G-matrix Elements Research Articles

Microscopic Calculation of the Optical Model Potential from One Boson Exchange Poten­tials

A new method for calculating the optical model potential from One Boson Exchange Potentials (OBEPs) is developed. The G-matrix is calculated by solving the Bethe-Goldstone equation in momentum space. Using vector brackets these G-matrix elements can be transformed from the center of mass representation into the laboratory system. This allows the evaluation of the (r-matrix interaction between nucleons in bound states and those in a plane-wave state. The lowest order contribution to the real part of the potential comes from the Hartree-Fock term, while the lowest order contribution to the imaginary part comes from the two-particle-one-hole (2p1h) diagram. Calculations for 16O and 40Ca have been carried out. Local approximations are obtained by describing the results for the central part in terms of a Woods-Saxon potential and those for the spin-orbit part in terms of the corresponding derivatives. The dependence of these potentials on energy and angular momentum is discussed. The parameters for these local approximations are in good agreement with empirical fits.

Design of specially adapted reactive coordinates to economically compute potential and kinetic energy operators including geometry relaxation.

Quantum dynamics simulations require prior knowledge of the potential energy surface as well as the kinetic energy operator. Typically, they are evaluated in a low-dimensional subspace of the full configuration space of the molecule as its dimensionality increases proportional to the number of atoms. This entails the challenge to find the most suitable subspace. We present an approach to design specially adapted reactive coordinates spanning this subspace. In addition to the essential geometric changes, these coordinates take into account the relaxation of the non-reactive coordinates without the necessity of performing geometry optimizations at each grid point. The method is demonstrated for an ultrafast photoinduced bond cleavage in a commonly used organic precursor for the generation of electrophiles. The potential energy surfaces for the reaction as well as the Wilson G-matrix as part of the kinetic energy operator are shown for a complex chemical reaction, both including the relaxation of the non-reactive coordinates on equal footing. A microscopic interpretation of the shape of the G-matrix elements allows to analyze the impact of the non-reactive coordinates on the kinetic energy operator. Additionally, we compare quantum dynamics simulations with and without the relaxation of the non-reactive coordinates included in the kinetic energy operator to demonstrate its influence.

Proton capture by 14N at astrophysical energies

Calculations are reported for low energy, proton capture by 14N with the recoil corrected continuum shell model. An interaction from a fit to Cohen and Kurath (6–16) p-shell matrix elements and Reid soft-core g-matrix elements is employed. The prediction for 14N(p, γ)15O, based on this model and available data, is that S(0) equals 1.676 and S(30) is 1.666 keV b. Good agreement with available cross section data support this result. No evidence is found for significant contributions from the subthreshold resonance.

The hypernucleus 5ΛHe in a two frequency model

Our previously developed two-frequency folded-diagram method is applied to calculate the binding energy of 5ΛHe starting with Nijmegen YN potentials NSC89 and NSC97(a-f). Since the masses of a hyeron and a nucleon are different and so may be their binding energies in a hypernucleus, wave functions of a hyperon and a nucleon are treated with different frequencies. The frequency ℏ ω N for a nucleon in 5ΛHe is set constant according to an empirical formula, while that for a hyperon ℏ ω Y remains as a parameter. The G-matrix elements are calculated exactly by using an exact Pauli exclusion operator. We then obtain the Λ's single particle energy in 5ΛHe as a function of ℏ ω Λ . A stable ℏ ω Λ can be found at the saturate minimum. Results from various modes of the realistic Nijmegen NSC89 and NSC97 YN potentials are compared to each other. Calculations by switching ℏ ω N to other values are also done. The result from NSC97f is found to be closest to the experimental value. Possible implications of the results are discussed.

First-Principle Calculation of Reduced Masses in Vibrational Analysis Using Generalized Internal Coordinates: Some Crucial Aspects and Examples

In this paper we present and analyze the most essential aspects of reduced masses along generalized internal coordinates. The definition of reduced masses in the internal coordinate formalism is established through the Wilson G-matrix concept and includes sophisticated relations between internal and Cartesian coordinates. Moreover, reduced masses in internal coordinates are, in general, no longer constant but coordinate-dependent. Based on the approach presented earlier [Stare, J.; Balint-Kurti, G. G. J. Phys. Chem. A 2003, 107, 7204-7214] and on our experience with reduced masses discussed in this paper, we have developed a robust program for the calculation of Wilson G-matrix elements and their functional coordinate dependence. The approach is based on the first principles and can be used in virtually any (internal) coordinate set. Since the program allows for projection of any kind of nuclear motion on the selected internal coordinates, the method is particularly suitable for ab initio or DFT potential energy functions calculated by partial geometry optimization. Moreover, reduced masses obtained by this program can be used as a decision tool for selecting the most appropriate internal coordinates for the considered vibrational problem and for the inclusion or omission of the kinetic coupling terms in the vibrational Hamiltonian.

Systematics of the α decay to vibrational2+states

We give a systematic analysis of \ensuremath{\alpha} decays to low-lying ${2}^{+}$ states in even-even nuclei. Collective excitations are considered within the spherical quasiparticle random-phase approximation. We use realistic G-matrix elements of the Bonn interaction as a residual two-body force. The only free parameters are the ratio between the isovector and isoscalar strengths and proton-neutron asymmetry. The formalism can reproduce the main experimental trends versus the excitation energy for both the $B(E2)$ values and the \ensuremath{\alpha}-decay hindrance factors. We reproduced most of the available data by using one common parametrization. It turns out that the fine structure of the \ensuremath{\alpha} decay is more sensitive than electromagnetic transitions as a tool for investigating nuclear interaction. With the adopted parametrization, we predict $B(E2)$ values and \ensuremath{\alpha}-decay hindrance factors in even-even nuclei.

Inclusion of nonspherical components of the Pauli blocking operator in (p,p') reactions

We present the first calculations of proton elastic and inelastic scattering in which the Pauli blocking operator contains the leading non-spherical components as well as the usual spherical (angle-averaged) part. We develop a formalism for including the contributions to the effective nucleon-nucleon interaction from the resulting new G-matrix elements that extend the usual two-nucleon spin structure and may not conserve angular momentum. We explore the consequences of parity conservation, time reversal invariance, and nucleon-nucleon antisymmetrization for the new effective interaction. Changes to the calculated cross section and spin observables are small in the energy range from 100 to 200 MeV.

Mass dependence ofM3Y-type interactions and the effects of tensor correlations

The mass dependence of the M3Y-type effective interactions and the effects of tensor correlations are examined. Two-body nuclear matrix elements are obtained by the lowest order constrained variational (LOCV) technique with and without tensor correlations. We have found that the tensor correlations are important especially in the triplet-even (TE) and tensor-even (TNE) channels in order to reproduce the G-matrix elements obtained previously. Then M3Y-type potentials for inelastic scattering are obtained by fitting our two-body matrix elements to those of a sum of Yukawa functions for the mass numbers A=24, A=40 and A=90.

α-nucleus optical potential in the double-folding model

The double-folding formalism for the $\ensuremath{\alpha}$-nucleus optical potential is revised to study the exchange effects and density dependence of the effective nucleon-nucleon $(\mathrm{NN})$ interaction in detail. A realistic density dependent M3Y interaction, based on the G-matrix elements of the Paris NN potential, has been used in the folding calculation. The local approximation for the nonlocal one-body density matrix in the calculation of the exchange potential was tested by using the harmonic oscillator representation of the nonlocal density matrices of the $\ensuremath{\alpha}$-particle and target nucleus. The inclusion of a realistic density dependence into the effective NN interaction was shown to be vital for a correct description of the refractive $\ensuremath{\alpha}$-nucleus scattering data. A high sensitivity of the density distributions of the $\ensuremath{\alpha}$-particle and target nucleus to the shape of the $\ensuremath{\alpha}$-nucleus potential was found, which can be used in the folding analysis to test various density models for the $\ensuremath{\alpha}$-particle and target as well as to choose the most realistic approximation for the overlap density in the dinuclear system. Our results also stress the importance of $\ensuremath{\alpha}$-nucleus scattering experiment in the nuclear structure study.

A local density approximation treatment for the Pauli exclusionoperator in hypernuclei

An effort is made to simplify the complications of computinghyperon-nucleon G-matrix elements arising from the Pauliexclusion operator contained in the integral equation. Weperform a two-frequency shell-model folded-diagram calculationon hypernuclei using two different treatments for the Pauliexclusion operator Q2, namely, a local densityapproximation whereQ2 is replaced by a nuclear-matter Pauli operator {Q̄}(ρ), with ρ being a density parameter, and an exactcalculation where Q2 is expressed in terms of shell-modelwavefunctions. With a proper choice of ρ, it is possibleto reach a reasonable accuracy by using a localdensity approximation for the Pauli operator. For heavy hypernuclei, this{Q̄}(ρ) can be used to savea tremendousamount of computing time.

Generalized folding model for elastic and inelastic nucleus–nucleus scattering using realistic density dependent nucleon–nucleon interaction

A generalized double-folding model for elastic and inelastic nucleus–nucleus scattering is presented. It is designed to accommodate effective nucleon–nucleon (NN) interactions that depend upon the density of nuclear matter in which the two nucleons are immersed. A recently parametrized density dependent M3Y interaction, based on the G-matrix elements of the Paris NN potential, has been used in the present folding calculation. The effects of knock-on exchange of the interacting nucleon pair are included in an accurate local approximation. Examples of the application of this model to study the refractive elastic and inelastic scattering data of 12 C+ 12 C and α+ 58,60Ni systems are presented. A detailed comparison of the use of deformed optical potential (DP) and microscopic folded potential in the analysis of inelastic scattering has shown that the use of DP fails to reproduce the inelastic 12 C+ 12 C scattering data measured over a wide angular range, which covers both the diffractive and refractive regions. The use of the DP model also significantly underestimates the nuclear deformation lengths of the quadrupole and octupole excitations under study.

Theoretical study of the O-H stretching band in 3-hydroxy-2-methyl-4-pyrone

The infrared spectrum of 3-hydroxy-2-methyl-4-pyrone reveals an O-H stretching frequency roughly 200 cm−1 lower than that of a typical alcohol group. The frequency lowering results from intramolecular hydrogen bonding between the alcohol and ketone groups. In this work, the stretching and bending vibrations of the O-H group in 3-hydroxy-2-methyl-4-pyrone are studied with a theoretical methodology more rigorous than the conventional harmonic approximation. A two-dimensional potential energy surface in internal coordinates corresponding to different hydrogen positions in the plane of the molecule is calculated with the use of the second order Mo/ller-Plesset perturbation theory. To include all possible variations in kinetic energy in a large amplitude vibrational mode, g-matrix elements with variable values are employed. The analytical expression for the Hamiltonian matrix elements of the two-dimensional vibrational problem in a basis of shifted Gaussian functions is derived. Expectation values for the O-H stretch nuclear states are variationally determined with the use of shifted Gaussian functions as the basis set. The results of the calculations are compared with the recent matrix-isolation infrared (IR) spectroscopic results. The calculated transition frequency corresponding to the in-plane O-H stretching is found to be in good agreement with the experimental value.

Nuclear G-Matrix Elements from Nonlocal Potentials

We study effects of nonlocality in the nuclear force on the G-matrix elements for finite nuclei. Nuclear G-matrix elements for 16O are calculated in the harmonic oscillator basis from a nonlocal potential which models quark exchange effects between two nucleons. We employ a simple form of potential that gives the same phase shifts as a realistic local nucleon potential. The G-matrix elements calculated from the nonlocal potential show moderate increase in repulsion from those derived from the local potential.

Projected quasi-particle calculations of 119Sn, 121Sn, and 123Sn using realistic interactions

We have performed quasi-particle calculations for the tin isotopes 119Sn, 121Sn and 123Sn, using realistic effective interactions derived from the Paris and Bonn-A NN potentials. A number-projected one- and three-quasi-particle formalism in the (g72, d52, d32, s12, h112) model space is employed. We have used a folded-diagram method to derive the corresponding model-space effective interaction, starting from the G-matrix elements derived from the above NN potentials. The G-matrix Pauli-exclusion operator Q2, specified by the shell-model orbitals (n1, n2, n3) = (11, 21, 36), has been treated accurately. An almost pure (h112)3 three-quasiparticle structure for the high spin states from J = 172− to 272− of the above nuclei has been observed. With practically no parameter adjustment, our calculated energies are in very good agreement with available experimental data.

Systematic study of beta and double beta decay to excited final states

A systematic study of the two-neutrino double beta (2νββ) decay to the final ground state and excited states is performed within an extension of the quasiparticle random-phase approximation (QRPA) model. In this extension, the multiple commutator model (MCM), a simultaneous treatment of the double-odd and double-even nuclei is possible by assuming their states to have the structure of one or two QRPA phonons. The MCM calculation is done for nine 2νβ−β− decays and six 2νECEC decays, and systematics of the relevant matrix elements and half-lives have been created. The study of the 2νββ-decay rates is complemented with the MCM study of the single-beta-decay properties of the relevant nuclei within the double-beta isobaric chains. The Woods-Saxon single-particle energies have been corrected near the Fermi surface by comparing the BCS quasiparticle energies with spectroscopic data of odd-mass nuclei. Pairing gaps, energy systematics of the Gamow-Teller states and the available beta-decay data have been used to obtain effective, model-space adapted, two-body matrix elements starting from the G-matrix elements of the Bonn one-boson-exchange potential. This enables a parameter-free calculation of the double Gamow-Teller matrix elements and theoretical prediction of double-beta half-lives.

Folding model description of fusion reaction

The shape, separation and orientation dependence of the interaction potential and fusion cross section of the system are investigated within the double-folding model of deformed nuclei. The effective nucleon - nucleon interaction is taken to be the M3Y, which has been derived from G-matrix elements and based upon the Reid soft-core potential. A deformed Fermi shape with quadrupole and hexadecapole deformations is considered for the density distribution of . It is found that both the interaction potential and fusion cross section strongly depend on the deformations and orientations. The integrated fusion cross section over all orientation angles yields an excellent agreement with the experiment data.

Refractive alpha-nucleus scattering: a probe for the incompressibility of cold nuclear matter

Refractive α-nucleus scattering observed at intermediate energies is of great help in determining the α-nucleus potential at small distances, where the overlap density is large. A systematic folding analysis of the elastic α+ 12C, 40Ca and 90Zr scattering data at E lab = 59.1–172.5 MeV has been performed using effective density dependent NN interactions based on the G-matrix elements of the Reid- and Paris NN potentials. The new interactions consistently reproduce the saturation properties of cold nuclear matter and generate, at the same time, different values of the nuclear incompressibility K. With the existing refractive scattering patterns, it is possible to conclude from our folding analysis that a nuclear incompressibility K ⋍ 220 ± 50 MeV is the most realistic one.

Simple approximation for the starting-energy-independent two-body effective interaction with applications to 6Li.

We apply the Lee-Suzuki iteration method to calculate the linked-folded diagram series for a new Nijmegen local NN potential. We obtain an exact starting-energy-independent effective two-body interaction for a multishell, no-core, harmonic-oscillator model space. It is found that the resulting effective-interaction matrix elements can be well approximated by the Brueckner G-matrix elements evaluated at starting energies selected in a simply way. These starting energies are closely related to the energies of the initial two-particle states in the ladder diagrams. The ``exact'' and approximate effective interactions are used to calculate the energy spectrum of $^{6}\mathrm{Li}$ in order to test the utility of the approximate form.

Energy-independent effective interactions: Analytic expressions for the required G-matrix energy derivatives

In a recent paper it has been shown that the successful application of the method of Suzuki and Lee for calculating an energy-independent effective interaction requires that the energy derivatives of the G-matrix be calculated accurately. In that work methods by which these derivatives may be calculated analytically in problems where P, the projection operator which defines the active space, can be written in the center-of-mass and relative coordinates were presented. In this work we extend the applicability of the theory by presenting analytic expressions for calculating the energy derivatives of the G-matrix elements obtained by the method of Barrett et al. Convergence properties of the iterative processes by which the effective interaction is calculated are investigated and guidelines for choosing the starting energy are given. As an example of its use, the theory is applied to the ground state and low-lying resonances of 4He.

G-matrix elements with effective masses for mesons and nucleons

G-matrix elements for finite nuclei are calculated using effective masses for mesons and nucleons at finite density, i.e., those in nuclear medium. We find that the central G-matrix elements are very sensitive to masses of the sigma and omega. By decreasing the mass of the rho, the LS matrix elements are generally enhanced, while the tensor matrix elements are suppressed. A uniform decrease in various masses yields G-matrix elements which are rather consistent with empirical matrix elements obtained by Brown, Richter, Julies and Wildenthal.