Formulation of the third-order Grüneisen parameter at extreme compression

We present a direct method using the basic principles of calculus to derive the expression for the third-order Grüneisen parameter in terms of the pressure derivatives of bulk modulus at extreme compression. The derivation presented here does not depend on the assumptions regarding the values of free-volume parameter and its variation with pressure. The identities used in the present analysis are valid at extreme compression for all physically acceptable equations of state.     

Pressure dependence of the optical-absorption edge of AlN and graphite-type BN

Effect of pressure on the band gaps on AlN and graphite-type BN (g-BN) has been studied up to 2.7 GPa at room temperature by measuring the optical-absorption edge of single crystals of each substance pressurized in a sapphire-anvil cell. The direct band gap of AlN shifted towards higher energy at a rate of 49±1 meV/GPa, whereas in g-BN the pressure dependence of the band gap was −36±1 meV/GPa. The results are compared with existing first-principles calculations.     

Pressure dependence study on the electron-phonon coupling of thulium hexachloroelpasolite

Raman spectra of Cs₂NaTmCl₆ have been recorded using a diamond anvil cell at ambient temperature. The vibrational energy of each of the Raman-active TmCl₆⁻³ moiety modes increases linearly with pressure. The integrated band areas of the ν₁(a_1g) and ν₂(e_g) modes are independent of applied pressure. However, the band area of the ν₅(t_2g) mode shows an anomalous behaviour, which has been qualitatively interpreted as due to electron-phonon coupling of the aΓ₅ electronic state with the Γ₁+ν₅(t_2g) vibronic state. This interaction between the coupled states is strongest between ca. 10 and 13 GPa at ambient temperature. The results serve to emphasize the specificity of the occurrence of strong electron-phonon coupling for particular transitions of a given rare earth ion.     

Pressure dependence of the Grüneisen parameter of sodium and potassium

Temperature changes associated with adiabatic pressure changes (?T/?P)_s have been measured for sodium and potassium at room temperature (297 K) under hydrostatic pressures up to 32 kbar. For both metals a large decrease in the value of (?T/?P)_s is observed as the pressure increases. The Grüneisen parameter γ is related to (?T/?P)_s by γ = B_s(?T/?T)_s/T where B_s is the adiabatic bulk modulus. Available data for the pressure dependence of B_s and the isothermal bulk modulus B_T are used to predict the pressure and volume dependence of γ.     

The volume dependence of the Grüneisen parameter and crystal stability

Utilizing only the general features of the lattice structure and potential, the explicit volume dependence of the entire frequency spectrum is derived in terms of its zero pressure values. It is shown that even though the frequencies and the mode Grüneisen parameters depend on the various details of the interactions and lattice structure, their variation with volume can be relatively simple functions of the volume. A criterion for mode instability with pressure is established involving only the measured value of γ(kλ). The volume dependence of the high temperature Grüneisen parameter is derived utilizing the characteristics of the full frequency spectrum through the first two moments of the mode gammas and the results are applied to the alkali-halides.     

Analysis of volume dependence of Grüneisen ratio

Various models on volume dependence of the Grüneisen ratio have been analyzed in the present study. The Sharma model [Mod. Phys. Lett. B 22/31 (2008) 3113] is found to be similar to that used by Nie [Phys. Stat. Sol. (b) 219 (2004) 241] on the basis of approximation made by Jeanloz [J. Geophys. Res. 94 (1989) 5929]. The Nie expression is amended in a manner so that the resulted expression follows the constraint of high pressure thermodynamics in the limit of infinite pressure. The newly developed relationship is applied successfully on materials for which experimental data are accessible such as epsilon-iron, NaCl, Li, Na, and K.     

Infinite pressure behaviour of certain inverted type equations of state for solids

It is shown that a generalized version of inverted type equation of state (EOS) yields the recently proposed Saxena EOS [J. Phys. Chem. Solids 65 (2004) 1561] for a particular value of the parameter related to , the pressure derivative of bulk modulus at extreme compression. It is found that some other inverted type EOS due to earlier workers can also be derived from a generalized version. A common feature of the inverted equations considered in the present paper is that they yield negative values for . This is in strong contradiction and violation of the theory of infinite pressure extrapolation due to Stacey.     

The temperature dependence of the equation of state at high pressures revisited: a universal model for solids

A general method to include temperature effects into the equation of state (EOS) of solids is discussed. A universal model based on a pseudo-spinodal approach is used to predict the pressure and temperature dependencies of the thermodynamic properties for a variety of solids: n-H₂, Ar, Kr, Xe, NaCl, LiF, NaF, KCl, CsCl, Li, Na, K, Rb, Cs, Al, Fe, Cu, Zn, Ag, Cd, Pt, Au, and Pb. The predictive capabilities of the complete EOS are discussed and compared with available models.     

Volume and pressure dependence of Grüneisen parameter γ for solids at high temperatures

The present paper simply proposes a computing model for the volume and pressure dependence of the Grüneisen parameter γ at high temperatures. Applying it to ε-Fe, NaCl, Li, Na and K in different pressure ranges, we find that the calculated results of γ are in good agreement with the experimental data. Based on this modified model the thermal pressure P_th of NaCl is also investigated.     

On the determination of the volume dependence of Grüneisen parameters in cubic and non-cubic solids

For a material of orthorhombic symmetry under hydrostatic stress, three generalized Grüneisen parameters relative respectively to the three principal directions of the deformation can be defined. Finite strain expressions for these parameters are derived in terms of the Lagrangian strain tensor and the frame-indifferent analogue of the Eulerian strain tensor. The expressions require for their evaluation the thermal expansion coefficients, the elastic moduli and their pressure and temperature derivatives, and the specific heat of the material, so that there are no arbitrary constants or “curve fitting.” In the case of cubic materials, it is possible to determine the unique Grüneisen parameter as a function of volume directly. For non-cubic materials, the volume dependence of the Grüneisen parameters is calculated using a quasi-harmonic finite strain model of equation of state. Numerical applications are given for some cubic and hexagonal metals and the results are discussed.     

Pressure induced phase transformation of semiconductor clathrates

Pressure induced phase transformation and amorphization for Ge-based type-I clathrates have been investigated by means of synchrotron XRD and Raman experiments under high pressure. The XRD results of Sr₈Ga₁₆Ge₃₀, Ba₈Ga₁₆Ge₃₀, and I₈Sb₈Ge₃₈ demonstrated volume collapse phase transitions at 18, 33, and 42 GPa, respectively. Reitveld analyses performed for I₈Sb₈Ge₃₈ reveal a deformation of six-member rings of 14-hedron cages with increasing pressure.     

Compression of silver in a diamond anvil cell: Pressure dependences of strength and grain size from X-ray diffraction data

Two silver samples, coarse grained (c-Ag, grain size 300±30 nm) and nanocrystalline (n-Ag, grain size 55±6 nm), are compressed in a diamond anvil cell in separate experiments. The pressure is increased in steps of ∼3 GPa and the diffraction pattern recorded at each pressure. The grain size and compressive strength are determined from the analysis of the diffraction line-widths. The grain size of c-Ag decreases rapidly from 300±30 nm at ambient pressure to 40±8 nm at 15 GPa, and then gradually to 20±3 nm at 40 GPa. After pressure release to ambient condition, the grain size is 25±4 nm. The strength at ambient pressure is 0.18±0.05 GPa and increases to 1.0±0.3 GPa at 40 GPa. The grain size of n-Ag decreases from 55±6 nm at ambient pressure to 17±4 nm at 15 GPa and to 14±3 nm at 55 GPa. After release of pressure to ambient condition, the grain size is 50±7 nm. The strength increases from 0.51±0.07 GPa at ambient pressure to 3.5±0.4 GPa at 55 GPa. The strength is found to vary as the inverse of the square-root of the grain size. The results of the present measurements agree well with the grain-size dependence of strength derived from the hardness versus grain size data at ambient pressure available in the literature.     

Second order and third order Grüneisen parameters at extreme compression

On the basis of the free volume theory of Grüneisen parameter (γ) and using the calculus of indeterminates, it is found that the second order Grüneisen parameter (q) and the second pressure derivative of bulk modulus (KK″) change in a similar manner in the limit of extreme compression. The ratio of q and KK″ becomes finite at infinite pressure. This finding has been used further to obtain a relationship for the third order Grüneisen parameter λ in terms of pressure derivatives of bulk modulus up to the third order. The results are found to be consistent with the identities obtained recently by Shanker et al. [14] using the free volume theory.     

Plastic-deformation dependence of the Grüneisen parameter strain derivatives

Summary The strain derivatives of the Grüneisen parameter ψ at room temperature are experimentally found to strongly depend on the plastic deformation ɛ_p imposed to the solid. ψ is very sensitive to the alterations of the microstructure, which in turn remarkably influences the lattice dynamics of the crystal. To test such a complex interaction, measurements were performed on α-titanium at values of ɛ_p up to 10⁻¹. The obtained exponential relations between the strain derivatives of ψ at the origin and ɛ_p are discussed for the first- and the second-order derivatives, ∂γ/∂ɛ|₀ and ∂²γ/∂ɛ²|₀, respectively. The results allow not only for a deeper knowledge of the ψ behaviour, but also for outlining the work-hardening evolution in metallic materials.

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Riassunto Le derivate rispetto alla deformazione elastica del parametro di Grüneisen ψ, misurate a temperature ambiente, mostrano una forte dipendenza dalla deformazione plastica ɛ_p imposta al solido: ψ è molto sensibile ai cambiamenti delle microstruttura, che a sua volta influisce notevolmente sulla dinamica reticolare del cristallo. Con le misure effettuate su titanio α, fino a deformazione plastica ɛ_p=10^-1, si è studiata tale complessa interazione. Si discutono le relazioni esponenziali ottenuate fra la derivata prima e, rispettivamente, la derivata seconda di ψ, nell'origine ed ɛ_p. I risultati sperimentali consentono di approfondire le conoscenze sul comportamento di ψ e di delineare l'evoluzione dell'indurimento da lavorazione nei materiali metallici.

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Резюме Экспериментально обнаружено, что пронзводные парамеетра Грюнайзена по деформацин при комнатной температире сильно зависят от пластической деформации ɛ_p, приложенной к твердому телу. Величина ψ очень чувствительна к изменениям микроструктуры, которая сушественно влияет на динамику решетки кристалла. Для при величинах ɛ_p вплоть до 10⁻¹. Обсуждаются полученные экспоненшиальные соотношения между производными ψ по деформации в начале при наложении пластнческих деформаший для производных первого и второго порядков, ∂γ/∂ɛ|₀ and ∂²γ/∂ɛ²|₀. Полученные результаты позволяут не только лучше понять поведение, ψ, а также эволюшию деформацноноого упрочнения в металлических материалах.

     

Two universal equations of state for solids satisfying the limiting condition at high pressure

In this paper it is shown that the relationship of bulk modulus with pressure, B=f(P), should be linear both at low and high-pressure limiting conditions. Because most of present equations of state (EOS) for solids cannot satisfy such linear relationship at high pressure, a new function f(P) is proposed to satisfy the linearity. By integrating the bulk modulus, an EOS with three parameters and satisfying the quantum-statistics limitation is derived. It is shown that the EOS can be reduced to two-parameter EOS approximately satisfying the limiting condition. By applying the two EOSs and other three typical EOSs to 50 materials, it is concluded that for materials at low and middle-pressure regimes, the limiting condition does not operate, the Baonza EOS gives the best results, but it cannot provide analytic expression for cohesive energy. The Vinet and our second EOSs are slightly inferior, both EOSs can provide analytic expression for cohesive energy, and for materials at high-pressure regimes our second EOS gives the best results. The Holzapfel and our first EOSs give the worst results, although they strictly satisfy the limiting condition. For practical applications, the limiting condition is not important because it only operates as V→0.     

In situ X-ray diffraction study of germanium at pressures up to 11 GPa and temperatures up to 950 K

In situ X-ray diffraction measurements on germanium were conducted in the pressure range of 5-11 GPa and temperatures up to 950 K. Using our data a better defined P-T diagram for germanium is presented. The coordinates of the triple point between Ge_I-Ge_II-Ge_L have been determined to a better degree of precision. The onsets of the Ge_I-Ge_II transition were found both under hydrostatic and non-hydrostatic conditions. Anisotropy of thermal expansion coefficient for the Ge_II is characterized from the c/a ratios in the temperature interval 473-823 K. Phases Ge_III and Ge_IV are shown to be metastable forms of germanium.     

Explicit Gibbs free energy equation of state for solids

Semi-empirical equations of state (EOS) are used for interpolation and extrapolation of experimental data and/or electronic structure calculations. For calculation of phase equilibria, it is preferable to use an explicit Gibbs free energy EOS, that is, to express the Gibbs free energy directly as a function of the pressure and temperature. Existing explicit Gibbs free energy EOS formulations often give unphysical predictions at high pressures. The origins of these problems are internal inconsistencies and uncontrolled extrapolations. A set of conditions is put forward, that should be fulfilled by semi-empirical EOS formulations in order to constrain them to known physical behaviour, e.g., to the Thomas-Fermi and quasi-harmonic models at high pressures. A new alternative integration path is devised that eliminates the need for the problematic extrapolation of the heat capacity to high temperatures at low pressures. Based on these developments, a new explicit Gibbs free energy EOS is formulated which is suitable for computational applications. The new EOS may be fitted to represent the thermophysical properties of solids with a reasonably small number of adjustable parameters. A sample application for MgO is presented.     

Heat capacity of CeIrSi3 under pressure

We measured the heat capacity of CeIrSi₃ (100 mK<T<6 K) under high pressure up to P=1.38 GPa. The measurements have been used a quasiadiabatic method utilizing a CuBe piston-cylinder pressure cell in a dilution refrigerator. At 0 GPa, a sharp anomaly which indicates the antiferromagnetically transition is observed at T_N=5 K. T_N decreases monotonically with increasing pressure up to P=1.38 GPa. The magnetic entropy is released below T_N only 19% of R ln 2 at 0 GPa. And the magnetic entropy decreases with increasing pressure up to 1.38 GPa, 64% compared to that at 0 GPa.     

Strong tolerance of blue-green alga Microcystis flos-aquae to very high pressure

It was shown in our previous reports that a few spores of moss Venturiella could tolerate the very high pressure of 20 GPa for 30 min and germinated a protonema to the length of 30 μm. However, these spores did not grow any further, and disappeared at around 30 days of incubation after seeded. On the other hand, colonies of blue-green alga Microcystis flos-aquae came to appear about 76 days after the moss spores were seeded. Many of these colonies appeared at the places where the moss spores had disappeared. These colonies were formed by the algae that had adhered to the spore cases of the moss and survived after exposure to the very high pressure of 20 GPa. Though the appearance of the colonies of high pressure exposed algae was delayed by about 50 days compared with that of the control group which was not exposed to high pressure, there seems no difference in their shape and color from those of the control group. The pressure tolerance of blue-green alga is found to be enormously strong, and it can survive after exposure to the high pressure which corresponds to the depth of about 550–600 km from the surface of the Earth, just above the lower mantle.     

Pressure and volume dependence of thelo-to phonons in InAs

The pressure dependence of thelo-to phonons in InAs has been investigated by Raman scattering using the diamond anvil cell. Indium arsenide transforms, presumably to the rock-salt structure at 70±1 kbar. The mode Grüneisen parameters for thelo-to phonons are γ _lo =0.99±0.03, γ _to =1.2±0.03 respectively. The effective charge,e* _T , for InAs decreases slightly with pressure and this trend is in accordance with the behaviour of other III–V zinc blende structured semiconductors: The structural phase transition is discussed in the light of theoretical calculations for phase stability of III–V compounds, as well as recent high pressure x-ray diffraction studies.