Brueckner Reaction Matrix Research Articles

Discrete wave-packet representation in nuclear matter calculations

The Lippmann-Schwinger equation for the nucleon-nucleon $t$-matrix as well as the corresponding Bethe-Goldstone equation to determine the Brueckner reaction matrix in nuclear matter are reformulated in terms of the resolvents for the total two-nucleon Hamiltonians defined in free space and in medium correspondingly. This allows to find solutions at many energies simultaneously by using the respective Hamiltonian matrix diagonalization in the stationary wave packet basis. Among other important advantages, this approach simplifies greatly the whole computation procedures both for coupled-channel $t$-matrix and the Brueckner reaction matrix. Therefore this principally novel scheme is expected to be especially useful for self-consistent nuclear matter calculations because it allows to accelerate in a high degree single-particle potential iterations. Furthermore the method provides direct access to the properties of possible two-nucleon bound states in the nuclear medium. The comparison between reaction matrices found via the numerical solution of the Bethe-Goldstone integral equation and the straightforward Hamiltonian diagonalization shows a high accuracy of the method suggested. The proposed fully discrete approach opens a new way to an accurate treatment of two- and three-particle correlations in nuclear matter on the basis of three-particle Bethe-Faddeev equation by an effective Hamiltonian diagonalization procedure.

Two-vacua RPA and the two-neutrino double-beta decay

Abstract A new method “two-vacua random-phase approximation” (TVRPA) for the description of the two-neutrino nuclear double-beta decay is developed. The nuclear wave functions of the intermediate odd-odd nucleus are constructed as linear combinations of proton-particle neutron-hole excitations from the ground state of the initial even-even ( N , Z ) nucleus and, at the same time, as neutron-particle proton-hole excitations from the ground state of the final ( N − 2, Z + 2) nucleus. The Ritz variational principle is used to obtain RPA-like equations. The ground state in the initial and final nucleus is approximated with HFB wave functions including proton-neutron pairing. A projection on numbers of protons and neutrons is performed additionally on the wave functions in both nuclei to restore the exact number of particles, broken by the quasi-particle transformation. The method is applied, as a preliminary test, to the 2ν double-beta decay in 20 48 Ca 28 . More realistic B (GT − ) and B (GT + ) Gamow-Teller strength distributions than QRPA and PQRPA are obtained. No dependence is found on the particle-particle strength parameter g pp , renormalizing Brueckner reaction matrix elements of the Bonn potential.

Realistic microscopic calculations for the light nuclei A = 2 to 7

The conventional theory of shell-model effective interactions encounters a divergence in its perturbation expansion owing to intruder states. By enlarging the model space to eliminate the core, and, hence, all core-polarization processes, we circumvent this problem. The perturbation expansion for the effective interaction can then be reasonably expressed in terms of only the Brueckner reaction matrix G in the no-core space plus all folded diagrams. The effective interaction for A = 2 is simply the Brueckner G-matrix. For A > 2 exact results for the eigenenergies are obtained, if the generalized, A-nucleon G matrix can be constructed. For A = 4 to 7, we approximate the A-nucleon G-matrix with the Brueckner G-matrix. Reasonable results can be obtained by treating the starting energy for the G matrix as a variable parameter to fix the binding energy.

Effects of the single-particle potential insertions in the effective interaction.

We investigate the effects of the single-particle potential insertions in the effective interaction by comparing energy spectra obtained from different treatments of these insertions as a function of the size of the no-core model space. The Brueckner reaction matrix used in the first calculation includes the single-particle insertions in ladder diagrams to all orders, while the Brueckner reaction matrix used in the second calculation only keeps the single-particle potential term in the lowest-order ladder diagram. The two calculations yield almost identical ground-state energies and low-lying excitation spectra for $^{4}\mathrm{He}$ and $^{6}\mathrm{Li}$ for large enough no-core model spaces, indicating that the effects of the single-particle potential insertions in second- and higher-order ladder diagrams are small. We explain the reason for the diminishing role of these insertions with increasing size of the model space. We also show that, through a standard method of instilling a single center-of-mass wave function into all low-lying states, the spurious center-of-mass kinetic-energy term shifts the energies of all the low-lying states by nearly a constant and, therefore, has little effect on the excitation spectrum.

Neutron-proton pairing in light nuclei and two-neutrino double-beta decay

We have investigated neutron-proton (np) pairing in light nuclei (2p-lf shell) using a specialized Hartree-Fock-Bogoliubov (HFB) method. For the pairing interaction we used the constant Kisslinger-Sorensen interaction and the realistic G-matrix obtained by solving the Bethe-Goldstone equation with the Bonn one-boson-exchange potential. Our results show that there are at least two minima of the total energy corresponding to two solutions of the HFB equation. The lower solution shows large np pairing. This is confirmed by a recent empirical determination of the np pairing gap, which turns out to be surprisingly large in agreement with our theoretical result. If one renormalizes the Brueckner reaction matrix elements of the Bonn potential by a factor g np multiplying the neutron-proton particle-particle matrix elements to reproduce the experimental np gaps δ np, one obtains large T = 1 np pairing. Preliminary results indicate that the np pairing increases even more if particle-number projection is included. These effects are expected to affect strongly the double-beta decay. The 2 v double-beta decay in 20 48Ca 28 → 22 48Ti 26 is calculated. The Gamow-Teller matrix elements are reduced remarkably by the large np pairing compared to the results without np pairing. The sensitivity of the 2 tv double-beta decay transition on the particle-particle force strength parameter g pp is also reduced compared with the results without np pairing correlations.

Mean field and effective cross sections in hot nuclei

Collisions between heavy nuclei produce nuclear matter of high density and excitation. Brueckner methods are used to calculate the momentum- and temperature-dependent mean field for nucleons propagating through nuclear matter during these collisions. The use of a potential model for the NN interactions is bypassed by calculating the Brueckner reaction matrix directly from the NN phase shifts using a version of Brueckner theory previously published by the author. Arndt phase-shift solutions up to 1600 MeV in energy are used. The binding energy of nuclear matter is normalized to −15.5 MeV by adjusting one free parameter. This gives a saturation density of 0.16 fm −3 and the compressibility is 200–250 MeV. The temperature and density dependence of the effective interaction is shown. The mean field is complex and the real part relates to “one-body” collisions while the imaginary part is related to the “two-body” collision term in transport theory. The error in using free cross-sections when calculating the imaginary part is in general less than 20% but can be at least as large as 75%. The influence of the mean field (effective mass) on the collision term is emphasized.

Skyrme force approach to intermediate energy proton scattering.

The optical model potential for medium energy nucleons is calculated from a momentum and density dependent force which is derived from Brueckner's reaction matrix in a nuclear matter approximation. It is found that Pauli blocking effects substantially increase the nonlocality of the reaction matrix. Calculated optical model parameters are compared with those obtained from the conventional approach of a localized effective two-nucleon force. Proton elastic scattering observables on $^{16}\mathrm{O}$ and $^{40}\mathrm{Ca}$ targets at ${E}_{\mathrm{p}=135--}$300 MeV are analyzed.

Approximations to Brueckner theory for nucleon-nucleus collisions

The concept of quasiparticles has been very fruitful for the development of theories of many-body systems of strongly interacting particles. A central problem for a quantitative treatment is the interaction between the quasiparticles (the effective interaction between particles). The Brueckner reaction matrix theory and the Jastrow variational approach have been used extensively, especially for nuclei. Our present concern is the collisions between nuclei. An exact calculation of the effective interaction between the nucleons is then extremely complicated. Only the simplest extreme, the nucleon-nucleus optical potential, can be treated fully. Approximate treatments are called for. The present paper investigates some approximations to the Brueckner theory. We are primarily interested in the imaginary part of the one-body (one-nucleon) mean field, as it relates to the damping of collective motion. Of special interest is the increase at low density. Comparisons with Skyrme forces show that they exaggerate this density dependence.

Surface and volume contributions to real and imaginary parts of the heavy-ion optical potential

Starting from the Feshbach expression for the optical potential, explicit formulae for the real and the imaginary parts of the optical potential between two heavy ions (HI's) are obtained. They are each composed of a volume and a surface term. The contributions to the volume term are calculated in two nuclear Fermi liquids which flow through each other starting from the realistic Reid soft core nucleon-nucleon (NN) interaction. Since the Fermi surface is formed by two spheres one obtains a complex Brueckner reaction matrix which is approximated by a complex, effective local interaction. It is used in a fully antisymmetrized double folding procedure to obtain the volume terms of the optical potential between the two HI's. The surface contributions are directly calculated in the collision of the two finite HI's. The collective surface vibrations (3 − octupole state and 2 +, 4 + ( T = 0) giant resonances for the 16O− 16O collision) are included as intermediate states. This yields especially an imaginary contribution at the surface which reduces the transparency found with the volume terms alone. The method is applied to 16O− 16O scattering at 83 and 332 MeV laboratory energy. The local approximations to the real and imaginary parts obtained in this way agree well with phenomenological fits. The surface terms improve the agreement of the differential cross section at 80 MeV where experimental data are available.

Comparison of reaction matrices for free scattering and nuclear matter

The matrix inversion method in momentum space provides a powerful tool for analysing numerically the correlation between the free scattering matrix and the Brueckner reaction matrix in nuclear matter. A study of this correlation is made for the 1S0 Reid soft-core and the de Tourreil-Rouben-Sprung super-soft-core potential. Reaction matrices and binding energies for all uncoupled channels with J less than two are studied. The effect of the important isobar virtual channels on these is also investigated.

Converged values of second-order core-polarization diagrams with orthogonalized-plane-wave intermediate states

We carry out an extended study of the Vary-Sauer-Wong effect on the second-order core-polarization diagrams ${G}_{3\mathrm{p}1\mathrm{h}}$ and ${G}_{3\mathrm{p}1\mathrm{h}}^{T}$ in the effective interaction between two valence nucleons. As in Vary-Sauer-Wong, we first calculate ${G}_{3\mathrm{p}1\mathrm{h}}$ using a harmonic oscillator propagator for the intermediate particle states $p$, including particle-hole excitations with energies up to $22\ensuremath{\hbar}\ensuremath{\omega}$. Results are in close agreement with those of Vary-Sauer-Wong. We then calculate ${G}_{3\mathrm{p}1\mathrm{h}}^{T}$ where a free particle propagator for $p$ is used. This is obtained by including the $\ensuremath{-}U$, the oscillator one-body potential, insertions of all orders to $p$ of ${G}_{3\mathrm{p}1\mathrm{h}}$. Although the resulting matrix elements are generally smaller in magnitude than those of Vary-Sauer-Wong, the qualitative feature of the Vary-Sauer-Wong effect is clearly maintained. Namely there are strong cancellations between the contributions from the low and high energy $p$ states. This makes the net effect from the core polarization diagrams significantly weaker than from ${G}_{3\mathrm{p}1\mathrm{h}}$ calculated with $2\ensuremath{\hbar}\ensuremath{\omega}$ excitations alone. We also study some intermediate choices for the propagator of $p$, where the free particle propagator is used only for high energy valence states. The Brueckner reaction matrix elements in a mixed representation where one particle is in a harmonic oscillator state and the other in a plane wave state are needed in our calculations. By using the vector transformation brackets of Wong and Clement and of Balian and Brezin and the Tsai-Kuo treatment of the Pauli exclusion operator, we have developed a technique for accurately calculating these matrix elements.NUCLEAR STRUCTURE Contributions to effective interaction for $A=18$ nuclei calculated from converged values of lowest order core polarization diagrams as function of intermediate state propagator using new momentum space techniques.

Shell-model studies for theSn132region. I. Few proton cases

The Brueckner $G$ matrix from the Reid soft-core potential is taken for the leading contribution of the effective two-proton interaction in the $^{132}\mathrm{Sn}$ region. Phenomenological two-body pairing and multipole forces are added to adjust for all neglected effect. Two model spaces are selected; a small space ($0{g}_{\frac{7}{2}}, 1{d}_{\frac{5}{2}}$) and a large space ($0{g}_{\frac{7}{2}}, 1{d}_{\frac{5}{2}}, 0{h}_{\frac{11}{2}}, 2{s}_{\frac{1}{2}}, 2{s}_{\frac{1}{2}}$). For each space the strengths of the phenomenological terms are adjusted to fit the $^{134}\mathrm{Te}$ spectrum. It is found that both of the total interactions have a high statistical correlation with the bare $G$ matrix. The total effective interactions are then used without further adjustment in shell model calculations to obtain good predictions for the spectra of $^{135}\mathrm{I}$, $^{136}\mathrm{Xe}$, and $^{137}\mathrm{Cs}$.NUCLEAR STRUCTURE Shell-model structure of $^{132}\mathrm{Sn}$ core plus valence protons: shell-model effective interactions consisting of Brueckner reaction matrix plus phenomenological multipole corrections.

Shell-model studies for theSn132region. II. Exact and statistical results for multi-proton cases

Level structures and level densities of nuclei consisting of a $^{132}\mathrm{Sn}$ core plus as many as nine valence protons are calculated using a realistic effective interaction ($G$ matrix) plus previously determined correction terms. The correction terms were determined so that the total interaction, when used in shell-model calculations, gave a fit to the observed $^{134}\mathrm{Te}$ spectrum. Recently developed trace reduction methods are employed to determine the energy moments of the level densities for each spin value. Comparisons between these moments and results of exact shell-model diagonalizations show excellent agreement. Level density comparisons demonstrate that the effect of the necessary correction terms on the spectra is small and decreases as the number of allowed orbits increases. This level density method is also used to investigate the effects of the Coulomb force on the multi-proton spectra.NUCLEAR STRUCTURE Exact and statistical shell-model structure of $^{132}\mathrm{Sn}$ core plus valence protons; shell-model effective interaction consisting of Brueckner reaction matrix plus phenomenological multipole corrections; level densities via moment expansions.

Effective shell-model interaction through second order for thesdshell

We evaluate the effective shell-model interaction v/sub eff/ for the sd shell, through second order as a function of an energy shift between the valence and unoccupied oscillator states. The Brueckner reaction matrix G for the Reid soft-core potential is obtained in a single-particle basis so the Pauli operator is treated exactly. In terms of second order in G, intermediate-state sums and the dependence of energy denominators on passive particles and holes are both treated with care. Significant dependence on the intermediate-state spectrum is found for v/sub eff/ in both the T = 0 and T = 1 components. An energy shift yielding a total gap of about 49 MeV is found to yield A = 18 two-particle spectra in reasonable agreement with experiment in view of the expected role of 4p-2h intruder states. On the other hand, a comparison of the matrix elements of v/sub eff/ with those determined by Chung and Wildenthal from spectra of A = 18 to 22 nuclei indicates a preference for a total gap between 35 and 40 MeV. These empirically determined gaps may be compared with a total gap of about 30 MeV, motivated by a study of the self-consistent particle spectrum bymore » Gambhir and McCarthy.« less

Semirealistic shell-model interaction for neutron hole states inPb208

The Brueckner $G$ matrix from the Reid soft-core potential is taken for the leading contribution to the effective two-neutron hole interaction in the $^{208}\mathrm{Pb}$ region. To accomodate the corrections expected from core polarization and other higher order effects, including truncation of the model space, we add phenomenological two-body pairing and multipole forces whose strengths are determined by fitting the $^{206}\mathrm{Pb}$ spectrum. A satisfactory fit is obtained with an interaction where the average total contribution of the added terms is less than 45% of the interaction. The same total effective interaction is then shown to yield a satisfactory spectrum for $^{204}\mathrm{Pb}$ which may indicate the neglected effective many-body forces are not important. A comparison with the results of McGrory and Kuo is presented to examine the impact on the effective interaction of limiting the model space. We conclude the prospects are good for extending this approach to other nuclei below $^{208}\mathrm{Pb}$ viewed as shell-model systems of valence holes in $^{208}\mathrm{Pb}$.NUCLEAR STRUCTURE Shell-model structure of Pb isotopes; theory of shell-model effective interaction with phenomenological corrections; multipole potentials added to Brueckner reaction matrix.

Ground state properties of medium-heavy nuclei with a realistic interaction

The Brueckner $G$ matrix appropriate for medium-heavy nuclei is obtained from the Reid soft-core nucleon-nucleon potential. The $G$ matrix is strongly affected by the Pauli operator $Q$, which is treated exactly (no angle averaging). Within the range of valence space energies $G$ has a weak dependence on the starting energy $\ensuremath{\omega}$. Ground state properties of deformed rare earth nuclei ($Z=64\ensuremath{-}76,N=90\ensuremath{-}102$) and spherical semimagic nuclei ($\mathrm{Sn},\mathrm{Pb},N=82,N=126$) have been calculated in the Hartree-Fock-Bogoliubov approximation with an inert core of 110 nucleons. Deformations and pair gaps are both determined by the $G$ matrix. The systematic experimental dependence of ${\ensuremath{\epsilon}}_{\mathrm{spherical}}$, ${\ensuremath{\beta}}_{2}$, ${\ensuremath{\beta}}_{4}$, ${\ensuremath{\Delta}}_{p}$, ${\ensuremath{\Delta}}_{n}$, and ${E}_{\mathrm{po}}$ (prolate-oblate energy difference) on $N$ and $Z$ is reproduced. However, the magnitudes of ${\ensuremath{\beta}}_{2}$, ${\ensuremath{\Delta}}_{n}$, and ${E}_{\mathrm{po}}$ are too small. This may be largely due to the lack of isospin dependence of the oscillator basis states. ${\ensuremath{\beta}}_{2}$ and ${E}_{\mathrm{po}}$ could receive additional significant contributions from core polarization of the 110 particle core which is neglected here. The Hartree-Fock-Bogoliubov ground states obtained with this realistic interaction provide a reasonable foundation for high spin calculations.NUCLEAR STRUCTURE Brueckner reaction matrix; Hartree-Fock-Bogoliubov theory; ground state properties of medium-heavy nuclei.

Density-dependent effective interactions in nuclei

Effective interactions chosen to have the same structure as the Brueckner reaction matrix were used in self-consistent field calculations on some closed shell nuclei. These finite range interactions are velocity-dependent and depend on the starting energy. The density dependence and starting energy dependence were determined by fitting the diagonal K-matrix elements of nuclear matter which were calculated with the Reid soft-core potential. Two interactions were derived: (i) U p = 0 from a K-matrix defined with a pure kinetic energy particle spectrum and (ii) U p ≠ 0 from a K-matrix with a self-consistent insertion in particle states just above the Fermi surface. The model U p ≠ 0 was also modified by changing the size of the correlation wound integral which determines the starting energy dependence. It was found that the removal of that part of the wound integral which is due to the repulsive core of the nucleon-nucleon interaction resulted in the best single-particle spectrum.

Photonuclear reactions from correlated nucleon pairs (II)

A derivation of an expression for the cross section of the photoexcitation of correlated nucleon pairs in a nucleus is given. The final-state wave functions which are obtained from a solution of the Brueckner reaction matrix can be related to the scattering of free nucleons in a vacuum. Some results are presented and include the energy distribution and cross sections of the excited nucleons. An expression for the cross section of the giant dipole resonance is derived purely from the presence of the microscopic two-body forces in nuclei. These cross sections agree with experiment for a particular choice of the final-state single-particle spectrum.

The overcounting problem for Brueckner reaction-matrix ladders

It is argued that, contrary to an earlier conclusion of Brown and Wong, ladders of Brueckner reaction matrices with low-lying intermediate states produce near-total overcounting and should not be included in perturbation theory calculations.

Effective Interactions in Nuclear Structure

The basic notions associated with the idea of an “effective interaction” in nuclear physics are sketched by first recalling isolated two-body scattering analyzed with integral equations and in which the transition operator appears naturally. Brueckner's reaction matrix operator is discussed in the context of both infinite nuclear matter and finite nuclei with doubly closed shells. Finally, open-shell nuclei are touched upon by way of a many-body effective interaction operator deduced in simple language and a possible connection between this and the reaction matrix operator is given. It is emphasized throughout that the basic merit of the vantage-point sketched here is its fundamental origin in the many-body Schrödinger equation.