. Three temperature-dependent parameters in the equation of state have been scaled based on the surface tension and the liquid density at room temperature. In order to improve the predictive power of the mentioned EOS for ILs, we have proposed using a simple modification. We have taken 1,228 experimental points to show the reliability of the improved EOS. The comparison of predicted densities with literature data over a broad range of temperature, 293–472 K, and pressures up to 200 MPa led to encouraging results. The average absolute deviation of calculated densities from literature values was found to be 0.75%.     

A search for mechanoluminophors capable of pressure-induced thermal population of excited states

During the process of deforming a crystal, a high pressure is developed near the tip of mobile cracks, which may in turn produce a new ground state by thermal electron transfer. Upon sudden release of pressure, the electron can either relax to one atmosphere ground state or remain in the excited state potential well long enough to relax to one atmosphere and radiatively transfer back to the ground state. For analysing the pressure induced thermal population of the excited state, the mechanoluminescence(ML) and high pressure photoluminescence(PL) of several organic and inorganic crystals were measured. The study indicated that usual pressure coefficient of energy shift of the order of 50–100 cm⁻¹/kbar and the stress at the crack-tip of the order of 5–10 kbar, are not sufficient to cause the thermal population of the excited state. If by any means the product of pressure coefficient and stress at the mobile crack-tip can be increased by 50 to 100 times, then the thermal population of the excited states may take place. Using the pressure coefficient of energy shift and the difference in ML and PL spectra, and using independently the change in relative intensities of the vibronic peaks, the pressure at the emitting mechanoluminescent crystal sites is evaluated and it is found to be of the order of several kbar which varies from crystal to crystal.     

High Pressure Instrumentation: Low and Negative Pressures

Much work on semiconductors, soft solids and biological materials does not require the megabar capability of the diamond anvil cell; a few accurate kbar being all that may be required. Work in this range poses its own challenges, to make the experiments routine, safe and reliable, and well-calibrated. We contrast diamond anvil cells working at what for them is very low pressure, with traditional bombs working at what for them is dangerously high pressure. We describe our preferred solution, a single-diamond cell, and demonstrate its use with Raman data from ethanol under low pressure. Negative hydrostatic pressure cannot be obtained by traditional methods. However, we present data showing the Raman spectrum of ethanol apparently at the negative pressure of m 3 kbar.     

Applicability of equation of state in extreme compression region and study of diatomic solids under pressure

In the present study, the expression of first pressure derivative of bulk modulus B′?earlier derived by Goyal and Gupta EoS [7] is corrected and the validity of the EoS is then verified in extreme compression region. The expressions of second and third order pressure derivative $(B^{{}^{″}},B^{″\prime })$of bulk modulus are obtained. The values of B′, B²B? to BB″,?Grunesien parameters?λ_∞,?γ_∞,?q_∞ at infinite pressure (P?→?∞)?are calculated using the identities [11–13] to check the validity of the equation in extreme compression region. The Goyal and Gupta EoS is then used to study the volume compression in diatomic solids, LiH and MgO. The values of pressure, bulk modulus and its first order pressure derivative at different compressions are calculated and compared with the results obtained from Hama–Suito EoS. The results justify the validity of the present EoS in high pressure region and the results for diatomic solids are also found in good consistency with the compared results.     

A high pressure,low temperature system for μSR studies

A high pressure, low temperature system for μSR studies is described which compresses gases like He or Ar up to 14 kbar at temperatures between 10K and room temperature. The present construction of the cell, made of hardened CuBe, limits the pressure range to 7 kbar. The sample space is approximately 4 cm³. Pressure generation up to 14 kbar is accomplished in about 15 minutes while cool-down to 10K requires about 2 hours.     

Magnetic quantum oscillations in the organic superconductor κ-(BEDT-TTF)2Cu[N(CN)2]Br

The interlayer magnetoresistance of the organic superconductor κ-(BEDT-TTF)₂Cu[N(CN)₂)Br is measured at ambient pressure and under pressures of up to 12.5 kbar. In addition to the slow Shubnikov-de Haas oscillations with a frequency of ≃150 T observed at P⩾5 kbar, rapid oscillations attributed to the magnetic breakdown orbit enveloping an area equal to 100% of the area of the Brillouin zone are found to emerge above B=20 T. The latter oscillations are observed at ambient pressure as well as under pressures of up to 9 kbar. Pis’ma Zh. éksp. Teor. Fiz. 66, No. 3, 190–194 (10 August 1997) Published in English in the original Russian journal. Edited by Steve Torstveit.     

Equations of State for Regular Solids

Equations of State (EOS) for solids under strong compression and wide ranges in temperature are most commonly represented by "parametric" EOS forms using temperature dependent parameters for the volume V 0 , the bulk modulus K 0 , and its pressure derivative K_{0}^{\prime} for the given ambient (or zero) pressure. Therefore, various common "parametric" EOS forms are compared at first with the recently proposed [1,2] A dapted P olynomical expansion AP2, and in the second part with the Mie-Grüneisen approach, which uses one of the common EOS forms for the pressure of the static lattice or for the zero temperature isotherm, p ZT ( V ), and a detailed modelling of the additional thermal pressure, p th ( V , T ), in the form p(V,T) = p_{\rm ZT}(V) + p_{\rm th}(V,T) . Thereby, it is shown, that "intrinsic" anharmonicity effects have to be taken into account and between the two schemes differences of the order of a few percent in pressure are noticed for regular solids, like Cu, Ag, and Au. These differences are discussed with respect to the present uncertainties in a practical pressure scale for wide ranges in pressure (up to several TPa) and in temperature (up to 1500 K and above).     

Perturbation approach for equation of state for hard-sphere and Lennard–Jones pure fluids

In this paper we have established the equation of state (EOS) for liquids. The EOS was established for hard-sphere (HS) fluid along with Lennard–Jones (LJ) fluid incorporating perturbation techniques. The calculations are based on suitable axiomatic functional forms for surface tension S _m (r), r ≥ d/2 with intermolecular separation r, as a variable, and m is an arbitrary real number (pole). The results for βP/ρ from the present EOS thus obtained are compared with Percus-Yevick (PY), scaled particle theory (SPT), and Carnahan–Starling (CS). In addition, we have found a simple EOS for the HS fluid in the region which represents the simulation data accurately.     

Thermodynamic properties of cubic boron nitride based on an analytic mean field approach

We present a detailed derivation of the analytic expressions for the equation of state (EOS) and internal energy of Morse model solids based on an analytic mean field potential (AMFP) method. The formalism is applied to cubic boron nitride (c-BN). One set of potential parameters are determined by fitting the experimental P-V-T data of c-BN up to 160 GPa at 295 K and 80 GPa in the range 500–900 K. Various physical quantities including the isothermals, thermal expansion, isochoric heat capacity, Helmholtz free energy and internal energy are calculated and analyzed. The theoretical results are consistent with the available experimental data and those calculated by others. These results presented in this paper verify that the AMFP method is a useful approach to consider the anharmonic effect at high temperature. Numerous reasonable predictions and the change trend of the properties for c-BN at extreme conditions have been given.