Thk Barron

Negative thermal expansion

There has been substantial renewed interest in negative thermal expansion following the discovery that cubic ZrW2O8 contracts over a temperature range in excess of 1000 K. Substances of many different kinds show negative thermal expansion, especially at low temperatures. In this article we review the underlying thermodynamics, emphasizing the roles of thermal stress and elasticity. We also discuss vibrational and non-vibrational mechanisms operating on the atomic scale that are responsible for negative expansion, both isotropic and anisotropic, in a wide range of materials.

The zero static internal stress approximation in lattice dynamics, and the calculation of isotope effects on molar volumes

For crystals in which relative positions of atoms within the unit cell are not wholly determined by symmetry, a complete application of the quasiharmonic approximation requires the minimization of the free energy with respect to both external (ηλ) and internal (ej) strains. The zero static internal stress approximation first minimizes the static lattice energy with respect to the ej for each state of external strain; the total free energy is then minimized only with respect to the ηλ. We show that although this gives an incorrect internal strain, to the first order it gives the correct external strain at each temperature; in principle, errors are thus of the same order as those due to the use of the quasiharmonic approximation. In particular, recent calculations by Lacks [D. J. Lacks, J. Chem. Phys. 103, 5085 (1995)] of the effect of deuteration on the molar volume of polyethylene are shown by the present analysis to include indirectly the effect of vibrational stretching of the C–H(D) bonds, and their re...

SHELL - a code for lattice dynamics and structure optimisation of ionic crystals

This paper describes Shell, a program which uses lattice statics and quasiharmonic lattice dynamics to calculate analytically the free energy of a crystal, and its derivatives with respect to both internal and external strains, at a given temperature and pressure. These quantities can be used to perform efficient fully dynamic structure optimisation of unit cells containing hundreds of ions. Interactions are via short-ranged spherically symmetric pairwise and three-body potentials as well as the usual Coulomb terms, and polarizability effects may be accounted for by use of the shell model. Application of the code to the rutile phase of MgF2 is briefly described.

Free energy of formation of defects in polar solids

A more exact method than hitherto available, based on lattice statics and quasi-harmonic lattice dynamics, is presented for the direct minimisation of the free energies of periodic solids with very large unit cells. This is achieved via the calculation of analytic derivatives of the vibrational frequencies with respect to all external and internal variables. The method, together with large defective supercells, is used to calculate the free energies of defects in MgO as a function of temperature. A major advantage of the supercell approach is that constant-volume and constant-pressure quantities are calculated independently. This allows a critical appraisal of the common approximations used for many years: (i) to convert constant-volume defect parameters to constant-pressure and (ii) to justify the use of static calculations at constant volume in the interpretation of experimental data obtained at constant pressure and at high temperatures. Defect enthalpies show only a small variation with temperature and differ by ca. 2% from the internal energy change in the static limit. An assessment is also made of the commonly used ZSISA approximation, in which the free energy at each temperature is minimised with respect to external strains only, simultaneously determining the internal strains by minimising the static lattice energy.

Ionic solids at elevated temperatures and high pressures: MgF2

A combination of periodic Hartree–Fock theory, quasiharmonic lattice dynamics, and molecular dynamics is used to study the behavior of MgF2 at elevated temperatures and/or high pressures. Particular attention is paid to the pressure-induced transition from the rutile to the fluorite structure in view of earlier theoretical estimates of the transition pressure, which differ widely. It is shown that previously reported potentials obtained by fitting to empirical data fail to reproduce thermodynamic properties. To rectify this, a new set of consistent two-body potentials has been derived from ab initio periodic Hartree–Fock calculations. Lattice dynamics calculations in the quasiharmonic approximation based on these potentials has been used to study the two phases of MgF2 at high T and P. The resulting transition pressure and that obtained directly from Hartree–Fock calculations in the static limit are both ⩽30 GPa, which is close to the experimental value but appreciably lower than a previous molecular dyna...

Lithium oxide and superionic behaviour—A study using potentials from periodic ab initio calculations

Abstract A simple general methodology for obtaining interionic potentials from periodic ab initio calculations is presented, using periodic Hartree-Fock theory as implemented in the program crystal . To test the approach, two-body potentials are generated for Li2O. Results obtained from our new potential are compared with those from previously suggested empirical potentials, paying most attention to the possibility of superionic behaviour in this material at high temperatures. The application of ab initio Hartree-Fock theory, lattice statics, lattice dynamics and molecular dynamics is able to provide a consistent picture of a superionic transition in lithium oxide at 1100 K. Details of the mechanism of the transition are discussed with the aid of the calculated dispersion curves at high temperature, and individual molecular dynamics trajectories.

Evaluation of thermodynamic properties of solids by quasiharmonic lattice dynamics

Quasiharmonic lattice dynamics is a simulation technique complementary to Monte Carlo and molecular dynamics. Quantum effects are readily taken into account, and high precision does not normally require long runs. Vibrational stability is a sensitive test of interatomic potentials, and details of the vibrational motion reveal mechanisms for phase transitions or for thermal expansion. The major computational task is usually to find the equilibrium geometry at a given T, P; this done, calculating free energy, heat capacity, thermal expansion, etc., is rapid and accurate. For three-dimensional ionic crystals and slabs, our code SHELL calculates analytically first derivatives of the free energy with respect to all strains, internal as well as external; this gives a full minimization of the free energy so efficient that large unit cells can be used, allowing applications to defects and disordered systems. Various applications are discussed: MgF2, including the rutile/fluorite transition; negative thermal expansion in ZrW2O8; anisotropic expansion of polyethylene at very low temperatures; surface free energies for MgO; defect energies and volumes in MgO; and a new method for obtaining free energies and phase diagrams of disordered solids and solid solutions, applied to MnO/MgO and CaO/MgO.

Simple refinements of Brillouin zone integration

Calculations of thermodynamic properties of crystals by means of quasi-harmonic lattice dynamics require numerical integrations over the Brillouin zone, using successively finer grids to achieve convergence to the required precision; but for complex crystals convergence may be uneconomically slow. A model for orthorhombic polyethylene is used to show how convergence may be improved (1) at low temperatures by taking successively finer grids close to the origin of reciprocal space, and (2) at all temperatures by using a three-dimensional Simpsons rule.

Pressure dependence at low temperatures of thermodynamic properties of the heavy fermion compound CeAl3

Values of ( delta beta / delta P)T derived thermodynamically from the zero pressure ultrasonic data of Niksch and co-workers (1980) indicate that the absolute magnitude of the volumetric thermal expansion coefficient beta should be initially enhanced by pressure both at very low temperatures, where beta is negative, and at higher temperatures when beta becomes positive. Values of ( delta 2(CP/T)/ delta P2)T, similarly derived, are in qualitative agreement with measurements of CP under pressure above 0.5 K, but at lower temperatures exacerbate an apparent inconsistency with thermal expansion data noted earlier by Brodale and co-workers (1986); it is unlikely that this anomaly is due solely to sample variation.

Thermoelastic fluctuations in solids

Adiabatic thermoelastic heating can be used to monitor stress fluctuations in solids. Previous studies of the effects on the temperature fluctuations both of applied static stress and of the finite amplitude of the stress fluctuations have used approximate theory. The present rigorous thermodynamic treatment distinguishes between adiabatic second-order derivatives needed for finite amplitude and mixed derivatives needed for static applied stress. A detailed analysis is given for purely compressive stress, followed by computations for KCl, NaCl, Al, Cu, Ti, and the alloy Ti-6Al-4V. Additional terms revealed by the new analysis prove to be substantial, including the difference between the adiabatic and mixed derivatives. Revised forms are then proposed for earlier approximations. For unidirectional stress, expressions are taken from an analysis given elsewhere; and computations made for Al, Cu, Ti and Ti-6Al-4V. Corrections to earlier approximations are relatively smaller than for compressive stress, and of opposite sign because the shear component of the unidirectional stress dominates the second order effects.