On the order of levels in charmonium

We prove a theorem concerning the energies of the 2S and 3D states in a potential $\text{V(r) =}\text{?g}{}^2\text{r}\text{+ V}{}_{\mathrm{<mtext>c</mtext>}}\text{(r)}$, where V_c is a non-singular confining potential. If $\text{(}\text{d}\text{d}\text{r)}{}^3\text{(r}{}^2\text{V}{}_{\mathrm{<mtext>c</mtext>}}\text{)}$ is positive, then the 3D state lies above the 2S state, provided

$\text{d}\text{d}\text{r}\text{1}\text{r}\text{d}\text{d}\text{r}\text{2V}{}_{\mathrm{<mtext>c</mtext>}}\text{+r}\text{d}\text{V}{}_{\mathrm{<mtext>c</mtext>}}\text{d}\text{r}\text{< 0, ?}{}_{\mathrm{r}}\text{>0.}$

For V_c = r^α, this corresponds to 0 < α < 2.     

Observation of new nuclides37Si,40P,41S,42S produced in deeply inelastic reactions induced by40Ar on238U

A non-relativistic quantum-mechanical system is studied which consists ofN identical bosons interacting by pair potentials of the form 〈r¦V¦r ¹〉=?π/2ν ₀ a ^?3 f(r/a)f ^*(r ¹/a). General upper and lower bounds to the ground-state energyE _N are provided for alla, V ₀ andN, and detailed results are given in the case of the Yamaguchi potential for whichf(x)=e ^?x/x. It is shown that the ratioE _N /E ₂ diverges both under the limit (i) a↓0,E ₂ =arbitrary constant <0, and (ii) (V ₀ a ²)↓(V ₀ a ²)_c, where (V ₀ a ²) _c corresponds toE ₂=0. The results complement recent studies of the Efimov effect via scattering theory.     

The pair distributions and the osmotic coefficient in anomalous electrolytes up to concentrations c≈ 1 mol/1

The osmotic coefficient of anomalous electrolytes up to concentrations c ≈ 1 mol/l is explained by the pair distributions $\text{n(r) =}\text{exp}\text{[-β(}\text{V}{}_{\mathrm{c}}\text{(r) + V(}{}^{\mathrm{hs}}\text{)(r) + V}{}_1\text{(r))]}$. Here $\text{V}$_c(r) is a screened Coulomb potential, V(^hs)(r) a hard sphere potential and V₁(r) = ?A/r⁶ a short range attractive potential. For the contact distances R₊₊, R_?? and R_+? of the hard sphere potentials between ions with the same sign of their charges (++,??) and ions of opposite charges (+?) the relations R₊₊ = R_?? = R and R_+? = q₁R with 0 < q₁ < 1 are assumed. In contrast to a previous paper the parameter q₁ takes a fixed value q₁ ≈ 0,8. The constant A is determined by the fraction q₂ defined by A/R⁶ = q₂(Z²e²/DR) where the positive integer Z is the charge number of the ions and D the dielectric constant of the solvent. The numerical calculation of the osmotic coefficient of 1-1-valent hydrous electrolytes in the range of temperature 273 K ? T ? 293 K shows that the anomalous electrolytes are described by fractions q₂ in the range 0,25 ? q₂ ? 0,5 if the contact distances R are in the range $\text{3}\text{A}\text{?}\text{? R ? 7}\text{A}\text{?}$.     

The p-40Ca and n-40Ca mean fields from the iterative moment approach

The real part V(r; E) of the p-⁴⁰Ca and n-⁴⁰Ca mean fields is extrapolated from positive towards negative energies by means of the iterative moment approach, which incorporates the dispersion relation between the real and imaginary parts of the mean field. The potential V(r; E) is the sum of a Hartree-Fock type component V^HF, (r; E) and a dispersive correction δV(r; E); the latter is due to the coupling of the nucleon to excitations of the ⁴⁰Ca core. The potentials V(r; E) and V_HF(r; E) are assumed to have Woods-Saxon shapes. The calculations are first carried out in the framework of the original version of the iterative moment approach, in which both the depth and the radius of the Hartree-Fock type contribution depend upon energy, while its diffuseness is constant and equal to that of V(r; E). The corresponding extrapolation towards negative energies is somewhat sensitive to the detailed parametrization of the energy dependence of the imaginary part of the mean field, which is the main input of the calculation. Moreover, the radius of the calculated Hartree-Fock type potential then increases with energy, in contrast to previous findings in ²⁰⁸Pb and ⁸⁹Y. A new version of the iterative moment approach is thus developed in which the radial shape of the Hartree-Fock type potential is independent of energy; the justification of this constraint is discussed. The diffuseness of the potential V(r; E) is assumed to be constant and equal to that of V_HF(r; E). The potential calculated from this new version is in good agreement with the real part of phenomenological optical-model potentials and also yields good agreement with the single-particle energies in the two valence shells. Two types of energy dependence are considered for the depth U_HF(E) of the Hartree-Fock type component, namely a linear and an exponential form. The linear approximation is more satisfactory for large negative energies (E < −30 MeV) while the exponential form is better for large positive energies (E > 50 MeV). This is explained by relating the energy dependence of U_HF(E) to the nonlocality of the microscopic Hartree-Fock type component. Near the Fermi energy the effective mass presents a pronounced peak at the potential surface. This is due to the coupling to surface excitations of the core and reflects the energy dependence of the potential radius. The absolute spectroscopic factors of low-lying single-particle excitations in ³⁹Ca, ⁴¹Ca, ³⁹K and ⁴¹Sc are found to be close to 0.8. The calculated p-⁴⁰Ca and n-⁴⁰Ca potentials are strikingly similar, although the two calculations have been performed entirely independently. The two potentials can be related to one another by introducing a Coulomb energy shift. Attention is drawn to the fact that the extrapolated energy dependence of the real part of the mean field at large positive energy sensitively depends upon the assumed behaviour of the imaginary part at large negative energy. Yet another version of the iterative moment approach is introduced, in which the radial shape of the HF-type component is independent of energy while both the radius and the diffuseness of the full potential V(r; E) depend upon E. This model indicates that the accuracy of the available empirical data is probably not sufficient to draw reliable conclusions on the energy dependence of the diffuseness of V(r; E).     

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).     

The vector coupling α V((r) and the scales r 0, r 1 in the background perturbation theory

We study the universal static potential V _st(r) and the force, which are fully determined by two fundamental parameters: the string tension σ = 0.18 ± 0.02 GeV² and the QCD constants \(\Lambda _{\overline {MS} } (n_f )\) , taken from pQCD, while the infrared (IR) regulator M _B is taken from the background perturbation theory and expressed via the string tension. The vector couplings α _V(r) in the static potential and α _F(r) in the static force, as well as the characteristic scales, r ₁(n _f = 3) and r ₀(n _f = 3), are calculated and compared to lattice data. The result \(r_0 \Lambda _{\overline {MS} } (n_f = 3) = 0.77 \pm 0.03\) , which agrees with the lattice data, is obtained for M _B = (1.15 ± 0.02) GeV. However, better agreement with the bottomonium spectrum is reached for a smaller \(\Lambda _{\overline {MS} } (n_f = 3) = (325 \pm 15)\) MeV and the frozen value of α _V = 0.57 ± 0.02. The mass splittings \(\bar M(1D) - \bar M(1P)\) and \(\bar M(2P) - \bar M(1P)\) are shown to be sensitive to the IR regulator used. The masses M(1 ³ D ₃) = 10169(2) MeV andM(1 ³ D ₁) = 10155(3) MeV are predicted.     

A non-coulombic effective power-law potential for the heavy quarkoniums

An effective power-law potential of the form V(r) = 6.08 r^0.106?6.41 is found to describe satisfactorily the gross features of the mass spectra and the leptonic width ratios of the c$\text{c}$ and b$\text{b}$ systems in a flavour-independent manner.     

An analysis of the new pion form factor data

A form factor F(t) for the pion is constructed which is compatible with analyticity and the data in the space-like and time-like region. For the mean square pion radius 〈r²〉 = (0.46_?0.08^+0.06) fm² is obtained. Typical errors of the extrapolated F(t) are given (e.g. F(t =?8 GeV²/c²) = 0.07_?0.10^+0.04). Assuming F(t) ≈ β/(?t)^α at the end of the space-like data region we obtain β = 0.31, α = 0.81 for t in GeV²/c² together with the error contours of (α, β). No conclusive answer on the existence of zeroes of F(t) can be given.     

A new treatment of singular potentials

For a broad class of strongly singular potentials, the effective (finite-dimensional) forms of the hamiltonian H are constructed by its “smooth” algebraic truncation in a large model space. The method is based on an asymptotic factorization of H into a product of matrices. Its efficiency (acceleration of convergence of the energies) is illustrated on the singularly anharmonic potential V(r) = r² + hr^?4.     

Stability of bipolarons in the presence of a magnetic field

The stability of large Fröhlich bipolarons in the presence of a static magnetic field is investigated with the path integral formalism. We find that the application of a magnetic field (characterized by the cyclotron frequence ω _c) favors bipolaron formation: (i) the critical electronphonon coupling parameter α _c (above which the bipolaron is stable) decreases with increasing ω _c and (ii) the critical Coulomb repulsion strength U _c (below which the bipolaron is stable) increases with increasing ω _c. The binding energy and the corresponding variational parameters are calculated as a function of α, U and ω _c. Analytical results are obtained in various limiting cases. In the limit of strong electron-phonon coupling (α ? 1) we obtain for ω _c ? 1 that E _estim ? E _estim(ω _c = 0) + c(u)ω _c/α ⁴ with c(u) an explicitly calculated constant, dependent on the ratio u = U/α where U is the strength of the Coulomb repulsion. This relation applies both in 2D and in 3D, but with a different expression for c(u). For ω _c ? α ²? 1 we find in 3D E _estim ? ω _c - α ² A(u) ln²(ω _c/α ²), (also with an explicit analytical expression for A(u)) whereas in 2D E _estim ^2D ? ω _c - α√ω _cπ(u-2-√2)/2. The validity region of the Feynman-Jensen inequality for the present problem, bipolarons in a magnetic field, remains to be examined.     

The reference potential in the ATA

Precise limitations on the choice of the reference potential for the ATA are investigated, and it is shown that V⁽ⁱ⁾_r?〈V_i〉=O(xy).     

On the normalization of the Gamov resonant wave function in the configuration space

The regularization of the normalization integral for the resonant wave function, proposed by Zeldovich, is valid only when |Req _res| > |Imq _res|. A new normalization procedure is proposed and implemented, which is valid when this condition fails. First, an arbitrarily normalized vertex function g(k) is calculated using the formula with the potential V(r) in the integrand. This Fourier integral converges for a potential with the asymptotics V(r) → constr ^?n exp(?μr) if |Imq _res| < μ/2. Then the function g(k) is normalized using the generalized normalization rule, which is independent of the resonance pole position. The proposed method is approved by the example of calculation for a virtual triton.     

Scalar Charged Particle in Presence of Magnetic and Aharonov–Bohm Fields Plus Scalar–Vector Killingbeck Potentials

The generalized form of Killingbeck potential is an attractive Coulomb term plus a linear term and a harmonic oscillator term, i.e. ?a/r + br + λr ², which has a useful application in quarkonium spectroscopy. The ground state energy with the corresponding wave function are obtained for any arbitrary m-state in two-dimensional Klein–Gordon equation with equal mixture of scalar–vector Killingbeck potentials in the presence of constant magnetic and singular Ahoronov–Bohm flux fields perpendicular to the plane where the interacting charged particle is confined. The analytical exact iteration method is used in our solution. We obtain the energy eigensolutions for particle and antiparticle corresponding to S(r) = V(r) and S(r) = ?V(r) cases, respectively. Some special cases like the Coulomb, harmonic oscillator potentials and the nonrelativistic limits are found in presence and absence of external fields.     

On the Martin potential

The interquark potential V(r) = A + Br^α accounts for all spin-averaged levels and ratios of leptonic widths within each heavy-quark spectroscopy. We find, however, that ratios of leptonic widths between different spectroscopies are inconsistent with flavor-independence. Modification of the Martin potential by a Coulomb short-range part restores complete flavor-independence.     

Nonlinear stability analysis of the Lorenz model by Lyapunov's direct method

The preturbulent region of the Lorenz model is investigated with the aim to determine the attractive region around the stable fixed points as a function of the Rayleigh parameterr. Close to the turbulent threshold, i.e. for small deviations ?=r _T?r, the attractive region is a highly anisotropic ellipsoid in the phase space. Its volumeV obays the power lawV(ε)=const ?^α with the critical exponent α=5/2. A Landau type approximation, on the other hand, leads to α-3/2. The reason for this discrepancy is discussed. In addition we develop a series expression which allows to calculate the maximal domain of attraction also for large deviations ε. Numerical results from the two lowest terms of the series are represented for 0<ε?6.     

Stereochemical studies of oligomers XXIX. Reinvestigation of the structure of N-m-tolyl phthalimide

C₁₅H₁₁NO₂, Mr 237.3, monoclinic, space groupC _c, a=8.539(2),b=19.865(4),c=7.599(2)?,β=111.44(2)°,V=1199.8 ?³,Z=4,D _c=1.31 gcm⁻³,λ(CuKα)=1.5418 ?,μ=6.74cm⁻¹,F(000)=496, room temperature. The structure was solved by direct methods with SHELX-86 and refined down to agreement valueR=0.046 for 1117 reflections above 2σ(I). The angle between the plane of the phthalimide group, which shows a little bent [1.2(2)°] between its two rings, and the tolyl group is 56.1(1)°. The packing of the molecules is stabilized by van der Waal’s forces only. Part XXVIII: Bocelli and Rizzoli (1990)     

A remarkable duality in one-particle quantum mechanics between some confining potentials and (R+L∞ϵ) potentials

The equivalence between the bound-state problem for the spatial harmonic oscillator and for the Coulomb potential is reexamined and extended: a similar duality is displayed between large classes of confining potentials and (R+L^∞_?) potentials, i.e., potentials for which the Schrödinger operator has among other properties a discrete spectrum bounded below, in (-∞, 0). The involved confining potentials U(r) must satisfy: $\text{r}{}^{\mathrm{-\alpha }}\text{U(r)}\textcolor{red}{}\text{g>0}{}_{\mathrm{r\to +\infty }}\text{, r}{}^{\mathrm{<mtext>-2\alpha </mtext><mtext>(2+\alpha )</mtext>}}\text{U (r}{}^{\mathrm{<mtext>2</mtext><mtext>(2+\alpha )</mtext>}}\text{) ? g}\text{is R}\text{+}\text{L}{}^{\mathrm{\infty }}{}_{\mathrm{?}}$.     

Interaction between a He atom and a graphite surface

A study is presented of the interaction V(r) between a He atom and a graphite surface. V(r) is assumed equal to a sum of pair interactions U(r ? R_i) between the He and C atoms. None of a set of isotropic potentials (dependent only on the magnitude ¦r ? R_i¦) is consistent with recent scattering data. Anisotropie pair potentials, in contrast, are found to yield good agreement. The origin of this anisotropy is analyzed in terms of the graphite dielectric function and charge density.     

Pionic x-rays and the neutron radius of 44Ca

Differences between strong interaction level shifts and widths for 2p states in pionic atoms of ^44,40Ca have been measured. Analysis in terms of an effective pion-nucleus potential leads to a difference in neutron rms radii of r_n(44)?r_n(40) = 0.05 ± 0.05 fm.     

Barrett moments and rms charge radii

An empirical relation is established between Barrett equivalent radii R _k,α and rms charge radii < r ² >^1/2 based on the results of model-independent and Fermi model analyses of 2p → 1s transitions in muonic atoms. This relation follows simple Z dependence, and can be usefully applied to derive rms radii < r ² >^1/2 or differences δ ^AA ′ < r ² >^1/2 in cases where only R _k,α data or isotope shifts δ ^AA ′ R _k,α are published. The atomic number dependence of the Barrett parameters k(Z) and α(Z) is also given by empirical formulae. It is shown that the Barrett moment can be expanded in a sum of integer moments < r ^m > (m ≥ 2) using an effective exponential parameter α _eff(Z). The moments < r ^m > and isotopic differences δ < r ^m > of the two-parameter Fermi distribution expressed in terms of the parameters c and a are given in the Appendix for m = 1 – 10.