Asymptotic Normalization Coefficients in a Potential Model Involving Forbidden States

It is shown that values obtained for asymptotic normalization coefficients by means of a potential fitted to experimental data on elastic scattering depend substantially on the presence and the number n of possible forbidden states in the fitted potential. The present analysis was performed within exactly solvable potential models for various nuclear systems and various potentials without and with allowance for Coulomb interaction. Various methods for changing the number n that are based on the use of various versions of the change in the parameters of the potential model were studied. A compact analytic expression for the asymptotic normalization coefficients was derived for the case of the Hulthén potential. Specifically, the d + α and α + ¹²C systems, which are of importance for astrophysics, were examined. It was concluded that an incorrect choice of n could lead to a substantial errors in determining the asymptotic normalization coefficients. From the results of our calculations, it also follows that, for systems with a low binding energy and, as a consequence, with a large value of the Coulomb parameter, the inclusion of the Coulomb interaction may radically change the asymptotic normalization coefficients, increasing them sharply.     

Quality of the three-nucleon bound-state wave function from a two-nucleon separable expansion method

The numerical quality of the³H wave function obtained by the separable expansion method of Ernst, Shakin, and Thaler is examined. Separable approximations to the Paris potential with increasing accuracy are used in the¹ S ₀ and³ S ₁-³ D ₁ partial waves to calculate the binding energy, wave function, wave-function component percentages, and theS- andD-wave asymptotic normalization constants of³H. The results are compared with existing five-channel calculations obtained directly (without expansion) from the Paris potential to determine convergence. It is found that the results converge rapidly to the right values, indicating that the³H wave function thus obtained is of high quality and essentially indistinguishable from that obtained directly from the Paris interaction.Dedicated to Profs. Erich Schmid and Ivo laus on the occasion of their 60th birthdays     

Triton binding energy and three-nucleon potential

The method of continued fractions proposed by Horáek and Sasakawa is applied to calculate the binding energy of the triton. By this method, the Faddeev equation is solved very quickly. We solved it with 34 channels for the super-soft core (TRS), Paris (PARIS) and Argonne (AV) potentials with Tucson-Melbourne (TM) three-nucleon potential. The binding energy obtained from these realistic two-nucleon potentials without a threenucleon potential is at most 7.7 MeV. If we include the TM three-nucleon potential taking a cutoff mass of =800 MeV for the dipole NN form factor, the triton is overbound (9.3 9.7 MeV), whereas for =700 MeV, we get binding energies which almost agree with the experimental value; TRS+TM, 8.47; PARIS + TM, 8.32; AV+TM, 8.42 MeV.     

Noncentral effects in two-nucleon transfer reactions

Using general forms of the bound tri-nucleon wave function includingS- andD-components and of the two body interaction containing the tensor force, noncentral effects in two-nucleon transfer reactions of the type (t, p) or (³He, p) are investigated. The discussion suggests that non-central effects could play an important role in these reactions. The central and noncentral parts contribute coherently in certain cases in the expression for the differential cross section even if the spin independent optical potentials are used.     

Hypernuclear interactions and multi-Λ matter

Infinite nuclear matter composed of nucleons and hyperons in various mixture ratios is considered with the Skyrme-like phenomenological potentials. Basic characteristics (energy, density, incompressibility, symmetry energy, chemical potentials) are calculated and their link with the potentials properties is studied. Conditions of stability with respect to baryon emission are analyzed and it is shown that neutron-rich systems are preferable for binding the highest possible number of hyperons.     

Precise deuteron binding energy calculations

A new minimization approach to two-body binding energy calculation is suggested and applied to deuteron. The results of calculations with very high accuracy (at least 10 eV) for several nucleon--nucleon potentials are given. The suggestion to use new precise experimental values of deuteron binding energyE _D and calculated one as a constraint on N-N potentials follows from a study ofE _D sensitivity on the potential parameters.The authors are indebted to dr. E. Truhlik for a loan of initial version of DEUT code and useful discussions. The advices of dr. J. Piút are acknowledged. The presented work is based on thesis of P.K.     

Quasi-free scattering in the 3-N continuum

The interaction of a nucleon with the deuteron leading to break-up at the quasi-free scattering (QFS) condition is studied atE _N ^lab =12.0, 22.7, 68.0, and 140.0 MeV. Results of rigorous Faddeev calculations and of the impulse approximation (IA) are compared at these energies for bothn-p andp-p QFS in order to study the importance of rescattering terms. For the two-nucleon interaction the Paris potential is used. It is restricted to act in two-body states with total angular momentumj2. Also we compare the Faddeev calculations with experimental data for the QFS cross section and analyzing powerA _y and find overall agreement. The behaviour of scattering observables turns out to be different forn-p andp-p QFS. Also the energy dependence of rescattering terms is different in the two cases. For all energies studied the IA is not reliable.     

Vertex constants (asymptotic normalization coefficients) and mean-square radii of hypernuclei in the potential model

The vertex constants, asymptotic normalization coefficients, and mean-square radii for a number of Λ hypernuclei have been calculated in a wide range of mass numbers. Interaction of a Λ hyperon with a core nucleus was described by the Woods-Saxon, Hultén, and Yukawa potentials. Geometric parameters of the Woods-Saxon potential were taken from the data in the literature, and the interaction force was fitted to the experimental binding energy of the ground state of this hypernucleus. The parameters of the Hulthén and Yukawa potentials were chosen so as to reproduce simultaneously the binding energy of the hyperon and its mean-square radius calculated with the Woods-Saxon potential. Within the effective-radius approximation, the low-energy parameters for hyperon scattering by the core nucleus are expressed in terms of vertex constants and the numerical values of these parameters are determined.     

Dominant resonance pole of the three-neutron system

The resonance behaviour of a system of three neutrons is considered. For the quantum numbers (T. S, L) = (2/3, 1/2, 1^–) and s-wave Yamaguchi two-particle interaction the dominating pole pair of the S-matrix is determined. To find the pole positions and the asymptotic normalization parameters contour deformation technique has been applied to solve the homogeneous Faddeev equation on the unphysical sheets of the S-matrix. For the positions and the asymptotic normalization constants of the dominating pole pair the values z_p=(–88·0i27·9) keV and C _p ² = –0.05±i0·1 are obtained. Although the pole pair is close to the real axis in the complex energy plane it is not expected to cause an observable experimental effect due to the smallness of the asymptotic normalization constant.Presented at the symposium Mesons and Light Nuclei, Liblice, Czechoslovakia, June 1981.     

Calculating the characteristics of neutron-deuteron and proton-deuteron systems in a two-body potential model

³H and ³He nuclei are considered in a two-body model (³H = n + d; ³He = p + d). Two independent approaches are used: in the first, interaction is described by the folding potential, while NN potentials are taken in the Hulthen form with allowance for violations of isotopic invariance. The second approach features phenomenological Hulthen and Yukawa Nd potentials used as Nd interaction. In both approaches, the binding energies, vertex constants, and asymptotic normalization coefficients in the N+d channel are calculated.     

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.     

Saturation of nuclear forces in Brueckner theory

The basic equations in the Brueckner theory of nuclear matter are solved for a two-nucleon potential taken as a rectangular well with a rectangular repulsive sphere and the properties of the solutions are investigated for various dimensions of the repulsive sphere. The two-nucleon interaction is considered to be non-vanishing only if the nucleons are in theS-state. Under such assumptions a two nucleon potential always gives saturation, i.e. a minimum of the mean binding energy per nucleon, at a finite nuclear density. It is shown that the nuclear density and the mean binding energy decreases if the height or the width of the repulsive sphere increases. If the repulsive sphere is infinitely high a nucleus can exist as a bound state only if the width of this sphere is sufficiently small. The limit value for the width of the sphere is given. It is shown in the conclusion how the solution of the basic equations will change if the two-nucleon potential does not contain an infinite repulsive sphere, but only a very high one.     

Non-relativistic bound states in a finite volume

We derive general results for the mass shift of bound states with angular momentum $?\ge 1$ in a periodic cubic box in two and three spatial dimensions. Our results have applications to lattice simulations of hadronic molecules, halo nuclei, and Feshbach molecules. The sign of the mass shift can be related to the symmetry properties of the state under consideration. We verify our analytical results with explicit numerical calculations. Moreover, we comment on the relations connecting the effective range parameter, the binding momentum of a given state and the asymptotic normalization coefficient of the corresponding wave function. We give explicit expressions for this relation in the shallow binding limit.     

Eigenvalues of the one-dimensional Smoluchowski equation

The eigenvalues and eigenfunctions of the Smoluchowski equation are investigated for the case of potentials withN deep wells. The small parameter =kT/V, which measures the ratio of the thermal energy to a typical well depth, is used in connection with the method of matched asymptotic expansion to obtained asymptotic approximations to all the eigenvalues and eigenfunctions. It is found that the eigensolutions fall into two classes, namely (i) the top-of-the-well and (ii) the bottom-of-the-well eigensolutions. The eigenvalues for both classes of solutions are integer multiples of the squqres of the frequencies at the top or bottom of the various wells. The eigenfunctions are, in general, localized to the top or bottom of the corresponding well. The very small eigenvalues require special consideration because the asymptotic analysis is incapable of distinguishing them from the zero eigenvalue with multiplicityN. Another approximation reveals that, in addition to the true zero eigenvalue, there areN-1 eigenvalues of order exp(–). The case of other possible multiple eigenvalues is also examined.     

Inequalities on the number of bound states in oscillating potentials

For short-range oscillating potentialsV(r), such that possesses some regularity properties we establish inequalities on the number of bound states. In particular we show that by replacingV(r) by –4(W(r))² in the classical inequalities we get bounds for this new class of potentials. Optimal bounds are also obtained. The behaviour for large coupling constants is studied.     

Change of energy profiles for muons upon lattice relaxation

Systematic calculations of potentials for muons and protons in various simple metals have been recently performed using local pseudopotentials in first-order perturbation theory. In the present contribution some results which are of particular interest forSR are discussed. The modification of the energy profiles with lattice relaxation is studied in Cu and Cs.     

Analysis of the Effects of Three-nucleon Forces in A = 3, 4 Systems

It is not possible to reproduce both the three- and four-nucleon binding energies using the available two-nucleon potentials. This is one manifestation of the need to include a three-nucleon force in the corresponding Hamiltonian. In this paper we will analyze the capability of a three-nucleon force model to describe not only the aforementioned binding energies but also some N ? d low energy scattering observables.     

Nuclear vertex constants and asymptotic normalization coefficients for the tritium nucleus

The properties of the nuclear vertex constant for virtual triton decay to a deuteron and a neutron (T → d + n) are investigated along with the properties of the asymptotic normalization coefficient defined for the triton wave function and related to this constant. These quantities are calculated numerically on the basis of an equation that relates the asymptotic normalization coefficient to the triton effective radius ρ _T, which was introduced in the present study. The values of G _T² = 1.244(68) fm and C _T² = 2.958(162) found from our calculations are in good agreement with experimental and theoretical estimates obtained for these quantities in other studies. Physical properties of the triton virtual state are also discussed.     

Polarization effects in dispersion relations for photoproduction of π-mesons on deuterons

The photoproduction amplitude of ⁰-mesons on deuterons is calculated for different cases of polarization of the spin of deuterons by means of impulse approximation. The calculations carried out in the two-nucleon approximation in the Breit system are valid up to a maximum photon energy 236 MeV. In the unobservable region only a free proton and a free neutron contribute to the continuous part of the spectrum.The dispersion relations are given for all cases of deuteron-spin polarization.This paper is a continuation of the author's papsr elaborated in the Laboratory of Theoretical Physics of the Joint Institute of Nuclear Research, Dubna USSR, published in Nuclear Physics26 (1961), 681, which is hereafter referred to as I.     

Semi-infinite matter and the nuclear surface

The surface energy and thickness of semi-infinite nuclear matter are calculated by Swiatecki's variational method. Smooth, effective two-body interactions are used to study the relationship between the characteristics of each interaction and the predicted bulk surface properties. Both local and non-local Gaussian interactions are considered and are adjusted to either saturate infinite nuclear matter or to fit two-nucleon (singlet-even) phase shifts. Purely local potentials are found to give an excessively large surface energy ( ), whereas potentials having some non-locality (even if the non-locality is confined to short distances) reduce the surface energy to a reasonable value of (20–30) . The separate potential and kinetic energy contributions are examined in detail and are used to explain how the character of each interaction affects the surface properties. This model calculation shows that the surface energy is greatly influenced by non-locality. Also, we find that the predicted surface thickness tends to be too large (t ≈ 3.5 fm) for potentials that are adjusted to saturate and too small (t ≈ 1.8 fm) for potentials that are fitted to two-nucleon phase-shifts.