Algunos terminos

Quantum Mechanics
Schrodinger wave equation: One dimensional problem, particle in a box, tunnelling through a potential barrier, linear harmonic oscillator, K-P model; Particle in a central potendial: Hydrogen atom; WKB approximation method; Perturbation theory for degenerate & non-degenerate cases: First and second order perturbation, applications-Zeeman effect & Stark effect; Time dependent perturbation theory; Variation method: Application to He atom & van der Waals interaction between two hydrogen atoms; Pauli spin materices; Dirac equation: System of identical particles; many electron system-Hatree & Hatree-Fock approximation.

Advanced Quantum Mechanics
Radiation Theory: Quantization of Schrodinger field, scattering in Born approximation, quantization of classical radiation field, Emission probability of photon, angular distribution of radiation, intensities of Lyman lines, Compton effect and Bremstrahlung.
Path Integral: Approach to quantum mechanics, the principle of least action, quantum mechanical amplitude, path integrals, the path integral as function, the Schrodinger equation for a particle ina field of potential V(x), the Schrodinger equation for the keruel.

Physics of Deformed Solids
Theory of matter transport by defect mechanism: Random walk theory and correlation effects in metals and alloys for impurity and self-diffusion: Theory of ionic transport process, impurity defect association, long range interactions, dielectric loss due to defect dipoles, Internal friction, Radiation damage in metals and semiconductors, colour centres: mechanism of production by various methods, Optical and magnetic properties and models of different colour centre; Theoretical calculation of atomic displacement and energies in defect lattices and amorphous solids, stress-strain and dislocations; Elasticity theory of strees field around edge and screw dislocations, Dislocation interactions and reactions effects on mechanical properties.

X-ray Crystallography

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

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

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

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

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

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

Magnetism-I
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.

Magnetism-II
Magnetic anisotropy: pair model and one ion model of magnetic anisotropy, Phenomenology of magnetostriction, volume amgnetostricition and form effect; Law of approach of saturation, Structure of domain Wall, Technological applications of magnetic materials.

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

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

Time-resolved X-ray diffraction on laser-excited metal nanoparticles

Abstract

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.

Introduction

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

Experimental

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

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

Results and discussion

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

Quantum Mechanics Predicts Unusual Lattice Dynamics Of Vanadium Metal Under Pressure

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

In present paper in PNAS, again a collaboration between Uppsala University and Carnegie's Geophysical Laboratory, USA, we have looked in to the lattice dynamics of vanadium metal and it shows a very unusual behavior under pressure.
A huge change in the electronic structure is driving force behind this unusual lattice dynamics. Moreover, our findings provide a new explanation for the continuous rising of superconducting temperature in high-pressure vanadium, and could lead us to the next breakthrough in superconducting materials.

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

Referencias Bibliograficas:

www.physorg.com/news111321025.html
http://www.sciencedaily.com/releases/2007/10/071011125338.htm

International Journal of Solids and Structures, submitted for publicationAtomistic Viewpoint of the Applicability of Micrcontinuum Theories

Abstract

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.


Introduction

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

The fundamental departure of microcontinuum theories from the classical continuum theories is that the former is a continuum model embedded with microstructures for the purpose to describe the microscopic motion or a nonlocal model to describe the long-range material interaction, so as to extend the application of continuum model to microscopic space and short time scales. Among them, Micromophic Theory (Eringen and Suhubi [1964], Eringen [1999]) treats a material body as a continuous collection of a large number of deformable particles, each particle possessing finite size and inner structure. Upon some assumptions such as infinitesimal deformation and slow motion, micromorphic theory can be reduced to Mindlin’s Microstructure Theory [1964]. When the microstructure of the material is considered as rigid, it becomes the Micropolar Theory (Eringen and Suhubi [1964]). Assuming a constant microinertia, micropolar theory is identical to the Cosserat Theory [1902]. Eliminating the distinction of macromotion of the particle and the micromotion of its inner structure, it results Couple Stress theory (Mindlin and Tiersten [1962], Toupin [1962]). When the particle reduces to the mass point, all the theories reduce to the classical or ordinary continuum mechanics.

The physical world is composed of atoms moving under the influence of their mutual interaction forces. These interactions at microscopic scale are the physical origin of a lot of macroscopic phenomena. Atomistic investigation would therefore help to identify macroscopic quantities and their correlations, and enhance our understanding of various physical theories. This paper aims to analyze the applicability of those well-established microcontinuum theories from atomistic viewpoint of lattice dynamics and molecular dynamics.

Applicability Analyses from the Viewpoint of Dynamics of Atoms in Crystal

Dynamics of Atoms in Crystal

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

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

Optical Phonons

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

Micropolar theory (Eringen and Suhubi [1964])

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

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

Referencia Bibliografica:

www.seas.gwu.edu/~jdlee/index_files/ijss-applicability.doc

Algorithms for dynamical fermions -- Hybrid Monte Carlo

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

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

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

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

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

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

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

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

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

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

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

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

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

Applying this to a symmetric product, one finds

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

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

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

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

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

Referencia Bibliografica:

http://latticeqcd.blogspot.com/2007/12/algorithms-for-dynamical-fermions.html

Applicability Analyses from the Viewpoint of Dynamics of Atoms in Crystal

Dynamics of Atoms in Crystal

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

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

.

Fig.1 Typical motions for two atoms in a unit cell,

where ‘L’ stands for longitudinal, ‘T’ transverse, ‘A’ acoustic, ‘O’ optical

Optical Phonons

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

Dynamic Feature of Various Types of Crystals

The dynamic characteristics of crystals depend on crystal structures, as shown in Fig.2, and the binding between the atoms. 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. 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. 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.

Phonon Dispersion Relations by Various Microcontinuum Theory

Micromorphic theory (Eringen and Suhubi [1964], Eringen [1999])

Micromophic theory 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).

Micropolar theory (Eringen and Suhubi [1964])

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

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

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.

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.

Applicability Analysis of Continuum Theories from the Viewpoint of Molecular Dynamics

Micromorphic theory

Atomistic flow mechanisms make it possible to define the fluxes as sums of one- and two-atom contributions. The internal energy e, heat flux q, and the three stress tensors, namely, Cauchy stress t, microstress average s, and moment stress m, are composed of kinetic part and potential part. With the definition of temperature, it is seen that the kinetic parts of t, s, m, q, and e 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. 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.

Cosserat theory

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. Hence, compared with micropolar theory, Cosserat theory is not suited for problems involve the significant change of the orientation of the microstructure.

Nonlocal Theory

For each atom (k, a), 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.

Couple stress theory

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.

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 and hence s = t and the moment stress m = 0. The higher order stress, m, 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.

REFERENCIAS:
www.seas.gwu.edu/~jdlee/index_files/ijss-applicability.doc