tag:blogger.com,1999:blog-44079312473575819372024-03-14T08:20:10.617-07:00Lattice DynamicsOrlaninghttp://www.blogger.com/profile/08859376411004440286noreply@blogger.comBlogger21125tag:blogger.com,1999:blog-4407931247357581937.post-14676162888840880982010-07-24T20:58:00.000-07:002010-07-24T21:04:38.068-07:00Algunos terminos<div align="justify"><strong></strong> </div><div align="justify"><strong>Quantum Mechanics</strong><br />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. </div><div align="justify"> </div><div align="justify"> </div><div align="justify"> </div><div align="justify"><strong>Advanced Quantum Mechanics</strong><br />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.<br />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.</div><div align="justify"> </div><div align="justify"> </div><div align="justify"> </div><div align="justify"><strong>Physics of Deformed Solids</strong><br />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. </div><div align="justify"> </div><div align="justify"><br /><strong>X-ray Crystallography</strong> </div><div align="justify">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. </div><div align="justify"> </div><div align="justify"><br /><strong>Low Temperature Physics and Vacuum Techniques<br /></strong>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. </div><div align="justify"> </div><div align="justify"><br /><strong>Physics of Semiconductors and Superconductors</strong><br />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.<br />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. </div><div align="justify"> </div><div align="justify"><br /><strong>Solid State Physics<br /></strong>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. </div><div align="justify"><br /><br /><strong>Polymer Physics</strong><br />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. </div><div align="justify"> </div><div align="justify"><br /><strong>Optical Crystallography<br /></strong>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.</div><div align="justify"> </div><div align="justify"><br /><strong>Magnetism-I</strong><br />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. </div><div align="justify"> </div><div align="justify"> </div><div align="justify"><strong>Magnetism-II</strong><br />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. </div><div align="justify"> </div><div align="justify"><br /><strong>Thermodynamics of Solids</strong><br />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. </div><div align="justify"> </div><div align="justify"><br /><strong>Experimental Techniques in Solid State Physics<br /></strong>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). </div>Orlaninghttp://www.blogger.com/profile/08859376411004440286noreply@blogger.com0tag:blogger.com,1999:blog-4407931247357581937.post-44126426440030364282010-07-24T20:43:00.000-07:002010-07-24T20:54:35.897-07:00Time-resolved X-ray diffraction on laser-excited metal nanoparticles<div align="justify">
<br /><strong>Abstract</strong>
<br /></div><div align="justify">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.
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<br /><a name="SECTION00010000000000000000"><strong>Introduction</strong></a> </div><div align="justify">
<br />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 [<a href="http://www.iop.org/EJ/article/0295-5075/61/6/762/node6.html#kara98">1</a>]. One interesting issue is what effects such different vibrational properties may have on the rate of heat transfer across nanostructure interfaces [<a href="http://www.iop.org/EJ/article/0295-5075/61/6/762/node6.html#majumdar02">2</a>]. 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 [<a href="http://www.iop.org/EJ/article/0295-5075/61/6/762/node6.html#cahill02">3</a>]. 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.
<br />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 [<a href="http://www.iop.org/EJ/article/0295-5075/61/6/762/node6.html#tokizaki94">4</a>,<a href="http://www.iop.org/EJ/article/0295-5075/61/6/762/node6.html#bigot95">5</a>,<a href="http://www.iop.org/EJ/article/0295-5075/61/6/762/node6.html#roberti95">6</a>,<a href="http://www.iop.org/EJ/article/0295-5075/61/6/762/node6.html#perner97">7</a>,<a href="http://www.iop.org/EJ/article/0295-5075/61/6/762/node6.html#Inouye98">8</a>,<a href="http://www.iop.org/EJ/article/0295-5075/61/6/762/node6.html#kaempfe99">9</a>,<a href="http://www.iop.org/EJ/article/0295-5075/61/6/762/node6.html#nisoli97">10</a>,<a href="http://www.iop.org/EJ/article/0295-5075/61/6/762/node6.html#hodak">11</a>,<a href="http://www.iop.org/EJ/article/0295-5075/61/6/762/node6.html#delfatti99">12</a>,<a href="http://www.iop.org/EJ/article/0295-5075/61/6/762/node6.html#qian99">13</a>,<a href="http://www.iop.org/EJ/article/0295-5075/61/6/762/node6.html#perner00">14</a>,<a href="http://www.iop.org/EJ/article/0295-5075/61/6/762/node6.html#harata00">15</a>,<a href="http://www.iop.org/EJ/article/0295-5075/61/6/762/node6.html#link00">16</a>]. 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 [<a href="http://www.iop.org/EJ/article/0295-5075/61/6/762/node6.html#hodak">11</a>,<a href="http://www.iop.org/EJ/article/0295-5075/61/6/762/node6.html#delfatti99">12</a>,<a href="http://www.iop.org/EJ/article/0295-5075/61/6/762/node6.html#qian99">13</a>,<a href="http://www.iop.org/EJ/article/0295-5075/61/6/762/node6.html#perner00">14</a>]. 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 [<a href="http://www.iop.org/EJ/article/0295-5075/61/6/762/node6.html#schotte02">17</a>]. 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 [<a href="http://www.iop.org/EJ/article/0295-5075/61/6/762/node6.html#reiss01">18</a>].
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<br /><a name="SECTION00020000000000000000"><strong>Experimental</strong></a>
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<br /></div><p align="center"><img id="BLOGGER_PHOTO_ID_5497685928816691410" style="DISPLAY: block; MARGIN: 0px auto 10px; WIDTH: 371px; CURSOR: hand; HEIGHT: 166px; TEXT-ALIGN: center" alt="" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgpo2wWB0uddTqekOhTk6oeQOPv3wugiIwPbXYkfEg9Ljo-VsVSdGRbDfljG7hO1nruaeDL9rZxgPFAq5mSNtbd9uFrt66qGt7X-loeSTipZ_RpRtCvSVuO1VBAY4OF6Y3fAUnzTXw-zbo/s320/2.bmp" border="0" /></p><div align="justify">
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<br />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).
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<br /></div><p align="justify">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. <a href="http://www.iop.org/EJ/article/0295-5075/61/6/762/node2.html#bragg">1</a>a)) and TEM analysis [<a href="http://www.iop.org/EJ/article/0295-5075/61/6/762/node6.html#berg91">19</a>]. 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</a>,<a href="http://www.iop.org/EJ/article/0295-5075/61/6/762/node6.html#schrof98">21</a>]. 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%.
<br />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, [<a href="http://www.iop.org/EJ/article/0295-5075/61/6/762/node6.html#schotte02">17</a>]. 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. <a href="http://www.iop.org/EJ/article/0295-5075/61/6/762/node2.html#bragg">1</a>). 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) [<a href="http://www.iop.org/EJ/article/0295-5075/61/6/762/node6.html#techert01">22</a>]. The resulting Debye-Scherrer rings are integrated azimuthally and corrected for polarization and geometry effects [<a href="http://www.iop.org/EJ/article/0295-5075/61/6/762/node6.html#hammersley96">23</a>]. 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. <a href="http://www.iop.org/EJ/article/0295-5075/61/6/762/node2.html#bragg">1</a>a) and b)). Grazing angles of 8 degrees for X-rays and 30 degrees for the laser are used in the latter geometry. </p><div align="justify">
<br /><strong>Results and discussion</strong> </div><div align="justify">
<br />Azimuthally integrated profiles of the Debye-Scherrer rings are presented in fig. <a href="http://www.iop.org/EJ/article/0295-5075/61/6/762/node2.html#bragg">1</a> 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 .
<br />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 [<a href="http://www.iop.org/EJ/article/0295-5075/61/6/762/node6.html#kaempfe99">9</a>] by an accumulative effect that can reduce the size of the particles and create small precipitates around them.
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<br /></div><div align="justify"><img id="BLOGGER_PHOTO_ID_5497686723052554978" style="DISPLAY: block; MARGIN: 0px auto 10px; WIDTH: 320px; CURSOR: hand; HEIGHT: 122px; TEXT-ALIGN: center" alt="" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg1EjzIKpbsBDEsRWf5H6ggdczeX2bVVe-IT0sj2ufBby-p_FJOtFPtkdljOKbmlqvgp7LDVsDKdc5hhQ_FBeawlvcL9J3ApYOnQNA8-CO01AwSIsExDWDQN2cg3CeaI0ChjmDx7F2k6oA/s320/3.bmp" border="0" /><a name="figure145"></a></div>Orlaninghttp://www.blogger.com/profile/08859376411004440286noreply@blogger.com0tag:blogger.com,1999:blog-4407931247357581937.post-22207183162108278852010-07-24T20:31:00.000-07:002010-07-24T20:37:57.417-07:00Quantum Mechanics Predicts Unusual Lattice Dynamics Of Vanadium Metal Under Pressure<div align="justify">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.<br />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.<br />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.<br />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.</div><div align="justify"> </div><div align="justify"><br />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.<br />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.</div><div align="justify"> </div><div align="justify"> </div><div align="justify">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.<br />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.<br />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.<br />"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."</div><br /><br /><div align="right"><strong>Referencias Bibliograficas:</strong></div><div align="right"> </div><div align="right">www.physorg.com/news111321025.html<br /><a href="http://www.sciencedaily.com/releases/2007/10/071011125338.htm">http://www.sciencedaily.com/releases/2007/10/071011125338.htm</a></div>Orlaninghttp://www.blogger.com/profile/08859376411004440286noreply@blogger.com0tag:blogger.com,1999:blog-4407931247357581937.post-76555850067470495322010-07-24T20:22:00.000-07:002010-07-24T20:27:43.716-07:00International Journal of Solids and Structures, submitted for publicationAtomistic Viewpoint of the Applicability of Micrcontinuum Theories<div align="justify"><br /><br /><strong>Abstract</strong><br /><br />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.<br /><br /><br /><strong>Introduction<br /></strong><br />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.<br /><br />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. </div><div align="justify"> </div><div align="justify">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.<br /><br /><strong>Applicability Analyses from the Viewpoint of Dynamics of Atoms in Crystal<br /><br />Dynamics of Atoms in Crystal</strong><br /><br />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.<br /><br />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. </div><div align="justify"> </div><div align="justify"><strong>Optical Phonons</strong><br /><br />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.<br /> </div><div align="justify"> </div><div align="justify"><strong>Micropolar theory (Eringen and Suhubi [1964])<br /></strong><br />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.<br /><br />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.<br /> </div><div align="justify"> </div><div align="right"><strong>Referencia Bibliografica:</strong></div><div align="right">www.seas.gwu.edu/~jdlee/index_files/ijss-applicability.doc</div>Orlaninghttp://www.blogger.com/profile/08859376411004440286noreply@blogger.com0tag:blogger.com,1999:blog-4407931247357581937.post-82020830725108164672010-06-27T16:45:00.000-07:002010-06-27T16:47:47.916-07:00Algorithms for dynamical fermions -- Hybrid Monte Carlo<div align="justify">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.<br /><br />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).<br /><br />To build such a Markov chain, one alternates two steps: Molecular Dynamics Monte Carlo (MDMC) and momentum refreshment.<br /><br />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.<br /><br />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.<br /><br />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.<br /><br />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.<br /><br />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).<br /><br />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.<br /><br />The BCH formula says that for two elements A,B of an associative algebra, the identity<br /><br />log(eAeB) = A + (∫01 ((x log x)/(x-1)){x=ead Aet ad B} dt) (B)<br /><br />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<br /><br />log(eAeB) = A + B + 1/2 [A,B] + 1/12 [A-B,[A,B]] - 1/24 [B,[A,[A,B]]] + ...<br /><br />Applying this to a symmetric product, one finds<br /><br />log(e1/2 AeBe1/2 A) = A + B + 1/24 [A+2B,[A,B]] + ...<br /><br />where in both cases the dots denote fifth-order terms.<br /><br />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<br /><br />(eδτ/2 Peδτ Qeδτ/2 P)τ/δτ = eτ(P+Q) + O((δτ)2)<br /><br />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. </div><div align="justify"> </div><div align="right">Referencia Bibliografica:</div><div align="right"><a href="http://latticeqcd.blogspot.com/2007/12/algorithms-for-dynamical-fermions.html">http://latticeqcd.blogspot.com/2007/12/algorithms-for-dynamical-fermions.html</a></div>Orlaninghttp://www.blogger.com/profile/08859376411004440286noreply@blogger.com0tag:blogger.com,1999:blog-4407931247357581937.post-63725468429845985552010-06-27T10:45:00.000-07:002010-06-27T11:11:09.270-07:00Applicability Analyses from the Viewpoint of Dynamics of Atoms in Crystal
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</span></span></span><!--[endif]--><b style="color: rgb(0, 0, 0);"><span lang="EN-US">Dynamics of Atoms in Crystal</span></b></h1>
<br /> <p style="color: rgb(0, 0, 0);" class="MsoFootnoteText"><span style="" lang="EN-US"><o:p> </o:p></span></p> <p style="text-align: justify; color: rgb(0, 0, 0);" class="MsoNormal"><span style="color: rgb(0, 0, 0);" lang="EN-US">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 </span><span style="color: rgb(0, 0, 0);"><span lang="EN-US">lattice dynamics</span></span><span style="color: rgb(0, 0, 0);" lang="EN-US">.</span><span style="" lang="EN-US"><span style="color: rgb(0, 0, 0);"> These include: </span><span style="color: rgb(0, 0, 0);" class="msoDel"><del cite="mailto:lee" datetime="2002-09-25T17:42">thermal properties,</del></span> <span style="color: rgb(0, 0, 0);" class="msoDel"><del cite="mailto:lee" datetime="2002-09-25T18:09">thermal conductivity, </del></span>temperature effect, energy dissipation, sound propagation, phase transition,<span class="msoIns"><ins cite="mailto:lee" datetime="2002-09-25T18:09"> thermal conductivity, </ins></span>piezoelectricity, dielectric and optical properties, thermo-mechanical-electromagnetic coupling properties. </span></p><div style="text-align: justify; color: rgb(0, 0, 0);"> </div><p style="text-align: justify; color: rgb(0, 0, 0);" class="MsoNormal"><span lang="EN-US" style="font-size:10pt;"><o:p> </o:p></span></p><div style="text-align: justify; color: rgb(0, 0, 0);"> </div> <p style="text-align: justify; color: rgb(0, 0, 0);" class="MsoNormal"><span style="" lang="EN-US">The atomic motion<span class="msoIns"><ins cite="mailto:lee" datetime="2002-09-25T17:43">s,</ins></span><span class="msoDel"><del cite="mailto:lee" datetime="2002-09-25T17:43"> that are</del></span> revealed by those features<span class="msoIns"><ins cite="mailto:lee" datetime="2002-09-25T17:43">,</ins></span> are not random<span class="msoIns"><ins cite="mailto:lee" datetime="2002-09-25T17:43">.</ins></span><span class="msoDel"><del cite="mailto:lee" datetime="2002-09-25T17:43">,</del></span> <span class="msoIns"><ins cite="mailto:lee" datetime="2002-09-25T17:43">I</ins></span><span class="msoDel"><del cite="mailto:lee" datetime="2002-09-25T17:43">i</del></span>n 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. </span><span lang="EN-US" style="font-size:10pt;">T</span><span style="" lang="EN-US">he quantum of energy in an elastic wave is called a <span style="color: rgb(0, 0, 0);">Phonon</span><span style="color: rgb(0, 0, 0);">;</span></span><span style="color: rgb(0, 0, 0);font-family:";" lang="EN-US"> </span><span style="color: rgb(0, 0, 0);" lang="EN-US">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. </span><span style="color: rgb(0, 0, 0);" lang="EN-US">The frequency-wave vector relationship of phonons is called </span><span style="color: rgb(0, 0, 0);"><span lang="EN-US">Phonon Dispersion Relation</span></span><span style="color: rgb(0, 0, 0);" lang="EN-US">, which is </span><span style="color: rgb(0, 0, 0);" lang="EN-US">the </span><span style="color: rgb(0, 0, 0);" lang="EN-US">fundamental ingredient in the theory of lattice dynamics</span><span style="" lang="EN-US"> and can be determined through experimental measurements, such as <span class="msoIns"><ins cite="mailto:lee" datetime="2002-09-25T17:44">n</ins></span><span class="msoDel"><del cite="mailto:lee" datetime="2002-09-25T17:44">N</del></span>eutron scattering, <span class="msoIns"><ins cite="mailto:lee" datetime="2002-09-25T17:44">i</ins></span><span class="msoDel"><del cite="mailto:lee" datetime="2002-09-25T17:44">I</del></span>nfrared 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.</span></p> <p style="text-align: justify; color: rgb(0, 0, 0);" class="MsoNormal"><span style="" lang="EN-US">.</span>
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class="MsoHeading7"><span lang="EN-US">Fig.1 Typical motions for two atoms in a unit cell, </span></p> <p style="color: rgb(0, 0, 0);" class="MsoHeading7"><span style="font-weight: normal;" lang="EN-US">where<span style=""> </span>‘L’ stands for longitudinal, ‘T’ transverse, ‘A’ acoustic, ‘O’ optical
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10]> <style> /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Tabla normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-priority:99; mso-style-qformat:yes; mso-style-parent:""; mso-padding-alt:0cm 5.4pt 0cm 5.4pt; mso-para-margin:0cm; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman","serif";} </style> <![endif]--> </p><p style="text-align: center; color: rgb(0, 0, 0);" class="MsoDate"><span lang="EN-US">Optical Phonons</span></p> <p style="color: rgb(0, 0, 0);" class="MsoFootnoteText"><span lang="EN-US" style="font-size:8pt;"><o:p> </o:p></span></p> <p class="MsoBodyText2" style="color: rgb(0, 0, 0); text-align: justify;">Optical phonon branches <span lang="EN-US">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 <span style="position: relative; top: 5pt;"><!--[if gte vml 1]><v:shapetype id="_x0000_t75" coordsize="21600,21600" spt="75" preferrelative="t" path="m@4@5l@4@11@9@11@9@5xe" filled="f" stroked="f"> <v:stroke joinstyle="miter"> <v:formulas> <v:f eqn="if lineDrawn pixelLineWidth 0"> <v:f eqn="sum @0 1 0"> <v:f eqn="sum 0 0 @1"> <v:f eqn="prod @2 1 2"> <v:f eqn="prod @3 21600 pixelWidth"> <v:f eqn="prod @3 21600 pixelHeight"> <v:f eqn="sum @0 0 1"> <v:f eqn="prod @6 1 2"> <v:f eqn="prod @7 21600 pixelWidth"> <v:f eqn="sum @8 21600 0"> <v:f eqn="prod @7 21600 pixelHeight"> <v:f eqn="sum @10 21600 0"> </v:formulas> <v:path extrusionok="f" gradientshapeok="t" connecttype="rect"> <o:lock ext="edit" aspectratio="t"> </v:shapetype><v:shape id="_x0000_i1025" type="#_x0000_t75" style="'width:27pt;" ole=""> <v:imagedata src="file:///C:\DOCUME~1\Parce\CONFIG~1\Temp\msohtmlclip1\01\clip_image001.wmz" title=""> </v:shape><![endif]--><!--[if !vml]--><!--[endif]--></span><!--[if gte mso 9]><xml> <o:oleobject type="Embed" progid="Equation.3" shapeid="_x0000_i1025" drawaspect="Content" objectid="_1339150213"> </o:OLEObject> </xml><![endif]--><span style=""> </span>ferroelectrics the anomalously large Born effective charges produce a giant LO-TO splitting in phonon dispersion relations. This feature is associated <span class="msoIns"><ins cite="mailto:lee" datetime="2002-09-25T17:46">with</ins></span><span class="msoDel"><del cite="mailto:lee" datetime="2002-09-25T17:46">to</del></span> the existence of an anomalously large destabilizing dipole-dipole interaction, sufficient to compensate the stabilizing short-range force and induce the ferroelectric instability. Optical phonon<span class="msoDel"><del cite="mailto:lee" datetime="2002-09-25T17:46">s</del></span>, therefore, appears as the key concept to relate the electronic and structural properties through Born effective charge (Ghosez <span class="msoIns"><ins cite="mailto:lee" datetime="2002-09-25T17:47">[</ins></span>1995,1997<span class="msoIns"><ins cite="mailto:lee" datetime="2002-09-25T17:47">]</ins></span>). 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. </span></p><p class="MsoBodyText2" style="color: rgb(0, 0, 0); text-align: justify;">
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{mso-style-name:"Tabla normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-priority:99; mso-style-qformat:yes; mso-style-parent:""; mso-padding-alt:0cm 5.4pt 0cm 5.4pt; mso-para-margin:0cm; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman","serif";} </style> <![endif]--> <p style="text-align: justify;" class="MsoNormal"><span style="" lang="EN-US">The dynamic characteristics of crystals depend on crystal structures, as shown in Fig.2, and the binding between the atoms.<span style=""> </span>In metals the atomic cores are surrounded by a more-or-less uniform density of free electrons. This gives metals their electrical conductivity and a nonlocal character of the interatomic potential. Its dynamic feature is represented by the dispersive acoustic vibrations. </span><span lang="EN-US">In ionic crystals, strong Coulomb forces and short-range repulsive forces operate between the ions, and the ions are polarizable. The covalent bond is usually formed from two electrons, one from each atom participating in the bond. These electrons tend to be partially localized in the region between the two atoms and constitute the bond charge. The phonon dispersion relations of ionic and covalent crystals have both acoustic and optical branches, their optical vibrations describe the internal motion of atoms within the primitive basis, as in Fig.1 and Fig.3.<span style=""> </span></span><span style="" lang="EN-US">In molecular crystal there is usually a large difference between the frequencies of modes in which the molecules move as a united units (the external modes) and the modes that involve the stretch and distortion of the molecules (the internal modes). The framework crystals are similar to molecular crystals in that they are composed of rigid groups. The units are very stiff but linked flexibly to each other at the corner atoms. The phonon dispersion relations, as in Fig.4, of molecular and framework crystals include both acoustic and optical vibrations, and the optical vibrations further include internal modes and external modes. <o:p></o:p></span></p><div style="text-align: justify;">
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Definitions */ table.MsoNormalTable {mso-style-name:"Tabla normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-priority:99; mso-style-qformat:yes; mso-style-parent:""; mso-padding-alt:0cm 5.4pt 0cm 5.4pt; mso-para-margin:0cm; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman","serif";} </style> <![endif]--> <h1 style="margin-left: 18pt; text-align: center;"><!--[if !supportLists]--><span lang="EN-US"><span style=""><span style="font-family: "Times New Roman"; font-style: normal; font-variant: normal; font-weight: normal; font-size: 7pt; line-height: normal; font-size-adjust: none; font-stretch: normal;"> </span></span></span><!--[endif]--><span lang="EN-US">Phonon Dispersion Relations by Various Microcontinuum Theory</span></h1><h1 style="margin-left: 18pt;">
<br /><span lang="EN-US"></span><span lang="EN-US"><o:p></o:p><span style=""><span style="font-family: "Times New Roman"; font-style: normal; font-variant: normal; font-weight: normal; font-size: 7pt; line-height: normal; font-size-adjust: none; font-stretch: normal;"></span></span></span></h1><h1 style="margin-left: 18pt;"><span lang="EN-US"><span style=""><span style="font-family: "Times New Roman"; font-style: normal; font-variant: normal; font-weight: normal; font-size: 7pt; line-height: normal; font-size-adjust: none; font-stretch: normal;"> </span></span></span><!--[endif]--><b><span lang="EN-US">Micromorphic theory<span style=""> </span></span></b><span style="" lang="EN-US">(Eringen and </span><span lang="EN-US">Suhubi [1964], Eringen [1999]</span><span style="" lang="EN-US">)</span></h1> <p class="MsoNormal"><span lang="EN-US"><o:p> </o:p></span></p> <p class="MsoNormal" style="text-align: justify;"><span style="" lang="EN-US">Micromophic theory </span><span style="" lang="EN-US">views a material as a continuous collection of deformable particles. Each particle is attached with a microstructure of finite size. The deformation of a micromorphic continuum yields both macro-strains (homogeneous part) and microscopic internal strains (discrete part).
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*/ table.MsoNormalTable {mso-style-name:"Tabla normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-priority:99; mso-style-qformat:yes; mso-style-parent:""; mso-padding-alt:0cm 5.4pt 0cm 5.4pt; mso-para-margin:0cm; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman","serif";} </style> <![endif]--> </p><p class="MsoSubtitle" style="margin-left: 36pt; text-indent: -36pt;"><!--[if !supportLists]--><b><span lang="EN-US"><span style=""><span style="font-family: "Times New Roman"; font-style: normal; font-variant: normal; font-weight: normal; font-size: 7pt; line-height: normal; font-size-adjust: none; font-stretch: normal;"></span></span></span></b><!--[endif]--><b><span lang="EN-US">Micropolar theory </span></b><span style="" lang="EN-US">(Eringen and </span><span lang="EN-US">Suhubi [1964]</span><span style="" lang="EN-US">)</span><b><span lang="EN-US"><o:p></o:p></span></b></p> <p class="MsoSubtitle"><span lang="EN-US"><o:p> </o:p></span></p> <p class="MsoSubtitle"><span lang="EN-US">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. </span><span style="" lang="EN-US">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.<span style=""> </span>It accounts for the dynamic effect of material with rather stiff microstructure.<o:p></o:p></span></p> <p class="MsoSubtitle"><span lang="EN-US"><span style=""> </span></span></p> <p class="MsoSubtitle"><span lang="EN-US">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.</span></p> <p class="MsoSubtitle"><span lang="EN-US"><o:p> 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mso-style-noshow:yes; mso-style-priority:99; mso-style-qformat:yes; mso-style-parent:""; mso-padding-alt:0cm 5.4pt 0cm 5.4pt; mso-para-margin:0cm; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman","serif";} </style> <![endif]--> <p class="MsoSubtitle"><span lang="EN-US">For isotropic material, the phonon dispersion relations based on a nonlocal theory have been obtained by Eringen [1992] as shown in Fig.7. Remarkable similarity to atomic lattice dynamics solution with Born-von Karman model, and to the experimental results for Aluminum has been reported.</span></p> <p class="MsoSubtitle"><span lang="EN-US"><o:p> </o:p></span></p> <p class="MsoSubtitle"><span lang="EN-US">Non local theory takes long-range interatomic interaction into consideration. As a consequence it yields results dependent on the size of a body. Similar to classical continuum theory, the lattice particles are taken without structure and idealized as point masses. Hence, the effect of microstructure does not appear. It is not a theory for material with microstructure, but for material involving long-range interaction. It can be applied to crystal that has only one atom per primitive unit cell at various length scales.</span></p><p class="MsoSubtitle">
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lang="EN-US">Applicability Analysis of Continuum Theories from the Viewpoint of Molecular Dynamics<o:p></o:p></span></b></p> <p class="MsoNormal" style="text-align: justify;"><span lang="EN-US"><o:p> </o:p></span></p> <h3><span style="font-size: 12pt;" lang="EN-US">Micromorphic theory<o:p></o:p></span></h3> <p class="MsoNormal" style="text-align: justify;"><b style=""><span lang="EN-US"><o:p> </o:p></span></b></p> <p class="MsoNormal" style="text-align: justify;"><span lang="EN-US">Atomistic flow mechanisms make it possible to define the fluxes as sums of one- and two-atom contributions. The internal energy </span><i style=""><span style="font-family: Symbol;" lang="EN-US"><span style="">e</span></span></i><span lang="EN-US">, heat flux <b style="">q</b>, and the three stress tensors, namely, Cauchy stress <b style=""><i style="">t</i></b>, microstress average <b style=""><i style="">s</i></b>, and moment stress <b style=""><i style="">m</i></b>, are composed of kinetic part and potential part. With the definition of temperature, it is seen that the kinetic parts of <b style=""><i style="">t, s, m, q</i></b>, and </span><i style=""><span style="font-family: Symbol;" lang="EN-US"><span style="">e</span></span><span lang="EN-US"> </span></i><span lang="EN-US">are caused by the thermal motion of atoms, and can be linked to the temperature. The potential parts are caused by the interatomic forces, and can be determined from the potential functions and can be written in terms of lattice strain and internal strains. This is consistent with the constitutive relations of micromorphic theory. Therefore, micromorphic theory, including the mechanical variables, balance laws, and constitutive relations, can be obtained based on the kinetics and interactions of atoms.<span style=""> </span>The correspondence between the molecular dynamics model and the micromorphic theory can be achieved whenever an ensemble average is meaningful. The applicability of micromorphic theory in microscopic time and length scales is confirmed from the viewpoint of molecular dynamics.</span></p> <p></p>
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3.0cm; mso-header-margin:36.0pt; mso-footer-margin:36.0pt; mso-paper-source:0;} div.Section1 {page:Section1;} --> </style><!--[if gte mso 10]> <style> /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Tabla normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-priority:99; mso-style-qformat:yes; mso-style-parent:""; mso-padding-alt:0cm 5.4pt 0cm 5.4pt; mso-para-margin:0cm; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman","serif";} </style> <![endif]--> </p><p class="MsoNormal" style="text-align: justify;"><b><span lang="EN-US">Cosserat theory <o:p></o:p></span></b></p> <p class="MsoNormal" style="text-align: justify;"><span lang="EN-US"><o:p> </o:p></span></p> <p class="MsoBodyText"><span lang="EN-US">Compared with micropolar theory, Cosserat theory does not have the balance law of microinertia. The absence of the balance law of microinertia tensor, implies that the microinertia tensor is assumed to be constant. This is the case when the deformation of the microstructure of the particle is very small, and the change of the orientation can be ignored.<span style=""> </span><span style="">Hence, compared with micropolar theory, Cosserat theory is not suited for problems involve the significant change of the orientation of the microstructure. <o:p></o:p></span></span></p> <p class="MsoBodyText"><span lang="EN-US"><o:p> </o:p></span></p> <p class="MsoBodyText"><b><span lang="EN-US">Nonlocal Theory<o:p></o:p></span></b></p> <p class="MsoBodyText"><span lang="EN-US"><o:p> </o:p></span></p> <p class="MsoNormal" style="text-align: justify;"><span lang="EN-US">For each atom (<i style="">k, </i></span><i style=""><span style="font-family: Symbol;" lang="EN-US"><span style="">a</span></span></i><span lang="EN-US">), the interatomic force is taken from all other atoms in the body considered. This action-at-a-distance interactions give the related quantities a nonlocal character. Hence, the molecular dynamics formulation is in the nonlocal arena. Even in the limit case when the unit cell or the material particle only consists of one atom, the expressions and derivation are still applicable to nonlocal phenomena.</span></p> <p class="MsoBodyText"><span lang="EN-US"><o:p> </o:p></span></p> <p class="MsoBodyText"><b><span lang="EN-US">Couple stress theory <o:p></o:p></span></b></p> <p class="MsoBodyText"><span lang="EN-US"><o:p> </o:p></span></p> <p class="MsoBodyText"><span lang="EN-US">The couple stress theory, by including higher order stress, provides a model that can yield results depending on the size of specimen. However, there is no distinction between the micromotion and the macromotion, and hence it is only suited for material without microstructure. </span></p> <p class="MsoBodyText"><span lang="EN-US"><o:p> </o:p></span></p> <p class="MsoBodyText"><span lang="EN-US">The material that does not have microstructure, is referred as microscopically homogeneous, and is corresponding to crystal with only one atom in the unit cell. This follows that<span style=""> </span><span style="position: relative; top: 3pt;"><!--[if gte vml 1]><v:shapetype id="_x0000_t75" coordsize="21600,21600" spt="75" preferrelative="t" path="m@4@5l@4@11@9@11@9@5xe" filled="f" stroked="f"> <v:stroke joinstyle="miter"> <v:formulas> <v:f eqn="if lineDrawn pixelLineWidth 0"> <v:f eqn="sum @0 1 0"> <v:f eqn="sum 0 0 @1"> <v:f eqn="prod @2 1 2"> <v:f eqn="prod @3 21600 pixelWidth"> <v:f eqn="prod @3 21600 pixelHeight"> <v:f eqn="sum @0 0 1"> <v:f eqn="prod @6 1 2"> <v:f eqn="prod @7 21600 pixelWidth"> <v:f eqn="sum @8 21600 0"> <v:f eqn="prod @7 21600 pixelHeight"> <v:f eqn="sum @10 21600 0"> </v:formulas> <v:path extrusionok="f" gradientshapeok="t" connecttype="rect"> <o:lock ext="edit" aspectratio="t"> </v:shapetype><v:shape id="_x0000_i1025" type="#_x0000_t75" style="'width:39pt;" ole="" fillcolor="window"> <v:imagedata src="file:///C:\DOCUME~1\Parce\CONFIG~1\Temp\msohtmlclip1\01\clip_image001.wmz" title=""> </v:shape><![endif]--><!--[if !vml]--><!--[endif]--></span><span style="position: relative; top: 5pt;"><!--[if gte vml 1]><v:shape id="_x0000_i1026" type="#_x0000_t75" style="'width:99.75pt;height:17.25pt'" ole="" fillcolor="window"> <v:imagedata src="file:///C:\DOCUME~1\Parce\CONFIG~1\Temp\msohtmlclip1\01\clip_image003.wmz" title=""> </v:shape><![endif]--></span>and hence <b style=""><i style="">s = t</i></b> and the moment stress <b style=""><i style="">m</i></b> = 0. The higher order stress, <b><i>m</i></b>, is thus removed from the atomic formulation. The couple stress theory then falls into the framework of nonlocal theory, with the strain gradients accounting for the effect of neighborhood. </span></p> <p class="MsoBodyText"><span lang="EN-US"><o:p> </o:p></span></p> <p></p> <div style="text-align: right;">REFERENCIAS:
<br /><span class="f"><cite>www.seas.gwu.edu/~jdlee/index_files/ijss-applicability.doc</cite><span class="gl"></span></span>
<br /></div></div>
<br /><h1 style="text-align: center;"><span style="" lang="EN-US"><o:p></o:p></span></h1> Orlaninghttp://www.blogger.com/profile/08859376411004440286noreply@blogger.com0tag:blogger.com,1999:blog-4407931247357581937.post-43694912808255345532010-06-19T17:08:00.000-07:002010-06-19T17:20:23.656-07:00Notices of Lattice Dynamics<div align="justify"><br /></div><p align="center"><strong>Nuclear Spin and Magnetic Resonance</strong></p><p align="justify"><strong></strong><br />The nuclear spin - Most elements have at least one isotope with a non-zero spin angular momentum, I, and an associated magnetic moment, µ, which are related by the gyromagnetic ratio, g I is a quantized characteristic of the nucleus, and its value describes the symmetry of the nuclear charge distribution. In this course we will limit the discussion to spin=1/2 nuclei for which the nuclear charge distribution is spherically symmetric. The nucleus then has the properties of a magnetic dipole (essentially a bar-magnet), whose strength is given above. All of the interaction of spin=1/2 nuclei are purely magnetic.<br />Spin =1/2 nuclei are the most often studied by NMR since they generally have both higher resolution and higher sensitivity spectra (therefore, it is perhaps easier to extract chemical information from these nuclei). </p><div align="justify"><br /><br /></div><p align="justify">Quantum mechanics tells us a few important points about nuclear spins,<br />1. the projection of the nuclear magnetic moment along any direction is quantized and for spin=1/2 nuclear is restricted to the to values of +/- .<br />2. the uncertainty principle applies and places a limit on the amount of information we can know about the orientation of a nuclear magnetic moment. At any time, we can only know the magnitude of the vector and its projection along one axis. The projections along the other two axis are indeterminate (they are in a superposition state).<br /></p><p align="center"><strong>The Zeeman interaction</strong><br /></p><p align="justify">In an NMR experiment we are interested in exploring the interaction of the nuclear magnetic moment and an external magnetic field. The energy of this magnetic dipole-dipole interaction is given classically as,<br />where Bo is the strength of the external magnetic field. This external field has a direction and so this provides a coordinate system for the NMR experiment. From here on, we will work in coordinate systems where the applied magnetic field is oriented along the z-axis.<br />We can now see that for a spin=1/2 nuclei the two values of the spin along the z-direction, Iz= +/- 1/2, correspond to the nuclear magnetic moment oriented along and against the magnetic field. Classically we may compare this to the two stable positions of a compass needle in the earth’s magnetic field, a low energy configuration with the needle aligned with the earth’s field, and a higher energy (unstable equilibrium point) with the needle aligned against the earth’s field. In the case of the nuclear spins, the nuclear moment can not be aligned exactly along the applied field, since this would violate the uncertainty principle, and so there are two states, both of which are represented by cones.</p><img id="BLOGGER_PHOTO_ID_5484643281269801938" style="DISPLAY: block; MARGIN: 0px auto 10px; WIDTH: 237px; CURSOR: hand; HEIGHT: 216px; TEXT-ALIGN: center" alt="" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj4h5us3a3jPNplVkm_px3ne11F8BwFEO_poWdBtq7SOOgCpiwtThCStykn7PoHt1Xh_kAH2azqjduaD3SxgkkLCRRIN96h57XuRLNn2chS1bUrEIJRZU617W80seRISBepAATbZ79vjw4/s320/1.bmp" border="0" /><br /><br /><p align="center">The nuclear spin is restricted to being on these two cones oriented along the z-axis.</p><p align="justify"><br />Let us explore the torque first, in the presence of an applied magnetic field, then the Larmor precession states that the bulk magnetic moment will revolve about the applied field direction. If the bulk magnetization is along the field direction, as it is at equilibrium, then there is no torque and hence no motion. As we expect, at equilibrium the system is stationary. Note, this is true of the detectable bulk magnetization, but is not true at the microscopic level. The dynamics of single spins can not be discussed in the classical terms that we are using.<br />If the system is away from equilibrium, if the bulk magnetization vector is oriented other than along the z-axis, then the magnetization presesses (rotates) about the z-axis with a angular velocity given by the energy separation of the two states (g B0). Notice that this torque will not change the length of the magnetization vector, it only varries its orientation.<br />This rotation can not be the only motion, sine then the system would never return to equilibrium. So along with the rotation, there is a relaxation of the vector to bring it back along the z-axis. Therefore the x and y-components of the nuclear magnetization decay towards zero, and the z-component decays towards the equilibrium value (typically called M0). </p><p><img id="BLOGGER_PHOTO_ID_5484643287112053506" style="DISPLAY: block; MARGIN: 0px auto 10px; WIDTH: 320px; CURSOR: hand; HEIGHT: 293px; TEXT-ALIGN: center" alt="" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgz8a4MIuBzqeeiUyQww7LcWGIINwef8KDvLe4iOllGT7YTbcWYV-htinR5jw8G8EWXUjx-o_ZAz2VHR0SAL0zwPC0L6kM7kXupFiQJj7RTJJ57rspHefDtsSDTaOpbJQdnxHb37VksqUs/s320/2.bmp" border="0" /></p><div align="justify">The above is a “quick-time” movie that shows the motion of the bulk magnetization vector starting from a position along the x-axis and then evolving towards its equilibrium position along the z-axis. The movie was created in Mathematica (see appendix 1-3) and may be run by double clicking on the figure. The red bar progressing across the figure is meant to represent the flow of time.<br />Latter we will show how a pulse of radio frequency radiation will tilt the bulk magnetization vector away from the z-axis and creat this non-equilibrium magnetization. For now, we are only interested in the spin’s return to equilibrium as shown in the above figure.</div><div align="justify"> </div><div align="justify"> </div><div align="right">REFERENCIAS BIBLIOGRAFICAS:</div><p align="right">web.mit.edu/22.058/www/documents/Fall2002/.../NMR.doc</p><p align="justify"> </p>Orlaninghttp://www.blogger.com/profile/08859376411004440286noreply@blogger.com0tag:blogger.com,1999:blog-4407931247357581937.post-36709172469932548432010-06-19T16:59:00.001-07:002010-06-19T16:59:25.228-07:00inelastic neutronINELASTIC NEUTRON SCATTERING AND<br />LATTICE DYNAMICS OF NOVEL COMPOUNDS<br />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.<br />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.<br />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).<br />Sections below give brief information about the experimental technique and the lattice dynamics<br />calculations respectively, while the results and discussion, and conclusions are presented later.<br />Experimental<br />Inelastic-neutron-scattering (INS) experiments [3] may be performed using both single crystals<br />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.<br />Lattice Dynamical Calculations<br />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.<br />Results and Discussion<br />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].<br />Silicate mineral zircon, ZrSiO4<br />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<br />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.<br />Conclusions<br />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.Orlaninghttp://www.blogger.com/profile/08859376411004440286noreply@blogger.com0tag:blogger.com,1999:blog-4407931247357581937.post-11127095484905783662010-06-19T16:53:00.000-07:002010-06-19T16:57:55.946-07:00comparisonCOMPARISON OF THE LATTICE-DYNAMICSTHERMODYNAMIC PROPERTIES<br /><br />Several paths are available for calculating macroscopic thermodynamic properties for a system with given interparticle forces. Most paths are approximate. The lattice dynamics approximationl) treats correctly all terms in the energy which are quadratic in the particle displacements. The cell-model approximation2) in which a single particle moves in the field of its fixed neighbors modifies the quadratic terms but includes an estimate of anharmonic' corrections. More sophisticated theories3) treat anharmonic perturbations analytically (the actual calculations require fast computers) but seem complicated enough to attract few follo\vers.<br />The approximate methods have the important advantage of being quick and inexpensive to calculate. The computer experiments giving exact thermodynamic properties, either by following the motion of the particles or by sampling the configuration space 4), are relatively expensive because they require so much computer time, particularly if high precision is necessary. To halve the statistical uncertainty in computer-experiment results requires quadrupling the computer time used up. The precision also depends on the sensitivity of the property measured to fluctuations in pressure and energy. Because successively higher derivatives of the free energy involve higher moments of the pressure-tensor component distributions and energy distributions, the time necessary to characterize derivatives increases rapidly with derivative order. Second-order elastic constants and the specific heat involve second moments; third-order elastic constants involve third moments; and so on. So far only first-and second-derivative quantities have been examined. Thus, if the approximations should prove to come close to exact results, they would give us a useful shortcut to accurate thermodynamic properties. It was in the hope of establishing their usefulness that we undertook these calculations.<br />In this paper we compare the results of solid-phase Monte Carlo experiments on 108 particles interacting with the Lennard-Jones and exponential-six potentials with the predictions of 108-particle lattice dynamics and the cell model. In addition to the energy and pressure, we compare the secondderivative quantities: specific heat, Griineisen y, and elastic constants, with approximate predictions. Although either computer method, molecular dynamics or Monte Carlo, can be extended to quantum systems by using the \Vigner-Kirkwood Planck's constant expansion of the free energy5), we have made our comparisons using classical mechanics. To find out how important the small size and classical nature of our systems are, we use the latticedynamics method to investigate the number dependence of all of the thermodynamic properties and quantum corrections to the elastic constants. Quan-t tum corrections to other thermodynamic properties have been calculated elsewhere6).<br />In section 2 we describe the lattice-dynamics calculations. The method has been in use for over 50 years, although many so-called lattice-dynamics calculations of elastic constants have actually been calculations of the elastic response of a static lattice. We take lattice vibrations explicitly into account. The first correct harmonic calculation using normal-mode vibrations was announced by Feldman7). He obtained expressions for the second-order elastic constants which involved the vibration frequencies and their strain derivatives. The frequencies were then obtained by the usual method of diagonalizing the dynamical matrix. The frequency-shift derivatives were calculated by means of perturbation theory. We calculate the work of deforming the crystal by the alternative procedure of computing the free energy numerically for several slightly different values of the strain and then fitting the results to a strain polynomial. Besides avoiding the tedious algebra of Feldman's analytic approach, the numerical method is more readily generalized to higher-order elastic constants.<br />In section 3 we describe the cell-model calculations. The cell model, although actually only appropriate for solids, was first used in an attempt to describe gases and liquids 8). In the cell model the effect of heating the crystal lattice is approximated by a "one-particle" model in which a single particle moves in the field of its fixed neighbors. Neglecting interparticle correlations by approximating an N-body problem by a one-body problem for classical systems, most reasonable at low temperatures and is exact only in the static-lattice limit.<br />The advantage of the cell model over the harmonic approximation lies in its ability to estimate anharmonic contributions from potential-energy terms beyond the quadratic ones. The cell model has often been used to calculate energy, pressure, and specific heat 9). Our elastic-constant calculations are a new use of this model. Henkel10) has studied a similar model, a quantum cell model in which the potential was expanded in powers of displacement and the quartic contributions were treated by perturbation theory. If the perturbations were ignored, Henkel's work would reduce to the usual harmonic Einstein model.<br />In section 4 we compare the tabulated results from both approximations and consider the dependence of the results on number of particles. Vie also discuss some interesting cancellations found in the course of the Monte Carlo calculations. In section 5 we assess the importance of quantum effects on the elastic constants.<br />2. Lattice-dynamics calculations.<br />The lattice dynamics calculations are based on the approximation of truncating a Taylor expansion of the lattice potential energy after the quadratic terms in the particle displacements. Sometimes this is called the "quasi-harmonic" approximation. The coefficients in the Taylor series expansion are calculated from the assumed force law, and the lattice sites are chosen to match the structure of the lattice being described. The expansion is made about a configuration in which each atom is fixed at its average position in a perfect crystal with fixed center of mass. Thus the truncated potential depends upon the size and shape of the assumed static lattice configuration. By changing to normal-mode coordinates the quasi-harmonic Hamiltonian can be rewritten as a sum of 3N 3 independent harmonic oscillator Hamiltonians. The partition function of an oscillator, either quantum or classical, is known11) so that the quasi-harmonic thermodynamic properties of the system can be calculated.<br />The partition function, Z = exp(-A/kT), where A is the Helmholtz energy and kT is Boltzmann's constant times the absolute temperature, can be written as a product of single oscillator partition functions:<br /><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg6yXn39l7TiBsZU6lM044dZ8vKjaREizwJqu7hQTgsjFfLnjngFluJJccCDr4BsZ0_7Y66yEVcFri7uiah8P0eRxmoOSlAw-p0Z2jGtMaRRkyovcw1eu4a-nItaV64fbq-ALxjufTAvts/s1600-h/33.png"></a><br />In the classical limit only the first term in the expansion, kT/hv, is kept. from the quadratic Hamiltonian; the center of mass contribution is Zcm. For accurate work on small crystals Zcm has to be included if comparisons are made with Monte Carlo calculations in which the center of mass is allowed to move.<br />Thermodynamic properties can all be derived from the partition function. Temperature derivatives can be evaluated explicitly to compute the energy and specific heat:<br /><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh79_ArSzwd2K0ZUBriK1zlo2It90cH9uAv54xjRjc8hPAiAKkUUtFuQjwdF-XmZzCUiobs_yLDn1Fq4BpgHWTUTsUR7bB2TSR1t0-2BFanRtvFe4lE7WTj9HeXShMMS8giLROWNZf3hHM/s1600-h/34.png"></a><br />"Strain" derivatives are harder to evaluate. The strains are defined in terms of three vectors colinear with the edges common to one corner of a parallelepiped produced by deforming a cube of crystal with sidelength a. If aI, a2, and a3 are the vectors, then<br /><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgD5gNUCEueJK0r1o0efk55opMGbB808rELcseIT95eHbgA5Smr-xY6LJuSLrSMf0RHYljkqBR97Tqft6plRvSTD7ZyO9LzAwsWodK00JtgiXsPBRh52V6Wlew3eeAxsypiMjUQS0u7hdg/s1600-h/35.png"></a><br />are the six independent strains. The dynamical matrix then gives a complicated implicit relation for the vibration frequencies as functions of the strains. Because there is no convenient expression for v(1]) it is easiest to proceed numerically. Both the average pressure tensor component<br /><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjftgwuNuXzA3tlp-8VM6_HSvnXV5z86eMvv7xEqULKjxkUEiColARymcm2QlQtPVVCbY_07CtHpqGynZWkyQIXuDKH4W4ZboZycK2qWa7ZCiayULMmZ7iSyBIS8HBf2g-Y5TZwdXlwkto/s1600-h/36.png"></a>where fi and ri are the Cartesian coordinates of rand r, and 1]1 is the only nonzero strain. In the strained configuration the Hamiltonian is again expanded, the quadratic terms kept, and the result diagonalized, giving a new set of frequencies and the free energy A (1]1). The diagonalization can be visualized in terms of plane-wave solutions of the classical equations of motion. The waves propagate through the crystal with wavelengths and directions imposed by the shape of the crystal and described by wave vectors chosen from a convenient Brillouin zone12). If y is a wave vector in the unstrained crystal then the corresponding y in the strained crystal is:<br /><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiJJR5_ORdf0xbMzLtgyqQ_Yuy9b_oKFbQsSy-l1GCVO3PsV8ao-R4z0H-AJ76HkrOUyFLtVyXJnVvEbOm1ih7S-feM93ZCuItKAX_gGJIoUI2ej_LITc98xQZQr-qebWhg99b9XnCepGY/s1600-h/37.png"></a><br />where Yi and Yi are the Cartesian coordinates of y and y. Applying the Born-von Karman normal-mode analysis for several values of 1]1, the first four or five coefficients in the expahsion,<br /><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiHn2-eN_bYTQ_xBlFphGYtGje7BOE-kgkxhFzapzrVp4dHnFjdp7jzebbt2Yt8tdHcvUPNrjMIxtRNRR-UJMf00m92sErwtDwAfwJLOUsMryQM45IRhC-8UvQQ5j9QjjGGcupLD3QV_Fs/s1600-h/38.png"></a><br />can be determined. By choosing values of 1J1 separated by 0.0001, both the pressure and Cil were determined with four-figure accuracy in this way. By considering two simultaneous strains along the one axis and the two axis, and analyzing the resulting free energy changes as a series in 1]1 and<br /><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjrK1vzft5ICVqZ7gIJQX27tJSHfzSsSfT8WV0xaqnpqXt9H2_5Kx6DD6YJR8fjkktXm8xr4SlE34rixvpEJ8W86NFs5zrmLj0S3y6kquIYVE4nKak0xCDXaWmmEq91RmZcobBt2gmoZ3U/s1600-h/39.png"></a><br />The generalization of this technique to calculation of higher-order elastic constants or to mixed strain-temperature derivatives is straightforward. For crystals of lower symmetry one needs to calculate more elastic constants and hence one considers more different combinations of strain. Other strains would also be needed for cubic crystals to determine the six nonzero thirdorder constants or the eleven nonzero fourth-order constants and the mixed strain-temperature derivatives.The adiabatic elastic constants<br /><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhCbyt9UCM4t2H-8NfRuMxxLBcO7vJ9FmH9bi3rssPIa_9Jr4QQVWWtfY8KPmVS3FUGyHuFVDngt1ykCBaCIyR_3Lh4MOm8reUyZJE6wpFVDo7ZY8LC69UBN0OrrmCNCa7lwTPeBuCS32k/s1600-h/40.png"></a><br />where 5 is the entropy of the crystal. Because 5(1]1) and (oT/o5)n can be calculated exactly for a quasi-harmonic crystal, the correction term can be evaluated numerically.In order to make a comparison with the 108-particle Monte Carlo calculations13, 14) we have calculated the thermodynamic properties for a 108particle system with the same periodic boundaries and Hamiltonian as those used in the Monte Carlo work. Using the nearest-image convention, each particle in the crystal interacts with 107 neighbors according to the LennardJones 6-12 potential<br /><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEic313TmyU3hF8n6KuVRWU2PjcspOj8HKsfz34aznSkhMNPlLbJRSvUaCPGTbcpk57kekPEo9-ecY0Lh6xgKGy-IJbocJ5l8ccpS-R4VCjCgnBR0OFa2a64BBTpVwD33QqvGyvBjI0f3DM/s1600-h/41.png"></a><br />These potentials have both been used principally to describe rare gases and have shown themselves to fit these rea(materials well. Although we picked these potentials because of their value in describing rare gases we expect<br /><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj76bU6kJEmiXBCFNbaHCiy5wyX3z_pSKOqfB7JE1f6_IQ9eWN5NEDnwNSTlWN8wOZW7xHq7ihdEURup8BYbo66zam-DehLbejQAB0Ak2EeeLV6cfrIMHy1VyNw5HS3VtxvCoPh6GePUNY/s1600-h/42.png"></a><br />that our general conclusions in comparing approximate calculations with exact computer experiments will be valid for potentials describing interactions in salts or metals as well.Table I gives both the Monte Carlo and the lattice-dynamics results. The three temperatures span the range from about 0.48TtriPIO to O.95TtriPle and the densities correspond closely to zero pressure. Because the static-lattice contributions to the thermodynamic properties present no theoretical problems (they are correctly calculated by any theory) we have tabulated separately the thermal contributions to the thermodynamic properties. The data in the table show that the lattice-dynamics elastic constants are quite· close to the Monte Carlo values at three different temperatures and for both potentials tested.Orlaninghttp://www.blogger.com/profile/08859376411004440286noreply@blogger.com0tag:blogger.com,1999:blog-4407931247357581937.post-37581017098207017762010-06-19T16:14:00.000-07:002010-06-19T16:44:58.402-07:00The Maxwell Hypothesis<div align="center"><strong>GENERALIZING THE MAXWELL HYPOTHESIS AS A<br />CONCEPT OF OPTIMAL ENTROPY<br /></strong><br />University of Hagen / Germany<br /> </div><div align="justify"><br />Let us take the combined work of Briton James Clark Maxwell and Austrian Ludwig Boltzmann from the second part of 19th century as starting point. In order to study the phenomena of Statistical Mechanics Boltzmann has created a model of gas molecules representing them as N hard discs -- comparable billiard balls – within a container being imposed on the Newtonian dynamics.<br /><br />The great perception of Maxwell, i.e. the so-called Maxwell Hypothesis, states that the equilibrium distribution of momenta is a normal one; its variance being determined – up to a multiplicative constant -- by the temperature in [K], we speak here as of the Maxwell-Boltzmann distribution.<br />Although generally accepted a direct examination of the Maxwell Hypothesis by laboratory physics is not possible; a way out is offered by computer experimentation.<br /><br />By a computer experiment we can show that the equilibrium distribution of momenta is indeed a normal one. The momentum distribution is estimated in the statistical word sense based on the Boltzmann model of moving gas molecules from a 2-dimensional space implemented on the computer.<br /><br />A remark concerning the estimation technique:<br />The momenta realized by the computer experiment in the momentum space IU = IR2 are projected onto the linear subspace Lß of IU with polar angle ß varying between 0 and 360 degree in order to estimate the density-graph of the momentum distribution by a non-parametric procedure and parallel to this the variance of the actual distribution is estimated parametrically, i.e. within the class of centered normal distributions, which yields a second estimate of the momentum distribution. If these two estimates coincide we know the type of the distribution on Lß but also the actual parameter, 0° <= ß <= 360°, which determines by a Corollary of a Theorem of Cramer and Wold, cf. Billingsley (1986) Theorem 29.4, also the momentum distribution on IU.<br /><br />The estimated variances of the projected normal distribution are used – according to the Maxwell Hypothesis – to calculate the ‘temperature’ of the virtual system of moving molecules implemented on the computer. By the rotational symmetry of the Maxwell-Boltzmann distribution in even higher dimensions than 1 and the relation between temperature and variance it is thereby clear, that temperature is a scalar quantity.<br /><br />The Maxwell-Boltzmann distribution reveals itself – by mathematical considerations – as the one having maximal entropy under the condition of conservation of energy – we speak here of the entropic momentum distribution.<br />If an equilibrium distribution coincides with the entropic distribution, then we always have -- as a necessary condition – that temperature is a scalar quantity. In the case of such a coincidence we say that the generalized Maxwell Hypothesis or the Entropy Principle holds true.<br /><br />As an important question we have: Is the (generalized) Maxwell Hypothesis resp. the Entropy Principle strictly confined to the standard Newtonian dynamics already treated by Maxwell and Boltzmann or does this fact open a window to a more general insight?<br /><br />To this end we examine based on computer experiments representing moving gas molecules being imposed on various other dynamics than the standard Newtonian one, whether the estimated momentum equilibrium distribution coincide with the entropic one or in other words we examine the validity of generalized Maxwell Hypothesis for various types of dynamics being different from the standard Newtonian one.<br /><br />What happens, if we substitute – in the sense of non-real physics – the mass matrix m I (m mass of a molecule, I identity matrix) typical for the standard Newtonian dynamics by a general positive definite matrix M with the consequence that the entropic momentum distribution is still a normal distribution but by contrast to the standard Newtonian dynamics now with elliptical contours. In other words: As the rotational symmetry is lost, as we have it for the standard Newtonian dynamics, the question arises, whether temperature is still a scalar quantity. The latter is a necessary condition for the validity of generalized Maxwell Hypothesis.<br />But even so -- by a computer experiment an affirmative answer can be given for the validity of the generalized Maxwell Hypothesis.<br /><br />The same affirmative answer can be given, when in a next experiment the causality concept as we have it in classical physics is given up. Till now an energy splitting of two molecules is caused by an impact of them. The link of the energy splitting and the collision of the molecules is broken now. Any two molecules can split their energies and exchange their momentum according to the laws of momentum and energy conservation. The partners of an energy splitting as well as the time points and the places of it are determined randomly.<br /><br />A quite new situation we have treating a system of moving molecules being imposed on the relativistic dynamics due to Albert Einstein. We have not only a totally new dynamics but also the entropic momentum distribution is of a total another statistical type.<br /><br />Following Werner Heisenberg we consider finally a discrete momentum space; i.e. the momentum space is – according to quantum mechanics – a lattice. This means energy and also the momentum components cannot assume any value of IR, the continuum of real numbers. These values are restricted now to a sub-lattice of IR.<br />Also in the following experiments causality is given up. Notice in this context, that that causality plays no role in quantum mechanics.<br /><br />Denote by e the unit vector of the (classical) momentum exchange direction, connecting the both molecules i and j.<br /><br />Conservation of momentum leads in classical physics to the following ansatz, relating the momenta u*i, u*j and u i , u j of the molecules i and j shortly after and shortly before momentum exchange, respectively:</div><div align="justify"> </div><div align="center"> u*i = u i + s e<br /> <br /> u*j = u j -- s e</div><div align="center"> </div><div align="justify">where the scalar s is determined by the condition of energy conservation.<br /><br />Implementing now a micro-model of moving gas molecules for the case of a discrete momentum space IU being a lattice a problem arises, because the unit vector e may fail to be an element of IU (being a sub--lattice of IR).<br />One may try to overcome the sketched problem determining a dynamics for which the unit vector e is substituted by a unit vector e* being an element of IU such that the angle between e and e* becomes minimal.<br /><br />If the generalized Maxwell Hypothesis (Entropy Principle) should be fulfilled for the described dynamics, then temperature should be a scalar quantity!<br />But the computer experiment shows an another result. There are at least two temperatures, a ‘horizontal’ and a ‘vertical’ one, depending on the fact on which subspace the momentum data are projected. In other words we have found a dynamics for which the generalized Maxwell Hypothesis, i.e. the Entropy Principle does not hold true.<br /><br />Is it possible to determine a slightly different dynamics with the same momentum space and the same Hamiltonian (not explicitely introduced here), such that the generalized Maxwell Hypothesis, i.e. the Entropy Principle, is fulfilled?<br /><br />To this end consider the set of all nodes of the momentum space of a pair of molecules for which the sums of energies and momenta remain constant; i.e. for which the laws of energy and momentum conservation are fulfilled; we speak of the set of ‘possible’ nodes.<br />This set is finite and not empty, so the dynamics for the discrete momentum space is defined in such a way, that for any energy splitting one of these ‘possible’ nodes is realized randomly according to the uniform distribution determining the next momentum configuration.<br /><br />With this additional rule to determine a dynamics for a discrete momentum space the computer experiments shows that the validity of the generalized Maxwell Hypothesis, i.e. the Entropy Principle, is fulfilled.<br /><br />The experiment is meaningful insofar as the presented micro-model by the chosen Hamiltonian (not made explicit here) supports the theoretically postulated probabilities of the excited energy levels of the harmonic oscillator from quantum mechanics.<br /><br />Physicists have always postulated, i.e. they have always – successfully believed – in the Entropy Principle, but a respective micro-model confirming this postulate for the harmonic oscillator did -- to our knowledge – not exist.<br /> </div><div align="justify"> </div><div align="right">REFERENCIAS BIBLIOGRAFICAS:</div><div align="right"> </div><div align="right">www.science.az/cyber/pci2006/2/moeschlin.doc</div>Orlaninghttp://www.blogger.com/profile/08859376411004440286noreply@blogger.com0tag:blogger.com,1999:blog-4407931247357581937.post-54225051873426471542010-05-22T18:35:00.000-07:002010-05-22T18:56:56.308-07:00Lattice dynamics<div align="justify">In this the final chapter of the thesis perhaps the most interesting results will be presented. Here we will start with a short review of harmonic lattice the-ory together with a brief discussion of how lattice dynamics can be calculated from ab-initio theory. Here special focus will be on the so-called supercell method, since this is the method that has been used through out this thesisto calculate phonons from ab-initio theory. After this brief introduction theresults obtained within the harmonic, or rather quasi harmonic, approxima-tion will be presented (see papers III, IV and VI). The chapter is ends with adiscussion of the anharmonic lattice and a presentation of the self-consistentab-initio lattice dynamical (SCAILD) approach, and the results obtained withthis novel approach (see paper V) will also be discussed.</div><div align="justify"><br /> </div><br /><div align="justify"><strong>The Born Oppenheimer approximation</strong></div><div align="justify"><strong></strong><br /> </div><div align="justify">Before discussing the theory of lattice dynamics and the associated calcula-tional methods, it is important to take a closer look at one of the fundamental approximations used in calculating phonons from first principles. This approx-imation is commonly known as the Born Oppenheimer approximation, and itassumes that the electronic response to an atomic displacement is instanta-neous, making it possible to separate the electronic and the ionic subsystems.To convince oneself of the soundness of this approximation one should remember that the typical ionic mass mi is ∼ 105 times bigger than the mass ofan electron me and that the typical kinetic energy of an electron Eke is 103 times bigger than the typical ionic kinetic energy Eki, implying that the ratiobetween the typical velocity of an electron ve and that of an ion vi becomes(ve/vi)=%Ekemi/(Ekime) ∼ 104. Thus from the "perspective of an electron",the ions will always seem to have fixed positions. Hence if U(R) are the de-viations of the ions from their equilibrium positions at a snapshot in time, itis always possible to retain the total energy of the system, at that snapshot,by means of a static electronic structure calculation. Thus, through a seriesof electronic structure calculations, the potential energy of the ionic subsys-tem can be parameterized in terms of ionic deviations. It is general practice toexpress the potential energy in the Hamiltonian of the ionic subsystem, as aTaylor expansion around the equilibrium ionic configuration.<br /></div><br /><div align="justify"><img id="BLOGGER_PHOTO_ID_5474275653174874594" style="DISPLAY: block; MARGIN: 0px auto 10px; WIDTH: 320px; CURSOR: hand; HEIGHT: 67px; TEXT-ALIGN: center" alt="" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgvI1N4W4Wli7zcymsHDlAn_pzbkUNAju2DHUt3Pmiuj28obaIsU1S2zyTI287gexAY0XAsxW55BPOwxp_pSJeDaKuUvryglMn8gFJAg_XPfVts15yWOd8aHE2va3-7_HZ-0UdvbDGi-jI/s320/29.bmp" border="0" /></div><br /><p align="justify"><img id="BLOGGER_PHOTO_ID_5474276006174208274" style="DISPLAY: block; MARGIN: 0px auto 10px; WIDTH: 320px; CURSOR: hand; HEIGHT: 73px; TEXT-ALIGN: center" alt="" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiwa92X9YzDzJ6Z4I3kQyZVNl4EqGfR7pvPtAN4gphswQBGZxE8quNC7R4WmCLULgVdQfSLofqrKK94Ep0s4ef5sVObB4l88qPXvSkunGk9iignZ7-4yo4Cp41NVjx5vk8jQMexZnOEjDY/s320/30.bmp" border="0" /><br /></p><p align="justify"><strong>The harmonic lattice</strong><br /><br />In the harmonic lattice approximation the atomic deviations are assumed to beso small that the potential energy is well described by the second order term in(8.1). This is generally a good approximation, at least at relatively low temper-atures. Later on in this chapter examples of situations will be given in whichthe harmonic approximation fails, such as the high temperature bcc phase ofTi, Zr and Hf. Furthermore, in order to make the notation more transparent,the notation of a monoatomic lattice will be adapted without any loss of gen-erality. The harmonic Hamiltonian in the case of a monoatomic lattice is given by</p><img id="BLOGGER_PHOTO_ID_5474276770473766962" style="DISPLAY: block; MARGIN: 0px auto 10px; WIDTH: 260px; CURSOR: hand; HEIGHT: 53px; TEXT-ALIGN: center" alt="" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiq6pUWA0mAcCNiz0egCboe9-8qeVMR6u1S22_5Q-lyyB_-WYMqfB5wes4kYgI267k-j4r74F14V4Up5lG6_ZgZIFC5NH2DngwHi1QgtTZwexMEPSK_zILRWcg1zlZDGk_RZLl-_J0Fn8A/s320/31.bmp" border="0" /><br /><p align="justify">In the harmonic approximation, the ionic displacements UR satisfy Bornvon Karman periodic boundary conditions. This means that the displacementscan be expressed as a superposition of plane waves with wavevectors k ∈ 1BZ. Hence the canonical coordinates UR and PR appearing in (8.2), can beexpressed in terms of a new set of canonical coordinates Qk,s andPk,s, i.e</p><br /><p align="justify"><br /></p><img id="BLOGGER_PHOTO_ID_5474277172966171714" style="DISPLAY: block; MARGIN: 0px auto 10px; WIDTH: 320px; CURSOR: hand; HEIGHT: 130px; TEXT-ALIGN: center" alt="" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi_eDBWA82zrPGOyL-LnyVjVeBYHgfbTYbjlg8oAKKCXTQa22DBX1eI023jdWIhNp8zgii8smwNp-WTpFmTorKkHH12MUpSnVUtreLLwG4mBvwYSTeioPZdD-cA4R-drVDm3oPdpnNwzFE/s320/32.bmp" border="0" /><br /><p align="justify"><strong>The supercell method</strong></p><p align="justify">In the previous section it was shown that once the force constant matrix has been calculated and Fourier transformed, the phonon frequencies are easilyaccessed by a simple diagonalization. Fortunately there exists a fairly simpleand straightforward method for calculating from first principles, namely theso-called supercell method. The foundation of the method is provided by theHellman-Feynman theorem, stating that the force FR acting on an atom withspatial coordinate R.</p><p align="justify">From the above linear relation and the symmetry of the crystal the forceconstant matrix can then be easily calculated. The number of displacementsneeded to retain depends on the symmetry of the crystal. For instance inthe case of the bcc or fcc structure one displacement is sufficient, while in thecase of the hcp structure two displacements are needed.However since, at least in principle, Φij(R)→0 only as R→∞, and sinceonly finite sized supercells can be used, the summation in (8.5) has to be trun-cated, and the dynamical matrix can only be approximately calculated. Fur-thermore, due to the periodic boundary conditions employed in the electronicstructure calculations, the linear relation (8.16) is only true if an infinite sizedsupercell is used. In real life all the periodic images of the displaced atomcontribute in the induction of the forces in the supercell. The correct linear re-lation between force and displacement(s), to be used in a supercell calculation.</p><p align="justify"><strong>Some thermodynamics and the quasi harmonic</strong></p><p align="justify">approximationIn this section relations between the harmonic phonon spectrum and differentthermodynamic quantities, such as the free energy, internal energy and meansquare atomic deviation, will be derived and briefly discussed. Furthermore ashort presentation of the quasi harmonic approximation will also be given.</p><p align="justify">The two above expressions for the internal- and free-energy have been used inthe context of the so-called quasi harmonic approximation to calculate Equations of state, Hugoniots and thermal expansions (see papers III, IV and VI).What now remains in this section is a short discussion of the quasi harmonic approximation. This is the most simple approximation dealing with theeffects of anharmonicity in which the anharmonicity related to the terms oorder > 2 in the Taylor expansion (8.1) is neglected, only taking into accounthe anharmonicity related to the force constants dependence upon symmetryconserving strain. The simplicity of this approximation lies in the fact thafor each symmetry conserving strain the lattice dynamics of the system isregarded as being harmonic, permitting the use of the supercell method separately for each symmetry conserving strain. In Fig. 8.2 the phonon density ostates for fcc Au calculated with the supercell method for three different volumes are displayed, as an example of the volume dependence of the phonon spectra.</p><p align="justify"> </p><p align="justify"><strong>Thermal expansion</strong></p><p align="justify">In this section the art of calculating thermal expansion coefficients from firstprinciples will be discussed. This discussion will be based on the work donein papers IV and VI of the these.</p><p align="justify"><strong>Thermal expansion of cubic metals</strong></p><p align="justify">The calculation of the thermal expansion of elements with cubic symmetry isvery straightforward when done in the quasi harmonic approximation. Firstthe phonon and electron density of states together with static lattice energy iscalculated for a number of volumes around the T = 0K equilibrium volume.Then using Eq. (8.28-8.30) the total free energy is calculated for the differentvolumes at constant temperature and fitted to some EOS. </p><p align="justify"><strong>Thermal expansion of hexagonal metals</strong></p><p align="justify">To calculate the thermal expansion of hexagonal metals, the free energy andstatic lattice energy have to be parameterized with respect to two degrees offreedom. The most general second order parameterization of the free latticeenergy, allowing only symmetry conserving strains, can be expressed with thesix dimensional strain vector ¯ ε =(ε1, ε1, ε3,0,0,0) and the elastic constants.Using this strain vector together with the definitions given in chapter 5, thestatic lattice energy U.</p><p align="right">BIBLIOGRAFIA:</p><p align="right">web.mac.com/petros...2/.../urn_nbn_se_uu_diva-8198-1__fulltext.pdf</p><p align="justify"> </p>Orlaninghttp://www.blogger.com/profile/08859376411004440286noreply@blogger.com0tag:blogger.com,1999:blog-4407931247357581937.post-6563846477277125922010-05-22T17:47:00.000-07:002010-05-22T18:32:38.882-07:00Inelastic neutron scattering and lattice dynamics<div align="justify"><br /><div align="justify">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:<br /><br /><br /></div><div align="justify"><img id="BLOGGER_PHOTO_ID_5474269137520092114" style="DISPLAY: block; MARGIN: 0px auto 10px; WIDTH: 320px; CURSOR: hand; HEIGHT: 94px; TEXT-ALIGN: center" alt="" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgqqJj8l4NK8dpZklCfhKZZKSJlR_IUutWGwBn0Nfa-BqVEBgmabiDVBydxbIbp2e4EfFeDWbYdhdLSscy7I1whnZIDzPgQlS5rzOy7-SpT2ecXRn0Nx8nSPZa3XVxCxufUgzkJJNxyQZk/s320/25.bmp" border="0" /></div></div><br /><br /><p align="justify">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.<br /></p><p align="justify">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 </p><br /><br /><p align="justify"><img id="BLOGGER_PHOTO_ID_5474270204215259410" style="DISPLAY: block; MARGIN: 0px auto 10px; WIDTH: 320px; CURSOR: hand; HEIGHT: 194px; TEXT-ALIGN: center" alt="" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgkqp__0SgXesoO_rBdgu-nls26XsBk44hww-uWjk1mgIqoKdfplq90Pbz339jl1d1z5B4DhnvjrlrfVC1RfvjhpUDafn_q1zQUsmu_QDavs3L3kJWRKAcfTSs8FuWQaylxKfUmLs9QQwQ/s320/26.bmp" border="0" /></p><br /><br /><p align="justify">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.<br /></p><br /><br /><p align="justify"><strong>Geophysically important minerals</strong><br /></p><br /><br /><p align="justify">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</p><img id="BLOGGER_PHOTO_ID_5474271111413001154" style="DISPLAY: block; MARGIN: 0px auto 10px; WIDTH: 320px; CURSOR: hand; HEIGHT: 317px; TEXT-ALIGN: center" alt="" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjPrKZ6KdzDlTOJJblgTS0XxOl9eXApfDtdErffnfkyRfs_0o5hn82eaN8-ZucqZgId7iU0VeUVg8ZDXPynqIbfIZAv6-x-nKRe8WuW6_fsUmcOuluy4yymGYsweJAe2HfmLLFx6CNCp_c/s320/27.bmp" border="0" /><br /><p align="justify">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</p><br /><p align="justify"><br /></p><img id="BLOGGER_PHOTO_ID_5474271830616595906" style="DISPLAY: block; MARGIN: 0px auto 10px; WIDTH: 320px; CURSOR: hand; HEIGHT: 202px; TEXT-ALIGN: center" alt="" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjdTDOH8h50WDIP6OcSCJLWTFzJjAnPjmZetfrMbLXQ4lCeY8YJfaS2YBdIziw5Ol15QUmoFf5E1bT8XEL0WRLC_kAm78TTHvF9YbaBSk1Ltft_B1P-pyZugzv7gjpAXwiPwEryPHSVk64/s320/28.bmp" border="0" /><br /><p align="justify"><strong>Conclusion</strong></p><p align="justify">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.</p><p align="justify"> </p><p align="right">BIBLIOGRAFIA:</p><p align="right"><a href="http://www.ias.ac.in/pramana/v63/p73/fulltext.pdf">http://www.ias.ac.in/pramana/v63/p73/fulltext.pdf</a></p>Orlaninghttp://www.blogger.com/profile/08859376411004440286noreply@blogger.com0tag:blogger.com,1999:blog-4407931247357581937.post-3510215181304709112010-05-22T17:02:00.000-07:002010-05-22T17:37:51.176-07:00Quantum Dynamics of Matter Waves Reveal Exotic Multibody Collisions<div align="justify">At extremely low temperatures atoms can aggregate into so-called Bose Einstein condensates forming coherent laser-like matter waves. Due to interactions between the atoms fundamental quantum dynamics emerge and give rise to periodic collapses and revivals of the matter wave field.<br />A group of scientists led by Professor Immanuel Bloch (Chair of Experimental Physics at the Ludwig-Maximilians-Universität München (LMU) and Director of the Quantum Many Body Systems Division at the Max Planck Institute of Quantum Optics in Garching) has now succeeded to take a glance 'behind the scenes' of atomic interactions revealing the complex structure of these quantum dynamics. By generating thousands of miniature BECs ordered in an optical lattice the researchers were able to observe a large number of collapse and revival cycles over long periods of time.<br /><br />The research is published in the journal Nature.<br /><br />The experimental results imply that the atoms do not only interact pairwise -- as typically assumed -- but also perform exotic collisions involving three, four or more atoms at the same time. On the one hand, these results have fundamental importance for the understanding of quantum many-body systems. On the other hand, they pave the way for the generation of new exotic states of matter, based on such multi-body interactions.<br /><br />The experiment starts by cooling a dilute cloud of hundreds of thousands of atoms to temperatures close to absolute zero, approximately -273 degrees Celsius. At these temperatures the atoms form a so-called Bose-Einstein condensate (BEC), a quantum phase in which all particles occupy the same quantum state. Now an optical lattice is superimposed on the BEC: This is a kind of artificial crystal made of light with periodically arranged bright and dark areas, generated by the superposition of standing laser light waves from different directions. This lattice can be viewed as an 'egg carton' on which the atoms are distributed. Whereas in a real egg carton each site is either occupied by a single egg or no egg, the number of atoms sitting at each lattice site is determined by the laws of quantum mechanics: Depending on the lattice height (i.e. the intensity of the laser beam) the single lattice sites can be occupied by zero, one, two, three and more atoms at the same time.<br /><br />The use of those "atom number superposition states" is the key to the novel measurement principle developed by the researchers. The dynamics of an atom number state can be compared to the dynamics of a swinging pendulum. As pendulums of different lengths are characterized by different oscillation frequencies, the same applies to the states of different atom numbers. "However, these frequencies are modified by inter-atomic collisions. If only pairwise interactions between atoms were present, the pendulums representing the individual atom number states would swing synchronously and their oscillation frequencies would be exact multiples of the pendulum frequency for two interacting atoms," Sebastian Will, graduate student at the experiment, explains.<br /><br />Using a tricky experimental set-up the physicists were able to track the evolution of the different superimposed oscillations over time. Periodically interference patterns became visible and disappeared, again and again. From their intensity and periodicity the physicists found unambiguous evidence that the frequencies are actually not simple multiples of the two-body case. "This really caught us by surprise. We became aware that a more complex mechanism must be at work," Sebastian Will recalls. "Due to their ultralow temperature the atoms occupy the energetically lowest possible quantum state at each lattice site. Nevertheless, Heisenberg's uncertainty principle allows them to make -- so to speak -- a virtual detour via energetically higher lying quantum states during their collision. Practically, this mechanism gives rise to exotic collisions, which involve three, four or more atoms at the same time."<br /><br />The results reported in this work provide an improved understanding of interactions between microscopic particles. This may not only be of fundamental scientific interest, but find a direct application in the context of ultracold atoms in optical lattices. Owing to exceptional experimental controllability, ultracold atoms in optical lattices can form a "quantum simulator" to model condensed matter systems. Such a quantum simulator is expected to help understand the physics behind superconductivity or quantum magnetism. Furthermore, as each lattice site represents a miniature laboratory for the generation of exotic quantum states, experimental set-ups using optical lattices may turn out to be the most sensitive probes for observing atomic collisions. </div><div align="justify"> </div><div align="justify"> </div><img id="BLOGGER_PHOTO_ID_5474257681654640514" style="DISPLAY: block; MARGIN: 0px auto 10px; WIDTH: 320px; CURSOR: hand; HEIGHT: 256px; TEXT-ALIGN: center" alt="" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgnZENyJrV5NgLU_yu_OgvOVh4713HvYropehRc9w2AFpCJ7f7a1ETCuHR9l1PfEN_4GFxAs36gbpwfYSZJqLdbaEUaDlCb7ZPdE1kBgvMppmWcq32FEmu9MGir1cXxAJSYBT0aLmRfKoY/s320/24.bmp" border="0" /><br /><p> </p><p align="center">Collapse and revival of the matter wave field: The quantum dynamics of Bose-Einstein condensates trapped in an optical lattice reveal exotic multi-body interactions. The image shows a sequence of interference patterns of the atomic samples recorded in steps of 40 microseconds. A single cycle of the dynamics is highlighted (blue-orange). (Credit: Max Planck Institute of Quantum Optics)</p><p align="center"> </p><p align="right">BIBLIOGRAFIA:</p><p align="right"><a href="http://www.sciencedaily.com/releases/2010/05/100514094836.htm">http://www.sciencedaily.com/releases/2010/05/100514094836.htm</a></p>Orlaninghttp://www.blogger.com/profile/08859376411004440286noreply@blogger.com0tag:blogger.com,1999:blog-4407931247357581937.post-77775150191084920992010-05-22T16:25:00.000-07:002010-05-22T16:59:18.193-07:00Lattice dynamics and correlated atomic motion from the atomic pair distribution function<div align="center"><strong>INTRODUCTION</strong></div>
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<br /><div align="justify">The pair distribution function ~PDF! obtained from thepowder x-ray and neutron diffraction experiments has beenshown to be of great value in determining the local atomic structure of materials.1The PDF results from a Fourier trans-form of the powder diffraction spectrum Bragg peaks 1diffuse scattering! into real-space.2For well ordered crystals,apart from technical details, this is similar to fitting theBragg peaks 1 thermal diffuse scattering in the powder pat-tern in a manner first discussed by Warren.3A PDF spectrumconsists of a series of peaks, the positions of which give thedistances of atom pairs in real space. The ideal width of these peaks ~aside from problems of experimental resolution! isdue both to relative thermal atomic motion and to static dis-order. Thus an investigation of the effects of lattice vibra-tions on PDF peak widths is important for at least two rea-sons: first, to establish the degree to which information onphonons and the interatomic potential! can be obtained frompowder diffraction data, and, second, to account for correla-tion effects in order to properly extract information on staticdisorder in a disordered system such as an alloy.</div>
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<br /><div align="justify">In general, powder diffraction is not considered a favor-able approach for extracting information about phononssince, not only is energy information lost in the measure-ment, but also the diffuse scattering is isotropically averaged.The lattice vibrations are best described from the phonondispersion curves determined using inelastic neutron scatter-ing and high-energy-resolution inelastic x-ray scattering onsingle crystals.4,5Nevertheless, with the advent of high-energy synchrotron x-ray and pulsed-neutron sources andfast computers, it is possible to measure data with unprec-edented statistics and accuracy. The PDF approach has beenshown to yield limited information about lattice vibrations in powders, though the extent of which this information can beextracted remains controversial.7–10Measuring powders has the benefit that the experimentsare straightforward and do not require single crystals. It isthus of great interest to characterize the degree to whichlattice vibrations are reflected in the PDF using simple mod-els, such as the Debye model, in situations where detailedinteratomic potential information is not available. In this pa-per we explore these issues by comparing both measuredPDFs and those calculated from realistic potential modelswith PDFs obtained through a single-parameter Debyemodel. This comparison is carried out as a function of atomicpair separation, temperature and direction in the lattice. Wefind that a single parameter Debye model explains much ofthe observed lattice vibrational effects on PDF peak widths,including the temperature dependence, in crystals like Ni,Ce, and GaAs. However, small but non-negligible deviationsfrom the Debye model calculation are evident in crystalwhich needs a long-range interaction to explain anomalies inthe dispersion curves. </div>
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<br /><div align="center"><strong>CORRELATED ATOMIC MOTION IN REAL SPACE</strong></div>
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<br /><div align="justify">The existence of interatomic forces in crystals results inthe motion of atoms being correlated. This is usually treatedtheoretically by transforming the problem to normal coordi-nates, resulting in normal modes ~phonons! that are non-interacting, thus making the problem mathematically tract-ible. Projecting the phonons back into real-space coordinatesyields a picture of the dynamic correlations. This situationcan be understood intuitively in the following way. Figure 1shows a schematic diagram of atomic motion in three differ-ent interatomic force systems, each with its correspondingideal PDF spectrum. In a rigid-body system Fig. 1-a, the</div>
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<br /><div align="justify"><img id="BLOGGER_PHOTO_ID_5474242322696750914" style="DISPLAY: block; MARGIN: 0px auto 10px; WIDTH: 271px; CURSOR: hand; HEIGHT: 320px; TEXT-ALIGN: center" alt="" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhjZ0yih8ETytQFpfbvb0XWaNMwCIJhuFbWcJhoRL2xVrbzaE-CidNDZ0XwGyqXes5fa5wk6eRtzP9B0x2BhUj6mv0vnk5u4PotRM9oHPAs8Pa0L1y_3GWNGq_p_PurEakdufvV3gZFEPU/s320/23.bmp" border="0" /></div>
<br /><p> </p><p> </p><p align="justify">interatomic force is extremely strong and all atoms move inphase. In this case, the peaks in the PDF are delta-functions.At the opposite extreme the atoms are non-interacting ~theEinstein model! and move independently as shown in Fig.1~b!. This type of atomic motion results in broad PDF peakswhose widths are given by the root mean-square displace-ment amplitude (A^u2&). In real materials, the interatomicforces depend on atomic pair distances, i. e., they are strongfor nearest-neighbor interactions and get weaker as theatomic pair distances increase. In fact, these interactions areoften quite well described with just nearest-neighbor or first-and second-nearest-neighbor coupling. The case of nearest-neighbor interactions is shown in Fig. 1~c!. In this ~Debye!model a single parameter corresponding to the spring con-stant of the nearest-neighbor interaction is used. Here, near-neighbor atoms tend to move in phase with each other, whilefar-neighbors move more independently. As a result, thenear-neighbor PDF peaks are sharper than those of far-neighbor pairs. This behavior was first analyzed by Kaplowand co-workers in a series of papers11–13for a number ofelemental metals.</p><p align="justify"> </p><p align="center"><strong>EXPERIMENTS AND ANALYSIS</strong></p><p align="justify">The experimental PDFs discussed here were measured us-ing pulsed neutrons and synchrotron x-ray radiation. Theneutron measurements were carried out at the NPD diffrac-tometer at the Manual Lujan, jr., Neutron Scattering Center~LANSCE! at Los Alamos and the x-ray experiments atbeam line A2 at CHESS ~Cornell!. Powder samples of Niand a polycrystalline Ce rod were loaded into a vanadiumcan for the neutron measurements, carried out at room temperature. Powdered GaAs was placed between thin foils ofkapton tapes for the x-ray measurements, measured at 10 Kusing 60 KeV (l50.206 Å) x rays. Due to the higher x-rayenergy at CHESS and relatively low absorption coefficient ofGaAs, symmetric transmission geometry was used.Both the neutron and x-ray data were corrected14,15forexperimental effects and normalized to obtain the total scat-tering function S(Q), using programs PDFgetN.</p><p align="center"><strong>DISCUSSION</strong></p><p align="justify">The mean-square relative displacements sij2and the cor-responding correlation parameter shown in Figs. 2, 4, 5, 8,and 9 present two interesting pieces of information about the atomic motions in a crystalline material. First of all, theyshow that nearest-neighbor atomic motion is significantlycorrelated. Second, the details of the motional correlations asa function of pair distance display structures which deviatefrom the predictions of the simple CD model. Here we canraise some interesting questions. How is this structure in themotional correlation of atom pairs related to the underlyinginteratomic potentials? Can one extract the potential param-eters using an inverse approach to model the PDF peakwidths with the potential parameters as input?Reichardt and Pintschovius8argued that the calculatedPDF peak widths as a function of pair distance are ratherinsensitive to the details of the lattice dynamics models usedto calculate sij2. They found that PDFs calculated using ei-ther very simple or complex models didn’t show significantdifferences. A similar conclusion has been reached by Graf etal.,10in contradiction to previous claims by Dimitrov et al.7Indeed, the magnitude of errors implicit in the measurementand data analysis appear to be comparable to the effects thatmust be measured to obtain quantitatively accurate potentialinformation using this approach.9The conclusions of Rei-chardt and Pintschovius and Graf et al. and Thorpe et al. arelargely borne out by the present work; e.g., the grossly over-simplified CD model, which neglects elastic anisotropy andparameterizes the dynamics with a single number u ,is rather successful at explaining the smooth rijdependence ofthe PDF peak widths.Thus, when the BvK force parameters are not available,we have shown that the correlated Debye ~CD! model is areasonable approximation to describe both the smoothrij-dependence and the temperature dependence of sij2insimple elements. Considering the poor correspondence be-tween the Debye phonon density of states and the BvK den-sity of states, the reasonable agreement between the BvKmodel calculations of sij2and that of the CD model is rathersurprising. This confirms that the PDF peak width is ratherinsensitive to the details of the phonon density of states andthe phonon dispersion curves, as suggested by Reichardt andPintschovius and by Graf et al. Any information about theinteratomic forces in the PDF peak widths is contained in thesmall deviations of the sij2from those of the CD model cal-culations. Therefore, extracting interatomic potential infor-mation from the PDF peak widths is unlikely. However,these deviations could possibly yield some average phononinformation. For example, recent calculations by Graf et al.10showed that one can obtain phonon moments within a fewpercent accuracy for most fcc and bcc crystals using thenearest-neighbor force parameters extracted from a theoreti-cal BvK PDF spectrum. This result indicates that the PDFspectrum contains some average phonon information, although it doesn’t provide detailed phonon dispersion infor-mation. The average phonon information, such as phononmoments from the PDF peak widths, will be a usefulcomplement to optical and acoustic techniques that yieldzone-center information in situations where single crystalmeasurements are not possible. This complementarity alsoextends to the extraction of Debye-Waller factors from pow-der diffraction measurements.Finally, a comparison of the CD model calculations of the PDF peak widths in GaAs with those of experimental PDFand Kirkwood model calculations shows additional limita-tions of the CD model. In the CD model calculation, thenear-neighbor PDF peaks below r<5><p align="justify"> </p>Orlaninghttp://www.blogger.com/profile/08859376411004440286noreply@blogger.com0tag:blogger.com,1999:blog-4407931247357581937.post-63830204822743158212010-05-22T16:11:00.000-07:002010-05-22T16:21:16.398-07:00Femtosecond Lattice Dynamics in Photoexcited Bismuth<div align="justify"><br /></div><div align="justify"><br /></div><div align="justify">One of the grand challenges of ultrafast science is to follow directly atomic motion of a photo-induced reaction on the fastest time-scales and the shortest distances—those associated with the atomic vibrations and the making and breaking of the interatomic bonds. This is the regime that ultimately governs chemistry and materials characteristics. X-ray bursts produced from a free electron laser promise to be an ideal probe to meet this challenge because of their atomic-scale structural sensitivity and ultra-short pulse duration, which can “freeze” the atomic motion stroboscopically [1]. However, significant technical advances are needed before these sources can be used to make an atomic movie of the fastest events. In particular, the optical laser pulse used to trigger the reaction in these classes of experiments must be precisely timed with the x-ray pulses that are used to take atomic “snap-shots”. </div><div align="justify"><br /><br /></div><div align="justify">Using the ultra-short x-ray pulses of the Sub-Picosecond Pulse Source (SPPS) and a novel timing method, we observed the femtosecond response of a bismuth solid following intense photoexcitation of charge carriers. Our results provide insight into the fundamental interaction between the electronic states and the microscopic atomic arrangements of the solid. Furthermore, we demonstrated the ability to synchronize an optical laser to a linear accelerator based x-ray source with femtosecond accuracy. Bismuth is a material that shows very strong coupling between electronic and ionic structure. It is a model system that demonstrates a rich variety of ultrafast dynamics in the limit of high density excitations, such as extremely large phonon amplitudes, electronic softening and phase transitions. Using time-resolved x-ray diffraction techniques, we monitored the atomic positions within the bismuth unit cell as a function of time in response to impulsive photoexcitation of carriers (Figure 1). Coherent lattice oscillations were observed similar to those previously seen in a pioneering laser plasma based x-ray diffraction experiment [2]. However, the comparatively large x-ray fluence of the SPPS </div><div align="justify"><br /></div><img id="BLOGGER_PHOTO_ID_5474237789618025314" style="DISPLAY: block; MARGIN: 0px auto 10px; WIDTH: 320px; CURSOR: hand; HEIGHT: 187px; TEXT-ALIGN: center" alt="" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiQ9_cC3a2Oe_dhyphenhyphenBvEdby9GEr-gyKQ8_zD9Z2EsSuJ40sUSS-eV0s6hXhzOsnYFiztDRAApYG0jCIKii1G_5Xj_s0iK2PNXSH7jOCJ2fyAulGcip39eRzjjq3-ZaajLoweD0CPLtedzEg/s320/22.bmp" border="0" /><br /><p align="justify">resulted in a significant improvement in data quality as well as enabled carrier density dependent studies. We were able to quantify the oscillation frequency and the lattice coordinate the oscillations are occurring about from the time-resolved data. With this information we extrapolated the curvature and minima positions of the double well interatomic potential of bismuth as a function of photoexcited carrier density. Our results were compared to previous density functional calculations of the photoexcited system and are in agreement [3]. Electro-optic sampling methods were used to time the excitation laser pulse with the x-ray probe pulse [4]. In this technique, the electric field of the electron bunch that generated x-rays at the SPPS is used to alter the optical properties of an electro-optic crystal (Figure 2). This alteration is probed with a portion of the optical laser that is used to photoexcite the bismuth sample in crossed-beam geometry. Only the portion of the laser that is propagating within the electro-optic crystal when the electric filed is present will be altered. In this manner, the arrival time of the electron bunch is encoded onto spatial profile of the optical laser. The centroid of the electro-optic feature is used to time stamp each x-ray pulse and the data is compiled accordingly. </p><img id="BLOGGER_PHOTO_ID_5474237239586144034" style="DISPLAY: block; MARGIN: 0px auto 10px; WIDTH: 382px; CURSOR: hand; HEIGHT: 194px; TEXT-ALIGN: center" alt="" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhVv2nUZ-Wvy31A2_gNyjdfK4dCB32c9YdamRPzPYRUJsh794cgxd5nezhX1E42CxuJ1LiJelBR1KtlpobUbPRC_HiSaiJ__0y9MKmeL2-RwF-gRUnzNB5CkPK4NKZIna9ATS72wUux9SM/s320/21.bmp" border="0" /> <p align="justify"><br />These measurements have furthered our understanding of bismuth dynamics far from equilibrium. Our experiments provide the first quantitative characterization of the curvature and quasi-equilibrium position of the interatomic potential of a solid close to a free-carrier induced phase transition. From this, we showed that the electronic softening of the potential is the primary factor determining the frequency of the lattice vibrations. The experiments also demonstrate the successful implementation of an electro-optic timing diagnostic. This technical advancement enabled us to perform femtosecond resolution experiments at a linear accelerator based x-ray source. The experiments were carried out by a collaborative team from 20 different institutions. Portions of this research were supported by the U.S. Department of Energy, Office of Basic Energy Science through direct support for the SPPS and the SSRL. Additional support was received by the Swedish Research Council for Science, the Irish Research Council for Science, the Keck Foundation, the Deutsche Forschungsgemeinschaft, the European Union RTN FLASH, the Austrian Academy of Science, the Stanford PULSE center and the NSF FOCUS frontier center.</p><p align="justify"> </p><p align="right"><a href="http://ssrl.slac.stanford.edu/research/highlights_archive/spps07.pdf">http://ssrl.slac.stanford.edu/research/highlights_archive/spps07.pdf</a><br /></p><p align="justify"></p>Orlaninghttp://www.blogger.com/profile/08859376411004440286noreply@blogger.com0tag:blogger.com,1999:blog-4407931247357581937.post-4813779347569238382010-05-22T10:07:00.000-07:002010-05-22T10:52:12.155-07:00investigation of lattice dynamics<div align="center"><strong></strong> </div><div align="center"><strong>Ab initio investigation of lattice dynamics of fluoride scheelite</strong></div><div align="center"><strong></strong> </div><div align="center"><strong></strong> </div><div align="center"><strong></strong> </div><div align="justify"><strong>Abstract</strong></div><div align="justify"> </div><div align="justify">We report on the phonon dynamics of LiYF4 obtained by direct method using first principle calculations. The agreement between experimental and calculated modes is satisfactory. An inversion between two Raman active modes is noticed compared to inelastic neutron scattering and Raman measurements. The atomic displacements corresponding to these modes are discussed. Multiple inversions between Raman and infrared active groups are present above 360 cm-1. The total and partial phonon density of state is also calculated and analyzed. </div><div align="justify"> </div><div align="justify"> </div><div align="justify"><strong>Introduction</strong> </div><div align="justify"> </div><div align="justify"> </div><div align="justify">Researches on YLiF4 crystals are strongly linked to laser technology. The first structural data obtained by Thomas et al. date from 1961[1], just one year after the demonstration of the first laser. At ambient pressure, the crystal cell of YLiF4 is tetragonal with space group I41/a (C4h6). This phase is commonly named the scheelite structure in reference of the CaWO4 crystal. Lithium ions (Li+) are in the center of tetrahedrons composed by 4 fluoride ions (F-). Yttrium ions (Y3+) are in the center of polyhedrons composed by 8 F-. Y3+ can be substituted by rare earth presenting an oxidation state of +3, such as Erbium (Er+3) [2] or Thulium (Tm+3) [3], providing good matrix for upconversion laser. The efficiency of this kind of laser relies on intraionic and interionic process of relaxation that strongly depends on the host matrix[4]. </div><div align="justify"> </div><div align="justify">This relationship is evidenced particularly by the multiphonon relaxation process implying electron-phonon coupling[5]. Consequently, a fair knowledge of the structural and dynamics properties of the host matrix is crucial for the development of host matrix. To this end many studies have been carried out on the subject. Phonon frequencies were measured by Raman and IR spectra [6][7][8][9][10]. These methods give information at the center of the Brillouin zone. Inelastic neutron scattering measurement is needed to obtain the complete phonon dispersion curves that are essential to a good understanding of the global vibrational and relevant properties. This was done for LiYF4 by Salaün et al.[11]. Besides experimental work, numerical methods have been developed. Among them we can notice empirical methods, such as rigid ion model (RIM). Using this method, Salaün et al[11] and Sen et al.[12] performed lattice dynamical calculations on LiYF4 providing a large number of interesting results about lattice vibration. Obviously, the correctness and precision of this model is limited by the empirical parameters. Density functional theory is an empirical free parameter methods whose usefulness and predictive ability in different fields[13][14] are known since a long time. Recently, the association of DFT with different techniques such as linear response method[15][16] or direct methods[17][18][19] allows to evaluate phonon dispersion curves without empirical parameter. In particular Parlinsky et al.[20][21] developed a direct method where the force constant matrices are calculated via the Hellmann-Feynman theorem in total energy calculations. </div><div align="justify"> </div><div align="justify">In this work we present a first principle investigation of YLiF4 in its scheelite phase. DFT associated with projector augmented wave (PAW) and direct method were used. Cell parameters, phonon dispersion curve, phonon density of state are discussed and compared with previous experimental or numerical results. To our knowledge, this is the first ab initio calculation of LiYF4 lattice dynamics. </div><div align="justify"> </div><div align="justify"> </div><div align="justify"> <strong>Methodology</strong></div><div align="justify"> </div><div align="justify"> </div><div align="justify">Cell parameter and atomic positions of the initial structure were obtained from experimental results by E. Garcia and R.R Ryan[22]. All calculations were carried out with the VASP[23] code, based on DFT [24][25], as implemented within MEDEA[26] interface. Here the generalized gradient approximation (GGA) through the Perdew Wang 91 (PW91)[27] functional and projector augmented wave (PAW)[28] were employed for all calculations. Electronic occupancies were determined according to a Methfessel-Paxton scheme[29] with an energy smearing of 0.2 eV. The crystal structure was optimized without the constraints of the space group symmetry at 0 Gpa until the maximum force acting on each atom dropped below 0.002 eV/Å. The self consistent field (SCF) convergence criterion was set at 10-6 eV. High precision calculations, as defined in VASP terminology, were performed with a basis set of plane wave truncated at a kinetic energy of 700eV. The Pulay stress[30] obtained on the unit cell was -4 MPa and the convergence of the total energy was within 0.4 meV/atom compared to an energy of 750 eV. Brillouin zone integrations were performed by using a 3X3X3 k-points Monkorst-Pack[31] grid leading to a convergence of the total energy within 0.1 meV/cell compared to a 7x7x3 k-point mesh. PHONON code[19], based on the harmonic approximation, as implemented within MEDEA[26] was used to calculate the phonon dispersion. From the optimized crystal structure, a 2X2X1 supercell consisting of 96 atoms, was generated from the conventional cell to account for an interaction range of about 10 Å. The asymmetric atoms were displaced by +/- 0.03 Å leading to 14 new structures. The dynamical matrix was obtained from the forces calculated via the Hellmann-Feynman theorem. G point and medium precision, as defined in VASP terminology, were used for theses calculations. The error on the force can perturb the translation-rotational invariance condition. Consequently, this condition has to be enforced. A strength of enforcement of the translational invariance condition was fixed at 0.1 during the derivation of all force constants. The longitudinal optical (LO) and transversal optical mode (TO) splitting was not investigated in this work. Consequently, only TO modes at the G point were obtained.</div><div align="justify"> </div><div align="justify"> </div><div align="justify"><strong>Results and discussion:</strong> </div><div align="justify"> </div><div align="justify"><strong>Structural parameters.</strong> </div><div align="justify"> </div><div align="justify"> </div><div align="justify">Shows calculated and previous experimental or numerical structural properties of LiYF4. Compared to the most recent experimental data[32][33] our calculated volume is over-estimated. Nevertheless, the c/a axial ratio, whose evolution is significant in pressure induced transition phase, is close to experimental results. DFT results are strongly dependent on the approximation of the exchange correlation term. It’s known that local density approximation (LDA) favorizes high electron densities resulting in short bonds prediction and so in low equilibrium volume. Results obtained by Li et al.[35] and Ching et al. [36] owing to LDA illustrate this behavior. GGA corrects and sometimes over-corrects the failures of the LDA. That’s why cell parameters obtained using PW91 differ from experimental results. At least two reasons explain why our results are at variance with Li et al.[35]. The first one is due to the utilization of different parameters such as the energy cut off. The other one can be attributed to the difference of method to evaluate the equilibrium volume. Indeed, during a structure optimization the convergence criterion is set on the stress.</div><div align="justify"><strong></strong> </div><div align="justify"><strong>Lattice dynamic.</strong> </div><div align="justify"> </div><div align="justify">The phonon dispersion curves along several lines of high symmetry for LiYF4 structure at zero pressure are shown in Figure 1. To evaluate our calculated phonon dispersion curves, the acoustic branches will be first compared to results extracted from ultrasonic measurements and rigid ion models (RIM). Then the modes at the G point will be compared to experimental results obtained from Raman, IR or neutron scattering and RIM. Velocities of sound following different directions of propagation have been evaluated from the slopes of acoustic branches. Our results and experimental ultrasonic velocities at 4.2 K are presented in Table 3. The difference between calculated and measured velocities lies within 5% for 7 velocities out of 8. The 9% of discrepancy is obtained for the acoustic branch following the [001] direction. In this direction the longitudinal acoustic branches are non-degenerate although the modes at the Z point are degenerate. This behavior has been observed on the two phonon dispersion curves calculated with RIM but seems absent from neutron scattering experiments. Concerning the phonon modes, the spectrum contains 36 phonons modes at the G point as expected from the number of atoms per primitive cell.</div><div align="justify"> </div><div align="justify"><strong></strong> </div><div align="justify"><strong></strong> </div><div align="justify"><strong>Conclusion </strong></div><div align="justify"> </div><div align="justify"> </div><div align="justify">This work presents at our knowledge, the first ab initio lattice dynamics calculation of fluoride scheelite. Concerning the phonon dispersion curves, satisfactory agreement with inelastic neutron scattering measurement was obtained. Discrepancies between sound velocities calculated from acoustic branches and ultrasonic measurement do not exceed 300m.s-1. Moreover, at the center of the Brillouin zone the error on Raman active modes calculated compared to experimental results does not exceed 9%, the most important error being 33 cm-1. One inversion between the last Bg mode and the fourth Eg mode was put in evidence in comparison with experimental results. Concerning infrared active modes error lies within 8%, the most important error being 25 cm-1. Below 360 cm-1, only one inversion can be notice compared to experimental results, which is less than in RIM calculations. Important differences between ab initio and RIM calculated DOS were put in evidence mainly above 500 cm-1. </div><div align="justify"> </div><div align="right"><a href="http://hal.archives-ouvertes.fr/docs/00/02/99/96/PDF/article2.pdf">http://hal.archives-ouvertes.fr/docs/00/02/99/96/PDF/article2.pdf</a></div>Orlaninghttp://www.blogger.com/profile/08859376411004440286noreply@blogger.com0tag:blogger.com,1999:blog-4407931247357581937.post-47036541515976063122010-05-21T08:28:00.000-07:002010-05-21T08:39:51.804-07:00Specific Heat Capacities (Continuacion)<div align="center"><strong>Historical</strong></div><div><br /><strong>(a) Classical</strong><br />Dulong and Petit (1819)<br />Cv=3Nk, Correct at high temperature</div><div><br /><strong>(b) Einstein</strong><br />Based on Planck’s quantum hypothesis (1901)<br />Quantised energy, Showed exponential dependence of Cv<br /><br /><strong>(c) Debye</strong><br />Showed complete dependence (1912)<br /></div><div><br /></div><div align="center"><a name="_Toc494527499"><strong>Debye Model</strong></a><br /></div><div><br /></div><div align="justify">In thermodynamics and solid state physics, the Debye model is a method developed by Peter Debye in 1912 for estimating the phonon contribution to the specific heat (heat capacity) in a solid. It treats the vibrations of the atomic lattice (heat) as phonons in a box, in contrast to the Einstein model, which treats the solid as many individual, non-interacting quantum harmonic oscillators. The Debye model correctly predicts the low temperature dependence of the heat capacity, which is proportional to T3 – the Debye T3 law. Just like the Einstein model, it also recovers the Dulong-Petit law at high temperatures. But due to simplifying assumptions, its accuracy suffers at intermediate temperatures.<br /><br /></div><div><br /></div><div align="justify"><br />· Uses wide spectrum of frequencies to describe the complicated pattern of lattice vibrations. [It is assumed that hypothetical oscillators generate simple sine waves throughout the crystal and these will displace the atoms away from their equilibrium positions by an amount equal to the amplitude of the sine wave at that point. If we have a whole set of such oscillators generating sine waves of certain frequencies and amplitudes then we might hope that the superposition of such waves will simulate the complicated pattern of the actual atomic vibrations.]<br />· Assume that the distribution of oscillators is quasi-continuous in w, and so we may use integration instead of summation. [This could not be done in the derivation of the mean energy of a single oscillator where the individual quantum steps might be large compared to kT.]<br />· Each frequency (mode) contributes an Einstein-like term to C.</div><div><br /></div><div align="justify"></div><div><br /></div><div align="justify">The Debye model is a solid-state equivalent of Planck's law of black body radiation, where one treats electromagnetic radiation as a gas of photons in a box. The Debye model treats atomic vibrations as phonons in a box (the box being the solid). Most of the calculation steps are identical.<br />Consider a cube of side L. From the particle in a box article, the resonating modes of the sonic disturbances inside the box (considering for now only those aligned with one axis) have wavelengths given by<br /><br />In order to go further there are two problems to solve:<br />· A density of states function is required<br />· Need to set the range of frequencies over which the integration is to be performed, i.e. the cut-off or limiting frequency needs to be determined<br />Neglecting the zero point energy, the mean thermal energy will be given by</div><div><img id="BLOGGER_PHOTO_ID_5473746141704381410" style="DISPLAY: block; MARGIN: 0px auto 10px; WIDTH: 145px; CURSOR: hand; HEIGHT: 37px; TEXT-ALIGN: center" alt="" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjcTKyzJB2nXqldnEDoPJtggOAThBRHIuY54X1DdKRaaIOD19NmOr9FvUp7UDKD5zvIFf5C8BVhFrFF6R7ePnZax14DjxDu_tKZMO7TtM-inZ2KqaFG7M0qTT6xhM7eL2xc037cY4dCWTE/s320/21.bmp" border="0" /></div><div>for each frequency. The Debye specific heat will take the form,<br /></div><div><br /></div><div align="justify"></div><img id="BLOGGER_PHOTO_ID_5473746563459936930" style="DISPLAY: block; MARGIN: 0px auto 10px; WIDTH: 320px; CURSOR: hand; HEIGHT: 279px; TEXT-ALIGN: center" alt="" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhiU-0rkuAWeUc-D2nxCYhDv-1dYhduAair1jNDVHBb6vb89Vi5rcFU2lCt24xGIYCNBeIHjPDWb8CsQvcIdqcG2uuhhcuixKFjDxlb6Tncob2N84-ZKoCEOH-mSmxejKU5vY9jqw9mraI/s320/22.bmp" border="0" /><br /><br /><img id="BLOGGER_PHOTO_ID_5473746767472877122" style="DISPLAY: block; MARGIN: 0px auto 10px; WIDTH: 205px; CURSOR: hand; HEIGHT: 54px; TEXT-ALIGN: center" alt="" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgqY_giNp75uiUUP8noCiOxNXcfQJo5N5UP-326Qk-xdq8D6bYG7Seg1JbTESNXflzel3U0tiYU790E5AvWGSuQWw_hnK1h0SLA6g73c5-cvh937tDqvAnF9TofLj2dj34UXwBVktqKfLs/s320/23.bmp" border="0" /><br /><img id="BLOGGER_PHOTO_ID_5473747745092567122" style="DISPLAY: block; MARGIN: 0px auto 10px; WIDTH: 230px; CURSOR: hand; HEIGHT: 320px; TEXT-ALIGN: center" alt="" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhPw6ia_lumJFIarzctxHSENF-kWdqjGUNbeh8Mhn9GrdAZbb4SPpc4vAETHDLFkE95CZxTsTVmfeBGsDwYHTKcSo-vHJN5SbDTYl1dWxlF79_9mMdqMbZljwKRi_0MksQo6UJPRudVP4s/s320/24.bmp" border="0" /><br /><div align="justify"><strong>Debye temperature table</strong></div><strong></strong><div align="justify"><br />Even though the Debye model is not completely correct, it gives a good approximation for the low temperature heat capacity of insulating, crystalline solids where other contributions (such as highly mobile conduction electrons) are negligible. For metals, the electron contribution to the heat is proportional to T, which at low temperatures dominates the Debye T3 result for lattice vibrations. In this case, the Debye model can only be said to approximate the lattice contribution to the specific heat.</div><div align="justify"> </div><div align="justify"><strong>Debye's Contribution to Specific Heat Theory</strong> </div><div align="justify"><br />Einstein's oscillator treatment of specific heat gave qualitative agreement with experiment and gave the correct high temperature limit (the Law of Dulong and Petit). The quantitative fit to experiment was improved by Debye's recognition that there was a maximum number of modes of vibration in a solid. He pictured the vibrations as standing wave modes in the crystal, similar to the electromagnetic modes in a cavity which successfully explained blackbody radiation. The density of states for these modes, which are called "phonons", is of the same form as the photon density of states in a cavity.<br />To impose a finite limit on the number of modes in the solid, Debye used a maximum allowed phonon frequency now called the Debye frequency uD.</div><div align="justify"> </div><div align="justify"> </div><div align="justify"> </div><div align="right"><a href="http://en.wikipedia.org/wiki/Debye_model">http://en.wikipedia.org/wiki/Debye_model</a></div><div align="right"><a href="http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/debye.html">http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/debye.html</a></div><div align="right"><a href="http://www.plato.ul.ie/academic/Vincent.Casey/PH4607SS1/LatticeDynamics.doc">www.plato.ul.ie/academic/Vincent.Casey/PH4607SS1/LatticeDynamics.doc</a></div><div align="justify"> </div>Orlaninghttp://www.blogger.com/profile/08859376411004440286noreply@blogger.com0tag:blogger.com,1999:blog-4407931247357581937.post-58353550114665679142010-05-21T07:38:00.000-07:002010-05-21T08:25:38.172-07:00Specific Heat Capacities<div align="center"><br /><strong>Historical</strong><strong></strong><br /></div><div align="justify"><br /><strong>(a) Classical</strong><br />Dulong and Petit (1819)<br />Cv=3Nk, Correct at high temperature</div><div align="justify"><br /></div><div align="justify"><br /><br /><strong>(b) Einstein</strong><br />Based on Planck’s quantum hypothesis (1901)<br />Quantised energy, Showed exponential dependence of Cv<br /></div><div align="justify"><br /></div><div align="justify"><br /><strong>(c) Debye</strong><br />Showed complete dependence (1912)</div><div align="justify"><br /></div><div align="justify"><br /><br /></div><div align="justify"><br /></div><div align="center"><a name="_Toc494527497"><strong>Classical Model</strong></a></div><div align="justify"><br />The classical model for specific heats considered the atoms as being simple harmonic oscillators vibrating about a mean position in the lattice. Each atom could be simulated by three simple harmonic oscillators (SHOs) vibrating in mutually perpendicular directions.<br />For a classical SHO:</div><div align="justify"><br />Average kinetic energy = ½ kT<br />Average potential energy = ½ kT<br />Total average energy per oscillator = kT<br />Total average energy per atom = 3kT<br />For N atoms the total average energy = U = 3NkT<br />The specific heat capacity is</div><p align="center">Cv=3*R</p><p align="justify"><br />Classical treatment - Dulong and Petit - the specific heat capacity of a given number of atoms of a given solid is independent of T and is the same for all solids.</p><div align="justify"><br /></div><div align="center"><br /><br /><a name="_Toc494527498"><strong>Einstein Model</strong></a></div><div align="justify"><br />So far, the treatment of the vibrational behaviour of materials has been entirely classical. For a harmonic solid, the vibrational excitations are the collective, independent normal modes, having frequencies w determined by the dispersion relationship w(k) with the allowed values of k set by the boundary conditions. In the classical limit, the energy of a given mode with frequency w, determined by the wave amplitude, can take any value.<br />· MB energy distribution: as T is raised F(Ehigh) increases!<br />· The energy of the atomic vibrations becomes greater as we go from low to high T</div><div align="justify"><br /></div><div align="justify"><br />Einstein produced a theory of heat capacity based upon Planck’s quantum hypothesis. He assumed that each atom of the solid vibrates about its equilibrium position with an angular frequency w. Each atom has the same frequency and vibrates independently of other atoms. The quantum mechanical result, treating each normal mode as an independent harmonic oscillator with frequency w, is that the energy is quantised and can only take values characterised by the quantum number n(k,p) for a particular branch p. A vibrational state of the whole crystal is thus specified by giving the excitation numbers n(k,p) for each of the 3N normal modes. Instead of describing the vibrational state of a crystal in terms of this number, it is more convenient and convenntional to say, equivalently, that there are n(k,p) phonons (i.e. particle like entities representing the quantised elastic waves).</div><div align="justify"><br /></div><div align="justify"><br />· Einstein replaced the classical SHO with a Quantum SHO: energy does not increase continuously but in discrete steps: 1D SHO,<br /></div><div align="justify"><br /></div><div align="center"><img id="BLOGGER_PHOTO_ID_5473740986316082946" style="DISPLAY: block; MARGIN: 0px auto 10px; WIDTH: 116px; CURSOR: hand; HEIGHT: 26px; TEXT-ALIGN: center" alt="" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj02LEOj6o8RUOiqZjkCJHCK9m-XT4UIt_0S2alLBmW-MWgFSm3jgdv4qIYObUi6hmcNFhAIchtqkcq1VhKaYupKiNtdj-mAeK5nUNBbLS5VMYJwh4EoXMJwsQCZ3vmCSXF9aPmzry8QP4/s320/17.bmp" border="0" />n=0,1,2…..</div><div align="justify"><br /> </div><div align="justify">· Note that the quantum mechanical expression for the energy implies that the vibrational energy of a solid is non zero even when there are no phonons present: the residual energy of a given mode, 0.5*h*w , is the zero point energy<br />· choose zero energy at 0.5 h * w<br />· take<br />The probability of occupation of this level is:<br /></div><div align="justify"><br /></div><img id="BLOGGER_PHOTO_ID_5473741878837461474" style="DISPLAY: block; MARGIN: 0px auto 10px; WIDTH: 116px; CURSOR: hand; HEIGHT: 81px; TEXT-ALIGN: center" alt="" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEinwaV7hDB_bvsodFRBOtmiJUW9zI0FwpHHNFli7MWDwAgIi6MnzhTtaVoFCLI5aVZHlUlzDky4DdpeU6i1L4_pDdI5islOiw_-t9IfoWtgxihPiZGGQU1hyphenhyphenXHHeugT2MJddZLQmyOINZs/s320/18.bmp" border="0" /><br /><div align="justify">The total energy of the solid becomes,<img id="BLOGGER_PHOTO_ID_5473742197679787634" style="DISPLAY: block; MARGIN: 0px auto 10px; WIDTH: 130px; CURSOR: hand; HEIGHT: 121px; TEXT-ALIGN: center" alt="" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgdHyf25RN4y5FPH0SvG2hShZdZeDmYwnt7a-ovWr71puyke5GF7DQY_2ZHF-eiLOfth86ODxFCJ4jJzg4Zd3aKFJlfCjBZ0M26rHpx2JDcT7IYWGR7bDyXbxgidmngWbrv-uMVg5189KM/s320/19.bmp" border="0" /></div><br /><p align="justify">Taking account of the zero point energy and using the above result, the mean energy is therefore,</p><p align="justify"></p><img id="BLOGGER_PHOTO_ID_5473743432858506978" style="DISPLAY: block; MARGIN: 0px auto 10px; WIDTH: 142px; CURSOR: hand; HEIGHT: 37px; TEXT-ALIGN: center" alt="" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiWuUZ2w8ueeudwe8pUZ4nCDhJWODON4j3qmap4k7d5x0pXt8XIwsTRK3S499dUGjjULwTFop-ELIVHTPouEbb89DAev49gL2Yfzwefx6agNqD7QQjnFrSBcivqFGUk-qp95NfAQaqKUqU/s320/20.bmp" border="0" />This energy may be considered either as the time-averaged energy for a particular atom, or it can be thought of as the average energy of all the atoms in the assembly at any instant in time.<br /><p align="justify"><br /><strong>Einstein model conlusion!</strong></p><strong></strong><p align="justify"><br />· successfully predicts that C falls with decreasing T<br />· however, exponential decrease is not observed; if low frequencies are present, then will be small, much smaller than kT even at low temperatures; C will remain at 3kT to much lower frequencies and the fall off is not as dramatic as predicted by the Einstein model<br />· assumption of ‘an average’ single frequency w is too simplistic<br />· need a spread of frequencies - a frequency spectrum!</p><p align="right"><a href="http://www.plato.ul.ie/academic/Vincent.Casey/PH4607SS1/LatticeDynamics.doc">www.plato.ul.ie/academic/Vincent.Casey/PH4607SS1/LatticeDynamics.doc</a></p><p align="right">xbeams.chem.yale.edu/~batista/vaa/node28.html </p><p align="right"> </p>Orlaninghttp://www.blogger.com/profile/08859376411004440286noreply@blogger.com0tag:blogger.com,1999:blog-4407931247357581937.post-40755726037313966392010-05-20T17:36:00.000-07:002010-05-20T17:48:44.350-07:00General Form<div align="center"><a name="_Toc494527495"><strong>General Form of the Density of States</strong></a></div><br /><br /><div align="justify"><br />The diagram below shows a portion of two adjacent surfaces of constant frequency corresponding to frequencies w and w+dw. Owing to dispersion these surfaces are not normal to the direction of the wavevector k. Consider a small pill box bounded by the surfaces w and w+dw centred around the point k. The unit vector normal to the frequency surfaces is n. The pill box has area dS n.<br />The number of allowed values of k for which the phonon frequency is between w and w+dw is<br /></div><br /><br /><br /><br /><p align="center"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEghW6L7n2a5dMEpAguMn0K68JfRoSi2XW4yn-oPNAcIX5Qn5zZbWQQvcPchEjxQSEvY34qsFfte1umxfHArTdQXFDTFsDgP08G1WKB2BNd3xu3PaFj-HRRj_zeQatrZNsp8aMeox83qZQc/s1600/12.bmp"><img id="BLOGGER_PHOTO_ID_5473516392738350866" style="WIDTH: 143px; CURSOR: hand; HEIGHT: 78px" alt="" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEghW6L7n2a5dMEpAguMn0K68JfRoSi2XW4yn-oPNAcIX5Qn5zZbWQQvcPchEjxQSEvY34qsFfte1umxfHArTdQXFDTFsDgP08G1WKB2BNd3xu3PaFj-HRRj_zeQatrZNsp8aMeox83qZQc/s320/12.bmp" border="0" /></a><br /></p><br /><br /><p align="justify">= Volume of k space/volume occupied by one state (mode)<br /></p><br /><br /><p align="center"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhZ3SBEDdC67vH_E1TgZwuCS7Q-on3qdyDUw5iUAoy1ZR1BJ1xv4EFLHHPqGVnyaXhejs4zJ9d29LXiFslv693HHknHS4bccxDyTG1ThUVRBn4efSMwEpndj9O42vQWKSNkZhGRE0G32yI/s1600/13.bmp"><img id="BLOGGER_PHOTO_ID_5473516748008577266" style="WIDTH: 162px; CURSOR: hand; HEIGHT: 54px" alt="" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhZ3SBEDdC67vH_E1TgZwuCS7Q-on3qdyDUw5iUAoy1ZR1BJ1xv4EFLHHPqGVnyaXhejs4zJ9d29LXiFslv693HHknHS4bccxDyTG1ThUVRBn4efSMwEpndj9O42vQWKSNkZhGRE0G32yI/s320/13.bmp" border="0" /></a><br /></p><p align="justify">The integral is extended over the volume of the shell in k space bounded by the two </p><br /><br /><br /><br /><br /><br /><p align="center"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgE0JIH-3luazQT966moxAS5V_hV5t-p8vqAFfLDVnGh3ZvictYAu8mbn1y1bqxX65Te9QehX9AB_AWCzd5nCsD1SEDAIQuOru32noOpwxrat5DRTS_6i4uLpohsRFRlY1lrvcqI8feawI/s1600/14.bmp"><img id="BLOGGER_PHOTO_ID_5473517057293124770" style="WIDTH: 274px; CURSOR: hand; HEIGHT: 191px" alt="" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgE0JIH-3luazQT966moxAS5V_hV5t-p8vqAFfLDVnGh3ZvictYAu8mbn1y1bqxX65Te9QehX9AB_AWCzd5nCsD1SEDAIQuOru32noOpwxrat5DRTS_6i4uLpohsRFRlY1lrvcqI8feawI/s320/14.bmp" border="0" /></a><br /></p><p align="center">Surfaces of equal w in k space.</p><br /><br /><p align="justify">surfaces on which the phonon frequency is constant, one surface on which the frequency is w and the other on which the frequency is w+dw. This is straight forward where k is small, i.e. no dispersion, since the constant frequency surfaces will be spherical. However, for the general situation where k may be large, one has to deal with a much more complicated shape. The problem for us is to evaluate the volume between these surfaces. Owing to dispersion these surfaces are not normal to the direction of the wave vector k.</p><br /><br /><p align="justify"></p><p align="center"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhe_lNR8LlerpGrjsjkKoJH9h1Hxh4RxatlopRnyXi7JLOOoxMd_-0qsHGRsMNt3jFlGSDnGI1nKEOgpKlDogxLt-lEvVfd5SreLx3mHdV2CZq_G_i_GJlKdYgFPOMs97GhEhyphenhyphenDT96OTls/s1600/15.bmp"><img id="BLOGGER_PHOTO_ID_5473518439311324354" style="WIDTH: 164px; CURSOR: hand; HEIGHT: 143px" alt="" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhe_lNR8LlerpGrjsjkKoJH9h1Hxh4RxatlopRnyXi7JLOOoxMd_-0qsHGRsMNt3jFlGSDnGI1nKEOgpKlDogxLt-lEvVfd5SreLx3mHdV2CZq_G_i_GJlKdYgFPOMs97GhEhyphenhyphenDT96OTls/s320/15.bmp" border="0" /></a></p><div align="right">www.plato.ul.ie/academic/Vincent.Casey/PH4607SS1/LatticeDynamics.doc<br /></div><p align="justify"></p>Orlaninghttp://www.blogger.com/profile/08859376411004440286noreply@blogger.com0tag:blogger.com,1999:blog-4407931247357581937.post-16101819964120088032010-05-20T16:32:00.000-07:002010-05-20T17:03:14.516-07:00Vibrations, Linear 1D Lattice<div align="center"><a name="_Toc494527493"><strong>Vibrations of periodic systems</strong></a><br /></div><div align="justify">The presence of translational periodicity has a profound effect on the vibrational behaviour when the wavelength of the vibrational excitations becomes comparable to the periodic repeat distance, a. For l>>a, however, the behaviour characteristic of an elastic continuum is recovered.<br />For a periodic array of atoms of length L, periodic (or Born-von Karman) boundary conditions are appropriate, i.e. </div><p align="center"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi2OwvvfSHtBUsvHsejfDuoG49qQ18WIlC_5H3Bf6cdjc7gyF8zD4REARd0zGJEygFTjeJz8gyIg2QWfaJNP6uLRPxlxco-B1v8Xv0_UFRfXzYkywycOZQwA9Wdrq1c_zvNWFb9QURMOaI/s1600/4.bmp"><img id="BLOGGER_PHOTO_ID_5473500384334761442" style="WIDTH: 102px; CURSOR: hand; HEIGHT: 27px" alt="" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi2OwvvfSHtBUsvHsejfDuoG49qQ18WIlC_5H3Bf6cdjc7gyF8zD4REARd0zGJEygFTjeJz8gyIg2QWfaJNP6uLRPxlxco-B1v8Xv0_UFRfXzYkywycOZQwA9Wdrq1c_zvNWFb9QURMOaI/s320/4.bmp" border="0" /></a><br /></p><p align="justify">Such a boundary condition can be envisged as follows. In the case of a linear chain of N particles, where the nearest neighbours are connected by springs (representing bonds between atoms), with equilibrium spacing a, periodic boundary conditions are achieved by connecting one end of the chain to the other to form a ring of length L=Na, Figure 4a. An integral number of wavelengths must fit into the length L, resulting in allowed K-values for running-wave (travelling wave) states: </p><br /><br /><p align="center"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEha7pXjlusBPDvae-3glj5RC4Rjab4jpoLvNFOCN8hpox4PBctO_95bvppqkDknt1FGPoy6cLbHfpSSvWfG_vaKsew2Z0elVL2Ab6Y4KGL7KjsMCFJpBHMN6IdeUaJWq7HjEWvI30wwOcs/s1600/5.bmp"><img id="BLOGGER_PHOTO_ID_5473500619812598546" style="WIDTH: 160px; CURSOR: hand; HEIGHT: 94px" alt="" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEha7pXjlusBPDvae-3glj5RC4Rjab4jpoLvNFOCN8hpox4PBctO_95bvppqkDknt1FGPoy6cLbHfpSSvWfG_vaKsew2Z0elVL2Ab6Y4KGL7KjsMCFJpBHMN6IdeUaJWq7HjEWvI30wwOcs/s320/5.bmp" border="0" /></a><br /></p><br /><br /><div align="justify">An equivalent and more realistic way of understanding periodic boundary conditions involves the imposition of a mechanical constraint forcing atom N to interact with atom 1 via a massless, rigid rod and a spring, Figure 4b.<br />In contrast to the case for fixed boundary conditions leading to stationary waves, both positive and negative integers are allowed for running wave solutions, and moreover the spacing between allowed k-values is Dkr=2p/L, twice that for standing wave states. Therefore, the number of k-values, corresponding to running wave states, contained in unit volume of k-space is now<br /></div><p align="center"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh1tYeGs9PlUN_CCfF-qfnpj8Gtkvr5qTR35ahDKIvVdHznvAF_Hm691cGTaTMzDsvKtSelA478TbJGb_iYx0jcd8FwUySOHWjl3DCnlUgbWZDBpeRCbU50xtruchF4lq2d_hQDoW10pdQ/s1600/6.bmp"><img id="BLOGGER_PHOTO_ID_5473501651289425618" style="WIDTH: 78px; CURSOR: hand; HEIGHT: 42px" alt="" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh1tYeGs9PlUN_CCfF-qfnpj8Gtkvr5qTR35ahDKIvVdHznvAF_Hm691cGTaTMzDsvKtSelA478TbJGb_iYx0jcd8FwUySOHWjl3DCnlUgbWZDBpeRCbU50xtruchF4lq2d_hQDoW10pdQ/s320/6.bmp" border="0" /></a><br /></p><br /><br /><div align="justify">The number of distinct states, for a given polarisation type i, having wavevectors between k and k + dk is this density multiplied by the volume of an entire spherical shell in k-space (since both positive and negative k-values are allowed)<br /></div><p align="center"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgDgdsNYam3lbfvTa1aOHPEHA43z9FuPCmj-muj4vcfny_o-MCCNFsiFdY9mUUSXtvustXrJMql-fyi1Lml9LoFkqDb-pMABD5joyqfoMU3nWqNcrqMVLOLI9EUIBk9HOTYUArTzmNihzo/s1600/7.bmp"><img id="BLOGGER_PHOTO_ID_5473502273120258386" style="WIDTH: 320px; CURSOR: hand; HEIGHT: 206px" alt="" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgDgdsNYam3lbfvTa1aOHPEHA43z9FuPCmj-muj4vcfny_o-MCCNFsiFdY9mUUSXtvustXrJMql-fyi1Lml9LoFkqDb-pMABD5joyqfoMU3nWqNcrqMVLOLI9EUIBk9HOTYUArTzmNihzo/s320/7.bmp" border="0" /></a></p><br />A linear chain connected to form a ring of length L=8a. For modes of the form us ~exp (iska), periodic boundary conditions lead to eight modes (one per atom) with k=0, ±2pi/L, ±4pi/L, ±6pi/L, ±8pi/L.<br /><br /><br /><br /><br /><br /><div align="center"><a name="_Toc494527494"><strong>Linear 1D Lattice</strong></a></div><br />· All atoms identical (mass m)<br />· Lattice spacing ‘a’<br /><br /><br /><br /><p align="center"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg-hB9ClorQOrvz0Un1sh58-Jt0f-Iuq70kBAYNMx2VCoWiEXtsGMavnDKvYWXKIhgUlrWL5cnwc64y737ICMuEC0Urr1Lpw5TaNoHhn_7yecMU5eLOiP70Q7tiiyKR24UfGr_oySihtzk/s1600/8.bmp"><img id="BLOGGER_PHOTO_ID_5473503464152681954" style="WIDTH: 320px; CURSOR: hand; HEIGHT: 117px" alt="" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg-hB9ClorQOrvz0Un1sh58-Jt0f-Iuq70kBAYNMx2VCoWiEXtsGMavnDKvYWXKIhgUlrWL5cnwc64y737ICMuEC0Urr1Lpw5TaNoHhn_7yecMU5eLOiP70Q7tiiyKR24UfGr_oySihtzk/s320/8.bmp" border="0" /></a></p><br />For small vibrations, the force on any one atom is proportional to its displacement relative to all the other atoms.<br /><br /><p align="justify">Choose atom s</p><p align="center"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhuDsy_JSvj0ZpSLQ1M8Irwz6H9kSGWClKtHGes5x3YDre6W1VCletkeOFBdpGJxUZAOLGZlVZ_t6pJjew5gc7DapSMR1TN3dfwn2QXxTCtCu1uxYrbEDKR0l3wSKpFXiYJJE_KfpiA0SQ/s1600/9.bmp"><img id="BLOGGER_PHOTO_ID_5473504557514564786" style="WIDTH: 143px; CURSOR: hand; HEIGHT: 44px" alt="" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhuDsy_JSvj0ZpSLQ1M8Irwz6H9kSGWClKtHGes5x3YDre6W1VCletkeOFBdpGJxUZAOLGZlVZ_t6pJjew5gc7DapSMR1TN3dfwn2QXxTCtCu1uxYrbEDKR0l3wSKpFXiYJJE_KfpiA0SQ/s320/9.bmp" border="0" /></a><br /></p><div align="left">· p takes on both positive and negative values<br />· c is the force constant and depends on p, i.e. is large for p=1, smaller for p=2, etc..<br /></div><p align="center"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiyV4QkAsupT2AKeKYDCAKuFSSGSPf6B717ahzhTSTrYCluY8CsNrOaC0pL56dlAaRfjhpTJ-_5_NL5oAkIs44YKzhmj8OUCEwCLouNacvP4VTMdQMEg8bywMniPIIG_UAhwuBaSHspBCU/s1600/10.bmp"><img id="BLOGGER_PHOTO_ID_5473505556015342690" style="WIDTH: 242px; CURSOR: hand; HEIGHT: 50px" alt="" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiyV4QkAsupT2AKeKYDCAKuFSSGSPf6B717ahzhTSTrYCluY8CsNrOaC0pL56dlAaRfjhpTJ-_5_NL5oAkIs44YKzhmj8OUCEwCLouNacvP4VTMdQMEg8bywMniPIIG_UAhwuBaSHspBCU/s320/10.bmp" border="0" /></a></p><br /><br /><div align="justify">The displacement for k’ is therefore the same as for k. k’ consists of a wave of smaller wavelength than that corresponding to k, passing through all the atoms, but containing more oscillations than needed for the description.<br />We can describe the displacement of the atoms in these vibrations most easily by looking at the limiting cases k = 0 and k = p/a.<br />· the situation at k = 0 corresponds to an infinite wavelength; this means that all of the atoms of the lattice are displaced in the same direction from their rest position by the same displacement magnitude. For long wavelength vibrations neighbouring atoms are displaced by the same amount in the same direction. Since the long-wavelength longitudinal vibrations correspond to sound waves in the crystal, all of these vibrations with a similarly shaped dispersion curve, whether transverse or longitudinal vibrations are involved, are called acoustical branches of the vibration spectrum. [When k @ 0, dw/dk = w/k = velocity of sound]<br /></div><br /><br /><div align="justify"></div><br /><br /><div align="justify">In addition to longitudinal vibrations, the linear lattice supports transverse displacements leading to two independent sets (in mutually perpendicular planes) of vibrations that can propagate along the lattice. The forces acting in a transverse displacement are weaker<br /></div><br /><br /><p align="center"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgIRCy2Oo-wCCUsO8wAguFvNICPiBskcMVOH0D2bPRZJOhIMjU-BBXWj4ym19uxrS2Cwx8XCkDLLlvc__QWBgKxRkdeuB87aUw9ywEvJjtipBMRFuSJe_ndadOy6kVo3hGXcRod5lsk1SI/s1600/11.bmp"><img id="BLOGGER_PHOTO_ID_5473506118883788802" style="WIDTH: 224px; CURSOR: hand; HEIGHT: 231px" alt="" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgIRCy2Oo-wCCUsO8wAguFvNICPiBskcMVOH0D2bPRZJOhIMjU-BBXWj4ym19uxrS2Cwx8XCkDLLlvc__QWBgKxRkdeuB87aUw9ywEvJjtipBMRFuSJe_ndadOy6kVo3hGXcRod5lsk1SI/s320/11.bmp" border="0" /></a><br />Longitudinal and transverse modes for a monatomic lattice</p><p align="center"> </p><p align="justify">than those in a longitudinal one. They give rise to a new branch of dispersive modes lying below the longitudinal branch.</p><p align="right">www.plato.ul.ie/academic/Vincent.Casey/PH4607SS1/LatticeDynamics.doc</p>Orlaninghttp://www.blogger.com/profile/08859376411004440286noreply@blogger.com0tag:blogger.com,1999:blog-4407931247357581937.post-74682462294426873862010-05-20T11:47:00.000-07:002010-05-20T12:18:49.838-07:00Lattice Dynamics and Specific Heats<div align="center"><a name="_Toc494527488"><strong>Introduction</strong></a></div><br /><br /><div align="justify"><br />The static lattice model which is only concerned with the average positions of atoms and neglects their motions can explain a large number of material features such as:</div><br />· chemical properties;<br /><br />· material hardness;<br /><br />· shapes of crystals;<br /><br />· optical properties;<br /><br />· Bragg scattering of R-ray, electron and neutron beams;<br /><div align="justify"><br />· electronics structure as well as electrical properties.<br /><br />There are, however, a number of properties that cannot be explained by a static model. These include: </div><div align="justify"><br />· thermal properties such as heat capacity; </div><div align="justify"><br />· effects of temperature on the lattice, e.g. thermal expansion;</div><div align="justify"> </div><div align="justify">· the existence of phase transitions, including melting; </div><div align="justify"><br />· transport properties, e.g. thermal conductivity, sound propagation; </div><div align="justify"><br />· the existence of fluctuations, e.g. the temperature factor;</div><div align="justify"><br />· certain electrical properties, e.g. superconductivity;</div><div align="justify"><br />· dielectric phenomenon at low frequencies; </div><div align="justify"><br />· interaction of radiation with matter, e.g. light and thermal neutrons.</div><br /><br /><div align="justify">Are the atomic motions that are revealed by these factors random, or can we find a good description for the dynamics of the crystal lattice? The answer is that the motions are not random but are constrained and determined by the forces that atoms exert on each other.</div><div align="justify"><br /></div><br /><br /><div align="center"><a name="_Toc494527489"><strong>Vibrations in solids.</strong></a></div><br /><br /><div align="justify"><br />We will examine the vibrational behaviour of atoms in solids. The vibrations are thermally activated with a characteristic activation energy kBT.<br />Vibrational excitations are collective modes: all atoms in the material take part in the vibrational mode. The influence of translational periodicity characteristics of the structure of crystals has a dramatic effect on the vibrational behaviour when the wavelength of the vibrations becomes comparable to the size of the unit cell. When the vibration wavelength is much larger than the structural variation of the material, the solid may be considered as an elastic continuum (continuum approximation).</div><div align="justify"><br /></div><div align="center"><br /><a name="_Toc494527490"><strong>Representation</strong></a><br /></div><br /><br /><div align="justify">How can we visualise a vibration wave travelling through a crystal, where the space that vibrates is not continuous (like a string on a musical instrument) but is composed of discrete atoms? The answer is to think of the wave as representing displacements, u(x,t), of the atoms from their equilibrium position.</div><br /><br /><br /><br /><div align="center"><a name="_Toc494527491"><strong>Continuous Media and Sound Waves</strong></a></div><br /><br /><div align="justify"><br />A sound wave is simply an elastic wave travelling in a medium. For a material regarded as an elastic continuum, the sound velocity is then directly related to the elastic modulus of the material. The sound produces a spatially varying stress s which in turn causes an instantaneous displacement u. If the sound is propagating in the x direction within a cube of material of mass density r, the net force acting on the volume element is:<br /></div><br /><br /><br /><br /><p align="center"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgygXEPPeIeCBz2LpCPM3-a8oaHLAvTeY85sWvLDRhRtEFxDtxIRc6pCLwzhKKsdVZkxoPim5FLG99RhsNB5YmQh8x3m6jAR7zEskWDBkA9CQTzbgpWdrvpEjWmfU_CGZ7uFckI9lKbeyE/s1600/1.bmp"><img id="BLOGGER_PHOTO_ID_5473431696138349970" style="WIDTH: 272px; CURSOR: hand; HEIGHT: 150px" alt="" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgygXEPPeIeCBz2LpCPM3-a8oaHLAvTeY85sWvLDRhRtEFxDtxIRc6pCLwzhKKsdVZkxoPim5FLG99RhsNB5YmQh8x3m6jAR7zEskWDBkA9CQTzbgpWdrvpEjWmfU_CGZ7uFckI9lKbeyE/s320/1.bmp" border="0" /></a><br /></p><div align="justify">is a measure of the velocity of a wave packet, composed of a group of plane waves, and having a narrow spread of frequencies about some mean value, w. For acoustic waves with long wavelengths (k » 0), i.e. in the elastic continuum limit, the phase and group velocities are equal. In a liquid, only longitudinal vibrations (modes) are supported (shear modulus is zero). The situation is more complicated in solids , where more than one elastic modulus is non-zero. As a consequence, both longitudinal and transverse acoustic modes exist even in isotropic solids, having in general different sound velocities. The situation is even more complicated for anisotropic crystals.<br /></div><br /><br /><div align="center"><a name="_Toc494527492"><strong>Counting vibrational states</strong></a></div><br /><br /><div align="justify"><br />The wavevector k characterises the vibrational wave. In the general solution to the wave equation, Eqn. 5, all k values are allowed. Restrictions on the allowed values of k appear through the imposition of boundary conditions. Two types of boundary conditions can be envisaged, depending on whether standing waves or propagating waves are involved.<br />For standing waves, and considering a cube of material of side L, the appropriate boundary condition for vibrational waves reflected from mechanically free surfaces is that an antinode of the vibration amplitude should exist at each surface. This corresponds to there being an integral number of half-wavelengths of the standing wave along the length of the cube. The allowed values of the standing wave vectors are given by<br /></div><div align="justify"></div><br /><br /><p align="center"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiQ6TawGbmb0h9-xYyOOajgQMYJVmka9Cbc2LI71wRSeIr_8mE-0114X3YVfbSfWaNIQLCLfozLvUlGhfgRg4fl6Jp4bjCKoeaAMFqX4BZlCidy3NyPxuZhVsTpfUnSSFJLikbNNB_FEWY/s1600/2.bmp"><img id="BLOGGER_PHOTO_ID_5473432309698561474" style="WIDTH: 273px; CURSOR: hand; HEIGHT: 291px" alt="" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiQ6TawGbmb0h9-xYyOOajgQMYJVmka9Cbc2LI71wRSeIr_8mE-0114X3YVfbSfWaNIQLCLfozLvUlGhfgRg4fl6Jp4bjCKoeaAMFqX4BZlCidy3NyPxuZhVsTpfUnSSFJLikbNNB_FEWY/s320/2.bmp" border="0" /></a></p><p align="center"><br /></p><p align="center"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjUEikXUREuK6pKE5kFZLteXZgtJxhYyQJpwOn6OYNCvsgCdMn8Zsf2SSRxgAhw3DKw4ZqjkeuE2vQxEdTKIESKfDiYIuEmV588FsV-nSzBLYx3RtGYR7PVRPpzPlte8OtpHd9rR4D7bXA/s1600/3.bmp"><img id="BLOGGER_PHOTO_ID_5473432716752295218" style="WIDTH: 163px; CURSOR: hand; HEIGHT: 165px" alt="" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjUEikXUREuK6pKE5kFZLteXZgtJxhYyQJpwOn6OYNCvsgCdMn8Zsf2SSRxgAhw3DKw4ZqjkeuE2vQxEdTKIESKfDiYIuEmV588FsV-nSzBLYx3RtGYR7PVRPpzPlte8OtpHd9rR4D7bXA/s320/3.bmp" border="0" /></a></p><div align="justify">Schematic 1D illustration of a standing wave set up between the free surfaces of a cube of an elastic continuum with antinodes at the free surfaces.<br />Each allowed standing-wave solution of the wave equation consistent with the boundary conditions is represented by a point in the reciprocal space containing the k-vectors. The spacing between allowed k-values is Dks=p/L, and so the volume of k-space corresponding to the one k-value (standing-wave state).</div><br /><br /><br /><br /><br /><br /><br /><br /><div align="right">www.plato.ul.ie/academic/Vincent.Casey/PH4607SS1/LatticeDynamics.doc</div>Orlaninghttp://www.blogger.com/profile/08859376411004440286noreply@blogger.com0