The experimental melting data for alkali metals reported in the literature reveal that the melting temperature becomes maximum at a particular value of high pressure and then decreases with the increase in pressure. We demonstrate in the present communication that the Lindemann law does not yield negative values for the melting slopes of solids at high pressures as far as the bulk modulus increases and the Grüneisen parameter decreases with the increase in pressure such that the pressure derivative of bulk modulus and the Grüneisen parameter remain positive finite. Arafin and Singh have produced an agreement with the experimental melting data for alkali metals with the help of the Lindemann law using quadratic equations in powers of pressure for the Grüneisen parameter and bulk modulus. These quadratic equations are shown here to be inadequate and invalid at high pressures. For explaining the negative slopes of the melting curves, we need a theory of melting which should take into account the structural and electronic changes at melting. We present a discussion of some very important and highly relevant studies (Martinez-Canales and Bergara; Deng and Lee) based on fundamental considerations and ab initio calculations used recently for explaining the turnover in the melting curves of alkali metals and ferropericlase, an important Earth lower mantle mineral. The structural and electronic changes are responsible for the anomalous thermoelastic behavior yielding negative values of elastic moduli and Grüneisen parameter both, thus making the Lindemann law applicable to materials under study.

Expressions have been obtained for the higher order derivatives of the Gruneisen parameter in the limit of infinite pressure. The usefulness of these expressions has been demonstrated by considering a relationship between reciprocal Gruneisen parameter and the pressure-bulk modulus ratio at finite pressures. It has been found that the relationship under study satisfies the boundary conditions at infinite pressure for the higher order Gruneisen parameters. In order to investigate the applicability of the present formulation at finite pressures, we have calculated the Gruneisen parameter and its higher order derivatives for the Earth lower mantle and core using the input data from Stacey and Davis (2004). (C) 2016 Elsevier B.V. All rights reserved.

Thermoelastic properties have been determined in the limit of infinite pressure by considering the material to remain in the same phase up to extreme compression. We have used some basic principles of calculus in various thermodynamic identities to extrapolate thermoelastic properties at infinite pressure. The results thus obtained are found to be consistent with the earlier investigations made by Stacey (Rep Frog Phys, 68 (2005)341). It has been proved in the present study that the logarithmic volume derivatives of the Anderson-Gruneisen parameters become zero at infinite pressure. This finding has further been used to obtain new results for higher order thermoelastic parameters at extreme compression.

The extreme compression limits of thermal expansivity, isothermal bulk modulus, adiabatic bulk modulus and products of thermal expansivity and bulk modulus have been obtained using the basic principles of calculus and some thermodynamic identities in terms of the Anderson-Gruneisen parameters and pressure derivatives of bulk modulus. The isothermal extrapolations are found to be different from the corresponding adiabatic extrapolations. The results have been obtained using the Stacey thermodynamics of materials at infinite pressure.

An analysis of the Gruneisen parameter and its volume derivatives has been presented using the generalized free-volume theory for materials at extreme compression. It has been re-established using a direct and simple method that the Slater's formula for the Gruneisen parameter has the status of an identity at infinite pressure as found earlier by Stacey and Davis (Phys Earth Planet Inter, 142 (2004) 137). The expressions based on the free-volume theory extrapolated to infinite pressure are found to be in agreement with the corresponding expressions for higher-order Gruneisen parameters obtained recently by Shanker et al. (Physica B, 404 (2009) 4083).

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.

The extreme compression (P→∞) behaviour of various equations of state with K′∞>0 yields (P/K)∞=1/K′∞, an algebraic identity found by Stacey. Here P is the pressure, K the bulk modulus, K′=dK/dP, and K′∞, the value of K′ at P→∞. We use this result to demonstrate further that there exists an algebraic identity also between the higher pressure derivatives of bulk modulus which is satisfied at extreme compression by different types of equations of state such as the Birch–Murnaghan equation, Poirier–Tarantola logarithmic equation, generalized Rydberg equation, Keane's equation and the Stacey reciprocal K-primed equation. The identity has been used to find a relationship between λ∞, the third-order Grüneisen parameter at P→∞, and pressure derivatives of bulk modulus with the help of the free-volume formulation without assuming any specific form of equation of state.

We have derived formulations for the pressure derivatives of bulk modulus up to the third order and for higher order Grüneisen parameters using the generalized free volume theory, and the generalized Rydberg equation of state. The properties derived in the present study are directly related to the understanding of thermoelastic properties of solids. The third order Grüneisen parameter (lambda λ) in the limit of infinite pressure has been found to approach a positive finite value for lambda infinity (λ∞) equal to 1/3. This is a result shown to be independent of the value of K-prime infinity, i. e., the pressure derivative of the bulk modulus at infinite pressure. The results based on other equations of state have also been reported and discussed. We find a relationship between λ∞ and pressure derivatives of bulk modulus at infinite pressure which is satisfied by different types of equations of state.

Thermodynamics of solids in the limit of infinite pressure formulated by Stacey reveals that the thermal expansivity (alpha) of solids tends to zero at infinite pressure. The earlier models for the volume dependence of thermal expansivity do not satisfy the infinite pressure behaviour of thermal expansivity. The expressions for the volume dependence of the isothermal Anderson- Grüneisen parameter (delta T) considered in the derivation of earlier formulations for alpha (V) have been found to be inadequate. A formulation for the volume dependence of delta T is presented here which is similar to the model due to Burakovsky and Preston for the volume dependence of the Grüneisen parameter. The new formulation for alpha (V) reveals that delta T infinity must be greater than zero for satisfying the thermodynamic result according to which alpha tends to zero at infinite pressure. It is found that our model fits well the experimental data on thermal expansivity alpha (V) for hcp iron corresponding to a wide range of pressures (0-360 GPa).

Relationships for the volume dependence of the Grüneisen parameter γ have been used to discuss the behaviour of solids in the limit of infinite pressure (P→∞). The model recently developed by Burakovsky and Preston (J. Phys. Chem. Solids 65 (2004) 1581) yields γ∞, q∞ and λ∞, the values of Grüneisen parameter γ and its logarithmic volume derivatives q and λ at P→∞, which are found to have fixed values, same for all the solids studied. On the other hand, the thermodynamics of solids at P→∞ formulated by Stacey (Geophys. J. Int. 143 (2000) 621) reveals that γ∞ and pressure derivative of bulk modulus K′∞ are different for different materials. The empirical formulation for the volume dependence of γ used by Stacey and Davis (Phys. Earth Planet. Intr. 142 (2004) 137) has been shown to be approximately equivalent to the relationship proposed earlier by Al’tshuler et al. (J. Appl. Mech. Tech. Phys. 28 (1987) 129). The shock-pressure data for iron have been used to discuss the maximum compression limit for iron and to emphasize the invalidity of our recent criterion based on the lattice potential energy (Physica B 364 (2005) 186). The Burakovsky–Preston model based on the Thomas–Fermi approximation (γ∞=1/2 and K′∞=5/3) has been found to be more consistent with the shock-compression data. The constraints γ∞>2/3 and K′∞>5/3 developed by Stacey are not in agreement with the strong shock compression limit reported for several materials. It is shown here that the Slater formula for γ which was found by Stacey to assume the status of an identity at P→∞ and used by him to derive the constraints for γ∞ and K′∞, is invalid when K′∞=5/3 It is also pointed out that γ∞=1/2 is to be preferred over γ∞=2/3 because of the thermodynamic constraint K′∞>1+γ∞ developed by Stacey.

We present a comparative study of Keane's and Stacey's equations of state (EOS), which are based on the variations of K′ with pressure. It is found that higher derivative properties, such as the pressure derivatives of the isothermal bulk modulus K, viz. K″=d2K/dP2 and K‴=d3K/dP3 calculated from the two equations, differ appreciably from each other. In the limit of infinite pressure, KK″ and K2K‴ both tend to zero, but the ratio K2K‴/KK″ remains finite for both the EOS. This property has been used to prove that the second volume derivative of the Grüneisen parameter γ, represented by λ, remains finite (λ→λ∞) in the limit of infinite pressure, a result consistent with thermodynamics of solids. Values of λ∞ for the lower mantle and the core of the Earth have been obtained using the generalized free-volume formula with the help of Keane's and Stacey's equations. The expressions for λ∞ based on the two EOS differ substantially from each other.

The issues of providing quality blood products and maintaining donor safety are primary aims of blood transfusion services. A comprehensive quality system should be in place to fulfill these aims, which can be attained through strict adherence to the established standard operating procedures (SOPs). The Drugs and Cosmetics Act of India, which controls the licensing of blood transfusion services, does not provide clear guidelines regarding plateletpheresis procedure. We, therefore, established our own SOP and operational flow chart for plateletpheresis that can be easily followed by other centers in India. A total of 100 plateletpheresis procedures performed using two cell separators (CS3000 Baxter Healthcare, Round Lake, IL; MCS3p, Haemonetics Corporation, Braintree, MA) were evaluated following our established SOP. The mean platelet yield in CS3000 was 2.9 +/- 0.84 x 10(11) and in MCS3p it was 2.88 +/- 0.75 x 10(11) per unit. However, only 4-7% of SDPs showed WBC levels < 5 x 10(6) due to lack of appropriate methods to quantitate residual WBC counts. Six of 100 donors complained of hypocalcemic symptoms. The operational flow chart designed in this study was found to be simple and easy to adapt by blood transfusion services in this country. J. Clin. Apheresis 20:81-85, 2005 (c) 2005 Wiley-Liss, Inc.

Equations of state (EOS) for solids based on exponential functions for potential energy are found to yield results for pressure-volume relationships in close agreement with the available experimental data as well as with the values obtained from the Hama and Suito EOS based on first-principles. We have used equations of state based on exponential functions due to (i) Morse, (ii) Rydberg and (iii) Davydov. It is shown in the present study that a limit of maximum compression exists for each solid corresponding to zero potential energy. This is a fundamental concept revealed from the nature of potential energy curve. The exponential functions for potential energy have been used to determine the compression limits for NaCl. MgO and iron. Strong evidence is given to support the predicted values of compression limits on the basis of seismological data and phase transformations. The results for pressure as a function of volume obtained for iron using the potential functions are found to present close agreement with the seismological data upto a pressure relevant to the inner core boundary of the earth.

It is found that there exists a limit of maximum compression for each material. The formulation of the potential energy of a compressed solid yields a value of maximum compression which is related to the stability of the solid in a given phase. The necessary condition for the stability of a compressed material used in the present study is based on the fundamental concept that its potential energy must remain attractive (negative). An expression for the potential energy of a compressed material is obtained by using the Taylor series expansion in powers of compression x=1-(V/V0) retaining terms up to fifth order (x5). The equilibrium value of potential energy for uncompressed material has been determined by using the potential function due to Rydberg. The results for the bulk modulus and its pressure derivatives derived from the seismological data have been used to obtain the compression limit for iron. This limit is found to be close to the value of compression for iron in the inner core of the earth.

Estimates of core density deficit (cdd) of the Earth's outer core recently reported by Anderson and Isaak [Another look at the core density deficit of Earth's outer core, Phys. Earth Planet Int. 131 (2002) 19–27] are questionable in view of the serious errors in the pressure–volume and bulk modulus data due to an inadequacy in the calibration process used by Mao et al. [Static compression of iron to 300GPa and Fe0.8Ni0.2 alloy to 200GPa: implications for the core, J. Geophys. Res. 94 (1990) 21737–21742]. The data used by Anderson and Isaak deviate significantly from the corresponding values derived from seismology. In the present study we have used the input data on density, isothermal bulk modulus and its pressure derivative from Stacey and Davis [High pressure equations of state with application to lower mantle and core, Phys. Earth Planet Int. 142 (2004) 137–184] which are consistent with the seismological data. Volumes of hexagonal close-packed iron have been calculated at different temperatures under isobaric conditions at P = 330GPa, the inner core boundary (ICB) pressure using the relationship between thermal pressure and volume expansion based on the lattice potential theory originally due to Born and Huang [Dynamical Theory of Crystal Lattices, Oxford University Press, Oxford, 1954, p. 50]. The formulation for thermal pressure used by Anderson and Isaak has been modified by taking into account the variations of thermal expansivity α and isothermal bulk modulus KT with temperature. Values of cdd are then estimated corresponding to different temperatures ranging from 4000 to 8000K. The results for cdd at different temperatures obtained in the present study are significantly higher than those estimated by Anderson and Isaak suggesting that the cdd for the Earth's outer core is nearly 10%. The effects of nickel when an Fe–Ni alloy replaces Fe are estimated and found to be insignificant.

A phenomenological isothermal equation of state (EOS) is presented which is more accurate than the Birch-Murnaghan EOS and the Vinet EOS in case of a material (LiH) for which the pressure derivative of the isothermal bulk modulus, K-0' < 4. The present EOS yields results in close agreement with the Hama-Suito universal EOS based on first-principle calculations. The inverted form of the EOS has been found to be useful for calculating the change in volume as a function of pressure. The densities for the entire depth of the lower mantle have been calculated with the help of the present EOS. The results for densities have been found to be in good agreement with seismological data for the lower mantle.

In our reply we have emphasized that the polarizabilities of ions depend on the environment in which the ions exist. The polarizabilities are intimately related to the potential energies in molecules. A ‘single polarizable particle model’ considered by Donald et al. in their comment does not appear to be adequate for molecules and crystals.

We present a test of universality for phenomenological isothermal equations of state by applying them in different types of solids. The pressure–volume relationship and the isothermal bulk modulus have been obtained using the Birch–Murnaghan third-order as well as fourth-order equation of state (EOS), the Vinet EOS and the Shanker EOS for rare gas solids (Ne, Ar), metals (Al, Cu) and diatomic solids (LiH, MgO) down to a compression of V/V0=0.2. It is found that the results obtained from the Shanker EOS compare well with the corresponding values obtained from the Vinet EOS for all the solids up to very large compressions. On the other hand, the Birch–Murnaghan EOS becomes less successful at high compressions in several cases. The variation of the pressure derivative of the isothermal bulk modulus for the entire range of compression has also been studied and used to discuss the suitability of the equations of state.