Very-short-wavelength collective modes in fluids

The existence of very-short-wavelength collective modes in fluids is discussed. These collective modes are the extensions of the five hydrodynamic (heat, sound, viscous) modes to wavelengths of the order of the mean free path in a gas or to a fraction of the molecular size in a liquid. They are computed here explicitly on the basis of a model kinetic equation for a hard sphere fluid. At low densities all five modes are increasingly damped with decreasing wavelength till each ceases to exist at a cutoff wavelength. At high densities the extended heat mode softens very appreciably for wavelengths of the order of the size of the particles and becomes a diffusion-like mode that persists till much shorter wavelengths than the other modes. Except for the shortest wavelengths these collective modes and in particular the heat mode dominate the dynamical structure factorS(k, ) for all densities. The agreement of the theory with experimentalS(k, ) of liquid Ar seems to imply that very-short-wavelength collective modes also occur in real fluids.     

Dynamic nuclear polarization system for the SANS-J-II spectrometer at JAEA

We have developed a dynamic nuclear polarization (DNP) system for the SANS-J-II spectrometer at the JRR-3 atomic research reactor of Japan Atomic Energy Agency (JAEA). The DNP system is composed of a split-type horizontal superconducting magnet (3.3 T), a Gunn oscillator as a microwave source (94 GHz), and a cryostat (1.2 K). In particular, a sample cell with 40 in inner diameter and the magnet with a field homogeneity of 5×10⁻⁵ in a volume of 25 mm×8 mm were employed to polarize samples with a diameter of 20 mm for the ultra small-angle scattering experiment using the magnetic lens installed at the SANS-J-II spectrometer [S. Koizumi, H. Iwase, J. Suzuki, T. Oku, R. Motokawa, H. Sasao, H. Tanaka, D. Yamaguchi, H.M. Shimizu, T. Hashimoto, J. Appl. Crystallogr. 40 (2007) s474]. We obtained the proton polarization |P|=32% in the polyethylene doped with 2,2,6,6-tetra-methyl-piperidine-1-oxyl (TEMPO).     

The incoherent scattering function and related correlation functions in hard sphere fluids at short times

For a classical fluid of hard spheres and hard disks exact expressions for all densities and wave vectors are derived for the coefficients oft ⁿ in the short-time expansion of the incoherent intermediate scattering function (n = 0, 1,..., 4) and the velocity correlation function (n=0,1,2). Similarly, we obtain the coefficient of the leading term in the short-time behavior of the cumulants of the displacements. Furthermore,S(k, ) has a high-frequency tail ^–4, characteristic for the hard-sphere fluid, which leads to a modification of the standard sum rules. We present estimates for the frequency range, in which this tail may be observed in neutron scattering off noble gases. The results are also compared with Enskog's theory and molecular dynamics calculations.     

Charge correlations in cobaltates La2–xSrx CoO4

We report on an elastic neutron scattering study of the charge correlations in La_2–xSr_x CoO₄ with x = 1/3, 0.4 and 0.5. We found that the checkerboard charge ordering correlations present in the x = 0.5 sample persist in the x = 0.4 and 1/3 materials. These checkerboard charge ordering correlations are robust and explain the occurrence of nano‐phase separation in layered cobaltates for Sr‐concentrations away from half‐doping. The half‐integer reflections then arise from the nanometer‐sized hole‐rich regions (blue areas in title figure) instead of the undoped ones (red areas in title figure). The appearance of nano‐phase separation is an important ingredient for understanding the formation of hour‐glass shaped magnetic excitation spectra in La_2–xSr_x CoO₄.

Nano‐phase separation in La_2–xSr_x CoO₄ (schematically). Red areas: undoped La₂CoO₄ islands, blue areas: checkerboard charge ordered regions; black, green and blue balls represent nonmagnetic Co³⁺ ions, magnetic Co²⁺ ions and oxygen ions, respectively; green arrows indicate Co²⁺ spins [1, 2].     

Exact Solutions of the Schrödinger Equation with Inverse-Power Potential

The Schrödinger equation for stationary states is studied in a central potential V(r) proportional to r ^– in an arbitrary number of spatial dimensions. The presence of a single term in the potential makes it impossible to use previous algorithms, which only work for quasi-exactly-solvable problems. Nevertheless, the analysis of the stationary Schrödinger equation in the neighbourhood of the origin and of the point at infinity is found to provide relevant information about the desired solutions for all values of the radial coordinate. The original eigenvalue equation is mapped into a differential equation with milder singularities, and the role played by the particular case = 4 is elucidated. In general, whenever the parameter is even and larger than 4, a recursive algorithm for the evaluation of eigenfunctions is obtained. Eventually, in the particular case of two spatial dimensions, the exact form of the ground-state wave function is obtained for a potential containing a finite number of inverse powers of r, with the associated energy eigenvalue.     

Neutron Scattering and a.c. Susceptibility Studies on the Geometrically Frustrated Tb2Ti2O7

We have performed neutron scattering and a.c. susceptibility _a.c. studies on polycrystallline Tb₂Ti₂O₇ to investigate the magnetic correlations in this antiferromagnetic pyrochlores. Evidence of changes in the magnetic correlation length has been observed by neutron diffraction in different parts of an applied field/temperature phase diagram. This cooperative paramagnet has been shown to increase its magnetic correlations, from nearest neighbour to 100 Å, in modest fields (<7 T) and temperatures (T>2 K); however, in zero field the system does not order above 15 mK, even with its large moment.     

Collective modes,damping, and the scattering function in classical liquids

An exact representation for the density-density response function is presented. This representation is a generalization of the result obtained in the mean field approximation and amounts to replacing the static, effective potential by one which is both wavenumber- and frequency-dependent. This potential possesses both a real and an imaginary part; the latter describes the collisional damping of collective modes. Analyticity and sum rule arguments are used to describe the basic properties of this complex potential. The formalism allows us to write an exact formula for the scattering functionS(k, ) in which the basic unknown is the collisional damping function. Using a small portion of the recent experimental data on coherent neutron scattering in liquid argon, we are able to calculateS(k, ) and other quantities of interest and to make comparisons with the rest of the data.     

Spatial modulation of a neutron beam by Larmor precession

We modulated the intensity of a neutron beam using Larmor precession techniques. We simulated triangular coils by using magnetised foils in electromagnets with rectangular pole shoes. Reversing the orientation of two of the magnetised foils in the spin-echo small-angle neutron scattering (SESANS)-instrument in Delft and changing the field settings was sufficient to obtain a modulation with a period in the order of 1 mm. We expect to be able to go to a modulation with a period of .This technique can be used to measure small-angle neutron scattering, analogous to the method used in SESANS, but with the advantage that all the manipulation of the neutron spin occurs before the sample. This means that the technique is insensitive to magnetic perturbations at the sample position. By positioning several of these devices after each other it will be possible to obtain a sharper modulation, or a modulation in two directions.     

Neutron diffraction study of quasi-one-dimensional spin-chain compounds Ca<Subscript>3</Subscript>Co<Subscript>2–<Emphasis Type="Italic">x</Emphasis></Subscript>Fe<Subscript><Emphasis Type="Italic">x</Emphasis></Subscript>O<Subscript>6</Subscript>

We report the results of the DC magnetization, neutron powder diffraction and neutron depolarization studies on the spin-chain compounds Ca₃Co_2–x Fe _x O₆ (x = 0, 0.1, 0.2 and 0.4). Rietveld refinement of neutron powder diffraction patterns at room temperature confirms the single-phase formation for all the compounds in rhombohedral structure with space group Rc. Rietveld refinement also confirms that Fe was doped at the trigonal prism site, 6a (0, 0, 1/4) of Co. The high temperature magnetic susceptibility obeys the Curie-Weiss law; the value of the paramagnetic Curie temperature (θ _p) decreases as the concentration of iron increases and it becomes negative for x = 0.4. No extra Bragg peak as well as no observable enhancement in the intensity of the fundamental (nuclear) Bragg peaks has been observed in the neutron diffraction patterns down to 30 K. No depolarization of neutron beam has been observed down to 3 K confirming the absence of ferro- or ferrimagnetic-like correlation.      

Computer simulation of some dynamical properties of the Lorentz gas

We carried out molecular dynamics simulations of a Lorentz gas, consisting of a lone hydrogen molecule moving in a sea of stationary argon atoms. A Lennard-Jones form was assumed for the H₂-Ar potential. The calculations were performed at a reduced temperatureK ^* =kT/_{H 2–Ar} = 4.64 and at reduced densities ^*= _{Ar Ar} ³ in the range 0.074–0.414. The placement of Ar atoms was assumed to be random rather than dictated by equilibrium considerations. We followed the trajectories of many H₂ molecules, each of which is assigned in turn a velocity given by the Maxwell-Boltzmann distribution at the temperature of the simulation. Solving the equations of motion classically, we obtained the translational part of the incoherent dynamic structure factor for the H₂ molecule,S _tr(q, ). This was convoluted with the rotational structure factorS _rot(q, ) calculated assuming unhindered rotation to obtain the total structure factorS(q, ). Our results agree well with experimental data on this function obtained by Egelstaffet al. At the highest density ( ^*=0.414) we studied the dependence ofS(q, ) on system size (number of Ar atoms), number of H₂ molecules for which trajectories are generated, and the length of time over which these trajectories are followed.     

Radiation Scattering under Laser Irradiation of a Target Blown by a Two-Phase Stream

We have investigated the spectral characteristics of scattered radiation ( = 0.69 m and = 1.06 m) under probing laser action (q = 10³ W/cm²) on a textolite target of streamline shape blown by a cold two-phase stream (with talc and black particles). By the measured transverse scattering indicatrices the dependences of the scattered radiation intensity on the probing angles and the sort of particles and their number density in the stream have been established. The relative contributions of particles of the stream and the target surface to the total scattering by the target–two-phase-stream system have been determined.     

Differential entropy and tiling

This paper relates the differential entropy of a sufficiently nice probability density functionp on Euclideann-space to the problem of tilingn-space by the translates of a given compact symmetric convex setS with nonempty interior. The relationship occurs via the concept of the epsilon entropy ofn-space under the norm induced byS, with probability induced byp. An expression is obtained for this entropy as approaches 0, which equals the differential entropy ofp, plusn times the logarithm of 2/, plus the logarithm of the reciprocal of the volume ofS, plus a constantC(S) depending only onS, plus a term approaching zero with. The constantC(S) is called the entropic packing constant ofS; the main results of the paper concern this constant. It is shown thatC(S) is between 0 and 1; furthermore,C(S) is zero if and only if translates ofS tile all ofn-space.This paper presents the results of one phase of research carried out at the Jet Propulsion Laboratory, California Institute of Technology, under Contract No. NAS 7-100, sponsored by the National Aeronautics and Space Administration.     

The Green-Kubo Formula for the Spin-Fermion System

The spin-fermion model describes a two level quantum system (spin 1/2) coupled to finitely many free Fermi gas reservoirs which are in thermal equilibrium at inverse temperatures β _j . We consider non-equilibrium initial conditions where not all β _j are the same. It is known that, at small coupling, the combined system has a unique non-equilibrium steady state (NESS) characterized by strictly sitive entropy production. In this paper we study linear response in this NESS and prove the Green-Kubo formula and the Onsager reciprocity relations for heat fluxes generated by temperature differentials. Dedicated to Jean Michel Combes on the occasion of his sixtyfifth birthday     

Completeness of inner product spaces and quantum logic of splitting subspaces

We show that an inner product space S is complete whenever its system E(S) of all splitting subspaces, i.e., of all subspaces E for which EE =S holds, is a quantum logic, that is, an orthocomplemented orthomodular -orthoposet. It is well known that the quantum logic is an important axiomatic model of quantum mechanics. This generalizes the result of G. Cattaneo and G. Marino (Lett. Math. Phys. 11, 15–20 (1986)) who required that E(S) be a lattice. Moreover, the conditions are weakened to show that S is complete whenever E(S) contains the join of any sequence of one-dimentional orthogonal subspaces.     

Braided structure of fractionalZ 3-supersymmetry

It is shown that fractionalZ ₃-superspace is isomorphic to theqexp(2/3i) limit of the braided line.Z ₃-supersymmetry is identified as translational invariance along this line. The fractional translation generator and its associated covariant derivative emerge as theqexp(2/3i) limits of the left and right derivatives from the calculus on the braided line.Presented at the 5th International Colloquium on Quantum Groups: Quantum Groups and Integrable Systems, Prague, 20–22 June 1996.This paper describes research supported in part by E.P.S.R.C and P.P.A.R.C. (UK) and by the C.I.C.Y.T (Spain). J.C.P.B. wishes to acknowledge an FPI grant from the CSIC and the Spanish Ministry of Education and Science.     

A uniqueness condition for Gibbs measures,with application to the 2-dimensional Ising antiferromagnet

A uniqueness condition for Gibbs measures is given. This condition is stated in terms of (absence of) a certain type of percolation involving two independent realisations. This result can be applied in certain concrete situations by comparison with ordinary percolation. In this way we prove that the Ising antiferromagnet on a square lattice has a unique Gibbs measure if (4–|h)<1/2ln(P _c /(1–P _c )), whereh denotes the external magnetic field, the inverse temperature, andP _c the critical probability for site percolation on that lattice. SinceP _c is larger than 1/2, this extends a result by Dobrushin, Kolafa and Shlosman (whose proof was computer-assisted).The research which led to this paper started while the author was at Cornell University, partly supported by the U.S. Army Research Office through the Mathematical Sciences Institute of Cornell University     

X‐ray and neutron scattering on disordered nanosize clusters: a case study of lead‐zirconate‐titanate solid solutions

Defects and frequently used defect models of solids are reviewed. Signatures for identifying the disorder from x‐ray and neutron scattering data are given. To give illustrative examples how technologically important defects contribute to x‐ray and neutron scattering numerical method able to treat non‐periodical solids possessing several simultaneous defect types is given for simulating scattering in nanosize disordered clusters. The approach takes particle size, shape, and defects into account and isolates element specific signals. As a case study a statistical approximation model for lead‐zirconate titanate [Pb(Zr_xTi)O₃, PZT] is introduced. PZT is a material possessing several defect types, including substitutional, displacement and surface defects. Spatial composition variation is taken into account by introducing a model in which the edge lengths of each cell depend on the distribution of Zr and Ti ions in the cluster. Spatially varying edge lengths and angles is referred to as microstrain. The model is applied to compute the scattering from ellipsoid shaped PZT clusters and to simulate the structural changes as a function of average composition. Two‐phase co‐existence range, the so called morphotropic phase boundary composition is given correctly. The composition at which the rhombohedral and tetragonal cells are equally abundant was . Selected x‐ray and neutron Bragg reflection intensities and line shapes were simulated. Examples of the effect of size and shape of the scattering clusters on diffraction patterns are given and the particle dimensions, computed through Scherrer equation, are compared with the exact cluster dimensions. Scattering from two types of 180° domains in spherical particles, one type assigned to Ti‐rich PZT and the second to the MPB and Zr‐rich PZT, is computed. We show how the method can be used for modelling polarization reversal.

     

Pulse broadening in near-optimum graded-index fibres

Impulse responses of a wide class of multimode optical fibres with near-optimum index profiles are investigated. The index profile is described by three parameters: the power law exponent, the magnitude and the width of a dip at the centre of the core. It is found that for overcompensated profiles (< _opt) a moderate dip decreases the r.m.s. dispersion. Leaky modes are found to improve the dispersion characteristics for fibres with a dip in the index profile. Impulse response measurements are interpretated to give information about the index profile.     

Scaling behavior of surface irregularity in the molecular domain: From adsorption studies to fractal catalysts

For an unexpected variety of solids, the surface topography from a few up to as many as a thousand angstroms is very well described by fractal dimension,D. This follows from measurements of the number of molecules in surface monolayers, as function of adsorbate or adsorbent particle size. As an illustration, we present a first case, amorphous silica gel, whereD has been measured independently by each of the two methods. (The agreement, 3.02±0.06 and 3.04±0.05, is excellent, and the result is modeled by a heavy generalized Menger sponge.) The examples as a whole divide into amorphous and crystalline materials, but presumably all of them are to be modeled as random fractal surfaces. The observedD values exhaust the whole range between 2 and 3, suggesting that there are a number of different mechanisms by which such statistically self-similar surfaces form. We show that fractal surface dimension entails interfacial power laws much beyond what is the source of theseD values. Examples are reactive scattering events when neutrons of variable flux pass the surface (this is of interest for locating fractal substrates that may support adlayer phase transitions); the rate of diffusion-controlled chemical reactions at fractal surfaces; and the fractal implementation of the traditional idea that the active sites of a catalyst are edge and apex sites on the surface.