Generalized Oscillator Strengths of the Helium Atom. I

The generalized oscillator strengths $f_n\mathrm{}\left(K\right)\mathrm{}$ ($\overrightarrow{\mathrm{K}}\hslash =\mathrm{momentum}\mathrm{transfer}$) for the transitions of He from its ground state to excited states $n=2\mathrm{}{}_{}{}^1{}_{}{}^{}P, 3{}_{}{}^1{}_{}{}^{}P, 2{}_{}{}^1{}_{}{}^{}S, \mathrm{and} 3{}_{}{}^1{}_{}{}^{}S$ are computed from the Weiss correlated wave functions of over 50 terms each. For ${\left(K{a}_0\right)}^2\lesssim 2$ ($a_0=\mathrm{the}\mathrm{Bohr}\mathrm{radius}$), the results by two alternative formulas, corresponding to the "length" and "velocity" formulas in the optical limit, agree with each other within 0.5% for the $2{}_{}{}^1{}_{}{}^{}P$ and $2{}_{}{}^1{}_{}{}^{}S$ excitations, and within 1.5% for the $3{}_{}{}^1{}_{}{}^{}P$ and $3{}_{}{}^1{}_{}{}^{}S$ excitations. Our $f_{2{}_{}{}^1{}_{}{}^{}P}\mathrm{}\left(K\right)\mathrm{}$ is in accord with electron-scattering experiments by Lassettre and his co-workers. For ${\left(K{a}_0\right)}^2\gtrsim 0.2$, our $f_{2{}_{}{}^1{}_{}{}^{}S}\mathrm{}\left(K\right)\mathrm{}$ departs from experimental data at 500 eV, but its slope at $K=0\mathrm{}$ is consistent with experiment. Our results are very probably accurate within a few percent, and thus should provide a sound basis to test the validity of the (first) Born approximation. The representation of the Born excitation cross section for charged-particle impact is greatly simplified by a generalization of the Bethe procedure; it is shown that a few definite parameters can convey the essential content of the Born approximation. As an illustration, the cross sections for the excitations to the four states in He are evaluated and compared with experiments.