Nonorthogonal Terms Research Articles

NEW. STRONGLY CONSERVATIVE FINITE-VOLUME FORMULATION FOR FLUID FLOWS IN IRREGULAR GEOMETRIES USING CONTRAVARIANT VELOCITY COMPONENTS: PART 1. THEORY

Abstract A strongly conservative finite-volume procedure is presented for flows in complex geometries. The technique is based on a complete transformation of the governing equations, and physical velocity components, rather than the traditionally used Cartesian velocity components, are used as primitive variables. It was found that projecting the discretized vector transport equation in the direction of the covariant base vectors eliminated two substantial difficulties associated with flows in complex geometries. These difficulties stem from the presence of cross-pressure gradient terms and the need for a transformation between the different types of curvilinear velocity components in the mass conservation equation. It is shown that the present formulation ensures that the computational scheme is diagonally dominant. It was found that partially implicit treatment of nonorthogonal diffusion terms improved the convergence rate primarily for high-cell-Reynolds-number values. For nonstaggered grids, a new sol...

Consistent description of the (t, 3He) reaction for A = 12−89

New differential cross section data for seven target nuclei were analysed using a microscopic model. Particular care was taken to ensure that the absolute magnitude of the cross sections was correctly measured. The predicted direct or one-step (t, 3He) reaction did not account for the magnitude of the data except for forward angles and A⩾40. The calculations were estended to include the contributions from two-step or sequential single-nucleon transfers. Calculations were made using the second-order DWBA and zero-range or full finite-range forms. In the latter the non-orthogonality between the basis states in the two-step transfers was included. The transfer amplitudes were deduced from comparisons of predicted angular distributions with data obtained from the (t, α) and (t, d) reactions, and from amplitudes determined by repeating previously reported shell-model calculations. The full finite-range DWBA calculations which included non-orthogonality terms, accounted for both the shape and magnitude of the data within a factor of 3 in the worst case. The (t, α) reaction angular distributions are, however, poorly predicted.

Pseudostate description of breakup in the coupled-reaction-channel method: Numerical study of nonorthogonality effects.

To simulate the effects of the breakup channel on rearrangement amplitudes, the conventional coupled-reaction-channel (CRC) expansion is augmented by pseudoreaction channels. The construction of the projector for the extended CRC space is discussed, and transition-operator equations on this space are given. By solving the full and post-approximation forms of the CRC equations for a model three-particle problem, the crucial role played by the nonorthogonality terms is demonstrated.

Bare Potential DWBA in Rearrangement Reactions

The validity of bare potential DWBA (BPDWBA) is examined for reactions 40 Ca( d , p ) 41 Ca(1 f , 6.14 MeV) at 26.7 MeV and 10.5 MeV and 16 O( d , p ) 17 O(2 s , 1.05 MeV) at 24.89 MeV and 10.5 MeV, and associated ( d , d ) and ( p , p ) scattering processes through comparison with coupled channel (CC) calculations. BPDWBA can reproduce the CC calculations well at the higher energies only if the non-orthogonality terms of coupling kernels are neglected, but not so well at 10.5 MeV. Channel coupling effects depend strongly on the transferred angular momentum.

The Strong Potential Approximation with Non-Orthogonality Terms

Non-orthogonality contributions are included in the Strong Potential Approximation (SPA) for electron capture. These non-orthoganality contributions occur because the off-energy-shell Coulomb wave functions, representing the intermediate states of the collision, are not orthogonal to the intial bound state wave functions. As a result of this non-orthogonality an extra term arises in the matrix element between initial and intermediate states. This term may be written as an addition to the interaction potential which is treated in first order perturbation theory in the SPA method. It is shown here that this nonorthogonality contribution is zero in the SPA when peaking approximations are applied.

Reduced matrix equations with reactive components for direct nuclear interactions

We present a critical comparison of the two distinct formulations of the nuclear direct interaction theory in which either the asymptotic channels or interacting composite states are emphasized. In the truncated bound-state approximation, they reduce to the channel coupling array (CCA) and coupled reaction channel (CRC) approaches. The unique role played by the distortion potentials Y in the CCA method is studied, and the importance of the non-orthogonality terms in the CRC is pointed out. To incorporate the advantages of both formulations, introduction of an auxiliary reactive channel to the matrix equation was proposed previously, and the structure of this modified set of scattering equations is analyzed in detail.

An Approach for the DWBA Calculation of the Two-Nucleon Transfer (p, t) Reaction

A new approach of the DWBA calculation for the (p, l) reaction is investigated. The DWBA term is calculated as the two-step non-orthogonality term for Ca(p, t)46Ca(O+, g.s.) reaction at E p =20 MeV and 40 MeV. This, together with the two-step (P-d-l) amplitude calculated by the quasi-particle method, gives the full reaction amplitude in the present scheme. Further, the coherence property of the two-step amplitude is discussed. 6 ),7) who insist that the absolute cross section and the angular distribution of the (p, t) reaction (or ( t, p) reaction) are well described by the one-step DWBA. They analyzed the DWBA term for some cases with a wave function for the triton, and pointed out the importance of improving the form factor for the transferred two neutrons and of using the wave function for the triton. Nowadays, it is still difficult to get a triton wave function, even by means of solving the Faddeev equations or using variational approaches with potentials. It is well known that the wave functions calculated by such methods do not fully reproduce the binding energy and the charge form factor, etc. Thus, the use of the realistic triton wave function for the DWBA calculation in the (p, t) reactions is, strictly speaking, not always appropriate, and, possibly the conclusion obtained by means of such a treatment will be altered in future, especially in the analyses of the polarization phenomena and the unnatural parity transitions. In this paper, we propose a method of calculating the DWBA term accurately and discuss the reaction mechanism. The idea of our approach is to use the mechanism of the cancellation between the non-orthogonality term and the DWBA term. 9 ) In the DWBA calculation of the two-nucleon transfer reaction, we generally encounter the difficulty that the form factor cannot be constructed

The coupled-channel method and the integral-equation formalism for nuclear rearrangement reactions

The standard coupled-reaction-channel formalism for nuclear rearrangement reactions is compared with bound-state approximations obtained by starting from modernN-body scattering theory. This comparison will allow us to clarify the origin of nonorthogonality terms in standard coupled-reaction-channel equations as well as the reason of their absence inN-body formalisms. Contributions neglected by the bound-state approximation are taken into account in a three-body context by introducing interacting-particle representations. Thus we can also establish interrelations between different three-body formalisms.

A note on non-orthogonality effects in coupled channel methods for rearrangement reactions

Arguments are given that the presence of non-orthogonality terms in coupled channel formulations of rearrangement scattering is the consequence of inconsistent treatment of multichannel dynamics.

The (p, d) (d, t) Process in Strong (p, t) Reactions

46 Ca(O+, g.s.: 2+, 1.34 MeV) at 20 MeV and 40 MeV and 116Sn(p, t) 114Sn(O', g.s.) at 20 MeV are investigated in terms of second order DWBA with finite-range calculations of the non-orthogonality term of the transition matrix element. In all the cases investigated, the contribution of the two-step process is as much as that of the one-step process. The cross sections of the Ca transitions arc well accounted for by the one-step plus two-step mechanism. For the Sn transition, a strong enhancement of the two-step amplitude is found. It is as much and clue to essentially the same mechanism as the well-known enhancement of the one-step amplitude caused by the pairing correlation in the BCS ground states of the Sn nuclei.

The effect of non-orthogonality and previously neglected diagonal terms in the tight-binding approximation for the electronic structure of liquid metals and alloys

The averaged Green's function for an electron in non-simple liquid metals and alloys is evaluated in the tight-binding approximation (LCAO) when the effect of non-orthogonality and previously neglected diagonal terms are taken into account. The actual formalism is given on the basis of self-consistent single-site approximations (SSSA)-particularly of the coherent potential approximation (CPA) for liquid metals and of the Ishida-Yonezawa theory. The averaged Green's function is well known to be related to one-electron properties such as the density of states and, within SSSA assumptions, even to the electric conductivity, the Hall coefficient, etc. Calculated densities of states with and without non-orthogonal terms are given for comparison and discussion.

Role of the 12C-core exchange in the 16O- 12C elastic scattering

The exchange of the 12C core is taken into account explicitly in 16O- 12C elastic scattering to explain the backward rise of the angular distributions. The equation for the relative motion of the colliding particles contains the non-local kernel due to the exchange effect. The equation is solved by the iterative method. The angular distributions are calculated at E L ( 16 O) = 24, 42, 65 and 80 MeV . The results are compared with those of finite-range DWBA calculations. The effects of the multi-step transfer process, the α ∗ transfer process and of the non-orthogonality term are studied. The equivalent local potential to the non-local kernel is derived under the adiabatic approximation. The resulting local potential shows the parity dependence with the exponentially decreasing behaviour at the asymptotic regions and is found to describe the low-energy scattering phenomena quite well.

Two-Step Analysis of theO18(p, t)O162−Excitation and Phase Relations in the Nonorthogonal Term

We perform second-order distorted-wave Born-approximation analyses of the $^{18}\mathrm{O}(p, t)^{16}\mathrm{O}$ ${2}^{\ensuremath{-}}$ excitation. The experimental angular distribution can be well reproduced by the simultaneous consideration of ($p, t, {t}^{\ensuremath{'}}$), ($p, {p}^{\ensuremath{'}}, t$), and ($p, d, t$) processes and the last two processes are found to be comparably important. The phase relations in the nonorthogonal term are also discussed and it is shown that this term gives a negligibly small contribution to such unnatural-parity-state excitation processes.

New coupled-reaction-channels formalism for nuclear reactions

A set of coupled integral equations (CIE) for multichannel nuclear reactions is considered which appears to have some advantages over the conventional coupled-reaction-channel (CRC) equations when transfer channels must be included. Both the CIE and CRC formalism are based on the assumption that the system has a limited number of cluster type intermediate states available to it. The CIE have a connected kernel while those of the CRC do not. In addition, the troublesome nonorthogonality terms of the CRC formalism have no counterpart in the CIE formalism. The CIE are compared with those of the CRC formalism. The CIE for a simple model for deuteron scattering from a structureless target are presented as an explicit example.NUCLEAR REACTIONS Coupled-reaction-channels formalism with no nonorthogonality term and having a connected kernel.

The electronic energy bands in diamond

A calculation of the valence bands for diamond using a linear combination of bond orbitals is described. The method of calculation is basically that of Cohan, Pugh and Tredgold (1963). Improvements in the method of calculating the interaction integrals have been made and non-orthogonality terms are included.