Study ofpf-shell nuclei with a symmetry conserving generator coordinate method

Bands based on the 0⁺ ground state and the first excited 0⁺ pairing vibrational state of⁴⁸Ti,⁵²Cr and⁵⁶Fe are studied with the generator coordinate method. The generating wave functions for each value of the angular momentumJ are angular momentum and particle number projected selfconsistent Hartree-Fock-Bogoliubov states where the constrained amount of pairing correlations serves as the generator coordinate. The interaction is given by reaction matrix elements derived from the Hamada-Johnston force. The basis includes the four lowest oscillator shells. The excitation energies of the pairing vibrational states can be reproduced fairly well by the present choice of the generating wave functions, whereas the ground band is not much improved compared to projected Hartree-Bogoliubov calculations. We find that the strength of the pairing correlations in the 0⁺ and 2⁺ states of the ground state and the pairing vibrational bands can be related to data of two-particle transfer reactions. The angular momentum dependence of the pairing correlations and of the moments of inertia are studied. The results show that for a strongly paired ground state the ground state band and the pairing vibrational band intersect. This may produce in the yrast band the anomaly of the moment of inertia known from rare earth nuclei.     

The generator coordinate method as an approximation to the shell model: A comparison on 24Ne

The applicability of the generator coordinate method as an approximation to the complete shell model diagonalization is tested in the sd shell on ²⁴Ne. We use the quadrupole moment and the pairing energy as generator coordinates and choose as generating function a constrained Hartree-Bogoliubov solution projected on the subspace of good angular momentum and good proton and neutron number. The spectra obtained by the generator coordinate method and the complete diagonalization show good overall agreement. Also we draw some conclusions about the nature of some low-lying states in ²⁴Ne, interpreted in terms of vibrations.     

On the Stability of the Kerr Metric

The reduced (in the angular coordinate ϕ) wave equation and Klein–Gordon equation are considered on a Kerr background and in the framework of C ⁰-semigroup theory. Each equation is shown to have a well-posed initial value problem, i.e., to have a unique solution depending continuously on the data. Further, it is shown that the spectrum of the semigroup's generator coincides with the spectrum of an operator polynomial whose coefficients can be read off from the equation. In this way the problem of deciding stability is reduced to a spectral problem and a mathematical basis is provided for mode considerations. For the wave equation it is shown that the resolvent of the semigroup's generator and the corresponding Green's functions can be computed using spheroidal functions. It is to be expected that, analogous to the case of a Schwarzschild background, the quasinormal frequencies of the Kerr black hole appear as resonances, i.e., poles of the analytic continuation of this resolvent. Finally, stability of the solutions of the reduced Klein–Gordon equation is proven for large enough masses. Received: 28 August 2000 / Accepted: 4 April 2001     

A generator coordinate technique for arbitrary cluster widths

A generator coordinate technique for translationally invariant cluster wave functions with arbitrary cluster-widths is applied to the calculation of the α-⁴⁰Ca system.     

Microscopic study of heavy-particle transfer reactions: The (16O, α) reactions on 16O and 28Si

The transfer reactions ²⁸Si(¹⁶O, α)⁴⁰Ca and ¹⁶O(¹⁶O, α)²⁸Si have been studied within the framework of the two-channel generator coordinate method. The geometry of the reactions has been discussed in terms of the cross-channel overlap kernels of the generating functions. A clear connection between maxima in the reaction rate and entrance channel resonances has been established. A two-step reaction mechanism involving quasimolecular resonances of inelastic ¹⁶O-¹⁶O channels is discussed as a possible explanation for narrow structures seen in the ¹⁶O(¹⁶O, α)²⁸Si reaction cross section.     

On the connection between the hyperspherical harmonic and generator coordinate methods for monopole vibrations

It is shown that, when the generating functions are Slater determinants of harmonic oscillator wave functions, the generator coordinate method for monopole vibrations is identical to the first approximation of the hyperspherical harmonic method.     

Generator coordinate calculations of monopole and quadrupole vibrations with Skyrme's interaction

Skyrme's interaction is shown to lead to significant simplifications in generator coordinate calculations. As an illustration monopole and quadrupole modes are calculated in ⁴He, ¹⁶O and ⁴⁰Ca using pure oscillator wave functions.     

A particle number projection study with the generator coordinate method

The generator coordinate method (GCM) is used in order to describe pairing vibrations. The generating wave functions are particle-number-projected BCS-wave functions. With the conventional pairing Hamiltonian the method is tested in some simple models and applied also to the even Nickel isotopes. The results obtained can be regarded as encouraging ones.     

A study of even-parity states of the nuclei 19F, 21Ne and 23Na with the generator coordinate method and with mixing of projected Hartree-Fock determinants in comparison with complete diagonalization

The energies and electromagnetic properties of the even-parity states of the nuclei ¹⁹F, ²¹Ne and ²³Na are calculated with the generator coordinate method and with mixing of projected HartreeFock determinants, using the Kuo and the Preedom-Wildenthal two-body interactions. The two parameters β and ? of Nilsson's potential are chosen as generator coordinates and the subspace is enlarged with a few different configurations, i.e. one for A = 19 and three for A = 21 and 23 nuclei. Special care is taken in choosing the appropriate Hartree-Fock solution corresponding to possible occupied single-particle states. For both methods six basic functions are enough to obtain a good approximation to complete diagonalization within the same model space. The collective features of these nuclei are also investigated.     

The generator coordinate method as an approximation to the exact diagonalization in a given model space

The low-lying states of the nuclei ¹⁶O and ²⁰Ne are calculated with the generator coordinate method. For ¹⁶O two generator coordinates are used, which are related to the quadrupole and to the octupole deformation of the single-particle potential. For ²⁰Ne the generator coordinates are related to the parameters β and D of the Nilsson potential. In both cases six pairs of coordinate values are chosen. The results are compared with those of the complete diagonalization. The agreement is, for both cases, better than 0.5 MeV for most low-lying states.     

Microscopic analysis of the 12C(α, γ)16O reaction

Electric transition probabilities in the ¹⁶O spectrum, and the ¹²C(α, γ_o,3¹⁶O capture cross sections are calculated with antisymmetric wave functions by the generator coordinate method. The influence of bound states on radiative capture is shown to be automatically included in the model. The reduced α-widths of the ¹⁶O bound states are discussed, and compared with previous theoretical and experimental estimates. The microscopic E2 capture cross sections to the O⁺₁ and 2₁⁺ states yield an astrophysical S-factor of 0.09 MeV · b at 0.3 MeV. An attempt to treat the El multipolarity by relaxing the long-wavelength approximation leads to a large underestimation of the γ-widths. Adopting the experimental γ- width and the theoretical reduced α-width of the 1₁^? state provides s_E1 = 0.30 MeV · b at 0.3 MeV.     

Pairing vibrational and isospin rotational states in a particle number and isospin projected generator coordinate method

Pairing vibrational and isospin rotational states are described in different approximations based on particle number and isospin projected, proton-proton, neutron-neutron and proton-neutron pairing wave functions and on the generator coordinate method (GCM). The investigations are performed in models for which an exact group theoretical solution exists. It turns out that a particle number and isospin projection is essential to yield a good approximation to the ground state or isospin yrast state energies. For strong pairing correlations (pairing force constant equal to the single-particle level distance) isospin cranking (-ωT_x) yields with particle number projected pairing wave function also good agreement with the exact energies. GCM wave functions generated by particle number and isospin projected BCS functions with different amounts of pairing correlations yield for the lowest T = 0 and T = 2 states energies which are practically indistinguishable from the exact solutions. But even the second and third lowest energies of charge-symmetric states are still very reliable. Thus we conclude that also in realistic cases isospin rotational and pairing vibrational states may be described in the framework of the GCM method with isospin and particle number projected generating wave functions.     

Microscopic description of collective motion in pf shell nuclei

Rotational and vibrational excitations in pf shell nuclei are studied by means of the generator coordinate method. The generator coordinates are the pairing energies and the quadrupole moments of constrained Hartree-Bogoliubov states, projected onto good angular momenta and particle numbers. The Kuo interaction and the one modified by McGrory are used. The vibrational character of the yrast energies appears to be produced by mixing prolate and oblate wave functions. Pairing correlations are essential for this mixing. In contrast to the yrast states the excitation energies of the higher states depend strongly on the interaction used. They show good agreement with experiment, particularly in the case of ⁴⁸Ti with the Kuo force. The calculated B(E2) values exhibit a rotational band structure in general, even if the energies look more vibrational. The force dependence of the excitation energies can qualitatively be understood by inspection of the intrinsic energy surface.     

Calculation of electron hydrogen scattering resonances by the coordinate-rotation method

Energies and widths of severalS andP type resonance ine-H scattering are determined in a calculation involving only square-integrable functions of the integral transform type obtained by the generator coordinate method. Several resonances are apparently calculated for the first time. Advantages and disadvantages of the coordinate-rotation method are critically examined and discussed. A restricted perturbation scheme for the calculation of resonances is proposed and applied to the first¹S resonance of the He⁺ —e system.     

Interpreting vibrationally resolved spectra of molecular dications (doubly positively charged molecules): HCl2+

Vibrationally resolved spectra of HCl²⁺ appear to show five vibrational levels for the X³Σ^? ground electronic state, whereas calculations of vibrational levels supported by ab initio potential energy curves have been able to locate only three vibrational levels below the barrier; this discrepancy is resolved by considering vibrational states that the potential function supports in the continuum above the barrier maximum. A low resolution spectrum produced from first principles is compared with a spectrum obtained with threshold photoelectrons in coincidence (TPEsCO) spectroscopy, with agreement sufficient to suggest that care must be taken in the inversion of vibrational spectroscopic data for molecular dications to avoid generating potential functions that are too strongly bound.     

Generating power of a thermoelectric generator under periodically alternating temperature gradients

The thermo-emf ΔV and thermoelectric current ΔI generated by imposing a temperature gradient alternating at a period of T on a thermoelectric (TE) generator were measured as a function of t, where t is the lapsed time and 1/T was varied from 0 to 1/30 s^-1. A TE generator was sandwiched between two Peltier modules connected in series. The alternating temperature gradients were produced by imposing an alternating voltage V on two Peltier modules, where V was varied from 1.0 to 3.7 V. Both ΔV and ΔI generated by the TE generator oscillate at a period of T but their amplitudes tend to increase monotonically with an increase of V. The effective thermo-emf ΔV_eff and current ΔI_eff calculated from ΔV and ΔI increase abruptly with an increase of 1/T and have a local maximum at 1/T=1/120 or 1/240 s^-1. The generating power ΔW_eff(=ΔV_effΔI_eff) tends to increase proportionally with an increase of input power W_input, owing to the increase in the temperature difference. The rate of ΔW_eff to W_input at 1/T=1/240 s^-1 reached approximately 3.2 times as large as that obtained for the steady temperature gradient corresponding to 1/T=0 s^-1. It was thus found that the generating power of the TE generator operating under the temperature gradient alternating at an optimum period is remarkably increased compared to that of a TE generator working under a conventional steady temperature gradient. PACS 72.15.Jf; 84.60.Rb; 85.30De     

What can we learn from the tension between PLANCK and BICEP2 data?

Recently BICEP2 collaboration has announced the detection of the primordial gravitational waves at high confidence level.In light of the results of B-modes power spectrum from BICEP2 and using the basedΛCDM,a constraint on the tensor-to-scalar ratio r=0.20+0.07-0.05(68%C.L.)can be obtained,however,this result is in apparent tension with the limit on standard inflation models from the recent PLANCK measurement,r0.11(95%C.L.).Herein we review the recent progress on the cosmological studies after BICEP2 and discuss on different ways of reconciling the tension between PLANCK and BICEP2 data.We will discuss possible modifications on the standard cosmological model,such as including the running of scalar spectral index or other cosmological parameters correlated with inflationary cosmological parameters,or tilting the primordial power spectrum at large scales by introducing a cut off which can be predicted by bouncing cosmology.We will also comment on another possibility of generating extra B-modes of CMB polarization,namely by a non-zero polarization rotation angle during its transferring from the last scattering surface.