On accurate computations of slowly convergent atomic properties in few-electron ions and electron-electron correlations

We discuss an approach to accurate numerical computations of slowly convergent propertiesin two-electron atoms/ions which include the negatively charged Ps^? (e^?e⁺e^?)and H^? ions, Heatom and positively charged, helium-like ions from Li⁺ to Ni²⁶⁺. All these ions areconsidered in their ground 1¹S-state(s). The slowly convergent properties selected inthis study include the electron-nulceus (r^2k_eN) and electron-electron (r^2k_ee) expectation values for k = 2, 3, 4 and 5.     

The bond force constants of graphene and benzene calculated by density functional theory

Stretching (k_r) and bending (k_θ) bond force constants appropriate to describe the bond stiffness of graphene and benzene are calculated using density functional theory. The effect of employing different exchange-correlation functionals for the calculation of k_r and k_θ is discussed using the generalised gradient approximation (GGA) and the local density approximation (LDA). For benzene, k_r = 7.93 mdyn Å^-1 and k_θ = 0.859 mdyn Å rad^-2 using LDA, while k_r = 7.67 mdyn Å^-1 and k_θ = 0.875 mdyn Å rad^-2 using GGA. For graphene, k_r = 7.40 mdyn Å^-1 and k_θ = 0.769 mdyn Å rad^-2 using LDA, while k_r = 6.88 mdyn Å^-1 and k_θ = 0.776 mdyn Å rad^-2 using GGA. This means the difference between the bond force constants for benzene and graphene can be as large as ～12%. The comparison between these two systems allows for elucidation of the effect of periodicity and substitution of carbon atoms by hydrogen in the stiffness of C–C bonds. This effect can be explained by a different redistribution of the charge density when the systems are subjected to strain. The parameters k_r and k_θ computed here can serve as an input to molecular mechanics or finite element codes of larger carbon molecules, which in the past had frequently assumed the same bond force constants for graphene, benzene or carbon nanotubes.     

Diagonal and Off-Diagonal Recursion Relation for the General Potential V(r) = Arp

A recursion relation is derived for the potential V(r) = Ar ^p. Generally, this connects off-diagonal matrix elements of r ^k–2, r ^k+p, r ^k, and r ^k+2. The diagonal case is obtained by setting m = n in this relation. The relation is derived by elementary methods and without recourse to specific properties of the eigenstates. Finally, this relation is studied for the familiar potentials p = –1, 1, 2.     

Dependence of the spontaneous emission of singlet oxygen on the refractive index and molecular polarizability of the surrounding dielectric media

The rate constants (k _r ) for singlet oxygen O₂ (a ¹Δ _g ) luminescence in several selected solvents and in binary solvent mixture (acetone-toluene) were measured. All data have been normalized such that k _r ^rel = 1.0 in toluene. It has been demonstrated that the changes in these rate constants were caused both by optical properties of a medium (the local field factor and density of photon states) and by an inherent property of the emitter of ¹O₂ (the square of transition moment). In its turn, the value of the transition moment is directly proportional to molecular polarizability of the medium molecules.     

Radiation transfer due to a point source in an isotropically scattering,inhomogeneous solid sphere

A method of analysis is presented for solving radiation transfer due to a point source in an absorbing, isotropically scattering, inhomogeneous sphere, which is subjected to a diffusely reflecting boundary. The space-dependent single-scattering albedo is represented in the form ω(r) = Σ^K_k=0 D_K r^k, where r is the optical variable and D_k are the known expansion coefficients. The effects of various functional forms of albedo on the incident radiation, angular distribution of radiation intensity and emissivity are examined.     

The Ruderman-Kittel-Kasuya-Yosida interaction in amorphous La80Au20 alloys with dilute Gd impurities

From magnetization measurements on some amorphous dilute La_80?xGd_xAu₂₀ alloys with x ? 1 we have shown that the magnetic behavior follows the scaling laws of a spin-glass system, characteristic of the 1/r³ dependence of the pairwise interaction. We have also determined the strength of the Ruderman-Kittel-Kasuya-Yosida interaction V(r) = (V₀cos 2k_Fr)/r³, to be V₀ = 0.20 × 10^?37 ergcm³. The corresponding value of the s-f exchange integral is |J_sf| = 0.14 eV, which is compared with values determined from other experiments.     

Systematics of nuclear charge radii of the stable molybdenum isotopes from muonic atoms

The results of precise measurements of the energies of the $\text{2}\text{p}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}\text{?1}\text{s}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{and}\text{2}\text{p}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{?1}\text{s}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ muonic X-ray transitions of ⁹²Mo, ⁹⁴Mo, ⁹⁵Mo, ⁹⁶Mo, ⁹⁷Mo, ⁹⁸Mo and ¹⁰⁰Mo are reported. The data were analyzed in terms of the Barrett moments 〈r^ke^?αr〉 of the nuclear charge distributions from which the equivalent nuclear radii R_k and the differences ΔR_k between neighboring isotopes were computed. Systematic shell effects have been observed at the neutron numbers N = 50 and N = 56.     

Calculation method for the continuum states of atomic systems

In the present work, we develop a calculational method of solving the scattering equations for spherically symmetric potentials by expanding the solutions on Coulomb functions. We utilize a multistep integration scheme together with the standard partial wave analysis in a region where the potential term dominates. The method applies to any physical problem expressed as [? ² + V(r) + k ²]ψ(r) = 0, while the extension of the method to more general scattering problems is briefly discussed. At present, we demonstrate a two-step Coulomb-fitted integration scheme by calculating the short-range scattering phase shifts for various potentials V (r).     

Neighborhood preferences in random matching problems

We consider a class of random matching problems where the distance between two points has a probability law which, for a small distance l, goes like l^r. In the framework of the cavity method, in the limit of an infinite number of points, we derive equations for p_k, the probability for some given point to be matched to its kth nearest neighbor in the optimal configuration. These equations are solved in two limiting cases: r = 0 -- where we recover p _k = 1/2^k, as numerically conjectured by Houdayer et al. and recently rigorously proved by Aldous -- and r→ + ∞. For 0 < r < + ∞, we are not able to solve the equations analytically, but we compute the leading behavior of p_k for large k. Received 14 February 2001     

Jack Polynomials as Fractional Quantum Hall States and the Betti Numbers of the (k + 1)-Equals Ideal

We show that for Jack parameter α = ?(k + 1)/(r ? 1), certain Jack polynomials studied by Feigin–Jimbo–Miwa–Mukhin vanish to order r when k + 1 of the coordinates coincide. This result was conjectured by Bernevig and Haldane, who proposed that these Jack polynomials are model wavefunctions for fractional quantum Hall states. Special cases of these Jack polynomials include the wavefunctions of Laughlin and Read–Rezayi. In fact, along these lines we prove several vanishing theorems known as clustering properties for Jack polynomials in the mathematical physics literature, special cases of which had previously been conjectured by Bernevig and Haldane. Motivated by the method of proof, which in the case r = 2 identifies the span of the relevant Jack polynomials with the S _n -invariant part of a unitary representation of the rational Cherednik algebra, we conjecture that unitary representations of the type A Cherednik algebra have graded minimal free resolutions of Bernstein–Gelfand–Gelfand type; we prove this for the ideal of the (k + 1)-equals arrangement in the case when the number of coordinates n is at most 2k + 1. In general, our conjecture predicts the graded S _n -equivariant Betti numbers of the ideal of the (k + 1)-equals arrangement with no restriction on the number of ambient dimensions.     

Waves in a superlattice with anisotropic inhomogeneities

Dependences of the dispersion laws and damping of waves in an initially sinusoidal superlattice on inhomogeneities with anisotropic correlation properties are studied for the first time. The period of the superlattice is modulated by the random function described by the anisotropic correlation function K_?(r) that has different correlation radii, k _∥ ^?1 and k _⊥ ^?1 , along the axis of the superlattice z and in the plane xy, respectively. The anisotropy of the correlation is characterized by the parameter λ=1?k_⊥/k_∥ that can change from λ=0 to λ=1 when the correlation wave number k⊥ changes from k_⊥=k_∥ (isotropic 3D inhomogeneities) to k_⊥=0 (1D inhomogeneities). The correlation function of the superlattice K(r) is developed. Its decreasing part goes to the asymptote L that divides the correlation volume into two parts, characterized by finite and infinite correlation radii. The dependences of the width of the gap in the spectrum at the boundary of the Brillouin zone δν and the damping of waves ξ on the value of λ are studied. It is shown that decreasing L leads to the decrease of δν, and increase of ξ, with the increase of λ.     

Photoelectron-photon coincidence studies of the A$̃and B$̃excited electronic states of X-CC-CN+, X = CH3, CD3, Cl,Br, I

Fluorescence quantum yields and lifetimes of the above given cations in selected levels within their lowest excited electronic states have been measured by a photoelectron—photon coincidence technique. These data, obtained under collision-free conditions, lead to the radiative and non-radiative rate constants as a function of the internal energy. The symmetry of the A$\text{\$}\text{?}$state is ²A₁ (X = CH₃, CD₃), ²Σ⁺ (X = Cl), but ²Π (X = Br, I) and the corresponding k_r values for these two groups, 1–2 × 10⁶s^?1 and 2 × 10⁷s^?1 respectively, reflect the different nature of the transitions. Other essential features of the results are discussed.     

A class of modified sum rules in QCD

We consider a class of modifications to the exponential QCD sum rules, replacing the energy variableq ² by (q ²) ^k withk positive integer. As a result of this modification, (i) the relative role of the condensates on the theoretical side is changed, (ii) the integral on the phenomenological side acquires a rapidly decreasing weight function and (iii) the way of approaching infinity in theq ² complex plane splits intok rays. We explicitly discuss the modified sum rules withk=1,k=2 andk=3 for the ρ meson and conclude that, unlike to thek=1 case, thek=2 andk=3 sum rules do not yield a satisfactory scheme for the prediction of the resonance parameters unless many higher condensates are known. We show that, for practical applications with the present knowledge of condensates, the standard exponential QCD sum rules are the most suitable ones out of the class considered.     

The theory of diffusion-limited recombination of donor—acceptor pairs

The asymptotics of the quasi-steady kinetics of diffusion-limited recombination of donor-acceptor (DA) pairs has been developed. The theory is based on the probability of DA pair recombination according to the relation w₀ exp [-(r/r_B)], where w₀ is constant, r is the separation between defects and r_B is the half Bohr radius of the more diffuse wavefunction. The reaction rate equation (for the thermal activation mechanism) includes the reaction constant as follows: k = 4πr_B[E_a/kT + ln (w₀r^2_B/D₀) + 1.54 + 2 K₀ (α)], α = 2r₀(w₀/D)$\text{1}\text{2}\text{,}$ where E_a is the activation energy, and K₀, I₀ are the modified Bessel functions. The quasi-steady recombination radius is introduced. The theory developed has been compared with the zone one. The interpretation of the activation energy and frequency factor have been presented for the diffusion model. An extension of the theory developed by taking into account Coulomb interaction between defects has also been presented. The applicability of the theory to the recombination of V_k centers with electronics centers is alkali halides has been shown. It is assumed that the theory is applicable to other centers of hole nature (H, proton, low symmetry exciton i.e. V_k + electron).     

Representation of zero curvature for the system of nonlinear partial differential equations x_{\alpha,z\bar z} = \exp (kx)_\alpha and its integrability

Representation of zero curvature for the system of essentially nonlinear partial differential equations \(x_{\alpha ,z\bar z} = exp\left( {\sum\nolimits_{\beta = 1}^r {k_{\alpha \beta } x_\beta } } \right),1 \leqslant \alpha \leqslant r\) , with an arbitrary numeral matrix k is constructed in an explicit form. On the basis of this representation we give an invariant integration method for the system when k coincides with the Cartan matrix of simple Lie algebra of rank r. The final solutions depend on 2r arbitrary functions.     

σ models in even-dimensional spaces

We propose a class of grassmannian models in 2k dimensions which for k = 1 reduce to CP^N?1 models and for k = 2 to composite SU(2) Yang-Mills models. We define and discuss the conditions of self-duality for these models and present corresponding one-instanton field configurations. Some properties of these configurations are briefly discussed.     

Resonance absorption and fluorescence in argon

Absorption cross sections of argon for argon resonance radiation have been measured by several techniques. The apparent cross sections are small (0·1 to 1·6 × 10^-18 cm²) for resonance absorption and the values depend on the technique used for measurement. These observations are interpreted in terms of extensive reversal and broadening in the source. The excitation and quenching of resonance fluorescence was studied to provide information about the rates of the processes

The rate constants were estimated relative to k _r, the rate constant for radiation. Radiation imprisonment leads to a reduction of k _r from its natural value and observations of the decay of resonance fluorescence suggest that k _r ～ 1·5 × 10⁵ s^-1 in our system at [Ar] = 2 × 10¹⁷ atom cm^-3. Combining this value with the relative values for the quenching rate constants gives k ₁ < 1·5 × 10^-13, k ₁′(M = N₂) ～ 6 × 10^-12, k ₁′(M = NO) ～ 4 × 10^-10, in units of cm³ s^-1 molecule^-1.     

Effects of linking chain length and magnetic field on the kinetics of photoprocesses in covalently bonded porphyrin—viologen dyads

The formation and decay kinetics of chain linked triplet radical pairs derived from photo-induced electron transfer reactions in a series of 21 zinc porphyrin-flexible spacer-viologen (ZnP-Sp _n -Vi²⁺) dyads containing 2–138 atoms (n) in the spacer, have been examined by nanosecond laser flash photolysis techniques in an external magnetic field. In non-viscous polar solvents (acetone and CHCl₃ plus CH₃OH = 1:1 v/v), the effect of the spacer length on the rate constant of forward electron transfer can be described by the equation: k _et = k ⁰ _et(n + 6)^?1.5, with k ⁰ _et = 3 × 10¹⁰ s^?1 and 1.2 × 10¹⁰ s^?1 for electron transfer from the singlet and triplet states of ZnP, respectively. In zero magnetic field, the value of the triplet radical pair recombination rate constant, k _r(0) = 0.7 × 10⁶-8 × 10⁶ s^?1, is significantly smaller than k _et. The dependence of k _r(0) on n has an extremum with the maximum near n = 20. In a strong magnetic field (B = 0.21 T), significant retardation of triplet radical pair recombination is observed. In strong magnetic fields the effect of the chain length on triplet radical pair recombination rates is rather small and k _r(B) may vary in the range 0.3 × 10⁶-1 × 10⁷ s^?1. The phenomena observed are discussed in terms of the interplay of molecular and spin dynamics in the limits of slow and fast encounters, taking into account the exchange-interaction.