H.A. Mavromatis

Hartree-fock contributions to the effective charge and interaction in the sd shell

Abstract The second-order perturbation theory diagrams for the effective charge which contain HF insertions have been evaluated — they are sizeable and negative for neutrons and quite state-dependent for protons. The total second-order result is generally about − 1 3 of the first-order value which suggests that the effective charge does not smoothly converge to the required value in terms of the number of interactions. As concerns the effective-interaction expansion ordered according to the number of G-matrices, the inclusion of the HF diagrams is not expected to improve the poor convergence. The bare and first-order perturbation theory diagrams have been evaluated in a HF basis for both the effective charge and interaction. The bare effective-charge diagram is enhanced for protons compared to the oscillator result. For the bare interaction the levels are significantly less bound when the HF basis is used, particularly when the s 1 2 2 configuration is involved. The first-order perturbation corrections are much reduced in a HF basis compared to the oscillator results — the core-polarisation diagram is reduced by factors of roughly 2 and 5 for the effective charge and interaction respectively. In all cases the diagrams containing HF insertions give non-negligible contributions which are as important as those from the other diagrams.

RELATION BETWEEN VARIOUS METHODS USED IN CALCULATING NUCLEAR REACTION MATRIX ELEMENTS.

Abstract Various methods for calculating nuclear reaction matrix elements from phase shifts are examined. Calculations are made for the singlet S channel using these various methods and the results are compared. It is shown that a close analogy exists between the auxiliary potential method of calculating nuclear reaction matrix elements and the Scott-Moszkowski separation method.

The linked-cluster theorem in finite model spaces

Abstract It is pointed out that the linked-cluster expansion and the Rayleigh-Schrodinger perturbation (for a non-degenerate system) are not identical in a finite model space. Judging from some calculations the complete Rayleigh-Schrodinger expansion appears to be more accurate than evaluating only linked graphs in a finite model space.

Calculation of the quadrupole moments and B(E2) values of 17O and 17F with the Sussex matrix elements

Abstract The ground state quadrupole moments and 1 s 1 2 → 0 d 5 2 B( E 2) values of 17O and 17F are calculated To obtain A = 17 wave functions the 1p-1h energy matrix for 16O is first diagonalized yielding 2+ vibrations of 16O These are then coupled to the single-particle states of the 1s-0d oscillator shell yielding 2p-1h states Diagonalization is then performed of the A = 17 matrix which includes single-particle and these 2p-1h states Two alternatives to diagonalizing the A = 17 energy matrix are also considered One involves obtaining the 17O and 17F wave functions by using perturbation theory to couple the 2+ vibrations to the 1s-0d single-particle states and the other, simpler still, consists in coupling non-interacting 1p-1h excitations via perturbation theory to the 1s-0d single-particle states Both these are shown to give smaller results than the diagonalization procedure The potential matrix derived at Sussex is used as the interaction and single-particle energies are calculated rather than taken from experiment Thus the only free parameter used is the oscillator length b which is taken to be 1 7 fm In the principal part of the calculation only the 0p holes are excited and these only through two oscillator shells The effect of 0s holes is estimated Comparisons are made with calculations involving the Kuo-Brown G-matrix and the G-matrix derived from the Tabakin interaction

Goldstone graph components in perturbation calculations

Abstract It is shown that dividing Goldstone graphs into components simplifies certain perturbation calculations. This technique is used to calculate the binding energy of 40Ca to fourth order including all 2 h ω excitations and the single-particle energies of 17O to fourth order. The resulting large fourth-order contributions are discussed.

A study of the ground-state energy of 40Ca

Abstract The 40 Ca ground state is studied in the 176-dimensional model space of 0 + 2 ħω excitations. In this space diagonalization is compared to the sum of linked graphs to all orders. Both the original and the saturating Sussex matrix elements are used and the effect on the order-by-order convergence of modifying the unperturbed single-particle energies is studied.

A procedure for checking energy and effective charge perturbation calculations

Abstract A method is outlined for checking energy and effective charge perturbation calculations applicable in principle to any order by using the c.m. potential energy written as a sum of twobody operators. This method, which is more useful for lower order calculations can be applied to any system and has been used to check the expressions for the binding energies and spectra to second order of several simple nuclear systems as well as the expressions for the effective charge to first order. Both cases where the lowest model state is completely non-spurious and where it is partly spurious are considered.

A study of 17O

Abstract Using Q-box techniques, linked expansions in terms of the number of folds are carried out for j = 5 2 , 1 2 and 3 2 states of 17O, where the complete 0 + 2ℏω space is included. These expansions are shown to converge very quickly and are compared to energy matrix diagonalizations. The series involved in obtaining the archetypal Q-boxes however converges rather slowly in this space. Calculations are performed for various shift operators with the original and modified Sussex matrix elements.

Convergence studies for the one-body mass radius operator of some closed-shell nuclei

Abstract The convergence properties of the perturbation series for the mass radius operator are studied in detail using a harmonic oscillator basis. The systems considered are the ground states of the closed shell nuclei 4 He, 16 O and 40 Ca. Calculations are carried out to third order in the nuclear interaction and include up to 4ħω excitations for 4 He and 16 O, and up to 2ħω excitations for 40 Ca. Goldstone graph components are used for graphs with at most 2p-2h intermediate states and Goldstone graphs for graphs with 3p-3h intermediate states. Both, the original and the saturating Sussex matrix elements (SME) are used. Some diagonalizations and inversions are also considered. The mass radius operator results with the original SME give large negative contributions that result in small rms radii. The results with the saturating SME indicate a slow order by order convergence at the value of the size parameter b that gives saturation in the first order energy. However, in this case, the individual perturbation orders and the total corrections, for the mass radius operator, are small for each of the above three nuclei at these values of b , and reasonably good agreement with experimental rms radii is obtained.

Fourth-order high-lying contributions to the ground-state energies of some closed-shell nuclei

Abstract We discuss a complete fourth-order calculation, in an oscillator basis, of the ground-state energies of closed-shell nuclei using graphical techniques. The calculations, carried out by evaluating all linked Goldstone graphs that contribute in the subspace of intermediate excitations up to 4 ω for 4 He and 16 O show that the linked fourth order 3p-3h and 4p-4h intermediate state graphs are small compared to the corresponding 1p-1h and 2p-2h intermediate state Goldstone graphs. Both the original and the saturating Sussex matrix elements are used and the effect on the order-by-order convergence of modifying the unperturbed single-particle energies is studied. The sum of linked graphs is extrapolated from the lowest orders in the interaction using Pade approximants, and encouraging results are obtained.