Harness Processes and Non-Homogeneous Crystals

We consider the Harmonic crystal, a measure on with Hamiltonian H(x)=∑ _i,j J _i,j (x(i)−x(j))²+h∑ _i (x(i)−d(i))², where x, d are configurations, x(i), d(i)∈ℝ, i,j∈ℤ ^d . The configuration d is given and considered as observations. The ‘couplings’ J _i,j are finite range. We use a version of the harness process to explicitly construct the unique infinite volume measure at finite temperature and to find the unique ground state configuration m corresponding to the Hamiltonian.     

Spherical harmonics and Cartesian tensor scalar product expansion in the Fokker-Planck equation

On the basis of the expansion of the distribution functionf(v, r,t) in a sum of spherical harmonics, which is equivalent to a Cartesian tensor scalar product expansion of the distribution function, i.e.,f(v, r, t)=f ₀(v,r,t)+v. f ₁(v,r,t)+vvf ₂(v,r,t)+vvvf ₃(v,r,t)+ wheref _k (k=2, 3) arek-th order irreducible tensors, the Rosenbluth potential functions and the Fokker-Planck collision term are expanded in a similar sum. Collisions termsJ _Fk (k=0, 1, 2) and the equations forf _k (k=0, 1, 2) for the case of the Coulomb interactions are also determined.Technická 2, Praha 6, Czechoslovakia.The autor wishes to express his thanks to Prof. J. Kracík, DrSc. for valuable advice and suggestion.     

Conditional expectation and commutativity in von Neumann algebras

LetH be a Hilbert space,A the von Neumann algebra of all bounded operators onH,B a von Neumann subalgebra ofA, andw a bounded linear functional onA. The functionalw is said to commute withBA ifw(AB)=w(BA) for allAA. It is shown that the mapBw (BAB) is a complex measure on the orthocomplemented partially ordered set of all orthogonal projections inB for everyAA if and only ifw commutes with all members ofB. For anyAA, the conditional expectation ofA with respect toB andw is defined and it is shown that this expectation exists for an Abelian separableB ifw commutes with all members ofB. Using Gleason's theorem it is shown thatw commutes withB if and only if the density operator ofw commutes withB.     

Stable potentials I

We discuss a conjecture of Ruelle concerningstable potentials on a group. For the groupsZ ₂,Z ₃, ₄, andZ ₆ any stable potential can be written as the sum of a non-negative function and a function of non-negative type. This is not true for the groupsZ _k (k odd, 5). For the Euclidean groupR ^v the question is open.Supported by NSF GP 7946.     

The ideal polymer chain near planar hard wall beyond the Dirichlet boundary conditions

We present a new ab initio approach to describe the statistical behavior of long ideal polymer chains near a plane hard wall. Forbidding the solid half-space to the polymer explicitly (by the use of Mayer functions) without any other requirement, we derive and solve an exact integral equation for the partition function G _D(r,r′, N) of the ideal chain consisting of N bonds with the ends fixed at the points r and r′ . The expression for G(r,r′, s) is found to be the sum of the commonly accepted Dirichlet result G _D(r,r′, N) = G ₀(r,r′, N) - G ₀(r,r”, N) , where r” is the mirror image of r′ , and a correction. Even though the correction is small for long chains, it provides a non-zero value of the monomer density at the very wall for finite chains, which is consistent with the pressure balance through the depletion layer (so-called wall or contact theorem). A significant correction to the density profile (of magnitude 1/is obtained away from the wall within one coil radius. Implications of the presented approach for other polymer-colloid problems are discussed.     

Quantum condensates classified in superconductivity with topology in the Minkowski space-time

A substantial problem in the macroscopic theory of pure superconductivity has been left forgotten for a long time since London and London in 1935. An impression survived that the Meissner effect is more substantial than the zero-resistivity. But, the London equation [I], the Newtonian equation of motion, was abandoned, whereas the London equation [II], derived from the Maxwell equations, was postulated. The London equation [II] included the logical gap [ α ] in real time, whereas the London equation [I] has been ignored without even noting the logical gap [ β ] in space. Microscopically, after the publication of F. London's book and the discovery of the isotope effect in 1950, the success of the Bardeen--Cooper--Schrieffer (BCS) theory in 1957 was likely to have finally given the definitive explanation on superconductivity by proving only the London equation [II] that claimed the coherent condensation of Cooper pairs in the momentum space. Since then, these arguments have been regarded to be a standard among various preceding theories. Meanwhile, the London equation [I] has faded away and has been long-forgotten. But we must not abandon the London equation [I], and, rather, retrieve it. We later recognized also that the DC-component of a persistent current can never be determined by using the Fourier transform analysis, because of its singularity at ω?=?0 and q ?=?0 with huge differences of space-time domain. Quite recently, in 2003, we first recognized a proper and harmonious view to simultaneously account for (i) the zero-resistivity in an open system with (i-c) the resultant persistent current in a closed system, and (ii) the perfect diamagnetism at T???0?K in the space-time aspects in terms of the gauge field theory. Here, we further clarify where and how we have lost and found a properly perspective view of the superconductivity. Here, we eliminate two logical gaps [ α ] and [ β ] by using the gauge field theory for further clarifying a position of the previous and present works. We especially classify superconductors with topology which eventually leads us such as (ii-2D) magnetic flux quantization in a ring. By projecting the 3-dimensional BCS-theory with the concept of ‘coherence’ among an enormous number of Bosons like Cooper pairs onto the (1?+?3)-dimensional Minkowski space-time [β?=?(v/c)?=?0], we clarify responses of the ground state Ψ _macro at T???0?K in a set of the basic equations, for (i) the zero-resistivity, [E _K ???qφ( R )]?=?0 at ω?=?0 and (ii) the perfect diamagnetism [?K ???qA ( R )]?=?0 at q ?=?0 as an inevitable consequence at the gauge fields in the proper theory of superconductivity.     

Nanosize Semiconductors for Photooxidation

Nanosized semiconductors (semiconductor clusters) have the potential to revolutionize the fields of photooxidation and photocatalysis through the combined effects of quantum confinement and their unique surface morphologies. Photocatalytic oxidation as applied to environmental remediation (i.e., detoxification of chemical wastes), green/sustainable chemistry, as well as alternative energy paths (i.e., splitting of H ₂ O to produce H ₂ ) has already experienced improvements in activity, efficiency, and stability through the use of semiconductor nanoclusters based on materials such as TiO ₂ , MoS ₂ , WS ₂ , MoSe ₂ , FeS ₂ , and SnO ₂ . Issues such as improved control of size and surface chemistry play an important role in the success of these semiconductor nanocatalysts. This review explores the effect of advances in the fields of nanoscience and photocatalysis for current and future applications.     

An investigation on the magneto-optic properties of terbium gallium garnet under high magnetic field

The superexchange interaction on a magnetic ion may be represented by an effective field H_m = $\lambda$M in some paramagnetic materials, here λ is the coefficient of effective field and M = $\chi$H_e with $\chi$ being the magnetic susceptibility and H_e being the applied field. The variation of the equivalent $\lambda_{\chi}$with the dynamic applied field is given and the crystal field-splitting levels of the excited configuration 4f⁷5d¹ of the Tb³⁺ ion are calculated in the Tb₃Ga₅O₁₂. By means of the effective field H_m and the applied field H_e, the Faraday rotation of Tb₃Ga₅O₁₂ at 6 K and 41 K, under the high magnetic field and at 0.63 μm wavelength, are presented. Our calculated results are in agreement with the experimental data.     

Nonadiabatic effects in the phonon spectra of superconductors

The nonadiabatic corrections to the self-energy part Σ_s(q, ω) of the phonon Green’s function are studied for various values of the phonon vectors q resulting from electron-phonon interactions. It is shown that the long-range electron-electron Coulomb interaction has no direct influence on these effects, aside from a possible renormalization of the corresponding constants. The electronic response functions and Σ_s(q, ω) are calculated for arbitrary vectors qand energy ω in the BCS approximation. The results obtained for q=0 agree with previously obtained results. It is shown that for large wave numbers q, vertex corrections are negligible and Σ_s(q, ω) possesses a logarithmic singularity at ω=2Δ, where Δ is the superconducting gap. It is also shown that in systems with nesting, Σ_s(Q, ω) (where Q is the nesting vector) possesses a square-root singularity at ω=2Δ, i.e., exactly of the same type as at q=0. The results are used to explain the recently published experimental data on phonon anomalies, observed in nickel borocarbides in the superconducting state, at large q. It is shown, specifically, that in these systems nesting must be taken into account in order to account for the emergence of a narrow additional line in the phonon spectral function S(q, ω)≈−π ⁻¹ Im D _s (q, ω), where D _s (q, ω) is the phonon Green’s function, at temperatures T<T _c . Zh. éksp. Teor. Fiz. 115, 1799–1817 (May 1999)     

Theory of ferromagnetic resonance in metallic and insulator films

Simple intuitive explanations of the frequencies and intensitiesI of magnetostatic modes (with negligible exchange energy) and exchange modes (negligible microwave-field demagnetization energy) are given. The effects of explicit boundary conditions on the transverse, time-varying componentm of the magnetization (i.e., the amount of pinning) and the effects of inhomogeneities in the internal fieldH _i and saturation magnetizationM _s are discussed. A bulk inhomogeneity inM _s changes the effective exchange constant, while a bulk inhomogeneity inH _i lowers the fields for resonance of all high-order exchange modes by the same amount. The pinning conditions affect the values of andI of the exchange modes and the mixed exchange-magnetostatic modes, but have little effect on the magnetostatic modes. A surface-imperfection source of pinning and of the inhomogeneities inH _i andM _s is discussed.Part of this work was performed while the author was atNorth American Rockwell Science Center, Thousand Oaks, California.     

From macroscopic to atomic motions in liquid metals

Summary In the present review of liquid dynamics studies on liquid metals are reported. Particularly the case of liquid lead is reviewed because this case was carefully studied by neutron scattering technique,S(Q,ω) being determined at two widely different temperaturesT=623 K andT=1170 K and therefore different densities. In addition extensive supplementary MD simulations were made using a 16 384-particle system. The simulations ranged from a determination of an effective pair potential for lead to simulation of the density correlation functionsF(Q,t) andF _s(Q,t), as well as the longitudinal and transversal current correlation functionsJ ₁(Q,t) andJ _T(Q,t). The MD simulation ?calibrated? via the experimentalS(Q) andS(Q,ω) was used to prolong the range of neutron data to draw conclusions regarding such quantities as dispersion relations for the current correlationsJ ₁(Q,t) andJ _T(Q,t), the generalized viscosity functions ν₁(Q,t), ν₁(Q) and ν_s(Q). Information regarding bulk viscosity ν_B(Q) is also gained. Conclusions are drawn regarding the relative importance of the derived pair potential form by comparison to corresponding hard-sphere data. The general framework of linearized hydrodynamic equations for the macroscopic situation transforming to visco-elastic equations of motion for finite wave-length and high frequency works well also for the case of a continuous potential. The region of transition from simple visco-elastic to hydrodynamic behaviour is occurring at wavelengths in the range (12÷20) ? for the cases studied. The spatial properties of the viscosity functions ν₁(r), ν_s(r) and ν_B(r) are found to correlate well with the range of the radial distribution function for the liquid. The general results for liquid lead probably have wide range of applicability to other simple liquids with similarS(Q) andg(r) properties. The authors have agreed not to receive proofs for correction.     

Crystal structures of high-T c oxides

A comprehensive review of structure work on high-T _c oxides as reported during the years 1987 and 1988 is given. Thirteen structures are refined from X-ray single-crystal and/or neutron powder diffraction data:I. (Ba_1–x ,K _x )BiO₃ (T _c =30 K),II. (La_2–x , Sr _x )CuO₄ (T _c =40 K),III. (Nd, Ce, Sr)₂CuO₄ (T _c =28 K),IV. (Nd_2–x , Ce _x )CuO₄ (T _c =24 K),V. YBa₂Cu₃O₇ (T _c =90 K),VI. YBa₂Cu₄O₈ (T _c =80 K),VII. Y₂Ba₄Cu₇O₁₄ (T _c =40 K),VIII. Pb₂Sr₂NdCu₃O₈ (T _c =70 K),IX. TlBa₂CaCu₂O₇ (T _c =103 K),X. TlBa₂Ca₂Cu₃O₉ (T _c =120 K),XI. Tl₂Ba₂CuO₆ (T _c =90 K),XII. Tl₂Ba₂CaCu₂O₈ (T _c =112 K),XIII. Tl₂Ba₂Ca₂Cu₃O₁₀ (T _c =125 K). Except forI (perovskite type),II (K₂NiF₄ type) andIV (Nd₂CuO₄ type) they represent new structure types. Structure data, bond distances, structure drawings and calculated X-ray powder diffraction patterns are given for each compound. Structural features and correlations with superconductivity are discussed. The review contains 301 citations.     

Predictability of ion transport properties from the structure of solid electrolytes

The concept of bond valence (BV) is widely used in crystal chemical considerations, e.g. to assess equilibrium positions of atoms in crystal structures from an empirical relationship between bond lengthR _M−X and bond valenceS _A−X =exp [(R ₀ −R _M−X ) /b] as sites where the BV sumV(A)=∑ s _M−X equals the formal valenceV _id of the cationM ⁺ . Our modified BV approach that systematically accounts for the softness of the bond may then be effectively used to study the interplay between structure and properties of solid electrolytes. This is exemplified for correlations to experimental data from IR, NMR, and impedance spectroscopy. Combining the bond valence approach with reverse Monte Carlo (RMC) modeling or molecular dynamics (MD) simulations provides a deeper understanding of ion transport mechanisms, especially in highly disordered or amorphous solids. Local structure models for crystalline electrolytes are derived by combining crystallographic structure information with simulations. A method for the prediction of the activation energy of the ionic conductivity from the bond valence analysis of the crystal structure is proposed. Taking into account the mass dependence of the conversion factor from bond valence mismatch into an activation energy scale, we could establish a correlation that holds for different types of mobile ions. The strong coupling of the H⁺ transfer to the anion motion in proton conductors requires a special treatment. For glassy solid electrolytes RMC structure models are BV-analyzed to assess the total number of equilibrium sites and to identify transport pathways for the mobile ions. Recently, we have reported a correlation between the pathway volume fraction and the transport properties that permits to predict both absolute value and activation energy of the dc ionic conductivities of disordered solids (including mixed alkali glasses) directly from their structural models. Here we discuss a corresponding BV analysis of molecular dynamics simulation trajectories that allows quantifying the evolution of pathways in time and the influence of temperature on the transport pathways. Paper presented at the Patras Conference on Solid State Ionics — Transport Properties, Patras, Greece, Sept. 14 — 18, 2004.     

The connection between photon mass andB (3) : The poincaré group

The longitudinal vacuum fieldB ⁽³⁾ is an experimental observable which produces by magnetization a well-defined square-root beam power density dependence. Its longitudinal polarization implies that the helicities of the photon are +1, 0, and –1, and that the little group of the Poincaré group is the rotation group 0(3) of a massive boson. The mass of the photon (m) is therefore related directly toB ⁽³⁾ through the Proca equation, and it is concluded that experimental evidence forB ⁽³⁾ is also evidence for finitem.     

Primitive ideals of C q [SL(3)]

The primitive ideals of the Hopf algebraC _q [SL(3)] are classified. In particular it is shown that the orbits in PrimC _q [SL(3)] under the action of the representation groupH C ^*×C ^* are parameterized naturally byW×W, whereW is the associated Weyl group. It is shown that there is a natural one-to-one correspondence between primitive ideals ofC _q [SL(3)] and symplectic leaves of the associated Poisson algebraic groupSL(3,C).Partially supported by a grant from the N.S.A.     

Magnetization of an electron plasma by microwave pulses: Faraday induction

It has been argued theoretically that the recently proposed vacuum fieldB ⁽³⁾ is not accompanied by a real electric fieldE ⁽³⁾ . Experimental evidence for this interence is available in the data reported by Deschampset al. [10], using microwave magnetization of an electron plasma set up in helium gas. Faraday induction due toB ⁽³⁾ does not occur in the inert gas and is not observed experimentally in the absence of free electrons. WheneverB ⁽³⁾ interacts with free electrons, however, Faraday induction occurs through a pulse of induced magnetization (i.e., induced orbital electronic angular momentum).     

Crystal Structures of Tetrakis‐(4‐chlorophenylthio)‐butatriene and Tetrakis‐(tert‐butylthio)‐butatriene

Tetrakis‐(4‐chlorophenylthio)‐butatriene (3a) and tetrakis‐(tert‐butylthio)‐butatriene (3b) were synthesized, and their crystal structures were determined. The compound 3a is monoclinic, space group P2₁/c, a=6.9785(8), b=8.6803(9), c=22.884(2) Å, β=93.887(6)^o, V=1383.0(3) ų, Z=2. The compound 3b is monoclinic, space group P2₁/n, a=11.0615(6), b=10.8507(4), c=11.2717(6) Å, β =116.427(2)^o, V=1211.5(1) ų, Z=4. The title compounds 3a and 3b reside on an inversion center so that only half of the molecule is crystallographically unique. Both compounds are not planar. The crystal structures of 3a and 3b have cumulated double bonds. The C7–C8–C8ⁱ and C5–C6–C6ⁱ angles that show the linearity in both structures, respectively, are 176.4(3)° in 3a and 175.6(2)° in 3b.     

A Fluorescence Study of Novel Styrylindoles in Homogenous and Microhetrogeneous Media

In this paper are presented absorption and fluorescence emission properties of 3-styrylindoles viz. 3-(2-phenylethenyl-E)-NH-indole (1), 3-[2-(4-nitrophenyl)ethenyl-E)-NH-indole (2), 3-[2-(4-cyanophenyl)ethenyl-E]-N-ethylindole (3) and 3-[2-(4-cyanophenyl)ethenyl-E]-NH-indole (4) in organic solvents, 1,4-dioxane-water binary mixtures and micelles (SDS, CTAB and Triton-X-100). The fluorescence properties of 2-4 have been utilized to probe the microenvironment (binding constant, CMC, micropolarity and solubilization site) of the micelles.     

Noether's theorem and the fieldB (3)

It is shown that the longitudinal, magnetic flux density,B ⁽³⁾ , of vacuum electromagnetic radiation can be accommodated rigorously within Noether's theorem, which relates fundamental spacetime symmetries to fundamental conservation laws. This demonstration linksB ⁽³⁾ to the canonical energy-momentum tensorT _µv that appears in Einstein's field equations of general relativity. Thus,B ⁽³⁾ provides a link between electromagnetism and gravitation which might eventually lead to an unified understanding of field theory.     

Proton and Carbon‐13 NMR Studies of the Phencyclone Adducts of Medium Alkyl Chain‐Length N‐n‐Alkylmaleimides. Hindered Rotations and Magnetic Anisotropic Effects. Ab initio Calculations for Optimized Structures

Abstract

The Diels–Alder adducts, 3a–e, of phencyclone, 1, have been prepared from a series of N‐n‐alkylmaleimides, 2, with medium chain‐length n‐alkyl groups. The maleimides were obtained by cyclodehydration of the N‐n‐alkylmaleamic acids, 4, formed from reaction of maleic anhydride with the corresponding n‐alkylamines. The five adducts prepared included derivatives from n‐heptyl, 3a; n‐octyl, 3b; n‐nonyl, 3c; n‐decyl, 3d; and n‐dodecyl, 3e. The NMR spectra of the adducts were studied in CDCl₃ at ambient temperatures at 300 MHz for proton and 75 MHz for carbon‐13, with full proton assignments achieved by high‐resolution COSY45 spectra for the aryl and the alkyl regions. Slow exchange limit (SEL) spectra were observed for both ¹H and ¹³C spectra showing slow rotation on the NMR timescales of the unsubstituted bridgehead phenyl groups. Endo Diels–Alder adduct stereochemistry was supported by striking magnetic anisotropic shielding effects in the ¹H spectra of the alkyl groups, with the NCH₂ CH ₂ signal of each adduct appearing upfield of tetramethylsilane (TMS) at ca. ?0.32 ppm. Proton NMR spectra for precursor maleamic acids and maleimides are reported, with some solvent effects found (CDCl₃ vs. d ₆‐acetone) for the carbon‐bound HC?CH protons of 4. Ab initio molecular modeling calculations at the Hartree‐Fock level using the 6‐31G* basis set have been performed for two key conformers of the phencyclone adduct of N‐n‐octylmaleimide, as a representative structure for these hindered adducts, to estimate geometric parameters for the adduct. A syn conformer, with the alkyl chain directed into the adduct cavity, was found to be ca. 0.23 kcal/mol lower energy than an anti conformer (in which the alkyl chain was directed away from the phenanthrenoid moiety).