A new representation of the generalized oscillator strength for bound-free transitions in hydrogenlike systems

The concept o f generalized oscillator strength (GOS) associated with atomic transitions was introduced by Bethe [ 1 ]. It has, since then, been employed extensively in the evaluation o f excitation and ionization cross sections at high projectile impact energies. The generalized oscillator strength is a function o f K2a 2 where ao is the first Bohr radius and hK is the change in momentum of the projectile due to scattering. For the optically allowed transitions, GOS reduces to the optical oscillator strength as Kao goes to zero. This fact has been utilized by a number of investigators to derive photoionization cross sections over a wide range o f photon energies for various atoms and molecules from the high energy electron scattering experimental data [2,3 ]. I f one expands GOS as a power series in K2a 2, the series has only a finite radius o f convergence. In particular, this same series representation o f the generalized oscillator strength in powers ofKEa 2 would not be a good representation at the large values o f K2a 2 ( ~> 1 ). This difficulty was overcome by Lassettre [ 4] who replaced, for the b o u n d bound transitions, K2a 2 by another variable ( which varied only from 0 to 1 as K2a 2 changed from 0 to ~ . He showed that the new series, in powers o f (, converges for all physically attainable values of K2a 2. Using this series expansion of GOS in powers o f ( and only a few available experimental values o f the generalized oscillator strengths for the weak transition 1 ~S-,2 zS in helium, he obtained a simple analytical expression for the generalized oscillator strength which was correct asymptotically, predicted a correct position of the peak of GOS and was in good agreement with the experiment. Lassettre's idea was extended by Vriens [ 5 ] to obtain a suitable power series representation o f GOS for the excitation process in which an electron jumps from 1 s to ns, np and nd orbitals. He also applied the method to study excitation o f helium by electron and proton impacts. Along the same line, Wadehra and Spruch [ 6 ] provided convergent power series representations for the matrix elements associated with elastic and discrete inelastic transitions in helium. For the case o f b o u n d free transitions, Vriens [ 5 ] discussed the position o f poles o f the generalized oscillator strengths in the complex K2a 2 plane. However, to the best of our knowledge n o further investigation in this direction for the boundfree transitions has been carried out. In the present paper we give a general expression for the bound-bound generalized oscillator strengths for hydrogenlike systems, valid for any n, as a product o f the corresponding optical oscillator strength and a second factor. This second factor, representing the ratio o f the generalized to the optical oscillator strengths for b o u n d -