Comparison of several equivalent local potentials for microscopic nonlocal Nα potentials

The phase shifts obtained in the Perey-Saxon, Horiuchi and Peierls-Vinh Mau approximations for the microscopic nonlocal Nα interaction of Lassaut and Vinh Mau have been compared at 10 and 20 MeV to the exact results, without the customary approximation of Perey and Buck, which neglects part of the angular dependence of the nonlocality and is inadequate. We have also compared the equivalent local potentials with the corresponding local wave functions and damping factors to their exact wronskian counterparts and obtained the corresponding nonlocal wave functions as a product of the approximate local wave function and the damping factor in all cases. The results show that of all the approximations the Peierls-Vinh Mau approximation is the most accurate one for s-waves and that the Perey-Saxon approximation is inadequate for Nα scattering. The accuracy of all the approximations is dependent on the degree of repulsiveness of the effective nuclear force. For the exact wronskian equivalent local potential and damping factor however, we reproduce the exact nonlocal wave function to high accuracy in all cases, confirming the accuracy of our numerical methods. The implications for nuclear reactions where the off-shell behaviour of the Nα interaction is important are discussed.     

An energy-independent nonlocal potential model for bound and scattering states

The general expression of the nucleon-nucleus optical potential has been obtained using Watson's multiple scattering theory and Wolfenstein's parametrization of the nucleon-nucleon scattering amplitude. The resulting theoretical potential is nonlocal and consists of an energy-independent central volume plus surface real and imaginary potential and of a Thomas-like spin-orbit term. The analysis has been restricted to N = Z spherical nuclei, so that neither isospin-isospin nor spin-spin interactions have been included. The widely used Perey-Buck, Greenlees, and Watson expressions of the optical potential are easily obtained as particular cases. For practical purposes, the nonlocal potential has been parametrized in the Frahn-Lemmer form, using Woods-Saxon radial form factors, and the equivalent local potential (ELP) has been calculated by a Perey-Buck-like transformation.The ELP has a radial behavior very similar to the original nonlocal one, but the potential depths and radii are energy dependent. The six free parameters in the ELP have been adjusted to fit the available experimental data in the ?70 to + 150 MeV range of interest in nuclear reactions, namely, energies of single hole and single particle states, charge distributions, proton elastic scattering cross sections, and polarizations. The fitted potential depths show an energy dependence in remarkable agreement with the model predictions with a central nonlocality range β ? 1 F and a spin-orbit nonlocality range β₃ ? 0.8 F. The relative importance of surface and volume dependence in the real central potential in also discussed.     

Tunneling and energy splitting in an asymmetric double-well potential

An asymmetric double-well potential is considered, assuming that the minima of the wells are quadratic with a frequency ω and the difference of the minima is close to a multiple of ?ω. A WKB wave function is constructed on both sides of the local maximum between the wells, by matching the WKB function to the exact wave functions near the classical turning points. The continuities of the wave function and its first derivative at the local maximum then give the energy-level splitting formula, which not only reproduces the instanton result for a symmetric potential, but also elucidates the appearance of resonances of tunneling in the asymmetric potential.     

Eikonal expansion

An eikonal expansion of the potential scattering T matrix is evaluated, without approximation, through third order in the inverse momentum. Based on the results, their correspondence with the WKB approximation and a new statement of the unitarity constraint, we propose a sequence of four approximations to the exact impact parameter (Fourier-Bessel) representation of the scattering matrix. The sequence consists of the Glauber approximation and three systematic corrections to the Glauber approximation. The corrections are analytic functions of the impact parameter for Yukawa and Gaussian potentials; they vanish for a Coulomb potential.The sequence of eikonal amplitudes is convergent at high energy and is clearly established for small momentum transfer. Validity for all momentum transfer is conjectured based on systematic cancellation, explicitly verified through third order in the expansion, of momentum transfer dependence in the eikonal impact parameter representation. Such cancellation is shown to occur in the explicit construction of the eikonal expansion of the second Born amplitude for a Yukawa potential.Numerical tests of the sequence of eikonal amplitudes show systematic increase of the angular range of validity by comparison with partial wave results for continuous potentials; the theory is not convergent for discontinuous potentials.The WKB phase shift formula is shown to produce a systematic connection with eikonal expansion results. From this we deduce a generating function for the eikonal phase corrections of arbitrary order and also conjecture a sum of the eikonal expansion valid in the limit of high energy and arbitrary potential strength.     

Exact density-functional potentials for time-dependent quasiparticles

We calculate the exact Kohn-Sham potential that describes, within time-dependent density-functional theory, the propagation of an electron quasiparticle wave packet of nonzero crystal momentum added to a ground-state model semiconductor. The potential is observed to have a highly nonlocal functional dependence on the charge density, in both space and time, giving rise to features entirely lacking in local or adiabatic approximations. The dependence of the nonequilibrium part of the Kohn-Sham electric field on the local current and charge density is identified as a key element of the correct Kohn-Sham functional.     

Short-range part of the NN interaction: Lowest-order relativistic corrections to equivalent local potentials from quark exchange kernels

The full Breit hamiltonian, including the Darwin term and relativistic kinetic-energy correction, has been used to construct resonating group kernels in analytic form for the six-quark NN system. The nonlocal kernels have been converted to equivalent local potentials using the WKB approximation based on their Wigner transforms. The equivalent local potentials have been confirmed to be rather insensitive to a choice of parameters and to be fairly accurate by comparing with the phase shifts and the wave functions obtained with the RGM equation. The results indicate the importance of the Darwin term. The term leads to a large reduction of the strong repulsive cores given by the color magnetic interaction and makes the resulting potentials less repulsive than those which neglect this term. The relative importance of the various components in the Breit interaction and of the quark-gluon exchange diagrams is made clear.     

Numerisch genaue Lösungen des nichtlokalen Schalenmodells

Nonlocal potentials can in general be transformed only approximately into local potentials with additional energy dependence. For the optical modelPerey andBuck did an exact nonlocal calculation with the Schrödinger-equation in integro-differential form and also a good local approximation. Certain systematic deviations in the results (concerning the wave-functions) of both processes are of special interest in the shell model. Therefore the eigenvalue problem has been solved now with the most successful nonlocal potential, also used byPerey andBuck, both exactly and approximately, that is with the potential, which is nondiagonal, and its approximation, which is diagonal in local space. The solution of this approximate nonlocal potential serves again as zero order starting value (E ₀,u ₀) for an iteration process, which now is twofold, in order to find that exact solution (E,u). Thus we get together with the exact solution also the deviation of the approximate nonlocal (local-equivalent) one.     

The generalized WKB method in the three-dimensional case. IV

A method of successive approximations is developed for obtaining solutions of the basic equations of the WKB method in the three-dimensional case. Explicit expressions are found for the first approximations to the energy levels and wave functions of the bound states of a particle in a three-dimensional potential well which were obtained in zeroth approximation in [1, 2].Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 53–60, November, 1976.     

Transient response of a spherical shell in an acoustic medium—Comparison of exact and approximate solutions

The transient response of a spherical shell in an acoustic medium is studied. The exact solution is obtained by expressing the classical spherical wave equation in terms of a residual potential. Approximate solutions are obtained from the use of eight functions; namely, doubly asymptotic approximations, δ-sequence functions, and spherical wave approximations. The advantages and disadvantages of these approximate solutions are pointed out. The third member of the class of the modal doubly asymptotic approximations for a spherical surface and an improved spherical wave approximation are introduced.     

Semiclassical scattering theory with complex trajectories. I. Elastic waves

The WKB approximation to the one-particle Schrödinger equation is used to obtain the wave function at a given point as a sum of semiclassical terms, each of them corresponding to a different classical trajectory ending up at the same point. Besides the usual, real trajectories, also possible complex solutions of the classical equations of motion are considered. The simplicity of the method makes its use easy in practical cases and allows realistic calculations. The general solution of the one-dimensional WKB equations for an arbitrary number of complex turning points is given, and the solution is applied to calculate the position of the Regge poles of the scattering amplitude. The solution of the WKB equations in three dimensions for a central analytical potential is also obtained in a way that can be easily generalized to N-dimensions, provided the problem is separable. A multiple reflection series is derived, leading to a separation of the scattering amplitude into a smooth “background” term (single reflection approximation) that can be treated using classical but complex trajectories and a second resonating term that can be treated using the Sommerfeld-Watson transformation. The physical interpretation of the complex solutions of the classical equations of motion is given: they describe diffractive effects such as Fresnel, Fraunhofer diffraction, or the penetration of the quantal wave into shadow regions of caustics. They arise also in the scattering by a complex potential in an absorptive medium. The comparison with exact quantal calculations shows an astonishingly good agreement, and establishes the complex semiclassical approximation as a quantitative tool even in cases where the potential varies rapidly within a fraction of a wavelength. An approximate property of classical paths is discussed. The general pattern of the trajectories depends only on the product ? = EΘ, and not on energy and angle separately. This property is confirmed by experiments and besides the signature it gives for the semiclassical behavior, it simplifies considerably the search for all trajectories scattering through the same angle. Finally, a general classification of the different types of elastic heavy ion cross sections is given.     

超对称WKB近似与一维无限深势阱

证明了一维无限深势阱为非形状不变势.采用超对称WKB近似,得到了一维无限深势阱的精确能谱公式.计算结果表明,把势场具有形状不变性作为超对称WKB近似能够给出精确能谱的充要条件,是一个过强的提法,需要加以弱化.     

Application of the real Weinberg states in potential scattering

This work is the first step in a systematic application of a resonant scattering theory due to Huby in which the real Weinberg state is the crucial feature. The scattering cross sections ofl=0,1, 2 partial waves in a square well potential are calculated with the aid of the real Weinberg states. Perfect agreement with the exact calculation is obtained. Approximate cross sections resulting from a schematic approximation in the formalism which, apart from other desirable features, display the Breit-Wigner energy factor in the phase shifts, are also found to agree perfectly with the exact calculation. Some simple approximations of the complex and real Weinberg states are investigated, but none is found to be satisfactory forl>0 partial waves in complete contrast to works of previous authors on thel=0 partial wave.     

Green's function and scattering matrix in a discrete oscillator basis

Convenient analytic finite-dimensional approximations for basic operators of scattering theory-specifically, the Green's function and the off-shell T matrix—are constructed in an oscillator basis for real-and complex-valued local and nonlocal interaction potentials. It is shown that the approximate operators converge smoothly to their exact counterparts as the dimensions of the oscillator basis are increased step by step. The simple and rather accurate formulas obtained in this study can be widely used in various applications of quantum scattering theory.     

Gradient-level and nonlocal density functional descriptions of Cu-Au intermetallic compounds

We use three gradient level and two nonlocal density functional approximations to study the thermodynamic properties of Cu-Au compounds. It is found that a well-designed gradient level approximation (quasi non-uniform approximation, QNA) reproduces the experimental equilibrium volumes and the formation energies of L12 and L10 phases. On the other hand, QNA predicts a non-existent β₂ phase, which can be remedied only when employing the nonlocal hybrid-level Heyd-Scuseria-Ernzerhof (HSE06) or Perdew-Burke-Ernzerhof (PBE0) approximations. Gradient-level approximations lead to similar electronic structures for the Cu-Au compounds whereas hybrids shift the d-band towards negative energies and account for the complex d-d hybridization more accurately.     

Properties of nuclear matter in the Λ00-approximation for local potentials

In the Λ⁰⁰-approximation of the Green's function theory the nuclear matter properties have been calculated for several local potentials. All these potentials contain a hard core repulsion and match the experimentalS-wave phase-shifts. Due to the analytical form of these potentials one can obtain many results analytically. Thus the whole structure of this approximation becomes relatively transparent and can easily be calculated. Therefore this model can be used as an input for more complicated approximations. Furthermore one can also treat the optical potential problem. Te results for nuclear matter are comparable with the results for nonlocal potentials. The momentum distribution deviates more strongly from the independent particle model as in the case of nonlocal potentials.     

Effects of barrier penetration on energy spectrum and wave functions in the path integral formalism

We present a systematic approximation scheme for the non-relativistic Green function in the path integral representation which takes into account barrier penetration effects on the energy spectrum and wave functions for potentials having several minima. The method is applied to the cases of a symmetric potential with two minima and to a periodic potential. The energy spectrum is shown to agree with the usual WKB results. For the first mentioned case we also determine the wave functions in the classically allowed and forbidden regions from the residues of the Green function. They agree with those obtained in the standard WKB approximation.     

Complex paths and uniform approximations in a semi-classical description of direct reactions

The semi-classical (WKB) approximation in three dimensions is considered for scattering processes. Complex solutions of the classical equations of motion are included. They represent contributions to the quantal wave in classically forbidden regions, and in particular describe diffractive effects in the cases where the scattering potential varies appreciably within a wavelength. Special emphasis is given to inelastic and transfer processes, including multistep transitions within a coupled channel frame. Uniform approximation techniques are discussed and widely generalized. Especially, it is shown that when the semi-classical approximations are done consistently at every step of the calculation, there is no need for uniform treatments in the intermediate stages of the calculation, but only at the far end. This simplifies considerably the calculations. Applications to nuclear physics are considered by comparing quantum mechanical and semi-classical cross sections for the scattering of two heavy ions. Excellent agreement is obtained between the two calculations. The new insight brought by the complex semi-classical approximation is used in order to obtain phase rules for the angular distribution, and matching rules. Transition probabilities in their dependence on energy and angle are discussed. They turn out to depend dominantly on ΘE and not on both variables separately.     

Numerisch genaue Lösungen des nichtlokalen Schalenmodells

The nonlocal Schrödinger-equation has been solved in the shell model case with a kernel-function, which is well known from optical model calculations. Now in the eigenvalue region the energies, as well as the wave-functions, of the exact and the approximate nonlocal calculations (E andE ₀) differ slightly on the average. The exact solutions show in general larger energy-level distances. Although these differences are small, fitting of experimental neutron binding-energies requires nevertheless a relatively big change of the set of physical parameters; the exact nonlocal calculations tend to larger potential depthV and smaller radius constantr ₀ than the approximate nonlocal ones. For the wave-functions (Φ andΦ ₀) it can be shown that an application of the Perey-effect leads to a wrong result in the shell model case. Here the difference function shows two inflection points, whereas there is only one in the optical model. The form of nuclear matter distribution thus becomes more potential-like. The eigenfunctions of the exact nonlocal calculation are orthogonal, whereas the local equivalent ones are not.     

Dynamic dielectric response of a quantum wire superlattice in interaction with a nonlocal host medium

We study nonlocality effects of a bulk plasmalike dielectric medium on the plasmon spectrum of a one-dimensional (1D) quantum wire superlattice in interaction with the 3D nonlocal host bulk plasma, by carrying out a closed-form analytic determination of the inverse dielectric function κ for the joint nanostructure system within the random phase approximation (RPA), in which we treat nonlocality of the 1D superlattice in the RPA and that of the bulk medium in the hydrodynamic model. By examining the frequency poles of κ (i.e., the dispersions relations), we show analytically that coupled plasmon modes of the interacting 1D superlattice-3D nonlocal host are damped in high frequencies (damping is pronounced near resonance region) and that nonlocality of the host medium introduces nonlocal low frequency (real) modes into the spectrum, which have cutoff frequencies for finite wave vector values. In order to describe the impact of nonlocality effects more clearly, we also examine the spectrum numerically.     

Influence of strongly coupled inelastic excitations on the optical potential for12C+12C

The contributions of some of the strongly coupled collective excitations in¹²C+¹²C to the heavy ion optical potential have been explicitly obtained by calculating the exact Green's function for this problem. The resulting nonlocal heavy ion potential and the trivially equivalent local potential are calculated and theirJ- andE-dependence are discussed. It is shown that the strong direct excitations contribute significantly to the absorptive part of the heavy ion potential in the surface region.