Notices of Lattice Dynamics

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.

REFERENCIAS BIBLIOGRAFICAS:

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

inelastic neutron

INELASTIC NEUTRON SCATTERING AND
LATTICE DYNAMICS OF NOVEL COMPOUNDS
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.
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.
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).
Sections below give brief information about the experimental technique and the lattice dynamics
calculations respectively, while the results and discussion, and conclusions are presented later.
Experimental
Inelastic-neutron-scattering (INS) experiments [3] may be performed using both single crystals
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.
Lattice Dynamical Calculations
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.
Results and Discussion
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].
Silicate mineral zircon, ZrSiO4
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
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.
Conclusions
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.

comparison

COMPARISON OF THE LATTICE-DYNAMICSTHERMODYNAMIC PROPERTIES

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.
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.
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).
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.
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.
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.
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.
2. Lattice-dynamics calculations.
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.
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:

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.
Thermodynamic properties can all be derived from the partition function. Temperature derivatives can be evaluated explicitly to compute the energy and specific heat:

"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

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
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:

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,

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

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

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

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

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.

The Maxwell Hypothesis

GENERALIZING THE MAXWELL HYPOTHESIS AS A
CONCEPT OF OPTIMAL ENTROPY

University of Hagen / Germany

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.

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.
Although generally accepted a direct examination of the Maxwell Hypothesis by laboratory physics is not possible; a way out is offered by computer experimentation.

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.

A remark concerning the estimation technique:
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.

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.

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.
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.

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?

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.

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.
But even so -- by a computer experiment an affirmative answer can be given for the validity of the generalized Maxwell Hypothesis.

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.

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.

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.
Also in the following experiments causality is given up. Notice in this context, that that causality plays no role in quantum mechanics.

Denote by e the unit vector of the (classical) momentum exchange direction, connecting the both molecules i and j.

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:

u*i = u i + s e

u*j = u j -- s e

where the scalar s is determined by the condition of energy conservation.

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).
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.

If the generalized Maxwell Hypothesis (Entropy Principle) should be fulfilled for the described dynamics, then temperature should be a scalar quantity!
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.

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?

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.
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.

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.

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.

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.

REFERENCIAS BIBLIOGRAFICAS:

www.science.az/cyber/pci2006/2/moeschlin.doc

Lattice dynamics

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.

The Born Oppenheimer approximation

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.

The harmonic lattice

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

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

The supercell method

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.

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.

Some thermodynamics and the quasi harmonic

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.

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.

Thermal expansion

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.

Thermal expansion of cubic metals

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.

Thermal expansion of hexagonal metals

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.

BIBLIOGRAFIA:

web.mac.com/petros...2/.../urn_nbn_se_uu_diva-8198-1__fulltext.pdf

Inelastic neutron scattering and lattice dynamics

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

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

The inelastic neutron scattering and lattice dynamics studies carried out at Dhruva reactor may be broadly classiffed into two categories: studies of geophys-ically important minerals (silicates and carbonates) and those of technologically

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

Geophysically important minerals

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

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

Conclusion

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

BIBLIOGRAFIA:

http://www.ias.ac.in/pramana/v63/p73/fulltext.pdf

Quantum Dynamics of Matter Waves Reveal Exotic Multibody Collisions

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.
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.

The research is published in the journal Nature.

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.

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.

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.

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."

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.

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)

BIBLIOGRAFIA:

http://www.sciencedaily.com/releases/2010/05/100514094836.htm