Narayani Choudhury

Lattice dynamics and inelastic neutron scattering from forsterite, Mg2SiO4: Phonon dispersion relation, density of states and specific heat

Magnesium-rich olivine (Mg0.9Fe0.1)2SiO4 is considered to be a major constituent of the Earths upper mantle. Because of its major geophysical importance, the temperature and pressure dependence of its crystal structure, elastic and dielectric constants, long-wavelength phonon modes and specific heat have been measured using a variety of experimental techniques. Theoretical study of lattice dynamics provides a means of analyzing and understanding a host of such experimental data in a unified manner. A detailed study of the lattice dynamics of forsterite, Mg2SiO4, has been made using a crystal potential function consisting of Coulombic and short-range terms. Quasiharmonic lattice dynamical calculations based on a rigid molecular-ion model have provided theoretical estimates of elastic constants, long-wavelength modes, phonon dispersion relation for external modes along the three high symmetry directions in the Brillouin zone, total and partial density of states and inelastic neutron scattering cross-sections. The neutron cross-sections were used as guides for the coherent inelastic neutron scattering experiment on a large single crystal using a triple axis spectrometer in the constant Q mode. The observed and predicted phonon dispersion relation show excellent agreement. The inelastically scattered neutron spectra from a powder sample have been analyzed on the basis of a phonon density of states calculated from a rigid-ion model, which includes both external and internal modes. The experimental data from a powder sample show good agreement with the calculated spectra, which include a multiphonon contribution in the incoherent approximation. The computed phonon densities of states are used to calculate the specific heat as a function of temperature using both the rigid molecular-ion and rigid ion models. These results are in very good agreement with the calorimetric measurement of the specific heat. The interatomic potential developed here can be used with some confidence to study physical properties of forsterite as a function of pressure and temperature.

Inelastic neutron scattering and lattice dynamics of minerals

This paper reviews the inelastic-neutron-scattering measurements and theoretical lattice-dynamics calculations, which have aimed at providing a microscopic understanding of the vibrational and thermodynamic properties of geophysically important minerals. In the last decade, detailed inelastic-neutron-scattering measurements supported by extensive model calculations have extended our knowledge of the nature of phonon-dispersion relations and density of states of minerals and their variations in various mineral phases in the Earths mantle. An accurate understanding of these vibrational properties of minerals is crucial for predicting the phase transitions and thermodynamic properties of minerals at the pressures and temperatures prevalent in the Earths mantle. The mineral studies reviewed here include the olivine end members forsterite and fayalite, the pyroxene end member enstatite, the garnet minerals pyrope, almandine, grossular and spessartine, the silicate perovskite MgSiO3, the mineral zircon, the aluminium-silicate minerals sillimanite, kyanite and andalusite, the layer silicates vermiculite and muscovite, the oxide minerals MgO, FeO, Al2O3 and the SiO2 polymorphs, and the carbonate minerals rhodochrosite and calcite. Inelastic-neutron-scattering measurements using reactors and spallation sources on single crystals and powder samples have provided data of their phonon-dispersion relations and density of states, which have been interpreted using theoretical calculations. While quantum mechanical ab initio calculations have been successfully employed to understand the vibrational properties of minerals like MgO, Al2O3, MgSiO3 perovskite etc ., theoretical studies of structurally more complex minerals have largely employed an atomistic approach involving semi-empirical interatomic potentials. The calculations enabled microscopic interpretations of the experimental data and have been very useful in providing an atomic-level understanding of the vibrational and thermodynamic properties of these minerals.

Large phonon band gap in SrTiO3 and the vibrational signatures of ferroelectricity in ATiO3 perovskites: First-principles lattice dynamics and inelastic neutron scattering

We report on first-principles density functional perturbation theory calculations and inelastic neutron scattering measurements of the phonon density of states, dispersion relations, and electromechanical response of PbTiO3, BaTiO3, and SrTiO3. The phonon density of states of the quantum paraelectric SrTiO3 is found to be fundamentally distinct from that of ferroelectric PbTiO3 and BaTiO3 with a large, 70‐ 90 meV, phonon band gap. The phonon dispersion and electromechanical response of PbTiO3 reveal giant anisotropies. The interplay of covalent bonding and ferroelectricity strongly modulates the electromechanical response and gives rise to spectacular signatures in the phonon spectra. The computed charge densities have been used to study the bonding in these perovskites. Distinct bonding characteristics in the ferroelectric and paraelectric phases give rise to spectacular vibrational signatures. While a large phonon band gap in ATiO3 perovskites seems to be a characteristic of quantum paraelectrics, anisotropy of the phonon spectra correlates well with ferroelectric strength. These correlations between the phonon spectra and ferroelectricity can guide future efforts at custom designing still more effective piezoelectrics for applications. These results suggest that vibrational spectroscopy can help design novel materials.

Ab initio linear response and frozen phonons for the relaxor PbMg1/3Nb2/3O3

We report first principles density functional studies using plane wave basis sets and pseudopotentials and all electron linear augmented plane wave (LAPW) of the relative stability of various ferroelectric and antiferroelectric supercells of PMN for 1:2 chemical ordering along [111] and [001]. We used linear response with density functional perturbation theory (DFPT) as implemented in the code ABINIT to compute the Born effective charges, electronic dielectric tensors, long wavelength phonon frequencies and LO-TO splittings. The polar response is different for supercells ordered along [111] and [001]. Several polar phonon modes show significant coupling with the macroscopic electric field giving giant LO-TO splittings. For [111] ordering, a polar transverse optic (TO) mode with E symmetry is found to be unstable in the ferroelectric P3m1 structure and the ground state is found to be triclinic. Multiple phonon instabilities of polar modes and their mode couplings provide the pathway for polarization rotation. The Born effective charges in PMN are highly anisotropic and this anisotropy contributes to the observed huge electromechanical coupling in PMN solid solutions.

Free energy and relative stability of the enstatite Mg2Si2O6 polymorphs

Abstract Detailed lattice dynamical studies of the equations of state and the phase diagram of the geophysically important mineral enstatite have been undertaken. There are several enstatite polymorphs: ortho, proto and clinoenstatite, whose structures are characterized by double MgO6 octahedral ribbons and single silicate chains. The computed equations of state are found to be in good agreement with available experimental data and ab initio results. The calculations reveal that the stable phase at ambient conditions is orthoenstatite, which transforms to the protoenstatite phase at high temperatures. The monoclinic C2/c clinoenstatite phase is found to be stable at high pressures. The computed phase diagram is in overall qualitative agreement with the experimental data. These studies have enabled a microscopic understanding of the factors contributing to the relative stability and indicate that while the orthoenstatite to protoenstatite transition is temperature driven, the orthoenstatite to clinoenstatite transition is pressure driven.

Equation of state and melting point studies of forsterite

We report in this paper results of lattice dynamical calculations of the thermodynamic properties, namely, the equation of state and the melting point of forsterite. The root mean square displacements of atoms and thermal parameters are also evaluated. It is observed from our results, that, a modified Lindemann criterion for melting may be proposed for an ionic ‘molecular’ solid like forsterite. Agreement among various theoretical and experimental results is satisfactory indicating that lattice dynamical studies based on microscopic quasiharmonic formulations help in understanding the macroscopic properties of forsterite, over a wide range of temperature and pressure.

Phonon Density of States and Specific Heat of Forsterite, Mg2SiO4

The phonon density of states of the geophysically important mineral forsterite has been calculated with a rigid-ion model, which gives good agreement with an experimental measurement by inelastic neutron scattering. The density of states has been used to calculate the specific heat as a function of temperature, the results of which are in excellent agreement with calorimetrically measured values. The rigid-ion model takes account of the interatomic interactions and normal modes of vibration on a detailed microscopic basis, and is therefore more realistic than the Debye and other empirical models used previously.

Phonon density of states in fayalite, Se2SiO4

Abstract Inelastic neutron scattering measurements of phonon spectra from a powder sample of fayalite at various temperatures is reported. The measured spectra have been satisfactorily analyzed on the basis of a lattice dynamical model. The phonon density of states is used to derive the specific heat, which are found to be in good agreement with experimental data.

Phonon Density of States and Thermodynamic Properties of Minerals

An important objective of the earth sciences is to develop the capability of predicting the thermodynamic properties of minerals and their phase relations under various pressure-temperature conditions in the earth, moon and the terrestrial planets. Experimentally, thermodynamic properties such as specific heat can be measured by adiabatic and differential scanning calorimetry. However, there are cases, in which the determination of the low temperature specific heat by adiabatic calorimetry is not feasible due to the paucity of materials. Such is the case for the high-pressure and high-temperature magnesium silicate phases: Mg2SiO4 (β- and γ-spinel) and MgSiO3 (ilmenite, perovskite and garnet) considered to be stable in the earth’s mantle. These phases are synthesized in cubic-anvil and split-sphere apparati in 10 to 20 mg quantities, that are barely adequate for the specific heat measurement by differential scanning calorimetry (DSC), usually in the range 300 to 900 K. Hence, the ability to correctly predict the low temperature specific heat of these phases would be extremely useful. Such a theoretical treatment should at the same time provide an understanding of the thermodynamic properties of minerals at the atomistic level. This was the goal aimed at the Mineralogical Society of America Short Course organized by S.W. Kieffer and A. Navrotsky on “Macroscopic to microscopic: Atomic environments to mineral thermodynamics” held at Washington College, Chestertown, Maryland in spring, 1985.

Study of phonon dispersion relations in forsterite, Mg2SiO4 by inelastic neutron scattering

Abstract Phonon dispersion relations in forsterite, Mg2SiO4 (orthorhombic, a = 10.190, b = 5.978, c = 5.753 A , space group PnmaZ = 4) have been determined under ambient conditions using inelastic neutron scattering (INS). Nine acoustic and seventeen optic branches have been measured along the three principal symmetry directions in the Brillouin zone. Seven zone center phonons have been observed with frequencies 104, 144, 184, 192, 200, 258, 315, and 325 cm−1, which are in good agreement with Raman and infra red measurements, except the lowest frequency phonon at 104 cm−1, which is Raman and IR inactive and has been observed for the first time in INS. In order to select suitable wave vector transfers for phonon measurements, the one-phonon neutron structure factors predicted from a lattice-dynamical rigid ion molecular model were used as guides for the INS experiment. The measured dispersion relations agree very well with those predicted from a rigid ‘molecular’ ion model, where the tetrahedral [SiO4] unit is assumed to be a rigid group. The elastic constants determined from the slopes of the acoustic branches show good agreement with those measured by ultrasonic techniques.