Condensed Matter Theories

The following sections are included:

• PREFACE

• List of Participants

• CONTENTS

A retrospective of the career of Charles E. Campbell in condensed matter physics is presented as a tribute to his pathbreaking contributions to quantum many-body theory and his selfless dedication to the advancement of the research community associated with the Condensed Matter Workshop series.

In this paper we present an overview of a systematic development of the linear equations of motion for a dynamically correlated wave function that moves beyond the previous theories that include time-dependent pair correlations at most. We argue that these time-dependent pair correlations are insufficient to describe important physical effects in the energy/momentum regime of the ⁴He roton; minimally, time-dependent three-body correlations are necessary to capture the relevant physics. For simplicity we illustrate this on the problem of atomic impurities in ⁴He.

We describe the implementation of a manifestly gauge invariant configuration space method for ab initio electronic structure calculations in an arbitrarily strong external magnetic field. To be able to reproduce empirical data for realistic systems, we will also formulate our real-space algorithm in magnetic fields for non-local ionic pseudo-potentials. Numerical applications focus on two issues, namely a careful assessment of the convergence properties of our algorithm and in particular the implications of our gauge invariant formulation, and the calculation of NMR shifts for a number of typical molecules.

We investigate the structure of strongly correlated normal Bose liquids and fluids by employing correlated density-matrix (CDM) theory, a generalization of correlated basis functions (CBF) theory to nonzero temperatures. The formalism is applied in a study of structural properties of various correlation functions that characterize the one-body and two-body reduced density matrix elements of a boson system and the associated Fourier transforms such as the static structure function, the exchange structure function, the momentum distribution, and related thermodynamic quantities. We perform a numerical analysis for liquid para-hydrogen, close to the triple point, supercritical ⁴He gas at temperatures T ≥ 12 K, and fluid and liquid ⁴He in the normal phase below T = 12 K. The results reveal that H₂ and ⁴He, at T > 12 K, are typical quantum Boltzmann systems following classical statistics. In contrast, liquid ⁴He exhibits non-classical effects of particle exchange correlations, when the temperature is lowered toward the Bose-Einstein transition regime.

We present results on the behavior of the dynamic structure function in the short wave length limit using the equation of motion method. Within this framework we study the linear response of a quantum system to an infinitesimal external perturbation by direct minimization of the action integral. As a result we get a set of coupled continuity equations which define the self-energy. We evaluate the self-energy and the dynamic structure function in the short wavelength limit and show that sum rules up to the third moment are fulfilled. This implies, for instance, that the self-energy at short wavelengths and zero frequency is proportional to the kinetic energy per particle. An essential feature in this derivation is that the short range behavior of the two-particle distribution and the long wavelength phonon induced scattering are exactly satisfied. We calculate the condensate fraction and show that our results agree very well with the Monte Carlo simulations.

We report studies of adsorption of helium in translationally invariant polygonal pores at zero temperature, with emphasis on the route to capillary condensation and the appearance of metastable states. We analyze hysteresis and hysterectic-like phenomena associated to the existence of multiple equilibrium states in a rhombic pore and examine the effects of the angular geometry, as opposed to the smooth curvature of cylindrical tubes.

The derivation for collective modes of an interacting Bose gas trapped by an isotropic harmonic oscillator potential is presented using field-theoretic method. The presence of the two-body scattering term beyond the mean-field is seen to appear inevitably in the calculations, even in the simplest approximation. As a result we see the occurrence of a small number of non-condensate atoms in the ground state density fluctuations.

We explore p-wave pairing in a single-channel two-component Fermi system with unequal population near Feshbach resonance. Our analytical and numerical study reveal a rich superfluid (SF) ground state structure as a function of imbalance. In addition to the state Δ_±1 ∝ Y_1±1, a multitude of “mixed” SF states formed of linear combinations of Y_1m's give global energy minimum under a phase stability condition; these states exhibit variation in energy with the relative phase between the constituent gap amplitudes. States with local energy minimum are also obtained. We provide a geometric representation of the states. A T = 0 polarization vs. p-wave coupling phase diagram is constructed across the BEC-BCS regimes. With increased polarization, the global minimum SF state may undergo a quantum phase transition to the local minimum SF state.

The role in superconductors of hole-Cooper-pairs (CPs) are examined and contrasted with the more familiar electron-CPs, with special emphasis on their "background" effect in enhancing superconducting transition temperatures T_c — even when electron-CPs drive the transition. Both kinds of CPs are, of course, present at all temperatures. An analogy is drawn between the hole CPs in any many-fermion system with the antibosons in a relativistic ideal Bose gas that appear in substantial numbers only at higher and higher temperatures. Their indispensable role in yielding a lower Helmholtz free energy equilibrium state is established. For superconductors, the problem is viewed in terms of a generalized Bose-Einstein condensation (GBEC) theory that is an extension of the Friedberg-T.D. Lee 1989 boson-fermion BEC theory of high-T_c superconductors in that the GBEC theory includes hole CPs as well as electron-CPs — thereby containing as well as further extending BCS theory to higher temperatures with the same weak-coupling electron-phonon interaction parameters. We show that the Helmholtz free energy of both 2e- and 2h-CP pure condensates has a positive second derivative, and are thus stable equilibrium states. Finally, it is conjectured that the role of hole pairs in ultra-cold fermionic atom gases will likely be negligible because the very low densities involved imply a "shallow" Fermi sea.

Experiments in thin films whose thickness can be modified and by this way induce a superconductor to insulator transition, seem to suggest that in the quantum critical regime of this phase transition there might be a Bose metal, i.e., uncondensed bosonic carriers with a finite dissipation. This poses a fundamental problem as to our understanding of how such a state could be justified. On the basis of a simple Boson-Fermion model, where bosonic and fermionic degrees of freedom are strongly inter-related via a Boson-Fermion pair exchange coupling g, we illustrate how such a bosonic metal phase could possibly come about. We show that, as we approach the quantum critical point at some critical g_c from the superfluid side, the superfluid phase locking is sustained only for longer and longer spatial scales. On a finite spatial scale, the boson have a quasi-free itinerant behavior with metallic features. At the quantum critical point the systems exhibits a phase separation which shows a ressemblance to that of a He³-He⁴ mixture. This could be the clue to the apparent dilemma of a Bose metal at zero temperature.

An analytic expression for the contribution σ_B(λ,T) to the conductivity from charged bosonic Cooper pairs (CPs) is derived via two-time Green function techniques as a function of the BCS interelectron interaction model parameter λ and temperature T. Within the framework of a binary boson-fermion gas mixture model, it is shown that a self-consistent description of the resistivity data observed in high-temperature superconductors is possible only by assuming the presence of a finite gap between the energy spectra of free fermions and bosonic CPs.

A modeled Bose system consisting of N particles with two-body interaction confined within volume V under inhomogeneity of the system is investigated using the Feynman path integral approach. The two-body interaction energy is assumed to be dependent on the two-parameter interacting strength a and the correlation length l. The inhomogeneity of the system or the porosity can be represented as density with interacting strength b and correlation length L. The mean field approximation on the two-body interaction in the Feynman path integrals representation is performed to obtain the one-body interaction. This approximation is equivalent to the Hartree approximation in the many-body electron gas problem. This approximation has shown that the calculation can be reduced to the effective one-body propagator. Performing the variational calculations, we obtain analytical results of the ground state energy which is in agreement with that from Bugoliubov's approach.

Beyond the second row of elements in the Mendeleev periodic table, the consideration of the relativistic effect is important in determining proper configurations of atoms and ions, in many cases. Many important quantities of interest in determining physical and chemical properties of matter, such as the effective charge, root mean square radii, and higher moments of radii used in many calculations, e.g. in the determinations of legend stabilization bond energies depend on whether the treatment is relativistic or not. In general, these quantities for a given l-orbital having two different j-values, e.g. and , differ from each other, hence, making it necessary to treat them as separate orbitals. This also necessitates characterizing bands with their j-values in many instants and not l-values, particularly for “d” and f-orbitals. For example, in Au, and are to be dealt with as two distinct bands. The observed enhancement of laser induced field emission in W, which is not understood in terms of non-relativistic band-structures, can be explained in terms of the expected relativistic band structure. Spin-orbit coupling, which is the manifestation of the relativistic effect, is a prime factor in facilitating intersystem crossing in bio-molecules.

We report on recent studies of the spin-half Heisenberg and the Hubbard model on the sawtooth chain. For both models we construct a class of exact eigenstates which are localized due to the frustrating geometry of the lattice for a certain relation of the exchange (hopping) integrals. Although these eigenstates differ in details for the two models because of the different statistics, they share some characteristic features. The localized eigenstates are highly degenerate and become ground states in high magnetic fields (Heisenberg model) or at certain electron fillings (Hubbard model), respectively. They may dominate the low-temperature thermodynamics and lead to an extra low-temperature maximum in the specific heat. The ground-state degeneracy can be calculated exactly by a mapping of the manifold of localized ground states onto a classical hard-dimer problem, and explicit expressions for thermodynamic quantities can be derived which are valid at low temperatures near the saturation field for the Heisenberg model or around a certain value of the chemical potential for the Hubbard model, respectively.

The motion of the guiding center of magnetic circulation generates a charge transport. By applying kinetic theory to the guiding center motion, an expression for the magnetoconductivity σ is obtained: σ = e²n_cτ/M*, where M* is the magnetotransport mass distinct from the cyclotron mass, n_c the density of the conduction electrons, and τ the relaxation time. The density n_c depends on the magnetic field direction relative to copper's fcc lattice, when Cu's Fermi surface is nonspherical with "necks". The anisotropic magnetoresistance is analyzed based on a one-parameter model, and compared with experiments. A good fit is obtained.

The far-field reflectivity of metallic nanowire arrays designed to show strong surface-plasmon-polariton (SPP) resonances is studied numerically. The results of calculations in time and frequency space as well as the results of semi-analytic theories using different approximative boundary conditions at the metal surfaces are evaluated and compared. Good agreement between all different methods is obtained in most cases. The SPP-related features are superimposed on a strongly varying background. Combining FDTD simulations, finite element results, and semi-analytical calculations, the microscopic origin of the background contribution is identified. Resonant transmission through sub-wavelength slits leads to pronounced oscillations in the far-field reflectivity as a function of the height of the nanowires.

When applied to a finite Fermi system having a degenerate single-particle spectrum, the Landau-Migdal Fermi-liquid approach leaves room for the possibility that different single-particle energy levels merge with one another. It will be argued that the opportunity for this behavior exists over a wide range of strongly interacting quantum many-body systems. An inherent feature of the mergence phenomenon is the presence of nonintegral quasiparticle occupation numbers, which implies a radical modification of the standard quasiparticle picture. Consequences of this alteration are surveyed for nuclear, atomic, and solid-state systems.

The problem of a correct definition of the charged carrier effective mass in superfluid helium is revised. It is shown that the effective mass of such a quasi-particle can be introduced without Atkins's idea about the solidification of liquid He⁴ in the close vicinity of an ion (the so-called "snowball" model). Moreover, in addition to generalization of the Atkins's model, the charged carrier effective mass formation is considered within the framework of the two-fluid scenario. The physical reasons of the normal fluid contribution divergency and the way of corresponding regularization procedure are discussed. Agreement between the theory and the available experimental data is found in a wide range of temperatures.

A matrix Berry phase can be generated and detected by all electric means in II-VI and III-V n-type semiconductor quantum dots by changing the shape of the confinement potential. This follows from general symmetry considerations in the presence of spin-orbit coupling terms. We explain these results and discuss how the matrix Berry phase depends on geometric properties of adiabatic paths. We suggest how the matrix Berry phase may be detected in transport measurements.

We propose a new theory of the human mind. The formation of human mind is considered as a collective process of the mutual interaction of people via exchange of opinions and formation of collective decisions. We investigate the associated dynamical processes of the decision making when people are put in different conditions including risk situations in natural catastrophes when the decision must be made very fast or at national elections. We also investigate conditions at which the fast formation of opinion is arising as a result of open discussions or public vote. Under a risk condition the system is very close to chaos and therefore the opinion formation is related to the order disorder transition. We study dramatic changes which may happen with societies which in physical terms may be considered as phase transitions from ordered to chaotic behavior. Our results are applicable to changes which are arising in various social networks as well as in opinion formation arising as a result of open discussions. One focus of this study is the determination of critical parameters, which influence a formation of stable mind, public opinion and where the society is placed “at the edge of chaos”. We show that social networks have both, the necessary stability and the potential for evolutionary improvements or self-destruction. We also show that the time needed for a discussion to take a proper decision depends crucially on the nature of the interactions between the entities as well as on the topology of the social networks.

The advancement in new materials processing and fabrication techniques has made it possible to better control the atomistic level of structures in a way, which was not feasible only a decade ago. If one can couple this atomic control with a good understanding of the relationship between structure and properties, this will in the future lead to a significant contribution to the synthesizing of tailor-made materials. In this paper we have focused on, the structurally related nanolayered ternary compounds M_N+1AX_N, (MAX) where N = 1, 2 or 3, M is an early transition metal, A is an A-group (mostly IIIA and IVA) element, and X is either C and/or N, which has attracted increasing interest owing to their unique properties. The general relations between the electronic structure and materials properties of MAX phases have been elaborated based on ab initio calculations.

Interacting two-component Fermi gases loaded in a one-dimensional (1D) lattice and subjected to a harmonic trapping potential exhibit interesting compound phases in which fluid regions coexist with local Mott-insulator and/or band-insulator regions. Motivated by experiments on cold atoms inside disordered optical lattices, we present a theoretical study of the effects of a correlated random potential on these ground-state phases. We employ a lattice version of density-functional theory within the local-density approximation to determine the density distribution of fermions in these phases. The exchange-correlation potential is obtained from the Lieb-Wu exact solution of Fermi-Hubbard model. On-site disorder (with and without Gaussian correlations) and harmonic trap are treated as external potentials. We find that disorder has two main effects: (i) it destroys the local insulating regions if it is sufficiently strong compared with the on-site atom-atom repulsion, and (ii) it induces an anomaly in the inverse compressibility at low density from quenching of percolation. For sufficiently large disorder correlation length the enhancement in the inverse compressibility diminishes.

A new look at super heavy nuclei is attempted by employing the Isomorphic shell Model or, in different wording, the Multiharmonic Shell Model. After ²⁰⁸Pb the same magic numbers, like the conventional Shell Model, are predicted, i.e., at Z = 126 and N = 184. Good results for charge radii and binding energies for a sample of nuclei are also presented.

We discuss here a self-consistent method to calculate the properties of the cold asymmetric nuclear matter. In this model, the nuclear matter is dressed with s-wave pion pairs and the nucleon-nucleon (N-N) interaction is mediated by these pion pairs, ω and ρ mesons. The parameters of these interactions are calculated self-consistently to obtain the saturation properties like equilibrium binding energy, pressure, compressibility and symmetry energy. The computed equation of state is then used in the Tolman-Oppenheimer-Volkoff (TOV) equation to study the mass and radius of a neutron star in the pure neutron matter limit.

It has been a natural desire for a long time to be able to describe nuclear physics in terms of the fundamental strong interaction. Recently some significant progress has been made in this area in terms of lattice QCD calculations of simple nuclear physics processes such as nucleon nucleon scattering. An attempt is made to introduce the progress made in this area, to an audience composed mainly of many-body theorists (non-lattice QCD and even non-particle/nuclear physics) interested in inter-disciplinary approaches.

Alpha particle condensation is studied in ¹²C and ¹⁶O with the orthogonality condition model (OCM). The OCM equation is an approximation of the equation of motion of α bosons based on microscopic theories. We demonstrate that 1) the Hoyle state (the state at 7:65 MeV in ¹²C), located just above the 3α disintegration threshold has a 3α-particle condensate character, in which 3α particles occupy an identical 0S orbit with 70 % occupancy, forming a dilute gas-like configuration, and 2) the state at E_x = 15.1 MeV in ¹⁶O, appearing above the 4α breakup threshold, is a strong candidate with dilute 4α structure of the condensate-type.

The derivation of a time-dependent Schrödinger equation (TDSE) from a time-independent Schrödinger equation (TISE) in the coherent state representation is considered for the special case of a simple coupled atom-field system described by the soluble Jaynes-Cummings model. The derivation shows why, from the outset, a linear combination of energy eigenstates, instead of a single state, must be used in order to obtain a TDSE for general states. Moreover, this study leads to a method of solving a TDSE by simply solving a TISE.

Motivated by recent surprising experimental results for temperature dependent resistivities in 2D mesoscopic electron systems, we investigate transport in these systems by percolation connected through a network of metallic domains. The size of the domains is determined by the level of disorder in the system and by the strength of the electron correlations. In the insulating phase the metallic domains are connected for transport by two competing mechanisms, thermally activated hopping and quantum tunneling. We calculate the transmission across the potential barriers that separate the metallic domains. Using recent data from transport measurements in mesoscopic 2D systems, we obtain the observed saturation of the temperature dependent resistivity at T ~ 1 K and consistent values for the size of the domains and for magnitude of the average variation in the random disorder potential.

There are two approaches to understanding Boltzmann's ergodic hypothesis in statistical mechanics. The first one, purely mathematical, goes by way of theorems while the second one relies on physical measurements. By its own nature the former is universal whereas the latter is specific to a system. By all account they seem orthogonal to each other. But should not they meet at the end? If, for example, both conclude that the hypothesis is not valid in a given system, should not their conclusions be compatible? We illustrate in this work how the two cultures meet in the physics of ergodicity.

Graphene is a fascinating material for exploring fundamental science questions as well as a potential building block for novel electronic applications. In order to realize the full potential of this material the fabrication techniques of graphene devices, still in their infancy, need to be refined to better isolate the graphene layer from the environment. We present results from a study on the influence of extrinsic factors on the quality of graphene devices including material defects, lithography, doping by metallic leads and the substrate. The main finding is that trapped Coulomb scatterers associated with the substrate are the primary factor reducing the quality of graphene devices. A fabrication scheme is proposed to produce high quality graphene devices dependably and reproducibly. In these devices, the transport properties approach theoretical predictions of ballistic transport.

It is here shown how to use pieces of macroscopic thermodynamics to generate microscopic probability distributions for generalized ensembles, thereby directly connecting macro-state-axiomatics with microscopic results.

While ferromagnetism has been obtained above 100 K in doped semiconductors with magnetic ions such as Ga_1-xMn_xAs, bulk doped semiconductors in the absence of magnetic ions have shown no tendency towards ferromagnetism. We re-examine the non-magnetic doped semiconductor system at low carrier densities in terms of a generalized Hubbard model. Using exact diagonalization of the many-body Hamiltonian for finite clusters, we find that the system exhibits significant ferromagnetic tendencies at nanoscales, in a region of parameter space not accessible to bulk systems, but achievable in quantum dots and heterostructures. Implications for studying these effects in experimentally realizable systems, as well as the possibility of true (macroscopic) ferromagnetism in these systems is discussed.

The shape of solid lipid monolayer domain surrounded by a fluid phase is of considerable interest from physical and mathematical points of view. Here we report two new results about this topic. First, we obtain an exact analytical solution to an approximated shape equation that was derived by us recently [Phys. Rev. Lett. 93, 206101 (2004)]. This solution can well describe the kidney- and boojum-like domains that abound in lipid monolayer. Second, we derive an exact domain shape equation by a direct variation of domain energy without any artificial cutoff. We find that no continuous solutions satisfies this shape equation due to the divergence of its coefficients, which is rooted in the continuous description of electrostatic dipoles.

A study of electronic processes in the chlorophyll and carothenoid molecules of the photoreaction center II is presented with the focus on the electronic excitations and charge transfer in the photosynthetic process. Several novel ideas are mentioned especially concerning the electron replenishment and nuclear vibrational excitations. The study is build mainly on numerical quantum calculations of electronic structures of molecules in the photo-reaction center.

The dielectric function of dense plasmas is treated within a many-particle linear response theory beyond the RPA. In the long-wavelength limit, the dynamical collision frequency can be introduced which is expressed in terms of momentum and force auto-correlation functions (ACF). Analytical expressions for the collision frequency are considered for bulk plasmas, and reasonable agreement with MD simulations is found. Different applications such as Thomson scattering, reflectivity, electric and magnetic transport properties are discussed. In particular, experimental results for the static conductivity of inert gas plasmas are now well described.

The transition from bulk properties to finite cluster properties is of particular interest. Within semiclassical MD simulations, single-time characteristics as well as two-time correlation functions are evaluated and analyzed. In particular, the Laplace transform of current and force ACFs show typical structures which are interpreted as collective modes of the microplasma. The damping rates of these modes are size dependent. They increase for the transition from small clusters to bulk plasmas.

We discuss two basic problems in the Hohenberg-Kohn-Sham version of density functional theory, HKS-DFT: the first, the N-representability of the functional F[ρ] and, the second, the universality of F[ρ]. In relation to the first, we show that F[ρ] must satisfy N-representability conditions that follow from those on the 2-matrix D²(r₁, r₂; r′, r′₂). In the case of the second, we provide arguments based on the equivalence between ab initio DFT and HKS-DFT to show that the functional F[ρ] is not universal.

We present a calculation of the excitation spectrum of the electron liquid that includes time-dependent pair correlations. For the charged boson fluid these correlations provide a major mechanism for lowering the plasmon energy; here we extend that study to the much more demanding fermionic case. Based on the formalism of correlated basis functions we derive coupled equations of motion for time-dependent 1- and 2-particle correlation amplitudes. Our solution strategy for these equations ensures the fulfillment of the first two energy–weighted sum rules and, in the appropriate limit, is consistent with the bosonic version. Results are presented for the dynamic structure factor with special emphasis being put on studying the double plasmon.

We discuss an extension of time dependent density functional theory by a self-interaction correction (SIC). A strictly variational formulation is given taking care of the necessary constraints. A manageable and transparent propagation scheme using two sets of wave-functions is proposed and applied to laser excitation with subsequent ionization of a dimer molecule.

We study a model for the emergence of collective decision making, consisting of N interacting agents, whose opinions are described by Ising spin variables. In particular, we present dynamical phase transitions from ordered to chaotic behavior in the space-time evolution of the binary choice network. One focus of this study is the determination of critical parameters, where the network is placed "at the edge of chaos," i.e., at a subtle compromise between stability and flexibility, where the system has both, the necessary stability and the potential for "evolutionary" improvements.

We present a brief overview of the light wave interference in the atmospheric rainbow and how a similar mechanism can be observed in the elastic nucleus-nucleus scattering which gives rise to the nuclear rainbow. The latter phenomenon, observed at energies of around few tens MeV/nucleon, has been well investigated based on the basic concepts of the nuclear optical model. Given a weak absorption associated with the nuclear rainbow scattering, the observed data can be used to probe the density dependence of the effective nucleon-nucleon (NN) interaction based on the folding model study of elastic scattering. Most of the rainbow scattering data were found to be best described by a density dependent NN interaction which gives a nuclear incompressibility K ≈ 230-260 MeV in the Hartree-Fock calculation of nuclear matter. This result implies a rather soft equation of state of nuclear matter.

The present status of the α-nucleus potential, generated from the energy density functional (EDF) formalism using a realistic two-nucleon potential, which incorporates the Pauli principle, is discussed. The EDF potentials, calculated using a density distribution of α-particle that yields a binding energy of 20 MeV with a reasonable root-mean-squared radius and observed density distributions of ⁶Li and various target nuclei, are found to be shallow and non-monotonic in character. This non-monotonic EDF potential reproduces satisfactorily the experimental elastic scattering data, particularly at energies above the Coulomb barrier. Since the elastic scattering data and the binding energies of all nuclei considered herein are well reproduced using the mean field generated from a realistic two-nucleon potential for nuclear and nucleonic matter, one may conclude to have reasonable information on the equation of states of nuclear and nucleonic matter from a very low to the saturation density from the present investigation.

Tunneling of α particles through the Coulomb barrier is consecutively treated. The effect of sharp peaks arising in the case of coincidence of the α energy with that of a quasistationary state within the barrier is elucidated. Peaks' energy depend on the α-nucleus potential. They can give rise to "anomalous" properties of some neutron resonances. The peaks can also be observed in the incoming α-nucleus channel. The method is also applied for calculation of the angular distribution of the emitted α particles from the α decay of a compound nucleus of ¹³⁵Pr. Next fundamental test of the theory is proposed to be the α decay of superheavies.

Single-phase cubic Ba(Fe,Nb)_0.5O₃ (BFN) powder was synthesized by solid-state reaction at 1443 K for 4 hour in air. X-ray diffraction indicated that the BFN oxide mixture calcined at 1200°C crystallizes to the pure cubic perovskite phase. BFN ceramics were produced from this powder by sintering at 1623–1673 K for 4 hrs in air. Samples prepared under these conditions achieved up to 94.7% of the theoretical density. The temperature dependence of their dielectric constant and loss tangent, measured at difference frequencies, shows an increase in the dielectric constant with temperature which is probably due to disorder on the B site ion of the perovskite. Non-Debye type of relaxation phenomena has been observed in the BFN ceramics as confirmed by Cole–Cole plots. The higher value of ε′ at the lower frequency is explained on the basis of the Maxwell–Wagner (MW) polarization model.

The structural, dielectric and piezoelectric properties of (1 - x)PbZr_0.52Ti_0.48O₃-xBaFe_0.5Nb_0.5O₃ ceramic system with the composition near the morphotropic phase boundary were investigated as a function of the BaFe_0.5Nb_0.5O₃ content by X-ray diffraction (XRD), dielectric measurement and piezoelectric measurement techniques. Studies were performed on the samples prepared by solid state reaction for x = 0.10, 0.12, 0.14, 0.16, 0.18 and 0.20. The XRD analysis demonstrated that with increasing BFN content in (1 - x)PZT_-xBFN, the structural change occurred from tetragonal to the mixture of tetragonal and cubic phase. Changes in the dielectric behavior and piezoelectric properties were found to relate with these structural changes depending on the BFN contents.

The purpose of this research work is to investigate the effects of heat treatment on spin Hamiltonian parameters of Cr³⁺ ions in natural pink sapphire sample. The pink sapphire were heated at different temperatures of 1200, 1300, 1400, 1500 and 1600°C in oxygen atmosphere for 12 h, respectively. Electron Spin Resonance (ESR) spectra of pink sapphire samples before and after heat treatment were recorded in X-band frequency by mounting the crystal sample with the c-axis perpendicular to the applied magnetic field direction. The spectra were recorded in the range of 0-180 degrees for every 15 degrees of rotation angle (ɸ). It was found that, five strong ESR absorption peaks at the magnetic fields of 97, 163, 312, 519 and 724 mT were obtained. The magnetic fields of 163 and 519 mT corresponding to Cr³⁺ ions replacing the Al³⁺ ions site of corundum structure (α-Al₂O₃), and those of 97, 312, and 724 mT were assigned to Fe³⁺ ions. Some conspicuous peaks of Cr³⁺ ion at various rotation angles were used to calculate the spin Hamiltonian parameters by least-squares fit method with the help of a computer program. The optimum spin Hamiltonian parameters of Cr³⁺ ions were then used to simulate the energy levels diagram of Cr³⁺ ions in sample before and after heat treatment.

The main purpose of this work is to present the ESR spectra and calculate the spin Hamiltonian parameters of ¹⁴N and ¹⁵N impurities in natural diamond. The ESR spectra of diamond crystal were measured on ESR spectrometer operating at X-band microwave frequency. The results of ESR spectra show that the diamond has a P1 center. This center gives rise to three strong resonance absorption peaks at θ = 90°, ɸ= 0° due to hyperfine interaction between electron spin and nuclear spin of ¹⁴N. The ESR spectra of ¹⁵N impurity consist of two satellites at the same rotation angle (ɸ). The effects of isolated substitution nitrogen on carbon atom produced a symmetric distortion from T_d to C_3V symmetry. According to this symmetry, the resonance magnetic field positions of ESR spectra for the rotation angles of 0°, 90° and 180° are almost overlap. The g-factor values and spin Hamiltonian parameters of ¹⁴N and ¹⁵N are: g = 2.0019, A_⊥ = 29.73, A_∥ = 40.24 and g = 2.0019, A_⊥ = -39.90, A_∥ = -57.05, respectively.

This research work is aimed at lowering the sintering temperature of 0.9PMN-0.1PT ceramics by adding oxide additives. The oxides used for this purpose were Bi₂O₃ and Li₂CO₃ with various amounts, following the formula of xBi₂O_3+yLi₂CO₃, where x+y = 10 mol% and x = 1, 3, 5 and the mixed oxide additive powders were then added to the dried powder of 0.9PMN-0.1PT with 1 wt% concentration. An excess PbO content of approximately 3 wtadded to all compositions to compensate the lead loss. After that, the mixed powders were pressed into pellets and subsequently sintered to form the ceramic samples. The results showed that the sintering temperature of 0.9PMN-0.1PT ceramics could be lowered down from 1250°C to 900–1000°C by the addition of small amount of the oxide additives, where the optimum composition was found in the sample with x : y = 1 : 9 at sintering temperature of 1000°C. Moreover, the densification, dielectric and ferroelectric properties of this sample remain acceptable in particular uses, promising positive future for reduction of lead loss from electrical and electronic industries.

We show that the environment affects a quantum system in the form of the constrained trajectory. Our result allows one to describe the quantum system in terms of stochastic state vector rather than quantum history. Moreover we can alternatively reduce the time evolution operator. Then the trajectory of system is constrained.

The following section are included:

• AUTHOR INDEX