Murnaghan Equation Of State Research Articles

An analytic model for the prediction of incoherent shaped charge jets

It is now a well recognized fact that the jet from a shaped charge can be overdriven in the sense that the fastest moving particles are not produced as a cohesive mass of material. Rather, the tip material may be produced as a number of discrete particles which possess different nonzero radial components of velocity and hence spread out from the axis of symmetry of the charge. Such a jet is classed as incoherent and when this incoherency occurs the jet’s target penetration capability is invariably degraded. This physical phenomenon is the subject matter of this article. Several experimental results using common shaped charge materials are presented first. An analytic model which predicts the jet speed at the transition point between a coherent and incoherent state is then described. This model is based on the assumption that a stagnant core with circular boundaries exists in the flow region. Further, the flow field is assumed to be compressible with circular streamlines. The Murnaghan equation of state is used to relate the pressure and density in the flow region where the jet is produced. It is postulated that the transition between a coherent and incoherent state occurs when the circular flow becomes wholly supersonic. The critical Mach number for coherency is shown to be approximated to high accuracy by a simple formula depending on the collapse angle of the flow and the incoming flow speed. Excellent agreement between the model predictions and the experimental data is demonstrated.

A thermodynamic analysis of the system LiAlSiO4-NaAlSiO4-Al2O3-SiO2-H2O based on new heat capacity, thermal expansion, and compressibility data for selected phases

Heat capacity, thermal expansion, and compressibility data have been obtained for a number of selected phases of the system NaAlSiO4-LiAlSiO4-Al2O3-SiO2-H2O. All Cp measurements have been executed by DSC in the temperature range 133–823 K. The data for T ≥ 223 K have been fitted to the function Cp(T) = a + cT −2 + dT −0.5 + fT −3, the fit parameters being The thermal expansion data (up to 525 °C) have been fitted to the function V0(T) = V0(T) [1 + v1 (T−T0) + v2 (T−T0)2], with T0 = 298.15 K. The room-temperature compressibility data (up to 6 GPa) have been smoothed by the Murnaghan equation of state. The resulting parameters are These data, along with other phase property and reaction reversal data from the literature, have been simultaneously processed by the Bayes method to derive an internally consistent thermodynamic dataset (see Tables 6 and 7) for the NaAlSiO4-LiAlSiO4-Al2O3-SiO2-H2O quinary. Phase diagrams generated from this dataset are compatible with cookeite-, ephesite-, and paragonite-bearing assemblages observed in metabauxites and common metasediments. Phase diagrams obtained from the same database are also in agreement with the cookeite-free, petalite-, spodumene-, eucryptite-, and bikitaite-bearing assemblages known to develop in the subsolidus phase of recrystallization of␣lithium-bearing pegmatites. It is gratifying to note that the cookeite phase relations predicted earlier by Vidal and Goffe (1991) in the context of the system Li2O-Al2O3-SiO2-H2O agree with our results in a general way.

Dependence of the superconducting transition temperature and lattice parameter on hydrostatic pressure for Rb 3C 60

The equation of state (EOS) of polycrystalline Rb 3C 60 was accurately determined to 1 GPa at 295 K and 14 K using high-pressure neutron diffraction. In a separate experiment on Rb 3C 60, the variation of the superconducting transition temperature T c with hydrostatic pressure was also measured. The resulting dependence of T c on lattice parameter a is almost linear, with initial slope dT c dlna ∼- +520 K , but is less steep than the T c ( a)-dependence obtained at ambient pressure if Rb is substituted with smaller alkali metal cations. T c ( a) for Rb 3C 60 thus does not follow a universal curve in high pressure and substitution experiments, as previously claimed. Starting with the measured EOS at room temperature, we use a Debye model to estimate the Murnaghan EOS for both NaCl (used here as a pressure calibrant) and Rb 3C 60 over the entire temperature region below 400 K.

Structural Studies of CuInS2 and CuInSe2 under High Pressure

AbstractCuInS2 and CuInSe2 chalcopyrites have been studied up to 29 GPa in a diamond anvil cell by energy dispersive X‐ray diffraction (EDX) using synchrotron radiation. Phase transitions have been observed at 9.5 and 7.6 GPa and the high‐pressure phases have been indexed as cubic. The pressure dependence of the volume per formula unit has been fitted with a first‐order Murnaghan equation of state. The bulk moduli are 75 and 72 GPa for the chalcopyrite phases and 123 and 114 GPa for the cubic phases of CuInS2 and CuInSe2, respectively. The volume reduction at the transition pressure is roughly 10%. The recovered phases are quite different from the initial ones. For CuInSe2 it is of the zincblende type, for CuInS2 it is an amorphous phase. Thus, the pressure‐induced phase changes are not reversible.

Pressure-induced phase transition in icosahedral Al–Li–Cu quasicrystals

Abstract We report in-situ extended X-ray absorption fine-structure (EXAFS) studies of icosahedral Al6Li3Cu quasicrystals, without any aluminium excess, under a high pressure in quasihydrostatic conditions. The EXAFS experiments have been performed at the Cu K absorption edge from ambient pressure up to 33 GPa and give the variation in the average distance of fist neighbours around copper atoms against pressure. This average distance decreases monotonically with increasing pressure up to 12 GPa, remains almost constant up to 20 GPa and then increases. This increase in the interatomic distance is interpreted as the fingerprint of a phase transition. The relative first-neighbour distance has been fitted, up to 12 GPa, by a Murnaghan equation of state so that the zero-pressure bulk modulus is B 0 = 71 GPa for a first derivative B′0 = 6. This B 0 local value found around copper atoms is close to the overall bulk modulus of pure aluminium. After release of the pressure, the Al6Li3Cu quasicrystal shows spectacular 'metastable' properties.

Volume behavior of hydrous minerals at high pressure and temperature; II, Compressibilities of lawsonite, zoisite, clinozoisite, and epidote

The pressure dependence of the lattice parameters of natural zoisite [Ca2A13Si3012(OH)], clinozoisite [Ca2A13Si30'2(OH)], and epidote [Ca2A12FeSi3012(OH)] as well as synthetic lawsonite [CaAl2Si207(OH)2' H20] have been measured at ambient temperatures by energy-dispersive X-ray diffraction in a diamond-anvil cell. The experimental results for each phase may be summarized concisely in terms ofthe ambient-temperature isothermal bulk modulus K298(using the Murnaghan equation of state and assuming K' = 4): Lawsonite: K298= 191 :t 5 GPa; zoisite: K298= 279 :t 9 GPa; clinozoisite: K298= 154 :t 6 GPa; epidote: K298= 162 :t4 GPa. These new measurements, together with the new thermal expansion data in the companion paper (Pawley et al. 1996), were used to calculate some phase equilibria for lawsonite dehydration to high pressures for comparison with experimental brackets. Important discrepancies between calculated and experimentally determined reactions become evident above 3 GPa.

MgAl2O4 spinel crystal structure. An ab initio perturbed ion study

An ab initio perturbed ion (aiPI) study has been carried out for pure and doped MgAl2O4 normal and inverse spinel crystal structures. Clusters containing 136 ions have been built up, using large Slater-type orbitals to represent each atomic center. Basis sets and geometry optimizations have been performed with the aim of determining the relative stability, cell parameters, bulk modulus, force constants, and vibrational frequencies of radial displacements associated with the local relaxation for pure and doped structures. Numerical results are confronted against experimental data and previous theoretical calculations. The bulk modulus of the pure structures has been calculated by means of the Birch--Murnaghan equation of state, the normal structure being less compressible than the inverse one. The optimized geometrical cell parameters of the structures obtained are compared with experimental results. This comparison allows us to analyze the validity of the aiPI methodology for the theoretical characterization of the local properties of complex ionic systems. The energy changes associated with the substitution of Co2+, Mn2+, Ni2+, and Fe2+ for Mg2+ and Cr3+ and Fe3+ for Al3+ in normal and inverse MgAl2O4 structures are evaluated from a direct solid state reaction. All substitutions are favorable, except the replacements of Fe3+ for Al3+ in the normal structure and Fe2+ for Mg2+ in the inverse one. However, defect reaction energies for the normal structure produce large positive values for the substitutions at the octahedral site, and only the replacement of Mg2+ for Co2+, Mn2+, and Ni2+ at the octahedral site given negative defect reaction energies for the inverse structure. The doping process produces a decrease of force constant (K) values associated with the breathing fundamental vibrational mode at tetrahedral site for the normal structure while an opposite effect appears in the inverse structure. © 1995 John Wiley & Sons, Inc.

Quantum-mechanical calculation of the solid-state equilibrium MgO+ alpha -Al2O3

The ground-state crystal energies of cubic ${\mathrm{MgAl}}_{2}$${\mathrm{O}}_{4}$ (spinel) and MgO (periclase) and of rhombohedral \ensuremath{\alpha}-${\mathrm{Al}}_{2}$${\mathrm{O}}_{3}$ (corundum) have been calculated at different volumes, relaxing the corresponding structures, by all-electron periodic Hartree-Fock methods (crystal program). Basis sets of contracted Gaussian-type functions are employed for the 18 atomic (including d) orbitals representing each of the Mg, Al, and O atoms. Mulliken net atomic charges ${\mathit{z}}_{\mathrm{Mg}}$=1.86\ensuremath{\Vert}e\ensuremath{\Vert} (MgO), ${\mathit{z}}_{\mathrm{Al}}$=2.30\ensuremath{\Vert}e\ensuremath{\Vert} (\ensuremath{\alpha}-${\mathrm{Al}}_{2}$${\mathrm{O}}_{3}$), ${\mathit{z}}_{\mathrm{Mg}}$=1.74\ensuremath{\Vert}e\ensuremath{\Vert}, and ${\mathit{z}}_{\mathrm{Al}}$=2.24\ensuremath{\Vert}e\ensuremath{\Vert} (spinel) are obtained. The elastic bulk modulus, the Murnaghan equation of state p(V) at the athermal limit, the Mg-O and Al-O bond compressibilities, and the binding energy have been derived for each phase (and the elastic constants ${\mathit{C}}_{11}$ and ${\mathit{C}}_{12}$ for spinel only). Comparison with existing experimental data is discussed. The enthalpy change for spinel decomposition into the simple oxides has been computed as a function of pressure, including a correction for the electron correlation energy based on local-density-functional theory. A decomposition pressure of 11 GPa at T=0 K is predicted, against values of 8 and 13 GPa derived from experimental thermodynamic data and from direct compression experiments, respectively.

A 109Pa high-pressure cell for X-ray and optical measurements. Erratum

In the paper by Leszczynski, Podlasin & Suski [J. Appl. Cryst. (1993), 26, 1–4], the value of the GaAs bulk modulus B0 = 1.1 (0.2) × 1011 Pa cited from the paper by Besson, ltie, Polian, Weill, Manssot & Gonzalez [Phys. Rev. B (1991), 44, 4214–4234] was misread from the figure. The correct value, as given by Itie [Phase Transit. (1992), 39, 81], is B0 = 0.85 × 1011 Pa and B′ = 4.5 in a Murnaghan equation of state.

Equation-of-state measurements for crude oils at pressures up to 1 GPa

Equation-of-state measurements of two crude oils with different compositions and viscosity were performed at room temperature in the pressure range 0<P<1.0 GPa. We found large compressibilities and a strong dependence of the compressibility on oil content and viscosity. The bulk modulus changed with pressure from 2.0 to 12.1 GPa for one oil and from 1.3 to 9.5 GPa for the other. We discuss the possibility of detecting phase transitions in the region under investigation. The Tait and Murnaghan equations of state were fitted to the data, and residuals are presented.

Cratering mechanics on Venus: Pressure enhancement by the atmospheric “ocean”

The mass per unit area of the Venusian atmosphere is ∼106 kg/m2, equivalent to a 1‐km‐deep water ocean. An important effect of this “ocean” on cratering is pressure enhancement due to multiple reverberations during the compression stage. We consider impactors with diameters, D, less than a scale height (∼15 km). Such impactors drive before them bow shocks with standoff distances of Δ ∼ 0.05 D into the lowermost atmosphere. Reflection of this shock at normal incidence from the basalt surface raises the pressure in the atmosphere and surface, but initially not as much as a direct impact would. Several shock reverberations should occur, however, as the shock crossing time in the trapped, compressed atmospheric layer is much less than either the shock crossing time in the impactor or the time for the impactor to penetrate the atmospheric layer. With each reverberation we estimate new Hugoniots for shocked atmosphere, impactor, and target. The first is treated as an ideal gas of γ= 1.2–1.3, and the latter are modeled with Murnaghan equations of state and Grüneisen Γs ∝ ρ−1. The asymptotic pressure enhancement for a 20 km/s carbonaceous chondrite impactor (modeled as serpentine) striking basalt at 20 km/s, relative to the atmosphereless 1‐D impedance match pressure (∼550 GPa), is a factor of ∼1.4. This maximum enhancement factor increases with velocity, reaching 1.6 for serpentine moving at typical Venus‐impacting cometary velocities (∼65 km/s), and is greater for more compressible impactors (comets). Each reverberation sends a shock pulse into the surface or impactor that overtakes the previous ones, hence at the end of the compression stage both impactor and target are shocked to a higher pressure than would occur in the absence of the atmosphere. The impactor is effectively denser or staffer than in the atmosphereless case, but as the energy and momentum delivered are essentially the same, entering efficiency should be unaffected to first order. However, fusion and vaporization efficiency should be enhanced for impactors whose impact velocities are below (whether initially or due to atmospheric drag) the threshold for complete melting or vaporization, respectively, in the absence of an atmosphere.

X-ray diffraction study of potassium metal under high pressure

Abstract The equation of state of potassium metal at ∼25°C was measured to 6.1 GPa using energy-dispersive synchrotron radiation diffraction and a Drickamer-type cell. Previous piston-cylinder, shock wave and x-ray measurements disagree and this study is intended to help resolve these disagreements. Our P-V data fit closely to a Murnaghan equation of state and least-squares refinement gives Bo = 3.35 GPa and Bo′ = 3.28 in good agreement with recent theoretical calculations based on accurate experimental results.

Bulk modulus and equation of state

By combining data from compressional and vibrational spectral studies, both the zero-pressure isothermal bulk modulus ( B T,0 ) and its first pressure derivative ( B′ T,0 ) can be directly and independently calculated for a material without employing an empirical equation of state. On the basis of the observed behavior of Raman frequencies as a function of pressure and volume, a series of equations of state are derived, the simplest being the Murnaghan equation of state. These equations of state are consistent not only with the data from compressional studies but also with those of spectral studies. Experimental data for forsterite and stishovite are employed as an illustration.

Hydrostatic pressure dependences of elastic constants and vibrational anharmonicity of uranium nitride

Measurements of the effect of hydrostatic pressure on ultrasonic wave velocities have been used to determine the pressure derivatives of the elastic stiffness of uranium nitride at room temperature.∂C44/∂P, and hence the Gruneisen parameter for the transverse mode propagating down an 〈001〉 axis, is negative; however, this mode softening is not anomalous for rocksalt structure crystals. The Gruneisen gammas of the acoustic modes obtained in the long wavelength limit have a pronounced anisotropy which accrues largely from the presence or absence of contributions to modes of vibration from nearest-neighbour repulsive forces. The compression of uranium nitride calculated from the Murnaghan equation of state is much smaller than those of the alkali halides or IV–VI compounds because this compound is much stiffer.

The compressibility of forsterite up to 300 kbar measured with synchrotron radiation

The effect of pressure on the lattice constants and therefore the volume of synthetic forsterite was determined by energy-dispersive X-ray diffraction from polycrystalline material using synchroton radiation from the storage-ring DORIS at the HASYLAB, DESY (Hamburg). Pressures up to 300 kbar were applied with a diamond-anvil cell. The pressures were calculated from the volume compression by internal standard method using MgO, Fe and NaCl as pressure calibrants. The isothermal bulk modulus K 0 and also its first pressure derivative K 0′ (at 25°C) of forsterite have both been obtained from the experimental data with the Murnaghan equation of state by a two parameter fit yielding K 0 = 1357(10) kbar and K 0′=3.98(10) (standard deviations in parentheses). We found no differences with the different marker.

The effect of hydrostatic and uniaxial pressure on the elastic constants of cadmium

The third order elastic constants of the hep metal cadmium have been determined from measurements of the hydrostatic and uniaxial pressure dependences of the velocities of ultrasonic waves propagated through single crystals. The hydrostatic pressure dependences of the second order elastic constants have also been obtained. The compression of cadmium has been estimated by extrapolation of the data up to high pressures by using the Murnaghan equation of state. Using the generalised Grüneisen theory in the quasiharmonic approximation, the long wavelength acoustic mode Grüneisen parameters have been obtained; the mean acoustic mode parameter is compared with the thermal Grüneisen parameter. In general the anisotropy of the vibrational anharmonicity of the acoustic modes is found to be consistent with the deviation of the c/a ratio of cadmium from that expected for an ideally close-packed metal.

Dynamic fission instability of dissipative protoplanets

All theories of fission require a catastrophic, dynamic phase in order to produce two separate bodies. We have used nonlinear numerical and linear analytical calculations to show that the dynamic fission instability probably does not occur in dissipative protoplanets. The numerical calculations were performed with a three-spatial-dimension hydrodynamical code, with the proto-planet represented by a fluid with a Murnaghan equation of state. The kinetic energy in the protoplanet (other than rigid body rotation) is dissipated throughout the evolution in order to simulate the effects of viscous dissipation. Protoplanets rotating above the limit for dynamic instability were given initial asymmetric density perturbations; in each case the asymmetry did not grow during a time on the order of the rotational period. This dynamical stability has been verified by including the dissipative terms in the tensor-virial equation analysis for the stability of a Maclaurin spheroid: the dynamic instability vanishes when the dissipative terms are included, while the secular instability (with a growth time much larger than the rotational period) remains. The result applies to bodies of radius R with a kinematic viscosity ν⪢ 4 × 10 13 ( R/6400 km) 2cm 2sec −1, and hence may be applicable to any terrestrial protoplanet which is not totally molten. Current thermal histories for the Earth predict a partially molten mantle with a viscosity greater than this critical value. Depending on the detailed rheology of the early Earth, our results appear to rule out the possibility of forming the Earth-Moon system through a dynamic fission instability.

The Planar Deformation Behavior of Skinned Striated Muscle Fibers

Abstract The osmotically produced planar deformation behavior of skinned striated muscle fibers (smectic B, lattice microstructure) is investigated using light microscopy and compared with analogous results determined using x-ray diffraction. The nonlinear pressure-volume relationship measured using light microscopy is shown to be essentially the same as that resultant of similar experiments using x-ray diffraction. The application of a continuum analysis based on the Murnaghan equation of state to this deformation behavior is examined. The Murnaghan equation is shown to approximate well the nonlinear planar deformation behavior of these muscle fibers.

Thermal expansion and compressibility of β-As 4S 3

Temperature and pressure variation of the lattice constants and unit cell volume have been studied with the help of low temperature Guinier-Simon film technique and energy dispersive high pressure X-ray diffraction. Volume thermal expansion coefficient at room temperature and its derivative have been obtained by fitting the experimental data with volume-temperature equation obtained from purely thermodynamic considerations. Similarly bulk modulus at ambient pressure and its derivative with respect to pressure have been obtained by fitting volume-pressure data with the Murnaghan equation of state. The anisotropy of thermal expansion coefficient and the compressibility have been discussed.

Structure and compression of crystalline argon and neon at high pressure and room temperature

Argon and neon crystallize in the face-centered cubic structure (Fm3m, z = 4) at 11.5±0.5 and 47.4±0.5 kbar, respectively, at 293 K. Single-crystal cell dimensions were obtained at 82 kbar for argon and 144 kbar for neon with high-pressure, x-ray diffraction techniques. The data were corrected to 0 K and fitted to a second-order Murnaghan equation of state with V0 and K0 constrained to results obtained in low-temperature experiments. No solid-solid phase transitions are observed to the highest pressures studied.