sábado, 22 de mayo de 2010

Inelastic neutron scattering and lattice dynamics

Inelastic neutron scattering (INS) is one of the experimental methods to studythe dynamics of materials, the complementary methods being Brillouin spec-troscopy, Raman scattering, infrared spectroscopy and inelastic x-ray scattering.Determination of the crystal structure through diffraction methods gives informa-tion of the atomic positional coordinates i.e. the positions where the inter-atomicpotential has minima. On the other hand, vibrations are related to the shape ofthe potential near these minima. A knowledge of the vibrations gives access to themicroscopic quantities (like inter-atomic interaction potential) involved in thermo-dynamic properties, phase transitions, electronic properties and many others.Collective vibrations (phonons) are the elementary excitations of any orderedsystem in condensed matter. Thermal neutrons, with energies of the order of a fewmeV and de Broglie wavelengths of the order of Angstrom units, are unique probesof these excitations. In principle, a complete determination of the phonon spectrumis possible through inelastic scattering of neutrons. In addition to determination ofthe phonon spectra through experimental methods, an understanding of these spec-tra through theoretical formalisms is essential, for interpretation of the results fromexperiments. Collective vibrations are investigated by means of coherent inelastic neutron scattering. On the other hand, incoherent inelastic neutron scattering ispredominantly employed to study single-particle motions, usually, as has been usedat Trombay, for investigation of materials containing hydrogen for example, ro-tational behaviour of ammonium ions in salts, water in hydrates and dynamics ofvarious subgroups in amino acids. This talk will focus only on experiments andresults from coherent inelastic neutron scattering.In Trombay, apart from certain measurements (for example, to determine thephonon dispersion relation in beryllium) which employed the lter detectorspectrometer (FDS), the triple-axis spectrometer (TAS) has been the instrumentof choice for these experiments, for determination of the phonon density of states(PDOS) or phonon dispersion relation (PDR). The TAS (invented by ProfessorBertram Brockhouse (in 1961) who was honoured with the award of the Nobel Prizein Physics in 1994) is a very important instrument for neutron spectroscopy sinceit allows for a controlled measurement of the scattering function S(Q;E) at anypoint in momentum (Q) and energy (E) space. In TAS, the monochromator singlecrystal (Cu (1 1 1) at Dhruva) determines the energy of the neutron incident onthe sample while the analyzer single crystal (pyrolytic graphite (0 0 0 2) at Dhruva)is used to analyze the spectrum of the neutrons scattered from the sample. Thelaws of momentum and energy conservation governing all scattering experimentsare well-known:



In these equations, the wave vector magnitude k = 2¼, where is the wavelengthof the neutron, and the momentum transferred to the crystal is Q. The subscripti refers to the beam incident on the sample and f to the (nal) beam scattered fromthe sample; G is a reciprocal-lattice vector. The energy transferred to the sampleis hº.At CIRUS reactor, numerous studies of incoherent scattering of neutrons fromhydrogenous materials were carried out; some of them being studies of ammoniumion dynamics in salts, the librational modes of water molecules in single crystalhydrates, amino acids. Most of these studies were carried out using the FDS. Onthe TAS, phonon dispersion relations of materials like magnesium, beryllium, zinc,potassium nitrate, and Sb2S3 were measured. In fact, the determination of thephonon dispersion curves of beryllium were, for the time, largely carriedout on a FDS and gave accurate results, comparable to those obtained on a TASand could extend measurements beyond what were accessible on a TAS. The PDRmeasurements on potassium nitrate (KNO3) were interpreted through latticedynamical computations on the basis of a rigid molecular ion model using theexternal mode formalism the first time that this was done for an ionic-molecular system.
The inelastic neutron scattering and lattice dynamics studies carried out at Dhruva reactor may be broadly classiffed into two categories: studies of geophys-ically important minerals (silicates and carbonates) and those of technologically


relevant materials (high-temperature superconductors, intermetallic superconductors, and ceramics). In the sections that follow, studies carried out on some of these would be described in brief, highlighting the significance of the results.

Geophysically important minerals

With the aim to provide a microscopic understanding of the vibrational and thermo-dynamic properties of geophysically important minerals, studies were carried out on a large number of silicate minerals including the olivine end members forsterite and fayalite, the pyroxene end member enstatite, the garnet mineral almandine,the mineral zircon and the aluminium silicate polymorphs sillimanite, andalusiteand kyanite. Detailed inelastic neutron scattering measurements of the PDR and PDOS supported by group theoretical selection rules and model calculationshave been instrumental in the prediction of the thermodynamic properties of min-erals corresponding to the pressure and temperature at which they are believedto occur in the Earth. All of these minerals have fairly complex structures but acomparatively simple interatomic potential model has been employed to providetheoretical estimates of several microscopic and macroscopic properties includingthe elastic constants, phonon frequencies, dispersion relations, density of states andthermodynamic quantities like specific heat, thermal expansion, equation of stateand melting. Forsterite and enstatite: In forsterite (Mg2SiO4), group theoretical selection ruleswere used as guides for coherent INS experiments on single crystals (carried outat the Brookhaven National Laboratory) to determine the phonon dispersion relations. The model calculations, in fact, reproduced both the phonon frequencies as

well as the neutron intensities (and hence, the polarization vectors) fairly well. Thestructure of forsterite consists of isolated silicate tetrahedra while that of orthoen-statite (Mg2Si2O6) contains chains of these tetrahedra. Measurement of density ofstates were carried out using powder samples at Argonne National Laboratory. INSmeasurements on polycrystalline samples of these minerals show features which area consequence of these structural differences the band gaps found in the phonondensity of states in forsterite are falled by the vibrations of the bridg-ing oxygens in the silicate chains in orthoenstatite. The calculated phonon spectra reproduce these differences. Al2SiO5 polymorphs: Phase transitions amongst the three aluminium silicatepolymorphs sillimanite, andalusite and kyanite have been studied both theoret-ically and experimentally. In the structure of these polymorphs, one aluminium ionis in octahedral coordination and forms edge-sharing chains, the other aluminumion is in tetrahedral coordination in sillimanite, ¯ve-coordinated in andalusite andin octahedral coordination in kyanite. The phonon dispersion curves of the low energy modes of andalusite (figure 1) have been measured on the TASat Dhruva and are complementary to previously reported data (phonon dis-persion curves along [0 0 1]) from measurements at the Paul Scherrer Institute,Switzerland. Measurements on polycrystalline samples of sillimanite and kyanite


Conclusion
This talk has reviewed the extensive work done on various materials (geophysicallyimportant minerals (Al2SiO5 polymorphs, zircon, MnCO3) and technologically im-portant materials (ZrW2O8, °uorohalides, high temperature superconductors)) andthus highlighted the complementary nature of coherent inelastic neutron scatter-ing experiments and lattice dynamical model computations leading to a completeunderstanding of the nature of dynamics of atoms in these materials, and in turn,explaining several data pertaining to macroscopic thermodynamic properties.


ORLANING COLMENARES
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