Multiple-Scattering Eikonal Expansion: Systematic Corrections to the Glauber Theory

A systematic investigation of the Glauber multiple-scattering theory is made using eikonal expansion methods developed in two previous papers and shown there to be valid for large-angle scattering. The multiple-scattering eikonal expansion of the $T$ matrix is developed for scattering of a fast projectile interacting with $A$ target particles through local, two-particle interactions. Throughout this study, the fixed scatterer form of the impulse approximation is employed. The expansion starts from the most naive form of the Glauber theory based on the simple eikonal approximation and is shown to produce an equivalent form of the sophisticated version of the Glauber theory in which empirically determined scattering amplitudes are generally used. The multiple-scattering eikonal expansion also produces corrections to the Glauber theory which arise from overlapping interactions, i.e., the fact that the projectile may interact with more than one target particle at a time during its passage through the target. All of these effects are summarized in terms of a generalized phase-shift function for multiple scattering which is a product of two-particle Fourier-Bessel impact parameter amplitudes, $\Pi {i=1\mathrm{}}^A\left[S_F\mathrm{}\right(|\mathrm{}\overrightarrow{\mathrm{b}}-{\overrightarrow{\mathrm{b}}}_i\mathrm{}|\left)\mathrm{}\right]$, one for each target particle, times an overlap factor, $\mathcal{O}\left(\overrightarrow{\mathrm{B}}\right)$, which contains multiparticle phase-shift contributions. The overlap factor depends on the set of coordinates $\overrightarrow{\mathrm{B}}=\{\overrightarrow{\mathrm{b}},{\overrightarrow{\mathrm{b}}}_1,\dots ,{\overrightarrow{\mathrm{b}}}_A\}$ of the projectile and the $A$ fixed scatterers and represents nonadditive contributions to the phase shift. Some approximate forms of the three-particle contribution to the phase shift in $\mathcal{O}\left(\overrightarrow{\mathrm{B}}\right)$ are developed for hadron-nucleus scattering and also for atomic collisions. For nonoverlapping interactions, we find that the sophisticated version of the Glauber theory appears as the sum of the eikonal expansion rather than its leading term, the eikonal approximation. This removes the usual small-angle limitation.