Thermal equation of state of goethite (α-FeOOH)

ABSTRACT

The pressure–volume–temperature (P–V–T) equation of state (EoS) of natural goethite (α-FeOOH) has been determined by an X-ray diffraction study using synchrotron radiation. Fitting the volume data to the third-order Birch–Murnaghan EoS yielded an isothermal bulk modulus, B₀ of 85.9(15)?GPa, and a pressure derivative of the bulk modulus, B′, of 12.6(8). The temperature derivative of the bulk modulus, (?B/?T)_P, was –0.022(9)?GPa?K^?1. The thermal expansion coefficient α₀ was determined to be 4.0(5)?×?10^?5?K^?1.     

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.      

Generalized pressure-volume equations mimicking the Stacey reciprocal K-primed equation of state

Two generalized polynomial expressions, one based on a logarithmic form and the other on an exponential form have been set up that give pressure-volume (P-V) relationship and higher derivative properties mimicking the Stacey reciprocal K-primed equation of state. The results have been obtained for pressure P, bulk modulus K and its pressure derivative K′ for six metals viz. Ag, Al, Au, Cu, Mo and W at different values of compression down to V/V₀=0.5. The zero-pressure values of input parameters K₀ and have been taken from the literature, whereas and have been fixed to match the Stacey reciprocal K-primed equation. The polynomial equations thus formulated can be used as a substitute for the Stacey equations of state (EOS) for determining P-V relationship and higher derivative properties such as K and K′.     

Analysis of K-prime equations of state

In the present communication, we introduce a new K-prime equation of state (EoS), which can be used to understand the interior of the Earth. The newly developed EoS is found to yield similar results as given by Stacey K-prime EoS and Keane EoS, in reference to the seismological data. However, the zero pressure and infinite pressure extrapolation of higher pressure derivatives of bulk modulus are found different for different K-prime EoS.     

Relaxation of finite, closed systems

On a semi-finite W^*-algebra together with a faithful normal semi-finite trace τ and a one-parameter group of trace preserving ^*-automorphism α_t, we study the limit as t → ∞ of α^*_tψ of a normal state ψ. It is shown that the existence of this limit in the weak sense is determined by the spectral properties of the evolution operator. These results are specialized to finite classical and quantum mechanical systems.     

Observation of χb production in the exclusive reaction ′→γγχb→γγ→γγ(eeorμμ−)

We report on the observation of 1 ³P_J (χ_b) production in the reaction ′→γχ_b→γγ→γγ(e⁺e⁻ or μ⁺μ⁻). The data were recorded with the nonmagnetic CUSB detector at the Cornell Electron Storage Ring, CESR. We observe 124 γγ events with either an electron or muon pair in the final state. In the γγ correlation plot about 40% of the events cluster around (120, 430) MeV.     

Zero Temperature Limits of Gibbs-Equilibrium States for Countable Alphabet Subshifts of Finite Type

Let Σ_A be a finitely primitive subshift of finite type on a countable alphabet. For appropriate functions f:Σ_A → IR, the family of Gibbs-equilibrium states (μ_tf)_t⩾1 for the functions tf is shown to be tight. Any weak^*-accumulation point as t→∞ is shown to be a maximizing measure for f.     

An in situ high-pressure X-ray diffraction experiment of hydroxyapophyllite

The compression behavior of a natural hydroxyapophyllite is investigated up to about 10.01 GPa at 300 K using in situ angle-dispersive X-ray diffraction and a diamond anvil cell at High Pressure Experiment Station, Beijing Synchrotron Radiation Facility (BSRF). Over this pressure range, no phase change or disproportionation is observed. The isothermal equation of state is determined for the first time. The values of zero-pressure volume V₀, isothermal bulk modulus K₀, and K₀' refined with a third-order Birch-Murnaghan equation of state are V₀=1276.3±0.9 ų, K₀=71±3 GPa, and K₀'=8±1. Furthermore, we confirm that the values of linear compressibility β along a and c directions of hydroxyapophyllite are elastically anisotropic.     

Analysis of K-prime equations of state

We present an analysis of the $K$-prime equations of state (EOS) due to Keane and Stacey. It is found that the two EOS differ significantly from each other in some important respects. $K$-prime represents the pressure derivative of the bulk modulus. It is shown that the volume dependence of $K$-prime and higher derivative properties predicted from the Keane EOS are compatible with those predicted from Stacey’s reciprocal $K$-prime EOS only when the Murnaghan approximation is valid. It has been emphasized that the Stacey EOS is more appropriate for describing the relationship between pressure and the bulk modulus and its pressure derivative. The results based on the two EOS have been compared and discussed.     

Compression behavior of nanocrystalline anatase TiO2

We present a synchrotron X-ray diffraction study of pressure-induced changes in nanocrystalline anatase (with a crystallite size of 30-40 nm) to 35 GPa. The nanoanatase was observed to a pressure above 20 GPa. Direct transformation to the baddeleyite-TiO₂ polymorph was seen at 18 GPa. A fit of the pressure versus volume data to a Birch-Murnaghan equation yielded the following parameters: zero-pressure volume, V₀=136.15 Å³, bulk modulus, K_T=243(3) GPa, and the pressure derivative of bulk modulus, K′=4 (fixed). The bulk modulus value obtained for the nanocrystalline anatase is about 35% larger than that of the macrocrystalline counterpart.     

High pressure X-ray diffraction study of Fe2B

We report the results of a synchrotron based X-ray diffraction study of bct-Fe₂B under quasi-hydrostatic conditions from 0 to 50 GPa. Over this pressure range, no phase change or disproportionation has been observed. A weighted fit of the data to the Birch-Murnaghan equation of state yields a value of the bulk modulus, K, of 164±14 GPa and the first pressure derivative of the bulk modulus, K′, of 4.4±0.5. The compression is found to be anisotropic, with the a-axis being more incompressible than the c-axis.     

Gauge theories in the many-gluon limit

We argue that the idea that the dynamics of a gauge theory simplifies in the limit N → ∞, where N is the number of colors, can be invoked even if the gauge group is an exceptional Lie group, rather than one of the classical groups. We also point out that quantum tunneling phenomena can in some cases survive in the N → ∞ limit, contrary to the usual claim that the N → ∞ limit is “classical.”     

Thermo-elastic property of Ca(OH)2 portlandite

An equation of state (EoS) for Ca(OH)₂ portlandite has been obtained through measurements of pressure and temperature dependence of volume by means of in-situ X-ray observation. The bulk modulus and its pressure derivative at zero pressure calculated using third-order Birch-Murnaghan's equation of state is 33.1 GPa and 4.2 at 300 K, respectively. The unit cell parameters and the volumes have been also determined at 573 K and 673 K. Temperature derivatives of the bulk modulus and its pressure derivative have been calculated to be ?0.022 GPa/K and 0.0072 K^?1, respectively. Thermal expansion coefficient of portlandite has been calculated from the EoS. The pressure dependence of entropy has been obtained from the present thermo-elastic parameters.     

Elastic, thermal and structural properties of platinum

The thermal equation of state (EOS) for platinum has been calculated to 300 GPa and 3000 K using ab initio molecular dynamics employing the local density approximation (LDA) and the projector augmented-wave methods (PAW). Direct ab initio molecular dynamics avoids the simplifying assumptions inherent in empirical treatments of thermoelasticity. A third-order Birch-Murnaghan equation EOS fitted to the 300 K data yielded an isothermal bulk modulus of B_T0=290.8 GPa and a pressure derivative of B_T′=5.11, which are in better agreement with the measured values than those obtained by previous calculations. The high-temperature data were fitted to a thermal pressure EOS and a Mie-Grüneisen-Debye EOS. The resulting calculated thermal expansion coefficient, α₀, temperature derivative of the isothermal bulk modulus, (∂B_T/∂T)_V, and second temperature derivative of the pressure, (∂²P/∂T²)_V, were 1.94×10⁻⁵ K⁻¹, −0.0038 GPa K⁻¹, and 1.7×10⁻⁷ GPa² K⁻², respectively. A fit to the Mie-Grüneisen-Debye EOS yielded values for the Grüneisen parameter, γ₀, and its volume dependence parameter, q, of 2.18 and 1.75, respectively. An analysis of our data revealed a strong volume dependence of the thermal pressure of platinum. We also present a qualitative analysis of the effects of intrinsic anharmonicity from the calculated Grüneisen parameter at high temperatures.     

Thermal equation-of-state of osmium: a synchrotron X-ray diffraction study

The pressure-volume-temperature behavior of osmium was studied at pressures and temperatures up to 15 GPa and 1273 K. In situ measurements were conducted using energy-dispersive synchrotron X-ray diffraction in a T-cup 6-8 high pressure apparatus. A fit of room-temperature data by the third-order Birch-Murnaghan equation-of-state yielded isothermal bulk modulus K₀=435(19) GPa and its pressure derivative K′₀=3.5(0.8) GPa. High-temperature data were analyzed using Birch-Murnaghan equation of state and thermal pressure approach. The temperature derivative of bulk modulus was found to be −0.061(9) GPa K⁻¹. Significant anisotropy of osmium compressibility was observed.     

In-situ X-ray diffraction study on β-CrOOH at high pressure and high-temperature

ABSTRACT

High pressure hydrous phases with distorted rutile-type structure have attracted much interest as potential water reservoirs in the Earth’s mantle. An in-situ X-ray diffraction study of β-CrOOH was performed at high pressures of up to 6.2?GPa and high-temperatures of up to 700?K in order to clarify the temperature effect on compression behaviors of β-CrOOH. The P-V-T data fitted to a Birch–Murnaghan equation of state yielded the following results: isothermal bulk modulus K_T0?=?191(4)?GPa, temperature derivative (?K_T/?T)_P?=??0.04(2)?GPa?K^?1, and volumetric thermal expansion coefficient α?=?3.3(2)?×?10^?5?K^?1. In this study, at 300?K, the a-axis became less compressible at pressures above 1–2?GPa. We found that the pressure where the slopes of a/b and a/c ratios turned positive increased with temperature. This is the first experimental study indicating the temperature dependence of the change in the axial compressibility in distorted rutile-type M³⁺OOH.     

P–V–T equation of state of rhodium oxyhydroxide

A high pressure X-ray diffraction study of RhOOH was carried out up to 17.44?GPa to investigate the compression behavior of an oxyhydroxide with an InOOH-related structure. A fit to the third-order Birch–Murnaghan equation of state gave K₀?=?208?±?6?GPa, and K′?=?9.4?±?1.3. The temperature derivative of the bulk modulus was found to be ?K/?T?=??0.06?±?0.02?GPa K^?1. The refined parameters for volume thermal expansion were α₀?=?2.7?±?0.3?×?10^?5 K^?1; α₁?=?1.7?±?1.1?×?10^?8 K^?2 in the polynomial form (α(T)?=?α_0?+?α₁(T?300)). Our results show that RhOOH is very incompressible, and has a higher bulk modulus than other InOOH-structured oxyhydroxides (e.g. δ-AlOOH, ε-FeOOH, and γ-MnOOH).     

Resonances in at rest

Data on at rest show two resonant processes: (a) f₀(1370)η,f₀(1370)→σσ and ρρ, (b) η(1440)σ, η(1440)→ηπ⁺π⁻. The branching ratio BR[f₀(1370)→ρρ]/BR[f₀(1370)→σσ]=0.98±0.25 in the mass range available here. Using data on , the ratio Γ_4π/Γ_2π5 for f₀(1370). The effects of the strongly s-dependent width of f₀(1370) are discussed in some detail.The η(1440) is observed decaying to ησ and a₀(980)π, with strong destructive interference between them. In its decay to a₀(980)π, a narrow peak appears in the ηπ mass spectrum, but 30–50 MeV above that usually attributed to a₀(980) and significantly above the KK threshold. This effect is explained naturally by a two-step process: η(1440)→K^*(890)K followed by rescattering of the two kaons through a₀(980) to ηπ above the KK threshold.     

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.     

The emission spectrum of HgI

The ultraviolet emission systems of HgI—C → X, D → X, F₃ → X, G → X, and H → X—are photographed and analyzed using tesla-discharge sources containing isotopically pure ²⁰⁰Hg. The previous assignment for F₃ → X is revised, the main change being a decrease by 3 units in the v″ numbering. Results for the other systems corroborate the existing interpretations, except that for C → X there are prominent intensity gaps in the unresolved rotational structure of the bands, not reported previously for “natural” HgI spectra. These gaps are attributed to perturbations of the C state by high levels (v ≈ 90) of the B state. The data for these systems are combined with existing B → X data for ²⁰⁰Hg¹²⁷I and ²⁰⁰Hg¹²⁹I and fitted simultaneously to yield optimal vibrational parameters for all states. In this analysis the X state is fitted to a mixed representation—a polynomial in ( ) for v ≤ 20 and a near-dissociation expansion for v ≥ 20, with G_v and its first derivative constrained to be continuous at v = 20. The revised estimate of _e for X is 2800 ± 40 cm⁻¹. The recommended vibrational parameters (cm⁻¹) for v ≤ 20 are ω_e = 125.41, ω_ex_e = 1.009, ω_ey_e = −0.0159.