Shell-model Wave Functions Research Articles

A comparative study of the 13C(p, p′) 13C and 13C(p, n) 13N reactions at Ep = 35 MeV

Differential cross sections were measured at E p = 35 MeV for the 13C(p, n) and 13C(p, p′) reactions leading to the four low-lying states in the mirror nuclei 13N and 13C. In addition, the analyzing powers were measured for the 13C(p, p′) reaction. The data are generally well accounted for by DWBA calculations except for the 13 C(p, p′) 13 C(3.09 MeV, 1 2 +) reaction, for which the calculations cannot even reproduce the qualitative features of the data. A comparison of the (p, n) and the (p, p′) results suggests that the isoscalar part of the 13 C( g.s., 1 2 −) → 13 C(3.09 MeV, 1 2 +) transition is not correctly described by currently available shell-model wave functions.

(p, α) Reactions on 9Be and 10B for Ep = 18–45 MeV

The 9Be(p, α) 6Li and 10B(p, α) 7Be reactions have been studied at incident energies between 18 and 45 MeV. At higher energies the integrated cross sections σ(0–90°) and σ(90–180°) for the observed transitions are well reproduced by relative spectroscopic factors for the light-particle and heavy-particle pickup processes respectively, calculated with current shell-model wave functions. No evidence has been found for the presence of a knockout process. At lower energies the measured cross section values are no longer characterized by the spectroscopie factors but tend to be proportional to (2 I + 1), with I being the spin of the residual state.

Elastic magnetic electron scattering from 19F

Elastic magnetic electron scattering from 19F was measured at 154° and 180°. The large momentum transfer range covered ( q = 0.4–2.8 fm −1) allowed us to perform a model-independent analysis of the magnetic dipole form factor. It indicates the presence of strong configuration mixing. Shell-model calculations in the full 2s1d configuration space are unable to account for the behaviour of the experimental data which exhibit an intermediate form factor maximum. A large-basis shell-model calculation using all orbits up to the 1f 7 2 shell produces such a maximum, as does a core-polarization calculation starting from shell-model wave functions in the full 2s1d space. In both cases excitations from the 1s to the 2s shell are mainly responsible for the appearance of the intermediate maximum. Non-nucleonic degrees of freedom do not disturb this conclusion since they yield only small corrections to the form factor.

Particle-hole strength excited in theCa40(p,n)40Sc reaction at 134 MeV

The $^{40}\mathrm{Ca}$(p,n${)}^{40}$Sc reaction was studied at 134 MeV. Neutron energy spectra were measured by the time-of-flight technique with resolutions of 220 keV at angles from 0\ifmmode^\circ\else\textdegree\fi{} to 41\ifmmode^\circ\else\textdegree\fi{} and 415 keV out to 62\ifmmode^\circ\else\textdegree\fi{}. The ${2}^{\mathrm{\ensuremath{-}}}$,${3}^{\mathrm{\ensuremath{-}}}$,${4}^{\mathrm{\ensuremath{-}}}$,${5}^{\mathrm{\ensuremath{-}}}$ band of states based on the (${f}_{7/2}$,${d}_{3/2}^{\mathrm{\ensuremath{-}}1}$) 1p1h structure was observed at low excitation energies, in good agreement with known analog states in $^{40}\mathrm{Ca}$ and $^{40}\mathrm{K}$. The shapes of the cross-section and analyzing-power angular distributions are in good agreement with distorted-wave impulse-approximation calculations using simple 1p1h (Tamm-Dancoff approximation) shell-model wave functions. A relatively strong transition to a state at ${E}_{x}$=2.3 MeV with L=3 is identified tentatively as a ${4}^{\mathrm{\ensuremath{-}}}$ state with the predominant 1p1h structure (1${f}_{7/2}$,2${s}_{1/2}^{\mathrm{\ensuremath{-}}1}$).The excitation of the (T=1) ${6}^{\mathrm{\ensuremath{-}}}$ (${f}_{7/2}$,${d}_{5/2}^{\mathrm{\ensuremath{-}}1}$) stretched state is observed near ${E}_{x}$=7 MeV, fragmented over approximately 3 MeV. The normalization factor required to make a distorted-wave impulse-approximation calculation agree in magnitude with the ${6}^{\mathrm{\ensuremath{-}}}$ excitation is 0.35, which is larger than the normalization factor for this excitation in mass 28, analyzed in a similar manner. The L=1 giant resonance is observed to be centered near ${E}_{x}$=10 MeV with a width (full width at half maximum) of about 5 MeV, and a distorted-wave impulse-approximation normalization factor of 0.16. Two ${1}^{+}$ excitations are observed at ${E}_{x}$=2.7 and 4.3 MeV which indicate directly the presence of ground-state correlations in the $^{40}\mathrm{Ca}$ target. The fact that the analog of the higher ${1}^{+}$ excitation is not seen in inelastic electron scattering indicates strong interference between spin and orbital current contributions. The effect of ground-state correlations on the 1p1h wave functions was investigated by performing calculations which allowed multiparticle-multihole correlations in the ${d}_{3/2}$ and ${f}_{7/2}$ orbitals. The distorted-wave impulse-approximation normalization factors obtained with these wave functions were found to increase by a factor of about 2 relative to those obtained with $^{40}\mathrm{Ca}$ assumed to be a closed core.

High-spin states in26Mg

Theγ-decays of levels in26Mg have been investigated up to 12.5 MeV excitation energy by proton-γ-ray coincidence measurements in the23Na(α, pγ) reaction at 14.2 and 16 MeV bombarding energy. Lifetime-measurements, made with the Doppler-shift attenuation method, and proton-γ-ray angular-correlation measurements were performed at Eα=14.2 MeV. Many high-spin states were observed, among them levels at 6,978 (5+), 7,283(4−), 7,395(5+), 7,953(5−), 8,202(6+), 8,472(6+), 9,065(5), 9,112(6+), 9,169(6−), 9,383 (6+), 9,542(5), 9,829(7+), 9,989(6+) and 12,479(8+, 7−) keV excitation energy. The spectrum of positive-parity states and their electromagnetic properties are reproduced with good accuracy by shell-model calculations which employ a unifieds-d shell Hamiltonian and the unrestricted configuration space of the 0d5/2 1s1/2 0d3/2 shell. Members of five inferred rotational bands, withKπ=0+, 2+, 3+, 0+ and 3− have been observed up to at leastI=6. TheKπ=2+ band shows strong anomalies of excitation energies andE2 transition rates near theI=6 state. The static intrinsic quadrupole moments calculated from the shell model wave functions indicate transitions from prolate to oblate deformation within theKπ=2+ band and also the ground state band. The lowest lyingIπ=4+ state appears to be “spherical” and cannot be associated with a rotational band.

Continuity-equation constraint in the two-vertex nuclear processes: Radiative muon capture on 16O and 40Ca

A modified form of the impulse-approximation effective hamiltonian of the nuclear radiative muon capture (RMC) is developed. In the derivation we have used the electromagnetic current-continuity equation; the procedure is actually an extension for the nonzero momentum transfer of the Siegert theorem. Radiative capture rates and spectra of the emitted photons for RMC on 16O and 40Ca are analysed by using the nuclear shell-model wave functions for the calculation of the partial RMC transition rates. The spectra obtained are strongly reduced in comparison with the standard impulse-approximation calculations and agree with the phenomenological results. The main contradiction of recent RMC theoretical works has been resolved in this way.

The ? particle as a probe of deeply bound single particle states in heavy nuclei

Decay rates of excited states in heavy hypernuclei have been calculated using shell model wave functions. The nuclear Auger effect determines the widths of the states in most cases. The exception is the radiative decay for theΛ particle in the 1p and in exceptional cases also 1d shell. In general, the natural widths of the highly excited states below the threshold for theΛ emission are smaller than the spacing between theΛ shells. Spectroscopy ofΛ bound states in heavy hypernuclei thus seems to be feasible.

Effect of nuclear correlations on the Λ decay width in nuclei

Because of nuclear short range correlations the ground state Λ wave function of light nuclei is pushed towards the surface of the nucleus with respect to the conventional Λ wave function of the Λ-nucleus potential. As a consequence the mesonic width increases while the nonmesonic width decreases. An extra and more important increase of the mesonic width is obtained when proper account is taken of the renormalized properties of the pion in the nuclear medium. If these pion properties are considered the mesonic width becomes less sensitive to the Λ wave function and the results are about the same with the shell model wave function or the correlated one. This lack of sensitivity to the nuclear wave function, if confirmed for heavier nuclei, together with the extreme sensitivity to the pion properties in the medium, would rend the measurement of the mesonic width in nuclei an excellent experimental method to test the pion dispersion relation ( ω = ω( q) ) of pions inside a nuclear medium.

Hybrid quark-hadron model of Lambda nonmesonic decay: Finite nuclei.

\ensuremath{\Lambda} hypernuclear nonmesonic lifetimes are calculated in the framework of the hybrid quark-hadron model, for both infinite nuclear matter and for a finite nucleus $_{\mathrm{\ensuremath{\Lambda}}}^{12}\mathrm{C}$. For nuclear matter, an approximate solution of the Bethe-Goldstone equation is used for the initial state \ensuremath{\Lambda}N cluster wave function. For $_{\mathrm{\ensuremath{\Lambda}}}^{12}\mathrm{C}$ a shell model wave function is used. An eikonal approximation is used to represent final state interactions. It is shown that with an effective weak quark Hamiltonian explicitly constructed to be consistent with the \ensuremath{\Delta}I=1/2 rule the $_{\mathrm{\ensuremath{\Lambda}}}^{12}\mathrm{C}$ nonmesonic width is about 1.3 times the free \ensuremath{\Lambda} width, which is consistent with a recent experimental measurement.

Shell-model analysis of high-resolution data for elastic and inelastic electron scattering on 19F.

A comprehensive theoretical and experimental investigation of electron scattering on /sup 19/F is presented. Theoretical procedures for calculating the various components of electron scattering form factors from multiparticle shell-model wave functions are summarized. These procedures are used to compare shell-model predictions with data from electron scattering on /sup 19/F. For the positive-parity states of /sup 19/F we use 1s0d shell-model wave functions obtained with a new ''universal'' Hamiltonian and for the negative-parity states we use 0p-1s0d shell-model wave functions based on the cross-shell Hamiltonian of Millener and Kurath. The comparisons are made with measured longitudinal and transverse form factors for momentum transfers up to 2.4 fm/sup -1/ which are extracted from experimental data obtained with electron energies from 78 to 340 MeV, and angles of 45/sup 0/, 90/sup 0/, and 160/sup 0/. The energy resolution was 25--50 keV and most of the known levels below 8 MeV in excitation were resolved and compared with theory.

Study of the two-nucleon mechanism of pion absorption in nuclei

The two-nucleon mechanism of pion absorption by nuclei is investigated in the energy region of the πN P 33 Δ-resonance. The basic absorption process is governed by a πNN … NΔ … NN transition matrix, derived from the phenomenological Hamiltonian of M. Betz and T.-S. H. Lee ( Phys. Rev. C, 23 (1981) , 375), which was constructed to describe NN scattering phase-shifts up to 1 GeV. The model allows a realistic description of pion absorption on a pair of bound nucleons with quantum numbers and relative radial wave functions different from those of the physical deuteron. The deuteron-like 3 S 1 ( T = 0) pairs are shown to play a privileged role, in accordance with the assumption underlying the conventional quasideuteron model. We then enbed the two-nucleon mechanism into complex nuclei, using the impulse approximation. The many-body effects on the two-body absorption mechanism are analyzed in detail by using the Faddeev wavefunction for 3He and harmonic oscillator shell-model wavefunction for 1 p-shell target nuclei in our calculations. All nonlocal effects owing to nucleon Fermi motion and NΔ off-shell propagation are treated rigorously. The main features of ( π +, p) reactions on 3He, 4He, and 12C are predicted correctly, when large pion distortion effects are taken into account by using the isobar-hole model with all of the parameters pradetermined from earlier studies of pion nucleus scattering. The predicted cross sections at the two-body absorption peaks overestimate the data by a factor ranging from 1.2 ∼ 2 for the ( π +, p) reactions, and as much as a factor of ∼ 4 for the coincidence ( π +, pp) on 12C. Our results indicate that a consistent description of all inclusive data can be obtained only when we further assume that the outgoing nucleons must be strongly rescattered by nuclear medium in 12C and heavier nuclei. When the calculated absorption cross sections are integrated over entire kinematic regions at each pion energy, we find that the calculated values are only about 1 3 to 1 2 of the total absorption cross section extracted from earlier measurements. By combining our results and calculations by K. Masutani and K. Yazaki ( Nucl. Phys. A, 407 (1983) , 309), it is concluded that a large part of total absorption cross section could originate from the inelastic absorption process: ( π, π′ N) nucleon knockout followed by pion absorption. We are, however, unable to calculate microscopically this inelastic absorption cross section unambiguously, and hence the existence of other absorption mechanisms and possible contributions to proton spectra from targer fragmentation cannot be excluded from our analysis. These possibilities become even more likely when the weaker ( π −, p) cross section is found to be undertestimated by the theory by a very large factor. The origin of this problem is identified to be the suppression, by nuclear geometry and isospin selection rules, of the two-body process πNN … NΔ … NN initiated by π −. We discuss the necessary experimental information for future improvements of the theory.

On parity violation in α-decay

The Fermi liquid model of α-decay and the shell-model wave functions of Zuker, Buck and McGrory are used to calculate the “regular” and “irregular” (parity non-conserving) α-widths to some J π T = 2±0 low-lying levels of 16O. The Pauli corrections in the α-channel wave functions are taken into account.

Intrinsic-state coefficients of fractional parentage in the oscillator shell model

Intrinsic-state c.f.p. allow access to the internal structure of shell-model wave functions without involving the c.m. coordinate. A method is presented which allows intrinsic-state c.f.p. to be calculated from known standard shell-model c.f.p. Sample results are tabulated and an example of the usage of such c.f.p. is given.

Electron scattering form factors for fp-shell nuclei

Elastic and inelastic electron-scattering form factors for multipolarities up to L = 7 and some transition-strength distributions are calculated with shell-model wave functions for about ten target nuclei in the mass range A = 52–62. The results are compared with the available measured transition strengths and (e, e') form factors.

A study of the 14C( 6Li, 6He) 14N reaction at 93 MeV

The reaction 14C( 6Li, 6He) 14N was investigated with 93 MeV 6Li ions in an angular interval of 7–26°. Angular distributions were analysed for the four most intense groups of 6He nuclei, corresponding to transitions to the ground (1 1 +) and the excited (1 2 +, 2 1 −, 4 1 −) states of 14N. In the theoretical analysis a mechanism of the spin-isospin excitation was suggested in the DWBA frame with the finite range of interaction and recoil in the light system ( 6Li 6He) taken into account. In the calculations both shell-model wave functions and transition densities obtained in the theory of finite Fermi systems (FFS) were used. From the comparison between theory and experiment the Landau-Migdal force constant g′ is estimated in order to obtain some information on the degree of nuclear proximity to the threshold of pion condensation.

Particle-hole strength excited in the 48Ca(p,n)48Sc reaction at 134 and 160 MeV: ( pi f7/2, nu f7/2-1) band.

The $^{48}\mathrm{Ca}$(p,n${)}^{48}$Sc reaction was studied at 134 and 160 MeV. Neutron energy spectra were measured by the time-of-flight technique with resolutions from 320 to 460 keV at 11 angles from 0\ifmmode^\circ\else\textdegree\fi{} to 60\ifmmode^\circ\else\textdegree\fi{} spaced approximately 6\ifmmode^\circ\else\textdegree\fi{} apart. The neutron spectra reveal strong excitation of the (\ensuremath{\pi}${f}_{7/2}$,\ensuremath{\nu}${f}_{7/2}^{\mathrm{\ensuremath{-}}1}$) band of states at low excitation energies; five of the known eight members of the band are excited strongly, generally consistent with distorted-wave-impulse-approximation calculations with 1f-2p shell-model wave functions. The normalization of the distorted-wave-impulse-approximation calculations to the extracted angular distributions at 135 MeV vary from 0.40 for the ${5}^{+}$ state to 0.93 for the ${0}^{+}$, isobaric analog state. The normalization factor of 0.60 required for the ${7}^{+}$, stretched-state transition is significantly larger than those required for similar distorted-wave-impulse-approximation calculations of various stretched-state excitations observed in inelastic proton scattering, which involve promoting a nucleon up into the next major shell (a ``1 \ensuremath{\Elzxh}\ensuremath{\omega}'' excitation). The larger normalization factor required for the (\ensuremath{\pi}${f}_{7/2}$,\ensuremath{\nu}${f}_{7/2}^{\mathrm{\ensuremath{-}}1}$) ${7}^{+}$ excitation (a ``0 \ensuremath{\Elzxh}\ensuremath{\omega}'' excitation) indicates that this strength is more highly concentrated in a single state. The ${4}^{+}$ and ${6}^{+}$ states of this band are observed to be only weakly excited, consistent with predictions that transitions dominated by the tensor term of the nucleon-nucleon interaction will predominantly excite non-normal parity states. The excitation of the ${2}^{+}$ member of this band shows both \ensuremath{\Delta}l=0 and \ensuremath{\Delta}l=2 contributions to the angular distribution and may indicate significant two-step processes for this transition.

Pion scattering to 4(-) states in 14C.

Angular distributions and excitation functions for inelastic scattering of ${\ensuremath{\pi}}^{+}$ and ${\ensuremath{\pi}}^{\mathrm{\ensuremath{-}}}$ from $^{14}\mathrm{C}$ were measured at incident pion energies near the ${\ensuremath{\Delta}}_{33}$ resonance. Three states at excitation energies 11.7, 15.2, and 17.3 MeV were identified as ${4}^{\mathrm{\ensuremath{-}}}$ states. Isovector and isoscalar spectroscopic amplitudes ${Z}_{0}$ and ${Z}_{1}$, and equivalently, neutron and proton amplitudes ${Z}_{\mathrm{n}}$ and ${Z}_{\mathrm{p}}$ were deduced by comparison with microscopic distorted wave impulse approximation calculations. The 11.7-MeV state was found to be excited with a ${Z}_{\mathrm{n}/{Z}_{\mathrm{p}}}$ amplitude ratio of -1/3, resulting in a complete cancellation of the ${\ensuremath{\pi}}^{+}$ cross section. A nearly pure proton excitation was observed for the transition to the 17.3-MeV state. Both results are in qualitative agreement with the presented shell-model calculations. A poor correspondence with theory is found for the 15.2-MeV state. Data and distorted-wave impulse approximation calculations using shell-model wave functions are presented for the first ${3}^{\mathrm{\ensuremath{-}}}$ state at 6.73 MeV as an example of a transition dominated by \ensuremath{\Delta}S=0 (no spin transfer). Its excitation function and angular-distribution shape contrast sharply with the transitions to the ${4}^{\mathrm{\ensuremath{-}}}$ states that proceed by \ensuremath{\Delta}S=1.

Direct observation of the alpha particle D-state in the (d, α) reaction

Evidence for the direct observation of effects arising from a D-state component of the alpha particle wave function is presented in an analysis of tensor analysing power data for the 40Ca(d, α) 38K reaction at 12.3 DWBA calculations and data are consistent with shell model wave functions for the target nuclei and simple alpha particle wave function models.

Calculation of the β + decay of proton-rich sd-shell nuclei into a Gamow-Teller resonance

The β + strength functions for the β + decay of the proton-rich nuclei 32Ar, 33Ar and 36Ca are studied using shell-model wave functions. A resonance-like structure in the strength function is calculated to be positioned about 3 MeV above the state that is the isobaric analogue state of the ground state of the decaying nucleus. A comparison with available data for A = 33 reveals a hindrance of strength in this resonance region of a factor of 2–3, in accordance with information from other spin-flip probes. The calculations for A = 32 and 36 indicate that these systems possess a J π ; T = 1 +; 1) Gamow-Teller resonance based on the J π ; T = 0 +; 2 state of the system.

Proton and neutron occupation numbers forfp-shell nuclei

Spectroscopic factors for single-particle transfer are calculated from many-particle shell-model wave functions for 23 different transfer reactions onfp-shell nuclei. A comparison of the present theory with the available experimental data is very successful. Subshell occupation numbers of target ground states are studied in detail. Some systematic deficiencies for low seniority shell-model wave functions are revealed. In several cases the experimental information appears to be rather incomplete.