Unified Reaction Theory Research Articles

Structure of the dynamical polarization potential.

A closed expression for the dynamical polarization potential is obtained starting from the unified reaction theory of Feshbach. This result is compared with a few methods available in the current literature to evaluate the optical potential. Two main features emerging from the comparison are that the propagator in the nonelastic channels is different from the one appearing in iterative approaches and that in the polarization potential the final matrix elements, which recouple the inelastic channels to the elastic channel, are not linear functions of the coupling interaction. \textcopyright{} 1996 The American Physical Society.

Interior dynamics in the unified connected reaction theory of Polyzou and Redish

We discuss the unified reaction theory of Polyzou and Redish [Ann. Phys. (N.Y.) 119, 1(1979)] with regard to the freedom to perturb the approximate Hamiltonian by a connected operator. A proper treatment of this freedom is crucial for a good treatment of the dynamics. We show how to construct an effective N-body force so the approximate Hamiltonian is a projection of the exact Hamiltonian on an appropriate infinite dimensional subspace. Because this operator is connected it should not alter the unitarity considerations in the above paper. The methods used in constructing this effective interaction can also be used to reformulate the scattering integral equations as equivalent equations with noncompact contractive kennels. This reduces the scattering problem to uniformly convergent perturbation theory.

Variational principle for the bloch unified reaction theory

The unified reaction theory formulated by Bloch uses a boundary value operator to write the Schr\"odinger equation for a scattering state as an inhomogeneous equation over the interaction region. As suggested by Lane and Robson, this equation can be solved by using a matrix representation on any set which is complete over the interaction volume. Lane and Robson have proposed, however, that a variational form of the Bloch equation can be used to obtain a "best" value for the $S$ matrix when a finite subset of this basis is used. We first consider the variational principle suggested by Lane and Robson, which gives a many-channel $S$ matrix different from the matrix solution on a finite basis, and show that the difference results from the fact that their variational principle is not, in fact, equivalent to the Bloch equation. We then present a variational principle which is fully equivalent to the Bloch form of the Schr\"odinger equation and show that the resulting $S$ matrix is the same as that obtained from the matrix solution of this equation.NUCLEAR REACTIONS Variational principle for the Bloch equation.

Form Factors for Proton Transfer Reactions to Unbound Isobaric Analog States

Proton form factors for isobaric analog resonances populated directly as residual states in ($d,n$) and ($^{3}\mathrm{He},d$) reactions are discussed and computed explicitly using a Lane-model Hamiltonian and the formalism of Feshbach's unified reaction theory. The resonance form factors thus obtained are compared in specific cases with equivalent parent bound-state form factors and with single-particle resonance wave functions. Numerical results and tests using data are presented for $^{92}\mathrm{Mo}(d,n)^{93}\mathrm{Tc}^{A}$, and it is shown that resonance form factors lead to a dramatic enhancement of the magnitude of the calculated cross section for some transitions, compared with calculations using conventional form factors. Such enhancements are consistent with the available data.

Singular Lippmann-Schwinger equations and the unified theory of nuclear reactions

State normalization for a special type of singular Lippmann-Schwinger equation is investigated. The unified reaction theory of MacDonald is based on equations of this type. It is shown that the resolvent method cannot be used to obtain the formal solutions of these equations. We further show that the expressions given by MacDonald are valid only in the simple case for which the targets and residual nuclei have a pure shell-model configuration or for elastic scattering.

Unified Theory of Alpha Decay

A new theory of $\ensuremath{\alpha}$ decay is presented in which the nuclear-radius parameter is not used explicitly. An expression is derived---similar to the one in Feshbach's unified reaction theory, and based on the time-dependent perturbation method---which overcomes the inadequacies of previous $\ensuremath{\alpha}$-decay theories. It is pointed out that the new decay formula makes it possible to calculate even the absolute value of the decay rate. In this theory, the $\ensuremath{\alpha}$-decay process is interpreted as the transition which is caused by the difference between four times the nucleon-nucleus and $\ensuremath{\alpha}$-nucleus potentials. The method of numerical calculation is illustrated for the case of a simple $\ensuremath{\alpha}$ decay.

Asymptotic Channel States and Matrix Equations in the Unified Reaction Theory

The multidimensional formulation of the unified reaction theory with asymptotic channel states of different cluster compositions is discussed and extended. Modified sets of projection operators and Hermitian orthoprojectors are derived using the channel-mixing operator and it is shown that the Hermiticity of the operators may be an unnecessarily stringent requirement for the formulation of the theory. The simple diagonal form of the orthoprojectors given earlier is shown, however, to be sufficient to construct a reaction theory consistent with the asymptotic boundary conditions if the matrix equations are suitably modified. Based on the many-particle scattering theory, a useful reduction method for the modification of the matrix equations is developed. Application of the method provides a series of sets of coupled equations for the description of direct processes, which contains a sufficient amount of desirable properties in the restricted direct-channel spaces to give consistent solutions.

Fine Structure in Nuclear Resonances

A K-matrix formulation is given for the shell model approach to a unified reaction theory. This K-matrix differs from the R-matrix of Wigner and Eisenbud in using distorted waves of a diffuse potential. The penetrabilities and channel phase shifts are therefore those for a diffuse potential rather than those for a square well with hard sphere boundary conditions at some radius. This K-matrix is used to discuss the fine structure produced by a doorway-hallway system and a result obtained earlier by Ferrell and MacDonald is rederived. A resonance expansion is then found which explicitly exhibits the widths and resonance energies in a multilevel formula. The distribution of the widths is found, a sum rule is derived, and the average cross section is derived. A general resonance formula is then derived for the case of any number of hallway and doorway states. This is used to generalize the previously obtained doorway-hallway results to the case of hallways coupled to the continuum. A characteristic asymmetry is shown to result in certain cases. 12 refs., 3 figs.

Excitation of the giant dipole resonance in μ-capture

A previously proposed approximate method of the unified reaction theory is applied to the problem of neutron emission in muon capture. The obtained total first-forbidden transition probability for the reaction 16O( μ −, νn) 15N is in agreement with the experiment. The resonance structure of the neutron spectrum appears as a consequence of the channel coupling.

Elastic neutron scattering on 12C

The elastic scattering of neutrons on 12C has been studied up to 4.5 MeV using Feshbach's unified reaction theory. The intermediate states were obtained from the shell-model calculation using a zero-range residual interaction. The coupling between the intermediate and continuum states was described by the same residual interaction. The calculated energies, widths and spin values of the resonances, as well as the absolute value of the cross section, all agree quite well with the experimental data. The only free parameter used in this calculation was the strength of the spin-orbit interaction. The lowest two intermediate states, which are approximately degenerate in the shell-model scheme, appear as two well separated resonances in the cross section due to the coupling between the intermediate and continuum states.

Projection Operators in the Unified Reaction Theory

Projection operators applicable for rearrangement and exchange processes with full recoil effect for unified reaction theory of Feschbach

Intermediate-State Theory of Single-Particle Resonances

The ${d}_{\frac{3}{2}}$ state in oxygen is observed as a very sharp scattering resonance in ${\mathrm{O}}^{16}(n,n){\mathrm{O}}^{16}$. The state is a "virtual" state, therefore, and its inclusion among the single-particle discrete states employed in the shell-model description of states of ${\mathrm{O}}^{16}$ requires some justification. In this paper a unified reaction theory recently given by one of the authors is used as the basis of a description in which the resonance is described as arising from the virtual transitions into a true bound state. The continuum states of this description are nonresonant because the continuum ${d}_{\frac{3}{2}}$ resonance is replaced by a bound ${d}_{\frac{3}{2}}$ state which plays the role of a new "doorway state." This description leads to the conventional shell-model description of the particle-hole resonances of ${\mathrm{O}}^{16}$ and simplifies the description of the interaction between virtual and bound states.