The direct reaction $A(a,b)B$ is described by exactly solving the Schrödinger equation for a few-body Hamiltonian within a restricted model space. The model space allows for coupling to rearrangement channels by including basis vectors classified according to different mass partitions. The nonorthogonality of basis vectors that correspond to different mass partitions is investigated in detail. A surface approximation is developed to understand the magnitudes of multistep amplitudes produced by channel coupling to rearranged partitions. The surface approximation uses a separable Green's function approximation to the multistep series, and gives a convenient closed-form expression. Finite-range coupled-channels calculations are presented for the reactions ($d,d$), ($d,p$), and ($p,p$) on oxygen and zirconium targets, and for (${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$, ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$), (${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$, ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$), and (${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$, ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$) on silicon and sulfur targets. To assess higher-order effects, these results are consistently compared with lowest-order results using folded distorting potentials. The calculations indicate that the effects of nonorthogonality are small for these reactions, but the light-ion reactions showed much larger effects. Theoretical explanations of the behavior of nonorthogonality and channel-coupling effects are given, and lead to criteria for predicting when such effects may be important. Channel-coupling effects are expected to be important whenever the distorted-wave Born-approximation transfer amplitude is unusually sensitive to the intermediate channel optical potential. Nonorthogonality effects are roughly proportional to channel-coupling effects, but are much smaller than coupling effects except when a small mass is transferred between two large masses.

NUCLEAR REACTIONS ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}(d,d)$, ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}(d,p)$, $E=10.49,14.8,20$ MeV, ${}_{}{}^{17}{}_{}{}^{}\mathrm{O}(p,p)$, $E=10.98\mathrm{}$ MeV, ${}_{}{}^{90}{}_{}{}^{}\mathrm{Zr}(d,d)$, ${}_{}{}^{90}{}_{}{}^{}\mathrm{Zr}(d,p)$, $E=12\mathrm{}$ MeV, ${}_{}{}^{28}{}_{}{}^{}\mathrm{Si}$(${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$, ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$), ${}_{}{}^{28}{}_{}{}^{}\mathrm{Si}$(${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$, ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$), $E=42, 48$ MeV, ${}_{}{}^{32}{}_{}{}^{}\mathrm{S}$(${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$, ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$), $E=41.71\mathrm{}$ MeV; calculated $\sigma \left(\theta \right)$. Folded-potential DWBA plus multistep contribution of rearranged intermediate channels, corrected for nonorthogonality. Surface approximation with separable Green's function.

• Received 1 March 1976

DOI:https://doi.org/10.1103/PhysRevC.14.1739