Surface Delta Interaction Research Articles

A shell model of the cobalt isotopes

Binding energies, excitation energies, and single nucleon transfer spectroscopic factors of the 55 ≤ A ≤ 60 cobalt isotopes have been calculated in a 1f–2p shell configuration space assuming a 56Ni core. Single-particle energies were determined empirically and two-body matrix elements were computed with the modified surface delta interaction (MSDI). The observed properties of the low-lying states are reasonably well reproduced but some low-spin states observed above 1.3 MeV are outside the model space. The theory predicts quasi-rotational bands of high-spin states in both odd- and even-A cobalt isotopes.

Shell-model calculations for50Ti and51V

Shell-model calculations provide valuable information regarding the structure of properties of nuclear states. One attempts to get the maximum information with a simple effective interaction. The surface delta interaction (SDI) is a simple effective interaction and it has been used in calculations in many regions of the periodic table. It is rather common in SDI calculations to assume that the radial integrals are equal at the nuclear surface. It is shown that radial integrals should be taken into account explicity in such calculations.

Semimicroscopic description of the odd iodine nuclei in the mass region123≤A≤133

A systematic study of the low-energy properties of odd-mass I nuclei is performed in terms of the Alaga model. Previous theoretical works are updated, according to the present level of experimental information, and extended to lighter isotopes. The residual interaction among the valence protons is approximated by both the pairing force and the surface delta interaction. We conclude that the refinements introduced by the last interaction are of little importance in the description of low-energy states. Excitation energies, one-body reaction amplitudes, dipole and quadrupole moments, and B (M1) and B (E2) values are calculated and compared with the corresponding experimental data. Also, a few allowed ..beta.. transitions are briefly discussed.

Shell-model calculations on Ni and Cu isotopes

Binding energies, excitation energies and spectroscopic factors have been calculated for57–67Ni and58–68Cu in an unrestricted (2p3/2, lf5/2,2p1/2) shell-model space. The effective two-body matrix elements are obtained from the modified surface delta interaction (MSDI) and from a least-squares fit to experimental binding and excitation energies (ASDI). The average deviation between about 100 experimental and calculated energies is 0.14MeV for MSDI and 0.08 MeV for ASDI. Excitation energies of high-spin states are given also. Spectroscopic factors have been calculated for all single-nucleon transfer reactions on stable Ni or Cu targets leading to Ni or Cu isotopes. For spectroscopic factors larger than 0.4 the average deviation between theory and experiment is about 30%. The experimentally observed and calculated spectroscopic strengths are compared by using sum rules and are found to be consistent. An extensive compilation has been made of experimental data on energies,Jπ assignments and spectroscopic factors.

A surface-delta description of analyzing power measurements for collective states of Ni and Zn isotopes

Analyzing powers and differential cross sections for the inelastic scattering of 15 MeV polarized protons to the stronger collective states of 58, 60Ni and 64Zn have been measured. The data were compared to detailed microscopic reaction calculations using a central plus spin-dependent effective interaction. The nuclear structure wave functions used were obtained from a quasiparticle random-phase appoximation calculation using a spin-dependent surface delta interaction and a basis set containing both neutron and proton configurations. The overall quality of the fits strongly suggests that the surface-delta wave functions provide a good representation for spherical nuclei in this intermediate mass range. The effects of exchange and further modifications to the microscopic effective interaction are discussed.

The calculation of logft values and spectroscopic factors for nuclei in the mass range 10?15

Wave functions obtained in previous shell-model calculations [1, 2] for states of nuclei with massesA=10–15 are used to calculate logft values and spectroscopic factors. These wave functions, which were based on a modified surface delta two-body interaction, comprise the following active shell-model spaces: the 1p3/2 and 1p1/2 orbitals for mass 10–14 nuclei and restricted combinations of the 1p1/2, 2s1/2, 1d5/2 and 1d3/2 orbitals for mass 15 nuclei, leading to four different calculations in the latter case. The calculated results support evidence that the modified surface delta interaction is a valid approximation to the effective interaction in light nuclei.

Validity of pseudo SU(3) for rare earth nuclei

The validity of pseudo SU(3) for nuclei in the rare earth region is investigated by diagonalizing the surface delta interaction in the complete basis of three and four particles in the N ̃ = 3 shell. Both SU(10) and SU(3) fail to provide a reasonable truncation scheme.

Shell-model calculations with an adjusted surface-delta interaction (I) application to spectroscopic factors inA=24?28 nuclei

Shell-model calculations have been performed for positive-parity levels of theA=24–28,T≦3/2 nuclei in a 1d5/22s1/21d3/2 configuration space. The space, for each (A, J, T) set, was restricted such that only mixing of less than about 300 low-lying pure configurations was allowed. An effective interaction (MSDI) was used of which the parameters have been determined by a fit of the calculated energies to the experimental values. This resulted in an averaged deviation of 0.52 MeV for 67 fitted levels. Spectroscopic factors have been calculated with the derived wave functions. In order to obtain better agreement for the energies, all of the 63 two-body matrix elements and three single-particle energies were adjusted systematically. The wave functions derived from this adjusted set of matrix elements denoted by ASDI yield considerably better spectroscopic factors. For comparison, the spectroscopic factors have also been calculated in a 1d5/22s1/2 model space. The omission of the 1d3/2 subshell was found to result in a substantially poorer agreement for the spectroscopic factors for 1d5/2 and 2s1/2 transfer.

Shell-model calculations for the 0f7/2proton-shell nuclei

The (-1)A energy states of 50Ti, 51V, 52Cr, 53Mn, 54Fe and 55Co are calculated within the spherical shell-model framework, assuming a 48Ca closed core and the (Z-20) valence protons to occupy orbitals in the complete 0f-1p major shell. Both the schematic surface delta interaction (SDI) and the realistic Kuo-Brown (KB) force are used as effective two-body interactions, the interaction strength in the first case being smoothly decreased with increasing mass number. At least 40 experimental levels are well accounted for and the resultant wavefunctions are used to calculate E2 transition rates. Although the two interactions render very similar results, the KB spectra are generally too expanded and the B(E2) values too small. The failure of the present calculations to account for the low-lying 0+ excited states in the even-mass nuclei points to the necessity to allow for 0f7/2 neutron-shell excitations.

The doubly odd nucleus 144Pm : (II). Interpretation

Results from the proton transfer reactions, 143Nd( 3He, d) and 143Nd(α, t), and γ-ray studies have been used to characterize the levels in 144Pm. The spins, ( 3He, d) (α, t) cross section ratios and γ-ray branching ratios for the states below 600 keV can be interpreted consistently in terms of the mixed π2 d 5 2 ν2 f 7 2 and π1 g 7 2 ν2 f 7 2 shell-model configurations. The γ-ray intensity ratios measured in the 141Pr(α, nγ) and 144Nd(p, nγ) reactions exhibit a strong and smooth dependence on level spin. Theoretical calculations employing a residual surface delta interaction are in reasonable agreement with the twelve observed low-lying negative parity states. The eight positive parity states between 841 and 1451 keV are presumed to arise from the π1 h 11 2 ν2 f 5 2 configuration.

Application of the boson-expansion method to even Se and Ru isotopes

The expansion coefficients of a fourth-order collective Hamiltonian for the low-lying quadrupole vibrations are derived from the microscopic fermion Hamilton operator by a modified Marumori boson-expansion method. Their dependence on the phonon structure, on the parameters of the two-body (surface delta) interaction and on the single-particle energies is numerically investigated. For the isotopes $sup 76$Se, $sup 78$Se, and $sup 100$Ru, $sup 102$Ru the results are compared with coefficients that are obtained from phenomenological fits to low-lying levels. Quadrupole moments and B (E2) values are calculated in lowest order.

Inelastic Proton Scattering as a Test of Surface-Delta Wave Functions

Analyzing power and differential cross sections for the inelastic scattering of protons to the collective states of $^{58,60}\mathrm{Ni}$ and $^{64}\mathrm{Zn}$ have been measured with 15-MeV polarized protons. The success in fitting the experimental data with a microscopic reaction analysis strikingly points out the usefulness of nuclear-structure calculations with a spin-dependent surface-delta interaction.

Semi-decoupled bands in the even Hg isotopes

Abstract The odd-parity yrast states of the isotopes 190–200 Hg are studied in a model of two quasiparticles coupled to an oblate rotor and interacting by a surface delta interaction. The experimental energy spectra and the enhancement of E2 transition probabilities are well explained by the model. The pure Coriolis coupling problem (i.e., the problem obtained when the residual interaction is suppressed) is investigated in detail. It is shown that the main effect of the Coriolis force is to decouple the high-spin quasiparticle originating in the neutron 1 i 13 2 subshell from the system of the rotor and the low-spin quasiparticle. We study the coupling of a particle to an axially symmetric rotor, which has an intrinsic spin along its symmetry axis, and derive a good approximate solution to the problem.

Shell-model calculations in A = 23−28 nuclei

Energy levels, electromagnetic transition rates and multipole moments have been calculated for positive parity states in A = 23−28 nuclei in an untruncated ld 5 2 2 s 1 2 . configuration space. The results obtained by considering all two-body matrix elements as free parameters (FP) are compared with those obtained with matrix elements derived from the modified surface-delta interaction (MSDI), as well as with matrix elements from a realistic Hamada-Johnston interaction (HJ). The binding and excitation energies as well as strong B(E2) values calculated with FP and MSDI are found to be rather similar and in good agreement with experiment, in contrast to the B(M1) values. Poor agreement with experiment is obtained for HJ matrix elements.

A gamma ray study of40K (using37Cl( ,n) and40Ar(p,n))

The electromagnetic decay scheme of 40K below an excitation of 3.2 MeV has been measured using the 37Cl( alpha ,n)40K reaction in conjunction with both a Si(Li)-Ge(Li) and a Ge(Li)-NaI(Tl) coincidence system. Gamma ray angular distribution and angular correlation measurements were also made using both the 40Ar(p,n)40K and 37Cl( alpha ,n)40K reactions. Four unique spin assignments were made to levels at 2398 keV (4-), 2419 keV (2-), 2787 keV (3) and 3147 keV (1). Spin restrictions were placed on other levels, and both branching ratios and multipole mixing ratios determined for several transitions. A shell model calculation using the effective surface delta interaction was performed. The predictions, in good agreement with the present experimental results, showed that the low lying states of positive parity are well described by two particle two hole excitations involving only the 1d3/2 and 1f7/2 orbits.

Binding energies in (1d, 2s) shell nuclei

Binding energies of nuclei throughout the (1d, 2s) shell were calculated with a modified surface delta interaction. The full energy matrices within the complete space of all configurations containing 1 d 5 2 , 2 s 1 2 and 1 d 3 2 protons and neutrons were diagonalized. In spite of the crudeness of the interaction used, fair agreement is obtained between calculated and experimental energies. In particular, the calculated curve of α-particle separation energies shows kinks closely following the experimental ones.

Étude des isotopes du plomb de masse 200, 198, 196 et 194 formés par réaction (ions lourds, xn)

The low-lying excited states of 194, 196, 198, 200Pb have been studied through the de-excitation γ-rays following the reactions Os( 12C, 4n), W( 16O, 4n) and Re( 14N, 5n). Prompt and delayed spectra, γ-γ coincidences, γ-ray angular distributions and γ-ray time distributions have been obtained. Half-lives were measured for the 12 + level in the four isotopes, for the 9 − level in 194, 196, 200Pb, for the 7− level in 200Pb and for the 5− level in 196, 198Pb. The results of microscopic calculations, performed with either a Gaussian interaction or a surface delta interaction, are compared to the experimental level schemes and transition probabilities, and discussed.

Shell model calculations on natural parity states in 10≦ A≦14 nuclei

Shell model calculations of natural parity states in the 10≦A≦14 mass region have been performed by assuming an inert4He core with the residual interaction in the 1p shell only. The modified surface delta interaction (MSDI) has been used as an effective two-body interaction. The MSDI parameters as well as the single-particle binding energies have been deduced from a least-squares fit to experimentally known levels in, firstly, the seperate10B,11B-C,12C,13C-N and14N nuclei, and secondly, the whole mass region 10≦A≦14. Multipole moments for ground states and M1 and E2 radiative widths for excited states have been calculated with the resultant wave functions.

États excités des isotopes de plomb de masse 202, 200 et 198

The low-lying excited levels of 198, 200, 202Pb have been studied by observing γ- and β-rays and γ−γ coincidences in bismuth decay. Decay schemes have been proposed for these nuclei. Half-lives were determined for the 9 −, 7 −, 5 − and 4 + levels in 200Pb, and for the 5 − level in 198Pb. Microscopic calculations have been made with Gaussian and surface delta interactions to account for level schemes. The results are discussed.

On the transition from shell structure to collective behavior: A simplified shell-model study

To study the feasibility of carrying out shell-model calculations in nuclei with active protons and neutrons in different major shells, the following simple idealized model has been studied: (i) Proton and neutron configurations are chosen to be ( f 5 2 p 3 2 p 1 2 ) n p and ( g 7 2 d 5 2 d 3 2 s 1 2 ) n n , so that results for the separate proton and neutron basis states to be used in any approximation scheme can be compared with the results for exact shell-model calculations, (ii) The proton and neutron single-particle energies for these active shells are separately taken to be degenerate. (iii) The two-body interaction is approximated by the simple surface delta interaction (SDI). To effect the severe truncation of the full shell-model space needed to make such a shell-model study possible the separate proton and neutron parts of the shell-model basis are built from a superposition of the favored pair states of the SDI (with J ≠ 0, as well as J = 0). In the neutron configuration ( g 7 2 d 5 2 d 3 2 s 1 2 ) n n = 4 , for example only three of the 94 shell-model states with J n = 2 are retained in the truncation scheme. In this highly truncated basis both the energies and the strong B(E k) values for the transitions from these states to similar favored states with other J-values are within a few percent (or better) of the results of exact shell-model calculations. A truncation of the shell-model space based on such superpositions of favored pair states leads to a manageable shell-model basis (dimensions ≦ 200). (a) The number of states in the separate proton and neutron parts of the basis are small enough (8–13 for the proton space, 15–30 for the neutron space). They are also the key states in the following sense (b) They include the low-lying energy eigenstates of the separate p-p and n-n parts of the interaction (c) They contain most of the collective coherence of the separate proton and neutron configurations. (d) The matrix elements of the n-p part of the interaction between the favored states are in general very large compared with the matrix elements between a favored and an excluded state. The latter effect is studied from several aspects, in particular in terms of sum rules for the matrix elements of the surface multipole operators from which the n-p part of the SDI is built. For most of the low-lying favored states the sum over all favored states gives more than 90 % of the total sum rule for the squares of matrix elements of the surface multipole operators. The results of shell-model calculations in this truncation scheme, with n p = 4 or 6, and n n = 4, show many of the features of a quadrupole vibrational spectrum. The presence and exact nature of a 0 + member of the 0 +, 2 +, 4 + “two-phonon triplet” is dependent on the inclusion of the key favored states with seniorities of 6.