Shell-model States Research Articles

The calculation of diagonal matrix elements of two-body relative state operators between many-body oscillator states

AbstractA general method is given for the determination of diagonal matrix elements of relative state operators in shell model states of a given supermultiplet and SU3 symmetry, for states in s, p and sd shells, and for cross-terms between these shells, enabling the excitation energies of particle–hole configurations to be evaluated.

Rotational bands of 20Ne

Using alpha-particle reduced-widths to distinguish alpha-cluster states from shell model states it is suggested that the lowest lying 8 + level of 20Ne belongs to the (8p-4h) rather than to the (4p) ground state band. Many new levels and level properties are predicted.

F19(d, p)F20and the Nuclear Structure ofF20

The reaction $^{19}\mathrm{F}(d, p)^{20}\mathrm{F}$ has been studied with 16-MeV deuterons. Outgoing protons were detected in photographic emulsions in a magnetic spectrograph. Spectroscopic factors were extracted and combined with previous information and compared with results of shell-model calculations performed in a complete $\mathrm{sd}$-shell basis. Of the previously known 25 states below ${E}_{x}=4.5$ MeV, angular distributions measured at 14 angles were obtained for all but the 5 at ${E}_{x}=1.824, 2.871, 3.761, 4.20, \mathrm{and} 4.21$ MeV. Strong stripping angular distributions were observed for 10 states-6 dominated by $l=2, \mathrm{and} 4$ by $l=0$. These 10 states agree reasonably well in position and strength with the 10 lowest shell-model states predicted to have appreciable amounts of the configuration [$^{19}\mathrm{F}(\mathrm{g}.\mathrm{s}.)\ensuremath{\bigotimes}1{d}_{\frac{5}{2}} or 2{s}_{\frac{1}{2}}$ neutron)].

Approximate treatment of correlations in nuclear spectroscopy: (II). Two-nucleon transfer in the lead region

The effect of ground state correlations on calculated two-nucleon transfer cross sections for the reactions (p, t), (t, p), ( 3He, p) an ( 3He, n) in the lead region is investigated. The sensitivity to ground state correlations is isolated by comparing calculated DWBA cross sections obtained by using two-nucleon shell-model (2n SM) states with those obtained by using two-nucleon random-phase approximation (2n RPA) state descriptions. The correlations obtained in a 2n RPA calculation produce strong enhancements of the calculated ground state to ground state transitions involving the transfer of two identical particles over the same transitions calculated without correlations. In addition, certain other transitions are substantially affected while others are unaltered. The sensitivity of the calculated cross sections to the model and the choice of effective interaction is investigated and compared with the sensitivity of other calculated quantities.

Solution of the Bethe–Goldstone Equation in Finite Nuclei from N–N Phase Shift Data

An approach is outlined wherein the Bethe–Goldstone wave function may be computed in a shell-model basis from a knowledge of the harmonic-oscillator matrix elements. These in turn may be derived from the N–N phase shifts by a number of different methods. The Pauli operator is treated exactly in the space of two-particle shell-model states. The method is applied to a calculation of the levels of 18O and 18F and results obtained using different two-body matrix elements are compared.

Effective single-particle magnetic moments around 208Pb

Abstract An effective magnetic-moment operator is empirically derived for shell-model states around 208Pb. It reproduces thirteen measured moments to a few tenths of a nuclear magneton, but predicts too high a hindrance for M1 transitions.

Shell-model bound state calculations including continuous spectra

The factorization method is used to calculate bound states of nuclei using single particle spectra including the continuum. The influence of direct channel-channel coupling and of the position of potential resonances is discussed. A comparison between spectra obtained by different methods is given.

Beta-gamma circular-polarization correlation in the decay of 141Ce

The βγ cp correlation of 141Ce has been investigated using the Compton-backscattering method and a Ge(Li) detector for the determination of the circular polarization of the γ-rays. Beta particles are detected by a silicon surface-barrier detector. The measured asymmetry coefficient is ω =−0.26±0.12. This value is independent of Rayleigh-scattering influences. A discussion of the result on the basis of the ξ-approximation formalism yields the two ranges of the nuclear matrix element ratio X: 0.3 ≦ X ≦ 0.7 and 2.1 ≦ X ≦ 7.1. Assuming pure shell-model states it is possible to interpret the result as a ( 2 f 7 2 ) neutron to (1 g 7 2 ) proton β-transition which favours for the zero-order matrix element ratio Λ 0 a value between 1.28 and 1.65.

Rotational states and rotation-particle coupling in 155Tb and 157Tb

For the investigation of excited states of the odd-proton nuclei 155Tb and 157Tb, studies have been made of the γ-radiation emitted in the reactions 153Eu(α, 2n), 155Gd(d, 2n) and 155Gd(p, n) as well as 157Gd(d, 2n) and 157Gd(p, n), respectively. Measurements of coincidence relations were performed in the reactions with α-partieles and deuterons. Furthermore, in the reaction with α-partieles excitation functions and angular distributions γ-rays were measured. The level schemes proposed for 155Tb and 157Tb are very similar. They contain the band heads (energies are given in keV for 155Tb and 157Tb, respectively) 3 2 +[411] (ground state), 5 2 +[413] (271.3; 327.6), 7 2 +[404] (466.9; 658.6), 5 2 −[532] (227.0; 326.4) and 7 2 −[523] (545.3; 571.7), as well as several rotational states. In both nuclei the distorted rotational band built on the 5 2 −[532] band head has been analyzed in terms of the Coriolis coupling between the Nilsson orbitals based on the H 11 2 . shell-model state. In 155Tb the Coriolis coupling (mainly with the 7 2 +[404] orbital) gives rise to an energy difference of only 63.7 keV between the 7 2 and 5 2 members of the 5 2 +[413] band.

(p, 2p) reactions at 600 MeV and the distorted-wave impulse approximation

The summed-energy spectra obtained in a 600 MeV (p, 2p) experiment on 12C, 28Si and 40Ca have been analysed so that a large amount of the information from a non-symmetric coplanar experiment is utilized. The experimental cross section is integrated over a selected interval in the summed-energy spectrum and presented as a function of the polar angle, θ R and of the momentum, p R, of the residual nucleus. By comparing the distributions of the experimental cross sections to p R and θ R distributions calculated in the distorted-wave impulse approximation, DWIA, information is obtained on distortion effects and on rms radii for the various shell-model states contained in the summed-energy spectra. The DWIA calculations reproduce quite well the experimental distributions for the outer shells, while for the 1s shell in 28Si and 40Ca too large values of the proton occupation numbers are found.

Direct Reduction to Talmi Integrals Without Use of Transformation Brackets

The two-body shell-model matrix elements for central and tensor forces can be expanded as linear combinations of Talmi integrals. A closed form, which does not involve vector-coupling coefficients or transformation brackets, is obtained for the coefficients appearing in this expansion. For the case that all the oscillator constants are the same, the result is simple enough to allow tabulation. A table of these coefficients for matrix elements involving the shell-model states $0s$, $0p$, and $0d$ is included. The main advantage of this method over the Talmi-Brody-Moshinsky method is discussed.

Non-adiabatic effects in rotational bands of the odd proton nucleus 155Tb

An anomalous rotational band found in 155Tb by in-beam γ-ray spectroscopy is interpreted in terms of the coriolis coupling between the Nilsson orbitals based on the h 11 2 proton shell model state.

Alpha-particle type four-body correlation in 212Po

It is shown that a configuration in which two protons are excited from the pure shell-model ground state configuration to the same orbit as the uppermost occupied neutron orbit contributes appreciably to the α-decay reduced width of 212Po. This configuration has been neglected in most shell-model calculations. It is noted that the four-body correlation is enhanced by this new configuration if the effective interaction is attractive in the singlet-even state; this condition coincides with that on the effective interaction in order that an α-cluster be formed. If the conventional low-lying configurations are included, the α-decay reduced width of 212Po becomes twice as large as that for the configuration mixing calculations by Harada. It is also suggested that there may be states with strong α-type correlation among excited states.

Pb208(d, α)Tl206Reaction and theπ−1ν−1Structure ofTl206

The proton-neutron hole states of ${\mathrm{Tl}}^{206}$ have been studied by the direct ${\mathrm{Pb}}^{208}(d, \ensuremath{\alpha}){\mathrm{Tl}}^{206}$ reaction with 17-MeV deuterons. Experimental resolutions of 12-15 keV for about 28-MeV $\ensuremath{\alpha}$ particles permitted the investigation of many two-hole states previously unknown. For most levels, $L$ values could be extracted from the comparison of microscopic two-nucleon transfer distorted-wave Born-approximation (DWBA) calculations with experimental angular distributions. The experimental level energies and ${J}^{\ensuremath{\pi}}$ assignments (or limits) were compared where possible to recent ($n, \ensuremath{\gamma}$) and ($t, \ensuremath{\alpha}$) data for ${\mathrm{Tl}}^{206}$ and to structure calculations by Kuo. It was possible to correlate about 20 low-lying levels (${E}_{x}\ensuremath{\lesssim}2.2$ MeV) with calculated shell-model states. In addition, Kuo's wave functions were used to derive detailed ($d, \ensuremath{\alpha}$) cross-section intensity distributions, and to check the dependence of the absolute normalization constant $N(d, \ensuremath{\alpha})$ on various DWBA parameters. Constructive coherence in the transition amplitudes predicted by the wave functions for the lowest states of a given ${J}^{\ensuremath{\pi}}$ was varified experimentally. Ten enhanced ($d, \ensuremath{\alpha}$) cross sections are observed, in good qualitative agreement with theoretical predictions. The predicted enhancement for the population of some of these states was occasionally in excess of the observed enhancement.

Five-dimensional quasispin toward a complete classification of the isospin characteristics of shell-model states in the seniority scheme

A solution is proposed for the inner multiplicity problem associated with the five-dimensional quasispin description of shell-model states. A classification scheme, in terms of an orthonormal basis, leads to tractable results with definite symmetry properties under particle-hole conjugation. Explicit constructions are given for the R(5) irreducible representations (ω 1 1), (ω 1 3/2), (t + 1, t) for which the inner multiplicities are never greater than two. For states of seniority v = 2, reduced isospin t = 1, of a configuration j n general expressions are given for the matrix elements of an arbitrary two-body interaction, to determine their n, T dependence, and to isolate those features of the actual interaction among nucleons which are most effective in splitting the isospin degeneracy of such states.

Some model calculations of Coulomb mixing

The nucleus 56Co offers a good case for testing various mechanisms of isospin mixing. The analog state is bound making the mixing calculation easier than for isobaric analog resonances. There is, however, a direct correspondence between the mechanisms in the bound case and in the continuum case. The analog states of this nucleus are at low excitation so there is hope of making detailed model calculations of all the states involved in the mixing. For these calculations we consider the J = 0 + T = 2 state of 56Co as a simple 2p-2h shell-model state. Four possibilities are offered for a nearby T = 1 0 + state. The external mixing mechanism of continuum calculations is a second-order perturbation in this case which gives matrix elements several keV in magnitude. The internal mixing caused by the off-diagonal matrix elements of the Coulomb force has been neglected in continuum calculations, but we find that in this case it can be comparatively large, of the same order of magnitude as the other mechanisms. There is sufficient disparity between the matrix elements calculated for the different models to distinguish among the models experimentally. For 56Co, experimental data can be interpreted to give a mixing of 33 keV with a nearby state. This is large enough to exclude two of the models.

Deformed state admixtures in mass-42 nuclei

We have attempted to explain the energy spectrum and transition rates in 42Ca and 42Sc by considering admixtures of 4p-2h deformed and 2p spherical shell-model states for the low-lying levels. Two methods of handling the 4p-2h deformed states have been considered; both treat the four-particle wave function in the same way but in the first method the two holes occupy a deformed orbit in a similar way to the four particles, while in the more extended second method the four-particle states are coupled to a complete set of two-hole states in the s-d shell. Using three variational parameters (the deformation parameters) and three fitted parameters (effective charge, f 7 2 − d 3 2 energy splitting and the overlap between spherical and deformed 40Ca cores), we have reproduced to a good approximation (in the case of the extended model) the positive-parity spectrum and the transition rates of 42Ca up to 3.4 MeV excitation energy. With this model we have also accounted for the two 1 + states recently observed at about 2 MeV excitation in the T = 0 spectrum of 42Sc, and have explained the large number of low-lying levels in the spectrum of this nucleus. The simpler model has been used to investigate the nature of some pure deformed states in the low-lying spectrum of 42Sc which have probably not yet been detected experimentally.

Self-Consistent Study of Nuclear Shapes

Self-consistent Hartree-Fock (HF) and Hartree-Fock-Bogoliubov (HFB) calculations have been performed for the even Ti, Cr, and Fe isotopes. The residual two-body interaction is assumed to be a central potential with a Yukawa radial dependence and an exchange mixture which has been adjusted to fit the low-lying shell-model states of ${\mathrm{O}}_{10}^{18}$ and ${\mathrm{F}}_{9}^{18}$. It is found that the HF minima occur for prolate shapes, and hence the HFB calculations are done only for prolate deformations. The HFB calculation yields normal solutions for $N=Z$ nuclei and spherical shapes for $Z<N=28$ nuclei. The effect of pairing on the ground-state properties is also studied. The value of the mass quadrupole moment for the HFB solution is smaller than that for the HF solutions for all the nuclei under consideration. A spherical BCS calculation has been performed to estimate the effect of deformation on pairing energy in the HFB calculation. We construct the excited states for the $Z<N=28$ nuclei from the spherical BCS ground-state solutions in the random-phase approximation. This gives the values for the ${0}^{+}$${\ensuremath{\rightarrow}2}^{+}$ energy separation and the $B(E2)$ rates which are in good agreement with experiment.

A projection operator for the spurious centre-of-mass motion

A simple projection operator K is introduced that projects spurious states from a general harmonic oscillator shell-model state. The operator is used to define effective Hamiltonians which, in the example of the J = 1 −, T = 0 states of 16O, appear capable of minimizing the contributions of spurious states to low-energy states in truncated bases.

Lifetime of the 3.19-MeV (6+) State inCa42

The lifetime of the (${6}^{+}$) 3.19-MeV state in $^{42}\mathrm{Ca}$ has been measured by obtaining the time distribution of coincidences between positrons forming the state, from the decay of the (${7}^{+}$) 615-keV state in $^{42}\mathrm{Sc}$, and 440-keV-decay $\ensuremath{\gamma}$ rays. The metastable (61-sec) state in $^{42}\mathrm{Sc}$ was generated by the $^{40}\mathrm{Ca}(^{3}\mathrm{He}, p)^{42}\mathrm{Sc}$ reaction using a 6.10-MeV ${\mathrm{He}}^{3}$ beam from the University of Iowa Van de Graaff. A 60-sec beam-on-beam-off cycle was employed. Positrons were detected by a 2-in.-diam \ifmmode\times\else\texttimes\fi{} 1-in.-high plastic scintillator, and $\ensuremath{\gamma}$ rays by a 1-in. \ifmmode\times\else\texttimes\fi{} 1-in. NaI(Tl) scintillator. Both scintillators were viewed by RCA-8575 photomultipliers whose dynodes were gated to avoid damage during the beam-on time. The value obtained for the mean lifetime was (7.72\ifmmode\pm\else\textpm\fi{}0.26) \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}9}$ sec which corresponds to a $B(E2, {{6}_{1}}^{+}\ensuremath{\rightarrow}{{4}_{1}}^{+})=(6.40\ifmmode\pm\else\textpm\fi{}0.22){e}^{2}$ ${\mathrm{F}}^{4}$. This value may be compared with the (${2}^{+}$) 1.52-MeV to (${0}^{+}$) ground-state decay, assuming both represent transitions between ${{f}_{\frac{7}{2}}}^{2}$ states. The result indicates that collective admixtures in the ${0}_{1}^{+}$ and ${2}_{1}^{+}$ wave functions are larger than for the ${6}_{1}^{+}$ and ${4}_{1}^{+}$ levels. This can be understood quantitatively if one considers the low-lying levels in $^{42}\mathrm{Ca}$ to be shell-model states mixed with rotational states arising from 4p-2h excitation according to a recent model by Gerace and Green.