Shell Model Analysis Research Articles

A shell-model analysis of Λ binding energies for the p-shell hypernuclei III. Further analysis and predictions

A simplified phenomenological analysis of the Λ binding energies for the p-shell hypernuclei is given on the basis of the shell-model configuration [(1 s) N 4(1p) N A−5 (1s) Λ ], using the intermediate-coupling formalism set up in Part I, assuming the ΛNN interaction Q 00 0 of central form (as motivated by the double-OPE mechanism), and taking into account the new data recently available about several p 1 2 -shell hypernuclei. The systematics of the mass matrix for p-shell Λ hypernuclei are discussed in some detail. The fit obtained has only moderate strengths for the (ΛN) spin-orbit and (ΛNN) three-body interactions and requires rather little mixing of different nuclear core states into the hypernuclear wavefunctions, except for the special case Λ 8Li. The implications of this fit are spelled out for the well-known hypernuclear species. A discrepancy between the wavefunction calculated here for Λ 8Li and the characteristics of the decay Λ 8Li → ααπ − is emphasized, as is also the need for Λ hypernuclear γ-ray data.

A shell-model analysis of a binding energies for the p-shell hypernuclei. III. Further analysis and predictions

A shell-model analysis of a binding energies for the p-shell hypernuclei. III. Further analysis and predictions

Finite Deformation Analysis of a Thick Spherical Shell Model for the Mechanical Behavior of the Left Ventricle

Finite Deformation Analysis of a Thick Spherical Shell Model for the Mechanical Behavior of the Left Ventricle

Sc45(α,d)Ti47reaction

Using a split-pole magnetic spectrograph, the $^{45}\mathrm{Sc}(\ensuremath{\alpha},d)^{47}\mathrm{Ti}$ reaction was studied at $E(\ensuremath{\alpha})=25$ Mev and with a range of angles from 7\ifmmode^\circ\else\textdegree\fi{} to 50\ifmmode^\circ\else\textdegree\fi{}. These measurements are compared with results we obtained using the $^{45}\mathrm{Sc}(^{3}\mathrm{He},p)^{47}\mathrm{Ti}$ reaction. One result of this comparison is the identification of candidates for states with high spin. The major emphasis of this paper is the study of such states. Two-particle spectroscopic amplitudes are calculated using an ${({f}_{\frac{7}{2}})}^{7}$ model, assuming an inert $^{40}\mathrm{Ca}$ core. Zero-range distorted-wave Born-approximation calculations are then performed with these amplitudes. A comparison between these shell-model calculations and the experimental results indicates that the states at 4.506 and 3.583 MeV have ${J}^{\ensuremath{\pi}}'\mathrm{s}$ of ${19/2}^{\ensuremath{-}}$ and ${17/2}^{\ensuremath{-}}$. A second result of the comparison between the ($\ensuremath{\alpha},d$) and ($^{3}\mathrm{He},p$) reactions concerns the nature of the 3.22-MeV state in $^{47}\mathrm{Ti}$. It is strongly populated with an orbital angular momentum transfer of 0 by the ($^{3}\mathrm{He},p$) reaction and only weakly populated by the ($\ensuremath{\alpha},d$) reaction, suggesting that it is formed by the transfer of a neutron-proton pair with $S=0$, $T=1$. This is similar to the transfer forming the state at 7.346 MeV, the analog of the $^{47}\mathrm{Sc}$ ground state.NUCLEAR REACTIONS $^{45}\mathrm{Sc}(\ensuremath{\alpha},d)$, $E=25$ MeV; measured $\ensuremath{\sigma}(\ensuremath{\theta})$. $^{47}\mathrm{Ti}$ deduced levels, $J$, $\ensuremath{\pi}$. Shell model and DWBA analyses.

Core polarization and exchange effects in the inelastic scattering of tritons from92,94,96Zr

A microscopic analysis of the inelastic scattering of 20 MeV tritons from 92Zr, 94Zr, and 96Zr is carried out which includes core polarization effects and exchange effects. A semi-realistic Gaussian effective interaction between the incident triton and the target nucleon is used. Inclusion of the core polarization contribution is found to decrease the necessary direct interaction strength of the collective states by at least a factor of two from those predicted by the shell-model analysis. An estimate of the relative importance of exchange effects is made using a simple pseudopotential. Exchange terms are found to be important for most of the single-particle shell-model transitions studied.

Shell-model analysis of helion and triton inelastic scattering from90Zr including core polarization effects

A microscopic analysis of the inelastic scattering of helions and tritons from90Zr is performed using the shell-model wavefunctions and phenomenological effective interactions. Effects of the core polarization are studied assuming a collective model for the core of closed shells. Core coupling parametersA L , are obtained from the measured electromagnetic transition rate,B(EL). For those transitions for which no meauredB(EL) values are available,A L values which fit the data have been extracted. The radial forms of both the Gaussian and Yukawa interactions were studied. Effects of the non-locality of the bound states on the predicted cross sections are found to be small. As in the case of proton inelastic scattering, it is found that the valence (direct) contributions are small and the transitions are dominated by core polarization. The strength of the effective interaction required to fit the data is reduced by a factor of two or more when the core polarization contributions are included.

Shell-model analysis of pairing excitations in the region 52 ≦ A ≦ 60

A shell-model diagonalization of a T = 1 residual pairing interaction is performed in the vicinity of 56Ni for J π = 0 + states. Energies and differential cross sections for two-body transfer processes are compared with experimental values and with the harmonic predictions. Special attention is paid to those reactions which are sensitive to the existence of a phase transition between normal and superconducting systems. Furthermore, it is shown that the assumption of a harmonic non-adiabatic addition phonon explains systematically a number of weak transitions. A list of spectroscopic amplitudes for relevant (and so far unreported) cross sections is included.

A shell-model analysis of Λ binding energies for the p-shell hypernuclei II. Numerical Fitting, Interpretation, and Hypernuclear Predictions

A phenomenological analysis of the Λ binding energies known for the p-shell hyper nuclei is given in terms of their representation by the shell-model configuration {(1s) N 4(1p) N A−5 (1s) Λ}, using the formalism set up in Part I. The ΛN interactions are described quite generally by five free parameters, and the inclusion of ΛNN interactions with the forms arising through the double-OPE mechanism requires five further parameters. Various constraints motivated by theoretical arguments concerning the nature of ΛN and ΛNN interactions are imposed on these parameters, and best fits are then obtained for them following a minimization procedure. A good and economical fit to these data is obtained for a parameter set in good accord with our theoretical notions about these interactions (although the fit otherwise most attractive does require ΛN spinorbit forces with sign opposite those known for the NN system). On the basis of these fits to the B Λ data, a discussion is given of the energy-level spectra for the p-shell hypernuclei, of isomeric states for Λ 7He ∗ and Λ 7Li ∗, and of the resulting predictions for the properties of the Λ hypernuclei corresponding to core nuclei assigned to the p 1 2 shell.

A Modified Valence Force Field Approach to Lattice Dynamics of Trigonal Selenium

A modified valence-force-field (VFF) model is applied to the lattice dynamics of trigonal selenium. Five adjustable parameters, four valence-force-constants ( K r , K θ , K r r ' , K r θ ) within intra-chain couplings and one inter-chain force constant ( K R ), are determined from the observed six elastic constants by mans of the least squares method. By using these fore constants, comparable fits to the \(\varGamma\)-point optical frequencies for the A 1 , E 1 and E 2 modes are obtained except that for the A 2 mode. However, the last can be also accounted for well if the contribution of about 50 cm -1 from the long-range electrostatic interaction is taken into account. The estimation of the above contribution seems reasonable from the view-point of the shell-model analysis due to Chen and Zallen [Phys. Rev. 173 (1968) 833]. In conclusion, the modified VFF employed here reflects the bonding in trigonal Se fairly well.

Shell model analysis of elastic e −- 4He scattering

A modified square well single particle potential is found to give as good a fit to the elastic e −- 4He scattering data as the harmonic oscillator model with short range nucleon-nucleon correlations;

Magnetization distribution of the 7Li nucleus as obtained from electron scattering through 180°. The electric quadrupole moment of 7Li

Cross sections for scattering of electrons from the ground state doublet of 7Li have been measured in the energy range of 25 to 90 MeV with an energy resolution better than 200 keV. From the cross sections for transverse elastic scattering measured at 180° the rms radius of the ground state magnetization density distribution has been determined: 〈r 2〉 M 1 2 = 2.98 ± 0.05 fm . The value of the reduced transition probability for M1 excitation of the first excited level has been determined from the ratio of transverse inelastic and elastic cross sections at 180°. The result [ B(M1, ↑; 478 keV)=2.50±0.12 μ N 2] agrees very well with the result of lifetime measurements. The cross sections for longitudinal (C2) excitation of this level have been extracted from scattering through angles of 90° to 150°. The measured reduced transition probability [ B(C2, ↑; 478 keV)=7±4 fm 4] is compared with the values obtained by several other experimental methods. In a shell-model analysis our data can be considered to measure the size of the 1p shell. The resulting value ( a 1p=1.90±0.03 fm) is much larger than the one determined from earlier electron scattering experiments, where the 478 keV level was not resolved. A relative measurement of the longitudinal elastic scattering from 7Li and from 6Li showed no isotopic effect on the rms charge radius: 〈r 2〉 7 1 2 /〈r 2〉 6 1 2 =1.001±0.008 . Accepting the measured rms charge radius for 6Li this corresponds to a 1s= a 1p=1.90 fm for 7Li reconfirming the above value. The data are compared with shell-model and projected Hartree-Fock calculations.

A shell-model analysis of Λ binding energies for the p-shell hypernuclei. I. Basic formulas and matrix elements for ΛN and ΛNN forces

A uniform intermediate-coupling shell model is formulated for the p-shell Λ hypernuclei, those belonging to the configuration {(1s) N 4(1p) N Λ−5(1s) Λ}, using NN parameters obtained from previous fits to the corresponding core nuclei. The physical origins of the ΛN forces are discussed, with special attention to the noncentral components, and the characteristics of the three-body double-OPE ΛNN interaction are reviewed briefly. Expressions are derived for all the ΛN and ΛNN matrix elements needed for the discussion of these configurations. Attention is drawn particularly to their qualitative characteristics. For the purpose of orientation, numerical estimates of them are given for all possible types of ΛN forces and for all the constituents of the double-OPE ΛNN interaction, using explicit forms for the radial wave functions, as a basis for the purely phenomenological analysis (now complete) of the data available on the p-shell hypernuclei to be discussed in Part II.

Binding Energies of thep-ShellΛΛHypernuclei

Using the results of a shell-model analysis of the $p$-shell hypernuclei by the authors, the binding energies of the two $\ensuremath{\Lambda}$ particles in the $p$-shell $\ensuremath{\Lambda}\ensuremath{\Lambda}$ hypernuclei are calculated. A consistent account of the binding energies of the two known $\ensuremath{\Lambda}\ensuremath{\Lambda}$ hypernuclei, $_{\ensuremath{\Lambda}\ensuremath{\Lambda}}^{ 6}\mathrm{He}$ and $_{\ensuremath{\Lambda}\ensuremath{\Lambda}}^{10}\mathrm{Be}$ (or $_{\ensuremath{\Lambda}\ensuremath{\Lambda}}^{11}\mathrm{Be}$) is obtained. Close agreement with the results of other investigators who used different methods is also pointed out. The effects of the noncentral potentials in the effective two-body $\ensuremath{\Lambda}\ensuremath{-}N$ interaction are discussed.

Electromagnetic transitions between odd parity states in 40K

Abstract Lifetimes of twelve levels in 40 K up to 2.63 MeV excitation have been measured. Effective moments and charges and configuration mixing are deduced from a shell model analysis of the transition rates between eight odd-parity states with comparatively simple particle-hole structure.

Optical Phonons and Dynamic Charge in Trigonal Se and Te

Lattice vibrations in elemental crystals can possess a first-order electric moment, and thus exhibit one-phonon infrared absorption (reststrahlen), by the mechanism of displacement-induced charge redistribution (dynamic charge). Trigonal Se and Te, with three atoms per unit cell, belong to the simplest class of reststrahlen-displaying elemental crystals. We have performed a group-theoretical, shell-model analysis of the zone-center optical phonons in these monatomic semiconductors using the most general dynamical matrix consistent with crystal symmetry and translational invariance. The dynamic charges associated with the infrared-active modes have been derived, and the effective-charge tensor ($\frac{\ensuremath{\partial}\mathbf{p}}{\ensuremath{\partial}\mathbf{u}}$) constructed. In this model the macroscopic polarization is associated entirely with the shell motion, and the three independent entries appearing in the effective-charge tensor correspond to coupling coefficients connecting acoustical shell displacements with optical core displacements. For the ${A}_{2}$ mode, the chain-twisting core motion induces an axial shell motion responsible for the electric moment so that a vibration with atomic displacements $\ensuremath{\perp}c$ interacts with radiation polarized $\ensuremath{\parallel}c$. For this mode both dynamic charge and restoring force depend upon next-nearest-neighbor interactions between chains, and vanish in the limit of isolated chains. The expressions obtained for the two $E$-mode effective charges provide a means for understanding the small oscillator strength observed for one $E$-mode pair in both Se and Te. The lattice sums entering into the long-range electrostatic interaction matrix have been evaluated numerically for the detailed geometry of both crystals. In addition, a simple central-force model for the short-range interactions is discussed to illustrate some of the more generally derived results.

Inelastic Scattering of 20-MeV Tritons fromZr90

The inelastic scattering of tritons from $^{90}\mathrm{Zr}$ was studied at a beam energy of 20 MeV, and angular distributions of states up to 5.22 MeV of excitation were obtained. The data are compared with distorted-wave calculations, employing both collective and shell-model form factors. Both calculations reproduce the shapes of the experimental angular distributions quite well. Values of deformation parameters obtained from the collective-model form-factor calculation are in good agreement with the results of inelastic scattering experiments employing beams of $^{3}\mathrm{He}$ and $^{4}\mathrm{He}$ ions, but are in poor agreement with the results of proton inelastic scattering measurements. The shell-model analysis assumed an effective two-body interaction between the triton projectile and the target nucleons, and the form-factor calculation employed a Yukawa well with a 1.0-F range. Under these conditions, an effective interaction strength averaging 660 MeV was obtained for the excitation of the well-known pure proton-configuration states. This value is approximately three times the strength obtained in the previous study of proton inelastic scattering from these states. This result supports the assumption that the optical potential has a real well depth approximately three times as great for the triton as for the single nucleon. The strengths of the transitions to the other observed states are also discussed in terms of configurations expected to contribute to the states. Characteristics of the ($t,{t}^{\ensuremath{'}}$) reaction are compared with those of the ($p,{p}^{\ensuremath{'}}$) reaction.

Shell model analysis of the magnetic hyperfine splitting in muonic thallium

The magnetic hyperfine splitting of the ground and first excited state in muonic203Tl and205Tl is calculated on the basis of the shell model with configuration mixing. Fair agreement with the recently measured ground state splitting is found.

Electron scattering and the spectroscopy of pseudo-nuclei

The method of Cohen, Lawson and Soper of comparing a shell-model study and an exact calculation for a two-level model (d- and f-states) is extended to the s-d shell and to the non-degenerate case. In contrast to the case of d-f states, concealed configuration interactions cannot easily occur in the s-d shell. It is shown that inelastic electron scattering provides, in principle, a sensitive test for the purity of the configurations occurring in a shell-model analysis of experimental results.

Energy levels, configuration mixing and proton neutron interaction in the Zr region

A shell model analysis of the interaction between configurations of protons in the 2 p 1 2 and 1 g 9 2 orbits is presented. Matrix elements of the effective interaction are determined which accurately reproduce a larger number of experimental data. These are binding energies and positions of excited states in Zr 90, Nb 91, Mo 92 and Tc 93. The interaction of a 2 d 5 2 neutron with 2 p 1 2 and 1 g 9 2 protons is determined form the levels of Y 90 and Nb 92. This interaction is used to calculate the spectra of nuclei with protons in the 2 P 1 2 −1 g 9 2 shell and neutrons in the 2 d 5/ 2 orbit. Good agreement is obtained between calculated and experimental energies. In particular, the calculated energy of the isomeric state in Mo 93 and the energies in the following cascade agree very well with the experimental energies.

Inelastic Proton Scattering from Nuclei with 28 Neutrons

The inelastic scattering of 17.45-MeV protons from ${\mathrm{Ti}}^{50}$, ${\mathrm{Cr}}^{52}$, ${\mathrm{Fe}}^{54}$, and ${\mathrm{V}}^{51}$ has been measured. Levels up to 5-MeV excitation were studied and the 30\ifmmode^\circ\else\textdegree\fi{}-90\ifmmode^\circ\else\textdegree\fi{} differential cross sections were measured for most of the levels. Spins and parities were assigned on the basis of the angular distributions and agree well with other experiments. The strength of the various inelastic cross sections were studied using a direct reaction theory with distorted waves. Both the collective model and the shell model of nuclear structure were used to describe the nuclear states. In describing the excitation of the strongly excited levels of the even-$A$ nuclei, the collective model picture yielded a strength parameter, ${\ensuremath{\beta}}_{l}$, which agreed within experimental error with the ${\ensuremath{\beta}}_{l}$ value extracted from Coulomb excitation experiments. The shell-model formulation described this data as well using a two-body Gaussian potential of finite range and depth of 45 MeV. The analysis of ${\mathrm{V}}^{51}$, however, was better explained using a shell-model analysis.