Hamada-Johnston Potential Research Articles

Generalized pairing inN=Zeven-even2p−1fshell nuclei

Hartree-Fock-Bogoliubov calculations have been performed for self-conjugate even-even nuclei in the $2p\ensuremath{-}1f$ shell using the central Yukawa interaction and Kuo-Brown reaction matrix elements for the Hamada-Johnston potential. The possibility of generalized pairing correlations is studied by considering $T=0$, $T=1$, and $T=0+T=1$ correlations and comparing the ground-state properties for axial and triaxial shapes. It is found that for nuclei studied neither the correlations $T=0$ nor $T=1$ play any important role in the ground state. In general $T=0+T=1$ pairing yields results which are identical with those obtained with either $T=0$ or $T=1$. In the excited states, however, the $T=0$ correlations are found to be more important than the $T=1$ correlations.NUCLEAR STRUCTURE $^{44}\mathrm{Ti}$, $^{48}\mathrm{Cr}$, $^{52}\mathrm{Fe}$, $^{56}\mathrm{Ni}$, $^{60}\mathrm{Zn}$; calculated binding energies, quadrupole moments, pickup strengths. Hartree-Fock-Bogoliubov method with generalized pairing.

New Results of a Proton-Proton Bremsstrahlung Experiment at 42 MeV

We report the results of new measurements and analysis of 42-MeV proton-proton bremsstrahlung cross sections. Comparison with the predictions of the Hamada-Johnston potential over proton polar-angle ranges from 16\ifmmode^\circ\else\textdegree\fi{} to 40\ifmmode^\circ\else\textdegree\fi{} is made and significant disagreement is observed.

Matrix solution of the Bethe-Faddeev equations

The Bethe-Faddeev equations are set up in matrix form and solved for the $^{3}\mathrm{H}$ nucleus. The two-body $g$ matrix is derived from a revised version of the Brueckner-Bethe-Goldstone formalism. Application to the Reid soft core potential indicates that this interaction yields an eigenvalue very close to the experimental binding energy of $^{3}\mathrm{H}$.NUCLEAR STRUCTURE Three-body correlations, Brueckner-Bethe-Goldstone method, binding energy, $^{3}\mathrm{H}$, Reid and Hamada-Johnston potentials.

Proton-proton bremsstrahlung: Coulomb effect

A nonrelativistic potential model calculation of proton-proton bremsstrahlung is carried out in which the Coulomb potential as well as double scattering are treated exactly. The conventional electromagnetic current operator for point protons with charge and magnetic moment is used. Cross sections differential with respect to the photon angle are calculated in the symmetric, coplanar Harvard geometry for proton angles of 10, 20, and 30\ifmmode^\circ\else\textdegree\fi{}, at laboratory energies of 5, 20, 42, 62, 99, and 158 MeV, using the Hamada-Johnston potential. Cross sections integrated over the photon angle are presented over a wider range of angles. Including the Coulomb potential lowers the cross section by amounts which are comparable with the errors of the most accurate experiments, and at proton angles smaller than any of the existing experiments the Coulomb effect becomes quite large. Cross sections for the asymmetric Orsay experiment are also computed, including the effect of noncoplanarity, and there is a large discrepancy at some of the experimental points.NUCLEAR REACTIONS $p(p,p\ensuremath{\gamma})$; calculated $\ensuremath{\sigma}({\ensuremath{\theta}}_{\ensuremath{\gamma}})$ for lab proton energies 5 to 158 MeV, including Coulomb effects.

Effective interaction calculations for nuclei of mass 6 to 9

Shell model calculations using the effective interaction approach of Talmi are reported for normal parity states in nuclei of mass number 6–9. To fit simultaneously the energies of levels in these nuclei, energy shifts of ground state rotational bands in 8 Be and 9 Be are introduced. These shifts have their origin in the deformations of nuclei. The obtained matrix elements fit the data better than those of earlier authors, and resemble the matrix elements obtained from realistic interactions such as the Hamada-Johnston potential. Theoretical predictions are compared with new experimental data. M1 radiative widths, log ft values, and spectroscopic factors for single- and two-nucleon transfers are given to help in the interpretation of new data. The need for carrying out more calculations for 1p shell nuclei with realistic forces is discussed.

Antipairing and antistretching in 22Ne

The ground state rotational band of 22Ne has been investigated with the angular momentum and particle number projected Hartree-Bogoliubov theory. Variation before projections was performed in the framework of the constrained Hartree-Bogoliubov theory with the quadrupole moment and the degree of pairing as constraints. It is shown that there is a clear decrease of the pairing correlations (antipairing) and a decrease of the quadrupole deformation (antistretching) with increasing angular momentum. The antipairing effect appears to be essential to reproduce the experimentally known deviation of the spectrum from the J( J + 1) rule. The antistretching indicated by the B(E2) values is much stronger than that indicated by the calculated static quadrupole moments. This amounts to a breakdown of the rotational picture which is possibly connected with the low band cut-off values in the sd shell. The antipairing effect decreases the antistretching only slightly. The interaction between the pairing and quadrupole degrees of freedom is found to be too weak to change the earlier conclusions concerning antipairing and antistretching in the whole sd shell. No inert core was assumed in the calculations. The effective G-matrix elements of the Hamada-Johnston potential were used as an interaction.

Effect of short-range correlations on Coulomb matrix elements

Correlated wave functions obtained by solving the Bethe-Goldstone equation with the Hamada-Johnston potential are used to calculate Coulomb matrix elements for use in the nuclear $1p$ shell. When proper attention is given to the Pauli operator we find that Coulomb matrix elements are not appreciably larger than those obtained using uncorrelated wave functions, in contrast to conclusions reached by previous investigators.

Meson-Exchange Effects in Neutron-Proton Bremsstrahlung

A prescription for including exchange effects in neutron-proton bremsstrahlung ($\mathrm{np}\ensuremath{\gamma}$) by requiring gauge invariance of the complete $\mathrm{np}\ensuremath{\gamma}$ amplitude is given entirely within a potential framework. To lowest order in the photon momentum (${K}^{0}$) this prescription is unambiguous and it includes, as well as exchange effects, any other nonlocal effects in the nuclear potential. The lowest-order contribution is written, with the use of the Schr\"odinger equation, so as to eliminate the necessity of intergrating over the nuclear potential. The higher-order exchange contributions are treated through order ${K}^{2}$ using the one-pion exchange (OPE) only. (Because the effective expansion is $\frac{K}{\ensuremath{\mu}}$, where $\ensuremath{\mu}$ is the mass of the exchanged meson, the pion is expected to dominate.) These higher-order terms are the same as the Feynman-diagram prescription would give for $\mathrm{np}\ensuremath{\gamma}$ due to OPE. Calculations with the Hamada-Johnston (HJ) and Bryan-Scott III (BS) potentials are compared to the experimental results at 208 MeV by Brady et al. and at 130 MeV by Edgington et al. This inclusion of the exchange bremsstrahlung (order ${K}^{0}$) increases the cross section by roughly a factor of 2, providing generally good agreement with both experiments. An estimate of the order-${K}^{2}$ terms indicates that they contribute 2%; possible implications of this result on order-$K$ contributions are discussed. Contributions arising from nonlocal terms other than exchange such as momentum-dependent and spin-orbit effects amount to about 1%. There is little difference between the $\mathrm{np}\ensuremath{\gamma}$ predictions for the HJ and BS potentials. Our $\mathrm{np}\ensuremath{\gamma}$ coplanar results for the HJ potential are compared to those obtained in a calculation, which uses the low-energy theorem for internal radiation, by Celenza et al.[NUCLEAR REACTIONS Neutron-proton bremsstrahlung ($\mathrm{np}\ensuremath{\gamma}$), $E=130,200$ MeV; calculated coplanar $\ensuremath{\sigma}({\ensuremath{\theta}}_{n},{\ensuremath{\theta}}_{p},{\ensuremath{\theta}}_{\ensuremath{\gamma}})$, $\ensuremath{\sigma}({\ensuremath{\theta}}_{n},{\ensuremath{\theta}}_{p})$ including meson-exchange contributions, comparison to experiment. Developed higher-order corrections arising from OPE.]

Pairing correlations and simultaneous projection of particle number and angular momentum in light nuclei

The Hartree-Bogolubov theory with simultaneous projection on sharp angular momentum quantum number and sharp particle number is presented as a description of the ground state rotational bands of doubly even nuclei. Calculations with effective G-matrix elements of the Hamada-Johnston potential and also of the Yale potential have been performed within this framework for the nuclei 20Ne, 22Ne, 24Ne, 24Mg, 26Mg, 28Si, 30Si, 32S and 34S. Rotational energies, moments of inertia, pairing correlations, quadrupole moments and B(E2) values are discussed and compared with available experimental data. The effect of particle number fluctuation on the ground state rotational spectrum and on the intrinsic wave function is investigated. The particle number projection turns out to be important for the description of the rotational levels, whereas it has only a negligible effect on the properties of the intrinsic wave function. Comparison with Hartree-Fock results and with experimental data shows that the inclusion of pairing correlations provides a significant improvement on the excitation energies, their relative ratios and the angular momentum dependence of the moments of inertia. All solutions show an antistretching effect in the quadrupole moments and especially in the B(E2) values. The rotational bands in 20Ne, 22Ne, 24Ne and 28Si turn out to be well described in the framework presented.

Effective interactions and the40Ca(p, p′) reaction

The excitation of the lowest 3− and 5− levels in40Ca is studied using a microscopic description. The long-range part of the Hamada-Johnston potential is shown to be a good effective interaction especially when supplemented by a phenomenological imaginary term. Transition densities are constructed from holeparticle RPA wave functions normalized to give the correct electric transition rates. Various wave functions and other interactions are studied. Other features of the calculations are discussed.

Shell-Model Calculations forA=18,19, and 20Nuclei with Core Excitation Included Explicity

Eigenvalues and eigenvectors for all low-lying positive- and negative-parity nuclear states of $A=18,19, \mathrm{and} 20$ are calculated in a shell-model basis of all Pauli-allowed $0{p}_{\frac{1}{2}}\ensuremath{-}0{d}_{\frac{5}{2}}\ensuremath{-}1{s}_{\frac{1}{2}}$ configurations outside an inert $^{12}\mathrm{C}$ core. Two different effective Hamiltonians are used. One is based on a reaction matrix treatment of the Hamada-Johnston potential. The second one is obtained by varying the 33 effective Hamiltonian matrix elements to reach a least-squares fit between 153 experimental level energies in nuclei in the $A=13\ensuremath{-}22$ region and the corresponding shell-model eigenvalues. The energy level spectra and single-nucleon spectroscopic factors from these calculations are compared to the available experimental data in this region. The calculations are also examined for the existence and characteristics of sequences of levels which might be called "rotational bands."

Off-the-Energy-ShellTMatrix. I

A simple method is presented to obtain the two-body off-energy-shell T matrix at a positive energy. As an example of a realistic potential, the Hamada-Johnston potential in the singlet state is assumed. In the present method, the off-shell Lippman-Schwinger equation is solved under suitable boundary conditions. From its solution, an expression for the off-shell element of T matrix is derived. The off-shell behaviour of reduced T matrix is investigated on the angular-momentum states J = 0 and J = 1 at energy parameter 75 MeV.

σ-Exchange Potential for Nucleon-Nucleon Scattering

A nucleon-nucleon interaction potential is derived which includes all exchanges between a pair of nucleons involving a single $\ensuremath{\sigma}$ meson and any number of pion-pair inserts on the $\ensuremath{\sigma}$-meson propagator and vertices. The reduction of the Bethe-Salpeter kernel to a potential follows the method of Partovi and Lomon. The $\ensuremath{\sigma}$-pion and $\ensuremath{\sigma}$-nucleon couplings are restricted by the Adler chiral condition at zero four-momentum transfer, renormalized. The $\ensuremath{\sigma}$-meson mass is taken from experimental evidence to be nearly ${M}_{\ensuremath{\rho}}$. Various $\ensuremath{\sigma}$-meson widths are considered. Widths greater than 300 MeV predict small $\ensuremath{\sigma}$-meson contributions to the potential due to persistent cancellations between graphs. In particular near the width of about 610 MeV predicted by Weinberg's asymptotic restrictions the $\ensuremath{\sigma}$-potential contribution is a minimum and nearly cancels the familiar one-plus-two nucleon-pair contribution to the potential. The corrections to the previously obtained single-pion, $\ensuremath{\eta}\ensuremath{-}$, $\ensuremath{\rho}\ensuremath{-}$, and $\ensuremath{\omega}\ensuremath{-}$ meson and two-pion potentials are such as to increase the resemblance to Hamada-Johnston and Reid potentials or increase the resemblance to the Feshbach-Lomon potential for $\ensuremath{\sigma}$-meson widths near 620 MeV and 660 Mev, respectively.

Study of Even Calcium and Nickel Isotopes with Self-Consistent Approximations

Employing the central Yukawa interaction, and renormalized matrix elements for the Hamada-Johnston and Tabakin potentials, self-consistent Hartree-Fock, Hartree-Fock-Bogoliubov (HFB), and spherical BCS calculations for even calcium and nickel isotopes are carried out. The existence of spherical $^{40}\mathrm{Ca}$ and $^{56}\mathrm{Ni}$ cores is assumed and the calcium and nickel isotopes are treated within the framework of single-closed-shell structure. It is found that for a given isotope and interaction the HFB and spherical BCS approximations always correspond to the same actual minimum energy solution. This feature is observed for all the isotopes and interactions considered here, though the quasiparticle energies and the occupation numbers are found to differ. For Ca isotopes the BCS results are also compared with the shell-model calculations of McGrory, Wildenthal, and Halbert, and are found to approximate the ground-state properties of all the even isotopes rather well.

P- p bremsstrahlung calculations and relativistic spin corrections

We discuss in detail the Hamiltonian for p- p bremsstrahlung, including relativistic corrections to the Pauli-Schrodinger equation. We derive the cross section for this process, and present calculations, based on the Hamada-Johnston potential with and without relativistic corrections. When the relativistic spin correction is neglected, our results, here found by calculating the amplitude in the c.m. frame, are quite different from results found previously using the lab frame. However, when the relativistic correction is included, then the lab and c.m. calculations are in agreement with each other, and also with experimental data. The correction is small in the c.m. frame and large at higher energies in the lab frame. Comparisons between theory and experiment are made at various energies, and for cases of unequal proton polar angles, and noncoplanar scattering.

Final-state interactions in the break-up reaction of deuterons by nucleons

The nucleon-nucleon correlation spectra of the cross sections for the break-up reaction of douterons by nucleons are calculated taking account of final-state interactions between nucleons in pairs. The higher partial waves in the interactions and the D-state wave function of deuterons are included. The partial waves are obtained by solving the Schrödinger equations with the Hamada-Johnston potential for two-body scattering states. The results are compared with experimental data for 198, 156, 100, 46, 45, 23 and 17 MeV incident protons.

The Λ 00 and Λ 10 calculations of the binding energy of nuclear matter with realistic hard-core nucleon-nucleon potentials

Realistic potentials have been used in a many-body Green function calculation of the properties of nuclear matter. The Green function method has the advantage that the single-particle energies are determined by the theory and there is no necessity to develop prescriptions for their calculation. Using the Λ 10 theory and the Hamada-Johnston potential a binding energy of −4.6 MeV per nucleon is obtained.

Two-particle states in 210Pb, 210Bi and 210Po with realistic forces

Two-particle states in 210Pb, 210Bi and 210Po have been calculated with reaction matrix elements deduced from the Hamada-Johnston potential. The inclusion of core polarization is essential for reproducing experimental spectra. Some qualitative aspects of the effective interaction are discussed.

A phenomenological effective two-body interaction for calcium isotopes

The 1]~-2pg shell has long been one of the most studied nuclear regions. Recently there has been an extensive effective-interaction calculation on calcium isotopes by F~I)~MAN~ and PITTnL (1). The single set of matrix elements given by them has a good convergence and appears to reproduce the energy spectra, (t,p) strengths and single-neutron spectroscopic factors with reasonable agreement. Moreover, the /~pg matrix elements have the desh'ed feature of repulsion on the average. This is in agreement with the findings obtained by analysing the data from (3He, d) and (d, p) measurements in this shell (~). These matrix elements, derived from Hamada-Johnston potential (a), are, however, attractive on the average. The effective-interaction method in the framework of the shell model has the disadvantage that one cannot use a large configuration space in such calculations. Then the number of variable two-particle matrix elements increases drastically and the method fails. Even in a reasonable configuration space, the two-particle matrix elements obtained from the least-squares fit to the experimental binding energies and level energies of calcium isotopes have large statistical errors (a-e), when all the required two-particle matrix elements are allowed to vary. On the other hand, a large configuration space can be used with a phenomenological effective two-body interaction with a few adjustable parameters. The wave functions thus obtained, in view of greater configurational freedom, are more realistic than those obtained by the effective-interaction method. But it is difficult to get detailed quanti tat ive fit to the binding energies and level energies like the effective-interaction method because there are only a few adjustable parameters in a phenomenological force. So an initial test of a phenomenological residual twobody interaction is to reproduce the matrix elements obtained by the effective-interaction method to get a good description of the experimental data.

Nuclear Matter Properties for the Reid, Bressel–Kerman–Rouben, and Hamada–Johnston Potentials

Nuclear matter calculations have been carried out for the Reid, Hamada–Johnston, and Bressel–Kerman–Rouben potentials. The convergence of the reference spectrum series for the G matrix is investigated, as is the approximation of choosing an average center of mass momentum. Higher-order cluster contributions are estimated from the scaling formulas of Day. The Reid potential predicts saturation at kF = 1.49 fm−1, with a binding energy −13.7 ± 1.5 MeV per particle.