Coordinate asymptotic behavior of the radial three-particle wave function for a bound state involving two charged particles

The asymptotic expression for the radial component of the wave function for a three-particle bound state involving two charged particles is derived in an explicit form. This expression contains a three-particle asymptotic normalization factor C(φ), where φ is a hyperangle in the six-dimensional space of intrinsic coordinates of the three-particle system. The resulting expressions are used to analyze the asymptotic behavior of the wave functions for the ⁹Be nucleus that were calculated within the α + α + n three-particle model for various forms of the an potential. A comparison of the asymptotic expression derived here and the asymptotic expressions for model wave functions makes it possible to extract C(φ) values, which turned out to be sensitive to the form of αn interaction. This permits deducing information about two-particle interaction from a comparison of the theoretical values of C(α) with their phenomenological counterparts found from an analysis of experimental differential cross sections for relevant nuclear reactions.     

Asymptotic Normalization Coefficients (Nuclear Vertex Constants), Three-Body Asymptotic Normalization Functions (On-Shell Vertex Functions) and Nuclear Astrophysics

In the present review work, the main methods of the determination of asymptotic normalization coefficients (or respective nuclear vertex constants) for the A + x → B channel and of threebody asymptotic normalization functions (or respective on-shell vertex functions) for the A + b + c → B channel are briefly discussed. The main attention is paid both to the use of the specific asymptotic normalization coefficients and three-body asymptotic normalization functions as a source of getting the valuable information about the pair (nucleon–nucleon, nucleon–cluster and cluster–cluster) nuclear interactions and to their application for the specific direct nuclear-astrophysical radiative capture and peripheral transfer reactions at low energies.     

The Coulomb-oscillator relation on <Emphasis Type="Italic">n</Emphasis>-dimensional spheres and hyperboloids

We establish a relation between Coulomb and oscillator systems on n-dimensional spheres and hyperboloids for n≥2. We show that, as in Euclidean space, the quasiradial equation for the (n+1)-dimensional Coulomb problem coincides with the 2n-dimensional quasiradial oscillator equation on spheres and hyperboloids. Using the solution of the Schrödinger equation for the oscillator system, we construct the energy spectrum and wave functions for the Coulomb problem.     

Elastic scattering of light nuclei through a simple potential model

The phase function method is adapted to deal with the scattering on our proposed interactions for α–α and α–³He systems. The effect of the electromagnetic interaction is included in terms of a screened Coulomb potential. Based on our proposed potential models we present results for α–α and α–³He elastic scattering phase shifts which compare well with more detailed calculations.     

Coulomb deexcitation of muonic hydrogen within the quantum close-coupling method

The Coulomb deexcitation of muonic hydrogen in collisions with the hydrogen atom has been studied in the framework of the fully quantum-mechanical close-coupling method for the first time. The calculations of the l-averaged cross sections of the Coulomb deexcitation are performed for (μp)_n and (μd)_n atoms in the initial states with the principal quantum number n = 3–9 and at relative energies E = 0.1–100 eV. The obtained results for the n and E dependences of the Coulomb deexcitation cross sections drastically differ from the semiclassical results. An important contribution of the transitions with Δn > 1 to the total Coulomb deexcitation cross sections (up to ～37%) is predicted.     

A statistical model for simulating the emission of light particles from excited nuclei

The algorithms and basic equations of a novel evaporation model that have been implemented in the program package EVAP15 are detailed. The level density of an excited nucleus is described by the composite Gilbert–Cameron formula with parameter values as suggested by the IAEA working group RIPL-3. Special attention is paid to the cross sections of inverse reactions and, in particular, to those for the interactions of low-energy neutrons with nuclei and for crossing of the Coulomb barrier by low-energy charged particles. The model predictions are compared with a large volume of experimental data on the spectra of particles emitted in the reactions (n, xn), (n, xp), and (n, xα) induced by neutrons with energy near 14 MeV and on the four spectra for the reaction (p, xp) induced by 62-MeV protons.     

Coulomb Rydberg electron <Emphasis Type="Italic">L</Emphasis>-mixing in collisions with atomic ions

The cross sections of the Rydberg electron L-mixing in a hydrogen atom and a hydrogen-like ion are calculated for slow collisions with atomic ions H*(n, L) + A⁺ = H*(n, L′) + A⁺ without variation of the principal quantum number n. The probability of the L-mixing L → L′ is associated with the quantum interference of the wave functions of adiabatic states, i.e., with the mixing of the time phases of these functions exp(?i∫E _k (t)dt). The effective cross section of such L-mixing for the states with n = 28 are 4–5 orders of magnitude greater than the cross sections determined in previous investigations. The expansion coefficients of spherical Coulomb wave functions in terms of parabolic ones and vice versa, which are necessary for determining cross sections, are calculated on the basis of a comprehensive analysis of the spatial properties of these functions.     

Orbital fractional parentage coefficients for the harmonic oscillator shell model

The Hamiltonian ofn particles moving in a common harmonic oscillator potential has as its symmetry group the unitary groupU(3n) in 3n dimensions,n particle states of the harmonic oscillator shell model can be characterized as bases of irreducible representations (BIR) of the groupU(3n) and of certain subgroups of this group. Use is made of these subgroups for the factorization and calculation of 2, 3, and 4 particle fractional parentage coefficients (fpc) of the harmonic oscillator shell model. Recoupling coefficients for subgroup chains of the symmetric groupS (n) appear as factors in the fpc. These coefficients are analyzed and calculated explicitly. The 2, 3 and 4 particle fpc of the 1s 1p shell configuration are obtained as products of these recoupling coefficients with known reduced Wigner coefficients of the unitary groupU(3) in 3 dimensions. Possible applications are indicated.     

Discretized kinetic theory on scale-free networks

The network of interpersonal connections is one of the possible heterogeneous factors which affect the income distribution emerging from micro-to-macro economic models. In this paper we equip our model discussed in [1, 2] with a network structure. The model is based on a system of n differential equations of the kinetic discretized-Boltzmann kind. The network structure is incorporated in a probabilistic way, through the introduction of a link density P(α) and of correlation coefficients P(β|α), which give the conditioned probability that an individual with α links is connected to one with β links. We study the properties of the equations and give analytical results concerning the existence, normalization and positivity of the solutions. For a fixed network with P(α) = c/α^q, we investigate numerically the dependence of the detailed and marginal equilibrium distributions on the initial conditions and on the exponent q. Our results are compatible with those obtained from the Bouchaud-Mezard model and from agent-based simulations, and provide additional information about the dependence of the individual income on the level of connectivity.     

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.     

Asymptotic three-particle approach to the Coulomb breakup process <Superscript>6</Superscript>Li + <Superscript>208</Superscript>Pb → <Superscript>208</Superscript>Pb + α + <Emphasis Type="Italic">d</Emphasis>

On the basis of the distorted-wave method, experimental data on the triple-differential cross section for the Coulomb breakup reaction ²⁰⁸Pb(⁶Li, αd)²⁰⁸Pb are analyzed by employing a correct expression for the final-state ²⁰⁸Pb–α–d three-particle Coulomb wave function. It is shown that the effect of final-state three-particle Coulomb dynamics can be used to assess the kinematical condition of clean Coulomb breakup processes. New values of the astrophysical S factor for the direct-radiative-capture reaction d(α, γ)⁶Li at ultralow energies in the range of 70 ≤ E _dα ≤ 600 keV were extracted from experimental data. The value of S(0) = 1.60 ± 0.17 MeV nb was obtained.     

<Emphasis Type="Italic">nd</Emphasis> scattering at low energies in the two-body potential model

The S-wave phase shift δ(E) for the spin-doublet nd scattering at low energy E is calculated in the framework of the two-body approach. The effective-range-theory formula k cot δ = (1+k²/k ₀ ² )^?1(?1/α+C₂k²+C₄k⁴) is used to obtain approximate analytical results with different potentials. The corresponding coefficients C₂ and C₄ are obtained from our previous calculations of the asymptotic normalization parameter function C _t ² (aκ), where κ is the triton wave number and a is the doublet nd scattering length. The model reasonably describes δ(E), the results being quite sensitive to the choice of the effective nd potential.     

Carcass functions in variational calculations for few-body systems

For variational calculations of molecular and nuclear systems involving a few particles, it is proposed to use carcass basis functions that generalize exponential and Gaussian trial functions. It is shown that the matrix elements of the Hamiltonian are expressed in a closed form for a Coulomb potential, as well as for other popular particle-interaction potentials. The use of such carcass functions in two-center Coulomb problems reduces, in relation to other methods, the number of terms in a variational expansion by a few orders of magnitude at a commensurate or even higher accuracy. The efficiency of the method is illustrated by calculations of the three-particle Coulomb systems μμe, ppe, dde, and tte and the fourparticle molecular systems H₂ and HeH⁺ of various isotopic composition. By considering the example of the _Λ ⁹ Be hypernucleus, it is shown that the proposed method can be used in calculating nuclear systems as well.     

Root mean square radii of heavy flavoured mesons in a quantum chromodynamics potential model

We report the results of root mean square (r.m.s.) radii of heavy flavoured mesons in a QCD model with the potential V(r)=?(4α_s/3r) + br + c. As the potential is not analytically solvable, we first obtain the results in the absence of confinement and Coulomb terms respectively. Confinement and Coulomb effects are then introduced successively in the approach using the Dalgarno’s method of perturbation. We explicitly consider the following two quantum mechanical aspects in the analysis: (a) The scale factor c in the potential should not effect the wave function of the system even while applying the perturbation theory. (b) Choice of perturbative piece of the Hamiltonian (confinement or linear) should determine the effective radial separation between the quarks and antiquarks. The results are then compared with the available theoretical values of r.m.s. radii.     

Correlation between the properties of the deuteron and the low-energy triplet parameters of neutron-proton scattering

The correlation between the asymptotic normalization constant for the deuteron, A_S, and the neutron-proton scattering length for the triplet case, a_t, is investigated. It is found that 99.7% of the asymptotic constant A_S is determined by the scattering length a_t. It is shown that the linear correlation between the quantities A _S ^?2 and 1/a_t provides a good test of correctness of various models of nucleon-nucleon interaction. It is revealed that, for the normalization constant A_S and for the root-mean-square deuteron radius r_d, the results obtained with the experimental value recommended at present for the triplet scattering length a_t are exaggerated with respect to their experimental counterparts. By using the latest experimental data obtained for phase shifts by the group headed by Arndt, it proved to be possible to derive, for the low-energy parameters of scattering (a_t, r_t, P_t) and for the properties of the deuteron (A_S, r_d) results that comply well with experimental data.     

On a Semiclassical Formula for Non-Diagonal Matrix Elements

Let H(?)=?? ²d²/dx ²+V(x) be a Schrödinger operator on the real line, W(x) be a bounded observable depending only on the coordinate and k be a fixed integer. Suppose that an energy level E intersects the potential V(x) in exactly two turning points and lies below V _∞=lim?inf?_|x|→∞ V(x). We consider the semiclassical limit n→∞, ?=? _n →0 and E _n =E where E _n is the nth eigenenergy of H(?). An asymptotic formula for 〈n|W(x)|n+k〉, the non-diagonal matrix elements of W(x) in the eigenbasis of H(?), has been known in the theoretical physics for a long time. Here it is proved in a mathematically rigorous manner.     

Die Coulomb-Energien leichter Atomkerne

A compilation of the known data on Coulomb energy differences of isobaric doublets and isobaric triplets is given. Plots of the Coulomb energy differences versus¯Z/A ^1/3 with¯Z=(Z ₁+Z ₂)/2 show an analogous shell structure behaviour for the three series with 2¯Z=A?1,A andA+1 (T=1, 1/2 and 1), i.e. discontinuities at the closed shells atA=4, 16 and 40 and the closed subshell atA=32 and oscillations mainly being due to Coulomb proton-proton pairing energy. A positive energy shift of the lowest states withT=1 of all self-conjugate nuclei withA=4n+2 seems to be indicated by the experimental data. A semi-empirical formula is given that describes the data.     

Über das Verhältnis der Wirkungsquerschnitte für den (e,n)- und (γ,n)-Prozeß

The ratio of the cross sections for the (e, n)-and (γ, n)-process is derived with the use of the relativistic Coulomb Eigenfunctions for the continuous spectrum, used bySommerfeld-Maue andBethe-Maximon. Only the electric and magnetic dipole transition are considered. For this transitions the Coulomb correction to the Born approximation is calculated. The result of an estimation of the electric quadrupole transition is given. The effect of screening and that of finite nuclear size is estimated. With the ratio thus obtained one is able to calculate the cross sections for the (e, n)- and (γ, n)-process using the experimental values of only one of this quantities.     

Asymptotic Properties of the Density of Particles in $$\beta $$-Ensembles

We extend recent results on the asymptotic equipartition property for the density of n particles in \(\beta \)-ensembles, as n tends to infinity. We prove the large deviation principle of the log-density for a general potential and the mod-Gaussian convergence in the classical examples.