Theory of electron capture from a hydrogenlike ion by a bare ion

We have systematically developed a new approximation for the calculation of the cross section for electron capture from a hydrogenlike ion of large nuclear charge ZTe by a bare ion of charge ZPe moving with speed v. The amplitude in the wave treatment is obobtained through consistent expansion in the small parameters ZPZT and ZPe²v; however, the ratio ZTe²v is not assumed small. Following a careful analysis of the approach to the energy shell, an off-shell factor is seen to arise which does not appear in the impulse approximation. The effects of this factor on the capture amplitude are explored. Using a peaking approximation, we derive a closed-form expression for the 1s→ns capture amplitude which includes the effects of the off-shell factor and is accurate to order (ZPe²v)². We tabulate the peaking-approximation cross section. The derivation of the asymptotic form of the cross section when ZPZT<<1 is justified. Using a realistic one-electron model, K-shell capture cross sections are calculated for protons on carbon, neon, and argon. These are seen to represent the data generally better than do the impulse approximation cross sections.