sábado, 24 de julio de 2010

Algunos terminos


Quantum Mechanics
Schrodinger wave equation: One dimensional problem, particle in a box, tunnelling through a potential barrier, linear harmonic oscillator, K-P model; Particle in a central potendial: Hydrogen atom; WKB approximation method; Perturbation theory for degenerate & non-degenerate cases: First and second order perturbation, applications-Zeeman effect & Stark effect; Time dependent perturbation theory; Variation method: Application to He atom & van der Waals interaction between two hydrogen atoms; Pauli spin materices; Dirac equation: System of identical particles; many electron system-Hatree & Hatree-Fock approximation.
Advanced Quantum Mechanics
Radiation Theory: Quantization of Schrodinger field, scattering in Born approximation, quantization of classical radiation field, Emission probability of photon, angular distribution of radiation, intensities of Lyman lines, Compton effect and Bremstrahlung.
Path Integral: Approach to quantum mechanics, the principle of least action, quantum mechanical amplitude, path integrals, the path integral as function, the Schrodinger equation for a particle ina field of potential V(x), the Schrodinger equation for the keruel.
Physics of Deformed Solids
Theory of matter transport by defect mechanism: Random walk theory and correlation effects in metals and alloys for impurity and self-diffusion: Theory of ionic transport process, impurity defect association, long range interactions, dielectric loss due to defect dipoles, Internal friction, Radiation damage in metals and semiconductors, colour centres: mechanism of production by various methods, Optical and magnetic properties and models of different colour centre; Theoretical calculation of atomic displacement and energies in defect lattices and amorphous solids, stress-strain and dislocations; Elasticity theory of strees field around edge and screw dislocations, Dislocation interactions and reactions effects on mechanical properties.

X-ray Crystallography
X-ray: production and properties of X-rays, continuous and discrete X-ray spectra, Reciprocal lattice, structure factor and its application, X-ray diffraction from a crystal, X-ray techniques: Weissangerg and precession methods, identification of crystal structure from powder photograph and diffraction traces, Laue photograph for single crystal, geometrical and physical factors affecting X-ray intensities, analysis of amorphous solids and fibre textured crystal.

Low Temperature Physics and Vacuum Techniques
Production of low temperature; Thermodynamics of liquefaction; Joule-Thompson liquefiers; Cryogenic system design: Cryostat design, heat transfer, temperature control, adiabatic demagnetization; Different types of pumps: rotary, diffusion and ion pumps, pumping speeds, conductance & molecular flow; Vacuum gauges: Mclead gauge, thermal conducitivity ionization gauges; Cryogenic thermometry: gas & vapour pressure thermometers, resistance, semiconductor and diode capacitance thermometers, thermocouples, magnetic thermometry.

Physics of Semiconductors and Superconductors
Intrinsic, extrinsic, and degnerate semiconductors; Density of states in a magnetic field; Transport properties of semiconductors; thermo-electric effect, thermomagnetic effect, Piezo-electric resistance, high frequency conductivity; contact phenomena in semiconductors: metal-semiconductor contacts, p-n junction, etc. Optical and photoelectrical phenomena in semiconductors: light absorption by free charge, charge carriers, lattices, and electrons in a localized states, photoresistive effect, Dember effect, photovoltaic effect, Faraday effect, etc.
Phenomena of superconductivity: Pippard?s non-local electrodynamics, thermodynamics of superconducting phase transition; Ginzburg-Landau theory; Type-I and type-II superconductors, Cooper pairs; BCS theory; Hubbard model, RVB theory, Ceramic superconductors: synthesis, composition, structures; Thermal and transport properties: Normal state transport properties, specific heat; role of phonon, interplay between magnetism and superconductivity: Possible mechanism other than electron-phonon interaction for superconductivity.

Solid State Physics
Lattice dynamics of one, two & three dimensional lattices, specific heat, elastic constants, phonon dispersion relations, localized modes; Dielectric and optical properties of insulators: a.c. conductivity dielectric constant, dielectric losses; Transport theory: Free electron theory of solids: density of states, Fermi sphere, Electrons in a periodic potential; Band theory of solids: Nearly free electron theory, tight binding approximation, Brillouin zones, effective mass of electrons and holes.

Polymer Physics
Introduction to macromolecular physics: The chemical structure of polymers, Internal rotations, Configurations, and Conformations, Flexibility of macromolecules, Morphology of polymers; Modern Concept of polymer structure: Physical methods of investigatiing polymer structure such as XRD, UV-VIS, IR, SEM, DTA/TGA, DSC, etc., the structure of crystalline polymers; The physical states of polymers: The rubbery state, Elasticity, etc.; The glassy state, Glass transition temperature, etc., Viscosity of polymers; Advanced polymeric materials: Plasma polymerization, Properties and application of plasma-polymerized organic thin films; Polymer blends and composites: Compounding and mixing of polymer, Their properties of application; Electrical properties of polymers: Basic theory of the dielectric properties of polymers, Dielectric properties of structure of cyrstalline and amorphous polymers.

Optical Crystallography
The morphology of crystals, the optical properties of crystals, the polarizing microscopy, general concept of indicatrix, isotropic and uni-axial indicatrix, orthoscopic and conscopic observation of interference effects, orthoscopic and conscopic examination of crystals. Optical examination of uni-axial and bi-axial crystals, determination of retardation and birefringence, extinction angles, absorption and pleochroism, determination of optical crystallographic properties.

Classification of magnetic materials, Quantum theory of paramagnetism, Pauli paramagnetism, Properties of magnetically ordered solids; Weiss theory of ferromagnetism, interpretation of exchange interaction in solids, ferromagnetic domains; Technical magnetization, intrinsic magnetization of alloys; Theory of antiferromagnetic and ferrimagnetic ordering; Ferrimagnetic oxides and compounds.
Magnetic anisotropy: pair model and one ion model of magnetic anisotropy, Phenomenology of magnetostriction, volume amgnetostricition and form effect; Law of approach of saturation, Structure of domain Wall, Technological applications of magnetic materials.

Thermodynamics of Solids
Properties at O.K, Gruneisen relation, Heat capacities of crystals, specific heat arising from disorder. Rate of approach of equality, Variation of compressibility with temperature, relation between thermal expansion and change of compressibility with pressure. Thermodynamics of phase transformation and chemical reactions. Thermodynamic properties of alloy system: Factors determining the crystal structure; The Hume-Rothery rule, the size of ions; Equilibrium between phases of variable composition, Free energy of binary systems; Thermodynamics of surface and interfaces, Thermodynamics of defects in solids.

Experimental Techniques in Solid State Physics
Measurement of D.C. conductivity, dielectric constant and dielectric loss as a function of temperature and frequency, Magnetization measurement methods (Faraday, VSM and SQUID) magnetic anisotropy and magnetostriction measurements, magnetic domain observation, optical spectroscopy (UV-VIS, IR, etc.), Electron microscopy; Differential thermal analysis (DTA) and thermogravimetric analysis (TGA), Deposition and Growth of thin films by vacuum evaporation Production of low temperature. Single crystal growth and orientation. Magnetic and non-magnetic annealing; Electron spin resonance (ESR), Ferromagnetic resonance (FMR) and nuclear magnetic resonance (NMR).
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Time-resolved X-ray diffraction on laser-excited metal nanoparticles


The lattice expansion and relaxation of noble-metal nanoparticles heated by intense femtosecond laser pulses are measured by pump-probe time-resolved X-ray scattering. Following the laser pulse, shape and angular shift of the (111) Bragg reflection from crystalline silver and gold particles with diameters from 20 to 100 are resolved stroboscopically using 100 X-ray pulses from a synchrotron. We observe a transient lattice expansion that corresponds to a laser-induced temperature rise of up to 200 , and a subsequent lattice relaxation. The relaxation occurs within several hundred picoseconds for embedded silver particles, and several nanoseconds for supported free gold particles. The relaxation time shows a strong dependence on particle size. The relaxation rate appears to be limited by the thermal coupling of the particles to the matrix and substrate, respectively, rather than by bulk thermal diffusion. Furthermore, X-ray diffraction can resolve the internal strain state of the nanoparticles to separate non-thermal from thermal motion of the lattice.

The vibrational properties of nanocrystalline materials, such as the vibrational density of states, can substantially differ from those of bulk crystals, with significant implications for their thermodynamics [1]. One interesting issue is what effects such different vibrational properties may have on the rate of heat transfer across nanostructure interfaces [2]. In comparison to macroscopic situations, heat transfer processes may be considerably modified as structure sizes approach the length scales of electron and phonon wavelengths and mean free paths. Relatively little is known experimentally on the rate of heat transfer from two- or three-dimensionally confined nanostructures, presumably due to difficulties in measuring such rates on extremely small length scales [3]. From an applied point of view, an improved knowledge and understanding of heat transfer processes from such nanostructures appears desirable, as feature sizes of microelectronic devices continue to shrink to nanometer dimensions, leading to increased power dissipation per unit volume and aggravated cooling problems, with the risk of device failure if local overheating occurs.
Here we investigate the thermal dynamics of metal nanoparticles that are heated by femtosecond laser pulses and subsequently cool down via heat transfer to the environment. The electron and lattice dynamics of this model system has previously been investigated in a number of time-resolved optical pump-probe experiments [
4,5,6,7,8,9,10,11,12,13,14,15,16]. It is known to be controlled, on femto- and picosecond time scales, by thermalization of the laser-excited electrons and subsequent electron cooling concomitant with lattice heating. The lattice expansion associated with the lattice heating triggers coherent particle vibrations observable as picosecond periodic signal modulations [11,12,13,14]. However, the heat transfer from the nanoparticles into the embedding material, which usually occurs on much longer time scales, has attracted little attention. For example, it is unclear whether the heat transfer rate is limited by the thermal coupling of the nanoparticles to the embedding matrix, or by bulk thermal diffusion in the embedding material. In the present work, we address this and related issues, using a novel time-resolved optical pump/X-ray probe technique [17]. It gives us much more direct access to the lattice dynamics in the nanoparticles than was available from previous all-optical experiments. The advantage of X-ray scattering methods is that they directly probe the lattice parameter and strain state of the metal particles. Therefore they give direct access to structural properties such as lattice temperature and coherent motion, as recently shown in the case of semiconductor surfaces [18].

Figure 1: Debye-Scherrer ring profiles for a) embedded silver particles of 79 diameter and b) supported gold particles of 20 diameter at different delay times after excitation. Full circles: non-excited profile; open circles in a): ; crosses: . Open circles in b): . Insets: absorbance of the samples. Sketches: experimental geometries, i.e. transmission geometry for embedded particles and reflection geometry for supported particles (X denoting incoming X-ray beam, L laser beam, S sample and X-rays scattered under twice the Bragg angle onto the area detector D).


We study spherical silver and gold nanoparticles of various sizes. The silver particles are prepared in flat glass by ion exchange and subsequent tempering. The particle size is controllable by the preparation conditions; it is derived from absorbance measurements (see inset of fig. 1a)) and TEM analysis [19]. We investigate mean diameters from 24 to 100 , with size dispersions of below 10%. The analysis of the Scherrer width of the particles reveals that the small particles (diameter < href="http://www.iop.org/EJ/article/0295-5075/61/6/762/node6.html#schmitt99">20,21]. Commercial solutions (BBInternational) containing spherical gold particles with defined diameters (20, 60, 80 and 100 ) and dispersion ( ) are used to deposit monolayered colloid films on polyelectrolyte-coated silicon substrates, with surface coverages of around 10%.
By synchronizing a femtosecond laser to the pulse structure of X-rays emitted from the synchrotron radiation source ESRF (Grenoble), we resolve the (111) Bragg reflection of the metal lattice as a function of delay time between exciting laser pulse and probing X-ray pulse, [
17]. The laser system at the station ID09B is an amplified Ti:sapphire femtosecond laser that is phase-locked to the RF clock of the storage ring. The laser delivers pulses of 150 duration at a wavelength of 800 , which are frequency doubled in a BBO crystal to excite the plasmon resonance of the particles (see insets of fig. 1). The chirped pulse amplifier runs at a repetition rate of 896.6 , the 392832th subharmonic of the RF clock. The X-ray pulses are diluted to the same 896.6 repetition rate by a ultrasonic mechanical chopper wheel. The powder scattering from the samples of the monochromatic X-rays (16.45 , (111) double monochromator, toroidal mirror) is collected on a two-dimensional CCD camera (Mar Research) [22]. The resulting Debye-Scherrer rings are integrated azimuthally and corrected for polarization and geometry effects [23]. The X-ray pulse length lies between 90 and 110 (FWHM), depending on the ring current. The delay time is varied by means of electronic delay units, with a typical jitter of 10 (RMS), which is small compared to the X-ray pulse duration. The scattering from X-ray probe pulses is accumulated on the detector at each . As the volume filling factor of the embedded particles is only of the order of 10-4, the Scherrer rings have an intensity of about 5 to 10% of the scattering from the glass matrix. This background is used for a normalization of the profiles prior to baseline subtraction. The embedded particles are excited and probed in transmission geometry through the 0.1-0.2 thick glass substrates, whereas the supported particles are excited and probed in reflection geometry (see insets of figs. 1a) and b)). Grazing angles of 8 degrees for X-rays and 30 degrees for the laser are used in the latter geometry.

Results and discussion

Azimuthally integrated profiles of the Debye-Scherrer rings are presented in fig. 1 for various time delays of the X-ray probe pulses with respect to the laser excitation pulses, . A shift of the peak position is observed for small positive . This shift is a direct measure of the lattice expansion caused by the laser heating of the particles. Peaks split in position at times around 0 , where the earlier part of the X-ray pulse probes the non-excited sample and the later part the excited sample. This splitting allows a determination of the shift even at times shorter than the X-ray pulse duration. The effective time resolution for measuring the onset of the laser-induced lattice expansion is therefore lowered to about 80 .
The laser fluence on the silver samples is optimized for highest lattice expansion without noticeable damage of the sample on the time scale of the experiment (several hours of exposure, corresponding to approximately 107 laser pulses). We note that irreversible damage at higher fluences shows itself as a gradual decrease of the Bragg intensity, followed by Scherrer profile changes. It is known that the particles can be deformed upon excitation with intense laser pulses [
9] by an accumulative effect that can reduce the size of the particles and create small precipitates around them.


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Quantum Mechanics Predicts Unusual Lattice Dynamics Of Vanadium Metal Under Pressure


ScienceDaily (Oct. 12, 2007) — A Swedish research team of Dr. Wei Luo & Professor Rajeev Ahuja and US team of Dr. Y. Ding & Prof. H.K. Mao have used theoretical calculations to understand a totally new type of high-pressure structural phase transition in Vanadium. This phase was not found in earlier experiments for any element and compound. These findings are being published in the Proceedings of the National Academy of Science.
The relation between electronic structure and the crystallographic atomic arrangement is one of the fundamental questions in physics, geophysics and chemistry. Since the discovery of the atomic nature of matter and its periodic structure, this has remained as one of the main questions regarding the very foundation of solid systems.
Scientists at Carnegie's Geophysical Laboratory, USA and Uppsala University, Sweden have discovered a new type of phase transition - a change from one form to another-in vanadium, a metal that is commonly added to steel to make it harder and more durable. Under extremely high pressures, pure vanadium crystals change their shape but do not take up less space as a result, unlike most other elements that undergo phase transitions. This work was appeared in the February 23, 2007 issue of Physical Review Letters.
Trying to understand why high-pressure vanadium uniquely has the record-high superconducting temperature of all known elements inspired us to study high-pressure structure of vanadium. Usually high superconductivity is directly linked to the lattice dynamics of material.

In present paper in PNAS, again a collaboration between Uppsala University and Carnegie's Geophysical Laboratory, USA, we have looked in to the lattice dynamics of vanadium metal and it shows a very unusual behavior under pressure.
A huge change in the electronic structure is driving force behind this unusual lattice dynamics. Moreover, our findings provide a new explanation for the continuous rising of superconducting temperature in high-pressure vanadium, and could lead us to the next breakthrough in superconducting materials.
The relation between electronic structure and the crystallographic atomic arrangement is one of the fundamental questions in physics, geophysics and chemistry. Since the discovery of the atomic nature of matter and its periodic structure, this has remained as one of the main questions regarding the very foundation of solid systems.
Scientists at Carnegie's Geophysical Laboratory, USA and Uppsala University, Sweden have discovered a new type of phase transition - a change from one form to another-in vanadium, a metal that is commonly added to steel to make it harder and more durable.
Under extremely high pressures, pure vanadium crystals change their shape but do not take up less space as a result, unlike most other elements that undergo phase transitions. This work appeared in the February 23, 2007 issue of Physical Review Letters.
"Trying to understand why high-pressure vanadium uniquely has the record-high superconducting temperature of all known elements inspired us to study high-pressure structure of vanadium," said Dr. Wei Luo. "In present paper we have looked into the lattice dynamics of vanadium metal and it shows a very unusual behavior under pressure. A huge change in the electronic structure is driving force behind this unusual lattice dynamics. Moreover, our findings provide a new explanation for the continuous rising of superconducting temperature in high-pressure vanadium, and could lead us to the next breakthrough in superconducting materials."

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International Journal of Solids and Structures,

Submitted for publicationAtomistic Viewpoint of the Applicability of Microcontinuum Theories


Microcontinuum field theories, including micromorphic theory, microstructure theory, micropolar theory, Cosserat theory, nonlocal theory and couple stress theory, are the extensions of the classical field theories for the applications in microscopic space and time scales. They have been expected to overlap atomic model at micro-scale and encompass classical continuum mechanics at macro-scale. This work provides an atomic viewpoint to examine the physical foundations of those well established microcontinuum theories, and to give a justification of their applicability through lattice dynamics and molecular dynamics.


Continuum theories describe a system in terms of a few variables such as mass, temperature, voltage and stress, which are suited directly to measurements and senses. Its success, as well as its expediency and practicality, has been demonstrated and tested throughout the history of science in explaining and predicting diverse physical phenomena.

The fundamental departure of microcontinuum theories from the classical continuum theories is that the former is a continuum model embedded with microstructures for the purpose to describe the microscopic motion or a nonlocal model to describe the long-range material interaction, so as to extend the application of continuum model to microscopic space and short time scales. Among them, Micromophic Theory (Eringen and Suhubi [1964], Eringen [1999]) treats a material body as a continuous collection of a large number of deformable particles, each particle possessing finite size and inner structure. Upon some assumptions such as infinitesimal deformation and slow motion, micromorphic theory can be reduced to Mindlin's Microstructure Theory [1964]. When the microstructure of the material is considered as rigid, it becomes the Micropolar Theory (Eringen and Suhubi [1964]). Assuming a constant microinertia, micropolar theory is identical to the Cosserat Theory [1902]. Eliminating the distinction of macromotion of the particle and the micromotion of its inner structure, it results Couple Stress theory (Mindlin and Tiersten [1962], Toupin [1962]). When the particle reduces to the mass point, all the theories reduce to the classical or ordinary continuum mechanics.
The physical world is composed of atoms moving under the influence of their mutual interaction forces. These interactions at microscopic scale are the physical origin of a lot of macroscopic phenomena. Atomistic investigation would therefore help to identify macroscopic quantities and their correlations, and enhance our understanding of various physical theories. This paper aims to analyze the applicability of those well-established microcontinuum theories from atomistic viewpoint of lattice dynamics and molecular dynamics.

Applicability Analyses from the Viewpoint of Dynamics of Atoms in Crystal

Dynamics of Atoms in Crystal

There are a number of material features, such as chemical properties, material hardness, material symmetry, that can be explained by static atomic structure. There are, however, a large number of technically important properties that can only be understood on the basis of lattice dynamics. These include: thermal properties, thermal conductivity, temperature effect, energy dissipation, sound propagation, phase transition, thermal conductivity, piezoelectricity, dielectric and optical properties, thermo-mechanical-electromagnetic coupling properties.

The atomic motions, that are revealed by those features, are not random., Iin fact they are determined by the forces that atoms exert on each other, and most readily described not in terms of the vibrations of individual atoms, but in terms of traveling waves, as illustrated in Fig.1. Those waves are the normal modes of vibration of the system. The quantum of energy in an elastic wave is called a Phonon; a quantum state of a crystal lattice near its ground state can be specified by the phonons present; at very low temperature a solid can be regarded as a volume containing non-interacting phonons. The frequency-wave vector relationship of phonons is called Phonon Dispersion Relation, which is the fundamental ingredient in the theory of lattice dynamics and can be determined through experimental measurements, such as nNeutron scattering, iInfrared spectroscope and Raman scattering, or first principle calculations or phenomenological modeling. Through phonon dispersion relations, the dynamic characteristics of an atomic system can be represented, the validity of a calculation or a phenomenological modeling can be examined, interatomic force constants can be computed, Born effective charge, on which the strain induced polarization depends, can be obtained, various involved material constants can be determined.
Optical Phonons

Optical phonon branches exist in all crystals that have more than one atom per primitive unit cell. In such crystals, the elastic distortions give rise to wave propagation of two types. In the acoustic type (as LA and TA), all the atoms in the unit cell move in essentially the same phase, resulting in the deformation of lattice, usually referred as homogeneous deformation. In the optical type (as LO and TO), the atoms move within the unit cell, leave the unit cell unchanged, contribute to the discrete feature of an atomic system, and give rise to the internal deformations. In an optical vibration of non-central ionic crystal, the relative displacement between the positive and negative ions gives rise to the piezoelectricity. Optics is a phenomenon that necessitates the presence of an electromagnetic field. In ferroelectrics the anomalously large Born effective charges produce a giant LO-TO splitting in phonon dispersion relations. This feature is associated withto the existence of an anomalously large destabilizing dipole-dipole interaction, sufficient to compensate the stabilizing short-range force and induce the ferroelectric instability. Optical phonons, therefore, appears as the key concept to relate the electronic and structural properties through Born effective charge (Ghosez [1995,1997]). The elastic theory of continuum is the long wave limit of acoustic vibrations of lattice, while optical vibration is the mechanism of a lot of macroscopic phenomena involving thermal, mechanical, electromagnetic and optical coupling effects.

Micropolar theory (Eringen and Suhubi [1964])

When the material particle is considered as rigid, i.e., neglecting the internal motion within the microstructure, micromorphic theory becomes micropolar theory. Therefore, micropolar theory yields only acoustic and external optical modes. They are the translational and rotational modes of rigid units. For molecular crystals or framework crystal, or chopped composite, granular material et al, when the external modes in which the molecules move as rigid units have much lower frequencies and thus dominate the dynamics of atoms, micropolar theory can give a good description to the dynamics of microstructure. It accounts for the dynamic effect of material with rather stiff microstructure.

Assuming a constant microinertia, micropolar theory is identical to Cosserat theory [1902], Compared with micropolar theory, Cosserat theory is limited to problems not involving significant change of the orientation of the microstructure, such as liquid crystal and ferroelctrics.
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themes of Lattice Dynamics

Comparison between the lattice dynamics and molecular dynamics methods: Calculation results for MgSiO3 perovskite

The lattice dynamics (LD) and molecular dynamics (MD) methods have been used to calculate the structure, bulk modulus, and volume thermal expansivity of MgSiO3 perovskite, in order to investigate the reliability of the two simulation techniques over a wide range of temperature and pressure conditions. At an intermediate temperature of 500 K and zero pressure, the LD and MD values are in exellent agreement for both the structure and bulk modulus of MgSiO3 perovskite. At high temperatures and zero pressure, however, the LD method, which is based on the quasi‐harmonic approximation, increasingly overestimates the molar volume of MgSiO3 perovskite because of the neglect of higher‐order anharmonic terms. At the high temperatures and high pressures prevailing in the lower mantle, the errors in the LD values for both the molar volume and bulk modulus, relative to the MD values, are generally small or negligible. However, since anharmonicity decreases substantially with pressure but increases rapidly with temperature, the error in the LD simulated volume thermal expansivity is serious, especially in the lower pressure region.

Ultrafast carrier and lattice dynamics in semiconductor and metal nanocrystals

This articule presents an experimental study of the time-resolved optical response of three different nanoscale systems: CdSe and PbSe quantum dots, and silver triangular nanoplates. The first part of the thesis is devoted to the understanding of the effects of quantum confinement on carrier-carrier interaction in a "model" system: CdSe quantum dots. This issue is addressed by investigating the evolution of the early-time fluorescence spectra of quantum dots of different sizes and lattice structure. The experiment is performed using a femtosecond photoluminescence up-conversion technique, with polychromatic detection. The transient photoluminescence spectra reveal the emission from short-lived multiexciton states. By combining a detailed spectral and kinetic analysis, it is possible to: (i) evaluate the binding energies of these states and therefore acquire insight on the strength of multi-particle interactions, (ii) understand how these interactions affect the lifetime of multiexciton states and, (iii) infer their mechanisms of formation upon optical excitation. We find that confinement-enhanced Coulomb interaction between carriers leads to large binding energies (> 20 meV) and activates efficient Auger-type recombination. This last mechanism points to somewhat different carrier interactions with respect to bulk semiconductors. Surprisingly, we observe that "tailoring" the lattice structure of the quantum dot does not significantly affect the spectral and dynamic properties of multiexciton states. The second part of the thesis addresses the effects of quantum confinement in semiconductor nanocrystals from a slightly different point of view, by investigating PbSe quantum dots. This material is supposed to exhibit a mirror-like, sparse, energetic structure due to extreme quantum confinement which should profoundly alter the carrier relaxation dynamics. We analyze the inter- and intra-band relaxation by combining several techniques. In order to characterize the evolution of the particles luminescence from the nanosecond to the femtosecond range, we perform time-correlated single photon counting and femtosecond near-infrared photoluminescence up-conversion measurements. The results are compared with near infrared, broadband transient absorption measurements. Overall, we observe extremely fast intraband relaxation times, on sub-ps time scales, slightly increasing with decreasing dot size. From our analysis we can estimate a weak electron-phonon coupling between excited states, and we observe that surface mediated relaxation does not play a relevant role in this system. The third part of this work concerns the investigation of the time-resolved optical response of silver triangular nanoplates. The optical response provides fundamental information about the relaxation mechanisms of plasmons, electrons and phonons in metal nanocrystals, and access to the mechanical properties of metal nanoparticles. The anisotropy of the system under study is found to influence the physical properties: we observe for the first time two different excitation mechanisms of mechanical vibrations. In order to disentangle homogenous and inhomogeneous contributions, we present a model which takes into account a realistic distribution of particle size and shape, and which is able to capture the relevant dynamics in these complex systems.
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Lattice Dynamics of MgO at High Pressure: Theory and Experiment


The experimental determination of the phonon dispersion at high pressure constitutes an important ingredient for the characterisation of the physical properties of materials at extreme conditions. It gives access to valuable quantitative information concerning elasticity, thermodynamic properties, and the dynamics of phase instabilities. Furthermore, the experimental data provide important tests for the accuracy of theoretical lattice dynamical models. Among these the most advanced ones are ab initio quantum mechanical calculations, using density functional perturbation theory. Critical inputs are the appropriate choice of the potential (all-electron or pseudopotential approaches) and the correct description of the exchange-correlation term. If a good agreement with the experimental phonon dispersion is observed, these calculations can then be used with increased confidence to describe the physical properties at very high pressures beyond the reach of current experimental methods.

Here we present experimental and theoretical results on MgO, a prototype oxide due to its simple structure and the large stability field (in pressure and temperature) of the NaCl structure. MgO is furthermore an important ceramic for industrial applications, and of great interest for Earth sciences, since it is a major mineral phase of the Earth's lower mantle. A doubly polished single crystal of MgO of (100) orientation, 30 x 50 µm size and a thickness of 20 µm was loaded in a diamond-anvil cell with He as pressure transmitting medium. The IXS experiment was performed on beamline ID28 with an overall energy resolution of 3 meV. Theoretical phonon dispersion curves were calculated using density-functional perturbation theory using the pseudopotential plane wave code ABINIT [1]. Details on the calculations can be found elsewhere.
The fact that characteristic features in the phonon dispersion are well reproduced by calculations gives confidence that ab initio predictions of thermodynamic properties of MgO at high pressure will be accurate. The determination of a thermodynamic property at high pressure requires experimental determination of the thermal expansion and bulk modulus, which are recast into an equation of state (EOS). Such EOS data are very few and when available, the data usually requires large extrapolation. Thermodynamic properties at high pressure may be calculated from a combination of calorimetric data at 1 bar and the volume integral with changing pressure and temperature [3]. Using the available thermodynamic data we obtain CV = 30 (+/-5) and S = 20.68 (+/-1) Jmol-1K-1. From the calculated phonon density-of-states at 35 GPa we determine CV = 31.71 and S = 20.04 Jmol-1K-1. The two data sets match within the errors of experimental data.
In summary, we demonstrate the ability of modern theory to reproduce experimental data on lattice dynamics of an inorganic compound at very high pressure. Expanding such tests to other, more complex systems could be beneficial for the development of both theory and experiment. These tests, validating the approximations done in the calculations, will allow the reliable determination of the thermodynamic properties of materials at high pressure, which are otherwise extremely difficult to assess by experimental methods.
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Pressure Dependence of Phonon Anomalies in Molybdenum

A collaborative group of researchers from Lawrence Livermore National Laboratory and the ESRF have been able to pin down the high-pressure lattice dynamics of the transition metal molybdenum by mapping its phonon energies under extremely high pressure. Using the inelastic X-ray scattering beamline ID28 at the European Synchrotron Radiation Facility (ESRF) and theoretical calculations, the team tracked the pressure evolution of a dynamical anomaly within molybdenum that has challenged scientists for over 40 years.
Much of the interest in the H-point phonon is derived from its anomalous increase in energy with increasing temperature. This observation stimulated numerous theoretical atte mpts to explain this strange behaviour. Changing the temperature or pressure is helpful in that it allows one to probe systems in different thermodynamic configurations. Indeed, the study of mater ials at high pressure is very useful for gaining insight into the nature of the chemical bonds in materials. Notably, the study of lattice dynamics at high pressures in general cannot be performed with neutrons due to the requirement of relatively large samples.

The group developed a new technique for preparing extremely small single Mo crystals of high crystalline quality [1]. These samples (40 micrometres in diameter by 20 micrometres thick) were placed into diamond anvil cells and taken to pressures as high as 40 GPa (400,000 atmospheres) to observe the evolution of the anomaly.

Fig: A small molybednum single crystal loaded in the helium pressure medium. The photomicrograph was taken of the sample in situ at high pressure in the diamond anvil cell.
The researchers observed strong changes in the phonon dispersions at high pressure [2]. The most significant was a large difference in the Gruneisen parameter of modes at the H-point and those around q=0.65 along [001]. These differences lead to a dramatic decrease in the magnitude of the H-point anomaly with increasing pressure. Using theoretical codes developed to model molybdenum, the group showed that there is strong sensitivity of the H-point phonon on the electronic band structure. In fact, the decrease in the H-point anomaly required significant pressure induced broadening to match the experimental data. This implied a strong coupling between electronic states and phonons. With compression, the combination of an increase in the Fermi energy together with a broadening of the electronic states, leads to a significant decrease in this electron-phonon coupling. Thus, molybdenum becomes a much more 'normal' bcc metal at high pressures possibly explaining it's extraordinary stability in the bcc structure to pressures in excess of 400 GPa.

Fig: Phonon dispersions in molybdenum at high pressure. The filled symbols show IXS data taken at 17 GPa at ID28, the open symbols are inelastic neutron scattering results at one atmosphere. Circles are longitudinal acoustic modes; squares transverse acoustic modes. Along [0] the triangles and squares show the two non-degenerate transverse acoustic modes TA[110]<-110> and TA[110]<001> respectively. The dashed lines show the calculations performed at one atmosphere, and the solid lines the calculations at 17 GPa.

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Lattice dynamics of lithium oxide

Li2O fnds several important technological applications, as it is used in solid- state batteries, can be used as a blanket breeding material in nuclear fusion reactors, etc. Li2O exhibits a fast ion phase, characterized by a thermally induced dynamic disorder in the anionic sub-lattice of Li+, at elevated temperatures around 1200 K. We have car- ried out lattice-dynamical calculations of Li2O using a shell model in the quasi-harmonic approximation. The calculated phonon frequencies are in excellent agreement with the reported inelastic neutron scattering data. Thermal expansion, speci¯c heat, elastic con- stants and equation of state have also been calculated which are in good agreement with the available experimental data.


Lithium oxide (Li2O) belongs to the class of superionics, which allow macroscopic movement of ions through their structure. This behavior is characterized by the rapid di®usion of a significant fraction of one of the constituent species within an essentially rigid framework of the other species. In Li2O, Li is the di®using species, while oxygens constitute the rigid framework [1,2].

This material finds several technological applications ranging from lightweight high power-ensity lithium-ion batteries to being a possible candidate for blanket material in future fusion reactors [3,4]. Li2O crystallizes in the anti-fuorite structure with a face-centered cubic lattice and belongs to the Fm3m (O5 h) space group [1,2], lithium being in the tetrahedral coordination. Like other fuorites [5], this also shows a decrease in the elastic constant C11 [6] with emperature around the transition the fast ion phase.In several other fuorites, the fast ion phase is characterized by a specific heat anomaly [7], a Schottky hump in the speci¯c heat. However, no such anomaly has been observed in Li2O [6,8,9].

This paper reports the lattice dynamics calculations done to understand dynamics of anti-fuorite Li2O. Lattice dynamics calculations have been done to calculate the phonon spectrum, specific heat and elastic constants of the oxide. These results are in very good agreement with the available experimental data [6,9{11].

Lattice dynamics calculations
Calculations have been carried out in the quasi-harmonic approximation using inter-atomic potential consisting of both long and short-range terms, using DISPR [12].The form of the potential is given below:

where a and b are empirical parameters [13], and a = 1822 eV and b = 12:364. Oxygen ions have been modeled using a shell model [13,14]. Group theoretical considerations classify the phonons in the entire Brillouin zone into the following representations:
The phonon dispersion relation at ambient conditions is given in figure 1. The zone center modes and the phonons in the entire Brillouin zone are in very good agreement with the available inelastic neutron scattering data [10,11]. The phonon density of states at ambient conditions along with the partial densities of lithium and oxygen is given in figure 2. Both lithium and oxygen contribute almost in the entire Brillouin zone. Lithium's contribution is higher on the higher energy side, with a prominent peak in the region between 50 and 75 meV. Oxygen contribution is greater on the lower energy side with prominent peaks below 60 meV. The specisic heat, CP(T) can be calculated from the knowledge of the phonon density of states. The calculated ratio CP(T)=T has been compared with the ex-perimental result in ¯gure 3. The variation of the Debye temperature (µD) with temperature is given in ¯gure 4. Table 1 gives the calculated values of the elas-tic constants and equilibrium lattice parameters using the model calculations as compared with the experimental results.
A shell model has been successfully used to study the phonon properties of Li2O. The interatomic potential is able to reproduce the equilibrium lattice constant

elastic constants (except C12) and phonon frequencies, which are in unison with the experimental data [1,6]. The phonon dispersion in the entire Brillouin zone agrees well with reported experimental data. The calculated specific heat is in good agreement with the experimental data. The interatomic potential formulated for Li2O oxide may be transferred to other similar °uorites and anti°uorites like Na2O, K2O, UO2, ThO2 etc., with suitable modifications.

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Lattice dynamics of rubidium-IV, an incommensurate host-guest system

In recent years, a number of surprisingly complex crystal structures have been discovered in the elements at high pressures, in particular incommensurately modulated structures and incommensurate host-guest composite structures (see [1] for a review). The crystal structure of the high-pressure phase rubidium-IV shown in Figure 11 belongs to the group of incommensurate host-guest structures that have also been observed in the elements Na, K, Ba, Sr, Sc, As, Sb, and Bi. The structure comprises a framework of rubidium host atoms with open channels that are occupied by linear chains of rubidium guest atoms, and the periodicities of the host and guest subsystems are incommensurate with each other (i.e., they have a non-rational ratio). Although considerable progress has been made in determining the detailed crystal structures of the complex metallic phases at high pressure, little is known about their other physical properties, and the mechanisms that lead to their formation and stability are not yet fully understood.
Fig: Inelastic X-ray scattering spectrum of Rb-IV at 17.0 GPa, with the scattering vector q = (0 0 3.2)h referring to the host lattice. The inset shows the composite crystal structure of Rb-IV with the rubidium host and guest atoms in blue and red, respectively.
We investigated the lattice dynamics in incommensurate composite Rb-IV by inelastic X-ray scattering (IXS) on beamline ID28. The focus was on the longitudinal-acoustic (LA) phonons along the direction of the incommensurate wavevector (parallel to the guest-atom chains). Calculations on simpler model systems predict these phonons to reflect the incommensurability most clearly. Phase IV of Rb is stable at pressures of 16 to 20 GPa at room temperature, and a high-quality single crystal of Rb-IV was grown in a diamond anvil high pressure cell. In the IXS experiment, the incident radiation was monochromatised at a photon energy of 17.8 keV, and two grazing-incidence mirrors focussed the X-rays onto the sample with a focal size of 25 x 60 µm. The spectrum of the scattered radiation was analysed by a high-resolution silicon crystal analyser to yield an overall energy resolution of 3 meV.
Figure 11 shows a typical IXS spectrum of Rb-IV along with its decomposition into the elastic line, the phonon excitation peaks and a constant background, which were obtained by least-squares fitting using the FIT28 software. From a series of IXS spectra collected for different momentum transfers Q, phonon dispersion curves were obtained as shown in Figure 12a. A central result of this study is the observation of two well-defined longitudinal-acoustic (LA)-type phonon branches along the chain direction. They are attributed to separate LA excitations in the host and the guest sublattices, which is a unique feature of an incommensurate composite crystal.
A series of dispersion curves was measured at different pressures, and from this the sound velocities of the host and guest excitations and their pressure dependences were determined (Figure 12b). While the absolute values of the sound velocities in the host and the guest are rather similar, their pressure dependences differ notably. A simple ball-and-spring model of Rb-IV with only one spring constant reproduces these observations semi-quantitatively. This suggests that the difference in the pressure dependences is determined largely by geometrical factors, i.e., by the spatial arrangement of the atoms rather than differences in the chemical bonding in the two subsystems.
There is only very weak coupling between the incommensurate host and the guest in Rb-IV, which raises a rather interesting question. Can the 1D chains of guest atoms in Rb-IV be considered a manifestation of the "monatomic linear chain" treated in solid-state physics textbooks to introduce the concepts of crystal lattice dynamics? The pressure dependence of the interatomic spacing in the guest-atom chains was measured in earlier structural studies and enables the spring constant in the linear chain model to be determined, and also its pressure dependence. On this basis, the sound velocity in the linear chains and its pressure dependence were modelled as shown in Figure 12b. The results are in excellent agreement with the IXS data for the guest-atom chains in the composite Rb-IV structure, which can thus be regarded as a manifestation of the monatomic linear chain model with regard to the LA phonons.

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KNb03 continues t o be of a p a r t i c u l a r i n t e r e s t among f e r r o e l e c t r i c (FE) materials because it undergoes three successive phase t r a n s ~ t i o n s , whose mechanism is t h e object of many controversies. I n e l a s t i c neutron scattering measurements1 performed i n the tetragonal phase (T = 245OC) reveal the presence of a low-frequency E(T0) zone center mode together with an high anisotropy in the lowest phonon branches which can be related t o the existence of strong correlations along the <100> axes between t h e l a r g e motions associated t o the soft modes. Recent infrared r e f l e c t i v i t y measurements clearly show the softening of the lowest-frequency Flu phonon with decreasing temperature in the e n t i r e range of the cubic phase (425"C<910°C).<>
Lattice-dynamical calculations are c a r r i e d out for the cubic (0; space group) and the tetragonal ( c : ~sp ace group) phases in order t o explain the main experimental features. The aim of t h i s paper is to show, f o r t h e f i r s t time, that the dispersion curves for both phases can be described s a t i s f a c t o r i l y by the same harmonic model.

Description of the model.
In t h i s work for the cubic phase we use the model developped by cowley3 for SrTi03 with axially symmetric short-range force constants A and B between nearest neighbours (K-0, Nb-0, 0-0). In addition we allow an anisotropy in the oxygen core-shell coupling constant k(O) as previously suggested by Migoni e t a l . for both SrTi03 and KTa03. Therefore the components of the tensor 5 (0) in the directions of neighbouring K and Nb ions are denoted k(0-K) and k(0-Nb) respectively. This leads to a 15 parameters model f o r t h e cubic phase (Table 1) .
Since the symmetry is lower in the tetragonal phase (Figure I ) , the number of parameters is, i n principle, larger than in the cubic phase. Using simple geometry arguments we are able to derive the parameters of the tetragonal phase from those of the cubic structure. Consequently, the short-range constants of the tetragonal phase are expressed as functions of the l a t t i c e parameters c and a, of the spontaneous ionic displacements 6, measured by ~ e w a t ~an,d of the s e t of parameters A and B, already defined i n the cubic phase. Furthermore, in the tetragonal phase, we distinguish two kinds of oxygen ions, 0(1) (located on the polar axis passing through the Nb ion) and 0(2) or 0(3) ( i n t h e plane perpendicular) . Consequently we allow the
parameter k (0(1) -Nb) to be different from the parameters k(0 -Nb) and k(0(3)-Nb) .

Results and discussion.
For the tetragonal phase, the dispersion curves are calculated with parameters adjusted to the results of i n e l a s t i c neutron1 and ama an^ scattering measurements. The experimental data are q u i t e s a t i s f a c t o r i l y reproduced by the model calculations (Fig.2) except for the A2 (TOhpranchnearthe zoneboundary This f a i l u r e may be due to the large anharmonic coupling between acoustic and s of toptic phonons in t h i s direction. The strong anisotropy in the lowest dispersion branches is interesting to notice together with an anticrossing and eigenvector exchange between optic and acoustic A2 branches, which take place approximately a t 1/3 of the Brillouin zone edge. The s o f t E(T0) phonon a t qZO is essentiallyckiaracterized by a large vibrationnal amplitude of the niobium ion relative to the oxygen ion located along the mode polarization direction. Moreover the Einstein o s c i l l a t o r response of t h i s mode in the whole Brillouin zone confirms the presence of dynamical correlation chains Nb-0-Nb directed along the [loo] and [010] axes1. This shows t h e v a l i d i -
ty of t h e l i n e a r chain model for describing the s o f t mode behaviour in perovskites, as emphasized by Bilz e t a1.7 for KTa03.

For the cubic phase, the phonon branches are obtained with the parameter values previously used in the tetragonal phase, except for the coupling k(0-Nb) (Table 1). In figure 2 the calculations are compared with the experimental data of Nunes e t a1.8 In addition to t h e l a r g e anisotropy in the phonon dispersion surface, we can also note the softening of the lowest Flu phonon with a r e l a t i v e l y s l i g h t change in the value of k(0-Nb). The other zone-centre modes are nearly insensitive to this variation.

At the cubic-tetragonal phase transition, therefore, the splitting and the frequency shift6 of these phonons depend only on the geometrical effect in the force constants. On the contrary, the large separation of the FE soft Flu phonon into a soft E component (1.55 THz) and a hardened A1 (8.35 Mz) essentially originates from a k (0 -Nb) value larger than k (0 -Nb) (Table 1) . (1) (2) All these results emphasize the role of the intraionic oxygen polarizability in the phase transition mechanism of KNb03. This polarizability is dynamically enhanced by the hybridization of oxygen 2p states with niobium 4d states9 in the directdn of the chain like coupling Nb-0-Nb. This leads to a softening of the lowest cubic Flu(TO) and tetragonal E(TO1 modes. At the phase transition, the disappearance of the [001] correlation due to the asymmetry of the Nb-0-Nb bound is related to the abrupt change in the value of k(0-Nb) along the polar direction, and therefore to the stabilization of the ferroelectric Al(T0) component.
In order to specify the behaviour of the oxygen polarizability, calculations with a model including the temperature dependence of the soft mode are actually in progress.
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domingo, 27 de junio de 2010

Algorithms for dynamical fermions -- Hybrid Monte Carlo


In the previous post in this series parallelling our local discussion seminar on this review, we reminded ourselves of some basic ideas of Markov Chain Monte Carlo simulations. In this post, we are going to look at the Hybrid Monte Carlo algorithm.

To simulate lattice theories with dynamical fermions, one wants an exact algorithm that performs global updates, because local updates are not cheap if the action is not local (as is the case with the fermionic determinant), and which can take large steps through configuration space to avoid critical slowing down. An algorithm satisfying these demands is Hybrid Monte Carlo (HMC). HMC is based on the idea of simulating a dynamical system with Hamiltonian H = 1/2 p2 + S(q), where one introduces fictitious conjugate momenta p for the original configuration variables q, and treats the action as the potential of the fictitious dynamical system. If one now generates a Markov chain with fixed point distribution e-H(p,q), then the distribution of q ignoring p (the "marginal distribution") is the desired e-S(q).

To build such a Markov chain, one alternates two steps: Molecular Dynamics Monte Carlo (MDMC) and momentum refreshment.

MDMC is based on the fact that besides conserving the Hamiltonian, the time evolution of a Hamiltonian system preserves the phase space measure (by Liouville's theorem). So if at the end of a Hamiltonian trajectory of length τ we reverse the momentum, we get a mapping from (p,q) to (-p',q') and vice versa, thus obeying detailed balance: e-H(p,q) P((-p',q'),(p,q)) = e-H(p',q') P((p,q),(-p',q')), ensuring the correct fixed-point distribution. Of course, we can't actually exactly integrate Hamilton's in general; instead, we are content with numerical integration with an integrator that preserves the phase space measure exactly (more about which presently), but only approximately conserves the Hamiltonian. We make the algorithm exact nevertheless by adding a Metropolis step that accepts the new configuration with probability e-δH, where δH is the change in the Hamiltonian under the numerical integration.

The Markov step of MDMC is of course totally degenerate: the transition probability is essentially a δ-distribution, since one can only get to one other configuration from any one configuration, and this relation is reciprocal. So while it does indeed satisfy detailed balance, this Markov step is hopelessly non-egodic.

To make it ergodic without ruining detailed balance, we alternate between MDMC and momentum refreshment, where we redraw the fictitious momenta at random from a Gaussian distribution without regard to their present value or that of the configuration variables q: P((p',q),(p,q))=e-1/2 p'2. Obviously, this step will preserve the desired fixed-point distribution (which is after all simply Gaussian in the momenta). It is also obviously non-ergodic since it never changes the configuration variables q. However, it does allow large changes in the Hamiltonian and breaks the degeneracy of the MDMC step.

While it is generally not possible to prove with any degree of rigour that the combination of MDMC and momentum is ergodic, intuitively and empirically this is indeed the case. What remains to see to make this a practical algorithm is to find numerical integrators that exactly preserve the phase space measure.

This order is fulfilled by symplectic integrators. The basic idea is to consider the time evolution operator exp(τ d/dt) = exp(τ(-∂qH ∂p+∂pH ∂q)) = exp(τh) as the exponential of a differential operator on phase space. We can then decompose the latter as h = -∂qH ∂p+∂pH ∂q = P+Q, where P = -∂qH ∂p and Q = ∂pH ∂q. Since ∂qH = S'(q) and ∂pH = p, we can immediately evaluate the action of eτP and eτQ on the state (p,q) by applying Taylor's theorem: eτQ(p,q) = (p,q+τp), and eτP = (p-τS'(q),q).

Since each of these maps is simply a shear along one direction in phase space, they are clearly area preserving; so are all their powers and mutual products. In order to combine them into a suitable integrator, we need the Baker-Campbell-Hausdorff (BCH) formula.

The BCH formula says that for two elements A,B of an associative algebra, the identity

log(eAeB) = A + (∫01 ((x log x)/(x-1)){x=ead Aet ad B} dt) (B)

holds, where (ad A )(B) = [A,B], and the exponential and logarithm are defined via their power series (around the identity in the case of the logarithm). Expanding the first few terms, one finds

log(eAeB) = A + B + 1/2 [A,B] + 1/12 [A-B,[A,B]] - 1/24 [B,[A,[A,B]]] + ...

Applying this to a symmetric product, one finds

log(e1/2 AeBe1/2 A) = A + B + 1/24 [A+2B,[A,B]] + ...

where in both cases the dots denote fifth-order terms.

We can then use this to build symmetric products (we want symmetric products to ensure reversibility) of eP and eQ that are equal to eτh up to some controlled error. The simplest example is

(eδτ/2 Peδτ Qeδτ/2 P)τ/δτ = eτ(P+Q) + O((δτ)2)

and more complex examples can be found that either reduce the order of the error (although doing so requires one to use negative times steps -δτ as well as positive ones) or minimize the error by splitting the force term P into pieces Pi that each get their own time step δτi to account for their different sizes.
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Lattice dynamics (of KI perturbed and Thermodynamics)


Of KI perturbed, The theoretical study of the physical properties of JT centres (impurities, defects, etc. )embedded in a crystal is usually done in the framework of the cluster model [I]. In such a model the localized electron of the centre, which is in a degenerate state by definition of the JT effect, is assumed to interact only with a single localized vibrational mode, any lattice dispersion of frequency being neglected. This model is known to take into account several important features of the JT induced properties [I] but not the details of the optical spectra (Raman, infrared, vibronic sidebands of the absorption/emission spectra), which are very sensitive to the dynamics of the host crystal.
In a recent paper [2] the problem of the lattice dynamics perturbed by a JT centre was studied and resolved in the weak interaction limit by using the Green function method. It was found that a JT centre generates a very peculiar perturbation on the dynamical matrix.
On the other hand, a similar result can be obtained when the absorption lineshape due to a singletmultiplet transition (the dynamic JT effect) is studied. In this case the JT perturbed lattice dynamics is that relative to the excited electronic degenerate state. In both cases we conclude that the JT interaction induces a perturbation on the force constant matrix, which can be eventually seen in the structures of the phonon densities. Such a result, which will be presented in details in a forthcoming paper, can be deduced in the limit of the weak interaction from the absorption lineshape theory proposed some years ago (ref. [3] hereafter called MT).
In the present paper, after some consideration on the previous point, we present the computed one-phonon spectra for a (e x E) JT centre. As usual, (e x E) means that the electron is in a e-symmetry degenerate state and interacts with the E-symmetry displacements of the neighbour ions. Our aim is to see to what extent the JT dynamical perturbation can qualitatively modify the host lattice dynamics. So we do not apply it to any real impurity.
In the following we assume : 1) The JT centre is embedded in an ionic crystal. 2) The JT interaction is weak. 3) Only the n.n. of the JT centre are involved in the JT interaction.
The shape of the optical spectra is related to the symmetrized (E-symmetry in this case) projected phonon density p,(w) relative to the lattice dynamics, either of the ground electronic state (Raman and infrared spectra), or of the excited electronic state (vibronic sidebands). We remember that p,(o) is given by the imaginary part of the E-symmetry phonon propagator DE(a). So we first show the relevant JT induced processes which dress the unperturbed onephonon
propagator, and then we evaluated p,(o) from D,(o). By unperturbed propagator we mean the onephonon propagator when all the dynamical perturbations induced by the centre (change of mass and of force constants) are considered, but the JT interaction.
In the cluster model, where one assumes that only the n.n. of the centre are involved in HL, as they are in He,, only two degenerate E-symmetry modes with the same frequency are representative of the whole lattice dynamics. So the commutator (6) loses its meaning. From the properties of the commutators (4) and (5) one can see that, while the electron and the
phonons are truly independent when He, = 0, they are so entangled for HeL # 0 to determined new coupled states, the so-called vibronic states. When the vibronic states are studied in the cluster model, the role played by HeL on the electronic states is emphasized but the possible modifications of HL are neglected [I].
The opposite approach is here follow ed. In fact we compute the modifications of the one-phonon projected density of states pE(02), when the electron is supposed to be degenerate but structureless. Without the JT interaction, the unperturbed phonon propagator D;(t) (a dotted line in the diagram of figure 1) is given at T = 0 K by

where p;(02) is the unperturbed projected one-phonon density. (By &(w2) we mean its Hilbert transform.) In the harmonic approximation a phonon does not change frequency when a linear electron-phonon interaction is switched on and the only effect is a generation of many unperturbed phonons. When the electron is in a degenerate state, since the usual separation of the phonons from the electron (adiabatic approximation) is not valid anymore, the linear JT interaction modifies the frequency of the phonons. By following MT [3], one can prove that in the diagrams accounting for the many-phonon process there is always a class of higher order graphs (the crossing graphs) which give a contribution to the lower order graphs, when the electron line is removed by assuming that the electron is structureless.
Lattice Dynamics and Thermodynamics of Molybdenum from First-Principles Calculations
of the ultrahigh pressure scale. The equation of state (EOS) of Mo at high pressure is being used as a calibration standard to the ruby fluorescence in diamond anvil compression (DAC) experiments.1 Because of its important position in the field of material science and condensed matter science, Mo has attracted tremendous experimental and theoretical interest in its wide range of properties recently. At ambient condition, Mo is in body-centered-cubic (bcc) structure and melts at 2890 K.2 But what is the most stable phase of Mo under ultrahigh pressure? Experimentally, the shock wave (SW) acoustic velocity measurements showed that there was a sharp break on the Hugoniot curve at about 210 GPa (at a calculated temperature of 4100 K), which indicated that a solid-solid phase transition occurred prior to melting at 390 GPa (at a calculated temperature of 10 000 K).3 To compare with the SW data, Vohra and Ruoff investigated the static compression of Mo by energy-dispersive X-ray diffraction and found that the bcc Mo was stable up to 272 GPa at 300 K.4 The phase transition in shock compression at 210 GPa was not observed. By further X-ray diffraction investigation, Ruoff et al. showed that Mo was stable in a bcc structure up to at least 560 GPa at room temperature.5 Theoretically, Moriarty suggested that Mo was stable in the bcc structure up to 420 GPa, where it transformed to a hexagonal close-packed (hcp) structure and then at 620 GPa to a face-centered closepacked (fcc) structure.6 Later, Boettger7 and Christensen et al.8 showed that the hcp phase of Mo was not stable, and the bcc phase transformed directly to the fcc phase at 700 GPa.
Belonoshko et al. confirmed the results of Boettger and showed the transition pressure was 720 GPa at zero pressure.9 By calculating the Gibbs free energies of the bcc and fcc Mo in the pressure range from 350 to 850 GPa at room temperatures up to 7500 K, Belonoshko et al. found that Mo had lower free energy in the fcc phase than in the bcc phase at elevated temperatures.10 Shortly after these new results were reported, Cazorla et al. found that the hcp Mo was noticeably more stable above 350 GPa at high temperature by calculating the Helmholtz and Gibbs free energies of the bcc, fcc, and hcp phases.11 The other intriguing problem is the melting properties. For the transition metals, such as Mo, Ta, and W, there are enormous discrepancies in melting curves between laser-heated DAC12-17 and SW3,18 methods. As for Mo (as well as Ta and W), several thousand degrees of discrepancies exist in extrapolating from DAC pressures of around 100 GPa12-15,17 to SW pressure of 390 GPa.18 As is known that the overestimation of the melting temperature exists in SW experiments, Errandonea13 corrected the SW data by considering 30% superheating. The revised melting temperatures are located at 7700 ( 1500 K (390 GPa), also much larger than the melting temperature (just above 4000 K) at this pressure extrapolated from DAC experiments. Results from the empirical and phenomenological melting models are dependent on the selection of the parameters.13,19-22 It is shown that the choice of different sets of parameters leads to huge differences in the melting temperatures at high pressure. In addition, theoretical results are consistent with the SW data at high pressure but diverge from DAC data below 100 GPa. All these results are inadequate to explain the extreme discrepancies in extrapolating the DAC data to the SW data.9-11,23 The highpressure melting curve of Mo still remains inconclusive up to now.

The low-temperature phonon spectrum of Mo as measured in inelastic neutron scattering experiments exhibited a variety of anomalies: the large softening near the H point, the T2 branch at the N point, and the longitudinal branch did
not display the rounded dip near q ) 2/3 [111], which was typical for "regular" monatomic bcc metals.24-26 With temperature increasing, the H point phonon displayed anomalous stiffening, which had been proposed to arise from either intrinsic anharmonicity of the interatomic potential or electron-phonon coupling.26 Theoretically, using the molecular dynamics (MD) simulations with environment-dependent tight-binding parametrization, Haas et al. reproduced the weakening of the phonon anomalies as the temperature increased.27 Later, Farber et al. determined the lattice dynamics of Mo at high pressure to 37 GPa using high-resolution inelastic X-ray scattering (IXS).28 Meanwhile, they calculated the quasiharmonic phonon spectrum up to the highest experimental pressure by linear response theory. Both the experimental and theoretical results showed an obvious decrease in the relative magnitude of the H point phonon anomaly under compression. Recently, Cazorla et al. obtained the phonon dispersion curves of the bcc Mo at zero pressure using the small displacement method,29 but under larger compression, there are no experimental and theoretical studies. The first-principles density functional theory is very successful in predicting the high-pressure behavior of phonon dispersion relations and their concomitant anharmonic effects.30 It is more effective to connect the phonon properties directly to the lattice dynamics under pressure, temperature, or their combination. One of the main purposes of this work is to investigate the lattice dynamics and thermodynamics of Mo under high pressure and temperature.
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