Exact second Born calculations for electron capture for systems with various projectile and target charges

Results of exact numerical calculations of differential and total $1s-1s$ electron-capture cross sections evaluated in the second Born approximation are presented for targets and projectiles of various charges $Z_T$ and $Z_P$ at velocities between 10 and 200 MeV/amu. For symmetric systems with $Z_P=Z_T=Z$ the Thomas peak in the differential cross section, characteristic of a free-wave second Born-approximation process, appears at velocities above $Z^2$×(5 MeV/amu), where $Z$ is the nuclear charge of the target (or projectile). The shape of this Thomas peak contains information about real and virtual intermediate states of the system. For total cross sections at velocities below $Z^2$×(2 MeV) the second Born-approximation cross section is larger than the first Born-approximation cross section indicating a breakdown of the second Born approximation using the free-wave Green's function. Results using the peaking approximation of Drisko converge to our exact second Born-approximation results only at velocities well above $Z^2$×(10 MeV/amu). For systems asymmetric in $Z_P$ and $Z^T$ no exact scaling is found, although the systematics are qualitatively similar to the symmetric case using $Z=\frac{1}{2}(Z_P+Z_T)$. For $p$ + Ne at 100 MeV, the exact Born-approximation results lie somewhat above exact impulse-approximation calculations. It is found that the peaking approximation of Briggs and Simony converges to exact second Born-approximation results as the asymmetry of the projectile and target charges increases. At very high velocities the peaking approximation of Drisko also converges slowly to the exact second Born-approximation result.