Low-lying nuclear deformed states and angular momentum projection

A fast computational method for approximate angular momentum projection is proposed for low-lying deformed nuclear states. The accuracy of the method is demonstrated by calculating the low-lying states of the ground-state band and the two 0⁺ excited bands in ¹⁶⁸Er with good experimental comparison.     

Options for angular momentum projection

A polynomial expansion of Löwdins projection operator is found to be suitable for state by state angular momentum projection using the techniques of the Lanczos algorithm. It does not seem suitable for approximations in the calculation of the projected states. For this case a variational derivation of a possible scheme is presented.     

A study of γ-decay modes from high angular momentum nuclear states by high order multiplicity measurements

Reactions between ¹⁸O and ¹⁶O ions, with energies well above the Coulomb barrier, and ^{148, 150}Nd target nuclei were used to study the cascades of γ-rays in the residual nuclei ^{161, 162}Er. A multi-counter setup was used in which the γ-rays were detected by a Ge(Li) counter coupled in coincidence with up to 9 NaI(Tl) counters. The γ-multiplicity has been studied for γ-rays cascading through states above and along the yrast line. Higher moments of the multiplicity distributions (shape parameters) were deduced from the data. These moments allow the construction of the spin distribution for the entry states. The difference between these spin distributions from reactions induced by ¹⁶O and ¹⁸O ions was studied. Also the fusion cross section was measured for the two systems as a function of bombarding energy and compared to the multiplicity results at two different excitation energies (E_x = 49.5 and 56.2 MeV). A subtraction technique has been applied in the data analysis with the intention to study the decay of a selected part of the high-spin region, and the results show besides the 4n channel an unexpected large contribution of the 5n channel from this region. The sidefeeding patterns of the multiplicity distributions are extracted and show a significant difference for the ¹⁶O and ¹⁸O induced reactions, especially at the lowest bombarding energies. The results are compared with statistical model calculations.     

Fission barriers of Pb nuclei at high angular momentum

Fission and evaporation residue excitation functions have been measured after the nuclei ^{192, 195, 198, 200}Pb were formed by the fusion of ^{28, 30}Si with ^{164, 167, 170}Er. The fission probabilities extracted were fitted using the rotating liquid drop/statistical model codes ORNL ALICE and MBII. The range of values of the mass asymmetry, $\text{(N?Z)}\text{A}$, of the fissioning systems allows some restrictions to be placed on the value of the surface asymmetry parameter K_s, used in the liquid drop model, despite the many uncertainties and approximations in the data analysis.     

Effects of particle evaporation on the angular momentum of the emitting nucleus for deep inelastic and compound nuclear reactions

A model is developed which allows one to calculate analytically the angular momentum removed, and the angular momentum misalignment created by the evaporation of light particles from an excited nucleus. The mass, temperature and angular momentum of the emitting nucleus are explicitly considered. The formalism applies equally well to heavy-ion and compound nuclear reactions.     

Bombarding energy dependence of angular momentum transfer in deep inelastic collisions

Measurements of first and second moments of γ-ray multiplicity distributions from deep inelastic collisions of ⁸⁶Kr + ¹⁵⁴Sm are reported. A global systematics of the angular momentum distributions from deep inelastic reactions with projectile masses ? 40 is presented. The average angular momentum is found to depend linearly on the incident channel average angular momentum, while no simple systematics for the second moment appears obvious. In order to illuminate the question whether the angular momentum transfer process reaches statistical equilibrium in deep inelastic collisions, numerical calculations have been performed on two models: a two-sphere classical model including the collective modes of twisting, bending, wriggling and tilting, and a statistical equilibrium Fermi-gas model. The two-sphere classical model is not able to account for the observed second moments, and neither does the Fermi-gas model give an explanation of the deep inelastic multiplicity data.     

The use of an angular momentum constraint in calculating rotational spectra

A Hartree-Fock method with a constraint on J², the square of the angular momentum, is proposed and applied to simple examples with encouraging results.     

On a RPA method with angular momentum projection

In view of projecting the angular momentum eigenstates out of the intrinsic RPA states, we first study linear relations between deformed and spherical phonons. By use of these relations, the deformed RPA Hamiltonian can be expressed in terms of the spherical phonons in the form of a displaced harmonic oscillator. The angular momentum projection can be applied to an intrinsic RPA state determined by the reduced Hamiltonian, and the correspondence with the macroscopic deformed phonon model of Lipaset al. can be done at this stage. Two important parameters, the constant term in the linear relations and the strength of the spherical phonon Hamiltonian, are evaluated and compared with the best-fit values obtained by Lipaset al.     

Study of continuum gamma rays following low angular momentum reactions

The continuum γ-rays following the reaction ¹⁶⁶Er(α, 2n)¹⁶⁸Yb have been studied at bombarding energies of 21, 27 and 36 MeV. The shape of the statistical γ-rays below 1 MeV was deduced. A low energy bump at ～300 keV was observed. This may arise from transitions along bands at and above the pairing gap.     

Numerical aspects of angular momentum projection for rotational nuclei

The question of complete and exact angular momentum projection from intrinsic Slater determinant representations is studied for the example ¹⁶²Er as a pilot case for the treatment of rotational nuclei. Adequate approximation schemes can be formulated.     

Exact angular momentum projection of cranked Hartree-Fock-Bogoliubov wave functions

The angular momentum projection of triaxial cranking model wave functions of realistic heavy nuclei is carried out exactly. Technical details of the procedure are discussed. The method is used to test the validity of the Kamlah expansion and of the angular momentum constraint in the cranking model. It turns out that the expansion up to the second order is already a very good approximation to the exact projection.     

On number and angular momentum projection from Hartree-Bogoliubov states

Rotational spectra, calculated by angular momentum projection from Hartree-Bogoliubov states may be completely distorted by particle number nonconservation. A simple method for correcting this is presented, which brings them close to the number projected spectra. It is also shown that by a slight modification of the usual number projection operator this projection may then be executed two times faster and with better numerical accuracy. We study the effect ofJ- and/orN projection before the variation in a constrained Hartree-Bogoliubov model. It is demonstrated that only simultaneous projection of both particle number and angular momentum before the variation is meaningful. Then a more gradual antipairing effect is found than known from previous work. We conclude however that the diagonalization of the Hamiltonian in a space of appropriately chosen generator wave functions is preferable to projection before the variation. In all cases the examples are nuclei in thesd-shell, calculated selfconsistently without separating off and inert core. The nucleon-nucleon force is the Hamada-Johnston potential.     

An analysis of odd-mass rare-earth nuclei using angular momentum projection

A systematic analysis of positive-parity rotational states in odd-mass rare-earth nuclei is presented. The theory used is microscopic in the true sense and conserves both the angular momentum and the particle number as the projection technique is applied. The hamiltonian contains the quadrupole pairing interaction (in addition to the conventional (monopole) pairing and Q · Q forces) which plays an essential role in odd-mass as well as in doubly-even nuclei. Agreement between theory and experimental data is excellent.     

Approximate projection of angular momentum and particle number in heavy nuclei

The problem of symmetry violations in Hartree-Fock-Bogoliubov wavefunctions is handled by projection techniques. For the cases of strong and weak symmetry violations, approximate formulas of the projected energy are given. They allow an approximate projection before the variation for heavy nuclei. The methods are applied numerically to some model calculations for strong and weak deformations.     

A finite sum representation of the angular momentum projection operator

Exact finite sum representations of the angular momentum projection operator in the following two cases are derived: i) when the intrinsic state is axially symmetric but the azimuthal quantum numberK is not equal to zero, ii) when the intrinsic state does not have axial symmetry. Advantages of such representations over projection via exact numerical quadrature are discussed.     

Induced asymmetric deformation by Coriolis anti-pairing and angular momentum projection

A simplified method is suggested for the study of the Coriolis anti-pairing effect of the high-spin rotational states in strongly deformed nuclei. This method allows for small asymmetric deformations and includes exact angular momentum and particle number projection. The angular momentum projection requires only the integration over one Eulerian angle, although small asymmetric deformations may be present.