Non-central Potential Research Articles

Noncentral potentials: The generalized Levinson theorem and the structure of the spectrum

Fredholm theory is applied to the Lippmann–Schwinger equation for noncentral potentials. For a specified wide class of potentials it is proved that the modified Fredholm determinant cannot vanish for real k≠0. The point k=0 is examined and the analog of the distinction between zero-energy bound states and zero-energy resonances for central potentials is found. A generalized Levinson theorem is proved.

Infinite-order sudden approximation for rotational excitation of hydrogen molecules by electrons in the energy range 10–40 eV

Electron scattering by H2 is treated using a noncentral interaction potential including short-range and long-range static contributions, exchange, and polarization effects. The molecule is treated as a rigid rotator and the scattering is treated in the infinite-order sudden approximation. The results show that the rotational excitation cross section exceeds the elastic scattering cross section at large scattering angles at intermediate energies but not at small angles at low energies.

Elastic constants for noncentral interatomic potentials with crystal symmetry

The method of homogeneous deformation has been modified and applied to the calculation of second-order elastic constants for a class of noncentral interatomic potentials with the symmetry of Bravais lattices. The conditions of rotational invariance of the potential energy and the zero initial stress conditions were found and their validity was verified. For cubic symmetry a method was elaborated which applies to the noncentral interatomic potentials proposed byJohnson andWilson. The presented method of homogeneous deformation is compared with the original method ofFuchs and with the method of long waves in calculations of elastic constants.

Exact motion in noncentral electric fields

We study the problem of the motion of a charged particle in noncentral potentials of the type f(θ)/r2 + V(r). Newton's and Schrödinger's mechanics are considered. Exact solutions exist if V(r)=−H/r or Kr2 (i.e., Coulomb or harmonic oscillator potentials) while f(θ) may have at least three different expressions as a function of θ if the problem is three-dimensional and seven expressions if it is two-dimensional. The classical trajectories are computed and the energy levels in the corresponding quantum problem are given. Analogies between the two treatments are discussed.

Phenomenological study of s-shell hypernuclei with ΛN and ΛNN potentials

The binding-energy data of the s-shell hypernuclei and the low-enery Λp scattering data are analysed in terms of a simple model which employs both a Λ-nucleon (ΛN) and a Λ-nucleon-nucleon (ΛNN) potential. The ΛN potential is chosen to be central, with a hard core of radius 0.45 fm and an intrinsic range of 2.0 fm, and to have a charge-symmetry-breaking component. Two types of ΛNN potential are considered in this investigation, one being a phenomenological central potential and the other being the strongly non-central potential obtained by Bhaduri, Loiseau and Nogami using a two-pion-exchange mechanism. The ΛN potential fitted to the B Λ values for Λ 3H, Λ 4H and Λ 4H and Λ 4He and to the Λp scattering data (total cross sections and forward-backward asymmetries) has a rather small degree of spin dependence and a strength in odd-parity states which is only 20–70% of that in even-parity states. This potential gives overbinding for Λ 5He by more than 2 MeV. The effect on this overbinding problem of introducing a ΛNN potential into the analysis is carefully examined. It is shown that a repulsive central ΛNN potential does reduce the calculated value for B Λ ( Λ 5He) but that this is only by a negligible amount, for general reasons which are discussed. For the non-central ΛNN potential, it is found that a proper account of nucleon-nucleon correlations in the hypernuclear wave functions can modify its effect greatly, to an extent depending on the cut-off radius d adopted in this potential. For small values of d(0.45 to 0.7 fm), this non-central ΛNN potential actually increases the overbinding; for d⪆0.7 fm , it does give net repulsion but the reduction found in B Λ ( Λ 5He) is far too small to yield agreement between the calculated and experimental values.

Determination of the Scattering Amplitude from the Differential Cross Section for Scattering by an Arbitrary Noncentral Potential

The unitarity relation of quantum-scattering theory is studied for the general case of scattering by an arbitrary noncentral force and it is shown how it can be utilized to determine the phase of the scattering amplitude from the measured differential cross section. It is found that the unitarity relation leads to a pair of nonlinear integral equations for the phase of the scattering amplitude if the cross section is known. It is shown that from these equations one can obtain, without introducing approximations, a system of linear algebraic equations for a certain set of scalars from the values of which one can readily calculate the phase. In general, the above system of linear equations may admit also redundant nonphysical solutions for the phase. However, when the magnitude of the cross-section data satisfies certain conditions the solution of the equations is unique and must equal the correct physical value of the phase. Unlike previous methods for phase determination, the present approach is based on a form of the unitarity equation which is not simplified by any of the assumptions of parity, rotational, or time-reversal invariance for the underlying scattering interaction. Therefore, it can be applied to the determination of the phase also in cases such as particle scattering by, or in the presence of, external fields which do not possess the above-mentioned symmetries. A generalization of this method to many-channel scattering is also provided. Similar results hold also for the determination of phases in the theory of electromagnetic-wave scattering by an obstacle of arbitrary shape.

Binding Energies of thep-ShellΛΛHypernuclei

Using the results of a shell-model analysis of the $p$-shell hypernuclei by the authors, the binding energies of the two $\ensuremath{\Lambda}$ particles in the $p$-shell $\ensuremath{\Lambda}\ensuremath{\Lambda}$ hypernuclei are calculated. A consistent account of the binding energies of the two known $\ensuremath{\Lambda}\ensuremath{\Lambda}$ hypernuclei, $_{\ensuremath{\Lambda}\ensuremath{\Lambda}}^{ 6}\mathrm{He}$ and $_{\ensuremath{\Lambda}\ensuremath{\Lambda}}^{10}\mathrm{Be}$ (or $_{\ensuremath{\Lambda}\ensuremath{\Lambda}}^{11}\mathrm{Be}$) is obtained. Close agreement with the results of other investigators who used different methods is also pointed out. The effects of the noncentral potentials in the effective two-body $\ensuremath{\Lambda}\ensuremath{-}N$ interaction are discussed.

Density Dependence of 129Xe Chemical Shifts in Mixtures of Xenon and Other Gases

Unlike other chemical shifts in gaseous systems which have been found to have strictly linear dependence on density, we have found the 129Xe chemical shift in pure xenon gas to have a quadratic and cubic dependence in addition to the dominant linear dependence on density. This implies the importance of three or more body interactions in xenon. In mixtures of xenon with another gas (Ar, Kr, CO2, HCl, CH4, CH3F, CH2F2, CHF3, CF4), the dependence of the 129Xe chemical shift on the density of the other gas is found to be linear within experimental error, and varying from 2300–11 700 ppm/mol cc−1. These shifts are orders of magnitude greater than the reported H and F shifts in gases. Analysis of the results show that the density dependence cannot adequately be reproduced by the contributions, Σ1 = Σb − B〈ε2〉 − B〈F2〉, which had been adequate for H and F shifts. The general formulation for calculation of the A and B parameters, the coefficients of the linear and quadratic electric field terms in the theory of chemical shift in gases, is developed. An approximate calculation of B for atoms is given, and the repulsive and anisotropic contributions are estimated. The sensitivity of the chemical shift to the form of the intermolecular potential is suggested in the case of Xe and the fluoromethanes where the results are consistent with a noncentral field potential but not with a central field potential like the Lennard-Jones.

Time-Correlation Functions, Memory Functions, and Molecular Dynamics

The memory functions for the velocity, angular-momentum, and dipolar autocorrelation functions from a series of molecular-dynamics studies of liquid carbon monoxide are examined. The velocity and angular-momentum memory functions decay initially almost to zero in a Gaussian fashion. However, their long-time behavior has a much slower time dependence. The dipolar memory function from a simulation using a strong noncentral potential is approximately this system's angular-momentum autocorrelation function. Approximate velocity and angular-momentum correlation functions are generated from approximate memory functions and the results are compared to experiment. Gaussian memories based on the second and fourth moments of the corresponding autocorrelation functions give the best agreement with experiment. However, none of the approximate memories examined adequately represents the long-time behavior of the experimental memories. The static atomic radial distribution functions are given and are shown to depend upon the strength of the orientational parts of the pair potential used in the dynamics calculations. The non-Gaussian characteristics of the Van Hove self-correlation functions are examined and shown to depend on the potential and number of particles used in the dynamics calculations. The intermediate scattering function and its memory are also examined.

An equation for the potential scattering amplitude

A nonlinear integral equation for the total scattering amplitude is derived. This equation is an analogue of the well-known phase equations for partial scattering amplitudes or phase shifts. It is valid for noncentral potentials as well as for central ones.

On the use of a noncentral velocity dependent potential for the interpretation of the nucleon-nucleon scattering data

A phenomenological nucleon-nucleon potential containing a noncentral velocity dependent term of the form (σ1 ·p)(σ2 ·p) is used to fit the solutions YLAM and YLAN 3 M of the phaseshift analysis ofBreit et al. andHull et al. for protonproton and neutron-proton scattering respectively. The potential contains altogether 20 parameters. The fit is carried through in the range from 10 to 300 MeV laboratory energy. While the results for thep-p phase parameters are useful only in some cases the results for then-p phase parameters are in general quite satisfactory.

Some Non-Born Phase Shifts

Over the past few years, there have been a number of independent attempts to improve on the Born approximation calculation of scattering phase shifts. The similarity of several of the first-order results for central potentials has led to misconceptions as to their interrelationship. The interpretation of the approximations has now been put on a firm basis (sometimes at variance with the previous literature). The alternatives in going to higher order are sorted out and their meaning is clarified. The extensions to noncentral potentials are given. In addition, a numerical exploration of the precision of different higher order results has been carried out.

Explicit non-central potentials and wave functions for givenS-matrices

Wave functions and potentials are obtained for non-central forces coupling angular momental andl+2, with the scattering matrix and bound state energies as a starting point. The method, an extension of the techniques developed by Bargmann for central forces, is based on the solution of an integral equation developed and discussed recently byGel’fand andLevitan, Jost, Kohn andNewton. The ensuing non-central potentials are the first spin-orbit and tensor potential combinations known for which the Schrodinger equation is solvable in closed form. The results allow the approximation of an experimentally givenS-matrix by one whose elements are rational functions of the wave numberk and subsequent determination of the wave function and potential by purely finite algebraic means.

Interactions between Neutron and Proton

An attempt has been made to find an interaction between neutron and proton which will account for not only the binding energy and quadripole moment of the deuteron and the low energy scattering data, but also the results of the experiments on the scattering of 90 MeV. neutrons by protons. Three types of modification of the triplet neutron-proton interaction have been used which embody the following features : (1) A non-central potential of spherical well form, whose radius of interaction is varied. (2) A non-central potential whose form is closer to that of the pseudo-scalar meson potential than the usual Rarita-Schwinger form, but which does not possess the objectionable singularities of the former. (3) The inclusion of a large short-range repulsion. In each case exchange forces of the usual types have been used. The results obtained, like those of other workers in this field using different forms of interaction, fail to agree with the high energy data.

Potential scattering and the continuity of phase-shifts

Let S(k) be the scattering matrix for a Schrodinger operator (Laplacian plus potential) on Rn with compactly supported smooth potential. It is well known that S(k) is unitary and that the spectrum of S(k) accumulates on the unit circle only at 1; moreover, S(k) depends analytically on k and therefore its eigenvalues depend analytically on k provided they stay away from 1. We give examples of smooth, compactly supported potentials on Rn for which (i) the scattering matrix S(k) does not have 1 as an eigenvalue for any k > 0, and (ii) there exists k0 > 0 such that there is an analytic eigenvalue branch e2iδ(k) of S(k) converging to 1 as k ↓ k0. This shows that the eigenvalues of the scattering matrix, as a function of k, do not necessarily have continuous extensions to or across the value 1. In particular this shows that a ‘micro-Levinson theorem’ for non-central potentials in R3 claimed in a 1989 paper of R. Newton is incorrect.