Algorithms for dynamical fermions -- Hybrid Monte Carlo

 

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

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

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

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

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

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

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

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

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

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

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

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

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

Applying this to a symmetric product, one finds

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

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

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

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

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

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Referencia Bibliografica:

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

Lattice dynamics (of KI perturbed and Thermodynamics)

 

Of KI perturbed, The theoretical study of the physical properties of JT centres (impurities, defects, etc. )embedded in a crystal is usually done in the framework of the cluster model [I]. In such a model the localized electron of the centre, which is in a degenerate state by definition of the JT effect, is assumed to interact only with a single localized vibrational mode, any lattice dispersion of frequency being neglected. This model is known to take into account several important features of the JT induced properties [I] but not the details of the optical spectra (Raman, infrared, vibronic sidebands of the absorption/emission spectra), which are very sensitive to the dynamics of the host crystal.

In a recent paper [2] the problem of the lattice dynamics perturbed by a JT centre was studied and resolved in the weak interaction limit by using the Green function method. It was found that a JT centre generates a very peculiar perturbation on the dynamical matrix.

On the other hand, a similar result can be obtained when the absorption lineshape due to a singletmultiplet transition (the dynamic JT effect) is studied. In this case the JT perturbed lattice dynamics is that relative to the excited electronic degenerate state. In both cases we conclude that the JT interaction induces a perturbation on the force constant matrix, which can be eventually seen in the structures of the phonon densities. Such a result, which will be presented in details in a forthcoming paper, can be deduced in the limit of the weak interaction from the absorption lineshape theory proposed some years ago (ref. [3] hereafter called MT).
In the present paper, after some consideration on the previous point, we present the computed one-phonon spectra for a (e x E) JT centre. As usual, (e x E) means that the electron is in a e-symmetry degenerate state and interacts with the E-symmetry displacements of the neighbour ions. Our aim is to see to what extent the JT dynamical perturbation can qualitatively modify the host lattice dynamics. So we do not apply it to any real impurity.
In the following we assume : 1) The JT centre is embedded in an ionic crystal. 2) The JT interaction is weak. 3) Only the n.n. of the JT centre are involved in the JT interaction.

The shape of the optical spectra is related to the symmetrized (E-symmetry in this case) projected phonon density p,(w) relative to the lattice dynamics, either of the ground electronic state (Raman and infrared spectra), or of the excited electronic state (vibronic sidebands). We remember that p,(o) is given by the imaginary part of the E-symmetry phonon propagator DE(a). So we first show the relevant JT induced processes which dress the unperturbed onephonon
propagator, and then we evaluated p,(o) from D,(o). By unperturbed propagator we mean the onephonon propagator when all the dynamical perturbations induced by the centre (change of mass and of force constants) are considered, but the JT interaction.

In the cluster model, where one assumes that only the n.n. of the centre are involved in HL, as they are in He,, only two degenerate E-symmetry modes with the same frequency are representative of the whole lattice dynamics. So the commutator (6) loses its meaning. From the properties of the commutators (4) and (5) one can see that, while the electron and the
phonons are truly independent when He, = 0, they are so entangled for HeL # 0 to determined new coupled states, the so-called vibronic states. When the vibronic states are studied in the cluster model, the role played by HeL on the electronic states is emphasized but the possible modifications of HL are neglected [I].

The opposite approach is here follow ed. In fact we compute the modifications of the one-phonon projected density of states pE(02), when the electron is supposed to be degenerate but structureless. Without the JT interaction, the unperturbed phonon propagator D;(t) (a dotted line in the diagram of figure 1) is given at T = 0 K by

where p;(02) is the unperturbed projected one-phonon density. (By &(w2) we mean its Hilbert transform.) In the harmonic approximation a phonon does not change frequency when a linear electron-phonon interaction is switched on and the only effect is a generation of many unperturbed phonons. When the electron is in a degenerate state, since the usual separation of the phonons from the electron (adiabatic approximation) is not valid anymore, the linear JT interaction modifies the frequency of the phonons. By following MT [3], one can prove that in the diagrams accounting for the many-phonon process there is always a class of higher order graphs (the crossing graphs) which give a contribution to the lower order graphs, when the electron line is removed by assuming that the electron is structureless.

Lattice Dynamics and Thermodynamics of Molybdenum from First-Principles Calculations

of the ultrahigh pressure scale. The equation of state (EOS) of Mo at high pressure is being used as a calibration standard to the ruby fluorescence in diamond anvil compression (DAC) experiments.1 Because of its important position in the field of material science and condensed matter science, Mo has attracted tremendous experimental and theoretical interest in its wide range of properties recently. At ambient condition, Mo is in body-centered-cubic (bcc) structure and melts at 2890 K.2 But what is the most stable phase of Mo under ultrahigh pressure? Experimentally, the shock wave (SW) acoustic velocity measurements showed that there was a sharp break on the Hugoniot curve at about 210 GPa (at a calculated temperature of 4100 K), which indicated that a solid-solid phase transition occurred prior to melting at 390 GPa (at a calculated temperature of 10 000 K).3 To compare with the SW data, Vohra and Ruoff investigated the static compression of Mo by energy-dispersive X-ray diffraction and found that the bcc Mo was stable up to 272 GPa at 300 K.4 The phase transition in shock compression at 210 GPa was not observed. By further X-ray diffraction investigation, Ruoff et al. showed that Mo was stable in a bcc structure up to at least 560 GPa at room temperature.5 Theoretically, Moriarty suggested that Mo was stable in the bcc structure up to 420 GPa, where it transformed to a hexagonal close-packed (hcp) structure and then at 620 GPa to a face-centered closepacked (fcc) structure.6 Later, Boettger7 and Christensen et al.8 showed that the hcp phase of Mo was not stable, and the bcc phase transformed directly to the fcc phase at 700 GPa.

Belonoshko et al. confirmed the results of Boettger and showed the transition pressure was 720 GPa at zero pressure.9 By calculating the Gibbs free energies of the bcc and fcc Mo in the pressure range from 350 to 850 GPa at room temperatures up to 7500 K, Belonoshko et al. found that Mo had lower free energy in the fcc phase than in the bcc phase at elevated temperatures.10 Shortly after these new results were reported, Cazorla et al. found that the hcp Mo was noticeably more stable above 350 GPa at high temperature by calculating the Helmholtz and Gibbs free energies of the bcc, fcc, and hcp phases.11 The other intriguing problem is the melting properties. For the transition metals, such as Mo, Ta, and W, there are enormous discrepancies in melting curves between laser-heated DAC12-17 and SW3,18 methods. As for Mo (as well as Ta and W), several thousand degrees of discrepancies exist in extrapolating from DAC pressures of around 100 GPa12-15,17 to SW pressure of 390 GPa.18 As is known that the overestimation of the melting temperature exists in SW experiments, Errandonea13 corrected the SW data by considering 30% superheating. The revised melting temperatures are located at 7700 ( 1500 K (390 GPa), also much larger than the melting temperature (just above 4000 K) at this pressure extrapolated from DAC experiments. Results from the empirical and phenomenological melting models are dependent on the selection of the parameters.13,19-22 It is shown that the choice of different sets of parameters leads to huge differences in the melting temperatures at high pressure. In addition, theoretical results are consistent with the SW data at high pressure but diverge from DAC data below 100 GPa. All these results are inadequate to explain the extreme discrepancies in extrapolating the DAC data to the SW data.9-11,23 The highpressure melting curve of Mo still remains inconclusive up to now.

The low-temperature phonon spectrum of Mo as measured in inelastic neutron scattering experiments exhibited a variety of anomalies: the large softening near the H point, the T2 branch at the N point, and the longitudinal branch did

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

ORLANING COLMENARES

C.I.V.- 18.991.089

CRF

Referencias Bibliograficas:

http://electronicadeestadossolidos.blogspot.com/2010/03/lattice-dynamics-of-ki-perturbed-by.html

http://electronicadeestadossolidos.blogspot.com/2010/03/lattice-dynamics-and-thermodynamics-of.html

http://electronicadeestadossolidos.blogspot.com/2010/03/specific-energy-of-crystal-deformed-in.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 motion

s,
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 <!--[if !vml]--><!--[endif]--> ferroelectrics the anomalously large Born effective charges produce a giant LO-TO splitting in phonon dispersion relations. This feature is associated

with
to 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.

ORLANING COLMENARES

C.I.V.- 18.991.89

Referencias Bibliograficas:

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

Phonon Anomalies of Pressure Dependence in Molybdenum and RUBIDIUM

 

A collaborative group of researchers from Lawrence Livermore National Laboratory and the ESRF have been able to pin down the high-pressure lattice dynamics of the transition metal molybdenum by mapping its phonon energies under extremely high pressure. Using the inelastic X-ray scattering beamline ID28 at the European Synchrotron Radiation Facility (ESRF) and theoretical calculations, the team tracked the pressure evolution of a dynamical anomaly within molybdenum that has challenged scientists for over 40 years.

Much of the interest in the H-point phonon is derived from its anomalous increase in energy with increasing temperature. This observation stimulated numerous theoretical atte mpts to explain this strange behaviour. Changing the temperature or pressure is helpful in that it allows one to probe systems in different thermodynamic configurations. Indeed, the study of mater ials at high pressure is very useful for gaining insight into the nature of the chemical bonds in materials. Notably, the study of lattice dynamics at high pressures in general cannot be performed with neutrons due to the requirement of relatively large samples.

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

Fig. 1: A small molybednum single crystal loaded in the helium pressure medium. The photomicrograph was taken of the sample in situ at high pressure in the diamond anvil cell.

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

Fig. 2: Phonon dispersions in molybdenum at high pressure. The filled symbols show IXS data taken at 17 GPa at ID28, the open symbols are inelastic neutron scattering results at one atmosphere. Circles are longitudinal acoustic modes; squares transverse acoustic modes. Along [0] the triangles and squares show the two non-degenerate transverse acoustic modes TA[110]<-110> and TA[110]<001> respectively. The dashed lines show the calculations performed at one atmosphere, and the solid lines the calculations at 17 GPa.

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

In recent years, a number of surprisingly complex crystal structures have been discovered in the elements at high pressures, in particular incommensurately modulated structures and incommensurate host-guest composite structures (see [1] for a review). The crystal structure of the high-pressure phase rubidium-IV shown in Figure 11 belongs to the group of incommensurate host-guest structures that have also been observed in the elements Na, K, Ba, Sr, Sc, As, Sb, and Bi. The structure comprises a framework of rubidium host atoms with open channels that are occupied by linear chains of rubidium guest atoms, and the periodicities of the host and guest subsystems are incommensurate with each other (i.e., they have a non-rational ratio). Although considerable progress has been made in determining the detailed crystal structures of the complex metallic phases at high pressure, little is known about their other physical properties, and the mechanisms that lead to their formation and stability are not yet fully understood.

Fig. 11: Inelastic X-ray scattering spectrum of Rb-IV at 17.0 GPa, with the scattering vector q = (0 0 3.2)h referring to the host lattice. The inset shows the composite crystal structure of Rb-IV with the rubidium host and guest atoms in blue and red, respectively.

We investigated the lattice dynamics in incommensurate composite Rb-IV by inelastic X-ray scattering (IXS) on beamline ID28. The focus was on the longitudinal-acoustic (LA) phonons along the direction of the incommensurate wavevector (parallel to the guest-atom chains). Calculations on simpler model systems predict these phonons to reflect the incommensurability most clearly. Phase IV of Rb is stable at pressures of 16 to 20 GPa at room temperature, and a high-quality single crystal of Rb-IV was grown in a diamond anvil high pressure cell. In the IXS experiment, the incident radiation was monochromatised at a photon energy of 17.8 keV, and two grazing-incidence mirrors focussed the X-rays onto the sample with a focal size of 25 x 60 µm. The spectrum of the scattered radiation was analysed by a high-resolution silicon crystal analyser to yield an overall energy resolution of 3 meV.

Figure 11 shows a typical IXS spectrum of Rb-IV along with its decomposition into the elastic line, the phonon excitation peaks and a constant background, which were obtained by least-squares fitting using the FIT28 software. From a series of IXS spectra collected for different momentum transfers Q, phonon dispersion curves were obtained as shown in Figure 12a. A central result of this study is the observation of two well-defined longitudinal-acoustic (LA)-type phonon branches along the chain direction. They are attributed to separate LA excitations in the host and the guest sublattices, which is a unique feature of an incommensurate composite crystal.

A series of dispersion curves was measured at different pressures, and from this the sound velocities of the host and guest excitations and their pressure dependences were determined (Figure 12b). While the absolute values of the sound velocities in the host and the guest are rather similar, their pressure dependences differ notably. A simple ball-and-spring model of Rb-IV with only one spring constant reproduces these observations semi-quantitatively. This suggests that the difference in the pressure dependences is determined largely by geometrical factors, i.e., by the spatial arrangement of the atoms rather than differences in the chemical bonding in the two subsystems.

There is only very weak coupling between the incommensurate host and the guest in Rb-IV, which raises a rather interesting question. Can the 1D chains of guest atoms in Rb-IV be considered a manifestation of the "monatomic linear chain" treated in solid-state physics textbooks to introduce the concepts of crystal lattice dynamics? The pressure dependence of the interatomic spacing in the guest-atom chains was measured in earlier structural studies and enables the spring constant in the linear chain model to be determined, and also its pressure dependence. On this basis, the sound velocity in the linear chains and its pressure dependence were modelled as shown in Figure 12b. The results are in excellent agreement with the IXS data for the guest-atom chains in the composite Rb-IV structure, which can thus be regarded as a manifestation of the monatomic linear chain model with regard to the LA phonons.

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Referencias Bibliograficas:

http://electronicadeestadossolidos.blogspot.com/2010/02/pressure-dependence-of-phonon-anomalies.html

http://www.esrf.eu/news/spotlight/spotlight36phonon/index_html/

http://www.esrf.eu/UsersAndScience/Publications/Highlights/2008/HRRS/hrrs4

MgO at High Pressure: Theory and Experiment

 

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

Here we present experimental and theoretical results on MgO, a prototype oxide due to its simple structure and the large stability field (in pressure and temperature) of the NaCl structure. MgO is furthermore an important ceramic for industrial applications, and of great interest for Earth sciences, since it is a major mineral phase of the Earth's lower mantle. A doubly polished single crystal of MgO of (100) orientation, 30 x 50 µm size and a thickness of 20 µm was loaded in a diamond-anvil cell with He as pressure transmitting medium. The IXS experiment was performed on beamline ID28 with an overall energy resolution of 3 meV. Theoretical phonon dispersion curves were calculated using density-functional perturbation theory using the pseudopotential plane wave code ABINIT [1]. Details on the calculations can be found elsewhere.

The fact that characteristic features in the phonon dispersion are well reproduced by calculations gives confidence that ab initio predictions of thermodynamic properties of MgO at high pressure will be accurate. The determination of a thermodynamic property at high pressure requires experimental determination of the thermal expansion and bulk modulus, which are recast into an equation of state (EOS). Such EOS data are very few and when available, the data usually requires large extrapolation. Thermodynamic properties at high pressure may be calculated from a combination of calorimetric data at 1 bar and the volume integral with changing pressure and temperature [3]. Using the available thermodynamic data we obtain CV = 30 (+/-5) and S = 20.68 (+/-1) Jmol-1K-1. From the calculated phonon density-of-states at 35 GPa we determine CV = 31.71 and S = 20.04 Jmol-1K-1. The two data sets match within the errors of experimental data.

In summary, we demonstrate the ability of modern theory to reproduce experimental data on lattice dynamics of an inorganic compound at very high pressure. Expanding such tests to other, more complex systems could be beneficial for the development of both theory and experiment. These tests, validating the approximations done in the calculations, will allow the reliable determination of the thermodynamic properties of materials at high pressure, which are otherwise extremely difficult to assess by experimental methods.

Ultrafast carrier and lattice dynamics in semiconductor and metal nanocrystals

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

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Nuclear Spin and Magnetic Resonance

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.
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).

Quantum mechanics tells us a few important points about nuclear spins,
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 +/- .
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).

The Zeeman interaction

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,
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.
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.

The nuclear spin is restricted to being on these two cones oriented along the z-axis.

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.
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.
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).

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.
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.

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REFERENCIAS BIBLIOGRAFICAS:

web.mit.edu/22.058/www/documents/Fall2002/.../NMR.doc