Calculation Of Bulk Modulus Research Articles

Calculation of bulk modulus for highly anisotropic materials

Bulk modulus is usually obtained from fitting experimental or theoretical energy–volume data to an empirical functional form of the equation of state. Most theoretical studies generate the energy–volume data by uniformly changing the unit cells in total-energy calculations. This approach works well for isotropic systems such as structures with cubic symmetries; but for noncubic systems with high anisotropy, full structural relaxation has to be carried out at each volume to ensure correct extraction of bulk modulus. In this Letter, we present an approach that uses uniaxial compression and the volume derivative (of energy) definition of bulk modulus. First principles pseudopotential calculations are carried out with the generalized gradient approximation applied to more localized core electrons while the local density approximation applied to valence electrons to improve the accuracy of calculated structural data. We have applied this approach to elemental hexagonal graphite and binary hexagonal boron nitride and have obtained results in excellent agreement with experiment. We also have used this approach to predict the value of bulk modulus for a graphitic phase of a ternary BC 2N structure.

Ab initiocalculations of bulk moduli and comparison with experiment

Bulk moduli appear readily accessible in electronic structure calculations, but the calculated values are often substantially greater than experimental bulk moduli. This discrepancy is the result of an unfair comparison of calculated and experimental results: many workers ignored the zero-point and finite-temperature effects that are present in experiments but absent from most calculations. These effects can alter bulk moduli by up to 20%. We show how good approximations to the required corrections may be obtained with little effort. We also deal with the statistical errors and biases in quantities derived from the noisy energy-volume curves produced by quantum Monte Carlo simulations.

Calculation of bulk moduli of semiconductor compounds

The bulk modulus has been calculated by using the energy gap along Γ→ X and the transition pressure for diamond and zinc-blende semiconductors. The obtained order of magnitude is in reasonable agreement with experimental and other calculations.

Nitrogen adsorption in compliant materials

The adsorption of nitrogen at 77.4 K, a standard tool for the characterization of porous materials, has been carefully reexamined for compliant mesoporous materials. Simultaneous measurements of nitrogen sorption and linear length change of silica aerogels with bulk moduli between 1.8 and 16 MPa reveal a characteristic shape of the isotherm that is due to contraction and reexpansion of the sample in different phases of adsorption and desorption. The capillary pressure upon filling and emptying of mesopores results in a volumetric contraction of up to 50%. Once the mesopores in the bulk of the contracted porous backbone are completely filled, the capillary pressure is released and the material slowly reexpands, accompanied by further adsorption until the sample is fully relaxed. This is the first time that the reexpansion close to saturation has been observed. The lack of this feature in previous experiments is due to insufficient equilibration of the sample upon adsorption. Analysis of the isotherm indicates that, even for compliant samples, a breakthrough radius can be extracted that is in excellent agreement with values derived from other methods. In addition, the isotherm of compliant samples allows for a calculation of bulk modulus.

Calculation of bulk modulus of titanium alloys by first principles electronic structure theory

A theoretical study of the electronic structure and binding energy of some hypothetical Ti-X alloys was carried out using a first-principles discrete variational cluster method. The formation energy of an alloying atom in solution of titanium was estimated based on such calculations, and the case of multi-constituent practical Ti alloys was considered in the dilute limit by a linear superimposition approach. The influences of alloying additions on the bulk modulus of the alloys were evaluated from the variation of the formation energy. The calculated moduli of the Ti alloys were found to vary linearly with the experimental values. This indicates that the present approach is appropriate for the simulation of modulus of titanium alloys.

Structure and Bulk Modulus of High-Strength Boron Compounds

The structures and homogeneity ranges of B6O1−xand B12S2−xwere studied using Rietveld analysis of powder X ray patterns. The oxygen content of boron suboxide decreases with temperature in the range 1250–1450°C. Stoichiometric boron suboxide cannot be prepared from amorphous orα-rh boron and B2O3at ambient pressure. Significantly higher pressures are required. The boron subsulfide was found to be stable from B12S<1to B12S1.3at 1400–1600°C. Semiempirical bulk modulus calculations are reported for hard icosahedral boron-rich compounds and diamond-like tetrahedrally coordinated boron compounds. In connection with this the structure of diamond-like B2O is discussed.

Bulk Modulus Calculations for Liquid Binary Alloys

In this work, the bulk modulus values as functions of concentration for K-Na, Cs-Na and Rb-Na liquid binary alloys have been studied in the framework of the Gibbs-Bogoliubov(GB) method[1-2]. We have used the local Heine-Abarenkov pseudopotential and Ichimaru-Utsumi screening function[3] and Percus-Yevick partial structure factors in these alloys. The results have been compared with the experimental values[4].

Bulk modulus calculations based on perturbation self-consistency

We propose a method for bulk modulus calculations of solids based on perturbation self-consistency within the local density functional. The essential assumption is that a finite scaling is applied to the one-electron wavefunction when the solid adjusts to a new distorted structure under pressure. Thus the one-electron potential of a deformed structure near equilibrium can be obtained by performing a scaling transformation to charge density directly. The method is formulated within the linear muffin-tin orbital method in the atomic sphere approximation and applied to the calculation of bulk moduli of , FCC Al, BCC Li, and an ordered FCC superlattice. The bulk modulus calculated from a single self-consistency is in reasonably good agreement with that of a full self-consistent calculation. Our results for Al - Li systems confirm that the addition of lithium to FCC Al causes the bulk modulus to decrease.

Infrared spectroscopy of yttrium aluminum, yttrium gallium, and yttrium iron garnets

All 17 infrared lattice modes for yttrium aluminum, yttrium gallium, and yttrium iron garnets Y3R2(RO4)3 are obtained from single-crystal reflection and absorption spectroscopy, and the powder technique. Assignments for 12 modes are firmly establish based on chemical substitution among these garnets and within related series. Frequencies of the modes associated with translation of yttrium are nearly constant across the series. Frequencies of all other modes decrease quasilinearly as cell volume increases. Values the high and low frequency dielectric constants are in good agreement with independent measurements; also, the Lyddane–Sachs–Teller relationship was found to hold. Calculations of bulk moduli from vibrational spectra using a modification of Brout’s formula linearly correlate with experimental values for KT.

Calculation of bulk modulus and its pressure derivatives from vibrational frequencies and mode Grüneisen Parameters: Solids with cubic symmetry or one nearest‐neighbor distance

The bulk modulus KT can be related to structural parameters and to the sum of the squares of the vibrational frequencies. ∑vi2, for N‐atom crystals of cubic symmetry or for any symmetry that contains only one nearest‐neighbor distance. Required assumptions are the existence of (1) electrostatic forces exclusively between atoms at equivalent positions in the primitive unit cell, (2) pair‐wise central repulsive potentials between all other atoms, and (3) rigid ions. Including pressure in the derivation requires a more stringent version of point 2, namely, (4) that the structure scales isotropically upon compression. This leads to structurally independent formulas for the pressure derivatives of bulk modulus at 1 atm: the only variables are KT and sums involving vi2 and the first or second pressurderivative of vi, i.e., the mode Grüneisen parameters. Implicit in all equations is the independence of the sums on wave vector. Thus, knowledge of zone center phonons (mostly infrared and Raman bands) is sufficient to calculate elastic properties. Investigating these relationships for 45 minerals with 10 different structures shows that KT(0) is predicted within 7% of experiment for 21 solids, all of which should lack strong interactions between atoms at equivalent sites in the Bravais unit cell (i.e., rutile, corundum, ilmenite, and spinel structures). The accuracy of the model strongly depends on the accuracy of the sum ∑vi2. Agreement for B1, B2, fluorite, and wurtzite structures is moderate to poor, as expected, because these structures place like atoms as second nearest neighbors and can thus violate assumption 1. For rocksalt, agreement varies, depends on the size of the cation, not on the polarizability (i.e. the ionic rigidity) as inferred earlier. Calculated KT for the garnets is 1.367 time experimental KT, instead of unity, which is related to the structure not scaling perfectly with pressure. Agreement is variable (3–20%) for the perovskite structure with large discrepancies associated with ill‐constrained sums. However, for all structures, the relative size of KT is predicted, so that the theory can be used for systematics for all minerals with the proper symmetry, reguardless of whether the assumptions are satisified. For all of the above mentioned structures, K′o and K″o are predicted within the experimental uncertainties from the available spectroscopic measurements of 13 solids for K′o and 6 solids for K″o at pressure. Based on this, the good agreement for ilmenite, which does not meet the symmetry requirements, and the structural independence of the formulas, the relations involving K′o and K″o are inferred to hold for structures more complex than cubic or AXBY.

Calculation of bulk moduli of diamond and zinc-blende solids.

Theoretical arguments on the role of covalency in determining the bulk moduli of diamond and zinc-blende semiconductors and insulators are shown to yield a surprisingly simple and accurate expression for determining the bulk moduli B of these materials. One resulting formula for compounds in the center of the Periodic Table depends only on the nearest-neighbor separation d. It has the form B=1761${d}^{\mathrm{\ensuremath{-}}3.5}$ for B in GPa and d in A\r{}.

Elastic moduli of polycrystalline cubic metals

The author discusses the problem of relating the fundamental elastic properties of anisotropic single crystals to those of polycrystalline bulk materials. It is shown that the orientation-independent relationships are not in fact independent of each other and are insufficient to enable the constants to be found. Some useful results do, however, follow. Thus, the methods of Reuss (1929), Voigt (1928), and Hill (1952), lead to the same calculation of bulk modulus for polycrystalline materials and show that it should be identical with the single-crystal value. It is also shown why polycrystalline cubic metals behave in an isotropic manner when the grains are completely randomly oriented. Departures from randomness (development of an orientation texture) render increasingly invalid the widely accepted relationship between the moduli, E and G, and v (Poisson's ratio), viz. E=2G(1+v).

Calculation of bulk modulus, shear modulus and Poisson's ratio of glass

Combining Grüneisen's equation with our Young's modulus equation of glass, new formulae were semi-empirically derived for the calculation of bulk modulus, shear modulus and Poisson's ratio of glass. Considering packing density of atoms and bond energy in unit volume the elastic moduli of glass can be calculated. The agreements between calculated and observed values of the moduli of glass are satisfactory for more than 30 glasses.