Particle-hole Matrix Elements Research Articles

Effect of Tensor Force on the Multipole Resonances of Finite Nuclei

The effect of the tensor force on the properties of finite nuclei is discussed by analyzing the multipole giant resonances in 208 Pb. It is found that the tensor force has a larger effect on the spin-flip magnetic dipole and Gamow-Teller resonances than on the natural parity low-lying isoscalar quadrupole (2 + ) and octupole (3 � ) states. By analyzing the modifications to the Hartree-Fock mean field induced by the tensor terms, and the specific features of the residual particle-hole tensor interaction, we find that the tensor force gives an attractive contribution to the particle-hole matrix elements.

Comment on “Critique of the foundations of time-dependent density-functional theory”

Two particular subjects tackled in the recent paper by Schirmer and Dreuw (SD) [J. Schirmer and A. Dreuw, Phys. Rev. A 75, 022513 (2007)] are commented on by us. By providing a convincing example, we answer positively the question posed by SD whether a local operator can be reconstructed from the knowledge of its particle-hole matrix elements (when the number of particle states exceeds one). In connection with the problem of the utmost reduction of the density-functional theory (DFT) to a solution of one equation for the density amplitude---the radical Kohn-Sham (rKS) approach---used by SD as a valuable tool to investigate the time-independent and time-dependent DFT, we demonstrate that the calculational scheme implementing the rKS formulation is rather uncompetitive with the traditional KS scheme.

Reaction mechanism for natural parity (p,p') transitions inB10

We report angular distribution measurements of the cross section, analyzing power, induced polarization, and all polarization transfer coefficients for the natural parity transitions in $^{10}\mathrm{B}$($\stackrel{\ensuremath{\rightarrow}}{p},{\stackrel{\ensuremath{\rightarrow}}{p}}^{'}){}^{10}$B at 197 MeV. All of the transitions closely follow the pattern for spin-1/2 scattering from spin-0, namely $A=P,{D}_{{\mathit{SS}}^{'}}={D}_{{\mathit{LL}}^{'}}$, and ${D}_{{\mathit{SL}}^{'}}=\ensuremath{-}{D}_{{\mathit{LS}}^{'}}$. Small deviations from ${D}_{{\mathit{NN}}^{'}}=1$ reflect spin-flip contributions to these transitions. Distorted wave impulse approximation calculations were generally successful, supporting the use of an empirical effective nucleon-nucleon interaction, the inclusion of channel-coupling in the ground state and the ${3}^{+}\ensuremath{\leftrightarrow}{4}^{+}$ transition, and the procedure of increasing the $L=2$ strength of shell-model particle-hole matrix elements to match the collective contribution to these transitions.

The effects of the spin-orbit and tensor interactions in nuclei

We devise a nucleon-nucleon interaction V c + xV so + yV t to study the effects of the spin-orbit and tensor interactions by varying x and y. For negative parity 1 p−1 h states in 16O, we find that in a TDA calculation, the weighted sum of the eigenenergies Σ J (2 J + 1) Σ i E i ( J, T) is zero for a pure two-body spin-orbit interaction or a pure tensor interaction. If a minor closed shell nucleus like 12C is taken as the core, the tensor interaction lowers the p 1 2 single-particle energy relative to p 3 2 . This is precisely opposite to what the two-body spin-orbit interaction does. For the 1 + states in 12C, a 1 p−1 h diagonalization leads to a near collapse with E ∗(T = 0) ∼ 0.9 MeV and E ∗(T = 1) ∼ 3.7 MeV as compared with experiment (12.7 MeV for T = 0 and 15.1 MeV for T = 1). Only a full scalar shell model calculation brings these states up to a respectable energy. We find that in a full shell model calculation, the excitation energies of these 1 + states are surprisingly insensitive to the spin-orbit interaction over a rather wide range of the parameter x and there does not appear to be any phase transition as we vary x. We reconsider the old problem of the effect of the spin-orbit and tensor interactions on the nearly vanishing Gamow-Teller matrix element 14C( J = 0, T = 1) → 14N( J = 1, T = 0). We calculate the single-particle energies with the same interaction that is used for the residual particle-particle or particle-hole matrix elements.

17O(d

Differential cross sections and analyzing powers were measured for $^{17}\mathrm{O}$(d\ensuremath{\rightarrow},t${)}^{16}$O with polarized deuterons of 89 MeV. States of $^{16}\mathrm{O}$ up to 22.89 MeV excitation energy have been studied with an overall resolution of 60 keV. Discrimination between analyzing powers (${\mathit{A}}_{\mathit{y}}$) for different j transfers was found to be very good. The unique ${\mathit{A}}_{\mathit{y}}$(\ensuremath{\theta}) shapes were used to deduce ${\mathit{p}}_{1/2}$-${\mathit{p}}_{3/2}$ mixing ratios for those levels where it was significant. We find unexpectedly strong mixing for the 6.130 MeV, ${3}^{\mathrm{\ensuremath{-}}}$, and 8.872 MeV, ${2}^{\mathrm{\ensuremath{-}}}$, levels. Exact finite-range distorted-wave Born approximation calculations were used to identify l transfers and to extract spectroscopic strengths. Up to 23 MeV excitation, pickup strength is dominated by 1p transfer; less than 2% of the observed particle-hole strength results from 2s, 1d, or 1f transfer. Twelve particle-hole matrix elements for (${\mathit{p}}_{1/2}^{\mathrm{\ensuremath{-}}1}$${\mathit{d}}_{5/2}$) and (${\mathit{p}}_{3/2}^{\mathrm{\ensuremath{-}}1}$${\mathit{d}}_{5/2}$) configurations have been deduced. Comparison with previous work shows general agreement except for the T=0,${\mathit{J}}^{\mathrm{\ensuremath{\pi}}}$=${3}^{\mathrm{\ensuremath{-}}}$ term.

A new prescription for determining particle-hole interactions near closed shells

Starting from one-nucleon separation energies near doubly-closed shells as well as of the experimental centroid for particle-hole excitations, a new prescription for obtaining residual particle-hole matrix elements is presented. It is shown that the major difference with the standard Blomqvist prescription is a shift in the energy that can be estimated in a graphical way.

Hartree-Fock Single-Particle Energy and Interaction Matrix Element for Density-Dependent Interactions in Superconducting Nuclei

The density-dependent interactions have been successfully used as effective interactions in Hartree-Fock (HF) and RPA calculations. Among them the Skyrme type interactions are simple and easy to treat, and were employed by many authors to explain properties of giant resonances as well as ground state properties of magic nuclei. 1) It is well known that in the case of a density-dependent interaction there occurs an extra term in the HF field. This term vanishes in the case of a density-independent interaction and we call it the rearrangement field. For the RPA particle-hole matrix elements, the Landau prescription ) has been utilized to take account of the rearrangement effects. The density dependence brings the symmetric contributions to the particle-hole interaction. In a previous shortnote) a self-consistent evaluation of particle-hole type matrix elements for density-dependent interactions is presented. We evaluated the expectation value of the Hamiltonian with respect to a Slater determinant which is not orthogonal to a reference Slater determinant. The particle-hole type matrix elements which correctly include the rearrangement effect were obtained, and the Landau prescription came out as the simplest case. The purpose of this letter is to generalize the method to treat the superconducting nuclei. The rearrangement effects on the Hartree-Bogoliubov (HB) condition and interaction matrix elements in the quasi-particle TDA equation are obtained for densitydependent interactions. We now take 1<1» to be a quasi-particle vacuum

Geometrical derivation of collective operators and paths in TDHF

It is first found that the intrinsic parity of an operator under time reversal and the interpretation of the operator as coordinate- or momentum-like in a TDHF calculation are not simply related. This is because the TDHF particle-hole basis is, in general, complex. The TDHF equation is then reformulated in the plane tangent to the Slater determinant manifold. This plane is spanned by the particle-hole basis. The particle-hole matrix elements of the Hartree-Fock Hamiltonian define the energy gradient vector in this tangent plane. This gradient is real when the Slater determinant is real. A TDHF calculation initiated from a real determinant induces, during the first infinitesimal time step, a purely imaginary variation of this determinant along the gradient. The gradient is thus identified with the matrix elements of a boost operator. The next infinitesimal time step defines, in turn, a displacement operator. These operators are retained as collective if the TDHF path is stable under changes of velocities. Various criteria are found for this stability condition. The theory cannot be applied straightforwardly to translations and rotations for there is no energy gradient to generate coordinate operators. Particle-hole matrix elements of boost operators can be obtained, however, by a multiplication by i of the matrix elements of displacement operators, since the latter are known explicitly. It is finally found that the rotation of a wavefunction is contradictory with angular momentum conservation in general. Conservation can be ensured by a rotation of the density only and a more elaborate evolution of the velocity field.

Nuclear vibrations with a zero-range interaction and the multipole condition

Isoscalar monopole, quadrupole, and octopole states are calculated in closed shell nuclei. A delta interaction is used, its strength determined by the multipole condition, namely that the mean single-particle-single-hole potential energy difference is equal to the corresponding kinetic energy difference. This interaction is used to obtain particle-hole matrix elements appropriate for a random-phase approximation calculation. The strengths given by the different multipole conditions are different but (except for $L=0$) they appear to approach each other as the mass number $A$ becomes large. The monopole mode has already collapsed somewhat before the strength implied by the quadrupole condition is reached. Some of the calculations were repeated using a zero-range Skyrme interaction. We observe a very high degeneracy in our calculation which we are able to explain in terms of $L\ensuremath{-}S$ coupling and the fact that the particle-hole matrix element of a delta interaction is of the form $\ensuremath{\int}{f}_{{P}_{A}{H}_{A}}(r){f}_{{P}_{B}{H}_{B}}(r){r}^{2}\mathrm{dr}$.

Collective effects in atoms via a schematic model

Linear response theory is used to investigate the collective excitation spectra of the outer shells of heavy atoms. A model, based on the approximate separability of particle-hole interaction matrix elements, is solved in closed form and found to be in semi-quantitative agreement with experiment for the vibrations of the 4d shell in Xenon. The separability of these matrix elements in the relevant energy region is shown to be due to a general property of wave functions of atomic potentials. A comparison is also made with full random phase approximation calculations for Xe. It is shown that the schematic model contains all the relevant features to describe the enhanced photo-absorption of this system in the far ultraviolet. The role of anf-wave resonance in the average atomic field is stressed in connection with this enhancement. Detailed agreement between the schematic model and experiment is less good than that obtained from the full random phase calculation, the differences arising due to an approximate treatment of exchange in the former calculation. The total dipole strength in the photo-electric region is approximately the same for both calculations however and in reasonable accord with experiment.

Octupole vibrations and ground state correlations

The relationship between ground-state correlations and collectiveness is investigated for the case of low-lying nonrotational states in the rare earth nuclei. Both octupole and quadrupole modes of excitation are studied and the quasiparticle virtual populations associated with each of them are discussed. The relative importance of particle-particle and particle-hole interaction matrix elements is also analyzed in connection with the shape of the correlation patterns. The fundamental role of the Nilsson + BCS scheme is emphasized and the consistency of the quasiparticle random-phase approximation is established.

Generalised quartet model for particle-hole excitations across major shells

The schematic quartet model of Arima et al. is generalised to include any number of particle-hole excitations. The assumption of constancy of particle-hole matrix elements with mass number is examined in the deformed-oscillator model and shown to be poor in certain cases. A revised picture for particle-hole excitations shows smoother trends than with the schematic model. Comparison with the Zamick model yields greater insight into the structure of two-particle excited states.

Self-consistency in the application of Brueckner's method to doubly-closed-shell nuclei

The relationships between various self-consistency conditions for the single-particle potential U are investigated. These conditions are the Brueckner-Hartree-Fock condition, the requirement that the single-particle orbitals be maximum overlap orbitals, or generalized natural orbitals. The implications of the use of the latter two sets of single-particle functions are examined in detail. It is shown that these two different choices lead to the cancellation of different sets of diagrams in the linked-cluster expansions of various quantities of physical interest. Both sets of basis functions lead to definitions of the particle-hole matrix elements of U which in lowest order coincide with each other, and with that obtained from the Brueckner-Hartree-Fock condition. The definition of the hole-hole matrix elements of U, on the other hand, as obtained from the Brueckner-Hartree-Fock condition is not related to these two sets.

Brueckner-Hartree-Fock Calculations of Spherical Nuclei in a Harmonic-Oscillator Basis. II

An approximation made in previous Brueckner-Hartree-Fock calculations is removed. Both the hole-hole and the particle-hole matrix elements of the single-nucleon potential are now calculated with the correct available energy of the pair. It is shown that, with a given $G$ matrix, the occupied states and energies do not depend on the prescription used for the particle-particle elements. The formalism is easily extended to include Brandow's occupation probabilities.

Particle-hole interaction and core polarization

The discrepancy between particle-hole matrix elements of the nuclear interaction as obtained using a realistic two nucleon potential and as derived from experiment is partly but not completely removed by considering the effects of core polarization. Simple formulae for the corrections to the monopole interaction between a particle and a hole are given.

On the Renormalization Effect due to the Ground State Correlation on the Nuclear Vibrational Motion

A method is given which takes into account the renormalization effects due to the ground state correlation in the nuclear vibrational motion of medium and heavy nuclei. The collective eigenmode is taken as a two-quasi-particle mode; this means that Bogoliubov bound pairs break up in a coherent manner. Self-consistent equations are derived which determine the vibrational eigenmodes and the Bogoliubov quasi-particles, both renormalized owing to the effects of the ground state correlations. Calculations are made for the simple case of a single j-shell with the aim of seeing the physical features of the effect, especially the dependence of the effects on the properties of the nuclear interaction, ~y changing the ratio of the magnitude of the particle-particle and particle-hole matrix elements of the interaction. According as the particle-particle or the particle-hole matrix elements of the nuclear interaction is dominant, we get different results for the renormalization effects on the quasi-particle energy and the E2 transition probability from the vibrational state to the ground state. In either case the renormalization effects reduce the ground state correlation by reducing the difference of the magnitude of the particle-particle and particle-hole matrix elements.

The importance of the strong isotopic spin dependence of particle-hole matrix elements on the many particle-many hole states in nuclei

The importance of the strong isotopic spin dependence of particle-hole matrix elements on the many particle-many hole states in nuclei

Composition of the giant dipole state

The structure of the giant dipole state is studied within the framework of the many-particle Green function theory. It is found that the configuration mixing from two-particle, two-hole states influences all particle-hole matrix elements. As this is believed small from general nuclear structure evidence, the particle-hole model of Brown and collaborators ought to be substantially correct for the dipole state. A specific exception might occur for the low-lying T = 0, 3 − states.

Inelastic electron scattering from Ca 40 in the giant resonance region

Matrix elements for the excitation of particle-hole states in Ca 40 by inelastic electron scattering are calculated. Electric dipole transitions to T = 1 states are considered, taking into account both transverse and Coulomb contributions. The particle-hole matrix elements are evaluated for a zero-range residual interaction. With the help of angular momentum identities, these matrix elements are expressed in a form involving only two Clebsch-Gordan coefficients, thus eliminating the usual 9- j symbols. Wave functions for the excited states are obtained from the diagonalization of the secular matrix. As functions of momentum transfer, some of the matrix elements for the electro-excitation of the different energy levels exhibit characteristically different behaviour, thus helping to identify these levels in the extended shell model scheme. Varying the parameters of the residual interaction over a reasonable range brings about minor modifications in this behaviour, but the qualitative results do not depend critically on the choice of these parameters. The sum of the contributions from the four energy levels above threshold, as a function of momentum transfer, proves to be extremely insensitive to changes in the parameters of the residual interaction, thus exhibiting the effects of closure.