Fragmentation of giant multipole resonances in deformed nuclei

The fragmentation of the giant monopole resonance in deformed nuclei is first studied by coupling the monopole oscillation with the quadrupole oscillation by means of the variational procedure for resonance frequencies. It is shown that, for non-axial symmetry, the monopole oscillation couples with both m = 0 and 2 modes of the quadrupole oscillation and the giant monopole resonance is split into three components, whereas for axial symmetry, the fragmentation is given by E₀(1 + 0.86δ² ± 1.25δ³) and E₀(0.74 ± 0.22δ ? 0.21δ² ± 0.57δ³), where E₀ is the g monopole resonance energy for spherical nuclei, δ is the deformation parameter, and the upper and lower signs stand for prolate and oblate deformations, respectively. The initial fragmentation of the giant quadrupole resonance is seen to be little modified by the coupling, except for the m = 0 mode which is split into two components. The variational method is extended to general multipoles for an ellipsoid and the fragmentation of giant multipole resonances in deformed nuclei is investigated for both axial and non-axial symmetries. A brief discussion is also made about the meaning of the energy eigenvalue involved in the model wave equation in terms of multipole sum rules. The giant dipole resonance for the static octupole deformation is shortly considered. The giant E0 and E3 resonances for largely deformed nuclei are finally examined by solving the spheroidal eigenvalue equation and they are compared with the results of the giant dipole and quadrupole resonances.     

Static and dynamic quadrupole moments of high-spin isomers in the Pb-region

Quadrupole moments of high-spin isomers in the Pb-region are studied by using core polarization charges induced by the coupling of single-particle states to giant resonances. We found that the core polarization charges are enough to reproduce the observed quadrupole moments of the isotones with neutron number N=126 quantitatively without any intrinsic deformation. The mass number dependence of the quadrupole moments in the Rn-isotopes is also discussed by taking into account the configuration mixing involving the neutron excited first 2⁺ state.     

Core polarization effects and giant quadropole resonances in the A ≈ 90 region

The existence, in A ≈ 90 nuclei, of large E2 core polarization effects evident from experimental and shell model fits to decay rates is examined in two independent model calculations: a linearized Hartree-Fock calculation studying the nuclear response function and a macroscopic model involving excitations of giant quadrupole resonances. Both investigations confirm the need for large renormalization of proton and neutron E2 effective charges consistent with the recently discovered isoscalar and isovector giant quadrupole resonances.     

Collective states in 48V

The low-lying positive parity states of ⁴⁸V have been studied in the framework of deformed configuration mixing model calculations based on projected Hartree-Fock theory, within the full fp shell space. The modified Kuo-Brown effective interaction has been used. The calculated spectrum and the electromagnetic properties of these states are in good agreement with the experiment. The calculation predicts an excited low-lying collective K = 2⁺ band in the spectrum of ⁴⁸V and accounts for the observed breakdown of the “signature” selection rule arising in the shell-model calculation within the ($\text{f}\text{7}\text{2}\text{)}{}^{\mathrm{nd}}$ space. It does not favour a 5⁺ assignment to the observed 1.099 MeV level. Two sets of proton and neutron effective charges (i) e_p = 1.32e and e_n = 0.89e and (ii) Kuo and Osnes charges e_p = 1.25e and e_n = 0.47e were employed. The observed decay properties appear to favour the latter charges. Our model also explains in a semiquantitative way the observed K-value, moment of inertia parameter and the intrinsic quadrupole moment of the K = 1^?1 rotational band.     

Giant monopole resonances in the statistical model

The strength functions fore ⁺ e ^– pair decay of the isoscalar and isovector giant monopole resonances in highly excited nuclei are derived and used in a statistical model calculation of thee ⁺ e ^– pair energy spectrum from compound nuclear decay in¹¹⁰Sn following a fusion evaporation reaction. This result is then compared to thee ⁺ e ^– spectrum derived from internal pair decay of the giant dipole and giant quadrupole resonances. The computation shows that the pair decay from the excited-state GDR dominates the pair spectrum over the region of all giant resonances, exceedingL=0 transitions by at least a factor of ten. We also compute the angular correlations betweene ⁺ ande ^– for theL=0, L=1 andL=2 transitions and estimate their power to discriminate between the various multipolarities.Dedicated to Prof. Dr. P. Kienle on the occasion of his 60th birthday     

Semi-empirical analysis of magnetic moments of single-particle (or-hole) states around the 208Pb core

Magnetic moments of single-particle (or-hole) states around the ²⁰⁸Pb core were fitted by three state-independent parameters δg_l^(p), δg_l⁽ⁿ⁾ and α, where α is a parameter that is connected closely with δg_S and g_p. The analysis was performed under the assumptions that (i) the state-dependence of radial integral I((nl)²(n'l')²) involved in δg_s and g_p is calculated by the harmonic-oscillator potential, and (ii) δg_s is taken as being equal to ?4g_p, as expected from the δ-force type core polarization. An excellent fit was obtained when δg_l^(p) = 0.10(2), δg_l⁽ⁿ⁾ = ?0.05(1) and α = 1.2(1), which show that the main contributor to δg_s is the M1 core polarization and δg_l⁽ⁿ⁾ is certainly smaller than 0. B(M1)'s for allowed transitions calculated with the use of the same parameters as above are smaller than the observed B(M1)'s whereas α = 1.01(1) explains well the observed B(M1)'s.     

Semimicroscopic description of giant octupole resonances in deformed nuclei

The strength functions b(E3, ω) are calculated, and the positions and widths of giant octupole resonances in deformed nuclei are found. It is shown that the giant octupole isoscalar resonances have energies (19–20) MeV for the rare-earth nuclei and (17–18) MeV for the actinides and the widths (5–7) MeV. The energies of the giant octupole isovector resonances are defined by the value of the isovector constant κ⁽³⁾₁.     

Description of radiative strength functions in deformed nuclei

Radiative strength functions ofE1- andM1-transitions from ground states of doubly even deformed nuclei to states near the neutron binding energyB _n are calculated within the quasiparticle-phonon nuclear model. The wave functions of excited states include one- and two-phonon components. The calculations were made with the Pauli principle being or not included in the two-phonon components of the wave functions. It is shown that the radiativeE1- andM1-strength functions as well as the widths of giant dipole resonances in deformed nuclei are slightly influenced by the two-phonon components of the wave functions and they can be calculated in the RPA. Thek _E1- andk _M1-values are calculated for some deformed nuclei of the rare-earth and actinide region. The calculated values ofk _E1 are 1.5–2 times larger and the values ofk _M1 are somewhat less than the average values obtained in [14] from the analysis of available experimental data.     

Nuclear moments as a test of shell-model and collective model descriptions of the nucleus

The distribution of electric and magnetic charges and currents in the atomic nucleus can be used to probe various nuclear structure aspects. We discuss, in particular, magnetic dipole and electric quadrupole moments as well as nuclear radii starting from shell-model and collective model approaches. We give attention to the relation between the free electromagnetic coupling strenghts (e_p, e_n, g_l, g_s) and the corresponding renormalized quantities and to the way in which polarization properties depend on the nuclear model space chosen. The above points are largely illustrated with recent experimental data on these various static moments (μ, Q, <r²>). At the same time, the limits of applicability of the various model calculations in extracting preclse nuclear structure information from the above data are discussed.     

Excitation of giant multipole resonancesL=1,2 and 3 in the16O(p,2p) reaction

The¹⁶O(p,2p) quasi elastic reaction at 45 MeV is investigated in the framework of the antisymmetrized distorted wave approximation (DWA). The inclusion of a direct and exchange two step core polarization mechanism, virtually exciting giant multipole resonances was consistently applied as in the (p, p′) non normal parity transition of¹⁶O. Thus utilizing the same renormalized effective two nucleont-matrix as in the inelastic transition a good fit to the data was obtained. It is suggested by the present analyses that corepolarization must be fully included for (p, 2p) like for inelastic scattering in the intermediate energy range.     

Radiative strength functions in odd-A spherical nuclei

The radiative strength functions for the partial γ-transitions from neutron resonances to the ground and low-lying states of odd-A spherical nuclei are calculated within the quasiparticle-phonon nuclear model. The fragmentation of one-quasiparticle and quasiparticle-plus-phonon states is calculated. This allowed one to calculate γ-transitions between the one-quasi-particle components (valence transitions) and γ-transitions between the quasiparticle-plus-phonon and one-quasiparticle components of the wave functions. The energy dependence of the strength functions $\text{C}$(E1, η) and $\text{C}$(M1, η) is calculated near the neutron binding energy B_n for ⁵⁵Fe and ^{59, 61}Ni. The corresponding experimental data are described qualitatively. The contribution of the valence E1 transitions to the strength function is shown to be from 20% to 90%, and M1 transitions about 1%. The influence of the M1 giant resonance is important for M1 transition probabilities.     

s- and p-wave neutron spectroscopy. Xe. Intermediate structure in 90Y and the generalized R-matrix theory

The resonance structure observed in the ⁸⁹Y(n, n)⁸⁹Y total cross section measurements in the range of 0.3 to 1.2 MeV incident energy was investigated using the generalized R-matrix theory of nuclear reactions and the doorway interpretation of intermediate structure. The energies and wave functions of the doorway resonances were calculated in a 2-particle and 3p-1h basis of the shell model. The model space and the parameters of the model calculation chosen were consistent with other shell model calculations in the mass-90 region. Several strong p-wave doorways with J^π = 0⁺, 1⁺, and 2⁺ were predicted by the model in the energy range studied. This is due to proximity of p-wave giant resonance. The escape widths Γ^↑ and the spreading widths Γ^↓ for these states were evaluated using the model wave functions and the R-matrix formalism. The calculated energy dependence of the total cross section shows that most of the predicted doorways are in general agreement with the observed anamolies with similar relative strength. More significantly, the underlying p-wave gross structure representing a grand average is of very similar shape in both theory and experiment. As expected in the mass 90-region, the s- and d-wave doorways contribute less significantly to the calculated resonance structure.     

Integral Hellmann-Feynman theorem as a criterion of the quality of wave functions of atomic-molecular systems

It is proposed to use the integral Hellmann-Feynman theorem for the quality control and refinement of atomic and molecular wave functions. Its validity is verified for variational wave functions of the positronium ion e ⁺ e ^? e ^?, the negative ion of the hydrogen atom p _∞ ⁺ e ^? e ^? (H^?), and mesomolecules μ^?π⁺π⁺ and μ^? p ⁺ p ⁺. The relative violation of this theorem (10^?2) is six orders of magnitude larger than the error of the energy calculation (10^?8), which demonstrates its high sensitivity to the quality of wave functions. A way of refining wave functions on the basis of combination of the integral Hellmann-Feynman theorem for exactly solvable model and real atomic-molecular systems is proposed. A rule for verification of the mutual consistency of the wave functions of any three quantum-mechanical systems is formulated.     

Virtual excitation of the GDR mode in the subbarrier23Na(p, γ)24Mg reaction

Differential cross sections for nonresonant radiative capture of low energy protons (E _p = 1,348 keV and 1,370 keV) by²³Na nuclei exhibit features pointing to the virtual excitation of the giant dipole resonance (GDR) mode. Theoretical analysis carried out within the framework of the direct — semidirect capture model reveals an enhanced coupling of the GDR with the incident protonf-wave consistent with the microscopic structure of the GDR in thes-d shell nuclei.     

The π N → e +e−N reaction close to the vector-meson production threshold

The π^? p→e ⁺ e ^? n and π⁺ n→e ⁺ e ^? p reaction cross sections are calculated below and in the vicinity of the vector-meson (?⁰,ω) production threshold. These processes are largely responsible for the emission of e ⁺e^? pairs in pion-nucleus reactions and contribute to the dilepton spectra observed in relativistic heavy ion collisions. They are dominated by the decay of low-lying baryon resonances into vector-meson-nucleon channels. The vector mesons materialize subsequently into e ⁺ e^? pairs. Using πN→?⁰ N and πN→ωN, amplitudes calculated in the center of mass energy interval 1.4 < √s<1.8 GeV, we compute the π^? p→e ⁺ e ^? n and π⁺ n→e ⁺ e ^? p reaction cross sections in these kinematics. Below the vector-meson production threshold, the π⁰?ω interference in the e ⁺ e^? channel appears largely destructive for the π^? p→e ⁺ e ^? n cross section and constructive for the π⁺ n→e ⁺ e ^? p cross section. The pion beam and the HADES detector at GSI offer a unique possibility to measure these effects. Such data would provide strong constraints on the coupling of vector-meson-nucleon channels to low-lying baryon resonances.     

Spin-dependent effects in inelastic scattering

The differential cross sections for the inelastic scattering of 10 MeV, 19.6 MeV, 30.4 MeV, 40 MeV and 49.35 MeV protons to the 2⁺ state (1.409 MeV) in ⁵⁴Fe and of 19.6 MeV protons to the 2⁺ state (0.846 MeV) in ⁵⁶Fe are analyzed in conjunction with the available data on the asymmetries and spin-flip probability amplitudes. The scattering amplitudes for both one step (valence plus core polarization) and two step (intermediate resonance) processes are evaluated using an antisymmetrized distorted wave approximation. Collective model representations for both the one step (core polarization) and two step (intermediate resonance) processes are used, and included are the effects of deforming the full Thomas spin-orbit potentials. The one step processes are fixed by the analyses of the scattering of 30.4, 40 and 49.35 MeV protons, with the core polarization contributions being constrained by the B(E2) values for the γ-ray deexcitation of the 2⁺ states. The analyses of the 19.6 MeV data demonstrates the need for an extra (two step) contribution to the reaction process and are consistent with the virtual formation of an L = 3 giant resonance. The 10 MeV data most certainly demonstrate compound nucleus effects but could also have some strength due to the virtual formation of an intermediate L = 2 giant resonance. The resonance parameters are consistent with recent information concerning the mass variation of giant resonances.     

Transition wavelengths for helium atom in weakly coupled hot plasmas

We have investigated the effect of surrounding plasmas on several singly excited and doubly excited meta-stable bound states of helium atom using highly correlated basis functions for singly excited S, P, D states and CI-type basis functions for doubly excited meta-stable D states. Plasma effect is taken care of by using a screened Coulomb (Yukawa) potential obtained from the Debye model that admits a variety of plasma conditions, and such a model plays an important role in plasma spectroscopy. The wavelengths for transitions from the 1snp ¹P° (n=2,3)→1s²¹S^e, 1snp ³P° (n=2,3)→1s2s ³S^e, 2pnp ¹P^e (n=3,4)→1s2p ¹P°, 2pnp ³P^e (n=2,3)→1s2p ³P°, 2pnd ¹D° (n=3,4)→1s3d ¹D^e, 2pnd ³D° (n=3,4)→1s3d ³D^e, 2p3p ¹P^e→2pnd ¹D° (n=3,4), 2pnd ¹D°(n=3, 4)→2p4p ¹P^e, 2pnp ³P^e (n=2,3)→2p3d ³D°, and 2pnp ³P^e (n=2,3)→2p4d ³D° of helium atom in plasmas for various Debye lengths are reported.     

Doubly excited resonances of positronium negative ions

Some lower-lying doubly excited ¹S^e and ³S^e resonances of positronium negative ions associated with N = 2 up to N = 5 positronium thresholds are calculated by a method of complex-coordinate rotation. Hylleraas-type wave functions with N = 203 terms are used to represent the system.     

Population dynamics of the ions SVI, ClVII and ArVIII interacting with solid surfaces

We used the recently developed time-symmetrized, two-state vector model to investigate the intermediate stages of the electron capture into the Rydberg states (n_A?1,l_A=0−3, m_A=0) of multiply charged ions SVI, ClVII and ArVIII, with polarized core charges Z=6, 7 and 8, respectively, escaping solid surfaces at low velocities. Within the framework of the model, the two wave functions are used to describe the system at intermediate stages; the corresponding probabilities and rates are based on the calculation of the mixed flux. The simple analytical formulae derived for the neutralization rates enable us to analyze the localization of the process and to calculate the neutralization distances. In the pointlike core approximation, the rates are comparable with the available coupled-angular-mode results. The neutralization distances follow the classical predictions for all considered Rydberg states with exception of the states with critical quantum numbers n_A=n_c, mainly populated via tunneling mechanism; the corresponding values are larger than it is predicted by the classical overbarrier model. Inclusion of core polarization significantly reduces the neutralization distances.     

Proton-neutron correlations in nuclei with N = 30

Structure of the nuclei with N = 30 and Z = 20–28 is investigated by the nuclear shell model within the proton-neutron configurations (1f_{$\text{7}\text{2}$})^z?20_p × (2p_{$\text{3}\text{2}$}, 2p_{$\text{case1}\text{2}$}, 1f_{$\text{case5}\text{2}$})²_n. Effective proton-neutron interactions determined by a least-squares fit to the observed spectra of N = 29 nuclei are adopted. Agreement of the calculated spectra with experimental spectra is satisfactory. Strong correlations between protons and neutrons break down the pairing scheme and lower the first J = 2 levels in doubly even nuclei, which is shown from the resultant wave functions. A relation between the shell model and collective rotational model is discussed concerning the calculated rotation-like spectrum of ⁵⁶Fe. Electromagnetic properties and spectroscopic factors of single-nucleon transfer reactions are calculated. They are in good agreement with experiments.