Connected kernel methods in nuclear reactions: (III). Effective interactions

The recently derived connected kernel equation (CKE) for N-body scattering operators is applied to direct nuclear reactions. A spectral representation is derived for the kernel of the CKE in order to obtain manageable approximations. This allows the kernel to be split into orders corresponding to the propagation of different numbers of bound clusters. By formally solving one part of the kernel at a time, the CKE is written as a hierarchy of nested equations in increasingly many variables. The first equation of this hierarchy is a set of coupled channel Lippmann-Schwinger equations coupling together all two-cluster channels. These equations reduce to the usual coupled channel equations for inelastic scattering and to the coupled channel Born approximation for rearrangement reactions when weak coupling assumptions are made. The second equation of the hierarchy is a two-variable integral equation for the effective interactions appearing in the coupled channel equations. The driving terms and kernel of this integral equation are obtained from the third equation of the hierarchy which is a three-variable integral equation and so forth. The use of the spectral expansion results in a renormalized theory in the sense that the bound state and reaction problems are separated. This permits the inclusion of nuclear models in the theory in a straightforward manner. The hierarchy is applied to a particular example, that of nucleon-nucleus scattering. For this case the hierarchy is truncated at the level allowing no more than three clusters in the continuum. By suppressing exchange and keeping only one-particle transfer and single-nucléon knockout channels, a set of equations for the optical potentials and transfer operators is obtained. These equations provide a three-body treatment of the single scattering approximation to the optical potential. Iteration of the equations yields the usual single scattering approximation in first order including three-body off-shell effects. After suppression of Fermi motion and off-shell effects, the standard impulse approximation is recovered. Modifications of the method for other cases are discussed and other possible applications suggested.     

Coupled channel T-operator equations with connected kernels: (I). Three-body problem with pairwise interactions

Previous work on T-operator coupled equations for two-channel systems is generalized and applied to the problem of three bodies interacting via pair potentials. Sets of coupled, integral equations for the two-body arrangement channel T-operators are derived using a channel coupling array W, and the connectedness properties of the kernels of these equations are discussed. It is shown that either disconnected or connected (iterated) kernels can be obtained by various choices of W. One particular realization of the coupled equations is seen to be similar but not identical to the Lovelace form of the Faddeev equations. Since the matrix form of the coupled equations is similar to the one-body Lippmann-Schwinger equation, the introduction of Møller wave operators is straightforward, and these are used to derive coupled integral equations for the channel state vectors.     

A set of Lippmann-Schwinger equations: (III). Complete continuity of the projected kernel

The disconnected Lippmann-Schwinger kernel for a three-body system is shown to be completely continuous on the real energy axis, when it is projected and operated on a subspace of continuous functions composed of a finite number of partial waves.     

Dynamics of nuclear fluid: (I). Foundation

Starting with the time-dependent Hartree-Fock (TDHF) formulation of the many-body problem, we cast the equation into a set of conservation laws of classical type. Besides the equation of continuity, TDHF leads to an equation of motion which is analogous to the Euler equation in classical fluid dynamics. The forces do not come from the collective kinetic stress alone, but also from a density-dependent chemical potential, the surface tensional force which depends on density differences and the Coulomb interaction. With an assumed Navier-Stokes generalization of the stress tensor, such a set of differential equations provides a powerful tool for the study of complicated collective motions of nuclear systems such as those involved in heavy-ion reactions and nuclear fission. In the static case, the equation of motion leads to the Thomas-Fermi model of a finite nucleus as formulated by Bethe.     

Methods in hamiltonian lattice field theory (I). Connected diagram expansions

Connected diagram expansions in hamiltonian lattice field theory are discussed, using a formalism due to Goldstone, together with the method of cumulant expansions. Connected diagram rules are derived for the ground-state energy, the axial string tension in a gauge model, and for excited state masses.     

Coupled channel T-operator equations with connected kernels: (II). N coupled two-body channels

A time-independent theory of rearrangement collisions involving transitions between two-body states is presented. It is assumed that the system of interest consists of particles that may be partitioned into two-body systems in N ways, including interchanges of particle labels without changing the kind of channel. An infinite family of sets of N coupled T-operator equations is derived by use of the channel coupling array, as in previous work on the three-body problem. Specialization to the channel-permuting arrays guaranteeing connected (N?1)th iterates of the kernel of the coupled equations is made in the N-channel case (N > 3) and the nature of the solutions to the coupled equations is discussed. Various approximation schemes to be used with numerical calculations are suggested. Since the transition operators for all rearrangement channels are coupled together, no problems concerning non-orthogonality of the eigenstates of different channel Hamiltonians are encountered; also the presence of the outgoing wave boundary condition in all channels is made explicit. The close resemblance of the equations in matrix form to those of one-channel scattering is exploited by introducing Møller wave operators and associated channel scattering states, an optical potential formalism that leads to rearrangement channel optical potential operators, and a variational formulation of the coupled equations using a Schwinger-like variational principle. A brief comparison with other many-body formalisms is also given.     

Gross properties of heavy-ion transfer reactions: (I). Formulation

A simple approximation to the semiclassieal theory of heavy-ion transfer reactions is presented. For one-nucleon transfer reactions it reduces to the matching condition model given by Brink. Inelastic scattering and multinucleon transfer reactions can also be treated for either case in which heavy ions follow a straight-line trajectory or temporarily form a dinuclear system. For multistep processes a binary-step approximation is introduced, in which some effects of the time differences between consecutive transfer events are taken into account. Finally the possibility of treating the multinucleon transfer reaction as a Markov process is suggested.     

Proton reactions with 17O: (I). Elastic and inelastic scattering reactions

Differential cross sections for the elastic and inelastic scattering of protons by ¹⁷O have been measured at 8.62, 9.45 and 10.5 MeV. Excitation functions at 110° and 140° were measured from 8.5 to 10.5 MeV. The elastic scattering angular distributions were used to find optical model parameters for the scattering. The angular distributions of inelastically scattered protons were analysed using the effective interaction method of Satchler, and also with the microscopic theory of Geramb and Amos.     

The quark propagator from the Dyson-Schwinger equations: (I). The chiral solution

Within the framework of the Dyson-Schwinger equations in the axial gauge, and using a truncation procedure which respects the Ward-Takahashi identities, we study the effect that nonperturbative glue has on the quark propagator. We show that within this truncation scheme, the requirement of matching perturbative QCD at high momentum transfer leads to a multiplicatively renormalisable equation. Technically, the matching with perturbation theory is accomplished by the introduction of a transverse part to the quark-gluon vertex. In the case of an analytic gluon propagator, this truncation scheme can lead to chiral symmetry breaking only after the introduction of such a transverse vertex: massless solutions do not exist beyond a critical value of as. Using the gluon propagator that we previously obtained, we obtain small corrections to the quark propagator, which keeps a pole at the origin in the chiral phase.     

Thomas-fermi approach to nuclear mass formula: (I). Spherical nuclei

With a view to having a more secure basis for the nuclear mass formula than is provided by the drop(let) model, we make a preliminary study of the possibilities offered by the Skyrme-ETF method. Two ways of incorporating shell effects are considered: the “Strutinsky-integral” method of Chu et al., and the “expectation-value” method of Brack et al. Each of these methods is compared with the HF method in an attempt to see how reliably they extrapolate from the known region of the nuclear chart out to the neutron-drip line. The Strutinsky-integral method is shown to perform particularly well, and to offer a promising approach to a more reliable mass formula.     

Polarization effects in nucleus-nucleus collisions: (I). Distortion of nuclear intrinsic states

Polarization effects due to the interaction between two complex nuclei are investigated in the rotating frame whose z-axis lies in the direction connecting the two nuclei. This allows one to define intrinsic states of the dinuclear system including these polarization effects. We present here a modified coupled-channel formalism whose basis states are these intrinsic states. Results on energy shifts and particle density and B(E2) value modifications in a schematic microscopic model are presented and compared with those of the phenomenological surface-vibrational model. In the case of the rotational model the influence of the nucleus-nucleus interaction on the directivity of nuclear symmetry axes is also analyzed.     

Derivation of the statistical model of nuclear reactions from the theory of stationary irreversible processes

Nuclear reactions via a compound nucleus are considered as an example of the quantum theory of stationary irreversible processes. A density matrix formalism with Liouville space methods is developed, including in a natural way the dynamical and statistical aspects of nuclear reactions. For densely spaced compound levels the statistical model is obtained as a limiting case.     

Microscopic analysis of nuclear collective motions in terms of the boson expansion theory: (I). Formulation

A normal-ordered linked-cluster boson expansion theory, previously worked out by one of the authors (T.K.) and Tamura, has been developed further by reformulating it in a “physical” quasiparticle subspace which contains no spurious particle-number excitation modes. The expansion coefficients of the collective hamiltonian for low-lying quadrupole motions are determined starting from a microscopic fermion hamiltonian including self-consistent higher-order (many-body) interactions derived in our previous work. The contributions from the non-collective states with all possible non-collective one-boson excitations having I^π = 0⁺− 4⁺, which can directly couple to the collective states with one or two phonons, are taken into account in a systematic and compact way.     

Heavy-ion one-neutron transfer reactions involving 13C: (I). Spectroscopic factor products

The total cross sections of some heavy-ion one-neutron transfer reactions involving ¹³C, which lead to excited states of one of the final nuclei, have been measured at energies below the Coulomb barrier. Products of the neutron spectroscopic factors in the initial and final states have been extracted using a Coulomb wave Born approximation, and have been compared with theory.     

On the analytic properties of non-relativistic two-body off-shell amplitudes: (I). Spectral function methods

Spectral function methods introduced in field theoretic schemes, are used for the study of nonrelativistic two-body off-shell amplitudes taken in a l-wave and generated by Yukawa-type interactions. The analytic properties of the basic components of such amplitudes are specified in a detailed way, which can provide a useful guide line for their parametrisations. Related integral equations are also examined, as well as some discrepancies with independent results concerning the half-shell case. The connections of such an approach with other r- and p-space formalisms are fully recognised in paper II, in which further unitary representations related with this analysis, are also considered.     

Kinetics of small disturbances in an isotropic universe I. Derivation of the equations for the disturbances

Gravitational disturbances in an isotropic cosmological model are considered within the bounds of relativistic kinetic theory. It is assumed that the collision frequency of the particles of the medium is much smaller than the frequency of the disturbances being studied. Equations are obtained for scalar, vector, and tensor disturbances in the case where the undisturbed solution describes a flat, isotropic cosmological model.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 30–35, March, 1978.     

Study of neutron induced charged particle reactions on 40K: (I). Thermal neutrons

The (n_th, α) (n_th, p) and (n_th, γα) reactions spectroscopical data are reported on ⁴⁰K and ¹⁴³Nd. For ⁴⁰K, values of σ_α = 0.39 ± 0.08 b, σ_p = 4.4 ± 0.9 b, σ_γα = 26 ± 4 mb are obtained and accurate measurements of E_α0 = 3491.7 ± 7 keV and E_po = 2232.9 ± 3 keV have been performed, α₁, and p₁ transitions and the ⁴⁰K(n_th, γp) reaction have not been observed. The ⁴⁰K(n_th, γα)³⁷Cl and ¹⁴³Nd(n_th, γα)¹⁴⁰Ce spectra are presented. The multipolarity of the primary low energy γ-rays is deduced from the γ-α spectra.     

Detailed study of nuclear charge and mass densities: (I). Surface and asymptotic properties

Theoretical and experimental densities are analyzed and compared in detail, in particular in the surface region. For this purpose nuclear size parameters are discussed and new sets of surface parameters are proposed. It is shown that the densities are very close to the error function in the external part of the surface and can be characterized there by two new parameters. For very large r the densities show an exponential behaviour which is analyzed in terms of single-particle density distributions. Furthermore, the effects of the asymmetry, spin-orbit and Coulomb forces on the density distributions are discussed.