inelastic neutron

INELASTIC NEUTRON SCATTERING AND
LATTICE DYNAMICS OF NOVEL COMPOUNDS
Acrystal is described as a perfect periodic three-dimensional array of atoms. However, the atoms are not static at their lattice sites but vibrate about their mean positions with energies governed by the temperature of the solid. The collective motions of atoms in solids form traveling waves (called lattice vibrations), which are quantized in terms of "phonons". The study of lattice vibrations is of considerable interest because several physical properties of crystals like their specific heat, thermal expansion, phase transitions are related to the vibrations of atoms in solids [1-3]. The experimental studies of lattice vibrations are carried out using techniques like Raman spectroscopy, infrared absorption (IR), inelastic neutron scattering, inelastic X-ray scattering, etc.
Unlike Raman and infrared studies which probe only the long wavelength excitations in onephonon scattering, inelastic neutron and X-ray scattering can directly probe the phonons in the entire Brillouin zone. While inelastic neutron scattering is widely used for such measurements, inelastic X-ray scattering has also been recently used at intense synchrotrons sources for the study of phonons in a few materials. Experimental studies at high pressures and temperatures are often limited and accurate models for the compounds are of utmost importance. A major goal of research therefore has been theoretical predictions of the thermodynamic properties. The success of the models in predicting thermodynamic properties depends crucially on their ability to explain a variety of microscopic and macroscopic dynamical properties. These include an understanding of the crystal structure, elastic constants, equation of state, phonon frequencies, dispersion relations, density of states and thermodynamic quantities like the specific heat and thermal expansion.
The experimental neutron and long wavelength optical data are used to test and validate models of interatomic potentials, which in turn have been used to predict thermodynamic properties at high pressures and temperatures. We have developed models of interatomic potentials for several novel compounds which allow to calculate the structural and dynamical properties as a function of pressure and temperature. In order to validate the interatomic potentials, we have carried out inelastic neutron scattering experiments on polycrystalline and single crystal samples at different facilities namely, Dhruva reactor, Trombay (India), ILL (France), ISIS (UK) and ANL (USA).
Sections below give brief information about the experimental technique and the lattice dynamics
calculations respectively, while the results and discussion, and conclusions are presented later.
Experimental
Inelastic-neutron-scattering (INS) experiments [3] may be performed using both single crystals
and polycrystalline samples, which provide complementary information. The single crystals may be used to obtain the details of the phonon dispersion relation (PDR), namely the relation between the phonon energies and their wavevectors, for selected values of the wavevectors. On the other hand, the polycrystalline samples provide the phonon density of state (PDOS) integrated over all wave vectors in the Brillouin zone. The inelasticneutron- scattering experiments require much larger-sized samples (single crystals of the order of 1 cm3 and powder samples of about 10 cm3 upwards) than those used in optical spectroscopies. Measurements of the phonon dispersion relations and density of states can in principle be carried out using both reactors as well as spallation sources. However, thermal neutrons (E ~ 25 meV) from a nuclear reactor are best suited for the measurements of the acoustic and low-frequency optic modes in a single crystal. On the other hand, the high energies of neutrons from a spallation source enable measurements over the entire spectral range and are best exploited for the measurements of the phonon density of states.
Lattice Dynamical Calculations
Lattice dynamical calculations [2] of the vibrational properties may be carried out using either a quantum-mechanical ab-initio approach or an atomistc approach involving semiempirical interatomic potentials. However, due to structural complexity of the compounds which we have studied, detailed calculations are carried out using semiempirical models. The interatomic potentials consist of Coulombic and short-ranged Born-Mayer type interactions. The parameters of the potentials have been evaluated using the structural and dynamical equilibrium conditions as well as other known experimental data. The optimized parameters are used for lattice dynamics studies of the system.
Results and Discussion
Negative thermal expansion compounds: ZrW2O8, HfW2O8 and ZrMo2O8 The compounds ZrW2O8, HfW2O8 and ZrMo2O8 are of considerable interest [4] due to their large isotropic negative thermal expansion (NTE) in their cubic phase over a wide range of temperatures up to 1443 K, 1050 K and 600 K, respectively. This remarkable feature makes these compounds potential constituents in composites to adjust thermal expansion to a desired value. Thermal expansion in insulating materials is related to the anharmonicity of lattice vibrations. We have carried out lattice dynamical calculations for these compounds using a transferable interatomic potential [4-8]. The phonon frequencies as a function of wave vectors in the entire Brillouin zone and its volume dependence in quasiharmonic approximation are calculated. The calculations predicted that large softening of the phonon spectrum involving librational and translational modes below 10 meV would be responsible for NTE in these compounds. In order to check our prediction we have carried out high-pressure inelastic neutron scattering experiments [8-10] at several pressures up to 2.5 kbar on polycrystalline samples of ZrW2O8 and ZrMo2O8 using IN6 spectrometer at ILL, France. In case of ZrW2O8 at 1.7 kbar, phonon softening of about 0.1-0.2 meV is observed (Fig. 1) for phonons below 8 meV. Similar shift is observed for ZrMo2O8 at 2.5 kbar. The Grüneisen parameters of phonon modes have been determined as a function of their energy. The experiments validate our lattice dynamical calculations (Fig. 1). In order to check the quality of interatomic potential model the phonon density of states data has also been recently obtained upto 160 meV for HfW2O8 using time of flight technique at IPNS (USA) in collaborative experiments [7].
Silicate mineral zircon, ZrSiO4
Zircon, ZrSiO4 is an important host silicate mineral for radioactive elements uranium and thorium in the earth's crust. Since it is a natural host for the radioactive elements in the crust, it is a potential candidate for nuclear waste storage. High pressure and temperature stability of zircon is therefore of considerable interest.The phonon dispersion relation has been measured (Fig. 2) in zircon (ZrSiO4) from neutron experiments at Dhruva reactor, Trombay, at low energies upto 32 meV [11]. The measurements at high energies require good resolution and high intensity of the neutron beam. We have further extended the measurements upto 70 meV (Fig. 2) using the time of flight technique [12] at ISIS, UK. These extensive phonon measurements upto 70 meV provide a rare example of such studies carried out using a pulse neutron sources on any material. Such extensive measurements have
been performed on only a few mineral systems even using a continuous reactor source. A lattice dynamical model was used to plan the experiments and analyze the data, as well as to calculate the elastic constants, long-wavelength phonon frequencies and thermal expansion [13]. The calculations are in good agreement with the experimental data.
Conclusions
A combination of lattice dynamics calculations and inelastic neutron scattering measurements have been successfully used to study the phonon properties and their manifestations in thermodynamic quantities like the specific heat, thermal expansion and equation of state. The experiments validate the models and the models in turn have been fruitfully used to calculate the phonon spectra and various thermodynamic properties at high pressures and temperatures. The calculations have been very useful in the planning, execution and analysis of the experiments and have enabled microscopic interpretations of the observed data. These studies have also been exploited to study the anomalous properties like large negative thermal expansion in various compounds.