Mixed valence states in cobalt iron cyanide

Cobalt iron cyanide with both Co and Fe in mixed valence states were prepared and characterized. In this mixed valence system the cobalt atom is found both as high spin Co(2+) and low spin Co(III) while iron always appears in low spin state to form two solid solutions: Co(2+)Co(III) hexacyanoferrates (II,III), and Co(2+)Co(III) hexacyanoferrate (II). Such solid solutions have the following formula units: (Co²⁺)_x(Co^III)_1−xK[(Fe^II)_1−x(Fe^III)_x(CN)₆]·H₂O and (Co²⁺)_1.5x(Co^III)_1−xK[Fe^II(CN)₆]·yH₂O (0?x?1, 1?y?14). Compounds within these two series were characterized from Infrared, Mössbauer, X-ray diffraction and thermo-gravimetric data, and magnetic measurements at low temperature. A model for their crystal structure is proposed and the structure for a representative composition refined from XRD powder patterns using the Rietveld method. A simple and reproducible procedure to prepare these solid solutions is provided. Within hexacyanoferrates, such mixed valence states system in both metal centres shows unique features, which are discussed from the obtained data.     

Description of ultracold atoms in a one-dimensional geometry of a harmonic trap with a realistic interaction

We compute the ground-state energy of two atoms in a one-dimensional geometry of a harmonic optical trap. We obtain a dependence of the energy on a one-dimensional scattering length, which corre-sponds to various strengths of the interaction potential V _int (x) = V ₀ exp {?2cx ²}. The calculation is performed by numerical and analytical methods. For the analytical method we choose the oscillator representation method (OR), which has been successfully applied to computations of bound states of various few-body systems. The main results of this paper are (1) a numerical investigation of the validity range of the previously used pseudopotential method and (2) an investigation of the validity range of the OR for the potential V(x) = V _conf (x) + V _int (x) = x ²/2 + V ₀ exp {?2cx ²}.     

Fredholm Determinants and Pole-free Solutions to the Noncommutative Painlevé II Equation

We extend the formalism of integrable operators à la Its-Izergin-Korepin-Slavnov to matrix-valued convolution operators on a semi–infinite interval and to matrix integral operators with a kernel of the form \fracE₁^T(l) E₂(m)l+m{\frac{E_1^T(\lambda) E_2(\mu)}{\lambda+\mu}}, thus proving that their resolvent operators can be expressed in terms of solutions of some specific Riemann-Hilbert problems. We also describe some applications, mainly to a noncommutative version of Painlevé II (recently introduced by Retakh and Rubtsov) and a related noncommutative equation of Painlevé type. We construct a particular family of solutions of the noncommutative Painlevé II that are pole-free (for real values of the variables) and hence analogous to the Hastings-McLeod solution of (commutative) Painlevé II. Such a solution plays the same role as its commutative counterpart relative to the Tracy–Widom theorem, but for the computation of the Fredholm determinant of a matrix version of the Airy kernel.     

Exact solutions for a forced Burgers equation with a linear external force

We investigate the solutions of the Burgers equation , where F(x,t) is an external force and Φ(x,t) represents a forcing term. This equation is first analyzed in the absence of the forcing term by taking F(x,t)=k₁(t)−k₂(t)x into account. For this case, the solution obtained extends the usual one present in the Ornstein-Uhlenbeck process and depending on the choice of k₁(t) and k₂(t) it can present a stationary state or an anomalous spreading. Afterwards, the forcing terms Φ(x,t)=Φ₁(t)+Φ₂(t)x and Φ(x,t)=Φ₃x−Φ₄/x³ are incorporated in the previous analysis and exact solutions are obtained for both cases.     

Realizations ofU q(su(1, 1)) by deformed quantum harmonic oscillator algebras and associated deformed Heisenberg algebras

Forsu(1, 1)-symmetric Hamiltonians of quantum mechanical systems (e.g. single-mode quantum harmonic oscillator, radial Schrödinger equation for Coulomb problem or isotropic quantum harmonic oscillator, etc.), the Heisenberg algebra of phase-space variables in two dimensions satisfy the bilinear commutation relation [ip,x]=1 (in normal units). Also there are different realizations ofsu(1, 1) by the generators of quantum harmonic oscillator algebra. We seek here the forms of deformed Heisenberg algebras (bilinear in deformedx and ip) associated with deformedsu(1, 1)-symmetric Hamiltonians. These forms are not unique in contrast to the undeformed case; and these forms are obtained here by considering different realizations of the deformedsu(1, 1) algebra by deformed oscillator algebras (satisfying different bilinear relations in deformed creation and annihilation operators), and then imposing different conditions (e.g. the deformed Heisenberg algebra of the form of the undeformed one, the form of realizations of the deformedsu(1, 1) algebra by deformed phase-space variables being the same as that ofsu(1, 1) algebra by undeformed phase-space variables, etc.), assuming linear relations between deformed phase-space variables and deformed creation-annihilation operators (as it is done in the undeformed case), we get different Heisenberg algebras. These facts are revealed in the case of a two-body Calogero model in its centre of mass frame (and for no other integrable systems in one-dimension having potential of the formV(x _i ? x_j).     

On-shell T-matrices in multiple scattering

The transition operator T for the scattering of a particle from N potentials V_j(x) can be expanded into a series featuring the transition operators t_j associated with the individual potentials. For V_j(x) both absolutely and square integrable in x, we show, using an analytic continuation argument, that if T is on-shell, i.e. in 〈k|T(k₀²±i0)|k′〉, |k| = |k′| = k₀, then each t_j is also on-shell.     

Influence of zirconium doping on structure,microstructure, dielectric and impedance properties of strontium bismuth niobate ceramics

Efforts have been made in this work to enhance the dielectric properties of SrBi₂Nb₂O₉ (SBN) by partial substitution of Zr⁴⁺ for Nb⁵⁺. Systematic investigations on structure, microstructure, dielectric and impedance properties of the SrBi₂(Nb_2−(4/5)xZr_x)O₉ [where, x = 0, 0.1 and 0.2] ceramic samples were carried out to understand the effect of substitution of Zr⁴⁺ for Nb⁵⁺ in SrBi₂Nb₂O₉. The X-ray diffraction (XRD) investigations indicated that the lattice volume of SrBi₂(Nb_{2− (4/5)x}Zr_x)O₉ with x = 0.1 and 0.2 decreases compared to SBN. The SEM investigations revealed an increase in the size of grains and the change on shape of grains to elongated plate shaped structure with the increase of x (x = 0.1 and 0.2) in SrBi₂(Nb_2−(4/5)xZr_x)O₉. Higher Curie temperature and enhanced peak dielectric constant at the Curie temperature were observed for both the SrBi₂(Nb_2−(4/5)xZr_x)O₉ with x = 0.1 and 0.2 ceramic samples compared to SBN. Among the investigated compositions the higher Curie temperature and enhanced peak dielectric constant at the Curie temperature was observed for SrBi₂(Nb_2−(4/5)xZr_x)O₉ with x = 0.1.     

SU(3)×SU(2)×U(1) next-to-leading corrections for proton decay in the SU(5) model

We compute in the standard model of SU(3)^c×(SU(2)×U(1)) with massless quarks and leptons the two-loop anomalous dimensions of the four-fermion operators relevant to proton decay in process involving (u, d, e, ν_e). The calculation is carried out by the use of dimensional reduction, a variant of dimensional regularization. Our aim is to give a complete calculation within the SU(5) GUT model of the next-to-leading enhancement-suppression factor for nucleon decay due to renormalization effects arising from hard gluons, W's and B's in process which involve (u, d, e, ν_e). It turns out that the result is sensitive to the ratios x_(i) = M_{H (i)}/M_X where M_H(i) are the masses of the twelve superheavy Higgs scalars in the $\text{24}$ multiplet which breaks SU(5) → SU(3)×SU(2)×U(1).     

Chiral-odd transversity spin structure function h1(x) of the nucleon in a constituent quark model

We study the chiral-odd transversity spin-dependent quark distribution function h₁(x) of the nucleon in a constituent quark model. The twist-2 structure functions, f₁(x), g₁(x) and h₁(x) are calculated within the diquark spectator approximation. Whereas an inequality f₁(x) > h₁(x) > g₁(x) holds with the interaction between quark and diquark being scalar, the axial-vector effective quark-diquark interaction, which contributes to the d-quark distribution, does not lead to such a simple relation. We find that h₁(x) for the d-quark becomes somewhat smaller than g₁^d(x), when we fix the model parameter to reproduce known other structure functions. We also include corrections due to the non-trivial structure of the constituent quark, which is modeled by the Goldstone boson dressing. This improves agreements of f₁(x) and g₁(x) with experiments, and brings further reduction of h₁^d(x) distribution. Consequences for semi-inclusive experiments are also discussed.     

Kinetic Theory and Lax Equations for Shock Clustering and Burgers Turbulence

We study shock statistics in the scalar conservation law ? _t u+? _x f(u)=0, x∈?, t>0, with a convex flux f and spatially random initial data. We show that the Markov property (in x) is preserved for a large class of random initial data (Markov processes with downward jumps and derivatives of Lévy processes with downward jumps). The kinetics of shock clustering is then described completely by an evolution equation for the generator of the Markov process u(x,t), x∈?. We present four distinct derivations for this evolution equation, and show that it takes the form of a Lax pair. The Lax equation admits a spectral parameter as in Manakov (Funct. Anal. Appl. 10:328–329, 1976), and has remarkable exact solutions for Burgers equation (f(u)=u ²/2). This suggests the kinetic equations of shock clustering are completely integrable.     

(1+1)-Dirac bound states in one dimension, with position-dependent Fermi velocity and mass

We extend Panella and Roy’s [17] work for massless Dirac particles with position-dependent (PD) velocity. We consider Dirac particles where the mass and velocity are both position-dependent. Bound states in the continuum (BIC)-like and discrete bound state solutions are reported. It is observed that BIC-like solutions are not only feasible for the ultra-relativistic (massless) Dirac particles but also for Dirac particles with PDmass and PD-velocity that satisfy the condition m(x) v _F ² (x) = A, where A ≥ 0 is constant. Dirac Pöschl-Teller and harmonic oscillator models are also reported.     

Analysis of the twist-four two-quark process by renormalization mixing

We discuss the renormalization of the twist-four operators (both the two-quark operator and the four-quark operator) and how the anomalous dimensions are calculated in a fairly compact way. Especially, we calculate large spin (n) behaviour of the anomalous dimensions which are relevant for the two-quark process. The most singular behaviour turns out to be 2″ up to some power of n. By using them, we find that, in the twist-four level, W_L ∞ (1 - x)⁰ and W_T(vW₃) ∞ (1 - x)¹ for the pion, and W_L ∞ (1 - x)² and W_T(vW₃) ∞ (1 - x)³ for the nucleon, up to radiative corrections. These are the results which remain after the kinematical target mass effects are taken into account. Target and current dependence is also discussed.     

Universality Limits of a Reproducing Kernel for a Half-Line Schrödinger Operator and Clock Behavior of Eigenvalues

We extend some recent results of Lubinsky, Levin, Simon, and Totik from measures with compact support to spectral measures of Schrödinger operators on the half-line. In particular, we define a reproducing kernel S _L for Schrödinger operators and we use it to study the fine spacing of eigenvalues in a box of the half-line Schrödinger operator with perturbed periodic potential. We show that if solutions u(ξ, x) are bounded in x by ${e^{\epsilon x}}We extend some recent results of Lubinsky, Levin, Simon, and Totik from measures with compact support to spectral measures of Schr?dinger operators on the half-line. In particular, we define a reproducing kernel S _L for Schr?dinger operators and we use it to study the fine spacing of eigenvalues in a box of the half-line Schr?dinger operator with perturbed periodic potential. We show that if solutions u(ξ, x) are bounded in x by e^ex{e^{\epsilon x}} uniformly for ξ near the spectrum in an average sense and the spectral measure is positive and absolutely continuous in a bounded interval I in the interior of the spectrum with x₀ ? I{\xi_0\in I}, then uniformly in I,

\fracSL(x0 + a/L, x0 + b/L)SL(x0, x0)? \fracsin(pr(x0)(a - b))pr(x0)(a - b),\frac{S_L(\xi_0 + a/L, \xi_0 + b/L)}{S_L(\xi_0, \xi_0)}\rightarrow \frac{\sin(\pi\rho(\xi_0)(a - b))}{\pi\rho(\xi_0)(a - b)},      16. Analytic Structure of Many-Body Coulombic Wave Functions      S?ren?Fournais  Maria?Hoffmann-Ostenhof  Thomas?Hoffmann-Ostenhof  Thomas??stergaard?S?rensen 《Communications in Mathematical Physics》2009,289(1):291-310 We investigate the analytic structure of solutions of non-relativistic Schrödinger equations describing Coulombic many-particle systems. We prove the following: Let ψ(x) with \({{\bf x} = (x_{1},\dots, x_{N})\in \mathbb {R}^{3N}}\) denote an N-electron wavefunction of such a system with one nucleus fixed at the origin. Then in a neighbourhood of a coalescence point, for which x 1 = 0 and the other electron coordinates do not coincide, and differ from 0, ψ can be represented locally as ψ(x) = ψ (1)(x) + |x 1|ψ (2)(x) with ψ (1), ψ (2) real analytic. A similar representation holds near two-electron coalescence points. The Kustaanheimo-Stiefel transform and analytic hypoellipticity play an essential role in the proof.      17. Perturbation of the Laplacian supported by null sets,with applications to polymer measures and quantum fields      Sergio Albeverio  Jens-Erik Fenstad  Raphael Høegh-Krohn  Witold Karwowski  Tom Lindstrøm 《Physics letters. A》1984,104(8):396-400 We discuss hamiltonians in L2(Rd, dx) of the form H = ?Δ + V, with V a potential supported by a zero measure set C. In particular if C is a path of a brownian motion b such that V(x) = ∫01λ(x, ω)δ(x-b(s, ω)) ds, we show that H exists as a nontrivial, self-adjoint, lower bounded perturbation of ?Δ when d ?5. We must choose λ to be an infinitesimal, negative function for d = 4,5, but for d ? 3 any bounded real-valued function λ will do. The connection with Edward's model of polymers as well as with quantum fields of the ?d4-type is also discussed. The proofs use methods of nonstandard analysis.      18. Control of electronic state in organic conductors by systematic band filling      《Journal of Physics and Chemistry of Solids》2002,63(6-8):1239-1243 Organic superconductors have been usually obtained by the bandwidth control of antiferromagnetic insulator. In order to find another parameter to control the electronic state, the band filling control has been applied to obtain the carrier-controlled organic conductors. As a result, the systematic band filling control for four kinds of organic conductors has been succeeded. The obtained crystals were (a) 2:1 salts, λ-ET2(GaCl41−)1−x(CoCl42−)x [ET=bis(ethylenedithio)tetrathiafulvalene], doping to the antiferromagnetic insulator, (b) 2:1 salts, δ′-ET2(GaCl4)1−x(CoCl4)x, doping to the non-magnetic insulator, (c) 3:1 salts, α-ET3(CoCl4)1−x(GaCl4)x(TCE), doping to the charge ordered salt, and (d) 3:2 salts, β′-ET3(CoCl4)2−x(GaCl4)x, doping to the half-filled band insulator. For both 2:1 salts, (a) and (b), the semiconducting behaviors have transferred to relatively conductive semiconducting ones by doping. The largest change of transport property is observed for (d) β′-phase from semiconducting behavior of parent material to metallic ones [β′-ET3(CoCl4)2−x(GaCl4)x (x=0.88, 0.66)] down to 140 and 100 K, respectively.These systematic band filling control suggests that the doping to the 1/2-filled band insulator is most effective.      19. Gauge symmetry and the EMC spin effect      S. D. Bass 《Zeitschrift fur Physik C Particles and Fields》1993,60(2):343-348 We emphasise the EMC spin effect as a problem of symmetry and discuss the renormalisation of theC=+1 axial tensor operators. This involves the generalisation of the Adler-Bell-Jackiw anomaly to each of these operators. We find that the contribution of the axial anomaly to the spin dependent structure functiong 1(x, Q 2) scales at O(αs). This means that the anomaly can be a largex effect ing 1. Finally we discuss the jet signature of the anomaly.      20. Operators in a translation invariant two-dimensional bag model      R.L Jaffe 《Annals of Physics》1981,132(1):32-52 Operator matrix elements are studied in a translation invariant version of the bag model in two-dimensional space time. In particular we calculate the matrix elements of quark fields Q(x), currents Jμ(x), and bilocal currents Q(x)γμλaQ(0) whose Taylor expansion yields the twist two operators of leptoproduction. The structure functions of this model are related to those of a naive cavity approximation by a particularly simple transformation.

Analytic Structure of Many-Body Coulombic Wave Functions

We investigate the analytic structure of solutions of non-relativistic Schrödinger equations describing Coulombic many-particle systems. We prove the following: Let ψ(x) with \({{\bf x} = (x_{1},\dots, x_{N})\in \mathbb {R}^{3N}}\) denote an N-electron wavefunction of such a system with one nucleus fixed at the origin. Then in a neighbourhood of a coalescence point, for which x ₁ = 0 and the other electron coordinates do not coincide, and differ from 0, ψ can be represented locally as ψ(x) = ψ ⁽¹⁾(x) + |x ₁|ψ ⁽²⁾(x) with ψ ⁽¹⁾, ψ ⁽²⁾ real analytic. A similar representation holds near two-electron coalescence points. The Kustaanheimo-Stiefel transform and analytic hypoellipticity play an essential role in the proof.     

Perturbation of the Laplacian supported by null sets,with applications to polymer measures and quantum fields

We discuss hamiltonians in L²(R^d, dx) of the form H = ?Δ + V, with V a potential supported by a zero measure set C. In particular if C is a path of a brownian motion b such that V(x) = ∫₀¹λ(x, ω)δ(x-b(s, ω)) ds, we show that H exists as a nontrivial, self-adjoint, lower bounded perturbation of ?Δ when d ?5. We must choose λ to be an infinitesimal, negative function for d = 4,5, but for d ? 3 any bounded real-valued function λ will do. The connection with Edward's model of polymers as well as with quantum fields of the ?_d⁴-type is also discussed. The proofs use methods of nonstandard analysis.     

Control of electronic state in organic conductors by systematic band filling

Organic superconductors have been usually obtained by the bandwidth control of antiferromagnetic insulator. In order to find another parameter to control the electronic state, the band filling control has been applied to obtain the carrier-controlled organic conductors. As a result, the systematic band filling control for four kinds of organic conductors has been succeeded. The obtained crystals were (a) 2:1 salts, λ-ET₂(GaCl₄¹⁻)_1−x(CoCl₄²⁻)_x [ET=bis(ethylenedithio)tetrathiafulvalene], doping to the antiferromagnetic insulator, (b) 2:1 salts, δ′-ET₂(GaCl₄)_1−x(CoCl₄)_x, doping to the non-magnetic insulator, (c) 3:1 salts, α-ET₃(CoCl₄)_1−x(GaCl₄)_x(TCE), doping to the charge ordered salt, and (d) 3:2 salts, β′-ET₃(CoCl₄)_2−x(GaCl₄)_x, doping to the half-filled band insulator. For both 2:1 salts, (a) and (b), the semiconducting behaviors have transferred to relatively conductive semiconducting ones by doping. The largest change of transport property is observed for (d) β′-phase from semiconducting behavior of parent material to metallic ones [β′-ET₃(CoCl₄)_2−x(GaCl₄)_x (x=0.88, 0.66)] down to 140 and 100 K, respectively.These systematic band filling control suggests that the doping to the 1/2-filled band insulator is most effective.     

Gauge symmetry and the EMC spin effect

We emphasise the EMC spin effect as a problem of symmetry and discuss the renormalisation of theC=+1 axial tensor operators. This involves the generalisation of the Adler-Bell-Jackiw anomaly to each of these operators. We find that the contribution of the axial anomaly to the spin dependent structure functiong ₁(x, Q ²) scales at O(α_s). This means that the anomaly can be a largex effect ing ₁. Finally we discuss the jet signature of the anomaly.     

Operators in a translation invariant two-dimensional bag model

Operator matrix elements are studied in a translation invariant version of the bag model in two-dimensional space time. In particular we calculate the matrix elements of quark fields Q(x), currents J_μ(x), and bilocal currents Q(x)γ_μλ^aQ(0) whose Taylor expansion yields the twist two operators of leptoproduction. The structure functions of this model are related to those of a naive cavity approximation by a particularly simple transformation.