Derivation of Hamaker Dispersion Energy of Amorphous Carbon Surfaces in Contact with Liquids Using Photoelectron Energy-Loss Spectra

Hamaker interaction energies and cutoff distances have been calculated for disordered carbon films, in contact with purely dispersive (diiodomethane) or polar (water) liquids, using their experimental dielectric functions ε (q, ω) obtained over a broad energy range. In contrast with previous works, a q-averaged <ε (q, ω) >  _q is derived from photoelectron energy-loss spectroscopy (XPS-PEELS) where the energy loss function (ELF) < Im[?1/ε (q, ω)] >  _q is a weighted average over allowed transferred wave vector values, q, given by the physics of bulk plasmon excitation. For microcrystalline diamond and amorphous carbon films with a wide range of (sp³/sp² + sp³) hybridization, non-retarded Hamaker energies, A ₁₃₂ (L < 1 nm), were calculated in several configurations, and distance and wavenumber cutoff values were then calculated based on A ₁₃₂ and the dispersive work of adhesion obtained from contact angles. A geometric average approximation, H _0?CVL ?=?(H _0?CVC H _0?LVL )^1/2, holds for the cutoff separation distances obtained for carbon-vacuum-liquid (CVL), carbon-vacuum-carbon (CVC) and liquid-vacuum-liquid (LVL) equilibrium configurations. The linear dependence found for A _CVL, A _CLC and A _CLV values as a function of A _CVC, for each liquid, allows predictive relationships for Hamaker energies (in any configuration) using experimental determination of the dispersive component of the surface tension, \( {\gamma}_{CV}^d \), and a guess value of the cutoff distance H _0?CVC of the solid.

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Inverse Scattering Problem for a Two Dimensional Random Potential

We study an inverse problem for the two-dimensional random Schrödinger equation (Δ + q + k ²)u = 0. The potential q(x) is assumed to be a Gaussian random function whose covariance operator is a classical pseudodifferential operator. We show that the backscattered field, obtained from a single realization of the random potential q, determines uniquely the principal symbol of the covariance operator of q. The analysis is carried out by combining harmonic and microlocal analysis with stochastic methods.     

A Minkowski Type Trace Inequality and Strong Subadditivity of Quantum Entropy II: Convexity and Concavity

We revisit and prove some convexity inequalities for trace functions conjectured in this paper’s antecedent. The main functional considered is

$ \Phi_{p,q} (A_1,\, A_2, \ldots, A_m) = \left({\rm Tr}\left[\left( \, {\sum\limits_{j=1}^m A_j^p } \, \right) ^{q/p} \right] \right)^{1/q} $

for m positive definite operators A _j . In our earlier paper, we only considered the case q = 1 and proved the concavity of Φ _p,1 for 0 < p ≤ 1 and the convexity for p = 2. We conjectured the convexity of Φ _p,1 for 1 < p < 2. Here we not only settle the unresolved case of joint convexity for 1 ≤ p ≤ 2, we are also able to include the parameter q ≥ 1 and still retain the convexity. Among other things this leads to a definition of an L ^q (L ^p ) norm for operators when 1 ≤ p ≤ 2 and a Minkowski inequality for operators on a tensor product of three Hilbert spaces – which leads to another proof of strong subadditivity of entropy. We also prove convexity/concavity properties of some other, related functionals.

     

Inverse Scattering at Fixed Energy on Surfaces with Euclidean Ends

On a fixed Riemann surface (M ₀, g ₀) with N Euclidean ends and genus g, we show that, under a topological condition, the scattering matrix S _V (λ) at frequency λ > 0 for the operator Δ+V determines the potential V if \({V\in C^{1,\alpha}(M_0)\cap e^{-\gamma d(\cdot,z_0)^j}L^\infty(M_0)}\) for all γ > 0 and for some \({j\in\{1,2\}}\) , where d(z, z ₀) denotes the distance from z to a fixed point \({z_0\in M_0}\) . The topological condition is given by \({N\geq \max(2g+1,2)}\) for j = 1 and by N ≥ g + 1 if j = 2. In \({\mathbb {R}^2}\) this implies that the operator S _V (λ) determines any C ^{1, α} potential V such that \({V(z)=O(e^{-\gamma|z|^2})}\) for all γ > 0.     

The Boundary Control Approach to the Titchmarsh-Weyl <Emphasis Type="Italic">m</Emphasis>–Function. I. The Response Operator and the <Emphasis Type="Italic">A</Emphasis>–Amplitude

We link the Boundary Control Theory and the Titchmarsh-Weyl Theory. This provides a natural interpretation of the A?amplitude due to Simon and yields a new efficient method to evaluate the Titchmarsh-Weyl m?function associated with the Schrödinger operator H = ?? _x ²  + q(x) on L ₂(0, ∞) with Dirichlet boundary condition at x = 0.     

Limit of Fluctuations of Solutions of Wigner Equation

We consider fluctuations of the solution W _ε (t, x, k) of the Wigner equation which describes energy evolution of a solution of the Schrödinger equation with a random white noise in time potential. The expectation of W _ε (t, x, k) converges as ε → 0 to \({\bar{W}(t,x,k)}\) which satisfies the radiative transport equation. We prove that when the initial data is singular in the x variable, that is, W _ε (0, x, k) = δ(x)f(k) and \({f\in {\mathcal{S}}(\mathbb{R}^d)}\), then the laws of the rescaled fluctuation \({Z_\varepsilon(t):=\varepsilon^{-1/2}[W_\varepsilon(t,x,k)-\bar{W}(t,x,k)]}\) converge, as ε → 0+, to the solution of the same radiative transport equation but with a random initial data. This complements the result of [6], where the limit of the covariance function has been considered.     

Effect of the hexagonal warping on the dynamical conductivity of surface states in a topological insulator

We report on the effect of hexagonal warping on the dynamical conductivity of the surface states of a topological insulator in the presence of nonmagnetic impurities. It is found that the photon energy dependent conductivities are determined by a polarization-function-liked term,  Π₂ (q,ω), which contains a velocity term corresponding to the difference of group velocities between the two states due to an electron-impurity scattering. This is different from the conductivity of 2-dimentional electron systems where the conductivity depends on the inverse imaginary part of the dielectric function Im [1/κ(q,ω)]. We present both the real part and imaginary part of the polarization function with different warping strength. It is found that the warping strength can both enhance single particle excitations (SPEs) and suppress the screening effect of electrons. As a result the inverse scattering time is enhanced by up to about two orders of magnitudes. The real part of the longitudinal conductivity of the intra-band process is analog to the case with a conductivity of σ ~ μδ(ω). The broadening of the spectrum in the low energy is not only determined by chemical potential, but also dependent on the warping strength. At higher frequency, the real part of conductivity shows a jump at the threshold photon energy of μ, where the inter-band contribution takes over.     

Delbrück-Streuung von 17 MeV-Photonen

Differential cross sections have been measured for the small angle scattering ofγ-rays by iron, silver, tantalum, lead and uranium with 17 MeV photons from Li⁷ (p, γ) Be⁸ at mean four momentum transfersq of 0·5mc and 1·3mc and with 7 MeV photons from F¹⁹(p, αγ)O¹⁶ at mean four momentum transfer of 0·5mc. Under these experimental conditions only Compton, Rayleigh and possibly Delbrück scattering are of importance. Extrapolation of known theoretical results to higher energies shows, that Rayleigh and Compton scattering from bound electrons should depend only onq for small angles, smallq and fixedZ. Using this it follows, that at 17 MeV andq≈0·5mc an additional scattering process must be present, which increases with growingZ and which is negligible in the measurements at 17 MeV withq≈1·3mc and at 7 MeV withq≈0·5mc. These results are in qualitative agreement with the approximate theory for Delbrück scattering ofBethe andRohrlich, however experimental cross sections at 17 MeV andq≈0·5mc are about a factor of 1·6 lower than those predicted by this theory. This discrepancy is not unexpected, since exact calculations of Delbrück scattering amplitude fromKessler andZernik at 2·62 MeV and 6·14 MeV show even greater deviations in the same sense.     

Noncompact <Emphasis Type="Italic">u</Emphasis><Subscript>q</Subscript>(2, 1) quantum algebra: Discrete series of highest weight irreducible representations

The structure of unitary irreducible representations of the noncompact u_q(2, 1) quantum algebra that are related to a negative discrete series is examined. With the aid of projection operators for the su_q(2) subalgebra, a q analog of the Gelfand-Graev formulas is derived in the basis corresponding to the reduction u_q(2, 1) → su_q(2)×u(1). Projection operators for the su_q(1, 1) subalgebra are employed to study the same representations for the reduction u_q(2, 1) → u(1)×su_q(1, 1). The matrix elements of the generators of the u_q(2, 1) algebra are computed in this new basis. A general analytic expression for an element of the transformation brackets <U∣T>_q between the bases associated with the above two reductions (the elements of this matrix are referred to as q Weyl coefficients) is obtained for a general case where the deformation parameter q is not equal to a root of unity. It is shown explicitly that, apart from a phase, the q Weyl coefficients coincide with the q Racah coefficients for the su_q(2) quantum algebra.     

Asymptotic Behavior of a Network of Oscillators Coupled to Thermostats of Finite Energy

We study the asymptotic behavior of a finite network of oscillators (harmonic or anharmonic) coupled to a number of deterministic Lagrangian thermostats of finite energy. In particular, we consider a chain of oscillators interacting with two thermostats situated at the boundary of the chain. Under appropriate assumptions, we prove that the vector (p, q) of moments and coordinates of the oscillators in the network satisfies (p, q)(t) → (0, q _c ) as t → ∞, where q _c is a critical point of some effective potential, so that the oscillators just stop. Moreover, we argue that the energy transport in the system stops as well without reaching thermal equilibrium. This result is in contrast to the situation when the energies of the thermostats are infinite, studied for a similar system in [14] and subsequent works, where the convergence to a nontrivial limiting regime was established.The proof is based on a method developed in [22], where it was observed that the thermostats produce some effective dissipation despite the Lagrangian nature of the system.     

Rigidity and Non-local Connectivity of Julia Sets of Some Quadratic Polynomials

For an infinitely renormalizable quadratic map \({f_c: z\mapsto z^2+c}\) with the sequence of renormalization periods {k _m } and rotation numbers {t _m  = p _m /q _m }, we prove that if \({\limsup k_m^{-1} \log |p_m| >0 }\), then the Mandelbrot set is locally connected at c. We prove also that if \({\limsup |t_{m+1}|^{1/q_m} <1 }\) and q _m → ∞, then the Julia set of f _c is not locally connected and the Mandelbrot set is locally connected at c provided that all the renormalizations are non-primitive (satellite). This quantifies a construction of A. Douady and J. Hubbard, and weakens a condition proposed by J. Milnor.     

The effect of non-linear capacitances in the localization properties of aperiodic dual electric transmission lines

This study investigates the localization properties of dual electric transmission lines with non-linear capacitances. The V_C,n voltage across each capacitor is selected as a non-linear function of the electric charge q_n, i.e., V_C,n = q_n(1/C_n -ε_n|q_n|²)where C_n is the linear part of the capacitance and ε_n the amplitude of the non-linear term. We follow a binary distribution of values of ε_n, according to the Thue-Morse m-tupling sequence. The localization behavior of this non-linear case indicates that the case m = 2 does not belong to the m ≥ 3, family because when m changes from m = 2 to m = 3, the number of extended states diminishes dramatically. This proves the topological difference of the m = 2 and m = 3 families. However, by increasing m values, localization behavior of the m-tupling family resembles that of the m = 2, case because the system begins to regain its extended states. The exact same result was obtained recently in the study of linear direct transmission lines with m-tupling distribution of inductances. Consequently, we state that the localization behavior of the m-tupling family as a function of the m value is independent of both the linear and the non-linear system under study, but independent of the kind of transmission line (dual or direct). This is curious behavior of the m-tupling family and thus deserves more scholarly attention.     

On the Negative Spectrum of the Two-Dimensional Schrödinger Operator with Radial Potential

For a two-dimensional Schrödinger operator H _{α V}  = ?Δ ?αV with the radial potential V(x) = F(|x|), F(r) ≥ 0, we study the behavior of the number N _?(H _{α V} ) of its negative eigenvalues, as the coupling parameter α tends to infinity. We obtain the necessary and sufficient conditions for the semi-classical growth N _?(H _{α V} ) = O(α) and for the validity of the Weyl asymptotic law.     

Internal friction phenomena in composites based on PZT-type ferroelectric powder and ferrites

The aim of the work was to determine the phenomena of internal friction (mechanical losses) occurring in ferroelectric-ferromagnetic composites created based on PZT-type ferroelectric powder and ferrite. The composites were obtained using ceramic powders – multi-component PZT-type solid solutions with ferroelectric properties. Their magnetic component included zinc-nickel powder Ni_0.64Zn_0.36Fe₂O₄. 30 × 10 × 1 mm³ test specimens were obtained using free sintering. Temperature Q^-1(T) and amplitude Q^-1(ε) internal friction dependencies were determined. Wide high temperature maxima were observed with regard to the internal friction temperature dependencies obtained for the tested specimens. The conducted measurements of amplitude (isothermal) dependencies of internal friction Q^-1(ε) for the tested composites have allowed for interpreting the previously observed maximum on the temperature dependencies of internal friction.     

Special case of sunset: Reduction and <Emphasis Type="Italic">ε</Emphasis> expansion

We consider two-loop sunset diagrams with two mass scales m and M at the threshold and pseudothreshold that cannot be treated by earlier published formulas. The complete reduction to master integrals is given. The master integrals are evaluated as a series in ratio m/M and in ε with the help of a differential-equation method. The rules of asymptotic expansion in the case when q² is at the (pseudo) threshold are given.     

New forms of the Cauchy operator and some of their applications

In this paper, we first construct the Cauchy q-shift operator T(a, b;D _xy ) and the Cauchy q-difference operator L(a, b; θ _xy ). We then apply these operators in order to represent and investigate some new families of q-polynomials which are defined in this paper. We derive some q-identities such as generating functions, symmetry properties and Rogers-type formulas for these q-polynomials. We also give an application for the q-exponential operator R(bD _q ).     

Spin-dimerization in rare-earth substituted La<Subscript>2</Subscript>RuO<Subscript>5</Subscript>

The Ru-Ru spin-singlet formation in La_2 ? x L n _x RuO₅ (Ln = Pr, Nd, Sm, Gd, Dy) was investigated by measurements of the specific heat and magnetic susceptibility. After subtraction of the lattice contribution from the specific heat (C _p ), similar excess entropy values were obtained for all compounds. These entropies can be explained by the formation of antiferromagnetic Ru-spin dimers at low temperatures and provide a lower estimate for the intradimer exchange strength. Pronounced changes in the transition temperatures and a broadening of the corresponding peak in C _p were observed. These changes depend on the rare-earth element and are due to local structural changes and heterogeneities caused by the substitution. The magnetic susceptibilities can be described by the sum of a rare-earth paramagnetic moment and the susceptibility of the unsubstituted La₂RuO₅. Density functional theory (DFT) calculations were performed for various compounds to investigate the origin of the magnetic transition and the relationship between structural changes and the spin-dimerization temperature. The combination of the present results with previous structural investigations supports the model of a spin-pairing of the Ru moments which occurs as a reason of the structural phase transition in La_2 ? x L n _x RuO₅.     

Dependence of structure factor and correlation energy on the width of electron wires

The structure factor and correlation energy of a quantum wire of thickness b ? a _B are studied in random phase approximation (RPA) and for the less investigated region r _s < 1. Using the single-loop approximation, analytical expressions of the structure factor are obtained. The exact expressions for the exchange energy are also derived for a cylindrical and harmonic wire. The correlation energy in RPA is found to be represented by ? _c (b, r _s ) = α(r _s )/b + β(r _s ) ln(b) + η(r _s ), for small b and high densities. For a pragmatic width of the wire, the correlation energy is in agreement with the quantum Monte Carlo simulation data.     

Ferrimagnetic nanoparticles for self-controlled magnetic hyperthermia

Based on the Heisenberg model including single-site uniaxial anisotropy and using aGreen’s function technique we studied the influence of size and composition effects on theCurie temperature T _C , saturationmagnetization M _S and coercivityH _C of spherical nanoparticles with astructural formulaM e _1?x Zn _x Fe₂O₄,Me = Ni, Cu, Co, Mn. It is shown that for x = 0.4–0.5and d = 10–20 nm these nanoparticles have aT _C  = 315 K and are suitable for aself-controlled magnetic hyperthermia.     

Photoionization of the Xe atom and Xe@C<Subscript>60</Subscript> molecule

Photoionization of the Xe atom and Xe@C₆₀ molecule have been studied usingthe random phase approximation with exchange (RPAE) method. The Xe atom was described byrelaxed orbitals including overlap integrals. The C₆₀ fullerene has beenrepresented by an attractive short range spherical well with potentialV(r), given byV(r) =  ?V ₀ forr _i  < r < r _o ,otherwise V(r) = 0 wherer _i andr _o are respectively, the inner and outerradii of the spherical shell. The time independent Schrödinger equation was solved usingboth regular and irregular solutions and the continuous boundary conditions atr _i andr _o . The results demonstrate improvementto previous calculations for both the Xe atom and Xe@C₆₀ molecule and comparevery well with the recent experimental data.